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[
"Mathematics Subject Classification. Primary 53C15 ; Secondary 53B15",
"Mathematics Subject Classification. Primary 53C15 ; Secondary 53B15"
]
| [
"Mattia Pujia ",
"Luigi Vezzoni "
]
| []
| []
| In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1, 1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long time solution to the pluriclosed flow in 2-step nilmanifolds. | 10.1016/j.crma.2018.01.002 | [
"https://arxiv.org/pdf/1707.08809v1.pdf"
]
| 119,657,228 | 1707.08809 | 39e6d962eb424026e0b73eb6d27d0006c2d2dc8f |
Mathematics Subject Classification. Primary 53C15 ; Secondary 53B15
July 28. 2017. 2000
Mattia Pujia
Luigi Vezzoni
Mathematics Subject Classification. Primary 53C15 ; Secondary 53B15
July 28. 2017. 2000This work was supported by by G.N.S.A.G.A. of I.N.d.A.M. 1 Here we adopt the convection ω(·, ·) = g(J·, ·), in contrast to the one adopted in [6].
In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1, 1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long time solution to the pluriclosed flow in 2-step nilmanifolds.
The Bismut Ricci form on 2-step SKT nilmanifolds
An Hermitian manifold is called SKT if its fundamental form is ∂∂-closed. The SKT condition can be described in terms of the Bismut connection by requiring that the torsion form is closed. Indeed, on any Hermitian manifold (M, g) there is a unique Hermitian connection ∇ such that the tensor c := g(T (·, ·),) is skew-symmetric in its entries [2], where T is the torsion of ∇. The metric g is SKT if and only if dc = 0. In this note we focus on the Ricci form of ∇. In analogy to the Kähler case, the form is defined by ρ B (X, Y ) = tr ω R B (X, Y, ·, ·) , ω being the fundamental form of g and R B the curvature tensor of ∇.
We consider as manifold M a 2-step nilpotent Lie group G equipped with an invariant Hermitian structure (J, g). Under these assumptions, the form ρ B takes the following expression
(1) ρ B (X, Y ) = i n r=1 g([X, Y ], [Z r ,Z r ]) , for every X, Y ∈ g ,
where {Z r } is an arbitrary g-unitary frame of the Lie algebra g of G 1 . More generally, if G is just a Lie group with an invariant Hermitian structure, ρ B takes the following expression
(2) ρ B (ω)(X, Y ) = −i n r=1 g([[X, Y ] 1,0 , Z r ],Z r ) − g([[X, Y ] 0,1 ,Z r ], Z r ) − g([X, Y ], [Z r ,Z r ]) .
We have the following Proposition 1.1. Let G be a 2n-dimensional 2-step nilpotent Lie group with a left-invariant SKT structure (J, g). Then
ρ B (Z,Z) = −i n r=1 [Z,Z r ] 2
for every Z ∈ g 1,0 , where {Z r } is an arbitrary unitary frame. In particular,
ρ B (X, JX) ≤ 0
for every X ∈ g. Here we adopt the convection ω(·, ·) = g(J·, ·), in contrast to the one adopted in [6].
Proof. Let Z and W be vector fields of type (1, 0) on g ⊗ C and let ω be the fundamental form of g.
= − ω([Z,Z] 0,1 , [W,W ] 1,0 ) + ω([Z, W ], [Z,W ]) − ω([Z,W ] 0,1 , [Z, W ] 1,0 ) − ω([Z, W ] 0,1 , [Z,W ] 1,0 ) + ω([Z,W ], [Z, W ]) − ω([W,W ] 0,1 , [Z,Z] 1,0 ) = + ig([Z,Z] 0,1 , [W,W ] 1,0 ) + ig([Z, W ], [Z,W ]) + ig([Z,W ] 0,1 , [Z, W ] 1,0 ) + ig([Z, W ] 0,1 , [Z,W ] 1,0 ) − ig([Z,W ], [Z, W ]) + ig([W,W ] 0,1 , [Z,Z] 1,0 ) = + ig([Z,Z], [W,W ]) + ig([Z,W ], [Z, W ]) .
The SKT assumption ∂∂ω = 0 implies
g([Z,Z], [W,W ]) = −g([Z,W ], [Z, W ]) .
Therefore in view of (1) we get
ρ B (Z,Z) = i n r=1 g([Z,Z], [Z r ,Z r ]) = −i n r=1 g([Z,Z r ], [Z, Z r ]) ,
being {Z r } an arbitrary unitary frame, and the claim follows.
Remark 1.2. Another description of the Bismut-Ricci form on 2-step nilmanifolds can be found in [1].
Next we observe that in general the form ρ B is not seminegative definite if we drop the the assumption on G to be nilpotent or on the metric to be SKT. By using (2) with respect to a unitary frame {Z r } we easily get
ρ B = 2 3 e 12 − 2 3 e 13 + 4 3 e 23 .
In particular Again by using (2) with respect to a unitary frame {Z r } we easily get
ρ B (e 2 , Je 2 ) = 4 3 and ρ B (4e 1 + e 2 , J(4e 1 + e 2 )) = − 4 3 which implies that ρ B is not seminegative definite as (1, 1)-form.ρ B = −e 12 − 1 2 e 23 ,
which implies that ρ B is not seminegative definite as (1, 1)-form.
2. Non-existence of invariant SKT metrics satisfying (ρ B ) 1,1 = λω
In this section we observe that our proposition 1.1 easily implies that on a 2-step nilpotent Lie groups there are no SKT invariant metrics such that (ρ B ) 1,1 = λω for some constant λ. This result is already known: the case λ = 0 was studied in [4], while the case λ = 0 follows from [5].
Indeed in the setting of proposition 1.1, if we assume (ρ B ) 1,1 = λω, then, taking into account that the center of G in not trivial, formula (1) implies λ = 0 and from proposition 1.1 it follows [g 1,0 , g 0,1 ] = 0. Therefore if {ζ k } is a unitary co-frame in g we havē
∂ t ω t = −(ρ B ωt ) 1,1 , ω |t=0 = ω , where ρ B
ωt is computed with respect to ω t and the superscript "1, 1" is the (1, 1)-component with respect to J. The flow was introduced in [10] and then investigated in [3,10,11,12,13] and it is a powerful tool in SKT geometry.
In [6] it is proved that on a 2-step nilpotent Lie group the flow has always a long-time solution for any initial invariant datum. The proof makes use of the bracket flow device introduce by Lauret in [9].
In our setting, let G be a 2-step nilpotent Lie group with a left-invariant complex structure J and consider the PCF equation starting form an invariant SKT form ω. The solution ω t holds invariant for every t and, therefore, the flow can be regarded as on ODE on Λ 2 g * ⊗ g, where g is the Lie algebra of G. The bracket flow device consists in evolving the Lie bracket on g instead of the form ω. For this purpose one considers the bracket variety A consisting on the elements λ ∈ Λ 2 g * ⊗ g such that
λ(λ(X, Y ), V )) = 0 ,(3)λ(JX, JY ) − Jλ(JX, Y ) − Jλ(X, JY ) − λ(X, Y ) = 0 ,(4)∂ λ∂λ ω = 0 .(5)
for every X, Y, V ∈ g, where the operators ∂ λ and∂ λ are computed by using the bracket λ. Any λ ∈ A gives a structure of 2-step nilpotent Lie algebra to g such that (J, ω) is a SKT structure. It turns out that the PCF is equivalent to a bracket flow type equation, i.e. an ODE in A. The equivalence between the two equations is obtained by evolving the initial bracket µ of g as
µ t (X, Y ) = h t µ(h −1 t X, h −1 t Y ) , X, Y ∈ g , being h t the curve in End(g) solving d dt h t = − 1 2 h t P ωt , h |t=0 = I
and P ωt ∈ End(g) is defined by
ω t (P ωt X, Y ) = 1 2 ρ B ωt (X, Y ) + ρ B ωt (JX, JY ) .
The form ω t reads in terms of h t as
ω t (X, Y ) = ω(h t X, h t Y ) .
Now in view of formula (1) ρ B ωt (X, ·) = 0 for every X ∈ ξ and then ω t (X, ·) = ω(X, ·) for every X ∈ ξ, where ξ is the center of µ. Let ξ ⊥ be the g-orthogonal complement of ξ in g and let g t be the Hermitian metric corresponding to the solution to the PCF equation starting from ω. Then d dt g t (X, ·) = 0 for every X ∈ ξ and g t preserves the splitting g = ξ ⊕ ξ ⊥ and the flow evolves only the component of g in ξ ⊥ × ξ ⊥ . It follows that h t preserves the splitting g = ξ ⊕ ξ ⊥ and h t|ξ = I ξ .
Since (g, µ) is 2-step nilpotent, then µ(X, Y ) ∈ ξ for every X, Y ∈ g and
µ t (X, Y ) = µ(h −1 t X, h −1 t Y ) . Therefore d dt µ t (X, Y ) = −µ(h −1 tḣ t h −1 t X, h −1 t Y ) − µ(h −1 t X, h −1 tḣ t h −1 t Y ) = −µ t (ḣ t h −1 t X, Y ) − µ t (X,ḣ t h −1 t Y ) = 1 2 µ t (P µt X, Y ) + 1 2 µ t (X, P µt Y ) ,
where for any λ ∈ A we set ω(P λ X, Y ) = i 1 2 n r=1 g(λ(X, Y ), λ(Z r ,Z r )) + g(λ(JX, JY ), λ(Z r ,Z r )) being {Z r } an arbitrary g-unitary frame and in the last step we have used h t P ωt = P µt h t .
Hence the bracket flow equations writes as d dt µ t (X, Y ) = 1 2 µ t (P µt X, Y ) + 1 2 µ t (X, P µt Y ) , µ |t=0 = µ and its solution satisfies d dt g(µ t , µ t ) = 2g(μ t , µ t ) = 4 2n r,s=1 g(µ t (P µt e r , e s ), µ t (e r , e s )) being {e r } an arbitrary g-orthonormal frame. In view of proposition 1.1 all the eigenvalues of any P µt are nonpositive. Fixing t and taking as {e r } an orthonormal basis of eigenvectors of P µt we get d dt g(µ t , µ t ) = 4 2n r,s=1 a r g(µ t (e r , e s ), µ t (e r , e s )) ≤ 0 .
Therefore d dt g(µ t , µ t ) ≤ 0 and the PCF is defined in [0, ∞).
Acknowledgements. The authors are grateful to Anna Fino for very useful conversations.
Date:
July 28, 2017. 2000 Mathematics Subject Classification. Primary 53C15 ; Secondary 53B15, 53C30. This work was supported by by G.N.S.A.G.A. of I.N.d.A.M. 1
Example 1 . 3 .
13Let g be the solvable unimodular Lie algebra with structure equations de 1 = 0, de 2 = −e 13 , de 3 = e 12 , de 4 = −e 23 , equipped with the complex structure Je 1 = e 4 and Je 2 = e 3 and the SKT metric 1 ⊗ e 3 + e 3 ⊗ e 1 ) − 1 2 (e 2 ⊗ e 4 + e 4 ⊗ e 2 ) .
Example 1. 4 .
4Let (g, J) be the 2-step nilpotent Lie algebra with structure equations de 1 = de 2 = de 3 = 0 , de 4 = e 12 , de 5 = −e 23 , de 6 = e 13 , and equipped with the complex structure Je 1 = e 2 , Je 3 = e 4 and Je 5 = e 6 and the non-3 ⊗ e 6 + e 6 ⊗ e 3 ) − 1 2 (e 4 ⊗ e 5 + e 5 ⊗ e 4 ) .
can write ∂ζ k = c k rs ζ r ∧ ζ s . for some c k rs in C. a ∧ ζ b ∧ζ r ∧ζ s and the SKT assumption implies that all the c k rs 's vanish in contrast to the assumption on G to be not abelian.3. Long-time existence of the pluriclosed flow on 2-step nilmanifoldsThe pluriclosed flow (PCF) is a parabolic flow of Hermitian metrics which preserves the SKT condition. The flow is defined on an SKT manifold (M, ω) as
The pluriclosed flow on nilmanifold and tamed symplectic forms. R Arroyo, R Lafuente, in preparationR. Arroyo and R. Lafuente, The pluriclosed flow on nilmanifold and tamed symplectic forms, in preparation.
A local index theorem for non-Kähler manifolds. J.-M Bismut, Math. Ann. 2844J.-M. Bismut, A local index theorem for non-Kähler manifolds. Math. Ann. 284 (1989), no. 4, 681-699.
Homogeneous Solutions of Pluriclosed Flow on Closed Complex Surfaces. J Boling, J. Geom. Anal. 263J. Boling, Homogeneous Solutions of Pluriclosed Flow on Closed Complex Surfaces. J. Geom. Anal. 26 (2016), no. 3, 2130-2154.
Static SKT metrics on Lie groups. N Enrietti, Manuscripta Math. 1403-4N. Enrietti, Static SKT metrics on Lie groups, Manuscripta Math. 140 (2013), no. 3-4, 557-571.
Tamed symplectic forms and strong Kähler with torsion metrics. N Enrietti, A Fino, L Vezzoni, J. Symplectic Geom. 102N. Enrietti, A. Fino and L. Vezzoni, Tamed symplectic forms and strong Kähler with torsion metrics. J. Symplectic Geom. 10, n. 2 (2012), 203-223.
The pluriclosed flow on nilmanifolds and Tamed symplectic forms. N Enrietti, A Fino, L Vezzoni, J. Geom. Anal. 252N. Enrietti, A. Fino and L. Vezzoni, The pluriclosed flow on nilmanifolds and Tamed symplectic forms, J. Geom. Anal. 25 (2015), no. 2, 883-909.
Families of strong KT structures in six dimensions. A Fino, M Parton, S M Salamon, Comment. Math. Helv. 792A. Fino, M. Parton and S. M. Salamon, Families of strong KT structures in six dimensions. Comment. Math. Helv. 79 (2004), no. 2, 317-340.
Hermitian connections and Dirac operators. P Gauduchon, Boll. Un. Mat. Ital. B. 7supplP. Gauduchon, Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B (7) 11 (1997), no. 2, suppl., 257-288.
The Ricci flow for simply connected nilmanifolds. J Lauret, Comm. Anal. Geom. 195J. Lauret, The Ricci flow for simply connected nilmanifolds. Comm. Anal. Geom. 19 (2011), no. 5, 831-854.
A parabolic flow of pluriclosed metrics. J Streets, G Tian, Int. Math. Res. Notices. J. Streets and G. Tian, A parabolic flow of pluriclosed metrics. Int. Math. Res. Notices (2010), 3101-3133.
Regularity results for pluriclosed flow. J Streets, G Tian, Geom. Topol. 174J. Streets and G. Tian, Regularity results for pluriclosed flow. Geom. Topol. 17 (2013), no. 4, 2389-2429 .
Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds. J Streets, Comm. Partial Differential Equations. 412J. Streets, Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds. Comm. Partial Differential Equations 41 (2016), no. 2, 318-374.
Pluriclosed flow on manifolds with globally generated bundles. J Streets, Complex Manifolds. 3J. Streets, Pluriclosed flow on manifolds with globally generated bundles, Complex Manifolds 3 (2016), 222-230.
A note on canonical Ricci forms on 2-step nilmanifolds. L Vezzoni, Proc. Amer. Math. Soc. 1411L. Vezzoni, A note on canonical Ricci forms on 2-step nilmanifolds. Proc. Amer. Math. Soc. 141 (2013), no. 1, 325-333.
Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail address: mattia. G Dipartimento Di Matematica, Peano, [email protected], [email protected] di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail address: [email protected], [email protected]
| []
|
[
"Vortex-like kinematic signal, spirals, and beam smearing effect in the HD 142527 disk",
"Vortex-like kinematic signal, spirals, and beam smearing effect in the HD 142527 disk"
]
| [
"Y Boehler \nUniv. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance\n",
"F Ménard \nUniv. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance\n",
"C M T Robert \nUniv. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance\n",
"A Isella \nDepartment of Physics and Astronomy\nRice University\nMain Street77005HoustonUSA\n",
"C Pinte \nMonash University\nWellington Rd, Clayton VIC 3800Australie\n",
"J.-F Gonzalez \nUniv Lyon\nUniv Claude Bernard Lyon 1Ens de Lyon\n\nCentre de Recherche Astrophysique de Lyon\nUMR5574\nCNRS\nF-69230Saint-Genis-LavalFrance\n",
"G Van Der Plas \nUniv. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance\n",
"E Weaver \nDepartment of Physics and Astronomy\nRice University\nMain Street77005HoustonUSA\n",
"R Teague \nCenter for Astrophysics\nHarvard & Smithsonian\n60 Garden Street02138CambridgeMAUSA\n",
"H Garg \nMonash University\nWellington Rd, Clayton VIC 3800Australie\n",
"H Méheut \nLaboratoire Lagrange\nUniversité Côte d'Azur\nObservatoire de la Côte d'Azur\nCNRS\nBd de l'Observatoire34229, 06304Nice cedex 4CSFrance\n"
]
| [
"Univ. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance",
"Univ. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance",
"Univ. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance",
"Department of Physics and Astronomy\nRice University\nMain Street77005HoustonUSA",
"Monash University\nWellington Rd, Clayton VIC 3800Australie",
"Univ Lyon\nUniv Claude Bernard Lyon 1Ens de Lyon",
"Centre de Recherche Astrophysique de Lyon\nUMR5574\nCNRS\nF-69230Saint-Genis-LavalFrance",
"Univ. Grenoble Alpes\nCNRS\nF-38000GrenobleIPAGFrance",
"Department of Physics and Astronomy\nRice University\nMain Street77005HoustonUSA",
"Center for Astrophysics\nHarvard & Smithsonian\n60 Garden Street02138CambridgeMAUSA",
"Monash University\nWellington Rd, Clayton VIC 3800Australie",
"Laboratoire Lagrange\nUniversité Côte d'Azur\nObservatoire de la Côte d'Azur\nCNRS\nBd de l'Observatoire34229, 06304Nice cedex 4CSFrance"
]
| []
| Vortices are one of the most promising mechanisms to locally concentrate millimeter dust grains and allow the formation of planetesimals through gravitational collapse. The outer disk around the binary system HD 142527 is known for its large horseshoe structure with azimuthal contrasts of ∼ 3-5 in the gas surface density and of ∼ 50 in the dust. Using 13 CO and C 18 O J = 3-2 transition lines, we detect kinematic deviations to the Keplerian rotation, which are consistent with the presence of a large vortex around the dust crescent, as well as a few spirals in the outer regions of the disk. Comparisons with a vortex model suggest velocity deviations up to 350 m s −1 after deprojection compared to the background Keplerian rotation, as well as an extension of ± 40 au radially and ∼ 200 • azimuthally, yielding an azimuthal-to-radial aspect ratio of ∼ 5. Another alternative for explaining the vortex-like signal implies artificial velocity deviations generated by beam smearing in association with variations of the gas velocity due to gas pressure gradients at the inner and outer edges of the circumbinary disk. The two scenarios are currently difficult to differentiate and, for this purpose, would probably require the use of multiple lines at a higher spatial resolution. The beam smearing effect, due to the finite spatial resolution of the observations and gradients in the line emission, should be common in observations of protoplanetary disks and may lead to misinterpretations of the gas velocity, in particular around ring-like structures. | 10.1051/0004-6361/202040089 | [
"https://arxiv.org/pdf/2103.13474v3.pdf"
]
| 232,352,800 | 2103.13474 | f632dc24ec56a4ecc3e1d3841816a1d0431ed22f |
Vortex-like kinematic signal, spirals, and beam smearing effect in the HD 142527 disk
April 23, 2021 April 23, 2021
Y Boehler
Univ. Grenoble Alpes
CNRS
F-38000GrenobleIPAGFrance
F Ménard
Univ. Grenoble Alpes
CNRS
F-38000GrenobleIPAGFrance
C M T Robert
Univ. Grenoble Alpes
CNRS
F-38000GrenobleIPAGFrance
A Isella
Department of Physics and Astronomy
Rice University
Main Street77005HoustonUSA
C Pinte
Monash University
Wellington Rd, Clayton VIC 3800Australie
J.-F Gonzalez
Univ Lyon
Univ Claude Bernard Lyon 1Ens de Lyon
Centre de Recherche Astrophysique de Lyon
UMR5574
CNRS
F-69230Saint-Genis-LavalFrance
G Van Der Plas
Univ. Grenoble Alpes
CNRS
F-38000GrenobleIPAGFrance
E Weaver
Department of Physics and Astronomy
Rice University
Main Street77005HoustonUSA
R Teague
Center for Astrophysics
Harvard & Smithsonian
60 Garden Street02138CambridgeMAUSA
H Garg
Monash University
Wellington Rd, Clayton VIC 3800Australie
H Méheut
Laboratoire Lagrange
Université Côte d'Azur
Observatoire de la Côte d'Azur
CNRS
Bd de l'Observatoire34229, 06304Nice cedex 4CSFrance
Vortex-like kinematic signal, spirals, and beam smearing effect in the HD 142527 disk
April 23, 2021 April 23, 2021Astronomy & Astrophysics manuscript no. HD142527-kinematicsProtoplanetary disks -stars: individual: HD 142527 -Methods: observational -submillimetre: planetary systems
Vortices are one of the most promising mechanisms to locally concentrate millimeter dust grains and allow the formation of planetesimals through gravitational collapse. The outer disk around the binary system HD 142527 is known for its large horseshoe structure with azimuthal contrasts of ∼ 3-5 in the gas surface density and of ∼ 50 in the dust. Using 13 CO and C 18 O J = 3-2 transition lines, we detect kinematic deviations to the Keplerian rotation, which are consistent with the presence of a large vortex around the dust crescent, as well as a few spirals in the outer regions of the disk. Comparisons with a vortex model suggest velocity deviations up to 350 m s −1 after deprojection compared to the background Keplerian rotation, as well as an extension of ± 40 au radially and ∼ 200 • azimuthally, yielding an azimuthal-to-radial aspect ratio of ∼ 5. Another alternative for explaining the vortex-like signal implies artificial velocity deviations generated by beam smearing in association with variations of the gas velocity due to gas pressure gradients at the inner and outer edges of the circumbinary disk. The two scenarios are currently difficult to differentiate and, for this purpose, would probably require the use of multiple lines at a higher spatial resolution. The beam smearing effect, due to the finite spatial resolution of the observations and gradients in the line emission, should be common in observations of protoplanetary disks and may lead to misinterpretations of the gas velocity, in particular around ring-like structures.
Introduction
For a couple of years, observations with the Atacama Large Millimeter Array (ALMA) have detected numerous deviations to the Keplerian rotation in protoplanetary disks, most of which are consistent with the presence of embedded Jupiter-mass planets interacting with the disk (Teague et al. 2018a;Pinte et al. 2018;Casassus & Pérez 2019). The first detections were found in the multi-ringed disk around HD 163296. Using azimuthal averaging, Teague et al. (2018a) studied the gas kinematics which is sensitive to radial pressure gradients and therefore a direct probe for local variations of the gas surface density. With this method, they found gaps in the gas, possibly carved by embedded planets and colocated with gaps previously discovered in the dust. Pinte et al. (2018) found a local deviation, also called velocity kink, in the outer regions of the gaseous disk at ∼ 260 au, which they interpreted as a perturbation due to a massive planet. Since then, numerous kinks, wiggles, and Doppler flips have been discovered in other protoplanetary disks Casassus & Pérez 2019;Pinte et al. 2020), suggesting the presence of numerous planets.
Regarding their formation, theories and simulations have long shown that large dust grains (i.e., ≥ 10-100 microns) must be trapped in local pressure maxima to avoid processes such as inward radial drift, bouncing, and grains fragmentation (Weidenschilling 1977;Birnstiel et al. 2010). When the dust-to-gas ratio in these traps reaches a value between 0.1 and 1 (with an initial value of 0.01), hydrodynamic instabilities such as the streaming instability can be triggered and potentially lead to the formation of planetesimals (Youdin & Goodman 2005;Johansen & Youdin 2007;Bai & Stone 2010;Raettig et al. 2015;Auffinger & Laibe 2018).
The process of dust trapping is likely common in protoplanetary disks. Comparisons of dust ring structures with gas emission suggest that dust grains are trapped in radial gas pressure maxima in several disks (Dullemond et al. 2018;Rosotti et al. 2020). Evidence of both radial and azimuthal dust trapping has also been found in approximately ten protoplanetary disks so far.
These dust concentrations can present large variations in magnitude, be single or multiple. They can be located in dust rings such as in V1247 Orionis (Kraus et al. 2017), MWC 758 Dong et al. 2018;, HD 143006 , and HD 135344B (van der Marel et al. 2016;Cazzoletti et al. 2018), or they can be characterized by a large horseshoe structure such as in HD 142527 (Casassus et al. 2013;Muto et al. 2015;Boehler et al. 2017;Soon et al. 2019), IRS 48 (van der Marel et al. 2013Calcino et al. 2019), AB Aur (Tang et al. 2012), and SZ91 (van der Marel et al. 2018).
Observations of these asymmetries at multiple wavelengths indicate that grains with a larger size a, proportional to λ/(2π), have a more compact spatial distribution (van der Marel et al. 2015;. Indeed, the efficiency of the dust trapping depends on the Stokes number (i.e., the ratio of the dust stopping time to their orbital period), which is proportional to the grain size over gas density. The strongest effect of drag is expected for a Stokes number of ∼ 1, while dust grains probed with ALMA have a Stokes number generally estimated around ∼ 10 −2 -10 −1 . It is therefore likely that the dust trapping process is much more frequent than currently observed with ALMA. Its study will require longer wavelengths to probe grains with a Stokes number close to 1, even in dense gas areas (van der Marel et al. 2020), and to be less limited by the dust optical depth.
One of the main theories regarding the production of azimuthal dust concentrations invokes anticyclonic vortices (Barge & Sommeria 1995;Lyra & Lin 2013; Baruteau & Zhu 2016;Sierra et al. 2017), which can form via the Rossby wave instability (Lovelace et al. 1999;Li et al. 2000), for instance in a steep gradient in density (de Val-Borro et al. 2007;Zhu & Stone 2014b) or in viscosity (Varnière & Tagger 2006;Regály et al. 2012). The edge of a disk cavity or of a ring is thus a favorable site for vortex formation. However, these vortices also require a low viscosity to form, with a turbulent parameter α 10 −3 (Shakura & Sunyaev 1973;Zhu & Stone 2014b).
More recently, numeric simulations have also shown that a companion with a mass ratio q ≥ 0.05 can create an eccentric cavity and trigger an azimuthal clump in the gas (Shi et al. 2012;Ragusa et al. 2017). For even larger mass ratios, the azimuthal gas contrast may reach a value between two and four in steady state, similar to what is observed in HD 142527 Ragusa et al. 2020). Performed with a relatively high turbulent viscosity (α ∼ 5 × 10 −3 ), these simulations did not produce vortices but were still able to trap dust particles in the gas clump, which is rotating at Keplerian velocity.
Apart from the presence of a massive companion, the main physical property that may favor one or the other scenario is thus the gas viscosity. With the level of turbulence in disks being poorly known (Flaherty et al. 2020), it is unclear whether vortices can actually develop. Only the detection of their kinematic signature will undoubtedly confirm their presence. The study of the gas kinematics is therefore the key element for understanding the origin of the dust concentrations in protoplanetary disks.
The binary system HD 142527 is located at 157 ± 2 pc, based on the stellar parallax (Gaia Collaboration et al. 2018), and consists of a ∼ 2.1 M Herbig star and of a ∼ 0.3 M companion in an eccentric and non-coplanar orbit, currently at a distance of about 12 au from the central star (Biller et al. 2012;Claudi et al. 2019). It is surrounded by a bright circumbinary disk with an extremely large extent, from 100 au to 300 au in the dust emission and up to 800 au in the gas emission. Observations and models of the outer disk reveal a horseshoe morphology with an azimuthal contrast of ∼ 3-5 in the gas surface density and of ∼ 50 in the dust surface density Boehler et al. 2017), leading to a gas-to-dust ratio of ∼ 1 at the north of the disk (Yen & Gu 2020).
Moreover, Soon et al. (2019) found an azimuthal variation of the spectral index β, consistent with a dichotomy between small micron-size grains, coupled to the gas, and large millimeter grains strongly concentrated in the gas horseshoe structure, in very good agreement with the process of dust trapping in an azimuthal pressure maximum. The first piece of evidence of non-Keplerian gas motions in the circumbinary disk was found by Yen & Gu (2020) using the C 18 O J=1-0 transition line. This suggests a radial gas pressure bump in the north of the disk, similarly to what has been observed across the rings in the HD 163296 and AS 209 systems (Teague et al. 2018a,c). All these indications of dust trapping, as well as the large size and brightness of the circumbinary disk around HD 142527, make of this system the perfect target to look for the existence of a vortex in a protoplanetary disk.
Here we present our data on the circumbinary disk around HD 142527 and perform a detailed analysis of the gas kinematics (Sect. 2). By comparison with models, we then show that the measured deviations are consistent with the presence of a large vortex (Sect. 3). In Sect. 4, we examine biases in the measurement of the gas velocity due to beam smearing and inspect how they may compare with the current observations around the horseshoe structure. In Sect. 5, we discuss the vortex scenario and develop strategies to distinguish true velocity signals from artifacts, both in HD 142527 and around kinks, spirals, and ringlike structures present in other disks. In sect. 6, we summarize our findings.
Observations
Morphology of the dust and gas emission
This work is based on the analysis of the observations of the HD142527 disk obtained with ALMA (project 2012.1.00725.S), and already published in Boehler et al. (2017). We refer the reader to this paper for a description of the observational setup and data calibration. For the sake of completeness, we show in Fig. 1 images of the dust continuum, as well as the spectrally integrated intensity (moment 0) maps of the 13 CO and C 18 O J=3-2 line emission obtained in ALMA band 7 (∼ 345 GHz). For each pixel in the integrated emission maps, we only kept the channels with a signal-to-noise higher than 5 σ. This procedure slightly underestimates the total emission but yields a better signal to noise, especially in the outer regions where line emission only comes from a few channels.
The major axis of the disk has a position angle (PA) of -19 • relative to celestial north. The disk is rotating in the clockwise direction and has an inclination of 27 • (Fukagawa et al. 2013), with the far side toward the east (i. e., on the left in the figures). The azimuthal angle θ starts from the major axis (north side) and is counted positively counterclockwise. It is measured in the disk plane and therefore slightly differs, due to the disk inclination, from the usual PA measured in the image plane. A few azimuthal angles are indicated in Fig. 1, spaced by 45 • .
The dust emission around the binary system HD 142527 has a horseshoe structure with a maximum in emission of 93 mJy beam −1 , or 34.0 K in brightness temperature taking the inverse of the Planck function. It is located at θ = 52 • and at a radius of ∼ 166 au, taking a distance of 157 pc for the system. The dust emission is not azimuthally symmetric around the maximum of emission but features a secondary maximum in the clockwise direction at θ = -20 • with a value of 71 mJy beam −1 (or 27.5 Fig. 1. From left to right: Dust emission, 13 CO and C 18 O J = 3-2 integrated line emission after continuum subtraction around the binary system HD 142527. The spatial resolution is indicated by the white ellipse at the bottom-left corner of the panels. Its value is 0."207 × 0."178 for the dust emission, obtained using super-uniform weighting, and 0."31 × 0."27 for the gas emission, obtained with the Briggs parameter fixed to 0.5. The rms noise is 90 µJy beam −1 for the dust emission, 5.9 mJy beam −1 per channel in 13 CO and 7.8 mJy beam −1 in C 18 O. The integrated images for 13 CO and C 18 O only take channels with a signal larger than 5 σ into account. K). This double-peaked structure may trace a similar structure in dust surface density or may only come from a local decrease in temperature due to the shadow, seen in infrared thermal emission and scattered light (Verhoeff et al. 2011;Marino et al. 2015), cast by the inner disk surrounding the main star on the circumbinary disk.
Another tracer of the disk density is the gas through the 13 CO and C 18 O emission lines. Both isotopologues are detected inside the disk cavity, and out to large radii, contrary to the millimeter dust emission that is concentrated in a smaller radial range due to radial drift and dust trapping Boehler et al. 2017). The apparent depletion of the gas emission in the north of the disk, at about 200 au from the central star, does not imply a local decrease in the gas density and/or temperature, but is an artifact mainly produced by the continuum subtraction method. This process overestimates the amount of dust emission to remove as it does not take into account that dust emission can be absorbed at the molecular line frequency (Boehler et al. 2017;Weaver et al. 2018). The 13 CO J=3-2 and C 18 O optical depths are estimated to be about 10-15 and ∼ 2-3 at the north side of the circumbinary disk. In our observations, the 13 CO and C 18 O emission decrease in the horseshoe structure by about 50% and 60% at the center of the line after continuum subtraction.
Additionally, part of the 13 CO and C 18 O emission from the back side of the disk can be absorbed by dust particles located in the midplane Rab et al. 2020). This process can theoretically delete up to half of the emission if the line is optically thin and the dust highly optically thick. It can have a similar decrease for optically thick molecules if the emission from the front and back molecular layers consist of two different lines. This happens when the disk is sufficiently inclined, with gas emission layers at high altitude, such that two different radii, and then two different velocities, are probed along the lineof-sight (Teague et al. 2018c). The absorption of the line emission by the dust is, however, probably not the dominant process around the horseshoe structure, given the magnitude of the continuum subtraction effect, and the fact that the considered transition lines are optically thick and the disk faintly inclined.
Overview of the gas kinematics
The velocity map from the 13 CO J = 3-2 is shown on the left panel of Fig. 2. The velocity was obtained using the intensity weighted method, or moment 1 of the velocity in CASA (Mc-Mullin et al. 2007). This is the method we favored in the present work as the peak emission method appeared more sensitive to the rms noise within our data. A comparison of both methods is, however, given in Appendix B and visible in Fig. B.1. A careful study of the velocity can be performed by comparing the gas kinematics with the projected Keplerian velocity along the lineof-sight :
V proj (r, θ) = (GM /r) 0.5 sin(i) cos(θ),(1)
where G is the Newtonian gravitational constant, M the stellar mass, i the disk inclination, r the orbital radius, and θ the azimuthal angle. We discarded the inner 100 au of the system because our current spatial resolution prevent us to sample precisely the disk velocity in the inner region of the disk, but also to avoid known perturbations of the gas flow due to the binary (Casassus et al. 2015). We used a geometrically thin disk for the fit because we could not precisely constrain the gas scale height, while including it in the global fit did not change the value of the other parameters by more than 1σ. This is explained by the low inclination of the disk and the moderate spatial resolution of the data, but also by the intensity weighted method that probes both the front and back molecular layers of the disk. Teague et al. (2018c) also showed that assuming a gas scale height only created very small differences in the measured gas velocity for the disk around AS 209 due to its moderate inclination of 35 • , even with the peak emission method, the 12 CO molecule that emits at a high altitude, and a spatial resolution of 0.2 . The 2D map of the velocity deviations from the Keplerian profile is shown in the right panel of Fig. 2 while radial cuts for different azimuthal angles are presented in Fig. 3. We used the software Eddy described in which is based on an MCMC method, and takes into account the signal-to-noise in each pixel of the image. We fixed the disk inclination to 27 • and Article number, page 3 of 15 A&A proofs: manuscript no. HD142527-kinematics fit the mass of the binary, the systemic velocity of the system, the PA of the disk, and the center of rotation. Large discrepancies were found depending on the radial distance range considered. For R between 100 and 200 au, we obtain a binary mass of 2.53 ± 0.04 M , where for R > 200 au, we obtain a binary mass of 2.29 ± 0.07 M . These variations are due to the radial Doppler shifted structure denoted 1 at the north of the disk in the right panel of Fig. 2, covering a large area, and which presents super-Keplerian velocities at the inner side of its structure and sub-Keplerian velocities at the outer side.
Performing a fit throughout the circumbinary disk favors the regions at small radii due to a significantly higher signal-tonoise, yielding a binary mass of 2.48 ± 0.03 M . Therefore, to avoid biasing toward any specific radii, we proceeded in two steps. First, we used the MCMC code Eddy well outside from the cavity (i. e., R ≥ 150 au) to avoid any potential perturbations by the companion (Casassus et al. 2015) and kept the value of the center of rotation (x 0 ,y 0 ) = (15h 56min 41.872s -42d 19min 23.694s) located 40 mas south and 20 mas west of the phase center, and of the PA = 161.1 • ± 0.4 • of the disk. Second, we used a chi-square method along the minor and major axes of the disk to fit the mass and the systemic velocity V lsr of the system between 100 and 400 au, without taking radial variations of the signal-tonoise ratio into account, and obtain a binary mass of 2.36 ± 0.07 M , and a V lsr of 3730 ± 20 m s −1 .
Aside from the inner region that we cannot precisely sample and which displays non-Keplerian flows due to the presence of the binary, four kinematic features are shown in the right panel of Fig. 2. They are present independently of the exact fitting procedure used, and consist in 1) a large radial Doppler-shifted structure in the north of the disk with projected velocities up to 160 m s −1 , roughly colocated with the dust crescent and possibly tracing an anticyclonic vortex (cf section 3), 2) a redshifted arc of ∼ 180 • in azimuth at the north of the disk, which may be related to the spiral S1 observed in near-IR scattered light, in 12 CO and in 13 CO by Fukagawa et al. (2013), Christiaens et al. (2014), and Garg et al. (2020), 3) a blueshifted arc of ∼ 180 • in azimuth at the east of the disk, which may be the dynamical counterpart of the spiral S4 recently observed by Garg et al. (2020), and 4) a smaller redshifted arc at the southwestern side of the disk. All these arcs and spirals have maximum projected velocity deviations on the order of 50 m s −1 . This supports the idea that the outer region of the disk may present a radial succession of spiral and inter-spiral structures, each of them with deviations to the Keplerian rotation. Such spirals have also been recently observed at larger radii by Garg et al. (2020) using the 12 CO J=2-1 emission line.
A vortex-like kinematic signal around the dust concentration
Preliminary analysis
In this study, we focus on the radial Doppler-shifted feature, denoted (1) in the right panel of Fig. 2, and whose projected deviations to the Keplerian rotation reach 160 m s −1 , compared to ∼ 50 m s −1 along the spiral arcs. Radial cuts of the velocity deviations at different azimuthal angles are shown in Fig. 3. The velocities measured using the 13 CO and C 18 O lines give very similar values, with differences no larger than 20-30 m s −1 between ∼ 100 and ∼ 350 au. This suggests that both isotopologues emit from a similar altitude and that the precision on the velocity measurement is only of a few tens of m s −1 , well below the channel width of ∼ 110 m s −1 in the observations. The gas velocity is correctly matched by a Keplerian profile, represented by the black dashed line, on the Southern side of the disk. On the contrary, the gas velocity along the northern major axis presents a clear distinct S-profile that cannot be approximated by a single power law. The sensitivity to azimuthal motions is maximized along the major axis due to projection ef- fects. On the contrary, we are blind to radial motions because they are perpendicular to the line-of-sight. We may also observe vertical motions as their projection along the line-of-sight is independent of the azimuthal angle. However, at first order, they can probably be neglected in the radial Doppler-shifted structure as 3D simulations have shown that vertical motions around vortices in steady state are negligible (Lin 2012;Richard et al. 2013) and while spirals can present vertical motions, their projected deviations are not higher than 50 m s −1 in the rest of the disk and with a different morphology.
Taking into account the clockwise rotation of the disk with the far side being in the east (i.e., on the left), the gas rotates at a super-Keplerian speed at radii between ∼ 80 and 185 au, and at a sub-Keplerian speed at radii between ∼ 185 and 300 au, confirming the deviations observed in Yen & Gu (2020). Projected velocities are of about ± 0.16 km s −1 , or of 0.35 km s −1 after deprojection, about 10% of the background Keplerian rotation. The transition radius between these two regimes is at 185 au and corresponds to the distance at which the dust is radially concentrated in the horseshoe structure (Boehler et al. 2017;Soon et al. 2019;Yen & Gu 2020).
These azimuthal velocity deviations may only come from radial gas pressure gradients due to the outer ring-like structure (Teague et al. 2018a). It may also trace the azimuthal deviations coming from a vortex centered near this position Robert et al. 2020). The vortex presence would imply radial deviations to the Keplerian rotation. At the adjacent angles θ = 45 • and 315 • , we are equally sensitive to radial and azimuthal motions. The interpretation of the projected velocities along the line-of-sight is therefore not as direct but the S-shape previously seen along the major axis is, however, still visible, suggesting that the azimuthal behavior of the gas described at θ = 0 • is still present. In addition, velocities are slightly blueshifted at both angles indicating, at θ = 45 • , that the gas is flowing inward in the radial direction while, at θ = 315 • , gas is flowing radially outward. Finally, along the minor axis, we are only sensitive to radial velocity deviations. At the near side, outward gas motion is measured at R ∼ 185 au while at the far side, at θ = 90 • , the gas motion does not display any radial deviations.
The preliminary analysis of the velocity deviations in the north of the disk shows then that, at first glance, they are compatible with the presence of a large vortex located at a radius of 185 au, centered between 315 • and 45 • , and extending azimuthally over about half of the circumbinary disk. A sketched view of this kinematic structure denoted (1) is shown in the right panel of Fig. 2. Alternatives to this scenario are also discussed in Sect. 4.
Vortex model
To better compare the deviations to the Keplerian rotation with the kinematic signature of a vortex, a model is required. Simulations by Huang et al. (2018) and Robert et al. (2020) have shown that the vortex size and its associated velocity can largely vary as a function of the process responsible for its origin (massive planet, binary, dead zones), the underlying gas surface density, or the disk viscosity. Our goal here is not to perform a full simulation able to reproduce the vortex-like kinematic signal but, on the contrary, to use a simple model to constrain the main parameters of a possible vortex, such as its position, size, aspect ratio, and velocity.
A sketch describing the vortex parameters is shown in Fig. 4. In the polar coordinates of the disk, the flow of the gas due to a vortex is often described with elliptic streamlines of constant velocity characterized by a central position (R 0 , θ 0 ) and an aspect ratio χ = b/a, with b the major axis in the azimuthal direction and a the minor axis in the radial direction (Kida 1981;Goodman et al. 1987;Chavanis 2000;Surville & Barge 2015). The flow of the gas is null at the vortex center (R 0 , θ 0 ), then it increases with distance to that center, up the velocity V max before decreasing again to a null value. The exact vortex velocity profile cannot be measured from our current data, and we assume it has a Gaussian profile. Along the radial axis, we note R v the distance from the vortex center at which the velocity reaches its maximum, V max , and w v the half-width of the Gaussian. Azimuthally, the maximum in velocity is reached at the distance of χR v from the vortex center. Fig. 4. Sketch describing the vortex and its parameters. Such a vortex may be located on the northern side of the disk, at a similar localization than the polar ellipse denoted (1) and drawn in the right panel of Fig. 2. For simplicity, we note in the following the distance to the vortex center in Cartesian coordinates (x,y) with x = R 0 × (θθ 0 ) and y = r -R 0 . Along the radial axis, the absolute value of the vortex velocity is then:
θ r V max R v V max /e (θ 0 ,RR 0 ) V max /e W v χ.R v (x,Ry) ϕ|V(0, y)| = V max exp − |y|−Rv wv 2 .
(2)
Thereafter, we can obtain the vortex velocity at any point (x,y) of the disk by finding the ellipse, of constant velocity, which crosses both this position (x,y) and the radial axis along the vortex eye. The elliptic streamlines of the vortex can be compared to circles inclined along the azimuthal axis. At the position (x,y) and angle φ of the vortex, we are at the position (x, χ y ) and at the angle φ of such a circle. φ is defined between the radial axis and the current position, and counted counterclockwise such that:
cos(φ ) = χy x 2 + (χy) 2 .(3)
The ellipse going through the position (x,y) will then cross the radial axis of the vortex at the position y/cos(φ ), with the absolute velocity:
|V(x, y)| = |V(0, y/ cos(φ ))| = V max exp − |y/ cos(φ )|−Rv wv 2 .(4)
The final step is to calculate the velocity along the line-of-sight; the only observable quantity. This requires us to decompose the vortex velocity into its radial (V r ) and azimuthal (V θ ) components. Using the inclination i of the disk, the disk azimuthal angle θ, and the angle φ in the vortex reference frame, the projected velocity along the line-of-sight is given by:
V proj = [V r sin(θ) + V θ cos(θ)] sin(i), = |V| − sin(φ ) sin(θ) + cos(φ ) cos(θ) sin(i), = |V| cos(θ + φ ) sin(i).(5)
Comparison of the observations with the vortex model
This comparison is focused in the circumbinary disk at a radius between 100 and 350 au, where the Doppler shifted structure denoted (1) in the right panel of Fig. 2 is present. In the polar 2D map (r, θ) shown in the top panel of Fig. 5, the observed kinematic feature consists in a redshifted clump located between -50 • to 50 • and at radii between 180 and 300 au surrounded by a large blueshifted arc located at PA between -100 • and 100 • and at radii between 120 and 200 au. This velocity structure has maximum projected kinematic deviations of ∼ 150 m s −1 and is centered at ∼ 185 au, at a radius similar to the dust and gas maximum surface densities inside the dust crescent Boehler et al. 2017).
In the bottom panels of Fig. 5, we applied our vortex prescription to three different possibilities, whose parameters are given in Table 1. In models A and B, we made the hypothesis that the dust concentration is due to a single large vortex. Its size corresponds to the extension of the kinematic structure, with R v = 42 au, W v = 26 au, and an azimuthal-to-radial aspect ratio of ∼ 5. The maximum velocity reached by the vortex is set to 350 m s −1 , a value constrained by the deprojection of the observed velocity. This value is comparable to the local sound speed for a temperature of 30-40 K at the horseshoe position using the formula k b T/(µm h ), with k b the Boltzmann constant, µ = 2.3 the mean molecular weight, and m h the hydrogen mass. The only difference between Models A and B is the azimuthal position with a vortex centered at the position of the dust emission maximum (r = 185 au, θ = 50 • ) in model A and at the position of the secondary maximum (r = 185 au, θ = -20 • ) in model B. We further tested a third model (Model C) where two smaller vortices are located at the two maxima in the dust emission.
From these three models, it appears that our comparison favors the option of a large and single vortex, located near the secondary maximum in dust emission at PA = -20 • . The model with two vortices, on the contrary, does not reproduce the morphology of the kinematic signal. Furthermore, the two vortices would have a small aspect ratio of ∼ 2 to cover the extent of the kinematic deviations, and would therefore probably not with- We show in the top-center and top-right panels of Fig. 6 that measurable deviations exist even in the case of ideal observations (i.e., without noise and at a velocity resolution of 25 m s −1 ). The velocity deviations are of a few percent and can be divided into two regions. At R ≤ 150-175 au, the velocity deviations appear sub-Keplerian, meaning redshifted in the north of the disk (at PA between -90 • and 90 • ) and blueshifted in the south of the disk. On the contrary, at R ≥ 150-175 au, the gas appears mainly super-Keplerian, with an inversion of the blueshifted and redshifted azimuthal locations. The only exception is the area near the north of the major axis, which will be detailed in section 4.3.
In the top-left panel of Fig. 6, we present the velocity deviations measured in the observations. At a radius larger than 150 au, the observed kinematic signal is similar to the velocity pattern seen in the pure Keplerian model at 0.3 , even if the velocity deviations reach a velocity of 150 m s −1 in our observations, instead of only 20-80 m s −1 in our models. We observe the rounded redshifted region at ∼ 230 au, close to PA = 0 • , sur- Fig. 6. Artificial velocity deviations measured in a perfectly Keplerian disk compared to the observations. Top: Differences between the measured 13 CO J=3-2 velocity and the Keplerian prescription in a 2D-map (R,θ). Blue and red colors indicate blue-and redshifted velocities along the lineof-sight compared to the Keplerian rotation. Middle: 13 CO integrated emission. Bottom: Flux ratio I out /I in of the emission at the outer and inner edges of the synthesized beam. The left panel is surrounded by a red rectangle because the flux ratio is only an approximation in observations, being measured after beam smoothing. Left column shows the observations. The central and the right columns are the models convolved at a spatial resolution of 0.3 and 0.1 . The vertical black bar on the top-left corner of the top panels is the averaged radial spatial resolution as a function of the azimuth. rounded by the blueshifted area at θ between -90 • and 90 • , and mainly a redshifted region in the south of the disk. The similarity of these characteristics can cast doubts about the correct physical interpretation of the kinematic deviations to the Keplerian rotation and might reveal a bias in the velocity measurements to start with. Inside the cavity, at R ≤ 100-150 au, the observed velocity deviations diverge from our models and probably trace perturbations of the Keplerian flow by the companion, as observed in Casassus et al. (2015).
Line intensity gradients skew measured velocities through beam smearing
Observations are naturally limited by characteristics such as the rms noise, and the spatial and spectral resolutions. Each of them can bring systematic biases in the measurements. In the HD 142527 disk models, the two main regions in terms of velocity deviations are separated at a radius of about 150 au, corresponding to the distance at which the 13 CO J=3-2 emission peaks. This is visible in the middle line in Fig. 6 which shows the spatial distribution of the 13 CO integrated emission. In the inner regions, as the 13 CO flux increases with radius, we collect more emission from large radii than from small radii inside the synthesized beam, leading to a sub-Keplerian profile. The opposite effect happens in the outer regions where the gas emission mainly decreases with radius, leading to a super-Keplerian profile. The spurious velocity deviations in the models are then explained by the finite spatial resolution of the observations and the radial gradient of the molecular line emission. This effect, also called "beam smearing," was discussed a first time in Keppler et al. (2019) for the kinematic analysis of the PDS 70's cavity (see the appendix A.2).
A useful parameter to understand these artifacts in the velocity measurements is the radial variation of the line emission. For a given spatial resolution, this variation can be estimated by the emission ratio I out /I in between the outer and inner edges of the synthesized beams. When this ratio is greater than 1, the measured velocity becomes sub-Keplerian, and vice versa. Starting from the initial non-convolved image, this ratio can be perfectly known in the models and is shown in the bottom-center and bottom-right panels of Fig. 6. As expected, the emission ratio I out /I in is larger at 0.3 than at 0.1 and lead to stronger velocity biases. In real observations, the emission ratio can only be approached because the image received is already convolved by the observational beam. We show, however, in the bottom-left panel of Fig. 6 that the general tendency of the flux variations can be recovered, even if their magnitude is probably underestimated.
In general, we find in our models that the flux ratio I out /I in is proportional to the beam size, with ratios at 0.3 about three times larger than at 0.1 , which is an expected result when the spatial variation of the line emission is relatively smooth compared to the synthesized beam size. Furthermore, biases are also sensitive to the velocity range ∆V probed by the synthesized beam, which is proportional to the beam size, and increases at smaller distances from the star due to the steeper gradient of the Keplerian velocity. This leads to velocity artifacts generally larger at 0.3 compared to 0.1 and particularly important in the inner regions of the disk. Along the northern major axis, as shown in the top panel of Fig. 7, the velocity deviations at 0.1 are about three times smaller than at 0.3 . Going to a higher spatial resolution will have then the double advantage of i) reducing velocity biases and of ii) allowing a better determination of the value and morphology of these biases, through a more precise knowledge of the disk structure.
Dust also affects the measured gas velocity
The circumbinary disk around HD 142527 is not only made of gas but also contains a large concentration of dust particles at the north of the disk. The presence of this dust concentration, which is optically thick at 0.9 millimeter with τ Dust ∼ 2-3 Boehler et al. 2017), has an important impact on the measured gas velocity at the north of the disk, as shown in the top panel of Fig. 7. The four radial regions indicated in this figure are delimited by the radii at which the velocity shifts between a sub-and a super-Keplerian velocity at the spatial resolution of 0.3 . Regions I and IV, which contain almost no millimeter dust emission, correspond to the two regions described previously in section 4.2. The 13 CO integrated emission uniformly increases for R ≤ 135 au and decreases for R ≥ 270 au, leading to the sub-Keplerian and to the super-Keplerian profiles. The existence of the additional regions II and III, located between ∼ 135 and ∼ 270 au, is of main interest in our analysis as it corresponds to the location of the potential vortex signature in the observations, around the dust horseshoe position. As shown in the middle panel of Fig. 7, velocity biases in the models are related to the artificial and local cavity in the gas emission in the north of the disk, visible both in the models and in the observations in Fig. 1. The gas cavity is colocated with the presence of dust emission and mainly arises from the continuum subtraction process, which over-estimates the dust contribution to subtract (Boehler et al. 2017;Weaver et al. 2018), as already discussed in Section 2.1.
Independently of its origin, real or artificial, the top and bottom panels in Fig. 7 reveal that the measured velocity deviations are in good agreement with the flux variations measured after continuum-subtraction. We obtain a super-Keplerian profile in Region II (135 < R < 215 au), and a sub-Keplerian profile in Region III (215 < R < 270 au), matching the flux ratio smaller and greater than 1, reciprocally. The presence of dust has then an important impact on the measured velocity.
These deviations are not due to the continuum subtraction process, however. The measured velocity does not change before or after this process, as shown by Teague et al. (2018c) using the peak emission method, which can be performed indistinctly with or without continuum subtraction. Indeed, the emission that we subtract is the interpolation of the dust emission from adjacent line-free channels and is essentially a flat spectrum compared to the width of the line. For a typical protoplanetary disk located at a distance of ∼ 150 pc, the 13 CO J=3-2 line has a spectral width of about 1-2 MHz at a spatial resolution of 0.1-0.3 . At a line frequency ν of ∼ 330 GHz, this gives a ratio ∆ν/ν of 6 × 10 −6 . Therefore, the dust thermal emission, with a spectral index between 2 and 3.7, only varies in amplitude by about 0.001% along the spectral width of the line and does not affect the measured velocity.
On the contrary, the fact that velocity biases follow flux variations after dust-subtraction indicates that the weight given to each regions inside the synthesized beam is not the absolute magnitude of the line but its excess over a "quasi-flat spectrum," which can be zero-emission (if there is no dust) or the dust emission (if dust is present). When the gas emission becomes optically thick, the combined dust and gas emission at the line frequency is smaller than the sum of the two components taken separately, due to optical depth effects. The prominence of the line over the quasi-flat spectrum decreases and the weight given to the regions with more dust emission is then reduced for the measurement of the gas velocity.
To illustrate the artifacts in the velocity measurement due to dust concentrations, we present in the left panel of Fig. 8 a simple sketch of a disk where the gas surface density is slowly decreasing with radius and which contains a dust ring or horseshoe structure. If the gas emission is optically thick, this will create a local and sharp decrease in the gas emission when measured after continuum subtraction. Two synthesized beams are represented at the inner and outer rim location of the dust structure where flux variations, and then velocity biases, are maximum. The right panel of Fig. 8 shows qualitatively how the spectral profile of the emission line is modified at these two positions.
Without the presence of the dust structure, the measured velocity profile is slightly super-Keplerian due to the slow decrease in the gas emission with radius. On the contrary, the steep gradient in the line emission at the rims position artificially creates strong deviations to the Keplerian rotation, with a super-Keplerian rotation at the inner rim and a sub-Keplerian rotation at the outer rim. We also note that the line profile is only modified by an offset after continuum subtraction, leaving the measured velocity unchanged. A similar effect on the measured velocity will happen if the line emission coming from the back side of the disk is absorbed by dust particles lying on the midplane. The integrated emission at the ring position, and then the weight given to this location, will also be attenuated in the measurement of the gas velocity. This effect can, however, be mitigated if the front molecular layer of the disk, usually brighter, can be differentiated spectrally, as shown in 12 CO for AS 209 (Teague et al. 2018c). Such an observation, however, will still be affected by the continuum subtraction effect if the molecular line is optically thick.
Beam smearing and gas pressure gradients may also explain the vortex-like kinematic feature
The main difficulty in identifying the origin of the radial Doppler-shifted structure at the north of the disk comes from the possible confusion with artifacts in the measurement of the velocity and other kinematic signals. We have shown in the top panel of Fig. 7 that beam smearing can produce along the north major axis a pattern similar in morphology to the azimuthal velocity deviations that a vortex would produce. However, these artifacts are of lower amplitude in our model, with velocities reaching 90 m s −1 on the inner side and 20 m s −1 on the outer side, while we measure about 150-160 m s −1 on both sides in the observations. Besides, the kinematic structure due to beam smearing is located at 215 au, at the outer edge of the artificial gas cavity due to continuum subtraction, while the radial Doppler-shift in the observations is centered at 185 au, at the radius where both dust and gas reach their maximum surface density. It is also more extended radially, from 85 to 300 au, compared to 135 to 270 au in the models. It therefore appears unlikely that the beam smearing effect alone can produce the azimuthal velocity deviations. However, azimuthal kinematic deviations are also expected on the inner and outer sides of the circumbinary disk in HD 142527 due to radial gas pressure gradients (Teague et al. 2018a;Yen & Gu 2020). This process may produce super-Keplerian velocities at the inner edge of the horseshoe structure and sub-Keplerian velocities outward, as for an anticyclonic vortex. On the south side of the disk, possible deviations are up to 30 m s −1 . With a gas surface density three to five times larger on the northern side of the disk, significant non-Keplerian motions may appear and add to the beam smearing effect to possibly produce the large velocity deviations that we observe. Therefore, the presence of a vortex should be more easily constrained through radial motions to not be mistaken with azimuthal motions due to steep gas pressure gradients. At the northern side of the circumbinary disk, we observe a shift toward blueshifted emission on the order of 50 m s −1 at θ around ∼ 45 • -90 • and ∼ 270 • -315 • , as already mentioned in Sect. 3.1. This blueshifted emission is visible in Figs. 2, 3, and 5 and may correspond to inward motions of the gas at θ ∼ 45 • -90 • and outward motions at θ ∼ 270 • -315 • , similar to what would be expected for an anticyclonic vortex. However, as shown in Fig. 6, the beam smearing effect also predicts blueshifted emission at these angles due to a global decrease in the emission with radii. It is then currently very difficult to assess if the signal is due to a real vortex or is produced by beam smearing associated with gas pressure gradients.
In addition, we note that some uncertainties also exist in the comparison of the measured velocity with the Keplerian rotation. First, as shown by Yen & Gu (2020) in their appendix, uncertainties in the disk inclination of 2 • may lead to errors in the measured velocity of about 25 m s −1 at intermediate angles. In our approach, the disk inclination was fixed but its uncertainty from previous measurements is on the order of 1 • (Fukagawa et al. 2013;Yen & Gu 2020). Possible errors due to this process are then limited for our measurements and the general pattern predicted by Yen & Gu (2020) is not visible in the residuals of the velocity deviations. Actually, most of the uncertainties in the measurement of the Keplerian rotation in HD 142527 come from the complexity of the kinematic signal that presents large deviations from a pure Keplerian profile, in particular at the north of the disk, as shown in Fig. 7. Our fit was done using the same weight to each radii between 100 and 400 au. If we would have given more weight to the inner region of the disk due to the higher local signal-to-noise, the blueshifted emission detected at PA around 45 • -90 • and 270 • -315 • would have been reduced. On the contrary, if velocity deviations due to gradients in the gas surface density or due to beam smearing are in fact larger on the inner side of the disk, as our model suggests, the true Keplerian rotation would be in reality slightly smaller and the blueshifted emission detected at these angles would then be more pronounced.
Discussion
Observations and theories regarding the possible existence of a large vortex around HD 142527
Based on 13 CO and C 18 O J=3-2 data at a spatial resolution of 0.3 , current observations of the gas kinematics in the circumbinary disk around HD 142527 is consistent with the existence of a large vortex, even if complementary observations would be necessary to distinguish it from other possibilities. When compared with a model, the center of this anticyclonic vortex is located at 185 au from the star, at the estimated radius where both dust and gas surface densities reach their maximum (Boehler et al. 2017), with maximum velocity deviations of ∼ 150-160 m s −1 in the azimuthal direction along the major axis, meaning ∼ 350 m s −1 after deprojection. It is extended radially on each side on about two pressure scale heights, and azimuthally over ∼ 200 • , yielding an azimuthal-to-radial aspect ratio χ of ∼ 5. Using 12 CO, 13 CO, and C 18 O J = 2-1 emission lines, Garg et al. (2020) could estimate the morphology of the disk cavity and revealed a steep radial gradient in gas surface density at the inner edge of the circumbinary disk, making it a favorable site for vortex formation. Radially wide vortices (i.e., larger than a gas pressure scale height radially) have been predicted in simulations that include the displacement of the system barycenter due to the lopsided structure of the disk (Mittal & Chiang 2015;Baruteau & Zhu 2016). Such vortices should also be azimuthally extended, with ratio between 4 and 6, to withstand the elliptic instability (Lesur & Papaloizou 2009). Recently, Robert et al. (2020) performed numerical simulations in cavity-hosting disks with the formation of very extended vortices. For vertical aspect ratios H/R 0.13, with H the pressure scale height in the disk, they predict maximum velocity deviations projected along the line-of-sight of 150 m s −1 for a disk inclination similar to the one in HD 142527. This vertical aspect ratio, while slightly larger than usually considered in protoplanetary disks, may correspond to the vertical geometry of the circumbinary disk around HD 142527 as the inner rim is directly illuminated by the Herbig star. A large vortex in HD 142527, with a kinematic signal similar to the one observed around the horseshoe structure, is then theoretically possible.
Another interesting result of the simulations performed by Mittal & Chiang (2015) and Baruteau & Zhu (2016) is that dust grains can concentrate at a different azimuth than the vortex due to the indirect force exerted by the disk self-gravity. In brief, only small grains with a Stokes number ∼ 0.01 concentrate in the eye of the vortex due to their strong coupling with the gas while larger grains will concentrate generally ahead of it, potentially with a difference of 90 • in azimuth, and giving a double peak structure to the continuum emission. At first glance, it is tempting to connect this scenario with the double-peaked structure observed in the dust emission in the circumbinary disk around HD 142527. Nevertheless, in the observations, the maximum in the dust emission is located at PA = 50 • , 70 • behind the azimuthal position of the vortex and of the secondary maximum in dust emission, both located at PA ∼ -20 • , contrary to the predictions in Mittal & Chiang (2015) and in Baruteau & Zhu (2016).
It is also possible that the presence of the companion, not present in the previous simulations, plays an important role in the location of the large dust grains. Hammer et al. (2019Hammer et al. ( , 2021 found that a large vortex characterized by a flat pressure bump would be sensitive to the overlapping of spiral density waves. Dust grains will then not concentrate in a small and centered area but in an elongated and complex structure, with a possible off-centered peak, a skewness around this peak, or even a doublepeaked structure. While elongated, the double peaked structure in the continuum emission may, however, only be due to the shadow cast by the inner disk (Verhoeff et al. 2011;Marino et al. 2015). Recent modelings of the dust distribution using wavelengths between 0.9 and 3 millimeter, and the temperature given by optically thick molecules, suggest that dust grains may in fact reach a density maximum near the major axis (Yen & Gu 2020), or even at the position of the secondary maximum (Soon et al. 2019), at approximately the vortex position. A solid understanding of the dust concentration process in the HD 142527 system will require a better mix of long wavelengths and high spatial resolution than the 0.30-0.43 currently available at λ > 1 mm.
Vortex VS binary
In an ALMA survey of disks presenting large central cavities, van der Marel et al. (2020) revealed that dust asymmetries were only present in disks with a sufficiently low gas surface density. This suggests that local gas pressure maxima are actually common in such disks but that only dust grains with a Stokes number 10 2 are efficiently dragged into them. One possibility to create a local gas pressure is through a vortex, as already discussed in Sect. 5.1. While not mutually exclusive, another theory suggests that an azimuthal gas pressure maximum can be formed by the interaction between a binary, with a mass ratio q 0.05 with the main star, and the circumbinary disk. Numerical simulations have shown that the massive companion would create a large and eccentric cavity, leading to a gas asymmetry in the gas (Shi et al. 2012;D'Orazio et al. 2016;Ragusa et al. 2017;Price et al. 2018) rotating at Keplerian velocity and able to trap dust particles (Calcino et al. 2019;Ragusa et al. 2020).
These studies have been performed using a relatively high viscosity α = 5×10 −3 , and therefore did not produce vortices as they require α 10 −3 (Zhu & Stone 2014b). It is, however, plausible that both processes can act together at a smaller viscosity. As shown by Price et al. (2018), the HD 142527 system is particularly well suited for the binary scenario with the only massive companion (q ∼ 0.1-0.15) directly imaged in a large cavity (van der Marel et al. 2020). Other systems like AB Aur (Tang et al. 2012), IRS 48 (van der Marel et al. 2013, or HD 135344B ) also present a large cavity with a horseshoe structure and may correspond to this scenario as well. On the contrary, asymmetries located across ring structures in the outer regions of the disks, like in MWC 758, HD 143006 and V1247 Orionis, probably require the presence of a vortex to form (Baruteau et al. 2019).
The main uncertainty about the formation and viability of vortices comes from the disk viscosity whose measurements are still scarce. To our knowledge, no estimation of the disk viscosity in the circumbinary disk around HD 142527 has been performed. Apart from the notable case of DM Tau where a turbulent viscosity α of ∼ 0.08 has been measured in the upper layers of the disk using 12 CO emission, only upper limits on the order of a few times 10 −3 were found using the turbulent broadening property of the gas emission lines in HD 163296 (Flaherty et al. 2017), TW Hya (Flaherty et al. 2018;Teague et al. 2018a), MWC 480 and V4046 Sgr (Flaherty et al. 2020). Other methods, more indirect because integrating hypotheses on the grains properties, have generally suggested α values from a few 10 −4 to a few 10 −3 by estimating the dust settling degree in HL Tau (Pinte et al. 2016), or by using the radial dust dispersion across rings (Dullemond et al. 2018).
Simulations performed using a lower viscosity would help to constrain the complex dust concentration process in the HD 142527 disk. Further observations of the gas kinematics at a higher spatial resolution, for instance of 0.1 instead of 0.3 , would allow velocity biases due to beam smearing to diminish and a possible vortex signal to be more clearly distinguished from the background rotation. With a dust absorption coefficient β estimated to 1.6 by Yen & Gu (2020) at the horseshoe position, it will also be judicious to choose a lower frequency, for example with the J=2-1 transitions lines of the CO isotopologues, to significantly reduce the specific beam smearing effect around the dust crescent. Indeed, the dust optical depth would decrease by a factor of ∼ 2, significantly diminishing the dust emission and the depth of the artificial gas cavity.
Beam smearing with kinks, spirals, and rings
During the analysis of the disk kinematics, we pointed out the beam smearing effect due to variations of the line emission inside of the synthesized beam and which may lead to a misinterpretation of kinematic signals. Indeed, local variations of a few tens to a few hundreds of m s −1 in the gas velocity have been used in the recent years to: i) probe radial pressure gradients and then gaps in the gas surface density (Teague et al. 2018a,c), ii) observe the kinematic signature of spirals (Teague et al. 2019b) and iii) to directly trace the presence of planets through the observation of Doppler flips or kinks Casassus & Pérez 2019;Pinte et al. 2020).
Velocity artifacts due to beam smearing present characteristics that may help in distinguishing them from real kinematic features. First, these artifacts should appear in regions where the line emission undergoes a steep spatial variation. This may be produced by a change in the gas surface density in the temperature, or processes such as photo-dissociation, freeze-out onto grains, chemical reactions, and optical depth effects around dust structures. Second, the artifacts will also increase in regions that present a strong gradient in the projected velocity. For a disk dominated by Keplerian motions, this will correspond to regions near the major axis with biases decreasing in azimuth with ∼ cos(θ), and in regions close to the star with biases decreasing with the radius in ∼ r −1.5 . In addition, similarly to the projected Keplerian rotation, biases will increase with the disks inclination, in sin(i), and with the system mass, in M 0.5 .
In comparison, kinks and Doppler flips are very localized features that do not reveal any preferential azimuthal locations Casassus & Pérez 2019;Pinte et al. 2020). Nor do they seem to be linked with a specific point-like cavity or source of emission in the lines considered. Therefore, these kinematic signals do not correspond to the expected velocity artifacts due to beam smearing but better agree with the presence of a planet, located potentially at any azimuthal angle and perturbing the gas motion in the radial, vertical, or azimuthal direction .
It is more difficult to directly interpret the gas kinematics around rings and spirals as both structures are generally accompanied by variations of the line emission, like around the horseshoe structure in HD 142527. The velocity artifacts created across dust rings have been shown in Fig 8 and in Sec 4.3. The possible decrease in the gas emission at the dust ring position, due to optical depth effects, will produce the measurement of a super-Keplerian and of a sub-Keplerian velocity at the inner and outer rims of the dust disk, respectively. This is similar to what is theoretically expected if dust grains are trapped in a radial pressure maximum (Kanagawa et al. 2015;Teague et al. 2018a). True kinematic signatures and artifacts being due to azimuthal motions, there will be both mostly visible around the major axis of the disk and their differentiation may require a careful analysis. Inversely, spirals would appear as maximum in the gas emission and the beam-smearing effect will artificially produce sub-Keplerian and super-Keplerian velocities on the inner and outer edges of these structures. However, compared to ring-like structures, spirals can be more easily distinguished by the variation of their kinematic signal as a function of the azimuth as significant non-azimuthal motions around spirals are also expected ).
Conclusion
We present a study focusing on the gas kinematics in the lopsided disk surrounding the binary system HD 142527 at a spatial resolution of 0.3 and at a wavelength of ∼ 0.9 millimeter. Our major findings are:
1. The main kinematic structure has a vortex-like morphology with potential azimuthal and radial motions. It is located at the north of the circumbinary disk, around the dust concentration, and has projected velocities up to 150 m s −1 compared to the Keplerian background. Three spiral-like kinematic features are also detected in the outer region of the circumbinary disk at R between 200 and 450 au. Two of them are probably related to the two most innermost spirals detected in this system (S1 and S4 following Garg et al. (2020)). 2. Using comparisons with a vortex model, the main kinematic feature is consistent with the existence of a large vortex located at R ∼ 185 au, at the radius where dust and gas reach their maximum surface densities, and at a PA of ∼ -20 • . Such a vortex may have formed through the Rossby-wave instability at the inner edge of the circumbinary disk, suggesting a relatively low turbulent viscosity α 10 −3 (Zhu & Stone 2014b). Its very large size, of about ± 40 au radially and ∼ 200 • azimuthally, suggests that the disk self-gravity plays an important role through the indirect force (Mittal & Chiang 2015;Baruteau & Zhu 2016). The relatively large velocities of approximately 350 m s −1 after deprojection, comparable to the local sound speed, have been predicted in simulations of vortices performed by Huang et al. (2018) and Robert et al. (2020).
3. Velocity measurements are, however, subject to artifacts due to variations of the line emission at a sub-synthesized beam scale. If associated with non-Keplerian motions due to gas pressure gradients at the inner and outer edges of the circumbinary disk, the beam smearing effect may also create a kinematic signal similar to an anticyclonic vortex around the horseshoe structure. Our current observations, at a spatial resolution of 0.3 , do not allow us to distinguish between these two possibilities. 4. Velocity deviations due to beam smearing should be common in protoplanetary disks and may lead to the misinterpretations of kinematic signals, in particular around ring-like structures. The principal method to reduce them is to perform observations at a higher spatial resolution. This would also allow, through a better knowledge of the disk structure, to constrain much more precisely the amplitude and localization of such artifacts. In HD 142527, a spatial resolution of 0.1 is reachable in 2-3 hours of telescope time on target and would decrease by a factor of ∼ 3 the current beam smearing effect along the major axis of the disk. where θ is the angle starting from the northern major axis and rotating counterclockwise. The red curve is the velocity measured using the intensity weighted method (mom1 in CASA) with 13 CO J=3-2. The orange and blue curves correspond to the velocity measured using the peak emission method with the 13 CO J=3-2 and C 18 O J=3-2 transition lines, respectively, with the procedure detailed in Teague & Foreman-Mackey (2018b).
Fig. 2 .
2Gas velocity of the circumbinary disk. Left: 13 CO J=3-2 velocity map performed using the intensity weighted method, and subtracted by the systemic velocity of 3.73 km s −1 . Contours are displayed from -2.75 to 2.75 km s −1 and are spaced by 0.5 km s −1 . Right: Blue-and redshifted velocities (indicated by blue and red colors) along the line-of-sight compared to the Keplerian rotation, as expressed by eq. 1. A polar ellipse represented by a black solid line denoted (1) indicates the potential presence of a anticyclonic vortex, and three spiral-like structures in black dotted-line denoted (2), (3), and (4) are superimposed.
Fig. 3 .
3Radial profiles of the velocity measured in 13 CO and C 18 O J = 3-2 subtracted by the systemic velocity displayed for various azimuthal angles θ, spaced by 45 • . The dashed line represents the Keplerian prescription following equation 1 for a 2.36 M binary system. The two dotted lines in the top central panel indicate the Keplerian velocity for a binary mass of 2.86 M and 1.86 M .
Fig. 5 .
5stand the elliptic instability (Lesur & Papaloizou 2009).4. Other origins of kinematic deviations around the horseshoe structure4.1. Velocity deviations are also observed in pure Keplerian disksTo probe the precision of the velocity measurements, we carried out a 3D model of the circumbinary disk around HD 142527, performed using the software RADMC3D (Dullemond et al. Comparison of the velocity deviations measured in the HD 142527 system with vortex models. Top: Difference between the 13 CO J=3-2 velocity measured from our observations and the Keplerian prescription in a 2D-map (R,θ). Blue and red colors indicate blue-and redshifted velocities along the line-of-sight compared to the Keplerian rotation. θ = 0 • represents the north of the major axis. Bottom: Three vortex prescriptions with their corresponding velocity amplitude in the plane of the disk (top), rotating clockwise around the vortex center on elliptical streamlines, and the projected velocity along the line-of-sight (bottom). Model A corresponds to a large vortex centered at the position of the continuum intensity maximum and Model B an equivalent vortex centered on the secondary maximum. Model C presents the kinematic signature for two smaller vortices at these positions. 2012) and based on the parameters previously determined in the studies ofBoehler et al. (2017),Soon et al. (2019), andYen & Gu (2020). A detailed description of the procedure is presented in Appendix A and images of the model are given inFig. A.1. In this model, the disk is rotating in pure Keplerian rotation. The resulting signal is then convolved by a Gaussian of 0.1 and 0.3 in order to reproduce typical spatial resolutions of molecular lines in ALMA observations.
Fig. 7 .
7Velocity deviations due to the presence of dust along the north of the major axis. Top: Velocity deviations measured in the pure Keplerian model at two different spatial resolutions; Middle: 13 CO integrated emission and dust emission in the original non-convolved model. By definition, the integrated emission is always measured after continuum subtraction. The cavity between 150 au and 260 au is due to the continuum subtraction procedure; Bottom: Flux ratio between the outer and inner edges of the synthesized beam.
Fig. 8 .
8Velocity artifacts created by a ring or horseshoe dust structure. Left panel: Sketch representing the integrated gas emission over the line width as a function of the disk radius. The black solid line is the gas emission without the dust ring, and the two red lines represent the emission in presence of the dust ring before and after continuum subtraction. The two ellipses indicate the position of two synthesized beams. Right panel: Line emission profile (black solid line) in the simplified case where the line emission is the sum of the emission at the inner (blue dashed line) and outer (red dashed lines) edges of the synthesized beam. The velocity v c , represented by the dotted black vertical line, is the velocity at the center of the beam while v m , represented by the solid black line, is the intensity weighted velocity.
Fig. A. 1 .Fig. B. 1 .
11From the left to the right: Dust emission, 13 CO J = 3-2 and C 18 O J = 3-2 integrated emission of the 3D toy model of HD 142527. The spatial resolution is 0.3 . Radial profiles of the velocity for different PAs θ, spaced by 45 • ,
Y.Boehler et al.: Vortex-like kinematic signal, spirals, and beam smearing effect in the HD 142527 diskDisk Orientation
N
W
Major axis
Near side
Back side
Table 1. Parameters used in the three models to describe the vortex properties.Model A Model B
Model C
V1
V2
R 0 (au)
185
185
185 185
θ 0 (degrees)
50 •
-20 •
50 • -20 •
χ A (aspect ratio)
5
5
2.0
2.0
V max (m s −1 )
350
350
350 350
R v (au)
42
42
35
35
w v (au)
26
26
22
22
A&A proofs: manuscript no. HD142527-kinematicsLine profile
Velocity
Velocity
Inner rim
Outer rim
Acknowledgements. We thank the referee whose comments improved the quality of the manuscript, and G. Lesur for useful discussions about vortex properties. The authors acknowledge funding from ANR (Agence Nationale de la Recherche) of France under contract number ANR-16-CE31-0013 (Planet-Forming-Disks). JFG thanks the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Université de Lyon for its financial support within the programme 'Investissements d'Avenir' (ANR-11-IDEX-0007) of the French government operated by the ANR. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2012.1.00725.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC,https://www.cosmos.esa.int/web/gaia/dpac/ consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.A&A proofs: manuscript no. HD142527-kinematics Σ 0 (g cm −2 ) R 0 (au) w in(au)Appendix A: A 3D toy model for HD 142527The model of the circumbinary disk around HD 142527 is built in a spherical framework (r, θ, φ) using cells of 1 au in radius, 2 • in azimuth and 1 • in elevation. The horseshoe structure in the dust and gas surface density is modeled using modified Gaussians along the radial and azimuthal directions following the formula:with Σ max the surface density at the position (R 0 (θ max ), θ max ), where the dust and gas surface densities are maximum, and Σ min the surface density at the position (R 0 (θ min ), θ min ) for the azimuth where the gas and dust surface densities are minimum. The parameters w in (θ) and w out (θ) are the radial half-widths of the Gaussian for r < R 0 or for r > R 0 , and w ctcl and w cl the azimuthal halfwidths of the Gaussian in the counterclockwise and clockwise directions, starting from θ max . To represent the eccentric aspect of the circumbinary disk, the parameters R 0 (θ), and w in,out (θ) vary as a function of the azimuth through the formula, for instance for R 0 (θ) :The value of the parameters are given inTable A.1. They were inspired by the analysis inBoehler et al. (2017), using the same grain properties, but also by the studies performed bySoon et al. (2019)andYen & Gu (2020), which took into account the azimuthal variation of the temperature, obtained using optically thick molecules, and then estimated that the gas surface density maximum was at θ ∼ 15 • . We then calculated the circumbinary disk temperature through radiative transfer by using RADMC-3D . No shadowing effect by the inner disk has been taken into account in this step. With the same code, we performed the ray-tracing in a square grid with a pixel size of 20 mas and a velocity resolution of 10 m s −1 , about ten times smaller than in our observations, to produce the final image before beam dilution. In this model, the gas is in pure Keplerian rotation around a single star of 2.36 M with v(r, z) = GM/(r 2 + z 2 ) 0.5 . We assume a disk inclination of 27 • and a PA of -19 • . Images of the circumbinary disk model for the dust intensity, and the 13 CO and C 18 O J=3-2 integrated emission are given inFig. A.1.Appendix B: Comparison of the "peak" and "intensity weighted" methodsWe have used in this study the intensity weighted method. Another popular method to estimate the gas velocity along the lineof-sight is the peak emission method. It consists in identifying for each pixel of the map the spectral position of the peak of the line emission. The precision of this method has recently been improved in Teague & Foreman-Mackey (2018b) by performing a polynomial fit on the three channels around the maximum in emission. This method is especially useful when the front and back molecular layers can be disentangled spatially and spectrally (see also Appendix A.3 inTeague et al. (2018c)). This allows the precise determination of the origin (r, θ, z) of the emission and the comparison of the front layer of the disk with the Keplerian velocity. However, such observations are still rare. In our observations, the low inclination of the circumbinary disk around HD 142527, the choice of the 13 CO and C 18 O transitions lines that emit at a lower altitude than 12 CO, and the moderate spatial resolution of our observations, do not give us the ability to distinguish the two molecular layers. We mainly privileged the intensity weighted method in our study because it appeared less affected by the rms noise than the peak emission approach, in particular with the 13 CO J=3-2 transition line. This is probably due to the high optical depth of the line and to the moderate spatial resolution of our observations, which give to the peak of the emission a flattened aspect. Another reason for the better precision of the intensity weighted method within our data is the good signal-to-noise in the circumbinary disk where about 10 channels are involved in the measurement of the gas velocity.We show inFig. B.1 the velocity measured from the 13 CO and C 18 O J=3-2 transition lines using the intensity weighted and the peak emission methods. For the intensity weighted method, we only indicate the velocity measured in 13 CO as both molecules show the same velocity (cfFig. 3). The results between both methods and molecules are in general agreement, with a Keplerian profile at the south of the disk and a super-Keplerian profile at the north for R 160-180 au and sub-Keplerian for R 160-180 au.There are also slight differences between the methods and the lines involved, which is in itself not a surprising result. First, the two methods would only give the same results if the line profile was symmetric around the peak of emission, what may not be the case in practice. For instance, spectral profiles similar to the examples shown in the right panel ofFig. 8would give very different values. Second, the two methods do not probe the same regions (r, θ, z) along the line-of-sight due to optical depth effects. In fact, the largest differences in the measured velocity inFig. B.1 are found between the peak emission of the 13 CO line, optically thick and only probing the front 13 CO layer, and the two other measurements, which are the peak intensity method applied to the moderately optically thick C 18 O and the intensity weighted method, which probes all the line profile including optically thin wings, therefore also sensitive to the back molecular layer. Finally, the optical depth can also modify the surface brightness, and thus the velocity artifacts due to beam smearing, as the weight given to each region of the synthesized beam would change.
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| []
|
[
"Entanglement in Four-Dimensional SU(3) Gauge Theory",
"Entanglement in Four-Dimensional SU(3) Gauge Theory"
]
| [
"Etsuko Itou \nKEK Theory Center\nHigh Energy Accelerator Research Organisation\n305-0801TsukubaJapan\n",
"Keitaro Nagata \nHigh Energy Accelerator Research Organisation (KEK)\n305-0801TsukubaJapan\n",
"Yoshiyuki Nakagawa \nRIISE\nHiroshima University\nHigashi-Hiroshima739-8521HiroshimaJapan\n",
"Atsushi Nakamura \nRIISE\nHiroshima University\nHigashi-Hiroshima739-8521HiroshimaJapan\n\nTheoretical Research Division\nNishina Center\nRIKEN\n351-0198WakoJapan\n\nResearch Center for Nuclear Physics (RCNP)\nOsaka University\n567-0047IbarakiOsakaJapan\n\nSchool of Biomedicine\nFar Eastern Federal University\nSukhanova 8690950VladivostokRussia\n",
"V I Zakharov \nSchool of Biomedicine\nFar Eastern Federal University\nSukhanova 8690950VladivostokRussia\n\nITEP\nB. Cheremushkinskaya 25Moscow117218 Russia\n\nMoscow Inst. Phys. & Technol\nDolgoprudnyMoscow Region141700 Russia\n"
]
| [
"KEK Theory Center\nHigh Energy Accelerator Research Organisation\n305-0801TsukubaJapan",
"High Energy Accelerator Research Organisation (KEK)\n305-0801TsukubaJapan",
"RIISE\nHiroshima University\nHigashi-Hiroshima739-8521HiroshimaJapan",
"RIISE\nHiroshima University\nHigashi-Hiroshima739-8521HiroshimaJapan",
"Theoretical Research Division\nNishina Center\nRIKEN\n351-0198WakoJapan",
"Research Center for Nuclear Physics (RCNP)\nOsaka University\n567-0047IbarakiOsakaJapan",
"School of Biomedicine\nFar Eastern Federal University\nSukhanova 8690950VladivostokRussia",
"School of Biomedicine\nFar Eastern Federal University\nSukhanova 8690950VladivostokRussia",
"ITEP\nB. Cheremushkinskaya 25Moscow117218 Russia",
"Moscow Inst. Phys. & Technol\nDolgoprudnyMoscow Region141700 Russia"
]
| []
| We investigate the quantum entanglement entropy for the four-dimensional Euclidean SU(3) gauge theory. We present the first non-perturbative calculation of the entropic c-function (C(l)) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For 0 l 0.7 fm, where l is the length of the subspace, the entropic c-function is almost constant, indicating conformally invariant dynamics. The value of the constant agrees with that perturbatively obtained from free gluons, with 20 % discrepancy. When l is close to the Hadronic scale, the entropic c-function decreases smoothly, and it is consistent with zero within error bars at l 0.9 fm.Quantum entanglement is a fascinating phenomenon that was first highlighted by the Einstein-Podolsky-Rosen paradox [1] and has remained a focus of research activity for decades. If there is a system in a pure quantum state, measurements on a subsystem A determine the results of measurements on its complement B, even if no causal communication is possible between the two measurements. The entanglement entropy S A of subsystem A is defined as von Neumann entropy corresponding to the reduced density matrix ρ A : | 10.1093/ptep/ptw050 | [
"https://arxiv.org/pdf/1512.01334v1.pdf"
]
| 118,869,036 | 1512.01334 | deb1e3162f571e6edaa3902f8c358260b2f74865 |
Entanglement in Four-Dimensional SU(3) Gauge Theory
4 Dec 2015
Etsuko Itou
KEK Theory Center
High Energy Accelerator Research Organisation
305-0801TsukubaJapan
Keitaro Nagata
High Energy Accelerator Research Organisation (KEK)
305-0801TsukubaJapan
Yoshiyuki Nakagawa
RIISE
Hiroshima University
Higashi-Hiroshima739-8521HiroshimaJapan
Atsushi Nakamura
RIISE
Hiroshima University
Higashi-Hiroshima739-8521HiroshimaJapan
Theoretical Research Division
Nishina Center
RIKEN
351-0198WakoJapan
Research Center for Nuclear Physics (RCNP)
Osaka University
567-0047IbarakiOsakaJapan
School of Biomedicine
Far Eastern Federal University
Sukhanova 8690950VladivostokRussia
V I Zakharov
School of Biomedicine
Far Eastern Federal University
Sukhanova 8690950VladivostokRussia
ITEP
B. Cheremushkinskaya 25Moscow117218 Russia
Moscow Inst. Phys. & Technol
DolgoprudnyMoscow Region141700 Russia
Entanglement in Four-Dimensional SU(3) Gauge Theory
4 Dec 2015numbers: 0365Ud1115Ha1238Aw
We investigate the quantum entanglement entropy for the four-dimensional Euclidean SU(3) gauge theory. We present the first non-perturbative calculation of the entropic c-function (C(l)) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For 0 l 0.7 fm, where l is the length of the subspace, the entropic c-function is almost constant, indicating conformally invariant dynamics. The value of the constant agrees with that perturbatively obtained from free gluons, with 20 % discrepancy. When l is close to the Hadronic scale, the entropic c-function decreases smoothly, and it is consistent with zero within error bars at l 0.9 fm.Quantum entanglement is a fascinating phenomenon that was first highlighted by the Einstein-Podolsky-Rosen paradox [1] and has remained a focus of research activity for decades. If there is a system in a pure quantum state, measurements on a subsystem A determine the results of measurements on its complement B, even if no causal communication is possible between the two measurements. The entanglement entropy S A of subsystem A is defined as von Neumann entropy corresponding to the reduced density matrix ρ A :
S A = − Tr HA ρ A log(ρ A ),(1)
where ρ A = Tr HB [ρ tot ], and it is assumed that the total Hilbert space is a direct product of two subspaces corresponding to the subsystems considered, H tot = H A ⊗H B . More generally, studies of the entanglement entropy become central in cases of complex systems with strong interactions, where the properties of the ground state cannot be evaluated directly. In particular, the notion of quantum entanglement is crucial for the theory of quantum phase transitions, i.e., non-thermal phase transitions at temperature T = 0 [2][3][4]. In physics of black holes, consideration of the quantum entanglement is central to discussions of the information paradox [5], which challenges the consistency of general relativity and quantum mechanics.
Applications to field theory are more recent. First of all, the entanglement entropy is ultraviolet divergent in field theory [6]. In more detail, one considers the vacuum state and defines the subsystem A as a slab of length l in one of the spatial dimensions, at a fixed time slice. Then, the entanglement entropy contains, as its most divergent term, a term that is proportional to |∂A|/a d−1 , where d is the number of spatial dimensions, a is the lattice spacing, and |∂A| is the area of the boundary surface between the slab and the rest of the space. To eliminate this divergence, which depends on details of the UV cutoff, one focuses on the entropic c-function [7]:
C(l) = l 3 |∂A| ∂S A ∂l ,(2)
where we choose d = 3. C(l) is a finite quantity even in the a → 0 limit, and it becomes constant as a function of l in the conformal case.
In the present work, we non-perturbatively obtain the entropic c-function of SU(3) gauge theory, which describes the dynamics of gluons and has a confinement property. Although no analytic proof exists yet, accumulated numerical evidences imply that quantum chromodynamics (QCD) has a finite mass gap. At zero temperature, we expect from the asymptotic freedom that the entropic c-function is approximated by the contribution of non-interacting gluons at short distances l. On the other hand, at the hadronic scale, l ∼ Λ −1 QCD , the entropic c-function captures the physics of confinement, or strong interactions. No analytical calculation of C(l) seems possible in this region. In the limit l ≫ Λ −1 QCD , the effective degrees of freedom responsible for the entanglement apparently reduce to non-interacting glueballs. It has been suggested [13] that the entropic c-function estimated by the correlation function of glueballs shows a Hagedrontype divergence. Furthermore, several works based on holographic and geometrical approaches found that S A undergoes a quantum phase transition at l cr ∼ Λ −1 QCD [7][8][9][10][11][12][13][14][15]. At this critical value of l, the entropic c-function in the large N c limit changes its behavior from S A ∼ N 2 c at short distance to S A ∼ N 0 c at long distance. Here N c is the number of colors.
There is a subtlety concerning the local gauge invariance of the entanglement entropy and of the Hilbert space for the subspace A on the lattice. Although several predictions and definitions for the entanglement entropy for the lattice gauge theory have been proposed [9,[16][17][18][19][20][21][22][23][24][25][26][27][28][29], it turns out that some definitions give different values for the entanglement entropy. Recently, a definition that emphasizes the maximally gauge invariance has been proposed in Refs. [26,28]. The entanglement entropy in the replica method [18][19][20][21] agrees with that of the maximal gauge-invariant definition. In our work, we utilize the replica method following Refs. [18][19][20][21], and we obtain the entropic c-function numerically.
The replica method is a powerful technique for calculating the entanglement entropy. Based on this method, the entanglement entropy is given by the following equation:
S A = lim n→1 − ∂ ∂n ln (Trρ n A ) .(3)
Here, n is an integer, and it is referred to as a replica number. The period in the temporal direction for field variables in subsystem A is n times as long as that of subsystem B. The trace of the n-th power of the reduced density matrix ρ A is given by the ratio of the partition functions:
Trρ n A = Z(l, n)/Z n .(4)
Here, Z is the original partition function for the whole system, and Z(l, n) is the partition function for the system with an n-sheeted Riemann surface. The subsystem B is a patch of the n-th Riemann surface while the subsystem A, whose length of one direction is l, is defined on a single Riemann surface.
The whole system is realized on a four-dimensional lattice of size N 3 s × N t , with lattice spacing a, where N s and N t are the spatial and temporal lattice sizes, respectively. The system is divided into two subsystems, A and B, in the x-direction, and the numbers of sites in x-direction of A and B are L and N s − L, respectively. We adopt periodic boundary conditions for all directions. As explained above, the period of the temporal direction depends on the x coordinate. We show an example of the boundary condition on the replica lattice with N t = 4, n = 2, in Fig. 1. In the figure, the boundaries with the same symbols in the t-direction are matched with each other via the periodic boundary condition. Thus, in subsystem B, the period of the temporal direction for the link variables is N t (= 4) in Fig. 1, while in A, it becomes (n · N t ). The boundary surface between A and B is extended in the y-z plane, and the area is given by |∂A| = (N s a) 2 in physical unit.
The entropic c-function is obtained as the derivative of Trρ n A with respect to l and n. These derivatives are approximated by finite differentials with (∆L) = 1 and (∆n) = 1. We introduce the interpolating action [30,31] given by where S L and S L+∆L represent the averaged action density on the replica lattices in which L and L + ∆L are the lengths of the subsystem A. Here, U denotes the link variables, which are related to SU(3) gauge fields as
S int = (1 − α)S L [U ] + αS L+∆L [U ],(5)U µ ( x, t) = exp(ig 0 A µ ( x, t)
) with a bare coupling constant g 0 . Now, we can rewrite Eq. (2) as
C(l) = L 3 N 2 s ∆L 1 0 dα S L+∆L [U ] − S L [U ] α ,(6)
where · α refers to the Monte Carlo average with the interpolating action S int at a fixed value of α. We regard l in C(l) as (L + ∆L/2)a. The strategy for calculating the entropic c-function using the lattice Monte Carlo simulation consists of five steps:
Step 1: Generate gauge configurations on the replica lattice using Monte Carlo simulation. The interpolating action S int is used as a weight for the probability.
Step 2: Measure S L+∆L −S L on each generated gauge configuration, and take an ensemble average of this for each value of α.
Step 3: Numerically integrate S L+∆L − S L α as a function of α.
Step 4: Take the continuum limit.
Step 5: Estimate the replica number dependence.
We utilize the standard Wilson plaquette action as an action, which has one coupling constant, namely the lattice bare coupling constant β = 6/g 2 0 . Gauge configurations are generated by using the pseudo-heatbath algorithm. Thus, the link variables (U ) are updated using the local interpolating gauge action S int at a fixed value of α.
The details of simulation are as follows. The simulations were performed with a replica lattice volume N 3 s × N t = 16 4 and n = 2. The simulation parameters and the number of generated configurations are summarized in Table I. Each configuration is separated by 100 sweeps to avoid the autocorrelation. We also show the value of the lattice bare coupling constant (β) and the corresponding length of the lattice spacing (a) in physical unit. The pure Yang-Mills gauge theory is an asymptotically free theory, and it has only one physical scale, Λ QCD . Once we fix a relation between a physical reference scale and a quantity in lattice unit, then all physical quantities can be obtained in physical unit. We use the relation between the lattice bare coupling constant and the lattice spacing given in Eq.(2.18) of Ref. [32]. Here, as a reference scale, we utilize the Sommer scale, in which the dimensionless static quark-antiquark force satisfies r 2 F (r)| r=r0 = 1.65. To convert a quantity in lattice unit into physical unit, we assume that r 0 = 0.5 fm.
The error is estimated using the bootstrap method. Firstly we calculate ∂S A /∂l for each bootstrap sample, and then estimate the statistical error from its distribution. The typical number of bootstrap samples constructed is O(10 3 − 10 5 ).
In Step 2 and Step 3, for each L, we take 11 points of α between α = 0.0 to α = 1.0, at intervals of ∆α = 0.1. The numerical integration is carried out using the cubic polynomial function. We also numerically investigated the dependence of C(l) on the number of α and the integration formula, and found that such effects are sufficiently smaller than the statistical uncertainty. Figure 2 shows the result for the entropic c-function of the pure Yang-Mills theory at zero temperature. We found that the c-function is almost constant in the small l region (l 0.7 fm); we fit the data with a constant for the data in the region 0 ≤ l ≤ 0.7, and obtain the best-fit value
C = 0.206 ± 0.007,(7)
where the chi-square of degrees of freedom is 0.88. Here the error bars denote 1σ statistical error. To estimate the systematic uncertainty, we change the range of fitting to 0 ≤ l ≤ 0.6 and 0 ≤ l ≤ 0.8, and obtained C = 0.208 (8) and C = 0.202 (7), respectively. The systematic uncertainty of C coming from the choice of the fit range is smaller than the statistical error. The data shows continuous decrease in the middle l regime, and becomes consistent with zero beyond l = 0.88 fm. The critical temperature of the pure SU (3) Yang-Mills theory determined by the center symmetry breaking is T c = 280 MeV, that is, 1/T c = 0.714 fm [33]. The Λ scale obtained from the running coupling constant based on the lattice simulation is r 0 Λ MS = 0.602(48) [32] and r 0 Λ MS = 0.613(2)(25) [34]. They correspond to Λ −1 MS ∼ 0.831 fm and Λ −1 MS ∼ 0.816 fm, respectively, when we set the Sommer scale r 0 = 0.5 fm. The length of l, for which the c-function starts decreasing, is in approximate agreement with these scales.
Next, let us compare our results with those found by other studies. A numerical simulation for the pure SU(2) gauge theory was carried out in Ref. [19]. In SU(2) gauge theory, the entropic c-function shows a clear discontinuity around l = 0.5 fm, and it shows an enhancement when the length is slightly less than this. These features are qualitatively different from those seen in SU(3).
Furthermore, concerning the existence of the discontinuity, several holographic models also show a clear phase transition of the confinement as a function of l [7,[11][12][13]. However, they are relevant to large-N c Yang-Mills theories only in the far infrared region.
Assuming that the monotonic decrease of the entropic c-function results from the confinement, it is worth comparing the continuous behavior of the entropic c-function with another observation of the confinement, namely, the static quark-antiquark potential. The lattice data for the static potential (see e.g. a review paper [35]) reproduces the Coulomb potential of a quark-antiquark pair for short distances, which is seen in the perturbative picture. On the other hand, it shows a linear potential for long distances that is a signal of confinement. For intermedi-ate distances, the lattice data smoothly connect the two regimes. Our results are analogous to the behavior of the static potential in the whole regime.
At short distance, the observed value of the entropic c-function, Eq. (7), can be understood reasonably well in terms of the degrees of freedom of gluons. First, we note in case of SU(2) gauge theory, most of the data for C(l) SU(2) at l ≤ 0.3 fm (Fig. 6 in Ref. [19]) are located in the range of C = 0.07-0.08 [44]. Scaling down our value for Eq. (7) proportional to (N 2 c − 1), we obtain C SU(2) ≈ 0.077.
Calculating the value of the entropic c-function for free gluon theory requires an independent quantitative discussion. The entanglement entropy in four-dimensional free theory is expressed as
S A (l) = K|∂A| 1 a 2 − 1 l 2 .(8)
The coefficient, K ∼ 0.0049 [9], is obtained for the free real scalar theory. Assuming that the contribution of a free gauge boson is approximated with two real scalars, and taking into account eight color degrees of freedom in the SU(3) gauge theory, we get the following estimate:
C(l) free ∼ 0.1568.(9)
Keeping in mind the approximations made, the prediction Eq. (9) falls remarkably close to our observed value, Eq. (7). The slight discrepancy may be also caused by additional degrees of freedom, such as those due to glueballs, other excited states or topological ground states in the lattice data, while Eq. (9) is obtained in a perturbative vacuum. We examine the validity of each analysis, in particular, we consider the finite volume effect, continuum extrapolation, and replica number dependence. To estimate the finite volume effect, we carried out a simulation with twice larger lattice extent in each direction with fixed (β, L, ∆L). We found that the finite volume effects are negligible compared to the statistical error. Next, to estimate the discretization effects, we investigate the a dependence of the entropic c-function. Clearly, there is no a dependence as shown in Fig. 2. Moreover, we carried out a simulation with a fixed physical lattice extent with half the lattice spacing. Although the statistical error is still large, the results with the halved spacing are consistent with the results shown in Fig. 2. The replica number dependence was studied by generating the data on n = 3 replica lattice. The next-to-leading correction for the nth derivative of the entanglement entropy is smaller than the statistical error of our main results.
In summary, we present the first non-perturbative determination of the entropic c-function, which is the ldependence of the entanglement entropy in the SU(3) pure gauge theory, by using lattice QCD simulations.
We utilize the replica method to obtain the entropic cfunction, which is consistent with the maximally gauge invariant definition. At short distances, the entropic c-function is proportional to the degree of freedom of gluons, and it vanishes at long distances. In addition, the change between those two regimes occurs smoothly around a distance that is consistent with the QCD scale. Several systematic uncertainties are under control in our results.
For our future work, we note that it will be straightforward to extend the present study to QCD with dynamical fermions. Although the exact order parameter for quark confinement is as yet unknown, the entropic c-function is expected to provide a new insight for the confinement from the viewpoint of the effective degrees of freedom. The application of this to finite temperature QCD will also be interesting. The entropic c-function at finite temperature gives the thermal entropy density in the long distance limit. Preliminary results have already been presented in Ref. [21], and as expected, these are roughly consistent with the thermal one obtained by the other approaches [36]- [41]. Furthermore, determining the length of l at which the entanglement entropy density becomes consistent with the thermal entropy density, would give a quantum correlation length for the quark-gluon plasma phase.
As an other direction, several recent studies have found various conformal field theories for nonabelian gauge theories by using a lattice numerical simulation to observe the non-perturbative running coupling constant [42,43]. Applying the present method to such conformal systems, the entropic c-function tells us the universal quantity related with the central charge for the four dimensional conformal field theories. grams for Innovative Research (SPIRE) Field 5.
The work was completed due to support of the RSF grant 15-12-20008.
FIG. 1 :
1Replica lattice and boundary conditions.
parameters and the number of configurations for (Ns, Nt, n) = (16, 16, 2) lattice.
AcknowledgementsWe would like to thank S. Aoki Numerical simulations were performed on Hitachi SR16000 at YITP, Kyoto University, on a NEC SX-8R and SX-9 at the Research Center for Nuclear Physics (RCNP) Osaka University, and on a Hitachi SR16000 at KEK under its Large-Scale Simulation Program (Nos. 14/15-05). We acknowledge the Japan Lattice Data Grid for data transfer and storage. The work of K. N. and A. N. are supported in part by a Grant-in-Aid for Scientific Researches No. 00586901, 26610072, and 15H03663. E. I. and K. N. are supported in part by Strategic Pro-
* Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]
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| []
|
[
"arXiv:hep-ph/0508126v1 11 Aug 2005 Quark Asymmetries and Intrinsic Charm in Nucleons 1",
"arXiv:hep-ph/0508126v1 11 Aug 2005 Quark Asymmetries and Intrinsic Charm in Nucleons 1"
]
| [
"Johan Alwall \nHigh Energy Physics\nUppsala University\nBox 535S-75121UppsalaSweden\n"
]
| [
"High Energy Physics\nUppsala University\nBox 535S-75121UppsalaSweden"
]
| [
"Quark Asymmetries in Nucleons\", given at the XIII Interna-tional Workshop on Deep Inelastic Scattering"
]
| We have developed a physical model for the non-perturbative xshape of parton density functions in the proton, based on Gaussian fluctuations in momenta, and quantum fluctuations of the proton into meson-baryon pairs. The model describes the proton structure function and gives a natural explanation of observed quark asymmetries, such as the difference between the anti-up and anti-down sea quark distributions and between the up and down valence distributions. We find an asymmetry in the momentum distribution of strange and anti-strange quarks in the nucleon, large enough to reduce the NuTeV anomaly to a level which does not give a significant indication of physics beyond the standard model. We also consider charmed fluctuations, and show that they can explain the excess at large x in the EMC F c 2 data. | null | [
"https://arxiv.org/pdf/hep-ph/0508126v1.pdf"
]
| 10,895,843 | hep-ph/0508126 | 832e46de6ea80a3664173bc44d0b81b58e5bd48a |
arXiv:hep-ph/0508126v1 11 Aug 2005 Quark Asymmetries and Intrinsic Charm in Nucleons 1
April 27-May 1, 2005
Johan Alwall
High Energy Physics
Uppsala University
Box 535S-75121UppsalaSweden
arXiv:hep-ph/0508126v1 11 Aug 2005 Quark Asymmetries and Intrinsic Charm in Nucleons 1
Quark Asymmetries in Nucleons", given at the XIII Interna-tional Workshop on Deep Inelastic Scattering
Madison, USAApril 27-May 1, 20051quark asymmetriesparton density distributionss-sbar asymme- tryNuTeV anomalyintrinsic charm PACS: 1239Ki1130Hv1240Vv1315+g1360Hb
We have developed a physical model for the non-perturbative xshape of parton density functions in the proton, based on Gaussian fluctuations in momenta, and quantum fluctuations of the proton into meson-baryon pairs. The model describes the proton structure function and gives a natural explanation of observed quark asymmetries, such as the difference between the anti-up and anti-down sea quark distributions and between the up and down valence distributions. We find an asymmetry in the momentum distribution of strange and anti-strange quarks in the nucleon, large enough to reduce the NuTeV anomaly to a level which does not give a significant indication of physics beyond the standard model. We also consider charmed fluctuations, and show that they can explain the excess at large x in the EMC F c 2 data.
The low-scale parton density functions give a description of the hadron at a nonperturbative level. The conventional approach to these functions is to make parameterizations using some more or less arbitrary functional forms, based on data from deep inelastic scattering and hadron collision experiments. Another approach, however, is to start from some ideas of the behavior of partons in the non-perturbative hadron, and build a model based on that behavior. The advantage with this approach is that the successes and failures of such a model allows us to get insight into the non-perturbative QCD dynamics. The model presented here, and described in detail in [1,2], describes the F 2 structure function of the proton, as well as sea quark asymmetries of the nucleon. Most noteworthy, our model predicts an asymmetry between the momentum distributions of strange and anti-strange quarks in the nucleon of the same order as the newly reported results from NuTeV [3]. The model also suggests an intrinsic charm component in the proton.
This work extends the model previously presented in [4]. The model gives the fourmomentum k of a single probed valence parton (see Fig. 1a for definitions of momenta) by assuming that, in the nucleon rest frame, the shape of the momentum distribution for a parton of type i and mass m i can be taken as a Gaussian which may be motivated as a result of the many interactions binding the parton in the nucleon. The width of the distribution should be of order hundred MeV from the Heisenberg uncertainty relation applied to the nucleon size, i.e. σ i = 1/d N . The momentum fraction x of the parton is then defined as the light-cone fraction x = k + /p + . We impose constraints on the final-state momenta in order to obtain a kinematically allowed final state, which also ensures that 0 < x < 1 and f i (x) → 0 for x → 1. Using a Monte Carlo method these parton distributions are integrated numerically without approximations.
f i (k) = N(σ i , m i ) exp − (k 0 − m i ) 2 + k 2 x + k 2 y + k 2 z /2σ 2 i (1) 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 (c) xu(x) xd(x) xg(x) xd -(x) xu -(x) xs(x) xs -(x) x xf(x) Q 2 =Q 2 0
To describe the dynamics of the sea partons, we note that the appropriate basis for the non-perturbative dynamics of the bound state nucleon should be hadronic. Therefore we consider hadronic fluctuations, for the proton |p = α 0 |p 0 + α pπ 0 |pπ 0 + α nπ + |nπ + + . . . + α ΛK |ΛK + + . . .
Probing a parton i in a hadron H of a baryon-meson fluctuation |BM (see Fig. 1b) gives a sea parton with light-cone fraction x = x H x i of the target proton. The momentum of the probed hadron is given by a similar Gaussian, but with a separate width parameter σ H . Also here, kinematic constraints ensure that we get a physically allowed final state. The procedure gives
x H ∼ M H /(M B +M M ), i.e.
the heavier baryon gets a harder spectrum than the lighter meson. The normalization of the sea distributions is given by the normalization coefficients α 2 BM of Eq. (2). These cannot be calculated from first principles in QCD and are therefore taken as free parameters to be fitted using experimental data.
The resulting valence and sea parton x-distributions apply at a low scale Q 2 0 , and the distributions at higher Q 2 are obtained using perturbative QCD evolution at next-toleading order.
The model has in total four shape parameters and three normalization parameters, plus the starting scale, to determine the parton densities u, d, g,ū,d, s,s. These are (with values resulting from fits to experimental data as described below):
σ u = 230 MeV σ d = 170 MeV σ g = 77 MeV σ H = 100 MeV α 2 pπ 0 = 0.45 α 2 nπ + = 0.14 α 2 ΛK = 0.05 Q 0 = 0.75 GeV(3)
The resulting parton densities are shown in Fig. 1(c). In order to fix the values of the model parameters, we make a global fit using several experimental data sets: Fixed-target F 2 data to fix large-x (valence) distributions (Fig. 2); HERA F 2 data for the gluon distribution width and the starting scale Q 0 ;d/ū-asymmetry data for the normalizations of the |pπ 0 and |nπ + fluctuations (see Fig. 4); and strange sea data to fix the normalization of fluctuations including strange quarks (see Fig. 5a). We have also compared with W ± charge asymmetry data as a cross-check on the ratio of Gaussian widths for the u and d valence quark distributions (Fig. 3). It is interesting to note that this simple model can describe such a wealth of different data with just one or two parameters per data set. . Comparison between our model and data from the E866/NuSea collaboration [8]:
(a)ū(x)/d(x) (b) xd(x) − xu(x).
The full line uses the physical pion mass, while the dashed line uses an effective pions mass m eff = 400 MeV as discussed in the text.
In our model, the shape difference between the valence u and d distributions in the proton, apparent from the W ± charge asymmetry data, is described as different Gaussian widths. This would correspond to a larger effective volume in the proton for d quarks than for u quarks, an effect which could conceivably be explained by Pauli blocking of the u quarks.
Since the proton can fluctuate to π 0 and π + by |pπ 0 and |nπ + , but to π − only by the heavier |∆ ++ π − , we get an excess ofd overū in the proton sea. Interestingly, the fit to data improves when we use a larger effective pion mass of 400 MeV (see Fig. 4). This might indicate that we have a surprisingly large coupling to heavier ρ mesons, or that one should use a more generic meson mass rather than the very light pion.
The lightest strange fluctuation is |ΛK + . If we let this implicitly include also heavier strange meson-baryon fluctuations, we can fit the normalization α 2 ΛK to strange sea data (see Fig. 5a). The result corresponds to This is especially interesting in connection to the NuTeV anomaly [10]. NuTeV found, based on the observable The resulting distributions of c andc at the starting scale are shown in Fig. 6. In Fig. 6a the distributions from the |Λ C D fluctuation are shown compared with the s and s distributions. The fluctuation parameters (σ H and σ q ) are taken to be the same as for the light quark fluctuations, just as for the strange sea. However, the sensitivity of the result on the precise values of these parameters is small. The normalization is in Fig. 6 taken to be ∝ 1/∆M BM (to be discussed below), in order to easily compare the shapes of the strange and charmed sea. In this case, there is an asymmetry between the c andc distributions, similar to that of the strange sea, but much smaller due to the similarity in mass between the Λ C (2285 MeV) and the D (1865 MeV). Fig. 6b shows the distributions from the |p J/ψ fluctuation. They are very similar to that of the |Λ C D distributions, except that the asymmetry is missing since both the c and thec are here in the meson. From Fig. 6 it is clear that an investigation of the intrinsic charm distributions from our . Data for the charm structure function F c 2 from the EMC experiment [12], compared to our model results with the best-fit normalization (solid curve). The dotted curves show the perturbative QCD prediction for γg → cc from [13], with three different choices for the renormalization and factorization scales. For comparison, the intrinsic charm distribution of [11] is also shown, with the 0.7% normalization allowed according to [13] (dashed curve).
R − = σ(νµN →νµX)−σ(νµN →νµX) σ(νµN →µ − X)−σ(νµN →µ + X) = g 2 L − g 2 R = 1 2 − sin 2 θ W ,
model is not much affected by the precise nature of the dominating fluctuation mode, and in the following we will use only the |Λ C D fluctuation. Note that the shape of our intrinsic charm component is somewhat different from that in the intrinsic charm model by Brodsky et al. [11], which is based on partonic fluctuations p → uudcc (see Fig. 7).
The normalization used in Fig. 6 corresponds to a relative importance between different mass states proportional to 1/∆M BM , as suggested by the strange sea fit to CCFR data. This normalization would give an intrinsic charm component ((c +c)/2 integrated number density) of 0.9%. However, it might be more appropriate to use a normalization ∝ 1/∆M 2 BM (as given by old-fashioned perturbation theory) compared to the strange fluctuations, corresponding to an intrinsic charm component of 0.18%.
The only experimental data for the large-x charmed structure function F c 2 = 4 9 x(c +c) (at leading order) comes from the EMC experiment [12], which measured charmed hadron production in muon-proton scattering. There, an intriguing excess was found in the largest x bins, compared to the perturbative photon-gluon fusion expectation. A later analysis gave further evidence that the excess cannot easily be attributed to standard perturbative production channels [13]. Intrinsic charm was immediately suggested as an explanation for the excess, but the shape of the charm distribution in the original intrinsic charm model is not optimal to explain the EMC excess. In Fig. 7, the EMC data is shown in bins of ν = Q 2 /2M p x, together with the result from our model, evolved in Q 2 using NLO QCD evolution [14]. Here we use the best-fit normalization, corresponding to 0.45% intrinsic charm. This lies between the two normalizations discussed above, which should not be surprising since the energy denominator only gives an order-of-magnitude estimate. For comparison, we also show the intrinsic charm distribution of [11] with the largest normalization allowed by the EMC data according to [13] (0.7% intrinsic charm), and the perturbative photon-gluon fusion results from [13].
As can be seen from Fig. 7, the shape of the intrinsic charm distribution in our model seems to fit the data very well, giving an enhancement at precisely the right values of x.
Unfortunately, the statistics of the EMC result is too small to allow any discrimination between different models for intrinsic charm, and measurements of the charm structure function at HERA (H1 [15] and ZEUS [16]) are at too low values of x to contribute to our understanding of intrinsic charm. If a future experiment would measure the large-x charm structure function with large statistics, it would be very interesting to get a decisive verification of the presence of intrinsic charm in the proton.
FIGURE 1 .
1Illustration of the processes probing (a) a valence parton in the proton and (b) a sea parton in a hadronic fluctuation (letters are four-momenta). (c) shows the resulting parton distributions at the starting scale Q 0 .
FIGURE 2 .
2The proton structure function F 2 (x, Q 2 ) for large x values; NMC and BCDMS data[5,6] compared to our model, also showing the results of ±20% variations of the width parameters σ u and σ d for the u and d valence distributions.
FIGURE 3 .
3The charge asymmetry for leptons from W ± -decays in pp collisions at the Tevatron[7] compared to our model, with best-fit parameters and a 20% reduced width of the valence d quark distribution.
FIGURE 4
4FIGURE 4. Comparison between our model and data from the E866/NuSea collaboration [8]: (a)ū(x)/d(x) (b) xd(x) − xu(x). The full line uses the physical pion mass, while the dashed line uses an effective pions mass m eff = 400 MeV as discussed in the text.
FIGURE 5 .
5xū + xd)dx ≈ 0.5, in agreement with standard parton density parameterizations. We note that this indicates a normalization ∝ 1/∆M BM = 1/(M B + M M − M p ) rather than ∝ 1/∆M 2 BM , as expected from old-fashioned perturbation theory. The fluctuation parameters are taken from the light sea results, σ H = 100 MeV and σ q = σ proton d = 170 MeV as discussed in[1]. Since the s quark is in the heavier baryon Λ and thes quark is in the lighter meson K + , (a) CCFR data[9] on the strange sea distribution (xs(x) + xs(x))/2 compared to our model based on |ΛK fluctuations with different normalizations. (b) The strange sea asymmetry s − (x) = xs(x) − xs(x) (at Q 2 = 20 GeV 2 ) from the model and combined with the function F (x) accounting for NuTeV's analysis giving ∆ sin 2 θ W = 1 0 dx s − (x)F (x) = −0.0017. The uncertainty bands correspond to the uncertainties for S − and ∆ sin 2 θ W quoted in the text.
FIGURE 6 .
6(a) Comparison between the strange and charm sea obtained from our model with the inclusion of the Λ C D fluctuation. The normalization is here taken to be ∝ 1/(M B + M M − M p ) as suggested by strange sea data. (b)The charm quark distributions from the p J/ψ fluctuation (the c andc distributions are identical with this fluctuation). Note that the distributions are very similar to those from the Λ C D distribution, except for the small asymmetry between c andc.we get a non-zero asymmetry S − = 1 0 dx[xs(x) − xs(x)] in the momentum distribution of the strange sea, as seen inFig. 5b and 6a. Depending on details of the model, we get the range 0.0010 ≤ S − ≤ 0.0023 for this asymmetry.
a 3σ deviation of sin 2 θ W compared to the Standard Model fit: sin 2 θ NuTeV W = 0.2277 ± 0.0016 compared to sin 2 θ SM W = 0.2227±0.0004. However, an asymmetric strange sea would change their result, since ν only have charged current interactions with s andν withs. Using the folding function provided by NuTeV to account for their analysis, the s-s asymmetry from our model gives a shift −0.0024 ≤ ∆ sin 2 θ W = 1 0 dx s − (x)F (x) ≤ −0.00097, i.e. the discrepancy with the Standard Model result is reduced to between 1.6σ and 2.4σ, leaving no strong hint of physics beyond the Standard Model. In our model, it is also natural to consider fluctuations involving heavy quarks. Assuming that the hadronic fluctuation description is still valid for the proton fluctuating into charmed baryon-meson pairs, the lightest such fluctuations are p → Λ C D and p → p J/ψ. Taking these fluctuations into account implies that there should be an intrinsic charm component in the proton at intermediate x ∼ 0.4. This component is quite different from the purely perturbative charm component from gluon splitting, which falls steeply with x (see Fig. 7). The intrinsic charm component should be present from the scale where the momentum transfer can realize the charmed fluctuation.
FIGURE 7. Data for the charm structure function F c 2 from the EMC experiment [12], compared to our model results with the best-fit normalization (solid curve). The dotted curves show the perturbative QCD prediction for γg → cc from [13], with three different choices for the renormalization and factorization scales. For comparison, the intrinsic charm distribution of [11] is also shown, with the 0.7% normalization allowed according to [13] (dashed curve).
This is an extended version of the talk "Quark Asymmetries in Nucleons", given at the XIII International Workshop on Deep Inelastic Scattering, Madison, USA, April 27-May 1, 2005
Acknowledgments: I would like to thank the organizers for the opportunity to talk at DIS'05, and Stan Brodsky and Gunnar Ingelman for very interesting discussions.
. J Alwall, G Ingelman, hep-ph/0503099Phys. Rev. 7194015J. Alwall and G. Ingelman, Phys. Rev. D71 (2005) 094015, hep-ph/0503099.
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| []
|
[
"Mean-field theory on a coupled system of ferromagnetism and electronic nematic order",
"Mean-field theory on a coupled system of ferromagnetism and electronic nematic order"
]
| [
"Hiroyuki Yamase \nMax-Planck-Institute for Solid State Research\nD-70569StuttgartGermany\n\nNational Institute for Materials Science\n305-0047TsukubaJapan\n"
]
| [
"Max-Planck-Institute for Solid State Research\nD-70569StuttgartGermany",
"National Institute for Materials Science\n305-0047TsukubaJapan"
]
| []
| We analyze an effective model on a square lattice with two types of forward scattering interactions, which, respectively, drive ferromagnetism (FM) and electronic nematic order via a d-wavePomeranchuk instability (dPI). The FM and dPI in general compete with each other and they are typically separated by a first order phase boundary in the plane of the chemical potential and temperature. Nevertheless there is a parameter region where the dPI occurs inside the FM phase, leading to their coexistence. We also study the effect of a magnetic field by choosing a chemical potential where the ground state is paramagnetic without a field. In this case, instead of FM, the dPI competes with a metamagnetic instability. The latter occurs above a threshold strength of the FM interaction and otherwise the dPI is stabilized with a dome-shaped phase diagram in the plane of a magnetic field and temperature. The FM interaction shifts the center of the dome to a lower field, accompanied by a substantial reduction of the field range where the dPI is stabilized and by an extension of the first order part of the transition line, although the maximal critical temperature does not change. The experimental phase diagram of the bilayer ruthenate Sr 3 Ru 2 O 7 can be well captured by the present theory. | 10.1103/physrevb.87.195117 | [
"https://arxiv.org/pdf/1212.4883v1.pdf"
]
| 118,308,691 | 1212.4883 | 7abe6e2ef4c2d328a8f883c1075016c2b41bce0d |
Mean-field theory on a coupled system of ferromagnetism and electronic nematic order
19 Dec 2012
Hiroyuki Yamase
Max-Planck-Institute for Solid State Research
D-70569StuttgartGermany
National Institute for Materials Science
305-0047TsukubaJapan
Mean-field theory on a coupled system of ferromagnetism and electronic nematic order
19 Dec 2012(Dated: May 5, 2014)numbers: 7127+a7118+y7525Dk7470Pq
We analyze an effective model on a square lattice with two types of forward scattering interactions, which, respectively, drive ferromagnetism (FM) and electronic nematic order via a d-wavePomeranchuk instability (dPI). The FM and dPI in general compete with each other and they are typically separated by a first order phase boundary in the plane of the chemical potential and temperature. Nevertheless there is a parameter region where the dPI occurs inside the FM phase, leading to their coexistence. We also study the effect of a magnetic field by choosing a chemical potential where the ground state is paramagnetic without a field. In this case, instead of FM, the dPI competes with a metamagnetic instability. The latter occurs above a threshold strength of the FM interaction and otherwise the dPI is stabilized with a dome-shaped phase diagram in the plane of a magnetic field and temperature. The FM interaction shifts the center of the dome to a lower field, accompanied by a substantial reduction of the field range where the dPI is stabilized and by an extension of the first order part of the transition line, although the maximal critical temperature does not change. The experimental phase diagram of the bilayer ruthenate Sr 3 Ru 2 O 7 can be well captured by the present theory.
I. INTRODUCTION
In the nematic liquid crystal, 1 rodlike molecules have a preferred orientation. This state is characterized by breaking of orientational symmetry, retaining other symmetries of the system. Electronic analogues of the nematic liquid crystal attract much interest. Electrons have spin and the direction is defined in spin space. Using spin degrees of freedom, a spin nematic state is studied in quantum spin systems. 2, 3 Electrons also have orbital degrees of freedom.
With orbital order such as an occupation difference between the d yz -and d zx -orbital in a d-electron system, electrons may break orientational symmetry without any additional symmetry breaking, leading to an orbital nematic state. 4,5 Ferropnictides are possible materials for such a state. 6,7 On the other hand, the orientation cannot be defined for charge itself.
However, a nematic state can be realized by using a charge degree of freedom. Two routes toward a charge nematic state are proposed. When the system is close to a charge stripe order, namely one-dimensional-charge order, where both translational and orientational symmetry are broken, fluctuations of charge stripes may restore the former but the latter may be still broken. 8 The charge nematic order can be obtained also without invoking charge stripes.
It was found theoretically that the two-dimensional t-J 9 and Hubbard 10 models exhibit a tendency toward a d-wave Pomeranchuk 11 instability (dPI). In this state, the Fermi surface expands along the k x direction and shrinks along the k y direction, or vice versa, whereas in a real space representation the nearest neighbor hopping is effectively enhanced along one direction and suppressed along the other direction.
The dPI was extensively studied not only in the t-J 9,12-14 and Hubbard 10,15-19 models, but also in phenomenological models, 20,21 a model with central forces, 22,23 general Fermi liquid schemes, 24,25 and continuum (not lattice) models. [26][27][28][29][30][31] Mean-field theory of the dPI 20,21 showed that the dPI occurs around van Hove filling with a dome-shaped transition line.
Typically the transition is second order at high temperature and changes to first order at lower temperature. The end points of the second order line are tricritical points. The meanfield phase diagram is characterized by a single energy scale, similar to the BCS theory of superconductivity, and thus various universal numbers were found. 21 Fluctuations of the dPI suppress the first order transition obtained in mean-field theory and when they are strong enough, the transition changes to be continuous even at zero temperature, leading to a quantum critical point. 32,33 At the quantum critical point, dPI fluctuations lead to a non-Fermi liquid ground state. 34,35 At finite temperatures close to the dPI, thermal fluctuations become dominant. They turned out to truncate the original Fermi surface, leading to a Fermi-arc-like feature. 36 Signatures of nematicity were observed in cuprate superconductors. Neutron scattering measurements revealed a strong anisotropy of magnetic excitations in momentum space. [37][38][39] The anisotropy showed strong temperature and doping dependences, which are well captured in terms of the competition of the dPI and singlet pairing formation. 40,41 Transport measurements also revealed a very strong anisotropy of the Nernst coefficient, 42 which was interpreted as a signature of charge nematic order. 43 There is growing evidence that the bilayer ruthenate Sr 3 Ru 2 O 7 (Sr327) exhibits a dPI in a strong magnetic field. [44][45][46] In fact, many features observed in experiments were well understood in terms of the dPI, for example, the metamagnetic transition, 47 the enhancement of the residual resistivity, 48 the bilayer effect, 49,50 the suppression of the critical temperature by impurities, 51 and the spin-orbit effect. 52 Furthermore, the experimental phase diagram is very similar to that obtained in mean-field theory. 53 In particular, it was found that the mean-field phase diagram is characterized by a single energy scale even in the presence of a magnetic field. 54 Therefore there exist various universal ratios for a given chemical potential, which can be compared directly with experimental data. Although several universal ratios agree with the experimental data, ratios of the characteristic temperature and field give one order of magnitude smaller than the experimental ones. 54 This apparent inconsistency cannot be resolved by invoking different choices of parameters. The key may lie in the set of experimental indications that Sr327 is located close to a ferromagnetic instability: a large Wilson ratio, 55 a uniaxial-pressure-induced ferromagnetic transition, 56 and the presence of ferromagnetic fluctuations observed by the inelastic neutron scattering, 57 the nuclear spin-lattice relaxation rate, 58 and thermal expansion measurements. 59 Moreover several band calculations 60,61 for Sr327 (without a field) suggested that the system is close to ferromagnetism (FM). Hence the presence of a ferromagnetic interaction is quite plausible in Sr327. In fact, early theoretical work 62,63 for Sr327 focused on the role of ferromagnetic interactions, especially in the context of a metamagnetic transition observed in experiments. 64 In this paper, we develop a mean-field theory by taking two types of forward scattering interactions, which drive the dPI and FM, respectively, into account. In the context of Sr327, it is interesting to explore how the mean-field phase diagram of the dPI obtained previously is modified by the presence of a ferromagnetic interaction and how well the experimental phase diagram of Sr327 is captured. Furthermore, the interplay of the dPI and FM is interesting in its own right. While FM is an instability in the spin channel whereas the dPI is in the charge channel, both are instabilities in the particle-hole channel of q=0 and do not break translational symmetry. Several theoretical analyses of microscopic models 52,65,66 actually suggested the presence of a ferromagnetic instability, which competes with the dPI.
Therefore in a more general setting we study the interplay of the dPI and FM, and clarify possible scenarios in such a coupled system.
We propose an effective model, suitable to address the interplay of the dPI and FM, and derive resulting phase diagrams. In Sec. II, we introduce a forward scattering model and present results in Sec. III by separating two cases: i) zero magnetic field (h = 0) and ii) finite magnetic field (h = 0). The latter case is relevant to Sr327. Conclusions follow in Sec. IV.
II. MODEL
To analyze a coupled system of the dPI and FM, we consider the following Hamiltonian on a square lattice,
H = H 0 + H φ + H m + H Z .
(1)
The first term H 0 is the kinetic term,
H 0 = kσ (ǫ 0 k − µ)c † kσ c kσ ,(2)
where c † kσ (c kσ ) is a creation (annihilation) operator of an electron with spin σ and momentum k; µ is the chemical potential. The electron dispersion is given by
ǫ 0 k = −2t(cos k x + cos k y ) − 4t ′ cos k x cos k y(3)
with t and t ′ being the nearest and second nearest neighbor hopping amplitudes, respectively.
The second term H φ is a forward scattering interaction driving a dPI,
H φ = − g φ 2N kk ′ σσ ′ d k d k ′ c † kσ c kσ c † k ′ σ ′ c k ′ σ ′ ,(4)
where the coupling constant g φ is positive, d k is a d-wave form factor such as d k = cos k x − cos k y , and N is the total number of lattice sites. This term describes the d-wave weighted density-density interaction with zero momentum transfer, which was obtained in microscopic models such as the t-J 9 and Hubbard 10,65 models.
The third term H m describes an Ising ferromagnetic interaction,
H m = − g m 2N kk ′ σσ ′ c † kσ σ 2 c kσ c † k ′ σ ′ σ ′ 2 c k ′ σ ′ ,(5)
where g m (> 0) is a coupling constant and σ = +1 and −1 for up-spin and down-spin, respectively. This interaction is obtained by focusing on the spin-spin interaction with a spin quantization axis parallel to the z direction and by extracting a scattering process with zero momentum transfer. Therefore the interaction described by H m is appropriate when the system has a strong spin anisotropy as well as dominant forward scattering processes of electrons. The interaction of H m is also obtained by considering a mean-field analysis of spin rotational invariant interactions. For instance, in the case of the Hubbard onsite interaction U i n i↑ n i↓ , our coupling constant is given by g m = 2U.
The last term H Z is the Zeeman energy,
H Z = − h 2 kσ σc † kσ c kσ .(6)
Here h is an effective magnetic field given by h = gµ B H, with g being a g factor, µ B is the Bohr magneton, and H is a magnetic field.
The terms of H φ and H m describe pure forward scattering interactions of electrons. Thus fluctuations around the mean-field vanish in the thermodynamic limit. In other words, mean-field theory solves our Hamiltonian exactly in the limit of N → ∞.
The order parameter of the dPI is defined by
φ = g φ N kσ d k c † kσ c kσ .(7)
This quantity becomes finite only if the system breaks square lattice symmetry because of the presence of the d-wave form factor. FM order is defined by
m = g m 2N kσ σ c † kσ c kσ ,(8)
where we include the coupling constant g m in the definition of m; while the magnetization is then given by m/g m , we may refer to m as magnetization, as long as no confusion occurs.
We decouple the interaction terms (4) and (5) by introducing the order parameters φ and m, and obtain the mean-field Hamiltonian,
H M F = kσ ξ kσ c † kσ c kσ + N 2g m m 2 + N 2g φ φ 2 ,(9)
where the renormalized dispersion is given by
ξ kσ = ǫ 0 k − σ 2 (m + h) − d k φ − µ .(10)
The grand canonical potential per site at temperature T is obtained as
ω = − T N kσ log(1 + e −ξ kσ /T ) + 1 2g m m 2 + 1 2g φ φ 2 .(11)
The stationary condition of ω with respect to φ and m leads to the self-consistent equations
φ = g φ N kσ d k f (ξ kσ ) ,(12)m = g m 2N kσ σf (ξ kσ ) ,(13)
which we solve numerically. Here f (ξ kσ ) is the Fermi function.
III. RESULTS
We fix g φ /t = 1 throughout this paper unless otherwise noted and explore how the phase diagram of the dPI changes with increasing the FM interaction g m . We first study the case of h = 0 and then that of h = 0. As a band parameter, we choose t ′ /t = 0.35, which was used for the study of Sr327. 53,54 Since the presence of t ′ turns out to play a crucial role to understand phase diagrams for h = 0, we also study the case of t ′ = 0 for h = 0. Hereafter we set t = 1 and all quantities with dimension of energy are in units of t. For g m = 6.5, the FM interaction becomes strong enough to realize FM near the edge on the side of a high chemical potential [ Fig. 1 (b)]. The transition from the paramagnetic to FM phase is second order at high temperature, but the second order line ends at a tricritical point and changes to a first order line at low temperature. This feature is the same as the transition between the paramagnetic and dPI phase. The boundary of the dPI and FM is characterized by a first order transition (T 1st φm ). As shown in Fig. 1 (c As expected, with further increasing g m , the first order boundary between the dPI and FM shifts to a lower chemical potential. In fact, as shown in Fig. 3 (a), the dPI is realized only near the edge of the dome for g m = 7.8. However, qualitative changes occur in the phase diagram. First, the coexistence of the dPI and FM is stabilized inside the FM phase near the edge of the first order line of the FM around µ = 2.04. This region is magnified in For g m = 8, as shown in Fig. 4 (a), the FM becomes dominant and a pure dPI phase is not stabilized. Instead the dPI is realized in coexistence with the FM around µ = 2.06, as magnified in Fig. 4 (b). In contrast to the case of g m = 7.8 [ Fig. 3 (b)], the phase boundary of the coexistence is well separated from the first order line of the FM, leading to a phase The magnitude of m decreases monotonically and vanishes at the band edge of µ = 2.6. The system becomes a band insulator for µ > 2.6.
With further increasing g m (Fig. 5), the band-edge FM is absorbed into the main FM phase. A first order phase transition then occurs only on the lower side of µ. Inside the FM, the coexistence of the dPI and FM is stabilized up to g m = 9.8. Figure 5 (a) is the representative phase diagram computed for g m = 9. In Fig. 5
Discussions
The coexistence of the dPI and FM is stabilized even for g m ≫ g φ (Figs. 3−5). This is because of the presence of the van Hove singularity. After performing explicit calculations up to g m = 10, we confirm the van Hove singularity due to the down-spin band (m > 0 is assumed) inside the FM phase for g m 7.8. Around the van Hove filling, either the dPI or a metamagnetic transition occurs in our model, depending on energetics. We find that the coexistence of the dPI and FM is more favorable for 7.8 g m 9.8 and the metamagnetic transition for g m 9.8. Our results for h = 0 are summarized as follows: i) in 0 ≤ g m ≤ g m1 , only the dPI phase is realized, ii) in g m1 ≤ g m ≤ g m3 , both dPI and FM are stabilized, but they are separated from each other by a first order transition line, iii) in g m2 ≤ g m ≤ g m4 , the coexistence with dPI occurs inside the FM phase, and iv) in g m4 ≤ g m , only the FM is realized. We have obtained g m1 ≈ 6.5, g m2 ≈ 7.8, g m3 ≈ 7.9, and g m4 ≈ 9.8 for t ′ = 0.35, leading to rich phase diagrams as shown in Figs. 1, 3, 4, and 5. For t ′ = 0, on the other hand, we have obtained g m1 ≈ 8.84, g m2 = g m3 = g m4 ≈ 8.87. As a result, a phase diagram is occupied by either the dPI or FM except for a tiny range of g m .
B. Results for h = 0
Next we examine the effect of a magnetic field, motivated by the experimental indication that Sr327 is paramagnetic in zero field and exhibits a nematic instability around 8 Tesla. [44][45][46] Fixing the chemical potential µ = 1 and taking the field as a tuning parameter, we study how the phase diagram of the dPI evolves with increasing the ferromagnetic interaction. To understand these features, we consider a magnetic field h vH , at which the σ-spin band touches the van Hove energy, and the dPI is expected around that. From Eq. (10), h vH fulfills for φ = 0 the relation,
σ(m + h vH ) 2 + µ = µ vH ,(14)
and the corresponding relation for the other spin band should be −σ(m + h vH )/2 + µ = 2µ − µ vH , where µ vH = 4t ′ . Since µ is fixed in our case, we obtain While the magnetization is not fully linear in field in the entire field range we consider, we may invoke the equation obtained in linear response theory,
h vH = 2|µ − µ vH | − m .(15)m/g m ≈ χh vH ,(16)= χ 0 1 − g m χ 0 h vH ,(17)
where χ is the full magnetic susceptibility, which is expressed by the non-interacting magnetic susceptibility χ 0 as shown in the second line; the presence of g m on the left-hand side is due to our definition of m [see Eq. (8)]. We then obtain
h vH = 2(1 − g m χ 0 )|µ − µ vH | ,(18)
that it, the value of h vH is reduced with increasing g m . Since the dPI occurs around the van
Hove energy, the dPI should be realized around a lower field with increasing g m .
Equation (16) The range of a magnetic field where the dPI is stabilized becomes narrower for a larger g m . As seen in Eq. (10), the sum of m and h plays a role as an effective field. Since m becomes more susceptible to a field as g m becomes larger and furthermore m is proportional to g m in our definition [Eq. (8)], the value of h to stabilize the dPI is necessarily reduced.
The first order transition line extends with increasing g m . To understand this, we expand the free energy Eq. (11) with respect to the order parameter of the dPI around φ = 0,
ω(φ; m) − ω(0; m) = 1 2 a 2 φ 2 + 1 4! a 4 φ 4 + · · · .(19)
The coefficients of a 2 and a 4 are obtained as
a 2 = 1 g φ 1 + g φ N kσ d 2 k f ′ (ξ 0 kσ ) ,(20)a 4 = 1 N kσ d 4 k f ′′′ (ξ 0 kσ ) − 3g m 1 2N kσ σd 2 k f ′′ (ξ 0 kσ ) 2 1 + gm 4N kσ f ′ (ξ 0 kσ ) ,(21)
where ξ 0 kσ = ǫ 0 k − σ(m+h) In Fig. 7 (b), the order parameter of the dPI is plotted as a function of h for a sequence of g m at low T . Because of two first order transitions at low T [ Fig. 7 (a)], the order parameter exhibits two jumps. Interestingly the maximal value of φ does not depend on g m . This feature is easily understood from Eqs. (10) and (12). The right-hand side of Eq. (12) depends on the quantityμ kσ = σ 2 (m + h) + d k φ for a fixed µ. Suppose the maximal value of φ, say φ max , is obtained at h = h max for g m = 0, namely for m = 0. Even when g m is turned on, the same value of φ max is obtained as long as m and h fulfills the equation
m + h = h max .(22)
This equation may hold unless the value of m becomes as large as h max . We can check that
Eq. (22) indeed holds up to g m ≈ 7, leading to the same maximal value of φ for g m = 0 − 7.
In Fig. 7 (c), the magnetization is plotted as a function of h at low T . Because of first order transitions at low T , the magnetization exhibits two successive jumps. It is instructive to recognize that there could occur a metamagnetic transition at h ≈ 0.12 if the coupling g φ would be turned off, indicating the underlying competition of the dPI and a metamagnetic transition. We can check that the dPI overcomes the metamagnetic transition up to g m = 7.9.
For g m ≥ 8, on the other hand, the metamagnetic transition becomes dominant and the magnetization exhibits a single jump as shown in Fig. 8 (a). The Landau free energy is plotted in Fig. 8 transition; the order parameter φ is optimized to minimize the free energy at each m.
There are three local minima. Two local minima, where φ = 0 is stabilized, are associated with the metamagnetic phenomenon. The other local minimum, at which φ becomes finite, corresponds to a solution of the dPI. This solution, however, does not give the absolute minimum and thus the dPI does not occur.
When g m exceeds 8.25, the FM occurs even for h = 0. In this case, neither a metamagnetic transition nor a dPI occurs by applying a magnetic field.
The effect of a ferromagnetic interaction on the dPI for h = 0 can be summarized as follows: i) the dPI occurs in a lower magnetic field, ii) the field range where the dPI is stabilized becomes narrower, iii) the first order part of the transition line extends, and iv) the dPI and a metamagnetic transition compete with each other and the former is realized up to g m ≈ 8, and the latter for 8 g m 8.25 for the present choice of parameters.
IV. CONCLUSIONS
We have studied a two-dimensional electron system, where electrons interact with each other via interactions favoring a dPI and FM. In the absence of a magnetic field, we have obtained rich phase diagrams. The dPI and FM typically compete with each other. In fact, while both dPI and FM can be realized simultaneously, they are separated by a first order phase boundary. Nevertheless it is possible that the dPI is stabilized inside the FM phase, leading to their coexistence. The presence of t ′ , leading to a breaking of particle-hole symmetry, plays an important role. For t ′ = 0, either the dPI or the FM is typically realized in the plane of the chemical potential and temperature, and coexistence is not stabilized.
We have also studied the effect of a magnetic field, motivated by the experimental indication that Sr327 is in the normal state without a magnetic field and exhibits a nematic instability by applying a field. In this case, instead of FM, the dPI competes with a metamagnetic transition. The latter occurs above a threshold strength of the FM interaction and otherwise the dPI is stabilized with a dome-shaped transition line around the van Hove energy in the plane of a field and temperature. With increasing the FM interaction, the center of the dome shifts to a lower field, accompanied by a substantial reduction of the field range where the dPI is stabilized and by an extension of the first order part of the transition line, although the maximal T c does not change.
It might seem that the interaction strength of g m is considered up to a too large value (g m ∼ 10) in our study. However, this seemingly large value is due to our definition of g m in Eq. (5) where a factor of (1/2) 2 originating from spin is not absorbed into the definition of g m .
A typical feature of the dPI is that its mean-field phase diagram is characterized by universal ratios. 21,54 In the model solved in Ref. 54, several universal ratios reasonably agree with experimental values, but ratios of temperature and a magnetic field come out one order of magnitude smaller than the experimental data. For example, in experiments, T tri c /h tri ∼ 0.6k B /(0.15gµ B ) = 6g −1 ≈ 3 if g = 2, whereas theoretically we obtain T tri c /h tri ∼ 0.3 for g m = 0 (Ref. 67); here h tri is the field at a tricritical point measured from the van Hove energy. However in the presence of a ferromagnetic interaction, we have found that only the scale of a magnetic field is substantially reduced while the temperature scale is not. As a result, from Fig. 7 (a), we obtain T tri c /h tri ∼ 2 and 7 (Ref. 67) for g m = 6 and 7, respectively.
The ratio of T tri c /h tri is substantially modified by a FM interaction to become comparable to the experimental one. The large value of g m indicates that the system is close to the FM instability for h = 0, the same situation as in Sr327. [55][56][57][58] The FM interaction pushes up T tri c to a higher temperature, but the maximal T c does not change. As a result, other ratios such as T tri c /T vH c , where T vH c is T c at the van Hove energy, now becomes slightly larger than the experimental value, although it showed better agreement with experimental data in the model with g m = 0. 54 However, this may be easily improved by invoking weak fluctuations associated with the dPI, since it was shown 32,33 that fluctuations suppress T tri c stronger than T vH c . Therefore the ratios in the experimental phase diagram of the dPI are well understood by the presence of a FM interaction tuning the system close to the FM instability, and by weak dPI fluctuations.
The lines of first order phase transitions tilt outward in the experimental phase diagram, 44 indicating that the entropy inside the dPI phase is larger than that in the normal state. 46 This counterintuitive phenomenon is not captured in the present theory. This inconsistency may be explored further in terms of the interplay of ferromagnetic fluctuations and the dPI by going beyond the mean-field model.
While we have analyzed a single band model, Sr327 is a t 2g system and orbital nematic order may provide another possible scenario. 4,5 Since there are interactions among different orbitals, the dPI is expected to generate orbital nematic order, or vice versa. It is an open question which is the driving force for nematicity observed in Sr327.
FIG. 1 :
1Evolution of phase diagrams with increasing FM interactionFigure 1shows a sequence of phase diagrams for g m ≤ 7.0 in the plane of the chemical potential µ and temperature T . Because of the competition with the dPI, no FM instability occurs at least up to g m = 6.0 [Fig. 1 (a)] and the phase diagram is occupied only by the dPI. As already clarified previously, 20,21 the dPI occurs below a dome-shaped transition line, with a maximal T c near the van Hove energy (µ vH = 4t ′ = 1.4); a deviation from µ vH is due to the presence of t ′ , which breaks particle-hole symmetry. The phase transition is of second order at high temperature (T 2nd c ) and of first order at low temperature (T 1st c ). The end points of the second order line are tricritical points (T tri c ). Phase diagram in the (µ, T ) plane for a sequence of couplings g m . Transition from the paramagnetic to ordered phase is a second order (T 2nd c ) at high T and a first order (T 1st c ) at low T ; T tri c is the temperature at a tricritical point. A dashed line (T 1st φm ) denotes the first order phase boundary between the dPI and FM, which appears in (b) and (c).
), this first order phase boundary shifts to the middle of the phase diagram for g m = 7.0 and the FM becomes more stable. The order parameters are plotted as a function of µ in Figs. 2 (a) and (b) at T = 0.01 and 0.20, respectively. At a low temperature (T = 0.01), φ and m show a jump at µ ≈ 1.05 and 1.81, respectively, because of a first order transition from the paramagnetic phase. The dPI changes to the FM via a first order transition at µ ≈ 1.45 and there is no mixing of φ and m. At a high temperature (T = 0.20), on the other hand, φ and m develop continuously at µ ≈ 1.10 and 1.72, respectively. The transition between the dPI and FM is however still of first order. online) µ dependence of φ and m at T = 0.01 (a) and 0.20 (b) for g m = 7.0.
Fig. 3
3(b). The transition from the FM to the coexistence is first order at low temperature and becomes second order at high temperature. While one end point of the second order line at µ ≈ 2.037 is a tricritical point, the other end point at µ ≈ 2.045 is just a point touching the first order line of the FM. There is a direct first order transition from the paramagnetic phase to the coexistence around µ = 2.05. Second, an additional FM phase appears in 2.52 µ ≤ 2.6 as shown in Figs. 3 (a) and (c). This FM comes from the enhancement of the density of state at the band edge of µ = 2.6. A first order transition occurs only on the side of a lower chemical potential and the second order line disappears at the band edge.This band-edge FM is realized for 7.6 g m 8.
FIG. 3 :
3Phase diagram in the (µ, T ) plane for g m = 7.8. The regions near µ = 2.04 and 2.55 are magnified in (b) and (c), respectively.
FIG. 4 :
4(a) Phase diagram in the (µ, T ) plane for g m = 8.0. The region of the coexistence around µ = 2.06 is magnified in (b). (c) µ dependence of m at T = 0.001. Two successive jumps around µ = 2.06 are magnified in the inset. diagram very similar to that of the pure dPI [Fig. 1 (a)], but with a significant extension of the first order portion of the transition line; the reason for this will be explained later in terms of Eq. (21). The magnetization m is plotted as a function of µ in Fig. 4 (c) at low temperature. After the first order FM transition at µ ≈ 1.05, the value of m increases with increasing µ and forms a cusp at µ ≈ 1.45 where the density of states of up-spin electrons is fully occupied and the system changes to a half-metallic state. For µ 1.45, m decreases since electrons with down-spin increase whereas the electron density of up-spin remains unity. At µ ≈ 2.05 and 2.07, m exhibits a jump [see the inset of Fig. 4 (c)] because of the presence of the coexistence of the dPI and FM, which occurs via a first order transition at low T . The magnetization m vanishes discontinuously at µ ≈ 2.15, but appears again with a jump at µ ≈ 2.42 because of a first order transition associated with the band-edge FM.
FIG. 5 :
5(b) the region of the coexistence of the dPI and FM is magnified. This phase diagram is very similar to that for g m = 8 [Fig. 4 (b)] with the same maximal T c , but with a further extension of the first order transition line. For g m 9.8, however, the coexistence is replaced by a first order transition associated with a jump of the magnetization, namely a metamagnetic transition inside the FM, as denoted by solid squares in Fig. 5 (c). The magnetization is potted as a function of µ at low T in Fig. 5 (d). The jump at µ ≈ 2.23 comes from the metamagnetic transition. The cusp at µ ≈ 0.81 indicates that the up-spin band is fully occupied in µ 0.81, where the system becomes half-metallic. (a) Phase diagram in the (µ, T ) plane for g m = 9.0. The region of the coexistence is magnified in (b). (c) Phase diagram for g m = 10. T meta denotes the position where a metamagnetic transition occurs. (d) µ dependence of m at T = 0.001 for g m = 10.
FIG. 6 :
6Phase diagram in the (µ, T ) plane for a sequence of couplings g m by setting t ′ = 0. The phase diagram is occupied by the dPI in g m ≤ 8.84 (a) and by the FM in g m ≥ 8.87 (d). In a tiny range of g m [(b) and (c)], both FM and dPI are realized, but separated from each other by a first order boundary; the line of T 1st φm appears only in (b) and (c). Our results shown in Figs. 1−5 are very asymmetric with respect to the van Hove energy of the bare dispersion, which is given by µ vH = 4t ′ = 1.4. This is because the presence of t ′ breaks particle-hole symmetry. In fact, for t ′ = 0, the phase diagram becomes symmetric with respect to the axis of µ = 0. For 0 ≤ g m ≤ 8.84, the dPI is stabilized and no FM is realized [Fig. 6 (a)]. For g m 8.85, however, the dPI starts to be replaced by the FM phase from a higher temperature [Fig. 6 (b)] and is stabilized only around µ = 0 at low T for g m = 8.86 [Fig. 6 (c)]. The dPI disappears already for g m = 8.87. The change from the dPI [Fig. 6 (a)] to the FM phase [Fig. 6 (d)] occurs in a very small range of g m . In contrast to the case of Figs. 3, 4, and 5, no coexistence of the dPI and FM is stabilized. Furthermore a band-edge FM does not appear.
Figure 7 (
7a) is a set of phase diagrams of the dPI in the plane of a magnetic field and temperature for a sequence of g m , showing four characteristic features: with increasing g m , i) the dPI occurs in a lower field, ii) the field range where the dPI is stabilized shrinks substantially, iii) the first order part of the transition line extends and tricritical points are pushed up to higher temperatures, but iv) the maximal T c does not change.
FIG. 7 :
7(Color online) (a) Phase diagram in the (h, T ) plane for a sequence of couplings g m ; the value of g m is indicated near the maximal T c . The dPI is stabilized inside the dome for each g m . (b) h dependence of φ at T = 0.001 for a sequence of g m . (c) h dependence of m for g m = 7 at T = 0.001. The corresponding result for g φ = 0 is also plotted.
is a rough approximation especially near a metamagnetic transition and the resulting Eq. (18) should be taken as qualitative understanding. To get quantitative understanding, we solve Eq. (13) numerically under the condition of φ = 0 and Eq. (14). We then obtain h vH ≈ 0.61, 0.41, 0.22, 0.12 for g m = 2, 4, 6, 7, respectively; for g m = 0, on the other hand, h vH = 2|µ vH − µ| = 0.8 since m = 0. The dPI indeed occurs around those fields in Fig. 7.
2 −
2µ and f ′ , f ′′ , f ′′′ are the first, second, third derivative of the Fermi function. When a 4 becomes negative, a first order transition can occur. The second term on the right-hand side of Eq. (21) originates from the φ dependence of m. The denominator of this term is positive close to the dPI and the numerator becomes in general finite when the spin symmetry is broken. Hence the second term is negative for h = 0. Furthermore the second term is proportional to g m . Therefore the presence of the second term in Eq. (21) leads to an extension of the first order transition line of the dPI and this effect becomes stronger for a larger g m . The same argument explains the extension of the first order portion of the transition line in Figs. 4 (b) and 5 (b), since the second term of Eq. (21) becomes negative also in the FM phase.A second order transition is given by the condition a 2 = 0. Since µ is fixed, the quadratic term a 2 is a function ofh = m + h. Suppose the maximal T c is obtained ath max , there can exist a field h and a magnetization m, which give the same value ofh max for a different g m , although the values of m and h themselves depend on g m . This is actually the case up to g m = 7.8, leading to the same maximal T c inFig. 7 (a). A similar consideration also explains the same maximal T c in Figs. 4 (b) and 5 (b). Keeping in mind that our system is half-metallic in the range of µ where the coexistence is stabilized [see the discussion about Fig. 4 (c)] and thus only the down-spin band is active, the coefficient a 2 becomes a function of the quantityμ = −m 2 + µ for h = 0. We confirm the same value ofμ at the maximal T c in Figs. 4 (b) and 5 (b), respectively, which necessarily yields the same maximal T c .
(b) as a function of m at h = 0.023, just below the metamagnetic online) (a) h dependence of m for g m = 8 at T = 0.001. (b) Free energy as a function of m at h = 0.023 and T = 0.001 for g m = 8. The value of φ which minimizes the free energy at each m is also plotted.
P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, UK, 1993). 2 A. F. Andreev and I. A. Grishchuk, Sov. Phys. JETP 60, 267 (1984).
AcknowledgmentsThe author thanks W. Metzner for a critical reading of the manuscript and valuable comments. Support by the Alexander von Humboldt Foundation and a Grant-in-Aid for Scientific Research from Monkasho is also gratefully acknowledged.
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| []
|
[
"M-theory/heterotic Duality: a Kaluza-Klein Perspective",
"M-theory/heterotic Duality: a Kaluza-Klein Perspective"
]
| [
"H Lü $ \nDepartment of Physics\nUniversity of Pennsylvania\n19104PhiladelphiaPA\n\nLaboratoire de Physique Théorique de l'École Normale Supérieure 2\nCEDEX 05 § SISSA\n3 24 Rue Lhomond -, Via Beirut No. 2-475231, 34013Paris, TriesteItaly\n",
"C N Pope \nCenter for Theoretical Physics\nTexas A&M University\n77843College Station, Texas\n",
"K S Stelle ⋆♯ \nThe Blackett Laboratory\nImperial College\n\n\nPrince Consort Road\nSW7 2BZLondonUK\n\nTH Division\nCERN\nCH-1211Geneva 23Switzerland\n",
"\nUnité Propre du Centre National de la Recherche Scientifique, associéeà l'École Normale Supérieure età l'Université de Paris-Sud\n\n"
]
| [
"Department of Physics\nUniversity of Pennsylvania\n19104PhiladelphiaPA",
"Laboratoire de Physique Théorique de l'École Normale Supérieure 2\nCEDEX 05 § SISSA\n3 24 Rue Lhomond -, Via Beirut No. 2-475231, 34013Paris, TriesteItaly",
"Center for Theoretical Physics\nTexas A&M University\n77843College Station, Texas",
"The Blackett Laboratory\nImperial College\n",
"Prince Consort Road\nSW7 2BZLondonUK",
"TH Division\nCERN\nCH-1211Geneva 23Switzerland",
"Unité Propre du Centre National de la Recherche Scientifique, associéeà l'École Normale Supérieure età l'Université de Paris-Sud\n"
]
| []
| We study the duality relationship between M-theory and heterotic string theory at the classical level, emphasising the transformations between the Kaluza-Klein reductions of these two theories on the K3 and T 3 manifolds. Particular attention is devoted to the corresponding structures of σ-model cosets and the correspondence between the p-brane charge lattices. We also present simple and detailed derivations of the global symmetries and coset structures of the toroidally-compactified heterotic theory in all dimensions D ≥ 3, making use of the formalism of solvable Lie algebras. | 10.1016/s0550-3213(99)00086-3 | [
"https://arxiv.org/pdf/hep-th/9810159v1.pdf"
]
| 119,434,050 | hep-th/9810159 | 1d4606de1c76bad7bbb00353a6e52b8986c37b2b |
M-theory/heterotic Duality: a Kaluza-Klein Perspective
arXiv:hep-th/9810159v1 22 Oct 1998 October 1998
H Lü $
Department of Physics
University of Pennsylvania
19104PhiladelphiaPA
Laboratoire de Physique Théorique de l'École Normale Supérieure 2
CEDEX 05 § SISSA
3 24 Rue Lhomond -, Via Beirut No. 2-475231, 34013Paris, TriesteItaly
C N Pope
Center for Theoretical Physics
Texas A&M University
77843College Station, Texas
K S Stelle ⋆♯
The Blackett Laboratory
Imperial College
Prince Consort Road
SW7 2BZLondonUK
TH Division
CERN
CH-1211Geneva 23Switzerland
Unité Propre du Centre National de la Recherche Scientifique, associéeà l'École Normale Supérieure età l'Université de Paris-Sud
M-theory/heterotic Duality: a Kaluza-Klein Perspective
arXiv:hep-th/9810159v1 22 Oct 1998 October 1998CERN-TH/98-303, CTP TAMU-36/98, Imperial/TP/97-98/77, LPTENS-98/39, UPR/0819-T hep-th/9810159
We study the duality relationship between M-theory and heterotic string theory at the classical level, emphasising the transformations between the Kaluza-Klein reductions of these two theories on the K3 and T 3 manifolds. Particular attention is devoted to the corresponding structures of σ-model cosets and the correspondence between the p-brane charge lattices. We also present simple and detailed derivations of the global symmetries and coset structures of the toroidally-compactified heterotic theory in all dimensions D ≥ 3, making use of the formalism of solvable Lie algebras.
Introduction
The duality relations [1,2,3,4] between the heterotic string theory and M-theory are perhaps the most surprising of the web of dualities that is now seen to underpin the search for a satisfactory nonperturbative formulation of a quantum theory of gravity and everything else. One of the most striking features of this web of dualities is the degree to which such nonperturbative relations can be seen already in nonlinear features of the classical or semiclassical field theory limits of the underlying quantum theories. In the present paper, we shall investigate in some detail the classical Kaluza-Klein relations between the heterotic theory and D = 11 supergravity, which is the field-theory limit of M-theory. Specifically, we shall consider the relation between D = 11 supergravity compactified on a K3 manifold and the heterotic theory compactified on T 3 .
A number of general features of the M-theory/heterotic correspondence were originally detailed in Refs [1,2]. In the present paper, we shall study some of the apsects of the Kaluza-Klein reduction procedure in more detail, to reveal further aspects of the correspondences between the two theories. We begin in sections 2 and 3 by considering the nonlinear σ-model structure of the dimensionally-reduced heterotic string, focusing in particular on the relation between group-theoretic coset descriptions of these σ-models and embeddings of the σ-models into linearly realised representations of the numerator symmetry groups, combined with appropriate invariant constraints. Generalising the Borel-subalgebra constructions of analogous σ-models in the maximally supersymmetric supergravities [5,6], we find that the appearance of solvable subalgebras [7,8,9,10,11,12] of the heterotic theory duality symmetries significantly simplifies the derivation of explicit parametrisations for the corresponding heterotic σ-model cosets. We include a relatively digestible discussion of solvable Lie algebras, taking examples from some of the simpler global symmetry groups arising in the heterotic compactifications to illustrate the essential ideas.
In section 5, we analyse the dimensional reduction of M-theory on a K3 manifold, using in particular the approximate description of a K3 manifold as T 4 /Z 2 , with the 16 orbifold singularities of this construction blown up by the cutting and pasting in of Hanson instantons [13]. This analysis will allow us to establish a detailed correspondence between the fields of M-theory and the heterotic theory, and in particular to establish the relationship between the couplings of dilatonic scalars in the two theories.
In section 6, we proceed to establish the correspondence between the lattices of pbrane charges in the two theories. This correspondence reconfirms the structure of the M-theory charge lattice that had originally been derived using duality relations [14,15] and the special properties of "scale-setting" p-brane species [16]. The paper concludes with a discussion of the charge orbits containing p-brane solutions supported purely by the Yang-Mills sector. Given the currently anticipated general relationship between string theory states and semiclassical solutions, such solutions are required to correspond to the short massive multiplets of Yang-Mills sector states associated to fields in the heterotic Lagrangian acquiring masses from the Higgs effect for a generic heterotic vacuum. A perhaps unsettling feature of the only available supersymmetric p-branes supported by the Yang-Mills sector is that they have naked singularities. We shall explain how these naked singularities appear as an artefact of dimensional reduction from wave-like solutions using singular Killing vectors.
In the appendices, we give some details of the toroidal dimensional reduction of the heterotic theory and also some additional embeddings of coset models into constrained linear realisations of symmetry groups.
2 Global symmetries of the heterotic string on T n 2.1 D = 9 heterotic string Taking the general T n reduction of the heterotic theory given in Appendix A, and specialising to the case of reduction to D = 9 on a single circle, we find, after rotating the dilatonic scalars so that
φ 1 = 1 2 √ 2 ϕ − 1 2 7 2 φ , φ 2 = 1 2 7 2 ϕ + 1 2 √ 2 φ ,(2.1)
that the nine-dimensional Lagrangian is given by
e −1 L 9 = R − 1 2 (∂φ) 2 − 1 2 (∂ϕ) 2 − 1 2 e √ 2ϕ I (∂B I (0) ) 2 − 1 12 e − 8 7 φ (F (3) ) 2 − 1 4 e − 2 7 φ e √ 2ϕ (F (2) ) 2 + e − √ 2ϕ (F (2) ) 2 + I (G I (2) ) 2 ,(2.2)
where the I index labels the 16 unbroken Cartan-subalgebra gauge-group generators for a generic "fully-Higgsed" vacuum configuration. From (A. 10), and dropping the primes on the potentials, the field strengths are given by (1) , F (2) = dA (1) , G I (2) = dB I (1) + B I (0) A (1) , F (2) = dA (1)
F (3) = dA (2) + 1 2 B I (1) dB I (1) − 1 2 A (1) dA (1) − 1 2 A (1) dA+ B I (0) dB I (1) + 1 2 B I (0) B I (0) dA (1) ,(2.3)
the nine-dimensional string coupling constant is given by λ 9 = e √ 7/8 φ .
The symmetry group of the scalar manifold, parametrised by the dilatons φ and ϕ, and the axions B I (0) , is easily analysed. Let us, for greater generality, consider the case where there are N abelian vectors potentials B I (1) (1 ≤ I ≤ N ) in D = 10, rather than the particular case N = 16 arising in the heterotic string. We now introduce the set of N + 2 fields X, Y, Z I in IR 1,N +1 , defined by
X + Y = e 1 √ 2 ϕ , X − Y = e − 1 √ 2 ϕ + 1 2 B I (0) B I (0) e 1 √ 2 ϕ , Z I = 1 √ 2 B I (0) e 1 √ 2 ϕ . (2.4)
It is evident that these satisfy the constraint
X 2 − Y 2 − Z I Z I = 1 ,(2.+ 1 4 e − 2 7 φ (dA X ) 2 − (dA Y ) 2 − (dB I (1) ) 2 − 2(X dA X + Y dA Y + Z J dB J (1) ) 2 ,
where we define (1) ) . (2.8) In terms of these redefined potentials, the 3-form field F (3) becomes (2.10)
A X = 1 √ 2 (A (1) + A (1) ) , A Y = 1 √ 2 (A (1) − AF (3) = dA (2) − 1 2 A X dA X + 1 2 A Y dA Y + 1 2 B I (1) dB I(
Note that F (3) is a singlet. In particular, this completes our demonstration that the dimensional reduction of the heterotic theory with U (1) 16 gauge fields gives a theory with an O (1,17) global symmetry in D = 9. There is in addition the previously-mentioned IR factor also, corresponding to a constant shift of the dilaton φ, accompanied by appropriate rescalings of the potentials.
Geometry of O(p, q)/(O(p) × O(q)) cosets
In the previous subsection, we showed how the global symmetry of the heterotic theory when reduced on S 1 could be understood from a geometrical point of view. In fact, more generally, we showed that the reduction of the bosonic sector of ten-dimensional N = 1 supergravity coupled to N abelian gauge fields gives rise to a nine-dimensional theory with , (2.11) where 0 ≤ m ≤ p and 0 ≤ n ≤ q.
To begin, we introduce the indefinite-signature flat metric (2.14)
Note that matrices W satisfying these two conditions do not form a group; rather, as we shall now show, they decompose into orbits described by cosets of the form (2.11).
To make this more precise, we shall consider the orbits of matrices W satisfying ( For a given fiducial matrix W 0 , the denominator group K is easily determined, since it is nothing but the stability subgroup of O(p, q) that leaves W 0 invariant. From (2.12) and
(2.16), we can see that W 0 will be left invariant by the subgroup
K = O(m, q − n) × O(p − m, n) (2.17)
of O(p, q). Thus the orbit for this particular fiducial matrix W 0 is the coset space (2.11).
Note that all points on the given orbit have matrices W that satisfy the trace condition and O(8, N + 8) respectively. 1 Here, we shall make use of the more general discussion of coset spaces given in subsection 2.2 in order to give a geometrical interpretation of the symmetries of the D-dimensional theories.
tr(W η) = q − p + 2(m − n) .(2.
Our starting point is the D-dimensional Lagrangian (A.6), obtained by dimensional reduction from D = 10. We use the expressions (A.10) for the 3-form and 2-form field strengths, which were obtained after making the field redefinitions (A.9). First, we note that the particular linear combination a 1 · φ of dilatons that couples to the 3-form field strength is decoupled from all the axionic scalars A (0)αβ , A α (0)β and B I (0)α . In other words, the dot products a 1 · a 1αβ , a 1 · b αβ and a 1 · c α all vanish. It is therefore natural to perform a rotation on the dilatons so that the combination φ = − (D − 2)/8 a 1 · φ is separated from the rest. After this rotation, one expects that the Lagrangian should be expressible in the (together with a further IR factor for the decoupled scalar φ), we need only show that it can indeed be written in the form (2.21), and that M satisfies the constraints (2.13, 2.14) and
form e −1 L D = R − 1 2 (∂φ) 2 + 1 4 tr(∂M −1 ∂M) − 1 12 e − √ 8/(D−2) φ F 2 (3) − 1 4 e − √ 2/(D−2) φ H T (2) M H (2) ,(2.
1 These symmetry enlargements in D = 4 and D = 3 result from dualising the 2-form or 1-form potentials to give additional axionic scalars. If one leaves them undualised instead, then the D ≥ 5 discussion of global symmetries extends uniformly to include these dimensions too.
it is equal to the identity for some special set of values of the scalar fields. Furthermore, the number of independent scalar fields in M is equal to the dimension of the coset. The conditions (2.13, 2.14) ensure that the symmetry is O(10 − D, 10 − D + N ), while the occurrence of the special point M = 1l ensures that the orbits are those containing the fiducial point W 0 given in (2.20), leading to the coset structure (2.20).
The easiest way to determine the M matrix from (A.6) is by studying the kinetic terms for the 2-form field strengths, and comparing them with the expression H T M H in (2.21).
From (A.6), we see that, after making the rotation of dilatons described above, we must
have H T (2) M H (2) = α e cα· φ (F (2)α ) 2 + α e − cα· φ (F α (2) ) 2 + I (G I (2) ) 2 , (2.22) since we have c α = a 1α − 1 2 a 1 = − b α + 1 2 a 1 .
It is convenient to express the matrix M, which should be symmetric, as M = V T V, so that the left-hand side of (2.22) can be written as (V H (2) ) T (V H (2) ). We can then read off the matrix V by inspecting the expressions in (A.10) for the 2-form field strengths. It is convenient to order the various 1-form potentials so that the column vector C (1) is given by
C (1) = A (1)α B I (1) A α (1) , H (2) = dC (1) ,(2.23)
where it is understood that the indices α and I increase as one descends downwards through the sets of fields in the column vector. We then find that V is given by
V = e 1 2 cα· φ γ β α e 1 2 cα· φ γ γ α B I (0)γ e 1 2 cα· φ γ γ α (A (0)γβ + 1 2 B I (0)γ B I (0)β ) 0 δ J I B I (0)α 0 0 e − 1 2 cα· φγα β . (2.24)
Here, the indices α and I label rows, while β and J label columns.
It follows from (A.10) that the 3-form F (3) can now be written as
F (3) = dA (2) + 1 2 C (1) Ω dC (1) ,(2.25)
where
Ω = 0 0 −1l n 0 1l N 0 −1l n 0 0 . (2.26)
Here, 1l m denotes the m × m unit matrix, and n = 10 − D. The matrix Ω has eigenvalues In principle, the proof that the D-dimensional Lagrangian (A.6) can be written in the form (2.21) could be completed by directly evaluating 1 4 tr(∂M −1 ∂M), where M = V T V and V is given by (2.24), and showing that it correctly reproduces the terms in the scalar sector of (A.6). However, some more insight into the structure of the theory can be obtained by following a slightly different approach, showing first that V can be written as the exponential of a Lie algebra. Specifically, this algebra is the solvable Lie subalgebra of the group O(10 − D, 10 − D + N ). It is to this topic that we shall turn in the next section, where we shall be able to complete the proof that the bosonic Lagrangians have O(10−D, 10−D +N ) global symmetries.
Scalar cosets in the T n -compactified heterotic theory
The construction of the cosets describing the scalar manifolds arising in the toroidal reductions of eleven-dimensional supergravity have been discussed in detail in [5]. It was shown that the scalar-field coset in D-dimensional maximal supergravity can be parametrised by the Borel subalgebra of E n , where n = 11 − D. In other words, there is a one-to-one correspondence between the scalar fields in the theory and the generators of the Borel subgroup. This shows that the scalar manifold is the coset E n /K(E n ), where K(E n ) is the maximal compact subalgebra of E n , and also shows that the group E n is of the maximally non-compact form E n(+n) [5]. This latter feature is a consequence of the fact that only the maximally non-compact form of a group allows an Iwasawa decomposition into the product of its maximal compact subgroup and its Borel subgroup. Although there are other ways to parameterise the scalar-field cosets, the Borel parameterisation is a particularly convenient one in this context because it is the one that arises naturally in the "step-by-step" dimensional reduction procedure.
In the toroidal reduction of the heterotic theory, one expects [17] the global symmetry group in D dimensions to be O(n, n + 16) × IR, where D = 10 − n > 4. In D = 4 the symmetry actually enlarges to O(6, 22) × SL(2, IR) [18,19,20,21], and in D = 3 it enlarges to O (8,24) [22,23]. In all of these cases, the symmetry group is not maximally non-compact, and hence a slightly different approach is necessary in order to parameterise the relevant scalar cosets O(p, q)/(O(p) × O(q)). This difference is reflected in the fact that the number of scalar fields is smaller than the dimension of the Borel subgroup of the relevant O(p, q) numerator group.
It is nonetheless convenient, in the context of dimensional reduction, to parameterise the scalar cosets in an analogous manner. The necessary generalisation of the Borel parameterisation is provided by the Iwasawa decomposition [7]. This decomposition is rather more subtle in the case of groups that are not maximally non-compact. One again has a unique factorisation of a group element g ∈ G into a product g = k a n, where k is in the maximal compact subgroup K, a is in the maximal non-compact Abelian subgroup A, and n is in the nilpotent subgroup N of G. (In the case where G is maximally non-compact, A is the entire Cartan subgroup and N is the strict Borel subgroup, so the product A N belongs to the standard Borel subgroup.) At the level of the algebra, the Iwasawa decomposition implies that The mathematical understanding of solvable Lie algebras relevant to supergravity stems from Ref. [8]. The application of solvable Lie algebras has been extensively studied recently in [9,10,11,12]. The exponential V = exp(G s ) gives a parameterisation of the coset G/K.
G = K ⊕ G s ,(3.
From this, one can construct the G-invariant scalar coset Lagrangian
e −1 L scalar = 1 4 tr(∂ µ M −1 ∂ µ M) = − 1 2 tr ∂V V −1 ∂V V −1 + (∂V V −1 ) # ,(3.
D = 9 coset
In D = 9 it is easy to see how to write the scalar sector of the Lagrangian (2.2) in a coset formulation. Let us, for this purpose, omit φ, since it decouples from the rest of the scalars, and plays no significant rôle in the discussion. We introduce generator matrices H and E I , associated with the scalars ϕ and B I (0) respectively, and we define the coset representative
V = e 1 2 ϕ H e B I (0) E I . (3.3)
The scalar Lagrangian for ϕ and B I (0) can then be written as This is a subalgebra of O(1, N + 1). To see this, we first need to establish conventions and notation for the generators and roots of the orthogonal algebras.
L = 1 4 tr(∂ µ M −1 ∂ µ M) , where M = V T V ,(3.
The orthogonal algebras O(p, q) divide into two cases, namely the D n series when p+q = 2n, and the B n series when p + q = 2n + 1. The positive roots are given in terms of an orthonormal basis e i as follows:
D n : e i ± e j , i < j ≤ n , B n : e i ± e j , i < j ≤ n , and e i ,(3.6)
where e i · e j = δ ij . It is sometimes convenient to take e i to be the n-component vector e i = (0, 0, . . . , 0, 1, 0, . . . , 0), where the "1" component occurs at the i'th position. However, we shall find later that a different basis is more suitable for our purposes. The
Cartan subalgebra generators, specified in a basis-independent fashion, are h e i , which sat-
isfy [h e i , E e j ±e k ] = (δ ij ± δ ik ) E e j ±e k , etc.
Of these, min(p, q) are non-compact, with the remainder being compact. It is convenient to take the non-compact ones to be h e i with 1 ≤ i ≤min(p, q).
Returning now to our algebra (3.5), we find that the generators H and E I can be expressed in terms of the O(1, N + 1) basis as follows:
H = √ 2 h e 1 , E 2k−1 = E e 1 −e 2k , E 2k = E e 1 +e 2k 1 ≤ k ≤ [ 1 2 + 1 4 N ] , (3.7) E 1+ 1 2 N = E e 1 , if N is even .
It is easily seen that h e 1 and E e 1 ±e i , together with E e 1 in the case of N even, are precisely Thus it follows from the general discussion at the beginning of this section that the scalar Lagrangian for the D = 9 theory is described by the coset 2 (O(1, N + 1)/O(N + 1)) × IR.
(Recall that there is the additional decoupled scalar field φ with an IR shift symmetry.)
D = 8 coset
Turning now to the reduction of the heterotic theory to D = 8, we begin from the general toroidal reduction given in Appendix A, and make the following orthogonal transformation 2 It should be emphasised that the mere fact that one can embed the algebra (3.5) into the Lie algebra of a larger Lie group G does not, of itself, mean that the group G acts effectively on the scalar manifold. Only when (3.5) is the solvable Lie algebra of the group G can one deduce that G has an effective group action on the scalar manifold.
of the dilatons:
φ 1 φ 2 φ 3 = − 3 4 1 8 1 8 3 28 − 1 56 7 8 1 7 6 7 0 φ ϕ 1 ϕ 2 .
(3.8)
In terms of these rotated fields, the Lagrangian for the scalar subsector of the eightdimensional theory becomes
e −1 L 8 = − 1 2 (∂φ) 2 − 1 2 (∂ϕ 1 ) 2 − 1 2 (∂ϕ 2 ) 2 − 1 2 e √ 2(ϕ 1 +ϕ 2 ) (∂A (0)23 + B I (0)2 ∂B I (0)3 ) 2 (3.9) − 1 2 e √ 2(ϕ 1 −ϕ 2 ) (∂A 2 (0)3 ) 2 − 1 2 e √ 2ϕ 2 (∂B I (0)2 ) 2 − 1 2 e √ 2ϕ 1 (∂B I (0)3 − A 2 (0)3 ∂B I (0)2 ) 2 .
For generality, we again allow the range of the index I to be 1 ≤ I ≤ N , rather than just the specific range 1 ≤ I ≤ 16 that arises in the heterotic theory. The eight-dimensional string coupling constant is given by λ 8 = e √ 3/4 φ . Note that the dilaton φ decouples from the rest of the scalars. We shall therefore temporarily suppress φ in the following discussion of the coset structure of the scalar manifold, with the understanding that its constant shift symmetry contributes an additional independent IR factor to the full global symmetry.
We can then show that the Lagrangian (3.9), with φ omitted, can be obtained by parameterising a coset as 10) and substituting this into the first line of (3.2), with M = V T V. The commutation relations for the various generators can then be read off by noting that the 1-form field strengths are given by [5]
V = e 1 2 ϕ· H e A 2 (0)3 E 2 3 e A (0)23 V 23 e B I (0)2 U 2 I e B I (0)3 U 3 I ,(3.G = dV V −1 = 1 2 d ϕ · H + F 2 (1)3 E 2 3 + F (1)23 V 23 + G I (1)2 U 2 I + G I (1)3 U 3 I . (3.11)
Comparing with the explicit expressions given in Appendix A and in (3.9), we find that
[H 1 , V 23 ] = √ 2 V 23 , [H 2 , V 23 ] = √ 2 V 23 , [H 1 , E 2 3 ] = √ 2 E 2 3 , [H 2 , E 2 3 ] = − √ 2 E 2 3 , [H 1 , U 2 I ] = 0 , [H 2 , U 2 I ] = √ 2 U 2 I , [H 1 , U 3 I ] = √ 2 U 3 I , [H 2 , U 3 I ] = 0 , [U 2 I , U 3 J ] = δ IJ V 23 , [E 2 3 , U 2 I ] = − U 3 I ,(3.12)
with all other commutators vanishing.
We shall now show that the algebra (3.12) is precisely the solvable Lie algebra for O(2, N + 2) (or, in other words, that the exponentiation of (3.12) gives a parameterisation we have four positive-root generators in this case, namely E 2 3 , V 23 , U 2 1 and U 3 1 . It is easy to see that the commutation relations in (3.12) lead to the identifications and e 2 ± e 3 . From the algebra (3.12), it is clear that we should take
E 2 3 = E e 1 −e 2 , V 23 = E e 1 +e 2 , U 2 1 = E e 2 , U 2 1 = E e 1 , H 1 = √ 2 h e 1 , H 2 = h e 2 .H 1 = √ 2 h e 1 , H 2 = √ 2 h e 2 E 2 3 = E e 1 −e 2 , V 23 = E e 1 +e 2 . (3.14)
It is then evident that in order for the remaining generators to have the proper weights under H 1 and H 2 , we must have and bearing in mind that the relations (3.15) should preserve the strengths of the generators, we find that, up to arbitrariness in the phases, the solution is is as follows. When N is even, we find
U 2 1 = α 1 E e 2 +e 3 + β 1 E e 2 −e 3 , U 3 1 = α 1 E e 1 +e 3 + β 1 E e 1 −e 3 , U 2 2 = α 2 E e 2 +e 3 + β 2 E e 2 −e 3 , U 3 2 = α 2 E e 1 +e 3 + β 2 E e 1 −e 3 ,(3.α 1 = β 1 = 1/ √ 2, α 2 = −β 2 = i/ √ 2. Thus we have U 2 1 = 1 √ 2 (E e 2 +e 3 + E e 2 −e 3 ) , U 3 1 = 1 √ 2 (E e 1 +e 3 + E e 1 −e 3 ) , U 2 2 = i √ 2 (E e 2 +e 3 − E e 2 −e 3 ) , U 3 2 = i √ 2 (E e 1 +e 3 − E e 1 −e 3 ) ,(3.H 1 = √ 2 h e 1 , H 2 = √ 2 h e 2 , E 2 3 = E e 1 −e 2 , V 23 = E e 1 +e 2 , U 2 2k−1 = 1 √ 2 (E e 2 +e k+2 + E e 2 −e k+2 ) , U 3 2k−1 = 1 √ 2 (E e 1 +e k+2 + E e 1 −e k+2 ) , U 2 2k = i √ 2 (E e 2 +e k+2 − E e 2 −e k+2 ) , U 3 2k = i √ 2 (E e 1 +e k+2 − E e 1 −e k+2 ) ,(3.18)
where k has the range 1
≤ k ≤ [ 1 2 N ]. If N is odd, in addition to the identifications (3.18) for 1 ≤ k ≤ [ 1 2 N ], we have U 2 N = E e 2 , U 3 N = E e 1 . (3.19)
(This embedding of the generators of the solvable Lie algebra in O(2, N + 2) was also encountered in [11].)
It is easy to see that the subset of O(2, N + 2) generators h e 1 , h e 2 , E e 1 ±e 2 , E e 1 ±e i and
E e 2 ±e i , with 3 ≤ i ≤ 2 + [ 1 2 N ], together with E e 1 and E e 2 if N is odd,
Cosets in D ≥ 5
Having seen how the coset construction works in the special cases in D = 9 and D = 8, we are now in a position to consider the general D-dimensional case. However, owing to the fact that higher-degree fields can be dualised to give additional scalars in D = 4 and D = 3, we shall treat these two dimensions separately, having first considered the more straightforward cases D ≥ 5.
From Appendix A, the scalar Lagrangian in D dimensions can be expressed as
e −1 L D = − 1 2 (∂ ϕ) 2 − 1 2 i<j e b ij · ϕ (F i (0)j ) 2 − 1 2 i<j e a ij · ϕ (F (1)ij ) 2 − 1 2 i,I e c i · ϕ (G I (1)i ) 2 , (3.20)
together with a free Lagrangian for the dilaton
φ = − D−2 8 a 123 · φ ,(3.21)
which is decoupled from (3.20). The 3-form field strength F (3) couples only to φ, and the string coupling constant is given by Λ D = exp( (D − 2)/8 φ). We shall, as usual, concentrate only on the sector with φ omitted during our discussion of the global symmetries.
Note that here the notation for the dilaton vectors used here is a little different from the one introduced in Appendix A. Since the dilaton φ has been truncated out, the dilaton vectors in (3.20) have (10 − D) components rather than (11 − D). They are given by
b ij = √ 2(− e i + e j ) , a ij = √ 2( e i + e j ) , c i = √ 2 e i . (3.22)
We have also changed from indices α, β, . . . which range from 2 to (11 − D) to i, j, . . . which range from 1 to (10 − D). Since there will be no confusion, we shall use the same symbols b ij and c i as in Appendix A, and a ij in place of a 1αβ . The 1-form field strengths in (3.20) are given by
F i (1)j = γ k j dA i (0)k , F (1)ij = γ k i γ ℓ j (dA (0)kℓ + B I (0)[k dB I (0)ℓ] ) , (3.23) G a (1)i = γ j i dB I (0)j .
We find that we can write the Lagrangian (3.20) in the form (3.2), where the coset
representative V is parametrised as [5] V = e 1 2 φ· H e A i (0)j E i j e 1 2 A (0)ij V ij e B I (0)i U I i . (3.24)
The commutation relations for the various generators can be determined by comparing the expression for the field strengths
dV V −1 = 1 2 d φ · H + i<j e 1 2 b ij · φ F i (1)j E i j + i<j e 1 2 a ij · φ F (1)ij V ij + i,I e 1 2 c i · φ G I (1)i U I i (3.25)
with the expressions given in (3.23). We find that the non-vanishing commutators are given by
[ H, E i j ] = b ij E i j , [ H, V ij ] = a ij V ij , [ H, U I i ] = c i U I i , [E i j , E k ℓ ] = δ j k E i ℓ − δ ℓ i E k j , [E i j , V kℓ ] = −δ k i V jℓ − δ ℓ i V kj , [E i j , U I k ] = −δ k i U I j , [U I i , U J j ] = δ IJ V ij . (3.26)
The way in which the multiple commutators arising in the evaluation of dV V −1 conspire to produce the precise expressions (3.23) is discussed in detail in [5,24].
We shall now show that the above set of generators and their commutation relations
H i = √ 2 hẽ i , E i j = E −ẽ i +ẽ j , V ij = Eẽ i +ẽ j , (3.28) where we have 1 ≤ i < j ≤ 10 − D.
For the generators U I i associated with the Yang-Mills axions, we find that we can write To do this, we need only look at the form of (2.24) in the neighbourhood of the identity, in which case it is easy to read off the generator matrices associated with each of the scalar fields. 3 By this means, we see that the generators are given as follows:
U i 2k−1 = 1 √ 2 (Eẽ i +e k+m + Eẽ i −e k+m ) , U i 2k = i √ 2 (Eẽ i +e k+m − Eẽ i −e k+m ) , (3.29) where m = 10 − D and 1 ≤ k ≤ [ 1 2 N ]. If N is odd, then in addition we have U i N = Eẽ i .H i = i c i e ii 0 0 0 0 0 0 0 − i c i e ii , E i j = −e ji 0 0 0 0 0 0 0 e ij , V ij = 0 0 e ij − e ji 0 0 0 0 0 0 , U i I = 0 e iI 0 0 0 e Ii 0 0 0 . (3.31)
Here, each e ab is defined to be a matrix of the appropriate dimensions that has zeroes in all its entries except for a 1 in the entry at row a and column b. These satisfy the matrix product rule e ab e cd = δ bc e ad . It is not hard to show that these matrices indeed satisfy the commutation relations (3.26). This completes our demonstration that the entire Lagrangian has a global O(10 − D, 10 − D + N ) symmetry.
D = 4 coset
In four dimensions there is an additional axion, over and above those of the generic Ddimensional discussion, which arises if the 2-form potential A (2) is dualised. If A (2) is left undualised, the scalar Lagrangian will have an O(6, N + 6) × IR global symmetry, as one would expect from the general results of the previous subsection. If A (2) is dualised, the symmetry group enlarges to O(6, 6 + N ) × SL(2, IR). We can see this very easily in the formalism that we have been using in this paper.
To include the effect of dualising A (2) to give an additional axion, we first add the kinetic term for A (2) to the scalar Lagrangian. Together with the kinetic term for φ, this extra term
gives e −1 L extra = − 1 2 (∂φ) 2 − 1 2 e −2φ F 2 (3) ,(3.32)
where φ = − 1 2 a 123 · φ. This is the linear combination of the dilatons which, as discussed in the previous subsection, is decoupled from the rest of the scalar Lagrangian. In the absence of the extra term (3.32), it would be responsible for contributing the extra IR factor in the global symmetry. If we now dualise A (2) , the term (3.32) gives the additional contribution
e −1 L (φ,χ) = − 1 2 (∂φ) 2 − 1 2 e 2φ (∂χ) 2 (3.33)
to the scalar Lagrangian, where
F (3) = e −2φ * dχ , (3.34)
and χ is the new axion dual to A (2) . Since the dilaton/axion system (φ, χ) is decoupled from the rest of the scalar Lagrangian, it follows that the total global symmetry group for the scalar sector is now the direct product O(6, 6 + N ) × SL(2, IR).
This global symmetry extends to the full four-dimensional theory. To see this, we note that since the Bianchi identity for (1) ) to the dualised Lagrangian. In the notation of section 2.3, the full Lagrangian can therefore be written as
F (3) gives dF (3) = 1 2 dB I (1) ∧ dB I (1) − dA (1)α ∧ dA α (1) , the dualisation of A (2) to χ will also give the extra contribution 1 2 χ * (dB I (1) ∧ dB I (1) − 2dA (1)α ∧ dA αe −1 L 4 = R − 1 2 (∂φ) 2 − 1 2 e 2φ (∂χ) 2 + 1 4 tr(∂M −1 ∂M) , − 1 4 e −φ H T (2) M H (2) + 1 2 χ * (H T (2) Ω H (2) ) ,(3.
D = 3 coset
In three dimensions, the discussion of subsection 3.3 shows that if one leaves the higherdegree fields in their undualised form, the global symmetry group will be O(7, 7 + N ) × IR.
If one dualises the vector potentials (A α (1) , A (1)α , B I (1) ) to give an additional 7 + 7 + N axions ( χ α , χ α , λ I ), then the global symmetry group enlarges to O(8, 8 + N ). Note that the entire bosonic sector is now composed only of scalar fields.
To see how the symmetry enlarges, we first perform the dualisations specified above. To do this, we begin by obtaining the Bianchi identities for the 2-form field strengths. From the results in Appendix A, we find the following:
dF α (2) = F α (1)β ∧ F β (2) , dF (2)α = −F β (1)α ∧ F (2)β − F (1)αβ ∧ F β (2) + G I (1)α ∧ G I (2) , (3.36) dG I (2) = G I (1)α ∧ F α (2) .
Adding the appropriate Lagrange multipliers to the original Lagrangian, namely the terms
L LM = χ α (dF α (2) − F α (1)β ∧ F β (2) ) + λ I (dG I (2) − G I (1)α ∧ F i (2) α) −χ α (dF (2)α + F β (1)α ∧ F (2)β + F (1)αβ ∧ F β (2) − G I (1)α ∧ G I (2) ) ,(3.37)
we find, after eliminating the 2-form field strengths by solving algebraically for them, that the dualised 1-form field strengths are given by
F (1)α ≡ −e bα· φ * F α (2) = d χ α + χ β F β (1)α − χ β F (1)αβ + λ I G I (1)α , F α (1) ≡ e a 1α · φ * F (2)α = dχ α − χ β F α (1)β , (3.38) G (1)I ≡ −e c· φ * G I (2) = dλ I − χ α G I (1)α .
(The sign differences in the Lagrange multiplier and definition of the field strength for the χ α terms are purely conventional, and have been introduced in order to simplify the form of the final result.)
Looking at the dilaton vectors for the full set of 1-form field strengths, we find that they are as follows:
F α (1)β : b αβ = − c α + c β , F α (1) : − a 1α = − c α + c 9 , F (1)αβ : a 1αβ = c α + c β , F (1)α : − b α = c α + c 9 , (3.39) G I (1)α : c α , G (1)I : − c = c 9 , where c A = ( c α , c 9 ) , c A · c B = 2 δ AB . (3.40)
We see that by writing the dilaton vector − c for the axions coming from the dualisation of G I (2) as c 9 , it is then natural to extend the index range from c α with 2 ≤ α ≤ 8 to c A with 2 ≤ A ≤ 9. The sets of field strengths on each line in (3.39) then naturally pair together to make an extended set.
Let us define new extended sets of potentialsĀ
(0)AA ,Ā A (0)B andB I (0)A bȳ A (0)αβ = A (0)αβ ,Ā (0)α9 =γ β α χ β + 1 2 B I (0)α λ I , A α (0)β = A α (0)β ,Ā α (0)9 = χ α , (3.41) B I (0)α = B I (0)α ,B I (0)9 = λ I .
We also can define an extended set of matricesγ A B , and their inverses γ
A B ≡ δ A B +Ā A (0)B . From the definitions (3.41), it follows that γ α β = γ α β ,γ α 9 = γ α β χ β ,γ 9 9 = 1 . (3.42) (As usual,γ A B is zero if A > B.)
From the above, we then find that the extended set of 1-form field strengthsF (1)AA ,F A (1)B andḠ I (1)A can be written as
F (1)AB =γ C Aγ D B (dĀ (0)CD −B I (0)[C dB I (0)D] ) , F A (1)B =γ C B dĀ A (0)C ,Ḡ I (1)A =γ B A dB I (0)B . (3.43)
In terms of these extended sets of fields, the fully-dualised three-dimensional Lagrangian is given by
e −1 L 3 = − 1 2 (∂ φ) 2 − 1 2 A<B e b AB · φ (F A (0)B ) 2 − 1 2 A<B e a AB · φ (F (1)AB ) 2 − 1 2 A,I e c A · φ (Ḡ I (1)A ) 2 , (3.44) where b AB = − c A + c B , a AB = c A + c B and c A · c B = 2δ AB .
Thus we see that the D = 3
Lagrangian has the identical form as (3.20), except that the index range of A is extended to 2 ≤ A ≤ 9, rather than the 1 ≤ i ≤ 10 − D range occurring in (3.20). It follows from the discussion in subsection 3.3 that the Lagrangian (3.44) is therefore described by the coset
O(8, 8 + N )/(O(8) × O(8 + N )).
Time-like reductions of the heterotic string theory
Dimensional reductions on Lorentzian tori have been discussed in [25,26,27,28,29,30]. It was shown that the global symmetry groups remain unchanged from those of the usual Euclidean-torus reductions, but the coset structure is changed by virtue of the fact that the previous compact denominator groups are replaced by certain non-compact versions of these groups.
In this section, we give explicit derivations of the coset structure in all dimensions 3 ≤ D ≤ 9, using the techniques that we have presented earlier in this paper. As we showed in section 2.2, the denominator groups K for cosets G/K, where G = O(p, q), are determined by the choice of a fiducial matrix W 0 lying on a particular orbit of the matrices W satisfying (2.13, 2.14). In the discussion in section 2.3, we saw that the matrices M that parameterise the scalar manifolds in dimensionally reduced Lagrangians played the rôle of the matrices W , and that the fiducial matrix W 0 could be read off simply by setting all the scalar fields to zero. In the usual reductions on Euclidean tori, the fiducial matrix is always just the identity, and hence it follows that the denominator group is the compact
form O(p) × O(q).
For reductions on Lorentzian tori, a general discussion can easily be given for the cases D ≥ 5. As in section 2.3, the M matrix can be read off from the kinetic terms of the 2-form field strengths. The fiducial matrix where all the scalars vanish is diagonal, with unit-magnitude components whose signs are determined by the signs of the kinetic terms for the corresponding 2-form field strengths. Without loss of generality, 4 let us consider the case where the time-like reduction step is the one from D = 10 to D = 9, corresponding, in our notation, to the internal index α taking the value 2. It follows that all lower-dimensional fields that have a single α = 2 internal index suffer a sign-reversal for their kinetic terms [29].
This means that the two 2-form field strengths F α (2) and F (2)α with α = 2 will acquire signreversed kinetic terms, while all the other 2-forms will retain their standard signs. This pair of 2-forms can be seen to be associated with one symmetric pair of off-diagonal −1 entries in the metric Ω given in (2.26), and thus they are associated with one eigenvalue +1 and
O(10 − D, 10 − D + N ) O(1, 9 − D) × O(1, 9 − D + N ) × IR . (4.1)
There is an alternative way to determine the fiducial matrix W 0 , which will prove to be useful later when we look at the time-like reduction down to D = 3. From the form of the Kaluza-Klein metric reduction ansatz (A.2), we see that the effect of making a time-like reduction in the step from D = 10 to D = 9 can be achieved by performing the complex field redefinition
φ −→ φ + i π 2 ( c 2 − c) . (4.2)
(Note that the ten-dimensional dilaton φ 1 is unchanged under this redefinition.) From (2.24), we see that upon setting all the axions to zero, the matrix M = V T V is given by
M = diag (e cα· φ δ β α , δ J I , e − cα· φ δ α β ) ,(4.3)
for the usual case of reduction on a Euclidean torus. The transformation (4.2) then implies that there is a sign reversal on the two components corresponding to α = 2. If we now set the dilatons to zero, we get precisely the same fiducial matrix as described above.
In Euclidean-torus reduction, will now instead be O(1, 1). The reason for this is that the kinetic term for the axion χ in (3.33) will be reversed when the dualisation from (3.32) is performed in the four-dimensional Euclidean-signatured space. Thus the coset for the scalar manifold in D = 4 will be
O(6, N + 6) O(1, 5) × O(1, N + 5) × SL(2, IR) O(1, 1) . (4.4)
In D = 3, the global symmetry group is O(8, N + 8). In this case, upon setting all the axions to zero, the (N + 16) × (N + 16) matrix M is given by
M = diag (e c A · φ δ B A , δ J I , e − c A · φ δ A B ) , (4.5)
where 2 ≤ A ≤ 9 (see section 3.5). The field redefinition (4.2) implies that there will now be four terms in (4.5) that undergo sign reversals, namely those corresponding to A = 2 and A = 9. Thus, setting the dilatons to zero, we obtain a fiducial matrix of the form (2.16) with m = n = 2. Consequently, the coset space describing the theory in D = 3 obtained by timelike reduction has the form
O(8, N + 8) O(2, 6) × O(2, N + 6)
.
(4.6)
The coset structures for Lorentzian torus reductions that we have derived in this section are all in agreement with those given in [28].
K3 compactifications of M-theory
Now let us consider, by contrast, the K3 reduction of D = 11 supergravity. We aim to show in some detail the relations between this theory and the T 3 reduction of the E 8 ×E 8 heterotic theory as discussed above, which are conjectured to be equivalent under duality [1,2]. The reduction of the heterotic theory we have a 2-form potential instead. This indicates that one needs to perform a dualisation of one of the two seven-dimensional theories in order to make contact with the other. In particular, this indicates that the relation between the two involves an interchange between strong and weak coupling regimes.
To make things more precise, let us begin by looking in detail at the T 3 reduction of the heterotic theory. In order to make contact with the K3 reduction of D = 11 supergravity, we will make a dualisation of the 2-form potential A (2) arising in the T 3 reduction of the heterotic theory. To do this, we need to know the Bianchi identity for the field strength F (3) .
From the results given in Appendix A, we find this to be dF 2) ) to the undualised Lagrangian. Treating F (3) now as an auxiliary field, we can solve its algebraic equation of motion, giving e a 1 · φ * F (3) = dA (3) ≡ F (4) .
(3) +F (2)α ∧F α (2) + 1 2 G I (2) ∧G I (2) = 0. To dualise A (2) ,
we introduce a 3-form A (3) as a Lagrange multiplier, adding the term
A (3) ∧(dF (3) +F (2)α ∧F α (2) + 1 2 G I (2) ∧G I(
Substituting this back into the Lagrangian, we obtain the dualised version
e −1 L 7 = R − 1 2 (∂ φ) 2 − 1 48 e − a 1 · φ (F (4) ) 2 − 1 4 α e a 1α · φ (F (2)α ) 2 − 1 2 α<β e a 1αβ · φ (F (1)αβ ) 2 − 1 2 I e c· φ (G I (2) ) 2 − 1 2 α,I e cα· φ (G I (1)α ) 2 − 1 4 α e bα· φ (F α (2) ) 2 − 1 2 α<β e b αβ · φ (F α (1)β ) 2 (5.1) + * A (3) ∧ (dA ′ (1)α ∧ d α (1) + 1 2 dB I ′ (1) ∧ dB I ′ (1) ) ,
where F (4) = dA (3) , and the other field strengths are given in (A.10).
To make comparison with the K3 reduction of D = 11 supergravity, we first need to discuss the nature of the K3 manifold. (A detailed account of its properties may be found in [31].) The required K3 metric is Ricci flat and Kähler. Although an existence proof for
Ricci-flat metrics on K3 has been given long ago [32], the explicit form of such metrics is still unknown, owing to the complexity of the Einstein equation. It is, however, possible to give an approximate construction of the Ricci-flat metrics (see, for example, [33,13]). A detailed discussion of this "physical" picture was given in [13]. One can construct K3 by beginning with the 4-torus T 4 , defined by identifying coordinates y i ∼ y i + 2π in IR 4 . Next, we make the identification y i ∼ −y i . This identification has 16 fixed points, located at What is needed is a smooth 4-space with curvature localised to a small region, and which then opens out into an asymptotic region that approaches flat Euclidean 4-space but with an antipodal identification so that its boundary is again RP 3 . Such a space is known; it is the Eguchi-Hanson instanton [34], which indeed approaches Euclidean space factored by Z 2 at infinity [35]. By taking the "size" of the instanton to be sufficiently small, one achieves -torus that will be present in the K3 itself will be the subset that survives the antipodal identification y i ∼ −y i . Thus the six harmonic 2-forms dy i ∧ dy j survive, whilst the four harmonic 1-forms dy i are projected out. Thus we may define three self-dual 2-forms J α + and three anti-self-dual 2-forms J α − on T 4 (with the triplet index α running over the values 2, 3, 4 in order to match with the notation arising in the T 3 compactification of the heterotic string):
y i = π n i ,J 2 ± = dy 1 ∧ dy 4 ± dy 2 ∧ dy 3 , J 3 ± = dy 2 ∧ dy 4 ± dy 3 ∧ dy 1 , (5.2) J 4 ± = dy 3 ∧ dy 4 ± dy 1 ∧ dy 2 ,
The second type of harmonic 2-forms are those associated with the Eguchi-Hanson
instantons. There is one of these for each of the sixteen instantons, corresponding to the fact that the Eguchi-Hanson solution has one normalisable harmonic 2-form. We shall represent these 2-forms by the symbols ω I (2) . Each of these 2-forms is strongly localised within a small region of the Eguchi-Hanson space itself. Concretely, the Eguchi-Hanson metric is given by [34]
ds 2 = 1 − a 4 r 4 −1 dr 2 + 1 4 r 2 1 − a 4 r 4 (dψ + cos θ dϕ) 2 + 1 4 r 2 (dθ 2 + sin 2 θ dϕ 2 ) . (5.3)
The radial coordinate has the range a ≤ r < ∞, with the bolt occurring at r = a. The coordinates θ and ϕ are angles on S 2 . The level surfaces r = constant have the topology of RP 3 , since the fibre coordinate ψ has period 2π rather than the 4π period that would occur on S 3 [35]. In the natural orthonormal frame e 0 = (1 − a 4 /r 4 ) −1/2 dt, e 1 = 1 2 r dθ, e 2 = 1 2 r sin θ dϕ, e 3 = 1 2 r(1−a 4 /r 4 ) 1/2 (dψ +cos θ dϕ), it is easy to see that the anti-self-dual 2-form
ω (2) = 1 r 4 (e 0 ∧ e 3 − e 1 ∧ e 2 ) (5.4)
is closed. This is the normalisable anti-self-dual harmonic 2-form on Eguchi-Hanson.
In total, we then have 22 harmonic 2-forms; six from T 4 plus 16 from the corks. These divide into 3 self-dual harmonic 2-forms from T 4 , plus 19 = 3 + 16 anti-self-dual harmonic 2-forms, coming from T 4 and from the 16 Eguchi-Hanson metrics. As we shall see, the ways in which the 2-forms from T 4 and from the Eguchi-Hansons contribute in the dimensional reduction procedure will be slightly different.
Upon performing the K3 reduction, the D = 11 fields give rise to the following D = 7 fields:
g M N −→ g µν , A i (0)j , A I (0)α , φ , A (3) −→ A (3) , A (1)ij , A (1)I . (5.5)
Here, as usual, the i, j, . . . indices range over the four internal coordinates y i . There are four dilatons φ, arising from the fact that in the construction of K3 that we are using here, there are the usual four circles making up the 4-torus. There are also six axions A i (0)j (i < j) that parameterise the angular deformations of the 4-torus. There are also 48 = 16 × 3 further scalars A I (0)α , corresponding to the remaining parameters that make up the total of 58 = 10 + 48 moduli of K3 [36,37]. These can be understood in the picture of K3 that we are using as follows. There are two parameters that characterise the orientation of each of the Eguchi-Hanson instantons, and a further parameter characterising its scale size. This gives 16 × (2 + 1) = 48 parameters in total that are associated with the Eguchi-Hanson corks. The 22 vector fields that we mentioned previously split as six vectors A (1)ij coming from the T 4 reduction of A 3 , plus 16 vectors A (1)I coming from the harmonic expansion involving the 16 harmonic 2-forms ω I (2) localised in the 16 Eguchi-Hanson instantons. The dilaton couplings for each field can be obtained straightforwardly, by examining the ansatz for the reduction of the eleven-dimensional metric on T 4 :
ds 2 11 = e 1 3 g· φ ds 2 7 + 4 i=1 e 2 γ i · φ (dz i + A i (0)j dz j ) 2 . (5.6)
The constant vectors g and γ i can be found in [5,39], and are given by
γ 1 = (− 2 3 , 0, 0, 0) , γ 2 = ( 1 12 , − √ 7 4 , 0, 0) , γ 3 = ( 1 12 , 1 4 √ 7 , − 3 7 , 0) , γ 4 = ( 1 12 , 1 4 √ 7 , 1 2 √ 21 , − 5 12 ) , (5.7) g = − 6 5 i γ i = ( 1 2 , 3 2 √ 7 , 3 7 , 3 5 ) .
Note that the radius R i of the i'th circle on T 4 , and the T 4 volume V 4 = i R i , are given by
R i = e γ i · φ , V 4 = e − 5 6 g· φ . (5.8)
Thus the combination of dilatons ϕ = 5/8 g · φ is the breathing mode of T 4 , and hence also of K3. It is also sometimes useful to present another form of the metric reduction ansatz that is applicable to computations that do not involve the "internal" structure of the compactifying 4-manifold, but only depend on the breathing mode. This is given by
ds 2 11 = V −2/5 4 ds 2 7 + V 1/2 4 ds 2 4 . (5.9)
Associating dilaton vectors with the various seven-dimensional fields as follows,
A i (0)j A I (0)α A (3) A (1)ij A (1)I b ij b α a a ij d ,(5.10)
we find that they can be expressed in terms of g and γ i as
b ij = 2 γ i − 2 γ j , b 2 = γ 1 + γ 2 − γ 3 − γ 4 , b 3 = γ 2 + γ 3 − γ 1 − γ 4 , b 4 = γ 3 + γ 1 − γ 2 − γ 4 , a = − g , a ij = −2 γ i − 2 γ j − 1 3 g , d = 1 2 g . (5.11)
We shall now show in detail how these dilaton couplings arise in the K3 reduction of M-theory. The dilaton vectors a, a ij and b ij , corresponding to the fields A (3) , A (1)ij and A i (0)j can be understood straightforwardly, since they are a subset of those one would obtain by dimensionally reducing M-theory on T 4 . As we shall see, they are in fact nothing but an SL(4, IR) truncation of maximal supergravity in D = 7, which has an SL(5, IR) global symmetry. To see this, consider the ansatz for the reduction of the 3-form potential
A (3) (x, y) = A (3) (x) + 1 2 A (1)ij (x) ∧ dy i ∧ dy j + A (1)I (x) ∧ ω I (2) . (5.12)
For now, it is only the first two terms here that concern us. From the metric ansatz (5.6),
we can see that the determinant of the vielbein reduces according to e → e e 1 3 g· φ , and thus we have that
− 1 48 e F 2 (4) −→ − 1 48 e e − g· φ F 2 (4) − 1 4 i<j e −(2 γ i +2 γ j + 1 3 g)· φ (F (1)ij ) 2 + · · · (5.13)
where · · · represents the F (2)I terms that we shall discuss presently, and in obtaining the exponential factors we have used the appropriate inverse metric components in D = 7
or D = 4, as given in (5.6). The exponents can indeed be seen to be a · φ and a ij · φ in (5.11). The dilaton couplings for the axions A i (0)j coming from the torus reduction of the metric follow from the standard Kaluza-Klein formulae as given, for example, in [39].
In fact we can also obtain the result for these axions by a simple linearised calculation, and it is useful to present this here because a similar argument will be used below in discussing the more difficult case of the other 48 axions coming from the K3 metric moduli.
If a metricḡ ij is subjected to a transverse traceless perturbation h ij , i.e. g ij =ḡ ij + h ij whereḡ ij h ij = 0 and∇ i h ij = 0, then the perturbed Ricci tensor will be of the form
R ij =R ij + 1 2 ∆ L h ij , where the Lichnerowicz operator ∆ L is defined by ∆ L h ij = −¯h ij − 2R ikjℓ h kℓ + 2R (i k h j)k .
The fluctuation h ij will therefore give rise to a contribution of the form 1 2 h ij h ij in the Einstein-Hilbert Lagrangian, where h ij =ḡ ikḡjℓ h kℓ . In the present context, we see from (5.6) that the metric fluctuation corresponding to the axion A i (0)j is given by h ij = e 2 γ i · φ A i (0)j when i < j (together with h ji = h ij ). Thus we will get a contribution of the form 1 2
e −2( γ i + γ j )· φ e 2 γ i · φ A i (0)j (e 2 γ i · φ A i (0)j ) ∼ − 1 2 e 2( γ i − γ j )· φ (∂A i (0)j ) 2 .
This result shows that indeed the dilaton vector describing the dilaton coupling is the one given by b ij in (5.11).
It is a little more involved to understand the dilaton vectors d and b α describing the couplings of the 16 vector potentials A (1)I and the 48 axions A I (0)α , since these arise from the sixteen 2-form harmonics ω I (2) that are intrinsic to K3. Let us begin by considering the vector fields A (1)I , which are the easier of the two sets to analyse. From the last term in (5.12), we see that these give the dA (1)I ∧ ω I (2) contributions to the elevendimensional 4-form F (4) . In the same spirit as above, we can calculate the dimensional reductions of these terms by making the necessary contractions of indices using the appropriate metric components as given by the Kaluza-Klein ansatz. For these fields, since their internal components involve ω I (2) , we should use the metric given by (5.9). Thus we find − 1 48 e F 2 (4) → − 1 4 e V The determination of the dilaton vectors for the A I (0)α is more complicated. One approach is to note that the subgroup GL(4, IR) ∼ O(3, 3) × IR of the O(3, 19) × IR global symmetry group of the K3 reduction corresponds precisely to the unbroken general coordinate symmetry on the T 4 . The antipodal identification in the T 4 described above preserves this global symmetry group. All of the fields should therefore form representations under this GL(4, IR). This can be seen in particular in the couplings of the dilatonic scalars.
Specifically, the dilaton vectors should form weight vectors under GL(4, IR) = IR×SL(4, IR).
Here the IR factor is generated by the breathing mode. Indeed, the dilaton couplings of F (4) and the F I In the approximate description of K3 that we are using here, we may note that each of the 16 fixed points under antipodal identification should be "patched" with an Eguchi-Hanson instanton. The 16 instantons are equivalent, and so we can discuss just a single one of them as a representative. Each instanton contributes three metric zero modes, described by three axions. The insertion of the Eguchi-Hanson instantons preserves the SL(4, IR) symmetry of the original T 4 , since, as we have observed, its asymptotic limit is the same as the antipodally-identified T 4 . Following the above discussion, the three dilaton vectors and their negatives form a six-dimensional representation of SL(4, IR). (The inclusion of the negatives of the dilaton vectors is clearly necessary since there exists no triplet representation of SL(4, IR).) We now find that the set ± b α form a six-dimensional representation of SL(4, IR), given the chosen basis b ij for the positive roots. This representation is not unique, however; another example is the set a ij discussed above. However, the set ± b α given in (5.11) forms the unique solution in which three vectors together with their negatives comprise a six-dimensional representation. Thus we see that the forms of these dilaton vectors are dictated by the global symmetry SL(4, IR).
An alternative way to understand these dilaton vectors is to note that the K3 metric has 3 self-dual and 3 + 16 anti-self dual harmonic 2-forms. In the approximate K3 construction, we see that for each Eguchi-Hanson instanton there is one localised anti-self-dual harmonic 2-form ω (2) , and three covariantly-constant self-dual 2-forms J α + . As shown in [36], one can use these to build three zero-mode deformations of the metric (i.e. Lichnerowicz zeromodes) of the form h ij = J a +ik ω k j . This gives a total of 3 × 16 = 48 metric zero-modes, which, together with the 10 coming from the 4-torus (4 dilatons φ plus 6 axions A i (0)j ), give the 58 metric zero-modes of K3. In the bulk T 4 /Z 2 part of K3, we have a total of six 1-form field strengths F i (1)j for six axions, forming three pairs 5
F 2 (1)± = F 1 (1)4 ± F 2 (1)3 , F 3 (1)± = F 2 (1)4 ± F 3 (1)1 = F 2 (1)4 ∓ F 1 (1)3 e − b 13 · φ , F 4 (1)± = F 3 (1)4 ± F 1 (1)2 .
On the other hand, for each instanton there are just three axions. This difference is associated with the fact that, whereas in the T 4 /Z 2 bulk there are 3 self-dual and 3 anti-self-dual constant harmonics associated with the above three pairs of axions, there are in each instanton just three selfdual covariantly constant 2-forms. The nature of the dilatonic couplings for the three axions associated with each instanton can be revealed by first studying the detailed structure of the dilaton couplings for the bulk T 4 /Z 2 pairs of field strengths F α (1)± . Let us consider the Lagrangian for the F 2 (1)± pair, which is given by
e −1 L = − 1 2 (∂ φ) 2 − 1 4 e b 14 · φ (F 1 (1)4 ) 2 − 1 4 e b 23 · φ (F 2 (1)3 ) 2 = − 1 2 (∂ φ) 2 − 1 4 e 1 2 ( b 14 + b 23 )· φ (F 2 (1)+ , F 2 (1)− ) c s s c F 2 (1)+ F 2 (1)− ,(5.14)
where c = cosh θ and s = sinh θ, with θ = 1 2 ( b 14 − b 23 ) · φ. Thus for F 2 (1)+ , the dilaton couplings are naturally described by 1 2
( b 14 + b 23 )
. This is because we can set consistently set both F 2 (1)− = 0 and θ = 0, and then cosh θ is replaced by unity in the dilaton coupling. Indeed, the natural augmentation of (5.14) to include additional axions ψ I gives gauge, the Lagrangian is expressed naturally in terms of F i (1)j with i < j and we have F i
e −1 L = − 1 2 (∂ φ) 2 − 1 4 e 1 2 ( b 14 + b 23 )· φ (F 2 (1)+ , F 2 (1)− ) c s s c F 2 (1)+ F 2(1)(1)j = −F j (1)i e b ji · φ .
In other words, the original pair of axions not only have an overall dilaton factor e
1 2 ( b 14 + b 23 )· φ
but also the c s s c matrix coupling, while all further axions ψ i are "unpaired," and have only the overall e 1 2 ( b 14 + b 23 )· φ factor. Thus we can argue that one of the three axions associated with a given Eguchi-Hanson instanton can naturally be grouped with the F 2 (1)± , with a Lagrangian contribution of the same form as those of the ψ I in (5.15). This is because the three axions can be approximately viewed as an internal self-dual truncation (analogous to setting F 2 (1)− to zero), since there are only three self-dual constant harmonic 2-forms in the Eguchi-hanson metric. This implies that the dilaton coupling for one of the three axions is given by 1 2
( b 14 + b 23 ) = γ 1 + γ 2 − γ 3 − γ 4 .
The other two sets of N = 16 axions are then associated with the F 3
(1)± and F 4 (1)± pairs, with dilaton couplings given by − γ 1 + γ 2 + γ 3 − γ 4 and γ 1 − γ 2 + γ 3 − γ 4 respectively.
Comparing with the T 3 reduction of the heterotic string that we obtained previously, we find that the correspondence between the fields in the two descriptions can be summarised in the following Table: M-theory on K3
Heterotic string on T 3 In fact the fields A (1)ij and A i (0)j in the K3 compactification can be expressed in terms of the associated fields of the heterotic theory in a more covariant fashion by making use of the self-dual and anti-self-dual 2-forms J α ± defined in (5.2):
D = 11 D = 7 Duality D = 7 D = 10 A (3) , a ←→ A (2) , a 1 A (2) A (1)14 , a 14 ←→ A (1)2 , a 12 A (1)24 , a 24 ←→ A (1)3 , a 13 A (1)34 , a 34 ←→ A (1)4 , a 14 A (3) A I (1) , d ←→ B I (1) , c B I (1) A (1)12 , a 12 ←→ A 4 (1) , b 4 A (1)13 , a 13 ←→ A 3 (1) , b 3 G µν A (1)23 , a 23 ←→ A 2 (1) , b 2 A I (0)α , b α ←→ B I (0)α , c α B I (1) A 1 (0)4 , b 14 ←→ A (0)34 , a 134 A 2 (0)4 , b 24 ←→ A (0)24 , a 124 A (2) G µν A 3 (0)4 , b 34 ←→ A (0)23 , a 123 A 2 (0)3 , b 23 ←→ A 3 (0)4 , b 34 A 1 (0)2 , b 12 ←→ A 2 (0)3 , b 23 G µν A 1 (0)3 , b 13 ←→ A 2 (0)4 , b 24A (1)ij = 1 2 (J α +ij + J α −ij ) A (1)α + 1 2 (J α +ij − J α −ij ) A α (1) , A i (0)j = 1 4 (J α +ij + J α −ij ) ǫ αβγ A (0)βγ + 1 4 (J α +ij − J α −ij ) ǫ αβγ A β (0)γ . (5.16)
The four dilatons in the heterotic and M-theory reductions are related by an orthonormal The matrix M also maps the dilaton vector a = − g of the 4-form field strength F (4) in the M-theory reduction to − a 1 of the 3-form field strength F (3) in the heterotic string reduction. The minus sign is consistent with the fact that a dualisation of the 4-form field strength is necessary in order to make the identification of the two theories. Since the effective string coupling λ 7 of the heterotic string on T 3 is given by λ 7 = e − 5 8 a 1 · φ , it follows from (5.8) that the seven-dimensional string coupling is [2]
transformation, φ H = M φ M , where M = λ 7 = V 3/4 4 . (5.18)
The complete set of mappings for all the dilaton vectors can be seen from Table 1
φ M + M 2 φ M ∼ (φ L ) 2 ,(5.φ M + M 2 φ M ∼ (∂φ L ) 2 ,(5.φ M ∼ 1 + M 2 −1 −1 −1 (∂φ L ) 2 ∼ −1 (∂φ L ) 2 + · · · . (5.22)
Thus, rather than having the situation sketched in (5.21) where the effect of "integrating out" the massive modes is to modify the effective low-energy action by higher-order derivative couplings that are damped by inverse powers of the Kaluza-Klein mass scale, the effect now in the small-M regime is to obtain non-derivative modifications with no damping.
It is therefore important in the context of M-theory/heterotic duality to try to establish whether or not the truncation to the massless sector is a consistent one. It is often asserted that the modulus space for Ricci-flat metrics on K3 is the coset space (19)), and that this endows the seven-dimensional theory following from the K3 reduction of M-theory with a scalar manifold having this same coset structure. Indeed, it appears to be the case that if one substitutes the Kaluza-Klein ansatz for the zero-mode sector into the D = 11 Lagrangian, and then integrates over K3, then the resulting seven-dimensional Lagrangian will have a scalar sector described by this (19)) coset. However, as we have discussed above, it is a much more exacting and stringent question as to whether instead the substitution of the zero-mode ansatz into the eleven-dimensional field equations will be consistent with these fields' own equations of motion. Furthermore, although the effects of integrating out massive fields in the low-energy approximation (5.21) would not upset the coset structure of the Lagrangian for the lower-dimensional scalar fields, it is not so clear that this sigma-model structure would survive unscathed in the small-M regime described by (5.22). As far as we are aware, this is a question that has not been addressed in the literature, and there appears to be no a priori argument that guarantees the consistency of the truncation. Although this could be argued to be a negligible problem in the context of low-energy phenomenology, it would seem to be a more significant one in the context of M-theory/heterotic duality, and it is deserving of further study. Indeed, one might argue that the consistency issue could provide a non-trivial test of the validity of the conjectured duality between M-theory and the heterotic string: Since the truncation of the T 3 compactification of the heterotic string to its massless sector is consistent, then the consistency of the truncation to the massless sector of the K3 compactification of M-theory would be a necessary consequence of the duality between the two theories.
O(3, 19)/(O(3) × OO(3, 19)/(O(3) × O
Charge lattice relations
In this section, we shall consider in detail the relation between the lattices of electric and magnetic charges that are allowed by the Dirac quantisation conditions in the K3 reduction of M-theory and the T 3 reduction of the heterotic string. It has been shown in Refs [14,15,16] that the minimum charges of M-branes can be fixed by invoking duality relations between M-theory, type IIA and type IIB string theories, together with the existence of certain "scale-setting" p-brane species [16]. In the case of M-theory, the charge units can also be fixed by consideration of the topological L F F A term in the Lagrangian [43]. We shall now
show that the M-brane charges can also be fixed by consideration of the conjectured duality relation between M-theory compactified on K3 and heterotic string theory compactified on
e −1 L 10 = κ −2 H R − 1 2 (∂φ 1 ) 2 − 1 2 e φ 1 (F (3) ) 2 − 1 2 e 1 2 φ 1 tr (G (2) ) 2 ,(6.1)
where the E 8 × E 8 gauge fields G (2) are written in terms of gauge potentials as Upon making a T 3 reduction, the 3-form gives rise to F (3) and F (2)α , together with 1-form field strengths for axions, which we shall not consider here. In addition, there are three Kaluza-Klein 2-forms F α (2) . The charges of these fields are given by
G (2) = dB (1) + 1 √ α ′ [B (1) , B (1) ] .Q e(3) = (2π) β κ 3/2 H L −3 2 , Q m(2)α = (2π) 1−β κ 1/2 H L −1 2 , Q KK m(2)α = L 2 . (6.7)
In each case, the corresponding magnetic or electric dual charges are related by the D = 7
Dirac quantisation conditions. Note that the magnetic Kaluza-Klein charge is associated with a NUT charge, and hence its charge unit is determined by topological considerations.
The duality between M-theory reduced on K3 and the heterotic theory reduced on T 3
implies that the charges carried by the various fields in the two pictures should be equated, in accordance with the equivalences of the corresponding fields as given in Table 1. This gives rise to three independent equations:
(2π) β κ 3/2 H L −3 2 = (2π) 1−α κ 2/3 11 , (2π) 1−β κ 1/2 H L −1 2 = L 2 = (2π) 1−α κ 2/3 11 L −2 1 . (6.8)
Together with the relation (6.3) between the gravitational constants, we find that
κ 2 11 = 1 2π L 6 1 L 3 2 , κ 2 H = 1 2π L 2 1 L 6 2 , α = 2 3 , (2π) 4β = (2π) 3 L 2 1 L −2 2 . (6.9)
From these, we see that the M-brane charges in D = 11 are completely determined, and must be given by Q e(4) = n (2πκ 2 11 ) 2/3 , Q m(4) = m (2πκ 2 11 ) 1/3 , (6.10)
where n and m are integers. This conclusion is the same as that obtained in [16], where T-duality between the type IIA and type IIB theories was invoked.
We now turn to a consideration of the Yang-Mills sector. The E 8 × E 8 gauge potentials may be expanded in terms of the Cartan-subalgebra generators H I and the non-zero root generators E a as B (1) = B I (1) H I + B a (1) E a . In particular we see that with respect to the Cartan subalgebra gauge potentials, the coupling of the non-zero root potentials takes the
form dB a (1) E a + (α ′ ) −1/2 B I (1) ∧ B a (1) [H I , E a ] = (dB a (1) + (α ′ ) −1/2 α Ia B I (1) ∧ B a (1) ) E a .
Thus the fields B a (1) interact with the U (1) 16 Cartan subalgebra potentials via minimal coupling, with the 16 electric charges given in terms of the components of the root vectors α a . The basic units of electric Yang-Mills charge in the ten-dimensional heterotic string are therefore given in terms of the 16 simple-root vectors α i of E 8 × E 8 , since all the other Yang-Mills charges are expressible in terms of linear combinations of these with integer coefficients. To make this more precise, we choose to define the magnetic charges by the integrals
Q I m ≡ G I (2) . (6.11)
It follows from the form of the covariant derivative DB i
(1) = dB i (1) + (α ′ ) −1/2 α Ii B I (1) ∧ B i
(1) that the Dirac quantisation conditions will require the magnetic charges to lie on the reciprocal lattice
Q m = m i √ α ′ µ i ,(6.12)
where m i are integers and µ i are the fundamental weight vectors, defined by α i · µ j = δ j i . For comparison, we now consider the charge lattice for the Yang-Mills sector coming from the reduction of M-theory on K3. In particular, we consider the charges under the 16 abelian 2-form fields arising from the 16 anti-self-dual harmonic 2-forms on K3 which, in the approximate discussion of section 5, were associated with the 16 Eguchi-Hanson instantons.
These are the fields that are conjectured to be related by duality to the Cartan subalgebra of the E 8 × E 8 symmetry of the heterotic string.
We may derive the charge lattice for these 16 fields in the M-theory picture by using an abstract description of the cohomology of K3. Specifically, one may introduce a set of sixteen 2-forms σ i (2) that are "dual" to the sixteen anti-self-dual harmonicsω (2)i , in the sense that that
K3 σ i (2) ∧ω (2)j = δ i j . (6.13)
The 2-forms σ i (2) are normalised so that their integrals over the relevant 16 2-cycles Σ i in K3 are given by
L 2 1 Σ i σ j (2) = δ j i .(6.14)
(We are taking the harmonic 2-formsω (2)i to have the dimensions of (Length) 2 , and the dual 2-forms σ i (2) to have dimensions (Length) −2 . We use the dimensionful parameter L 1 = V 1/4 4 that we introduced earlier, where V 4 is the volume of K3, in order to balance the dimensions.)
Then, one has the result that [31] K3ω i ∧ω j = L 4 1 M ij , (6.15) where M ij is the Cartan matrix of E 8 × E 8 . This is given in terms of the simple root vectors α i by M ij = α i · α j . From the above equations, we easily see that if we expandω (2)i in terms of σ i (2) , we haveω (2)i = M ji σ j (2) , and hence that Σ iω
(2)j = L 2 1 M ij . (6.16)
The 2-formsω (2)i do not have the appropriate normalisation for giving canonical diagonal kinetic terms for the associated spacetime 2-form field strengths in the Kaluza-Klein reduction of F (4) . To obtain the proper kinetic terms, we should now define new linear
combinations ω I (2) by ω I (2) = µ Iiω (2)i (6.17)
where µ Ii denotes the set of 16 components of the fundamental weight vectors µ i that we introduced previously. The 2-forms ω I (2) therefore satisfy the relations
K3 ω I (2) ∧ ω J (2) = L 4 1 δ IJ , Σ i ω I (2) = L 2 1 α I i . (6.18)
Performing the Kaluza-Klein reduction
F (4) = F I (2) ω I (2) ,(6.19)
we see from (6.18) that the normalisation of the ω I (2) implies that the kinetic terms for the spacetime 2-forms F I (2) will be canonical. The magnetic charge Q m(4) in D = 11 for a 5-brane wrapped around the 2-cycle Σ i in K3 will therefore be given by
Q m(4) = F (4) = Q I m Σ i ω I (2) = L 2 1 Q I m α I i ,(6.20)
where Q I m = F I (2) is the magnetic charge of the resulting 3-brane in D = 7. (Recall that L 1 is the length scale-factor for K3 with volume V 4 = L 4 1 .) The 5-brane charges in D = 11 have already been determined in (6.10), however. Thus it follows that consistency 6 with (6.20) requires that the magnetic 3-brane charges in D = 7 lie on the lattice Q m = m i µ i Q 0 , (6.21) 6 It is worth noting that there is an alternative basis for the 16 Eguchi-Hanson type anti-self-dual harmonic 2-formsωi, in which we have Σ iω j = L 2 1 δij and (6.15), rather than (6.16) and (6.15). In this alternative basis, (6.21) is replaced by Qm = ni ai Q0, where ni are integers. This is entirely equivalent to (6. tion that is not supersymmetric. 7 The reason why such a p-brane is non-supersymmetric is that the supersymmetry transformation rule for the gauginos is of the form δλ I ∼ G I µν Γ µν ǫ, and the matrix G I µν Γ µν is non-degenerate in the case of the standard ansatz for one of the fields G I (2) . On the other hand, we know that the Cartan-subalgebra U (1) 16
D = 9 black holes and Yang-Mills wave excitations
The key issues can be adequately illustrated by considering the nine-dimensional theory arising from the dimensional reduction of N = 1 supergravity in D = 10 coupled to a single D = 10 Maxwell multiplet. This subset of the heterotic-theory fields has a global O (1,2) invariance in D = 9. The bosonic sector of the Lagrangian is given by (2.2), with the index I on the Yang-Mills fields B I (1) and B I (0) taken to have one value only. We shall denote the associated potentials by B (1) and B (0) . The Lagrangian is
e −1 L 9 = R − 1 2 (∂φ) 2 − 1 2 (∂ϕ) 2 − 1 2 e √ 2ϕ (∂B (0) ) 2 − 1 12 e − 8 7 φ (F (3) ) 2 7
By the "standard construction" we mean that one starts with a Lagrangian of the form e −1 L = R − 1 2 (∂φ) 2 − e aφ F 2 , and makes a metric ansatz of the usual form ds 2 = e 2A(r) dx µ dxµ + e 2B(r) (dr 2 + r 2 dΩ 2 ), with F either of the form F = Q Ω (for a magnetic solution) or of the dual form (for an electric solution).
− 1 4 e − 2 7 φ e √ 2ϕ (F (2) ) 2 + e − √ 2ϕ (F (2) ) 2 + (G (2) ) 2 , (7.1)
where the various field strengths, following (A.10), are given by
F (2) = dA (1) , G (2) = dB (1) + B (0) dA (1) , F (2) = dA (1) + B (0) dB (1) + 1 2 (B (0) ) 2 dA (1) , F (3) = dA (2) + 1 2 B (1) dB (1) − 1 2 A (1) dA (1) − 1 2 A (1) dA (1) . (7.2)
We can construct extremal 2-charge electric 0-brane or magnetic 5-brane solutions using the Kaluza-Klein and winding vectors A (1) and A (1) . In this subsection, we shall look at the electrically-charged situation, and shall consider extremal black-hole solutions. The non-vanishing fields in this case are given by
ds 2 9 = (H 1 H 2 ) −6/7 dx µ dx µ + (H 1 H 2 ) 1/7 (dr 2 + r 2 dΩ 2 7 ) , e − √ 14 φ = H 1 H 2 , e √ 2 ϕ = H 1 H 2 , (7.3) A (1) = ±H −1 1 dt , A (1) = ±H −1 2 dt , B (1) = 0 ,
where the harmonic functions are given by 8
H 1 = 1 + q 1 r 6 , H 2 = 1 + q 2 r 6 . (7.4)
The ± signs on the 1-form potentials in (7.3) are independent, and reflect the fact that the bosonic equations are quadratic in field strengths, and hence solutions exist for any choice of signs. The electric charges Q 1 and Q 2 are given by
Q 1 = ±q 1 , Q 2 = ±q 2 . (7.5)
The mass of the black hole is given by
m = q 1 + q 2 .µ = m ± (Q 1 + Q 2 ) ,(7.7)
with each sign choice occurring with multiplicity 8. This means that in the N = 1 theory, only two out of the four sign choices in (7.5) give solutions that preserve supersymmetry,
namely (Q 1 , Q 2 ) = (q 1 , q 2 ) , or (Q 1 , Q 2 ) = (−q 1 , −q 2 ) . (7.8)
In each of these cases, the 2-charge solution preserves 1 2 of the N = 1 supersymmetry. For the remaining two sign choices in (7.5), all of the supersymmetry is broken. From now on, we shall consider just the supersymmetric solutions, and shall concentrate on the first of the two cases listed in (7.8). Note that when Q 1 = Q 2 the two harmonic functions H 1 and H 2 become equal, and the solution reduces to a "single-charge" solution supported by the single 2-form field strength of pure N = 1 supergravity in D = 9 [44].
There are in total three electric 2-form charges Q 1 , Q 2 and Q 3 in the nine-dimensional theory that we are considering, associated with the vectors A (1) , A (1) and B (1) respectively.
As we saw in section 2.1, these vectors form a triplet under the O(1, 2) global symmetry group. The associated invariant quadratic form constructed from the charges is
I = Q 2 3 + Q 2 Y − Q 2 X = Q 2 3 − 2Q 1 Q 2 ,(7.9)
where Q X = 1 To see the nature of the three types of orbit in detail, we first note that a single-charge solution supported either by Q 1 or by Q 2 belongs to a light-like orbit of O(1, 2), for which I vanishes. On the other hand, if Q 1 and Q 2 are both non-vanishing, and of the same sign, the orbit is time-like, with I < 0. If instead these two charges have opposite signs, then the orbit is space-like, with I > 0.
√ 2 (Q 1 + Q 2 ), Q Y = 1 √ 2 (Q 1 − Q 2 ),
Such solutions with light-like or time-like orbits have curvature singularities which, for positive-mass solutions, lie on the horizon at r = 0, but they are free of singularities outside the horizon. On the other hand solutions lying on space-like orbits necessarily have naked singularities that lie outside the horizon. This is because the charges are of opposite sign, and hence the supersymmetry requirement (7.8) implies that one of the harmonic functions will have a negative coefficient for its r −6 term. Although one might be tempted to dismiss such solutions from consideration, their presence is necessary in order to complete the charge lattice. Indeed, a single-charge solution supported exclusively by the Yang-Mills charge Q 3 would lie on such a spacelike orbit. Moreover, the fundamental D = 9 Yang-Mills field excitations of the theory corresponding to spontaneously broken Yang-Mills generators must lie in short massive supermultiplets that carry such charges. Given the presumed correspondence between such excitations and classical particle or wave solutions, one would expect to find classical solutions with these charges. The fact that massive supermultiplets carrying Yang-Mills charges must necessarily be short multiplets gives a hint that their origin may be sought in massless wave-like super Yang-Mills solutions in ten dimensions.
Indeed, as we shall now show, they do originate from such wave-like solutions, and moreover, this helps explain the origin of the naked singularities.
In fact, the naked singularity in the 2-charge solution (7.3) with Q 1 = q 1 > 0 and Q 2 = q 2 < 0 is purely an artefact of dimensional reduction. If we oxidise the solution back to D = 10, the metric becomes where tan 1 2 θ = (−q 1 /q 2 ) 1/2 , we obtain a new solution with the same metric as in (7.3), but with the other fields now given by
e − √ 14 φ = H 1 H 2 , e − √ 2 ϕ = H 1 H 2 , B 0 = 1 √ 2 (H 2 − H 1 ) sin θ , A (1) = ((H 1 H 2 ) −1 − H −1 1 − H −1 2 ) dt , A (1) = −(H 1 H 2 ) −1 dt , B (1) = 1 √ 2 sin θ (H −1 1 − H −1 2 ) dt . (7.12)
Oxidising this up to D = 10, we obtain the solution
ds 2 10 = −(H 1 H 2 ) −1 dt 2 + (H 1 H 2 )(dz + (H 1 H 2 ) −1 dt) 2 + d y 2 , φ 1 = 0 , B (1) = 1 √ 2 sin θ (H 2 − H 1 ) dz , A (2) = pure gauge . (7.13) Note in particular that F (3) = 0.
This solution describes a wave-like excitation of the Yang-Mills field, propagating along the z direction. It can be compared with the general class of wave-like solutions in the heterotic string, described in [48]. There, solutions were sought of the form
ds 2 10 = −2du (dv − W (u, y) du) 2 + d y 2 , B 1 = M (u, y) du , A (2) = b(u, y) ∧ du , φ 1 = 0 . (7.14)
For the class of solutions that are of interest to us, b(u, y) is zero, and M (u, y) depends only on y. From the results of [48], one has it that the equations of motion are satisfied provided
∂ i ∂ i M = 0 , ∂ i ∂ i W = −∂ i M ∂ i M . (7.15)
Thus, M is harmonic on the transverse space, and W can be solved by taking W = α M − 1 2 M 2 . If M is the harmonic function c r −6 , then the wave solution (7.14) is equivalent to (7.13) under the following identifications: It is worth emphasising that the solutions with unavoidable naked singularities in D = 9 arise in the case where the quadratic invariant (7.9) is positive, i.e. where the charge orbit is space-like. In [21], it was shown that for perturbative fundamental single-string excitations, the above quadratic invariant I is bounded above, lying in the interval 9 −∞ < I < 1. At the level of supergravity particle solutions, on the other hand, there is no such upper bound.
u = z √ 2 , v = 2t − z √ 2 , c = −2q 1 q 2 , α = q 1 + q 2 √ −2q 1 q 2 .(7.ds 2 10 = (a 2 H 1 + b 2 H 2 ) −3/4 − 2dt + H 1 H 2 a 2 H 1 + b 2 H 2 dz dz + (a 2 H 1 + b 2 H 2 ) 1/4 ,
This is because supergravity solutions can describe not only single-string excitations but also multiple-string excitations. The algebraic classification of heterotic string states into single-string and multiple-string spectra remains an interesting open problem.
5-branes in D = 9
Now we turn to the nine-dimensional 5-brane solution, which is given by
ds 2 9 = (H 1 H 2 ) −1/7 dx µ dx µ + (H 1 H 2 ) 6/7 (dr 2 + r 2 dΩ 2 2 ) , e √ 14 φ = H 1 H 2 , e √ 2 ϕ = H 2 H 1 , (7.18) A (1) = Q 1 ω , A (1) = Q 2 ω , B (1) = 0 ,
where the harmonic functions are given by
H 1 = 1 + q 1 r , H 2 = 1 + q 2 r ,(7.19)
and dω = Ω (2) is the volume form on the unit 2-sphere. (Again, for simplicity we are considering single-centre isotropic solutions here, which could easily be generalised to multicentre solutions.) The mass per unit 4-volume of the 5-brane is given by
m = q 1 + q 2 . (7.20)
In terms of the two individual parameters q 1 and q 2 , there exist 5-brane solutions where the magnetic charges under the Kaluza-Klein and winding vectors are
Q 1 = ±q 1 , Q 2 = ±q 2 . (7.21)
As in the case of electric black holes discussed earlier, the supersymmetry requires that
(Q 1 , Q 2 ) = (q 1 , q 2 ) or (Q 1 , Q 2 ) = (−q 1 , −q 2 ).
To construct a 5-brane solution supported purely by the Yang-Mills 2-form in D = 9, let us begin with the solution (7.18), for which Q 1 = q 1 and Q 2 = q 2 are arbitrary, with Q 1 Q 2 < 0, while Q 3 = 0. In order to map the charges to a configuration where Q 1 = Q 2 = 0, while Q 3 is non-zero, we must act with a matrix in an O(1, 1) subgroup of O(1, 2) as follows:
1 √ 2 (Q 1 + Q 2 ) 1 √ 2 (Q 1 − Q 2 ) 0 −→ 0 0 Q 3 = cosh t sinh t 0 0 0 −1 sinh t cosh t 0 1 √ 2 (Q 1 + Q 2 ) 1 √ 2 (Q 1 − Q 2 ) 0 . (7.22)
The parameter t is therefore given by
e 2t = − Q 2 Q 1 . (7.23)
Again, we see that Q 1 and Q 2 must have opposite signs in order for the mapping to be possible. Applying this transformation to the fields X, Y and Z introduced in section 2.1, one can determine the transformed expressions for the dilaton ϕ and the axion B (0) .
Similarly, by transforming the column vector (A (1) , A (1) , B (1) ), one obtains the expressions for the transformed vector potentials. Upon doing this, we find that the solution (7.18) becomes
ds 2 9 = (H 1 H 2 ) −1/7 dx µ dx µ + (H 1 H 2 ) 6/7 (dr 2 + r 2 dΩ 2 2 ) , e √ 14 φ = H 1 H 2 , e √ 2 ϕ = 1 2 − Q 1 H 2 Q 2 H 1 + 1 2 − Q 2 H 1 Q 1 H 2 , (7.24) 1 √ 2 B (0) = Q 1 H 2 + Q 2 H 1 Q 1 H 2 − Q 2 H 1 , A (1) = 0 , A (1) = 0 , B (1) = Q 1 − 2Q 2 Q 1 ω .
The harmonic functions retain their original form, and can in general describe multi-centre solutions. In the single-centre isotropic case, they are given by (7.19). In this special case, the dilaton ϕ and the axion B (0) can be rewritten as
e −2 √ 2ϕ = − 4 Q 1 Q 2 (Q 1 − Q 2 ) 2 H 1 H 2 , 1 √ 2 B (0) = Q 1 H 2 + Q 2 H 1 Q 1 − Q 2 . (7.25)
In line with our previous discussion we see indeed that this solution, which carries only the charge Q 3 of the Yang-Mills field strength G (2) , is not of the form of a "standard" single-charge p-brane solution. It is nevertheless, of course, supersymmetric, since we have obtained it by performing an O(1, 2) rotation on a standard supersymmetric solution.
Although it involves only a single Yang-Mills charge Q 3 = q 1 −2q 2 /q 1 , it has two independent parameters q 1 and q 2 associated with the two harmonic functions H 1 and H 2 given in (7.19). However, the two parameters q 1 and q 2 must both be non-zero and of opposite signs, implying that there is again a naked singularity, at r = |min (q 1 , q 2 )|. If q 1 = −q 2 , the 5-brane has zero mass. The connection between naked singularities and masslessness has been extensively discussed in the context of BPS black holes of the D = 4 toroidallycompactified heterotic string in Refs [49,50]. A similar phenomenon also occurs in p-brane solitons in maximal supergravity [45], including massless dyonic strings in D = 6 [51]. The occurrence of a naked singularity means that the naive relation between zero eigenvalues of the Bogomol'nyi matrix and unbroken supersymmetries no longer holds, and in particular the apparent enhancement of supersymmetry in the massless limit does not in actuality occur [52].
It is interesting to look at the form of the solution (7.24) when oxidised back to D = 10.
We find that it becomes
ds 2 10 = e 1 4 √ 2 ϕ (H 1 H 2 ) −1/8 (dx µ dx µ + e − √ 2ϕ dz 2 ) + (H 1 H 2 ) 7/8 (dr 2 + r 2 dΩ 2 2 ) , e 2φ 1 = e 1 √ 2 ϕ (H 1 H 2 ) −1/2 , A (2) = Q 1 − Q 2 Q 1 Q 1 H 2 + Q 2 H 1 Q 1 H 2 − Q 2 H 1 ω ∧ dz , B (1) = Q 1 − 2Q 2 Q 1 ω + √ 2 Q 1 H 2 + Q 2 H 1 Q 1 H 2 − Q 2 H 1 dz ,(7.26)
where e √ 2ϕ is the function given in (7.24). Note that in D = 10 the solution also involves the NS-NS 3-form field strength. As usual, we give the metric in the Einstein frame here.
In terms of the string frame, this becomes
ds 2 str = dx µ dx µ + e − √ 2ϕ dz 2 + (H 1 H 2 ) (dr 2 + r 2 dΩ 2 2 ) . (7.27)
In the above discussion, we have considered the case where we began with a 2-charge solution with Q 1 = q 1 and Q 2 = q 2 opposite in sign. By doing so, we ensured that the solution was supersymmetric, but at the price of its having a naked singularity. We could instead have started from a 2-charge solution with charges Q 1 = q 1 and Q 2 = −q 2 , again taken to be opposite in sign. In this case, the solution is non-supersymmetric, however, since now all the Bogomol'nyi eigenvalues in (7.7) are non-zero. However, if Q 1 and −Q 2 are both positive then the solution will be free from naked singularities. This solution can also be rotated to one that carries only the Yang-Mills charge Q 3 . The form of this solution is identical to (7.24), except that the replacement of (Q 1 , Q 2 ) = (q 1 , q 2 ) by (Q 1 , Q 2 ) = (q 1 , −q 2 ) implies that the harmonic function H 2 is given by 1 − Q 2 /R rather than 1 + Q 2 /r. In particular, this means that when Q 1 = −Q 2 , and hence H 1 = H 2 , the solution (7.24) is now a "standard" single-charge 5-brane solution, supported by the Yang-Mills field strength G (2) . In line with our earlier discussion, this is indeed non-supersymmetric.
This analysis of p-brane orbits supported by 2-form field strengths in the toroidally- The solution space in this case has been extensively studied [53,54,55,56,57].
In this section, we have been principally interested in supersymmetric solutions supported purely by the Yang-Mills fields, in order to make contact with M-theory compactified on K3. As we have discussed, such solutions suffer from naked singularities. The origin of these naked singularities in the M-theory picture is less clear. We have seen that when an M-brane wraps around any of the sixteen Eguchi-Hanson 2-cycles in K3, it gives rise to a p-brane which, in the heterotic picture, is supported by a Yang-Mills field. As we have seen in section 5, these 2-cycles can shrink to zero size, in which case the wrapped M-brane will become massless [58]. Indeed, the p-brane supported by the Yang-Mills field in the heterotic picture can also become massless by adjusting the Yang-Mills moduli, i.e. the expectation values of the Cartan-subalgebra scalar fields.
quantities h α are given by h α = dz α + A α (1) + A α (0)β dz β =γ α β (dz β +Â β (1) ) , (A.5)
whereγ α β = δ α β + A α (0)β , and γ α β is the inverse ofγ α β [39,5]. We have also introduced the redefined Kaluza-Klein vectors α 1 = γ α β A β (1) . Note that γ a β andγ α β are non-zero only when α ≤ β.
Applying the above reduction ansatz, we obtain the D-dimensional Lagrangian
e −1 L D = R − 1 2 (∂ φ) 2 − 1 12 e a 1 · φ (F (3) ) 2 − 1 4 α e a 1α · φ (F (2)α ) 2 − 1 2 α<β e a 1αβ · φ (F (1)αβ ) 2 − 1 4 I e c· φ (G I (2) ) 2 − 1 2 α,I e cα· φ (G I (1)α ) 2 − 1 4 α e bα· φ (F α (2) ) 2 − 1 2 α<β e b αβ · φ (F α (1)β ) 2 . (A.6)
The indices α, β . . . run from 2 to 11 − D. The various field strengths here are given by IIA supergravity, which arises as the first-step reduction of F (4) .) The dilaton vectors b α and b αβ , for the Kaluza-Klein vectors A α (1) and the Kaluza-Klein axions A α (0)β also coincide precisely with the ones introduced in [39,5]. The reason for these relations to the dilaton vectors arising from the reduction of D = 11 supergravity is that the pure supergravity sector of the heterotic theory can be obtained as a truncation of type IIA supergravity where the R-R fields are set to zero. Finally, the dilaton vectors c and c α for the Yang-Mills fields G I (2) and their dimensional reductions G I (1)α come from reducing 2-form field strengths in D = 10 to 2-forms and 1-forms in D dimensions.
F (3) = dA (2) + 1 2 B I (1) dB I (1) − (dA (1)α + 1 2 B I (0)α dB I (1) + 1 2 B I (1) dB I (0)α )Â α (1) + 1 2 (dA (0)αβ − B I (0)[α dB I (0)β] )Â α (1)Â β (1) , F (2)α = γ β α dA (1)β + 1 2 B I (0)β dB I (1) + 1 2 B I (1) dB I (0)β + (dA (0)βγ − B I (0)[β dB I (0)γ] )Â γ
It is advantageous at this stage to perform some field redefinitions on the 2-form and 1-form vector potentials, in order to simplify the structure of the equations. Thus we define
B I (1) = B ′ I (1) + B I (0)α α (1) , A (1)α = A ′ (1)α − A (0)αβ β (1) + 1 2 B I α B ′ I (1) , (A.9) A (2) = A ′ (2) + 1 2 A (0)αβ α (1) β (1) + 1 2 B I (0)α B ′ I (1) α (1) + 1 2 A ′ (1)α α (1) .
In terms of these, the 3-form and 2-form field strengths become I + B I (0)α d α (1) , F α (2) =γ α β d β (1) .
F (3) = dA ′ (2) + 1 2 B ′ I (1) dB ′ I (1) − 1 2 α (1) dA ′ (1)α − 1 2 A ′ (1)α d α (1) , F (2)α = γ β α dA ′(
The expressions for the 1-form field strengths F (1)αβ , F α (1)β and G I (1)α are unchanged, and still given by (A.7).
B Further coset geometry examples
In section 2.2 of this paper, we gave a geometrical construction of the class of cosets (2.11), which includes those arising in the toroidal dimensional reduction of ten-dimensional simple supergravity coupled to N abelian gauge fields. In this appendix, we extend this construction to several further classes of cosets. Since the principles of the construction are closely parallel to those discussed in section 2.2, we shall just present the essential features of the examples.
We begin with an additional class of cosets in which the matrix W is real, and satisfies the pseudo-orthogonality condition W T η W = η. Then we can have an antisymmetricmatrix embedding:
W T η W = η , W T = −W
The numerator group here is generated by matrices Λ satisfying Λ T η Λ = η, whose action on W is W → Λ τ W Λ. It is easily seen that, without loss of generality, the fiducial matrix W 0 can be taken to have the antisymmetric block diagonal form Thus we see that the numerator group is O(2p, 2q), while the stability subgroup is given by O(2p, 2q) matrices Λ that satisfy not only Λ T η Λ = η but also Λ T W 0 Λ = W 0 . It is not hard to see that with these conditions, the matrices Λ lie in U (p, q). Thus we obtain that where σ is given by (B.2). In these cases the numerator group is generated by matrices Λ that satisfy Λ T Ω Λ = Ω, and whose action on W is W → Λ T W Λ. We first consider a symmetric-matrix embedding:
W T Ω W = Ω , W T = W
In this case, the numerator group is defined by matrices Λ satisfying Λ T Ω Λ = Ω, and thus it is Sp(2n, IR). It is easy to see that the fiducial matrix W 0 can be taken to be diagonal, and of the form (The individual numbers of − and + signs must necessarily both be even, in order that W 0 be able to satisfy the given conditions.) The stability subgroup is given by matrices Λ that satisfy both Λ T Ω Λ = Ω and Λ T W 0 Λ = W 0 . These are the same conditions as in the previous example, and so again we find that the stability subgroup to be U (p, q). Thus we obtain the coset Sp(2p + 2q, IR) U (p, q) .
(B.7)
Next, consider the antisymmetric-matrix embedding:
W T Ω W = Ω , W T = −W
The numerator group is the same as for the symmetric-matrix embedding above. The fiducial matrix should be taken to be antisymmetric in this case; we may pick Continuing on with metric-preserving numerator groups, we can now consider cases where the numerator group is generated by complex matrices Λ that preserve the sesquilinear form Λ † η Λ = η. These act on complex matrices W according to W → Λ † W Λ. Without loss of generality, η can be taken to be symmetric. Thus we may consider a Hermitean embedding:
W † η W = η , W † = W :
Here, the metric η has the form Note that there is no loss of generality in taking W to be Hermitean rather than anti-Hermitean, since the latter is related to the former by a multiplication by i. The fiducial matrix can be taken to be The numerator group is U (p, q), and its stability subgroup is U (m, q − n) × U (p − m, n).
Thus we have the coset U (p, q) U (m, q − n) × U (p − m, n) . (B.12)
As a final set of examples, we consider two cases where the numerator group is not metric-preserving. Specifically, we shall take it to be SL(n, IR), defined by n × n real matrices that satisfy det Λ = 1. Their action on W is W → Λ T W Λ. We can then take the coset matrices W to satisfy either W T = W or W T = −W :
W T = W
The fiducial matrix W 0 can be taken to be It follows that the stability subgroup will be the subgroup of SL(p + q, IR) matrices that satisfy Λ T W 0 Λ = W 0 ; in other words it will be SO(p, q). Thus we have the coset SL(p + q, IR) SO(p, q) .
(B.14)
W T = −W :
The fiducial matrix W 0 can be taken to be W 0 = diag (σ, σ, . . . , σ) .
(B.15)
The stability subgroup will therefore be generated by the subset of SL(2n, IR) matrices that satisfy Λ T W 0 Λ = W 0 ; in other words it will be Sp(2n, IR). Thus we have the coset structure SL(2n, IR) Sp(2n, IR) .
(B.16)
the scalar part of the Lagrangian (2.2) can be written ase −1 L scalar = (∂X) 2 − (∂Y ) 2 − (∂Z I ) 2 − 1 2 (∂φ) 2 ,(2.6) subject to the constraint (2.5). Thus we see that the Lagrangian and the constraint are invariant under global O(1, N + 1) transformations, which act by matrix multiplication on the column vector (X, Y, Z I ), and also that the Lagrangian is invariant under constant shifts of φ. Thus the symmetry of the scalar Lagrangian is O(1, N + 1) × IR.We find that in terms of the fields (X, Y, Z I ), the Lagrangian (2.2) can be written as e −1 L 9 = R + L scalar −
now manifest that if the O(1, N + 1) transformations act on the column co-vector (A X , A Y , B I (1) ) of 1-form potentials at the same time as they act on (X, Y, Z I ), then the entire Lagrangian (2.7) is invariant. To be precise, if Λ is an O(1, N + 1) group element then the Lagrangian is invariant under the global transformation
an O(1, N + 1) × IR global symmetry. The O(1, N + 1) symmetry could be understood geometrically as a set of linear transformations on an (N +2)-vector of coordinates (X, Y, Z I ) on IR 1,N +1 with the Minkowski metric η = diag (−1, 1, 1, . . . , 1). The scalar manifold is therefore the coset space O(1, N + 1)/O(N + 1) (together with an extra trivial IR factor). In this subsection, we shall consider the geometry of the coset spaces that arise in general toroidal dimensional reductions of the heterotic theory. In fact, we shall consider more generally the geometrical construction of the entire class of coset spaces O(p, q) O(m, q − n) × O(p − m, n)
W in O(p, q) then, by definition, satisfyW T η W = η .(2.13)One can also then impose the additional O(p, q)-covariant constraint W T = W .
≤ m ≤ p and 0 ≤ n ≤ q. The specific distributions of the +1 and −1 eigenvalues within the p × p timelike and q × q spacelike subspaces may be modified by further O(p) × O(q) transformations. The fiducial matrix W 0 in (2.16) is therefore representative of a class of equivalent matrices. Note, however, that no O(p, q) transformation is able to exchange eigenvalues between the timelike and spacelike sectors, since the corresponding eigenvectors would also have to be exchanged, and this is impossible since O(p, q) transformations preserve the norms of vectors, while timelike and spacelike vectors have norms of opposite signs. Thus, the numbers m and n in (2.16) are O(p, q)-invariant. These numbers will determine the denominator groups K in the coset spaces O(p, q)/K.
T
10−D instead. In these cases, one expects the global symmetry group to be O(10 − D, 10 − D + N ) × IR when D ≥ 5, while in D = 4 and D = 3 one expects O(6, N + 6) × SL(2, IR)
21) where M is a square matrix of dimension(20 − 2D + N ) that is parametrised by the rest of the dilatonic scalars and the axionic scalars, and H (2) = dC (1) is a column vector formed from the exterior derivatives of the 1-form potentials. The NS-NS 3-form F (3) is coupled only to the dilaton φ, implying that the D-dimensional string coupling is λ D = exp( (D − 2)/8 φ). To show that the Lagrangian (A.6) is invariant under O(10 − D, 10 − D + N ), and that its scalar manifold is the coset space O(10 − D, 10 − D + N )/(O(10 − D) × O(10 − D + N ))
±1, of which (10 − D) are negative and (10 − D + N ) are positive. In fact Ω is nothing but a metric on the indefinite-signature flat space IR 10−D,10−D+N . It is straightforward to see that V satisfies V T Ω V = Ω . (2.27) This implies that V lies in the group O(10 − D, 10 − D + N ). It follows that M = V T V satisfies the two conditions given in (2.13, 2.14). Furthermore, it is evident from (2.24) that if one sets all the axions and dilatons to zero, then V, and hence M, becomes the identity. Thus, from the discussion at the end of subsection 2.2, it follows that V and M give parameterisations of the coset O(10 − D, 10 − D + N )/(O(10 − D) × O(10 − D + N )). (Indeed, the number of independent scalar fields in V is equal to the dimension of the coset.) Note that F (3) as given in (2.25) is a singlet under the O(10−D, 10−D+N ) transformations.
2) where M = V # V. Here V # = τ (V −1 ) where τ denotes the Cartan involution which reverses the sign of all the non-compact generators, while leaving the sign of the compact generators unchanged (see, for example,[5]). (For orthogonal groups, V # is just equal to the transposeV T , and for unitary groups it is the Hermitean conjugate V † .) The Lagrangian is invariant under the global symmetry transformations V −→ V ′ = O V Λ, where Λ is any element of the group G, and O is a field-dependent compensating transformation that is used to bring the transformed coset representative V ′ back to the form V ′ = exp(G ′ s ). The Iwasawa decomposition guarantees the existence of O, and the fact that it is contained in the maximal compact subgroup K. It then follows that M is transformed to M ′ = Λ # M Λ, and hence that the Lagrangian (3.2) is invariant. In this section, we shall show that the scalar Lagrangians for the toroidally-reduced heterotic theory can be written in the form (3.2), where V ∼ e 1 2 φ· H+χa E a . We shall obtain the explicit forms of the algebras for the generators H and E a , and we shall show that they are the solvable Lie algebras associated with the global symmetry groups. This provides an explicit derivation of the global symmetries of the scalar sectors of the toroidally-compactified heterotic theory. We show that the M obtained from the scalar sector using the Solvable Lie Algebra technique and the one obtained by studying the coupling of the scalars with the vector potentials are equivalent, hence completing the proof that the full Lagrangian has an O(10 − D, 10 − D + N ) global symmetry.
the generators of the solvable Lie algebra of O(1, N + 1). In other words, h e 1 is the noncompact Cartan generator of O(1, N +1), while the other generators in (3.7) are precisely the subset of positive-root O(1, N + 1) generators that have strictly positive weights under h e 1 .
roots for O(2, 3) are e 1 − e 2 , e 1 + e 2 , e 1 and e 2 . On the other hand, from(3.12)
in this case, because O(2, 3) is maximally non-compact, all of the Borel generators of O(2, 3) occur in the associated solvable Lie algebra. For the case O(2, 4), we have generators E 2 3 , V 23 , U 2 I and U 3 I in our coset parameterisation (3.10), where 1 ≤ I ≤ 2. From (3.6), the positive roots of O(2, 4) are e 1 ± e 2 , e 1 ± e 3
precisely constitute the set of generators in the solvable Lie algebra of O(2, N + 2). This is because h e 1 and h e 2 are the two non-compact Cartan generators of O(2, N + 2), and the positive-root generators we have just listed are the full set that have strictly positive weights under h e 1 and h e 2 . Thus it follows that the quantity V defined in (3.10) gives a parameterisation of the coset O(2, N +2)/(O(2)×O(N +2)). Together with the shift symmetry of the dilaton φ, this shows that the scalar Lagrangian (3.9) is invariant under global O(2, N + 2) × IR transformations.
can be embedded into those of O(10 − D, 10 − D + N ), and that in fact they precisely correspond to the solvable lie algebra of O(10 − D, 10 − D + N ). To see this, it is useful first to introduce the set of orthonormal vectorsẽ i , related to e i bỹ e i = e 11−D−i , 1 ≤ i ≤ 10 − D . (3.27) Let us, for definiteness, first consider the case where N is even. We then find that the above generators can be written in terms of those of O(10 − D, 10 − D + N ) as follows. The generators H, E i j and V ij are written as
easily seen that this set of generators comprises the solvable Lie algebra of O(10 − D, 10−D+N ). In other words, they are written in terms of the complete set of non-compact Cartan generators of O(10 − D, 10 − D + N ), together with all the positive-root generators that have strictly positive weights under the non-compact Cartan generators. So far in this subsection, we have constructed the coset representative V for the purpose of writing the scalar Lagrangian in the form 1 4 tr(∂M −1 ∂M), where M = V T V. Using the solvable Lie algebra formalism, we have accordingly shown that the scalar Lagrangian is described by the coset O(10 − D, 10 − D + N )/((O(10 − D) × O(10 − D + N )) (together with an extra IR factor for the scalar field φ). This construction is abstract, in the sense that we have not taken any specific realisation for the generators; they are simply required to satisfy (3.26). On the other hand, in section 2 we have also obtained an expression for a coset parameterisation V, by considering the coupling of the scalars to the 1-form potentials. Since these potentials form a fundamental representation of O(10 − D, 10 − D + N ), the representation for V that we obtained there was necessarily given in terms of matrices of dimension (20 − 2D + N ). To complete the proof that the entire D-dimensional Lagrangian has a global O(10 − D, 10 − D + N ) symmetry, we need to make contact between the two descriptions, by showing explicitly that we can write V as given by (2.24) in the form (3.24), and by showing that the generator matrices satisfy the commutation relations (3.26).
35) where M is a parameterisation of the coset O(6, N + 6)/(O(6) × O(N + 6)). As well as the manifest global O(6, N +6) symmetry of the Lagrangian, there is also an SL(2, IR) symmetry of the equations of motion, under which V H(2) and e −φ * (V H (2) ) T form an SL(2, IR) doublet.
one eigenvalue −1 in the O(10 − D, 10 − D + N )-invariant metric Ω. This means that in terms of the diagonal invariant metric that we used in section 2.2, the fiducial matrix W 0 is given by (2.16) with m = n = 1. Thus the coset spaces describing scalar manifolds in the D-dimensional theories obtained by time-like reductions to D ≥ 5 are
D = 4, the global symmetry group is O(6, 6+N )×SL(2, IR). The same considerations as given above show that the stability subgroup of the O(6, 6+N ) factor is O(1, 5)×O(1, 5+ N ). The stability subgroup of the SL(2, IR) factor, which would be O(2) in a standard
numbers of zero modes in the two theories may be compared straightforwardly by counting the harmonic forms corresponding to the various D = 7 spin sectors in each theory.Reducing D = 11 supergravity on K3, one obtains a D = 7 metric and 58 scalars from the reduction of the D = 11 metric, plus a D = 7 three-form antisymmetric tensor gauge potential plus 22 one-form gauge potentials from the reduction of the D = 11 three-form gauge potential. The 58 scalars arise from the 57 shape-determining plus one volume-setting moduli of the internal K3 manifold, while the 22 one-forms arise from the 22 harmonic twoforms occurring on K3.In the supergravity sector of the heterotic theory compactified on T 3 , one has for comparison: a D = 7 metric plus a triplet of dilatonic scalars, together with a triplet of one-form Kaluza-Klein gauge potentials and a triplet of axionic scalars, all descending from the D = 10 metric; one additional dilatonic scalar descending from the D = 10 dilaton; and also a D = 7 two-form gauge potential, three one-form gauge potentials and three axionic scalars, all descending from the D = 10 two-form gauge potential. The Yang-Mills sector of the heterotic theory contributes a number of D = 7 zero-modes as well. We shall assume for
where n i are any integers. One then cuts out a small 4-ball around each of the 16 fixed points. Had we not performed the identifications, the boundaries of the 4-balls would each have been a 3-sphere. Because of the identification, the boundaries are instead copies of RP 3 , the real projective plane. (This is S 3 with an antipodal identification.) One now patches up the manifold, by "plugging in" an appropriate space into each of the 4-balls.
an almost-smooth join between the torus and the Eguchi-Hanson instanton, which splays out like a champagne cork and plugs into the excised 4-ball in the 4-torus. Inserting a total of 16 such corks, i.e. one for each excised 4-ball, one achieves an approximation to a Ricciflat K3 manifold that becomes arbitrarily precise as the scale sizes of the Eguchi-Hanson instantons are taken to zero [13]. Using the above construction, it is possible to give a fairly explicit construction of the K3 compactification of D = 11 supergravity. In particular the harmonic 2-forms on K3, in terms of which all of the non-trivial zero-modes are described, can be seen to fall into two different categories. First of all, there are those that can be viewed as being harmonic 2-forms on the 4-torus. In fact, one can easily see that the set of harmonic forms on the 4
( 2 )
2depend only on the breathing mode, and they are accordingly singlets under SL(4, IR). On the other hand, the dilaton vectors a ij of F (2)ij are precisely the weight vectors of the six-dimensional representation of SL(4, IR), after the subtraction of a universal constant vector associated with the breathing mode. The dilaton vectors b ij for the scalars A i (0)j form the positive roots of SL(4, IR), with simple roots b 12 , b 23 and b 34 . To see that these axionic scalars, taken together with the dilatonic degrees of freedom orthogonal to the breathing mode, realise the full SL(4, IR), it is necessary to include the negatives of the dilaton vectors, i.e. − b ij , since the set of vectors ± b ij form the complete root system of SL(4, IR). Note, however, that dilatonic couplings with − b ij do not occur in the Lagrangian.This is a reflection of the fact that the scalars parameterise the coset SL(4, IR)/O(4), and thus provide a non-linear realisation of SL(4, IR). The negative roots are generated[38] by the non-linear (Weyl group) transformation A i (0)j → e − b ij · φ A i (0)j + . . .. Thus for the scalar sector, both the dilaton vectors and their negatives should be included in discussing the global symmetry.
(
∂ψ I ) 2 . (5.15) 5 Note that the indices 1, 2 and 3 on the F(1) fields are cyclic. In the Borel-type (Solvable Lie Algebra)
Figure 1 .
1satisfies M 5 = 1l. To understand the nature of this transformation, we note that both M-theory on K3 and the heterotic theory on T 3 have an O(3, 3) ∼ SL(4, IR) global symmetry as a subgroup of the full O(3, 19). The simple roots of O(3, 3) are given by the dilaton vectors of the underlined axionic fields listed in Table 1. Simple roots of the O(3, 3) subgroup The O(3, 3) ∼ SL(4, IR) group can be also viewed as a subgroup of the SL(5, IR) global symmetry group of maximal supergravity in D = 7, which has simple roots b 12 , b 23 , b 34 and a 123 . Thus we see that M-theory on K3 and the heterotic string on T 3 make two different truncations of SL(5, IR). From the maximal supergravity point of view, the two sets of simple roots of O(3, 3) are related by the Weyl group of SL(5, IR), which is S 5 , the permutation group of five objects. Thus we would naturally expect that M 5 = 1l.
. Note that we have concentrated so far on establishing the relation between the dilaton couplings in the two theories; the detailed matching of the Kaluza-Klein modifications to the field strengths requires a more detailed analysis.It is appropriate at this point to make a few remarks about the nature of the Kaluza-Klein reduction procedure in the K3 compactification, and in particular to address the issue of the consistency of the reduction. In principle, the first step in any Kaluza-Klein compactification is to perform a harmonic expansion of all the higher-dimensional fields in terms of appropriate complete sets of scalar and tensor harmonics on the internal space, thereby arriving at a lower-dimensional theory with infinite towers of massive fields, together with finite numbers of massless fields. At this stage the reduction is guaranteed to be consistent, since one has done nothing more than a generalised Fourier expansion of the higher-dimensional fields.In practice, one is usually interested in retaining only the finite number of massless fields arising from the Kaluza-Klein reduction. In other words, one would ideally wish to be able to set the infinite towers of massive fields to zero. The question then arises as to whether this is a consistent truncation of the lower-dimensional theory. In other words, is the setting to zero of the massive fields consistent with their own equations of motion?The dangers of inconsistency all stem from the non-linear interaction terms in the theory, which have the possibility in general of including "source terms" for the massive fields, built purely from the massless fields that are to be retained. Thus if we denote the massless fields generically by φ L , and the massive ones by φ M , a typical inconsistency would be signalled by the occurrence in the Lagrangian of non-linear interactions of the form φ M (φ L ) 2 , leading to equations of motion of the form
(
20) and that there accordingly exists a regime of excitations of the massive fields where the energies are small compared to the Kaluza-Klein mass scale M , with the consequence that the inconsistencies in such reductions can then be neglected. One can expand (5.∂φ L ) 2 ∼ M −2 (∂φ L ) 2 + · · · , (5.21)Thus, at low energies the φ M term in the massive field equation can be dropped, and the massive field φ M can effectively be "integrated out," by substituting the solutionφ M ∼ M −2 (∂φ L ) 2 intothe lower-dimensional Lagrangian. A related approach is to substitute the Kaluza-Klein reduction ansatz for the massless fields into the higher-dimensional Lagrangian, and then to integrate over the internal compactifying manifold. Such an approximate discussion is applicable to situations where one is seeking to extract an effective low-energy "phenomenological" theory from the string compactification, where the mass scale M of the Kaluza-Klein massive modes is very large compared with the energies of interest. However, it is not clear that this applicability extends to the regime of interest for non-perturbative duality symmetries. In particular, the conjectured duality between the heterotic string compactified on T 3 and M-theory compactified on K3 involves an inverse relation between the scale sizes of the T 3 and K3. Thus, to make any meaningful statements it is necessary to consider M-theory compactified on a large K3, where the Kaluza-Klein mass scale M tends to zero, and here the neglect of kinetic terms for massive fields such as that in (5.20) becomes less and less innocent. Indeed, as M tends to zero it is presumably more appropriate to expand (5.20) using not (5.21), but rather
(
first consider the subset of the theory corresponding to the O(3, 3)-invariant subsector of the complete D = 7 reduction; in other words, in the heterotic picture the Yang-Mills fields are not yet to be included. Equivalently, in the M-theory picture, the fields in D = 7 associated with the "Eguchi-Hanson harmonics" on K3 are not yet to be included. Let us assume that the volume of K3 is given by V 4 = L 4 1 , whilst the volume of the T 3 in the heterotic picture is V 3 = L 3 2 . The duality of the two theories implies that e and m subscripts indicate electric and magnetic charges, and the (4) subscript indicates that they are carried by the 4-form field strength. The constant α is as yet arbitrary. Upon reduction to D = 7 on K3, in the O(3, 3) subset the 4-form reduces to F We need only list either electric or magnetic charges, since these are related by the Dirac quantisation condition.) In the heterotic picture, on the other hand, the O(3, 3) subset of fields is obtained from the T 3 reduction of the pure N = 1 supergravity multiplet. The most general solutions for the D = 10 string and 5-brane charges are given by Q e(3) = (2π)
21), since the integers ni are related to the integers mi in (6.21) by ni = (M −1 ) ij mj, and both Mij , the Cartan matrix of E8 × E8, and its inverse (M −1 ) ij , have entirely integer-valued components.
the lattice of magnetic charges(6.21), derived from the K3 compactification of Mtheory, is of the same form as the lattice (6.12) that we obtained in the T 3 compactification of the heterotic string. A precise identification then requires that α ′ = Q 2 0 .7 p-brane orbits with Yang-Mills chargesIn the previous section, we looked at the duality relation between the heterotic string compactified on T 3 and M-theory compactified on K3. As well as being a relation that holds in the low-energy effective field theories, this should also be seen at the level of BPSstates in the two full theories. For example, the M5-brane wrapped around the K3 manifold is dual to a vertical reduction of the NS-NS string of the heterotic theory, while the vertical reduction of the M2-brane is dual to the NS-NS 5-brane wrapped around the volume of T 3 . To study this, let us focus on the spectrum of particles, and their 3-brane duals, in seven dimensions. From the M-theory point of view, they arise from membranes and 5-branes wrapping around the 2-cycles of K3. In our discussion given in section 5, we encountered two different types of 2-cycle in the construction that we were using for K3, namely six 2-cycles corresponding to the usual non-contractible 2-surfaces in T 4 , and sixteen additional 2-cycles, each associated with one of the sixteen Eguchi-Hanson instantons used to smooth out the conical singularities on the identified 4-torus. ¿From the field-theoretic point of view, the concept of "wrapping" a p-brane soliton around a particular m-cycle in the internal space essentially translates into the idea of constructing a (p − m)-brane soliton supported by the lower-dimensional field arising in the Kaluza-Klein expansion from the harmonic form associated to the given m-cycle on the internal space. When m is less than the dimension of the internal manifold, part of the transverse space of the p-brane also becomes internal. In a toroidal reduction, this notion of vertical reduction can be made mathematically precise, by "stacking" p-branes along the reduction axes. No such analogous procedure has yet been implemented for K3 or Calabi-Yau compactifications, where the internal coordinates correspond both to the world-volume and the transverse space of the wrapped p-brane. (Such an implementation would presumably first of all require that one know the explicit metric on the K3 or Calabi-Yau internal space.) When the membrane wraps around any of the six 4-torus 2-cycles, it gives a corresponding set of six 0-branes that are supported by the six 2-form field strengths coming from the Kaluza-Klein reduction of the D = 11 4-form on T 4 . These correspond, in the heterotic picture, to the six 0-branes that describe the three winding and three Kaluza-Klein modes on the compactifying 3-torus.When the membrane wraps instead around any of the sixteen 2-cycles associated with the Eguchi-Hanson manifolds, it should give rise to sixteen 0-branes that are supported by the associated 2-form field strengths which, in the heterotic picture, come from the T 3 dimensional reduction of the Yang-Mills fields of the D = 10 heterotic string. However, a standard construction of a p-brane supported by one of the Yang-Mills fields leads to a solu-
subset of the Yang-Mills fields, together with the Kaluza-Klein and winding vector fields, form an irreducible multiplet in the fundamental representation of O(10 − D, 26 − D). Thus naively one might expect that starting from a supersymmetric p-brane supported either by a Kaluza-Klein vector or by a winding vector, one could rotate to a solution supported purely by the Yang-Mills fields. However, it turns out that this is not the case; the reason is that solutions supported only by the Yang-Mills fields would lie on different O(10 − D, 26 − D) orbits from those supported by a single Kaluza-Klein or winding vector field. In this section we shall study the various possible types of orbit that can arise, in order to see on which orbits supersymmetric solutions supported purely by the Yang-Mills fields can lie.
bosonic theory has four solutions, corresponding to the four different sign choices in (7.3), the supersymmetry transformations depend linearly on the field strengths, and, consequently, not all of the four sign choices for the charges need yield supersymmetric solutions. In the case of maximal supergravities, on the other hand, the fraction of preserved supersymmetry is independent of the sign choices for similar 2-charge solutions. This can be seen from the fact that the 32 eigenvalues of the Bogomol'nyi matrix are [39] µ = m ± Q 1 ± Q 2 , with each of the four sign combinations occurring with multiplicity 8. Thus in maximal supergravity, the 2-charge solutions always preserve 1 4 of the supersymmetry, regardless of the choice of signs. On the other hand, in N = 1 nine-dimensional supergravity the 16 eigenvalues of the corresponding Bogomol'nyi matrix are a subset of the 32 given above. Specifically, they are given by
as can be seen from the discussion in section 2.1. The orbits on the three-dimensional charge lattice that are filled out by acting on a given solution are thus characterised by the quadratic invariant I. In particular, there are three inequivalent types of orbit, corresponding to the cases where I is positive, negative or zero.
the intersection of a string, associated with the harmonic function H 1 , and a gravitational pp-wave, associated with H 2 . The string has no associated naked singularity since H 1 has a positive coefficient in its r −6 term. Although r −6 in H 2 has a negative coefficient, this does not necessarily imply the existence of a naked singularity in the associated wave solution.To study this issue, let us concentrate on a particular solution lying on the space-like charge orbit which has a particularly simple ten-dimensional interpretation. If we act on the 2-charge solution (7.3) with the O(1,
d y 2 ,
2(7.17) where a and b are constants arising from the O(1, 2) rotation. For generic a and b, this solution describes the intersection of a wave and a string. The linear combination a 2 H 1 + b 2 H 2 becomes the harmonic function associated with the string.
compactified heterotic string theory can be easily generalised to lower dimensions. In D ≥ 6, the global symmetry group is O(10−D, 26−D), and the vector potentials form a (36−2D)dimensional representation. The orbits are characterised by an invariant quadratic form that generalises (7.9), and fall into the three categories of time-like, space-like and light-like orbits. Any single-charge solution supported by a Kaluza-Klein vector or a winding vector lies on a light-like orbit. A two-charge solution involving both a Kaluza-Klein vector and a winding vector can lie either on a time-like or a space-like orbit: When the two charges are of the same sign, and hence the solution has no naked singularity, the orbit is time-like. It does not cover the points in the charge lattice where the solution involves only charges that are carried by the Yang-Mills 2-forms. On the other hand, when the two charges are of opposite signs, and so the solution suffers from a naked singularity, the orbit is spacelike and the solution can be rotated to one supported purely by Yang-Mills field strengths. All the above solutions preserve half of the supersymmetry of the theory. (There are also time-like and space-like orbits for non-supersymmetric two-charge solutions. They can be obtained from the supersymmetric 2-charge solutions by reversing the relative sign of the two charges but keeping the mass unchanged. This phenomenon of supersymmetry being broken as a consequence of a sign change of a charge, while keeping the mass fixed, occurs also in maximal supergravities, but only for solutions with more than 3 charges [45,46,47].) In D = 5 there is an additional 2-form field strength that comes from the dualisation of the NS-NS 3-form field. This is a singlet under the O(5, 21) global symmetry, and there exists a 3-charge solution involving this singlet, a Kaluza-Klein vector, and a winding vector. In D = 4, the global symmetry is enlarged to O(6, 24) × SL(2, IR), and the theory allows 4-charge solutions.
In the Lagrangian (A.6), we have augmented the (10−D)-component vector φ introduced in (A.2) by appending φ 1 as its first component, so that now φ = (φ 1 , φ 2 , . . . , φ 11−D ). The dilaton vectors a 1 , . . . characterise the couplings of the dilatonic scalars φ to the various field strengths; their details can be found in[39,5], and they are given bya 1 = (1, −2 s) , a 1α = (1, f α − 2 s) , a 1αβ = (1, f α + f β − 2 s) , b α = (0, − f α ) , b αβ = (0, − f α + f β )vectors c α satisfy the orthogonality property c α · c β = 2 δ αβ . The dot product of a 1 or c with b αβ vanishes. In fact b αβ can also be written as b αβ = − c α + c β . For the 3-form F (3) and its dimensional reductions F (2)α and F (1)αβ , the dilaton vectors a 1 , a 1α and a 1αβ are precisely those introduced in [39,5] for the dimensional reduction of the 4-form field F (4) in D = 11 supergravity. (The 3-form F (3) here coincides with the NS-NS 3-form of type
1)β − A (0)βγ d γ (1) + B I (0)β dB ′ I
then evident that this matrix can only satisfy the given conditions if η has the form η = diag (−1, −1, . . . ,
the O(2p, 2q) orbits span the coset spaceO(2p, 2q) U (p, q).(B.4)For the next examples, we consider two classes of coset where W is real, but now leaves invariant the antisymmetric metric Ω defined by Ω = diag (σ, σ, . . . , σ) , (B.5)
W 0 =
0(−1, −1, . . . , −1 2p , 1, 1, . . . , 1 2q) .(B.6)
W 0
0= diag (−σ, −σ, . . . , subgroup is defined by matrices Λ satisfying both Λ T Ω Λ = Ω and Λ T W 0 Λ = W 0 , and it is thus clearly SP (2p, IR) × SP (2q, IR). The coset in this case is thereforeSp(2p + 2q, IR) Sp(2p, IR) × Sp(2q, IR). (B.9)
W 0
0= diag (−1, −1, . . . . , −1 m , 1, 1, . . . , 1 p−m , −1, −1, . . . . , −1 n , 1, 1, . . . , 1 q−m ) . (B.11)
Noting that any non-degenerate symmetric matrix S can be diagonalised under the action S → Λ T S Λ where Λ is some O(p, q) matrix, we see that every O(p, q) orbit for W2.13,
2.14), under the O(p, q) transformations
W −→ Λ T W Λ .
(2.15)
satisfying (2.13, 2.14) passes through a point where W is a diagonal matrix. For diagonal
matrices, (2.13) implies directly that the diagonal elements are all ±1. Thus we may
characterise the O(p, q) orbits by a fiducial matrix
It is cosets of this type that will be principally relevant in our subsequent discussion. We discuss some further examples of classes of cosets in Appendix B.2.3 Global O(10 − D, 10 − D + N) symmetries from dimensional reductionWe saw in subsection 2.1 that the bosonic sector of ten-dimensional simple supergravity coupled to N abelian gauge multiplets gives rise, when reduced on S 1 , to a theory with a18)
This follows from the fact that tr(W η) is manifestly O(p, q)-invariant, and that tr(W 0 η) can
be determined by inspection from (2.12) and (2.16). Note that if we consider the particular
fiducial matrix
W 0 = diag (1, 1, . . . , 1) ,
(2.19)
corresponding to m = n = 0, then the O(p, q) orbits will describe the coset
O(p, q)
O(p) × O(q)
.
(2.20)
global O(1, N + 1) × IR symmetry in nine-dimensions. In order to bring out the geometrical
structure, we have exploited the fact that the coset O(1, N + 1)/O(N + 1) could be viewed
as a hypersurface in the flat Minkowski-signature space IR 1,N +1 . In this subsection, we
shall extend the discussion to the lower-dimensional theories obtained by compactifying on
1 )
1where K denotes the generators of the maximal compact subalgebra of G, and G s is a so-called Solvable Lie Algebra, comprising a subset of the Borel generators of G. To be specific, it comprises the non-compact Cartan generators H nc , together with the subset of the positive-root generators that has strictly positive weights under H nc . (Clearly if G weremaximally non-compact, in which case all the Cartan generators would be non-compact,
G s would comprise the entire Borel subalgebra.)
this accounting that the Yang-Mills gauge symmetry is "fully Higgsed," i.e. that the gauge symmetry is maximally broken by giving vacuum expectation values to the various E 8 × E 8 adjoint representation scalar fields. This breaks the E 8 × E 8 group down to its Cartan subgroup, (U (1))16 . Thus we get 16 further vector potentials and 16 × 3 = 48 axionic scalars from the Yang-Mills sector. Comparing with the K3 reduction of M-theory as described above, we see that we have a total of 22 vector potentials in each case, and 58 scalars in each case. From the K3 reduction we have a 3-form potential in D = 7, while from the T 3
Table 1 :
1The correspondence between M-theory and the heterotic fields in D = 7
19 )
19which would not allow the massive fields to be set to zero.In the simplest cases, such as Kaluza-Klein reduction on a torus, such dangerous terms cannot occur and so the truncation is indeed consistent. This follows from a simple grouptheoretic argument: if all the massless fields are singlets under some global symmetry group, while all the massive fields are non-singlets, then the massless fields cannot provide sources for the massive ones[40]. In the toroidal case, the group in question is the global U (1) n symmetry of the n-torus. The massless fields are all uncharged, and are hence singlets underthe U (1) factors, while the massive fields are charged. To put it another way, products of the
zero-mode harmonics on the torus (which are all constant) cannot generate non-zero-mode
harmonics.
For the K3 reduction, the situation is less clear-cut. It would seem now to be quite
conceivable that the product of zero-mode harmonics could generate non-zero-mode ones,
since even the zero-mode harmonics are not now in general covariantly-constant. Corre-
spondingly, one might expect that the massive fields could now have equations of motion
of the form (5.19). It was argued in [41, 42] that, in K3 or Calabi-Yau compactifications of
supergravities, the source terms must in fact necessarily involve derivative couplings, of the
generic form
T 3 ,
3and we shall show that the results are consistent with the previous ones. We shall use the relations between charges under Kaluza-Klein dimensional reduction given in Ref./ [16]. To begin with, let us consider the heterotic string in ten dimensions. Making the gravitational constant κ H and the string tension α ′ explicit, we may write the low-energy effective action as
16 )
16Although the O(1, 2) global symmetry can rotate the charges so that the Kaluza-Klein electric charge Q 2 vanishes, one finds that the Kaluza-Klein vector A (1) can never become of pure gauge form upon rotation from any non-trivial starting value. This implies that if a generic O(1, 2) rotation of the solution (7.3) is oxidised to D = 10, the metric will necessarily have a wave-like character; it is given by
It is necessary, once again, to translate between the 2 ≤ α ≤ 11 − D notation and the 1 ≤ i ≤ 10 − D notation.
It was shown in[29] that time-like and space-like reduction steps commute.
For simplicity, we are considering single-centre isotropic solutions here. The discussion that follows can be immediately generalised to multi-centre solutions, where Hi = 1+ a q a i | y− y i a | , with the potential ω written in terms of * dHi in the standard way.
Note that our sign convention is the opposite of the one used in[21].
AcknowledgmentsWe are grateful to P.S. Aspinwall, G.W. GibbonsAppendices A Toroidal reduction of the heterotic stringOur starting point is the low-energy effective Lagrangian for the bosonic sector of the tendimensional heterotic string. In this paper, we have considered the duality relations of the dimensionally reduced and "fully Higgsed" heterotic string. That is, after dimensional reduction, we have considered vacuum configurations in which the adjoint-representation scalars arising in the dimensional reduction are assigned general expectation values corresponding to vanishing potential energy. Such a general Higgs vacuum causes a spontaneous breakdown of the E 8 × E 8 Yang-Mills gauge symmetry to its maximal torus subgroup U (1) 16 , whose algebra coincides with the Cartan subalgebra of E 8 × E 8 . It is thus the set of dimensional reductions of this U (1) 16 Abelian Yang-Mills supergravity that principally concern us here. For simplicity, we may begin by retreating to D = 10 and replacing the full Yang-Mills-supergravity theory by its Abelian U (1) 16 maximal-torus sub-theory. The bosonic sector of this D − 10 Lagrangian is given by[59]where G I (2) = dB I (1) are the sixteen U (1) gauge field strengths, and F (3) = dA (2) + 1 2 B I (1) ∧dB I(1). The dilaton φ in the standard convention is in fact equal to −φ 1 .We perform the toroidal reduction using the notation and conventions of[39,5]. The metric is reduced according to the ansatzThe vectors s and f α satisfyHere, we are using φ to denote the (10 − D)-component vector φ = (φ 2 , φ 3 , . . . , φ 11−D ).Note that we have reserved φ 1 as the (negative of) the original dilaton in D = 10. The
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| []
|
[
"Synchronous Hybrid Message-Adversary",
"Synchronous Hybrid Message-Adversary"
]
| [
"Danny Dolev [email protected].†email:[email protected]. ",
"Huji ",
"Eli Gafni "
]
| []
| []
| The theory of distributed computing, lagging in its development behind practice, has been biased in its modelling by employing mechanisms within the model mimicking reality. Reality means, processors can fail. But theory is about predicting consequences of reality, hence if we capture reality by "artificial models," but those nevertheless make analysis simpler, we should pursue the artificial models.Recently the idea was advocated to analyze distributed systems and view processors as infallible. It is the message delivery substrate that causes problems. This view not only can effectively emulate reality, but above all seems to allow to view any past models as synchronous models. Synchronous models are easier to analyze than asynchronous ones. Furthermore, it gives rise to models we haven't contemplated in the past. One such model, presented here, is the Hybrid Message-Adversary. We motivate this model through the need to analyze Byzantine faults. The Hybrid model exhibits a phenomenon not seen in the past. * Recently, in[2], it was shown that a read-write wait-free asynchronous system can be modelled by a synchronous message passing system with a message adversary. This was a step beyond [5, 6] that equated shared-memory and message-passing when at most a minority of the processors are faulty. The work in[2], not only equated message-passing and shared-memory for the potential of n − 1 faults, but in addition the message-passing system was synchronous. And, it paid handsomely with an almost trivial derivation of the necessity condition of asynchronous computability[23].Here, as a first step, we extend the study of [2] to the m-resilient read-write asynchronous system, and find the message adversary m-AD in the synchronous message passing model that makes the synchronous system equivalent to the asynchronous one. m-AD is a system in which in each synchronous round m-AD chooses m processors and can remove any of the messages they send, subject to the condition that for the two messages exchanged between a pair of processors from the chosen m processors in the round, it removes at most one message.This follows from the simple realization that in [2], when we take a system of two processors, instead of viewing the adversary as one that at each round can remove one message of the two sent, we could equivalently say that the adversary chooses a processor and can remove the message it sends. A simple semantic difference, that was not realized by us for a while.The structure of 1-AD was realized 3 decades ago in[28]but not pursued beyond m = 1. The celebrated transformation of shared-memory to message-passing in[5,6], which holds only for m < (1/2)n, is hard to cast as a message-adversary that chooses a set of processors and removes some of its messages. It is more than a decade later that the second author quoted in [4] discovered that after 3 rounds of the transformed system it can be viewed as a message-adversary.When solving colorless tasks ([9]), if m < (1/2)n, we can do away with this last condition that prevents m-AD from removing the two messages exchanged by faulty processors. For the rest of the paper we consider only colorless tasks, hence we allow m-AD a complete freedom in removing in a round any messages from the m processors it chose for the round. We call this type of adversary m-mobile omission faults.This paper deals with just a single colorless task. The colorless task we consider is the binary-vectorconsensus-task. To explain this task we first put it within the historical perspective of the development of the various versions of the set-consensus task.The ingenious original version of set-consensus task invented by Chaudhuri[13], was a version we call here, set-election: n processors start each with their own identifier (id) as its input, and each outputs a participating id, such that the number of distinct outputs should be smaller than n.Using topological arguments this problem was later proved to be unsolvable read-write wait-free[7,23,27]. About a decade and a half later a technically simple but profound paper [3] formulated the task of vector-set-consensus. In the vector-set-consensus task, the n processors are faced with n − 1 independent consensuses. Each processor outputs for at least one of the consensuses, such that, two processors that output for the same consensus satisfy validity and agreement for that consensus. A simple reduction to set-election showed the vector-set-consensus to be unsolvable read-write wait-free, and to be equivalent to set election.But maybe, had we restricted each of the consensuses in the vector-set-consensus task to be just binaryconsensuses, rather than multi-valued consensus, then the resulting binary-vector-set-consensus is solvable read-write wait-free? Precluding this possibility is equivalent to proving Generalized Universality [18], a major result, which succeeded only few years after the formulation of the vector-set-consensus task. | null | [
"https://arxiv.org/pdf/1605.02279v2.pdf"
]
| 8,848,598 | 1605.02279 | f421d6673a43570b6b6374c1f448cfbda22dc6ff |
Synchronous Hybrid Message-Adversary
5 Jul 2016
Danny Dolev [email protected].†email:[email protected].
Huji
Eli Gafni
Synchronous Hybrid Message-Adversary
5 Jul 2016regular submission, not eligible for the best student paper award
The theory of distributed computing, lagging in its development behind practice, has been biased in its modelling by employing mechanisms within the model mimicking reality. Reality means, processors can fail. But theory is about predicting consequences of reality, hence if we capture reality by "artificial models," but those nevertheless make analysis simpler, we should pursue the artificial models.Recently the idea was advocated to analyze distributed systems and view processors as infallible. It is the message delivery substrate that causes problems. This view not only can effectively emulate reality, but above all seems to allow to view any past models as synchronous models. Synchronous models are easier to analyze than asynchronous ones. Furthermore, it gives rise to models we haven't contemplated in the past. One such model, presented here, is the Hybrid Message-Adversary. We motivate this model through the need to analyze Byzantine faults. The Hybrid model exhibits a phenomenon not seen in the past. * Recently, in[2], it was shown that a read-write wait-free asynchronous system can be modelled by a synchronous message passing system with a message adversary. This was a step beyond [5, 6] that equated shared-memory and message-passing when at most a minority of the processors are faulty. The work in[2], not only equated message-passing and shared-memory for the potential of n − 1 faults, but in addition the message-passing system was synchronous. And, it paid handsomely with an almost trivial derivation of the necessity condition of asynchronous computability[23].Here, as a first step, we extend the study of [2] to the m-resilient read-write asynchronous system, and find the message adversary m-AD in the synchronous message passing model that makes the synchronous system equivalent to the asynchronous one. m-AD is a system in which in each synchronous round m-AD chooses m processors and can remove any of the messages they send, subject to the condition that for the two messages exchanged between a pair of processors from the chosen m processors in the round, it removes at most one message.This follows from the simple realization that in [2], when we take a system of two processors, instead of viewing the adversary as one that at each round can remove one message of the two sent, we could equivalently say that the adversary chooses a processor and can remove the message it sends. A simple semantic difference, that was not realized by us for a while.The structure of 1-AD was realized 3 decades ago in[28]but not pursued beyond m = 1. The celebrated transformation of shared-memory to message-passing in[5,6], which holds only for m < (1/2)n, is hard to cast as a message-adversary that chooses a set of processors and removes some of its messages. It is more than a decade later that the second author quoted in [4] discovered that after 3 rounds of the transformed system it can be viewed as a message-adversary.When solving colorless tasks ([9]), if m < (1/2)n, we can do away with this last condition that prevents m-AD from removing the two messages exchanged by faulty processors. For the rest of the paper we consider only colorless tasks, hence we allow m-AD a complete freedom in removing in a round any messages from the m processors it chose for the round. We call this type of adversary m-mobile omission faults.This paper deals with just a single colorless task. The colorless task we consider is the binary-vectorconsensus-task. To explain this task we first put it within the historical perspective of the development of the various versions of the set-consensus task.The ingenious original version of set-consensus task invented by Chaudhuri[13], was a version we call here, set-election: n processors start each with their own identifier (id) as its input, and each outputs a participating id, such that the number of distinct outputs should be smaller than n.Using topological arguments this problem was later proved to be unsolvable read-write wait-free[7,23,27]. About a decade and a half later a technically simple but profound paper [3] formulated the task of vector-set-consensus. In the vector-set-consensus task, the n processors are faced with n − 1 independent consensuses. Each processor outputs for at least one of the consensuses, such that, two processors that output for the same consensus satisfy validity and agreement for that consensus. A simple reduction to set-election showed the vector-set-consensus to be unsolvable read-write wait-free, and to be equivalent to set election.But maybe, had we restricted each of the consensuses in the vector-set-consensus task to be just binaryconsensuses, rather than multi-valued consensus, then the resulting binary-vector-set-consensus is solvable read-write wait-free? Precluding this possibility is equivalent to proving Generalized Universality [18], a major result, which succeeded only few years after the formulation of the vector-set-consensus task.
Introduction
The various tasks above have the obvious analogue when we change n − 1 to k < n − 1. Indeed, Chaudhuri [13] proposed the problem in the context of m-resilient system, and asked whether the system can solve m-set consensus. The BG-simulation in [7,9] showed that had her problem been solvable readwrite m-resiliently, then set-election could have been solved for m + 1 processors read-write wait-free, which is impossible. On the flip-side, a m-resilient system can trivially solve m + 1 election! But all these analyses were conducted in the context of benign failures. What if we have an asynchronous system with n processors and f < (1/3)n Byzantine failures? Obviously, Byzantine failures are more serious failures than omission failures, and since we cannot solve f -binary-vector-set-consensus in the n processors f -resilient model we cannot solve it in the asynchronous n processors system with f Byzantine faults. Although the Byzantine failures are on a fixed set of processors, if these processors just lie about say their inputs, they cannot be detected, and a processor has to move on after receiving n − f messages, lest the f it missed are the Byzantine set, which omitted messages. Hence f + 1-set consensus is a lower bound in this case too. But can we solve the f + 1-vector-binary-set-consensus problem, in n processors system with f < (1/3)n Byzantine faults like we could do it for the analogue f -resilient system?
To our knowledge it is the first time this question has been asked. This is not surprising since the notion of binary-vector-set-consensus can only really be understood on the background of [18]. The question was asked with respect to the election version [14], but not for the vector-set-consensus task, let alone the binary-vector-set consensus task, since these tasks had not been introduced as yet, then. Why wasn't it asked with respect to the vector-set-consensus? Since these are reducible to each other without introducing the ideas in [18]. It looked meaningless to recast the results in [14] for vector-set-consensus. We do not believe that the general multi-valued set consensus task with its 24 (at the time!) possible versions, or for that matter even the general vector-set-consensus has any bearing on the vector-binary-set-consensus, when we consider Byzantine faults. Hence we do not pursue this multi-valued route, since we do not know a reasonable formulation of it in the Byzantine case. Thus, we are left with the question of the vector-binaryset-consensus power of asynchronous Byzantine n, f system.
We thus investigate this new general type of system. A synchronous system with f -fixed Byzantine faults and in addition m-mobile omission faults. We show this system to be equivalent in its vector-binary-setconsensus power to the same system of less severe failure of omission rather than Byzantine. We therefore investigate the general model of f -fixed omission faults and m-mobile omission faults. We call such a system that intermingles fixed omission failure and mobile omission failure, an hybrid adversary.
We completely characterize the vector-set-consensus power of the synchronous hybrid fix/mobile omission system with respect to all set-consensus versions, since in the non-hybrid benign faults all these versions are equivalent.
We do not show a reduction between the two synchronous systems one with the f -fixed Byzantine failures and m-mobile omissions, and the other with both types of failures being omission failures. Byzantine processors can always alter their input and behave correctly there on. Thus, when inputs conflict there is no resolution as to who are the "good ones" and who are the "bad ones." Rather, we should consider only contexts in which such resolution is not needed. To show such a result, the way we speculate to do it is first to show that the two systems have the same vector-binary-set-consensus power, which we do here.
Obviously the set consensus power of the benign system with fixed omissions and mobile omissions is stronger than the analogue synchronous hybrid Byzantine system. Thus a lower bound to the former is a lower bound to the latter. To show equivalence we present an algorithm for the Byzantine system that gets the same consensus power as that of the benign system, explicitly.
Our way of proceeding first with the binary-vector-set-consensus-power is in line with our recent think-ing that systems that agree on set consensus tasks have the same power when other tasks are concerned. That is the thinking of: "set consensus tasks are the coordinates of any (reasonable) system," [15]. We obtain the result that an n, f, m omission system requires n ≥ (m + 1)f + 1 in order to solve the best value of set-consensus it can solve, m + 1. For lesser values of n it can solve set consensus m + j, when n ≥ f (m + j)/j + 1.
This is the first time that we see a model where its set-consensus power changes gradually with n. In the m-resilient shared-memory case, we can always solve m + 1-set consensus. In the message passing system when m > n/2 − 1 we cannot solve anything. Once we are below that threshold we can solve m + 1-set consensus. It is the combination of faults that give rise to this gradually changing power phenomenon.
Outline of the Paper
We first show and prove the synchronous analogue of the asynchronous m-resilient model, and show that some restrictions on the adversary can be removed in the case of colorless-tasks ( [9]).
We then characterize the hybrid n, f, m omission adversary for different combinations of these values. We use the [7] simulation to show that if a set of less than (m + 1)f + 1 processors can do m + 1 set consensus then m + 2 processors can do set-consensus wait-free contradicting [7,23,27].
Finally we show that the upper bound algorithm holds for the hybrid synchronous Byzantine system when dealing with binary-vector-set-consensus. For the upper bound we use the rotating coordinator algorithm [26,12].
Problem Statement and Models
We assume a set of n processors Π = {p 1 , p 2 , ..., p n }. The paper focuses on solving the k-vector-set consensus problem in the hybrid omission model SMPfm, to be explained below. , and it is one of the inputs at entry j for some participating processor. vset3: There exists an index j ∈ [1..k] such that for every two processors, p, and q, o
p [j] = o q [j].
Notice that requiring all to output for the same index j, is equivalent to asking just to output for some j, since processors can write their outputs, and then read and adopt outputs it sees. The first output written will be adopted by all.
Definition 2 (binary-k-vector-set consensus problem). Same as the above only that the vector of k initial inputs each processor has is a binary vector of 0's and 1's.
The model SMPfm, called hybrid omission n, f, m, is a synchronous point-to-point message passing system. We consider an adversary that can remove messages. Before the start of the algorithm the adversary chooses a set S f of f processors. In each round the adversary can remove some or all of the messages sent by S f . In addition in each synchronous round it can choose a set S m of m processors and remove some or all of the messages sent from S m . We call S f , the fixed set, and S m the mobile set.
Presenting models as "message adversary" enables us to easily deal with dynamic systems in which processors that presented an external erroneous behavior at one point to start behaving correctly later on, without the need to discuss what is their internal state when this happens.
Where is SMPfm coming from. The method in [2] transformed asynchronous wait-free SWMR shared memory into a synchronous message-passing adversary. This message passing adversary can be viewed, in hindsight, as an adversary that at each round chooses n − 1 processors and removes some of their messages so that between two processors it chose, at least one of the two messages sent between them is not removed.
In the asynchronous m-resilient SWMR iterated shared memory, which is equivalent to the classical m-resilient non-iterated model [8,21,22], at each iteration processors are presented with a fresh SWMR memory initialized to ⊥, and all write their cell and then snapshot the memory. The m-resiliency assumption entails that for each processor the snapshot returns at most m cells with the value ⊥.
We now claim this model is equivalent to the synchronous message passing system in which a message adversary chooses a set S m of m processors at a round and removes some of the messages they send. But, nevertheless, this removal is constrained by the condition that the adversary removes at most one of the messages sent between two members of S m . We call this system constrained-SMPm.
Obviously an iteration of the asynchronous SWMR iterated shared memory, simulates a synchronous round of constrained-SMPm.
In the reverse direction we notice that constrained-SMPm can simulate an iterated SWMR collect step in 2n − 1 rounds [2], since a round of constrained-SMPm is obviously a round of constrained-SMP n−1 . Hence we can now simulate the Atomic-Snapshot algorithm in [1]. Since in each round of constrained-SMPm processors "read" from at least n − m processors, the minimum size snapshot will be n − m.
Can we remove the constrain on constrained-SMPm, so that the message adversary chooses S m and can possibly remove all their messages violating the requirement that among pairs in S m at least one message survives? We can, when all we want is to solve colorless tasks and m < (1/2)n. When m < (1/2)n a processor that through the rounds hears that at least n/2 + 1 processors have heard from it, can be sure that in the next round all will, since one of these n/2 + 1 processors will not be chosen by the adversary in the next round. Hence we can see that at least n − m processors can progress simulating read write. What we lose is that if the adversary sticks with a fixed set S m those processors cannot communicate. But at least n − m processors will have outputs and S m can adopt theirs.
Our last system of interest is SBMPfm. This system is like SMPfm only that the adversary can now tamper with messages from the fixed set S f . Thus, it is, in a sense, like having f fixed byzantine faults and m mobile omission faults.
The Lower Bound
Theorem 1. For the hybrid omission n, f, m, if n < f (m + 1) + 1, there is no algorithm to solve (m + 1)vector-set-consensus.
Proof. The proof is based on a simulation that uses constructions similar to those used in [19,20]. The details appear in the references, but nevertheless we sketch the construction of the simulation.
W.l.o.g. by way of contradiction an algorithm exists with all processors sending messages to all. Take m + 2 BG processors [19,20] simulating one round before moving to the next. In each round a simulator decides by safe-agreement [7,9] whether a message sent is received or not. A BG processor will claim that a message was removed if it does not know the simulated local state of the processor that sends the message.
Since we are dealing with SMPfm we can equivalently deal with the election-version of set-consensus. Initially, every simulator tries to install its input value as the input to all simulated processors. At most m+1 safe agreements modules may be blocked and the corresponding processors cannot be simulated as sending messages. Thus, to proceed, when a simulator does not know the local state of a processor, then it will try to reach agreement that the message was removed.
But in the meantime, the safe agreement for this processor might be resolved, hence other processors may contend that a message was sent. If we do these message delivery safe agreements for a processor proceeding negatively (a message was not sent) from the highest index receiver to the lowest, and in the opposite positive direction, from low index to high, for the message sent, at most one safe-agreement about a single message from this processor may be blocked at the index that the positives and the negatives meet. This will manifest itself as a message to some processor that we do not know whether it was removed or received, and therefore we do not know the local state of this potentially receiving processor.
Thus, the initial possible lack of knowledge about the m + 1 inputs may propagate to at most m + 1 omission failures from round to round. Thus we get an execution of a synchronous system n, m + 1 with n processors and m + 1-omission faults.
Now we move to the ramification of this simulation. If there exists an algorithm for the system n, f, m such that each run of n, m can be viewed as a run of n, f, m, then the algorithm that obtains m + 1-vectorset consensus for n, f, m will be an algorithm for m + 1-vector-set consensus for the system n, m + 1, contradicting [7,23,27].
Hence, it must be that there exits a run of n, m + 1 that cannot be explained as a run of an algorithm for m + 1-vector-set consensus of a synchronous hybrid system n, f, m.
The system n, f, m can have m+1 omission failure in a round by choosing each round m+1 processors from which to remove messages as to simulate a round of n, m + 1. Observe that at least one of these m + 1 processor chosen at a round has to be on the account of the f fixed processors, since in n, f, m we have only m-mobile omission faults at a round rather than m + 1.
Thus, we want to show that for n small enough, n < f (m + 1) + 1, given a run of n, m + 1 we can allocate f -fixed processors that explain the run as a run of n, f, m.
We take an infinite run of n, m + 1 and divide it into chunks of n rounds. In the first round processors 1, 2, . . . , m + 1 omit, in the second round 2, 3, . . . m + 2 omit, etc, with wrap around at round n. Then repeat the same for the second chunk etc.
Thus, if we take any k chunks like this, a processor appears in all the chunks (m + 1)k times each at a different round. Thus, since we have f fixed faults, the largest number of rounds we can justify with these fixed faults is f (m + 1)k. But if nk > f (m + 1)k we will not be able to justify it as an n, f, m run.
Obviously for any n ≤ f (m + 1) we will be able to justify it as an n, f, m run. Our example made the repetition of each processor equal. If it is unequal we will attribute the processors that are at the top f in ranking of repetitions as the fixed set and will able to justify any run of n, m + 1 as a run of n, f, m.
The lower bound proof implies that if we take n ≥ f (m + 1) + 1 we will not be able to explain the run as an n, m + 1 run. What is left is to show that indeed for n ≥ f (m + 1) + 1 we have an algorithm that obtains (m + 1)-vector-set consensus.
An easy repetition of the lower bound arguments above show that if n < f (m + j)/j + 1 we cannot solve m + j-set consensus.
Binary-k-vector-set Consensus
For simplicity of exposition we will use only the binary-k-vector-set consensus even for SMPfm as we know we can solve the multi-valued one using [18]. To show the phenomenon of consensus power growing gradually with n given fixed f and m, we assume for convenience that m + f < n/2 and f < n/4. The algorithm rely on the idea of the rotating coordinator [26,12]. Only that in hindsight we know that each phase of the implementation hides a COMMIT ADOPT protocol [17]. 1 Hence, we first pause the presentation and show how we solve COMMIT ADOPT in SMPfm and then in SBMPfm.
Commit-Adopt implementation for SMPfm, and SBMPfm
In the COMMIT ADOPT protocol each processor invokes COMMIT ADOPT with an initial value. Each processor, p, returns as an output a pair v p , e p , where e p ∈ {COMMIT, ADOPT}. COMMIT ADOPT ensures that:
CA1: If all processors invoke COMMIT ADOPT with the same value, then every processor, p, returns with that value and with e p = COMMIT. CA2: If a processor, p, returns with e p = COMMIT, then for any processor q, v q = v p .
A COMMIT ADOPT algorithm in [17] is given in a wait-free shared-memory model. To use this algorithm in different models we either implement the shared-memory in the model, or just show an implementation that comply with the properties that make COMMIT ADOPT algorithm in shared-memory work. The properties are to have two iteration where at most one value will be observed as a proposal to commit in the second round, and if a processor views only commit proposal in the second round, then any other processor in the second round will observe a proposal to commit.
In SMPfm if all processors start with the same bit, every processor will get all messages of the same bit and there will be at least n/2 + 1 of them. A processor that receives all same bit, will propose in the second round to commit that bit. Obviously, since majority sent same bit, no other processor will propose to commit a different bit.
In the second round, a processor commits if it obtains at least n/2 + 1 proposals to commit. A processor that does not propose to commit does not send a message. Obviously if one processor receives at least n/2 + 1 proposals to commit, others cannot miss all these proposals and will see at least one proposal to commit, and hence will adopt that bit.
In SBMPfm we have to worry about the f processors whose messages the adversary is allowed to tamper with. Thus, we cannot require to see all messages of the same bit, since then the adversary can prevent commit in the case that all started with the same bit. Nevertheless we know that if all started with the same bit then a processor will get at least n/2 + 1 of that bit, and at most f of the complement bit.
Hence, we use this test in order to propose commit at the second round. Since in the worst case the complement bit was send by "correct" processors, we nevertheless are left with more than quarter of correct processors whose input is that bit. Hence, this set of processors in the first round will prevent any other processor to propose to commit the complement bit, as the size of the set is greater than f . A processor that does not propose to commit does not send anything in the second round.
In the second round for a processor to "read" a commit proposal it is enough if it obtains at least n/4 commit proposals of the same bit. To commit, a processor needs to receive again at least n/2 + 1 proposals of commit of the same bit (We can now ignore commit of the different bit since there can be anyway at most f of them). Again we can argue that in the second round, if any processors commits, we must have at least n/4 correct processors which send commit proposal of that bit and hence all will at least adopt that bit. Figure 1 and Figure 2 present both versions of the COMMIT ADOPT protocol.
Binary-k-vector-set Consensus Protocol
We first focus on the case of k = m+1. For the discussion below assume that n = f (m+1)+1. The idea of the protocol is to run in parallel the basic process for each of the m + 1 entries in the binary-(m + 1)-vectorset consensus. The process below will ensure that in each entry in the output vector different processors never produce conflicting outputs, and that for at least one entry all processors report an output. We assign m+1 coordinators to each phase of the protocol, one per entry in the vector. The coordinators play a role in a specific round of sending messages in each phase, as described below. We run the protocol for f (m + 1) + 2 phases, each takes three rounds of message exchange.
For a given entry all processors repeatedly exchange their values in each phase. Each phase begins with concurrently running a COMMIT ADOPT on the current values of all processors. In the first phase processors use their initial input values, and later phases the values computed by the end of the previous phase.
Following the COMMIT ADOPT step the coordinator of the current phase broadcasts the value it obtained from the recent COMMIT ADOPT.
A processor that completed the recent COMMIT ADOPT with COMMIT ignores the coordinator's message and updates its value to be the committed value of the COMMIT ADOPT. A processor that did not complete the recent COMMIT ADOPT with COMMIT adopts the value it receives from the coordinator, if it received a value, if no value was received it remains with its original value.
By the end of this value updating we are guaranteed that if the coordinator was correct when it sent its coordinator's value, then all processors will end up holding identical values. The reason is that the COM-MIT ADOPT properties imply that if a processor returns from the COMMIT ADOPT with COMMIT, all processors return from the COMMIT ADOPT with identical values, so this is also the value the correct coordinator sends. If this is not the case, every processor adopts the coordinator's value, and again they hold identical values.
Observe that our assumptions are that all processors receive the values from all correct processors, even when the adversary chooses to change their messages. Therefore, the current coordinator received the correct value from the COMMIT ADOPT as every other processor.
Once all processors hold identical values, in all future phases the COMMIT ADOPT at each processor will return COMMIT with that value, no matter who the rest of the coordinators are.
The above basic process is repeated for f (m + 1) + 2 phases for all the (m + 1) entries of the (m + 1)vector-set consensus. After the end of the last phase each processor reports output for every entry in which the latest COMMIT ADOPT returned COMMIT. The COMMIT ADOPT properties imply that there will not be any conflict on output values in any index. Moreover, for each entry for which in one of the first f (m + 1) + 1 phases there happened to be a correct coordinator sending its value, all processors return that value for that entry.
What we are left to discuss is why there would always be at least one correct coordinator in at least one entry in at least one phase. Although this argument is repetition of the argument in the lower-bound section, we repeat it here. Observe that we assign to each phase m + 1 different coordinators. The assignment of coordinators to phases is such that for n = f (m + 1) + 1 each one appears in exactly f + 1 different phases. This implies that there can be at most f (m + 1) phases in which at least one of the coordinators assigned to entries in that phase is from the fixed set f . Look at a phase in which no coordinator is from the fixed set. The adversary can drop messages from at most m of the coordinators that send their coordinators' values in that phase. Therefore, there should be an entry at which the coordinator sending the coordinator's value is correct.
Observe that for binary values one can replace the condition in Line 10 of Figure 2 to #maj > 0, since if no processor returns with COMMIT then non-Byzantine processors have sent both 0 and 1. For non-binary values, instead of testing for #min we need to test for non-max values, and can replace the condition in Line 10 of Figure 2 to #maj > f . Moreover, one can add a filtering in Line 3 to filter out values that do not conform with what one expects to receive, since they are clearly being sent by Byzantine processors. Similar filter can be used in Line 9 of Figure 3. We do not have any use for such a filtering in the protocols of the current paper.
One can generalize the lower bound proof of Section 3 for m + k-vector-set consensus algorithm for k ≥ 1 to obtain a lower bound of n > f (m+k) k . The DYNAMIC (m + 1)-VECTOR-SET CONSENSUS protocol of Figure 3 can be changed accordingly and will run in f (m+k) k + 2 phases. When increasing the number of of processors by m one can device a protocol that runs for only f (k−m) + 2 phase. Thus for let v(j) be the initial input to consensus index j, 1 ≤ j ≤ m + 1; /* the input values*/ 3: /* the permutation over the set of n processors */ 4: let si,j = p , where = (i + j − 1) mod n, for 1 ≤ i ≤ f (m + 1) + 2, 1 ≤ j ≤ m + 1; v(j),ê(j) =COMMIT ADOPT(v(j)); 8: if p = si,j then sendv(j) to all; /* the rotating coordinator for index j sends its value */ 9:
let v (j) be the value received from si,j; /* ⊥ if no value was received */ n ≥ f (m + 1) + 1 we can solve m + 1-set consensus. for n ≥ f (m + 2)/2 + 1 we can solve m + 2 set consensus, etc. All the formal proofs appear in the appendix.
Conclusions
We introduced a new type of distributed-system call Hybrid-Message-Adversary. It gives rise to phenomenon never seen before of set-consensus power changing gradually even though the various types and number of faults do not change. In our mind the only notion of set consensus that makes sense in the Byzantine setting is that of binary-vector-set-consensus. To our knowledge we are the first to ask this question, and in fact we are still at loss but not far, we suspect, from an answer. Next, we can imagine message adversary with mobile Byzantine faults and combinations thereof with omission fixed or mobile faults etc.. In fact, the analogue of message adversary with mobile Byzantine faults was studied in the domain of Cryptography under the name of mobile viruses, transient or proactive faults [24,10,25,11], but none looked at the relative power of tasks, let alone the set-consensus power.
Why should we? We recently [15] started to suspect that "natural systems" can be characterized by their set consensus power. Thus if this is proved and we equate the set-consensus power of synchronous Byzantine of f faults and SBMPfm with m = f , then they will be equivalent.
Definition 1
1(k-vector-set consensus problem). Every processor has a vector of k initial inputs. Each processor, p, returns a vector o p [1..k] that satisfies: vset1: If for any index j ∈ [1..k] the inputs of all processors are the same, then every processor returns that value for index j. vset2: For every entry j ∈ [1..k] and for every two processors, p, and q, if o p [j] = ⊥ and o q [j] = ⊥ then o p [j] = o q [j]
Algorithm 1 :
1COMMIT ADOPT (v p ): The Commit Adopt protocol for omission faults 1: round 1: send vp to all; /* executed by processor p */ 2: round 2: if all "bits" received are the same then send the "bit" to all else do not sendbe the bit received and let #maj be the number of processors that sent it; > n/2 then set ep :=COMMIT else set ep :=ADOPT; 7: if #maj > 0 then set vp := maj; /* otherwise remain with the original vp */ 8: return vp, ep . Algorithm 2: COMMIT ADOPT (v p ): The Commit Adopt protocol for Byzantine faults 1: round 1: send vp to all; /* executed by processor p be the bit received the most and let#maj be the number of processors that sent it; 4: let min be the bit received the least and let its number be #min; 5: round 2: if #min ≤ f and #maj > n/2 then send maj else do not send; be the bit received the most and let #maj be the number of processors that sent it; > n/2 then set ep :=COMMIT else set ep :=ADOPT; 10: if #maj ≥ n/4 then set vp := maj; /* otherwise remain with the original vp */ 11: return vp, ep .
loop for each of the k indices, 1 ≤ j ≤ m + 1, all of them in parallel
v (j) = ⊥ then set v(j) := v (j); /* adopt coordinator si,j value*j, 1 ≤ j ≤ m + 1: ifê(j) = COMMIT then op(j) := v(j) else op(j) := ⊥; 14: return op[1..m + 1].
The COMMIT ADOPT protocol is the concept behind last two rounds of the original Gradecast[16] protocol.
Appendix A ProofsWe prove the correctness of COMMIT ADOPT for the SBMPfm model. The proof for the SMPfm is much simpler.Claim 1. Assume n > max(2f + 2m, 4f ) and SBMPfm adversary. The protocol ofFigure 2meets the COMMIT ADOPT requirements.Proof. To prove property CA1 observe that if all processors send an identical value in Line 1 ofFigure 2then each processor can receive at most f different values, from processors the adversary tampers with their messages. Therefore, at every processor the test in Line 5 will bring it to propose this identical value. Exactly for the same reasons each processor, p, will set up e p to be COMMIT and will return the identical value.For CA2, assume that processor p proposes a value in the second round of the protocol. Thus, the test in Line 5 is true. Denote by maj p , #maj p andn p the parameters processor p used in Line 5 and V p the multiset ofn p values it received, and respectively for another arbitrary processor q. Consider two cases. The first case is when m ≥ f . In our model V p may not contain at most the m values from omission faults and some of the values from the Byzantine processors. Since #maj p > n/2 and #maj p ≥n p − f there are at least m + 1 values in V p from processors that are correct at the current sending step and sent v p to all. V q should contain all these processors' values. The assumption that m ≥ t implies that V q contains at least f + 1 value of v p , therefore, Line 5 ofFigure 2implies that if decides to propose a value it should be that v q = v p . Now consider the case that m < f . In this case the condition n > 4f implies that n/2 > 2f. This implies that V p contains at least n/2 − f values from processors that are correct at the current sending step that sent v p , thus at least f + 1, and the rest of the above arguments hold.The above argument implies that no two non-Byzantine processors send different values in the second round. Assume that processor p commits in Line 9. Let v be a value committed to. Assume the above notations for the messages received in the 2nd round. Neither V q nor V p can contain more than f non ⊥ values the are not v. Thus, the protocol implies that maj p = v. By definition we know that |V q ∩ V p | ≥ n/2, all of which are correct at the current message sending step. Since #maj p ≥n p − f and #maj p > n/2, V q contains at least n/2 − f copies of v, thus more than n/4 and more than f . Claim 2. Assume n > max(2f +2m, 4f ) and SBMPfm adversary. In the DYNAMIC (m+1)-VECTOR-SET CONSENSUS protocol, if all processors' initial values to a given index j are the same, they output that value for that index at the end.Proof. In DYNAMIC (m + 1)-VECTOR-SET CONSENSUS processors update their initial values either in Line 10 or Line 11 ofFigure 3. By Claim 1 we know that any invocation of it will result with all processors obtaining the identical value and with evaluation "commit", therefore Line 11 will never be executed, and at the end of the protocol all will produce an identical value.Claim 3. Assume n > max(2f +2m, 4f ) and SBMPfm adversary. In the DYNAMIC (m+1)-VECTOR-SET CONSENSUS protocol, if for some index j at some phase the coordinator p is correct when executing Line 8, then after executing Line 10 or Line 11 ofFigure 3in that phase, v(j) is identical at all processors.
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If there is any processor that completes the COMMIT ADOPT in Line 7 of phase r with "commit" and the other if none. In the first case, by the COMMIT ADOPT properties we know that all processors complete the COMMIT ADOPT with the same value (including the faulty processors). Therefore, the value p will send in Line 8 is the same, and therefore all processors will complete Line 9 with the same value. This implies that no matter which of the two lines, Line 10 or Line 11, any processor executes, all obtain the same value. some phase r. Consider two cases. Let p be the correct processor executing Line 8 of some index j. In the second case, no processor completes Line 7 with "commit. and therefore all will execute Line 11 and will obtain the value the correct sender sent when it was correct while executing Line 8Proof. Let p be the correct processor executing Line 8 of some index j, in some phase r. Consider two cases. If there is any processor that completes the COMMIT ADOPT in Line 7 of phase r with "commit" and the other if none. In the first case, by the COMMIT ADOPT properties we know that all processors complete the COMMIT ADOPT with the same value (including the faulty processors). Therefore, the value p will send in Line 8 is the same, and therefore all processors will complete Line 9 with the same value. This implies that no matter which of the two lines, Line 10 or Line 11, any processor executes, all obtain the same value. In the second case, no processor completes Line 7 with "commit", and therefore all will execute Line 11 and will obtain the value the correct sender sent when it was correct while executing Line 8.
If n > max f (m + 1), 4f, 2(m + f ) and assuming SBMPfm adversary, the DYNAMIC (m + 1)-VECTOR-SET CONSENSUS protocol satisfies the properties of k-vector-set consensus. Theorem 2. for k = m + 1Theorem 2. If n > max f (m + 1), 4f, 2(m + f ) and assuming SBMPfm adversary, the DYNAMIC (m + 1)-VECTOR-SET CONSENSUS protocol satisfies the properties of k-vector-set consensus, for k = m + 1.
The definition of s i,j = p , where = (i + j − 1) mod n, for 1 ≤ i ≤ f (m + 1) + 2, 1 ≤ j ≤ m + 1, assigns each processor p to exactly m + 1 times in the first f (m + 1) + 1 phases. The fix set of f processors appear in at most f (m + 1) phases. Therefore, there is a phase in which none of them appear. Since omission faults can silence at most m processors in that phase, there is a correct processor executing Line 8 of Figure 3 in that phase. Let j be the index of that processor. By Claim 3 we know that by the end of that phase all processors hold the same values in their v(j). order to prevent having any correct coordinator in a phase all m + 1 processors assigned to be coordinators in that phase need to either be in the fixed set of f faulty, or one of the m processors that suffer from omission in that phase. From the next phase on, until the end of phase f (m + 1) + 2 all will complete Line 7 in Figure 3 with "commit", and will end up having the same value in the j-th index. Moreover, for any other index j. two processors that assign a value to that index, assign the same value, since it is the value they completes Line 7 of index j with "commit. and it is an identical value. The remaining property of DYNAMIC k-VECTOR-SET CONSENSUS obviously holds as wellProof. In order to prevent having any correct coordinator in a phase all m + 1 processors assigned to be coordinators in that phase need to either be in the fixed set of f faulty, or one of the m processors that suffer from omission in that phase. The definition of s i,j = p , where = (i + j − 1) mod n, for 1 ≤ i ≤ f (m + 1) + 2, 1 ≤ j ≤ m + 1, assigns each processor p to exactly m + 1 times in the first f (m + 1) + 1 phases. The fix set of f processors appear in at most f (m + 1) phases. Therefore, there is a phase in which none of them appear. Since omission faults can silence at most m processors in that phase, there is a correct processor executing Line 8 of Figure 3 in that phase. Let j be the index of that processor. By Claim 3 we know that by the end of that phase all processors hold the same values in their v(j). From the next phase on, until the end of phase f (m + 1) + 2 all will complete Line 7 in Figure 3 with "commit", and will end up having the same value in the j-th index. Moreover, for any other index j , two processors that assign a value to that index, assign the same value, since it is the value they completes Line 7 of index j with "commit", and it is an identical value. The remaining property of DYNAMIC k-VECTOR-SET CONSENSUS obviously holds as well.
| []
|
[
"SCATTERING STATES AND SYMMETRIES IN THE MATRIX MODEL AND TWO DIMENSIONAL STRING THEORY",
"SCATTERING STATES AND SYMMETRIES IN THE MATRIX MODEL AND TWO DIMENSIONAL STRING THEORY"
]
| [
"Antal Jevicki ",
"João P Rodrigues ",
"André J Van Tonder ",
"\nPhysics Department\nPhysics Department and Center for Nonlinear Studies\nBrown University\n02912ProvidenceRIUSA\n",
"\nPhysics Department\nUniversity of the Witwatersrand\n2050WitsRSA\n",
"\nand Physics Department and Center for Nonlinear Studies\nBrown University\n02912ProvidenceRIUSA\n",
"\nUniversity of the Witwatersrand\n2050WitsRSA\n"
]
| [
"Physics Department\nPhysics Department and Center for Nonlinear Studies\nBrown University\n02912ProvidenceRIUSA",
"Physics Department\nUniversity of the Witwatersrand\n2050WitsRSA",
"and Physics Department and Center for Nonlinear Studies\nBrown University\n02912ProvidenceRIUSA",
"University of the Witwatersrand\n2050WitsRSA"
]
| []
| We study the correspondence between the linear matrix model and the interacting nonlinear string theory. Starting from the simple matrix harmonic oscillator states, we derive in a direct way scattering amplitudes of 2-dimensional strings, exhibiting the nonlinear equation generating arbitrary N-point tree amplitudes. An even closer connection between the matrix model and the conformal string theory is seen in studies of the symmetry algebra of the system. | 10.1016/0550-3213(93)90474-4 | [
"https://arxiv.org/pdf/hep-th/9209057v1.pdf"
]
| 16,264,926 | hep-th/9209057 | 39ea470bc491555d605c2db2c1b7b9b46a428e29 |
SCATTERING STATES AND SYMMETRIES IN THE MATRIX MODEL AND TWO DIMENSIONAL STRING THEORY
arXiv:hep-th/9209057v1 16 Sep 1992 September 1992
Antal Jevicki
João P Rodrigues
André J Van Tonder
Physics Department
Physics Department and Center for Nonlinear Studies
Brown University
02912ProvidenceRIUSA
Physics Department
University of the Witwatersrand
2050WitsRSA
and Physics Department and Center for Nonlinear Studies
Brown University
02912ProvidenceRIUSA
University of the Witwatersrand
2050WitsRSA
SCATTERING STATES AND SYMMETRIES IN THE MATRIX MODEL AND TWO DIMENSIONAL STRING THEORY
arXiv:hep-th/9209057v1 16 Sep 1992 September 1992
We study the correspondence between the linear matrix model and the interacting nonlinear string theory. Starting from the simple matrix harmonic oscillator states, we derive in a direct way scattering amplitudes of 2-dimensional strings, exhibiting the nonlinear equation generating arbitrary N-point tree amplitudes. An even closer connection between the matrix model and the conformal string theory is seen in studies of the symmetry algebra of the system.
Introduction
Matrix models provide not only a novel formulation of low dimensional string theory but one which is integrable and exactly solvable. They lead to exact string equations in D < 1 and a wealth of results for the free energy and correlation functions [1]. The largest model understood so far is the two dimensional string theory, described by a simple dynamics of a (matrix) harmonic oscillator [2][3][4][5][6][7][8].
While the numerical results follow straightforwardly, the physical picture encoded in the matrix model is however not seen directly. It is exhibited once appropriate physical observables (collective fields) [3] are identified. For the tachyon one has the bosonic collective field defining perturbative states. While the matrix model is linear, the collective field exhibits a nonlinear interaction which leads to nontrivial physical scattering processes [4]. A fermi liquid description can be used to give a semiclassical picture of the scattering [5]. The field theory is integrable:
it exhibits an infinite sequence of conserved charges and an even larger symmetry of W ∞ generators [6].
String theory is however most naturally described in terms of the world sheet string coordinates and associated conformal vertex operators [9]. These indeed exhibit similar symmetries [10] and can be seen to give the same correlation functions. Except for the coincidence of various results a closer connection between the matrix model description and the string language is still lacking.
It is the purpose of this paper to address this problem and give a more direct relationship between linear states of the matrix model and nonlinear scattering states of string theory. One has the matrix harmonic oscillator The question is then how this simple set of exact matrix model states translates into nontrivial string scattering states. Continuing on the constructions begun in [6], we shall explain a correspondence in section 2 and describe a simple derivation of general string scattering amplitudes using the integrable states. As such we exhibit how the nonlinear string dynamics follows from the linear and integrable matrix dynamics.
L = 1 2 Tr Ṁ 2 + M(t) 2 ,(1.
In section 3 we discuss the symmetry algebra of the theory. We demonstrate there a close connection between the matrix W ∞ generators and those of the conformal string theory. In particular we shall see that the collective (tachyon) field representation of these operators is nothing but the representation defined in the conformal approach by Klebanov in [11].
From States to Scattering
Strings in two dimensions are described by the coordinates X µ ≡ (X, φ), where X is (usually) taken as spacelike and φ is the nontrivial Liouville coordinate [9].
One has translation invariance in the X direction (this is the time coordinate of the matrix model, i.e.X = it) and only asymptotic translation invariance in φ due to an exponential wall µ e − √ 2φ . The vertex operators of the lowest string modes (massless tachyons) are
V ± = e ipX+β±φ , β ± = − √ 2 ± |k|.
(2.1)
Only the + branch describes physical scattering states. The − operators grow at φ → −∞ and are termed "wrongly dressed". For scattering one has left movers (as initial states) and right movers (as final states) respectively denoted by
T (±) k = e ikX+(− √ 2+|k|)φ ,(2.2)
where ± = sign k.
States of the matrix model can be seen to be in close correspondence. In particular, of (1.2) half of the states have a scattering interpretation as
B (−) −k |0 = Tr (P − M) −ik = |k ; in , B (+) k |0 = Tr (P + M) ik = |k ; out ,(2.3)
This physical interpretation will arise once the spatial (Liouville) coordinate is identified. This was understood to be related to the eigenvalue index of the matrix variable. The physical world is the positive real axis with a barrier at the origin, and so one only considers an in state that is left moving and an out state that is right moving.
The identification of physical states and of the extra Liouville momentum is seen in a transition to the collective field theory language [3]. This transition can be summarized [3][4][5][6] by the following set of replacement rules:
M → x, P → α(x, t), Tr → dx 2 π α+ α− dα.
(2.4)
The matrix hamiltonian then becomes
H = 1 6 dx 2 π α + 3 − α − 3 − 1 2 dx 2 π x 2 α + − α − ,(2.5)
describing a scalar field φ(x, t) and its conjugate Π(x, t), with α ± = ∂ x Π ± πφ.
The collective representation exhibits in addition to the time t a spatial dimension x. One has a classical background field πφ 0 = x 2 − 2µ, which induces a reparametrization of the new spatial coordinate to τ = dx πφ0(x) , or
x(τ ) = 2µ cosh(τ ), πφ 0 (τ ) = 2µ sinh(τ ).
(2.6)
Asymptotic translations in τ are scale transformations of x since
x(τ ) ∼ µ 2 e τ .(2.7)
Indeed, in addition to time translation the collective Lagrangian transforms covariantly under scale transformations
x → λ x, α(x, t) → 1 λ α(λ x, t), H → 1 λ 4 H. (2.8)
This symmetry is the origin of a second (spatial momentum) quantum number p τ .
In linearized approximation with
φ(x) = φ 0 (x) + ∂ x ψ(x), p(x) = −∂ x Π(x), ψ(τ ) = ψ(x), p(τ ) = πφ 0 p(x), α ± (x) = ± πφ 0 + 1 πφ 0α ± (τ ). (2.9)
one has right-left moving massless modes (tachyons)
α ± (τ, t) = f (t ∓ τ ) = ± ∞ −∞ dk α ± k e −ik(t∓τ ) ,(2.10)
satisfying (∂ t ± ∂ τ )α ± = 0. (2.11) with the energy momentum values α ± −k : p 0 = k, p τ = ±k. (2.12) The exact states of the matrix model are directly translated into the field theoretic representation. We have as exact tachyon eigenstates 13) introduced by Avan and one of the authors in [6]. Using the Poisson brackets 2.14) and one has eigenstates with ip 0 = ±n. Defining
T (±) n = dx 2 π dα (α ± x) n = dx 2 π (α ± x) n+1 n + 1 ,(2.{α(x), α(y)} = 2π δ ′ (x − y) one easily shows {H, T (±) n } = ±n T (±) n(p τ = scale dimension − 4,(2.15)
one has p τ = −2 + n.
These states stand in comparison with the vertex operators of conformal field
theory T (±) p ≡ e i p X+(− √ 2+|p|)ϕ ↔ dx 2 π (α ± x) n+1 n + 1 ↔ Tr (P ± M) n .
(2.16)
The tachyon vertex operators with opposite (Liouville) dressing correspond to singular operators in the matrix model
e i p X+(−2−|p|) ϕ ↔ dx 2 π (α ± x) 1−n 1 − n ↔ Tr (P ± M) −n .
(2.17)
We have now described a one to one correspondence between the matrix model states and string states. Scattering amplitudes can be derived immediately once this correspondence is understood.
We note that the collective field theory seemingly introduces a degeneracy. For each state of the matrix model one can define two states in collective field theory since we can replace P → α ± (x, t). Each of the separate fields α + or α − can be used to define states with the above quantum numbers. In particular
dx 2 π (α + ± x) 1±ik 1 ± ik and dx 2 π (α − ± x) 1±ik 1 ± ik
both have the same quantum numbers
p 0 = k, p τ = −2 ± ik.
These have to be identified, up to a phase factor. It can be shown (below) that boundary conditions fix the phase factor to be −1. So one has
dx 2 π (α ∓ ± x) 1∓ik 1 ∓ ik = − dx 2 π (α ± ± x) 1∓ik 1 ∓ ik ,(2.α ± (x) = ± x ∓ 1 x µ ∓α ± (τ ) + 1 x 2 terms, (2.19) we shall find the relation ∞ −∞ dτ e ±ikτα ± µ = − ∞ −∞ dτ ik ± 1 e ∓ikτ 1 +α ∓ µ ik±1 − 1 . (2.20)
This functional equation relating left and right moving waves of the collective field was shown to represent a solution to the scattering problem in [8]. Here we exhibited how this nonlinear scattering equation emerges directly from the exact oscillator states. The fact that the left and the right hand side of the equation are interpreted as eigenstates of collective field theory implies also the following: a complete quantization procedure was given [4] for the field theory Hamiltonian, involving normal ordering and the subtraction of counterterms. The same procedure can be applied to the states and will lead to a fully quantum version of the scattering equation.
The main ingredient in obtaining the scattering equation are the proper boundary conditions. Let us now elaborate on this question. The issue of boundary conditions is of paramount importance in a correct treatment of the spectrum within the collective approach. In QCD-like unitary matrix models, it is well known that as the system moves from a strong coupling regime to a weak coupling regime where the classical density of states φ 0 has only finite support, Dirichlet boundary conditions must be imposed on the shifted field ψ(τ ). This is essentially due to the fact that φ 0 (τ = 0) = φ 0 (τ = L → ∞) = 0, and in this way the time independence of the original constraint condition dx φ = N is preserved [3]. For c = 1 strings, this "constraint" equation determines the value of the cosmological constant. Therefore, apart from problems of consistency, a choice other than Dirichlet boundary conditions would result in a time dependent cosmological constant. Notice that this implies that in a density variable description of "wall" scattering, the "wall" at τ = 0 is rigid. A creation-annihilation basis that automatically enforces Dirichlet boundary conditions on the scalar field ψ is defined by the expansioñ
α ± (τ ) = ± ∞ −∞ dk |k| e ± i kτ a k , a −k ≡ a † k , [a k , a † k ] = δ(k − k ′ ).
(2.21)
We could equally well have chosen the "left-right" basis (2.10). Once one expresses a scalar theory with fields satisfying boundary conditions in a left-right basis, there is a standard problem, also present in the critical open string: the functions e ikτ are not orthogonal over the half line, and therefore the computation of Fourier coefficients require some modification. To this standard problem there is a standard solution [12]: one notices that the definitions of all the fields in (2.10) naturally extend to negative values of τ . Therefore we extend the definition of the fields
from 0 ≤ τ < ∞ to −∞ < τ < ∞ by requiring ψ(−τ ) = − ψ(τ ), α ± (−τ ) = −α ∓ (τ ). (2.22)
In other words, the fields of interest to us are the fields defined on the full line which are odd (in coordinate free form) under the involution τ → −τ . This point of view has been extensively used in works relating critical open string amplitudes to those of the closed string [13]. One can then compute Fourier coefficients ofα + , say:
∞ 0 dτ 2π e ∓ ikτα ± (τ ) − ∞ 0 dτ 2π e ± ikτα ∓ (τ ) = ±α k . (2.23)
We can now reformulate the problem as follows: suppose we introduce the arbitrary left-right expansion (2.10). Equation (2.22) is then equivalent to
α − k = −α + k . (2.24)
Physically, this simply means that in order to preserve the boundary conditions of the system, if a right mover is created then a left mover must also be created with amplitude minus one, and similarly for annihilation operators. This means that the Dirichlet boundary conditions cause the left and right movers to combine into standing waves, which are perturbative tachyon states in the matrix model. Now, in terms of the matrix variables (2.21) described above, this condition is immediately built into the expansion of the fields. However, for asymptotic incoming and outgoing states, which are naturally defined on the full line, the analogue of condition (2.24), imposed on the the exact states of the system, leads to the nonlinear scattering matrix.
We now concentrate on T (+) ik and introduce the following notation to represent the two degenerate states described previously
T (+) ik = dx 2 π α+ α− dα (α + x) ik ≡ T + ik (+) − T + ik (−) = dx 2 π (α ∓ + x) ik+1 ik + 1 − dx 2 π (α ± + x) ik+1 ik + 1 .
(2.25)
The equation relating left and right moving fields reads
T (+) ik,+ = 1 2 ∞ −∞ dτ 2π πφ 0 ik + 1 (πφ 0 + x) + α + (τ ) πφ 0 ik+1 − x ik+1 = − 1 2 ∞ −∞ dτ 2π πφ 0 ik + 1 (−πφ 0 + x) + α − (τ ) πφ 0 ik+1 − x ik+1 = −T (+) ik,− .
(2.26)
The range of integration has been extended as described above equation (2.23).
This is a restatement of equation ( As x ≫ √ 2µ we wish to express this condition in terms of the asymptotic fieldŝ α ± defined in equation (2.19), using the asymptotic behaviour (2.7). We remind ourselves thatα − (t + τ ) is an incoming left-moving wave andα + (t − τ ) is the outgoing, right-moving wall scattered wave. In the asymptotic description the fact that, from the collective field theory point of view, the "wall" at τ = 0 is rigid, is not immediately built into the definition of the fields. This condition has to be imposed on the exact states of the system, i.e., equation (2.27) must be satisfied.
Expressing the exact states in terms of the variables (2.19) we get
T (+) ik,+ − C (+) + = 1 ik + 1 dx 2 π (α + + x) ik+1 − (πφ 0 + x) ik+1 = √ 2µ ik+1 ik + 1 ∞ −∞ dτ 8π e ikτ e 2τ ∞ j=0 Γ(ik + 2) Γ(ik + 2 − j) j! (−) j e −2jτ 1 −α + µ j − 1 , (2.28) − T (+) ik,− − C (+) − = − 1 ik + 1 dx 2 π (α − + x) ik+1 − (−πφ 0 + x) ik+1 = − √ 2µ ik+1 ik + 1 ∞ −∞ dτ 8π e −ikτ ∞ p=1 Γ(ik + 2) Γ(ik + 2 − p) p! α − µ p . (2.29)
Equating these expressions as required by the condition (2.27), and applying partial integrations and a Fourier transform, we obtain
∞ j=0 e −2(j−1)τ Γ(−∂ + 2) Γ(−∂ + 2 − j) j! (−) j e −2jτ 1 −α + (τ ) µ j − 1 = − ∞ p=1 Γ(∂ + 2) Γ(∂ + 2 − p) p! α − (−τ ) µ p .
(2.30)
We can now extract the asymptotic limit by letting τ → ∞. We find that on the left hand side only the j = 1 term contributes (lower order terms would correspond, on the right hand side, to terms dropped in the asymptotic definition (2.7)). As
τ → ∞ (−∂ + 1)α + (τ ) µ = − ∞ p=1 Γ(−∂ + 2) Γ(−∂ + 2 − p) p! ᾱ − (τ ) µ p ,(2.31)
whereᾱ − (τ ) ≡α − (−τ ). It follows that
α + (τ ) = − ∞ p=1 Γ(−∂ + 1) Γ(−∂ + 2 − p) p! 1 µ p−1ᾱ − (τ ) p .
(2.32)
This relation expressing left moving fields in terms of right moving ones is the result (2.20) for the scattering problem [8].
Symmetries
The spacetime field theory given by the collective field exhibits a large (W ∞ ) spacetime symmetry of 2-dimensional string theory [6]. The generators of this symmetry can be directly found or simply induced from the matrix model. There one has the invariant operators Tr (P r M s ), (3.1) which are closed under commutation. The field theory operators read
H n m = dx 2 π α m−n + m − n x m−1 (3.2)
and can be shown to satisfy the w ∞ algebra
[H n1 m1 , H n2 m2 ] = i [(m 2 − 1) n 1 − (m 1 − 1) n 2 ] H n1+n2 m1+m2−2 .(3.3)
Of particular relevance to us are the spectrum generating operators
O JM ≡ Tr (P + M) J−M (P − M) J+M ,(3.4)
which become, in the collective field theory representation
O JM = dx 2 π α+ α− dα (α + x) J+M +1 (α − x) J−M +1 . (3.5)
One sees that these are linear combinations of the basic w ∞ operators 3.6) and it follows that the spectrum generating algebra is precisely a w ∞ , i.e.,
O JM = H −2J−2 1 + 2M H −2j 2 + (2M 2 − J − 1) H 2J+2 3 + . . . .([ O J1,M1 , O J2,M2 ] = i [(M 2 − 1) J 1 − (M 1 − 1) J 2 ] O J1+J2,M1+M2 . (3.7)
This can also be shown directly from (3.5) by doing partial integrations [6].
There is a close connection between these w ∞ operators and the operators describing exact tachyon states of the field theory. Recall the one parameter family of operators
T (±) n = dx 2 π α+ α− dα (α ± x) n . (3.8)
Their commutators give the generators of the w ∞ algebra, i.e., one can show that
O JM = 1 2i (J − M + 2) (J + M + 2) [ T + J+M +2 , T − J−M +2 ]. (3.9)
The symmetry generators were also written down in the conformal field theory approach [10]. There is a close parallel with all of the matrix model relationships and the commutators are simply replaced by operator products. The above implies for example that the w ∞ generators are obtained as operator products of basic tachyon vertex operators. A closer correspondence is seen by comparing the above field theory forms with the representations deduced for the action of the symmetry generators on the tachyon module [11].
Fourier Expansion
We now consider the spectrum generating operators O JM of (3.5) in more detail.
We shall see that the correspondence with the conformal field theory results of [11] will then follow. To expand in terms of creation-annihilation operators we make the substitutions 3.10) in the spectrum generating operators (3.5). Applying partial integration to (3.5) and inserting the limits (3.10), one finds
α + = x +ᾱ + , α − = −x +ᾱ −(O JM = dx 2 π J+M +1 k=0 (−) k (J − M + 1)! (J + M + 1)! (J − M + 2 + k)! (J + M + 1 − k)! × × ᾱ J−M +2+k + (ᾱ + + 2x) J+M +1−k −(ᾱ − − 2x) J−M +1+kᾱJ+M +2−k − . (3.11)
The leading term inᾱ + is of orderᾱ J−M +2 + . The leading term inᾱ − seems to be linear inᾱ − . However, this is not true, as careful consideration shows that there are two terms linear inᾱ − which cancel. One might expect that in general something similar happens also for higher order terms inᾱ − . This indeed turns out to be the case. The easiest way to see this, is to do the partial integration of (3.5) in the other "direction". One finds
O JM = dx 2 π J−M +1 k=0 (−) k (J + M + 1)! (J − M + 1)! (J + M + 2 + k)! (J − M + 1 − k)! × × ᾱ J−M +1−k + (ᾱ + + 2x) J+M +2+k −(ᾱ − − 2x) J−M +1−kᾱJ+M +2+k − .
(3.12)
The terms inᾱ + andᾱ − in (3.11) and (3.12) must separately be equal, up to c-number terms of the form dx 2 π x 2J+3 . Substituting the change of variables (2.7), this becomes ∼ ∞ −∞ dτ 2π e (2J+3)τ , which can in general be argued to vanish after an analytic continuation τ → iτ (see below). It therefore follows that we can write the expansion
O JM = dx 2 π J+M +1 k=0 (−) k (J − M + 1)! (J + M + 1)! (J − M + 2 + k)! (J + M + 1 − k)!ᾱ J−M +2+k + × × (ᾱ + + 2x) J+M +1−k − dx 2 π J−M +1 k=0 (−) k (J + M + 1)! (J − M + 1)! (J + M + 2 + k)! (J − M + 1 − k)!ᾱ J+M +2+k − × × (ᾱ − − 2x) J−M +1−k .
(3.13)
Thus to lowest order in the fields, one finds
O JM = 1 J − M + 2 dx 2 π (2x) J+M +1ᾱJ−M +2 + − 1 J + M + 2 dx 2 π (−2x) J+M +1ᾱJ+M +2 − .
(3.14)
Now, applying the change of variables (2.7), i.e.,
x = µ 2 e τ , α ± → dτ dxᾱ ± ,(3.15)
one finds that the leading order expression for the charges is given by
O JM = 2 J+1 µ M J − M + 2 dτ 2 π e 2M τᾱJ−M +2 + − (−) J−M +1 2 J+1 µ −M J + M + 2 dτ 2 π e −2M τᾱJ+M +2 − . (3.16)
Expanding in right and left moving modes
α + = ∞ −∞ dkᾱ(k) e −ik(t−τ ) , α − = ∞ −∞ dkβ(k) e −ik(t+τ ) (3.17)
and applying the rotation τ → iτ , k → −ik, we find that in terms of the analytically continued oscillators
α(k) ≡ᾱ(−ik), β(k) ≡β(−ik) (3.18)
the charges have the form
O JM = 2 J+1 µ M J − M + 2 i dk 1 . . . dk J−M +2 × × α(k 1 ) . . . α(k J−M +2 ) δ k i + 2M − (−) J−M +1 2 J+1 µ −M J + M + 2 i dp 1 . . . dp J+M +2 × × β(p 1 ) . . . β(p J+M +2 ) δ p i + 2M . (3.19)
We emphasize that this is the expression for the charges to lowest order in the fields, which corresponds to the leading order in µ. The full expression (3.13) has corrections in 1/µ that are higher order polynomials in the fields. In the remainder of the discussion we do not consider these corrections.
Defining
a(k) ≡ α(k), b(p) ≡ β(p), a † (k) ≡ α(−k)/k, b † (p) ≡ β(−p)/p (3.20) satisfying [a(k), a † (k ′ )] = δ(k −k ′ ), [b(p), b † (p ′ )] = δ(p−p ′ ),
we have the expressions of [11] (up to an inessential difference in normalization), plus additional contributions. To see these, note that in addition to the term
2 J+1 µ M i ∞ 0 dk ∞ 0 dk 1 . . . dk J−M +1 × × k a † (k) a(k 1 ) . . . a(k J−M +1 ) δ k i − k + 2M (3.21)
found in [11], we in general also have terms of higher order in the creation operators.
The next term would be, for example
2 J+1 µ M (J − M + 1) i ∞ 0 dk dk ′ ∞ 0 dk 1 . . . dk J−M × × kk ′ a † (k) a † (k ′ ) a(k 1 ) . . . a(k J−M ) δ k i − k − k ′ + 2M . (3.22)
If M < 0, we also get an additional contribution of the form
2 J+1 µ M J − M + 2 i ∞ 0 dk 1 . . . dk J−M +2 × × a(k 1 ) . . . a(k J−M +2 ) δ k i + 2M .
(3.23)
These additional contributions have to be included in order to obtain a representation of the algebra (3.7). The reason for this is that terms of the type (3.22), commuted with terms of the type (3.23), give additional contributions of the type (3.21), which are needed to again obtain a member of the algebra on the right hand side. This effect cannot be produced by only using terms of the type (3.21).
To see where the representation (3.21) fails, one has to take careful account of the regions of momentum integration. For example, if one were to use only terms of the type (3.21) one would find for the commutator
O M M , O 1 2 ,− 1 2 = ∞ 0 dk 1 dk 2 (−2k 1 − 2k 2 − 4M) (k 1 + k 2 + M − 1 2 )× × a † (k 1 + k 2 + M − 1 2 ) a(k 1 ) a(k 2 ) + ∞ 0,k1+k2> 1 2 dk 1 dk 2 (2k 1 + 2k 2 − 1) (k 1 + k 2 + M − 1 2 )× × a † (k 1 + k 2 + M −1 2
) a(k 1 ) a(k 2 ). , were it not for the fact that the regions of integration do not match. It is now not difficult to see how to fix the representation (3.21). Simply remove the restrictions on the ranges of integration, i.e., take them to be ∞ −∞ dk instead of ∞ 0 dk. This solves the problem on a formal level, and imposing the reality conditions a −n = na † n ≡ α −n , we recover our full representation (3.19).
One can now ask whether the Ward identities derived in [11] for the tachyon scattering amplitudes will be affected by these corrections. As we will show in the next section, they will not be affected.
Ward Identities
One can now identify the spectrum generating operators as we did for the tachyons by comparing quantum numbers as in (2.18), or alternatively, by imposing Dirichlet boundary conditions as in (2.26). One simply requires 3.25) which implies that to leading order (3.26) This identification will, in practice, be very useful in explicit calculations of Ward identities, as will be seen below.
O JM,+ = −O JM,− ,(O JM = 2i 2 J+1 µ M J − M + 2 dk 1 . . . dk J−M +2 × × α(k 1 ) . . . α(k J−M +2 ) δ k i + 2M = 2i (−) J−M +1 2 J+1 µ −M J + M + 2 dp 1 . . . dp J+M +2 × × β(p 1 ) . . . β(p J+M +2 ) δ p i + 2M .
The "bulk" scattering amplitudes only involve fixed, discrete values of the outgoing momenta. This can be interpreted in our formalism as follows: Imposing the above identification of quantum numbers, one has
T (−) 2M ≡ O M −1,M = 2i (−) 2M +1 2 µ M β(2M) = 2i (2µ) M dk 1 . . . dk 2M +1 α(k 1 ) . . . α(k 2M +1 ) δ( k i − 2M).
( 3.27) Thus, in terms of the oscillators α(k) and β(p) defined in (3.10) and (3.18), we find an S-matrix that is different from the one we previously calculated in terms of the "asymptotic" variables (2.7). In particular, to leading order an out state 0|β(2M) is (2M+1)-linear in α, so that a correlation function β(p)α(k 1 ) . . . α(k N )
can only be nonvanishing to this order if
p = N − 1. (3.28)
Except for an overall factor of 1 2 , due to different normalization of the momentum, this agrees with the "sum rule" stated in [11]. Now, to see how the Ward identities can be derived in our formalism, note that if one has an operator O that annihilates the vacuum from the left and the right, i.e., were considered in the analysis of [11], one might expect corrections to the Ward identities derived in [11] due to our extra terms such as (3.22) and (3.23 ∼ a N (aaa + a † aa + a † a † a) (a † ) N +1 . (3.33) It immediately follows that only the term linear in a † contributes. This argument generalizes to the identity for expressing N → 1 amplitudes directly in terms of 2 → 1 amplitudes, where the relevant charge is Q N/2−1,−N/2−1 . The conclusion is therefore that for the purpose of deriving these Ward identities, it is sufficient to consider only the terms (3.21) linear in the creation operators, as was done in [11].
0|O = 0 = O|0 ,(3.
Finally, as an example, we calculate the Ward identity relating 3 → 1 amplitudes to 2 → 1 amplitudes. Using the identification (3.26), we have
Q1 2 ,− 1 2 = 4 √ 2 i √ µ ∞ 0 dk 1 dk 2 dk 3 k 1 a † (k 1 ) a(k 2 ) a(k 3 ) δ(−k 1 + k 2 + k 3 − 1)
= 4 2µ i ∞ 0 dp 1 dp 2 p 1 b † (p 1 ) b(p 2 ) δ(−p 1 + p 2 − 1). (3.34) up to terms that we have argued to be irrelevant. Inserting this into the general formula (3.31), for p = 1 and k 1 + k 2 + k 3 = 2, we obtain the Ward identity
b(2) a † (k 1 ) a † (k 2 ) a † (k 3 ) = 1 µ (k 1 + k 2 − 1) b(1) a † (k 1 + k 2 − 1) a † (k 3 ) + cyclic.
It is also possible to derive recursion relations in our formalism using methods similar to those used in [11]. The argument roughly goes as follows: The operators
O N N ≡ dx dα (α + x) 2N +1 (α − x)
have quantum numbers p x = N = p τ , while T k has p x = k, p τ = −1 + k. Adding these, it follows that the commutator [ O N N , T k ] has quantum numbers p x = N +k, p τ = −1 + (n + k), and should therefore be identified with T k+N , i.e.,
[ O N N , T k ] ∼ T k+n . (3.36)
In conlusion, we stress that the matrix model w ∞ generators, when expanded,
give the conformal field theory expressions of [11] and the associated bulk Ward identities. But in addition they also contain higher corrections in 1/µ, which could be used to derive improved Ward identities which would give the full amplitudes.
A
± ≡ P ± M =Ṁ ± M, A ± (t) = A ± (0) e ±t being standard creation-annihilation operators. In terms of these one easily writes down the eigenstates of the hamiltonian A + A − . For example, the one-parameter set A ± n = Tr (P ± M) n gives imaginary eigenvalues with energies ǫ n = ∓i n. Real energy states are obtained by analytic continuation n = ik: B (±) k = Tr (P ± M) ik . (1.2)
2.18). Since the c-number contributions C (+) ± to the above operators are are the same, we rewrite this condition as T is a necessary consequence of Dirichlet boundary conditions. To linear order, it is straightforward to show that equation(2.27) is equivalent to equation (2.24).
the charge O N +M,M has quantum numbers p x = M, p τ = N + M. Thus it follows that [ O N +M,M , T k1 . . . T kN+1 ] has quantum numbers p x = M + k i ,p τ = −1 + (m + k i ), so that we have to identify [ O N +M,M , T k1 . . . T kN+1 ] ∼ T M + N+1 i=1 ki .
one can write, commuting O respectively to the left and to the right[β(p), O] α(k 1 ) . . . α(k N ) = β(p) [O, α(k 1 ) . . . α(k N )] .This equation expresses the Ward identities, and for suitable choices of the charges O, can be used to derive recursion relations relating scattering amplitudes.In general, however, our representation(3.19) of the charges O JM have terms of the type aa . . . a or a † a † . . . a † , and therefore would fail to annihilate the vacuum from either the left or the right. Also, in addition to the terms (3.21), which29)
then, starting from the expectation value
β(p) O α(k 1 ) . . . α(k N )
(3.30)
(3.31)
). However, counting numbers of creation and annihilation operators, one sees that indeed only terms of the type (3.21) contribute to the Ward identities. For example, consider the Ward identity relating N + 1 → 1 amplitudes to N → 1 amplitudes. The relevant charge is O 1 ∼ aaa + a † aa + a † a † a, and we should take the momentum of the out state to be p = N −1. The out state is then, from our previous discussion, α(k 1 ) . . . α(k N +1 )2 ,− 1
2
of order
0| β(p) ∼ 0|a N ,
(3.32)
so that one can write (3.30) as
β(p) O 1
2 ,− 1
2
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| []
|
[
"Adversarial-Residual-Coarse-Graining: Applying machine learning theory to systematic molecular coarse-graining",
"Adversarial-Residual-Coarse-Graining: Applying machine learning theory to systematic molecular coarse-graining"
]
| [
"Aleksander E P Durumeric \nDepartment of Chemistry\nInstitute for Biophysical Dynamics, and Computation Institute\nJames Franck Institute\nThe University of Chicago\n60637IllinoisUSA\n",
"Gregory A Voth \nDepartment of Chemistry\nInstitute for Biophysical Dynamics, and Computation Institute\nJames Franck Institute\nThe University of Chicago\n60637IllinoisUSA\n"
]
| [
"Department of Chemistry\nInstitute for Biophysical Dynamics, and Computation Institute\nJames Franck Institute\nThe University of Chicago\n60637IllinoisUSA",
"Department of Chemistry\nInstitute for Biophysical Dynamics, and Computation Institute\nJames Franck Institute\nThe University of Chicago\n60637IllinoisUSA"
]
| []
| We utilize connections between molecular coarse-graining approaches and implicit generative models in machine learning to describe a new framework for systematic molecular coarse-graining (CG). Focus is placed on the formalism encompassing generative adversarial networks. The resulting method enables a variety of model parameterization strategies, some of which show similarity to previous CG methods. We demonstrate that the resulting framework can rigorously parameterize CG models containing CG sites with no prescribed connection to the reference atomistic system (termed virtual sites); however, this advantage is offset by the lack of explicit CG free energy at the resolution obtained after integration over the virtual CG sites. Computational examples are provided for cases in which these methods ideally return identical parameters as Relative Entropy Minimization (REM) CG but where traditional REM CG is infeasible. | 10.1063/1.5097559 | [
"https://arxiv.org/pdf/1904.00871v1.pdf"
]
| 90,243,502 | 1904.00871 | 1fe51c3f8d200858e0a01aa7058426fa4ac4d8a1 |
Adversarial-Residual-Coarse-Graining: Applying machine learning theory to systematic molecular coarse-graining
Aleksander E P Durumeric
Department of Chemistry
Institute for Biophysical Dynamics, and Computation Institute
James Franck Institute
The University of Chicago
60637IllinoisUSA
Gregory A Voth
Department of Chemistry
Institute for Biophysical Dynamics, and Computation Institute
James Franck Institute
The University of Chicago
60637IllinoisUSA
Adversarial-Residual-Coarse-Graining: Applying machine learning theory to systematic molecular coarse-graining
(Dated: 2 April 2019)
We utilize connections between molecular coarse-graining approaches and implicit generative models in machine learning to describe a new framework for systematic molecular coarse-graining (CG). Focus is placed on the formalism encompassing generative adversarial networks. The resulting method enables a variety of model parameterization strategies, some of which show similarity to previous CG methods. We demonstrate that the resulting framework can rigorously parameterize CG models containing CG sites with no prescribed connection to the reference atomistic system (termed virtual sites); however, this advantage is offset by the lack of explicit CG free energy at the resolution obtained after integration over the virtual CG sites. Computational examples are provided for cases in which these methods ideally return identical parameters as Relative Entropy Minimization (REM) CG but where traditional REM CG is infeasible.
I. INTRODUCTION
Classical atomistic molecular dynamics (MD) simulation has provided significant insight into many biological and materials processes. [1][2][3][4] However, its efficacy is often restricted by its computational cost: for example, routine atomic resolution studies of biomolecular systems are currently limited to microsecond simulations of millions of atoms. Phenomena that cannot be characterized in this regime often require investigation using modified computational approaches. Coarse-grained (CG) molecular dynamics can be effective for studying systems where the motions of nearby atoms are highly interdependent. [5][6][7][8][9] By simulating at the resolution of CG sites or "beads", each associated with multiple correlated atoms, biomolecular processes at the second timescale and beyond can be accurately probed. High-fidelity CGMD models often depend on flexible parameterizations; as a result, the design of systematic parameterization strategies is an active area of study (e.g., methods and applications in references .
The CGMD models considered in this article are similar to their atomistic counterparts. They are comprised of point-mass CG beads, a corresponding CG forcefield, and a simulation protocol that produces Boltzmann statistics in the long-time limit. We restrict the bulk of our study to the parameterization of the CG effective force-field. Here, and in the remainder of the article, we refer to these models simply as CG models. We only consider the static equilibrium properties of these models, and not their dynamics. There are two nonexclusive classes of parameterization strategies for CG models of interest to this article: top-down and bottom-up approaches. [5][6][7] Top-down approaches aim to parameterize CG models to recapitulate specific macroscopic propa) [email protected] b) [email protected] erties, such pressure and partition coefficients, 33 while bottom-up methods attempt to parameterize CG models to reproduce the multidimensional distribution given by explicitly mapping each atomistic configuration (produced by a suitable reference simulation) to a specific CG configuration. [13][14][15][16][17] The distribution of this mapped system is produced via a Boltzmann distribution with respect to an effective CG Hamiltonian referred to as the many-body Potential of Mean Force (PMF).
Certain scientific inferences could be informally drawn from the fit CG force-field itself, assuming that the forcefield is constrained to intuitive low dimensional contributions (e.g., pairwise forces, such as in ref 34). For example, one could attempt to infer the effect of an amino acid mutation on protein behavior by considering how the approximated PMF differs when fit on reference wild type and mutant proteins simulations, similar to the analysis of low dimensional free energy surfaces. However, the primary use of CG models is typically based on their ability to generate CG configurations of a system of interest using their approximate force-field. 20,27,35,36 The computational similarity of CG models with their atomistic counterparts also allows CG models to be simulated using the same high performance software packages as those used in atomistic simulation. [37][38][39][40][41][42][43] As a result, the computational profile of CG models is often controlled by the same dominating factor as atomistic models: the calculation of the force-field at each timestep. 37,44,45 This cost provides additional motivation for specific low dimensional force-field contributions. However, there is no guarantee that a force-field characterized solely by traditional bonded and simple pairwise nonbonded terms either describes the true PMF of the CG variables or can accurately reproduce all observables of interest to the parameterization. [5][6][7] In the case of bottom-up methods, while typical approaches will produce the PMF in the infinite sampling limit when they are capable of representing any CG force-field, in practice each method creates a characteristic approximation (e.g., reproducing two-body at the expense of higher order correlations).
The compromises invoked by various bottom-up CG methods in realistic applications are critical to the utility of the resulting models. Certain methods focus on reproducing correlations dual to the potential form used; 11,12,46,47 for example, when using a pairwise nonbonded potential these methods recapitulate the radial distribution function of the target system. Other specific methods are characterized by attempting to reproduce both these dual correlations along with certain higher order correlations intrinsically connected to the CG potential. [13][14][15]18,46 The nature of the distributions approximated suggests three natural approaches for improving an inaccurate model: improve the CG force-field basis used, modify the CG representation, or select a different procedure to generate the CG force-field. The first two options are often a central part of the design of a systematic CG model; however, realistic systems, such as proteins, may not be well described by correlations which are typically connected to computationally efficient CG potentials coupled with appropriate CG representations. 7 More generally, the specific correlations critical to a reasonably accurate CG model may depend on the study at hand, and may be representable by simple force-fieldsbut only at the expense of other correlations connected to that potential form as dictated by a particular method. As a result, the diversity of possible applications motivates the creation of additional strategies for bottom-up CG modeling, each of which has different biases in the approximations it produces.
The task of generating examples (such as images) similar to a known empirical sample is of significant interest to the Machine Learning (ML) community. [48][49][50][51] The creation of an artificial process that can produce realistic samples often entails encoding an understanding of the true mechanism underlying the real world distribution; internal representations of an accurately parameterized generative model, such as neural network parameters, can be transferred for use in secondary tasks such as classification 52 or image retrieval. 53 The artificial samples produced by the models themselves have additionally shown value by providing novel molecular targets for synthesis 54,55 or as labeled images for training in classification or regression. 56,57 A substantial number of these complex applications utilize implicit generative models. 51,58-60 Implicit generative models, such as Generative Adversarial Networks (GANs), 58 are characterized by their lack of an explicit probability distribution, or an associated free energy, at the resolution they produce examples. 51 For example, a GAN may be trained to generate pictures containing human faces. 58 Each picture that could be generated has a parameterization specific probability of being a reasonable picture of a human face (admittedly, this probability is often very close to one or zero); however, the GAN itself does not have explicit knowledge of this probability. Instead, the GAN is simply characterized as a procedure which transforms random numbers from a simple noise distribution to images which follow the probability distribution of plau-sibly images. The methods used to parameterize (i.e., train) GANs therefore focus on the ability to critique a model distribution against reference samples without knowledge of the probability density function characterizing the model. This is in strong contrast to typical molecular simulation 1,61,62 which traditionally requires a known free energy surface to produce samples through molecular dynamics or Markov Chain Monte Carlo-and whose systematic parameterization techniques often naturally explicitly involve evaluation of the corresponding model free energy surface. 10,11,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] However, both methods are focused on accurately producing samples, or configurations, as their primary goal.
This article focuses on making this intuitive connection between GANs and molecular models explicit, allowing us to apply established insight from the adversarial community to bottom-up CG modeling, giving rise to new strategies for CG parameterization we term Adversarial-Residual-Coarse-Graining (ARCG). By doing so we facilitate the use of additional classes of CG model quality measures which may show promise in modifying the approximations characterizing the optimal CG model when using a constrained set of candidate potentials to represent the CG force-field. We additionally find that it is possible to decouple the resolution at which one critiques the behavior of the CG model and the resolution at which a CG force-field is required: as an example we describe a novel rigorous avenue to increase the expressiveness of bottom-up CG models through the use of virtual sites. Critically, we do not utilize a full GAN architecture to generate CG samples; rather, we utilize the supporting theory 58,63-68 to optimize traditional CG force-fields.
In this work we discuss formal connections between CG and GAN-type implicit generative models and provide an initial implementation of the resulting ARCG framework. Section II provides both an informal and a formal summary of the theoretical underpinnings, while section III provides details on a particular instance of ARCG and a public computational implementation. Section IV then provides results on three simple test systems, and section V outlines the consequences of the results and possible future studies. Section VI provides concluding remarks.
II. THEORY
The purpose of this section is to both informally describe and formally define ARCG, and to summarize connections between ARCG and previous CG parameterization methods. We begin by presenting an intuitive understanding of a specific form of ARCG to provide clarity for the subsequent mathematical description. We then follow by defining notation and the fine-grained/CG systems to which ARCG applies. We define ARCG and describe its estimation and optimization. We then move to decouple the resolution at which one critiques the CG model from the resolution native to the CG Hamiltonian, thereby generalizing our application to systems contain-ing virtual CG sites. We continue by discussing the corresponding challenges with momentum consistency, and we finish by summarizing ARCG's relationship to previous CG methods.
A. Informal Description of ARCG
Bottom-up CG models are parameterized to approximate the free energy surface implied by mapping finegrained (FG) configurations to the CG resolution. 6,7 Generally, this entails considering many different possible CG models (each, for example, characterized by a different pair potential) and their relationship to the reference FG data. Often, this is operationalized by creating a variational statement and searching for the CG model which minimizes it (for example, minimizing the empirical relative entropy between the CG model and FG data 17 ). After such a procedure is complete the modeler is well advised to visually inspect and compare the configurations produced by the selected CG model to the reference FG model. If the configurations are dissimilar, then the CG model is likely not adequate, and aspects of the variational statement or set of initial models considered must be modified and the parameterization process repeated.
It is natural to ask whether the final inspection of configurations produced by the FG and CG models can be intrinsically linked to the variational statement parameterizing the CG model. It is intuitive that for systematic CG parameterization methods derived from thermodynamic consistency 10,11,[13][14][15][16][17][18][19][20][21][22][23][24] that when an indefinite amount of samples are used and all possible CG models are considered that the optimized CG model will perfectly reproduce the mapped FG statistics, and as a result, the configurations produced by the FG and CG models will be indistinguishable. 69 However, in cases where perfectly reproducing the FG statistics is infeasible, it seems natural to ask if a model could be trained using this criteria of distinguishability directly.
While it could be possible in simple situations to use a human observer to intuitively rank CG models by considering the configurations they produce, this procedure quickly becomes subjective and untenable for complex models. A natural progression in method design is then to train a computer to distinguish CG models via their samples. One appropriate statistical procedure is classification, 70 where a computer attempts to differentiate individual configurations based on whether they are more likely drawn from either the CG or mapped FG data sets. The implied procedure for CG parameterization is then to optimize the CG model such that it is intrinsically difficult to complete this task: as a result, the computer will inevitably make many mistakes on average when attempting to isolate configurations characteristic to only the FG and CG data. One possible in-tuitive manifestation of ARCG theory concretely implements this procedure while maintaining clear connection to CG methods such as relative entropy minimization (REM). 17 Previous CG parameterization methods have used similar, but not identical, motivations to produce parameterization strategies. 24,28 ARCG theory serves to connect, clarify, and reframe these methods where possible while extending beyond the classification metaphor.
It is important to note that the task of classification is a variational procedure itself: 63,70 the ideal estimate of the true sources of a set of molecular configurations has a lower error than all other estimates. The optimization in classification searches over these various possible hypotheses. As a result, at each step of force-field optimization ARCG must perform this variational search over possible hypotheses, resulting in two nested variational statements in the full model optimization procedure: one required for classification, and the other for choosing the resulting CG model. Importantly, the irreducible error of the classifier can be explicitly linked to various f -divergences (e.g., relative entropy) between the mapped FG and CG distributions. 63 This suggests an equivalent formalism with which to view ARCG: the variational estimation of divergences. This alternate interpretation additionally illustrates how additional divergences, such as the Wasserstein distance, 68 can be estimated, even without a clear connection to classification. As a result ARCG theory is primarily treated through the lens of variational divergence estimation in the following sections.
The variational estimation intrinsic to ARCG affords an interesting extension to traditional parameterization strategies. Specifically, training a classifier requires only samples from the reference FG and CG model, and makes no reference to the CG force-field used to create said samples. As a result, CG samples can be mapped before being compared to the mapped reference FG samples. For example, additional particles may be introduced to facilitate complex effective interactions between the CG particles, and then may be mapped out before comparing to the mapped reference FG samples. Traditionally, applying such a mapping creates difficulties in parameterizing CG models as the free energy surface of the mapped CG system is unknown. However, this difficultly is sidestepped via the variational estimation of ARCG.
B. Model Definitions and Selection
We consider a FG probability density p FG ref and a mapping operator M which maps a FG configuration to a CG one. The FG simulation is constructed such that it produces samples from the Boltzmann distribution with respect to a FG Hamiltonian giving the following probability density:
p FG ref (r 3n , p 3n ) := Z FG ref,r −1 Z FG ref,p −1 exp −β n i=1 p 2 i 2m i + U FG ref (r 3n ) = p FG ref,p (p 3n )p FG ref,r (r 3n )(1)
where β is 1 k b T with the temperature T set by the simulation protocol, m i are the FG masses, r 3n and p 3n are the FG positions and momenta variables, and our partition functions are defined as expected 71 such that
Z FG ref,r = X FG r exp −βU FG ref (r 3n ) dr 3n (2) Z FG ref,p = X FG p exp −β n i=1 p 2 i 2m i dp 3n(3)
where the integrals are taken over the full domains of the position and momentum variables (denoted via X FG r and X FG p ). The application of the CG map M produces CG configurations which follow an implied probability distribution. M is constrained such that it is linear and can be decomposed into momentum and position components, i.e., M (r 3n , p 3n ) = [M r (r 3n ); M p (p 3n )], 72 implying a factorizable probability density
p ref (R 3N , P 3N ) := p ref,R (R 3N )p ref,P (P 3N ) over the CG variables defined as p ref,R (R 3N ) := X FG r p FG ref (r 3n )δ(M r (r 3n ) − R 3N )dr 3n (4) p ref,P (P 3N ) := X FG p p FG ref (p 3n )δ(M p (p 3n ) − P 3N )dp 3n .(5)
Bottom-up CG models aim to directly produce samples from the distribution described by p ref . 14,15 Ideally, this is achieved by defining a model CG Hamiltonian
N i=1 P 2 i 2Mi + U mod (R 3N ) such that the corresponding Boltzmann statistics p mod (R 3N , P 3N ) := Z mod,R −1 Z mod,P −1 exp −β N i=1 P 2 i 2M i + U mod (R 3N ) = p mod,R (R 3N )p mod,P (P 3N )(6)
are ideally identical to the mapped FG statistics, a criteria expressed with the following CG consistency equations 15
p ref,R (R 3N ) = p mod,R (R 3N )(7)
p ref,P (P 3N ) = p mod,P (P 3N ).
Momentum and configurational consistency are generally treated separately, with momentum consistency exactly satisfied through direct definition of CG masses M i and configurational consistency approximated through a variational statement (as the corresponding integral is not generally tractable). 15 We defer further discussion of momentum consistency until subsection II E 1. The configurational variational statement is specific to the particular bottom-up CG method chosen and utilizes a variety of information depending on the method considered. Gen-
erally, knowledge of U FG ref , U ref , M
, and a FG sample are generated in some fashion (e.g., through MD simulation) are used. In many cases the corresponding variational principle can be considered in the following form
θ † := argmin θ F[p mod,R,θ , p ref,R ](9)
where θ denotes our finite parameterization of our CG potential, θ † parameterizes our ideal model, and F is a function characterizing the quality of our model. This is often 10,13-18,24 operationalized as a empirical varia-tional principle, which is numerically formulated and minimized. Importantly, while the models discussed in the remainder of this article fit into this framework, they differ in two important respects to many previous CG parameterization strategies. First, they introduce a variational definition of F itself, resulting in two nested variational statements in the numerical optimization procedure. Second, unlike some current CG methods, 13-17 estimation of F does not require us to evaluate the CG Hamiltonian on any configurations, although the numerical optimization procedure does require parameters related to the distribution as described in subsection II C. The utility of not requiring the evaluation of the Hamiltonian will underpin extension of the method to CG virtual sites as described in section II E, but has no direct utility for traditional CG model parameterization as in this case said Hamiltonian is often required for the production of CG samples.
C. Adversarial-Residual-Coarse-Grained Models
The class of ARCG models considered in this paper are characterized by a set of possible F which are defined variationally as the difference in ensemble averages of a pair of coupled functions. The functions, f and g, are found as producing the maximum of the following variational definition (10) leading to a minimax variational statement for the fit model itself
F[p mod,R,θ , p ref,R ] := max (f,g)∈Q f p mod,θ − g p ref ,θ † = argmin θ max (f,g)∈Q f p mod,θ − g p ref .(11)
In other words, for a specific choice of p mod and p ref the numerical value of our residual is determined by a specific (f, g) pair; all other choices of pairs of observables in Q produce a more optimistic estimate of the quality of our model. These observables are only evaluated via their configurational average at the CG resolution. As we update θ, the optimal choice of (f, g) will change. Critically, each (f, g) present in Q has a specific relationship between f and g which is characteristic to the particular variation of ARCG used, examples of which are explicitly demonstrated in the next section. This relationship, along with further limitations on Q itself, fully characterize methods expressible in the form of Eq. (11). Importantly, the variational residual itself in Eq. (11) only makes reference to p mod,θ through an ensemble average, which can be estimated without knowledge of the effective CG Hamiltonian at that resolution. However, it is clear that we must minimize over θ to satisfy the external minimization in Eq. (11). This optimization is amenable to methods using the first derivative with respect to θ: in this case, due to the envelope theorem (see Appendix A), derivatives with respect to θ only include terms related to the ensemble average over the model distribution, p mod
d dθ i F[p mod,θ , p ref ] = d dθ i max (f,g)∈Q f p mod,θ − g p ref (12) = ∂ ∂θ i f † p mod,θ(13)
where f † represents one of the optimal observables found at the internal maximum. This can be expressed using the log trick and simple substitution providing a covariance expression for estimation
∂ ∂θ i f † p mod,θ = β f † ∂U θ ∂θ i − β f † ∂U θ ∂θ i .(14)
These results suggest a straightforward numerical optimization of Eq. (11) using gradient descent and related first order methods (e.g., RMSprop 73 ). We represent Q by indexing with a finite dimensional vector ψ. At each iteration of optimization, holding θ constant, we maximize over ψ; then, holding ψ constant, we take a single step on the estimated gradient of θ. This two step process is completed until convergence of θ.
Not all definitions of Q produce meaningful procedures for creating CG models. Generally, particular forms of F are derived individually, each of which is amenable to the procedures outlined here. We continue by investigating an informative subset of possible F, characterized via f -divergences, which will provide functionality directly encompassing REM CG, 17 as well as previous approaches by Stillinger 10 and Vlcek and Chialvo 24 .
D. f -divergences
The f -divergences are a category of functions which characterize the difference between two distributions. 63 When probability density functions are available, we can express this family of divergences as
I f (p ref , p mod ) := χ p mod (x)f p ref (x) p mod (x) dx(15)
where each member of the family is indexed by a convex function f which satisfies f (1) = 0. Relative entropy, the divergence central to REM CG, can be obtained by defining f (x) := x log x, 74 and the Hellinger distance, central to previous methods by Stillinger 10 and Vlcek and Chialvo 24 can be obtained by via f (
x) := ( √ x − 1) 2 .
The f -divergence between p mod and p ref can be expressed in multiple variational statements. [63][64][65]67 We here utilize its duality with the difficulty of class probability estimation, 75 which is mathematically expressed in the following formulation where we have used • to denote function compositions, e.g., f • g(x) := f (g(x)), giving the following form
I f (p mod , p ref ) = max η − 1 2 l mod •η p mod − 1 2 l ref •η p ref(16)
where
L(x) := −2(1 − x)f x 1 − x (17) l mod (h) := L(h) − h ∂L ∂x h (18) l ref (h) := L(h) + (1 − h) ∂L ∂x h .(19)
The functionη is here itself a function of, for example, a CG configuration, mapping each configuration to a positive real number between 0 and 1. Substitution into Eq. (11) (along with the removal of prefactors) provides us with our training residual (20) where the optimalη, denoted η, is known to be 63
θ † = argmin θ max η − l mod •η p mod,θ − l ref •η p refη(x) = p ref (x) p mod (x) + p ref (x) .(21)
We provide concrete expressions for calculating relative entropy in section III and in appendix D.
Despite the seemingly opaque form of Eq. (20), the variational statement provided has a notable intuitive description, which will be useful when considering implementation and connections to similar methods. Consider an external observer which has access to a mixture of molecular configurational samples, some of which are produced by our mapped reference simulation and others from our CG model (termed our reference and model samples, respectively). The observer is faced with the following task: they must distinguish which examples came from which source based solely on configurational details. If the model is very poor, this will likely be easy-the configurations from the model will be distinct from the reference configurations. However, for higher quality models, many of their configurations will plausibly come from either the model or the reference simulation. As a result, the observer is allowed to guess the probability each example came from either the reference sample or the model sample, and is penalized less if they guess a higher probability of the correct answer. It will likely be impossible to perfectly guess the origin of all the samples, but there is a unique best answer in some circumstances; 70 this inability to perfectly distinguish samples is directly related to our f -divergences (e.g., relative entropy). 63 Modifying the manner in which we penalize incorrect predictions specifies which divergence is produced.
Importantly, this fictional game is highly similar to classification in ML when utilizing specific training loss functions. Classification rests on an algorithm's ability to determine the class of samples based on their features. There exists a specific loss function for each f -divergence such that the training loss experienced by the ideal classifier is directly related to the f -divergence between the class conditional distributions. 63 This loss function is asymmetric depending on the true origin of the sample: l mod penalizes a prediction on a sample gained from the model, while l ref penalizes a prediction on the reference sample. As a result, we simply need to train a classifier with a loss on our samples and consider its probabilistic predictions, represented asη. An extension of this intuitive explanation is found in the appendix, and a formal description is presented in Reid and Williamson 63 . This interpretation is central to the term adversary in the name of Generative Adversarial Networks: 58 the adversary attempts to complete this task, and we wish to make its task as difficult as possible.
E. Virtual Sites
The ARCG framework can be lightly generalized to decouple the resolution at which the CG potential acts and the resolution at which we compare our CG and reference systems. More specifically, we see that we can apply a distinct mapping operator to our CG system before it is compared to the mapped FG samples. To better illustrate the practical use of this extension we begin by providing a motivating example.
As previously discussed, many bottom-up CG methods are shown to produce the ideal PMF when they are allowed to adopt any force-field in the ideal sampling limit. However, CG models are often limited to molecular mechanics type potentials (e.g., pairwise nonbonded potentials), which often do not contain the ideal PMF as a possible parameterization. For example, one might use Multiscale Coarse-Graining 13-16 (MS-CG) to parameterize a CG lipid bilayer in which all of the solvent and some of the lipid degrees of freedom have been removed. Upon generating samples using the CG model we may find that certain properties of the membrane, such as its thermodynamic force of bilayer assembly, are poor. However, MS-CG method has likely provided one with its correct characteristic approximation; in order to improve the model with the same parameterization method one must either increase the complexity of the CG forcefield via higher order terms or through changing the CG representation retain more FG details via modification of M , the CG map seen in Eqs. (4) and (5). Here, we discuss a third option: augmenting the CG representation directly, without modifying M . As a simplistic example consider modeling the interaction of two benzene molecules using a CG pairwise potential. It may be difficult to capture the π-stacking effect using this type of potential at the CG resolution. As a remedy one could add particles normal to the plane containing the benzene molecule, as shown in Fig. 1, without associating these additional CG sites to FG sites via M . Importantly, however, we will only critique the behavior of our CG model after these virtual sites have been integrated out: for example, the CG model is optimized to minimize the relative entropy between the mapped FG reference and CG model after the integration over the possible positions of these virtual CG sites.
Description of the formalism encompassing these situations requires us to suitably expand our notation. We still consider all distributions described previously but use the following modifications: first, samples from p mod are no longer generated by a simulation protocol using the approximated PMF as its Hamiltonian. Instead, these samples are produced via a new mapping operator G and simulation of a new finer grained representation characterized by p pre mod via its own Hamiltonian
ν i=1 p 2 i /2m i + U pre mod (r 3ν )
where m i are the masses at the pre-CG resolution. As a result, p mod is redefined FIG. 1. An example of virtual particle usage. The atomistic representation of benzene (left) is mapped via M to a CG representation (center) only preserving three carbons (red). The full CG representation (right) of the same configuration has these three carbons and two additional virtual sites (purple) to help a pairwise potential capture the correct PMF. These sites are removed upon application of the virtual particle map G. These virtual sites have no atomistic counterpart.
with the following relations.
p mod,R (R 3N ) := X pre r p pre mod,r (r 3ν )δ(G r (r 3ν ) − R 3N )dr 3ν (22) p mod,P (P 3N ) := X pre p p pre mod,p (p 3ν )δ(G p (p 3ν ) − P 3N )dp 3ν(23)
The resulting relations between resolutions are summarized in Fig. 2 Importantly, our training procedure needs two minor modifications. First, the variational estimation of divergences presented in Eq. (13) is comprised solely of ensemble averages, which are approximated via sample averages; these averages can be evaluated by generating empirical samples from p mod via samples drawn from p pre mod and application of G. This is a consequence of Eq. (24). (24) Second, the gradients required for optimization of the parameters of the variational search (θ) are calculable again through Eq. (24), allowing us to utilize our previous expression Eq. (14) at the resolution native to our new pre-CG Hamiltonian by minimizing the variationally optimized observable composed with G.
f p mod = f • G p pre mod
Importantly, while our examples in this section have primarily concerned situations in which fictional particles are added to the CG representation and then completely integrated over before calculating divergences, G can easily be generalized. Fundamentally, it has the full flexibility of M ; similarly, additional constraints are born from maintaining momentum consistency via methods described in the next subsection. However, if one discards momentum consistency, it is possible to maintain an intuitive pre-CG representation while nonlinearly modifying M and G to represent custom high-dimensional observables. In this case these mapped distributions are used for determining the quality of the CG model. We reserve the bulk of our discussion and investigation of this more complex option to a future article.
Momentum Consistency
Previous sections have discussed the configurational variational statement central to ARCG; here we discuss how to ensure momentum consistency. In the case that no pre-CG resolution is considered, momentum consistency in ARCG may be achieved through identical methods as stated in previous approaches, such as MS-CG. 15 However, when considering three distinct resolutions momentum consistency takes on a slightly modified form. We provide suitable constraints for a common case below, although extensions are straightforward.
Momentum consistency is characterized by the following equation:
p ref,P (P 3N ) = p mod,P (P 3N ).(25)
We here consider the specific case where both M r and G r are linear functions which satisfy the constraints defined in the MS-CG work: 15 G r is limited to associate each CG site in X pre unambiguously to a single site in X and has imposed translational and positivity constraints, and analogous constraints are placed on M r (see appendix for more details). The momentum map M p (and G p with appropriate modifications) are assumed to take the following form as in reference 15:
M p I (p 3n ) := M M I i∈I M I c M Ii 2 p i m i ,(26)
In this case, previous work 15 has shown that the constants defining M p (and similarly G p ) can be combined with the masses of the sites contributing to a mapped site to provide a definition of the mapped masses Eq. (27) which define a Boltzmann distribution equal to the mapped momentum distribution
M M I −1 := i∈I M I c M Ii 2 m i ,(27)
where M M I is the mass of CG particle I as implied by map M , I M I is the set of all atoms which map to CG site I according to map M , and c M Ii is the coefficient describing how the positions of FG particle i contribute to CG particle I according to map M . More generally, this implies that we can explicitly characterize the mapped momentum distributions for both the mapped FG and mapped CG systems, which when combined with Eq. (25) provides the following relation implying momentum consistency in a system with virtual particles
exp −β N I=1 P 2 I 2M G I ∝ exp −β N I=1 P 2 I 2M M I (28) M G I −1 := i∈I G I c G Ii 2 m i .(29)
An attractive solution is to set M G I = M M I for each CG site I; in this case we find a set of equations implying consistency Eq. (30).
0 = i∈I M I M c 2 Ii m i − i∈I G I G c 2 Ii m i ∀ CG sites I(30)
Note that these equations are positively constrained with respect to masses and mapping constants (along with the previously stated constraints on the mapping constants). This provides a simple condition connecting our FG masses, pre-CG masses, M , and G, and allows one to check for momentum consistency if all the relevant variables are defined. It is important to note that I indexes the CG sites at the resolution of p ref and p mod -that is, without the virtual particles. As such, in the case of G simply dropping virtual particles consistency is trivially satisfied by simply matching the masses of the non dropped particles to those implied by the FG system with M . Additional details may be found in the appendix.
F. Related Methods
Despite differences in representation, ARCG can be formulated to elucidate connections to a variety of previous CG parameterization strategies, some of which have been mentioned in previous sections. This is performed via the appropriate design of the characteristic function space Q in Eq. (13). Additionally, ARCG bears resemblance to a recent CG method based on distinguishability and classification. 28 In this section we make explicit connections between the f -divergence implementation presented in this article and such external methods. The applications of the f -divergence duality presented here are in the infinite sampling limit with a fully expressive variational search; in practice, significant differences in seemingly equivalent methods may easily arise.
Classification has been recently used to train a CG model by using the resulting decision functionη † to directly update the CG configurational free energy. 28 This is motivated by noticing that the η which satisfies the variational bound in Eq. (16) with an unbounded Q can be related to the pointwise free energy difference as
log 1 − η η = log p ref − log p mod ,(31)
suggesting a procedure where log(1 −η † ) − log(η † ) is scaled and used as an additive update to the CG potential. This procedure is similarly valid using any of the f -divergence losses discussed in this article. 63 However, beyond the differing update rules, the variational divergence approach presented in this article is differentiated by a subtle but important difference in characteristic assumptions. The divergence interpretations of ARCG rely on the completeness of Q, but place no constraint on p mod . In contrast, the interpretation of the method of Lemke and Peter 28 also requires an fully expressive Q; however, as the update to p mod inherently utilizes members of Q, the method naturally also forces the {p mod,θ } θ to be fully expressive. In other words, Q and {p mod,θ } θ are directly coupled. As a result, in the case that the classifier used in the additive update method similarly has a relation to a specific f -divergence, an ideal model would always be chosen, rending the specific choice of f -divergence inconsequential. Beyond this, it is unclear how to expand the update rule of Lemke and Peter 28 to apply to virtual sites, as the classifier is only directly present at the resolution of p mod and extension of the update to the resolution of p pre mod is unclear. REM CG proposes that approximate CG models should be parameterized by minimizing the relative entropy, 17 or KL-divergence, between the distributions produced at the FG resolution:
X FG p FG ref (x) log p FG ref (x) p FG mod (x) dx(32)
where we have introduced a new quantity, p FG mod , defined to be the probability density implied by the CG model over FG space (which is not used in ARCG theory). Operationally, this differs by a constant (when considering CG force-field optimization) from the relative entropy considered at resolution of the CG model, given by
X p ref (x) log p ref (x) p mod (x) dx.(33)
KL-divergence is an f -divergence (generated by f (x) :=
x log x) and in the case of Eq. (33) can resultingly be formulated and solved for in the current framework, providing the following losses through Eq. (16)
l RE ref (h) = 2 h 1 − h (34) l RE mod (h) = 2 log 1 − h h − 1 .(35)
We utilize this method for the computational examples presented in Sec. III. Importantly, the full specification of REM CG considers comparing a coarser CG model to a finer FG model at the FG resolution by defining a new model density at the FG resolution, denoted FG mod p(x). This fundamentally differs with respect to the conceptual approach in this work, where we only calculate residuals at the resolution of p mod and p ref . However, in practice, these both produce the same statement for optimizing the CG force-field in the case of relative entropy. Additionally, the reasoning applied in the full REM CG formulation can be used to extend models parameterized via relative entropy ARCG in a similar manner.
Alternatively, recent work by Vlcek and Chialvo 24 (as well as previous work by Stillinger 10 ) suggests that the Bhattacharyya distance (BD) Eq. (37) is a natural metric to judge approximate models.
BC(p mod , p ref ) := X p mod (x)p ref (x)dx (36) BD(p mod , p ref ) := − log BC(p mod , p ref )(37)
While the Bhattacharyya distance is not an f -divergence, it is related to one via a monotonic transformation: the Hellinger distance (H)
H(p mod , p ref ) := 1 − BC(p mod , p ref ) (38) = I ( √ t−1) 2 (p mod , p ref ).(39)
This can be variationally approximated in the same framework as REM CG, resulting in the following losses
l H mod (h) = 2 h 1 − h (40) l H ref (h) = 2 1 − h h .(41)
Justification of the Bhattacharyya distance may be grounded in information geometry and the distinguishability of samples produced by the FG and CG models. Despite the apparent similarity to the fictional game described earlier, the justification of Vlcek and Chialvo 24 is grounded in distinguishing populations via their collective empirical samples, while our game focuses on distinguishing individual configurations. The stated connection simply occurs through our duality with fdivergences. Inverse Monte Carlo (IMC), 11 also known as Newton Inversion (NI), minimizes an observable which characterizes the difference distributions between the mapped FG and CG systems (often through their radial distribution functions). The distributions utilized for this comparison are often low dimensional and are calculated via traditional binning approaches. ARCG may be viewed similarly as minimizing the expected value of observables; however in ARCG the observable minimized at each step of optimization must be variationally found, and subsequently changes from step to step. However, due to the envelope theorem, the derivatives calculated for both ARCG and IMC/NI share a similar covariance form shown in Eq. (14). Additionally, the typical approach in IMC/NI requires histograms to calculate the desired empirical correlation functions, limiting the metric to low dimensional distributions; ARCG does not perform binning of any kind.
There exist additional CG methods which are difficult to directly compare to ARCG (e.g., references 11, [13][14][15][16]18). However, in general, most methods considered make assumptions which strongly inhibit virtual site application. Specifically, methods often assume that the CG potential (or its derivatives) can be applied at the resolution of the CG samples acquired (either through calculation of the residual or the update strategy facilitating optimization), although extensions are sometimes feasible. For example, traditional MS-CG force-matching optimizes the CG force-field to optimally match mapped forces; with a general linear G and U pre mod this would likely require an iterative procedure to determine the mean force implied at the CG resolution by G and U pre mod . Alternatively, gYBG inverts two-and three-body CG correlations to produce a force-field at the corresponding resolution of the observed correlations; similarly, Iterative Boltzmann Inversion requires an map to define the iterations which connect modifications in the potential to changes in the observed correlations (which is nonintuitive when considering parameters associated with general virtual sites). These limitation often do not appear to be fundamental ones, but rather one of operationalization; extensions to these methods which circumvent this limitation are likely possible. There are three straightforward strategies to remove this limitation, the first two of which the authors know are in current use. First, several methods such as binning or kernel density estimation are used to approximate the probability density at a resolution differing from the CG configurational Hamiltonian (e.g., the radial distribution approach in reference 24). This approach is often limited to lower dimensional spaces when comparing models. Second, constraints are placed on virtual sites such that U pre mod may be related via closed expression to U mod . 76 This approach inherently requires limiting the type of virtual site considered. Third, methods which allow the observed mapped FG sample to be backmapped to the pre-CG domain are applied and then traditional approaches are used on the backmapped sample. In contrast, ARCG is well suited to higher dimensions, imposes no constraint on the virtual sites, and does not require backmapping; however, it incurs increased training complexity.
Finally, we note that while there is significant overlap between ARCG and GANs with respect to the residual calculation and optimization, the method by which samples are produced in the models is conceptually distinct. GANs are characterized by transforming noise to a fit a desired empirical distribution; the optimization of the model parameters modifies the nature of this transformation. In contrast, the transformation present in ARCG is held constant, while the underlying sample generating process is modified.
III. IMPLEMENTATION
Previous sections have provided abstract descriptions of the ARCG method, including the specific form with connection to f -divergences. In this section we provide the corresponding concrete expressions for optimizing models using relative entropy by implementing the classification based approach described in Sec. II D. Additional practical points on implementation, relaxations of the method for stability, and the specification of Q are also discussed.
As previously noted, the relative entropy between p ref and p mod is an f -divergence and is obtained by by setting f (x) := x log x. This implies equivalence with a class probability estimation task with the aforementioned losses in Eq. (34), from which we derive the model optimization statement using Eq. (20) and associated gradients using Eq. (14), such that
F RE p pre modθ , p ref ; G = max η − log 1 −η η p ref − η • G 1 −η • G p pre mod,θ (42) d dθ i F RE p pre mod,θ , p ref ; G = −β η † • G 1 −η † • G p pre mod,θ ∂U pre modθ ∂θ i p pre mod,θ + β η † • G 1 −η † • G ∂U pre mod,θ ∂θ i p pre mod,θ(43)
This comprises a full residual and associated gradient for optimization. However, in practice, the loss functions are poorly behaved: pointwise values ofη = 1 easily create a divergent residual value (identical to the corresponding situation with the traditional relative entropy estimation methods). Fortunately, the optimal η is shared among all proper losses. 63 As a result,η † can be similarly discovered with the corresponding statement using the log-loss 63,70
η † = argmin η logη p ref + log(1 −η • G) p pre mod,θ(44)
while the gradient estimation remains unchanged. To summarize, the models trained in this article indirectly minimize Eq. (42) by producing derivatives over θ via Eq. (44) and Eq. (43), whereη † retains the same meaning across equations. This equality only holds assuming thatη † = η; imperfect Q can cause the resultingη † 's to differ.
In some cases of ARCG, including the case of fdivergence estimation, the functions achieving the inner maximum with an ideal Q can be expressed as a pointwise functions of the mapped distributions. Specifically, as noted in Eq. (21) the optimal witness function η in the case of relative entropy is expressible as a function of the conditional class densities. This can guide how elements of a tractable Q are parameterized. When the algebraic forms of p ref and p mod are known to be functions of summary statistics of their respective systems (e.g., the inverse 6 and 12 moments in a traditional Lennard-Jones potential 77 ), we can often express an ideal Q exactly with a manageable number of terms per member; however, this is not true of practical bottom-up CG application: the form of the mapping operator does not provide us with an algebraic understanding the implied mapped free energy surfaces. However, the resulting η does share invariances with the free energy surfaces it is comprised of (e.g., rotational and translational invariances).
The variational search over possibleη was performed via a neural network outputting class probability predictions penalized via the log-loss. The classifier was optimized to convergence at each step of gradient estimation. All neural networks used in examples in this paper utilized 3 hidden layers each with 10 nodes. All internal nodes used rectified linear activation functions with the output normalized via softmax. The duality with class probability estimation underpins the utility such traditional choices have in our variational search.
In practice, we have noticed that ARCG optimization may suffer from instability, especially when optimizing the parameters of a model which produces a distribution significantly different than its optimization target. This issue can be noted by observing that the classifier achieves 100 In these cases we find that an effective strat-egy is to introduce standard Gaussian noise into both the model and reference samples; the variance of this noise is gradually reduced to zero as the optimization progresses. It is likely that a correct local minima is achieved in this case as the optimization appears stationary at the end of optimization, but it is unclear if the selection of a specific local minima is biased using this strategy.
A public proof-of-concept python/Lammps based implementation is available at the weblink https://github.com/uchicago-voth/ARCG. This codebase makes extensive use of the theano, theanets, pyLammps, numpy, and dill libraries. All computational examples presented in this paper may be found in the test portion of this code, which includes the complete settings used to generate the data used. Visualizations were performed with the matplotlib and seaborn libraries, as well as the base plotting system and rgl package in R. Extensions providing scalability for more complex systems and potentials will be considered in future work.
IV. RESULTS
The relative entropy approach described in section III was applied to three test systems. First, a simple single component 12-6 Lennard-Jones system was optimized; no virtual particles were present, and no coarse-graining of the reference system was performed. Second, a system representing bonded real particles where force is partially mediated by a bonded virtual particle was optimized. Finally, a binary Lennard-Jones liquid was simulated and optimized after particles of a single type had been integrated out. In these cases we observed good convergence of suitable correlation functions; however, in cases with virtual particles we found that numerically recovering the known parameters of the reference system is difficult; in other words, it seems likely that the parameter space is either redundant or sloppy, 78 with similar correlation functions arising from distinct parameter sets.
All models considered here are theoretically fully able to achieve the reference distributions provided (i.e., the model optimized is never misspecified). This is ensured by forcing M and G to be the same function. Additionally, both the pre-CG resolution and the "atomistic" resolution are the same. For example, in the case of the virtual solvent Lennard-Jones system, two systems of binary Lennard-Jones particles were simulated, each with differing parameter sets. Both systems then had the particles of a specific shared type integrated out. The resulting integrated distributions were then the basis of comparison used to train the model parameters. In general application this will not be the case: the FG and pre-CG system will almost always differ.
A. Lennard-Jones Fluid
A single component 12-6 Lennard-Jones liquid was simulated with 864 particles at 300K (the potential form is given in Eq. (45) with r ij denoting the Euclidean distance between particles i and j).
U (R 3N ) = 4 i>j σ r ij 12 − σ r ij 6(45)
The system was simulated at constant NVT conditions using a Langevin thermostat with coupling parameter set to 100.0 fs and a timestep of 1.0 fs. No virtual particles were present; i.e., G and M are set to be the identity function. Inverse sixth and twelfth moments were used as input to the variational estimator (in this case, this set of features is known to be complete). System A initial with A initial = 0.6kcal/mol and σ A initial = 3.5Å was optimized to match the statistics of system B characterized by B = 0.75kcal/mol and σ B = 3.0Å. Upon optimization, the parameters of A were seen to quantitative converge to those of B: Aopt = 0.744kcal/mol and σ Aopt = 3.00Å. Additionally, convergence of the pairwise correlation functions (Fig. 3) was observed. During training Gaussian noise was used to smooth out initial gradients to resolve initial soft wall differences; this noise is reduced to zero by the end of optimization. Optimization was performed using RMSprop 73 with individual rates for each parameter. These results demonstrate relatively good convergence properties with small parameter sets when no virtual particles are considered in the pre-CG resolution.
FIG. 3. Radial distribution functions calculated for the unoptimized system A initial , the reference system B, and the optimized system Aopt.
B. Virtual Bond Site
A system of three particles completely connected via harmonic bonds was simulated at 300 K. The system was propagated in constant NVT conditions using a Langevin thermostat with coupling parameter set to 100.0 fs and a timestep of 1.0 fs. Two types of particles are present; we denote the types of the particles X, Y, X. Upon application of M and G, the Y particle is removed, resulting in a system comprised of two particles of type X. This mapped system is optimized using the distance between the two X particles as input to the discriminator; in this case, this feature set is complete. Initial, optimized and reference parameters are seen in x denotes the zero energy point of the bond while k denotes bond strength. Subscripts specify the particle types between which the bond acts. System A initial was optimized to match system B, resulting in Aopt.
timization was performed using stochastic gradient descent with momentum. Again, Gaussian noise is found to be useful when optimizing systems which initially significantly differ from the target system. Convergence to a specific parameter set which reproduce observed correlations (Fig. 4) is fast; however these parameters differ from the parameters of the reference system. Additional simulations were run where the CG model was initialized with parameters set to those of the reference system (results not shown); in this case, we observed local diffusion around a small set of parameters including the true set. This suggests that virtual particles may create degeneracy in model specification in practice (i.e., even if the model parameters are identifiable, the specification is sloppy). This case represents an application where a FIG. 4. Bond distance distribution functions calculated for the unoptimized system A initial , the reference system B, and the optimized system Aopt.
pairwise force-field may be augmented via bonded virtual particles to create modified correlations. For example, a heterogeneous elastic network 79 may be augmented by introducing virtual particles to facilitate higher order correlations.
C. Virtual Solvent Lennard-Jones Fluid
A binary system comprised of 864 Lennard-Jones particles of types X and Y was simulated at 300 K. The system was simulated at constant NVT conditions using a Langevin thermostat with coupling parameter set to 100.0 fs and a timestep of 1.0 fs. Equal numbers of X and Y particles were present prior to the application of mapping operators; upon application all particles of type Y were removed. The target system was parameterized to undergo phase coexistence, while the unoptimized CG model was well mixed. Parameters are found in table II. Optimization was performed using RMSprop with rates adjusted for each parameter. Gaussian noise was used to stabilize initial training. Visual inspection of representative molecular configurations showed greatly improved similarity for the optimized parameter set (Fig. 5). Again, while convergence of correlation functions is readily observed (Fig. 6), parameters do not converge to those of the reference system, likely due to sloppiness in specification. x denotes the zero energy point of the bond while k denotes bond strength. Subscripts specify the particle types between which the bond acts. System A initial was optimized to match system B, resulting in Aopt.
System σXX /Å XX / kcal mol σY Y /Å Y Y / kcal
This case is representative of the situation where higher order correlations may be captured by the addition of virtual solvent particles. For example, the hydrophobic driving force underlying a CG lipid slab could be facilitated by a virtual solvent. This is distinct from using traditional explicit solvent where each solvent molecule is directly connected to the FG reference system: there, the behavior of the solvent is incorporated into the quality of the model, as where the approach of ARCG ignores the direct solvent behavior.
V. DISCUSSION
In previous sections we have described a broad new class of variational statements for optimizing CG models and described methods for their optimization by utilizing the theory underpinning adversarial models in ML. Subsequently, we have shown that it is possible to parameterize a CG model via ARCG at a coarser resolution than that native to the CG Hamiltonian. A clear application of ARCG is the parameterization of models which contain virtual sites; however, the CG distribution may be critiqued at any coarser resolution, providing the intriguing ability to control what aspects of a CG model are visible for optimization purposes. In the process of doing so we showed that gradients needed at each step of divergence minimization can be reformulated as modifying the system Hamiltonian to minimize the value of a specific observable, but that this observable depends on the exact distributions being considered at that step of optimization. We note that more generally the method pre- sented can be used to calculate the KL divergence (and any of the other divergences discussed) between distributions for which no probability density/mass is known and for which one cannot be approximated via kernel density approximation or binning. Beyond our central results we have provided work and discussion on two supporting topics.
1. We have provided comparisons to multiple contemporary methods for CG parameterization. In certain cases we have shown that divergences characterizing their configuration variational principles can be used in ARCG modeling. In one case we showed that classifier based approaches bear striking but not complete similarity to the presented approach. In the remaining cases we have discussed how decoupling the resolution at which we critique a model from the resolution of the CG Hamiltonian creates difficulties in said approaches.
2. We have provided a set of sufficient conditions for momentum consistency in the case of virtual sites, and described how these conditions may be extended. These are closely related to consistency requirements for traditional bottom-up CG models.
Additionally, we have provided simple numerical examples (and a public computational implementation) for which we have optimized CG potentials to match specific distributions, some of which utilize CG virtual particles.
The results show quantitative agreement for calculated correlations, visual agreement, and qualitative agreement in matching exact coefficients when the answer is known (quantitative agreement is seen when virtual particles are not present). Difficulties in convergence appear to be either due to instability in the parameterization process or sloppiness in the model specifications. The manner in which this will affect realistic systems is yet to be seen, but may present a significant challenge. It is clear that in the most general case parameter uniqueness is not guaranteed: if CG consistency can be obtained without virtual particles, then a model which can both decouple the virtual particle interaction from the real particles and modify the behavior of the virtual particles independently of said coupling will inherently be nonidentifiable. Additionally, it is likely that in the case of f -divergence based ARCG optimization that a relatively good initial hypothesis for the CG potential may be necessary, or significant amounts of noise must be added initially during optimization.
There are multiple additional studies that could naturally expand and clarify the results presented.
1. The methods provided can be applied to approximate nontrivial molecular systems without virtual particles. This will require multiple steps: first, the proof-of-concept software framework presented will have to be expanded for larger system sizes. Second, the training method used will have to be developed such that it remains stable, whether through the systematic addition of noise or the use of enhanced sampling techniques. Third, the featurespace used to index Q will likely have to be correctly engineered based on knowledge of the FG and CG Hamiltonians. All three of these are tractable challenges.
2. The effect of using virtual particles should be investigated both computationally and theoretically, as previous analysis on incomplete basis sets (e.g., that on relative entropy and MS-CG 46 ) does not apply transparently. In the process of doing so a better theoretical understanding of how to utilize these methods to capture specific higher order correlations in the training data should additionally be investigated, possibly leading to new ways in which bottom-up CG parameterization may be tuned to reproduce specific novel correlation functions.
3. The effect of various divergences on training approximate CG models should be investigated theoretically and through simulation. This will facilitate the design of CG parameterization methods which have different biases in the approximations they produce when coupled with realistic CG potentials. This applies to not only to various fdivergences but also the wider set of divergences not heavily discussed in this article, such as the Wasserstein, 68 Sobolev, 80 Energy, 81 , and maximum mean descrepancy 82 distances. The Wasserstein and Energy distances share the interesting property of taking into account the spatial organization of the domain of the probability distributions considered through a separate spatial metric. Combined with kinetically informed coordinate transforms such as TICA 83 and variants, 84,85 it may be possible to parameterize models to have stationary distributions which are kinetically close to one another.
4. The effect of an incomplete Q should continue to be investigated. 86 In this case the presented divergence based interpretation is not trivially accurate. Understanding of how imperfect classifiers affect the parameterization of approximate models may have large implications on the optimization of complex multicomponent systems; overly expressive Q will likely impede model parameterization as more sampling of the CG and FG system may be required.
VI. CONCLUDING REMARKS
In this article we discussed a new class of methods for the systematic bottom-up parameterization of a CG model. In doing so we illustrated concrete connections between CG models and algorithms such as generative adversarial networks. Utilizing these connections we both decoupled the resolution at which we critique our CG model from the CG potential itself, and enabled the use of a variety of novel measures of quality for CG model parameterization. We provided a proof of concept implementation and several numerical examples. Additionally, we illustrated precise connections to several previous methods for CG model parameterization. Finally, we noted multiple future branches of studies that can now be pursued. Together, these results open a new conceptual basis for future systematic CG parameterization strategies.
The envelope theorem can be informally stated as follows. Suppose we have the following unconstrained maximization problem
f † (r) := max x f (x, r). (A1)
and define x † (r) to be the value of x that gives the maximum for a specific r (we assume all such maxima exist). Then, if df † (r)/dr exists,
df † (r) dr = ∂f (r, x † (r)) ∂r . (A2)
where we have noted the dependence of x † on r, but the partial derivative does not take into account this dependence (this notation is continued in the following informal proof). This can be seen via the following algebra. We know that
f (x † (r), s) − f † (s) ≤ 0 ∀s;(A3)
additionally, we have that
f (x † (r), r) − f † (r) = 0.(A4)
Thus, r maximizes f (x † (r), r) − f † (r). We additionally have by assumption ∂f (x † , r)/∂r i and ∂f † (r)/∂r i exist. We then have that ∂f (x † , r)/∂r i − ∂f † (r)/∂r i = 0. A more formal presentation may be found in Bottou et al. 67 .
Appendix B: Class Probability Estimation Duality
The variational relationship used to estimate the fdivergence in subsection II D is central to the current computational implementation. A formal introduction may be found in Reid and Williamson 63 ; however, to facilitate intuition about the method used, we provide an expanded informal description of task of the discriminator and pertinent underlying equations.
Continuing in the context of the fictional task given in subsection II D, consider the simpler task of considering whether a single, specific molecular configuration is produced by our model or our reference distribution. Again, the participant, or adversary, is allowed to provide a real number between zero and one to signify their confidence that the specific configuration came from either the reference sample or the CG model; zero corresponds to the CG model, where one corresponds to the configuration coming from the reference sample. The participant is reprimanded less when they provide a real number closer to the correct label of zero or one. Suppose there is a fixed probability of the specified configuration having a true label of zero or one. In this case, if the participant is allowed to perform this task many times and must give the same numerical guess every time, they will naturally have an average reprimand depending on their guess. This reprimand is additionally determined by the fraction of times the molecular configurations provided have true values of zero or one. If our loss functions encoding the penalty of each guess are chosen carefully, the optimal guess is coded by the η, which is a function of these probabilties (in this specific case, η only takes on a single value as we are only judging a single configuration). We denote the average loss is given by the ideal hypothesis by L(η).
Up to now, we have considered the task of a judging a single molecular configuration. However, we can consider a large number of configurations, each drawn from a large distribution, for which we have to complete this task. In this case, our optimal guess, η, is now a function of each configuration judged and takes the form discussed earlier. We again wish to minimize the average loss. Interestingly, the lowest expected loss we can achieve is directly related to a specific f -divergence. More specifically, given a convex f with f As before, this type of map transforms global consistency into constituent momentum and position space components, i.e., p mod (R 3N , P 3N ) = dr 3ν dp 3ν p pre mod (r 3ν , p 3ν )δ(M r (r 3ν ) − R 3N )δ(M p (p 3ν ) − P 3N ) = p mod,R (R 3N )p mod,P (P 3N ) (C4)
where the vector valued delta functions are understood to be products of scalar delta functions. If M does not associate any individual atoms to more than a single CG site, then We will additionally assume that analogous constraints are put on G r when considering momentum consistency below.
exp −β N I=1 P 2 I 2M I ∝ dp 3ν exp −β ν i=1 p 2 i 2m i ×δ(M p (p 3ν ) − P 3N )(C5)
Momentum Consistency
Using these points we now move forward directly discussing momentum consistency. As stated previously, by constraining G and M to as above, and assuming the underlying systems are characterized by separable probability densities, we find p mod (R 3N , P 3N ) = p mod,R (R 3N )p mod,P (P 3N ) (C7)
p ref (R 3N , P 3N ) = p ref,R (R 3N )p ref,P (P 3N ) (C8)
As a result, we split up our consistency statement (omitting arguments for clarity) (p mod,R = p ref,R ∧ p mod,P = p ref,P ) =⇒ p mod = p ref (C9) Configurational consistency is handled via divergence matching as described in the main article; we here consider momentum consistency algebraically.
p mod,P = p ref,P ⇐⇒ dp 3ν exp −β ν i=1 p 2 i 2m i δ(G p (p 3ν ) − P 3N ) ∝ dp 3n exp −β n i=1 p 2 i 2m i δ(M p (p 3n ) − P 3N ).(C10)
We substitute these using two sets of properly designed CG masses, each set implied by a mapping operator and the masses at resolution it maps
exp −β N I=1 P 2 I 2M G I ∝ exp −β N I=1
open for modification; here, Eq. (C13) describes linear equality and positivity constraints on 1/m i . It is critical to note that I indices the CG sites at the resolution of p ref and p mod -that is, without the virtual particles. As such, in the case of G simply dropping virtual particles consistency is trivially satisfied by simply matching the masses of the non dropped particles to those implied by the FG system with M .
FIG. 5 .
5Sample configurations of the unoptimized model (green), the optimized model (blue) and the reference data (red). Configurations are shown at the resolution of comparison, i.e., after the application of M and G. Slab type formation, similar to that present in the optimized model, is seen after parameter optimization.
FIG. 6 .
6Probability densities across the slab type formations present in the integrated binary LJ systems (along the z axis of the simulation box). No slab structure is present in the initial model.
f (p mod , p ref ) = −L(η, p mod , p ref ). (B2)where L is the mean reprimand taken over the combination of all configurations from p mod and p ref . Finally, we can additionally explicitly relate our pointwise average loss L to the specific penalties for each prediction (e.g., l ref ) through the relationships in the main text Eq.(16). The variational nature of this task arises from the fact that we are searching over possibleη.of the output of M r , i iterates over the particles which contribute to site I, and c denotes positive constants. we allow M r to imply M p up to the factor of the CG masses {M I } I as stated in MS-CG.
.FIG. 2. The relationship between resolutions when comparing
FG and CG systems at a custom resolution, such as the case
of virtual sites. Samples from the pre-CG domain X pre (e.g.,
a CG configuration including virtual sites) are mapped to the
CG domain X (e.g., a CG configuration without virtual sites)
via G; samples from the FG domain X FG (e.g., atomistic)
are mapped to the same CG domain X via M . The mapped
samples are then compared via F.
table I .
IOp-System xXY /Å kXY / kcal
mol xXX /Å kXX / kcal
mol
B
2
2.7
2.3
0.4
A initial
0.75
2.5
1.8
0.2
Aopt
1.932
2.443
2.361
0.430
TABLE I. Parameters for systems with virtual bonded sites.
ACKNOWLEDGMENTSThis material is based upon work supported by the National Science Foundation (NSF Grant CHE-1465248). This research was conducted with Government support under and awarded by DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. Simulations were performed using the resources provided by the University of Chicago Research Computing Center (RCC).Appendix A: Envelope TheoremAppendix C: Momentum ConsistencyThe described approach to achieve momentum consistency requires that we put more specific constraints on G. This is needed due to our minimal strategy for providing sufficiency conditions for consistency: primarily, we utilize arguments in previous work to provide sufficient constraints. The resulting conditions given suffice for the case of virtual particles which are simply dropped from the system by G. Generalizations to linear mappings which share particles between sites can additionally be inferred. First we discuss the approach of previous work on momentum consistency as is relevant to our work, and then concisely give a route to momentum consistency.MS-CGGenerally, we will here assume that M r satisfies specific properties. Once M r is defined, we construct an appropriate M p . First, M r must be expressible in the following linear form, where M rI denotes the Ith particle entry The basis of the duality central to f -divergences is translated from thm. 9 in Reid and Williamson 63 . The equations relating loss functions l from the combined loss L. May be confirmed via algebra after the two following identities are note, both of which may be found in Reid and Williamson 63 .Using these we can now derive the losses for relative entropy. First, note that the function generating the appropriate relative entropy is x log x (not log x). From this we find (only in the case of relative entropy)Through substitution we then arrive at Eq.(34).
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| [
"https://github.com/uchicago-voth/ARCG."
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[
"HOMOLOGY REPRESENTATIONS ARISING FROM THE HALF CUBE",
"HOMOLOGY REPRESENTATIONS ARISING FROM THE HALF CUBE"
]
| [
"R M Green [email protected] \nDepartment of Mathematics\nUniversity of Colorado\nCampusBox 39580309-0395BoulderCOUSA\n"
]
| [
"Department of Mathematics\nUniversity of Colorado\nCampusBox 39580309-0395BoulderCOUSA"
]
| [
"Advances in Mathematics"
]
| We construct a CW decomposition C n of the n-dimensional half cube in a manner compatible with its structure as a polytope. For each 3 ≤ k ≤ n, the complex C n has a subcomplex C n,k , which coincides with the clique complex of the half cube graph if k = 4. The homology of C n,k is concentrated in degree k − 1 and furthermore, the (k − 1)-st Betti number of C n,k is equal to the (k − 2)-nd Betti number of the complement of the k-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type D n act naturally on the complexes C n,k , and thus on the associated homology groups. | 10.1016/j.aim.2008.11.019 | [
"https://arxiv.org/pdf/0806.1503v2.pdf"
]
| 17,448,852 | 0806.1503 | 176a6ff9ae31744e49f0856b3d3675d85b713c30 |
HOMOLOGY REPRESENTATIONS ARISING FROM THE HALF CUBE
4 Dec 2008
R M Green [email protected]
Department of Mathematics
University of Colorado
CampusBox 39580309-0395BoulderCOUSA
HOMOLOGY REPRESENTATIONS ARISING FROM THE HALF CUBE
Advances in Mathematics
4 Dec 2008
We construct a CW decomposition C n of the n-dimensional half cube in a manner compatible with its structure as a polytope. For each 3 ≤ k ≤ n, the complex C n has a subcomplex C n,k , which coincides with the clique complex of the half cube graph if k = 4. The homology of C n,k is concentrated in degree k − 1 and furthermore, the (k − 1)-st Betti number of C n,k is equal to the (k − 2)-nd Betti number of the complement of the k-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type D n act naturally on the complexes C n,k , and thus on the associated homology groups.
Introduction
The half cube, also known as the demihypercube, may be constructed by selecting one point from each adjacent pair of vertices in the n-dimensional hypercube and taking the convex hull of the resulting set of 2 n−1 points. Although the resulting polytope is not a regular polytope like the hypercube, it still has a large symmetry group, and its k-faces are of two types: regular simplices and isometric copies of half cubes of lower dimension.
1991 Mathematics Subject Classification. 05E25, 52B11, 57Q05.
Typeset by A M S-T E X
In this paper, we will give a detailed description of the faces of the half cube (Theorem 2.3.6) and their intersections with each other (Theorem 2.3.8). Using these results, we show (Theorem 3.1.2) that the faces of the half cube may be assembled into a regular CW complex, C n , in a natural way. (Such a structure is sometimes called a polytopal complex.)
It is not very difficult to show that the half cube is contractible, and therefore that the reduced homology of the complex C n is trivial. However, C n has some topologically interesting subcomplexes C n,k for 3 ≤ k ≤ n. The complex C n,k is not a truncation of C n ; rather, it is constructed from C n by deleting the interiors of all the half cube shaped faces of dimensions l ≥ k. If k = 3 or k = 4, the complexes C n,k are simplicial; indeed, the case k = 4 gives the clique complex of the half cube graph 1 2 H n . Using a combination of Forman's discrete Morse theory [11] and the theory of CW complexes, we prove (Theorem 3.3.2) that the reduced homology of C n,k is concentrated in degree k − 1. We also give an explicit formula for the nonzero Betti numbers (Theorem 4.1.2), which also appear as the entries T (n, n − k) in a certain Pascal-like triangle (Definition 4.1. 3). Although this triangle is not particularly well known, its entries appear in a surprisingly diverse range of contexts, including:
(i) in the problem of finding, given n real numbers, a lower bound for the complexity of determining whether some k of them are equal [5,6,7, §1], (ii) as the (k − 2)-nd Betti numbers of the k-equal real hyperplane arrangement in R n [7], (iii) as the ranks of A-groups appearing in combinatorial homotopy theory [1,2], (iv) as the number of nodes used by the Kronrod-Patterson-Smolyak cubature formula in numerical analysis [19, Table 3], and (v) (when k = 3) in engineering, as the number of three-dimensional block structures associated to n joint systems in the construction of stable underground structures [18].
Although the relationships between (i), (ii) and (iii) above are well understood (see the remarks in §4.3 below), the connections with the half cube polytope, numerical analysis and engineering appear not to have been noticed before. Curiously, although explicit formulae for the numbers T (n, n − k) are given (or implicit) in applications (ii), (iii) and (iv) above, these formulae all differ from each other and from our formula in Theorem 4.1.2.
In this paper, we concentrate mostly on case (ii) above. The k-equal real hyperplane arrangement V R n,k is the set of points (x 1 , . . . , x n ) ∈ R n such that x i 1 = x i 2 = · · · = x i k for some set of indices 1 ≤ i 1 < i 2 < · · · < i k ≤ n.
The complement R n − V R n,k , denoted by M R n,k , is a manifold whose homology is concentrated in degrees t(k − 2), where t ∈ Z satisfies 0 ≤ t ≤ ⌊ n k ⌋ (see [7, Theorem 1.1(b)]). We will prove in Theorem 4.1.5 and Corollary 4.1.6 that the (k − 1)-st
Betti number of C n,k is equal to the (k − 2)-nd Betti number of the complement of the k-equal real hyperplane arrangement.
The n-dimensional half cube has a large symmetry group G n of orthogonal transformations acting on it. This group always contains the Coxeter group W (D n ), which has order 2 n−1 n!, although this containment is proper if n = 4. Our final main result, Theorem 4.2.3, describes the orbits of these groups acting on the kfaces of hγ n ; this is useful because it induces an action of W (D n ) on the nonzero homology groups.
The results of this paper raise various interesting questions, which we discuss in the concluding section.
The geometry of the half cube
The purpose of §2 is to obtain a detailed understanding of the geometry of the half cube, meaning a classification of its vertices, edges, and other k-dimensional faces (Theorem 2.3.6). This enables us to verify that the faces of the half cube intersect in a nice way (Theorem 2.3.8). Both these results play a key role in the topological constructions of §3. A related combinatorial problem that we solve along the way (Proposition 2.2.6) is the classification of the cliques in the half cube graph, which can be characterized as the 1-skeleton of the half cube.
Polytopes.
Following Coxeter [9, §7.4], we define an n-dimensional (Euclidean) polytope Π n to be a closed, bounded, convex subset of R n enclosed by a finite number of hyperplanes. (The polytopes in this paper are all Euclidean.) The part of Π n that lies in one of the hyperplanes is called a facet, and each facet is an (n − 1)dimensional polytope. (Coxeter [9] uses the term "cell" to mean "facet".) Iterating this construction gives rise to a set of k-dimensional polytopes Π k (called k-faces) for each 0 ≤ k ≤ n. The elements of Π 0 are called vertices and the elements of Π 1 are called edges. Two vertices are called adjacent if they share an edge. The cardinality of the set Π i is denoted by N i .
It is immediate from the definitions that a polytope is the convex closure of its facets, and iteration of this observation shows that a polytope is the convex hull of its set of vertices; in particular, the set of vertices determines the polytope. We can therefore speak of "the polytope on the set V " to refer to the polytope Π(V ) whose vertex set is V .
Another immediate consequence of the definitions is that the facets of a polytope lie in the boundary. Conversely, the fact that there is a finite number of bounding hyperplanes means that the boundary of a polytope consists precisely of the union of its facets; in other words, if a point of the polytope lies in no facet, then it must be an interior point.
Example 2.1.1. Let x 1 , . .
. , x n be the usual coordinate functions in R n . The 2n hyperplanes of the form x i = ±1 for 1 ≤ i ≤ n bound a closed convex subset of R n containing the origin. The corresponding polytope is the hypercube or measure polytope, denoted by γ n in Coxeter's notation [9]. The vertices of γ n are the 2 n points of the form (±1, ±1, . . . , ±1).
If every 2-dimensional face of a polytope has an even number of sides, as is the case with γ n , one can apply a general procedure (described in detail in [9, §8.6]) known as alternation. This involves selecting precisely half the vertices of γ n in such a way that one vertex is selected from each adjacent pair. (For our purposes, we will include a vertex in our selection if the entry −1 occurs an even number of times in its position vector.) One can then introduce a new bounding hyperplane for each rejected vertex, and the part of the original polytope on the same side of the hyperplane as the rejected vertex is discarded. This procedure does not introduce any new vertices.
In the case of γ n , the resulting polytope is called the half cube or demihypercube, and denoted by hγ n . It is the convex hull of the 2 n−1 vertices of γ n that contain an even number of minus signs.
Half cube graphs.
Let n ≥ 4 be an integer. We define the set Ψ n to be the set of 2 n vertices of the polytope γ n , Ψ + n to be the set of 2 n−1 vertices with an even number of negative coordinates, and Ψ − n to be Ψ n \Ψ + n . The Hamming distance, d(x, y), between two elements x, y ∈ Ψ n , is defined to be the number of coordinates at which the n-tuples x and y differ; in other words,
we have 2d(x, y) = n i=1 |x i − y i |.
There is an equivalence relation ∼ on Ψ n given by the conditions that x ∼ y if and only if d(x, y) is even, and the equivalence classes are precisely Ψ + n and Ψ − n . The half cube graph, 1 2 H n , is the simple undirected graph whose vertices are the elements of Ψ + . There is an edge between vertices x and y if and only if d(x, y) = 2.
(Note that we could equally well have defined this graph using Ψ − .)
A k-clique (or clique for short) in a graph G is a set of k vertices of G, any two of which are mutually adjacent. The clique complex of G is the abstract simplicial complex whose k-faces are the k-cliques of G.
Definition 2.2.1. Let n = {1, 2, . . . , n}, v ∈ Ψ − and S ⊆ n. If |S| = k > 0, we define a k-clique K(v, S) of 1 2 H n by the condition that v ′ ∈ K(v, S) if and only if v and v ′ differ only in the i-th coordinate, and i ∈ S. (It is immediate from the definitions that K(v, S) is a k-clique.) We call S a mask of K(v, S). We say that
K(v, S) is opposite v, or that v is an opposite point of K(v, SK(v, S).
Proof. Suppose K is a set of the required form with |K| ≥ 3. For each 1 ≤ i ≤ n, it is the case that a majority of the points in K (all but one if i ∈ S, and all of them if i ∈ S) contain the entry a i in the i-th coordinate for some a i ∈ {±1}. We define v = (a 1 , a 2 , . . . , a n ).
The subset S ⊆ n is then the set of all j for which some point of K differs from v only in the j-th coordinate. In particular, if |S| = 3, then the elements of L(v, S) are v itself, together with the three vectors that differ from v in precisely two coordinates indexed by S, and it follows from the definitions that L(v, S) is a 4-clique.
Note that a 4-clique of the form L(v, S) cannot be of the form K(v ′ , S ′ ) for any v ′ and S ′ : the elements of a 4-clique of the form K(v ′ , S ′ ) (respectively, L(v, S)) agree at all but precisely four (respectively, three) coordinates.
Proposition 2.2.6. Every clique in the graph 1 2 H n is of the form K(v, S) for some S, or of the form L(v, S) for some S with |S| = 3.
Proof. It is enough to show that if C is a k-clique of one of the above forms, then any extension of C to a (k + 1)-clique C ′ is also of one of the above forms.
If k = 1, then C is necessarily a single point v 1 , and C ′ consists of two points v 1 and v 2 which differ from each other in precisely two coordinates. This satisfies the required condition.
If k = 2, then C = {v 1 , v 2 } consists of two points differing from each other in precisely two coordinates, a and b. Suppose that C ′ = {v 1 , v 2 , v 3 } is an extension of C to a 3-clique. Since {v 1 , v 3 } is a 2-clique, it must be the case that v 3 differs from v 1 in precisely two coordinates, c and d. However, since {v 2 , v 3 } is a 2-clique, it follows that v 2 and v 3 differ from each other in precisely two coordinates, and thus that the set {a, b, c, d} has cardinality 3. Without loss of generality, we may assume that d = a and b = c. Letting v ′ ∈ Ψ − n be the unique element that differs from v 1 only in the a-th coordinate, we find that C ′ = K(v ′ , {a, b, c}), which satisfies the Suppose that k = 4. We will show that, in the above notation, if We may therefore assume that C = {v 1 , . . . , v 4 } is of the form K(v ′ , {a, b, c, d}), and that each of the points v 2 , v 3 , v 4 differs from v 1 in two coordinates, one of which is a. Suppose again that the 5-clique C ′ = {v 1 , . . . , v 5 } is an extension of C. Since {v 1 , v 2 , v 3 , v 5 } is a clique of the form K(v ′′ , S), the argument used to deal with the case k = 3 shows that v 5 differs from v 1 in two coordinates, one of which is a.
C = {v 1 , . . . , v 4 } is of the form L(v 1 , {a, b, c}),
We define e to be the other coordinate at which these points differ; since C ′ has cardinality 5, it must be the case that the set S = {a, b, c, d, e} also has cardinality 5. This proves that C ′ = K(v ′ , S), and completes the proof of the case k = 4.
The case of k > 4 proceeds similarly. Let C = {v 1 , . . . , v k } be a clique of size k > 4; by the arguments above, it follows that C = K(v ′ , S) for suitable v ′ and S. Moreover, we can arrange things so that each of v 2 , v 3 and v 4 differ from v 1 in the a-th coordinate and one other coordinate (the b-th, c-th and d-th, respectively), and that v ′ differs from v 1 only in the a-th coordinate. Suppose that C ′ = {v 1 , v k+1 } is an extension of C to a (k + 1)-clique, and consider the
4-subclique C ′′ = {v 1 , v 2 , v 3 , v l } for some 4 < l ≤ k + 1. Since C ′′ must be of the form K(v ′′ , S ′ )
, it must be the case that v l differs from v 1 in two coordinates, one of which is the a-th. Since this is true for all 4 < l ≤ k + 1, it follows that
C ′ = K(v ′ , S ′′ ),
where v ′ is as before, and S ′′ consists of all coordinates at which some pair of elements in C ′ disagree. This completes the proof.
Faces of the half cube.
In §2.3, we give a more explicit description of the half cube hγ n . The vertices of this polytope were described in §2.1 above, and we will now describe the k-faces for 0 < k < n. Although such a description is not new, the only reference we know for the result of Theorem 2.3.6 is the proof of [10, Proposition 4.2], and unfortunately, the latter paper does not give a detailed enough statement of the result for our purposes, nor does it give the details of the proof.
Recall in the sequel that a polytope is the convex hull of its set of vertices.
Lemma 2.3.1. (i) Let v ′ = (v 1 , . . . , v n ) ∈ Ψ −
n and S ⊆ n. The convex hull of the set K(v ′ , S) consists precisely of the points (x 1 , . . . , x n ) ∈ R n satisfying the following three conditions: Proof. The convex hull of the set K(v ′ , S) is the set of convex linear combinations of K(v ′ , S), that is, the set of all points of the form
(a) x i = v i for all i ∈ S; (b) sgn(v i )(x i − v i ) ≤ 0 for all 1 ≤ i ≤ n, where sgn(v i ) = v i /|v i |; and (c) i∈S sgn(v i )(x i − v i ) = −2. (ii) Let x = (x 1 , . . . , x n ) beu∈K(v ′ ,S) λ u u such that λ u ≥ 0 for all u and such that u∈K(v ′ ,S) λ u = 1.
It is convenient to shift the origin of the problem by subtracting v ′ from all points.
The problem then becomes to determine the convex hull of the points
{−2 sgn(v i )e i : i ∈ S},
where {e 1 , . . . , e n } is the standard basis for R n . This second convex hull is equal
to (x 1 , . . . , x n ) ∈ R n : x i = 0 for all i ∈ S and i∈S −1 2 sgn(v i ) x i = 1 ,
where each term in the sum is nonnegative. An equivalent description of this set is
(x 1 , . . . , x n ) ∈ R n : x i = 0 for all i ∈ S and i∈S sgn(v i )x i = −2 ,
where sgn(v i )x i ≤ 0 for all i ∈ S, and assertion (i) follows.
For (ii), consider a convex combination
x = u∈L(v,S) λ u u such that λ u ≥ 0 for all u and such that u∈L(v,S) λ u = 1.
The fact that there are only two possible entries in the i-th coordinate for each u,
namely ±1, means that the only way we can have x i = c is for all u = (u 1 , . . . , u n ) such that λ u = 0 to satisfy u i = c. The assertion of (ii) follows from this.
Lemma 2.3.2.
(i) If n = 4, the half cube hγ n has precisely 16 facets: eight of these are the polytopes
Π(K(v ′ , {1, 2, 3, 4})) arising with 4-cliques (where v ′ ∈ Ψ − n )Lemma 2.3.3. Let V ′ = K(u ′ , n), where u ′ ∈ Ψ − n , so that Π(V ′ ) is a facet of hγ n . Let V be a set of vertices of the form K(v ′ , S), where v ′ ∈ Ψ − n and S ⊆ n. (i) The intersection Π(V ) ∩ Π(V ′ ) is either empty, or is another polytope of the form Π(K(x ′ , S ′ )), where x ′ ∈ Ψ − n and S ′ ⊆ n. In either case, we must have K(x ′ , S ′ ) = K(v ′ , S) ∩ K(u ′ , n). (ii) If V = V ′ and some interior point of Π(V ) lies in Π(V ′ ), then it must be the case that V ⊂ V ′ and that Π(V ) is contained in the boundary of Π(V ′ ).
Proof. Suppose that x = (x 1 , . . . , x n ) is a point in the intersection of the two polytopes. Since v ′ and u ′ lie in Ψ − n , they must differ in an even number of coordinates. If v ′ = u ′ , then K(v ′ , S) ⊆ K(u ′ , n) and (i) and (ii) follow. If v ′ = u ′ , let i and j be two positions at which v ′ and u ′ differ. In the notation of Lemma 2.3.1 (i) with u ′ = (u 1 , . . . , u n ), this means that sgn(v i ) = − sgn(u i ) and sgn(v j ) = − sgn(u j ). It follows from this that we have
sgn(v i )(x i − v i ) + sgn(u i )(x i − u i ) = −2 and sgn(v j )(x j − v j ) + sgn(u j )(x j − u j ) = −2.v i )(x i − v i ) = −2 and 1≤i≤n sgn(u i )(x i − u i ) = −2 and the sign constraints mentioned in Lemma 2.3.1 (i) is to have both sgn(v i )(x i − v i ) + sgn(v j )(x j − v j ) = −2(1)
and
sgn(u i )(x i − u i ) + sgn(u j )(x j − u j ) = −2.(2)
In order for x to exist, that is, for (1) and (2) to have any common solutions, we We may now assume that {i, j} ⊆ S, in which case (1) and (2) force x k = u k = j k for all 1 ≤ k ≤ n other than k = i and k = j. Let us define x 1 (respectively, x 2 )
need either i ∈ S or j ∈ S or both. If S ∩ {i, j} = {i}, the only solution is x j = v j and x i = u i , meaning that Π(V ) ∩ Π(V ′ ) = x, where x is the common point of V and V ′ that differs from v ′
to be the element of Ψ + n that agrees with v ′ except in the i-th (respectively, j-th) coordinate, so that x 1 and x 2 lie in K(v ′ , S) ∩ K(u ′ , n). We have shown that x 1 and x 2 are the only two vertices that lie in both polytopes, and it follows that x lies on the edge containing x 1 and x 2 . This edge is therefore the intersection of the two polytopes; we thus have Π Note. Note that the inclusion Π
(V ) ∩ Π(V ′ ) = Π(V ∩ V ′ ),(V ∩ V ′ ) ⊆ Π(V ) ∩ Π(V ′ ) in (i) is immediate, but
that the reverse inclusion is not.
Proof. By Lemma 2.3.2, any facet of hγ n is of the form Π(K(u ′ , n)) or of the form Π(L(u, T ′ )), where u ′ ∈ Ψ − n , u ∈ Ψ + n and T ′ ⊂ n satisfies |T ′ | = n − 1. There are four cases to consider. The first, and most complicated case is an intersection of the form
Π(K(v ′ , S)) ∩ Π(K(u ′ , n)),
and this follows from Lemma 2.3.3.
The second of the four cases is an intersection of the form Π(L(v, T )) ∩ Π(K(u ′ , n)).
Suppose that x is an element of the intersection. If i is a coordinate not indexed by T , all the points in Π(L(v, T )) must agree with v in the i-th coordinate, and the same is true for x. We have now reduced to the case where u ′ agrees with v at all coordinates not indexed by T . In order for x to exist, we also need x to agree with v at all coordinates not indexed by T . Since K(u ′ , T ) = L(v, T ) ∩ K(u ′ , n), Lemma 2.3.1 (i) and the fact that x ∈ Π(K(u ′ , n)) now show that x ∈ Π(K(u ′ , T )), which proves (i). Lemma 2.3.2 shows that Π(K(u ′ , T )) lies in the boundary of L(v, T ), from which assertion (ii) follows.
The third case involves an intersection of the form
Π(K(v ′ , S)) ∩ Π(L(u, T ′ )),
which is similar to, but easier than, the second case. If v ′ disagrees with u at the
(v ′ , S) or L(v, T ), where v ′ ∈ Ψ − n , v ∈ Ψ +
n , S ⊆ n and T n, is equal to an intersection of subsets of Ψ + n of the form K(u ′ , n) and/or L(u, T ′ ), where u ′ ∈ Ψ − n , u ∈ Ψ + n and |T ′ | = n − 1.
Proof. Consider a subset of Ψ of the form K(v ′ , S), where v ′ ∈ Ψ − n and S ⊆ n, and observe that K(v ′ , S) is a subset of K(v ′ , n). Suppose that there exists i ∈ n\S.
Let u i ∈ Ψ + be any element agreeing with v ′ in the i-th coordinate (note that such a u does exist), and let S i = n\{i}. It follows that
K(v ′ , S) = K(v ′ , n) ∩ i∈n\S L(u i , S i ).
The assertion for the sets K(v ′ , S) follows by intersecting the simplex shaped facet corresponding to K(v ′ , n) with the bounding hyperplanes corresponding to the sets L(u i , S i ).
The assertion for the sets L(v, T ) follows similarly from the fact that
L(v, T ) = i∈n\T L(v, S i ),
where S i is as above.
Theorem 2.3.6. The k-faces of hγ n for k < n are as follows:
(i) 2 n−1 0-faces (vertices) given by the elements of Ψ + n ; (ii) 2 n−2 n 2 1-faces Π(K(v ′ , S)), where v ′ ∈ Ψ − n and |S| = 2; (iii) 2 n−1 n 3 simplex shaped 2-faces Π(K(v ′ , S)), where v ′ ∈ Ψ − n and |S| = 3; (iv) 2 n−1 n k+1 simplex shaped k-faces Π(K(v ′ , S)), where v ′ ∈ Ψ − and |S| = k + 1 for 3 ≤ k < n;
(v) 2 n−k n k half cube shaped k-faces Π(L(v, S)), where v ∈ Ψ + n and |S| = k for 3 ≤ k < n.
Proof. The numbers in (i), (ii) and (iii) are the numbers of 1-cliques, 2-cliques and 3-cliques in 1 2 H n , which correspond to simplex-shaped faces. Suppose that k ≥ 3. There are 2 n−1 n k+1 (k + 1)-cliques of the form K(v, S) in (iii) Every point of hγ n is either an interior point of hγ n , or a vertex of hγ n , or is an interior point of a (unique) k-face of hγ n for some 0 < k < n.
Proof. We first prove (i). If V ′ is the whole of Ψ + n , then the claim follows immediately, so we may suppose that this is not the case.
Theorem 2.3.6 shows that V ′ is of the form K(v ′ , S) or L(v, T ), where v ′ ∈ Ψ − n , v ∈ Ψ +
n , S ⊆ n and T n. Lemma 2.3.5 then shows that V ′ can be written as a finite intersection
V ′ = r i=1 V i for some r ≥ 1, where each V i is of the form K(u ′ , n) or L(u, T ′ ) for u ′ ∈ Ψ − n , u ∈ Ψ + n and |T ′ | = n − 1.
The proof proceeds by induction on r, and the base case, r = 1, is Lemma 2.3.4 (i). The inductive step follows by writing
V ∩ V ′ = V ∩ r−1 i=1 V i ∩ V r
and appealing to the base case. This completes the proof of (i).
We now turn to (ii). Suppose for a contradiction that x is a common interior point of the distinct faces Π(V ) and Π(V ′ ) (not necessarily of the same dimension).
If Π(V ′ ) is the whole of hγ n then it follows that Π(V ) is not, and hence that Π(V ) is contained in a face of hγ n . This is a contradiction because all points of Π(V ) are contained in the boundary of hγ n , and thus cannot be interior points of
Π(V ′ ).
We may therefore assume that Π(V ′ ) is not the whole of hγ n , and we may write
V ′ = r i=1 V i
as in the proof of (i), so that r ≥ 1. By part (i), we have
Π(V ′ ) = r i=1 Π(V i ),
where the Π(V i ) are facets. Since, for each i, x is an interior point of Π(V ) that lies in Π(V i ), Lemma 2.3.4 (ii) shows that V ⊆ V i for all i, and hence that V ⊆ V ′ .
Reversing the roles of V and V ′ in the argument then shows that V = V ′ , which is a contradiction establishing (ii).
To prove (iii), we first remark that the analogous result for simplices is well known (see [13, Definition 6.2 (2 ′ )]). The uniqueness part of (iii) is immediate from part (ii), so we concentrate on establishing existence, and the proof is by induction on the dimension of the largest face containing the point x in question (where we consider hγ n to be an n-face). The base case, involving a polytope of dimension 0, is trivial because it forces x to be a vertex. For the inductive step, assume x is a point of hγ n . If x is an interior point, we are done; if not, x lies in some facet of hγ n . By Lemma 2.3.2, this facet is either a half cube of lower dimension, in which case we are done by induction, or it is a simplex of lower dimension, in which case we appeal to the known corresponding result for simplices.
The topology of the half cube
In §3, we examine some topological properties of the half cube. The half cube itself is not interesting topologically, since it is homeomorphic to a ball. However, the geometric results of §2 can be used to assemble the k-faces of the half cube hγ n into a CW complex C n in a manner that mirrors its geometric properties (Theorem 3.1.2). The complex C n has a subcomplex C n,k for each 3 ≤ k ≤ n, which is obtained by deleting the interiors of all half cube shaped faces of dimension at least k. Using a combination of discrete Morse theory and the theory of CW complexes, we show (Theorem 3.3.2) that the reduced homology of C n,k is concentrated in degree k − 1.
The half cube as a CW complex.
We now recall the definition of a finite regular CW complex; full details may be found in [20, §8].
If X and Y are topological spaces with A ⊂ X and B ⊂ Y , we define a continuous map g : (X, A) −→ (Y, B) to be a continuous map g : X −→ Y such that g(A) ⊆ B.
If, furthermore, g| X−A : X −A −→ Y −B is a homeomorphism, we call G a relative homeomorphism.
An n-cell, e = e n is a homeomorphic copy of the open n-disk D n − S n−1 , where D n is the closed unit ball in Euclidean n-space and S n−1 is its boundary, the unit (n − 1)-sphere. We call e a cell if it is an n-cell for some n.
If a topological space X is a disjoint union of cells X = {e : e ∈ E}, then for each k ≥ 0, we define the k-skeleton X (k) of X by
X (k) = {e ∈ E : dim(e) ≤ k}.
The CW complexes we consider in this paper are all finite, which means that we can give the following abbreviated definition. (ii) for each k-cell e ∈ E, the map Φ e : (D k , S k−1 ) −→ (e ∪ X (k−1) , X (k−1) ) is a relative homeomorphism.
A subcomplex of the CW complex (X, E, Φ) is a triple (|E ′ |, E ′ , Φ ′ ), where E ′ ⊂ E, |E ′ | := {e : e ∈ E ′ } ⊂ X, Φ ′ = {Φ e : e ∈ E ′ }, and Im Φ e ⊂ |E ′ | for every e ∈ E ′ .
The Euler characteristic χ(X) of a finite CW complex (X, E, Φ) is given by
χ(X) = i≥0 (−1) i α i ,
where α i is the number of i-cells in E.
The complexes considered here have the property that the maps Φ e (regarded as mapping to their images) are all homeomorphisms. Such CW complexes are called regular. Note that the condition onē in the above definition excludes precisely those elements e for whichē = Π(L(v, S)) for some v ∈ Ψ + n and |S| ≥ k. Proof. Suppose that Π(V ) and Π(V ′ ) are faces of hγ n with respective vertex sets V and V ′ . Suppose in addition that V ⊂ V ′ . If V is of the form L(v, S) for some v ∈ Ψ + n and S ⊂ n, then V ′ must also be of the form L(u, S ′ ), where |S ′ | > |S|, because the boundary of a simplex shaped face consists of simplex shaped faces of lower dimension. It follows that if the interior of Π(V ) is missing from E ′ , then the interior of Π(V ′ ) is also missing from E ′ , which implies the statement.
Discrete Morse Theory.
Discrete Morse theory, which was introduced by Forman [11], is a combinatorial technique for computing the homology of CW complexes. By building on work of Chari [8], Forman later produced a version of discrete Morse theory based on acyclic matchings in Hasse diagrams [12]. This version of the theory plays a key role in computing the homology of C n,k . We call a cell of K paired if it lies in (a unique) one of the above pairs, and unpaired otherwise.
If V is a discrete vector field on a regular CW complex K, we define a V -path to be a sequence of cells α 0 , β 0 , α 1 , β 1 , α 2 , . . . , β r , α r+1 such that for each i = 0, . . . , r, (a) each of α i and α i+1 is a codimension 1 face of β i , (b) each (α i , β i ) belongs to V and (c) α i = α i+1 for all 0 ≤ i ≤ r. If r ≥ 0, we call the V -path nontrivial, and if α 0 = α r+1 , we call the V -path closed. Note that all the faces α i have the same dimension, p say, and all the faces β i have dimension p + 1.
Let P be the set of cells of K, together with the empty cell ∅, which we consider to be a cell of dimension −1. The set P becomes a partially ordered set under inclusion. Let H be the Hasse diagram of this partial order. We regard H as a directed graph, in which all edges point towards cells of larger dimension.
Suppose now that V is a discrete vector field on K. We define H(V ) to be the directed graph obtained from H by reversing the direction of an arrow if and only if it joins two cells K 1 ⊂ K 2 for which (K 1 , K 2 ) is one of the pairs of V . If the graph H(V ) has no directed cycles, we call V an acyclic (partial) matching of the Hasse diagram of K. (ii) C n,k is homotopic to a CW complex with no cells in dimension p for any p ≥ k;
in particular, C n,k has zero homology in dimensions p ≥ k;
(iii) the homology of C n,k over Z is concentrated in degree k − 1 and is free.
Proof. We have already seen that C n,k is a regular CW complex. The hypothesis |S| ≥ k together with Lemma 2.2.2 show that V is a discrete vector field. To complete the proof of (i), it suffices by Theorem 3.2.2 (i) to show that there are no nontrivial closed V -paths.
Suppose for a contradiction that α 0 , β 0 , α 1 , β 1 , α 2 , . . . , β r , α r+1 is such a path, where α r+1 = α 0 , r ≥ 0, and for all 0 ≤ i ≤ r we have α i = α i+1 and α i ⊂ β i . By the definition of V -path, each of the (α i , β i ) is a pair of V . By definition of V , we have α 0 = K(v ′ , S) and β 0 = K(v ′ , S ∪ {n}), where v ′ ∈ Ψ − n and n ∈ S. Since α 1 is a codimension one face of β 0 that is different from α 0 , we must have α 1 = K(v ′ , T ), where T = S ∪ {n}\{i} for some 1 ≤ i < n. The fact that n ∈ T then shows that (α 1 , β 1 ) cannot lie in V , which is the contradiction required to prove (i).
To prove (ii), we first observe that every cell Π(K(v ′ , S)) for which |S| ≥ k + 1 is involved in a pair of V , whether or not n is an element of S. It follows that every cell of C n,k of dimension at least k is paired. Since the empty set (of dimension −1) is unpaired in V , the conclusion of (ii) now follows from (i) and Theorem 3.2.2 (ii).
Part (iii) follows from (ii) together with [20, Corollary 8.40 (iii)].
Cellular homology.
Cellular homology is a convenient theory for computing homology groups of CW complexes. We do not recall the full definition here, but instead refer the reader to [20, §8]. However, the basic idea is to introduce, for a CW complex X, a free R-module W k , where R may be the integers or a field. The rank of the R-module W k is the number of k-cells in X, and the complex W * may be equipped with differentials ∂ * so that the homology of W * is the (singular) homology, H * (X), of X. One may optionally consider ∅ to be the unique (−1)-cell of X, which gives rise to reduced cellular homology. Proof. This holds because the chain groups and differentials of the two complexes agree in degrees k and k + 1.
Theorem 3.3.2.
For all 3 ≤ k ≤ n, the reduced homology of the CW complex C n,k is concentrated in degree k − 1, and is free over Z in degree k.
Proof. By Proposition 3.2.3 (iii), the reduced homology of C n,k vanishes in degrees k and higher and is free in degree k. The complex C n,k agrees with X in degrees k − 1 and lower, so Lemma 3.3.1 shows that the homology (and, therefore, the reduced homology) of C n,k agrees with that of X in degrees k − 2 and lower.
Since X is closed, bounded and convex, X is contractible and has trivial reduced homology.
Nonvanishing homology
In §4, we use combinatorial techniques to compute the ranks of the nonzero homology groups of C n,k (Theorem 4.1.2). These ranks can be readily expressed in terms of generating functions, or by more explicit formulas (Theorem 4.1.5), and using these, we can verify the coincidence of the (k − 1)-st Betti number of C n,k with the (k − 2)-nd Betti number of a manifold arising in the theory of hyperplane arrangements (Corollary 4.1.6). We then show that the complex C n,k admits a group of cellular automorphisms arising from the action of the Coxeter group G of type D n (Theorem 4.2.3), which endows the nonzero homology groups with the structure of G-modules.
Betti numbers.
In §4.1, we determine the dimensions of the nonzero homology groups of C n,k appearing in Theorem 3.3.2. This is made easy by the following well-known result. where the extra "1" appears because we are considering nonreduced homology.
By the proof of Theorem 3.3.2, the reduced homology of C n,n+1 is trivial, which implies that χ(n, n + 1) = 1. Let n(i, k) be the number of i-cells in C n,n+1 that do not lie in C n,k . If i < k, we have n(i, k) = 0. On the other hand, if i ≥ k, we have n(i, k) = 2 n−i n i , by Theorem 2.3.6 (v). By comparing χ(n, k) with χ(n, n + 1), we then have
χ(n, k) + n i=k (−1) i n(i, k) = 1.
The result now follows by equating the two above expressions for χ(n, k).
Recall from [7] that the k-equal real hyperplane arrangement V R n,k is the set of points (x 1 , . . . , x n ) ∈ R n such that x i 1 = x i 2 = · · · = x i k for some set of indices 1 ≤ i 1 < i 2 < · · · < i k ≤ n. The manifold M R n,k is defined to be the complement R n −V R n,k . The Betti numbers of M R n,k are special cases of the entries of a particular Pascal-like triangle, T (n, k), defined as follows. (ii) for 0 < k < n, T (n, k) = 2T (n − 1, k − 1) + T (n − 1, k).
We interpret T (n, k) = 0 if k < 0 or k > n, which implies that the recurrence relation in (ii) also holds for k = 0. and this holds by definition of T (n, n − k) since 0 < n − k < n. It is clear that the
coefficient of x k in x k (1 − 2x) k (1 − x)
is 1, which shows that T (k, 0) = 1 for all k, as required. Finally, if k = 0, the coefficient of x n in (1 − x) −1 is 1, showing that T (n, n) = 1 for all n, completing the proof of (i).
Let T 1 (n, k) (respectively, T 2 (n, k)) denote the sequence defined by the statement of (ii) (respectively, (iii)).
We prove that T 1 (n, k) = T (n, k) by induction on n. The base case, n = k = 0, is trivial, so suppose that n > 0. It is then enough to prove that
T 1 (n, k) − T 1 (n − 1, k) = 2T 1 (n − 1, k − 1).
If L is the left hand side of this equation, we have
L = n i=n−k (−1) n−k−i 2 n−i n i − n−1 i=n−1−k (−1) n−1−k−i 2 n−1−i n − 1 i = n i=n−k (−1) n−k−i 2 n−i n i − n i=n−k (−1) n−k−i 2 n−i n − 1 i − 1 = n i=n−k (−1) n−k−i 2 n−i n − 1 i = 2 n−1 i=(n−1)−(k−1) (−1) (n−1)−(k−1)−i 2 n−1−i n − 1 i , which equals 2T 1 (n − 1, k − 1), proving (ii).
To prove (iii), we show that T 1 (n, k) = T 2 (n, k) by induction on k. The base case, k = 0, follows easily. For the inductive step, we recall that Björner and Welker [7, 7.5 (ii)] show that T 2 (n, n − (k + 1)) + T 2 (n, n − k) = 2 n−k n k ; they attribute this observation to V. Strehl. (Note that this identity also holds for k = n − 1, because T 2 (n, 0) = 1 and T 2 (n, 1) = 2n − 1.) Equivalently, we may write T 2 (n, k − 1) + T 2 (n, k) = 2 k n k .
The corresponding identity T 1 (n, k − 1) + T 1 (n, k) = 2 k n k is immediate from (ii), and this completes the proof. (i) The group W (D n ) acts as signed permutations on each of the sets Ψ + n and Ψ − n . This action induces an embedding of W (D n ) into Aut( 1 2 H n ). (ii) There is an injective group homomorphism φ from W (D n ) to the orthogonal group O n (R) with the property that whenever g ∈ W (D n ) and ψ ∈ Ψ n , we have φ(g).ψ = g(ψ).
Proof. Part (i) holds because the signed permutation action (a) fixes each of the two sets Ψ ± n setwise, and (b) respects Hamming distance. For part (ii), recall from [15,Proposition 5.4] that the two sets Ψ ± n are the weights of the two spin modules for the simple Lie algebra of type D n over C. (iv) The groups W (D n ) act naturally on the nonzero homology groups H k−1 (C n,k ) for all 3 ≤ k ≤ n.
Proof. Let g ∈ W (D n ) and consider g acting as a signed permutation. If v ′ ∈ Ψ − n , v ∈ Ψ + n and S ⊆ n, we have
g(K(v ′ , S)) = K(g.v ′ , g(S))
and g(L(v, S)) = L(g.v, g(S)),
where g(S) refers to g acting on the set S as an ordinary permutation, ignoring the sign changes. Since g.v ′ ∈ Ψ − n and g.v ∈ Ψ + n , g induces a permutation of the subsets of Ψ n indexing the faces of hγ n , as in Theorem 2.3.6. In particular, we see that the action of W (D n ) both (a) permutes the k-faces of hγ n for each k and (b) sends faces to faces of the same type (K-type or L-type).
We observe that the vertices of hγ n are the only points x in hγ n for which
x · x = n; this property is inherited from the hypercube, for which it is obvious.
Since the group G n acts by orthogonal transformations, it must therefore permute the vertices of hγ n . In turn, g induces a permutation of the faces of hγ n that respects dimensions, and this completes the proof of (i).
It remains to verify the assertions about the orbits. Observe that if Π(K(v ′ , S)) and Π(K(u ′ , S ′ )) are two faces of hγ n for which |S| = |S ′ |, then we can find an element x ∈ S n for which x.S = S ′ . Furthermore, the fact that v ′ and u ′ lie in Ψ − n means that the elements x.v ′ and x.u ′ of Ψ − n differ by an even number of sign changes. It follows that Π(K(v ′ , S)) and Π(K(u ′ , S)) are conjugate under the action of W (D n ), and a similar remark applies to faces Π(L(v, S)) and Π(L(u, S ′ )) in the case where |S| = |S ′ |.
To complete the proof, we must show that if |S| = |T | = r, then two k-faces of the form Π(K(v ′ , S)) and Π(L(v, T )) cannot be conjugate by an element g ∈ G n .
Since the action of g respects convex hulls, it must induce a bijection between the vertex sets K(v ′ , S) and L(v, T ). These two sets have cardinalities k + 1 and 2 k−1 , respectively, and the only integer solution to the equation k + 1 = 2 k−1 for k ≥ 3 is k = 3. However, the assumption that n ≥ 5 means that the 3-face Π(K(v ′ , S)) occurs as a codimension 1 face of the 4-face K(v ′ , S ∪ {i}), where i ∈ S, whereas the 3-face L(v, T ) is not a codimension 1 face of any 4-face. This shows that no such g can exist, completing the proof of (ii).
The proof of (iii) follows the same line of argument as the proof of (ii), except when k = 3. If k = 3, it can be checked by hand that the two W (D 4 )-orbits of 3-faces are as claimed. However, if g is the orthogonal transformation given by reflection perpendicular to the vector (1, 1, 1, 1), we find that g stabilizes the set Ψ + n , and thus the half cube hγ n . However, we also have g(L ((1, 1, 1, 1), {1, 2, 3})) = K((−1, −1, −1, 1), {1, 2, 3, 4}), which proves that there is one G 4 -orbit of 3-faces.
To prove (iv), we first observe that any g ∈ W (D n ) maps the i-skeleton of the CW complex C n,k to itself for all 0 ≤ i ≤ n; this follows from the definition of C n,k together with fact that the action of W (D n ) preserves the type of each face. The filtration of a CW complex by its sequence of i-skeletons is an example of a cellular filtration (see [20,Theorem 8.38]), and the property of W (D n ) just mentioned means that g acts on C n,k by a cellular map, which is a homeomorphism by invertibility of g. The remarks before [20,Theorem 8.38] also show that g induces an automorphism g * on the nonzero homology groups of C n,k .
Remark 4.2.4.
The unexpectedly large symmetry group of hγ n for n = 4 occurs because in this case, hγ n is the 4-dimensional hyperoctahedron, also known as the "16-cell". This is a regular polytope, called a "cross polytope" in [9] and denoted by β 4 .
Concluding remarks.
We conclude with a number of problems and questions which we believe would be interesting topics for future research. One obvious natural question is the following. Barcelo et al [2] introduced a combinatorial homotopy theory for simplicial complexes and graphs, known as A-theory. Work of Babson et al [1] and unpublished work of Björner shows that the nonzero Betti numbers for M R n,k also show up in the context of A-theory as the ranks of A-groups. Furthermore, the coincidence between these numbers is fairly well understood, and in the simplest case (k = 3), the problem of computing the Betti numbers reduces to the computation of the fundamental group of the permutahedron after attaching a 2-cell to each square; see [3,Theorem 5.4] for more details. The connection between A-theory and hyperplane arrangements gives rise to another formula for the numbers T (n, n − k) in Theorem 4.1.5. Like the formula in part (iii) of that result, it consists of a sum of products of positive numbers. We did not present it here in order to save space; however, Barcelo and Smith [4, §5] give complete details in the case k = 3, and their arguments adapt easily to the case of general k. [14]. The symmetric group S n , which is a less complicated group than W (D n ), acts on the homology of the manifold M R n,k , but (to the best of our knowledge) not much is known about the representations that arise in this way. It is possible that progress could be made on Problem 4.3.4 via the next question. Since the (k − 1)-st homology of C n,k is free over Z, a natural question is Question 4.3.6. Is C n,k homotopic to a wedge of (k − 1)-spheres?
It seems likely that this question could be answered in the affirmative using the homology version of Whitehead's Theorem [16,Proposition 4C.1].
Example 2 .2. 3 .
23Let n = 5 and v 1 = (1, −1, −1, 1, −1). Then we have K(v 1 , {3}) = {(1, −1, 1, 1, −1)}, K(v 1 , {3, 4}) = {(1, −1, 1, 1, −1), (1, −1, −1, −1, −1)},K(v 1 , {2, 3, 4}) = {(1, 1, −1, 1, −1), (1, −1, 1, 1, −1), (1, −1, −1, −1, −1)}. In contrast to Lemma 2.2.2, we have K(v 1 , {3, 4}) = K(v 2 , {3, 4}), where v 2 = (1, −1, 1, −1, −1), so that K(v 1 ,{3, 4}) has two opposite points, but only one mask.There are a total of 5 pairs, (v 1 , S) giving rise to the singleton set K(v 1 , {3}), and the set has 5 opposite points, each with a different mask.
Definition 2 .2. 4 .
24Let n = {1, 2, . . . , n}, v ∈ Ψ + and let S ⊆ n. We define the subset L(v, S) of 1 2 H n by the condition that v ′ ∈ L(v, S) if and only if v and v ′ agree in the i-th coordinate whenever i ∈ S. The set S is called the mask of L(v, S); it can be characterized as the set of coordinates at which not all points of L(v, S) agree.
.
Let n = 5, v = (1, −1, −1, 1, 1) and S = {1, 3, 4}. Then we have L(v, S) = {(1, −1, −
k = 3, then the preceding arguments show that C = {v 1 , v 2 , v 3 } is of the form K(v ′ , {a, b, c}), where a, b and c are as in the previous paragraph. Suppose that C ′ = {v 1 , . . . , v 4 } is a 4-clique extending C. If v 4 differs from v 1 in the a-th coordinate, then it also differs from v 1 in the d-th coordinate for some d ∈ {a, b, c}, and we have C ′ = K(v ′ , {a, b, c, d}). The other possibility is that v 4 agrees with v 1 in the a-th coordinate, which means that it disagrees with each of v 2 and v 3 in the a-th coordinate. Because v 4 = v 1 and {v 2 , v 4 } is a 2-clique, it must be the case that v 4 agrees with v 2 in the b-th coordinate, and a similar argument shows that v 4 agrees with v 3 in the c-th coordinate. The fact that {v 1 , v 4 } is a 2-clique then shows that v 4 agrees with v 1 at all but two coordinates: b and c. In summary, we have C ′ = L(v 1 , {a, b, c}), completing the analysis of the case k = 3.
then C cannot be extended to a 5-clique C ′ = {v 1 , . . . , v 5 }. If this were the case, then the 4-clique {v 1 , v 2 , v 3 , v 5 } would be an extension of {v 1 , v 2 , v 3 }, and the argument of the previous paragraph combined with the fact that v 4 = v 5 then shows that v 5 would differ from v 1 in two coordinates, a and e, where e ∈ {a, b, c}. This would imply that v 4 and v 5 would differ in four coordinates, a, b, c and e, contradicting the requirement that {v 4 , v 5 } be a 2-clique.
Remark 2 . 2 . 7 .
227The above proof shows that, if n > 4, a 4-clique is of the form K(v, S) if it can be extended to a 5-clique, and is of the form L(v, S) otherwise. In particular, there can be no automorphism of the graph 1 2 H n exchanging 4-cliques of different types.
a point in the convex hull of the set L(v, S) for some v ∈ Ψ + n and S ⊆ n. If x i = c for c ∈ {1, −1} then x is a convex combination of a subset of points of L(v, S) all of which agree with x in the i-th coordinate.
and the other eight are the polytopes L(v, S) associated with the 4-cliques obtained as S ranges over the eight subsets of n and v ∈ Ψ + n . (ii) Suppose that n > 4. Then the number of facets of the half cube hγ n is 2 n−1 + 2n. Of these, 2 n−1 facets are the polytopes Π(K(v ′ , {1, . . . , n})) arising from n-cliques (where v ′ ∈ Ψ − n ) and the other 2n are the polytopes Π(L(v, S)), where v ∈ Ψ + n and S ⊂ n satisfies |S| = n − 1. Proof. The proof of part (i) can be regarded as special case of the proof of part (ii), if we define hγ n for n = 3 in the obvious way as the convex hull of alternating points on a regular cube (namely L(v, S) for suitable v and S). We will therefore only prove (ii). Coxeter [9, §8.6] proves that the half cube hγ n has 2 n−1 facets that are regular simplices, and 2n facets that are half cubes of smaller dimension. The alternation construction described in §2.1 creates no new vertices and shows that there are no other facets. General properties of facets show that the vertices of facets must be vertices of the original polytope. This reduces the problem to finding 2 n−1 different regular simplices and 2n different half cubes, all of dimension n − 1, whose vertices are contained in the set of vertices for hγ n .For the simplices, note that the cliques K(v ′ , {1, . . . , n}) as v ′ ranges over the points of Ψ − n are distinct by Lemma 2.2.2. They exhaust the possible simplices on the vertex set of hγ n by Proposition 2.2.6, so these must be the 2 n−1 faces described in[9, §8.6].
For the half cubes, note that there are 2n distinct sets of the form L(v, S) as S ranges over the subsets of n of size n − 1, and each of these 2n sets contains 2 n−2 points of Ψ + n . (Note that for each set, there are 2 n−1 possible choices for v.) If one deletes the coordinate of the points of L(v, S) that is not indexed by S, then one obtains either Ψ + n−1 or Ψ − n−1 , which shows that L(v, S) is isometric (by applying a reflection in one coordinate if necessary) to a half cube of dimension n − 1. The mask S of L(v, S) is characterized as the set of coordinates at which not all 2 n−2 elements of L(v, S) agree, from which it follows that there are 2n distinct subsets of the form L(v, S). However, the sets L(v, S) can also be characterized as those vertices that lie in some fixed coordinate plane of the form x i = ±1, and these are bounding faces of hγ n , so by Lemma 2.3.1 (ii), the convex hulls of the 2n distinct subsets are faces of hγ n that are also half cubes of dimension n − 1. They are therefore equal to the 2n faces described in [9, §8.6], and the statement follows.
that all four terms being added are nonpositive, and the only way this can be compatible with the inequalities 1≤i≤n sgn(
only in the i-th coordinate, and differs from u ′ only in the j-th coordinate. This satisfies (i), and (ii) holds vacuously because x is not an interior point of Π(V ). A similar argument deals with the case S ∩ {i, j} = {j}.
satisfying (i). Either this edge is the whole of Π(V ), or it lies entirely within the boundary of Π(V ), and in either case, (ii) holds.
.
Let Π(V ′ ) be a facet of hγ n with vertex set V ′ . Let V be a set of vertices of the form K(v ′ , S) or L(v, T ), where v ′ ∈ Ψ −n , v ∈ Ψ + n , S ⊆ n and T ⊂ n satisfies 2 ≤ |T | < n.
( i )
iThe intersection Π(V )∩Π(V ′ ) is either empty, or is another polytope of the formΠ(V ′′ ), where V ′′ is of the form K(t ′ , S ′ ) or L(t, T ′ ), and we have t ′ ∈ Ψ − n , t ∈ Ψ +n , S ′ ⊆ n and T ⊂ n. In either case, we must have V ′′ = V ∩ V ′ .(ii) If V = V ′ and some interior point of Π(V ) lies in Π(V ′ ), then it must be the case that V ⊂ V ′ and that Π(V ) is contained in the boundary of Π(V ′ ).
If u ′ disagrees with v in the i-th coordinate, the only solution for x compatible with Lemma 2.3.1 (i) is for x to be the point of K(u ′ , n) that disagrees with u ′ only in the i-th coordinate. Even this can only happen if x agrees with v in all other coordinates not indexed by T . This proves that x is the unique element of V ∩ V ′ in this case, and (i) and (ii) follow because x cannot be an interior point of either polytope.
unique coordinate not indexed by T ′ , then the intersection of the two polytopes is the unique point of intersection of the sets V and V ′ , which proves (i) and(ii) because this point cannot be interior to Π(V ).Suppose on the other hand that v ′ agrees with u at the coordinate not indexed by T ′ , and let x be a point in the intersection of the two polytopes. Since x lies in Π(L(u, T ′ )), x agrees with u at the coordinate not indexed by T ′ , and it follows that x agrees with v ′ at the coordinate not indexed by T ′ . Lemma 2.3.1 (i) thenshows that x lies in Π(K(v ′ , S ∩ T ′ )). Assertion (i) follows from the observation thatK(v ′ , S) ∩ L(u, T ′ ) = K(v ′ , S ∩ T ′ ),and assertion (ii) follows from the fact that either S ⊆ T ′ or Π(K(v ′ , S ∩ T ′ )) lies in the boundary of the simplex Π(K(v ′ , S)).The fourth and final case deals with the intersectionΠ(L(v, T )) ∩ Π(L(u, T ′ ));recall that |T ′ | = n − 1 in this case. If i ∈ T ∪ T ′ and v and u disagree at the i-th position, then the intersection is empty and we are done, so suppose that x is a point in the intersection.Suppose first that T ⊆ T ′ . In this case, x must agree with each of v at all coordinates outside T ′ , and x must agree with u at all coordinates outside T . Letx ′ be a point of Ψ + n that agrees with v (respectively, u) at all points outside T ′ (respectively, T ); such a point exists because |T | ≥ 2 by assumption, and thus |T ∩ T ′ | ≥ 1. Lemma 2.3.1 (ii) now shows that x lies in Π(L(x ′ , T ∩ T ′ )). Since L(v, T ) ∩ L(u, T ′ ) = L(x ′ , T ∩ T ′ ), we have shown that Π(L(v, T )) ∩ Π(L(u, T ′ )) = Π(L(x ′ , T ∩ T ′ )), which proves (i) in this case. If i is the unique coordinate in T \(T ∩ T ′ ), then the i-th coordinate of x is ±1. This means that x is not an interior point of Π(V ), and (ii) follows. The other possibility is that T ⊆ T ′ , and that v and u agree at all coordinates not indexed by T ′ . In this case, we have L(v, T ) ⊆ L(v, T ′ ) = L(u, T ′ ) and Π(L(v, T )) ∩ Π(L(u, T ′ )) = Π(L(v, T )), which proves (i). Assertion (ii) now follows from Lemma 2.3.2. Lemma 2.3.5. Any subset of Ψ + n of the form K
1 2
1H n : they are determined by Lemma 2.2.2 by their n k+1 possible masks and by their 2 n−1 possible opposite points. These are the simplex-shaped faces. The half cube shaped faces of dimension k have n k possible masks. For each mask there are 2 n−k values for coordinates outside the mask. This proves (iv) and (v) and completes the proof.
Remark 2.3. 7 .
7If k = 3, parts (iv) and (v) of Theorem 2.3.6 give two types of simplex shaped 3-faces, corresponding to the two types of 4-cliques in hγ n . If n > 4, they can be distinguished by the fact that the first type of 4-clique appears as the face of a 5-clique, and the other does not. Theorem 2.3.8.
( i )
iIf Π(V ) and Π(V ′ ) are faces of hγ n , where V and V ′ are as in Theorem 2.3.6, then Π(V ) ∩ Π(V ′ ) = Π(V ∩ V ′ ).
( ii )
iiNo point of hγ n can be an interior point of more than one face.
Definition 3.1.1. A CW complex is an ordered triple (X, E, Φ), where X is aHausdorff space, E is a family of cells in X, and {Φ e : e ∈ E} is a family of maps, such that (i) X = {e : e ∈ E} is a disjoint union;
.
Let X be the half cube hγ n regarded as a subspace of R n , and let E be the union of the following three sets:(a) the set of vertices of hγ n ;(b) the set of interiors of all k-faces of hγ n for all 0 < k < n;(c) the interior of hγ n itself.Then (X, E, Φ) is a regular CW-complex, where the maps Φ e are the natural identifications.Proof. It is a standard result that the faces of a convex polytope form a regular CW complex. The result now follows from Theorem 2.3.8.
.
Let E be as in Theorem 3.1.2, and let k ≥ 3. We define the subset E(k) of E to consist of all elements e ∈ E except those for whichē is an isometric copy of an l-dimensional half-cube for some l ≥ k.
Example 3 .1. 4 .
34If k = 3 in Definition 3.1.3, the effect is to delete the interiors of all the faces of hγ n that are isometric to k-dimensional half cubes, including the tetrahedral faces of this type. However, the tetrahedral faces corresponding to 4-cliques of the form K(v ′ , S) are retained, as are all simplex shaped faces of higher dimensions. The case k = 4 differs only from the case k = 3 in that it includes the tetrahedral faces corresponding to 4-cliques of the form L(v, S). Proposition 2.2.6 then shows that C n,4 is the clique complex of the half cube graph 1 2 H n . At the other extreme, if k > n in Definition 3.1.3, we have E(k) = E. If k = n, then E(k) includes all but one element of E, namely the n-dimensional interior of hγ n . Proposition 3.1.5. For each 3 ≤ k ≤ n, the CW complex (X, E, Φ) of Theorem 3.1.2 has a subcomplex (|E ′ |, E ′ , Φ ′ ), where E ′ = E(k). We denote this subcomplex by C n,k .
.
Let K be a finite regular CW complex. A discrete vector field on K is a collection of pairs of cells (K 1 , K 2 ) such that (i) K 1 is a face of K 2 of codimension 1 and (ii) every cell of K lies in at most one such pair.
). Let V be a discrete vector field on a regular CW complex K.
( i )
iThere are no nontrivial closed V -paths if and only if V is an acyclic matching of the Hasse diagram of K.(ii) Suppose that V is an acyclic partial matching of the Hasse diagram of K in which the empty set is unpaired. Let u p denote the number of unpaired p-cells. Then K is homotopic to a CW complex with exactly u p cells of dimension p for each p ≥ 0. Proof. Part (i) is [12, Theorem 6.2] and part (ii) is [12, Theorem 6.3].
.
Let V k be the set of pairs (Π(K(v ′ , S)), Π(K(v ′ , S ′ ))) of cells in C n,k such that |S| ≥ k, S ⊂ S ′ and S ′ \S = {n}. Then:(i) V k is an acyclic matching on the Hasse diagram of C n,k ;
.
Let X be a CW complex and let Y be a subcomplex of X. Suppose that k ≥ 0 and that for i ∈ {k, k + 1}, every i-cell of X is also an i-cell of Y . Then we have H k (X) ∼ = H k (Y ).
Lemma 4.1.1 [16, Theorem 2.44]. Let (X, E, Φ) be a finite CW complex. Then we haveχ(X) = i≥0 (−1) i rankH i (X).
.
The rank of the (k − 1)-st homology group of C n,k is given by Let χ(C n,k ) denote the Euler characteristic of the complex C n,k . From Theorem 3.3.2 and Lemma 4.1.1, we see that the (k − 1)-st homology of C n,k satisfies 1 + (−1) k−1 rankH k−1 (C n,k ) = χ(n, k),
Definition 4.1.3 [21, sequence A119258]. Let n ≥ 0 and 0 ≤ k ≤ n be integers.We define the sequence T (n, k) by the conditions (i) T (n, 0) = T (n, n) = 1;
Corollary 4 .1. 6 .
46If 3 ≤ k ≤ n, the number T (n, n − k) is equal both to (i) the (k − 1)-st Betti number of the complex C n,k and (ii) the (k−2)-nd Betti number of the complement M R n,k of the k-equal real hyperplane arrangement. Proof. Part (i) follows from comparing Theorem 4.1.5 (ii) with Theorem 4.1.2. Part (ii) follows from [7, §7.5]. 4.2. Homology representations.
.
Let Aut( 1 2 H n ) be the automorphism group of the half cube graph1 2 H n defined in §2.2. Regard the group Z 2 ≀ S n as acting as signed permutations on n objects. Let W (D n ) be the subgroup of Z 2 ≀ S n consisting of all elements that effect an even number of sign changes.The group W (D n ) is isomorphic to the Coxeter group of type D n ; see[17, §2.10] for more details.
The action of W (D n ) on these sets via orthogonal transformations in R n is given by[15, Proposition 3.6, Lemma 5.3]. This action restricts to the action as signed permutations by the remarks in [17, §2.10].
.
Let G n be the full subgroup of O(R n ) stabilizing hγ n setwise, and consider W (D n ) as a subgroup of G n as in Lemma 4.2.2.(i) The groups W (D n ) and G n permute the k-faces of hγ n for each k.
( ii )
iiIf n > 4, the G n -orbits of k-faces coincide with the W (D n ) orbits. There are two orbits of k-faces for 3 ≤ k < n, and one orbit of k-faces otherwise. In the case where there are two orbits, the orbits are distinguished by whether they correspond to a set K(v ′ , S) or a set L(v, S) in Theorem 2.3.6. (iii) If n = 4, there is one G n -orbit of k-faces for each 0 ≤ k ≤ n. The W (D n )-orbits coincide with the G n -orbits, except that there are two W (D n )-orbits of 3-faces; these are distinguished by whether they correspond to a set K(v ′ , S) or a set L(v, S) in Theorem 2.3.6.
Problem 4 .3. 1 .
41Give a conceptual explanation for the numerical coincidence of Betti numbers in Corollary 4.1.6.
.
Find a direct relationship between A-theory and the complexes C n,k .
Find an explicit basis for the nonzero homology groups of C n,k .The two identities mentioned above that express the Betti numbers as sums of products of positive integers may give a clue to the nature of such a basis. An answer to Problem 4.3.3 would also help with the following problem.
Find the characters (over C) of the representations of W (D n ) afforded by the nonzero homology groups.
likely to be difficult in general, even though the character theory of the group W (D n ) is well understood
Does the recurrence relationT (n, k) = 2T (n − 1, k − 1) + T (n − 1, k)correspond to a branching rule obtained by restricting the homology representation arising from C n,k to the subgroup W (D n−1 )?
).Lemma 2.2.2. Let K(v, S) be a set of the form given in Definition 2.2.1. If
|K(v, S)| ≥ 3, then the point v and the set S are each determined by the set
T (n, n − k) − 2T (n − 1, n − 1 − k) = T (n − 1, n − k),
AcknowledgementsI thank the referee for reading the paper quickly and carefully, for simplifying and correcting some of the arguments, and for pointing out that the homology of C n,k is in fact free over Z. I also thank Tim Penttila for some helpful conversations.Example 4.1.4. The entries of T (n, k) for 0 ≤ n ≤ 6 and 0 ≤ k ≤ n are as follows: (i) The generating function ∞ n=0 T (n, n − k)x n is given by.(ii) We havewhere we interpret −1 −1 to mean 1.Proof. To prove (i), we note that if k > 0, we have the identityIf n − k > 0, this expresses the recurrence relation
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[
"Hidden Simplicity of Gauge Theory Amplitudes",
"Hidden Simplicity of Gauge Theory Amplitudes"
]
| [
"J M Drummond \nUniversité de Savoie\nCNRS\nB.P. 110F-74941Annecy-le-Vieux CedexFrance\n"
]
| [
"Université de Savoie\nCNRS\nB.P. 110F-74941Annecy-le-Vieux CedexFrance"
]
| []
| These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N = 4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries. | 10.1088/0264-9381/27/21/214001 | [
"https://arxiv.org/pdf/1010.2418v2.pdf"
]
| 118,478,665 | 1010.2418 | 5320b6a8341f9be834a6fc98a5bd2f3b3147a7d6 |
Hidden Simplicity of Gauge Theory Amplitudes
18 Jan 2011
J M Drummond
Université de Savoie
CNRS
B.P. 110F-74941Annecy-le-Vieux CedexFrance
Hidden Simplicity of Gauge Theory Amplitudes
18 Jan 2011
These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N = 4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries.
Introduction
There are many reasons for studying scattering amplitudes in gauge theories. An obvious current motivation is the need to understand QCD processes in sufficient detail to distinguish new physics from the otherwise overwhelming background at the Large Hadron Collider. The study of the S-matrix can help in the search for new tools for realising this program. Many of the techniques for efficient calculation of scattering amplitudes were first developed using the maximally supersymmetric Yang-Mills theory as a testing ground (see e.g. [1]). The amplitudes of this theory are similar in general structure to QCD amplitudes but simpler. Moreover they are sufficiently non-trivial to reveal many interesting and surprising mathematical structures governing the behaviour of the S-matrix.
In fact the planar N = 4 theory exhibits an infinite-dimensional symmetry, known as Yangian symmetry which is generated by two distinct versions of superconformal symmetry. The existence of an infinite-dimensional symmetry is a reflection of the integrable structure which is believed to govern physical quantities of the planar theory. This leads to the hope that a solution of the planar S-matrix might be found in this theory which would be a remarkable example of solvability in an interacting four-dimensional gauge theory. This in itself provides another reason for studying the S-matrix of gauge theories.
These lectures will focus on the second motivation. The aim is to review some of the developments that have occurred in recent years with the focus on the symmetries of the S-matrix of N = 4 super Yang-Mills theory. However some of the techniques that we will discuss in these notes have wider application than the N = 4 theory, in particular the use of recursion relations to derive tree-level amplitudes. Moreover it is to be hoped that the observed symmetries underlying planar N = 4 super Yang-Mills theory will lead to a better understanding of the S-matrix in gauge theory more generally. One of the main lessons learned from the existence of recursive techniques and extended symmetries is that manifest locality obscures the general underlying structure of the S-matrix. This is a statement which is not tied to the supersymmetry of the underlying gauge theory.
We will present the solution for the tree-level S-matrix obtained from recursion relations and discuss the appearance of non-trivial symmetries. We will then go on to review the structure of the planar perturbative expansion. Much progress has also been made on the structure of the S-matrix at strong coupling which is accessible in this theory via the AdS/CFT conjecture. There is much overlap between the two regimes of the theory, in particular in the way extended symmetries make an appearance, constraining the form of the S-matrix, and in a relation between the amplitudes and certain light-like Wilson loops. We refer the reader to [2,3,4,5] for more details of the progress made in solving for the amplitudes at strong coupling.
We will begin by reviewing a few basic properties of tree-level scattering amplitudes in gauge theories in general in Section 2. We will discuss the colour structure, helicity structure and general analytic properties of such amplitudes. More details on these topics can be found in the lecture notes of Dixon [6]. We will also introduce N = 4 on-shell supersymmetry as we will focus on this theory for most of the course. We will then go on to show in Section 3 how amplitudes can be solved for recursively by exploiting their analytic structure. As we will see, in the N = 4 theory this will lead to a complete solution for the tree-level S-matrix. The explicit form of the S-matrix of N = 4 super Yang-Mills theory will reveal an unexpected symmetry, namely dual superconformal symmetry which we discuss in detail in Section 4. We will see that the full symmetry is the Yangian of the superconformal algebra. Finally Section 5 will be a brief review of amplitudes at loop level. In particular we discuss the relation to Wilson loops and the way the extended symmetry is exhibited beyond tree level.
2 Tree-level gauge theory scattering amplitudes We will begin by considering pure Yang Mills theory. The action of this theory is
S = − d 4 xTr 1 4 F µν (x)F µν (x) ,(1)
where
F µν = ∂ µ A ν − ∂ ν A µ − ig 2 [A µ , A ν ].(2)
Here A µ = A a µ T a and F a µν T a are the Yang-Mills connection and curvature respectively, while T a are the generators of the gauge group. The action of course has a gauge symmetry -this is the price of making the locality of the theory manifest. In order to write down Feynman rules the gauge should be fixed. In the end we will be interested in gauge-invariant on-shell amplitudes so the choice of gauge will not matter, however it does mean that individual Feynman diagrams are not gauge invariant. In order to compute amplitudes using Feynman diagrams we must therefore carry around a lot of intrinsically unphysical information which in the end will cancel out. The consequence is that intermediate expressions for even the tree-level amplitudes are much more complicated than the final results.
All of this suggests that there is a better way of expressing scattering amplitudes which does not refer directly to the existence of the gauge-invariant action (1). We will indeed see that once we drop the requirement that amplitudes are built from rules derived from a local action we will gain a huge simplicity in the form of the scattering amplitudes. Moreover we will find that new symmetries are revealed which are simply invisible at the level of the Yang-Mills action.
Let us begin by thinking about the analytic structure of the tree-level diagrams generated by the Yang-Mills action. For simplicity we will choose to work in Feynman gauge ∂ µ A µ = 0. In this gauge the propagators and vertices are particularly simple. We show them in Fig. 1.
In order to compute a leading-order on-shell gluon scattering amplitude we then consider all amputated tree-level diagrams contributing to the correlation function, A a 1 µ 1 (p 1 ) . . . A an µn (p n ) . There are various aspects of the calculation which need to be organised. Firstly let us deal with the colour structure. We will be interested in SU(N) gauge groups. Every instance of the SU(N) structure constants f abc can be written in terms of the generators
f abc = Tr(T a T b T c ) − Tr(T a T c T b ).(4)
When two adjoint indices a and b are contracted via a propagator we can use the relation
T a iī T a jj = δj i δ¯i j − 1 N δ¯i i δj j .(5)
These replacements can be represented diagrammatically as in Fig. 2.
Now if we take into account all propagators and vertices and vertices in a diagram we find that we end up with one term which is a single trace going round the whole tree-level diagram plus other terms which are related to this one by swapping over all external legs in all possible ways (the 1/N terms from (5) always drop out). Thus in the end the amplitude can be expressed as a sum over non-cyclic permutations of a single cyclically ordered partial amplitude A µ 1 ,...,µn (p 1 , . . . , p n ),
A a 1 µ 1 . . . A an µn = σ∈Sn/Zn
Tr(T a σ (1) . . . T a σ(n) )A µ 1 ...µn (p σ(1) , . . . , p σ(n) ).
Let us consider the physical on-shell degrees of freedom of Yang-Mills theory. This is most easily captured by the spinor helicity formalism. A four-momentum p µ i can be thought of as a bispinor after contraction with spin matrices (σ µ ) αα ,
p αα i = (σ µ ) αα p µ i .(7)
The square p 2 i is then the determinant of the 2 × 2 matrix p αα . The on-shell condition then says that the momentum can be expressed as the product of two commuting spinors
det(p αα ) = 0 ⇐⇒ p αα i = λ α iλα i .(8)
Hereλα = ±λα = ±(λ α ) * with the sign given by the sign of the energy component of p. Note that the λ andλ are not unambiguously fixed by the definition (8). The ambiguity is given by the freedom to rescale by a phase,
λ −→ e iφ λ,λ −→ e −iφλ .(9)
This rescaling is generated by the helicity operator,
h = 1 2 −λ α ∂ ∂λ α +λα ∂ ∂λα .(10)
By convention we have chosen λ to have helicity − 1 2 andλ to have helicity 1 2 .
The free Yang-Mills field equations ∂ µ F µν = 0 and Bianchi identity ∂ [µ F νρ] = 0 are expressed in the two-component notation as
∂ αα F αβ = 0, ∂ αα Fαβ = 0,(11)
where F αβ and Fαβ are the self-dual and anti-self-dual parts of the field strength respectively,
F ααββ = F αβ ǫαβ + Fαβǫ αβ .(12)
Note that F αβ and Fαβ are symmetric in their indices.
The equations (11) can be written in terms of the momentum p αα = λ αλα as
λ αλα F αβ = 0 =⇒ F αβ = λ α λ β G + ,(13)λ αλα Fαβ = 0 =⇒ Fαβ =λαλβG − .(14)
Thus we see that the Yang-Mills equations admit two on-shell solutions described by G + and G − , the positive and negative helicity gluon states, carrying helicity 1 and −1 respectively (so that F has no helicity).
We are interested in the scattering of these on-shell states so our amplitudes will be characterised by an ordered sequence of + and − signs and we will have an on-shell momentum for each particle,
p αα i = λ α iλα i .(15)
Our conventions will be that all particles are incoming. Of course an incoming positive helicity particle is equivalent to an outgoing negative helicity particle and vice-versa.
The colour-ordered, partial amplitudes can be obtained from A µ 1 ...µn of (6) by contracting each of the Lorentz indices with the appropriate polarisation vector. The polarisation vectors can be defined with the help of auxiliary light-like momenta l αα
i = µ α iμ iα , ǫ αα +,i =λα i µ α i λ i µ i , ǫ αα −,i = λ α iμ iα [λ iμi ] .(16)
Here we have introduced the notation for the spinor scalar products,
ab = a α b α = a α b β ǫ βα , [ab] = aαbα = aαbβǫαβ.(17)
So an ordered amplitude is given by, for example
A(+, +, −, . . . , +, −) = ǫ µ 1 +,1 ǫ µ 2 +,2 ǫ µ 3 −,3 . . . ǫ µ n−1 +,n−1 ǫ µn −,n A µ 1 ...µn (p 1 , . . . , p n ).(18)
The amplitude does not depend on the auxiliary momenta µ i used to define the polarisation vector. A shift in µ i is simply a gauge transformation. For example, under the shift µ −→ µ + δµ we have
δǫ αα + =λα δµ α λµ −λαµ α λδµ λµ 2 =λα δµ α λµ −λαµ α λδµ λµ 2 .(19)
Using the cyclic identity a α bc + b α ca + c α ab = 0 we have
δǫ αα + = λ αλα δµµ λµ 2 = p αα δµµ λµ 2 .(20)
The overall factor of p αα means that the variation of the polarisation vector contributes nothing to the amplitude due to the Ward identity
p µ A µ (p) . . . = 0.(21)
Thus the amplitudes depend on the variables {λ i ,λ i } only.
When expressed in momentum space, amplitudes will always have an overall factor of δ 4 (p) = δ 4 (p 1 + . . . + p n ) as a consequence of translation invariance,
A n = δ 4 (p)A n .(22)
After stripping off the overall momentum conserving delta function, we can think of the scattering amplitudes as being given by a single Lorentz-invariant rational function A n of the ordered set of spinors {λ 1 ,λ 1 . . . , λ n ,λ n } with only local poles of the form, 1
(p i + p i+1 + . . . + p j ) 2 .(23)
The poles originate from the propagators in the Feynman diagram expansion. The fact that the momenta in the denominator form an ordered sum (p i + p i+1 + . . . + p j ) is due to the fact that we are considering the ordered partial amplitude. The presence and structure of the poles are crucial analytic properties which we will need in order to be able to solve for all tree-level amplitudes.
We can classify amplitudes according to their helicity structure. At tree level the amplitudes with no negative helicity gluon or only one negative helicity gluon vanish. This can be seen by making a suitable choice of polarisation vectors but we will shortly see a symmetry-based argument for why this is the case. Those with two negative helicity gluons and the rest positive helicity are called the maximally helicity-violating (MHV) amplitudes. Those with three negative helicities are called next-to-MHV (NMHV) and so on. The names come from the fact that a helicity-conserving amplitude would have the same incoming and outgoing helicity structure. In the convention with all incoming particles this means an equal number of positive and negative helicity particles (obviously this would require an even number of particles in total). Therefore the helicity configuration furthest away from an equal number is called maximally-helicity-violating. By parity the amplitudes must also have at least two positive helicity gluons (those with exactly two are called the anti-MHV or MHV amplitudes). The simplest non-trivial amplitude is therefore the four-particle amplitude which is both MHV and MHV. The classification is illustrated in Fig. 3.
So far we have discussed pure Yang-Mills theory where the only scattering particles are the two gluon states. More general gauge theories will have additional particles describing the particular matter content of the theory. At tree level the pure gluon scattering amplitudes however are common to every gauge theory, regardless of the matter content. This is simply because the only Feynman diagrams which arise in their calculation are the ones with gluons on every line. There is one gauge theory which surpasses all others in its remarkable properties which is the maximally supersymmetric one, N = 4 super Yang-Mills theory. The theory has sixteen onshell states, eight bosons and eight fermions. The bosonic states are the two helicity states of the gluon and six real scalars which transform in the adjoint representation of su(4) (or vector of so (6)). The fermionic states are the four gluinos and four anti-gluinos transforming in the anti-fundamental and fundamental representation of su(4) respectively,
bosons: G + , G − , S AB = 1 2 ǫ ABCD S CD , fermions: Γ A , Γ A .(24)
The N = 4 theory is unique in that all on-shell states arrange themselves into a single PCT self-conjugate multiplet. We can describe this multiplet by a superfield depending on four Grassmann variables η A , transforming in the fundamental representation of su(4), Moreover, if we assign helicity 1 2 to the variable η we see that the whole superfield Φ has helicity 1. In other words we have added a term to the helicity generator,
Φ(η) = G + + η A Γ A + 1 2! η A η B S AB + 1 3! η A η B η C ǫ ABCD Γ D + 1 4! (η) 4 G − .(25)h = 1 2 −λ α ∂ ∂λ α +λα ∂ ∂λα + η A ∂ ∂η A ,(26)
so that hΦ(η) = Φ(η).
We have made a choice in writing (25) by putting the positive helicity gluon in the bottom component of the multiplet. We could equally well have written the multiplet the other way round by expanding in a Grassmann variable in the anti-fundamental representation of su (4),
Φ(η) = G − +η A Γ A + 1 2!η AηB S AB + 1 3!η AηBηC ǫ ABCD Γ D + 1 4! (η) 4 G + .(28)
The two superfields are related by a Grassmann Fourier transform,
Φ(η) = d 4 η e η·η Φ(η).(29)
The supersymmetry generators take the form
p αα = λ αλα , q αA = λ α η A ,qα A =λα ∂ ∂η A .(30)
It is straightforward to see that these generators, together with the Lorentz and su(4) generators,
M αβ = λ (α ∂ ∂λ β) , Mαβ =λ (α ∂ ∂λβ ) , R A B = η A ∂ ∂η B − 1 4 δ A B η C ∂ ∂η C ,(31)
do indeed give a representation of the super Poincaré algebra.
The fact that the N = 4 theory is PCT self-conjugate means that all n-particle amplitudes arrange themselves into a single superamplitude. All of the component amplitudes can then be obtained by expanding in the Grassmann variables,
A(Φ 1 ,
We have suppressed the su(4) indices in the second term for simplicity. The amplitudes have helicity 1 in each particle,
h i A(Φ 1 , . . . , Φ n ) = A(Φ 1 , . . . , Φ n ).(33)
When we consider the superamplitudes the symmetry generators are simply the sums of the single-particle representations (30,31). For example we have
p αα = i λ α iλα i , q αA = i λ α i η A i ,qα A = iλα i ∂ ∂η A i .(34)
What can the symmetries of the theory tell us about the scattering amplitudes? Assuming that the momentum variables λ andλ are not constrained, translation invariance tells us there is an overall momentum conserving delta function δ 4 (p). Similarly supersymmetry tells us that there should be a delta function δ 8 (q). We recall that the delta function of a Grassmann quantity δ(ψ) is simply ψ itself. Thus we have that the amplitude can be written,
A(Φ 1 , . . . , Φ n ) = δ 4 (p)δ 8 (q) 12 . . . n1 P n (λ,λ, η).(35)
Here we have put the factor 12 23 . . . n−1 n n1 in the denominator to carry the helicities of the superparticles so that the function P n has helicity zero for every particle, i.e. it is annihilated by each h i .
The fact that there is a factor δ 8 (q) in (35) due to supersymmetry means that that certain component amplitudes (e.g. pure gluon amplitudes with fewer than two negative helicity gluons) must vanish because there at least eight η variables in (35). This statement about the vanishing of certain pure gluon scattering amplitudes is true in all gauge theories at tree level because such amplitudes are common to every gauge theory regardless of the matter content. This shows that all gauge theories exhibit the effects of N = 4 supersymmetry, even if they are not supersymmetric theories. Of course the statement is true in N = 4 super Yang-Mills even beyond tree-level since it is a consequence of supersymmetry while in other gauge theories the all-plus amplitudes, for example, are non-vanishing at loop level.
The function P n is also constrained by symmetry. In particular the su(4) symmetry means that Grassmann variables always appear in multiples of four so that P n (λ,λ, η) can be expanded into terms of Grassmann degree 0,4,8 etc. These terms correspond to the classification of amplitudes as MHV, NMHV, NNMHV and so on,
P n (λ,λ, η) = P (0) n + P (4) n + P (8) n + . . . + P (4n−16) n .(36)
The final term in the expansion of P n corresponds to the MHV amplitudes. Theq supersymmetry also imposes constraints on the form of P n . Indeed we haveqα A P n = 0. Note that we can use theq supersymmetry to fix any two of the η variables to zero. The finiteq transformation with parameter given by
ξ Ȧ α =λ iα η A j −λ jα η A i [λ 1λn ](37)
will set η i and η j to zero. We will use this fact when we consider recursive relations among tree-level amplitudes in the N = 4 theory.
The symmetries (30,31) are not the only symmetries of the the theory. N = 4 super Yang-Mills theory is a superconformal field theory and the dilatation generator,
d = 1 2 i λ α i ∂ ∂λ α i +λα i ∂ ∂λα i ,(38)
and the special conformal and superconformal generators,
k αα = i ∂ 2 ∂λ α i ∂λα i , s αA = i ∂ 2 ∂λ α i ∂η A i ,s Ȧ α = i η A i ∂ ∂λα i(39)
are also symmetries of the tree-level amplitudes [7]. We will see that even this large symmetry algebra is not the end of the story. When written in the most compact way the tree-level amplitudes reveal another superconformal symmetry, called dual superconformal symmetry.
BCFW recursion relations
We will now see how we can reconstruct the entire tree-level S-matrix from the simple analytic structure that we have described in the previous section. We will first present the general recursive method which is due to Britto, Cachzo, Feng and Witten [8,9]. The presentation of the method will essentially follow that of [9]. The arguments can be framed in a very general form, in particular they can be applied to theories in any number of dimensions [10]. Here we will focus on the case of four dimensions and make direct use of the spinor helicity formalism. We will then formulate the method in a supersymmetric fashion as in [11,12] and then use it to solve for the tree-level S-matrix of N = 4 super Yang-Mills [13].
We will consider a tree-level gluon amplitude with incoming massless momenta p αα i = λ α iλα i . We will consider deforming the momenta by making the following shift of the spinor variables,
λ 1 −→λ 1 (z) = λ 1 − zλ n , λ n −→λ n (z) =λ n + zλ 1 ,(40)
where z is a complex number 1 . Under this shift the momenta p 1 and p n are deformed in a complex direction by an amount proportional to z,
p αα 1 −→p αα 1 (z) = (λ α 1 − zλ α n )λα 1 , p αα n −→p αα n (z) = λ α n (λα n + zλα 1 ).(41)
Note that by construction the shifted momenta are still light-likep 2 1 =p 2 n = 0 and that momentum is still conserved,p 1 +p n = p 1 + p n .
As we have deformed the momenta p 1 and p n the amplitude will also be deformed to become a function of z. What can we say about the analytic structure of the amplitude as a function of z? We have already seen that we can write a given amplitude as
A n = δ 4 (p)A n ,(42)
where A n is a rational function of the spinor variables {λ i ,λ i } with only local poles of the form
1 (p i + p i+1 + . . . p j ) 2 .(43)
This implies that the deformed amplitude A(z) will only have simple poles as a function of z.
The only propagator factors which will exhibit a dependence on z are those of the form
1 (p 1 + p 2 + . . . + p i−1 ) 2 ≡ 1 P 2 i .(44)
For simplicity of notation we will simply write P instead of P i until we need to remember that there are many such poles. The propagators which are affected by the shift in one example are shown in Fig. 4.
Under the z-shift such a pole will become
1 P 2 −→ 1 P (z) 2 = 1 (p 1 (z) + p 2 + . . . p i−1 ) 2 = 1 P 2 − z n|P |1] .(45)
Here we adopt the notation that n|P |1] = λ α n P ααλα 1 . We have found that the amplitude has a pole at
z P = P 2 n|P |1](46)
1n shifted propagators for every possible propagator of the form (44). At this value of z the complex shift that we have performed is such that the internal propagator carrying momentumP (z) has gone on shell,
P (z P ) 2 = 0.(47)
Near the pole the amplitude behaves as follows
A n (z) ∼ 1 (z − z P ) −1 n|P |1] s A s L 1 (z P ), . . . , i − 1, −P (z P ) As R P (z P ), i, . . . ,n(z P ) .(48)
This notation means that when the intermediate propagator goes on-shell, every diagram factorises into left and right pieces, with every external leg on shell. Summing up all diagrams which possess the internal propagator in question one obtains tree-level subamplitudes A s L and As R on either side of the intermediate on-shell propagator. Since one adds all possible Feynman diagrams, the state s exchanged between the two subamplitudes can be anything and one must sum over all possible states. In pure Yang-Mills theory this means that either a positive or a negative helicity gluon can be exchanged over the internal line. In N = 4 super Yang-Mills theory then the exchanged state can be any of the sixteen on-shell states of the theory. The sum over subamplitudes contributing to a particular residue is illustrated in Fig. 5.
Let us now consider the function A(z)/z. Near the pole at z = z P we find
A(z) z ∼ − 1 z − z P A L (z P ) 1 P 2 A R (z P ).(49)
The initial amplitude we are interested in can be written as
A = A(0) = C dz 2πi A(z) z .(50)
Here we have chosen the contour to be a small circle around the origin. By deforming the contour we can write the amplitude as a sum over residues from the other poles plus a potential contribution from infinity. This contour deformation is pictured in Fig. 6.
Thus we obtain
A(0) = i s A s L (z P i ) 1 P 2 i As R (z P i ) + res(z = ∞),(51)
where we have restored the notation P i to refer to the different possible poles of the form (44) which can arise. For the amplitudes we are interested in we will find that the contribution from z = ∞ vanishes and so we have the BCFW recursion relation
A(0) = i s A s L (z P i ) 1 P 2 i As R (z P i ).(52)
Let us now consider how the function A(z) behaves as z goes to infinity. To address this question we again need to refer back to the Feynman rules. Let us consider how a typical Feynman diagram behaves under the shift we are performing. Between the legs carrying the shifted momentap 1 andp n there will be a succession of internal propagators joining vertices which connect to the unshifted part of the diagram. Each internal propagator which feels the shift will contribute a negative power of z as z −→ ∞. Any three-point vertices on the line of the shifted momenta contribute a positive power of z (due to the fact that the three-point gluon coupling contains a derivative of the gluon field). Any four point vertices are milder, contributing no z dependence. Therefore the dominant behaviour as z −→ ∞ comes from diagrams where the vertices along the line of shifted propagators are all three-point interactions, as in Fig. 7. There is always one more vertex than internal propagator so we conclude that the dominant Feynman diagrams scale like z as z −→ ∞.
We must also include the effects of the polarisation vectors. Since they are momentum dependent, the polarisation vectors for particles 1 and n can also contribute to the large z behaviour. The scalings of the possible choices for positive and negative helicity gluons are summarised below,
ǫ αα 1,+ =λα 1 µ α λ 1 (z)µ ∼ 1 z , ǫ αα n,+ =λα n (z)µ α λ n µ ∼ z, ǫ αα 1,− =λ α 1 (z)μα [λ 1μ ] ∼ z, ǫ αα n,− = λ α nμα [λ n (z)μ] ∼ 1 z .(53)
Thus we see that the dominant contributions to each amplitude scale differently depending on the choice of the helicities of the shifted legs. In summary we have the following limits on the large z behaviour (noting only the helicities for particles 1 and n),
A(+−) ∼ 1 z , A(++) ∼ z, A(−−) ∼ z, A(−+) ∼ z 3 .(54)
The fact that the (+−) amplitude falls off as z −→ ∞ means that it is always possible to choose legs corresponding to this helicity configuration in order to get a recursive relation to lower-point amplitudes. In fact we will need a stronger result than this in order to proceed with finding the solution for tree-level scattering amplitudes in the N = 4 theory. We would like to show that the large z limiting behaviour (54) is improved for the case of the (++) and (−−) amplitudes. Note that in (54) we only have limits on the asymptotic behaviour because it could happen that in certain cases there is a cancellation among the contributions coming from different diagrams resulting in much softer large z behaviour. Indeed we will now argue that in the case of the (++) and (−−) amplitudes this is exactly what happens, leading to a suppression by two extra powers of z. The argument closely follows the discussion of Arkani-Hamed and Kaplan [10].
In the limit of large z we can think of the scattering amplitude as the amplitude for a single particle at very large (complex) momentum scattering of some soft background describing the other particles. So let us consider the Lagrangian expanded around some soft background. We will write the gauge field as A µ = B µ + a µ where B µ is the soft background field and a µ is the fluctuation. Adding the gauge-fixing term (D µ a µ ) 2 to the Lagrangian we find the terms quadratic in a µ are given by
L quad = − 1 4 TrD µ a ν D µ a ν + i 2 TrG µν [a µ , a ν ].(55)
Here G µν is the field strength for the background field B µ . The first term contains the derivative couplings and is responsible for the leading behaviour at large z. It also has a symmetry which is broken only by the second term in the quadratic Lagrangian. The symmetry is a Lorentz symmetry which acts only on the indices of the fluctuation field a µ but not on the indices of the background fields or derivatives. This symmetry is referred to as 'spin-Lorentz' symmetry in [10]. To make it explicit we will use Latin characters for the relevant indices,
L quad = − 1 4 TrD µ a a D µ a b η ab + i 2 TrG ab [a a , a b ].(56)
Here we see that the leading term is invariant under Lorentz transformations of the Latin indices while the second term breaks this symmetry due to the presence of the antisymmetric tensor G ab . Thus the leading contribution to the two-point function will be proportional to η ab while the next correction will be given when there is exactly one coupling to the background field G ab and hence will be antisymmetric in the spin-Lorentz indices a and b. Corrections given by two or more couplings to the background will have a generic tensor structure in the spin-Lorentz indices.
In summary the two-point function for the hard particle in the soft background will behave as follows,
A ab = η ab (cz + . . .) + A ab + 1 z B ab + . . . ,(57)
where A ab is an antisymmetric tensor in the spin-Lorentz indices. In two-component notation we can write this as follows,
A ααββ = ǫ αβ ǫαβ(cz + . . .) + (ǫ αβsαβ + ǫαβs αβ ) + 1 z B ααββ + . . . ,(58)
where s αβ andsαβ are both symmetric in their indices. If we now contract this expression with the polarisation vectors ǫ αα 1,+ and ǫ ββ n,+ from (53) we can see that the leading term in (58) does not contribute. The subleading term in (58) contributes the following term to A(++),
[λ 1λn (z)]s αβ µ α µ β λ 1 (z)µ λ n µ .(59)
The z dependence in the numerator actually drops out because [λ 1λn (z)] = [λ 1λn ]. Thus we see that in fact the (++) amplitude is suppressed by two powers of z relative to the worst Feynman diagrams. The same happens for the (−−) amplitude. Recall that an incoming positive-helicity particle can be thought of as an outgoing negative-helicity one and vice versa. So if one thinks of one of the particles as incoming and the other as outgoing then the (++) amplitudes and (−−) amplitudes we are discussing actually correspond to a single particle which flips its helicity when scattering off the soft background. Physically we can therefore think of the extra suppression by two powers of z as a penalty for the hard particle for flipping its helicity while scattering off the soft background.
In summary we have found the following improved behaviour for the scattering amplitudes at large z,
A(+−) ∼ 1 z , A(++) ∼ 1 z , A(−−) ∼ 1 z , A(−+) ∼ z 3 .(60)
The fact that the (++) amplitude falls off as z goes to infinity is crucial. Recall that in the N = 4 theory we can use aq-supersymmetry transformation to shift the η variables for any two legs to zero. If we use this transformation to shift the η variables associated to the shifted legs we can relate the full superamplitude to the (++) amplitude. In order to make use ofqsupersymmetry we should perform the shift in the N = 4 theory in a way which is compatiblē q transformations. Recall thatq-supersymmetry relates the η andλ variables so if we are to respectq-supersymmetry then we should shift η whenever we shiftλ. Thus the full shift we will perform in the N = 4 theory is the following [11,12],
λ 1 −→λ 1 (z) = λ 1 − zλ n , λ n −→λ n (z) =λ n + zλ 1 , η n −→η n (z) = η n + zη 1 .(61)
With this definition of the shift we can see that the parameter of the finiteq transformation required to set η 1 andη n (z) to zero is independent of z,
ξ Ȧ α =λ 1αη A n (z) −λ nα (z)η A 1 [λ 1λn (z)] =λ 1α η A n −λ nα η A 1 [λ 1λn ] .(62)
Thus by usingq-supersymmetry we can relate the z-dependence of the whole superamplitude to that of its (−−) component (where the helicities relate to particles 1 and n as before). Thus we see that with the correct supersymmetric definition of the shift the whole superamplitude falls off like 1/z as z goes to infinity [12].
So for the superamplitudes in the N = 4 theory we have a recursion relation with no contribution from z = ∞. Furthermore the sum over states can be replaced by a single Grassmann integral over the η variable associated to the internal line joining the two subamplitudes in the recursion relation. In summary the recursion relation for N = 4 super Yang-Mills theory is
A n = i d 4 ηP i P 2 i A L 1 (z P i ), 2, . . . , i − 1, −P (z P i ) A R P (z P i ), i, . . . , n − 1,n(z P i ) .(63)
We will proceed to solving the recursion relation to obtain the full tree-level S-matrix for N = 4 super Yang-Mills theory. To start the recursion we will need on-shell three-point vertices. The reason that these are needed is that it can happen that the internal propagator closest to leg 1 (or leg n) is actually attached to a three point vertex with both external legs 1 and 2 (or n − 1 and n). The contribution from the pole when this type of internal propagator goes on shell will involve on-shell three-point vertices with complex incoming momenta. The fact that the momenta are complex is important because for real momenta the three-point amplitude vanishes. Let us consider on-shell three-point kinematics. Momentum conservation reads
λ α 1λα 1 + λ α 2λα 2 = −λ 3λα 3 .(64)
Taking the square of both sides tells us 12 [12] = 0 (65) and hence 12 = 0 or [12] = 0.
If 12 = 0 then λ 1 ∝ λ 2 and then (64) tells us that λ 1 ∝ λ 2 ∝ λ 3 and hence all of the possible angle brackets vanish. Likewise if [12] = 0 all the square brackets vanish so we have two distinct possibilities for three-point vertices, which we will call MHV and MHV,
12 = 23 = 31 = 0 (MHV),(67)[12] = [23] = [31] = 0 (MHV).(68)
Note that for real momenta both conditions are satisfied because the λ i andλ i are related by complex conjugation thus the particle momenta would all have to be collinear. This is why the amplitude vanishes for real momenta.
For complex momenta we can construct the three-point amplitudes from their symmetries. Let us consider the MHV case first. We need to find a three-point superamplitude with helicity 1 on each leg. Translation invariance and q-supersymmetry tell us that there are factors of δ 4 (p) and δ 8 (q) as before while Lorentz invariance and the helicity conditions uniquely fix the other factors so that we obtain
A MHV 3 = δ 4 (p)δ 8 (q) 12 23 31 .(69)
The MHV case is related to this one by parity. To find the amplitude we interchange λ i and λ i and replace η i byη i . Then to express the amplitude back in terms of the η i we perform the Grassmann Fourier transform (29) for each leg. The result is [11,12] A MHV
3 = δ 4 (p)δ 4 (η 1 [23] + η 2 [31] + η 3 [12]
) [12][23] [31] .
It may be slightly surprising that for this amplitude q-supersymmetry does not imply that there is a factor of δ 8 (q) as usual. The reason is that the three-point kinematics in the MHV case are such that all λ variables are parallel. This means that q αA itself factorises, q αA = λ α F q A F for some λ α F and q A F and the requirement of q-supersymmetry is only that the amplitude contain a factor of δ 4 (q F ) as we indeed find in (70). Thus the MHV 3 amplitude has Grassmann degree four while all other amplitudes have Grassmann degree at least eight.
The recursion relation (63) can be decomposed into contributions of various Grassmann degrees. Since there is a Grassmann integral on the RHS the sum of the degrees of the two subamplitudes A L and A R must be four more than the degree of the amplitude we are solving for. The us we find
A N p MHV n = d 4 ηP P 2 A MHV 3 (z P )A N p MHV n−1 (z P ) + p−1 m=0 i d 4 ηP i P 2 i A N m MHV i (z P i )A N (p−m−1) MHV n−i+2 (z P i ) .(71)
Note that we have not allowed for the left subamplitude to be A MHV
3
. This cannot happen because in the MHV case the square bracket [12] vanishes. For the left subamplitude theλ variables are unshifted and hence this would imply that [12] and hence (p 1 +p 2 ) 2 vanishes for the full amplitude as well. This is a restriction on the kinematics which is not true in general and hence such a term does not contribute to the recursion relation. Similarly the right subamplitude can never be A MHV
3
.
We can now begin to construct amplitudes recursively using the three-point amplitudes as the starting point. The simplest amplitude is the four-point amplitude. There is only one pole which arises under the z shift and hence only one term in the recursion relation which is given by
A 4 = d 4 ηP P 2 A MHV 3 (1, 2, −P )A MHV 3 (P , 3,4).(72)
This BCFW term is represented in Fig. 8.
Using the form of the three-point amplitudes we find
A 4 = d 4 ηP P 2 δ 4 (η 1 [2P ] + η 2 [P 1] + ηP [12]) [12][2P ][P 1] δ 8 (λP ηP + λ 3 η 3 + λ 4η4 ) P 3 34 4P(73)
The δ 4 factor tells us that ηP can be expressed in terms of the other η variables,
ηP = − 1 [12] (η 1 [2P ] + η 2 [P 1]).(74)
Examining the δ 8 factor we see that it can then be written
δ 8 −λP 1 [12] (η 1 [2P ] + η 2 [P 1]) + λ 3 η 3 + λ 4η4 = δ 8 λ 1 η 1 + λ 2 η 2 + λ 3 η 3 + λ 4η4 = δ 8 (q). (75)
where in the first step we have used momentum conservation to write λPλP =λ 1λ1 + λ 2λ2 and in the second we cancel the z dependence between the first and last terms in the argument. Thus we obtain the expected supersymmetry delta function δ 8 (q).
All that remains is to collect the bosonic factors together and simplify them. The δ 4 factor soaks up the Grassmann integration, producing a factor of [12] 4 due to the coefficient in front of ηP in the argument. The factor of P 2 in the denominator can be written (p 1 + p 2 ) 2 = 12 [12].
Thus in total we obtain the expected expression for the four-point amplitude 23 34 41 .
A 4 = δ 8 (q) 12
An almost identical calculation shows that n-point MHV amplitude takes the form
A MHV n = δ 8 (q) 12 . . . n1 .(79)
Again there is only a single BCFW term, represented in Fig. 9. This formula first appeared in [14], supersymmetrising the Parke-Taylor formula [15,16] for MHV gluon amplitudes.
Let us now consider the next-to-MHV (NMHV) case. The recursion relation has two types of terms, one where the subamplitudes are A MHV 3 and A NMHV n−1 and the other where they are both MHV,
A NMHV n = d 4 ηP P 2 A MHV 3 (z P )A N p MHV n−1 (z P ) + n−1 i=3 d 4 ηP i P 2 i A MHV i (z P i )A MHV n−i+2 (z P i ) .(80)
The two kinds of terms are represented in Fig. 10.
Let us look at the second type of term first. We call this the inhomogeneous term because it only involves the MHV amplitudes which we have already solved for. Writing out the ith term in the sum in more detail we have One of the two δ 8 factors can immediately be exchanged for the overall supersymmetry conserving delta function with the help of the other δ 8 ,
I i = d 4 ηP i P 2 i δ 8 λ 1 η 1 + i−1 2 λ j η j − λP i ηP i 1 2 23 . . . i − 1P i P i1 δ 8 λP i ηP i + n−1 i λ j η j − λ nηn P i i i i + 1 . . . nP i .(81)I i = δ 8 (q) d 4 ηP i P 2 i δ 8 λ 1 η 1 + i−1 2 λ j η j − λP i ηP i 1 2 23 . . . i − 1P i P i1 P i i i + 1 i + 2 . . . nP i .(82)
We must now perform the Grassmann integral and simplify the remaining bosonic factors. In order to organise the calculation it will be very helpful to introduce 'dual' coordinates x i and θ i . These are defined so that their differences give the momenta and supercharges associated to the external legs,
x αα i − x αα i+1 = λ α iλα i = p αα i , θ αA i − θ αA i+1 = λ α i η A i = q αA i .(83)
We will give a more geometric interpretation to these dual coordinates in the next section where we will see that they are very helpful in revealing a new symmetry (dual superconformal symmetry). Here we are just using them for notational convenience. We will use the shorthand notation
x ij ≡ x i − x j = p i + p i+1 + . . . + p j−1 , θ ij ≡ θ i − θ j = q i + q i+1 + . . . q j−1 .(84)
Note that as a consequence of their definitions the dual coordinates satisfy some useful relations,
i|x ij = i|x i+1,j , i|θ ij = i|θ i+1,j(85)
and similarly with angle brackets replaced by square ones.
Returning to (82) we can split the δ 8 into a product of two δ 4 factors,
δ 8 λ 1 η 1 + i−1 2 λ j η j − λP i ηP i = 1P i 4 δ 4 η 1 + i−1 2 P i j P i1 η j δ 4 i−1 2 1 j 1P i η j − ηP i .(86)
The second δ 4 soaks up the Grassmann integration. The remaining δ 4 and the angle bracket factors combine to give
δ 4 P i1 η 1 + i−1 2 P i j η j = δ 4 i−1 1 P i j η j − z P i n η 1 .(87)
Noting that the numerator and denominator of I i are now homogeneous of degree 4 in λP i , we can simplify by multiplying both by ( n1 [1P i ]) 4 . Then every λP i now appears in the combination n1 [1P i ] P i |... . We can make use of the dual coordinates to remove all dependence on the hatted quantities as follows,
n1 [1P i ] P i |... = n1 [1|P i ... = n1 [1|P i ... = n1 [1|x 1i ... = n1 [1|x 2i ... = n|x 12 x 2i ... = n|x n2 x 2i ... .(88)
We can use these identities directly on the first term in the argument of the δ 4 in (86)
so that it becomes − n|x n2 x 2i |θ i1 .(89)
For the second term in the argument it is convenient to use the last form in the first line of (88) and to recall that the value of z is fixed to be
z = P 2 i n|P i |1] .(90)
This term then becomes
−z n1 [1|P i |n η 1 = −P 2 i n1 η 1 = −x 2 1i n1 η 1 = n|x 1i x i1 |1 η 1 = − n|x ni x i2 |θ 21 .(91)
Thus the δ 4 factor becomes
δ 4 n|x n2 x 2i |θ i1 + n|x ni x i2 |θ 21 = δ 4 n|x n2 x 2i |θ i + n|x ni x i2 |θ 2 + x 2 2i nθ 1(92)
from which we can see that θ 1 can be exchanged for θ n as it comes projected with λ n .
Similar manipulations lead to the simplification of the bosonic factors in I i . Finally we arrive at the result
I i = A MHV n R n,2i ,(93)
where
R n,2i = 21 i i − 1 δ 4 n|x n2 x 2i |θ in + n|x ni x i2 |θ 2n x 2 2i n|x n2 x 2i |i n|x n2 x 2i |i − 1 n|x ni x i2 |2 n|x ni x i2 |1 .(94)
Note that for the five-particle amplitude this is the only term in the amplitude because the first term in (80) vanishes. For n > 5 we can postulate the final form of the result and verify by induction that it is correct. The final form obtained for the NMHV amplitudes is
A NMHV n = A MHV n P NMHV n ,(95)
where
P NMHV n = 2≤a<b≤n−1 R n,ab .(96)
Here R n,ab is the natural generalisation of (94), R n,ab = a a − 1 b b − 1 δ 4 n|x na x ab |θ bn + n|x nb x ba |θ an x 2 ab n|x na x ab |b n|x na x ab |b − 1 n|x nb x ba |a n|x nb x ba |a n|x nb x ba |a − 1 (97) and the sum over a and b in (96) is performed such that a < b − 1. The final result (96) is remarkably simple. The six-particle case for example is expressed as sum of only three terms.
Note that in the result for the NMHV amplitudes, non-local poles make a prominent appearance. Poles of the form 1 n|x na x ab |b
are not (except for special values of a and b) expressible in terms of the local poles of the type
1 (p i + . . . + p j ) 2 .(99)
Of course the theory is local and the spurious poles of the form (98) cancel between different terms in the sum over the labels a and b. However it is notable that the simple way of expressing the amplitude involves terms which are necessarily non-local. We will see a deeper reason for this in the next section.
The process of solving the recursion relation can be continued to higher levels in the MHV degree. We will not give the details of the calculations here but instead refer the reader to [13] for the explicit derivations. The amplitudes are expressed in terms of new quantities that have many pairs of labels,
R n;b 1 a 1 ;b 2 a 2 ;...;brar;ab = a a − 1 b b − 1 δ (4) ( ξ|x ara x ab |θ bar + ξ|x arb x ba |θ aar ) x 2 ab ξ|x ara x ab |b ξ|x ara x ab |b − 1 ξ|x arb x ba |a ξ|x arb x ba |a − 1 ,(100)
where the chiral spinor ξ| is given by
ξ| = n|x nb 1 x b 1 a 1 x a 1 b 2 x b 2 a 2 . . . x brar .(101)
In the case where there is only one pair of labels ab after the initial label n, (100) is just the quantity R n,ab we have already seen appearing in the NMHV amplitudes. The cases where there is more than one pair are generalisations.
As an example we quote the result for NNMHV amplitudes here.
P NNMHV n = 2≤a 1 ,b 1 ≤n−1 R n;a 1 b 1 a 1 +1≤a 2 ,b 2 ≤b 1 R 0;a 1 b 1 n;b 1 a 1 ;a 2 b 2 + b 1 ≤a 2 b 2 ≤n−1 R a 1 b 1 ;0 n;a 2 b 2 .(102)
We see that the generalised objects (100) begin appearing at the NNMHV level in the first term in the brackets. The superscripts on the Rs in the brackets mean that the boundary terms in the sum should be treated differently. The right superscript on R n,b 1 a 1 ,a 2 b 2 indicates that when the upper boundary b 2 = b 1 is reached in the sum the explicit appearance of the spinor b 1 | should be replaced by n|x na 1 x a 1 b 1 . Similarly the left superscript on R n,a 2 b 2 means that when the lower boundary a 2 = b 1 is reached the explicit appearance of b 1 − 1| is replaced by n|x na 1 x a 1 b 1 . The 0 superscripts indicate that nothing happens at the other boundaries. The formulas for all N p MHV amplitudes can be found in [13].
As we have already discussed, at the level of pure gluon scattering, the fact that we have solved the amplitudes in the N = 4 theory is no restriction at all; the gluon amplitudes are the same in any gauge theory. Thus the simplicity of the expressions arising from the recursive structure is universal for gluon amplitudes in all gauge theories, as is the associated presence of spurious non-local poles. The explicit expressions for the pure-gluon amplitudes can be derived from (79) and (95,96,97) etc. by reading off the coefficient of the relevant combination of η variables.
The same recursive technique is also valid for gravitational theories [17]. Again it can be made manifestly supersymmetric and becomes much simpler for the maximally supersymmetric theory N = 8 supergravity [12], admitting an explicit solution [18].
Dual superconformal symmetry
Here we will see that the non-manifestly local form of the amplitudes arising from the solution of the BCFW recursion relation is very natural from the point of view of symmetry. Indeed the different terms arising in the BCFW expansion are all invariant under a very large symmetry algebra.
Let us focus on the form of the MHV and NMHV amplitudes which we found in the previous section. The dual coordinates x which are used to express the amplitudes can be taken seriously as the coordinates of a dual copy of spacetime. The amplitudes are trivially invariant under translations of the dual coordinates as they were introduced only through their differences. Lorentz transformations of the dual coordinates are the same as Lorentz transformations of the particle momenta and so are also a symmetry of the scattering amplitudes. The surprise comes when one examines conformal transformations of the dual coordinates. It turns out that these transformations are also a symmetry of the scattering amplitudes. Since the symmetry acts canonically on the dual coordinates and these are linearly related to the particle momenta the generator is first order acting on the momenta. Note that such a conformal transformation is not related to the conformal symmetry of the Lagrangian, which is rather related to the second-order generators k αα of (39). The conformal symmetry acting in the dual space is referred to as dual conformal symmetry.
The dual coordinates x i define a closed polygon with light-like edges in the dual space, as represented in Fig. 11. The contour is closed because we identify x n+1 with x 1 . This statement reflects the total momentum conservation of the scattering process p 1 + p 2 + . . . + p n = 0. The edges of the polygon are light-like because the particles in the scattering process are on-shell. The role of the polygon was first made clear at strong coupling [2] where it becomes the boundary for a minimal surface in AdS space, leading to a relation between scattering amplitudes and Wilson loops. We will discuss this relation in more detail in the next section. For now we would just like to note that a light-like polygon maps into another such polygon under conformal transformations of the dual space. Indeed under conformal inversions
x µ −→ − x µ x 2 ,(103)
we have that
x 2 ij −→ x 2 ij x 2 i x 2 j .(104)
Thus if two points x i and x j are light-like separated they will remain so after a conformal inversion. The conformal group is generated by Lorentz transformations, translations and conformal inversions so the light-like nature of the polygon is invariant under the action of the whole conformal group.
To see that dual conformal transformations are actually a symmetry of the scattering amplitudes we need to define their action on all of the variables in the problem. The helicity variables λ andλ must also transform under dual conformal transformations as they are related to the dual coordinates via the defining relations,
x αα i − x αα i+1 = λ α iλα i .(105)
Indeed we see that x αα ij transforms as follows
x αα ij −→ − x αα i x 2 i + x αα j x 2 j = −x αα i x 2 j + x 2 i x αα j x 2 i x 2 j = (x i (x i − x j )x j ) αα x 2 i x 2 j = (x −1 i x ij x −1 j ) αα .(106)
Choosing
j = i + 1 we find λ α iλα i −→ (x −1 i λ iλi x −1 i+1 ) αα .(107)
The transformations λ i andλ i can therefore be defined as follows [19],
λ α i −→ (x −1 i λ i )α,λα i −→ (x −1 i+1λ i ) α ,(108)
so that they are compatible with the defining relations (105).
The superpartners of the dual coordinates also transform canonically under conformal inversions,
θ αA i −→ (x −1 i θ i )α A ,(109)
which implies that the variables η i must also transform in analogy with λ i andλ i . One can derive the transformation of the η i by performing an inversion on both sides of the relation
θ αA i − θ αA i+1 = λ α i η A i .(110)
It will not be necessary for us to give the transformation as we can always use (110) to eliminate the η i in favour of the θ i (we can similarly eliminate theλ i in favour of the x i using (105)).
If we look at the MHV amplitude,
A MHV n = δ 4 (p)δ 8 (q) 12 . . . n1 ,(111)
we can see that it is in fact covariant under dual conformal transformations. Firstly, if we drop the requirement that x n+1 ≡ x 1 and θ n+1 ≡ θ 1 then the delta functions can be written as
δ 4 (p)δ 4 (q) = δ 4 (x 1 − x n+1 )δ 8 (θ 1 − θ n+1 ).(112)
This combination is dual conformally invariant as the bosonic delta function has conformal weight 4 at the point x 1 (which is identified with x n+1 under the delta function) as can be seen from d 4 x 1 δ 4 (x 1 −x n+1 ) = 1. The Grassmann delta function has the opposite conformal weight (which is only true because of maximal supersymmetry) and the hence the product (112) is invariant.
The denominator in the MHV amplitude is covariant under dual conformal transformations because factors of the form i i + 1 transform as follows,
i i + 1 −→ i|x −1 i x −1 i+1 |i + 1 = i|x i x i+1 |i + 1 x 2 i x 2 i+1 = i|x i+1 x i+1 |i + 1 x 2 i x 2 i+1 = i i + 1 x 2 i .(113)
Thus we find that the MHV tree-level amplitude is covariant with weight 1 at each point,
A MHV n −→ (x 2 1 . . . x 2 n )A MHV n .(114)
If we now look at the NMHV amplitude defined by equations (95), (96) and (97) we find that it is similarly covariant. The reason is that each term R n,ab in P NMHV n is invariant under dual conformal transformations. Indeed returning to the formula (97) we see that it is made of dual conformally covariant factors. The spurious poles are covariant following a similar analysis to (113). For example we have
n|x na x ab |b −→ n|x na x ab |b x 2 n x 2 a x 2 b , n|x na x ab |b − 1 −→ n|x na x ab |b − 1 x 2 n x 2 a x 2 b−1 .(115)
The Grassmann delta function is also covariant as can be see when written in a form similar to (92),
δ 4 n|x na x ab |θ bn + n|x nb x ba |θ an = δ 4 n|x na x ab |θ b + n|x nb x ba |θ a + x 2 ab nθ n(116)
Checking all the factors in (97) one can see that the weights cancel and thus R n,ab is invariant under dual conformal transformations. In fact one can show from the recursion relation itself that all terms produced this way will respect dual conformal symmetry [11]. One can also check directly that dual conformal symmetry is present for the generalisations of R n,ab appearing in the N p MHV amplitudes.
The dual conformal symmetry we have seen actually extends to dual superconformal symmetry. Dual superconformal symmetry has a canonical action on the coordinates of the dual superspace x i , θ i . It is very helpful to express the symmetry in terms of the generators of infinitesimal transformations. For example the generator of special conformal transformations of the dual coordinates is
K αα = i [x iαβ x iα β ∂ iββ + x iα β θ B iα ∂ iβB ].(117)
Just as we have seen for dual conformal inversions the transformation must act on the on-shell superspace variables {λ,λ, η} in order to respect the relations between them (constraints). In terms of the generators this means we must add terms so that the generator commutes with the constraints modulo constraints. One can perform this process for all generators of the superconformal algebra. The result is the following set of generators,
P αα = i ∂ iαα , Q αA = i ∂ iαA , Q Ȧ α = i [θ αA i ∂ iαα + η A i ∂ iα ], M αβ = i [x i(αα ∂ iβ)α + θ A i(α ∂ iβ)A + λ i(α ∂ iβ) ] , Mαβ = i [x i(α α ∂ iβ)α +λ i(α ∂ iβ) ] , R A B = i [θ αA i ∂ iαB + η A i ∂ iB − 1 4 δ A B θ αC i ∂ iαC − 1 4 δ A B η C i ∂ iC ] , D = i [−xα α i ∂ iαα − 1 2 θ αA i ∂ iαA − 1 2 λ α i ∂ iα − 1 2λα i ∂ iα ] , C = i [− 1 2 λ α i ∂ iα + 1 2λα i ∂ iα + 1 2 η A i ∂ iA ] , S A α = i [−θ B iα θ βA i ∂ iβB + x iαβ θ βA i ∂ iββ + λ iα θ γA i ∂ iγ + x i+1 αβ η A i ∂ iβ − θ B i+1 α η A i ∂ iB ] , Sα A = i [x iα β ∂ iβA +λ iα ∂ iA ] , K αα = i [x iαβ x iα β ∂ iββ + x iα β θ B iα ∂ iβB + x iα β λ iα ∂ iβ + x i+1 αβλiα ∂ iβ +λ iα θ B i+1 α ∂ iB ] .(118)
Here we have employed the following shorthand notation
∂ iαα = ∂ ∂x αα i , ∂ iαA = ∂ ∂θ αA i , ∂ iα = ∂ ∂λ α i , ∂ iα = ∂ ∂λα i , ∂ iA = ∂ ∂η A i .(119)
It is simple to check that the generators in (118) do obey the commutation relations of the superconformal algebra. There are several remarks worth making at this point. Firstly it is clear from the first-order form of the generators that the dual superconformal symmetry is distinct from the ordinary superconformal symmetry generated by (34), (39) etc. Secondly, we note that the su(4) nature of the fermionic generators is swapped between the original superconformal algebra and the dual superconformal algebra. For example the supersymmetry generator q αA is in the fundamental representation of su(4) while the dual supersymmetry generator Q αA is in the anti-fundamental. Similarly, on the on-shell superspace variables, the two dilatation generators coincide up to a sign because dual coordinates are actually related to particle momenta. Finally we should note that the two superconformal algebras overlap non-trivially. That is, the fermionic superconformal generatorS coincides with the original supersymmetry generatorq on the on-shell superspace, while the dual supersymmetry generatorQ coincides with the original superconformal generators. The definitions of the dual variables manifestly respect covariance under the Lorentz, dilatation and su(4) symmetries and so these symmetries are shared between the two copies of the superconformal algebra. The overlap is schematically represented in the following picture.
s =Q
A similar picture also arises from considering the symmetries of the string sigma model (see [20,21,22]) which can be used to describe scattering amplitudes at strong coupling.
With all the generators of the superconformal algebra to hand we can now verify that the quantity R n,ab is actually a dual superconformal invariant. We have already verified that it is invariant under dual conformal inversions. Since dual translation invariance and Lorentz invariance are manifest the inversion symmetry is equivalent to invariance under the dual special conformal transformations, generated by K αα . It remains to show that it is invariant under the fermionic generators. Invariance under the chiral dual supersymmetry Q αA is manifest (the θ variables only appear as differences in R n,ab ) and hence by commutation with K αα we know that Sα A is a symmetry. The non-trivial symmetry to verify is the anti-chiral dual supersymmetrȳ Q Ȧ α .
To show thatQ Ȧ α is indeed a symmetry of R n,ab we can exploit Q andS by using a finite transformation made from these generators to fix a frame where θ a = θ b = 0. Since R n,ab is invariant under Q andS and all generators which arise through commutation of these withQ we know thatQR n,ab is invariant under Q andS. So we can evaluateQR n,ab in the frame where θ a = θ b = 0,
Q Ȧ α R n,ab = θ αA n ∂ ∂x αα n a a − 1 b b − 1 δ 4 nθ n x 2 ab n|x na x ab |b n|x na x ab |b − 1 n|x nb x ba |a n|x nb x ba |a − 1 .
∝ nθ n δ 4 nθ n = 0.
(120)
Thus we can see that the nilpotent nature of R n,ab is crucial in satisfying the invariance.
As we have seen the higher R-invariants appearing in the tree-level S-matrix are also dual conformal invariants. They are not dual superconformal invariants, as they are not annihilated byQ. However they always appear in a nested fashion. For example at NNMHV level the quantity R n,b 1 a 1 ,a 2 b 2 always appears multiplied by R n,a 1 b 1 . TheQ variation of R n;b 1 a 1 ,a 2 b 2 vanishes when multiplied by R n,a 1 b 1 so that the product is again dual superconformal invariant. Again we refer the reader to [13] for more details.
The end result for the full superamplitude is that the function P n is invariant under dual superconformal symmetry while the MHV prefactor is covariant under D, C, K and S and invariant under all other dual superconformal transformations. Thus we have
DA n = nA n , CA n = nA n , K αα A n = − i x αα i A n , S αA A n = − i θ αA i A n (121)
In addition we have invariance under the standard superconformal symmetry (see [7,23,24,25] for a fuller discussion), j a A n = 0.
We can get some insight into the nature of the symmetries we have found by combining the dual superconformal symmetry with the original one. In order to put the dual superconformal symmetry on the same footing as invariance under the standard superconformal algebra (122), the covariance (121) can be rephrased as an invariance of A n by a simple redefinition of the generators [26],
K ′αα = K αα + i x αα i ,(123)S ′αA = S αA + i θ αA i ,(124)D ′ = D − n.(125)
With this redefinition all generators of the dual superconformal algebra annihilate A n .
In order to have both symmetries acting on the same space it is useful to restrict the dual superconformal generators to act only on the on-shell superspace variables (λ i ,λ i , η i ). Then one finds that the generators P αα , Q αA become trivial while the generators {Q, M,M , R, D ′ ,S} coincide (up to signs) with generators of the standard superconformal symmetry. The non-trivial generators which are not part of the j a are K ′ and S ′ . In [26] it was shown that the generators j a and S ′ (or K ′ ) together generate the Yangian of the superconformal algebra, Y (psu(2, 2|4)). The generators j a form the level-zero psu(2, 2|4) subalgebra 2 ,
[j a , j b ] = f ab c j c .(126)
In addition there are level-one generators j (1) a which transform in the adjoint under the level-zero generators,
[j a , j b (1) ] = f ab c j c (1) .(127)
Higher commutators among the generators are constrained by the Serre relation 3 ,
[j (1) a , [j (1) b , j c ]] + (−1) |a|(|b|+|c|) [j (1) b , [j (1) c , j a ]] + (−1) |c|(|a|+|b|) [j (1) c , [j (1) a , j b ]] = h 2 (−1) |r||m|+|t||n| {j l , j m , j n }f ar l f bs m f ct n f rst .(128)
The level-zero generators are represented by a sum over single particle generators,
j a = n k=1 j ka .(129)
The level-one generators are represented by the bilocal formula [27,28],
j a (1) = f a cb k<k ′ j kb j k ′ c .(130)
Thus finally the full symmetry of the tree-level amplitudes can be rephrased as
yA n = 0,(131)
for any y ∈ Y (psu(2, 2|4)).
One can naturally describe the symmetry in terms of twistor variables. In (2, 2) signature the twistor variables are simply related to the on-shell superspace variables (λ,λ, η) by a Fourier transformation λ −→μ [7]. Twistor space linearises the action of superconformal symmetry. Expressed in terms of the twistor space variables Z A = (μ α ,λα, η A ), the level-zero and level-one generators of the Yangian symmetry assume a simple form
j A B = i Z A i ∂ ∂Z B i ,(132)j (1)A B = i<j (−1) C Z A i ∂ ∂Z C i Z C j ∂ ∂Z B j − (i, j) .(133)
Both of the formulas (132) and (133) are understood to have the supertrace removed. In this representation the generators of superconformal symmetry are first-order operators while the level-one Yangian generators are second order.
In fact one can also phrase things the other way round. In [29] it was demonstrated that there exists an alternative T-dual representation of the symmetry, where it is the dual superconformal generators J a which play the role of the level-zero generators, while the generators k and s of the original superconformal symmetry form part of the level-one generators. The generators take the same form as (132) and (133) but with the twistor variables replaced by momentum twistor variables [30]. Momentum twistors are the twistors associated to the dual space and linearise the dual superconformal symmetry. They are defined as
W A i = (λ α i , µα i , χ A i ) with µ α i = x αα i λ iα , χ A i = θ αA i λ iα .(134)
The generators then take the form
J A B = i W A i ∂ ∂W B i ,(135)J (1)A B = i<j (−1) C W A i ∂ ∂W C i W C j ∂ ∂W B j − (i, j) ,(136)
again with supertraces removed. In this representation the generators annihilate the amplitude with the MHV prefactor dropped, i.e. J a P n = 0, J (1) a P n = 0.
So we have seen that there are two equivalent ways of looking at the full symmetry of theory. One can choose either version of the superconformal symmetry to be fundamental. This effectively means choosing one of the superconformal symmetries to be realised locally. In so doing the remaining symmetries are necessarily non-local. Thus the non-local poles found in the expression for the tree-level amplitudes are inevitable if the amplitudes are to be expressed in a form which reveals the full symmetry.
The fact that the full symmetry is the Yangian Y (psu(2, 2|4)) is certainly not accidental. The planar limit of the N = 4 gauge theory is known to possess an integrable structure in other regimes. In particular the believed integrability of the spectrum of anomalous dimensions (see e.g. [31,32,33,34,35]) can be traced to the fact that there is an underlying Yangian symmetry. Moreover at strong coupling the theory is related via the AdS/CFT correspondence to the AdS 5 sigma model which is classically integrable [36]. Indeed the integrability of this model has been used to calculate the scattering amplitudes via a relation to minimal surfaces in AdS which we will describe in the next section.
Having described the symmetry of the theory, one might naturally ask how one can produce invariants. This question has been addressed in various papers [25,29,37,38]. It turns out to be intimately connected to another conjecture about the leading singularities of the scattering amplitudes of N = 4 super Yang-Mills theory. In [39] Arkani-Hamed et al proposed a formula for all leading singularities of the planar N = 4 S-matrix. A leading singularity of a loop amplitude is obtained by evaluating the loop integration via compact contours instead of the usual non-compact Minkowski space integration. An example is given by the four-particle cuts of [40]. Here one takes the one-loop amplitude and evaluates it by choosing a contour which localises the loop integration to a point where four internal propagators go on-shell. There are four parameters in the loop integration variable which are fixed by choosing four constraints to be satisfied. There are in general two solutions to these conditions, each of which is a leading singularity. See [41,42,39] for more discussions of leading singularities.
The formula of [39] takes the form of an integral over the Grassmannian G(k, n), where k is the level in the MHV expansion and the n is the number of particles. It can be expressed either in the original twistor space [39] or in momentum twistor space [43]. The formula possesses one manifest superconformal symmetry and one non-manifest one and hence it produces (for different choices of the contour of integration) different Yangian invariants, including the ones we have seen appearing from the BCFW expansion. The twistor and momentum twistor versions of the formula can be directly related [44] showing that both versions do possess both copies of the superconformal symmetry. In fact this formula is the most general way of producing such an invariant [37,38]. If the conjecture of [39] is correct then we see that the Yangian symmetry also plays a role at loop level by constraining the form of the leading singularities. In the next section we will see how at least part of the symmetry also constrains the form of the loop amplitudes themselves and not just the leading singularities.
Loop corrections
Having examined in detail the structure of the amplitudes and symmetries at tree-level it is natural to ask what happens when perturbative corrections are taken into account. In this section we will review some of the features of scattering amplitudes in perturbation theory. As our main motivation is to understand the extended symmetries we discussed in the previous section, we will be concerned entirely with planar amplitudes in the N = 4 theory. Many of the developments we have already outlined at tree-level were in fact preceded by various observations for loop corrections to scattering amplitudes. As we will see the dual conformal symmetry can be directly observed in the form of the loop integrals appearing in the planar scattering amplitudes of N = 4 super Yang-Mills theory. A lot can be learned from the simplest case, namely the fourgluon scattering amplitudes. As we have seen, these amplitudes are examples of the so-called maximally-helicity-violating or MHV amplitudes.
MHV amplitudes are particularly simple in that they can naturally be written as a product of the rational tree-level amplitude and a loop-correction function which is a series in the 't Hooft coupling a,
A MHV n = A MHV
n,tree M n (p 1 , . . . , p n ; a).
By parity we could of course equivalently study the MHV amplitudes 4
In perturbation theory the function M n is expressed in terms of scalar loop integrals. Let us consider the four-gluon amplitude at one loop. The only contribution to M 4 at this order is given by the scalar box integral,
I (1) = d 4 k k 2 (k − p 1 ) 2 (k − p 1 − p 2 ) 2 (k + p 4 ) 2 .(139)
This integral formally exhibits dual conformal symmetry. To make this apparent we will employ our usual change of variables from momenta to dual coordinates,
p µ i = x µ i − x µ i+1 ≡ x µ i,i+1 ,(140)
with x 5 ≡ x 1 . After the change of variables (140) the integral can then be written as a four-point star diagram (the dual graph for the one-loop box) with the loop integration replaced by an integration over the internal vertex x 5 (see Fig. 12).
x 1
x 2
x 3 If we consider conformal inversions of the dual coordinates,
x 4 p 1 p 2 p 3 p 4 x 5x µ i −→ − x µ i x 2 i ,(141)
then we see that the integrand, including the measure factor d 4 x 5 , is covariant.
d 4 x 5 x 2 15 x 2 25 x 2 35 x 2 45 −→ (x 2 1 x 2 2 x 2 3 x 2 4 ) d 4 x 5 x 2 15 x 2 25 x 2 35 x 2 45 .(142)
If we had normalised the integral with an extra factor of x 2 13 x 2 24 then it would actually be invariant.
The property of dual conformal covariance of the integral form is not restricted to one loop but continues to all loop orders so far explored [45,46]. In Fig. 13 we show the integral topologies occurring in the four-point amplitude up to three loops. All of these integrals exhibit dual Figure 13: Integral topologies appearing in the four particle amplitude up to three loops.
conformal symmetry in the same sense as the one-loop scalar box.
The dual conformal property is rather restrictive in the kinds of integrals it allows to appear. At two loops there is again only one topology, the two-loop scalar box. At three loops there are two topologies, the three-loop box and the so-called 'tennis court'. The tennis court requires a precise numerator factor to be dual conformally covariant (see Fig. 14). Note that the operation
x 1 x 1 x 2 x 2 x 3
x 3
x 4 x 4 x 5
x 5
x 6
x 6
x 7 of drawing the dual graph is only possible for planar diagrams. This fits with our expectation that the symmetry is related to the integrable structure of the planar theory.
The integrals, e.g. as defined in (139), are infrared divergent. This is to be expected as amplitudes in massless theories inevitable exhibit infrared divergences. One therefore need to consistently introduce a regulator in order to talk about the S-matrix beyond tree level. A common choice is to use dimensional regularisation, by taking the original Lagrangian in 4 − 2ǫ dimensions, with ǫ < 0 regulating the infrared divergences. This breaks the dual conformal symmetry slightly since the measure is then no longer in four dimensions,
d 4 x 5 −→ d 4−2ǫ x 5 .(143)
Another choice for dealing with the infrared divergences is to study the theory on the Coulomb branch [47]. Now the VEVs of the scalar fields regulate the infrared divergences by introducing masses in a particular way for the virtual particles propagating in the loops. One can think of this regularisation geometrically; the dual Minkowski space is the boundary of a five-dimensional AdS space and one moves the points x 1 , x 2 , x 3 and x 4 slightly off the boundary into the bulk. The masses are interpreted as radial coordinates in the AdS 5 space and the corresponding action of dual conformal symmetry leaves the correctly normalised integral invariant 5 . One can therefore say that it is a function of the invariants one can construct from the dual x i and the masses (which also transform under the action of the dual conformal symmetry).
With either choice of regularisation the consequences of the new symmetry as the regulators are taken to zero are not immediately apparent. In dimensional regularisation one needs to know more precisely how the symmetry is broken by the O(ǫ) effects. In the AdS 5 regularisation one needs to know precisely how the amplitudes are allowed to depend on the radial coordinates (or masses) in the limit in which they become small.
Fortunately, there is a dual picture which allows us to understand the breaking of dual conformal symmetry in a precise way and also potentially sheds light on its origin. In the dual description planar MHV amplitudes are related to Wilson loops defined on a piecewise light-like contour in the dual coordinate space. The contour is none other than the light-like polygon in Fig. 11 that we have seen arising from the definition of the dual coordinates. In a gauge theory it is very natural to associate a Wilson loop to this contour,
W n = Pexp Cn A .(144)
Here, in contrast to the situation for the scattering amplitude, the dual space is being treated as the actual configuration space of the gauge theory, i.e. the theory in which we compute the Wilson loop is local in this space.
So let us consider the general structure of MHV amplitudes and light-like Wilson loops in N = 4 super Yang-Mills theory. We will begin with the MHV amplitude. As we have discussed we can naturally factorise MHV amplitudes into a tree-level factor and a loop-correction factor M n . The factor M n is infrared divergent and will therefore depend on the regularisation parameters.
We will use dimensional regularisation so M n depends on the regulator ǫ and an associated scale µ.
Since we are discussing planar colour-ordered amplitudes it is clear that the infrared divergences will involve only a very limited dependence on the kinematical variables. Specifically, the exchange of soft or collinear gluons is limited to sectors between two adjacent incoming particles and thus the infrared divergences will factorise into pieces which depend only on a single Mandelstam variable s i,i+1 = (p i + p i+1 ) 2 . Moreover the dependence of each of these factors is known to be of a particular exponentiated form [50,51,52,53,54,55,56,57,58,59,60,61,62] where there is at most a double pole in the regulator in the exponent. It is therefore natural to consider the logarithm of the loop corrections M n ,
log M n = ∞ l=1 a l Γ (l) cusp (lǫ) 2 + Γ (l) col lǫ i µ 2 IR −s i,i+1 lǫ + F MHV n (p 1 , . . . , p n ; a) + O(ǫ).(145)
The leading infrared divergence is known to be governed by Γ cusp (a) = a l Γ (l) cusp , the cusp anomalous dimension [63,64], a quantity which is so called because it arises as the leading ultraviolet divergence of Wilson loops with light-like cusps. This is a first hint at the connection between scattering amplitudes and Wilson loops.
In [65] Bern-Dixon and Smirnov (BDS) made an all order ansatz for the form of the finite part of the n-point MHV scattering amplitude. Their ansatz had the following form,
The notable feature of this ansatz is that the dependence on the coupling arises only via a single function, the cusp anomalous dimension, while the momentum dependence is contained in the coupling-independent function F n . The latter could therefore be defined by the one-loop amplitude, making the ansatz true by definition at one loop and highly non-trivial at higher loops. The formula (146) was conjectured after direct calculations of the four-point amplitude to two loops [66] and three loops [65]. It was found to be consistent with the five-point amplitude at two loops [67,68] and three loops [69]. Now let us consider Wilson loops on the polygon contour (144). A lot is known about the structure of such Wilson loops. In particular, due to the cusps on the contour at the points x i the Wilson loop is ultraviolet divergent. The divergences of such Wilson loops are intimately related to the infrared divergences of scattering amplitudes [63,64,70]. Indeed the leading ultraviolet divergence is again the cusp anomalous dimension and one can write an equation very similar to that for the loop corrections to the MHV amplitude,
log W n = ∞ l=1 a l Γ (l) cusp (lǫ) 2 + Γ (l) lǫ i (−µ 2 U V x 2 i,i+2 ) lǫ + F WL n (x 1 , . . . , x n ; a) + O(ǫ).(147)
The objects of most interest to us here are the two functions F MHV n from (145) and F WL n from (147) describing the finite parts of the planar MHV amplitude and Wilson loop respectively. In fact there is by now a lot of evidence that in the planar theory, the two functions are identical up to an additive constant, F MHV n (p 1 , . . . , p n ; a) = F WL n (x 1 , . . . , x n ; a) + d n (a) (148) once one changes variables from p i to x i as in (140).
The relation between amplitudes and Wilson loops first appeared at strong coupling [2]. In this regime the identification is a consequence of a particular T-duality transformation of the string sigma model which maps the AdS background into a dual AdS space [71]. The calculation of the scattering amplitude is then reduced to a minimal surface calculation in AdS space with the boundary of the surface being the light-like polygon on the boundary of AdS. The infinite coupling set up does not distinguish between different helicity configurations; the distinction should become apparent when subleading corrections are taken into account. The identification between amplitudes and Wilson loops is also true in the perturbative regime, however here it is important that the amplitudes are MHV. This is why we have referred only to the MHV amplitudes in the description of the duality. The fact that the Wilson loop is dual to the MHV amplitudes is in some sense natural because both are described by a single scalar function of the kinematic variables, making the identification possible. Non-MHV amplitudes have a richer structure which is so far not incorporated into the duality. The fact that the duality between planar MHV amplitudes and Wilson loops holds at both strong and weak coupling suggests that it should hold non-perturbatively. The explicit matching of the two quantities was observed at four points and one loop [72] and generalised to n points in [73]. Two loop calculations then followed [74,75,76,77,78]. In each case the duality relation (148) was indeed verified.
A point to be stressed here is that dual conformal symmetry finds a natural home within the duality between amplitudes and Wilson loops. It is simply the ordinary conformal symmetry of the Wilson loop defined in the dual space. Moreover, since this symmetry is a Lagrangian symmetry from the point of view of the Wilson loop, its consequences can be derived in the form of Ward identities [74,75]. We gave the form of the generator of special conformal transformations in (117). Here we keep only the part acting on the x i as the Wilson loop is only a function of these variables.
K µ = i x iµ x i · ∂ ∂x i − 1 2 x 2 i ∂ ∂x µ i .(149)
The analysis of [75] shows that the ultraviolet divergences induce an anomalous behaviour for the finite part F WL n which is entirely captured by the following conformal Ward identity
K µ F WL n = 1 2 Γ cusp (a) i (2x µ i − x µ i−1 − x µ i+1 ) log x 2 i−1,i+1 .(150)
A very important consequence of the conformal Ward identity is that the finite part of the Wilson loop is fixed up to a function of conformally invariant cross-ratios,
u ijkl = x 2 ij x 2 kl x 2 ik x 2 jl .(151)
In the cases of four and five edges, there are no such cross-ratios available due to the light-like separations of the cusp points. This means that the conformal Ward identity (150) has a unique solution up to an additive constant. Remarkably, the solution coincides with the Bern-Dixon-Smirnov all-order ansatz for the corresponding scattering amplitudes (here we give the formulas in terms of the x i variables),
F (BDS) 4 = 1 4 Γ cusp (a) log 2 x 2 13 x 2 24 + const ,(152)F (BDS) 5 = − 1 8 Γ cusp (a) 5 i=1 log x 2 i,i+2 x 2 i,i+3 log x 2 i+1,i+3 x 2 i+2,i+4 + const .(153)
Thus, taking the Ward identity over from the Wilson loops to the MHV amplitudes, we see that the agreement of the amplitudes with the BDS ansatz for four and five points can be explained by dual conformal symmetry.
In fact the BDS ansatz provides a particular solution to the conformal Ward identity for any number of points. From six points onwards however the functional form is not uniquely fixed as there are conformal cross-ratios available. Thus the general solution of the Ward identity contains an arbitrary function of the cross-ratios (which can also depend on the coupling),
F (WL) n = F (BDS) n + f n (u i ; a).(154)
At six points, for example there are three cross-ratios,
The solution to the Ward identity is therefore
F (WL) 6 = F (BDS) 6 + f 6 (u 1 , u 2 , u 3 ; a) .(156)
Here, upon the identification
p i = x i − x i+1 , F (BDS) 6 = 1 4 Γ cusp (a) 6 i=1 − log x 2 i,i+2 x 2 i,i+3 log x 2 i+1,i+3 x 2 i,i+3 + 1 4 log 2 x 2 i,i+3 x 2 i+1,i+4 − 1 2 Li 2 1 − x 2 i,i+2 x 2 i+3,i+5 x 2 i,i+3 x 2 i+2,i+5 + const ,(157)
while f 6 (u 1 , u 2 , u 3 ; a) is some function of the three cross-ratios and the coupling. The function f 6 is not fixed by the Ward identity and has to be determined by explicit calculation of the Wilson loop. The calculation of [73] shows that at one loop f 6 is constant (recall that at one loop the BDS ansatz is true by definition and the Wilson loop and MHV amplitude are known to agree for an arbitrary number of points). At two loops, direct calculation shows that f 6 is a non-trivial function [76,77]. Moreover the calculation [78] of the six-particle MHV amplitude shows explicitly that the BDS ansatz breaks down at two loops and the same function appears there,
F MHV 6 = F WL 6 + const, F MHV 6 = F BDS 6 .(158)
The agreement between the two functions F MHV 6 and F WL 6 was verified numerically to within the available accuracy. Subsequently the integrals appearing in the calculation of the finite part of the of the hexagonal Wilson loop have been evaluated analytically in terms of multiple polylogarithms [79].
Further calculations of polygonal Wilson loops have been performed. The two-loop diagrams appearing for an arbitrary number of points have been described in [80] where numerical evaluations of the seven-sided and eight-sided light-like Wilson loops were made. These functions have not yet been compared with the corresponding MHV amplitude calculations [81,82] due the difficulty of numerically evaluating the relevant integrals. However given the above evidence it seems very likely that the agreement between MHV amplitudes and light-like polygonal Wilson loops will continue to an arbitrary number of points, to all orders in the coupling.
While the agreement between Wilson loops is fascinating it is clearly not the end of the story. In particular the duality applies only to the MHV amplitudes. In the strict strong coupling limit this does not matter since all amplitudes are dominated by the minimal surface in AdS, independently of the helicity configuration [2]. At weak coupling that is certainly not the case and non-MHV amplitudes reveal a much richer structure than their MHV counterparts which is so far not captured by any dual object like a Wilson loop.
Despite the absence of such a dual model, one may still ask what happens to dual conformal symmetry for non-MHV amplitudes. Based on analysis of the one-loop NMHV amplitudes it was conjectured in [19] that the dual conformal anomaly is universal, i.e. is independent of the MHV degree. This is very natural because we have seen that the anomaly arises due to the infrared divergences (or ultraviolet divergences of the Wilson loop) and these are known to be independent of the helicities of the incoming particles. This means that one can write the all-order superamplitude as a product of the MHV superamplitude and an infrared finite ratio function,
A n = A MHV n R n .(159)
The conjecture of universality of the anomaly states that, setting the regulator to zero, R n is dual conformally invariant, K µ R n = 0.
In [83] it was argued that this conjecture holds for NMHV amplitudes at one loop, based on explicit calculations up to nine points using supersymmetric generalised unitarity. Subsequently [84,85,86] it has been argued to hold for all one-loop amplitudes by analysing the dual conformal anomaly arising from infrared divergent two-particle cuts.
Note that the conjecture (160) makes reference only to the dual conformal generator K and not to the full set of dual superconformal transformations. The reason is that some of these transformations overlap with the broken part of the original superconformal symmetry. In 1n i − 1 i Figure 15: A unitarity cut of a one-loop amplitude is given by the product of two tree-level amplitudes integrated over the phase space of the two exchanged on-shell particles.
particular the generatorQ does not annihilate the ratio function R n . This fact is related to the breaking of the original superconformal invariance by loop corrections sinceQ is really the same symmetry ass. Indeed, even at tree-levels is subtly broken by contact term contributions [87,88,89]. At one loop unitarity relates the discontinuity of the amplitude in a particular channel to the product of two tree-level amplitudes integrated over the allowed phase space of the exchanged particles (as illustrated in Fig. 15). The subtle non-invariance of the trees therefore translates into non-invariance of the discontinuity and therefore of the loop amplitude itself [88,89,90]. This breaking of thes =Q symmetry is in addition to that induced by the IR divergences as can be seen by considering a unitarity cut with more than two particles on each side. Such a cut is IR finite but notQ invariant as discussed in [88,90].
Summary
We have seen how the use of general considerations about the analytic structure of scattering amplitudes has led to a lot of insights into the nature of the S-matrix of gauge theories. The BCFW recursion relations allow for simple explicit expressions for tree-level scattering amplitudes. With the addition of maximal supersymmetry they become an extremely powerful tool. The expressions obtained contain non-local poles despite the fact that the underlying gauge theory is local. The non-local poles are spurious, cancelling between the different terms in the full amplitude. Their presence is connected to the fact that there are non-local symmetries at work. Each term in the BCFW expansion is an invariant of two copies of superconformal symmetry, the original symmetry of the Lagrangian and a new dual superconformal symmetry. Together these symmetries generate an infinite-dimensional Yangian symmetry. On the S-matrix the original superconformal symmetry acts locally but the extra charges generated by the dual superconformal symmetry act non-locally. Thus to express the amplitude in a way which reveals the symmetry inevitably means that locality is not manifest. As discussed in [12] there are good reasons for wanting a formulation of the S-matrix which does not refer to spacetime locality in an intrinsic way, in particular if one wants to include gravitational physics in the picture. These ideas suggest that the nicest formulation of the S-matrix of quantum field theories in general should not be intrinsically local and that the observation of the additional symmetries in the N = 4 case is perhaps the most concrete manifestation of a general principle.
Beyond tree level the dual conformal symmetry continues to provide strong constraints on the form of planar scattering amplitudes. Indeed the MHV amplitudes are equivalently described by light-like Wilson loops in the dual space whose conformal symmetry is the dual conformal symmetry of the amplitudes. Since the Wilson loops obey a conformal Ward identity, they and the corresponding MHV amplitudes are fixed up to a function of (dual) conformal invariants. This is sufficient to fix the four-point and five-point Wilson loops and amplitudes to all orders.
There are many open directions to pursue. A general picture that has emerged over recent years is that planar N = 4 super Yang-Mills theory is governed by some integrable structure. Indeed there is strong evidence that the spectrum of anomalous dimensions in the theory can be obtained from a nested Bethe ansatz [32] or Y-system [34]. These systems of equations arise in physical models from some underlying quantum group structure. It is known that the classical sigma model does indeed exhibit an infinite-dimensional symmetry algebra [36], arising from the existence of a one-parameter family of flat connections. In the study of scattering amplitudes (or Wilson loops) at strong coupling this structure has been exploited to derive systems of equations similar to those arising in the spectral problem [5]. While the existence of extra symmetries has been observed at weak coupling, as we have discussed in these notes, it is not yet known how to tie it in concretely with the above ideas. Thus one of the most central questions is how to exploit the integrability of theory to tell us more about the loop corrections. For example can we fix the remainder function f n of (154) using the symmetries? Perhaps one should talk directly about the ratio function R n of (159). Both of these quantities are infrared finite and are presumably controlled by the integrable structure lying behind the theory. At present there is not a complete understanding of the nature of the breaking of the original conformal symmetry by the amplitudes beyond tree-level. This has prevented it being used as a predictive tool for understanding the form of the remainder function f n or the ratio function R n . A related question is whether there is a generalisation of the amplitude/Wilson loop duality beyond the MHV amplitudes. Such a model should provide some understanding of the breaking of theQ supersymmetry observed in the amplitudes. For recent developments on these issues see [91,92,93,94,95,96,97].
A Conventions
We use the following conventions for the spinor contractions. A vector can be exchanged for a bispinor making use of the spin matrices σ µ ,
x αα = (σ µ ) αα x µ , x 2 = x µ x µ = 2x αα x αα . (A.1)
Derivatives are defined by
∂ αα x ββ = δ β α δβα, ∂ α λ β = δ β α etc. (A.2)
The spinor scalar products are defined as follows, The remaining non-trivial commutation relations are,
{Q αA , Q Ḃ α } = δ B A P αα , {S A α , Sα B } = δ A B K αα , [P αα , S βA ] = δ β α Q Ȧ α , [K αα , Q β A ] = δ β α Sα A , [P αα , Sβ A ] = δβαQ αA , [K αα , Qβ A ] = δβαS A α , [K αα , P ββ ] = δ β α δβαD + M α β δβα + Mαβδ β α , {Q α A , S B β } = M α β δ B A + δ α β R B A + 1 2 δ α β δ B A (D + C), {Qα A , Sβ B } = Mαβδ A B − δα β R A B + 1 2 δα β δ A B (D − C). (A.7)
Removing the hypercharge yields su(2, 2|4). Setting the central charge to zero gives pu(2, 2|4). Doing both gives the superconformal algebra psu(2, 2|4).
Figure 1 :Figure 2 :
12Schematic form for the propagators and vertices for Yang-Mills theory. The detailed index structure is not important for us. We just note that the colour structure enters only via the structure constants, while the momentum dependence appears in the propagators as 1/p 2 and in the three-point vertices as a positive power of p. Colour rules.
Figure 3 :
3Different amplitudes classified according to their helicity structures. Parity acts as reflection about the vertical axis of the diagram, swapping MHV and MHV amplitudes for example.
Figure 4 :
4An example of a Feynman diagram showing the propagators affected by the BCFW shift.
Figure 5 :Figure 6 :
56The sum over states giving a particular residue. The contour deformation giving a sum over residues.
Figure 7 :
7An example of a Feynman diagram contributing to the leading z dependence as z goes to infinity. The vertices joining the line of shifted propagators are all three-point vertices.
Figure 8 :
8The single BCFW diagram contributing to the four-point amplitude.
Figure 9 :
9The single BCFW diagram contributing to the n-point MHV amplitude.
Figure 10 :
10The two contributions to the supersymmetric recursion relation for NMHV amplitudes. We call them homogeneous and inhomogeneous respectively.
Figure 11 :
11The light-like polygon in dual coordinate space defined by the particle momenta.
Figure 12 :
12Dual diagram for the one-loop box.
Figure 14 :
14Dual diagrams for the three-loop box and for the 'tennis court' with its numerator denoted by the dashed line corresponding to a factor x 235 .
F
BDS n (p 1 , . . . , p n ; a) = Γ cusp (a)F n (p 1 , . . . , p n ) + c n (a).
ab = a α b α = a α b β ǫ βα , [ab] = aαbα = aαbβǫαβ (A.3)Longer spinor contractions are also useda|P |b] = a α P αα bα, a|xy|b = a α x αα yα β b β ,give the commutation relations for the superconformal algebra. We begin by listing the commutation relations of the algebra u(2, 2|4). The Lorentz generators M αβ , Mαβ and the su(4) generators R A B act canonically on the remaining generators carrying Lorentz or su(4) indices. The dilatation D and hypercharge B act via [D, J] = dim(J) J, [B, J] = hyp(J) J.
We are choosing to shift λ 1 andλ n for later convenience but one can derive relations for amplitudes by shifting any pair of legs.
We use the symbol [O 1 , O 2 ] to denote the bracket of the Lie superalgebra, [O 2 , O 1 ] = (−1) 1+|O1||O2| [O 1 , O 2 ]. 3 The symbol {·, ·, ·} denotes the graded symmetriser.
As we saw already, the four-particle amplitudes are actually both MHV and MHV.
In addition to this conceptual advantage this regularisation also has practical advantages, see[48,49] for more discussion.
AcknowledgementsI would like to thank Livia Ferro, Johannes Henn, Gregory Korchemsky, Jan Plefka, Vladimir Smirnov and Emery Sokatchev for collaboration on the topics presented in these notes. I also thank Johannes Henn for comments on the manuscript and the organisers of the CERN winter school for the opportunity to lecture at such an enjoyable school.
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A Brandhuber, P Heslop, G Travaglini, arxiv:0906.3552Proof of the Dual Conformal Anomaly of One-Loop Amplitudes in N = 4 SYM. A. Brandhuber, P. Heslop and G. Travaglini, "Proof of the Dual Conformal Anomaly of One-Loop Amplitudes in N = 4 SYM", arxiv:0906.3552.
T Bargheer, N Beisert, W Galleas, F Loebbert, T Mcloughlin, arxiv:0905.3738Exacting N = 4 Superconformal Symmetry. T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, "Exacting N = 4 Superconformal Symmetry", arxiv:0905.3738.
G P Korchemsky, E Sokatchev, arxiv:0906.1737Symmetries and analytic properties of scattering amplitudes in N = 4 SYM theory. G. P. Korchemsky and E. Sokatchev, "Symmetries and analytic properties of scattering amplitudes in N = 4 SYM theory", arxiv:0906.1737.
A Sever, P Vieira, arxiv:0908.2437Symmetries of the N = 4 SYM S-matrix. A. Sever and P. Vieira, "Symmetries of the N = 4 SYM S-matrix", arxiv:0908.2437.
One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills. N Beisert, J Henn, T Mcloughlin, J Plefka, arxiv:1002.1733JHEP. 100485N. Beisert, J. Henn, T. McLoughlin and J. Plefka, "One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills", JHEP 1004, 085 (2010), arxiv:1002.1733.
From correlation functions to Wilson loops. L F Alday, B Eden, G P Korchemsky, J Maldacena, E Sokatchev, arxiv:1007.3243L. F. Alday, B. Eden, G. P. Korchemsky, J. Maldacena and E. Sokatchev, "From correlation functions to Wilson loops", arxiv:1007.3243.
From correlation functions to scattering amplitudes. B Eden, G P Korchemsky, E Sokatchev, arxiv:1007.3246B. Eden, G. P. Korchemsky and E. Sokatchev, "From correlation functions to scattering amplitudes", arxiv:1007.3246.
N Arkani-Hamed, J L Bourjaily, F Cachazo, S Caron-Huot, J Trnka, arxiv:1008.2958The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, "The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM", arxiv:1008.2958.
J M Drummond, J M Henn, arxiv:1008.2965Simple loop integrals and amplitudes in N=4 SYM. J. M. Drummond and J. M. Henn, "Simple loop integrals and amplitudes in N=4 SYM", arxiv:1008.2965.
The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space. L Mason, D Skinner, arxiv:1009.2225L. Mason and D. Skinner, "The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space", arxiv:1009.2225.
B Eden, G P Korchemsky, E Sokatchev, arxiv:1009.2488More on the duality correlators/amplitudes. B. Eden, G. P. Korchemsky and E. Sokatchev, "More on the duality correlators/amplitudes", arxiv:1009.2488.
S Caron-Huot, arxiv:1010.1167Notes on the scattering amplitude / Wilson loop duality. S. Caron-Huot, "Notes on the scattering amplitude / Wilson loop duality", arxiv:1010.1167.
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"On Tverberg partitions",
"On Tverberg partitions"
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"Moshe J White \nInstitute of Mathematics\nHebrew University\nJerusalemIsrael\n"
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"Institute of Mathematics\nHebrew University\nJerusalemIsrael"
]
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| A theorem of Tverberg from 1966 asserts that every set X ⊂ R d of n = T (d, r) = (d+1)(r−1)+1 points can be partitioned into r pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of n into r parts (that is, r integers a 1 , . . . , a r satisfying n = a 1 + · · · + a r ), in which the parts a i correspond to the number of points in every subset. In this paper, we prove that for any partition of n where the parts satisfy a i ≤ d + 1 for all i = 1, . . . , r, there exists a set X ⊂ R of n points, such that every Tverberg partition of X induces the same partition on n, given by the parts a 1 , . . . , a r . | 10.1007/s11856-017-1490-2 | [
"https://arxiv.org/pdf/1508.07262v2.pdf"
]
| 119,161,593 | 1508.07262 | f13f93f2e07cf319cae446a9ad8e12078a3022e7 |
On Tverberg partitions
15 May 2017
Moshe J White
Institute of Mathematics
Hebrew University
JerusalemIsrael
On Tverberg partitions
15 May 2017
A theorem of Tverberg from 1966 asserts that every set X ⊂ R d of n = T (d, r) = (d+1)(r−1)+1 points can be partitioned into r pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of n into r parts (that is, r integers a 1 , . . . , a r satisfying n = a 1 + · · · + a r ), in which the parts a i correspond to the number of points in every subset. In this paper, we prove that for any partition of n where the parts satisfy a i ≤ d + 1 for all i = 1, . . . , r, there exists a set X ⊂ R of n points, such that every Tverberg partition of X induces the same partition on n, given by the parts a 1 , . . . , a r .
Introduction
Tverberg's theorem [Tve66] from 1966 asserts the following: For any two integers d, r define n = T (d, r) = (d + 1)(r − 1) + 1. Then every X ⊂ R d of n points can be partitioned into r disjoint subsets, X 1 , . . . , X r , so that ∩ r i=1 conv(X i ) = ∅. Such a partition is known as a Tverberg partition of X, and the points in ∩ r i=1 conv(X i ) are known as Tverberg points of X.
In this paper, we prove the following theorem:
Theorem 1. Given integers d, r and n = T (d, r) = (d+1)(r−1)+1, let a 1 , . . . , a r be integers satisfying a i ≤ d + 1 and r i=1 a i = n . Then there exists a set X ⊂ R d of n points, such that for any X 1 , . . . , X r that form a Tverberg partition of X, the cardinalities of X 1 , . . . , X r are a permutation of the integers a 1 , . . . , a r .
Remark 1.1. The requirement a i ≤ d + 1 for all i = 1, . . . , r is necessary: Suppose X 1 , . . . , X r is a Tverberg partition of X with some Tverberg point p, and |X 1 | > d + 1 (without loss of generality). By Caratheodory's theorem, there exists x ∈ X 1 so that p ∈ conv(X 1 \{x}). Therefore if we move the point x from X 1 to the smallest subset among X 2 , . . . , X r , we obtain a new Tverberg partition of X, which induces a different integer partition on n.
We will construct the family of sets X that will be used in this proof, follow by proving some properties that apply to such sets, prove the theorem, and conclude by showing that the number of Tverberg partitions for these sets is always [(r − 1)!] d .
Construction
Assume d, r, n ∈ N satisfy n = (d + 1)(r − 1) + 1. We construct the set X as a union of d + 1 disjoint sets, A, A 1 , . . . , A d , defined as follows: Denote by e 1 , . . . , e d the standard basis
in R d . For 1 ≤ j ≤ d, A j = {e j , 2e j , . . . , (r − 1)e j }.
Let a 1 , . . . , a r be integers as in Theorem 1. Note that:
r i=1 (d + 1 − a i ) = r(d + 1) − r i=1 a i = r(d + 1) − n = d. Therefore for each 1 ≤ j ≤ d we can choose 1 ≤ i(j) ≤ r, so that for all 1 ≤ i ≤ r, #{1 ≤ j ≤ d : i(j) = i} = d + 1 − a i .(1)
For every 1 ≤ i ≤ r and every 1 ≤ j ≤ d, define:
x i j = 0 if i = i(j) −i otherwise. The set A is now defined as A = {x 1 , . . . , x r }, where x i = (x i 1 , . . . , x i d ) for all i = 1, . . . , r. Note that X = A ∪ A 1 ∪ · · · ∪ A d ,
where A contains r points and each A j contains r − 1 points, therefore |X| = r + d(r − 1) = n as required.
Remark 2.1. The specific values of the coordinates of the points in X are not crucial. It can be shown that if we construct X and then move all the points very slightly, the theorem can be proved with only slight modifications. Therefore it is always possible to find a set X that satisfies the theorem and is also in general position or even strong general position [DV77], [PS14]. On the other hand, we could allow X to be a multiset, and let A j consist of r − 1 copies of the same point e j (without any change to the proof below).
Proof of Theorem 1
Throughout the proof, assume X is a set constructed as above, X 1 , . . . , X r is a Tverberg partition of X, and p = (p 1 , . . . , p d ) ∈ r i=1 conv(X i ) is a Tverberg point of X. Lemma 3.1. p = 0. In other words, 0 is the only Tverberg point of X.
Proof. Choose any 1 ≤ j ≤ d. There are exactly r − 1 points x ∈ X satisfying x j > 0 (these are the points in A j ). Therefore for some 1 ≤ i ≤ r, X i is contained in the closed half-space {x ∈ R d : x j ≤ 0}. Since p ∈ conv(X i ), we must have p j ≤ 0.
On the other hand, there are exactly r − 1 points x ∈ X satisfying x j < 0 (these are the points x i ∈ A, for i = i(j)). Thus by a similar argument, p j ≥ 0. Therefore p j = 0 for all 1 ≤ j ≤ d.
Corollary 3.2. For all 1 ≤ i ≤ r, X i ∩ A contains exactly one point.
Proof. Otherwise, there is some 1 ≤ i ≤ r such that X i ∩ A = ∅ (since A contains r points). This implies that p ∈ conv(X i ) ⊂ conv(∪ d j=1 A j ). Since every point x ∈ ∪ d j=1 A j satisfies d j=1 x j > 0, p must also satisfy d j=1 p j > 0, but this contradicts Lemma 3.1.
Corollary 3.3. Let X 1 , . . . , X r be some Tverberg partition of X. We may always assume that x i ∈ X i for all 1 ≤ i ≤ r (otherwise we use Corollary 3.2 and renumber the sets of the partition). Then for every 1 ≤ i ≤ r and every 1 ≤ j ≤ d:
|X i ∩ A j | = 0 if i = i(j) 1 otherwise.
In particular, |X i | = a i , and X satisfies the requirements of Theorem 1.
Proof. Fix 1 ≤ i ≤ r and 1 ≤ j ≤ d. From Lemma 3.1, we deduce that 0 ∈ conv(X i ). Since x i is the only point in X i ∩ A, there is some λ ∈ (0, 1] and some y i ∈ conv(X i \A), satisfying
λx i + (1 − λ)y i = 0.
Note that λ cannot be 0, because 0 / ∈ conv(X\A). If i = i(j), we have x i j < 0. Looking at the j-th coordinate of the above equation, we deduce that y i j > 0. This is only possible if X i ∩ A j is non-empty (as only points from A j have positive values in the j-th coordinate). Thus
|X i ∩ A j | ≥ 0 if i = i(j) 1 otherwise.(2)
Since A, A 1 , . . . , A d is a partition of X, we obtain (using eq. (1)):
|X i | = 1 + d j=1 |X i ∩ A j | ≥ 1 + (d − #{1 ≤ j ≤ d : i(j) = i}) = a i .(3)
However, X 1 , . . . , X r is also a partition of X, thus n = |X| = r i=1 |X i | ≥ r i=1 a i = n. This leads us to conclude all the inequalities in (2) and (3) are in fact equalities.
The number of Tverberg partitions of X Sierksma conjectured that any set X ⊂ R d of n = T (d, r) points has at least [(r − 1)!] d different Tverberg partitions (for known lower bounds, see [VŽ93] and [Hel07]). In the above construction, we can account for all Tverberg partitions of X, and their number is exactly [(r − 1)!] d . We state and prove this as the following proposition:
Proposition 3.4. Let d, r, n and X be as above. Then X has exactly [(r − 1)!] d different Tverberg partitions.
Proof. Assume X 1 , . . . , X r is a Tverberg partition of X, such that x i ∈ X i for all 1 ≤ i ≤ r (we may always assume that x i ∈ X i , due to Corollary 3.2). For each 1 ≤ j ≤ d, define a bijection σ j : A j → {1, . . . , r}\{i(j)}, satisfying x ∈ X σ j (x) for all x ∈ A j (σ j is a bijection due to Corollary 3.3). Note that any Tverberg partition X 1 , . . . , X r is uniquely determined by the d bijections σ 1 , . . . , σ d .
On the other hand, for any d bijections σ j : A j → {1, . . . , r}\{i(j)}, define for each 1 ≤ i ≤ r:
X i = {x i } ∪ {σ −1 j (i) : 1 ≤ j ≤ d, i = i(j)}.
It is easy to verify that 0 ∈ conv(X i ) for all 1 ≤ i ≤ r, therefore X 1 , . . . , X r is a Tverberg partition of X, which is uniquely determined by the bijections σ 1 , . . . , σ d . This shows that the number of Tverberg partitions of the set X is equal to the number of unique choices of bijections σ j : A j → {1, . . . , r}\{i(j)} for 1 ≤ j ≤ d. For each 1 ≤ j ≤ d, A j contains r − 1 points, so there are (r − 1)! possible choices for σ j , and a total of [(r − 1)!] d unique Tverberg partitions of X.
Acknowledgments I am very grateful to my advisor Gil Kalai, for his guidance and support. I would also like to thank Imre Bárány for a fruitful discussion.
Radon partitions and a new notion of independence in affine and projective spaces. J.-P Doignon, G Valette, Mathematika. 24J.-P. Doignon and G. Valette, Radon partitions and a new notion of independence in affine and projective spaces, Mathematika 24 (1977), 86-96.
On the number of Tverberg partitions in the prime power case. S Hell, European Journal of Combinatorics. 28S. Hell, On the number of Tverberg partitions in the prime power case, European Journal of Combinatorics 28 (2007), 347-355.
M A Perles, M Sigron, arXiv:1409.2899Strong general position. arXiv preprintM. A. Perles and M. Sigron, Strong general position, arXiv preprint arXiv:1409.2899 (2014).
A generalization of Radons theorem. H Tverberg, J. London Math. Soc. 41H. Tverberg, A generalization of Radons theorem, J. London Math. Soc 41 (1966), 123-128.
A Vučić, R T Živaljević, Note on a conjecture of Sierksma, Discrete & Computational Geometry. 9A. Vučić and R. T.Živaljević, Note on a conjecture of Sierksma, Discrete & Com- putational Geometry 9 (1993), 339-349.
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"\nSchool of Information Technologies\nUniversity of Sydney\nNSW\nAustralia\n"
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"School of Information Technologies\nUniversity of Sydney\nNSW\nAustralia"
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| The availability of large-scale annotated image datasets coupled with recent advances in supervised deep learning methods are enabling the derivation of representative image features that can potentially impact different image analysis problems. However, such supervised approaches are not feasible in the medical domain where it is challenging to obtain a large volume of labelled data due to the complexity of manual annotation and inter-and intra-observer variability in label assignment. Algorithms designed to work on small annotated datasets are useful but have limited applications. In an effort to address the lack of annotated data in the medical image analysis domain, we propose an algorithm for hierarchical unsupervised feature learning. Our algorithm introduces three new contributions: (i) we use kernel learning to identify and represent invariant characteristics across image sub-patches in an unsupervised manner; (ii) we leverage the sparsity inherent to medical image data and propose a new sparse convolutional kernel network (S-CKN) that can be pre-trained in a layer-wise fashion, thereby providing initial discriminative features for medical data; and (iii) we propose a spatial pyramid pooling framework to capture subtle geometric differences in medical image data. Our experiments evaluate our algorithm in two common application areas of medical image retrieval and classification using two public datasets. Our results demonstrate that the medical image feature representations extracted with our algorithm enable a higher accuracy in both application areas compared to features extracted from other conventional unsupervised methods. Furthermore, our approach achieves an accuracy that is competitive with state-ofthe-art supervised CNNs. | 10.1016/j.media.2019.06.005 | [
"https://arxiv.org/pdf/1807.05648v2.pdf"
]
| 51,646,820 | 1807.05648 | 9a46d4c52548eba1fe58fef7b3289c6d55a21276 |
School of Information Technologies
University of Sydney
NSW
Australia
Index Terms-Unsupervised Feature LearningMedical Image RetrievalMedical Image ClassificationConvolutional Neural NetworkTransfer Learning
The availability of large-scale annotated image datasets coupled with recent advances in supervised deep learning methods are enabling the derivation of representative image features that can potentially impact different image analysis problems. However, such supervised approaches are not feasible in the medical domain where it is challenging to obtain a large volume of labelled data due to the complexity of manual annotation and inter-and intra-observer variability in label assignment. Algorithms designed to work on small annotated datasets are useful but have limited applications. In an effort to address the lack of annotated data in the medical image analysis domain, we propose an algorithm for hierarchical unsupervised feature learning. Our algorithm introduces three new contributions: (i) we use kernel learning to identify and represent invariant characteristics across image sub-patches in an unsupervised manner; (ii) we leverage the sparsity inherent to medical image data and propose a new sparse convolutional kernel network (S-CKN) that can be pre-trained in a layer-wise fashion, thereby providing initial discriminative features for medical data; and (iii) we propose a spatial pyramid pooling framework to capture subtle geometric differences in medical image data. Our experiments evaluate our algorithm in two common application areas of medical image retrieval and classification using two public datasets. Our results demonstrate that the medical image feature representations extracted with our algorithm enable a higher accuracy in both application areas compared to features extracted from other conventional unsupervised methods. Furthermore, our approach achieves an accuracy that is competitive with state-ofthe-art supervised CNNs.
automatically analyse, categorise or retrieve images. These CAD systems often relate low-level image features to highlevel semantic concepts or expert domain knowledge using machine learning techniques.
Learning-based algorithms take advantage of the incorporation of prior knowledge derived from labelled training data. Deep learning approaches, such as convolutional neural networks (CNNs), produce impressive results in natural (photographic) image classification [2], [3], [4]. CNNs learn image features in a hierarchical fashion. Each deeper layer of the network learns a representation of the image data that is high-level and semantically more meaningful, e.g., in a classification application the learned features can be a classspecific representation [5] that enables better discrimination between different image classes [2], [3]. CNNs require a large number of annotated training images (e.g., ImageNet with over 1 million natural images). In the medical imaging domain, however, there is a limited availability of large annotated datasets; medical images are complex to interpret, requiring clinicians to label the images, which is a costly exercise that is further hindered by inter-and intra-observer variability among clinicians [6].
The concept of transfer learning has been used to overcome the lack of available labelled medical image data, either by using a model that was pre-trained on a different domain (e.g., natural images [7]) as a generic feature extractor, or alternatively, using a relatively small set of medical images to optimise a pre-trained model from a different domain (i.e., finetuning) [8], [9], [10], [11]. Unfortunately, since these approaches rely upon more general image features they may potentially be unable to capture the high-level semantic features that are most relevant for a specific dataset. Consequently, they are unable to match the overall accuracy of learning image features directly from large annotated data of a specific type. An emerging approach to tackle this limitation is to use unsupervised feature learning algorithms to build features from unlabelled data, which allows large unannotated medical image collections to be used. Many of these approaches are based on algorithms such as sparse coding [12], sparse auto-encoders [13], and Restricted Boltzmann Machines (RBMs) [14]. However, many of these Medical School, University of Sydney, Camperdown, NSW, Australia (E-mail: [email protected]). methods have only shown strong performance in learning lowlevel features such as 'blobs' or 'edges'. Only a few methods have succeeded in extracting semantic high-level features, such as the stacked sparse autoencoder (SSAE) presented by Le et al. [5], which learned image features in a hierarchal manner. The SSAE was used to pre-train a model that was later coupled to supervised deep learning (i.e., fine-tuning) and achieved improved results in object recognition [5]. Such unsupervised pre-training learned useful priors that acted as an initialisation point for the supervised fine-tuning, making the supervised model less prone to overfitting or being trapped in local minima [15], [16].
Hierarchical convolutional kernel-based networks (CKNs) have recently been introduced to generate multi-layer image representations in an unsupervised manner [17], with state-ofthe-art performance in natural image classification [17] and retrieval [18]. The CKN architecture is capable of learning the local geometry of the data without reliance on labels. The kernel learning is a function that describes an inner product of any two training samples in some induced Hilbert space [19]. It formalises the notion of similarity and provides a representation of the data that can better reconstruct training samples. CKNs incorporate these characteristics and learn data representations in a non-linear hierarchical manner. However, they are prone to overfitting when the number of training data is small. That is, the learning cost function often gets stuck in local minima.
The concept of sparsity has been widely used in computer vision and has proven to be effective in image compression [20], denoising [21], and tomographic reconstructions. Sparsity can be used to derive compact and optimal representations of image data, where a number of trivial information or parameters can be ignored without compromising image quality or characteristics [22]. For example, the temporal resolution of magnetic resonance (MR) imaging was greatly improved [23] with the use of an additional sparsity constraint, thereby allowing for the development of a number of novel CAD systems in cardiac and brain imaging. Our hypothesis is that if we use such sparsity constraints, we will potentially enable the derivation of compact and optimal representations of medical images.
In this paper, we propose a new sparse convolutional kernel network (S-CKN) that leverages the sparsity inherent to medical image data to address the current limitations outlined above. Our S-CKNs can be pre-trained in a layer-wise fashion to extract sparse features that are more relevant to medical image data, i.e., are more discriminative for medical images. In addition, we couple our S-CKN to a spatial pyramid pooling (SPP) framework that enables a better characterisation of the local geometry of the medical image data. We validate our proposed method on two public datasets with comparisons to other unsupervised feature learning algorithms as well as supervised CNNs. The main contributions of our work can be summarized as follows:
1) A new approach to characterise medical images by combining kernel learning and CNNs to learn the local geometry of the medical data in a hierarchical manner.
2) An unsupervised sparse feature learning algorithm which effectively learns initial discriminative features. Specifically, the algorithm is used to initialise the weights of a CKN, which can be pre-trained in a layer-wise fashion. We emphasize that our proposed method learns image features completely unsupervised, which is more challenging than the standard use of supervised CNNs.
3) A spatial pyramid pooling framework that provides more discriminative and geometrically invariant local feature representations of medical image data. 4) A comprehensive comparison of our approach to state-ofthe-art methods.
II. RELATED WORK
A. X-ray Image Retrieval
Plain X-ray images are commonly performed and wellunderstood medical images. The retrieval of X-ray image is a critical step for imaging-based clinical decision support systems [24], [25], [7]. Automated X-ray image retrieval, however, is challenging due to irregular brightness and contrast, and artifacts caused by prostheses and other implants. There are also high intra-class variability and inter-class similarity among images in X-ray repositories.
Prior studies have used hand-crafted descriptors and as such the similarity of classes was only measured within the specific feature space. Moreover, these features were often represented as a collection of unordered local features and disregarded the local geometry of the features. For example, the scale-invariant feature transform (SIFT) was used to extract image features invariant to changes in scaling orientation and illumination and coupled with the bag-of-visual words (BoVW) model to form a histogram representation of the image [26,27]. Other common approaches used a combination of multi-visual features including local binary patterns, texture and shape [28]. Avni et al. [26] presented densely sampled normalised features coupled with spatial information for X-ray categorisation and retrieval.
Anavi et al. [29] recently benchmarked hand-crafted descriptors against features extracted using pre-trained CNNs, and concluded that the pre-trained CNNs produced state-of-theart results for chest X-ray classification. Recently, Ahn et al. [7] combined hand-crafted local features with learned features transferred from a different domain. However, as mentioned previously, such features cannot currently match the overall accuracy of directly learning features from a large dataset that is specific to the problem domain [9,10].
B. Medical Imaging Modality Classification
While a multitude of different types of images have been collected to assist in the development of more advanced CAD systems, the labelling of the collated image data remains problematic [30], [31], [32], [33]. In cases where appropriate labels are absent, automatic identification of the imaging modality is an initial important step because the semantics and content of an image can vary greatly depending on the modality.
In prior research, a variety of algorithms have been used to extract and fuse a range of image features [34]. These features were often designed by humans to derive particular underlying image characteristics such as colour, texture, local binary patterns, and spatial orientation. The performance of these algorithms was limited by the quality of the features. Kumar et al. recently proposed an ensemble of fine-tuned CNNs to learn different levels of semantic image representations [8].
III. METHODS
A. Background: Convolutional Neural Networks
CNNs process an input image using multiple layers and learn features in a hierarchical manner. CNN layers generally have: 1) convolutional layers to learn weights (i.e., filters) that can be used to extract features from the input; 2) a linear operation followed by a pointwise non-linearity such as Sigmoid or rectified linear units, which prevents explosion of gradients and speeds up training; and 3) pooling layers to aggregate features that are in spatial proximity (down-sampling the data in the process). The output of single layer CNN can be represented as:
( ) = ( (W⨂ + b)),(1)
where is the input feature vector, σ(•) is the pointwise nonlinear function and = {W, b} are the set of parameters (i.e., weights and biases). The function denotes a downsampling operation and is the size of pooling region. The symbol indicates the linear convolution. When a convolutional layer is dense and unstructured, it is called "fully connected". For example, the well-established AlexNet [35] CNN has 8 trainable layers comprising five convolutional layers followed by three fully connected layers. Training such a CNN, however, is challenging because of the number of hyperparameters that need to be carefully tuned. Some major hyperparameters include the size of learnable filters, the number of layers, the number of outputs per layer, and the size of the down-sampling factor. Sub-optimal hyperparameter choice leads to overfitting and an inability to derive optimal high-level semantic image features. Some supervised CNNs have exploited unsupervised layerwise pre-training schemes to render better generalisation of image data [5], [16]. The pretraining acts as a form of regularisation which minimises variance and restricts the range of the parameter values for subsequent supervised training [15]. Layerwise unsupervised pre-training allows all the available unlabelled image data to be used to pre-train the network's local parameters, which potentially provides a good initialisation point for further supervised training.
B. Convolutional Kernel Networks
CKNs have the classic hierarchical architecture of CNNs but use kernel maps to represent image features. A kernel map is used to understand the local geometry of the image data by modelling invariance [17]. We suggest that kernels coupled with a hierarchical architecture allow the effective learning of image features without a reliance on labels. The architecture of a two-layer CKN is shown in Fig 1. Let us consider two image patches and ′ of an image of size × ( = 200 in this paper), with Ω being a set of pixel coordinates (Ω = {1, … } 2 ). Given the location in Ω, and the sub-patches ∈ and ′ ∈ ′ of the image feature map, a single layer convolutional kernel network is then defined as follows:
( , ′ ) = ∑ ‖ ‖ ℋ ‖ ′ ‖ ℋ − 1 2 2 ‖ − ′ ‖ 2 2 − 1 2 2 ‖̃−̃′‖ ℋ 2 , ′ ∈Ω (2)
The kernel is a positive definitive kernel that consists of a sum of pairwise comparisons between image features of subpatches. The ‖•‖ ℋ denotes the Hilbertian norm and the term ‖ ‖ ℋ ‖ ′ ‖ ℋ acts to emphasise the spatial and feature similarity (captured by the exponential terms) for non-small intensityvalued patches. The term measures the feature similarity between sub-patches. These two terms work in conjunction with the Hilbertian norm terms to create a kernel that gives larger values for patches that are close in both space and intensity.
We used two different types of input feature maps; 1) Patch map: the is an image sub-patch size × centred at . The sub-patch is simply ℝ × and ̃ denotes a contrast-normalised version of the sub-patch.
2) Gradient map: the sub-patch is the two-dimensional gradient of the image at pixel , which is computed with firstorder differences along each dimension. In this formulation, ‖ ‖ ℋ is the gradient intensity and ̃ denotes its orientation defined as an angle with [cos , sin ] [36]. When the input data is in a compact set ( ℝ , ≤ 2) , Equation (2) The coefficients and are smoothing Gaussian kernel parameters that control spatial distances between and ′ and the feature closeness between ̃ and ̃′ in Hilbert space respectively. The corresponding kernel map is formalised as a weighted match kernel between all sub-patches from training samples, which defines a feature representation of the image.
C. Unsupervised Feature Learning via CKNs
Match kernels are expensive to compute when the input data has high dimensionality ( ℝ , > 2 ). The computational complexity also grows quadratically with increasing sample Fig 1. A two-layer convolutional kernel network; each layer is a weighted match kernel between all sub-patches of the previous layer. The image is adapted from [17] sizes. To prevent the curse of dimensionality, we used a fast approximation approach with finite-dimensional embedding proposed by Marial et al. [17]. For all and ,
( , ′ ) ≈ ∑ ( ; ) ( ; ′ ) ∈Ω 1 (3) where ( ; ) ≔ ∑ − 1 2 2 ‖ − ‖ 2 2 ℎ( ; ) ∈Ω (4) and ℎ( ; ) ≔ ‖ ‖ 2 [√b − 1 2 ‖W −̃‖ 2 2 ] =1 1 ,(5)
where Ω 1 is a subset of Ω, 1 denotes number of filters, and b and W are learned parameters. This operation can be considered to be similar to a spatial convolution of the feature map followed by a pointwise non-linearity. Since ( , ′ ) is a sum of the match kernel terms, we can learn to approximate the kernel using training data. The parameters b and W are learned at the sub-patch level by solving an optimisation problem:
min W ,b ∑ ( − ‖̃−̃′ ‖ 2 2 2 2 − ∑ b − ‖W −̃‖ 2 2 2 =1 − ‖W −̃′ ‖ 2 2 2 ) =1(6)
We randomly selected 400,000 pairs of sub-patches from the training data and used the standard Limited memory Broyden Fletcher Goldfarb Shanno with Bounds (L-BFGS-B) [37] optimiser to solve Equation (6) [17]. The L-BFGS-B requires less parameters and can be superior to the conjugate gradient (CG) or stochastic gradient descent (SGD) in many applications such as image classification [38].
D. Layerwise Unsupervised Pre-training with Sparsity
It has been widely demonstrated that feature representations of medical image data have an intrinsic sparse structure under certain fixed bases (e.g., Fourier) [23,39]. This intrinsic sparsity often comes in two complementary forms [40]: population and lifetime sparsity. Population sparsity refers to the activation of small subsets of the bases (i.e., a sparse set of the population) to encode different information; only a small subset of the coding outputs (feature maps or bases) are active for any given stimulus (input images), and different subsets are active for different stimuli. This ensures that the activation of different bases is a discriminator for different image data. In contrast, lifetime sparsity refers to the short frequency of activation of bases for different inputs (i.e., each base has a sparse lifetime); different bases are active very rarely and each activation has a high response. This ensures that the strong rare activations are indicators for high degree of information (the higher the information, the higher the entropy) about the underlying image data. A number of investigators have successfully applied sparsity to many situations including medical image segmentation and classification [7], [41], [42], [43]. Motivated by these findings, we hypothesise that incorporating sparsity into layerwise unsupervised pretraining could allow the extraction of more discriminative features for medical image data.
We formulated a layerwise unsupervised feature learning algorithm that efficiently enforces population and lifetime sparsity (EPLS) [44]. Our approach learns convolutional sparse features in reproducing kernel Hilbert space (RKHS), in contrast to the original EPLS algorithm by Romero et al. [44] that learns sparse features from decomposed raw image patches. The convolutional sparse features learned in the unified feature space are often more discriminative and therefore allow us to build more class-specific representations [45]. Furthermore, the convolutional features learned by our method preserve the relationships between neighbourhood pixels so as to learn local structures and reduce redundancy in the parameters [44,46]. The learned parameters are used as initialisation points in CKNs learning (i.e., the initial value of = {W, b} of each layer). The algorithm iteratively creates a layer-specific sparse target of the input data and optimises the dictionary by minimising the error between the output of the layer and the sparse target. The degree of sparsity is therefore controlled and learned differently at each layer. The parameters of the layer are then calculated as follows [44]:
= arg min‖ − ‖ 2 2 ,(7)
where ∈ ℛ N ×N ℎ are the data vectors in RKHS, which are represented as a weighted combination of the training samples used to construct the kernel matrix at layer , and denotes the = + N ℎ / 8: end for 9: Remap to active/inactive values is the output that has to be activated in the -th row of and is an accumulator that counts the number of times an output has been selected. sparse target of the layer that addresses population and lifetime sparsity.
Algorithm 1 is the pseudocode of the single layer EPLS derivation. Let us define as an element of row vector and denote as N output vectors of dimensionality N ℎ , where N is the size of mini-batch. Starting with no activation in (line 1), input patches of are normalised between 0 to 1 (line 2). The algorithm iteratively processes a row of by selecting the th element of the -th row of that has the maximal activation value minus an inhibitor (line 5). Here, the inhibitor is an accumulator that counts the number of times an output has been selected, increasing its inhibition by N ℎ /N until reaching maximal inhibition, where N is the total number of training patches. This enforces the lifetime sparsity and prevents the selection of an output that has already been activated N ℎ /N times. The th element of -th row of target matrix is then activated as in line 6 (i.e., by assigning 1), considering population sparsity. The inhibitor is progressively updated and finally the output target is remapped to active and inactive values of corresponding non-linearity. The optimisation in relation to Equation 7 is performed using standard stochastic gradient descent (SGD) with adaptive learning rates [47].
E. Multi-layer Convolutional Kernel Networks
A CKN kernel can be learned in a hierarchical fashion for a deeper and potentially improved high-level semantic feature representation. Essentially: 1) the input feature map of layer + 1 can be computed by applying the convolution operation, learned weights and biases to kernel maps from layer ; 2) EPLS is then used to learn initial sparse features used as a starting point of CKN learning; 3) a multi-layer CKN is learned in a feedforward manner, using a given input sub-patch of size , and kernel parameters and for each layer. Fig 3 is an overview of our S-CKN framework.
F. The Spatial Pyramid Pooling (SPP) Layer
SPP is widely used in computer vision and has proven efficacy in representing the spatial layout of image features [48]. It partitions the image into multi-level regions and aggregates local features by taking spatial information into account. Hence, a number of researchers have successfully applied SPP to image classification [49], [50] and object detection [51]. Our assessment is that SPP can effectively characterise subtle geometric differences (e.g., size of similar bones or organs) in medical image data.
We added SPP as the last feature pooling layer to extract a final image representation that also captures subtle geometric variations. The outputs of the SPP layer are • dimensional vectors with multi-level spatial bins ( is the filter size). We determined the window size of each pyramid level ( ) based on the last feature maps ( × ) generated from S-CKN, as = / . We then pooled and aggregated the responses of each filter by selecting the maximum values (max pooling) across different locations and over different spatial scales of the kernel map. This provides invariant image representations that is more robust to local transformations. Fig 2 and Fig 3 show a SPP layer combined with our S-CKN.
IV. EXPERIMENTAL SETUP AND RESULTS
A. IRMA X-ray Dataset
The Image Retrieval in Medical Application (IRMA) dataset comprises 14,410 gray-scale X-ray images with 193 hierarchical classes; the dataset has been used for many years for CAD development [24], [25]. IRMA is a challenging dataset because it contains images with irregular contrast, brightness, and artifacts. There is also high intra-class variability and interclass similarity among the classes. We used the standard pre-
IRMA Code
1121-420-212-700
Technical Code X-ray, Plain radiography, Overview Image
Directional Code
Other orientation, occipitofrontal
Anatomical Code
Facial cranium, eye area
Biological Code
Musculoskeletal system defined training set of 12677 images and test set of 1733 images [25]. The images were annotated according to the IRMA coding system with four different axes, as described by Lehmann et al. [25]: 1) a technical code that describes imaging modality, 2) a directional code for imaging orientation, 3) an anatomical code for body region examined, and 4) a biological code for biological system examined. Fig 4 illustrates a sample X-ray image and the corresponding labels from the IRMA code.
B. ImageCLEF Dataset
We used the medical Subfigure Classification dataset used in the Image Conference and Labs of the Evaluation Forum (ImageCLEF) 2016 competition [52]. We used the standard pre-defined training set of 6776 images and test set of 4166 images from 30 different image modalities. Ground truth annotations are available for both image datasets. A detailed description of the datasets can be found in the ImageCLEF 2016 overview paper [52].
C. Experimental Setup
We evaluated our method in comparison to several unsupervised and supervised learning methods:
• conventional unsupervised feature learning approaches: SIFT+BoVW, Independent Component Analysis (ICA), and sparse coding [12]. We implemented the SIFT descriptor together with BoVW model (SIFT+BoVW). We used a patch size of 16x16 pixels with spacing of 8 pixels in the extraction of SIFT descriptors. We used the standard codebook size of 1000 [26]. The number of filters (i.e., weights) for the first layer of the ICA, and sparse coding was all set to 1600, which was consistent with other research [44]. • state-of-the-art unsupervised learning methods: SSAE [6,13] and CKN [17] . The number of filters for the first layer of the SSAE was set to 1600, which was consistent with the conventional baselines above; We set the number of filters for the second layer to 1024. For the purpose of comparison, we trained the CKN using the same parameters as our proposed S-CKN (see Section IV.D). • state-of-the-art supervised pre-trained CNNs (with natural images). We used the AlexNet [35], VGG [2], GoogLeNet [4], and ResNet [3], which have achieved high rankings in object recognition and localisation from the ImageNet Challenge. For all pre-trained CNNs models, the final fully-connected layers were used as the feature extractors. • state-of-the-art supervised fine-tuned CNNs. We used the same models as in the pre-trained baselines above: AlexNet [35], VGG [2], GoogLeNet [4], and ResNet [3]. For medical image analyses, these fine-tuned CNNs have been shown to perform as well as fully trained CNNs or even outperform when there is limited training data [8], [9], [10]. All of the models were trained for 60 epochs with the IRMA dataset. We used a batch size of 128 and an initial learning rate of 10 -4 . We used learning rate annealing, decaying the rate by a factor of 10 when the error plateaued. All the networks (our S-CKN, the SSAE, the baseline CKN, and the fine-tuned CNNs) were trained with a GeForce GTX 1080 Ti GPU (11GB memory). It took 8 hours for our S-CKN to be trained with this GPU on a machine with Intel Core i7-6800K 3.40 GHz (6 cores) processor. For the results of the supervised CNNs models with ImageCLEF dataset, we used the results reported in their papers [8].
We conducted medical image retrieval experiments on the IRMA dataset [26] and classification experiments on the ImageCLEF dataset [52]. For the medical image retrieval experiments, each of the test images was used as a query image and the training images were ranked according to the Euclidean distance from the query image. For quantitative comparisons, we used precision estimates at Q = 1, 5, 10, and 30 as follows:
Precision@Q = # # (8).
For the classification experiments, we used the Top 1 accuracy (the correctness of the predicted label), which is the standard performance measure adopted in recent CNN studies for the classification of medical image modalities [8]. For all learned features, we used the setup of the multi-class linear SVM introduced by Yang et al. [49], who used a differentiable quadratic hinge loss so that the training could easily be done with simple gradient-based optimisation methods. We used LBFGS with a learning rate of 0.1 and a regularization parameter of 1, consistent with the parameters specified by Yang et al. [49].
D. Implementation Details
S-CKNs have four parameters that need to be determined for each layer: the size of sub-patch the coefficients α and β, and the pooling factor or filter size . The parameters of our Gaussian kernel α and β are automatically determined for each layer. The coefficient β was set to be the pooling factor divided by √2 consistent with the work reported by Mairal et al. [17]. The coefficient α was set to be the 0.1 quantile of the distribution of pair-wise distances between sub-patches, as reported by Mairal et al. [17]. In our settings, the final results were insensitive to the use of smaller quantiles such as 0.01 and 0.001. This is also consistent with other research studies [18]. For the IRMA plain X-ray images, we adopted a two-layer architecture that was shown to perform better on gray-scale images [18]. We used the gradient map (defined in Section III.B) as the input of the initial layer of our architecture; the gradient map as input has been shown to perform better than raw patches [17]. Our parameter selection process searched within a restricted space to find the optimal values of the parameters, following other research studies [17]. We used values in the range 2 to 8 for sub-patch sizes and pooling factors of 100, 256, 512, 800 and 1024. For the ImageCLEF dataset, we used the same settings as the IRMA dataset but used raw patches instead of gradient maps as the input, as the raw patches performed better when working with RGB patches. We then empirically chose the remaining parameters as shown in Table 1. For the SPP layer, we used a 4-level spatial pyramid (1x1, 2x2, 3x3, 6x6) of 50 spatial bins in all of our experiments.
E. Results
The results of image retrieval experiments are shown in Table 2. We show sample results of the query and retrieval of varying structures in Fig 5. The query images are the shoulder of the scapulo-humeral joint (top row), shoulder of the acromioclavicular joint (middle row), and (bottom left) forearm (bottom row), with artifacts including plates, screws and wires. The retrieved images are ranked by order of similarity from left to right (top 1 to 3).
Our approach had greater accuracy than other unsupervised feature learning algorithms as well as other pre-trained CNN models. Furthermore, our unsupervised S-CKN outperformed all the fine-tuned CNNs, achieving a top 1 precision of 52.97%. The fine-tuned GoogLeNet method achieved the best precision when considering the top 5, 10, and 30 retrieved images.
The results of image modality classification experiments are shown in Table 3. We compared our approach with several conventional unsupervised feature learning methods as well as the supervised image-based methods presented in the competition held in 2016. Our S-CKN had greater accuracy than other unsupervised approaches, achieving a top 1 accuracy of 70.99%. The second best unsupervised method was SSAE with an accuracy of 65.17%. The best performing supervised method was the fine-tuned ResNet-152 with an accuracy of 85.38%. Fig 6 shows how our sparsity-based pre-training improves the feature representation of medical images compared to other standard pre-training methods including random initialisation and the K-mean algorithm. We also show the improvement made by SPP.
V. DISCUSSION
A. Comparisons to Other Methods
Our findings show that our S-CKN outperforms other conventional unsupervised approaches and achieved competitive accuracy with the state-of-the-art supervised CNNs. We attribute this to: 1) the hierarchical kernel learning deriving semantically more meaningful image features (see Table 2 and 3); 2) sparse feature representation as part of layerwise pre-training, which extracts discriminative initial features for medical images (see Section V. B and V.C); and 3) spatial pyramid pooling that effectively characterises the local Fig 5. Sample results of query and retrieval of X-ray images using S-CKN. geometry information in medical image data (see Section V.C). X-ray image retrieval -Our unsupervised S-CKN learned and extracted data-specific features and achieved a high accuracy (52.97%) (see Table 2). Our results show that the qualities of the features learned with conventional unsupervised hand-crafted features such as SIFT coupled with BoVW model, sparse coding, and ICA were not as robust as that of the SSAE. The accuracy of pre-trained CNNs was lower than our method as these approaches extracted features that were not tuned to a particular dataset or application, and as such have limited capacity to extract the most meaningful or discriminative features. The deeper network of pre-trained CNNs had higher accuracy (e.g., VGG-16 to ResNet-152 layers) and the finetuned GoogLeNet had the highest accuracy in top 5, 10, 30, and we attribute this to its network architecture exploiting the local sparse structure of a convolutional network [4]. Our method was designed to learn class-specific image features for better discrimination in an unsupervised fashion but this means it can be sensitive to subtle inter-class variations which is why accuracy drops faster as more subtly similar images are retrieved. For medical image retrieval applications, the five most similar images (i.e., top 5) for a query are commonly used for comparative analysis [53]. Our S-CKN achieved a competitive top 5 accuracy (44.18%), which was the second best after the fine-tuned GoogLeNet (44.61%).
Medical image modality classification -Our unsupervised S-CKN outperformed all other unsupervised approaches and achieved a comparable accuracy (70.99%) compared to all supervised CNNs that were part of the ImageCLEF 2016 challenge. Our results show that sparse coding and ICA could not learn and build discriminative image features, consistent with X-ray image retrieval results. Unlike sparse coding and ICA, the SSAE learned image features in a hierarchal manner and hence was the closest method to our approach. The top performing methods were all based on well-established supervised CNNs including AlexNet [54], VGG [55], GoogLeNet [56], and ResNet [56]. These CNNs were trained from scratch or fine-tuned with medical images to derive highlevel data specific features. The deeper CNNs also had higher accuracy than shallower CNNs (see Table 3). Our unsupervised S-CKN (accuracy of 70.99%) performed better than supervised VGG-like CNNs (65.31%) [55] with over 5% improvement in modality classification. While most of referenced methods used the same training data, the method by Koitka et al. [56] with the best performance in the competition, added extra data from additional sources.
The ImageCLEF dataset also contains different generic biomedical illustrations such as gene sequence or chemical structure and so, in comparison to the X-ray IRMA dataset, there were more diverse and complex variations in image characteristics. As a consequence, the overall performance of our approach was lower on the ImageCLEF dataset than the the IRMA dataset. Nevertheless, our method was able to derive discriminative medical image features from a variety of image modalities without reliance on labels, and its accuracy was better than that of supervised VGG-like CNNs [55].
B. Discovering the Structure of Medical Image Patches
Unsupervised learning is capable of discovering the underlying structure of image patches [57]. The learned weight parameters can be visualised using raw pixel data, and welltrained networks generally display some structure such as edges, lines, and ridges. The visualisation of the learned weights from the first layer of our S-CKN is shown in Fig 7. We used 400,000 image patches of size 12x12 and learned 256 filters [57]. Our S-CKN not only learned common structures such as lines and edges but also identified spatial patterns. Unlike the structure of natural image patches where lines, edges or blobs are dominant, our results show that the structure of medical image patches also contains spatially localised patterns, such as corners and sparse regions. These findings indicate that our S-CKN is able to learn the complex and diversified characteristics of medical image data.
C. The Role of Sparsity-based Pre-training and SPP Layer
The results from Fig 6 suggest that sparsity-based pretraining improves the feature representation of medical images compared to other standard pre-training methods including random initialisation and K-means algorithm. We attribute this to our robust pre-training scheme which provided good initialisation points for subsequent convolutional kernel learning. It acts as a form of regularisation that restricts parameters into certain spaces that are more discriminative for medical image data [15], [58]. The spatial pyramid pooling framework also improves feature representation in medical images (see Fig 6) through a multi-level spatial feature pooling technique that effectively characterises the local geometry information in the image data. To the best of our knowledge, this is the first research to couple unsupervised pre-training with unsupervised learning frameworks, which is dissimilar to conventional approaches that combine unsupervised pretraining with subsequent supervised learning [15], [59].
We also experimented with deeper architectures to further exploit the possibilities of extracting more high-level semantic image features. Our experiments using 3 and 4 layer S-CKN architectures did not result in any significant performance gain, which is consistent with other research [18] (see Fig 8).
We suggest that our unsupervised initialisation will benefit supervised learning approaches when there are limited labelled training data. We suggest that our S-CKN, when used to initialise a CNN for supervised fine-tuning, could potentially enable the derivation of semantically more meaningful representations of the image data than traditional CNN finetuning approaches that are initialised with natural images. The investigation of impact on fine-tuning is a substantial research study and it is something that we will pursue in future work. We suggest that our S-CKN could provide an important first step to accessing the large volume of unannotated data in medical imaging repositories. We note that compared to other supervised CNNs, our S-CKN requires learning fewer parameters across fewer layers (two layers in this paper), and therefore, can be efficiently coupled with subsequent supervised learning approaches without a large computational cost.
D. Limitations
Although our approach improves the ability to learn feature representations of medical images without reliance on labelled data, some of the parameters (including sub-patch size, subsampling factor, or pooling factor (i.e., filter size) for each layer), must be empirically derived (see Section IV.D). Generally, smaller subsampling factors and larger pooling factors (i.e., filter size) led to better performance at the cost of increased computational complexity. Nevertheless, our results show that sparsity-based pre-training and SPP pooling consistently improved overall feature representation even when different parameters were used.
Our S-CKN is currently restricted to use an integral form of the Gaussian Radial Basis Function (RBF) kernel to approximate a kernel map (image feature representation in a RKHS). Other type of kernels or multiple kernels were not considered in this paper and we will explore such approaches in future work.
VI. CONCLUSION
In this paper, we proposed a new unsupervised sparsity-based feature learning architecture to enable better characterisation of medical image data. Our layerwise pre-training, using convolutional sparse features, improves the learning outcome and feature representation in image retrieval and classification. We compared our approach to other unsupervised and supervised methods on two large public datasets and showed that our approach was competitive with the state-of-the-art supervised CNNs. Our approach shows the feasibility of using large collections of unlabelled medical data to characterise medical image features and offers the opportunity to access the large volume of unannotated data that are available in medical imaging repositories. In future work we will explore the use of our S-CKN combined with subsequent supervised deep learning to optimise the ability to derive semantically more meaningful representations of the image data.
Fig 2 .
2The spatial pyramid pooling layer on top of S-CKN.
Fig 3 .
3The architecture of our proposed algorithm.
Fig 4 .
4Sample X-ray images (Face) and the corresponding labels from IRMA code.
Fig 6 .
6Top 1 average precision or accuracy of CKN with random and K-mean initialisation, and our Improved S-CKN with SPP.
Fig 7 .
7The visualisation of learned weights by the first layer of the S-CKN using ImageCLEF dataset (gray-scale).
TABLE 1 FOR
1EACH LAYER, THE SUB-PATCH SIZE, SUB-SAMPLING FACTOR, AND THE NUMBER OF POOLING FACTOR ARE SHOWN. FOR INITIAL GRADIENT MAP, THE VALUES 16 INDICATES THE NUMBER OF ORIENTATIONS.Dataset
Layer
Sub-patch
Size
Sub-
sampling
Factor
Pooling
Factor
IRMA
Layer 1
1x1
4
16
Layer 2
3x3
4
1024
ImageCLEF
Layer 1
2x2
2
100
Layer 2
2x2
4
800
TABLE 2 AVERAGE
2IMAGE RETRIEVAL PRECISION ESTIMATES (%) AT Q = 1,5,10,
AND 30 (BASED ON THE IRMA DATASET).
Type
Methods/
Average Q
1
5
10
30
Unsupervised
SIFT+BoVW
34.21
25.42
21.78
16.32
Unsupervised
SSAE (2 layers)
38.54
31.74
27.71
20.57
Unsupervised
ICA
33.92
26.10
22.42
16.69
Unsupervised
Sparse Coding
31.27
23.85
20.64
15.32
Supervised
Pre-trained
AlexNet
37.91
30.46
26.72
20.90
Supervised
Pre-trained
VGG-16
39.29
32.39
29.25
24.17
Supervised
Pre-trained
VGG-19
38.83
32.46
29.54
24.20
Supervised
Pre-trained
GoogLeNet -22
40.39
33.90
31.09
26.10
Supervised
Pre-trained
ResNet-152
41.31
34.48
31.06
24.80
Supervised
Fine-tuned
AlexNet
44.48
36.93
32.87
26.73
Supervised
Fine-tuned
VGG-16
48.75
43.73
40.40
34.59
Supervised
Fine-tuned
VGG-19
49.45
43.94
40.98
34.87
Supervised
Fine-tuned
GoogLeNet
49.39
44.61
43.12
38.70
Supervised
Fine-tuned
ResNet
47.20
41.66
39.11
34.56
Unsupervised
Our S-CKN
52.97
44.18
39.87
31.59
TABLE 3
3TOP 1 IMAGE CLASSIFICATION ACCURACY (%) USING IMAGECLEF DATASET.Type
Methods
Accuracy
(%)
Unsupervised
Sparse Coding
57.08
Unsupervised
ICA
58.79
Unsupervised
SSAE (2 layers)
65.17
Supervised
VGG-like CNN (500 epochs) [55]
65.31
Unsupervised
Our S-CKN
70.99
Supervised
Fine-tuned AlexNet (100 epochs) with
data augmentation [54]
77.55
Supervised
Modified GoogLeNet (60 epochs) with
additional data [56]
81.03
Supervised
Ensemble of CNNs (50 epochs) with data
augmentation [8]
82.48
Supervised
Fine-tuned ResNet-152 with additional
data [56]
85.38
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| []
|
[
"SINGULAR INTEGRAL OPERATORS ON TENT SPACES",
"SINGULAR INTEGRAL OPERATORS ON TENT SPACES"
]
| [
"Pascal Auscher ",
"Christoph Kriegler ",
"Sylvie Monniaux ",
"Pierre Portal "
]
| []
| []
| We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly appropriate condition on the kernel is time-space decay measured by off-diagonal estimates with various exponents.Date : revised, March 15, 2012. 1991 | 10.1007/s00028-012-0152-4 | [
"https://arxiv.org/pdf/1112.4292v2.pdf"
]
| 22,895,855 | 1112.4292 | 7137e985e3ccafcd83112c86b3e6846ad2bbf9e6 |
SINGULAR INTEGRAL OPERATORS ON TENT SPACES
19 Mar 2012
Pascal Auscher
Christoph Kriegler
Sylvie Monniaux
Pierre Portal
SINGULAR INTEGRAL OPERATORS ON TENT SPACES
19 Mar 2012
We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly appropriate condition on the kernel is time-space decay measured by off-diagonal estimates with various exponents.Date : revised, March 15, 2012. 1991
Introduction
Let −L be a densely defined closed linear operator acting on L 2 (R n ) and generating a bounded analytic semigroup (e −tL ) t≥0 . Consider the maximal regularity operator originally defined for f ∈ L 2 (R + , dt; D(L)), R + = (0, +∞), by the Bochner integral
(1.1) M L f (t) = t 0 Le −(t−s)L f (s) ds.
This is an example of a singular integral operator with operator-valued kernel. The bounded extension of this operator to L 2 (R + , dt; L 2 (R n )) = L 2 (R n+1 + , dtdx), R n+1 + = R + × R n , was established by de Simon in [26]. The maximal regularity operator plays a crucial role in evolution equations, where its boundedness is used to deduce a priori estimates, which, in turn, can be used to solve non-autonomous and/or non linear problems (see the lecture notes [21]). It has thus been the source of intense study, especially in the past 10 years, in L p , and in Besov spaces. As part of the recent development of an evolution equation approach to boundary value problems on the upper half-space with data in L 2 (R n ), based on the functional calculus of appropriate Dirac operators, a weighted version of de Simon's theorem is proven in [3] and [4,Theorem 1.3], but can be essentially attributed to the earlier work [15] (see below). Theorem 1.1. With L as above, M L extends to a bounded operator on L 2 (R n+1 + , t β dtdx) for all β ∈ (−∞, 1).
This was proven before in [25] for β ∈ [0, 1) and a more general class of operators akin to the ones we introduce next, and then for β ∈ (−1, 1) in [15,Theorem 1.13] when L has dense range. The range of β is shown in [3] to be optimal. Values such as β = −1 and also an endpoint result for β = 1 were central for applications to the boundary value problems in [3]. It should be noted, however, that while the statement of [15,Theorem 1.13] does not include the case β = −1, its proof via their Proposition 1.14 actually gives Theorem 1.1.
The articles [15] and [25] actually prove weighted L p estimates for 1 < p < ∞ and show that weighted maximal regularity is equivalent to the unweighted one. However, the L p analogue of Theorem 1.1 needed in the applications we have in mind does not involve weighted L p (R n+1 + ) spaces for p = 2, but more appropriate spaces of functions on the upper half space R n+1 + . Let us explain this fact.
Traditionally, an evolution problem of the form u t + Lu = g, with initial value u 0 = f ∈ L p (R n ), is seen as an ordinary differential equation for L p (R n )-valued functions. One assumes that −L generates an analytic semigroup on L p (R n ), and looks for maximal regularity in spaces such as L p (R + ; L p (R n )). However, if L = −divA∇ is a second order, divergence form elliptic operator on R n with bounded measurable complex valued coefficients, −L only generates an analytic semigroup on L p (R n ) for p in an interval (p − (L), p + (L)) including 2, but not always equal to (1, ∞) (see [1]). In this range, maximal regularity results can be proven using the extrapolation method pioneered by Blunck and Kunstmann in [12], and developed in [1]. Outside of that range, however, maximal regularity in L p (R n+1 + ) spaces, weighted or not, cannot hold. In this paper, we prove maximal regularity results on the (unweighted) tent space T p,2,2 for all p ∈ ( n n+1 , ∞] (see Proposition 1.6 below), even though, for small p, −L does not even generate a C 0 -semigroup on L p (R n ).
Moreover, even when L = −∆, the free evolution (t, y) → e t∆ f (y) does not belong to L p (R n+1 + ) when f ∈ L p (R n ). This can be compensated by assuming more regularity on f , or by using a weighted L p (R n+1 + ) space with an appropriate weight. However, when dealing with L p initial data (in boundary value problems, or evolution problems with rough data, for instance), it is desirable to use a norm of the heat extension (t, y) → e t∆ f (y) that is equivalent to the L p norm of f for p ∈ (1, ∞), and to its H p norm for p ∈ (0, 1]. Weighted L p (R n+1 + ) norms do not have this property, but classical harmonic analysis gives many different norms that do.
The one which is of interest to us is given by the following area integral:
f p R n R n+1 + 1 B(x,t 1 2 ) (y) t n 2 ∆e t∆ f (y) 2 tdtdy p 2 dx 1 p .
Such a characterisation of the L p (or H p ) norm of a function in terms of its heat extension originates from the work of Fefferman-Stein [14]. In more recent terminology, this says that ∆e t∆ f belongs to a parabolic version of one of the tent spaces introduced by Coifman-Meyer-Stein [13]. Now, if one considers the "mild solution" u of u t − ∆u = g and u 0 = 0, given formally by the integral formula t 0 e (t−s)∆ g(s) ds, one is led to consider the boundedness of the maximal regularity operator M −∆ in the norm above. Having such a priori estimates in the same space as the free evolution (t, y) → e t∆ f (y) is a first step towards solving, for example, non-linear problems with L p data. Remark that this solution space has, a priori, nothing to do with the space of continuous functions of t with values in L p . We thus depart from the tradition of looking at evolution problems for functions on R n+1 + as Banach space valued ODE, and work on spaces where the time and space variables are intrinsically linked.
We refer to [10], and the forthcoming [9], for more on the PDE aspect of such questions via a tent space approach. We just mention here that this idea goes back (at least) to Koch-Tataru's work on Navier-Stokes equations [20].
We introduce the alluded variants of the tent spaces as follows. For 0 < p < ∞, m ∈ N * , β ∈ R, define the tent space T p,2,m (t β dtdy) as the space of all locally square integrable functions on R n+1
+ such that g T p,2,m (t β dtdy) = R n R n+1 + 1 B(x,t 1 m ) (y) t n m g(t, y) 2 t β dtdy p 2 dx 1 p < ∞.
The classical case is β = −1, m = 1, in which case, the space is simply denoted by T p,2 . Since g T p,2,m (t β dtdy) = g T p,2 , whereg(s, y) = √ m g(s m , y)s m(β+1) 2 , T p,2,m (t β dtdy) is isometric to T p,2 . However, the parameter m is needed to handle different homogeneities (corresponding to differential operators of different orders), and the parameter β is used to handle different applications (e.g. different degree of smoothness for initial data in evolution problems). We also remark that a simple use of Fubini's theorem shows that g 2 T 2,2,m (t β dtdy) = b n g 2
L 2 (R n+1 +
,t β dtdy) , whatever the parameter m is, with b n being the volume of the Euclidean unit ball. Therefore, for p = 2, tent spaces agree with weighted L 2 spaces. But it is easy to show that it is not true when p = 2.
The nature of the norm (a quasi-norm when p < 1), makes local square integrability a requirement. As already showed in [7] (and subsequently in [19]) for different types of operators, a pertinent notion for boundedness of the maximal regularity operator on tent spaces is a measure of decay of the semigroup called (L 2 − L 2 ) off-diagonal estimates.
Definition 1.2. A family of bounded linear operators (T t ) t≥0 ⊂ B(L 2 (R n )) is said to satisfy off-diagonal estimates of order M , with homogeneity m ∈ N * , if, for all Borel sets E, F ⊂ R n , all t > 0, and all f ∈ L 2 (R n ):
1 F T t 1 E f 2 1 + dist(E, F ) m t −M 1 E f 2 .
Here, and in what follows, · 2 denotes the norm in L 2 (R n ).
This property is a replacement for pointwise kernel estimates, which is satisfied, for instance, by heat semigroups generated by elliptic operators with rough coefficients. Note that we allow a polynomial decay.
With the definition above, the following result was proved in [8].
Theorem 1.3. Let m ∈ N * , β ∈ (−∞, 1), p ∈ 2n n+m(1−β) , ∞ ∩ (1, ∞)
, and τ = min(p, 2). If (tLe −tL ) t≥0 satisfies off-diagonal estimates of order M > n mτ , with homogeneity m, then M L extends to a bounded operator on T p,2,m (t β dtdy).
The surprise is to obtain results for p < 2. This is particularly true in applications to stochastic parabolic PDEs. Results in this context have been developed in parallel to this article in [10], which contains lighter versions of some of the material presented here. In the present paper we concentrate more on the abstract theory, and try to weaken assumptions as much as possible. This is important even for maximal regularity operators, see Section 5.
An end-point result, for p = ∞, was also obtained in [8]. In this context, the appropriate tent space consists of functions such that |g(t, y)| 2 dtdy t is a Carleson measure, and is defined as the space of all locally square integrable functions such that
g 2 T ∞,2 = sup (x,r)∈R n ×R+ r −n B(x,r) r 0 |g(t, y)| 2 dtdy t < ∞.
The weighted version (defined through a change of variable as above) is given by
g 2 T ∞,2,m (t β dtdy) := sup (x,r)∈R n ×R+ r −n B(x,r) r m 0 |g(t, y)| 2 t β dtdy.
Theorem 1.4. Let m ∈ N * and β ∈ (−∞, 1). If (tLe −tL ) t≥0 satisfies off-diagonal estimates of order M > n 2m , with homogeneity m, then M L extends to a bounded operator on T ∞,2,m (t β dtdy).
Note that the backward maximal regularity operator
M − L f (t) = ∞ t Le −(s−t)L f (s) ds,
can be studied on tent spaces, either by duality as M − L = (M L * ) * , or directly. Here, we continue the development of such tent space boundedness results, and we obtain three-fold improvements. The main statements are in the core of the article. We give here our motivation and extract sample new results as illustrations.
The first observation is that the conclusion of Theorem 1.3 is far from optimal in concrete situations. For instance, for −∆ (heat semigroup), and its square root √ −∆ (Poisson semigroup), or even −∆ + V with V ∈ L 1 loc (R n ), V ≥ 0, and its square root, or −div A∇ a second order divergence form elliptic operator on R n with bounded, measurable, real -valued coefficients, and its square root, the range of p can be much improved.
Proposition 1.5.
(1) M −∆+V and M −div A∇ (with real-valued coefficient matrix A) extend to bounded operators on T p,2,2 (dtdy) when n n+1 < p ≤ ∞. (2) M √ −∆+V and M √ −div A∇ (with real-valued coefficient matrix A) extend to bounded operators on T p,2,1 (t −1 dtdy) when n n+1 < p ≤ ∞. The range of p is a consequence of the pointwise decay of the corresponding heat kernels. However, not all semigroups obey pointwise decay. In that case, one can use intermediate conditions between pointwise decay and L 2 − L 2 off-diagonal estimates such as L q − L r offdiagonal estimates with q ≤ r and q = 2 or r = 2 (see Definition 2.4). This information can then be used to quantify the range of p for tent space boundedness. This is the case for −div A∇ with complex-valued coefficients. Here, the decay is Gaussian but the range of q or r may be limited as dimension increases. Proposition 1.6. For a complex-valued coefficient matrix A, M −div A∇ extends to a bounded operator on T p,2,2 (dtdy) when
1 2 < p ≤ ∞ if n = 1, 2 3 < p ≤ ∞ if n = 2, 6 7 − ε < p ≤ ∞ if n = 3, and 2 − 4 n − ε < p ≤ ∞ if n ≥ 4.
The ε > 0 depends on the operator but the lower bound is at least n n+1 . These two propositions (see Section 5 for their proofs) follow from general statements (proved in Sections 3 and 4) in which one requires a lower bound on the polynomial decay exponent M of Definition 2.4. Note that this lower bound increases with dimension. As the decay here is exponential, the exponent M can be as large as one wants, and the results apply.
We now consider the case of polynomial decay. This is our second point. In this case, the value of M is to be compared with the lower bound in our statements for applicability. For example, one has M = 1 in the L 2 − L 2 off-diagonal estimates with homogeneity m = 1 for √ −div A∇ (even for √ −∆). Theorem 1.3 requires M > n/τ , but one can take advantage of the fact that the exponent M in the L q − L 2 off-diagonal estimates grows linearly in 1/q (see Proposition 5.3). However, the range of q may be limited as well, which is the case for −div A∇ operators with complex-valued coefficients, and again we may not have a large enough value of M .
On the other hand, with no decay at all, the p = 2 boundedness follows from Theorem 1.1. So it seems reasonable to expect a range of p near 2 depending on q and M , when q ∼ 2 and M > 0 is small, by some kind of interpolation procedure. We will obtain (see Section 5) a general result in this direction, which gives, for this particular operator, the proposition below.
Proposition 1.7. For a complex-valued coefficient matrix A, M √ −div A∇ extends to a bounded operator on T p,2,1 (t −1 dtdy) when 1 2
< p ≤ ∞ if n = 1, 2 3 < p ≤ ∞ if n = 2, 6 7 − ε < p ≤ ∞ if n = 3, 1 − ε < p ≤ ∞ if n = 4 and 2 − 4 n − ε < p < 2n−4 n−4 + ε ′ if n ≥ 5.
The ε, ε ′ > 0 depend on the operator but the lower bound is at least n n+1 . To do this interpolation procedure, we view the maximal regularity operator within a family of operators of the same nature. Thus, and this is the third point, it becomes interesting and convenient to develop an abstract formulation that is not restricted to the maximal regularity operator. We introduce, in the next section, a class of singular integral operators in the context of tent spaces. Sufficient conditions for their boundedness are given in Sections 3 and 4. We remark that, in contrast to the usual L p theory for Calderón-Zygmund operators, no regularity of the kernel is necessary. In a sense, despite the fact that tent spaces, for 1 < p < ∞, can be seen as subspaces of Hilbert-valued L p spaces ( [18]), Calderón-Zygmund theory does not seem to be an appropriate machinery to study singular integral operators in this context. We depart from the usual treatment of maximal regularity through a singular integral operator acting on some Banach-valued functions. Here, we start from the "easy" Hilbert (L 2 ) space theory, and then move on to tent spaces, using the notion of L q − L r off-diagonal decay, which extends the notion of L 2 − L 2 off-diagonal decay defined above. Remark 1.8. Our results can, nevertheless, be extended to the context of tent spaces of Banach space-valued functions (provided the Banach space X is UMD, and 1 < p < ∞). This is done by adapting the arguments of [19] to take advantage of L q − L 2 (for q ≤ 2, resp. L 2 − L q for q ≥ 2), rather than L 2 − L 2 , off-diagonal estimates, in the same way it is done in this paper. However, the obvious adaptation does not seem to produce optimal relationships between p, q, M , and the geometry of X. We choose not to attempt to address this issue here.
Singular integral operators
2.1. Abstract setup. Consider a separable complex Hilbert space H. For β ∈ R, set H β = L 2 (R + , t β dt; H) . We consider the following classes of operators SIO ± ⊂ B(H 0 ).
Definition 2.1.
(1) We say T ∈ SIO + if T ∈ B(H 0 ) and there exist a strongly measurable family of operators K(t, s) ∈ B(H), t, s ∈ R + and a constant C < ∞ such that
K(t, s) ≤ C|t − s| −1 and (2.1) T f (t) = t 0 K(t, s)f (s) ds for all f ∈ H 0 with bounded support in R + and almost all t ∈ R + not in the support of f . (2) We say T ∈ SIO − if T ∈ B(H 0 )
and there exist a strongly measurable family of operators K(t, s) ∈ B(H), t, s ∈ R + and a constant C < ∞ such that K(t, s) ≤ C|t − s| −1 and T has the representation
(2.2) T f (t) = ∞ t K(t, s)f (s) ds
for all f ∈ H 0 with bounded support in R + and almost all t ∈ R + not in the support of f .
We remark that K(t, s) need only be defined on s < t for T ∈ SIO + and on t < s for T ∈ SIO − and the value at t = s is irrelevant. With this precaution, we say that T ∈ SIO ± is associated to the operator-valued kernel K(t, s) and that such kernels belong to the class SK ± of singular kernels.
Our terminology follows in part that of singular integrals (here with operator-valued kernels) but we assume a sign condition on s − t.
Let us make a few remarks. The representation (2.1) of T f above is a Bochner integral and the equality holds in H. It is clearly equivalent to
(2.3) T f, g = s<t K(t, s)f (s), g(t) dsdt
for f, g ∈ H 0 having bounded disjoint support. The inner product on the left is the canonical one in H 0 , and on the right the canonical one in H. It is clear that T ∈ SIO + if and only if T * ∈ SIO − , with associated kernel K(s, t) * . Hence, similar comments apply to (2.2).
The basic examples are of course M L ∈ SIO + and M − L ∈ SIO − . For M L the boundedness on H 0 is given by de Simon's theorem. Then the formula (2.3) holds for all f ∈ L 2 (R + , dt; D(L)) and all g ∈ H 0 with continuous kernel K(t, s) = Le −(t−s)L on s < t. If now, f, g have disjoints supports, one can argue by density of D(L) in H. For M − L , we simply use
M − L = (M L * ) * .
There is a natural splitting of operators T ∈ SIO + into an integral part plus a singular part. Let K be the associated kernel. Using that t − s ∼ t when s < t/2 and Hardy inequality,
one has ∞ 0 t 2 0 K(t, s) f (s) ds 2 dt ∞ 0 1 t t 2 0 f (s) ds 2 dt f 2 H0 .
Hence, the integral part of T is the operator defined for f ∈ H 0 for almost all t > 0 by the Bochner integral
(T 2 f )(t) = t 2 0 K(t, s)f (s) ds,
and T 2 ∈ SIO + as well. The singular part is T 1 := T − T 2 ∈ SIO + , and carries the singularity at s = t. Its associated kernel is K(t, s)1 t/2<s<t . Note that, for the integral part, the integral representation is valid without restriction on f and t.
For T ∈ SIO − , one has the same splitting with
T 2 f (t) = ∞ 2t
K(t, s)f (s) ds as the integral part, and T 1 = T − T 2 as the singular part. Theorem 1.1 and its proof carry to this abstraction.
Theorem 2.2. Let β ∈ (−∞, 1)
. Any operator in T ∈ SIO + extends to a bounded operator on H β which is denoted by T as well. Furthermore, for any kernel K ∈ SK + and
f ∈ H β , t 2 0 K(t, s) f (s) ds is an element of L 2 (R + , t β dt), so that for almost all t > 0, t 2 0 K(t, s)f (s) ds is a Bochner integral in H. If, in particular, K is the kernel of T then this integral agrees with (T 2 f )(t).
The same statement holds for T ∈ SIO − and −β ∈ (−∞, 1).
We include a quick argument. For α = β/2 < 1/2, we have that
∞ 0 t 0 K(t, s) |t α − s α | s −α s α f (s) ds 2 dt f 2 H β
using the Schur test and the bound on K. Hence, the integral operator
f → [t → t 0 K(t, s)(t α − s α )f (s) ds] is bounded from H β to H 0 . For f ∈ H β with compact support in R + , it agrees with t α (T f )(t) − (T (s α f ))(t).
Since T ∈ B(H 0 ), this readily gives the result by density. The second part follows from the weighted Hardy inequalities [22] when β < 1
∞ 0 1 t t 2 0 f (s) ds 2 t β dt f 2 H β .
The proof for SIO − is left to the reader.
2.2.
Concrete situation. Now, in order to get tent space results, we specialise to H = L 2 (R n ), and introduce subclasses. First recall that H β can be identified with L 2 (R n+1 + , t β dtdy). Hence, we now write f (s) as f or f (s, ·) if we want to specialise the s variable. Using that we have a spatial variable, we extend (2.1) as follows.
open intervals in R + . Assume that f → [(t, y) → t 0 |(K(t, s)f (s, ·))(y)| ds] is bounded from the space of functions f ∈ L 2 (s β dsdx) with support in I × E into L 2 (J × F, t β dtdy).
Then the representation T f (t, y) = t 0 (K(t, s)f (s, ·))(y) ds holds for all such f with equality in L 2 (J × F, t β dtdy).
The corresponding statement holds for T ∈ SIO − and −β < 1.
Remark that this lemma is only needed for singular parts. For regular parts, the representation is valid without support conditions.
Proof. Both terms are defined in L 2 (J × F, t β dtdy) by assumption so that it suffices to prove the following claim:
T f, g = J×F t 0 (K(t, s)f (s, ·))(y) ds g(t, y)dtdy for all f ∈ C ∞ 0 (I; L 2 (E)) and g ∈ C ∞ 0 (J; L 2 (F )).
We implicitly extend f (s, ·) by 0 outside E and g(t, ·) by 0 outside F . Remark that, from the assumption, (s, t, y) → (K(t, s)f (s, ·))(y)g(t, y)1 s<t is integrable with integral bounded by f L 2 (s β dsdx) g L 2 (t β dtdy) , hence, by Fubini's theorem, we only have to show
T f, g = s<t K(t, s)f (s, ·), g(t, ·) dsdt.
Choose orthonormal bases (e j ) of L 2 (E) and (ε k ) of L 2 (F ). By a limiting argument for each term, it is enough to assume that f (s, ·) and g(t, ·) take values in finite dimensional linear spans of the respective bases. Indeed, use boundedness of T in the left hand side and the integrability assumption in the right hand side. By linearity, it is enough to assume that f (s, ·) = f j (s)e j and g(t, ·) = g k (t)ε k for scalar test functions f j , g k . In this case, there is a distribution S j,k ∈ D ′ (I × J) such that T f, g = (S j,k (t, s), f j (s)g k (t)). It follows from (2.3) and decomposing on the orthonormal bases that K(t, s)e k , ε j is the restriction to 0 < s < t < ∞, s ∈ I, t ∈ J of S j,k . Thus the desired equality holds for such f, g and we are done.
We skip the similar proof for T ∈ SIO − .
In applications, it suffices to show (absolute) convergence of the integral
t 0 K(t, s)f (s, ·) ds
in the norm L 2 (J × F, t β dtdy) to obtain an estimate of T f in that norm, when f is supported in I × E. We shall use this when E and F are at positive distance and K(t, s) satisfies certain decay estimates. We thus introduce subclasses of SIO ± , where the size estimate K(t, s) |t − s| −1 is complemented by the following time-space estimates.
Definition 2.4. Let 1 ≤ q ≤ r ≤ ∞. An operator-valued kernel K = (K(t, s)) t,s>0 ⊂ B(L 2 (R n )) is said to satisfy L q − L r decay of order M > 0, with homogeneity m ∈ N * , if, for all Borel sets E, F ⊂ R n , all t = s, and all f ∈ L 2 (R n ) ∩ L q (R n ): 1 F K(t, s)1 E f r |t − s| −1− n m ( 1 q − 1 r ) 1 + dist(E, F ) m |t − s| −M 1 E f q .
Here, and in what follows · q denotes the norm in L q (R n ).
Note that, in the proofs, one only needs this property for sets of the form E = B(x, r) and F = B(x, 2 k+1 r)\B(x, 2 k r) (or vice versa). For this restricted property, L q − L r decay implies Lq − Lr decay for q ≤q ≤r ≤ r (by Hölder's inequality), but the order of decay changes. See [6] for more on this issue. We do not, however, use this fact in this paper.
We need only two specific cases: 1 ≤ q ≤ 2 and r = 2, and q = 2 and 2 ≤ r ≤ ∞. In certain cases, the decay is actually exponential, so the polynomial decay defined above holds for all M > 0, in which case we say that the order is ∞. In this paper, we are particularly interested in obtaining results under minimal values of polynomial decay.
Definition 2.5. Let 1 ≤ q ≤ ∞ and M ∈ R + ∪ {∞}. We say that T ∈ SIO ± m,q,M if T ∈ SIO ± and the associated operator-valued kernel K(t, s) ∈ SK ± satisfies L q − L 2 (resp. L 2 − L q ) decay of order M , with homogeneity m, when q ≤ 2 (resp. q ≥ 2).
The value of m is dictated by the situation, and q and M are the most important parameters. Let us point out that all calculations work with m being any positive real number, rather than just integer. We mention this for potential development towards fractal situations where fractional homogeneity can occur.
3. Role of L q − L 2 decay
The range of p below 2 for which T p,2 boundedness results hold can be quantified by L q −L 2 decay. Some technical conditions are also required. In particular the order M should not be too small.
Theorem 3.1. Let T ∈ SIO + m,q,M with 1 ≤ q ≤ 2, M > n 2m and let p M < 1 be defined by M = n 2m ( 2 pM − 1)
. Let q ′ be the dual exponent to q and β < 1.
(1) If q ′ ≤ 2n m(1−β) or equivalently n 2m ≥ − β−1 2 + n m ( 1 q − 1 2 )
then T extends to a bounded operator on T p,2,m (t β dtdy) when 2 ≥ p > p c , where
p c = 2 n 2m − n m 1 q − 1 2 n 2m − n m 1 q − 1 2 + 1−β 2 = 4n 2n + m(1 − β)q ′ ≥ 1 . (2) If q ′ > 2n m(1−β) or equivalently − β−1 2 + n m ( 1 q − 1 2 ) > n 2m then T extends to a bounded operator on T p,2,m (t β dtdy) when 2 ≥ p > sup(p M ,p c ), wherẽ p c = 2n 2n q + m(1 − β) < 1 .
Let us say a word on the exponents p c ,p c . In the first case, p c ≥ 1. In the second case, p c < 1. It is consistent as
p c = p c ⇐⇒p c = 1 ⇐⇒ p c = 1 ⇐⇒ n 2m = − β − 1 2 + n m 1 q − 1 2 .
When q is small, we thus get results for p below 1 provided M is not too small (e.g. in the case of exponential decay).
As a function of q, the exponents p c ,p c are increasing. When q = 2,p c = 2n n+m(1−β) which is the exponent found in Theorem 1.3.
Remark that we improve over the lower bound: M > n 2m suffices here instead of M > n pm when p ≤ 2. In [8], Theorem 1.3 was proved using comparison of tent space norms under change of apertures, i.e. B(x, t 1 m ) changed to B(x, ct 1 m ) for c > 1. The sharp behavior of these comparisons was obtained in [2] using atomic decompositions and interpolation. It is thus natural to use atoms here as well to prove our results. Furthermore, it simplifies the proofs greatly.
Recall that for 0 < p ≤ 1, the tent space T p,2 has an atomic decomposition [13]: A T p,2 atom is a function a(t, y) supported 1 in a region (0, r] × B where B is a (closed) ball on R n of radius r, satisfying . Any T p,2 function g can be represented as a series g = λ j a j where a j is a T p,2 atom and |λ j | p ∼ g p T p,2 . Here the series converges in the tent space quasi-norm, and, in particular, in L 2 loc (R n+1 + ). Translating this to our setting, T p,2,m (t β dtdy) atoms are functions A(t, y) with support in (0, r m ] × B, where B is a (closed) ball in R n of radius r, satisfying B r m 0 |A(t, y)| 2 t β dtdy ≤ r −n( 2 p −1) , and the decomposition theorem holds in T p,2,m (t β dtdy). Remark that atoms are also special elements of L 2 (R n+1 + , t β dtdy) = T 2,2,m (t β dtdy) which is helpful for representation purposes of SIO ± operators acting on them. 1 The support is a relatively closed subset of R n+1 + .
Remark 3.2. Recall that the map j : T p,2,m (t β dtdy) → T p,2,1 (t −1 dtdy) defined by j(f )(t, y) = √ mt m(1+β) 2 f (t m , y) is an isometry; it also sends T p,2,m (t β dtdy) atoms to T p,2,1 (t −1 dtdy) atoms. Lemma 3.3. Let p ≤ 1 and T a linear operator bounded on T 2,2,m (t β dtdy). Then T has a bounded extension from T p,2,m (t β dtdy) ∩ T 2,2,m (t β dtdy) to T p,2,m (t β dtdy) if it is uniformly bounded on T p,2,m (t β dtdy) atoms.
Proof. Adapt to p ≤ 1 the argument in Step 3 of the proof of Theorem 4.9 in [7] done for T p,2,1 (t −1 dtdy) (without loss of generality, one can take m = 1, and β = −1 by Remark 3.2). This argument also furnishes the extension procedure. Theorem 3.1 follows immediately from the two lemmas below applied to the decomposition of T ∈ SIO + m,q,M into its singular part T 1 plus its integral part T 2 . Recall that M > n 2m . Lemma 3.4. The operator T 1 extends to T p,2,m (t β dtdy) for p > p M . Proof of Lemma 3.4. By interpolation (see [13] for the case m = 1, β = −1, and apply Remark 3.2 to deduce the general case) it suffices to consider p M < p ≤ 1. By Lemma 3.3, it is enough to show that T 1 A ∈ T p,2,m (t β dtdy) if A is a T p,2,m (t β dtdy) atom, with a uniform bound. Since the proofs are scale invariant, we assume that A is supported in (0, 1] × B(0, 1). Then we remark that if t > 2, T 2 A(t, ·) = T A(t, ·) because of the definition of T 2 and the support of A. Hence (T 1 A)(t, ·) = 0 for t > 2. We let f j (t, y) = (T 1 A)(t, y) if 2 j ≤ |y| < 2 j+1 , 0 elsewhere, and f 0 (t, y) = (T 1 A)(t, y) if |y| ≤ 2, 0 elsewhere. We show that f j = λ j A j with A j a T p,2,m (t β dtdy) atom and |λ j | p 1. For j = 0, this follows from the boundedness of T 1 on T 2,2,m (t β dtdy) as β < 1. For j ≥ 1, we argue as follows:
B(0,2 j+1 ) 2 (j+1)m 0 |f j (t, y)| 2 t β dtdy = 2 0 2 j ≤|y|<2 j+1 |(T 1 A)(t, y)| 2 dy t β dt = 2 0 2 j ≤|y|<2 j+1 t t 2 t − s t − s ε− 1 2 (K(t, s)A(s, ·))(y) ds 2 dy t β dt 2 0 2 j ≤|y|<2 j+1 t t 2 t 2ε (t − s) 1−2ε |(K(t, s)A(s, ·))(y)| 2 dsdy t β dt 2 0 t t 2 t 2ε 1 (t − s) 1+2ε+ 2n m ( 1 q − 1 2 ) 1 + 2 jm t − s −2M A(s, .) 2 q t β ds dt 1 0 A(s, .) 2 2 s β s 2ε 2s s 1 (t − s) 1+2ε+ 2n m ( 1 q − 1 2 ) 1 + 2 jm t − s −2M dtds 2 −2jmM 1 0 A(s, .) 2 2 s β ds.
We used Cauchy-Schwarz inequality in the fourth line and t 2ε t t 2 (t − s) 2ε−1 ds when ε > 0.
In the next to last line, we impose ε < M − n m ( 1 q − 1 2 ), which is possible as M > n 2m and q ≥ 1. The estimate A(s, .) q A(s, .) 2 uses the fact that A(s, ·) is supported in B(0, 1). As γ = 2mM − n( 2 p − 1) > 0, we thus get the desired estimate with λ j = C2 −jγ/2 . We also remark that we implicitly used Lemma 2.3, which is possible since the last four lines yield the required estimate to write T 1 A(t, y) = t t 2 (K(t, s)A(s, ·))(y) ds on the support of f j .
Proof of Lemma 3.5. We imbed T 2 into an analytic family of integral operators J α defined for α ∈ C by
J α f (t, y) = t 2 0 s t α (K(t, s)f (s, ·))(y)ds. Observe that R n+1 + |J α f (t, y)| 2 t β dtdy = R n+1 + t 2 0 s t α− β−1 2 (tK(t, s)(s β+1 2 f (s, ·)))(y) ds s 2 dtdy
t .
An application of Schur's lemma, using that t ∼ t−s and the uniform boundedness of tK(t, s), shows that, provided ℜe α − β−1 2 > 0, the last integral is bounded by
C ℜe α − β − 1 2 R n+1 + |s β+1 2 f (s, x)| 2 dsdx s = C ℜe α − β − 1 2 R n+1 + |f (s, x)| 2 s β dsdx.
Hence, J α is well-defined for ℜe α > β−1 2 and bounded on T 2,2,m (t β dtdy) for all m. Notice that β < 1 implies that this domain contains α = 0 and J 0 = T 2 . Now we let A be a T p,2,m (t β dtdy) atom and estimate J α A. Since the proof below is scale invariant, we assume that A is supported in (0, 1] × B(0, 1). We let
f j (t, y) = (J α A)(t, y) if 2 j ≤ |y| < 2 j+1
and t < 2 jm , (J α A)(t, y) if |y| < 2 j+1 and 2 jm ≤ t < 2 (j+1)m , 0 otherwise, for j = 0 and f 0 (t, y) = (J α A)(t, y) if |y| ≤ 2 and t < 2 m , 0 elsewhere, so that J α A = f 0 + f 1 + . . .
By the boundedness property of J α , we get
B(0,2) 2 m 0 |f 0 (t, y)| 2 t β dtdy ≤ C ℜe α − β − 1 2 B(0,1) 1 0 |A(s, x)| 2 s β dsdx ≤ C ℜe α − β − 1 2 .
Next,
B(0,2 j+1 ) 2 (j+1)m 0 |f j (t, y)| 2 t β dtdy = 2 j <|y|<2 j+1 2 jm 0 |f j (t, y)| 2 t β dtdy + |y|<2 j+1 2 (j+1)m 2 jm |f j (t, y)| 2 t β dtdy.
Call I j and J j the square roots of the first and second integrals. For I j , we split the integral in s defining J α A(t, y) as k≥1 2 −k t 2 −k−1 t s t α (K(t, s)A(s, ·))(y) ds so that by Minkowski inequality I j ≤ k≥1 I j,k with
I 2 j,k = 2 j <|y|<2 j+1 2 jm 0 2 −k t 2 −k−1 t s t α (K(t, s)A(s, ·))(y) ds 2 t β dtdy.
Using Cauchy-Schwarz inequality in the s integral and then the L q − L 2 decay with t ∼ t − s, we get
I 2 j,k 2 jm 0 2 −k t 2 −k t 2 −k−1 t s t 2ℜe α 1 t 2+ 2n m ( 1 q − 1 2 ) 1 + 2 jm t −2M A(s, ·) 2 q ds t β dt 2 −2jmM 2 jm 0 2 −k t 2 −k t 2 −k−1 t 2 −2kℜe α 1 t 2+ 2n m ( 1 q − 1 2 )−2M A(s, ·) 2 2 ds t β dt 2 −2jmM 2 k(−2ℜe α+β−1) 2 jm−k 0 A(s, ·) 2 2 s β (2 k s) 2M− 2n m ( 1 q − 1 2 ) ds.
Recall that the support condition on A forces s ≤ 1. Also M > n 2m ≥ n m ( 1 q − 1 2 ). Using also the size requirement on A we obtain
I 2 j,k 2 −2jmM 2 k(−2ℜe α+β−1) 2 inf(k,jm)(2M− 2n m ( 1 q − 1 2 )) . Hence, k≥1 I j,k is controlled by 2 −jm inf(M,v(α,q)) with v(α, q) = ℜe α − β−1 2 + n m ( 1 q − 1 2 ) if M = v(α, q) and by jm2 −jmM if M = v(α, q).
Next, for the second integral, we remark that the support of A forces s ≤ 1 while t ∼ 2 jm ≥ 2. Hence
J 2 j ≤ |y|<2 j 2 (j+1)m 2 jm 1 0 s t 2ℜe α−(β−1) t(K(t, s)s β+1 2 A(s, ·))(y) 2 ds s dt t 2 (j+1)m 2 jm 1 0 s t 2ℜe α−(β−1) t 2 t 2n m ( 1 q − 1 2 )+2 s β+1 2 A(s, ·) 2 q ds s dt t 2 −j(2(ℜe α− β−1 2 )+ 2n m ( 1 q − 1 2 ))m = 2 −2jmv(α,q) .
We used Hölder's inequality, the size requirement on A, and also s 2ℜe α−(β−1) ≤ 1. In all |x|<2 j+1
2 (j+1)m 0 |f j (t, y)| 2 t β dtdy 1 2 (1 + jm)2 −jm inf(M,v(α,q)) .
We now start the discussion. Case (2) corresponds to v(0, q) > n 2m . The exponentp c is such that v(0, q) = n 2m ( 2 pc − 1). By Lemma 3.3, J 0 extends to a bounded map on T p,2,m (t β dtdy) for any p ≤ 1 with n 2m ( 2 p − 1)≥ inf(M, v(0, q)), which means 1 ≥ p > sup(p M ,p c ). By interpolation with the p = 2 result, J 0 extends to a bounded map on T p,2,m (t β dtdy) for sup(p M ,p c ) < p ≤ 2.
Case (1) corresponds to v(0, q) ≤ n 2m . Let α 1 > 0 be such that v(α 1 , q) = n 2m . As in the preceding case, for any α with ℜe α > α 1 , J α extends to a bounded map on T 1,2,m (t β dtdy) and by checking the proof above, the bound does not depend on ℑm α. By the p = 2 case, if α 2 = β−1 2 < 0, then for any α with ℜe α > α 2 , J α extends to a bounded map on T 2,2,m (t β dtdy) and the bound does not depend on ℑm α. Hence, by Stein's interpolation theorem for analytic families extended to tent spaces (see [18] for its extension to the tent spaces T p,2 with p ≥ 1), J 0 extends to a bounded map on T p,2,m (t β dtdy) for p c < p < 2 and p c is the exponent with 1 pc = θ 1 + 1−θ 2 when 0 = θα 1 + (1 − θ)α 2 . A calculation yields the explicit formula of the statement.
Remark 3.6. Note that the most restrictive conditions on p come from the tail operator T 2 , not the singular one T 1 , which is contrary to usual feeling for singular integral operators. This can be understood by noticing that this tail operator contains the terms where s is close to 0, and some decay is required to control the tent space norms near this boundary.
We next give a result for operators in SIO − m,q,M when q ≤ 2.
Proposition 3.7. Let β > −1, m ∈ N * , T ∈ SIO − m,q,M with 1 ≤ q ≤ 2 and M > n 2m . Let p M < 1 be such that M = n 2m ( 2 pM −1 )
. Then T extends to a bounded operator on T p,2,m (t β dtdx) for p M < p < 2.
Proof. By interpolation, it suffices to treat the case p M < p ≤ 1. Take such a p. Let A be a T p,2,m (t β dtdy) atom, i.e. a function supported in some (0, r m ] × B(x 0 , r), and satisfying
R n+1 + |A(s, x)| 2 s β dsdx ≤ r −n( 2 p −1) . For j ∈ N, let B j = (0, (2 j r) m ] × B(x 0 , 2 j r) ⊂ R n+1 + and C j = B j \B j−1 (with B −1 = ∅). For k, j ∈ N, and (k, j) = (0, 0) we let T k,j A(t, y) = 1 Cj (t, y) 2 k+1 t 2 k t (K(t, s)A(s, ·))(y)ds and (T 0,0 A)(t, y) = 1 B0 (t, y)(T 1 A)(t, y)
where T 1 is the singular part of T .
We claim that, for a sequence λ k,j > 0, which is independent of A and satisfies ∞ k,j=0 λ k,j < ∞, we have Bj |T k,j A(t, y)| 2 t β dtdy ≤ (2 j r) −n( 2 p −1) λ 2 k,j , so λ −1 k,j T k,j A is a T p,2,m (t β dtdy) atom. Note that k≥1,j≥0 T k,j A = T 2 A. Using Lemma 2.3 a posteriori, we have T 1 A = j≥0 T 0,j A. Hence k≥0,j≥0 T k,j A = T A and thus T A T p,2,m (t β dtdy) ∞ k,j=0 λ k,j . By Lemma 3.3, we are then able to conclude the proof. It remains to prove the claim.
The proof is scale and translation invariant so we assume that x 0 = 0 and r = 1. For j ≥ 1, we have
Bj |T k,j A(t, y)| 2 t β dtdy ≤ Cj (2 k t) 2ǫ 2 k+1 t 2 k t (s − t) 1−2ǫ |(K(t, s)A(s, ·))(y)| 2 ds t β dtdy.
Here we have used the Cauchy-Schwarz inequality as in the proof of Lemma 3.4 and the parameter ǫ > 0 will be determined later.
Write C j = (0, 2 (j−1)m ] × B(0, 2 j )\B(0, 2 j−1 ) ∪ [2 (j−1)m , 2 jm ] × B(0, 2 j ) =: C (1) j ∪ C(2)
j . If (t, y) ∈ C
j , then t ≥ 2 (j−1)m ≥ 1, and if s > 2 k t ≥ 1, then A(s, ·) = 0. Thus, we can replace C j by C (1) j in the above multiple integral and impose t ≤ 1. Then we can apply the L q − L 2 decay with F = B(0, 2 j )\B(0, 2 j−1 ) and E = B(0, 1) to continue estimating as follows
≤ 1 0 (2 k t) 2ǫ 2 k+1 t 2 k t 1 (s − t) 1+2ε+ 2n m ( 1 q − 1 2 ) (1 + 2 jm s − t ) −2M A(s, ·) 2 q ds t β dt ∼ = 1 0 (2 k t) 2ǫ 2 k+1 t 2 k t 1 (s − t) 1+2ε+ 2n m ( 1 q − 1 2 ) (1 + 2 jm s − t ) −2M A(s, ·) 2 2 ds t β dt = 2 2kǫ 2 k+1 0 2 −k s 2 −k−1 s t β+2ε (s − t) 2n m ( 1 q − 1 2 )+1+2ε (1 + 2 jm s − t ) −2M dt A(s, ·) 2 2 ds ∼ = 2 2kǫ 1 0 (2 −k s) β+2ǫ 2 −k s 2 −k−1 s 1 (s − t) 2n m ( 1 q − 1 2 )+1+2ε ( 2 jm s − t ) −2M dt A(s, ·) 2 2 ds.
We take ǫ ∈ (0, M − n m ( 1 q − 1 2 )) so that the integral with respect to t converges. Indeed, M > n 2m ≥ n m ( 1 q − 1 2 ) and the calculation continues as follows:
∼ = 2 2kǫ 2 −k(β+2ǫ) 2 −2Mmj 2 −k 1 0 s 2M− 2n m ( 1 q − 1 2 )−2ǫ A(s, ·) 2 2 s β+2ǫ ds 2 −k(β+1) 2 −2Mmj 1 0 A(s, ·) 2 2 s β ds ≤ 2 −jn( 2 p −1) λ 2 k,j with λ k,j ∼ = 2 ( n 2 ( 2 p −1)−Mm)j 2 − k 2 (β+1)
, and we used M > n 2m ≥ n m ( 1 q − 1 2 ). If j = 0 and k ≥ 1, we do not use the decay but rather the fact that (t − s)K(t, s) is uniformly bounded on L 2 (R n ). Then we can repeat the above calculation literally taking q = 2 and M = 0.
If k = 0 and j = 0, using the boundedness of T 1 since β > −1,
B(0,2) 2 m 0 |(T 0,0 A)(t, y)| 2 t β dtdy ≤ C B(0,1) 1 0 |A(s, x)| 2 s β dsdx.
We conclude that λ k,j ∼ = 2 ( n 2 ( 2
p −1)−Mm)j 2 − k 2 (β+1) is summable for β > −1 and M > n 2m ( 2 p − 1). 4. Role of L 2 − L q decay
When q ≥ 2, L 2 − L q decay can be used to quantify T p,2 results for p above 2. Clearly the adjoint class to SIO ± m,q,M is SIO ∓ m,q ′ ,M with respect to the inner product
f, g = R n ∞ 0 f (t, y)g(t, y)dtdy.
It is easy to deduce from [13, Section 5] that for p ∈ (1, ∞), m ∈ N * and β ∈ R, we have T p,2,m (t β dtdy) ′ = T p ′ ,2,m (t −β dtdy), with duality given by f, g , i.e.
f T p,2,m (t β dtdy) ∼ sup
g T p ′ ,2,m (t −β dtdy) ≤1 | f, g |.
Thus, we obtain results for 2 < p < ∞ by dualizing Theorem 3.1 and Proposition 3.7 in the classes SIO ± m,q,M with 2 ≤ q ≤ ∞ and M > n 2m . In addition, the results for p = ∞ also hold.
Theorem 4.1. Let T ∈ SIO − m,q,M with 2 ≤ q ≤ ∞ and M > n 2m . Let β > −1. (1) If q ≤ 2n
m(1−β) or equivalently n 2m ≥ − β−1 2 + n m ( 1 2 − 1 q ) then T extends to a bounded operator on T p,2,m (t β dtdy) when 2 ≤ p < p ′ c , where
p c = 2 n 2m − n m 1 2 − 1 q n 2m − n m 1 2 − 1 q + 1−β 2 = 4n 2n + m(1 − β)q .
(2) If q > 2n m(1−β) or equivalently − β−1 2 + n m ( 1 2 − 1 q ) > n 2m then T extends to a bounded operator on T p,2,m (t β dtdy) when 2 ≤ p ≤ ∞. It is enough to prove the result for p = ∞. The extension is done by taking f ∈ T ∞,2,m (t β dtdy), truncating f on (0, k m ) × B(0, k) and letting k go to infinity.
Proof of Proposition 4.2. This is very similar to [8]. Pick a point x 0 ∈ R n and r > 0. Let the sets B j and C j be defined as in the proof of Proposition 3.7. Set
I 2 = B(x0,r) r m 0 |(T f )(t, y)| 2 t β dtdy.
We want to show that I 2 r n f 2 T ∞,2,m (t β dtdy) . We set
I 2 j = B(x0,r) r m 0 |(T f j )(t, y)| 2 t β dtdy
where f j (s, x) = f (s, x)1 Cj (s, x)1 (0,r m ] (s) for j ≥ 0. Thus by Minkowski inequality, I ≤ I j .
Since the proofs are scale and translation invariant, we assume x 0 = 0 and r = 1 for simplicity. For I 0 we use again Theorem 2.2 which implies
I 2 0 B(0,2) 2 m 0 |f (s, x)| 2 s β dsdx f 2 T ∞,2,m (t β dtdy) .
Next, for j = 0, we proceed as in the proof of Proposition 3.7 by representing T f j (t, y) through a kernel (which is justified by the calculation below and Lemma 2.3 for the singular part) but using this time L 2 − L q decay (after using Hölder inequality for the integral with respect to y on B(0, 2)) to obtain
I 2 j ∞ k=1 1 0 2 −k t 2 −k−1 t 2 −k t |t − s| 2n m ( 1 2 − 1 q )+2 1 + 2 jm t − s −2M f j (s, .) 2 2 ds t β dt + 1 0 t t 2 t β+2ε |t − s| 2n m ( 1 2 − 1 q )+1+2ε 1 + 2 jm t − s −2M f j (s, .) 2 2 ds dt.
Exchanging the order of integration, and using the fact that t ∼ t − s in the first part and that t ∼ s in the second, and M > n m ( 1 2 − 1 q ) + ε for small enough ε, and β < 1, we have the following.
I 2 j ∞ k=1 2 −k 2 −2jmM 2 −k 0 2 k+1 s 2 k s t β−1+2M− 2n m ( 1 2 − 1 q ) f j (s, .) 2 2 dtds + 1 0 2s s t β+2ε |t − s| 2n m ( 1 2 − 1 q )+1+2ε 1 + 2 jm t − s −2M f j (s, .) 2 2 s β dtds ∞ k=1 2 −k 2 −2jmM 2 −k 0 (2 k s) β f j (s, .) 2 2 ds + 2 −2jmM 1 0 f j (s, .) 2 L 2 s β ds 2 −2jmM 2 jm 0 f j (s, .) 2 2 s β ds.
We thus have
I 2 j 2 −2jmM 2 jn f 2 T ∞,2,m (t β dtdy)
, and the condition M > n 2m allows us to sum these estimates. Proof of Theorem 4.1. The proof is almost entirely similar to the above one. Set I j as in the proof of Proposition 3.7. I 0 is estimated as before. When j ≥ 1, the inner term in I j can be expressed using the kernel representation from t to +∞, which is split into I j,k on the dyadic intervals (2 k t, 2 k+1 t) for k ∈ N, using Minkowski inequality. The k = 0 term is estimated as was the term corresponding to (t/2, t). For k ≥ 1, the kth term is controlled by
1 0 2 k+1 t 2 k t 2 k t |t − s| 2n m ( 1 2 − 1 q )+2 1 + 2 jm t − s −2M f j (s, .) 2 2 ds t β dt.
Exchanging order, we obtain the bound
2 −2jmM 2 k(1−β+2M− 2n m ( 1 2 − 1 q )) 2 k 0 f j (s, .) 2 2 s β ds.
Note that the support of f j forces s ≤ 2 (j+1)m in the integral, which is bounded by C2 jn . The series for I j,k is summable in k under the condition in the statement and summable in j if M > n 2m .
Maximal regularity operators
Let us come back to our original motivation which is to bound maximal regularity operators on tent spaces.
Definition 5.1. Let 1 ≤ q ≤ r ≤ ∞. A family of bounded linear operators (T t ) t>0 ⊂ B(L 2 (R n )) is said to satisfy L q − L r off-diagonal estimates of order M , with homogeneity m, if, for all Borel sets E, F ⊂ R n , all t > 0, and all f ∈ L 2 (R n ) ∩ L q (R n ):
1 F T t 1 E f r t − n m ( 1 q − 1 r ) 1 + dist(E, F ) m t −M 1 E f q .
With this definition we have the following simple fact.
Proposition 5.2. Let 1 ≤ q ≤ 2 (resp. 2 ≤ q ≤ ∞) and assume that (tLe −tL ) t≥0 satisfies L q − L 2 (resp. L 2 − L q ) off-diagonal estimates (of order M ), with homogeneity m. Then M L ∈ SIO + m,q,M and M − L * ∈ SIO − m,q ′ ,M .
Indeed, the operator-valued kernel Le −|t−s|L has L q − L 2 (resp. L 2 − L q ) decay (of order M ), with homogeneity m so that it suffices to apply Definition 2.5.
To illustrate our results so far, let us prove Proposition 1.5.
Proof of Proposition 1.5. Let L = −∆ + V or −div A∇ with real coefficients. Then, the kernel of the semigroup (e −tL ) t≥0 satisfies pointwise Gaussian estimate (see e.g. [24, Theorem 6.10]), hence L 1 − L 2 and L 2 − L ∞ off-diagonal estimates with homogeneity m = 2 of order ∞. Therefore we have that M L ∈ SIO + 2,1,∞ ∩ SIO + 2,∞,∞ . We now apply the second case of Theorem 3.1 and Proposition 4.2 with β = 0 to conclude that T p,2,m (dtdy) boundedness of M L holds for ∞ ≥ p >p c = n n+1 . Using the subordination formula, the Poisson semigroup associated with √ L satisfies L 1 − L 2 and L 2 − L ∞ off-diagonal estimates with homogeneity m = 1 and order n 2 + 1. Thus
M √ L ∈ SIO + 1,1, n 2 +1 ∩ SIO + 1,∞, n 2 +1
. From M = n 2 + 1 and m = 1, we have p M = n n+1 . As β = −1, m = 1 and q = 1, n 2 < − β−1 2 + n m 1 q − 1 2 = 1 + n 2 and we are in the second case of Theorem 3.1. Applying this result and Proposition 4.2, we conclude that T p,2,m (t −1 dtdy) boundedness of M √ L holds for ∞ ≥ p > sup(p M ,p c ) = n n+1 . As explained in the introduction, applications of our results require M to be sufficiently large, namely M > n 2m , whatever the value of q. Of course, with exponential decay, this is not a problem. Semigroups generated by elliptic operators of even order m ≥ 2 have, in general, such an exponential off-diagonal decay. However, in the case of Poisson type semigroups, small polynomial decay is to be expected. This application suggests that the lower bound on M should be kept as low as possible. Looking at the proof of Lemma 3.4, there seems to be unavoidable restrictions if we are only given M without further information. However, the decay of the semigroup is usually computed from the decay of the resolvent and integration on a contour. This is the point of view we shall take.
We consider the following conditions on resolvent estimates for fixed 1 ≤ q ≤ r ≤ ∞. 1) There exists a bisectorial operator L of angle ω ∈ [0, π 2 ) having a bounded H ∞ functional calculus on L 2 (R n ) such that L = | L|(= L 2 = √ L 2 ), and for any K ∈ N and ω < ν < π/2,
(H1) 1 F (1 − z L) −1 1 E f r ≤ c(K, ν)|z| − n m ( 1 q − 1 r ) 1 + dist(E, F ) m |z| −K 1 E f q .
for all f ∈ L 2 (R n ) ∩ L q (R n ), E, F Borel subsets of R n , z = e ±iθ t, t > 0 and |θ − π 2 | < π 2 − ν.
2) The operator L 2 is sectorial in L 2 (R n ) of angle 2ω < π and for any K ∈ N and ω < ν < π/2,
(H2) 1 F (1 − zL 2 ) −1 1 E f r ≤ c(K, ν)|z| − n 2m ( 1 q − 1 r ) 1 + dist(E, F ) 2m |z| −K 1 E f q for all f ∈ L 2 (R n ) ∩ L q (R n )
, E, F Borel subsets of R n , z = e ±iθ t, t > 0 and 2ν < θ ≤ π.
Operators of Dirac type satisfying (H1) with m = 1 appear in [11,Proposition 5.2]. See also [7] and [19].
(H1) and (H2) are closely related and, in fact, (H1) implies (H2). Indeed, it follows from the resolvent formula
(1 − z 2 L 2 ) −1 = 1 2 (1 − z L) −1 + 1 2 (1 + z L) −1
for z as in (H1). Remark that, in (H2), 2w may be greater than or equal to π/2, in which case −L 2 may not generate a semigroup.
Proposition 5.3. Let L be a sectorial operator of angle ω < π/2 with an H ∞ functional calculus on L 2 (R n ). Assume that (H1) or (H2) is satisfied and fix ω < ν < π/2. Then for any 0 < ǫ < R < ∞ and any α ∈ C with ℜe α ∈ [ǫ, R], 1 F (tL) α e −tL 1 E f r has bound c(ǫ, R, q, r, ν)e ν|ℑm α| · t − n m ( 1 q − 1
r ) 1 + dist(E, F ) m t −ℜe α− n m ( 1 q − 1 r ) 1 E f q .
A result in this spirit is in [16] for q = r = 2.
Proof. It is enough to assume (H2). In this case, fix ω < ν ′ < θ < ν, and let φ t (λ) = (tλ 1 2 ) α e −tλ 1 2 which is holomorphic and bounded for | arg λ | < π − 2ν ′ . The Cauchy integral formula for sectorial operators implies that
(tL) α e −tL = 1 2πi Γ φ t (λ)(1 − λ −1 L 2 ) −1 dλ λ
holds with Γ the oriented contour {|s|e isign(s)2θ : s ∈ R}. We write c θ = ℜe (e iθ ) = cos θ > 0. Fix f with 1 E f q = 1. In the following, we write a = ℜe α, d = dist(E, F ), γ = n m ( 1 q − 1 r ). Then (H2) gives us
1 F (tL) α e −tL 1 E f r c(K) ∞ 0 t a s a 2 e θℑm α e −c θ ts 1 2 s γ 2 (1 + d 2m s) −K ds s ∼ = c(K, ν)e ν|ℑm α| ∞ 0 t a s a s γ e −c θ ts (1 + d 2m s 2 ) −K ds s c(K, ν)e ν|ℑm α| t −γ ∞ 0 s a s γ e −c θ s 1 + d 2m s 2 t 2 −K ds s c(ǫ, K, q, r, ν)e ν|ℑm α| t −γ 1 + d m t −a−γ .
provided 2K > R+γ. We used the fact that 1 ≤ 2(1+x) −1 when x ≤ 1, and x −1 ≤ 2(1+x) −1 when x ≥ 1. The parameter ǫ > 0 is only needed when q = r.
It is clear that similar results hold for fractional powers of sectorial operators. We shall not get into this here. Note also that an exponential decay in the resolvent estimates would not yield a better conclusion in general.
Definition 5.4. Let L be a sectorial operator of type ω < π/2 and having a bounded holomorphic functional calculus on a Hilbert space H. For ℜe α > 0, we define the operator M α acting on L 2 (R + , dt; D(L α )) by
M α f (t) = t 0 (t − s) α−1 L α e −(t−s)L f (s) ds. Clearly M 1 = M L .
Proposition 5.5. Let α ∈ {z ∈ C ; a ≤ ℜe z ≤ b} for some a, b ∈ R + . Then M α extends boundedly to L 2 (R + , dt; H), with a bound not exceeding ce ν|ℑm α| for any ω < ν < π/2, and some constant c dependent on a, b.
Proof. Using operational calculus as in [3], which is possible since L has bounded holomorphic functional calculus on H, it is enough to prove the same thing for L = zI on L 2 (R + , dt; C) for | arg z | < ν. In this case, we use Schur's lemma for the complex-valued kernel (t − s) α−1 z α e −(t−s)z 1 s<t . For w = ℜe z, |z| ≤ w cos ν , hence
t 0 |(t − s) α−1 z α e −(t−s)z | ds ≤ e ν|ℑm α| (cos ν) ℜe α t 0 |(t − s) ℜe α−1 w ℜe α e −(t−s)w | ds ≤ Γ(ℜe α)e ν|ℑm α| (cos ν) ℜe α and ∞ s |(t − s) α−1 z α e −(t−s)z | dt ≤ e ν|ℑm α| (cos ν) ℜe α ∞ s |(t − s) ℜe α−1 w ℜe α e −(t−s)w | dt ≤ Γ(ℜe α)e ν|ℑm α| (cos ν) ℜe α ,
with Γ being the Euler Gamma function.
Corollary 5.6. Let H = L 2 (R n ). If 1 ≤ q ≤ ∞ and (H2) holds for (q, 2) if q ≤ 2 or (2, q) if q ≥ 2 then M α ∈ SIO + m,q,Mq with M q = ℜe α + n m | 1 q − 1 2 |. Observe that the order of decay becomes a function of q, hence the notation M q . M q increases as q moves away from 2: this is the interesting point for us. As mentioned in the introduction, M 2 = 1 is best possible for the Poisson semigroup of −∆, so it seems one cannot improve this conclusion.
Proof. The fact that M α ∈ SIO + is contained in Proposition 5.5. The decay of the kernel (t − s) α−1 L α e −(t−s)L with s < t is clear from Proposition 5.3.
Corollary 5.7. Let H = L 2 (R n ).
A] Assume (H2) holds for (q, 2) with q ≤ 2. Then M L extends to a bounded operator on T p,2,m (t β dtdy) for p L < p < 2 with p L calculated as follows:
(1) If n mq ′ < 1 and β ≤ −1, p L = p Mq . (2) If n mq ′ < 1 and −1 < β < 1, p L = inf(p c , p c ). (3) If n mq ′ ≥ 1 then 1 pL − 1 2 = mq ′ n ( 1 inf(pc,1) − 1 2 ). B] Assume (H2) holds for (2, q) with q ≥ 2. Then M L extends to a bounded operator on T p,2,m (t β dtdy) for 2 < p < p L with p L = 2n n−mq if mq ≤ n and for 2 < p ≤ ∞ if mq > n. Note that the result for p ≥ 2 does not depend on β. The exponents p Mq ,p c , p c are explicitely defined in Theorem 3.1. The last two depend on β.
Proof. A] The condition n mq ′ < 1 is equivalent to M q = 1 + n m ( 1 q − 1 2 ) > n 2m . Cases (1) and (2) thus follow from Theorem 3.1. In the third case, Theorem 3.1 does not apply to M L but to M α for any α with ℜe α > α 1 and α 1 + n m ( 1 q − 1 2 ) = n 2m which implies that M α is bounded for inf(p c , 1) < p < 2. At the same time, M α is bounded for p = 2 when ℜe α > 0. The third case follows by complex interpolation for the analytic family M α (since the growth in ℑm α is admissible) in tent spaces. B] The condition mq > n means M q = 1 + n m ( 1 2 − 1 q ) > n 2m . So we apply Proposition 4.2 to M L . If mq ≤ n, then we apply this result not to M L but to M α for ℜe α > α 1 and α 1 + n m ( 1 2 − 1 q ) = n 2m and the p = 2 result for ℜe α > 0 and conclude by interpolation for analytic families again.
Proof of Propositions 1.6 and 1.7. Write L = −div A∇. We have that (e −tL ) t≥0 satisfies pointwise Gaussian estimates if n = 1, 2. Hence the conclusion of the first part of Proposition 1.5 applies. For n ≥ 3, let 1 ≤ p − (L) < p + (L) ≤ ∞ be the numbers introduced in [1] such that for p − (L) < q ≤ r < p + (L), (e −tL ) t≥0 satisfies L q − L r off-diagonal estimates with homogeneity m = 2. As the decay is Gaussian, the order is ∞. Moreover, p − (L) < 2n n+2 , p + (L) > 2n n−2 and, by [17], this is sharp for this class of complex operators. Taking q < 2n n+2 , we use the second item in Corollary 5.7, A] when n = 3, 4 and the third one when n ≥ 5 to get the lower bound on p. For the upper bound p = ∞ included, we use B]. Now for the semigroup associated to √ L. When n = 1 or 2, we have the pointwise Poisson kernel estimate, hence L 1 − L 2 and L 2 − L ∞ off-diagonal estimates with order n 2 + 1 and homogeneity m = 1. Hence the conclusion of the second part in Proposition 1.5 applies since m = 1 and β = −1. For n ≥ 3, with the same numbers p − (L), p + (L) as above, the resolvent estimate (H2) holds with m = 1 and p − (L) < q ≤ r < p + (L). Taking q < 2n n+2 , we use the first item in Corollary 5.7, A] when n = 3, 4 and the third one when n ≥ 5 to get the lower bound on p. For the upper bound, we use B] with q > 2n n−2 and find ∞ included if n = 3, 4, and the proposed value if n ≥ 5.
Corollary 5.8. Let H = L 2 (R n ).
A] Assume (H2) holds for (2, q) with 2 ≤ q. Then M − L extends to a bounded operator on T p,2,m (t β dtdy) for 2 < p < p L with p L calculated as follows:
(1) If n mq < 1 and β ≥ 1, p L = ∞ (and boundedness holds at ∞). (2) If n mq < 1 and −1 < β < 1, p L = ∞ (and boundedness holds at p = ∞) if p c < 1 and p L = p ′ c if p c ≥ 1. (3) If n mq ≥ 1 then 1 pL − 1 2 = mq n (− 1 2 ) = − mq 2n . B] Assume (H2) holds for (q, 2) with q ≤ 2. Then M − L extends to a bounded operator on T p,2,m (t β dtdy) for p L < p < 2 with p L = 2n n+mq ′ if mq ′ ≤ n and p L = p Mq if mq ′ > n. This time, this follows from Proposition 3.7 and Theorem 4.1 where one finds the value of p c , using the operators
Lemma 2 . 3 .
23Let β < 1 and T ∈ SIO + . Let E, F be two Borel sets of R n and I, J two
Lemma 3 . 5 .
35The statement of Theorem 3.1 holds for T 2 .
Proposition 4 . 2 .
42Let T ∈ SIO + m,q,M with 2 ≤ q ≤ ∞ and M > n2m . For all β < 1, T extends to a bounded operator on T p,2,m (t β dtdy) for 2 ≤ p ≤ ∞.
− t) α−1 L α e −(s−t)L f (s) ds and the interpolation procedure of Corollary 5.7. Details are left to the reader.
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UMR 8628, F-91405 Orsay; CNRS, F-91405. Pascal Auscher Univ. Paris-Sud, laboratoire de MathématiquesPascal Auscher Univ. Paris-Sud, laboratoire de Mathématiques, UMR 8628, F-91405 Orsay; CNRS, F-91405
Campus des Cézeaux, 63177 Aubière Cedex christoph. Clermont-Ferrand 2Christoph Kriegler Laboratoire de Mathématiques (UMR 6620), Université Blaise-PascalChristoph Kriegler Laboratoire de Mathématiques (UMR 6620), Université Blaise-Pascal (Clermont-Ferrand 2), Campus des Cézeaux, 63177 Aubière Cedex [email protected]
. Sylvie Monniaux, Latp-Umr 7353, F-13397 Marseille Cédex 20. [email protected] Saint-Jérôme, Univ. Aix-MarseilleSylvie Monniaux LATP-UMR 7353, FST Saint-Jérôme, Univ. Aix-Marseille, F-13397 Marseille Cédex 20. [email protected]
. Ac- ton ACT 0200John Dedman Building. Pierre Portal Permanent Address: Université Lille 1, Laboratoire Paul Painlevé, F-59655 Villeneuve d'Ascq. Current Address: Australian National University, Mathematical Sciences InstitutePierre Portal Permanent Address: Université Lille 1, Laboratoire Paul Painlevé, F-59655 Villeneuve d'Ascq. Current Address: Australian National University, Mathematical Sciences Institute, John Dedman Building, Ac- ton ACT 0200, Australia. [email protected]
| []
|
[
"SIMPLICIAL DOLLAR GAME",
"SIMPLICIAL DOLLAR GAME"
]
| [
"Jesse Kim ",
"David Perkinson "
]
| []
| []
| The dollar game is a chip-firing game introduced by Baker as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these generalizes the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second generalizes the fact that trees (graphs of genus 0) are exactly the graphs on which every divisor of degree 0, interpreted as an instance of the dollar game, is winnable.UCSD, | 10.37236/9821 | [
"https://arxiv.org/pdf/1908.09350v3.pdf"
]
| 202,152,019 | 1908.09350 | 619d84e610c59b8b305759da789a04afe8ec7a76 |
SIMPLICIAL DOLLAR GAME
23 May 2022
Jesse Kim
David Perkinson
SIMPLICIAL DOLLAR GAME
23 May 2022arXiv:1908.09350v3 [math.CO]
The dollar game is a chip-firing game introduced by Baker as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these generalizes the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second generalizes the fact that trees (graphs of genus 0) are exactly the graphs on which every divisor of degree 0, interpreted as an instance of the dollar game, is winnable.UCSD,
Introduction
Let G = (V, E) be a finite, connected, undirected graph with vertex set V and edge set E. To play the dollar game on G, assign an integer number of dollars to each vertex. Negative integers are interpreted as debt. A lending move consists of a vertex giving one of its dollars to each of its neighboring vertices, and a borrowing move is the opposite, in which a vertex takes a dollar from each neighbor. Vertices may lend or borrow, regardless of the number of dollars they possess. The goal of the game is to bring all vertices out of debt through a sequence of such moves.
The dollar game was introduced in Riemann-Roch and Abel-Jacobi theory on a finite graph, by Baker and Norine ( [2]) as a variant of an earlier version due to Biggs ([4]). Baker and Norine's work develops the divisor theory of graphs, which views a graph as a discrete version of an algebraic curve or Riemann surface. The assignment of a v dollars to each vertex v is formally a divisor D = v∈V a v v in the free abelian group Div(G) := ZV . The net amount of money on the graph is deg(D) := v∈V a v , the degree of D. Divisors D and D ′ are linearly equivalent, denoted D ∼ D ′ , if one may be obtained from the other via lending and borrowing moves. The group of divisors modulo linear equivalence is the Picard group Pic(G). Since lending and borrowing moves conserve net wealth, Pic(G) is graded by degree. Its degree zero component is the Jacobian group Jac(G), which is a finite group with size equal to the number of spanning trees of G. A divisor is effective if its coefficients are nonnegative. Thus, in the language of algebraic geometry, an instance of the dollar game is a divisor D ∈ Div(G), and the game is won by finding a linearly equivalent effective divisor.
A fundamental concept introduced in [2] is the notion of the rank of a divisor. If there is no effective divisor linearly equivalent to D, then the rank of D is r(D) = −1. Otherwise, the rank is the maximum integer k such that D − E is linearly equivalent to an effective divisor for all effective divisors E of degree k. In terms of the dollar game, the rank is a measure of robustness of winnability: the dollar game D is winnable if and only if r(D) ≥ 0, and if r(D) = k > 0, it is winnable even after removing k dollars arbitrarily.
The Riemann-Roch theorem for graphs ([2, Theorem 1.12]) has a form nearly identical to that for algebraic curves. It says that for all D ∈ Div(G), Here, g = |E| − |V | + 1 and K = v∈V (deg G (v)v − 2) v where deg G (v) is the number of edges incident on v. These play the role of the genus and the canonical divisor, respectively, for an algebraic curve.
Since the rank is at least −1,
r(D) = deg(D) + 1 − g + r(K − D) ≥ deg(D) − g.
A consequence is that if deg(D) ≥ g, then the dollar game D is winnable. This result is sharp, too: there are always unwinnable divisors of degree g − 1 ([2, Theorem 1.9]). It follows that all divisors of degree 0 are winnable if and only if g = 0, i.e., G is a tree. In summary, the dollar game has a minimal "winning degree" g, and that minimal degree is 0 exactly when the game is played on a tree. Our main goal is to generalize these results to a dollar game played on a simplicial complex of any dimension. Lending moves are sometimes called vertex-firings or chip-firings (and borrowing moves are reverse firings). They arise naturally as an encoding of the discrete Laplacian operator for the graph. Duval, Klivans, and Martin ( [8], [9], [10]) use a version of a combinatorial Laplacian to generalize the divisor theory of graphs to higher-dimensional simplicial (and cellular) complexes. In this theory, an i-chain-a formal integer sum of idimensional faces-of a complex ∆ may be thought of as an assignment of an integer "flow" to each i-face. Firing an i-face f then diverts flow around the (i + 1)-faces incident on f . The group of i-cycles modulo these firing moves is the i-th critical group of the complex, K i (∆), generalizing the Jacobian group of a graph. By [8,Corollary 4.2], under certain restrictions on ∆, the size of the torsion part of K i (∆) is the number of torsion-weighted (i + 1)-dimensional spanning trees of ∆.
In this paper, we interpret Duval, Klivans, and Martin's theory as a higher-dimensional dollar game. A chain on a simplicial complex is thought of as a distribution of wealth among the faces. The goal of the game is to use face-firings to redistribute wealth, leaving no face in debt. For this purpose, the naive version of degree as the net wealth of the system is not appropriate: using that notion of degree, there would be simplicial complexes with chains of arbitrarily negative degree that are winnable and arbitrarily positive degree that are unwinnable. The root of the problem is that, unlike for graphs, lending and borrowing moves on simplicial complexes are not necessarily conservative. Instead, in Definition 4 we introduce a natural generalization of the degree of a divisor on a graph to one that is invariant under firing moves on the chains of a complex. Our main results generalize the properties of divisors on graphs discussed in connection with the Riemann-Roch theorem, above: Theorem 18 shows that if the degree of a chain is sufficiently large, then it is winnable, and Corollary 34 shows that for each i, all (i − 1)-chains of degree 0 are winnable if and only if the i-skeleton of the complex is a spanning forest, torsion-free in codimension one. Section 2 sets notation and presents required background on (abstract) simplicial complexes and polyhedral cones. In particular, ∆ always denotes a d-dimensional simplicial complex. In Section 3, we recall the definition of the i-dimensional Laplacian L i and critical group K i (∆) for ∆ and use these to carefully define the dollar game determined by each i-chain. Two i-chains are linearly equivalent if their difference is in the image of L i . Section 4 defines the degree of each i-chain σ of ∆ and relates it the winnability of the dollar game, generalizing results from graphs (the special case d = 1) to higher dimensions. Let H be the minimal additive basis, i.e., the Hilbert basis, for the monoid of nonnegative integer points in the kernel of L i . Using H, we define the degree of σ as an integer vector deg(σ) ∈ Z |H| . By Proposition 6, the degree of a chain is invariant under linear equivalence, with the immediate consequence (Corollary 7) that if the dollar game determined by the chain σ is winnable, then deg(σ) ≥ 0. Lemma 10 is a key technical result showing there is a strictly positive element in the kernel of L i . By Theorem 13, the group of degree zero i-chains modulo linear equivalence is isomorphic to the torsion part of the i-th critical group. In the special case where d = 1, this result generalizes the fact that the Jacobian group of a connected graph is the torsion part of the Picard group (in accordance with isomorphism (1)). Theorem 18 achieves one of our main goals: it says that if the degree of a chain is sufficiently large, its corresponding dollar game is winnable.
Section 5 considers the case where ∆ is a pseudomanifold. We compute the critical group of an oriented pseudomanifold (Proposition 21), generalizing [8, Theorem 4.7 and subsequent remarks]. Our main result on pseudomanifolds is a combinatorial description of the Hilbert basis H, described above, in codimension one (Theorem 22). The section ends with an example of calculating minimal degrees δ such that every chain of degree at least as large as δ is winnable. Section 6 builds on the work of Duval, Klivans, and Martin ( [8], [9], [10]) on higher-dimensional forests and critical groups. Our main result is Corollary 34, which shows that all (i − 1)-chains of degree zero are winnable if and only if the i-skeleton is an i-dimensional spanning forest, torsion-free in codimension one. We also generalize Theorem 3.4 of [8], which for each dimension gives an isomorphism between the critical group and the cokernel of the reduced Laplacian-a submatrix of the Laplacian determined by a spanning forest. In Section 6.1, we consider an alternative generalization of the set of divisors of nonnegative degree on a graph due to Corry and Keenan ([6]). We use it to characterize higher-dimensional spanning trees that are acyclic in codimension one in terms of winnability of the dollar game.
Section 7 poses some open questions. Finally, the proofs of Proposition 21 and Theorem 30 are relegated to an appendix to avoid distraction from our main line of argument.
Readers interested in learning more about chip-firing on graphs and its relation to a diverse range of mathematics may wish to consult the textbooks [7] and [15]. complexes in terms of the dollar game and for sharing some of his unpublished joint work with Liam Keenan, motivating the results in Section 6.1. We thank Collin Perkinson and our anonymous referee for comments on the exposition.
Preliminaries
2.1. Simplicial complexes. Throughout this paper, ∆ is a d-dimensional simplicial complex on the set V = [n] := {1, . . . , n} for some integer n. A subset of V of cardinality i + 1 that is an element of ∆ is an i-dimensional face or i-face of ∆, and the collection of all i-faces is denoted ∆ i . Let f i = f i (∆) := |∆ i | be the number of faces of dimension i. The empty set is the single face of dimension −1. The elements of V are called vertices. The set of all faces forms a poset under inclusion, graded by dimension, and its maximal elements are the facets of ∆. To say that ∆ has dimension d means that its highest-dimensional facet has dimension d. The complex ∆ is pure if all of its facets have dimension d, which we do not assume. If R is a commutative ring, the module of i-chains, C i (∆, R), is the free R-module with basis ∆ i . In particular, let C i (∆) denote the integral i-chains, C i (∆, Z). Take C i (∆, R) = 0 for i > d and i < −1, whereas C −1 (∆, R) ≈ R. Given an i-chain σ = f ∈∆i a f f , we write σ(f ) := a f and define the support of σ to be supp(σ) := {f ∈ ∆ i : σ(f ) = 0}.
In general, our results will depend on the choice of an orientation of ∆ (cf. Example 2). In order for the dollar game to be sensible, this orientation must be acyclic, i.e., for all i, every positive sum of i-faces has nonzero boundary. Since any such orientation induces an acyclic orientation on the 1-skeleton of ∆, every acyclic orientation is the standard orientation up to renumbering of the vertices, so we fix the standard orientation on ∆ induced by the natural ordering on the vertex set V = [n]. Thus, each i-face is represented by the list of its vertices v 0 · · · v i with v 0 < · · · < v i . We fix the lexicographic total ordering on each ∆ i and the corresponding induced isomorphism C i (∆) ≃ Z fi . If π is a permutation, we write v π(0) · · · v π(i) = sgn(π) v 0 · · · v i as chains.
For each i, there is a boundary mapping
∂ i : C i (∆, R) → C i−1 (∆, R) defined by ∂ i (v 0 · · · v i ) := i j=0 (−1) j v 0 · · · v j · · · v i ,
where v j indicates that v j is omitted. We have ∂ i • ∂ i+1 = 0. The elements of ker ∂ i are the i-cycles and elements of im ∂ i are i-boundaries. The i-th reduced homology group is
H i (∆, R) := ker ∂ i / im ∂ i+1 .
The ordinary homology groups H i (∆, R) use the same definition, with one change: ∂ 0 is taken to be the zero mapping, or equivalently, C −1 (∆) is defined to be the trivial group. We write simply H i (∆) and H i (∆) in the case R = Z. The i-th reduced Betti number is
β i (∆) = rank Z H i (∆) = dim Q H i (∆, Q).
Applying the functor Hom( · , R), we get the dual mapping ∂ t i+1 : C i (∆, R) → C i+1 (∆, R) identifying chain modules with their duals using our fixed orderings of the faces of ∆.
If Σ is a subcomplex of ∆, we assume it has the orientation inherited from ∆ (induced by the natural ordering on V ) and may write ∂ Σ,i for its i-th boundary mapping. The i-skeleton of ∆, denoted Skel i (∆), is the subcomplex consisting of all faces of ∆ of dimension i or less.
Relative homology is mentioned in Section 5. The relative chain complex (with Z-coefficients) for a nonempty subcomplex Σ of ∆ is the complex
· · · → C i (∆)/C i (Σ) ∂i − → C i−1 (∆)/C i−1 (Σ) → · · · ,
where ∂ i is induced by ∂ i . The i-th relative homology group is
H i (∆, Σ) := ker ∂ i / im ∂ i+1 .
If Σ = ∅, we take H i (∆, Σ) := H i (∆).
Polyhedral cones.
We recall some facts about polyhedral cones, using [11], [13], and [18] as references. Let Q be a cone in R n . For us, this means Q is a subset of R n closed under nonnegative linear combinations: if x, y ∈ Q and α, β ∈ R ≥0 , then αx + βy ∈ Q. The cone Q is pointed if Q \ {0} is contained in an open half-space in R n , i.e., there exists z ∈ R n such that x · z > 0 for all x ∈ Q \ {0} (using the ordinary dot product on R n ). We say Q is polyhedral if it is finitely generated, i.e., if there exist x 1 , . . . , x ℓ ∈ R n such that
Q = Span R ≥0 {x 1 , . . . , x ℓ } := ℓ i=1 α i x i : α i ≥ 0 for 1 ≤ i ≤ ℓ .
If the generators x 1 , . . . , x ℓ can be taken to be integral, then Q is a rational polyhedral cone.
Let Q be a rational polyhedral cone. Then the semigroup of its integral points, Q Z := Q∩Z n , has a Hilbert basis H, defined to be a set of minimal cardinality such that every point of Q Z is a nonnegative integral combination of elements of H. If Q is pointed, then H is unique, determined by the property that x ∈ H if and only if x ∈ Q Z \ {0} and there do not exist y, z ∈ Q Z \ {0} such that x = y + z. If Q is integrally generated by x 1 , . . . , x ℓ , let
Π := Π(x 1 , . . . , x ℓ ) := ℓ i=1 α i x i : 0 ≤ α i < 1 for 1 ≤ i ≤ ℓ ⊂ R n
be the corresponding fundamental parallelepiped. Then
H ⊂ {x 1 , . . . , x ℓ } ∪ Π.
The dual of Q is the rational polyhedral cone Q * := {x ∈ R n : x · q ≥ 0 for all q ∈ Q}, and we have (Q * ) * = Q. The Minkowski sum of two rational polyhedral cones Q 1 and Q 2 is the rational polyhedral cone Q 1 + Q 2 := {x + y : x ∈ Q 1 , y ∈ Q 2 }. We will need the following well-known fact:
(Q 1 ∩ Q 2 ) * = Q * 1 + Q * 2 .
2.3. Partial order. Throughout this paper, fix the following "component-wise" partial order on the i-chains
of ∆: write σ ≥ τ if σ(f ) ≥ τ (f ) for all faces f ∈ ∆ i . We say σ is nonnegative if σ ≥ 0, where 0 denotes the zero i-chain. Fix a similar partial order on R k : write v ≥ w if v i ≥ w i for all i; and v is nonnegative if v ≥ 0,
where 0 denotes the zero vector.
The dollar game
The i-th Laplacian of ∆, also know as the i-th up-down combinatorial Laplacian, is the mapping
L i := ∂ i+1 • ∂ t i+1 : C i (∆) → C i (∆)
. The isomorphism C i (∆) ≃ R fi identifies L i with an f i × f i matrix whose rows and columns are indexed by the i-faces.
Think of σ = f ∈∆i σ(f )f ∈ C i (∆) as a distribution of wealth to the i-faces of ∆: face f has σ(f ) dollars, interpreted as debt if σ(f ) is negative. A borrowing move at an i-face f redistributes wealth by replacing σ by the i-chain σ + L i f.
A lending move at f replaces σ by
σ − L i f.
The goal of the dollar game for σ is to bring all faces out of debt through a sequence of lending and borrowing moves. In detail, say σ is linearly equivalent to the i-chain σ ′ and write σ ∼ σ ′ if there exists v ∈ Z fi such that
(2) σ ′ = σ + L i v.
Call σ ′ effective if σ ′ ≥ 0. Then σ is winnable if there exists an effective σ ′ linearly equivalent to σ, and winning the dollar game determined by σ means finding such a σ ′ . The i-chain class group is
J i (∆) := C i (∆)/∼ = C i (∆)/ im L i .
So an i-chain σ is winnable if and only if there is an effective chain in its class [σ] ∈ J i (∆). The image of the i-th Laplacian is contained in the kernel of the i-th boundary mapping, which allows us to define the i-th critical group of ∆ introduced by Duval, Klivans, and Martin in [8]:
K i (∆) := ker ∂ i / im L i .
Choosing a splitting ρ : im ∂ i → C i (∆) of the exact sequence of free abelian groups
0 → ker ∂ i → C i (∆) → im ∂ i → 0
gives a corresponding isomorphism
J i (∆) → K i (∆) ⊕ im ∂ i (3) [σ] → ([σ − ρ(σ)], ∂ i (σ)).
The torsion part of J i (∆) is thus the torsion part of the critical group, T(K i (∆)), (which, itself, is sometimes called the critical group of ∆ (e.g., in [9])). There is a natural surjection K i (∆) → H i (∆) which is an isomorphism when restricted to the free parts of each group (Corollary 14). Example 1. Figure 1 illustrates an instance of the dollar game determined by a 1-chain σ on the simplicial complex with two facets: 123 and 234. Calling the winning chain on the right σ ′ , Equation (2) in this case takes the form
1 0 0 1 0 = −1 2 −3 2 −1 + 1 −1 1 0 0 −1 1 −1 0 0 1 −1 2 −1 1 0 0 −1 1 −1 0 0 1 −1 1 0 −1 1 0 0
13 23 24 34
σ ′ = σ + L 1 v.
Note that in moving from σ to σ ′ , money has been introduced from nowhere: the net amount in σ is −$1, while in σ ′ it is $2. While the simplicial dollar game does not conserve the net amount of money, other quantities are conserved, and we will discuss this at length starting in the next section. For now, as an example, it is easy to check that the sum of the amount of money on just the edges 12 and 13 is conserved under lending and borrowing moves. Thus, for instance, if we change the amount of money on 12 in σ from −$1 to −$3, the resulting game could never be won. And that statement would continue to hold no matter how much money we added to the edges 23, 24, and 34.
Example 2.
Here we show that winnability depends on the orientation of the simplicial complex. Figure 2 depicts two dollar games on the 2-simplex (the simplicial complex with the single facet 123). The first can be won by lending at the edge 13. The second is not winnable. To see this, note that the sum of the 13 and 23 components of a 1-chain on this complex-which is −$2 for the second game-is invariant under lending and borrowing moves. So one of these games is winnable and the other is not, yet they are the same up to a relabeling of the vertices (which amounts to a change in orientation). Example 3 (Graphs). Let ∆ = G be a connected, undirected graph as in the introduction. In that case, the dollar game for 0-chains on ∆ we just defined is the same as the dollar game for graphs from [2]. If the vertices of G are v i = i for i = 1, . . . , n, then the 0-th Laplacian is the usual discrete Laplacian for a graph:
1 2 3 −$1 $1 −$1 1 2 3 $1 −$1 −$1L 0 = diag(deg G (v 1 ), . . . , deg G (v n )) − A,
the difference of the diagonal matrix of vertex degrees and the adjacency matrix of G. The 0-chain class group and 0-th critical group are the Picard group and Jacobian group, respectively, described in the introduction: J 0 (∆) = Pic(G) and K 0 (∆) = Jac(G). Isomorphism (3) specializes to the usual isomorphism (1) for graphs.
Degree
The naive way of generalizing the degree of a divisor on a graph to the degree of an i-chain on a simplicial complex ∆, by simply summing up the coefficients of the i-faces, fails to retain many of the useful properties of the graph-theoretic degree. Under this naive definition of degree, as shown in Example 1, linearly equivalent ichains can fail to have the same degree, i-chains with negative degree can be winnable, and for a fixed complex, there can exist i-chains of arbitrarily large degree that are unwinnable. This section will introduce a better generalization of degree, avoiding these problems. To summarize the rest of this section: Theorem 13 shows that the group of i-chains of degree zero modulo firing rules is exactly the torsion part of the i-th critical group, as it is in the usual case of connected graphs. Our main result is Theorem 18, which states that i-chains of large enough degree are winnable. Unlike for graphs, it turns out that all i-chains of a given degree may be winnable even though there exists an i-chain of larger degree that is not (cf. Example 37). Corollary 20 says this will not occur if the Hilbert basis H i consists of 0-1 vectors.
For divisors on a graph, the degree function, deg : ZV → Z, is a linear function with the following two properties:
invariance under linear equivalence:
D ∼ D ′ ⇒ deg(D) = deg(D ′ ), nonnegativity on effective divisors: E ≥ 0 ⇒ deg(E) ≥ 0.
To generalize the notion of degree to higher dimensions, for each i, we look for a linear function deg : C i (∆) → Z with the above two properties. Any such linear function can be represented by σ → σ, σ ′ for a fixed
σ ′ ∈ C i (∆), where σ, σ ′ := f ∈∆i σ(f )σ ′ (f ).
To have invariance under linear equivalence, σ ′ must lie in the kernel of L i . For the function to be nonnegative on effective chains, σ ′ must itself be effective. Thus, an integer-valued linear function has our two desired properties if and only if it is expressible as the inner product with an effective i-chain in ker L i . But no particular one of these functions stands out as a preferred choice. Instead, we will take our generalization to contain the information of the output of all such functions, as we now describe.
The set C := v ∈ R fi : L i v ≥ 0 and v ≥ 0 is a pointed, rational, polyhedral cone. Therefore, its set of integer points, C ∩ Z fi , has a unique Hilbert basis H ( [14], [18]). This means that C ∩ Z fi is exactly the set of nonnegative integer linear combinations of H, and H is the smallest subset of C ∩ Z fi with this property. We can now give our definition of degree:
Definition 4. Let i ∈ Z. The i-th nonnegative kernel for ∆ is the monoid ker + L i := {σ ∈ ker L i : σ(f ) ≥ 0 for all f ∈ ∆ i } .
Fix an ordering
H i = H i (∆) = (h 1 , . . . , h ℓi )
for the elements of the Hilbert basis for ker
+ L i . The degree of σ ∈ C i (∆) is deg(σ) := deg i (σ) := (σ · h 1 , . . . , σ · h ℓi ) where σ · h j := f ∈∆i σ(f )h j (f ).
Remark 5. Another possible definition for the degree function is to replace H i in the definition with a list of only those elements of the Hilbert basis that are rays of the cone L + i ⊗ R. Denoting this variant of the definition of degree by rdeg, we have
deg(σ) ≥ deg(σ ′ ) ⇐⇒ rdeg(σ) ≥ rdeg(σ ′ ) for σ, σ ′ ∈ C i (∆)
. This means that all our results relating winnability of the dollar game to the degree of a chain will hold using either definition. One advantage of rdeg over deg is that it is easier to compute.
For each i, our definition of degree is a linear function into Z ℓi , where ℓ i is the number of elements in H i , and satisfies the two essential properties described earlier: invariance under linear equivalence is shown below, and nonnegativity on effective chains is obvious. It also specializes to the usual definition of degree in the case of a connected graph, as the Hilbert basis in that case is the sum of all of the vertices of the graph. Proposition 6. The degree of an i-chain depends only on its linear equivalence class.
Proof. It suffices to show that every element of im L i has degree zero. If τ ∈ ker L i and σ ∈ C i (∆), then
τ, L i σ = L t i τ, σ = L i τ, σ = 0, since L i is symmetric. In particular, τ, L i σ = 0 for all τ ∈ ker + L i . Corollary 7. If an i-chain σ is winnable, then deg(σ) ≥ 0.
Proof. If σ is winnable, then σ ∼ τ for some τ ≥ 0. Then deg(σ) = deg(τ ), and since each element of the Hilbert basis H i (∆) has nonnegative coefficients, deg(τ ) ≥ 0.
Remark 8. Using (4), below, the proof of Proposition 6 is easily modified to show that every element of im ∂ i+1 has degree zero. Thus, we get the stronger result that degree is a homology invariant.
Definition 9. A vector δ ∈ Z |Hi| is a realizable i-degree if there exists an i-chain σ such that deg(σ) = δ.
It is typically the case that not all degrees are realizable. For instance, consider the 3-simplex with single facet 1234. In this case, the Hilbert basis for ker + L 2 , computed by Sage ( [20]), is {123 + 124, 123 + 234, 134 + 124, 134 + 234}.
Ordering these elements as listed, it is easy to check that there are no 2-chains of degree (0, 0, 0, 1).
In general, the set of realizable i-degrees forms an additive monoid M i (∆), and Proposition 6 says that
the i-class group J i (∆) is graded by M i . Given δ ∈ M i (∆), let J δ i (∆) denote the δ-th graded part of J i (∆)
. Then there is a faithful action of the group J 0 i (∆) on J δ i (∆) given by addition of i-chains.
4.1. The group of chain classes of degree zero. Our next goal is Theorem 13, identifying the group of degree zero i-chains modulo firing rules with the torsion part of the critical group K i (∆), and thus generalizing a well-known result from the divisor theory of graphs (cf. Example 16). Letting K = Z, Q, or R, we use the standard notation X ⊥ = {y ∈ K : x · y = 0 for all x ∈ X} for the perpendicular space for a subset X ⊆ K n . By standard linear algebra,
(4) ker L i = ker ∂ i+1 ∂ t i+1 = ker ∂ t i+1 .
Using the chain property of boundary maps, we identify a useful subset of the kernel:
im ∂ t i ⊆ ker ∂ t i+1 = ker L i . If f is an (i − 1)-face of ∆, the element ∂ t i (f ) is called the star of f ; it is a signed sum of the faces radiating from f . If f = v 0 · · · v i−1 , then each element in the support of its star has the form v 0 · · · v k vv k+1 · · · v i−1 for some vertex v. The set of stars generates im ∂ t i . Lemma 10. For each i, there is a strictly positive element τ ∈ ker L i , i.e., such that τ (f ) > 0 for all f ∈ ∆ i .
Proof. For the sake of contradiction, assume no such element τ exists. Then for every σ ∈ ker L i , let m σ denote the least (in lexicographic ordering) i-face such that σ(m) ≤ 0. Choose a σ ∈ ker L i with maximal m σ . Say m := m σ = v 0 · · · v i , and consider the star S := ∂ t i (v 1 · · · v i ). The coefficient of m in S is 1, and if m 0 is an i-face such that m 0 < m, then m 0 begins with a vertex v smaller than v 1 , meaning one of two cases occurs: either m 0 = vv 1 · · · v i , in which case the coefficient of m 0 in S is 1, or m 0 does not contain v 1 · · · v i as a subface, and the coefficient of m 0 in S is 0. Either way, if m 0 < m, then the coefficient of m 0 in S is nonnegative. Now consider σ ′ := σ + (1 − σ(m))S. Then σ ′ ∈ ker L i , and σ ′ (f ) > 0 for all faces f ≤ m, contradicting the maximality of m. So our assumption must be false.
The following is an immediate consequence: Corollary 11. If σ is an effective i-chain and deg(σ) = 0, then σ = 0.
Corollary 12. For each i, the Z-span of ker + L i is ker L i . Hence, (ker + L i ) ⊥ = (ker L i ) ⊥ = (ker ∂ t i+1 ) ⊥ . Proof.
Take a strictly positive element τ ∈ ker L i that is primitive, i.e., it is not an integer multiple of any other element. We can then complete {τ } to a basis {τ, σ 1 , . . . , σ k } for ker L i . (To see this, consider the exact sequence 0 → Zτ → Z n → Z n /Zτ → 0.
Since Z n /Zτ is torsion-free, the sequence splits.) Then, for each nonzero N ∈ Z, the set
{τ, σ 1 + N τ, . . . , σ k + N τ }
is still a basis for ker L i . By taking N ≫ 0, this basis will consist solely of elements ker + i L i .
Theorem 13. For each i, the group of i-chains of degree zero modulo firing rules is isomorphic to the torsion part of the i-th critical group of ∆:
(ker L i ) ⊥ / im(L i ) = T(K i (∆)). Proof. To see that im L i ⊆ (ker L i ) ⊥ , let σ ∈ Z∆ i and τ ∈ ker L i = ker ∂ t i+1 . Then τ, L i σ = τ, ∂ i+1 ∂ t i+1 σ = ∂ t i+1 τ, ∂ t i+1 σ = 0, ∂ t i+1 σ = 0. We also have (im ∂ t i ) ⊥ ⊆ ker ∂ i . To see this, take σ ∈ (im ∂ t i ) ⊥ and τ ∈ Z∆ i−1 . Then 0 = σ, ∂ t i τ = ∂ i σ, τ . Since τ is arbitrary, ∂ i σ = 0. Next, im ∂ t i ⊆ ker ∂ t i+1 ⇒ (ker L i ) ⊥ = (ker ∂ t i+1 ) ⊥ ⊆ (im ∂ t i ) ⊥ ⊆ ker ∂ i . Hence, (ker L i ) ⊥ / im L i ⊆ ker ∂ i / im L i =: K i (∆). Since dim Q (ker L i ) ⊥ = dim Q (im L i ), the group (ker L i ) ⊥ / im L i is finite, and hence torsion. So it is a subset of T(K i (∆)).
To show the opposite inclusion, let σ ∈ ker ∂ i , and suppose there exists a positive integer k such that kσ ∈ im L i . Say kσ = L i τ , and let ν ∈ ker L i = ker ∂ t i+1 . Then k ν, σ = ν, kσ = ν, L i τ = ∂ t i+1 ν, ∂ t i+1 τ = 0. Therefore, ν, σ = 0. So each torsion element of K i (∆) is an element of (ker L i ) ⊥ / im(L i ).
Corollary 14. The natural surjection
K i (∆) → H i (∆)
is an isomorphism when restricted to the free parts of K i (∆) and H i (∆) and a surjection when restricted to the torsion parts.
Proof. Consider the exact sequence
0 → im ∂ i+1 / im L i → K i (∆) → H i (∆) → 0. We have im L i ⊆ im ∂ i+1 ⊆ (ker L i ) ⊥ ,
where the second inclusion follows by an argument similar to that given for im L i at the beginning of the proof of Theorem 13. From Theorem 13, it follows that im ∂ i+1 / im L i is finite. Tensoring the sequence by Q then gives the result about the free parts, and since the torsion functor T( · ) is left-exact, there is a surjection for the torsion parts.
Remark 15. Let δ be a realizable i-degree, and fix any σ ∈ C i (∆) such that deg(σ) = δ. Then there is a bijection of chain class groups J 0 i (∆) → J δ i (∆) given by ω → ω +σ for each ω ∈ J 0 i (∆). By Theorem 13, the group J 0 i (∆) is the torsion part of the (finitely-generated abelian group) K 0 (∆) and hence is finite. Thus, there are only finitely many chains to check to determine whether all chains of a given degree are winnable.
Example 16 (Graphs). Consider again how our structures generalize those on graphs. In the case d = 1, the simplicial complex ∆ is determined by its 1-skeleton, a graph G. We have two notions of degree for an element σ ∈ C i (∆): as a 0-chain on ∆, there is the degree determined by dot products with elements of the Hilbert basis H 0 ; and as a divisor on a graph, there is the usual degree given by ∂ 0 (σ) = v∈V σ(v). Call the former the ∆-degree, deg(∆, σ), of σ, and call the latter the G-degree, deg(G, σ).
By definition, the Picard group Pic(G) is the set of 0-chains modulo the image of L 0 , and hence, coincides with the 0-th class group J 0 . Now, Pic(G) is graded by G-degree, and its G-degree zero part is by definition the Jacobian group Jac(G). Hence,
Jac(G) = K 0 (∆) = ker ∂ 0 / im L 0 .
On the other hand, J 0 (∆) is graded by ∆-degree. While Pic(G) = J 0 (∆) as groups, in the case where G is not connected, their gradings differ.
If G is connected or, equivalently,β 0 (∆) = 0, the Hilbert basis H 0 consists of the all-ones vector 1, and deg(∆, σ) = σ · 1 = ∂ 0 (σ) = deg(G, σ). Thus, Pic(G) = J 0 as graded groups, and Jac(G) is the collection of ∆-degree zero 1-chains. As is well-known, the matrix-tree theorem implies that | Jac(G)| is the number of spanning trees of G. So Jac(G) is finite, hence torsion, in agreement with Theorem 13. Now consider the case where G is not connected. To fix ideas, say G is the graph consisting of the disjoint union of two triangles, one with vertices 1, 2, 3 and the other with vertices 4, 5, 6. In this case,
Jac(G) = K 0 (∆) ≃ Z/3Z ⊕ Z/3Z ⊕ Z.
The Hilbert basis H 0 consists of two elements h 1 = (1, 1, 1, 0, 0, 0) and h 2 = (0, 0, 0, 1, 1, 1). So if σ ∈ C 0 (∆), 0, 0, −1, 0, 0), then deg(G, σ) = 0 while deg(∆, σ) = (1, −1) = (0, 0). The ∆degree zero part of J 0 is isomorphic to the direct sum of two copies of the Jacobian group of a triangle, i.e., to Z/3Z ⊕ Z/3Z.
then deg(G, σ) = 6 i=1 σ i and deg(∆, σ) = ( 3 i=1 σ i , 6 i=4 σ i ). For instance, if σ = 1 − 4 = (1,
4.2.
Degree/winnability condition. We now show that if the degree of an i-chain is sufficiently large, it is winnable. The proof requires the following lemma:
Lemma 17. For each integer i, there exists a finite set of i-chains P i such that any σ ∈ C i (∆) with deg(σ) ≥ 0 can be written as σ = ζ + τ + φ where deg(ζ) = 0, τ is effective, and φ ∈ P i . Proof. Having ordered ∆ i lexicographically, we make the identification
C i (∆, R) ≃ R fi where f i := |C i (∆)|. Let L R i := L i ⊗ R : R fi → R fi ,
and let O + be the nonnegative orthant of R fi . Using dual cones, the fact that σ has degree at least 0 can be expressed as follows:
σ ∈ ((ker L R i ) ∩ O + ) * ∩ Z fi = ((ker L R i ) * + (O + ) * ) ∩ Z fi = ((ker L R i ) * + O + ) ∩ Z fi .
We can split both (ker L R i ) * and O + into the Minkowski sum of the integer points they contain and their respective fundamental parallelepipeds P 1 and P 2 (with respect to any choice of integral generators), to get
((ker L R i ) * + O + ) ∩ Z fi = (((ker L R i ) * ∩ Z fi + P 1 ) + (O + ∩ Z fi + P 2 )) ∩ Z fi = (ker L R i ) * ∩ Z fi + O + ∩ Z fi + (P 1 + P 2 ) ∩ Z fi . Since ker L R i is a linear space, (ker L R i ) * = (ker L R i ) ⊥ .
Hence, (ker L R i ) * ∩ Z fi is the set of all i-chains of degree 0, and O + ∩ Z fi is the set of effective i-chains. So letting P i = (P 1 + P 2 ) ∩ Z fi , which is a finite set since P 1 and P 2 are bounded, completes the proof.
Theorem 18. If the degree of a chain is sufficiently large, then it is winnable: for each integer i there exists a realizable i-degree δ ∈ Z |Hi| such that for all σ ∈ C i (∆), if deg(σ) ≥ δ, then σ is winnable.
Proof. Let S be a set of representatives for T(K i (∆)), and let P i be as in Lemma 17. By finiteness of S and P i , there exists an i-chain ω such that the chain ω + γ + φ is effective for all γ ∈ S and φ ∈ P i . Set δ = deg(ω), and let σ be an i-chain such that deg(σ) ≥ δ. Then deg(σ − ω) ≥ 0, so by Lemma 17 we can write
σ − ω = ζ + τ + φ
where deg(ζ) = 0, τ is effective, and φ ∈ P i . Since deg(ζ) = 0, we have ζ ∈ (ker + L i ) ⊥ = (ker L i ) ⊥ by Corollary 12. So by Theorem 13, there exists γ ∈ S such that ζ ∼ γ. It follows that σ is winnable:
σ ∼ (ω + γ + φ) + τ ≥ 0.
Let W i be the set of all δ satisfying the conditions in Theorem 18. Then W i is partially ordered ( §2.3) and bounded below by 0 ∈ Z |Hi| . So it is natural to consider its set of minimal elements, min(W i ). To see that min(W i ) is finite, consider the polynomial ideal generated by the monomials x δ := i x δi i as δ varies over W i . By the Hilbert basis theorem, this ideal is finitely generated, and its minimal set of generators corresponds with min(W i ). See Example 27 for the computation of min(W 1 ) for a hollow tetrahedron.
Intuition coming from the dollar game on graphs may not apply to W i on a general simplicial complex. For instance, as in Example 27, there are typically infinitely many nonnegative realizable degrees that are not in W i . Further, as will be demonstrated in Example 37, it may be the case that all i-chains of a particular realizable degree δ are winnable even though there exists an unwinnable i-chain σ with deg(σ) ≥ δ.
To finish this section, we describe conditions under which δ ∈ W i if and only if δ is realizable and all i-chains of degree exactly δ are winnable. Proof. Suppose the result is false, and let σ be a counterexample of minimal degree deg(σ) ≥ 0 (using the component-wise partial order defined in Section 2.3). Note that deg(σ) = 0. Using notation for dual cones from the proof of Lemma 17, we have
σ ∈ (ker L R i ∩ O + ) * = (ker L R i ) * + O + = (ker L R i ) ⊥ + O + .
The last equality follows because ker L R i is a linear space. Therefore, over R,
we have σ = ν + τ where ν ∈ (ker L R i ) ⊥ and τ = f ∈∆i τ (f )f with τ (f ) ≥ 0 for all f ∈ ∆ i . So τ · h = σ · h for all h ∈ H i ,(σ − f ′ ) · h = (τ − f ′ ) · h = f ∈∆i τ (f )h(f ) − h(f ′ ) = f =f ′ τ (f )h(f ) + (τ (f ′ ) − 1)h(f ′ ) > −1. Since (σ − f ′ ) · h ∈ Z for all h ∈ H i , it follows that deg(σ − f ′ ) ≥ 0.
On the other hand, by Lemma 10, there exists some h ∈ H i such that h(f ′ ) > 0, and therefore deg(σ − f ′ ) is strictly smaller than deg(σ). By minimality, there exists an effective integral i-chain ρ with deg(ρ) = deg(σ − f ′ ). But then ρ + f ′ is an effective divisor of degree deg(σ), contradicting the fact that σ is a counterexample. Corollary 20. Suppose H i consists of 0-1 vectors and that there exists a realizable i-degree δ such that every i-chain of degree δ is winnable. Then every i-chain with degree at least δ is winnable.
Proof. Let σ ∈ C i (∆) with deg(σ) ≥ δ. By Corollary 19, there exists an effective chain τ ∈ C i (∆) of degree deg(σ) − δ. Since σ − τ has degree δ, by hypothesis it is linearly equivalent to an effective chain ρ. Therefore, σ ∼ τ + ρ ≥ 0, and σ is winnable.
Pseudomanifolds
In this section we take ∆ to be a d-dimensional orientable pseudomanifold. References for pseudomanifolds include [16] and [19]. To say that ∆ is a pseudomanifold means that it is (1) pure: each facet has dimension d;
(2) non-branching: each (d − 1)-face is a face of at most two facets; and (3) strongly connected: if σ and σ ′ are facets, there exists a sequence of facets σ 0 , . . . , σ k with σ 0 = σ and σ k = σ ′ such that each pair of consecutive facets σ i and σ i+1 share a (d − 1)-face.
The boundary ∂∆ of ∆ is the collection of (d − 1)-faces of ∆ that are faces of exactly one facet. Since ∆ is a pseudomanifold, it is a standard result that exactly one of the following must hold in relative homology: In our case, we are assuming that ∆ is an orientable pseudomanifold, which by definition means that (i) holds. It is then possible to orient the facets of ∆ so that the sum of their boundaries is supported on the boundary of ∆. Letting f (1) , . . . , f (m) ∈ C d (∆) be the facets of ∆, this means that for each i we can choose γ i ∈ ±f (i) and define γ = γ 1 + · · · + γ m so that ∂ d (γ) is supported on ∂∆. (In particular, if ∆ has no boundary, then ∂ d (γ) = 0.) We call the relative cycle γ a pseudomanifold orientation for ∆. Its class [γ] ∈ H d (∆, ∂∆) is a choice of generator for the top relative homology group. Recall that the simplicial complexes studied in this paper all come with a fixed underlying orientation as a simplicial complex, upon which the dollar game depends. The orientations of the facets γ i need not agree with those given by that fixed orientation. The proof of the following is in the appendix. It was proved in [8] for the case H d−1 (∆) = 0 and ∂∆ = ∅.
Proposition 21. Suppose ∆ is a d-dimensional orientable pseudomanifold. If ∂∆ = ∅,
K d−1 (∆) = H d−1 (∆)
and otherwise, if ∆ has no boundary,
K d−1 (∆) ≃ (Z/mZ) ⊕ H d−1 (∆)
where m = f d is the number of facets of ∆.
To define the degree of a (d − 1)-chain on a pseudomanifold ∆, we need to compute the Hilbert basis for ker + L d−1 . Our main goal for this section is a combinatorial description of this basis. We start by defining the γ-incidence graph Γ = Γ(∆, γ) as a directed graph whose vertices are the oriented facets {γ i }. If ∂∆ = ∅, let γ 0 := 0 ∈ C d (∆), and include it, too, as a vertex of Γ. The edges of Γ are in bijection with the codimension-one faces of ∆. To describe them, let σ be any (d − 1)-face and write ∂ t d (σ) = γ j − γ i for uniquely determined i and j. (If σ ∈ ∂∆, then one of i or j will be 0.) Let σ − := i and σ + := j. The directed edge corresponding to σ then starts at γ σ − and ends at γ σ + . See Figures 3 and 4 for examples.
Theorem 22 (Hilbert basis for an orientable pseudomanifold). Let ∆ be a pseudomanifold with pseudomanifold orientation γ. Then the Hilbert basis for the nonnegative kernel ker + L d−1 is the set of incidence vectors for the simple directed cycles of Γ(∆, γ).
Proof. Let τ = σ a σ σ ∈ C d−1 (∆) = 0. Then τ ∈ ker L d−1 = ker ∂ t d if and only if 0 = ∂ t d (τ ) = σ a σ (γ σ + − γ σ − ).
Requiring τ ∈ ker + L d−1 adds the restriction that a σ ≥ 0 for all σ, which is equivalent to saying that τ is a directed cycle in Γ. Then τ is simple if and only if it is not the sum of two other non-trivial directed cycles, which is exactly the requirement that τ belong to the Hilbert basis.
Corollary 23. Suppose δ is a realizable (d − 1)-degree on the orientable pseudomanifold ∆ of dimension d and that every (d − 1)-chain of degree δ is winnable. Then every (d − 1)-chain with degree at least δ is winnable.
Proof. The result follows immediately from Theorem 22 and Corollary 20. Example 25. Figure 4 shows a triangulated annulus ∆ in the plane and its γ-incidence graph for the counter-clockwise orientation, γ = 125 + 143 + 154 + 236 + 265 + 346.
The boundary is ∂∆ = 12, 13, 23, 45, 46, 56 . Since the boundary is nonempty, the γ-incidence graph includes the vertex * , representing 0 ∈ C d (∆). The Hilbert basis for ker + L 1 has ten elements, two of which are displayed below: Example 27 (Computing minimal winning degrees). Let ∆ be the hollow tetrahedron in Example 24, and use lexicographic ordering of the edges of ∆ to identify C 1 (∆) with Z 6 , as usual. For the purpose of computing degrees, we can order the elements of the Hilbert basis H 1 for ∆, computed in Example 24, as 1, 1, 0, 0, 0), h 2 = (0, 0, 1, 0, 1, 1), h 3 = (0, 1, 1, 1, 1, 0).
h 1 = (1,
By Theorem 18, there exists an effective 1-chain τ ∈ Z 6 such that every 1-chain of degree at least δ := deg(τ ) is winnable. In this example, we compute all minimal such δ (the set min(W i ), using earlier notation). We then exhibit an infinite family of nonnegative realizable 1-degrees that are not realizable by winnable 1-chains. Choose an effective τ ∈ C 1 (∆) = Z 6 with deg(τ ) = δ, and suppose that every 1-chain of degree at least δ is winnable. Let σ (0) , σ (1) , σ (2) , σ (3) be representatives for the elements of K 1 (∆) ≃ Z/4Z. Then the equivalence classes of 1-chains of degree δ in J 1 := C 1 (∆)/ im L 1 are τ + σ (i) for i = 0, . . . , 3 (cf. Remark 15). Each σ (i) has degree 0 by Theorem 13. By assumption τ + σ (i) is winnable, so working modulo im L 1 , we can choose the σ (i) so that each τ + σ (i) is effective. In order to minimize δ, we minimize τ .
First, suppose δ 1 = 0. Since τ is effective and τ ·h 1 = τ 1 +τ 2 +τ 3 = δ 1 = 0, it follows that τ 1 = τ 2 = τ 3 = 0. Using this, it similarly follows that σ
(i) 1 = σ (i) 2 = σ (i) 3
= 0 for i = 0, 1, 2, 3. Some linear algebra shows that K 1 (∆) is generated by (0, 0, 0, 1, −1, 1) and 1-chains in the image of the Laplacian which are 0 in the first three components are exactly those of the form (0, 0, 0, 4k, −4k, 4k) for some integer k. So up to reindexing, σ (i) = (0, 0, 0, i + 4k i , −i − 4k i , i + 4k i ) for some integers k i . Now consider the conditions on τ , besides τ ≥ 0, required to ensure each τ + σ (i) is effective. These are
τ 4 τ 5 τ 6 ≥ −i i −i + k i −4 4 −4
for some integer k i and for i = 0, . . . , 3. For i = 0, we take k i = 0 and see there is no additional condition imposed on τ ; for i = 1, either τ 5 ≥ 1 or both τ 4 and τ 6 are at least 3; for i = 2, either τ 5 ≥ 2 or both τ 4 and τ 6 are at least 2; and for i = 3, either τ 5 ≥ 3 or both τ 4 and τ 6 are at least 1. Thus, to minimize τ , there are eight cases to consider. In all of these, deg(τ ) ≥ (0, 3, 3).
Next, suppose δ 2 = 0. By a similar argument (or by symmetry, swapping vertex 1 with 4 and vertex 2 with 3), we find minimal τ have degree at least (3, 0, 3). Finally, suppose δ 3 = 0. In that case, τ 2 = τ 3 = τ 4 = τ 5 = 0 and σ to represent an element in K 1 (∆)-and hence be in the kernel of ∂ 1 -forces a = b = 0. That is not possible since the σ (i) are a full set of representatives for K 1 (∆). So we must have δ 3 ≥ 1.
Combining the above, we conclude δ is greater than or equal to one of (0, 3, 3), (3, 0, 3), or (1, 1, 1). In fact, these three degrees are minimal winning degrees for ∆ since there exist four effective 1-chains of each degree that are pairwise not linearly equivalent. We list these chains in the table below: degree δ representatives for J 1 (∆) (0, 3, 3) (0, 0, 0, 3, 0, 3), (0, 0, 0, 2, 1, 2), (0, 0, 0, 1, 2, 1), (0, 0, 0, 0, 3, 0) (3, 0, 3) (3, 0, 0, 3, 0, 0), (2, 1, 0, 2, 0, 0), (1, 2, 0, 1, 0, 0), (0, 3, 0, 0, 0, 0) (1, 1, 1) (1, 0, 0, 1, 0, 1), (1, 0, 0, 0, 1, 0), (0, 1, 0, 0, 0, 1), (0, 0, 1, 0, 0, 0) . On a graph, there are only finitely many nonnegative degrees realizable by unwinnable divisors. That is not usually the case for a general simplicial complex. For instance, on our current ∆, consider the family of 1-chains σ = (a, −b, b, 0, 0, 0) where a ≥ 0 and b > 0. We have deg(σ) = (a, b, 0) ≥ 0 = (0, 0, 0). Let τ be any effective 1-chain of degree (a, b, 0). Taking the dot product of τ with each h i , it follows that τ = (a, 0, 0, 0, 0, b), and thus σ − τ = (0, −b, b, 0, 0, −b). However, computing the Hermite normal form for L 1 , we see that im L 1 is spanned by (1, 0, −1, 3, −2, 3), (0, 1, −1, 1, −1, 2), and (0, 0, 0, 4, −4, 4). It is straightforward to check that σ − τ ∈ im L 1 , and hence σ ∼ τ . Hence, σ is not winnable.
Forests
It is well-known that the dollar game on a graph is winnable for all initial configurations of degree zero if and only if the graph is a tree (e.g., cf. [2]). In this section, that result is extended to higher dimensions. We first recall the basics of trees on simplicial complexes as developed by Duval, Klivans, and Martin in [8] and [9]. In [8], it is shown that under certain circumstances, each critical group is isomorphic to the cokernel of a certain submatrix of the corresponding Laplacian matrix called the reduced Laplacian. Theorem 30 generalizes that result by loosening the hypotheses.
Definition 28. A spanning i-forest of ∆ is an i-dimensional subcomplex Υ ⊆ ∆ with Skel i−1 (Υ) = Skel i−1 (∆) and satisfying the three conditions
(1) H i (Υ) = 0; (2)β i−1 (Υ) =β i−1 (∆); (3) f i (Υ) = f i (∆) −β i (Skel i (∆)).
In the case whereβ i−1 (∆) = 0, a spanning i-forest is called a spanning i-tree. The complex ∆ is a forest if it is a spanning forest of itself, i.e., if H d (∆) = 0. If, in addition,β d−1 (∆) = 0, then ∆ is a tree.
Remarks. Let Υ be an i-dimensional subcomplex of ∆ sharing the same (i − 1)-skeleton.
(1) For a graph G, the above definition says that a (one-dimensional) spanning forest contains all of the vertices of G and: (i) has no cycles, (ii) has the same number of components as G, and (iii) has m − c edges, where m is the number of edges and c is the number of components of G. (2) The condition H i (Υ) = 0 is equivalent to the elements of the set
A := {∂ Υ,i (f ) : f ∈ Υ i }
being linearly independent (over Z or, equivalently, over Q).
(3) Since Υ and ∆ have the same (i − 1)-skeleton, ∂ ∆,i−1 = ∂ Υ,i−1 , and hence,β i−1 (Υ) =β i−1 (∆) is equivalent to rank im ∂ Υ,i = rank im ∂ ∆,i . (4) It follows from the previous two remarks that Υ is a spanning i-forest if and only if A, defined above, is a basis for im ∂ ∆,i over Q, i.e, the columns of the matrix ∂ ∆,i corresponding to the i-faces of Υ are a Q-basis for the column space of ∂ ∆,i . In particular, spanning i-forests always exist. (5) Since ∂ ∆,j = ∂ Skel i (∆),j for all j ≤ i, it follows the j-th reduced homology groups, Betti numbers, and critical groups for ∆ and for Skel i (∆) are the same for all j < i. In particular, this implies that the j-forests (resp., j-trees) of ∆ are the same as those for Skel i (∆) for all j ≤ i. The proof of the following is in the appendix. It generalizes a result in [8], where it is proved with the assumptions that ∆ is pure, thatβ i (∆) = 0 for all i < d, and that H i−1 (Υ) = 0.
Theorem 30. Suppose that Υ is an i-dimensional spanning forest of ∆ such that H i−1 (Υ) = H i−1 (∆). Let Θ := ∆ i \ Υ i . Define the reduced LaplacianL of ∆ with respect to Υ to be the square submatrix of L i consisting of the rows and columns indexed by Θ. Then there is an isomorphism Proof. Suppose that τ i (∆) = 1. Then ∆ possesses a unique spanning i-forest Υ, and H i−1 (Υ) is torsion-free. Considering ∂ i as a matrix, it follows that its set of columns has a unique maximal linearly independent subset: those columns corresponding to the faces of Υ. Since the columns of ∂ i are all nonzero, it must be that the columns corresponding to Υ are the only columns, i.e., f i (Υ) = f i (∆), and hence Υ = Skel i (∆). It follows that H i−1 (∆) = H i−1 (Υ) and hence is torsion-free. Now suppose Skel i (∆) is a spanning i-forest and let Υ ⊆ ∆ be any spanning i-forest. Since H i (Skel i (∆)) = 0, it follows from condition 3 of Definition 28 that
K i (∆) ∼ − → ZΘ/ imLf i (Υ) = f i (∆) −β i (Skel i (∆)) = f i (∆).
Hence, Υ = Skel i (∆). So Skel i (∆) is the unique spanning i-forest of ∆. Further, if H i−1 (Skel i (∆)) is torsion free, then τ i (∆) = |T( H i−1 (∆))| 2 = 1. Remark 35. As discussed in the introduction, Corollary 34 generalizes the result that all divisors of degree 0 on a graph are winnable if and only if the graph is a tree. However, for graphs, Corollary 34 says that all divisors of degree 0 on a forest are winnable. This apparent contradiction is resolved by the fact that for unconnected graphs, our simplicial notion of degree differs from the usual one for graphs. See Example 16.
Example 36. Simply being a spanning tree is not enough to guarantee winnability of all degree 0 divisors. Figure 6 illustrates a two-dimensional complex P which is a triangulation of the real projective plane. We have H 0 (P ) = H 2 (P ) = 0, and H 1 (P ) ≈ Z/2Z. Therefore, P is a spanning tree with tree number τ 2 (P ) = 4. The cycle σ := 12 + 23 − 13 is a 1-chain in the image of ∂ 2 and hence, by Remark 8, has degree 0. As argued in the first line of the proof of Corollary 34, if σ were winnable, it would be linearly equivalent to the zero chain. We used Sage ( [20]) to find that K 1 (P ) ≈ Z/2Z × Z/2Z and σ / ∈ im L 1 . Hence, 2σ is winnable, but σ is not. Example 37. This example demonstrates that all i-chains of degree 0 of a complex can be winnable, even though there are unwinnable i-chains of nonnegative degree. Let ∆ be the three-dimensional simplicial complex with facets (1,2,3,4), (1,2,3,6), (1,2,3,7), (1,2,4,6), (1,2,5,7), (1,3,4,7), (1,3,5,7), (1,4,5,6), (1,4,5,7), (1,4,6,7), (2,3,4,7), (2, 3, 5, 6), (2, 3, 5, 7), (2,4,5,6), (3,4,5,7), (3,5,6,7), (4, 5, 6, 7).
We haveH 3 (∆) ∼ = 0 andH 2 (∆) ∼ = Z; so by Proposition 32, it follows that ∆ is a forest with τ 3 (∆) = 1.
Corollary 34 then implies that all 2-chains on ∆ of degree 0 are winnable. The Hilbert basis of ker + L 2 for ∆ has 445 elements. 2 Let A be the matrix whose rows are these Hilbert basis elements. Each 2-face of ∆ may be considered as a chain and, thus, has a degree. These degrees form the 33 columns of A. It follows that the degrees of all effective 2-chains are precisely the nonnegative integer linear combinations of the columns of A. The Hilbert basis for the polyhedral cone generated by the columns of A consists of the columns of A and one other element δ. By the characterization of the Hilbert basis, δ cannot be realized by any effective two-chain, but using linear algebra it is possible to find non-effective two-chains of degree δ, one of which is (1, 2, 3) − (1, 2, 7) + (1, 3, 5) + (1, 3, 6) + (1, 4, 6) + (1, 6, 7) + (2, 4, 5).
Thus, the above 2-chain is unwinnable but has nonnegative degree.
6.1. Spanning trees acyclic in codimension one.
Definition 38. For each integer i, let
Λ i (∆) = Span Z ≥0 {∂ i+1 (f ) : f ∈ ∆ i+1 } ⊂ C i (∆) := Z∆ i . and X i (∆) := {σ ∈ C i (∆) : ∂ i (σ) ∈ Λ i−1 (∆)} .
The above definition was introduced by S. Corry and L. Keenan ( [6]). Since Λ −1 (∆) = Z ≥0 and, therefore, X 0 (∆) = {σ ∈ C 0 (∆) : ∂ 0 (σ) ≥ 0}, they regarded the sets X i (∆) as generalizing the notion of divisors of nonnegative degree on a graph and explored their relation to the winnability of the dollar game. They conjectured the equivalence of (1) and (2) in the following proposition and proved it in the case i = 2 on a simplicial surface.
Proposition 39. The following are equivalent for i ≤ d:
(1) Every σ ∈ X i−1 (∆) is winnable. (2) K i−1 (∆) = 0. (3) Skel i (∆) is a spanning i-tree of ∆ and H i−1 (∆) = 0.
In particular, when i = d, the three conditions are equivalent to ∆ being a tree, acyclic in codimension one.
Proof. We first note that since ∆ has the standard orientation, the only nonnegative element of ker ∂ i−1 is 0. To see this, suppose σ = f ∈∆i a f f = 0 with a f ≥ 0 for all f . Let v 0 · · · v i be the lexicographically largest element in the support of σ (with v 0 < · · · < v i ). For each v ∈ V such that v ≤ v 0 , let g v := vv 1 · · · v i . Then the coefficient of v 1 · · · v i in ∂ i−1 (σ) is v∈V a gv > 0. Hence, σ / ∈ ker ∂ i−1 . We will need this fact later in the proof.
Letting E denote the set of effective (i − 1)-chains, we can write X i−1 (∆) = E + ker ∂ i−1 . Thus, (1) is equivalent to E + ker ∂ i−1 ⊆ E + im L i−1 , which in turn is equivalent to
(1) ′ E + ker ∂ i−1 = E + im L i−1 since im L i−1 ⊆ ker ∂ i−1 . Now, if K i−1 (∆) = 0, then im L i−1 = ker ∂ i−1 ,
and (1) ′ holds. Conversely, suppose (1) ′ holds, and let σ ∈ ker ∂ i−1 . By (1) ′ , there exist τ ∈ E and φ ∈ im
L i−1 ⊆ ker ∂ i−1 such that σ = τ + φ. But then σ − φ ∈ E ∩ ker ∂ i−1 = {0}, which implies σ = φ ∈ im L i−1 .
It follows that K i−1 (∆) = 0. Therefore, (1) is equivalent to (2). We now prove the equivalence of (2) and (3) using Proposition 32. If K i−1 (∆) = 0, then 1 = |T(K i−1 )| = τ i (∆) by Theorem 33. Further, the natural surjection
K i−1 (∆) → H i−1 (∆) implies H i−1 (∆) = 0. Hence, Skel i (∆)
is a spanning i-tree of ∆. Conversely, suppose that Skel i (∆) is a spanning i-tree and H i−1 (∆) = 0. Then τ i (Skel i (∆)) = 1, which implies that K i−1 (∆) is free by Theorem 33. However, the free part of K i−1 (∆) is the same as the free part of H i−1 (∆) by Corollary 14. Therefore, K i−1 (∆) = 0.
Example 40. This example shows that condition H i−1 (∆) = 0 in part (3) of Proposition 39 is necessary. Consider the simplicial complex ∆ pictured in Figure 7. By inspection, H 2 (∆) = 0 and H 1 (∆) ≃ Z = 0. So the complex is a forest but not a tree.
One may compute directly that K 1 (∆) ≃ Z or argue as follows. By Proposition 32, we have τ 2 (∆) = 1. By Theorem 33, it follows that |T (K 1 (∆))| = 1. Then Corollary 14 says
Further work
There is still much to be learned about winnability of the dollar game on a simplicial complex. Here, we will present three general open areas of investigation: computation of minimal winning degrees, algorithms for determining winnability, and generalization of the rank function.
Theorem 18 says there exists a realizable degree δ such that all i-chains of degree at least δ are winnable. Call any minimal such δ a minimal winning degree for i-chains on ∆. For divisors on connected graphs, there is one minimal winning degree, g = |E| − |V | + 1. We know of no such formulas in higher dimensions.
(1) Is there a simple combinatorial description of the set of minimal winning degrees for the i-chains of a simplicial complex? (2) It would be nice to compute minimal winning degrees for a class of simplicial complexes. For example, what are the minimal winning degrees for (d − 2)-chains on the d-dimensional simplex? On a graph, there are three standard methods of determining whether the dollar game is winnable, and if it is winnable, finding a sequence of moves leading to a winning position. One of these is a greedy algorithm. It proceeds as follows:
(i) Check if the divisor is effective. If so, the divisor is winnable.
(ii) Modify the divisor by borrowing at any vertex with a negative amount of dollars, prioritizing vertices that have borrowed earlier in the algorithm. (iii) If all vertices have been forced to borrow, the original divisor is unwinnable. Otherwise, return to step (i). The proof of the validity of this greedy algorithm (cf. [7, Section 3.1]) relies on two main facts. First, a vertex cannot be brought out of debt by only borrowing at other vertices, and second, the only way to leave a divisor unchanged through a series of borrowing moves is to borrow at every vertex an equal number of times. Neither of these two facts remains true for chains on a simplicial complex, so an immediate translation of the greedy algorithm fails in higher dimensions. The ideas in this paper suggest possible fixes for the second fact. For instance, one might attempt to modify the algorithm to avoid borrowing at any combination of vertices forming an element of the Hilbert basis H i (∆) of the nonnegative kernel ker + i L i . Our attempts in this direction have failed due to the first fact. So we propose the question: (3) Can the greedy algorithm for the dollar game on graphs be generalized to one for simplicial complexes?
Another method for determining winnability of the dollar game on a graph is through q-reduction of a divisor ( [2], [3]). In this method, given a divisor, one computes a linearly equivalent standard form for the divisor with respect to a chosen vertex q. The game is winnable if and only if q is out of debt in this standard form. Knowing whether q-reduction generalizes to chains on a simplicial complex would be of general interest to the chip-firing community ([1, Problem 17], [12]). Perhaps the methods of [17] could be employed. In that work, q-reduction is interpreted as an instance of Gröbner reduction of the lattice ideal of the graph Laplacian. We formulate the general question in the context of the dollar game:
(4) Can one define an efficiently computable standard representative of the equivalence class of a chain on a simplicial complex which is effective if and only if the chain is winnable?
A third way of computing winnability for graphs is to determine whether a certain simplex, defined using the columns of the Laplacian matrix, contains integer points (cf. [7, Section 2.3] or [5]). This method easily extends to the dollar game on a simplicial complex, and it is the one we use in our own computations. However, the general problem of determining whether a simplex has integer points is NP-hard unless the dimension is fixed. Even so, for graphs, q-reduction provides a method of determining winnability of a divisor that is polynomial in the size of the divisor and the size of the graph ( [3]).
(5) Is there any efficient algorithm for determining winnability of the dollar game on a simplicial complex?
The rank function, discussed in the introduction, is a measure of the robustness of winnability of a divisor on a graph. As noted in [2, Remark 1.13], for a divisor D on an algebraic curve, the same definition for rank would give r(D) = ℓ(D) − 1, where ℓ(D) is the dimension of the vector space of global sections of the line bundle associated with D, appearing in the standard formulation of the Riemann-Roch theorem for curves. The Riemann-Roch theorem for divisors D on an algebraic surface can be thought of as a refinement of a lower bound on ℓ(D) in terms of data associated with D and the structure of the surface (by dropping the superabundance term). This motivates the following: Proof of Proposition 21. The projection mapping from the critical group to the relative homology group in codimension one gives the short exact sequence (6) 0
→ im ∂ d / im L d−1 → K d−1 (∆) → H d−1 (∆) → 0.
Let γ = γ 1 + · · · + γ m be as in the statement of Lemma 41, and first consider the case where ∂∆ = ∅.
Reasoning as in the beginning of the lemma, we still have
X := Span Z {∂ d (γ i ) − ∂(γ j ) : 1 ≤ i, j ≤ m} ⊆ im L d−1 .
Given any f ∈ ∂∆, there exists a unique γ k whose boundary contains f in its support. Hence, L d−1 (f ) = ±∂ d (γ k ). Since im L d−1 contains X and ∂ d (γ k ), it contains all of the im ∂ d (γ i ). So im L d−1 = im ∂ d , and hence, K d−1 (∆) = H d−1 (∆), as claimed. Now consider the case where ∂∆ = ∅. Since ∆ is an orientable pseudomanifold, H d−1 (∆) is torsion-free, and thus sequence (6) splits. By the lemma, the mapping
Z/mZ → im ∂ d / im L d−1 k → k∂ d (γ 1 )
is an isomorphism. The result follows.
Our proof of Theorem 30 follows the general outline of that in [8] with substantial modifications.
Proof of Theorem 30. Considering the commutative diagram
ZΥ i ZΥ i−1 ZΥ i−2 Z∆ i Z∆ i−1 Z∆ i−2 ∂ Υ,i ∂ Υ,i−1 ∂∆,i ∂∆,i−1 ,
we see im ∂ Υ,i ⊆ im ∂ ∆,i ⊆ ker ∂ ∆,i−1 = ker ∂ Υ,i−1 . Thus, there is a short exact sequence 0 → im ∂ ∆,i / im ∂ Υ,i → H i−1 (Υ) → H i−1 (∆) → 0.
By hypothesis, H i−1 (Υ) = H i−1 (∆), and hence (7) im ∂ Υ,i = im ∂ ∆,i .
We now describe a basis for ker ∂ ∆,i . For each θ ∈ Θ, since im ∂ Υ,i = im ∂ ∆,i ,
(8) ∂ ∆,i (θ) = τ ∈Υi a θ (τ )∂ Υ,i (τ )
for some a θ (τ ) ∈ Z. Since H i (Υ) = 0, the boundary mapping ∂ Υ,i is injective, and thus the coefficients a θ (τ ) are uniquely determined. Define α(θ) := τ ∈Υi a θ (τ )τ and extend linearly to get a well-defined mapping α : ZΘ → ZΥ i . For each θ ∈ Θ, let θ := θ − α(θ).
We claim ker ∂ ∆,i = {θ : θ ∈ Θ}.
Theθ are linearly independent elements of the kernel. To show they span, suppose γ = σ∈∆i b σ σ ∈ ker ∂ ∆,i . Consider
γ ′ := γ − σ∈Θ b σσ = σ∈Υi b σ σ + σ∈Θ b σ (σ −σ) = σ∈Υi b σ σ + σ∈Θ b σ α(σ).
Then since γ and theσ are in ker ∂ ∆,i , so is γ ′ . Further, since each α(σ) ∈ ZΥ i , so is γ ′ . But ∂ ∆,i restricted to Υ i is equal to ∂ Υ,i , which is injective. It follows that
γ = σ∈∆i b σ σ = σ∈Θ b σσ .
We thus have an isomorphism π : ZΘ ∼ − − → ker ∂ ∆,i determined by σ →σ with inverse given by setting elements of Υ i equal to 0:
σ∈∆i b σ σ −−→ σ∈Θ b σ σ.
Next, we claim there is a commutative diagram with exact rows
ZΘ ZΘ cokL 0 Z∆ i ker ∂ ∆,i K i (∆) 0 L ι π ∼ Li
where ι is the natural inclusion. To check commutativity of the square on the left, let θ ∈ Θ. Then by definition of L and the fact that ι(θ) is supported on Θ, L i ι(θ) = ρ +Lθ for some ρ ∈ ZΥ i . We then have π −1 (ρ +Lθ) =Lθ, as required. Hence, there is a well-defined vertical mapping cok L → K i (∆) on the right. By the snake lemma, that mapping is an isomorphism if and only if the mapping ZΘ → Z∆ i / ker L i given by composing ι with the quotient mapping is surjective. Therefore, to finish the proof, it suffices to show that for all γ ∈ Υ i , there exists δ ∈ ZΘ such that γ + δ ∈ ker L i (so then γ = −δ mod ker L i ). Now ker L i = ker ∂ ∆,i+1 ∂ t ∆,i+1 = ker ∂ t ∆,i+1 . To get a description of ker ∂ t ∆,i+1 , consider the exact sequence
Z∆ i+1 ∂∆,i+1 − −−− → Z∆ i → cok ∂ ∆,i+1 → 0.
Applying the left-exact functor Hom( · , Z), gives the exact sequence
(9) Z∆ i+1 ∂ t ∆,i+1 ← −−− − Z∆ i ← (cok ∂ ∆,i+1 ) * ← 0,
where we have identified Z∆ i and Z∆ i+1 with their duals (using the bases ∆ i and ∆ i+1 , respectively). There is an exact sequence,
0 → ker ∂ ∆,i / im ∂ ∆,i+1 → Z∆ i / im ∂ ∆,i+1 → Z∆ i / ker ∂ ∆,i → 0, i.e,(10)0 → H i (∆) → cok ∂ ∆,i+1 → Z∆ i / ker ∂ ∆,i → 0.
However, Z∆ i / ker ∂ ∆,i ∼ − − → im ∂ ∆,i = im ∂ Υ,i ≃ ZΥ i using (7) and the fact that ∂ Υ,i is injective. Since ZΥ i is free, sequence (10) splits: (11) cok ∂ ∆,i+1 ≈ H i (∆) ⊕ ZΥ i , with each γ ∈ Υ i identified with its class in cok ∂ ∆,i+1 . Given γ ∈ Υ i , let γ * : ZΥ i → Z be the dual function. Then use isomorphism (11), to identify γ * with an element of (cok ∂ ∆,i+1 ) * . The image of γ * in Z∆ i under the mapping in (9) is γ + θ∈Θ a θ (γ)θ, which by exactness of (9) is an element of ker ∂ t ∆,i+1 . Letting δ := θ∈Θ a θ (γ)θ, we see that γ+δ ∈ ker ∂ t ∆,i+1 , as required.
Remark 42. Theorem 30 generalizes Theorem 3.4 of [8]. Remark 3.5 of [8] considers the case where ∆ is the 6-vertex simplex, i = 2, and Υ is a certain triangulation of the real projective plane (shown in Fig. 3 of [10]). In this case, H 1 (∆) = 0 = H 1 (Υ) = Z/2Z, and K 2 (∆) = (Z/6Z) 4 ≃ ZΘ/ im L ≃ (Z/12Z) ⊕ (Z/6Z) 3 ⊕ (Z/2Z).
] → ([D − deg(D)v], deg(D)).
r(D) − r(K − D) = deg(D) + 1 − g.2010 Mathematics Subject Classification. 05E45.
Figure 1 .
1Winning the dollar game σ = −12+2·13−3·23+2·24−34 on the 2-dimensional simplicial complex with facets 123 and 234.
Figure 2 .
2Two dollar games on the edges of a 2-simplex. Only the first is winnable.
Proposition 19 .
19Suppose the i-th Hilbert basis H i of ∆ consists of 0-1 vectors, and let σ be an i-chain such that deg(σ) ≥ 0. Then there exists an effective i-chain τ (not necessarily linearly equivalent to σ) such that deg(τ ) = deg(σ).
and since deg(σ) = 0, there exists a face f ′ such that τ (f ′ ) > 0. To compute the degree of the integral chain σ − f ′ , let h = f ∈∆i h(f )f be an arbitrary element of H i . Since h(f ′ ) ∈ {0, 1}, taking dot products,
( i )
iH d (∆, ∂∆) ≈ Z and H d−1 (∆, ∂∆) is torsion-free. (ii) H d (∆, ∂∆) = 0 and H d−1 (∆, ∂∆) has torsion subgroup T(H d−1 (∆, ∂∆)) ≈ Z/2Z.
Example 24 .
24Let ∆ be the hollow tetrahedron with facets 123, 124, 134, and 234. A pseudomanifold orientation is given by γ = 132 + 124 + 143 + 234 = −123 + 124 − 134 + 234.Both ∆ and its associated γ-incidence graph Γ(∆, γ) appear inFigure 3. The edges of Γ(∆, γ) are labeled by the corresponding 1-faces of ∆. The incidence vectors for the three simple directed cycles of Γ(∆,
Figure 3 .
3The hollow tetrahedron and its γ-incidence graph (cf. Example 24).hence the elements of the Hilbert basis for ker + L 1 , are listed as rows inthe table below
Figure 4 .Figure 5 .
45A triangulated annulus and its γ-incidence graph (cf. Example 25). in the Hilbert basis for ker + L 1 .Example 26. The condition of being orientable as a pseudomanifold is necessary in both Proposition 21 and Theorem 22. The Klein bottle simplicial complex inFigure 5is a non-orientable pseudomanifold of dimension 2. Computing with Sage ([20]), we find K 1 (∆) ≃ Z/2Z ⊕ Z/2Z ⊕ Z and that the Hilbert basis for ker + L 1 has 14 elements. Three of these basis elements are not 0-1 vectors and, thus, are not incidence vectors of simple cycles in a directed graph. Triangulation of a Klein bottle (cf. Example 26).
all i. However, requiring a chain of the form (a, 0, 0, 0, 0, b)
Proposition 29 ([ 8 ,
298Prop 3.5], [9]). Any two of the three conditions defining a spanning i-forest implies the remaining condition.
obtained by setting the faces of Υ i equal to 0.Definition 31. Define the i-complexity or i-forest number of ∆ to beτ := τ i (∆) := Υ⊆∆ |T( H i−1 (Υ))| 2where the sum is over all spanning i-forests Υ of ∆.Proposition 32. τ i (∆) = 1 if and only if Skel i (∆) is a spanning i-forest of ∆ and H i−1 (∆) is torsion-free. If Skel i (∆) is a spanning i-forest, regardless of whether H i−1 (∆) is torsion-free, then Skel i (∆) is the unique spanning i-forest of ∆.
Theorem 33 ([ 9 ,
339Theorem 8.1]). |T(K i−1 (∆))| = τ i (∆). 1 Corollary 34. All (i − 1)-chains of degree 0 on ∆ are winnable if and only if τ i (∆) = 1. Proof. By Proposition 6 and Corollary 11, an (i − 1)-chain of degree 0 is winnable if and only if it is linearly equivalent to the zero chain. The (i − 1)-chains of degree 0 are the elements (ker + L i−1 ) ⊥ = (ker L i−1 ) ⊥ . Hence, by Theorem 13, all (i − 1)-chains of degree 0 are winnable if and only if T(K i−1 (∆)) = 0. The result then follows from Theorem 33.
Figure 6 .
6A triangulation of the real projective plane.
.Figure 7 .
7K 1 (∆) = H 1 (∆) ≃ Z. Now consider a generator for the first homology such as σ = (0, 0, 0, 1, −1, 1) = 23 − 24 + 34. The Hilbert basis H 1 for ker + L 1 , computed by Sage ([20]), is given by the rows of Ordering the elements of H 1 as they appear in the table, top-to-bottom, we have deg(σ) = (1, −1, 1, 0) ≥ 0. So σ is not winnable even though ∂ 1 (σ) = 0 ∈ Λ 0 (A simplicial complex with facets 123, 124, and 34 (cf. Example 40).
( 6 )
6Is there a generalization of the rank function to 1-chains on a simplicial complex of dimension 2, measuring robustness of winnability and perhaps related to the Riemann-Roch theorem for algebraic surfaces? If so, can one find a combinatorial lower bound for it? i + ℓm for some integer ℓ, set a i := s i − t i − ℓ for all i. Then (5) holds, and so σ and τ are linearly equivalent.
In[9], this theorem is stated only for i = dim(∆). The version stated here follows by restricting to Skel i (∆) (cf. Remark 5).2 We used the PyNormaliz package in Sage ([20]) for the Hilbert basis computations in this example.
Acknowledgments. We would like to thank Scott Corry for the idea of thinking of chip-firing on simplicial This example is given in[8]to show that the condition H i−1 (∆) = H i−1 (Υ) = 0 in Theorem 3.4 cannot be dropped. Here, it serves the same purpose for the more relaxed hypothesis H i−1 (∆) = H i−1 (Υ) of Theorem 30.AppendixIn this appendix, we prove Proposition 21 and Theorem 30. The proof of Proposition 21 requires the following lemma.Lemma 41. Let ∆ be a d-dimensional orientable pseudomanifold without boundary. Let γ 1 , . . . , γ m be the facets of ∆ oriented so that γ = γ 1 +· · ·+γ m is a pseudomanifold orientation for ∆, i.e., such that ∂ d (γ) = 0. Let σ, τ be two (d − 1)-chains in the image of ∂ d , and writefor some integers {s i } and {t i }. Then σ and τ are linearly equivalent if and only ifThen ξ is contained in exactly two facets, say γ i and γ j , and L d−1 (ξ) = ±(∂ d (γ i ) − ∂ d (γ j )). By strong connectivity, it follows that ∂ d (γ i ) − ∂ d (γ j ) is in the image of L d−1 for any pair 1 ≤ i, j ≤ m, and thus,So linear equivalence of σ and τ is equivalent to being able to writefor some integers a i summing to 0. Since the ∂ d (γ i ) do not form a basis for the image of ∂ d , we cannot directly conclude something about the relation between the coefficients on both sides of equation (5). However, note that the existence of arbitrary integers a i (not necessarily summing to 0) such that equation (5) holds is equivalent tobeing in ker ∂ d = H d (∆) = Zγ, and thus to the existence of an integer ℓ such that ρ = ℓ(γ 1 + · · · + γ m ). In this case, since the γ i form a basis for C d (∆), we conclude s i − t i − a i = ℓ for i = 1, . . . , m. Summing, we
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| []
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[
"PROPERTIES OF MODIFIED BESSEL FUNCTIONS AND COMPLETELY MONOTONIC DEGREES OF DIFFERENCES BETWEEN EXPONENTIAL AND TRIGAMMA FUNCTIONS",
"PROPERTIES OF MODIFIED BESSEL FUNCTIONS AND COMPLETELY MONOTONIC DEGREES OF DIFFERENCES BETWEEN EXPONENTIAL AND TRIGAMMA FUNCTIONS"
]
| [
"Feng Qi "
]
| []
| []
| In the paper, the author establishes inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and compute the completely monotonic degrees of differences between the exponential and trigamma functions. | 10.7153/mia-18-37 | [
"https://arxiv.org/pdf/1302.6731v2.pdf"
]
| 56,156,092 | 1302.6731 | 011e4fb0de9df43cc41cc165a94f80d349de2bf4 |
PROPERTIES OF MODIFIED BESSEL FUNCTIONS AND COMPLETELY MONOTONIC DEGREES OF DIFFERENCES BETWEEN EXPONENTIAL AND TRIGAMMA FUNCTIONS
27 Feb 2013
Feng Qi
PROPERTIES OF MODIFIED BESSEL FUNCTIONS AND COMPLETELY MONOTONIC DEGREES OF DIFFERENCES BETWEEN EXPONENTIAL AND TRIGAMMA FUNCTIONS
27 Feb 2013
In the paper, the author establishes inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and compute the completely monotonic degrees of differences between the exponential and trigamma functions.
on (0, ∞) was obtained and applied, where ψ(t) stands for the digamma function which may be defined by the logarithmic derivative
ψ(t) = [ln Γ(t)] ′ = Γ ′ (t) Γ(t)
and Γ(t) is the classical Euler gamma function which may be defined for ℜz > 0 by
Γ(z) = ∞ 0 t z−1 e −t d t.
The derivatives ψ ′ (z) and ψ ′′ (z) are respectively called the tri-and tetra-gamma functions.
In [17,Theorem 3.1] and [18,Theorem 1.1], among other things, the inequality (1.1) was generalized to a complete monotonicity respectively by different and elementary approaches, which reads that the difference
h(t) = e 1/t − ψ ′ (t) (1.2)
is completely monotonic, that is, (−1) k−1 h (k−1) (t) ≥ 0 for k ∈ N, on (0, ∞) and for ν ∈ R and z ∈ C, the hypergeometric series p F q (a 1 , . . . , a p ; b 1 , . . . , b q ; x) = ∞ n=0 (a 1 ) n · · · (a p ) n (b 1 ) n · · · (b q ) n x n n! (1.7)
for b i / ∈ {0, −1, −2, . . . }, and the shifted factorial (a) 0 = 1 and (a) n = a(a + 1) · · · (a + n − 1) (1. 8) for n > 0 and any real or complex number a. This gives an answer to an open problem posed in [22]. See also [21,Chapter 6]. By virtue of (1.4) or (1.5) for k = 0 and the well known formula
F β (u) = u 1 − e −u √ βu I 1 2 √ βu (1.12)
is decreasing on (0, ∞); when 0 < β < 1, it is unimodal (that is, it has a unique maximum) and 1 F β (u) is convex on (0, ∞).
1.4.
Necessary and sufficient conditions. For α, β > 0, let h α,β (t) = αe β/t − ψ ′ (t) (1. 13) on (0, ∞). In [17], among other things, the following necessary and sufficient conditions for the function h α,β (t) to be completely monotonic on (0, ∞) were obtained. If β ≥ 1 and αβ ≥ 1, the function h α,β (t) is completely monotonic on (0, ∞). A necessary condition for the function h α,β (t) to be completely monotonic on (0, ∞) is αβ ≥ 1.
If 0 < β < 1, the condition
αβ ≥ max u∈(0,∞) F β (u) > 1 (1.14)
is necessary and sufficient for h α,β (t) to be completely monotonic on (0, ∞), where
lim u→0 + F β (u) = 1 and lim u→∞ F β (u) = 0 (1.15)
for all β > 0.
1.5. Completely monotonic degree. The notion "completely monotonic degree" was created in [6, Definition 1], which may be regarded as a slight but essential modification of [11,Definition 1.5] and may be reformulated as follows.
Definition 1.1. Let f (t) be a function defined on (0, ∞) and have derivatives of all orders. If for some r ∈ R the function t r f (t) is completely monotonic on (0, ∞) but t r+ε f (t) is not for any positive number ε > 0, then we say that the number r is the completely monotonic degree of f (t) with respect to t ∈ (0, ∞); If for any r ∈ R none of t r f (t) is completely monotonic on (0, ∞), then we say that the completely monotonic degree of f (t) with respect to t ∈ (0, ∞) is −∞; If for all r ∈ R each and every t r f (t) is completely monotonic on (0, ∞), then we say that the completely monotonic degree of f (t) with respect to t ∈ (0, ∞) is ∞.
In [6, p. 9890], the notation deg t cm [f (t)] was designed to denote the completely monotonic degree r of f (t) with respect to t ∈ (0, ∞). We can redevelop the above Definition 1.1 as follows:
If f : (0, ∞) → [0, ∞) is a C ∞ -function, then deg t cm [f (t)] = sup{r ∈ R | t r f (t) is completely monotonic}; (1.16) if t r f (t) is not completely monotonic for any r ∈ R, then deg t cm [f (t)] = −∞. We claim that f (t) = 0 is the only function with deg t cm [f (t)] = ∞. It is clear that deg x cm [0] = ∞, as defined in [6, p. 9891, (4)]. Conversely, if deg x cm [f (x)] = ∞, then x r f (x)
is always completely monotonic on (0, ∞) for any positive number r, which means that [5, p. 98] asserts that for a completely monotonic function f on (0, ∞) the strict inequality
[x r f (x)] ′ = rx r−1 f (x) + x r f ′ (x) = x r−1 [rf (x) + xf ′ (x)] ≤ 0, that is, rf (x) + xf ′ (x) ≤ 0 (1.17) is valid on (0, ∞) for all r ≥ 0. A result on(−1) k−1 f (k−1) (t) > 0 for k ∈ N holds unless f (x) is constant. This implies that if f (x) = 0 then f (x) > 0 and f ′ (x) < 0 on (0, ∞).
Consequently, the inequality (1.17) may be rearranged as
r < − xf ′ (x) f (x) , x ∈ (0, ∞).
This leads to a contradiction to the arbitrariness of r ≥ 0. As a result, it holds that
f (x) ≡ 0 on (0, ∞). For x ∈ (0, ∞), let Ψ(x) = [ψ ′ (x)] 2 + ψ ′′ (x(t) = h α,β (t) − α (1.20) on (0, ∞).
Our main results in this paper are the following theorems in sequence.
Theorem 1.2. When 1 ≤ k ≤ 5, the inequality
I k 2 √ u u k/2 ≥ u 1 − e −u (k−1) (1.21)
is valid on (0, ∞).
When β ≥ 1, the function
G β (u) = βu I 2 2 √ βu u 1 − e −u ′ (1.22)
is decreasing on (0, ∞); when 0 < β < 1, it is unimodal and 1 G β (u) is convex on (0, ∞).
Theorem 1.3. Let α, β > 0. (1) If (α, β) = (1, 1), then deg t cm [H 1,1 (t)] = 4; (1.23) (2) if β > 1, then deg t cm H 1/β,β (t) = 2;
(1.24) (3) if αβ > 1 and β ≥ 1, or if αβ > 1, 0 < β < 1, and
αβ 2 ≥ max u∈(0,∞) {G β (u)}, then we have deg t cm [H α,β (t)] = 1.
(1.25)
Lemmas
We need the following lemmas.
c k (n) = n k (2n − k)! n! and Q n (x) = n k=0 c k (n)x n−k . (2.1)
If m, n ∈ N, then the polynomial Q 2n has no real root, the polynomial Q 2n+1 has a unique real root in (−∞, 0), and
Q 2n (x) Q 2n (−x) < e 1/x < − Q 2m+1 (x) Q 2m+1 (−x) , x ≥ 1 2(m + 1)
. a k x k be a real polynomial of degree n ≥ 0. Then the inequality
(2.2)min{b k | 0 ≤ k ≤ n} ≤ P (x) ≤ max{b k | 0 ≤ k ≤ n} (2.3) holds for 0 ≤ x ≤ 1, where b k = k ℓ=0 a ℓ k ℓ n ℓ for 0 ≤ k ≤ n.
Lemma 2.3 ([2]
). Let a k and b k for k ∈ {0} ∪ N be real numbers and the power series
A(x) = ∞ k=0 a k x k and B(x) = ∞ k=0 b k x k (2.4)
be convergent on (−R, R) for some R > 0. If b k > 0 and the ratio a k b k is (strictly) increasing for k ∈ N, then the function A(x) B(x) is also (strictly) increasing on (0, R).
I ν (z) ∼ e z √ 2πz 1 + ∞ ℓ=1 (−1) ℓ ℓ!(8z) ℓ ℓ j=1 µ − (2j − 1) 2 , | arg z| < π 2 .(2.ψ (n) (z) = (−1) n−1 (n − 1)! z n + n! 2z n+1 + ∞ k=1 B 2k (2k + n − 1)! (2k)!z 2k+n , (2.6)
where B n for n ≥ 0 stand for Bernoulli numbers which may be generated by
x e x − 1 = ∞ n=0 B n n! x n = 1 − x 2 + ∞ j=1 B 2j x 2j (2j)! , |x| < 2π. (2.7)
Lemma 2.6 ([20, p. 161, Theorem 12b]). A necessary and sufficient condition for f (x) to be completely monotonic on (0, ∞) is that
f (x) = ∞ 0 e −xt d µ(t), (2.8)
where µ is a positive measure on [0, ∞) such that the integral converges on (0, ∞).
Proof of Theorem 1.2
For proving Theorem 1.3, we need at first to verify Theorem 1.2 as follows.
Proof of the inequality (1.21). Taking ν = 5 and z = 2 √ u in (1.6) lead to
I 5 2 √ u u 5/2 = ∞ k=0 u k k!(k + 5)! ≥ 1 k=0 u k k!(k + 5)! = 6 + u 720 . (3.1)
Hence, in order to prove the inequality (1.21) for k = 5, it is sufficient to show
u + 6 720 ≥ u 1 − e −u (4) =
e u (u − 4)e 3u + (11u − 12)e 2u + (11u + 12)e u + u + 4 (e u − 1) 5 which can be rewritten as
F 1 (u) (u + 6)(e u − 1) 5 − 720e u (u − 4)e 3u + (11u − 12)e 2u + (11u + 12)e u + u + 4 ≥ 0.
By direct calculations, we have
F ′ 1 (u) = (31 + 5u)e 5u + 25(427 − 116u)e 4u + (18190 − 23730u)e 3u − 10(2533 + 1586u)e 2u − 5(713 + 143u)e u − 1, F ′′ 1 (u) = 5e u (32 + 5u)e 4u + 40(199 − 58u)e 3u + (6168 − 14238u)e 2u − 8(1663 + 793u)e u − 143u − 856 5e u F 2 (t), F ′ 2 (u) = (133 + 20u)e 4u + 40(539 − 174u)e 3u − 6(317 + 4746u)e 2u − 8(2456 + 793u)e u − 143, F ′′ 2 (u) = 8e u (69 + 10u)e 3u + (7215 − 2610u)e 2u − 3(1345 + 2373u)e u − 793u − 3249 8e u F 3 (u), and F 3 (0) = F ′′ 2 (0) = F ′ 2 (0) = F 2 (0) = F ′′ 1 (0) = F ′ 1 (0) = F 1 (0) = 0. As a result, when F 3 (u) ≥ 0 on (0, ∞), it follows that F 1 (u) ≥ 0 on (0, ∞).
It is easy to see that the function F 3 (u) can be rearranged as
F 3 (u) = 10u e u − 261 e 2u + 3 23e 2u − 2373u − 1345 e u + 7215e 2u − 793u − 3249 (3.2)
and that (1) the term e u − 261 is positive when u > ln 261 = 5.56 . . . , Consequently, when u ≥ 6, the function F 3 (u) is positive.
Applying Lemma 2.1 to m = 2 and n = 3 derives
x 6 + 42x 5 + 840x 4 + 10080x 3 + 75600x 2 + 332640x + 665280 x 6 − 42x 5 + 840x 4 − 10080x 3 + 75600x 2 − 332640x + 665280 < e x < x 5 + 30x 4 + 420x 3 + 3360x 2 + 15120x + 30240 30240 − 15120x + 3360x 2 − 420x 3 + 30x 4 − x 5 , 0 < x ≤ 6 and the function F 3 (u) can be written as F 3 (u) = 69e 3u + 7215e 2u − 4035e u − 3249 + 10e 3u − 2610e 2u − 7119e u − 793 u > 69 665280 + 332640u + 75600u 2 + 10080u 3 + 840u 4 + 42u 5 + u 6 665280 − 332640u + 75600u 2 − 10080u 3 + 840u 4 − 42u 5 + u 6 3 + 7215 665280 + 332640u + 75600u 2 + 10080u 3 + 840u 4 + 42u 5 + u 6 665280 − 332640u + 75600u 2 − 10080u 3 + 840u 4 − 42u 5 + u 6 2 − 4035 30240 + 15120u + 3360u 2 + 420u 3 + 30u 4 + u 5 30240 − 15120u + 3360u 2 − 420u 3 + 30u 4 − u 5 − 3249 + u 10 665280 + 332640u + 75600u 2 + 10080u 3 + 840u 4 + 42u 5 + u 6 665280 − 332640u + 75600u 2 − 10080u 3 + 840u 4 − 42u 5 + u 6 3 − 2610 30240 + 15120u + 3360u 2 + 420u 3 + 30u 4 + u 5 30240 − 15120u + 3360u 2 − 420u 3 + 30u 4 − u 5 2 − 7119 30240 + 15120u + 3360u 2 + 420u 3 + 30u 4 + u 5 30240 − 15120u + 3360u 2 − 420u 3 + 30u 4 − u 5 − 793 6uF 4 (u) F 5 (u) on (0, 6], where F 4 (u) = 621u 28 − 94751u 27 + 6068010u 26 − 218010408u 25 + 4603805304u 24 − 34479103680u 23 − 1412513172000u 22 + 69509082484800u 21 − 1835940264439680u 20 + 36018288433370880u 19 − 572728070517926400u 18 + 7670777548637952000u 17 − 88289462254568601600u 16 + 882241752079928217600u 15 − 7678232793596974694400u 14 + 58018545121380802560000u 13 − 376773142967398969344000u 12 + 2061377592729654140928000u 11 − 9156137875572402634752000u 10 + 30521365267364424843264000u 9 − 59514097618800165519360000u 8 − 48948451585366441328640000u 7 + 852510196971380523663360000u 6 − 3082810530742053482004480000u 5 + 5063190015183760203448320000u 4 − 350858087962497987379200000u 3 − 10582859071799067200716800000u 2 + 8481790289232945532108800000u + 4038947756777593110528000000 and F 5 (u) = u 5 − 30u 4 + 420u 3 − 3360u 2 + 15120u − 30240 2 × u 6 − 42u 5 + 840u 4 − 10080u 3 + 75600u 2 − 332640u + 665280 3 .
By Lemma 2.1, it follows that the function
F 6 (u) = u 5 Q 5 − 1 u = u 5 − 30u 4 + 420u 3 − 3360u 2 + 15120u − 30240
has a unique positive zero. Since F 6 (6) = −864 and F 6 (8) = 608, the function F 6 (u) is negative on (0, 6). By Lemma 2.1 once again, the function
u 6 Q 6 − 1 u = u 6 − 42u 5 + 840u 4 − 10080u 3 + 75600u 2 − 332640u + 665280
has no any zero. In a word, the function The positivity of F 4 (u) on the interval [1,2], [2,3], [3,4], [4,5], or [5,6] can be respectively transformed into the positivity of the function In conclusion, the function F 3 (u), and so F 1 (u), is positive on (0, ∞). This means that the inequality (1.21) for k = 5 is valid on (0, ∞).
F 5 (u) = F 4 (u + 1), F 6 (u) = F 5 (u + 1) = F 4 (u + 2), F 7 (u) = F 6 (u + 1) = F 5 (u + 2) = F 4 (u + 3),
The inequality (1.21) for 1 ≤ k ≤ 4 may be verified by similar arguments as above.
Proof of monotonicity of the function (1.22). For simplicity, we consider
1 G β (u) = (e u − 1) 2 (e u − 1 − u)e u I 2 2 √ βu βu Q β (u) P (u) , where P (u) = (e u − 1 − u)e u = ∞ k=2 2 k − k − 1 k! u k = ∞ k=2 p k−2 u k and Q β (u) = (e u − 1) 2 I 2 2 √ βu βu = ∞ k=2 2 k − 2 k! u k ∞ k=0 β k k!(k + 2)! u k = ∞ k=2 q k−2 (β)u k with p k = 2 k+2 − k − 3 (k + 2)! and q k (β) = 1 (k + 2)! k ℓ=0 k + 2 ℓ 2 k−ℓ+2 − 2 (ℓ + 2)! β ℓ for k ≥ 0. Hence, Q β (u) P (u) = ∞ k=0 q k (β)u k ∞ k=0 p k u k . When β ≥ 1, let c k (β) = q k (β) p k = 1 2 k+2 − k − 3 k ℓ=0 k + 2 ℓ 2 k−ℓ+2 − 2 (ℓ + 2)! β ℓ for k ∈ {0} ∪ N. It is clear that c 0 (β) = 1, c 1 (β) = 3 + β 4
, and c 2 (β) = 14 + 8β + β 2 22
satisfy c 0 (β) ≤ c 1 (β) < c 2 (β) for β ≥ 1. For k ≥ 2, c k+1 (β) − c k (β) = k ℓ=0 (b k+1,ℓ − b k,ℓ ) β ℓ (ℓ + 2)! + β k+1 (k + 1)!(2 k+3 − k − 4) , where b k,ℓ = 2 k−ℓ+2 − 2 2 k+2 − k − 3 k + 2 ℓ .
An easy computation yields
b k+1,0 − b k,0 = − 2 1 + 2 k+1 k (2 k+2 − k − 3)(2 k+3 − k − 4) < 0, b k+1,1 − b k,1 = 2 1 + 2 k 2 k+3 − k 2 + 2k + 6 (2 k+2 − k − 3)(2 k+3 − k − 4) ≥ 2 1 + 2 k 2 3 1 + k + k(k − 1)/2 − k 2 + 2k + 6 (2 k+2 − k − 3)(2 k+3 − k − 4) = 2 1 + 2 k 3k 2 + 2k + 2 (2 k+2 − k − 3)(2 k+3 − k − 4) > 0, and b k+1,0 − b k,0 2! + β b k+1,1 − b k,1 3! ≥ b k+1,0 − b k,0 2! + b k+1,1 − b k,1 3! = 2 k 2 k+3 − k 2 + 8k + 6 − 2 3(2 k+2 − k − 3)(2 k+3 − k − 4) > 2 k 2 3 [1 + k + k(k − 1)/2] − k 2 + 8k + 6 − 2 3(2 k+2 − k − 3)(2 k+3 − k − 4) = 2 k (3k − 4)k + 2 k+1 − 2 3(2 k+2 − k − 3)(2 k+3 − k − 4) > 0 for k ≥ 2. Further, when k ≥ ℓ ≥ 2, we have b k+1,ℓ − b k,ℓ k+2 ℓ = (k + 3) 2 k−ℓ+3 − 2 (k − ℓ + 3)(2 k+3 − k − 4) − 2 k−ℓ+2 − 2 2 k+2 − k − 3 = 2 k−ℓ+2 − 2 2 k+3 − k − 4 (k + 3) 2 k−ℓ+3 − 2 (k − ℓ + 3)(2 k−ℓ+2 − 2) − 2 k+3 − k − 4 2 k+2 − k − 3 = 2 k−ℓ+2 − 2 2 k+3 − k − 4 2(k + 3) k − ℓ + 3 1 + 1 2 k−ℓ+2 − 2 − 2 k+3 − k − 4 2 k+2 − k − 3 ≥ 2 k−ℓ+2 − 2 2 k+3 − k − 4 2(k + 3) k + 1 1 + 1 2 k − 2 − 2 k+3 − k − 4 2 k+2 − k − 3 = 2 k−ℓ+2 − 2 2 k+3 − k − 4 k + 3 k + 1 2 k+1 − 2 2 k − 2 − 2 k+3 − k − 4 2 k+2 − k − 3 = 2 k−ℓ+2 − 2 2 k 2 k+4 − k 2 − k + 22 + 2(k + 5) (k + 1)(2 k − 2)(2 k+2 − k − 3)(2 k+3 − k − 4) > 0,
where in the last line we used the inequality 2 k+4 > 2 4
1 + k + k(k − 1) 2 = k 2 − k + 22 + 7k 2 + 9k − 6 > k 2 − k + 22.
Consequently, when β ≥ 1 and k ≥ 2,
c k+1 (β) − c k (β) > b k+1,0 − b k,0 2! + b k+1,1 − b k,1 3! β + k ℓ=2 b k+1,ℓ − b k,ℓ (ℓ + 2)! β ℓ > 0.
Therefore, when β ≥ 1, we have c k+1 (β) − c k (β) > 0. Equivalently speaking, when β ≥ 1, the sequence c k (β) is increasing with respect to k ≥ 0. From this and Lemma 2.3, it follows that, when β ≥ 1, the function Q β (u) P (u) is increasing on (0, ∞). As a result, when β ≥ 1, the function G β (u) is decreasing on (0, ∞). The proof of monotonicity of the function (1.22) is complete.
Proof of unimodality and convexity of the function (1.22). By (1.6), it is straightforward to obtain d d z
I ν (z) (z/2) ν = I ν+1 (z) (z/2) ν . (3.3)
Making use of (3.3) and differentiating lead to
d d u 1 G β (u) = e u − 1 e u (1 − e u + u) 2 (e u − 1)(e u − 1 − u) d d u I 2 2 √ βu βu − [e u (u − 2) + u + 2] I 2 2 √ βu βu = 1 e u (e u − 1 − u) 2 β(e u − 1) 2 (e u − 1 − u) I 3 2 √ βu (βu) 3/2 − [e u (u − 2) + u + 2](e u − 1) I 2 2 √ βu βu R β (u) S(u) , where S(u) = (e u − 1 − u) 2 e u = u 4 ∞ k=0 3 k+4 − (k + 6)2 k+4 + k 2 + 9k + 21 (k + 4)! u k u 4 ∞ k=0 λ k u k , (e u − 1) 2 (e u − 1 − u) = ∞ k=4 3 k − (k + 6)2 k−1 + 2k + 3 k! u k , [e u (u − 2) + u + 2](e u − 1) = ∞ k=4 (k − 4)2 k−1 + 4 k! u k , and R β (u) = β(e u − 1) 2 (e u − 1 − u) I 3 2 √ βu (βu) 3/2 − [e u (u − 2) + u + 2](e u − 1) I 2 2 √ βu βu = ∞ k=4 3 k − (k + 6)2 k−1 + 2k + 3 k! u k ∞ k=0 β k+1 k!(k + 3)! u k − ∞ k=4 (k − 4)2 k−1 + 4 k! u k ∞ k=0 β k k!(k + 2)! u k = ∞ k=0 k ℓ=0 k + 4 ℓ 3 k−ℓ+4 − (k − ℓ + 10)2 k−ℓ+3 + 2(k − ℓ) + 11 (k + 4)!(ℓ + 3)! β ℓ+1 u k+4 − u 4 ∞ k=0 u k (k + 4)! k ℓ=0 k + 4 ℓ (k − ℓ)2 k−ℓ+3 + 4 (ℓ + 2)! = u 4 ∞ k=0 u k (k + 4)! k ℓ=0 k + 4 ℓ β ℓ (ℓ + 3)! 3 k−ℓ+4 − (k − ℓ + 10)2 k−ℓ+3 + 2(k − ℓ) + 11 β − (ℓ + 3) (k − ℓ)2 k−ℓ+3 + 4 u 4 ∞ k=0 ξ k (β)u k . When 0 < β < 1, let C k (β) = ξ k (β) λ k , that is, C k (β) = 1 U k k + 4 2 × k! β k+1 − 2 2 k+1 k + 1 + k ℓ=1 (k + 4)! ℓ!(ℓ + 2)!(k − ℓ + 5)! V k (ℓ)β ℓ k+1 ℓ=0 θ k,ℓ β ℓ for k ≥ 0, where U k = 3 k+4 − (k + 6)2 k+4 + k 2 + 9k + 21 and V k (ℓ) = 3 k−ℓ+5 ℓ + 2 k−ℓ+3 ℓ 2 − 17ℓ − 5k − k 2 − 2ℓ 2 + 2kℓ + 17ℓ − 4k − 20 = (k − m)3 m+5 + m 2 + (17 − 2k)m − 22k 2 m+3 − 2m 2 + (2k − 17)m + 13k − 20 W k (m) with 0 ≤ m = k − ℓ < k. It is not difficult to obtain that C 0 (β) = β − 1 3 , C 1 (β) = β 2 + 4β − 4 20 , C 2 (β) = 3β 3 + 35β 2 + 80β − 68 520 , C 3 (β) = β 4 + 22β 3 + 150β 2 + 256β − 168 1872 ,
and C 4 (β) = β 5 + 35β 4 + 448β 3 + 2268β 2 + 3220β − 1548 24444 .
Therefore, the differences
C 1 (β) − C 0 (β) = 3β 2 + 8(1 − β) 60 , C 2 (β) − C 1 (β) = 3 β 3 + 3β 2 + 4(3 − 2β) 520 , C 3 (β) − C 2 (β) = 5β 4 + 56β 3 + 120β 2 + 32(12 − 5β) 9360 , C 4 (β) − C 3 (β) = 52β 5 + 1141β 4 + 8358β 3 + 16086β 2 + 24(1399 − 266β)
1271088 are all positive for 0 < β < 1.
For k ≥ 4 and 0 < β < 1, we have
C k+1 (β) − C k (β) = k+1 ℓ=0 (θ k+1,ℓ − θ k,ℓ )β ℓ + θ k+1,k+2 β k+2 .
Since U k > 2 k+4 + (k + 4)2 k+3 + k + 4 2 2 k+2 − (k + 6)2 k+4 + k 2 + 9k + 21 = k 2 + 3k − 12 2 k+1 + k 2 + 9k + 21
> 0 for k ≥ 4, we easily obtain that θ k+1,k+2 > 0 and
C k+1 (β) − C k (β) > k+1 ℓ=0 (θ k+1,ℓ − θ k,ℓ )β ℓ (3.4)
for k ≥ 4 and 0 < β < 1. The inequality θ k+1,0 ≥ θ k,0 (3.5) may be rewritten as
3 k+4 (k − 2)2 k + 1 + 2 k 192 × 2 k − k 3 − 9k 2 − 37k − 106 + k + 5 ≥ 0,
which may be deduced from
192 × 2 k − k 3 − 9k 2 − 37k − 106 > 192 3 ℓ=0 k ℓ − k 3 − 9k 2 − 37k − 106 = (31k − 9)k 2 + 123k + 86 > 0.
Thus, the inequality (3.5) must be valid for k ≥ 2. The inequality θ k+1,ℓ ≥ θ k,ℓ (3.6)
for k ≥ 4 and k ≥ ℓ ≥ 1 can be rearranged as
U k+1 (k + 5)U k ≤ V k+1 (ℓ) (k − ℓ + 6)V k (ℓ) = W k+1 (m + 1) (m + 6)W k (m) ,
where 0 ≤ m = k − ℓ < k. Furthermore, for k ≥ 4 and 0 ≤ m ≤ k − 2, the inequality
W k+1 (m + 1) (m + 6)W k (m) ≥ W k+1 (m + 2) (m + 7)W k (m + 1) (3.7)
may be rearranged as
M m (k) A(m)k 2 + B(m)k + C(m) ≥ 0,(3.+ 66 + 215m + 30m 2 + m 3 2 3+m 3 m+4 + 4m 2 + 64m + 249 > 0, B(m) = 2 8m 3 + 92m 2 + 282m + 207 3 m+5 − (2m + 1)9 m+6 − 6m 4 + 145m 3 + 839m 2 + 592m − 1524 2 m+3 − m 3 + 31m 2 + 234m + 108 4 m+5 − m 3 + 5m 2 − 96m − 12 2 m+3 3 m+6 − 8m 3 + 148m 2 + 794m + 1065 ,
and C(m) = 2m 5 + 57m 4 + 388m 3 + 585m 2 + 480m + 2988 2 m+3 + m 4 + 36m 3 + 323m 2 + 12m − 756 4 m+4 + 4m 4 + 84m 3 + 569m 2 + 1401m + 1152
+ m 4 + 10m 3 − 99m 2 + 24m + 252 2 m+3 3 m+5 + m(m + 1)9 m+6 − 2 4m 4 + 52m 3 + 207m 2 + 327m + 288 3 m+5 .+ 2m 4 + 43m 3 + 389m 2 + 1728m + 2628 2 m+3 − 4m 2 − 40m − 69 > 3 8 3 ℓ=0 m ℓ 2 m−ℓ + m 3 − 3m 2 − 112m − 780 2 m+3 3 m+5 + m 3 + 23m 2 + 166m + 492 3 m+5 − 2 4m 2 + 52m + 141 3 m+5 + 2m 4 + 43m 3 + 389m 2 + 1728m + 2628 − 4m 2 − 40m − 69 = 5136 + 29404m + 6177m 2 + 2315m 3 2 m−4 3 m+5 + m 3 + 15m 2 + 62m + 210 3 m+5 + 2m 4 + 43m 3 + 385m 2 + 1688m + 2559 > 0.
This contradiction shows that, when k ≥ 4 and k ≥ m + 2 ≥ 2, the quantity M m (k) can be regarded as a quadratic polynomial of k and it has no any minimum. Combining this with the fact that A(m) > 0 concludes that the quadratic polynomial M m (k) of k is increasing with respect to k. A direct computation reveals that M 2 (k) = 336 750942 − 549881k + 95837k 2 are positive for k ≥ 4 and that for m ≥ 3 and k ≥ m + 2 M m (k) ≥ M m (m + 2) = 2 9 + m 4 + 20m 3 + 155m 2 + 508m + 780 2 2m+7 + 2m 4 + 61m 3 + 703m 2 + 3692m + 7140 2 m+2 + 3 m+5 3 m+7
− 492 + 172m + 29m 2 + m 3 2 m+2 − 2 111 + 32m + 2m 2 > 2 9 + m 4 + 20m 3 + 155m 2 + 508m + 780 2 2m+7 + 2m 4 + 61m 3
+ 703m 2 + 3692m + 7140 2 m+2 + 3 m+5 3 7 3 ℓ=0 m ℓ 2 m−ℓ − 492 + 172m + 29m 2 + m 3 2 m+2 − 2 111 + 32m + 2m 2 = 2 3 m+5 16 665m 3 2 m + 331 × 2 m − 64 m 2 + 4 893 × 2 m − 256 m + 48 73 × 2 m − 74 + m 4 + 20m 3 + 155m 2 + 508m + 780 2 2m+7 + 2m 4 + 61m 3 + 703m 2 + 3692m + 7140 2 m+2 + 9 > 0.
Accordingly, the inequality (3.8), and so the inequality (3.7), holds for all 0 ≤ m ≤ k − 2 and k ≥ 4. This means that the sequence W k+1 (m+1) (m+6)W k (m) is decreasing with respect to m, and so that the sequence V k+1 (ℓ) (k−ℓ+6)V k (ℓ) is increasing with respect to ℓ. Therefore, in order to show the inequality (3.6) for k ≥ 4 and k ≥ ℓ ≥ 1, it is sufficient to prove the inequality
U k+1 U k ≤ V k+1 (1) V k (1) (3.9)
for k ≥ 4, which is equivalent to k 2 − 3k − 12 2 k+1 3 k+4 + k 2 + 10k + 20 3 k+4 + k 2 + 6k + 4 + k 2 + 13k + 20 2 k+5 − k 4 + 16k 3 + 118k 2 + 435k + 540 2 k+1 ≥ 0 for k ≥ 4. Since k 2 + 13k + 20 2 k+5 − k 4 + 16k 3 + 118k 2 + 435k + 540
> 2 5 k 2 + 13k + 20 2 ℓ=0
k ℓ − k 4 + 16k 3 + 118k 2 + 435k + 540 = 15k 4 + 208k 3 + 442k 2 + 301k + 100 ≥ 0 and k 2 − 3k − 12 is positive for k ≥ 6, the inequality (3.9) is valid for k ≥ 6. By a straightforward computation, it is easy to see that the inequality (3.9) is also valid for k = 4, 5. Therefore, the inequality (3.9) is valid for all k ≥ 4. In conclusion, the inequality (3.6) holds for k ≥ 4 and k ≥ ℓ ≥ 1.
Substituting (3.5) and (3.6) into (3.4) reveals that C k+1 (β) − C k (β) > 0 is valid for k ≥ 4 and 0 < β < 1. Hence, the sequence C k (β) = ξ k (β) λ k is increasing with respect to k ≥ 0 for 0 < β < 1. By Lemma 2.3, it follows that the derivative
d d u 1 G β (u) = ∞ k=0 ξ k (β)u k ∞
k=0 λ k u k is increasing and that the function 1 G β (u) is convex on (0, ∞). The proof of the convexity of the function (1.22) is complete.
It is easy to obtain
lim u→0 + d d u 1 G β (u) = lim u→0 + ∞ k=0 ξ k (β)u k ∞ k=0 λ k u k = ξ 0 (β) λ 0 = β − 1 3 < 0
and, by Lemma 2.4,
d d u 1 G β (u) = β(e u − 1) 2 I 3 2 √ βu e u (e u − 1 − u)(βu) 3/2 − (e u − 1)[e u (u − 2) + u + 2]I 2 2 √ βu βue u (e u − 1 − u) 2 ∼ β I 3 2 √ βu (βu) 3/2 − 1 βe u I 2 2 βu → ∞
as u → ∞ for 0 < β < 1. Consequently, from its monotonicity on (0, ∞), the derivative d d u 1 G β (u) has a unique zero, and so the function 1 G β (u) has a unique minimum, and so the positive function G β (u) has a unique maximum, on (0, ∞). The proof of the unimodality of the function (1.22) is complete.
Proof of Theorem 1.3
With the help of Theorem 1.2, we now start off to prove Theorem 1.3. If the function t q H 1,1 (t) is completely monotonic on (0, ∞), then its first derivative is non-positive, that is, −t q−2 qt ψ ′ (t) − e 1/t + 1 + t 2 ψ ′′ (t) + e 1/t ≤ 0, which can be formulated as q ≤ By the integral representation (1.10), the formula (3.3), the definition of H 1,1 (t), and integration by part, we have
t 4 H 1,1 (t) = t 4 ∞ 0 I 1 2 √ u √ u − u 1 − e −u e −tu d u = −t 3 ∞ 0 I 1 2 √ u √ u − u 1 − e −u d e −tu d u d u = t 3 ∞ 0 d d u I 1 2 √ u √ u − u 1 − e −u e −tu d u = t 2 ∞ 0 d 2 d u 2 I 1 2 √ u √ u − u 1 − e −u e −tu d u = t ∞ 0 d 3 d u 3 I 1 2 √ u √ u − u 1 − e −u e −tu d u = 1 24 + ∞ 0 d 4 d u 4 I 1 2 √ u √ u − u 1 − e −u e −tu d u = 1 24 + ∞ 0 I 5 2 √ u u 5/2 − u 1 − e −u(H α,β (z) = ∞ 0 αβ I 1 2 √ βu √ βu − u 1 − e −u e −zu d u (4.4)
for ℜz > 0. Employing (3.3) and integrating in part as in (4.2) yield
tH α,β (t) = − ∞ 0 αβ I 1 2 √ βu √ βu − u 1 − e −u d e −tu d u d u = αβ − 1 + ∞ 0 αβ 2 I 2 2 √ βu βu − u 1 − e −u ′ e −tu d u.
Consequently,
(1) when αβ = 1, the function tH α,β (t) becomes
tH α,β (t) = ∞ 0 β I 2 2 √ βu βu − u 1 − e −u ′ e −tu d ut 2 H α,β (t) = − ∞ 0 β I 2 2 √ βu βu − u 1 − e −u ′ d e −tu d u d u = β − 1 2 + ∞ 0 β I 3 2 √ βu (βu) 3/2 − u 1 − e −u ′′ e −tu d u;
(a) if β > 1, by virtue of the fact that the function I3(2u) u 3 is strictly increasing on (0, ∞) and by the inequality (1.21) for k = 3, it is not difficult to see that the completely monotonic degree of the function H 1/β,β (t) for β > 1 is 2; (b) if 0 < β < 1, by the necessary condition (1.14), the function H α,β (t)
is not completely monotonic; (c) if β = 1, the discussing question goes back to the proof of (1.23);
(2) when αβ > 1 and
αβ 2 ≥ βu I 2 2 √ βu u 1 − e −u ′ = G β (u),(4.
Remarks
Finally we list some remarks on something to do with our lemmas and theorems. Remark 5.2. The function F 3 (u) defined by (3.2) can also be decomposed as Remark 5.3. In the draft of this manuscript, we ever used Theorem 2 in [15, p. 22] to prove the positivity of the function F 3 (u) defined in (3.2) on (0, 6). But, the inequality (14) stated in [15,p. 22,Theorem 2], and then the inequality (18) in [15, p. 22], is wrong. So we have to give up using [15,
F 3 (u) = f 1 (u) + f 2 (u) + f 3 (u),0 ≥ e x − S n (x) − α n (b)x n+1 ≥ (n + 1)!α n (b) − e b (n + 1)!(n + 1)b (b − x)x n+1 , (5.1) where S n (x) = n k=0 x k k! and α n (b) = e b − S n (b) b n+1 . (5.2)
The equalities in (5.1) are valid if and only if x = 0, b. This theorem was proved once again in [8] and was collected in the monograph [13, p. 290] and its older and subsequently revised version.
Remark 5.4. By Descartes' Sign Rule, it follows that (1) the polynomial P 1 (m) = m 4 + 36m 3 + 323m 2 + 12m − 756 has one possible positive zero; since P 1 (0) = −756 and P 1 (2) = 864, this zero belongs to the interval (0, 2), so P 1 (m) > 0 for m ≥ 2; (2) the polynomial P 2 (m) = m 4 + 10m 3 − 99m 2 + 24m + 252 has two possible positive zeros; since P 2 (0) = 252, P 2 (3) = −216, and P 2 (6) = 288, these two zeros locate in the interval (0, 6), so P 2 (m) > 0 for m ≥ 6. Furthermore, we have m(m + 1)9 m+6 − 2 4m 4 + 52m 3 + 207m 2 + 327m + 288 3 m+5 = m(m + 1)3 m+7 − 2 4m 4 + 52m 3 + 207m 2 + 327m + 288 3 m+5 > m(m + 1) These functions can be calculated for small values of ℓ and K 4 (7) = 0.0009 · · · < 1 720 . Therefore, the inequality (5.3) holds for u ≥ 7. As a result, it is sufficient to prove the inequality (5.3) on the interval [0, 7]. Remark 5.6. By a result in [19] (or see [11, p. 35 B 2k (2k)! u 2k + (−1) n u 2(n+1) V n (u), (5.4) where V n (u) = ∞ k=1 2 (u 2 + 4π 2 k 2 )(2πk) 2n (5.5) and it was proved in [11, Lemma 2.3] that u 2ℓ V n (u) (k) ≥ 0 (5.6) for u > 0 and 0 ≤ k ≤ ℓ. By (5.4) for n = 1, it follows that
u 4 V 1 (u) = 1 + u 2 + u 2 12 − u 1 − e −u ,(5.7)
hence, by (1.9) for n = 1,
∞ 0 u 4 V 1 (u)e −ux d u = 1 x + 1 2x 2 + 1 6x 3 − ψ ′ (x). (5.8)
Since the inequality (5.6) holds for 0 ≤ k ≤ 2, we obtain by [11,Theorem 1.3] that
x 2 1 x + 1 2x 2 + 1 6x 3 − ψ ′ (x)
is completely monotonic on (0, ∞). If we add the completely monotonic function
x 2 e 1/x − 1 − 1 x − 1 2x 2 −
1 6x 3 to the above, we find that the function
x 2 e 1/x − 1 − ψ ′ (x)
is completely monotonic on (0, ∞), that is, Since the inequality (5.6) holds for 0 ≤ k ≤ 4, we acquire by [11,Theorem 1.3] that
x 4 1 x + 1 2x 2 + 1 6x 3 − 1 30x 5 + 1 42x 7 − ψ ′ (x)
is completely monotonic on (0, ∞). If we add the completely monotonic function
x 4 e 1/x − 1 − 1 x − 1 2x 2 − 1 6x 3 − 1 4!x 4 − 1 5!x 5 − 1 6!x 6 − 1 7!x 7
to the above, we gain that the function
x 4 e 1/x − 1 − ψ ′ (x) − 1 24 − 1 24x
− 1 6!x 2 + 17 6!x 3 is completely monotonic on (0, ∞). By further adding three completely monotonic terms we finally earn that the function
x 4 e 1/x − 1 − ψ ′ (x) − 1 24 + 17 6!x 3
.
Integral representations. In [18, Theorem 1.2], among other things, 2 (1; k + 1, k + 2; t)t k e −zt d t (1.5) for ℜz > 0 and k ∈ {0} ∪ N, where the modified Bessel function of the first kind
17 ,
171 − e −u e −zu d u (1.10) for ℜz > 0. See the equation (4.3) in [18]. 1.3. Lower bounds for a modified Bessel function of the first kind. By the complete monotonicity obtained in [17, Theorem 3.1] and [18, Theorem 1.1] for h(t), by the integral representation (1.10), and by Lemma 2.6 below, it was deduced in [Theorem ∞) if and only if α ≥ 1 and β ≥ 1. More strongly, it was discovered in [17, Theorem 5.1] that, when β ≥ 1, the function
Theorem 1. 1 ([ 17 ,
117Theorem 4.1]). The function h 1,β (t) is completely monotonic on (0, ∞) if and only if β ≥ 1.
.
When ν is fixed, |z| is large, and µ = 4ν 2 ,
the term 23e 2u − 2373u − 1345 has a unique minimum at u = 1 2 ln 2373 46 = 1.97 . . . on (0, ∞) and equals 23 e 6 − 368 = 814.86 . . . at the point u = 3, (3) and 7215e 2u − (3249 + 793u) has a unique minimum at u = − 1 2 ln 1110 61 on (−∞, ∞) and is positive on (0, ∞).
F 5 (u) > 0 on (0, 6). A straightforward computation shows that the sequence {b k | 0 ≤ k ≤ n} in Lemma 2.2 applied to F 4 (u, by virtue of Lemma 2.2, that F 4 (u) is positive on [0, 1].
F 8
8(u) = F 7 (u + 1) = F 6 (u + 2) = F 5 (u + 3) = F 4 (u + 4), or F 9 (u) = F 8 (u + 1) = F 7 (u + 2) = F 6 (u + 3) = F 5 (u + 4) = F 4 (u + 5) on the unit interval [0, 1], which can be respectively verified by Lemma 2.2 as done in the proof of the positivity of the function F 4 (u) on [0, 1].
It is clear that M m (k) may be regarded as a quadratic polynomial of k and it has a unique possible minimum point − B(m) 2A(m) , which, due to k ≥ 4 and 0 ≤ m ≤ k − 2, should satisfy − B(m) 2A(m) ≥ m + 2. But, the fact is that − B(m) 2A(m) < m + 2, that is, 2(m + 2)A(m) + B(m) = 3 m+8 + m 3 − 3m 2 − 112m − 780 2 m+3 3 m+5 + m 3 + 23m 2 + 166m + 492 4 m+5 − 2 4m 2 + 52m + 141 3 m+5
and, by integration by part and the recursion (3.3),
where f 1 (u) = [10u(e u − 261) + 3966]e 2u , f 2 (u) = 69e 2u − 7119u − 4035 e u , f 3 (u) = 3249e 2u − 793u − 3249, and these three functions are all increasing respectively on the intervals [5, ∞), [3, ∞), and [0, ∞). Then it follows that F 3 (u) is increasing and positive on [5, ∞).
5 + 58m 4 + 278m 3 + 407m 2 − 825m − 1728 3 m+4and, by Descartes' Sign Rule, the polynomial P 3 (m) = 4m 5 + 58m 4 + 278m 3 + 407m 2 − 825m − 1728 has one possible positive zero. Since P 3 (0) = −1728 and P 3 (1) = 1530, it follows that P 3 (m) is positive for m ≥ 2. Consequently, we obtain that C(m) defined in (3.8) is positive for m ≥ 6. that C(m) are positive for all nonnegative integers m ≥ 0.By similar argument to above, we can determine that B(m) < 0 for all nonnegative integers m ≥ 0.
(ku − 4)e −ka = uK 4 (a) − 4K 3
Recently it was proved in[16] that the completely monotonic degree of Ψ(x) with respect to x ∈ (0, ∞) is 4, that is,).
(1.18)
deg x
cm [Ψ(x)] = 4.
(1.19)
For more results on complete monotonic degrees, please refer to [6, 9, 10, 16].
1.6. Main results of this paper. Since the complete monotonicity of h(t) and
the limit (1.3) have been verified in [17, Theorem 3.1] and [18, Theorem 1.1], we
naturally consider to compute the completely monotonic degrees of the completely
monotonic function
H α,β
5 )
5the completely monotonic degree of H α,β (t) is 1; By virtue of the monotonicity and unimodality of the function (1.22) obtained in Theorem 1.2, the quantity (1.25) follows. The proof of Theorem 1.3 is complete.
Remark 5.1. We note that Lemma 2.3 has been generalized in [12, Lemma 2.2].
p. 22, Theorem 2] to prove the inequality (1.21). By the way, we can reformulate [15, Theorem 1] as follows: For x ∈ [0, b] and n ≥ 0, we have
M 0 (k) = 3360 54 − 137k + 74k 2 , M 1 (k) = 1568 6480 − 7306k + 1909k 2 ,
t 2 ψ ′′ (t) + e 1/t t e 1/t − ψ ′ (t) − 1 p(t).By virtue of (2.6) for n = 1, 2 and the expansione 1/t = 1 + ∞ m=1 1 m! 1 t m , t = 0, we have p(t) ∼ 3 k=0 1 k!t k + O 1 t 3 − t 2 1 t 2 + 1 t 3 + 1 2t 4 − 1 6t 6 + O 1 t 6 t 4 k=0 1 k!t k + O 1 t 4 − 1 t + 1 2t 2 + 1 6t 3 − 1 30t 5 + O 1 t 5 − 1 ∼ 1 3!t 3 + O 1 t 3 t 1 4!t 4 + O 1t 4 → 4 as t → ∞. This implies that deg t cm [H 1,1 (t)] ≤ 4. (4.1)
Acknowledgements. The author would like to express heartfelt thanks to Professor Christian Berg at Copenhagen University for his valuable comments and helpful suggestions to the draft of this manuscript.is completely monotonic on (0, ∞).If one can manage to remove the last term176!x 3 , then the first result (1.23) in Theorem 1.3 follows.Remark 5.8. Motivated by properties of the functions F β (u) and G β (u) defined in (1.12) and (1.22) respectively, we conjecture that (1) when 3 ≤ k ≤ 5 and β ≥ 1, the functionis decreasing on (0, ∞);(2) when 3 ≤ k ≤ 5 and 0 < β < 1, the function H k,β (u) is unimodal on (0, ∞); (3) when 3 ≤ k ≤ 5 and 0 < β < 1, the function 1 H k,β (u) is convex on (0, ∞).Remark 5.9. We conjecture that for all k ≥ 6 the inequality (1.21) does not hold on (0, ∞).
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China E-mail address: [email protected], [email protected], qifeng618@qq. 454010Henan ProvinceSchool of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo CitySchool of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China E-mail address: [email protected], [email protected], [email protected] URL: http://qifeng618.wordpress.com
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"SOFTWARE-PRACTICE AND EXPERIENCE Automation of Application-level Caching in a Seamless Way",
"SOFTWARE-PRACTICE AND EXPERIENCE Automation of Application-level Caching in a Seamless Way"
]
| [
"Jhonny Mertz \nUniversidade Federal do Rio Grande do Sul (UFRGS)\nPorto AlegreBrazil\n",
"Ingrid Nunes \nUniversidade Federal do Rio Grande do Sul (UFRGS)\nPorto AlegreBrazil\n\nDortmund, DortmundGermany\n"
]
| [
"Universidade Federal do Rio Grande do Sul (UFRGS)\nPorto AlegreBrazil",
"Universidade Federal do Rio Grande do Sul (UFRGS)\nPorto AlegreBrazil",
"Dortmund, DortmundGermany"
]
| [
"Softw. Pract. Exper"
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| Meeting performance and scalability requirements while delivering services is a critical issue in web applications. Recently, latency and cost of Internet-based services are encouraging the use of applicationlevel caching to continue satisfying users' demands and improve the scalability and availability of origin servers. Application-level caching, in which developers manually control cached content, has been adopted when traditional forms of caching are insufficient to meet such requirements. Despite its popularity, this level of caching is typically addressed in an ad-hoc way, given that it depends on specific details of the application. Furthermore, it forces application developers to reason about a crosscutting concern, which is unrelated to the application business logic. As a result, application-level caching is a time-consuming and error-prone task, becoming a common source of bugs. Among all the issues involved with application-level caching, the decision of what should be cached must frequently be adjusted to cope with the application evolution and usage, making it a challenging task. In this paper, we introduce an automated caching approach to automatically identify application-level cache content at runtime, by monitoring system execution and adaptively managing caching decisions. Our approach is implemented as a framework that can be seamlessly integrated into new and existing web applications. In addition to the reduction of the effort required from developers to develop a caching solution, an empirical evaluation showed that our approach significantly speedups and improves hit ratios, with improvements ranging from 2.78% to 17.18%. Prepared using speauth.cls [Version: 2010/05/13 v3.00] arXiv:2011.00247v1 [cs.SE] 31 Oct 2020 2 J. MERTZ AND I. NUNESworkload characteristics and access patterns. Consequently, initial caching decisions may become obsolete over time[2,3].Because application-level caching is essentially a manual task, its design and implementation are time-consuming and error-prone. Moreover, because its implementation is often interleaved with the business logic, it decreases code understanding, thus being a common source of bugs. Existing research addresses these limitations through static and dynamic analyses that identify caching opportunities, such as web pages[4], and database queries and objects[2]. By focusing on particular bottlenecks, proposed approaches help developers while addressing caching problems, but complex logic and personalized web content remain unaddressed. Recent work focuses on identifying cacheable methods that repeatedly perform the same computation [5], but it is limited to the suggestion of performance fixes, and developers should review suggestions and manually refactor the code, inserting cache logic into the application.We thus, in this paper, propose a novel seamless and automated approach that chooses and manages cacheable content according to observations made by monitoring web applications at runtime, adding automatic and adaptive caching that leads to statistically significant speedups and hit-ratios. The automatically selected cache configuration reflects the monitored application workload, thus being sensitive to the application changing dynamics and self-optimizing caching decisions. As opposed to traditional caching approaches, our proposal focuses on caching methodlevel content. Moreover, we use application-specific information to make caching decisions, such as the user session, which are information that developers take into consideration while developing a caching solution. Thus, our approach can potentially reduce the reasoning and effort required from developers. In addition to the reduction of the effort required from developers to develop a caching solution, our approach is implemented as a framework, named APLCache, which seamlessly integrates the proposed solution to web applications. Consequently, our approach and framework can prevent code tangling and raise the abstraction level of caching as well as detach caching concerns from the application.We evaluated our approach empirically with three open-source web applications, which have different domains and sizes. Obtained results indicate that our approach can identify adequate caching opportunities by improving application throughput by factors between 2.78%-17.18%. Our approach can thus support developers by providing an automated approach to address issues related to the development of an application-level caching solution. As opposed to related work that addresses solely specific content, such as database-related methods or web page content, our approach can be applied to cache results of any computation done by a web application, which includes complex logic and personalized web content. Alternatively, it can be used as a decisionsupport tool to help developers in the process of deciding what to cache, guiding them while manually implementing caching.In summary, we provide the following contributions: (i) a caching approach focused on integrating caching into web applications in a seamless and straightforward way, providing an automated and adaptive management of cacheable methods; (ii) a framework that detaches caching concerns from application code; and (iii) an empirical evaluation of our automated approach that indicates that it can effectively identify cacheable opportunities in web-based applications.We next provide background on application-level caching. We detail the proposed approach in Section 3, and give details of its implementation as a framework in Section 4. An empirical evaluation of our approach is presented in Section 5. Limitations and related work are discussed in Sections 6 and 7, respectively. Finally, we conclude in Section 8. | 10.1002/spe.2571 | [
"https://arxiv.org/pdf/2011.00247v1.pdf"
]
| 21,699,382 | 2011.00247 | 88be42a91b54b7d460e3be3c2e2188d7cdf2fcd6 |
SOFTWARE-PRACTICE AND EXPERIENCE Automation of Application-level Caching in a Seamless Way
2010
Jhonny Mertz
Universidade Federal do Rio Grande do Sul (UFRGS)
Porto AlegreBrazil
Ingrid Nunes
Universidade Federal do Rio Grande do Sul (UFRGS)
Porto AlegreBrazil
Dortmund, DortmundGermany
SOFTWARE-PRACTICE AND EXPERIENCE Automation of Application-level Caching in a Seamless Way
Softw. Pract. Exper
00201010.1002/speReceived . . .application-level cachingweb applicationcacheframeworkadaptive systems
Meeting performance and scalability requirements while delivering services is a critical issue in web applications. Recently, latency and cost of Internet-based services are encouraging the use of applicationlevel caching to continue satisfying users' demands and improve the scalability and availability of origin servers. Application-level caching, in which developers manually control cached content, has been adopted when traditional forms of caching are insufficient to meet such requirements. Despite its popularity, this level of caching is typically addressed in an ad-hoc way, given that it depends on specific details of the application. Furthermore, it forces application developers to reason about a crosscutting concern, which is unrelated to the application business logic. As a result, application-level caching is a time-consuming and error-prone task, becoming a common source of bugs. Among all the issues involved with application-level caching, the decision of what should be cached must frequently be adjusted to cope with the application evolution and usage, making it a challenging task. In this paper, we introduce an automated caching approach to automatically identify application-level cache content at runtime, by monitoring system execution and adaptively managing caching decisions. Our approach is implemented as a framework that can be seamlessly integrated into new and existing web applications. In addition to the reduction of the effort required from developers to develop a caching solution, an empirical evaluation showed that our approach significantly speedups and improves hit ratios, with improvements ranging from 2.78% to 17.18%. Prepared using speauth.cls [Version: 2010/05/13 v3.00] arXiv:2011.00247v1 [cs.SE] 31 Oct 2020 2 J. MERTZ AND I. NUNESworkload characteristics and access patterns. Consequently, initial caching decisions may become obsolete over time[2,3].Because application-level caching is essentially a manual task, its design and implementation are time-consuming and error-prone. Moreover, because its implementation is often interleaved with the business logic, it decreases code understanding, thus being a common source of bugs. Existing research addresses these limitations through static and dynamic analyses that identify caching opportunities, such as web pages[4], and database queries and objects[2]. By focusing on particular bottlenecks, proposed approaches help developers while addressing caching problems, but complex logic and personalized web content remain unaddressed. Recent work focuses on identifying cacheable methods that repeatedly perform the same computation [5], but it is limited to the suggestion of performance fixes, and developers should review suggestions and manually refactor the code, inserting cache logic into the application.We thus, in this paper, propose a novel seamless and automated approach that chooses and manages cacheable content according to observations made by monitoring web applications at runtime, adding automatic and adaptive caching that leads to statistically significant speedups and hit-ratios. The automatically selected cache configuration reflects the monitored application workload, thus being sensitive to the application changing dynamics and self-optimizing caching decisions. As opposed to traditional caching approaches, our proposal focuses on caching methodlevel content. Moreover, we use application-specific information to make caching decisions, such as the user session, which are information that developers take into consideration while developing a caching solution. Thus, our approach can potentially reduce the reasoning and effort required from developers. In addition to the reduction of the effort required from developers to develop a caching solution, our approach is implemented as a framework, named APLCache, which seamlessly integrates the proposed solution to web applications. Consequently, our approach and framework can prevent code tangling and raise the abstraction level of caching as well as detach caching concerns from the application.We evaluated our approach empirically with three open-source web applications, which have different domains and sizes. Obtained results indicate that our approach can identify adequate caching opportunities by improving application throughput by factors between 2.78%-17.18%. Our approach can thus support developers by providing an automated approach to address issues related to the development of an application-level caching solution. As opposed to related work that addresses solely specific content, such as database-related methods or web page content, our approach can be applied to cache results of any computation done by a web application, which includes complex logic and personalized web content. Alternatively, it can be used as a decisionsupport tool to help developers in the process of deciding what to cache, guiding them while manually implementing caching.In summary, we provide the following contributions: (i) a caching approach focused on integrating caching into web applications in a seamless and straightforward way, providing an automated and adaptive management of cacheable methods; (ii) a framework that detaches caching concerns from application code; and (iii) an empirical evaluation of our automated approach that indicates that it can effectively identify cacheable opportunities in web-based applications.We next provide background on application-level caching. We detail the proposed approach in Section 3, and give details of its implementation as a framework in Section 4. An empirical evaluation of our approach is presented in Section 5. Limitations and related work are discussed in Sections 6 and 7, respectively. Finally, we conclude in Section 8.
INTRODUCTION
With the increasing popularity of web applications and software systems distributed on top of the web, it is crucial to improve their performance and scalability due to a large number of users. When traditional caching solutions are unable to meet performance requirements, applicationlevel caching is adopted to store content at a granularity that is possibly best suited to the application, thus allowing developers to separate common from customized content at a finegrained level. It thus has become a popular technique to reduce the workload on content providers, which can thus decrease the user perceived latency. However, deciding the right content to cache and the best moment of caching is a challenging task, given that it depends on extensive knowledge of application specificities to be done efficiently. Otherwise, caching may not improve application performance or even may lead to a performance decay [1]. Furthermore, developers must continuously inspect application performance and revise caching design choices, due to changing In short, the client component is essentially the client's computer and web browser; while the Internet component contains a wide range of different, interconnected mechanisms to enable the communication between client and server. The server can include multiple and different servers that are collectively seen as the web server by the client [8].
Each caching location has its benefits, challenges, and issues that together lead to trade-offs to be resolved when choosing a caching solution. For instance, particular forms of content can be cached according to a selected choice of location as well as the abstraction level provided, which can be fully transparent to the application or tightly integrated into the application code. In addition, hit and miss probabilities vary across different locations [4,9,10,1,11,12]. Due to the variety of application domains, workload characteristics, and access patterns, no universal web caching solution outperforms other caching options in all possible scenarios. Therefore, caching at these different locations is complimentary.
Application-level caching resides on the server side. As opposed to traditional caching alternatives, which are transparent to the application, application-level caching is tightly integrated into the application base code. It becomes needed in modern web applications because they manipulate and process customized content and, in this case, caching final web pages provides limited benefits. Therefore, application-level caching can be used to separate generic from customized content at a fine-grained level.
As an example, we present in Figure 2 a scenario in which application-level caching is used to lower the database workload. In this example, an e-commerce application has a ProductsRepository class, which is responsible for loading products from the database. First, the web application receives a request for all the products from a user (step a), which eventually leads to an invocation to the ProductsRepository class. However, such class makes a database query on DBAccess, and calling and executing DBAccess may imply an overhead regarding computation or bandwidth. Therefore, ProductsRepository manages to cache DBAccess results and, for every request, ProductsRepository verifies whether DBAccess should be called or there are previously computed results already in the cache that can be used (step b). Then, the cache component performs a look up for the requested data and returns either the cached result or a not found error. If a hit occurs, it means the content is cached, and ProductsRepository can avoid calling DBAccess. However, when a miss occurs and then a not found error is returned, it means that DBAccess computation is required (steps c and d). The newly fetched result of DBAccess can then be cached to serve future requests faster. These steps, described in Figure 2(b), are those typically performed in any cache implementation. The key difference is that, in application-level caching, the responsibility of managing the caching logic is entirely left to application developers, who must manually handle the cache.
In this scenario, caching decisions are made explicit, which involves many issues [13]. The first is that developers must manually develop and insert the caching logic into the application base code, which involves content retrieval and translation as well as key assignment and consistency maintenance. Such logic is usually tangled with the business logic-as illustrated in Figure 2(a)and making the caching code a cross-cutting concern, i.e. it is spread all over the application base code, resulting in increased complexity and maintenance time [1]. Furthermore, such implementation requires developers to make caching decisions, such as choosing which objects to get, put or remove from the cache. Such decisions demand a significant effort and reasoning from developers because they need to understand what are the typical usage scenarios, how often the content selected to be cached is going to be requested, how much memory it consumes, and how often it is going to be updated. These are all non-trivial decisions [13].
Caching implementation can be supported by available libraries and frameworks, which provide implementations of a cache system, e.g. Caffeine * , EhCache † and Memcached ‡ . Although there are existing tool-supported approaches that raise the abstraction level of caching and prevent adding much cache-related code to the base code [14,1], caching decisions such as determining what should be cached remains as a developer responsibility.
A fundamental problem of application-level caching is that all issues above usually demand extensive knowledge of the application to be properly solved. Consequently, developers manually design and implement solutions for all these mentioned tasks. However, even when adopting a caching solution to improve the application performance, the issues and challenges concerning maintenance remain unaddressed. While designing and implementing application-level caching, developers usually specify and tune cache configurations and strategies according to application specificities. Nevertheless, an unpredicted or unobserved usage scenario may eventually emerge. As the cache is not optimized for such situations, it would likely perform sub-optimally [3]. As a result, to achieve caching benefits so that the application performance is improved, it is necessary to tune cache decisions continuously [15].
This shortcoming motivates the need for adaptive caching solutions, which could overcome these problems by automatically adjusting caching decisions according to the application specificities to maintain a required performance level. Moreover, an adaptive caching solution minimizes the challenges faced by developers, requiring less effort and providing a better experience with caching.
Selection of Cacheable Content
Although application-level caching is commonly being adopted, the selection of cacheable content is typically an ad hoc and empirical process. To find caching best practices, developers can make use of widespread knowledge, consult development blogs, or simply search online for tips. Nevertheless, 5 Figure 3. Cacheability Pattern [13]. this knowledge is unsupported by concrete data or theoretical foundations that demonstrate its effectiveness in practice. Thus, developers usually implement the necessary cache logic for the assumed cache opportunities and evaluate the performance improvement empirically based on benchmarks and application executions.
Given this scenario, in previous work [13], we analyzed how developers deal with applicationlevel caching. This work allowed us to derive a set of patterns that capture criteria used to make cache-related decisions, giving practical guidelines for developers to appropriately design caching in their applications. One of such patterns, namely the Cacheability Pattern, focuses on the selection of cacheable content, more specifically method calls, and is specified in a flowchart, presented in Figure 3.
Although such pattern is conceived to reuse, it is an abstract solution that requires specific reasoning and coding to make use of it. In real-world situations, developers should reason about application specificities and decide whether to cache or not a specific method regarding the criteria. No objective measurement to evaluate whether a method satisfies each criterion has been provided. Therefore, although such an approach systematizes the decision regarding content cacheability, developers must still have knowledge of the application requirements, workload, domain, access patterns and business logic as well as manually select content to be cached.
AUTOMATED APPLICATION-LEVEL CACHING APPROACH
Based on the structured and documented knowledge captured by the Cacheability Pattern, we propose an approach that can automatically make decisions regarding cacheable content. In this section, we first present an overview of our approach. Next, we detail each of its key activities.
Approach Overview
Our approach is based on the decision process presented in Figure 3, which gives the criteria to be taken into consideration to make caching decisions. For each criterion, we propose objective means of measuring application methods and evaluating whether they should be cached. This evaluation is based on data collected by monitoring the workload of web applications at runtime. Furthermore, our automated approach takes into account application-specific details in caching decisions, which is, in fact, the information considered by developers in the development of a caching solution, thus providing a solution that is specific and optimized to the problem of selecting cacheable content. By providing an automated approach, we relieve developers from the process of instantiating that abstract solution according to their specific domains or characteristics.
Our proposed solution can be incorporated into web applications so that it can monitor and choose content to be cached according to changing usage patterns. It consists of two complementary asynchronous parts: (a) a reactive model applying, responsible for monitoring traces of the application execution and caching method calls (previously identified as caching opportunities) at runtime; and (b) a proactive model building, which analyzes on the background the behavior of the application, taking into account application-specific information, and finds cacheable opportunities. Figure 4 presents an overview of the dynamics of our approach, indicating the activities that comprise its running cycle. These asynchronous parts involve performing three different activities: (1) data tracking, (2) data mining and (3) cache management.
Because our approach monitors the application at runtime and self-optimize caching decisions, it is sensitive to the evolution of application workload and access patterns. This addresses the main problem regarding cache maintenance because it forces developers to revise their cache decisions constantly. Moreover, the integration between our approach and the application is seamless and does not require manual inputs, detaching caching concerns from the application and providing higher cohesion and lower coupling. As result, our approach can reduce the reasoning, time and effort required from developers to develop a cache solution, allowing them to dedicate time to write the most relevant code (i.e. business logic). If full automation is not desired, our approach can alternatively serve as a decision-support tool to help developers in the process of deciding what to cache, guiding them while manually implementing caching.
We next conceptually describe each activity of our approach. Later, we show how they are operationalized within an implemented framework.
Data Tracking: Monitoring Execution Traces
As opposed to traditional caching approaches, which cache content such as web pages or database queries, our approach focuses on caching method-level content. Moreover, we use applicationspecific information to make caching decisions, such as the user session, execution time of method calls, and data and cache sizes.
To monitor the application behavior, we collect method invocations or calls (i.e. application execution traces) at runtime. Related to this monitoring process, two issues must be addressed. First, we must specify what information should be recorded when invoking methods. Second, given that the monitored and recorded information may be a complex structure, it is essential to provide means of dealing with such complexity.
Concerning the first issue, we adopt a lightweight and conservative approach. It is lightweight because it is based on recording just the input and output of each method call. As stated by Toffola et al. [5], recording detailed information before and after each method call does not scale to large web applications with a high number of concurrent users. Thus, we focus on information regarding the arguments passed to the call and the return value of the call (if any), because this information is sufficient to describe the relevant input and output state for most of the methods. The monitoring process is also conservative in the sense that the recorded information is the complete method call, consisting of the method identification (i.e. its signature), the values of all method inputs, its output (i.e. its returned value), and additional information, namely cost and user session.
Each recorded method call is a tuple s, P, r, c, u where s is the representation of the target of the call, P = [p 1 , . . . , p n ] is a list of parameters of the call, r is the returned value, c is the cost of computing the method, and u is the user session associated with the method call. The cost can be the time taken to execute the method, memory consumed, or any other developer-specified resource. Finally, these method calls are mapped into a generic representation that is saved as a string. The representation describes the data itself and the shape of the data item. The representation captures the structure of objects and is independent of the memory locations where objects are stored or other globally unique identifiers of objects. Figure 5 gives an example of how a method call is mapped into a record.
Data Mining: Identifying Cacheable Method Calls
The second activity of our approach takes as input the output of the previous activity (data tracking), which provides us with information needed to identify methods to be cached. To decide what should be cached, the second activity is based on the mining of such information.
As said, the reasoning part of our automated approach is based on the decision process presented in Figure 3. This reasoning model specifies a sequence of questions to be answered that are associated with criteria to be analyzed. By chaining different decisions taking into account each criterion, an importance relationship among them is established. Content changeability is the first analyzed criterion, followed by usage frequency, shareability, retrieval complexity, and size properties. However, given the pattern was conceived to be used at design time, using it at runtime requires adaptations to be made.
These adaptations concern the size-related criteria, which make more sense at design time, because developers may not have enough information to predict the size required to store a particular piece of content and the size available in the cache. Consequently, in this case, some data may be chosen as not cacheable. However, at runtime, it is possible to know the size of the content to be cached as well as the available cache size and occupation ratio.
Therefore, using size-related criteria to decide, at runtime, what to cache (as suggested by the Cacheability pattern) can lead the cache to a non-maximum utilization, i.e. cacheable methods could be classified as uncacheable even with enough free space in the cache. To avoid this, we use the sizerelated information to decide whether a cacheable method should be put in the cache. Consequently, we do not consider the size-related criteria in the identification of cacheable method calls in the data mining activity, but they are taken into account in the next activity. As result, we obtain a simplified reasoning process to be automated and used while identifying cacheable content, which is presented in Figure 6.
To be automated, this process needs to have its decision criteria analyzed to make decisions. However, in the Cacheability Pattern, there are no objective definitions. This means that, when adopting this pattern, developers must provide a meaning for each criterion. We thus, as part of our approach, propose objective measurements to evaluate each criterion. There are five specified measurements, which are detailed in Table I.
Essentially, the objective evaluation is a statistical analysis of collected traces. All the information required by each criterion is presented in Table II. Note that due to a limited amount of data-such as considering a method with only a few executions as static, frequent, or less changing-wrong conclusions can be reached. Therefore, in situations in which we have an insufficient amount of data, we assume Undefined as the result of the criterion analysis.
Based on our objective criteria evaluation and the simplified decision process, our approach identifies cacheable methods. It is important to note that one method may result in many cacheable opportunities (and consequently many entries in the cache) because our approach distinguishes and analyzes method calls, which are specified as a combination of the method signature and parameter values. In the next activity, described as follows, we detail how we cache method calls defined as cacheable opportunities, using the two remaining criteria, not taken into account in this activity. 9 Table I. Objective Evaluation of the Cacheability Pattern Criteria.
Question Criterion Meaning
Is the data completely static?
Staticity A method staticity is associated with how many times a method returns the same value when it receives the same parameters. Staticity is given by staticity(m) = |P Set |/|P r Set |, where P Set is the set of different lists P of parameter values received by a method m, and P r Set is the set of different tuples P, r , where P is a list of parameters values and r is the returned value. A method m is said to be completely static if staticity(m) = 1. Does the data change more than it is used?
Changeability
Static methods tend to achieve the highest hit ratio when cached. However, caching methods that often do not change can still bring benefits. Therefore, the changeability of a method is the dual of staticity, i.e. changeability(m) = 1 − staticity(m). To evaluate whether a method does not often change, we use as a reference value µ ch + k × σ ch , where µ ch and σ ch are the changeability mean and standard deviation, respectively, and k is a given number. The changeability criterion has a yes answer when the method changeability is below the reference value, i.e. when it is k standard deviations below the changeability mean. Is the data used by many requests?
Frequency
A method frequency is associated with how many times a method is called, and this criterion is used in our approach also to assess whether the collected trace sample is large enough to make decisions. Therefore, we use a specified threshold to distinguish frequent from unfrequent methods. The threshold is the sample size size(c, e), where c is a confidence level e e is a margin of error. If the number of collected traces of a method is above the required sample size, it is said to be frequent. Is the data userspecific?
Shareability
The method shareability gives how much the results of a method call are shared among different users because if results of a method are shared among many users, caching this method may potentially increase the hit ratio. A method shareability shareability(m) is the percentage of different user sessions in which requests lead to a method call with the same parameter values. Similarly to frequency, a method is said shared if its shareability is k standard deviations σ sh above the shareability mean µ sh , that is, shareability(m) > µ sh + k × σ sh . Anonymous method calls (not associated with any user) are not taken into consideration in this criterion. Is the data expensive to compute? Expensiveness A method expensiveness is associated with the cost cost(m) for computing it, which can be the time taken to compute it or consumed memory, for example. As above, a method is said expensive if cost(m) > µ ct + k × σ ct , where µ ct and σ ct are the cost mean and standard deviation, respectively.
Cache Management: Caching Identified Opportunities
With the previous activities, we identify a set of cacheable method calls. We now discuss how our approach manages the cache component to cache the selected method calls as well as to keep consistency. Similarly to the data tracking activity, through monitoring the application execution, we intercept calls to cacheable methods and assess whether the content associated with the call is in the cache.
As our approach solely learns whether method executions should be cached, other cache concerns, such as eviction and consistency, were addressed with standard solutions. To make our approach less dependent on the effectiveness of alternative cache policies and algorithms, as a default configuration, we suggest a conservative approach that caches content only when there is enough space in the cache, considering the data size and cache size, the two remaining decisions of the Cacheability pattern, which are presented in Table III. Moreover, to periodically free space in the cache and remove outdated content, a time-to-live (TTL) should be specified. With TTL, cached methods expire after a time in cache, regardless of possible changes. This frees cache space for caching new content associated with cacheable methods. In addition, TTL is a popular solution to deal with consistency issues, which requires to relax freshness and admit potential staleness to increase performance and scalability. During the execution, when a cacheable method is called and the returned content is not in the cache, we estimate how much space of the cache this method call requires and verify whether the cache has the corresponding free space to allocate such content. If there is no enough space in the cache, no content is cached until TTL expires cached data. It is important to note that cache size and TTL are both domain-specific and have no relation to the identification of cacheable content. Thus, these values should be manually specified when instantiating our approach.
APLCACHE FRAMEWORK
The three main activities of our approach, conceptually described above, were implemented as a framework, namely APLCache, that can be instantiated to integrate web applications. Our framework is implemented in Java, thus can be used with Java web applications. This choice is due to our previous programming experience and tools available that were adopted as part of our implementation. Moreover, we adopted a set of technologies that provide an appropriate infrastructure for the framework. Used technologies are highlighted in Figure 7, which presents the APLCache architecture, with its modules and communication among them.
To collect data to be analyzed and manage cacheable methods, we intercept method executions using aspect-oriented programming (AOP), more specifically the AspectJ * implementation. AOP 11 provides an easy way to weave code at specified points, without changing the base code. Considering the explained generic representation to save application traces, to compare input and output data of method calls and ensure that structurally equivalent objects have the same representation, objects are loaded and compared by using implementations of the equals and hashCode methods. Saving actions are performed in an execution thread separate from the one that is processing the request, minimizing response delays. Our framework also provides means for developers to make customizations using hints, indicating possible locations of cacheable methods, which can improve the set of caching candidates as well as exclude methods that should not be cached, thus saving the time of tracking them.
Available solutions of caching components were also adopted. APLCache automates the decision of what to cache, while these caching components provide APIs that allow us to manipulate data and access metadata information about the cache, such as statistics and configurations. Our framework is decoupled from particular caching components, and thus supports the most popular distributed cache systems and libraries, which can be configured through property files and annotations. Therefore, APLCache provides a fully customizable environment where different cache policies and algorithms can be configured and used along with the proposed approach to detect cacheable content. Nevertheless, although we provide an off-the-shelf solution, we strongly encourage developers to customize and tune components according to their needs and preferences to achieve improved results.
The collected data are analyzed offline, separately from the web application, to prevent impact in the application performance-it can even run on a dedicated machine. To evaluate shared execution traces (accessed by multiple users), we obtain the user session, which is an application-specific information thus taken into account only in application-level caching. Our framework provides a set of alternative implementations to obtain this information from the most popular web frameworks, such as Java EE and the Spring Framework * . In case alternative ways of managing user sessions are adopted, developers should implement interfaces provided by our framework.
APLCache not only can be used to incorporate our proposal to web applications but was also used in our evaluation, which is presented next.
EVALUATION
In this section, we proceed to the evaluation of our automated caching approach, measuring two aspects: (i) performance; and (ii) differences between developers' caching decisions and our approach. Given that collecting data to make caching decisions is required either if developers manually analyze such information, or automatically by an algorithm, our evaluation focuses on the caching decision rather than the monitoring process. We first describe our evaluation procedure and then discuss obtained results and threats to validity.
Goal and Research Questions
As stated in the introduction, our primary objective while proposing this approach is to provide automation and guidance to developers when adopting application-level caching in their applications. This approach aims to provide such guidance using an automated and seamless application-centric approach, i.e. by identifying cacheable opportunities in their applications, taking into account application details. The evaluation of our approach is based on the goalquestion-metric (GQM) approach proposed by Basili et al. [16]. The GQM was adopted to define the goal of the evaluation, the research questions to be answered to achieve the goal and metrics for responding to these questions. Following this approach, we present in Table IV the description of the evaluation, following the GQM template. To achieve our goal, we investigated different aspects of our automated approach, which are associated with three key research questions presented in Table V along with their metrics. Given that caching is essentially a technique to improve performance and scalability, RQ1 concerns evaluating this. Although our approach relieves the developer from this task, it still needs to provide performance improvements. Determining the cacheable content and the right moment of caching or clearing the cache content are a developer's responsibility and might not be trivial in complex applications. In RQ2, we aim to identify what methods were considered caching opportunities and how they can be compared to the choices manually made by developers.
Procedure
To evaluate our approach, we used performance test suites, which aim to simulate realworld workloads and access patterns [17]. Furthermore, these tests have been used to evaluate improvements of caching in web applications [5,2].
Simulations were performed using three different caching configurations: (i) no application-level caching (NO), (ii) application-level caching manually designed and implemented by developers (DEV); and (iii) our approach (AP). To assess performance, we used three metrics: throughput (number of requests handled per second), hit ratio and the total number of hits, because they are well-known in the context of web applications and cache performance tests.
Our simulation emulated client sessions to exercise applications and evaluate decision criteria. Simulations consisted of variations of 1, 5, 10, 25 and 50 simultaneous users constantly navigating through the application, always at a limit of 500 requests per user. Each simulation was repeated ten times, and the mean of each metric was collected. Each emulated client navigates from an application page to another, selecting the next page from those accessible from the current page, to better represent a real user. The navigation process starts on the application home page and follows a non-uniform random selection that falls into a distribution where 80% of the requests are readonly, while the remaining 20% perform at least one write operation. This distribution is mentioned in standard performance benchmarks such as TPC-W * and RUBiS † . The evaluation of different criteria requires different parameters. We used a web application, not used in our evaluation, to empirically choose these parameters. As result, we adopted 99% and 3% as the confidence level and margin of error, respectively, for the frequency criterion. For shareability and expensiveness, we adopted k = 1, while for changeability, k = 0. For expensiveness, the cost corresponds to the method execution time.
13
For all the executions of AP, caches were bounded by a cache size and configured with a TTL, which were both specified and used by developers of our target systems (DEV). As the results may be influenced by the cache policy adopted, in addition to our standard technique based on TTL, we also evaluated our approach combined with popular replacement policies, namely least recently used (LRU) or least frequently used (LFU). Finally, we used information collected over 2 minutes to build the caching decision model in AP scenario.
Our evaluation was performed with three open-source web applications * , presented in Table VI, which summarizes the general characteristics of each target system. To prevent application bias in our results, we selected applications with different sizes (6.3-111.3 KLOC) and domains. Cloud Store, in particular, is developed mainly for performance testing and benchmarking, and follows the TPC-W performance benchmark standard. It is important to highlight that the DEV configuration was implemented by developers independently from the results of our previous work [13] and, therefore, the Cacheability Pattern was not considered in this implementation.
For Pet Clinic and Cloud Store, we used test cases written by their developers and developed test cases for Shopizer, for which they are unavailable. For the latter, we created test cases to cover searching, browsing, adding items to shopping carts, checking out, and editing products.
We used Tomcat † with 2G RAM dedicated its JVM as our web server and MySQL ‡ as the DBMS. We used two machines located within the same network, one machine for the DBMS and web server (16G RAM, Intel i7 2GHz), and one machine for JMeter § (32G RAM, Intel i7 3.4GHz). The selected underlying caching framework is EhCache because it is used in all target applications. Moreover, we also used the same cache component configurations (TTL and in-memory cache size) as specified by developers in each application. Regarding the TTL implementation and its operation, we rely on the default behavior provided by the chosen cache provider.
Results
RQ1. What is the performance improvement provided by our automated caching approach?
Based on our simulation, we observed that our approach (AP) improves the throughput of all target applications, in comparison with no use of caching (NO). Moreover, when compared with the application-level caching manually implemented by developers (DEV), our approach achieves at least similar performance. After a manual investigation, we concluded that our approach caches methods that lead to a higher number of hits than those chosen by developers. Consequently, caching them significantly reduced the network transfer time, and thus resulted in a substantial performance improvement. Thus, even if results were not as good as those obtained with DEV, they could be considered good because our approach automates an error-prone and time-consuming task performed by developers. Table VII shows the throughput obtained for each target application in simulations with five simultaneous users.
As conclusion, AP provides higher performance improvements (2.78%-17.18%), when compared to the improvements provided by developers with manual caching implementation. Figures 8, 9 and 10 further complement the results with the throughput obtained with different amount of simultaneous users for each application. As can be seen, for the three studied applications, the throughput achieved by AP is higher or at least similar to all DEV executions even considering different levels of stress on the application server. However, in general, only a few methods are usually cached through application-level techniques. Thus, the advantage of caching is usually limited in scope, but yet beneficial to the overall performance of the system.
Regarding different cache policies, the evaluation with LRU and LFU resulted in similar throughput and hits as our standard configuration, for all the applications. This is explained by the fact that in all executions of our approach no eviction was necessary because the number of cacheable items never reached the maximum cache size before the TTL of existing content expired (which has a value specified by developers). Thus, we limit ourselves to present only the results achieved by our standard configuration, i.e. a TTL-based eviction. The complete results are available elsewhere * .
Table VIII presents the hit ratio and the total number of hits for each target application in simulations with five simultaneous users. When considering these metrics, DEV is used as the baseline for AP, and we observed that AP achieves good results in comparison with DEV. First, our approach shows a high hit ratio improvement for Cloud Store and Shopizer. This improvement is related to caching search operations because developers cache all search combinations, and our approach only caches searches that can potentially improve application performance (according to our evaluation criteria). For Pet Clinic, AP achieves a lower hit ratio than DEV. However, the total number of hits is 43.14% higher. Despite the decrease in hit ratio, it means that AP identified cacheable opportunities that in general provides more hits than misses, and thus it indicates that AP identifies good cacheable opportunities.
As conclusion, our approach was able to discover cacheable method calls at runtime, based on the application workload, with an improvement of our baseline. Therefore, it can relieve the burden from developers of identifying and implementing caching. Furthermore, if there is no enough trust to allow our approach to automatically manage the cache in a production environment, its caching decisions can be used as a guideline to developers while developing application-level caching, which is not trivial mainly in large applications. Our approach takes about 1 minute to analyze 2 million application traces in the machines with the described configuration. It is only required to derive the set of cacheable methods. Such process is executed in background and thus has minimal impact on the application. Note that it can alternatively be configured to run on a separate machine. Monitoring all method calls can also minimize the performance improvement of our approach. However, this depends on when and how traces are collected from the application. For example, samples can be collected during off-peak periods to not compromise the overall application performance.
RQ2
. What are the similarities and differences between decisions made by our automated approach and by developers? The results of our simulation showed that our approach can improve the performance of web applications. However, it is interesting to understand the causes of this improvement. Therefore, we now compare the number of caching opportunities that were selected and managed by our approach with the choice made by developers, implemented in the target applications. By making this comparison, we observed that our approach not only caches the methods selected by developers but many others, as shown in Table IX. However, the number of selected methods to be cached is small in comparison with all possible methods, as shown in the column AP.
Results indicate that developers may be conservative while identifying cacheable methods and select only those that lead to a strong confidence that caching them increases will result in cache hits. For instance, in Pet Clinic the number of veterinaries is often the same, and thus it is the only cacheable method identified by developers. Such opportunities are always detected by our approach. However, our approach was able to identify more cacheable opportunities, which justifies the performance improvement. As conclusion, our approach identifies a higher number of cacheable methods (46.66%-300%), when compared to the total number of cached methods identified by developers. For large applications like Shopizer, manually analyzing and identifying possible cacheable methods may be time-consuming or even infeasible in practice.
Furthermore, while implementing a cache solution, developers usually select cacheable methods, and thus any call to a cacheable method is cached. This can lead to a higher and inefficient use of the cache size because not all method calls are frequently called or expensive. Our approach, in contrast, can deal with specific method calls, leading to an optimal utilization of the cache infrastructure. Moreover, this makes the cache effectiveness less dependent on the cache replacement policy, because fewer method calls are added to the cache and, with adequate TTL and size, less eviction is needed to free space in the cache.
17 assumption regarding the workload when conducting our experiments and rely on the randomness added to the tests. Nevertheless, our approach does not depend on a particular workload and can find cacheable methods with any pre-specified workload, which may evolve over time in real world scenarios. Therefore, even if the workload changes substantially and initial cacheable methods are no longer useful, our approach can adapt itself, automatically discarding old caching configurations and discovering a new set of cacheable methods. Second, our evaluation involves only three target applications. Therefore, results may not be generalizable. To address this threat, we selected opensource applications, with different sizes and domains, implemented by different developers.
LIMITATIONS
Providing a caching solution requires dealing with many challenges other than that addressed in this paper, such as consistency management, replacement policies, and distributed infrastructures. Therefore, some challenges are out of the scope. Moreover, considering our challenge of deciding what to cache, we are aware of shortcomings of our approach. We next discuss these limitations of our work, and how they can be addressed.
Given that we monitor method inputs and outputs to make caching decisions, we assume that the output depends only on the provided inputs, i.e. the output is a function of the input. For example, an invoked method may not only provide an output but also change other objects, i.e. the state of the application. Therefore, if we cache this method, it will not be invoked but its result will be obtained from the cache, and these side effects of the method call will not be achieved, possibly leading the application to an inconsistent state. In this hidden state cases, developers could annotate the code to guide the approach towards avoiding tracking and caching such methods. It is reasonable to assume that the identification of methods that cannot be cached is easier than those that can be cached to improve performance. A static analysis of the source code may be enough to detect such methods.
Many popular caching frameworks adopt a weak consistency approach as default and use a timebased expiration policy to invalidate cached methods. Such approach favors performance [18] and is easier than defining a hard-to-maintain but more robust invalidation process. Our approach does not cover the challenge of when to expire cached content in the cache, so our implementation currently provides weak consistency, but this should be customized by developers if a more robust policy is required.
The overhead of the data tracking activity was not the focus of our evaluation. However, its impact did not prevent our approach to deliver cacheable opportunities in a timely fashion. However, if this activity has an impact on the application execution that is unacceptable, it can be configured to collect only samples or be enabled only at specific times. For example, after the analysis and identification of cacheable methods, the monitoring can remain disabled until the performance decreases, which means that the previously identified opportunities are not useful anymore.
Other caching issues not addressed in this paper, such as concurrency, scheduling, and replacement policies, are part of our future work. Given that our goal is to identify and cache content, we rely on the underlying caching frameworks for solving these issues. We use default configurations of caching providers and provide means for developers to customize such configurations.
RELATED WORK
In this section, we present work focused on supporting the implementation of a caching solution, which is one of our challenges, followed by automated and adaptive approaches to assess the cacheability level of data at the application-level.
Cache Implementation
A research challenge in application-level caching is how to reduce the burden of implementing application-level caching from developers. As a consequence, implementation-focused approaches have been extensively proposed. These solutions can raise the abstraction level of caching and reduce a significant amount of cache-related code to be added to the base code.
These approaches take the form of supporting libraries and frameworks, providing useful and ready-to-use cache-related features. However, such solutions require code changes and a manual integration with web applications to exploit caching benefits i.e. developers are responsible for managing the cache. Examples of such solutions are distributed cache systems, e.g. Redis * and Memcached, and libraries that cache content locally, e.g. Spring Caching † , EhCache, Infinispan ‡ , Caffeine and Rails low-level caching § . In addition, as caching is essentially a cross-cutting concern [19], aspect-oriented programming (AOP) [20] have been explored in order to provide a flexible and easy-to-use solution, such as Jcabi-aspects ¶ .
The drawback of traditional supporting libraries and frameworks is partially addressed by approaches that provide developers with ways to declare knowledge associated with the semantics of application code and data through annotations in the code, and then specific caching tasks are automated. Such solutions have a lower impact on the application, as it does not require to introduce code interleaved with its base code. For example, CacheGenie [14] and TxCache [1] provide cache abstractions based on a specific and simple declarative programming model where developers can indicate the methods that should be cached. Then the proposed approaches can automatically cache and invalidate the results. Similarly, Huang et al. [21] proposed a browser-side caching framework that allows developers to customize their caching strategies, such as expiration times, cache granularity, and replacement policies.
Totally seamless and transparent solutions are usually coupled to the application as a surrounding layer, such as database [7] or proxy-level [22] caching approaches. In this context, EasyCache [23] is a hybrid solution that combines properties of database caching and applicationlevel caching to provide transparent cache pre-loading, access and consistency maintenance without extensive modifications to the application or a complete redesign of the database. Similarly, AutoWebCache [19] can also be seen as a hybrid approach that associates the back-end databases and the dynamic web pages at the front-end. AutoWebCache uses AOP with pre-defined caching aspects to add caching of dynamic web pages to a servlet-based web application that interfaces a database with JDBC, managing consistency between such components through effective cache invalidation policies.
Although implementation-centered approaches can raise the abstraction level of caching, they still demand cache reasoning on developers, such as deciding whether to cache content.
Identification of Caching Opportunities
As mentioned before, application-level caches allow arbitrary content to be cached, and opportunities for caching thus emerge in the most diverse parts of the application. A popular approach to support developers while admitting content to the cache is to recommend caching opportunities. In this context, application profiling is usually the technique adopted, such as in MemoizeIt [5], which compares inputs and outputs of method calls and gives a set of redundant operations. To avoid comparing objects in detail, MemoizeIt first compares objects without following any object references, and then iteratively increases the depth of exploration while shrinking the set of considered methods. Also by analyzing method calls, Maplesden et al. [24] 19 proposes an approach that analyzes the smallest parent distance among common parents of a method to identify repeated patterns, which are named subsuming methods.
Xu [25] addresses the problem of frequent creation of data structures, whose the lifetimes are not connected, but the content is always the same. Then, a list of top allocation sites that create such data structures are reported. Similarly, Cachetor [26] addresses repeatedly computations by providing a runtime profiling tool that uses a combination of dynamic dependency and value profiling to suggest spots of invariant data values.
Although these approaches can potentially relieve reasoning burden from developers, such recommendations should be analyzed and implemented by them, integrating the appropriate cache logic into the application. Thus, approaches that identify but also automate the caching of cacheable content have been proposed. However, besides the analysis of the application behavior, such approaches also employ mechanisms to manage cache and application at runtime.
In this context, IncPy [27] achieve this by implementing a technique as a custom open-source Python interpreter. IncPy explores the repetitive creation and processing of intermediate data files, which should be properly managed by developers to multiple dependencies between their code and data files. Otherwise, their analyses produce wrong results. To enable developers to iterate quickly without needing to manage intermediate data files, they added a set of dynamic analyses to the programming language interpreter. IncPy then automatically caches the results of long-running pure method calls to disk, manages dependencies between code and on-disk data, and later reuses results, rather than re-executing those methods. Furthermore, this approach also allows developers to customize the execution by inserting annotations, which can force IncPy to always or never cache particular methods. Our framework is similar in this sense because it also provides annotations to developers aggregate domain-specific knowledge.
Another approach that deals with the same problem is CacheOptimizer [2], which is based on two analysis parts. First, it performs a static code analysis to identify possible caching spots in database queries. Then, CacheOptimizer monitors readable weblogs and creates mappings between current workload and database access to decide which and when database access is cacheable. Although this approach addresses method caching opportunities, it is focused on database-centric web applications; thus, only database-related methods are supported. Our approach share commonalities with CacheOptimizer; however, we focus on general application methods while searching for cacheable options.
Baeza-Yates et al. [12] addressed the identification of cacheable content in a different way, by filtering infrequent queries of web search engines, which cause a reduction in hit ratio because caching them often does not lead to hits. Therefore, this approach can prevent infrequent queries from taking space of more frequent queries in the cache. The proposed cache monitors stateless and stateful information of query executions and has two fully-dynamic parts. The first part is an admission controlled cache that only admits queries that the admission policy classifies as future cache hits. All queries that the admission policy rejects are admitted to the second part of the cache, an uncontrolled cache. Both caches implement a regular cache policy, more specifically, LRU. The uncontrolled cache can, therefore, manage queries that are infrequent but appear in short bursts, considering that the admission policy will reject queries that it concludes to be infrequent. Thus, the uncontrolled cache can handle cases in which infrequent queries may be asked again by the same user or within a short period. It guarantees that fewer infrequent queries enter the controlled cache, which is expected to handle temporal locality better.
Also focusing on filtering content, TinyLFU [28] uses an approximate LFU structure, which maintains a representation of the access frequency of recently accessed contents, to boost the admission effectiveness of caches. TinyLFU acts reactively when the cache is full and decides whether it is worthwhile admitting content, considering the cost of an eviction and the usefulness of the new content. This approach is currently available to developers as a replacement policy of the Caffeine caching framework.
Sajeev and Sebastian [29] proposed a semi-intelligent admission technique using the multinomial logistic regression (MLR) classifier. They used previously collected traces to build an MLR model, which can classify the content worthiness class. Such class is achieved by computing six different parameters, which refers to the traffic and the content properties. Then, at runtime, every incoming content to the cache has its worthiness class computed or updated, and it is used in an admission control mechanism, based on thresholds, to decide whether the content should be admitted.
Discussion
Different from programmatic solutions to application-level caching implementation, such as traditional caching libraries and frameworks, our proposed caching approach does not require any additional implementation, detaching caching concerns from the application. Furthermore, our framework can automatically capture all the application-specific information needed to achieve its objectives (i.e. select and cache cacheable content), as opposed to other solutions [1,14], which demand input and configuration from developers. Despite not being required, our framework also allows developers to provide additional knowledge by using a declarative approach, which can improve the solution.
In addition to the traditionally explored access history and cache-related statistics in existing admission solutions [12,29,28], such as frequency and recency, our approach also considers caching metadata retrieved from the application during its execution, such as cost to retrieve, user sessions, cache size and data size. Such metadata are in fact information that developers use while designing and implementing application-level caching, and thus enrich the application model with valuable application-specific information regarding the applicability of caching. Our results provide evidence of the value of this kind of information.
Apart from exploring new kinds of metadata, our approach also differs from these admission solutions in the granularity of cached content. We address complex logic and personalized content, which are produced and handled by method calls of applications, as opposed to whole web pages or database queries. Moreover, we consider all method calls as cacheable options and do not focus on specific methods or types of web applications, as some approaches do [27,24,5,2].
CONCLUSION
Web developers often make use of different application-level caching frameworks to improve performance. However, they need to reason about what to cache and continually revise caching decisions. Otherwise, the designed caching may not achieve performance requirements. In this paper, we proposed an automated approach that manages the cache according to data collected at runtime. We also presented a seamless framework that implements our approach and detaches caching concerns from the application. Our approach combines a monitoring and analysis of system execution, and a runtime control loop to deal with caching concerns. As a result, we can provide an application-specific solution without pre-defined thresholds or assumptions.
We evaluated our approach with three open source applications, and results indicate it may improve their throughput up to 17.18%, in comparison with the caching configuration designed by developers. Alternatively, our approach can be used as a supporting tool to help developers select cacheable content, given that developing a caching solution may require extensive and scattered code changes, which can be an error-prone and time-consuming task for developers. Although our approach was implemented specifically for Java-based web applications, it is generic enough to be used with any programming language.
Future work involves extending the approach to deal with other caching issues towards reducing the developers' effort when designing and implementing application-level caching. Although the processing phase of our approach seems to be fast enough to provide cacheable opportunities in a timely fashion, the overhead of the data tracking activity should be further evaluated regarding scalability. Therefore, techniques for reducing the amount of data required to identify cacheable methods should be investigated, towards reducing the overhead and providing a faster model building. Finally, we intend to study the execution of developer-written tests, such as unit and acceptance tests to find a less confident but yet representative list of cacheable methods. Thus, the application can benefit from caching since its first releases.
Figure 1 .
1Traditional Web Application Architecture with Associated Caching Locations.
Figure 2 .
2Application-level Caching Overview.
Figure 4 .
4Overview of our Application-level Caching Framework.
7 Figure 5 .
75Example of a Method Calls translated to an Execution Trace.
Figure 6 .
6Criteria associated with the Cacheability Pattern.
Figure 7 .
7APLCache Architecture and Technologies.
Figure 8 .
8Throughput by Caching Approach for Cloud Store.
Figure 9 .
9Throughput by Caching Approach for Pet Clinic.
Figure 10 .
10Throughput by Caching Approach for Shopizer.
Table II .
IIRequired Information for Evaluating Cacheability. Staticity Input and output of the method calls Changeability Input and output of the method calls Frequency Input and output of the method calls Shareability Session user that lead to the method calls Expensiveness Execution time of the method callsCriterion
Required information
Table III .
IIIObjective Evaluation of the Size-related Criteria.A method is only considered to be cached if size(m) < f ree(c), where size(m) and f ree(c) are the estimated size of the result of a method call and the available space in the cache, respectively.Question
Criterion
Meaning
Is the cache size large?
Is the data size large?
Cache size
Data size
Table IV .
IVGoal Definition (GQM Template).Definition Element
Evaluation Goal
Motivation
To assess the improvements provided by our automated application-
level caching approach,
Purpose
evaluate
Object
the effectiveness of the automated caching approach
Perspective
from a perspective of the researcher
Domain: web-based applications
when compared to manually developed caching solutions
Scope
in the context of 3 software projects, obtained from open-source
repositories.
Table V .
VResearch Questions and Metrics.Research Question
Metric
RQ1. What is the performance improvement
provided by caching method calls selected by
our automated caching approach?
M1-1. Throughput.
M1-2. Hit ratio.
M1-2. Total number of cache hits.
RQ2. What are the similarities and differences
between decisions made by our automated
approach and by developers?
M2-1. Number of caching opportunities that were
identified by both developers and our automated approach.
M2-2. Number of caching opportunities identified only by
developers.
M2-3. Number of caching opportunities identified only by
our automated approach.
Table VI .
VITarget systems of our study.Project
Domain
LOC # Files Database Properties
Cloud Store e-Commerce based on
TPC-W benchmark
7.6K
98 300K customer data and 10K items
Pet Clinic
Sample application
6.3K
72 6K vets, 10K owners and 13K pets
Shopizer
e-Commerce
111.3K
946 300K customer data and 10K items
Table VII .
VIISimulation Results: Throughput for Five Simultaneous Users.NO
DEV
AP
Application Throughput
Throughput
Throughput
Cloud Store
22.28 22.73
(+2.01%) 22.90
(+2.78%)
Pet Clinic
7.68
8.47 (+10.28%)
9.01 (+17.18%)
Shopizer
15.92 16.25
(+2.07%) 16.96
(+6.53%)
Throughput (TR): requests handled per second
8.37 8.37
22.28 22.28
28.84 28.84
30.42 30.42
30.45 30.45
9.13 9.13
22.73 22.73
29.21 29.21
30.49 30.49
30.50 30.50
9.15 9.15
22.90 22.90
29.35 29.35
30.57 30.57
30.59 30.59
NO
DEV
AP
1 user
5 users
10 users
25 users
50 users
Table VIII .
VIIISimulation Results: Hit Ratio and Total Number of Hits for Five Simultaneous Users. Table IX. Number of Cacheable Methods: Our Approach vs. Human-made Decisions.DEV
AP
Application Hit Ratio Total Hits
Hit Ratio
Total Hits
Cloud Store
74.25%
1179 97.42% (+31.20%)
2416 (+104.90%)
Pet Clinic
99.58%
1122 92.70%
(-6.90%)
1606
(+43.14%)
Shopizer
73.44%
8991 99.98% (+36.13%) 35259 (+292.11%)
Application Total DEV AP Intersection DEV Only AP Only
Cloud Store
812
4
8
4
0
4
Pet Clinic
205
1
4
1
0
3
Shopizer
5212
15
22
15
0
7
ACKNOWLEDGEMENTWe thank the anonymous reviewer who were asked by Software: Practice and Experience to review this article, as well as the editor. They provided constructive feedback that was used to expand our research and improve this article extensively. Jhonny Mertz
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| []
|
[
"Retracing some paths in Process Algebra",
"Retracing some paths in Process Algebra"
]
| [
"Samson Abramsky \nFoundations of Computer Science\nLaboratory for\nUniversity of Edinburgh\n\n"
]
| [
"Foundations of Computer Science\nLaboratory for\nUniversity of Edinburgh\n"
]
| []
| The semantic universe: transducersMilner's starting point was the classical automata-theoretic notion of trans-where Q is a set of states, q 0 ∈ Q the initial state, X the set of inputs, Y the set of outputs, and δ : Q × X ⇀ Y × Q 1 Similar ideas appeared independently in the work of Hans Bekić [Bek71]. | 10.1007/3-540-61604-7_44 | [
"https://arxiv.org/pdf/1401.5113v1.pdf"
]
| 1,181,387 | 1401.5113 | 7456806e8550d375d523a6292fa77c27b3e0befd |
Retracing some paths in Process Algebra
20 Jan 2014
Samson Abramsky
Foundations of Computer Science
Laboratory for
University of Edinburgh
Retracing some paths in Process Algebra
20 Jan 2014
The semantic universe: transducersMilner's starting point was the classical automata-theoretic notion of trans-where Q is a set of states, q 0 ∈ Q the initial state, X the set of inputs, Y the set of outputs, and δ : Q × X ⇀ Y × Q 1 Similar ideas appeared independently in the work of Hans Bekić [Bek71].
Introduction
The very existence of the concur conference bears witness to the fact that "concurrency theory" has developed into a subject unto itself, with substantially different emphases and techniques to those prominent elsewhere in the semantics of computation.
Whatever the past merits of this separate development, it seems timely to look for some convergence and unification. In addressing these issues, I have found it instructive to trace some of the received ideas in concurrency back to their origins in the early 1970's. In particular, I want to focus on a seminal paper by Robin Milner [Mil75] 1 , which led in a fairly direct line to his enormously influential work on ccs [Mil80,Mil89]. I will take (to the extreme) the liberty of of applying hindsight, and show how some different paths could have been taken, which, it can be argued, lead to a more unified approach to the semantics of computation, and moreover one which may be better suited to modelling today's concurrent, object-oriented languages, and the type systems and logics required to support such languages.
is the transition function (here a partial function). If we supply a sequence of inputs x 0 , . . . , x k to such a transducer, we obtain the orbit
q 0 x 0 −→ y 0 , q 1 x 1 −→ y 1 , q 2 x 2 −→ · · · x k −→ y k , q k+1
if δ(q i , x i ) = y i , q i+1 , 0 ≤ i ≤ k. This generalizes to non-deterministic transducers with transition function
δ : Q × X −→ P(Y × Q)
in an evident fashion.
The key idea in [Mil75] is to give a denotational semantics for concurrent programs as processes, which were taken to be extensional versions of transducers. There are two ingredients to this idea:
1. Instead of modelling programs by functions or relations, to model them by entities with more complex behaviours, taking account of the possible interactions between a program and its environment during the course of a computation.
"The meaning of a program should express its history of access to resources which are not local to it." [Mil75] 2. Instead of modelling concurrent programs by automata, with all the intensionality this entails, to look for a more extensional description of the behaviours of transducers.
To obtain this extensional view of transducers, consider the recursive definition
R = X ⇀ Y × R.
This defines a mathematical space of "resumptions" in which the states of transducers are "unfolded" into their observable behaviours. Milner solved equations such as this over a category of domains in [Mil75], but in fact it can be solved in a canonical fashion over Set-in modern terminology, the functor
T X,Y : Set −→ Set T X,Y (S) = X ⇀ Y × S has a final coalgebra R ∼ = −→ T X,Y (R)
. Indeed, Milner defined a notion ∼ of behavioural equivalence between transducers, and for any transducer (Q, X, Y, q 0 , δ) a map h δ : Q −→ R which is in fact the final coalgebra homomorphism from the coalgebrâ
δ : Q −→ T X,Y (Q)
to R (whereδ is the exponential transpose of δ), and proved that
(Q, X, Y, q 0 , δ) ∼ (Q ′ , X, Y, q ′ 0 , δ ′ ) ⇐⇒ h δ (q 0 ) = h δ ′ (q ′ 0 ).
From a modern perspective, we can also make light of a technical problem which figured prominently in [Mil75], namely how to model non-determinism. Historically, this called forth Plotkin's work on powerdomains [Plo76], but for the specific application at hand, the equation
R = X −→ P(Y × R)
has a final coalgebra in the category of classes in Peter Aczel's non-wellfounded set theory [Acz88], and if we are content to bound the cardinality of subsets by an inaccessible cardinable κ, then the equation
R = X −→ P <κ (Y × R)
has a final coalgebra in Set [Bar93b]. Moreover, the equivalence induced by this model coincides with strong bisimulation [Acz88]. However, this is not central to our concerns here. Rather, we want to focus on three important choices in the path followed by Milner from this starting point:
• Type-free vs. typed
• Extrinsic vs. intrinsic interaction
• Names vs. information paths.
We want to examine the consequences of making different choices on these issues.
Typed vs. type-free
Rather than looking at a single type-free space of resumptions as above, and trying to invent some plausible operations on this space, we will focus instead on the category of resumptions, and try to identify the structure naturally present in this category.
The category R of resumptions (we will for simplicity confine ourselves to the deterministic resumptions) has as objects sets, and as morphisms
R(X, Y ) = X ⇀ Y × R(X, Y )
i.e. the space of resumptions parameterized by the sets of "inputs" X and "outputs" Y . The composition of resumptions f ∈ R(X, Y ) and g ∈ R(Y, Z) is defined (coinductively [Acz88]) by:
f ; g(x) = (z, f ′ ; g ′ ) f (x) = (y, f ′ ), g(y) = (z, g ′ ) undefined otherwise.
The identity resumption id X ∈ R(X, X) is defined by
id X (x) = (x, id X ).
We can picture this composition as sequential (or "series") composition of transducers. We can define a monoidal structure on R by
X ⊗ Y = X + Y (disjoint union of sets) and if f ∈ R(X, Y ), g ∈ R(X ′ , Y ′ ), f ⊗ g ∈ R(X ⊗ X ′ , Y ⊗ Y ′ ) is defined by: f ⊗ g(inl(x)) = (inl(y), f ′ ⊗ g), f (x) = (y, f ′ ) undefined otherwise f ⊗ g(inr(x ′ )) = (inr(y ′ ), f ⊗ g ′ ), g(x ′ ) = (y ′ , g ′ ) undefined otherwise.
This is (asynchronous) parallel composition of transducers: at each stage, we respond to an input on the X "wire" according to f , with output appearing on the Y wire, and to an input on the X ′ wire according to g, with output appearing on the Y ′ wire. The remaining definitions to make this into a symmetric monoidal structure on R are straightforward, and left to the reader. Note that the associativity and symmetry isomorphisms, like the identities, have just one state; they are "history-free".
Finally, there is a feedback operator: for each X, Y , U a function
Tr U X,Y : R(X ⊗ U, Y ⊗ U ) −→ R(X, Y ) defined by Tr U X,Y (f )(x) = (y, f ′ ), ∃k. f (x) = (u 0 , f 0 ), f 0 (u 0 ) = (u 1 , f 1 ), . . . f k (u k ) = (y, f ′ ) undefined otherwise.
One should picture a token entering at the X wire, circulating k times around the feedback loop at the U wire, and exiting at Y .
This feedback operator satisfies a number of algebraic properties (to simplify the statement of these properties, we elide associativity isomorphisms, i.e. we pretend that R is strict monoidal):
Naturality in X Tr U X,Y ((g ⊗ id U ); f ) = g; Tr U X ′ ,Y (f ) where f : X ′ ⊗ U −→ Y ⊗ U , g : X −→ X ′ . Naturality in Y Tr U X,Y (f ; (g ⊗ id U )) = Tr U X,Y ′ (f ); g where f : X ⊗ U −→ Y ′ ⊗ U , g : Y ′ −→ Y . Naturality in U Tr U X,Y (f ; (id Y ⊗ g)) = Tr U ′ X,Y ((id X ⊗ g); f ) where f : X ⊗ U −→ Y ⊗ U ′ , g : U ′ −→ U .
Vanishing
Tr I X,Y (f ) = f where f : X −→ Y , and Tr U ⊗V X,Y (f ) = Tr U X,Y (Tr V X⊗U,Y ⊗U (f )) where f : X ⊗ U ⊗ V −→ Y ⊗ U ⊗ V . Superposing Tr U X⊗Z,Y ⊗W ((id X ⊗ sym Z,U ); (f ⊗ g); (id Y ⊗ sym U,W )) = Tr U X,Y (f ) ⊗ g where f : X ⊗ U −→ Y ⊗ U , g : Z −→ W . Yanking Tr X X,X (sym X,X ) = id X .
This says that R is a traced (symmetric) monoidal category in the sense of [JSV95] (cf. also [Has96] for the symmetric and cartesian cases, and [BE93] for related axioms).
Intrinsic vs. extrinsic interaction: paths vs. names
Why this apparent digression into the structure of the category of resumptions? Our aim is to address the question of how to model interaction between processes, which is surely the key notion in concurrency theory, and arguably in the semantics of computation as a whole. Resumptions as they stand model a single process in terms of its potential interactions with its environment. To quote Robin Milner again:
"A crucial feature is the ability to define the operation of binding together two processes (which may represent two cooperating programs, or a program and a memory, or a computer an an input/output device) to yield another process representing the composite of the two computing agents, with their mutual communications internalized." [Mil75] The route Milner followed to define this binding was in terms of the use of "names" or "labels": in terms of resumptions, one modifies their defining equation to
R(X, Y ) = X ⇀ Y × L × R(X, Y )
where L is a set of labels, so that output is tagged with a label, which can then be used by some "routing combinator" to dispatch the output to its destination process. This led in a fairly direct line of descent to the action names α, β, γ of ccs [Mil80,Mil89], and the names of the π-calculus [MPW92] and action structures [MMP95]. Clearly a great deal has been achieved with this approach. Nevertheless, we wish to lodge some criticisms of it.
• interaction becomes extrinsic: we must add some additional structure, typically a "synchronization algebra" on the labels [Win83], which implicitly refers to some external agency for matching up labels and generating communication events, rather than finding the meaning of interaction in the structure we already have.
• interaction becomes ad hoc: because it is an "invented" additional structure, many possibilities arise, and it is hard to identify any as canonical.
• interaction becomes global: using names to match up communications implies some large space in which potential communications "swim", just as the use of references in imperative languages implies some global heap. Although the scope of names may be delimited, as in the π-calculus, the local character of particular interactions is not immediately apparent, and must be laboriously verified. This appears to account for many of the complications encountered in reasoning about concurrent object-oriented languages modelled in the π-calculus, as reported in [Jon93,Jon96].
We will now describe a construction which appears in [JSV95], and which can be seen as a general form of the "Geometry of Interaction" [Gir88], and also as a general but basic form of game semantics [Abr96b]. This construction applies to any traced monoidal category C, i.e. to any calculus of boxes and wires closed under series and parallel composition and feedback, and builds a compact closed category G(C), into which C fully and faithfully embeds.
(It is in fact the unit of a (bi)adjunction between the categories of traced monoidal and compact closed categories.) Its significance in the present context is that it gives a general way of introducing a symmetric notion of interaction which addresses the issues raised above:
• interaction is intrinsic: it is found from the basic idea that processes are modelled in terms of their interactions with their environment. Building in the distinction between "process" and "environment" at a fundamental level makes interaction inherent in the model, rather than something that needs to be added.
• interaction is modelled as composition in the category G(C). Thus interaction is aligned with the computation-as-cut-elimination paradigm, and hence a unification of concurrency with other work in denotational semantics, type theory, categorical logic etc. becomes possible. See [AGN96a, Abr93, Abr95b] for a detailed discussion of this point.
• interaction is local. The dynamics of composition traces out "information paths", which are closely related to the types of the processes which interact. There is no appeal to a global mechanism for matching names. As we will see, this is general enough to model λ-calculus, state and concurrency, but, we believe, carries much more structure than the use of names to mediate interactions.
The G construction
Given a traced monoidal category C, we define a new category G(C) as follows:
• The objects of G(C) are pairs (A + , A − ) of objects of C. The idea is that A + is the type of "moves by Player (the System)", while A − is the type of "moves by Opponent (the Environment)".
•
A morphism f : (A + , A − ) −→ (B + , B − ) in G(C) is a morphism f : A + ⊗ B − −→ A − ⊗ B + in C.
• Composition is defined by symmetric feedback (cf. [AJ94b,AJ94a]):
✲ ✛ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ C + C − B − B + B + B − A − A + g f If f : (A + , A − ) −→ (B + , B − ) and g : (B + , B − ) −→ (C + , C − ) then f ; g : (A + , A − ) −→ (C + , C − ) is defined by f ; g = Tr B − ⊗B + A + ⊗C − ,A − ⊗C + (α; f ⊗ g; γ)
where α :
A + ⊗ C − ⊗ B − ⊗ B + ∼ = −→ A + ⊗ B − ⊗ B + ⊗ C − and γ : A − ⊗ B + ⊗ B − ⊗ C + ∼ = −→ A − ⊗ C + ⊗ B − ⊗ B +
are the canonical isomorphisms defined using the symmetric monoidal structure. (Again, we have elided associativity isomorphisms.)
• The identities are given by the symmetry isomorphisms in C:
id (A + ,A − ) = sym A + ,A − : A + ⊗ A − ∼ = −→ A − ⊗ A + .
There is an evident involutive duality on this category, given by
(A + , A − ) * = (A − , A + ).
There is also a tensor structure, given by
(A + , A − ) ⊗ (B + , B − ) = (A + ⊗ B + , A − ⊗ B − ).
G(C) is a compact-closed category [KL80], with internal homs given by
(A + , A − ) −• (B + , B − ) = (A − ⊗ B + , A + ⊗ B − ).
Examples
From resumptions to strategies
To interpret the category G(R), think of an object (X + , X − ) as a rudimentary two-person game, in which X + is the set of moves for Player, and X − the set of moves for Opponent. A resumption f : X − −→ X + is then a strategy for Player. Note that we can represent such a strategy by its set of plays:
P (f ) = {x 1 y 1 · · · x k y k | f (x 1 ) = (y 1 , f 1 ), . . . , f k−1 (x k ) = (y k , f k )}.
One can then show that composition in G(R) is given by "parallel composition plus hiding" [Abr94, AJ94a, Abr96b]:
P (f ; g) = {s ↾ X, Z | s ∈ P (f )||P (g)} S||T = {s ∈ L(X, Y, Z) | s ↾ X, Y ∈ S ∧ s ↾ Y, Z ∈ T } where X = X + + X − , Y = Y + + Y − , Z = Z + + Z − , and L(S 1 , S 2 , S 3 ) = {s ∈ (S 1 + S 2 + S 3 ) * | s i ∈ S j ∧ s i+1 ∈ S k =⇒ |j − k| ≤ 1}.
The identities are the "copycat" strategies as in [AJ94a,Abr96b]. We can then obtain the simple category of games described in [Abr96b] by applying a specification structure in the sense of [AGN96b] to G(R), in which the properties over (X + , X − ) are the prefix-closed subsets of (X − X + ) * , i.e. the "safety properties" [AP93], which in this context are the game trees.
Some geometries of interaction
Suppose we begin with the simpler category Pfn of sets and partial functions (which is a lluf sub-category of R). This is easily seen to be a sub-tracedmonoidal category of R, with tensor as disjoint union, and the trace given by a sum-of-paths formula (cf. [AM82]). That is, if
f : X + U ⇀ Y + U
is a partial function, then
Tr U X,Y (f ) = k∈ω f k ,
where f k (x) is defined and equal to y iff starting from x we perform exactly k iterations of the feedback loop around U before exiting at Y with result y:
f k = inl X,U ; (f ; [0, inr X,U ]) k ; f ; [id Y , 0]
where 0 is the everywhere undefined partial function. We can think of this sub-category of R as the "one-state resumptions", so that, applying the G construction to Pfn we get a category of history-free strategies [AJ94a]. As a minor variation, we could start with the category PInj of sets and partial injective maps. Then G(PInj) is essentially the original Geometry of Interaction construction of Girard, as explained in [AJ94a,AJM96]. In particular, the composition in G(PInj) corresponds exactly to the Execution Formula. This category can be lifted to the setting of Hilbert spaces by applying the free construction described in [Bar93a], which sends a set X to the Hilbert space l 2 (X) of square summable families {a x | x ∈ X}.
As a final variation, we could start with Rel, the category of sets and relations. This yields a non-deterministic version of the Geometry of Interaction, which can be generalized via non-deterministic resumptions to a category of non-deterministic strategies. G(Rel) is the example mentioned at the end of [JSV95].
Stochastic interaction
As a more substantial variation of the above, consider the following category of stochastic kernels [Law62,Gir81]. Objects are structures (X, M(X)), where M(X) is a σ-algebra of subsets of X. Identities are given by point measures:
A morphism f : X −→ Y is a function f : X × M(Y ) −→ [0, 1] such that for each x ∈ X f (x, ·) : M(Y ) −→ [0, 1]id X (x, M ) = 1, x ∈ M 0, x ∈ M.
Tensor product is given by disjoint union; note that M(X + Y ) ∼ = M(X) × M(Y ). Feedback is given by a sum-over-paths formula. Given f : X ⊗ U −→ Y ⊗ U , and x ∈ X, we define for each k ∈ ω a measure µ k on M(U ) which gives the probability that we will end up in M starting from x after exactly k traversals of the feedback loop:
µ 0 (M ) = f (inl(x), (∅, M )) µ k+1 (M ) = U f (inr(·), (∅, M ))dµ k .
The probability that we will end up in M ∈ M(Y ) starting from x after exactly k iterations of the feedback loop is given by:
f 0 (x, M ) = f (inl(x), (M, ∅)) f k+1 (x, M ) = U f (inr(·), (M, ∅))dµ k .
Finally, the trace is defined by summing over all paths:
Tr U X,Y (f )(x, M ) = Σ k∈ω f k (x, M ).
4.4 From particles to waves: the "New Foundations" version of Geometry of Interaction
All the above models can be thought of as dynamical systems in which an information "token" or "particle" traces some path around a network. This particulate interpretation of diagrams of boxes and wires is supported by the "additive" (disjoint union) interpretation of the tensor. It is also possible to give an interpretation in which an information "wave" travels through the network; formally, this will be supported by a "multiplicative" (cartesian product) interpretation of the tensor. Specifically, we can define a traced monoidal structure on the category Cpo of cpo's and continuous functions, in which the tensor is given by the cartesian product, and feedback by the least fixpoint operator: that is, if
f : D × A −→ E × A, then Tr A D,E (f ) = λd : D. f (d, Y(f (d, ·); snd)); fst.
The category G(Cpo) is then exactly the category GI(C) described in [AJ94b]. A sub-category of this category will consist of dataflow networks, built up from objects which are domains of streams. The symmetric feedback operator giving the composition in G(Cpo) has been used in this context [SDW96,GS96], inter alia in developing assumption/commitment style proof rules for dataflow networks.
The continuous case?
One final "example" should be mentioned, although we have not as yet succeeded in working out the details. The operations of series and parallel composition and feedback are standard in continuous-time control systems, electronic circuits and analogue computation. In particular, feedback is interpreted by solving a differential equation. There should then presumably be a traced monoidal category C of manifolds and smooth maps, for which G(C) would give an "infinitesimal" model of interaction. Such a category might be relevant to the study of hybrid systems [PS95].
Consequences
We shall, very briefly, sketch some further developments from this point.
Correctness issues
We can associate correctness properties with the rudimentary types of G(C), in the setting of specification structures [AGN96b]. Types can then carry strong correctness information, and the type inference rule for composition
f : A → B g : B → C f ; g : A → C
becomes a compositional proof rule for process interaction. See [Abr93, Abr95b, AGN96a, AGN96b] for further discussion and applications. We shall mention some particular cases for the examples described above.
Resumptions In this case, we can get the structure of games as safety properties, and of winning strategies as liveness properties, as described in [AJ94a,Abr96b]. In particular, the fact that winning strategies are closed under composition corresponds to a guarantee that there is no "infinite chattering" [Hoa85] in interaction.
Geometry of Interaction
In this case, we can focus on nilpotency as a semantic analogue of normalization, as in [Gir88], or instead proceed as in the previous example, as in [AJ94a], where a Full Completeness Theorem for Multiplicative Linear Logic is obtained.
Modelling types and functions
The divide between concurrency theory and denotational semantics, type theory and categorical logic is bridged in our approach, since the categories we construct, or derivatives thereof, have the right structure to model typed, higher-order programming languages. The key point is that we are now modelling functions as processes, and function application as a particular form of process interaction, as advocated in [Mil92], but in a highly structured, syntax-free and compositional fashion. Moreover, the quality of these process models of functional computation is high: the models based on games yielded the first syntax-independent constructions of fully abstract models for PCF [AJM96,HO96], and this has been followed by a number of further results [AM95,McC96b,McC96a]. The degree of mathematical structure in these models is also witnessed by the axiomatic treatment of full abstraction it has been possible to extract from them [Abr96a].
is a measure, and for each M ∈ M(Y ), f (·, M ) : X −→ [0, 1] is a measurable function. One can think of stochastic kernels as "probabilistic transition functions". Note that we do not require that each f (x, ·) is a probability measure, i.e. that f (x, Y ) = 1, since we wish to allow for "partial" transition functions.Composition is by integration: if f : X → Y and g : Y → Z, then f ; g(x, M ) = Y g(·, M )df (x, ·).
This model of Idealized Algol extends smoothly to incorporate concurrency[Abr95a]. It remains to be seen how accurate the model of the concurrent language is, but the situation looks quite promising: moreover, Idealized Parallel Algol is rich enough to represent rather directly many of the features of today's concurrent object-oriented languages.State and concurrencyIt has also proved possible to give a game semantics for Idealized Algol[Abr95a], which is a clean integration of higher-order functional programming with imperative features and block structure[Rey81,Ten94]. Again, this has led to the first syntax-independent construction of a fully abstract model[AM96]. The treatment of local variables is process-based, following the line of[Mil75,Mil80,Red96]; but with the the right mathematical tools now available, a more definitive treatment can be given, as confirmed by the results on full abstraction.
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Assumption/commitment rules for networks of asynchronously communicating agents. Formal Aspects of Computing. K Stølen, F Dederichs, R Weber, K. Stølen, F. Dederichs, and R. Weber. Assump- tion/commitment rules for networks of asynchronously commu- nicating agents. Formal Aspects of Computing, 1996.
Denotational semantics. R D Tennent, Handbook of Logic in Computer Science. S. Abramsky, D. M. Gabbay, and T. S. E. MaibaumOxford University Press3R. D. Tennent. Denotational semantics. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 3, pages 169-322. Oxford University Press, 1994.
Synchronization trees. G Winskel, Automata, Languages and Programming: 10th International Colloquium. Springer-VerlagG. Winskel. Synchronization trees. In Automata, Languages and Programming: 10th International Colloquium, pages 695- 711. Springer-Verlag, 1983.
| []
|
[
"Inclusive approach to hunt for the beauty-charmed baryons Ξ bc",
"Inclusive approach to hunt for the beauty-charmed baryons Ξ bc"
]
| [
"Qin Qin \nSchool of Physics\nHuazhong University of Science and Technology\n430074WuhanChina\n",
"Yu-Ji Shi \nHelmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nUniversität Bonn\n53115BonnGermany\n",
"Wei Wang \nSchool of Physics and Astronomy\nMOE Key Laboratory for Particle Physics, Astrophysics and Cosmology\nShanghai Key Laboratory for Particle Physics and Cosmology\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Guo-He Yang \nSchool of Physics\nHuazhong University of Science and Technology\n430074WuhanChina\n",
"Fu-Sheng Yu \nSchool of Nuclear Science and Technology\nLanzhou University\n730000LanzhouChina\n\nCenter for High Energy Physics\nPeking University\n100871BeijingChina\n",
"Ruilin Zhu \nDepartment of Physics\nInstitute of Theoretical Physics\nNanjing Normal University\n210023NanjingJiangsuChina\n"
]
| [
"School of Physics\nHuazhong University of Science and Technology\n430074WuhanChina",
"Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nUniversität Bonn\n53115BonnGermany",
"School of Physics and Astronomy\nMOE Key Laboratory for Particle Physics, Astrophysics and Cosmology\nShanghai Key Laboratory for Particle Physics and Cosmology\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics\nHuazhong University of Science and Technology\n430074WuhanChina",
"School of Nuclear Science and Technology\nLanzhou University\n730000LanzhouChina",
"Center for High Energy Physics\nPeking University\n100871BeijingChina",
"Department of Physics\nInstitute of Theoretical Physics\nNanjing Normal University\n210023NanjingJiangsuChina"
]
| []
| With a distinctive internal structure from all established hadrons, the beauty-charmed baryons Ξ bc can provide us with new points of view to decipher the strong interaction. In this work, we point out that the inclusive Ξ bc → Ξ ++ cc + X decay is a golden channel for the experimental discovery of Ξ bc at the LHC. A unique feature of this process is that the Ξ ++ cc is displaced, which greatly reduces the combinatorial background. A feasibility analysis is performed on the Ξ + bc search, which is expected to have a longer lifetime than Ξ 0 bc and thus a better displacement resolution. The Ξ + bc → Ξ ++ cc + X branching ratio is calculated within the heavy diquark effective theory. Combining the Ξ bc production rate and the Ξ ++ cc detection efficiency, we anticipate that hundreds of signal events will be collected during LHCb Run 3. | 10.1103/physrevd.105.l031902 | [
"https://arxiv.org/pdf/2108.06716v2.pdf"
]
| 246,664,248 | 2108.06716 | 1f8ddca4c94cb6da261bab0cdb9d5e0c315b88ad |
Inclusive approach to hunt for the beauty-charmed baryons Ξ bc
Qin Qin
School of Physics
Huazhong University of Science and Technology
430074WuhanChina
Yu-Ji Shi
Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics
Universität Bonn
53115BonnGermany
Wei Wang
School of Physics and Astronomy
MOE Key Laboratory for Particle Physics, Astrophysics and Cosmology
Shanghai Key Laboratory for Particle Physics and Cosmology
Shanghai Jiao Tong University
200240ShanghaiChina
Guo-He Yang
School of Physics
Huazhong University of Science and Technology
430074WuhanChina
Fu-Sheng Yu
School of Nuclear Science and Technology
Lanzhou University
730000LanzhouChina
Center for High Energy Physics
Peking University
100871BeijingChina
Ruilin Zhu
Department of Physics
Institute of Theoretical Physics
Nanjing Normal University
210023NanjingJiangsuChina
Inclusive approach to hunt for the beauty-charmed baryons Ξ bc
With a distinctive internal structure from all established hadrons, the beauty-charmed baryons Ξ bc can provide us with new points of view to decipher the strong interaction. In this work, we point out that the inclusive Ξ bc → Ξ ++ cc + X decay is a golden channel for the experimental discovery of Ξ bc at the LHC. A unique feature of this process is that the Ξ ++ cc is displaced, which greatly reduces the combinatorial background. A feasibility analysis is performed on the Ξ + bc search, which is expected to have a longer lifetime than Ξ 0 bc and thus a better displacement resolution. The Ξ + bc → Ξ ++ cc + X branching ratio is calculated within the heavy diquark effective theory. Combining the Ξ bc production rate and the Ξ ++ cc detection efficiency, we anticipate that hundreds of signal events will be collected during LHCb Run 3.
With a distinctive internal structure from all established hadrons, the beauty-charmed baryons Ξ bc can provide us with new points of view to decipher the strong interaction. In this work, we point out that the inclusive Ξ bc → Ξ ++ cc + X decay is a golden channel for the experimental discovery of Ξ bc at the LHC. A unique feature of this process is that the Ξ ++ cc is displaced, which greatly reduces the combinatorial background. A feasibility analysis is performed on the Ξ + bc search, which is expected to have a longer lifetime than Ξ 0 bc and thus a better displacement resolution. The Ξ + bc → Ξ ++ cc + X branching ratio is calculated within the heavy diquark effective theory. Combining the Ξ bc production rate and the Ξ ++ cc detection efficiency, we anticipate that hundreds of signal events will be collected during LHCb Run 3. Introduction. -Doubly heavy baryons, especially the Ξ ++ cc , have recently received much attention [1]. Different from other baryons with one or zero heavy quarks, doubly heavy baryons resemble a 'double-star' core surrounded by a light 'planet'. The Ξ ++ cc discovery also motivated studies to probe the nature of exotic four quark states or structures, e.g. cusps or true resonances (see e.g. [2][3][4]). Very recently, a first doubly heavy tetraquark candidate T + cc was observed by the LHCb [5,6]. However unlike in doubly charmed systems, the 'double-star' core in the beauty-charm baryons Ξ bc is imbalanced, resulting in diverse features. Compared to the charm-charm binary, the beauty-charm core is expected to have a smaller size, behaving more like a point particle. Moreover, the beauty-charmed baryons involve more energy scales, the beauty mass, the charm mass and the nonperturbative QCD scale Λ QCD , so they implicate more affluent dynamics. Thereby, the beauty-charmed baryons would provide a unique new hadronic platform to decode the strong interaction.
Experimentalists have made abundant efforts to search for the beauty-charmed baryons Ξ bc . However, such searches are much more difficult than those for Ξ ++ cc . For example, the exclusive channels Ξ 0 bc → D 0 pK − [7] and Ξ 0 bc → Ξ + c π − [8] were used to search for Ξ bc at the LHCb, but no definitive evidence was established. With some theoretical and experimental inputs, the experimental upper limit on the Ξ 0 bc → Ξ + c π − branching ratio can be ‡ corresponding authors: Qin Qin, Yu-Ji Shi and Ruilin Zhu FIG. 1: Sketch of Ξ bc production and decay at the LHC. The secondary decay vertex of Ξ bc → Ξ ++ cc + X is displaced from the proton-proton collision vertex, which produces a unique signal: a displaced Ξ ++ cc .
extracted from Ref. [8] to be O(10 −4 ). Comparing it to the theoretical prediction [9], we find a big gap of about 3 orders of magnitude. One difficulty in such exclusive searches lies in the limited production rate for Ξ bc at the LHC, but a bigger challenge is due to the very low reconstruction efficiency, because a beauty typically decays with fractions of O(10 −3 ) even to the most abundant exclusive final states [9][10][11]. To overcome this difficulty, we propose an approach to search for Ξ bc via an inclusive decay channel Ξ bc → Ξ ++ cc + X, where X stands for all possible particles.
This inclusive approach to search for Ξ bc has multiple advantages. Firstly, it has a much larger branching ratio than any exclusive decay channel. Secondly, the detection efficiency is greatly improved because only Ξ ++ cc needs to be reconstructed. Lastly but very importantly, because the weakly decaying Ξ bc has a relatively long lifetime and can typically form a sub-millimeter displaced secondary decaying vertex, the Ξ ++ cc 's generated from Ξ bc do not draw back to the primary proton-proton collision vertices. This feature characterized by a nonzero impact parameter (IP) can clearly distinguish the signal events from the main background, strongly produced Ξ ++ cc 's from the primary vertices. To clarify this point, a diagrammatic sketch is displayed in Fig. 1. The use of the IP has been applied at the LHCb for a long time (see e.g. [12]), and it was proposed that displaced B c mesons can be used to search for Ξ bb in [13], which greatly inspired the original idea of this work. According to [13], the relatively long lifetime of Ξ + bc [14] can ensure it a good IP resolution, while the situation will be worse for Ξ 0 bc . Apart from the IP, the distance between the Ξ ++ cc decay vertex and the primary collision vertex can also be used in the search to improve the sensitivity.
As the Ξ bc production rates and lifetimes have been evaluated in e.g. [15,16] and [14] (see also references therein), respectively, the remaining key issue is the Ξ bc → Ξ ++ cc + X branching ratio. Based on the heavy diquark effective theory [17][18][19], we calculate the branching ratio and find B(Ξ + bc → Ξ ++ cc + X) ≈ 7%. Moreover, because the signal mainly characterizes in a displaced Ξ ++ cc , the reconstruction efficiency of the signal events is close to the detection efficiency of Ξ ++ cc , which can be reliably extracted from previous experiments. Combining all this information, we find that hundreds of signal events are expected during LHCb Run 3, with an integrated luminosity of 23 fb −1 by 2024. Consequently, the proposed inclusive approach for the Ξ bc search is most feasible and also timely for the LHCb study.
Decay rate. -In the feasibility analysis of this approach, the inclusive Ξ bc → Ξ ++ cc + X decay rate was calculated with the following several steps. Based on the heavy quark symmetry and the heavy diquark symmetry, each step of the calculation is trustworthy. Firstly, under the heavy (di)quark symmetry, it can be demonstrated that the leading contribution to the inclusive Ξ bc → Ξ ++ cc + X decay is from X bc → X cc +f f , where X QQ stands for a heavy diquark constituted by the heavy Q and Q quarks and f ( ) can be any possible quarks or leptons. Subsequently, the unknown X bc → X cc diquark transition current was evaluated by matching from the b → c transition current. Afterwards, the decay rate of Ξ bc → Ξ ++ cc +X was numerically calculated, with possible theoretical uncertainties taken into account.
We first validate the treatment of the two heavy quarks QQ as a point-like object in a doubly heavy baryon.The QQ form a color anti-triplet and have an attractive potential. As illustrated by [20][21][22], the distance between the two heavy quarks is estimated as
r QQ ∼ 1/(m Q v)
with v being the heavy quark velocity in the baryon rest frame, while the spatial size of the light quark in the baryon is r Qq ∼ 1/Λ QCD . Furthermore, it can be deduced that v is small if m Q is heavy enough [23]. Numer-ical calculations confirm the hierarchy by giving v 2 c ∼ 0.3 and v 2 b ∼ 0.1 [23,24]. It indicates that
m b v 2 b ∼ m c v 2 c ∼ Λ QCD , so r QQ /r Qq ∼ Λ QCD /(m Q v)
1. In conclusion, the two heavy quarks can be treated as a point-like diquark compared to the baryon size. This greatly simplifies the structure of a three-quark system to a bound state of a heavy diquark and a light quark.
Benefitting from the quark-diquark picture, we can formulate the inclusive decay of a doubly heavy baryon within the heavy diquark effective theory [17][18][19]. Performing the operator product expansion, the inclusive Ξ bc → H cc + X decay rate can be expanded in inverse powers of the diquark mass M X , with the leading-power contribution given by the free diquark decay rate,
Γ(Ξ bc → H cc + X) = f,f Γ(X bc → X ccf f ) + O 1 M X .(1)
The fermion pairsf f includev − ( = e, µ, τ ) and ud,ūs,cd,cs. The H cc represents all doubly charmed hadrons, including the ground-state baryons Ξ
++(+) cc
, Ω + cc and the tetraquarks T cc , and their excited states as well.
As the fragmentation rates to strange baryons and to tetraquarks are much smaller than those to non-strange baryons (see, e.g., [25,26]), the fragmentations to Ω cc and T cc and their excited states are neglected in the following discussions. For the non-strange baryons, the excited states eventually decay strongly (or electromagnetically) into Ξ cc . Therefore, all the Ξ bc → H cc + X decay processes produce a displaced Ξ cc , half Ξ ++ cc and half Ξ + cc by the isospin symmetry, i.e., Γ(Ξ bc → H cc + X) ≈ Γ(Ξ bc → Ξ cc + X) ≈ 2Γ(Ξ bc → Ξ ++ cc + X). In the evaluation of Γ(X bc → X ccf f ) induced by the weak interaction vertices such ascγ µ P L bf γ µ P L f with the left-handed projector P L ≡ (1 − γ 5 )/2, thef f part can be factorized out at the leading order of the strong coupling constant α s , and the key issue in the calculation is the remaining diquark current X i cc |cγ µ P L b|X l bc , where i, l are color indices. The S-wave diquark X bc is either a scalar or axial-vector, but as implied by studies of the beauty-charmed baryon spectroscopy [27][28][29][30][31], an axial-vector X bc state is dominant in the Ξ bc baryons. On the other hand, X cc can only be an axial-vector due to the flavor and spin symmetries. The calculation of the diquark current is performed in two different kinematic regions, the large recoil region and the small recoil region. In the former, perturbative calculation is applicable because typically a hard gluon exchange between the spectator quark and the weak interacting quarks is required, as displayed in FIG. 2. In practice, we adopt the nonrelativistic QCD (NRQCD) factorization for this calculation, which were applied to calculate the B c → η c , J/ψ form factors [32][33][34]. In the small recoil region, the socalled soft overlap contribution is dominant and the perturbative QCD expansion is less trustworthy. However, the heavy quark symmetry determines the form of the diquark current at the zero recoil point. For the intermediate region, we use a simplified z-series expansion [35] to perform the interpolation.
The vector and axial-vector diquark currents can be parametrized as
X i cc (v, )|cγ µ b|X l bc (v , ) = δ il 2M cc M bc − a 0 * · v µ − a 1 * · v µ +a 2 * · v µ + a 3 v · * µ , X i cc (v, )|cγ µ γ 5 b|X l bc (v , ) = δ il 2M cc M bc − ib 0 * v µ − ib 1 * vµ ,(2)
where v ( ) , ( ) and M cc(bc) are the 4-velocity, polarization vector and mass of X cc(bc) . The functions a i (q 2 )'s and b i (q 2 )'s of the transfer momentum squared q 2 are to be determined. At the zero recoil (maximal q 2 = (M bc − M cc ) 2 ) point, they can be obtained by taking the heavy quark limit, and the results read
a 0,1,2,3 (q 2 max ) = b 0,1 (q 2 max ) = 1 .(3)
It can be derived in the following way. Due to the heavy quark symmetry, the ground state QQ diquark is represented by a Lorentz bilinear field (see e.g. (9) of [19])
D QQ v (x) = 1 + / v 2 [γ µ A µ (x) + iγ 5 S(x)]C ,(4)
where C ≡ iγ 0 γ 2 , the axial-vector field A µ (x) annihilates an axial-vector diquark with a polarization vector µ and the scalar field S(x) annihilates a scalar diquark. All of the color indices are hidden for convenience. Its Lorentz transformation property is
D v (x) → D v (x ) = D(Λ)D v (Λ −1 x)D(Λ) T ,v = γ 0 D † v γ 0 which transforms as D v (x) →D v (x ) = [D(Λ) −1 ] TD v (Λ −1 x)D(Λ) −1 .
Then, we can match the quark transition currents to the corresponding diquark transition currents viā
cΓb = tr L TDcQ v ΓD bQ v ,(5)
which is determined by the heavy quark spin symmetry and Lorentz covariance. The Γ matrix represents a general 4 × 4 matrix, and only γ µ and γ µ γ 5 are involved in this work. The Lorentz bispinor L only depends on v and v . The general expression for L with the correct parity and time-reversal properties is L = proper normalization gives the matrix elements
L 0 +L 1 / v+L 2 / v +L 3 / v/ v ,1 √ 2M bc M cc X cc (v, )|cγ µ b|X bc (v , ) = ξ(w) − * · v µ − * · v µ + * · v µ + v · * µ , 1 √ 2M bc M cc X cc (v, )|cγ µ γ 5 b|X bc (v , ) = ξ(w) − i * v µ − i * vµ ,(6)
where the symmetry factor √ 2 is due to the identical c quarks in X cc , and the masses are introduced to make ξ(w) dimensionless. Replacing X cc (v, ) → X bc (v , ) and c → b, the above vector-current expression leads to X bc (v , )|bγ µ b|X bc (v , ) /M bc = 2ξ(1) v µ = 2v µ . It determines that ξ(w) = 1 at the zero recoil w = 1, leading to the final result in (2) and (3).
In the large-recoil (small-q 2 ) region, the diquark currents are induced by exchanges of hard gluons. At the leading order with one hard gluon exchange, a sample Feynman diagram is shown in FIG. 2. The hard gluon leads to the large recoil, while the soft gluon exchanges can be absorbed into the initial and final diquark wave functions in the NRQCD framework. The NRQCD calculation formulates the diquark currents as nonperturbative matrix elements along with the corresponding Wilson coefficients as
a i [χ bc (v ) → χ cc (v)] = jk c jk i (µ) m d j −4 2 b m d k −4 2 c × 0 K j (µ) χ bc (v ) χ cc (v) |K k (µ)| 0 ,(7)
where the K ( ) (µ) are all possible independent bilinear combinations of two component operators which can be power counted by the velocity v ( ) . The c jk i (µ) are the short-distance Wilson coefficients which can be calculated order by order in series of α s . Following an analogous procedure to obtain (4) in [32], we calculate the quark-level hard kernel with one hard gluon exchange, convolute it with the diquark nonperturbative matrix elements, and obtain the leading-order result
a 2,3 (q 2 ) = α s 2(1 − w) 2 √ w N c + 1 N c 1 m 3 c R bc (0)R * cc (0) , a 0 (q 2 ) = b 0 (q 2 ) =ξ 2 a 2,3 (q 2 ) , a 1 (q 2 ) = b 1 (q 2 ) =ξ 1 a 2,3 (q 2 ) ,(8)
whereξ 1 ≡ m b /M bc ,ξ 2 ≡ m c /M cc , and the number of colors N c = 3. The diquark wave functions at the origin are defined through the nonperturbative matrix elements
ε ijk 0|ψ T c,i iσ 2 σψ b,j |X k bc ( ) √ 2M bc = N c ! R bc (0) √ 4π , ε ijk X k cc ( )|ψ † c,i i σσ 2 ψ * c,j |0 √ 4M cc = N c ! R * cc (0) √ 4π * ,(9)
where ψ's are two-component spinor fields, σ's are Pauli matrices and ε ijk is the Levi-Civita symbol in the color space. In principle, both the next-to-leading order α s corrections and the subleading power corrections to the diquark transition amplitudes can be calculated as the calculation for B c → J/ψ in [33,34]. We leave these calculations to future works. To obtain the numerical result, it requires the input of the diquark wave functions at the origin R bc,cc (0). They are obtained by solving the nonrelativistic Schrödinger equations, with the potential V (r) = − 2 3 αs(ν lat ) r + c2r+c1 c3r+1 + σr [36] with ν lat = 2.16 GeV , σ = 0.21GeV 2 , c 1 = 1.948 GeV , c 2 = 15.782 GeV , c 3 = 9.580 GeV, which were fitted from the lattice calculation [37,38]. The quark masses take values of m c = 1.392(11) GeV and m b = 4.749(18) GeV. The ground-state solutions to the Schrödinger equations give R cc (0) = (0.66 ± 0.06) GeV 3/2 and R bc (0) = (0.87 ± 0.09) GeV 3/2 . The uncertainties were estimated from the differences between the results obtained with the above lattice potentials and the Cornell potentials [39], though the real uncertainties could be larger.
To interpolate the diquark current in the whole range from the above results in the small-and large-recoil regions, a simplified z-series expansion [35] is adopted with the formulation
f (q 2 ) = f (0)/[1 − q 2 /m 2 Bc ] 1 + bζ(q 2 ) + cζ 2 (q 2 ) ,(10)for a 0 (= b 0 ), a 1 (= b 1 ) and a 2 (= a 3 ), where ζ(q 2 ) = z(q 2 )−z(0), z(q 2 ) = ( t + − q 2 − √ t + − t 0 )/( t + − q 2 + √ t + − t 0 ), t ± = (M bc ± M cc ) 2 , t 0 = t + (1 − 1 − t − /t + )
and the free parameters f (0), b, and c are to be determined. The value of t 0 is chosen to minimize |z| to improve the convergence. Unlike Ref. [35], we use ζ(q 2 ) instead of z(q 2 ) for the expansion, though these two parameterizations are equivalent. The expansion in ζ(q 2 ) ensures that f (0) is exactly the value of the form factor at q 2 = 0. Fitting the points of q 2 = 0, 0.1, 0. Finally, with the numerical results for the diquark currents, the inclusive Ξ bc → H cc + X decay rate (1) can be calculated. The leading power free diquark decay rates were calculated by phase space integration of the amplitude squares. For example, for the electron channel contribution Γ(X bc → X cc e −ν e ), the amplitude is given by the product of 4G F / √ 2V cbū (p e )γ µ P L v(p ν ) and the diquark current X cc |cγ µ P L b|X bc , where G F is the Fermi constant and V qq is the corresponding Cabbibo-Kobayashi-Maskawa (CKM) matrix element. The calculation is similar for the other leptonic channels; for the hadronic channels, the replacement |V cb | 2 → |V cb V * U D | 2 (3C 2 1 +2C 1 C 2 +C 2 2 ) should be performed at the level of amplitude squares with U = u, c and D = d, s, where the Wilson coefficients are defined in Ref. [40]. Summing over the contributions from all possible channels withf f = eν, µν, τν,ūd,ūs,cd,cs, the numerical result for the inclusive decay rate reads Γ(Ξ bc → H cc + X)
= (1.9 ± 0.1 ± 0.3 ± 0.4) × 10 −13 GeV . (11)
Most numerical inputs have been given previously, except that the Wilson coefficients took values from Ref. [40] and the Fermi constant and the CKM matrix elements took values from Ref. [41]. The uncertainties in order are from the quark mass variation, the diquark wave functions at the origin, and the scale dependence, respectively. The former two were obtained by varying the values of the quark masses and the diquark wave functions at the origin as listed below (9). As for the scale, we chose µ = m b for the central value calculation, and doubled and halved it for the uncertainty estimation. In addition, one would expect more uncertainties induced by unknown power corrections. The dominant v 2 corrections are expected to potentially modify the result by ∼ 30% [34]. The decay rate translates to the branching ratios as
B(Ξ +(0) bc → H cc + X) ≈ 14% (3%) ,(12)
where we have taken τ (Ξ
+(0)
bc ) ≈ 508 (105) fs [14]. As analyzed before, the Ξ +,0 bc → Ξ ++ cc + X branching ratio is approximately 1/2 of B(Ξ +,0 bc → H cc + X). Phenomenology. -Based on the Ξ bc → Ξ ++ cc + X branching ratio calculated above, as well the information on Ξ bc production and the Ξ ++ cc detection efficiency, the number of signal events containing a displaced Ξ ++ cc can be estimated.
In practice, the inclusive approach to search for Ξ bc depends crucially on the lifetimes of the Ξ bc baryons. Only if they fly far enough from the collision vertices before decaying, the displacement of the decay products Ξ ++ cc 's can be clearly distinguished. According to the study of Ξ bb → B − c + X [13], with the vertex resolution of the LHCb detector, the Ξ bb particles with lifetimes above about 500 fs can lead to displaced B c 's with significantly higher IP values than those of the prompt B c 's. In contrast, if their lifetimes are much below 500 fs, the IP values will hardly help separate their decaying B c 's from the prompt ones. Therefore, we will focus on the Ξ + bc , which is expected to have a sufficiently long lifetime [14].
The Ξ bc production cross section at the LHC has been theoretically evaluated in Refs. [15,16]. To reduce systematic uncertainties, instead of the direct result for the cross section we adopt the cross section ratio σ(Ξ bc )/σ(Ξ cc ) ≈ 40% [15]. The signal is determined by a displaced Ξ ++ cc , so its detection efficiency is expected to be identical to that of a normal Ξ ++ cc , (Ξ ++ cc ). With these inputs, the expected signal yield N s is expressed as
N s = N p (Ξ + bc ) · B(Ξ bc → Ξ ++ cc + X) · (Ξ ++ cc ) = N d (Ξ ++ cc ) · σ(Ξ bc ) σ(Ξ cc ) · B(Ξ + bc → Ξ ++ cc + X),(13)
where N p,d are the number of produced and detected particles. Quantitatively, it is expected that LHCb Run 3 will collect approximately 10 4 Ξ ++ cc 's through the Λ + c K − π + π + [42] and Ξ + c π + [43] reconstruction. Combining the inclusive decay branching ratio (12) and the Ξ bc production information σ(Ξ bc )/σ(Ξ cc ) ≈ 40% [15], one finally arrives at the signal yield at the LHCb Run 3, N s ≈ 300. In a real measurement, some of these events will get swamped by the background of the primarily produced Ξ ++ cc 's, if, for example, the Ξ + bc 's do not fly far enough before decaying into Ξ ++ cc 's. Although it will lose some efficiencies, the Ξ bc discovery will still be hopeful during LHCb Run 3, and will be very promising during LHCb Run 4 and at the high-luminosity LHC.
As for the background, a displaced Ξ ++ cc is also possibly produced from B + c decays, B + c → Ξ ++ cc + X. However, such background is negligible because the branching ratio is expected to be tiny due to the phase-space suppression. The dominant quark-level transition for such decays isb → ccs, so the least massive final state is Ξ ++ ccΞ − c , with ∼0.18 GeV phase space. Considering a similar decay channel B → Λ c +Ξ c with ∼0.5 GeV phase space having an O(10 −3 ) branching ratio [41], the B + c → Ξ ++ ccΞ − c branching ratio is expected to be even smaller. It also allows decay processes with some other final states such as Ξ ccΞc π and Ξ ccΞ * c , but all of them are expected to have similar or smaller branching ratios due to even smaller phase spaces compared to Ξ ++ ccΞ − c . As B c and Ξ bc have production cross sections of the same order at the LHC [16], the number of the displaced Ξ ++ cc 's produced via B c decays is smaller than that of the signal by at least 1 to 2 orders of magnitude. Therefore, this background source can be safely neglected.
With the above analyses/calculations about aspects of the experimental Ξ bc search, we can conclude that it will be very hopeful to discover Ξ bc during LHCb Run 3 via the inclusive approach that we proposed. The inclusive approach should be more efficient than searches using exclusive decays. The exclusive channels induced by b quark decaying typically have branching ratios smaller than B(Ξ bc → Ξ ++ cc + X) by more than 1 or 2 orders of magnitude [9]. The bc annihilation channels are power suppressed and are thus even rarer. The c quark decay channels suffer low reconstruction efficiencies of the bhadrons in their final states [11].
Conclusion. -We have proposed that the inclusive Ξ bc decay channel -or, more explicitly, Ξ + bc → Ξ ++ cc +X-can be used to search for the Ξ bc baryons, with a very clean and simple signal: a displaced Ξ ++ cc . By making use of effective theories of QCD, we have calculated its branching ratio at the leading order and found that it is approximately 7%, while the radiative and power corrections are left for future studies. Based on the result for the Ξ + bc → Ξ ++ cc + X branching ratio, the Ξ bc production rate, and the Ξ ++ cc detection efficiency extracted from previous experiments, we estimated that LHCb Run 3 can accumulate approximately 300 such signal events. The possible background, the B + c → Ξ ++ cc + X decay, has been demonstrated to be negligible. In conclusion, the inclusive Ξ + bc → Ξ ++ cc + X decay is very likely to serve as the discovery channel for the Ξ bc baryons.
PACS numbers: 13.30.-a;12.39.Hg;12.39.St
where the coefficients L i are functions of w ≡ v · v . The property / vD v = D v together with v· = v · = 0 simplifies the current (5) such that L T can be replaced by a scalar function ξ(w) = L 0 +L 1 +L 2 +L 3 . Evaluating the trace with Γ = γ µ , γ µ γ 5 under (5) with aFIG. 2: A sample Feynman diagram of the X bc → Xcc diquark transition induced by the V − A current at large recoil. The double lines denote the heavy quarks. The gluon line close to the weak vertex denotes a hard gluon. The dashed lines denote any number of soft gluons which can be absorbed into the initial and final diquark wave functions. The velocity of X cc(bc) is v ( ) . The relative momentum of the two heavy quarks in X cc(bc) is k ( ) .
FIG. 3 :
3The central values (solid curves) and the ±1σ uncertainties (shadow areas) of the numerical results for a0,1,2(q 2 ) appearing in the diquark currents.and q 2 max ≈ 11.3 GeV 2 , the parameters are extracted as: f (0) = 0.247, b = −58.6, c = 238.2 for a 2 ; f (0) = 0.124, b = −52.9, c = 1898.2 for a 0 ; f (0) = 0.191, b = −57.0, c = 388.0 for a 1 . The corresponding results are plotted in FIG. 3, with the uncertainties transferred from the diquark wave functions at origin.
Acknowledgement. -The authors are grateful to Ji-Bo He, Xiang-Peng Wang and Yan-Xi Zhang for useful discussions on the experimental search at the LHC and the theoretical framework. This work is supported by Natural Science Foundation of China under Grants No. 12005068, 11735010, U2032102, 11975112, 12075124 and 12125503, and by the DFG and the NSFC through funds provided to the Sino-German Collaborative Research Center TRR110 "Symmetries and the Emergence of Structure in QCD" (NSFC Grant No. 12070131001, DFG Project-ID 196253076).
with Λ and D(Λ) the Lorentz transformation matrices for Lorentz vectors and Dirac spinors, respectively. The conjugate field of D v can be introduced asD
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| []
|
[
"Some aspects of electronic topological transition in 2D system on a square lattice. Excitonic ordered states",
"Some aspects of electronic topological transition in 2D system on a square lattice. Excitonic ordered states"
]
| [
"M Kiselev \nLaboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance\n\nRussian Research Center \"Kurchatov Institute\"\n123 182MoscowRussia\n",
"F Bouis \nLaboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance\n",
"F Onufrieva \nLaboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance\n",
"P Pfeuty \nLaboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance\n"
]
| [
"Laboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance",
"Russian Research Center \"Kurchatov Institute\"\n123 182MoscowRussia",
"Laboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance",
"Laboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance",
"Laboratoire Leon Brillouin\n91191Gif-sur-YvetteCE-SaclayFrance"
]
| []
| We study the ordered "excitonic" states which develop around the quantum critical point (QCP) associated with the electronic topological transition (ETT) in a 2D electron system on a square lattice. We consider the case of hopping beyond nearest neighbors when ETT has an unusual character. We show that the amplitude of the order parameter (OP) and of the gap in the electron spectrum increase with increasing the distance from the QCP, δc − δ, where δ = 1 − n and n is an electron concentration. Such a behavior is different from the ordinary case when OP and the gap decrease when going away from the point which is a motor for instability. The gap opens at "hot spots" and extends untill the saddle points (SP) whatever is the doping concentration. The spectrum gets a characteristic flat shape as a result of hybrydization effect in the vicinity of two different SP's. The shape of the spectrum and the angle dependence of the gap have a striking similarity with the features observed in the normal state of the underdoped high-Tc cuprates. We discuss also details about the phase diagram and the behaviour of the density of states. PACS. 7 4.25. -q 74.72.-h 74.25.Dw 74.25.Ha | 10.1007/s100510070178 | [
"https://arxiv.org/pdf/cond-mat/9804263v2.pdf"
]
| 17,879,182 | cond-mat/9804263 | 0835cd881f446645fdfabcffe28c081b79b097bd |
Some aspects of electronic topological transition in 2D system on a square lattice. Excitonic ordered states
28 Jun 1999 November 20, 2018
M Kiselev
Laboratoire Leon Brillouin
91191Gif-sur-YvetteCE-SaclayFrance
Russian Research Center "Kurchatov Institute"
123 182MoscowRussia
F Bouis
Laboratoire Leon Brillouin
91191Gif-sur-YvetteCE-SaclayFrance
F Onufrieva
Laboratoire Leon Brillouin
91191Gif-sur-YvetteCE-SaclayFrance
P Pfeuty
Laboratoire Leon Brillouin
91191Gif-sur-YvetteCE-SaclayFrance
Some aspects of electronic topological transition in 2D system on a square lattice. Excitonic ordered states
28 Jun 1999 November 20, 2018EPJ manuscript No. (will be inserted by the editor)
We study the ordered "excitonic" states which develop around the quantum critical point (QCP) associated with the electronic topological transition (ETT) in a 2D electron system on a square lattice. We consider the case of hopping beyond nearest neighbors when ETT has an unusual character. We show that the amplitude of the order parameter (OP) and of the gap in the electron spectrum increase with increasing the distance from the QCP, δc − δ, where δ = 1 − n and n is an electron concentration. Such a behavior is different from the ordinary case when OP and the gap decrease when going away from the point which is a motor for instability. The gap opens at "hot spots" and extends untill the saddle points (SP) whatever is the doping concentration. The spectrum gets a characteristic flat shape as a result of hybrydization effect in the vicinity of two different SP's. The shape of the spectrum and the angle dependence of the gap have a striking similarity with the features observed in the normal state of the underdoped high-Tc cuprates. We discuss also details about the phase diagram and the behaviour of the density of states. PACS. 7 4.25. -q 74.72.-h 74.25.Dw 74.25.Ha
Many experiments performed for high T c cuprates provide an evidence for the existence of a pseudogap in the underdoped regime above T c and below some temperature T * (δ) which value increases with increasing the doping distance from the optimal doping, δ opt −δ [1]- [8]. The pseudogap is observed directly by angle-resolved photoemission spectroscopy (ARPES) measurements [9]- [13]. The striking about this gap is its increase with increasing δ opt − δ [12] while the critical temperature of superconducting (SC) transition, T sc , decreases. Another prominent feature is the so-called (π, 0) feature discovered by ARPES: the electron spectrum around the saddle-point (SP) is flat and disappears above some threshold value of wavevector [9]. Several hypothesis exist about possible origin of the pseudogap [14]- [17]. In this paper we present another explanation of this phenomenon in the framework of the model developed in [18]- [20]. In these works the concept of the Electronic Topological Transition in 2D system is developed and applied for the explanation of various effects experimentally observed in High -T c cuprates.
In the present paper we consider various ordered states appearing in the vicinity of ETT point. We show that the ordered "excitonic" phase [19] formed in a proximity of QCP in a 2D fermion system on a square lattice is characterized (on one side of it, δ < δ c ) by the electron spectrum strikingly similar to that observed in the underdoped cuprates. The mentioned QCP corresponds to the electron concentration n c = 1−δ c at which Fermi level (FL) crosses saddle points (SP) in the bare spectrum. As shown in [19], in the case of hopping between more than nearest neighbors this point is a point of a fundamental electronic topological transition (ETT) for which singularities in thermodynamical properties (and the logarithmic divergence in density of states at ω = 0) is only first quite trivial aspect (related to the local change in topology of the Fermi surface (FS)). The other aspects of criticality related to the mutual change in topology of FS near two different SP's lead to a very asymmetric behaviour of the noninteracting and interacting system on two sides of ETT being quite anomalous on the side δ < δ c . On the other hand, for realistic for the high-T c cuprates ratios of hopping parameters t ′ /t, δ c is given by : δ c = 0.27 for t ′ /t = −0.3 and δ c = 0.17 for t ′ /t = −0.2, i.e. the anomalous regime δ < δ c occurs in the doping range where the experimentally observed "strange metal" behaviour takes place. Moreover, δ = δ c corresponds to a maximum of T sc (δ) (as discussed in [19]) and therefore the latter regime can be considered as an underdoped regime. Some anomalies concerning the ordered "excitonic" phase have been discussed in [18]- [21]. Namely, it was shown that the line of the "excitonic" instability grows from QCP to the side δ < δ c instead of having the form of a bell around QCP as it usually happens for an ordinary QCP. Other anomalies which exists in the ordered phase are considered in the present paper. [We call this phase "excitonic" ordered phase because the discussed instabil-ity has the same origin as the classical "excitonic" instability intensively discussed in the 60-70 [22]- [27]. Namely it is related to the opposite curvature of two parts of electron spectrum in a proximity of FL. In the case considered they correspond to spectra in vicinities of two SP's.]
We consider various possibilities for the ordered state, namely, Spin and Charge Density Wave orderings with different types of the order parameter symmetries (s-wave, d-wave) depending on the effective interaction between the quasiparticles. Despite of the different symmetries, the properties of such ordered states resemble an "excitonic" states [22]- [27] are quite similar. The explanation of this phenomenon is in the fundamental role of ETT and effects of criticality in the vicinity of the corresponding QCP [20]. We show that the electron spectrum in the ordered phase is characterized by a gap on FL which opens at "hot spot" and extends until SP whatever is the doping concentration. Therefore there is always a gap at the SP wavevectors (0, ±π), (±π, 0). This remarkable feature is related, as we show in the paper, to a quite nontrivial aspect of ETT: it is the end point of two critical lines for the "polarization operator" characterizing a behaviour of the free electron system. The other side of the same effect is an increase of the amplitude of the order parameter (and of the gap) with increasing the distance from QCP on the underdoped side. We show also that the electron spectrum in a vicinity of SP gets a specific "flat" form which on one hand is typical for an "excitonic" phase (see for example [25]) being a result of a hybridization of two parts of the bare spectrum with the opposite curvature and on the other hand has a striking similarity with the form of the spectrum observed by ARPES [9]- [13]. We show that the spectrum "disappears" above some threshold value of wavevector in the direction (π, 0) − (π, π) that is also an effect of the same hybridization. We briefly discuss also features related to strong-coupling limit of the model and effects of strong electron correlations.
A starting point is a 2D system of free fermions on a square lattice with hopping between nearest (t) and next nearest (t ′ ) neighbors
ǫ k = −2t(cos k x + cos k y ) − 4t ′ cos k x cos k y(1)
(as in [19] we consider t > 0 and t ′ < 0) and |t ′ /t| not too small and not too close to the limit |t ′ /t| = .5. The dispersion law (1) is characterized by two different saddle points (SP's) located at (± π, 0) and (0, ±π) (in the first Brillouin zone (−π, 0) is equivalent to (π, 0) and (0, −π) is equivalent to (0, π)) with the energy ǫ s = 4t ′ . When we vary the chemical potential µ or the energy distance from the SP, Z, determined as
Z = µ − ǫ s = ǫ F − 4t ′ ,(2)
the topology of the Fermi surface changes when Z goes from Z < 0 to Z > 0 through the critical value Z = 0, see Fig.1.
In [19] we have shown that such a system undergoes a fundamental ETT at the electron concentration corresponding to Z = 0. The corresponding quantum critical point is quite rich. It combines several aspects of criticality. The first standard one is related to singularities in thermodynamic properties, in density of states at ω = 0 (Van Hove singularity), to additional singularity in the superconducting (SC) response function, all reflect a local change in the topology of FS. This aspect is not important for the properties we are interested in the present paper. Important aspects which reflect a mutual change in the topology of FS in the vicinities of two SP's are the following. First of all, it is a logarithmic divergence of the polarizalibility of noninteracting electrons
χ 0 (k, ω) = 1 N q n F (ǫ q ) − n F (ǫ q+k ) ǫ q+k −ǫ q − ω − i0 + ,(3)
as k = Q = (π, π), ω = 0 and Z → 0 :
χ 0 (Q, 0) ∝ ln ω max |Z| .(4)
which has an "excitonic" origin (ω max ∼ t is a cutoff energy). By "excitonic" origin we mean that two branches of the spectrum corresponding to vicinities of two SP's (
a = t − 2t ′ , b = t + 2t ′ ) ǫ 1 (k) = ǫ 1 (k) − µ = −Z + ak 2 x − bk 2 y , ǫ 2 (k) = ǫ 2 (k) − µ = −Z + ak 2 y − bk 2 x(5)
have such a form (see Fig.2) that at Z = 0 the chemical potential lies on the bottom of one "band" and on the top of the another for the given directions (0, π) − (π, π) and (π, 0)−(0, 0), (see Fig.2). Therefore, no energy is needed to excite the electron-hole pair. It is this divergence that is at the origin of density wave (DW) instability. The DW instability can be of Spin Density Wave (SDW), Charge Density Wave (CDW), Spin Current Density Wave (SCDW) or Orbital Current Density Wave (OCDW) instability [28]) of interacting electron system depending on a nature of interaction. The nontriviality stems from the aspect of criticality related to the effect of Kohn singularity in 2D system : the point Z = 0, T = 0 is the end of the critical line Z < 0 each point of which is a point of static Kohn singularities in polarizability of noninteracting electrons. As shown in [19], the latter aspect is a motor for the anomalous behaviour of the system on the side Z > 0 of ETT. One among the anomalies found in [19] concerns the ordered DW phases. We have obtained that the line of DW "excitonic" instability T DW (Z) has the anomalous form on the side Z > 0 : it grows from QCP instead of having the form of a bell around QCP as it usually happens in the case of ordinary QCP and as it indeed happens on the side Z < 0. Below we show that this latter aspect is also at the origin of anomalous behaviour of the order parameter and of some other anomalies in the ordered state in the same regime Z > 0.
As shown in [19], on the side Z > 0 of the electronic topological transition, a maximum of the static electronhole susceptibility occurs at the wavevector q = Q. Therefore in a presence of q independent interaction or q dependent interaction negative for q = Q, the DW instability happens at q = Q and this is the wavevector of ordering in the DW phase. As usual for such phases, one should consider a matrix electron Green function containing as components the normal and anomalous Green functions in terms of operators a + k,σ and a kσ which are the creation and annihilation electron's operators respectively:
K 11 (k, iω n ) = − β 0 dτ e iωnτ < T τ a kσ (τ )|a + kσ (0) > K 22 (k, iω n ) = − β 0 dτ e iωnτ < T τ a k+Qσ (τ )|a + k+Qσ (0) > K σσ ′ 12 (k, iω n ) = − β 0 dτ e iωnτ < T τ a k+Qσ (τ )|a + kσ ′ (0) > .(6)
[Below we will omit spin indices in the Green functions keeping in mind that K 12 = K σ−σ 12 for CDW and OCDW states and K 12 = K σσ 12 for SDW and SCDW states.] If the anomalous Green function K 12 is nonzero (that should be found selfconsistently) the explicit expressions for the two Green functions are as follows
K 11 (k, iω n ) = u 2 (k) iω n − ε 1 + v 2 (k) iω n − ε 2 K 22 (k, iω n ) = u 2 (k) iω n − ε 2 + v 2 (k) iω n − ε 1 , K 12 (k, iω n ) = K 21 (k, iω n ) = = u(k)v(k) 1 iω n − ε 1 − 1 iω n − ε 2 ,(7)
where u, v -coefficients have a standard form :
u 2 (k) = 1 2 1 + ǫ A (k) − ǫ B (k) 2E(k) , v 2 (k) = 1 2 1 − ǫ A (k) − ǫ B (k) 2E(k) , E(k) = ǫ A − ǫ B 2 2 + |∆(k)| 2 .(8)
The spectrum in the ordered state is given by
ε 1,2 = ǫ A + ǫ B 2 ± ǫ A − ǫ B 2 2 + |∆(k)| 2 , ǫ A (k) ≡ ǫ(k) ǫ B (k) = ǫ(k + Q),(9)
where ǫ(k) is defined by (1). The equation for the gap is
∆(k) = −T ωn 1 N p Γ 12 (k, k + Q, p)K 12 (p, iω n ) (10)
where Γ 12 is a vertex which in mean field approximation coincides with the bare interaction: (11) where V k = 2V (cos(k x ) + cos(k y )). The type of the interaction and therefore, type of the excitonic phase depend on the model. The SDW and OCDW instabilities occur in the case of a positive interaction in the triplet channel (exchange interaction), the CDW and SCDW instabilities take place for positive interaction in the singlet channel (density-density interaction). We will not fix for the moment a type of interaction and therefore a nature of the ordered phase assuming that there exists either the first or the second interaction.
Γ 12 (k, k + Q, p) = V Q , f or SDW (CDW ) Γ 12 (k, k + Q, p) = V k−p , f or OCDW (SCDW )
The equation (10) is reduced to the following equation
1 = 4|V |Π DW k=0 (Q, Z, ∆)(12)
where the "polarization operators" Π DW (Q, Z, ∆) are given by one of the following equations [30]:
Π SDW,CDW (Q, Z, ∆) = = 1 4N p 1 E(p) [tanh( ε 1 2T ) − tanh( ε 2 2T )],(13)Π OCDW,SCDW (Q, Z, ∆) = = 1 4N p (cos p x − cos p y ) 2 4E(p) [tanh( ε 1 2T ) − tanh( ε 2 2T )](14)
The expressions (12) are the equation for the SDW, CDW, OCDW or SCDW gap which should be solved selfconsistently. We emphasis that for V > 0 only SDW (OCDW) solution is possible whereas CDW (SCDW) solution takes place for V < 0.
The solution of (12)- (14) is given by one of the following expressions
∆ = ∆ SDW (k) = ∆ SDW 0 , ∆ = ∆ CDW (k) = ∆ CDW 0 , ∆ = ∆ OCDW (k) = ∆ OCDW 0 (cos k x − cos k y )/2, ∆ = ∆ SCDW (k) = ∆ SCDW 0 (cos k x − cos k y )/2(15)
The equations (10)-(15) are quite standard. A nontriviality, as we show below, is related to the behaviour of the "polarization operator" in a proximity of ETT. As we have shown in [19], the effect that the point of ETT is the end point of the critical line Z < 0 leads to the anomalous behaviour of the electron-hole susceptibility χ 0 (Q, Z, ω) on the side Z > 0. Below we show that a similar effect takes place for the "polarization operator" (14). The two functions coincides in the limit cases: χ 0 (Q, Z, ω = 0) = Π DW (Q, Z, ∆ = 0). [It is important to emphasize that the behaviour of the "polarization operator" depends only on properties of the system of noninteracting electrons, namely on the topology of FS.] Calculated for T = 0 "polarization operators" Π DW (Q, Z, ∆ 0 ) as a function of ∆ 0 for fixed Z (in the regime Z > 0) are shown in Fig.3. Since the properties of the "polarization operators" are similar in many aspects we shall omit later the indices (SDW, CDW, OCDW or SCDW) except for the cases when it will be necessary to emphasize the difference.
One can see that there is a singularity at some point
∆ 0 = ∆ c (Z). The value of ∆ c (Z) increases with increasing Z.
The situation is quite similar to that analyzed in [19] for χ 0 as a function of ω for fixed Z and T = 0. In the latter case we have found a square-root singularity at
ω = ω c = 2Z 1 − 2t ′ /t ,
which is the dynamic Kohn singularity. As we see in Fig.3, for the polarization operator Π(Q, Z, ∆) the singularity is weaker, while ∆ c (Z) also scales with Z.
Analytical estimations show that ∆ c (Z) is given by
∆ c (Z) = Z(16)
while the asymptotic form of Π(Q, Z, ∆) near the singularity is given by :
tΠ(Q, Z, ∆) = A 1 |1 − ∆/∆ c | + B, ∆ < ∆ c A 2 |1 − ∆/∆ c | + B, ∆ > ∆ c .(17)
The jump in the derivative, A 1 − A 2 , depends only on t ′ /t [31] and is proportional to
A 1 − A 2 ∝ 1 |t ′ /t| ln | 4 t ′ /t | − A 0 ,(18)
where A 0 is a constant (for the spectrum (5) A 0 = π/8).
The critical line (16) is clearly seen in Fig.4a where we present the calculated Π(Q, Z, ∆ 0 ) as a function of Z and ∆ 0 . From the point of view of the behaviour of the "polarization operator", the ETT point is the end point of two critical lines. The first is the semiaxis Z < 0 each point of which corresponds to the square-root singularity in Π(q, Z, ∆) occurring as ∆ 0 → 0 and q → q m , where the latter is the characteristic for this regime wavevector of incommensurability (see [19] where the q dependence of Π(q, Z, 0)= χ 0 (q, Z) is analyzed in details.) The second is the line Z = ∆ 0 each point of which corresponds to the kink in Π(q, Z, ∆(q)) occurring at T = 0 as q → Q where the latter is the characteristic wavevector for the regime Z > 0. At the point of intersection of these lines, Z = 0, the two types of singularities are transforming into the logarithmic singularity ;
Π(q, 0, ∆) ∝ ln | max(q − Q, ∆)|.
The existence of the growing with Z critical line determines a quite unusual form of the lines Π(Q, Z, ∆) = constant which develop around the critical line ∆ c (Z) and grow with increasing Z, (see Fig.4(b)).
In preceeding discussion we presented some general analysis which does not depend on details of interaction considered but only on the topology of the FS. To provide the calculations, let us consider a particular case of interaction resulting in Spin Density Wave (11), (13). The solution of corresponding eq. (12) for t/V = 1.8 is shown in Fig.5. Two branches of the solution have an anomalous dependence of the gap on Z reproducing the form of the lines Π(Q, Z, ∆ 0 ) = constant in Fig.4(b). The anomaly is that for both solutions gap increases with increasing the distance from the quantum critical point, i.e. from the point which is at the origin of the ordered phase.
[For an ordinary QCP the gap is maximum at the electron concentration corresponding to QCP and decreases monotonously with increasing the distance from QCP. For example such a picture takes place for DW phase on both sides from QCP in the case of t ′ = t ′′ = ... = 0; as we discussed in [19] in the latter case all anomalies in the regime δ < δ c disappear. In the case considered in the paper it happens on the overdoped side of the QCP.]
The difference between two solutions for the gap presented in Fig.5 is that
∆ 1 (Z) > Z(19)
while
∆ 2 (Z) < Z(20)
for any Z, any t/V , any t ′ /t since the two lines, ∆ 1 (Z) and ∆ 2 (Z) are attached to the critical line ∆ = ∆ c (Z) = Z from above and from below. For the most range of the existence of the ordered phase Z < Z
cr , see Fig.5, only one solution exists, the one corresponding to eq. (12). In the hyperbolic approximation and under the condition |t ′ /t| not too small Z (1) cr is given by:
Z (1)
cr ∝ ω max exp(−π 2 t/(V ln |t/t ′ |)).
For this solution one has
∆ 1 (Z) ≡ ∆ 0 (Z) = f (Z) + ∆(0)(21)
where ∆(0) is given by
∆(0) ∝ |t ′ | exp(− 2π 2 |t ′ /V | 1 − (2t ′ /t) 2 ),(22)
and f (Z) is an increasing function of Z, linear under the condition ∆(Z) ≫ ∆(0). The expression (22) is valid under condition π 2 |t ′ /V |/ 1 − (2t ′ /t) 2 ≫ 1. For the narrow Z range of the coexistence of the two solutions Z
cr < Z < Z (2) cr it is the solution ∆ 1 which is favorable (see Appendix). Therefore, the value of the gap increases with increasing Z being always larger than Z. As we have shown, this is a consequence of the effect that the point of ETT is the end point of two critical lines.
Let's analyze now the form of the spectrum in the DW phase. The spectrum given by (9) is plotted in Fig.6. for three important directions : (π, π) − (π, 0) − (0, 0) and (0, 0) − (π, π). The spectrum in the vicinity of SP has the following prominent features: The first is a characteristic "flat" shape (very close to the experimental shape [9], see Fig.8(a) being a consequence of the hybridization of the two branches of the bare spectrum in the vicinity of two different SP's with the opposite curvatures, (see Fig.7).
The second: the spectrum in the direction (π, π)−(0, π) "disappears" above some threshold value of wavevector since the residue v 2 k tends to zero (that is also an ordinary consequence of the hybridization). On the other hand, since
ε 1 (k SP ) = −Z + ∆, ε 2 (k SP ) = −Z − ∆
(see, e.g. Eq. (9)) and ∆ > Z, the chemical potential always lies in the gap for the part of Brillouin zone (BZ) starting from the "hot spot" until SP that is a consequence of the existence of the critical line ∆ = ∆ c related to the discussed above aspect of criticality of the QCP. For the direction (0, 0) − (1, 1), Fermi level lies in the lower branch of the spectrum, (see Fig.6(b)), i.e. the system remains metallic. The theoretical spectrum has a striking similarity with the anomalous experimental electron spectrum in the underdoped cuprates observed below the characteristic line T * (δ) by ARPES [9], we reproduce it in Fig.8. We remind that ARPES measures a spectral function only below FL. Then in Fig.9 we present the angle dependence of the value of ǫ k − µ, i.e. of the gap calculated from FL, in the same way as it is done in ARPES experiments [10]. Namely we plot the minimal value of |ε k − µ| for each given direction. The dependence is of a "d-wave type" in a sense that the gap increases with increasing the argument (cos k x − cos k y ) almost linearly in the proximity of SP. However the dependence is flat (not linear as it happens in the d-wave case) when approaching the direction (1, 1). Such a behaviour is also close to the experimentally found behaviour above T c [10] reproduced in Fig.9(b). [Although the authors of [10] claim that the behaviour observed above and below T c is the same, what one sees in the experimental plot is not exactly this : the behaviour above and below T c is similar in the vicinity of SP and different when approaching the (1, 1) direction and this occurs quite systematically, see also the plots in [10] for other samples.]
We considered the particular case of SDW as an example of ordreed "excitonic" state. Nevertheless, all aforesaid is true for any other types of ordered states since the existence of such states is determined only by topology of FS.
Let us study now the one particle density of states (DOS) given by the expression
N(ε) = − 1 π 1 N p [ImK R 11 (p, ε) + ImK R 22 (p, ε)] = = 1 N p [δ(ε − ε 1 (p)) + δ(ε − ε 2 (p))],(23)
Numerical calculations with the spectrum (1) give the picture shown in Fig.10.
The density of states of SDW (CDW) states deviates from the DOS in the initial metallic state in two ε ranges notated as A and B. For OCDW (SCDW) states only feature A survives. Analytical calculations show that the A-feature is related to the existence of the discussed above QCP (which we call below QCP1). Calculations of the integral in (23) performing with the hyperbolic spectrum (5) valid in the vicinities of SP's show that in the A range DOS is characterized by three singularities (instead of one logarithmic singularity in the bare density of states N 0 (ε) as ε → −Z). Those are a logarithmic singularity at
ε 1 = −Z − ∆ 0 1 − 4(t ′ /t) 2 [32] N (ε → ε 1 ) ∼ 1 t 1 − 4(t ′ /t) 2 ln( ω max |ε − ε 1 | )(24)
and jumps at two energies
ε 2,3 = −Z ± ∆ 0 .
The distance between two jumps is equal to 2∆. The B feature is related to the existence of the second quantum critical point in the system (QCP2) discussed in [29]. This point corresponds to the electron concentration when the chemical potential is equal : µ = µ c2 = 0 or by other words when the wavevector connecting two parts of FS in the direction (1, 1) is equal to Q AF = (π, π). In this case two "hot spots" on FS come together at the singular position (±π/2, ±π/2) before disappearing. The calculations of the integral in (23) with the spectrum taken around (π/2, π/2) give a logarithmic divergence at the point ε 4 = −Z − 4t ′ + ∆(π/2, π/2): (25) and a jump at the point ε = −Z − 4t ′ − ∆(π/2, π/2). This feature does not exist for OCDW (SCDW) states since ∆(k) = 0 along the diagonal of BZ. The B feature is important in the case when the chemical potential lies close to the pseudogap in the B part that should take place in the electron-doped cuprates. For the hole-doped cuprates we are interested in the present paper, it is QCP1 which determines properties of the system. In this case the chemical potential lies in the "pseudo-gap" A according to the properties of the electron spectrum in the vicinity of SP discussed above.
N (ε → ε 4 )−N 0 (ε 4 ) ∼ 1 t ∆(π/2, π/2) |t ′ | ln( ω max |ε − ε 4 | )
Let's analyze now the range of the existence of the ordered phase in the T − Z plane. For this sake let's analyze the behaviour of Π(Q, Z, ∆ 0 ) as a function of Z at finite temperature. [We again consider SDW state for certainity.] Results of calculations are presented in Fig.11. The first observation is that the gap changes only little with T at low T . The second is that the behaviour at finite temperature as a function of Z is qualitatively the same as for T = 0 and it is anomalous: the value of the gap increases with increasing Z.
The phase diagram in T − Z plane obtained for SDW (CDW) instability based on the analysis of the gap behaviour at finite T is presented in Fig.12. It is worthwhile to note that the polarization operator Π (Q, Z, ∆) (14) calculated for OCDW (SCDW) ordered states has essentially more abrupt behavior as a function of Z in comparison with those for Π SDW,CDW (Q, Z∆) (13). Such behaiour appears due to additional factor (cos(p x )−cos(p y )) 2 in the integral (12). As a result, the domain of existence of OCDW (SCDW) solutions for equation (12) at various doping concentrations is substantially narrower than for SDW (CDW) case. Nevertheless, it does not affect on the qualitative shape of phase diagram Fig.12.
The solid line is the line where ∆ 1 (T ) = 0. The dashed line is the line where ∆ 2 (T ) = 0. These two lines are at the same time the lines of instabilities of the undistorted metallic state. The line ∆ 2 (T ) = 0 is not however a line of a phase transition since the nonzero solutions for the gap exist on the left of this line until the dot-dashed line. Along the latter line corresponding to the disappearance of the "ordered" solution, the gap is finite and the two solutions coincide : ∆ 0 (T ) = ∆ 1 (T ) = ∆ 2 (T ). The situation is clearly seen from Fig.13 where we present the lines Π(Q, Z, ∆) = const for different Z and fixed t/J which in fact give the full picture of the behaviour of the DW gap as a function of Z and T .
As we discuss in the Appendix, in the region between the dot-dashed and dashed line, where three solutions ∆ 0 = ∆ 1 , ∆ 0 = ∆ 2 and ∆ = 0 coexist, it is the solution ∆ 0 = ∆ 1 which is energetically favorable.
Thus, the dot-dashed line in the phase diagram in Fig.12 is the line of the first-order phase transition. The gap along this line changes only little at low temperature and tends to zero rapidly in the vicinity of the point O. The latter is a tricritical point. The range in T − Z plane in the vicinity of this point corresponds to a strongly fluctuating regime which we will consider elsewhere. It is important to add also that at the point Z = Z (2) cr of the appearance of the ordered phase at T = 0, the gap is exactly equal to Z that means that the upper branch of the spectrum in Fig.6(a) touches FL. Then when moving inside the ordered phase the gap ∆ becomes larger than Z and this branch goes up leaving the FL.
Above we have considered the critical temperatures and the gap behaviour as functions of the energy distance from the QCP, Z. It is worth for applications to cuprates to change the description and to consider physical properties as functions of electron concentration n e or of hole doping δ = 1 − n e . To do this we use the relation between Z (or the chemical potential µ) and the hole doping which for T = 0 is given by :
1 − δ = ω N (ω)dω.(26)
So far as
Z ∝ δ c − δ,(27)
all dependencies considered above can be rewritten as functions of doping distance from QCP. For example, the phase diagram in the plane T −δ calculated for t ′ /t = −0.3 for which δ c = 0.27 gets the form shown in Fig.14.
One can easily obtain values of doping for all plots presented in Fig.4-Fig.11 when comparing the phase diagram in T − Z plane in Fig.12, and in T − δ plane in Fig.14. Obviously, the gap ∆ 0 (δ) increases with δ c − δ in the same way as it increases with Z, see Fig.5 and Fig.11 for ∆ 0 = ∆ 1 .
All features discussed above do not depend on the nature of the ordered phase, SDW, CDW, OCDW or SCDW since they reflect the topological aspects of ETT. The type of the excitopic phases developing around ETT point depend on the type of interaction. It is the SDW or OCDW state in the case of a positive interaction in the triplet channel (exchange interaction) and the CDW or SCDW state in the case of a positive interaction in a singlet channel (density-density interaction). The ordered SDW phase is characterized by spin ordering with momentum < S z Q >= 1/2(< n σσ (Q) > − < nσσ(Q) >) = ∆ 0 and the CDW phase by the charge ordering. In the SCDW (OCDW) the staggered magnetization (density) is equal to zero. Nevertheless, the spin-current (charge-current) correlation functions survive.
In our opinion for the case of high-T c cuprates it is the interaction in the triplet channel which determines the behaviour of the system and the nature of DW phase. From the theoretical point of view it is this situation which corresponds to the strong-coupling limit models : the Hubbard model and the t − J model. For example for the latter with the J term written as
H J = ij J ij {aS i S j − (b/4)n i n j } one has V SDW q = aJ q while V CDW q = − b 4 J q ,
i.e. the interaction in the triplet channel is positive while in the singlet channel is negative. This version is supported also by experiments in the high-T c cuprates : observed experimentally (by INS, see for example [33] and NMR) strong magnetic response around q = Q is a phenomenological argument in a favor of a strong momentum dependent interaction in a triplet channel, i.e. of V q = J q (J > 0). However, we can not exclude an importance of an interaction leading to the CDW (SCDW) order.
Another point concerning the interaction is its strength. Depending on the ratio |V |/W (where W is an energy bandwidth), maximal T max DW can be high or low. Respectively, the DW phase can lean out of SC state or can be hidden under it. [In the presence of the interaction in the triplet channel, J q , both SDW and SC instabilities occur around QCP1 under the same condition : J > 0, for the SC instability see [34]]. It is temptating to identify the properties obtained for the DW state with the properties observed experimentally in the underdoped cuprates above T sc (δ) and below T * (δ). Indeed they have a striking resemblance, as one can see when comparing Fig.6 and Fig.8, Fig.9(a) and Fig.9(b) and when comparing the behaviour of the gap as a function of Z (or doping, δ c − δ) with the experimental behaviour [12]. Our calculations (when considering both d-wave SC and DW instabilities in the presence of interaction J in the triplet channel) show that the answer is quite subtle. When t ′ /t = −0.2 the ordered DW phase leans out of the SC phase for t/J < 1.90, for t ′ /t = −0.3 this happens when t/J < 1.55. So far as realistic value of t/J for cuprates is estimated to be in the interval t/J = 1 − 3, both variants (when the DW phase leans out or is hidden under the SC phase) are possible.
In this case two scenarios can be discussed. First, the DW state of s-or d-wave symmetry can coexist with dwave SC state (this situation is considered in [36]) resulting in appearance of supplementary π-triplet state. Sec-ond, the DW state can be suppressed under the SC state. Nevertheless, the strong "fluctuation" memory of this ordering will affect on the behaviour of disordered metallic state.
Even in the case if the long-range ordered DW phase is hidden under SC phase it is this type of ordering which determines short range correlations in the disordered metallic state above T c and below T * (δ). This point is discussed in [19], [18]. The state below T * (δ) ∝ δ c −δ and above T c is quite exotic. It is almost frozen in both temperature and doping. By this we mean that the parameter κ 2 which determines a proximity to the ordered DW phase does almost not change neither with T nor with doping remaining therefore quite low in a wide region in T − δ plane below T * (δ). Such a quasiordered state keeps a strong memory about the ordered phase. Therefore, electron properties in this state should be close to those in the DW ordered state being however characterized by strong damping. [By the way it is exactly what is observed by ARPES. The experimental electron spectrum has a form shown in Fig.8. being however characterized by a spectral function of a very damped form. Explicit consideration of the electron spectrum in this state will be presented elsewhere].
Summarizing, we have studied the DW phase which is formed around QCP1 (associated with ETT) and we have shown that this phase is characterized by the following prominent features:
(i) the specific "flat" shape of the spectrum in the vicinity of SP, (ii) "disappearance" of the spectrum above some threshold value of wavevector in the direction (π, 0) -(π, π), (iii) pseudogap in DOS with FL lying in it, (iv) increasing of the gap in the spectrum around SP and of the pseudogap in DOS with decreasing doping for δ c − δ (v) angle dependence of the gap calculating from FL which is of a d-wave type close to SP and flat close to the direction (1, 1).
All these features have a striking similarity with the experimental features revealed by ARPES in the normal state of the underdoped hole-doped cuprates.
Appendix A
The free energy density in the approximation corresponding to considered in the paper is given by:
F = −T 1 N k α=1,2 [ln(2 cosh( ε α (k, ∆ k ) 2T )) + ∆ 2 k 4
] + µn.
(28) [Note that the equation (10) corresponds to ∂F/∂∆ = 0.] Therefore, the difference between free energies corresponding to ∆ = ∆ 1 and ∆ = ∆ 2 is given by
F 1 −F 2 = ∆ 2 1 − ∆ 2 2 4V −T 1 N k α=1,2 ln cosh( ε α (k, ∆ 1 ) 2T ) cosh( ε α (k, ∆ 2 ) 2T ) .(29)
One can check by numerical calculations that F 1 − F 2 < 0 for the whole range of the coexistence of the two solutions.
Some analytical estimations can be also done for low T based on the well-known expression [35] for the difference between thermodynamic potentials of the ordered and disordered states:
δΩ = Ω(∆ 1 ) − Ω(0) = ∆1 0 d(1/V ) d∆ ∆ 2 d∆.(30)
When substituting the expressions for ∆ 1 (21), (22) one gets
δF/t = δΩ/t ∼ − 1 − 4|t ′ /t| 2 |t ′ /t| ∆ 3 1 t 2 ∆(Z = 0) ∼ − (Z (1) cr ) 3 t 2 ∆(Z = 0) .(31)
One can see that this correction is negative. Therefore, the solution ∆ = ∆ 1 is favorable with respect to the solution ∆ = 0 for any Z Fig. 6. One particle spectra along (π, π) − (π, qy/π) and (π − qx/π, 0) − (0, 0) symmetry lines (a) and in (0, 0) -(π, π) direction (b), t ′ /t = −0.3, t/V = 1.8, Z/t = 0.03. Long dashed line is the bare spectrum, dot-dashed line corresponds to the spectrum when the residue of the Green function (7) less than 0.1. (0,0) (π,π) (π/2,π/2) Energy relative to E
. * Present address: Institüt für Theoretische Physik, Universität Würzburg, D-97074 Würzburg, Germany
Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .
12345Fermi surface of the electron system with the dispersion law (1) for different Z and t ′ /t = −0.3. The thick line corresponds to Z = 0. Schematic presentation of the electron spectrum in a vicinity of two SP's for Z = Calculated "polarization operator" Π(Q, Z, ∆(Q)) as a function of ∆0 for fixed Z and T = 0. Π(Q, Z, ∆0) as a function of ∆0 and Z at T = 0 and lines Π zz (Q) = const in Z − ∆0 coordinates. Gap ∆ obtained by solving equations (12), (14) as function of Z (t/V=1.8, t ′ /t = −0.3). The solid line corresponds to ∆1(Z), the dot-dashed line to ∆2(Z), the dashed line to ∆c(Z).
Fig. 7 .
7Schematic representation of the bare spectrum in the vicinity of the two saddle points for Z = 0.
Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .
89101112Experimental one particle spectra along (π, π) − (π, 0) − (0, 0) symmetry lines (a) and in (0, 0) -(π, π) direction (b) measured in the overdoped regime of BSCO. The data are taken from[10]. Theoretical angle dependence of the SDW gap calculated from FL in the underdoped regimeZ > 0 (t ′ /t = −0.3, t/V = 1.7, Z/t = 0.3) (a)and the experimental leading edge midpoint measured by ARPES in the underdoped BSCO [11] (b). Density of states in the ordered "excitonic" phase calculated for Z/t = 0.03 (t/V = 1.8, t ′ /t = −0.3). Dashed line corresponds to the DOS in the initial metallic state. The DW gap in t units as a function of Z for increasing temperature : T /t = 0.005, 0.1, 0.2 (t ′ /t = −0.3, t/V = 1.8). Phase diagram around QCP1 in T-Z coordinates (t/V = 1.8, t ′ /t = −0.3). We show only the regime Z > 0 corresponding to the anomalous behavior. The solid line is a line of second-order phase transition, the dot-dashed is a line of first-order phase transition and the dashed line is a line of instability of the disordered metal state (spinodal). The point O is a tricritical point.
Fig. 13 .
13Lines of Π(Q, Z, ∆) = const for fixed t/V = 1.8 and different Z. The plot (b) is a zoom of the plot (a) corresponding to the coexistence of the two solutions for the gap.
Fig. 14 .
14Phase diagram around QCP1 in T-δ coordinates (t/V = 1.8, t ′ /t = −0.3)
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We write down the expression for SDW and OCDW polarization operator just as an example. The pair of equations for CDW and SCDW is different only due to definition of Π DW. We write down the expression for SDW and OCDW polar- ization operator just as an example. The pair of equations for CDW and SCDW is different only due to definition of Π DW .
This is true under the condition of |t ′ /t| not too small. This is true under the condition of |t ′ /t| not too small.
We are grateful to D.N. Aristov for paying our attention on the complicated structure of DOS in the vicinity of ε1. We are grateful to D.N. Aristov for paying our attention on the complicated structure of DOS in the vicinity of ε1.
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| []
|
[
"BL Lacertae: the multiwavelength campaign of 2000",
"BL Lacertae: the multiwavelength campaign of 2000"
]
| [
"M Ravasio [email protected] ",
"G Tagliaferri ",
"Ghisellini G ",
"M Böttcher ",
"H D Aller ",
"M F Aller ",
"O Mang ",
"L Maraschi ",
"E Massaro ",
"C M Raiteri ",
"M Villata ",
"H Teräsranta ",
"\nDepartment of Physics and Astronomy\nOsservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly\n",
"\nAstronomy Department\nRice University\nMS 108, 6100 S. Man street Houston77005-1892TXUSA\n",
"\nInstitut für Experimentelle und Angewandte Physik\nUniversity of Michigan\n48109Ann ArborMIUSA\n",
"\nUniversität Kiel\nLeibnizstraβe 15-1924118KielGermany\n",
"\nDipartimento di Fisica\nOsservatorio Astronomico di Brera\nVia Brera 28, Universitá La Sapienza, P.le A. Moro 2, II-20121, 00185Milano, RomaItaly, Italy\n",
"\nOsservatorio Astronomico di Torino\nVia Osservatorio 2010025Pino TorineseItaly\n",
"\nHelsinki University of Technology\nMetsähovintie 11402540KylmäläFinland\n"
]
| [
"Department of Physics and Astronomy\nOsservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly",
"Astronomy Department\nRice University\nMS 108, 6100 S. Man street Houston77005-1892TXUSA",
"Institut für Experimentelle und Angewandte Physik\nUniversity of Michigan\n48109Ann ArborMIUSA",
"Universität Kiel\nLeibnizstraβe 15-1924118KielGermany",
"Dipartimento di Fisica\nOsservatorio Astronomico di Brera\nVia Brera 28, Universitá La Sapienza, P.le A. Moro 2, II-20121, 00185Milano, RomaItaly, Italy",
"Osservatorio Astronomico di Torino\nVia Osservatorio 2010025Pino TorineseItaly",
"Helsinki University of Technology\nMetsähovintie 11402540KylmäläFinland"
]
| []
| We present two BeppoSAX observations of BL Lacertae as part of a multiwavelength radio-to-TeV campaign. During the first observation we observe a faint Compton spectrum, while during the second, we detect a synchrotron spectrum with the highest [2-10] keV flux ever measured; above 10 keV an inverse Compton component begin to dominate. The synchrotron flux is very variable with time scales of ∼ 1 hr. We describe four different SED shifting the synchrotron peak both in frequency and flux intensity and we sketch a scenario in which a blob moves along a jet and can be located in or outside the BLR. This implies different radiative mechanism at work, SSC or external Compton, producing different high energy spectra. | null | [
"https://arxiv.org/pdf/astro-ph/0210415v1.pdf"
]
| 12,645,659 | astro-ph/0210415 | 0a18d898b22f7fc5525ee88a8e8da4ff9eb5324b |
BL Lacertae: the multiwavelength campaign of 2000
18 Oct 2002
M Ravasio [email protected]
G Tagliaferri
Ghisellini G
M Böttcher
H D Aller
M F Aller
O Mang
L Maraschi
E Massaro
C M Raiteri
M Villata
H Teräsranta
Department of Physics and Astronomy
Osservatorio Astronomico di Brera
Via Bianchi 46I-23807MerateItaly
Astronomy Department
Rice University
MS 108, 6100 S. Man street Houston77005-1892TXUSA
Institut für Experimentelle und Angewandte Physik
University of Michigan
48109Ann ArborMIUSA
Universität Kiel
Leibnizstraβe 15-1924118KielGermany
Dipartimento di Fisica
Osservatorio Astronomico di Brera
Via Brera 28, Universitá La Sapienza, P.le A. Moro 2, II-20121, 00185Milano, RomaItaly, Italy
Osservatorio Astronomico di Torino
Via Osservatorio 2010025Pino TorineseItaly
Helsinki University of Technology
Metsähovintie 11402540KylmäläFinland
BL Lacertae: the multiwavelength campaign of 2000
18 Oct 2002
We present two BeppoSAX observations of BL Lacertae as part of a multiwavelength radio-to-TeV campaign. During the first observation we observe a faint Compton spectrum, while during the second, we detect a synchrotron spectrum with the highest [2-10] keV flux ever measured; above 10 keV an inverse Compton component begin to dominate. The synchrotron flux is very variable with time scales of ∼ 1 hr. We describe four different SED shifting the synchrotron peak both in frequency and flux intensity and we sketch a scenario in which a blob moves along a jet and can be located in or outside the BLR. This implies different radiative mechanism at work, SSC or external Compton, producing different high energy spectra.
Introduction
Blazars are radio loud active galactic nuclei characterized by an extremely wide spectral range, from radio to γ-ray (sometimes up to TeV frequencies) and by fast and large variability: simultaneous multiwavelength observations are therefore the most powerful tool to reveal the underlying mechanisms. During the last 20 years, BL Lacertae, the BL Lac prototype, has been the target of many multiwavelength campaigns (Bregman et al., 1990;Kawai et al., 1991;Sambruna et al., 1999;Madejski et al., 1999;Ravasio et al., 2002). X-ray observations are particularly interesting for this source, because they have revealed the transition from synchrotron, which is rapidly varying, to the more quiet inverse Compton emission (Ravasio et al., 2002). Therefore during the second half of 2000 a new multiwavelength campaign was organized, ranging from radio to TeV energies: the X-ray band was covered by BeppoSAX, with a 10 5 sec run in the core of the campaign (July 17 -August 11; Ravasio et al., in prep.) and by RXTE, which assured 3 short pointings per week (Marscher et al., in prep.). Besides X-ray, the campaign comprised radio
X-ray observations
BeppoSAX observed the source while in two different optical states: during October-November BL Lac was 1.5-2 times brigher than in July (M R ∼ 14).
Date
Instrument 1). PDS data lie above the described model leaving positive residuals becoming larger towards higher energies: this can be explained as the transition from a steepening synchrotron component to an hard inverse Compton, dominating above ∼ 10 keV. BeppoSAX data are confirmed by the simultaneous RXTE [3][4][5][6][7][8][9][10][11][12][13][14][15] keV spectra: fitting them with power law models, we obtained α = 0.88 and α = 1.45 for the summer and autumn observations respectively (see table 1). The temporal behaviour of the source confirmed the high state of activity during the autumnal observation : BeppoSAX detected flux variations of more than a factor 3 in time scales of 1 hr (see fig. 1). This is similar to the event flare detected during the observation of July 1999 (Ravasio et al., 2002). In that occasion, the BL Lac X-ray spectrum was displaying the transition between the two emission mechanisms: the flare was visible only in the energy range where synchrotron radiation was dominating. In autumn 2000, instead, both LECS and MECS were seeing synchrotron emission: the fast and large variability is found in the full 0.1-10 keV energy range. This extreme behaviour is not surprising since we are observing the emission of very energetic electrons (γ ∼ 10 5 − 10 6 ) that cool very quickly (∼ 10 3 sec). Furthermore, the spectrum is steep: a small change in the spectral slope will produce large flux variations.
α 1 E b α 2 F 2−10keV (keV) (ergs cm −2 s −
Discussion
The great differencies discussed above are clearly evidenced in the Spectral Energy Distributions reported in fig. ??. To show the complex behaviour of BL Lac we plotted also two historical simultaneous SEDs, relative to the faint state of November 1995 and to the big flare of July 1997: the high energy peak of this first SED was interpreted by Sambruna et al. (1999) as SSC emission (Maraschi, Ghisellini & Celotti, 1992), while the hardness of the X-to-γ-ray flaring spectrum was attributed by Madejski et al. (1999) to an External Compton mechanism (Sikora, Begelmann & Rees, 1994). Shifting the synchrotron component in frequencies and fluxes we can phe-nomenologically reproduce all the observed SEDs, except the one of the big flare of July 1997: in that case, the X-ray Compton spectra was extremely higher (more than a factor 4) than all the other Compton spectra, while the synchrotron component was similar. This uniqueness can be accounted for using a simple scenario: a synchrotron emitting blob moving along a jet can be inside or outside the Broad Line Region (see also Ravasio et al., 2002). If outside, the synchrotron photons will be the only available targets for inverse Compton scattering; if inside, otherwise, there will be also the disk emission reprocessed by the BLR. In the special case in which a BLR cloud is present along the jet, there would be a futher target radiation field, composed by the synchrotron radiation reprocessed by the cloud (Ghisellini & Madau, 1996). This latter case could explain the extraordinary X-to-γ-ray spectra seen during July 1997. We are not able to distinguish the engine producing the spectra of 2000, since we lack γ-ray informations. Anyway the fast variability suggest the compactness of the emitting region: during the second observation, the blob could be inside the BLR, producing an hard high energy component.
(4.8, 8, 14.5 GHz: Michigan Radio Astronomy Observatory; 22, 37 GHz: Metsähovi Radio Telescope) , optical (WEBT collaboration, Villata et al., 2000) and VHE γ-ray observations (CAT; HEGRA). Because of an increase in the activity of the source (Villata et al., 2002), the campaign was prolonged until the end of 2000, with a second BeppoSAX observation started at the end of October. During the autumn, the HEGRA team accumulated 10.5 hrs on-source time and was able to set an upper limit of 25% of the Crab flux above 0.7 TeV (Mang et al., 2001). For a more detailed description of the campaign, we refer to Böttcher et al. (in prep).
Figure 1 :
1Left: October-November 2000 BeppoSAX light curves. Right top panel: simultaneous July 2000 and October-November 2000 observations. The arrow on the right represent the HEGRA upper limit, calculated at 0.7 TeV. The butterflies represent RXTE data simultaneous to BeppoSAX. Right low panel: the best sampled BL Lac simultaneous SEDs: the November 1995 faint state and the great July 1997 flare.
Table 1 :
1Best fit spectral parameters: N H is fixed at the value 2.5 × 10 21 cm −2 . The October-November analysis is performed only on LECS-MECS data. The differencies are further evident in the X-ray: when optically faint the source was not detected by the PDS experiment and the [0.6-10] keV LECS-MECS spectrum can be interpreted as inverse Compton emission, since it is well fitted by a hard power law model of index α = 0.8 (N H fixed to 2.5 × 10 21 cm −2 ; Sambruna et al., 1999; Ravasio et al., 2002). During the second observation, the PDS detected the source up to ∼ 50 keV; the [0.3-10] keV flux was the highest recorded for BL Lac and the spectrum was well fitted by a convex broken power law, softening at E b ∼ 2.2 keV (see table
AcknowledgmentsThis research was financially supported by the MURST and by the Italian Space Agency.
. M Böttcher, in prepBöttcher M. et al., in prep.
. J N Bregman, A E Glassgold, P J Huggins, ApJ. 352574Bregman J.N., Glassgold A.E., Huggins P.J. et al., 1990, ApJ, 352, 574
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. O Mang, 2658in proc. of the 27 th ICRCMang O. et al., 2001, in proc. of the 27 th ICRC, 2658
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. A P Marscher, in prepMarscher A.P. et al., in prep.
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. M Ravasio, G Tagliaferri, G Ghisellini, in prepRavasio M., Tagliaferri G., Ghisellini G. et al., in prep.
. R M Sambruna, G Ghisellini, E Hooper, R I Kollgard, J E Pesce, C M Urry, ApJ. 515140Sambruna R.M., Ghisellini G., Hooper E., Kollgard R.I., Pesce J.E., Urry C.M., 1999, ApJ, 515, 140
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. M Villata, C M Raiteri, A&A. 390407Villata M., Raiteri C.M. et al., 2002, A&A, 390, 407
| []
|
[
"Spatial distribution of electric field of equal probability quantum walks based on three-level quantum system",
"Spatial distribution of electric field of equal probability quantum walks based on three-level quantum system"
]
| [
"Xiaoguang Chen \nDepartment of communications science and engineering\nSchool of information science and engineering\nFudan University\n200433ShanghaiChina\n"
]
| [
"Department of communications science and engineering\nSchool of information science and engineering\nFudan University\n200433ShanghaiChina"
]
| []
| Based on the three-level quantum system, when it is in resonance, according to any two lattice points closest to Hamiltonian coupling, electrons transition from high energy level to low energy level and release photons; Or absorb photons and transition from low energy level to high energy level, thus obtaining the physical process of quantum walking along a straight line under the condition of equal probability. Then, the optical radiation in the quantum walk is mapped into a Gaussian pulse of the electric field, and the Maxwell's equation is solved by the threedimensional finite-difference time-domain method to obtain the spatial electric distributio. Finally, the physical process of quantum walking on two parallel lines is further discussed, involving some physical properties such as electromagnetic coupling or coherence, quantum state exchange and so on. The electric field coupling between two lines can be calculated by FDTD, which provides a useful tool for the design and analysis of quantum devices. | null | [
"https://arxiv.org/pdf/2203.06346v1.pdf"
]
| 247,447,507 | 2203.06346 | d2d848a88539690f03f0d91e733a3ae540e54c46 |
Spatial distribution of electric field of equal probability quantum walks based on three-level quantum system
Xiaoguang Chen
Department of communications science and engineering
School of information science and engineering
Fudan University
200433ShanghaiChina
Spatial distribution of electric field of equal probability quantum walks based on three-level quantum system
Based on the three-level quantum system, when it is in resonance, according to any two lattice points closest to Hamiltonian coupling, electrons transition from high energy level to low energy level and release photons; Or absorb photons and transition from low energy level to high energy level, thus obtaining the physical process of quantum walking along a straight line under the condition of equal probability. Then, the optical radiation in the quantum walk is mapped into a Gaussian pulse of the electric field, and the Maxwell's equation is solved by the threedimensional finite-difference time-domain method to obtain the spatial electric distributio. Finally, the physical process of quantum walking on two parallel lines is further discussed, involving some physical properties such as electromagnetic coupling or coherence, quantum state exchange and so on. The electric field coupling between two lines can be calculated by FDTD, which provides a useful tool for the design and analysis of quantum devices.
Introduction
Quantum walk is the counterpart of classical random walk in quantum mechanics. Quantum walk first appeared in Feynman's article on quantum mechanical computer [1]. Today, it can be regarded as the earliest continuous quantum walk [2] model. Later, the quantum random walk proposed by Aharonov et al. [3]. can be regarded as the earliest discrete quantum walk model. Quantum walk can also be divided into atomic [4,5], Xiaoguang Chen: [email protected] ionic [6], and photonic [7]. When single photons and entangled photons operate in the integrated circuit of the chip, their behavior can be coupled into harmonic oscillator or waveguide structures [8,9], which may be placed on a chip. This will greatly encourage people to study the transmission of non-classical light and the interference in the transmission process [10]. . In particular, for example, some structures can be regarded as the nearest Hamiltonian coupling, which is similar to the tightly constrained Hamiltonian, which is very famous in the field of condensed matter physics. Moreover, the integrated structure also can be allowed to do the operation of quantum logic. In addition, quantum electromagnetism, or the quantum effect in electromagnetism, or the quantization of electromagnetic field, has been given new significance by W. C. Chew et al. [11,12]. At the same time, it is driven by the single photon source and measurement, the effectiveness of Bell theory, and the rapid development of nanofabrication technology. We know that in computational theory, three state units are the most effective for classical computers [13]. In the quantum world, multi-level systems are very common. We conjecture that the future quantum processor is a quantum system based on multiple energy levels. Therefore, based on the three-level quantum system, this paper will analyze the specific physical process of equal probability quantum walking. As a preliminary knowledge, we simple review the quantum walks [2,14,15,16], the three-dimensional finite-difference time-domain method [17], and then construct the Hamiltonian operator of the open quantum system [13], mapping the trajectory of the quantum walk into the displacement of the electric field pulse. Through the three-dimensional finite-difference time-domain method, the Maxwell equation is solved, and then the spatial electromagnetic distribution is obtained.
Quantum walks
Quantum walks are typically introduced by analogy with classical random walks. The discrete quantum walks are usually called coined quantum walks [2,25], if considering walk on an infinite line as an example, one can define an amplitude of shifting to the left adjacent site or to the right adjacent site. In this case the wave function of a walking particle, initially localized at site "0" is
|ψ(t+∆t) = A|−1 +B|+1 , |A| 2 +|B| 2 = 1 (1)
Where time interval ∆t counts steps. One way to achieve this starting with state |0 is to use the state of a qubit (two-quantum system), typically referred to as "quantum coin", to supply amplitudes for the two different directions
|ψ(t + ∆t) = A| − 1, 0 + B| + 1, 1(2)
Here the second state index denotes the basis states of the coin. In the paper, we mainly consider the discrete quantum walk and replaces the coin operator with equal probability, it will be explained in detail later. Continuous time quantum walks do not rely on auxiliary quantum coins to propagate. It is evolution due to dynamic change of a unitary. By the processing method of graph theory and introducing a scalar incoherent parameter, and discussing a mixed quantum continuum and related classical dynamics, and solves the quantum random standard equations [14,15].
Three dimensional finite difference time domain FDTD
In 1966, Yee established a set of finite difference equations for the time-dependent Maxwell's curl equations system [18]. These equations can be expressed in discrete form in space and time, using the second-order accurate central difference formula. The discrete positions of electric and magnetic field components in time and space are sampled. FDTD technology divides the threedimensional problem geometry into cells to form a grid, which is composed of N x × N y × N z elements. The algorithm samples and calculates the field at discrete time points. The material parameters (dielectric constant and permeability) are distributed on the FDTD grid and are related to the field component; thus, their numbers are the same as their respective field components [17].
When the wavelength of the electromagnetic field is larger than the atomic size or the spacing of lattice points, the macro electromagnetic theory is still valid, just as expressed by classical electromagnetism [19,20]. Under the same conditions, the finite-difference time-domain method is also applicable to the quantum world [21]. As mentioned in reference [22], quantum information brings vitality to computational electromagnetism.
For lossless, non-dissipative and non-uniform media, we can obtain the discrete expression of quantum FDTD:
E n+1 x (i, j, k) =Ê n x (i, j, k) + ∆t ε x (i, j, k)∆y (Ĥ n+ 1 2 z (i, j, k) −Ĥ n+ 1 2 z (i, j − 1, k)) − ∆t ε x (i, j, k)∆z (Ĥ n+ 1 2 y (i, j, k) −Ĥ n+ 1 2 y (i, j, k − 1)) (3) H n+ 1 2 x (i, j, k) =Ĥ n− 1 2 x (i, j, k) + ∆t µ x (i, j, k)∆z (Ê n y (i, j, k + 1) −Ê n y (i, j, k)) − ∆t µ x (i, j, k)∆y (Ê n z (i, j + 1, k) −Ê n z (i, j, k)) (4)
Here, the classical electromagnetic field is promoted to the quantum world. Here, only the component representation of the x-axis direction of the quantum electromagnetic field in the rectangular coordinate system is given, and the components in other directions can be obtained similarly. ∆t represents the time step, ∆x, ∆y, ∆z which is the spatial step in the x, y, z three directions respectively. ε is dielectric constant and µ is permeability of the material and n in the equation represents the number of iterations.The "∧"in Eq.(3) and Eq.(4) represents the variables of the quantum world.
Hamiltonian operator of a threelevel quantum system
For a three-level system, 1, 2 and 3 are used to represent the energy of each energy level, namely E 1 , E 2 and E 3 . The resonant monochromatic field generated by two wave envelopes, j = 1, 2, has frequencies of and respectively. Under the rotating wave approximation (RWA) [13,24,25,26], its Hamiltonian can be written as:
H = E 1 u 1 (t)e iω 1 t 0 u 1 (t)e −iω 1 t E 2 u 2 (t)e iω 2 t 0 u 2 (t)e −iω 2 t E 3 (5)
Here,
ω 1 := E 2 −E 1 , ω 2 := E 3 −E 2 , the control quantity u j (t), j = 1, 2.
The above formula can be further written as:
H = D + 0 Ω 1 (t) 0 Ω * 1 (t) 0 Ω 2 (t) 0 Ω * 2 (t) 0 (6)
Here,
Ω i (t)=u i (t)e iω i t , i = 1, 2. D : =diag(E 1 , E 2 , E 3 ).
That is, the energy state of the system only appears on the diagonal. This item D is called "drifting" and can be eliminated by U transformation [23]. For a three-level system, throw away the fundamental terms, there are
H = 0 Ω 1 (t) 0 Ω * 1 (t) 0 Ω 2 (t) 0 Ω * 2 (t) 0 (7)
For complex three-level system, the system is in resonant state if and only if the following conditions are satisfied [24]:
Ω 1 (t) = cos(t/ √ 3)e i [(E 2 −E 1 )t+ϕ 1 ] Ω 2 (t) = sin(t/ √ 3)e i [(E 3 −E 2 )t+ϕ 2 ](8)
Here ϕ 1 and ϕ 2 are two arbitrary phases.
Quantum walking on one line
Suppose that in a system of three-level atoms, the electron has a transition (such as the L structure in Fig. 1), and the release of photons and is the time required for the electron to transition from the third energy level of the atom to the middle energy level, and the time required for the electron to transition from the middle (or second) energy level of the atom to the first (or lowest) energy level. According to the nearest neighbor Hamiltonian coupling principle [10], the following quantum walking can be obtained: It is assumed that there are seven atomic lattice points on a straight line. . Ω a and Ω b are the frequency of photons, respectively [13]. In this paper, we use L-type configuration to analyze the problem. As shown in Figure 2, the atomic lattice points are located at the coordinate position points on the one-dimensional x-axis, i.e.
x 1 = −3, x 2 = −2, x 3 = −1, x 4 = 0, x 5 = 1, x 6 = 2, x 7 = 3.
Step 1: Assuming that at the atomic lattice point of x 4 = 0, the electron transitions from the higher energy level (E 3 ) to the middle energy level (E 2 ) , and releases the photon ω 1 , consuming time is T 1 . In the resonant state, the electron transitions from the middle energy level (E 2 ) to the lower energy level (E 1 ), consuming time is T 2 . At this moment, the atomic wave function can be written as:
|ψ x 4 = |E 1(9)
At the same time, the electrons located in atomic lattice x 3 = −1 and x 5 = 1, absorb photons ω 1 and ω 2 with a probability of 1/2 respectively, completing the transition from the low energy level to the high energy level. Its atomic wave function can be written as:
|ψ x 3 = |ψ x 5 = c 2 |E 2 + c 3 |E 3(10)
Step 2: As shown in Fig.3, when the atomic lattice point is at the x 3 = −1 and x 5 = 1, the electron transitions from the higher energy level (E 3 ) to the middle energy level (E 2 ) and releases the photon ω 1 , consuming time is T 1 . In the resonant state, the electron transitions again from the middle energy level (E 2 ) to the lower energy level (E 1 ) and releases the photon ω 2 , consuming time is T 2 . At the moment, the atomic wave function can be written as:
|ψ x 3 = |ψ x 5 = |E 1(11)
When the atom lattice point is located in x 2 = −2 and x 6 = 2, the electron absorbs the photon ω 1 with a quarter probability and transitions from the intermediate level (E 2 ) to the higher energy level (E 3 ), which takes time T 1 . Then, the electron absorbs photon ω 2 with a quarter probability and transitions from lower energy level (E 1 ) to intermediate energy level (E 2 ), which takes time T 2 . At this moment, the atomic wave function can be written as:
|ψ x 2 = |ψ x 6 = c 2 |E 2 + c 3 |E 3(12)
The atomic lattice point located in x 4 = 0, the electron absorbs photon ω 1 with a probability of (1/4 + 1/4) and transitions from the intermediate level E 2 to the high-energy level E 3 , which takes time T 1 . Similarly, the electron absorbs photon ω 2 with a probability of (1/2 + 1/2) and transition from the lowest energy level E 1 to the middle energy level E 2 , which takes time T 2 . At this moment, the atomic wave function can be written as:
|ψ x 4 = c 2 |E 2 + c 3 |E 3(13)
Step 3: Located in the atom lattice point x 4 = 0, electrons release the photons ω 1 and ω 2 , and the total consuming time is T 1 + T 2 . At the same time, the electrons on the atomic lattice points x 2 = −2 and x 6 = 2, and release photons and, and the total consuming time is still T 1 + T 2 . At the atomic lattice points x 3 = −1 and x 5 = 1, the electron absorbs the photon ω 1 with a probability of 1/4 and 1/8, respectively. Similarly, the photon ω 2 is absorbed with 1/4 probability and 1/8 probability. At last, located in x 1 = −3 and x 7 = 3 atomic lattice points, electrons absorb photons with a probability of 1/8. In addition, the atomic wave function at this step is omitted here and will not be repeated.
Numerical calculation
Suppose a beam of light, with a wavelength of 804nm and a frequency of 3.7 × 10 14 Hz, passes through the center of a 900nm × 90nm × 190nm quartz crystal, as shown in Fig.5. The first step of quantum walking is realized in the center of the crystal, the electrons transition from the high energy level to the middle energy level, releasing photon ω in the form of electromagnetic pulse, while the electrons in the surrounding atomic lattice absorb photons. We can regard this process as electric field energy storage, that is
1 2 εE 2 0 = ω(14)
From this, we can calculate the average electric field intensity of the energy level, further write a simple Gaussian form of electromagnetic pulse, namely:
E(t) = E 0 cos(ω c (t − t 0 ))e − (t−t 0 ) 2 τ 2(15)
That is, the cosine modulated Gaussian wave packet is used to represent photons. The above formula is also the mathematical expression of the transient electric field of photons in time domain. The mathematical expression of photons in frequency domain can be obtained through Fourier transform: This is consistent with the mathematical expression [10] of the electric field when the electron is in the transition between the atomic energy levels, and also satisfies the resonance condition-Eq. (8).
E(ω) = E 0 ( τ √ π 2 e − τ 2 (ω−ω 0 ) 2 4 + τ √ π 2 e − τ 2 (ω+ω 0 ) 2 4 )(16)
To select the time delay t 0 = 4.5τ , and τ = nc∆Smax 2c
, where ∆S max is the maximum value of FDTD grid step (∆x, ∆y, ∆z), and n c is the number of grids per wavelength, and c is the speed of light.
In addition, in FDTD simulation [17], the initial condition of the field is zero, so the electric field of the excitation source must also be zero. This can move the time Term in the Gaussian waveform by one time unit, so that the instantaneous value of the electric field at the beginning is zero. As shown in Fig. 6, when the number of iterations is n = 100, a complete Gaussian electric field pulse can be obtained. Assuming that the distance between atomic lattice points is 40nm, the calculation results of the upper half of the first step of quantum walking in Fig. 2 are shown in Fig.7. Fig.8 is the second step of quantum walking, corresponding to the upper half of Fig.3, in which, the electric field value is only 0.707 of the corresponding value in Fig.7. Fig.9 corresponds to the upper half of Fig.4. It can be seen from the above figures that the quantum electromagnetic field at this time is instantaneous and ultra-short Figure 7: The first step of quantum is the spatial electric field distribution when the electron completes the transition, in which the abscissa is in the x direction, the ordinate is in the z direction, and the unit is nm. The electric field is polarized along the z direction. distance. Each step of quantum walking will randomly add an excitation source.
7 Description of quantum walking process and electric field distribution of two parallel beams of light As shown in Fig.10, the distances between adjacent crystal lattice points are equal. It is assumed that the initial excitation source is generated at the origin 0 of the coordinate axis, that is, the electron transitions from the third energy level (E 3 ) to the intermediate energy level (E 2 ) and releases the photon ω 1 , takes time T 1 . As shown in Fig.11, that is, they located above x 1 and x 2 axes. The double arrow straight line on the left between x 1 = 0 and x 2 = 0, it indicates that the electron between the lattice points of the two lines transitions and absorbs the photon ω 1 with a probability of 1/3. That is, in the resonant state, the electron transitions from the intermediate energy level (E 2 ) to the lowest energy level (E 1 ) again. The time taken is T 2 , as shown below the x 1 and x 2 axis in Fig.11, and the double arrow straight line on the right between x 1 = 0 and x 2 = 0, and indicates that the electron between the atomic lattice points of the two lines transitions and absorbs the photon ω 2 with 1/3 probability. In time T 1 or T 2 , the absorption and release of photons by electrons between x 1 = 0 and x 2 = 0 atomic lattices, and it can be regarded as the exchange of quantum states.
The electric field distribution corresponding to Fig.11 is shown in Fig.12. At this moment, as can be seen from Fig.12, the electric field is strongly coupled between x 1 = 0 and x 2 = 0. Quantum walking of two parallel line photons takes the second step, as shown in Fig.13. In time T 1 , the electrons at x 1 = 1 and x 1 = −1, as well as x 2 = 1 and x 2 = −1 begin to transition from high energy level to intermediate energy level, releasing the photon ω 1 with a probability of 1/3, while x 1 = 2 and x 1 = −2 as well as x 2 = 2 and x 2 = −2, the electrons absorb photon ω 1 with a probability of 1/9 respectively. At this time t = T 1 , the electrons at x 1 = 1 and x 1 = −1 as well as x 2 = 1 and x 2 = −1, absorb the photon ω 1 released by the electron transition at x 1 = 0 and x 2 = 0 with a probability of 1/9, that is, at this time, the photons are exchanged with the nearest atomic lattice around x 1 = 0 and x 2 = 0 with a probability of 1/9, that is, the exchange of quantum states. At this point, the atomic wave function can be written as: Similarly, when time t is in the time period T 2 , quantum walking path of photon ω 2 is the same as the above. As can be seen from Fig. 14, the electric field coupling between atomic lattice points is very close, and three excitation sources are generated on each line. As shown in Fig.15, the electrons on the lattice points of x 1 = 3 and x 1 = −3 as well as x 2 = 3 and x 2 = −3 absorb the photon ω 1 with a probability of 1/27, while the electrons on the lattice points of x 1 = 2 and x 1 = −2 as well as X 2 = 2 and X 2 = −2, transition from high energy level to intermediate energy level with a probability of 1/9, releasing the photon ω 1 . The electrons on the lattice points of x 1 = 1 and x 1 = −1 as well as x 2 = 1 and x 2 = −1, absorb photon ω 1 with the probability of (1/9 + 1/27) and release photon ω 1 with the probability of 1/9, that is, the quantum states with the probability of 1/9 exchange with each other. Finally, the electrons on the lattice point at x 1 = 0 and x 2 = 0, release the photon ω 1 from the high energy level to the intermediate energy level with a probability of1/9. At the same time, they exchange photons with the electrons on the nearest lattice point with a probability of 1/9, that is, quantum exchange. The electric field distribution and its evolution with time are shown in Fig.16. The evolution process of electric field with a time T 1 . As can be seen from Fig.16, the coupling of electric field between grid points begins to weaken and spread along the straight line, and five excitation sources are generated on each line. At this time, the wave function on the atomic lattice point can be written as
ψ x 1 =0 = 3 9 |E 3(17)ψ x 1 =1 = ψ x 1 =−1 = 2 9 |E 3(18)ψ x 1 =2 = ψ x 1 =−2 = 1 9 |E 3(19)ψ x 1 =0 = 1 3 |E 3(20)ψ x 1 =1 = ψ x 1 =−1 = 1 9 + 1 27 |E 3(21)ψ x 1 =2 = ψ x 1 =−2 = 2 27 |E 3(22)ψ x 1 =3 = ψ x 1 =−3 = 1 27 |E 3(23)
Similarly, when time t is in the time period T 2 , the quantum walks of photons ω 2 is the same as the above.
In the above discusstion, we suppose that, two quantized harmonic oscillator modes interact with three energy levels, these modes intersect with only one quantized harminic oscillator every two energy levels, and there is only one direct interaction between two of the three possible energy levels. Only for a quantum walk on a straight line, its physical process is similar to the traditional discrete-time quantum walk [27]. However, the quantum walk based on two parallel lines has rich physical content. As can be seen from Fig.14 and Fig.16, the third step of quantum walking produces two pairs of excitation sources than the second step of quantum walking, and the excitation sources between the two lines interact with each other and produce the exchange of quantum states. In addition, the time evolution reflects the radiation process of light pulse, that is, from weak to strong, and finally disappear. At the same time, it also shows the instantaneity and ultrashort distance of electromagnetic field.
Conclusions
Based on the three-level atomic structure and the geometric control theory of quantum computing, this paper gives the equal probability model of quantum walking on a straight line in the resonant state of the system. Through the quantum finite-difference time-domain method, the spatial electric field distribution generated by the first step, the second step and the third step of quantum walking are calculated respectively, and the physical process of each step and the mathematical expression of atomic wave function are given. Next, we give the physical evolution process of the quantum walk on two parallel lines under the condition of synchronization at the same frequency. Using the quantum electromagnetic calculation tool FDTD, the coupling of quantum evolution at a time can be obtained. However, the electric field at this time is instantaneous and ultra-short distance. Moreover, the degree of electromagnetic coupling or electromagnetic coherence between the two wires and the exchange of quantum states can be obtained. The combination of quantum walking and quantum finite-difference timedomain method brings convenience to quantum information processing and provides a practical tool for the development of quantum devices.
Figure 1 :
1Three possible configurations of atomic energy levels, when the system is in a resonant state, ∆a = ∆b = 0
Figure 2 :
2The first quantum walk in a three-level atomic system, where 1/2 represents the probability of transition
Figure 3 :
3The second step of quantum walking in a three-level atomic system, in where 1/4 represents the probability of transition.
Figure 4 :
4The third step of quantum walking in a threelevel atomic system, where 1/4 and 1/8 represent the probability of transition.
Figure 5 :
5A beam of red light passes through a quartz crystal, assuming that it passes through seven atomic lattice points.
Figure 6 :
6The Gaussian pulse waveform.
Figure 8 :
8The distribution of electric field in space when the electron completes the transition in the second step of quantum walking.
Figure 9 :
9The distribution of electric field in space when the electron completes the transition in the third step of quantum walking.
Figure 10 :
10Two parallel beams passing through a quartz crystal. The x 1 expresses photon ω 1 propagates along the x-axis. In the same way, x 2 represents photon ω 2 propagates along the x-axis.
Figure 11 :
11The first step of quantum walking of four photons on two parallel lines.
Figure 12 :
12Quantum walking of four photons on two parallel lines takes the first step. At t = T 1 , the electric field distribution in space is shown.
Figure 13 :
13Quantum walking of four photons on two parallel lines takes the second step.
Figure 14 :
14In the second step of the quantum walking of four photons on two parallel lines, the distribution of electric field in space is from top to bottom, and the time step is from ndt = 40dt to ndt = 100dt. The evolution process of electric field with time is shown.
Figure 15 :
15Quantum walking of four photons on two parallel lines takes the third step.
Figure 16 :
16In the third step of the quantum walking of four photons on two parallel lines, the distribution of electric field in space is from top to bottom, and the time step is from ndt = 40dt to ndt = 100dt.
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[
"VANISHING COEFFICIENTS IN SOME q-SERIES EXPANSIONS",
"VANISHING COEFFICIENTS IN SOME q-SERIES EXPANSIONS"
]
| [
"Dazhao Tang "
]
| []
| []
| Motivated by the recent work of Hirschhorn on vanishing coefficients of the arithmetic progressions in certain q-series expansions, we study some variants of these q-series and prove some comparable results. For instance, leta 1 (n)q n , then a 1 (5n + 3) = 0. | 10.1142/s1793042119500398 | [
"https://arxiv.org/pdf/1912.11185v1.pdf"
]
| 125,467,970 | 1912.11185 | 3084dab2dc726cee0d3ccd2ff30733cf4cfafe2e |
VANISHING COEFFICIENTS IN SOME q-SERIES EXPANSIONS
24 Dec 2019
Dazhao Tang
VANISHING COEFFICIENTS IN SOME q-SERIES EXPANSIONS
24 Dec 2019
Motivated by the recent work of Hirschhorn on vanishing coefficients of the arithmetic progressions in certain q-series expansions, we study some variants of these q-series and prove some comparable results. For instance, leta 1 (n)q n , then a 1 (5n + 3) = 0.
Introduction
The study of vanishing coefficients in infinite product expansions, which was derived from the Hardy-Ramanujan-Rademacher expansions for quotients of certain infinite products, can be traced back to Richmond and Szekeres [9]. Soon after, Andrews and Bressoud [2] proved a general theorem for certain Rogers-Ramanujan type infinite products, which contains the results of Richmond and Szekeres as special cases. Later, Alladi and Gordon [1] arrived at a more general result with some restrictions. In a recent paper, among other things, McLaughlin [8] further generalized the results of Alladi and Gordon.
Quite recently, Hirschhorn [5] studied the following two q-series:
(−q, −q 4 ; q 5 ) ∞ (q, q 9 ; q 10 ) 3 ∞ = ∞ n=0 a(n)q n , (1.1)
(−q 2 , −q 3 ; q 5 ) ∞ (q 3 , q 7 ; q 10 ) 3 ∞ = ∞ n=0 b(n)q n . (1.2)
He proved that a(5n + 2) = a(5n + 4) = 0, b(5n + 1) = b(5n + 4) = 0.
D. TANG
Here and in what follows, we adopt the following customary q-series notations:
(a; q) ∞ := ∞ n=0
(1 − aq n ), (a 1 , a 2 , · · · , a m ; q) ∞ := (a 1 ; q) ∞ (a 2 ; q) ∞ · · · (a m ; q) ∞ , for |q| < 1.
In this paper, we consider some variants of (1.1) and (1.2), and obtain some comparable results on vanishing coefficients in these q-series expansions.
Define
(−q, −q 4 ; q 5 ) 2 ∞ (q 4 , q 6 ; q 10 ) ∞ = ∞ n=0 a 1 (n)q n , (−q 2 , −q 3 ; q 5 ) 2 ∞ (q 2 , q 8 ; q 10 ) ∞ = ∞ n=0 b 1 (n)q n , (−q, −q 4 ; q 5 ) 3 ∞ (q 2 , q 8 ; q 10 ) ∞ = ∞ n=0 a 2 (n)q n , (−q 2 , −q 3 ; q 5 ) 3 ∞ (q 4 , q 6 ; q 10 ) ∞ = ∞ n=0 b 2 (n)q n , (−q, −q 4 ; q 5 ) 3 ∞ (q 3 , q 7 ; q 10 ) ∞ = ∞ n=0 a 3 (n)q n , (−q 2 , −q 3 ; q 5 ) 3 ∞ (q, q 9 ; q 10 ) ∞ = ∞ n=0 b 3 (n)q n .
Theorem 1.1. For any integer n ≥ 0,
a 1 (5n + 3) = 0, (1.3) b 1 (5n + 1) = 0, (1.4) a 2 (5n + 4) = 0, (1.5) b 2 (5n + 1) = 0, (1.6) a 3 (5n + 3) = a 3 (5n + 4) = 0, (1.7) b 3 (5n + 3) = b 3 (5n + 4) = 0. (1.8)
Proofs
Ramanujan's theta function is defined by
f (a, b) := ∞ n=−∞ a n(n+1)/2 b n(n−1)/2 ,
where |ab| < 1. The function f (a, b) enjoys the well-known Jacobi triple product identity [4, p. 35, Entry 19]:
f (a, b) = (−a, −b, ab; ab) ∞ .
For notational convenience, denote
E j := (q j ; q j ) ∞ .
Let k, l be positive integers and G(q) = ∞ n=0 g(n)q n be a formal power series. Define an operator H k,l by
H k,l (G(q)) := ∞ n=0 g(kn + l)q kn+l .
Recall that Ramanujan's classical theta functions ϕ(q) and ψ(q) are given by [7, Eqs.
ϕ(q) := f (q, q) = ∞ n=−∞ q n 2 = E 5 2 E 2 1 E 2 4 , ψ(q) := f (q, q 3 ) = ∞ n=0 q n(n+1)/2 = E 2 2 E 1 . Lemma 2.1. Define S 1 = ∞ m,n=−∞ q 20m 2 +2m+20n 2 +6n , S 2 = ∞ m,n=−∞ q 20m 2 +18m+20n 2 +6n , S 3 = ∞ m,n=−∞ q 20m 2 +2m+20n 2 +14n , S 4 = ∞ m,n=−∞ q 20m 2 +18m+20n 2 +14n , S 5 = ∞ m,n=−∞ q 20m 2 +2m+20n 2 +4n , S 6 = ∞ m,n=−∞ q 20m 2 +18m+20n 2 +4n , S 7 = ∞ m,n=−∞ q 20m 2 +2m+20n 2 +16n , S 8 = ∞ m,n=−∞ q 20m 2 +18m+20n 2 +16n .
Then
H 5,3 S 1 − q 4 S 2 = H 5,3 q 2 S 3 − q 6 S 4 = H 5,3 qS 5 − q 5 S 6 = H 5,3 q 4 S 7 − q 8 S 8 = 0.
Proof. We only prove H 5,3 (S 1 − q 4 S 2 ) = 0, and the remaining cases are similar. In S 1 , if 2m + 6n ≡ 3 (mod 5), then 2m + n ≡ −2 (mod 5). Of course, m − 2n ≡ −1 (mod 5). Assume 2m + n = 5r − 2 and m − 2n = −5s − 1, it follows that m = 2r − s − 1
and n = r + 2s. Thus H 5,3 (S 1 ) equals ∞ r,s=−∞ q 20(2r−s−1) 2 +2(2r−s−1)+20(r+2s) 2 +6(r+2s) = q 18 ∞ r,s=−∞ q 100r 2 +100s 2 −70r+50s = q 18 ∞ r,s=−∞ q 100r 2 +100s 2 +70r+50s . In S 2 , 18m + 6n ≡ −1, −2m + n ≡ −1, that is, 2m − n ≡ 1, m + 2n ≡ −2. Let 2m − n = 5r + 1 and m + 2n = −5s − 2, then m = 2r − s, n = −r − 2s − 1. Therefore, H 5,3 (q 4 S 2 ) is q 4 ∞ r,s=−∞ q 20(2r−s) 2 +18(2r−s)+20(−r−2s−1) 2 +6(−r−2s−1) = q 18 ∞ r,s=−∞ q 100r 2 +100s 2 +70r+50s , as desired.
The above q-series manipulation was developed by Hirschhorn [5], therefore we called it Hirschhorn's operation in the sequel.
We have
(−q, −q 4 ; q 5 ) 2 ∞ = f (q, q 4 ) 2 E 2 5 = 1 E 2 5 ∞ m,n=−∞ q (5m 2 +3m+5n 2 +3n)/2 = 1 E 2 5 ∞ r,s=−∞ q (5(r+s) 2 +3(r+s)+5(r−s) 2 +3(r−s))/2 + ∞ r,s=−∞ q (5(r+s−1) 2 +3(r+s−1)+5(r−s) 2 +3(r−s))/2 = E 5 10 E 4 5 E 2 20 ∞ n=−∞ q 5n 2 +3n + 2q E 2 20 E 2 5 E 10 ∞ n=−∞ q 5n 2 +2n = E 5 10 E 4 5 E 2 20 ∞ n=−∞ q 20n 2 +6n + q 2 ∞ n=−∞ q 20n 2 +14n + 2 E 2 20 E 2 5 E 10 q ∞ n=−∞ q 20n 2 +4n + q 4 ∞ n=−∞ q 20n 2 +16n .
Moreover,
(q 4 , q 6 ; q 10 ) ∞ = 1 E 10 ∞ m=−∞ (−1) m q 5m 2 +m = 1 E 10 ∞ m=−∞ q 20m 2 +2m − q 4 ∞ m=−∞ q 20m 2 +18m .
We then get
∞ n=0 a 1 (n)q n = 1 E 10 ∞ m=−∞ q 20m 2 +2m − q 4 ∞ m=−∞ q 20m 2 +18m × E 5 10 E 4 5 E 2 20 ∞ n=−∞ q 20n 2 +6n + q 2 E 5 10 E 4 5 E 2 20 ∞ n=−∞ q 20n 2 +14n + 2q E 2 20 E 2 5 E 10 ∞ n=−∞ q 20n 2 +4n + 2q 4 E 2 20 E 2 5 E 10 ∞ n=−∞ q 20n 2 +16n = E 4 10 E 4 5 E 2 20 S 1 − q 4 S 2 + q 2 S 3 − q 6 S 4 + 2E 2 20 E 2 5 E 2 10 qS 5 − q 5 S 6 + q 4 S 7 − q 8 S 8 .
In view of Lemma 2.1, we obtain (1.3). The proof of (1.4) is similar so is omitted here. Now, we are ready to prove (1.5)-(1.8).
We start with
∞ n=0 a 2 (n)q n = f (q, q 4 ) E 5 E 10 ∞ m=−∞ q 20m 2 +6m − q 2 ∞ m=−∞ q 20m 2 +14m × E 5 10 E 4 5 E 2 20 ∞ n=−∞ q 20n 2 +6n + q 2 E 5 10 E 4 5 E 2 20 ∞ n=−∞ q 20n 2 +14n + 2q E 2 20 E 2 5 E 10 ∞ n=−∞ q 20n 2 +4n + 2q 4 E 2 20 E 2 5 E 10 ∞ n=−∞ q 20n 2 +16n = E 4 10 E 5 5 E 2 20 f (q, q 4 ) ∞ r,s=−∞ q 20(r+s) 2 +6(r+s)+20(r−s) 2 +6(r−s) + ∞ r,s=−∞ q 20(r+s−1) 2 +6(r+s−1)+20(r−s) 2 +6(r−s) − q 4 ∞ r,s=−∞ q 20(r+s) 2 +14(r+s)+20(r−s) 2 +14(r−s) − q 4 ∞ r,s=−∞ q 20(r+s−1) 2 +14(r+s−1)+20(r−s) 2 +14(r−s) 6 D. TANG + 2E 2 20 E 3 5 E 2 10 f (q, q 4 ) q ∞ r,s=−∞ q 20(r+s) 2 +6(r+s)+20(r−s) 2 +4(r−s) + q ∞ r,s=−∞ q 20(r+s−1) 2 +6(r+s−1)+20(r−s) 2 +4(r−s) + q 4 ∞ r,s=−∞ q 20(r+s) 2 +6(r+s)+20(r−s) 2 +16(r−s) + q 4 ∞ r,s=−∞ q 20(r+s−1) 2 +6(r+s−1)+20(r−s) 2 +16(r−s) − q 3 ∞ r,s=−∞ q 20(r+s) 2 +14(r+s)+20(r−s) 2 +4(r−s) − q 3 ∞ r,s=−∞ q 20(r+s−1) 2 +14(r+s−1)+20(r−s) 2 +4(r−s) − q 6 ∞ r,s=−∞ q 20(r+s) 2 +14(r+s)+20(r−s) 2 +16(r−s) − q 6 ∞ r,s=−∞ q 20(r+s−1) 2 +14(r+s−1)+20(r−s) 2 +16(r−s) = E 4 10 E 5 80 f (q, q 4 ) E 5 5 E 2 20 E 2 40 E 2 160 ∞ n=−∞ q 40n 2 +12n − q 4 ∞ n=−∞ q 40n 2 +28n + 2E 4 10 E 2 160 f (q, q 4 ) E 5 5 E 2 20 E 80 q 14 ∞ n=−∞ q 40n 2 +28n − q 10 ∞ n=−∞ q 40n 2 +12n + 2E 2 20 f (q 30 , q 50 )f (q, q 4 ) E 3 5 E 2 10 q ∞ n=−∞ q 40n 2 +2n − q 10 ∞ n=−∞ q 40n 2 +38n + q 4 ∞ n=−∞ q 40n 2 +22n − q 3 ∞ n=−∞ q 40n 2 +18n + 2E 2 20 f (q 10 , q 70 )f (q, q 4 ) E 3 5 E 2 10 q 15 ∞ n=−∞ q 40n 2 +38n − q 6 ∞ n=−∞ q 40n 2 +2n + q 8 ∞ n=−∞ q 40n 2 +18n − q 9 ∞ n=−∞ q 40n 2 +22n . Denote A 1 := f (q, q 4 ) ∞ n=−∞ q 40n 2 +12n − q 4 f (q, q 4 ) ∞ n=−∞ q 40n 2 +28n , A 2 := q 14 f (q, q 4 ) ∞ n=−∞ q 40n 2 +28n − q 10 f (q, q 4 ) ∞ n=−∞ q 40n 2 +12n , A 3 := qf (q, q 4 ) ∞ n=−∞ q 40n 2 +2n − q 10 f (q, q 4 ) ∞ n=−∞ q 40n 2 +38n , A 4 := q 4 f (q, q 4 ) ∞ n=−∞ q 40n 2 +22n − q 3 f (q, q 4 ) ∞ n=−∞ q 40n 2 +18n , A 5 := q 15 f (q, q 4 ) ∞ n=−∞ q 40n 2 +38n − q 6 f (q, q 4 ) ∞ n=−∞ q 40n 2 +2n , A 6 := q 8 f (q, q 4 ) ∞ n=−∞ q 40n 2 +18n − q 9 f (q, q 4 ) ∞ n=−∞ q 40n 2 +22n .
Next, we prove that
H 5,4 (A i ) = 0, for 1 ≤ i ≤ 6.
We only prove the case A 1 here because the proofs of remaining cases are similar. Also,
f (q, q 4 ) = ∞ m=−∞ q (5m 2 +3m)/2 = ∞ m=−∞ q 10m 2 +3m + q ∞ m=−∞ q 10m 2 +7m = ∞ m=−∞ q 40m 2 +6m + q 7 ∞ m=−∞ q 40m 2 +34m + q ∞ m=−∞ q 40m 2 +14m + q 4 ∞ m=−∞ q 40m 2 +26m .
Therefore,
A 1 = P 1 − P 2 + P 3 − P 4 + P 5 − P 6 + P 7 − P 8 , where P 1 = ∞ m,n=−∞ q 40m 2 +6m+40n 2 +12n , P 2 = q 8 ∞ m,n=−∞ q 40m 2 +26m+40n 2 +28n , P 3 = q ∞ m,n=−∞ q 40m 2 +14m+40n 2 +12n , P 4 = q 11
∞ m,n=−∞ q 40m 2 +34m+40n 2 +28n ,
P 5 = q 4 ∞ m,n=−∞ q 40m 2 +26m+40n 2 +12n , P 6 = q 4 ∞ m,n=−∞ q 40m 2 +6m+40n 2 +28n , P 7 = q 7 ∞ m,n=−∞ q 40m 2 +34m+40n 2 +12n , P 8 = q 5 ∞ m,n=−∞ q 40m 2 +14m+40n 2 +28n .
By Hirschhorn's operation, we have
H 5,4 (P 2i−1 − P 2i ) = 0, for 1 ≤ i ≤ 4.
This proves (1.5).
Similarly, we get ∞ n=0 a 3 (n)q n = f (q, q 4 ) E 3 5 E 10 ∞ m=−∞ q 20m 2 +4m − q 3 ∞ m=−∞ q 20m 2 +16m × E 5 10 E 2 5 E 2 20 ∞ n=−∞ q 20n 2 +6n + q 2 E 5 10 E 2 5 E 2 20 ∞ n=−∞ q 20n 2 +14n + 2q E 2 20 E 10 ∞ n=−∞ q 20n 2 +4n + 2q 4 E 2 20 E 10 ∞ n=−∞ q 20n 2 +16n = E 4 10 f (q 30 , q 50 )f (q, q 4 ) E 5 5 E 2 20 ∞ n=−∞ q 40n 2 +2n + q 2 ∞ n=−∞ q 40n 2 +18n − q 3 ∞ n=∞ q 40n 2 +22n − q 9 ∞ n=∞ q 40n 2 +38n + E 4 10 f (q 10 , q 70 )f (q, q 4 ) E 5 5 E 2 20 q 8 ∞ n=−∞ q 40n 2 +22n + q 14 ∞ n=−∞ q 40n 2 +38n − q 7 ∞ n=∞ q 40n 2 +18n − q 5 ∞ n=∞ q 40n 2 +2n + 2E 2 20 E 5 80 f (q, q 4 ) E 3 5 E 2 10 E 2 40 E 2 160 q ∞ n=−∞ q 40n 2 +8n − q 7 ∞ n=−∞ q 40n 2 +32n + 4E 2 20 E 2 160 f (q, q 4 ) E 3 5 E 2 10 E 80 q 17 ∞ n=−∞ q 40n 2 +32n − q 11 ∞ n=−∞ q 40n 2 +8n .
Now, we discuss the following two cases: 1) Denote
B 1 := f (q, q 4 ) ∞ m=−∞ q 20n 2 +2n − q 9 f (q, q 4 ) ∞ n=∞ q 40n 2 +38n , B 2 := q 2 f (q, q 4 ) ∞ n=−∞ q 40n 2 +18n − q 3 f (q, q 4 ) ∞ n=∞ q 40n 2 +22n , B 3 := q 8 f (q, q 4 ) ∞ n=−∞ q 40n 2 +22n − q 7 f (q, q 4 ) ∞ n=∞ q 40n 2 +18n , B 4 := q 14 f (q, q 4 ) ∞ n=−∞ q 40n 2 +38n − q 5 f (q, q 4 ) ∞ n=∞ q 40n 2 +2n , B 5 := qf (q, q 4 ) ∞ n=−∞ q 40n 2 +8n − q 7 f (q, q 4 ) ∞ n=−∞ q 40n 2 +32n , B 6 := q 17 f (q, q 4 ) ∞ n=−∞ q 40n 2 +32n − q 11 f (q, q 4 ) ∞ n=−∞ q 40n 2 +8n .
By reasoning as above, we obtain
H 5,3 (B i ) = 0, for 1 ≤ i ≤ 6.
Therefore, a 3 (5n + 3) = 0.
2) Denote
C 1 := f (q, q 4 ) ∞ m=−∞ q 20n 2 +2n − q 3 f (q, q 4 ) ∞ n=∞ q 40n 2 +22n , C 2 := q 2 f (q, q 4 ) ∞ n=−∞ q 40n 2 +18n − q 9 f (q, q 4 ) ∞ n=∞ q 40n 2 +38n , C 3 := q 8 f (q, q 4 ) ∞ n=−∞ q 40n 2 +22n − q 5 f (q, q 4 ) ∞ n=∞ q 40n 2 +2n , C 4 := q 14 f (q, q 4 ) ∞ n=−∞ q 40n 2 +38n − q 7 f (q, q 4 ) ∞ n=∞ q 40n 2 +18n , C 5 := qf (q, q 4 ) ∞ n=−∞ q 40n 2 +8n − q 7 f (q, q 4 ) ∞ n=−∞ q 40n 2 +32n , C 6 := q 17 f (q, q 4 ) ∞ n=−∞ q 40n 2 +32n − q 11 f (q, q 4 ) ∞ n=−∞ q 40n 2 +8n .
Thus, as above, This establishes (1.7).
H 5,4 (C i ) = 0, for 1 ≤ i ≤ 6.
The proofs of (1.6) and (1.8) are similar to those of (1.5) and (1.7).
Final remarks
On one hand, there are more analogous results on vanishing coefficients in other infinite product expansions beyond this paper. Define
(−q r , −q t−r ; q t ) 3 ∞ (q s , q 2t−s ; q 2t ) ∞ := ∞ n=0
a r,s,t (n)q n , (3.1)
(−q r , −q t−r ; q t ) ∞ (q s , q 2t−s ; q 2t ) 3 ∞ := ∞ n=0 b r,s,t (n)q n (3.2)
where t ≥ 5 is a prime, r, s are positive integers and r < t, s = t. There are other equalities similar to (3.3)-(3.9) for t = 7 or 11. However, there are no similar results for t = 13 or 17. It is natural to ask whether or not there exists a criterion which can be used for searching for vanishing coefficients of the arithmetic progressions in a r,s,t (n) and b r,s,t (n).
On the other hand, the product representation of the Rogers-Ramanujan fractions is given by [7, Eq. (16.2.1)]:
R(q) = (q, q 4 ; q 5 ) ∞ (q 2 , q 3 ; q 5 ) ∞ = ∞ n=0 u(n)q n .
(3.10)
Richmond and Szekeres [9] also examined asymptotically the power series coefficients of a large class of infinite products including (3.10). They proved that, for sufficiently large n, u(5n), u(5n + 2) > 0, and u(5n + 1), u(5n + 3), u(5n + 4) < 0.
(3.11)
A similar result was also obtained for the coefficients of 1/R(q).
In 1981, Andrews [3] used partition-theoretic interpretations of these coefficients, to prove (3.11) holds for all n ≥ 0, except that u(3) = u(8) = u(13) = u(23) = 0. Hirschhorn [6] later provided an elementary proof of (3.11) using only the quintuple product identity.
With the aid of computer, the signs of coefficients in q-series (3.1) and (3.2) appear to be periodic from some large n for t = 5, 7, and 11. For example, b 1 (5n), b 1 (5n + 2), b 1 (5n + 3) > 0, and b 1 (5n + 4) < 0, a(10n), a(10n + 3), a(10 + 6) > 0, and a(10n + 1), a(10n + 5), a(10n + 8) < 0, a 1,1,7 (7n + 1), a 1,1,7 (7n + 6) > 0, and a 1,1,7 (7n + 3), a 1,1,7 (7n + 4) < 0, b 3,3,11 (22n + 2) > 0, and b 3,3,11 (22n + 13) < 0.
However, it is unclear how these inequalities could be proved by q-series.
Furthermore, the numerical evidence suggests the following conjecture.
Conjecture 3.1. For given r, s, and t, the signs of a r,s,t (n)'s and b r,s,t (n)'s are periodic with period t or 2t from sufficiently large n.
AcknowledgementThe author is indebted to Shishuo Fu for his helpful comments on a preliminary version of this paper. The author would like to acknowledge the referee for his/her careful reading and helpful comments on an earlier version of the paper. This work was supported by the National Natural Science Foundation of China (No. 11501061) and the Fundamental Research Funds for the Central Universities (No. 2018CDXYST0024).
Vanishing coefficients in the expansion of products of Rogers-Ramanujan type. K Alladi, B Gordon, Proc. Rademacher Centenary Conference. G. E. Andrews and D. BressoudRademacher Centenary Conference166K. Alladi and B. Gordon, Vanishing coefficients in the expansion of products of Rogers-Ramanujan type, Proc. Rademacher Centenary Conference, (G. E. Andrews and D. Bressoud, Eds.), Contemp. Math. 166 (1994), 129-139. 1
Vanishing coefficients in infinite product expansions. G E Andrews, D M Bressoud, J. Austral. Math. Soc. Ser. A. 272G. E. Andrews and D. M. Bressoud, Vanishing coefficients in infinite product expansions, J. Austral. Math. Soc. Ser. A 27 (1979), no. 2, 199-202. 1
Ramanujan's "lost" notebook III. The Rogers-Ramanujan continued fraction. G E Andrews, Adv. Math. 4111G. E. Andrews, Ramanujan's "lost" notebook III. The Rogers-Ramanujan continued fraction, Adv. Math. 41 (1981), 186-208. 11
Ramanujan's Notebooks, Part III. B C Berndt, Springer-VerlagNew YorkB. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991. 3
Two remarkable q-series expansions. M D Hirschhorn, Ramanujan J. in press. 1, 4M. D. Hirschhorn, Two remarkable q-series expansions, Ramanujan J., in press. 1, 4
On the expansion of Ramanujan's continued fraction. M D Hirschhorn, Ramanujan J. 511M. D. Hirschhorn, On the expansion of Ramanujan's continued fraction, Ramanujan J., 5 (1998), 521-527. 11
The Power of q. M D Hirschhorn, Developments in Mathematics. 4910M. D. Hirschhorn, The Power of q, Developments in Mathematics Vol. 49, Springer 2017. 3, 10
Further results on vanishing coefficients in infinite product expansions. J Mclaughlin, J. Austral. Math. Soc. Ser. A. 981J. McLaughlin, Further results on vanishing coefficients in infinite product expansions, J. Austral. Math. Soc. Ser. A 98 (2015), 69-77. 1
The Taylor coefficients of certain infinite products. B Richmond, G Szekeres, Acta Sci. Math. (Szeged). 13-410B. Richmond and G. Szekeres, The Taylor coefficients of certain infinite products, Acta Sci. Math. (Szeged) 40 (1978), no. 3-4, 347-369. 1, 10
Huxi Campus LD206, Chongqing 401331, P.R. China E-mail address: dazhaotang@sina. Dazhao Tang) College of Mathematics and Statistics, Chongqing University(Dazhao Tang) College of Mathematics and Statistics, Chongqing University, Huxi Cam- pus LD206, Chongqing 401331, P.R. China E-mail address: [email protected]
| []
|
[
"The Problem with False Vacuum Higgs Inflation",
"The Problem with False Vacuum Higgs Inflation"
]
| [
"Malcolm Fairbairn \nPhysics\nKings College London\nWC2R 2LSStrand, LondonUK\n",
"Philipp Grothaus \nPhysics\nKings College London\nWC2R 2LSStrand, LondonUK\n",
"Robert Hogan \nPhysics\nKings College London\nWC2R 2LSStrand, LondonUK\n"
]
| [
"Physics\nKings College London\nWC2R 2LSStrand, LondonUK",
"Physics\nKings College London\nWC2R 2LSStrand, LondonUK",
"Physics\nKings College London\nWC2R 2LSStrand, LondonUK"
]
| []
| We investigate the possibility of using the only known fundamental scalar, the Higgs, as an inflaton with minimal coupling to gravity. The peculiar appearance of a plateau or a false vacuum in the renormalised effective scalar potential suggests that the Higgs might drive inflation. For the case of a false vacuum we use an additional singlet scalar field, motivated by the strong CP problem, and its coupling to the Higgs to lift the barrier allowing for a graceful exit from inflation by mimicking hybrid inflation. We find that this scenario is incompatible with current measurements of the Higgs mass and the QCD coupling constant and conclude that the Higgs can only be the inflaton in more complicated scenarios. * [email protected] † [email protected] ‡ [email protected] 1 This requirement is to fit the perturbations for N = 60 e-folds before the end of inflation. This model is also in tension with Planck's n S − r plane constraints [1], where n S is the spectral index and r is the tensor-to-scalar ratio 1 arXiv:1403.7483v1 [hep-ph] | 10.1088/1475-7516/2014/06/039 | [
"https://arxiv.org/pdf/1403.7483v1.pdf"
]
| 119,235,365 | 1403.7483 | 5c786271ac76fa51e513733aa5a01800b2580604 |
The Problem with False Vacuum Higgs Inflation
Malcolm Fairbairn
Physics
Kings College London
WC2R 2LSStrand, LondonUK
Philipp Grothaus
Physics
Kings College London
WC2R 2LSStrand, LondonUK
Robert Hogan
Physics
Kings College London
WC2R 2LSStrand, LondonUK
The Problem with False Vacuum Higgs Inflation
We investigate the possibility of using the only known fundamental scalar, the Higgs, as an inflaton with minimal coupling to gravity. The peculiar appearance of a plateau or a false vacuum in the renormalised effective scalar potential suggests that the Higgs might drive inflation. For the case of a false vacuum we use an additional singlet scalar field, motivated by the strong CP problem, and its coupling to the Higgs to lift the barrier allowing for a graceful exit from inflation by mimicking hybrid inflation. We find that this scenario is incompatible with current measurements of the Higgs mass and the QCD coupling constant and conclude that the Higgs can only be the inflaton in more complicated scenarios. * [email protected] † [email protected] ‡ [email protected] 1 This requirement is to fit the perturbations for N = 60 e-folds before the end of inflation. This model is also in tension with Planck's n S − r plane constraints [1], where n S is the spectral index and r is the tensor-to-scalar ratio 1 arXiv:1403.7483v1 [hep-ph]
Introduction
A period of exponential expansion in the early Universe solves the horizon, flatness and monopole problem as well as sourcing the seeds of structure formation. The spectrum of scalar perturbations predicted from such inflationary theory has been measured many times, most recently to an impressive accuracy by the Planck satellite [1].
The recently reported observation of primordial B-modes in the polarization of the CMB by the BICEP-2 experiment [2] may turn out to be the most convincing evidence of inflation to date. Although the Planck data has made some steps in selecting from the various models that can produce inflation we are still a long way from pinning down what features the precise microscopic mechanism responsible for inflation would have to have.
What is, however, common to almost all models is the presence of a scalar inflaton. The discovery of the Higgs boson, h, by the ATLAS [3] and CMS [4] collaborations is the first (seemingly [5]) fundamental scalar we have detected. It is therefore natural to ask whether the Higgs can play the role of the inflaton. A naive first answer would be that it cannot because it is well known that for V (φ) 1 4 λφ 4 the measured spectrum of perturbations requires 1 the quartic coupling λ 10 −13 whereas the measured Higgs mass requires λ ∼ 0.13. This, however, neglects the effect of quantum corrections. Properly considered, these effects can lead to substantial modifications to the tree-level potential and a significant scale dependence of λ.
For a finely chosen mass of the top quark it is possible, as shown in [6], that the effective Higgs potential develops a flat part at large field values or even a second, local minimum, also called a false vacuum. Remarkably these features appear at approximately the correct scale to generate the observed perturbations which suggests the Higgs does indeed have a role to play in inflation.
Recently there has been a lot of interest in using the Higgs as the inflaton in the context of a nonminimal coupling to gravity [7][8][9][10][11][12][13][14][15]. It is worth noting that quantum corrections may reduce the predictiveness of such models [16] and should be taken into account. Additionally, if the recent measurement by the BICEP collaboration proves to be true then these models will be put under pressure [17] (for a possible way out see the recent works [18,19] that rely on similar tunings of the Higgs potential). Here we don't consider any such coupling and so refer to it as minimal Higgs inflation.
In this paper we will investigate how the plateau or the false vacuum could be used to explain the inflationary phase of the universe. To do so we will first look at the situation where there is a plateau in the potential and see whether the Higgs can inflate the universe on its own by slowly rolling down the plateau. The case of a false vacuum in the potential demands a mechanism for a graceful exit from inflation. Therefore, we extend the model and add an additional scalar field, s, which can lift the Higgs out of its local minimum. The strong CP-problem motivates the existence of such an additional scalar field and it is worth investigating if such a mechanism can give successful inflation.
Our calculation improved upon a previous treatment in [20] by considering the full 3-loop renormalisation group equation (RGE) improved 2-loop effective potential [21,22], including the 1-loop RGE's for the new scalar field and its threshold effect at the matching scale. Also, we account for the movement of the Higgs during inflation and further address a degeneracy in the initial depth of the false vacuum. We will see that these improvements can dramatically affect the conclusions.
The structure of the paper is as follows. In section 2 we discuss the RGE improved effective potential and attempt to use the resulting plateau for inflation. In section 3 we discuss the possibility of false vacuum inflation which is the main focus of this paper. Finally, in section 4 we present our conclusions.
Plateau Inflation
In [21,22] a state of the art 3-loop RGE improved 2-loop effective potential for the Standard Model Higgs was presented and discussed. This calculation showed that, within the current experimental errors on the Higgs and top masses, we appear to be living in a very special Universe. In figure 5 of [21] we can see that the current experimental data places the electroweak vacuum at the boundary between stable and meta-stable. The possibility of instability/meta-stability, as has been much discussed [6,[21][22][23][24][25][26], is largely a result of the sizeable negative contribution of the top yukawa coupling to the beta function of the Higgs quartic coupling, λ h , which can cause λ h to become negative at some high scale, creating an additional AdS vacuum into which we might tunnel. We appear to be safe from a catastrophic tunnelling event, however, because the lifetime of our vacuum is much greater than the current age of the Universe [21] (note that when the Higgs is not the inflaton, its dynamics during inflation might drastically reduce this lifetime [27][28][29][30]). The proximity of the current experimental data to the boundary between stable and meta-stable is a result of the peculiar fact that both λ h and β λ h can vanish at the same scale, which is highly non-trivial and merits investigation. Here M pl = 2.345 × 10 18 GeV is the reduced Planck mass.
10 −1 10 0 h/Mpl 10 −2 10 −1 V 1/4 /Mpl M t =171.0569 M t =171.0574 M t =171.0578 M t =171.0584
One consequence is the development of plateau in the Higgs potential that could lead to slow roll inflation [6] and, remarkably, it appears at approximately the correct scale to generate the observed perturbations. In figure 1 we can see the effect of tuning the top mass on the effective potential. The figure shows that tuning on the order of 0.1-1 MeV can interpolate between and stable and meta-stable vacuum. At the boundary of this transition we see the appearance of a plateau. In order to test the suitability of this scenario for inflation we can start the field above the plateau and let it roll down the potential and calculate the e-folds. The field, h, will evolve according to the field equations,
h + 3Hḣ = dV eff dh ,(1)
with
H = 1 M pl ρ 3 and ρ = 1 2ḣ 2 + V eff .(2)
Here we have
V eff = 1 4 λ eff h 4 ,(3)
where λ eff contains the 3-loop RGE's and the 2-loop corrections to the effective potential such that when we choose h as the renormalization scale λ eff becomes a function of h. The total number of e-folds is then given by,
N tot = t f ti Hdt. (4)
The result is shown in figure 2. We see that in order to get the required e-folds (50-60) to solve the horizon problem we need M h 129 GeV which is inconsistent with the value observed at the LHC. It is possible to try to relieve this constraint by, say, introducing another scalar that mixes with the Higgs such that our input λ h is smaller for the same M h . This will delay the appearance of a plateau and push it to higher scales, allowing more e-folds for lower M h values. This does not resolve the matter, however, because in both cases the inflationary scale is too high to fit the amplitude of the scalar perturbations. In the slow roll regime this amplitude is given by
A s = V 24π 2 M 4 pl 2 × 10 −9 ,(5)
where for slow roll max = 1 so we can put and upper bound on the inflationary scale of
max V 1/4 M pl ∼ 2.5 × 10 −2 .(6)
We find that whenever enough e-folds are generated by a plateau this upper bound is exceeded. It is also possible to consider very careful choices of initial conditions such that a large number of efolds could be generated by the field rolling very slowly passed the inflection point. This was addressed in the slow roll regime in [31] and it was found that satisfying both the perturbations and the e-folds simultaneously is impossible. Finally, you could imagine repeating the above calculation and allowing for a shallow well to slow the Higgs as it rolls passed, producing more e-folds. It was found that in order to avoid being trapped in the minimum by Hubble friction, the Higgs can only be slowed by a tiny amount. We found that this impacted the e-folds less than varying α s (M Z ) by 1σ.
False Vacuum Inflation
Although successful inflation cannot be achieved in the simple case of a plateau it may still be possible that the Higgs may be connected to inflation in a slightly less minimal way. To see this we can imagine starting with the plateau situation and increasing the top mass by a few×0.1 MeV. In this way we can create a false vacuum with large positive energy density that can be used to inflate the universe. This is then the scenario of old inflation and we are therefore burdened with problem of graceful exit. One possible solution [32] is to extend general relativity to a scalar-tensor theory. This allows the expansion rate of the universe with a constant inflaton energy density to decrease with time, eventually becoming slow enough to allow successful exit through tunnelling. In this paper we revisit an alternative solution, proposed in [20], that introduces an extra scalar whose dynamics during inflation slowly lifts the Universe out of the false vacuum such that is can roll classically down to the true vacuum. At this point the reader may worry that if we are introducing an extra scalar why we are not just letting that extra scalar to be the inflaton with, say, a quadratic potential. While this is a reasonable position to take, it ignores presence of the false vacuum in the Higgs potential. We also expect that the Higgs will be coupled to any additional scalars (e.g. the scalar responsible for Peccei-Quinn symmetry breaking) that appear above the electroweak scale through the Higgs portal coupling regardless of whether these scalars can achieve inflation on their own. We therefore consider the approach that the Higgs is responsible for inflation and the additional scalar merely facilitates the graceful exit. For a recent update on false vacuum Higgs inflation see [33], in which the dynamics for the removal of the barrier are left undiscussed.
The tree-level potential in terms of the real-fields is given by,
V = 1 4 λ s s 2 − w 2 2 + 1 4 λ h h 2 − v 2 2 + 1 4 λ hs s 2 − w 2 h 2 − v 2 ,(7)
where s is the real part of a possibly complex standard model singlet scalar field and respects a global Z 2 (real field) or U (1) (complex field) symmetry. Such a complex field, S, arises in the context of invisible axion models, where the symmetry is identified with the U (1) Peccei-Quinn that solves the strong CP problem. The phase of S then becomes the QCD axion.
During inflation the tachyonic s field will roll towards its minimum s and the mixing term between h and s will grow and lift the false vacuum as shown in figure 3. The end of inflation is taken as the point at which the false minimum disappears. In reality, tunnelling will become highly probable when the well depth is sufficiently small (when Γ tunnel H) causing inflation to end slightly earlier. Additionally the subsequent free rolling of the field down to the global minimum can still produce some inflation. Both of these effects however are small and change the calculation by a negligible number of efolds which will not alter our conclusions and so we neglect these contributions. We therefore have a setup similar to hybrid potential [34] in which the rolling of s triggers the waterfall field h but in this case the false vacuum is created purely by quantum effects. It is possible to treat this as an approximately single field model because h is trapped in the false minimum throughout inflation. The renormalized potential can then be written 2 as a function of s
V s = 1 4 λ s s 2 − s 2 2 + 1 4 λ eff − λ 2 hs 4λ s h 2 − v 2 2 ,(8)with s = 1 √ 2λ s M 2 s + λ hs (v 2 − h 2 ) ,(9)
where M s is the mass of the new scalar and v = 246 GeV is Higgs vev in the true vacuum. The position, h , of the false vacuum will change during inflation (see figure 3) so we may treat it as a function of s.
As h rolls to the global minimum s relaxes to its ground state value given by
f a = s | h =v = M s √ 2λ s ,(10)
where f a can be interpreted as the axion decay constant for the case where S is a complex field charge under U (1) P Q .
The total number of e-folds is then calculated using
N = 1 M pl s end s=0 ds √ 2 ,(11)with = M 2 pl 2 V s V s 2 .(12)
The amplitude of the scalar perturbations are then calculated N * = 50 − 60 e-folds before the end of inflation using equation (5). The introduction of the new scalar will modify the low energy Higgs parameters and the RGE's of standard model. Firstly, it was shown in [35] that when the mass of the extra scalar is much larger than the electroweak scale (as will always be the case here) it can be integrated out to yield an effective theory below M s . In this effective theory the Higgs quartic coupling will be modified to that of the standard model as a result of the λ hs mixing term. Below M s we must replace
λ h → λ = λ h − λ 2 hs 4λ s ,(13)
where λ ∼ 0.129 is what is inferred from the Higgs mass measurement and is what enters the SM running below M s . At M s we must therefore apply a threshold effect to match to the full theory by replacing λ with λ h = λ + λ 2 hs 4λs . Above M s we must also include the s field in the RGE's
(4π) 2 β λ h = (4π) 2 β SM λ h + 1 2 λ 2 hs ,(14)
(4π) 2 β λ hs = 1 4 λ hs 12y 2 t −
9 5 g 2 1 − 9g 2 2(15)+ λ hs (6λ h + 4λ s ) + 2λ 2 hs , (4π) 2 β λs = λ 2 hs + 10λ 2 s .(16)
In order for this mechanism to work both λ s and λ hs will need to be very small so the RGE contribution will be minor. The threshold effect however can still be significant because even small changes in λ h can substantially change the position of the false vacuum.
In order to test this model we select as inputs {M h , α s (M Z ), λ s , λ hs , M s }. Requiring the presence of the false vacuum then largely determines M t . There is, however, a degeneracy in such an approach resulting from the freedom of tuning the initial depth of the well using M t . Different well depths will result in different s end , and hence A s , values for the same set of input parameters. To resolve this we tune M t for each set of inputs such that the resulting well depth generates the best possible fit to A s . We therefore choose only the best possible point in the degenerate set of outputs given our 5 inputs.
The result of an extensive nested sampling scan using MultiNest [36] is shown in figure 4. The χ 2 was derived from fitting the experimental value of M h = 125.66 ± 0.34 GeV (see [21] and references therein) and M t = 173.35±0.76 GeV [37], the world average of α s (M Z ) = 0.1184 ± 0.0007 [38], and observed value of A s = (2.196 ± 0.060) × 10 −9 [1]. It is clear from figure 4 that the best fit point is inconsistent with the 3 sigma contours in (M h , α s (M Z )).
Conclusion
In this paper we have considered two possible implementations of minimal Higgs inflation. In section 2 we tuned the Higgs potential in such a way that a plateau appears and investigated whether this plateau can be used to inflate the universe via a slow-rolling of the Higgs alone. We considered the full 3-loop RGE improved 2-loop effective potential.
A simultaneous fit of the number of e-foldings and the scalar perturbations turned out to be impossible, such that an extension of the Standard Model is necessary, compare figure 2. The most minimal extension was investigated in section 3 where we introduced an additional singlet scalar field s and looked at a hybrid scenario. Such a scalar field is motivated by the strong CP-problem. In this case, the Higgs sits in a local minimum of the potential and s slowly rolls towards the minimum of its potential. The mutual coupling between s and the Higgs field removes the barrier during the rolling of s such that the Higgs can then roll towards its global minimum and successful exit is guaranteed. To ensure a correct treatment, we included the 1loop RGE's for the new scalar, the threshold effect in the Higgs potential occurring at the mass of the singlet scalar, the movement of the Higgs field during inflation and the degeneracy in the well depth.
Our results are summarised in figure 4 where one can see that those sets of parameters that give a good fit to the inflationary observables are clearly excluded by measurements of the Higgs mass and the strong coupling constant.
With standard General Relativity and Quantum Field Theory with minimal couplings between the particles and gravity it has been shown that one cannot obtain inflation using only the standard model Higgs. In this work we show that even with an additional field allowing the Higgs to become the waterfall field of a hybrid inflation model, the coupling between the two fields conspires to prevent good inflationary parameters. Inflation can only be explained using either a more complicated scenario or an entirely separate field such that the Higgs plays no role in the process.
Figure 1 :
1The figure shows the effective potential for M h = 125 GeV. The top mass is tuned in order to show the appearance of a plateau or an instability. The four curves plotted differ by 0.5 MeV in M t .
Figure 2 :
2This figure shows the total number of efolds of inflation caused by a Higgs rolling from rest at 10 M pl . The thickness in the band is set by the ±1σ error on α s (M Z ) and the color bar indicates the value of M t required for a plateau. For smaller M h the plateau is shorter and occurs at a lower scale and so has only a very small effect on the rolling of the field. For larger M h the plateau is significant enough to cause an extended period of slow roll.
Figure 3 :
3This plot shows the effect of the mixing between the singlet, s, and the Higgs, h, on the Higgs contribution to V as s rolls towards its minimum. The singlet field manages to successfully remove the false vacuum allowing the Higgs to roll down to true vacuum
Figure 4 :Figure 5 :
45This plot shows the binned best fit point in the M h -α s (M Z ) plane for N * = 50. Also depicted are the 1,2, and 3σ experimental contours. The global best fit point, marked with a yellow star, is inconsistent with experiment at more that 3σ.There is a clearly visible sharp boundary in figure 4. This marks the point when the scale of inflation exceeds the maximum value allowed by equation (6) preventing any chance of achieving a good fit to A s by tweaking . The corresponding distributions for λ s , λ hs , and M s and their best fit value are shown infigure 5. It is clear that in order to fit the scalar perturbations and the e-folds simultaneously, quite small values for the couplings are needed, which would result in a very large (Planckian) expectation value s | h =v for the extra scalar field, making the case where we identify it with the real part of the Peccei-Quinn scale less phenomenologically interesting. The sharp line in the λ s -λ hs plane marks the region above which the threshold effect becomes too large and results in a push of the false vacuum to too large scales. This figure shows the binned best fit point for λ s , λ hs , and M s with N * = 50. The global best fit point is marked with a yellow star.
Here we neglect the h 2 term because we are interested in the large h behaviour.
AcknowledgementsMF is funded by the STFC, PG is funded on an ERC Senior Fellowship and RH receives funding from a KCL graduate school award.
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| []
|
[
"Population inference of spin-induced quadrupole moments as a probe for non-black hole compact binaries",
"Population inference of spin-induced quadrupole moments as a probe for non-black hole compact binaries"
]
| [
"Muhammed Saleem \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA\n\nChennai Mathematical Institute\n603103SiruseriTamilnaduIndia\n",
"N V Krishnendu \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nCallinstr. 3830167HannoverGermany\n\nLeibniz Universitat Hannover\nD-30167HannoverGermany\n",
"Abhirup Ghosh \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476PotsdamGermany\n",
"Anuradha Gupta \nDepartment of Physics and Astronomy\nThe University of Mississippi\n38677OxfordMSUSA\n",
"W Del Pozzo \nDipartimento di Fisica \"Enrico Fermi\"\nUniversita' di Pisa\nINFN sezione di Pisa\nI-56127PisaItaly\n\nInstitute of Gravitational Wave Astronomy\nUniversity of Birmingham\nEdgbastonB15 2TTBirminghamUnited Kingdom\n",
"Archisman Ghosh \nGhent University\nProeftuinstraat 869000GentBelgium\n",
"K G Arun \nChennai Mathematical Institute\n603103SiruseriTamilnaduIndia\n"
]
| [
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA",
"Chennai Mathematical Institute\n603103SiruseriTamilnaduIndia",
"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nCallinstr. 3830167HannoverGermany",
"Leibniz Universitat Hannover\nD-30167HannoverGermany",
"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476PotsdamGermany",
"Department of Physics and Astronomy\nThe University of Mississippi\n38677OxfordMSUSA",
"Dipartimento di Fisica \"Enrico Fermi\"\nUniversita' di Pisa\nINFN sezione di Pisa\nI-56127PisaItaly",
"Institute of Gravitational Wave Astronomy\nUniversity of Birmingham\nEdgbastonB15 2TTBirminghamUnited Kingdom",
"Ghent University\nProeftuinstraat 869000GentBelgium",
"Chennai Mathematical Institute\n603103SiruseriTamilnaduIndia"
]
| []
| Gravitational wave (GW) measurements of physical effects such as spin-induced quadrupole moments can distinguish binaries consisting of black holes from non-black hole binaries. While these effects may be poorly constrained for single-event inferences with the second-generation detectors, combining information from multiple detections can help uncover features of non-black hole binaries. The spin-induced quadrupole moment has specific predictions for different types of compact objects, and a generalized formalism must consider a population where different types of compact objects co-exist. In this study, we introduce a hierarchical mixturelikelihood formalism to estimate the fraction of non-binary black holes in the population. We demonstrate the applicability of this method using simulated GW signals injected into Gaussian noise following the design sensitivities of the Advanced LIGO Advanced Virgo detectors. We compare the performance of this method with a traditionally-followed hierarchical inference approach. Both the methods are equally effective to hint at inhomogeneous populations, however, we find the mixture-likelihood approach to be more natural for mixture populations comprising compact objects of diverse classes. We also discuss the possible systematics in the mixture-likelihood approach, caused by several reasons, including the limited sensitivity of the second-generation detectors, specific features of the astrophysical population distributions, and the limitations posed by the waveform models employed. Finally, we apply this method to the LIGO-Virgo detections published in the second GW transient catalog (GWTC-2) and find them consistent with a binary black hole population within the statistical precision. | 10.1103/physrevd.105.104066 | [
"https://arxiv.org/pdf/2111.04135v1.pdf"
]
| 243,847,375 | 2111.04135 | dc3eb9d4f05cf2b04497c297199f4d6ec0d0f04a |
Population inference of spin-induced quadrupole moments as a probe for non-black hole compact binaries
Muhammed Saleem
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMNUSA
Chennai Mathematical Institute
603103SiruseriTamilnaduIndia
N V Krishnendu
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Callinstr. 3830167HannoverGermany
Leibniz Universitat Hannover
D-30167HannoverGermany
Abhirup Ghosh
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 114476PotsdamGermany
Anuradha Gupta
Department of Physics and Astronomy
The University of Mississippi
38677OxfordMSUSA
W Del Pozzo
Dipartimento di Fisica "Enrico Fermi"
Universita' di Pisa
INFN sezione di Pisa
I-56127PisaItaly
Institute of Gravitational Wave Astronomy
University of Birmingham
EdgbastonB15 2TTBirminghamUnited Kingdom
Archisman Ghosh
Ghent University
Proeftuinstraat 869000GentBelgium
K G Arun
Chennai Mathematical Institute
603103SiruseriTamilnaduIndia
Population inference of spin-induced quadrupole moments as a probe for non-black hole compact binaries
(Dated: November 9, 2021)
Gravitational wave (GW) measurements of physical effects such as spin-induced quadrupole moments can distinguish binaries consisting of black holes from non-black hole binaries. While these effects may be poorly constrained for single-event inferences with the second-generation detectors, combining information from multiple detections can help uncover features of non-black hole binaries. The spin-induced quadrupole moment has specific predictions for different types of compact objects, and a generalized formalism must consider a population where different types of compact objects co-exist. In this study, we introduce a hierarchical mixturelikelihood formalism to estimate the fraction of non-binary black holes in the population. We demonstrate the applicability of this method using simulated GW signals injected into Gaussian noise following the design sensitivities of the Advanced LIGO Advanced Virgo detectors. We compare the performance of this method with a traditionally-followed hierarchical inference approach. Both the methods are equally effective to hint at inhomogeneous populations, however, we find the mixture-likelihood approach to be more natural for mixture populations comprising compact objects of diverse classes. We also discuss the possible systematics in the mixture-likelihood approach, caused by several reasons, including the limited sensitivity of the second-generation detectors, specific features of the astrophysical population distributions, and the limitations posed by the waveform models employed. Finally, we apply this method to the LIGO-Virgo detections published in the second GW transient catalog (GWTC-2) and find them consistent with a binary black hole population within the statistical precision.
I. INTRODUCTION
Gravitational-wave (GW) observations are slated to unravel a plethora of compact binaries in the coming years. The LIGO-Virgo-KAGRA Collaboration has already observed several binary black holes (BBHs) [1][2][3][4][5][6][7][8][9][10], binary neutron stars [11,12] and neutron star -black hole mergers [13]. However, one may wonder if there are compact objects other than black holes (BHs) and neutron stars (NSs) that are made of exotic matter or described by some unknown physics 1 . There are theoretical predictions of exotic compact objects which can mimic properties of BHs and are referred as BH mimickers [14,15] (a.k.a non-BH compact objects). Some examples include boson stars [16][17][18][19][20][21][22][23][24], fermionic stars [25][26][27][28][29][30], multicomponent stars [31,32], dark energy stars [33][34][35][36][37][38] and dark matter stars [31,[39][40][41][42][43]. The unknown intrinsic properties of these objects are expected to be imprinted in the GWs they emit and hence GW observations of such objects provide a unique way to probe their presence [44][45][46][47].
The compact binary mergers observed by the advanced LIGO [48,49] and the advanced Virgo detectors [50] in the a [email protected] b [email protected] c [email protected] 1 White dwarfs, whose sizes are larger than their gravitational radii, are not considered as compact objects in current gravitational wave astronomy.
first three observing runs are all consistent with mergers composed of BHs and NSs [1][2][3][4][5][6][7][8][9][10][11][12]51]. The current sensitivity of detectors is insufficient to rule out the presence of signals from exotic compact objects in the data. With more of such detections in the future [48,[52][53][54][55][56][57][58][59][60], one of the important science goals would be to look for the existence of exotic compact objects in the data [61]. These observations may allow us to constrain what fraction of the detected events could be exotic compact objects. In turn, this can shed light on some of the unexplored physics realms concerning exotic particles and dark matter physics [34,[45][46][47].
There have been remarkable progress in the modeling of BBH waveforms in the inspiral [87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103], merger [104] and ringdown [105,106] regimes in general relativity (GR). There has also been much progress in modeling binaries containing neutron stars [107][108][109]. But the same is not true for binaries of BH mimickers, where the progress has been slow, primarily due to the mathematical complications these objects pose in the modeling [46,[110][111][112][113]. Therefore, looking for the exact imprints of the BH mimicker models in the observed GW signals is difficult.
In the post-Newtonian (PN) formalism, the effects distinguishing BHs from BH mimickers are well studied. These effects include the deformations of the compact objects due to the tidal field of the companion [45,47], or its spinning motion and the effects of tidal heating [79][80][81][82]. A number of tests have been proposed to distinguish BH mimickers from BHs using arXiv:2111.04135v1 [gr-qc] 7 Nov 2021 parametrizations of such physical effects. Examples include the tests based on the tidal deformability measurements [11, 45-47, 66-68, 73-78, 114-116], tidal heating parameter estimations [79][80][81][82]117], etc. There are also methods based on the inference of the late-ringdown echo parameters [118].
In this work, we follow a method that uses spin-induced quadrupole moment parameter to distinguish BBHs from binaries of BH mimickers, as outlined in [44,119]. The spininduced quadrupole moment parameter has a unique value, unity, for Kerr BH according to the no-hair conjecture [120][121][122], whereas, for any other compact object, its value can be different from unity. A Bayesian framework to measure this parameter has been comprehensively demonstrated in a previous study [61] using simulated GW signals. Moreover, the constraints on the spin-induced quadrupole moment parameters from the GWTC-2 events are reported in [123]. In this work, our focus is on methods to combine their measurements from multiple GW detections, which would be key in enhancing statistical evidence in favor of or against BH mimickers. In particular, the fact that BHs and BH mimickers can co-exist in the universe, points to the need for having a generic framework that can unravel how various compact objects are distributed in the universe.
We discuss two methods to combine measurements from multiple detections. The first is a so-called hierarchical combining approach in which the spin-induced quadrupole moment parameters of the detected population are assumed to follow a Gaussian distribution and use a hierarchical framework to infer the moments of the distribution. This approach is similar to the one proposed in [124] and has been used in [125] to infer the distribution of spin-induced quadrupole moment parameters for the LIGO-Virgo detected events. The second method, namely the mixture-likelihood approach, explicitly assumes the population to be a mixture of BBH and non-BBH mergers by parametrizing the fraction of events in the respective categories. Specifically, we use a parameter f nbh to quantify the fraction of non-BBH mergers in the detected population. Note that the mixture-likelihood approach is also hierarchical in nature, however, for the sake of name distinction, the notion of hierarchical is henceforth used only for the former method.
The applicability of the two approaches are demonstrated using simulated GW signals from binaries of compact objects of diverse classes. The hyperparameters that are used in both the methods can effeciently signal at the non-BH sub-populations that are present in the detected population. Furthermore, the mixture-likelihood approach captures the complexity in the population powerfully and can capture the fraction of events that are from non-BH sub-populations. However, some systematics are noticed with the mixture-likelihood approach, which could be attributed to the limitations posed by the current detector sensitivities and some of the intrinsic properties of the astrophysical population.
The rest of this paper is organized as follows. We review the GW measurements of spin-induced quadrupole moment parameters in Sec. II. Sec. III describes the statistical methods employed in this study, including the hierarchical and the mixture-likelihood approaches. In Sec. IV, we detail the properties of simulated population of compact binary mergers used to for this analysis. We discuss the main findings in Sec. V and discuss the systematic effects in Sec. VI. We conclude the study in Sec. VII with a discussion on the future aspects. We also provide an appendix on how the mixture-likelihood approach would benefit by including astrophysical models of spin-induced quadrupole moments into the framework.
II. REVIEW: GRAVITATIONAL-WAVE MEASUREMENTS OF SPIN-INDUCED MULTIPOLE MOMENTS
The spin-induced multipole moments arise due to the spinning motion of the compact objects in the binary and these effects appear in the gravitational waveform along with the selfspin terms. The leading order effect [44,93,126] at the second post-Newtonian (2PN) order can be schematically represented in the following form,
Q = −κ χ 2 m 3 ,(1)
where Q is the spin-induced quadrupole moment scalar, m is the mass and χ is the dimensionless spin parameter, defined as χ = S /m 2 where S is the spin angular momentum of the compact object. The spin-induced quadrupole moment parameter κ has a unique value, unity, for Kerr BH according to the no-hair conjecture [120][121][122], whereas, for any other compact object, its value can be different from unity. For example, for spinning NSs, the value of κ varies between ∼ 2 − 14 depending upon the internal structure of the star [83][84][85]. Also, calculations show that the value of κ can vary roughly between 10 to 150 for boson stars [23] while for gravastars [127] κ can be negative as well [128,129]. In Fig. 1 we compare gravitational waveforms of BBHs (i.e., κ = 1) and non-BBHs (κ 1) for two different values of spin parameters. Both the binaries have same masses (m 1 , m 2 ) = (20, 10)M while non-BBHs have (κ 1 , κ 2 ) = (40,25). Highly spinning binaries have (χ 1 , χ 2 ) = (0.6, 0.5) while slowly spinning binaries have (χ 1 , χ 2 ) = (0.15, 0.1), assuming the spins aligned to the orbital angular momentum axis. We used IMRPhenomPv2 [130][131][132] waveform model to simulate the time-domain GW signal. We see that as the spins of the binary components increase, the dephasing between the BBH and non-BBH waveforms increases.
A Bayesian framework to measure the κ parameters was demonstrated in [61], to constrain the nature of the stellar mass compact binaries detected by Advanced LIGO and Advanced Virgo detectors. It was shown that the spin-induced quadrupole moment measurements can be used to distinguish the observed BBHs from non-BBHs for inspiral dominated systems with moderate to high spins [61].
In this framework, one uses BBH waveforms which allow the spin-induced quadrupole moment coefficient κ to vary around the expected Kerr value as κ = 1 + δκ. Here the parametrized deformations (labeled as δκ) represents the deviations from the BH nature. It is pointed out in [61] that the simultaneous measurement of δκ 1 and δκ 2 is difficult due to strong correlations between binary parameters in the gravitational waveform. In order to capture the deviation from the BBH nature Ref. [61] proposed to measure their symmetric combination δκ s = (δκ 1 + δκ 2 )/2, assuming the anti-symmetric combination δκ a = (δκ 1 − δκ 2 )/2 vanishes for a BBH signal. The δκ a = 0 assumption also implies that the individual compact objects in the binary system are of the same nature. For cases with any violation of this assumption, we would expect an offset in δκ s posterior distribution, and such cases need more investigations keeping δκ 1 and δκ 2 as separate parameters. (See [119] for a more detailed discussion). The applicability of this test has been further explored in the context of expected detections from the third generation GW detectors [133].
III. FORMALISM
In a universe where all the massive compact objects are BBHs, δκ s assumes the unique and universal value δκ s = 0 . However, if we admit the possibility that compact objects come in many flavors, this universality assumption would be wrong, and the inference on δκ s obtained by naively multiplying each observation's likelihood would lead to erroneous conclusions. Even for the binaries that are made up of some specific class of exotic compact objects with a unique equation of state, the value of δκ s could vary depending on the intrinsic properties such as masses and spins as is found to be the case for boson stars [23]. In such a case where the value of δκ s can vary from event to event, one would aim to infer the underlying distribution of δκ s associated with the compact binary population. This is the context in which we discuss the two combining approaches and their applicability.
Below, we introduce our notations for the Bayesian inference variables, followed by the formalisms for hierarchical combining and mixture-likelihood approach.
A. Bayesian inference: basic notations
The first step in our formalism is to perform the Bayesian parameter estimation of all the detected GW events. Here, we briefly overview the Bayesian inference method employed to estimate the spin-induced quadrupole moment parameter, δκ s . We define θ as the vector representing the set of parameters that describes a BBH merger on quasi-circular orbits. This includes masses, spins, luminosity distance, time and phase of arrival, and the angles describing the sky-location and binary orientation. The data from the j th event is labeled as d j , and the set of data from N events together is denoted as d. H is our hypothesis that the data d j carries a signal h j ( θ, δκ s ) plus colored Gaussian random noise. Under this hypothesis, the posterior for the binary parameters can be written as,
p( θ, δκ s |d j , H) = π( θ, δκ s |H) L(d j | θ, δκ s , H) Z nbh j ,(2)
where L(d j | θ, δκ s , H) is the likelihood of d j being the data given the parameters { θ, δκ s }, and π( θ, δκ s |H) is the prior probability of parameters { θ. δκ s }. The evidence Z nbh j = P(d j | H) is obtained by marginalizing the likelihood over the prior,
Z nbh j = π( θ, δκ s ) L(d j | θ, δκ s ) d θ dδκ s ,(3)
where, the superscript 'nbh' stands for the non-BBH hypothesis and we have dropped H for brevity. The BBH hypothesis is a special case obtained by fixing δκ s = 0 in the likelihood L(d j | θ, δκ s = 0) which we simply write as L(d j | θ) and the corresponding evidence can be expressed as,
Z bh j = π( θ) L(d j | θ) d θ .(4)
The posterior on δκ s can be obtained by marginalizing Eq. (2) over the BBH parameters as
p(δκ s |d j , H) = p( θ, δκ s |d j , H) d θ.(5)
The IMRPhenomPv2 signal model [130][131][132] used for the Bayesian analysis includes δκ s as a free parameter along with the BBH parameters θ. We assume a uniform prior on δκ s in [−500, 500]. For the component masses, we consider uniform priors on in the range [4,100]M . The priors on component spin-magnitudes are uniform in [0,1] and their orientations assumed to be isotropic. We choose uniform in co-moving volume ranging between [10,5000]. The parameter estimation is performed using the lalinference_nest sampler available in the LALInference library package [134]. The posterior samples as well as the Bayesian evidences in Eq. (3) and (4) are obtained as raw outputs from lalinference_nest.
Below, we discuss the two different approaches for combining δκ s measurements from the Bayesian analysis of individual events.
B. Hierarchical combining approach: population distribution of δκ s
In this approach, we assume that δκ s follows some underlying distribution governed by a set of hyper-parameters α, similar to the method demonstrated in [124]. The posterior on α given the data d can be written as
p( α| d) ∝ L( d| α) p( α),(6)
where the proportionality becomes equality by normalizing the right-hand side to unity. The prior p( α) is taken to be flat assuming no prior knowledge of the underlying distribution of the hyper-parameters α. Here L( d| α) is the likelihood function which can be obtained as a product of likelihoods of α from individual events, as
L( d| α) = N j=1 L(d j | α).(7)
which can be further expanded by re-writing the likelihood for the j th event as a marginalization over the δκ s parameter,
L( d| α) = N j=1 L(d j |δκ s ) p(δκ s | α) dδκ s .(8)
In the above equation, the term L(d j |δκ s ) in the integral is the likelihood of δκ s for the j th event, marginalized over the BBH parameters. Since we use a uniform prior on δκ s in the singleevent analyses, this likelihood will be the same as the posterior given in Eq. (5). The other term in the integral of Eq. (8), p(δκ s | α) is the predicted distribution of δκ s given the hyperparameters α. In this study, as mentioned earlier, we assume a Gaussian distribution with hyper-parameters α = {µ, σ} which implies to,
p(δκ s | α) = N(µ, σ 2 ) .(9)
In this study, we assume µ and σ to have uniform priors in the ranges [−150, 150] and [0, 300] respectively and we use the Dynesty sampler from Bilby to sample over µ and σ as per the likelihood given in Eq. (8).
Once the posterior of µ and σ (or α in general) is computed, the population distribution of δκ s can be obtained as,
p(δκ s | d) = p(δκ s | α) p( α| d)d α ,(10)
where we have marginalized δκ s over the inferred distributions of the hyper-parameters α.
C. Mixture-likelihood approach: Estimating the fraction of non-BBH events
Unlike the hierarchical approach in the preceding section, here we ask a more generic question "what fraction of the detected population are from non-BBH events?". We try to answer this with a mixture-likelihood which parametrizes the presence of non-BBH events as f nbh , defined as the fraction of total detected signals that are from non-BBH events. Mixture likelihoods have been used in literature for various problems, see e.g., Ref. [135].
Let's start with the single-event likelihood expression for the non-BBH model,
L(d j | θ, δκ s ) ∝ exp − 1 2 d j − h j ( θ, δκ s ) d j − h j ( θ, δκ s ) . (11)
Here (|) represents the noise-weighted inner product, defined as (x|y) = 4
∞ 0 x( f ) * y( f )/S n (f) df, where the * indicates complex conjugate and S n (f) is the one-sided power spectral density (PSD) of the noise. Suppose a fraction f nbh of the overall detectable signals are from non-BBH events, then the probability of any single event being a non-BBH will be equal to f nbh . On the complementary side, the probability of any event being a BBH will be equal to (1 − f nbh ), as BBH and non-BBH are two mutually exclusive and exhaustive 2 cases. To take into account these possibilities, we can re-write the likelihood as a sum of the BBH and non-BBH likelihoods weighted by their respective probabilities as
L(d j | θ, δκ s , f nbh ) = (1 − f nbh ) L(d j | θ) + f nbh L(d j | θ, δκ s ) . (12)
Equation (12) is the single-event mixture-likelihood. Marginalizing over θ and δκ s , the above expression becomes, (13) where the integrals on the R.H.S are the evidences for the BBH and non-BBH models, defined in Eq. (4) and (3). Equation (12) then becomes,
L(d j | f nbh ) = (1 − f nbh ) × π( θ) L(d j | θ) d θ + f nbh × π( θ, δκ s ) L(d j | θ, δκ s ) d θ dδκ s ,L(d j | f nbh ) = (1 − f nbh ) Z bh j + f nbh Z nbh j .(14)
Note that in going from Eq. 13 to Eq. 14, we have used a uniform prior on δκ s as described in Section III A. In Section VI, we will investigate how the prior choices would affect the results. For a population of N detected events, the combined likelihood can be written as,
L pop ( d| f nbh ) = N j=1 (1 − f nbh ) Z bh j + f nbh Z nbh j .(15)
Equation (15) is the mixture-likelihood for a population and can be evaluated for any value of f nbh , by only knowing the evidences of BBH and non-BBH models for all the events in the population. With the above likelihood evaluated, we can express the posterior on f nbh as,
p( f nbh | d) ∝ π( f nbh ) L pop ( d| f nbh ) ,(16)
where the prior on f nbh can be taken as uniform, U(0, 1) owing to the most generic and uninformative case. Additionally, one can also define three mutually exclusive population-hypotheses based on the f nbh values.
1. f nbh = 0: "all the events are BBHs", 2. 0 < f nbh < 1: "the population is a mixture of BBH and non-BBH events", 3. f nbh = 1: "all the events are non-BBHs".
It is straightforward to compute the Bayes factors between any of these two hypotheses. The Bayes factor between "all are BBH" and "mixture of BBH and non-BBH events" can be obtained as
B BBH mix = L pop ( d| f nbh = 0) 0< f nbh <1 L pop ( d| f nbh ) π( f nbh ) d f nbh ,(17)
where the denominator included marginalizing the likelihood in Eq. (15) over the relevant range of f nbh . Similarly, the Bayes factor between "all are BBH" and "all are non-BBH" events can be obtained as,
B BBH non−BBH = L pop ( d| f nbh = 0) L pop ( d| f nbh = 1) .(18)
IV. SIMULATED COMPACT BINARY POPULATION
To demonstrate the performance of different combining approaches, we simulate a set of compact binary populations that include only BBH signals, only non-BBH signals, and different types of mixtures that include BBH and non-BBH signals at different proportions. In the subsections below, we describe the steps we followed to construct the populations.
A. Masses and spins 1. We first choose a representative mass model from which we draw the component masses. The primary masses (m 1 ) follow a distribution what is referred to as Model-C in [136] which is a power-law function smoothened at the lower mass end and embedded with a Gaussian peak towards the higher mass end. The secondary masses (m 2 ) are drawn from a smoothened power-law conditional on the primary masses such that m 1 ≥ m 2 .
2. For each binary, the magnitudes of the two componentspins are drawn according to the Default Model as named in [137]. In this model, the spin magnitudes a i (i = 1, 2) are drawn from a beta distribution
p(a i |α a , β a ) ∝ a α a −1 i (1 − a β a −1 i ) ,(19)
where α a and β a are shape parameters. We choose α a = 2.75 and β a = 6.00 to make sure that we do not have sources with a i ∼ 0, as non-spinning compact objects do not carry imprints of spin-induced multipole moments.
3. The spin orientations are randomly drawn from a mixture of isotropic and aligned-to-orbital-angular-momentum orientations. In other words, the populations include binaries with precessing spins and binaries with nonprecessing spins.
Note that there are several mass and spin models in literature [137] which can explain the current GW data, and our choice here is arbitrary since they do not affect the conclusions of this study.
B. Source selection based on signal-to-noise-ratio
The sources with masses and spins as described above are distributed uniformly in co-moving volume up to a redshift of 0.5 [7,137]. The inclination and polarisation angles are chosen so that the binary orientations are isotropically distributed w.r.t detectors.
We construct our populations from sources that pass the following two criteria:
SNR ≥ 10, SNR insp ≥ 2 SNR post−insp ,(20)
where, SNR is the optimal network signal-to-noise ratio with the HLV network, assuming all of them at their designed sensitivity [50,[138][139][140]. SNR insp and SNR post−insp are the signal-to-noise ratio in the inspiral and post-inspiral (mergerringdown) regimes of the signal, respectively, determined by an inspiral cut-off frequency given by the inspiral to intermediate transition frequency of phenomenological waveform models. This cut-off frequency is calculated, given the total mass of the binary (M) as, f cut = 0.018/M [130][131][132].
The second criterion imposes the inspiral SNR to be at least twice the post-inspiral SNR, which makes sure that there are enough number of waveform cycles in the inspiral phase. This is because the spin-induced quadrupole moment effects predominantly affect the inspiral phase, as modeled in the current waveform models. However, with the advancements in numerical relativity simulations, future waveform models might accurately account for their evolution in the post-inspiral phase, which in turn would allow us to test higher-mass binaries whose SNR dominates in the post-inspiral phase.
C. Distribution of spin-induced quadrupole moment parameters
We simulate six random instances of the BBH populations with parameters as described in the preceding sections and apply the SNR criterion of Eq. (20). We keep the first of these as a BBH population but turn the other five into either a non-BBH or a mixture population by associating spin-induced quadrupole moment parameters (δκ 1 and δκ 2 ) to each of them. Addition of these parameters would in principle change the SNR of the signals, however, the changes in our case are small 3 (≤3%) and all the signals still survive the SNR criterion.
The δκ 1 and δκ 2 distributions for every populations are made by mixing three components: a uniform distribution U(−40, 40) (Uniform), a positive Gaussian with a mean value 25 and standard deviation of 5, N(25, 5 2 ) (GausPos) , and a similar Gaussian with a negative mean, N(−25, 5 2 ) (GausNeg). We chose a few representative values for the mixing proportions and the total number of sources for the various populations. The boundaries of the uniform distribution or the mean and variance of the two Gaussian distributions do not carry any direct physical significance, rather these are chosen for the sake of diversity in the non-BBH signals.
All the six populations and their δκ 1 and δκ 2 distributions are summarised in Table I. Note that we do not impose δκ 1 = δκ 2 (or δκ a = 0) for the simulated signals, to keep them as generic as possible, though our analysis framework makes this assumption. In general, the compact binary distribution in the universe could be diverse, characterized by different values of δκ s and δκ a . We create various population models to mimic such scenarios by allowing the fraction of non-BBH systems to differ from model to model. We list all the six population below. You may refer to Table I for the details of their ingredients.
5.
MixtureAll: A population containing 50% BBH signals and 50% non-BBH signals whose δκ 1 and δκ 2 are drawn from Uniform, GausPos and GausNeg.
6.
MixturePos: A population containing 50% BBH signals and 50% non-BBH signals whose δκ 1 and δκ 2 are taking only positive values, from GausPos, and Uniform with a restriction that δκ 1,2 > 10.
We use the IMRPhenomPv2 waveform model [130][131][132]141] for simulating all the signals, which is the same as the model used for the Bayesian analysis as well, as mentioned before.
V. RESULTS
A. Hierarchical combining approach
We apply the method described in section III B on all the six simulated populations described in Table I. The results are shown as violin plots in Figure 2. The top two rows (silver) show the posterior distributions of the hyper-parameters µ and σ for each population and the bottom panel shows the δκ s distributions reconstructed from µ and σ according to Eq. (10). For BBH, the distributions of both µ and σ peak at their true values (i.e., zero) with narrow error bars and so is the δκ s distribution which is constrained to [-10.7, 8.8] at 90% credibility. However, this is not the case with the rest of the populations, as we discuss below.
For the NonBBH, the µ posterior is consistent with zero which is expected because it has equal number of sources from both sides of δκ s = 0. The σ posterior has increasing support towards the higher end of the prior and there is little to no support for σ = 0. This implies that the population could not be fit with a simple Gaussian and can be taken as an indication For the NonBBHPos and NonBBHNeg, the µ tends to peak at their true values again but the σ posteriors do not exclude zero. This is because the injected distributions of δκ s are one-sided for these two (∼ [0, 40] and ∼ [−40, 0]) unlike NonBBH (∼ [ −40, 40]) and hence are relatively better candidates for finitewidth Gaussians. Furthermore, we see that the peaks of the δκ s distributions are shifted to positive and negative values respectively for NonBBHPos and NonBBHNeg, as one would expect, though they do not exclude zero.
For the mixture populations (MixtureAll and MixturePos), the σ posteriors completely excludes zero and peaks at the highest value allowed by the prior. This again shows that a Gaussian of finite width could not be used to represent the underlying δκ s distribution and hence one could conclude the population to have complex components. The reconstructed δκ s distributions are, as expected, uninformative.
In summary, the hierarchical combining approach provides reasonably good estimates when we have only BBH signals or only non-BBH signals with all of them belonging to similar nature, i.e., δκ s having only positive/negative values in a small range of values like the ones we considered. For mixture population containing both BBH and non-BBH signals, the method can indicate to the complexity of the underlying distributions however can not realize whether it is full of non-BBH or a mixture of both BBH and non-BBH. Posterior distributions of the f nbh parameter -the fraction of non-BBH signals in the population -estimated using the mixturelikelihood approach. For a population with only BBH signals, the posterior is expected to peak at zero. A peak at unity would imply to a population with only non-BBH signals while a peak in between indicating a mixture population with BBH and non-BBH signals. The posteriors shown here are for the six simulated compact binary populations described in Table I. The vertical dashed lines represent the 90% credible intervals.
B. Mixture likelihood approach
Now, we apply the Mixture likelihood method ( section III C) on all the six simulated populations of Table I. Fig. 3 shows the posteriors on f nbh , the fraction of non-BBH events in the population. We see that, for the BBH-only population (BBH), the f nbh distribution peaks at zero, as expected. For the non-BBH populations with no BBH sources at all (NonBBH, NonBBHPos and NonBBHNeg), we expect f nbh to peak at unity. Table I. Of course, the ensembles are not mutually exclusive as we have a limited number of sources in the population. Nevertheless, the averaging helps reduce the outlier effects, which would be expected with a population size as small as 30.
However, the peaks occur between 0.1 and 0.5. Similarly, for the mixture populations with an equal number of BBH and non-BBH sources (MixtureAll and MixturePos), the peaks are expected to be at 0.5, but we obtain the peaks between 0.2 and 0.4. This shows that there is an overall tendency for f nbh to lean towards the BBH value. These offsets can be understood as systematics due to multiple reasons and have been discussed in detail in Sec. VI. Nevertheless, for all the non-BBH populations we have chosen, the f nbh posteriors exclude zero at 90% credibility.
In Fig. 4, we show the log e Bayes factor between the "All BBH" vs the "Mixture of BBH and non-BBH" hypotheses as derived in Eq. (17), as a function of the injected value of f nbh . Note that we do not use any of the populations in Table I for this plot. Rather we follow an averaging procedure using the sources in all those populations. We first construct two pools of BBH and non-BBH sources by collecting all available sources from the six populations in Table I. Now, for a given value of f nbh on the x-axis, for example, f nbh = 0.4, we randomly pick 12 BBH and 18 non-BBH sources from the respective pools, with a total of 30 sources. By repeating this 500 times, we compute log e Bayes factor for all these 500 cases and then take the average of them, and is shown on the y-axis of Fig. 4. The averaging helps remove the fluctuations due to any outliers when considering a population size as small as 30.
The log e Bayes factor favors the "only BBH" hypothesis when we inject only BBH signals (at f nbh = 0), whereas it rules out the "only BBH" model for all the populations that included non-BBH signals (as we move towards larger values of f nbh ). Fig. 3 but the posterior of f nbh for the compact binary detections of Advanced LIGO and Advanced Virgo reported in the GWTC-2 considered for testing GR analysis [125]. The peak of the posterior, as obtained here, is consistent with a population of only BBH signals.
C. Estimation of fraction of non-BBH signals from real LIGO-Virgo observations
The first bounds on spin-induced quadrupole moment parameter from observed gravitational wave events are reported in [61]. Furthermore, Ref. [125] provided the combined posterior distributions on δκ s obtained from the O1, O2, and the first half of O3 observing runs of LIGO-Virgo detectors along with the individual bounds. In [125], the combined bounds and Bayes factors are calculated following two methods: multiplying likelihoods (universality assumption on δκ s ) and hierarchical combining. Here we demonstrate the applicability of the mixture likelihood approach on all the events considered in [125] and infer f nbh for this population. We use the public data available from [142]. The result is shown in Fig. 5. Our analysis confirms that the detections reported in GWTC-2 [123] are consistent with a BBH population.
VI. UNDERSTANDING THE SYSTEMATIC BIASES
The bias in the f nbh posterior towards the BBH value ( f nbh = 0) as discussed above and shown in Fig. 3 could be a sum of various effects in play. We discuss these effects in the present section.
A. Biased single-event inferences
One of the possible reasons for the systematic bias in the inference of f nbh is the choices of spins and SNRs of the individual events in the populations. It has been shown that the single-event δκ s inferences are better when the spins are higher, and of course, improves further with higher SNR [119] while our populations include many events with low SNRs and low spins.
To investigate this, we collect events from our simulations that have effective spins |χ eff | ≥ 0. 15 SNR insp ≥ 20 4 . Out of all the BBH and non-BBH simulations that span over six populations, we obtain 16 BBH and 8 non-BBH events satisfying the above criterion. We construct two populations out of them: (1) All the 8 non-BBH events together, with a true value of f nbh = 1, and (2) all the 16 BBH and 8 non-BBH events together, with f nbh = 0.33.
We estimate f nbh for these two populations using the mixturelikelihood approach and find that the posteriors are consistent with the true values at very high confidences, as illustrated on the left panel of Fig. 6. The fact that we only had 8 and 24 events for these two populations has been reflective in the respective statistical uncertainties in Fig. 6 (left panel). This finding indicates that the bias in f nbh primarily arises because our population contains many low-spin-low-SNR events.
B. Prior effects
Throughout the Bayesian inference, we have assumed the prior on δκ s as uniform in [-500,500]. The injected populations, on the other hand, have δκ s drawn from narrower ranges with non-uniform distributions. Prior ranges that are wider than required will be penalized by Ocaam's factor. Below, we show that our prior choices can partly account for the bias in f nbh estimates. The right panel of Fig. 6 shows how the f nbh posterior would change if we were to perform the analysis with the same prior as that of the injected δκ s distribution. In order to achieve this, we did a prior re-weighting as described in Appendix A. We have taken an example from our populations with f nbh = 1 and find that the posterior on f nbh after prior re-weighting (dashed curves) is closer to the true values.
Of course, in reality, we can not pre-acquire the knowledge of the underlying distribution. However, we can always assume a hyper-parametrized prior model characterizing the underlying 4 χ eff is the effective inspiral spin parameter captures the spin effects of nonprecessing binary system which is defined in terms of component masses m i and dimensionless spins χ i = S i ·L/m 2 i as, χ eff = m 1 χ 1 +m 2 χ 2 m 1 +m 2 [130].
distribution and marginalize the likelihood over the hyperparameters to obtain the posteriors on f nbh . In Appendix A, we have derived the formalism of performing this hyperparametermarginalization though its detailed demonstration is differed for a future work.
C. Non-identical nature of the binary components
Another possible, though minor, reason is our assumption of δκ a = 0 (i.e., δκ 1 = δκ 2 ) in the analysis while our injections did not assume this. In other words, we have injected different values for δκ 1 and δκ 2 while in the Bayesian sampling, we assumed them to have equal values. This can lead to a bias in the estimated δκ s and the Bayes factor. Indeed, this is a prospect to explore in detail in a future study. Regardless of the δκ a = 0 assumption, the posterior could capture the true value of f nbh , if the SNR and spins are high enough, as shown in the left panel of Fig. 6.
At higher detector sensitivity and with better waveform models such as those with higher modes [143][144][145][146][147][148], some further correlations between binary parameters are expected to break, leading to improved estimates of κ parameters and improved Bayes factors, for the individual events. This will eventually improve the measurement of f nbh .
VII. CONCLUSION AND FURTHER REMARKS
We have discussed how to effectively combine gravitationalwave data from multiple detections to probe sub-populations that include non-BH compact binaries. Our method is based on using spin-induced quadrupole moment as a physical parameter that distinguishes BBH from non-BBHs. We first analyzed the efficacy of the previously-employed hierarchical combining approach that relies on a Gaussian assumption for the population distribution of the spin-induced quadrupole moment parameter. Next, we introduced a mixture-likelihood approach (Sec. III C) that estimates the fraction ( f nbh ) of non-BBH signals present in the observed population.
We simulated various populations that included BBH and non-BBH signals at different proportions. Our results show that both approaches are good for homogeneous populations like a BBH-only population. Also, both the approaches would signal if the population has complex nature with subpopulations being present. The mixture-likelihood approach is a natural choice to capture such complex distributions of non-BBH sub-populations. We applied the method on the LIGO-Virgo detected GW events from the GWTC-2 catalog and found them consistent with a BBH population.
Though the mixture-likelihood approach effectively rules out the BBH population hypothesis for the simulated populations that included some fraction of non-BBH signals, we notice that the method suffers from systematics in measuring the fraction of non-BBH signals precisely. We investigated possible reasons for these biases in Sec. VI.
In the future, with more realistic astrophysical population models, our estimates may improve. This is anticipated primarily because the spin-induced quadrupole moment estimates from individual events and, hence, the population can change according to the intrinsic mass-spin distributions. Also, with more accurate waveform models being available, our measurements could further improve as they could help break some of the degeneracies that lead to the systematics in this study. We would also include the effect of selection bias in the inference of f nbh which would allow re-defining the fraction f nbh as the fraction of non-BBH signals in the universe, rather than the fraction in the observed signals, in the future.
Finally, though we have demonstrated the mixturelikelihood approach using the measurement of spin-induced quadrupole moment parameters, the method is generic enough to include in or to be applied for the other physical parametrizations that distinguish BBH from non-BBH signals (as mentioned in Sec. I).
ACKNOWLEDGMENTS
The authors are grateful to Frank Ohme for useful comments on the manuscript. The authors acknowledge the use of LIGO LDG clusters for the computational work done for this study. This research has made use of data obtained from the Gravitational Wave Open Science Center (www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the US NSF. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, This document has LIGO preprint number LIGO-P2100382.
Appendix A: Marginalizing the mixture likelihood over the hyper parameters of δκ s distribution In Eq. (16), we have considered a uniform prior on δκ s in estimating the evidence Z nbh . In a more generic treatment, we should hyper-parametrize this prior and marginalize over, as the underlying δκ s distribution in the universe is unknown (See Ref. [149] for a comprehensive treatment of marginalizing over the prior hyper-parameters, though in a different context). Thus, we first rewrite Eq. (16) with a generic prior on δκ s , labelled as Λ. That means, Eq. (15) takes the form,
L pop ( d| f nbh , Λ) = N i=1 (1 − f nbh ) Z bh i + f nbh Z nbh i (Λ) , (A1)
where Z nbh i (Λ) is the generalized version of Eq. (3), Z nbh (Λ) = π( θ, δκ s |Λ) L(d| θ, δκ s ) d θ dδκ s .
The only difference being that the prior on δκ s is now informed by the hyper parameters Λ. Also, this hyper parameter space Λ could in general be a complex multi-dimensional parameter space. Perhaps, the simplest case one could consider is a zero-centered top-hat function, with only one parameter, namely λ. This would be a prior pretty much like the one we have used in our analysis, π(δκ s |λ) ∼ U(−λ, λ). Alternatively, if one assumes that all the non-BBH signals are from a certain class of exotic compact objects and have their δκ s distribution coming from a localized distribution, then a Gaussian with unknown mean and width would be a good representation, i.e., Λ = {µ, σ 2 } Given a hyper-parameter model, the corresponding evidence Z nbh (Λ) at any given point in the Λ-space can be evaluated by re-weighting the prior, as discussed in Appendix A 1).
Once we have the tools to evaluate the likelihood, we can sample from the likelihood using any sampler, over the parameters, { f nbh , Λ}. This would give us the posterior probability density function as,
p( f nbh , Λ|d j ) = L pop (d j | f nbh , Λ) π( f nbh ) π(Λ) Z pop Λ ,(A3)
with Z pop Λ = L pop (d j | f nbh , Λ) π( f nbh ) π(Λ) d f nbh dΛ.
(A4)
Now we can obtain the posterior on f nbh marginalized over Λ as, p( f nbh |d j ) = p( f nbh , Λ|d j )dΛ .
Employing a nested sampling algorithm for the sampling, from the likelihood given in Eq. (A2) the output will by default provide the probability of fraction defined in Eq. (A3). The priors on f nbh and Λ can be taken as uniform. Given the posterior distribution on δκ s , we can also reconstruct the population distribution of δκ s as, p(δκ s |d j ) = p(δκ s |Λ) p(Λ|d j ) dΛ .
Evidence estimation by prior re-weighting
Suppose we have the evidence estimated already assuming one prior, in our case a prior on δκ s assuming U[-500, 500]. Let us call this prior as Φ. Now the evidence for this prior Z nbh (Φ), according to the Bayes theorem, can be written as, Z nbh (Φ) p( θ, δκ s |d j , Φ) = π( θ, δκ s |Φ) L(d j | θ, δκ s ), (A7)
With the prior Λ, the new evidence will be as given by Eq. (A2). Plugging Eq. (A7) to Eq. (A2), we get Z nbh (Λ) = Z nbh (Φ) π( θ, δκ s |Λ) π( θ, δκ s |Φ) p( θ, δκ s |d j , Φ)d θ dδκ s , (A8) where the probability distribution p( θ, δκ s |d j , Φ) is known and the above integral can be approximated as a Monte-Carlo average over the posterior samples, i.e.,
Z nbh (Λ) = Z nbh (Φ) × 1 n n k=1
π( θ, δκ s |Λ) π( θ, δκ s |Φ)
,(A9)
where n is the number of samples in the δκ s posterior distribution. Both π( θ, δκ s |Λ) and π( θ, δκ s |Φ) must be normalized distributions. Throughout the analysis, we assume that the priors on δκ s and θ are uncorrelated and hence we can decouple π( θ) from both numerator and denominator so that they cancel each other. This leaves,
Z nbh (Λ) = Z nbh (Φ) × 1 n n k=1
π(δκ s |Λ) π(δκ s |Φ)
,(A10)
Equation (A10) makes it easier to compute the likelihood in Eq. (A1) for any instance of Λ. The reader may refer to Ref. [150] for a detailed treatment of the prior-re-weighting procedure.
Figure 1 .
1Time-domain gravitational waveforms for a fast-spinning (top) and slowly-spinning (bottom) compact binary mergers with component masses (m 1 , m 2 ) = (20, 10)M . The spins are aligned with the orbital angular momentum vectors (no precession modulations) and have the dimensionless spin magnitudes (χ 1 , χ 2 ) = (0.6, 0.5) for the top and (χ 1 , χ 2 ) = (0.15, 0.1) for the bottom panels. The black (dash-dot) traces are BBH waveforms, and the red (solid) traces are non-BBH waveforms with spin-induced quadrupole moment parameters assuming δκ 1 = 40 and δκ 2 = 25 for component compact objects. The time-domain waveforms are generated using backward fast Fourier transform (FFT) of the frequency-domain waveform model, IMRPhenomPv2. For each waveform, the time t (x axis) is set to zero at a point when the instantaneous frequency of the waveform is 40 Hz, and the waveforms are also aligned to be in phase at that point.
1 .
1BBH: A fully BBH population. 2. NonBBH: A fully non-BBH population with δκ 1 and δκ 2 drawn from Uniform, GausPos and GausNeg. . Details of the simulated compact-binary populations used in this study. Our six populations are labelled as written in the left-most column. The second column provides the fraction ( f nbh ) of non-BBH signals contained in each population. The third column provides the number of BBH signals in each population. The δκ 1 and δκ 2 values of the non-BBH signals in each population are distributed as a mixture of three statistical models: a uniform distribution U(−40, 40) (Uniform), a positive Gaussian N(25, 5 2 ) (GausPos), and a negative Gaussian N(−25, 5 2 ) (GausNeg). The columns 4-6 describes respective numbers drawn from each of these model. Finally, the right-most column gives the total size of the population (N tot ).
Figure 2 .
2Violin plots showing the results from hierarchical combining formalism applied on the six simulated populations as labelled on the x-axis. The top two rows (silver) are the posterior densities of the hyper parameters µ and σ characterising the Gaussian that models the population distribution δκ s (see Sec. III B and Eq. (9) for details). The bottom row (blue) shows the δκ s distributions obtained by marginalizing over the µ and σ posteriors (as per Eq. (10)). We assumed µ and σ to have uniform priors in the ranges [−150, 150] and [0, 300] respectively and sampled using the Dynesty sampler from Bilbyto the presence of one or more of sub-population with different values of δκ s . The marginalized δκ s distribution for this case is a wide distribution with no insight, which results from the behaviour of the σ posterior.
Figure 3 .
3Figure 3. Posterior distributions of the f nbh parameter -the fraction of non-BBH signals in the population -estimated using the mixturelikelihood approach. For a population with only BBH signals, the posterior is expected to peak at zero. A peak at unity would imply to a population with only non-BBH signals while a peak in between indicating a mixture population with BBH and non-BBH signals. The posteriors shown here are for the six simulated compact binary populations described in Table I. The vertical dashed lines represent the 90% credible intervals.
Figure 4 .
4Figure showinghow the log e Bayes factor between the population-hypotheses "All are BBH" and "A mixture of BBH and non-BBH" (as derived in Eq.(17)) varies as a function of the true value of the f nbh of the population. The log e Bayes factors shown on the y-axis, at each f nbh on the x-axis, are ensemble-averaged over 500 population instances, where each instance has N tot = 30 sources. The 500 instances are constructed by collecting the simulated events across the six populations of
Figure 5 .
5Same as
Figure 6 .
6[Left] Posterior distribution of f nbh for populations that include simulated sources with |χ eff | ≥ 0.15 and network SNR insp ≥ 20. These populations are constructed by collecting sources from all the six populations that satisfy the SNR and spin criterion. [Right] The posterior distribution on the fraction f nbh with and without prior-reweighting discussed in Sec. VI and Appendix A. On both the panels the dashed vertical lines indicate the injected value of f nbh .
with contributions by Polish and Hungarian institutes. M. Saleem acknowledges support from NSF grants PHY-00090754, PHY-1806630, and PHY-2010970, and the support from the Infosys Foundation, the Swarnajayanti fellowship grant DST/SJF/PSA-01/2017-18. N. V. Krishnendu acknowledges support from the Max Planck Society's Independent Research Group Grant. The research of Archisman Ghosh is supported by Ghent University's BOF Starting Grant BOF/STA/202009/040. K. G. Arun partially supported by a grant from Infosys Foundation. K. G. Arun also acknowledges the Swarnajayanti fellowship grant DST/SJF/
A6) is identical to Eq. (10) with the only visible difference being the hyper parameter vector α replaced by Λ. The different notions of hyper parameters in Eq. (A6) and Eq. (10) are purposeful, while α represents the distribution of δκ s from the entire population, Λ by definition represents the distribution of f nbh , the fraction of non-BBHs in the population. The remaining (1 − f nbh ) fraction of sources have a delta function prior, given by δ(δκ s − 0).
Alternatively, one can consider specific non-BH models such as boson stars and/or gravastars. However, these models, together with the BH model, would still be non-exhaustive as yet unknown non-BH models could possibly exist.
. NonBBHPos: A fully non-BBH population with δκ 1 and δκ 2 taking only positive values, i.e., from GausPos and Uniform with a restriction that δκ 1,2 > 10.4. NonBBHNeg: Afully non-BBH population with δκ 1 and δκ 2 taking only negative values, i.e., from GausNeg and Uniform with a restriction that δκ 1,2 < −10.3 Due to the moderate to low spin values of the sources in our populations, we do not expect the power in the signals to change significantly, which is also apparent in the bottom panel ofFig. 1.
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| []
|
[]
| [
"Fumiko Okiharu \nDepartment of Physics\nFaculty of Science\nNihon University\n1-8 Kanda-Surugadai101Chiyoda, TokyoJapan\n",
"Hideo Suganuma \nYukawa Institute for Theoretical Physics\nTokyo Institute of Technology\n2-12-1 Ohokayama152-8551TokyoJapan\n",
"Toru T Takahashi \nKyoto University\n606-8502KitashirakawaSakyo, KyotoJapan\n"
]
| [
"Department of Physics\nFaculty of Science\nNihon University\n1-8 Kanda-Surugadai101Chiyoda, TokyoJapan",
"Yukawa Institute for Theoretical Physics\nTokyo Institute of Technology\n2-12-1 Ohokayama152-8551TokyoJapan",
"Kyoto University\n606-8502KitashirakawaSakyo, KyotoJapan"
]
| []
| The static penta-quark (5Q) potential V5Q is studied in SU(3) lattice QCD with 16 3 × 32 and β=6.0 at the quenched level. From the 5Q Wilson loop, V5Q is calculated in a gauge-invariant manner, with the smearing method to enhance the ground-state component. V5Q is well described by the OGE plus multi-Y Ansatz: a sum of the OGE Coulomb term and the multi-Y-type linear term proportional to the minimal total length of the flux-tube linking the five quarks. Comparing with QQ and 3Q potentials, we find a universality of the string tension, σ QQ ≃ σ3Q ≃ σ5Q, and the OGE result for Coulomb coefficients. | 10.1103/physrevlett.94.192001 | [
"https://arxiv.org/pdf/hep-lat/0407001v1.pdf"
]
| 5,665,603 | hep-lat/0407001 | 02411c26bc40581e3016286443160cb8449cbc99 |
2 Jul 2004
Fumiko Okiharu
Department of Physics
Faculty of Science
Nihon University
1-8 Kanda-Surugadai101Chiyoda, TokyoJapan
Hideo Suganuma
Yukawa Institute for Theoretical Physics
Tokyo Institute of Technology
2-12-1 Ohokayama152-8551TokyoJapan
Toru T Takahashi
Kyoto University
606-8502KitashirakawaSakyo, KyotoJapan
2 Jul 2004(Dated: November 1, 2018)First Study for the Pentaquark Potential in SU(3) Lattice QCD
The static penta-quark (5Q) potential V5Q is studied in SU(3) lattice QCD with 16 3 × 32 and β=6.0 at the quenched level. From the 5Q Wilson loop, V5Q is calculated in a gauge-invariant manner, with the smearing method to enhance the ground-state component. V5Q is well described by the OGE plus multi-Y Ansatz: a sum of the OGE Coulomb term and the multi-Y-type linear term proportional to the minimal total length of the flux-tube linking the five quarks. Comparing with QQ and 3Q potentials, we find a universality of the string tension, σ QQ ≃ σ3Q ≃ σ5Q, and the OGE result for Coulomb coefficients.
The inter-quark force is one of the elementary quantities for the study of the multi-quark system in the quark model. As for baryons, our group recently studied the three-quark (3Q) potential V 3Q in detail with lattice QCD, and clarified that it obeys the Coulomb plus Ytype linear potential [1]. However, no one knows the interquark force from QCD in the exotic multi-quark system such as tetra-quark mesons (QQ-QQ), penta-quark baryons (4Q-Q), dibaryons (6Q) and so on.
Very recently, an exotic anti-strange baryon Θ + (1540) with S = +1 was experimentally discovered at SPring-8 (LEPS), and was confirmed by ITEP(DIANA), JLab(CLAS) and ELSA(SAPHIR) [2]. The Θ + (1540) was theoretically predicted in the Skyrme model [3], and is regarded as a penta-quark (5Q) baryon of u 2 d 2s in the valence-quark picture. Another 5Q baryon Ξ −− (1862) was found by CERN(NA49) [4], and also an anti-charmed 5Q baryon Θ c (3099) was found by HERA(H1) [5].
Accordingly, many theoretical studies [6] have been done for the 5Q baryon using various approaches such as lattice QCD [7,8], the constituent quark model [9], the diquark model [10], the QCD sum rule [11], the flux-tube model [12], the string theory [13] and so on. However, there are several puzzling problems on the Θ + (1540): its mass seems to be rather small and its decay width is extremely narrow [2]. To solve them, one encounters the many-body problem of quarks, and therefore it is quite desired to clarify the inter-quark force in the multi-quark system based on QCD.
In this paper, motivated by the recent discovery of the penta-quark baryons, we perform the first study of the static penta-quark (5Q) potential V 5Q , i.e., the interquark force in the 5Q system, in SU(3) lattice QCD with β=6.0 and 16 3 × 32 at the quenched level. Note that the lattice QCD result of V 5Q presents a key information in modeling the multi-quark system based on QCD.
For the penta-quark system, we investigate the QQ-Q-QQ type configuration with the two "QQ clusters" belonging to the 3* representation of the SU(3) color as shown in Fig.1, since this type of the 5Q configuration is expected to have a small energy and seems to be natural as a realistic candidate of the Θ + (1540). Indeed, in the perturbative sense, an attractive (repulsive) force acts between two quarks, when their total SU(3) color belongs to 3* (6). Therefore, the nearest QQ cluster tends to form 3* rather than 6 in the low-lying 5Q system, which leads to the 3*-diquark model [10]. Similar to the derivation of the Q-Q (3Q) potential from the (3Q) Wilson loop, the 5Q static potential V 5Q can be calculated with the 5Q Wilson loop W 5Q , which is defined in a gauge-invariant manner as shown in Fig.2.
We define the 5Q Wilson loop W 5Q [8] as
W 5Q ≡ 1 3! ǫ abc ǫ a ′ b ′ c ′ M aa ′ (L 3L12L4 ) bb ′ (R 3R12R4 ) cc ′ ,(1)
whereM ,L i ,R i (i = 1, 2, 3, 4) are given bỹ
M ,L i ,R i ≡ P exp{ig M,Li,Ri dx µ A µ (x)} ∈ SU(3) c .(2)
As shown in Fig.2
L 12 ,R 12 ∈ SU(3) c are defined as L a ′ a 12 ≡ 1 2 ǫ abc ǫ a ′ b ′ c ′L bb ′ 1L cc ′ 2 ,R a ′ a 12 ≡ 1 2 ǫ abc ǫ a ′ b ′ c ′R bb ′ 1R cc ′ 2 .(3)
Note that the 5Q Wilson loop W 5Q is gauge invariant, and its gauge invariance is owing to the nontrivial assignment of the color indices ofL 12 andR 12 in Eq.(3).
(Recall that the "two quark lines" combining into the 3* representation correspond to an "antiquark line" as the color current.)
In principle, the ground-state 5Q potential V 5Q is obtained from the 5Q Wilson loop W 5Q as V 5Q = − lim T →∞ 1 T ln W 5Q .
However, the practical lattice calculation is done with a finite region of T , where excitedstate contributions remain. For the accurate measurement of V 5Q in lattice QCD, we use the gauge-covariant smearing method [1] to enhance the ground-state component of the 5Q state in the 5Q Wilson loop.
The smearing is known to be a powerful method for the accurate measurement of the Q-Q and the 3Q potentials [1], and is expressed as the iterative replacement of the spatial link variables
U i (s) (i=1,2,3) by the obscured link variablesŪ i (s) ∈ SU(3) c which maximizes Re tr {Ū † i (s)V i (s)} with V i (s) ≡ αU i (s) + j =i ± {U ±j (s)U i (s ±ĵ)U † ±j (s +î)}(4) with the simplified notation of U −j ≡ U † j (s −ĵ)
. We here adopt α = 2.3 and the iteration number N smr = 40, which lead to a large enhancement of the ground-state component in the 5Q Wilson loop at β=6.0. Now, we proceed the actual lattice QCD calculation for the 5Q potential V 5Q [8]. We generate 150 gauge configurations using SU(3) c lattice QCD with the standard action with β = 6.0 and 16 3 × 32 at the quenched level. The gauge configurations are taken every 500 sweeps after a thermalization of 5000 sweeps using the pseudoheat-bath algorithm. The lattice spacing a is estimated as a ≃ 0.104fm from the string tension σ=0.89 GeV/fm in the Q-Q potential V QQ [8].
As for the 5Q configuration, we consider the QQ-Q-QQ type configuration as shown in Fig.1. Note that the multi-quark system including four or more quarks can take a three-dimensional shape, while the QQ and the 3Q systems can take only planar configuration [8,12]. Then, we investigate both the planar 5Q configuration as shown in Fig.3 and the twisted 5Q configuration as shown in Fig.4. In this paper, we take For these types of 5Q configurations, we calculate the 5Q potential V 5Q from the 5Q Wilson loop W 5Q using the smearing method. Owing to the smearing, the ground-state component is largely enhanced, and therefore the 5Q Wilson loop W 5Q composed with the smeared link variable exhibits a single-exponential behavior as W 5Q ≃ e −V5QT even for a small value of T . Then, for each 5Q configuration, we extract V 5Q from the least squares fit with the single-exponential form
d 1 = d 2 = d 3 = d 4 ≡ d,W 5Q =Ce −V5QT(5)
in the range of T min ≤ T ≤ T max listed in Table I. The prefactorC physically means the ground-state overlap, andC ≃ 1 corresponds to the quasi-ground-state. Here, we choose the fit range of T such that the stability of the "effective mass" V (T ) ≡ ln{ W 5Q (T ) / W 5Q (T + 1) } is observed in the range of T min ≤ T ≤ T max − 1. For the lattice calculation of W 5Q , we use the translational and the rotational symmetries on lattices.
For 56 different patterns of the 5Q configurations as shown in Figs.3 and 4, we present the lattice QCD data for the 5Q potential V 5Q together with the ground-state Fig.3. We list also the ground-state overlapC, the fit range of T and the theoretical form V theor 5Q of the OGE plus multi-Y Ansatz (8) with (A5Q,σ5Q) fixed to be (A3Q,σ3Q) in V3Q in Ref. [1]. All the data are measured in the lattice unit. overlapC in Table I and II. The statistical errors are estimated with the jackknife method. We find a large ground-state overlap asC > 0.85 for almost all 5Q configurations. Next, we consider the theoretical form of the 5Q potential V 5Q . The lattice QCD studies [1] at the quenched level show that the Q-Q potential V QQ takes a form of
V QQ (r) = − A QQ r + σ QQ r + C QQ ,(6)
and the 3Q potential V 3Q takes a form of
V 3Q = −A 3Q i<j 1 |r i − r j | + σ 3Q L min + C 3Q ,(7)
where L min denotes the minimal value of total length of color flux tubes linking the three quarks. In fact, both V QQ and V 3Q are described by a sum of the short-distance one-gluon-exchange (OGE) result and the long-distance flux-tube result [1,14]. For the static penta-quark (5Q) system, we find that the lattice QCD results are well described by the OGE plus multi-Y Ansatz: a sum of the OGE Coulomb term and the multi-Y type linear term [8],
V 5Q = g 2 4π i<j T a i T a j |r i − r j | + σ 5Q L min + C 5Q = −A 5Q {( 1 r 12 + 1 r 34 ) + 1 2 ( 1 r 15 + 1 r 25 + 1 r 35 + 1 r 45 ) + 1 4 ( 1 r 13 + 1 r 14 + 1 r 23 + 1 r 24 )} + σ 5Q L min + C 5Q (8)
with r ij ≡ |r i − r j | and ith quark location r i in Fig.1.
Here, L min is the minimal length of the flux-tube linking five quarks as shown in Fig.1. (For the extreme case, e.g., d > √ 3h 1 , we here assume that the flux-tube is formed as the two straight lines on Q 1 Q 5 and Q 2 Q 5 , considering the color combination, although there may appear several possibilities as the "flip-flop".) Note that there appear three kinds of Coulomb coefficients (A 5Q , 1 2 A 5Q , 1 4 A 5Q ) in the penta-quark system, while only one Coulomb coefficient, A QQ or A 3Q , appears in the QQ or the 3Q system. Here, the Coulomb coefficient A 5Q in Eq.(8) corresponds to A 3Q or 1 2 A QQ in terms of the OGE result.
We add in Table I and II the theoretical form V theor 5Q of the OGE plus multi-Y Ansatz (8) with (A 5Q ,σ 5Q ) fixed to be (A 3Q ,σ 3Q ) in the 3Q potential V 3Q obtained in Ref. [1], i.e., A 5Q = A 3Q ≃ 0.1366, σ 5Q = σ 3Q ≃ 0.046a −2 and C 5Q ≃ 1.58a −1 . (Note that there is no adjustable parameter for the theoretical form V theor 5Q besides an irrelevant constant C 5Q , since A 5Q and σ 5Q are fixed to be A 3Q and σ 3Q , respectively.) Thus, the 5Q potential V 5Q is found to be well described by the OGE Coulomb plus multi-Y-type linear potential.
We show in Fig.5 typical examples of the lattice QCD data for the penta-quark potential V 5Q . The symbols denote the lattice data, and the curves denote the theoretical form of the OGE plus multi-Y Ansatz with (A 5Q ,σ 5Q ) fixed to be (A 3Q ,σ 3Q ). One finds a good agreement between the lattice QCD data and the theoretical curves. Note that the planar and the twisted 5Q configurations with the same (d, h 1 , h 2 ) are almost degenerate, although the energy of the planar one is slightly smaller. In terms of the OGE plus multi-Y Ansatz, the only energy difference between the two states originates from a small difference of the Coulomb interaction between Q i (i = 1, 2) and Q j (j = 3, 4), where the Coulomb coefficient is reduced as 1 4 A 5Q (≃ 1 8 A QQ ). Then, no special configuration is favored in the 5Q system in terms of the energy. This fact also indicates that the 5Q system is unstable against the twisted motion of the two QQ clusters as shown in Fig.4. In fact, general 5Q systems tend to take a three-dimensional configuration [8,12] in terms of the entropy.
From the comparison with the QQ and the 3Q potentials, the universality of the string tension and the OGE result are found among QQ, 3Q and 5Q systems as
σ QQ ≃ σ 3Q ≃ σ 5Q , 1 2 A QQ ≃ A 3Q ≃ A 5Q .(9)
This result supports the flux-tube picture on the confinement mechanism even for the multi-quark system [8].
To conclude, we have performed the first study of the penta-quark potential in lattice QCD, and have found that the 5Q potential is well reproduced by the OGE Coulomb plus multi-Y-type linear potential.
H.S. was supported in part by a Grant for Scientific Research (No.16540236) from the Ministry of Education, Culture, Science and Technology, Japan. T.T.T. was supported by the Japan Society for the Promotion of Science. The lattice QCD Monte Carlo calculations have been performed on NEC-SX5 at Osaka University.
FIG. 1 :
1The QQ-Q-QQ type configuration for the pentaquark system. The two QQ clusters belong to the 3* representation of the color SU(3).
FIG. 2 :
2,M ,L i ,R i (i = 3, 4) are line-like variables andL i ,R i (i = 1, 2) are staple-like variables. Here, The penta-quark (5Q) Wilson loop W5Q for the 5Q potential V5Q. The contours M, Li, Ri(i = 3, 4) are line-like and Li, Ri(i = 1, 2) are staple-like. The 5Q gauge-invariant state is generated at t = 0 and is annihilated at t = T with the five quarks (4Q-Q) being spatially fixed in R 3 for 0 < t < T .
and present the lattice QCD result of V 5Q in terms of (d, h 1 , h 2 ).FIG. 3:A planar configuration of the penta-quark system. Q1Q2 is parallel to Q3Q4, and H1H2 is perpendicular to Q1Q2 and Q3Q4. Here, we take d1 = d2 = d3 = d4 ≡ d.FIG. 4:A twisted configuration of the penta-quark system. Q1Q2 is perpendicular to Q3Q4, and H1H2 is perpendicular to Q1Q2 and Q3Q4. Here, we take d1 = d2 = d3 = d4 ≡ d.d
d 1
d
d
h
h
1
4
3
2
Q
Q 3
1
Q 2
Q 4
Q
2
5
H 1
2
H
d
d 1
h
h
1
2
2
d 4
d 3
Q
Q
Q
3
Q 1
Q 2
4
5
H 1
H 2
TABLE I :
ILattice QCD results for the penta-quark potential V5Q for the planar 5Q configuration labeled by (d, h1, h2) as shown in
TABLE II :
IILattice QCD results for the penta-quark potential
V5Q for the twisted 5Q configuration labeled by (d, h1, h2) as
shown in Fig.4. The notations are the same in Table I.
(d, h1, h2)
V5QC
Tmin-Tmax V theor
5Q
(1,1,1)
1.4476(23) 0.9378(64)
3-8
1.4458
(1,1,2)
1.5438(14) 0.9528(25)
2-8
1.5419
(1,1,3)
1.6155(17) 0.9459(31)
2-6
1.6148
(1,1,4)
1.6767(21) 0.9370(37)
2-6
1.6767
(1,1,5)
1.7365(22) 0.9357(42)
2-5
1.7332
(1,1,6)
1.7912(26) 0.9297(42)
2-4
1.7866
(1,1,7)
1.8337(88) 0.8933(231)
3-5
1.8380
(1,2,2)
1.6302(16) 0.9472(29)
2-8
1.6324
(1,2,3)
1.7022(18) 0.9445(32)
2-4
1.7022
(1,2,4)
1.7657(25) 0.9427(44)
2-5
1.7624
(1,2,5)
1.8232(30) 0.9385(51)
2-7
1.8177
(1,2,6)
1.8728(32) 0.9230(58)
2-8
1.8704
(1,3,3)
1.7710(24) 0.9376(42)
2-6
1.7702
(1,3,4)
1.8326(27) 0.9335(46)
2-4
1.8293
(1,3,5)
1.8952(32) 0.9394(58)
2-7
1.8839
(1,4,4)
1.8950(30) 0.9315(52)
2-4
1.8877
(2,1,1)
1.7735(23) 0.9377(41)
2-6
1.7615
(2,2,2)
1.8832(30) 0.9279(54)
2-7
1.8899
(2,3,3)
2.0011(37) 0.9233(64)
2-5
2.0100
(2,4,4)
2.1155(49) 0.9167(85)
2-6
2.1212
(3,1,1)
2.0049(37) 0.9032(67)
2-5
2.0008
(3,2,2)
2.0873(38) 0.8987(69)
2-4
2.0973
(3,3,3)
2.1870(54) 0.8912(92)
2-6
2.2055
(3,4,4)
2.3021(68) 0.9019(113)
2-5
2.3108
(4,1,1)
2.2141(65) 0.8741(107)
2-6
2.2155
(4,2,2)
2.2874(64) 0.8768(107)
2-5
2.2879
(4,3,3)
2.3715(70) 0.8577(118)
2-4
2.3880
(4,4,4)
2.4680(94) 0.8459(149)
2-4
2.4890
FIG. 5: Lattice QCD results of the penta-quark potential V5Q for the planar 5Q configuration with h1 = h2 ≡ h inFig.3in the lattice unit. Each 5Q system is labeled by d and h. The symbols denote the lattice data, and the curves the theoretical form of the OGE plus multi-Y Ansatz.1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
1
2
3
4
5
6
7
8
9
10
h
d=1
d=2
d=3
d=4
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| []
|
[
"Note on Identities Inspired by New Soft Theorems",
"Note on Identities Inspired by New Soft Theorems"
]
| [
"Junjie Rao [email protected] \nZhejiang Institute of Modern Physics\nZhejiang University\n310027HangzhouP. R. China\n",
"Bo Feng [email protected] ",
"\nCenter of Mathematical Science\nZhejiang University\n310027HangzhouP. R. China\n"
]
| [
"Zhejiang Institute of Modern Physics\nZhejiang University\n310027HangzhouP. R. China",
"Center of Mathematical Science\nZhejiang University\n310027HangzhouP. R. China"
]
| []
| The new soft theorems, for both gravity and gauge amplitudes, have inspired a number of works, including the discovery of new identities related to amplitudes. In this note, we present the proof and discussion for two sets of identities. The first set includes an identity involving the half-soft function which had been used in the soft theorem for one-loop rational gravity amplitudes, and another simpler identity as its byproduct. The second set includes two identities involving the KLT momentum kernel, as the consistency conditions of the KLT relation plus soft theorems for both gravity and gauge amplitudes. We use the CHY formulation to prove the first identity, and transform the second one into a convenient form for future discussion. | 10.1007/jhep04(2016)173 | [
"https://arxiv.org/pdf/1604.00650v2.pdf"
]
| 118,436,464 | 1604.00650 | cd257bc8a1a44a87ea365a960215800b8c9a2701 |
Note on Identities Inspired by New Soft Theorems
4 May 2016
Junjie Rao [email protected]
Zhejiang Institute of Modern Physics
Zhejiang University
310027HangzhouP. R. China
Bo Feng [email protected]
Center of Mathematical Science
Zhejiang University
310027HangzhouP. R. China
Note on Identities Inspired by New Soft Theorems
4 May 2016Preprint typeset in JHEP style -HYPER VERSION * The unconventional order is to let authors get proper recognition of contributions under the outdated practice in China. † Emails:Amplitudes, Soft Theorem
The new soft theorems, for both gravity and gauge amplitudes, have inspired a number of works, including the discovery of new identities related to amplitudes. In this note, we present the proof and discussion for two sets of identities. The first set includes an identity involving the half-soft function which had been used in the soft theorem for one-loop rational gravity amplitudes, and another simpler identity as its byproduct. The second set includes two identities involving the KLT momentum kernel, as the consistency conditions of the KLT relation plus soft theorems for both gravity and gauge amplitudes. We use the CHY formulation to prove the first identity, and transform the second one into a convenient form for future discussion.
Introduction
Scattering amplitudes often have an universal soft behavior when the momentum of one external leg tends to zero. This soft limit can be traced back to the works [1,2,3]. Recently, a new soft theorem for tree level gravity amplitudes was studied in [4]. By using the on-shell recursion relation [5,6] and imposing the holomorphic soft limit, Cachazo and Strominger have proved that M n (λ n → ελ n ) = 1 ε 3 n−2 a=1 n − 1, a 2 [na] n − 1, n 2 na M n−1 λ n−1 →λ n−1 + ε an a, n − 1 λ n ,λ 1 →λ 1 + ε n − 1, n n − 1, a λ n + O(ε 0 ), (1.1) here for M n and M n−1 , the unmentioned external kinematic data are un-deformed and we prefer to suppress them for conciseness. Taylor expansion in ε exhibits three singular terms in orders ε −3 , ε −2 and ε −1 , while higher order terms in ε will be mixed with the less interesting O(ε 0 ) parts. A similar relation for tree level Yang-Mills amplitudes using the on-shell recursion relation, proved by Casali [7], takes the form A n (λ n → ελ n ) = 1 ε 2 n − 1, 1 n − 1, n n1 A n−1 λ n−1 →λ n−1 + ε 1n 1, n − 1 λ n ,λ 1 →λ 1 + ε n − 1, n n − 1, 1 λ n + O(ε 0 ), (1.2) where two singular terms in orders ε −2 and ε −1 appear after Taylor expansion. The mixing between higher order terms from the deformed A n−1 and O(ε 0 ) parts also persists to this case.
Based on this new discovery, many related studies have been done. In [20,21,22,23,24,25,26,27,28,29,30,31,32,33], the soft theorem has been generalized to arbitrary dimensions and other theories or categories: string theory, ABJM theory, theories with fermions or massive particles, and form factors. In [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55], the theorem has been understood from various perspectives, especially those of symmetries and invariance. In [8,25,56,57,58,59,60], its generalization to loop level has been discussed. In [61,62,63,64,65,66,67,68], the relevant double (or multiple) soft theorem has also been discussed.
Among these studies, we have met two sets of identities which have not been proved so far. We will present the proof in this note.
One identity of the first set was mentioned in [8], which explored loop correction to the soft theorem. It involves the so-called half-soft function h (first defined in [9] and reinterpreted in [10]), which appears naturally for all-plus one-loop gravity amplitude. Its general proof was not given in [8], but explicit checks up to 12 points had been done. The identity reads
b =n bn 2 M,N h(b, n, M )h(b, n, N ) b|K M |n] n|K N |b] 3 = 0, (1.3)
where M, N are two nonempty partition sets of the (n − 2) particles other than b and n, and K M and K N are the corresponding total momenta. During the proof, we had also discovered another simpler identity, which can serve as its logical preliminary. It reads
[1n] 1n |ψ N ∪M | w w w1 wn + N 1|K N |n] n|K N |1] |ψ N | x x x1 xn |ψ M | y y y1 yn = 0, (1.4)
where the ψ matrix is related to h, and other symbols above will be explained shortly. The second set of identities was conjectured in [11], which is a consequence of consistency conditions between the soft theorems for gravity and gauge amplitudes, under the well-known KLT relation [12]. It involves the KLT momentum kernel [9,13,14,15], and the transformation matrices (D and C below) between BCJ basis of gauge amplitudes [16]. These two identities are
α t ′ ,β t ′ ∈S n−3 D[t, α t , n − 1, n|t ′ , α t ′ , n − 1, n]S[α t ′ |β t ′ ] p n−1 C[t ′ , n − 1, β t ′ , n|t, n − 1, β t , n] = S[α t |β t ] p n−1 , (1.5) n−2 t ′ =1 α t ′ ,β t ′ ∈S n−3 D[t, α t , n − 1, n|t ′ , α t ′ , n − 1, n]S[α t ′ |β t ′ ] p n−1 · J t ′ C[t ′ , n − 1, β t ′ , n|t, n − 1, β t , n] = 0, (1.6)
where S[α t |β t ] p n−1 is the KLT momentum kernel of pivot p n−1 , and J t ′ ≡ J t ′ ,αβ is the anti-holomorphic angular momentum operator. We will use the CHY formulation [17,18,19] to prove the first identity and discuss the second one.
This note is organized as follows. In section 2, we prove identity (1.3) of the half-soft function, and also the byproduct identity (1.4). In section 3, we prove identity (1.5) of the KLT momentum kernel by using the CHY formulation, while we transform identity (1.6) into a convenient form for possible future attempts and end with some discussion.
Two Identities of the Half-soft Function
In this section we will prove (1.3) and (1.4), first let's set up a bit convenient facilitation. For reader's reference, we write (1.
3) again below b =n bn 2 M,N h(b, n, M )h(b, n, N ) b|K M |n] n|K N |b] 3 = 0, (2.1)
where M, N are two non-overlapping nonempty sets satisfying M ∪ N = {1, . . . , n − 1} \ b, and momentum conservation enforces k b + k n + K M + K N = 0. The half-soft function h above is defined as [10] h(b, n, N ) = 1
N i ib 2 in 2 |Ψ| r r = 1 N i ib 2 in 2 ||Ψ||, (2.2)
where |Ψ| r r denotes the determinant of matrix Ψ after deleting its r-th row and r-th column, and ||Ψ|| indicates this quantity is independent of the choice r ∈ N . If there is only one row and one column, the determinant is 1 after deletion. The matrix Ψ is defined as
Ψ ij (b, n) = − [ij] ij ib in jb jn for i = j, Ψ ii = N j =i Ψ ij ,(2.3)
where b and n serve as auxiliary spinors. The sum of each row is zero, so Ψ is degenerate. Observe that the summand in (1. for brevity N stands for K N in spinorial products (and later N also represents the number of elements in the set N , depending on the context).
To simplify the proof, we define the matrix ψ as
ψ ij (b, n) = − [ij] ij jb jn for i = j, ψ ii = N j =i ψ ij ,(2.5)
where the common factor ib in of the i-th row in Ψ has been stripped off. One can easily verify that
h(b, n, N ) = 1 N i ib 2 in 2 ||Ψ|| = 1 N i ib in |ψ N | x x xb xn ,(2.6)
where N has been added to ψ to label the corresponding set, note that |ψ N | x x / xb xn is independent of the choice x ∈ N . Then we have
h(b, n, N )h(b, n, M ) = 1 N +M i ib in |ψ N | x x xb xn |ψ M | y y yb yn = bn i =n in N +M i ib |ψ N | x x xb xn |ψ M | y y yb yn , (2.7)
where i =n in is a common factor independent of b so it can be dropped, hence (2.
A simpler byproduct identity
In the proof of (2.8), we happened to discover (1.4). For reader's reference, it is given below
[1n] 1n |ψ N ∪M | w w w1 wn + N 1|N |n] n|N |1] |ψ N | x x x1 xn |ψ M | y y y1 yn = 0, (2.9)
where N, M are two non-overlapping nonempty sets satisfying N ∪ M = {2, . . . , n − 1}, and the auxiliary spinors are 1 and n. Also note that w ∈ N ∪ M , x ∈ N , y ∈ M and it is free to switch the choices w, x, y within each set. Since this is mandatory for (2.8) to hold, we will prove it first as the tricks used here are analogous to those for (2.8).
Now we will adopt the BCFW deformation and reduce it into an identity of the same form, but with one particle removed, in other words, we will perform an inductive proof. Before induction, the identity is confirmed analytically at lower points for n = 4, 5, 6. For later convenience, we multiply it by a non-zero factor, yields
1 i =1,n i1 [1n] 1n |ψ N ∪M | w w w1 wn + N 1|N |n] n|N |1] |ψ N | x x x1 xn |ψ M | y y y1 yn = 0, (2.10)
which is of course equivalent to (1.4). But now there are two advantages: The large z behavior of its LHS is improved, and it has the desired simple pole for residue evaluation, as we will soon see.
For generic n, consider BCFW deformation 1|n] and a particular pole 21 . Note that particles 1 and n are special while the rest (n − 2) ones are symmetric, so it is sufficient to consider the residue of 21 only, as all i1 's with i ∈ {2, . . . , n − 1} behave similarly. At 21 = 0, we have
|1 = |1 − |n 12 n2 = |2 1n 2n , |n] = |n] + |1] 12 n2 , (2.11) and |1 [1| + |2 [2| ≡ |2 [2|, [2| = [2| + [1| 1n 2n , (2.12)
by which we mean to combine the momenta of particle1 and 2 into that of particle2, or more physically, particles1 and 2 merge into particle2. Including the deformed particlen, the set {1, 2, . . . , n} now shrinks into {2, . . . ,n} while momentum conservation still holds, as what induction requires.
To locate pole 21 in (2.10), we immediately find one in the overall factor. Naively, there might be another one under |ψ N | x x if we take x = 2, for example. However, the expansion of |ψ N | x x in terms of 21 will cancel this pole. In other words, |ψ N | x x / x1 xn is a polynomial of 21 (one may also choose x = 2 to invalidate this pole), that's why the overall factor is mandatory.
The next step is to analyze the large z behavior of the LHS in (2.10) before evaluating its residues at finite locations. To clarify the analysis, we further separate the second term in the parenthesis, and from now on we redefine N, M to exclude particle 2 from them while N ′ , M ′ denote the original sets. Depending on whether N ′ or M ′ contains particle 2, the set {2, . . . , n − 1} has three types of splitting: [12] |ψ N ∪M | w w w1 wn .
{{2} ∪ N, M }, {N, {2} ∪ M } and {{2}, N ∪ M }, where N ∪ M = {3, . . . , n − 1}. So the second term becomes N ′ 1|N ′ |n] n|N ′ |1] |ψ N ′ | x x x1 xn |ψ M ′ | y y y1 yn = {2}∪N,M ( 1|N |n] + 12 [2n])( n|N |1] + n2 [21]) |ψ N ∪{2} | 2 2 21 2n |ψ M | y y y1 yn + N,{2}∪M 1|N |n] n|N |1] |ψ N | x x x1 xn |ψ M ∪{2} | 2 2 21 2n + [n2]
(2.13)
Also, the first term in (2.10) can be written as
[1n] 1n |ψ N ′ ∪M ′ | w w w1 wn = [1n] 1n |ψ N ∪M ∪{2} | 2 2 21 2n .
(2.14)
Since the three ψ's in (2.14) and the first and second terms of (2.13) contain particle 2, we can choose to delete its corresponding row and column. Large z power counting shows that all four terms in (2.13) and (2.14) behave as z N +M −1 = z n−4 under 1|n], but the overall factor in the front of (2.10) behaves as z −(n−2) , which renders the entire expression as z −2 , so there is no boundary contribution. Therefore, via contour integration, the LHS of (2.10) (denoted I below) can be expressed as
z=0 dz z I(z) = − 21 =0 dz z I(z) − . . . − n−1,1 =0 dz z I(z),(2.15)
if the residue at 21 = 0 vanishes, by the symmetry among particles {2, . . . , n − 1} the entire un-deformed expression must also vanish. Note the contribution from the overall factor in (2.10) is universal, so it can be dropped. At 21 = 0, after some algebra, the residue evaluation gives after assuming the identity of (n − 1) particles holds. This finishes the inductive proof of (1.4).
|ψ N ∪{2} | 2 2 21 2n → 1n 2n N −1 (− n|N |2]) |ψ N | x x x2 xn ,(2.
Proof of the first identity
Now we move to prove (
{{1, 2} ∪ N b , M b }, {N b , {1, 2} ∪ M b }, {{1} ∪ N b , {2} ∪ M b } and {{2} ∪ N b , {1} ∪ M b }, where N b ∪ M b = {3, .
. . , n − 1} \ b, but the last two will not contribute to the residue of 21 and hence the corresponding terms are neglected, which will be explained shortly.
According to the splittings above, we can write After the separation, we now analyze the large z behavior. Under 1|n], large z power counting shows that I 1 ∼ z −2 , I 2 ∼ z −1 and I b =1,2,n ∼ z −1 , so there is no boundary contribution. Then we can repeat the contour integration (2.15). Again, thanks to the symmetry among particles {2, . . . , n − 1}, it is sufficient to consider the residue of 21 only.
I 1 1n 3 = 1 i =1,I b =1,2,n = b =1,2,n bn 3 i =1,b,n ib 1b × N b b|N b + 1 + 2|n] n|N b + 1 + 2|b] 3 |ψ N b ∪{1,2} | 1 1 1b 1n |ψ M b | y y yb yn + b|N b |n] n|N b |b] 3 |ψ N b | x x xb xn |ψ M b ∪{1,
Recalling (2.16) and (2.17), at 21 = 0 the residue evaluation gives 21 To settle this leftover, we look back to I b =1,2,n in (2.23) and find
1n 3 I 1 → 2n 3 1n 3 1 i =1,2,n i2 × N 1 |N + 2|n] n|N + 2|1] 3 (− n|N |2]) + 1 |N |n] n|N |1] 3 (− n|M |2]) |ψ N | x x x2 xn |ψ M | y y y2 yn − 2n 4 1n 2 [12] 3 [2n] i =1,2,n i2 |ψ N ∪M | w w w2 wn ,(2.|ψ N b ∪{1,2} | 1 1 1b 1n → [21] 21 |ψ N b ∪{2} |2 2 = − [12] 21 2b 2n |ψ N b ∪{2} | x x xb xn , (2.31)
where again we have used the independence of choice x to switch the deleted row and column. Now 21 1n
2n 2 I b =1,2,n → − [12] b =1,2,n bn 3 i =1,b,n ib × N b b|N b +2|n] n|N b +2|b] 3 |ψ N b ∪{2} | x x xb xn |ψ M b | y y yb yn + b|N b |n] n|N b |b] 3 |ψ N b | x x xb xn |ψ M b ∪{2} |+ I 2 + I b =1,2,n ) → 2n 3 i =1,2,n i2 N 2|N |n] n|N |2] 3 |ψ N | x x x2 xn |ψ M | y y y2 yn + b =1,2,n bn 3 i =1,b,n ib N ′ b b|N ′ b |n] n|N ′ b |b] 3 |ψ N ′ b | x x xb xn |ψ M ′ b | y y yb yn = b =1,n bn 3 i =1,b,n ib N ′ b|N ′ |n] n|N ′ |b] 3 |ψ N ′ | x x xb xn |ψ M ′ | y y yb yn = 0,
(2.33) which returns to the form of (2.8) for the set {2, . . . ,n}! It vanishes after assuming the identity of (n − 1) particles (without particle 1) holds. Similar to N ′ , M ′ , here N ′ b , M ′ b denote the sets including2 but not b. This finishes the inductive proof of (1.3).
Two Identities of the KLT Momentum Kernel
In this section we will prove (1.5) and (1.6) as conjectured in [11]. To understand these relations, we must first define the transformation matrices D and C between BCJ basis of gauge amplitudes via
A n (t, α t , n − 1, n) = α t ′ ∈S n−3 A n (t ′ , α t ′ , n − 1, n)D[t ′ , α t ′ , n − 1, n|t, α t , n − 1, n], (3.1) A n (t, n − 1, β t , n) = β t ′ ∈S n−3 C[t, n − 1, β t , n|t ′ , n − 1, β t ′ , n] A n (t ′ , n − 1, β t ′ , n),(3.2)
where α t ′ and β t ′ denote the permutations of (n − 3) particles other than t ′ , (n − 1) and n. In a tensorial sense, D and C are the transformation matrices with respect to the summation of all (n − 3)! permutations, which is defined as the inner product. For reader's reference, we write (1.5) and (1.6) again below
α t ′ ,β t ′ ∈S n−3 D[t, α t , n − 1, n|t ′ , α t ′ , n − 1, n]S[α t ′ |β t ′ ] p n−1 C[t ′ , n − 1, β t ′ , n|t, n − 1, β t , n] = S[α t |β t ] p n−1 , (3.3) n−2 t ′ =1 α t ′ ,β t ′ ∈S n−3 D[t, α t , n − 1, n|t ′ , α t ′ , n − 1, n]S[α t ′ |β t ′ ] p n−1 · J t ′ C[t ′ , n − 1, β t ′ , n|t, n − 1, β t , n] = 0, (3.4)
where S[α t |β t ] p n−1 is the KLT momentum kernel of pivot p n−1 , and J t ′ ≡ J t ′ ,αβ is the anti-holomorphic angular momentum operator. Here we follow the convention of S in [13,14,15], namely
S[α 1 , . . . , α k |β 1 , . . . , β k ] p n−1 = k i=1 s α i ,n−1 + k j=i+1 θ(α i , α j )s α i ,α j ,(3.5)
where s ij is each Mandelstam variable, and θ(α i , α j ) is zero when the pair (α i , α j ) has the same ordering at both sets {α 1 , . . . , α k } and {β 1 , . . . , β k }, and unity otherwise. For the first identity, its physical interpretation is straightforward: If we regard the KLT momentum kernel S as the metric, it is simply the tensorial transformation rule for metric. In fact, such a tensorial formulation had been established in [17,19] (known as the KLT orthogonality or the CHY formulation) and we will use it to formally prove the first identity shortly. The second identity is however more intricate, as it roughly represents angular momentum conservation in an entangled way. The CHY formulation can help transform it into a relation that may reveal very nontrivial properties of scattering process, while to prove it directly is yet beyond our understanding.
Proof of the first identity
Before the proof, we must first rewrite gauge amplitudes in the CHY formulation [19] which is based on the scattering equations [18]. It tells that
A n (t, α t , n − 1, n) = (n−3)! i=1 1 det ′ (Φ)(σ (i) ) Σ (i) (t, α t , n − 1, n) Pf ′ Ψ(σ (i) ),(3.
6)
A n (t, n − 1, β t , n) =
(n−3)! i=1 1 det ′ (Φ)(σ (i) ) Σ (i) (t, n − 1, β t , n) Pf ′ Ψ(σ (i) ), (3.7)
where σ (i) denotes the i-th solution to the scattering equations b =a s ab σ ab = 0, (3.8) with σ ab = σ a − σ b , and there are (n − 3)! solutions in total. The definitions of det ′ (Φ) and Pf ′ Ψ, namely the reduced determinant of Jacobian Φ and the reduced Pffafian of antisymmetric matrix Ψ, can be found in [19]. The object mainly concerns us is
Σ (i) (α) ≡ 1 σ (i) α(1),α(2) . . . σ (i) α(n−1),α(n) σ (i) α(n),α(1)
.
(3.9)
On the other hand, the KLT relation gives 11) or more compactly,
(−) n+1 M n (1, . . . , n) = αt,βt∈S n−3 A n (t, α t , n − 1, n)S[α t |β t ] p n−1 A n (t, n − 1, β t , n) = (n−3)! i=1 Pf ′ Ψ(σ (i) )Pf ′ Ψ(σ (i) ) det ′ (Φ)(σ (i) ) ,(3.Σ (i) (t, α t , n − 1, n)S[α t |β t ] p n−1 Σ (j) (t, n − 1, β t , n) = det ′ (Φ)(σ (i) )δ ij ,(3.G iαt S[α t |β t ] (H jβt ) T = I (n−3)!×(n−3)! ,(3.12)
which is the KLT orthogonality, if we define matrices
G iαt ≡ Σ (i) (t, α t , n − 1, n) det ′ (Φ)(σ (i) ) , H jβt ≡ Σ (j) (t, n − 1, β t , n) det ′ (Φ)(σ (j) ) . (3.13)
From this matrix relation we immediately get
S[α t |β t ] = (G iαt ) −1 (H iβt ) T −1 .
(3.14)
Back to (3.6) and (3.7), if we further define the row vector
Θ i ≡ Pf ′ Ψ(σ (i) ) det ′ (Φ)(σ (i) ) , (3.15) then A n (t, α t , n − 1, n) = Θ i G iαt , A n (t, n − 1, β t , n) = Θ i H iβt . (3.16)
Plugging them back into (3.1) and (3.2), and assuming their independence of basis Θ i , we get 17) or equivalently,
G iαt = G iα t ′ D[t ′ , α t ′ , n − 1, n|t, α t , n − 1, n], H iβt = H iβ t ′ C[t, n − 1, β t , n|t ′ , n − 1, β t ′ , n] T ,(3.D[t ′ , α t ′ , n − 1, n|t, α t , n − 1, n] = G iα t ′ −1 G iαt , C[t, n − 1, β t , n|t ′ , n − 1, β t ′ , n] = (H iβt ) T (H iβ t ′ ) −1 T .
(3.18) Finally we plug them back into the LHS of (1.5) and interchange t and t ′ , together with (3.14) we get
α t ′ ,β t ′ ∈S n−3 D[t, α t , n − 1, n|t ′ , α t ′ , n − 1, n]S[α t ′ |β t ′ ] p n−1 C[t ′ , n − 1, β t ′ , n|t, n − 1, β t , n] = (G iαt ) −1 G iα t ′ G jα t ′ −1 H jβ t ′ T −1 (H kβ t ′ ) T (H kβt ) −1 T = (G iαt ) −1 (H iβt ) T −1 = S[α t |β t ] p n−1 ,(3.19)
which is exactly the RHS of (1.5), hence the proof is finished.
Discussion of the second identity
Now we move to prove (1.6). Equipped with the matrices defined in the previous subsection, the LHS of (1.6) can be simplified as
n−2 t ′ =1 (G iαt ) −1 G iα t ′ G jα t ′ −1 H jβ t ′ T −1 · J t ′ (H kβ t ′ ) T (H kβt ) −1 T = n−2 t ′ =1 (G iαt ) −1 H iβ t ′ T −1 · J t ′ (H jβ t ′ ) T (H jβt ) −1 T = n−2 t ′ =1 (G iαt ) −1 H iβ t ′ T −1 (H jβ t ′ ) T · J t ′ (H jβt ) −1 T + H iβ t ′ T −1 · J t ′ (H jβ t ′ ) T · (H jβt ) −1 T = n−2 t ′ =1 (G iαt ) −1 J t ′ (H iβt ) −1 T · (H jβt ) T + H iβ t ′ T −1 · J t ′ (H jβ t ′ ) T (H jβt ) −1 T ,(3.
20) assuming the two matrices in the front and end of the last line are non-degenerate, we should prove
n−2 t ′ =1 J t ′ (H iβt ) −1 T · (H jβt ) T + H iβ t ′ T −1 · J t ′ (H jβ t ′ ) T = 0. (3.21)
For the first term above, the summation over t ′ is trivial since the matrix product involves t only, so it is in fact
n−2 t ′ =1 J t ′ (H iβt ) −1 T · (H jβt ) T = 0,(3.22)
due to angular momentum conservation, as the absence of J n−1 and J n does not matter sinceλ n−1 andλ n have been solved by momentum conservation (see [11] for more details). Therefore we are left with
n−2 t=1 (H iβt ) T −1 · J t (H jβt ) T = 0, (3.23)
where the dummy variable t ′ has been replaced by t. We can continue to transform it into a convenient form for further attempts to prove, by isolating its real matrix content. Let's define
Σ jβt ≡ Σ (j) (t, n − 1, β t , n), W ij ≡ 1 det ′ (Φ)(σ (i) ) δ ij ,(3.24)
then it is clear that H iβt = W ij Σ jβt . While W ij is a trivial diagonal matrix, Σ jβt encodes the real matrix content. Now we can write the LHS of (3.23) as
n−2 t=1 W −1 (Σ iβt ) T −1 · J t (Σ jβt ) T W = n−2 t=1 W −1 (Σ iβt ) T −1 · J t (Σ jβt ) T · W + n−2 t=1 W −1 · J t W = n−2 t=1 (Σ iβt ) T −1 · J t (Σ jβt ) T + W −1 · n−2 t=1 J t W = n−2 t=1 (Σ iβt ) T −1 · J t (Σ jβt ) T ,(3.25)
where in the third line, the second term vanishes again due to angular momentum conservation. Finally, we are left with
n−2 t=1 (Σ iβt ) T −1 · J t (Σ jβt ) T = 0,(3.26)
which can no longer be further simplified.
To get some sense of this very nontrivial identity, it is helpful to see the first nontrivial case n = 4, which corresponds to the first nonempty β t . Recall that Σ (i) (α) = 1 σ (i) α(1),α (2) . . . σ which trivially holds by the antisymmetry of σ ab ! But as n increases, even for n = 5 this identity will be much more entangled and simple antisymmetry is insufficient for its proof. The potential toolkit for this purpose includes: (1) relations of spinor derivatives on scattering equations; (2) KK and BCJ relations of Σ jβt ; (3) induction, which may involve contour integration. We will come back to this point in the future after better understanding the scattering equations and their solutions.
A last comment is that in (1.6), the anti-holomorphic angular momentum operator J t ′ ,αβ should be generalized to J t ′ ,µν in arbitrary dimensions. Since in 4-dimension J µν ∼ ε αβ Jαβ + εαβJ αβ , and the soft theorem must hold for both holomorphic and anti-holomorphic soft limits, it is more natural to use J µν as all other quantities are already defined for arbitrary dimensions.
3) has even power of K M and K N , by momentum conservation this sum is symmetric between M and N , then we can replace K M by −K N and rewrite (1.b, n, N )h(b, n, M ) b|N |n] n|N |b]
i
=b,n ib N b|N |n] n|N |b] 3 |ψ N | x x xb xn |ψ M | y y yb yn = 0. (2.8)
2.8) by applying the similar pack of tricks: to consider deformation 1|n] acting on its LHS, and the pole 21 . First, we separate the expression into three parts corresponding to b = 1, b = 2 and b = 3, . . . , n − 1, namely b =n bn 3 i =b,n ib N b|N |n] n|N |b] 3 |ψ N | = I 1 + I 2 + I b =1,2,n . (2.20) Similarly, we now redefine N and M to exclude particles 2 and 1, with respect to I 1 and I 2 . For I 1 , the set {2, . . . , n − 1} has three types of splitting: {{2} ∪ N, M }, {N, {2} ∪ M } and {{2}, N ∪ M }, where N ∪ M = {3, . . . , n − 1}. For I 2 , we have {{1} ∪ N, M }, {N, {1} ∪ M } and {{1}, N ∪ M }. For I b =1,2,n , there are four types:
10) where the second line results from the CHY formulation. There is a subtle issue of the sign above, due to the different conventions M n = −M CHYn and S[α t |β t ] = S CHY [β t |α T t ]. Plugging (3.6) and (3.7) into this relation, yields αt,βt∈S n−3
16 )
16recall that 21 above is not a pole, while the real pole comes from the overall factor. Here |1 is replaced by |2 up to a factor, after recalling (2.11). By expanding the determinant to the first order of 21 , then using the independence of choice x to switch the deleted row and column for each term, we can collect a factor (− n|N |2]) as above. The similar (and simpler) story happens for|ψ M | y
y
y1 yn
→
1n
2n
M −2 |ψ M | y
y
y2 yn
.
(2.17)
Plugging them back, up to a factor ( 1n / 2n ) N +M −2 , the sum of (2.13) and (2.14) becomes
(− n|N + M |2])
|ψ N ∪M | w
w
w2 wn
[1n]
2n
+
N
2|N |n]( n|N |1] + n2 [21])(− n|N |2])
|ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
+
N
2|N |n] n|N |1](− n|M |2])
|ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
+ [n2][12]
|ψ N ∪M | w
w
w2 wn
.
(2.18)
By momentum conservation, up to a factor [12], it can be simplified into
1n
2n
[n1] + [n2]
|ψ N ∪M | w
w
w2 wn
+ n2
N
2|N |n] n|N |2] + n|N |1]
1n
2n
|ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
= n2
[2n]
2n
|ψ N ∪M | w
w
w2 wn
+
N
2|N |n] n|N |2]
|ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
= 0,
(2.19)
2|N + 1|n] n|N + 1|2] 3 |ψ N ∪{1} | 12,n i1 21
N
1|N + 2|n] n|N + 2|1] 3 |ψ N ∪{2} | 2
2
21 2n
|ψ M | y
y
y1 yn
+ 1|N |n] n|N |1] 3 |ψ N | x
x
x1 xn
|ψ M ∪{2} | 2
2
21 2n
+
1
i =1,2,n i1 21
1|2|n] n|2|1] 3
1
21 2n
|ψ N ∪M | w
w
w1 wn
,
(2.21)
I 2
2n 3 =
1
i =1,2,n i2 12
N
1
12 1n
|ψ M | y
y
y2 yn
+ 2|N |n] n|N |2] 3 |ψ N | x
x
x2 xn
|ψ M ∪{1} | 1
1
12 1n
+
1
i =1,2,n i2 12
2|1|n] n|1|2] 3
1
12 1n
|ψ N ∪M | w
w
w2 wn
,
(2.22)
For I b =1,2,n one can verify that, only terms for which 1 and 2 are in the same splitting set, have pole 21 and hence contribute to the residue, which explains why we only need the first two terms. Moreover, N b in {{1, 2} ∪ N b , M b } can be empty (similarly for M b ). While for I 1 , N in {{2} ∪ N, M } cannot be empty, otherwise such a splitting belongs to type {{2}, N ∪ M } (similarly for I 2 ).2} | 1
1
1b 1n
+ (two neglected terms).
(2.23)
2|N |n] n|N |1] n|N + 1|2] 3 − n|N + 2|1] n|N |2] 3 |ψ N | x − n|N |2] n|N + 2|1] 3 + n|N + 1|2] n|N |1] 3 + 2n 2 n|N |1] n|N + 1|2] 3 − n|N + 2|1] n|N |2] 324)
or after a bit simplification,
21 1n
2n 2 I 1 →
1n 2
i =1,2,n i2
N
2|N |n] − n|N |2] n|N + 2|1] 3 + n|N + 1|2] n|N |1] 3 |ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
− 1n 2 2n 2 [12] 3 [2n]
i =1,2,n i2
|ψ N ∪M | w
w
w2 wn
.
(2.25)
Similarly for I 2 ,
21 1n
2n 2 I 2 →
2n 2
i =1,2,n i2
N
x
x2 xn
|ψ M | y
y
y2 yn
− 1n 3 2n
[12] 3 [1n]
i =1,2,n i2
|ψ N ∪M | w
w
w2 wn
.
(2.26)
Combining I 1 and I 2 , we find
21 1n
2n 2 (I 1 + I 2 ) → −
[12] 2n 3
i =1,2,n i2
N
2|N |n] n|N |2] 3 |ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
−
[12] 3 1n 2 2n 3
i =1,2,n i2
N
2|N |n] n|N |2]
|ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
− 1n 2 2n 2 [12] 3 [2n]
i =1,2,n i2
|ψ N ∪M | w
w
w2 wn
,
(2.27)
after using the following identity
1n 2 = − [12]( 1n n|N |1] + 2n n|N |2]) 3 − [12] 3 1n 2 2n 2 ( 1n n|N |1] + 2n n|N |2])
= − [12] 2n 3 n|N |2] 3 − [12] 3 1n 2 2n 3 n|N |2].
(2.28)
Now note the second and third terms in (2.27) can be regrouped as
−
[12] 3 1n 2 2n 3
i =1,2,n i2
N
2|N |n] n|N |2]
|ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
+
[2n]
2n
|ψ N ∪M | w
w
w2 wn
= 0,
(2.29)
which is exactly identity (1.4) for the set {2, . . . ,n}! Therefore we are left with
21 1n
2n 2 (I 1 + I 2 ) → −
[12] 2n 3
i =1,2,n i2
N
2|N |n] n|N |2] 3 |ψ N | x
x
x2 xn
|ψ M | y
y
y2 yn
.
(2.30)
AcknowledgmentsThe authors would like to thank Qingjun Jin for reading the manuscript. This work is supported by Qiu-Shi Funding and Chinese NSF funding under contracts No.11135006, No.11125523 and No.11575156.
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| []
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[
"LEAVE-ONE-OUT SINGULAR SUBSPACE PERTURBATION ANALYSIS FOR SPECTRAL CLUSTERING",
"LEAVE-ONE-OUT SINGULAR SUBSPACE PERTURBATION ANALYSIS FOR SPECTRAL CLUSTERING"
]
| [
"Anderson Y Zhang \nDepartment of Statistics and Data Science\nUniversity of Pennsylvania\n\n",
"Harrison Y Zhou \nDepartment of Statistics and Data Science\nYale University\n\n"
]
| [
"Department of Statistics and Data Science\nUniversity of Pennsylvania\n",
"Department of Statistics and Data Science\nYale University\n"
]
| []
| The singular subspaces perturbation theory is of fundamental importance in probability and statistics. It has various applications across different fields. We consider two arbitrary matrices where one is a leave-one-column-out submatrix of the other one and establish a novel perturbation upper bound for the distance between two corresponding singular subspaces. It is well-suited for mixture models and results in a sharper and finer statistical analysis than classical perturbation bounds such as Wedin's Theorem. Powered by this leaveone-out perturbation theory, we provide a deterministic entrywise analysis for the performance of the spectral clustering under mixture models. Our analysis leads to an explicit exponential error rate for the clustering of sub-Gaussian mixture models. For the mixture of isotropic Gaussians, the rate is optimal under a weaker signal-to-noise condition than that ofLöffler et al. (2021). | 10.48550/arxiv.2205.14855 | [
"https://arxiv.org/pdf/2205.14855v1.pdf"
]
| 249,192,260 | 2205.14855 | 3c3a818c5b68ade8c31acd94588567185be4b0eb |
LEAVE-ONE-OUT SINGULAR SUBSPACE PERTURBATION ANALYSIS FOR SPECTRAL CLUSTERING
May 2022
Anderson Y Zhang
Department of Statistics and Data Science
University of Pennsylvania
Harrison Y Zhou
Department of Statistics and Data Science
Yale University
LEAVE-ONE-OUT SINGULAR SUBSPACE PERTURBATION ANALYSIS FOR SPECTRAL CLUSTERING
May 2022Submitted to the Annals of Statistics
The singular subspaces perturbation theory is of fundamental importance in probability and statistics. It has various applications across different fields. We consider two arbitrary matrices where one is a leave-one-column-out submatrix of the other one and establish a novel perturbation upper bound for the distance between two corresponding singular subspaces. It is well-suited for mixture models and results in a sharper and finer statistical analysis than classical perturbation bounds such as Wedin's Theorem. Powered by this leaveone-out perturbation theory, we provide a deterministic entrywise analysis for the performance of the spectral clustering under mixture models. Our analysis leads to an explicit exponential error rate for the clustering of sub-Gaussian mixture models. For the mixture of isotropic Gaussians, the rate is optimal under a weaker signal-to-noise condition than that ofLöffler et al. (2021).
1. Introduction. The matrix perturbation theory [36,4] is a central topic in probability and statistics. It plays a fundamental role in spectral methods [10,18], an umbrella term for algorithms involving eigendecomposition or singular value decomposition. It has a wide range of applications including principal component analysis [1,7], covariance matrix estimation [14], clustering [38,33,34,29], and matrix completion [27,13], throughout different fields such as machine learning [5], network science [31,2], and genomics [19].
Perturbation analysis for eigenspaces and singular subspaces dates back to seminal works of Davis and Kahan [11] and Wedin [40]. Davis-Kahan Theorem provides a clean bound for eigenspaces in terms of operator norm and Frobenius norm, and Wedin further extends it to singular subspaces. In recent years, there has been growing literature in developing finegrained ℓ ∞ analysis for singular vectors [2,14] and ℓ 2,∞ analysis for singular subspaces [24,9,6,3], which often lead to sharp upper bounds. For clustering problems, they can be used to establish the exact recovery of spectral methods, but are usually not suitable for low signal-to-noise ratio regimes where only partial recovery is possible.
In this paper, we consider a special matrix perturbation case where one matrix differs from the other one by having one less column and investigate the difference between two corresponding left singular subspaces. Consider two matrices Y = (y 1 , . . . , y n−1 ) ∈ R p×(n−1) andŶ = (y 1 , . . . , y n−1 , y n ) ∈ R p×n ,
where Y is a leave-one-column-out submatrix ofŶ with the last column removed. Let U r and U r include the leading r left singular vectors of Y andŶ , respectively. The two corresponding left singular subspaces are span(U r ) and span(Û r ), where the former one can be interpreted as a leave-one-out counterpart of the latter one.
We establish a novel upper bound for the Frobenius norm ofÛ rÛ T r − U r U T r to quantify the distance between the two singular subspaces span(U r ) and span(Û r ). A direct application of the generic Wedin's Theorem leads to a ratio of the magnitude of perturbation (I − U r U T r )y n to the corresponding spectral gap σ r − σ r+1 . We go beyond Wedin's Theorem and reveal that the interplay between U r U T r y n and (I − U r U T r )y n plays a crucial role. Our new upper bound is a product of the aforementioned ratio and a factor determined U T r y n . That is, informally (see Theorem 2.1 for a precise statement),
Û rÛ T r − U r U T r F (I − U r U T r )y n σ r − σ r+1
× a factor from U T r y n .
When this factor is smaller than some constant, it results in a sharper upper bound than Wedin's Theorem. The established upper bound is particularly suitable for mixture models where the contributions of U T r y n are well-controlled, and consequently provides a key toolkit for the follow-up statistical analysis on spectral clustering.
Spectral clustering is one of the most popular approaches to group high-dimensional data. It first reduces the dimensionality of data by only using a few of its singular components, followed by a classical clustering method such as k-means to the data of reduced dimension. It is computationally appealing and often has remarkably good performance, and has been widely used in various problems. In recent years there has been growing interest in theoretical properties of spectral clustering, noticeably in community detection [2,23,17,32,33,43,15,30,22]. In spite of various polynomial-form upper bounds in terms of signal-to-noise ratios for the performance of spectral clustering, sharper exponential error rates are established in literature only for a few special scenarios, such as Stochastic Block Models with two equalsize communities [2]. Spectral clustering is also investigated in mixture models [29,25,1,12,39,35]. For isotropic Gaussian mixture models, [25] shows spectral clustering achieves the optimal minimax rate. However, the proof technique used in [25] is very limited to the isotropic Gaussian noise and it is unclear whether it is possible to be extended to either sub-Gaussian distributed errors or unknown covariance matrices. Spectral clustering for sub-Gaussian mixture models is studied in [1] but only under special assumptions on the spectrum and geometry of the centers. It requires eigenvalues of the Gram matrix of centers to be all in the same order and sufficiently large, which rules out many interesting cases.
We study the theoretical performance of the spectral clustering under general mixture models where each observation X i is equal to one of k centers plus some noise ǫ i . The spectral clustering first projects X i ontoÛ T 1:r X i whereÛ 1:r includes the leading r left singular vectors of the data matrix, and then performs k-means on this low-dimensional space. Powered by our leave-one-out perturbation theory, we provide a deterministic entrywise analysis for the spectral clustering and show that whether X i is correctly clustered or not is determined bŷ U T −i,1:r ǫ i whereÛ −i,1:r is the leave-one-out counterpart ofÛ 1:r that uses all the observations except X i . The independence betweenÛ −i,1:r and ǫ i enables us to derive explicit error risks when the noises are randomly generated from certain distributions. Specifically:
1. For sub-Gaussian mixture models, we establish an exponential error rate for the performance of the spectral clustering, assuming the centers are separated from each other and the smallest non-zero singular value is away from zero. Our conditions are more general than those needed in [1]. To remove the spectral gap condition, we further propose a variant of the spectral clustering where the number of singular vectors used is selected adaptively. 2. For Gaussian mixture models with isotropic covariance matrix, we fully recover the results of [25]. Empowered by the leave-one-out perturbation theory, our proof is completely different and is much shorter compared to that of [25]. In addition, the signal-to-noise ratio condition of [25] is improved.
3. For a two-cluster symmetric mixture model where coordinates of the noise ǫ i are independently and identically distributed, we provide a matching upper and lower bound for the performance of the spectral clustering. This sharp analysis provides an answer to the optimality of the spectral clustering in this setting: it is in general sub-optimal and is optimal only if each coordinate of ǫ i is normally distributed.
Organization. This paper is organized as follows. In Section 2, we first establish a general leave-one-out perturbation theory for singular subspaces, followed by its application in mixture models. In Section 3, we use our leave-one-out perturbation theory to provide theoretical guarantees for the spectral clustering under mixture models. The proofs of main results in Section 2 and Section 3 are given in Section 4 and in Section 5, respectively. The remaining proofs are included in the supplement [42].
Notation. For any positive integer r, let [r] = {1, 2, . . . , r}. For two scalars a, b ∈ R, denote a ∧ b = min{a, b}. For two matrices A = (A i,j ) and B = (B i,j ), we denote A, B = i,j A i,j B i,j to be the trace product, A to be its operator norm, A F to be its Frobenius norm, and span(A) to be the linear space spanned by columns of A. If both A, B are symmetric, we write A ≺ B if B − A is positive semidefinite. For scalars x 1 , . . . , x d , we denote diag(x 1 , . . . , x d ) to be a d × d diagonal matrix with diagonal entries being x 1 , . . . , x d . For any integers d, p ≥ 0, we denote 0 d ∈ R d to be a vector with all coordinates being 0, 1 d ∈ R d to be a vector with all coordinates being 1, and O d×p ∈ R d×p to be a matrix with all entries being 0. We denote I d×d and I d to be the d × d identity matrix and we use I for short when the dimension of clear according to context. Let O d×p = V ∈ R d×p : V T V = I be the set of matrices in R d×p with orthonormal columns. We denote I {·} to be the indicator function. For two positive sequences {a n } and {b n }, a n b n , a n = O(b n ), b n a n all mean a n ≤ Cb n for some constant C > 0 independent of n. We also write a n = o(b n ) when lim sup n→∞ an bn = 0. For a random variable X, we say X is sub-Gaussian with variance proxy σ 2 (denoted as X ∼ SG(σ 2 )) if Ee tX ≤ exp σ 2 t 2 /2 for any t ∈ R. For a random vector X ∈ R d , we say X is sub-Gaussian with variance proxy σ 2 (denoted as X ∼ SG d (σ 2 )) if u T X ∼ SG(σ 2 ) for any unit vector u ∈ R d .
2.
Leave-one-out Singular Subspace Perturbation Analysis. In this section, we establish a general matrix perturbation theory for singular subspaces. In particular, we consider two arbitrary matrices with one having a less column than the other and study the difference between two corresponding left singular subspaces. We will first develop a general theory and then apply it to mixture models.
2.1. General Results. Consider two matrices as in (1) such that they are equal to each other except thatŶ has an extra last column. Let the Singular Value Decomposition (SVD) of these two matrices be to include the leading r left singular vectors of Y andŶ , respectively. Since Y can be viewed as a leave-one-out submatrix ofŶ that is without the last column y n , U r can be interpreted as a leave-one-out counterpart ofÛ r .
Y = i∈[p∧(n−1)] σ i u i v T i andŶ = i∈[p∧n]σ iûiv T i , where σ 1 ≥ . . . ≥ σ p∧(n−1) andσ 1 ≥ . . . ≥σ p∧n . Consider any r ∈ [p ∧ (n − 1)]. Define
The two matrices U r ,Û r correspond to two singular subspaces span(U r ), span(Û r ), respectively. The difference between these two subspaces can be captured by sin Θ distances, sin Θ(Û r , U r ) or sin Θ(Û r , U r ) F , where
Θ(Û r , U r ) := diag(cos −1 (α 1 ), cos −1 (α 2 ), . . . , cos −1 (α r )) with α 1 ≥ α 2 ≥ . . . ≥ α r ≥ 0 being the r singular values ofÛ T r U r . It is known (cf. Lemma 1 of [8]) that Û rÛ T r − U r U T r F = √ 2 sin Θ(Û r , U r ) F .
Throughout this section, we will focus on establishing sharp upper bounds for Û rÛ T r − U r U T r F , i.e., the Frobenius norm of the difference between two corresponding projection matrices U r U T r andÛ rÛ T r . Since the augmented matrix Y ′ := (Y, U r U T r y n ) ∈ R p×n concatenated by Y and U r U T r y n has the same leading r left singular subspace and projection matrix as Y , a natural idea is
to relate Û rÛ T r − U r U T r F with the differenceŶ − Y ′ .
The classical spectral perturbation theory such as Wedin's Theorem [41,8]
leads to that if σ r − σ r+1 > 2 (I − U r U T r )y n , then Û rÛ T r − U r U T r F ≤ 2 (I − U r U T r )y n σ r − σ r+1 .(2)
See Proposition D.1 in the supplement for its proof. The upper bound in (2) requires the spectral gap σ r − σ r+1 is away from zero. It also indicates the magnitude of the difference Ŷ − Y ′ = (I − U r U T r )y n plays a crucial role. In spite of its simple form, (2) comes from generic spectral perturbation theories not specifically designed for the setting (1).
In the following Theroem 2.1, we provide a deeper and finer analysis for Û rÛ T r − U r U T r F , utilizing the fact thatŶ and Y differ by only one column and exploiting the interplay between U r U T r y n and (I − U r U T r )y n .
THEOREM 2.1. If ρ := σ r − σ r+1 (I − U r U T r )y n > 2,(3)
we have
Û rÛ T r − U r U T r F ≤ 4 √ 2 ρ r i=1 u T i y n σ i 2 .(4)
Theorem 2.1 gives an upper bound on Û rÛ T r − U r U T r F that is essentially a product of ρ −1 and some quantity determined by
{σ −1 i u T i y n } i∈[r] . Since (σ −1 i u T i y n ) 2 ≤ σ −2 r (u T i y n ) 2 for each i ∈ [r], (4) leads to a simpler upper bound Û rÛ T r − U r U T r F ≤ 4 √ 2 ρ U r U T r y n σ r .
The condition (3) in Theorem 2.1 can be understood as a spectral gap assumption as it needs the gap σ r − σ r+1 to be larger than twice the magnitude of the perturbation (I − U r U T r )y n . This condition can be slightly weakened into σ 2 r − σ 2 r+1 − (I − U r U T r )y n 2 > 0, though resulting in a more involved upper bound. See Theorem 4.1 in Section 4.1 for details.
We are ready to have a comparison of our result (4) and (2) that is from Wedin's Theorem. Under the assumption (3), the upper bound in (2) can be written equivalently as 2ρ −1 . As a result, the comparison is about the magnitude of ( i∈[r] (σ −1 i u T i y n ) 2 ) 1/2 . If it is smaller than 1/(2 √ 2), then (4) gives a sharper upper bound than (2). To further compare these two bounds, consider the following examples.
• Example 1. When U T r y n = 0 and (3) is satisfied, (4) gives the correct upper bound 0. That is,Û rÛ T r = U r U T r . On the contrary, (2) gives a non-zero bound 2/ρ −1 . To be more concrete, let Y = σ 1 (p −1/2 1 p )((n − 1) −1/2 1 n−1 ) T be a rank-one matrix and y n be some vector that is orthogonal to 1 p . Then if σ 1 > 2 y n , we haveû 1 = u 1 = p −1/2 1 p up to sign. (4) gives the correct answer û 1û T 1 − u 1 u T 1 F = 0 as u T 1 y n = 0, while (2) leads to a loose upper bound 2 y n /σ 1 . • Example 2. Let Y be a matrix with two unique columns such that y j is equal to either θ or −θ for all j ∈ [n − 1] and for some vector θ ∈ R p . Then Y is a rank-one matrix with σ 1 = θ √ n − 1. Let y n = θ + ǫ. As long as θ √ n − 1 > 2 ǫ , we have û 1û (4). If we further assume θ = 1 and ǫ ∼ N (0, I p ) with p ≪ n, we have û 1û
T 1 − u 1 u T 1 F ≤ 4 √ 2ρ −1 ( θ + ǫ )/σ 1 fromT 1 −u 1 u T 1 F p/nρ −1 = o(ρ −1 ) with high probability. In contrast,(2)
only gives 2ρ −1 .
In the next section, we consider mixture models where the magnitude of ( i∈[r] (σ −1 i u T i y n ) 2 ) 1/2 is well-controlled and (4) leads to a much sharper upper bound compared to (2).
2.2.
Singular Subspace Perturbation in Mixture Models. The general perturbation theory presented in Theorem 2.1 is particularly suitable for analyzing singular subspaces of mixture models.
Mixture Models. We consider a mixture model with k centers θ * 1 , θ * 2 , . . . , θ * k ∈ R p and a cluster assignment vector z * ∈ [k] n . The observations X 1 , X 2 , . . . , X n ∈ R p are generated from
X i = θ * z * i + ǫ i ,(5)
where ǫ 1 , . . . , ǫ n ∈ R p are noises. The data matrix X := (X 1 , . . . , X n ) ∈ R p×n can be written equivalently in a matrix form
X = P + E,(6)
where P := (θ * z * 1 , θ * z * 2 , . . . , θ * z * n ) is the signal matrix and E := (ǫ 1 , . . . , ǫ n ) is the noise matrix. Define β := 1 n/k min a∈[k] |{i : z * i = a}| such that βn/k is the smallest cluster size.
We are interested in the left singular subspaces of X and its leave-one-out counterparts. For each i ∈ [n], define X −i to be a submatrix of X with its ith column removed. That is,
X −i := (X 1 , . . . , X i−1 , X i+1 , . . . , X n ) ∈ R p×(n−1) .(7)
Let their SVDs be X = j∈[p∧n]λ jûjv T j and 1) . Note that the signal matrix P is at most rank-k. Then for any r ∈ [k], definê
X −i = j∈[p∧(n−1)]λ −i,jû−i,jv T −i,j , wherê λ 1 ≥λ 2 ≥ . . . ≥λ p∧n andλ −i,1 ≥λ −i,2 ≥ . . . ≥λ −i,p∧(n−U 1:r := (û 1 ,û 2 , . . . ,û r ) ∈ O p×r andÛ −i,1:r = (û −i,1 , . . . ,û −i,r ) ∈ O p×r
to include the leading r left singular vectors of X and X −i , respectively. We are interested in controlling the quantity Û 1:rÛ
T 1:r −Û −i,1:rÛ T −i,1:r F for each i ∈ [n].
In Theorem 2.2, we provide upper bounds for Û 1:κÛ
T 1:κ −Û −i,1:κÛ T −i,1:κ F for all i ∈ [n]
where κ ∈ [k] is the rank of the signal matrix P . In order to have such a uniform control across all i ∈ [n], we consider the spectrum of the signal matrix P . Let λ 1 ≥ λ 2 ≥ . . . ≥ λ p∧n be the singular values of P and κ be the rank of P such that κ ∈ [k], λ κ > 0, and λ κ+1 = 0.
ρ 0 := λ κ E > 16.(8)
For any i ∈ [n], we have
Û 1:κÛ T 1:κ −Û −i,1:κÛ T −i,1:κ F ≤ 128 ρ 0 kκ βn + Û −i,1:κÛ T −i,1:κ ǫ i λ κ .(9)
Theorem 2.2 exploits the mixture model structure (5) that the signal matrix P has only k unique columns with each appearing at least βn/k times. The assumption βn/k ≥ 16 helps ensure that spectrum and singular vectors of P are not much changed if any column of P is removed. We require the condition (8)
so thatλ −i,κ −λ −i,κ+1 > 2 Û −i,1:κÛ T −i,1:κ X i holds for each i ∈ [n]
, and hence Theorem 2.1 can be applied uniformly for all i ∈ [n]. The upper bound (9) is a product of ρ −1 0 and a summation of two terms. The second term Û −i,1:κÛ T −i,1:κ ǫ i /λ κ can be trivially upper bounded by E /λ κ ≤ ρ −1 0 . The first term kκ/(βn) = o(1) if βn/k 2 ≫ 1, for example, when β is a constant and k ≪ √ n. Then (9) leads to Û 1:κÛ
T 1:κ −Û −i,1:κÛ T −i,1:κ F o(1)ρ −1 0 + ρ −2 0
, superior to the upper bound (2) obtained from the direct application of Wedin's Theorem that is in an order of ρ −1 0 . Theorem 2.2 studies the perturbation for the leading κ singular subspaces where κ is the rank of P . In the following Theorem 2.3, we consider an extension to Û 1:rÛ
T 1:r −Û −i,1:rÛ T −i,1:r F
where r is not necessarily κ.
ρ 0 := λ r − λ r+1 max E , k 2 βn λ r+1 > 16.(10)
For any i ∈ [n], we have
Û 1:rÛ T 1:r −Û −i,1:rÛ T −i,1:r F ≤ 128 ρ 0 √ kr √ βn + Û −i,1:rÛ T −i,1:r ǫ i λ r .(11)
In Theorem 2.3, r ∈ [k] is any number such that (10) is satisfied. When r is chosen to be κ, (10) is reduced to (8), and (11) leads to the same upper bound as (9). When r < κ, λ r+1 is non-zero and in (10) it needs to be smaller than the spectral gap λ r − λ r+1 after some scaling factor.
To provide some intuition on the condition (10) when r < κ, let the SVD of the signal matrix P be P = j∈[p∧n] λ j u j v T j and define U 1:r := (u 1 , u 2 , . . . , u r ) ∈ O p×r and U (r+1):κ := (u r+1 , u r+2 , . . . , u κ ) ∈ O p×(κ−r) . Then the data matrix (6) can be written equivalently as
X = P ′ + E ′ , where P ′ := U 1:r U T 1:r P and E ′ := E + U (r+1):κ U T (r+1):κ P.(12)
Since it is still a mixture model, Theorem 2.2 can be applied. Nevertheless, the condition (8) essentially requires λ r /( E + λ r+1 ) > 16 as E ′ ≤ E + U (r+1):κ U T (r+1):κ P = E + λ r+1 , which is stronger than the condition (10). In order to weaken the requirement on the spectral gap into (10), we study the contribution of U (r+1):κ U T (r+1):κ P towards to the leading r singular subspaces perturbation of E. It turns out that its contribution is roughly k 2 /(βn)λ r+1 instead of λ r+1 , due to the fact that U (r+1):κ U T (r+1):κ P has at most k unique columns with each one appearing at least βn/k times.
Spectral Clustering for Mixture Models.
3.1. Spectral Clustering and Polynomial Error Rate. Recall the definition of the mixture model in (5) and also in (6). The goal of clustering is to estimate the cluster assignment vector z * from the observations X 1 , X 2 , . . . , X n . Since the signal matrix P is of low rank, a natural idea is to project the observations {X i } i∈[n] onto a low dimensional space before applying classical clustering methods such as variants of k-means. This leads to the spectral clustering presented in Algorithm 1.
Algorithm 1: Spectral Clustering
Input: Data matrix X = (X 1 , . . . , Xn) ∈ R p×n , number of clusters k, number of singular vectors r Output: Cluster assignment vectorẑ ∈ [k] n 1 Perform SVD on X to have
X = p∧n i=1λ iûiv T i , whereλ 1 ≥λ 2 ≥ . . . ≥λ p∧n ≥ 0 and {û i } p∧n i=1 ∈ R p , {v i } p∧n i=1 ∈ R n . Let U 1:r := (û 1 , . . . ,ûr) ∈ R p×r .
2 Perform k-means on the columns ofÛ T 1:r X. That is,
ẑ, ĉ j j∈[k] = argmin z∈[k] n ,{c j } j∈[k] ∈R r i∈[n] Û T 1:r X i − cz i 2 .(13)
In (13), the dimensionality of each data pointÛ T 1:r X i is r, reduced from original dimensionality p. This is computationally appealing as r can be much smaller than p. The second step of Algorithm 1 is the k-means on the columns ofÛ T 1:r X, which is equivalent to performing k-means onto the columns ofÛ 1:rÛ T 1:r X ∈ R p×n . That is, defineθ a =Û 1:rĉa for each a ∈ [k]. It can be shown that (cf., Lemma 4.1 of [25])
ẑ, θ j j∈[k] = argmin z∈[k] n ,{θj} j∈[k] ∈R p i∈[n] Û 1:rÛ T 1:r X i − θ zi 2 ,(14)
due to the fact thatÛ 1:r has orthonormal columns. As a result, in the rest of the paper, we carry out our analysis onẑ using (14).
Before characterizing the theoretical performance of the spectral clusteringẑ, we give the definition of the misclustering error which quantifies the distance between an estimator and the ground truth z * . For any z ∈ [k] n , its misclustering error is defined as
ℓ(z, z * ) = min φ∈Φ 1 n i∈[n] I {z i = φ(z * i )}, where Φ = {φ : φ is a bijection from [k] to [k]}.
The minimization of Φ is due to that the cluster assignment vector z * is identifiable up to a permutation of the labels [k]. In addition to β that controls the smallest cluster size, another important quantity in this clustering task is the separation of the centers. Define ∆ to be the minimum distance among centers, i.e.,
∆ := min a,b∈[k]:a =b θ * a − θ * b .
As we will see later, ∆ determines the difficulty of the clustering task and plays a crucial role. In Proposition 3.1, we give a rough upper bound on the misclustering error ℓ(ẑ, z * ) that takes a polynomial expression (16) . It is worth mentioning that Proposition 3.1 is deterministic with no assumption on the distribution or the independence of the noises {ǫ i } i∈ [n] . In fact, the noise matrix E can be an arbitrary matrix as long as the data matrix has the decomposition (6) and the separation condition (15) is satisfied. In addition, it requires no spectral gap condition. Proposition 3.1 is essentially an extension of Lemma 4.2 in [25] which is only for the Gaussian mixture model and needs r = k. We include its proof in Appendix D for completeness. Recall κ is the rank of the signal matrix P . PROPOSITION 3.1. Consider the spectral clusteringẑ of Algorithm 1 with κ ≤ r ≤ k. Assume
ψ 0 := ∆ β −0.5 kn −0.5 E ≥ 16. (15) Then ℓ(ẑ, z * ) ≤ β/(2k). Furthermore, there exists one φ ∈ Φ such thatẑ satisfies ℓ(ẑ, z * ) = 1 n |{i ∈ [n] :ẑ i = φ(z * i )}| ≤ C 0 k E 2 n∆ 2 ,(16)and max a∈[k] θ φ(a) − θ * a ≤ C 0 β −0.5 kn −0.5 E ,(17)
where C 0 = 128. Proposition 3.1 provides a starting point for our further theoretical analysis. In the following sections, we are going to provide a sharper analysis for the spectral clusteringẑ that is beyond the polynomial rate stated in (16), with the help of singular subspaces perturbation established in Section 2.
Entrywise Error Decompositions.
In this section, we are going to develop a finegrained and entrywise analysis on the performance ofẑ. Proposition 3.1 points out that there exists a permutation φ ∈ Φ such that nℓ(ẑ, z * ) = |{i ∈ [n] :ẑ i = φ(z * i )}| ≤ nβ/(2k). Since the smallest cluster size in z * is at least βn/k, such permutation φ is unique. With φ identified,ẑ i = φ(z * i ) means that the ith data point X i is incorrectly clustered inẑ, for each i ∈ [n]. The following Lemma 3.2 studies the eventẑ i = φ(z * i ) and shows that it is determined by the magnitude of Û 1:rÛ
T 1:r ǫ i . LEMMA 3.1. Consider the spectral clusteringẑ of Algorithm 1 with κ ≤ r ≤ k. Assume (15) holds. Let φ ∈ Φ be the permutation such that ℓ(ẑ, z * ) = 1 n |{i ∈ [n] :ẑ i = φ(z * i )}|.
Then there exists a constant C > 0 such that for any i ∈ [n],
I {ẑ i = φ(z * i )} ≤ I 1 − Cψ −1 0 ∆ ≤ 2 Û 1:rÛ T 1:r ǫ i .(18)
To understand Lemma 3.1, recall that in (14)ẑ is obtained by k-means on {Û 1:rÛ
T 1:r X i } i∈[n] .
Since we have the decompositionÛ 1:rÛ
T 1:r X i =Û 1:rÛ T 1:r θ * z * i +Û 1:rÛ T 1:r ǫ i for each i ∈ [n],T 1:r X i is closer tô U 1:rÛ T 1:r θ * z *
i than any other centers, and thus z * i can be correctly recovered. Lemma 3.1 itself is not sufficient to obtain explicit expressions for the performance of spectral clustering when the noises {ǫ i } i∈[n] are assumed to be random. The entrywise upper bound (18) shows that the eventẑ i = φ(z * i ) is determined by the Û 1:rÛ T 1:r ǫ i , but the fact thatÛ 1:rÛ T 1:r depends on ǫ i makes any follow-up probability calculations challenging. The key to make use of Lemma 3.1 is our leave-one-out singular subspace perturbation theory, particularly, Theorem 2.2. To decouple the dependence betweenÛ 1:rÛ T 1:r and ǫ i , we replace the former quantity by its leave-one-out counterpartÛ −i,1:rÛ T −i,1:r . Take r to be κ. Note that
Û 1:κÛ T 1:κ ǫ i ≤ Û −i,1:κÛ T −i,1:κ ǫ i + U 1:κÛ T 1:κ −Û −i,1:κÛ T −i,1:κ F ǫ i .(19)
The perturbation U 1:κÛ (15)
T 1:κ −Û −i,1:κÛ T −i,1:κ Fhold. Let φ ∈ Φ be the permutation such that ℓ(ẑ, z * ) = 1 n |{i ∈ [n] :ẑ i = φ(z * i )}|.
Then there exists a constant C such that for any i ∈ [n],
I {ẑ i = φ(z * i )} ≤ I 1 − C ψ −1 0 + ρ −2 0 ∆ ≤ 2 Û −i,1:κÛ T −i,1:κ ǫ i .
Consequently, if the noises {ǫ i } i∈[n] are random, we have the risk ofẑ satisfy
Eℓ(ẑ, z * ) ≤ n −1 i∈[n] EI 1 − C ψ −1 0 + ρ −2 0 ∆ ≤ 2 Û −i,1:rÛ T −i,1:r ǫ i .
Lemma 3.2 needs three conditions. The first one βn/k 2 ≥ 10 is on the smallest cluster sizes and can be easily satisfied if both β, k are constants. The second condition (8) is a spectral gap condition on the smallest non-zero singular value λ κ . The third one is for the separation of the centers ∆. With all the three conditions satisfied, Lemma 3.2 shows that the entrywise clustering error for X i boils down to Û −i,1:κÛ T −i,1:κ ǫ i . When the noises {ǫ j } j∈ [n] are assumed to be random and independent of each other, the projection matrixÛ −i,1:κÛ
T −i,1:κ is independent of ǫ i for each i ∈ [n]
, a desired property crucial to our follow-up investigation on the risk Eℓ(ẑ, z * ). When {X i } i∈[n] are generated randomly as in the following sections, Lemma 3.2 leads to explicit expressions for the performance of the spectral clustering.
The key towards establishing Lemma 3.2 is Theorem 2.2. Without Theorem 2.2, if the classical perturbation theory such as Wedin's theorem is used instead, then in order to obtain similar upper bounds in Lemma 3.2, the second term on the RHS of (19) needs to be much smaller than ∆. This essentially requires max i∈[p] ǫ i 2 λ κ ∆, in addition to (8) and (15). As we will show in the next section, for sub-Gaussian noises, this additional condition requires p log n √ n in regimes where Lemma 3.2 only needs p n.
, we provide an upper bound for
ÛÛ T −Û −iÛ T −i showing that it is essentially deter- mined by U −i U T −i ǫ i under an eigen-gap condition.
3.3. Sub-Gaussian Mixture Models. In this section, we investigate the performance of the spectral clusteringẑ for mixture models with sub-Gaussian noises. Theorem 3.1 assumes that each noise ǫ i is an independent sub-Gaussian random vector with zero mean and variance proxy σ 2 and establishes an exponential rate for the risk Eℓ(ẑ, z * ). THEOREM 3.1. Consider the spectral clusteringẑ of Algorithm 1 with r = κ. Assume ǫ i ∼ SG p (σ 2 ) independently with zero mean for each i ∈ [n]. Assume βn/k 2 ≥ 10. There exist constants C, C ′ > 0 such that under the assumption that
ψ 1 := ∆ β −0.5 k 1 + p n σ > C (20)
and
ρ 1 := λ κ √ n + √ p σ > C,(21)
we have
Eℓ(ẑ, z * ) ≤ exp − 1 − C ′ ψ −1 1 + ρ −2 1 ∆ 2 8σ 2 + exp − n 2 .
Under this sub-Gaussian setting, standard concentration theory shows that the noise matrix E has its operator norm E σ( √ n + √ p) with high probability (cf. Lemma D.1). Under this event, (20) and (21) are sufficient conditions for (8) and (15), respectively. The risk in Theorem 3.3 has two terms, where the first term takes an exponential form of ∆ 2 /(8σ 2 ) and the second term exp(−n/2) comes from the aforementioned event of E . The first term is the dominating one, as long as ∆ 2 /σ 2 , which can be interpreted as the signal-to-noise ratio, is smaller than n/2. In fact, ∆ 2 /σ 2 log n is the most interesting regime as otherwiseẑ already achieves the exact recovery (i.e.,ẑ = z * ) with high probability, since E{ℓ(ẑ, z * ) = 0} = o(1). Theorem 3.1 makes a substantial improvement over Proposition 3.1. Using the aforementioned with-high-probability event on E , (16) only leads to Eℓ(ẑ, z * ) (1 + p/n) 2 σ 2 /∆ 2 + exp (−n/2) which takes a polynomial form of the ∆ 2 /σ 2 . On the contrary, Theorem 3.1 provides a much sharper exponential rate.
Our leave-one-out singular subspace perturbation theory and its consequence Lemma 3.2 provide the key toolkit towards Theorem 3.1. SinceÛ T −i,1:κ is independent of ǫ i , we havê U T −i,1:κ ǫ i ∼ SG κ (σ 2 ) being another sub-Gaussian random vector. This makes it possible to control the tail probabilities of Û −i,1:κÛ
T −i,1:κ ǫ i 2 = Û T −i,1:κ ǫ i 2
which is a quadratic form of sub-Gaussian random vectors. Without using our perturbation theory, if the classical perturbation bounds such as Wedin's Theorem is used instead, the previous section shows that max i∈[p] ǫ i 2 λ κ ∆ is additionally needed to obtain results similar to Lemma 3.2. This equivalently requires λ κ ∆/(σ 2 p log n) 1. When ∆/σ, k, β are constants, this additional condition essentially requires p log n √ n. In contrast, Theorem 3.1 only needs p n.
Theorem 3.1 gives a finite-sample result for the performance of spectral clustering in sub-Gaussian mixture models. In the following Corollary 3.1, by slightly strengthening conditions (20) and (21), it immediately yields an asymptotic error bound with the exponent being
(1 − o(1))∆ 2 /(8σ 2 ).1, if ψ 1 , ρ 1 → ∞ is further assumed, we have Eℓ(ẑ, z * ) ≤ exp − (1 − o(1)) ∆ 2 8σ 2 + exp − n 2 .
If ∆/σ ≥ (1 + c)2 √ 2 log n is further assumed where c > 0 is any constant,ẑ achieves the exact recovery, i.e., EI {ℓ(ẑ, z * ) = 0} = o(1).
In the exponents of Theorem 3.1 and Corollary 3.1, we are able to obtain an explicit constant 1/8. In addition, we obtain an explicit constant 2 √ 2 for the exact recovery in Corollary 3.1. These constants are sharp when the noises are further assumed to be isotropic Gaussian, as we will show in Section 3.5.
The recent related paper by [1] develops a ℓ p perturbation theory and applies it to the spectral clustering for sub-Gaussian mixture models. It obtains exponential error rates but with unspecified constants in the exponents and under special assumptions on the spectrum and geometric distribution of the centers. It first assumes both β and k are constants. Let G ∈ R k×k be the Gram matrix of the centers such that
G i,j = θ * T i θ * j for each i, j ∈ [k]
. It requiresλI ≺ G ≺ cλI for some constant c > 1, i.e., all k eigenvalues of G are in the same order. It implies that the maximum and minimum distances among centers are comparable. This rules out many interesting cases such as all the centers are on one single line. In addition, [1] needsλ/σ → ∞. Equivalently it means that the leading k singular values λ 1 , λ 2 , . . . , λ k of the signal matrix P not only are all in the same order, but also λ k /( √ nσ) ≫ max{1, p/n}. As a comparison, we allow collinearity of the centers such that the rank of G (and P ) can be smaller than k. We allow the singular values λ 1 , λ 2 , . . . , λ κ not in the same order as long as the smallest one satisfies (21), which can be equivalently written as λ κ /( √ nσ) max{1, p/n}. The distances among the centers are also not necessarily in the same order as long as the smallest distance satisfies (20). Hence, our conditions are more general than those in [1]. The spectral gap condition (21) ensures that singular vectors corresponding to small nonzero singular values are well-behaved. It is not needed in Section 3.4 where we propose a variant of spectral clustering with adaptive dimension reduction. It can also be dropped in Section 3.5 when the noise is isotropic Gaussian. When the mixture model is symmetric with two components (for example, the model considered in Section 3.6), the signal matrix P is rank-one. Hence, (21) is also no longer needed as it can be directly implied from (20).
3.4.
Spectral Clustering with Adaptive Dimension Reduction. The theoretical analysis for the spectral clusteringẑ of Algorithm 1 that is carried out in Lemma 3.2 and Theorem 3.1 requires the use of all the κ singular vectors where κ is the rank of the signal matrix P . Nevertheless, not all singular components are equally useful towards the clustering task and the importance of an individual singular vector can be characterized by its corresponding singular value. This motivates us to propose the following algorithm where the number of singular vectors used is carefully picked.
Algorithm 2: Spectral Clustering with Adaptive Dimension Reduction
Input: Data matrix X = (X 1 , . . . , Xn) ∈ R p×n , number of clusters k, threshold T Output: Clustering label vectorz ∈ [k] n 1 Perform SVD on X same as Step 1 of Algorithm 1. 2 Letr be the largest index in [k] such that the difference between the corresponding two neighboring singular values is greater than T , i.e.,
r = max{a ∈ [k] :λa −λ a+1 ≥ T }.(22)
LetÛ 1:r := (û 1 , . . . ,ûr) ∈ R p×r .
3 Perform k-means on the columns ofÛ T 1:r X. That is,
z, c j k j=1 = argmin z∈[k] n ,{c j } k j=1 ∈Rr i∈[n] Û T 1:r X i − cz i 2 .(23)
Algorithm 2 is a variant of Algorithm 1 with the number of singular vectors selected by (22), wherer is the largest integer such that the empirical spectral gapλr −λr +1 is greater or equal to some threshold T . The choice of the threshold T matters. When T is small,r might be even bigger than the rank κ. When T E , it guarantees that the singular values of the signal matrix P satisfy λr − λr +1 T and λr +1 T . When T is too large, the singular subspaceÛ 1:r misses singular vectors such asûr +1 whose importance scales with λr +1 that can not be ignored. This in turn deteriorates the clustering performance ofz.
A rule of thumb for the threshold T is that T / E is at least in a constant order. It is allowed to grow but not faster thanφ 0 defined in (24). The precise description of the choices of T needed is given below in Lemma 3.3, which provides an entrywise analysis ofz that is analogous to Lemma 3.2. LEMMA 3.3. Consider the estimatorz from Algorithm 2. Assume βn/k 4 ≥ 400. Let φ ∈ Φ be the permutation such that ℓ(ẑ, z * ) = 1 n |{i ∈ [n] :
ẑ i = φ(z * i )}|. Definẽ ψ 0 := ∆ β −0.5 k 2 n −0.5 E(24)
andρ := T / E . Assume 256 <ρ <ψ 0 /64. There exist constants C, C ′ such that ifφ 0 > C, then
I {ẑ i = φ(z * i )} ≤ I 1 − C ′ ρψ −1 0 +ρ −1 ∆ ≤ 2 Û −i,1:rÛ T −i,1:r ǫ i .
Consequently, we have
Eℓ(ẑ, z * ) ≤ n −1 i∈[n] EI 1 − C ′ ρψ −1 0 +ρ −1 ∆ ≤ 2 Û −i,1:rÛ T −i,1:r ǫ i .
With a proper choice of the threshold T , Lemma 3.3 only poses requirements on the smallest cluster size βn/k and minimum separation among the centers ∆. Compared to Lemma 3.2 and Theorem 3.1, it removes any condition on the smallest non-zero singular value such as (8) or (21). In addition, it requires no knowledge on the rank κ.
With Lemma 3.3, we have the following exponential error bound on the performance ofz on sub-Gaussian mixture models, analogous to Theorem 3.1 and Corollary 3.1 forẑ. THEOREM 3.2. Consider the estimatorz from Algorithm 2. Assume ǫ i ∼ SG p (σ 2 ) independently with zero mean for each i ∈ [n]. Assume βn/k 4 ≥ 400. There exist constants C, C ′ , C 1 , C 2 > 0 such that under the assumption that
ψ 2 := ∆ β −0.5 k 2 1 + p n σ > C and ρ 2 := T /(σ( √ n + √ p)) satisfies C 1 ≤ ρ 2 ≤ ψ 2 /C 2 , we have Eℓ(z, z * ) ≤ exp − 1 − C ′ ρ 2 ψ −1 2 + ρ −1 2 ∆ 2 8σ 2 + exp − n 2 .
If ψ 2 , ρ 2 → ∞ and ρ 2 /ψ 2 = o(1) are further assumed, we have
Eℓ(z, z * ) ≤ exp − (1 − o(1)) ∆ 2 8σ 2 + exp − n 2 .
3.5. Isotropic Gaussian Mixture Models. In this section, we consider the isotropic Gaussian mixture models where the noises are sampled from N (0, σ 2 I p ) independently. As a special case of the sub-Gaussian mixture models, Theorem 3.1 can be directly applied. Nevertheless, the isotropic Gaussian noises make it possible to remove the spectral gap condition (21). In addition, we study the performance of the spectral clusteringẑ from Algorithm 1 with exactly the leading k singular vectors, regardless of κ the rank of matrix P . As a result, it requires no knowledge on κ and needs no adaptive dimension reduction such as Algorithm 2. We have the following theorem on its performance.
We have
Eℓ(ẑ, z * ) ≤ exp − 1 − C ∆ k 3.5 β −0.5 1 + p n σ −0.25 ∆ 2 8σ 2 + 2e −0.08n ,(26)
where C > 0 is some constant. (1))∆ 2 /(8σ 2 )) + 2 exp (−0.08n) where the first term dominates when ∆ 2 /σ 2 = o(n). The minmax lower bound for recovering z * under the given model is established in [26]: (1))∆ 2 /(8σ 2 )) as long as ∆ 2 /σ 2 ≫ log(kβ −1 ). This immediately implies that the considered estimator is minimax optimal. Theorem 3.3 also impliesẑ achieves the exact recovery E{ℓ(ẑ, z * ) = 0} = o(1) when ∆/σ ≥ (1 + c)2 √ 2 log n for any small constant c > 0. When ∆/σ ≤ (1 − c)2 √ 2 log n, no algorithm is able to recover z * exactly with high probability according to the minimax lower bound.
Theorem 3.3 shows that asymptotically
Eℓ(ẑ, z * ) ≤ exp(−(1 − oinfẑ sup (θ * 1 ,...,θ * k ),z * Eℓ(ẑ, z * ) ≥ exp(−(1 + o
It is worth mentioning that Theorem 3.3 requires no spectral gap condition such as (8) or (21). The purpose of such conditions is to ensure that singular vectors of X are well controlled, especially those corresponding to small non-zero singular values of the signal matrix P . When the noises are isotropic Gaussian, the distribution of each right singular vectorv j is well-behaved for any j ∈ [p ∧ n]. Lemma 4.4 of [25] shows that each (I − V 1:κ V T 1:κ )v j is Haar distributed on the sphere spanned by (I − V 1:κ V T 1:κ ), where V 1:κ := (v 1 , v 2 , . . . , v κ ) ∈ O n×κ is the right singular subspace of the signal matrix P . Theorem 3.3 is about the singular sub-spaceÛ 1:k . In its proof, we decompose it intoÛ 1:r andÛ (r+1):k , for some index r ∈ [κ] with sufficient large spectral gap λ r − λ r+1 so that the contribution ofÛ 1:r can be precisely quantified following similar arguments used to establish Lemma 3.3 and Theorem 3.1. The contribution of eachû j where j ∈ {r + 1, . . . , k} is eventually connected with properties of the corresponding right singular vectorv j , particularly, the distribution of (I − V 1:κ V T 1:κ )v j . These two sources of errors together lead to the upper bound (26).
The performance of Algorithm 1 with r = k under the same isotropic Guassian mixture model is the main topic of [25] which derives a similar upper bound for Eℓ(ẑ, z * ) assuming ∆/(β −0.5 k 10.5 (1 + p/n)) → ∞. The key technical tool used in [25] is spectral operator perturbation theory of [20,21] on the difference between empirical singular subspaces and population ones, which works for the Gaussian noise case and it is not clear whether it is possible to be extended to other distributions including sub-Gaussian distributions. In this paper, the proof of Theorem 3.3 is completely different, using Theorem 2.3 on the difference between empirical singular subspaces and their leave-one-out counterparts. We not only recover the main result of [25] with a much shorter proof, but also improve the dependence of k. Despite that Theorem 3.3 needs an extra condition βn/k 4 ≥ 100, it only requires k 3.5 to satisfy (25), while [25] needs k 10.5 instead which is a stronger condition.
3.6. Lower Bounds and Sub-optimality of Spectral Clustering. In the above sections, we focus on quantifying the performance of spectral clustering under mixture models. An interesting question is whether the spectral clustering is optimal or not. When the noise is the isotropic Gaussian, Theorem 3.3 matches with the minimax rate assuming (25) holds, showing that the spectral clustering is indeed optimal in this case. It remains unclear whether the spectral clustering is optimal or not when the noise is beyond the isotropic Gaussian model.
To answer this question, in this section we consider a two-cluster symmetric mixture model whether the centers are proportional to 1 p and the noises have i.i.d. entries. This setup makes it possible to apply the central limit theorem to characterize the performance of the spectral clustering with sharp upper and lower bounds, as 1 T p ǫ i is asymptotically normal for each i ∈ [n] when p is large.
A Two-cluster Symmetric Mixture Model. Consider a mixture model (5) with two clusters such that
θ * 1 = −θ * 2 = δ1 p , and {ǫ i,j } i∈[n],j∈[p] iid ∼ F,(27)
for some δ ∈ R and some distribution F , where {ǫ i,j } j∈[p] are entries of ǫ i for each i ∈ [n].
Under the above model (27), we have k = 2, ∆ = 2 √ pδ and the largest singular value λ 1 = δ √ np. Since the signal matrix matrix P is rank-one (i.e., κ = 1) with u 1 = (1/ √ p)1 p , a natural idea is to cluster using the first singular vector only. Define
ž, {č j } 2 j=1 = argmin z∈[2] n ,{cj} 2 j=1 ∈R i∈[n] û T 1 X i − c zi 2 .(28)
The performance of the spectral estimatorž will be the focus in this section. Note thatû T 1 X = λ 1v T 1 wherev 1 is the leading right singular vector of X, sož equivalently performs clustering on {v 1,i } i∈[n] , the entries ofv 1 . This is closely related to the sign estimator {sign(v 1,i )} i∈[n] , which estimates the cluster assignment by the signs of {v 1,i } i∈ [n] .
Sincež is exactly the spectral clusteringẑ of Algorithm 1 with r = 1, Theorem 3.1 can be directly applied when noises are sub-Gaussian and yields the following result. Under the model (27), assume that F is a SG(σ 2 ) distribution with zero mean and βn > 40. There exist constants C, C ′ > 0 such that under the assumption that
ψ 3 := ∆ β −0.5 1 + p n σ > C, we have Eℓ(ž, z * ) ≤ exp(−(1 − C ′ ψ −1 3 )∆ 2 /(8σ 2 )
) + exp(−n/2). The special structure of (27) makes it possible to derive a sharper upper bound than the above one and a matching lower bound on the performance ofž with some additional assumption on the distribution F . Instead of directly using Lemma 3.2 (which leads to Theorem 3.1 and then the above upper bound), we can further connect the clustering error with u T 1 ǫ i where u T 1 ǫ i = p −1/2 p j=1 ǫ i,j is approximately normally distributed when p is large. On the other hand, the structure of (27) enables us to have a lower bound for I {ẑ i = φ(z * i )} that is in an opposite direction of Lemma 3.2. See Lemma 5.1 for details. The key technical tool used is Theorem 2.2 on the perturbation |û 1û
T 1 −û −i,1û T −i,1 | for all i ∈ [n]
. These together give a sharp and matching lower bound for Eℓ(ž, z * ) where the clustering error is essentially determined by ∆ and the varianceσ 2 . THEOREM 3.4. Consider the model (27). For any ξ ∼ F , assume Eξ = 0, Var(ξ) =σ 2 , and ξ ∼ SG(σ 2 ) where σ ≤ Cσ for some constant C > 0. Assume βn > 40. Then there exist constants C ′ , C ′′ , C ′′′ > 0 such that if ψ 3 ≥ C ′ , we have
Eℓ(ž, z * ) ≤ exp − 1 − C ′′ ψ −1 3 2 ∆ 2 8σ 2 + exp −C ′′ √ p + exp − n 2 , and Eℓ(ž, z * ) ≥ exp − 1 + C ′′′ ψ −1 3 2 ∆ 2 8σ 2 − exp −C ′′′ √ p − exp − n 2 .
In Theorem 3.4, the term exp(−C ′′ √ p) is due to the normal approximation of u T 1 ǫ i and decays when the dimensionality p increases. The term exp(−n/2) is due to a with-highprobability event on E . If additionally ∆/σ ≪ max{p 1/4 , n 1/2 } is assumed, Theorem 3.4 concludes asymptotically
Eℓ(ž, z * ) = exp − (1 + c)∆ 2 8σ 2 ,(29)
for some small constant c.
The upper and lower bounds in Theorem 3.4 give a sharp characterization on the performance ofž. To answer the question of whether it is optimal or not, we need to establish the minimax rate for the clustering task under the model (27). Since the model (27) is essentially about a testing between two parametric distributions, the optimal procedure is the likelihood ratio test. According to the classical asymptotics theory [37], the likelihood ratio behaves like a normal random variable as p → ∞ under some regularity condition. This leads to an error rate determined by ∆ and the Fisher information. LEMMA 3.4. Consider the model (27). Assume the distribution F has a positive, continuously differentiable density f with mean zero and finite Fisher information I :=
(f ′ /f ) 2 f dx. Assume ∆ is a constant. We have
C 1 exp − ∆ 2 8I −1 ≤ lim p→∞ inf z sup z * ∈[2] n Eℓ(z, z * ) ≤ C 2 exp − ∆ 2 8I −1 ,(30)
for some constants C 1 , C 2 > 0.
With Lemma 3.4, the question of whetherž is optimal or not boils down to a comparison of the varianceσ 2 and the inverse of the Fisher information I −1 . Due to the fact that I −1 ≤ σ 2 and the equation holds if and only if F is a normal distribution, we have the following conclusion. THEOREM 3.5. Consider the model (27). Assume all the assumptions needed in Theorem 3.4 and Lemma 3.4 hold. Then the spectral clusteringž is in general suboptimal, i.e., it fails to achieve the minimax rate (30). It is optimal if and only if the noise distribution F is N (0,σ 2 ). Theorem 3.5 establishes the sub-optimality of the spectral clusteringž under the model (27). Thoughž achieves an exponential error rate, it has a fundamentally sub-optimal exponent involvingσ 2 instead of I −1 . This is due to the factž clusters data points based on Euclidean distances while the optimal procedure is the likelihood ratio test. Only when the noise is normally distributed, the likelihood ratio test is equivalent to a comparison of two Euclidean distances, leading to the optimality ofž in the Gaussian case. Despite that Theorem 3.5 is only limited to the model (27), the above reasoning suggests the spectral clustering is generally sub-optimal under mixture models beyond (27)
− σ 2 r+1 − (I − U r U T r )y n 2 > 0 instead of assuming ρ > 2.
We defer the proof of Theorem 2.1 to the end of this section, which is an immediate consequence of Theorem 4.1.
THEOREM 4.1. If σ 2 r − σ 2 r+1 − (I − U r U T r )y n 2 > 0, we have Û rÛ T r − U r U T r F ≤ 2 √ 2σ r (I − U r U T r )y n σ 2 r − σ 2 r+1 − (I − U r U T r )y n 2 r i=1 u T i y n σ i 2 .
PROOF. Decompose y n into y n = θ + ǫ with θ := U r U T r y n and ǫ := (I − U r U T r )y n . Then we have u T i θ = u T i y n for each i ∈ [r]. Throughout the proof, we denote
α 2 = Û rÛ T r − U r U T r 2 F .
Denote d = p ∧ (n − 1). If p ≤ n − 1, we have d = p and denote U := (u 1 , . . . , u p ) ∈ R p×p which is an orthogonal matrix. If p > n − 1, we let U ∈ R p×p be an orthogonal matrix with the first p ∧ (n − 1) columns being u 1 , . . . , u p∧(n−1) . In both cases, we have U being an orthogonal matrix. ThenÛ r can be written asÛ r = UB for someB = (B i,j ) ∈ R p×r . LetB i,· be the ith row ofB for each
i ∈ [p]. Define b 2 i = 1 − B i,· 2 for each i ∈ [r] and b 2 i = B i,· 2
for each i > r. Then we have
α 2 = Û rÛ T r 2 F + U r U T r 2 F − 2 Û rÛ T r , U r U T r = 2k − 2 U T rÛ r 2 F = 2k − 2 i∈[r] j∈[r]B 2 i,j = 2 i∈[r] b 2 i = 2 p i=r+1 b 2 i ,(31)
where in the last equation we use the fact that B 2 F = r. Note thatÛ rÛ T rŶ is the best rank-r approximation ofŶ . We have
I −Û rÛ T r Ŷ 2 F ≤ I − U r U T r Ŷ 2 F .
Due to the factŶ = (Y, y n ), we have
I −Û rÛ T r Y 2 F + I −Û rÛ T r y n 2 ≤ I − U r U T r Y 2 F + I − U r U T r y n 2 ,
which implies
I −Û rÛ T r Y 2 F − I − U r U T r Y 2 F ≤ I − U r U T r y n 2 − I −Û rÛ T r y n 2 .(32)
We are going to simplify terms in (32).
(Simplification of the LHS of (32)).
Recall the decomposition Y = i∈[d] σ i u i v T i . Since I − U r U T r Y = d i>r σ i u i v T i , we have I − U r U T r Y 2 F = d i>r σ 2 i . Since U T Y = U T i∈[d] σ i u i v T i = σ 1 v T 1 . . . σ d v T d 0 p−d = diag(σ 1 , . . . , σ d , 0 p−d ) v T 1 . . . v T d O (p−d)×n ,
we have
I −Û rÛ T r Y 2 F = U I − U TÛ rÛ T r U U T Y 2 F = I −BB T diag(σ 1 , . . . , σ d , 0 p−d ) v T 1 . . . v T d O (p−d)×n 2 F = tr diag(σ 1 , . . . , σ d , 0 p−d ) I −BB T diag(σ 1 , . . . , σ d , 0 p−d ) I d×d O (p−d)×(p−d) ,
where in the last equation we use the following facts: (1) for any two square matrices of the same size A, D, we have AD
I −Û rÛ T r Y 2 F = tr diag(σ 1 , . . . , σ d , 0 p−d ) I −BB T diag(σ 1 , . . . , σ d , 0 p−d ) = d i=1 σ 2 i 1 − B i,· 2 F . Then we have LHS of (32) = r i=1 σ 2 i 1 − B i,· 2 F − d i>r σ 2 i B i,· 2 F = r i=1 σ 2 i b 2 i − d i>r σ 2 i b 2 i ≥ r i=1 σ 2 i b 2 i − σ 2 r+1 α 2 2 ,
where we use d i>r b 2 i ≤ p i>r b 2 i = α 2 /2 from (31) in the last inequality .
(Simplification of the RHS of (32)). Recall thatÛ r = UB. We decompose it intoB = (B T 1 ,B T 2 ) T whereB 1 ∈ R r×r are the first r rows andB 2 ∈ R (p−r)×r . We have RHS of (32) = y T n I − U r U T r y n − y T n I −Û rÛ T r
y n = y T n Û rÛ T r − U r U T r y n = y T n U B 1B T 1 − I r×rB1B T 2 B 2B T 1B 2B T 2 U T y n .
DefineB ⊥ ∈ R p×(p−r) to be the matrix such that (B,B ⊥ ) ∈ R p×p is an orthonormal matrix. We can further decompose it intoB ⊥ = (B ⊥T 1 ,B ⊥Y 2 ) T whereB ⊥ 1 ∈ R r×(p−r) including the first r rows andB ⊥ 2 ∈ R (p−r)×(p−r) . Since (B,B ⊥ ) has orthogonal columns, we have
(B 1 ,B ⊥ 1 )(B 1 ,B ⊥ 1 ) T =B 1B T 1 +B ⊥ 1B ⊥T 1 = I r×r , and (B 1 ,B ⊥ 1 )(B 2 ,B ⊥ 2 ) T = O r×(p−r) , which implieŝ B 1B T 2 = −B ⊥ 1B ⊥T 2 .
We also decompose the matrix U =: (U r , U ⊥ ). Then (31). We also have
RHS of (32) = y T n (U r , U ⊥ ) −B ⊥ 1B ⊥T 1 −B ⊥ 1B ⊥T 2 −B ⊥ 2B ⊥T 1B 2B T 2 (U r , U ⊥ ) T y n = −y T n U rB ⊥ 1B ⊥T 1 U T r y n − 2y T n U rB ⊥ 1B ⊥T 2 U T ⊥ y n + y T n U ⊥B2B T 2 U T ⊥ y n ≤ − B ⊥T 1 U T r y n 2 + 2 B ⊥T 1 U T r y n B ⊥T 2 U T ⊥ y n + B T 2 2 U T ⊥ y n 2 . Note that B ⊥T 2 ≤ 1 and B T 2 2 ≤ B T 2 2 F = p i>r B i,· 2 = α 2 /2 which is byU T ⊥ y n = ǫ . Since B ⊥ 1 2 F = r i=1 1 − B i,· 2 = α 2 /2 according to (31), we have B ⊥ 1 ≤ α/ √ 2. Thus, using U T r ǫ = 0, we have B ⊥T 1 U T r y n = B ⊥T 1 U T r θ . Then, RHS of (32) ≤ 2 B ⊥T 1 U T r θ ǫ + α 2 2 ǫ 2 .
To simplify B ⊥T 1 U T r θ , denote w i = u T i θ and s i = |w i | /σ i for each i ∈ [r]. Recall that u T i θ = u T i y n for each i ∈ [r]. We have
s i = u T i y n σ i , ∀i ∈ [r].
We then have
B ⊥T 1 U T r θ = r i=1 w iB ⊥ i,· ≤ r i=1 |w i | B ⊥ i,· = r i=1 s i σ i |b i | ≤ s r i=1 σ 2 i b 2 i ,
where we denote the ith row ofB ⊥ 1 asB ⊥ i,· and we use the fact that B ⊥ i,·
2 = 1 − B i,· 2 = b 2 i for each i ∈ [r]
. As a result,
RHS of (32) ≤ 2 s r i=1 σ 2 i b 2 i ǫ + α 2 2 ǫ 2 .
(Combining the above simplifications for (32)). From the above simplifications on the LHS and RHS of (32), we have
r i=1 σ 2 i b 2 i − σ 2 r+1 α 2 2 ≤ 2 s r i=1 σ 2 i b 2 i ǫ + α 2 2 ǫ 2 . Define t = r i=1 σ 2 i b 2 i .
Then after arrangement, the above display becomes
t 2 − 2 s ǫ t ≤ σ 2 r+1 α 2 2 + α 2 2 ǫ 2 .
Note that the function t 2 − 2 s ǫ t is increasing as long as t ≥ t 0 where we define t 0 := s ǫ . On the other hand, from (31), we have the domain t ≥ ασ r / √ 2. We consider the following two scenarios.
If
ασ r / √ 2 ≤ t 0 , we have α ≤ √ 2t 0 σ r = √ 2 s ǫ σ r . (33) If ασ r / √ 2 > t 0 , we have t 2 − 2 s t ≥ α 2 σ 2 r 2 − √ 2 s ǫ ασ r .
Hence, we have an inequality of α:
α 2 σ 2 r 2 − √ 2 s ǫ ασ r ≤ σ 2 r+1 α 2 2 + α 2 2 ǫ 2 ,
which can be arranged into
α 2 σ 2 r − σ 2 r+1 − ǫ 2 ≤ √ 2 s σ r ǫ .
Hence, under the assumption σ 2
r − σ 2 r+1 − ǫ 2 > 0, we have α ≤ 2 √ 2σ r s ǫ σ 2 r − σ 2 r+1 − ǫ 2 .(34)
Since 2σ 2 r > σ 2 r − σ 2 r+1 − ǫ 2 , the upper bound in (33) is strictly below that in (34). Hence, (34) holds for both scenarios. The proof is complete. PROOF OF THEOREM 2.1. Since we assume ρ > 2, we have
σ 2 r − σ 2 r+1 − (I − U r U T r )ǫ i 2 ≥ σ r (σ r − σ r+1 ) − (σ r − σ r+1 ) 2 /4 ≥ σ r (σ r − σ r+1 )/2 = ρσ r (I − U r U T r )ǫ i /2.ρ −i :=λ −i,κ −λ −i,κ+1 I −Û −i,1:κÛ T −i,1:κ X i .
We need to show ρ −i > 2. In the following, we provide a lower bound for the numerator
λ −i,κ −λ −i,κ+1 .
Define λ −i,1 ≥ λ −i,2 ≥ . . . ≥ λ −i,p∧(n−1) to be singular values of P −i , the leave-one-out counterpart of the signal matrix P where
P −i := (θ * z * 1 , . . . , θ * z * i−1 , θ * z * i+1 , . . . , θ * z * n ) ∈ R p×(n−1) .(35)
We are interested in the value of λ −i,κ . Recall that λ κ is the κth largest singular value of P which is rank-κ. Since P has k unique columns {θ * a } a∈[k] , its left singular vectors u j ∈ Θ for each j ∈ [k] where Θ := span({θ * a } a∈ [k] ). Note that each θ * a appears at least βn/k times in the columns of P . Then P −i also has these k unique columns with each appearing at least βn/k − 1 times. This concludes that P −i has the same leading left singular vector space as P . We then have
λ 2 −i,κ = min w∈Θ: w =1 w T P −i 2 = min w∈Θ: w =1 j∈[n]:j =i (w T θ * z * j ) 2 ≥ βn k − 1 βn k min w∈Θ: w =1 j∈[n] (w T θ * z * j ) 2 = 1 − k βn min w∈Θ: w =1 w T P 2 ≥ 1 − k βn λ 2 κ .(36)
We also have λ −i,κ+1 = 0 as P −i is rank-κ.
Next, we are going to analyzeλ −i,κ andλ −i,κ+1 , the κth and (κ + 1)th largest singular values of X −i . Recall the SVD of X −i in Section 2.2. Define
E −i := (ǫ 1 , . . . , ǫ i−1 , ǫ i+1 , . . . , ǫ n ) ∈ R p×(n−1) ,(37)
so that X −i = P −i +E −i . By Weyl's inequality, we have
|λ −i,κ −λ −i,κ |, |λ −i,κ+1 −λ −i,κ+1 | ≤ E −i ≤ E . Then we havê λ −i,κ ≥ λ −i,κ − E ≥ 1 − k βn λ κ − E (38) andλ −i,κ −λ −i,κ+1 ≥ λ −i,κ − λ −i,κ+1 − 2 E ≥ 1 − k βn λ κ − 2 E .(39)
Next, we study (I −Û −i,1:κÛ
T −i,1:κ )X i . SinceÛ −i,1:κÛ T −i,1:κ X −i is the best rank-κ ap- proximation of X −i , we have Û −i,1:κÛ T −i,1:κ X −i − X −i ≤ P −i − X −i = E −i ,
where we use the fact that P −i is rank-κ. Then by the triangle inequality, we have
I −Û −i,1:κÛ T −i,1:κ P −i = Û −i,1:κÛ T −i,1:κ P −i − P −i ≤ Û −i,1:κÛ T −i,1:κ (P −i − X −i ) + Û −i,1:κÛ T −i,1:κ X −i − X −i + X −i − P −i ≤ 3 E −i .
Using the fact P −i is rank-κ again, we have
I −Û −i,1:κÛ T −i,1:κ P −i F ≤ √ κ I −Û −i,1:κÛ T −i,1:κ P −i ≤ 3 √ κ E −i ≤ 3 √ κ E .
Since P −i has at least βn/k − 1 columns being exactly θ * z * i , we have
I −Û −i,1:κÛ T −i,1:κ θ * z * i ≤ I −Û −i,1:κÛ T −i,1:κ P −i F βn k − 1 ≤ 3 √ κ E βn k − 1 ,(40)
and consequently,
I −Û −i,1:κÛ T −i,1:κ X i ≤ I −Û −i,1:κÛ T −i,1:κ θ * z * i + I −Û −i,1:κÛ T −i,1:κ ǫ i ≤ 3 √ κ E βn k − 1 + E .(41)
From (39) and (41), we have
ρ −i ≥ 1 − k βn λ κ − 2 E E + 3 √ κ E √ βn k −1 ≥ ρ 0 8 > 2,(42)
where the last inequality is due to the assumption ρ 0 > 16 and βn/k 2 ≥ 10.
The next thing to do is to study {û T −i,a X i } a∈ [κ] . Denote the columns of P −i and E −i as
{(P −i ) ·,j } j∈[n−1] and {(E −i ) ·,j } j∈[n−1] , respectively. Define S := {j ∈ [n − 1] : (P −i ) ·,j = θ * z * i }.
Then for any a ∈ [κ], by the SVD of X −i , we havê
u T −i,a θ * z * i = 1 |S| j∈Sû T −i,a (P −i ) ·,j = 1 |S| j∈Sû T −i,a (X −i ) ·,j + 1 |S| j∈Sû T −i,a (E −i ) ·,j = 1 |S| j∈Sλ −i,a (v −i,a ) j + 1 |S|û T −i,a j∈S (E −i ) ·,j .
Hence, by Cauchy-Schwarz inequality and the fact that v −i,a = 1, we have
û T −i,a θ * z * i ≤λ −i,a |S| |S| + |S| E −i |S| ≤λ −i,a βn k − 1 + E βn k − 1 . (43) Since |û T −i,a X i | ≤ |û T −i,a θ * z * i | + |û T −i,a ǫ i |, we have |û T −i,a X i | λ −i,a ≤ 1 βn k − 1 + 1 λ −i,a E βn k − 1 + |û T −i,a ǫ i | ≤ 1 βn k − 1 + 1 λ −i,κ E βn k − 1 + 1 λ −i,κ |û T −i,a ǫ i |.
Consequently,
a∈κ û T −i,a X î λ −i,a 2 ≤ √ κ βn k − 1 + 1 λ −i,κ E √ κ βn k − 1 + 1 λ −i,κ Û −i,1:κÛ T −i,1:κ ǫ i ,
where we use the fact Û −i,1:κÛ
T −i,1:κ ǫ i = Û T −i,1:κ ǫ i = ( i∈[κ] (û T −i,a ǫ i ) 2 ) 1/2 .
Lastly, by Theorem 2.1, we have
Û 1:κÛ T 1:κ −Û −i,1:κÛ T −i,1:κ F ≤ 4 √ 2 ρ −i √ κ βn/k − 1 + 1 λ −i,κ √ κ E βn/k − 1 + Û −i,1:κÛ T −i,1:κ ǫ i .
Since βn/k 2 ≥ 10 and ρ 0 > 16 are assumed, we haveλ −i,κ ≥ λ κ /2 by (38). Then together with (42), the above display can be simplified into
Û 1:κÛ T 1:κ −Û −i,1:κÛ T −i,1:κ F ≤ 32 √ 2 ρ 0 2 √ kκ √ βn + 2 Û −i,1:κÛ T −i,1:κ ǫ i λ κ ≤ 128 ρ 0 √ kκ √ βn + Û −i,1:κÛ T −i,1:κ ǫ i λ κ .
This concludes the proof of Theorem 2.2.
Proof of Main Results in Section 3.
In this section, we include proofs of main results in Section 3 except Lemma 3.3, Theorem 3.2, and Theorem 3.3. Their proofs are included in the supplement [42] due to page limit.
Proof of Lemma 3.1 and Lemma 3.2.
PROOF OF LEMMA 3.1. For simplicity, we denoteÛ to be short forÛ 1:r throughout the proof. From (14), we knowẑ i must satisfŷ
z i = argmin a∈[k] ÛÛ T X i −θ a ,
where {θ a } a∈[k] satisfies (17) according to Proposition 3.1. Hence, we have
I {ẑ i = φ(z * i )} = I min a∈[k]:a =φ(z * i ) ÛÛ T X i −θ a ≤ ÛÛ T X i −θ φ(z * i )
.
Consider a fixed a ∈ [k] such that a = φ(z * i ). Note that for any vectors x, y, w of same dimension, if x − y ≤ x − w , then we must have y − w /2 ≤ x − w . Hence, we have
I ÛÛ T X i −θ a ≤ ÛÛ T X i −θ φ(z * i ) = I 1 2 θ φ(z * i ) −θ a ≤ ÛÛ T X i −θ φ(z * i ) ≤ I 1 2 θ φ(z * i ) −θ a ≤ ÛÛ T ǫ i −θ φ(z * i ) + ÛÛ T θ * z * i −θ φ(z * i ) ≤ I θ φ(z * i ) −θ a − 2 θ * z * i −θ φ(z * i ) ≤ 2 ÛÛ T ǫ i −θ φ(z * i ) ,
where we use the fact that
X i = θ * z * i + ǫ i and ÛÛ T θ * z * i −θ φ(z * i ) ≤ θ * z * i −θ φ(z * i ) . Sincê θ φ(z * i ) −θ a =θ φ(z * i ) − θ * z * i + θ * z * i − θ * φ −1 (a) + θ * φ −1 (a) −θ a , we have I ÛÛ T X i −θ a ≤ ÛÛ T X i −θ φ(z * i ) ≤ I θ * z * i − θ * φ −1 (a) − θ φ(z * i ) − θ * z * i − θ * φ −1 (a) −θ a − 2 θ * z * i −θ φ(z * i ) ≤ 2 ÛÛ T ǫ i ≤ I θ * z * i − θ * φ −1 (a) − 4 max b∈[k] θ * b −θ φ(b) ≤ 2 ÛÛ T ǫ i ≤ I 1 − 4C 0 β −0.5 kn −0.5 E ∆ ∆ ≤ 2 ÛÛ T ǫ i ,(44)
where in the last inequality, we use the fact that max b∈
[k] θ * b −θ φ(b) ≤ C 0 β −0.5 kn −0.5 E from Proposition 3.1 and min b,b ′ ∈[k]:b =b ′ θ * b − θ * b ′ = ∆.
Since the above display holds for each a ∈ [k] that is not φ(z * i ), we have
I {ẑ i = φ(z * i )} ≤ I 1 − 4C 0 β −0.5 kn −0.5 E ∆ ∆ ≤ 2 ÛÛ T ǫ i = I 1 − 4C 0 ψ −1 0 ∆ ≤ 2 ÛÛ T ǫ i ,
where in the last inequality we use the definition of ψ 0 in (15).
PROOF OF LEMMA 3.2. For simplicity, throughout the proof we denoteÛ andÛ −i to be short forÛ 1:κ andÛ −i,1:κ , respectively. We have the following decomposition forÛÛ T ǫ i ,
ÛÛ T ǫ i ≤ Û −iÛ T −i ǫ i + ÛÛ T −Û −iÛ T −i F ǫ i .
Using the fact that ǫ i ≤ E and Theorem 2.2, after rearrangement, we have
ÛÛ T ǫ i ≤ 128k E √ nβρ 0 + 1 + 128 E ρ 0 λ k Û −iÛ T −i ǫ i = 128ψ −1 0 ρ −1 0 ∆ + 1 + 128 ρ 2 0 Û −iÛ T −i ǫ i .
In Lemma 3.1 we establish (18). From there we have
I {ẑ i = φ(z * i )} ≤ I 1 − Cψ −1 0 ∆ ≤ 256ψ −1 0 ρ −1 0 ∆ + 2 1 + 128 ρ 2 0 Û −iÛ T −i ǫ i ≤ I 1 − C ′ ψ −1 0 + ρ −2 0 ∆ ≤ 2 Û −iÛ T −i ǫ i ,
for some constant C ′ > 0, where in the last inequality we use the assumption ρ 0 > 16 from (8). The upper bound on Eℓ(ẑ, z * ) is an immediate consequence as Eℓ(ẑ,
z * ) = n −1 i∈[n] EI {ẑ i = φ(z * i )}.
Proofs of Theorem 3.1.
PROOF OF THEOREM 3.1. For simplicity, we denoteÛ −i to be short forÛ −i,1:κ throughout the proof. Define ψ := ψ −1 1 + ρ −2 1 . Then ψ < 2/C. Since E is a random matrix with independent sub-Gaussian columns, we have
P E ≤ 8σ( √ n + √ p) ≥ 1 − e −n/2 ,(45)
by Lemma D.1. Denote F to be this event. Under F , as long as ψ 1 , ρ 1 ≥ 128, we have both (15) and (8) hold. Let φ ∈ Φ satisfy ℓ(ẑ, z * ) = n −1 i∈[n] I {ẑ i = φ(z * i )}. Consider a fixed i ∈ [n]. Then from Lemma 3.2, we have
I {ẑ i = φ(z * i )}I {F} ≤ I (1 − C 1 ψ) ∆ ≤ 2 Û −iÛ T −i ǫ i I {F} ≤ I (1 − C 1 ψ) ∆ ≤ 2 Û −iÛ T −i ǫ i , where C 1 > 0 is some constant that does not depend on C. Then, Eℓ(ẑ, z * ) ≤ EI F ∁ + Eℓ(ẑ, z * )I {F} ≤ e −n/2 + n −1 i∈[n] EI (1 − C 1 ψ) ∆ ≤ 2 Û −iÛ T −i ǫ i .(46)
Since ǫ i ∼ SG p (σ 2 ) and it is independent ofÛ −iÛ T −i , we can apply concentration inequalities
for Û −iÛ T −i ǫ i from Lemma D.2. Define t = (1 − C 2 ψ)∆ 2 /(8σ 2 ) where C 2 = C 1 + 16.
Since C 2 does not depend on C, we can let C > max{4C 2 , 128} such that 1 − C 2 ψ > 1/2. Then we have k/t ≤ 16k 2 σ 2 /∆ 2 ≤ 16ψ 2 1 where we use the fact that ∆ kσ > ψ −1 1 from (20) as β ≤ 1. Then we have
σ 2 (κ + 2 √ κt + 2t) = 2σ 2 t 1 2 κ t + κ t + 1 ≤ 2σ 2 t 8ψ 2 1 + 4ψ 1 + 1 ≤ 2σ 2 t (1 + 8ψ 1 ) ≤ (1 − C 2 ψ)∆ 2 /(8σ 2 ) (1 + 8ψ) ≤ (1 − C 1 ψ)∆ 2 /(8σ 2 ),
where we use that ψ 1 < 1/128 and ψ < 1/64 as we let C > 128. Then from Lemma D.2, we have
EI (1 − C 1 ψ) ∆ ≤ 2 Û −iÛ T −i ǫ i ≤ exp (−t) = exp −(1 − C 2 ψ) ∆ 2 8σ 2 .
5.3. Proof of Theorem 3.4. The proof of Theorem 3.4 relies on the following entrywise decomposition that is analogous to Lemma 3.2 but in an opposite direction. Note the the singular vectorsû 1 , and {û 1,−i } i∈ [n] are all identifiable up to sign. Without loss of generality, we assume û 1 , u 1 ≥ 0 and û 1,−i , u 1 ≥ 0 for all i ∈ [n].
LEMMA 5.1. Consider the model (27). Let φ ∈ Φ be the permutation such that ℓ(ž, z * ) = 1 n |{i ∈ [n] :ž i = φ(z * i )}|. Then there exists a constants C,
C 1 > 0 such that if ∆ β −0.5 n −0.5 E ≥ C,(47)
then for any i ∈ [n],
I {ž i = φ(z * i )} ≥ I 1 + C 1 β −0.5 n −0.5 E ∆ ∆ ≤ −2(û T 1,−i ǫ i )sign(u T 1 θ φ(z * i ) ) .(48)
PROOF. The proof mainly follows the proofs of Lemma 3.1 and Lemma 3.2 with some modifications such as adding a negative term instead of a positive term in order to obtain a lower bound.
We first writež equivalently as
ž, θ j 2 j=1 = argmin z∈[2] n ,{θj} 2 j=1 ∈R p i∈[n] û 1û T 1 X i − θ zi 2 ,
whereθ a =û 1ča for each a ∈ [2]. Note that k = 2. From Proposition 3.1, we have
1 n |{i ∈ [n] :ž i = φ(z * i )}| ≤ C 0 k E 2 n∆ 2 , and max a∈[2] θ φ(a) − θ * a ≤ C 0 β −0.5 kn −0.5 E ,(49)
for some permutation φ : [2] → [2] and some constant C 0 > 0. Without loss of generality, assume φ = Id.
Recall that θ *
1 = −θ * 2 = δ1 p , u 1 = 1/ √ p1 p , λ 1 = δ √ np = ∆ √ n 2 , and |u T 1 (θ * z * i − (−θ * z * i ))| = 2δ √ p = ∆. By Davis-Kahan Theorem, we have min s∈±1 û 1 − su 1 ≤ E λ 1 = 2 E √ n∆ ≤ 1/16,
where the last inequality is due to the assumption (15). Since we assume û 1 , u 1 ≥ 0, we have û 1 − su 1 = min s∈±1 û 1 − su 1 . Consider any i ∈ [n] and any a ∈ [2] such that a = z * i . Note that for any scalars x, y, w, if |x − y| ≤ |x − w|, we have equivalently sign(w − y)(y + w)/2 ≥ sign(w − y)x. Since (y + w)/2 = (y − w)/2 + w, a sufficient condition is |w − y| /2 + |w| ≤ (−sign(w − y))x. Hence, we have Following the same analysis as in the proof of Lemma 3.1, we can get the following result that is analogous to (44):
I û 1û T 1 X i −θ a ≤ û 1û T 1 X i −θ z * i ≥ I 1 + 4C 0 β −0.5 kn −0.5 E ∆ ∆ ≤ −2(û T 1 ǫ i )sign(u T 1 (θ * z * i − θ * a )) .
Next, we are going to decomposeû T 1 ǫ i following the proof of Lemma 3.2. Denoteû 1,−i be the leave-one-out counterpart ofû 1 , i.e.,û 1,−i is the leading left singular vector of X −i . Since we assume
û 1,−i , u 1 ≥ 0, we have û 1,−i − u 1 ≤ 2 E /( √ n − 1∆). As a result, we have û 1,−i −û 1 ≤ 4 E /( √ n − 1∆) which leads to û 1,−i ,û 1 ≥ 1 − 4 E /( √ n − 1∆) > 0.(50)
We have the following decomposition:
(û T 1 ǫ i )sign(u T 1 (θ * z * i − θ * a )) = û 1 ,û 1û T 1 ǫ i sign(u T 1 (θ * z * i − θ * a )) = û 1 , (û 1,−iû T 1,−i )ǫ i sign(u T 1 (θ * z * i − θ * a )) + û 1 , (û 1û T 1 −û 1,−iû T 1,−i )ǫ i sign(u T 1 (θ * z * i − θ * a )) = û 1 ,û 1,−i (û T 1,−i ǫ i )sign(u T 1 (θ * z * i − θ * a )) + û 1 , (û 1û T 1 −û 1,−iû T 1,−i )ǫ i sign(u T 1 (θ * z * i − θ * a )) ≤ û 1 ,û 1,−i (û T 1,−i ǫ i )sign(u T 1 (θ * z * i − θ * a )) + û 1û T 1 −û 1,−iû T 1,−i ǫ i . Note that λ 1 / E = ∆ √ n/(2 E ) isû 1û T 1 −û 1,−iû T 1,−i ≤ 128 λ 1 / E k √ βn + û 1,−iû T 1,−i ǫ i λ 1 .
Then,
(û T 1 ǫ i )sign(u T 1 (θ * z * i − θ * a )) ≤ û 1 ,û 1,−i (û T 1,−i ǫ i )sign(u T 1 (θ * z * i − θ * a )) + 128k √ nβ(λ 1 / E ) + 128 û 1,−iû T 1,−i ǫ i λ 2 1 / E E = û 1 ,û 1,−i (û T 1,−i ǫ i )sign(u T 1 (θ * z * i − θ * a )) + 256n −0.5 kβ −0.5 E 2 ∆ + 512 û T 1,−i ǫ i n −1 E 2 ∆ 2 .
So far we have obtained
I û 1û T 1 X i −θ a ≤ û 1û T 1 X i −θ z * i ≥ I 1 + 4C 0 β −0.5 kn −0.5 E ∆ ∆ ≤ −2 û 1 ,û 1,−i (û T 1,−i ǫ i )sign(u T 1 (θ * z * i − θ * a )) − 256n −0.5 kβ −0.5 E 2 ∆ − 512 û T 1,−i ǫ i n −1 E 2 ∆ 2 = I 1 + 4C 0 β −0.5 kn −0.5 E ∆ + 256n −0.5 kβ −0.5 E 2 ∆ 2 ∆ ≤ −2 û 1 ,û 1,−i (û T 1,−i ǫ i )sign(u T 1 (θ * z * i − θ * a )) −
Then,
EI 1 − (C 1 + C 2 )ψ −1 3 ∆ ≤ 2 u T 1 ǫ i = EI 1 − (C 1 + C 2 )ψ −1 3 ∆ σ ≤ 2 |Y | ≤ EI 1 − (C 1 + C 2 )ψ −1 3 ∆ σ ≤ 2 |Z| + 2DY 2 √ p + 2D √ p + EI {|Y | > η √ p} ≤ EI 1 − (C 1 + C 2 + C 3 + 2D)ψ −1 3 ∆ σ ≤ 2 |Z| + EI 2DY 2 √ p ≥ C 3 + EI {|Y | > η √ p},
where C 3 > 0 is a constant. Using the fact that Y ∼ SG(1) with zero mean, we have
EI 1 − (C 1 + C 2 )ψ −1 3 ∆ ≤ 2 u T 1 ǫ i ≤ 2 exp − 1 − (C 1 + C 2 + C 3 + 2D)ψ −1 3 2 ∆ 2 8σ 2 + 2 exp − C 3 √ p 4D + 2 exp − η 2 p 2 .
Then we have
Eℓ(ž, z * ) ≤ 1 n n i=1 EI 1 − (C 1 + C 2 )ψ −1 3 ∆ ≤ 2 u T 1 ǫ i + 1 n n i=1 EI C 2 ∆ ≤ 2 (û 1,−i − s i u 1 ) T ǫ i + e −0.5n ≤ 2 exp − 1 − (C 1 + C 2 + C 3 + 2D)ψ −1 3 2 ∆ 2 8σ 2 + 2 exp − C 2 2 ∆ 2 128σ 2 + 2 exp − C 3 √ p 4D + 2 exp − η 2 p 2 + e −0.5n ,
where e −0.5n is the probability that (45) does not hold. Since σ ≤ Cσ, when C 2 is chosen to satisfy C 2 2 /(128C 2 ) ≥ 16, we have
Eℓ(ž, z * ) ≤ 2 exp − 1 − C ′′ ψ −1 3 2 ∆ 2 8σ 2 + exp −C ′′ √ p + e −0.5n ,
for some constant C ′′ > 0. For the lower bound, from (48) we know
I {ž i = φ(z * i )} ≥ I 1 + C 4 ψ −1 3 ∆ ≤ −2(û T 1,−i ǫ i )sign(u T 1 (θ φ(z * i ) − θ 3−φ(z * i ) ))
, for some constant C 4 > 0 assuming ψ 3 is large. Using the same argument as in the upper bound, we are going to decomposeû T 1,−i ǫ i into u T 1 ǫ i and (û 1,−i − y 1 ) T ǫ i . Hence,
I {ž i = φ(z * i )} ≥ I 1 + C 4 ψ −1 3 ∆ ≤ −2(u T 1 ǫ i )sign(u T 1 (θ φ(z * i ) − θ 3−φ(z * i ) )) − 2 (û 1,−i − s i u 1 ) T ǫ i ≥ I 1 + (C 4 + C 5 ) ψ −1 3 ∆ ≤ −2(u T 1 ǫ i )sign(u T 1 (θ φ(z * i ) − θ 3−φ(z * i ) )) − I C 5 ψ −1 3 ∆ ≤ 2 (û 1,−i − s i u 1 ) T ǫ i ,
for some constant C 5 > 0 whose value to be chosen. Let
Y ′ d =σ −1 (u T 1 ǫ i )sign(u T 1 (θ φ(z * i ) − θ 3−φ(z * i ) )) = sign(u T 1 (θ φ(z * i ) − θ 3−φ(z * i ) ))σ −1 p − 1 2 p j=1 ǫ i,j .
Then using the same argument above, there exists some Z ′ ∼ N (0, 1) such that whenever
Y ′ ≤ η ′ √ p, we have |Y ′ − Z ′ | ≤ D ′ Y ′2 √ p + D ′ √ p where D ′ , η ′ > 0 are constants. Then EI 1 + (C 4 + C 5 ) ψ −1 3 ∆ ≤ −2(u T 1 ǫ i )sign(u T 1 (θ φ(z * i ) − θ 3−φ(z * i ) )) = EI 1 + (C 4 + C 5 ) ψ −1 3 ∆ σ ≤ −2Y ′ ≥ EI 1 + (C 4 + C 5 ) ψ −1 3 ∆ σ ≤ −2Z ′ − 2DY ′2 √ p − 2d √ p I Y ′ ≤ η ′ √ p ≥ EI 1 + (C 4 + C 5 + 2D + C 6 ) ψ −1 3 ∆ σ ≤ −2Z ′ − EI 2DY ′2 √ p ≥ C 6 − EI Y ′ > η ′ √ p ,
where C 6 > 0 is a constant. Then following the proof of the upper bound, and by a proper choice of C 5 , we have
Eℓ(ž, z * ) ≥ 2 exp − 1 + C ′′′ ψ −1 3 2 ∆ 2 8σ 2 − exp −C ′′′ √ p − e −0.5n ,
for some constant C ′′′ > 0.
Proofs of Lemma 3.4 and Theorem 3.5.
PROOF OF LEMMA 3.4. For the upper bound, we consider the following likelihood ratio test. For any x ∈ R p , define the two log-likelihood functions as
l 1 (x) = p j=1 log f (x j − δ), and l 2 (x) = p j=1 log f (x j + δ).
Then for each i ∈ [n], define the likelihood ratio test aŝ
z LRT i = 1, if l 1 (X i ) ≥ l 2 (X i ), 2, otherwise.
Then for any i ∈ [n] such that z * i = 1, we have
EI ẑ LRT i = 2 = P (l 2 (X i ) > l 1 (X i )) = P p j=1 log f (2δ + ǫ i,j ) f (ǫ i,j ) > 0 = P p j=1 log f ∆ √ p (ǫ i,j ) f 0 (ǫ i,j ) > 0 ,
where we use the fact 2δ = ∆ √ p . Since ∆ is a constant, by local asymptotic normality (c.f., Chapter 7, [37]), we have
p j=1 log f ∆ √ p (ǫ i,j ) f 0 (ǫ i,j ) d → N − I∆ 2 2 , I∆ 2 .
Then, lim p→∞ EI ẑ LRT i = 2 ≤ C 1 exp −I∆ 2 /8 for some constant C 1 > 0. We have the same upper bound if z * i = 2 instead. Hence,
lim p→∞ inf z sup z * ∈[2] n Eℓ(z, z * ) ≤ lim p→∞ sup z * ∈[2] n Eℓ(ẑ LRT , z * ) ≤ exp − I∆ 2 8 .
For the lower bound, instead of allowing z * ∈ [2] n , we consider a slightly smaller parameter space. Define Z = {z ∈ [2] n : z i = 1, ∀1 ≤ i ≤ n/3, z i = 2, ∀n/3 + 1 ≤ i ≤ 2n/3}. The proof idea is similar to that of Theorem 2.2 but with more involved calculation as r is not necessarily κ. Consider any i ∈ [n]. Definẽ
ρ −i :=λ −i,r −λ −i,r+1 I −Û −i,1:rÛ T −i,1:r X i .
We need to verifyρ −i > 2 first in order to apply Theorem 2.1. Recall the definition of P −i in (35) and E −i in (37). Let the SVD of P −i be
P −i = p∧(n−1) j=1 λ −i,j u −i,j v T −i,j , where λ −i,1 ≥ λ −i,2 ≥ . . . ≥ λ −i,p∧(n−1) . Denote U −i,1:r = (u −i,1 , u −i,2 , . . . , u −i,r ) ∈ O p×r .
Then by Weyl's inequality, we have
|λ −i,r − λ −i,r |, |λ −i,r+1 − λ −i,r+1 | ≤ E −i ≤ E .(51)
Then the numeratorλ
−i,r −λ −i,r+1 ≥ λ −i,r − λ −i,r+1 − 2 E .(52)
In the following, we are going to connect λ −i,r − λ −i,r+1 with λ r − λ r+1 .
To bridge the gap between λ −i,r , λ −i,r+1 and λ r , λ r+1 , definẽ P −i := (θ * z * 1 , . . . , θ * z * i−1 , U −i,1:r U T −i,1:r θ * z * i , θ * z * i+1 , . . . , θ * z * n ) ∈ R p×n . Letλ −i,1 ≥λ −i,2 ≥ . . . ≥λ −i,p∧n be its singular values. Note that U −i,1:r U T −i,1:rP −i is the best rank-r approximation ofP −i . This is because for any rank-r projection matrix M ∈ R p×p such that M 2 = M , we have where we use the fact U −i,1:r U T −i,1:r P −i is the best rank-r approximation of P −i . Hence, span(U −i,1:r ) is exactly the leading r left singular space ofP −i . It immediately implies:
•λ −i,j = λ −i,j for any j ≥ r + 1, including
λ −i,r+1 = λ −i,r+1 .(53)
• Since U −i,1:r U T −i,1:rP −i and U −i,1:r U T −i,1:r P −i only differ by one column where the latter one can be seen as the leave-one-out counterpart of the former one, using the same argument as in (36), we have
λ 2 −i,r ≥ 1 − k βn λ 2 −i,r .(54)
Then from (52), we havê
λ −i,r −λ −i,r+1 ≥ 1 − k βnλ −i,r −λ −i,r+1 − 2 E .(55)
where in the last two inequalities we use the assumption that d/k ≥ 10. As a consequence, we haveρ
* z * i + κ j=r+1 û T −i,j θ * z * i 2 ≤ 3 √ κ E βn k − 1 + κ j=r+1 λ −i,j βn k − 1 + E βn k − 1 2 ≤ 3 √ κ E βn k − 1 + √ κ λ −i,r+1 βn k − 1 + E βn k − 1 ,
where the second to the inequality is due to (40) and (43). By (57) and the Weyl's inequality, we haveλ
−i,r+1 ≤ λ −i,r+1 + E ≤ 1 1 − √ κ √ βn k −1 λ r+1 + E .
Then, with the assumption βn/k 2 ≥ 10, we have For any a, b ∈ [k], U 1:r U T 1:r θ * a −U 1:r U T 1:
I −Û −i,1:rÛ T −i,1:r θ * z * i ≤ 3 √ κ E βn k − 1 + √ κ λ r+1 βn k − 1 − √ κ + 2 E βn k − 1 ≤ √ kκ √ βn (6 E + 2λ r+1 ).r θ * b = (θ * a −θ * b )−U (r+1):k U T(U (r+1):k U T (r+1):k θ * a ≥ ∆ − 2 √ k(kρ + 1) E βn/k .(62)
Then from Proposition 3.1, as long as (which will be verified later) θ φ(z) − U 1:r U T 1:r θ * a ≤ C 0 β −0.5 kn −0.5 Ě .
where C 0 = 128.
(Entrywise Decomposition forž). Next, we are going to have an entrywise decomposition for I {ẑ i = φ(z * i )} that is analogous to that of Lemma 3.2. When (63) is satisfied, from Lemma 3.1, we have Hence,
I {ž i = φ(z * i )} ≤ I 1 − C 0ψE ≤ 4 max u ′ ∈U ,v ′ ∈V u ′ T Ev ′ .
For any u ′ ∈ U , v ′ ∈ V, we have each u ′T ǫ i being an independent SG(σ 2 ) and then u ′T Ev ′ ∼ SG(σ 2 ). Note |U | ≤ 9 p ≤ e 3p and similarly |V | ≤ e 3n . Then by the tail probability of sub-Gaussian random variable and by the union bound, we have
P E ≤ 4tσ( √ n + √ p) ≤ P max u ′ ∈U ,v ′ ∈V u ′T Ev ′ ≤ tσ( √ n + √ p) ≤ |U | |V | exp − t 2 √ n + √ p 2 2 ≤ exp − (t 2 − 3)n 2 ,
for any t ≥ 2.
LEMMA D.2. Let X ∼ SG d (σ 2 ). Consider any k ≤ d. For any matrix U = (u 1 , . . . , u k ) ∈ R d×k that is independent of X and is with orthogonal columns {u i } i∈ [k] . We have
P U U T X 2 ≥ σ 2 (k + 2 √ kt + 2t) ≤ e −t .
PROOF. Note that tr(U U T ) = tr((U U T ) 2 ) = k and U U T = 1. This is a direct consequence of Theorem 1 in [16] for concentration of quadratic forms of sub-Gaussian random vectors. PROOF OF PROPOSITION 3.1. DefineP = i∈[r]λ iûiv T i . Due to the fact thatP is the best rank-r approximation of X in spectral norm and P is rank-κ, under the assumption that κ ≤ r, we have that P − X ≤ P − X = E .
Since r ≤ k is assumed, the rank ofP − P his at most 2k, and we have
P − P F ≤ √ 2k P − P ≤ √ 2k P − X + P − X ≤ 2 √ 2k E(72)
Now, denoteΘ := (θẑ 1 ,θẑ 2 , . . . ,θẑ n ). SinceΘ is the solution to the k-means objective (14), we have that
Θ −P F ≤ P −P F .
Hence, by the triangle inequality, we obtain that
Θ − P F ≤ 2 P − P F ≤ 4 √ 2k E .
Now, define the set S as
S = i ∈ [n] : θẑ i − θ * z * i > ∆ 2 .
U
r := (u 1 , . . . , u r ) ∈ O p×r andÛ r := (û 1 , . . . ,û r ) ∈ O p×r
Assume βn/k 2 ≥ 10. Assume
THEOREM 2 . 3 .
23Assume βn/k 2 ≥ 10. Assume there exists some r ∈ [k] such that
is well-controlled by Theorem 2.2, which shows the second term on the RHS of the above display is essentially O(ρ −2 0 ) Û −i,1:κÛ T −i,1:κ ǫ i . This leads to the following Lemma 3.2 on the entrywise clustering errors. LEMMA 3.2. Consider the spectral clusteringẑ of Algorithm 1 with r = κ. Assume βn/k 2 ≥ 10, (8), and
COROLLARY 3 . 1 .
31Under the same setting as in Theorem 3.
THEOREM 3. 3 .
3Consider the spectral clusteringẑ of Algorithm 1 with r = k. Assume ǫ i iid ∼ N (0, σ 2 I p ) for each i ∈ [n]. Assume βn/k 4 ≥ 100 and ∆ k 3.5 β −0.5 1 + p n σ → ∞.
2 F
2= tr(D T A T AD) = tr(A T ADD T ); (2)B has orthogonal columns such that (I −BB T ) 2 = I −BB T ; and (3) {v 1 , . . . , v d } ∈ R n−1 are orthogonal vectors. Since the diagonal entries ofBB T are { B i,· 2 } i∈[p] , we have
we are going to simplify the denominator of the above display. Using the orthogonality of the singular vectors, we have û −i,r+1 , . . . ,û −i,κ ) (û −i,r+1 , . . . ,û −i,κ )
=
βn (6 E + 2λ r+1 ) + E ≥ρ 0 8 > 2,under the assumption that βn/(k 2 ) ≥ 10 and (10).whereǫ i := U (r+1):k U T (r+1):k θ * z * i + ǫ i .In this way, we have a new mixture model with the centers being {U 1:r U T 1:r θ * a } a∈[k] and the additive noises being {ǫ i }. DefineĚ := (ǫ 1 , . . . ,ǫ n ).Then Ě ≤ E + U (r+1)E + U (r+1):k U T (r+1):k P = E + λ r+1 ≤ (kρ + 2) E .(60)The separation among the new centers is no longer ∆.
r ǫ i + U (r+1):k U T (r+1):k θ * z * i ≤ Û 1:rÛ T 1:r ǫ i + √ k(kρ + 1) E βn/k .
the data points {Û 1:rÛ T 1:r X i } i∈[n] follow a mixture model with centers {Û 1:rÛ :r ǫ i } i∈[n] . In the proof of Lemma 3.1 we can show these k centers preserve the geometric structure of {θ * a } a∈[k] with minimum distance around ∆. Intuitively,T
1:r θ *
a } a∈[k]
and noises {Û 1:rÛ
T
1if Û 1:rÛ
T
1:r ǫ i is smaller than half of the minimum distance,Û 1:rÛ
unless the noise is Gaussian. 4. Proof of Main Results in Section 2. In this section, we give the proofs of Theorem 2.1 and Theorem 2.2. The proof of Theorem 2.3 is included in the supplement [42] due to page limit. 4.1. Proof of Theorem 2.1. Before giving the proof of Theorem 2.1, we first present and prove a slightly more general perturbation result, Theorem 4.1, which only requires σ 2 r
Together with Theorem 4.1, we obtain the desired bound. PROOF OF THEOREM 2.2. Consider any i ∈ [n]. In order to apply Theorem 2.1, we need to verify that the spectral gap assumption (3) is satisfied. That is, define4.2. Proof of Theorem 2.2.
greater than 16 under the assumption (47) holds for a large constant C. From Theorem 2.2 we have
SUPPLEMENT TO "LEAVE-ONE-OUT SINGULAR SUBSPACE PERTURBATION ANALYSIS FOR SPECTRAL CLUSTERING" BY Anderson Y. Zhang and Harrison H. Zhou University of Pennsylvania and Yale University APPENDIX A: PROOF OF THEOREM 2.3
Hence, we havě ∆ ≥ min a,b∈[k]:a =b θ * a − θ * b − 2 max a∈[k]r+1):k θ *
a +U (r+1):k U T
(r+1):k θ *
b .
Also,
max
a∈[k]
U (r+1):k U T
(r+1):k θ *
a = max
a∈[k]
i∈[n]:z *
i =a U (r+1):k U T
(r+1):k θ *
a
2
|{i ∈ [n] : z *
i = a}|
≤
U (r+1):k U T
(r+1):k P
F
βn/k
≤
2
√
kλ r+1
βn/k
≤
√
k(kρ + 1) E
βn/k
.
(61)
ǫ i )sign(u T 1 (θ * z * i − θ * a )) .
From (50) we haveFor any x, y, z, w ∈ R such that x ≥ 0, 1 ≥ z ≥ 0, and z |y| > w ≥ 0, we have I {x ≤ zy − w} ≥ I {x ≤ (z − w/|y|) y}. We then have,.The proof is complete. PROOF OF THEOREM 3.4. Recall that λ 1 = ∆ √ n/2. Same as the proof of Theorem 3.1,we work on the with-high-probability event (45).For the upper bound, from Lemma 3.2, there exists some φ ∈ Φ such that for any i ∈ [n],∆ ≤ 2 û T 1,−i ǫ i , for some C 1 > 0, where the last inequality is due to that ψ 3 is large. By Davis-Kahan Theorem, we know there exists someThen0 is a constant whose value will be determined later. Due to the independence ofand thenOn the other hand, u Twith varianceσ 2 , which can be approximated by a normal distribution. Since the distribution F is sub-Gaussian, its moment generating function exists. Then we can use the following KMT quantile inequality (cf., Proposition [KMT] of[28]There exist some constants D, η > 0 and Z ∼ N (0, 1), such that whenever |Y | ≤ η √ p, we haveThen for any z, z ′ ∈ Z we have ℓ(z,where it is reduced into a testing problem on whether X n has mean θ * 1 or θ * 2 . According to the Neyman-Pearson Lemma, the optimal procedure is the likelihood ratio testẑ LRT n defined above. By the same argument, we havefor some constant C 2 > 0.PROOF OF THEOREM 3.5. First, we have the following connection between the Fisher information I and the varianceσ 2 :where we use Cauchy-Schwarz inequality and the integral by partThe equation holds if and only if f ′ /f ∝ x, which is equivalent to F being normally distributed.SUPPLEMENTARY MATERIALSupplement A: Supplement to "Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering" (url to be specified). In the supplement[42], we first provide the proof of Theorem 2.3 in Appendix A, followed by the proofs of Lemma 3.3 and Theorem 3.2 in Appendix B. The proof of Theorem 3.3 is given in Appendix C. Auxiliary lemmas and propositions and their proofs are included in Appendix D.For the difference betweenλ −i,r ,λ −i,r+1 and λ r , λ r+1 , we use the Weyl's inequality again:In the proof of Theorem 2.2, we showFor any a ∈ [κ] such a ≥ r + 1, we haveHence, we obtain θ *Then together with (53), we have |λ −i,r+1 − λ r+1 | ≤ √ κλ −i,r+1 / βn/k − 1 and henceThe remaining part of the proof is to study {û T −i,a X i } a∈[r] and then apply Theorem 2.1. Following the exact argument as in the proof of Theorem 2.2, we haveUnder the assumption that βn/(k 2 ) ≥ 10 and(10), (58) is lower bounded by λ r /2. This also impliesλ −i,r ≥ λ r /2. Then a direct application of Theorem 2.1 leads toAPPENDIX B: PROOFS OF LEMMA 3.3 AND THEOREM 3.2 PROOF OF LEMMA 3.3. Note thatr ∈ [k] is a random variable. We are going to associate it with some deterministic index in [k]. Recall λ 1 ≥ λ 2 ≥ . . . ≥ λ p∧n are singular values of the signal matrix P and κ is the its rank. Let its SVD be P = i∈[p∧n]By the definition ofr in(22)and the definition ofρ, we knowλr −λr +1 ≥ρ E and λr +1 ≤ kρ E . By Weyl's inequality, we have |λ a − λ a | ≤ E for all singular values of X and P . Then we have λr − λr +1 ≥ (ρ − 2) E and λr +1 ≤ (kρ + 1) E . Note that (ρ − 2) E > 0 is as long asρ > 2. Definewhich is a deterministic subset of [κ]. Thenr ∈ R.Consider an arbitrary r ∈ R and defineÛ 1:r := (û 1 , . . . ,û r ) ∈ R p×r . Perform k-means on the columns ofÛ 1:rÛ T 1:r X and let the output beIn the following, we are going to establish statistical properties forž(r) and eventually obtain a desired upper bound for ℓ(ž(r), z * ). Since performing k-means on the columns ofÛ T 1:r X is equivalent to k-means on the columns ofÛ 1:rÛ T 1:r X, and sincer ∈ R, we havez =ž(r) and thus the desired upper bound also holds for ℓ(z, z * ).In the rest of the proof we are going to analyzež(r) for any r ∈ R. For simplicity, we use the notationž, {θ j } j∈[n] instead ofž(r), {θ j (r)} j∈[n]. The remaining proof can be decomposed into several parts.(Preliminary Results forž, {θ j } j∈[n] ). We are going to use Proposition 3.1 to have some preliminary results. Define U 1:r := (u 1 , . . . , u r ) and U (r+1):k := (u r+1 , . . . , u k ). Instead of the decomposition (5), we can writeThen, we haveFrom(59), under the assumption thatρ > 4 and βn/k 4 > 400, we haveρ 0 defined as in (10) to satisfyρThen Theorem 2.3 can be applied, with which we haveThen following the proof of Lemma 3.2, we havewhere in the last inequality we use λ r ≥ (ρ − 2) E > 0 (as long asρ > 2) from (59). The last step of the proof is to simplify the above display using ∆ instead of∆. Then, under the assumption thatρ > 256, we have (1 + 256/(ρ − 2)) −1 ≤ (1 − 512/ρ). Recall the definition ofψ 0 in(24). Under the assumption thatρ ≤ψ 0 /64, we havěaccording to (62). Then together with (60), we can verify (63) holds due tǒRearranging all the terms with the help of (64), we can simplify67)hold, under the assumption βn/k 4 ≥ 100, we haveApplying Theorem 2.3, we haveHence,where in the second to the last inequality, we use (66) for λ r and the event F for E . Then (71) leads towhere c 3 , c 4 > 0 are some constants. As long as 1 − c 4 (k 2 τ β −0.5 (1 + p/n)σ/∆ + τ −1 ) > 1/2, we can use Lemma D.2 to calculate the tail probability of Û −i,1:rÛ T −i,1:r ǫ i . Following the proof of Theorem 3.1, we havefor some constant c 5 > 0. Then we have,(Since we assume βn/k 4 ≥ 100, we have (n − k)/n > 0.99. Hence, under the assumption that ∆/(k 3.5 β −0.5 (1 + p n )σ) → ∞, we can take τ, τ ′′ to be τ = τ ′′−1 := ∆ k 3.5 β −0.5 1 + p n σ 0.25 such that τ → ∞ and τ ′′ = o(1). Then for some constant c 6 > 0, we have(1), we have (2) holds assuming σ r − σ r+1 > 2 (I − U r U T r )y n .PROOF. Recall the definition of the augmented matrix Y ′ . Note that U r U T r Y is the best rank-r approximation of Y . Sincer Y ′ also being the best rank-r approximation of Y ′ . This proves that span(U r ) and U r U T r are also the leading r left singular subspace and projection matrix of Y ′ . Then U rÛ T r − U r U T r is about the perturbation betweenŶ and Y ′ . Let σ ′ r , σ ′ r+1 be the rth and (r + 1)th largest singular values of Y ′ , respectively. By Wedin's Thereom (cf. Section 2.3 of[8]), if σ ′ r −σ r+1 > 0, then we have.Regarding the values of σ ′ r and σ ′ r+1 , first we have σ ′ r ≥ σ r . This is becauseIn addition, we have σ ′ r+1 = σ r+1 , due to the fact that (I − U r U T r )Y ′ = ((I − U r U T r )Y, 0). By Weyl's inequality, we haveHence, if σ r − σ r+1 > 2 (I − U r U T r )y n is further assumed, we have σ ′ r −σ r+1 ≥ σ r − σ r+1 − (I − U r U T r )y n ≥ 1 2 (σ r − σ r+1 ) .The proof is complete.LEMMA D.1. Let E = (ǫ 1 , . . . , ǫ n ) ∈ R p×n be a random matrix with each column ǫ i ∼ SG p (σ 2 ), ∀i ∈ [n] independently. Thenfor any t ≥ 2.PROOF. We follow a standard ǫ-net argument. Let U and V be a 1/4 covering set of the unit sphere in R p and in R n , respectively. That is, for any u ∈ R p such that u = 1, there exists a u ′ ∈ U such that u ′ = 1 and u − u ′ ≤ 1/4. Similarly, for any v ∈ R n such that v = 1, there exists a v ′ ∈ V such that v ′ = 1 and v − v ′ ≤ 1/4. ThenMaximizing over u, v on both sides, we have
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[
"arXiv:hep-ph/0507004v1 1 Jul 2005 Final results for the SM Higgs-boson production at the Photon Collider",
"arXiv:hep-ph/0507004v1 1 Jul 2005 Final results for the SM Higgs-boson production at the Photon Collider"
]
| [
"P Nieżurawski \nInstitute of Experimental Physics\nWarsaw University\nul. Hoża 6900-681WarsawPoland\n"
]
| [
"Institute of Experimental Physics\nWarsaw University\nul. Hoża 6900-681WarsawPoland"
]
| []
| Feasibility of the precise measurement of the SM Higgs-boson production cross section γγ → h → bb at the Photon Collider is studied in detail for M h = 120-160 GeV. All relevant experimental and theoretical issues, which could affect the measurement, are taken into account. The study is based on the realistic γγ-luminosity spectra simulation. The heavy-quark background γγ → QQ(g) is estimated using the NLO QCD results. Other background processes, which were neglected in earlier analyses, are also studied: γγ → W + W − , γγ → τ + τ − and light-quark pair production γγ → qq. The contribution from the so-called overlaying events, γγ → hadrons , is taken into account. The nonzero beam crossing angle and the finite size of colliding bunches are included in the event generation. The detector simulation and realistic b-tagging are used. Criteria of event selection are optimized separately for each considered Higgs-boson mass. In spite of the significant background contribution and deterioration of the invariant mass resolution due to overlaying events, precise measurement of the Higgs-boson production cross section is still possible. For the Standard-Model Higgs boson with mass of 120 to 160 GeV the corresponding partial width Γ(h → γγ)BR(h → bb) can be measured with a statistical accuracy of 2.1-7.7% after one year of the Photon Collider running. The systematic uncertainties of the measurement are estimated to be of the order of 2%. | null | [
"https://arxiv.org/pdf/hep-ph/0507004v1.pdf"
]
| 118,326,650 | hep-ph/0507004 | 618f5b49f4d457507f36b434b2b7f25894ad821f |
arXiv:hep-ph/0507004v1 1 Jul 2005 Final results for the SM Higgs-boson production at the Photon Collider
P Nieżurawski
Institute of Experimental Physics
Warsaw University
ul. Hoża 6900-681WarsawPoland
arXiv:hep-ph/0507004v1 1 Jul 2005 Final results for the SM Higgs-boson production at the Photon Collider
2005 International Linear Collider Workshop -Stanford, U.S.A.
Feasibility of the precise measurement of the SM Higgs-boson production cross section γγ → h → bb at the Photon Collider is studied in detail for M h = 120-160 GeV. All relevant experimental and theoretical issues, which could affect the measurement, are taken into account. The study is based on the realistic γγ-luminosity spectra simulation. The heavy-quark background γγ → QQ(g) is estimated using the NLO QCD results. Other background processes, which were neglected in earlier analyses, are also studied: γγ → W + W − , γγ → τ + τ − and light-quark pair production γγ → qq. The contribution from the so-called overlaying events, γγ → hadrons , is taken into account. The nonzero beam crossing angle and the finite size of colliding bunches are included in the event generation. The detector simulation and realistic b-tagging are used. Criteria of event selection are optimized separately for each considered Higgs-boson mass. In spite of the significant background contribution and deterioration of the invariant mass resolution due to overlaying events, precise measurement of the Higgs-boson production cross section is still possible. For the Standard-Model Higgs boson with mass of 120 to 160 GeV the corresponding partial width Γ(h → γγ)BR(h → bb) can be measured with a statistical accuracy of 2.1-7.7% after one year of the Photon Collider running. The systematic uncertainties of the measurement are estimated to be of the order of 2%.
INTRODUCTION
The neutral Higgs boson, h, couples to the photon pair only at the loop level, through loops of all massive charged particles. In the Standard Model (SM) the dominant contribution is due to W and t loops. This loop-induced hγγ coupling is sensitive to contributions of new particles which may appear in various extensions of the SM. Hence, the precise measurement of the Higgs-boson partial width Γ(h → γγ) can indicate existence of very heavy particles even if their direct production is not possible. A photon-collider option of the e + e − collider offers a unique possibility to measure Γ(h → γγ) as the Higgs boson can be produced in the s-channel process γγ → h. The SM Higgs boson with the mass below ∼ 140 GeV is expected to decay predominantly into the bb final state. Therefore, we consider the measurement of the cross section for the process γγ → h → bb, for M h = 120-160 GeV. The aim of this study is to estimate the precision of the Γ(h → γγ) measurement obtainable after one year of the Photon Collider running, taking all relevant experimental and theoretical effects into account.
PHOTON-PHOTON COLLISIONS
The analysis is based on the realistic γγ-luminosity simulation for the Photon Collider at TESLA [1]. The simulated photon-photon events were directly used in this analysis for generation of the so-called overlaying events γγ → hadrons where a proper description of the low energy tail of the spectrum was crucial. In case of other processes, for which only the high-energy part of the γγ spectrum is important, the subroutines of the CompAZ package [2] were used. We assume that the center-of-mass energy of colliding electron beams, √ s ee , is optimized for the production of a Higgs boson with a given mass. Presented results are obtained for the total integrated luminosity between 400 and 500 fb −1 , corresponding to the luminosity expected after one year of the TESLA Photon Collider running [1].
For our analysis the longitudinal size of the collision region is most important. As this is of the order of 100 µm, we can expect that additional tracks and clusters due to overlaying events (resulting in additional vertexes, changed jet characteristics etc.) can influence the flavour-tagging algorithm and affect the event selection. Therefore, generation of all event samples used in the described analysis took into account the Gaussian smearing of primary vertex and the beams crossing angle in horizontal plane, α c = 34 mrad.
EVENT GENERATION AND SIMULATION
Total widths and branching ratios of the Higgs boson were calculated with the program Hdecay [3], where higher order QCD corrections are included. Event generation for Higgs-boson production process, γγ → h → bb, was done with the Pythia program [4]. A parton shower algorithm implemented in Pythia was used to generate the finalstate partons. The fragmentation into hadrons was also performed using the Pythia program, both for Higgs-boson production and for all background event samples.
The main background for the considered signal process is the heavy-quark pair production, γγ → QQ. Events of 'direct', nonresonant bb production contribute to the irreducible background. In LO approximation the cross section for J z = 0 is suppressed and the dominant contribution is due to the |J z | = 2 state. This is very fortunate as the γγ-luminosity spectrum is optimized to give highest J z = 0 luminosity and the |J z | = 2 component is small in the higgs-production region. Unfortunately, NLO corrections compensate partially the m 2 Q /s suppression and, after taking into account luminosity spectra, both contributions (for J z = 0 and |J z | = 2) become comparable.
The other processes γγ → qq(g), where q = u, d, s, c, and γγ → τ + τ − contribute to the reducible background. One has to consider these processes due to the non-zero probability of wrong flavour assignment by the reconstruction procedure. Events with cc(g) in the final state have the highest mistagging probability. In comparison to the γγ → bb process there is an enhancement factor of (e c /e b ) 4 = 16 in the γγ → cc cross section. It turns out that after flavour tagging both processes give similar contribution to the background. The background events due to processes γγ → bb(g), cc(g) were generated using the program written by G. Jikia [5], where a complete NLO QCD calculation for the production of massive quarks is performed in the massive-quark scheme.
In cases of M h = 150 and 160 GeV also the pair production of W bosons, γγ → W + W − , is considered as a possible background. For generation of γγ → W + W − events the Pythia program is used with polarized differential cross section formulae from [6] to obtain correct distributions for J z = 0 and |J z | = 2 contributions.
Because of the large cross section and huge γγ-luminosity at low W γγ , about one γγ → hadrons event is expected on average per bunch crossing. Such events can contribute to the background on their own and may have a great impact on the reconstruction of other events produced in the same bunch crossing, by changing their kinematical and topological characteristics.
We generate γγ → hadrons events with Pythia, using luminosity spectra from the full simulation of the photonphoton collisions [1], rescaled to the chosen beam energy. For each considered e − e − energy, √ s ee , an average number of the γγ → hadrons events per bunch crossing is calculated. Then, for every signal or background event, the γγ → hadrons events are overlaid (added to the event record) according to the Poisson distribution. Fortunately, the γγ → hadrons cross section is very forward-peaked. A cut on the polar angle of tracks and clusters measured in the detector can greatly reduce contribution of particles from γγ → hadrons processes to selected events. For more details concerning γγ → hadrons overlaying events and their influence on the reconstruction see [7].
The fast simulation program for the TESLA detector, Simdet version 4.01 [8], was used to model the detector performance. Because two forward calorimeters, Low Angle Tagger and Low Angle Calorimeter, cannot be installed in the detector at the Photon Collider, they are not used in our simulation setup. To take into account the modified mask setup for the photon-photon option, all generator-level particles are removed from the event record, before entering the detector simulation, if their polar angle is less than θ mask = 130 mrad.
RESULTS
The contribution from overlaying events is expected to affect observed particle and energy flow mainly at low polar angles. Therefore, we introduce an angle θ T C defining the region strongly contaminated by this contribution; tracks and clusters with polar angle less than θ T C are not taken into account when applying energy-flow algorithm. We decided to use the value θ T C = 0.85 which results in the best final cross-section measurement precision. In the presented study jets are reconstructed using the Durham algorithm [9], with y cut = 0.02. Higgs-boson decay events are expected to consist mainly of two b-tagged jets with large transverse momentum and nearly isotropic distribution of the jet directions. The significant number of events (∼ 25%) contains the third jet due to the real-gluon emissions which are approximated in this analysis by the parton shower algorithm, as implemented in Pythia. The following cuts are used to select properly reconstructed bb events coming from Higgs decay.
1. Number of selected jets should be 2 or 3.
2. The condition | cos θ jet | < C θ is imposed for all jets in the event where θ jet is the jet polar angle, i.e. the angle between the jet axis and the beam line. This cut should improve signal-to-background ratio as the signal is almost uniform in cos θ, while the background is peaked at | cos θ| = 1.
3. Since the Higgs bosons are expected to be produced almost at rest, the ratio of the total longitudinal momentum calculated from all jets in the event, P z , to the total energy, E, should fulfill condition |P z |/E < C Pz .
The cut parameter values C θ and C Pz were optimized for each considered Higgs boson mass value to obtain the best statistical precision of the cross section measurement. For M h = 120 GeV the optimized values are C θ = 0.725 and C Pz = 0.1.
For b-tagging the Zvtop-B-Hadron-Tagger package was used [10][11][12], based on the the neural-network algorithm trained on the Z decays. For each jet the routine returns a "b-tag" value -the number between 0 and 1 corresponding to "b-jet" likelihood. In order to optimize the signal cross-section measurement, the two-dimensional cut on b-tag values for two jets with highest transverse momentum is used. The selection criterion is found by considering the value of the signal to background ratio S/B, where S and B denote the expected numbers of events for the signal γγ → h → bb and for the sum of background contributions from processes γγ → QQ(g) (Q = c, b) and γγ → qq (q = u, d, s), respectively. Obtained S/B distribution in the b-tag(jet 1 )⊗b-tag(jet 2 ) plane for Higgs-boson production with M h = 120 GeV is shown in Fig. 1. The selection region which results in the best precision of the Γ(h → γγ)BR(h → bb) measurement corresponds to the condition S/B > 0.19 as indicated in the figure (stars).
The invariant-mass distributions for signal events passing all optimized selection cuts, before and after taking into account the overlaying events γγ → hadrons, are compared in Fig. 2 (left). The overlaying events and cuts suppressing their contribution significantly influence the mass reconstruction and result in the increase of distribution width by about 2 GeV, and in the shift of the mean value, µ, by about 3 GeV. A drop in the selection efficiency, resulting in the reduced number of events expected after selection cuts is also observed. The tail towards low masses is due to events with energetic neutrinos coming from semileptonic decays of D and B mesons (see [13] for more details). To compensate for this effect we use the corrected invariant mass defined as [13]:
W corr ≡ W 2 rec + 2P T (E + P T ).
In Fig. 2 (right) the distributions of W corr for the selected signal events, without and with overlaying events, are presented. The tail of events with invariant masses below ∼ 110 GeV is much smaller than for the W rec distributions (compare with the left figure). The final W corr distributions for the signal and background events (with overlaying events included) are shown in Fig. 3. For M h = 120 GeV the most precise measurement of the Higgs-boson production Final results of analysis [7] are compared with our earlier results, which did not take into account all aspects of the measurement.
cross section is obtained for the mass window between 108 and 133 GeV, as indicated by arrows. In the selected W corr region one expects, after one year of the Photon Collider running at nominal luminosity, about 4900 reconstructed signal events and 5400 background events (i.e. µ S /µ B ≈ 0.9). This corresponds to the statistical precision of:
∆ Γ(h → γγ)BR(h → bb) Γ(h → γγ)BR(h → bb) = 2.1%.
The systematic uncertainty of the total background contribution is estimated to be about 2%, and the J z = 0 luminosity contribution will be measured with precision of around 1% [14]. Using maximal likelihood method to take these uncertainties into account we obtain precision of 2.7% for σ(γγ → h → bb) measurement at M h = 120 GeV, corresponding to the systematic error of the measurement of 1.8%.
We have performed the full simulation of signal and background events for M h = 120 to 160 GeV choosing optimal e − e − beam energy for each Higgs-boson mass. Statistical precision of Γ(h → γγ)BR(h → bb) measurement was estimated in each case. Results are presented in Fig. 4. For comparison, our earlier results obtained without overlaying events, without various background contributions or without distribution of interaction point are also shown. For M h = 160 GeV, after the full optimization of the selection cuts, better precision is obtained than in earlier analyses, which did not take into account some background contributions.
SUMMARY
One of the measurements crucial for understanding of the Higgs sector and for the verification of the particle physics models is the measurement of Γ(h → γγ). The Photon Collider, which has been proposed as an extension of the e + e − linear collider project, is considered the best place to do this measurement.
We present the first fully realistic estimates for the precision of γγ → higgs → bb cross-section measurement at the Photon Collider with parameters of the TESLA project. The analysis is based on the realistic γγ-luminosity spectrum simulation. Due to the high beam intensity, resulting in high γγ-luminosity per bunch crossing, the contribution of overlaying events γγ → hadrons turns out to be sizable and affects the event reconstruction. Crossing angle between beams resulting in the significant broadening of the interaction region is also taken into account. These two factors have significant impact on the performance of the b-tagging algorithm. It is shown that the contamination of γγ → hadrons overlaying events in the signal can be reduced by rejecting low-angle tracks and clusters in the event. Additional cuts are proposed to suppress contributions from other background sources.
After optimizing selection cuts and applying correction for escaping neutrinos from D-and B-meson decays the quantity Γ(h → γγ)BR(h → bb), for the SM Higgs boson with mass around 120 GeV, can be measured with the precision of about 2% already after one year of the Photon Collider running. The systematic uncertainties of the measurement are estimated to be of the order of 2%. The statistical precision of the measurement decreases up to 7.7% for the SM Higgs boson with mass M h = 160 GeV. For higher masses of the SM Higgs boson other decay channels are expected to give better precision of Γ(h → γγ) measurement, see e.g. [15]. Presented results are consistent with earlier studies [16,17], which however did not take into account all aspects of the measurement considered here.
The measurement discussed in this paper can be used to derive the partial width Γ(h → γγ), taking BR(h → bb) value from precise measurement at the e + e − International Linear Collider. With 2% accuracy on Γ(h → γγ)BR(h → bb), as obtained in this analysis, and assuming BR(h → bb) will be measured to 1.5% [18], Higgs-boson partial width Γ(h → γγ) can be extracted with accuracy of about 2.5%. With this precision the measurement will be sensitive to the deviations from the SM coming from loop contributions of new heavy charged particles. For example, heavy charged higgs contribution in the SM-like 2HDM is expected to change Γ(h → γγ) by 5-10% [19]. Using in addition the result from the e + e − Linear Collider for BR(h → γγ) [20], one can also extract the total width Γ h with precision of about 10%.
Figure 1 :
1The expected ratio of signal (γγ → h → bb) to background (γγ → QQ(g), Q = c, b, and γγ → qq, q = u, d, s) event distributions in the plane btag(jet1) ⊗ btag(jet2). The region which results in the best precision measurement for the cross-section measurement is indicated by stars.
Figure 2 :
2Reconstructed invariant-mass, Wrec, (left) and corrected invariant-mass, Wcorr, (right) distributions for selected γγ → h → bb events, for M h = 120 GeV. Distributions obtained without and with overlaying events (OE) are compared. Results for the mean µ and dispersion σ from the Gaussian fit in the region from µ − 1.3σ to µ + 1.3σ, are also shown.
Figure 3 :Figure 4 :
34Distributions of the corrected invariant mass, Wcorr, for selected bb events. Contributions of the signal, for M h = 120 GeV, and of the background processes, i.e. γγ → QQ(g) for Q = c, b, γγ → qq for q = u, d, s, γγ → τ + τ − , and γγ → hadrons (as a separate contribution with hadron-like×hadron-like interactions only, indicated as 'resolved'), are shown separately. Arrows indicate the mass window, 107.5 to 132.5 GeV, optimized for the measurement of the Γ(h → γγ)BR(h → bb), which leads to the statistical precision of 2.1%. Statistical precision of Γ(h → γγ)BR(h → bb) measurement for the SM Higgs boson with mass 120-160 GeV.
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| []
|
[]
| [
"Adi Nusser ",
"Marc Davis ",
"\nPhysics Department\nDepartments of Astronomy & Physics\nAsher Space Science Institute-Technion\n32000HaifaIsrael\n",
"\nUniversity of California\n94720BerkeleyCA\n"
]
| [
"Physics Department\nDepartments of Astronomy & Physics\nAsher Space Science Institute-Technion\n32000HaifaIsrael",
"University of California\n94720BerkeleyCA"
]
| []
| The cosmological bulk flow: consistency with ΛCDM and z ≈ 0 constraints on σ 8 and γ ABSTRACT We derive estimates for the cosmological bulk flow from the SFI++ Tully-Fisher (TF) catalog. For a sphere of radius 40h −1 Mpc centered on the MW, we derive a bulk flow of 333±38 km s −1 towards Galactic (l, b) = (276 • , 14 • ) within a 3 • 1σ error. Within a 100h −1 Mpc we get 257±44 km s −1 towards (l, b) = (279 • , 10 • ) within a 6 • error. These directions are at a 40 • with the Supergalactic plane, close to the apex of the motion of the Local Group of galaxies after the Virgocentric infall correction. Our findings are consistent with the ΛCDM model with the latest WMAP best fit cosmological parameters. But the bulk flow allows independent constraints. For WMAP inferred Hubble parameter h = 0.71 and baryonic mean density parameter Ω b = 0.0449, the constraint from the bulk flow on the matter density Ω m , the normalization of the density fluctuations, σ 8 , and the growth index, γ, can be expressed as σ 8 Ω γ−0.55 m (Ω m /0.266) 0.28 = 0.86 ± 0.11 (for Ω m ≈ 0.266). Fixing σ 8 = 0.8 and Ω m = 0.266 as favored by WMAP, we get γ = 0.495 ± 0.096. The constraint derived here rules out popular DGP models at more than the 99% confidence level. Our results are based on a method termed ASCE (All Space Constrained Estimate) which reconstructs the bulk flow from an all space three dimensional peculiar velocity field constrained to match the TF measurements. At large distances ASCE generates a robust bulk flow from the SFI++ that is insensitive to the assumed prior. For comparison, a standard straightforward maximum likelihood estimate leads to very similar results.Subject headings: Cosmology: large-scale structure of the Universe, dark matter, cosmological parameters | 10.1088/0004-637x/736/2/93 | [
"https://arxiv.org/pdf/1101.1650v3.pdf"
]
| 117,128,737 | 1101.1650 | 9530aa5abfc3be4bef43be152c0c01672ff6609a |
8 Jun 2011
Adi Nusser
Marc Davis
Physics Department
Departments of Astronomy & Physics
Asher Space Science Institute-Technion
32000HaifaIsrael
University of California
94720BerkeleyCA
8 Jun 2011
The cosmological bulk flow: consistency with ΛCDM and z ≈ 0 constraints on σ 8 and γ ABSTRACT We derive estimates for the cosmological bulk flow from the SFI++ Tully-Fisher (TF) catalog. For a sphere of radius 40h −1 Mpc centered on the MW, we derive a bulk flow of 333±38 km s −1 towards Galactic (l, b) = (276 • , 14 • ) within a 3 • 1σ error. Within a 100h −1 Mpc we get 257±44 km s −1 towards (l, b) = (279 • , 10 • ) within a 6 • error. These directions are at a 40 • with the Supergalactic plane, close to the apex of the motion of the Local Group of galaxies after the Virgocentric infall correction. Our findings are consistent with the ΛCDM model with the latest WMAP best fit cosmological parameters. But the bulk flow allows independent constraints. For WMAP inferred Hubble parameter h = 0.71 and baryonic mean density parameter Ω b = 0.0449, the constraint from the bulk flow on the matter density Ω m , the normalization of the density fluctuations, σ 8 , and the growth index, γ, can be expressed as σ 8 Ω γ−0.55 m (Ω m /0.266) 0.28 = 0.86 ± 0.11 (for Ω m ≈ 0.266). Fixing σ 8 = 0.8 and Ω m = 0.266 as favored by WMAP, we get γ = 0.495 ± 0.096. The constraint derived here rules out popular DGP models at more than the 99% confidence level. Our results are based on a method termed ASCE (All Space Constrained Estimate) which reconstructs the bulk flow from an all space three dimensional peculiar velocity field constrained to match the TF measurements. At large distances ASCE generates a robust bulk flow from the SFI++ that is insensitive to the assumed prior. For comparison, a standard straightforward maximum likelihood estimate leads to very similar results.Subject headings: Cosmology: large-scale structure of the Universe, dark matter, cosmological parameters
Introduction
Cosmological bulk flows are the peculiar velocities of whole spherical regions around us. Bulk flows are usually considered for sufficiently large spheres where linear expressions for the velocity and density power spectra are valid. This greatly facilitates the calculation of expected bulk flows in cosmological models, in contrast to analyzing the full field field which may involve non-linear effects on small scales (Feldman et al. 2010;Abate & Erdogdu 2009;Zaroubi et al. 2001;Freudling et al. 1999). In linear theory, the bulk flow of a sphere is solely determined by the gravitational pull of only the dipole component of the external mass distribution. Bulk flows are, therefore, an unmistakable indicator of distant large mass concentrations should they exist. The exact expression of the bulk flow, B(r), of a sphere of radius r is,
B(r) = 3 4πr 3 x<r v v v(x x x)d 3 x .(1)
where v v v(x x x) is the 3D peculiar velocity field as a function of the comoving coordinate x x x. Beneath this innocuous expression lie a multitude of nuisances. An unbiased estimate of B requires knowledge of v v v sampled uniformly overall the volume. However, observational probes of peculiar velocities measurements are available only for a few thousand galaxies with a patchy coverage of the local Universe. Further, peculiar velocity probes such as the TF relation allow us to constrain only the radial component of the peculiar velocities of galaxies.
Recently compiled data on peculiar velocities has triggered renewed interest in the analysis of large scale flows, including the bulk flow (Davis et al. 2011;Lavaux et al. 2010;Erdogdu et al. 2006). Feldman et al. (2010) report an unusually large bulk flow of 416 ± 78 km s −1 in a sphere of 100h −1 Mpc which is at odds with the ΛCDM model with the best fit parameters of the Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP7) (e.g. Jarosik et al. 2010;Larson et al. 2010).
Here we provide an alternative estimate of the bulk using a single data set of TF measurements of galaxies, trimmed at faint magnitudes to ensure the linearity of the TF relation. The estimate is based on a method which we term ASCE for All Space Constrained Estimate. The method computes B(r) using (1) from a three dimensional v v v(x x x) defined everywhere in a large region of space and constrained to match the TF data. For the analysis below, we use the SFI++ survey of spiral galaxies with I-band Tully-Fisher distances, (Masters et al. 2006;Springob et al. 2007), which builds on the original Spiral Field I-band Survey (Giovanelli et al. 1994(Giovanelli et al. , 1995Haynes et al. 1999) and Spiral Cluster I-band Survey (Giovanelli et al. 1997a,b). We use the published SFI++ magnitudes and velocity widths, and derive our own peculiar velocities, rather than taking the published distances as given. We shall use the inverse of the Tully-Fisher (ITF) relationship. The main advantages of ITF methods is that samples selected by magnitude, as most are, will be minimally plagued by Malmquist bias effects when analyzed in the inverse direction (Schechter 1980;Aaronson et al. 1982). We assume that the circular velocity parameter, η ≡ log(line width), of a galaxy is, up to a random scatter, related to its absolute magnitude, M , by means of a linear inverse Tully-Fisher (ITF) relation, i.e.,
η = sM + η 0 .(2)
The preparation of the data is done following Davis et al. (2011). We include all field, group, and cluster galaxies. Galaxies in groups and clusters are treated as individual objects, though the redshifts for template cluster galaxies are replaced by the systematic redshift of the cluster. Galaxies fainter the an estimated magnitude of -20 were removed from the sample as those showed significant deviations from a linear TF relation. In order to get a cleaner TF sample we select only objects with inclination i > 45 • to ease problems with inclination corrections. All this leaves us with a sample of 2859 galaxies with redshifts less than 100h −1 Mpc. The effective depth of the sample defined as the error weighted mean redshift of galaxies is ∼ 40h −1 Mpc.
We will refer with ΛCDM7 to the ΛCDM cosmological model with the WMAP7 best fit parameters (Larson et al. 2010) for a flat Universe, i.e. the total mass density parameter Ω m = 0.266, baryonic density parameter Ω b = 0.0449, a Hubble constant h = 0.71 in units of 100 km s −1 Mpc −1 , a scalar spectral index n s = 0.963, and σ 8 = 0.8 for the rms of linear density fluctuations in spheres of 8h −1 Mpc. Throughout the paper, variants of ΛCDM7 with different Ω m and σ 8 will be considered. All other parameters will be fixed at their WMAP7 values.
The outline of the paper is as follows. Details of the ASCE method are described in §2, while the more standard Maximum Likelihood Estimate (MLE) outlined in §3. Tests of the methods using mock catalogues designed to match the SFI++ catalogue are presented in §3. Results for the bulk flows from the SFI++ data are given in §5 with the subsection §5.1 providing a comparison with the ΛCDM models. Finally, §6 discusses the results and some of their cosmological implications.
The All Space Constrained Estimate (ASCE)
Observations of distance (peculiar velocity) indicators, such as the SFI++ TF survey, are available for only a small fraction of galaxies in the local Universe (out to ∼ 100h −1 Mpc). The absence of uniformly distributed data prevents a direct application of equation (1). To circumvent this problem, the ASCE method effectively uses (1) to reconstruct the bulk flow from a 3D field v v v(x x x) which satisfies two conditions: a) at sufficiently large distances from the observed galaxies in the TF data, it has a power spectrum that is dictated by a cosmological model such as the ΛCDM, and b) it has radial peculiar velocities at the positions of the observed galaxies, which are consistent with the TF measurements. The approach is similar to that of constrained realization from noisy data (Hoffman & Ribak 1991;Zaroubi et al. 1995), but it is more general and easier to implement. Assume that the TF catalogue contains i = 1 . . . N g galaxies with measured redshifts (in km s −1 ), cz i , apparent magnitudes, m i , and line width parameters, η i . We write the absolute magnitude of a galaxy,
M i = M 0i + P i ,(3)
where
M 0i = m i + 5log(cz i ) − 15(4)
and
P i = −5log(1 − u i /cz i )(5)
with u i the radial peculiar velocity of the galaxy. Both cz i and u i are defined in the frame of the cosmic microwave background radiation (CMB). Assume that an estimate of the underlying cosmological velocity field, v v v(x x x), can be written as a linear combination of
v v v(x x x) = Na α a α v v v α (x x x) ,(6)
where the N fields, v v v α (x x x) (α = 1 · · · N a ), are gaussian random velocity fields generated using a cosmologically viable power spectrum. In practice these basis velocity fields will be extracted from a linear cosmological velocity field generated in a very large box using the power spectrum of the ΛCDM model. We then compute u α i , the radial component of the v v v α at the redshift space positions of the observed galaxies, and define the P −basis functions, P α i = −5log(1−u α i /cz i ) for the observed galaxies only. The model P is then written as
P M i = α a α P α i .(7)
The best fit mode coefficients a α , the slope, s, and the zero point η 0 , are found by minimizing the χ 2 statistic
χ 2 = 1 σ 2 η Ng i=1 sM 0i + sP M i + η 0 − η i 2 + Na α=1 (a α ) 2 ,(8)
where σ 2 η is the rms of the intrinsic scatter in η about the ITF relation, and N g is the number of galaxies in the sample. The second term of the sum over the squares of a α is introduced in order to regularize the solution especially in regions of poor data coverage. In the appendix we derive this term from a Bayesian formulation. The solution to the equations ∂χ 2 /∂a α = 0, ∂χ 2 /∂s = 0 and ∂χ 2 /∂η 0 = 0 is straightforward. The coefficients a α will be used in equation (6) to get v v v(x x x) everywhere in a region of space large enough to contain the data. For each field v v v α (x x x) we compute its corresponding bulk flow, B α (r), according to equation (1) and write our ASCE bulk flow as
B ASCE (r) = α a α B α (r) .(9)
The Maximum Likelihood Estimate (MLE)
For comparison we will present estimates of the bulk flow obtained with the standard MLE (Kaiser 1988). This method approximates the bulk flow of a sphere of radius r as the vector B MLE which renders a minimum in
χ 2 = 1 σ 2 η cz i <r sM 0i + 2.17s B ·r r r i cz i + η 0 − η i 2(10)
with respect to the three components of B. The sum is over galaxies within r andr r r i is unit vector in the direction of galaxy i. Further, in this expression we have approximated
P = −5log(1 − B · r r r i /cz i ) ≈ 2.17B ·r r r i /cz i .
Tests
In order to test the performance of ASCE and the MLE reconstructions of the bulk flow from the SFI++ TF data we use 2200 mock catalogs of TF measurements. In each of the catalogues, galaxies with the same positions as in the real SFI++ data are assigned absolute magnitudes, M i , and line width parameters, η i , following an artificial ITF relation with slope s = −0.12 and intrinsic scatter σ η = 0.057 (e.g. Davis et al. 2011). The peculiar velocities of galaxies in each mock are taken from a linear gaussian random velocity field in a cubic box of 1454h −1 Mpc on the side. Each mock is placed randomly in this large box and the peculiar velocity of each galaxy is then obtained by interpolating the velocity field on the position of the galaxy. A gaussian random realizaton of the velocity field is generated for ΛCDM7 using the COSMICS package (Ma & Bertschinger 1995). Further, we work with a parametric form of the power spectrum taken from Eisenstein & Hu (1998) (eqs. 29-31 in their paper) For ASCE we still need to construct the basis velocity fields v v v α (x x x). Here we use N a = 120 basis functions, extracted in a similar way to the mocks, but from a completely different random realisation of a velocity field in a very large box. Thanks to the regularization term in (8), the ASCE inferred bulk flow has very little dependence on the actual value of N a as long as it is large enough to capture the main features of the 3D flow: very similar results are obtained with N a = 50 and N a = 120. Each of these 120 velocity fields v v v α (x x x) are further smoothed with a gaussian window of 10h −1 Mpc in width. The purpose of this small scale smoothing is to filter out low frequency modes which would be over-fitted by the data especially at large distances. This smoothing, however, has very little effect on the bulk flows reconstructed by ASCE. We emphasize that a basis function v v v α (x x x) is not only defined at the galaxy positions, but also at any point in a sufficiently large volume (radius 100h −1 Mpc) which contains all the galaxies used in the analysis. Hence, for each basis function we can measure its corresponding bulk flow, B α (r), and once coefficients a α have been determined from the data by minimization of (8) then the ASCE reconstructed bulk, B ASCE (r), is readily given by (9). Figure 1 shows scatter plots of the estimated versus true bulk flows of a spherical region of 60h −1 Mpc in radius. For clarity, results from only 400 randomly selected mock catalogues are shown. Blue dots and red plus signs correspond to B MLE and B ASCE , respectively. Because of the anisotropic distribution of the observed galaxies, the methods may not reconstruct the three cartesian components equally well. Hence, x, y and z bulk flow components in Supergalactic coordinates are, respectively, shown in the top, middle and bottom panels. Blue and red lines in each panel are linear regressions of the estimated on true bulk flows. The corresponding mathematical expressions of the regressions are indicated in each panel.
The regularization term in (8) naturally tends to underestimate the coefficients a α and subsequently the reconstructed bulk flow. However, the agreement between the ASCE reconstructed and the true bulk flows seen in figure (1) clearly demonstrates that the effect is meagre. To further explore the quality of the ASCE and MLE reconstructions and to ascertain that the regularization term does not cause a significant reduction in the amplitude of the bulk flow, we apply ASCE and MLE to the mock data but with true velocities amplified by a factor of 1.5. Everything else, including the regularization term in (8), remained the same. The reconstructed B ASCE and B MLE versus true amplified bulk flow are shown in figure (2). Both ASCE and MLE perform well even with this amplification of the bulk flow in the mocks.
In both ASCE and MLE, the slopes of the regression lines plotted in (1) are close, but not equal to unity. The deviation from unity is significant (compared to the scatter of the points) and persists when the regression is done using all the 2200 mock points. This small but statistically significant bias depends on the radius of the sphere for which bulk flow is computed. The bias can easily be calibrated using the mock catalogues. Hereafter, all reconstructed bulk flows, from ASCE and MLE, are corrected for the systematic bias in the mean of the estimated bulk given the mean of the true value. In practice we write the corrected estimate of the bulk flow B corr (r) from the raw bulk B raw (r) (directly reconstructed by either ASCE or MLE) as B corr (r) = C 1 B raw (r) + C 2 where C 1 is the ratio of the rms values of the true to raw bulk flows and C 2 is a constant term which accounts for the offset between the true and raw bulk flows.
In all panels of figure (1), the mock B ASCE are tightly scattered around their corresponding regression lines. The scatter in B MLE appears to be more significant. To quantify the scatter between the reconstructed and true bulk flows, we plot, in figure (3), the cumulative fraction, P < (θ), of mock catalogues for which the angle between estimated and true bulk flows is less than θ. The solid blue and red dashed curves refer to ASCE and MLE, respectively, while thick and thin to bulk flows within 40h −1 Mpc and 100h −1 Mpc, also respectively. The curves are computed after employing the correction to the systematic bias as explained above. The performance of ASCE is excellent. For 40h −1 Mpc, the direction of B ASCE is recovered within 3 • for about 68% of the mocks.
For 100h −1 Mpc this uncertainty increased to 7 • . The ASCE method is significantly superior to MLE. The thin blue and thick dashed lines almost overlap, meaning that the performance of ASCE for 100h −1 Mpc is as as good as that of MLE for 40h −1 Mpc. For r = 100h −1 Mpc, MLE recovers the direction only within 27 • for 68% of the mocks. Figure (4) plots the differential probability distribution function, P (δB), where δB refers to the difference in all cartesian components between estimated and true bulk flows, from the 2200 mocks. The notation of the lines is the same as in the previous figure and as displayed in the figure. The figure also indicates σ, the rms value of δB for the plotted cases. The low values of σ corresponding to ASCE demonstrate its excellent ability at recovering the true bulk. The performance of MLE is good, but less satisfactory.
Results
The results of the application of the ASCE and MLE methods to recover the bulk flow from the real SFI++ TF catalogue are summarized in figures (5)-(6). The bulk flows are reconstructed for spheres centered on the Milky-Way and of radii from r = 20h −1 Mpc to 100h −1 Mpc in steps of 10h −1 Mpc. The smallest radius is chosen large enough so that nonlinear effects are not expected to be important (Nusser et al. 1991), facilitating the comparison with cosmological models. The largest radius corresponds to the distance within which the data are used. Figure (6). The 1σ errorbars in both figures are based on the 2200 mocks. The component B y is clearly the most significant. This is just a coincident and has no bearing on the statistical analysis of the results as one can always choose a coordinate system such that the bulk is along a given axis.The solid curve in this figure is the theoretical expectation of ΛCDM7 with but with σ 8 = 0.85 instead of the default σ 8 = 0.8. The theoretical curve is computed given the density power spectrum p δ (k, Ω m , Ω b , h, n s ) by
σ 2 v (r) = H 2 0 f 2 2π 2 dk p δ (k)W 2 (kr)(11)
where f = Ω γ m with a growth index γ ≈ 0.55 for a flat Universe (Linder 2005), and W = W TH (kr)exp(−k 2 R 2 s /2) with W TH is the top-hat window function and the gaussian window takes care of the fact that the basis functions v v v α used in ASCE are smoothed with a gaussian window of R s = 10h −1 Mpc in width. The expression (11) is obtained assuming the linear relation
H 0 f δ = −∇ · v v v(x x x)
between the density contrast, δ, and v v v(x x x) Peebles (1980).
The MLE and ASCEreconstructed bulks are similar especially at large distances r > 40h −1 Mpc. This is because, the data covers space more isotropically at larger distances. Figure (2) clearly demonstrates that ASCE will not cause a significant artificial under-estimation of large bulk flows such as reported in Feldman et al. (2010). To further, ascertain that our ASCE derived bulk flow is robust, the blue circles in figure (7) the amplitude of the ASCE bulk flow reconstructed using basis functions generated from a ΛCDM7 power spectrum but with a scalar index n = 0.75 and σ 8 = 1. The results are very similar to the ASCE bulk flow shown in figure (6) despite the significantly enhanced large scale power. The agreement is particularly striking at large distances.
Comparison with cosmological models
Figure (6) indicates that the estimated bulk flows are consistent with the theoretical expectations. But the errors are strongly correlated and a proper statistical analysis must take into account the covariance of the errors. Our large number (2200) of mock catalogues allows a robust determination of the error covariance function between the bulk flow estimates at different radii. Since ASCE is significantly superior to MLE, we restrict the comparison with models to ASCE reconstruction. We use all components of B ASCE estimated at 8 values of r ranging from r = 30h −1 Mpc to 100h −1 Mpc in steps of 10h −1 Mpc. The reason for not considering smaller radii is that the bulk is most robustly constrained independent of the assumed basis functions at r > 30h −1 Mpc. We denote the set of ASCE reconstructed cartesian components at these 8 values of r by B t and the corresponding underlying true quantities by B t . We write the probability for observing the set B o as
P (B o ) = dB t P (B o |B t )P (B t )(12)
where the probability P (B t ) for the underlying B t is computed within the framework of a cosmological model. Here, we adopt the ΛCDM model. For gaussian velocity fields, the calculation of P (B t ) is easily done by integrating standard analytic expressions involving the power spectrum. We assume that the probability P (B o |B t ) for B o given B t is gaussian with error covariance matrix computed from the 2200 mocks. Under these assumptions, the expression (12) yields Fig. 7.-The same as the previous figure with ASCE basis functions generated using a ΛCDM7 power spectrum but with scalar index n = 0.75 and σ 8 = 1, which has more power on large scales compared to our standard choice n = 0.963.
P (B o ) = 1 (2π) d |Σ| exp − 1 2 B T o Σ −1 B o ,(13)
Consistency with ΛCDM7
We begin by assessing how well ΛCDM7 is consistent with the data. To do that we generated 10 7 sets, B rnd , each containing d = 18 numbers selected at random from a gaussian distribution given by (13) computed with Σ t for ΛCDM7. For each of those 10 7 sets of B rnd we compute the corresponding P (B rnd ) using (13) and tabulate the negative of the log of the probability, nlP rnd = −lnP (B rnd ). We also compute nlP o = −lnP (B o ) for the observed B o also using ΛCDM7. We find that only 26% of the 10 7 values of nlP rnd exceed nlP o . Therefore, the ΛCDM7 cannot be rejected by the bulk flow results.
Independent constraints on σ 8 and γ
The ΛCDM expected amplitude of the bulk flow depends separately on the cosmological parameters (see equation 11 and the parametric form for the power spectrum in Eisenstein & Hu (1998)). But the most significant dependence is on σ 8 and Ω m and hence we restrict ourselves here to deriving constraints on these two parameters only. We compute nlP o for a grid of values of Ω m and σ 8 used in Σ t , maintaining all other parameters at their default ΛCDM7 values. (Press et al. 1992). The ΛCDM7 point is well within the 68% confidence level. The shape of the contours implies the correlation σ 8 ∼ Ω −0.28 m . This reflects the dependence of the shape of the density power spectrum p δ on Ω m and from the factor f (Ω) ≈ Ω 0.55 (see eq. 11). Only if we neglect the dependence of the shape of p δ on Ω m we get σ 8 ∼ Ω 0.55 m . It is of interest to inspect the constraints when either of the parameters Ω m or σ 8 is fixed at certain values. Figure (9) shows two curves of ∆χ 2 versus σ 8 corresponding to the WMAP7 Ω m = 0.266 and 0.236 giving a minimum of ∆χ 2 in the (Ω m , σ 8 ) plane as seen in figure (8). Figure (10) plots ∆χ 2 as a function of Ω m , for σ 8 at the WMAP7 value of 0.8 and at 0.88 corresponding to the minimum of ∆χ 2 in figure (8). In each of the curves in figures (9) and (10), the value of ∆χ 2 at the minumum of the curve is set to zero. Hence, the δχ 2 = 1 and 4 correspond to 68% (1σ) and 95% (2σ) CLs, respectively (Press et al. 1992). These curves assume a growth index γ = 0.55 as is appropriate for a ΛCDM model. Hence figure (9) gives σ 8 = 0.86 ± 0.11 (1σ) for Ω m = 0.266 and γ = 0.55. However, we see from (11) that by varying γ alone we get the scaling σ 8 (γ = 0.55) = σ 8 (γ)Ω γ−0.55 m . We can use this to set a constraint on γ if we adopt σ 8 = 0.8 and Ω m = 0.266 (Larson et al. 2010). Demanding that σ 8 (γ) = 0.8, the scaling gives γ = 0.496 ± 0.096. Figure (11) confirms this result. The figure plots ∆χ 2 as a function of γ for the adopted values of σ 8 and Ω m as indicated. The left and right arrows mark the values γ = 0.42 and 11/16. The lower value is expected in f (R) models (e.g. Gannouji et al. 2009) and the highest corresponds to a Dvali-Gabadadze-Porrati (DGP) (Dvali et al. 2000;Wei 2008) flat braneworld cosmology. We could also substitute the scaling with γ in the correlation σ 8 ∼ Ω −0.28 m obtained from the contour plot to get the constraint σ 8 Ω γ−0.55 m (Ω m /0.266) 0.28 = 0.86 ± 0.11 between γ, Ω m and σ 8 .
Discussion
The analysis presented here uses a trimmed version of the SFI++ in which galaxies fainter than M = −20 are removed. This ensures the linearity of the TF relation. Further, to avoid dealing with selection effects imposed on the magnitudes we use the inverse TF relation (see Strauss & Willick (1995) for a thorough review of this issue). Further, to minimize inhomogeneous Malmquist bias (Lynden-Bell et al. 1988), we do not place galaxies at their TF inferred distances, but at their measured redshifts which has significantly smaller observational errors. We also collapse the main known galaxy clusters.
The bulk flows estimated here are remarkably featureless and do not seem to reflect the gravitational effects of any of the individual main nearby clusters. Bulk flow estimates from TF-like relations are traditionally featureless (e.g. Dekel 1994), in contrast to velocity dipoles estimated by from the distribution of galaxies in redshift surveys (e.g. Nusser & Davis 1994). In the analysis here, we have collapsed clusters and therefore, signatures of individual clusters, in our estimated bulk flows, could be smeared out. It is instructive to explore how much we are missing by collapsing clusters and wether signature of infall on clusters of nearby galaxies can actually be clearly seen in SFI++ or similar data. As an illustrative representative case we plot in figure 12 individual peculiar velocities of 54 SFI++ galaxies contained in a cylinder of 6 • in radius and of 2600 km s −1 centered on the Virgo cluster. The individual distances, d TF to galaxies are obtained from the observed galaxy deviation from the straight line describing the inverse TF relation as determined by the 54 galaxies. The individual radial peculiar velocities are then given by V TF = cz − H 0 d TF . Red circles in this figure show V TF versus cz while the blue plus signs correspond to V TF versus d TF . The solid line is obtained by statistical regression of V TF on cz, i.e. the red pints. The two straight dashed lines correspond to 95% confidence levels of this regression. The blue points show a pattern that could mistakenly be confused with actual galaxy infall onto Virgo. However, this pattern is entirely due to inhomogeneous Malmquist bias: Galaxies scattered to large estimated d TF beyond the cluster, will also have a negative inferred V TF . The effect of this Malmquist bias will be more pronounced for more distant clusters which have larger absolute errors on d TF . The red points do not show a clear infall (on Virgo) of galaxies in the immediate vicinity of Virgo. Karachentsev & Nasonova (2010) presents an impressive study of the observed flow of 1792 galaxies near Virgo. Taken into account the inhomogeneous Malmquist bias, it is hard to detect a clear infall signature in the near vicinity of Virgo in this study as all (although the focus of their paper is different).
The constraints given in figures (8)-(11) show that the bulk flow alone provides useful additional constraints on the cosmological parameters. To achieve tighter constraints one must investigate the full information in the peculiar velocity measurements. This could be done by analysis of power spectra and correlation functions by maximum likelihood techniques (e.g. Gorski et al. 1989;Jaffe & Kaiser 1995;Zaroubi et al. 1997;Juszkiewicz et al. 2000;Bridle et al. 2001;Abate & Erdogdu 2009). However, the bulk flow is particularly appealing because of its simplicity and the fact that it is entirely linear for sufficiently large spheres. The constraints from peculiar velocities, including the bulk flow, are unique since they are local at redshifts very close to zero and they directly probe the growth index γ = dlnf dΩm where f is the linear growth factor (Peebles 1980;Linder 2005). Adopting the WMAP7 cosmological parameters (Larson et al. 2010), we derive a local constraint γ = 0.495 ± 0.096. This constraint is completely independent of the biasing relation between galaxies and mass. Further, it is essentially a constraint at z = 0. In contrast, the lowest redshift constraint obtained from a study of redshift distortions in the 2dF galaxy redshift survey is at z ≈ 0.15 (Hawkins et al. 2003). Our constraint significantly improves on previous constraints on γ (Dossett et al. 2010;Wei 2008) derived at higher redshifts. This result could help us distinguish between alternative theories for structure formation (e.g. Amendola et al. 2005;Guzzo et al. 2008;Keselman et al. 2010). For σ 8 = 0.8, the constraint on γ disfavors DGP models at ∼ 2σ level, but it is consistent with f (R) gravity models (e.g. Starobinsky 2007;Gannouji et al. 2009;Wu et al. 2009;Fu et al. 2010). But σ 8 in these models should be computed self-consistently, assuming the same normalization at the recombination epoch. Based on WMAP7, this implies σ 8 = 0.63 and 0.855 for DGP and f (R) models, respectively. Adopting these σ 8 values for these models we get γ = 0.315 ± 0.091 and 0.55 ± 0.098, respectively, for SGP and f (r) (Cinzia Di Porto, priv. comm). In the DGP model, the expected value for γ at z = 0 is 0.664 (Wu et al. 2009), which is ruled by our constrain at more than the 3 σ level. The f (R) model cannot be ruled out at high confidence level by the constraint derived here.
Our results are in agreement with the analysis of Sandage et al. (2010). Peculiar velocities from supernovae, although very sparse, yield bulk flows that are consistent with WMAP7 (Dai et al. 2011;Colin et al. 2011), as we do. The results are in agreement with the WMAP7. The analysis of Bilicki et al. (2011) of the 2MASS dipole from galaxy fluxes is also in agreement with the WMAP7 LCDM. The direction of the bulk flow is robust and agrees with the direction of the motion of the local group (LG) after correcting for the Virgocentric infall (Sandage et al. 2010). But we disagree strongly with the bulk flows Feldman et al. (2010) who find a significant large bulk flow at r = 100h −1 Mpc, using the untrimmed SFI++ survey, other individual data sets and also using a composite catalogue. They also use the TF estimated distances instead of the redshifts in their analysis of the bulk flow, leading to results which are highly susceptible to Malmquist bias. We have opted to use a single uniformly calibrated catalog, namely the SFI++, excluding faint galaxies which spoil the linearity of the ITF (Davis et al. 2011). We have also refrained from using composite data since minor mis-calibration errors between different catalogs could lead to large artificial flows when these catalogues are combined. Further, we place galaxies at their measured redshifts rather than estimated distances from the TF relation. This greatly suppresses inhomogeneous Malmquist bias which is known to lead to significant spurious signal especial at large distances. We also refrained from using gaussian window so that the bulk flow within a certain radius is completely unaffected by the increasing uncertainties at large distances.
We have seen that the MLE and the ASCE methods give very similar results. Further, the ASCE bulk flow at r > 30h −1 Mpc is almost completely independent of the cosmological model used in generating the basis functions. This is clearly demonstrated by the comparison of figures 6 and 7. However, in ASCE even if the results turned out to be sensitive to the assumed model used in generating the basis function, the validity of the model can still be confidently assessed. The reason is that the sensitivity would imply that the data are insufficient for constraining the bulk flow within the framework of the assumed model used in generating the basis function. Fortunately, this ambiguity is irrelevant for the SFI++ used here since the corresponding bulk flow is extremely insensitive to the model used in generating the basis functions.
Acknowledgments
Special thanks are due to Enzo Branchini for many stimulating discussions. We thank Cinzia Di Porto for providing σ 8 for the DGP and f (R) models. This work was supported by THE ISRAEL SCIENCE FOUNDATION (grant No.203/09), the German-Israeli Foundation for Research and Development, the Asher Space Research Institute and by the WINNIPEG RESEARCH FUND. MD acknowledges the support provided by the NSF grant AST-0807630. -Individual peculiar velocities, V TF , of galaxies in the line-of-sight to Virgo, plotted against the redshift (red circles) and the estimated distance d TF (blue plus signs). The centroid of these galaxies is at cz ∼ 0 and V TF ∼ 0.
Fig. 1 .
1-Scatter plots of estimated versus true bulk flows in 400 mock catalogues. The top, middle and bottom panels correspond to the Supergalactic x, y and z components of the bulk flow. Plotted are bulk flows in a spherical region of r = 60h −1 Mpc centered at the origin. Bulk flows from ASCE and MLE are represented as blue dots and red crosses, respectively. The overlaid lines in each panel are linear regressions.
Fig. 2 .
2-The same as the previous figure but where the peculiar velocities in the mocks have been amplified by a factor of 1.5.
Fig. 3 .
3-The cumulative fraction of mock catalogues with estimated bulks directed within angle smaller than θ from the direction of the true bulk flow. Blue solid lines and red dashed lines, respectively, correspond to the ASCE and MLE reconstructions. Thick and thin lines, respectively, refer to bulk flows of spheres of radii 40h −1 Mpc and 100h −1 Mpc centered on the observer.
Fig. 4 .
4-The differential distribution functions of the difference between estimated and true respective cartesian components. The notation of the lines is the same as in the previous figure.
(5) shows the Galactic x (blue dotted), y (black solid), and z (red dot-dashed) components of B ASCE (top panel) and B MLE (bottom) as a function of r. The magnitudes of B ASCE and B MLE versus radius are plotted, respectively, as the blue circles and plotted in figure
Fig. 5 .Fig. 6 .
56where d is the number of elements in B o , i.e. d = 24 = 8 × 3; for 8 values of r and 3 cartesian components. The d × d covariance matrix Σ = Σ o + Σ t , where Σ o is the covariance of the errors on B o and Σ t describes the covariance of the underlying quantities B t . The dependence on the cosmological models comes through Σ t . -The three Galactic cartesian components of the bulk flow as a function of radius. Top and bottom panels correspond to ASCE and MLE estimation, respectively. -The amplitude of the bulk as a function of distance for ASCE and MLE as indicated in the figure. The solid curve shows the rms value of the bulk flow as expected in a flat Universe ΛCDM model with Ω m = 0.266, h = 0.71 and σ 8 = 0.85.
Confidence levels (CLs) on Ω m and σ 8 are obtained by inspecting the contours of ∆χ 2 (Ω m , σ 8 ) = 2(nlP o − min(nlP o )) in the (Ω m , σ 8 ) plane. The minimum of nlP o (i.e. ∆χ 2 = 0 ) is at (Ω m , σ 8 ) = (0.236, 0.88), marked by the plus sign in the figure. The ΛCDM7 default values (Ω m , σ 8 ) = (0.266, 0.8) are indicated by the circle. The inner and outer contours of ∆χ 2 shown in figure (8) correspond to 68% and 95% CLs for two degrees or freedom
Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .
891011-Contour plot of the 68% and 95% confidence levels in the Ω m − σ 8 place. The plus sign marks the maximum of the probability distribution function at (Ω m , σ 8 = (0.236, 0.88), while the circle indicates (0.266, 0.8), corresponding to the best fit WMAP7 values. -Curves of ∆χ 2 as a function of σ 8 for Ω m = 0.266 (blue solid line) and Ω m = 0.236 (red dot-dashed). -Curves of ∆χ 2 as a function of Ω m for σ 8 = 0.8 (blue solid) and σ 8 = 0.88 (red dot dashed). -Curves of ∆χ 2 as a function of the growth index γ given σ 8 = 0.8 and Ω m = 0.266. The left and right arrows, respectively, indicate γ values obtained in f (R) and flat DGP models.
Fig. 12.-Individual peculiar velocities, V TF , of galaxies in the line-of-sight to Virgo, plotted against the redshift (red circles) and the estimated distance d TF (blue plus signs). The centroid of these galaxies is at cz ∼ 0 and V TF ∼ 0.
This preprint was prepared with the AAS L A T E X macros v5.2.
A. The regularization termThe motivation for the second term on the r.h.s in equation(8)can be found in a a Bayesian statistical formulation. Our model for the three dimensional velocity field defined at any point in space) each corresponding to a realization of a random gaussian field with a cosmologically motivated power spectrum. Given the data, the probability Pwhere(5). This probability function gives rise to the first term on the r.h.s in equation(8). The prior function P (v v v M ) is the probability for the realization of the particular velocity field model v v v M independent of the data. For a gaussian random fieldwhere the indices J and J ′ refer to the three velocity components and Ξ(x x x, x x x ′ , J, J ′ ) is the velocity correlation function. Substituting the expansion vSince v v v α are all space independent fields, the terms with α = β will be negligibly small. Since all of v v v α are generated with the same power spectrum, the termIf the model is also to represent a random realization of the same power spectrum as each of the basis functions v v v α then the sum in (A2) must also be equal to S. Hence P (v v v M ) ∝ P ({a α }) ∝ exp(− α (a α ) 2 /2) which leads to the second term in (8).
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| []
|
[
"Quantum-gravitational trans-Planckian energy of a time-dependent black hole",
"Quantum-gravitational trans-Planckian energy of a time-dependent black hole"
]
| [
"A J Nurmagambetov \nAkhiezer Institute for Theoretical Physics of NSC KIPT\n1 Akademicheskaya St61108KharkovUAUkraine\n",
"I Y Park \nDepartment of Applied Mathematics\nPhilander Smith College Little Rock\n72223ARUSA\n\nKarazin Kharkov National University\n4 Svobody Sq61022KharkovUA\n",
"\nUsikov Institute for Radiophysics and Electronics\n12 Proskura St61085KharkovUA\n"
]
| [
"Akhiezer Institute for Theoretical Physics of NSC KIPT\n1 Akademicheskaya St61108KharkovUAUkraine",
"Department of Applied Mathematics\nPhilander Smith College Little Rock\n72223ARUSA",
"Karazin Kharkov National University\n4 Svobody Sq61022KharkovUA",
"Usikov Institute for Radiophysics and Electronics\n12 Proskura St61085KharkovUA"
]
| []
| We continue our recent endeavor in which a time-dependent black hole solution of a one-loop quantum-corrected Einstein-scalar system was obtained and its near-horizon behavior was analyzed. The energy analysis led to a trans-Planckian scaling behavior near the event horizon. In the present work the analysis is extended to a rotating black hole solution of an Einstein-Maxwell-scalar system with a Higgs potential. Although the analysis becomes much more complex compared to that of the previous, we observe the same basic features, including the quantum-gravitational trans-Planckian energy near the horizon. | 10.3390/sym11101303 | [
"https://arxiv.org/pdf/1909.10054v1.pdf"
]
| 208,224,606 | 1909.10054 | a02009c516fe10601e9f9c45455ae0a6bf87c80a |
Quantum-gravitational trans-Planckian energy of a time-dependent black hole
22 Sep 2019
A J Nurmagambetov
Akhiezer Institute for Theoretical Physics of NSC KIPT
1 Akademicheskaya St61108KharkovUAUkraine
I Y Park
Department of Applied Mathematics
Philander Smith College Little Rock
72223ARUSA
Karazin Kharkov National University
4 Svobody Sq61022KharkovUA
Usikov Institute for Radiophysics and Electronics
12 Proskura St61085KharkovUA
Quantum-gravitational trans-Planckian energy of a time-dependent black hole
22 Sep 2019quantum gravityloop effectsFirewalltrans-Planckian energy
We continue our recent endeavor in which a time-dependent black hole solution of a one-loop quantum-corrected Einstein-scalar system was obtained and its near-horizon behavior was analyzed. The energy analysis led to a trans-Planckian scaling behavior near the event horizon. In the present work the analysis is extended to a rotating black hole solution of an Einstein-Maxwell-scalar system with a Higgs potential. Although the analysis becomes much more complex compared to that of the previous, we observe the same basic features, including the quantum-gravitational trans-Planckian energy near the horizon.
Introduction
We are entering an era of muti-messenger astrophysics, and a substantial amount of new data is being collected for various astronomical objects. It is by now firmly established that diverse ultra-high-energy cosmic rays (UHE-CRs) of extra-galactic origins constantly bombard Earth's atmosphere. Since the energy scale of these particles -∼ 10 19 eV -far exceeds that of LHC, study of their origin may well allow us to take a leap in solving some of as-yet unsolved problems in the field.
Although the accumulated data indicate that active galactic nuclei (AGNs) should be largely responsible for the generation of UHECRs, the precise mechanism is yet to be understood. There is wide consensus that the UHE-CRs must be the work of the super-massive black holes at the centers of the active galaxies. Therefore the focus of one's quest should be the physics that can produce various extreme-high-energy particles -such as gamma ray photons, protons, heavier ions, and neutrinos -in massive volume. Motivated by this and more theoretically-oriented issues, such as black hole information (see, e.g., [1] [2] for reviews) and Firewall [3] [4], we have initiated in [5][6][7][8] the study of quantum gravitational effects as the potential agent behind certain astrophysical phenomena, including the generation of the UHECR particles.
It is conventionally believed that the quantum gravitational effects are largely negligible. (The same has been believed in the astrophysical situations such as in astrophysical black hole environs.) This view has been challenged through a series of recent works, according to which the quantumgravitational effects may not only be observable but may also be behind some of the spectacular astrophysical phenomena. In this work we continue our recent endeavor in which a time-dependent black hole solution of a one-loop quantum-corrected Einstein-scalar system was obtained and its near-horizon behavior was analyzed. The energy analysis led to a trans-Planckian energy behavior near the event horizon, which should lead to observable effects. We extend the analysis to an Einstein-Maxwell-scalar system in the present work.
To put things in an orderly perspective, let us summarize the status of the matters surrounding the new gravity quantization approach proposed in [9]. A gravity theory can be shown to be one-loop offshell renormalizable in the the conventional covariant quantization framework: for instance it was shown in the classic paper by 't Hooft and Veltman [11] that pure Einstein gravity is one-loop renormalizable. Once matter fields are included, the one-loop offshell renormalizability is lost. (However, the renormalizability is restored once one includes a cosmological constant that provides more leverage to absorb the one-loop ultraviolet divergences.) At two-loop, things become worse and the conventional offshell renormalizability is lost. Therefore, although one could conduct various quantum-level studies, the nonrenormalizability would force one to stipulate that the results be taken up to the issue of renormalizability, which has been a frustrating setback to further progress. Motivated by the holographical reduction of the physical states [9], 1 the covariant quantization has recently been revisited [12]: a way out of the longstanding nonrenormalizability has been proposed, establishing the renormalizability of the physical states that are a certain subset of the perturbative offshell states. Explicit one-loop renormalization procedures have been worked out for several gravity-matter systems [13]. With the new scheme of quantization, one can be assured that the one-loop analysis will remain valid even to higher loops. Further out, the development has provided a stage for further investigation of quantum-gravitational effects and their applications. Through our recent works it has been shown that the quantum-gravitational effects are of "order-1" in the sense to be reviewed below.
What sets the present work apart from the previous works is that the system being considered is more realistic: we consider an Einstein-Maxwellscalar system with a Higgs-type potential and its rotating black hole solution. As in [6] [7] 2 , the analysis leads to a trans-Planckian energy. The trans-Planckian energy behavior may well be a generic feature of a time-dependent black hole configuration at the quantum level.
The rest of the paper is organized as follows.
In section 2, before embarking on the technical analysis, we give a qualitative reasoning on why there ought to be a trans-Planckian energy behavior near the event horizon. In section 3, after reviewing the Einstein-scalar system analyzed in [7], we obtain a time-dependent quantum-level solution of an Einstein-Maxwell-scalar system. We consider the Λ 0 = 0 case -where Λ 0 = 0 denotes the classical part of the cosmological constant -for a reason to be explained; extension to the Λ 0 = 0 case is left for future. The classical part of the quantum-level solution is required to settle down to a Kerr geometry. A novel feature observed in [15,16] is shared: the quantum effects remove the time-dependence of the classical part, which is crucial for the subsequent energy analysis. In section 4, we analyze the energy observed near horizon by an infalling observer and are led to a trans-Planckian energy. Although the quantum-induced trans-Planckian energy may sound radical, the result is obtained within the norm of quantum field-theoretic techniques. 3 Toward the end of section 4, we briefly comment on the boundary conditions. We reason that the perfect infall boundary condition that is used in the context of the quasi-normal modes is rather restrictive and that more general boundary conditions must be considered to describe the physics of the ring-down phase of a black hole. In section 5, which is the concluding section, we summarize the results and list future directions.
Physical origin of trans-Planckian energy
The present work is motivated in part by the Firewall proposal; let us briefly review the Firewall argument. The backbone of the Firewall argument is as follows. For simplicity let us take a Schwarzschild black hole. Consider the Kruskal observer and the corresponding vacuum. The Kruskal vacuum must not be a vacuum to a Schwarzschild observer and should appear to be radiating -which is nothing but the Hawking radiation -to a Schwarzschild observer. Now let us consider things in 'reverse': consider a Schwarzschild observer and the corresponding vacuum (or an eigenstate of the observer's number operator). Similarly as before, the Schwarzschild vacuum (or the eigenstate) must not be a vacuum to a Kruskal observer. What makes this part of the physics more dramatic is that the Kruskal observer is infalling so the radiation the observer will encounter is highly blue-shifted near the horizon, a Firewall.
One of the goals of the present work (and its sequels) is to back up the Firewall proposal by a quantitative analysis in a more realistic astrophysical environment. What we set out to check in the present work is conceptually simple but highly complicated technically: for one thing, we intend to calculate the quantum-corrected energy measured by an infalling observer near the horizon of a time-dependent rotating black hole of a Einstein-Maxwellscalar system. To this end, one needs the quantum-corrected action and the field equation with its solution: in particular, a time-dependent solution. Afterwards, one needs to work out the four-velocity of the observer, U µ , in the quantum-corrected background. The stress-energy tensor T µν is obtained by taking the functional derivative of the matter part of the action with respect to the metric. We quote the classical part of it here for convenience:
T µν = − 2 κ 2 Λg µν + g µν − |∂ ρ ψ − iqA ρ ψ| 2 − m 2 |ψ| 2 − 1 4 F 2 ρσ + [(∂ µ ψ − iqA µ ψ)(∂ ν ψ * + iqA ν ψ * ) + (µ ↔ ν)] + F µρ F ν ρ + O( ),(1)
where g µν , A ρ , ψ denote the metric, vector field, and scalar, respectively. The energy density ρ measured by the observer is given by ρ ≡ U µ U ν T µν where the full quantum-level solution is to be substituted into T µν . Finally, one evaluates the contributions from each term in T µν . As we will see, some of the terms in ρ exhibit a trans-Planckian behavior, which is also an overall behavior of the entire energy density ρ.
There is a clear qualitative way to see how the presence of the quantum correction part leads to a trans-Planckian energy (whereas the purely classical analysis doesn't): when computing the energy density, evaluate the righthand side of the metric field equation, T µν = Gµν 8πG , and contract with the four-velocities. As we will see, the classical metric comes to take the same form as the classical Kerr form that yields a finite energy (zero energy, more precisely). Being an additional contribution to the stress-energy tensor, the presence of the quantum modes changes this status (as we will see in more detail) by directly analyzing the stress-energy tensor in section 4.
One may wonder about the physical origin of such a non-smooth structure in the vicinity of the horizon. Consider a particle heading toward the black hole. It will produce other particles through the quantum-field-theoretic chain reactions on its way to the event horizon. 4 As an infalling particle accelerates toward the black hole, the acceleration would increase without bound and the particle will release the energy though various radiation channels such as synchrotron radiation and bremsstrahlung.
One thing worth noting is that the quantum-gravitational effects introduce non-minimal coupling terms such as ψψ * R. Although the effect of such nonminimal terms is small (and will not play a role in this work), they will lead to violation of the Equivalence Principle. To recap, on one hand, the quantum gravitational effects are responsible for the trans-Planckian energy, and on the other hand, they produce such non-minimal couplings, which are at odds with the classical-level understanding of the Equivalence Principle and/or should limit the range of validity thereof.
Time-dependent solutions
Although it is conventionally believed that the quantum gravitational effects are largely negligible, in general there are circumstances, as our recent works have revealed, in which the quantum-gravitational effects are of "order-1". Such effects may be behind some of the highly energetic astrophysical phenomena. In the previous works [6] [7], we considered an Einstein-scalar system and examined the possibility that the quantum corrections may produce a violent energy behavior in the vicinity of the event horizon. It was shown that for a time-dependent solution, one indeed encounters such a behavior. In the present work we consider an Einstein-Maxwell-scalar system in extension of the previous works.
In section 3.1 we start by reviewing our earlier work of [7] on an Einsteinscalar case. We highlight some salient results of the analysis and point out an undesirable feature of the solution therein obtained. This motivates introduction of a potential in the scalar sector; we consider a Higgs-type potential. In section 3.2, a time-dependent solution of the Einstein-Maxwell-scalar system with a Higgs potential is obtained. For a reason we will detail, we set the classical part of the cosmological constant to zero, Λ 0 = 0; extension to Λ 0 = 0 case will be pursued elsewhere.
black hole while the other one escapes, thereby causing the information loss at the end.
When the BH undergoes an active accretion, such effects must produce a sufficient amount of the outgoing flux for detection. In other words, the quantum-gravitational effects should lead to mass production of cascading particles, some of which escape the black hole.
Review of Einstein-scalar case
Let us warm up by reviewing the case considered in [7], in which an Einsteinscalar system was considered. The classical action of [7] is
S = 1 κ 2 d 4 x √ −g R − 2Λ − d 4 x √ −g 1 2 (∂ µ ζ) 2 + 1 2 m 2 ζ 2 .(2)
It admits an AdS black hole solution,
ds 2 = − 1 z 2 F dt 2 + 2dtdz + Φ 2 (dx 2 + dy 2 )(3)
with ζ = 0, and
F = − Λ 3 − 2M z 3 , Φ = 1 z .(4)
The one-loop 1PI effective action is given by [12] S = 1
κ 2 d 4 x √ −g R − 2Λ − d 4 x √ −g 1 2 (∂ µ ζ) 2 + 1 2 m 2 ζ 2 + 1 κ 2 d 4 x √ −g e 1 κ 4 Rζ 2 + e 2 κ 2 R 2 + e 3 κ 2 R µν R µν + e 4 κ 6 (∂ζ) 4 + e 5 κ 6 ζ 4 + · · · ,(5)
where the e's are numerical constants that can be determined with the chosen renormalization conditions. The quantum-level field equations can be obtained by varying the action of eq. (5). 5 The quantum system admits the following form of the time-dependent metric solution [17]:
ds 2 = − 1 z 2 F (t, z)dt 2 + 2dtdz + Φ 2 (t, z)(dx 2 + dy 2 )(6)
with the quantum-corrected series
F (t, z) = F 0 (t) + F 1 (t)z + F 2 (t)z 2 + F 3 (t)z 3 + ... + κ 2 F h 0 (t) + F h 1 (t)z + F h 2 (t)z 2 + F h 3 (t)z 3 + ... , Φ(t, z) = 1 z + Φ 0 (t) + Φ 1 (t)z + Φ 2 (t)z 2 + Φ 3 (t)z 3 + ... + κ 2 Φ h −1 (t) z + Φ h 0 (t) + Φ h 1 (t)z + Φ h 2 (t)z 2 + Φ h 3 (t)z 3 + ... .(7)
Similarly, for the scalar:
ζ(t, z) = ζ 0 (t) + ζ 1 (t)z + ζ 2 (t)z 2 + ζ 3 (t)z 3 + ... + κ 2 ζ h 0 (t) + ζ h 1 (t)z + ζ h 2 (t)z 2 + ζ h 3 (t)z 3 + ... ,(8)
where the modes with superscript 'h' represent the quantum modes. The cosmological constant is set to
Λ ≡ Λ 0 + κ 2 Λ 1 ,(9)
where has been explicitly displayed for convenience. The order-by-order analysis of the field equations in z and , for the classical modes, leads to
m 2 = 2Λ 0 3 , ζ 0 = 0, F 0 = − Λ 0 3 , Φ 1 = 0, F 1 = −F 0 Φ 0 − Λ 0 Φ 0 , W 2 = 0, F 2 = 1 4 4F 0 Φ 0 2 − 8∂ t Φ 0 , ζ 3 = 0, Φ 3 = 0, F 3 = const, ζ 4 = 0, Φ 4 = 0, F 4 = −F 3 Φ 0 ;(10)
and for the quantum modes, to
ζ h 0 = 0, F h 0 = − 1 3 Λ 1 , Φ h 1 = 0, F h 1 = 2 3 3F h 0 Φ 0 + Λ 0 Φ 0 Φ h −1 − Λ 0 Φ h 0 − 3∂ t Φ h −1 , Φ h 2 = 0, F h 2 = 1 3 − Λ 1 Φ 0 2 + 2Λ 0 Φ 0 2 Φ h −1 − 2Λ 0 Φ 0 Φ h 0 + 6Φ h −1 ∂ t Φ 0 − 6∂ t Φ h 0 , ζ h 3 = − 1 Λ 0 Λ 0 ζ h 1 Φ 0 2 + 2Λ 0 ζ h 2 Φ 0 + 3Φ 0 ∂ t ζ h 1 + 3ζ h 1 ∂ t Φ 0 + 3∂ t ζ h 2 , Φ h 3 = 0, ∂ t F h 3 = −3F 3 ∂ t Φ h −1 , F h 4 = F 3 Φ 0 Φ h −1 − F 3 Φ h 0 − F h 3 Φ 0 , Φ h 4 = −3e 2 F 3 Φ 0 2 + 3e 2 F 5 − 2e 3 F 3 Φ 0 2 + 2e 3 F 5 , ζ h 4 = F 3 ζ h 1 2Λ 0 + 12∂ t ζ h 1 ∂ t Φ 0 Λ 2 0 + 6Φ 0 ∂ 2 t ζ h 1 Λ 2 0 + 6ζ h 1 ∂ 2 t Φ 0 Λ 2 0 + 6∂ 2 t ζ h 2 Λ 2 0 + 9Φ 0 2 ∂ t ζ h 1 Λ 0 + 9Φ 0 ∂ t ζ h 2 Λ 0 + 9ζ h 1 Φ 0 ∂ t Φ 0 Λ 0 +2ζ h 1 Φ 3 0 + 3ζ h 2 Φ 2 0 .(11)
The subsequent analysis then leads to a trans-Planckian energy for which the quantum modes played a crucial role (see [7] for more details; the corresponding analysis will be carried out in section 4). As noted in [15], consideration of the field equations at the quantum level leads to additional constraints among some of the classical modes as well, and in particular yields
ζ 1 = 0 = ζ 2 .(12)
Because these two modes serve as the building blocks of the higher modes, the entire tower of the classical modes comes crumbling down. This shows that the quantum-level field equations deform the classical part as well, although one may naively expect that the classical part will remain intact. Many features of the analysis in [7], including the one just mentioned, are present in the Einstein-Maxwell-scalar system, as we will see. The solution above has an undesirable feature. Because the quantum effects force the entire classical part of the scalar field to vanish, the classical part of the cosmological constant Λ 0 must vanish as well. With 1 Λ 0 appearing in some of the mode relationships above, this can potentially be a problem. It turns out that this is not a genuine problem: one can set Λ 0 = 0 from the beginning. 6 Not unrelated to this, one of the mode relations, m 2 = 2Λ 0 3 , does not look natural. These observations motivate introduction of the scalar potential and, while doing so, we also introduce a Maxwell's field to make the system even more realistic. We will come back to these issues in more detail toward the end of the next subsection.
Einstein-Maxwell-scalar system
With the review of an Einstein-scalar system, we now turn to a more realistic system of an Einstein-Maxwell-scalar system with a Higgs potential. The action is given by [13]
S = 1 κ 2 √ −g R − 2Λ + d 4 x √ −g c 1 R 2 + c 2 R µν R µν + · · · − 1 4 √ −g F µν F µν − d 4 x √ −g |∂ µ ψ − iqA µ ψ| 2 + λ |ψ| 2 + 1 2λ ν 2 2 .(13)
6 If one actually sets Λ 0 = 0 from the beginning, one gets a different solution. This means that the classical limit approaches the usual Kerr as opposed to the dS/AdS Kerr.
The metric and scalar field equations are
R µν − Λg µν − κ 2 2 g µν λ |ψ| 2 + 1 2λ ν 2 2 − 1 4 F αβ F αβ +c 1 R 2 + (2c 1 + c 2 )∇ 2 R + c 2 R αβ R αβ + · · · +κ 2 − 1 2 ((∂ µ ψ − iqA µ ψ)(∂ ν ψ * + iqA ν ψ * ) + (µ ↔ ν)) − 1 2 F µρ F ν ρ +2c 1 RR µν − (2c 1 + c 2 )∇ µ ∇ ν R − 2c 2 R κ 1 µνκ 2 R κ 1 κ 2 + c 2 ∇ 2 R µν + · · · = 0,(14)∇ µ F µν + iqψ(∂ ν + iqA ν )ψ * − iqψ * (∂ ν − iqA ν )ψ + · · · = 0, (∇ µ − iqA µ )(∇ µ − iqA µ )ψ − ν 2 ψ − 2λψ|ψ| 2 + · · · = 0.
Below we will obtain, in a series form, a time-dependent solution that settles down to the standard Kerr geometry as the time-dependence fades out. It is thus useful to have the following series expansion of the standard Kerr geometry (note that it is a Kerr geometry but not an (A)dS Kerr for a reason to be explained),
ds 2 = − 1 − 2M z 1 + a 2 z 2 cos 2 θ (dt + a sin 2 θdφ) 2 +2(dt + a sin 2 θdφ) − dz z 2 + a sin 2 θdφ + 1 z 2 + a 2 cos 2 θ (dθ 2 + sin 2 θdφ 2 ),(15)
where the factor in front of (dt + a sin 2 θdφ) 2 can be expanded:
1 − 2M z 1 + a 2 z 2 cos 2 θ = 1 − 2M z + 2a 2 M cos 2 θz 3 − 2a 4 M cos 4 θz 5 + 2a 6 M cos 6 θz 7 + · · · .(16)
For a time-dependent solution, let us try the following ansatz:
ds 2 = − F (t, z, θ) z 2 (dt + a sin 2 θdφ) 2 + 2(dt + a sin 2 θdφ) − dz z 2 + a sin 2 θdφ +Φ 2 (t, z, θ)(dθ 2 + sin 2 dφ 2 ) = − F (t, z, θ) z 2 dt 2 − 2 z 2 dtdz + 2a − F (t, z, θ) z 2 + 1 sin 2 θ dtdφ − 2a z 2 sin 2 θdzdφ +Φ 2 (t, z, θ)dθ 2 + − a 2 F (t, z, θ) z 2 sin 2 θ + 2a 2 sin 2 θ + Φ 2 (t, z, θ) sin 2 θdφ 2 .(17)
For the field variables, let us take the following ansatze: for the scalar field, ψ(t, z, θ, φ) = ψ 0 (t, θ) + ψ 1 (t, θ)z + ψ 2 (t, θ)z 2 + ψ 3 (t, θ)z 3 + ...
+ κ 2 ψ h 0 (t, θ) + ψ h 1 (t, θ)z + ψ h 2 (t, θ)z 2 + ψ h 3 (t, θ)z 3 + ... .(18)
For the vector field, 7
A µ (t, z, θ, φ) = (0, A 1 (t, z, θ), A 2 (t, z, θ), A 3 (t, z, θ))(19)
with
A 1 (t, z, θ) = A z0 (t, θ) + A z1 (t, θ)z + A z2 (t, θ)z 2 + A z3 (t, θ)z 3 + ... + κ 2 A h z0 (t, θ) + A h z1 (t, θ)z + A h z2 (t, θ)z 2 + A h z3 (t, θ)z 3 + ... , A 2 (t, z, θ) = A θ0 (t, θ) + A θ1 (t, θ)z + A θ2 (t, θ)z 2 + A θ3 (t, θ)z 3 + ... + κ 2 A h θ0 (t, θ) + A h θ1 (t, θ)z + A h θ2 (t, θ)z 2 + A h θ3 (t, θ)z 3 + ... , A 3 (t, z, θ) = A φ0 (t, θ) + A φ1 (t, θ)z + A φ2 (t, θ)z 2 + A φ3 (t, θ)z 3 + ... + κ 2 A h φ0 (t, θ) + A h φ1 (t, θ)z + A h φ2 (t, θ)z 2 + A h φ3 (t, θ)z 3 + ... .(20)
For the metric,
F (t, z, θ) = F 0 (t, θ) + F 1 (t, θ)z + F 2 (t, θ)z 2 + F 3 (t, θ)z 3 + ... + κ 2 F h 0 (t, θ) + F h 1 (t, θ)z + F h 2 (t, θ)z 2 + F h 3 (t, θ)z 3 + ... , Φ(t, z, θ) = 1 z + Φ 0 (t, θ) + Φ 1 (t, θ)z + Φ 2 (t, θ)z 2 + Φ 3 (t, θ)z 3 + ... + κ 2 Φ h −1 (t, θ) z + Φ h 0 (t, θ) + Φ h 1 (t, θ)z + Φ h 2 (t, θ)z 2 + Φ h 3 (t, θ)z 3 + ...(21)
where the modes with superscript 'h' represent the quantum modes. The quantum corrections of the metric imply a deformation of the geometry by quantum effects [18]. The cosmological constant Λ is set to Λ = Λ 0 + κ 2 Λ 1 with vanishing Λ 0 , Λ 0 = 0, to prevent occurrence of the undesirable feature noted for the Einstein-scalar system. One can consider the first several zpowers. For each z-power, expand the coefficients to the first order of . Our analysis yields the following results: for the classical modes,
ψ 0 (t, θ) = ψ 0 , ψ 0 ψ * 0 = − ν 2 2λ , ψ 1 (t, θ) = 0, ψ 2 (t, θ) = 0, ψ 3 (t, θ) = 0, A z0 (t, θ) = 0, A θ0 (t, θ) = 0, A φ0 (t, θ) = 0, A z1 (t, θ) = 0, A θ1 (t, θ) = 0, A φ1 (t, θ) = 0, A z2 (t, θ) = 0, A θ2 (t, θ) = 0, A φ2 (t, θ) = 0, A θ3 (t, θ) = 0, A φ3 (t, θ) = 0, A φ4 (t, θ) = 0, F 0 (t, θ) = 0, F 1 (t, θ) = 0, F 2 (t, θ) = 1, F 3 (t, θ) = const, F 4 (t, θ) = 0, Φ −1 (t, θ) = 1, Φ 0 (t, θ) = 0, Φ 1 (t, θ) = 1 2 a 2 cos 2 θ, Φ 2 (t, θ) = 0, Φ 3 (t, θ) = − 1 8 a 4 cos 4 θ;(22)
for the quantum modes,
∂ t ψ h 0 (t, θ) = 0, ψ h * 0 (t, θ) = ν 2 ψ h 0 (t, θ) 2λψ 2 0 , ψ h * 1 (t, θ) = ν 2 ψ h 1 (t, θ) 2λψ 2 0 , ∂ t ψ h 1 (t, θ) = 0, ∂ t ψ h 2 (t, θ) = 0, ψ h * 2 (t, θ) = ν 2 ψ h 2 (t, θ) 2λψ 2 0 , ψ h * 3 (t, θ) = ν 2 ψ h 3 (t, θ) 2λψ 2 0 , A h z0 (t, θ) = − iψ h 1 (t, θ) qψ 0 , A h θ0 (t, θ) = − i∂ θ ψ h 0 (t, θ) qψ 0 , A h φ0 (t, θ) = 0, A h θ1 (t, θ) = − i∂ θ ψ h 1 (t, θ) qψ 0 , A h φ1 (t, θ) = 0, A h z1 (t, θ) = − 2iψ h 2 (t, θ) qψ 0 , A h z2 (t, θ) = − 3i 2λψ 2 0 ψ h * 3 (t, θ) + ν 2 ψ h 3 (t, θ) 2ν 2 qψ 0 , A h φ2 (t, θ) = 0, A h θ2 (t, θ) = − i ν 2 q 2 ∂ θ ψ h 2 (t, θ) + λ∂ t ∂ θ ψ h 1 (t, θ) ν 2 q 3 ψ 0 , A h φ3 (t, θ) = 0, F h 0 (t, θ) = − 1 3 Λ 1 , F h 1 (t, θ) = −2∂ t Φ h −1 , F h 2 (t, θ) = − 5 3 a 2 Λ 1 cos 2 θ + 2Φ h −1 (t, θ) − 2(cos 2θ + 2) csc θ sec θ ∂ θ Φ h −1 (t, θ), ∂ 2 θ Φ h −1 (t, θ) = 1 4 − cot 2 θ 3F 3 (t, θ)∂ t Φ h −1 (t, θ) + a 2 Λ 1 (cos 2θ + 3) −2(cos 2θ + 3) csc θ sec θ ∂ θ Φ h −1 (t, θ) , ∂ t Φ h 0 (t, θ) = −2Φ h −1 (t, θ) + 2 cot θ ∂ θ Φ h −1 (t, θ), ∂ θ Φ h 0 (t, θ) = −2a 2 sin θ cos θ ∂ t Φ h −1 (t, θ), Φ h 1 (t, θ) = − 3 2 a 2 cos 2 θ Φ h −1 (t, θ), Φ h 2 (t, θ) = − 1 2 a 2 cos 2 θ Φ h 0 (t, θ), Φ h 3 (t, θ) = 7 8 a 4 cos 4 θ Φ h −1 (t, θ).(23)
Several remarks are in order. Although the field equations are more complex and entangled, there exists, as in [7], a robust pattern in the manner in which the mode relationships above are obtained. The lowest -and κ-order terms in the action (13) are important in determining the building blocks of the higher modes. Some of the leading -correction parts introduce additional constraints among the classical modes -which is thus of "order-1" effect, and thereby qualitatively change the classical part of the solution. As the higherorder terms (that are not explicitly shown in (13)) are added to the action (13) and thus to the field equations, their effects are limited to the newly introduced higher modes that are then determined in terms of the lower modes. Some mode results in (22) deserve specific comments: one particularly novel feature is that, just as in the Einstein-scalar case analyzed in [7], the classical modes of the matter fields (i.e., the scalar field and Maxwell's field) are removed by the quantum-level constraints. There are differences as well.
One of the differences is that, unlike in [7], in which the mode Φ 0 (t, θ) is not constrained, it is constrained to vanish here. The field equations constrain F 3 to be a constant; with the requirement that the solution settles down to the usual Kerr geometry, it is determined to be F 3 = −2M . The results above have been obtained by setting Λ 0 to Λ 0 = 0 from the beginning. Let us clarify this. We suspect that the undesirable feature noted in the Einstein-scalar case should be due to the inadequacy of the ansatz (17)
Near-horizon dynamics
The upshot of the previous section is that the essential quantum-level physics can be captured by the action eq. (13), and the system admits the following time-dependent solution
g µν = − F (t,z,θ) z 2 − 1 z 2 0 a 1 − F (t,z,θ) z 2 sin 2 θ − 1 z 2 0 0 − a sin 2 θ z 2 0 0 Φ 2 (t, z, θ) 0 a 1 − F (t,z,θ) z 2 sin 2 θ − a sin 2 θ z 2 0 2a 2 sin 4 θ − a 2 F (t,z,θ) sin 4 θ z 2 + Φ 2 (t, z, θ) sin 2 θ ,(24)A µ (t, z, θ, φ) = (0, A 1 (t, z, θ), A 2 (t, z, θ), A 3 (t, z, θ)), ψ(t, z, θ, φ) = ψ(t, z, θ),
with each field component expanded in terms of the modes that satisfy the relationships given in (22). In this section, we complete the rest of the steps of the energy computation.
To compute the energy in the leading order in and κ, it suffices to compute only the classical geodesic, a feature shared by the Einstein-scalar system considered in the previous work, [7]. Since the time-dependence of the classical part of the solution is removed by the quantum effects, the classical geometry is that of the Kerr. It turns out that it is the boundary modes that are the building blocks of the time-dependence and represent the deformations. They also take a part in the trans-Planckian energy.
In section 4.1, we review the computation of the geodesic in the Kerr background. In section 4.2, we consider reexpansion of the solution around the classical location of the event horizon. In the analysis analogous to that in [7], we show that the energy measured by an infalling obsever is trans-Planckian. What we called the "horizon quantum modes" in [7] leads to the trans-Planckian energy. In section 4.3, we comment on the boundary conditions.
Four-velocity of an infalling observer
One of the ingredients needed to compute the local energy measured by an infalling observer is the four-velocity vector (see e.g. [19] [6] [7]). As seen in the previous section, the time-dependent pieces of the classical part of the quantum-level solution become constrained to vanish: the time-dependent part of the solution for the field equations is only the quantum correction piece. This implies that the classical part of the stress-energy is that of a Kerr geometry. Since the stress-energy tensor vanishes for a Kerr geometry, one can use the geodesic analysis of Kerr spacetime in order to compute the leading quantum-gravitational correction of the energy.
Let us review the geodesic analysis of Kerr spacetime [22]. The metric admits two Killing vectors:
k µ t = (1, 0, 0, 0), k µ ϕ = (0, 0, 0, 1),(25)
which leads to two integrals to the geodesic equations: the energy
E = −g µν k µ t U ν ,(26)
and angular momentum projection
l = g µν k µ ϕ U ν ,(27)
where U ν is the four-velocity U µ ≡ dx µ dλ with λ being the proper-time parameter along the geodesic. It is normalized according to
g µν dx µ dλ dx ν dλ = −µ 2 .(28)
The time-like and light-like geodesics correspond to µ = 1, 0, respectively. One can show that the four-velocity components are given by (the dot rep-
resents d dλ )φ = 1 1 z 2 + a 2 cos 2 θ aE + l sin 2 θ − a −1 (P + √ R) ,(29)t = 1 1 z 2 + a 2 cos 2 θ − a(l + aE sin 2 θ) + 1 z 2 + a 2 −1 (P + √ R) , (30) z = − z 2 1 z 2 + a 2 cos 2 θ √ R , θ = 1 1 z 2 + a 2 cos 2 θ √ Θ (31) with = a 2 + z −2 − 2M z ,(32)P = al + (a 2 + z −2 )E,(33)Θ = K − (l + Ea) 2 − cos 2 θ a 2 (µ 2 − E 2 ) + l 2 0 sin 2 θ ,(34)R = P 2 − K + µ 2 z 2 ,(35)
where K is another integral of motion called the Carter constant.
Trans-Planckian energy near horizon
The energy density as measured by a free-falling observer is given by
ρ ≡ T µν U µ U ν .(36)
The stress-energy tensor 8 is obtained by taking the functional derivative of the matter part of the action with respect to the metric:
T µν = − 2 κ 2 Λg µν + g µν − |∂ ρ ψ − iqA ρ ψ| 2 − λ |ψ| 2 + 1 2λ ν 2 2 − 1 4 F 2 ρσ + c 1 R 2 − (4c 1 + c 2 )∇ 2 R + c 2 R ρσ R ρσ + · · ·(37)+ ((∂ µ ψ − iqA µ ψ)(∂ ν ψ * + iqA ν ψ * ) + (µ ↔ ν)) + F µρ F ν ρ −2 2c 1 RR µν − (2c 1 + c 2 )∇ µ ∇ ν R − 2c 2 R κ 1 µνκ 2 R κ 1 κ 2 + c 2 ∇ 2 R µν + · · · .
For the leading-order energy correction 9 , one can use the classical form of the stress-energy tensor,
T µν = − 2 κ 2 Λg µν + g µν − |∂ ρ ψ − iqA ρ ψ| 2 − λ |ψ| 2 + 1 2λ ν 2 2 − 1 4 F 2 ρσ + [(∂ µ ψ − iqA µ ψ)(∂ ν ψ * + iqA ν ψ * ) + (µ ↔ ν)] + F µρ F ν ρ .(38)
Note that although the classical form of the stress-energy tensor is used, the full quantum-level solution is to be substituted into the tensor. An additional simplifying feature is that the terms with g µν are bound by the geodesic normalization, U µ U µ = −µ 2 , and thus unimportant. 10 Thus the leading-order quantum correction to the classical energy is given by
ρ ∼ [(∂ µ ψ − iqA µ ψ)(∂ ν ψ * + iqA ν ψ * ) + (µ ↔ ν)] + F µρ F ν ρ U µ U ν .(39)
8 See [23][24][25][26] for reviews on the quantum-level stress tensor. 9 The leading correction is of second power in and of inverse second power in κ (after the κ-rescaling of the matter fields to be discussed below). 10 The four-velocity reviewed in section 4.1 was at the classical level and, in particular, so is the normalization condition. However, in anticipation of the full quantum-level normalization condition, we omit the terms with g µν from the present leading-order energy correction computation.
Pole terms -which later yield trans-Planckian scaling -arise from the scalar or vector kinetic term above. More specifically, the ∂ t ψ∂ t ψ * ṫṫ and F tρ F t ρṫṫ terms produce the pole terms 11 :ṫ scales asṫ ∼ 1 z−z EH where the classical horizon z EH is located at the vanishing of a 2 + z −2 − 2M z −1 .
Since the four-velocity has a pole at the classical location of the event horizon z = z EH , a more transparent understanding of the behavior of the matter fields near the horizon can be gained by considering reexpansion of the z-series solution in
Y ≡ z − z EH .(40)
Let us suppose the following form of expansion of the matter fields around z EH 12 :
ψ(t, z, θ) =ψ 0 (t, θ) +ψ 1 (t, θ)Y +ψ 2 (t, θ)Y 2 +ψ 3 Y 3 + · · · +κ 2 ψ h 0 (t, θ) +ψ h 1 (t, θ)Y +ψ h 2 (t, θ)Y 2 +ψ h 3 Y 3 + · · · , A 1 (t, z, θ) =Ã z0 (t, θ) +Ã z1 (t, θ)Y +Ã z2 (t, θ)Y 2 +Ã z3 (t, θ)Y 3 + ... +κ 2 Ã h z0 (t, θ) +Ã h z1 (t, θ)Y +Ã h z2 (t, θ)Y 2 +Ã h z3 (t, θ)Y 3 + ... , A 2 (t, z, θ) =Ã θ0 (t, θ) +Ã θ1 (t, θ)Y +Ã θ2 (t, θ)Y 2 +Ã θ3 (t, θ)Y 3 + ... +κ 2 Ã h θ0 (t, θ) +Ã h θ1 (t, θ)Y +Ã h θ2 (t, θ)Y 2 +Ã h θ3 (t, θ)Y 3 + ... , A 3 (t, z, θ) =Ã φ0 (t, θ) +Ã φ1 (t, θ)Y +Ã φ2 (t, θ)Y 2 +Ã φ3 (t, θ)Y 3 + ... +κ 2 Ã h φ0 (t, θ) +Ã h φ1 (t, θ)Y +Ã h φ2 (t, θ)Y 2 +Ã h φ3 (t, θ)Y 3 + ... .(41)
The 'tilded' modes will be given as sums of the original modes. To the orders that we have checked in section 3, all of the classical modes (except ψ 0 , which is irrelevant for the energy computation) vanish; because of this the mode 11 As a matter of fact,φ has a pole too. We will focus onṫ. 12 At least to the orders analyzed, the time-dependence of the classical parts of the matter fields is absent. In [7], the similar feature was checked to remain true to all orders in z, not just to the first several orders. That was done by solving the field equations with the expansion given in eq. (41). Although the corresponding task for the present system turns out to be too involved, we expect that the feature remains true. expansion above gets simplified to
ψ(t, z, θ) =ψ + κ 2 ψ h 0 (t, θ) +ψ h 1 (t, θ)Y +ψ h 2 (t, θ)Y 2 +ψ h 3 Y 3 + · · · , A 1 (t, z, θ) = κ 2 Ã h z0 (t, θ) +Ã h z1 (t, θ)Y +Ã h z2 (t, θ)Y 2 +Ã h z3 (t, θ)Y 3 + ... , A 2 (t, z, θ) = κ 2 Ã h θ0 (t, θ) +Ã h θ1 (t, θ)Y +Ã h θ2 (t, θ)Y 2 +Ã h θ3 (t, θ)Y 3 + ... , A 3 (t, z, θ) = κ 2 Ã h φ0 (t, θ) +Ã h φ1 (t, θ)Y +Ã h φ2 (t, θ)Y 2 +Ã h φ3 (t, θ)Y 3 + ... .(42)
Before getting to the final-stage energy analysis, let us note that rescaling of the matter fields is necessary for correct κ-scaling of various physical quantities, including the energy. The fact that the matter part of the action comes at higher order of κ 2 implies [21] [7] that the solution generically takes the form of
ψ = ξ κ , A m = a m κ , m = 1, 2, 3 ,(43)
where ξ, a m represents the rescaled scalar and vector fields; they will have series expansions -which are similar to those in eq. (41) -in terms of the modes with tildes. In particular, the modes (ξ h 0 (t, θ),ã h z0 (t, θ),ã h θ0 (t, θ),ã h φ0 (t, θ)) play an important role in the energy, as we will now see. The location of the horizon at the quantum level, z h EH , (whose precise determination we do not pursue in the work) will take the form of
z h EH = z EH + O(κ 2 )(44)
and this impliesṫ
∼ O(κ −2 )(45)
at z = z h EH . With this scaling one gets, for the leading behavior of ρ,
T µν U µ U ν ∼ κ 2 f (ξ h 0 ,ã h z0 ,ã h θ0 ,ã h φ0 ) κ 4 ∼ 1 κ 2 ,(46)
where f (ξ h 0 ,ã h z0 ,ã h θ0 ,ã h φ0 ) is a quantity that is proportional to T 00 . A direct calculation yields 13 T 00 = 1 sin 2 θ (a 2 z 2 EH cos 2 θ + 1)
a 2 z 6 EH sin 4 θ(∂ tã h z0 ) 2 + z 2 EH (∂ tã h φ0 ) 2 + sin 2 θ z 4 EH (∂ tã h z0 ) 2 a 2 z 2 EH cos 2 θ − 2M z EH + 1 + 2∂ tξ h * 0 ∂ tξ h 0 a 2 z 2 EH cos 2 θ + 1 +2az 4 EH ∂ tã h z0 ∂ tã h φ0 + z 2 EH (∂ tã h θ0 ) 2 + · · · ,(48)
where the explicitly shown terms represent the expression inside the parentheses in (39).
On the boundary conditions
Dirichlet boundary conditions have been widely considered in quantum and gravitational field theories. The recent works show, however, that a complete description of a gravitational system requires extension of the Hilbert space by including other boundary conditions [13]. Let us examine the mode expansions (18), (20), and (21). The presence of the dynamic boundary modes such as Φ h −1 (t, θ), Φ h 0 (t, θ)) implies that the solution satisfies a certain Neumann-type boundary condition but not a Dirichlet boundary condition. One noteworthy point is that the boundary condition at the asymptotic infinity closely controls what's happening at the event horizon, as one can see by examining the reexpansions eq. (42) (and those of the metric fields).
In certain circumstances such as a black hole merger, a fixed boundary condition is considered at the horizon. The most widely used one is a perfectinfall boundary condition in the context of the quasi-normal modes. However, the quantum effects obtained in [7] and the present work seem to suggest more general and inclusive boundary conditions as the natural ones at the horizon. The presence of the aforementioned quantum boundary modes will imply that both transmitted and reflected waves will be present. When 13 Similarly, one gets
T 33 = 2q 2ξ 0ξ * 0 (ã h φ0 ) 2 + z 2 EH a 2 z 2 EH cos 2 θ + 1 a 2 sin 2 θ(∂ tã h φ0 ) 2 + (∂ θã h φ0 ) 2 −2∂ tã h φ0ã h φ1 [a 2 z 2 EH cos 2 θ + a 2 z 2 EH sin 2 θ + 1] +z 2 EH (ã h φ1 ) 2 [a 2 z 2 EH cos 2 θ + a 2 z 2 EH sin 2 θ − 2M z EH + 1] . (47)
the time-dependence fades out, the boundary modes become die out and a Dirichlet boundary condition naturally arises.
Conclusion
In this work we have considered a time-dependent quantum-corrected black hole solution of an Einstein-Maxwell-scalar with a Higgs potential. We have computed the near-horizon energy measured by an infalling observer. The analysis consists of three components: computation of the quantum action (for which we referred to the previous works) and its time-dependent solution, computation of the four-velocity of the observer, as well as evaluation of the energy after expansion around the location of the horizon. As in the previous works [6] [7], a trans-Planckian energy has resulted. Concerning the physical origin of the trans-Planckian energy behavior, the loop effects become important near the horizon. The power of the previous works and that of the present is the generality of the analyses and the quantitative conclusions drawn from them. In other words, there exists a robust pattern in determination of the higher order modes in terms of the lower ones: the solutions are built out of several lower modes for which only the first two leading quantum correction terms in the action, R 2 , R ρσ R ρσ , are important.
The present work is motivated in part by recent developments in astrophysics, in particular, ultra high energy cosmic rays (UHECRs). Although further work is required, the recent observations indicate active galactic nuclei (AGNs) -the central supermassive black holes of active galaxies -as the candidates of the steady sources of the UHECRs. The relatively new paradigm in the field is the view of the black hole at the center of an AGN as a highly efficient engine that converts the gravitational infall energy into outgoing radiation energy. The quantum gravitational effects may well be the mechanism of generating the energy that feeds various outgoing radiation, especially the UHECRs. Being a more realistic system, we expect that the result of the present work will have applications to the physics of γAGNs and UHECRs.
Before embarking on such an enterprise, there are several more urgent and immediate issues to settle. We present them as near-future directions:
An undesirable feature of the solution of the free scalar sector has motivated the introduction of the potential for the scalar sector and consideration of the vanishing classical part of the cosmological constant. Because of the vanishing cosmological constant, our time-dependent solution settles down to the usual Kerr black hole as opposed to a dS/AdS Kerr black hole. In order to relax the restriction Λ 0 = 0, one should first consider the (A)dS-Kerr solution in the present Eddington-Finkelstein-type coordinates. The (A)dS-Kerr solution was long known in the Boyer-Lindquist coordinates. It appears that the solution has recently been converted in [27] into the Eddington-Finkelsteintype coordinates that we have employed in the present work.
Another relatively urgent direction is a charged black hole case. Taking the temporal gauge, the solution obtained in this work cannot cover the timedependent extension of a charged black hole, since the standard charged black hole has a nonzero A 0 component. It will be of some interest to study how to incorporate the charged case. Presumably a different gauge choice will have to be made.
Still another direction that lies directly ahead of the path of the present work is curved space electrodynamics. The present setup lays necessary foundations for carrying out quantum-gravitational scalar electrodynamics. In particular, it should be possible to compute the quantum-gravitational Poynting vector, which should be useful in comparing with the present and future observations made for astronomical black holes.
for an (A)dS case. In the literature, the (A)dS Kerr solution is long known in the Boyer-Lindquist coordinates. The first step in proper handling of a timedependent (A)dS Kerr solution should be to write down the ansatz based on the (A)dS Kerr solution in the Eddington-Finkelstein-type coordinates that we have employed. (More on this in the conclusion.)
A related discussion can be found in[10]. 2 See[14] for a related result: there it was observed in a time-dependent setup that the quantum stress-energy tensor inside the black hole reaches a near-Planckian value.
Although the framework has radically new ingredients, it is only two-and higherloop renormalizability that requires such ingredients: the one-loop renormalizability can be established within the conventional framework. The subsequent techniques of finding the time-dependent solution and analyzing the curved space scalar electrodynamics are standard.
In the conventional picture, pair creation process has been argued to be responsible for the Hawking radiation. Although the pair-creation process is expected to be one of the main channels of the quantum-gravitational effects, our picture posits a much more complex process than the conventional one where one of the pair particles falls into the
As a matter of fact, the boundary conditions must be considered before varying the action. We refer to[13] for potential issues associated with the boundary conditions in varying a gravitational action.
This form of the ansatz does not cover the charged black hole case. We leave the charged case for the future investigation.
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| []
|
[
"Ultrafast sequential charge transfer in a double quantum dot",
"Ultrafast sequential charge transfer in a double quantum dot"
]
| [
"A Putaja \nNanoscience Center\nDepartment of Physics\nUniversity of Jyväskylä\nFI-40014JyväskyläFinland\n",
"E Räsänen \nNanoscience Center\nDepartment of Physics\nUniversity of Jyväskylä\nFI-40014JyväskyläFinland\n"
]
| [
"Nanoscience Center\nDepartment of Physics\nUniversity of Jyväskylä\nFI-40014JyväskyläFinland",
"Nanoscience Center\nDepartment of Physics\nUniversity of Jyväskylä\nFI-40014JyväskyläFinland"
]
| []
| We use optimal control theory to construct external electric fields which coherently transfer the electronic charge in a double quantum-dot system. Without truncation of the eigenstates we operate on desired superpositions of the states in order to prepare the system to a localized state and to coherently transfer the charge from one well to another. Within a fixed time interval, the optimal processes are shown to occur through several excited states. The obtained yields are generally between 99 % and 99.99 % depending on the field constraints, and they are not dramatically affected by strict frequency filters which make the fields (e.g., laser pulses) closer to experimental realism. Finally we demonstrate that our scheme provides simple access to hundreds of sequential processes in charge localization while preserving the high fidelity. | 10.1103/physrevb.82.165336 | [
"https://arxiv.org/pdf/1006.5615v2.pdf"
]
| 118,704,186 | 1006.5615 | 62e6b4417b93603afe064a6354f30f9b5236d4c4 |
Ultrafast sequential charge transfer in a double quantum dot
31 Aug 2010 (Dated: September 1, 2010)
A Putaja
Nanoscience Center
Department of Physics
University of Jyväskylä
FI-40014JyväskyläFinland
E Räsänen
Nanoscience Center
Department of Physics
University of Jyväskylä
FI-40014JyväskyläFinland
Ultrafast sequential charge transfer in a double quantum dot
31 Aug 2010 (Dated: September 1, 2010)arXiv:1006.5615v2 [quant-ph]numbers: 7867Hc7321La7820Bh0367Bg
We use optimal control theory to construct external electric fields which coherently transfer the electronic charge in a double quantum-dot system. Without truncation of the eigenstates we operate on desired superpositions of the states in order to prepare the system to a localized state and to coherently transfer the charge from one well to another. Within a fixed time interval, the optimal processes are shown to occur through several excited states. The obtained yields are generally between 99 % and 99.99 % depending on the field constraints, and they are not dramatically affected by strict frequency filters which make the fields (e.g., laser pulses) closer to experimental realism. Finally we demonstrate that our scheme provides simple access to hundreds of sequential processes in charge localization while preserving the high fidelity.
I. INTRODUCTION
During the past few years, coherent control of charge in double quantum dots (DQDs) has been a subject of active experimental 1-4 and theoretical [5][6][7][8][9][10] research. Here one of the long-term aims is the design of a solid-state quantum computing scheme. 11 It is still to be seen whether the optimal control mechanism DQDs turns out to operate through magnetic fields, 12,13 gate voltages, 4 or optimized laser pulses. 7,9 Dynamical control of charge in DQDs has been a popular application for few-level schemes 6,10,[14][15][16][17][18][19] (modeling DQDs as two-, three-, or four-level systems), which have demonstrated ultrafast high-fidelity processes. However, a physical DQD has, in principle, infinitely many levels, and in fast processes a considerable number of states might have practical relevance. For example, a two-level approximation is exact only in the limit of using an infinitely long resonant continuous wave with an infinitely small amplitude. A linear field (bias) is an appealing and simple alternative to control charge in DQDs, 5 but it is not applicable to fast processes.
With quantum optimal control theory 20,21 (OCT) it is possible to find optimized external fields driving the system -having an arbitrary number of states -from the initial state to the desired target state without any approximations, apart from a possible model potential to describe the physical apparatus. OCT has been used to analyze the general controllability criteria of twodimensional single-electron DQDs and to optimize interdot charge transfer. 7 Optimal control of two-electron DQDs has been obtained in an extensive work of Nepstad at al. 9 addressing various control schemes 22 and hyperfine interactions. 23 In this work we apply OCT to construct external electric fields that lead to fast sequential charge transfer processes in single-electron DQDs. To obtain high fidelity we operate on the superpositions of the lowest states corresponding to the charge localization in left or right well. We show that hundreds of sequential charge transfer processes can be achieved without a significant loss of the yield. To make the experimental production of the obtained fields more realistic, we cut off the high-frequency components already during the optimization procedure.
The use of such filters does not dramatically affect the fidelity.
II. MODEL
We use a one-dimensional (1D) model describing a single-electron semiconductor DQD. The external potential has a form
V c (x) = ω 2 0 2 min x − d 2 2 , x + d 2 2(1)
in effective atomic units (a.u.), see below. Here d = 6 is the interdot distance and ω 0 = 0.5 is the confinement strength. The potential is visualized in Fig. 1. We consider typical GaAs material parameters within the effective-mass approximation, i.e., m * = 0.067 and ǫ = 12.7. Now, the energies, lengths, and times scale as E * h = (m * /m 0 )/(ε/ε 0 ) 2 E h ≈ 11 meV, a * 0 = (ε/ε 0 )/(m * /m 0 )a 0 ≈ 10 nm, and t * 0 = /E * h ≈ 60 fs, respectively. We emphasize that below the abbreviation a.u. refers to these effective atomic units.
It should be noted that a harmonic potential in Eq. (1) is, in its two-dimensional (2D) form, a general model for realistic semiconductor quantum-dot structures. 24 Since the first Coulomb-blockade experiments it has been shown that the harmonic model is essentially valid up to dozens of electrons confined in the dot, and thus up to a large number of levels. The validity is clear, e.g., in recent works combining experiments and theory in the spin-blockade regime. 25,26 The precise energy-level spectrum in a given device can be explicitly obtained through single-electron transport experiments, and this information can be utilized to reconstruct the particular form of the extenal potential. For example, in Ref. 27 it was explicitly shown that measured energy-level spectrum can be well reproduced by a harmonic model potential upon slight refinements. Hence, when necessary, Eq. (1) can be tuned to match a particular device. Regarding the results below, the 1D model does not yield a qualitative difference from a more realistic 2D potential, but it significantly speeds up the calculations.
Electronic states localized to left and right dots can be expressed as superpositions of the two lowest (gerade and ungerade) states |0 and |1 as follows:
|L = 1 √ 2 ( |0 + |1 ) (2) |R = 1 √ 2 ( |0 − |1 )(3)
If the system is prepared in either of the superpositions, the occupation probabilities of |L and |R oscillate with the resonance frequency ω 01 = E 1 − E 0 ≈ 0.0135 (see Ref. 33). For instance, if the system is first prepared at |L , it reaches the state |R at t = T /2 = π/ω 01 ≈ 232.87. As discussed in detail below, we aim at controlling this charge-transfer procedure in an arbitrary way.
III. METHOD
In OCT the objective is to find an external timedependent field ǫ(t) that drives the system into the predefined state through the solution of the Schrödinger equation,
i ∂ ∂t Ψ(r, t) =Ĥ[ǫ k (t)]Ψ(r, t).(4)
Here ǫ(t) is an electric field (e.g., laser pulse) dealt with the dipole approximation, so that the Hamiltonian has the form,Ĥ
=T +V c −μǫ(t),(5)
where the (static) external potential is that of Eq. (1) whereμ = −r is the dipole operator. Starting with an initial guess for the electric field ǫ(t), we maximize the expectation value of the target operator O:
J 1 [ψ] = Ψ(r, T )|Ô|Ψ(r, T ) ,(6)
whereÔ = |Φ F Φ F | is now a projection operator, since we aim at maximizing the occupation of the target state Φ F at the end of the field at time T :
J 1 = Ψ(r, T )|Φ F 2 .(7)
In the following, this quantity is referred to the yield. As a constraint, avoiding fields with very high energy, the fluence (time-integrated intensity) of the field is limited by a second functional,
J 2 [ǫ] = −α T 0 dt ǫ 2 (t) − E 0 ,(8)
where E 0 is the fixed fluence [see Eq. (13) below] and α is a time-independent Lagrange multiplier. 21 Finally, the satisfaction of the time-dependent Schrödinger equation [Eq. (4)] introduces yet another functional,
J 3 [ǫ, Ψ, χ] = −2 Im T 0 χ(t)|i∂ t −Ĥ(t)|Ψ(t) ,(9)
where χ(t) is a time-dependent Lagrange multiplier. Variation of J = J 1 + J 2 + J 3 with respect to Ψ, χ, ǫ, and α lead to the control equations
i∂ t Ψ(t) =Ĥ(t)Ψ(t), Ψ(0) = Φ I ,(10)i∂ t χ(t) =Ĥ(t)χ(t), χ(T ) =ÔΨ(T ), (11) ǫ(t) = − 1 α Im χ(t)|µ|Ψ(t) ,(12)T 0 dt ǫ 2 (t) = E 0 .(13)
which can be solved iteratively. 21, 28 We apply a numerically efficient forward-backward propagation scheme introduced by Werschnik and Gross. 29 When solving the control equations, the Lagrange multiplier α is calculated through the fixed fluence E 0 as explained in detail in Ref. 21. The field is constrained by an envelope function of a form
f (t) = 1 2 Erf a T t − T b + Erf − a T t − T + T b (14)
with a = 100 and b = 20. This corresponds to a step function ascending (descending) rapidly at t ∼ b/4 (t ∼ T − b/4). The scheme also allows straightforward inclusion of spectral constraints discussed in the following section. In the numerical calculations we have used To approximate the time-propagator we applied the time-reversal symmetry, i.e., propagating Ψ(t) forward by ∆t/2 should correspond to propagating Ψ(t + ∆t) backward by ∆t/2. This condition leads to an approximation for the propagator, 31 which can be further improved by extrapolating the time-dependent potentials. In octopus 30 the used method is called Approximated Enforced Time-Reversal Symmetry (AETRS).
IV. RESULTS
First, the system is prepared from the ground state |0 to the desired superposition. Hence, we simply define the target wave function in Eq. (6) as |Φ F = |L . We set the field length to T = 100 (∼ 6 ps) and the initial frequency to 0.5 corresponding to the oscillator frequency ω 0 of the DQD. Unless stated otherwise, the fluence is fixed to E 0 = 0.3, so that the average intensities are of the order of 10 3 W/cm 2 (note the units given in Sec. II).
The optimized field in Fig. 1(a) looks rather complicated with distinct high-frequency components, whose role and possible removal is discussed in detail below. The occupations of the states, i.e., their overlaps with the time-propagated wave function, are plotted in Fig. 1(b). The ground state (initially occupied) and the first excited state (initially empty) get half populated, so that their superposition |L becomes fully populated and the electron is localized in the left well. The obtained yield is as high as 0.99985.
After the preparation of the localized state we optimize a transition from |L to |R , i.e., a charge transfer between the quantum wells. The result of the optimization is summarized in Fig. 3. The optimal field having a fixed duration of T = 100 [thin blue line in Fig. 4(a)] leads to an extremely high yield of 0.9992. In Fig. 3(b) and (c) we plot the occupations of the five lowest states during the charge-transfer process. Each of these states reach a maximum occupancy of more than 10 % during the process. The tenth lowest state still obtains ∼ 1 % of the occupation. Thus, with the present length of the field, the inclusion of several states seems to be crucial for the success of the optimization. Consequently, an alternative OCT procedure for a few-level model system (higher levels omitted) would lead to a completely different solution field, which most likely would perform poorly when applied to the "full" system (as here) due to the leaking of the occupancy to higher states. 32 Similarly to the preparation field in Fig. 2(a), the op- timized charge-transfer field in Fig. 4(a) shows abrupt peaks corresponding to high frequencies. Hence, the field would be practically impossible to construct, e.g., with the present pulse-shaping techniques. To relieve these limitations, we apply a spectral constraint cutting off the high-frequency components beyond a selected threshold frequency ω th . The thick red line in Fig. 3(a) shows the field obtained using ω th = 0.817 (∼ 14 THz) in the optimization. The Fourier spectra of both fields are shown in Fig. 4(a). Both fields have a peak at at ω = 0.5, which in fact corresponds to the oscillator frequency ω 0 in Eq. (1).
It is interesting to note that despite the relatively strong frequency constraint at ω th = 0.817, leading to a considerable smaller search space for the optimization, the obtained yield is reduced only down to 0.9986. This is a significant result in view of the fact that the original field has a large fraction of high frequencies as shown in Fig. 4(a). Nevertheless, using a frequency filter does not considerably reduce the importance of higher states in the optimization: in this particular case the fifth lowest state still gains a maximum occupancy of ∼ 10 %. In any case, further tightening of the threshold to smaller values leads to decrease in the overlap as demonstrated in Fig. 4(b). The dependency is nonmonotonic due to numerical variation (note the logarithmic scale) and has a step structure resulting from the discrete Fourier transform. Below ω th ∼ 0.5 corresponding to the oscillator frequency the fidelity collapses from 96 % to 42 %. If the fluence of the field is increased from 0.3 to 1, the critical threshold remains at 0.5, at which the fidelity decreases from 92 % to 74 %.
Besides the threshold frequency, the main constraints in the field to be optimized are the length and the fluence. In Fig. 5(a) we show the yield, again for the process |L → |R , as a function of the field length for fixed fluence values E 0 = 0.3 and 1, respectively. Both cases show some saturation around T 100 although, as expected, the smaller fluence allows longer fields with even higher fidelities. However, increasing the yield above 0.9999 is difficult in this fluence range unless relatively long fields are required. Here, the chosen length T = 100 seems an appropriate compromise between T and the obtained yield. Figure 5(b) shows the yield as a function of the fluence for three fixed field lengths. We remind that the fluence is a time-integrated quantity [see Eq. (13)] so that the curves correspond to different distributions of the energy in the field. In all cases the yield first increases exponentially with the fluence until a point of saturation is reached. When T = 200 the slight decrease in the overlap at fluences above ∼ 0.2 might be due to numerical constraints: in that regime higher and higher states (with an increasing number of nodes) are required, and they have a finite accuracy on the numerical grid. Finally we consider sequential charge-transfer processes by merging optimized fields together. For the process |L → |R → |L → . . . we combine, in turns, the optimized field ǫ L→R (see above) with its time-inversion corresponding to ǫ R→L . The combined field with a threshold frequency ω th = 0.817 is visualized in the upper panel of Fig. 6. The lower panel shows the occupations of the states |R and |L by solid and dotted lines, respectively. The final yield after the five-fold process is 99.46 %.
A more complete view on the results of up to 100 sequential processes is given in Table I. We consider fluences E 0 = 0.3 and 1 for a single process, respectively, and different threshold frequencies as well as the case without a filter, i.e., ω th → ∞. The total yield shown in the table can be expected to (roughly) follow a power law, J 1,tot = J n 1,single , where n is the number of processes (charge transfers). In this respect, the fidelity for a single transfer is essential for the quality of the final result. Indeed, the computational result follows the trend of the power law, but we find also significant differences: most importantly, in all cases the computational result is better than the prediction of the power law. The most dramatic discrepancy can be found in the last example with E 0 = 1 and ω th = 0.817, where after 100 pulses the yield is still almost 99 %, whereas the power law predicts is only 64 %. The reason behind the robustness of the yield in a sequential process is in the identity of the frequency components between the original and inverted fields, so that the population "lost" in higher states is partially attained back in the inverse process. There is, however, no clear trend in Table I indicating which field parameters are particularly favorable for robust sequential processes. Construction of such population-preserving, yet well optimized sequential fields is a subject of future work.
We point out that a critical aspect in the feasibility of the present approach is the sensitivity to decoherence. Typical decoherence mechanisms in semiconductor quantum dots are the hyperfine effects and interactions with optical and acoustic phonons. Their interplay and significance are largely dependent on the external conditions in a particular device. Detailed assessment of these mechanisms is beyond the scope of this work. We only mention that typical decoherence times in semiconductor quantum dots have been measured to be relatively large, even up to the millisecond scale, 34 which in fact has been one of the main motivations of utilizing quantum dots in solid-state quantum computing. 11 In view of the time scales considered here (up to hundreds of picoseconds) we believe that our approach is robust against the essential sources of decoherence, although further analysis is in order.
V. SUMMARY
Here we have numerically constructed optimal fields for charge-transfer processes in single-electron double quantum dots. The only approximation has been the model potential for the device, so that no truncation of eigenstates in terms of N -level approximations have been used. We have found that optimal control theory provides an efficient way to operate on desired superpositions of the eigenstates regarding both the preparation of the localized state as well as coherent charge transfer between the quantum wells. We have analyzed the interplay between different field constraints including the frequency filter, fluence, and the field length. Relatively strict frequency filters can be used without losing the extremely high yields obtained in the processes. Combination of the optimized pulses can be used in sequential charge transfers while preserving the high fidelity.
FIG. 1 :
1(Color online) Model potential for the quantum dot (black solid line), the ground state (green line), the first excited state (black dash-dotted line), and their superpositions corresponding to left (red dotted line) and right (blue dashed line) states.
online) (a) Optimized field to prepare the system to the superposition |L (see text) from the ground state. (b) Occupations of the ground state |0 , first excited state |1 and their superposition |L . the octopus code 30 which solves the control equations in real time on a real-space grid.
FIG
. 3: (Color online) (a) Optimized fields without (thin blue line) and with spectral constraints (thick red line) for transition |L → |R . (b-c) Occupations of the five lowest eigenstates during the process. (d) Occupations of the initial and target superposition states.
online) (a) Spectrum of the optimized field for the process |L → |R without (blue thin line) and with a spectral constraint (red thick line) at ω th = 0.817. (b) Occupation of target state as the function of the frequency threshold ω th used as the filter. The step-like form of the curve is a consequence of the discrete Fourier transform.
online) (a) Obtained yield for the process |L → |R as a function of the field length. (b) Yield for the same process as a function of the fluence for three field lengths.
FIG. 6 :
6(Color online) Optimized field (upper panel) for the five-fold charge-switch process |L → |R → |L ... → |R (lower panel) in a double quantum dot. Here we have used a threshold frequency of ω th = 0.817 leading to the target state occupation of 99.46 % at the end of the total five-fold process.
TABLE I :
IFinal-state occupations after n-fold sequential
charge-switch processes obtained when merging the optimized
fields. They are compared with an estimate based on the
power of the yield given by the original (not inverted) field
(see text).
AcknowledgmentsThis work has been funded by the Academy of Finland. A. P. acknowledges support by the Finnish Academy of
Vilho, Yrjö and Kalle Väisälä Foundation. Science and Letters, Vilho, Yrjö and Kalle Väisälä Foun- dation.
* Electronic address: [email protected]. * Electronic address: [email protected]
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| []
|
[
"Siamese Generative Adversarial Privatizer for Biometric Data",
"Siamese Generative Adversarial Privatizer for Biometric Data"
]
| [
"Witold Oleszkiewicz [email protected] \nWarsaw University of Technology\n\n",
"Peter Kairouz \nStanford University\n3 Tooploox\n",
"Karol Piczak \nWarsaw University of Technology\n\n",
"Ram Rajagopal \nStanford University\n3 Tooploox\n",
"Tomasz Trzciński \nWarsaw University of Technology\n\n"
]
| [
"Warsaw University of Technology\n",
"Stanford University\n3 Tooploox",
"Warsaw University of Technology\n",
"Stanford University\n3 Tooploox",
"Warsaw University of Technology\n"
]
| []
| State-of-the-art machine learning algorithms can be fooled by carefully crafted adversarial examples. As such, adversarial examples present a concrete problem in AI safety. In this work we turn the tables and ask the following question: can we harness the power of adversarial examples to prevent malicious adversaries from learning identifying information from data while allowing non-malicious entities to benefit from the utility of the same data? For instance, can we use adversarial examples to anonymize biometric dataset of faces while retaining usefulness of this data for other purposes, such as emotion recognition? To address this question, we propose a simple yet effective method, called Siamese Generative Adversarial Privatizer (SGAP), that exploits the properties of a Siamese neural network to find discriminative features that convey identifying information. When coupled with a generative model, our approach is able to correctly locate and disguise identifying information, while minimally reducing the utility of the privatized dataset. Extensive evaluation on a biometric dataset of fingerprints and cartoon faces confirms usefulness of our simple yet effective method. | 10.1007/978-3-030-20873-8_31 | [
"https://arxiv.org/pdf/1804.08757v3.pdf"
]
| 13,745,279 | 1804.08757 | cfdc00afbfa7cfd7ad20a5b0578d320d89d33130 |
Siamese Generative Adversarial Privatizer for Biometric Data
Witold Oleszkiewicz [email protected]
Warsaw University of Technology
Peter Kairouz
Stanford University
3 Tooploox
Karol Piczak
Warsaw University of Technology
Ram Rajagopal
Stanford University
3 Tooploox
Tomasz Trzciński
Warsaw University of Technology
Siamese Generative Adversarial Privatizer for Biometric Data
State-of-the-art machine learning algorithms can be fooled by carefully crafted adversarial examples. As such, adversarial examples present a concrete problem in AI safety. In this work we turn the tables and ask the following question: can we harness the power of adversarial examples to prevent malicious adversaries from learning identifying information from data while allowing non-malicious entities to benefit from the utility of the same data? For instance, can we use adversarial examples to anonymize biometric dataset of faces while retaining usefulness of this data for other purposes, such as emotion recognition? To address this question, we propose a simple yet effective method, called Siamese Generative Adversarial Privatizer (SGAP), that exploits the properties of a Siamese neural network to find discriminative features that convey identifying information. When coupled with a generative model, our approach is able to correctly locate and disguise identifying information, while minimally reducing the utility of the privatized dataset. Extensive evaluation on a biometric dataset of fingerprints and cartoon faces confirms usefulness of our simple yet effective method.
Introduction
Large-scale datasets enable researchers to design and apply state-of-the-art machine learning algorithms that can solve progressively challenging problems. Unfortunately, most organizations release datasets rather reluctantly due to the excessive amounts of sensitive information about participating individuals.
Ensuring the privacy of subjects is done by removing all personally identifiable information (e.g. names or birthdates) -this process, however, is not foolproof. Correlation and linkage attacks [25,15] often identify an individual by combining anonymized data with personal information obtained from other sources. Several such cases have been presented in the past, e.g. deanonymization of users' viewing history that was published in the Netflix Prize competition [25], identifying subjects in medical studies based on fMRI imaging data [9], and linking DNA profiles of anonymized participants with data from the Personal Genome Project [32].
Typical approaches to countering the shortcomings of anonymization techniques leverage data randomization. While randomizing datasets with differen-arXiv:1804.08757v3 [cs.CV] 8 Oct 2018 tial privacy [7] provides much stronger privacy guarantees, the utility of machine learning models trained on such randomized data is often significantly impaired [30,16,18]. We therefore believe that there is an ever increasing need for new privatization methods that preserve the value of the data while protecting the privacy of individuals.
The above privacy problem becomes critical when dealing with sensitive biometric and medical images. Several breakthrough applications of computer vision have been proposed in this domain: [12] used machine learning algorithms to parcellate human cerebral cortex, [29] utilized convolutional networks to detect arrhythmia, and [8] used machine learning to realize a precision medicine system. These applications, though critical for the advancement of the domain, rely on the access to highly sensitive data. This calls for novel privatization schemes that allow for the publication of images containing medical and biometric information without sacrificing the utility of the applications discussed above.
Our contributions
In this work, we take a new approach towards enabling private data publishing. Instead of adopting worst case, context-free notions of statistical data privacy (such as differential privacy), we present a novel framework that allows the publisher to privatize images in a context-aware manner (Fig. 1). Our framework builds up on the recent work [17] where they propose a Generative Adversarial Privacy (GAP) method that casts the privatization as a constrained minimax game between a privatizer and an adversary that tries to infer private data. The approach we propose here is focused on biometric images and exploits a Siamese neural network architecture to identify image parts that bear the highest discriminative power and perturb them to enforce privatization. Contrary to other works that quantify privacy in a subjective manner using user surveys [28], we define here empirical conditions our privatizer needs to fulfill and propose metrics that allow to evaluate the privacy-utility trade-off we aim to explore. Finally, we present the results of our experiments on datasets of fingerprints and cartoon faces. Our results show that the proposed framework prevents an attacker from re-identifying privatized data while leaving other important image features intact. We call this approach Siamese Generative Adversarial Privatizer (SGAP).
To summarize our contributions are twofold:
a novel privatization method that uses a Siamese architecture to identify identity-discriminative image parts and perturbates them to protect privacy, while preserving the utility of the resulting data for other machine learning tasks, and an empirical data-driven privacy metric (c.f. Section 4.2) based on mutual information that allows to quantify the privatization effects on biometric images.
Paper outline
The remainder of this paper is organized as follows. In Sec. 2, we provide a brief survey of recent relevant works. In Sec. 3, we present the architectural details of our SGAP model. The main results of our paper are presented in Sec. 4. We conclude our paper in Sec. 5.
Related Work
Privatization of data has been an active area of research with multiple works touching on this subject [18,30,16,1]. Our approach extends the concept of contextindependent data privatization by incorporating context-dependent information as an input to the privatization algorithm. More precisely, it identifies the discriminative characteristics of the data and distorts them to enforce privacy. Although standard methods of protecting privacy based on erasing personal information have been widely used, correlation and linkage attacks allow to reidentify the users, even when explicitly identifying information is not present in the released datasets [25]. Those kinds of attacks pose an even greater threat to individual privacy when used against publicly available medical databases [14]. [15] show that using publicly available genotype-phenotype correlations, an attacker can statistically relate genotype to phenotype and therefore re-identify individuals. Publicly available profiles in the Personal Genome Project can be linked with names by using demographic data [32]. Also, when considering fRMI imaging data, individual variability across individuals is both robust and reliable, thus can be used to identify single subjects [9].
Although numerous works are focused on finding discriminative patterns within the data [10,34], we use a Siamese neural network architecture [4] since it allows us to learn a discriminant data embedding in an end-to-end fashion. Contrary to the typical goal of a Siamese architecture, i.e. learning similarity, we use it to identify discriminant parts of a pair of images and alter those parts with minimal impact on other useful features. When both examples come from the same individual, this setup allows us to learn a perturbation that carefully disguises the individual's identity, hence protecting their privacy.
One can consider the problem of data anonymization to be conceptually similar to the idea of adversarial examples in neural network architectures [21,19,20,3,33]. In the case of adversarial examples, the adversary wants to trick the neural network into misclassifying a slightly perturbed input of a given class. Similarly, our goal is to modify the data points in such a way that the identity of the individual corresponding to the data cannot be correctly classified. The most relevant work is [20], where they use a Generative Adversarial Network (GAN) [13] framework to create adversarial examples and use them in training to increase the robustness of the classifier.
Similar to us, [28] analyses the trade-off between data privacy and utility. In their work, however, privacy and utility metrics are defined based on a userstudy, where the users were asked to assess the usefulness of the anonymized images in the context of social media distribution. The privacy, on the other hand, was measured by first enlisting a number of attributes linked to privacy (e.g. passport number or registration plates) and then asking the users to validate if a given privacy attribute is visible in the photo or not. We argue that this way of measuring both privacy and utility is limited to a very specific subset of applications. In our work we propose fundamentally different metrics for both privacy and utility that have backing in information theory and machine learning.
Another relevant and recent works [33], [5] address the privatization problem using a generative adversarial approach while providing theoretical privacyutility trade-offs. The work of [5], which is the most similar to our work, proposes an architecture combining Variational Autoencoder (VAE) and GAN to create an identity-invariant representation of a face image. Their approach differs from ours as they use an additional discriminator, which explicitly controls which useful features of the images are to be preserved, whereas in our approach the model has no information about other features of the images, except that it knows whether a pair of images belongs to the same person or different people. This is a significant contribution because in practice, one cannot expect to know all potential applications of the privatized images. Therefore our approach proves to be more robust towards real-life applications.
[27] presents a similar game-theoretic perspective on image anonymization. However, the difference is that it focuses on adversarial image perturbations (carefully crafted perturbations invisible to human), while our privatizer introduces structural changes to the image. In [31], a head inpainting obfuscation technique is proposed by generating a realistic head inpainting using facial landmarks. On contrary, our goal is to hide the identity of a person without knowing which part of the image is responsible for identity. Thanks to this, our framework is more universal and has a much wider field of application, not only to hide face identity, but also hide identity in cases where there is no prior knowledge of which part of the image should be obfuscated. [23] and [24] are relevant to our work and deal with a problem similar to ours. However, the formulation of the problem is different from ours. [23] and [24] transform an input face image in a way such that the transformed image can be successfully used for face recognition (so the identity is preserved) but not for gender classification. Our goal is the opposite, we want to hide identity while maintaining as much other features as possible, without explicitly modeling the non-malicious classification tasks. Another difference is that our model requires only identity labels. The architecture of the models presented in [23] and our work are similar, however we use Siamese discriminator what makes our approach advantageous when applied to large datasets with thousands or even millions of people, since this architecture reduces the output of the discriminator to a binary output rather than create a long list of individual class predictions.
Method
The goal of our approach is to develop a privatizer that converts an input image into its privatized version in such a way that: (1) the privacy of the subject is preserved by making sure that the identifying features are hidden, (2) the utility of the original image is maintained by preserving the non-identifying features that are vital for other machine learning tasks, and (3) the privacy-utility tradeoff can be adjusted.
Proposed approach
To enforce the above conditions, we will use a custom neural network architecture, dubbed Siamese Generative Adversarial Privatizer, that consists of two tightly coupled models: a generator G(θ g ) and a discriminator D(θ d ). This coupling is inspired by Generative Adversarial Networks (GANs) [13]. Two neural networks compete with each other: the discriminator tries to predict the identity of the person in the image, while the generator tries to generate such an image which fools the discriminator and thus hides the identity of the person.
We use a Siamese architecture [4] for the discriminator. This allows us to extract discriminative and identifying features from images. More importantly, this architecture reduces the output of the discriminator to a single value (from 0 to 1) rather than create a long list of individual class predictions, an approach which would be prohibitive when applied to large datasets with thousands or even millions of people. In this case, we use pairs of images (instead of single images) to train the neural network, and the goal of the Siamese discriminator is to classify whether the two images belong to the same person or to different people.
Furthermore, the above problem is subjected to a distortion constraint, which ensures that the privatized images are not too different from the original images.
We did not use L 2 since it is sensitive to small changes (e.g. shift, rotation, etc.) which do not significantly affect the content of the image. Instead we chose SSIM (structural similarity index) [35] which is sensitive to the structural changes of images, not pixel-by-pixel differences like L 2 [36]. We enforce Discriminator's architecture. We use a Siamese neural network to verify the identities of people in the images. The discriminator classifies whether a pair of images belongs to the same person or to different people. We get the output from the range between 0 and 1 applying distance-based loss function to the output of the last fully connected layer of the Siamese discriminator. a constraint on SSIM which allows us to control the level of distortion added to protect identity, and thus ensure that the quality of privatized images is not substantially degraded. The architecture overview can be seen in Fig. 2.
Architecture
Our discriminator is a Siamese convolutional neural network, which consists of two identical branches with shared weights, as shown in Fig. 3. Each branch consists of 3 blocks of the following form: (1) Convolutional layer (mask 3 × 3, stride=1, padding=0), (2) Leaky rectified linear unit (α = 0.1), (3) Batch normalization, (4) Dropout (p = 0.2). The blocks are followed by 2 dense layers (500 neurons, leaky rectified linear unit, α = 0.1) and an output layer (15 neurons). A discriminator network converts two input images to two output representations (embeddings) D(X 1 , X 2 ) → (o 1 , o 2 ). At each upsampling step we halve the number of feature channels. Also we concatenate the feature maps of the decoder part with the corresponding feature map from the encoder part (these are bypasses). Last deconvolutional layer is followed by a hyperbolic tangent activation function.
A noise matrix Z is added to the bottleneck part of the generator, i.e. to the latent space variable representing input image in a low-dimensional space. We use a noise matrix instead of a vector, as we do not use a standard fully-connected layers in our generator and retain convolutional layers instead. The output of generator network is a privatized version of original image: G(Z, I) →Ĩ.
Training
When iterating over training dataset we get tuples: (I i , I i , l i ), where I i and I i is a pair of images and l i is a binary label where l i = 0 if the images have the same identity and l i = 1 for different identities. There are two types of pairs in the training set. Firstly, when the generator is turned off, I i , I i are images from the original training set. Secondly, when the generator is turned on,Ĩ i = G(Z i , I i ) is the privatized version of the image I i from the original training set. I i is the reference image, also from the original training set. In both cases mentioned above we use stratified random sampling in order to balance two classes: l = 0 and l = 1.
The discriminator D takes a pair of images I, I and outputs a probability that both images come from the same person, i.e. l = 0, based on a distancebased metric: We train our model similarly to GAN. When the generator training is frozen, our goal is to train the discriminator to recognize whether a pair of images belongs to the same person or to different people. When the generator is trained, there is a minmax game between the generator and the discriminator in which the generator is trying to fool the discriminator and generate an image that hides the identity of the subject. The training equation for our privatization task is:
D(I, I ) → 1 + e −m 1 + e d(min D max G 1 N N −1 i=0 L(l i , D(I i , I i )) + 1 N N −1 i=0 L(0, D(I i , G(Z i , I i )))
Furthermore, the above minimax optimization problem is subject to the following critical constraint: 1 y) is a distortion metric and δ is a distortion threshold. The distortion constraint is used to limit all the other image changes except for hiding identity and therefore the utility of the images is preserved. We use Structural Similarity Index as the distortion metric. The above constraint can be incorporated into the main minimax objective function as follows:
N N −1 i=0 d(I i , G(Z i , I i )) < δ, where d(x,min D max G N −1 i=0 L(l i , D(I i , I i )) + N −1 i=0 L(0, D(I i , G(Z i , I i ))) + λ N −1 i=0 d(I i , G(Z i , I i ))
(1) Our Siamese Generative Adversarial Privatizer network is trained for 100 epochs using ADAM optimizer with β 1 = 0.9 and β 2 = 0.999.
Results
In this section, we present the results of evaluation of our method. We first present the datasets and evaluation metrics. Then we show qualitative and quantitative results of our evaluation that confirm usefulness of our approach in the context of data privatization.
Datasets
Fingerprints To validate how well our method performs in terms of identity privatization, we evaluate it on a dataset of fingerprints. Although the main purpose of fingerprint datasets is to identify people and therefore their privatization may not be needed in their real-life use cases, we treat this dataset as our toy example and evaluate how well we can hide the privacy of the fingerprint owner. Since there exists a trade-off between the privatization and the utility of the resulting data, we refer to a proxy task of finger type classification to validate how useful our privatization method is. In other words, we try to classify the type of the finger (e.g. middle finger, index finger, ring finger) while gradually increasing the privacy of the dataset. Sec. 4.4 presents the results of this experiment.
We use NIST 8-Bit Gray Scale Images of Fingerprint Image Groups [26]. This database contains 4000 8-bit grayscale fingerprint images paired in couples. This way we have a 50%/50% split over similar/dissimilar pairs and the dataset loader is quasi-deterministic (for a given index i the first image is guaranteed to be constant).
Animated faces
The second dataset that we use is FERG dataset [2]. FERG is a dataset of cartoon characters with annotated facial expressions. It contains 55769 annotated face images of six characters. The images for each identity are grouped into 7 types of facial expressions, such as: anger, disgust, fear, joy, neutral, sadness and surprise.
In each epoch the dataset is iterated over 10000 pairs of images. For the first half of the pairs we use different randomly selected images of the same person. In this case l = 0. For the second half of the pairs we use randomly selected images of different people. In this case l = 1. This way we have a 50%/50% split over similar/dissimilar pairs and the dataset loader is quasi-deterministic.
Evaluation metrics
To evaluate the performance of our SGAP model and show that it learns privacy schemes that are capable of hiding biometric information even from computationally unbounded adversaries, we propose computing the mutual information between: (a) X = (X 1 , X 2 ) where X 1 is a privatized image and X 2 is an original image, and (b) Y where Y = 0 when X 1 and X 2 belong to the same person and Y = 1 when they belong to different people. X 1 is privatized using the scheme that is learned in a data-driven fashion using SGAP. By Fano's inequality, if I(X; Y ) is low then Y cannot be learned from X reliably (even under computationally infinite adversaries) [6]. In other words, if I(X; Y ) is sufficiently small, there's no way we can reliably learn whether or not a privatized image belongs to the same person in another non-privatized image. This ensures that privacy is guaranteed in a strong sense.
In practice, we do not have access to the joint distribution P (X, Y ). We instead have access to a dataset of i.i.d observations
D = {(X i , Y i } n i=1 }.
Here, the X i 's are computed after the SGAP training phase is over by applying the learned privacy scheme on a separate test set. We are thus interested in empirically estimating I(X; Y ) from D. We will call this estimate "empirical mutual information" and denote it byÎ(X; Y ). To computeÎ(X; Y ), we can use the following formula:Î
(X; Y ) =Ĥ(X) −Ĥ(X|Y )
whereĤ(X) andĤ(X|Y ) are the empirical entropies of X and X given Y . To compute these empirical entropies, we use the Kozachenko-Leonenko entropy estimator [11] which we briefly explain next. Letting R i = min j,j =i X i − X j , for j = 1, . . . , n, we get
H(X) = 1 n n i=1 log (n − 1)R d i + constant = d n n i=1 log R i + 1 n n i=1 log(n − 1) + constant
where d is the dimension of X, i.e. X i ∈ R d . Assuming we have a two-class problem (Y = 0 for same identities, Y = 1 for different identities), the conditional entropy is given bŷ
H(X|Y ) =Ĥ(X|Y = 0)P (Y = 0) +Ĥ(X|Y = 1)P (Y = 1)
Notice thatP (Y = 0) = n0 n ,P (Y = 1) = n1 n , where n 0 and n 1 are the counts of samples with label Y equals 0 and 1 respectively. We divide sample X into two partitions. Letting i 1 , i 2 , . . . , i n0 be the indices corresponding to Y i = 0, we have a set X 0 = {X i1 , X i2 , . . . , X in 0 }. Automatically we have i 1 , i 2 , . . . , i n0 , the indices of samples associated with Y i = 1. Thus, we get X 1 = {X i 1 , X i 2 , . . . , X i n 1 }. Therefore we calculate the nearest neighbor distance for each sample within the particular set as follows:
R i k = min l =k,l=1,...,n0 X i k − X i l R i k = min l =k,l=1,...,n1 X i k − X i l H(X|Y = 0) = 1 n 0 n0 k=1 log (n 0 − 1)R d i k + constant H(X|Y = 1) = 1 n 1 n1 k=1 log (n 1 − 1)R d i k + constant
Then the empirical mutual information can be expressed aŝ
I(X, Y ) =Ĥ(X) − Ĥ (X|Y = 0)P (Y = 0) +Ĥ(X|Y = 1)P (Y = 1) = 1 n n i=1 log (n − 1)R d i + − 1 n 0 n0 k=1 log (n 0 − 1)R d i k n 0 n + 1 n 1 n1 k=1 log (n 1 − 1)R d i k n 1 n
To estimate values of R i k and R i k we use L 2 norm between image pixels projected to a 3-dimensional space via t-SNE [22]. We reduce the dimensionality to increase the efficiency of computation, but our metric remains agnostic to image distance calculation and other methods can also be used here.
The second approach to quantify privacy is by measuring an identity misclassification rate. We measure what percentage of privatized images effectively fool our Siamese discriminator.
To quantify utility of privatized dataset we measure accuracy of the proxy classification task (finger type classification for fingerprint dataset and facial expression classification for faces dataset). More precisely, we evaluate how good in terms of accuracy a separate independent method can be trained for using a privatized dataset. We use fine-tuned ResNet architecture, pre-trained on ImageNet without freezing. In addition we split the dataset into training and validation. The accuracy is measured using k-fold validation (k = 4).
Qualitative results
In this section, we present the qualitative results of our evaluation, demonstrating the ability of our network to increase the privacy of input data. Fig. 5 and 6 show sample results obtained as an output of our privatization. In Fig. 6 we see that the identities of people have been hidden, while other useful features, in this case facial expressions, have been preserved. Fig. 7, 8 and 9 illustrate the trade-off between utility and privacy while tuning λ distortion metric constraint. We see that by tuning the λ parameter we can adjust the level of privacy and utility, finally finding the optimal value for both privacy and utility. Fingerprints with added artifacts that fool identity discriminator in the middle row. Structural Similarity difference [35] of the original and privatized images is presented in the bottom row. Our Siamese Generative Adversarial Privatizer learns to locate discriminant image features, such as fingerprint minutiae, and substitutes them with anonymizing artifacts. Although in practice fingerprints are used for person identification, we validate if privatized images can be useful (i.e. if they can retain utility) for a proxy task of finger type classification. Since our method does not add noise arbitrarily across the image, but only focuses on hiding sensitive personal information, the resulting dataset can be published and used by machine learning for other tasks, e.g. finger type classification or skin disease detection.
Quantitative results
To obtain quantitative results we train our SGAP model with different values of maximal distortion constraint λ (see Eq. 1) in order to adjust the privacy level of the dataset. The goal of our generator is to add such noise to the latent space that privatized image fools the discriminator, which the discriminator in turn has to verify if the pair of images comes from the same person. After SGAP is trained, the generator part can be used to privatize datasets. To measure the utility of the privatized fingerprints dataset, we refer to a proxy task of finger type classification. Although in fingerprints are typically used to identify the identity of an individual, in our case we use the proposed privatization method to hide the identity and anonymize the dataset. The objective of this experiment is to evaluate how increasing data privacy effects the utility of the resulting dataset when used as training data for a machine learning algorithm. Hence, we use a proxy machine learning task, finger type classification. To measure the utility of the privatized cartoon faces dataset, we use facial expression classification as machine learning task. As a classifier, trained on privatized datasets, we use fine-tuned ResNet architecture, pre-trained on ImageNet without freezing. For each dataset generated using different maximal distortion constraint, we calculate classification accuracy and quantify the privacy by estimation of mutual information (fingerprint dataset) or using identity misclassification rate (faces dataset). Fig. 10 and 11 show the results. In both cases we see a significant drop in privacy metric, while for the same range of parameters, the accuracy of the classifier remains stable, indicating that the utility of the dataset is not decreased.
Conclusions
We presented the Siamese Generative Adversarial Privatizer (SGAP) model for privacy-preserving of biometric data. We proposed a novel architecture combining Siamese neural network, autoencoder, and Generative Adversarial Network to create a context-aware privatizer. Experimental results on two public datasets demonstrate that our approach strikes a balance between privacy preservation and dataset utility. Preserve utility while improving privacy Further privatization destroys utility Fig. 10. Graph of mutual information estimation and the accuracy of a classifier trained with fingerprint dataset privatized with different maximal constraint distortion thresholds. In green the region where the utility of dataset is preserved while the likelihood of classifying the privatized version of the image as belonging to a given person is reduced. This result proves that by using our privatization method we are able to significantly increase the privacy of the biometric dataset, while not reducing its utility for a proxy task of finger type classification. Preserve utility while improving privacy Further privatization destroys utility Fig. 11. Graph of identity misclassification rate and the accuracy of a classifier trained with cartoon faces dataset privatized with different maximal constraint distortion thresholds. In green the region where the utility of dataset is preserved while the likelihood of classifying the privatized version of the image as belonging to a given person is reduced. This result proves that by using our privatization method we are able to significantly increase the privacy of the biometric dataset, while not reducing its utility for a task of facial expression classification.
Fig. 1 .
1Basic functionality of the proposed Siamese Generative Adversarial Privatizer: given an original face image, the privacy filter generates a privatized image. The original identity is hidden, at the same time other useful features, e.g. facial expression, are preserved. Siamese discriminator identifies the discriminative features of the images.
Fig. 2 .
2Overview of our Siamese Generative Adversarial Privatizer model. The generator acts as a privacy filter, which hides the identity of the person in the original images. The Siamese discriminator recognizes whether the person in the privatized image is the same person as in the reference image.
Fig. 3 .
3Fig. 3. Discriminator's architecture. We use a Siamese neural network to verify the identities of people in the images. The discriminator classifies whether a pair of images belongs to the same person or to different people. We get the output from the range between 0 and 1 applying distance-based loss function to the output of the last fully connected layer of the Siamese discriminator.
Fig. 4 .
4Generator's architecture. We use Variational Autoencoder-like architecture to generate a privatized image in a context-aware manner based on the original image. At the bottleneck of the generator we get a compressed representation of the image without identity features, and thanks to the bypasses between the layers we preserve other useful features of the original image. The generator network, as presented in Fig. 4, consists of two parts: the encoding part and the decoding part. The encoder follows the typical architecture of a convolutional neural network. It consists of 5 blocks of the following form: (1) Convolutional layer (mask 4 × 4, stride=2, padding=1), (2) Leaky rectified linear unit (α = 0.1), (3) Batch normalization. At each downsampling step we double the number of feature channels. The decoder consists of 5 blocks of the following form: (1) Transpose convolutional layer (mask 4 × 4, stride=2, padding=1), (2) Leaky rectified linear unit (α = 0.1), (3) Batch normalization, (4) Dropout (p = 0.5).
∼
I ) where m is a predefined margin and d(o, o ) is an Euclidean distance between embeddings o and o in the last fully connected layer of the discriminator. Given this formulation of the discriminator we use a cross entropy loss for training: L(l, D(I, I )) = −(1 − l) log D(I, I ) − l log 1 − D(I, I )
Each image is 512-by-512 pixels with 32 rows of white space at the bottom. We use only one image of each pair in our experiments. The dataset contains images for 2000 individuals. For each person there are two different fingerprint shots of the same finger (denoted as: f , s). Our method requires pairs of images as input. In each epoch the dataset is iterated over 4000 pairs of images. For the first half of the pairs when index of a pair is i < 2000 we return a label l = 0 and a pair of images (f , s) belonging to the person with ID = i. For the second half of the pairs when index i >= 2000 we return a label l = 1 and two images. First image is image f of person with ID = i − 2000. Second image is an image (f or s) of a different person (selected at random).
Fig. 5 .
5A toy example of how our privatization method can hide identities of the fingerprint owners. Original fingerprints in the upper row.
Fig. 6 .
6Original cartoon faces in the upper row. Privatized versions of cartoon faces in the bottom row. Our Siamese Generative Adversarial Privatizer learns to hide the identity of the people, while other important image features, such as facial expression remain intact.
Fig. 7 .
7Too much privacy, utility is not preserved. Original cartoon faces in the upper row. Privatized versions of cartoon faces in the bottom row. Our model has been tuned too much towards ensuring privacy, so that the utility of the images has not been preserved, facial expressions are hard to recognize.
Fig. 8 .
8Not enough privacy, utility is preserved. Original cartoon faces in the upper row. Privatized versions of cartoon faces in the bottom row. Our model has been tuned too much towards preserving utility, so that the identities of the people in the images are not hidden, only minor changes have been added to the images.
Fig. 9 .
9Images in the first column are the original ones, next there are privatized images generated for different values of distortion constraint λ ∈ {10, 8, 6, 4, 2, 1, 0.7}. Original images of different identities collapse into an anonymous identity with the expression preserved from the original image.
AcknowledgmentThe work was partially supported as RENOIR Project by the European Union Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 691152 (project RENOIR) and by Ministry of Science and Higher Education (Poland), grant No. W34/H2020/2016. We thank NVIDIA Corporation for donating Titan Xp GPU that was used for this research.
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| []
|
[
"Robust Landmark Detection for Alignment of Mouse Brain Section Images",
"Robust Landmark Detection for Alignment of Mouse Brain Section Images"
]
| [
"Yuncong Chen \nUCSD CSE Department\n\n",
"David Kleinfeld \nUCSD Physics Department\n\n",
"Martyn Goulding \nSalk Institute for Biological Studies\n\n",
"Yoav Freund \nUCSD CSE Department\n\n"
]
| [
"UCSD CSE Department\n",
"UCSD Physics Department\n",
"Salk Institute for Biological Studies\n",
"UCSD CSE Department\n"
]
| []
| Brightfield and fluorescent imaging of whole brain sections are fundamental tools of research in mouse brain study. As sectioning and imaging become more efficient, there is an increasing need to automate the post-processing of sections for alignment and three dimensional visualization. There is a further need to facilitate the development of a digital atlas, i.e. a brain-wide map annotated with cell type and tract tracing data, which would allow the automatic registration of images stacks to a common coordinate system. Currently, registration of slices requires manual identification of landmarks. In this work we describe the first steps in developing a semi-automated system to construct a histology atlas of mouse brainstem that combines atlas-guided annotation, landmark-based registration and atlas generation in an iterative framework. We describe an unsupervised approach for identifying and matching region and boundary landmarks, based on modelling texture. Experiments show that the detected landmarks correspond well with brain structures, and matching is robust under distortion. These results will serve as the basis for registration and atlas building. | null | [
"https://arxiv.org/pdf/1803.03420v1.pdf"
]
| 3,790,944 | 1803.03420 | 024ab22e4ce3ee286bd7b87598045e45730ee867 |
Robust Landmark Detection for Alignment of Mouse Brain Section Images
Yuncong Chen
UCSD CSE Department
David Kleinfeld
UCSD Physics Department
Martyn Goulding
Salk Institute for Biological Studies
Yoav Freund
UCSD CSE Department
Robust Landmark Detection for Alignment of Mouse Brain Section Images
landmark detectionatlas buildingmouse brainregistrationauto- mated annotation
Brightfield and fluorescent imaging of whole brain sections are fundamental tools of research in mouse brain study. As sectioning and imaging become more efficient, there is an increasing need to automate the post-processing of sections for alignment and three dimensional visualization. There is a further need to facilitate the development of a digital atlas, i.e. a brain-wide map annotated with cell type and tract tracing data, which would allow the automatic registration of images stacks to a common coordinate system. Currently, registration of slices requires manual identification of landmarks. In this work we describe the first steps in developing a semi-automated system to construct a histology atlas of mouse brainstem that combines atlas-guided annotation, landmark-based registration and atlas generation in an iterative framework. We describe an unsupervised approach for identifying and matching region and boundary landmarks, based on modelling texture. Experiments show that the detected landmarks correspond well with brain structures, and matching is robust under distortion. These results will serve as the basis for registration and atlas building.
Introduction
In this paper we describe a method for the automatic detection of landmarks in histology images of mouse brain sections. The purpose is to facilitate image registration and atlas generation. By aligning images of nissl and fluorescent-stained brain sections, we aim to create a digital atlas for the mouse brainstem, which incorporates cell type, tract tracing and other physiological data. Unlike most brain atlases which are intended to be viewed by a human, the atlas we aim to build is intended to be used by a computer. Specifically, the goal is to represent the landmarks in a datastructure that would be used by an algorithm to automatically align a whole stack to a common coordinate system. The novelty of the work described here is the use of unsupervised learning to find regions with distinct texture and clear boundaries. This will reduce the manual work needed to identify reliable landmarks. Figure 1 shows a typical image of nissl-stained mouse brainstem section. Although the brainstem does not have salient edges like the cortex, it contains many compact neuron clusters (nuclei) and striated regions (fiber tracts). Both types of structure have distinct texture that can be detected and modelled.
We use semi-supervised learning to create a library of landmark detectors based on texture and shape. They can be applied to identify landmarks from new images and also be updated by incorporating new data. This is at the center of our vision of integrating atlas-guided annotation, landmark-based registration and atlas generation into an iterative framework.
The remaining of this paper is organized as the follows. Section 3 describes how we represent textures using Gabor filters and superpixels. Section 4 and 5 explains how we identify potential landmark regions based on texture distinctiveness. Section 6 describes how we detect stable boundary segments to complement region landmarks. Section 7 describes how landmarks are matched between sections.
Related Work
Existing work on automatically reconstructing 3D volume from histology section include the Allen Mouse Brain Atlas [6,10] and the Waxholm Space [4], both of which use intensity-based methods such block-matching [7,8] and mutual information maximization. Landmark-based methods take advantage of details in the histology image using descriptors such as SIFT [9] and binarized gradient orientation histogram [5].
Representing Texture using Histograms of Gabor Textons
Images are first filtered using Gabor filters [3] with 9 orientations and 11 scales, resulting in 99 dimensional feature vectors. We reduce the data by quantizing these feature vectors into 100 clusters using K-Means. Clusters with close centroids are then merged, and the final cluster centroids are the textons. In our experiment, we have 14 textons.
We further reduce the data by describing texture at the level of superpixels. For each superpixel, texture is represented by the histogram of the textons it contains ( Figure 1). We denote the texton histogram of the i'th superpixel by h i . Superpixels are obtained using the SLIC algorithm [1].
Since different sections may be oriented at different angles, we need the features to be rotation-invariant. For this purpose, before doing K-Means, we compute the energy distribution over orientations for each Gabor feature vector, and shift all feature vectors such that the modes of the directional energy distributions are aligned. In this way, the texture is decoupled from directionality.
Evaluating Region Significance using Statistical Test
We define a region landmark as a connected set of superpixels. Before describing how to find significant regions, we need a scoring function that evaluates the significance of a region. We propose the significance score, F (S), of a region S, to be a linear combination of three terms.
The first term is the distinctiveness of this region's texture, or in other words, how different its texture is from the surrounds. Since textures are represented as histograms, we formulate this problem as a statistical test on whether the region's average texton histogram and the texton histograms of the surrounding superpixels are sampled from the same distribution. We employ the chi-squared test for independence and the resulting p-value serves as a quantitative measure of the distinctiveness. The smaller the p-value, the more significant the region is. Since a region often has neighbours with different textures, the test is made between the region's histogram and that of every surrounding superpixel. To be conservative, the largest p-value among all tests is used. Denote the set of surrounding superpixels by T (S), the first term is,
F cont (S) = − max j∈T (S) pval(h S , h j )
where pval(·, ·) is the p-value of applying the chi-square test for independence to two histograms.
The second term is the texture homogeneity within the region, defined as the mean p-value of chi-square tests between all superpixels in the region and the region's average:
F coh (S) = 1 |S| i∈S pval(h S , h i )
The third term stems from the fact that most region landmarks have compact shapes. A common measure for the compactness of a closed curve is the isoperimetric quotient, defined as the enclosing area divided by the square of the circumference. We use the number of surrounding superpixels as a proxy for the circumference, and the number of superpixels in the region as that for the area.
F comp (S) = |T (S)| |S| 2
The overall significance score F (S) = w 1 ·F cont (S)+w 2 ·F coh (S)+w 3 ·F comp (S), with weights chosen empirically.
Finding Significant Regions with Region Growing and Clustering
To find potential regions, we perform region growing for every superpixel. Starting with a singleton containing the seed superpixel, this greedy procedure considers all neighbours of the current set, and iteratively adds the one whose texture is closest to the current region's texture, measured by the χ 2 distance between texton histograms. The growing stops when the region reaches 10% of the total area. Eventually, the region at the point when the significance score is the largest is returned. Figure 2 shows examples of this procedure. We call the set of superpixels that grows from a seed superpixel k the region proposal of the seed, denoted by S k . If a region has distinct texture, the proposals of superpixels inside this region should be very similar, while proposals from within an inhomogeneous region are random (see Figure 3). In other words, the region proposals form clusters. The denser a cluster is, the more distinctive the region it represents. Using Jaccard index as pairwise distance, we group the region proposals using hierarchical clustering. Then within each group whose size is large enough, we select the proposal with the highest significance score to be the representative of that group. All representative proposals are ranked according to their significance scores.
Identify Robust Boundaries by Region Consensus
Sometimes a region gradually transitions into neighboring texture on one side, but has a clear boundary on the other side. The method described in the previous section may not capture this inhomogeneous region, but the open boundary is nonetheless a perfect landmark. Figure 4 shows such an example. Notice how much the proposals from seeds in such region vary, but many of them still agree on the clear boundary. This motivates a consensus-based approach for evaluating boundary robustness, in which each region proposal votes for the segments on its boundary. We represent a boundary segment between two superpixels by an ordered tuple (i, j), where i is the interior superpixel and j is the exterior superpixel. We also denote by δS k , the set of segments on the boundary of a region proposal S k . The vote a region proposal casts to a boundary segment depends on how contrasty the segment is, measured by the texture histogram distance between the region's average and the segment's exterior superpixel. We call the set of superpixels that vote for a segment the supporter set of the segment, and denote it by R (i,j) = {k : (i, j) ∈ δS k }. The total score received by a segment (i, j) is then:
b (i,j) = k∈R(i,j) χ 2 (h S k , h j )
We discard segments whose vote is lower than a threshold. Figure 5 shows a vote map. Instead of modelling individual segments, we combine segments with similar supporter sets into groups, again using hierarchical clustering with Jaccard index between supporter sets as pairwise similarity. Each segment group represents a boundary. All boundaries are ranked according to the total vote received by its segments.
Matching Landmarks from Different Sections
By representing region landmarks using closed boundaries and merging coinciding boundaries, we unify the two types of landmarks into a single set consisting of both open and closed boundaries. In order to use the landmarks for registration, correspondences must be made between them. In this section, we design a distance function for comparing two boundaries. This function is a weighted combination of the differences in four aspects, with weights chosen empirically:
D(B 1 , B 2 ) = w int D int (B 1 , B 2 ) + w shape D shape (B 1 , B 2 ) + w ext D ext (B 1 , B 2 ) + w loc D loc (B 1 , B 2 )
The first term is the difference of interior textures. For a region landmark S, the interior texture is simply the average texton histogram h S . For a boundary B, the interior texture is the average texton histogram of the union of all segments' supporter sets. Denote this union by
Q B = ∪ (i,j)∈B R (i,j) , then h int B = h Q B .
The difference is computed using the χ 2 distance:
D int (B 1 , B 2 ) = χ 2 (h int B1 , h int B2 )
The second term measures the similarity of boundary shapes. We reduce the boundary to a point set consisting of midpoints of the segments. The shape distance between two point sets can be computed using shape context descriptors [2]. The shape context descriptors characterize the organization of other points around each point using a histogram. Two sets of points are matched by finding the minimum bipartite matching, where the edge weights are the chi-square distances between shape context descriptors. The Hungarian algorithm is used to find the minimum matching. The average cost of this matching, is used as the second term:
D shape (B 1 , B 2 ) = 1 |M | (i,j),(p,q)∈M (B1,B2) χ 2 (c i,j , c p,q )
where M (B 1 , B 2 ) is the minimum matching, c i,j and c p,q are the shape context descriptors of segment (i, j) and segment (p, q) respectively. After the matching is made, we compute the third term, defined as the total distance between exterior textures of all matched segments:
D ext (B 1 , B 2 ) = (i,j),(p,q)∈M (B1,B2) χ 2 (h j , h q )
The fourth term measures the spatial proximity. This is the thresholded Euclidean distance between the center of mass of the boundaries' midpoint sets:
D loc (B 1 , B 2 ) = max(0, m B1 , m B2 2 − l)
where m B1 and m B2 are the center of mass of boundaries B 1 and B 2 , and l is a tolerance within which the position deviation is not penalized (set to 1 mm in our experiment).
Using this distance function, we compute the pairwise distances for two sets of landmarks detected from different sections. Two landmarks are matched if they are simultaneously the closest landmark to each another. Figure 6 shows the landmarks detected from one image. Also shown is a version created by a human labeller who annotated for nuclei and fiber tracts with the help of a printed atlas. Most significant structures from the human labelling are detected by the algorithm.
Robustness of Matching
The landmark matching algorithm is applied to all 30 pairs of consecutive images. In order to test the robustness of matching under large displacement, we remove the spatial proximity term from the landmark distance function, leaving only the texture and shape terms. A human evaluator then judges whether a matching is correct, partially correct, or incorrect. Among all 166 matchings returned by the algorithm, 106 are correct (63%), 34 are partially correct (22%) and 26 are wrong (15%). One example is shown in Figure 7. This demonstrates the effectiveness of texture modelling. Even though some landmarks change shape significantly, the algorithm still finds the correct matchings. In this paper we described algorithms for detecting region and boundary landmarks from histology images of mouse brainstem. Region proposals are grown from superpixels, based on which significant regions such as nuclei and fiber tracts are identified using clustering. Detected regions are shown to correspond well with real anatomical structures. To complement region landmarks, robust boundary segments are also found by consensus voting. Landmarks from different sections can be matched using a distance function that is robust under distortion.
Our next step is to use the matched landmarks for intra-specimen registration, and further co-register multiple specimens to generate the atlas. Meanwhile, we also plan to include human feedback in the learning loop. Although the advantage of our approach is reducing human supervision, it is important for human experts to correct wrong labellings made by the algorithm in order to learn accurate models for landmarks. Instead of requiring them to label every region of interest, our method only needs them to validate our library of detectors. Once those have been validated, the process is completely automatic.
Fig. 1 .
1Left: A typical image of mouse brainstem section. Right top: part of the original image with superpixels overlaid. Middle: texton map of the same area. Bottom: texton histograms of three superpixels with different textures.
Fig. 2 .
2Regions during growing, from left to right, at iteration 1, 10, 50, 96 (most significant), 150 (last iteration).
Fig. 3 .
3Top row: the region proposals of four different seeds (in red) in the facial motor nucleus. They are very similar. Bottom row: the proposals of four neighboring seeds in an inhomogeneous region. They are very random.
Fig. 4 .
4The three region proposals in an inhomogeneous area are not consistent as a whole, but they all agree on a robust boundary segment (highlighted).
Fig. 5 .
5Left: thresholded boundary vote map. Right: grouped boundary segments
our algorithm on a series of 30 section images of a nissl-stained mouse brainstem. The images are scanned at 2 microns per pixel, showing individual neuronal cell bodies. Both types of landmarks are detected on all images. For each image we use the top 20 closed boundary and the top 10 open boundary.
Fig. 6 .
6Left: top 20 landmarks detected by the algorithm. Right: human labelling for recognizable nuclei and white matter.
Fig. 7 .
7Landmark matching example. Matched landmarks are marked with the same color and number. Note that matchings are found based only on texture and shape. Matching 13 is made possible by modelling both open boundaries and close boundaries. Landmarks such as 12 and 9 show considerable shape change, but are still matched.
Slic superpixels compared to state-of-the-art superpixel methods. Pattern Analysis and Machine Intelligence. R Achanta, A Shaji, K Smith, A Lucchi, P Fua, S Susstrunk, IEEE Transactions on. 3411Achanta, R., Shaji, A., Smith, K., Lucchi, A., Fua, P., Susstrunk, S.: Slic superpixels com- pared to state-of-the-art superpixel methods. Pattern Analysis and Machine Intelligence, IEEE Transactions on 34(11), 2274-2282 (2012)
Shape context: A new descriptor for shape matching and object recognition. S Belongie, J Malik, J Puzicha, In: NIPS. 23Belongie, S., Malik, J., Puzicha, J.: Shape context: A new descriptor for shape matching and object recognition. In: NIPS. vol. 2, p. 3 (2000)
Unsupervised texture segmentation using gabor filters. A K Jain, F Farrokhnia, Systems, Man and Cybernetics. IEEEJain, A.K., Farrokhnia, F.: Unsupervised texture segmentation using gabor filters. In: Sys- tems, Man and Cybernetics, 1990. Conference Proceedings., IEEE International Conference on. pp. 14-19. IEEE (1990)
Waxholm space: an image-based reference for coordinating mouse brain research. G A Johnson, A Badea, J Brandenburg, G Cofer, B Fubara, S Liu, J Nissanov, Neuroimage. 532Johnson, G.A., Badea, A., Brandenburg, J., Cofer, G., Fubara, B., Liu, S., Nissanov, J.: Wax- holm space: an image-based reference for coordinating mouse brain research. Neuroimage 53(2), 365-372 (2010)
Landmark/imagebased deformable registration of gene expression data. U Kurkure, Y H Le, N Paragios, J P Carson, T Ju, I A Kakadiaris, Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on. IEEEKurkure, U., Le, Y.H., Paragios, N., Carson, J.P., Ju, T., Kakadiaris, I.A.: Landmark/image- based deformable registration of gene expression data. In: Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on. pp. 1089-1096. IEEE (2011)
Genome-wide atlas of gene expression in the adult mouse brain. E S Lein, M J Hawrylycz, N Ao, M Ayres, A Bensinger, A Bernard, A F Boe, M S Boguski, K S Brockway, E J Byrnes, Nature. 4457124Lein, E.S., Hawrylycz, M.J., Ao, N., Ayres, M., Bensinger, A., Bernard, A., Boe, A.F., Bo- guski, M.S., Brockway, K.S., Byrnes, E.J., et al.: Genome-wide atlas of gene expression in the adult mouse brain. Nature 445(7124), 168-176 (2007)
Reconstructing a 3d structure from serial histological sections. S Ourselin, A Roche, G Subsol, X Pennec, N Ayache, Image and vision computing. 191Ourselin, S., Roche, A., Subsol, G., Pennec, X., Ayache, N.: Reconstructing a 3d structure from serial histological sections. Image and vision computing 19(1), 25-31 (2001)
Toward routine use of 3d histopathology as a research tool. N Roberts, D Magee, Y Song, K Brabazon, M Shires, D Crellin, N M Orsi, R Quirke, P Quirke, D Treanor, The American journal of pathology. 1805Roberts, N., Magee, D., Song, Y., Brabazon, K., Shires, M., Crellin, D., Orsi, N.M., Quirke, R., Quirke, P., Treanor, D.: Toward routine use of 3d histopathology as a research tool. The American journal of pathology 180(5), 1835-1842 (2012)
Nearly rigid descriptor-based matching for volume reconstruction from histological sections. S Sun, N De Magalhaes, N D Cahill, SPIE Medical Imaging. Sun, S., De Magalhaes, N., Cahill, N.D.: Nearly rigid descriptor-based matching for volume reconstruction from histological sections. In: SPIE Medical Imaging. pp. 831417-831417. International Society for Optics and Photonics (2012)
mouse brain reconstruction from histology using a coarse-to-fine approach. P A Yushkevich, B B Avants, L Ng, M Hawrylycz, P D Burstein, H Zhang, J C Gee, Biomedical Image Registration. SpringerYushkevich, P.A., Avants, B.B., Ng, L., Hawrylycz, M., Burstein, P.D., Zhang, H., Gee, J.C.: 3d mouse brain reconstruction from histology using a coarse-to-fine approach. In: Biomedi- cal Image Registration, pp. 230-237. Springer (2006)
| []
|
[
"HYPERCRITICAL ACCRETION ONTO A MAGNETIZED NEUTRON STAR SURFACE: A NUMERICAL APPROACH",
"HYPERCRITICAL ACCRETION ONTO A MAGNETIZED NEUTRON STAR SURFACE: A NUMERICAL APPROACH"
]
| [
"C G Bernal ",
"W H Lee ",
"Dany Page "
]
| []
| [
"Revista Mexicana de Astronomía y Astrofísica"
]
| RESUMENLa acreción sobre una proto-estrella de neutrones en las horas que siguen al colapso del núcleo de una estrella masiva que le dio origen puede afectar sus propiedades observables. Este fenómeno se da en el regimen denominado hipercrítico (Chevalier 1989), donde el enfriamiento por neutrinos es crucial para la evolución termodinámica. En este trabajo presentamos un estudio en este contexto en una dimensión con simetría esférica y llevamos a cabo simulaciones numéricas en dos dimensiones dentro de una columna de acreción sobre una estrella de neutrones. Consideramos procesos microfísicos detallados, enfriamiento por neutrinos y la presencia de campos magnéticos en la aproximación de magnetohidrodinámica ideal. Comparamos nuestros resultados numéricos con las soluciones analíticas e investigamos como las soluciones, tanto hidrodinámicas como magnetohidrodinámicas, difieren deéstas. Iniciamos también una exploración de como este proceso puede afectar la aparición del remanente como un pulsar típico en el radio.ABSTRACTThe properties of a new-born neutron star, produced in a core-collapse supernova, can be strongly affected by the possible late fallback which occurs several hours after the explosion. This accretion occurs in the regime dominated by neutrino cooling, explored initially in this context byChevalier (1989). Here we revisit this approach in a 1D spherically symmetric model and carry out numerical simulations in 2D in an accretion column onto a neutron star considering detailed microphysics, neutrino cooling and the presence of magnetic fields in ideal MHD. We compare our numerical results to the analytic solutions and explore how the purely hydrodynamical as well as the MHD solutions differ from them, and begin to explore how this may affect the appearance of the remnant as a typical radio pulsar. | null | [
"https://arxiv.org/pdf/1006.3003v1.pdf"
]
| 55,114,460 | 1006.3003 | 69817512ffa8aa3246b1b1d3e9e4785509f1e923 |
HYPERCRITICAL ACCRETION ONTO A MAGNETIZED NEUTRON STAR SURFACE: A NUMERICAL APPROACH
15 Jun 2010
C G Bernal
W H Lee
Dany Page
HYPERCRITICAL ACCRETION ONTO A MAGNETIZED NEUTRON STAR SURFACE: A NUMERICAL APPROACH
Revista Mexicana de Astronomía y Astrofísica
15 Jun 2010Accretion -Hydrodynamics -Magnetic Fields -Supernovae: indi- vidual (SN1987A) -Stars: Neutron
RESUMENLa acreción sobre una proto-estrella de neutrones en las horas que siguen al colapso del núcleo de una estrella masiva que le dio origen puede afectar sus propiedades observables. Este fenómeno se da en el regimen denominado hipercrítico (Chevalier 1989), donde el enfriamiento por neutrinos es crucial para la evolución termodinámica. En este trabajo presentamos un estudio en este contexto en una dimensión con simetría esférica y llevamos a cabo simulaciones numéricas en dos dimensiones dentro de una columna de acreción sobre una estrella de neutrones. Consideramos procesos microfísicos detallados, enfriamiento por neutrinos y la presencia de campos magnéticos en la aproximación de magnetohidrodinámica ideal. Comparamos nuestros resultados numéricos con las soluciones analíticas e investigamos como las soluciones, tanto hidrodinámicas como magnetohidrodinámicas, difieren deéstas. Iniciamos también una exploración de como este proceso puede afectar la aparición del remanente como un pulsar típico en el radio.ABSTRACTThe properties of a new-born neutron star, produced in a core-collapse supernova, can be strongly affected by the possible late fallback which occurs several hours after the explosion. This accretion occurs in the regime dominated by neutrino cooling, explored initially in this context byChevalier (1989). Here we revisit this approach in a 1D spherically symmetric model and carry out numerical simulations in 2D in an accretion column onto a neutron star considering detailed microphysics, neutrino cooling and the presence of magnetic fields in ideal MHD. We compare our numerical results to the analytic solutions and explore how the purely hydrodynamical as well as the MHD solutions differ from them, and begin to explore how this may affect the appearance of the remnant as a typical radio pulsar.
INTRODUCTION
The neutrino signal (Hirata et al. 1987;Bionta et al. 1987) detected from the supernova SN1987A clearly demonstrated the birth of a neutron star (Burrows & Lattimer 1986). Identification of the progenitor as the blue supergiant Sanduleak −69 • 202a (Gilmozzi et al. 1987) and modeling of the early light curve (Hillebrandt et al. 1987;Shigeyama et al. 1987) proved that the supernova resulted from the core-collapse of a massive, ∼ 20 M ⊙ , star. However, to date, there is no evidence for the presence of a pulsar, or even a quiet neutron star, in the remnant (see, e.g., discussion in Haberl et al. 2006 andShternin &Yakovlev 2008). Several solutions to this dilemma have been proposed as, e.g., the delayed collapse of the neutron star into a black-hole (Ellis et al. 1996;Brown & Bethe 1994) or a delayed turn-on of the pulsar (Michel 1994;Muslimov & Page 1995). The latter case is just an extreme case of the more mundane possibility that the neutron star is weakly magnetized and/or slowly rotating, resulting in a spin-down energy that is low enough so as to be undetectable. Fig. 1. Schematic evolution of a collapsing stellar core. A, the Chandrasekhar-mass Iron core collapses; B, upon formation of a proto-neutron star, the equation of state stiffens and produces a bounce, launching an outwardly propagating shock; C, a reverse shock is formed at the He-H interface in the stellar envelope, moving back towards the neutron star; D, an accretion shock is established, interior to which an atmosphere close to hydrostatic equilibrium in the neutrino-cooled regime deposits mass and energy onto the neutron star in the hours following the explosion.
In recent years, several measurements of pulsar masses point toward large values approaching, or even exceeding, 2 M ⊙ (see, e.g., Freire 2008) which would strongly disfavor the black-hole explanation. On the other side, timing of radio quiet compact stars in young supernova remnants, usually dubbed CCOs ("Central Compact Objects", Pavlov et al. 2002) recently unveiled at least three case of weakly magnetized young neutron stars (Gotthelf & Halpern 2008): PSR J0821-4300 (in the SNR Puppis A) with a rotational period P = 112 ms, an upper limit on its spin-down powerĖ < 2.3 × 10 35 erg s −1 , and a surface dipolar magnetic field strength B dip < 9.8 × 10 11 G (Gotthelf & Halpern 2009); PSR 1E1207.4-5209 (in the SNR PKS 1209-51/52) with P = 424 ms,Ė < 1.3 × 10 32 erg s −1 , and B dip < 3.3 × 10 11 G (Gotthelf & Halpern 2007), and finally PSR J1852+0040 (in the SNR Kes 79) with P = 424 ms, and measurements ofĖ = 3.0 × 10 32 erg s −1 , and B dip = 3.1 × 10 10 G (Halpern & Gotthelf 2010). The last two of these hence have an energy output well below the 0.2-10.0 keV luminosity of the SN 1987A remnant, L < 5.7 × 10 34 erg s −1 (Haberl et al. 2006). If the neutron star produced by SN 1987A has similar characteristics it would presently be undetectable.
In the present paper we consider the scenario in which the initial magnetic field of the new-born neutron star is strongly modified by a phase of late, and intense, accretion, occurring a few hours after the initial explosion (Geppert et al. 1999). When a massive star explodes as a supernova, following the core-collapse scenario (Woosley & Janka 2005;Mezzacappa 2005;Janka et al. 2007), a large fraction of its mass expands freely and interacts with the interstellar medium. However, the central compact remnant also interacts with the inner envelope through its gravitational field. In Type II SNe (see Figure 1), the initial core-collapse (panel A) produces a proto-neutron star when the equation of state stiffens close to nuclear density. A highvelocity (10 4 km s −1 ) expansive shock then starts moving outward (panel B). Flow lines bifurcate and some part of the matter falls back onto the central remnant. The rest is unbound and ends up being ejected in the explosion. This scenario produces a low density region in near free fall between the surface of the compact object and the extended atmosphere in near hydrostatic equilibrium that has formed. In case the progenitor star had a low density envelope surrounding the He core a reverse shock (panel C) decelerates the matter and causes a late fallback onto the compact object, depositing great amounts of matter onto the surface of the new-born neutron star in the hours following the explosion (panel D). Following the ideas of Blondin (1986), Chevalier (1989), Houck & Chevalier (1991), and Brown & Weingartner (1994) about the accretion of matter onto compact objects, it is possible to develop an analytical model of accretion following core-collapse, and particularly in the case of SN1987A. One of the salient features of this analysis is that the gas, being quite dense, is unable to cool by photon emission, and the mass accretion rates are highly super-Eddington in that sense. However, at sufficiently high temperatures, cooling through neutrinos sets in, mostly through pair annihilation and pair capture onto free nucleons, and given their much lower interaction cross section with matter, they are able to remove enough energy from the flow for accretion to take place. This regime is usually termed "hypercritical" accretion, and is common in the inner collapsing stellar cores and is likely to drive the central engines of Gamma Ray Bursts (Lee & Ramirez-Ruiz 2007). With this model it is formally possible to obtain the radial position of the shock as a function of the mass accretion rate from fallback, assuming steady state in spherical symmetry, as well as the structure of the envelope. Chevalier (1989) and Houck & Chevalier (1991) computed such solutions in the context of SN1987A. Here we wish to explore the behavior of the flow under more general conditions, and present solutions for an accretion column in two dimensions, which we compare with the analytical scalings. Neutrino cooling is a crucial ingredient in the relevant density and temperature regimes, and we consider it along with a detailed equation of state. In addition and more importantly, we begin to explore the effects of the magnetic field on the accumulation of matter onto the neutron star surface. This is only possible through 2D simulations of the kind shown here, and we make a comparative analysis between the analytical and numerical approaches to consider the submergence of the magnetic field in the crust of the neutron star and the piling up of matter on its surface. Previously, Muslimov & Page (1995) and Geppert et al. (1999) considered how such accretion might delay the switch-on of a pulsar following its formation in one-dimensional calculations, computing the ohmic diffusion time of the magnetic field through the accreted matter. Fryer et al. (1996) studied the twodimensional accretion dynamics onto new-born neutron stars in the neutrino cooled regime, finding that in some cases, neutrino-driven convection can significantly modify the simple one-dimensional steady state solution.
Here we report on preliminary, two-dimensional numerical calculations which aim to determine if hypercritical, neutrino-cooled accretion can submerge the magnetic field into the crust of the neutron star, and if it plays an important role in the dynamics in this regime. In § 2 we develop the analytical model of the hypercritical accretion process and calculate the structure of the envelope for a two dimensional accretion column. We build a numerical model, based on these analytical consideration, in § 3. In § 4, we show numerical results for various configurations including magnetic fields at several accretion rates and present a comparative analysis between the numerical and analytical solutions for the scenario of SN1987A. Finally, in § 5 we present some preliminary conclusions.
ANALYTICAL MODELS
As a benchmark against which to compare our numerical simulations, we summarize below the basic results of an analytical model, based on the one developed by Chevalier (1989), and adapt them to the case of an accretion column. The essential assumptions of the model are that the neutron star is at rest within the expanding medium at infinity and rotation is neglected. For this analytical approach, we also neglect the effect of a possible magnetic field. Matter is described by a polytropic equation of state, P = Kρ γ with an index γ = 4/3, and is assumed to evolve adiabatically except at the shock interfaces and close to the neutron star surface where neutrino emission (through e ± pair annihilation) assures that the accretion energy is properly removed from the system.
The Initial Late-Accretion Rate in SN1987A
Spherically symmetric accretion by a compact star in an initially static, infinite, background was described by Bondi (1952) (see also, e.g., Shapiro & Teukolsky 1983). The mass accretion rateṀ is obtained from the density and sound velocity at infinity, ρ ∞ and c ∞ respectively, aṡ
M B = 4πλ GM c 2 ∞ 2 ρ ∞ c ∞ ,(1)
where the numerical constant λ = 1/ √ 2 ≃ 0.707 for the case of an ideal gas with adiabatic index γ = 4/3. In our case the medium is not strictly initially at rest but has been set into expansion by the supernova shock wave. At early times the core is in homologous expansion, with a velocity v = r/t and density ρ such that ρt 3 = ρ a t 3 a is constant, in terms of a reference density ρ a at time t a . As long as the time t is smaller than the Bondi accretion time scale τ B ≃ GM/c 3 ∞ , one can still estimateṀ with Eq. (1) by allowing ρ ∞ and c ∞ to be time dependent, and obtain (Chevalier 1989)
M B = 5.77 (GM ) 2 K 3/2 (ρ a t 3 a ) 1/2 t −3/2 .(2)
According to Woosley (1988) and Shigeyama et al. (1988), ρ a t 3 a ∼ 10 9 g cm −3 s 3 . Now the density and sound velocity at infinity have been estimated, for SN1987A, by Woosley (1988) and Bethe & Pizzochero (1990) as being of the order of
ρ ∞ = 2.5 M ⊙ (4/3)π(v f t) 3 ≃ 1.78 × 10 −13 t yr −3 g cm −3 ,(3)c 2 ∞ = γk B T ∞ µm H ≃ 1.24 × 10 12 t yr − 3 4 cm 2 s −2 ,(4)
where v f ≃ 600 km s −1 is the final expansion velocity and T ∞ ≃ 70 keV(4 × 10 9 cm/v f t) 3/4 for a radiation dominated shock as in a supernova like SN1987A. Bethe & Pizzochero (1990) calculated that the temperature for a shock radius of 4 × 10 9 cm is T sh ≃ 70 keV. The mass of the expanding CO core is ∼ 4 M ⊙ , but here we considered only 2.5 M ⊙ because 1.5 M ⊙ were taken to make the compact object at the center of the supernova. We hence haveṀ Woosley (1988) calculated the time that the reverse shock takes to return to the surface of the neutron star for SN1987A as t ≃ 7 × 10 3 s, giving us, for the accretion rate in SN1987A in the hypercritical regimė M ≃ 1.57 × 10 29 g s −1 = 2500 M ⊙ yr −1 .
≃ 2.23 × 10 22 t yr − 15 8 g s −1 ≃ 3.5 × 10 −4 t yr − 15 8 M ⊙ yr −1 .(5)
This accretion rate exceeds by an order of magnitude the value calculated by Chevalier (1989),Ṁ = 2.2 × 10 28 g s −1 = 340 M ⊙ yr −1 because of different assumed values at infinity. Now the Eddington mass accretion rate when considering photon radiation isṀ Edd = 3.77 × 10 18 g s −1 , if one considers electron scattering in pure ionized Hydrogen as the source of opacity, k es = 0.4 cm 2 g −1 . WhenṀ >>Ṁ Edd the flow is what we described above as Hypercritical Flow, studied by Blondin (1986). For the case of SN1987A, we havė Ṁ M Edd ≃ 10 9 − 10 10 ,
for the two values given above, placing such flows clearly in the hypercritical, neutrino cooled regime. Henceforth we adopt as our fiducial accretion rate the valuė M 0 ≃ 2.2 × 10 28 g s −1 = 340 M ⊙ yr −1 .
The Envelope and the Shock Radius: Spherical Case
When the reverse shock bounces against the surface of the neutron star, a third expansive shock is formed, which propagates through the infalling matter. Thus, eventually an atmosphere in quasi-hydrostatic equilibrium is formed around the compact object (see Panel D in Figure 1), whose general structure can be calculated analytically under some simplifying assumptions.
Following the formulation of Chevalier (1989), the structure of the envelope is calculated and an expression for the pressure at the surface of the neutron star, P ns , in terms ofṀ and the shock radius, r sh , is derived. Cooling by neutrinos close to the neutron star surface, which depends on P ns , is introduced and this ultimately determines the shock radius solely as a function of the accretion rate. From the condition of hydrostatic equilibrium, dP/dr = −ρGM/r 2 , we obtain the integrated values for the pressure, density and velocity as a funtion of the distance from the neutron star surface. This is possible because neutrino cooling is only important near the surface of the neutron star and we can consider that the post-shock flow is adiabatic over the greater part of the volume. In addition, the flow is highly subsonic except close the shock. With this we obtain,
P ∝ r −γ/(γ−1) ∝ r −4 , ρ ∝ r −1/(γ−1) ∝ r −3 , v ∝ r (3−2γ)/(γ−1) ∝ r,
where we have used γ = 4/3. These results also are valid in the shock and the envelope structure in hydrostatic equilibrium is,
P = P sh r r sh −4 ,(9)ρ = ρ sh r r sh −3 ,(10)v = v sh r r sh ,(11)
where P sh , ρ sh , v sh and r sh are the values at the shock. These values can be obtained from the jump conditions when P sh >> P 0 , where P 0 and ρ 0 the pre-shock pressure and density. Under these considerations we obtain
P sh = (7/8)ρ 0 v 2 0 , ρ sh = 7ρ 0 and v sh = −(1/7)v 0 .
The pre-shock velocity is that of free-fall, v 0 = 2GM/r sh , and the density is ρ 0 =Ṁ /4πr 2 sh v 0 . From equation (8) we obtain the pressure at the surface, P ns = 1.36 × 10 −12Ṁ r 3/2 sh , with M ∼ 1.44 M ⊙ and r ns ∼ 10 6 cm. On the other hand, the energy loss by neutrinos (only pair production) by unit volume can be estimated as (Dicus 1972),
ε n = 1.83 × 10 −34 P 2.25 erg cm −3 s −1 .(12)
In this case, we consider that e ± pairs also contribute to the pressure. Now, this cooling is operative only in a small volume close to the neutron star surface, ∼ πr 3 ns since it is a sensitive function of temperature. So, from energy conservation, the shock radius is obtained as,
r sh ≃ 7.58 × 10 8 Ṁ M ⊙ yr −1 − 10 27 cm.(13)
With this we have the structure of the envelope and the shock radius as a function of the accretion rate. For the case of SN1987A with our fiducial accretion rateṀ =Ṁ 0 = 340 M ⊙ yr −1 , the shock radius is r sh ≃ 8.77 × 10 7 cm.
The Envelope and the Shock Radius: The Accretion Column
If we consider a small rectangular accretion column of area A col onto a fraction of the neutron star surface, with area A = 4πR 2 NS , we can take it to be a plane-parallel surface (see Figure 2). In this case, the spherical mass accretion rate must be scaled to its value in the column. Since in the spherical case the area depends on the distance to the neutron star, while for the case of an accretion column the area is constant, the structure of the envelope and the shock radius are modified. Note that this modification causes the mass accretion rate per unit area to be independent of height in the domain. Also, since it is smaller than in the spherical case, the analytical estimate of the shock radius decreases significantly, and is now given by
y sh ≃ 7.61 × 10 6 Ṁ /A M ⊙ yr −1 /A col −0.16 cm,(14)
where y measures the height above the neutron star surface. With these considerations and taking A col = (3 × 10 5 ) 2 cm 2 , the shock radius for the fiducial accretion rate, Eq. (8), is y sh = 6.92 × 10 6 cm, and the structure of the envelope is,
P = P sh y y sh −4 ,(15)ρ = ρ sh y y sh −3 ,(16)v = v sh y y sh 3 .(17)
The velocity profile is different for the accretion column as well because the column area is constant as a function of height above the neutron star. The conditions in the shock in the SN1987A scenario are thus
v 0 = 2GM y sh ≃ 7.53 × 10 9 cm s −1 ,(18)ρ 0 =Ṁ v 0 × A ≃ 2.31 × 10 5 g cm −3 ,(19)ρ sh = 7ρ 0 ≃ 1.62 × 10 6 g cm −3 ,(20)v sh = − 1 7 v 0 ≃ −1.07 × 10 9 cm s −1 ,(21)P sh = 7 8 ρ 0 v 2 0 ≃ 1.14 × 10 25 dyn cm −2 .(22)
Now we can build a numerical model with more refined physics to perform 2D hydrodynamics (HD) and magnetohydrodynamics (MHD) simulations, which we can compare with the 1D analytical results.
NUMERICAL APPROACH
For the work shown in this paper we used the numerical code AMR FLASH2.5 (Fryxell et al. 2000) to perform the 2D simulations. FLASH (http://flash.uchicago.edu/website/home/) is a modular, portable, highly scalable, adaptive-mesh simulation code for astrophysical hydrodynamics problems. It was originally developed at the DOE ASCI Alliances Center for Astrophysical Thermonuclear Flashes at the University of Chicago for the purpose of simulating Type Ia supernovae, novae, and X-ray bursts. It has since evolved to handle more general astrophysical problems, including those involving collisionless particle dynamics. FLASH is freely available from the ASCI Flash Center. This code is designed to allows users to configure initial and boundary conditions, change algorithms, and add new physics modules with minimal effort. It uses the PARAMESH library to manage a block-structured adaptative grid, placing resolution elements where they are needed most.
The Numerical Method
FLASH2.5 provides two main types of modules: Physics and Infrastructure Modules. In our model we used the hydro-mhd, eos-helmholtz, gravity and neutrino-cooling custom modules.
The FLASH code solves the the equations of a magnetized fluid (ideal or non-ideal), described by
∂ρ ∂t + ∇ · (ρv) = 0,(23)∂ρv ∂t + ∇ · (ρvv − BB) + ∇P * = ρg + ∇ · τ,(24)∂ρE ∂t + ∇ · [v (ρE + P * ) − B (v · B)] = ρv · g+O(τ, η),(25)∇· (v · τ +σ∇T ) + ∇ · (B × (η∇ × B)) = O(τ, η),(26)∂B dt + ∇ · (vB − Bv) = −∇ × (η∇ × B) ,(27)
where
P * = P + B 2 2 ,(28)E = 1 2 v 2 + ε + B 2 2ρ ,(29)τ = µ (∇v) + (∇v) T − 2 3 (∇v) .(30)
Here P * , E and τ are the total pressure, total specific energy and stress tensor, respectively, and the remaining symbols have their usual meaning. Units in these equations are such that no 4π and µ 0 factors appear. We have simplified the above set of equations by restricting ourselves to the ideal hydro and MHD cases. Setting the thermal conductivity, σ, and electrical resistivity, η, to zero is justified by the fact that time scales for heat and magnetic field diffusion are many orders of magnitude larger than our simulation times. The inviscid (µ = 0) approximation, i.e., neglect of momentum diffusion, is acceptable because we have not considered rotation in our models. Note that when B = 0, the Euler equations are then obtained.
A particular complication associated with solving the MHD equations numerically lies in the solenoidality of the magnetic field. The non-existence of magnetic monopoles, ∇ · B = 0 is difficult to satisfy in discrete computations. Being only an initial condition of the MHD equations, it enters the equations indirectly and is not, therefore, guaranteed to be generally satisfied unless special algorithmic provisions are made. FLASH2.5 uses a simple yet very effective method to destroy the magnetic monopoles on the scale on which they are generated. In this method, a diffusive operator proportional to ∇∇ · B is added to the induction equation, so that the equations become
∂B dt + ∇ · (vB − Bv) = −∇ × (η∇ × B) − v∇ · B+η a ∇∇ · B,(31)
with the artificial diffusion coefficient η a chosen to mach that of grid numerical diffusion. In the FLASH code, η a = (λ/2)(1/∆x + 1/∆y + 1/∆z) −1 , in 3D where λ is the largest characteristic speed in the flow. Since the grid magnetic diffusion Reynolds number is always on the order of unity, this operator locally destroys magnetic monopoles at the rate which they are created. All our simulations are in cartesian coordinates: in the presence of a magnetic field polar/spherical coordinates are very troublesome and, presently, the MHD version of FLASH does not support them.
The Physics Ingredients
In the analytical approach an ideal gas equation of state has been used. This allows much simplification in the structure of the envelope and in addition, the flow can be managed like an adiabatic fluid with γ = 4/3. The gas is dominated by radiation, which is trapped within the flow. Also, the neutrino losses depend on a high power of the pressure, but are only important at the base of the envelope. Nevertheless, to account for the thermodynamics more accurately and for the consequent piling up of matter on the star, it is advisable and necessary to work with a more complete and realistic equation of state. The Helmholtz EOS provided with the FLASH2.5 distribution contains more physics and is appropiate for addressing astrophysical phenomena in which electrons and positrons may be relativistic and/or degenerate and in which radiation may significantly contribute to the thermodynamic state. This EOS thus includes contributions from black-body radiation, completely ionized ideal nuclei, and free electrons and positrons. The pressure and internal energy are calculated as the sum over the components, P tot = P rad + P ion + P ele + P pos + P coul (32)
ε tot = ε rad + ε ion + ε ele + ε pos + ε coul .(33)
Here the subscripts "rad", "ion", "ele", "pos" and "coul" represent the contribution from radiation, nuclei, electrons, positrons, and Coulomb corrections, respectively. The radiation portion assumes a blackbody in local thermodynamic equilibrium, the ion portion (nuclei) is treated as an ideal gas with γ = 5/3, and the electrons and positrons are treated as a non-interacting Fermi gas of arbitrary degeneracy and relativity. Under the physical conditions of interest for the set of simulations presented here, the gas is dense enough that the optical depth for photons is τ γ ≫ 1, and they are fully trapped in the flow. Adding the corresponding term to the pressure as P rad = aT 4 /3 is thus entirely appropriate. We note that more recent versions of FLASH (upwards of 3.2) include modules for radiation transport, making them useful for a wider range of studies.
The gravity module suplied with FLASH2.5 computes gravitational source terms for the code. These can take the form of the gravitational potential φ(x) or the gravitational acceleration,
g(x) = −∇φ(x).(34)
The gravitational field can be externally imposed or self-consistently computed from the gas density via the Poisson equation,
∇ 2 φ(x) = 4πGρ(x),(35)
where G is Newton's gravitational constant. In the latter case, either periodic or isolated boundary conditions can be applied. In our case, we used an externally applied gravitational field (plane-parallel gravitational field ), where the acceleration vector is parallel to one of the coordinate axes, and its magnitude drops with distance along that axis as the distance squared. Its magnitude and direction are independent of the other two coordinates.
In the conditions present in both the high density part of the accretion flow and the underlying envelope neutrino emission occurs essentially through neutral currents processes. The five processes we included in the models are analogous to standard photon emission processes where the γ emission is replaced by a ν − ν pair. They are: PAIR ANNIHILATION: e − + e + → ν + ν, PHOTONEUTRINOS: γ + e ± → e ± + ν + ν, the analogous of Compton scattering, PLASMON DECAY: Γ → ν + ν, where Γ is a plasmon, BREMSSTRAHLUNG: e ± + N → e ± + N + ν + ν, where N is a nucleus, and SYNCHROTRON: e ± + B → e ± + B + ν + ν, where B represents the magnetic field. For the first four processes we used the calculations of Itoh et al. (1996) and for the synchrotron emission we followed Bezchastnov et al. (1997). Pair annihilation is the dominant process but synchrotron can make some significant contribution when the magnetic field becomes strongly compressed. As noted above, the density in the flow is typically high enough that photons are trapped, but not neutrinos. As a rough guide, the optical depth for neutrinos under coherent scattering off free nuclei is τ ν ≃ 1 when ρ ≃ 10 11 g cm −3 , which is several orders of magnitude higher than the maximum values studied here. Thus neutrino cooling can be implemented simply as a sink in the energy equation.
The Initial and Boundary Conditions
We simulated a small 2D accretion column in cartesian coordinates anchored onto the surface of the neutron star, and considered various accretion rates and magnetic field configurations. This set of simulations allows us to compare numerically obtained results in the pure hydrodynamical and MHD case with the proposed analytical approach, as well as to analyze the reaction of the magnetic field to the infalling gas. The computational domain covers 0 ≤ x ≤ 3×10 5 cm, 0 ≤ y ≤ 10 7 cm. The dimensions for the column are: A col = L x ×L z = (3×10 5 ) 2 cm 2 (for the base) and height col = L y = 10 7 cm for the height. This height is adequate because it is below the analytical shock radius value calculated for the accretion rate of SN1987A. The fluid is initially in free fall and we set a constant temperature in the gas. We considered horizontal (B x = 10 12 G, B y = 0), vertical (B x = 0, B y = 10 12 G, mimicking accretion onto the magnetic pole of the neutron star), diagonal (B x = B y = 10 12 G) and dipolar (B x = 2µ/y 3 , B y = 0, representing accretion onto the neutron star equator) cases, where µ = 5 × 10 29 is the dipolar moment, fixed so that B x = 10 12 G at the neutron star surface. With these considerations, the initial conditions in the column for velocity, temperature and density are:
ρ =Ṁ col v f f × A col ,(36)T = 10 9 K,(37)v f f = 2GM y ,(38)B = 10 12 G,(39)
whereṀ col =Ṁ A col /A is the scaled accretion rate in the column making up the domain. For the vertical boundaries of the accretion column, x = 0 and x = 3 × 10 5 cm, parallel to the y-axis, we implement standard periodic boundary conditions. Thus, any fluid element moving out of the computational domain on the right (left) boundary re-enters the domain on the left(right) edge with the same thermodynamical properties and velocity.
For the top and bottom of the computational domain, parallel to the x-axis, we implemented custom boundary conditions. The gravity vector is along the y-axis, and we want the lower boundary at y = 0 to support the fluid above against infall, mimicking the hard surface of the neutron star, and in addition to have the magnetic field anchored to it. In order to establish this boundary, we use "guard", or "ghost" numerical cells. These are cells outside the formal computational domain (e.g., at y ≤ 0 or y ≥ 10 7 ) for which we can fix the hydrodynamical and thermodynamical properties and that are not evolved along with the rest of the flow. They are useful precisely to guarantee boundary conditions of interest, depending on the setup of the problem. A layer of at least 2 such cells along the top and bottom of the domain can thus be used to compute proper gradients at the edge of the flow (e.g., a pressure or temperature gradient). In this case, along the bottom edge of the column, we fix the velocities to be null in all guard cells, (v x = v y = v z = 0), keeping them at rest, and copy the density and the pressure of the first zone of the numerical domain to mimick the the neutron star surface: ρ = ρ(1), P = P (1) + ρv 2 + ρgh , where the label 1 refers to the first cell in the computational domain. The magnetic field is put in this boundary in such form that it is continuous from the guard cell to the physical domain, i.e, we anchor the magnetic field onto the neutron star surface and in the rest of the guard cells it is null. The other thermodynamics variables are calculated from the equation of state. At the top of the column, y = 10 7 cm, we set the velocity to be that of free fall, v y = − 2GM/y in all the guard cells, and set the density to fix a constant inflow mass accretion rate, ρ =Ṁ col /(|v y | × A col ). As in the computational domain initially, the temperature in the guard cells is set to T = 10 9 K (at all times). The remaining variables are calculated from the equation of state.
RESULTS AND DISCUSSION
We now present results obtained from the 2D hydrodynamical simulations (HYDRO) as well as for the MHD case for an accretion column in cartesian coordinates, and compare these to the analytical scalings. We varied the accretion rate and magnetic field configuration (for the MHD case). The chosen rates were one, two and three orders of magnitude above our fiducial rateṀ 0 = 340 M ⊙ yr −1 .
Comparison of the HYDRO and MHD solvers
For the assumed physical parameters of SN1987A, in 200 ms the system reaches a quasi-stationary state, whereas for higher rates of accretion, this drops substantially: 60 ms at 10Ṁ 0 , 20 ms at 100Ṁ 0 and 5 ms at 1000Ṁ 0 . We set a level of refinement of 4, with 2 blocks along the x-axis and 18 along y-axis. This implies an effective resolution of 128 × 1152 zones in the computational domain.
In Fig. 3 we show the density contrast for the HYDRO and MHD cases with null magnetic field (MHD 0), forṀ = 100Ṁ 0 . We choose this accretion rate as being representative since its associated shock radius is much smaller than forṀ =Ṁ 0 , and is therefore easier to visualize. In addition, it is possible to both do a comparative analysis of solvers (HYDRO and MHD) and of their response to the imposed initial conditions. This comparison allows us to determine whether the equations are being solved in both modules to a comparable accuracy. In principle, the MHD module with null magnetic field should reproduce exactly the results obtained with module HYDRO. The constrasts of pressure, specific total energy, velocity and neutrino cooling per unit volume for all the cases (at t = 10 ms), are shown in Fig. 4. The radial profiles of density, pressure and velocity for the SN1987A accretion rate are given in Fig. 5.
We note that although the system reaches a quasi-stationary state in t = 20 ms, there is remnant noise in the radial profile of the velocity due to the interaction of the matter with the lower boundary condition and to the fact that horizontal motions are allowed because of the periodic boundary condition. On the other hand, only the bottom section, 4 × 10 6 cm, of the entire accretion column, with height 10 7 cm is shown, where the most interesting processes occur. We note that the profiles, while not identical in all respects, are indeed very similar, showing that the HYDRO and MHD solvers are giving essentially the same final state, both in space and time evolution. There is some convection in the early stages of the evolution, and Rayleigh-Taylor instabilities are present, but are quickly damped as the system approaches the stationary solution. Deviations from this are most evident when one examines variations in the velocity field.
The location of the shock is reproduced quite well, to within 5% when compared to the analytical calculation. Moreover, both solvers place it essentially at the same height, indicating that the quantitative aspects are not affected from one to the other. Since the code is able to model the bottom of the column self-consistently within the imposed boundary condition, the numerical solution deviates from the self-similar scaling once neutrino cooling becomes important, and matter starts piling up near the surface.
Hereafter, unless otherwise noted we refer to calculations withṀ =Ṁ 0 . The adiabatic and radiative gradients can be calculated from the simulations, when the system is relaxed. We find ∇ ad = 1 − 1/γ c ≃ 0.26, ∇ rad = (d ln T /d ln P ) ≃ 0.24. In this case, the value of the adiabatic index γ c has been taken directly from the simulation (γ c = 1.35), and the radiative gradient was calculated by building a plot of temperature vs. pressure. These gradients have almost constant values within the envelope, except in the region close to the neutron star surface. Since ∇ ad > ∇ rad , the system is manifestly stable to convection. Nevertheless, being so close numerically is probably indicative of marginal stability. Within the envelope the flow is fully subsonic, as expected after passing through the accretion shock front: the sound speed is c s = γ c P/ρ ≃ 6.9 × 10 9 cm s −1 , and v ≃ 1.24 × 10 7 cm s −1 , giving a Mach number m = v/c s ≃ 10 −3 . Therefore, besides confirming that the HYDRO and MHD solvers give accurate and consistent results, we are able to study the global structure of the accretion column in detail and compare it with the analytical approach, particularly in the region where the approximations in the latter break down. It is worthy to note the thermodynamical conditions the fluid is in as it accretes towards the proto-neutron star. The Fermi temperature can be computed from the Fermi energy E F at the base of the flow, where p F = (3π 2 n e ) 1/3h is the Fermi momentum. The temperature obtained from the simulation, close to the bottom of the accretion column in quasi-stationary state is T ≃ 4.54 × 10 10 K, so T /T F ≃ 0.7. It is thus clear that assuming that the e ± pairs are entirely degenerate is not a proper approximation, and a full expression such as the one in the Helmholtz equation of state is required if one wishes to compute the evolution of the flow accurately. It is also clear that neutrino cooling effectively turns The accretion rate isṀ =Ṁ0. The HYDRO (red), MHD 0 (green) and analytical solution (blue) are shown together. The location of the shock is reproduced with very good agreement in the two numerical cases with respect to the analytical solution. In the shocked region, the velocity shows significantly higher behavior due in part to lateral motions of the gas. At small radii, the numerical solutions deviate from the analytical curve, since the assumption of self-similarity is no longer valid as material piles up near the star and cooling becomes relevant in the energy balance.
T F = E F k B = p 2 F c 2 + m 2 e c 4 − m e c 2 k B ≃ 6.48 × 10 10 K,(40)
on at a scale height L y ∼ 2 × 10 5 cm. For the simulation withṀ =Ṁ 0 , the integrated neutrino luminosity, shown in Fig. 6, is L ν ≃ 2.51 × 10 46 erg s −1 , close to the value estimated with the cooling function of Dicus (1972) scaled to the column: L ν =ε n × V ≃ 1.83 × 10 46 erg s −1 , with V ≃ (2 × 10 5 ) × (3 × 10 5 ) 2 cm 3 .
Once the system reaches the quasi-stationary state, radial profiles can be compared for different accretion rates. Four different rates for each initial condition were computed. In all of these, the piling up of material close to the neutron star surface is seen. The velocity profiles remain noisy and turbulent in the shocked region, but on average the analytical profile is globally recovered. In Fig. 7 these are plotted, along with density and pressure, for case MHD 0. Note also that at greater accretion rates the shock is located at lower height, as expected. Foṙ M /Ṁ 0 = 1, 10, 100, 1000, the position of the shock in the simulation is at R sh /10 6 cm = 7.06, 4.96, 3.74, 2.68, in excellent agreement with the analytical values given by R sh /10 6 cm = 6.92, 4.80, 3.33, 2.31, respectively (see Fig. 8).
Magnetic field submergence
We now consider the case with non-zero magnetic field strength. Fig. 9 shows the radial profiles of density, pressure and magnitude of the velocity forṀ =Ṁ 0 with several field configurations: null (MHD 0), constant horizontal (MHD H), constant vertical (MHD V), constant diagonal (MHD D) and dipolar (MHD DIP), for comparative effects. The initial intensity of the magnetic field in all cases is 10 12 G, except in the dipole configuration, where it is 10 12 G at the neutron star surface. We also overplot the hydrodynamical solution for comparison. Note that the profiles are practically the same at this accretion rate indicating that the magnetic field is not playing an important role as far as the dynamics are concerned.
In all simulated cases, regardless of the magnetic field configuration, when the system has relaxed and reached the quasi-stationary state, the field is completely submerged in the neutron star crust. Its intensity rises accordingly, by up to two orders of magnitude for the highest accretion rates. Fig. 10 shows the distribution of magnetic field strength after the system has relaxed, whenṀ = 1000Ṁ 0 , for our four initial magnetic field configurations. It is only within the first km in the column, where the matter piles up, that the magnetic field is at or above the initial value in the calculation, and the compression is quite clear.
The initial dynamics in the MHD case are somewhat more violent than in the pure hydrodynamical case. The infalling gas quickly drags the initial field towards the neutron star surface since the ram pressure, P ram = ρv 2 /2 is substantially greater than the magnetic pressure P mag = B 2 /8π, even for the smallest accretion rate, M =Ṁ 0 . The increased magnetic pressure as compression takes place is insufficient to overcome this flow, and large field strengths close to the surface result. The effect on the large scale dynamics is thus of a more transitory nature, and sensitive to the initial conditions, than a permanent feature. As a second point, we note Pressure (dyne/cm ) Density (g/cm ) 3 Fig. 7. Radial profiles of density (top), pressure (middle) and velocity (bottom) for the MHD 0 case after a stationary state has been reached. The accretion rates areṀ /Ṁ0 = 1, 10, 100, 1000 (cyan, blue, green and red, respectively).
that the magnetic field, advected along with the flow, fluctuates in strength strongly in the shocked region as it piles up against the lower boundary, where neutrino cooling is efficient. The additional piling up of material makes it even harder for the field to rise to significant levels as the evolution proceeds further. Nevertheless, as the system evolves the turbulent structures that form initially begin to smooth themselves until they disappear completely in the hydrodynamical case, but some small scale structure remains when magnetic fields are present. Once the accretion rate drops significantly, it is in principle possible that the field will rise buoyantly through the envelope, playing some dynamical role as the accretion time becomes long and the balance between ram and magnetic pressure is reversed. This will occur on a much longer time scale than simulated here, and its modeling requires a different set of assumptions in terms of the present set of calculations.
CONCLUSIONS
We have presented the results of two-dimensional simulations of accretion in the hypercritical, neutrinocooled regime onto the surface of a neutron star, using the FLASH code. The flow in accretion columns for a variety of initial accretion rates was simulated until a steady state was reached. We find that at this stage, the location of the accretion shock, where the flow transitions from free fall to subsonic settling onto the neutron star surface, is well reproduced when compared with the analytical estimates of Chevalier (1989). However, close to the surface, matter piles up, the solution is no longer adiabatic, and the self-similar character of the flow breaks down as expected.
We performed a detailed comparison of the hydrodynamical and ideal MHD routines in FLASH, and found excellent agreement between the two when the initial field is null. For various finite field configurations (initially horizontal, vertical, diagonal and dipolar), we find that performing the calculations in two dimensions does not allow for any additional buoyancy effects of the field to be manifested: for all accretion rates simulated, the initial field is entirely advected by the flow and submerged close to the neutron star surface. Its intensity rises accordingly, by up to two orders of magnitude in some cases. In principle, thus, it is possible for such an accretion episode following core collapse and the formation of a proto-neutron stars to effectively bury the initial field and delay the appearance of a classical radio pulsar (Muslimov & Page 1995). The simulated time scales at present do not allow us to place hard constraints on the re-diffusion of the field at late times, and a more quantitative estimation of this is left for future work.
CGB acknowledges support from a DGEP-UNAM scholarship. Financial support for this work was provided in part by CONACyT (45845E) and DGAPA-UNAM (IN 122609). The software used in this work was in part developed by the DOE-supported ASC / Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago. The numerical calculations were carried out on the KanBalam Supercomputer at DGSCA, UNAM, whose support team is gratefully acknowledged. We thank the anonymous referee for comments and criticism which helped improve this final version.
Fig. 2 .
2Schematic geometry of the accretion column onto the neutron star surface.
Fig. 3 .
3Color maps of density for cases HYDRO (top) and MHD 0 (bottom) at t = 5, 10, 20 ms from left to right. The accretion rate isṀ = 100Ṁ0.
Fig. 4 .
4Color maps of pressure, total specific energy, magnitude of velocity and neutrino emissivity, for cases HYDRO (top) and MHD 0 (bottom) at t = 10 ms. The accretion rate isṀ = 100Ṁ0.
Fig. 5 .
5Radial profiles of density (top), pressure (middle) and magnitude of velocity (bottom) at t = 200 ms.
Fig. 6 .
6Neutrino luminosity integrated over the computational domain for the fiducial accretion rate,Ṁ =Ṁ0, up to t = 200 ms. Note how after an initial transient, the power output is leveling off as the system reaches a quasi-stationary state.
Fig. 8 .
8Accretion rate as a function of shock radius for the hydrodynamical simulations withṀ /Ṁ0 = 1000, 100, 10, 1 after a stationary state has been reached. The analytical solution given by equation 14 is shown for comparison.
Fig. 9 .Fig. 10 .
910Radial profiles of density (top), pressure (middle) and velocity (bottom) forṀ =Ṁ0 and various field configurations (labeled). Maps (left half-panel) and horizontally-averaged vertical profiles (right half-panel) of magnetic field strength for a magnetic field that is initially horizontal, case MHD H (top left), vertical, case MHD V (top right), diagonal, case MHD D (bottom left), and dipolar, case MHD D (bottom right), after the system has relaxed forṀ = 1000Ṁ0. Note the submergence of the magnetic field close to the neutron star surface.
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. T Shigeyama, K Nomoto, M Hashimoto, D Sugimoto, Nature. 328320Shigeyama, T., Nomoto, K., Hashimoto, M., & Sugimoto, D. 1987, Nature, 328, 320
. P S Shternin, D G Yakovlev, Astronomy Letters. 34675Shternin, P. S. & Yakovlev, D. G. 2008, Astronomy Letters, 34, 675
. S Woosley, T Janka, Nature Physics. 1147Woosley, S. & Janka, T. 2005, Nature Physics, 1, 147
S E Woosley, Atmospheric Diagnostics of Stellar Evolution. Berlin Springer Verlag305361Lecture Notes in PhysicsWoosley, S. E. 1988, in Lecture Notes in Physics, Berlin Springer Verlag, Vol. 305, IAU Colloq. 108: Atmospheric Diagnostics of Stellar Evolution, ed. K. Nomoto, 361-+
. C G Bernal, W H Lee, D Page, México D.F, CPInstituto de Astronomía, Universidad Nacional Autónoma de México ; Ciudad UniversitariaApdo. Postal 70 -26404510 (bernalcg,wlee,[email protected]. G. Bernal, W. H. Lee, D. Page: Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo. Postal 70 -264, Ciudad Universitaria, México D.F, CP 04510 (bernalcg,wlee,[email protected]).
| []
|
[
"Scalar Charges in Asymptotic AdS Geometries",
"Scalar Charges in Asymptotic AdS Geometries"
]
| [
"Hai-Shan Liu \nInstitute for Advanced Physics & Mathematics\nZhejiang University of Technology\n310023HangzhouChina\n",
"H Lü \nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n"
]
| [
"Institute for Advanced Physics & Mathematics\nZhejiang University of Technology\n310023HangzhouChina",
"Department of Physics\nBeijing Normal University\n100875BeijingChina"
]
| []
| We show that for n-dimensional Einstein gravity coupled to a scalar field with masssquared m 2 0 = −n(n − 2)/(4ℓ 2 ), the first law of thermodynamics of (charged) AdS black holes will be modified by the boundary conditions of the scalar field at asymptotic infinity.Such scalars can arise in gauged supergravities in four and six dimensions, but not in five or seven. The result provides a guiding principle for constructing designer black holes and solitons in general dimensions, where the properties of the dual field theories depend on the boundary conditions. | 10.1016/j.physletb.2014.01.056 | [
"https://arxiv.org/pdf/1401.0010v2.pdf"
]
| 84,179,419 | 1401.0010 | e4c8131d7618f7828f447ab8a5571015191e2810 |
Scalar Charges in Asymptotic AdS Geometries
13 Jan 2014
Hai-Shan Liu
Institute for Advanced Physics & Mathematics
Zhejiang University of Technology
310023HangzhouChina
H Lü
Department of Physics
Beijing Normal University
100875BeijingChina
Scalar Charges in Asymptotic AdS Geometries
13 Jan 2014Introduction: Black holes are one of the most important classes of object predicted by
We show that for n-dimensional Einstein gravity coupled to a scalar field with masssquared m 2 0 = −n(n − 2)/(4ℓ 2 ), the first law of thermodynamics of (charged) AdS black holes will be modified by the boundary conditions of the scalar field at asymptotic infinity.Such scalars can arise in gauged supergravities in four and six dimensions, but not in five or seven. The result provides a guiding principle for constructing designer black holes and solitons in general dimensions, where the properties of the dual field theories depend on the boundary conditions.
Introduction: Black holes are one of the most important classes of object predicted by Einstein's theory of gravity. Many important properties of black holes have been studied and established. Black holes that are asymptotic to anti-de Sitter spacetime (AdS) are particularly useful in the AdS/CFT correspondence [1], for studying quantum field theory or even condensed matter physics at finite temperature. Numerical evidence suggests that black holes can develop scalar "hair" in both asymptotic AdS or flat spacetimes [2]. Recently, many explicit examples of scalar hairy black holes in four and higher dimensions have been found [3]. This leads to a natural question as to whether there exist scalar charges, analogous to those associated with electromagnetic fields. The subject of scalar charge and its contribution to the first law of black hole thermodynamics is not well understood.
If a theory with a scalar φ has a global symmetry with a constant shift, φ → φ + c, as in the case of the dilaton in ungauged supergravities, the asymptotic value φ 0 at infinity is an integration constant, and can be treated as a scalar charge [4]. If the scalar has a potential with a stationary point φ = φ 0 , the global symmetry disappears and φ takes the fixed value φ 0 at infinity. In this situation, there is no obvious definition of a scalar charge and one might expect that the first law would not be modified by the contribution from the scalar.
In four dimensions, it was shown that a massless scalar (in the AdS sense) with the large-r boundary behavior
φ = φ 1 r + φ 2 r 2 + · · ·(1)
can break some of the boundary AdS symmetries unless one of the following three conditions is satisfied: (1) φ 1 = 0, or (2) φ 2 = 0 [5], or (3) φ 2 /φ 2 1 is some fixed constant [6]. (See also [7].) Solitons that violate all these three conditions were constructed numerically in [8], giving rise to "designer gravity", where the properties of the field theory depend on the boundary conditions. In [9], a Kaluza-Klein dyonic AdS black hole in four-dimensional maximal gauged supergravity was constructed, for which the scalar boundary behavior also violates these three criteria. The consequence is that the naively-expected first law of thermodynamics dM = T dS + Φ e dQ e + Φ p dQ p for the dyon does not hold. Instead dM in the first law is shifted by a 1-form −XdY ≡ Z, given by [9]
Z = 1 12ℓ 2 (2φ 2 dφ 1 − φ 1 dφ 2 ) ,(2)
where ℓ is the radius of the asymptotic AdS. The first law becomes
dM = T dS + Φ e dQ e + Φ p dQ p + XdY .(3)
It reduces to the standard one if any of the three criteria mentioned above is met.
In this paper, we show that this phenomenon can occur also in higher dimensions, and we determine the conditions for a non-vanishing Z. We begin by considering n-dimensional
Einstein gravity coupled to a scalar field with a generic potential
L = √ −g R − 1 2 (∂φ) 2 − V (φ) ≡ √ −gL 0 .(4)
The equations of motion are
E µν ≡ R µν − 1 2 ∂ µ φ∂ ν φ − 1 n−2 V g µν = 0 , φ = dV dφ .(5)
Many explicit examples of exact solutions for hairy black holes have been obtained for some specific choices of the scalar potential [3]. We would like to examine whether these solutions admit a non-vanishing Z.
Wald's canonical charge: We begin by reviewing Wald's formalism for deriving the first law of thermodynamics by the Noether procedure [10]. We first consider a generic variation of the Lagrangian (4):
δL = e.o.m. + √ −g ∇ µ J µ ,(6)
where e.o.m. denotes the equations of motion for the fields, and
J µ = g µρ g νσ (∇ σ δg νρ − ∇ ρ δg νσ ) − ∇ µ φδφ .(7)
From this one can define a 1-form J (1) = J µ dx µ and its Hodge dual Θ (n−1) = (−1) n+1 * J (1) .
We now specialize to a variation that is induced by an infinitesimal diffeomorphism δx µ = ξ µ .
One can show that
J (n−1) ≡ Θ (n−1) − i ξ * L 0 = e.o.m. − d * J (2) ,(8)
where i ξ denotes a contraction of ξ µ on the first index of the n-form * L 0 , and J (2) = dξ (1) with
ξ (1) = ξ µ dx µ . One can thus define an (n − 2)-form Q (n−2) ≡ * J (2) , such that J (n−1) = dQ (n−2) .
Note that we use the subscript notation "(p)" to denote a p-form. To make contact with the first law of black hole thermodynamics, we take ξ µ to be the time-like Killing vector that is null on the horizon. Wald shows that the variation of the Hamiltonian with respect to the integration constants of a given solution is [10]
δH = 1 16π δ c J (n−1) − c d(i ξ Θ (n−1) ) = 1 16π Σ (n−2) δQ (n−2) − i ξ Θ (n−1) ,(9)
where c denotes a cauchy surface and Σ (n−2) is its two boundaries, one at infinity and one on the horizon.
Application in Einstein-scalar theory: We now apply Wald's formalism to the Einstein-scalar theory (4). In this paper, we shall mainly consider static and sphericallysymmetric solutions. For our purpose, it is convenient to write the metric ansatz in Schwarzschild-like coordinates, with
ds 2 n = −h(r)dt 2 + dr 2 h(r) + r 2 dΩ 2 n−2 ,(10)
where dΩ 2 n−2 is the metric on the unit S n−2 . Assuming that the metric is well behaved at asymptotic infinity with h ∼h ∼ ℓ −2 r 2 , the properly-normalized time-like Killing vector is ξ = ∂/∂t. We then find
Q (n−2) = r n−2 h ′ h h Ω (n−2) ,(11)
where a prime denotes a derivative with respect to r and Ω (n−2) is the volume form of the unit S n−2 . The quantity i ξ Θ (n−1) has contributions from both the gravity sector and the scalar sector:
i ξ Θ grav (n−1) = r n−2 δ h ′ h h + n−2 r h h δh Ω (n−2) , i ξ Θ φ (n−1) = r n−2 hh φ ′ δφΩ (n−2) .(12)
It is clear that the scalar contribution vanishes on the horizon where h andh vanish, and so δH evaluated on the horizon is simply T dS. It then remains to evaluate the contributions from the sphere at infinity. We consider only the case where the mass contributes the leading-order deviation of g tt from the AdS, namely
h = ℓ −2 r 2 + 1 − m r n−3 + m 1 r n−2 + · · · .(13)
We find
δH = 1 16π r→∞ (δQ − i ξ Θ) = − ω n−2 16π lim r→∞ r n−2 n−2 r δh + hh φ ′ δφ = δM + Z ,(14)
where
M = (n − 2)ω n−2 16π m , Z = − ω n−2 16π lim r→∞ r n−2 n−2 r δ(h − h) + ℓ −2 r 2 φ ′ δφ .(15)
Here ω n−2 = Ω (n−2) and M is the mass. Thus whether the quantity Z diverges, converges or vanishes depend on the specific falloffs of φ andh − h. For solutions with no scalar, such as the Schwarzschild or Reissner-Nordstrøm AdS black holes, h =h and hence Z vanishes identically. It was shown that the quantity Z for the Kaluza-Klein dyonic AdS black hole in four dimensions is finite and non-vanishing [9]. More general solutions involving multiple dyonic charges in the STU gauged supergravity model were constructed in [11]. The nonvanishing of Z was interpreted in [11] as indicating that the mass is not well defined. We prefer to take the view proposed in [9], that one can still give a meaningful definition of mass, but with the first law modified by the addition of Z = −XdY .
At the first sight, one may think that the quantity Z could only be evaluated in explicit solutions, where the falloffs of φ and the metric functions are known. In fact rather general statements can be made without knowing the explicit solutions. Two linear combinations of the Einstein equations (5) do not involve the scalar potential. In particular, the combination E t t − E i i = 0, where i denotes any specific sphere direction, does not involve the scalar at all:
h ′′ h − h ′2 2h 2 + h ′h′ 2hh + (n − 3)h ′ rh ′ −h ′ rh − 2(n − 3)(h − 1) r 2h = 0 .(16)
Thus we find that for (13), the leading falloff forh − h is given by ∆ 1 r n−4 , where ∆ 1 is the integration constant, proportional to m 1 in (13).
From E t t − E r r = 0, we have φ ′2 = (n − 2)(hh ′ − hh ′ ) rhh .(17)
Thus the leading falloff for φ is φ 1 r (n−2)/2 . Assuming that the solutions are well defined at infinity and can be expanded as:
h = ℓ −2 r 2 + 1 − m r n−3 + m 1 r n−2 + · · · ,h − h = ∆ 1 r n−4 + ∆ 2 r n−3 + · · · ,(18)φ = φ 1 r (n−2)/2 + φ 2 r n/2 + · · · ,(19)
we find that ∆ 1 and ∆ 2 are related to φ 1 and φ 2 by
∆ 1 = 1 4 ℓ −2 φ 2 1 , ∆ 2 = n 2(n−1) ℓ −2 φ 1 φ 2 .(20)
Note that the relations (20) are fixed by equation (17). The coefficient of the 1/r 2(n−3) falloff in h turns out always to be zero in the absence of electromagnetic charges. Substituting these asymptotic behaviors into (15), we find
Z = ω n−2 16π − (n − 2)δ∆ 2 + 1 2 ℓ −2 (nφ 2 δφ 1 − (n − 2)φ 1 δφ 2 ) = ω n−2 32π(n−1)ℓ 2 nφ 2 δφ 1 − (n − 2)φ 1 δφ 2 .(21)
It is important to note that the ∆ 1 contribution to Z yields a divergent result, but it cancels precisely the contribution from the leading term from φ ′ δφ. Thus we conclude that Z is finite, and in general non-vanishing unless either (1) φ 1 = 0, or (2) φ 2 = 0, or (3) φ 2 /φ n/(n−2) 1 is a fixed constant. In four dimensions, we recover the result obtained in [9]. Note that if we have multiple scalars in a linear σ-model, the quantity Z is then simply the summation of the contributions of each scalar as in (21). Although we focused on solutions with spherical symmetry, the formula (21) applies also to AdS black holes with planar horizon geometries.
Scalar properties:
We define the mass of scalar in the linearized equation in the AdS background as follows
+ 2(n − 3) ℓ 2 − M 2 φ = 0 .(22)
In this definition, scalars in gauged supergravities are massless. From the leading falloffs of the scalar in (19), we find that
M 2 = − (n − 4)(n − 6) 4ℓ 2 .(23)
Thus the required mass of the scalar is zero in four and six dimensions. A massless scalar has a typical falloff of 1/r n−3 , which coincides with the required falloff (19) in D = 4. In six dimensions, the other falloff of a massless scalar is 1/r 2 , which coincides also with (19).
The required mass (23) implies that the leading-order expansions of the scalar potential must be
V = −(n − 1)(n − 2)ℓ −2 − 1 8 n(n − 2)ℓ −2 φ 2 + · · · .(24)
The "bare" mass-squared is thus m 2 0 = −n(n − 2)/(4ℓ 2 ), corresponding to a conformally massless scalar. The conformal dimension of the dual (relevant) operator of the asymptotic AdS boundary field theory is therefore ∆ = n/2. Interestingly, for this particular conformal dimension, the coefficients φ 1 and φ 2 are the zero modes of the two boundary fields A(x) and B(x) at asymptotic infinity:
φ(x, r) ∼ A(x) r d−∆ + B(x) r ∆ ,(25)
where d = n − 1 is the dimension of the boundary field theory. This implies that the falloff coefficients φ 1 and φ 2 in (19) coincide with the boundary values of the boundary fields A and B.
Since scalars in gauged supergravities are massless, our results imply that they cannot contribute a non-vanishing Z in five and seven dimensions. In six dimensions, the massless scalar can be embedded in gauged supergravity, but only the solutions with both the slower scalar falloff 1/r 2 and the typical 1/r 3 can give rise to non-vanishing Z. In four dimensions, the requirement selects the "normal" massless scalar, and indeed dyonic AdS black holes of gauged supergravity with a non-vanishing Z were constructed [9,11]. On the other hand, the quantity Z vanishes for all the hairy black holes recently constructed in [3].
Charged system: We can also add a Maxwell field to the system, with the Lagrangian
e −1 L A = − 1 4 e aφ F 2 , F = dA .(26)
The electric ansatz that satisfies both the Bianchi identity and the Maxwell equation is
F = Qe −aφ r n−2 h h dr ∧ dt .(27)
Equation (16) is now modified by the charge, and it is straightforward to verify that the parameter Q enters first at the 1/r 2(n−3) level in the large-r expansion of h. However,
Equation (17) is unchanged. It is then clear that the relations (20) between ∆ i and φ i (i = 1, 2), which are determined by (17), will not be modified by the charge. Thus we conclude that the formula (21) for Z holds also for charged solutions.
Conclusions: In this paper, we obtained the formula for calculating the modification of the first law of thermodynamics for asymptotic AdS black holes due to scalar charges.
In n dimensions, a scalar with bare mass-squared m 2 0 = −n(n − 2)/(4ℓ 2 ) falls off at large radius as in (19). We find that the first law will be modified, with dM replaced by dM + Z where Z is given by (21). This scalar has the unique property that the two decay modes are separated in order by one inverse power of r, and hence can conspire to modify the variation of the Hamiltonian in the Wald formalism. The dual operator in the boundary field theory is relevant, with a conformal dimension ∆ = n/2. In four and six dimensions, such a scalar can arise in gauged supergravities, whilst in five and seven dimensions, it cannot. Our results provide a guiding principle for constructing new designer black holes and solitons, which can be used to compute certain effective potentials of the dual field theories, whose properties depend on the boundary conditions of the scalar field.
AcknowledgementWe are grateful to Chris Pope for useful discussions. H-S.L. is supported in part by the NSFC grant 11305140 and SFZJED grant Y201329687. H.L. is supported in part by the NSFC grants 11175269 and 11235003.
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| []
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[
"The Charge Conjugation Quantum Number in Multiquark Systems",
"The Charge Conjugation Quantum Number in Multiquark Systems"
]
| [
"Fl Stancu [email protected] \nInstitut de Physique B5\nUniversité de Liège\nSart TilmanB-4000Liège 1Belgium\n"
]
| [
"Institut de Physique B5\nUniversité de Liège\nSart TilmanB-4000Liège 1Belgium"
]
| []
| We discuss the charge conjugation quantum number for tetraquarks and meson-meson molecules, both seen as possible interpretations of the newly found XY Z charmonium-like resonances. | 10.1063/1.2987177 | [
"https://arxiv.org/pdf/0809.0408v2.pdf"
]
| 10,336,720 | 0809.0408 | c4c841e03b9902cfaef3e49eba14c35f30505ef3 |
The Charge Conjugation Quantum Number in Multiquark Systems
22 Oct 2009
Fl Stancu [email protected]
Institut de Physique B5
Université de Liège
Sart TilmanB-4000Liège 1Belgium
The Charge Conjugation Quantum Number in Multiquark Systems
22 Oct 2009Charge ConjugationExotic Resonances PACS: 1130Er1420Kp
We discuss the charge conjugation quantum number for tetraquarks and meson-meson molecules, both seen as possible interpretations of the newly found XY Z charmonium-like resonances.
INTRODUCTION
The discovery by Belle [1] of the very narrow narrow X(3872) resonance has revived the interest in heavy quarkonium, both experimentally and theoretically. The observation of X(3872) has been confirmed by CDF [2], D0 [3] and Babar Collaborations [4]. Other charmonium-like states, as for example X(3940) [5], Y (4260) [6] and Z + (4430) [7] have been uncovered in B-factory experiments. A partial list is shown in Table 1. Several review papers, as for example [8,9], discuss the difficulty of interpreting these resonances as charmonium states. Seen as exotics, they can possibly be tetraquarks, meson-meson molecules, hybrids, glueballs, etc.
J P + D * D * Y(4260) 4264± 12 83 ± 22 1 −− π + π − J/Ψ Z + (4430) 4433 ± 5 45 +35 −18 ? π ± Ψ ′ Y(4660) 4664 ± 12 48 ± 15 1 −− π + π − Ψ ′
Whatever picture is adopted, it is important to properly determine the charge conjugation number C of the resonance. From Table 1 one can see that C is known in nearly all cases. Here we shall discuss the tetraquark option, for example [10], and the molecular option, for example [11] or [12], where we noticed that some difficulty has been encountered in defining it.
THE BASIS STATES
We recall the definition of tetraquark basis states as given in Refs. [13,14,15]. This basis can provide a direct connection with the diquark-antidiquark states [16] and also with the molecular states [11,12,17]. In Fig. 1 we suppose that the particle 1 is the charmed quark c, the particle 2 a light quark q = u, d, particle 3 the anticharmed quarkc and particle 4 a light antiquarkq =ū,d.
The total wave function of a tetraquark is a linear combination of products of orbital, spin, flavour and colour parts. We shall successively introduce the orbital, colour and spin parts of the wave function. The flavor part is fixed.
The orbital part
There are at least three possible ways to define the relative coordinates. The three relevant possibilities for our problem are shown in Fig. 1. In the cases (a), (b) and (c) the internal coordinates are
σ = 1 √ 2 ( r 1 − r 2 ), σ ′ = 1 √ 2 ( r 3 − r 4 ), λ = 1 2 ( r 1 + r 2 − r 3 − r 4 ),(1)ρ = 1 √ 2 ( r 1 − r 3 ), ρ ′ = 1 √ 2 ( r 2 − r 4 ), x = 1 2 ( r 1 − r 2 + r 3 − r 4 ),(2)α = 1 √ 2 ( r 1 − r 4 ), α ′ = 1 √ 2 ( r 2 − r 3 ), x = 1 2 ( r 1 − r 2 − r 3 + r 4 ).(3)
The first system of coordinates is convenient when the quarks or antiquarks are correlated to form diquarks, as in the diquark-antidiquark model. The coordinates (2) and (3) called direct and exchange meson-meson channels, are useful in describing strong decays or introduce a molecular picture [17], provided a quark structure is imposed at short separations. One should use the system which is more appropriate for a given problem. But in specific calculations one can pass from one coordinate system to the other by orthogonal transformations [14].
The colour part
In the colour space one can construct a colour singlet tetraquark state using intermediate couplings associated to the three coordinate systems defined above. In this way one obtains three distinct bases [15] |3 12 3 34 , |6 12 6 34 ,
|1 13 1 24 , |8 13 8 24 ,(4)|1 14 1 23 , |8 14 8 23 .(5)
In the basis (4) the states 3 and 3 are antisymmetric and 6 and 6 are symmetric under interchange of quarks or antiquarks. This basis is convenient in a diquarkantidiquark picture. In Ref. [16] only the |3 12 3 34 state has been considered which restricts the spectrum to half of the allowed states [18]. The sets (5) or (6), contain a singlet-singlet colour and an octet-octet colour state. The amplitude of the latter vanishes asymptotically when a tetraquark decays into two mesons. These are called hidden colour states, by analogy to states which appear in the nucleon-nucleon problem. When the amplitude of a hidden colour state is far the most dominant in the wave function, the corresponding open channel acquires, in exchange, a tiny value for its amplitude and in this way the authors of Ref. [10] explained the small width of X(3872), when interpreted as a ccqq tetraquark.
The spin part
As the quarks and antiquarks are spin 1/2 particles, the total spin of a tetraquark can be S = 0, 1 or 2. This can be obtained by first coupling two particles to each other and then couple the two subsystems together. Let us denote the intermediate coupling states between two quarks (antiquarks) by S ij for spin 0 (Scalar) and by A ij for spin 1 (Axial). For a quark-antiquark pair we denote the states by P ij for spin 0 (Pseudoscalar) and by V ij for spin 1 (Vector). We need their permutation symmetry properties under a given transposition (ij). For quark (antiquark) pairs we have (12)|S 12 = −|S 12 , (12)|A 12 = +|A 12 ,
and (34)|S 34 = −|S 34 , (34)|A 34 = +|A 34 ,
For quark-antiquark pairs we have (13)|P 13 = −|P 13 , (13)|V 13 = +|V 13 ,
and (24)|P 24 = −|P 24 , (24)|V 24 = +|V 24 .
For S = 0 there are two independent basis states for each channel. The bases associated to (4), (5) and (6) are
|A 12 A 34 , |S 12 S 34 ,(11)|P 13 P 24 , |(V 13 V 24 ) 0 ,(12)|P 14 P 23 , |(V 14 V 23 ) 0 ,(13)
For S = 1 there are three independent states in each channel, to be identified by three distinct Young tableaux. As an example we give the basis for the direct meson-meson channel [14] |(P 13
V 24 ) 1 , |(V 13 P 24 ) 1 , |(V 13 V 24 ) 1 .(14)
The lower index indicates the total spin 1.
The case S = 2 is trivial. There is a single state
χ S = |(V 13 V 24 ) 2 ,(15)
which is symmetric under any permutation of particles.
CHARGE CONJUGATION
We deal with the ground state, i.e. we have J = S. Making the identification 1 = c, 2 = q, 3 = c and 4 = q, introduced above, it is convenient to first couple 1 to 3 and 2 to 4 and then the subsystem 13 to 24, as in Fig. 1b. The charge conjugation is equivalent to applying the permutation (13)(24) to the wave function.
Tetraquarks
The colour state (5) does not change under the permutation (13)(24). The discussion holds for spin states only.
From the properties (9) and (10) one can see that the spin states (12) and (15) remain unchanged under the permutation (13)(24). It follows that the states J P = 0 + and J P = 2 + have charge conjugation C = +.
For J P = 1 + the situation is slightly more complicated. There are six linearly independent basis vectors built as products of colour (5) and spin (14) states. α 1 = |1 13 1 24 (P 13 V 24 ) 1 , α 2 = |1 13 1 24 (V 13 P 24 ) 1 , α 3 = |1 13 1 24 (V 13 V 24 ) 1 , α 4 = |8 13 8 24 (P 13 V 24 ) 1 , α 5 = |8 13 8 24 (V 13 P 24 ) 1 , α 6 = |8 13 8 24 (V 13 V 24 ) 1 .
Under the permutation (13)(24) the basis vectors |α 1 , |α 2 , |α 4 , |α 5 change sign thus have charge conjugation C = −. On the other hand |α 3 and |α 6 do not change sign, thus have charge conjugation C = +. Also, by construction, they correspond to tetraquarks where the spin of the cc pair is S cc = 1. All states α i of Eq. (16) used to construct Table 1 of Ref. [10] have inadvertently been associated to C = +, as explained in Ref. [18].
FIGURE 1 .
1Three independent relative coordinate systems. Solid and open circles represent quarks and antiquarks respectively: (a) diquark-antidiquark channel, (b) direct meson-meson channel, (c) exchange meson-meson channel.
TABLE 1 .
1Properties of newly discovered charmonium-like resonancesResonance
Mass
Width
J P C
Decay modes
(MeV)
(MeV)
X(3872)
3871.4 ± 0.6
< 2.3
1 ++ π + π − J/Ψ, γJ/Ψ
X(3940)
3942 ± 9
37 ± 17 J P +
DD *
Y(3940)
3943 ± 17
87 ± 34 J P +
ωJ/Ψ
Z(3930)
3929 ± 5
29 ± 10 2 ++
DD
X(4160)
4156 ± 29
139 +113
−65
Acknowledgements I am grateful to Yoshi Fujiwara for pointiong out an anomaly in the relations presently numbered as Eq.(18).Meson-meson moleculesThe only interesting case is J P = 1 + . Here we need the basis states in the exchange channel corresponding toFig. 1c. As for the direct channel there are six linearly independent basis vectorsUsing the Appendix C of Ref.[14]and the relations [15]one can express β i in terms of α i or vice-versa. In particular, one hasFrom the properties of α i one can infer that β 1 , β 2 , β 4 and β 5 do not have a definite charge conjugation. Contrary, β 3 and β 6 have C = −, because they are linear combinations of states with C = −. Inverting the above relations one can obtain α i in terms of β i . We have, for exampleSection 3.1 implies that both β 1 − β 2 and β 4 − β 5 have C = +. From the identification 1 = c, 2 = q, 3 = c and 4 = q and the definitions (17) it follows thatThus one haswhere the second component is a hidden colour state with the same spin structure as (D 0 D * 0 − D * 0 D 0 ). It follows that a molecular structure with C = + can be obtained only from the difference β 1 −β 2 plus its hidden colour counterpart. Except for this part, our result is in agreement with the discussion given in Ref.[12], but in disagreement with the molecular interpretation of Ref.[11]. The hidden colour component with C = + can influence the energy of the system at short separations[13]. Such contributions have been ignored in the simple molecular picture, because the molecules are point-like particles[11,12,17].One can also find thatwhich show that β 1 +β 2 and β 4 +β 5 have C = −. Thus the (open channel) molecular state with C = − reads β 1 + β 2 = D 0 D * 0 + D * 0 D 0 (30) again in agreement with Ref.[12]. Interestingly, its hidden channel counterpart, β 4 + β 5 is completely decoupled from the open channel (30).
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"EVOLUTION OF DARK-MATTER HALOES IN A VARIETY OF DARK-ENERGY COSMOLOGIES"
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| High-resolution, numerical simulations of 17 cluster-sized dark-matter haloes in eight different cosmologies with and without dynamical dark energy confirm the picture that core halo densities are imprinted early during their formation by the mean cosmological density. Quite independent of cosmology, halo concentrations have a log-normal distribution with a scatter of ∼ 0.2 about the mean. We propose a simple scaling relation for halo concentrations in dark-energy cosmologies.IHaving to accept that the expansion of the Universe is accelerating today and that only ∼ 30% of its content is contributed by matter, we need to search for what may be driving the accelerated expansion. Friedmann's equations require the dominant form of matter to have a pressure p < −ρc 2 /3, where ρ is its density and c is the speed of light. The cosmological constant has p = −ρc 2 . Generalising this, the equation of state is modified to p = wρc 2 , with w < −1/3. In the simplest of these models, w is constant, but it is more natural to assume that w is a function of time, scale factor or redshift. One possible, admittedly hypothetical form of matter with such an equation of state is a self-interacting scalar field with an interaction potential which is sufficiently larger than its kinetic energy (e.g.Wetterich, 1988;Ratra and Peebles, 1988;Peebles and Ratra, 2002)Replacing the cosmological constant by such a hypothetical "dark energy" has consequences for structure growth and the properties of dark-matter haloes(Bartelmann et al., 2002;Weinberg and Kamionkowski, 2003;Klypin et al., 2003). We report here on our studies of how halo concentrations change in a variety of dark-energy models(Dolag et al., 2004). This leads us to suggest a remarkably simple scaling of halo concentrations with the linear growth factor in dark-energy models. We indicate consequences for strong lensing by galaxy clusters, which offer one possibility for constraining dark-energy models. Throughout, we use present-day matter and dark-energy density parameters of Ω m0 = 0.3 and Ω Q0 = 0.7, and a Hubble constant of h = 0.7 in units of 100 km s −1 Mpc −1 . | 10.1016/j.newar.2005.01.014 | [
"https://arxiv.org/pdf/astro-ph/0404489v1.pdf"
]
| 119,340,060 | astro-ph/0404489 | f60a00721e9d3f32a4c13287b92389e0db75678a |
EVOLUTION OF DARK-MATTER HALOES IN A VARIETY OF DARK-ENERGY COSMOLOGIES
26 Apr 2004
M B
K D
F P
C B
L M
M M
G T
EVOLUTION OF DARK-MATTER HALOES IN A VARIETY OF DARK-ENERGY COSMOLOGIES
26 Apr 2004Proceedings contribution for "Dark Matter/Dark Energy 2004"1 ITAU HG 2 D AU PI 3 SISSATI 4 D AU BI
High-resolution, numerical simulations of 17 cluster-sized dark-matter haloes in eight different cosmologies with and without dynamical dark energy confirm the picture that core halo densities are imprinted early during their formation by the mean cosmological density. Quite independent of cosmology, halo concentrations have a log-normal distribution with a scatter of ∼ 0.2 about the mean. We propose a simple scaling relation for halo concentrations in dark-energy cosmologies.IHaving to accept that the expansion of the Universe is accelerating today and that only ∼ 30% of its content is contributed by matter, we need to search for what may be driving the accelerated expansion. Friedmann's equations require the dominant form of matter to have a pressure p < −ρc 2 /3, where ρ is its density and c is the speed of light. The cosmological constant has p = −ρc 2 . Generalising this, the equation of state is modified to p = wρc 2 , with w < −1/3. In the simplest of these models, w is constant, but it is more natural to assume that w is a function of time, scale factor or redshift. One possible, admittedly hypothetical form of matter with such an equation of state is a self-interacting scalar field with an interaction potential which is sufficiently larger than its kinetic energy (e.g.Wetterich, 1988;Ratra and Peebles, 1988;Peebles and Ratra, 2002)Replacing the cosmological constant by such a hypothetical "dark energy" has consequences for structure growth and the properties of dark-matter haloes(Bartelmann et al., 2002;Weinberg and Kamionkowski, 2003;Klypin et al., 2003). We report here on our studies of how halo concentrations change in a variety of dark-energy models(Dolag et al., 2004). This leads us to suggest a remarkably simple scaling of halo concentrations with the linear growth factor in dark-energy models. We indicate consequences for strong lensing by galaxy clusters, which offer one possibility for constraining dark-energy models. Throughout, we use present-day matter and dark-energy density parameters of Ω m0 = 0.3 and Ω Q0 = 0.7, and a Hubble constant of h = 0.7 in units of 100 km s −1 Mpc −1 .
D-E M
The continuity equation requires that the density of dark energy changes with the scale factor a as Ω Q (a) = Ω Q0 exp −3 1 a [1 + w(a ′ )]d ln a ′ .
(1)
This term replaces the usual cosmological-constant term in Friedmann's equation. For w = const. = −1, the cosmologicalconstant behaviour is retained. For w = const., the scale-factor dependence simplifies to
Ω Q (a) = Ω Q0 a −3(1+w) .(2)
If w = −1/3, the model behaves like an open CDM model without cosmological constant because then Ω Q mimics the curvature term in Friedmann's equation. We use one model with w = −0.6 = const. for reference, and two models with time-varying w. These are the Ratra-Peebles model (Ratra and Peebles, 1988) in which the scalar-field potential is a power law, and the SUGRA model (Brax and Martin, 2000) which has an additional exponential factor in the potential. Both are normalised such that w 0 = −0.83 at a = 1. While w is almost constant for the Ratra-Peebles model in the relevant redshift range, it increases from −0.83 to ∼ −0.4 between redshifts 0 and 2 in the SUGRA model.
If normalised to its amplitude today, structure grows earlier in dark-energy compared to cosmological constant models. Since numerical simulations demonstrate that dark-matter haloes keep a memory in their cores of the mean cosmic density at their formation times (Navarro et al., 1997), haloes forming earlier are expected to have higher core densities. Thus, at fixed mass and redshift, haloes are expected to be more concentrated in dark-energy than in cosmological-constant models. This was expected from analytic considerations. The work reported here aims at testing this expectation with numerical simulations.
N S
Using the Gadget code (Springel et al., 2001), we ran a largescale cosmological simulation of the ΛCDM model, identified massive haloes within it, identified their Lagrangian volumes at the initial redshift, added small-scale power and re-simulated them at much increased resolution (Tormen et al., 1997). The particle mass is 5 × 10 9 h −1 M ⊙ . We thus created a sample of 17 clusters with final masses between 3 × 10 14 and 2 × 10 15 h −1 M ⊙ . The ΛCDM simulation was normalised to σ 8 = 0.9.
We then re-simulated this same cluster sample in dark-energy cosmologies. For doing so, we shifted the initial redshift of the simulation to higher values such that the earlier structure growth in the dark-energy models was compensated. For achieving the same density-fluctuation normalisation today, the initial redshift z ini of the dark-energy simulations needs to satisfy
D + (z ini ) D + (0) = D +,ΛCDM (z ini ΛCDM ) D +,ΛCDM (0) ,(3)
where D + (z) is the linear growth factor as a function of redshift, and z ini ΛCDM is the initial redshift of the ΛCDM simulation. It is important to also rescale the initial velocities of the simulation particles to the higher initial redshift.
The normalisation of the models is an open issue. The earlier structure growth in dark-energy models increases the Integrated Sachs-Wolfe effect and thus the amplitude of large-scale secondary fluctuations in the CMB. A smaller fraction of the observed fluctuations can then be attributed to the primordial CMB, thus the normalisation of the power spectrum should be lowered.
However, we are interested in the properties of dark-matter haloes whose linear scale is much smaller than that of the Sachs-Wolfe tail in the CMB power spectrum. How the large-scale amplitude measured by on CMB translates to small scales depends sensitively on the large-scale slope of the density power spectrum. We argue that weak gravitational lensing directly measures the power-spectrum amplitude at the scales relevant here and should thus develop into the prime method for normalising the power spectrum at small scales. For now, we study two sets of normalisations. One has constant σ 8 = 0.9 for all cosmological models, the other has reduced σ 8 such as to take the enhanced ISW effect into account.
In total, we simulate our sample of 17 clusters in eight different cosmologies: ΛCDM, Ratra-Peebles, SUGRA, and w = −0.6, the latter three with two different normalisations each, and an open-CDM model with Ω Λ = 0 for comparison.
R
We fit the NFW density profile to all clusters and obtained concentration parameters for all of them at 50 output redshifts between z = 3 and today. For all these cluster snapshots, we also compute analytically expected concentrations according to the algorithms proposed by Navarro et al. (1997);Bullock et al. (2001);Eke et al. (2001). These algorithms implement in different ways essentially the same idea: The core halo density is determined by the mean cosmological density at the halo formation redshift, which is typically defined as the redshift when the most massive progenitor of the final halo reaches a certain small fraction of the final halo mass. The factor between the mean cosmological density and the halo core density, and the fraction of the final halo mass used for defining the halo formation redshift, are two free parameters in the algorithms by Navarro et al. (1997) and Bullock et al. (2001). The algorithm by Eke et al. (2001) has only one free parameter. We determine these parameters by minimising the squared deviation between the analytic halo concentrations expected for the given halo masses and redshifts, and the numerically-determined halo concentrations.
We find that excellent agreement between the numerical halo concentrations and the predictions by the algorithm of Bullock et al. (2001) can be achieved if haloes are assumed to form very early, or, in other words, if we assume that halo core properties are imprinted when only a very small fraction of the final halo mass is already in place. The results from the algorithm of Eke et al. (2001) agree very well with the numerical concentrations without any parameter adaptation, while the Navarro et al. (1997) algorithm predicts too shallow redshift evolution (cf. Fig. 1).
Our results admit a power-law fit of the form
c(M, z) = c 0 1 + z M 10 14 h −1 M ⊙ α ,(4)
where c 0 is a constant for each cosmology. The exponent α ≈ −0.1 for all cosmologies tested, thus the mass-dependence of the Table 1 summarises the parameters we find. The scatter among the concentrations is large, but welldescribed by a log-normal about the mean given by (4). Quite independent of the cosmological model, the standard deviation of ln(c/c) is ≈ 0.22 (cf. Fig. 2), see also Jing (2000).
Interestingly, it turns out that a simple description can also be given for the cosmology-dependence of the parameter c 0 . We find c ΛCDM 0 ≈ 9.6, and for the other cosmological models
c 0 = c ΛCDM 0 D + (z coll ) D ΛCDM + (z coll ) ,(5)
i.e. the concentrations scale in proportion to the linear growth factor, provided that the halo collapse redshift z coll is chosen high enough; in fact, even z coll → ∞ produces good agreement with the numerical simulations. This reinforces the impression that halo properties are imprinted at very early times, when only a very small fraction of the final halo mass is already in place. The higher halo concentrations found in dark-energy cosmologies are expected to have pronounced consequences for strong gravitational lensing by galaxy clusters. This is indicated by an analytic study we have carried out, and demonstrated by a numerical study using the same cluster sample described here (cf. the contribution of Meneghetti et al. to these proceedings).
C
Using numerical simulations of 17 cluster-sized haloes in eight different cosmologies, we have tested and confirmed analytic expectations for the dependence of halo concentrations on models for the dark energy. Earlier structure growth in dark-energy models yields more concentrated haloes. If it is assumed that core halo properties are imprinted at very early times, when only a very small fraction of the final halo mass is already in place, the halo-concentration algorithms proposed by Bullock et al. and Eke et al. turn out to work remarkably well. Halo concentrations have a log-normal distribution with a scatter of ≈ 0.22 about their mean values, quite independent of cosmology. For all cosmologies tested, the mean halo concentration at fixed mass and redshift scales in proportion to the linear growth factor at the halo formation time if the latter is defined to be very early.
A
We are grateful to Volker Springel for his code and support, and to Simon White for his constructive comments. The simulations were carried out on the IBM-SP4 machine at the "Centro Interuniversitario del Nord-Est per il Calcolo Elettronico" (CINECA, Bologna), with CPU time assigned under an INAF-CINECA grant. K. Dolag acknowledges support by a Marie Curie Fellowship of the European Community program "Human Potential under contract number MCFI-2001-01227.
F
. 1.-Example for the agreement between analytically expected and numerically determined halo concentrations in the SUGRA model normalised to σ 8 = 0.9. The solid line with the error bars shows our numerical results in eight redshift bins between redshifts 3 and 0. "NFW" and "ENS" stand for Navarro et al. (1997) and Bullock et al. (2001), respectively. T 1.-Parameters c 0 and α for the fit formula (4) for halo concentrations in our eight cosmological models. 11.32 ± 0.09 −0.092 ± 0.005 w = −0.6 0.86 10.44 ± 0.08 −0.066 ± 0.005 halo concentrations is quite shallow for massive haloes.
F
. 2.-Quite independent of cosmology, the halo concentrations fall on a log-normal distribution with a scatter of ≈ 0.22.
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]
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| In systems with local gauge symmetries, the memory effect corresponds to traces inscribed on a suitable probe when a pure gauge configuration at infinite past dynamically evolves to another pure gauge configuration at infinite future. In this work, we study the memory effect of 2-form gauge fields which is probed by strings. We discuss the "string memory effect" for closed and open strings at classical and quantum levels. The closed string memory is encoded in the internal excited modes of the string, and in the open string case, it is encoded in the relative position of the two endpoints and the noncommutativity parameter associated with the D-brane where the open string endpoints are attached. We also discuss 2-form memory with D-brane probes using boundary state formulation and, the relation between string memory and 2-form soft charges analyzed in[1]. | 10.1007/jhep02(2019)053 | [
"https://arxiv.org/pdf/1811.07368v2.pdf"
]
| 118,901,685 | 1811.07368 | dce67740a602e45f20a1b0c30bb5f24db87f73cf |
Prepared for submission to JHEP String Memory Effect
13 Dec 2018
Hamid Afshar [email protected]
School of Physics
Institute for Research in Fundamental Sciences (IPM)
P.O.Box19395-5531TehranIran
Erfan Esmaeili [email protected]
School of Physics
Institute for Research in Fundamental Sciences (IPM)
P.O.Box19395-5531TehranIran
M M Sheikh-Jabbari
School of Physics
Institute for Research in Fundamental Sciences (IPM)
P.O.Box19395-5531TehranIran
Prepared for submission to JHEP String Memory Effect
13 Dec 2018
In systems with local gauge symmetries, the memory effect corresponds to traces inscribed on a suitable probe when a pure gauge configuration at infinite past dynamically evolves to another pure gauge configuration at infinite future. In this work, we study the memory effect of 2-form gauge fields which is probed by strings. We discuss the "string memory effect" for closed and open strings at classical and quantum levels. The closed string memory is encoded in the internal excited modes of the string, and in the open string case, it is encoded in the relative position of the two endpoints and the noncommutativity parameter associated with the D-brane where the open string endpoints are attached. We also discuss 2-form memory with D-brane probes using boundary state formulation and, the relation between string memory and 2-form soft charges analyzed in[1].
Introduction
Conserved charges associated with global symmetries through the celebrated Noether theorem have been the cornerstone of analyzing physical processes, especially in scatterings. Some of these physical processes, while can be understood as scatterings of matter fields (probes) off massless gauge fields (e.g. photons or gravitons), may also be analyzed as memory effects. In these specific cases, the initial and the final state of the gauge field or the metric differ by a pure gauge or coordinate transformation, while there is a non-zero field strength in the middle of the process, due to the radiation. The memory effect is the imprint of this evolution of the gauge field from far past to far future on an appropriate probe. These processes involve very low energy soft photons/gravitons, and one may hence expect them to be closely related to other IR effects in gauge theories, like (Weinberg's) soft theorems. One may wonder whether these IR effects can also be analyzed using the notion of conserved charges. We are therefore, dealing with the "IR triangle" [2][3][4][5][6] which relates soft theorems, asymptotic charges and memory effects.
Asymptotic symmetries involve a specific set of gauge transformations remaining after certain gauge fixing and preserving certain boundary conditions. Asymptotic charges associated with these symmetries can differentiate the initial and final photon/graviton states. Therefore, the change in the value of the charge can be stored in the probe as memory effect.
Depending on the quantity imprinted on the probe we may have different types of memory effects. Historically, the first memory effect was discussed in a gravitational context where passage of gravity waves was imprinted in displacements of a geodesic congruence [7,8]. This memory effect in the language of soft charges is associated with supertranslations. There are also two other kinds of gravitational memories: spin and refraction memory effects, respectively associated with superrotations [9] and superboosts [10]; see also [11][12][13]. In electrodynamics the leading memory effect [14] is the shift of the velocity of a charged point particle under the 1-form gauge field radiation (the kick memory effect) -see also [15][16][17][18].
Antisymmetric gauge fields which are ubiquitous in theories of supergravity and string theory also come with a gauge symmetry; a (p + 1)-form (with a (p + 2)-form field strength) enjoys a gauge symmetry generated by p-forms. Unlike the electromagnetic case, objects which are charged under (p + 1)-forms are spatially extended p-branes with a (p + 1)-dimensional worldvolume. Generically, p-branes (with p > 0) have internal degrees of freedom which can be degenerate, so that the transition between these internal modes has no energy cost. In particular, there could be transitions between these modes by the exchange of soft (p + 1)-form modes. This latter yields a new type of memory effect, the "internal memory effect", which is associated with the change in internal excitations of an extended object, like a p-brane. The internal memory effect may be put in contrast with other so-far discussed memories, the "external memory", which are associated with the change in a spacetime property of the probe like position, momentum or spin.
In this work we intend to study such an internal memory effect by focusing on the 2-form theory (the p = 1 case) which is naturally coupled to strings. This specific example, while carries aspects of the higher form cases, has its own unique features. In particular, due to the conformal symmetry of the worldsheet theory, we completely know the spectrum of free strings and have much better control on the analysis. Moreover, the 2-form gauge field background as well as the fundamental string probes, are both contents of the string theory setup and prepare a framework to study "string memory effect". We also connect the string memory effect to those soft charges and asymptotic symmetries recently studied [1] (see also [19,20]) to construct one edge of the 2-form soft triangle.
In order to study the string memory effect, we revisit the problem of strings in a slowly varying Neveu-Schwarz-Neveu-Schwarz (NSNS) 2-form B-field background. It is known that the B-field affects closed and open strings in different ways and hence we consider these two cases separately. As we show, for the closed string case the string memory effect is encoded in the transition of the massless states of the closed string into each other. Thus, in the string memory effect the probes themselves can be massless states, in contrast with other usual memories in which the probe is a massive state. 1 In the open string, the constant part of the B-field affects string dynamics through the boundary conditions. It is known that the endpoints of open strings attached to a D-brane in the B-field background parameterize a noncommutative space [21][22][23][24][25]. The open string memory effect is then encoded in the change in the noncommutativity of the open string endpoints. As we will discuss, this memory can hence be observed through the effective noncommutative field theory residing on the brane where the open string endpoints attach. Alternatively, one may use the D-branes as probes of the background time-varying B-field, using boundary state formulation for D-branes [26,27]. In this system, the change in the B-field is encoded in the mass density or other Ramond-Ramond (RR) charges of the D-brane probe. The similar problem of strings in a gravitational background pulse was considered in [28] and the possibility of permanent shift in brane separation was discussed, which resembles point particle gravitational memory effects.
Organization of the paper. In section 2, to set the stage we review the memory effect especially in the case of electromagnetism. We discuss the electromagnetic kick memory effect at classical and quantum levels. As we will review, the memory on quantum probes can be expressed by an S-matrix. The quantum memory effect is what is relevant to our string theory analysis. In section 3, we study the closed string memory effect and discuss how the passage of a B-field wave causes a shift in the internal mode of the closed string, without affecting its center of mass motion. In section 4, we analyze the open string memory effect caused by the passage of an NSNS B-field wave packet. This memory is imprinted in the noncommutativity parameter associated with the open string endpoints. In the last section, we discuss physical implications and possible extensions of our analysis here, as well as other open questions for future analysis. In an appendix, we have discussed the absence of electromagnetic memory on a harmonic oscillator probe.
Notation and conventions:
Spacetime coordinates. We are mainly interested in an experiment performed at the radiation zone of a scattering process which involves the emission of 2-form soft photons, see [1] for more details. In the radiation zone, if the size of the measurement apparatus is much smaller than the radius of the sphere surrounding the scattering region, wavefronts can be well approximated as plane waves. In the vicinity of a point on the sphere the line element is
ds 2 = −du 2 − 2dudr + δ ij dX i dX j ,(1.1)
where u = t − r is the retarded time, X i are local Cartesian coordinates in vicinity of a point p on the (D − 2)-sphere with i = 1, 2, · · · , D − 2 and r is the radial outward pointing coordinate.
For transverse Cartesian coordinates we use i, j, m, l, etc. Finally Greek indices µ, ν, α etc. go on all coordinates including u.
The fall-off behavior of fields are most often expressed in spherical coordinates θ A with metric r 2 q AB dθ A dθ B , which covers the whole sphere. It is always presumed that, if the large-r behaviour of (say) a 1-form in spherical coordinates is
T A ∼ O(r −n ), then ∂ B T A ∼ O(r −n ).
The implicit premise here is that T A does not vary too fast on the sphere, so its value and its derivative have the same magnitude. This will be our working premise too. 2 One should, however, note that when working in Cartesian coordinates on a small patch of the sphere, one has
T A ∼ O(r −n ) → T i = e A i T A ∼ O(r −n−1 ) , (1.2) ∂ B T A ∼ O(r −n ) → ∂ j T i = e B j e A i ∂ B T A ∼ O(r −n−2 ) , (1.3)
where e A i are changing the Cartesian to spherical basis. That is, each Cartesian derivative lowers the r-power by one unit.
Worldsheet. Worldsheet coordinates are (τ, σ) and derivatives with respect to them are denoted by dot and prime respectively. For closed strings σ ∈ [0, 2π] while for open strings σ ∈ [0, π]. The string length is s ≡ √ 4πα . The left/right worldsheet derivatives are defined as
∂ + = ∂ τ + ∂ σ ∂ − = ∂ τ − ∂ σ . (1.4)
We will work in light-cone gauge u = s τ throughout. Embedding coordinates of the string into the target space will be denoted by X µ (τ, σ), which are divided into u, r(u, σ), X i (u, σ) in light-cone gauge. When dealing with D p -branes we use X I , I = 0, · · · , p for directions along the brane.
Spacetime fields. The fields in spacetime are denoted by curly letters, while for the coefficients of asymptotic expansions, normal fonts are used. The 1-form electromagnetic gauge field is denoted by A and its field strength by F = dA. The 2-form gauge field is shown by B and its field strength by H = dB.
Memory effect: review and remarks
In this section we discuss three issues. First, we give a definition for memory effect with the focus on Hamiltonian formulation. Second, we revisit electromagnetic memory effect on a free particle. Third, we discuss memory effect for the quantized version of the same system. Consider a physical system with classical phase space coordinates (q n , p n ) and the Hamiltonian H. Let the system interact with the environment through a gauge interaction with a gauge potential that we schematically denote by A. In typical examples, the classical equations of motion are gauge invariant, depending exclusively on a gauge invariant field strength F. For instance, the form of Lorentz force in electromagnetism is proportional to F µνẊ ν . Also, observable quantities O[q, p; A] in systems of physical interest are always gauge invariant. In other words, O[q, p; A] = O[q, p;Ã] where A andà are related by a gauge transformation generated by Λ. Since gauge transformations of the external field leave all observables of the system invariant, one expects them to be equivalent to a canonical transformation U [q, p; Λ, A] on phase space (q n , p n ). In particular, if the field strength F is vanishing, one can find a canonical transformation U to set A = 0 (up to possible topological obstructions). We will find such transformations in the examples we analyze.
We will assume that the external field A is only a function of the retarded time; justified physically as follows. If the radius of the sphere on which the probes (point particle, strings, etc.) are located is large enough compared to the characteristic length of the probe (i.e. x 0 , r x 0 ), then A is almost constant in the vicinity of the probe. This assumption fails if A is highly oscillating on the celestial sphere which will be disregarded in current calculation. Proposition 1. Let a classical system (p n , q n ; H) interact through an external gauge potential A with time-dependent field strength F(u) which is vanishing at early and late times:
lim u→±∞ F(u) = 0 , (2.1)
so the system evolves with free Hamiltonian in the limit u → ±∞. 3 Defining the pure gauge configurations,
A + ≡ A(u = +∞), A − ≡ A(u = −∞),(2.A + , A − ].
The time evolution of the system is free at large |u|. As a result, the phase space and the Hamiltonian at u → ±∞ are related by a canonical transformation U [q, p; A + , A − ] and free solutions are mapped to free solutions by the same operator.
In the above setting, we propose the following definition for the memory effect:
Definition 1 (Memory effect). Let O[q, p; A]
be an observable of the system described above, for which the following quantities are well-defined:
O + = lim u→+∞ O[q, p; A], O − = lim u→−∞ O[q, p; A]. (2.3)
If O + and O − are not related to each other by the canonical transformation defined in proposition 1, we say that gauge field A has induced a memory effect on the system.
Classical treatment
Although most of the work on memory effect has been done for classical systems, one can figure out the effect on the state of a quantum system. We first consider the classical approach, and analyze memory effect in Hamiltonian formulation. This latter sets the stage for formulating memory effect at quantum level.
Electromagnetic Memory Effect. Before analyzing the memory effect exerted by a 2-form gauge field on a string probe, we first review the memory effect in the case of electromagnetic 1-form gauge field coupled to a point particle of charge q and mass m. One may analyze the problem in action or Hamiltonian formulations. Let us start with the action and equations of motion:
S = dτ η 1 2 η −2 ∂ τ X µ ∂ τ X ν g µν + q η −1 ∂ τ X µ A µ − 1 2 m 2 ,(2.4)
where η(τ ) is the einbein of the metric on the worldline ds 2 = η 2 (τ )dτ 2 . We take X µ = (u, r, X i ) as Minkowski coordinates defined in (1.1). 4 This action is gauge invariant up to a boundary term and we can fix the radial gauge A r = 0. To fix the reparametrization freedom we use the light-cone gauge which sets u as the clock, i.e. u = τ . The action is then
S = dτ 1 2η (−1 − 2ṙ +Ẋ iẊ j δ ij ) + q Ẋ i A i + A u − 1 2 η m 2 . (2.5)
Assuming that the gauge field components A have very mild r dependence (according to fall-off behavior given below), the equation of motion for r yields η ≈ const and we must choose this constant equal to 1/m to be consistent with non-relativistic limit, taken below. 5 Equations of motion for X i then reads asẌ
i = q m (F i jẊ j + F i u ) . (2.6)
We now focus on cases where the particle is coupled to the radiation photon field through A i with the boundary conditions at large r,
A i ∼ O(1/r) , A u ∼ O(1/r) , A r = 0 . (2.7)
The field strength component F ui is at order r −1 , while the other components of the field strength F ij and F ur are subleading. Integrating (2.6) along the retarded time u, we are left with an electric kick memory effect,
∆Ẋ i =Ẋ i (u = ∞) −Ẋ i (u = −∞) q m ∞ −∞ F i u du = − q mr (A i (u → ∞) − A i (u → −∞)) , (2.8)
in which A is the leading term in the asymptotic falloff (2.7) and terms of O(r −2 ) are ignored.
In order to apply and use the results of the proposition 1, we present these results in the Hamiltonian formulation. From (2.5) the canonical momentumP i andP r conjugate to the position coordinates X i and r are,
P i = 1 ηẊ i + q A i ,P r = − 1 η . (2.9)
The Hamiltonian is
H = − m 2 +P 2 r + (P i − q A i ) 2 2P r − q A u . (2.10) The constraint equation (c.f. footnote 5), η 2 m 2 − 1 − 2ṙ − (P i − qA i ) 2 =
0 associated with the reparametrization invariance of the probe action can be solved for η. Thus, for a non-relativistic particle, deviation of η from 1/m is small, suggestive of defining a shifted radial momentum P r ≡P r + m 1. Finally, the Hamiltonian for a non-relativistic charged particle of rest mass m moving in the light-cone gauge τ = u for the particle and in the radial gauge A r = 0 for the gauge field, takes the familiar form
H = m + P 2 r + (P i − qA i ) 2 2m − q A u , i = 1, 2 . (2.11)
We drop the rest mass constant term m from now on. Clearly, the Hamiltonian (2.11) takes different forms at u → ±∞. Recalling the falloff behavior of the gauge field at large r (2.7), the particle is "free" at order r −1 and one expects early and late Hamiltonians to be related by a canonical transformation. To show this, consider new dynamical variables (P, X) related to old variables (P, X) by the canonical transformation on momenta
P r =P r + m , P i =P i − q A i , (2.12)
while the coordinates remain unaltered. The new Hamiltonian K is
K = P 2 r + P 2 i 2m + q X i ∂ u A i , i = 1, 2 . (2.13)
To verify this statement, we note that new and old variables in the large r limit are related aṡ 14) and the generating function 6 for the transformation is
X aP a − H =Ẋ a P a − K + dG du , a = 1, 2, 3,(2.G = qx i A i + q u u 0 A u (u )du . (2.15)
The integral term in G does not appear in transformation relations (2.12); it becomes subleading in r when Cartesian derivatives are performed.
The new Hamiltonian K has the same form at u → ±∞ since the electromagnetic radiation
F ui = ∂ u A i + O(r −2 )
by assumption vanishes at both temporal limits. We are now ready to find an observable in the new coordinate system which has different asymptotic values at early and late times. The asymptotic form of equations of motion in the new basis arė
X a = [X a , K] = P a m ,Ṗ i = [P i , K] = −qȦ i ,Ṗ r = 0 . (2.16)
Proposition 2 (Memory effect). Electromagnetic Memory effect on a free charged non-relativistic particle is the difference between late and early transverse momenta:
∆P i = −q ∆A i , (2.17)
where for a generic variable V , ∆V measures the difference between the late and early values,
∆V ≡ V (u = +∞) − V (u = −∞) . (2.18)
Eq. (2.17) is the analogue of (2.8) in the Hamiltonian formulation. The above example provides a smooth transition to a quantum treatment of the memory effect.
Quantum treatment
The analysis of previous section in Hamiltonian formulation can be quantized in a straightforward way, leading to electromagnetic memory effect on the momentum shift of quantum charged particles on the corresponding Hilbert space. The canonical transformation which brings the Hamiltonian to the convenient form becomes a unitary transformation on the Hilbert space. Consider the quantized Hamiltonian (2.11) 19) and the unitary operator 7
H =P 2 r + P i − qA i (X) 2 2m − qA u (X) ,(2.U = exp −iqX i A i (X) − iq u u 0 du A u (X) . (2.20)
Neglecting subleading terms inr, A i (X) = A i (u) and A u (X) = A u (u), so the unitary transformation will asymptotically act as follows:
UX a U † =X a , (2.21) UP i U † =P i + q A i (u) , (2.22) UP r U † =P r , (2.23)
while transforming the Hamiltonian tô
K =ÛĤÛ † + i dÛ du U † =P iPi +P 2 r 2m + qX i dA i (u) du . (2.24)
Time evolution of a state vector |Ψ in this canonical frame is given by
|Ψ(u) = e −i(u−u 0 )K |Ψ(u 0 ) . (2.25)
In the u → ±∞ limit, the in/out Hilbert spaces can be written in the basis of eigenstates of the free Hamiltonian (K at q = 0). We want to find the S-matrix operatorŜ : H in → H out which quantifies the overlap of in and out states.
Proposition 3. If a quantum particle with Hamiltonian (2.24) is prepared in an energy eigenstate |in = |n at u → −∞, then its state at u → +∞ is given by
|out =Ŝ|in ,Ŝ = exp −iqX i int ∆A i , (2.26)
where ∆A i is defined in (2.18), andX int =X −P u/m is the interaction-picture position operator. The quantum memory effect can also be expressed as an operator equation in Heisenberg pictureŜ
−1P iŜ = −q ∆A i . (2.27) 7
Hatted symbols refer to quantum operators.
To show (2.26), we start with the more precise statement of adiabatic evolution by solving the problem for a particle in a large 3D box of dimension L. The free spectrum iŝ
P aPa 2m |n = E n |n , E n = π 2 2mL 2 n · n (2.28)
Let |n be the interaction picture state, so that its time evolution be given by the interaction term in the Hamiltonian:
i d du |n = q e −iK 0 uX iȦi e iK 0 u |n = q(X −P u/m) iȦ i |n ≡ qX i intȦ i |n . (2.29)
The solution is
|n(u) = exp −iqX i int A i u u 0 |n(u 0 ) ,(2.30)
being in particular true for in/out states.
As the above indicates, the momentum kick and memory is present for free particles. Nevertheless, one can show that (see appendix A), for bounded particles (e.g. in harmonic oscillator) the memory for both classical and quantum systems, is averaged out in time.
Memory effect and conserved charges
In this subsection we relate the 'memory effect' to a change in (soft) charges associated to nontrivial large gauge transformation from space-like infinity i 0 ≡ I + − to future time-like infinity i + ≡ I + + as two endpoints of future null infinity 8 . Conserved charges associated with large gauge transformations can be obtained as integrals of the time component of the current J µ , which can in turn be (locally) written as divergence of a two-form J ν = ∇ µ J µν , on a spacelike Cauchy surface Σ. In the Bondi-slicing of flat spacetime we have
Q = i 0 dΩ d−2 J ru .
(2.31)
Once the boundary conditions are imposed appropriately, this boundary charge is finite, conserved and integrable. In the case of electromagnetism the current associated with the gauge parameter λ is J µν [Λ] = λ F νµ .
Soft charges at past I + − and future I + + of future null infinity are sensitive to the value of gauge field A(u = −∞) and A(u = +∞) there. Since memory effect depends on their difference ∆A, it will be related to the charge difference at two ends of null infinity. In the electromagnetic case, this difference is
Q λ [I + + ] − Q λ [I + − ] = I + + λ F ur − I + − λ F ur = I + λ ∂ u F ur = S 2 λ D i A i u=+∞ u=−∞ (2.32)
where in the last equality we used the field equation (d F) u = 0 asymptotically. We also saw in (2.8) that electromagnetic memory effect is also given asymptotically by ∆A i with definition as in (2.18). This is the general pattern. Figure 1. Observation of memory effect due to the out-going radiation by massive probes. The 1,2 symbols show detection points, located at constant r. One could set the detection times earlier to observe the in-going memory effect. The dashed lines depict some Cauchy surfaces and as we see they all intersect at spatial infinity i 0 . In the figure we have implicitly imposed the antipodal matching
Σ 2 1 I − I + i 0 i − i +through i 0 = I + − = I − + .
The above equation may be read as follows: While I + is not a Cauchy surface, Σ + = I + ∪i + is so and i 0 is the boundary of this and all other constant time Cauchy surfaces, cf. dashed lines in Figure 1. 9 Therefore, one expects that the soft charge Σ + λF ur to be conserved. However, massless particles can reach I + and the massive probes to i + . So, the change in the photon soft charge can be attributed to the change in the soft charge of the probe particle, a.k.a the memory.
Two-form soft charges. In the case of 2-form theory, the value of the Noether charge associated to large gauge transformation at space-like infinity (past boundary of future null infinity) was derived in [1] in six dimensions where it was shown that the conserved charges of the theory split into two separately conserved exact and coexact pieces associated to the exact and the coexact parts of the one-form gauge transformation parameter on the sphere S 4 . In [1], the expression of the charge was derived in de Sitter slicing, while in the analysis for the memory effect we need the expressions on the null infinity and the Bondi slicing. The details of the derivation of the charges in the latter slicing will be given elsewhere [29] here we show the final result:
Q coexact [λ] = I + − J ru [λ] , Q exact [λ] = I + − K ru [λ] ,(2.33)
where
J µν [Λ] = Λ α H ανµ , K µν [Λ] = 2(dΛ) α[µ (dC) ν] α (2.34)
are the corresponding currents associated to the coexact and exact charges. These expressions are for a two-form theory with Lagrangian L = 1 3! H 2 , H = dB and Λ is the one-form gauge parameter, associated with gauge transformation B → B + dΛ. In the case of exact currents, we have B = dC and obviously C is not constrained by field equations, thus the corresponding charge is conserved off-shell. Using the stokes theorem similar to (2.32) we have,
Q coexact [i + ] − Q coexact [i 0 ] = I + ∂ u J ru = S 4 λ j D i B ij u=+∞ u=−∞ (2.35) Q exact [i + ] − Q exact [i 0 ] = I + ∂ u K ru = S 4 λ u D i (D i C r ) u=+∞ u=−∞ (2.36)
where B and C are the leading in 1/r contributions of B and C while λ is the leading contribution of the gauge parameter Λ. In the last equality of (2.35) we used the field equations (d H) u = 0. As we will discuss, the coexact charges are relevant to the string memory effect. As we see in (2.36), the difference of the exact charges at past and future of null infinity reduces to the case of electromagnetism (2.32) with λ u and D i C r as new gauge parameter and potential.
Closed string memory effect
In this section, we consider a closed string in flat spacetime located at the radiation zone of a scattering process, which involves 2-form soft radiation 10 and calculate the response of the string to the 2-form soft radiation. The Polyakov action for a closed string, coupled to a background 2-form field B µν is:
S = − 1 4πα dτ dσ √ −γ γ ab ∂ a X µ ∂ b X µ + ab ∂ a X µ ∂ b X ν B µν . (3.1)
Variations w.r.t. the worldsheet metric γ ab and the target space coordinates X µ give the following constraint and equations of motion,
(∂ ± r) 2 − 2 s ∂ ± r + (∂ ± X i ) 2 = 0 , (3.2) ∂ + ∂ − X i = − s ∂ u B ij X j . (3.3)
where we used the light-cone gauge u = s τ for the coordinate system (u, r, X i ), i = 1, 2, · · · , D− 2 with u = t − r, the conformal gauge choice γ ab = e ω η ab for the worldsheet metric and the radial gauge B rµ = 0 for the background 2-form field. The B-term is a topological term (it is independent of the worldsheet metric γ ab ) and does not contribute to the constraint (3.2) which determines the non-zero-mode part of the r-coordinate.
In our analysis, we neglect variation of B around the string; we assume ∂ σ B = 0, that is, the string length is much smaller than variation of B. The solution to equations of motion (3.3) in presence of small B-field can be related to a solution with vanishing B-field,
∂ − ∂ + Y i = 0 , (3.4) whose general solution is Y i (τ, σ) = Y i L (τ − σ) + Y i R (τ + σ) ,(3.
5) 10 We note that while the soft charge analysis of [1] is made for 2-form theory in six dimensions, our string probe analysis here is valid for strings in target space in any dimensions. This target space may be a six dimensional one associated with a ten dimensional critical superstrings on a four dimensional compact Ricci flat manifold.
where Y i L,R (x) = Y i L,R (x + 2π). The general solution (3.5) subject to these boundary conditions, takes the following form,
Y i (τ, σ) = y i 0 + α p i τ + i α 2 n =0 1 n α i n e −in(τ −σ) +α i n e −in(τ +σ) . (3.6)
Next, consider
X i (τ, σ) ≡ [δ ij + 1 2 B ij (u)]Y j L + [δ ij − 1 2 B ij (u)]Y j R ,(3.7)
which satisfies
∂ + ∂ − X i = − s ∂ u B ij X j + 2 s 2 ∂ 2 u B ij (Y j L − Y j R ) + O(B 2 ) . (3.8)
Assuming that B is soft, i.e. its frequency is much smaller than the string frequency −1 s , the second term is negligible and X i satisfy equations of motion (3.3). Therefore, the closed string solution to (3.3) (with periodic boundary conditions) for a slowly varying background s ∂ u B B and neglecting O(B 2 ) terms is
X i = x i 0 + α p i τ + i α 2 n =0 1 n (δ ij + 1 2 B ij (u))α j n e −in(τ −σ) + (δ ij − 1 2 B ij (u))α j n e −in(τ +σ) . (3.9)
where α i n andα i n are the oscillator modes of Y i L and Y i R in (3.6), respectively.
Classical closed string memory effect. The memory effect is the imprint of the background field on the string, found by comparing the state of the string at far future and far past.
Defining ∆B ij = B ij (u → +∞) − B ij (u → −∞) ,(3.10)
the closed string memory effect is
α i n (u → +∞) = (δ ij + 1 2 ∆B ij )α j n (u → −∞) (3.11) α i n (u → +∞) = (δ ij − 1 2 ∆B ij )α j n (u → −∞) (3.12) p i (u → +∞) = p i (u → −∞) (3.13)
plus terms of O(B 2 ) and O( s ∂ u B) which are negligible. The center of mass motion is unaltered, while oscillation modes are rotated, with a relative minus sign between left-and right-modes.
Quantum closed string memory
To study the quantum closed string memory, we start from (3.11)-(3.13) and quantize the system 11 . We employ a Hamiltonian approach, perform the calculations in the light-cone gauge and treat the soft background as a perturbation. Using proposition 1, we first introduce a canonical transformation that turns the initial and the final Hamiltonian into similar forms. The light-cone Hamiltonian for the action (3.1) is
H = 2 s 4 2π 0 dσ (P i − 2 2 s X j B ij )(P i − 2 2 s X k B ik ) + 4 whereP i = ∂L ∂Ẋ i = 2 s Ẋ i + X j B ij is the canonical momentum of X i . Consider the canonical transformation on the momenta P i =P i − 2 2 s X j B ij ,(3.15)
and leave the coordinates unaltered. The transformed Hamiltonian K is
K = 2 s 4 2π 0 dσ P i P i + 4 4 s X i X i + 4 4 s X i X jḂ ij . (3.16)
The generating function for the transformation is
G = X i (P i + 1 2 s X j B ij ) . (3.17)
To quantize, as usual, we promote the phase space coordinates to quantum operators and impose the commutation relations
[X i (σ), P j (σ )] = iδ ij δ(σ − σ ) , [r, pr] = i ,(3.18)
wherer = dσr(σ) . The free string mode expansion (3.6) then yields the following brackets
[α i m , α j n ] = mδ ij δ m+n , [α i m ,α j n ] = mδ ij δ m+n , [x j 0 , p j ] = i . (3.19)
The Hamiltonian is 12
K = K 0 + n =0 i 2n (α i −n α j n −α i −nα j n + 2α i nα j n )Ḃ ij (3.20)
where K 0 = 1 2 n α i n α i −n +α i nα i −n is the unperturbed Hamiltonian with α i 0 =α i 0 = α /2 p i . In our analysis we will drop the intercept (zero point energy) as we implicitly assume that our X modes are a part of a superstring theory where the zero point energy cancels out by the worldsheet superpartners.
For a slowly-varying small B-field background (i.e. the adiabatic evolution, in which the perturbation varies much slower than the energy gap among states), the transition amplitude between states of different energy is vanishing. As a result, the very last term in (3.20), which does not commute with the free Hamiltonian K 0 gives no contribution to the evolution of 12 One could perform the canonical transformation after quantization, which would be a unitary transformation on Hilbert space and operators. Consider the following unitary transformation
U [B] = exp − i 2 s 2π 0 dσ X i X j B ij .
with its action on momentum operator
P i → U P i U † = P i + 2 2 s X j B ij − 1 2 s X j ∂ σ B ij
with the transformed Hamiltonian K = U HU † + iU U † .
Assuming the string length is much smaller than the variation of B, ∂ σ B 0, K takes the same form as (3.16), but now as a quantum operator.
states. (The fact that adiabatic evolution can give rise to transitions only in degenerate states has a simple derivation; see appendix A for the example of a harmonic oscillator subject to soft electromagnetic radiation.). The first couple of terms are identified as the left and right components of the angular momentum operators 13 (3.21) and commute with the Hamiltonian. The Hamiltonian is then easily integrated to give the time evolution operator
E ij ≡ n =0 i n α i n α j −n ,Ẽ ij ≡ n =0 i nα i nα j −n ,U (u 2 , u 1 ) = exp − i 4 (B ij (u 2 ) − B ij (u 1 )(E ij −Ẽ ij ) exp (−iK 0 (u 2 − u 1 )) . (3.22)
The S operator which maps in and out states is thus identified as
S = exp − i 4 ∆B ij (E ij −Ẽ ij ) , (3.23)
and ∆B ij is defined in (3.10). The 2-form memory effect on a closed string is then given by the Heisenberg-picture evolution of the operators, The memory effect on this state is the transition in its polarization tensor according to (3.24):
S −1 α k m S = + 1 2 ∆B ik α k m + O(B 2 ) , S −1αk m S = − 1 2 ∆B ikα k m + O(B 2 ) ,ζ ij → δ im + 1 2 ∆B im ζ mn δ jn − 1 2 ∆B jn , (3.26)
where as usual the dilaton, graviton and b-field states are respectively associated with trace, symmetric-traceless and antisymmetric parts of polarization tensor ζ ij :
√ D − 2ϕ ≡ δ ij ζ ij , b ij = 1 2 (ζ ij − ζ ji ) , h ij = 1 2 (ζ ij + ζ ji ) − 1 √ D − 2 δ ij ϕ .
(3.27) 13 We are using the notation of [30], where the angular momentum operator is defined as
J µν = 2 2π 0 dσX [µ P ν] = µν + E µν +Ẽ µν ,
and µν = 2x
[µ 0 p ν] .
Variation of different components are
∆ϕ = − 1 √ D − 2 ∆B mn b mn , (3.28a) ∆b ij = ϕ √ D − 2 ∆B ij − ∆B n[i h j]n , (3.28b) ∆h ij = 1 D − 2 δ ij ∆B mn b mn + ∆B n(i b j)n . (3.28c)
Variations of ϕ and h ij depend exclusively on the initial 2-form state b ij . On the other hand, variation of the 2-form field comes from other components, i.e. ϕ and h ij . This is of course quite expected from group theory viewpoint; the contracted product of an antisymmetric tensor and symmetric one is an antisymmetric tensor and the contracted product of two antisymmetric tensors is a symmetric one. That is, the passage of a soft 2-form converts the massless b state to dilaton or graviton and vice versa, without changing its momentum or energy.
Effective field theory analysis
In this subsection, we will provide a field theoretic explanation of the closed string memory effect. We write the effective field theory for the graviton h µν , the dilaton ϕ and the Kalb-Ramond 2-form field b µν , on a slowly varying 2-form background. We calculate the transition amplitudes of previous sections by reading the vertices in Feynman rules.
The effective action of NSNS sector of string theory in string frame is [31]
S = 1 2κ 2 d D x √ −Ge −2Φ R + 4∇ µ Φ∇ µ Φ − 1 12 H µνλ H µνλ (3.29)
where κ 2 = 8πG N . For each field we consider a perturbation on top of a background field
G µν =Ḡ µν + h µν , B µν =B µν + b µν , Φ =Φ + ϕ (3.30)
We assume that background fields satisfy their own classical equations of motion, thus, only terms of second and higher order in perturbation fields appear in the action. The second-order terms are kinetic terms. We assume thatḠ µν = η µν andΦ = 0 and neglect second order terms in background B-field. The second order action in Einstein frame 14 becomes 14 The Einstein frame action is There are also a couple of kinetic mixing terms (interaction with background B-field) at second order due to the background H-field, as depicted in Figure 2 L
S (2) = 1 2κ 2 d D x L (2) h + L (2) b + L (2) ϕ (3.31) where L (2) h = − 1 2 ∇ µ h νρ ∇ µ h νρ + ∇ µ h νρ ∇ ν h µρ − ∇ µ h∇ ρ h µρ + 1 2 ∇ µ h∇ µ h (3.32a) L (2) b = − 1 12 (db) µνλ (db) µνλ (3.32b) L (2) ϕ = − 4 D − 2 ∇ µ ϕ∇ µ ϕ (3.32c)S = 1 2κ 2 d D x √ −G R − 4 D − 2 ∇ µ Φ∇ µ Φ − 1 12 e −8Φ/(D−2) H µνλ H µνλ . ⊗ b µν (p) ϕ(p) ∆B µν ⊗ b µα (p) h ν α (p) ∆B µν(2) bϕ = 4 3(D − 2) ϕH µνλ (db) µνλ (3.33) L (2) bh = − 1 6 3h αβH ανλ (db) β νλ + 1 2 hH µνλ (db) µνλ (3.34)
Gauge fixed action; TT gauge. We impose transversality condition both on graviton and 2-form field:
∇ µ h µν = 0 , ∇ µ b µν = 0 . (3.35)
In addition, the temporal components can be set to zero by using the residual gauge symmetry while the metric perturbation is set to traceless: We implement canonical quantization of the fields on constant-u surfaces. Although u is a timelike coordinate, constant-u surfaces, where outgoing wavefronts lie, are null hyperplanes. The canonical momenta are the following
η µν h µν = 0 , h uα = 0 , b uα = 0 .Π (ϕ) (x) = ∂L ∂φ = 4 κ 2 (D − 2) ∂ r ϕ (3.37) Π ab (b) (x) = ∂L ∂ḃ ab = − 1 2κ 2 H uab (3.38) Π ab (h) (x) = ∂L ∂ḣ ab = 1 2κ 2 ∇ r h ab (3.39)
The interaction-picture free fields are 15 The quantum operators satisfy
ϕ(x, t) = κ 2 √ D − 2 d D−1 p 1 √ 2p r ϕ p e ip·x + ϕ † p e −ip·x (3.40) b ab (x, t) = κ s d D−1 p 1 √ 2p r ζ ab (p, s) b s p e ip·x + ζ * ab (p, s) b s † p e −ip·x , (3.41) h ab (x, t) = κ s d D−1 p 1 √ 2p r ξ ab (p, s) h s p e ip·x + ξ * ab (p, s) h s † p e −ip·x .[ϕ p , ϕ † q ] = δ D−1 (p − q) (3.43a) [b r p , b s † q ] = δ rs δ D−1 (p − q) , s = 1, · · · (D − 2)(D − 3)/2 (3.43b) [h r p , h s † q ] = δ rs δ D−1 (p − q) , s = 1, · · · , D(D − 3)/2 . (3.43c)
The free Hamiltonian is
H = d D−1 p ω p ϕ † p ϕ p + s b s † p b s p + s h s † p h s p , ω p = p u = p 2 r + p 2 i 2p r . (3.44)
Two-form-dilaton conversion amplitude. The time evolution operator (interaction Hamiltonian) at first order in B is
− i duH I (u) = − 2i 3(D − 2)κ 2 d D xϕ(x)H µνλ (x)(db) µνλ (x) .
(3.45)
Fourier transform on spatial dimensions and using free field expansions gives
3 √ D − 2 duH uij (u) s d D−1 p 2p r p [u ζ ij] * (p, s)ϕ p b s † p − p [u ζ ij] (p, s)ϕ † p b s p . (3.46)
The first integral picks out the soft mode of the background field
duH uij =H uij (ω → 0) = ∆B ij . (3.47)
The transition amplitude is hence proportional to the intensity of the soft 2-photon.
The simplest example the matrix element M(b µν → ϕ), between an in-going 16 2-form particle with momentum p µ = (p, 2p, 0, · · · , 0) and polarization tensor ζ ij (p, s), and a scalar particle of the same momentum. The amplitude according to (3.46) is
M(b µν → ϕ) = −1 √ D − 2 ∆B ij ζ ij (p, s) . (3.48)
Two-form-graviton conversion amplitude. The time evolution operator (interaction Hamiltonian) is 49) and in terms of quantum operators it is
−i duH I (u) = i 4κ 2 d D xh αβ (x)H ανλ (x)(db) β νλ (x) ,(3.3 4 duH uij (u)× s,w d D−1 p 2p r ξ β [u (p, w)p i ζ j]β * (p, s) h w p b s † p − ξ β * [u (p, w)p i ζ j]β (p, s) h w † p b s p . (3.50)
The amplitude for incoming graviton of momentum p, to convert to two-form b is
M(h µν → b µν ) = −∆B ij ξ m[i (p, s)ζ j]m * (p, w). (3.51)
The above of course matches with our string theory computation and result (3.28). 16 Outgoing radial particles are beyond our scope. They move on constant u world-lines, where "time stops". In other words, the out-going field is only a function of u = t − r, whereas the in-going field is only a function of v = t + r = u + 2r. So, for the out-going particle p r = 0 and the conjugate momentum vanishes.
Open string memory effect
In the previous section we showed how the closed string memory effect was a consequence of invariance of the worldsheet or the corresponding low energy effective action, under Λ-gauge transformations, B → B + dΛ. For the open string case and due to the presence of worldsheet endpoints the Λ gauge symmetry should be modified by the addition of the boundary 1-form gauge field [32,33]. The action (3.1) augmented by the boundary term
S b = 2 2 s dτẊ I A I ,(4.1)
where I, J = 1, · · · , p denote the spatial directions along the D p -brane and A I is the U(1) gauge field on it, leads to the total open string world-sheet action. Variation of this action, and then fixing the light-cone gauge yields equations of motion (3.3) and a boundary term,
− 2 2 s dτ γ σb (δX I ∂ b X I ) + σb δX I (∂ b X J B IJ + B Iu ) + δX I d dτ A I , b = τ, σ . (4.2)
which results in the mixed (Neumann and Dirichlet) boundary conditions for directions along the brane [34,35] X I + F IJẊ J + s F Iu = 0 . In other words, the combination F ≡ B − dA (4.5)
is invariant under Λ-gauge transformation as well as under λ-gauge transformation, A → A+dλ.
As in the closed string case, in the light-cone gauge u = s τ by fixing the radial gauge B rµ = 0, we find
X I = x I 0 + 2α p I τ + w I σ + i α 2 n =0 1 n δ IJ + B IJ (u) 2 α J n e −in(τ −σ) + δ IJ − B IJ (u) 2 α J n e −in(τ +σ) (4.6)
Next, we impose the boundary condition (4.3): 17
(1 − F) IJ 1 + B 2 JL α L n = (1 + F) IJ 1 − B 2 JLα L n ,(4.7)
Since B IJ and F IJ have time dependence, (4.7) is in general not consistent with constant α i n andα i n (which is required by equations of motion). In a slowly varying and small 2-form B IJ background, this can, however, be remedied if
F IJ (u) − 1 2 B IJ (u) = C IJ ,(4.8)
where C is a constant 2-form. We can further choose C = 0 by making an appropriate Λ-gauge transformation on B such that, the boundary condition (4.3) yields α I n =α I n .
(4.9)
Eq.(4.3) also fixes the zero mode part, and the mode expansion becomes
X I (τ, σ) = x I 0 + 2α p I τ − 2α F IJ p J σ − 2α s F Iu σ + √ 2α n =0 e −inτ n i cos nσδ IJ − sin nσF IJ α J n , (4.10)
where the O(B 2 ) and higher order terms are neglected.
The analysis above and in particular (4.9) and (4.10) shows that the effects of adiabatically time-varying B-field on open string is completely different than on closed string. Specifically, the mode expansion coefficients α i n and the Hamiltonian of the system in the appropriate canonical frame are both time independent. This is essentially the same as the open string mode expansion in a constant B-field background [21-24, 33, 34]. The effects of the time-variation of the B-field may be seen in the center of mass motion of the strinḡ 11) in the last two terms. We will discuss this further in the following subsection.
X I (u) = 1 π π 0 dσ X I (τ, σ) = x I 0 + 2α p I τ − πα F IJ p J − πα s F Iu ,(4.
Quantum treatment
Having the mode expansion we can readily quantize the open string by imposing [21][22][23][24][25]
[x I 0 , x J 0 ] = iπα F IJ , [x I 0 , p J ] = iδ IJ , [α I n , α J m ] = nδ IJ δ m+n,0 . (4.12)
As we see the effects of the adiabatically changing background B-field has appeared only in the noncommutativity of the x i 0 coordinates. Note that x i 0 are basically the coordinates of the D p -brane the open string endpoint is attached to. Since the Hamiltonian is not affected by the background B-field in the open string case we do not have a memory effect in the usual sense. Nonetheless, open strings in the adiabatically changing B-field background behaves as an electric dipole [22], whose dipole moment is changing in time:
∆d I ≡ d I (u → +∞) − d I (u → −∞) = 2πα ∆B IJ p J ,(4.13)
where d I (u) = X I (σ = π, u) − X I (σ = 0, u) . This result can be also understood from the effective field theory of the open strings, which is the Born-Infled theory residing on the brane. In presence of an adiabatically changing B-field we are dealing with a noncommutative gauge theory [36,37] with a slowly varying noncommutativity parameter (see also [28] for a related analysis). As it is known in the noncommutative field theories the kinetic term is not affected by noncommutativity, e.g. see [38] and the effects of noncommutativity appear only in interaction terms which is not captured in the usual memory effect described in previous sections.
D-brane probes and boundary states
D-barnes are part of the spectrum of the string theory. They appear by requiring T-duality in theories of open strings [39,40] by the fact that under T-duality, Neumann and Dirichlet boundary conditions are interchanged. Besides this open string description [41], it is known that D-branes can be represented as a coherent (bound) state of closed strings they can emit. The amplitude of the closed string emission is given through the boundary state [26], for a review of constructing these states see [42,43] and references therein.
The open string satisfies the boundary condition (4.3) at its end along the D-brane worldvolume and along the transverse directions we have X i (σ = 0) = x i 0 for i = p + 1, · · · , d − 1. Boundary state for D p -branes in a constant B-field background has also been worked out and studied [27,44]. One may generalize the analysis of [27] to adiabatically changing B-field background, which should satisfy 18
(∂ τ X µ + F µν ∂ σ X ν ) τ =τ 0 |B(τ 0 ) = 0 .
(4.14)
We note that the boundary state |B(τ 0 ) should satisfy the above condition at the given arbitrary time τ 0 , which in the light-cone gauge τ 0 = u 0 / s . That is, |B(τ 0 ) is giving the amplitude of closed string mode emissions from a D-brane at time τ 0 . In a static background like the cases analyzed in [27,44], the τ 0 dependence of the boundary state appears simply through e inτ 0 dependence of the string oscillator modes, while in our case there is an extra dependence due to the slowly varying background B field. Since the analysis is similar to the static case where F is constant, we skip the details of computation and quote the final result: where the braces denote higher orders in B or F. Having the D p -brane boundary state we can use it as the probe to explore the memory effect associated with passage of a B-field. As the first example, let us compute the action of the closed string memory S-matrix (3.23) on this state:
|B(τ 0 ) = exp − n=1 e 2inτ 0 n α µ −n D µν (τ 0 )α ν −n |0 , D µν = (Q αβ (τ 0 ), −δ AB ) .S|B = 1 + 1 2 ∞ n=1 e 2inτ 0 n α α −n (∆B) αβα β −n − α α −n (∆B) βαα β −n − (2∆B) αβ α α −nα β −n + · · · |B ,(4.17)
where braces denote terms second or higher order in B, F. We therefore have
S|B = |B + O(B 2 ) . (4.18)
The boundary state is unchanged at first order in background fields. This result may be understood as follows. The D p -brane in a slowly-varying B-field background is a non-marginal bound 18 Here we prefer covariance and do not to go to the light-cone gauge.
state of D p and lower dimensional D (p−2n) -branes where n rank of the B field along the brane, much like the constant B-field case [34]. The mass density of the brane is 1
gs det (1 + F) 1 gs (1 + O(F 2 ))
where g s is the string coupling. This is compatible with the softness of the passing B-field wave. The (p − 2n)-form RR charge density of carried by the bound state is proportional to F n . Therefore, in our approximation only n = 1 case is remaining. The change in this D (p−2) -brane charge density is then proportional to ∆B. We note, however, that to see this RR charge density from the boundary state one needs to go beyond the bosonic sector discussed above, and to consider the superstring case and include fermionic degrees of freedom. Moreover, using the boundary state (4.15) one can study scattering of two such D p -branes off each other, where the |in and |out boundary states differ in their value of the F field.
Discussion
In this work, we continued our analysis of the p-form soft charges focusing on the 2-form case and studied how these charges can be probed. In particular, we considered the natural probes of a 2-form, the strings, and studied the 2-form memory imprinted on them due to the passage of a 2-form wave-packet. The string memory effect we analyzed has some particular features not shared by previously studied memory effects: the memory is encoded in an internal excitation mode of the probe, here the strings, and that the probe itself can be a massless object. 19 We discussed string memory effect for closed and open strings and mentioned that the latter can also be analyzed through D-brane probes, using boundary state formulation. In our analysis here, we demonstrated how the basic idea works for bosonic strings and of course we expect the same analysis to be worked through, almost verbatim, for superstring case, either in RNS or GS formulations [45].
While the p-form soft charge analysis of [1] was carried out for 2p + 4 dimensional theories, our analysis here was made for strings in generic dimension D. The six dimensional strings (for p = 2 case) may then be viewed as a critical 10 dimensional superstring compactified on a 4d manifold. In this case, the strings may have winding modes and the B-field may have legs along the compact direction. It is interesting to check how these may affect/appear in the string memory.
As discussed in [1], there are two classes of p+1-form charges for p ≥ 1, the coexact charges and the exact soft charges. The former is a direct generalization of the p = 0 electromagnetic soft charges case and as we discussed here these are the charges relevant for the closed string memory effect. To discuss relevance of exact soft charges to the memory effect, we recall that the exact configurations B = dC have vanishing field strength but can in principle be probed by open strings where the worldsheet has a boundary. In the case that open strings end on D-branes, cf. section 4, the brane may be viewed as the "boundary of the spacetime" and the C field which may be viewed as the edge-mode associated to the two-form, can be identified with the gauge field on the brane. The corresponding memory effect is then expected to essentially reduce to the electromagnetic memory effect on the brane.
Our discussions here can be readily extended to generic p-form memory using (p − 1)-brane probes. Again we expect to have a similar memory effect sharing the features of string memory effect. This p-brane memory effect can then be of relevance for black hole information problem within the membrane paradigm viewpoint [46].
Here and in [1] we studied two corners of the p-form "IR-triangle" [6]. It is desirable to complete this by studying (1) more direct relation of p-form soft charges of [1] and the memory effect and, (2) study the p-form soft theorems and connecting it to the other two corners. For some relevant analysis especially within string theory framework see [47][48][49][50][51]. As general comment on soft theorems we note that for any observable X on the celestial sphere, the difference in its early and late retarded time values ∆X is given by This shows that requiring a nontrivial ∆X implies a nontrivial simple pole inX.
Equations of motion for transverse momenta arë
p i + ω 2p i = qȦ i (A.3)
The homogeneous solution isp i (u) = β(u) + c.c. where β(u) =βe iωu andβ is a fixed complex number. Suppose that we are interested in the solution in interval u ∈ [−L, L]. Divide the interval into N segments and define u n = L(2n/N − 1). Evolution of the homogeneous solution is given by the phase shift β(u n+1 ) = U N β(u n ), where U N = exp(2iLω/N ). The first term is the free evolution of a neutral oscillator and the second term decays as 1/L. The zero-frequency mode (L → ∞) of the radiation gives no contribution to the time evolution of a charged oscillator. In other words, the momentum kick is averaged out. We will show below that if the quantum harmonic oscillator with Hamiltonian (A.8) is prepared in a energy eigenstate |in = |n at u → −∞, then its state at u → +∞ is given by |out =Ŝ|in ,Ŝ = 1 (A.9)
The free spectrum (when q = 0) consists of states |n = |n 1 , n 2 , n 3 with energy E n = ω(n 1 + n 2 + n 3 + 3/2) and n a ≥ 0. Let |n evolve with unperturbed Hamiltonian (in the interaction picture). The Schrödinger equation for an arbitrary state can be easily seen to take the following form i d du n|Ψ = In the adiabatic approximation, whereȦ ωA, the quantity above can be nonvanishing only if E n = E m . The position operator, however, has no diagonal element in energy eigen-state representation, n|qx i |m = 0 and n|Ψ is constant in time. Therefore, the zero-frequencymode of radiation has no effect on a harmonic oscillator, in contrast to the free particle.
the classical result (3.11). Eqs.(3.24) are our main result of this section. To explore it further, consider the general massless state in bosonic string theory
Figure 2 .
2Kinetic mixing vertices.
3.35) and(3.36) comprise 2D conditions on graviton and 2D − 3 conditions on the 2-form field to give the right degrees of freedom.
ab (p)ζ s * cd (p) = δ a[c δ d]b − 2p [a δ b][d p c] /|p| 2 , s ξ ab (p, s)ξ s * cd (p, s) = δ a(c δ d)b − 2p (a δ b)(d p c) /|p| 2 .
action introduces a coupling of the boundary ∂Σ of the open string to the gauge field on the D-brane which is invariant under the new Λ-gauge transformations, δB IJ = 2∂ [I Λ J] , δA I = Λ I (4.4)
, β label all directions parallel to the brane, while those normal to the brane are labelled by A, B. The matrix Q(τ 0 ) is defined as Q(τ 0 ) = (1 − F(τ 0 ) + B(τ 0 )/2) −1 (1 + F(τ 0 ) − B(τ 0 )/2) = 1 + 2F(τ 0 ) − B(τ 0 ) + · · · , (4.16)
Suppose
A i has a small shift at each step, such thatȦi = i δ(u − u n )/N , where i = q (A + i − A − i )/ √ mω.Since the momentum shift in each interval is at the following recursive formulaβ(u n+1 ) = U N β(u n ) − i i N → ∞limit the denominator becomes 1 − U N → −2iωL/N and β(L) = exp(2iωL)β(−L) + i 2ωL (1 − e 2iωL ) . (A.7)
x)|m e −iu(En−Em) m|Ψ . (A.10)
Quantum treatment. Consider Hamiltonian (2.24) augmented by the harmonic oscillator potential termK=p
ipi +p 2
3
2m
+
1
2
mω 2x2
a + qx iȦ
i (x).
(A.8)
Note that in the internal string memory effect the probe can also be one of the massive modes of string excitations.
This means that we are neglecting large components of the field in a spherical harmonics expansion.
This is typically the case when F(u) is a radiative field which is strong only in a finite duration.4 In order to capture the physics of radiation at null infinity we expand all quantities in inverse powers of r in (1.1) with expansion coefficients which are functions of retarded time u and Cartesian coordinates X i .
Variation of the action (2.5) with respect to η givesẊ µẊ µ + η 2 m 2 = 0 as a constraint.
The function G as appears in (2.14) does not generate the transformation. Different kinds of generating functions are obtained only by adding certain combinations of old and new coordinates and momenta (like XP ) to G.
More precisely: spacelike infinity (i 0 ) is the destination of all spacelike radial curves. I + − is a boundary of i 0 where the spacelike curves tend to be null. The same statement holds for timelike infinity (i + ) and I + + .
Note the I + is defined at r → ∞ with arbitrary u, and I + ± is defined as r → ∞ and then u → ±∞.
s (X i X i ) .(3.14)11 Quantum operators are not hatted in this section.
Here we considered the case with both ends of the open string with the same boundary conditions. In principle one can consider cases where the two ends have different Neumann, Dirichlet or mixed boundary conditions, e.g. as in[34].
While in this work we focused on the internal string memory effect, we note that the center of mass of massive string modes can exhibit the same external memory effects as usual particle probes under gravitational waves can have, e.g. the displacement[28] or super-boost memories.
AcknowledgmentsWe would like to thank Paolo Di Vecchia, Vahid Hosseinzadeh, Reza Javadinezhad, Massimo Porrati, Hesam Soltanpanahi and Alexander Zhiboedov for fruitful discussions. H.A. and M.M.Sh-J. are partially supported by the junior research chair in black hole physics of Iranian NSF, project no 951024. We also acknowledge the ICTP NT-04 network scheme. H.A. would like to gratefully acknowledge the support of the CERN Theory Department and the Iran-Austria IMPULSE project grant during his stay at CERN and TU Wien while this work was being prepared.A Absence of electromagnetic memory for harmonic oscillator probes As we reviewed in section 2, adiabatically changing background electric field is encoded as the change in momentum of a charged free particle probing the background, the electromagnetic kick memory. The kick on the particle is accumulated over the particle trajectory from far in the past to far in future. One may wonder whether this momentum shift accumulation also happens for a probe which has perioidic trajectories, like a harmonic oscillator. As we show below the answer is negative.Classical treatment. The classical Hamiltonian of a charged harmonic oscillator in the radial gauge A r = 0 is,where H 0 is the Hamiltonian for the free charged particle(2.11). We now want to show that the zero-frequency mode of electromagnetic radiation leaves no memory on a charged harmonic oscillator. By
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| []
|
[
"THE BISHOP-PHELPS-BOLLOBÁS THEOREM ON BOUNDED CLOSED CONVEX SETS",
"THE BISHOP-PHELPS-BOLLOBÁS THEOREM ON BOUNDED CLOSED CONVEX SETS"
]
| [
"Hoon Dong ",
"Yun Sung Cho ",
"Choi "
]
| []
| []
| This paper deals with the Bishop-Phelps-Bollobás property (BPBp for short) on bounded closed convex subsets of a Banach space X, not just on its closed unit ball BX . We firstly prove that the BPBp holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces X and Y the pair (X, Y ) has the BPBp on every bounded closed convex subset D of X, and also that for a Banach space Y with property (β) the pair (X, Y ) has the BPBp on every bounded closed absolutely convex subset D of an arbitrary Banach space X. For a bounded closed absorbing convex subset D of X with positive modulus convexity we get that the pair (X, Y ) has the BPBp on D for every Banach space Y . We further obtain that for an Asplund space X and for a locally compact Hausdorff L, the pair (X, C0(L)) has the BPBp on every bounded closed absolutely convex subset D of X. Finally we study the stability of the BPBp on a bounded closed convex set for the 1-sum or ∞-sum of a family of Banach spaces. | null | [
"https://arxiv.org/pdf/1409.3008v1.pdf"
]
| 119,667,590 | 1409.3008 | b6355aa8fc038ab24c330168a029401b6fe85615 |
THE BISHOP-PHELPS-BOLLOBÁS THEOREM ON BOUNDED CLOSED CONVEX SETS
Hoon Dong
Yun Sung Cho
Choi
THE BISHOP-PHELPS-BOLLOBÁS THEOREM ON BOUNDED CLOSED CONVEX SETS
This paper deals with the Bishop-Phelps-Bollobás property (BPBp for short) on bounded closed convex subsets of a Banach space X, not just on its closed unit ball BX . We firstly prove that the BPBp holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces X and Y the pair (X, Y ) has the BPBp on every bounded closed convex subset D of X, and also that for a Banach space Y with property (β) the pair (X, Y ) has the BPBp on every bounded closed absolutely convex subset D of an arbitrary Banach space X. For a bounded closed absorbing convex subset D of X with positive modulus convexity we get that the pair (X, Y ) has the BPBp on D for every Banach space Y . We further obtain that for an Asplund space X and for a locally compact Hausdorff L, the pair (X, C0(L)) has the BPBp on every bounded closed absolutely convex subset D of X. Finally we study the stability of the BPBp on a bounded closed convex set for the 1-sum or ∞-sum of a family of Banach spaces.
Introduction
A remarkable result so called the Bishop-Phelps theorem [8] came out in 1961, which states that for every Banach space X, every linear functional on X can be approximated by norm attaining ones. In fact, they showed a more general results: Let D be a closed bounded convex subset of a real Banach space X. Then the set of support functionals of D is a norm dense subset of its dual space X * . In other words, the set of all elements of X * that attain their suprema on D is a norm dense subset of X * . However, Lomonosov [20] showed in 2000 that this statement cannot be extended to general complex spaces by constructing a closed bounded convex set with no support points. From now on, we assume that X and Y are real Banach spaces without any other comment.
After a while, J. Lindenstrauss [19] studied in 1963 the denseness of norm attaining linear operators between Banach spaces, which has been a classical research topic in functional analysis since then. In particular, Bourgain [10] obtained in 1976 such a surprising results that a Banach space X has the Bishop-Phelps property if and only if it has the Radon-Nikodym property(RNP for short). We recall that a Banach space X is said to have Bishop-Phelps property if for every bounded closed and absolutely convex subset D of X and for every Banach space Y , the subset of L(X, Y ) attaining their suprema in norm on D is dense in the space L(X, Y ), where L(X, Y ) is the Banach space of bounded linear operators from X into Y .
In 1977 Stegall [23] obtained a nonlinear form of Bourgain's result: Let X be a Banach space with RNP, D be a bounded closed convex subset of X and f : D → R be an upper semicontinuous bounded above function. Then for > 0, there exists x * ∈ X * such that x * < and f + x * , f +|x * | strongly expose D. Applying this result to a vector-valued case, he showed the following. Let X be a Banach space with RNP, D be a bounded closed convex subset of X, and Y be a Banach space. Suppose that ϕ : D → Y is a uniformly bounded function such that the function x → ϕ(x) is upper semicontinuous. Then, for δ > 0, there exist T : X → Y a bounded linear operator of rank one, T < δ such that ϕ + T attains its supremum in norm on D and does so at most two points.
We refer to [1] surveying most of recent results on the denseness of norm attaining linear or nonlinear mappings such as multilinear mappings, polynomials or holomorphic mappings.
On the other hand Bollobás [9] sharpened in 1970 the Bishop-Phelps theorem by dealing simultaneously with norm attaining linear functionals and their norming points, which is stated as follows. We denote by B X and S X the closed unit ball and sphere of X, respectively. Theorem 1.1. [9] Let X be a Banach space and 0 < < 1. Given x ∈ S X and x * ∈ S X * with |1 − x * (x)| < 2 2 , there are elements y ∈ S X and y * ∈ S X * such that y * (y) = 1, x − y < , and y * − x * < + 2 .
He also showed that this theorem is best possible in the following sense. For any 0 < < 1 there exist a Banach space X, point x ∈ S X and functional f ∈ S X * such that f (x) = 1 − ( 2 /2), but if y ∈ S X , g ∈ S X * and g(y) = 1, then either f − g or x − y .
Since this theorem of Bollobás is stated explicitly, we have referred it more often than the theorem of Brønsted and Rockafellar [11], a more general and earlier result than Bollobás. Using the concept of the subdifferential of a convex function it is written as follows: Suppose that f is a convex proper lower semicontinuous function on a Banach space X. Then given any point
x 0 ∈ dom(f ), > 0, λ > 0 and any x * 0 ∈ ∂ f (x 0 ), there exist x ∈ dom(f ) and x * ∈ X * such that x * ∈ ∂(f ), x − x 0 λ , and, x * − x * 0 λ.
In particular, the domain of ∂f is dense in dom(f ).
Acosta et al. [2] introduced in 2008 the following definition to study this property for linear operators between Banach spaces.
Definition 1.2. [2]
A pair of Banach spaces (X, Y ) is said to have the Bishop-Phelps-Bollobás property (BPBp for short) if for every > 0 there are 0 < η( ) < 1 and β( ) > 0 with lim →0 β( ) = 0 such that for all T ∈ S L(X,Y ) and x 0 ∈ S X satisfying T (x 0 ) > 1 − η( ), there exist a point u ∈ S X and an operator S ∈ S L(X,Y ) that satisfy the following conditions:
Su 0 = 1, u 0 − x 0 < β( ), and T − S < .
Since they characterized in [2] the Banach space Y for which the BPBp holds for operators from 1 into Y , lots of interest has been caused in this property (for instance see [3,4,5,6,7,14,17,18] So far, the BPBp has been studied on the closed unit ball B X , but in this paper we deal with this property on bounded closed convex subsets D of a Banach space X, not just on B X . We introduce the following more general definition. Let
T D = sup{ T x : x ∈ D} for T ∈ L(X, Y ).
Definition 1.3. Let X and Y be Banach spaces. Let D be a bounded closed convex subset of X. We say that (X, Y ) has the Bishop-Phelps-Bollobás property on D (BPBp on D for short) if for every > 0, there is η D ( ) > 0 such that for every T ∈ L(X, Y ), T D = 1 and every x ∈ D satisfying
T (x) > 1 − η D ( ), there exist S ∈ L(X, Y ) and z ∈ D such that S(z) = 1 = S D ,
x − z < and T − S < .
Similarly we say that (X, Y ) has the Bishop-Phelps property on D (BPp on D for short) if for every > 0 and for every T ∈ L(X, Y ) with T D = 1, then there exist S ∈ L(X, Y ) and z ∈ D such that S(z) = 1 = S D and T − S < .
In general, we cannot expect the same results in the BPBp on a closed bounded convex set D as those on B X . For a uniformly convex space X the pair (X, Y ) has the BPBp on B X for every Banach space Y ( [6,18]). However, there is a Banach space Y such that ( 2 2 , Y ) fails to have the BPBp on D = B 2 1 , even though 2 2 is a uniformly convex space of dimension 2. We can actually show this fact by just considering the bounded operators T k defined on 2 1 in [3, Example 4.1] as those on 2 2 . In fact, for k ∈ N, consider Y k = R 2 with the norm (x, y) = max |x|, |y| + 1 k |x| ,
and Y = [ ∞ k=1 Y k ] ∞ . Define T k ∈ L( 2 2 , Y k ) by T k (e 1 ) = −1, 1 − 1 k and T k (e 2 ) = 1, 1 − 1 k . Since B 2 1 = D ⊂ B 2 2 , we have T k D = 1 for all k ∈ N.
It follows from the same argument as in [3,Example 4.1] with Theorem 3.2 and Proposition 4.3 that ( 2 2 , Y ) fails to have the BPBp on D = B 2 1 . In Section 2, we show that the BPBp holds for bounded linear functionals on arbitrary bounded closed convex sets. Using this, we sharpen Stegall's optimization principles [23] in the sense of the BPBp. In Section 3, we show that for all finite dimensional Banach spaces X and Y the pair (X, Y ) has the BPBp on every bounded closed convex subset D of X, and also that for a Banach space Y with property (β) the pair (X, Y ) has the BPBp on an every bounded closed absolutely convex subset D of an arbitrary Banach space X. For a bounded closed convex subset D of X with positive modulus convexity we get that the pair (X, Y ) has the BPBp on D for every Banach space Y . We further prove that for an Asplund space X and for a locally compact Hausdorff L, the pair (X, C 0 (L)) has the BPBp on every bounded closed absolutely convex subset D of X. In Section 4, we study the stability of the BPBp for the 1 -sum or ∞ -sum of a family of Banach spaces.
Linear functionals attaining their suprema on bounded closed convex sets
We begin by recalling Ekeland's variational principle [13,15], which can be stated as follows: (Ekeland). Let f : X → R ∪ {∞} be a proper lower semicontinuous and bounded below function on a Banach space X. Then given > 0 and δ > 0, there exists
Theorem 2.1x 1 ∈ X such that f (x 1 ) < f (x) + x − x 1 for every x ∈ X with x = x 1 . Moreover if f (x 0 ) < b + δ 2 , where b = inf {f (x) : x ∈ X}, then x 1 can be chosen so that x 0 − x 1 < δ .
The proof of Theorem 7.41 in [13] actually gives the following:
Theorem 2.2. Let D be a bounded convex closed subset of a Banach space X. Given > 0 and δ > 0, if f ∈ X * and x 0 ∈ D such that f (x 0 ) > sup{f (x) : x ∈ D} − δ 2 ,
then there exist g ∈ X * and x 1 ∈ D satisfying
g(x 1 ) = sup{g(x) : x ∈ D}, f − g and x 1 − x 0 δ .
It is trivial that Theorem 2.2 is not true any more for a complex Banach space ( [20]). Corollary 2.3. Let > 0 be given. If f ∈ S X * and x 0 ∈ S X satisfy that
|1 − f (x 0 )| < 2 4 ,
then there exist g ∈ S X * and z ∈ S X such that g(z) = 1, f − g and x 0 − z .
Proof. Apply Theorem 2.2 with δ = 2 2 and = 2 and we can choose x * 1 2 and z ∈ S X so that f + x * attains its norm at z.
Set g = (f + x * )/ f + x * . Then f − g f − (f + x * ) + (f + x * ) − g 1 2 + f + x * · 1 − 1 f + x * 1 2 + | f + x * − 1| 1 2 + 1 2 = .
Also we get |g(z)| = 1 and x 0 − z
2 2 / 2 = .
We can also obtain the following theorem for a bounded linear functional, which is analogous to Stegall's nonlinear form [23] of Bourgain's result mentioned in Introduction.
Theorem 2.4. Let D be a bounded closed convex set in a Banach space X. Given 0 < < 1/4 and f ∈ X * , there exist x * ∈ X * and x 0 ∈ D such that both f + x * and f + |x * | attain their suprema simultaneously at x 0 and x *
.
Moreover (f + x * )(x 0 ) = (f + |x * |)(x 0 ).
Proof. We may assume D ⊂ B X and f D = 1. By the Bishop-Phelps theorem, there exists x * ∈ X * such that f + x * attains its supremum at
x 0 ∈ D and x * 2 . If f (x 0 ) + x * (x 0 ) f (x) + |x * (x)| for every x ∈ D, we are done. Otherwise, there exists y ∈ D such that f (y) + |x * (y)| > f (x 0 ) + x * (x 0 ). Clearly x * (y) < 0, and f (y) − x * (y) > f (x 0 ) + x * (x 0 ). Let s = sup x∈D {f (x) − x * (x)} and α = s − (f (x 0 ) + x * (x 0 )) < (1 + 2 ) − (1 − 2 ) = . Choose y 0 ∈ D so that f (y 0 ) − x * (y 0 ) > s − α 2 2 2 . By Theorem 2.2, there exists x * 1 such that (f − x * ) + x * 1 attains its supremum at z 0 ∈ D, x * 1 α and y 0 − z 0 α . Then, f (z 0 ) − x * (z 0 ) + x * 1 (z 0 ) f (y 0 ) − x * (y 0 ) − x * 1 z 0 − y 0 + x * 1 (z 0 ) > s − 3α 2 2 2 − α > f (x 0 ) + x * (x 0 ) + α ,
where the last inequality follows from the definition of α and 0
< α < < 1/4. Set x * 2 = −x * +x * 1 . Clearly, x * 2 . We can see that f + |x * 2 | also attains its supremum f (z 0 ) + z * 2 (z 0 ) at z 0 on D. Otherwise, there exists w ∈ D such that f (w) + |x * 2 |(w) > f (z 0 ) + x * 2 (z 0 ), which implies that x * 2 (w) < 0. Therefore, we have f (w) − x * 2 (w) > f (z 0 ) + x * 2 (z 0 ) > f (x 0 ) + x * (x 0 ) + α f (w) + x * (w) + x * 1 , which implies that −x * 1 (w) > x * 1 . This contradiction shows that f (z 0 ) + x * 2 (z 0 ) = f (z 0 ) + |x * 2 (z 0 )| f (x) + |x * 2 (x)| for every x ∈ D.
By the same argument as in the proof of Theorem 2.2 we can obtain the following: If f (x 0 ) < inf{f (x) : x :∈ D} + δ 2 , then there exist g ∈ X * and x 1 ∈ D such that
g(x 1 ) = inf{g(x) : x ∈ D}, f − g and x 1 − x 0 δ .
Further, we can show in the following that for a bounded closed convex set D the set {f : |f | attains its supremum on D} is dense in X * .
Theorem 2.5. Let D be a bounded closed convex set in a Banach space X. Given f ∈ X * and > 0, there exists x * ∈ X * such that |f + x * | attains its supremum on D and x * . Moreover, if D is symmetric, and f (x 0 ) > f D − δ 2 for some x 0 ∈ D and δ > 0, then x * and x 1 ∈ D can be chosen so that x * , x 0 − x 1 δ , and |f + x * | attains its supremum at x 1 on D.
Proof. We may assume that D is a bounded closed convex subset of B X . Let s = sup D f . We now consider three cases.
1 • . Suppose that s > | inf D f |. Set η = s − | inf D f | > 0. By Theorem 2.4, we can choose x * min{ η 2 , } so that both (f + x * ) and (f + |x * |) attain their suprema at x 0 ∈ D and (f + x * )(x 0 ) = (f + |x * |)(x 0 ). Since f (x 0 ) + x * (x 0 ) f (x) + |x * (x)| for every x ∈ D, we have f (x 0 ) + x * (x 0 ) s. Therefore, for every x ∈ D we have −f (x) − x * (x) −f (x) + |x * (x)| s − η + η 2 < s f (x 0 ) + x * (x 0 ), which implies that |f (x) + x * (x)| f (x 0 ) + x * (x 0 ) = |(f + x * )(x 0 )| for every x ∈ D. 2 • . Suppose that s = | inf D f |. By Theorem 2.4, we can choose x * so that f + x * attains its supremum at x 0 ∈ D and x * 2 . If |f +x * |(x) f (x 0 )+x * (x 0 ) for every x ∈ D, we are done. Otherwise, there exists y ∈ D such that |f + x * |(y) > f (x 0 ) + x * (x 0 ). Clearly, (f + x * )(y) < 0 and −f (y) − x * (y) > f (x 0 ) + x * (x 0 ) f (x) + x * (x) for every x ∈ D. Set g = −f − x * . Then we have g(y) > −g(x 0 ) −g(x)
for every x ∈ D, which means that sup D g > | inf D g|. It follows from the case 1 • that there exist y * ∈ X * and y 0 ∈ D such that y * 2 and |g + y * |(x) (g + y * )(y 0 ) for every x ∈ D. Therefore,
|f + (x * − y * )|(x) = |g + y * |(x) (g + y * )(y 0 ) = (−f − x * + y * )(y 0 ) |f + (x * − y * )|(y 0 ).
for every x ∈ D and x * − y *
1 2 + 1 2 . 3 • Suppose that sup D f < | inf D f |.
We can prove this case by applying the case 1 • to (−f ).
Further, if D is symmetric, we note that f D = sup D f = | inf D f |.
From the assumption and Theorem 2.2, we can choose x * ∈ X * and x 1 ∈ D so that
sup D (f + x * ) = (f + x * )(x 1 ), x * and x 1 − x 0 δ . Since (f + x * )(x 1 ) = f + x * D , it means that |f + x * | attains its supremum at x 1 on D.
If D is not symmetric in the above theorem, we can hardly choose x 1 ∈ D satisfying x 0 −x 1 δ and also that |f + x * | attains its supremum at x 1 on D. Indeed, for f ∈ S X * which does not attain its norm and for 0 < < 1 3 , we let
S 1 = {x ∈ B X : f (x) 1 − 2 }, T = B X ∩ ker f, and S 2 = (−S 1 + T ).
Note that S 2 is a closed bounded convex subset of 2B X . We set D = co(S 1 ∪ S 2 ). Clearly, f D = 1 and we can see easily that there exists
x 0 ∈ D ∩ S X such that f (x 0 ) > 1 − 2 2
. We claim that for every x * the function |f + x * | cannot attain its supremum on D at
any point z ∈ D with z − x 0 . Otherwise, there exists x * such that the function |f + x * | attains its supremum on D at some point z ∈ D with z − x 0 . Since (f + x * )(z) = (f + x * )(x 0 ) + (f + x * )(z − x 0 ) 1 − 2 2 − − (1 + ) > 0, we can deduce that (f + x * )(z) = |(f + x * )(z)| is sup D (f + x * )
. Choose a sequence {z n } in co(S 1 ∪ S 2 ) converging to z. For each n ∈ N we can write z n = (1 − λ n )x n + λ n y n , where x n ∈ S 1 , y n ∈ S 2 and 0 λ n ≤ 1. An easy computation shows that
(f + x * )(z) 1 − 2 − 3 2 2 and (f + x * )(y n ) −1 + 2 + 2 .
It follows from these inequalities that for each n ∈ N
(1 + ) z − z n > |(f + x * )(z − z n )| = |(f + x * )(z) − (f + x * )((1 − λ n )x n + λ n y n )| = (1 − λ n ) ((f + x * )(z) − (f + x * )(x n )) + λ n ((f + x * )(z) − (f + x * )(y n )) λ n (1 − 2 − 3 2 2 ) − (−1 + 2 + 2 ) = λ n (2 − 4 − 5 2 2 ) 0.
Therefore, we have that for each n
0 λ n (1 + ) z − z n (2 − 4 − 5 2 2 )
, which implies that λ n → 0 as n → ∞. It means that z ∈ S 1 .
Now we recall that f + x * attains its supremum on D, but f doesn't attain its supremum on D because sup B X f = sup D f and f doesn't attain its norm. Hence x * = αf for any α ∈ R, and there exists w ∈ T with x * (w) < 0 and w < .
Since −z + w ∈ (−S 1 + T ) ⊂ D, we obtain sup D |f + x * | |(f + x * )(−z + w)| > (f + x * )(z) = |(f + x * )(z)|,
which is a contradiction.
Operators attaining their suprema in norm on bounded closed convex sets
It was shown in [2] that for all Banach spaces X and Y of finite dimension (X, Y ) has the BPBp for B X . We show that it is still true for arbitrary bounded closed convex subsets which are not necessarily symmetric. Proof. Otherwise, there exists a bounded closed convex subset D in X satisfying following condition: For some 0 , we can find T n ∈ L(X, Y ) such that for every T ∈ L(X, Y ) with
T n − T 0 , there is x T n ∈ D satisfying T n (x T n ) > T n D − 1 n and dist(x T n , N A D (T )) 0 , where N A D (T ) = {z ∈ D : T (z) = T D }.
By finite dimensionality, we can choose T n converging to T 0 ∈ L(X, Y ) and x T 0 n converging to x 0 ∈ D. Then we can easily show that
T 0 (x 0 ) = T 0 D and x T 0 n − x 0 0 , which contradicts x T 0 n − x 0 → 0.
J. Lindenstrauss [19] introduced the notion of property β: A Banach space Y is called to have property β if there is 0
λ < 1 and a family {(y α , f α ) ∈ B Y × B * Y } such that (i) f α (x α ) = x α = 1 (ii) |f α (y β )| λ for α = β (iii) y = sup α |f α (y)|.
He showed that if a Banach space Y has property (β), then the set of all norm attaining operators from X into Y is dense in L(X, Y ) for every Banach space X. In 1982, J. Partington [21] proved rather a surprising result that every Banach space can be renormed to have property (β). Acosta et al. [2] showed that if Y has property (β), then (X, Y ) has the BPBp on B X for every Banach space X. Now we prove that it is still true for bounded closed convex subsets of X.
Theorem 3.2. Let Y be a Banach space with property (β) and D be a bounded closed convex set in X. Then (X, Y ) has the BPp on D. Moreover, if D is symmetric, then (X, Y ) has the BPBp on D.
Proof. Without loss of generality we may assume D ⊆ B X , and first consider the case where D is symmetric. Let
0 < < 1 − λ 2 + λ and (2 + λ) 1 − 2 − λ − λ η 2 (2 + λ) 1 − 2 − λ − λ . Assume that T ∈ L(X, Y ), T D = 1, T = M and T x 0 > 1 − 2 2 for some x 0 ∈ D. Then we can choose α 0 so that |(T * f α 0 )(x 0 )| = |f α 0 (T x 0 )| > 1 − 2 2
. By Theorem 2.5, there exist g ∈ X * and z 0 ∈ D such that
|g(z 0 )| = g D , g − T * f α 0 , and x 0 − z 0 . Since g − T * f α 0 D g − T * f α 0 , we have 1 − 2 1 − 2 2 − g D 1 + . Define T 0 ∈ L(X, Y ) by T 0 (x) = T (x) + ((1 + η)g(x) − T * f α 0 (x))y α 0 . Clearly T * 0 f α 0 D = (1 + η) g D (1 + η)(1 − 2 ). For α = α 0 , T * 0 f α D T * f α D + λ( g − T * f α 0 D + η g D ) 1 + λ( + η(1 + )) (1 + η)(1 − 2 ),
where the last inequality follows from η (λ+2)
1−2 −λ−λ . Since T * 0 f α D T * 0 f α 0 D for all α = α 0 , we have T 0 D = sup α { T * 0 f α D } = T * 0 f α 0 D = f α 0 (T 0 z 0 ) T 0 z 0 T 0 D .
Therefore, T 0 attains its supremum at z 0 ∈ D with x 0 − z 0 and
T − T 0 g − T * f α 0 + η g + η(M + ) 1 + 2(2 + λ)(M + ) 1 − 2 − λ − λ .
For a bounded closed convex subset D, by Theorem 2.5, we can choose g ∈ X * and z 0 ∈ D so that g D = |g(z 0 )| and g − T * f α 0 . The rest of proof follows similarly.
Recall that the modulus of convexity of a Banach space X is defined on B X by
δ( ) = inf 1 − x + y 2 : x, y ∈ B X , x − y .
We can naturally extend this notion for a bounded closed absorbing convex set D. We define δ D ( ) for 0 < < 1 by
δ D ( ) = inf 1 2 ρ D (x) + 1 2 ρ D (y) − ρ D x + y 2 : x, y ∈ D, ρ D (x − y) ,
where ρ D is the Minkowski functional of D, that is, ρ D (x) = inf{λ > 0 : x ∈ λD}.
In the following we get such a general result that for a bounded closed absorbing convex subset D of X with positive modulus convexity, the pair (X, Y ) has BPBp on D for every Banach space Y . Theorem 3.3. Let X and Y be (real or complex) Banach spaces and D be a bounded closed absorbing convex subset of B X such that δ D ( ) > 0 for every 0 < < 1 2 . If T ∈ S L(X,Y ) and x 1 ∈ D satisfy T x 1 > T D − 3 δ D ( ), for sufficiently small relatively to T D , then there exist S ∈ L(X, Y ) and z ∈ D such that
Sz = S D , S − T < 4 2 1− and x 1 − z ρ D (x 1 − z) < 1− . Proof. Let T 1 = T . Choose f 1 ∈ S Y * so that f 1 (T 1 x 1 ) = T 1 x 1 > T 1 D − 3 δ D ( ). Inductively choose {T k } ∞ k=2 , {x k } ∞ k=2 ⊆ D, and {f k } ∞ k=2 ⊆ S Y * satisfying T k (x) = T k−1 (x) + k f k−1 (T k−1 x)T k−1 x k−1 , T k x k > T k D − k+2 δ D ( k ), ρ D (x k ) = 1, f k−1 (T k−1 x k ) = |f k−1 (T k−1 x k )|, and f k (T k x k ) = T k x k . Since T k < 2 and T k+1 − T k 2 k+1 T k for every k, {T k } ∞ k=1 converges to S and T − S 4 2 1− . An upper bound of T k+1 D is T k+1 D < T k+1 x k+1 + k+3 δ D ( k+1 ) T k D + k+1 |f k (T k x k+1 )| · T k D + k+3 δ D ( k+1 ).
A lower bound is
T k+1 D T k+1 x k = T k x k · |1 + k+1 f k (T k x k )| > ( T k D − k+2 δ D ( k ))(1 + k+1 ( T k D − k+2 δ D ( k ))) > T k D + k+1 T k 2 D − 2 2k+3 δ D ( k ) T k D − k+2 δ D ( k )
. Combining these two bounds yields
|f k (T k x k+1 )| > T k D − 2 k+2 δ D ( k ) − δ D ( k ) + 2 δ D ( k+1 ) T k D . Since δ D ( k ) δ D ( k+1 ) and T x ρ D (x) T D for every x ∈ D, we have ρ D x k+1 + x k 2 T k D T k x k+1 + x k 2 1 2 Re(f k T k x k+1 + f k T k x k ) T k D − 1 2 2 k+2 δ D ( k ) + δ D ( k ) + 2 δ D ( k+1 ) T k D + k+2 δ D ( k ) T k D − δ D ( k ) 3 2 k+2 + + 2 2 T k D . It follows from T k − T < 4 2 1− < 4 , T D − 4 < T k D < T D + 4 ρ D x k+1 + x k 2 > 1 − δ D ( k ) 3 2 k+2 + + 2 2 T D −8 T D − 4 .
Since is sufficiently small relatively to T D , then we have
ρ D x k+1 + x k 2 > 1 − δ D ( k ) = 1 2 ρ D (x k+1 ) + 1 2 ρ D (x k ) − δ D ( k ), which implies that ρ D (x k+1 − x k ) k . Hence {x k } ∞ k=1 converges in norm to z ∈ D.
We can also see easily that
x 1 − z ρ D (x 1 − z) 1− , T − S 4 2 1− and Sz = S D .
Corollary 3.4 ([18])
. Let X be a uniformly convex Banach space and Y be a Banach space.
Then (X, Y ) has the BPBp on B X .
Remark 3.5.
It is easy to notice that the BPBp is an isometric property, but not isomorphic. On the other hand, the BPBp still holds on the image of an isomorphism in the following sense. Let Ψ be an isomorphism from a Banach space X into a Banach space Z. We can see that if (X, Y ) has the BPBp on B X , then (Ψ(X), Y ) has the BPBp on D = Ψ(B X ). Further, if Y is injective, then (Z, Y ) has the BPBp on D. We recall that a Banach space Z is called injective if for every Banach space X and for every subspace W of X, every operator from W into Z can be extended to an operator from X into Z preserving its norm.
A Banach space X is called an Asplund space if the set of all points of U where f is Fréchet differentiable is dense G δ -subset of U for every real-valued convex continuous function f defined on an open convex subset U ⊆ X. Equivalently every w * -compact subset of (X * , w * ) is ·fragmentable. Here we say a subset C of (X * , w * ) is · -fragmentable if for every nonempty bounded subset A ⊂ C and for every > 0, there is a nonempty w * -open neighborhood V ⊂ X * such that A ∩ V is nonempty and has · -diameter less than ( [13]). Recall that an operator T ∈ L(X, Y ) is called by an Asplund operator if it factors through an Asplund space. That is, there are an Asplund space Z and operators T 1 ∈ L(X, Z) and T 2 ∈ L(Z, Y ) such that T = T 2 • T 1 . For example, every weakly compact operator is an Asplund operator since it factors through a reflexive space, so that a rank one operator is an Asplund operator. We also note that the family of Asplund operators is an operator ideal, hence the sum of two Asplund operators or the composition of an operator with an Asplund operator is again an Asplund operator.
It was shown in [5] that the BPBp on B X holds for an Asplund operator from X into C 0 (L). We can extend this result to a symmetric bounded closed convex subset D ⊂ B X . Some modifications of [5, Lemma 2.3 and Theorem 2.4] are just needed, but we give the details for the sake of completeness. Lemma 3.6. Let D be a symmetric bounded closed convex subset of B X and T ∈ L(X, Y ) be an Asplund operator with T D = 1 and T M 1. If
T (x 0 ) > 1 − 2 4M ,
for some x 0 ∈ D, then for every norming set B ⊆ B Y * and 0 < M 2 , there exist x * ∈ X * , u 0 ∈ D and a w * -open neighborhood U in X * such that
|x * (u 0 )| = 1 = x * D , x 0 − u 0 < and z * − x * < 4 ,
for every z * ∈ U ∩ T * (B).
Proof. Since B is a norming set, we can choose y * 0 ∈ B such that
|y * (T x 0 )| = |T * y * 0 (x 0 )| > 1 − 2 4M .
Define a w * -open neighborhood in X * by
U 1 = z * ∈ X * : |z * (x 0 )| > 1 − 2 4M . Since T * y * 0 ∈ T * (B) ∩ U 1 , we get T * (B) ∩ U 1 = ∅. Since T * (B) is · -fragmentable, ([5]), we can find a w * -open neighborhood U 2 ⊂ X * such that U ∩ T * (B) = ∅ and diam(U ∩ T * (B)) < , where U = U 1 ∩ U 2 . Now fix z * 0 ∈ U ∩ T * (B). Write z * 0 = T * (w * 0 ) for some w * 0 ∈ B ⊂ B Y * .
We can see z * 0 M and
z * 0 D = sup x∈D |T * w * 0 (x)| = sup x∈D |w * 0 (T x)| T D 1, which implies that |z * 0 (x 0 )| > 1 − 2 4M z * 0 D − 2 4M
. By Theorem 2.5, we can choose x * ∈ X * and u 0 ∈ D such that
x * − z * 0 < 2M , x 0 − u 0 < and |x * (u 0 )| = x * D .
It also follows from an easy computation that
1 − M x * D 1 + 2M . Let k = 1 x * D . Clearly, 2M 2M + k M M − and kx * − z * 0 kx * − x * + x * − z * 0 (k − 1) x * + 2M M − M + 2M + 2M 2 + 2 + 2 3 ,
Since |kx * (u 0 )| = 1 from the choice of k, kx * satisfies the desired properties. Moreover,
kx * − z * 3 + = 4 ,
for every z * ∈ U ∩ T * (B).
Theorem 3.7. For a symmetric bounded closed convex subset D ⊆ B X and a locally compact Hausdorff space L, let T : X → C 0 (L) be an Asplund operator with T D = 1 and
T D = M 1. Given 0 < < M 2 , if x 0 ∈ D satisfies that T x 0 > 1 − 2 4M ,
then there exist an Asplund operator S : X → C 0 (L) and a point u 0 ∈ D such that
S D = Su 0 = 1, x 0 − u 0 < and T − S < 4 .
Proof. Define δ : L → C 0 (L) * by δ(s)(f ) = f (s) for f ∈ C 0 (L) and s ∈ L. It is easy to check that φ = T * • δ : L → X * is w * -continuous. Since {δ(s) : s ∈ L} is a norming set in B C * 0 (L) , we can find a w * -open neighborhood U and x * ∈ X * by Lemma 3.6. Here we have T (x)(s) = φ(s)(x) and T * (B) = φ(L). Since U ∩ T * (B) = ∅, there is s 0 ∈ L such that φ(s 0 ) ∈ U . Consider the set Define a linear operator S : X → C 0 (L) by
W = {s ∈ L : φ(s) ∈ U },S(x)(s) = f (s)x * (x) + (1 + f (s))T (x)(s).
Define S 2 ∈ L(C 0 (L), C 0 (L)) by S 2 (h) = (1 + f )h. Then S(x) = f · x * (x) + S 2 (T (x)), hence S is an Asplund operator. It follows easily that S D 1 and |S(u 0 )(s 0 )| = |y * (u 0 )| = 1, which shows that S attains its supremum at u 0 on D and u 0 −x 0 < . For an upper bound of T −S ,
T − S = sup x∈B X T x − Sx = sup x∈B X sup s∈L |f (s)| · |T (x)(s) − x * (x)| = sup s∈W sup x∈D |φ(s)(x) − x * (x)| sup s∈W φ(s) − x * 4 ,
where the last inequality is derived from φ(s) ∈ U ∩ T * (B) and Lemma 3.6. This completes the proof.
Corollary 3.8. For any (real) Asplund space X and any locally compact Hausdorff space L, the pair (X, C 0 (L)) has the BPBp on a symmetric bounded closed convex set D of X.
Stability of the Bishop-Phelps-Bollobás property on direct sums
In order to compare the function η( ) appearing in the definition of the BPBp for different pairs (X, Y ), the notion of η(X, Y )( ) was introduced in [3]. We now generalize it to a bounded closed convex subset D of B X . Definition 4.1. Let X and Y be (real or complex) Banach spaces. For a bounded closed convex subset D ⊆ B X and T ∈ L(X, Y ),
Π D (X, Y ) = {(x, T ) : x ∈ D, T (x) = T D = 1} η D (X, Y )( ) = inf{1 − T x : x ∈ D, T D = 1, dist ((x, T ), Π D (X, Y )) }, where dist ((x, T ), Π D (X, Y )) = inf{max{ x − y , T − S } : (y, S) ∈ Π D (X, Y )}.
It Let {X i : i ∈ I} and {Y j : j ∈ J} be families of Banach spaces, X = ( i∈I X i ) 1 and Y = ( j∈J Y j ) ∞ . Let E i and F j be the natural isometric embeddings of X i and Y j into X and Y , respectively and let P i and Q j be the canonical projections of norm one from X and Y onto
X i and Y j , respectively. For D ⊂ X we let D i = P i (D) X i for each i ∈ I.
Payá and Saleh [22] studied the denseness of norm attaining operators from the 1 -sum of domain space into the ∞ -sum of range spaces. Their methods in [22] were applied in studying the Bishop-Phelps-Bollobás property for operators on those spaces ( [3,12]). With some suitable condition on D, we have the following analogous results to [3, Theorem 2.1].
Proposition 4.2. Let X = ( i∈I X i ) 1 , Y = ( j∈J Y j ) ∞ and D be a bounded closed convex subset of B X . Suppose that D = co(∪E i D i ) ⊂ B X . If the pair (X, Y ) has the BPBp with η D ( ) on D, then the pair (X i , Y j ) has the BPBp on D i for every i ∈ I and for every j ∈ J. More precisely,
η D (X, Y )( ) η D i (X i , Y j )( ), (i ∈ I, j ∈ J).
Proof. Fix h ∈ I and k ∈ J. Suppose that T (x h ) > 1 − η D ( ) for T ∈ L(X h , Y k ), T D h = 1 and x h ∈ D h . Define an operator T = F k T P h ∈ L(X, Y ). It is easy to check that
T D = T D h , so that T (E h x h ) > 1 − η D ( ).
The assumption gives us (u, S) ∈ Π D (X, Y ) such that
S − T < and u − E h x h < . Define S = Q k SE h ∈ L(X h , Y k ). Then S − T S − T < .
For j = k, we have Q j T = 0 by the definition of T , which implies that
Q j S D = Q j S − Q j T D S − T .
Since the range of S is the ∞ -sum of Y j 's, we have Q k S D = S D = 1 = S(u) = Q k Su . It follows from the assumption D = co(∪E i D i ) that every u ∈ D can be written by
u = ∞ i=1 λ i E i u i , where u i ∈ D i and ∞ i=1 λ i 1. Indeed, choose v n ∈ co(∪E i D i ) converging to u. We can write v n = ∞ i=1 λ n i E i v n i (v n i ∈ D i ),
where v n i = 0 and λ n i = 0 except finitely many i's. Since E i D i ∩ E j D j = {0} for i = j, we have that λ n i v n i → P i u for every i as n → ∞. By the diagonal argument, up to a subsequence, there exists a sequence {λ i } such that λ n i → λ i 0 for every i as n → ∞. By the Fatou lemma we obtain that ∞ i=1 λ i=1 1. We can also see that there exists u i ∈ D i for every i such that v n i → u i as n → ∞ and
λ i u i = P i u, hence u = ∞ i=1 λ i E i u i . Since Q kS (u) = 1 and E i u i ∈ D for every i, we have that Q kS (E i u i ) = 1 for every i where λ i = 0 and also that ∞ i=1 λ i = 1. It follows fromT E i = 0 for i = h that Q kS E i T −S for i = h. Therefore, 1 = Q kS u λ h Q kS E h u h + i =h λ i λ i = 1, which implies that λ i = 0 for i = h, λ h = 1 and S(u h ) = Q kS E h u h = 1 = S D h . Further, u h − x h = P h (u − E h x h ) u − E h x h .
If we fix the domain space X, then the reverse inequality also holds for the ∞ -sum of range spaces.
Proposition 4.3. η D (X, Y ) = inf j∈J η D (X, Y j )
Proof. It is enough to prove that η D (X, Y ) inf j∈J η D (X, Y j ). Fix ∈ (0, 1). Let 0 α = inf j∈J η D (X, Y j )( ) < 1. For 0 < α < 1, suppose that T x 0 > 1−α for x 0 ∈ D and T ∈ L(X, Y ) with T D = 1. We can choose k ∈ J so that Q k T x 0 > 1 − α. Then there exist S k : X → Y k and u ∈ D such that S k u = S k D = 1, S k − Q k T < and x 0 − u < .
Define S : X → Y by S = j =k F j Q j T + F k S k .
It is easy to check that (u, S) ∈ Π D (X, Y ). Moreover
T − S = sup j∈J Q j (T − S) = Q k T − S k < , which means that η D (X, Y )( ) α.
We now consider the case where X is the ∞ -sum of domain spaces X i .
Proposition 4.4. Let X = [ i∈I X i ] ∞ .
Assume that D is a bounded closed convex subset of B X , D = Π i∈I D i and that there exists 0 > 0 such that λx i x i ∈ D i for every x i ∈ D i and for every 0 λ 0 . If the pair (X, Y ) has the BPBp on D with η( ), then the pair (X i , Y ) has the BPBp on D i with η( ) for every i ∈ I. More precisely,
η D i (X i , Y )( ) η D (X, Y )( ) for every i ∈ I. Proof. Fix h ∈ I. Suppose that T (x h ) > 1 − η( ) for some T ∈ L(X h , Y ) with T D h = 1 and x h ∈ P h (D). DefineT ∈ L(X, Y ) by T (u h , z) = T (u h ), where (u h , z) ∈ P h X ⊕ (I − P h )X.
Then T D = 1 and T (E h x h ) > 1 − η( ). Since E h x h ∈ D and the pair (X, Y ) has the BPBp on D, for 0 < < 0 there exist S ∈ L(X, Y ) with S D = 1 and u ∈ D such that S(u) = 1, S − T < , and E h x h − u < .
Now we define an operator S ∈ L(X h , Y ) for u h ∈ X h S(u h ) = S(E h u h ).
From E h x h − u < , we get P i u < for i = h. The assumption yields that 0 P i (u) ∈ D i for i = h. Let w be the element in D such that P h (w) = P h (u) and P i (w) = 0 P i u for i = h. Then
S(w) = S(E h P h u) + i =h 0 S(E i P i u),
hence,
S(u) = 1 − 0 S(E h P h u) + 0 S(w).
It follows from S D = 1 = S (u) that S(E h P h u) = S(w) = 1. Since S D h S D = 1, S attains its maximum at P h u on D h . Moreover,
S − T < and x h − P h u = E h x h − E h P h u E h x h − u < ,
so that the proof is completed.
Examples satisfying the above assumption on D include i∈I λ i B X i with inf i∈I λ i > 0 as well as B X . The case of the 1 -sum of range spaces follows immediately from [3, Proposition 2.7], so we omit the proof. Proposition 4.5. Let D be a bounded closed convex subset of B X and Y = [ j∈J Y j ] 1 . If the pair (X, Y ) has the the BPBp on D with η( ), then the pair (X, Y j ) also has the the BPBpon D with η( ) for every j ∈ J. More precisely, for every j ∈ J, η D (X, Y ) η D (X, Y j ) A Banach space X is called a universal BPB domain space if for every Banach space Z, the pair (X, Z) has the BPB p on B X . It was proved in [3] that the base field R or C is the unique Banach space which is a universal BPB domain space in any equivalent renorming. Its proof follows immediately from [3, Lemma 3.2]: Let X be a Banach space containing a non-trivial L-summand and Y be a strictly convex Banach space. If the pair (X, Y ) has the BPB p on B X , then Y is uniformly convex. We can extend this result to a bounded closed convex subset D of X. With proposition 4.6 and remark 3.5, using similar argument in [3], we can say that the the base field R or C is the unique BPB domain on every bounded closed convex subset D.
Proposition 4.6. Let X be a (real) Banach space containing a nontrivial L-summand, i.e. X = X 1 ⊕ 1 X 2 for some non trivial subspaces X 1 and X 2 . Let D be a bounded closed convex subset of B X such that D = co(E 1 D 1 ∪ E 2 D 2 ). If Y is a strictly convex space and if the pair (X, Y ) has the BPBp on D, then Y is a uniformly convex space.
Proof. To prove that Y is uniformly convex, for every 0 < < 1/2 we need to find δ( ) > 0 such that y 1 = 1 = y 2 and y 1 +y 2 and since S D = 1, the inequalities S x 1 λ , 0 1 and S 0, + 4) , where the last inequality follows from S − T < 2M and T < 2M . Therefore Y is uniformly convex.
Theorem 3. 1 .
1Let X and Y be finite dimensional Banach spaces. Then the pair (X, Y ) has the BPBp on every bounded convex closed subset D of X.
which is an open neighborhood of s 0 due to the w * -continuity of φ. By Urysohn's lemma there exists a continuous function f : L → [0, 1] satisfying f (s 0 ) = 1 and supp(f ) ⊂ W.
Remark 3. 9 .
9By the remark after Theorem 2.5, without the symmetry of D we can show only that there exists an Asplund operator X → C 0 (L) such that u 0 ∈ D, S D = 1 = S(u 0 ) and T − S < 4 , by modifying the proof of Lemma 3.6 and by the first part of Theorem 2.5.
is clear the pair (X, Y ) has the BPBp on D if and only if η D (X, Y )( ) > 0 for every 0 < < 1. If a function −→ η D ( ) is valid in the definition of the BPBp on D for the pair (X, Y ), then η D ( ) η D (X, Y )( ). In other words, η D (X, Y )( ) is the largest function we can find to ensure that (X, Y ) has the BPBp on D. It is clear that η D (X, Y )( ) is increasing with respect to .
of Y yields that S x 1 λ , 0 = S 0, x 2 1−λ . Moreover y 1 − y 2 = T (e 1 , 0) − T (0, e 2 ) T (e 1 , 0) − S(e 1 , 0) + S(e 1 , 0) − S x 1 λ , 0 + S 0, x 2 1−λ − S(0, e 2 ) + S(0, e 2 ) − T (0,
) .
)We note that the BPBp is not so closely related with RNP as the Bishop-Phelps-Bollobás property. For example, 1 has RNP, but there exists a Banach space Y such that the pair ( 1 , Y ) does not have the BPBp ([2]). On the other hand, the pair (L 1 [0, 1], L ∞ [0, 1]) has the BPBp ([7]), but L 1 [0, 1] does not have RNP.
and S(x 1 , x 2 ) = 1.We can see that x 1 = 0 = x 2 . Indeed, if x 1 = 0, thenwhich is a contradiction.Choose {z n } ⊂ co(E 1 D 1 ∪ E 2 D 2 ) converging to (x 1 , x 2 ). Since E 1 D 1 and E 2 D 2 are convex sets, z n can be written as λ n u n + (1 − λ n )v n , where u n ∈ E 1 D 1 and v n ∈ E 2 D 2 . Passing to a subsequence we may assume λ n → λ as n → ∞. Then it is easy to check that λu n → E 1 x 1 and (1 − λ)v n → E 2 x 2 as n → ∞, which implies 0 < λ < 1 because x 1 = 0 = x 2 . Since E 1 D 1 and E 2 D 2 are closed, we can see that E 1 x 1 λ ∈ E 1 D 1 and E 2 x 2 1−λ ∈ E 2 D 2 . We claim that |λ − 1 2 | 1 2 . Suppose λ < 1 2 − 1 2 . Let δ = 1 2 − λ > 2 and x 0 = 1 2 e 1 − x 1 . ThenSince x 1 λ ∈ D 1 , this contradicts to e * 1 D 1 = 1. We can use the same argument for the case λ > 1 2 + , which completes the claim. ThenSince 1 = S(x 1 , x 2 ) = λS x 1 λ , 0 + (1 − λ)S 0,
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Republic of Korea E-mail address: [email protected] (Choi) Department of Mathematics, POSTECH, Pohang. Republic of Korea E-mailCho) Department of Mathematics, POSTECH, PohangCho) Department of Mathematics, POSTECH, Pohang (790-784), Republic of Korea E-mail address: [email protected] (Choi) Department of Mathematics, POSTECH, Pohang (790-784), Republic of Korea E-mail address: [email protected]
| []
|
[
"OBDD-Based Representation of Interval Graphs",
"OBDD-Based Representation of Interval Graphs"
]
| [
"Marc Gillé \nTU Dortmund\nLS2InformatikGermany\n"
]
| [
"TU Dortmund\nLS2InformatikGermany"
]
| []
| A graph G = (V, E) can be described by the characteristic function of the edge set χ E which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using Ordered Binary Decision Diagrams (OBDDs) to store χ E can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e. g., quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is O(|V |/ log |V |) and the OBDD size of interval graphs is O(|V | log |V |) which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is Ω(|V | log |V |). We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using O(log |V |) operations and a coloring algorithm for unit and general intervals graphs using O(log 2 |V |) operations and evaluate the algorithms empirically. | 10.1007/978-3-642-45043-3_25 | [
"https://arxiv.org/pdf/1305.2772v1.pdf"
]
| 17,727,399 | 1305.2772 | 44c36db2ecf421481e25154d6d8cadc39cc3eefd |
OBDD-Based Representation of Interval Graphs
May 14, 2013
Marc Gillé
TU Dortmund
LS2InformatikGermany
OBDD-Based Representation of Interval Graphs
May 14, 2013
A graph G = (V, E) can be described by the characteristic function of the edge set χ E which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using Ordered Binary Decision Diagrams (OBDDs) to store χ E can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e. g., quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is O(|V |/ log |V |) and the OBDD size of interval graphs is O(|V | log |V |) which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is Ω(|V | log |V |). We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using O(log |V |) operations and a coloring algorithm for unit and general intervals graphs using O(log 2 |V |) operations and evaluate the algorithms empirically.
Introduction
The development of graph algorithms is a classic and intensively studied area of computer science. But the requirements on graph algorithms have changed by the emergence of massive graphs, e. g., the internet graph or social networks. The representation size of a graph with N nodes and M edges given as adjacency matrix or lists is Θ(N 2 ) or Θ(N + M ). There are applications, e. g., dealing with a state transition graphs in circuit verification, where even polynomial running time may not be feasible or the input does not fit into the main memory. In order to deal with such massive graphs, symbolic or implicit graph algorithms have been investigated, where the input is represented by the characteristic function of the edge set. Identifying the nodes by binary numbers, the characteristic function becomes a Boolean function, which can be represented by Ordered Binary Decision Diagrams (OBDDs), which are a well known and commonly used data structure for Boolean functions. OBDDs were introduced by Bryant [8] and support many important functional operations on Boolean functions efficiently. Therefore, a research area came up concerning the design and analysis of implicit/symbolic (graph) algorithms on OBDD represented inputs ( [9,18,14,31,32,33,37,15]). A motivation of this line of research is that implicit representations can be significantly smaller than explicit representations on structured graphs, thus enabling the algorithms to process larger amounts of data. In particular, processing data in a more compact form might speed up the time needed for the algorithms.
In theory, problems on implicitly represented inputs become harder than their explicit equivalent. Even the s-t-connectivity problem, i. e., the decision whether two nodes s and t of an undirected graph are connected, is PSPACE-complete on OBDD-represented graphs [13] while the explicit variant is in L, the complexity class consisting of all problems decidable by a logspace Turing machine. Nevertheless, implicit algorithms are successful in many practical applications, e. g., model checking [10], integer linear programming [23] and logic minimization [12], and can be seen as a kind of heuristic (regarding time and space) to compute an optimal solution for problems on very large instances. One of the first implicit algorithms for a classical graph problem was the maximum flow algorithm on 0-1-networks presented in [18]. There, Hachtel and Somenzi were able to solve instances up to 10 36 edges and 10 27 nodes in reasonable time. Sawitzki [31] described another implicit algorithm for the same problem, which uses O(N log 2 N ) functional operations.
The number of operations used in an implicit algorithm is an important measure of difficulty [2] but it is also known [2,20] that an implicit algorithm computing the transitive closure, that uses an iterative squaring approach and a polylogarithmic number of operations, is often inferior to an implicit sequential algorithm, which needs in worst case a linear number of operations. In this case the advantage of the small number of functional operations is canceled out by the probably large sizes of the intermediate OB-DDs. In order to analyze the actual running time of an OBDD-based graph algorithm, it is crucial to determine both the number of functional operations and the sizes of the OBDDs which are generated during the computation.
Sawitzki [32,34] showed that all problems which are decidable by polynomially many processors using polylogarithmic time (i. e., which are in N C) are computable by an implicit algorithm using a polylogarithmic number of functional operations on a logarithmic number of Boolean variables. This is a quite structural result and does not lead to either an efficient transformation of parallel algorithms into implicit algorithms or a guarantee that implicit algorithms using a polylogarithmic number of functional operations perform well in practice.
Implicit algorithms using a polylogarithmic number of operations were designed for instance for topological sorting [37], maximal matching [6] and minimum spanning tree [3] where a matching M , i. e., a set of edges without a common vertex, is called maximal if M is no proper subset of another matching. However, non trivial bounds on the sizes of the OBDDs are hard to determine and, with it, the actual running time of an implicit algorithm. Only on very structured graphs like grid graphs good analysis of the running time are known, e. g., for the maximum flow algorithm [31], topological sorting [37] and maximal and maximum matching [6,4] (a matching is called maximum matching if there is no matching consisting of a larger number of edges). As a consequence, the practical performance of implicit algorithms is often evaluated experimentally, e. g., for the maximum matching problem in bipartite graphs [4] or for the maximum flow problem [18,31].
For a good running time the input size of an implicit algorithm, i. e., the size of the OBDD representing the input graph, should be small. Nunkesser and Woelfel [27] showed that the size of an OBDD representing an arbitrary graph is O(N 2 / log N ), which is similar to the space needed for a representation by adjacency matrix. For bipartite graphs they were able to show a lower bound of Ω(N 2 / log N ), which means that there is a bipartite graph which OBDD size is bounded below by Ω(N 2 / log N ). Notice that a lower bound of the OBDD size of a graph class does not mean that the OBDD size of every graph from this class is bounded below by this value.
Nunkesser and Woelfel [27] also investigated other restricted graph classes such as interval graphs. An interval graph is an intersection graph of intervals on the real line, i. e., two intervals (nodes) are adjacent iff they have a nonempty intersection. If these intervals have a length of 1, then the graph is called unit interval graph. (Unit) Interval graphs were extensively studied and have many applications, e. g., in genetics, archaeology, scheduling, and much more [16]. Nunkesser and Woelfel [27] proved that general interval graphs can be represented by OBDDs of size O(N 3/2 log 3/4 N ) and O(N/ √ log N ) for unit interval graphs. Due to counting arguments, they proved a lower bound of Ω(N ) for general interval graphs and Ω(N/ log N ) for unit interval graphs.
As in [27], we use n = log N bits, i. e., the minimal number of bits, to encode the nodes of a graph. Since the worst-case OBDD size is exponentially large in the number of input bits, using χ E in an implicit algorithm motivates to use a minimal amount of input bits to avoid a large worst-case OBDD size. Using a larger domain for the labels also possibly increases the size of the data structure storing the valid labels which is often needed in implicit algorithms. Aiming for a good compression of χ E , Meer and Rautenbach [24] investigated graphs with bounded clique-width or tree-width and increases the number of bits used for the node labeling to c · log N with constant c and were able to improve for instance the OBDD size of cographs from O(N log N ) [27] to O(N ).
We investigate implicit algorithms for coloring interval graphs, i. e., coloring the nodes of an interval graph such that all adjacent nodes have different colors and the number of used colors is minimal, and for maximum matching on unit interval graphs. Coloring of interval graphs has applications in VLSI design and scheduling [17] and there is an optimal greedy coloring algorithm [28] which can be implemented in linear time. The first parallel matching algorithm for (general) interval graphs was given by a parallel algorithm for two processor scheduling [19] using N 10 processors and O(log 2 N ) time. It was improved step-by-step until in [11] the current best parallel algorithm was presented using N 3 / log 4 N and O(log 2 N ) time. Furthermore, they showed that it is possible to compute a maximum matching on unit interval graphs in parallel by O(N/ log N ) processors using O(log N ) time.
Our Contribution
In Section 3 we present a new method to show upper and lower bounds of the size of an OBDD representing a graph. We sort the rows and columns of the adjacency matrix of a graph in such a way that each node of the OBDD (labeled with the same input variable) corresponds to a distinct block of this adjacency matrix. Thus, by counting the number of different blocks, which is probably easier than counting different subfunctions, we can bound the number of nodes of the OBDD.
Using this method and some known structure of the adjacency matrix of interval graphs [26], we improve the upper bound on general interval graphs to O(N log N ) while using a more convenient way to label the nodes than in [27]. Using a probabilistic argument, we prove that the worst-case OBDD size is Ω(N log N ) if we use the same labeling and variable order as in our upper bound. We can close the gap of the upper bound and the lower bound in the case of unit interval graphs and show that the OBDD size is O(N/ log N ).
In Section 4 we present two implicit algorithms for (unit) interval graphs: A maximum matching algorithm for unit interval graphs using only O(log N ) functional operations and a coloring algorithm for interval graphs using O(log 2 N ) functional operations. The matching algorithm takes advantage of the information given by the labels of the nodes. Furthermore, we were able to compute the transitive closure of a unit interval graph using only O(log N ) operations instead O(log 2 N ) operations, which are needed in general. In order to implement this algorithm efficiently, we have to extend a known result due to Woelfel [37] to a different variable order for constructing OBDDs representing multivariate threshold functions. For the coloring algorithm we show how to get a total order on the right endpoints (given that the labels of the nodes respect the order of the left endpoints) and how to compute a minimal coloring of the nodes by using these orders based on a optimal greedy algorithm [28]. To the best of the author's knowledge, this is the first time that the labeling of nodes is used to speed up an implicit algorithm for a large graph as interval graphs and to improve the number of functional operations. In Section 5 we evaluate the implicit algorithms experimentally and see that the matching algorithm is both very fast and space efficient while, unfortunately, the coloring algorithm does not perform well on both unit and general interval graphs. The poor performance of the coloring algorithm is very likely due to the fact that the implicit algorithm is simulating the sequential coloring algorithm. Nevertheless, it uses some nice ideas to accomplish this which differ from the parallel implementation ideas.
A simple implicit representation of an interval graph with N nodes is a list of N inter-vals and needs Θ(N ) space. Our result on the OBDD size of interval graphs shows that in the worst case the OBDD representation is almost as good as the interval representation with the advantage that it is possible to use o(N ) space for some instances. Together with the experiments, this shows that the representation of at least unit interval graphs with OBDDs enables a good compression without loosing the usability in algorithms.
Preliminaries
OBDDs
We Let G = (V, E) be a directed graph with node set V = {v 0 , . . . , v N −1 } and edge set E ⊆ V × V . Here, an undirected graph is interpreted as a directed symmetric graph. Implicit algorithms are working on the characteristic function χ E ∈ B 2n of E where n = log N is the number of bits needed to encode a node of V and χ E (x, y) = 1 if and only if (v |x| , v |y| ) ∈ E. In order to deal with Boolean functions, OBDDs were introduced by Bryant [8] to get a compact representation (see Fig. 1), which supports a bunch of functional operations efficiently. Representation. A π-OBDD is a directed, acyclic and rooted graph G with two sinks labeled by the constants 0 and 1. Each inner node is labeled by an input variable from X and has exactly two outgoing edges labeled by 0 and 1. Each edge (x i , x j ) has to respect the variable order π, i. e., π(i) < π(j).
Evaluation. An assignment a ∈ {0, 1} n of the variables defines a path from the root to a sink by leaving each x i -node via the a i -edge. A π-OBDD G f represents f iff for every a ∈ {0, 1} n the defined path ends in a sink with label f (a).
Complexity. The size of a π-OBDD G, denoted by |G|, is the number of nodes in G. The π-OBDD size of a function f is the minimum size of a π-OBDD representing f . The OBDD size of f is the minimum π-OBDD size over all variable orders π. The width of G is the maximum number of nodes labeled by the same input variable.
In the following we describe a list of important operations on Boolean functions which we will use in this paper and give the time requirements in the case of OBDDs (see, e. g., Section 3.3 in [36] for a detailed list). Let f and g be Boolean functions in B n on the variable set X = {x 0 , . . . , x n−1 } and let G f and G g be OBDDs representing f and g, respectively. In the rest of the paper quantifications over k Boolean variables Qx 1 , . . . , x k : f are denoted by Qx : f , where x = (x 1 , . . . , x k ). The following operation (see, e. g., [33]) is useful to reverse the edges of a given graph, i. e., let χ E (x, y) be a directed graph and we want to compute χ E (y, x) which represents the edge set {(v |y| , v |x| ) | (v |x| , v |y| ) ∈ E} consisting of the reverse edges of E.
Definition 2.2:
Let k ∈ N, ρ be a permutation of {1, . . . , k} and f ∈ B kn be defined on Boolean variable vectors x (1) , . . . , x (k) of length n. The argument reordering R ρ (f ) ∈ B kn with respect to ρ is defined by R ρ (f )(x (1) , . . . , x (k) ) := f (x (ρ(1)) , . . . , x (ρ(k)) ).
This operation can be computed by just renaming the variables and repairing the variable order using 3(k − 1)n functional operations (see [5]).
An important variable order is the interleaved variable order which is defined on vectors of length n where the variables with the same significance are tested one after another.
Definition 2.3:
Let x (1) , . . . , x (k) ∈ {0, 1} n be k input variable vectors of length n. Let π be a permutation of {0, . . . , n − 1}. Then
π k,n = (x (1) π(0) , x (2) π(0) , . . . , x (k) π(0) , . . . , x (1) π(n−1) , . . . , x (k) π(n−1) )
is called k-interleaved variable order for x (1) , . . . , x (k) . If π = (n − 1, . . . , 0) then we say that the variables are tested with decreasing significance.
Choosing an interleaved variable order in OBDD-based algorithms is common practice, since auxiliary functions, e. g., the equality or the greater than function, and multivariate threshold functions, which we will define in section 4.1, have to use an interleaved variable order for a compact OBDD representation.
The input of an OBDD-based graph algorithm consists of the characteristic function χ E represented by an OBDD and the output is a characteristic function χ O of a set O which is computed by mainly using functional operations. As we can see in the above listing of the operations, the running time of such algorithms depends on the actual size of the OBDDs which are used for an functional operation during the computation. In general, it is difficult to prove a good upper bound on the running time because we have to know a good upper bound on the size of every OBDD used as an input for an operation.
However, if the size of the OBDD representing the input graph is large, any implicit algorithm using this OBDD is likely to have an inadequate running time. Beside the variable order, the labeling of the nodes is another optimization parameter with huge influence on the input size. For OBDDs representing state transitions in finite state machines, Meinel and Theobald [25] showed that there can be an exponential blowup of the OBDD size from a good labeling to a worst-case labeling. Nevertheless, a small input OBDD size, i. e., a good labeling of the nodes for some variable order, does not guarantee a good performance of the implicit algorithm since the sizes of the intermediate OBDDs do not have to be small. Indeed, an exponential blowup from input to output size is possible [33,3].
We denote by
f |x π(0) =a π(0) ,...,x π(i−1) =a π(i−1) the subfunction where x π(j) is replaced by the constant a π(j) for 0 ≤ j ≤ i − 1. The function f depends essentially on a variable x i iff f |x i =0 = f |x i =1 .
A characterization of minimal π-OBDDs due to Sieling and Wegener [35] can be often used to bound the OBDD size.
Theorem 2.4 ([35]):
Let f ∈ B n and for all i = 0, . . . , n − 1 let s i be the number of different subfunctions which result from replacing all variables x π(j) with 0 ≤ j ≤ i − 1 by constants and which essentially depend on x π(i) . Then the minimal π-OBDD representing f has s i nodes labeled by x π(i) , i. e., the minimal π-OBDD has size n−1 i=0 s i .
Basic Functions and Implicit Algorithms
It is well known that the OBDD size of the equality EQ(x, y) and greater than function GT (x, y) with EQ(x, y) = 1 ⇔ |x| = |y| and GT (x, y) = 1 ⇔ |x| > |y| is linear in the number of input bits for an interleaved variable order (see, e. g., [36]). It is also possible to construct the representing OBDDs for these functions in linear time. For the sake of code readability, we use |x| = |y| and |x| > |y| to denote EQ(x, y) and GT (x, y) in our algorithms. Furthermore, by |x| > c (|x| = c) for some constant c we denote the function
GT (x, y) |y=[c] 2 (EQ(x, y) |y=[c] 2 )
where the y-variables are replaced by constants corresponding to the binary number [c] 2 of c.
Let R(x, y) ∈ B 2n be a Boolean function. R(x, y) can be seen as a binary relation R on the set {0, 1} n with x R y ⇔ R(x, y) = 1. The transitive closure of this relation is the function
R * (x, y) with R * (x, y) = 1 iff there is a sequence x = x 1 , . . . , x l = y with R(x i , x i+1 ) = 1 for all i = 1, . . . , l − 1. E. g., let R(x, y) = χ E (x, y) ∨ (|x| = |y|)
be the function that returns 1 iff there is an edge between v |x| and v |y| or (|x| = |y|), then is R * (x, y) = 1 iff there the nodes v |x| and v |y| are in the same connected component. The transitive closure can be computed implicitly by O(n 2 ) functional operations using the so called iterative squaring or path doubling technique (see Algorithm 1).
Algorithm 1 T ransitiveClosure(R(x, y)) Input: Boolean function R(x, y) ∈ B 2n Output: Transitive closure R * (x, y) of R(x, y) 1: R * (x, y) = R(x, y) 2: for i = 0 to n do 3: R * (x, y) = ∃z : R * (x, z) ∧ R * (z, y) 4: end for 5: return R * (x, y)
Let O(x, y) represent a total order ≺ on the input bitstrings, i. e., O(x, y) = 1 ⇔ x ≺ y (e. g., O(x, y) = 1 ⇔ |x| ≤ |y|). Since ≺ is a total order, the input bitstrings can be sorted in ascending order according to ≺. Let EO(x, l) = 1 iff x is in the |l|-the position in this sorted sequence. Similar to the transitive closure, it is known (see, e. g., [34]) that EO(x, l) can be computed using O(n 2 ) functional operations (see Algorithm 2).
Algorithm 2 EnumarateOrder(O(x, y)) Input: Total order O(x, y) ∈ B 2n
Output: EO(x, l) with EO(x, l) = 1 iff the rank of x is |l| in the ascending order // Compute the pairs of direct successors
1: DS(x, y) = O(x, y) ∧ ∃z : O(x, z) ∧ O(z, y)
// EO i (x, y, l) = 1 iff |l| ≤ 2 i // and the distance between the position of x and y is equal to |l| 2: EO 0 (x, y, l) = ((|l| = 0) ∧ (|x| = |y|) ∨ ((|l| = 1) ∧ DS(x, y))) // Divide and conquer approach: If 2 i−1 < |l| ≤ 2 i then there has to be // an intermediate bitstring z with distance 2 i−1 to x and |l| − 2 i−1 to y 3: for i = 1 to n do 4:
EO i (x, y, l) = ((|l| ≤ 2 i−1 ) ∧ EO i−1 (x, y, l)) ∨ [(2 i−1 < |l| ≤ 2 i ) ∧ ∃l 1 , z : EO i−1 (x, z, 2 i−1 ) ∧ EO i−1 (z, y, l 1 ) ∧ (|l 1 | + 2 i−1 = |l|)] 5: end for
// Compute the rank according to the distance to the first element
6: EO(x, l) = ∃z : EO n (z, x, l) ∧ ∃z : O(z , z) 7: return EO(x, l)
Interval Graphs
We start with a formal definition of (unit) interval graphs.
Definition 2.5 (Interval Graph): Let I = {[a i , b i ] | a i < b i and 0 ≤ i ≤ N − 1} be a set of N ∈ N intervals on the real line.
The interval graph G I = (V, E) has one node for each interval in I and two nodes v = w are adjacent iff the corresponding intervals intersect. If no interval is properly contained in another interval, G I is called proper interval graph. If the length of every interval in I is equal to 1 then G I is called unit interval graph (see Fig. 2).
Notice that the set of all interval graphs does not change if we restrict ourselves to sets I where all endpoints are different because if there are two intervals with a shared endpoint then there exists an > 0 such that moving the shared endpoint of one of the two intervals by generates the same interval graph. The definitions of proper and unit interval graphs are equivalent in the sense that they generate the same class of interval graphs [29]. Hence, in the following we only use the term of unit interval graphs. An undirected graph H is a (unit) interval graph iff there is a set of (unit) intervals I such that H = G I . Due to the one-to-one correspondence of the nodes of G I and the elements of I, we use the notion of node and interval synonymously.
OBDD Size of Interval Graphs
In order to bound the size of a function f using Theorem 2.4, we have to count different subfunctions of f . We present a way to count the subfunctions of the characteristic This can give us a better understanding what a subfunction looks like in the graph scenario and get a more graph theoretic approach to subfunctions. The adjacency matrix of graphs from special graph classes (e. g., for interval graphs) yields some structural properties which we use to bound the size of the π-OBDD. As we know from the last section, the OBDD size is dependent on the labeling of the nodes. So if we use the knowledge about the structure of the adjacency matrix for a fixed labeling to bound the number of different subfunctions for a variable order π, we can show an upper and/or lower bound of the π-OBDD size.
The rows (columns) of an adjacency matrix correspond to the x-variables (y-variables) of χ E (x, y). We can sort the rows of the adjacency matrix according to a variable order π by connecting the i-th row to the input x with n−1 l=0 x π(n−l−1) · 2 l = i, i. e., we let the l-th x-variable in π have significance 2 n−l−1 to sort the rows. This can be done analogously to sort the columns. Thus, the variable order π defines a permutation of the rows and columns of the adjacency matrix resulting in a new matrix which we call π-ordered adjacency matrix.
Definition 3.1:
Let G = (V, E) be a graph and π := π 2,n be a 2-interleaved variable order for the characteristic function f := χ E . The π-ordered adjacency matrix A π of G is defined as follows: a ij = 1 iff f (x, y) = 1 with n−1 l=0 x π(n−l−1) · 2 l = i and n−1 l=0 y π(n−l−1) · 2 l = j.
Notice that the π-ordered adjacency matrix is equal to the "normal" adjacency matrix where the rows and columns are sorted by the node labels iff the variables in π are tested with decreasing significance. The π-ordered adjacency matrix gives us a visualization of the subfunctions in terms of blocks of the matrix.
Definition 3.2:
Let n ∈ N and A be a 2 n × 2 n matrix. For 0 ≤ k ≤ n and 0 ≤ i, j ≤ 2 k − 1 the block B k ij of A is defined by the quadratic submatrix of size 2 n /2 k × 2 n /2 k which is formed by the intersection of the rows i · 2 n /2 k , . . . , (i + 1) · 2 n /2 k − 1 and the columns j · 2 n /2 k , . . . , (j + 1) · 2 n /2 k − 1.
Recall Theorem 2.4 that we want to count the number of different subfunctions which result from replacing the first i variables according to π by constants. We will see later that for an upper bound it is enough to consider only the case when i is even, i. e., the For instance, let say that the variables are tested with decreasing significance. Then a ij = 1 iff f (x, y) = 1 with n−1 l=0 x l · 2 l = |x| = i and n−1 l=0 y l · 2 l = |y| = j, i. e., the π-ordered adjacency matrix A π of G is the standard adjacency matrix where the labeling of the columns and rows is ordered by the node labels. In Fig. 3 we can see that for every k each subfunction of f where the first k bits (according to π 2,n ) are replaced by constants corresponds to a block of this adjacency matrix.
Bollig and Wegener [7] use a similar approach to visualize subfunctions of a storage access function by building a matrix whose columns and rows are sorted according to the variable order and correspond to variables (not assignments as in our π-ordered matrix). Notice that A π is not the communication matrix which is often used to show lower bounds of the OBDD size.
Next, we use the π-ordered adjacency matrix and count the number of different blocks to improve the bounds of the OBDD size of interval graphs.
Theorem 3.3:
Let π := π 2,n be the interleaved variable order with decreasing significance and G = (V, E) be an interval graph with N := |V | nodes. The π-OBDD size of χ E can be bounded above by O (N log N ).
Proof. Let f := χ E , 1 ≤ k ≤ n and s k be the number of different subfunctions f |α,β of f where α ∈ {0, 1} k is an assignment to the variables x n−1 , . . . , x n−k and β ∈ {0, 1} k is an assignment to the variables y n−1 , . . . , y n−k , respectively. The number of different subfunctions where the variables x n−1 , . . . , x n−k and y n−1 , . . . , y n−k−1 are replaced by constants can be bounded by 2 · s k because one additional variable can at most double the number of subfunctions. We label the nodes according to their position in the sorted sequence of interval starting points (as for example in Fig. 2). Recall that the interleaved variable order with decreasing significance means that a i,j is one if and only if interval i intersects interval j. Now, notice that if a i,j is zero for j > i, i. e., interval j has a larger starting point than interval i and does not cut interval i, then no interval j > j with a larger starting point can cut interval i. Thus, for every column i ∈ {0, . . . , N −1}, the sequence (a i+1,i , . . . , a N −1,i ) is zero or starts with a continuous sequence of ones followed by only zeros, i. e., there exists a j such that a k,i = 1 for i < k ≤ j and a k,i = 0 for k > j.
As seen in the beginning of this section, every subfunction f |α,β corresponds to a block of A π . Let β = 0 k and |α| ≥ 1, i. e., we consider the blocks B k |α|,0 of size 2 n−k × 2 n−k (see Fig. 4). As we observed, every column of A π has (below the diagonal) at most one possible changing position k such that a k,i = 1 and a k+1,i = 0. Looking at the sequence (B k 1,0 , . . . , B k 2 k −1,0 ) of blocks, this fact implies that a block B k i,0 can only form a new block, i. e., all previous block in the sequence are different to this block, if there is a changing position in one column inside of B k i,0 or inside the block B k i−1,0 or between these two blocks. Therefore, every changing position can induce at most two different blocks and, therefore, we can bound the number of different blocks by two times the number of possible changing positions which is at most the number of columns of a block, i. e., 2 · 2 n−k . Since the graph is symmetric and the blocks containing the diagonal can only add 2 k additional distinct blocks, we can bound the overall number of different blocks by O(2 n−k · 2 k + 2 k ) = O(2 n ) and thus s k = O(2 n ). Summing this up over all possible values of k we get O(2 n · n) = O(N log N ) as an upper bound on the size of the π-OBDD.
In the case of unit interval graph Nunkesser and Woelfel [27] proved that the OBDD size is bounded below by Ω(N/ log N ) and above by O(N/ √ log N ). We can close this gap by using the π-ordered adjacency matrix to get a better upper bound on the number of subfunctions of χ E for large values of k.
Theorem 3.4:
Let π be the interleaved variable order with decreasing significance. The π-OBDD size of χ E for a unit interval graph G = (V, E) is O (N/ log N ).
Proof. Again, let f := χ E and s k be the number of different subfunctions f |α,β of f where α is an assignment to the variables x n−1 , . . . , x n−k and β is an assignment to the variables y n−1 , . . . , y n−k respectively. As we have seen in the last proof, the number of different subfunctions where the first k x-variables and k + 1 y-variables are replaced by constants can be bounded by 2 · s k . We label the nodes according to their interval starting points. We know that f |α,β corresponds to the block B k |α|,|β| of the π-ordered adjacency matrix of G. Let |α| > |β|. Every column of these blocks consists of a beginning sequence of ones of length l ≥ 0 and an ending sequence of zeros of length C − l, where C = 2 n−k is the number of rows and columns of B k |α|,|β| . Let l 1 , . . . , l C be the lengths of the beginning sequence consisting of ones of every column in the block B k |α|,|β| . Recall that the intervals are labeled according to their interval starting point. Since G is a unit interval graph, this is equivalent to labeling them according to their interval ending points, i. e., if j > i, then interval j starts and ends after interval i. This implies that the sequence l 1 , . . . , l C is monotonically increasing, i. e., l 1 ≤ l 2 ≤ . . . ≤ l C . How many different blocks of this form can be constructed? We can construct such a block by drawing C numbers between 0 and C and sorting them, i. e., it is equivalent to selecting C numbers out of {0, . . . , C} with repetition, where order does not matter. The number of C-combinations with repetition is equal to (C+1)+C−1 C = 2C C and this can be bounded above by 2 2C . Since G is symmetric, this is also a bound on the number of different blocks above the diagonal. The omitted blocks on the diagonal can be constructed in a similar way: At first, the diagonal of these blocks is zero and the blocks are symmetric. Below the diagonal the blocks also consist of a sequence of ones probably followed by a sequence of zeros. So the number of different blocks is bounded above by the number of different blocks, which are not on the diagonal, i. e., by 2 2C . Hence, for C = 2 n−k we can bound s k above by 3 · 2 2 n−k+1 . Nunkesser and Woelfel [27] also showed that s k ≤ 2 k+2 − 2. Therefore, the OBDD size is at most
n−1 k=0 min{2 k+2 , 3 · 2 2 n−k+1 } ≤ n−log n+1 k=0 2 k+2 + 3 · n−1 k=n−log n+2 2 2 n−k+1 ≤ 2 n−log n+4 + 3 · 2 2 log n−1 · (log n − 2) = O(N/ log N ) + O( √ N · log log N ) = O(N/ log N ).
The difference between unit and general interval graphs is that in general interval graphs there is no dependence between the columns of the π-ordered adjacency matrix, which is important for our lower bound, while in unit interval graphs, the row number of the last 1 entry in a column is increasing from left to right. The proofs of the upper bounds suggest that the number of blocks B k i,j with a changing position roughly determines the number of OBDD nodes labeled by x n−k−1 . We know that every layer of the OBDD, i. e., every set of OBDD nodes labeled by the same variable, has size O(N ) which means that there has to be Ω(n) layers of the OBDD of size Ω(N ) to show a lower bound of Ω(N log N ). Explicitly constructing a worst-case interval graph with OBDD size of Ω(N log N ) is difficult because Ω(n) layers correspond to Ω(n) values of k and, since the block B k i,j results from dividing a block B k−1 i ,j , many dependencies have to be considered to ensure that Ω(N ) blocks are different for all the possible values of k.
In order to overcome these dependencies, we look at a random interval graph and compute the expected value of the number of different blocks for Ω(n) values of k. Intuitively, in the worst-case the lengths of the 1-sequences of the columns are uniformly distributed such that there are many blocks with a small number of changing positions inside which maximizes the possibility that there are many different blocks. Choosing an appropriate distribution on the set of interval graphs, we show that the expected number of different blocks with one changing position is Ω(N ) for Ω(n) values of k. Due to the linearity of expectation, the expected value of the sum of the number of different blocks over all values of k is Ω(N n) = Ω(N log N ), i. e., there is an interval graph whose OBDD size is also Ω(N log N ).
Theorem 3.5:
The worst-case π-OBDD size of an interval graph is Ω (N log N ) where the nodes are labeled according to the interval starting points and π is an interleaved variable order with decreasing significance.
Proof. We describe a random process to generate an interval graph where the adjacency matrix is constant except the N/2 × N/2 lower left submatrix which we denote by R (see Fig. 5). For this, we choose the length of the 1-sequence of column j for all 0 ≤ j ≤ N/2 − 1 uniformly at random from {N/2 − j, . . . , N − 1 − j} and for all N/2 ≤ j ≤ N − 1 the length of column j is equal to N − 1 − j. As a result, the length of the 1-sequence of each column within R is uniform at random in {1, . . . , N/2}.
Let G = (V, E) be a random interval graph generated by the above process and f := χ E . Let 1 ≤ k ≤ n and s k be the number of different subfunctions f |α,β of f where α ∈ {0, 1} k is an assignment to the variables x n−1 , . . . , x n−k and β ∈ {0, 1} k is an assignment to the variables y n−1 , . . . , y n−k , respectively, and f |α,β is essentially dependent on x n−k−1 . We show that the expected value of s k with n/2 + 1 ≤ k ≤ (3/4)n is Ω(2 n ) = Ω(N ). Therefore, there has to be an interval graph with π-OBDD size Ω (N log N ).
We known that k induces a grid in R consisting of 2 n−k × 2 n−k blocks. At first, we calculate the expected number of blocks with exactly one changing position. The probability that a fixed block of size L × L with L ≤ 2 n/2−1 has exactly one changing position is
L i=1 L − 1 2 n−1 · 1 − L − 1 2 n−1 L−1 ≥ L · (L − 1) 2 n−1 · 1 − L 2 n−1 L−1 ≥ L · (L − 1) 2 n−1 · 1 − 2 n/2 2 n−1 2 n/2−1 −1 ≥ L · (L − 1) 2 n−1 · e −1 .
Let n/2 + 1 ≤ k ≤ (3/4)n be fixed. Since we have 2 k−1 · 2 k−1 blocks of size 2 n−k × 2 n−k in R, the expected value of the number of blocks with exactly one changing position is at least 1 2e · 2 k · 2 k · 2 n−k · (2 n−k − 1) 2 n = 1 2e · 2 k · (2 n−k − 1) = Ω(2 n ). Now, we have to ensure that these blocks correspond to different subfunctions which are also essentially dependent on x n−k−1 . The subfunctions, where, additionally, x n−k−1 is replaced by 0 and 1, correspond to a half of the blocks. Thus, a block is symmetric iff the corresponding subfunction is not essentially dependent on x n−k−1 . Due to the one changing position in each block, this is not possible. Blocks B k i,j and B k i ,j with exactly one changing position and i = i clearly correspond to different subfunctions because they are in the same block column. But blocks B k i,j and B k i ,j with j = j , i. e., from different block columns, do not have to be different. By replacing some columns of the matrix by constants, we ensure that this also holds.
Consider the case k = (3/4)n, i. e., the finest grid of R made by 2 n−k × 2 n−k blocks with n/2 + 1 ≤ k ≤ (3/4)n. For every block column 0 ≤ j ≤ 2 k − 1 we fix the first k columns of B k i,j with 0 ≤ i ≤ 2 k −1 such that they represent the binary number [j] 2 of the column index. Thus, we have that blocks B k i,j and B k i ,j with j = j are always different. Since we looked at the finest grid, this also holds for smaller values of k because every larger block is equal to a union of small blocks. The probability that a block contains exactly one changing position is smaller than before, since we fix some columns. For k = (3/4)n the number of fixed columns is (3/4)n and in each k → k − 1 step this number is doubled, i. e., for n/2 + 1 ≤ k ≤ (3/4)n the number of "free" columns is 2 n−k − 2 (3/4)n−k · (3/4)n = 2 n−k − 2 (3/4n−k+log((3/4)n)) = Ω(2 n−k ) for n large enough. Replacing L = 2 n−k by Ω(2 n−k ) in the calculation of the expectation does not change the asymptotic behavior. Thus, the expected number of blocks with exactly one changing position remains Ω(2 n ) for every n/2 + 1 ≤ k ≤ (3/4)n.
Implicit Algorithms on Interval Graphs
In this section, we want to develop a maximum matching algorithm on unit interval graphs and a coloring algorithm on unit and general interval graphs. Before we start with the algorithms, we have to investigate a special function class, which we will use in our algorithms, so called multivariate threshold functions. This function class was investigated in [37] to analyze the running time of an implicit topological sorting algorithm on grid graphs and Woelfel [37] looked into the OBDD size of these functions for the interleaved variable order with increasing significance, i. e., just the reverse of our variable order. Hosaka et al. [21] showed that the difference of the OBDD sizes for this two orders is at most n − 1. We can show that an OBDD using our variable order is not only small but can also be constructed efficiently which is important in view of the implementation.
Constructing OBDDs for Multivariate Threshold Functions
We start with a definition of multivariate threshold functions [37].
f (x (1) , . . . , x (k) ) = 1 ⇔ k j=1 w j · |x (j) | ≥ T.
The set of k-variate threshold functions f ∈ B kn with weight parameter W is denoted by T W k,n .
Woelfel [37] showed that there exists an OBDD representing a multivariate threshold function f ∈ T W k,n of size O(k 2 W n) and such an OBDD can be constructed efficiently. Our proof for our variable order is similar to the proof in [37]: It is sufficient to look at the carry values of the sum k j=1 w j · |x (j) | − T and, especially, at the carry value generated at the position with the most significance. Reading the bits with increasing significance, Woelfel showed that after each bit it is enough to store a number with absolute value O(kW ) to compute the carry values. Here, we show that the influence of input bits with lower significance is small such that we can also bound the number which we have to store after each bit while we read the bits with decreasing significance.
Theorem 4.2:
Let f ∈ T W k,n be a k-variate threshold function with weight bound W ∈ N and π k,n be the k-interleaved variable order where the variables are tested with decreasing significance. Then we can construct a π k,n -OBDD representing f with width O(kW ) and size
O(k 2 W n) in time O(k 2 W n).
Proof. Similar to the proof in [37], we choose T 0 , . . . , T n−1 ∈ {0, 1} and T n ∈ Z such that −T = n i=0 T i · 2 i . Notice that the T i are unique, and that T 0 , . . . , T n−1 are the n least significant coefficients of |T | in binary representation and T n is the number of times that we need to add 2 n in order to make up for the missing coefficients in this binary representation. The function value of f is determined by the sign of S :
= −T + k j=1 w j · |x (j) | = n−1 i=0 (T i + k j=1 w j · x (j)
i ) · 2 i + T n · 2 n . Now, we represent S in the same way as T , i. e., we define S 0 , . . . , S n−1 ∈ {0, 1} and S n ∈ Z as the unique coefficients satisfying S = n i=0 S i · 2 i . We want to compute S i step-by-step: Notice that S i results from adding T i + k j=1 w j · x (j)
i and the carry value which is generated at position i − 1, and taking the remainder of the division of this sum by two. In particular, S i is only influenced by factors of 2 j for j ≤ i, and it holds that for 0 ≤ i ≤ n − 1
S i := c i−1 + T i + k j=1 w j x (j) i mod 2 and c i := c i−1 + T i + k j=1 w j x (j) i /2
with c −1 = 0. Finally, we compute S n = c n−1 + T n . Now, we have f (x (1) , . . . , x (k) ) = 1 ⇔ c n−1 ≥ −T n , i. e., it is sufficient to compute the c i values. We rewrite the c i to have them in a more convenient form. Notice that for m, n ∈ N and x ∈ R it holds that x + m n = x + m n (see, e. g., [22]). So we have
c 1 = c 0 + T 1 + k j=1 w j x (j) 1 /2 = T 0 + k j=1 w j x (j) 0 /2 + T 1 + k j=1 w j x (j) 1 /2 = T 0 + k j=1 w j x (j) 0 /2 + T 1 + k j=1 w j x (j) 1 /2 = T 0 + k j=1 w j x (j) 0 /4 + T 1 + k j=1 w j x (j) 1 /2 . Let c i = T i + k j=1 w j x (j) i 2 n−i .
Applying the above observation iteratively, we have
c n−1 = n−1 i=0 c i .
According to our variable order, we have to compute c i backwards from n − 1 to 0. This is possible because each c i only depends on i. We describe an algorithm that is divided into phases. In each phase, the algorithm is in a state Q, reads all k bits of the input variable vectors of the same significance and changes the state depending on the former state and the read bits. After phase i, the algorithm has the correct sum of the summands from n − 1 to i. Notice that the bits with lesser significance can only add a value to S with bounded absolute value, so if the accumulated sum has a large enough absolute value, then we can already decide which sign S has.
Let us start with phase n − 1 and state Q = 0. In phase 1 ≤ i ≤ n − 1 we compute the value of c i by reading x 1. If c i + Q ≥ −T n + (kW + 1)/2 n−i then change into the accepting state Q acc .
2. If c i + Q < −T n − (kW + 1)/2 n−i then change into the rejecting state Q rej .
3. Otherwise update the state Q = c i + Q and go to phase i − 1.
In phase 0 we compute c 0 and accept iff c 0 + Q ≥ −T n .
If we reach phase 0 then the output is correct due to our above observations. So we have to show that we correctly accept/reject within phase i with 1 ≤ i ≤ n − 1. For i = 0, . . . , n − 1 it is |c i | ≤ kW +1 2 n−i because T i ∈ {0, 1} and all weights are bounded by W and therefore Based on this algorithm the construction of the π k,n -OBDD is easy: Assume that we update the state immediately after reading an input variable. Then each state is represented by an OBDD node labeled by the variable which the algorithm will read next. The states for accepting and rejecting are represented by the sinks. The edges correspond to the state transition of the algorithm. If we are not in an accepting or rejecting state, we know that the state value is between −T n − kW +1 2 n−i−1 and −T n + kW +1 2 n−i−1 −1. We also know that |c i | ≤ kW +1 2 n−i , i. e., the values computed in phase i has to be between
−T n − kW + 1 2 n−i−1 − kW + 1 2 n−i and − T n + kW + 1 2 n−i−1 − 1 + kW + 1 2 n−i .
So all values are in the interval
I = [−T n − 3 2 kW +1 2 n−i−1 , −T n + 3 2 kW +1 2 n−i−1 )
. The denominator of c i is an integer, i. e., only at most 2 n−i · |I| = O(kW ) values of I are possible during the computation. Therefore, we have an OBDD width of O(kW ) and overall an OBDD size of O(k 2 W n). The construction algorithm is straightforward and has a running time which is linear to the OBDD size.
The proof of Theorem 4.2 also showed that the complete-OBDD width is bounded by O(kW ) where an OBDD is called complete if the length of every path from the root to a sink is equal to the number of input bits, i. e., all variables are tested on the path. A binary synthesis of two functions with complete-OBDD width w 1 and w 2 has a complete-OBDD width of at most w 1 ·w 2 [33]. Since the complete-OBDD size is an upper bound on the general OBDD size, we can compute a sequence of O(1) binary synthesis of multivariate threshold functions efficiently using the interleaved variable order with decreasing significance if k and W are constants. We use the arithmetic notation in our algorithm instead of the functional notation whenever we use multivariate threshold functions or simple combination of multivariate threshold functions, e. g., we denote by |x| − |y| = 1 the conjunction of the multivariate threshold functions f (x, y) = 1 ⇔ |x| − |y| ≥ 1 and g(x, y) = 1 ⇔ |y| − |x| ≥ −1.
Maximum Matching on Unit Interval Graphs
Let G = (V, E) be a unit interval graph and the nodes are labeled according to the sorted sequence of starting points. Our maximum matching algorithm is based on a simple observation that was also used in a parallel algorithm for this problem [11]: Assume that the unit interval graph is connected (otherwise this observation holds for every connected component). Then we have {v i , v i+1 } ∈ E for i = 0, . . . , N − 2. Assume that there is an i ∈ {0, . . . , N − 2} such that {v i , v i+1 } ∈ E, then due to the connectivity there has to be another interval with starting point left of v i or right of v i+1 , which intersects both intervals v i and v i+1 . The length of this interval would be larger than 1 which is a contradiction.
Algorithm 3 uses besides the characteristic function χ E also the characteristic function of the set of nodes. This is important if the number of nodes is not a power of two and we have assignments to the input variables which do not represent a node. Since we label our node according to their interval starting point, we have that the characteristic function of the node set is equal to f (x) = 1 ⇔ |x| < N .
Algorithm 3 Implicit maximum matching algorithm for unit interval graphs Input: Unit interval graph χ E Output: Matching χ M // Compute path graph 1:
χ − → E (x, y) = χ E (x, y) ∧ (|y| − |x| = 1) // Compute set of starting nodes 2: F irst(z) = (|z| < N ) ∧ ∀x : χ − → E (x, z) // Compute set of reachable nodes 3: S(z) = ∃z : χ − → E (z, z ) 4: Reachable(x, y) = (|x| ≤ |y|) ∧ ∀z : (|x| ≤ |z| < |y|) ⇒ S(z) 5: Reachable(x, y) = Reachable(x, y) ∧ (|x| < N ) ∧ (|y| < N ) // Compute matching 6: F (x) = ∃z, d : F irst(z) ∧ Reachable(z, x) ∧ (|x| − |z| = 2|d|) 7: M (x, y) = χ − → E (x, y) ∧ F (x) 8: χ M (x, y) = M (x, y) ∨ M (y, x) 9: return χ M
At first, the algorithm computes a directed path graph, i. e., a union of paths, which is a subgraph of the input graph and consists of the edges (x, y) with |x| − |y| = 1. As we have seen, for every connected component this path consists of all nodes within the component. Maximum matchings on vertex disjoint paths can be computed with O(log 2 N ) functional operations [6]. Here, we know that every path P consists of a consecutive sequence of nodes, i. e., P = (v i , . . . , v k ) for 0 ≤ i ≤ k ≤ N − 1. We can use this information to lower the number of functional operations: We compute the set of nodes which are starting nodes of the paths. Then we want to compute the connected components of the graph. Usually, this is done by computing the transitive closure, which needs O(log 2 N ) operations. Here, we can do it better: Two nodes x and y of the unit interval graph are connected iff every node z with |x| ≤ |z| < |y| has a successor, i. e., there is an edge (v |z| , v |z|+1 ) ∈ − → E . Having this information, we can compute the matching by adding every second edge of a path to the matching beginning with the first edge. To compute this set of edges on general paths, the distance of every node to the first node has to be computed which can be done by an iterative squaring approach with O(log 2 N ) functional operations [6]. Here, we can easily determine the set of edges by comparing the difference of two node labels due to the structure of the paths.
Implicit Coloring of Interval Graphs
Coloring refers to the task to color the nodes of a graph using the least number of colors, such that two adjacent nodes have different colors. Colorings of interval graphs are for example used in VLSI design (where it is called channel assignment). In the case of interval graphs, there is an easy greedy coloring algorithm: Sort the endpoints of the intervals (i. e., both left and right endpoints) in ascending order. At the beginning all colors are on a stack. Then color the intervals sequentially by traversing the sorted list and using the first color available on the stack when the current element is a left endpoint. As soon as we visit an right endpoint, we push the used color onto the top of the stack. This greedy algorithm is optimal and can be implemented to run in linear time by determining the order without sorting [28]. The parallel algorithm in [38] assigns weights to the endpoints and computes prefix sums to simulate the stack. In our implicit algorithm we can do the simulation in a more direct manner: We call two intervals I i = [a i , b i ] and I j = [a j , b j ] related iff b i < a j and I j is the first interval with the same color as I i in the greedy algorithm. The following easy observation helps us to compute this "related" relation implicitly.
Observation 4.4:
The intervals I i and I j are two related iff the number of right endpoints r with b i < r < a j is equal to the number of left endpoints l with b i < l < a j and for all intervals I j with b i < a j < a j the number of right endpoints r with b i < r < a j is not equal to the number of left endpoints l with b i < l < a j Now, in the case of unit intervals we want to show, how we can compute a function RELAT ED(x, y), which is 1 iff the interval I |x| and I |y| are related. The general case is discussed later in this section. As before, the intervals are labeled according to their left endpoints. Let RE(x, y, l) = 1 iff |x| ≤ |y| and the number of right endpoints between b |x| and a |y| is equal to |l|. Similarly, let LE(x, y, l) = 1 iff |x| ≤ |y| and the number of left endpoints between b |x| and a |y| is equal to |l|. Let χ E (x, y) be the characteristic function of the edge set of a unit interval graph G = (V, E) and χ E c (x, y) the characteristic function of the edge set of the complement graph, i. e., E c = {(u, v) | u = v and (u, v) ∈ E}. Then we can compute RE(x, y, l) and LE(x, y, l) in the following way:
H 1 (x, y, z) = (|x| ≤ |z| < |y|) ∧ χ E c (z, y) RE(x, y, l) = (|x| ≤ |y|) ∧ ∃z : H 1 (x, y, z) ∧ (|z| − |x| = |l|) ∧ ∃z : H 1 (x, y, z ) ∧ (|z | > |z|) H 2 (x, y, z) = (|x| < |z| ≤ |y|) ∧ χ E c (x, z) LE(x, y, l) = (|x| ≤ |y|) ∧ ∃z : H 2 (x, y, z) ∧ (|y| − |z| = |l|) ∧ ∃z : H 2 (x, y, z ) ∧ (|z | < |z|) .
The right endpoint of an interval I |z| is greater or equal than b |x| and less than a |y| iff |x| ≤ |z| < |y| and I |z| does not intersect I |y| . Since we are dealing with unit interval graphs, if for some z with |x| ≤ |z| < |y| the intervals I |z| and I |y| do not intersect, then it holds also for all z with |x| ≤ |z | < |z|. I. e., the maximal value of |z| − |x| over all z with the above property is equal to the number of right endpoints between b |x| and a |y| and, therefore, we compute the function RE(x, y, l) correctly. Similar arguments show that LE(x, y, l) is computed correctly, too.
Together with Observation 4.4, we can compute the function RELAT ED(x, y) as follows:
RELAT ED(x, y) = ∃l : RE(x, y, l) ∧ LE(x, y, l) ∧ ∃z, l : (|z| < |y|) ∧ RE(x, z, l ) ∧ LE(x, z, l ) Now, we have to compute the sequence of related intervals, which is nothing more than the transitive closure of the related relation, which can be computed with O(log 2 N ) functional operations. Finally, we have to assign a color to each interval, such that all intervals in a sequence of related intervals are getting the same color. In order to do this, we compute an order on the sequences of related intervals and assign the colors to the sequences according to that order. The order on the sequences is given by the order on the minimal interval number within the sequences. Putting all together, algorithm 4 computes a coloring on a unit interval graph. Proof. That the output is a coloring with the minimal number of colors follows directly from correctness of the greedy algorithm. The number of functional operations is dominated by the T ransitiveClosure and EnumerateOrder procedures. As we have seen in section 2.2, both procedures need O(log 2 N ) functional operations.
Algorithm 4 Implicit coloring algorithm for unit interval graphs Input: Unit interval graph (χ E , χ V ) Output: Coloring COLOR(x, l) with COLOR(x, l) = 1 iff I |x| has color |l|.
// Complement graph 1: χ E c (x, y) = χ V (x) ∧ χ V (y) ∧ χ E (x, y) ∧ (|x| = |y|)
// Auxiliary functions to compute the number of right/left endpoints // between two intervals 2: LE(x, y, l) = (|x| ≤ |y|) ∧ ∃z : H 2 (x, y, z) ∧ (|y| − |z| = |l|) ∧ ∃z : H 2 (x, y, z ) ∧ (|z | < |z|) // Compute related intervals 6: RELAT ED(x, y) = ∃l : RE(x, y, l) ∧ LE(x, y, l) ∧ ∃z, l : (|z| < |y|) ∧ RE(x, z, l ) ∧ LE(x, z, l ) // Compute set of intervals with the same color 7: SAM ECOLOR(x, y) = T ransitiveClosure(RELAT ED(x, y) ∨ (|x| = |y|)) // Order these sets 8: F IRST (x) = ∃x : SAM ECOLOR(x , x) ∧ (|x | < |x|) 9: COLORORDER(x, y) = ∃x , y : SAM ECOLOR(x , x) ∧ F IRST (x ) ∧ SAM ECOLOR(y , y) ∧ F IRST (y ) ∧ (|x | < |y |) // Assign the colors 10: COLOR(x, l) = EnumerateOrder(COLORORDER(x, y)) 11: return COLOR(x, l)
H 1 (x, y, z) = (|x| ≤ |z| < |y|) ∧ χ E c (z, y) 3: H 2 (x, y, z) = (|x| < |z| ≤ |y|) ∧ χ E c (x, z) //
The only difference between the unit interval and the general case is the computation of the functions LE and RE (this is the only place where we need the unity property). What we actually need, is an order on the sequence of right endpoints to compute RE and an order on the left endpoints of the intervals to compute LE (and in the case of unit intervals both orders are the same). Assuming that we label the intervals according to their left endpoints, we only need to compute the order on the right endpoints. Let EO(x, y) be this order, i. e., EO(x, y) = 1 iff b |x| ≤ b |y| . Remember the adjacency matrix of an interval graph from section 3 and assume that the left points of the intervals are the integers 0, . . . , N −1. We know that the interval with left endpoint i ∈ {0, . . . , N −1} has a maximal value j such that I i and I k intersect for all i ≤ k ≤ j. Therefore, the right endpoint of I i has to be in [j, j + 1). Let j and j be the maximal values such that I i intersects all I k with i ≤ k ≤ j and I i intersects all I k with i ≤ k ≤ j , respectively. If j < j (j > j ), then b i < b i (b i > b i ). If j = j , then we can break ties arbitrary Since all additional operations are dominated by the EnumarateOrder procedure, we get the same result as for unit intervals.
Experimental Evaluation
We evaluated the implicit maximum matching algorithm on unit interval graphs and the implicit coloring algorithm on unit and general interval graphs. Unfortunately, the implicit coloring algorithm performed poorly even on instances of size around 2000. Therefore, we only show the results for the maximum matching algorithm but want to begin with a discussion of this performance difference: At a first glance, this might not be surprising due to the more complex coloring algorithm but having a closer look we see that the implicit matching algorithm is optimized for the implicit setting while the implicit coloring algorithm uses some nice ideas to simulate the sequential algorithm. Hence, these results do not rule out the possibility of an efficient implicit coloring algorithm but suggest that there have to be new ideas to benefit more from the strengths of implicit algorithms.
Unit interval graphs can be represented as balanced nonnegative strings over {'[', ']'} (see, [30]) and such strings are created randomly using the algorithm in [1]. We generated 35 random graphs of size 2 i for i = 10, . . . , 23. The nodes of the graphs are encoded as in Section 3. We compare the OBDD-based algorithm to the algorithm which gets the interval representation as an input, sort the intervals according to their starting point and compute a maximum matching by scanning this sorted sequence with the same idea used in the implicit algorithm.
Experimental Setup
We implemented the implicit algorithm with the BDD framework CUDD 2.5.0 1 by F. Somenzi. The algorithms are implemented in C++ and were compiled with Visual Studio 2012 in the default release configuration. All source files, scripts and random seeds are publicly available 2 . The experiments were performed on a computer with a 2.6 GHz Intel Core i5 processor and 4 GB main memory running Windows 7. The runtime is measured by used processor time in seconds and the space usage of the implicit algorithm is given by the maximum SBDD size which came up during the computation, where a SBDD is a collection of OBDDs which can share nodes. Due to the small variance of these values, we only show the mean in the diagrams.
Results
The implicit matching algorithm outperforms the explicit matching algorithm on unit interval graphs (see Fig. 6). Even on graphs with more than 8 million nodes the implicit algorithm computes a maximum matching within 1 seconds. Storing a SBDD of size S needs O(S log S) bits. The memory diagram shows that the asymptotic space usage of the implicit algorithm on these instances is close to O(N ). Recall that the unit interval representation needs Θ(N log N ) space since log N bits are needed to represent the starting points. I. e., the implicit algorithm needs less space and can compute a maximum matching on larger instances than the explicit one. An interesting consequence of these results is that the submodules of our maximum matching algorithm, namely computing the connected components, a Hamiltonian path in every connected component and a maximum matching on these paths, are also very fast and space efficient which is surprising, since especially the computation of the transitive closure is often a bottleneck in implicit algorithms.
Conclusion and Open Questions
In this paper, we presented a method to show upper bounds of the size of OBDDs representing a graph by using the adjacency matrix. Using this method, we could improve known results on the OBDD size of interval graphs and we think that it is possible to show similar results for other graph classes with a well structured adjacency matrix, e. g., convex graphs where the nodes can be ordered such that the neighborhood of every node consists of nodes which are consecutive in this order. The gap between the upper and lower bound (using another labeling or variable order) of the OBDD size of interval graphs is O(log N ). It is an interesting open question whether there is another labeling and/or variable order such that the OBDD size is O(N ) or the general lower bound can be increased to Ω(N log N ), which we believe is more likely, since the observation that the columns of the π-ordered adjacency matrix are independent also holds for an arbitrary labeling. Even for a fixed variable order, the complexity of computing a node labeling for a given graph, such that the representing OBDD has minimal size, is unknown. The πordered adjacency matrix seems to help to prove upper/lower bounds on the OBDD size for a fixed labeling. Using this matrix to bound the size of OBDDs for every labeling could be object of further research.
The parallel maximum matching algorithm on general interval graphs [11] is more complex than the parallel maximum matching algorithm on unit interval graphs and the algorithm does not seem to be directly applicable to develop an implicit algorithm as for unit intervals.
The investigation of implicit algorithms on special graph classes seems quite promising and it would be interesting if the good performance can also be achieved for other large graph classes.
denote the set of Boolean functions f : {0, 1} n → {0, 1} by B n . Let (x 0 , . . . , x n−1 ) = x ∈ {0, 1} n be a binary number of length n and |x| := n−1 i=0 x i · 2 i the value of x. Further, let l ∈ N be a natural number then we denote by [l] 2 the corresponding binary number of l, i. e., |[l] 2 | = l.
Definition 2 . 1 (
21Ordered Binary Decision Diagram (OBDD)): Order. A variable order π on the input variables X = {x 0 , . . . , x n−1 } of a Boolean function f ∈ B n is a permutation of the index set I = {0, . . . , n − 1}.
Figure 1 :
1A minimal π-OBDD representing the function GT (x, y) with GT (x, y) = 1 iff |x| > |y| and π = (x 1 , y 1 , x 0 , y 0 )1. Negation: Given G f , compute an OBDD representing the function f ∈ B n . Time: O(1) 2. Replacement by constant: Given G f , an index i ∈ {0, . . . , n−1}, and a Boolean constant c i ∈ {0, 1}, compute an OBDD representing the subfunction f |x i =c i . Time: O(|G f |) 3. Synthesis: Given G f and G g and a binary Boolean operation ⊗ ∈ B 2 , compute an OBDD representing the function h ∈ B n defined as h := f ⊗g. Time: O(|G f |·|G g |) 4. Quantification: Given G f , an index i ∈ {1, . . . , n} and a quantifier Q ∈ {∃, ∀}, compute an OBDD representing the function h ∈ B n defined as h := Qx i : f where ∃x i : f := f |x i =0 ∨ f |x i =1 and ∀x i : f := f |x i =0 ∧ f |x i =1 . Time: see replacement by constant and synthesis
Figure 2 :
2Example of a set of unit intervals and the corresponding unit interval graph function f := χ E of the edge set of a graph using the adjacency matrix of the graph.
Figure 3 :
3The π-ordered adjacency matrix of the unit interval graph from Fig. correspond to the subfunctions where the first bit is replaced by a constant. number of replaced x-and y-variables is exactly i/2. Now, we can see that the block B i/2 |a|,|b| represents the function table of the subfunction which results from replacing the x-variables by a ∈ {0, 1} i/2 and the y-variables by b ∈ {0, 1} i/2 . Therefore, counting the number of different blocks B i/2 |a|,|b| is equivalent to counting the number of different subfunctions.
Figure 4 :
4Possible adjacency matrix with 8 nodes and framed subfunctions f |α,β with β = 0 k , |α| ≥ 1, and k = 2.
Figure 5 :
5Random interval graph where only R is generated randomly
function f : {0, 1} kn → {0, 1} with k input variable vectors x (1) , . . . , x (k) ∈ {0, 1} n of length n is called k-variate threshold function, if there exist a threshold T ∈ Z and W ∈ N and weights w 1 , . . . , w k ∈ {−W, . . . , W } such that
I
. e., if in phase i it is either c i +Q ≥ −T n +(kW +1)/2 n−i or c i +Q < −T n −(kW +1)/2 < −T n respectively. So our algorithm works correctly.
Algorithm 3 computes a maximum matching for unit interval graphs using O(log N ) functional operations.Proof. As we have seen in the beginning of this section, every connected component has always a path which consists of consecutive nodes and visits every node in this component. The algorithm computes such paths and construct a maximum matching of each path. Clearly, the union of these matchings is a maximum matching of the complete graph.The number of functional operations is determined by the lines 2-4 where we use quantifications over O(log N ) variables. Otherwise, there is only a constant number of operations.
Theorem 4. 5 :
5Algorithm 4 computes a coloring of a unit interval graph using the minimal number of colors and O(log 2 N ) functional operations.
Number of right endpoints between b |x| and a |y| 4: RE(x, y, l) = (|x| ≤ |y|) ∧ ∃z : H 1 (x, y, z) ∧ (|z| − |x| = |l|) ∧ ∃z : H 1 (x, y, z ) ∧ (|z | > |z|) // Number of left endpoints between b |x| and a |y| 5:
Algorithm 4 with the modified computation of RE(x, y, l) outputs a coloring of an interval graph using the minimal number of colors and O(log 2 N ) functional operations.
Figure 6 :
6Runtime and memory of the matching algorithms on random unit interval graphs. Memory plot shows the ratio of S log S (space usage of the OBDDbased algorithm) and number of nodes.
http://vlsi.colorado.edu/~fabio/CUDD/ 2 http://ls2-www.cs.uni-dortmund.de/~gille/
AcknowledgementsI thank Beate Bollig, Melanie Schmidt and Chris Schwiegelshohn for the valuable discussions and, together with the anonymous referees, for their comments on the presentation of the paper.
b i ≤ b i iff i ≤ i ). Now, we can compute EO. x, y) as followse. g., b i ≤ b i iff i ≤ i ). Now, we can compute EO(x, y) as follows:
= (|x| ≤ |x |) ∧ (|y| ≤ |y |) ∧ χ E (x, x ) ∧ χ E (y, y ) EO(x, y) = ∃x. ) ∧ ((|x | > |x |) ∨ (|y | > |y |). H(x, y, x , y; y : H(x, y, x , y; y : H(x, y, x , y) ∧ (|x | < |y | ∨ (|x | = |y | ∧ |x| < |y|)) ∧ ∃xH(x, y, x , y ) = (|x| ≤ |x |) ∧ (|y| ≤ |y |) ∧ χ E (x, x ) ∧ χ E (y, y ) EO(x, y) = ∃x , y : H(x, y, x , y ) ∧ (|x | < |y | ∨ (|x | = |y | ∧ |x| < |y|)) ∧ ∃x , y : H(x, y, x , y ) ∧ ((|x | > |x |) ∨ (|y | > |y |))
Notice that this order on the right endpoints does not have to be the same order on the original right endpoints. But, as we have shown, there is an interval representation of the graph, such that the left and right endpoints are ordered according to labels of the nodes and EO(x, y), respectively. Finally. we have to compute EEO(x, l) =Notice that this order on the right endpoints does not have to be the same order on the original right endpoints. But, as we have shown, there is an interval representation of the graph, such that the left and right endpoints are ordered according to labels of the nodes and EO(x, y), respectively. Finally, we have to compute EEO(x, l) =
we get RE(x, y, l) for general interval graphs: H 1 (x, y, z) = EO(z, x) ∧ EO(z, y) ∧ χ E c (z, y) RE(x, y, l) = (|x| < |y|) ∧. ( Enumerateorder, Eo(x, ∃z, l 1 , l 2 : H 1 (x, y, z) ∧ EEO(x, l 1 ) ∧EnumerateOrder(EO(x, y)) and, with it, we get RE(x, y, l) for general interval graphs: H 1 (x, y, z) = EO(z, x) ∧ EO(z, y) ∧ χ E c (z, y) RE(x, y, l) = (|x| < |y|) ∧ [∃z, l 1 , l 2 : H 1 (x, y, z) ∧ EEO(x, l 1 ) ∧
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| []
|
[
"Electroluminescence from multi-particle exciton complexes in transition metal dichalcogenide semiconductors",
"Electroluminescence from multi-particle exciton complexes in transition metal dichalcogenide semiconductors"
]
| [
"Matthias Paur \nInstitute of Photonics\nVienna University of Technology\nGußhausstraße 27-291040ViennaAustria\n",
"Aday J Molina-Mendoza [email protected] \nInstitute of Photonics\nVienna University of Technology\nGußhausstraße 27-291040ViennaAustria\n",
"Rudolf Bratschitsch \nInstitute of Physics and Center for Nanotechnology\nUniversity of Münster\nWilhelm-Klemm-Strasse 1048149MünsterGermany\n",
"Kenji Watanabe \nNational Institute for Materials Science\n1-1 Namiki305-0044Tsukuba\n",
"Takashi Taniguchi \nNational Institute for Materials Science\n1-1 Namiki305-0044Tsukuba\n",
"Thomas Mueller \nInstitute of Photonics\nVienna University of Technology\nGußhausstraße 27-291040ViennaAustria\n"
]
| [
"Institute of Photonics\nVienna University of Technology\nGußhausstraße 27-291040ViennaAustria",
"Institute of Photonics\nVienna University of Technology\nGußhausstraße 27-291040ViennaAustria",
"Institute of Physics and Center for Nanotechnology\nUniversity of Münster\nWilhelm-Klemm-Strasse 1048149MünsterGermany",
"National Institute for Materials Science\n1-1 Namiki305-0044Tsukuba",
"National Institute for Materials Science\n1-1 Namiki305-0044Tsukuba",
"Institute of Photonics\nVienna University of Technology\nGußhausstraße 27-291040ViennaAustria"
]
| []
| Light emission from higher-order correlated excitonic states has been recently reported in hBN-encapsulated monolayer WSe 2 and WS 2 upon optical excitation. These exciton complexes are found to be bound states of excitons residing in opposite valleys in momentum space, a promising feature that could be employed in valleytronics or other novel optoelectronic devices. However, electrically-driven light emission from such exciton species is still lacking. Here we report electroluminescence from bright and dark excitons, negatively charged trions and neutral and negatively charged biexcitons, generated by a pulsed gate voltage, in hexagonal boron nitride encapsulated monolayer WSe 2 and WS 2 with graphene as electrode. By tailoring the pulse parameters we are able to tune the emission intensity of the different exciton species in both materials. We find the electroluminescence from charged biexcitons and dark excitons to be as narrow as 2.8 meV. | 10.1038/s41467-019-09781-y | null | 85,459,555 | 1812.03832 | 26bff6210c1867cb9be0bff394780eb675485971 |
Electroluminescence from multi-particle exciton complexes in transition metal dichalcogenide semiconductors
Matthias Paur
Institute of Photonics
Vienna University of Technology
Gußhausstraße 27-291040ViennaAustria
Aday J Molina-Mendoza [email protected]
Institute of Photonics
Vienna University of Technology
Gußhausstraße 27-291040ViennaAustria
Rudolf Bratschitsch
Institute of Physics and Center for Nanotechnology
University of Münster
Wilhelm-Klemm-Strasse 1048149MünsterGermany
Kenji Watanabe
National Institute for Materials Science
1-1 Namiki305-0044Tsukuba
Takashi Taniguchi
National Institute for Materials Science
1-1 Namiki305-0044Tsukuba
Thomas Mueller
Institute of Photonics
Vienna University of Technology
Gußhausstraße 27-291040ViennaAustria
Electroluminescence from multi-particle exciton complexes in transition metal dichalcogenide semiconductors
10.1038/s41467-019-09781-yARTICLE OPEN These authors contributed equally: Matthias Paur, Aday J. Molina-Mendoza. Correspondence and requests for materials should be addressed to A.J.M-M. ( 1
Light emission from higher-order correlated excitonic states has been recently reported in hBN-encapsulated monolayer WSe 2 and WS 2 upon optical excitation. These exciton complexes are found to be bound states of excitons residing in opposite valleys in momentum space, a promising feature that could be employed in valleytronics or other novel optoelectronic devices. However, electrically-driven light emission from such exciton species is still lacking. Here we report electroluminescence from bright and dark excitons, negatively charged trions and neutral and negatively charged biexcitons, generated by a pulsed gate voltage, in hexagonal boron nitride encapsulated monolayer WSe 2 and WS 2 with graphene as electrode. By tailoring the pulse parameters we are able to tune the emission intensity of the different exciton species in both materials. We find the electroluminescence from charged biexcitons and dark excitons to be as narrow as 2.8 meV.
M onolayer transition metal dichalcogenide (TMD) semiconductors provide a unique platform to study light-matter interaction and many-body effects at the atomic scale. The strong Coulomb interaction in these materials leads to the formation of tightly bound electron-hole pairs (excitons) with binding energies of hundreds of millielectronvolts [1][2][3] . Moreover, excitons residing in the two different valleys at the K points of the Brillouin zone can also interact [4][5][6][7][8][9][10] , giving rise to four-and five-particle states, which have been recently identified in tungsten diselenide (WSe 2 ) as neutral and charged biexcitons [11][12][13][14] , as well as biexcitons in WS 2 15,16 . The radiative emission from such exciton complexes can further be tuned by an external electric field, modifying the doping of the material and thus favoring the formation of either charged or neutral complexes. This rich and complex excitonic scenario could be exploited to develop applications such as valleytronics and (quantum) optoelectronic devices, especially in the case of electrically-driven light emitters. Electroluminescence (EL) from monolayer TMDs has been widely studied in the past, especially at room temperature, where the emission originates from neutral excitons [17][18][19][20][21][22][23][24][25][26] . At cryogenic temperatures, the emission can also be due to localized exciton states or negatively charged trions [27][28][29][30][31][32][33] . However, electrically driven emission from more complex exciton species has not yet been observed.
Here, we report on EL from monolayer WSe 2 and WS 2 by pulsed transient EL 34 , which triggers the formation of exciton complexes and thus their light emission. The high sample quality, enabled by encapsulating the monolayer semiconductor in hexagonal boron nitride (hBN), allows for the observation of EL from multi-particle exciton complexes, including neutral and negatively charged biexcitons, with narrow emission linewidths down to ∼2.8 meV (full-width-at-half-maximum (FWHM)). Furthermore, the formation of different exciton species is tunable by tailoring of the electrical pulse parameters. Our results provide crucial insights into electrically-driven light emission in monolayer TMDs and open a new route to applications in novel optoelectronic devices.
Results
Sample structure and light emission at room temperature. The sample structure used to study EL in monolayer TMDs is schematically depicted in Fig. 1a. A microscope image of the WS 2 sample presented in this article is shown in Fig. 1b (a microscope image of the WSe 2 sample is shown in Supplementary Fig. 1): the monolayer TMD (namely WSe 2 or WS 2 ) and graphene are encapsulated in between two multilayer hBN flakes. The graphene layer is used as electrical contact to the monolayer semiconductor. Part of the graphene flake is left un-encapsulated in order to make an electrical contact with a Pd/Au electrode. The top hBN serves as encapsulation 35 to avoid environment related effects (residues left after sample processing, adsorbents) on the light emission from TMDs, and the bottom hBN also provides an insulating layer to apply a gate voltage to the sample by means of a prepatterned Ti/Au back-gate electrode, on top of which the entire heterostructure is placed. Graphene was chosen as electrode for two reasons: (i) it is a layered material that can be easily integrated in a van der Waals heterostructure, and (ii) it has been shown that charge carrier injection from graphene for transient EL is more efficient than from standard metals 34 , resulting in enhanced emission. Details of the sample fabrication can be found in the Methods section.
This sample structure (metal-insulator-semiconductor-graphene) allows for electrically driven light emission by the recently reported method of transient EL in transition metal dichalcogenides 34 , in which light emission is achieved by alternated injection of electrons and holes into the semiconductor by means of a pulsed gate voltage. In Fig. 1c we show band diagrams that schematically depict the emission process. The shape of the (~9 ns long) electrical pulses used for EL is schematically shown in the inset of Fig. 1c. The gate voltage sweeps from a negative value (V 0 ), where the Fermi level in graphene is below the Dirac point and holes are accumulated in the valence band of the semiconductor (Fig. 1c left), to a positive value (V 1 ), where the Fermi level in graphene shifts upwards in energy and electrons are injected into the conduction band (Fig. 1c right). The steep (lateral) band bending in the TMD due to the large voltage drop at the Schottky contact leads to a transient tunneling current. As a result, injected electrons diffuse in the semiconductor while holes (radiatively) recombine with incoming electrons or partially drift out through the contact 34 . This process is known as transient EL, as the injected carriers recombine only during the rise and fall periods of the pulse. The light generation can be depicted from Fig. 1d, where we show an optical microscopy picture of the sample presented in Fig. 1b, with a clear red-orangish light emission originating from the WS 2 flake.
The samples are characterized at room temperature by their PL spectrum ( Fig. 1e) upon optical excitation with a continuouswave laser (λ = 532 nm, P d = 2 × 10 3 Wcm −2 ), where we observe an emission maximum centered at ∼1.653 eV for WSe 2 and at ∼2.003 eV in the case of WS 2 , corresponding to their respective free exciton emission (comparison between spatial resolution of PL and EL is presented in Supplementary Figs. 1a and 2). Similar spectra are observed when an AC voltage is applied between the back-gate and the graphene layer, inducing transient EL near the interface between the TMD monolayer and graphene.
Electro-and photoluminescence at T = 5 K. At room temperature, only one emission maximum can be identified in the PL and EL spectra of both WSe 2 and WS 2 . However, the scenario becomes more complex when the sample is studied at cryogenic temperatures. At T = 5 K different emission peaks can now be distinguished in the EL spectra of both materials. In WSe 2 ( Fig. 2a), apart from the well-known bright exciton (X 0 ) centered at 1.7315 eV, the EL spectrum reveals the emission lines associated with other exciton complexes, previously observed upon optical excitation [11][12][13][14] . On one hand, we are able to distinguish the two negatively charged trions at 1.7007 and 1.6921 eV, attributable to the intravalley (X À 1 ) and intervalley trions (X À 2 ), respectively 36 , and the spin-forbidden dark exciton (X D ) at 1.6864 eV, which is the two-particle ground state exciton 6,37,38 . Furthermore, the neutral biexciton (XX) centered at 1.7121 eV, a four-particle complex, is formed from a long-lived dark (spinforbidden) and a bright (spin-allowed) exciton, while the charged biexciton centered at 1.6808 eV (XX − ), a five-particle complex, is formed from a negatively charged trion and a neutral exciton in different K and K′ valleys 8,[11][12][13][14] . It is worth mentioning that the pulsed EL from WSe 2 shows a narrow emission linewidth of ∼2.8 meV FWHM only for both the charged biexciton and the dark exciton. Finally, we also observe a broad peak at ∼1.675 eV which may be assigned to a positively or negatively charged dark trion (X D± ) 39 . At energies lower than ∼1.67 eV, additional emission peaks (labelled L 1 , L 2 , and L 3 ) emerge that most probably stem from localized states 37,[40][41][42][43][44] . In Table 1, we summarize the energy of the different exciton complexes with respect to the bright exciton X 0 .
A similar scenario is uncovered in the EL spectrum of WS 2 , shown in Fig. 2b (Table 1 provides again a summary of the exciton energies). The bright exciton X 0 appears at an energy of 2.075 eV, as well as the charged trions X À 1 and X À 2 at 2.045 and 2.038 eV, respectively. The peak appearing at 2.023 eV has previously been associated to a biexciton 15,16 , although its charged or neutral nature remains still unclear. Here we anticipate that it is due to a negatively charged biexciton (XX − ), since its peak height (emission intensity hereafter) increases when a positive gate voltage is applied to the sample, as will be discussed later. Furthermore, we are able to observe additional emission lines corresponding to a dark exciton (X D ), centered at 2.028 eV (only observed so far in the PL of non-encapsulated samples 45 ) and to a neutral biexciton (XX), centered at 2.056 eV, which, to our knowledge, has not been previously reported. Importantly, the latter two exciton complexes are hardly visible in the PL spectrum, but can be clearly resolved in the respective EL. To our knowledge, this is the first evidence of EL of such multi-particle complexes in both WSe 2 and WS 2 .
The rich EL scenario in our samples, based on charged and neutral excitonic complexes, motivates the operation of such heterostructures as tunable light emitters by just tailoring the pulse parameters. By shifting the offset of the pulsed voltage, it is possible to create either an electron-rich or a hole-rich environment in the 2D semiconductor, consequently enhancing or diminishing EL from the different exciton complexes in a similar way as it is commonly observed in gated PL measurements. To this end, according to the schematic pulse diagram shown in Fig. 1c, the EL experiment is performed as follows (see also Supplementary Fig. 3): the pulse amplitude V p = |V 1 − V 0 | is kept constant in all measurements, while the offset voltage V 0 is varied over a certain range. In this situation, an offset of V 0 = 0 indicates that the voltage applied to the back-gate sweeps in a pulse cycle from 0 to V p , while with an offset of V 0 = −V p , the voltage applied to the back-gate sweeps from −V p to 0. Therefore, in the former case an electron-rich (p − /n + ) environment is induced in the semiconductor, while in the latter the injected charge carrier density will be hole-rich (p + /n − ). Finally, in the symmetric case, V 0 = −V p /2, the amount of injected electrons and holes is balanced (p/n). Note that this picture applies to the case of undoped devices with contacts that are equally transparent for both electrons and holes. In realistic samples the different regimes may occur at other offset voltages, but the qualitative picture remains the same.
In Fig. 2c, d we show colormaps representing the EL emission from WSe 2 and WS 2 , respectively, for different V 0 values and with fixed pulse amplitude of V p = 7.0 V. We observe in both materials that for higher offset values (p − /n + region), i.e. higher electron injection, the EL is dominated by negatively charged exciton species such as XX − , X À 1 , and X À 2 , where the charged biexciton is the predominant emission and the free exciton is much weaker. For lower offset values (p/n region), the emission from negatively charge exciton complexes vanish and the spectrum is then dominated by neutral species such as the dark and bright excitons and neutral biexciton, as well as localized excitons. This indicates that the charge carrier injection is balanced between electrons and holes, favoring the formation of neutral excitons. Finally, when the offset voltage has a more negative value (p + /n − region), the The bottom picture schematically depicts the shape of the pulse. d Optical microscopy image of the sample while emitting light. The EL emission is generated near the interface between graphene and monolayer WS 2 due to carrier diffusion from graphene. Scale bar, 20 μm. e Photoluminescence (PL) and electroluminescence (EL) emission spectrum (excitation λ = 532 nm, P d = 2 × 10 3 Wcm −2 ). The EL spectrum closely matches the PL spectrum induced charge density in the semiconductor is then hole-rich and the emission from positively charged species such as the positive dark trion in WSe 2 becomes stronger. We are not able to resolve positively charged trions in EL neither in WSe 2 or WS 2 , possibly because the induced hole concentration is too low to favor trion formation.
The tunable EL can be compared to the respective PL colormaps (Fig. 2e, f), where the PL emission is measured for different DC gate voltages (V g ). In a similar manner as for EL, negative exciton species become the dominant emission for positive applied gate voltages (V g > 0), i.e. for n-doping, although in the PL spectra we are able to additionally resolve the negative dark trion in WSe 2 and the double-negatively charged trion (X −− ), the next charging state of the trion 13 , in WS 2 . In the WS 2 spectra we also observe XX − emission only in the n-doped region, supporting the negatively charged nature of this biexciton as concluded above from the EL measurements. Around V g = 0, the negatively charged species vanish and only the neutral excitons and biexciton, as well as the localized states, show emission, again in accordance with the EL spectra. Here it is worth noting that we are not aware of any previous reports on the PL emission of neutral biexcitons in WS 2 , which we are able to resolve both in EL and PL. Finally, for V g < 0, the positive dark trion appears in the WSe 2 PL spectrum, but not in the one of WS 2 , suggesting that WS 2 is unintentionally n-doped and we therefore cannot achieve sufficient p-doping in this material. We refer the reader to the Supplementary Figs. 4 Table 1 Energy of the different exciton complexes with respect to the bright exciton X 0 appearing in the EL spectra for a further analysis of the PL spectra as a function of the backgate voltage.
X 0 (eV) X 0 − XX (meV) X 0 − X À 1 (meV) X 0 − X À 2 (meV) X 0 − X D (meV) X 0 − XX − (meV)
Discrimination between different multi-particle excitons. The electrical tunability of the EL and PL spectra permits the discrimination between charged and neutral exciton species. Whether they are formed by two, three, four or five particles cannot be distinguished by this method. We therefore studied the dependence of the emission intensity with the excitation intensity. While the emission intensities of dark and bright excitons, composed by one electron and one hole, follow a linear dependence with the excitation power, biexcitons are expected to show a quadratic behavior [11][12][13][14][15][16] . In Supplementary Figs. 6, 7, and 8 we show the PL spectra of WSe 2 and WS 2 , at different excitation powers and fits of the respective intensities to the power-law I PL ∝P α , where I PL is the integrated PL emission and P the incident laser power. These fits yield the exponent values α = 1.01 for X 0 , α = 1.82 for XX, and α = 1.44 for XX − in WSe 2 , and similar values in WS 2 (see Supplementary Table 1). The deviation from the quadratic growth for the charged biexciton can be attributed to the lack of equilibrium between exitonic states 46,47 .
We have also studied the dependence of the emission intensity on the excitation power in EL by keeping a fixed offset voltage and changing the electrical pulse amplitude. For increasing pulse amplitude, the Fermi level shifts deeper in the conduction band and, therefore, more charge carriers are injected, resulting in increased EL emission. The measured spectra are shown in Fig. 3a, c for WSe 2 and WS 2 , respectively, and the corresponding plots of EL intensities against pulse amplitude are presented in Fig. 3b, d. On a semi-logarithmic scale, the dependence of EL intensity versus voltage shows a linear behavior for all exciton species. This observation motivates us to empirically model the EL intensity by the expression I EL / expðακV p Þ, where V p is the applied voltage, κ is a constant related to the injected carrier concentration in the 2D semiconductor, and α is a coefficient that indicates the nature of the exciton, as in the PL measurements. We fix κ to fit the data for the neutral exciton X 0 with α ≡ 1, which then serves as reference for determining the α values of all other exciton species.
The fit of the measurement data to this empirical expression yields for WSe 2 a coefficient of α = 1.75 for XX − , which is significantly larger than that of the X 0 and X D excitons (α ≡ 1 and α = 1.03, respectively) and it is close to a quadratic dependence. For the neutral biexciton XX we obtain a coefficient of α = 1.44. For WS 2 , the coefficients for X 0 , X À 1 , and X À 2 are α ≡ 1, α = 1.32, and α = 1.33, respectively, and α = 2.33 and α = 2.31 for the neutral biexciton XX and charged biexciton XX − . In Supplementary Table 1 we summarize the obtained coefficients for the different exciton complexes. The emission of all exciton species in WS 2 saturate at voltage amplitudes larger than~8 V, in accordance with measurements performed at room temperature by Lien et al. 34 . We do not observe such saturation in WSe 2 , most probably because of the lower peak voltage applied in these measurements. Finally, we present EL spectra for both WSe 2 and WS 2 at different pulse repetition rates in Supplementary Fig. 9.
Discussion
In summary, we have studied transient EL of monolayer TMD semiconductors (in particular WSe 2 and WS 2 ) sandwiched between two layers of hBN and one graphene sheet at low temperatures. The high material quality of our samples, together with the strong Coulomb interaction between electrons and holes in 2D semiconductors, permits the observation of electrically driven light emission of higher-order correlated excitonic states, including bright and dark excitons, negatively charged trions and neutral and negatively charged biexcitons. By tailoring the pulse parameters, it is possible to create either an electron-or a holerich environment in the 2D semiconductor, consequently favoring the enhanced or diminished EL from the different exciton species. Our technique extends and complements gate-dependent PL spectroscopy and will enable further investigations of manybody phenomena in 2D materials. From an applied point of view, our devices may find application as wavelength tunable light emitters or furnish new opportunities for quantum light sources, e.g. by quantum-confinement of electrically induced biexcitons.
Methods
Sample fabrication. Bulk WSe 2 (grown by vapor phase transport; see ref. 48 for details) and WS 2 (purchased from HQ Graphene), graphite and hBN were exfoliated by an adhesive tape and transferred onto a SiO 2 substrate thermally grown on Si. The substrates were cleaned by an acetone/isopropanol bath and oxygen plasma treatment. They were then annealed at 300°C for 2 h in a dry-air atmosphere prior to the transfer of the exfoliated materials to remove possible adsorbates. The tape was peeled-off afterward and the atomically-thin flakes were selected with optical microscopy. A pick-up and place technique 49 was used to pick up the flakes by means of a polypropylene carbonate/polydimethylsiloxane (PPC/ PDMS) stamp, starting with the top hBN flake. The heterostructure (hBN/graphene/WSe 2 /hBN) was then transferred onto a Ti/Au (5/100 nm) electrode, prepatterned on a SiO 2 /Si substrate. The entire exfoliation and transfer process was performed in a nitrogen-purged glovebox. The remaining PPC residues were removed by introducing the sample in an acetone/chloroform bath. Finally, an additional electrode was fabricated by means of electron-beam lithography and metal evaporation to make an electrical contact to the graphene flake (Pd/Au, 60/ 140 nm).
PL and EL measurements. For EL measurements a pulse generator (Agilent 8114A) in combination with a source-meter unit (Keithley 2612A) was used. The pulses were applied to the bottom electrode, while the top contact was grounded.
Optical and electrical investigations were carried out in a liquid-helium exchange gas cryostat (Oxford Instruments) under high vacuum (10 −7 mbar), integrated in a scanning micro-PL setup. The setup includes a dichroic mirror at λ = 550 nm to separate the PL emission from the excitation. PL and EL were collected using a ×50 long working distance objective lens (NA = 0.5). In PL experiments, the sample was optically excited non-resonantly using a continuous-wave 532 nm laser; the diameter of the laser beam on the sample was~1 μm. The PL/EL emission was spectrally filtered with a monochromator (Horiba iHR320) and detected with a liquid nitrogen cooled charge-coupled device (CCD). EL imaging was performed by using a CCD camera (Thorlabs).
Data availability
The data that support the findings of this study are available from the authors upon reasonable request; see authors contributions for specific data.
Received: 6 December 2018 Accepted: 1 April 2019
Fig. 1
1Sample structure and light emission at T = 300 K. a Schematic drawing of the sample. The monolayer TMD and graphene are sandwiched between two multilayer hBN flakes. b Optical microscope image of a typical sample, in this case based on WS 2 . The cross marks the position at which the PL spectra were taken. Scale bar, 5 μm. c Band diagram of transient EL. On the left, the pulse voltage is at a negative value, while on the right it is at a positive value.
Photon energy (eV)
Fig. 2
2Emission spectra and tunability at T = 5 K. Pulsed EL from a WSe 2 and b WS 2 . In WSe 2 , a narrow emission (linewidth ∼2.8 meV FWHM) is found for both the charged biexciton and the dark exciton. Colormap of EL spectra at 5 K plotted as function of pulse offset V 0 in c WSe 2 and d WS 2 (V p = 7.5 V). Colormap of charged doping control of PL emission in e WSe 2 and f WS 2 , for different exciton species acquired at an excitation of P d = 5 × 10 3 Wcm −2
Fig. 3
3EL intensity dependence with the excitation power. a Pulsed EL emission spectra of WSe 2 for different pulse amplitudes and fixed offset voltage. b Semi-logarithmic plot of the integrated EL intensity as a function of the pulse amplitude. The circles represent the experimental data, while the lines depict the fit to an exponential function as discussed in the main text. c, d Same as in a, b but for | (2019) 10:1709 | https://doi.org/10.1038/s41467-019-09781-y | www.nature.com/naturecommunications
NATURE COMMUNICATIONS | (2019) 10:1709 | https://doi.org/10.1038/s41467-019-09781-y | www.nature.com/naturecommunications
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09781-y
© The Author(s) 2019
AcknowledgementsWe acknowledge financial support by the Austrian Science Fund FWF (START Y 539-N16) and the European UnionAuthor contributionsAdditional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-019-09781-y.Competing interests: The authors declare no competing interests.Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ Journal peer review information: Nature Communications thanks Matteo Barbone and the other anonymous reviewers for their contribution to the peer review of this work.Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
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"Effects of high order interatomic potential on elastic phonon scatterings",
"Effects of high order interatomic potential on elastic phonon scatterings"
]
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"Hangbo Zhou \nInstitute of High Performance Computing\nA*STAR\n138632Singapore\n",
"Gang Zhang \nInstitute of High Performance Computing\nA*STAR\n138632Singapore\n",
"Yong-Wei Zhang \nInstitute of High Performance Computing\nA*STAR\n138632Singapore\n"
]
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"Institute of High Performance Computing\nA*STAR\n138632Singapore",
"Institute of High Performance Computing\nA*STAR\n138632Singapore",
"Institute of High Performance Computing\nA*STAR\n138632Singapore"
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| []
| Interatomic potentials beyond quadratic order provide scattering sources for phonon transport in lattice. By using a weakly-interacting interface model, we investigated the relation between the order of interatomic potential and the multiple-phonon scattering process. We find that high order interatomic potential not only causes multiple-phonon scattering processes, but also has significant impacts on elastic phonon scattering processes. Using fourth order potential as an example, we show that it can significantly affects elastic phonon scatterings, through the formation of localized phonons. Such impact is closely related to the correlations of interfacial atoms and it becomes more significant with increasing temperature. Our work suggests that it is insufficient to consider only quadratic potential to investigate elastic phonon transport. | null | [
"https://arxiv.org/pdf/2202.11037v1.pdf"
]
| 247,026,072 | 2202.11037 | 0e5b54f043d70568fc11c2f5e6e0046c014a80e7 |
Effects of high order interatomic potential on elastic phonon scatterings
22 Feb 2022
Hangbo Zhou
Institute of High Performance Computing
A*STAR
138632Singapore
Gang Zhang
Institute of High Performance Computing
A*STAR
138632Singapore
Yong-Wei Zhang
Institute of High Performance Computing
A*STAR
138632Singapore
Effects of high order interatomic potential on elastic phonon scatterings
22 Feb 2022(Dated: February 23, 2022)
Interatomic potentials beyond quadratic order provide scattering sources for phonon transport in lattice. By using a weakly-interacting interface model, we investigated the relation between the order of interatomic potential and the multiple-phonon scattering process. We find that high order interatomic potential not only causes multiple-phonon scattering processes, but also has significant impacts on elastic phonon scattering processes. Using fourth order potential as an example, we show that it can significantly affects elastic phonon scatterings, through the formation of localized phonons. Such impact is closely related to the correlations of interfacial atoms and it becomes more significant with increasing temperature. Our work suggests that it is insufficient to consider only quadratic potential to investigate elastic phonon transport.
INTRODUCTION
In many materials, phonon transport is responsible for heat conduction [1,2]. However, along the path of travelling, phonons will experience various scatterings due to various reasons, which can significantly influence the heat conductance. Such scatterings include, for example, elastic scatterings [3,4] due to the mismatches of phonon spectral density and inelastic scatterings [5][6][7][8][9] due to the anharmonic lattice vibrations. Across an interface, heat conductance across an interfaces is traditionally modelled phenomenologically by acoustic mismatch model (AMM) and diffusion mismatch model (DMM) . They provide two extreme cases that heat are either completely carried by elastic waves (AMM) or diffusive phonons (DMM), respectively. However, phonons will experience both elastic and inelastic scatterings [10,11].
Rigorous developement of quantum theory of phonon transport based on atomistic model has been established in the last two decades using Non-equilibrium Green's function (NEGF) technique [12,13]. NEGF predicts that for a harmonic lattice (the interatomic potential is quadratic with respect to vibrational displacement), phonon experience only elastic scatterings, which means the phonon energy is conserved . Since then, many studies of elastic phonon scatterings have been reported, applied to many materials or nanostructures [14][15][16][17][18][19]. In this approach, the lattice potential is approximated by quadratic potential and this assumption is justified at low temperature. Often, Elastic phonon transport also serves as a foundation to understand more-involved multiplephonon scattering process [6].
So far the the analysis of elastic scattering is still limited to interatomic potentials within quadratic order. For potentials beyond quadratic order, it turns out to be extremely challenging to solve exactly and quantum mechanically [8,20]. As a result, the relation between higher order potential and multiple-phonon scattering processes is much less understood. From the Fermi' Golden rule we can understand that if the order of potential reaches n, the maximum phonons involved in the scattering is n. For instance, a cubic order potential is able to cause three-phonon process and a fourth order potential is able to cause four-phonon processes [21][22][23]. However, whether a high order potential has impacts on elastic phonon scatterings has not been addressed.
In this work we will investigate the relation between higher order potentials and elastic phonon scattering process. In order to bypass the difficulties of exactly solving high order potential problems, we introduce an anharmonic interface and limit the coupling of the interface to be weak, so that it can be treated perturbatively. Such perturbation treatment will not obscure the rendering of phonon scatterings and thus it provides a opportunity to discover the role higher order potentials to phonon scattering process. Furthermore, heat transfer such weakly interacting interface has important applications such as the in-plane heat conductance through van der Waals heterostructures [24,25]. To our surprise, we find that a n-th order potential will not only cause n-phonon process, but also takes important roles to elastic phonon scattering processes. Depending on the details of potential, it can either enhance or suppress the elastic scatterings.
THEORETICAL DERIVATION
We model our interface by connecting two harmonic thermal baths via interfacial couplings. In general, the Hamiltonian can be written as
H = H L + H R + H int ,(1)
where
H L = q (p L q ) 2 2m + 1 2 ω 2 q (x L q ) 2 and H R = q (p R q ) 2 2m + 1 2 ω 2 q (x R q ) 2
are collections of harmonic oscillators. The interfacial couplings consist of both quadratic couplings and higher order couplings. To be specific, we used the potential up to fourth order of interatomic forces,
V = 1 2! ij K i,j x i x j + 1 3! ijk (V ij,k x i x j x k + V i,jk x i x j x k ) + 1 4! ijkl (T ijk,l x i x j x k x l + T ij,kl x i x j x k x l + T i,jkl x i x j x k x l ),
where K i,j are interatomic force constants (IFCs) of quadratic coupling, V are the IFCs of cubic couplings and T are the IFCs of the fourth order couplings. The displacement, for example x L i , can be expanded with respect to the displacement of the normal modes of phonons with wave vector q in L as x L i = q c q ix L q . For the calculations of thermal current we employ the formalism developed in ref [26]. In the weak interaction regime, The thermal current is determined by the correlations of the operators that is involved in the interface coupling. It is well-understood that the quadratic coupling between the two baths only causes elastic scattering processes. In other words, the phonons are transmitted through without changing their energies. For the elastic scatterings caused by quadratic coupling, its contribution to thermal conductance can be written as [26]
I 2p = − 1 4h ij,kl K i,j K k,l ∞ −∞ Ψ ik (t)Φ jl (t)dt,(2)where Ψ ij (t) = dΦij (t) dt , Φ ij (t) = x i (t)
x j are the twopoint displacement correlation functions. The two-point correlation functions can be written in terms of the spectral densities of the left part Γ L and right part Γ R as Φ ij (t) = ∞ −∞ dω π Γ ij (ω)n(ω)e iωt and Ψ ij (t) = i ∞ −∞ dω π Γ ij (ω)ωn(ω)e iωt . A straightforward derivation will show that it can be cast into Landauer formula,
I 2p = 1 (2!) 2 ijkl K i,j K k,l H ikjl ,(3)
where
H ikjl = 4 h ∞ 0 dω π ωJ L ik (ω)J R jl (−ω)[n L (ω) − n R (ω)]. (4)
This results can also be derived from NEGF approach with bath-bath coupling developed in the literature [27]. Such elastic scattering processes are phenomenological shown in Fig. 1(a) and Fig. 1(b). In the figure we focus on a phonon with specific frequency ω, since the phonon energy and phonon frequency will not change during the elastic scattering. The atoms labelled with 1 and 2 are the interfacial atoms that involve interactions with the atoms of the other side of interface. The coupling between atom 1 and 2 are in quadratic order. Through the coupling, a phonon of energy (hω) can be annihilated at atom 1 and simultaneously a phonon of same energy can be created at atom 2. through this process, a energy of amounthω is transmitted across the interface.
! " !" # " $ % # % $ & !" #$ % &' () &! (* + # $ " ! % ' % ( & !"#$%&'()&!(*+ % # % $ & % # & % $ )" # " $ " ' " ( ,&- ,.- ,/- ,#- ,0- ,1- ! " !" # " $ % # & % $ !" #$ % &' () &! (* + # $ " ! % ' % ( & !"#$%& '()&!(*+ % # % $ & % # & % $ )" # " $ " ' " ( ! # $ " % ( % # & !"#$%&'()&!(*+ % $ % ' & % $ & % ' )" # " $ " ' " ( ! # $ " % # % ( & !"#$%& '()&!(*+ % $ % ' & % $ & % ' )" # " $ " ' " ( 2"#3%*0#' 4*$5&$036$*/#77 8&/95&$036$*/#77 :*/&'()#036$*/#77
FIG. 1. Phenomenological illustration of elastic phonon scattering caused by atoms that involves quadratic couplings (a and b) and fourth order couplings (c-f). The origin symbols shows the model, the red lines show the forward scattering process, the blue lines show the backward scattering process and the brown lines shows localized processes that does not involve energy transmission from left side to right side.
Upon transmission, this phonon will be dissipated and thermalized into the thermal bath of the right side. As required by detailed balance, phonons will also enter a backward scattering process shown in blue in Fig. 1(b), where a phonon of energyhω is transmitted from atom 2 to 1. During the cycles of phonon creation and annihilation between atom 1 and 2, phonons are either dissipated into or emitted from baths of both sides. The mount of net heat flow is determined by the competition between the phonon emission and dissipation rates at both baths. For example, if the left bath has larger emission ratio ( Fig. 1(a)) than dissipation ratio ( Fig. 1(b)), then the right bath should have larger dissipation ratio than emission ratio. As a result, heat will flow from the left side to the right side. However, the magnitude of heat conductance will be determined by the amount of energy carried by the phonon and the occurrence probability of the scattering processes. So in detail, they are determined the phonon spectra density, the phonon occupation number (temperature) and the strength of the quadratic coupling. Mathematically they are summarized in Eq. (2). For the third order coupling, we have shown that it contributes to three-phonon processes, which consists of phonon splitting, merging, partial reflection and partial transmission [26]. However, in the weak coupling regime, its existence will not affect the elastic scattering processes. In other words, the quadratic coupling contributes elastic scatterings and cubic coupling contributes to three-phonon scatterings. Their effects on thermal conductance are separable are additive.
The interesting roles come from the fourth order coupling at the interface. Firstly, we find that, the fourth order coupling do contribute to the four-phonon processes as expected, which involve processes of a single phonon splitting into three phonons, or three phonons merging into one, or two phonons merging together with emission of two new phonons. However, surprisingly, in addition to the four-phonon processes, the fourth order coupling also affects the elastic scattering processes.
Mathematically, such effects come from both the cross term between quadratic and fourth order coupling, and the fourth order couplings alone. In the following, we will analyze it in detail.
We first analyze the contribution of the correlation between term ij K i,j x i x j and term
1 4! ijkl T ijk,l x i x j x k x l ,
where the forward process is mediated via the coupling of ij K i,j x i x j while the backward process is mediated via the coupling
1 4! ijkl T ijk,l x i x j x k x l .
The evaluation of this term involve the calculation of four-point correlation function. By using Wick's theorem we can find that
φ L ijkl (t) = x i (t)x j x k x l = c L ij (t)Z L kl +c L ik (t)Z L jl +c L il (t)Z L jk .
Here we have defined Z ij = c ij (t = 0) as the correlation function at equal time. If i = j, it is the expectation value of square of the amplitude of atomic vibration. Therefore, it increases with temperature as well as the spectral density of that atom. In the high temperature limit, it should be proportional to temperature according to equal partition theorem.
With the correlation function, we find that its contribution to thermal current is
I h = 1 2! 1 4! ijklmn 6K i,j T klm,n Z L lm H ikjn(5)
We immediately find that its contribution to thermal current is proportional to Z L , which provides a extra temperature-dependent components. As we know H ikjn will saturate in the high temperature limit. So this term will eventually be linearly increasing with temperature of left part. This phonon scattering process can be phenomenological explained through the scattering process shown in Fig. 1. It describes a combination of two cycles of scattering processes. In the first cycle, the forward process is carried via the quadratic coupling Fig. 1(a), the backward scattering is through the fourth order coupling Fig. 1(d). In this backward scattering process, the transmitted phonon maintain the same energy across the interface. So it is regarded as elastic scattering. It happens when the other two atoms that involved in the fourth order coupling, atom 1 and 2, forms a localized phonon mode, such that the phonon forms a closed cycle in the left bath, and it is not travelling to the other side of interface. However, whenever a elastic scattering from 4 to 3 is happened, a phonon conversion is simultaneously occurred between atom 1 and 2, due to the fact that their interatomic coupling is in fourth order. In such a way, the localized phonons between atom 1 and 2 will significantly affects the scattering probability between 3 and 4, and such affects the total heat conduction. Such effects is quantitatively described by the quantity Z L .
In the other cycle, on the contrast, the forward process is carried by the fourth order coupling Fig. 1(c) and the backward process is carried by the quadratic coupling Fig. 1(b). Similarly, the direction of net heat flow is determined by the phonon emission and dissipation ratio of the two baths.
In the above, we have shown a typical example that a high order potential can cause elastic phonon scattering, through the formation of localized phonons. In a similar manner, we can also calculate the other contributions. Next we consider the cross term between ij K i,j x i x j and 1 4! ijkl T i,jkl x i x j x k x l . It turns out to be
I h = 1 2! 1 4! ijklmn 3K i,j T k,lmn Z R mn H ikjl(6)
Its phenomenological illustration can be described by two cycles. One of which is illustrated by 1(a) and Fig. 1(f), and the other is illustrated by Fig. 1(b) Fig. 1(e). We can find that this term depends on the Z R . This term and previous term together cause asymmetry between the left and right bath and thus they will result in thermal rectification effect under temperature bias. For the cross term between ij K i,j x i x j and mnop T mn,op,kl x m x n x o x p , it will not contribute to thermal current, because a closed cycle is not able to be formed.
So far we have analyze the cross term between the quadratic and fourth order coupling. We show that the cross term will increase linearly with temperature in the high temperature limit. Next we will show the terms coming from solely the fourth order coupling. We first analyze the term from coupling between T ijk,l and T mno,p . We find that such coupling not only contributes to fourphonon process, but also to elastic scattering process. Its contribution on elastic scattering process is illustrated through cycles formed by Fig. 1(c) and Fig. 1(d). In this case, two localized phonons are formed at left size, which affecting both the forward and backward processes. Mathematically, its contribution to thermal current is given by
I h(2p) = 1 (4!) 2 ijklmnop 9T ijk,l T mno,p Z L ij Z L no H kmlp (7)
We find that it depends on the second order of Z and increase quadratically in the high temperature limit. Similarly, the contribution from coupling between T l,ijk and T p,mno is (8) and it is illustrated by Fig. 1(e) and Fig. 1(f). The contribution from coupling between T ijk,l and T m,nop is (9) and it is illustrated in Fig. 1(f). By defining
I h(2p) = 1 (4!) 2 ijklmnop 9T l,ijk T p,mno Z R ij Z R no H lpkmI h(2p) = 1 (4!) 2 ijklmnop 9T ijk,l T m,nop Z L ij Z R op H kmlnS L ij = mn∈L T imn,j Z L mn , S R ij = mn∈R T i,mnj Z R mn(10)
We also find that cross term between ijkl T ij,jk x i x j x k x l and mnop T mn,op,kl x m x n x o x p will not contribute to the elastic scatterings. In this case, only the four-phonon processes contribute to the phonon transport. The localized phonons, even formed, are not able to affect the phonon transport processes. By summerizing all the contributions, we find that the total elastic scattering processes can be written in a concise form as
I h(2p) = ij,kl H ijkl × 1 2! K ik + 3 4! (S L ik + S R ik ) 1 2! K jl + 3 4! (S L jl + S R jl )(11)
Hence we have shown that the fourth order coupling at interface will have an impact on elastic scattering process. We found a temperature dependence quantity S that can be regarded as a effective quadratic force constant. The value of S will increase linearly with temperature. Therefore, such impacts will increase with increasing of temperature and it will eventually dominate at high temperature regime. This formalism suggest that even in the evaluation of elastic scattering process, it is insufficient to consider only the quadratic interatomic force constant K. One need to evaluate the effective force constant S from fourth order potential. With even higher temperature, there should be even contributions from higher order potentials.
NUMERICAL RESULTS OF AN APPLICATION
Next we use one-dimension chain to demonstrate such effects. In the simplest model, the interface is comprised by connecting to Rubin baths as shown in Fig. 2. At the interface, only the nearest atoms are interacting with each other. The interatomic force constant within the bath are characterized by k. At the interface, we use Morse potential V (r) = De(e −2a(r−re) − 2e −a(r−re) ) to simulate the coupling potential between two baths. The interatomic potential within the lead K is significantly larger interface coupling.
In our setup, we allow to add external forces to adjust the interatomic distance at the interface, and hence will adjust the interatomic potential. If we stretch the two leads, then both the interatomic distance between the leads and within the leads will increase. They will reach a new equilibrium position at the point where V ′ (r) = kr.
The interfacial atoms at this new position is balanced by both the Morse potential and the quadratic potential within the lead. However, with regarding to the interface coupling, both the second order and higher order force constants are adjusted. The quadratic and fourth-order IFCs can be calculated via derivatives of Morse potential with respect that new equilibrium position. Fig. 3 shows the temperature dependence of Z, which can be regarded as the mean square of vibrational amplitude of interfacial atoms. It shows Z increase with increase of temperature. Theoretically it will eventually linear increase with respect to temperature in the high temperature regime. The figure also shows Z is larger when k is smaller. The magnitude of Z in comparison with the ratio of fourth order and second order IFCs (η = T /K) will determine how important will the fourth order potential be on elastic scattering processes. If Z is comparable to η, the impact of 4th order IFCs will have comparable effects on elastic scattering pro- cesses with respect to quadratic IFCs. As a result, we can conclude that 1) 4th order potential is less important in low temperature regime and increasingly important with increase of temperature. This result is consistent with previous findings in the literature. [7] 2) 4th order potential is more important when the bonds in the baths are weaker (smaller k) but less important for stronger bonds. This is consistent with literature that elastic scatterings normally dominates for phonons in graphene, which has strong carbon-carbon bonds [28]. Fig. 4 shows the temperature dependence of the contribution of elastic scattering processes to the thermal conductance, with and without considering of 4th order IFCs. We used the interatomic distance at interface to adjust the potential. We find that 4th order IFC can both enhance or suppress the elastic scattering process at different distance and temperatures. When r = 0.4nm, the 4th order potential suppress elastic scattering process at low temperature but enhance it at high temperature with crossover at around T = 300K. When r = 0.5nm, 4th order IFCs will suppress the elastic scattering process while at t = 0.6nm and r = 0.7nm it will enhance the elastic scattering processes. In this particular one-dimensional case, the effects is determined by the sign of the 2nd and 4th order IFC. Specifically, when r = 0.5nm, The 4th order potential will suppress elastic scattering process when the sign of 4th IFC is different from that of 2nd order IFC. For r = 0.4nm, 0.6nm and 0.7nm, The 4th order potential will enhance the elastic scattering elastic scattering since the sign of 4th IFC is the same as that of 2nd IFC CONCLUSION During heat conduction in lattice, a quadratic potential can cause elastic scatterings for the travelling phonons. It will manifest as elastic scattering processes at interface. Higher order nonlinear potentials will responsible for multiple phonon scattering processes. In this work, we find that the 4th order potential has significant effect on elastic scattering process as well. This effect will be more significant with increase of temperature. From our model calculation, it shows that the 4th order potential can either enhance or suppress the elastic scattering process, depending to the coupling coefficients as well as temperature regime. This work suggests that in order to completely evaluate the elastic scattering process of phonon transport in lattice, one need to consider quadratic potential as well as higher order potentials, especially when the temperature is not sufficiently low.
FIG. 2 .FIG. 3 .
23Illustration of the setup used in our calculation. The interface is modelled by Morse potential, which has minimum energy of De at equilibrium position re. The left and right lead are Rubin baths. The mean square of vibrational amplitude Z is plotted against temperature under different interatomic force constants k = 50N/m, k = 100N/m and k = 350N/m.
FIG. 4. Temperature dependence of thermal conductance with or without 4th order coupling under different distances. Parameters: a=1/A. De=0.5eV. k=350N/m. re=0.3nm.w/ 4th IFC
w/o 4th IFC
r=0.4nm
G(µW )
T(K)
w/ 4th IFC
w/o 4th IFC
r=0.5nm
G(µW )
T(K)
w/ 4th IFC
w/o 4th IFC
r=0.6nm
G(µW )
T(K)
w/ 4th IFC
w/o 4th IFC
r=0.7nm
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| []
|
[
"Quenching of light hadrons at RHIC in a collisional energy loss scenario",
"Quenching of light hadrons at RHIC in a collisional energy loss scenario"
]
| [
"Pradip Roy \nSaha Institute of Nuclear Physics\n1/AF BidhannagarKolkataIndia\n",
"Jan-E Alam \nVariable Energy Cyclotron Centre\n1/AF BidhannagarKolkataIndia\n",
"Abhee K Dutt-Mazumder \nSaha Institute of Nuclear Physics\n1/AF BidhannagarKolkataIndia\n"
]
| [
"Saha Institute of Nuclear Physics\n1/AF BidhannagarKolkataIndia",
"Variable Energy Cyclotron Centre\n1/AF BidhannagarKolkataIndia",
"Saha Institute of Nuclear Physics\n1/AF BidhannagarKolkataIndia"
]
| []
| We evaluate the nuclear suppression factor, R AA (p T ) for light hadrons by taking into account the collisional energy loss. We show that in the measured p T domain of RHIC the elastic process is the dominant mechanism for the partonic energy loss. | 10.1088/0954-3899/35/10/104047 | [
"https://arxiv.org/pdf/0806.0446v1.pdf"
]
| 15,429,015 | 0806.0446 | 0469e0e627de3ccedba78cefb268c9f8f48fa30a |
Quenching of light hadrons at RHIC in a collisional energy loss scenario
3 Jun 2008
Pradip Roy
Saha Institute of Nuclear Physics
1/AF BidhannagarKolkataIndia
Jan-E Alam
Variable Energy Cyclotron Centre
1/AF BidhannagarKolkataIndia
Abhee K Dutt-Mazumder
Saha Institute of Nuclear Physics
1/AF BidhannagarKolkataIndia
Quenching of light hadrons at RHIC in a collisional energy loss scenario
3 Jun 2008
We evaluate the nuclear suppression factor, R AA (p T ) for light hadrons by taking into account the collisional energy loss. We show that in the measured p T domain of RHIC the elastic process is the dominant mechanism for the partonic energy loss.
Introduction
The suppressions of high p T hadrons and unbalanced back-to-back azimuthal correlations of the dijet events in Au+Au collisions measured at RHIC [1] provide experimental evidence in support of the jet quenching. Most of the calculations (see [2] for a review) considers the energy loss due to induced bremsstrahlung radiation and reproduces the observed nuclear suppression of light hadrons (π, η) in Au + Au collisions for centre of mass energy √ s NN = 62 − 200 GeV at RHIC. The effect of collisional loss was completely ignored in most of the previous calculations. However, the non-photonic single electron spectrum from heavy meson decays measured by PHENIX Collaboration [3] shows much larger suppression than expected. By considering radiative energy loss for heavy quarks the data cannot be reproduced as radiation is suppressed for heavy quarks due to dead-cone effects. Thus, there has been a renewed interest to revisit the importance of collisional energy loss both for light as well as heavy quarks. The partonic energy loss due to collisional processes was revisited in [4] and its importance was demonstrated in the context of RHIC in [5]. It is the purpose of the present work to show that the omission of the collisional energy loss to explain the RHIC data is not justified. To this end, we calculate the nuclear suppression factor (R AA ) for pions (η) considering only the collision energy loss.
Theoretical framework
In order to calculate the p T distribution of hadrons from parton fragmentation we need the phase space distribution of partons, f (p, t). The dynamical evolution of f (p, t) is obtained by solving the Fokker-Planck (FP) equation which reads,
∂ ∂t − p t ∂ ∂p f (p, t) = ∂ ∂p i [p i ηf (p, t)] + 1 2 ∂ 2 ∂p 2 [B (p)f (p, t)] + 1 2 ∂ 2 ∂p 2 ⊥ [B ⊥ f (p, t)],(1)
where the second term on the left hand side arises due to expansion [6]. Bjorken hydrodynamical model [7] has been used here for space time evolution. In Eq.
(1) f (p, t) represents the distribution function of the partons under study, Figure 1. Collisional versus radiative energy loss. The radiative loss is taken from [12] (see [4] for details) represent diffusion constants along parallel and perpendicular directions of the propagating partons.
η = (1/E)dE/dx, denotes drag coefficient, B = d (∆p ) 2 /dt, B ⊥ = d (∆p ⊥ ) 2 /dt, ‡ email: [email protected]
The matrix elements required to calculate the transports coefficients include diagrams involving exchange of massless gluons which render dE/dx and B ,⊥ infrared divergent. Such divergences can naturally be cured by using the hard thermal loop (HTL) [8] corrected propagator for the gluons, i.e. the divergence is shielded by plasma effects. For jet with energy E >> T (see [4] for details) the energy loss is given by
dE dx ∼ α 2 s T 2 C R ln E g 2 T(2)
Having known the drag and diffusion [4], we solve the FP equation using Green's function techniques with the initial condition
P ( p, t = t i | p 0 , t i ) = δ (3) ( p − p 0 )(3)
along the line of Refs. [9,10]. The solution with an arbitrary initial momentum distribution can now be written as [9,10],
E dN d 3 p j | y=0 = d 3 p j 0 P (p j , t|p j 0 , t i )E 0 dN d 3 p j 0 | y0=0(4)
where j stands for any parton species. In order to take into account the jet production geometry we assume that all the jets are not produced at the same point and the path length traversed by these partons before fragmentation are not the same. It is also assumed that the jet initially produced at (r, φ) leave the plasma after a proper time (t L ) or equivalently after traversing a distance L (for light quarks t L ∼ L) given by t L = R 2 − r 2 sin φ 2 − R cos φ. As this is not a measurable quantity, we have to average it out to obtain the p T spectra of hadrons:
dN π 0 (η) d 2 p T dy = f d 2 rP(r) tL ti dt t L − t i dz z 2 × D π 0 (η)/j (z, Q 2 )| z=pT /p j T E dN d 3 p f | yj =0 ,(5)
where t c is the time when temperature cools down to the transition temperature T c (=190 MeV) [11]. The temperature profile is taken as in Ref. [9]. The nuclear suppression factor, R AA is defined as
R AA (p T ) = dN π 0 (η) AA d 2 pT dy dN π 0 (η) AA d 2 pT dy 0 (6)
where the suffix '0' in the denominator indicates that energy loss has not been
Results
To understand the relative importance of the energy loss mechanisms we plot the contributions from radiative and collisional processes in Fig. 1. It is observed that collisional energy loss is the dominant mechanism of energy loss for parton energy up to E = E c ∼ 85(60) GeV for quark (gluon). Nuclear suppression factor, R AA for neutral pions is plotted as a function of transverse momentum in Fig. 2 which describes the PHENIX data [13] for Au + Au at √ s = 200 GeV reasonably well. It should be noted here that the R AA (p T ) with collisional loss has a tendency to increase for higher p T , indicating diminishing importance of collisional loss at this domain, where the radiative loss may become important. Therefore, a detailed calculation with both collisional and radiative loss may be useful to delineate the importance of individual mechanism.
It is important to note that the result for R AA is very sensitive to the initial temperature (T i ), equation of state(EOS) (through velocity of sound (c s )) and the thermalization time (t i ). This is demonstrated in Fig. (3). This aspect, has not been received much attention in the literature. Our calculations show that the data can be reproduced reasonably well with T i = 450 MeV, c 2 s =0.2 and t i =0.15 fm/c similar to those required to reproduce the single photon data [14].
Summary
In conclusion, we have used the R AA (p T ) for π 0 measured by PHENIX collaboration to characterize the QCD medium created after the Au + Au collisions at RHIC. The data can be be explained with T i ∼ 450 − 500 MeV depending upon the EOS, i.e. velocity of sound, c s in the medium. For a lower value of c s the expansion is slow. Consequently, the energy loss process occurs for a longer duration in the medium for a given T c . Therefore, for smaller values of c s the corresponding T i is smaller.
Our investigations suggest that in the measured p T range of light hadrons at RHIC collisional, rather than the radiative, is the dominant mechanism of jet quenching. This is in sharp contrast to all the previous analyses.
coll.) L=2 fm gluon (coll.) L=2 fm quark (rad.) L=2 fm gluon (rad.) L= 2 fm
Figure 2 .
2Nuclear suppression factor for pion. Experimental data are taken from PHENIX collaboration[13] for Au + Au collisions at √ s = 200 GeV. Solid line indicates result from the present calculation with collisional energy loss of the partons propagating through the plasma before fragmenting into pions. We have taken T i =450 MeV, t i =0.15 fm and c 2 s =0.2
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"arXiv:astro-ph/0004348v1 26 Apr 2000 Stellar Populations and Galaxy Morphology at High Redshift",
"arXiv:astro-ph/0004348v1 26 Apr 2000 Stellar Populations and Galaxy Morphology at High Redshift"
]
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"Andrew Bunker [email protected] \nDepartment of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA\n\nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUK\n",
"Hyron Spinrad \nDepartment of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA\n",
"Daniel Stern \nDepartment of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA\n\nJet Propulsion Laboratory\nCalifornia Institute of Technology\nMS 169-32791109PasadenaCAUSA\n",
"Rodger Thompson \nSteward Observatory\nUniversity of Arizona\n85721TucsonAZUSA\n",
"Leonidas Moustakas \nDepartment of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA\n\nAstrophysics Department\n1 Keble RoadOX1 3RHOxfordUK\n",
"Marc Davis \nDepartment of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA\n",
"Arjun Dey \nKitt Peak National Observatory\n950 N. Cherry Ave85726TucsonAZUSA\n\nDepartment of Physics & Astronomy\nThe John Hopkins University\n21218BaltimoreMDUSA\n"
]
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"Department of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA",
"Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUK",
"Department of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA",
"Department of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA",
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\nMS 169-32791109PasadenaCAUSA",
"Steward Observatory\nUniversity of Arizona\n85721TucsonAZUSA",
"Department of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA",
"Astrophysics Department\n1 Keble RoadOX1 3RHOxfordUK",
"Department of Astronomy\nUniversity of California at Berkeley\n601, 94720Campbell Hall, BerkeleyCAUSA",
"Kitt Peak National Observatory\n950 N. Cherry Ave85726TucsonAZUSA",
"Department of Physics & Astronomy\nThe John Hopkins University\n21218BaltimoreMDUSA"
]
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| In this article we investigate the morphology and stellar populations of high-redshift galaxies through multi-waveband HST imaging and ground-based spatiallyresolved spectroscopy. We study the redshift evolution of galaxy morphology in the Hubble Deep Field, using the deep IDT-NICMOS near-infrared HST imaging coupled with spectroscopic and photometric redshifts. Using the multi-waveband data to compare the appearance of galaxies at the same rest-frame wavelengths reveals that morphological k-corrections (the change in appearance when viewing high-z objects at shorter rest-frame wavelengths) are only important in a minority of cases, and that galaxies were intrinsically more peculiar at high redshift. One example of significant morphological k-corrections is spiral galaxies, which often show more pronounced barred structure in the near-infrared than in the optical. Therefore, the apparent decline in the fraction of barred spirals at faint magnitudes in the optical HDF may be due to band-shifting effects at the higher redshifts, rather than intrinsic evolution.Using such features as the age-sensitive Balmer+4000Å break, the spatially-resolved colours of distant galaxies in optical/near-infrared imaging can also be used to study their component stellar populations. We supplement this with deep Keck/LRIS spectroscopy of two extended sources: a chain galaxy at z = 2.8 (HDF 4-555.1, the "Hot Dog" -the brightest U -drop Lyman-break galaxy in the HDF) and a pair of z = 4.04 gravitationally lensed arcs behind the cluster Abell 2390. The absence of measurable rotation across the z = 2.8 chain galaxy implies that it is unikely to be a disk viewed edge on. With the resolution enhancement from lensing, we detect stellar populations of different ages in the z = 4 arcs. The Ly-α emission powered by the HII regions is spatially offset from the star-forming knots in these arcs, possibly as a result of resonant scattering by neutral hydrogen.2 Bunker et al. | null | [
"https://arxiv.org/pdf/astro-ph/0004348v1.pdf"
]
| 16,224,310 | astro-ph/0004348 | 6c29a56e0b5448905175afd034a322138f50d080 |
arXiv:astro-ph/0004348v1 26 Apr 2000 Stellar Populations and Galaxy Morphology at High Redshift
Andrew Bunker [email protected]
Department of Astronomy
University of California at Berkeley
601, 94720Campbell Hall, BerkeleyCAUSA
Institute of Astronomy
Madingley RoadCB3 0HACambridgeUK
Hyron Spinrad
Department of Astronomy
University of California at Berkeley
601, 94720Campbell Hall, BerkeleyCAUSA
Daniel Stern
Department of Astronomy
University of California at Berkeley
601, 94720Campbell Hall, BerkeleyCAUSA
Jet Propulsion Laboratory
California Institute of Technology
MS 169-32791109PasadenaCAUSA
Rodger Thompson
Steward Observatory
University of Arizona
85721TucsonAZUSA
Leonidas Moustakas
Department of Astronomy
University of California at Berkeley
601, 94720Campbell Hall, BerkeleyCAUSA
Astrophysics Department
1 Keble RoadOX1 3RHOxfordUK
Marc Davis
Department of Astronomy
University of California at Berkeley
601, 94720Campbell Hall, BerkeleyCAUSA
Arjun Dey
Kitt Peak National Observatory
950 N. Cherry Ave85726TucsonAZUSA
Department of Physics & Astronomy
The John Hopkins University
21218BaltimoreMDUSA
arXiv:astro-ph/0004348v1 26 Apr 2000 Stellar Populations and Galaxy Morphology at High Redshift
In this article we investigate the morphology and stellar populations of high-redshift galaxies through multi-waveband HST imaging and ground-based spatiallyresolved spectroscopy. We study the redshift evolution of galaxy morphology in the Hubble Deep Field, using the deep IDT-NICMOS near-infrared HST imaging coupled with spectroscopic and photometric redshifts. Using the multi-waveband data to compare the appearance of galaxies at the same rest-frame wavelengths reveals that morphological k-corrections (the change in appearance when viewing high-z objects at shorter rest-frame wavelengths) are only important in a minority of cases, and that galaxies were intrinsically more peculiar at high redshift. One example of significant morphological k-corrections is spiral galaxies, which often show more pronounced barred structure in the near-infrared than in the optical. Therefore, the apparent decline in the fraction of barred spirals at faint magnitudes in the optical HDF may be due to band-shifting effects at the higher redshifts, rather than intrinsic evolution.Using such features as the age-sensitive Balmer+4000Å break, the spatially-resolved colours of distant galaxies in optical/near-infrared imaging can also be used to study their component stellar populations. We supplement this with deep Keck/LRIS spectroscopy of two extended sources: a chain galaxy at z = 2.8 (HDF 4-555.1, the "Hot Dog" -the brightest U -drop Lyman-break galaxy in the HDF) and a pair of z = 4.04 gravitationally lensed arcs behind the cluster Abell 2390. The absence of measurable rotation across the z = 2.8 chain galaxy implies that it is unikely to be a disk viewed edge on. With the resolution enhancement from lensing, we detect stellar populations of different ages in the z = 4 arcs. The Ly-α emission powered by the HII regions is spatially offset from the star-forming knots in these arcs, possibly as a result of resonant scattering by neutral hydrogen.2 Bunker et al.
Abstract. In this article we investigate the morphology and stellar populations of high-redshift galaxies through multi-waveband HST imaging and ground-based spatiallyresolved spectroscopy. We study the redshift evolution of galaxy morphology in the Hubble Deep Field, using the deep IDT-NICMOS near-infrared HST imaging coupled with spectroscopic and photometric redshifts. Using the multi-waveband data to compare the appearance of galaxies at the same rest-frame wavelengths reveals that morphological k-corrections (the change in appearance when viewing high-z objects at shorter rest-frame wavelengths) are only important in a minority of cases, and that galaxies were intrinsically more peculiar at high redshift. One example of significant morphological k-corrections is spiral galaxies, which often show more pronounced barred structure in the near-infrared than in the optical. Therefore, the apparent decline in the fraction of barred spirals at faint magnitudes in the optical HDF may be due to band-shifting effects at the higher redshifts, rather than intrinsic evolution.
Using such features as the age-sensitive Balmer+4000Å break, the spatially-resolved colours of distant galaxies in optical/near-infrared imaging can also be used to study their component stellar populations. We supplement this with deep Keck/LRIS spectroscopy of two extended sources: a chain galaxy at z = 2.8 (HDF 4-555.1, the "Hot Dog" -the brightest U -drop Lyman-break galaxy in the HDF) and a pair of z = 4.04 gravitationally lensed arcs behind the cluster Abell 2390. The absence of measurable rotation across the z = 2.8 chain galaxy implies that it is unikely to be a disk viewed edge on. With the resolution enhancement from lensing, we detect stellar populations of different ages in the z = 4 arcs. The Ly-α emission powered by the HII regions is spatially offset from the star-forming knots in these arcs, possibly as a result of resonant scattering by neutral hydrogen.
Introduction
Until recently, the study of the most distant galaxies was very restricted: unresolved ground-based imaging gave measurements of the global colours; noisy spectra revealed AGN-or starburst-driven emission lines; and WFPC 2 imaging with the Hubble Space Telescope (HST) showed the shape of high-redshift galaxies in their unfamiliar rest-frame ultraviolet light. However, our knowledge has radically increased through the light-gathering power of the Keck telescopes, and the availability of the near-infrared NICMOS camera on HST. Deep Keck spectra showing the rest-ultraviolet continuum and stellar-and ISM-absorption features in z ∼ 3 galaxies (e.g., Steidel et al. 1996b;Pettini et al. 2000) has opened the door to spectroscopic study of stellar populations in the distant Universe. A complementary approach to exploring the composition of high-redshift objects comes from multi-waveband HST imaging from the ultraviolet to the near-infrared, which provides the resolution necessary to study their spatiallyresolved stellar populations.
Morphology offers a window on the evolutionary status of galaxies. However, because of band-shifting effects, the interpretation of the shape of galaxies hinges on knowledge of their redshifts and spectral energy distributions (SEDs). In this article we explore this through high resolution optical/near-infrared imaging and deep spectroscopy of distant galaxies. To resolve the issue of whether the peculiar galaxies which dominate the number counts at faint magnitudes are the counterparts of local irregulars or whether their morphological peculiarity is due mainly to band-shifting effects at high redshift, an unbiased study of the rest-frame optical morphological properties is demanded, where the effects of dust and recent star formation are less dominant than in the rest ultraviolet. In Section 2 we address this by analysing the IDT-NICMOS images of the northern Hubble Deep Field (HDF). In Section 3 we present a case study of the brightest z ∼ 3 galaxy in the HDF. This example, HDF 4-555.1 (known as the "Hot Dog"), is a highly elongated system and is sufficiently extended to allow resolved ground-based spectroscopy, which we have obtained with Keck/LRIS (Oke et al. 1995). We use a similar approach to explore the stellar populations in a pair of z ≈ 4 gravitationally-lensed arcs behind the cluster Abell 2390, which is described in Section 4. Throughout, we assume a cosmology where h 50 = H 0 / 50 km s −1 Mpc −1 , q 0 = 0.5 and Λ = 0, unless otherwise stated. All magnitudes in this paper are with respect to the AB system (Oke & Gunn 1983) where m AB = −48.57 − 2.5 log 10 f ν /(erg cm −2 s −1 Hz −1 ).
Galaxy Morphology and its Redshift Evolution
With ground-based seeing, the study of galaxy morphology was restricted to redshifts of no more than a few tenths. The advent of HST, and its resolution of ∼ 0.1 ′′ , has revolutionized this field. Results from projects such as the HST Medium Deep Survey (MDS, Griffiths et al. 1994ab) have shown that at faint magnitudes (I AB > 21) an increasing fraction of galaxies do not conform to the traditional categories (e.g., Glazebrook et al. 1995;Driver et al. 1995). The first Hubble Deep Field (Williams et al. 1996) dramatically pushed this study to even lower fluxes, tracing sub-L * galaxies to high redshift. The optical images of the HDF show that by I AB > ∼ 24, the conventional Hubble sequence no longer provides an adequate description of many or most galactic systems Driver et al. 1998). Indeed, at higher redshifts we may be seeing new classes of galaxy emerge with no local counterpart, such as the 'chain galaxies' (Section 3 and Cowie, Hu & Songaila 1995) and 'tadpoles' .
Some of these faint sources are intrinsically under-luminous peculiar galaxies at modest redshift. However, the median redshift has risen to z > ∼ 1 for a limiting magnitude of I AB = 26 (Lanzetta, Fernández-Soto & Yahil 1997). Hence, in the faint magnitude régime, band-shifting effects become important: the optical passbands sample shorter rest-frame wavelengths in galaxies at the higher redshifts, and large "morphological k-corrections" can arise (e.g., Odehahn et al. 1996). At z > ∼ 1, the appearance in the observed optical is dominated by regions of recent star formation, luminous in the rest-frame ultraviolet on account of the massive, short-lived OB stars. Indeed, Colley et al. (1996) suggest that the observed peak in the two-point angular correlation function of the optical HDF at ≈ 0. ′′ 3 is due to mis-classifying multiple compact star-forming regions within larger high-redshift galaxies as separate systems, exacerbated by the cosmological (1 + z) −4 bolometric surface-brightness dimming which boosts the contrast between the compact star-forming knots and the more diffuse host galaxy.
High-Resolution Imaging in the Near-Infrared
The rest-optical is a far better tracer of the dynamical mass of a galaxy than the ultraviolet. This suggests a strategy of high-resolution imaging in the nearinfrared; the V -and R-bands in the rest-frame of a z ≈ 1 galaxy are well approximated by the J-and H-passbands, and multi-colour imaging out to the H-band can trace the rest-frame B-band morphology of galaxies as far as z ≈ 3. However, until recently there has been no high-resolution infrared data set which reaches a limiting flux comparable to the optical HDF.
The Instrument Development Team (IDT) of the HST NICMOS camera (Thompson et al. 1998) have imaged an area of the northern HDF to unprecedented depth in the near-infrared, observing for 49 orbits in each of the F110W and F160W filters (centered at 1.1 µm and 1.6 µm and similar to the groundbased J-and H-bands). The widest-field NIC 3 camera was used to survey a ∼ 1 arcmin 2 portion of the HDF. A detailed description of the observations and data reduction are given by Thompson et al. (1999). Once we correct for different resolutions of NIC 3 and WFPC 2 (through "PSF matching"), we can use the spatially-resolved colours to study different stellar populations and/or dust-reddening within a galaxy (see Figs. 2 & 5).
The Transformation of Spiral Galaxies with Wavelength
One of the most visually striking differences between the optical and nearinfrared HDF images are spiral galaxies at moderately-high redshift (z ∼ 1). At NICMOS wavelengths (the rest-optical), many of these are clearly classic spirals, and therefore dynamically-evolved stable systems which certainly should not fall under the banner of morphological peculiars. However, as illustrated in Fig. 1, moving to the rest-UV shifts the classification toward a much later Hubble type i.e., becoming more irregular (Bunker, Spinrad & Thompson 1999). In extreme cases, the galaxy appearance is such a strong function of wavelength that some systems which resemble small groups of tidally-interacting sub-galactic clumps in the WFPC 2 optical images are only unveiled as nucleated spirals by the infrared observations. A classic example is the galaxy HDF 4-474.0 at z = 1.059 (Cohen et al. 1996) which is totally dominated by an off-centre star forming H II region in the U -and B-images, but transforms into a 'grand design' face-on spiral in the near-infrared (Fig. 1a). Spiral bulges are dominated by cool giants, and so brighten at the redder wavelengths; in the case of HDF 4-378 (at an estimated photometric redshift of z = 1.20, Fernández-Soto, Lanzetta & Yahil 1999) the bulge is totally absent from the observed optical passbands, but dominates the infrared light (Fig. 1b). This is reminiscent of the far-UV 1500Å imaging with UIT of the local spiral, M81, presented in O'Connell (1997).
From the optical HDF, there also appears to be strong redshift evolution in the relative fraction of galactic bars. Indeed, van den Bergh et al. (1996) report just one barred spiral in the whole of HDF-North. More recently, Abraham et al. (1999) have found similar evolution in the WFPC 2 images of HDF-South (Williams et al. 1999), with a marked decline at z > 0.5 in the proportion of barred spirals in both fields. If this is a truly evolutionary effect, then it has great significance for the physics of disk formation. However, once again the effects of large morphological k-corrections at higher-redshifts makes the case for evolution inferred from the apparent decline of barred spirals at faint optical magnitudes less clear cut. Bars are dominated by older stellar populations, with similar colors to bulges (de Vaucouleurs 1961), and so are prominent at redder wavelengths. In the rest ultraviolet, the star forming regions in the disk will typically dominate the light, and a spiral which would be identified as being barred when viewed in the rest optical may be (mis-)classified as unbarred at shorter wavelengths. Examination of the IDT-NICMOS images reveals bars in the near-infrared which are undetected in the WFPC 2 images (e.g., Fig. 3b at z ≈ 1); hence, claims of evolution in the frequency of galactic bars based on optical data alone should be treated with some caution.
The Redshift Evolution in the Fraction of Truly Peculiar Systems
Using the six wavebands from the WFPC 2 and IDT-NICMOS imaging of the Hubble Deep Field, we have compared galaxy morphology at the same rest-frame wavelengths. Where available, we use the spectroscopically-measured redshifts (from Cohen et al. 1996 unless otherwise noted). Where no published spectroscopic redshift exists, we adopt the photometric redshift estimate of Fernández-Soto, Lanzetta & Yahil (1999). Figure 3 of shows the rest-frame B-band of all the galaxies in the IDT-NICMOS field brighter than I AB = 25, which extends out to z ≈ 3.
Down to I AB ≈ 25.5 (the brightest 100 galaxies in IDT-NICMOS field), only about 1/6 of galaxies change their appearance greatly between the WFPC 2 and NICMOS images -these have large morphological k-corrections. Of the remaining number, about half of the galaxies retain the same morphology in all wavebands (above the redshifted Lyman break) and are "true peculiars". Hence, the increased fraction of unusually-shaped systems at faint optical magnitudes is largely due to evolution rather than simply band-shifting effects. The remaining third of galaxies are too compact for changes in morphology to be ascertained (limited by the NIC 3 PSF, which has a FWHM of ≈ 0.25arcsec), and this fraction increases greatly at magnitudes fainter than I AB = 25. For most cosmologies, the higher-redshift systems are on average more compact, once allowance has been made for the fact that the higher-redshift systems are intrinsically more luminous in this apparent-magnitude limited sample.
3 The "Hot Dog" -A Study of a z = 2.8 Chain Galaxy in the HDF Some high-redshift galaxies which fall outside the traditional Hubble tuningfork diagram belong to new morphological groups, such as tadpoles and 'bow-shock' systems ( Fig. 4). A class which has received much attention is that of 'chain galaxies' (Cowie, Hu & Songaila 1995). It has been widely speculated that chain galaxies are linearly-organized giant starforming regions, although some have argued that they might be galactic disks viewed edge-on (e.g., Dalcanton & Shectman 1996). It is unlikely that most of the chain galaxies are gravitational lensing phenomena, as the incidence of potential foreground lenses is small.
The brightest 'U -drop' galaxy in the HDF is a chain galaxy, quite unlike the bulk of the Lyman-break population which are compact and isolated (Giavalisco, Steidel & Macchetto 1996). This z = 2.80 galaxy, HDF 4-555.1 (see Fig. 4 and Steidel et al. 1996a, source C4-06), has been dubbed "the Hot Dog" because of its highly-elongated morphology -it is extended over ≈ 2. ′′ 5, and clearly resolved even in ground-based seeing.
Examining the spatially-resolved colors of the Hot Dog reveals that both of the prominent lobes are well fit with young stellar populations (< 100 Myr), although the southern component is bluer on average and exhibits a much smaller dispersion in colors than the northern (Fig. 5). This indicates that the star formation history of the northern component is more extended in time, or that the dust extinction along its length varies more than for the southern lobe. There is little evidence for a significant underlying older stellar population, which might be expected in a disk viewed edge-on.
We have obtained deep, spatially-resolved optical spectroscopy with Keck/LRIS, using a long slit aligned along the major axis of the Hot Dog (Bunker et al. 1998b). Our 14 ksec spectra (λ/∆λ FWHM ∼ 1000) sample the rest-frame ultraviolet (1090 − 1890Å), a region devoid of strong forbidden emission lines. However, many of the resonance lines associated with hot stellar winds do appear, although their P Cygni profiles are less extended in velocity than those observed in some local star-bursts. The overall rest-ultraviolet continuum also indicates that the Hot Dog is an actively star-forming galaxy, albeit with internal dust extinction of E(B − V ) ≈ 0.1 m . The presence of high-ionization N V and He II in emission demands at least some O-stars. Ly-α emission is completely suppressed, and the absorption profile can be fit by a combination of stellar photospheric absorption and a modest interstellar hydrogen column of N (H I) ≈ 10 20 cm −2 (a borderline damped system). The Hot Dog exhibits some of the strongest interstellar absorption features seen in the Lyman-break population. However, a search for velocity gradients in these lines along the major axis of this chain galaxy revealed that any systemic rotation must be small (< 100 km s −1 ), inconsistent with an edge-on rotating disk.
We also serendipitously discovered in our long-slit spectroscopy a compact companion galaxy, HDF 4-497.0, with Ly-α emission (W 0 ≈ 30Å) at a redshift within 1000 km s −1 of the Hot Dog, and a projected separation of 35 h −1 50 kpc. The flux in the Ly-α line of this companion galaxy is 3 × 10 −17 erg cm −2 s −1 , and the rest-ultraviolet continuum suggests a star formation rate of SF R UV ≈ 5.6 h −2 50 M ⊙ yr −1 . The existence of this companion closely aligned along the major axis of the chain galaxy offers support to the contention that this is an example of star formation triggered by collapse along a filament.
As we have spatially-resolved spectroscopy of an extended high-redshift source, we can also use our data as a probe of the dimensions of intervening absorbers without being restricted to the one-dimensional sightlines avalable from QSOs. Our spatially-resolved spectroscopy reveals Mg II λλ 2796/2803Å absorption by a foreground system at z = 1.239. This absorption is most likely associated with the galaxy HDF 4-516.0 which has a photometric redshift consistent with the z = 1.239 absorption lines, and is at a projected distance of 2 arcsec from the Hot Dog. The Mg II absorption is only pronounced in the northern component of the Hot Dog, and the absence of strong absorption in the southern component enables us to constrain the physical size of the Lyman limit system (the optically-thick halo) of this galaxy to be r < 27 h −1 50 kpc.
Resolving the Stellar Populations in a Lensed Galaxy
Gravitational lensing can be used as a tool to increase the resolution attainable in studies of distant galaxies. Although morphological information is hard to disentangle because of the geometric distortions and uncertainties in lens modelling, the amplification afforded by strong lensing can allow the stellar populations to be mapped on the sub-kpc scale, as well as magnifying the total flux.
Optical/Near-Infrared Imaging of Lensed Arcs at z = 4.04
Combining archival HST/WFPC 2 data with deep near-infrared imaging taken with Keck/NIRC (Matthews & Soifer 1994) in good seeing, we have measured the spatially-resolved colours in a z = 4.04 galaxy, gravitationally lensed by the rich cluster Abell 2390 (z ≈ 0.23) into a pair of highly-magnified near-linear arcs 3-5 ′′ in length (Frye & Broadhurst 1998). At the redshift of these arcs, the H (λ cent ≈ 1.65 µm) and K (λ cent ≈ 2.2 µm) near-infrared pass-bands straddle the agesensitive rest-frame 4000Å + Balmer break (Fig. 6). Comparison of the optical and near-infrared photometry with a suite of spectral evolutionary models (the latest version of Bruzual & Charlot 1993) has enabled us to map the underlying stellar populations and differential dust extinction (Bunker et al. 1998a). The WFPC2 images clearly reveal several knots, bright in the rest-ultraviolet, which correspond to sites of active star formation. However, there are considerable portions of the arcs are significantly redder, consistent with being observed > 100Myr after star formation has ceased, with modest dust extinction of E(B − V ) ≈ 0.1 m . There is degeneracy in the models between dust reddening and age for the optical/near-infrared colours, but the most extreme scenario where the colour gradients are solely due to heavy dust reddening of an extremely young stellar population are strongly ruled out by upper limits in the far-infrared/submm from ISO/SCUBA (Lémonon et al. 1998;Blain et al. 1998).
Keck/LRIS Spectroscopy
We have obtained optical spectroscopy from Keck/LRIS at moderate dispersion (λ/∆λ FWHM ≈ 1000) with a long slit aligned along the major axis of the arcs (Fig. 7). Our 4 ksec spectrum shows regions with Ly-α in emission that are adjacent to some of the bright knots seen in the optical HST images which sample the rest-frame ultraviolet (Figs. 7 & 9). The non-detections of N V 1240Å, C IV 1549Å & He II 1640Å strongly favor the interpretation that the Ly-α arises from the Lyman continuum flux produced by OB stars, rather than the harder ultraviolet spectrum of an AGN. We see the Ly-α line morphology extending ≈ 1 ′′ beyond the ultraviolet continuum, which we attribute to resonant scattering from H I (Bunker, Moustakas & Davis 2000). In the bright knots, the SEDs are consistent with a very young stellar population (< 10 Myr) or ongoing star formation.
Evolutionary Status of the z = 4 Galaxy
We have evidence for both ongoing star formation and regions of older stellar populations in the lensed arcs. It is therefore unlikely that this z = 4 system in a true 'primaeval' galaxy, viewed during its first major burst of star formation.
Rather, our results suggest that the star formation history of this system has not been coeval, with current activity concentrated into small pockets within a larger, older structure. Correcting for the gravitational amplification (estimated to be ≈ 10 from lens models), the intrinsic properties of the z = 4.04 galaxy are comparable to the Lyman-break selected z ≈ 3 − 4 population of Steidel et al. (1996bSteidel et al. ( ,1999. The current extinction-corrected star formation rate (≈ 15 h −2 50 M ⊙ yr −1 for q 0 = 0.5) may be adequate to 'build' an L * galaxy over a Hubble time, but a more likely scenario may be the creation of a sub-unit which will undergo subsequent merging with nearby systems (such as the other z = 4.04 galaxy identified in this field by Pelló et al. 1999) to assemble hierarchically the massive galaxies of today.
Conclusions
In this article, we have explored spatially-resolved stellar populations at highredshift, and addressed the impact on studies of galaxy morphology. The deep, high-resolution IDT-NICMOS near-infrared imaging of a portion of the northern Hubble Deep Field has been combined with the WFPC 2 data and photometric redshift estimates to study the redshift evolution of morphology, comparing galaxy appearance at the same rest-wavelengths. Some Hubble tuning-fork galaxies only reveal their true morphology in the near-infrared images. This is particularly so for galaxies with a large dispersion in stellar ages and spatially-distinct stellar populations, such as spiral galaxies which sometimes exhibit galactic bars in the NICMOS images which are invisible at shorter wavelengths. However, galaxies which do undergo a morphological metamorphosis from the WFPC 2 to NIC 3 images are in the minority; most galaxies retain the same appearance in all wavebands, or are too compact for the structural parameters to be determined. Once the morphological k-corrections have been accounted for, it appears that the fraction of galaxies falling outside the Hubble sequence does increase at faint magnitudes/high-z. Many of these "true peculiars" show evidence of being dynamically disturbed (possibly through mergers) with recent star formation activity. From the HST imaging and resolved spectroscopy with Keck/LRIS, we have shown that a z = 2.8 chain galaxy in the HDF has a predominantly young stellar population and no significant rotation, and is thus unlikely to be an edge-on disk galaxy. Using gravitational amplification to increase our resolution, we have also resolved the stellar populations on sub-kpc scales in a system of z = 4.04 lensed arcs.
The analysis of galaxy morphologies and colours in multi-waveband imaging, coupled with resolved spectroscopy, provides a valuable probe into the stellar populations and evolution of galaxies. A natural progression is to use the integral field unit spectrographs currently being developed. Spectroscopy of spatiallyresolved stellar populations in the high-redshift Universe will be a major scientific goal for NGST and the next generation of large ground-based telescopes with adaptive optics. et al. 1996), and this seems to be through chance alignment of a swath of young stars with the approximate axis of the true bar. The galactic bar in the spiral displayed in the right panel is only recognizable at infrared-wavelengths -at its redshift of z ≈ 1, the optical wavebands only sample the rest-ultraviolet, where the older & redder bulge/bar stellar populations are not prominent. Note the bow-shock area itself is comparatively blue, implying a young stellar population with star formation presumably triggered by the shock front, whereas the redder (older) core of the galaxy is more prominent in the near-infrared. The chain galaxy (the two-component U -drop Lyman-break galaxy called "the Hot Dog"; Steidel et al. 1996a, Bunker et al. 1998b appears the same at all wavelengths and is blue, implying a relatively homogeneous, young population (a primaeval galaxy candidate?). Fig. 4). The northern and southern components exhibit subtly different colours, attributable to either different stellar populations or non-uniform dust extinction. Adopting the approach of Abraham (1997), we also plot the evolution in the (V − I) and (J − H) colours with time for a Salpeter IMF and an exponentially-decaying star formation rate, with e-folding times ranging from 0.1 Gyr to 1 Gyr. At z = 2.8, (J − H) straddles the age-sensitive 4000Å break. Fig. 6. Left: An illustration of the unreddened rest-frame optical spectra of two galaxies, one observed only 3 Myr after the end of an instantaneous burst of star formation (long-dash curve) and the other seen after 400 Myr have elapsed (solid line). We also show the 3 Myr model with dust extinction of AV = 0.5 m , typical of high-z star-forming galaxies (e.g., Pettini et al. 1998;Steidel et al. 1999). Note the strong Balmer + 4000Å break due to the older stars. Also plotted ( N2 N1 S1 S2 S3
[OII] g Fig. 7. Left: the F814W image with elliptical galaxy model subtracted (note the counter arcs perpendicular to the axis of the main arcs, predicted by the lens model of Frye & Broadhurst 1998). The area covered by the long-slit optical spectroscopy is shown (slit axis is vertical). The right panel is this elliptical-subtracted image, smoothed to ≈ 0.6 ′′ seeing. Center: the LRIS spectrum, with the long-slit aligned along the arcs. The dispersion axis is horizontal, with wavelength increasing from left to right. The image has been smoothed by convolving with a Gaussian kernel of σ = 1 pixel. Note the spatial range of Ly-α emission, which extends well beyond the detectable continuum of the arcs. The positions of the closest continuum knots N4 (−7.6 ′′ ) and S1 (4.3 ′′ ) are indicated by the horizontal bars. The knot N4 (top) lies on the edge of the slit, hence the slight spatial offset of the Ly-α line centroid and the lower flux compared to the line from S1 (bottom). Also indicated on the left panel is the z = 1.129 [O II] λ 3727Å galaxy also falling on our spectroscopic long-slit (see Fig. 9). .9Å galaxy at z = 1.1293, 3 ′′ below the southern arc and intercepted by our spectroscopic long-slit (Fig. 7). The extraction width is 7 pixels (1.5"). Note that the emission-line doublet is clearly resolved. We do not see this structure in the emission lines from the arcs, implying that their origin is not [O II] 3727Å at z = 0.64. Right: one-dimensional spectral extraction of the southern arc, showing the region around Ly-α at z = 4.04. The extraction width is 8 pixels (1.7 ′′ ), and encompasses both line-and continuum-emission regions. The asymmetric emission line profile is readily apparent, with the sharp decline on the blue side due to absorption by H I within the galaxy -a blueshifted Ly-α absorption trough is visible from the outflowing H I. Fig. 10. The broad-band optical/near-infrared flux from the entire northern arc. Also plotted are reddened instantaneous-burst stellar population models viewed at various ages, arbitrarily normalized to the flux measured from the HST/WFPC 2 F814W image. The flux in F555W is severely attenuated by the opacity of the intervening Ly-α forest. The colours are best reproduced by a stellar population ∼ 50 Myr old, with in situ dust reddening of E(B − V ) ≈ 0.1 m . Note that at z = 4.04, the strong Balmer + 4000Å break due to the older stars lies between the H-and K ′ -filters.
Fig. 1 .Fig. 2 .Fig. 3 .
123AcknowledgmentsAJB acknowledges a NICMOS postdoctoral fellowship while at Berkeley, and a U.K. PPARC observational rolling grant at the Institute of Astronomy inCambridge (ref. no. PPA/G/O/1997/00793). The observations were obtained in part with the NASA/ESA Hubble Space Telescope operated by the Space Telescope Science Institute manged by the Association of Universities for Research in Astronomy Inc. under NASA contract NAS 5-26555. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. We are grateful to Mark Dickinson, Chuck Steidel, Lisa Storrie-Lombardi & Ray Weymann for useful discussions, and BrendaFrye and Tom Broadhurst for providing details of their z = 4 arcs in advance of publication. We have made use of the spectral evolutionary models of Gustavo Bruzual and Stéphane Charlot. We thank Hans Hippelein and Klaus Meisenheimber at the Max-Planck-Institut für Astronomie for organizing an enjoyable and informative meeting at Ringberg. Spiral galaxies at z ≈ 1, showing the great change in apparent morphology going from the optical (the rest-ultraviolet, where the appearance is irregular) to the near-infrared, where their true spiral nature is revealed. In the case of HDF 4-474 (left), the WFPC 2 images are dominated by a star forming knot, and for HDF 4-378 (right) the older/redder population of the bulge is only visible at infrared wavelengths. Stellar population fits to two spatially resolved regions of the z ≈ 1 spiral HDF 4-474 (seeFig. 1a), using the latest version of theBruzual & Charlot (1993) models. The bulge (left panel) is clearly very much older than the star-forming HII region in one of the spiral arms (right panel). The left panel shows the only optically-selected barred spiral in HDF-North (van den Bergh
Fig. 4 .
4Examples of a bow-shock interacting system (left) and a chain galaxy (right).
Fig. 5 .
5Spatially-resolved colours of the northern and southern components of the chain galaxy called "the Hot Dog" (HDF 4-555.1;
dotted lines) are the H and K filters in the rest-frame of a z = 4.04 galaxy, straddling the break. The SEDs come from the latest Bruzual & Charlot models. Right: The evolution of the (H − K) colour of a galaxy at z = 4.04 as a function of the time elapsed since an instantaneous burst of star formation. The solid curve is the Bruzual & Charlot model for Solar metallicity (Z = 0.020), with the dashed line showing lower metallicity, 1 5 solar (Z = 0.004). For this redshift, the (H − K) colour is an excellent tracer of the time elapsed since the end of star formation. The dotted curve is the solar-metallicity model with dust reddening of AV = 0.5 m .
Fig. 8 .Fig. 9 .
89The top panels show archival HST/WFPC 2 imaging of the cluster Abell 2390. The z = 4.04 galaxy is the arclet at PA=+23 • that is bisected by the elliptical. Top left is the HST V -band (F555W, 8400 s) which encompasses Ly-α, with the HST I-band (F814W, 10500 s) top right. The knots which are bright in the rest-ultraviolet (and so are presumably sites of recent star formation) are indicated. Our Keck/NIRC images were obtained in good seeing (0.4 − 0.5 ′′ FWHM) and are shown lower left (H, 2280 s) and lower right (K ′ , 2880 s). Left: one-dimensional spectral extraction of the [O II] λλ 3726.1,3728
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| []
|
[
"Theoretical proposal of a revolutionary water-splitting photocatalyst: The monolayer of boron phosphide",
"Theoretical proposal of a revolutionary water-splitting photocatalyst: The monolayer of boron phosphide"
]
| [
"Tatsuo Suzuki [email protected] \nTokyo Metropolitan College of Industrial Technology\n8-17-1, Minami-Senju, Arakawa-ku116-8523TokyoJapan\n"
]
| [
"Tokyo Metropolitan College of Industrial Technology\n8-17-1, Minami-Senju, Arakawa-ku116-8523TokyoJapan"
]
| []
| Recently, hydrogen generation by water-splitting photocatalysts is attracting attention as a sustainable and clean energy resource. Photocatalytic hydrogen-generation systems are much simpler, cheaper, and easier to scale up than the coupled systems of electrolysis and solar cells, wind-power generation, etc. However, photocatalytic hydrogen generation is currently inefficient. This paper proposes the monolayer of boron phosphide as a stable highly-efficient water-splitting photocatalyst by high-precision density-functional theory calculations using a HSE06 functional with a solvent effect. The monolayer of boron phosphide has a direct allowed energy gap of about 1.4 eV, and functions as a one-step excitation photocatalyst. It absorbs sunlight with wavelengths below about 890 nm (ultraviolet, visible, and near-infrared light) and produces both hydrogen gas and oxygen gas from water at a suitable pH condition. By calculating the overpotentials of hydrogen and oxygen evolution reactions, its photocatalytic effectiveness was confirmed. The monolayers of boron phosphide will realize green hydrogen revolution. | 10.1016/j.apsusc.2022.153844 | [
"https://arxiv.org/pdf/2204.06525v2.pdf"
]
| 248,157,627 | 2204.06525 | 5019387fee990bb7384a11c2e02e2f4ef9e718e6 |
Theoretical proposal of a revolutionary water-splitting photocatalyst: The monolayer of boron phosphide
Tatsuo Suzuki [email protected]
Tokyo Metropolitan College of Industrial Technology
8-17-1, Minami-Senju, Arakawa-ku116-8523TokyoJapan
Theoretical proposal of a revolutionary water-splitting photocatalyst: The monolayer of boron phosphide
1
Recently, hydrogen generation by water-splitting photocatalysts is attracting attention as a sustainable and clean energy resource. Photocatalytic hydrogen-generation systems are much simpler, cheaper, and easier to scale up than the coupled systems of electrolysis and solar cells, wind-power generation, etc. However, photocatalytic hydrogen generation is currently inefficient. This paper proposes the monolayer of boron phosphide as a stable highly-efficient water-splitting photocatalyst by high-precision density-functional theory calculations using a HSE06 functional with a solvent effect. The monolayer of boron phosphide has a direct allowed energy gap of about 1.4 eV, and functions as a one-step excitation photocatalyst. It absorbs sunlight with wavelengths below about 890 nm (ultraviolet, visible, and near-infrared light) and produces both hydrogen gas and oxygen gas from water at a suitable pH condition. By calculating the overpotentials of hydrogen and oxygen evolution reactions, its photocatalytic effectiveness was confirmed. The monolayers of boron phosphide will realize green hydrogen revolution.
Introduction
Fossil fuels cause global warming and will dry up in the future. Nuclear fuel has a risk of radioactive contamination such as Fukushima or Chernobyl. Recently, hydrogen generation by water-splitting photocatalysts 1 is attracting attention as a sustainable and clean energy resource to replace fossil fuels and nuclear fuel. Photocatalytic hydrogen-generation systems are much simpler, cheaper, and easier to scale up than the coupled systems of electrolysis and solar cells, wind-power generation, etc. Recently, Nishiyama et al. 2 reported solar hydrogen production from water on a 100 m 2 -scale using an aluminum-doped strontium titanate particulate photocatalyst. They demonstrated that safe, large-sale photocatalytic water splitting and gas collection and separation were possible; however, the hydrogen production was inefficient.
Finding a stable highly-efficient photocatalyst is an urgent task for humankind to realize green hydrogen revolution.
In 2012, Sun et al. 3 reported the fabrication of large-area freestanding single layers of ZnSe with four-atomic thickness, which have an enhanced water-splitting efficiency and photostability.
The ZnSe single layers exhibit a photocurrent density of 2.14 mA cm −2 at 0.72 V versus Ag/AgCl under 300W Xe lamp irradiation, 195 times higher than that of bulk ZnSe. Similarly, the efficiency of SnS2 single layers is 72 times higher than bulk SnS2 4 , and SnS monolayers is 104 times higher than bulk SnS 5 . These reports reveal that two-dimensional materials can be highly-efficient water-splitting photocatalysts. The possible reason is as follows. In the case of bulk photocatalysts, light penetrates deep inside the photocatalyst, where it generates electronhole pairs. The generated electrons and holes must move to the surface of the photocatalysts against the mutual Coulomb attraction in order to react with atoms or ions adsorbed on the surface. Therefore, bulk photocatalysts require mechanisms such as depletion layers or heterojunctions to separate electrons and holes; nevertheless, the electrons and holes still recombine, resulting in energy loss. However, in the case of two-dimensional photocatalysts, the generated electrons and holes can instantly react with atoms or ions adsorbed on the surface with little movement because two-dimensional photocatalysts are surface-only materials. Since Sun's reports, many two-dimensional photocatalysts have been developed, and details are found in review articles. 6
Proposal
This paper theoretically proposes the monolayer of boron phosphide (BP) as a stable highlyefficient water-splitting photocatalyst (cf. Fig. 1(a)). As a result of searching for various twodimensional materials by high-precision density-functional theory (DFT) calculations, the revolutionary property of the BP monolayer was discovered. At present, there are no reports that BP monolayers were synthesized. However, in 2019, on the surface of a cubic zinc-blend BP nanocrystal grown at 1250 ºC through a solid state reaction route, graphite-like layers with the lattice spacing of 0.35 nm were observed in high-resolution TEM images. 9 These layers may become the precursors of BP monolayers. In 2020, Hernández et al. 10 theoretically proposed that BP monolayers can be exfoliated by incorporating arsenic in the (1 1 1) surface of a cubic zinc-blend BP.
Below, we investigate the property of the BP monolayer as a photocatalyst by two types of high-precision DFT calculations: a plane-wave (PW) basis calculation using a pseudo-potential and a Gaussian type orbital (GTO) basis calculation using all electrons. The PW basis calculation also include a solvent effect, i.e., a water polarization effect by an implicit solvation model based on the Poisson-Boltzmann equation. All calculations use Heyd-Scuseria-Ernzerhof (HSE06) hybrid density functionals 11 12 for both structural optimizations and energy bands calculations because the HSE06 functional is one of the most reliable calculation methods, and the error between the calculated energy gaps and the experimental values is less than 10%. 13
Important conditions
First, we confirm four important conditions that two-dimensional photocatalysts must satisfy for large-scale practical use; (a) stable in water against long exposure to strong sunlight, (b) made from earth-abundant elements, (c) one-step excitation using a single semiconductor because Z-scheme using two connected semiconductors requires double photons of one-step excitation, and PEC water splitting using two electrodes is difficult to deploy on a large scale due to the complexity of equipment, and (d) a direct allowed transition semiconductor with an energy gap g that is larger than the theoretical limit ∆ limit but as small as possible. In order to achieve one-step excitation of (c), band edges (i.e., conduction band minimum C and valence band maximum V ) should straddle water redox potentials H + /H 2 and O 2 /H 2 O ; that is,
C ≥ H + /H 2 and O 2 /H 2 O ≥ V .
(1)
And, ∆ limit of (d) is the redox potential difference; that is,
g = C − V ≥ ∆ limit = H + /H 2 − O 2 /H 2 O .(2)
Water molecules H2O and hydrogen ions H + adsorbed on the surface of the BP monolayer are transformed into various intermediates on the surface (cf. Fig. 1(b)-(h)), and are finally decomposed into hydrogen molecules H2 and oxygen molecules O2, as described in detail below.
Here, ° is the standard electrode potential relative to the standard hydrogen electrode (SHE).
Reviews of previous researches
First, we review the bulk crystal of BP, which has a cubic zinc-blend structure. The cubic zinc-blend BP is quite chemically stable and is resistant to chemical corrosion. It is not attacked by hot concentrated mineral acids or aqueous alkali solution. 14 It is also resistant to oxidation in air up to about 800-1000 ºC. 15 The cubic zinc-blend BP is an indirect transition semiconductor with an energy gap of 2.0 eV, and functions as a photocatalyst for H2 evolution. 9 14 16 The photocatalytic H2 evolution reactions continue even in strong acid or strong alkaline, and the BP photocatalyst is stable under these extreme conditions. 14 Next, we review BP monolayers. The most stable structure of a BP monolayer is a hexagonal planar structure like a graphene or a h-BN monolayer (cf. Fig. 1(a)). 19 Şahin et al. 20 performed DFT calculations using a PW basis and local-density approximation, and reported the bond length = 1.83 Å and the direct energy gap g = 0.82 eV or 1.81 eV which is corrected by frequency-dependent GW0 calculations. They calculated the phonon dispersion of a BP monolayer also, and confirmed the dynamical stability of a BP monolayer. Suzuki & Yokomizo 21 calculated =1.87Å and g = 1.912 eV using a GTO basis and a B3LYP/6-31G(d) functional.
After that, the high-precision calculation method using a HSE06 functional became widespread. showed that the optical absorption of a BP monolayer was strong over a wide energy range between 1.37 and 4 eV. They performed a BOMD simulation by using the Nosé-Hoover method at 2500K for 5 ps, and indicated the high thermal stability of BP monolayers. Furthermore, they reported the chemical stability of BP monolayers in environment, such as N2, O2, H2O, H2, and 7 CO2. However, they did not mention the photocatalytic application of BP monolayers. Shu et al. 24 stated that BP monolayers were suitable for a water-splitting photocatalyst based on their calculations using a PW basis, a PBE functional, and the extrapolation of G0W0 method.
However, their low-precision calculation overestimated the energy gap as g = 1.833 eV.
Therefore, the intrinsic high-efficiency of BP monolayers was not reported correctly, and the photocatalytic effectiveness of BP monolayers has been buried. There are no other papers reporting photocatalytic applications of BP monolayers, except for van der Waals heterostructures with other monolayers. 25
Calculation methods
PW basis calculations use a plane-wave basis set, the projector augmented wave (PAW) potentials, 30 PWvac and GTOvac, there is 0.4% difference in bond lengths due to the different calculation methods. These bond lengths are in good agreement with previous researches. 22 23 Next, we consider the results of PWsol in Fig. 2. An infinitely wide BP monolayer is placed on the x-y plane at z=0. ( ) is the potential which is averaged within the unit cell.
( ) = ∬ sol ( , , ) unit cell ∬ unit cell ⁄ ,(7)
where Their experiments show that electron exchange systematically occurs between diamond and the oxygen redox couple; that is, the electrochemical potential (Fermi energy) of the diamond is pinned by the oxygen redox potential. By the way, on the BP monolayer side of water-solid interface there is little charge to compensate for ions adsorbed on the water side because the BP monolayer does not have free electrons like metals nor does it form a depletion layer like bulk semiconductors. Furthermore, the covalent bonds between elements B and P are so strong that B or P does not dissolve as ions. Therefore, the BP monolayer will not be charged very much.
Then, we calculate the overpotentials of HER and OER by Nørskov approach. 40 41 42 Here, we consider an acidic reaction involving H + ; however, the results are the same for the alkaline reaction involving OH -. 43 For each reaction (a)-(i), we calculate the Gibbs free energy change ∆ . Because the reaction does not proceed unless ∆ < 0, we find the threshold potential e,th when ∆ = 0. The difference between this threshold potential and the redox potential is defined as ; that is, = e,th − H + /H 2 for HER and = O 2 /H 2 O − e,th for OER. Note that is pH independent because the threshold potential and the redox potential are the same pH dependent. For all reactions included in a pathway, the maximum value of those is the overpotential HER or OER . Fig. 4 shows Gibbs free energy change ∆ in each pathway. Therefore, HER =0.45 eV along a pathway (a and b) with a H*2 adsorbed-proton; and OER = 0.06 eV along a pathway (d, e, g, and i) or (d, f, g, and i). By the way, the difference between the energy gap and the theoretical limit g − ∆ limit are 0.10, 0.12, and 0.25 eV for PWsol, PWvac, and GTOvac, respectively.
Although these values are a little smaller than HER + OER , the BP monolayer will function well as a photocatalyst under sunlight. The reason is as follows. When a photon with an energy larger than the energy gap is absorbed, an electron with an energy larger than C and a hole with an energy smaller than V are generated. Before the electron falls to the bottom of the conduction band, the hole rises to the top of the valence band, or the electron and the hole recombine or form an exciton, the electron and the hole can react with atoms or ions adsorbed on the surface of the BP monolayer because the BP monolayer is a surface-only material. This is a great feature that is different from bulk photocatalysts.
And finally, Fig. 5 shows the calculated energy gaps of BP monolayers and solar radiation spectrum (air mass 1.5 global). 47 The BP monolayer has an energy gap of about 1.4 eV. It absorbs sunlight with wavelengths below about 890 nm (ultraviolet, visible, and near-infrared light), and uses about 48% of the photon flux from the sun effectively.
Conclusions
This paper theoretically proposes the BP monolayer as a highly-efficient water-splitting photocatalyst. It is a stable semiconductor with a direct allowed energy gap of about 1.4 eV, and functions as a one-step excitation photocatalyst. It absorbs sunlight with wavelengths below about 890 nm (ultraviolet, visible, and near-infrared light) and produces both H2 and O2 from water at a suitable pH condition. By calculating the overpotentials of hydrogen and oxygen evolution reactions, its photocatalytic effectiveness was confirmed. BP monolayers will realize the hydrogen economy as a sustainable and clean energy resource; therefore, we hope that BP monolayers will be synthesized.
Supplementary Information
Theoretical proposal of a revolutionary water-splitting photocatalyst:
The monolayer of boron phosphide Tatsuo Suzuki (鈴木 達夫)
H + /H 2
2and O 2 /H 2 O are expressed by the Nernst equation in Eqs. (5) and (6), respectively. definition of the Nernst equation, H + /H 2 and O 2 /H 2 O are the potential energies of the electron in the BP monolayer, not the potential energies in the solution. The origin of these equations is the vacuum level, and -4.44 eV is the energy level of SHE relative to the vacuum level. B , , and [H + ] are Boltzmann constant, elementary charge, and the molar concentration of H + ions, respectively. Temperature T = 298.15 K, and = − log 10 [H + ]. The reactionvessel is filled with generated gas; therefore, the partial pressures of H2 and O2 are H
Zhuang & Hennig 22 calculated = 1.855Å and g = 1.36 eV using a PW basis and a HSE06 functional. Wang & Wu 23 calculated the band edges of a BP monolayer in a vacuum using a PW basis and a HSE06 functional: = 1.855Å, C = −3.96 eV, and V = −5.33 eV; and
26 27 28 Wu et al.29 performed DFT calculations combined with ab initio molecular dynamics (AIMD) simulations, and demonstrated that the physical and chemical stabilities of BP monolayers in a vacuum, and in oxygen, water, and oxygen-water environment.
31 a HSE06 functional, a 27×27×1 Monkhorst-Pack k-point mesh, the super-cell width of 18Å, the cutoff energy of 800 eV for the PW basis, and VASP 5.4.1 package 32 33 . The convergence criteria of electronic self-consistent calculations and ionic relaxations are 10 -6 and 10 -5 eV, respectively. A solvent effect by the polarization of water is implemented as an implicit solvation model based on the Poisson-Boltzmann equation, and is performed by using VASPsol package. 34 35 GTO basis calculations use a Gaussian type orbital 6-311G(d,p) basis set and a HSE06 functional under the periodic boundary condition. A pseudo-potential is not used because of all-electron calculations. About 2000 k-points are requested in a unit cell. Theconvergences of electronic self-consistent calculations are that the maximum and the root mean square (RMS) of the variations of the density matrix are less than 10 -5 and 10 -7 , respectively, and that the variation of the total-energy is less than 10 -5 Hartree. The convergences of ionic relaxations are that the maximum and RMS of force are less than 2×10 -6 and 10 -6 Hartree/Bohr, respectively, and that the maximum and RMS of displacement are less than 6×10 -6 and 4×10-6 Bohr, respectively. The GTO basis calculation is performed by using Gaussian 09 package. 368 6. Results and considerationsThis paper shows three types of calculated results: (a) PWsol using a PW basis with a solvent effect, (b) PWvac using a PW basis in a vacuum, and (c) GTOvac using a GTO basis in a vacuum.First, we consider the shapes of BP monolayers after structural optimizations. The bond lengths of PWsol, PWvac, and GTOvac are 1.845, 1.846, and 1.854 Å, respectively. Comparing PWsol and PWvac, the bond lengths are almost the same with and without a solvent effect. Comparing
sol ( , , ) is the entire local potential (ionic plus Hartree plus exchange correlation) with the solvent effect. The potential value at the midpoint of the spacer is vacuum level, and it is aligned with the origin of energy; that is, ( = 9) = 0. ( ) is the averaged number density of valence electrons of the BP monolayer with the solvent effect, and b ( ) is the averaged bound charge density of water. Drawn for comparison, vac ( ) is the averaged potential of PWvac, i.e., the averaged potential without the solvent effect. vac ( ) at the midpoint of the spacer is also aligned with the origin of energy; that is, vac ( = 9) = 0. We define d ( ) ≡ ( ) − vac ( ), which is the potential difference caused by the polarization of water. Here, we notice that d ( )is the vacuum level when the solvent effect exists; that is, d ( ) corresponds to the vacuum level of Anderson model 37 which explains the energy band profiles of semiconductor heterostructures. Therefore, H ≡ d ( = 0) = 0.21 eV is the electrical double layer voltage. C = −3.85 eV and V = −5.17 eV are band edges with the solvent effect. H + /H 2 = −3.90 eV and O 2 /H 2 O = −5.12 eV are the redox potentials in Eqs. (5) and (6) at = 9.2. Here, H + /H 2 and O 2 /H 2 O are based on the vacuum level d ( > 3) outside the electric double layer. Under the condition in this figure, the band edges straddle the redox potentials, and Eqs. (1) and (2) are satisfied. Then, we consider energy bands in the left panel of Fig. 3. Here, the energy bands of PWsol and PWvac are drawn under the same condition that ( = 9) = vac ( = 9) = 0, as in Fig. 2. The energy bands of GTOvac are also drawn so that the vacuum level is the origin of energy. First, we compare PWsol and PWvac. The energy gaps of PWsol and PWvac are 1.32 and 1.34 eV, respectively. We understand that the difference of energy gaps between with and without the solvent effect is small. The energy gap of PWvac is in good agreement with previous researches. 22 23 The band edges ( C, , V ) of PWsol and PWvac are (-3.85, -5.17) and (-4.06, -5.39) eV, respectively. ( C, , V ) are shifted only by the electrical double layer voltage H = 0.21 eV. Next, we compare PWvac and GTOvac. The energy gap of GTOvac is 1.47 eV, which is 10% larger than PWvac. This discrepancy is due only to the different calculation methods. ( C, , V ) of GTOvac is (-4.13, -5.60) eV. Despite the different calculation methods, the band edges ( C, , V ) are located at almost the same energy levels, which are in good agreement with a previous research. 23 Then, we consider the relationship between band edges and redox potentials. The band edges in this paper are calculated under the condition that the BP monolayer is not charged, i.e., the pH of the solution is at the point of zero charge pHpzc. It is unclear how the band edges of the BP monolayer depend on pH; that is, whether the band edges show a Nernstian dependence of 0.059 eV/pH like semiconducting metal oxides (e.g. TiO2), 38 or are pH-independent like a hydrogenterminated diamond. 39 Therefore, we cannot estimate pHpzc. In the right panel of Fig. 3, the redox potentials are drawn according to Eqs. (5) and (6). If the pHpzc of PWsol is in the range of 8.4 -10.0, the BP monolayer functions as a photocatalyst because it satisfies Eq. (1). However, even if the pHpzc is out of this range, the BP monolayer still functions as a photocatalyst. This is because the band edges C and V are very close to the redox potentials H + /H 2 and O 2 /H 2 O ,respectively; then, either band edge is pinned to the corresponding redox potential. The basis for this idea is the experiments by Chakrapani et al.39 They added hydrogen-terminated diamond particles into aqueous solutions and measured the changes in pH and oxygen concentrations.
site on the BP monolayer. In the following, temperature T = 298.15 K. . Similarly, OH * , OOH * , and H * are calculated; however, for H * we consider four types: H * 0 , H * 1 , H * 2 , and H * 3 . H*0 is a proton adsorbed on the pure surface of the BP monolayer. H*1, H*2, and H*3 are protons adsorbed on the back surface of O*, OH*, and OOH*, respectively (cf. Fig. 1). Spin-polarized calculations were performed by using VASP 5.4.4 package, PAW potentials, a PBE functional 45 , a DFT-D3 method for vdW interactions 46 , and the cutoff energy of 520 eV for the PW basis. The convergence criteria of self-consistent calculations and ionic relaxations were 10 -5 and 10 -4 eV, respectively. The adsorbed atoms were calculated with a 2×2×1 supercell, a 7×7×1 Monkhorst-Pack k-point mesh, and the super-cell width of 18Å. H2(g) and H2O(g) were calculated in a box of 20Å×20Å×20Å at Γ k-point. As a result, H 2 (g) , O 2 (g) , H 2 O(l) , O * , OH * , OOH * , H * 0 , H * 1 , H * 2 , H * 3 and H + + e − are -6.827, -9.911, -14.216, -6.403, -9.938, -15.932, -2.557, -2.703, -3.862, -4.324, and 1.032 − 0.0592 + e eV, respectively (see the supplementary information for computational details).
Figure 1 .
1Optimized structures of (a) BP monolayer, (b) O*, (c) OH*, (d) OOH*, (e) H*0, (f) H*1, (g) H*2, and (h) H*3. Black, green, red, and blue balls are boron, phosphorus, oxygen, and hydrogen atoms, respectively.
Figure 2 .
2Potentials, charge densities, and band edges vs. a z-coordinate. The inset is an overall view.
Figure 3 .
3(Left panel) energy bands. (Right panel) band edges and redox potentials vs. pH.
Figure 4 .
4Gibbs free energy change ∆ in each pathway. Here, a0 means a reaction with a H*0 adsorbed-proton. a1, a2, …, c3 are similar.
Figure 5 .
5Calculated energy gaps, solar radiation spectrum (air mass 1.5 global), and photon flux vs. wavelength.
These durable properties of the cubic zinc-blend BP originate in the strong covalent bonds between elements B and P. This durability will be carried over to BP monolayers. Recently, Li et al. 17 prepared cubic zinc-blend BP nanosheets with a thickness of about 4 nm, and demonstrated that the BP nanosheets were well dispersed in water with a concentration of 0.2 mg/mL. Liang et al. 18 synthesized cubic zincblend BP single crystals larger than 1mm, and evaluated their excellent thermal stability up to 1200 ºC.
TABLE I .
ICalculated results: Gibbs free energy , total energy calculated by DFT DFT , zeropoint energy calculated by DFT ZPE , internal energy change ∆°(0K → ), enthalpy change ∆°(0K → ), standard entropy °, and Gibbs free energy at standard conditions °. The unit is eV.DFT
ZPE
∆°(0K → ) −°
or
∆°(0K → ) −°
°
*
-49.059
O*
-6.403
-55.545
0.094
-0.011
OH*
-9.938
-59.335
0.374
-0.036
OOH*
-15.932
-65.375
0.439
-0.055
H*0
-2.557
-51.850
0.235
-0.001
H*1
-2.703
-58.494
0.247
-0.001
H*2
-3.862
-63.427
0.232
-0.002
H*3
-4.324
-69.940
0.243
-0.001
H 2 (g)
-6.827
-6.771
0.271
-0.316
-6.816
H 2 O(g)
-14.219
0.572
-0.481 -14.128
H 2 O(l)
-14.216
-14.216
O 2 (g)
-9.911
-9.883
H + + e -1.032 − 0.0592
+ e
∑ ℎ O * , , and ∆ O * °( 0 → ) − O * °= B ∑ ln{1 −
AcknowledgementsThis work was partially supported by Tokyo Metropolitan College of Industrial Technology.Competing interestsThe author declares no competing interests.
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"Ambipolar electric field and potential in the solar wind estimated from electron velocity distribution functions",
"Ambipolar electric field and potential in the solar wind estimated from electron velocity distribution functions"
]
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"Laura Berčič [email protected] \nMullard Space Science Laboratory\nUniversity College London\nRH5 6NTDorkingUK\n",
"Milan Maksimović \nLESIA\nObservatoire de Paris\nUniversité PSL\nCNRS\nSorbonne Université\nUnversité de Paris\n5 place Jules Janssen92195MeudonFrance\n",
"Jasper S Halekas \nDepartment of Physics and Astronomy\nUniversity of Iowa\n52242Iowa CityIAUSA\n",
"Simone Landi \nPhysics and Astronomy Department\nUniversity of Florence\nSesto FiorentinoItaly\n\nINAF -Osservatorio Astrofisico di Arcetri\nFirenzeItaly\n",
"Christopher J Owen \nMullard Space Science Laboratory\nUniversity College London\nRH5 6NTDorkingUK\n",
"Daniel Verscharen \nMullard Space Science Laboratory\nUniversity College London\nRH5 6NTDorkingUK\n\nSpace Science Center\nUniversity of New Hampshire\n8 College Road03824DurhamNHUSA\n",
"Davin Larson \nPhysics Department\nUniversity of California\nBerkeleyCAUSA\n",
"Phyllis Whittlesey \nPhysics Department\nUniversity of California\nBerkeleyCAUSA\n",
"Samuel T Badman \nPhysics Department\nUniversity of California\nBerkeleyCAUSA\n",
"Stuart D Bale \nPhysics Department\nUniversity of California\nBerkeleyCAUSA\n\nSpace Sciences Laboratory\nUniversity of California\n94720-7450BerkeleyCAUSA\n",
"Anthony W Case \nSmithsonian Astrophysical Observatory\nCambridgeMAUSA\n",
"Keith Goetz \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA\n",
"Peter R Harvey \nSpace Sciences Laboratory\nUniversity of California\n94720-7450BerkeleyCAUSA\n",
"Justin C Kasper \nSmithsonian Astrophysical Observatory\nCambridgeMAUSA\n\nUniversity of Michigan\nAnn ArborMIUSA\n",
"Kelly E Korreck \nSmithsonian Astrophysical Observatory\nCambridgeMAUSA\n",
"Roberto Livi \nPhysics Department\nUniversity of California\nBerkeleyCAUSA\n",
"Robert J Macdowall \nSolar System Exploration Division\nNASA/Goddard Space Flight Center\n20771GreenbeltMDUSA\n",
"David M Malaspina \nAstrophysical and Planetary Sciences Department\nUniversity of Colorado\nBoulderCOUSA\n",
"Marc Pulupa \nSpace Sciences Laboratory\nUniversity of California\n94720-7450BerkeleyCAUSA\n",
"Michael L Stevens \nSmithsonian Astrophysical Observatory\nCambridgeMAUSA\n",
"Laura Berčič "
]
| [
"Mullard Space Science Laboratory\nUniversity College London\nRH5 6NTDorkingUK",
"LESIA\nObservatoire de Paris\nUniversité PSL\nCNRS\nSorbonne Université\nUnversité de Paris\n5 place Jules Janssen92195MeudonFrance",
"Department of Physics and Astronomy\nUniversity of Iowa\n52242Iowa CityIAUSA",
"Physics and Astronomy Department\nUniversity of Florence\nSesto FiorentinoItaly",
"INAF -Osservatorio Astrofisico di Arcetri\nFirenzeItaly",
"Mullard Space Science Laboratory\nUniversity College London\nRH5 6NTDorkingUK",
"Mullard Space Science Laboratory\nUniversity College London\nRH5 6NTDorkingUK",
"Space Science Center\nUniversity of New Hampshire\n8 College Road03824DurhamNHUSA",
"Physics Department\nUniversity of California\nBerkeleyCAUSA",
"Physics Department\nUniversity of California\nBerkeleyCAUSA",
"Physics Department\nUniversity of California\nBerkeleyCAUSA",
"Physics Department\nUniversity of California\nBerkeleyCAUSA",
"Space Sciences Laboratory\nUniversity of California\n94720-7450BerkeleyCAUSA",
"Smithsonian Astrophysical Observatory\nCambridgeMAUSA",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA",
"Space Sciences Laboratory\nUniversity of California\n94720-7450BerkeleyCAUSA",
"Smithsonian Astrophysical Observatory\nCambridgeMAUSA",
"University of Michigan\nAnn ArborMIUSA",
"Smithsonian Astrophysical Observatory\nCambridgeMAUSA",
"Physics Department\nUniversity of California\nBerkeleyCAUSA",
"Solar System Exploration Division\nNASA/Goddard Space Flight Center\n20771GreenbeltMDUSA",
"Astrophysical and Planetary Sciences Department\nUniversity of Colorado\nBoulderCOUSA",
"Space Sciences Laboratory\nUniversity of California\n94720-7450BerkeleyCAUSA",
"Smithsonian Astrophysical Observatory\nCambridgeMAUSA"
]
| []
| The solar wind escapes from the solar corona and is accelerated, over a short distance, to its terminal velocity. The energy balance associated with this acceleration remains poorly understood. To quantify the global electrostatic contribution to the solar wind dynamics, we empirically estimate the ambipolar electric field (E ) and potential (Φ r,∞ ). We analyse electron velocity distribution functions (VDFs) measured in the near-Sun solar wind, between 20.3 R S and 85.3 R S , by the Parker Solar Probe. We test the predictions of two different solar wind models. Close to the Sun, the VDFs exhibit a suprathermal electron deficit in the sunward, magnetic field aligned part of phase space. We argue that the sunward deficit is a remnant of the electron cutoff predicted by collisionless exospheric models(Lemaire & Scherer 1970, 1971Jockers 1970). This cutoff energy is directly linked to Φ r,∞ . Competing effects of E and Coulomb collisions in the solar wind are addressed by the Steady Electron Runaway Model (SERM) (Scudder 2019a). In this model, electron phase space is separated into collisionally overdamped and underdamped regions. We assume that this boundary velocity at small pitch angles coincides with the strahl break-point energy, which allows us to calculate E . The obtained Φ r,∞ and E agree well with theoretical expectations. They decrease with radial distance as power law functions with indices α Φ = −0.66 and α E = −1.69. We finally estimate the velocity gained by protons from electrostatic acceleration, which equals to 77% calculated from the exospheric models, and to 44% from the SERM model. | 10.3847/1538-4357/ac1f1c | [
"https://arxiv.org/pdf/2108.08528v1.pdf"
]
| 237,213,604 | 2108.08528 | 8cfbada960c1bb9ffd707012072d8da18beb0426 |
Ambipolar electric field and potential in the solar wind estimated from electron velocity distribution functions
August 20, 2021
Laura Berčič [email protected]
Mullard Space Science Laboratory
University College London
RH5 6NTDorkingUK
Milan Maksimović
LESIA
Observatoire de Paris
Université PSL
CNRS
Sorbonne Université
Unversité de Paris
5 place Jules Janssen92195MeudonFrance
Jasper S Halekas
Department of Physics and Astronomy
University of Iowa
52242Iowa CityIAUSA
Simone Landi
Physics and Astronomy Department
University of Florence
Sesto FiorentinoItaly
INAF -Osservatorio Astrofisico di Arcetri
FirenzeItaly
Christopher J Owen
Mullard Space Science Laboratory
University College London
RH5 6NTDorkingUK
Daniel Verscharen
Mullard Space Science Laboratory
University College London
RH5 6NTDorkingUK
Space Science Center
University of New Hampshire
8 College Road03824DurhamNHUSA
Davin Larson
Physics Department
University of California
BerkeleyCAUSA
Phyllis Whittlesey
Physics Department
University of California
BerkeleyCAUSA
Samuel T Badman
Physics Department
University of California
BerkeleyCAUSA
Stuart D Bale
Physics Department
University of California
BerkeleyCAUSA
Space Sciences Laboratory
University of California
94720-7450BerkeleyCAUSA
Anthony W Case
Smithsonian Astrophysical Observatory
CambridgeMAUSA
Keith Goetz
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMNUSA
Peter R Harvey
Space Sciences Laboratory
University of California
94720-7450BerkeleyCAUSA
Justin C Kasper
Smithsonian Astrophysical Observatory
CambridgeMAUSA
University of Michigan
Ann ArborMIUSA
Kelly E Korreck
Smithsonian Astrophysical Observatory
CambridgeMAUSA
Roberto Livi
Physics Department
University of California
BerkeleyCAUSA
Robert J Macdowall
Solar System Exploration Division
NASA/Goddard Space Flight Center
20771GreenbeltMDUSA
David M Malaspina
Astrophysical and Planetary Sciences Department
University of Colorado
BoulderCOUSA
Marc Pulupa
Space Sciences Laboratory
University of California
94720-7450BerkeleyCAUSA
Michael L Stevens
Smithsonian Astrophysical Observatory
CambridgeMAUSA
Laura Berčič
Ambipolar electric field and potential in the solar wind estimated from electron velocity distribution functions
August 20, 2021Draft version Typeset using L A T E X twocolumn style in AASTeX631 Corresponding author:Solar wind (1534)Space plasmas (1544)Interplanetary particle acceleration (826)Colli- sion processes (2065)Space vehicle instruments (1548)
The solar wind escapes from the solar corona and is accelerated, over a short distance, to its terminal velocity. The energy balance associated with this acceleration remains poorly understood. To quantify the global electrostatic contribution to the solar wind dynamics, we empirically estimate the ambipolar electric field (E ) and potential (Φ r,∞ ). We analyse electron velocity distribution functions (VDFs) measured in the near-Sun solar wind, between 20.3 R S and 85.3 R S , by the Parker Solar Probe. We test the predictions of two different solar wind models. Close to the Sun, the VDFs exhibit a suprathermal electron deficit in the sunward, magnetic field aligned part of phase space. We argue that the sunward deficit is a remnant of the electron cutoff predicted by collisionless exospheric models(Lemaire & Scherer 1970, 1971Jockers 1970). This cutoff energy is directly linked to Φ r,∞ . Competing effects of E and Coulomb collisions in the solar wind are addressed by the Steady Electron Runaway Model (SERM) (Scudder 2019a). In this model, electron phase space is separated into collisionally overdamped and underdamped regions. We assume that this boundary velocity at small pitch angles coincides with the strahl break-point energy, which allows us to calculate E . The obtained Φ r,∞ and E agree well with theoretical expectations. They decrease with radial distance as power law functions with indices α Φ = −0.66 and α E = −1.69. We finally estimate the velocity gained by protons from electrostatic acceleration, which equals to 77% calculated from the exospheric models, and to 44% from the SERM model.
INTRODUCTION
arXiv:2108.08528v1 [astro-ph.SR] 19 Aug 2021
The solar wind is a continuous outflow of plasma from the hot solar corona (Parker 1958). The particles escaping the Sun are mostly electrons and protons, with a smaller population of heavier ions. Over a small radial distance, these particles reach bulk velocities of order a few 100 km s −1 . The nature of the acceleration mechanisms converting the coronal thermal energy to the solar wind kinetic energy remains one of the most important open questions in heliophysics.
The terminal velocity of the solar wind is closely related to the density and temperature of the solar coronal plasma. These can be estimated remotely through spectroscopy and multi-frequency radio imaging (Mercier & Chambe 2015). In coronal holes, which are regions of open magnetic field lines along which plasma can flow freely in the radial direction, the typical electron temperature is 0.79 MK (David et al. 1998;Cranmer 2002), while much higher temperatures are found on the edges of coronal holes and in active regions (Stansby et al. 2020).
In coronal holes, the proton distributions appear hotter than that of electrons, and anisotropic with T ⊥ > T at radial distance ∼ 3 R S (Cranmer 2002). Heavier ion distributions are strongly anisotropic at these distances with T ⊥ /T ranging between 10 and 100 (Kohl et al. 1998). Preferential perpendicular ion heating is believed to contribute to the solar wind acceleration (Munro & Jackson 1977) through mechanisms like stochastic heating (Chandran et al. 2010;Bourouaine & Chandran 2013), ion-cyclotron resonance (Dusenbery & Hollweg 1981;Hollweg 1999;Li et al. 1999; or the dissipation of turbulence (Bieber et al. 1996;Oughton et al. 2001;Verdini et al. 2010;Karimabadi et al. 2013;Matthaeus et al. 2015;Agudelo Rueda et al. 2021).
In the case of electrons, a non-Maxwellian coronal VDF with an excess of high-energy electrons can alone accelerate the solar wind protons to velocities above 700 km s −1 Zouganelis et al. 2004). The radial evolution of the collisionless, expanding solar wind is captured by the exospheric solar wind models (Lemaire & Scherer 1970, 1971Jockers 1970;Pierrard et al. 1999;Maksimovic et al. 2001;Zouganelis et al. 2005). The drivers of the solar wind in these models are the solar wind electrons. Due to their small mass, their thermal velocity just above the solar surface is large enough for the majority of electrons to escape the Sun's gravity. However, this is not the case for the heavier protons. A global electric polarization field, also referred to as the ambipolar electric field (E ), builds up, accelerating the protons and decelerating the electrons. It preserves equal ambipolar diffusion of ions and electrons in the radial direction. This study focuses on the quantification of E in the solar wind and thus its contribution to the total solar wind acceleration.
E decreases with radial distance from the Sun, and has a magnitude of order a few nV/m in the inner heliosphere (e.g. Berčič et al. 2021), thus it is practically undetectable by spacecraft electric field antennas. However, electron VDFs measured in the near-Sun solar wind are highly affected by E and thus can tell us something about its properties.
Electron VDFs in the solar wind have a complex structure and are commonly modelled with three components. Low energy electrons belong to the almost isotropic core population and are well represented by a Maxwellian distribution. Higher energy electrons belong to either the isotropic halo population, or the magnetic field aligned, beam-like population, called the strahl (Feldman et al. 1975;Schwartz & Marsch 1983;Pilipp et al. 1987;Maksimovic et al. , 2005Štverák et al. 2008;Štverák et al. 2009;Tao et al. 2016). Another electron feature is often observed in the near-Sun solar wind -a relative deficit of electrons compared to the Maxwellian core model appears in the suprathermal energy range in the portion of phase space, opposite to the strahl direction (Halekas et al. 2019;Halekas et al. 2020;Berčič et al. 2020;Bercic, L. et al. 2021;Berčič et al. 2021). The statistical properties of this sunward deficit are presented by Halekas et al. (2021).
The exospheric prediction
A deficit of sunward moving electrons is also a feature of collisionless exospheric models, where it is referred to as the "electron cutoff". The electron VDF at any radial distance in these models is separated into two parts: anti-sunward moving electrons with energy greater than the ambipolar electric potential energy (E Φ ) represent the escaping electrons, which focus towards the magnetic field direction and form the strahl population; ballistic electrons with energies less than E Φ represent the core population. In the sunward direction, these electrons are limited by the cutoff energy (E C ) corresponding to the ambipolar potential between their location and the asymptotic value at large heliocentric distances, where r → ∞ (Jockers 1970;Maksimovic et al. 2001):
Φ r,∞ = E C /e,(1)
where e is the electron charge and E C is defined in the Sun's rest frame. In this paper, we identify E C related to the sunward deficit in the electron VDFs observed by Parker Solar Probe (PSP), and use the exospheric prediction to estimate Φ r,∞ .
The electron runaway model prediction
A different theoretical description of the solar wind electrons is proposed by Scudder (1996Scudder ( , 2019a termed the Steady Electron Runaway Model (SERM). SERM accounts for the behaviour of weakly-collisional electrons in the large scale ambipolar electric field. Instead of assuming local thermodynamic equilibrium, it proposes a steady electron runaway effect. The Dreicer electric field (E D ) (Dreicer 1959(Dreicer , 1960) is used to compare the strength of E to the collisionality of the system. E D is defined as the constant electric field strength needed to accelerate a thermal particle to twice its velocity in one collision time (Dreicer 1959(Dreicer , 1960:
E D = 2k B T c eλ mfp ,(2)
where k B is the Boltzmann constant, T c is the core electron temperature parallel to the magnetic field and λ mfp is the collisional electron mean-free path. The resulting electron VDFs consist of collisionally overdamped and underdamped regions, separated by a 2D separatrix in phase space (Fuchs et al. 1986). The overdamped region corresponds to the core population, and the underdamped region to the suprathermal populations. The boundary for small pitch angles can be related to the energy at which the core transitions to the strahl, the strahl break-point energy (E BP ), which we also identify in the PSP electron VDFs. E then follows from (Scudder 2019a)
E = αk B T c E BP E D ,(3)
where α = 3 (Dreicer 1960) and E BP is defined in the plasma frame.
In Sec. 2 we describe the data set and the method we use to obtain Φ r,∞ and E . Sec. 3 presents our results, which we discuss in Sec. 4. We summarise our findings and draw conclusions in Sec. 5.
DATA ANALYSIS
Data Set
This work is based on the analysis of the electron VDFs measured by PSP (Fox et al. 2016), a heliospheric mission exploring the young solar wind near the Sun. The SPAN Electron (SPAN-E) instrument , part of the SWEAP investigation package on PSP (Kasper et al. 2016), measures the solar wind electrons. SPAN-E consists of two toroidal top-hat analysers, SPAN-A and SPAN-B, which together cover almost a full sky field of view (FOV). A small portion of the combined FOV in the direction of the Sun is blocked by the spacecraft's heat shield, which protects the payload from direct solar radiation. During encounter periods this FOV obstruction affects measurements taken within ∼ 10 • from the radial direction. Each of the tophat analysers measures electron velocity directions with 8 small (6 • ) and 8 large (24 • ) azimuth anodes, and 16 elevation deflection states, which vary in angular width from 10 • to 15 • . The electron energy is sampled in 32 log-spaced bins covering the energy range between 2 eV and 2 keV with a ∆E/E of 7%. The duration of a full 3D sweep over all energy and deflection bins is 0.218 s.
We use electron VDFs collected in Survey Mode during encounter periods, which consist of multiple full 3D sweeps integrated over time. The presented data was collected during PSP's perihelion passages 4 (Jan 23 2020 -Feb 3 2020) and 5 (May 30 2020 -June 15 2020), with the closest approach at 27.8 R S , and 6 (Sep 16 2020 -Oct 5 2020) and 7 (Jan 10 2021 -Jan 28 2021), with the closest approach at 20.3 R S . We thus investigate the region between 20.3 R S (0.10 au) and 85.3 R S (0.40 au). For encounters 4, 5 and 7 the integration time is set to 13.95 s, and for encounter 6 it is set to 3.49 s. Detailed descriptions of the SPAN-E instruments and their operating modes are provided by Whittlesey et al. (2020).
In our analysis we also use the magnetic field vector measured by the triaxial fluxgate magnetometer (MAG) part of the FIELDS investigation suite , and the proton velocity moment derived from the proton VDFs detected by the SPAN Ion (SPAN-I) instrument (Kasper et al. 2016). Both magnetic field and proton velocity are available with higher or equal cadence than the electron VDFs and are thus averaged over the SPAN-E integration periods to match the electron data.
Our main reason for using the data from four orbits of PSP, out of eight in total to date, is the availability of the SPAN-I data, which improves significantly after encounter 3. A Faraday cup instrument (SPC) (Case et al. 2019) also measures solar wind proton velocity on PSP, but discrepancies between the two instruments exist (Woodham et al. 2021). We choose to use SPAN-I data which provides more accurate data closer to the Sun, where the aberration allows the solar wind protons to fly into the instrument protected by the heat shield. Other reasons for the data selection are the changes made in the integration time and elevation deflection tables over the course of the first three orbits, ensuring optimal operation of SPAN-E during the later encounters. We also exclude the electron VDFs obtained at larger distances from the Sun during PSP cruise phase, which are integrated over larger time intervals. Our goal is to obtain a consistent data set of electron VDFs and Figure 1. Parallel and perpendicular cuts through an electron VDF in the magnetic field aligned, instrument centred frame. Blue and red dots are the measured points, while blue and red lines denote the parallel and perpendicular cuts through the 3D bi-Maxwellian fit to the core electron population. The blue and black dashed lines mark the transitions between the suprathermal deficit and the core (EC), and between the strahl and the core (EBP) in the parallel direction.
use it to investigate features typical for the near-Sun environment.
Method
Following the example of previous studies (e.g. Berčič et al. 2019;Halekas et al. 2019;Berčič et al. 2020) we analyse electron VDFs in the magnetic-field aligned, plasma rest frame. We rotate the VDFs given in the SPAN-A and SPAN-B instrument frames, using the magnetic field vector, and shift them using the spacecraft and the solar wind velocities. We show an example of an observed VDF in Fig. 1 as cuts along and perpendicular to the magnetic field direction. We already note the features investigated in this work along the parallel direction: the suprathermal deficit and the strahl. Our aim is to identify the energies at which the electron VDF starts to depart from the Maxwellian core in the directions parallel and anti-parallel to the magnetic field.
We fit the core with a bi-Maxwellian distribution function:
f c (v ⊥1 , v ⊥2 , v ) = A c exp − (v ⊥1 − v c⊥1 ) 2 w 2 c⊥ − (v ⊥2 − v c⊥2 ) 2 w 2 c⊥ − (v − v c ) 2 w 2 c ,(4)
where A c is the normalisation factor, w c⊥ is the perpendicular core thermal velocity, w c is the parallel core thermal velocity, v c⊥1,2 are the two core perpendicular drift velocities, and v c is the core parallel drift velocity. These quantities are our fit parameters, from which we obtain the core density as n c = A c π 3/2 w 2 c⊥ w c (5) and the core parallel and perpendicular temperatures as
T c = m e w 2 c 2k B and T c⊥ = m e w 2 c⊥ 2k B ,(6)
where m e is the mass of an electron. Even though v c⊥1,2 are expected to be 0 (Pilipp et al. 1987, e.g.), we allow for perpendicular core drifts to correct for possible errors in the measured solar wind velocity vector. v c⊥1,2 we obtain are small, and the resulting fit parameters are not strongly affected by it.
For the core fit, we only use the data points belonging to the core population, which we determine according to the electron energy and pitch angle. Firstly, we avoid the inclusion of photo-electrons and secondary electrons reflected from the spacecraft by setting a lower energy limit to 35.7 eV 1 .
Then, we avoid the inclusion of the strahl population through a two-step fitting procedure, which is based on the expected strahl break-point energy (E BP ) following from the kinetic solar wind model BiCoP (Binary Collisions in Plasmas) Berčič et al. (2021). We use this technique because we find that the core fit along the parallel direction is very sensitive to the selection of data points at small pitch angles. Assuming that E BP coincides with the separatrix between the collisionally overdamped and underdamped regions (Scudder 2019a), E BP is related to T c as
E BP = E D E 3k B T c .(7)
We perform the first fit to all measurements with pitch angles greater than 60 • and energies less than 132 eV. The parameters related to the first fit are marked with a tilde. T c,⊥ obtained from this first fit is already very accurate as the strahl is narrow near the Sun and mainly affects the core electron fit along the parallel direction. To obtain a zero-order estimation of E BP , we assume E ≈ E D and T c,⊥ ≈ T c , and calculate E BP as E BP = 3k B T c⊥ . This energy is then used for the second (final) fit as an upper energy limit for the data points with pitch angles less than 60 • .
To define the boundaries between the deficit and the core, and between the core and the strahl, we look for departures of the measured VDF from the fitted bi-Maxwellian core distribution. We calculate the normalised difference between the two in each measured point as
∆f data,fit = f data − f fit f data ,(8)
where f data are the measured values and f fit are the core fit values corresponding to the centres of the measurement bins in phase space. We bin ∆f data,fit into 20 •wide pitch-angle bins and calculate the median value in each bin (med {∆f data,fit }). This way we avoid the possible effects of the FOV blockage by the heat shield, which is ∼ 10 • wide in pitch angle, when magnetic field is aligned with radial direction (Kasper et al. 2016). Figure 2 shows these values for separate instrument energy bins on the example VDF from Figure 1. The value of ∆f data,fit at low energies, represented by the blue part of the colour-scale spectra, remains around 0, which means that the bi-Maxwellian fit represents well the electron VDF in this energy range. High energies are plotted in red and reach 1, which indicates that the measured VDF is much greater than the obtained core fit. The departure from the core fit at high energies is expected due to presence of the halo population. For the energies in between we observe a pitch-angle dependent evolution of the departures from the bi-Maxwellian core. We define the strahl break-point energy (E BP ) as the lowest energy at which med {∆f data,fit } exceeds the value of 0.15 in the 0 -20 • pitch-angle bin. If this energy is greater than 40 eV and less than 700 eV, we consider it a successful determination of E BP .
We define the electron cutoff energy (E C ) as the lowest energy at which med {∆f data,fit } decreases below the value of −0.15 in the 160 -180 • pitch-angle bin. Black dashed lines mark the limit values in Figure 2. The cut-off is only considered for instances when med {∆f data,fit } at any energy is less than −0.5. We move E C to the Sun rest frame using the solar wind velocity. The method is considered successful if the identified energy lies within an interval from 60 eV to 400 eV.
We calculate λ mfp used in Eq. 2 from the ratio between the electron parallel core thermal velocity (w c ) and the electron collision frequency, which we obtain from the relation between the electron density and temperature (e.g. Salem et al. 2003)
ν e = 2.9 · 10 −6 n c T −3/2 c ln Λ,(9)
where ln Λ is the Coulomb logarithm defined as
ln Λ = ln 12π( 0 k B T c ) 3/2 n 1/2 c e 3 ,(10)
0 is the vacuum permittivity.
RESULTS
The Figure 3 shows the same electron VDF as shown in Figure 1 but plotted against v and v ⊥ . We show a single VDF using four different 2D representations, which we obtain by the integration along the angle perpendicular to the magnetic field. All plots show the distribution in the magnetic-field-aligned frame centred on the core parallel velocity. The representation marked original shows the measured VDF values with a logarithmic colour-scale. In the representation marked scaled, each energy bin -i.e., each circular belt in (v , v ⊥ ) parameter space -is scaled to a value between 0 and 1, where 1 corresponds to the maximum value of the VDF in the given energy belt. This representation removes the information about the absolute value of the VDF and its strong gradient in energy. The benefit of the scaled VDF is the exposure of smaller anisotropic features at all energies. In cases for which two features arise in the same energy bin, the scaled VDFs can be misleading though as they highlight only the stronger feature. The representation marked normalised is obtained by dividing each of the VDF values with f (v ⊥ , v = 0) of the associated energy bin. Pitchangle directions in which the distribution function is less than f (v ⊥ , v = 0), appear in green, and those in which the distribution function is greater than f (v ⊥ , v = 0) appear in red. The representation marked fit-normalised shows the logarithm of electron VDF divided by the core fit. Yellow phase space regions are well represented by a bi-Maxwellian distribution while departures are seen in red and blue colours.
We use these representations to obtain a better understanding of the 2D shape of the features in the electron VDF. Electrons at energies below ∼ 100 eV are predominantly members of the almost isotropic core population, which shows almost no pitch-angle variation in all representations. At positive v , we observe a distinct strahl, the shape of which is most clearly defined in the scaled VDF. Its width in terms of perpendicular velocity appears almost constant, which gives a decreasing angular pitch-angle width with increasing electron energy. The sunward deficit is present at negative v , and shows in green and blue colours in the fit-normalised VDF. This feature not only persists at high pitch angles (close to 180 • ), but it forms a circular belt in phase space at some energies reaching to the strahl population at small pitch angles. The absence of the deficit in the normalised VDF tells us that this feature is close to isotropic, existing also at pitch angles around 90 • .
In total, we successfully fit 510,610 full 3D electron VDFs, out of which we determine E BP in 98.3 % and E C in 55.4 % of the cases. We statistically visualise the energy at which these two boundaries occur in a histogram in Figure 4, where the bin sizes correspond to the instrument's energy resolution. E BP and E C are strongly correlated and the mean ratio between the two (E C /E BP ) equals to 1.42.
We plot Φ r,∞ obtained from E C through Eq. 1 on a 2D histogram against radial distance in Figure 5a. The red dots mark the median values, and the corresponding error bars show the intervals of one standard deviation of the data set in each radial bin. Φ r,∞ decreases with radial distance and takes the values between 300 and 60 V. The step histograms above the plot compare the number of available data points in each radial bin to the number of data points for which E C is found. According to these histograms, the proportion of the electron VDFs with sunward deficit decreases with increasing radial distance. We fit all data points with a power law distribution of the form Φ r,∞ = Φ 0 r αΦ ,
where Φ 0 is a constant and α Φ is the power law index. The best fit is plotted with the black dashed line in Figure 5a and the fitting parameters noted in the legend. The black dotted line shows a fit with α Φ fixed to the value obtained from the radial evolution of the ambipolar potential in BiCoP simulations ). Figure 5. A 2D histogram showing the radial evolution of (a) Φr,∞, (b) E , (c) the ratio between EC and T c , and (d) the ratio between EBP and T c . The colour-scale represents the logarithm of the number of instances in each bin (log(#)). The red dots represent the median of samples in each radial bin with an error bar showing one standard deviation from the mean value in each bin. The dashed line represents the best power law fit (Eq. 11 and 12) to the data points, with the fitted parameters marked in the legend. The dotted line represents a fit where the power law index was fixed to equal the results from BiCoP . Above the main plot, 1D histograms show the number of data points in each of the radial bins. The number of all available data points is shown in red and the number of the data points used in the 2D histogram in blue.
We show a similar plot, but for E calculated from E BP through Eq. 3 in Figure 5b. The absolute values of E span between 0.5 to 10 nV/m and decrease faster with radial distance than Φ r,∞ . We fit the data points with a power law
E = E 0 r αE ,(12)
and mark the values of the fitting parameters E 0 and α E in the legend. As for the case of the ambipolar potential, we show a second fit as a dotted line, representing a power law function with an index equal to the one obtained from the BiCoP model. In the bottom row of Figure 5, we explore how the boundary energies E C and E BP compare to T c for different radial distances. The ratio E C /T c slightly decreases with radial distance, spanning from a mean value of 5.7 close to the Sun, to 3.3 farther away. The ratio E BP /T c exhibits an opposite trend, increasing from the mean value of 3.0 to 7.8. The electron deficit in the suprathermal energy range is reported already by Pilipp et al. (1987), analysing Helios data which covers distances down to 65 R S from the Sun. In the PSP data, the sunward deficit is a common feature (Halekas et al. 2019;Berčič et al. 2020), which contributes significantly to the net electron heatflux (Halekas et al. 2020). The characteristics of the sunward deficit and their relation to different solar wind parameters are investigated by Halekas et al. (2021).
In the collisionless exospheric models (Lemaire & Scherer 1970, 1971Jockers 1970;Pierrard et al. 1999;Maksimovic et al. 2001;Zouganelis et al. 2004) no electrons with energies greater than the electric potential energy (E Φ ) exist in the sunward portion of the electron VDF. Therefore, knowing the electron cutoff energy, we can obtain the value of the electric potential at any radial distance in the exosphere (Eq. 1). In kinetic models accounting additionally for Coulomb collisions, the electron cutoff is smoothed, appearing more like a deficit compared to the expected Maxwellian core VDF Berčič et al. 2021). These models reproduce the observed radial profiles of the electron core properties, like density, temper-ature and anisotropy. They also reproduce the strahl; however, they fail to produce the halo population. Electron VDFs observed close to the Sun (see example in Figures 1 and 3) exhibit only a tenuous halo population and are thus very similar to the VDFs from BiCoP simulation (see Figures 7 and 8 by Berčič et al. 2021). Comparing the normalised VDF in Figure 3 with the normalised VDF in Figure 7 in Berčič et al. (2021), we see however, that the shape of the sunward deficit is slightly different. In PSP data the deficit exists at pitch angles 45 • , while in BiCoP it only takes the angles 135 • . In the near-Sun solar wind (at ∼ 34 R S ), Coulomb collisions only effectively scatter the strahl electrons with energies smaller than 250 eV (Horaites et al. 2018;Boldyrev & Horaites 2019;Berčič et al. 2021). The scattering of the strahl at higher energies and the creation of the halo population are therefore attributed to other phenomena, including wave-particle interactions (Vocks et al. 2005;Kajdič et al. 2016;Verscharen et al. 2019;Jeong et al. 2020;Jagarlamudi et al. 2020;Cattell et al. 2021) and scattering by background turbulence (Pagel et al. 2007;Saito & Gary 2007). Observational studies byŠtverák et al. (2009) and Halekas et al. (2019) suggest that the halo is more prominent farther from the Sun, which could be the reason why the sunward deficit has not been observed at larger radial distances ( Figure 5a).
A recent study by Bercic, L. et al. (2021), who analyse Solar Orbiter (Müller et al. 2020) in situ measurements of solar-wind electrons, shows that the sunward deficit can drive the growth of quasi-parallel whistler waves, leading to quasi-linear electron diffusion in velocity space. This new proposed instability tends to fill the sunward deficit and may thus also be the reason why the electron cutoff ceases to exist at larger radial distances.
Another possibility for the disappearance of the deficit could simply be the Coulomb collisions: as Φ r,∞ decreases with radial distance it moves to the energy range where electron collisions are frequent. They could completely erase the residue of the cutoff.
Ballistic electrons, in exospheric models representing the electron core population, are limited in energy to a range E Φ . Therefore, we expect that the core electron temperature (T c ) follows the radial evolution of Φ r,∞ . This would show as a constant ratio between E C and T c in Figure 5c. The observed variation in the ratio is not large, but a slight decreasing trend is present, mostly for radial distances below 35 R S . The increase in E C /T c with decreasing heliocentric distance suggests the increasing importance of Coulomb collisions, which smear the exospheric VDF features and raise E C .
We compare the measured Φ r,∞ to the results of a kinetic numerical model (BiCoP) , which builds up a supersonic radially expanding solar wind taking into account binary particle collisions and the self-consistent E (Landi & Pantellini 2001, 2003. The simulation box spans from 3 to 49 R S , thus overlapping with approximately half of the radial interval shown in this study. The self-consistently obtained Φ r,∞ from BiCoP is added to Figure 5a and evolves with radial distance as a power law with an index α Φ,BCP = −0.55. This result is close to the power law index obtained in our observational study, α Φ = −0.66. In Figure 5a, we show a second fit to the data points with fixed α Φ = −0.55 (dotted line) to emphasise that the power law index is a sensitive fitting coefficient, and may vary across different solar wind streams. For the scope of this work, we only calculate the average properties of all of the solar wind measured during the four PSP encounters.
Our experimentally determined α Φ diverges from the power law index in collisionless exospheric models, α Φ,Exo = −1.33 (e.g. Meyer-Vernet & Issautier 1998; Zouganelis et al. 2004), indicating that collisions play an important role in the radial evolution of Φ r,∞ and in the ambipolar solar wind acceleration. An analytical solution of the drift-kinetic equation including the effects of Coulomb collisions (Boldyrev et al. 2020) gives a power law with α Φ,DK = −0.4, which is closer to our observations.
We add Φ r,∞ obtained with a medium-collisional Bi-CoP run (MC in Berčič et al. 2021) to Figure 5a. The modelled values are within the range of the observed values, even though the BiCoP boundary parameterselectron and proton temperature at 3 R S set to 121 eV -differ from the expected coronal temperatures. Electrons in the corona are observed to be colder, T e ∼ 86 eV (1 MK) (Cranmer 2002;Berčič et al. 2020;Stansby et al. 2020), while the proton temperature is expected to be greater. This difference in temperature between the two species and the preferential perpendicular heating of solar wind protons potentially result in the observed Φ r,∞ , even when the electron temperature at the origin is less than the electron temperature assumed in BiCoP simulations. Initialising BiCoP runs with different coronal temperatures for electrons and protons, and with varying proton anisotropies would give further insight into this phenomenon.
Ambipolar electric field (E )
The SERM (Scudder 1996(Scudder , 2019a accounts for the effects of the global ambipolar electric field (E ) in the presence of Coulomb collisions. The Dreicer electric field (E D , Eq. 2) serves as a measure of the importance of these two competing phenomena (Dreicer 1959(Dreicer , 1960. We use the measured E BP to estimate the ambipolar electric field in the solar wind (Eq. 3). Before discussing the properties of the observed E , we compare E BP in the near-Sun solar wind to already existing studies.
The idea that the separatrix between the thermal and suprathermal electron populations contains information about the electron kinetics is discussed in early observational studies (Feldman et al. 1975;Pilipp et al. 1987). Scudder & Olbert (1979) theoretically predict that the break-point energy scales with the local electron temperature as E BP = 7k B T c . This value agrees with the breakpoint between the core and the halo obtained byŠtverák et al. (2009), who analyse electron VDFs from Helios, Cluster, and Ulysses. However, the ratio E BP /T c corresponding to the strahl population assumes slightly lower values, between 2 (at 0.3 au) and 5 (at 3 au). Similar values are obtained by Landi et al. (2012) using a kinetic BiCoP simulation. At radial distances between 1 and 3 au, they find that E BP /T c varies between 1 and 4 and depends mainly on the density of the modelled solar wind. In a more recent study, including Cluster data, Bakrania et al. (2020) obtain the ratio of 5.5 at 1 au as well as an anti-correlation between the strahl-E BP and the solar wind velocity.
The ratio E BP /T c obtained from the PSP data shown in our work ( Figure 5d) agrees with previous Helios results. Its median value is approximately constant, ∼ 3, up to a radial distance of 50 R S . At greater distances, it approaches ∼ 6.
For the majority of the samples, we find E BP < E C , which suggests that electrons with energy less than E C are not limited to Maxwellian core electrons but include a small part of strahl electrons. The same is seen in the BiCoP kinetic solar wind model .
We present the first observational estimates of E in the solar wind (Fig. 5b). Its strength is of order a few nV/m, and, as expected, decreases with radial distance. The radial evolution is best represented by a power law function with an index α E = −1.69. This index is very close to the power index resulting from BiCoP simulations, α E,BCP = −1.55 . Following the same method as for Φ r,∞ , the dotted curve in Figure 5b shows a power law fit to the data points with a fixed index of −1.55. For comparison, we plot E from the medium-collisional BiCoP run to Figure 5b as a black line.
The parameters Φ r,∞ and E are related to each other as
Φ r,∞ = ∞ r E (r) dr.(13)
If we assume that Φ r,∞ and E follow power laws in r, then the difference between the power law indices of each of the quantities should be equal to 1 (α E − α Φ = −1).
Our results agree with this theoretical relation within the measurement uncertainty, as the difference between the fitted power law indices is 1.02.
Ambipolar contribution to the acceleration of the solar wind
The total solar wind proton potential energy Ψ(r) is the sum of the repulsive electric potential energy E Φ (r) and the attractive gravitational potential energy E G (r). We use the fitted curve from Figure 5a to
calculate E Φ (r) from Φ r,∞ as E Φ (r) = eΦ 0 r αΦ(14)
based on our determination of E C . Likewise, we use the fitted curve from Figure 5b to calculate E Φ (r) from E as
E Φ (r) = − eE 0 r 0 (α E + 1) r α E +1(15)
based on our determination of E BP . The gravitational energy of a proton is defined as
E G (r) = GM S r m p ,(16)
where G is the gravitational constant, M S the mass of the Sun and m p the mass of a proton. The results are shown in the top row of Figure 6. E G decreases with radial distance faster than E Φ , which means that E Φ dominates at larger radial distances, and Ψ(r) peaks at a radial distance, which we denote r max . The kinetic theory (Scudder 1996) and BiCoP numerical simulations (Landi & Pantellini 2003) predict a maximum of Ψ(r) near the proton sonic point. The bottom row of Figure 6 shows the radial evolution of the energy gradients, corresponding to the electric force F E and the gravitational force F G . At small radial distances, F G > F E , so that the net force on protons points towards the Sun. The radial distance at which F G = F E is marked with a dashed line and corresponds to the location of the total energy peak, r max . All protons with v ≥ 0 present at r max can escape the Sun's gravitational potential, as the net force on them above this distance is only positive. We calculate the radial evolution of the velocity gained by a test proton (v pΨ (r)) moving in the total potential Ψ(r) through integration of the net force, F (r) = F E (r) − F G (r), above
F E F G (N) from E C from E BP
The terminal test proton velocities v pΨ (∞) result in 164 km s −1 from the method using E C , and 54 km s −1 from the method using E BP .
To calculate the bulk velocity of the protons v p (r), we follow the exospheric approach. We assume a Maxwellian proton distribution at r max , with a parallel temperature T p = 0.7 MK. This is an estimation of the T p at 7 R S following from the extrapolation of the radial trends presented by Maksimovic et al. (2020). The proton parallel temperature in this simple approach does not vary with radial distance, so v p (r) is
v p (r) = v pΨ (r) + v p (r max ),(18)
where
v p (r max ) = 2w p √ π .(19)
w p is the proton parallel thermal velocity defined as
w p = 2k B T p m p .(20)
For simplicity we use the same v p (r max ) = 121 km s −1 for both obtained solutions, even though they exhibit different r max . Figure 7 shows the obtained velocity curves together with their asymptotic values, marked with blue and pink dashed lines. Black dashed line denotes v p (r max ). v p (r) resulting from E C is greater than v p (r) resulting from E BP , reaching a terminal velocity of 286 km s −1 . At the radial distance of 45 R S the average observed proton velocity is 303 km s −1 . The resulting v p (45 R S ) = 233 km s −1 corresponds to 77% of the observed velocity, or 59% of the proton kinetic energy. This means that 23% of the measured solar wind velocity must be gained through other solar wind acceleration mechanisms.
v p (r) obtained from E BP related to the SERM model is smaller, with a terminal velocity of 175 km s −1 . This result at first appears unphysical, since r max = 53 R S , which would suggest that below this distance we should not observe supersonic protons at all. However, depending on the location of the solar wind acceleration by mechanisms other than E , the contribution of the ambipolar acceleration could increase. Additional kinetic energy close to the Sun could produce a positive net force at smaller radial distances, creating more space for the ambipolar acceleration. Comparing the obtained terminal speed with the typical solar wind speed at 1 au, we find that the E is responsible for 44% of the solar wind velocity and 19% of proton kinetic energy.
Note that this is only a crude estimation, as the model we use is simplified and includes strong assumptions. On the other hand, the statistical errors arising from the data analysis and the fits in Figures 5a and 5b are small. We do not include them in Figure 7, because this could be misleading for the reader.
Since this is the first effort to empirically quantify the acceleration of the solar wind by E , it is difficult to make conclusions on which of the separately obtained results is more valid. Φ r,∞ calculated from E C is potentially overestimated, because we can not be sure that the energy associated with the sunward deficit directly corresponds to the electron cutoff in collisionless exospheric models. The boundary could be, as a consequence of Coulomb collisions, pushed towards higher energies.
On the other hand, E may be underestimated. In our analysis, we use the assumption that the separatrix between the overdamped and underdamped region is the same in the direction along with and opposite to the direction of the electric force on the electrons. We use this approximation because it appears consistent with the BiCoP simulations ), but theoretical work by Dreicer (1960) and Fuchs et al. (1986) suggests that the boundary is asymmetric and appears at higher energy in the direction opposite to the electric force. This would in our analysis lead to a multiplication factor in Eq. 3 and consequentially higher terminal solar wind velocities. Further investigations are needed to relate the 2D shape of the separatrix in the observed VDFs to the 2D shape predicted by theoretical models.
CONCLUSIONS
We analyse electron VDFs measured by PSP in the near-Sun solar wind during its orbits 4 to 7. We identify the electron energies at which the measured distribution departs from the bi-Maxwellian core electron fit in the direction parallel and anti-parallel to the magnetic field. In the sunward part of phase space, we define the cutoff energy (E C ) that marks the appearance of the sunward electron deficit. In the anti-sunward portion of phase space, we define the strahl break-point energy (E BP ) that marks the start of the strahl population. While the strahl is detected in almost all of the analysed distributions, the sunward deficit is only found in 56.8% of the cases. The relative amount of electron VDFs with a sunward deficit decreases for larger radial distances.
We relate E C to the electron cutoff in exospheric solar wind models, which allows us to estimate the ambipolar potential between the observation point and the asymptotic potential at large heliocentric distances (Φ r,∞ ). The resulting Φ r,∞ decreases with radial distance as r −0.66 . Its radial trend agrees with the results of the kinetic BiCoP model , while its magnitude is slightly smaller than Φ r,∞ obtained numerically.
We assume that E BP represents the separatrix between collisionally overdamped and underdamped regions of phase space, defined in the Steady Electron Runaway Model (Scudder 2019a). This allows us to estimate the ambipolar electric field in the solar wind (E ). The estimated E is of order 1 nV/m and decreases with radial distance as r −1.69 .
We finally calculate the total proton potential energy Ψ(r), separately from Φ r,∞ and E to estimate the contribution of the ambipolar acceleration to the total solar wind acceleration. From the approach following the exospheric models and E C we find a terminal solar wind velocity 286 km s −1 . Following the SERM model and E BP we find a terminal solar wind velocity 175 km s −1 . In the first case we are able to directly compare the observed solar wind velocity with the calculated ambipolar contribution, which amounts to 77% at the radial distance of 45 R S .
The SWEAP and FIELDS experiments on the Parker Solar Probe spacecraft were designed and developed under NASA contract NNN06AA01C. L. B., C. J. O., and D. V. are supported by STFC Consolidated Grant ST/S000240/1. D. V. is supported by STFC Ernest Rutherford Fellowship ST/P003826/1.
Figure 2 .
2The normalised difference between the measured VDF and the core fit (∆f data,fit ) binned in 20 • pitch-angle bins. Different colours mark separate instrument energy bins. Black dashed line shows the criteria used in the determination of EC and EBP.
Figure 3 .
3The same electron VDF as shown inFigures 1 and 3, plotted as a function of parallel and perpendicular velocity. The VDF is shown in the magnetic field aligned frame, centred on the core parallel velocity. Plots from left to right present: the original VDF; the scaled VDF, where values in each energy bin are scaled between 0 and 1; the normalised VDF, where the original VDF is divided by the perpendicular VDF cut (f (v ⊥ , v = 0)); and the fit normalised VDF, where the VDF is divided by the core electron fit.
Figure 4 .
4A 2D histogram showing the relation between EBP and EC . The bin size corresponds to the energy resolution of the SPAN-E instrument. The colour-scale represents the logarithm of the number of instances in each bin (log(#)). The dashed line denotes EC = 1.42EBP.
.
Ambipolar electric potential (Φ r,∞ )
Figure 6 .Figure 7 .
67(a) Radial evolution of the proton gravitational energy (EG) and the electric potential energy (EΦ); (b) Solar wind proton energy balance; (c) Radial evolution of the gravitational force (FG) and the electric force (FE) for a proton; (d) The resulting net force on a proton. In all panels, blue colour corresponds to the solution obtained from EC, and red colour to the solution obtained from EBP. Vertical dashed lines mark rmax. Radial evolution of the solar wind proton velocity (vp(r)) obtained via Eq. 18. The blue and pink lines mark the solutions resulting from different models and different features of the electron VDFs. The dashed blue and pink lines mark their asymptotic values. The black dashed line marks the bulk velocity vp(rmax) corresponding to T p = 0.7 MK. The black dots with belonging error bars show the mean value and the standard deviation of the observed solar wind velocity binned along the radial direction in the same way as inFigure 5.
The same method is used byBerčič et al. (2020); however,Halekas et al. (2019) fit and subtract the secondary electron peak. Both methods produce similar results.
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| []
|
[
"Detection of relic gravitational waves in the CMB: Prospects for CMBPol mission",
"Detection of relic gravitational waves in the CMB: Prospects for CMBPol mission"
]
| [
"Wen Zhao \nInternational Center for Astrophysics\nKorea Astronomy and Space Science Institute\n305-348DaejeonKorea\n\nSchool of Physics and Astronomy\nCardiff University\nCF24 3AACardiffUnited Kingdom\n\nDepartment of Physics\nZhejiang University of Technology\n310014HangzhouP.R.China\n"
]
| [
"International Center for Astrophysics\nKorea Astronomy and Space Science Institute\n305-348DaejeonKorea",
"School of Physics and Astronomy\nCardiff University\nCF24 3AACardiffUnited Kingdom",
"Department of Physics\nZhejiang University of Technology\n310014HangzhouP.R.China"
]
| []
| Detection of relic gravitational waves, through their imprint in the cosmic microwave background radiation, is one of the most important tasks for the planned CMBPol mission. In the simplest viable theoretical models the gravitational wave background is characterized by two parameters, the tensor-to-scalar ratio r and the tensor spectral index nt. In this paper, we analyze the potential joint constraints on these two parameters, r and nt, using the potential observations of the CMBPol mission, which is expected to detect the relic gravitational waves if r 0.001. The influence of the contaminations, including cosmic weak lensing, various foreground emissions, and systematical errors, is discussed. PACS numbers: 98.70.Vc, 98.80.Cq, 04.30.-w * Electronic address: [email protected] detection of primordial gravitational waves is rightly considered a highest priority task for the upcoming observation missions[1]. A stochastic background of the relic gravitational waves (RGWs), produced in the very early Universe due to the superadiabatic amplification of zero point quantum fluctuations of the gravitational field, is a necessity dictate by general relativity and quantum mechanics[2]. So the detection of RGWs maybe provide the unique way to study the birth of the Universe, and test the applicability of general relativity and quantum mechanics in the very high energy scale[3].In a whole range of scenarios of the early Universe, the primordial power spectrum of the RGWs can be well described by a power-law form in a fairly large frequency range[4][5][6][7][8][9]. Thus, the RGW backgrounds are conventionally simply characterized by two parameters, the so-called tensor-to-scalar ratio r and the primordial power spectral index of RGWs n t , where r describes the amplitude of the primordial spectrum, and n t denotes the tilt of the spectrum. So the constraints on r and n t will give us a direct glimpse into the physical conditions in the early Universe. In particular, they will allow us to place the constraints on the Hubble parameter of the early Universe, and time evolution of this Hubble parameter. Unfortunately, the models of the early Universe cannot give a definite prediction for the values of r and n t , i.e., different inflationary models predict the quite different values of r and n t [10], especially some string motivated inflationary models predict a very small gravitational waves with r ≪ 10 −4[11]. So, the only way to determine them is by the observations.The RGWs leave well understood imprints on the anisotropies in temperature and polarization of cosmic microwave background radiation (CMB)[12][13][14][15][16][17][18]. More specifically, RGWs produce a specific pattern of polarization in the CMB known as the B-mode polarization[13]. Moreover, RGWs produce a negative cross-correlation between the temperature and polarization known as the T E-correlation at the low multipoles ℓ 50[17,18]. The theoretical analysis of these imprints along with the data from CMB experiments allows to place constraints on the parameters r and n t describing the RGW background, which provides the unique way to detect the RGWs at the very low frequencies (10 −17 ∼ 10 −15 Hz).The current CMB experiments are yet to detect a definite signature of RGWs[19], although a hint of RGWs is found in the WMAP data[20]. A number of authors have discussed the possibility of RGW detection by the launched Planck satellite[20][21][22][23]. The results show that, due to the fairly large instrumental noises, only if the tensor-to-scalar ratio is large (r 0.05), the Planck satellite is expected to have a detection. In the previous paper [24], we also found that the Planck mission cannot give a good constraint for the spectral index n t , even if the tensor-to-scalar ratio is as large as r = 1. In addition, various ground-based [25] and balloon-borne [26] CMB experiments are expected to have the better detection abilities, which can constrain the parameter r fairly well when r 0.01. However, an accurate constraint of n t is still unexpected, due to the small partial sky observation or the short time observation[27].The accurate determination of the RGWs requires the full sky and the long time observations by the CMB experiment with the quite small instrumental noises. The future space-based mission focus on the CMB polarization[28](here and in the following, we use the label 'CMBPol' to refer it) provides an excellent opportunity to realize it (the similar projects, such as B-Pol[29]and LiteBird [30], are also proposed). The instrumental noises of CMBPol mission are more than 100 times smaller than those of Planck mission. If the foreground contaminations and the systematical errors can be well controlled, the signature of RGWs can be well detected, as long as r 0.001[28]. This will provide an observational tool to distinguish the different inflationary-type models.In this paper we shall analyze the joint constraints on two parameters r and n t that would be feasible with the analysis of the observations from the planned CMBPol mission. We shall detailedly discuss the constraints of r, n t and the best-pivot wavenumber k * t depending on the input (or true) value of the tensor-to-scalar ratio r. We discuss the effects of various contaminations from the cosmic weak lensing, foreground radiations and the beam systematics.The outline of the paper is as follows. In Sec. II we shall introduce and explain the notations for the power spectra of gravitational waves, density perturbations and various CMB anisotropy fields. Furthermore, in this section, we shall briefly introduce the existence of the best-pivot wavenumber k * t for the detection of RGWs in the CMB. The analytical formulae and explanation for the k * t , ∆r and ∆n t will also be discussed. In Sec. III, by using the analytical formulae and only considering the instrumental noises, we shall discuss the values of k * t , ∆r and ∆n t for different input (or true) value of r. Sec. IV is contributed to show the effect of the cosmic lensing contamination, and Sec. V is contributed to show the effect of the foreground radiations contamination. In Sec. VI, we discuss the effects of various beam systematics for the determination of the parameters r and n t . We also discuss the requirement of the CMBPol's systematics, if the biases of the parameters r and n t are ignorable. Finally, Sec. VII is dedicated to a brief discussion and conclusion. | 10.1088/1475-7516/2011/03/007 | [
"https://arxiv.org/pdf/1102.4908v1.pdf"
]
| 118,397,927 | 1102.4908 | 6483d667ebd1c7d954527ac0daa4ba9ca105fc2e |
Detection of relic gravitational waves in the CMB: Prospects for CMBPol mission
24 Feb 2011
Wen Zhao
International Center for Astrophysics
Korea Astronomy and Space Science Institute
305-348DaejeonKorea
School of Physics and Astronomy
Cardiff University
CF24 3AACardiffUnited Kingdom
Department of Physics
Zhejiang University of Technology
310014HangzhouP.R.China
Detection of relic gravitational waves in the CMB: Prospects for CMBPol mission
24 Feb 2011(Dated: January 13, 2013)numbers: 9870Vc9880Cq0430-w * Electronic address: wzhao7@kasirekr
Detection of relic gravitational waves, through their imprint in the cosmic microwave background radiation, is one of the most important tasks for the planned CMBPol mission. In the simplest viable theoretical models the gravitational wave background is characterized by two parameters, the tensor-to-scalar ratio r and the tensor spectral index nt. In this paper, we analyze the potential joint constraints on these two parameters, r and nt, using the potential observations of the CMBPol mission, which is expected to detect the relic gravitational waves if r 0.001. The influence of the contaminations, including cosmic weak lensing, various foreground emissions, and systematical errors, is discussed. PACS numbers: 98.70.Vc, 98.80.Cq, 04.30.-w * Electronic address: [email protected] detection of primordial gravitational waves is rightly considered a highest priority task for the upcoming observation missions[1]. A stochastic background of the relic gravitational waves (RGWs), produced in the very early Universe due to the superadiabatic amplification of zero point quantum fluctuations of the gravitational field, is a necessity dictate by general relativity and quantum mechanics[2]. So the detection of RGWs maybe provide the unique way to study the birth of the Universe, and test the applicability of general relativity and quantum mechanics in the very high energy scale[3].In a whole range of scenarios of the early Universe, the primordial power spectrum of the RGWs can be well described by a power-law form in a fairly large frequency range[4][5][6][7][8][9]. Thus, the RGW backgrounds are conventionally simply characterized by two parameters, the so-called tensor-to-scalar ratio r and the primordial power spectral index of RGWs n t , where r describes the amplitude of the primordial spectrum, and n t denotes the tilt of the spectrum. So the constraints on r and n t will give us a direct glimpse into the physical conditions in the early Universe. In particular, they will allow us to place the constraints on the Hubble parameter of the early Universe, and time evolution of this Hubble parameter. Unfortunately, the models of the early Universe cannot give a definite prediction for the values of r and n t , i.e., different inflationary models predict the quite different values of r and n t [10], especially some string motivated inflationary models predict a very small gravitational waves with r ≪ 10 −4[11]. So, the only way to determine them is by the observations.The RGWs leave well understood imprints on the anisotropies in temperature and polarization of cosmic microwave background radiation (CMB)[12][13][14][15][16][17][18]. More specifically, RGWs produce a specific pattern of polarization in the CMB known as the B-mode polarization[13]. Moreover, RGWs produce a negative cross-correlation between the temperature and polarization known as the T E-correlation at the low multipoles ℓ 50[17,18]. The theoretical analysis of these imprints along with the data from CMB experiments allows to place constraints on the parameters r and n t describing the RGW background, which provides the unique way to detect the RGWs at the very low frequencies (10 −17 ∼ 10 −15 Hz).The current CMB experiments are yet to detect a definite signature of RGWs[19], although a hint of RGWs is found in the WMAP data[20]. A number of authors have discussed the possibility of RGW detection by the launched Planck satellite[20][21][22][23]. The results show that, due to the fairly large instrumental noises, only if the tensor-to-scalar ratio is large (r 0.05), the Planck satellite is expected to have a detection. In the previous paper [24], we also found that the Planck mission cannot give a good constraint for the spectral index n t , even if the tensor-to-scalar ratio is as large as r = 1. In addition, various ground-based [25] and balloon-borne [26] CMB experiments are expected to have the better detection abilities, which can constrain the parameter r fairly well when r 0.01. However, an accurate constraint of n t is still unexpected, due to the small partial sky observation or the short time observation[27].The accurate determination of the RGWs requires the full sky and the long time observations by the CMB experiment with the quite small instrumental noises. The future space-based mission focus on the CMB polarization[28](here and in the following, we use the label 'CMBPol' to refer it) provides an excellent opportunity to realize it (the similar projects, such as B-Pol[29]and LiteBird [30], are also proposed). The instrumental noises of CMBPol mission are more than 100 times smaller than those of Planck mission. If the foreground contaminations and the systematical errors can be well controlled, the signature of RGWs can be well detected, as long as r 0.001[28]. This will provide an observational tool to distinguish the different inflationary-type models.In this paper we shall analyze the joint constraints on two parameters r and n t that would be feasible with the analysis of the observations from the planned CMBPol mission. We shall detailedly discuss the constraints of r, n t and the best-pivot wavenumber k * t depending on the input (or true) value of the tensor-to-scalar ratio r. We discuss the effects of various contaminations from the cosmic weak lensing, foreground radiations and the beam systematics.The outline of the paper is as follows. In Sec. II we shall introduce and explain the notations for the power spectra of gravitational waves, density perturbations and various CMB anisotropy fields. Furthermore, in this section, we shall briefly introduce the existence of the best-pivot wavenumber k * t for the detection of RGWs in the CMB. The analytical formulae and explanation for the k * t , ∆r and ∆n t will also be discussed. In Sec. III, by using the analytical formulae and only considering the instrumental noises, we shall discuss the values of k * t , ∆r and ∆n t for different input (or true) value of r. Sec. IV is contributed to show the effect of the cosmic lensing contamination, and Sec. V is contributed to show the effect of the foreground radiations contamination. In Sec. VI, we discuss the effects of various beam systematics for the determination of the parameters r and n t . We also discuss the requirement of the CMBPol's systematics, if the biases of the parameters r and n t are ignorable. Finally, Sec. VII is dedicated to a brief discussion and conclusion.
I. INTRODUCTION
II. OPTIMAL PARAMETERS AND THEIR DETERMINATIONS
The main contribution to the observed temperature and polarization anisotropies of the CMB comes from two types of the cosmological perturbations, density perturbations (also known as the scalar perturbations) and RGWs (also known as the tensor perturbations) [6,7,12,13], which are generally characterized by their primordial power spectra. These power spectra are usually assumed to be power-law, which is a generic prediction of a wide range of scenarios of the early Universe, including the inflationary models. In general there might be deviations from a power-law, which can be parameterized in terms of the running of the spectral index (see for example [10,31]), but we shall not consider this possibility in the current paper. Thus, the power spectra of the perturbation fields have the form
P s (k) = A s (k 0 )(k/k 0 ) ns−1 , P t (k) = A t (k 0 )(k/k 0 ) nt ,(1)
for density perturbations and RGWs respectively. In the above expression k 0 is an arbitrarily chosen pivot wavenumber, n s is the primordial power spectral index for density perturbations, and n t is the primordial power spectral index for RGWs. A s (k 0 ) and A t (k 0 ) are normalization coefficients determining the absolute values of the primordial power spectra at the pivot wavenumber k 0 . The choices of n s = 1 and n t = 0 correspond to the scale-invariant power spectra for density perturbations and gravitational waves respectively. The relative contribution of density perturbations and gravitational waves is described by the so-called tensor-toscalar ratio r, which is defined as follows
r(k 0 ) ≡ A t (k 0 ) A s (k 0 ) .(2)
Note that, in defining the tensor-to-scalar ratio r, we have not used any inflationary formulae which relate r with the physical conditions during inflation and the slow-roll parameters (see for example [32]). Thus, our definition depends only on the power spectral amplitudes of density perturbations and RGWs, and does not assume a particular generating mechanism for these cosmological perturbations. The RGW amplitude A t (k 0 ) = r(k 0 )A s (k 0 ) provides us with direct information on the Hubble parameter of the very early Universe [31]. More specifically, this amplitude is directly related to the value of the Hubble parameter H at a time when wavelengths corresponding to the wavenumber k 0 crossed the horizon [33] A 1/2
t (k 0 ) = √ 2 M pl H π k0/a=H ,(3)
where M pl = 1/ √ 8πG is the reduced Planck mass. If we adopt A s = 2.445 × 10 −9 as predicted by the WMAP5 observations [34], the Hubble parameter is H ≃ 2.67r 1/2 ×10 14 GeV, only depending on the value of r. In the canonical single-field slow-roll inflationary models, the Hubble parameter directly relates to the energy scale of inflation V 1/4 . The relation (3) follows that V 1/4 ≃ 3.35r 1/4 × 10 16 GeV, which has been emphasized by a number of authors.
Assuming that the amplitude of density perturbations A s (k 0 ) is known, taking into account the definitions (1) and (2), the power spectrum of the RGW field may be completely characterized by tensor-to-scalar ratio r and the spectral index n t . It is important to mention that, for spectral indices different from the scale-invariant case (i.e., when n s = 1 or/and n t = 0), the definition of the tensor-to-scalar ratio depends on the pivot wavenumber k 0 . If we adopt a different pivot wavenumber k 1 , the tensor-to-scalar ratio at this new pivot wavenumber r(k 1 ) is related to original ratio r(k 0 ) through the following relation (which follows from the definitions (1) and (2))
r(k 1 ) = r(k 0 ) k 1 k 0 nt−ns+1 .(4)
Let us now turn our attention to CMB. Density perturbations and gravitational waves produce temperature and polarization anisotropies in the CMB, which are characterized by four angular power spectra C T ℓ , C C ℓ , C E ℓ and C B ℓ as functions of the multipole number ℓ. Here C T ℓ is the power spectrum of the temperature anisotropies, C E ℓ and C B ℓ are the power spectra of the so-called E and B modes of polarization (note that, density perturbation do not generate B-mode of polarization [13]), and C C ℓ is the power spectrum of the temperature-polarization cross correlation. In general, the power spectra C Y ℓ (where Y = T, E, B or C) can be presented in the following form
C Y ℓ = C Y ℓ (dp) + C Y ℓ (gw),(5)
where C Y ℓ (dp) is the power spectrum due to the density perturbations, and C Y ℓ (gw) is the power spectrum due to RGWs. In the case of RGWs, the CMB power spectra can be presented in the following form [14,15] C Y ℓ (gw) = (4π) 2 dk k P t (k) ∆ (T ) Y ℓ (k) 2 , for Y = T, E, B,
C C ℓ (gw) = (4π) 2 dk k P t (k) ∆ (T ) T ℓ (k)∆ (T ) Eℓ (k) .(6)
The transfer functions ∆ (T ) Y ℓ (k) (see [14,15] for details) in the above expressions translate the power in the metric fluctuations (gravitational waves) into corresponding CMB power spectrum at an angular scale characterized by multipole ℓ. In this work, for numerical evaluation of the CMB power spectra due to density perturbations and gravitational waves, we use the publicly available CAMB code [35].
Since we are primarily interested in the parameters of the RGW field, in the analytical and numerical analysis below we shall work with a fixed cosmological background model. More specifically, we shall work in the framework of ΛCDM model, and keep the background cosmological parameters fixed at the values determined by a typical model [34] h = 0.705, Ω b h 2 = 0.02267, Ω m h 2 = 0.1131, Ω k = 0, τ reion = 0.084, A s = 2.445 × 10 −9 .
Furthermore, the spectral indices of density perturbations and gravitational waves are adopted as follows for the simplicity,
n s = 1, n t = 0.(8)
Note that throughout this paper, we have considered the simplest cosmological model. In the more general consideration, one should also include the running of the spectral indices [31], the details of the reionization history [36] and so on, which have been ignored in this paper.
The CMB power spectra C Y ℓ are theoretical constructions determined by ensemble averages over all possible realizations of the underlying random process. However, in real CMB observations, we only have access to a single sky, and hence to a single realization. In order to obtain information on the power spectra from a single realization, it is required to construct estimators of power spectra. In order to differentiate the estimators from the actual power spectra, we shall use the notation D Y ℓ to denote the estimators while retaining the notation C Y ℓ to denote the power spectrum. It is important to keep in mind that the estimators D Y ℓ are constructed from observational data, while the power spectra C Y ℓ are theoretically predicted quantities. The probability distribution functions for the estimators are described in detail in [18] (see also [22,37,38]), which predicts the expectation values of the estimators
D Y ℓ = C Y ℓ ,(9)
and the standard deviations
(σ D X ℓ ) 2 = 2(C X ℓ + N X ℓ ) 2 (2ℓ + 1)f sky , (X = T, E, B) (σ D C ℓ ) 2 = (C T ℓ + N T ℓ )(C E ℓ + N E ℓ ) + (C C ℓ + N C ℓ ) 2 (2ℓ + 1)f sky ,(10)
where f sky is the sky-cut factor. In this paper, we use f sky = 0.8 for the CMBPol survey. N Y ℓ are the noise power spectra, which are all determined by the specific experiments. In this formulas, the possible bias generated by the beam systematics has not been considered (see Sec. VI for details).
In order to estimate the parameters r and n t characterizing the RGW background, we shall use an analysis based on the likelihood function [39]. The likelihood function is just the probability density function of the observational data considered as a function of the unknown parameters (which are r and n t in our case). Up to a constant, independent of its arguments, the likelihood function is given by
L = ℓ f (D C ℓ , D T ℓ , D E ℓ , D B ℓ ),
where the function f (D C ℓ , D T ℓ , D E ℓ , D B ℓ ) is explained in detail in the previous works [22,24]. In the previous works [22,24], we have discussed how to constrain the parameters of the RGWs, r and n t , by the CMB observation. In [24], we found that in general, the constraints on r and n t correlate with each other. However, if we consider the tensor-to-scalar ratio at the best-pivot wavenumber k * t , i.e. r * ≡ r(k * t ), the constraints on r and n t becomes independent of each other, and the uncertainties ∆r and ∆n t have the minimum values. We have derived the formulae to calculate the quantities: the best-pivot wavenumber k * t , and the uncertainties of the parameters ∆r and ∆n t , which provides a simple and quick method to investigate the detection abilities of the future CMB observations. We shall briefly introduce these results in this section.
It is convenient to define the quantities as below,
a Y ℓ ≡ C Y ℓ (gw) σ D Y ℓ , b * ℓ ≡ ln ℓ ℓ * t , d Y ℓ ≡ D Y ℓ − C Y ℓ (dp) σ D Y ℓ ,(11)
where C Y ℓ (gw) is the CMB power spectrum generated by RGWs, and σ D Y ℓ is the standard deviation of the estimator D Y ℓ , which can be calculated by Eq. (10). We should notice that, the quantity d Y ℓ is dependent of random date D Y ℓ . By considering the relations in (9) and (5), we can obtain that d Y ℓ = a Y ℓ , which shows that d Y ℓ is an unbiased estimator of a Y ℓ . ℓ * t is the so-called best-pivot multipole, which is determined by solving the following equation [24]:
ℓ Y a Y 2 ℓ b * ℓ = 0.(12)
So the value of ℓ * t depends on the cosmological model, the amplitude of RGWs, and noise power spectra by the quantity a Y ℓ . The best-pivot wavenumber k * t relates to ℓ * t by the approximation relation [24],
k * t ≃ ℓ * t × 10 −4 Mpc −1 .(13)
The numerical factor here mainly reflects the angular-diameter distance to the last scattering surface.
Once the value of ℓ * t is obtained, the uncertainties ∆r * and ∆n t can be calculated by the following simple formulae
∆r * = r * / ℓ Y a Y 2 ℓ , ∆n t = 1/ ℓ Y (a Y ℓ b * ℓ ) 2 .(14)
As usual, we can define the signal-to-noise ratio S/N ≡ r * /∆r * . Using (14), we get
S/N = ℓ Y a Y 2 ℓ .(15)
In the previous work [24], we found the uncertainty of r(k 0 ), the tensor-to-scalar ratio at the pivot wavenumber k 0 = k * t , is larger than ∆r * . The value of ∆ ln r(k 0 ) is fairly well approximated by the following formula ∆ ln r(k 0 ) = (∆r * /r * ) 2 + (ln(k 0 /k * t )∆n t ) 2 .
The smallest uncertainty on tensor-to-scalar ratio r is achieved for the choice of the pivot scale at k 0 = k * t . This justified the title 'best' pivot wavenumber for k * t . We should notice that the values of k * t , S/N , ∆ ln r and ∆n t only depend on the input (or true) cosmological model, but not on the data D Y ℓ . In Fig. 1, we plot the value of ∆ ln r(k 0 ) as a function of the pivot scale k 0 , where the input model has r = 0.3, and the Planck instrumental noises are considered (see [24] for details). As expected, when k 0 ≫ k * t or k 0 ≪ k * t , the uncertainty becomes much larger than ∆r * . The likelihood function in (11) has the maximum value at (r * ML , n tML ). The values of r * ML and n tML depend on the data D Y ℓ , different from the value of ∆r * and ∆n t . In the previous work [24], we found that the values of r * ML and n tML can be very well approximated by the follows
r * ML = r * ℓ Y a Y ℓ d Y ℓ ℓ Y a Y 2 ℓ , n tML = n t + ℓ Y a Y ℓ d Y ℓ b * ℓ ℓ Y (a Y ℓ b * ℓ ) 2 ,(17)
which depend on the data by the quantity d Y ℓ . If the CMB estimator D Y ℓ is unbiased for C Y ℓ , as discussed above, we have r * ML = r * and n tML = n t , where Eq. (12) is used. These show that r * ML and n tML are the unbiased estimators for r * and n t , respectively. However, when D Y ℓ is a biased estimator for C Y ℓ , r * ML and n tML will also be the biased estimators for r * and n t , respectively (see Sec. VI for details), which brings the errors for the detection of RGWs.
The detection ability of the CMB experiment strongly depends on the noise levels, which include the instrumental noises, cosmic lensing contaminations, foreground radiation contaminations and the beam systematics. In the following sections, we shall discuss these effects separately. In addition, due to the partial sky survey, the leakage from the E-polarization into the B-polarization could be another kind of contamination. However, it was found that, this E-B mixture can be properly avoided (or deeply reduced) by constructing the pure E-mode and B-mode polarization fields (see [40][41][42] for details). So we shall not discuss this topic in this paper.
III. CMBPOL INSTRUMENTAL NOISES' CONTAMINATIONS
In this section, we shall discuss the determination of RGWs, when only taking into account the instrumental noises of the CMBPol mission.
For a single frequency channel i, we assume Gaussian beams. The noise power spectrum (after deconvolution of the beam window function) is
N T ins,ℓ (i) = (∆ T ) 2 exp ℓ(ℓ + 1)θ 2 F 8 ln 2 , N C ins,ℓ (i) = 0,(18)
and
N E ins,ℓ (i) = N B ins,ℓ (i) = (∆ P ) 2 exp ℓ(ℓ + 1)θ 2 F 8 ln 2 ,(19)
where θ F is the full width at half maximum (FWHM) of the beam i. ∆ T and ∆ P are the noises for the temperature and polarizations, which relate by ∆ P = √ 2∆ T . The values of the ∆ T and ∆ P depend on the number of the detectors, the integration time and the survey area. If the experiment includes several different channels, we need to generalize the above considerations. The optimal channel combination of these channels gives the total effective instrumental noise [28],
[N Y ins,ℓ ] −1 = i N Y ins,ℓ (i) −1 ,(20)
where i runs though the channels, N Y ins,ℓ (i) is the instrumental noise bias of the channel ν i . In this section, we shall only consider the instrumental noises, i.e.
N Y ℓ → N Y ins,ℓ .(21)
Since the precise experimental specifications of CMBPol have not yet been defined, we will consider two different cases (EPIC-2m) and (EPIC-LC) suggested by [28] (recently, an EPIC-Intemediate Mission is also suggested by the CMBPol team [43]). The experimental specifications are given in Table I, where 2-year design life is assumed [61]. In Fig. 2, we plot the polarization noise spectra N B ins,ℓ of EPIC-2m and EPIC-LC, respectively [62]. For EPIC-2m, when ℓ < 100, we find that N B ins,ℓ ∼ 2.7 × 10 −7 µK 2 , which is nearly 400 times smaller than that of the Planck mission (7 frequency channels from 30GHz to 353GHz and 28-month surveying time are assumed [21]). Even for the EPIC-LC, when ℓ < 100, we have N B ins,ℓ ∼ 6.2 × 10 −7 µK 2 , 200 times smaller than that of the Planck mission. So comparing with Planck mission, CMBPol is much more sensitive for the detection of CMB polarization. From Fig. 2, we also find that even for the model with quite small r = 0.01, the value of N B ins,ℓ is smaller than that of C B ℓ when ℓ < 120 for EPIC-2m, and when ℓ < 80 for EPIC-LC. So the CMBPol mission provides an excellent opportunity to detailedly observe the peak of C B ℓ at ℓ ∼ 80. Let us discuss the constraints on the gravitational waves by the potential observations of CMBPol mission. We shall discuss the values of the best-pivot scale k * t , the signal-to-noise ratio S/N and the uncertainty of the spectral index ∆n t , by considering the CMBPol instrumental noises.
The value of k * t directly relates to the best-pivot multipole ℓ * t by Eq. (13), and the value of ℓ * t is obtained by solving the equation in (12). By using (21), we obtain the value of ℓ * t as a function of the input (or true) value of the tensor-to-scalar ratio r * for EPIC-2m and EPIC-LC, which are plotted in Fig. 3 (left panel). We find that, in both cases, the value of ℓ * t becomes larger with the increasing of r * . For EPIC-2m, we have ℓ * t = 43 for r * = 0.001, and ℓ * t = 137 for r * = 0.1. For EPIC-LC, the value of ℓ * t is smaller than that of EPIC-2m, due to the larger noise level and the larger beam FWHM of EPIC-LC. When r = 0.001, we have ℓ * t = 26, and when r = 0.1, we have ℓ * t = 87. These reflect that gravitational waves in the frequency range k ∼ 0.01Mpc −1 will be best constrained by the future CMBPol observations, unless the value of r is extremely small. This is because the main contribution comes from the observation of the peak of B-polarization at ℓ ∼ 80. We should remember that this is different from the Planck case, where ℓ * t ∼ 10, due to the main contribution of the reionization peak of B-polarization [18,22,24]. The signal-to-noise ratio is calculated by Eq. (15). By using Eq. (21), we get the value of S/N as a function of r * for both EPIC-2m and EPIC-LC, which are shown in Fig. 3 (middle panel). As expected, the signal of RGWs can be very well determined by the CMBPol mission. Even for the model with r = 0.001, we can have S/N = 19 for EPIC-2m and S/N = 10 for EPIC-LC, when only considering the corresponding instrumental noises. When r = 0.1, we have S/N = 122 for EPIC-2m and S/N = 81 for EPIC-LC.
We can also calculate the value of ∆n t by using Eq. (14) and the value of ℓ * t given in left panel of Fig. 3. The results are shown in Fig. 3 (right panel). As expected, the value of ∆n t decreases with the increasing of r * . For EPIC-2m, we have ∆n t = 0.06 for the model with r * = 0.001, and ∆n t = 0.01 for the model with r * = 0.1. This uncertainty is about 20 times smaller than that given by Planck satellite [24,27]. This constraint, combining with ∆r * , will give a quite sensitive way to differentiate various inflationary type models. For the EPIC-LC, the uncertainty of n t is about 2 times larger that of EPIC-2m. When r = 0.001, we have ∆n t = 0.10, and when r = 0.1, we have ∆n t = 0.02.
It is necessary to discuss the contributions of S/N and ∆n t from every multipole, which can be very easily analyzed by the analytical formulae. From Eqs. (14) and (15), we find these two quantities can be rewritten as follows
(S/N ) 2 = ℓ Y a Y 2 ℓ , (1/∆n t ) 2 = ℓ Y (a Y ℓ b * ℓ ) 2 .(22)
They are the simple sums of the contributions from each multipole ℓ and CMB information channel Y . We plot the functions of Y a Y 2 ℓ and Y (a Y ℓ b * ℓ ) 2 as a function of ℓ for two different models (r = 0.01 and r = 0.1). The results are shown in Fig. 4. Left panel shows that all these four lines are peaked at ℓ ∼ 100, which is close to the peak of B-polarization. This reflects that when r > 0.01, the main contribution comes from the observation in the range ℓ ∼ 100, consistent with our previous discussion. This is different from the case of Planck satellite [18,22], where the reionization peak at ℓ ∼ 6 is extremely important. In the CMBPol case, the contribution from the largest scale ℓ < 20 is unimportant due to the cosmic variance, and the contribution from the small scale ℓ > 300 is also unimportant for the large instrumental noises. However, it is important to mention that if r ≪ 0.01, similar to Planck satellite, the reionization peak at ℓ < 10 again becomes the main contribution for the total S/N . However, it is different for Y (a Y ℓ b * ℓ ) 2 , which stands for the individual contribution for 1/∆n t . From the right panel of Fig. 4, we find that this function is sharply peaked at the largest scale ℓ < 30, and the contribution from intermedial scale around the best-pivot multpole is very small. These can be easily understood, the quantity b *
ℓ ≡ ln(ℓ/ℓ * t ) is zero when ℓ = ℓ * t , which follows that Y (a Y ℓ b * ℓ ) 2 = 0 at ℓ = ℓ * t . Only if ℓ ≪ ℓ * t or ℓ ≫ ℓ * t , b * ℓ has a large value, and follows a large Y (a Y ℓ b * ℓ ) 2 .
Especially, the contribution from ℓ ≪ ℓ * t is very important. For example, when ℓ * t = 137, b * 2 ℓ=2 is 30 times large than b * 2 ℓ=300 . This reflects that the constraint on the tilt of the primordial gravitational waves power spectrum strongly depends on the observations in a large scale range. The cosmic reionization is very important for the constraint of n t , although it might not be so important for the constraint of r for the CMBPol observations.
IV. COSMIC WEAK LENSING CONTAMINATION
In [13], it was pointed out that the gravitational waves result in CMB polarization with a B-mode, whereas density perturbations do not. Thus, the signal of gravitational waves could not be confused with density perturbations by detecting the B-polarization. Although, the amplitude of the B-polarization is expected to be quite small, it gives a clear information for gravitational waves. However, when taking into account the second-order effect, the B-mode can also arise from the lensing of the E-mode by density perturbations along lines-of-sight between the observer and the last-scattering surface [44]. The scalar contribution to the B-mode power spectrum is shown in Fig. 2 (grey dashed line). When ℓ < 200, it is nearly a white spectrum with the amplitude C B lens,ℓ ≃ 2 × 10 −6 µK 2 , which is 7 times larger than the instrumental noises of EPIC-2m, and 3 times larger than that of EPIC-LC.
When the instrumental noise of the CMB experiment is sufficiently small, as the CMBPol mission, the gravitational lensing contribution to the large-scale B-mode becomes one of the limiting sources of contamination for constraining the RGWs. High-sensitivity measurements of small-scale B-modes can reduce this contamination through a lens reconstruction technique, which has been discussed by a number of authors (see for instance [45][46][47]).
The effect of cosmic lensing contamination for the detection of RGWs can be easily discussed. The reduced lensed B-mode polarization can be treated simply as a well-known noise for gravitational waves in the likelihood analysis, i.e.
N B ℓ → N B ins,ℓ + C B lens,ℓ × σ lens ,(23)
where we have defined the residual factor σ lens for the lensed B-polarization. Note that in the real situation, the lens-induced B-modes are non-Gaussian, so we should not behave exactly it as the additional Gaussian noise (see [47] and references therein). However, on the scales relevant for B-mode detection in this paper, the non-Gaussianity has only a minor effect, which has been ignored in our discussion. In general, we have σ lens ≤ 1, with the equality holding for the lensed B-mode is not reduced. The reduction of gravitational lensing contribution strongly depends on the instrumental noise level, the beam FWHM, the foregrounds and the instrumental systematics. In the work [48], the authors found that, based on the noise level of EPIC-2m, one can expect to have σ lens ∼ 0.5. However, for EPIC-LC, it is very difficult to reduce the cosmic lensing contamination due to the large beam FWHM. We should mention that since the value of C B lens,ℓ is much larger than the instrumental noises of CMBPol mission, in the total effective noise N B ℓ , the cosmic lensing contamination becomes the dominant portion.
We have calculated the constraints of the gravitational waves by taking into account the cosmic lensing contaminations. The values of ℓ * t , S/N and ∆n t are shown in Fig. 3, where σ lens = 0.5 and σ lens = 1 are considered for EPIC-2m, and σ lens = 1 is considered for EPIC-LC. The left panel shows that, the best-pivot multipole is shifted to smaller scale by the lensing contamination. We find the value of S/N is much reduced by the lensing contamination, especially for the case with small tensor-to-scalar ratio. When r = 0.001, we have S/N = 7 for EPIC-2m (with σ lens = 0.5) and S/N = 4 for EPIC-LC, which are much smaller than the corresponding values with only instrumental noises. The uncertainty of n t is also much increased by the cosmic lensing, especially for the case with small r. When r = 0.001, EPIC-2m with σ lens = 0.5 can give ∆n t = 0.12, and EPIC-2m with σ lens = 1 can give ∆n t = 0.16, which are more than 2 times larger than those in the case without cosmic lensing, and are fairly loose to differentiate various inflationary models. However, when the tensor-to-scalar ratio is r = 0.1, we have ∆n t = 0.02 for EPIC-2m, which is still a quite tight constraint.
α -3 2.2 β E -2.6 -1.3 β B -2.6 -1.4 β C -2.6 -1.
We have also investigated the contribution of every multipole for S/N and ∆n t . The results can be found in Fig. 5, where we have focused on the EPIC-2m mission and σ lens = 0.5 is used for the case with cosmic lensing contamination. We find that the peak in each case is much reduced by the lensing contamination.
It is interesting to mention that for the high-sensitivity detectors the residual lensing noise dominates over the instrumental noises, and place the detection limit for CMB experiments [49]. In [46], the authors claimed that, for a extreme high-sensitivity detector, a reduction in lensing power by a factor 40 is possible using approximate iterative maximum-likelihood method. If we consider this residual as the lower limit of the reduced lensing noises, we find that r > 3.7×10 −6 can be detected at more than 2-σ level in absence of sky cuts, foregrounds and instrumental systematics [24]. This can be treated as the detection limit of the CMB experiments for gravitational waves. This lower limit corresponds to the Hubble parameter H ≃ 3.1 × 10 11 GeV, and the energy scale of inflation V 1/4 ≃ 1.5 × 10 15 GeV. In this limit case, the uncertainty of spectral index also becomes very small. When r = 0.1, we have ∆n t = 0.007 (see Fig. 2 in [24] for details), placing a very tight constraint on the inflationary models.
V. FOREGROUND CONTAMINATIONS
In this section, besides the instrumental noises and cosmic lensing contaminations, we shall take into account the impact of polarized foregrounds on the future CMBPol mission. In this paper, we shall neglect the effect of foregrounds on the CMB temperature, as the foreground cleaning is expected to leave a negligible contribution in the temperature [50]. CMB polarized foregrounds arise due to free-free, synchrotron, and dust emission, as well as due to extra-galactic sources such as radio sources and dusty galaxies. In this paper, we shall only consider only synchrotron and dust emission, which are expected to be dominant in the CMBPol frequency range [51].
The synchrotron emission results from the acceleration of cosmic-ray electrons in the magnetic field of Galaxy, which has been well measured on large angular scale at 23 GHz by WMAP. Following [28,51,52], for the frequency ν, the scale-dependence of the synchrotron signal may be parameterized as
C Y S,ℓ (ν) = A S ν ν 0 2αS ℓ ℓ 0 β Y S .(24)
The parameters in this formula for the various power spectra are all listed in Table II, where α S = −3 is assumed, β E , β B and β C are the corresponding β Y S for synchrotron emissions. This choice matches the synchrotron emission at 23GHz observed and parameterized by WMAP [53], and agrees with the DASI measurements [54].
Galactic emission in the 100−6000 GHz frequency range is dominated by the thermal emission from warm interstellar dust grains. Our knowledge of polarized dust emission is relatively poor, which is expected to be characterized by the Planck satellite in the near future. In this paper, we shall adopt the parameterized formula for the dust emission at frequency ν as follows, as suggested by [28,52],
C Y D,ℓ (ν) = p 2 A D ν ν 0 2αD ℓ ℓ 0 β Y D e hν0/kT − 1 e hν/kT − 1 2 ,(25)
where p = 5% and T = 18K. We list the other parameters for the various power spectra in Table II. Various methods have been discussed to subtract the foregrounds by their frequency-dependence (see for instance [55]). In this paper, we shall not discuss the subtraction of the foreground from the signal. Instead, as the previous works [28,52] we assume that the foreground substraction can be done correctly down to a given level, and treat these residual foregrounds as a kind of known gaussian noises in the data analysis. However, here we should mention that in this case we have assumed one can model and subtract the power spectra of residuals perfectly, and avoid the possible issues of bias. This might be a huge challenge for the future polarization observation.
If we consider the CMB experiment, including several frequency channels, and the different channels have different noise levels, the optimal channel combination gives the effective noise power spectra [28,52]
[N Y eff,ℓ ] −1 = i N Y fg,ℓ (i) + N Y ins,ℓ (i) −1 ,(26)
where i runs though the channels. N Y ins,ℓ (i) is the instrumental noise power spectra of channel ν i . N Y fg,ℓ (i) is the residual foreground noises of channel ν i , which is
N Y fg,ℓ (i) = fore=S,D C Y fore,ℓ (ν i )σ fg + N Y fg,ℓ (ν i ).(27)
Here, C Y fore,ℓ is the model for the power spectrum of the synchrotron and dust signals at the frequency ν i given by Eqs. (24) and (25), and σ fg is the assumed residual factor. N Y fg,ℓ (ν i ) is the noise power spectrum of the foreground template map, as foreground templates are created by effectively taking map differences and thus are somewhat affected by the instrumental noise. This term can be calculated by [28,52]
N Y fg,ℓ (ν i ) = N ′ Y ins,ℓ (ν ref ) n chan (n chan − 1)/4 ν i ν ref 2α ,(28)
where n chan is the total number of channels used, and the reference channel ν ref is the highest and lowest frequency channel included in the cosmological analysis for dust and synchrotron respectively, i.e., that listed in Table I. The parameters α for the foregrounds under consideration are defined in Table II, i.e. α = −3 for the synchrotron emissions and α = 2.2 for the dust emission. The quantity N ′ Y ins,ℓ (ν ref ) is the white instrumental noise (without the beam window function) of the corresponding template channel [52].
Thus the total noise power spectra, by combining the multipole-frequency instrumental noises and the residual foregrounds, as well as the residual cosmic lensing contamination, are given by
N X ℓ → N X eff,ℓ (X = T, C, E), N B ℓ → N B eff,ℓ + C B lens,ℓ × σ lens ,(29)
where σ lens is the residual factor for cosmic lensing contamination. In this and the following sections, we adopt σ lens = 0.5 for EPIC-2m, and σ lens = 1 for EPIC-LC. The effective noise power spectra N Y eff,ℓ strongly depend on the residual factor σ fg for the foregrounds. When no foreground subtraction is assumed, we have σ fg = 1. In this paper, we also consider two assumed residual cases, suggested by CMBPol team [28]: σ fg = 0.01 for the optimistic case, and σ fg = 0.1 for the pessimistic case. In Fig. 6, we plot the effective noise power spectrum N B eff,ℓ with different σ fg for EPIC-2m (left panel) and EPIC-LC (right panel). We find that for both EPIC-2m and EPIC-LC, the foregrounds increase the effective noise power spectrum in all the multipole range when σ fg = 1. However, when the foregrounds can be well subtracted, the residual foregrounds only increase the noise in the large scale. For σ fg = 0.1, the effective noise is increased in the range ℓ < 300, and for σ fg = 0.01, the effective noise is increased in the range ℓ < 100. We find that even if the optimistic case with σ fg = 0.01 is realized, the effective noise is much larger in the reionization peak (ℓ < 20) comparing with the no foreground case. Especially, when r is small, this increased noise is larger than the signal C B ℓ , and decreases the contribution of the reionizaiton peak. We have emphasized above, the reionization peak is very important for the constraint of spectral index n t for the CMBPol mission, it is predictable that the value of ∆n t would become much larger due to the foreground contaminations, even if the optimistic case is considered. This will be clearly shown in the following discussion.
By using the total effective noise power spectra in (29), we can calculate the values the best-pivot multipole ℓ * t , the signal-to-noise ratio S/N , and the uncertainty of the spectral index ∆n t . In Fig. 7 and Fig. 8, we show the results for EPIC-2m and EPIC-LC, respectively. We find that so long as r < 0.3, the value of ℓ * t is increased with the increasing of the residual foregrounds. This is because that, the contaminations from foreground are mainly in the low multipole (see Fig. 6). Increasing the foregrounds, the contribution for the detection of RGWs in the low multipoles becomes less and less important, induces an increasing of ℓ * t . From Fig. 7 and Fig. 8, we find that when σ fg = 0.01, the optimistic case we considered, the foregrounds decrease the S/N and increase ∆n t , when the tensor-to-scalar ratio r < 0.03. However, when r > 0.03, the effect of this residual foregrounds is negligible. This is also easily understood. The residual foregrounds with σ fg = 0.01 only increase the total noise power spectra in the largest scale ℓ < 100. This increased total noises are beyond the signals when r is small. We also find that, when σ fg = 0.1 or σ fg = 1, i.e. the foregrounds are not well subtracted, the effect of the foreground contaminations are quite important, especially for the determination of n t . With the decreasing of r, the contaminations become more and more important. For the EPIC-2m and the input model with r = 0.1, when σ fg = 0.01, we have ∆n t = 0.020. When σ fg = 0.1, the value becomes ∆n t = 0.024, and when σ fg = 1, the value becomes ∆n t = 0.045, two times larger than that in the optimistic case. We can investigate another case, for the EPIC-2m and the input model with r = 0.01, when σ fg = 0.01, we have ∆n t = 0.054. When σ fg = 0.1, the value becomes ∆n t = 0.098, and when σ fg = 1, the value becomes ∆n t = 0.330, six times larger than that in the optimistic case. So we conclude that if the value of r is not too small, such as r = 0.1, we do not need to remove the foreground to a very high level. The difference between optimistic case and pessimistic is very small. However, if the value of r is smaller than 0.01, very detailed removal for the foregrounds is very important for the determination of RGWs. Fig. 7 and Fig. 8 also show that in the optimistic case, EPIC-2m can detect the signal of RGWs with r = 0.001 at 5-σ level, and EPIC-LC can detect it at 3-σ level. However, in the pessimistic case, EPIC-2m can only detect this signal at 2.8-σ level, and EPIC-LC can detect it at 1.6-σ level.
As in the previous sections, we can discuss the contribution to the S/N and ∆n t from the individual multipole, by investigating the functions Y a Y 2 ℓ and Y (a Y ℓ b * ℓ ) 2 . They are plotted in Fig. 9, where we have considered the EPIC-2m, and the model with r = 0.1. We find that the foreground contamination mainly affects the S/N by decreasing the value of Y a Y 2 ℓ around the peak at ℓ ∼ 100. However, it affects the value of ∆n t mainly by decreasing of Y (a Y ℓ b * ℓ ) 2 at the largest scale ℓ < 50 and the intermedial range ℓ ∼ 200. Now, let us investigate the possible application of the CMBPol mission to differentiate the different inflationary models, which plays a role for the future inflation researches. As well known, one of the most important ways to distinguish different classes of inflations is to test the so-called inflationary consistency relations. This testing strongly depends on the determination of the parameters specifying the relic gravitational waves, i.e. the tensor-to-scalar ratio r and the spectral index n t . Now, let us focus on the possible testing of the consistency relation for the canonical single-field slow-roll inflationary models [10]. This testing might provide the unique model-independent criteria to confirm or rule out this class of models. The possible testing for other inflationary models by the CMBPol mission and the ideal CMB experiment can be found in the recent work [56].
The consistency relation for the canonical single-field slow-roll inflationary models can be written as [10] r = −8n t .
We find that this relation only depends on the parameters r and n t . Since the absolute value of n t is expected to be one order smaller than that of r, and also the measurement of n t is much more difficult than r, how well we can measure the spectral index n t plays a crucial role for testing the consistency relation in Eq. (30).
To access whether CMBPol mission might achieve the consistency relation test goal, in Fig. 7 and Fig. 8 (right panels), we compare the values of |n t | = r/8 with ∆n t . If ∆n t < |n t |, then the constraint on n t is tight enough to allow for the testing. From Fig. 7, we find that for the EPIC-2m mission, ∆n t < |n t | is satisfied only if r > 0.14 for the optimistic case with σ fg = 0.01. In the pessimistic case with σ fg = 0.1, it becomes r > 0.15. Similar results for the EPIC-LC can be found in Fig. 8. ∆n t < |n t | is satisfied only if r > 0.18 for the optimistic case, and r > 0.20 for the pessimistic case. So we conclude that the testing of the consistency relation for the canonical single-field slow-roll by the CMBPol mission is quite hard. The testing is possible only for some large-field inflationary models. However, we should mention that the situation could become quite promising for the general Lorentz-invariant single-field inflations and the two-field inflations (see [56] for the details).
VI. SYSTEMATICS CONTAMINATIONS
Beyond raw sensitivity requirements for the instrumentals, and the removal of the astrophysical foregrounds, much attention has already been given in the literature to the instrumental systematics for the constraints of the cosmological parameters and the cosmic weak lensing reconstruction [57][58][59][60]. The main goal of this section is to illustrate the effect of the instrumental systematics and systematically study the impact on the gravitational waves detection for the CMBPol mission.
All the effects of the beam systematics are associated with beam imperfections or beam mismatch in dual beam experiments. Several of these effects (e.g. differential gain, differential beam width and the first order pointing error) are reducible with an ideal scanning strategy and otherwise can be cleaned from the data set. Other spurious polarization signals, such as those due to differential ellipticity of the beam, second order pointing errors and the differential rotation, persist even in the case of ideal scanning strategy and perfectly mimic CMB polarization.
The beam systematics due to optical imperfections are dependent of the underlying sky, the properties of the polarimeter and the scanning strategy. If the outputs of two beams with orthogonal polarization-sensitive directions are slightly different, the temperature anisotropy can leak to the polarization or the E-mode polarization can leak to the B-mode and vice verse. (see [59] for the details). For example, if two beams are exactly same but the overall response, this difference of the measured intensity can generate a non-vanishing polarization signal. Another typical example is effect of the beam rotation, which is caused by the uncertainty in the overall beam orientation. This effect mixes the Stokes parameters Q and U , and induces a E-mode and B-mode leakage.
The CMB power spectra C Y sys,ℓ generated by these systematics are discussed in details by a number of authors. In the work [59], the authors discussed these effects separately, and got the simple analytical formulas to calculate the leading order of the generated power spectra, which are listed in Table III. The formulae in Table III separately describe the effects of the following instrumental systematics for experiments with the elliptical gaussian beams: differential gain effect, monopole effect, differential pointing effect, quadrupole effect, differential rotation effect. Differential gain can induce spurious polarization singles from temperature leakage due to beam mismatch. This effect is described by the parameter g ≡ g 1 − g 2 , where g 1 and g 2 refer to the gain factors of first and second beams. The differential rotation effect is due to uncertainty in the overall beam orientation. This mixes the Q and U Stokes parameters and as a result leaks E to B and vice verse. We describe this effect by the parameter ε ≡ (ε 1 + ε 2 )/2, where ε 1 and ε 2 are rotation errors of first and second beams. We note that these two parameters g and ε are not related to the beam shape. The monopole effect arises from circular beams with unmatched main-beam full width at half maximum, which is described by the parameter µ ≡ (σ 1 − σ 2 )/(σ 1 + σ 2 ), where σ 1 and σ 2 are the mean beamwidthes of first and second beams. The quadrupole effects arises from beams with differential ellipticities, and described by the ellipticity parameter e ≡ (σ x − σ y )/(σ x + σ y ), where σ x and σ y are the major and minor axes of the beam. Also, differential pointing, i.e. the dipole effect, is described the parameter ρ ≡ ρ 1 − ρ 2 , where ρ 1 and ρ 2 are the circular positions of first and second beams. These three effects can induce future spurious polarization singles from temperature leakage.
In the formulae in Table III, the functions f 1 , f 2 and f 3 are experiment-specific and encapsulate the information about the scanning strategy which couples to the beam mismatch parameters to generate spurious polarization. The exact definitions of f 1 , f 2 and f 3 are given in Eq. (27) in [59], i.e.
f 1 = 1 2 |h + (−1, 0)| 2 , f 2 = 1 2 |h + (−1, −1)| 2 + 1 2 |h + (−1, 1)| 2 , f 3 = 1 2 f (0, 1)h * − (1, −1) , where f (m, n) ≡ e −i(2m+n)α , h ± ≡ 1 D [f (m, n) − f (m ± 2, n) e ±4iα ], D ≡ 1 − e 4iα e −4iα .
f (m, n) andh ± (m, n) are the Fourier transformations of f (m, n) and h ± (m, n). Exact definition of α can be found in [59]. Angular brackets represent average over measurements of a single pixel, averaged over time. In general, these functions are spatially-anisotropic but for simplicity, and to obtain a first-order approximation, we consider them constants in general.
In the real data analysis, these generated CMB power spectra may be treated as a part of the signals, as well as the real signals generated by the perturbations fields, i.e.
C Y ℓ → C Y ℓ + C Y sys,ℓ .(31)
Thus the estimator of this contamination D Y ℓ becomes biased for the true power spectra C Y ℓ , by a term C Y sys,ℓ . And this biased estimator will induce the biased estimator for the cosmological parameters in the likelihood analysis. As in general, we can use the r * ML and n tML as the best estimator for the parameters r * and n t . Following the previous works [60], we define the bias of the tensor-to-scalar ratio δr and the spectral index δn t as follows,
δr ≡ r * ML − r * , δn t ≡ n tML − n t .(32)
Given the beam systematics the bias of the tensor-to-scalar ratio δr and the spectral index δn t can be calculated by the following formulae (similar to the previous works [60]),
δr = r * ℓ Y a Y ℓ e Y ℓ ℓ Y a Y 2 ℓ , δn t = ℓ Y a Y ℓ e Y ℓ b * ℓ ℓ Y (a Y ℓ b * ℓ ) 2 ,(33)
where e Y ℓ ≡ C Y ℓ (sys)/σ D Y ℓ . Notice that, for the requirement of the beam systematics of CMBPol mission (for instant, the parameter g, associated with the differential gains, satisfies g ≪ 0.01% [48]), the power spectra generated by the beam systematics are expected to be much smaller than those generated by gravitational waves with r > 10 −3 but the very large multipole range, where the noises are dominant. So beam systematics cannot change the value of the best-pivot multipole.
Considering the CMB experiment with multi-frequency channels, the effective combined noise power spectra in Eq. (26) can be extended to the follows, considering the contribution of the systematics,
[N Y eff,ℓ ] −1 = i N Y fg,ℓ (i) + N Y ins,ℓ (i) + C Y sys,ℓ (i) −1 ,(34)
where i runs though the channels. Throughout this section, we shall assume the optimistic foreground removal with the residual factor σ fg = 0.01 for both EPIC-2m and EPIC-LC missions. We should remember that, similar to the previous discussion, the B-mode contamination (σ lens = 0.5 for EPIC-2m, and σ lens = 1 for EPIC-LC) by weak lensing effect is also considered throughout this section. The total CMB power spectra generated beam systematics can be calculated by C Y ℓ (sys) = N Y eff,ℓ (with sys) − N Y eff,ℓ (no sys). In the simplest case with single frequency channel, this term returns to C Y ℓ (sys) = C Y sys,ℓ . Let us separately investigate the five systematical effects. Fig. 10 shows C B ℓ (sys) for different values of the parameter g = 0.002%, 0.005% and 0.01%. In all these figures, f 1 = 2π is used as the worst case. The values of ℓ(ℓ + 1)C B ℓ (sys) only weekly depends on the multipole ℓ. As long as r > 0.01, we find C B ℓ (sys) are all smaller than those of the signals C B ℓ or the noises N B eff,ℓ in the range ℓ < 200. In Fig. 11, we plot the values of biases δr and δn t induced by the differential gains. We find that the values of δr and δn t strongly depends on the value of the parameter g. A larger g follows a larger bias. However, the uncertainties ∆r and ∆n t are nearly independent of g unless the value of g is too large. We also find that, given the parameter g the value of the ratio δr/∆r also strongly depend on the tensor-to-scalar ratio r. For example, the EPIC-2m experiment with g = 0.01%, δr is larger than ∆r when r < 0.04, the bias is very obvious. However when r > 0.04, we have δr < ∆r, the bias is smaller than the uncertainty.
Similar to the previous work [60], we can define the critical value g c , which is the largest value of g as long as the the condition δr/∆r < 0.1 is satisfied. In Table IV, we list the critical values of g c for EPIC-2m and EPIC-LC, where r = 0.001, 0.01 and r = 0.1 are considered. Since Fig. 11 shows that for a given g value, both ∆r and δr increase with the increasing of r. So the tendency of the critical g c for different input r is not trivial. After analysis, we find that for the assumed g value, with the increasing of r, if the increasing of ∆r is more rapid than that of δr, thus the case with larger r corresponds to a larger g c . This is clearly shown in Table IV. On the other hand, if the increasing of δr is more rapid than that of ∆r, thus the case with larger r corresponds to a smaller g c . Similar discussion is also applied to the n t case, as well as the cases for the other four systematics contaminations. From Table IV, we find the severest constraints are obtained from the requirement of the small r case. The requirement for EPIC-2m is quite close to that of EPIC-LC.
Let us discuss the effect of beam gain on the determination of spectral index n t . From Eq. (33), we find that the bias e Y ℓ in the lower multipole range ℓ < ℓ * t contributes a negative δn t , and the bias e Y ℓ in the high multipole range ℓ > ℓ * t contributes a positive δn t . These two components are cancelled by each other and total bias δn t is expected to be very small, which is clearly shown in Fig. 11 (right panel). Comparing with the bias of r, the bias δn t is much smaller than that of ∆n t . So the constraint on g obtained from the requirement of the n t is much larger than that from the parameter r.
Now, let us turn to the effect of the differential monopole effect. From the formulae in Table III we know the power spectra generated by differential monopole effect strongly depend on the beam size. Larger beam size follows the larger power spectra C Y sys,ℓ . This can be seen clearly in Fig. 12. Given µ = 0.1%, the value of C B ℓ (sys) is much larger in EPIC-LC than that in EPIC-2m. So in order to achieve a same value of δr/∆r = 0.1, the requirement for EPIC-LC is much severer than that for EPIC-2m, which are clearly shown in Table IV. Fig. 13 shows that, for a given µ, a larger r follows a larger ratio value δr/∆r, which is correct for both EPIC-2m and EPIC-LC. In order to keep the tensor-to-scalar ratio in the range r ∈ (0.001, 0.1) unbiased, we should have µ c = 0.029% for EPIC-2m, and µ c = 0.005% for EPIC-LC, where f 1 = 2π is adopted. we can also discuss the effect of differential monopole on the determination of spectral index n t . From Fig. 12, we find that C Y sys,ℓ is sharply peaked at the high multipole, where ℓ > ℓ * t . Thus the total contribution to δn t is always positive (see Fig. 13), especially when the value of µ is not too small, the effect of differential monopole at the small scale is very important, and follows a fairly large bias for spectral index. Let us define the value of µ c , where δn t /∆n t = 0.1 is satisfied. From Table IV we find that the constraint on µ is a little severer from the parameter n t than that from the parameter r. This table shows that, the most severe constraint on µ is obtained from the requirement of n t in the case of r = 0.1.
In Figs. 14 and 15, we show the effect of differential pointing on the determination of gravitational waves, where f 2 = 2π is adopted. These figures show that similar with the case of differential monopole effect, the function C B ℓ (sys) generated by differential pointing is also sharply peaked at the high ℓ, which follows that the bias of the spectral index δn t is positive. Comparing the left panel with the right panel in Fig. 14, we find that for a given ρ, the effect of the differential pointing is more important for the EPIC-2m, due to the smaller instrumental noises. So more severe constraint on the differential pointing is followed for the EPIC-2m. In Table IV, we find that for both EPIC-2m and EPIC-LC, the most severe on ρ comes from the requirement of parameter n t at r = 0.1.
We have also investigated the effect of the differential quadrupole in Figs. 16 and 17. The effects are similar to those of the differential monopole. We find that EPIC-LC needs the much stricter requirement than EPIC-2m. For each experiment, the most severe constraint on the parameter e is obtained from the requirement of n t in the case with r = 0.1.
At last, we shall discuss the effect of the differential rotation, and the results are shown in Figs. 18 and 19. We The leading order contributions of the systematic effects to the CMB power spectra, assuming the underlying sky is not polarized (except for the rotation signal) [59], where z ≡ (ℓσ) 2 e and σ is mean beamwidth of the beam, which is calculated by σ = θF / √ 8 ln 2. c θ ≡ cos(θ), where θ is the angle between ellipse major axis of the elliptical gaussian beam and the horizontal x-axis of the fixed focal plane. c ψ ≡ cos(ψ) and s ψ ≡ sin(ψ), where ψ is the angle between the axis of polarization sensitivity and the major axis of the elliptical beam. A clear show of the angles θ and ψ can be found in Fig. 1 in [59]. The definitions of the other parameters can be found in the text, see also [59].
Effect
Parameter find that for a given ε, the ratios δr/∆r and δn t /∆n t only weakly depend on the tensor-to-scalar ratio. And the ratio for EPIC-2m is a little smaller than that of EPIC-LC. In order to keep the parameters r and n t unbiased, the requirement ε < 0.09 o is needed for EPIC-2m, and ε < 0.15 o is needed for EPIC-LC.
C C sys,ℓ C E sys,ℓ C B sys,ℓ Gain g 0 g 2 f1C T ℓ g 2 f1C T ℓ Monopole µ 0 4µ 2 (ℓσ) 4 C T ℓ f1 4µ 2 (ℓσ) 4 C T ℓ f1 Pointing ρ −c θ J 2 1 (ℓρ)C T ℓ f3 J 2 1 (ℓρ)C T ℓ f2 J 2 1 (ℓρ)C T ℓ f2 Quadrupole e −I0(z)I1(z)c ψ C T ℓ I 2 1 (z)c 2 ψ C T ℓ I 2 1 (z)s 2 ψ C T ℓ Rotation ε 0 4ε 2 C B ℓ 4ε 2 C E
As a conclusion, by analyzing the effects of the systematics on the determination of gravitational waves, we find that the requirement of n t unbiased follows the similar or even more severe constraints for the beam systematical parameters. For the effects of differential monopole, pointing and quadrupole, a larger r follows a more severe constraint for the systematics. The critical values for the systematical parameters are listed in Table IV, where the bold entries denote the most severe constraint in each case. We also find that comparing with EPIC-2m, the low cost EPIC-LC experiment has a much high requirement for the systematical parameters µ and e.
VII. CONCLUSION
The proposed CMBPol mission is the next generation of the space-based CMB experiment, which will survey the full sky and has the much smaller instrumental noises than Planck satellite. As one the most important tasks of this mission, detecting relic gravitational waves will be achieved if the tensor-to-scalar ratio r 0.001, which will provide a great opportunity to study the physics in the early Universe, especially in the inflationary stage. In this paper, we have detailedly discussed the detection of relic gravitational waves by focusing on the constraints of the parameters r and the spectral index n t , which are always used to describe the primordial power spectrum of gravitational waves. In our discussion, we deeply investigate various contaminations for the detection, including the instrumental noises of CMBPol mission, the cosmic lensing contamination, the foreground contaminations, and the effect of various beam systematics. We found that the cosmic lensing becomes the dominant noise sources, comparing the instrumental noises for both EPIC-2m and EPIC-LC projects. Different from Planck satellite, if r > 0.01, the detection of gravitational waves mostly depends on the observation at multipole ℓ ∼ 100, the peak of the B-mode polarization. However, the reionization peak at ℓ ∼ 10 still plays a crucial role for the determination of spectral index n t . We also found that if the foreground contaminations cannot be well controlled, the reionization peak may be unobservable, which could deeply increase the uncertainty of n t . At the same time, we have investigated the effect of various beam systematics on the detection of gravitational waves, which mainly cause a bias on the cosmological parameters. In order to keep these biases small enough, the requirements for the beam systematical parameters are quite severe, especially for the EPIC-LC mission.
FIG. 1 :
1The uncertainty of r(k0) for the different pivot wavenumber k0.
FIG. 2 :
2The instrumental noise power spectra N B ins,ℓ of EPIC-2m (black line) and EPIC-LC (red line).
FIG. 3 :
3The figures show the values of the best-pivot multipole ℓ * t (left panel), signal-to-noise ratio S/N (middle panel) and the uncertainty of the RGW spectral index ∆nt (right panel) as functions of r * . The black solid (dash-dotted, dashed) lines correspond to the EPIC-2m instrumental noises (instrumental noises + the reduced cosmic lensing contaminations, instrumental noises + cosmic lensing contaminations), and the red solid (dashed) lines correspond to the EPIC-LC instrumental noises (instrumental noises + the cosmic lensing contaminations).
FIG. 4 :
4The figures show the values of Y a Y 2 ℓ (left panel) and Y (a Y ℓ b * ℓ ) 2 (right panel)as functions of multipole ℓ for the cases with the different r * . Here, we have not considered the contaminations from cosmic weak lensing and foreground emissions.
FIG. 5 :
5For EPIC-2m, the figures show the values of Y a Y 2 ℓ (left panel) and Y (a Y ℓ b * ℓ ) 2 (right panel)as functions of multipole ℓ for the cases with and without the reduced cosmic lensing contamination with the residual factor σ lens = 0.5.
FIG. 6 :
6The total effective noise power spectra N B eff,ℓ when considering the foreground contaminations. The left panel is for EPIC-2m and the right panel is for EPIC-LC.
FIG. 7 :FIG. 8 :FIG. 9 :
789This figure shows the values of ℓ * t , S/N and ∆nt depend on the foreground contaminations for EPIC-2m. This figure shows the values of ℓ * t , S/N and ∆nt depend on the foreground contaminations for EPIC-LC. The figures show the values of Y a Y 2 ℓ (left panel) and Y (a Y ℓ b * ℓ ) 2 (right panel) as functions of multipole ℓ for the cases with different σ fg .
FIG. 10 :
10The contribution of differential gain to the B-polarization, comparing with the signal of C B ℓ for different r, the noise of CMBPol mission, and the lensed B-polarization. Left panel is for EPIC-2m, and the right panel is for EPIC-LC.
FIG. 12 :
12The contribution of monopole effect to the B-polarization, comparing with the signal of C B ℓ for different r, the noise of CMBPol mission, and the lensed B-polarization. Left panel is for EPIC-2m, and the right panel is for EPIC-LC.
FIG. 13 :
13The values of δr (left panel) and δnt (right panel) from the monopole effect for different tensor-to-scalar ratio r. For the comparison, we plot the corresponding ∆r and ∆nt in solid lines in the panels.
FIG. 14 :
14The contribution of differential pointing to the B-polarization, comparing with the signal of C B ℓ for different r, the noise of CMBPol mission, and the lensed B-polarization. Left panel is for EPIC-2m, and the right panel is for EPIC-LC.
FIG. 15 :
15The values of δr (left panel) and δnt (right panel) from the differential pointing for different tensor-to-scalar ratio r. For the comparison, we plot the corresponding ∆r and ∆nt in solid lines in the panels.
FIG. 16 :
16The contribution of quadrupole effect to the B-polarization, comparing with the signal of C B ℓ for different r, the noise of CMBPol mission, and the lensed B-polarization. Left panel is for EPIC-2m, and the right panel is for EPIC-LC.
FIG. 17 :
17The values of δr (left panel) and δnt (right panel) from the quadrupole effect for different tensor-to-scalar ratio r. For the comparison, we plot the corresponding ∆r and ∆nt in solid lines in the panels.
FIG. 18 :
18The contribution of differential rotation to the B-polarization, comparing with the signal of C B ℓ for different r, the noise of CMBPol mission, and the lensed B-polarization. Left panel is for EPIC-2m, and the right panel is for EPIC-LC.
FIG. 19 :
19The values of δr (left panel) and δnt (right panel) from the differential rotation for different tensor-to-scalar ratio r. For the comparison, we plot the corresponding ∆r and ∆nt in solid lines in the panels.
& --0.0050 & 0.0040 0.13 & 0.11 0.07 & 0.06 0.18 & 0.16
TABLE I :
IExperimental specifications for the mid-cost (EPIC-2m) CMBPol mission and the low-cost (EPIC-LC) CMBPol
mission. [28]
Frequency [GHz]
45
70
100
150 220
EPIC-2m
θF [arcmin]
17
11
8
5
3.5
∆T [µK-arcmin]
5.85
2.96
2.29
2.21 3.39
Frequency [GHz]
40
60
90
135 200
EPIC-LC
θF [arcmin]
116
77
52
34
23
∆T [µK-arcmin] 15.27
8.23
3.56
3.31 3.48
TABLE II :
IIAssumptions about foreground emissions.[28] Parameter
Synchrotron
Dust
AS,D
4.7 × 10 −5 µK 2 1.2 × 10 −4 µK 2
ν0
30 GHz
94 GHz
ℓ0
350
900
TABLE III :
III
FIG.11: The values of δr (left panel) and δnt (right panel) from the differential gain for different tensor-to-scalar ratio r. For the comparison, we plot the corresponding ∆r and ∆nt in solid lines in the panels.1E-3
0.01
0.1
1
1E-5
1E-4
1E-3
0.01
0.1
1
g=0.01%
g=0.01%
g=0.005%
g=0.005%
g=0.002%
g=0.002%
Red lines:
EPIC-LC
Black lines: EPIC-2m
n
t
and | n
t
|
r
1E-3
0.01
0.1
1
1E-5
1E-4
1E-3
0.01
g=0.01%
g=0.005%
g=0.002%
g=0.01%
g=0.005%
g=0.002%
Red lines:
EPIC-LC
Black lines: EPIC-2m
r* and
r*
r
TABLE IV :
IVSystematics tolerance for EPIC-2m and EPIC-LC, where different nominal values of r are considered. In each box, the left value is for the parameter r and the right value is for the parameter nt.Nominal value
gc
1%
f 1
2π
µc
1%
f 1
2π
ρc
1 ′′
f 2
2π
ec
1%
εc[deg]
AcknowledgementsThe author is partially supported by Chinese NSF Grants No. 10703005, No. 10775119, No. 11075141. We thank the anonymous referee for the useful comments and suggestions.
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V, we will show that the total noise level will be increased if the reduced foreground contaminations are considered. However, the noise level with high multipole ℓ > 100 nearly keeps same. SecNote that, in Sec. V, we will show that the total noise level will be increased if the reduced foreground contaminations are considered. However, the noise level with high multipole ℓ > 100 nearly keeps same.
| []
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[
"ON THE MAXIMAL DIRECTIONAL HILBERT TRANSFORM IN THREE DIMENSIONS",
"ON THE MAXIMAL DIRECTIONAL HILBERT TRANSFORM IN THREE DIMENSIONS"
]
| [
"Francesco Di ",
"Ioannis Parissis "
]
| []
| []
| A. We establish the sharp growth rate, in terms of cardinality, of the L p norms of the maximal Hilbert transform H Ω along nite subsets of a nite order lacunary set of directions Ω ⊂ R 3 , answering a question of Parcet and Rogers in dimension n = 3. Our result is the rst sharp estimate for maximal directional singular integrals in dimensions greater than 2.The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.2010 Mathematics Subject Classi cation. Primary: 42B20. Secondary: 42B25. | 10.1093/imrn/rny138 | null | 119,151,975 | 1712.02673 | 90fdb31b890c50dd4200317dfa87973703661d3f |
ON THE MAXIMAL DIRECTIONAL HILBERT TRANSFORM IN THREE DIMENSIONS
Francesco Di
Ioannis Parissis
ON THE MAXIMAL DIRECTIONAL HILBERT TRANSFORM IN THREE DIMENSIONS
A. We establish the sharp growth rate, in terms of cardinality, of the L p norms of the maximal Hilbert transform H Ω along nite subsets of a nite order lacunary set of directions Ω ⊂ R 3 , answering a question of Parcet and Rogers in dimension n = 3. Our result is the rst sharp estimate for maximal directional singular integrals in dimensions greater than 2.The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.2010 Mathematics Subject Classi cation. Primary: 42B20. Secondary: 42B25.
I
Let n ≥ 2. The Hilbert transform along a direction ω ∈ S n−1 acts on Schwartz functions on R n by the principal value integral
H ω f (x) p.v. ∫ R f (x + tω) dt t ,
x ∈ R n .
If Ω ⊂ S n−1 , we may de ne the corresponding maximal directional Hilbert transform
(1.1) H Ω f sup ω∈Ω |H ω f |.
The main result of this paper is the following sharp estimate in the three-dimensional case.
Theorem 1.1. Let Ω ⊂ S 2 be a nite order lacunary set [28]. Then for all 1 < p < ∞
(1.2) sup O ⊂Ω #O ≤N H O L p (R 3 )→L p (R 3 ) ≤ C log N .
The positive constant C may depend on 1 < p < ∞ and on the lacunary order of Ω only.
We stress that the supremum in Theorem (1.2) is taken over all subsets O having nite cardinality N of a given nite order lacunary set Ω, which may be in nite. Theorem 1.1 is in fact the Lebesgue measure case of a more general sharp weighted norm inequality which is a natural byproduct of our proof techniques, and is detailed in Corollary 1 for the interested reader. In [22] Laba, Marinelli, and Pramanik have extended to dimensions n ≥ 2 the lower bound (due to Karagulyan [19] in the case n = 2) (1. 3) inf #Ω=N H Ω L 2 (R n )→L 2 (R n ) = c n log N ,
where the in mum is taken over all sets Ω ⊂ S n−1 of nite cardinality N . A comparison with the upper bound of Theorem 1.1 and interpolation reveals that the dependence on the cardinality of the set of directions in our theorem is sharp for all 1 < p < ∞. In fact, [22] proves the analogue of (1.3) for all 1 < p < ∞.
1.2.
Maximal and singular integrals along sets of directions. The study of cardinalityfree, or sharp bounds for the companion directional maximal operator to (1.1)
(1.4) M Ω f (x) = sup ω∈Ω M ω f (x), M ω f (x) sup ε>0 1 2ε ∫ ε −ε | f (x + tω)|dt, x ∈ R n ,
is a classical subject in real and harmonic analysis, with deep connections to multiplier theorems, Radon transforms, and the Kakeya problem, to name a few. The seminal article by Nagel, Stein and Wainger [26] contains a proof that the projection on S n−1 of the set of directions
Ω λ {(λ kα 1 , . . . , λ kα n ) : k ≥ 1}, 0 < λ < 1,
gives rise to a bounded maximal operator M Ω λ in any dimension n ≥ 2. Besides providing the rst higher dimensional example of such a set of directions, the article [26] contains the important novelty of treating the geometric maximal operator M Ω through Fourier analytic tools. This allowed the authors to break the barrier p = 2 that was present in previous work of Córdoba and Fe erman [7], and Strömberg [31], where the authors used mostly geometric arguments. Sjögren and Sjölin [29] proved that, in dimension n = 2, a su cient condition for the L pboundedness of M Ω for some (equivalently for all) 1 < p < ∞ is that Ω is a lacunary set of nite order; loosely speaking, in dimension n = 2, a lacunary set Ω of order L is obtained from a lacunary set Ω of order L −1 by inserting within each gap between two consecutive elements a, b ∈ Ω two subsequences of suitably rotated copies of Ω 1 having a, b as limit points.
Bateman [2] subsequently showed that (up to nite unions) nite order lacunarity of Ω is necessary in order for M Ω to admit nontrivial L p -bounds when Ω is an in nite set. While the counterexample by Bateman is highly nontrivial and employs a probabilistic construction based upon tree percolation, it is rather easy to see that the L p norm of M Ω must depend on N if Ω is, say, the set of N -th roots of unity. In fact, the sharp dependence (1.5) M Ω L 2 (R 2 )→L 2 (R 2 ) ∼ log N for sets of this type was proved by Strömberg [31]; in [20], the structural restriction on Ω was lifted and the upper bound in (1.5) was shown to hold for all nite Ω ⊂ S 1 with cardinality N . Further results concerning maximal operators along directions coming from sets with intermediate Hausdor and fractal dimension can be found in [15,17]. We already reviewed that in all dimensions n ≥ 2, [26] provides us with an example of a lacunary set of directions Ω λ for which M Ω λ is bounded on L p (R n ) for all 1 < p < ∞. Another signi cant higher dimensional example is given in the article of Carbery, [5], where the author considers the projection on S n−1 of the in nite set Ω n−1 (2 k 1 , . . . , 2 k n ) : (k 1 , . . . , k n ) ∈ Z , and proves that M Ω n−1 is bounded on L p (R n ) for all 1 < p < ∞. In dimension n = 2 the set Ω 1 is the paradigmatic example of lacunary subset of S 1 . By the same token, the Carbery set Ω n−1 can be considered as the canonical example of a higher order, higher dimensional lacunary set. More precisely (cf. De nition 2.2) Ω n−1 is a lacunary set of order n − 1 with exactly one direction in each cell of the dissection. In dimensions n ≥ 3, a general su cient characterization of those in nite Ω ⊂ S n−1 giving rise to bounded directional maximal operators, subsuming those of [5,26], was recently established by Parcet and Rogers [28] via an almost-orthogonality principle for the L p norms of M Ω , resembling in spirit that of [1] by Alfonseca, Soria and Vargas in the two-dimensional case. This principle leads naturally to the notion of a lacunary subset of S n−1 when n ≥ 3, which is a su cient condition for nontrivial L p -bounds of (1.4). Again loosely speaking, Ω ⊂ S n−1 is lacunary of order 1 if there exists a choice of orthonormal basis-in the language of [28], a dissection of the sphere S n−1 -such that for all pairs of coordinate vectors e j , e k the projection of Ω on the linear span of e j , e k is a two-dimensional lacunary set; higher order lacunary sets are de ned inductively in the natural way. The authors of [28] also provide a necessary condition which is slightly less restrictive than nite order lacunarity; we send to their article for a precise de nition.
As (1.3) shows, if Ω ⊂ S n−1 is in nite, H Ω is necessarily unbounded. Therefore, the question of sharp quantitative bounds for H Ω in terms of the ( nite) cardinality of the set Ω arises as a natural substitute of uniform bounds. In dimension n = 2, several sharp or near-sharp results of this type have been obtained by Demeter [8], Demeter and the rst author [9], and the authors [11]. We choose to send to these references for detailed statements and just mention the quantitative bounds which are the closest precursors of our Theorem 1.1. To begin with, the two-dimensional analogue of (1.2), with the same O( log N ) quantitative dependence, was proved by the authors in [11]. The methods of [11] are essentially relying on the fact that (lacunary) directions in n = 2 can be naturally ordered, and that this order yields a telescopic representation of H Ω as a maximal partial sum of Fourier restrictions to disjoint (lacunary) cones: see also [8,19]. These methods do not extend to dimensions three and higher, where no ordering is possible in general. On the other hand, in [28,Corollary 4.1], following the ideas of [9, Theorem 1], the authors derive the quantitative estimate
(1.6) sup O ⊂Ω #O ≤N H O L p (R n )→L p (R n ) ≤ C log N M Ω L p (R n )→L p (R n )
at the root of which lies Hunt's exponential good-λ comparison principle between maximal and singular integrals [18]. Coupling (1.6) with the main result of [28] yields that the norms of the maximal Hilbert transform over nite subsets of a given nite order lacunary set in any dimension grow at most logarithmically with the cardinality of the subset. When n = 3, Theorem 1.1 improves this result to the sharp O( log N ) quantitative dependence, answering the question posed by Parcet and Rogers in [28,Section 4]. The estimate of Theorem 1.1 appears to be the rst sharp quantitative estimate for directional singular integrals in dimension n ≥ 3.
Techniques of proof.
The key observation leading to the Parcet-Rogers theorem [28] is that the Fourier support of the single scale distribution
f → ∫ R f (x − tω)ψ (t) dt,
where ψ is a Schwartz function on R, is covered by a union of two dimensional wedges Ψ σ ,ω over pairs σ of coordinate directions, provided a suitable smooth n-dimensional average of f at the same scale is subtracted o ; the latter piece is controlled by the strong maximal function of f . While these wedges heavily overlap with respect to σ , see e.g. Figure 3.2, the authors use the inclusion-exclusion principle to reduce to a square function estimate for compositions of two-dimensional multipliers adapted to the wedges Ψ σ ,ω . The fact that this square function is a bounded operator on L p follows from the bounded overlap, for xed σ , as ω ranges over a lacunary set Ω, of the associated wedges Ψ σ ,ω . The proof of our Theorem 1.1 is also based on a representation of the directional Hilbert transform H ω involving two-dimensional wedge multipliers, which splits H ω into an inner and an outer part: cf. Lemma 3.2. The inner part, which is supported on the union of the wedges Ψ σ ,ω , is amenable to a square function treatment; however, additional di culties are encountered in comparison to [28] as H Ω is not a positive operator and does not obey a trivial L ∞ -estimate. We circumvent this di culty by aiming for the stronger L 2 -weighted norm inequality and relying on extrapolation theory for suitable weights in the natural directional A 2 classes. This requires extending the maximal inequality of [28] to the weighted setting; while this extension does not require substantial additional e orts we wrote out the proofs in detail for future reference. As we previously remarked, it also has the pleasant e ect of giving a much more general weighted version of Theorem 1.1: see Corollary 1, Section 5.
Unlike the single scale operator, the outer part of the decomposition is nontrivial, and is actually the one introducing the dependence on the cardinality of the set of directions. It is a signed sum of 2 n terms which are compositions of two-dimensional angular multipliers; in general we cannot do better than estimating the maximal operator associated to each summand. The key observation of our analysis at this point is that these compositions can be bounded pointwise by (compositions of) strong maximal operators, upon pre-composition with at most n 2 directional Littlewood-Paley projections; see Lemma 3.3 and Remark 6.4. An application of at most n 2 Chang-Wilson-Wol decouplings, see Proposition 5.2, then reduces the maximal estimate to a square function estimate upon loss of n 2 factors of order log N . This is enough to obtain the sharp result for n = 2, 3 (and, less interestingly, recover (1.6) when n = 4, 5), hinting on the other hand that this approach is not feasible in general dimensions.
In fact, perhaps surprisingly, we show with a counterexample that this growth rate, worse than that of H Ω whenever n ≥ 6, is actually achieved by the maximal operator associated to the outer parts. This phenomenon displays how the model operator of Lemma 3.2, based on the combinatorics of two-dimensional wedges, is not subtle enough to completely capture the cancellation present in H Ω .
1.4. Relation to the Hilbert transform along vector elds. In addition to their intrinsic interest, Theorem 1.1 and predecessors may be seen as building blocks towards the resolution of the following question, apocryphally attributed to E. Stein and often referred to as the vector eld problem: if : R n → S n−1 is a vector eld with Lipschitz constant equal to 1 and pointing within a small neighborhood of (1/ √ n, . . . , 1/ √ n), prove or disprove that the truncated directional Hilbert transform along
H f (x) = p.v. ∫ |t |<ε 0 f (x − t (x)) dt t
for ε 0 > 0 small enough, is a bounded operator from L 2 (R n ) into L 2,∞ (R n ). The partial progress in dimension n = 2, beginning with the work of Lacey and Li [23,25] and continued in e.g. [3,14,10] by several authors, rests upon using the Lipschitz property to achieve decoupling of the full maximal operator into a Littlewood-Paley square function similar in spirit to the one appearing in (5.6). The estimation of a single Littlewood-Paley piece in the vector eld case is more di cult than the pointwise estimate available to us in Lemma 3.3 and involves, in dimension n = 2, time-frequency analysis of roughly the same parametric complexity as of that appearing in the Lacey-Thiele proof of Carleson's theorem [24]. Lemma 3.3 in this context may be interpreted as a single tree estimate (cf. [24,25]), showing that the annular estimate for n = 3 might display the same essential complexity as the n = 2 case.
1.5. Plan of the article. In the forthcoming Section 2, we set up the notation for the remainder of the article and provide the precise de nition of nite order lacunary sets in R n . Section 3 contains the reduction of H Ω to the above mentioned model operators, Lemma 3.2 as well as their single tree estimate of Lemma 3.3. In Section 4, after the necessary setup for directional weighted classes, we prove a weighted version of the Parcet-Rogers maximal estimate in Theorem 4.6 which, together with the extrapolation techniques of Lemma 4.3, is relied upon in the proof of our main result. Theorem 1.1 is derived in Section 5 as the Lebesgue measure case of a more general sharp weighted estimate, Corollary 1. This corollary in turn descends from Theorem 5.1, a L 2 -weighted almost-orthogonality principle for H Ω in the vein of [1,28]. The nal Section 6 contains the above mentioned sharp counterexamples for the model operator of Lemma 3.2 in dimension 4 and higher: the main result of this section is the lower bound of Theorem 6.3. on the subject of completion of a lacunary set. We would also like to thank Maria J. Carro for helpful discussion related to weighted norm inequalities for directional operators. We are indebted to Keith Rogers for an expert reading and insightful comments that helped us improve the presentation. Finally, we would like to thank the anonymous referees for providing helpful comments and references.
L :
In this section, we give a rigorous de nition of nite order lacunary sets which will be used throughout the article. In essence, our de nition is the same as the one given by Parcet and Rogers in [28].
2.1. Lacunary sets of directions of nite order. For convenience we keep most of the notational conventions of [28]. Throughout the paper we work in R n and consider sets of directions Ω ⊂ S n−1 . We allow the possibility that span(Ω) = R d for some non-negative integer d ≤ n and write Σ(d)
{(j, k) : 1 ≤ j < k ≤ d}; we will drop the dependence on d and just write Σ when there is no ambiguity. We typically denote the members of Ω as ω and the members of Σ as σ = (j, k).
Note that |Σ(d)| = d(d − 1)/2.
With the roles of n, d, and Ω as above we assume that for each σ = (j, k) ∈ Σ(d) we are given a sequence {θ σ , : ∈ Z} with the property that there exists λ σ ∈ (0, 1) such that
(2.1) θ σ , +1 ≤ λ σ θ σ , , θ σ ,0 = θ 0 , ∀σ .
Here we set λ max σ ∈Σ λ σ and throughout the paper we will x a numerical value of λ ∈ (0, 1) and we will adopt the convention that all sequences θ σ ,λ have lacunarity constants uniformly bounded by the same number λ. A choice of orthonormal basis (ONB) of span(Ω) ≡ R d
(2.2) B {e j : j = 1, . . . , d }
and of lacunary sequences {θ σ , } as above induces for each σ ∈ Σ(d) a partition of the sphere S d−1 into sectors S σ , :
(2.3) S d−1 = ∈Z S σ , , S σ , = S (j,k), ω ∈ S d−1 : θ σ , +1 ≤ |ω · e k | |ω · e j | < θ σ , .
We will henceforth write ω j ω · e j for 1 ≤ j ≤ d once the coordinate system is clear from context. The partition above is completed by adding the set
S σ ,∞ = S (j,k),∞ S d−1 ∩ (e ⊥ j ∪ e ⊥ k ). We henceforth write Z * Z ∪ {∞}.
Now such a partition of the sphere immediately gives a partition of Ω by setting
(2.4) Ω σ , Ω ∩ S σ , , σ ∈ Σ, ∈ Z * . The family of d 2 = d(d − 1)/2 partitions indexed by σ ∈ Σ(d) , Ω = ∈Z Ω σ , ,
will be called a lacunary dissection of Ω, with parameters an ONB B as in (2.2) and a choice of sequences {θ σ , } as in (2.1). Note that {S σ , } ∈Z * : σ ∈ Σ(d) is a lacunary dissection of S d−1 .
We will refer to sets of the type S σ , and Ω σ , as sectors of the lacunary dissection. We will also work with the partition of Ω into disjoint cells induced by a dissection, namely intersections of sectors Ω σ , σ . More precisely, let B be a choice of ONB as in (2.2). Given
= { σ : σ ∈ Σ(d)} ∈ Z Σ we de ne the -cell of the dissection corresponding to B as S σ ∈Σ S σ , σ , Ω σ ∈Σ Ω σ , σ .
Observe that this provides the ner partition of S d−1 and Ω, respectively, into cells
S d−1 = ∈Z Σ S , Ω = ∈Z Σ Ω .
The following de nition, which is the principal assumption in our main results, was given in [28, p. 1537].
De nition 2.2 (Lacunary set). Let Ω ⊂ S n−1 be a set of directions with span(Ω) = R d . Then
· Ω is a lacunary set of order 0 if it consists of a single direction;
· if L is a positive integer, then Ω is lacunary of order L if there exists an ONB B as in (2.2) and a choice of sequences {θ σ , } as in (2.1) with the property that for each σ ∈ Σ(d) and each ∈ Z * the sector Ω σ , in (2.4) is a lacunary set of order L − 1. A set Ω will be called lacunary if it is a nite union of lacunary sets of nite order.
For example, Ω is 1-lacunary if there exists a dissection such that, for each σ ∈ Σ(d) and ∈ N the set Ω σ , contains at most one direction.
Remark 2.3.
Let Ω be a lacunary set of directions and β ∈ (0, 1). Then Ω is a lacunary set of directions with respect to dissections given by the sequence θ σ , β . This is automatic if β ≥ λ while in the case β < λ it follows easily by suitably splitting the set Ω into O(log β/min σ log λ σ ) congruence classes. Unless explicitly mentioned otherwise, all lacunary sets in this paper are given with respect to the sequence
θ σ , = 2 − , ∈ Z.
As our choice of sequences {θ σ , } is universal, prescribing a lacunary dissection amounts to xing an orthonormal basis B as in (2.2). It is also clear that for all proofs in this paper it su ces to consider the case that Ω is contained in the open positive 2 d -tant of the sphere
S d−1 + S d−1 ∩ R d + .
While there are di erent coordinate systems involved in the de nition of a lacunary set Ω, by splitting any lacunary set into nitely many pieces we can assume this property for all dissections that come into play. Furthermore, by standard approximation arguments (e.g. monotone convergence) we can assume that Ω has empty intersection with all coordinate hyperplanes. These conventions allow us to only consider sectors S σ , , Ω σ , with ∈ Z instead of ∈ Z * . On the other hand, and in contrast with the previous conventions concerning the proofs, in the statements of our theorems we always assume that the set Ω is closed. Furthermore, the basis vectors of any dissection used in the de nition of a lacunary set of any order are assumed to be contained in the set. We adopt these conventions throughout the paper without further mention.
Remark 2.4. Although it is necessary to distinguish the case spanΩ = R d with d < n in the de nitions, in the proofs of our estimates we will argue with d = n without explicit mention; by Fubini's theorem, this is without loss of generality.
M
For ω ∈ S n−1 (re)de ne the directional Hilbert transform on R n
(3.1) H ω f (x) = ∫ R n f (ξ )sign(ξ · ω)e ix ·ξ dξ .
In this section we set up a representation formula for (3.1). The central result is Lemma 3.2 below. Before the statement we need to introduce some additional notation and auxiliary functions. For ∈ Z and γ > 0 we consider the two-dimensional wedges
Ψ σ , ,γ ξ ∈ R n \ (e σ (2) ) ⊥ : 2 −( +1) γ ≤ − ξ σ (1) ξ σ (2) < γ 2 − .
We are interested in the particular cases γ ∈ {n, n + 1} for which we use the special notations
(3.2) Ψ σ , ,n Ψ σ , , Ψ σ , ,n+1 Ψ σ , . Furthermore, let ϕ + , ϕ − : R → [0, 1] be smooth functions satisfying ϕ + (x) 0, x < −(n + 1), 1, x > −n, ϕ − (x) 1, x < − 1 2n , 0, x > − 1 2(n+1) .
We now use the functions ϕ + , ϕ − in order to de ne the essentially two-dimensional angular Fourier multiplier operators
K ± σ , σ (ξ ) = κ ± σ , σ (ξ σ (1) , ξ σ (2) ) ϕ ± 2 σ ξ σ (1) ξ σ (2) , K • σ , σ (ξ ) = κ • σ , σ (ξ ) κ + σ , σ (ξ )κ − σ , σ (ξ ),(3.3)
and their compositions
(3.4) K ε U , σ ∈U K ε σ σ , σ , U ⊆ Σ, ε ∈ {+, •, −} U ; when ε σ = • for all σ ∈ U we simply write K U , in place of K ε U , . Remark 3.1. Let ε ∈ {+, −, •}.
We record the support conditions (see Figure 3.1)
(3.5) ∇ ξ κ ε σ , σ 1 Ψ σ, ≡ ∇ ξ κ ε σ , σ 1 R n \ Ψ σ, ≡ 0, κ ε σ , σ 1 Ψ σ, ≡ 1, κ • σ , σ 1 R n \ Ψ σ, ≡ 0. Moreover we have the derivative estimates (3.6) sup |α |≤10n sup ξ ∈R n |ξ σ (1) | α 1 |ξ σ (2) | α 2 ∂ α 1 ξ σ (1) ∂ α 2 ξ σ (2) κ ε σ , σ (ξ ) 1, |α | = α 1 + α 2 .
We will also use below that if ξ Ψ σ , σ , then κ ε σ , σ is constant in a neighborhood of ξ .
|H ω f | | f | + sup U ⊆Σ H ω K U , f + sup ε∈{+,−} Σ sup U ⊆Σ K ε U , f . Proof. As Id = U ⊆Σ (−1) #U +1 K U , + σ ∈Σ Id − K σ , σ we write (3.7) H ω f = U ⊆Σ (−1) #U +1 H ω K U , f + T f
where T is the Fourier multiplier with symbol
(3.8) m(ξ ) = T (ξ ) = sign(ω · ξ ) σ ∈Σ 1 − κ • σ , σ (ξ ) .
We have to treat the term T . First of all, we check that
(3.9) C ω ξ ∈ R n : |ξ · ω | < 1 n max 1≤j≤n |ω j ξ j | ⊂ D σ ∈Σ Ψ σ , σ .
This is essentially depicted in Figure 3.2 and is a sharpening of the argument in [28, Proof of Theorem A]. We prove (3.9) by showing that R n \ D ⊆ R n \ C ω . To that end let ξ ∈ R n \ D .
Writing η j ω j ξ j and remembering the convention ω j > 0 for all j we then have that
− η σ (1) η σ (2) 1 n , n ∀σ ∈ Σ.
Choose j such that |η j | = max 1≤j≤n |η j |. Now we note that if η j η j ≥ 0 for all j ∈ {1, . . . , n} \ {j } then ξ C ω so we are done. Otherwise we de ne k by means of |η k | max j:η j η j <0 |η j |; as |η j | ≥ n|η k | we end up with
|ξ · ω | = 1≤j≤n η j ≥ |η j | − (n − 1)|η k | ≥ |η j | n = max 1≤j≤n |ω j ξ j |
which is the claim (3.9). Noting that
supp m = R n \ σ ∈Σ Ψ σ , σ = R n \ D
this claim tells us that supp m ∩ C ω = ∅ whence if ξ ∈ supp m the signum of (ω · ξ ) is constant in a neighborhood of ξ . Now using the easy to verify fact that (1 − ϕ + ϕ − ) = (1 − ϕ + ) + (1 − ϕ − ) and the two summand are supported in disjoint intervals we can rewrite (3.8) as
m(ξ ) = ε∈{+,−} Σ sign(ω · ξ )κ ε (ξ ), κ ε (ξ ) σ ∈Σ 1 − κ ε σ σ , σ ,
for ε = {ε σ : σ ∈ Σ}. As supp κ ε is a connected set not intersecting C ω we conclude that sign(ω · ξ ) is constant on supp κ ε . Therefore if T ε is the Fourier multiplier with symbol κ ε
(3.10) |T f | ≤ ε∈{+,−} Σ |T ε f |.
Now we observe that the symbol of Id − T ε is equal to
1 − σ ∈Σ 1 − κ ε σ σ , σ = U ⊆Σ (−1) #U +1 σ ∈U κ ε σ σ , σ ,
and putting together the last display with (3.7) and (3.10) we achieve the pointwise estimate claimed in the Lemma.
In the next lemma we prove an annular estimate for the multiplier operators of (3.4). To do so we will need to precompose these operators with suitable Littlewood-Paley projections which we now de ne. Let p, q be smooth functions on R with
supp p ⊂ ξ ∈ R : 1 2 < |ξ | < 2 , t ∈Z p(2 −t ξ ) = 1, ξ 0, supp q ⊂ ξ ∈ R : 1 4 <|ξ | < 4 , q = 1 on ξ ∈ R : 1 2 < |ξ | < 2 . Now for υ ∈ {1, .
. . , n} we de ne the Fourier multiplier operators on R n
P υ t f (ξ ) f (ξ )p(2 −t ξ · e υ ), Q υ t f (ξ ) f (ξ )q(2 −t ξ · e υ ).
Thus {P υ t } t is a one-dimensional Littlewood-Paley decomposition, acting on the υ-th variable only, and being the identity with respect to all other frequency variables. Here and in the rest of the paper we write M s for the strong maximal function and M 2
s M s • M s . Lemma 3.3. Let supp f ⊂ Q where Q is any of the 2 3 octants of R 3 . Let U ⊆ Σ, ε ∈ {+, −} U .
There is a choice υ = υ(U , ε, Q) ∈ {1, . . . , n} such that the pointwise estimate
K ε U , (P υ t f )(x) M 2 s (P υ t f )(x), x ∈ R 3 ,
holds uniformly over all t ∈ R.
Proof. As ε, are xed throughout the proof, and in order to avoid proliferation of indices, we shall write below κ ε σ σ , σ = κ σ , K ε σ σ , σ = K σ , K ε U , = K U , when these parameters are unimportant. As we are working with the strong maximal function, by rescaling on the sphere we may assume σ = 0 for all σ ∈ Σ; this is just for convenience of notation as we shall see. We divide the proof to di erent cases according to the cardinality of the set U ⊆ Σ.
Case #U = 1. In this case there exists σ ∈ Σ(3) such that U = {σ }, K U = K σ , and we may choose either υ = σ (1) or υ = σ (2). The choice does not depend on the quadrant Q. To x ideas, we work with σ = (1, 2) and choose υ = 1. By the observation (3.5) of Remark 3.1, we know that κ σ is constant in a neighborhood of ξ unless ξ ∈ Ψ σ ,0 , in which case |ξ σ (1) | ∼ |ξ σ (2) |. Therefore if ξ ∈ Ψ σ ,0 , and 2 t−2 < |ξ 1 | < 2 t+2 , there holds 2 t ∼ |ξ υ | ∼ |ξ σ (2) | and
∂ α 1 ξ σ (1) ∂ α 2 ξ σ (2) κ σ (ξ ) |ξ σ (1) | −α 1 |ξ σ (2) | −α 2 2 −tα , α = α 1 + α 2 .
Using the above inequality for α = 0, . . . , 10 · 3, it follows that
(3.11) Φ σ (x σ (1) , x σ (2) ) ∫ R 2 κ σ (ξ σ (1) , ξ σ (2) )q(2 −t ξ υ )e i(x σ (1) ξ σ (1) +x σ (2) ξ σ (2) ) dξ σ (1) dξ σ (2)
satis es
(3.12) |Φ σ (x σ (1) , x σ (2) )| 2 2t 1 + 2 t |x σ (1) | + 2 t |x σ (2) | −(3+1) .
We now write f t = P υ t f . Denoting convolution in the variables σ (1), σ (2) by * σ we have that
K σ f t = (K σ Q υ t )(f t ) = Φ σ * σ f t .
Hence using (3.12) we see that
|K U f t (x)| ≤ ∫ R 2 | f t (x 1 − 1 , x 2 − 2 , x 3 )||Φ σ ( 1 , 2 )| d M s (f t )(x)
as claimed.
Case #U = 2. In this case U = {σ , τ } for some σ , τ ∈ Σ(3) and necessarily σ , τ must have a common component. We choose υ to be this common component. This choice also does not depend on the quadrant Q. To x ideas σ = (1, 2), τ = (1, 3) and we choose υ = 1. Note that in this case K U = K (1,2) K (1,3) . With the same notation of (3.11) from the previous case we have the equality
K U f t = K (1,2) Q υ t • K (2,3) Q υ t (f t ) = Φ (1,2) * (1,2) Φ (1,3) * (1,
3) f t so using (3.12) again we see that
|K U f t (x)| ≤ ∫ R 2 ×R 2 | f t (x 1 − 1 − z 1 , x 2 − 2 , x 3 − z 3 )||Φ (1,2) ( 1 , 2 )||Φ (1,3) (z 1 , z 3 )| d dz M 2 s (f t )(x) as claimed.
Case #U = 3. We show that this case reduces to the preceding ones, with choice of υ depending on the quadrant Q. Let
Q σ = {ξ ∈ R 3 : ξ σ (1) ξ σ (2) ≥ 0}.
Notice that the constraints on the supports of ϕ ± imply that
κ − σ , σ 1 Q σ ≡ 0, κ + σ , σ 1 Q σ ≡ 1, ∀σ ∈ Σ.
As for each of the 8 quadrants Q of R 3 there exists (at least one) σ Q ∈ Σ such that Q ⊂ Q σ Q , we see that
K ε U , 1 Q = 0, if ∃σ ∈ U with ε σ = −, K ε U \{σ Q }, 1 Q , otherwise.
As #{U \ {σ Q }} = 2 for each quadrant Q the proof follows by the cases #U ∈ {1, 2} considered above.
W
We dedicate this section to the discussion of weighted norm inequalities for the maximal directional operator. These will serve as a tool for the proof of Theorem 1.1; in fact, they will be used to prove a weighted almost orthogonality principle that subsumes both Theorem 1.1 and its weighted analogue, which will be stated at the end of this section. However, we do think they are also of independent interest.
The weighted theory of the directional maximal operator has been studied, at least in the two-dimensional case, in [13], for the case of 1-lacunary sets of directions. Here we recall all the basic de nitions and tools, and then proceed to prove weighted norm inequalities for the directional maximal function M Ω associated to a nite order lacunary set Ω ⊂ S n−1 . In essence, the main result of this section, Theorem 4.6, is a weighted generalization of the main result of [28] by Parcet and Rogers. 4.1. Directional A p weights. We begin by de ning the appropriate directional A p classes. The easiest way to de ne the appropriate class is to ask for non-negative, locally integrable functions w (we will refer to such functions as weights) such that for all nice functions f we have
M Ω f L p (w) f L p (w) , f L p (w) ∫ | f | p w 1 p , 1 < p < ∞,
where Ω is a set of directions such that M Ω is bounded on L p (R n ). Without explicit mention, we work under the purely qualitative assumptions that all weights appearing below will be continuous and nonvanishing functions on R n ; this assumption may be removed via a standard approximation procedure which we omit. We will very soon specialize to sets Ω which are lacunary of nite order so we encourage the reader to keep this example in mind. Note that for smooth functions f we have M Ω f = M Ω f . We can then assume that Ω is closed when deriving necessary conditions for w.
For ω ∈ Ω, x ∈ R n and η > 0 we then de ne segments and corresponding one-dimensional averages of f ∈ C(R d ) as follows
I (x, η, ω) {x + tω : |t | < η}, f I (x,η,ω) 1 2η ∫ η −η f (x + tω) dt .
We set I Ω {I (x, η, ω) : x ∈ R d , η > 0, ω ∈ Ω} and for p ∈ (1, ∞) we adopt the usual notation for the dual weight σ w − 1 p−1 . Now the L p (R n )-boundedness of M Ω clearly implies the boundedness of M ω on L p (I (x, ω)), where I (x, ω)
{x + tω : t ∈ R}, uniformly in x ∈ R n and ω ∈ Ω. Now testing this one-dimensional boundedness property M ω for some xed p ∈ (1, ∞) against functions of the form σ 1 I (x,η,ω) shows the necessity of the directional A p condition
[w] A Ω p sup I ∈I Ω ∫ I w ∫ I σ p−1 < ∞;
here we remember that we have made the qualitative assumption that w is a continuous nonvanishing function. Note that if we write w(x) = w(x · ω, x · ω ⊥ ), the previous condition means that for almost every x ∈ R n and ω ∈ Ω, the one-dimensional weight x,ω (s) w(s, x · ω ⊥ ), s ∈ R, is in A p (R), with uniformly bounded A p constant:
sup x ∈R n , ω∈Ω [ x,ω ] A p = [w] A Ω p < ∞.
We complete the set of de nitions by de ning A Ω 1 to be the class of weights w such that
[w] A Ω 1 sup x ∈R n M Ω w(x) w(x) < ∞.
A well known class of Muckenhoupt weights is produced be considering Ω = {e 1 , . . . , e n }; then A Ω p is just the class A * p of strong or n-parameter Muckenhoupt weights. We also note that an obvious corollary of one dimensional theory is that
M ω L p (w)→L p (w) [w] 1 p−1 A Ω p ,
ω ∈ Ω, and the implicit constant is independent of w and ω. We refer to [4] for the sharp onedimensional weighted bound for M ω .
Extrapolation for
A Ω p weights. Having established the appropriate A Ω p classes, we now proceed to proving one of the most useful properties of weighted norm inequalities, that of extrapolation.
We begin by noting that, as in the case of classical A p weights, it is easy to create A Ω p -weights by using the Rubio de Francia method and factorization; see [12,Lemmata 2.1,2.2]. We omit the proofs which are essentially identical to the one-directional case.
Lemma 4.3. Let w ∈ A Ω p . For a nonnegative function ∈ L p (w) we de ne E ∞ k=0 M (k) Ω 2 k M Ω k L p (w)
Then E satis es the following properties
(i) ≤ E . (ii) For every ∈ L p (w) we have E L p (w) ≤ 2 L p (w) . (iii) If M Ω L p (w)→L p (w) < ∞ then E is an A Ω 1 weight with constant [E ] A Ω 1 ≤ 2 M Ω L p (w)→L p (w) .
Furthermore for all exponents 1 ≤ p < ∞, 1 < p 0 < ∞ and weights u, w there holds
[wu p−p 0 ] A Ω p 0 ≤ [w] A Ω p [u] A Ω 1 , p ≤ p 0 w p 0 −1 p−1 u p−p 0 p−1 A Ω p 0 ≤ [w] p 0 −1 p−1 A Ω p [w] p−p 0 p−1 A Ω p , p > p 0 .
We now provide the basic extrapolation result for A Ω p weights which will be our main tool for passing from L 2 (w)estimates to L p (w) estimates for all p ∈ (1, ∞). This result and its proof are completely analogous to [12, where P : R + → R + is a an increasing function and C > 0 does not depend on w or the pairs (f , ). Then for all 1 < p < ∞ we have
∫ R n p w 1 p ≤ CK(w) ∫ R n f p w 1 p , with K(w) P([w] A Ω p (2 M Ω L p (w) ) p 0 −p ), p < p 0 , P([w] p 0 −1 p−1 A Ω p (2 M Ω L p (σ ) ) p−p 0 p−1 ), p > p 0 .
4.5.
Weighted inequalities for the lacunary directional maximal operator. In this subsection, we consider directional maximal operators associated to lacunary sets of order L.
According to the previous discussion, the condition w ∈ A Ω p is necessary for the boundedness property M Ω : L p (w) → L p (w). In this paragraph we also show the su ciency of condition A Ω p , thus giving a characterization of the A Ω p class in terms of M Ω .
Theorem 4.6.
Let Ω ⊂ S n−1 be a lacunary set of directions of order L, where L is a positive integer, and w be a weight. For every 1 < p < ∞, the following are equivalent.
(i) For all f ∈ L p (w) we have M Ω f L p (w) f L p (w) , with implicit constant depending on w, the dimension, and the lacunarity constants of Ω.
(ii) We have that w ∈ A Ω p . Furthermore, if w ∈ A Ω p then we have the estimate M Ω L p (w)→L p (w) [w] δL
A Ω p for some exponent δ = δ (p, n) > 0 and implicit constant independent of w. .3) such that for each σ ∈ Σ(n) and ∈ N, the sets Ω σ , are lacunary of order L − 1. As metnioned before, cf. Remark 2.3, we assume that Ω is closed and that the axes {e 1 , . . . , e n } of the dissection of order L are contained in Ω. Then, the inclusion A Ω p ⊂ A * p holds, the latter being the class of strong A p weights with respect to these coordinate axes. In consequence, the strong maximal function M s is automatically bounded on L p (w) for w ∈ A Ω p . As in the proof of [27, Theorem A] we rely on the covering of the singularity hyperplane ξ · ω = 0 by nitely overlapping unions of two dimensional wedges {Ψ σ , σ : σ ∈ Σ} de ned in (3.2), where = ( σ : σ ∈ Σ) is the unique index in Z Σ such that ω belongs to the cell Ω . The core of the proof is contained in the following two lemmata which are weighted versions of the corresponding results from [27].
The rst result we need is a weighted analogue of [27, Lemma 1.1]. Note that it does not require the lacunarity assumption on Ω and the weight class needed is just the usual class of strong Muckenhoupt weights A * p .
Lemma 4.9. Let p > 1 and w ∈ A * p be a weight. There holds
M Ω f L p (w) [w] n p−1 A * p sup U ⊆Σ sup ∈Z Σ M Ω K U , f L p (w) ,
with the implicit constant depending upon dimension and p.
Proof. The proof follows from the arguments in the proof of [27, Lemma 1.1]. Indeed one just needs to note that the corresponding unweighted estimate in [27] is proved via the use of pointwise estimates, which of course are independent of the underlying measure, and the boundedness of the strong maximal function M s f on L p (R n ). The latter fact is replaced by the observation that M s maps L p (w) to itself whenever w ∈ A * p , and satis es the quantitative norm estimate
M s L p (w)→L p (w) [w] n p−1 A * p .
Here again we use the one-dimensional sharp weighted estimate for the Hardy-Littlewood maximal operator from [4].
The second result is a weighted square function estimate for the angular multipliers K U , associated to a lacunary dissection of the sphere. Lemma 4.10. Let 1 < p < ∞ and Σ corresponding to a given dissection of the sphere. Then for all w ∈ A * p we have
sup U ⊆Σ ∈Z U K U , f 2 1 2 L p (w) [w] β A * p f L p (w) .
The implicit constant depends upon dimension n, p, and β > 0 depends on p and n.
Proof. As in the proof of [27, Lemma 1.2] we note that it will be enough to prove the L p (w)boundedness of the randomized map T given as
f → ∈Z S ε σ ∈U κ • σ , σf ∨ (mf ) ∨ ,
uniformly over choices of signs {ε } ∈Z U . The unweighted L 2 (R n )-boundedness of this map follows simply by Plancherel and the nite overlap property of the supports { Ψ σ , : ∈ N}, which shows that m ∈ L ∞ , uniformly over choices of signs. For L p (w)-bounds, we need an A * p -weighted version of the standard Marcinkiewicz multiplier theorem. This can be found for example in [21,Theorem 3] so the proof of the lemma reduces to checking a number of conditions on averaged derivatives of m. In fact these conditions are identical to the hypothesis of the unweighted Marcinkiewicz multiplier theorem, as can be found for example in [30, p. 109] and can be veri ed by using estimates (3.6) for each single multiplier K • σ , . An inspection of the proof, which relies on the weighted vector valued boundedness of frequency projections on rectangles, and the weighted multiparameter Littlewood-Paley inequalities, shows that there exists a constant β depending on n and p such that T L p (w)→L p (w) [w] β A * Proof of (4.1). We rst perform the proof in the case p ≥ 2. Let us for a moment x a U ⊆ Σ and write Z Σ = Z U ⊗Z Σ\U so that given = { σ } σ ∈Σ we decompose = τ ×t with τ = {τ σ : σ ∈ U }. Replacing the supremum by an p function gives
(4.2) sup ∈Z Σ M Ω f τ L p (w) ≤ sup σ ∈Σ sup ∈Z M Ω σ, L p (w)→L p (w) τ ∈Z U | f τ | p 1 p L 2 (w)
.
As p ≥ 2 estimate (4.2) implies
sup ∈Z Σ M Ω f τ L p (w) ≤ sup σ ∈Σ sup ∈N M Ω σ, L p (w)→L p (w) τ ∈Z S | f τ | 2 1 2 L p (w)
. Now (4.1) follows by taking f τ K U ,τ where τ = {τ σ : σ ∈ U } and bounding the right hand side in the last display, from above, by Lemma 4.10, and the left hand side of the last display, from below, by Lemma 4.9. For 1 < p < 2 we note that, by monotone convergence, it su ces to show the estimate for every nite subset of Ω, which we still call Ω. Then M Ω L p (w)→L p (w) < ∞, w ∈ A Ω p , so we can interpolate between the estimates
sup ∈Z Σ M Ω f τ L p (w) ≤ M Ω L p (w)→L p (w) sup ∈Z Σ | f τ | L p (w)
and (4.2) to conclude
sup ∈Z Σ M Ω f τ L p (w) ≤ M Ω 1− p 2 L p (w)→L p (w) sup σ ∈Σ sup ∈Z M Ω σ, L p (w)→L p (w) p 2 τ ∈Z S | f τ | 2 1 2 L p (w)
.
Taking again f τ = K U ,τ an application of Lemmata 4.10 and 4.9 yields
M Ω L p (w)→L p (w) [w] γ A * p M Ω 1− p 2 L p (w)→L p (w) sup σ ∈Σ sup ∈Z M Ω σ, L p (w)→L p (w) p 2
with γ = β + n/(p − 1). As we have assumed that M Ω L p (w)→L p (w) < ∞ we may rearrange and complete the proof of the theorem.
A H
We now prove an almost orthogonality principle for the maximal Hilbert transform of a set Ω ⊂ S 2 . In the statements below it is convenient to write for all nonnegative integers N , weights w on R 3 , and Ω ⊂ S 2
Θ N (Ω, w) sup O ⊂Ω #O ≤N H O L 2 (R 3 ;w)→L 2 (R 3 ;w) .
Theorem 5.1. There exist C, γ ≥ 1 such that the following holds. Let N be a positive integer, B be a choice of ONB, Ω ⊂ S 2 a set of directions containing B and w ∈ A Ω 2 . Then
Θ N (Ω, w) ≤ C[w] γ A Ω 2 log N + sup σ ∈Σ sup ∈Z Θ N (Ω σ , , w)
where the lacunary dissection is taken with respect to B as in (2.3).
By iterative application of the almost orthogonality principle, and extrapolation, we obtain the following corollary, of which Theorem 1.1 is the particular case w = 1. Corollary 1. Let 1 < p < ∞ and L ≥ 0. There exists constants C = C p,L , γ = γ p,L such that for any Ω ⊂ S 2 lacunary set of order L and w ∈ A Ω
p sup O ⊂Ω #O ≤N H O L p (R 3 ;w)→L p (R 3 ;w) ≤ C[w] γ A Ω p log N .
The proof of Theorem 5.1 rests upon the results of the previous sections, as well as on the proposition below, a weighted version of the Chang-Wilson-Wol principle, which we state and prove before the main argument. In the statement of the proposition below we remember that A * p = A Ω p with Ω being the canonical basis of R n , namely Ω = {e 1 , . . . , e n }. We also use the standard notation A * ∞ ∪ p>1 A * p .
Proposition 5.2. Let {K 1 , . . . , K N } be Fourier multiplier operators on R n with uniform bound
sup 1≤j≤N K j L 2 (w)→L 2 (w) ≤ [w] α A * 2
for some α > 0. Let {P υ t } t ∈Z be a smooth Littlewood-Paley decomposition acting on the υ-th frequency variable, where 1 ≤ υ ≤ n. For w ∈ A * p and 1 < p < ∞ we then have
sup 1≤j≤N |K j f | L p (w) [w] γ A p f L p (w) + (log(N + 1)) 1 2 t ∈Z sup 1≤j≤N |K j P υ t f | 2 1 2 L p (w)
for some exponent γ = γ (α, p, n) and implicit constant depending on α, p, n.
Proof. To simplify the notation we work with υ = 1 and set
K f sup 1≤j≤N |K j f |, DK t ∈Z sup 1≤j≤N |K j P 1 t f | 2 1 2 .
Let {D j : j ∈ Z} be the standard dyadic ltration on R, E j be the associated sequence of conditional expectations, and ∆f denote the associated martingale square function. Let E 1 j be the sequence of conditional expectations on L 1 (R n ) acting on tensor products f (x) = (x 1 ) ⊗ h(x 2 , . . . , x n ) by E 1 j f E j ⊗ h and denote by ∆ 1 f the associated martingale square functions. The Chang-Wilson-Wol inequality [6] tells us that if w is an A * ∞ -weight then
w x ∈ R n : | (x) − E 1 0 (x)| > 2λ, |∆ 1 f (x)| ≤ γ λ ≤ A exp − b [w] (1) A ∞ γ 2 w x ∈ R n : |M e 1 (x)| > λ , (5.1)
where A, b are absolute positive constants and
[w] A (1) ∞ sup x ∈R n [w(x + ·e 1 )] A ∞ ;
here [·] A ∞ denotes the Wilson A ∞ constant of a weight on the real line, see [32]. The inequality (5.1) for n > 1 is in fact obtained from the one dimensional version of [6] and Fubini. As
[w] A (1) ∞ ≤ [w] A *
p and proceeding exactly as in in the proof of [10, Corollary 1.14] we can use the above inequality to reach
K f L p (w) M e 1 f L p (w) + sup 1≤j≤N M e 1 K j f L p (w) + log(N + 1) M e 1 L p (w) DK f L p (w) (5.2)
where M e 1 f (M e 1 | f | r ) 1 r , and r > 1 can be chosen arbitrarily close to 1; the implicit constant depends on p, r , and polynomially on [w] A * p . Since our weight w ∈ A * p , M e 1 is a bounded operator on L p (w). Furthermore, using the reverse Hölder property for A * ∞ weights, see e.g. [16,Theorem 1.4], we actually have the openness property
[w] A * p r ≤ 2[w] A * p , r ≤ p p − c([w] A * ∞ ) −1 ,
where the positive constant c = c(p, n) ≤ 1 can be explicitly computed. Therefore M e 1 is also a bounded operator on L p (w) provided r is chosen small enough to comply with the restriction in the last display. Making use of these L p (w)-bounds in (5.2) nally yields the proposition.
|H O f | | f | + sup U ⊆Σ sup ∈L O H O∩S K U , f + sup U ⊆Σ sup ε∈{+,−} U sup ∈L O K ε U , f . (5.3)
We may ignore the rst summand on the right hand side. We bound the norm of the second summand on the right hand side by a constant multiple of
sup U ⊆Σ ∈L O H O∩S K U , f 2 1 2 L 2 (w) ≤ B sup U ⊆Σ ∈Z U K U , f 2 1 2 L 2 (w) [w] γ A Ω 2 sup σ ∈Σ sup ∈Z Θ N (Ω σ , , w) f L 2 (w) ,(5.4)
where in the last step we have used the weighted estimate of Lemma 4.10, and we have also used the easy estimate
B sup L 2 (w ; 2 ) =1 ∈Z Σ H O∩S 2 1 2 L 2 (w) ≤ sup σ ∈Σ sup ∈Z Θ N (Ω σ , , w).f = Q f Q , f Q L 2 (w) [w] 3 A * 2 f L 2 (w) ,
where each f Q has frequency support in one of the octants Q of R 3 . By virtue of the norm estimate of the above display, we may x one of these octants Q and prove (5.5) for functions f whose frequency support is contained in Q, which we do here onwards. Now we remember that by Lemma 4.10 the multiplier operators {K ε U , : ∈ L} satisfy weighted L 2 bounds with weighted operator norms bounded polynomially in [w] A * 2 , uniformly in U and . This allows us to use Proposition 5.2 on the N = #L Fourier multiplier operators {K ε U , : ∈ L}, to get that
sup ∈L K ε U , f L 2 (w) [w] γ A * 2 f L 2 (w) + log(N + 1) t ∈Z sup ∈L |K ε U , P υ t f | 2 1 2 L 2 (w) (5.6)
for any υ ∈ {1, 2, 3}. We make the choice υ = υ(U , ε, Q) ∈ {1, 2, 3} according to Lemma 3.3, so that based on supp
f ⊂ Q K ε U , (P υ t f )(x) M 2 s (P υ t f )(x)
. Combining the last two inequalities followed by weighted Fe erman-Stein and Littlewood-Paley estimates
sup ∈L K ε U , f L 2 (w) [w] γ A * 2 f L 2 (w) + log(N + 1) t ∈Z M 2 s (P υ t f ) 2 1 2 L 2 (w) [w] γ A * 2 log(N + 1) f L 2 (w)
which is the claimed (5.5).
6.
In this section, we show that sharp higher dimensional (n ≥ 4) analogues of Theorem 1.1 cannot be attacked by means of the model operators of Section 3, which are essentially compositions of smooth two-dimensional lacunary cuto s. To wit, we show that the maximal operators
sup ∈L σ ∈Σ Id − K ε σ σ , σ f , sup ∈L K ε U , f ,
intervening in the decomposition of the maximal Hilbert transform induced by Lemma 3.2, have operator norms which grow at order (log #L) 1 2 n 2 . For n ≥ 4, this is unfavorable compared to the maximal Hilbert transform over nite subsets O of a ( nite order) lacunary set Ω, whose operator norm is of order at most log(#O); see [28,Corollary 4.1]. Our counterexamples are obtained by careful tensoring of the lower bound for the two-dimensional case Σ = {(1, 2)} which in turn descends from the main theorem of [19].
We use the notation of Section 3 and in particular of (3.3). However in this section it will be more convenient to use the equivalent (up to identity) de nition
H ω f (x) ∫ R n f (ξ )1 (0,∞) (ξ · ω)e ix ·ξ dξ .
6.1. A lower bound in n = 2. The lower bound for p = 2 of Karagulyan [19] combined with the upper bound for all 1 < p < ∞ of [9,11] tells us that for all L ≥ 0 and 1 < p < ∞ there exists c p,L > 0 such that the following holds: Whenever Ω ⊂ S 1 is a lacunary set of order L and O ⊂ Ω is nite there exists a Schwartz function f O with
(6.1) f O L p (R 2 ) = 1, H O f O L p (R 2 ) ≥ c p,L log #O.
Let now Ω be a lacunary set of order 1 with Ω ⊂ {ω ∈ S 1 : ω 1 , ω 2 > 0}. We can take f O to be frequency supported in the quadrants {ξ ∈ R 2 : ξ 1 ξ 2 < 0} as H O acts trivially on the remaining frequency plane. By a symmetry argument we can actually take
supp f O ⊂ Q (1,2) ξ ∈ R 2 : ξ 1 > 0, ξ 2 < 0 .
Rewriting (3.7) in this particular case we see that if ω ∈ Ω ∩ S (1,2), (ω) and supp f ⊂ Q (1,2) then ≤ C p f O L p (R 2 ) = C p ; this L p -boundedness is most easily seen by proving the weighted L 2 -bound as in Section 5 rst. Comparing this last display with (6.1) we obtain that sup ω∈O K + (1,2), (ω) f O L p (R 2 ) ≥ c p log #O provided #O is large enough, with c p = c p,1 /2. The arguments of Section 3 and symmetry considerations nally show that there exist positive absolute constants c p , C p such that for ε ∈ {+, •, −} and all nite index sets L ⊂ Z we have
H ω f (ξ ) = F H ω K • (1,2), (ω) f (ξ ) + 1 (0,∞) (ξ · ω) 1 − κ • (1,2), (ω) (ξ ) f (ξ ) = F H ω K • (1,2), (ω) f (ξ ) + 1 − κ + (1,2), (ω) (ξ ) f (ξ(6.2) c p ≤ 1 log #L f → sup ∈L K ε (1,2), f L p (R 2 ) ≤ C p .
Remark 6.2. Just like the maximal Hilbert transform, the maximal operators de ned in (6.2) are invariant under dilation and re ection through the frequency origin, and act trivially on functions supported outside ±Q (1,2) . For any xed L ⊂ Z with #L = N , using the lower bound in (6.2), the re ection symmetry and an approximation argument we may nd M > 0 and a Schwartz function f L with where A (1,2) (a, b) (ξ 1 , ξ 2 ) ∈ R 2 : a < ξ 2 1 + ξ 2 2 < b . Given any s ∈ R, the dilation invariance can then be used to nd f s,L with the same properties as f L in (6.3) but supp f s,L ⊂ Q (1,2) ∩ A (1,2) (2 s , 2 s+M ) .
The next result is the anticipated counterexample to estimate (5.5) in dimensions 4 and higher.
Theorem 6.3. Let n ≥ 2 be the dimension of the ambient space. Then (6.4) inf
ε∈{+,−} Σ sup L⊂Z Σ #L=N sup U ⊆Σ f → sup ∈L K ε U , f L p (R n )→L p (R n ) ≥ c p log N n 2 .
Proof. It su ces to prove the statement for even n = 2d and for N > 10d, say. By symmetry considerations we may argue in the case where ε = (+, . . . , +). Let L be the set of N d indices such that S ∩ Ω ∅, where Ω is the set of vectors on S 2d−1 obtained by normalizing the vectors (x 1 , . . . , x n ) with components
x 2k−1 = 2 −2kN , x 2k = 2 −2kN −m k , m k ∈ {1, .
. . , N }, k = 1, . . . , d.
In practice = (m 1 , . . . , m d ) ∈ L is completely determined by the 2d − 1 conditions Here Q (2k−1,2k) ξ ∈ R 2 : ξ 2k−1 > 0, ξ 2k < 0 . We now de ne
f (x) d k=1 f k (x 2k−1 , x 2k ).
The point of this choice is that if σ = (σ (1), σ (2)) is such that σ (1), σ (2) have the same parity then ξ σ (1) , ξ σ (2) have the same sign on the frequency support of f , so that Id − K + σ , σ f = f . Also, unless σ = (2k − 1, 2k) for some k = 1, . . . , d, there holds supp f ⊂ ξ ∈ R n : |ξ σ (1) | |ξ σ (2) | ≥ 2 3M , Id − K + σ , σ = Id on the cone |ξ σ (1) | > (2d + 1)2 − σ |ξ σ (2) |, which is a larger cone than the one where f is supported, as σ ≥ N in this case. Summarizing we may delete from the composition in (6.5) all the σ which are not of the form σ = (2k −1, 2k), and we have for all m = 1, . . . , N that 5.4 will thus lead to the estimate
sup ε∈{+,−} U sup ∈L |K ε U , f | L p (R n ) (log #L) s 2 f L p (R n ) ,
which, together with the previously made observation that s may be taken ≤ n/2 shows the sharpness of Theorem 6.3; in general the worst case is U = {(1, 2), (3, 4), ..., (2 n/2 − 1, 2 n/2 )}. We leave the details to the interested reader.
F 3. 1 .
1The Fourier support of the multipliers K • (1,2),σ , K − (1,2),σ , and K + (1,2),σ .
F 3. 2 .
2Suppose ω belongs to the cell S . The red line is the intersection with the sphere S 2 of the singularity ξ ·ω = 0 of H ω . The blue and yellow wedges are respectively Ψ (1,2),(1,2) and Ψ (2,3),(2,3) from (3.2). As in the depicted octant ξ 1 and ξ 3 have the same sign, Ψ (1,3),(1,3) is not visualized.
Lemma 3. 2 .
2Suppose ω ∈ S , the cell of S n−1 with lacunary parameters = { σ : σ ∈ Σ}. Then we have the pointwise bound
Lemma 4 . 4 .
44Let Ω ⊂ S n−1 be a set of directions such that for all 1 < p < ∞ and for all w ∈ A Ω p we have the weighted boundedness property M Ω : L p (w) → L p (w). Assume that for some family of pairs of nonnegative functions, (f , ), for some p 0 ∈ [1, ∞] and for all w ∈ A p
Remark 4. 7 .
7We note here that in dimension n = 2 and for L = 1 this theorem was known and contained in[13, Theorem 4]. 4.8. Proof of Theorem 4.6. Recall from §2 that a set Ω ⊂ S n−1 is called lacunary of order L, where L ≥ 1 is a positive integer, if there exists a dissection as in (2
5. 3 .
3Proof of Theorem 5.1. In this proof the implicit constants occurring in the inequalities as well as the exponent γ are meant to be absolute and are allowed to vary without explicit mention. Let Ω ⊂ S 2 and an ONB B ⊂ Ω be given. Fix a subset O ⊂ Ω with #O = N . Of course the set of addresses of the cells whose intersection with O is nonempty, in symbols L O { ∈ Z Σ : O ∩ S ∅}, has cardinality at most N . We use the pointwise estimate of Lemma 3.2 for each ω ∈ O to obtain that
(6. 3 )
3supp f L ⊂ Q (1,2) ∩ A (1,2) 2 − M 2 , 2 M 2 , sup ∈L |K ε (1,2), f L | L p (R 2 ) ≥ C p log N f L L p (R 2 ) ,
(2k− 1 ,
12k) = m k , k = 1, . . . , d; (2k−1,2k+1) = 2N , k = 1, . . . ,Id − K + σ , σ f L p (R n )→L p (R n ) ≥ c p log N d ,where product denotes composition. Now for each k = 1, . . . , d de ne the function of two variables f k = f k (x 2k−1 , x 2k ) given by f s,L in Remark 6.2 with the pair (2k − 1, 2k) in the place of (1, 2), with L = {1, . . . , N }, and with s chosen so that supp f k ⊂ Q (2k−1,2k) ∩ A (2k−1,2k) (2 −3kM , 2 −(3k−1)M ).
Theorem 3.1], making use of Lemma 4.3 as the analogous of [12, Lemmata 2.1, 2.2].
The third summand in(5.3) is treated in the next Proposition. In fact, coupling the bounds (5.4) above, and (5.5) below, with the pointwise estimate (5.3), and noticing that [w] A * 2 ≤ [w] A Ω since the coordinate basis vectors are contained in Ω, completes the proof of Theorem 5.1. Proposition 5.4. Let L be a nite subset of Z 3 . Then Proof. Fix U ⊆ Σ, ε ∈ {+, −} U throughout the proof. By means of compositions of Hilbert transforms along the coordinate directions we may decompose2
(5.5)
sup
U ⊆Σ
sup
ε∈{+,−} U
sup
∈L
K ε
U , f
L 2 (w)
[w]
γ
A *
2
log(#L + 1) f L 2 (w) .
). We notice that, for some absolute constant C pω∈Ω
|H ω K •
(1,2), (ω) f O | 2
1
2
L p (R 2 )
USA E-mail address: [email protected] E-mail address: [email protected]
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V
, B 400137, C
, VA 22904, D
M
, U
P V
, A
. 644, 48080 B
, S
I
, B
F
S
, B
, S
. An application of the bound (6.6) as in Proposition
Acknowledgments. The authors are deeply grateful to Sara Maloni for fruitful discussionsWe now give the conclusion of the proof of Theorem 4.6.Conclusion of the proof of Theorem 4.6. The key step is the estimatefor some exponent δ > 0 depending on the dimension n and on p. Indeed, if L = 1, each sector Ω σ , contains at most one direction, whence using the well-known weighted maximal inequality for each such direction and the obvious inequalityCoupling the latter display with (4.1) yields the claimed estimate in Theorem 4.6. We now proceed by induction and derive the L-lacunary case assuming the L − 1 holds true. Estimate (4.1) and the inductive assumption readwhere in the last inequality we have used the obvious fact that sup σ , [w]This completes the proof of the theorem up to showing estimate (4.1) holds true.with the caveat that the product sign on the left hand side denotes composition while the product sign on the right hand side denotes pointwise product. Therefore using (6.3) for the lower bound in the third lineThis proves (6.5) and thus completes the proof of the theorem. with k odd. In the case that U has odd cycles, in each given quadrant of R n at least one of the multipliers K ε σ , is trivial for both ε = ±; we can thus reduce to the case that U has no odd cycles. This case is treated below.Suppose that {υ 1 , . . . , υ s } ⊂ {1, . . . , n} are such that for all σ ∈ U there exists j such that υ j ∈ σ : in this case {υ 1 , . . . , υ s } is called spanning set of U . Notice that for every U ⊂ Σ we may nd a spanning set with s ≤ n/2 . Arguing in similar fashion as in the proof of Lemma 3.3 we may obtain the pointwise estimatex ∈ R n , uniformly over all t 1 , . . . , t s ∈ R. Now, we may use an s-parametric version of the Chang-Wilson-Wol inequality to reduce estimates for the maximal operator associated to the multipliers K ε U , over ∈ L to an s-fold Littlewood-Paley square function estimate involving the left hand side of (6.6) with a loss of (log #L) s
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[
"Search for CP violating top quark couplings in pp collisions at √ s = 13 TeV",
"Search for CP violating top quark couplings in pp collisions at √ s = 13 TeV"
]
| [
"\nThe CMS Collaboration\nEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)\n\n"
]
| [
"The CMS Collaboration\nEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)\n"
]
| []
| Results are presented from a search for CP violation in top quark pair production, using proton-proton collisions at a center-of-mass energy of 13 TeV. The data used for this analysis consist of final states with two charged leptons collected by the CMS experiment, and correspond to an integrated luminosity of 35.9 fb −1 . The search uses two observables, O 1 and O 3 , which are Lorentz scalars. The observable O 1 is constructed from the four-momenta of the charged leptons and the reconstructed top quarks, while O 3 consists of the four-momenta of the charged leptons and the b quarks originating from the top quarks. Asymmetries in these observables are sensitive to CP violation, and their measurement is used to determine the chromoelectric dipole moment of the top quark. The results are consistent with the expectation from the standard model. Submitted to the Journal of High Energy Physics © 2022 CERN for the benefit of the CMS Collaboration. CC-BY-4.0 license arXiv:2205.07434v1 [hep-ex] 16 May 2022 | null | [
"https://arxiv.org/pdf/2205.07434v1.pdf"
]
| 248,811,631 | 2205.07434 | 2e5b1f8f56595f4acc4e5b6a84283bd02e400e58 |
Search for CP violating top quark couplings in pp collisions at √ s = 13 TeV
2022/05/17
The CMS Collaboration
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
Search for CP violating top quark couplings in pp collisions at √ s = 13 TeV
2022/05/17
Results are presented from a search for CP violation in top quark pair production, using proton-proton collisions at a center-of-mass energy of 13 TeV. The data used for this analysis consist of final states with two charged leptons collected by the CMS experiment, and correspond to an integrated luminosity of 35.9 fb −1 . The search uses two observables, O 1 and O 3 , which are Lorentz scalars. The observable O 1 is constructed from the four-momenta of the charged leptons and the reconstructed top quarks, while O 3 consists of the four-momenta of the charged leptons and the b quarks originating from the top quarks. Asymmetries in these observables are sensitive to CP violation, and their measurement is used to determine the chromoelectric dipole moment of the top quark. The results are consistent with the expectation from the standard model. Submitted to the Journal of High Energy Physics © 2022 CERN for the benefit of the CMS Collaboration. CC-BY-4.0 license arXiv:2205.07434v1 [hep-ex] 16 May 2022
Introduction
The combination of charge conjugation and parity transformation (CP) is a symmetry known to be violated in the standard model (SM). CP violation was initially observed in kaon decays [1]. Many experiments have studied CP violation in other processes for the last three decades [2][3][4]. However, the magnitude of CP violation observed so far is not enough to accommodate the matter-antimatter asymmetry in the universe [5,6], motivating searches for additional CP violation.
It has been proposed that CP violation also takes place in the production and decay of top quark pairs (tt) [7,8]. Several models predict CP violations, including the multi-Higgs-doublet model and the minimal supersymmetric SM; in the 1st model, for example, the exchanges of neutral and charged Higgs bosons can generate dipole moments from the top quark. In the SM, only a very small amount of CP violation is expected to occur at the production vertex, as this requires at least three loops [7,8]. If a non-zero chromoelectric dipole moment (CEDM) is observed at the tt production vertex and the top quarks decay into Wb following the SM process, this can be a signal of CP violation in top quark pair events.
The flavor-diagonal dipole couplings between top quarks and gluons of magnetic and electric origin would be defined through a Lagrangian density
L = g S 2 t T a σ µν (a g t + iγ 5 √ 2v Λ 2 Im(d tG ))tG a µν ,(1)
where Im(d tG ) is the CP-odd CEDM, and a g t , g S , T a , and G µν represent the chromomagnetic dipole moment, the strong coupling (g S = √ 4πα S ), Gell-Mann matrices, and the field strength tensor, respectively. The scale Λ is the scale of the new physics responsible for the CP violation of CEDM, and v ≈ 246 GeV is the vacuum expectation value of the Higgs field. The discussion in Ref. [8] introduces various observables that probe such CP violation effects.
The studies presented in this analysis define two observables with CP-odd correlation in tt events in the dilepton final state, where both W bosons decay leptonically:
O 1 = (p t , p t , p + , p − ) = E t p t ,x p t ,y p t ,z E t p t ,x p t ,y p t ,z E + p + ,x p + ,y p + ,z E − p − ,x p − ,y p − ,z ,(2)
and
O 3 = (p b , p b , p + , p − ) = E b p b ,x p b ,y p b ,z E b p b ,x p b ,y p b ,z E + p + ,x p + ,y p + ,z E − p − ,x p − ,y p − ,z .(3)
These observables are obtained contracting the Levi-Civita tensor [8] with four 4-momenta, which results in the determinant of a 4×4 matrix. The components of O 1 and O 3 are the fourmomenta p of the charged leptons ( ± ), the reconstructed top quarks and antiquarks, and the b quark and antiquark jets. The elements in the 4×4 matrices are therefore the energy E and three-momentum components p x , p y , p z of these objects. Since these observables are scalars under Lorentz transformations, they do not depend on the reference frame.
The observables O 1 and O 3 are odd under the CP transformation. Hence, CP violation in the production of tt can be tested by a measurement of the asymmetries A i of the number of produced events, N, with a positive and negative value in the observable O i :
A i = N(O i > 0) − N(O i < 0) N(O i > 0) + N(O i < 0) .(4)
As argued in Ref. [8], the asymmetries of O 1 and O 3 are the observables with highest sensitivity to CP violation, and are linearly correlated to d tG . While the lack of a sizable asymmetry in tt production and decay would be consistent with the SM, a significant nonzero asymmetry would indicate the existence of CP violation and hence, new physics [9,10]. In certain models predicting a CEDM (see Ref. [8]), the asymmetries in O 1 and O 3 can be as large as 15 and 9%, respectively. Asymmetries have been measured by the CMS Collaboration at √ s = 8 TeV in the lepton+jets channel [11]. In this paper the asymmetries of the observables O 1 and O 3 measured in the dilepton channel at 13 TeV, as well as the Im(d tG ) derived from those asymmetries, are presented. The tabulated results are provided in HEPData [12].
The CMS Detector
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity (η) coverage provided by the barrel and endcap detectors. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid. Events of interest are selected using a two-tiered trigger system [13]. The first level, composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events at a rate of around 100 kHz within a fixed time interval of less than 4 µs. The second level, known as the high-level trigger, consists of a farm of processors running a version of the full event reconstruction software optimized for fast processing, and reduces the event rate to around 1 kHz before data storage. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [14].
Simulated events
The tt event candidates in the dilepton final state are selected and compared with simulated events to investigate the composition of processes in the data.
Simulated tt events are generated at next-to-leading order (NLO) using the POWHEG generator (v. 2) [15][16][17][18][19][20][21] at a top quark mass (m t ) of 172.5 GeV. The events are simulated with PYTHIA (v. 8.219) [22], referred to as PYTHIA 8 in the following, to simulate parton showers (PS) and hadronization using the underlying-event (UE) tune CUETP8M2 [23]. The NNPDF3.0 parton distribution function (PDF) set is used to describe the proton structure. In this analysis, tt events with two oppositely charged leptons (e + e − , e ± µ ∓ , and µ + µ − ) originating from W boson decays are considered as signal ("tt signal"). All other tt events (tt originating from leptonic decays of τ leptons, all-jet final states, or lepton+jet events) are regarded as a background (referred to as "tt other"). The impact of not including the events where charged leptons originate from τ decays as "tt signal" is small since the information contained in the lepton from a τ decay is diluted relative to the τ lepton.
Drell-Yan (qq → Z * /γ → + − , referred to as DY), W+jets, WW, WZ, ZZ, tt+W, tt+Z, and the tW channel of single top quark production regarded as the processes that can have similar decay topologies to tt in the dilepton final state. The DY process is simulated with up to four additional partons at leading-order (LO) precision using the MADGRAPH5 aMC@NLO event generator (v. 2.2.2) with the MLM jet merging prescription [24] and the CUETP8M1 tune [25]. The tt+V (where V refers to a W or Z) processes are simulated using MADGRAPH5 aMC@NLO, with PYTHIA 8 for parton showering and hadronization, and the CUETP8M1 tune. Events from WW, WZ, and ZZ diboson processes are generated at LO using PYTHIA 8 and the CUETP8M1 tune. The tW channel is simulated with POWHEG (v. 1) interfaced to PYTHIA 8 using the UE tune CUETP8M2T4. The W+jets events are generated at NLO using MADGRAPH5 aMC@NLO and interfaced to PYTHIA 8 with the UE tune CUETP8M1.
The interactions of particles within the CMS detector are modeled using the detector simulation GEANT4 [26].
To model the effect of additional pp interactions within the same or nearby bunch crossing (pileup), simulated minimum-bias interactions are inserted into the simulated events. The distribution of pileup in simulation is matched to the observed data by reweighting the simulated pileup distributions, assuming a total inelastic pp cross section of 69.2 mb [27].
The simulations are normalized to the theoretical cross sections and the integrated luminosity of the data. The cross sections for the DY and W+jets processes are normalized to their next-to-NLO (NNLO) prediction obtained from the FEWZ framework (v. 3.1.b2) [28]. Single top quark production is normalized to the approximate NNLO prediction [29], and diboson and tt+V processes to the NLO prediction [30][31][32]. The tt cross section, computed at NNLO with a next-to-next-to-leading logarithmic (NNLL) accuracy for soft gluon resummation, is 832 +20 −29 (scale) ± 35 (PDF+α S ) pb, where the scale uncertainty comes from varying the normalization and factorization scales and the PDF+α S uncertainly is obtained from different PDF sets and varying α S [33].
Event reconstruction and selection
The data used for this analysis are from proton-proton (pp) collisions collected at √ s = 13 TeV in the CMS detector, and correspond to an integrated luminosity of 35.9 fb −1 . The event selection is optimized to select tt events in which both top quarks decay to a leptonically decaying W boson.
The events are further selected and categorized into channels by lepton flavor (e + e − , e ± µ ∓ , and µ + µ − ). The selected events are required to pass a combination of single-lepton and dilepton triggers. In each event, at least one of the following triggers must be satisfied. The singlelepton trigger requires at least one isolated electron or muon with a transverse momentum p T > 27 or 24 GeV and with |η| < 2.4, respectively. The same-flavor dilepton triggers require a leading-p T lepton with p T > 23 or 17 GeV, and a subleading-p T lepton with p T > 12 or 8 GeV. The different-flavor dilepton triggers require either an electron with p T > 12 GeV and a muon with p T > 23 GeV, or an electron with p T > 23 GeV and a muon with p T > 8 GeV.
The selected events are reconstructed offline using a particle-flow (PF) algorithm [34] that aims to reconstruct and identify each individual particle (PF candidate) in an event, with an optimized combination of information from the various elements of the CMS detector.
Electron candidates are reconstructed using tracking and ECAL information [35] and are excluded when their ECAL clusters are located in the transition region between the barrel and endcap detectors (1.4442 < |η| < 1.5660) because the reconstruction efficiency in this region is reduced. A relative isolation criterion I rel < 0.0588 (0.0571) is applied for an electron candi-date in the barrel (endcap), where the I rel is defined as the p T sum of all neutral and, charged hadron, and photon candidates within a distance ∆R = √ (∆η) 2 + (∆φ) 2 = 0.3 (where φ is the azimuthal angle in radians) from the electron candidate in η-φ space, divided by the p T of the electron candidate, with a correction that suppresses the residual effect of pileup collisions [36]. In addition, electron identification requirements are applied to reject misidentified electron candidates and candidates originating from photon conversions. The electron candidates must have p T > 25 and 20 GeV for the leading and subleading candidate, respectively, within |η| < 2.4.
Muon candidates are reconstructed from the track information in the tracker and muon system [37]. They must be in the same p T and |η| ranges as the electron candidates above.
A relative isolation requirement of I rel < 0.15 is applied to muon candidates and lie within ∆R = 0.4 of the muon in η-φ space. In addition, muon identification requirements are used to reject muon candidates that are misidentified hadrons or come from the decay of charged pions or kaons [37].
Jets are reconstructed by clustering the PF candidates using the anti-k T algorithm with a distance parameter R = 0.4 [38,39]. The momentum of the jet is defined as the vector sum of all particle momenta in the jet cone, and is found from simulation to be within 5 to 10% of the true momentum over the entire p T range of interest and detector acceptance. Since pileup collisions contribute calorimetric energy depositions and extra tracks, tracks identified as originating from pileup vertices are discarded [34]. In addition, an offset correction is applied to jet energies in order to remove the residual energy contribution from pileup [40]. The energy scale corrections of jets are obtained from simulation and used to bring the mean measured response to that of jets at particle level. In situ measurements of the momentum imbalance in dijet, pho-ton+jets, Z+jets, and multijet events are used to account for any residual differences between jet energy in data and simulation [40]. Jets are required to have p T > 30 GeV and |η| < 2.4. If the separation ∆R between the jet and the closest lepton is <0.4, the jet is discarded.
Jets originating from the fragmentation and hadronization of b quarks are identified ("b tagged") using the combined secondary vertex algorithm that uses information from track impact parameters and secondary vertices identified within a given jet. The chosen working point provides a b jet identification efficiency of approximately 63% with a probability to misidentify light-flavor jets as b jets of ≈1% in tt events [41].
The missing transverse momentum vector p miss T is defined as the projection on the plane perpendicular to the beam axis of the negative vector momenta sum of all reconstructed PF candidates in an event. Its magnitude is referred to as p miss T . The selected events are required to have two oppositely charged isolated leptons (e + e − , e ± µ ∓ , or µ + µ − ) and at least two jets. At least one of the jets is required to be b tagged. Events with additional loosely isolated electron (muon) of p T > 20 GeV are vetoed. An electron is considered loosely isolated if I rel < 0.175 (barrel) or 0.159 (endcaps). A muon is loosely isolated with I rel < 0.25. The dilepton invariant mass must be greater than 20 GeV to suppress contributions from heavy-flavor resonance decays and low-mass DY events. In same-flavor lepton channels, the backgrounds from Z+jets are further suppressed by rejecting events with a dilepton invariant mass within 76 and 106 GeV (Z boson mass window) and requiring p miss T > 40 GeV. To estimate the remaining background contribution from Z+jets events in the same-flavor lepton channel, the events outside the Z mass window are normalized to the event yield in simulation within the Z boson mass window. A contamination from non-DY background can be present in the Z boson mass window, and this is taken into account in the estimation in the e + e − and µ + µ − channels. These are estimated using the e ± µ ∓ events, which are rescaled, and subtracted in the e + e − and µ + µ − channels. Other sources of background, such as single top quark production, diboson, tt+V, tt other, misidentified leptons, and leptons within jets are estimated using simulations.
Since the observables in Eq. (2) contain the four-momenta of the top quark and antiquark or of the b quark and antiquark, a kinematic event reconstruction [42,43] is necessary. The algorithm takes into account all combinations of jets and leptons and solves a system of equations based on the following constraints: p miss T is assumed to originate solely from the two neutrinos; the invariant mass of the reconstructed W boson (m W ) is constrained to 80.4 GeV [44]; and the invariant mass of each reconstructed top quark is constrained to 172.5 GeV. Effects of detector resolution are taken into account through a random smearing of the measured energies and directions of the reconstructed lepton and b jet candidates according to their simulated resolutions. In addition, the assumed invariant mass of the W boson is varied according to the simulated Breit-Wigner distribution of W boson masses [44]. For a given smearing, we choose the solution of the equations for the neutrino momenta yielding the smallest invariant mass of the tt system. For each solution, a weight is calculated based on the spectrum of the true invariant mass of the lepton and b jet system from top quark decays at the particle level [43]. The weights are summed over 100 smearings for each combination, and the top quark and antiquark four-momenta are calculated as the weighted average. Among the combinations, we choose the assignment of jets and leptons that yields the maximum sum of weights. The efficiency of the kinematic reconstruction is defined as the fraction of the selected tt events where a solution is found, and is about 90% in both data and simulation. Events without solutions for the neutrino momenta are excluded from further analysis. Using this method, the fourmomenta of the top quark and antiquark are reconstructed, as are those of the b and b quark jets. Event yields for the individual dilepton channels after event selection are shown in Table 1 for the prediction of the tt signal, various background processes, and the observed data. In Fig. 1, distributions in the observables of the selected leptons and jets are shown for the e ± µ ∓ channel. The events in these distributions satisfy all event selection criteria. predicted distributions correspond to the total uncertainty, summing in quadrature the effect of the systematic uncertainties discussed in Section 5.2, and the statistical uncertainties in each bin.
Results
Extraction of asymmetries
The distributions in Figs. 3 and 4 are split into positive and negative regions based on the algebraic sign of the observable. The corresponding Poisson probability density functions for the observed and predicted numbers of events is used to construct a likelihood function that essentially reflects Eq. (4). Using a maximum likelihood fit, the asymmetry A i of the observable O i and the tt production cross section, σ tt , are extracted simultaneously with their statistical uncertainties. The maximum likelihood function is defined as
L(A i , σ tt ) = P (N obs + , N pred + ) P (N obs − , N pred − ),(5)
where the variables N obs ± and N pred ± are are the respective numbers of observed and predicted events in the positive and negative regions of O i , and P (N obs ± , N pred ± ) denotes their Poisson probability density functions. The predicted number of events N pred ± is assumed to be a function of the two fitted parameters, A i and σ tt :
N pred ± = N tt 1 ± A i 2 + N bkg f bkg ± ,(6)
where N tt = LBε sig σ tt , with L the integrated luminosity, B the dileptonic branching fraction, and ε sig the signal efficiency. The first term in Eq. Table 1. Finally, the CP asymmetry and σ tt are extracted simultaneously by minimizing the negative log-likelihood function
− log L(A i , σ tt ) ∝ (−N obs + log N pred + + N pred + − N obs − log N pred − + N pred − ).(7)
The resulting asymmetries with their statistical uncertainties are shown in Table 2, and the extracted cross section is consistent with previous CMS results [43]. The asymmetries of O 1 and O 3 are statistically correlated, and the correlation factor is determined from pseudo-experiments to be 46%. The asymmetries measured in the three dilepton channels are combined using the best linear unbiased estimator method [45], taking into account the correlation of the systematic uncertainties across channels. A more detailed description of the combination can be found in Section 5.2.
Systematic uncertainties
Systematic uncertainties may affect the asymmetries A i and the CEDM values. The effect of each systematic uncertainty is estimated by shifting the nominal prediction by the uncertainty and repeating the asymmetry measurement. The uncertainty is taken as the average of the absolute value of the shifts of asymmetries in the up and down directions. This allows the reduction in the statistical fluctuations in the variations. The total systematic uncertainty is calculated by adding the effect of the individual variations in quadrature. In this section, we discuss each of the sources assessed to be relevant in the analysis.
The uncertainty in the integrated luminosity determination is 2.5% [46,47]. The uncertainty from the modeling of the number of pileup interactions is obtained by changing the inelastic pp cross section assumed in the simulation by ±4.6%, consistent with the cross section uncertainty presented in Ref. [27].
The efficiencies of the triggers in data are measured as the fraction of events passing alternative triggers based on a p miss T requirement that also satisfies the criteria of the trigger of interest [43,48]. As the efficiency of the p miss T requirement is only weakly correlated with the dilepton trigger efficiencies, the bias introduced by the p miss T requirement is negligible. The efficiencies are close to unity in both data and simulation, as are the corresponding data-tosimulation scale factors. These scale factors are changed within their uncertainties to take into account the corresponding uncertainties in the efficiency. The scale factors for the lepton identification and isolation efficiencies are determined using a tag-and-probe method [49,50], and the uncertainties are estimated by varying the scale factors by their uncertainties.
The calibration of the electron and muon momentum scales are varied within their uncertainties, and separately for each lepton [35,37].
The uncertainty arising from b tagging is estimated by varying the measured b tagging scale factors within one standard deviation, depending on the p T and η of the b jets [41]. The b tagging uncertainties for heavy-flavor (b and c) and light-flavor (u, d, s, and gluon) jets are calculated separately, and combined in quadrature to provide the total b tagging uncertainty.
The uncertainty arising from the jet energy scale (JES) is determined by changing the individual sources of uncertainty in the JES in bins of jet p T and η, and taking their sums in quadrature [40]. These changes are then propagated to the uncertainties in p miss T . An additional uncertainty in the calculation of p miss T is estimated by varying the energies of reconstructed particles not clustered into jets.
The uncertainty originating from the jet energy resolution (JER) is determined by changing the JER in simulation within its uncertainty within different η regions [40].
The simulated background samples were generated with a limited number of events. Therefore, the total number of background events is affected by statistical fluctuations in these samples. The fluctuations in asymmetries originating from such limitations are evaluated through nuisance parameters in the likelihood function associated with the numbers of the background events in the negative and positive regions of the observables. The likelihood function in Eq. (5) can be modified as:
L(A i , σ tt ) = P (N obs + , N pred + ) P (N obs − , N pred − ) G(N bkg − |µ bkg − , σ bkg − ) G(N bkg + |µ bkg + , σ bkg + ),(8)
where µ bkg ± and σ bkg ± are the mean and the statistical uncertainty of the expected background processes. Gaussian constraints are imposed on the nuisance parameters.
The uncertainty in the normalization of the expected background processes other than tt is estimated through scaling the background yield in simulation up and down by 30% [50], and then extracting the asymmetries.
The uncertainty in the PDFs used to simulate the tt production is obtained using the NNPDF3.0 PDFs [51]. The impact of the uncertainty in the renormalization and factorization scale in the tt simulation is estimated by varying the factorization and renormalization scales used during the generation of the simulated sample independently by factors of 0.5 and 2. The extreme cases where one scale is varied up, while the other one is varied down, are not considered.
The dependence of the asymmetries and of the CEDM on the assumed value of m t is estimated by varying the generated m t in the simulation by ±1 GeV with respect to the default value of 172.5 GeV.
Previous studies have shown that the p T distribution of the top quark in data is softer than expected in the NLO simulation of tt production [42,43,[52][53][54]. A reweighting procedure based on the p T spectrum of the top quark is applied to the nominal POWHEG prediction at NLO on an event-by-event basis so as to match the simulated spectrum of the top quark p T to data. The change in the result is taken as the systematic uncertainty.
The uncertainty in the b jet modeling has three components. The fragmentation into b hadrons is varied in simulation within the uncertainties of the Bowler-Lund fragmentation function tuned to ALEPH [55] and DELPHI [56] data. In addition, the difference between the Bowler-Lund [57] and the Peterson [58] fragmentation functions is included in the uncertainty. Lastly, the uncertainty from the semileptonic b hadron branching fraction is obtained by varying it by −0.45 and +0.77%, which is the range of the measurements from B 0 /PBp decays and their uncertainties [44].
The uncertainty in the matching scale between the matrix element (ME) and the parton shower is evaluated by varying the h damp parameter that regulates the emissions in POWHEG. The nominal value of h damp in the simulation is 1.58m t , and the modified values are 0.99m t and 2.24m t , obtained from tuning the parameter using tt data at √ s = 8 and 13 TeV [59].
The default setup in PYTHIA includes a multiple-parton interaction (MPI) scheme of the color reconnection (CR) model with "early" resonance decays switched off. To estimate the uncertainty from this choice of model, the analysis is repeated using three other CR models within PYTHIA: the MPI-based scheme with early resonance decays switched on, a gluon-move scheme [60], and a scheme [61] inspired by quantum chromodynamics (QCD). The total uncertainty from CR modeling is estimated by taking the maximum deviation from the nominal result.
The uncertainty from modeling of the underlying event is estimated by varying the parameters that govern the underlying event modelling of the PS tune CUETP8M2T4 in Pythia8 are simultaneously varied up and down within their uncertainties [23,59].
The renormalization scale for initial-and final-state gluon radiations (ISR and FSR) is varied up and down by a factor of 2 (for ISR) and √ 2 (for FSR, scaled to account for NLO compensation terms), to account for the uncertainties in the QCD scale in the parton shower description in the tt simulation [23].
The observables O 1 and O 3 in Eqs. (2) and (3) are related to Levi-Civita tensors contracted with the four-momenta of t, t, b and b quark jets, and two leptons. These four-momenta and the observables are affected by the uncertainties in detector measurements. A fluctuation in a measurement caused by detector response effects can change the sign of the observables in an event, and as a consequence dilute the asymmetries. We have considered the jet angular resolutions and the charge misidentification as additional systematic sources that can dilute the asymmetries.
Jet φ resolution: the limited tower size of the CMS calorimeter gives rise to an uncertainty in the measured jet position. A change in the position can change the sign of an observable, leading to a change in the asymmetry. The jet φ information is recalculated by adding or subtracting the φ angle corresponding to a change by one standard deviation based on the angular resolution. Using the modified four-momenta of the jets, the observables O 1 and O 3 are rederived. The O 1 and O 3 can be recalculated using the modified four momenta of the jets.
Charge misidentification: the tracks of charged particles at high momenta become almost straight lines in the 3.8 T magnetic field. Fluctuations in the measured hit positions along the track can change the sign of the sagitta and thereby change the sign of the charged track. In such cases, the lepton charge can be misidentified, which would result in a dilution of the asymmetry. We identify two ways of misidentifying lepton charge. In the first case, the charge of one lepton is assumed to be correctly identified while the charge of the other lepton is misidentified. Since the leptons would have the same charge, events would be rejected by the tt signal selection requirement of oppositely charged lepton pairs. The probability of the charge of one lepton being misidentified is calculated separately for the positive and negative region of the observables. Based on this probability, events are vetoed and the asymmetry measurements is repeated. In the second case, the charges of both leptons are assumed to be misidentified. In this case, the positions of the four-momenta of + and − are swapped in the Levi-Civita tensor. Consequently, the sign of the tensor changes because of the properties of the matrix determinant. The probability that both lepton charges are misidentified is calculated separately for the positive and negative region of the observables. Assuming the worst-case scenario in which events in one of the regions contain two leptons with misidentified charge and migrate into the other region, the number of events in the regions can be re-estimated. The changes in asymmetries in the individual channels are taken as uncertainty on the asymmetry arising from charge misidentification.
In the combination, the systematic uncertainties are assumed to be fully correlated across channels, except for the uncertainty in the limited number of simulated background events and in the normalization of the background models. Since these systematic sources are statistically independent, they can be taken as uncorrelated uncertainties. The statistical uncertainties are assumed to be uncorrelated.
The dominant contributions of systematic uncertainties and additional sources are summarized in Table 3. The uncertainty is listed only if it is greater or equal than 0.4 × 10 −3 for O 1 and O 3 in the e ∓ µ ± channels. The uncertainties in the integrated luminosity, pileup, lepton identification and isolation, trigger, b tagging, m t , top quark p T , the normalization of background models, and PDFs all have small contributions and are not listed.
Extraction of CEDM
As indicated in Eq. (1), the tt production vertex is modified by the presence of CEDMs, leading to CP violation. According to Ref. [8], the asymmetry defined in Eq. (4) is linearly proportional to the CEDM. Dedicated tt events with such CP modifications in Eq. (1) are generated using MADGRAPH5 aMC@NLO interfaced to PYTHIA 8 [22] using the UE tune CUETP8M2.
Seven samples are generated with different values of the dimensionless CEDM d tG parameter. Figure 5 shows the expected asymmetries as a function of Im(d tG ). Within one standard devi- Figure 5: Asymmetries as a function of Im(d tG ) for O 1 (left) and O 3 (right), for the combined dilepton channels. The inner and outer bands correspond to the uncertainties at the 68 and 95% confidence levels, respectively, of the linear fits. The square markers are the asymmetries measured using simulated samples corresponding to the different Im(d tG ) values. The horizontal line indicates the measured asymmetry, and the shaded region reflects the total statistical and systematic uncertainty.
Uncertainty (×10 −3 ) e + e − e ∓ µ ± µ + µ − Combined Source O 1 O 3 O 1 O 3 O 1 O 3 O 1 O 3
ation, the measured asymmetries are linearly proportional to the input CEDM values for both O 1 and O 3 . The relation between the asymmetry A and the CEDM can be written as A = a Im(d tG ) + b.
The parameters a and b are obtained from a least-squares fit to the values obtained from the simulated samples. From this relation, Im(d tG ) is extracted from the measured asymmetry as
Im(d tG ) = A measured − b a .(10)
The uncertainty in the measured dimensionless CEDM Im(d tG ) is calculated using the full covariance matrix from the fit as
∆ 2 Im(d t G ) = ∂Im(d tG ) ∂A ∂Im(d tG ) ∂b ∂Im(d tG ) ∂a ∆ 2 A 0 0 0 ∆ 2 b cov(b, a) 0 cov(a, b) ∆ 2 a ∂Im(d tG ) ∂A ∂Im(d tG ) ∂b ∂Im(d tG ) ∂a ,(11)
with ∆ Im(d t G ) being the uncertainty in Im(d tG ), and ∆ A , ∆ a , and ∆ b the uncertainties in A, a, and b. The measured asymmetries for the combined dilepton channel are used to extract the CEDM for O 1 and O 3 . The measured values for Im(d tG ) and CEDM with their statistical and systematic uncertainties are given in Table 4. 0.00 ± 0.13 (stat) ± 0.10 (syst)
The measured asymmetries and CEDMs are consistent with the expectation from the SM of a negligible CP asymmetry. The measurement of the tt spin correlation in the CMS experiment [62] also provided a CEDM value usingd t proposed in Ref. [63]. The valued t from the measured tt spin correlation is −0.020 <d t < 0.012 at the 95% confidence level. The parameter d t is related to Im(d tG ) asd t m t = √ 2v Λ 2 Im(d tG ).
At a new physics scale Λ = 1 TeV, using Eqs. (1) and (12), the measured Im(d tG ) values can be converted intod t , resulting in −0.014 <d t < 0.027 and −0.019 <d t < 0.019 at the 95% confidence level, extracted from the asymmetries of O 1 and O 3 , respectively. Hence, the sensitivity of this analysis to the CEDMd t value is similar to that of the tt spin correlation analysis.
Summary
Violations of CP symmetry are studied in top quark pair production in the dilepton final state. The analysis is based on proton-proton collisions at a center-of-mass energy of 13 TeV, collected by the CMS experiment and corresponding to an integrated luminosity of 35.9 fb −1 . The analysis uses two observables, O 1 and O 3 , which are related to the Levi-Civita tensor contracted with the four-momenta of the leptons, the jets originating from the b quarks, and the top quarks. Asymmetries are measured in these observables and converted to measurements of the chromoelectric dipole moment (CEDM) of the top quark, represented by the dimensionless CEDM Im(d tG ). In the SM prediction the size of CP violation and the CEDM is negligible. The measured Im(d tG ) based on the asymmetries of the O 1 and O 3 observables in the combined dilepton channels are 0.10 ± 0.12 (stat) ± 0.12 (syst), and 0.00 ± 0.13 (stat) ± 0.10 (syst), respectively. These results are consistent with the expectation of the standard model (SM), since in the SM prediction, the size of CP violation and the CEDM (Im(d tG )) is negligible. In this analysis, the extracted CEDMs are compared with the CEDM measured in the tt spin correlation analysis [62], and the sensitivity is found to be similar.
Figure 2 Figure 1 :Figure 2 :
212shows the p T distributions of top quark and antiquark reconstructed in the dilepton channel. For each selected event, O 1 and O 3 are computed from the kinematic information of the reconstructed top quarks, b quark jets, and leptons. Figures 3 and 4 present the comparison of data and prediction for the observables O 1 and O 3 . In these figures, the values of O 1 and O 3 are divided by the fourth power of m t to provide better visibility. In Figs. 1-4, the shaded bands in the The comparisons of the predictions and observed data in the kinematic distributions of the p T of the leading lepton (upper left), subleading lepton (upper right), leading jet (lower left), and subleading jet (lower right) in the e ± µ ∓ channels. The vertical bars on the markers of the observed data represent the statistical uncertainties. The shaded band in the predicted distributions includes statistical and systematic uncertainties. The last bin in each plot includes overflow events. The ratio of the data to the predictions from simulation is presented in the lower panel of each figure. The comparisons of the predictions and observed data in the p T distributions in the top quark (left) and antiquark (right) in the e + e − (upper), e ± µ ∓ (middle) and µ + µ − (lower) channels. The vertical bars on the markers of the observed data represent the statistical uncertainties. The shaded band in the predicted distributions includes statistical and systematic uncertainties. The last bin in each plot includes overflow events. The ratio of the data to the predictions from simulation is presented in the lower panel of each figure.
Figure 3 :
3The comparisons of the predictions and observed data in O 1 in the e + e − (upper left), e ± µ ∓ (upper right), and µ + µ − (lower) channel. The vertical bars on the markers of the observed data represent the statistical uncertainties. The shaded band in the predicted distributions includes statistical and systematic uncertainties. The first and last bins in each plot includes underflow and overflow events, respectively. The ratio of the data to the predictions from simulation is presented in the lower panel of each figure. The solid blue shows the ratio (Im(d tG ) = 2.6)/(Im(d tG ) = 0), and the dashed red line represents the ratio (Im(d tG ) = −2.6)/(Im(d tG ) = 0), using the CEDM samples.
Figure 4 :
4The comparisons of the predictions and observed data in O 3 in the e + e − (upper left), e ± µ ∓ (upper right), and µ + µ − (lower) channel. The vertical bars on the markers of the observed data represent the statistical uncertainties. The shaded band in the predicted distributions includes statistical and systematic uncertainties. The ratio of the data to the predictions from simulation is presented in the lower panel of each figure. The first and last bins in each plot includes underflow and overflow events, respectively. The solid blue line shows the ratio (Im(d tG ) = 2.6)/(Im(d tG ) = 0), and the dashed red line represents the ratio (Im(d tG ) = −2.6)/(Im(d tG ) = 0), using the CEDM samples.
( 6 )
6reflects the number of tt signal events, where N tt includes the positive and negative regions. In the second term, N bkg is the total number of events for all background processes combined, while f bkg + and f bkg − is the fraction of the number of background events in the positive (negative) regions of observable O i . Both f bkg and N bkg are fixed based on the background contributions in
Table 1 :
1Simulated event yields with their statistical uncertainties for the three dilepton channels, after implementing event selection criteria, and normalized as described in the text. Observed selected events are also shown.Process
e + e −
e ± µ ∓
µ + µ −
tt signal
22 216 ± 64 104 051 ± 140 45 818 ± 93
tt other
3 425 ± 25
16 787 ± 56
7 502 ± 38
Single top quark
899 ± 13
4 265 ± 28
1 793 ± 18
DY
700 ± 57
381 ± 26
1 627 ± 95
tt+V
72 ± 2
302 ± 4
144 ± 3
Diboson
37 ± 4
100 ± 7
70 ± 6
Total prediction 27 350 ± 90 125 878 ± 155 56 954 ± 140
Data
26 961
126 549
55 993
Table 2 :
2Measured asymmetries of O 1 and O 3 with statistical uncertainties.Asymmetry and uncertainty (×10 −3 )
Observable
e + e −
e ± µ ∓
µ + µ −
Combined
A O 1
8.8 ± 7.5 0.6 ± 3.4 6.9 ± 5.3 2.4 ± 2.8
A O 3
4.1 ± 7.5 −1.7 ± 3.4 6.1 ± 5.3 0.4 ± 2.8
Table 3 :
3Systematic uncertainties in the measured asymmetries of O 1 and O 3 , for the individual and combined channels.
Table 4 :
4The measured dimensionless CEDM Im(d tG ), extracted using the asymmetries in O 1 and O 3 , with their uncertainties.Observable
Im(d tG )
A O 1
0.10 ± 0.12 (stat) ± 0.12 (syst)
A O 3
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[
"Demographic stochasticity and resource autocorrelation control biological invasions in heterogeneous landscapes",
"Demographic stochasticity and resource autocorrelation control biological invasions in heterogeneous landscapes"
]
| [
"Andrea Giometto [email protected] \nSchool of Architecture\nCivil and Environmental Engineering\nLaboratory of Ecohydrology\nÉcole Poly-technique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n\nDepartment of Aquatic Ecology\nSwiss Federal Institute of Aquatic Science and Technology\nCH-8600Eawag, DübendorfSwitzerland\n",
"Florian Altermatt [email protected]. \nDepartment of Aquatic Ecology\nSwiss Federal Institute of Aquatic Science and Technology\nCH-8600Eawag, DübendorfSwitzerland\n\nDepartment of Evolutionary Biology and Environmental Studies\nUniversity of Zürich\nCH-8057ZürichSwitzerland\n",
"Andrea Rinaldo \nSchool of Architecture\nCivil and Environmental Engineering\nLaboratory of Ecohydrology\nÉcole Poly-technique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n\nDipartimento di Ingegneria Civile\nEdile ed Ambientale\nUniversità di Padova\nI-35131PaduaItaly\n"
]
| [
"School of Architecture\nCivil and Environmental Engineering\nLaboratory of Ecohydrology\nÉcole Poly-technique Fédérale de Lausanne\nCH-1015LausanneSwitzerland",
"Department of Aquatic Ecology\nSwiss Federal Institute of Aquatic Science and Technology\nCH-8600Eawag, DübendorfSwitzerland",
"Department of Aquatic Ecology\nSwiss Federal Institute of Aquatic Science and Technology\nCH-8600Eawag, DübendorfSwitzerland",
"Department of Evolutionary Biology and Environmental Studies\nUniversity of Zürich\nCH-8057ZürichSwitzerland",
"School of Architecture\nCivil and Environmental Engineering\nLaboratory of Ecohydrology\nÉcole Poly-technique Fédérale de Lausanne\nCH-1015LausanneSwitzerland",
"Dipartimento di Ingegneria Civile\nEdile ed Ambientale\nUniversità di Padova\nI-35131PaduaItaly"
]
| []
| Classical models of biological invasions assess species spread in homogeneous landscapes by assuming constant growth rates and random local movement. Mounting evidence suggests, however, that demographic stochasticity, environmental heterogeneity and non-random movement of individuals affect considerably the spread dynamics. Here, we show that the dynamics of biological invasions are controlled by the spatial heterogeneity of the resource distribution. We show theoretically that increasing the landscape resource autocorrelation length causes a reduction in the average speed of species spread. Demographic stochasticity plays a key role in the slowdown, which is streghtened when individuals can actively move towards resources. The reduction in the front propagation speed is verified in laboratory microcosm experiments with the flagellated protist Euglena gracilis by comparing spread in habitats characterized by different resource heterogeneity. Our theoretical and experimental findings highlight the need to account for the intrinsic stochasticity of population dynamics to describe spread in spatially extended landscapes, which are inevitably characterized by heterogeneous spatial distributions of resources controlling vital rates. Our work identifies the resource autocorrelation length as a key modulator and a simple measure of landscape susceptibility to biological invasions, with implications for predicting the characters of biological invasions within naturally heterogeneous environmental corridors. | 10.1101/043331 | [
"https://arxiv.org/pdf/1602.08700v1.pdf"
]
| 4,075,459 | 1602.08700 | c936517a35d6b66f18d408b5049f9e8b0f3461e3 |
Demographic stochasticity and resource autocorrelation control biological invasions in heterogeneous landscapes
28 Feb 2016
Andrea Giometto [email protected]
School of Architecture
Civil and Environmental Engineering
Laboratory of Ecohydrology
École Poly-technique Fédérale de Lausanne
CH-1015LausanneSwitzerland
Department of Aquatic Ecology
Swiss Federal Institute of Aquatic Science and Technology
CH-8600Eawag, DübendorfSwitzerland
Florian Altermatt [email protected].
Department of Aquatic Ecology
Swiss Federal Institute of Aquatic Science and Technology
CH-8600Eawag, DübendorfSwitzerland
Department of Evolutionary Biology and Environmental Studies
University of Zürich
CH-8057ZürichSwitzerland
Andrea Rinaldo
School of Architecture
Civil and Environmental Engineering
Laboratory of Ecohydrology
École Poly-technique Fédérale de Lausanne
CH-1015LausanneSwitzerland
Dipartimento di Ingegneria Civile
Edile ed Ambientale
Università di Padova
I-35131PaduaItaly
Demographic stochasticity and resource autocorrelation control biological invasions in heterogeneous landscapes
28 Feb 2016Environmental stochasticityBiological invasionsTraveling wavesFront propagationFisher- KolmogorovDispersal 1
Classical models of biological invasions assess species spread in homogeneous landscapes by assuming constant growth rates and random local movement. Mounting evidence suggests, however, that demographic stochasticity, environmental heterogeneity and non-random movement of individuals affect considerably the spread dynamics. Here, we show that the dynamics of biological invasions are controlled by the spatial heterogeneity of the resource distribution. We show theoretically that increasing the landscape resource autocorrelation length causes a reduction in the average speed of species spread. Demographic stochasticity plays a key role in the slowdown, which is streghtened when individuals can actively move towards resources. The reduction in the front propagation speed is verified in laboratory microcosm experiments with the flagellated protist Euglena gracilis by comparing spread in habitats characterized by different resource heterogeneity. Our theoretical and experimental findings highlight the need to account for the intrinsic stochasticity of population dynamics to describe spread in spatially extended landscapes, which are inevitably characterized by heterogeneous spatial distributions of resources controlling vital rates. Our work identifies the resource autocorrelation length as a key modulator and a simple measure of landscape susceptibility to biological invasions, with implications for predicting the characters of biological invasions within naturally heterogeneous environmental corridors.
Introduction
Environmental fluctuations and heterogeneity are ubiquitous in nature and are thought to affect nearly all aspects of ecology, ranging from species coexistence to population synchrony, driving range shifts and potentially causing abrupt biotic change (e.g., Nelson 2012;With 2002;With and Crist 1995). Local population dynamics in fluctuating and heterogeneous environments have been studied extensively in recent years (Duncan et al. 2013;Gonzalez and Holt 2002), mainly with respect to population synchrony (Benton et al. 2001;Fox et al. 2011;Vasseur and Fox 2009). Both theoretical (Roy et al. 2005;Vasseur 2007) and experimental (Fontaine and Gonzalez 2005;Gonzalez and Holt 2002;Massie et al. 2015) studies have highlighted the relevance of the temporal autocorrelation structure of environmental fluctuations for ecological dynamics. The study of ecological processes in the presence of environmental stochasticity at different levels of autocorrelation is of interest not only because environmental fluctuations are typically positively correlated (Benincà et al. 2011), but also in view of the global shift towards 'bluer' climate variables (i.e., more fluctuating) across most continents (e.g., García-Carreras and Reuman 2011). Whereas most experimental investigations focused on temporal environmental fluctuations, spatial heterogeneity received surprisingly little attention (Melbourne et al. 2007;With 2002). Accordingly, the study of the implications of environmental fluctuations for spatial dynamics (Duncan et al. 2013;Gonzalez and Holt 2002) and especially for the propagation of biological invasions (Méndez et al. 2011;Neubert et al. 2000;With 2002) is a challenging avenue for experimental research.
The effect of environmental fluctuations and spatial heterogeneity may be especially relevant in the context of biological invasions and range shifts, which are seen as some of the most relevant current dynamics across all ecosystems (Hastings et al. 2005). The spatial spread of invasions has been investigated extensively in the literature, starting with the pioneering works of Fisher, Kolmogorov and Skellam (Fisher 1937;Kolmogorov et al. 1937;Skellam 1951). Traditionally, the propagation of invasive fronts has been modeled with the Fisher-Kolmogorov equation (Fisher 1937;Kolmogorov et al. 1937) that predicts a linear rate of spread in homogeneous environments. Such equation was applied extensively to describe field data (Andow et al. 1990; Lubina and Levin 1988) and found applications also in other disciplines, for example physics and chemistry (Méndez et al. 2010). Comprehensive reviews of mathematical modeling and empirical studies of species spread exist (Hastings et al. 2005). Stochastic generalizations of the Fisher-Kolmogorov equation showed that demographic stochasticity affects the propagation dynamics, causing a reduction in the front propagation speed (Hallatschek and Korolev 2009). Other modeling approaches have shown that temporal fluctuations in mean dispersal distances can increase invasion speed, while temporally uncorrelated fluctuations in demographic parameters typically decrease the front propagation velocity (Ellner and Schreiber 2012;Méndez et al. 2011). Despite the fact that most natural environments are inevitably heterogeneous (e.g., Holyoak et al. 2005), however, much of the current understanding of species spread is based on theoretical models (Hastings et al. 2005;Méndez et al. 2010) that considered homogeneous landscapes. Only in recent years, progress has been achieved in the theoretical understanding of species spread in more complex, heterogeneous or fluctuating landscapes (Dewhirst and Lutscher 2009;Melbourne et al. 2007;Méndez et al. 2010;Nelson and Schnerb 1998;Neubert et al. 2000;Pachepsky and Levine 2011), and such progress calls for experimental verification (Seymour and Altermatt 2014). For example, thresholds for the minimal percentage of favorable habitat that can support spread have been studied (Dewhirst and Lutscher 2009;With 2002;With and Crist 1995). Dewhirst and Lutscher (2009), for example, derived quantitative relationships for the invasion threshold and spread rates in integro-differential equation models in fragmented landscapes. The speed of biological invasions has been claimed to be affected by environmental stochasticity (Méndez et al. 2011) and an extensive line of research addressed the contribution of geometrical heterogeneities of the landscape to the propagation of invading fronts (Bertuzzo et al. 2007;Campos et al. 2006;Méndez et al. 2004;Méndez et al. 2003;Seymour and Altermatt 2014) suggesting that, in general, geometrical heterogeneities slow the speed of front propagation.
Integrating environmental heterogeneity in models of spread is a challenging task and a modeling framework that allows drawing general conclusions is lacking to date (Hastings et al. 2005;Urban et al. 2008). In the search for such a framework, the study of biological invasions in heterogeneous and fluctuating environments has been addressed in the context of the Fisher-Kolmogorov equation (Fisher 1937;Kolmogorov et al. 1937) either by embedding various sources of environmental stochasticity in the original deterministic equation (Méndez et al. 2003(Méndez et al. , 2011Shigesada et al. 1986) or by considering spread in spatially heterogeneous media (Bertuzzo et al. 2007;Campos et al. 2006;Méndez et al. 2004;Méndez et al. 2003). Environmental stochasticity and spatial heterogeneity (Nelson 2012;Nelson and Schnerb 1998) have been incorporated in the Fisher-Kolmogorov equation through noise terms that were uncorrelated in space, periodic in space (Kinezaki et al. 2003;Shigesada et al. 1986) or else characterized by a gaussian spatial correlation function with a fixed correlation length (Méndez et al. 2011). Whereas the importance of the autocorrelation structure of temporal environmental fluctuations for local ecological processes is now widely recognized (Fontaine and Gonzalez 2005;García-Carreras and Reuman 2011;Gonzalez and Holt 2002;Vasseur 2007), the effect of the spatial autocorrelation of environmental fluctuations on biological spread rates has just begun to be explored (Urban et al. 2008). The experimental study of species spread has recently started to test theoretical predictions of the Fisher-Kolmogorov model in homogeneous habitats (Croze et al. 2011;Giometto et al. 2014;Korolev et al. 2011;Simpson et al. 2013). A limited number of empirical works has measured spread rates in heterogeneous and diverse habitats and compared realized spread distances in patchily distributed sites (Bailey et al. 2000;Bergelson et al. 1994;Williamson and Harrison 2002). However, the results of these studies were not linked to Fisher-Kolmogorov-like models embedding environmental stochasticity or heterogeneity. In particular, experimental studies investigating the role of the resource autocorrelation structure in driving the spread of species are lacking.
Here, we study biological invasions in the presence of spatially heterogeneous resource distributions, which could, for example, reflect the spatial composition and quality of soil or topographically determined habitat elements such as exposure or elevation, or habitat fragmentation due to human land-use (e.g., With 2002;With and Crist 1995). Motivated by previous research on environmental fluctuations mentioned above, we focus on the effect of the spatial autocorrelation structure of the resource distribution on the propagation speed of biological invasion fronts. The distribution of resources is assumed to affect both the growth dynamics and movement behavior of individuals. Giometto et al. (2014) showed that, in homogeneous landscapes, demographic stochasticity introduces a noise term in the reaction-diffusion equation describing the front propagation, leading to a quantifiable variability of the process across replicated experimental invasions. Therefore, our tenet is that both environmental and demographic stochasticity jointly affect biological invasions and thus the interplay between these two sources of stochasticity is specifically investigated here.
We first show theoretically that the speed of species spread decreases when the resource autocorrelation length increases, all other conditions being equal. Second, we verify such prediction in a microcosm experiment with the flagellated protist Euglena gracilis, by manipulating light intensity profiles along lin-ear landscapes (light is an energy source for E. gracilis, as it has chloroplasts and can photosynthesize). Third, we discuss the contribution of each process included in the model to the propagation of biological invasions. We show theoretically that demographic stochasticity is necessary to produce the slowdown, which is more pronounced if individuals can direct their movement towards resources.
Methods
Model
Species spread in heterogeneous linear landscapes is modeled via a stochastic generalization of the Fisher-Kolmogorov equation including demographic stochasticity (Bonachela et al. 2012;Dornic et al. 2005;Giometto et al. 2014):
∂ρ ∂t = D ∂ 2 ρ ∂x 2 + r(I)ρ 1 − ρ K + σ √ ρ η,(1)
where ρ(x, t) is the density of individuals, D is the diffusion coefficient of the species driven by the active movement of individuals, r is the growth rate, K is the carrying capacity, σ is a parameter describing the amplitude of demographic stochasticity and η is a gaussian, zero-mean white noise (i.e., the noise has
correlations η(x, t)η(x , t ) = δ(x − x )δ(t − t ),
where δ is the Dirac's delta function). Itô's stochastic calculus is adopted, as appropriate for the demographic noise term (Giometto et al. 2014). The growth rate r(I) = r 0 I is assumed to be a function of the local amount of resources I(x), which can assume two values: I(x) = 1 or I(x) = 0. Landscape heterogeneity is thus embedded in the resource profile I(x). We studied the dimensionless form of equation (1), which reads (see app. A available online):
∂ρ ∂t = ∂ 2 ρ ∂x 2 + χ I ρ 1 − ρ + σ ρ η,(2)
where t = r 0 t, x = D r 0 x, ρ = ρ/K, σ = σ √ K(rD) 1/4 and χ I (x ) is the indicator function of the set of x for which I(x ) = 1. In the following we drop primes for convenience: one can recover the original dimensions by multiplying t by r 0 , x by √ r 0 /D and rescaling ρ and σ as indicated above. Numerical integration of stochastic partial differential equations with square root noise terms require ad hoc numerical methods, as standard approaches such as the first-order explicit Euler method inevitably produce unphysical negative values for the density ρ (Dornic et al. 2005). Therefore, equation (2) was integrated with the split-step method proposed in Dornic et al. (2005), see app. A for details.
We generated landscapes with various resource autocorrelation lengths by imposing I(x) to be composed of subsequent independent patches of suitable (I(x) = 1 and r = r 0 ) or unsuitable (I(x) = 0 and r = 0) habitats ( fig. 1A). The length of each patch was drawn from an exponential distribution with rate µ. Therefore, each landscape was a stochastic realization of the so-called telegraph process with rate µ and autocorrelation length c L = 1/(2µ). The mean extent of suitable and unsuitable patches in such landscapes is 1/µ. Because simulated landscapes were finite, we only accepted landscapes with mean resources equal toĪ = L −1 L 0 I(x)dx = 1/2 and autocorrelation length confined to a narrow window around 1/(2µ). Examples of landscapes used in the simulations are shown in fig. 1A.
We generated 96 landscapes for each value of resource autocorrelation length c L and integrated equation (2) numerically for each landscape and for each value of σ ∈ {0.1, 0.2, 0.4, 0.6} ( fig. 1B), with initial density profiles localized at the origin. To avoid the extinction of the whole population, we fixed the left boundary at ρ = 1. For each numerical integration, we measured the position of the front by fixing a threshold value of the density (ρ = 0.15) and recording the furthest point from the origin where the cell density was higher than such value. The mean propagation speed for each value of the resource autocorrelation length was computed by fitting a straight line (least-squares fit) to the mean front position versus time in the asymptotic propagation regime ( fig. A6), before any of the replicated invasions reached the end of the landscape.
We derived a theoretical approximation to the mean front propagation speed, valid for large autocorrelation lengths c L and σ, by characterizing the mean time taken to cross a patch of unfavorable habitat (where I = r = 0) of length z. Such mean time is shown (app. A) to depend on z and σ as
τ (z, σ) = Cz 2 e d(zσ b ) a
, where C, a, b and d are constants, independent of z and σ. Additionally, we characterized the functional dependence of the variance of τ on z and σ and derived an approximation to the variance of the total time taken by a front to colonize completely a landscape of finite length L (app. A). Our approximation is in good agreement with numerical integrations of equation (2) To test whether deterministic models predict a slowdown of the invading front for increasing resource autocorrelation length, we numerically integrated equation (2) with σ = 0. Additionally, we numerically integrated equation (2) with σ = 0 and imposing a negative growth rate r in unfavorable patches where I = 0 (app. A).
Experiments
We performed experiments with the flagellated protist Euglena gracilis, acquired from Carolina Biological Supply (NC, USA). A culture of E. gracilis was initialized two weeks prior to the start of the experiment and kept at 22 • C under constant LED (Light Emitting Diode, model SMD 5050) light of wavelength 469 nm (emission width approximately 10 nm), in a filtered (0.2 µm filter) nutrient medium composed of sterilized spring water and Protozoan Pellets (Carolina Biological Supply, NC, USA) at a density of 0.45 g·l −1 in a 500 ml Schott flask .
In our experiment, light was used as the energy source for E. gracilis. To demonstrate that light was crucial for the growth of E. gracilis in our experimental setting, we measured E. gracilis' growth curves (fig. 4A) by initializing eight low-density cultures in 10 ml cell culture flasks. Half of such cultures were placed on top of two LEDs (for each culture) operated at a total flux of 1 mW each. The other half of the cultures were placed on top of two LEDs (for each culture) operated at the same power, but covered with black tape so that no light would penetrate. The spatial arrangement of illuminated and non-illuminated cell culture flasks was randomized.
Light also affects the movement behavior of E. gracilis individuals through a process known as phototaxis, the directed movement of cells towards or away from light (Drescher et al. 2010;Giometto et al. 2015). Specifically, at low to intermediate light intensities, E. gracilis swims towards the light source at a time scale much shorter than the typical generation time. At very high light intensities, negative phototaxis can also be observed, and the plastic reaction of phototaxis can be induced very reliably . The light intensity value used in our experiments is smaller than the light intensity value at which negative phototaxis occurs.
The front propagation experiment was performed in linear landscapes, which were channels drilled on a plexiglass sheet (5 mm wide, 3 mm deep and 1.9 m long, respectively, 300, 200 and 10 5 times the size of an individual, see Giometto et al. 2013), filled with filtered nutrient medium ( fig. 2A). A gasket avoided water spillage and a plexiglass lid was used to seal the system. The experimental replicates were kept in a climatized room at 22 • C for the whole duration of the experiment. Heterogeneous distributions of resources were generated via linear arrays of LEDs ( fig. 2B) controlled via Arduino Uno boards. LEDs in the array were separated by a distance of ∆L = 3.12 cm from each other and could be switched on or off individually. Switched-on LEDs emitted light with an intensity of 5.2 W·m −2 within the plexiglass channel, immediately above the LED. The linear landscapes were placed on top of the LED array at a distance of 4.5 mm. The light intensity profile generated by one LED was measured by placing a white paper sheet inside the plexiglass channel and by measuring the irradiance on the sheet with a digital camera operated in grayscale. The total radiant flux of the LED was measured via a calibrated photodiode. Light intensity profiles with the desired autocorrelation length were designed by imposing the probability λ of the LED number i + 1 in the LED array to be switched-on if the LED number i was switched-off, that is, P[LED(i + 1) =ON | LED(i) =OFF] = λ. Such Markov Chain was imposed to be symmetric, that is, P[LED(i + 1) =OFF | LED(i) =ON] = λ. Small and large values of λ generate resource distributions with long and small autocorrelation lengths (approximately equal to ∆L/(2λ)), respectively. Because landscapes were of finite total length, the above procedure could generate by chance resource profiles with autocorrelation length different from the desired one and with a mean frequency of switched-on LEDs different from 1/2. Therefore, the set of resource profiles obtained with the above Markov Chain procedure was restricted to those with a mean frequency of switched-on LEDs equal to 1/2 and in a narrow window of autocorrelation length around the desired one. Therefore, all replicates had the same mean light intensityĪ(x) = L −1 L 0 I(x)dx. We compared two treatments in the experiment. Treatment 1 consisted of landscapes with identical small autocorrelation length (c L 2 cm) but different switched-on LED sequences, generated via the Markov Chain procedure with λ = 0.75. Treatment 2 consisted of landscapes with identical large autocorrelation length (c L 6 cm) but different switched-on LED sequences, generated via the Markov Chain procedure with λ = 0.25. The choice of the large autocorrelation length value in the experiment was limited by the total finite length of the experimental setup and was chosen to be less than 1/20 of the total setup length. We initially had six landscape replicates of each treatment, but lost one replicate of Treatment 1 due to leakage. All 11 landscapes had the same total number of switched-on LEDs and the experimental light intensity profiles are shown in fig. 3. The stated values of autocorrelation length are based on the first-order autocorrelation of the Markov Chain that generated the landscape. The first three LEDs in every landscape were switched-on to allow the local establishment of the inoculated E. gracilis population and to avoid differences between the two treatments in the initial establishment dynamics. Thus, the landscapes generated via the Markov Chain procedure described in the text started at the fourth LED. In Treatment 2 (large autocorrelation length), three landscapes were chosen so that the fourth LED was switched on and the other four were chosen so that the fourth LED was switched off. In other words, the realized Markov Chain started from its stationary distribution. The spatial arrangement of landscapes belonging to the two treatments on the experimental bench was randomized.
At the start of the experiment, we introduced an ensemble of E. gracilis individuals at one end of the linear landscapes. Following the inoculation, we measured for eight consecutive days the density of E. gracilis throughout all replicates by taking pictures with a stereomicroscope (model Olympus SZX16 with the digital camera Olympus DC72) and counting individuals via image analysis .
Statistical analysis
We used a mixed effect model to compare the speed of the propagating E. gracilis among the two different treatments. Thereby, the autocorrelation treatment was included as a fixed effect, while day and replicate were included as random effect. We repeated this analysis using different choices of threshold values used for determining the front position. The minimum and maximum threshold values employed in the statistical analysis were chosen such that no replicate displayed a retreating front between successive measurements (caused by noise in the density profiles). The test statistics are reported in table 1 for the density threshold valueρ = 60 cm −1 and in table A1 for all values ofρ considered. We did not include the first timepoint in the analysis because it was measured immediately after the inoculation of E. gracilis in the landscape and thus was identical for all replicates. Because the propagating front reached the end of the landscape at day 4 in some replicates, the front propagation analysis was performed only with the data up to day 3 (included) to avoid spurious border effects due to the finite size of the system.
Model with directed movement towards resources
Equation (1) does not assume directed movement of individuals towards resources; such directed movement, however, occurs in our experiment and is likely to occur in nature (Andow et al. 1990;Fronhofer et al. 2013). Additionally, the experimental resource distributions (i.e., the light intensity profiles I(x)) were not simply sequences of illuminated and non-illuminated spatial patches with sharp edges, but, rather, smooth light intensity profiles alternating between well-lit and dark regions of the landscape according to the spatial arrangement outlined above. Because E. gracilis is capable to detect light intensity gradients and to move towards well-lit regions of the landscape, such directed movement may affect the invasion dynamics. To assess the net contribution of the directed movement of individuals towards resources, we incorporated in equation (1) the model for phototaxis derived in Giometto et al. (2015). The phototactic term was inferred from measurements of stationary density distributions of E. gracilis in the presence of light gradients and was shown to reproduce the accumulation dynamics of E. gracilis populations accurately in Giometto et al. (2015). The model equation reads:
∂ρ ∂t = ∂ ∂x D ∂ρ ∂x − dφ dx (I)ρ + r(I)ρ 1 − ρ K + σ √ ρ η,(3)
where φ = a(I − I c )/(I + I r ) is the phototactic potential describing E. gracilis' attraction towards (or against) light . The parameters describing φ were estimated and were set equal to a = 1.4 · 10 −8 m 4 ·W −1 ·s −1 , I r = 1.7 W·m −2 and I c = 28 W·m −2 . We assumed that r follows Monod kinetics (the assumption is customary for phytoplankton, Diehl 2002), that is, r(I) = r 1 I/(I + K I ), where K I is the half-saturation constant. The model (equation 3) was integrated with parameters suitable to describe the experimental system, r 1 = 6 · 10 −3 min −1 , K I = 1 W·m −2 , K = 300 cm −1 , D = 0.08 cm 2 ·min −1 (estimated in Giometto et al. 2015), various values of σ ( fig. A8) and initial condition localized at the origin. See app. A for details on the numerical integration scheme adopted. The slowdown effect caused by the resource autocorrelation structure is also found with other choices of the growth rate dependence on the resource density. In fact, we found that results do not change qualitatively by assuming a linear dependence of r on I. We used equation (3) to simulate biological invasions in linear landscapes with resource distributions I(x) exhibiting various autocorrelation lengths. To mimic the experimental setup ( fig. 3), such landscapes were generated with the same Markov-chain procedure used to design the experimental landscapes (see Experiment section), where the light intensity profile generated by a single LED (centered in x = 0) was assumed equal to the best fit of the equation I(x) = c 0 /(c 2 1 + x 2 ) 2 to the measured light intensity profile (see fig. S1 of Giometto et al. 2015). The total light intensity was kept constant for all landscapes. To further mimic the experiment, we set reflecting boundary conditions for the integration of equation (3) and simulations in which the population went extinct were excluded from the analysis. Therefore, the model equation (3) was specifically derived to reproduce as closely as possible the experimental system at hand. Landscapes used in the simulations were much longer (18 m) than those used in the experiment in order to avoid border effects. Such numerical settings allowed a clear identification of the invasion front and allowed simulating species spread in landscapes with very large autocorrelation length, which could not be investigated experimentally because of the finite size of the experimental setup.
Results
Our generalization of the Fisher-Kolmogorov equation (equations 1 and 2) includes demographic stochasticity and resource heterogeneity. Such resource heterogeneity affects the spread dynamics through the dependence of the growth rate r on the local amount of resources I (Methods). We found that the speed of invasion in the model equation (2) decreases with increasing resource autocorrelation length ( fig. 1B). The mean front propagation speed, in heterogeneous landscapes where resource patch lengths are distributed exponentially with rate µ, depends on c L and σ asymptotically (i.e., for large c L and σ) as:
v = L µL 2 L 0 dz τ (z, σ)µe −µz 8c 2 L ∞ 0 dz τ (z, σ)e −z/(2c L ) .(4)
Figs. 1B and A2 show that equation (4) correctly predicts the speed of invasion at large values of c L and σ. In heterogeneous landscapes with different spatial arrangements of favorable and unfavorable patches, if the percentage of space occupied by unfavorable patches is f 0 ∈ (0, 1) and the distribution of such patches lengths is p 0 (z), with mean dzzp 0 (z) = 1/µ, the asymptotic invasion velocity can be approximated as:
v = 1 µ f 0 ∞ 0 dz τ (z, σ)p(z) .(5)
We show in the app. A that equation (5) correctly predicts the speed of invasion in landscapes with percentages of unfavorable habitat different from f 0 = 1/2 ( fig. A5). Note that the speed of invasion according to equations (4,5) is a function of the autocorrelation length if the landscapes consist of favorable and unfavorable patches generated through the telegraph process outlined in the Methods section. In general, however, the speed of invasion is not a univocal function of the resource autocorrelation length (or of other characteristic length scales of the landscape), but it rather depends on the whole distribution of unfavorable patch lengths through equation (5). The slowdown effect is due to the fact that, in the presence of demographic stochasticity, long patches of unfavorable habitat act as obstacles for the spread of populations. The larger the extent of the unfavorable patch, the longer it takes for a population to cross it. The front propagation speed is also found to be a monotonically decreasing function of the amplitude of demographic stochasticity ( fig. 1B). Accordingly, integrating the model without demographic stochasticity (σ = 0 in equation 2, gray dots in fig. 1B) leads to no discernible slowdown of the front in strongly autocorrelated versus weakly autocorrelated landscapes, even when imposing negative values of the growth rate r in unfavorable patches where I = 0 (app. A). Such results demonstrate that the local extinctions caused by demographic stochasticity in unfavorable patches are responsible for the observed front slowdown. Numerical integration of equation (2) shows that the variability of the front position increases for larger values of c L and σ. Such increased variability is caused by two factors: i) two landscapes with identical resource autocorrelation lengths appear increasingly dissimilar for increasing values of the typical patch length 1/µ; ii) the variance of the distribution of waiting times (i.e., the times to cross an unfavorable patch of length z) increases (approximately) quadratically with the mean time τ (z, σ) ( fig. A3). These two observations can be used to approximate the fluctuations of the total time spent by the front to colonize a landscape of length L ( fig. A4), as shown in the app. A.
The model (equations 1 and 2) assumes random local movement of individuals. Although such assumption may be appropriate to describe spread in homogeneous landscapes (Andow et al. 1990;Giometto et al. 2014), individuals might be able to exploit local information on the availability of resources to direct their movement towards more favorable regions (Andow et al. 1990;Fronhofer et al. 2013Fronhofer et al. , 2015. We studied the effect of biased movement towards resources by including an advection term (towards regions endowed with more resources) in equation (1), leading to equation (3). The latter model predicts again that the front propagation speed decreases for increasing resource autocorrelation length, in accordance with the former model (equation 1). Integrating equation (3) with and without the advection term shows that the biased local movement towards resources causes an increased slowdown of the invasion front in strongly (compared to weakly) autocorrelated landscapes ( fig. A8). In other words, the biased movement towards resources acts as a spring that keeps the population in favorable patches and works against the exploration of unfavorable ones. Excluding demographic stochasticity from the model equation (3) leads again to the elimination of the slowdown effect (inset of fig. A8).
We designed an experiment with E. gracilis to test the slowdown effect on the front propagation caused by the spatial resource autocorrelation length. We observed a steady front propagation across all landscapes with a mean front propagation speed of 54 ± 9 cm/d (mean±SE). The mean total number of individuals was 2420 ± 110 (mean±SE) at the start of the experiment (day 0), 15000 ± 800 (mean±SE) at the end of the front propagation phase (day 4) and 27000 ± 4500 (mean±SE) at the end of the experiment (day 8). Thus, the invasion process was a combination of active, directed movement of individuals as well as reproduction. We found a significantly slower front propagation in landscapes in which the resources were strongly spatially autocorrelated (mixed effect model p = 0.027, see also table 1). The result is robust to changes of the threshold value at which the front position is evaluated (table A1, figs. 4C and A10). The slowdown effect is visible in fig. 4C, which shows the mean front position across replicated invasions in the two treatments.
Discussion
Our experiments show that the slowdown effect predicted by the stochastic models equations (1), (2) and (3) is found in microcosm experimental systems, which can be used to bridge theoretical models and natural systems (Benton et al. 2007). In these experiments the demographic and movement traits of the study species were fixed and dictated by the species. The accompanying models additionally allowed to single out the individual role and the mutual interconnections of all processes included in the equations to the propagation dynamics in landscapes with different resource autocorrelation lengths.
Our theoretical and experimental investigation advances our current understanding of the spread of invading organisms in heterogeneous landscapes by addressing the joint effect of spatial environmental autocorrelation and demographic stochasticity on the spread dynamics. As arguably all natural landscapes are characterized by heterogeneous distributions of resources and all populations are subject to demographic stochasticity, our model incorporates two key elements hitherto often overlooked in the modeling of biological spread. A major result of our work is that demographic stochasticity is a key factor in the slowdown of front propagation in heterogeneous landscapes. Such finding highlights the importance of including demographic stochasticity in theoretical models because of the many facets through which it affects species spread (Giometto et al. 2014;Hallatschek and Korolev 2009). The implications of the above results challenge the standard approach as stochastic effects are neglected by deterministic, Fisher-Kolmogorov-like models. Because the slowdown effect is only observed when demographic stochasticity is included in the models, our theoretical investigation suggests that the stochastic birth-anddeath dynamics are the main drivers of the observed reduction in propagation speed, rather than the movement behavior of individuals in heterogeneous landscapes that has received so far most attention in the literature (Morales and Ellner 2002;Van Dyck and Baguette 2005). Previous studies have investigated the minimum percentage of suitable habitat that allows invasions to spread (Dewhirst and Lutscher 2009;With 2002;With and Crist 1995), suggesting that invasions cannot propagate in landscapes with mean resource density below a critical threshold. Our work shows, complementarily, that the spatial arrangement of resources affects species spread even if the total amount of available resources is kept constant. Thus, it is not only the mean resource density that matters for the front propagation dynamics, because the autocorrelation structure of landscape heterogeneity alone also affects species spread. Our investigation extends previous works that addressed the effect of temporal environmental fluctuations on species spread (Ellner and Schreiber 2012;Méndez et al. 2011) by showing that the autocorrelation length of the resource distribution should be added to the environmental factors that can slow species spread, along with temporal fluctuations of vital rates (Ellner and Schreiber 2012;Neubert et al. 2000), geometrical heterogeneities of the substrate (Bertuzzo et al. 2007;Méndez et al. 2004;Méndez et al. 2003) and demographic stochasticity (Hallatschek and Korolev 2009).
Our finding that larger autocorrelation lengths reduce the spread rate of invading species is compatible with the results of Bergelson et al. (1994), who performed a field experiment with the invading weed Senecio vulgaris and found that the average spatial distance between two generations along linear transects increased when favorable patches were uniformly distributed in space (in the parlance of our work, the transect featured a small autocorrelation length), compared to transects with clumped patches (i.e., endowed with large autocorrelation length). Bailey et al. (2000) performed spread experiments with the fungal plant pathogen Rhizoctonia solani. Such work provides a complementing view to our investigation by evaluating the effect of the inter-distance between favorable patches on the spread and identifying experimentally the existence of a percolation threshold at a critical level of inter-patch distance. In the framework addressed here, the analog of such percolation threshold corresponds to an autocorrelation length much larger than the average distance traveled by the front during one generation. There exist considerable differences in the experimental setup and the study system between this investigation and those in Bergelson et al. (1994) and Bailey et al. (2000). Most importantly, biased active movement towards favorable patches was present in the experiment performed here and embedded in equation (3), while passive dispersal was implemented in Bergelson et al. (1994). Both Bailey et al. (2000) and Bergelson et al. (1994) differ from this study because the landscape and the distribution of resources herein are continuous, whereas they adopted discrete spatial distributions of favorable patches. Although such discrete distributions might provide a good approximation to some fragmented landscapes, continuous heterogeneous distributions may be equally likely to occur in nature. Compared to previous experimental efforts, we provide a general theoretical framework to interpret the dynamical processes underlying the realized invasions. The theoretical investigation of equations (1), (2) and (3), in fact, allowed isolating the net effect of each process embedded therein. Furthermore, the theoretical approximation to the mean speed of invasion in the model (equation 2) derived here allows to quantitatively predict the dependence of such mean speed on the resource autocorrelation length c L , the strength of demographic stochasticity σ and the other species traits.
Our results have important implications for species spread in natural environments, which are generally characterized by resources (seen as any field controlling vital rates, especially reproductive ones) being heterogeneously distributed. The typical autocorrelation length of the resource distribution can be inferred from environmental data (Turner 2005;Urban et al. 2008) and can be used as a concise indicator for the propagation success of a species of interest. Furthermore, the spatial availability of resources is often altered by human activities, reinforcing the fragmentation of landscapes. In fact, habitat fragmentation may decrease significantly the autocorrelation length of the landscape through the introduction of qualitatively different patches in the natural environment (Holyoak et al. 2005;With 2002). Our results give quantitative grounds to field observations on the effect of environmental heterogeneity on species spread. For instance, Lubina and Levin (1988) observed pauses in the spread of the California sea otter (Enhydra lutris) in the presence of habitat discontinuities. Such pauses and the corresponding piecewiselinear propagation of the front (see fig. 2 of Lubina and Levin 1988) are also found in our model ( fig. A7), which enables to relate the mean spatial extent of habitat discontinuities to the average speed of invasion through equations (4) and (5). An alternation between phases of halt and spread was also found in the range expansion of the cane toad (Chaunus marinus) in Australia ( fig. 2 of Urban et al. 2008). Urban et al. (2008) performed an in-depth analysis of the effect of environmental heterogeneity on the spread of the cane toad in the field and found a statistically significant effect of environmental heterogeneity and, most importantly, of the spatial autocorrelation of environmental variables on the realized patterns of invasion speed. They found such effect in nature in a realized (not replicable) invasion, and thus they could only correlate the realized spread dynamics and its reduction with the landscape autocorrelation. Here, we have given a mathematical framework and an experimental proof showing that the slowdown effect caused by the spatial autocorrelation structure of the landscape is not an artifact of the mathematical model.
Conclusion
In conclusion, our work demonstrates the need to account for the intrinsic stochasticity of population dynamics to broaden our understanding of ecological processes occurring in spatially extended natural landscapes, which typically display various degrees of heterogeneity. Further work should be dedicated to the modeling and experimentation of species spread in temporally-varying landscapes and, possibly, spatially-heterogeneous landscapes that fluctuate in time. Drawing from the literature on population dynamics in temporally-fluctuating environments, understanding the causal link between the autocorrelation structure of fluctuations and the dynamics of species spread is a promising direction for future research in this area. Mixed-effect test statistics testing the speed of front propagation, with the autocorrelation length treatment as single fixed effect and time/replicate as random effect. The treatment with small autocorrelation length had 5 replicates, the treatment with large autocorrelation length had 6 replicates. The front position was measured at the density threshold valueρ = 60 cm −1 . x (cm) x (cm) Figure 3: Light intensity profiles used in the experiment (Methods). One spread experiment was performed for each landscape. The total light intensity is the same for each landscape. Landscapes with the same color have identical small (blue) or large (red) autocorrelation length of the resource distribution I(x), but different LED on-off sequences. Table A1.
Tables
Figures
0 1 0 1 I(x) x I(x) Resource autocorrelation length c L =1/(2μ) A B σ=0 σ=0.1 σ=0.2 σ=0.4 σ=0.6 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
Online Appendix A: Additional Methods and Results
A.1 Additional Methods
A suitable spatial discretization of equation (2) reads (Dornic et al. 2005;Giometto et al. 2014):
dρ i dt (t) = 1 (∆x) 2 [ρ i+1 (t) + ρ i−1 (t) − 2ρ i (t)] + r i ρ i (t) [1 − ρ i (t)] + σ √ ∆x ρ i (t)η i (t),(A1)
where i identifies the lattice site, the term √ ∆x ensures proper normalization in the continuum limit (Doering et al. 2005) and r i = δ I i ,1 depends on the local value of the resource profile I (here, δ is the Kronecker's delta). The split-step method proposed in Dornic et al. (2005) was used to solve equation (A1). The spatial step in the numerical integration of equation (A1) was set to ∆x = 0.5, while the temporal step was chosen equal to ∆t = 0.1. The Courant-Friedrichs-Lewy condition for the diffusion equation 2∆t/∆x 2 < 1 was thus satisfied and ∆t/∆x < 1. The numerical integration of equation (A1) with σ = 0 was performed using the same numerical scheme, modified in the choice of ρ * (we refer the reader to Dornic et al. 2005 for notation and symbols), which in the deterministic case is ρ * = α/β e β∆t − 1 + ρe β∆t . The deterministic equation was integrated with three choices of the growth rate r in unfavorable regions of the landscape (where I = 0), specifically r = 0, r = −0.01 and r = −0.1. None of these choices for r produced a slowdown of the front at large resource autocorrelation lengths, compared to small ones. The spatial discretization of equation (3) reads:
dρ i dt (t) = D (∆x) 2 [ρ i+1 (t) + ρ i−1 (t) − 2ρ i (t)] − 1 2∆x [g i+1 ρ i+1 (t) − g i−1 ρ i−1 (t)] + r i ρ i (t) 1 − ρ i (t) K + σ √ ∆x ρ i (t)η i (t),(A2)
where g = dφ/dx[I(x)]. The split-step method proposed in Dornic et al. (2005) was modified to solve equation (A2), which contains an advection term that might cause an artificial loss of mass if the step sizes are too coarse. Such issue does not occur with the step sizes ∆x = 0.6 cm and ∆t = 0.5 min −1 chosen here. The Courant-Friedrichs-Lewy condition for the diffusion equation 2D∆t/∆x 2 < 1 was satisfied and ∆t/∆x < 1.
A.2 Additional Results
A.2.1 Mean front propagation speed in heterogeneous landscapes
Here we derive an approximation to the front propagation speed in the model equation (1), valid for large autocorrelation lengths and σ. We divide equation (1) by K and r 0 and rescale time as t = r 0 t, which gives:
∂ρ (x, t ) ∂t = D r 0 ∂ 2 ρ (x, t ) ∂x 2 + χ I (x)ρ (x, t ) 1 − ρ (x, t ) + σ √ r 0 ρ (x, t ) η(x, t ),(A3)
where ρ = ρ/K, σ = σ/ √ K and χ I is the indicator function of the set of x for which I(x) > 0. We can further rescale space as x = D r 0 x and rewrite equation (1) as:
∂ρ (x , t ) ∂t = ∂ 2 ρ (x , t ) ∂x 2 + χ I (x )ρ (x , t ) 1 − ρ (x , t ) + σ ρ (x , t ) η(x , t ),(A4)
where σ = σ (rD) 1/4 . In the following we will study the front propagation speed in the rescaled equation (2), where we drop primes for convenience; one can recover the original dimensions by multiplying t by r 0 and x by √ r 0 /D. The rationale for our approximation of the mean front propagation speed is as follows. Let L be the finite length of a landscape and T the time taken by the population to reach the end of such landscape (x = L), starting from a localized initial condition at x = 0. For large values of autocorrelation length c L and large enough σ, due to the local extinctions caused by demographic stochasticity, most of the time T is spent by the population trying to cross long patches of the landscape where r = I = 0. We can therefore approximate the mean front propagation speed for large c L by computing the mean time that the front takes to cross an unfavorable patch of finite length z. Of course, such approximation is only valid when the waiting times dominate over the typical time scale of front propagation in favorable regions of the landscape. Therefore, the approximation can only hold for large enough values of the strength of demographic stochasticity σ.
A.2.1.1 Propagation past a patch of unfavorable landscape
We computed numerically the mean time τ taken by the front to cross a region of landscape where I = 0, for different spatial extents of such region and different values of σ. We integrated numerically equation (2) in landscapes with resource profile I(x) = θ(x − z), where θ is the Heaviside step function. Such landscapes consist of a resource profile I(x) = r(x) = 1, except for x ∈ (0, 1], that is a finite patch of spatial extent z at the left end of the landscape, where I(x) = 0. The initial condition was ρ(x, 0) = 0 for x > 0 and ρ(0, 0) = k, where k is the mean population density computed numerically by integrating equation (2) in a landscape of spatial extent L = 100 with growth rate profile r(x) = 1 for all x ∈ [0, L]. We fixed the Dirichlet boundary condition ρ(0, t) = k and reflecting boundary conditions in x = L. We computed the mean time taken by the front to cross such unfavorable patch by measuring the first occurrence of ρ(z) > 10 −3 k in time. Fig. A1A shows the mean time τ taken by the population to cross unfavorable patches of various extents z, computed for various values of σ. Such mean time τ is a monotonically increasing function of both z and σ. To characterize the functional dependence of τ (z, σ) on z and σ, we note that in the limit σ = 0 the dependence of τ on z is that of the deterministic diffusion equation with boundary condition ρ(0, t) = 1, that is, τ(z, 0) = Cz 2 , where C is the solution of erfc 1 2 √ C = 10 −3 , where erfc is the complementary error function. We assume that τ (z, σ) depends on z and σ through the functional form:
τ (z, σ) = Cz 2 F(zσ b ),(A5)
where F(x) is a function that goes to the constant 1 for x → 0. We can verify the validity of equation (A5) by plotting z −2 τ versus zσ b and varying b. Because we are able to find a value of b = b * for which data from the numerical integrations collapse onto one single curve ( fig. A1B), the assumption on the functional form of τ is verified. To further identify the functional dependence of τ on z we plotted log[log(z −2 τ ) − log C] vs log(zσ b ) and observed that simulation data aligned along a straight line. Therefore, our numerical analysis suggests that the functional dependence of τ on z and σ is given by:
τ (z, σ) = Cz 2 e d(zσ b ) a .(A6)
We estimated b by maximizing the R 2 (coefficient of determination) of the least-squares linear fit of log[log(z −2 τ ) − log C] versus log(zσ b ). The slope and intercept of the linear fit with maximum R 2 gave
•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• -2 0 2 4 6 -3 -2 -1 0 1 2 log (σ b z) log (z
A.2.1.2 Approximation for the mean front propagation speed in heterogeneous landscapes
For large values of the autocorrelation length c L = 1/(2µ) (µ is the rate of the telegraph process used to generate the heterogeneous landscapes, see Methods), most of the time taken by the front to propagate through a landscape of length L is spent trying to cross finite stretches of the landscape where r = I = 0. We can therefore approximate the front propagation speed as v = L/T = L/ ∑ N i=1 τ (z i , σ) (black dots in fig. 4), where N is the number of unfavorable patches in x ∈ [0, L] (of extent z i ) and τ is the mean time taken to cross a patch of spatial extent z i , estimated via equation (A6). In landscapes where unfavorable patches of length z occur with probability µe −µz dz, one can approximate the mean front propagation speed for large autocorrelation length c L as:
v = L µL 2 L 0 dz τ (z, σ)µe −µz = c 2 L 2 L 0 dz τ (z, σ)e −z/(2c L ) ,(A7)
where τ (z, σ) is given by equation (A6) and µL/2 at the denominator is the average number of unfavorable patches in the landscape. If L is comparable to c L , one can substitute µL/2 with a more precise estimate, which is given in the next section. Fig. 4 shows that equation (A7) gives a good approximation to the front propagation velocities computed in the numerical integrations, for large values of c L .
v = L/T = L/ ∑ i∈Z τ (z i , σ),
where Z is the set of unfavorable windows in the numerical landscapes. Dashed lines and black dots may differ because the numerical landscapes were finite, thus the distribution of unfavorable window lengths may differ slightly from the exponential pdf with typical length 1/µ = 2c L .
A.2.1.3 Correction to the average number of patches if L is comparable to c L
We provide here a correction to the term µL/2 at the denominator of equation (A7), which is relevant when L c L . If the first patch at x = 0 is favorable (i.e., r > 0), the average number of unfavorable patches in a landscape of length L can be computed as follows. Let z i be the rightmost coordinate of each patch in the landscape. The average number of unfavorable patches is equal to:
N = ∞ ∑ n=1 nP [z 2n < L ∩ z 2n+1 ≥ L] + ∞ ∑ n=1 nP[z 2n−1 < 2 ∩ z 2n ≥ L].
Using properties of the exponential distribution of patch lengths one has:
P [z 2n < L ∩ z 2n+1 ≥ L] = µ 2n+1 L 0 dz 1 e −µz 1 L z 1 dz 2 e −µ(z 2 −z 1 ) · · · L z 2n−1 dz 2n e −µ(z 2n −z 2n−1 ) ∞ L dz 2n+1 e −µ(z 2n+1 −z 2n ) = e −µL (2n)! (µL) 2n , P[z 2n−1 < 2 ∩ z 2n ≥ L] = µ 2n L 0 dz 1 e −µz 1 L z 1 dz 2 e −µ(z 2 −z 1 ) · · · L z 2n−2 dz 2n−1 e −µ(z 2n−1 −z 2n−2 ) ∞ L dz 2n e −µ(z 2n −z 2n−1 ) = e −µL (2n − 1)! (µL) 2n−1
and therefore:
N = ∞ ∑ n=1 n e −µL (2n)! (µL) 2n + e −µL (2n − 1)! (µL) 2n−1 = µL 2 + e −µL 2 sinh(µL),
where sinh is the hyperbolic sine function. One can repeat the same analysis in the case where the first patch at x = 0 is unfavorable (i.e., r = 0). In this case one finds:
N = ∞ ∑ n=1 n e −µL (2n − 2)! (µL) 2n−2 + e −µL (2n − 1)! (µL) 2n−1 = µL 2 + 3 4 + e −µL .
Finally, if the first patch is favorable or unfavorable with equal probabilities, then: where S(x) is a function that goes to 0 for x → 0. In fact, data from the numerical integrations of equation (2) in landscapes with resource profile I(x) = θ(x − z) (θ is the Heaviside step function, the same numerical data were used to derive equation A6) collapse on the same curve when z −2 σ τ is plotted against z −2 τ (z, σ) ( fig. A3B). The functional form:
N = 1 2 µL 2 + e −µL 2 sinh(µL) + 1 2 µL 2 + 3 4 + e −µL = 1 2 + µL 2 . If L 2 µ = 4c L ,σ τ (z, σ) = τ (z, σ) 1 − e −kz −2 τ (z,σ)(A9)
is found to provide a good fit to the numerical data, with the best-fit estimate of the coefficient k = 4.17 (dashed lines in fig. A3).
A.2.2.2 Fluctuations of the total invasion time in heterogeneous landscapes
We can use equation (A9) to approximate the variance of the total invasion time T (i.e., the time after which the density ρ(L, T) is larger than a threshold density value) in heterogeneous landscapes composed of favorable and unfavorable patches. In fact, the variance of the total invasion time in our simplified model, where we neglect the time spent by the front in propagating through favorable patches, and further assuming that the times spent to cross each unfavorable patch are independent from each other, is given by:
Var[T] = N ∑ i=1 σ 2 τ [ τ (z i , σ)],(A10)
where N is the number of unfavorable patches in x ∈ [0, L] (patches of extent z i ) and σ τ is given by equation ( Var
a [T] • 1/μ=80 • 1/μ=100 • 1/μ=120 • 1/μ=140 • 1/μ=160 • 1/μ=180 • • • • • • • • • • • • • •
A.2.3 Front propagation at different mean resource densities
Other works (e.g., Dewhirst and Lutscher 2009) have studied the propagation of invasion fronts in landscapes with different average amounts of resources. One may wonder whether the slowdown effect caused by varying resource autocorrelation lengths of the resource distribution might also be found in landscapes endowed with mean percentages of suitable habitat different from f 1 = f 0 = 1/2. To show that such slowdown effect occurs also when the suitable and unsuitable habitats occur at different frequencies throughout the landscape, we have integrated equation (2) on landscapes endowed with various resource autocorrelation lengths and mean frequency of suitable (i.e., r > 0) and unsuitable (i.e., r = 0) habitat equal to f 1 = 1/3 and f 0 = 2/3, respectively. Such landscapes were generated as follows: we extracted the length of each favorable and unfavorable patch from exponential distributions with rate µ 1 = 3/(4c L ) and µ 0 = 3/(8c L ), respectively, so that the resource autocorrelation length was c L and the frequencies of favorable/unfavorable habitat were as desired. Additionally, we have integrated equation (2) on the same landscapes switching each favorable patch of the landscape with an unfavorable one, so that favorable habitats occurred with frequency f 1 = 2/3 (and thus unfavorable habitats with frequency f 0 = 1/3). Fig. A5 shows that increasing the mean frequency of suitable habitat increases the invasion speed, but the slowdown effect caused by varying resource autocorrelation lengths is also present when favorable and unfavorable habitats occur at frequencies different from 1/2. Furthermore, equation (5) can be used to approximate the mean speed of invasion for large c L at values of f 0 different from 1/2, as shown by the agreement between dashed lines and simulation data points in fig. A5. : Mean front propagation speed in landscapes with favorable and unfavorable habitats occurring at frequencies different from f 1 = f 0 = 1/2. Red dots display the mean front speed in 96 replicated invasions in different landscapes with frequency of unsuitable habitat f 0 = 1/3. favorable patches lengths were distributed exponentially with rate µ 1 = 3/(8c L ) and unfavorable ones with rate µ 0 = 3/(4c L ). Blue dots display the mean front speed in 96 replicated invasions in different landscapes with frequency of unsuitable habitat f 0 = 2/3. Error bars display the 95% confidence interval for v, computed with 2 · 10 3 bootstrap samples. favorable patches lengths were distributed exponentially with rate µ 1 = 3/(4c L ) and unfavorable ones with rate µ 0 = 3/(8c L ). Dashed lines show mean front speeds approximated via equation (5) of the main text. (3) (with spatial discretization equation A2). The mean invasion speed decreases with increasing resource autocorrelation length c L = ∆L/(2λ) (λ is the transition probability of the Markov Chain used to generate the heterogeneous landscapes and ∆L is the experimental distance between LEDs, see Methods) and is a decreasing function of the amplitude of demographic stochasticity σ (log-linear plot; black dots: σ = 0.4 min −1/2 ; red triangles: σ = 0.7 min −1/2 ). The mean speed of invasion is larger in the absence of directed movement towards resources (blue diamonds computed with σ = 0.4 min −1/2 and φ = 0). Invasion speeds are reported here divided by the mean front speed v 0 at σ = 0 min −1/2 , that is constant for different values of c L (inset). The mean front speed for each value of c L and σ was calculated by integrating equation (3) along 150 different landscapes with identical c L and fitting the mean front position versus time in the asymptotic propagation regime. Threshold: 90 cm -1 p=0.0433
A.3 Additional Tables
Threshold: 105 cm -1 p=0.0276
Threshold: 105 cm -1 p=0.0276 Figure A10: Experimental spread in autocorrelated landscapes. Left: position of the front in each experimental replicate, identified by different symbols. Red and blue lines and symbols refer to replicates with identical large (red) or small (blue) resource autocorrelation length. Right: mean (±SE) position of the front, calculated among replicates with identical large (red) or small (blue) resource autocorrelation length. Different rows refer to different threshold density values used to identify the position of the front. The gray shaded regions identify data points collected when at least one replicate had colonized the whole landscape. To avoid border effects, we excluded such points from the statistical analysis. In fact, at least one replicate with small autocorrelation length had reached the end of the landscape at time t = 4 d, and might have spread even further in a longer landscape. The reported p-values show that the autocorrelation treatment had a significant effect on the front propagation regardless of the choice of denstiy threshold.
(fig. A4).
Figure 1 :
1Mean front propagation in the model (dimensionless equation 2). (A) Examples of landscapes with different resource autocorrelation length c L , generated via the telegraph process with rate µ (Methods). (B) The mean invasion speed computed in numerical integrations of the model (equation 2) decreaseswith increasing resource autocorrelation length c L (log-linear plot) for σ > 0 and is a decreasing function of the amplitude of demographic stochasticity σ (different colors according to legend). With σ = 0 the dynamics is deterministic and the mean front propagation speed does not decrease with z (gray dots). Error bars display the 95% confidence interval for log v, computed with 2 · 10 3 bootstrap samples. Error bars for σ = 0 are smaller than symbols. Dashed lines show the mean front propagation speed computed according to the theoretical approximation (equation 4).
Figure 2 :
2Experimental setup. (A) Linear landscapes used in the experiments were channels drilled on a plexiglass sheet. A gasket (orange rubber band) avoided water spillage. (B) Photograph of the LED strips used to control the distribution of resources for E. gracilis. The red and blue lines show the paths of landscapes with large and small resource autocorrelation length, respectively.
Figure 4 :
4Experimental spread in autocorrelated landscapes. (A) Light was used as energy resource for E. gracilis. Replicated measured growth curves show that E. gracilis grows in the presence of light (blue symbols and lines) and does not grow in its absence (black symbols and lines). (B) Replicated measurements (gray lines) of E. gracilis density profiles (normalized by the value at the edge of the imaging window) in the presence of a LED at x = 0 cm show that E. gracilis populations accumulate around light sources through phototaxis. The blue line denotes the mean density profile across replicates (panel B is redrawn fromGiometto et al. 2015). (C) Mean (±SE) position of the front, calculated among replicates with identical large (red) or small (blue) resource autocorrelation length at the threshold density valuē ρ = 60 cm −1 . The inset shows mean front positions calculated at different threshold density valuesρ as indicated. The slowdown effect is significant with all choices ofρ, see
Figure A1 :
A1Mean time τ taken by a diffusing population subject to demographic stochasticity to cross patches of length z, calculated for different values of z and σ across 192 integrations of equation (2). (A) τ is a monotonically increasing function of z and σ. Dots of identical color were computed with identical z = 21, 29, 49, 57, 79, 111, 156 and 218, from bottom to top. Lines are computed via equation (A6), the color code identifies the value of z as for the dots. (B) Simulation data collapse onto the same curve when z −2 τ is plotted against σ b z, proving the assumption made in equation (A5). Dots are color-coded as in panel (A), the dashed black line shows the function F computed according to equation (A6).
Figure A2 :
A2The mean front speed v decreases with increasing resource autocorrelation length c L = (2µ) −1 (µ is the rate of the telegraph process used to generate the heterogeneous landscapes) and can be approximated by equation(A7)for large c L (dashed lines). Colored data points highlight the mean speed v computed by numerically integrating equation(2)and by fitting the mean front position versus time to a straight line. Different colors refer to different values of σ according to the legend. Error bars display the 95% confidence interval for v, computed with 2 · 10 3 bootstrap samples. Error bars for σ = 0 are smaller than symbols. Dashed lines are the mean front speed computed according to equation (A7). Black dots are the approximation to the mean front speed computed as
Figure A3 :
A3the average number of unfavorable patches in a landscape of length L tends to µL 2 . (A) Standard deviation σ τ of the time taken by a diffusing population subject to demographic stochasticity to cross patches of length z, calculated for different values of z and σ across 96 integrations of equation (2) (double logarithmic plot). Different colors refer to different values of σ, from σ = 0.1 (blue dots at the bottom left corner) to σ = 1.4 (violet dots at the top right corner). Dashed lines are computed with equation (A9). (B) Simulation data collapse onto the same curve when z −2 σ τ is plotted against z −2 τ , proving the validity of equation (A8). Dots are color-coded as in panel (A), the dashed black line shows the function S computed according to equation (A9).
A9). We show in fig. A4 that equation (A10) gives a good estimate of the variance of the total invasion time in heterogeneous landscapes. Details are provided in the figure caption.
Figure A4 :
A4(A) Mean total time T (black dots) of invasion and its standard deviation (blue dots) in numerical integrations of equation (2) in square-wave landscapes of length L = 1400, that is, landscapes composed of alternated favorable and unfavorable patches of length 1/µ (means and standard deviations were computed across 200 integrations for each value of 1/µ). The numerical estimates for T and Var[T] are well approximated by the approximations T = µL 2 τ(1/µ, σ) (black dashed line) and by equation (A10) (blue dashed line). (B) Numerically computed standard deviations Var n [T] (double logarithmic plot) of the total time T of invasion in numerical integrations of equation (2) in landscapes with exponentially-distributed favorable and unfavorable patches are well approximated by the theoretical approximation Var a [T], computed according to equation (A10). Each dot represents one landscape of length L = 2000 and mean patch length 1/µ according to the legend. Such landscapes were generated with the same procedure outlined in the Methods. To compute Var n [T], we performed 96 numerical integrations for each landscape. The dashed black line is the 1:1 line. Numerical estimates and theoretical approximations are calculated with σ = 0.4 in both panels.
Figure A5
A5Figure A5: Mean front propagation speed in landscapes with favorable and unfavorable habitats occurring at frequencies different from f 1 = f 0 = 1/2. Red dots display the mean front speed in 96 replicated invasions in different landscapes with frequency of unsuitable habitat f 0 = 1/3. favorable patches lengths were distributed exponentially with rate µ 1 = 3/(8c L ) and unfavorable ones with rate µ 0 = 3/(4c L ). Blue dots display the mean front speed in 96 replicated invasions in different landscapes with frequency of unsuitable habitat f 0 = 2/3. Error bars display the 95% confidence interval for v, computed with 2 · 10 3 bootstrap samples. favorable patches lengths were distributed exponentially with rate µ 1 = 3/(4c L ) and unfavorable ones with rate µ 0 = 3/(8c L ). Dashed lines show mean front speeds approximated via equation (5) of the main text.
Figure A6 :Figure A7 :Figure A8 :
A6A7A8Mean position of the front (blue lines) and 68% confidence interval (shaded regions) in numerical integrations of the model equation (2) with σ = 0.1 and resource autocorrelation lengths c L = 5 (A) and c L = 20 (Examples of front propagation in numerical integrations of the model (equation 2) in landscapes with different resource autocorrelation lengths c L and fixed amplitude of demographic stochasticity Front propagation computed in numerical integrations of the model equation
Figure A9 :
A9Experimental spread in autocorrelated landscapes. Left: position of the front in each experimental replicate, identified by different symbols. Red and blue lines and symbols refer to replicates with identical large (red) or small (blue) resource autocorrelation length. Right: mean (±SE) position of the front, calculated among replicates with identical large (red) or small (blue) resource autocorrelation length. Different rows refer to different threshold density values used to identify the position of the front. The gray shaded regions identify data points collected when at least one replicate had colonized the whole landscape. To avoid border effects, we excluded such points from the statistical analysis. In fact, at least one replicate with small autocorrelation length had reached the end of the landscape at time t = 4 d, and might have spread even further in a longer landscape. The reported p-values show that the autocorrelation treatment had a significant effect on the front propagation regardless of the choice of density threshold.
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Table 1 :
1Mixed-effect test statisticsValue Std. Error df t-value
p-value
Intercept
45.98
3.27 44
14.04 p < 10 −4
Autocorrelation length −11.61
4.43
9
−2.62
0.0279
Table A1 :
A1Mixed-effect test statistics for all choices of density thresholdρMixed-effect test statistics testing the speed of front propagation, with the autocorrelation length treatment as single fixed effect and time/replicate as random effect. The treatment with small autocorrelation length had 5 replicates, the treatment with large autocorrelation length had 6 replicates. Different lines refer to different threshold valuesρ at which the front position was measured.Thresholdρ
Value Std. Error df t-value
p-value
45 cm −1
Intercept
57.15
3.65 44
15.65
p < 10 −4
Autocorrelation length −11.31
4.94
9
−2.29 p = 0.0480
60 cm −1
Intercept
45.98
3.27 44
14.04
p < 10 −4
Autocorrelation length −11.61
4.43
9
−2.62 p = 0.0279
75 cm −1
Intercept
45.27
2.88 44
15.70
p < 10 −4
Autocorrelation length
−9.65
3.90
9
−2.47 p = 0.0355
90 cm −1
Intercept
36.65
2.84 44
12.91
p < 10 −4
Autocorrelation length
−9.04
3.85
9
−2.35 p = 0.0433
105 cm −1
Intercept
35.91
3.04 44
11.83
p < 10 −4
Autocorrelation length −10.79
4.11
9
−2.62 p = 0.0276
AcknowledgmentsWe thank Enrico Bertuzzo, Francesco Carrara, Lorenzo Mari and Amos Maritan for many useful discussions. We gratefully acknowledge the support by Swiss Federal Institute of Aquatic Science and Technology (Eawag) discretionary funds and Swiss National Science Foundation Projects 200021 157174 and PP00P3 150698.
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A1 shows that equation (A6) reproduces the numerical data satisfactorily with the parameters d = 0.74, a = 0.34 and b = 2.25. identified as outlined abovethe estimate of a and d. Fig. A1 shows that equation (A6) reproduces the numerical data satisfactorily with the parameters d = 0.74, a = 0.34 and b = 2.25, identified as outlined above.
In this section we study the fluctuations of the total invasion time in heterogeneous landscapes of finite size L. To this end, we first characterize the standard deviation σ τ of the time τ taken by a diffusing population subject to demographic stochasticity to cross an unfavorable patch (r = 0) of spatial extent z. Inspection of the numerical results shows (fig. A3B) that z −2 σ τ is a function of z −2 τ (z, σ). that is: σ τ (z, σ) = z 2 S z −2 τ (z, σ) , (A8)In this section we study the fluctuations of the total invasion time in heterogeneous landscapes of finite size L. To this end, we first characterize the standard deviation σ τ of the time τ taken by a diffusing population subject to demographic stochasticity to cross an unfavorable patch (r = 0) of spatial extent z. Inspection of the numerical results shows (fig. A3B) that z −2 σ τ is a function of z −2 τ (z, σ), that is: σ τ (z, σ) = z 2 S z −2 τ (z, σ) , (A8)
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"Lensless coherent diffraction imaging based on spatial light modulator with unknown modulation curve",
"Lensless coherent diffraction imaging based on spatial light modulator with unknown modulation curve",
"Lensless coherent diffraction imaging based on spatial light modulator with unknown modulation curve",
"Lensless coherent diffraction imaging based on spatial light modulator with unknown modulation curve"
]
| [
"Hao Sha \nSchool of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina\n",
"Chao He \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Shaowei Jiang \nDepartment of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA\n",
"Pengming Song \nDepartment of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA\n",
"Shuai Liu \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Wenzhen Zou \nSchool of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina\n",
"Peiwu Qin \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Haoqian Wang \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Yongbing Zhang [email protected] \nSchool of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina\n",
"Hao Sha \nSchool of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina\n",
"Chao He \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Shaowei Jiang \nDepartment of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA\n",
"Pengming Song \nDepartment of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA\n",
"Shuai Liu \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Wenzhen Zou \nSchool of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina\n",
"Peiwu Qin \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Haoqian Wang \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina\n",
"Yongbing Zhang [email protected] \nSchool of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina\n"
]
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"School of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"Department of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA",
"Department of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"School of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"School of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina",
"School of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"Department of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA",
"Department of Biomedical Engineering\nUniversity of Connecticut\n06269StorrsCTUSA",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"School of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518006ShenzhenGuangdongChina",
"School of Computer Science and Technology\nHarbin Institute of Technology (Shenzhen)\n518006ShenzhenGuangdongChina"
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| Lensless imaging is a popular research field for the advantages of small size, wide field-of-view and low aberration in recent years. However, some traditional lensless imaging methods suffer from slow convergence, mechanical errors and conjugate solution interference, which limit its further application and development. In this work, we proposed a lensless imaging method based on spatial light modulator (SLM) with unknown modulation curve. In our imaging system, we use SLM to modulate the wavefront of object, and introduce the ptychographic scanning algorithm that is able to recover the complex amplitude information even the SLM modulation curve is inaccurate or unknown. In addition, we also design a split-beam interference experiment to calibrate the modulation curve of SLM, and using the calibrated modulation function as the initial value of the expended ptychography iterative engine (ePIE) algorithm can improve the convergence speed. We further analyze the effect of modulation function, algorithm parameters and the characteristics of the coherent light source on the quality of reconstructed image. The simulated and real experiments show that the proposed method is superior to traditional mechanical scanning methods in terms of recovering speed and accuracy, with the recovering resolution up to 14 . | null | [
"https://arxiv.org/pdf/2204.03947v1.pdf"
]
| 248,069,258 | 2204.03947 | ee652fc6a44991c3b05b36c65ca0568cd104195d |
Lensless coherent diffraction imaging based on spatial light modulator with unknown modulation curve
Hao Sha
School of Computer Science and Technology
Harbin Institute of Technology (Shenzhen)
518006ShenzhenGuangdongChina
Chao He
Tsinghua Shenzhen International Graduate School
Tsinghua University
518006ShenzhenGuangdongChina
Shaowei Jiang
Department of Biomedical Engineering
University of Connecticut
06269StorrsCTUSA
Pengming Song
Department of Biomedical Engineering
University of Connecticut
06269StorrsCTUSA
Shuai Liu
Tsinghua Shenzhen International Graduate School
Tsinghua University
518006ShenzhenGuangdongChina
Wenzhen Zou
School of Computer Science and Technology
Harbin Institute of Technology (Shenzhen)
518006ShenzhenGuangdongChina
Peiwu Qin
Tsinghua Shenzhen International Graduate School
Tsinghua University
518006ShenzhenGuangdongChina
Haoqian Wang
Tsinghua Shenzhen International Graduate School
Tsinghua University
518006ShenzhenGuangdongChina
Yongbing Zhang [email protected]
School of Computer Science and Technology
Harbin Institute of Technology (Shenzhen)
518006ShenzhenGuangdongChina
Lensless coherent diffraction imaging based on spatial light modulator with unknown modulation curve
4 These authors contributed equally *
Lensless imaging is a popular research field for the advantages of small size, wide field-of-view and low aberration in recent years. However, some traditional lensless imaging methods suffer from slow convergence, mechanical errors and conjugate solution interference, which limit its further application and development. In this work, we proposed a lensless imaging method based on spatial light modulator (SLM) with unknown modulation curve. In our imaging system, we use SLM to modulate the wavefront of object, and introduce the ptychographic scanning algorithm that is able to recover the complex amplitude information even the SLM modulation curve is inaccurate or unknown. In addition, we also design a split-beam interference experiment to calibrate the modulation curve of SLM, and using the calibrated modulation function as the initial value of the expended ptychography iterative engine (ePIE) algorithm can improve the convergence speed. We further analyze the effect of modulation function, algorithm parameters and the characteristics of the coherent light source on the quality of reconstructed image. The simulated and real experiments show that the proposed method is superior to traditional mechanical scanning methods in terms of recovering speed and accuracy, with the recovering resolution up to 14 .
Introduction
According to Huygens-Fresnel principle, diffraction occurs from each point in space. In lensless imaging system, the main task is to recover the optical field distribution of object from mixed signal without any optical lens for imaging or amplification [1]. Because lensless imaging system simplifies the illumination and imaging optical path and makes the overall system develop toward miniaturization and lightness, many research achievements have been made in the field of macro imaging [2][3][4][5]. In terms of microscopic imaging, conventional microscope system is unable to achieve both high resolution and wide field-of-view (FOV), and the spatial bandwidth product (SBP) is limited by the FOV, while lensless cameras can theoretically reach the diffraction limit resolution with the advantages of both high SBP and low aberration [6,7]. In incoherent imaging system, the light intensity satisfies a linear relationship, so the image captured by sensor is the result of convolution of the object with the point spread function (PSF) of the system. The PSF can be calibrated by modulating the wavefront of the object with an encoding element (also known as mask). There are lots of researches on the design of mask in incoherent imaging system. R. Horisaki et al. achieved single-shot phase imaging using amplitude mask with compressed sensing method [8]. V. Ashok et al. proposed to mount high-precision amplitude mask made by 3D printing onto sensors, enabling highly flexible macro and micro imaging [9]. T. Shimano et al. used Fresnel diffraction grating as mask to calibrate the PSF [10]. L. Waller et al. utilized diffuser as the scatterer, which greatly reduced the complexity of calibration and was able to recover the 3D information of the object by calibrating at different depths [11]. Although incoherent lensless modulation imaging is able to reconstruct the final image using deconvolution algorithms, mask-based reconstruction algorithm still suffers from insufficient luminous flux, poor frequency domain characteristics and slow convergence speed. In addition, incoherent imaging cannot recover the phase information of the object, limiting its further application.
Different from incoherent imaging, the complex amplitude is linear during the propagation of coherent light, making it possible to recover the phase information. Phase recovery can be divided into interference-based methods [12][13][14] and intensity-based methods [15,16]. The main application of the interference-based method is holographic imaging, where the principle is to convert phase information into observable intensity information by introducing a reference light that interferes with the light carrying object information [17][18][19]. Although the holographic imaging method requires a relatively low amount of data, it is also susceptible to the interference of conjugate solutions. Besides, the experiment is difficult to carry out due to the high requirements for the stability of the optical path. The intensity-based method applies phase recovery such as Gerchberg-Saxton (GS) algorithm or its variants for image reconstruction [20][21][22]. However, the GS algorithm requires objects to satisfy the sparsity condition and has relatively slow convergence. In contrast, ptychography iterative engine (PIE) algorithm solves the complex amplitude distribution of high-resolution samples by acquiring a series of low-resolution images [23]. G. Zheng et al. placed the diffuser on the sensor surface and moved it through a two-dimensional high-precision displacement platform to modulate the object wavefront, enabling high-resolution pathology imaging with a wide field-of-view [24]. C. Lu et al. replaced mechanical scanning devices with LED arrays, to reduce the impact of mechanical errors [25]. However, these methods rely on the movement of light sources or sensors, which will inevitably introduce the mechanical errors.
Another method of wavefront modulation is to use a spatial light modulator (SLM), which is capable of modulating light waves according to a given pattern. M. Deweert et al. used SLM as a programmable amplitude mask to achieve incoherent imaging in natural light [26]. Y. Wu et al. achieved lensless high-resolution microscopic dynamic imaging of multiple samples and multiple scenes using high performance SLM [27]. The reconstruction performance of these methods depends on the accuracy of the modulation pattern, which imposes a high demand on the SLM hardware.
To address these problems, in this paper, we report a lensless imaging method based on SLM and ptychographic algorithm. We employ a low-cost SLM to modulate the phase of the object wavefront and design an interferometric method to calibrate the modulation curve of the SLM. Combining with the ePIE algorithm, our method can recover the amplitude and phase information simultaneously, which improves the robustness and convergence speed of the system. The contributions of our work mainly include:
(1) We replace the mechanical displacement platform with a programmable random pattern and recover the amplitude and phase of the object utilizing ptychography scanning algorithm.
(2) We calibrate the modulation curve of low-cost SLM using split-beam interference experiment.
(3) We build a real coherent lensless imaging system and evaluate the effects of modulation function, algorithm parameters, and the light source on the quality of reconstructed image.
Methods
Forward lensless imaging model
The schematic diagram of proposed lensless imaging system is shown in Fig.1 (a). In this system, we utilize a laser as coherent light source and convert it to parallel light by a collimator. Inspired by [28], we also insert two polarizers in the front and rear of the sample to fix the wavelength and polarization state of the light source to obtain stable modulation. The light of the object diffracts from the object plane to SLM plane after propagating 1 distance. The SLM changes the complex amplitude state of the unit by controlling the directions of liquid crystal molecules through voltage, thus realizing the modulation. Finally, the modulated complex amplitude is propagated 2 distance to the sensor plane. Since there is no lens in the imaging system, the object cannot be discerned from the image captured by CCD. A series of coherent diffraction images can be acquired by changing the modulation mode of the SLM. For the system shown in Fig.1(a), the intensity image produced by -th modulated pattern on CCD can be summarized as:
(r ccd ) = 2 * 1 (r sample ) 2 ,(1)
where (r ccd ) is the -th intensity measurement on CCD, (r sample ) is the complex amplitude distribution of the object, r denotes the coordinates on the corresponding plane, is the modulation function, also named pattern, corresponding to the -th measurement, represents the diffraction process with propagation distance . 1 is the distance between object and SLM, and 2 is the distance between SLM and CCD. As shown in Fig. 1(b), when the diffraction distance is relatively small, the diffraction process can be approximated as Fresnel diffraction, and has the following expression:
( ( , )) = j j ∬ ∞ −∞ ( , ) exp j 2 ( − ) 2 + ( − ) 2 ,(2)
where the (·) is the complex amplitude distribution of the plane, ( , ) and ( , ) represent the coordinates of different diffraction planes respectively, = 2 / is the wave vector, is the wave length, and is the distance between the two diffraction planes. Further, we define an intermediate variable:
ℎ( , ) = j j exp j 2 2 + 2 ,(3)
then the Fresnel diffraction process ( ( , )) can be expressed as :
( ( , )) = ( , ) ⊗ ℎ( , ),(4)
where "⊗" denotes convolution. Similarly, the image of the SLM plane can be calculated from the sensor plane according to the Fresnel inverse diffraction formula:
− ( ( , )) = j j ∬ ∞ −∞ ( , ) exp −j 2 ( − ) 2 + ( − ) 2 .(5)
The modulation function is controlled by a grayscale image, where the intensity of each pixel represents a complex value + j. The amplitude of this complex value represents the transmittance, ranged from 0 to 1, and its phase represents the phase modulation, ranged from −1.2 to 1.2 . To meet the requirements of the ptychography scanning algorithm, we use a random uniformly distributed image of size of 768 × 768 as a pattern, and a series of modulation functions with overlapping regions are generated by translating and stitching this pattern, as shown in Fig.2 and Visualization 1.
Image reconstruction algorithm
The ePIE algorithm [29] is not only able to reconstruct the unknown illumination function and the complex amplitude of object, but also has much higher robustness than traditional GS-based algorithms. In this paper, we employ the ePIE algorithm to jointly reconstruct the modulation function of SLM and the object function, which can greatly reduce the impact of SLM modulation errors.
In this algorithm, the recovered object wavefront on SLM plane and the initial random uniformly pattern of the SLM for the first measurement are 0 (r SLM ) and 0 (r SLM ), respectively. Since there is only translational relations between different modulation function, the pattern of the -th measurement can be expressed as (r SLM − s n ), where s n denotes the pixel displacement between different patterns. The modulated complex amplitude Ψ(r SLM ) is then propagated to the CCD plane by the Fresnel diffraction formula, and replace its amplitude with the square root of real image captured by digital sensor, √︁ (r ccd ). The complex amplitudes after replacement in the CCD plane is further back propagated to the SLM plane, denoted by Ψ (r SLM ), according to the Fresnel inverse diffraction formula. The inputs of next iteration, +1 (r SLM ) and +1 (r SLM ), can be updated as:
+1 (r SLM ) = (r SLM ) +¯( r SLM − s ) | (r SLM − s )| 2 max Ψ (r SLM ) − Ψ (r SLM ) , (6) +1 (r SLM − s ) = (r SLM − s ) +¯( r SLM ) (r SLM ) 2 max Ψ (r SLM ) − Ψ (r SLM ) ,(7)
where and are the iterative update coefficients, usually taken as 1, and¯and¯denote the conjugate of the corresponding values respectively. The +1 (r SLM − s ) also needs to be re-shifted by s pixels to get the initial pattern, +1 (r SLM ). To clearly demonstrate the ePIE-based lensless reconstruction algorithm, the pseudocode is given as below:
Ψ (r SLM ) = (r SLM ) * (r SLM − s n ) 5: Φ (r ccd ) = 2 (Ψ (r SLM )) 6: Φ (r ccd ) = √︁ (r ccd ) exp ( · ∠Φ (r ccd )) 7: Ψ (r SLM ) = − 2 (Φ (r ccd )) 8:
Update the +1 (r SLM ) using Eq. (6) 9:
Update the +1 (r SLM − s ) using Eq. (7) 10:
+1 (r SLM ) = +1 (r SLM − s + s ) 11:
end for 12: end for 13:
(r sample ) = − 1 ( (r SLM ))
The simulation result is shown in Fig.3, where the red curve is the real SLM phase/amplitudegrayscale curve, and the blue one is the initial modulation function used for reconstruction. When there are errors between these two curves, the original image can be recovered by ePIE algorithm while traditional GS algorithms such as the Amplitude-phase retrieval (APR) algorithm cannot converge.
SLM calibration
Theoretically, the ePIE can recover the complex amplitude of the object even if the modulation function is unknown, but the selection of the initial value can greatly affect the convergence speed. Practically, the modulation function can be approximated from the grayscale image via the calibration curve, which is usually given at the factory. So, the modulation function of the SLM is usually taken as the initial pattern for ePIE. However, such SLM is usually more expensive, and the polarization state error also affects the accuracy of the calibration curve to some extent. Therefore, we approximately measure the amplitude-grayscale and the phase-grayscale curve through the optical power meter and light interference.
The amplitude-grayscale calibration curve is relatively simple to measure by replacing the sensor with an optical power meter. In the measurement, the color range of [0-255] is scaled into [0,7,15,...,255], and the intensity corresponding to each grayscale image with a single value is recorded. The amplitude-grayscale calibration curve can thus be determined by interpolation. It should be noted that the phase-grayscale calibration curve records the phase difference under interference. The split beam interference optical path is shown in Fig.4(a). One beam of laser light reaches the sensor directly through the reflecting prism and the beam splitter, while the other beam passes through the SLM in its corresponding optical path. Since the SLM will modulate the complex amplitude of the light, there will be an optical path difference between two beams on the CCD plane, resulting in interference. The input grayscale image of SLM is shown in Fig.4(b), which consists of two different color blocks at the top and bottom. In the grayscale image, the upper half is filled with black as reference, and the lower varies from 0 to 255. When the grayscale value changes, the position of the interference fringe will also be shifted. By comparing the offset of the interference fringe corresponding to different grayscale values, the phase-grayscale calibrate curve is determined.
Experiments and Results
Experimental set-up
We build up a lensless coherent imaging system based on SLM for performance verification in real experiments. The laser generator (THORLABS S4FC520) generates a stable 520 nm coherent light source, which is then transformed into parallel light by a collimator. To verify the proposed method, we select a low-cost SLM ($1500) with unknown modulation curve. The size of liquid crystal unit of SLM (tSLM-III) is 12.5 and the number of pixel is 1024 × 768. After propagating a certain distance, the image sensor captures the modulated diffraction image. The image contains 2448 × 2048 pixels, and the pixel size is 3.45 . The hardware configuration for reconstruction is as follows: GPU of GeForce GTX Titan X, RAM of 32GB, and CPU of Intel(R) Core(TM) i9-9820X. The number of iterations is set to 10, and the reconstruction takes about 3 minutes.
In terms of modulation function, we use a noisy image conforming to a Gaussian distribution as the initial pattern and generate a sequence of 169 grayscale images based on a spiral scan with 2 pixels offset. The distance between the sample and the sensor ranges from 7 to 13 cm. In Fig.5 (a), it is able to resolve (6,2) sets of the USAF target with resolution of 14 . Here (6,2) denotes the 6th group and the 2nd element in USAF, respectively, and the object is about 9 cm away from the sensor. The process of back propagation is also available in the Visualization 2. It can be seen that the resolution of our proposed system is very close to the size of liquid crystal unit of SLM. Similarly, in Fig.5 (b), We can clearly distinguish the veins of the leave. Fig.5 (c) and (d) show the results of the reconstruction with and without staining the cells, respectively. In the amplitude image of Fig.5 (c), the shape of the cell can be distinguished, and the dark region is the stained cytoplasm. While in the phase image, the nucleus is distinct for its higher refractive index. Further, we evaluated the recovery quality of unstained biological sample. As shown in Fig.5 (d), the distribution of unstained cell is not visible in the amplitude image, but it can be clearly observed in the phase image (seeing red arrows in phase details). This reflects the characteristics of phase imaging, which enables imaging of samples with high transparency.
Reconstruction results
Another application of lensless imaging is digital refocusing [30]. According to the diffraction equation Eq. (5), we can back-propagate the lensless imaging result to any depth and focus on a local region of the object. As shown in Fig.6, we conducted digital refocusing on samples tilted to the CCD plane, with the right side 1 cm farther from the sensor than the left side. Since the depth of the tilted USAF resolution target is very large, for a specific depth, the reconstructed image contains both in-focus and out-of-focus regions. For example, the regions within the red and yellow rectangles in Fig.6 (a) correspond to the out-of-focus and in-focus, respectively. In contrast, the regions within the red and yellow rectangles in Fig.6 (b) correspond to the in-focus and out-of-focus, respectively.
Robustness analysis
By incorporating the ePIE into our imaging system, our method can recover the complex amplitude of the object with high accuracy and fast speed, even if the modulation function is not exactly known. In this section, we discuss the effects of different factors on the reconstruction quality.
Effect of modulation function
In other SLM-based phase recovery algorithms, the calibrated curve is assumed to known and are generally reconstructed using 16 or 32 patterns that are independent of each other. In this paper, we use a series of patterns with a certain translation relationship for image reconstruction, and utlizing the ePIE algorithm can reduce the errors introduced by SLM itself. The effect of the relationship between different patterns on the reconstruction is shown in Fig.7. The reconstruction is performed using 32 intensity images in all experiments. It can be found only the combination of ptychographic patterns and ePIE algorithm can recover the information of the object, and when the SLM calibration curve has errors or using random patterns, the complex amplitude information cannot be reconstructed by either APR or ePIE algorithm.
It is stated the design of the pattern affects the reconstruction of the complex amplitude [31], and we carried out an experiment to analyze the relationship between pattern modes and reconstruction quality. Since the resolution of SLM is 12.5 , which is much larger than the 3.45 of the sensor, the randomness of the modulation function can greatly affect the reconstruction results, as shown in Fig.8. The bottom modulation function in Fig.8 is obtained by upsampling a random noise image of size 96 × 96, and the upper one is satisfied with independent random distribution condition. It can be seen that the pattern with high randomness will have better reconstruction results.
Effect of algorithm parameters
The parameters that affect the reconstruction quality of ePIE algorithm mainly include the number of iterations and the number of captured intensity images. In Fig.9 (a), we compared the effect of the number of intensity images on the reconstruction results. For USAF resolution target, the reconstruction quality does not get improved with the increase of captured intensity images, but stabilizes after a certain number. Besides, the fewer the intensity images used, the more iterations are needed. As shown in Fig.9(b), when 169 images are used, 30 iterations are sufficient to achieve the desired result, but 64 images require 50 iterations to achieve the corresponding result.
On the other hand, the modulation curve determined by the calibration experiment can significantly improve the convergence speed of the algorithm, as shown in Fig.10. Fig.10 (a) uses a random pattern as the initial pattern, while Fig.10 (b) uses the modulated grayscale image of SLM as the initial pattern. In both Fig.10 (a) and (b), 24 images are employed and 50 iterations are implemented. The result shows that using the calibrated pattern can reduce the interference of noise and improve the reconstruction quality.
Effect of light source
Since the laser in our experiment is not an exactly parallel light source, it will interfere with the phase of the object in CCD plane, resulting in regular fringes in the phase image and affecting the final phase recovery quality. In this regard, we first measured the complex amplitude distribution of the light source with the same experimental parameters, and then put the sample into the optical path to eliminate the effect of the light source on phase recovery by phase filtering. As can be seen from Fig.11 (a), the difference between the angle of light incidence causes periodically varying tilted fringes in the phase image. These disturbances can couple with the complex amplitude of the object and affect the recovery quality, as shown in Fig.11 (b). The decoupled phase image is shown in Fig.11(c), which significantly eliminates interference from laser sources.
Discussion
The experimental results have proven that our proposed method is able to reconstruct the complex amplitude information of the object without mechanical devices. We believe that the proposed method can be further applied in other tasks with the following factors considered: (1) The acquisition rate and reconstructed resolution of the system in this study are still constrained by the sensor and the SLM itself. With high-precision hardware, the system has the potential to achieve higher resolution (close to 1 ) and real-time dynamic imaging at ms level. (2) The proposed lensless coherent imaging system can recover the wavefront of the object, and therefore has potential applications in the field of tomography. Conventional optical tomographic imaging often requires image acquisition at multiple focal planes to improve resolution. Our proposed system does not require any mechanical scanning and can achieve refocus at any depths according to the inverse propagation formula.
Conclusion
In this paper, we report a low aberration and wide FOV lensless imaging system based on SLM. First, we build a forward diffraction model of the object in coherent diffraction scenario. The simulation experimental results show that the object information can be recovered using the ePIE algorithm even the modulation function is not exactly known. Second, we design an optical interference experiment to coarsely calibrate the modulation curve of SLM, which can accelerate the convergence speed of the algorithm. Finally, we carry out a series of comparative experiments to evaluate the performance of our system. The reconstructed results show our lensless system can achieve to 14 resolution and is able to reduce the requirement of the accuracy of SLM. In addition, we also discuss the effect of different parameters on the reconstruction quality. We believe this system can be further applied in other tasks such as tomography and microscopy in the future.
Fig. 1 .
1The schematic diagram of proposed lensless imaging system. (a) The optical path of our system and (b) The Fresnel diffraction process.
Fig. 2 .
2The modulation process of SLM. (a)The random modulation function of the SLM, where the intensity of each pixel represents a complex value. (b) The scanning sequence of the pattern. (c) A series of diffraction intensity images of USAF resolution target captured by digital sensor under different patterns.
Algorithm 1 ePIE-based lensless reconstruction algorithmInput: Diffraction images ( = 1, 2, ..., ) with the translational shift of the pattern Output: Complex amplitude of the object (r sample ) and the modulation function of SLM (r SLM ) 1: Initialize the complex amplitude on SLM plane 0 (r SLM ) and 0 (r SLM ), and specify the number of iterations 2: for m=1:
Fig. 3 .
3The process of lensless imaging reconstruction based on ePIE algorithm.
Fig. 4 .
4The calibration experiment. (a) The optical path of split beam interference experiment. (b) The grayscale image and its corresponding interference fringe.
Fig. 5 .
5The reconstructed results of the complex amplitude of (a) USAF-1951 resolution target, (b) leaf specimen, (c) stained cell slide, and (d) unstained cell slide.
Fig. 6 .
6Reconstruction results at different back-propagation distances: (a) back propagation 63 mm, and (b) back propagation 64.5 mm.
Fig. 7 .
7The reconstructed results using (a) spiral scan path and APR algorithm, (b) random scan path and ePIE algorithm, and (c) spiral scan path and ePIE algorithm.
Fig. 8 .
8The effect of the randomness of the modulation function on the reconstruction. (a) Different modulation function. The randomness of the upper modulation function is higher than the bottom. (b) The intensity image corresponding to the pattern. (c) The reconstructed results.
Fig. 9 .
9The effect of the algorithm parameters on the reconstruction. (a) Reconstruction using different numbers of intensity images with 50 iterations. (b) Reconstruction using different numbers of iterations with 169 intensity images.
Fig. 10 .
10The effect of the initial modulation function on the reconstruction.(a) Using random pattern as initial pattern. (b) Using calibrated pattern as initial pattern.
Fig. 11 .
11Phase reconstruction results of (a) light source only, (b) object mixing with light source, and (c) decoupled object.
Funding.
This work was supported in part by the National Natural Science Foundation of China under Grants 61922048 and 62031023, in part by the Shenzhen Science and Technology Project (JCYJ20200109142808034, JCYJ20180508152042002, and JSGG20191129110812708), and in part by Guangdong Special Support under Grant 2019TX05X187.
Disclosures. The authors declare no conflicts of interest.Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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"Ezio Vasselli \nAlcune Note di Analisi Matematica\nDipartimento di Matematica\nUniversitá di Roma \"\nLa Sapienza\" P.le Aldo Moro, 200185Rome -Italy\n"
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"Alcune Note di Analisi Matematica\nDipartimento di Matematica\nUniversitá di Roma \"\nLa Sapienza\" P.le Aldo Moro, 200185Rome -Italy"
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| Metrizzabilitá e compattezza.Sia X uno spazio topologico con topologia τ X . Un intorno di x ∈ Xè un sottoinsieme U di X , tale che x ∈ U ′ ⊆ U per qualche aperto U ′ ∈ τ X . Diciamo che Xè separabile se esiste una successione X 0 := {x n ∈ X} densa in X , il che vuol dire che per ogni apertoData una metrica,è possibile definire su X la topologia avente come sottobase la famiglia dei dischiUna successione {x n } ⊆ X si dice di Cauchy se per ogni ε > 0 esiste n 0 ∈ N tale che d(x n , x m ) < ε , ∀n, m ≥ n 0 . Diremo che {x n } converge ad x ∈ X se per ogni ε > 0 esiste n 0 ∈ N tale che d(x, x n ) < ε , ∀n > n 0 . Uno spazio metrico (X, d) si dice completo se ogni successione di Cauchy converge ad un elemento di X .Proposizione 2.1. Uno spazio metrico compatto (X, d)è separabile. | 10.13140/rg.2.1.5017.4569/1 | [
"https://arxiv.org/pdf/1104.3699v6.pdf"
]
| 117,644,051 | 1104.3699 | 4bfa1a84520db58b25fcdcd63b9efa4daa3c1d82 |
14 Nov 2012 January 19, 2013
Ezio Vasselli
Alcune Note di Analisi Matematica
Dipartimento di Matematica
Universitá di Roma "
La Sapienza" P.le Aldo Moro, 200185Rome -Italy
14 Nov 2012 January 19, 2013
Metrizzabilitá e compattezza.Sia X uno spazio topologico con topologia τ X . Un intorno di x ∈ Xè un sottoinsieme U di X , tale che x ∈ U ′ ⊆ U per qualche aperto U ′ ∈ τ X . Diciamo che Xè separabile se esiste una successione X 0 := {x n ∈ X} densa in X , il che vuol dire che per ogni apertoData una metrica,è possibile definire su X la topologia avente come sottobase la famiglia dei dischiUna successione {x n } ⊆ X si dice di Cauchy se per ogni ε > 0 esiste n 0 ∈ N tale che d(x n , x m ) < ε , ∀n, m ≥ n 0 . Diremo che {x n } converge ad x ∈ X se per ogni ε > 0 esiste n 0 ∈ N tale che d(x, x n ) < ε , ∀n > n 0 . Uno spazio metrico (X, d) si dice completo se ogni successione di Cauchy converge ad un elemento di X .Proposizione 2.1. Uno spazio metrico compatto (X, d)è separabile.
In queste note vengono discussi argomenti si solito trattati in corsi di Analisi reale e complessa, Analisi Funzionale ed Analisi superiore. Viene assunta la conoscenza dei fondamenti di Topologia, Algebra lineare ed Analisi (nozioni elementari sulla continuitá, calcolo differenziale ed integrale in una e piú variabili reali).
Essendo mia intenzione restare intorno a 200 pagine di lunghezza penso di avere abbondato in concisione, il che spero abbia agevolato la chiarezza dell'esposizione piuttosto che pregiudicarla. D'altra parte ho cercato di essere auto-contenuto nei limiti del possibile; in particolare, le dimostrazioni dei risultati principali sono svolte in modo completo, e quelle volte in cui, invece, viene fornito solo un argomento di massima cióè debitamente evidenziato. A volte vi sono accenni ad argomenti non prettamente attinenti l'analisi (ad esempio, l'omologia in relazione al teorema di Brouwer in §2.3 o la dualitá di de Rham in §6.4): cióè voluto edè inteso a stimolare la curiositá di chi legge, nonché a supportare il punto di vista che la matematica nonè divisa in compartimenti stagni.
Ho tenuto quel livello di astrazione che considero utile per la comprensione dei risultati. Ad esempio, penso sia controproducente trattare i teoremi di passaggio al limite sotto il segno di integrale limitandosi al caso della retta reale, quando le stesse dimostrazioni si applicano a piú generici spazi misurabili. A parte l'ovvio vantaggio di avere teoremi validi in un ambito piú generale, muovendosi ad un livello piú astratto si ha la possibilitá di capire quali sono le proprietá dell'oggetto "concreto" (nella fattispecie, la retta reale) cruciali per la dimostrazione dei risultati. D'altro canto, spero di aver inserito un numero accettabile di esempi.
Ci sono argomenti che mi riprometto di includere in una futura versione, pur cercando di non valicare i limiti di cui ho scritto nelle righe precedenti. Tra questi ci sono senz'altro un'esposizione completa dei teoremi di Uryshon, Tietze e Stone-Weierstrass ( §2.2) ed una discussione sulla monodromia nell'ambito delle funzioni di variabile complessa (in particolare il logaritmo). Ovviamente la scelta degli argomenti trattati, e la misura del loro approfondimento, sono del tutto personali e quindi opinabili.
Molti esercizi sono ripresi da altre fonti. Nella maggiorparte dei casi ho inserito la referenza originale, dove spesso (e volentieri, suppongo) il lettore potrá trovare la relativa soluzione. Alcuni esercizi provengono da prove di esame per concorsi di ricercatore, e sono stati inseriti in quanto mi sono sembrati interessanti a livello pedagogico. In altri casi gli esercizi sono dei veri e propri complementi, come ad esempio il lemma di Borel-Cantelli (Es.4.5), il lemma di Riesz (Es.7.4), e le convoluzioni di misure (Es.6.7).
A prescindere dalle fonti utilizzateè possibile che queste note non siano scevre di inesattezze, errori matematici o di esposizione, ed ogni segnalazione in meritoè benvenuta.
Segue un elenco dei capitoli con relativi commenti e referenze. In §2 richiamiamo alcuni risultati di topologia generale di interesse in analisi, inclusi i teoremi di Tietze e Stone-Weierstrass, e dimostriamo i teoremi delle contrazioni e di Ascoli-Arzelá.
Rudimenti sulle equazioni differenziali ordinarie vengono dati in §3. Per questa sezione mi sono basato su [12,29], ad eccezione dell'Esempio 3.1, ripreso da vecchie dispense del Prof. P. Acquistapace, e del Teorema di Peano.
In §4 vengono trattati argomenti classici quali la teoria della misura e dell'integrazione secondo Lebesgue, i teoremi sul passaggio al limite sotto il segno di integrale e le funzioni AC-BV. Fonti principali sono [25,17,9]. La sezione 5 contiene una discussione sugli spazi L p , con particolare attenzione alla completezza (Teorema di Fischer-Riesz) ed alla dualitá di Riesz.
§6è una raccolta piuttosto eterogenea di appunti su funzioni di piú variabili: include una dimostrazione del Teorema della funzione implicita, una breve discussione sulle forme differenziali, i fondamenti del calcolo variazionale e, ad un livello un pó piú avanzato, i teoremi di Fubini e le convoluzioni, queste ultime (brevemente) discusse anche dai punti di vista dell'analisi funzionale e dell'analisi armonica astratta. Fonti principali sono [12,5].
La sezione 7è sicuramente quella piú approfondita e concerne l'analisi funzionale. Accanto ai rudimenti sugli spazi di Banach e di Hilbert, ed a risultati orientati verso la teoria delle equazioni alle derivate parziali (Teoremi di Stampacchia-Lax-Milgram, teorema di Schauder), il lettore troverá un'esposizione dei fondamenti delle algebre di operatori e delle distribuzioni, argomenti, questi ultimi, di interesse in fisica matematica ed in meccanica quantistica in particolare. Particolarmente corposaè la sezione degli esercizi, dove vengono approfonditi gli aspetti inerenti la teoria spettrale e le connessioni tra spazi di Hilbert, teoria della misura ed algebre di operatori. Le fonti principali sono [5,17,24,19].
L'analisi di Fourier viene trattata in §8. Oltre agli argomenti classici, vengono dati alcuni accenni ai gruppi topologici ed alla trasformata di Fourier astratta. Per le serie di Fourier ho seguito [12], mentre per la trasformata di Fourier mi sono basato su [10].
§9 concerne i fondamenti dell'analisi complessa edè fortemente debitrice degli appunti di un corso di Istituzioni di geometria superiore tenuto dal Prof. E. Arbarello alcuni anni fa (metá anni novanta). Una versione "ufficiale" di tali appunti, scritta dai Prof. Arbarello e Salvati-Manni,è reperibile alla pagina web 1 [4].
In §10 vengono dati alcuni accenni sugli spazi di Sobolev e dimostrati risultati di esistenza ed unicitá per problemi alle derivate parziali. Ho seguito in modo piuttosto pedissequo [5].
2 Alcuni risultati di topologia generale.
In questa sezione richiamiamo, senza pretese di esaustivitá, alcune nozioni di topologia generale che verranno usate spesso nel seguito. Successivamente dimostriamo alcuni risultati, a priori prettamente topologici, quali il teorema delle contrazioni e quello di Ascoli-Arzelá, i quali hanno peró importanti applicazioni in analisi.
Alcune proprietá delle funzioni continue.
Sia X uno spazio topologico e C(X) l'algebra delle funzioni continue su X a valori reali 2 . Un ben noto teorema di Weierstrass afferma che se Xè compatto allora f ∞ := sup x∈X |f (x)| < +∞ ; l'applicazione · ∞ soddisfa le proprietá
f ∞ = 0 ⇒ f = 0 , f + g ∞ ≤ f ∞ + g ∞ , ∀f, g ∈ C(X) ,
dunqueè una norma nel senso di §7 (o, equivalentemente, C(X)è uno spazio normato). Assumiamo ora, piú in generale, che X sia uno spazio localmente compatto, il che vuol dire che ogni x ∈ X ammette un intorno compatto; diciamo che f ∈ C(X) si annulla all'infinito se per ogni ε > 0 esiste un compatto K ε ⊂ X tale che sup x∈X−Kε |f (x)| < ε .
(2.1)
Denotiamo con C 0 (X) l'insieme delle funzioni continue che si annullano all'infinito, il quale, al pari di C(X),è un'algebra. Ora, (2.1) ed il teorema di Weierstrass implicano che
f ∞ = sup x∈Kε |f (x)| + sup x∈X−Kε |f (x)| < sup x∈Kε |f (x)| + ε < +∞ , ∀f ∈ C 0 (X) ,
per cui · ∞è ben definita su C 0 (X), il qualeè quindi uno spazio normato. Nel risultato seguente stabiliamo che C(X), C 0 (X) sono completi, ovvero spazi di Banach nel senso di §7. Allora esiste edè unica f ∈ C 0 (X) tale che lim n f − f n ∞ = 0 , ed {f n } converge uniformemente ad f . Lo stesso vale per successioni in C(X) nel caso in cui X sia compatto, con f ∈ C(X).
Dimostrazione. Scelto ε > 0 abbiamo che esiste n ε ∈ N tale che f m − f n ∞ < ε , ∀n, m ≥ n ε ⇒ |f n (x) − f m (x)| < ε , ∀x ∈ X .
(2.2)
Dunque ogni successione {f n (x)} , x ∈ X ,è di Cauchy in R ed esiste il limite f (x) := lim n f n (x). Ció definisce un'unica funzione f : X → R, alla quale {f n } converge uniformemente grazie al fatto che n ε non dipende da x. Grazie all'uniformitá della convergenza troviamo
|f (x) − f n (x)| < ε , ∀n ≥ n ε , x ∈ X ⇔ f − f n ∞ < ε , ∀n ≥ n ε ,
e quindi lim n f − f n ∞ = 0 . Per verificare che fè continua scegliamo ε > 0 ed osserviamo che esiste n ε ∈ N tale che f − f nε ∞ < ε/3 ; per continuitá di f nε esiste un intorno U ε ∋ x tale che |f nε (x) − f nε (x ′ )| < ε/3 per ogni x ′ ∈ U ε , per cui (sommando e sottraendo f nε (x), f nε (x ′ )) abbiamo la stima
|f (x) − f (x ′ )| ≤ 2 f − f nε ∞ + |f nε (x) − f nε (x ′ )| < ε .
Dunque fè continua. Verifichiamo infine, nel caso localmente compatto, che f si annulla all'infinito: scelto ε > 0 sappiamo che esistono n ε ∈ N con f − f nε ∞ < ε/2 ed un compatto K ε con sup x∈X−Kε |f nε (x)| < ε/2 . Per cui, (ovvero, se f ∈ A allora f * ∈ A); per una loro dimostrazione, peraltro "elementare" nel senso che non richiede nozioni non standard, rimandiamo a [19, §4.3].
Osservazione 2.1. La condizione (2.3)è indispensabile per ottenere la densitá di A in C(X, C) nel caso complesso. Ad esempio, prendiamo la palla unitaria ∆ := {z ∈ C : |z| ≤ 1} e l'algebra O(∆) delle funzioni analitiche in ∆ (vedi §9.2). E' chiaro che O(∆) contiene le costanti e separa i punti di ∆ (infatti, presi z = z ′ ∈ ∆ la funzione f (z) := z , z ∈ ∆,è analitica e tale che f (z) = f (z ′ )). Tuttavia, O(∆)è ben lungi dall'essere denso in C(∆, C); fosse cosí troveremmo O(∆) = C(∆, C), essendo O(∆) completo rispetto alla norma · ∞ (vedi Teo.9.11). Ma cióè assurdo, in quanto la funzione f * (z) := z , z ∈ ∆,è continua ma non analitica, come si dimostra usando le equazioni di Cauchy-Riemann (Lemma 9.1); l'esempio di f * mostra anche che O(∆) nonè chiuso rispetto al passaggio al coniugato, il che spiega il motivo per cui il teorema di Stone-Weierstrass nonè applicabile a O(∆), il qualeè quindi un sottospazio proprio di C(∆, C).
Il teorema di Stone-Weierstrass generalizza il classico teorema di densitá di Weierstrass (che si ritrova per X = [0, 1] ed A l'algebra dei polinomi, vedi [12, Teo.2.8.1]), nonché il teorema di densitá dei polinomi trigonometrici nell'algebra delle funzioni continue e periodiche su [0, 2π] (vedi [24,Thm.4.25]). Osserviamo che le dimostrazioni dei due risultati di cui sopra, a differenza di Teo.2.7, si basano sull'uso delle convoluzioni ( §6.7). Il Lemma seguente verrá utilizzato nel seguito (Prop.6.1): Lemma 2.9. Per ogni spazio localmente compatto di Hausdorff X , valgono le seguenti proprietá: (1) Per ogni x ∈ X esiste f ∈ C 0 (X) tale che f (x) = 0 ; (2) C 0 (X) separa i punti di X .
Dimostrazione. Sia Y ⊂ X compatto e tale che x appartenga alla parte internaẎ . Essendo X di Hausdorff, Yè anche chiuso (vedi [6,Prop.10.6] o [19, 1.6.5]). Essendo Y compatto e di Hausdorff, essoè anche normale. Possiamo ora dimostrare i due punti dell'enunciato: (1) Sia U ⊂Ẏ , U ∋ x; allora W = Y − Uè chiuso sia in Y che in X , e chiaramente disgiunto da {x} . Per il Lemma di Uryshon esiste f ∈ C(Y ) tale che f (x) = 1 e f | W = 0 . Del resto, per costruzione f si annulla sulla frontiera di Y , dunque estendiamo f ad X definendo f | X−Y := 0 e cosí otteniamo la funzione cercata. (2) Se x ′ = x, allora possiamo assumere che sia x che x ′ siano contenuti in un compatto Y . Scegliendo un intorno U ⊂Ẏ , U ∋ x, tale che x ′ / ∈ U e ragionando come nel caso precedente concludiamo che esiste f ∈ C 0 (X) tale che f (x) = 1 , f (x ′ ) = 0 .
Teoremi di punto fisso.
Il teorema seguenteè il piú classico tra quelli noti come teoremi di punto fisso. Motivato dalla questione dell'esistenza di soluzioni di equazioni differenziali (vedi §3), esso ha segnato un importante passo in avanti dal punto di vista concettuale, quello per il quale una funzione si puó riguardare come un "punto" in uno spazio topologico. Tra le varie applicazioni menzioniamo il teorema di Cauchy (Teo.3.1), il teorema delle funzioni implicite (Teo.6.6), ed i teoremi di Stampacchia-Lax-Milgram (Teo.7.27). Teorema 2.10 (Teorema delle contrazioni, Banach-Caccioppoli). Sia (X, d) uno spazio metrico completo e T : X → X un'applicazione continua tale che esista α ∈ (0, 1) con d(T x, T x ′ ) ≤ αd(x, x ′ ), ∀x, x ′ ∈ X . Allora esiste edè unico x ∈ X tale che T x = x .
Dimostrazione. Poniamo x n := T n x e stimiamo d(x n+1 , x n ) ≤ αd(x n , x n−1 ) ≤ . . . ≤ α n d(x 1 , x) .
Inoltre, per diseguaglianza triangolare,
d(x m+1 , x n ) ≤ m+1 k=n+1 d(x k , x k−1 ) ≤ m+1 k=n+1 α k d(x 1 , x) = α n m−n k=1 α k d(x 1 , x) ≤ d(x 1 , x) α n 1 − α ,
dunque (avendosi 0 < α < 1 ) {T n x}è di Cauchy. Il punto limite x soddisfa per costruzione l'uguaglianza T x = T lim n T n x = lim n T n+1 x = x , dunqueè un punto fisso. Inoltre, se
x ′ ∈ X soddisfa T x ′ = x ′ allora troviamo d(x, x ′ ) = d(T x, T x ′ ) ≤ αd(x, x ′ ), per cui d(x, x ′ ) = 0 .
Nelle righe che seguono discutiamo un altro risultato di punto fisso, il Teorema di Brouwer, il quale ha conseguenze importanti sia in analisi che in geometria. Nella sua forma piú semplice, quella in dimensione uno, essoè conseguenza di un teorema di Bolzano, il teorema del valore intermedio, il quale afferma che se f : Ci limiteremo qui ad esporre l'idea della dimostrazione di un caso particolare del teorema di Brouwer, la quale fa uso dei spazi di omologia, riguardo i quali rimandiamo a §6.4 e, piú in dettaglio, [4,Cap.5] (per un approccio diverso si veda [2, §6.8]). Per ogni n ∈ N, denotiamo con D n ⊂ R n la palla unitaria (chiusa) e con S n−1 ⊂ R n la sfera unitaria, che identifichiamo con il bordo ∂D n ⊂ D n . Teorema 2.13. Sia n ∈ N, ed f : D n → D n un'applicazione continua. Allora esiste x ∈ D n tale che f (x) = x.
Sketch della dimostrazione. Supponiamo per assurdo che f non abbia punti fissi. Allora per ogni x ∈ D nè ben definito il punto F (x) ∈ ∂D n ≃ S n−1 come l'intersezione tra ∂D n e la retta passante per x ed f (x). Otteniamo cosí un'applicazione continua F : D n → S n−1 tale che F (x) = x , ∀x ∈ S n−1 .
Ora, l'ideaè quella di dimostrare che Fè un ritratto per deformazione 4 , il che implica, per proprietá generali degli spazi di omologia, che si ha un'applicazione lineare iniettiva F : H n−1 (S n−1 ) → H n−1 (D n ) , n ∈ N . 4 In generale, dato uno spazio topologico X ed S ⊂ X , un ritratto per deformazioneè un'applicazione continua F : X → S tale che: (1) F | Sè l'identitá di S ; (2) esiste un'applicazione continua (detta omotopia) H : X ×[0, 1] → X tale che H(x, 0) = F (x) , H(x, 1) = x , ∀x ∈ X . Su tali argomenti rimandiamo ancora a [4,Cap.5].
che {g n } converge uniformemente in X . Consideriamo ε > 0 ; per equicontinuitá esiste δ ε > 0 tale che d(x, x ′ ) < δ implica d(g n (x), g n (x ′ )) < ε/3 , ∀n ∈ N .
Scegliamo quindi m ε ∈ N tale che m ε > 1/δ ε , cosicché ∆ f 1/mε ricopre X (vedi dimostrazione di Prop.2.1). Osserviamo che poiché {g n } converge puntualmente in X mε ⊂ X 0 , esiste n ε ∈ N tale che per ogni n, m > n ε risulta d(g n (y), g m (y)) < ε/3 , ∀y ∈ X mε (osservare che X mεè finito, altrimenti avremmo dei problemi inerenti la convergenza non uniforme di {g n } ). Sottolineiamo che il nostro n ε ∈ N dipende, in ultima analisi, solo da ε . Ora, se x ∈ X troviamo x ∈ ∆(y, δ ε ) per qualche y ∈ X mε (infatti, 1/m ε < δ ε ), e d(g n (x), g m (x)) ≤ d(g n (x), g n (y)) + d(g n (y), g m (y)) + d(g m (y), g m (x)) < ε .
Esercizi.
Esercizio 2.1. Fissato α > 0 , denotiamo con L α l'insieme delle funzioni f : [0, 1] → R continue a tratti e tali che sup x∈[0, 1] |f (x)| ≤ α . Si mostri che la famiglia (Suggerimenti: per il secondo quesito ovviamente si applica il teorema delle contrazioni. Riguardo il terzo quesito si applichi il teorema fondamentale del calcolo derivando membro a membro l'uguaglianza f 0 = T f 0 . Riguardo il quarto quesito, si osservi che derivando membro a membro l'uguaglianza f 0 = T f 0 in questo caso si ottiene la piú semplice delle equazioni differenziali ordinarie).
Esercizio 2.3. Sia X uno spazio metrico compatto. Si mostri che se F ⊂ C(X)è precompatto allora essoè limitato.
(Suggerimento: ragionando per assurdo, si assuma che F sia non limitato e si deduca che esiste una successione {f n } ⊂ F con f n ∞ ≥ n, ∀n ∈ N; si osservi infine che tale successione non puó avere sottosuccessioni convergenti).
Esercizio 2.4. Sia X uno spazio metrico compatto e K ∈ C(X × X). Per ogni x ∈ X si definisca κ x (y) := K(x, y), y ∈ X , e si dimostrino le seguenti proprietá: (1) κ x ∈ C(X) per ogni x ∈ X ;
(2) La famiglia F := {κ x } x∈X ⊂ C(X)è equicontinua.
(Suggerimenti: si osservi che X × Xè metrico 6 e compatto, per cui Kè uniformemente continua; si osservi quindi che d 2 ((x, y), (x, y ′ )) = d(y, y ′ ), ∀x, y, y ′ ∈ X ). 3 Equazioni differenziali ordinarie.
La ricerca di soluzioni di equazioni differenzialiè una delle questioni caratterizzanti dell'analisi. A livello storico, impulso determinante per lo studio delle equazioni differenzialiè stata la meccanica newtoniana, nell'ambito della quale esse hanno il ruolo di tradurre in termini matematici princípi fondamentali, come ad esempio quelli della dinamica. Un'equazione differenziale si puó pensare come un insieme di relazioni (algebriche, nei casi piú elementari) che legano la funzione incognita alle sue derivate. La funzione incognita si interpreta tipicamente come una grandezza che evolve nel tempo, si pensi ad esempio alla posizione nello spazio di un corpo materiale; dunque la sua conoscenza implica, nella misura in cui l'equazione (ben) descriva un sistema fisico, la possibilitá di prevederne il comportamento.
Diamo ora alcune definizioni piú precise. Sia n ∈ N ed A ⊆ R n+1 aperto; un'equazione differenziale ordinaria si presenta come un'espressione del tipo f (u, u ′ , u ′′ , . . . , u (n) ) = 0 , dove f : A → Rè una funzione (solitamente) continua, ed u : I → R, I ⊆ R aperto,è la funzione incognita. Osserviamo cheè sempre possibile ricondursi ad un problema di primo grado (n = 1 ), sostituendo u con la funzione dove u : I → R n , f : A → R n , con I ⊆ R, A ⊆ R n+1 aperti. Il problema (3.1) puó essere arricchito con ulteriori condizioni che u , e/o la sua derivata, devono soddisfare. In questa sezione consideriamo il Problema di Cauchy:
u ′ = f (t, u) u(t 0 ) = u 0 ,(3.2)
dove f : A → R n , (t 0 , u 0 ) ∈ A ⊆ R n+1 . Ora, in generale il calcolo esplicito della soluzione di (3.2)è un compito impossibile;è allora importante produrre teoremi che ne assicurino l'esistenza e (possibilmente) l'unicitá. Per costruzione Xè uno spazio metrico completo, una volta equipaggiato della distanza indotta dalla norma dell'estremo superiore. Il contenuto intuitivo della definizione precedenteè che stiamo considerando funzioni la cui immagine sia contenuta nel disco J all'interno del quale fè Lipschitz. Introduciamo ora l'operatore di Volterra Come prima cosa, verifichiamo che F sia ben definito, ovvero che si abbia effettivamente F u ∈ X . Chiaramente F uè continua in I 0 ; inoltre
F u − u 0 ∞ ≤ t t0 sup I0
|f (s, u(s))| ds ≤ M r 0 < r ′ , dunque F u ∈ X . Verifichiamo ora che Fè una contrazione:
F u − F z ≤ t t0
|f (s, u(s)) − f (s, z(s))| ds ≤ L u − z ∞ r 0 < u − z ∞ (infatti, Lr 0 < 1 ). Dunque, per Teo.2.10 esiste edè unica u tale che F u = u , ovvero
u(t) = u 0 + t t0
f (s, u(s)) ds , t ∈ I 0 .
L'espressione precedente ci dice che u ∈ C 1 (I 0 ), essendo essa una primitiva di f (·, u(·)) ∈ C(I 0 ). Derivando membro a membro concludiamo che uè la soluzione del problema (3.2).
La condizione di locale lipschitzianitá per f (t, ·) (ovvero il punto (2) dell'enunciato)è soddisfatta se fè convessa come funzione della variabile u (infatti ogni funzione convessaè localmente Lipschitz, vedi §4.7). Un'altra condizione sufficiente per la condizione di Lipschitz (nel caso n = 1 ) e che la derivata parziale di f rispetto ad u sia continua in un intorno di u 0 , come si verifica facilmente applicando il teorema del valor medio ad f (t, ·).
Infine, osserviamo che esplicitando la successione introdotta nella dimostrazione del teorema delle contrazioni in (3.3), si ottiene la successione di Peano-Picard y 0 := u 0 , y n+1 (t) := u 0 + t t0 f (s, y n (s)) ds , n ∈ N , (3.4) la quale fornisce una successione convergente alla soluzione del problema di Cauchy. Tuttavia, ad eccezione di casi particolarmente favorevoli, il calcolo degli integrali nell'espressione precedenteè in genere difficoltoso, ed in tal casoè conveniente procedere per approssimazione. Ad esempio, quando fè di classe C ∞ si puó procedere attraverso sviluppi in serie di Taylor, in modo da ridursi ad integrare dei polinomi; per esempi in tal senso rimandiamo a [26].
Prolungamento delle soluzioni.
Consideriamo il problema di Cauchy (3.2) definito su un intervallo I := (t 0 − δ, t 0 + δ). Una soluzione locale di (3.2)è il dato di una coppia (J, u), dove Jè un intervallo aperto contenuto in I ed u ∈ C 1 (J, R n )è una soluzione di (3.2); per brevitá, talvolta nel seguito scriveremo semplicemente u in luogo di (J, u). Denotiamo con C l'insieme di tali soluzioni locali.
Definizione 3.2. Siano (J 1 , u 1 ), (J 2 , u 2 ) ∈ C . Diciamo che (J 2 , u 2 )è un prolungamento di (J 1 , u 1 ) se J 1 ⊂ J 2 e u 2 | J1 = u 1 , ed in tal caso scriveremo (J 1 , u 1 ) ≺ (J 2 , u 2 ) o (piú brevemente)u 1 ≺ u 2 .
In base alla definizione precedente, (C, ≺)è un insieme parzialmente ordinato. Una soluzione locale u di (3.2) si dice massimale seè massimale rispetto alla relazione d'ordine ≺ (ovvero v ∈ C , u ≺ v ⇒ u = v ). Dimostrazione. Sia (J, u) una soluzione locale di (3.2), C J,u := {(J 1 , u 1 ) ∈ C : u ≺ u 1 } e (J 1 , u 1 ), (J 2 , u 2 ) ∈ C J,u . Allora u 1 (t 0 ) = u 2 (t 0 ) = u 0 e per unicitá della soluzione troviamo u 1 | J = u 2 | J = u . Ora, avendosi ∅ = J ⊂ J 1 ∩J 2 , abbiamo che J := J 1 ∪J 2è un intervallo, e definendo u ∈ C 1 ( J, R n ), u| J1 = u 1 , u| J2 = u 2 , troviamo facilmente che ( J, u) ∈ C J,u . Dunque C J,uè un insieme diretto, e quindi ammette un elemento massimale.
Sia (J, u), J := (a, b), una soluzione locale e (J 1 , u 1 ) un prolungamento con J ⊂ J 1 . Poiché u 1 ∈ C 1 (J 1 , R n ) troviamo che deve essere necessariamente L := lim t→b − u(t) = ±∞ , (3.5) in quanto L = u 1 (b). In effetti, (3.5)è anche condizione sufficiente affinché esista un prolungamento di (J, u): infatti, considerando il problema di Cauchy u ′ = f (t, u) u(b) = L ed una sua soluzione (J 2 , u 2 ), possiamo facilmente costruire il prolungamento J 1 := J ∪ J 2 , u 1 ∈ C 1 (J 1 , R n ), u 1 | J = u , u 1 | J2 = u 2 . Conseguenza di quanto appena affermatoè il seguente teorema. Dimostrazione. Se (t 0 , u 0 ) appartiene alla frontiera di K possiamo risolvere il problema di Cauchy con dato iniziale u(t 0 ) = u 0 , la cui soluzione u fornisce dei punti (t, u(t)) ∈ A − K . Del resto, u deve essere restrizione della soluzione massimale u , per cui (t, u(t)) ∈ A − K .
Dalle considerazioni precedenti segue cheè interessante stabilire quando una soluzione locale soddisfa (3.5) 7 , visto che in tal caso essaè prolungabile. Il seguente Lemma fornisce una condizione sufficiente per evitare esplosioni di soluzioni locali nel caso n = 1 .
Lemma 3.5. Sia u ∈ C 1 (a, b) con ε ≥ 0 , L > 0 tali che |u ′ (t)| ≤ ε + L|u(t)| , t ∈ (a, b) .
(3.6)
Allora per ogni t, t 0 ∈ (a, b) risulta
|u(t)| ≤ ε L + |u(t 0 )| e L|t−t0| . (3.7)
Dimostrazione. Usiamo il seguente trucco: preso λ > 0 , definiamo
z(t) := λ 2 + u(t) 2 ⇒ z ′ (t) = u ′ (t)u(t) λ 2 + u(t) 2 .
Usando (3.6) e |u(t)| ≤ z(t) troviamo
z ′ (t) ≤ |u(t)| λ 2 + u(t) 2 |u ′ (t)| ≤ |u ′ (t)| ≤ ε + L|u(t)| ≤ ε + Lz(t) ,
per cui
Lz ′ (t) ε + Lz(t) ≤ L ⇒ ln ε + Lz(t) ε + Lz(t 0 ) ≤ L|t − t 0 | ⇒ ε + Lz(t) ≤ (ε + Lz(t 0 ))e L(t−t0) ,
e ancora (avendosi chiaramente Lz(t) ≤ ε + Lz(t)) z(t) ≤ 1 L (ε + Lz(t 0 ))e L(t−t0) λ→0 ⇒ |u(t)| ≤ 1 L (ε + L|u(t 0 )|)e L(t−t0) .
Teorema 3.6. Sia I := (t 0 −r, t 0 +r) ed f ∈ C(I ×R) localmente Lipschitz nella seconda variabile. Supponiamo che per ogni compatto K ⊂ I × R esistano ε K , L K ≥ 0 tali che |f (t, u)| ≤ ε K + L K |u| , (t, u) ∈ I × R . (3.8) Allora il problema di Cauchy (3.2) ammette una soluzione massimale u ∈ C 1 (I).
Dimostrazione. Applicando il lemma precedente ad u ′ (t) = f (t, u(t)) otteniamo la stima (3.7), la quale implica che non si hanno esplosioni in nessun punto di I .
u ′ = 2tu 2 u(t 0 ) = u 0 (3.9)
al variare di (t 0 , u 0 ) ∈ R 2 . Innanzitutto, osserviamo che f (t, x) := 2tx 2è definita su tutto R 2 edè localmente Lipschitz, per cui possiamo applicare il teorema di Cauchy per ogni condizione iniziale (t 0 , u 0 ) ∈ R 2 . Tuttavia le soluzioni u variano sensibilmente al variare di (t 0 , u 0 ) in R 2 . Innanzitutto escludiamo il caso u 0 = 0 , poiché fornisce la soluzione banale u = 0 . Alché, risolvendo per separazione di variabili troviamo
t 0 = 0 , u 0 > 0 ⇒ u(t) = u 0 1 − u 0 t 2 ⇒ lim |t|→u −1/2 0 |u(t)| = +∞ ⇒ I max (0, u 0 ) = (−u −1/2 0 , u −1/2 0 ) ,
dove, in generale, I max (t 0 , u 0 ) denota il dominio della soluzione massimale del problema (3.9) con condizione iniziale (t 0 , u 0 ). Se u 0 < 0 allora la situazione cambia drasticamente:
t 0 = 0 , u 0 < 0 ⇒ u(t) = u 0 1 + |u 0 |t 2 ⇒ I max (0, u 0 ) = R .
Passiamo ora ad analizzare la situazione per t 0 = 0 :
u 0 > 0 ⇒ u(t) = u 0 1 − u 0 (t 2 − t 2 0 ) ⇒ lim |t|→ √ t 2 0 +u −1 0 |u(t)| = +∞ ⇒ I max (t 0 , u 0 ) = − t 2 0 + u −1 0 , + t 2 0 + u −1 0 . D'altro canto u 0 < 0 ⇒ u(t) = u 0 1 + |u 0 |(t 2 − t 2 0 )
; ora, abbiamo due casi:
1 − |u 0 |t 2 0 > 0 ⇔ t 2 0 − |u 0 | −1 < 0 ⇒ I max (t 0 , u 0 ) = R , altrimenti t 2 0 − |u 0 | −1 ≥ 0 ⇒ lim |t|→ √ t 2 0 −|u0| −1 |u(t)| = +∞ ⇒ I max (t 0 , u 0 ) = t 2 0 − |u 0 | −1 , +∞ , t 0 > 0 −∞ , − t 2 0 − |u 0 | −1 , t 0 < 0 .
Dipendenza continua dai dati iniziali.
Nelle applicazioni fisiche il ruolo della condizione iniziale di un problema di Cauchyè quello del valore assunto da una determinata grandezza fisica, diciamo u 0 , misurata al tempo t = t 0 . Poiché la misura di una grandezza fisica comporta inevitabilmente un errore di rilevazione (per quanto piccolo),è importante caratterizzare quei problemi di Cauchy tali che a fronte di piccole discrepanze u 0 = v 0 producano soluzioni (I, u), (J, v) che differiscano "poco" in I ∩ J . In termini rigorosi, consideriamo il problema differenziale
u ′ = f (t, u) , A ⊂ R n+1 , f ∈ C(A, R n ) (3.10)
e, fissato t 0 ∈ R, consideriamo un disco ∆ ⊂ R n tale che (t 0 , x) ∈ A per ogni x ∈ ∆. Assumiamo che esiste un intervallo I ⊆ R, I ∋ t 0 , tale che ogni problema di Cauchy
u ′ = f (t, u) , u(t 0 ) = x , x ∈ ∆ ,
abbia soluzione unica (I x , u x ) con I x ⊇ I . Diciamo che si ha dipendenza continua dai dati iniziali in I se l'applicazione ∆ → C 1 (I, R n ) , x → u x | I , è continua, dove C 1 (I, R n )è equipaggiato con la topologia della convergenza uniforme; questa condizione ci assicura che per ogni ε > 0 esiste un δ tale che |x − y| < δ , x, y ∈ ∆ ⇒ |u x (t) − u y (t)| < ε , ∀t ∈ I .
Nelle applicazioni, per parlare di dipendenza continua occorre anche richiedere una proprietá non prettamente matematica, ovvero che I , il quale si interpreta come l'intervallo temporale nel quale la soluzione di (3.10)è attendibile nel descrivere il comportamento del sistema fisico in oggetto, sia fisicamente significativo. Chiaramente, il termine fisicamente significativoè ambiguo: un intervallo I di pochi secondi puó essere ben accettabile in problema d'urto, e totalmente inadeguato, invece, in un modello astronomico. I risultati classici che permettono di dedurre la proprietá di dipendenza continua sono noti come il Lemma di Gronwall. Fenomeni fisici che conducono a problemi di Cauchy che non hanno dipendenza continua dai dati iniziali sono di solito associati ai cosiddetti sistemi caotici, e formano a tutt'oggi un'area di ricerca molto importante. Come esempio di sistema caotico portiamo il celebre modello di Lorenz correlato alle previsioni metereologiche ([29, §9.2], [13]): presi σ, ρ, β > 0 , esso si ottiene come il problema differenziale (3.10) associato alla funzione allora u(t) ≤ αe βt + γβ −1 (e βt − 1), t ∈ [0, T ].
f : R × R 3 → R 3 , f (t, u) := (−σ(u 1 − u 2 ) , ρu 1 − u 2 − u 1 u 3 , u 1 u 2 − βu 3 ) .
Dimostrazione. Definendo u := u + γβ −1 troviamo che (3.14) si scrive
u(t) ≤ α + γβ −1 + t 0 β u(s) ds , t ∈ [0, T ] ,
dunque possiamo applicare il Lemma precedente. Calcolando esplicitamente i (semplici) integrali coinvolti otteniamo la tesi.
Teorema 3.9 (Dipendenza continua). Siano f, g ∈ C(A, R n ) con f avente costante di Lipschitz L . Se (I x , u x ), (J y , v y ) sono soluzioni rispettivamente dei problemi di Cauchy
u ′ = f (t, u) u(t 0 ) = x , v ′ = g(t, v) v(t 0 ) = y , (3.15) allora |u x (t) − v y (t)| ≤ |x − y|e L(t−t0) + M L −1 (e L(t−t0) − 1) , t ∈ I x ∩ J y , (3.16) dove M := f − g ∞ .
Dimostrazione. Possiamo supporre senza ledere la generalitá che t 0 = 0 . Integrando e sottraendo le (3.15) abbiamo
|u x (t) − v y (t)| ≤ |x − y| + t 0 |f (s, u x (s)) − g(s, v y (s))| ds ,
e stimando le funzioni integrande troviamo (sommando e sottraendo f (s, v y (s)))
|f (s, u x (s)) − g(s, v y (s))| ≤ L|u x (s) − v y (s)| + M , da cui |u x (t) − v y (t)| ≤ |x − y| + t 0 (L|u x (s) − v y (s)| + M ) ds .
La tesi segue dunque applicando il Corollario precedente.
Poniamo ora f = g e consideriamo un disco ∆ ⊂ R n tale che {t 0 } × ∆ ⊂ A. Supposto che ∩ x∈∆ I x contenga un intervallo I non banale e di lunghezza |I| finita, in conseguenza del teorema precedente troviamo, per ogni x, y ∈ ∆, t ∈ I ,
|u x (t) − u y (t)| ≤ |x − y|e L|t−t0| ≤ |x − y|e L|I| ,
ovvero la dipendenza continua nel senso della definizione data all'inizio della sezione. Osserviamo che l'ipotesi di lipschitzianitá per f puó essere rilassata ad una lipschitzianitá locale se ci si restringe a domini compatti K ⊂ A (si veda [12]); tuttavia, occorre fare attenzione al fatto che se si ottengono costanti di Lipschitz L K che tendono ad infinito al crescere di K , la stima precedente diventa inutile e non si puó parlare di dipendenza continua in I .
Il teorema di Peano.
In effetti l'esistenza di una soluzione locale del problema di Cauchyè dimostrabile con la sola ipotesi di continuitá per f , attraverso il teorema di Peano. Il prezzo da pagare per la maggiore generalitá di questo risultatoè la perdita dell'unicitá della soluzione. Dal punto di vista del metodo della dimostrazione, segnaliamo che questa si basa sul teorema di Ascoli-Arzelá (Teo.2.15).
Teorema 3.10 (Peano). Sia dato il problema (3.2) con A := [t 0 − r, t 0 + r] × R n per qualche r > 0 , n ∈ N. Se f ∈ C(A, R n )è continua e limitata allora esiste una soluzione del problema (3.2) definita sull'intervallo [t 0 , t 0 + r].
Dimostrazione. L'ideaè quella di applicare il teorema di Ascoli-Arzelá ad un'opportuna successione di tipo Peano-Picard. A tale scopo, per ogni m ∈ N consideriamo la partizione P m := t (m) k
:= t 0 + kr/m di [t 0 , t 0 + r], e definiamo y m (t) := u (m) k + f (t (m) k , u (m) k )(t − t (m) k ) , t ∈ [t (m) k , t (m) k+1 ) , k = 0, 1, . . . , m , dove i coefficienti u (m) k , k ∈ N, sono definiti per iterazione, u (m) k+1 := u (m) k + f (t (m) k , u (m) k ) r m , k = 1, 2, . . . , m − 1 .
Le funzioni y m cosí costruite sono lineari a tratti e derivabili in (t 0 , t 0 + r) − P m , con derivata
y ′ m (t) = f (t (m) k , u (m) k ), t ∈ (t (m) k , t (m) k+1 ). Cosicché introducendo le funzioni costanti a tratti φ m : [t 0 , t 0 + r] → R n , φ m (t) := f (t (m) k , u (m) k ) , t ∈ [t (m) k , t (m) k+1 ) , troviamo φ m ∞ ≤ f ∞ , φ m = y ′ m in [t 0 , t 0 + r] − P m ,(3.17)
e, applicando il teorema fondamentale del calcolo,
y m (t) = u 0 + t t0 φ m (s) ds , t ∈ [t 0 , t 0 + r] . (3.18)
Ora,è un fatto generale che una funzione regolare a tratti F : [t 0 , t 0 + r] → R n avente derivata limitata ammette costante di Lipschitz F ′ ∞ √ n . La disuguaglianza in (3.17) assicura che ció e vero per le nostre y m , che si trovano cosí ad avere la stessa costante di Lipschitz f ∞ √ n. In tal modo, abbiamo verificato che {y m }è una successione equicontinua; poiché questaè anche equilimitata (infatti (3.18) implica y m ≤ u 0 + r f ∞ ), concludiamo per Ascoli-Arzelá che esiste una sottosuccessione, che denotiamo per brevitá sempre con {y m } , convergente ad un limite y ∈ C([t 0 , t 0 + r], R n ).
Vogliamo ora mostrare che yè effettivamente una soluzione del nostro problema di Cauchy. A tale scopo osserviamo che, preso t ∈ [t 0 , t 0 + r], una volta scelta la partizione P m , m ∈ N, esso apparterrá ad uno, ed uno solo, intervallo [t
|t − t (m) k | ≤ rm −1 ; (3.19) definiamo allora s t,m := t (m) k ed u t,m := u (m) k
= y m (s t,m ). Usando (3.19) concludiamo che lim m s t,m = t, e che la convergenza di tali successioniè uniforme al variare di t in [t 0 , t 0 + r] (infatti il termine rm −1 in (3.19) non dipende da t!). Inoltre, abbiamo la stima
|u t,m − y(t)| = |y m (s t,m ) − y(t)| ≤ |y m (s t,m ) − y m (t)| + |y m (t) − y(t)| . Essendo {y m } equicontinua abbiamo che, scelto ε > 0 , esiste δ > 0 tale che |y m (t ′ ) − y m (t ′′ )| < ε per |t ′ − t ′′ | < δ , uniformemente in m ∈ N; d'altro canto esiste certamente un m 0 ∈ N tale che |s t,m − t| ≤ rm −1 < δ per ogni m > m 0 , per cui lim m |y m (s t,m ) − y m (t)| = 0 uniformemente in t. D'altra parte lim m y m − y ∞ = 0 , cosicché |u t,m − y(t)| m −→ 0 , edè importante osservare che la convergenza di {u t,m } mè uniforme al variare di t in [t 0 , t 0 + r]. Dunque abbiamo (s t,m , u t,m ) m −→ (t, y(t)) , (3.20) uniformemente in t grazie alle considerazioni precedenti. Ora, poiché |u t,m | ≤ y m ∞ ≤ f ∞ ogni successione (s t,m , u t,m )è contenuta nel compatto K := [t 0 , t 0 + r] × ∆(0, f ∞ ) , ed fè uniformemente continua in K (per Heine-Cantor). Per cui, preso t ∈ [t 0 , t 0 + r] troviamo, usando (3.20), |φ m (t) − f (t, y(t))| = |f (s t,m , u t,m ) − f (t, y(t))| m −→ 0 , uniformemente al variare di t in [t 0 , t 0 + r]; dunque possiamo passare al limite sotto il segno di integrale in (3.18) e concludere y(t) = u 0 + t t0 f (s, y(s)) ds , t ∈ [t 0 , t 0 + r] .
Infine, lo stesso argomento usato nella dimostrazione del Teorema di Cauchy permette di concludere che y ∈ C 1 ([t 0 , t 0 + r], R n ), e che essaè soluzione cercata.
La dimostrazione precedente in teoria potrebbe essere facilmente tradotta in un algoritmo, visto che la soluzione y viene trovata per mezzo di una successione costruita per iterazione; si ha peró un inconveniente, derivante dal fatto che non abbiamo una costruzione esplicita della sottosuccessione convergente {y m } , la cui esistenza viene dedotta con un argomento di compattezza (Ascoli-Arzelá). Da qui l'utilitá di algoritmi, come quello di Runge-Kutta, che permettono di costruire (sempre per iterazione) successioni convergenti alla soluzione cercata.
La condizione di limitatezza per f puó essere rimossa, se ci restringiamo a cercare soluzioni in piccolo:
Corollario 3.11. Sia A ⊂ R n+1 aperto, (t 0 , u 0 ) ∈ A, f ∈ C(A, R n ). Allora il problema di Cauchy (3.2) ammette una soluzione in piccolo u definita su [t 0 , t 0 + τ ], τ ≤ r .
Dimostrazione. E' sufficiente considerare una funzione continua e limitataf che coincida con f in un rettangolo chiuso della forma [t 0 , t 0 + τ ] × ∆(u 0 , R) ⊆ A, ed applicare il teorema precedente al problema di Cauchy associato.
Esempio 3.2. Il problema
u ′ = 3u 2/3 u(0) = 0 ammette le soluzioni u 0 ≡ 0 e u(t) = t 3 .
Invitiamo il lettore a verificare che f (s, v) = 3v 2/3 e continua ma non Lipschitz in un intorno di 0 , cosicché nonè possibile applicare il teorema di Cauchy. Un'altra proprietá di facile verifica (che lasciamo per esercizio)è che ogni soluzione del precedente problemaè ≥ u 0 , la qualeè quindi il minimo dell'insieme delle soluzioni.
Cenni sul pennello di Peano. Consideriamo il nostro problema di Cauchy (3.2) con f : A → R n soddisfacente le condizioni di Teo.3.10 (ovvero f continua e limitata, A :
= [t 0 − r, t 0 + r] × R n ⊂ R n+1 per qualche r > 0 ). Se uè una soluzione di tale problema, allora abbiamo una costante di Lipschitz f ∞ √ n indotta dalla diseguaglianza |u ′ (t)| ≤ f ∞ , t ∈ [t 0 − r, t 0 + r] ; (3.21) dalla condizione di Lipschitz segue una condizione di limitatezza |u(t)| − |u 0 | ≤ f ∞ √ nr . (3.22) Osserviamo che né (3.21) né (3.22) dipendono dalla particolare soluzione u . Ne consegue che l'insieme delle soluzioni del problema (3.2)è equicontinuo ed equilimitato in [t 0 , t 0 + r].
Lemma 3.12. Sia I ⊂ R un intervallo chiuso e limitato, ed F ⊂ C(I, R) un insieme equicontinuo, equilimitato, e totalmente ordinato rispetto all'ordinamento standard di C(I, R). Posto φ : I → R, φ(t) := sup φ∈F φ(t), t ∈ I , risulta che per ogni ε > 0 esiste φ ε ∈ F tale che φ − φ ε ∞ < ε .
Dimostrazione. Omessa, ma semplice.
Teorema 3.13 (Peano). L'insieme delle soluzioni del problema (3.2) ammette un elemento massimale u ed un elemento minimale u . Per ogni punto p := (t, y) appartenente alla regione R compresa tra i grafici di u e u esiste almeno una soluzione u tale che u(t) = y .
Sketch della dimostrazione. Applicando il lemma precedente, ed un passaggio al limite sotto il segno dell'integrale di Volterra, concludiamo che ogni catena contenuta nell'insieme F delle soluzioni di (3.2) ammette un elemento massimale appartenente ad F . Applicando Zorn, concludiamo che esistono u ed u come nell'enunciato. Infine, l'affermazione inerente p si dimostra risolvendo il problema di Cauchy "all'indietro" con dato iniziale u 0 = y .
Osserviamo che qualora si abbia anche unicitá della soluzione (ad esempio, quando fè localmente Lipschitz nella variabile u ), allora per ogni p = (t, y) ∈ R esiste edè unica la soluzione u tale che u(t) = y . Dunque Rè "spazzata" dai grafici delle soluzioni di (3.2), i quali non si intersecano. E' questa l'idea alla base del concetto di foliazione.
Esercizi.
Esercizio 3.1. Effettuare uno studio qualitativo della soluzione del problema di Cauchy
u ′ = |u| + u 2 , u(0) = u 0 , al variare della condizione iniziale u 0 ∈ R. Esercizio 3.2. Sia f : R → R una funzione continua tale che xf (x) ≥ 0 , ∀x ∈ R , +∞ 0 1 1 + f (x) dx = +∞ ,(3.23)
e si consideri il problema di Cauchy
u ′ = f (t + |u|) u(0) = 0 . (3.24) (1) Si mostri che (3.24) ammette almeno una soluzione u ; (2) si verifichi che si ha 0 = f (0) = u ′ (0); (3) si mostri che u ′ (t) ≥ 0 , u(t) ≥ 0 , ∀t ≥ 0 ; (4) posto v(t) := t + u(t), si verifichi che v ′ (t) = 1 + f (v(t)) , v(0) = 0 , v(t) ≥ 0 ,
per ogni t ≥ 0 appartenente al dominio di u ; (5) usando il punto precedente, e (3.23), si verifichi che u non esplode in nessun T ∈ R.
(Suggerimenti: per (1) si usi il teorema di Peano; per (2) e (3) si usi la prima delle condizioni in (3.23), la quale implica che f (0) = 0 ; per (4) si osservi che, grazie ai punti precedenti,
f (t+|u(t)|) = f (t + u(t)), cosicché v ′ (t) = 1 + f (v(t))
; per (5) si osservi che, supponendo per assurdo che
∞ = lim t→T − u(t) = lim t→T − v(t) ,
integrando per sostituzione si trova la contraddizione
T = T 0 v ′ (t) 1 + f (v(t)) dt = ∞ 0 1 1 + f (v) dv ( 3.23) = ∞ .
Esercizio 3.3. Discutere esistenza ed unicitá della soluzione dell'equazione di Liouville
u ′′ + e λu = 0 , λ ∈ R , al variare della condizione iniziale u(0) = u 0 ∈ R. Esercizio 3.4. Sia T > 0 e φ ∈ C 2 (R) una funzione T -periodica, strettamente positiva e tale che sia φ ′ che φ ′′ si annullino solo due volte (ciascuna) in [0, T ). Si supponga per semplicitá che min φ = φ(0) e che max φ = φ(a) con a ∈ (0, T ). (a) Si tracci un grafico approssimativo di φ. (b) Posto N := {(x, y) ∈ R 2 : φ(x) = φ(y)} , N 0 := N ∩ [0, T ) 2 , si provi che Nè [0, T ) 2 -periodico in R 2 . (c)
Si provi che N 0è composto da due archi di classe C 1 che si intersecano ad angolo retto. (d) Si consideri il problema di Cauchy
y ′ = φ(x) φ(y) − 1 , y(0) = y 0 . (3.25)
Si provi che per ogni y 0 esiste un'unica soluzione definita su tutto R. (e) Si provi che ogni soluzionè e limitata su R.
(Suggerimenti: per il punto (c) si usi il Teorema delle funzioni implicite, mentre per i punti (d)-(e) si metta in relazione il grafico di N 0 con lo studio qualitativo delle soluzioni di (3.25)).
Esercizio 3.5. Si mostri che il problema di Cauchy
u ′ = u 2 − e t 2 + 1 , u(0) = 0 ,
ha soluzione massimale definita su tutto R. (Suggerimento: si riscontri che non si hanno esplosioni).
Esercizio 3.6. Si verifichi che la funzione di Lorenz (3.11) nonè lipschitziana nella variabile u ∈ R 3 .
4 Teoria della misura e dell'integrazione.
La teoria dell'integrazione siè emancipata dalla topologia grazie all'approccio di Lebesgue, attraverso il qualeè possibile integrare, a differenza di quanto accade nella teoria di Riemann, funzioni altrimenti intrattabili dal punto di vista della continuitá. Nelle sezioni seguenti tratteggeremo le proprietá elementari degli spazi di misura con particolare cura, ovviamente, del caso della retta reale, e poi passeremo a trattare la teoria dell'integrale.
Spazi misurabili.
σ -algebre e funzioni semplici. Sia X un insieme. Un'algebra di Boole su Xè il dato di un sottoinsieme R dell'insieme delle parti 2 X , chiuso sotto le operazioni di unione e passaggio al complementare (e ció implica che Rè chiusa anche rispetto all'intersezione), e tale che ∅, X ∈ R. Una σ -algebraè un'algebra di Boole M chiusa rispetto ad unioni numerabili, il che implica che M e chiusa anche rispetto ad intersezioni numerabili. Dato un insieme Y ed un sottoinsieme N ⊆ 2 Y , definiamo la σ -algebra generata da N come l'intersezione di tutte le σ -algebre che contengono N . Denotiamo tale σ -algebra con il simbolo σN .
Definizione 4.1. Siano X, Y insiemi ed M ⊆ 2 X , N ⊆ 2 Y σ -algebre. Un'applicazione f : X → Y si dice misurabile se f −1 (B) ∈ M per ogni B ∈ N .
Per ogni A ∈ M , introduciamo la funzione caratteristica
χ A (x) := 1 , x ∈ A 0 , x / ∈ A ; osserviamo che χ A∩B = χ A χ B , χ A∪B = sup{χ A , χ B } , χ A c = 1 − χ A ,
dove A c := X − A, cosicché Mè fedelmente rappresentata in termini di funzioni su X . Una funzione sempliceè una combinazione lineare finita a coefficienti in R di funzioni caratteristiche:
ϕ := i λ i χ Ai , λ i ∈ R , A i ∈ M .
Osserviamo che la rappresentazione di ϕ come combinazione lineare di funzioni caratteristiche noǹ e unica, ad esempio λχ A = λχ U + λχ V per ogni λ ∈ R ed A = U ∪ V con U ∩ V = ∅ . L'insieme delle funzioni sempliciè chiuso rispetto a moltiplicazioni scalari, prodotti e combinazioni lineari, dunque costituisce un'algebra che denotiamo con S(X).
Esempio 4.1. Sia X uno spazio topologico con topologia τ X ⊆ 2 X . La σ -algebra dei borelianiè la σ -algebra βX := σ(τ X), e per costruzione contiene sia gli aperti che i chiusi di X . Se Xè di Hausdorff ed a base numerabile allora ogni x ∈ X ammette un sistema numerabile di intorni {A n } tale che ∩ n A n = {x} , dunque {x} ∈ βX ; di conseguenza, ogni sottoinsieme numerabile di Xè boreliano. Una funzione f : X → R si dice boreliana se f −1 (I) ∈ βX per ogni aperto I ⊂ R. Ad esempio, ogni funzione continuaè boreliana, ed ogni funzione caratteristica χ A , A ∈ τ X ,è boreliana (ma, in genere, non continua: si prenda ad esempio X = R con la topologia usuale e si verifichi che χ I , I := (0, 1),è boreliana, oltre che, ovviamente, discontinua).
Misure. E' conveniente introdurre ora l'insieme dei reali estesi, o retta reale estesa
R := R ∪ {−∞, ∞} .
Gran parte della usuale struttura algebrica dei reali puó essere esportata alla retta reale estesa,
a ± ∞ := ±∞ , b · (±∞) := ±∞ , c · (±∞) := ∓∞ , a/ ± ∞ := 0 , ∀a ∈ R , b > 0 , c < 0 , tranne le operazioni ∞ − ∞, ±∞/ ± ∞,(1) µ∅ = 0 ; (2) µ soddisfa la proprietá di additivitá numerabile µ (∪ n A n ) = n µA n , ∀A n ∈ M , A n ∩ A m = ∅ , n = m . (4.1) Osservazione 4.1. (1) Considerando successioni {A n } tali che A n = ∅ , ∀n > 2 , troviamo che μ e additiva, ovvero µ(A 1∪ A 2 ) = µA 1 + µA 2 , ∀A 1 , A 2 ∈ M , A 1 ∩ A 2 = ∅ . (2) Presi A, A ′ ∈ M con A ⊂ A ′ abbiamo A 0 := A ′ − A ∈ M e µA ′ = µA + µA 0 ; dunque µè monotóna, ovvero µA ≤ µA ′ , ∀A ⊆ A ′ . (3) Sia (X, M, µ) uno spazio misurabile ed A ∈ M . Definiamo M A := {E ∩ A, E ∈ M} e µ A : M A → R, µ A (E ∩ A) := µ(E ∩ A). Allora (A, M A , µ A )è uno spazio misurabile, che chiamiamo la restrizione di (X, M, µ) ad A. Lemma 4.3. Sia (X, M, µ) uno spazio di misura. Se {E n } ⊆ Mè una successione tale che µE 1 < +∞ e E n+1 ⊆ E n per ogni n ∈ N, allora µ(∩ n E n ) = lim n µE n . Dimostrazione. Posto E := ∩ n E n abbiamo E 1 = E∪˙ n (E n − E n+1 ) e quindi, per additivitá numerabile, µE 1 = µE + n µ(E n − E n+1 ). Del resto E n = E n+1∪ (E n − E n+1 ), per cui µE n − µE n+1 = µ(E n − E n+1 ) ,
e concludiamo che
µE 1 = µE + n (µE n − µE n+1 ) = µE + µE 1 − lim n µE n .
Nei punti seguenti introduciamo alcune terminologie.
• Diciamo che uno spazio misurabile (X, M, µ) ha misura finita se µX < ∞; in particolare, diremo che µè una misura di probabilitá se µX = 1 .
• Uno spazio misurabile (X, M, µ) si dice σ -finito se esiste una successione {A n } di insiemi di misura finita tali che X = ∪ n A n 8 .
• Uno spazio misurabile (X, M, µ) si dice completo se per ogni A ∈ M con µA = 0 e B ⊆ A risulta B ∈ M . Chiaramente in tal caso µB = 0 . Osserviamo che, qualora (X, M, µ) non sia completo,è sempre possibile definire la σ -algebra [25,Prop.11.1.4]).
M * := σ(M ∪ M 0 ), dove M 0 := {A ⊂ X : A ⊆ E ∈ M, µE = 0} , ed estendere µ ad M * ponendo µA := 0 , A ∈ M 0 (vedi
• Uno spazio misurabile (X, M, µ) si dice localmente finito se per ogni x ∈ X esiste V ∈ M tale che x ∈ V e 0 < µV < +∞.
• Sia X uno spazio topologico. Una misura µ : M → R si dice di Borel se τ X ⊂ M (ed in tal caso M contiene la σ -algebra dei boreliani).
• Due misure µ, ν : M → R + sono mutualmente singolari se esistono A, B ∈ M tali che A ∪ B = X e µA = νB = 0 . Ed in tal caso, scriviamo µ ⊥ ν .
Esempio 4.2 (La misura di enumerazione). Consideriamo l'insieme N dei naturali, la σ -algebra 2 N dei sottoinsiemi di N, e la funzione µ : 2 N → R + : µA := |A| , A f inito , +∞ , altrimenti , dove |A|è la cardinalitá di A ⊆ N. Allora µè una misura localmente finita, ma non finita. L'unico sottoinsieme di N di misura nullaè l'insieme vuoto, per cui µè anche completa.
Misure con segno. Sia X un insieme ed M una σ -algebra su X . Una misura con segnoè una funzione µ : M → R tale che: (1) µ assume solo uno tra i valori +∞, −∞; (2) µ∅ = 0 ; (3) se E =∪ n E n , con {E n } ⊆ M , allora µE = n µE n , e la convergenza della serieè assoluta quando µE = ±∞.
Diciamo che A ∈ Mè positivo (negativo) se µA ′ ≥ 0 (≤ 0 ) per ogni A ′ ⊆ A, A ′ ∈ M .
Lemma 4.4. Sia (X, M, µ) uno spazio di misura con segno. Allora:
(1) Se E ⊂ Mè positivo allora ogni E ′ ⊂ E , E ′ ∈ M ,è positivo; (2) E := ∪ n E nè positivo per ogni successione {E n } di insiemi positivi; (3) Se µE ∈ (0, ∞) allora esiste A ⊂ E positivo con µA > 0 . Dimostrazione. (1) E' del tutto ovvia. (2) Per ogni A ⊆ E , A ∈ M , poniamo A n := A ∩ E n ∩ (∩ n−1 i=1 E c i ) .
Chiaramente ogni A nè misurabile e contenuto nel positivo E n , per cuiè esso stesso positivo. Inoltre, essendo gli A n disgiunti troviamo µA = n µA n ≥ 0 , per cui Eè positivo. (3) Se Eè positivo allora non viè nulla da dimostrare, per cui assumiamo che esiste E 1 ⊂ E tale che µE 1 < 0 . Definiamo n 1 ∈ N come il piú piccolo intero tale che
µE 1 < −1/n 1 . Procedendo ricorsivamente, se E − ∪ k i E i nonè positivo definiamo n k+1 come il piú piccolo intero tale che esista E k+1 ∈ M con E k+1 ⊂ E − ∪ k i=1 E i , µE k+1 < −1/n k+1 . Osserviamo che gli E k sono mutualmente disgiunti e definiamo A := E −∪ i E i ; allora E = A∪(∪ k E k ) ⇒ µE = µA + k µE k ∈ (0, ∞) .
La condizione precedente ci dice che k 1/n k converge, per cui lim k n k = ∞. Vogliamo ora mostrare che Aè positivo. A tale scopo prendiamo ε > 0 ed osserviamo che esiste k ∈ N tale che 1/(n k − 1) < ε ; poiché per costruzione A ⊂ E − ∪ k i E i , troviamo che A non puó contenere insiemi di misura minore di −1/(n k − 1) > −ε , e per arbitrarietá di ε concludiamo che A non contiene insiemi con misura negativa.
Proposizione 4.5 (La decomposizione di Hahn). Sia (X, M, µ) uno spazio di misura con segno. Allora esistono un insieme positivo X + ed un insieme negativo
X − tali che X + ∩ X − = ∅ e X + ∪ X − = X .
Dimostrazione. Possiamo supporre che µ non assuma il valore +∞ (altrimenti il ragionamento che segue si applica con ovvie modifiche). Sia λ := sup{µA : A positivo } < ∞; allora esiste una successione {A n } di insiemi positivi tali che µA n n → λ, e definiamo X + := ∪ n A n , X − := X − X + . E' chiaro che X + ∩X − = ∅ e X + ∪X − = X , per cui rimane da verificare che X +è positivo ed X − negativo. Che X + sia positivo segue dal Lemma precedente, punto (2), il che implica µX + ≤ λ; ma del resto
µX + = µA n + µ(A − A n ) ≥ µA n , ∀n ∈ N ⇒ µX + ≥ λ , e quindi µX + = λ. Passando a X − , supponiamo per assurdo che esista B ⊆ X − con µB > 0 . Per il Lemma precedente, punto (3), esiste B ′ positivo con B ′ ⊆ B ⊆ X − , e per costruzione B ′ ∩ X + = ∅ . Ora, B ′∪ X +è positivo (ancora per il Lemma precedente), e quindi λ ≥ µ(B ′∪ X + ) = µB ′ + µX + = µB ′ + λ ,
il cheè assurdo. Dunque X −è negativo.
Osserviamo che la decomposizione di Hahn nonè unica, in quanto se ne puó perturbare la costruzione con insiemi di misura nulla. Definendo invece
µ + A := µ(A ∩ X + ) , µ − A := −µ(A ∩ X − ) , A ∈ M ,
otteniamo due misure mutualmente singolari ed univocamente definite. Abbiamo cosí dimostrato: Proposizione 4.6 (La decomposizione di Jordan). Sia (X, M, µ) uno spazio di misura con segno. Allora esistono, e sono uniche, due misure µ + e µ − mutualmente singolari tali che µ = µ + − µ − .
La variazione totale di µ si definisce come la misura (positiva)
|µ| : M → R + , |µ|A := µ + A + µ − A , A ∈ M . (4.2)
Sia ora X un insieme e β ⊆ 2 X una σ -algebra (in effetti, abbiamo in mente il caso in cui Xè uno spazio topologico e β la σ -algebra dei boreliani). Definiamo lo spazio delle misure con segno finite
Λ 1 β (X) := {µ : M → R misura con segno : M ⊇ β e |µ|(X) < ∞} . (4.3) Prese µ, µ ′ ∈ Λ 1 β (X), µ : M → R, µ ′ : M ′ → R, osserviamo che β ⊆ M ∩ M ′ = ∅ , per cui ha senso considerare, preso λ ∈ R, l'applicazione µ + λµ ′ : M ∩ M ′ → R , {µ + λµ ′ }E := µE + λµ ′ E .
Una verifica immediata mostra che µ + λµ ′ appartiene in effetti a Λ 1 β (X), il quale diventa cosí uno spazio vettoriale. Definendo
µ := |µ|(X) , µ ∈ Λ 1 β (X) ,(4.4)
otteniamo una norma nel senso di §7 (per i dettagli si veda l'Esercizio 4.7), dunque Λ 1 β (X)è uno spazio normato.
Cenni sulle misure complesse. Sia X un insieme ed M una σ -algebra su X . Una misura complessaè un'applicazione del tipo µ = µ 1 + iµ 2 : M → C , dove µ 1 , µ 2 sono misure con segno finite definite su M . Diremo che µè completa, boreliana, di Radon, etc., se lo sono sia µ 1 che µ 2 . Analogamente al caso reale, fissata una σ -algebra β ⊆ 2 X troviamo che l'insieme delle misure complesse con dominio ⊇ βè uno spazio vettoriale complesso, che denotiamo con Λ 1 β (X, C). E' possibile definire il valore assoluto della misura complessa µ, nel modo che segue:
|µ|E := sup{ n |µE n | : {E n } ⊆ M,∪ n E n = E} , ∀E ∈ M .
(4.5)
Nonè difficile verificare che |µ| : M → Rè una misura finita nel senso usuale, cosicché definendo
µ := |µ|(X) , ∀µ ∈ Λ 1 β (X, C) , otteniamo una norma.
La costruzione di Carathéodory. Nelle righe che seguono riportiamo uno dei metodi piú comuni per costruire misure, di fatto un'astrazione di alcuni dei passi necessari per la definizione della misura di Lebesgue sulla retta reale, come vedremo nel seguito ( §4.2). Sia X un insieme; una misura esternaè un'applicazione µ * : 2 X → R + tale che:
(1) µ * (∅) = 0 ; (2) µ * A ≤ µ * B , ∀A ⊆ B ; (3) µ * (∪ k A k ) ≤ k µ * A k , ∀{A k } k∈N ⊆ 2 X (subadditivitá).
Un insieme E ⊆ X si dice misurabile secondo Carathéodory se
µ * A = µ * (E ∩ A) + µ * (E c ∩ A) , ∀A ∈ 2 X . (4.6)
Denotiamo con M * la classe degli insiemi misurabili secondo Carathéodory e definiamo µ * := µ * | M * . Osserviamo che per dimostrare che un generico E ∈ 2 X appartiene ad M * è sufficiente verificare, per subadditivitá, che
µ * A ≥ µ * (E ∩ A) + µ * (E c ∩ A) , ∀A ∈ 2 X . (4.7)
Lemma 4.7. M * è una σ -algebra, e (X, M * , µ * )è uno spazio misurabile completo.
Dimostrazione. L'insieme vuoto appartiene chiaramente ad M * ; inoltre, il complementare di ogni elemento di M * appartiene ad M * , per cui anche X appartiene ad M * . Presa una successione {E n } ⊆ M * , per ricorsivitá otteniamo, per ogni A ∈ 2 X ,
µ * A = µ * (A ∩ E 1 ) + µ * (A ∩ E c 1 ) = µ * (A ∩ E 1 ) + µ * (A ∩ E c 1 ∩ E 2 ) + µ * (A ∩ E c 1 ∩ E c 2 ) = . . . = µ * (A ∩ E 1 ) + n k=2 µ * A ∩ j<k E c j ∩ E k + µ * (A ∩ n k=1 E c k ) .
(4.8)
Ora, possiamo assumere che E n ⊆ (∪ k<n E k ) c , infatti ció non altera l'unione n-esima E (n) := ∪ n k E k . Per cui risulta
n k=1 E c k = E (n),c , A ∩ j<k E c j ∩ E k = A ∩ E k ∩ E (k−1),c = A ∩ E k ;
usando monotonía e subadditivitá numerabile di µ * , concludiamo che
µ * A = µ * (A ∩ E 1 ) + n k=2 µ * (A ∩ E k ∩ E (k−1),c ) + µ * (A ∩ E (n),c ) = n k=1 µ * (A ∩ E k ) + µ * (A ∩ E (n),c ) ≥ µ * (A ∩ E (n) ) + µ * (A ∩ E (n),c ) .
Per cui (4.7)è verificata e M * è una σ -algebra. Dimostriamo che µ * è numerabilmente additiva:
preso E :=∪ n E n , {E n } ⊂ M * , sostituendo A con E in (4.8) otteniamo µ * E = n µ * E n . Infine verifichiamo che µ * è completa. A tale scopo consideriamo E ∈ M * tale che µ * E = 0 e mostriamo che se E ′ ⊂ E allora E ∈ M * ; preso A ∈ 2 X osserviamo che E ′ ∩ A ⊂ E ∩ A e E c ∩ A ⊂ E ′ c ∩ A, cosicché, usando (4.6), la monotonía di µ * ed il fatto che µ * E = 0 , µ * (E ′ ∩ A) ≤ µ * (E ∩ A) = 0 , µ * A = µ * (E c ∩ A) ≤ µ * (E ′ c ∩ A) ≤ µ * A . Concludiamo che µ * (E ′ c ∩A) = µ * A, per cui µ * A = µ * (E ′ ∩A)+µ * (E ′ c ∩A), ovvero E ′ ∈ M * .
Ulteriori strutture su spazi misurabili. Nelle applicazioni capita spesso che un insieme X sia equipaggiato di ulteriore struttura (di spazio topologico, vettoriale, ...). Per cui, qualora si considerino delle misure definite su X ,è opportuno che esse presentino delle proprietá di compatibilitá rispetto a tale struttura; nelle righe seguenti diamo una breve rassegna delle costruzioni piú importanti in questo senso.
1. Misure di Borel e regolaritá esterna. Sia (X, M, µ) uno spazio misurabile di Borel. Diciamo che µ soddisfa la proprietá di regolaritá esterna se
µA = inf{µU : A ⊂ U, U ∈ τ X} , A ∈ M .
(4.9)
Nella definizione precedente gli aperti giocano il ruolo di una famiglia di insiemi "aventi delle buone proprietá", la cui misura sia facilmente calcolabile (si pensi alla lunghezza degli intervalli, come vedremo in seguito), ed attraverso i quali sia possibile "approssimare" la misura di un arbitrario A ∈ M nei termini dell'estremo inferiore (4.9).
Osservazione 4.2. Segue banalmente da (4.9) che µA = µA per ogni A ∈ M ; infatti se un aperto U contiene A allora esso contiene anche chiusura di A. (1) µC < +∞, ∀C ∈ K ; (2) µE = sup{µC : C ⊂ E, C ∈ K} , ∀E ∈ M 9 .
Analogamente al caso delle misure di Borel abbiamo una "famiglia privilegiata" di insiemi, quella dei compatti, a partire dalla quale calcolare per approssimazione la misura di un generico A ∈ M . Osserviamo che richiediamo che un compatto abbia misura finita, cosicché ogni misura di Radonè localmente finita (infatti, ogni x ∈ X possiede un intorno compatto). Esempio 4.3. Sia X uno spazio localmente compatto di Hausdorff equipaggiato con una σ -algebra M ⊇ τ X . Per ogni x ∈ X , definiamo la misura di Dirac
µ x A := 1 , x ∈ A 0 , x / ∈ A , A ∈ M .
Allora µ xè una misura di Radon, e chiaramente anche una misura di probabilitá.
Assumiamo ora che X sia, in particolare, compatto. Diciamo che una misura con segno µ su Xè di Radon se |µ|è di Radon, e denotiamo con R(X) l'insieme delle misure di Radon con segno su X . Per definizione R(X) ⊆ Λ 1 β (X) (vedi (4.3)), dove β := βXè la σ -algebra dei boreliani. Visto che Λ 1 β (X)è uno spazio vettoriale troviamo λµ + ν ∈ Λ 1 β (X) per ogni λ ∈ R, µ, µ ′ ∈ R(X), e si verifica facilmente che |λµ + µ ′ |E = sup{|λµ + µ ′ |C : C ⊂ E, C ∈ K} , E ∈ M ∩ M ′ ; 9 Questa proprietáè detta regolaritá interna. dunque λµ + µ ′ ∈ R(X) ed R(X)è uno spazio normato (con la stessa norma di Λ 1 β (X)). In effetti, R(X)è addirittura uno spazio di Banach (ció in conseguenza del teorema di Riesz-Markov, vedi Esempio 7.3). Per ulteriori dettagli sulle misure di Radon si veda [19, §6.3].
3. Misure di Haar. Sia G un gruppo, nonché uno spazio topologico localmente compatto e di Hausdorff. Diciamo che Gè un gruppo topologico se l'applicazione G × G → G , g, g ′ → g −1 g ′ e continua. Esempi di gruppi topologici sono gli spazi euclidei R n , C m , equipaggiati con l'operazione di somma, il toro complesso T := {z ∈ C : |z| = 1} , equipaggiato con l'operazione di moltiplicazione, ed i gruppi di matrici GL(d, R), U(d) , ..., d ∈ N (ovvero, i cosiddetti gruppi di Lie classici), tutti equipaggiati con l'operazione del prodotto matriciale. Una misura di Radon µ ∈ R(G), µ : M → R + , si dice misura di Haar se verifica una delle due proprietá di invarianza per traslazione:
µA = µ(Ag) , µA = µ(gA) , A ∈ M ,(4.µ δ ε (A) := inf { k r δ k : A ⊆ ∪ k∈N ∆(x k , r k ) , x k ∈ X , r k < ε } . (4.11) Ora, µ δ ε (A)è una funzione decrescente in ε , per cui esiste il limite µ δ, * (A) := lim ε→0 µ δ ε (A) , A ∈ 2 X .
Si puó verificare che µ δ, * : 2 X → Rè una misura esterna dalla quale, con il metodo di Carathéodory, possiamo costruire una σ -algebra M δ ed una misura µ δ : M δ → R nota con il nome di misura di Hausdorff. Una nozione correlata alla costruzione precedenteè la dimensione di Hausdorff, nota anche come dimensione di Hausdorff-Besicovitch, la qualeè definita dall'espressione dim H X := inf{δ > 0 : µ δ (X) = 0} .
Si verifica che se Xè uno spazio vettoriale a dimensione finita n, o una varietá di dimensione n, allora dim H X = n. Piú in generale, la dimensione di Hausdorff trova applicazioni nella teoria dei frattali. Per ulteriori dettagli si veda [25, §12.9] e [10, §2.6].
Funzioni misurabili. Passiamo ora ad introdurre l'importante concetto di funzione misurabile. Lemma 4.9. Sia X un insieme ed M ⊆ 2 X una σ -algebra. Presa una funzione f : X → R , le seguenti affermazioni sono equivalenti:
(1) Per ogni α ∈ R, f −1 ((α, +∞)) ∈ M ; (2) Per ogni α ∈ R, f −1 ([α, +∞)) ∈ M ; (3) Per ogni α ∈ R, f −1 ((−∞, α)) ∈ M ; (4) Per ogni α ∈ R, f −1 ((−∞, α]) ∈ M .
Dimostrazione. Se vale (1) allora, presa una successione monotóna crescente α n → α , abbiamo ∩ n f −1 ((α n , +∞)) ∈ M , ma del resto ∩ n f −1 ((α n , +∞)) = f −1 ([α, +∞)) , per cui vale (2). Supposto che valga (2), troviamo
X − f −1 ([α, +∞)) ∈ M , ma del resto X − f −1 ([α, +∞)) = f −1 ((−∞, α)) , e quindi vale (3). L'implicazione (3) ⇒ (4)è analoga a (1) ⇒ (2), mentre (4) ⇒ (1)è analoga a (2) ⇒ (3).α ∈ R si ha f −1 (α) = f −1 ((−∞, α]) ∩ f −1 ([α, +∞)) ∈ M .
Definizione 4.10. Sia X un insieme ed M ⊆ 2 X una σ -algebra. Una funzione f : X → R si dice misurabile seè verificata una delle proprietá del lemma precedente. Denotiamo con M (X) l'insieme delle funzioni misurabili su X .
Per illustrare la connessione tra la definizione precedente e Def.4.1 consideriamo, in particolare, una funzione a valori reali f ∈ M (X); segue allora banalmente dal Lemma 4.9 che in effetti fè misurabile secondo Def.4.1 se equipaggiamo R con la σ -algebra dei boreliani βR (si noti infatti che βRè generata dagli intervalli).
Usando la stabilitá di M rispetto all'unione ed all'intersezioneè facile verificare che M (X)è un'algebra ( [25,Prop.3.5.18]). Chiaramente ogni funzione caratteristica χ E , E ∈ M ,è misurabile, e quindi ogni funzione sempliceè misurabile, ovvero S(X) ⊆ M (X). Se Xè uno spazio topologico ed M contiene i boreliani allora ogni funzione borelianaè misurabile; in particolare, ogni funzione continuaè misurabile.
Teorema 4.11. Sia X un insieme ed M ⊆ 2 X una σ -algebra. Date f, g ∈ M (X) ed {f n } ⊂ M (X), si ha h := sup{f, g} ∈ M (X) , h := inf{f, g} ∈ M (X) , (4.12) f (x) := lim n sup f n (x) , x ∈ X ⇒ f ∈ M (X) , (4.13) f (x) := lim n inf f n (x) , x ∈ X ⇒ f ∈ M (X) . (4.14) Dimostrazione. Per ogni α ∈ R troviamo h −1 ((α, +∞)) = f −1 ((α, +∞)) ∪ g −1 ((α, +∞)) ∈ M ,
dunque hè misurabile. Analogamente hè misurabile. Ora, essendo M chiusa per unioni ed intersezioni numerabili abbiamo che estremi superiori ed inferiori di successioni di funzioni misurabili sono funzioni misurabili; per cui, concludiamo che f = inf n sup k≥n f k , f = sup n inf k≥n f k sono misurabili.
Proposizione 4.12. Sia (X, M, µ) uno spazio misurabile ed f ∈ M (X), f ≥ 0 . Allora esiste una succesione monotóna crescente {ψ n } ⊂ S(X) che converge puntualmente ad f . Se Xè σ -finito allora si puó scegliere ogni ψ n in maniera tale che il supporto abbia misura finita.
Dimostrazione. Per ogni n ∈ N e t ∈ R + esiste un unico k t,n tale che t ∈ k t,n 2 −n , (k t,n + 1)2 −n , cosicché definiamo
ψ n (x) := k f (x),n 2 −n , f (x) ∈ [0, n) n , f (x) ∈ [n, ∞] .
La successione {ψ n } chiaramente soddisfa le proprietá desiderate. Se Xè σ -finito allora abbiamo X = ∪ n A n con µA n < ∞, e ponendo A (n) := ∪ n m A m , ϕ n := χ A (n) ψ n otteniamo, come desiderato, che ogni ϕ n ha supporto con misura finita e ϕ n → f . Equivalenza e convergenza q.o.. Introduciamo ora un'importante terminologia: diciamo che una proprietá vale quasi ovunque in x ∈ X (q.o.) se la misura del sottoinsieme di X nel quale essa nonè verificata ha misura nulla. Ad esempio, useremo spesso l'espressione f = g q.o., intendendo con ció che µ{x : f (x) = g(x)} = 0 .
E' chiaro che l'uguaglianza q.o. di due funzioni definisce una relazione di equivalenza su M (X). Nel seguito accadrá spesso che identificheremo f ∈ M (X) con la sua classe di equivalenza q.o..
Proposizione 4.13. Sia (X, M, µ) completo, g ∈ M (X) ed f : X → R tale che f = g q.o.. Allora fè misurabile. Dimostrazione. Posto E := {x ∈ X : f (x) = g(x)} abbiamo µE = 0 e, preso α ∈ R, f −1 (α, +∞) = (g −1 (α, +∞) ∩ E c ) ∪ (f −1 (α, +∞) ∩ E) . L'insieme (g −1 (α, +∞)∩E c )è misurabile, essendo esso intersezione di insiemi misurabili. L'insieme (f −1 (α, +∞) ∩ E)
, essendo contenuto nell'insieme di misura nulla E ,è anch'esso misurabile, per cui concludiamo che f −1 (α, +∞)è misurabile. ad una funzione f . Allora per ogni ε > 0 esistono un insieme A ⊂ X con µA < ε ed n 0 ∈ N tali che sup X−A |f n (x) − f (x)| < ε per ogni n ≥ n 0 .
Dimostrazione. Scelto ε > 0 poniamo
A n := {x ∈ X : |f n (x) − f (x)| ≥ ε} e B N := ∞ n=N A n = {x ∈ X | ∃n ≥ N : |f n (x) − f (x)| ≥ ε} . Osservando che B N +1 ⊆ B N , usando la convergenza di {f n } q.o. troviamo 0 = µ N B N ( * ) = lim N →∞ µB N . (4.15)
Per l'uguaglianza (*) abbiamo usato il Lemma 4.3, per applicare il qualeè necessario che X (e quindi B 1 ) abbia misura finita (a tal proposito si veda l'Esercizio 4.6). Concludiamo che esiste N 0 ∈ N tale che µB N0 < ε , e posto A := B N0 risulta
x ∈ X − A ⇔ |f n (x) − f (x)| < ε , ∀n ≥ N 0 .
La misura di Lebesgue sulla retta reale.
In questa sezione esponiamo la costruzione della misura di Lebesgue sulla retta reale, con l'idea di base che questa debba coincidere con la lunghezza di un intervallo qualora sia valutata su questo.
1. µ * A ≥ 0 , A ∈ 2 R ; 2. µ * ∅ = 0 ; 3. A ⊆ B ⇒ µ * A ≤ µ * B , A, B ∈ 2 R ; 4. µ * {x} = 0 , x ∈ R; 5. µ * (A + x) = µ * A, A ∈ 2 R , A + x := {a + x : a ∈ A} (invarianza per traslazioni).
Dimostrazione. (1) Segue dal fatto che l(I) > 0 , ∀I ∈ I aa ; (2) Segue dal fatto che ∅è contenuto in intervalli di lunghezza piccola a piacere; (3) Segue dal fatto che se B ⊂ ∪ n I n allora A ⊂ ∪ n I n ;
(4) Poiché {x} ⊂ (x − ε, x + ε) abbiamo µ * {x} < 2ε , con ε > 0 piccolo a piacere; (5) Segue dall'uguaglianza l(I + x) = l(I), ∀I ∈ I aa .
µ * ([a, b]) ≤ l((a − ε, b + ε)) = b − a + 2ε ,
per cui, data l'arbitrarietá di ε , concludiamo che µ * ([a, b]) ≤ b − a. D'altra parte, per definizione di µ * esiste una successione {I n := (a n , b n )} che ricopre [a, b] e tale che
µ * ([a, b]) ≥ n l(I n ) − ε . (4.19) Essendo {I n } un ricoprimento aperto di [a, b] possiamo estrarre un sottoricoprimento finito di intervalli distinti { I k := ( a k , b k )} N k=1 . Chiaramente, possiamo ordinare { I k } in maniera tale che a k ≤ a k+1 , k = 1, . . . , N , per cui b N − a 1 = N k=1 ( b k − a k ) − N −1 k=1 ( b k − a k+1 ) ≤ N k=1 l( I k ) ≤ n l(I n ) .
(4.20)
Concludiamo che
µ * ([a, b]) ( 4.19) ≥ n l(I n ) − ε ( 4.20) ≥ b n − a 1 − ε ≥ b − a − ε . Lemma 4.18 (Subadditivitá). Sia {A n } una successione di sottoinsiemi di R. Allora µ * n A n ≤ n µ * A n . (4.21)
Dimostrazione. Possiamo assumere che n µ * A n < ∞, altrimenti non viè nulla da dimostrare. Per ogni ε > 0 ed n ∈ N esiste una successione {I (n) k } ⊂ I aa tale che
A n ⊆ ∪ k I (n) k , µ * A n ≥ k l(I (n) k ) − 2 −n ε . (4.22) Inoltre ∪ n A n ⊆ ∪ n ∪ k I (n) k , per cui, per definizione di µ * , µ * (∪ n A n ) ≤ n,k l(I (n) k ) ( 4.22) ≤ n µ * A n + ε . (4.23) Lemma 4.19. (Regolaritá esterna.) Sia A ∈ 2 R . Allora per ogni ε > 0 esiste un aperto U ⊆ R, U ⊇ A, tale che µ * U ≤ µ * A + ε .
Dimostrazione. E' sufficiente considerare il caso µ * A < ∞, cosicché per definizione esiste una successione {I n } ⊂ I aa tale che A ⊆ ∪ n I n e n l(I n ) ≤ µ * A + ε . Posto U := ∪ n I n , applicando la subadditivitá numerabile troviamo µ * U ≤ n l(I n ) e quindi otteniamo quanto volevasi dimostrare. Ora, grazie ai Lemmi precedenti sappiamo che µ * è una misura esterna nel senso di §4.1 per cui, specializzando la nozione di misurabilitá di Carathéodory al caso X = R, diciamo che un insieme E ⊆ Rè misurabile secondo Lebesgue se
µ * A = µ * (E ∩ A) + µ * (E c ∩ A) , ∀A ⊆ R .
Denotiamo con L ⊂ 2 R la classe degli insiemi misurabili secondo Lebesgue.
µ * (A ∩ J) + µ * (A ∩ J c ) ≤ µ * (A ∩ n I + n ) + µ * (A ∩ n I − n ) ≤ n l(I + n ) + n l(I − n ) ≤ n l(I n ) ≤ µ * A + ε ;
per arbitrarietá di ε concludiamo µ * (A ∩ J) + µ * (A ∩ J c ) ≤ µ * A e quindi (essendo la diseguaglianza opposta sempre verificata) che Jè misurabile. Dimostrazione. Il fatto che µè una misura segue dal Lemma 4.7. La regolaritá esterna, il fatto che µ estende la funzione lunghezza, e l'invarianza per traslazioni seguono dai risultati precedenti. Infine, la completezza segue sempre dal Lemma 4.7.
Il risultato seguente implica (essendo evidente che µC < ∞ per ogni compatto C ⊂ R) che la misura di Lebesgue soddisfa anche la proprietá di regolaritá interna, per cui essaè una misura di Radon. Avendosi anche invarianza per traslazioni concludiamo che µè la misura di Haar su R inteso come gruppo topologico rispetto all'operazione di somma.
Corollario 4.22. Sia E ∈ L. Per ogni ε > 0 esiste un chiuso W ε ⊆ E tale che µE − µW ε < ε .
Dimostrazione. Applicando il Lemma 4.19 ad E c troviamo che per ogni ε > 0 esiste un aperto
U ε ⊇ E c tale che µU ε − µE c < ε . Del resto µU ε − µE c = µ(U ε − E c ) = µ(E − U c ε ), ed essendo U c ε chiuso troviamo quanto volevasi dimostrare.
Osservazione 4.6 (Misura di Lebesgue e misure di Hausdorff). Confrontando la definizione di misura esterna di Lebesgue (4.17) con quella di Hausdorff (4.11), ed osservando che i dischi nello spazio metrico R sono proprio gli intervalli, concludiamo che la misura di Lebesgueè la misura di Hausdorff su R di dimensione δ = 1 .
Esempio 4.5 (Insiemi di Cantor). Sia θ ∈ (0, 1/3]. Togliendo da [0, 1] l'intervallo aperto I 1 1 di centro 1/2 e lunghezza θ otteniamo due intervalli chiusi J 2 1 , J 2 2 ; da questi togliamo i due intervalli aperti I 2 1 , I 2 2 di lunghezza θ 2 centrati nei due punti di mezzo di J 2 1 , J 2 2 . Iterando il procedimento, al passo k -esimo toglieremo da 2 k−1 intervalli chiusi i 2 k−1 intervalli aperti di mezzo di lunghezza θ k . Nel limite k → ∞ otteniamo l'insieme
W θ := [0, 1] − ∪ ∞ k=1 ∪ 2 k−1 n=1 I k n .
Chiaramente W θè chiuso e la sua misuraè facilmente calcolabile per additivitá numerabile,
µW θ = 1 − k 2 k−1 n θ k = 1 − 1 2 k (2θ) k = 1 − 3θ 1 − 2θ ,
ció nonostante le ben note proprietá "mostruose" degli insiemi W θ a livello topologico (vedi [25, §12.9]). Per θ = 1/3 otteniamo il classico insieme di Cantor, il quale ha quindi misura nulla. In realtá la misura piú adatta per trattare W 1/3è la misura di Hausdorff di cui abbiamo giá accennato; la dimensione di Hausdorff di W 1/3 risulta essere dim H W 1/3 = ln 2/ ln 3 (vedi [10, §2.6]).
Cenni sul Lemma di Vitali. Sia I una collezione di intervalli. Preso A ∈ 2 R , diciamo che I ricopre A nel senso di Vitali se per ogni ε > 0 ed x ∈ A esiste I ∈ I tale che x ∈ I e l(I) < ε . Sketch della dimostrazione. Possiamo assumere che ogni intervallo di I sia chiuso (altrimenti passiamo alla chiusura). Preso un aperto U di misura finita che contiene A, possiamo assumere che ogni I ∈ I sia contenuto in U . Costruiamo ora per induzione l'insieme finito cercato, nel seguente modo: (1) Scegliamo un arbitrario I 1 ∈ I ; (2) Assumendo di aver scelto i primi k intervalli disgiunti Poiché ogni I ∈ Iè contenuto in U abbiamo λ(k) < ∞. Ora, a meno che non sia A ⊆∪ k j I k , possiamo trovare I ∈ I disgiunto da I 1 , . . . , I k e tale che l(I) > 1/2λ(k); per cui abbiamo una successione {I k } di intervalli disgiunti tali che k l(I k ) < µU < ∞; (3) Scegliamo ora ε > 0 ; poiché la serie k l(I k )è convergente troviamo n ∈ N tale che ∞ k=n+1 l(I k ) < ε .
Per cui il Lemma sará dimostrato se verifichiamo che µ * (A −∪ n k=1 I k ) < ε .
Il Lemma di Vitali si puó generalizzare ad R n , n > 1 (Teorema di Besicovitch), e ad arbitrari spazi metrici. In tal caso il ruolo degli intervalli viene assunto da palle di raggio opportuno. Su questo argomento si veda [31] e referenze.
Funzioni misurabili sulla retta reale. Funzioni a gradini. Passiamo ora a studiare le funzioni misurabili sulla retta reale. In particolare, introduciamo un'interessante classe di funzioni semplici, le funzioni a gradini
ψ = n k λ k χ I k , λ k ∈ R , I k := (a k , b k ) , a k , b k ∈ R .
Denotiamo con G(A) l'insieme delle funzioni a gradini definite su un insieme misurabile A ⊆ R 11 . Osserviamo che G(R) rispecchia proprietá topologiche di R; infatti essoè definito per mezzo della base degli intervalli, a prescindere dalla struttura di R inteso come spazio misurabile. Chiaramente G(R)è una sottoalgebra di S(R), ed entrambe sono sottoalgebre di M (R). Usando la regolaritá esterna di µ, si dimostra il seguente Lemma 4.24. Siano a, b ∈ R e ϕ ∈ S([a, b]). Allora per ogni ε > 0 esiste una funzione a gradini ψ ε ∈ G([a, b]) tale che µ{x : |ϕ − ψ ε | ≥ ε} < ε .
Sketch della dimostrazione. Osserviamo che ϕè, per definizione, limitata, per cui esiste M ∈ R + tale che ϕ ∞ ≤ M . Scegliamo ε > 0 e partizioniamo [−M, M ] in intervalli J n := [α n −ε/2 , α n + ε/2], n = 1, . . . , N ; poiché ogni ϕ −1 (J n )è misurabile esiste un aperto U n ⊇ ϕ −1 (J n ) tale che U n − ϕ −1 (J n ) ha misura minore di ε/N . Ora,è semplice verificare che U n puó essere espresso come unione disgiunta di intervalli aperti, U n =∪ i I n i , per cui definiamo ψ ε (x) := α n , x ∈ I n i . Ripetiamo quindi il procedimento al variare di n = 1, . . . , N .
Proposizione 4.25. Sia f ∈ M ([a, b]) tale che µ{x : f (x) = ±∞} = 0 . Allora per ogni ε > 0 esistono una funzione a gradini ϕ ε : [a, b] → R ed una funzione continua, lineare a tratti g ε ∈ C([a, b]), tali che µ{x : |f (x) − ϕ ε (x)| ≥ ε} < ε , µ{x : |f (x) − g ε (x)| ≥ ε} < ε .
Inoltre, se m ≤ f ≤ M , allora g ε , ϕ ε possono essere scelte in maniera tale che m ≤ g ε , ϕ ε ≤ M . Dimostrazione. Si usa un argomento di tipo " 3 -ε ". Osserviamo che avendo f −1 (±∞) misura nulla, per regolaritá esterna per ogni ε > 0 esiste M ∈ R tale che |f | ≤ M tranne che su un insieme di misura ε/3 . Come primo passo, partizioniamo [−M, M ) in intervalli semiaperti a sinistra di lunghezza < ε , [−M, M ) =∪ k I k , I k = [y k , y k+1 ), y k+1 − y k < ε , ed introduciamo la funzione semplice
ϕ ε (x) := y k , x ∈ f −1 (I k ) 0 , se |f (x)| > M ,
cosicché |f − ϕ ε | < ε tranne che sull'insieme dove |f | > M (che ha misura ε/3 ). Come secondo passo, usando Lemma 4.24 otteniamo una funzione a gradini ψ ε tale che µ{x : |ϕ ε −ψ ε | ≥ ε} < ε/3 . Come terzo ed ultimo passo, osserviamo che l'insieme {x n } dei punti di discontinuitá di ψ εè al piú numerabile, per cui esiste una successione {J n := [x n − δ n , x n + δ n ]} di intervalli tra loro disgiunti e tali che n (2δ n ) < ε/3 . Definiamo quindi la funzione continua Il seguente risultato fornisce una caratterizzazione delle funzioni integrabili secondo Riemann dal punto di vista della misura di Lebesgue; si noti come la nozione di continuitá giochi un ruolo importante.
g ε (x) := ψ ε (x) , x ∈ [a, b] −∪ n J n ψ ε (x n − δ n ) + ψε(xn+δn)−ψε(xn−δn) 2δn (x − x n + δ n ) , x ∈ J n . Per costruzione µ{x : |g ε − ψ ε | ≥ ε} < ε/3 . Stimando |f − g ε | ≤ |f − ϕ ε | + |ϕ ε − ψ ε | + |g ε − ψ ε |, ed, analogamente, |f − ψ ε |, otteniamo il risultato cercato.
L'integrale di Lebesgue.
Sia (X, M, µ) uno spazio di misura, ϕ := i a i χ Ai , µA i < +∞ una funzione semplice ed A ⊆ X un insieme misurabile (non necessariamente di misura finita). Definiamo rispettivamente l'integrale di ϕ e l'integrale di ϕ su A
X ϕ := i a i µA i , A ϕ := X (ϕχ A ) .
(4.24)
Si verifica banalmente che la precedente definizione di integrale non dipende dalla particolare decomposizione di ϕ come combinazione lineare di funzioni caratteristiche. Abbiamo cosí definito un funzionale lineare su S(X). Nel seguito, quando sará chiaro dal contesto, ometteremo il simbolo " X " dalla notazione di integrale. Vogliamo ora estendere la precedente definizione di integrale alle funzioni misurabili. Come primo passo ci occuperemo del caso in cui X ha misura finita, e successivamente generalizzeremo a spazi (quasi) qualsiasi.
Proposizione 4.28. Sia (X, M, µ) di misura finita ed f : X → R limitata. Se fè misurabile allora ≥ f := inf f ≤ψ∈S(X) ψ = sup f ≥ϕ∈S(X) ϕ =: ≤ f . (4.25)
Se (X, M, µ)è completo, alloraè vero anche il viceversa.
Dimostrazione. Per semplicitá di notazione poniamo M := f ∞ . Se fè misurabile, allora gli elementi (mutualmente disgiunti) della successione
E k := { x ∈ X : (k − 1)M n < f (x) ≤ kM n } , −n ≤ k ≤ n ,(k − 1)χ E k troviamo ϕ n ≤ f ≤ ψ n , e 0 ≤ ≥ f − ≤ f ≤ ψ n − ϕ n ≤ M n n k=−n µE k = M n µX n → 0 .
Ció implica che f soddisfa (4.25). Supponiamo ora che (X, M, µ) sia completo e che valga (4.25), e mostriamo che fè misurabile. A tale scopo, osserviamo cheè sufficiente verificare che f = g q.o. per qualche funzione misurabile g (vedi Prop.4.13, edè qui che sfuttiamo la completezza). Consideriamo, per ogni n ∈ N, funzioni semplici ϕ n , ψ n tali che ϕ n ≤ f ≤ ψ n , ψ n − ϕ n < n −1 . Allora, le funzioni ψ := inf n ψ n , ϕ := sup n ϕ n sono misurabili (Teo.4.11) e ϕ ≤ f ≤ ψ . Ora, posto ∆ := {x ∈ X : ϕ < ψ} abbiamo, per ogni n ∈ N,
∆ ⊆ ∪ k ∆ k,n , ∆ k,n := {x ∈ X : ϕ n < ψ n − k −1 } .
Ora, usando (4.26) troviamo
k −1 µ∆ k,n < ∆ k,n (ψ n − ϕ n ) ≤ n −1 ⇒ µ∆ k,n < kn −1 .
Per cui (potendo scegliere n arbitrariamente grande) concludiamo che µ∆ = 0 . Dunque ψ , ϕ , e quindi f , coincidono quasi ovunque.
In accordo alla proposizione precedente, definiamo l'integrale di f e l'integrale di f su A ∈ M rispettivamente come
f := ≥ f = ≤ f , A f := X f χ A , A ∈ M .
(4.27)
Osservazione 4.7. Le seguenti proprietá sono banali da verificare nel caso di funzioni semplici, per cui nel caso generale seguono applicando la definizione precedente:
(1) Se A ha misura nulla ed f ∈ M (X) allora A f = 0 ; (2) Se f ≥ 0 allora f ≥ 0 . Riemann vs. Lebesgue. Siano a, b ∈ R ed f : [a, b] → R. Allora fè integrabile secondo Riemann se e solo se R ≥ f := inf f ≤ψ∈G([a,b]) ψ = sup f ≥ϕ∈G([a,b]) ϕ =: R ≤ f , (4.28) ed in tal caso poniamo R f := R ≤ f = R ≥ f .
La precedente definizione possiede un contenuto intuitivo, il quale evidenzia la filosofia di base dell'integrale di Riemann: se fè Riemann-integrabile, allora R f deve essere approssimabile a mezzo di integrali di funzioni a gradini, edè facilmente verificabile che, presa ϕ := i a i χ (xi,xi+1) ≤ f , l'errore
δ(ϕ) := R f − φ e migliorabile raffinando la partizione P := {x i } , ovvero considerando P ′ := {x ′ j } ⊃ P e definendo ϕ ′ := j a ′ j χ (x ′ j ,x ′ j+1 ) tale che f (x) ≥ a ′ j ≥ a i , ∀x ∈ (x ′ j , x ′ j+1 ) ⊆ (x i , x i+1
). Infatti ció garantisce che 0 ≤ δ(ϕ ′ ) ≤ δ(ϕ). Dunque l'idea di baseè : raffina le partizioni, e spera che, al restringersi degli intervalli, f sia abbastanza regolare da poter essere ben approssimata da una costante negli intervalli stessi.
Equipaggiamo ora [a, b] con la misura di Lebesgue,
cosicché G([a, b]) ⊂ S([a, b]); allora {ψ ∈ G([a, b]) : ψ ≤ f } ⊂ {ψ ∈ S([a, b]) : ψ ≤ f } , {ψ ∈ G([a, b]) : ψ ≥ f } ⊂ {ψ ∈ S([a, b]) : ψ ≥ f } , per cui concludiamo che R ≤ f ≤ ≤ f ≤ ≥ f ≤ R ≥ f .
Dunque se fè integrabile secondo Riemann alloraè integrabile secondo Lebesgue, ed i due integrali coincidono. D'altra parte, dall'Esempio 4.6 abbiamo una funzione integrabile secondo Lebesgue che nonè integrabile secondo Riemann.
Misure non finite. Sia (X, M, µ) uno spazio di misura non necessariamente finita. Iniziamo denotando con L ∞ µ,0 (X) l'algebra delle funzioni misurabili su X , limitate, e con supporto a misura finita. Chiaramente, possiamo definire l'integrale di f ∈ L ∞ µ,0 (X) su X semplicemente come l'integrale sul suo supporto, come dalla definizione precedente. Inoltre,è chiaro che tale integraleè finito:
| X f | ≤ f ∞ µ(supp(f )) < +∞. Denotiamo con M + (X) l'insieme delle funzioni definite su X , misurabili e non negative. Diciamo che f ∈ M + (X)è integrabile se X f := sup X h , h ∈ L ∞ µ,0 (X) , h ≤ f < +∞ . (4.29)
Passiamo infine al caso generale f ∈ M (X). Per prima cosa, osserviamo che
f = f + − f − , |f | = f + + f − , f + := max{f, 0} ≥ 0 , f − := − min{f, 0} ≥ 0 .
Per (4.12), fè misurabile se e soltanto se f + , f − sono misurabili. Diciamo che fè integrabile se sia f + che f − sono integrabili su X , ed in tal caso, definiamo
X f := X f + − X f − .
Osserviamo che, in base alla definizione precedente, se fè integrabile allora anche |f |è integrabile. Qualora si renderá necessario specificare la misura µ rispetto alla quale stiamo integrando, scriveremo X f dµ := X f .
Denotiamo con L 1 µ (X) lo spazio vettoriale delle funzioni integrabili su X 12 . Osserviamo che segue banalmente dalla definizione che
f ≤ g ⇒ f ≤ g , cf = c f , ∀c ∈ R ; (4.30)
per dimostrare invece che l'integrale rispetta l'operazione di somma sará conveniente usare i teoremi di passaggio al limite sotto il segni d'integrale (vedi sezione seguente).
Proposizione 4.29. Ogni f ∈ L 1 µ (X) ha supporto σ -finito; se f ≥ 0 , allora essaè limite puntuale di una successione monotóna crescente {ϕ n } di funzioni semplici aventi supporto con misura finita.
Dimostrazione. Per ogni n ∈ N definiamo E n := {x ∈ X : |f (x)| ≥ n −1 } ed osserviamo che n −1 µE n = n −1 χ En ≤ En |f | < ∞ ;
cosicché µE n < ∞, e del resto supp(f ) = ∪ n E n . Riguardo la seconda affermazione, prendiamo la successione {ψ n } di Prop.4.12 e definiamo A n := ∪ n i E i , ϕ n := χ An ψ n , cosicché f (x) = lim n ϕ n (x), ∀x ∈ X .
Osservazione 4.8. La nostra definizione di integrale funziona bene solo nel caso in cui (X, M, µ) abbia misura localmente finita 13 . Ad esempio, si prenda un qualsiasi insieme X , M := {X, ∅} con µX := +∞, µ∅ := 0 , cosicché le funzioni misurabili sono le costanti. In base alla nostra definizione di integrale f = 0 per ogni f ∈ M (X), mentre invece ci aspetteremmo, ad esempio, che 1 = µX = +∞. Questa difficoltáè aggirabile definendo f come l'estremo superiore degli integrali di generiche funzioni semplici ≤ f , non necessariamente con supporto di misura finita. In conseguenza di ció andrebbe modificata la dimostrazione del Lemma di Fatou rispetto a quella che daremo nella prossima sezione (vedi [25, §11.3]). Gli altri risultati non necessitano modifiche.
Misure con segno. Se (X, M, µ)è uno spazio di misura con segno allora, grazie alla decomposizione di Jordan, esistono uniche le misure µ ± tali che µ = µ + −µ − . Presa una funzione misurabile
f ∈ M (X) diciamo che essaè µ-integrabile se f ∈ L 1 µ + (X) ∩ L 1 µ − (X), ed in tal caso poniamo f dµ := f dµ + − f dµ − .
Denotiamo con L 1 µ (X) lo spazio delle funzioni µ-integrabili.
Il caso complesso. Sia (X, N , ν) uno spazio di misura (eventualmente con segno). Ogni funzione f :
X → C si puó scrivere come f = f 1 + if 2 , dove f 1 , f 2 : X → R. Diciamo che fè integrabile se f 1 , f 2 ∈ L 1 ν (X); in tal caso, poniamo f dν := f 1 dν + i f 2 dν ,
e scriviamo f ∈ L 1 ν (X, C). Ora, se (X, M, µ)è uno spazio di misura complesso allora per definizione µ = µ 1 + iµ 2 , con µ 1 , µ 2 misure con segno. Presa f : X → C, diciamo che essaè µ-integrabile se f ∈ L 1 µ1 (X, C) ∩ L 1 µ2 (X, C); in tal caso, poniamo
f dµ := f dµ 1 + i f 2 dµ 2 , e scriviamo f ∈ L 1 µ (X, C).
Limiti sotto il segno d'integrale.
Nelle pagine che seguono approcceremo la questione del passaggio al limite sotto il segno di integrale, premettendo che questa operazione non sempre funziona (si veda Esempio 4.8 alla fine della sezione). D'altro canto, se {f n }è una successione uniformemente convergente ad una funzione f , allora una semplice stima mostra come giá nell'ambito dell'integrale di Riemann risulti f n → f . La teoria di Lebesgue migliora tale risultato: vedremo infatti come sia sufficiente, sotto opportune ipotesi, una convergenza q.o.. 13 Osserviamo comunque che gli spazi di misura che si incontrano nella vita sono in genere localmente finiti.
µA < ε 4M ed n 0 ∈ N tale che (f n − f )| X−A ∞ < ε/(2µX) per ogni n > n 0 . Per cui X f n − X f ≤ X |f n − f | = X−A |f n − f | + A |f n − f | ≤ µX ε 2µX + 2M ε 4M . Teorema 4.31 (Lemma di Fatou). Sia (X, M, µ) uno spazio di misura ed {f n } ⊂ M + (X) una successione di funzioni convergente q.o. in x ∈ X ad una funzione f . Allora f ∈ M + (X), e X f ≤ lim n inf X f n . (4.31)
Dimostrazione. L'idea della dimostrazione consiste nel ridursi al caso della convergenza limitata, nel seguente modo:
consideriamo h ∈ L ∞ µ,0 (X) tale che f ≥ h ≥ 0 . Posto h n := min{f n , h} , osserviamo che deve essere h(x) = lim n h n (x) q.o. per x ∈ supp(h) (comunque, al di fuori di supp(h) si trova h ≡ h n = 0 ). Per convergenza limitata (Prop.4.30) si trova X h = supp(h) h = lim n supp(h) h n ≤ lim n inf X f n .
Passando all'estremo superiore per h ≤ f , h ∈ L ∞ µ,0 (X), per definizione di integrale su un insieme di misura non necessariamente finita si trova quanto volevasi dimostrare. Grazie al teorema precedente possiamo dimostrare alcune proprietá di base dell'integrale:
Corollario 4.33. Valgono le seguenti proprietá: (1) Se f, g ∈ M + (X) allora (f +g) = f + g ; (2) Se f ∈ M + (X) allora f = 0 se e solo se f = 0 q.o.; (3) Se f, g ∈ L 1 µ (X) e a ∈ R allora (af + g) = a f + g ; (4) Se f, g ∈ L 1 µ (X) e f ≥ g (f = g ) q.o. allora f ≥ g ( f = g ).f − g = (f − g) con f − g ≥ 0 (f − g = 0 ) q.o.
. Dunque l'asserto per f = g segue dal punto (2), mentre Dimostrazione. Definiamo f n (x) := inf{f (x), n} , n ∈ N, cosicché la successione {f n } converge puntualmente ad f edè monotóna crescente. Dal teorema di convergenza monotóna segue che preso ε > 0 , esiste n 0 ∈ N tale che X f n > X f − ε per ogni n ≥ n 0 . D'altra parte, scelti A, δ tali che µA < δ < ε n 0 ,
(f − g) = A (f − g) + B (f − g) Oss. 4.7 (1) = A (f − g) Oss. 4.7 (2) ≥ 0 , avendo posto A := {x : f (x) − g(x) ≥ 0} , B := {x : f (x) − g(x) < 0} con µB = 0 .ed usando il fatto che |f n0 (x)| ≤ n 0 , x ∈ X , troviamo A f ≤ A (f − f n0 ) + A f n0 ≤ ε + ε .
Applicando il corollario precedente ad X = [a, b], a, b ∈ R, troviamo:
Corollario 4.35. Sia f : [a, b] → R una funzione integrabile. Allora F (x) := x a f (t) dt, x ∈ [a, b], e continua.f n (x) q.o. in x ∈ X . Se esiste g ∈ L 1 µ (X) tale che |f n | ≤ g , ∀n ∈ N, allora X f = lim n X f n . (4.34)
Dimostrazione. Innanzitutto osserviamo che essendo |f n | ≤ g , troviamo che f , f n , n ∈ N, sono integrabili, per cui esistono X f n , X f < +∞. Consideriamo la successione {g − f n } , con g − f n ≥ 0 . Per il Lemma di Fatou (Teo.4.31) abbiamo
X (g − f ) ≤ lim n inf X (g − f n ) . Del resto lim n inf X (g − f n ) = lim n inf X g − X f n = X g − lim n sup X f n e quindi − X f ≤ − lim n sup X f n ⇔ lim n sup X f n ≤ X f .
Applicando lo stesso argomento alla successione {g + f n } otteniamo la diseguaglianza
lim n inf X f n ≥ X f ,
per cui il teoremaè dimostrato.
Esempio 4.8. (vedi [12]). Per ogni λ ∈ R, consideriamo la successione
{f n } ⊂ C([0, 1]) definita da f n (x) := n λ xe −nx , x ∈ [0, 1] .
Ora, f n converge puntualmente alla funzione nulla. Semplici verifiche mostrano che la convergenzà e anche uniforme se e soltanto se λ < 1 (infatti, f n ∞ = n λ−1 e −1 ). Restringiamo ora il campo delle nostre argomentazioni al caso λ ≥ 0 , e studiamo il passaggio al limite sotto il segno di integrale. Un calcolo esplicito mostra che Due applicazioni del teorema di Lebesgue. Consideriamo la retta reale equipaggiata con la misura di Lebesgue µ ed una successione {f n } ⊂ L 1 µ (R) con
f (x) := n f n (x) , q.o. in x ∈ R ; supponiamo che esista g ∈ L 1 µ (R) tale che m n f n (x) ≤ g(x) , q.o. in x ∈ R , ∀m ∈ N .
Allora il teorema di convergenza dominata ci dice che
f = n f n ,
il che fornisce un metodo di calcolo dell'integrale di f nel caso in cui siano facilmente calcolabili quelli delle funzioni f n , n ∈ N. Ció accade tipicamente quando n f nè uno sviluppo in serie di Taylor (f n (x) = c n x n ) o di Fourier (f n (x) = a n sin nx+b n cos nx); per esempi si veda [10, §3.3], [12,Es.6.4.4].
Sempre in conseguenza del teorema di Lebesgue, si ottiene il seguente risultato concernente la derivazione sotto il segno d'integrale:
Teorema 4.37. Sia (X, M, µ) uno spazio di misura, U ⊆ R k aperto ed f : X × U → R tale che: (1) f (·, t) ∈ L 1 µ (X) per ogni t ∈ U ; (2) f (x, ·) ∈ C 1 (U ) per ogni x ∈ X ; (3) Esistono g 1 , . . . , g k ∈ L 1 µ (X) tali che ∂f ∂t i (x, t) ≤ g i (x) , ∀x ∈ X , i = 1, . . . , k . Allora F (t) := X f (·, t), t ∈ U ,è di classe C 1 in U e ∂F ∂t i (t) = X ∂f ∂t i (x, t) dx .
Dimostrazione. Tenendo fisse le variabili t j , j = i , possiamo riguardare F come funzione della sola variabile t i e limitarci a dimostrare il teorema per k = 1 . Presi t, t ′ ∈ U , usando il teorema di Lagrange stimiamo il rapporto incrementale
F (t) − F (t ′ ) t − t ′ ≤ X f (x, t) − f (x, t ′ ) t − t ′ dµ = X ∂f ∂t i (x, ξ) dµ ≤ X g i , t ≤ ξ ≤ t ′ ,
cosicché possiamo applicare il teorema di convergenza dominata e passare al limite t → t ′ sotto il segno d'integrale.
Il teorema di Radon-Nikodym.
Sia M una σ -algebra definita su un insieme X . Date due misure µ, ν : M → R , diciamo che ν e assolutamente continua rispetto a µ se µA = 0 implica νA = 0 per ogni A ∈ M . In tal caso, scriviamo ν ≺ µ. Se µ, ν : M → R sono misure con segno, allora diciamo che νè assolutamente continua rispetto a µ se |µ|A = 0 implica νA = 0 . Lo stesso risultato vale nel caso in cui µ, ν siano misure con segno (con f ∈ M (X)).
Esempio 4.9. (1) Sia (X, M, µ) uno spazio misurabile ed f ∈ M + (X). Definendo νA := A f dµ, A ∈ M , otteniamo una misura assolutamente continua rispetto a µ (vedi Oss.4.7); (2) Sia (X, M, µ) uno spazio misurabile tale che {y} ∈ M e µ{y} = 0 , ∀y ∈ X . Preso x ∈ X , con- sideriamo la misura di Dirac µ x : M → R (vedi Esempio 4.3). Visto che µX = µ(X − {y}) per ogni y ∈ X , troviamo µA = 0 per ogni insieme finito A. D'altra parte µ x A = 1 per ogni A ∋ x, per cui µ x nonè assolutamente continua rispetto a µ.
Dimostrazione. Iniziamo assumendo che X abbia misura finita. Definiamo
C := {g ∈ M + (X) : g dµ ≤ νE , ∀E ∈ M} , α := sup g∈C g .
Poiché µX < ∞ abbiamo che α < ∞; inoltre, per definizione di estremo superiore, esiste una successione g n ∈ C tale che α = lim n g n dµ. Definiamo
f n := sup{g 1 , . . . , g n } ⇒ 0 ≤ f n ր f ; grazie a Teo.4.11 abbiamo f ∈ M + (X). Ora, per ogni E ∈ M troviamo facilmente E =∪ n i=1 E i : f n | Ei = g i , i = 1, . . . , n .
Per cui
E f n dµ = n i Ei g i dµ ≤ n i νE i = νE (ovvero f n ∈ C ), e per convergenza monotóna troviamo E f dµ = lim n E f n dµ ≤ νE , E ∈ M . Dunque f ∈ C e µ{x : |f (x)| = +∞} = 0 . Inoltre troviamo α ≥ f dµ = lim n f n dµ ≥ lim n g n dµ = α ⇒ f dµ = α .
Definiamo ora la misura (non negativa)
ν 0 E := νE − E f dµ , E ∈ M , e mostriamo che ν 0 ≡ 0 . Se per assurdo esistesse A ∈ M con ν 0 A > 0 , allora troveremmo, grazie alla decomposizione di Hahn, un B ⊆ A tale che (ν 0 − εµ)B = νB − B (f + ε1) dµ > 0 . (4.36) Poniamo ora f := f + εχ B ed osserviamo che E f dµ = E f dµ + εµ(E ∩ B) = E−B f dµ + B (f + ε1) dµ ( 4.36) < ν(E − B) + νB = νE .
La diseguaglianza precedente implica che f ∈ C . D'altra parte,
f dµ = f dµ + εµB > α , e quindi f / ∈ C .
Cióè assurdo e concludiamo che ν 0 = 0 . L'unicitá di f si dimostra in modo banale per assurdo, e ció mostra il teorema per X a misura finita. La generalizzazione al caso in cui Xè σ -finito si effettua decomponendo X =∪ n X n , µX n < ∞, ed applicando il risultato nel caso finito ad ogni X n . Ció produce funzioni f n ∈ M + (X), n ∈ N, tali che ν(E ∩ X n ) = E f n dµ, E ∈ M . Ponendo f := n f n otteniamo, per convergenza monotóna, che f soddisfa la proprietá desiderata. La generalizzazione al caso in cui µ, ν sono misure con segno si ottiene applicando la decomposizione di Hahn (vedi [10, §8.3]).
Funzioni BV ed AC.
In questa sezione consideriamo funzioni sullo spazio di misura [a, b], a, b ∈ R, equipaggiato della misura di Lebesgue.
Lo studio delle funzioni assolutamente continue (AC) ed a variazione limitata (BV)è motivato dalla questione della formulazione del teorema fondamentale del calcolo nell'ambito dell'integrale di Lebesgue. In particolare, attraverso tali classi di funzioni determineremo l'immagine dell'applicazione
L 1 ([a, b]) → C([a, b]) , f → F : F (x) := x a f ,(4.37)
che associa ad f ∈ L 1 ([a, b]) la sua primitiva (osservare che giá sappiamo che Fè continua, vedi Cor.4.35). Iniziamo approcciando la questione della derivabilitá.
Sia f : [a, b] → R una funzione. Per ogni x ∈ (a, b), definiamo le quattro derivate D ± f (x) := lim h→0± sup f (x + h) − f (x) h , D ± f (x) := lim h→0± inf f (x + h) − f (x) h . e diciamo che fè derivabile in x se D + f (x) = D − f (x) = D + f (x) = D − f (x) = ±∞. In tal caso, definiamo f ′ (x) := D + f (x).
Osserviamo che se f ∈ C([a, b]) e una delle quattro derivateè non negativa in (a, b), allora fè monotóna crescente. D'altra parte, abbiamo il seguente notevole risultato.
Teorema 4.39 (Lebesgue). Sia f : [a, b] → R monotóna. Allora fè derivabile q.o. in [a, b]. Inoltre, f ′è misurabile, e b a f ′ (x) dx ≤ f (b) − f (a) .
(4.38)
Dimostrazione. Per fissare le idee assumiamo che f sia non decrescente e consideriamo le derivate
D + f e D − f . L'obiettivoè quello di mostrare che la misura di E := {x : D + f (x) = D − f (x)}è nulla. A tale scopo definiamo, per ogni u < v ∈ Q , E uv := {x : D + f (x) > v > u > D − f (x)} , ed osserviamo che E = ∪ u,v E uv . Mostriamo dunque che ogni E uv ha misura nulla (il che implica che E , essendo unione numerabile degli E uv , ha misura nulla). Innanzitutto osserviamo che E uv , essendo contenuto in [a, b], ha certamente misura esterna finita. Applicando il Lemma di Vitali troviamo una collezione di intervalli disgiunti {I 1 , . . . , I N } , I n := (x n − h n , x n ), tale che A ′ :=∪ n I n ⊆ E uv , µA ′ > µ * E uv − ε . Ora, per ogni x ∈ A ′ esiste un intervallo [x − h, x] tale che f (x) − f (x − h) < uh ⇒ n (f (x n ) − f (x n − h n )) < u n h n < u(µ * E uv + ε) .
D'altro canto, ogni y ∈ A ′è tale che esiste k > 0 con (y, y + k) ⊆ I n ⊆ E uv . Applicando ancora il Lemma di Vitali, troviamo una collezione {J 1 , . . . , J M } , J m := (y m , y m + k m ), tale che
A ′′ :=∪ m J m , µA ′′ > µ * E uv − 2ε . Per y ∈ A ′′ troviamo f (y + k) − f (y) > vk , m (f (y m + k m ) − f (y m )) > v(µ * E uv − 2ε) .
Ora, per monotonía abbiamo
m : Jm⊆In (f (y m + k m ) − f (y m )) ≤ f (x n ) − f (x n − h n ) , da cui v(µ * E uv − 2ε) < m (f (y m + k m ) − f (y m )) ≤ n (f (x n ) − f (x n − h n )) < u(µ * E uv + ε) .
Per cui E uv ha misura nulla ed fè derivabile q.o.. Infine, (4.38) si dimostra applicando il Lemma di Fatou alla successione g n (x) := n(
f (x + 1/n) − f (x)), x ∈ [a, b], che converge q.o. a f ′ (x). Funzioni BV. Una partizione di [a, b]è una successione finita del tipo P := x 0 = a, x 1 , . . . , x n(P ) = b .
Denotiamo con P l'insieme delle partizioni di [a, b];è chiaro che Pè un insieme parzialmente ordinato rispetto all'inclusione; se P ⊂ P ′ , allora diremo che P ′è un raffinamento di P . E' ovvio che ogni partizione P ammette un raffinamento. Per ogni f :
[a, b] → R, definiamo V b a f (P ) = n(P )−1 k=0 |f (x k+1 ) − f (x k )| . Applicando la diseguaglianza triangolare troviamo V b a f (P ) ≤ V b a f (P ′ ) , P ⊆ P ′ . Definiamo ora la variazione totale V b a f := sup P ∈P V b a f (P ) = sup P ∈P n(P )−1 k=0 |f (x k+1 ) − f (x k )| . (4.39) Definizione 4.40. Una funzione f : [a, b] → R si dice a variazione limitata (BV) se V b a f < +∞. Denotiamo con BV ([a, b]) l'insieme delle funzioni a variazione limitata su [a, b].
Osservazione 4.10. La nozione di funzione BV puó essere interpretata anche in termini geometrici, nel senso che la condizione V b a f < +∞ indica che la curva definita dal grafico di f in [a, b]è rettificabile.
Una semplice applicazione della diseguaglianza triangolare implica che BV ([a, b])è uno spazio vettoriale. Per dare un'idea del contenuto intuitivo di (4.39), osserviamo che se fè in
C 1 ([a, b]), allora V b a f = b a |f ′ (x)| dx . Esempio 4.10. Consideriamo le funzioni f k : 0, 2 π → R, k = 0, 1, 2 , f k (x) := 0 , x = 0 x k sin(1/x) , x = 0 Risulta che f 0 , f 1 non sono BV, mentre f 2è BV.
Il seguente teorema fornisce una caratterizzazione delle funzioni BV. Dimostrazione. Iniziamo introducendo la seguente notazione: se a ∈ R, allora a + := sup{a, 0} ≥ 0 , 1
a − := − inf{a, 0} ≥ 0 . Cosicché, |a| = a + + a − , a = a + − a − . Sia f BV; definiamo V b a f ± := sup P ∈P n(P )−1 k=0 (f (x k+1 ) − f (x k )) ± , cosicché dalla relazione |a| = a + + a − si ottiene facilmente V b a f = V b a f + + V b a f − . (4.40) Inoltre, dalla relazione a = a + − a − segue n(P )−1 k=0 (f (x k+1 ) − f (x k )) + − n(P )−1 k=0 (f (x k+1 ) − f (x k )) − = f (b) − f (a) per cui otteniamo f (b) − f (a) = V b a f + − V b a f − . (4.41) Definiamo allora f ± (x) := V x a f ± (osservare che f BV implica |f ± (x)| < V b a f < +∞, cosicché f ± (x)è ben definita per ogni x). Da (4.41) segue f (x) = f + (x) − f − (x) + f (a), x ∈ [a, b], ed e chiaro che f ± sono monotóne crescenti. Viceversa, se f = g − h con g, h monotóne crescenti, abbiamo n(P )−1 k=0 |f (x k+1 )−f (x k )| ≤ n(P )−1 k=0 [(g(x k+1 ) − g(x k )) + (h(x k+1 ) − h(x k ))] ≤ g(b)−g(a)+h(b)−h(a), e ció implica f ∈ BV ([a, b]).−1 f ′ = 0 < f (1) − f (−1) = 1 . Definizione 4.42. Una funzione F : [a, b] → R si dice una primitiva di f ∈ L 1 ([a, b]) se F (x) = F (a) + x a f (t)dt , x ∈ [a, b] . Proposizione 4.43. Sia f ∈ L 1 ([a, b]) ed F una sua primitiva. Allora F ∈ BV ([a, b]) ∩ C([a, b]). Dimostrazione. Fè continua grazie a Cor.4.35. Ora se P una partizione di [a, b] troviamo n(P )−1 k=0 |F (x k+1 ) − F (x k )| = n(P )−1 k=0 x k+1 x k f (t) dt ≤ b a |f (t)| dt . Dunque F ∈ BV ([a, b]).f : [a, b] → R e tale che sup n V b a f n < +∞. Allora f ∈ BV ([a, b]). Funzioni AC. Una funzione f : [a, b] → R si dice assolutamente continua (AC), se per ogni ε > 0 esiste un δ > 0 tale che, presa una qualsiasi collezione finita C := {(x k , x ′ k )} k di intervalli disgiunti con l(C) := k (x ′ k − x k ) < δ , risulta k |f (x ′ k ) − f (x k )| < ε . Denotiamo con AC([a, b]) l'insieme delle funzioni AC su [a, b]. Si dimostra facilmente che AC([a, b])
e uno spazio vettoriale. Inoltre,è ovvio che ogni funzione ACè anche (uniformemente) continua.
Esempio 4.12. Sia f : [a, b] → R una funzione con costante di Lipschitz L > 0 . Scegliendo δ < ε/L nella definizione precedente, concludiamo che f ∈ AC([a, b]). Lemma 4.45. AC([a, b]) ⊆ BV ([a, b]), cosicché ogni funzione ACè derivabile q.o. in [a, b].
Dimostrazione.
Sia f ∈ AC([a, b]). Fissiamo ε > 0 tale che k |f (x ′ k ) − f (x k )| < ε per ogni collezione C := {(x k , x ′ k )} con l(C) < δ e δ > 0 opportuno. Sia P := {. . . , x k , . . .} una partizione di (a, b). Effettuando eventualmente un raffinamento di P , possiamo costruire N collezioni C i := {(y ij , y ′ ij )} j tali che [a, b] = ∪ i C i , P ⊆ ∪ ij {y ij , y ′ ij } ij , δ/2 < l(C i ) < δ .
Ora, il numero N di collezioni che soddisfano le condizioni precedenti rimane limitato, a prescindere dalla partizione P :
i δ/2 ≤ i l(C i ) = b − a ⇒ δ N (N − 1) 4 ≤ b − a ⇒ N ≤ N (N − 1) ≤ 4 δ (b − a)
.
Inoltre, essendo l(C i ) < δ troviamo k |f (x k ) − f (x k+1 )| ≤ i,j |f (y ij ) − f (y ′ ij )| ≤ i ε = N (N − 1) 2 ε .
Poiché la maggiorazione precedente non dipende da P , concludiamo che fè BV.
Osservazione 4.11. L'inclusione AC([a, b]) ⊆ BV ([a, b])èf 0 (t) := t , f n+1 (t) := 1/2f n (3t) , t ∈ [0, 1/3] 1/2 , t ∈ (1/3, 2/3] 1/2 1 + f n 3 t − 2 3 , t ∈ (2/3, 1] . Si verifica facilmente che f n+k − f n ∞ ≤ 2 −n , ∀n, k ∈ N, per cui esiste il limite uniforme f ∈ C([0, 1])
. A livello intuitivo, possiamo visualizzare f come una funzione costante sugli "intervalli di mezzo" che appaiono nella costruzione dell'insieme di Cantor W 1/3 e crescente in W 1/3 , il quale ha misura nulla (vedi Es.4.5). Ora, essendo ogni f n monotóna crescente abbiamo che fè monotóna crescente (e quindi BV) ed uniformemente continua (Heine-Cantor). Si noti che
f ′ = 0 q.o. in t ∈ [0, 1] e quindi 1 0 f ′ = 0 < f (1) − f (0) = 1 . Ció implica, in conseguenza del Lemma seguente, che f nonè AC.
A differenza dell'esempio precedente, abbiamo: 15 Detta altresí la scala del Diavolo.
Lemma 4.46. Sia f ∈ AC([a, b]). Se f ′ = 0 q.o., allora fè costante. Dimostrazione. Mostriamo che f (a) = f (c) per ogni c ∈ (a, b]. A tale scopo definiamo E c := {t ∈ (a, c) : f ′ (t) = 0} ⇒ µE c = c − a , e scegliamo arbitrari ε, ε ′ > 0 . Ora, per ogni t ∈ E c esiste h > 0 tale che [t, t + h] ⊆ [a, c] e |f (t + h) − f (t)| < ε ′ h. Sia ora δ > 0 il numero reale associato ad ε nella definizione di assoluta continuitá; applicando il Lemma di Vitali, possiamo estrarre da {[t, t + h]} una collezione finita {[t k , t ′ k ]} di intervalli disgiunti tali che µ (E c −∪ k [t k , t ′ k ]) < δ . (4.42) Osserviamo che si ha l'ordinamento t ′ 0 := a < t 1 < t ′ 1 ≤ t 2 < . . . < t ′ n < t n+1 := c, cosicché ogni intevallo (t ′ k , t k+1 )è contenuto in E c −∪ k [t k , t ′ k ]. Per cui n k=0 |t k+1 − t ′ k | < δ ⇒ n k=0 |f (t k+1 ) − f (t ′ k )| < ε . D'altra parte, n k=1 |f (t ′ k ) − f (t k )| ≤ ε ′ n k=0 (t ′ k − t k ) ≤ ε ′ (c − a) . Concludiamo quindi |f (c) − f (a)| = | k f (t k+1 ) − f (t ′ k ) + f (t ′ k ) − f (t k )| ≤ k |f (t k+1 ) − f (t ′ k )| + k |f (t ′ k ) − f (t k )| ≤ ε + ε ′ (c − a) .
Il seguente risultato risolve completamente la questione della determinazione dell'immagine dell'applicazione integrale (4.37):
Teorema 4.47. Una funzione F : [a, b] → Rè primitiva di una qualche f ∈ L 1 ([a, b]) se e soltanto se F ∈ AC([a, b]).
Dimostrazione. Sia F ∈ AC([a, b]). Allora Fè BV, derivabile q.o. (Lemma 4.45) e differenza di due funzioni monotóne g ed h (Teorema 4.41). Per cui, per disuguaglianza triangolare,
|F ′ (x)| ≤ g ′ (x) + h ′ (x) , q.o. in x ∈ [a, b] . Quindi, integrando membro a membro troviamo b a |F ′ (x)| dx ≤ g(b) − g(a) − h(b) + h(a) , il che implica che F ′è integrabile. Definendô F (x) := F (a) + x a F ′ (t) dt , x ∈ [a, b] , troviamo F ′ −F ′ = 0 q.o. e quindi (
Funzioni convesse e diseguaglianza di Jensen.
Una funzione ϕ : (a, b) → R si dice convessa se per ogni λ ∈ [0, 1] risulta ϕ((1 − λ)x + λy) ≤ (1 − λ)ϕ(x) + λϕ(y) , a < x, y < b .
(4.43)
Ció significa che per ogni x, y ∈ (a, b) il grafico di ϕ rimane confinato nella regione al di sotto della retta a secondo membro di (4.43). Una semplice verifica mostra che vale la diseguaglianza
ϕ(y) − ϕ(x) y − x ≤ ϕ(y ′ ) − ϕ(x ′ ) y ′ − x ′ , x ≤ x ′ < y ′ , x ′ < y ≤ y ′ . (4.44) Proposizione 4.48. Sia ϕ : (a, b) → R convessa. Allora 1. ϕè localmente Lipschitz in (a, b); 2. ϕè AC in ogni sottointervallo chiuso di (a, b); 3. D + ϕ = D + ϕ, D − ϕ = D − ϕ sono monotóne crescenti; 4. Se {ϕ i } i∈Iè una famiglia di funzioni convesse, allora ϕ := sup i ϕ iè convessa; 5. ϕè derivabile in (a, b) tranne che in un sottoinsieme numerabile. Sketch della dimostrazione. Punto 1: Se [c, d] ⊂ [a ′ , b ′ ] ⊂ (a, b), allora per ogni c < x, y < d troviamo ϕ(c) − ϕ(a ′ ) c − a ′ ≤ ϕ(y) − ϕ(x) y − x ≤ ϕ(b ′ ) − ϕ(d) b ′ − d ,(4.ϕ i ((1 − λ)x + λy) ≤ (1 − λ)ϕ i (x) + λϕ i (y) ≤ (1 − λ)ϕ(x) + λϕ(y) , i ∈ I ,
il che implica che ϕ soddisfa la condizione di convessitá. Punto 5: L'insieme di discontinuitá di una funzione monotóna puó essere al piú numerabile: applicando questo principio a D + ϕ, otteniamo che ϕè derivabile nei punti in cui D + ϕè continua.
La costante di Lipschitz associata a ϕ nel senso del Punto 2 dipende solo dai valori assunti da ϕ agli estremi c, d, a ′ , b ′ (vedi (4.45)), e questoè un fattore di cui tenere conto nel caso in cui si abbia necessitá di effettuare stime connesse all'equicontinuitá di famiglie di funzioni convesse.
Osserviamo inoltre cheè semplice dimostrare l'affermazione reciproca del Punto 3: se ϕè continua in (a, b) e se una delle sue derivateè crescente, allora ϕè convessa.
La diseguaglianza di Jensen. Una retta di supporto in x 0 ∈ (a, b) per ϕè una retta y(x) = m(x − x 0 ) + ϕ(x 0 ) , x ∈ (a, b) , tale che y(x) ≤ ϕ(x), x ∈ (a, b
). E' banale verificare che (proprio grazie alla convessitá di ϕ) esiste sempre una retta di supporto per ϕ in ogni x 0 ∈ (a, b).
Proposizione 4.49 (Diseguaglianza di Jensen). Sia ϕ : R → R convessa, (X, M, µ) uno spazio di misura di probabilitá ed f ∈ L 1 µ (X). Allora ϕ X f dµ ≤ X ϕ • f dµ . (4.46)
Dimostrazione. Innanzitutto osserviamo che, essendo ϕ continua, ϕ • fè misurabile. Definiamo α := f e consideriamo una retta di supporto in α ,
y(t) = m(t − α) + ϕ(α) , t ∈ R , cosicché valutando su valori del tipo f (x), x ∈ X , otteniamo ϕ(f (x)) ≥ m(f (x) − α) + ϕ(α). Integrando su X (tenuto conto che µX = 1 ) si trova X ϕ • f dµ ≥ m X (f − α) dµ + ϕ(α) = ϕ(α) ,
e quindi otteniamo quanto volevasi dimostrare.
Esempio 4.13. Per ogni f ∈ L 1 ([0, 1]) si trova e 1 0 f ≤ 1 0 e f (x) dx. 4.8 Esercizi. Esercizio 4.1 (Il Lemma di Riemann-Lebesgue). Sia f : R → R integrabile e ϕ limitata, misurabile e tale che esista π > 0 con ϕ(x + π) = −ϕ(x), ∀x ∈ R. (1) Preso k ∈ N si mostri che R f (x)ϕ(kx) dx = − R f x + π k ϕ(kx) dx .
(2) Si assuma come noto che
lim h→0 R |f (x + h) − f (x)| dx = 0 (4.47) (vedi Esercizio 5.2), e, usando il punto (1), si mostri che lim k→∞ R f (x)ϕ(kx) dx = 0 . (4.48)
Soluzione.
(1) Usando la sostituzione x → x + πk −1 e la periodicitá di ϕ si trova
I k := f (x)ϕ(kx) dx = f x + π k ϕ(kx + π) dx = − f x + π k ϕ(kx) dx .
(2) Usando il punto precedente e (4.47) otteniamo le stime
2|I k | ≤ f x + π k − f (x) |ϕ(kx)| dx ≤ ϕ ∞ f x + π k − f (x) dx k → 0 . Esercizio 4.2. Sia {f n } ∪ {f } ⊂ C 0 (R) ∩ L 1 (R), e lim n |f n − f | = 0 . Allora f (x) = lim n f n (x), ∀x ∈ R.
Soluzione. Supponiamo per assurdo che esistano x 0 ∈ R ed ε > 0 tali che esiste una sottosuccessione
{x n k } con |f (x 0 ) − f n k (x 0 )| ≥ ε .
Per continuitá, esiste un aperto A ∋ x 0 con µ(A) > 0 tale che
|f (x) − f n k (x)| ≥ ε 2 , x ∈ A .
Per cui,
|f n k (x) − f (x)| dx ≥ A |f n k (x) − f (x)| dx ≥ ε 2 µA ,
il che contraddice l'ipotesi fatta.
Esercizio 4.3. Sia f ∈ BV ([a, b]). Presa una funzione reale g , indicare le condizioni che g deve soddisfare affinché g • f sia BV . Esercizio 4.4. Sia A ⊂ [0, 1] tale che µA = 0 , dove µè la misura di Lebesgue. Dimostrare che: (1) Esiste una successione di aperti [0, 1] ⊃ A 1 ⊃ A 2 ⊃ . . . tale che A ⊂ A n e µA n ≤ 2 −n per ogni n ∈ N; (2) g := n χ An appartiene ad L 1 ([0, 1]); (3) f (x) := x 0 g , x ∈ [0, 1], appartiene ad AC([0, 1]), ma nonè derivabile in nessun punto di A.
(Suggerimenti: per (1) si usi la regolaritá esterna; per (2) si osservi che g 1 = n nµ(A n −A n+1 ); per (3) si osservi che g| A = +∞.)
Esercizio 4.5 (Il Lemma di Borel-Cantelli). Sia (X, M, µ) uno spazio di misura finita ed {A n } ⊆ M tale che m µA n < +∞. Posto B N := ∪ ∞ n=N A n , dimostrare che µ N B N = 0 . (4.49) Soluzione. Per ogni N ∈ N, si ha µ (∩ N B N ) ≤ µB N ≤ ∞ n=N µA n .
Esercizio 4.6 (Riguardo (4.15)).
Sia A n := [n, +∞), n ∈ N. Si verifichi che posto B N := ∪ ∞ n=N A n , risulta µ N B N = 0 , lim N µB N = +∞ . Esercizio 4.7 Sia X un insieme e β una σ -algebra su X . Date µ, ν ∈ Λ 1 β (X) (vedi (4.3)), si mostri che: (1) |µE| ≤ |µ|E , ∀E ∈ M µ ; (2) µ = 0 ⇒ µ = 0 (ovvero, µE = 0 per ogni E ∈ M ); (3) µ + ν ≤ µ + ν . Soluzione. Consideriamo decomposizioni di Hahn {X ± µ } per µ e {X ± µ+ν } per µ + ν . Riguardo (1), si osservi che |µE| = |µ(E ∩ X + µ ) + µ(E ∩ X − µ )| ≤ µ(E ∩ X + µ ) − µ(E ∩ X − µ ) = |µ|E , E ∈ M µ , cosicché (2) segue per monotonía di |µ|; riguardo (3), usando (1) troviamo µ + ν = |µ + ν|X = = {µ + ν}X + µ+ν − {µ + ν}X − µ+ν = = µX + µ+ν − µX − µ+ν + νX + µ+ν − νX − µ+ν ≤ ≤ |µ|X + µ+ν + |µ|X − µ+ν + |ν|X + µ+ν + |ν|X − µ+ν = = |µ|X + |ν|X .
Esercizio 4.8 Sia X uno spazio topologico. Si mostri che per ogni funzione boreliana f , λ ∈ R, e misure con segno, boreliane, finite µ, ν , risulta
f d{λµ + ν} = λ f dµ + g dν .
(Suggerimento: si inizi dimostrando l'uguaglianza precedente su funzioni semplici e poi si usi la definizione di integrale.)
Esercizio 4.9 Sia X uno spazio topologico localmente compatto e di Hausdorff. Preso x ∈ X ed una misura di Dirac µ x : M → R (vedi Esempio 4.3), si mostrino i seguenti punti: (1) Due funzioni misurabili f, g coincidono q.o. rispetto a µ x se e solo se f (x) = g(x); (2) f dµ x = f (x), per ogni f ∈ L 1 µx (X).
(Suggerimento: si osservi che, essendo X di Hausdorff, l'insieme {x}è boreliano e quindi misurabile; per cui,
µ x E = µ x (E − {x}) + µ x {x} = µ x {x} = 1 se E ∋ x, mentre µE = 0 se x / ∈ E . Di conseguenza, prese f, g ∈ M (X) si trova µ x {y : f (y) = g(y)} = µ x {x} = 1 , se f (x) = g(x) 0 , se f (x) = g(x) .
Per quanto riguarda il punto (2), si verifichi sulle funzioni semplici e poi si applichi la definizione di integrale).
Esercizio 4.10 Sia (X, M, µ) uno spazio misurabile ed f ∈ M + (X). Presa la misura
µ f E := E f dµ , E ∈ M , si mostri che g dµ f = gf dµ per ogni g ∈ L 1 µ f (X).
(Suggerimento: si verifichi -al solito -sulle funzioni semplici e poi si applichi la definizione di integrale).
Esercizio 4.11. Richiamando la notazione (4.16), si consideri l'applicazione
µ * ac A := inf n l(J n ) : A ⊂ ∪ n J n , {J n } ⊂ I ac , ∀A ⊆ R ,
e si mostri che essa coincide con la misura esterna di Lebesgue.
(Suggerimenti: si inizi mostrando che µ * ac = µ * sugli intervalli, verificando che, dati a < b ∈ R e I := (a, b), J := (a, b], J ′ := [a, b), risulta µ * ac (I) = µ * ac (I) = µ * ac (J) = µ * ac (J ′ ) = b − a , sulla linea del Lemma 4.17. Si usi il fatto che per ogni ε > 0 si ha µ * ac (a, b + ε] − µ * (a, b) = ε ). Esercizio 4.12 (Funzioni di distribuzione). Sia µ una misura boreliana finita su R. La funzione di distribuzione di µ si definisce come µ(x) := µ(−∞, x] , x ∈ R .
(1) Si mostri che µè :
(1.1) monotóna crescente; (1.2) tale che lim x→−∞ µ(x) = 0 ; (1.3) continua a destra, ovvero lim δ→0 + µ(x + δ) = µ(x), ∀x ∈ R; (2) Si mostri che µ(a, b] = µ(b) − µ(a) , ∀a < b .
(3) Si mostri che µè limitata. (4) Si mostri che µè continua se e solo se µ{x} = 0 , ∀x ∈ R. (2) Si mostri che la misura associata µ ωè boreliana. (3) Supposto che µ ω sia finita, si mostri che la relativa funzione di distribuzione µ ω coincide con ω .
(Suggerimenti: (1.1) si usi la monotonía di µ; (1.2) si noti che ∩ n∈N (−∞, −n] = ∅ ; (1.3) si ha lim n µ(x + 1/n) = lim n µ{∩ n k (−∞, x + 1/k]} Lemma 4.3 = µ(−∞, x] . (2) Si ha [a, b) = (−∞, b) − (−∞, a); (3) Si usi il fatto che µè monotóna crescente e si osservi che sup µ = lim x→∞ µ(x) = µ(R) < ∞. (4) Si ha µ{x} = µ(−∞, x] − ∪ n µ(−∞, x − 1/n] = µ(x) − lim n µ(x − 1/n)).(4) Si assuma che ω| [α,β] ∈ AC([α, β]) per qualche α, β ∈ R e si mostri che [α,β] f dµ ω = β α f (t)ω ′ (t) dt , ∀f : [α, β] → R boreliana
(il secondo integraleè rispetto alla misura di Lebesgue).
Gli spazi L p .
Esistono ampie ed interessanti classi di funzioni misurabili che non sono né continue né integrabili, ed un loro studio sistematico viene effettuato introducendo opportune norme, generalmente non euclidee, che generalizzano in modo naturale quelle definite su R n , n ∈ N, per p ∈ [1, +∞):
v p := n 1 |v i | p 1/p , v ∈ R n .
Ció conduce alla teoria degli spazi L p , la quale fornisce tra l'altro un'importante motivazione per la sistematizzazione di concetti (quali quelli di norma, funzionale, dualitá) che troveranno poi naturale collocazione nell'ambito degli spazi di Banach e di Hilbert, e quindi dell'analisi funzionale.
Proprietá generali.
Sia (X, M, µ) uno spazio misurabile completo. Preso p ∈ (0, +∞) denotiamo con L p µ (X) l'insieme delle funzioni misurabili su X tali che X |f | p < +∞ :
poiché |f + g| p < 2 p (|f | p + |g| p ) ,
concludiamo che ogni L p µ (X)è uno spazio vettoriale. Nel caso in cui X ⊆ R n , n ∈ N, sia equipaggiato con la misura di Lebesgue ometteremo il simbolo µ, per cui scriveremo ad esempio
f p := X |f | p 1/p , f ∈ L p µ (X) . (5.1) Denotiamo inoltre con L ∞ µ (X) lo spazio vettoriale delle funzioni limitate e misurabili su X , e definiamo f ∞ := sup x |f (x)|, f ∈ L ∞ µ (X)
. Come suggerito dalla notazione, la nostra intenzionè e quella di interpretare · p , p ∈ (0, +∞], come una norma su L p µ (X). Tuttavia,è chiaro che · p non puó essere una norma, in quanto svanisce su ogni funzione nulla q.o. su X . Per ovviare a ció, con un abuso di terminologia identificheremo funzioni L p con le corrispondenti classi di equivalenza q.o. su X 16 . Nel caso p = +∞ il passaggio a classi di equivalenza q.o. richiede un piccolo aggiustamento, che consiste nel definire l'estremo superiore essenziale,
f ∞ := inf {M ∈ R : µ{x ∈ X : |f (x)| > M } = 0} , f ∈ L ∞ µ (X) , (5.2) in maniera tale che se g = f q.o. allora g ∞ = f ∞ .
Osserviamo che ancora non sappiamo se · pè una norma, in quanto dobbiamo dimostrare la diseguaglianza triangolare; iniziamo osservando che, grazie a proprietá elementari dell'integrale, essaè verificata nei casi p = 1, ∞. Inoltre, una semplice stima mostra che
X |f g| ≤ f 1 g ∞ , f ∈ L 1 µ (X) , g ∈ L ∞ µ (X) .
Proposizione 5.1 (Diseguaglianza di Holder). Siano p, q ∈ [0, +∞] tali che
1 p + 1 q = 1 . (5.3) Allora X |f g| ≤ f p g q , f ∈ L p µ (X) , g ∈ L q µ (X) . (5.4)
Dimostrazione. Come primo passo, osserviamo che per ogni a, b > 0 risulta
a 1/p b 1/q ≤ a p + b q , (5.5)
come si dimostra facilmente usando la convessitá di exp:
e 1 p log a+ 1 q log b ≤ a p + b q .
Ora, se f, g sono come da ipotesi (e non nulle q.o., altrimenti non c'è niente da dimostrare), poniamo
a := |f | p f −p p , b := |g| q g −q q ed otteniamo, usando (5.5), |f g| f p g q ≤ 1 p |f | p f p p + 1 q |g| q g q q ,
ed integrando otteniamo (5.4).
Reali estesi p, q ∈ [0, +∞] che soddisfano (5.3) si dicono coniugati. Chiaramente qè univocamente determinato da p, ed in tal caso sará denotato con p.
Proposizione 5.2 (Diseguaglianza di Minkowski, ovvero la disegueglianza triangolare). Siano p ∈ [1, +∞] ed f, g ∈ L p µ (X). Allora f + g p ≤ f p + g p .
Dimostrazione. Essendo il caso p = 1 ovviamente verificato, supponiamo p > 1 , poniamo q := p, ed osserviamo che q(p − 1) = p, cosicché
X |f | p−1 q = f p p < +∞ , f ∈ L p µ (X) ⇒ f p−1 ∈ L q µ (X)
.
Poi, stimiamo f + g p p ≤ X |f ||f + g| p−1 + X |g||f + g| p−1 Holder ≤ f p (f + g) p−1 q + g p (f + g) p−1 q = ( f p + g p )( f + g p/q p ) . Corollario 5.3. L p µ (X)è uno spazio normato per ogni p ∈ [1, +∞]. Corollario 5.4. Sia (X, M, µ) uno spazio di misura finita e p ∈ [1, +∞]. Se f ∈ L p µ (X) allora f ∈ L 1 µ (X).
Dimostrazione. Basta osservare che la funzione costante 1 appartiene ad L q µ (X), q := p, cosicché
f 1 = |f 1| ≤ f p 1 q = f p (µX) 1/q .
Passiamo ora a dimostrare la completezza degli spazi L p . Per le nozioni di spazio di Banach e di Hilbert si veda §7.
Teorema 5.5 (Fischer-Riesz). L p µ (X)è uno spazio di Banach per ogni p ∈ [1, +∞]. Inoltre, se {f n } ∪ {f } ⊂ L p µ (X) con lim n f n − f p = 0 , allora esiste una sottosuccessione {n k } tale che f (x) = lim k f n k (x) q.o. in x ∈ X .
Dimostrazione. Il caso p = +∞è banale, per cui assumiamo p ∈ [1, +∞). L'ideaè quella di applicare il criterio delle serie convergenti in norma (Prop.7.1), per cui consideriamo una successione
{f i } ⊂ L p µ (X) tale che esista finito M := ∞ i f i p con lo scopo di dimostrare che esiste la somma i f i ∈ L p µ (X); grazie al giá menzionato criterio, ció sará sufficiente a concludere che L p µ (X)è uno spazio di Banach. Definiamo g n (x) := n i=1 |f i (x)| , x ∈ X , ed osserviamo che g n p ≤ n i f i p ≤ M . Ció implica che g n ∈ L p µ (X), ∀n ∈ N. Ora, per monotonía per ogni x ∈ X esiste g(x) := lim n g n (x) ∈ [0, +∞]. La funzione g cosí ottenutaè misurabile e, applicando la diseguaglianza di Minkowski ed il Lemma di Fatou (Teo.4.31), otteniamo X g p ≤ lim n inf n i X |f i | p ≤ M p , (5.6) per cui g ∈ L p µ (X). Ció implica che q.o. in x ∈ X la serie i f i (x)è assolutamente convergente per cui (essendo R completo) esiste finito s(x) := lim n n i f i (x). Definendo s(x) := 0 per ogni x tale che g(x) = +∞ otteniamo una funzione s tale che s = i f i q.o. Ció implica che sè misurabile (Teo.4.11), ed utilizzando (5.6) si trova s ∈ L p µ (X). Concludiamo che sè la somma cercata e quindi L p µ (X)è uno spazio di Banach. Sia ora lim n f − f n p = 0 . Scegliamo una sottosuccessione {n k } tale che f n k+1 − f n k < 2 −n k e definiamo δ m (x) := m k=1 |f n k+1 (x) − f n k (x)| .
La successione precedenteè monotóna crescente, per cui esiste il limite δ(x) per ogni x ∈ X ; per verificare che essoè finito q.o., osserviamo che
δ m p ≤ k f n k+1 − f n k p ≤ k 2 −n k ≤ 1 . Dunque δ ∈ L p µ (X) (teorema di convergenza monotóna), e quindi δ(x) < +∞ q.o. in x ∈ X . Usando ripetutamente la diseguaglianza triangolare, nell'insieme dove g(x) < +∞ troviamo |f m (x) − f l (x)| ≤ δ(x) − δ l−1 (x) (5.7) e quindi {f m (x)}è di Cauchy. Definendof (x) := lim m f m (x) otteniamo una funzionef definita q.o. in x ∈ X , e vogliamo mostrare che in effettif = f q.o. in x ∈ X . A tale scopo, osserviamo che (5.7) implica, essendo δ l (x) > 0 , che |f (x) − f l (x)| ≤ δ(x) , q.o. in x ∈ X ; dunque, avendosi δ ∈ L p µ (X), possiamo applicare il teorema di convergenza dominata e concludere lim l f − f l p = f − f p = 0 . Cosicchéf = f q.o. in x ∈ X .
Osservazione 5.1. La convergenza q.o.è garantita soltanto per la sottosuccessione {f n k } . A questo proposito si veda l'Esercizio 5.6.
Osservazione 5.2. Se f ∈ L p µ (X) allora certamente l'insieme {x ∈ X :
|f (x)| = ∞} ha misura nulla, per cui ogni funzione in L p coincide q.o. con una funzione a valori in R. Nell'ambito degli spazi L p possiamo quindi sempre assumere di avere a che fare, a meno di equivalenza q.o., con funzioni a valori reali (e non a valori nei reali estesi).
Osservazione 5.3. Sia E uno spazio di Banach (vedi §7). Allora possiamo introdurre su E la σ -algebra βE generata dai dischi ∆(v, r), v ∈ E , r > 0 (in altri termini, βEè la σ -algebra dei boreliani associata alla topologia della norma di E ). Per cui, dato il nostro spazio di misura (X, M, µ) ha senso considerare l'insieme M (X, E) delle funzioni misurabili f : X → E nel senso di Def.4.1. Presa f ∈ M (X, E) denotiamo con f (x) ∈ R la norma di f (x) in E ; la funzione {X ∋ x → f (x) }è chiaramente misurabile nel senso usuale, per cui definiamo L p (X, E) := {f ∈ M (X, E) : f (x) p dµ < ∞} , p ∈ [1, ∞] .
In particolare, per E = C (che identifichiamo con R 2 come spazio di Banach), otteniamo gli spazi L p complessi L p µ (X, C) , p ∈ [1, +∞] . Tutti i risultati precedenti (e successivi) rimangono validi per gli spazi L p µ (X, C), L p µ (X, E).
Approssimazione in L p .
Diamo ora alcuni utili risultati di approssimazione.
Proposizione 5.6. Dati p ∈ [1, ∞] ed f ∈ L p µ (X), per ogni ε > 0 esistono una funzione semplice ψ ε e g ε ∈ L ∞ µ (X), entrambe in L p µ (X) e tali che f − ψ ε p < ε , f − g ε p < ε . Dimostrazione. Effettuando la solita decomposizione f = f + − f − possiamo ridurci al caso f ≥ 0 . Applicando Prop.4.12 otteniamo una successione crescente di funzioni semplici e non negative {ψ n } convergente puntualmente ad f . Per p < ∞ abbiamo (f − ψ n ) p → 0 ed (f − ψ n ) p ≤ f p , con f p ∈ L 1 µ (X). Dunque applicando il teorema di convergenza dominata concludiamo che f − ψ n p p = (f − ψ n ) p → 0 . Nel caso p = ∞ osserviamo che per ipotesi l'insieme E := {x : |f (x)| > f ∞ } ha misura nulla.
Per costruzione (vedi ancora Prop.4.12) la successione {ψ n } converge uniformemente in X − E , e ció conclude la dimostrazione per quanto riguarda le funzioni semplici. Riguardo l'analoga affermazione per funzioni L ∞ consideriamo la succcessione g n (x) := inf{f (x), n} , ∀x ∈ X , cosicché 0 ≤ g n ≤ f , g n → f puntualmente e g n ∞ ≤ n, g n p ≤ f p . Possiamo quindi applicare il precedente argomento di convergenza dominata e concludere g n − f p → 0 .
Nella proposizione seguente ci specializziamo al caso in cui X sia uno spazio topologico normale (il che include gli spazi metrici).
Proposizione 5.7. Sia X uno spazio normale e µ : M → R + una misura di Borel e regolare esterna su X . Presi p ∈ [1, ∞) ed f ∈ L p µ (X), per ogni ε > 0 esiste una funzione ϕ ε ∈ C(X) ∩ L ∞ µ (X) con µ{supp(ϕ ε )} < ∞, tale che f − ϕ ε p < ε .
Dimostrazione. Al solito assumiamo f ≥ 0 . Applicando Prop.4.29 ad f p concludiamo che supp(f ) e σ -finito ed fè limite puntuale di una successione monotóna di funzioni semplici, non negative e con supporto di misura finita. Usando l'argomento della proposizione precedente troviamo che tale successione approssima f in norma · p , per cui possiamo assumere che f sia una funzione semplice, non negativa e con supporto di misura finita. Ma ancora, per linearitá ci possiamo ridurre al caso in cui f sia una funzione caratteristica χ E , dove E ⊆ X ha misura finita. Ora, per regolaritá esterna esiste un aperto U ⊂ X con misura finita tale che
E ⊂ U , µ(U − E) < ε (riguardo l'aver considerato la chiusura E vedi Oss.4.2). Applicando il Lemma di Uryshon (Teorema 2.4) troviamo ϕ ε ∈ C(X) tale che ϕ ε ∞ = 1 , ϕ ε | E = 1 e ϕ ε | X−U = 0 ; cosicché ϕ ε ha supporto con misura finita e χ E − ϕ ε p p = E |1 − 1| p + U−E |χ E − ϕ ε | p = U−E |ϕ ε | p ≤ µ(U − E) < ε .
Come applicazione del risultato precedente consideriamo lo spazio euclideo R d , d ∈ N, equipaggiato con la misura prodotto di Lebesgue (vedi §6.6), e diamo un risultato di approssimazione con le funzioni continue a supporto compatto.
Corollario 5.8. Per ogni p ∈ [1, ∞), lo spazio C c (R d )è denso in L p (R d ) nella norma · p . Dimostrazione. Grazie alla proposizione precedenteè sufficiente dimostrare che, presa f ∈ C(R d ) ∩ L ∞ (R d ) avente supporto con misura finita, e scelto ε > 0 , esiste f ε ∈ C c (R d ) tale che f −f ε p < ε . A tale scopo consideriamo la successione di funzioni ϕ n (x) := 1 , |x| < n n + 1 − |x| , |x| ∈ [n, n + 1) 0 , |x| ∈ [n + 1, ∞) , x ∈ R d . Ovviamente troviamo ϕ n , f ϕ n ∈ C c (R d ) per ogni n ∈ N, nonché |f (x) − f (x)ϕ n (x)| p → 0 ∀x ∈ R d , |f − f ϕ n | p ≤ |f | p . Essendo f ∈ L p (R d ) concludiamo, per convergenza dominata, che f ϕ n − f p → 0 .
Osserviamo che usando il teorema di Stone-Weierstrass possiamo approssimare in norma · ∞ ogni funzione in C c (R d ) con una funzione C ∞ c (R d ), e da ció segue che C ∞ c (R d )è denso in L p (R d ). D'altra parte, lo stesso risultato si puó dimostrare usando i mollificatori (vedi Prop.6.26).
Il teorema di Riesz-Fréchet-Kolmogorov.
Diamo ora una versione del Teorema di Ascoli-Arzelá per spazi L p nel caso in cui il soggiacente spazio misurabile sia R d , d ∈ N, equipaggiato con la misura (prodotto) di Lebesgue; tale risultato, il teorema di Riesz-Fréchet-Kolmogorov, fornisce un criterio di compattezza per famiglie di funzioni rispetto alla topologia della norma · p . Per esporne l'enunciato richiamiamo la nozione di funzione traslata
f h (x) := f (x + h) , f : R d → R , x, h ∈ R d . Inoltre, per ogni famiglia F di funzioni da R d in R ed Ω ⊆ R d , definiamo F Ω := {f | Ω : f ∈ F } .
Per la dimostrazione del teorema seguente occorrono le nozioni di convoluzione ( §6.7) e mollificatore (Def.6.25).
Teorema 5.9 (Riesz-Fréchet-Kolmogorov). Sia p ∈ [1, +∞) ed F ⊂ L p (R d ) limitato tale che lim h→0 f h − f p = 0 uniformemente in f ∈ F . Allora F Ωè precompatto in L p (Ω) per ogni Ω ⊂ R n con misura finita. Dimostrazione. L'ideaè quella di approssimare funzioni in F Ω con funzioni continue usando una successione di mollificatori {ρ n } ⊆ C ∞ c (R d ) e quindi usare il Teorema di Ascoli-Arzelá. Passo 1. Mostriamo che scelto ε > 0 esiste un δ > 0 tale che f − ρ n * f p < ε , ∀f ∈ F , n > δ −1 . (5.8)
Usando il fatto che ρ n (y)dyè una misura di probabilitá (infatti ρ n = 1 per ogni n ∈ N) troviamo
|ρ n * f (x) − f (x)| ≤ |f (x − y) − f (x)|ρ n (y) dy Holder ≤ |f (x − y) − f (x)| p ρ n (y) dy 1/p , per cui ρ n * f − f p p ≤ |f (x − y) − f (x)| p ρ n (y) dy dx F ubini = ρ n (y) f −y − f p p dy . Riguardo l'ultimo integrale osserviamo che preso il δ dell'uniforme continuitá di F abbiamo che il supporto di ρ n sará contenuto in ∆(0, δ) per n > δ −1 ; per cui, avendosi y ∈ ∆(0, δ) troviamo f −y − f p p < ε p e quindi ρ n * f − f p p ≤ ρ n (y)ε p dy ≤ ε p .
Notare che, grazie a Prop.6.24, abbiamo ρ n * f ∈ C ∞ (R d ) per ogni n ∈ N. Passo 2. Mostriamo che posto c n := ρ n q risulta
ρ n * f ∞ ≤ c n f p , ∀f ∈ L p (R d ) . (5.9)
Infatti si trova, posto q := p,
f (x − y)ρ n (y) dx ≤ |f (x − y)ρ n (y)| dx Holder ≤ f −y p ρ n q = f p ρ n q . (5.10) Passo 3. Esiste c ′ n > 0 tale che |{ρ n * f }(x 1 ) − {ρ n * f }(x 2 )| ≤ c ′ n f p |x 1 − x 2 | , ∀f ∈ F , x 1 , x 2 ∈ R d . (5.11) Per dimostrare questa stima osserviamo che, essendo ρ n ∈ C ∞ c (R d ), abbiamo ∇ρ n ∈ L q (R d , R d ), ∀q ∈ [1, ∞], e quindi, applicando Prop.6.24, ∇(ρ n * f ) = (∇ρ n ) * f ∈ L p (R d , R d ) , ∀f ∈ F . (5.12)
Ora, per il teorema di Lagrange abbiamo che per ogni
x 1 , x 2 , y ∈ R d esiste ξ ∈ R d tale che {ρ n * f }(x 1 − y) − {ρ n * f }(x 2 − y) = {∇(ρ n * f )}(ξ) · (x 1 − x 2 ) (qui con · intendiamo il prodotto scalare in R d ). Per cui, |{ρ n * f }(x 1 ) − {ρ n * f }(x 2 )| * ≤ |{∇(ρ n * f )}(ξ)| |x 1 − x 2 | (5.12) = |{(∇ρ n ) * f )}(ξ)| |x 1 − x 2 | * * ≤ ∇ρ n q f p |x 1 − x 2 | ,
avendo usato le diseguaglianze di Cauchy-Schwarz per (*) e di Holder per (**), quest'ultima usata in modo analogo a (5.10).
Passo 4. Preso ε > 0 esiste Ω ε ⊆ Ω limitato e misurabile tale che f | Ω−Ωε p < ε , ∀f ∈ F . (5.13)
Per questo basta osservare che
f | Ω−Ωε p ≤ f − ρ n * f p + (ρ n * f )| Ω−Ωε p , cosicché per n > δ −1 da (5.8) deduciamo f | Ω−Ωε p ≤ ε + (ρ n * f )| Ω−Ωε p ≤ ε + ρ n * f ∞ vol(Ω − Ω ε ) 1/p ,
e grazie a (5.9), ed alla limitatezza di F , basta scegliere Ω ε in maniera tale che vol(Ω − Ω ε ) sia sufficientemente piccolo. Passo 5. Fissato Ω ε ed n > δ −1 come nei passi precedenti consideriamo la famiglia
F n,ε := {(ρ n * f )| Ωε : f ∈ F } ⊂ C ∞ (Ω ε ) ;
grazie a (5.9) e (5.11) concludiamo che F n,εè limitata ed equicontinua in C(Ω ε ). Per Ascoli-Arzelá F n,εè precompatto nella topologia della norma dell'estremo superiore, e quindi loè anche in
L p (Ω ε ) (infatti la convergenza in · ∞ in C(Ω ε ) implica quella in · p ). Conclusione. Sia {f i } ⊆ F . Fissati ε, n come sopra, grazie al passo precedente esiste una sottosuccessione {(ρ n * f i k )| Ωε } di Cauchy in L p (Ω ε ) (per brevitá scriviamo f k ≡ f i k ). Usando (5.8), (5.9), (5.13), abbiamo le stime (f h − f k )| Ω p ≤ (f h − ρ n * f h )| Ω p + (ρ n * f h − ρ n * f k )| Ω p + (ρ n * f k − f k )| Ω p ≤ 2ε + (ρ n * f h − ρ n * f k )| Ω p ≤ 2ε + (ρ n * f h − ρ n * f k )| Ω−Ωε p + (ρ n * f h − ρ n * f k )| Ωε p ≤ 3ε + (ρ n * f h − ρ n * f k )| Ω−Ωε p ≤ 3ε + c n ( f h | Ω−Ωε p + f k | Ω−Ωε p ) ≤ 3ε + c n · 2ε .
Concludiamo quindi che {f k | Ω }è di Cauchy in L p (Ω), e ció mostra il teorema.
Gli spazi
L p loc . Sia (X, M, µ) uno spazio misurabile boreliano e p ∈ [1, +∞]. Una funzione misurabile f : X → R si dice localmente L p se per ogni x ∈ X esiste un intorno misurabile U ∋ x tale che f | U ∈ L p µU (U ) ⇔ f χ U ∈ L p µ (X) (vedi Oss.4.1). Denotiamo con L p µ,loc (X) l'insieme delle funzioni localmente L p (nel caso della misura di Lebesgue seguiremo le analoghe convenzioni degli spazi L p ).
Poiché per ipotesi ogni apertoè misurabile, nella definizione precedente nonè restrittivo considerare intorni aperti (se l'intorno U nonè aperto consideriamo un aperto U ′ ⊂ U ). E' ovvio che se f ∈ L p µ (X) allora fè localmente L p . Il viceversaè falso, e da ció sorge l'interesse nelle funzioni localmente L p .
Esempio 5.1. Sia f (x) := x −1 , x ∈ (0, 1). Alloraè chiaro che f / ∈ L p (0, 1) per ogni p ∈ [1, +∞]. Invece troviamo f ∈ L p loc (0, 1), visto che, preso x ∈ (0, 1) e 0 < a < x < b ≤ 1 , +∞ > b a |f (x)| p dx = 1 p−1 b 1−p − a 1−p , p > 1 log b − log a , p = 1 . Proposizione 5.10. Sia (X, M, µ) uno spazio di misura di Radon, p ∈ [1, +∞] ed f ∈ L p µ,loc (X). Allora per ogni compatto K ⊆ X risulta che f χ K ∈ L p µ (X). Dimostrazione. Per ogni x ∈ K esiste un intorno aperto U ∋ x con f χ U ∈ L p µ (X), per cui la famiglia {U } costituisce un ricoprimento aperto di K . Consideriamo un sottoricoprimento finito {U k } n k=1 ed osserviamo che X |f χ K | p = X |f | p χ K ≤ n k X |f | p χ U k < +∞ .
Corollario 5.11. Per ogni p ∈ [1, +∞], lo spazio L p µ,loc (X) ha una struttura di spazio localmente convesso 17 rispetto alle seminorme
η U (f ) := U |f | p 1/p , U ∈ τ X : U compatto . Dimostrazione. Se f, g ∈ L p µ,loc (X) allora per ogni x esistono intorni U, V di x con f χ U ∈ L p µ (X), gχ V ∈ L p µ (X). Preso un aperto W ⊆ U ∩ V con x ∈ W , troviamo (f + g)χ W ∈ L p µ (X), per cui concludiamo che L p µ,loc (X)è uno spazio vettoriale. Inoltre, dalla proposizione precedente segue che η U (f ) < +∞ per ogni intorno aperto U a chiusura compatta. Chiaramente ogni η Uè una seminorma, e se η U (f ) = 0 per ogni precompatto U allora f = 0 q.o..
Diamo ora un'applicazione del concetto di funzione localmente L p . Denotato con L p c (R d ) lo spazio delle funzioni in L p (R d ) a supporto compatto, osserviamo che ha senso definire la convoluzione (vedi §6.7)
f * g(x) := R f (x − y)g(y) dy , x ∈ R d , f ∈ L 1 loc (R d ) , g ∈ L p c (R d ) , (5.14) 17 Vedi §7.9. infatti f * g(x) = f | suppg * g(x) ed f | suppg ∈ L 1 (R d ); usando il Teorema 6.23, concludiamo che f * g ∈ L p (R d ).
La dualitá di Riesz.
In accordo con la notazione che verrá introdotta in §7, per ogni p ∈ [1, +∞] denotiamo con L p, * µ (X) lo spazio duale di L p µ (X). L'osservazione alla base dei risultati che esporremo in questa sezioneè la seguente: presi p ∈ [1, +∞], q := p e g ∈ L q µ (X) abbiamo l'applicazione lineare
F g : L p µ (X) → R , F g (f ) := F g , f := X f g , ∀f ∈ L p µ (X) ; (5.15)
con la precedente notazione la diseguaglianza di Holder implica, chiaramente, che
| F g , f | ≤ g q f p ,
per cui, nella terminologia di §7, abbiamo F g ∈ L p, * µ (X) con norma F g ≤ g q . In realtá si verifica che la norma di F gè proprio g q (vedi Esercizio 5.7), per cui abbiamo un'applicazione lineare isometrica
F : L q µ (X) → L p, * µ (X) , g → F g . (5.16)
Nel caso complesso, si puó analogamente definire
F : L q µ (X, C) → L p, * µ (X, C) , F g , f := X f g . Lemma 5.12. Sia (X, M, µ) uno spazio di misura finita e p ∈ [1, +∞). Se g ∈ L 1 µ (X) ed esiste M ∈ R tale che f g ≤ M f p , ∀f ∈ L ∞ µ (X) , (5.17) allora g ∈ L q µ (X) con q = p, e g q ≤ M .
Dimostrazione. Iniziamo considerando il caso p > 1 . Definiamo
g n (x) := g(x) , |g(x)| ≤ n 0 , |g(x)| > n ⇒ g n ∈ L ∞ µ (X) .
Osserviamo che gg n = g 2 n e sgn(g n ) = sgn(g) 18 . Definiamo quindi f n := |g n | q/p sgn(g n ), cosicché
f n p = g n q/p q = g n q−1 q . Dunque per costruzione g n ∈ L q µ (X) ∩ L ∞ µ (X) ed f n ∈ L p µ (X) ∩ L ∞ µ (X), e possiamo stimare g n q q = |g n | q/p+1 = |g n | q/p |g| = |f n ||g| = f n g ≤ M f n p = M g n q−1 q , per cui g n q ≤ M . Ora, {g n }è una successione monotóna crescente e convergente a g q.o. in x ∈ X (ed analogamente |g n | q ր |g| q q.o.); per cui, applicando il teorema di Beppo Levi concludiamo M q ≥ |g n | q n → |g| q ⇒ |g| q ≤ M q .
Discutiamo ora il caso p = 1 . Per ogni ε > 0 , poniamo
A := {x ∈ X : |g(x)| ≥ M + ε} , e, posto f := sgn(g)χ A , otteniamo f 1 = µA. Ora, f g ≤ M f 1 = M µA , f g ≥ A |g| ≥ (M + ε)µA .
Dalle due precedenti diseguaglianze, concludiamo che µA = 0 e quindi g ∞ ≤ M .
Teorema 5.13 (Dualitá di Riesz). Sia (X, M, µ) uno spazio di misura σ -finito. Per ogni p ∈ [1, +∞), l'applicazione (5.16) definisce un isomorfismo di spazi di Banach L q µ (X) → L p, * µ (X).
Dimostrazione. Sappiamo che (5.16)è isometrica (e quindi iniettiva), per cui dobbiamo verificarne soltanto la suriettivitá. Consideriamo allora F ∈ L p, * µ (X) e mostriamo che appartiene all'immagine di (5.16). Come primo passo, assumiamo che X abbia misura finita, e definiamo
νA := F, χ A , A ∈ M . Poiché µX < ∞ abbiamo χ A p = (µA) 1/p < ∞, per cui χ A ∈ L p
µ (X) e la definizione precedentè e ben posta. La strategiaè ora quella di mostrare che νè una misura con segno. A tale scopo,
osserviamo che ν∅ = F, 0 = 0 ; inoltre, se A∩B = ∅ allora χ A∪B = χ A +χ B e ν(A∪B) = νA+νB . Per mostrare l'additivitá numerabile, consideriamo una successione {A k } di insiemi misurabili mutualmente disgiunti, ed osserviamo che per il teorema di Lebesgue risulta 0 = lim n χ E − χ En p = lim n µ(E − E n ) 1/p , E := ∪ ∞ k A k , E n := ∪ n k A k .
Essendo F continuo, concludiamo che
0 = lim n F, χ E − χ En = νE − lim n νE n .
Dunque ν : M → Rè una misura con segno. Ora, se µA = 0 , A ∈ M , allora χ A p = 0 e νA = F, χ A = 0 . Dunque νè assolutamente continua rispetto a µ, ed il teorema di Radon-Nikodym implica che esiste g ∈ L 1 µ (X) tale che
F, χ A = A g dµ , A ∈ M . (5.18) Per cui, per ogni ϕ ∈ S(X) troviamo F, ϕ = ϕg dµ. Poiché S(X)è denso in L ∞ µ (X) in norma · ∞ , concludiamo che F, f = f g dµ , f ∈ L ∞ µ (X) ⇒ f g dµ ≤ F f p .
Applicando il Lemma precedente concludiamo che F = F g . Passiamo ora al caso in cui Xè σ -finito: consideriamo una successione {X i } di insiemi a misura finita e mutualmente disgiunti tali che X = ∪ i X i ; osserviamo quindi che -grazie al passo precedente -per ogni n ∈ N esistono g 1 , . . . , g n ∈ L q µ (X) (con supporti contenuti rispettivamente in X 1 , . . . , X n ) tali che
F, f χ An = f n i g i dµ , f ∈ L p µ (X) , A n := ∪ n i X i .
In altri termini, ogni F (n) := F | L p µ (An) , n ∈ N,è il funzionale associato a
n i g i ∈ L q µ (A n ) attraverso l'applicazione (5.16) 19 . Poiché chiaramente F (n) ≤ F per ogni n ∈ N, concludiamo che n i g i q = F (n) ≤ F , n ∈ N . Definiamo ora g(x) := ∞ i g i (x), x ∈ X . Usando il Lemma di Fatou, per ogni n ∈ N troviamo g q ≤ lim inf n n i g i q = lim inf n F (n) ≤ F . Dunque g ∈ L q µ (X). Ora, per il teorema di convergenza di Lebesgue troviamo f − f χ An p n → 0 , per cui, per continuitá di F abbiamo che F, f = lim n F, f χ An = lim n F (n) , f χ An = lim n f n i g i dµ .
D'altro canto, usando ancora Lebesgue troviamo g − n i g i q n → 0 ; per la diseguaglianza di Holder, concludiamo . Preso F ∈ L p, * µ (X), l'ideaè quella di costruire un insieme misurabile e σ -finito X 0 ⊆ X tale che
f g − n i g i dµ ≤ f p g − n i g i q n → 0 ⇒ f g dµ = lim n f n i g i = F, f .f | X0 = 0 ⇒ F, f = 0 , ∀f ∈ L p µ (X) ; osserviamo che in tal modo ogni f 0 ∈ L p µ (X 0 ) si scrive come f 0 = f | X0 , f ∈ L p µ (X), avendo posto f | X−X0 := 0 , f | X0 := f 0 , cosicchéè ben definito il funzionale F 0 ∈ L p, * µ (X 0 ) , F 0 , f 0 := F, f , ∀f 0 = f | X0 ∈ L p µ (X 0 ) . Scelta una successione {f n } ⊂ L p µ (X), f n p ≡ 1 , tale che F (1 − 1/n) ≤ F, f n , n ∈ N , il nostro candidatoè X 0 := n {x ∈ X : f n (x) = 0} , che sappiamo essere σ -finito grazie a Prop.4.29. Alché si applica il teorema del caso σ -finito, il che fornisce una funzione g ∈ L q µ (X 0 ) tale che F 0 = F g . Estendendo g ad X ponendo g ≡ 0 in X − X 0 otteniamo la funzione g ∈ L q µ (X) cercata. Alla regola L p, * µ (X) = L q µ (X), p ∈ [1, +∞), q = p, fa eccezione lo spazio L ∞ µ (X), il cui duale contiene strettamente L 1 µ (X) (vedi Esercizio 7.2).
In effetti, il duale L ∞, * µ (X) si puó caratterizzare in termini di misure di Radon (vedi §4.1); tale caratterizzazione richiede due risultati fondamentali: il teorema di Riesz-Markov (Esempio 7.3), ed il teorema di Gel'fand-Naimark (Teo.7.48):
Proposizione 5.14. Sia (X, M, µ) uno spazio misurabile. Allora esiste uno spazio compatto e di Hausdorff X µ con un isomorfismo L ∞, * µ (X) ≃ R(X µ ).
Dimostrazione. Immergiamo L ∞ µ (X) in L ∞ µ (X, C) ed osserviamo che quest'ultimaè una C*-algebra commutativa con identitá (Esempio 7.14). Denotato con
X µ lo spettro di L ∞ µ (X, C) nel senso di Def.7.45, abbiamo un isomorfismo di spazi di Banach (reali) L ∞ µ (X) ≃ C(X µ ). La tesi segue dunque dal teorema di Riesz-Markov. Sull'argomento precedente, si veda anche [5, §4.3.C] (e referenze).
La dualitá di Riesz su [0, 1]. E' possibile dimostrare la dualitá di Riesz sull'intervallo [0, 1] usando delle tecniche diverse rispetto alla sezione precedente, che fanno ricorso alle funzioni assolutamente continue piuttosto che al teorema di Radon-Nicodym:
Teorema 5.15 (Teorema di rappresentazione di Riesz sugli intervalli). Per ogni p ∈ [1, +∞) e q := p si ha l'isomorfismo di spazi di Banach L q → L p, * , g → F g . (5.19) Dimostrazione. Preso F ∈ L p, * dimostriamo che F = F g per qualche g ∈ L q . A tale scopo consideriamo le funzioni caratteristiche χ s := χ [0,s] e definiamo G(s) := F, χ s , s ∈ [0, 1] . Volendo mostrare che G ∈ AC([0, 1]), consideriamo una collezione C := {(x i , x ′ i )} di intervalli disgiunti con lunghezza totale l(C) minore di δ > 0 ed osserviamo che i |G(x i ) − G(x ′ i )| = i | F, χ (xi,x ′ i ) | = F, f , dove f := i χ (xi,x ′ i ) sgn F, χ (xi,x ′ i ) ⇒ f p = δ 1/p . Per cui, i |G(x i ) − G(x ′ i )| ≤ F f p ≤ F δ 1/p . Dunque Gè AC e quindi G(x) = x 0 g per qualche g ∈ L 1 (Teo.4.47), il che fa di g il nostro candidato per avere F = F g . Per definizione troviamo G(s) = F, χ s = 1 0 gχ s , il che mostra che F | G = F g , dove Gè lo spazio delle funzioni a gradini. Ora, ogni funzione f ∈ L ∞ e limite q.o. di una successione {ψ n } di funzioni a gradini (Prop.4.25), equilimitata da f ∞ . Per cui, il teorema di convergenza limitata (Prop.4.30) implica lim n f − ψ n p = 0 . Poiché Fè limitato, troviamo F, f − ψ n p ≤ F f − ψ n p n → 0 ⇒ F, f = lim n 1 0 gψ n ( * ) = f g dove in (*) siè usato il teorema di convergenza di Lebesgue per la successione |gψ n | ≤ |gf | ∈ L 1 . Dunque F | L ∞ = F g . Possiamo applicare ora Lemma 5.12, dal quale concludiamo g ∈ L q . I risultati di approssimazione in L p ( §5.2) permettono infine di dimostrare F = F g su tutto L p .
5.6 Esercizi.
Esercizio 5.1. Sia f ∈ L ∞ . Si mostri che lim p→∞ f p = f ∞ . Esercizio 5.2 (Continuitá delle convoluzioni). Sia f ∈ L p (R), p ∈ [1, +∞), ed f h (x) := f (x + h), x, h ∈ R. Si provi che la funzione n f (h) := |f (x + h) − f (x)| p dx 1/p ≡ f h − f p , ∀h ∈ R ,
e continua. Inoltre, presi q := p e g ∈ L q (R), si provi cheè ben definita, continua e limitata la funzione (detta convoluzione, si veda §6.7)
f * g(x) := f (x − y)g(y) dy , x ∈ R .
Soluzione. n fè ben definita in quanto la funzione integranda appartiene ad L p (R) per ogni h ∈ R; inoltre, osserviamo che
|n f (h) − n f (k)| ≤ f h − f k p = f h−k − f p = n f (h − k) , per cuiè sufficiente verificare la continuitá soltanto in h = 0 . A tale scopo, assumiamo inizialmente che f ∈ C c (R), cosicché fè uniformemente continua; scelto ε > 0 esiste δ > 0 tale che |f h (x) − f (x)| < ε per |h| < δ , e si trova n f (h) < εµ(supp(f )) 1/p , |h| < δ .
Per cui, n fè continua. Per una generica f ∈ L p (R), osserviamo che per ogni ε > 0 esiste
g ε ∈ C c (R) con f − g ε p < ε . Per cui, per h abbastanza piccolo troviamo n f (h) ≤ f h − g ε,h p + g ε,h − g ε p + g ε − f p ≤ 3ε . Infine, valutiamo |f * g(x)| ≤ |f (x − y)||g(y)| dy Holder ≤ f x p g q = f p g q . (5.20)
Per cui f * g(x)è ben definita e limitata per ogni x ∈ R. Per quanto riguarda la continuitá, abbiamo
|f * g(x)− f * g(x 0 )| ≤ |f (x− y)− f (x 0 − y)||g(y)| dy Holder ≤ g q f x − f x0 p = g q n f (x− x 0 ) ,
per cui, grazie alla continuitá di n f , concludiamo che f * gè continua.
Esercizio 5.3. Sia f ∈ L p (R), p ∈ [1, +∞). Allora, f n := f χ [−n,n] , n ∈ N,è una funzione in L 1 (R) ∩ L p (R).
Soluzione. E' ovvio che f n ∈ L p (R). Poi, si osservi che (1)) .
|f n | = n −n |f | Holder ≤ n −n |f (x)| p 1/p · χ [−n,n] q ≤ f p (2n) 1/q . Esercizio 5.4. Sia X := {u ∈ C 1 ([0, 1]) : u(0) = 0} . Per ogni λ ≥ 0 si consideri funzionale F λ : X → R , F λ (u) = arctan 1 0 [u ′ (t)] 2 dt − λ arctan(u
(1) Usando la diseguaglianza di Holder si mostri che
F λ (u) ≥ arctan(u(1) 2 ) − λ arctan(u(1)) , ∀u ∈ X ;
(2) Si trovi una famiglia M di funzioni in X tali che la diseguaglianza precedente si riduce ad un'eguaglianza;
(3) Si calcoli min u∈M F λ (u) al variare di λ. (Suggerimento: la diseguaglianza di Holder implica che (u ′ ) 2 ≥ ( u ′ ) 2 = u(1) 2 .
Una classe di funzioni tali che la diseguaglianza precedente si riduce ad un'eguaglianzaè data da quelle del tipo u α (t) := αt, t ∈ [0, 1], dove α ∈ R, e ció consente di ridurre il nostro problema allo studio della funzione g(λ, α) := F λ (u α )).
Esercizio 5.5. Dimostrare che la naturale inclusione (con
L p := L p ([0, 1]), p ∈ [1, ∞]) L ∞ ⊂ p∈[1,∞)
L ṕ e stretta, ovvero che esistono funzioni non limitate ma in L p per ogni p ∈ [1, ∞).
Esercizio 5.6 ([7, Ex.3.12]). Si consideri X := [0, 1) e la successione {f in } i,n∈N ⊂ L p (X), p ∈ [1, +∞], definita da f in (x) := 1 , x ∈ [ (i − 1)n −1 , in −1 ) 0 , altrimenti . Si verifichi che {. . . , f n1 , f n2 , . . . , f nn , f n+1,1 . . .} converge a 0 in L p (X), ma non q.o.. Si verifichi invece che f 1n n → 0 q.o.. Esercizio 5.7. Sia (X, M, µ) uno spazio di misura e g ∈ M (X).
(1) Si consideri la funzione segno
{sgn(g)}(x) := 1 , g(x) ≥ 0 , −1 , g(x) < 0 , e si mostri che sgn(g) ∈ M (X). (2) Si verifichi che se g ∈ L q µ (X) allora f := |g| q/p sgn(g) ∈ L p µ (X), dove p := q . (3) Si mostri che f p = g q/p q . (4) Si mostri che F g , f = g q f p . (Suggerimento: si osservi che |f | p = |g| q , q = 1 + q/p, e F g , f = f g = |g| q = g q q .) Esercizio 5.8 (Delta-approssimanti). Sia x := {x k } ⊂ [0, 1] una successione monotóna cres- cente tale che x 1 = 0 e lim k x k = 1 . (1) Presa f ∈ L p , si mostri che la funzione a gradini {S x f }(t) := 1 x k+1 − x k x k+1 x k f (s) ds , ∀k ∈ N , t ∈ [x k , x k+1 ) , appartiene ad L p e si concluda che S x : L p → L p , f → S x f , e un'applicazione lineare tale che S x f p ≤ f p ; (2) Diciamo che x ha passo δ se x k+1 −x k < δ per ogni k ∈ N. Si mostri che per ogni f ∈ C([0, 1]) ed ε > 0 esiste δ > 0 tale che, per ogni successione x con passo δ , risulta sup t∈[0,1]−x |f (t) − {S x f }(t)| < ε ;
(3) Si concluda che lo spazio vettoriale delle funzioni a gradiniè denso in L p .
(Suggerimenti: per il punto (1) si osservi che per t ∈ [x k , x k+1 ) |{S x f }(t)| p = 1 (x k+1 − x k ) p x k+1 x k f p = x k+1 x k f dµ p Jensen ≤ 1 (x k+1 − x k ) x k+1 x k |f | p , avendo definito la misura di probabilitá dµ(s) = ds/(x k+1 − x k ), per cui S x f p p = k (x k+1 − x k ) · |S x f | p | [x k ,x k+1 ) ≤ k x k+1 x k |f | p = f p p .
Per (2) si usi il teorema fondamentale del calcolo ed il teorema di Lagrange, che implicano
{S x f }(t) = F (x k+1 ) − F (x k ) x k+1 − x k = f (ξ k ) , dove ξ k ∈ [x k , x k+1 ) ed Fè una primitiva di f .).
Esercizio 5.9 (Basi di Haar-Schauder per L p ). Si consideri la successione di partizioni
x n := {x n,1 := 0 < . . . < x n,k < . . . < x n,i(n) := 1} , n ∈ N , e si assuma che lim n δ n := sup k (x n,k+1 − x n,k ) = 0 . (1) Presa una successione {τ h } ⊂ L ∞ tale che κ n (t, s) := n h τ h (t)τ h (s) = (x n,k+1 − x n,k ) −1 , ∀s ∈ (x n,k , x n,k+1 ) , 0 , ∀s ∈ [0, x n,k ) ∪ (x n,k+1 , 1] , si mostri che per ogni p ∈ [1, ∞) ed f ∈ L p risulta 1 0 κ n (s, t)f (s) ds = 1 x n,k+1 − x n,k x n,k+1
x n,k f , ∀t ∈ (x n,k , x n,k+1 ) .
(2) Si mostri che le applicazioni lineari
K n : L p → L p , K n f (t) := 1 0 κ n (s, t)f (s) ds , t ∈ [0, 1] , sono tali che K n f p ≤ f p , ∀f ∈ L p . (3) Si mostri che per ogni f ∈ C([0, 1]) risulta lim n sup t∈[0,1]−x |K n f (t) − f (t)| n → 0 , x := ∪ n x n . (4) Si mostri che per ogni f ∈ L p esiste una successione {a h } ⊂ R tale che f = lim n n h a h τ h in norma · p , e che {a h }è unica 20 se valgono le relazioni di ortogonalitá τ h τ m = 0 , ∀h = m.
Funzioni di piú variabili.
Da un punto di vista astratto possiamo pensare alle funzioni di piú variabili come quelle definite su prodotti cartesiani del tipo X × Y . Nelle sezioni seguenti approcceremo questioni relative alle ulteriori strutture con le quali possiamo arricchire il nostro insieme X × Y , quali quella topologica ( §6.1), quella differenziale con le sue applicazioni all'esistenza di funzioni implicite ed al calcolo variazionale ( §6.2, §6.3, §6.4, §6.5), e poi quella di spazio misurabile con l'applicazione dei prodotti di convoluzione ( §6.6, §6.7).
Topologie prodotto e prodotti tensoriali.
Le proprietá di base delle topologie prodotto sono ben note ed in questa sede ci limitamo a segnalare alcune referenze ( [27,6]). Passando ad un punto di vista piú analitico vogliamo invece dimostrare un importante risultato di approssimazione.
Siano X, Y spazi localmente compatti e di Hausdorff; allora X ×Y , equipaggiato della topologia prodotto,è uno spazio localmente compatto e di Hausdorff.
Date f ∈ C 0 (X), g ∈ C 0 (Y ), definiamo la funzione prodotto f ⊗ g : X × Y → R , f ⊗ g(x, y) := f (x)g(y)
.
Una verifica immediata mostra che f ⊗g ∈ C 0 (X×Y ). Denotiamo con C 0 (X)⊗C 0 (Y ) la sottalgebra di C 0 (X × Y ) generata dalle funzioni del tipo f ⊗ g . Proposizione 6.1. Siano X, Y spazi localmente compatti di Hausdorff. Allora C 0 (X) ⊗ C 0 (Y )è densa in C 0 (X × Y ) nella topologia della convergenza uniforme.
Dimostrazione. L'ideaè quella di applicare il teorema di Stone-Weierstrass per gli spazi localmente compatti (Cor.2.8). Per cui, posto A := C 0 (X)⊗C 0 (Y ), per dimostrare la proposizioneè sufficiente verificare che:
(1) A separa i punti di X × Y ; (2) Per ogni (x, y) ∈ X × Y esiste f ⊗ g ∈ A tale che f ⊗ g(x, y) = 0 . Ora grazie a Lemma 2.9 abbiamo che, dati x ∈ X ed y ∈ Y , esistono f ∈ C 0 (X) e g ∈ C 0 (Y ) tali che f (x) = 0 , g(y) = 0 . Dunque f ⊗ g(x, y) = 0 , e ció dimostra il punto (2). Riguardo il punto (1), se (x, y) = (x ′ , y ′ ) sono elementi distinti di X × Y , allora almeno una delle due coordinate di questi deve essere distinta, diciamo x = x ′ . Applicando ancora Lemma 2.9 troviamo f ∈ C 0 (X) tale che f (x) = f (x ′ ); aggiungendo eventualmente una costante possiamo assumere che f (x ′ ) = 0 . Presa infine g ∈ C 0 (Y ) tale che g(y) = 0 concludiamo che f ⊗ g(x, y) = 0 , mentre f ⊗ g(x ′ , y ′ ) = 0 .
Derivabilitá e differenziabilitá.
Come vedremo nelle righe che seguono la nozione di derivabilitá per funzioni di piú variabili reali, ed il suo rapporto con la continuitá,è una questione piú delicata rispetto al caso ad una variabile. Iniziamo dando la seguente terminologia: dato n ∈ N, una direzioneè un vettore v ∈ R n con norma 1 .
Definizione 6.2. Sia A ⊆ R n aperto. Una funzione f : A → R si dice derivabile in a ∈ A lungo la direzione v se esiste finito il limite ∂f ∂v (a) := lim t→0 f (a + tv) − f (a) t . (6.1)
Nel caso in cui v sia uno degli elementi e i , i = 1, . . . , n, della base canonica di R n , adottiamo la classica notazione
∂f ∂x i (a) := ∂f ∂e i (a) . (6.2)
Ora, sappiamo bene che se una funzione di una variabileè derivabile in a ∈ R, alloraè anche continua. Il seguente esempio mostra invece come una funzione di piú variabili f : A → R possa essere derivabile in ogni direzione in a ∈ A, e tuttavia essere discontinua in A.
Esempio 6.1. Si consideri f (x, y) := x 2 y x 4 +y 2 2 , (x, y) = 0 0 , (x, y) = 0 Si trova lim n→∞ f 1 n , 1 n 2 = 1 4 , il che implica che fè discontinua in 0 . D'altro canto, per ogni direzione v := (v 1 , v 2 ) si trova ∂f ∂v (0) = lim t→0 t −1 f (tv 1 , tv 2 ) = 0 .
Una nozione piú naturale per le proprietá inerenti la differenziazione di fè invece la seguente:
Definizione 6.3. Una funzione f : A → R si dice differenziabile in a ∈ A se esiste un'applicazione lineare df a ∈ R n, * , df a : R n → R , chiamato il differenziale di f in a, tale che risulti lim x→a f (x) − f (a) − df a (x − a) |x − a| = 0 .
Se fè differenziabile per ogni a ∈ A, allora si dice differenziabile in A.
In termini piú geometrici, potremmo riguardare lo spazio vettoriale generato dalle direzioni (isomorfo ad R n ) come il tangente ad A nel punto a. Da (6.1), segue immediatamente che fè differenziabile in a se e solo se
lim t→0 f (a + tv) − f (a) − df a (tv) t = 0 , ∀v ⇒ ∂f ∂v (a) = df a (v) .
Dunque possiamo riguardare df a come un elemento dello spazio dello spazio duale R n, * (in termini geometrici, lo spazio cotangente), il quale, valutato su v :
= i v i e i ∈ R n , fornisce il valore della derivata parziale di f lungo v . In particolare, per linearitá di df a troviamo, ricordando (6.2), df a (v) = ∂f ∂v (a) = i v i df a (e i ) = i v i ∂f ∂x i (a) . (6.3)
D'altro canto, denotando con dx i ∈ R n, * gli elementi della base canonica dello spazio cotangente, abbiamo per definizione
df a (v) = i (df a ) i dx i (v) = i (df a ) i v i ,
per cui, confrontando con (6.3) troviamo (df a ) i = ∂f ∂xi (a), e quindi la familiare espressione
df a = i ∂f ∂x i (a) dx i , a ∈ A . (6.4) Introducendo il gradiente di f in a ∇f (a) := ∂f ∂x i (a) i ∈ R n possiamo esprimere il differenziale in termini del prodotto scalare df a (v) = (∇f (a), v) , v ∈ R n .
I due seguenti teoremi sono ben noti e reperibili su ogni testo di Analisi II, per cui ne omettiamo la dimostrazione.
Teorema 6.4. Ogni funzione differenziabileè continua. D'altro canto, se una funzione continua ha derivate parziali continue in un intorno di a ∈ A, alloraè differenziabile in A.
Teorema 6.5 (Schwartz). Se esistono e sono continue le derivate parziali miste di f in un intorno di a ∈ A, allora ∂ 2 f ∂x i ∂x k (a) = ∂ 2 f ∂x k ∂x i (a) , i, k = 1, . . . , n .
Massimi e minimi. Lo studio dei punti di massimo e minimo relativi di una funzione f ∈ C 1 (A) si effettua in primo luogo analizzando i punti stazionari di f , ovvero cercando le soluzioni a ∈ A dell'equazione df a = 0 ⇔ ∇f (a) = 0 .
Se f ∈ C 2 (A), allora la ricerca di massimi e minimi relativi si effettua considerando la matrice Hessiana
H(a) := ∂ 2 f ∂x i ∂x j (a) ij ,
la qualeè autoaggiunta in conseguenza del teorema di Schwartz. Se aè di minimo relativo, allora H(a) ≥ 0 ; viceversa, se H(a) > 0 , allora aè di minimo relativo. In modo analogo si studia il comportamento dei punti di massimo.
La matrice jabobiana. Sia U ⊂ R m ed F : U → R n un'applicazione. Chiaramente possia- mo descrivere F in termini delle funzioni coordinate F 1 , . . . , F n : U → R definite da F (u) = (F 1 (u), . . . , F n (u)), cosicché diciamo che Fè differenziabile in U se lo sono le sue componenti F i , ∀i = 1, . . . , n. Denotiamo con C 1 (U, R n ) l'insieme delle applicazioni differenziabili da U ⊆ R m in R n ; ad uso futuro, introduciamo la matrice jacobiana di F , ∂F ∂w (u) := ∂F i ∂w j (u) ij ∈ M n,m (R) , u ∈ U .
Il teorema delle funzioni implicite.
In geometriaè usuale presentare un luogo, sia esso una curva o una superficie, in termini dell'annullarsi di un'espressione del tipo
F (x, y) = 0 , (x, y) ∈ U ⊆ R m+n ,(6.5)
dove F : U → R né un'applicazione che svolge il ruolo di una relazione tra le variabili x ed y . Il teorema delle funzioni implicite permette di esprimere curve e superfici in termini di grafici di funzioni, almeno a livello locale. Ció vuol dire che a partire da (6.5) saremo in grado di esibire,
in un intorno opportuno A ⊆ R m , una funzione f : A → R n tale che F (x, f (x)) = 0 , ∀x ∈ A .
Nel seguito denoteremo con x, a ∈ R m i vettori delle prime m variabili di F , e con y, b ∈ R n i vettori delle rimanenti n variabili. Se F ∈ C 1 (U ), allora spezziamo la jacobiana di F come segue:
∂F ∂x : U → M n,m (R) , ∂F ∂x (a, b) := ∂Fi ∂xj (a, b) ij ∂F ∂y : U → M n,n (R) , ∂F ∂y (a, b) := ∂Fi ∂y h (a, b) ih (6.6) Teorema 6.6 (Teorema delle funzioni implicite, o del Dini). Sia U ⊆ R m+n aperto ed F : U → R n un'applicazione di classe C 1 . Se (a, b) ∈ Uè tale che F (a, b) = 0 , det ∂F ∂y (a, b) = 0 , (6.7) allora esistono intorni A ∋ a, B ∋ b , con A ⊆ R m , B ⊆ R n , A × B ⊆ U , ed un'applicazione f : A → B , f ∈ C 1 (A) , tale che F (x, f (x)) = 0 , x ∈ A .
Inoltre, la matrice Jacobiana di fè data da
∂f ∂x (ξ) = − ∂F ∂y (ξ, f (ξ)) −1 ∂F ∂x (ξ, f (ξ)) , ξ ∈ A . (6.8)
Dimostrazione. Passo 1. Per semplicitá di notazione assumiamo (a, b) = 0 ∈ R m+n , come del restoè lecito fare applicando una traslazione all'aperto U . Inoltre, osserviamo che grazie a Teo.6.4 Fè differenziabile in U . Per economia di notazione, scriviamo Ció implica che esistono dischi chiusi ∆ m ⊂ R m , ∆ n ⊂ R m di centro l'origine e raggio r > 0 , tali che per a ∈ ∆ m , b ∈ ∆ n ,
T := ∂F ∂y (0) ∈ GL(n, R) (l'invertibilitá di Tè assicurata da (6.7)). Per differenziabilitá di F possiamo scrivere F (x, y) = T y + ∂F ∂x (0)x + R(x, y) (6.9) dove il resto R : U → R nè di classe C 1 (U ) e tale che lim (x,y)→0 R(x, y) |x| + |y| = 0 . (6.10) Da (6.9) otteniamo y = T −1 F (x, y) + Lx +R(x, y) (6.11) doveR : U → R n ,R(x, y) := −T −1 R(x, y) , ed L : R m → R nè l'applicazione lineare Lx := −T −1 ∂F ∂x (0)x , x ∈ R m . Per cui, concludiamo che F (x, y) = 0 ⇔ y = Lx +R(x, y) , (6.12) cosicché dimostrare l'esistenza della funzione implicita f equivale a mostrare che esistono intorni A ∋ 0 ∈ R m , B ∋ 0 ∈ R n tali che per ogni x ∈ A esista un solo y ∈ B che verifichi (6.12). Passo 2. Dimostriamo che esistono A ∋ 0 ∈ R m , B ∋ 0 ∈ R n tali che, per ogni x ∈ A, l'applicazione φ x : B → R n , φ x (y) := Lx +R(x, y) ,∂R i ∂x (a, b) ≤ 1 2 √ m , ∂R i ∂y (a, b) ≤ 1 2 √ n .
Siano ora y 1 , y 2 ∈ ∆ n ; applicando il teorema di Lagrange troviamo che esistono ξ 1 , . . . , ξ n ∈ ∆ n appartenenti al segmento che congiunge y 1 con y 2 , tali che 21
R i (x, y 1 ) −R i (x, y 2 ) = ∂R i ∂y (x, ξ i )(y 1 − y 2 ) , i = 1, . . . , n , da cui |R i (x, y 1 ) −R i (x, y 2 )| ≤ 1 2 √ n |y 1 − y 2 | e |R(x, y 1 ) −R(x, y 2 )| ≤ 1 2 |y 1 − y 2 | .
21 Per comoditá di notazione, nel corso della dimostrazione omettiamo il simbolo (·, ·) di prodotto scalare; cosicché ad esempio ∂R i ∂y (x, ξ i )(y 1 − y 2 ) denota il prodotto scalare di ∂R i ∂y (x, ξ i ) ∈ R n ed y 1 − y 2 ∈ R n .
Usando le relazioni precedenti troviamo
|φ x (y 1 ) − φ x (y 2 )| = |R(x, y 1 ) −R(x, y 2 )| ≤ 1 2 |y 1 − y 2 | . (6.14)
Per cui φ x decrementa la distanza. Troviamo ora un dominio compatto B ⊂ ∆ n tale che φ x (B) ⊆ B . A tale scopo, applicando (6.13) e ponendo L := sup |x|=1 |Lx|, osserviamo che
|φ x (y)| ≤ L |x| + |R(x, y)| ≤ L |x| + 1 2 (|x| + |y|) .
Per cui, scegliendo opportunamente r 1 , r 2 > 0 e dischi chiusi A := ∆(0,
r 1 ) ⊂ R m , B := ∆(0, r 2 ) ⊂ R n , otteniamo |φ x (y)| < r 2 . Dunque abbiamo una contrazione φ x : B → B .
Passo 3. Applicando il teorema delle contrazioni (Teo.2.10), otteniamo che per ogni x ∈ A esiste edè unico f (x) ∈ B punto fisso per φ x , ovvero f (x) soddisfa (6.12). In tal modo,è definita l'applicazione f : A → B , che chiaramenteè la nostra candidata a funzione implicita per F .
Passo 4 (Continuitá di f ). Come in (6.14), troviamo opportuni η ∈ A, ξ ∈ B tali che |f (x 1 ) − f (x 2 )| ≤ L |x 1 − x 2 | + |R(x 1 , f (x 1 )) −R(x 2 , f (x 2 ))| ≤ L |x 1 − x 2 | + |R(x 1 , f (x 1 )) −R(x 1 , f (x 2 ))| + |R(x 1 , f (x 2 )) −R(x 2 , f (x 2 ))| ≤ L |x 1 − x 2 | + 1 2 (|f (x 1 ) − f (x 2 )| + |x 1 − x 2 |) ; dunque, |f (x 1 ) − f (x 2 )| ≤ (2 L − 1)|x 1 − x 2 | ,(6.15)
ed fè continua.
Passo 5 (Differenziabilitá e Jacobiana di f ). Essendo F di classe C 1 , abbiamo che esistono opportuni (η 1 , ξ 1 ), . . . , (η n , ξ n ) appartenenti al segmento che congiunge (x 1 , f (x 1 )) ed (x 2 , f (x 2 )), tali che per ogni i = 1, . . . , n
0 = F i (x 1 , f (x 1 )) − F i (x 2 , f (x 2 )) = ∂F i ∂x (η i , ξ i )(x 1 − x 2 ) + ∂F i ∂y (η i , ξ i )(f (x 1 ) − f (x 2 )) ;
introducendo (per somma e sottrazione) nell'equazione precedente le matrici
M 1 := ∂F i ∂x j (η i , ξ i ) − ∂F i ∂x j (x 2 , f (x 2 )) ij , M 2 := ∂F i ∂y k (η i , ξ i ) − ∂F i ∂y k (x 2 , f (x 2 )) ik , otteniamo 0 = ∂F ∂x (x 2 , f (x 2 ))(x 1 − x 2 ) + ∂F ∂y (x 2 , f (x 2 ))(f (x 1 ) − f (x 2 )) + M 1 (x 1 − x 2 ) + M 2 (f (x 1 ) − f (x 2 )) . Ora M 1 , M 2 → 0 per x 1 → x 2 ; d'altro canto ∂F ∂y (x 2 , f (x 2 ))è invertibile, cosicché f (x 1 ) − f (x 2 ) = − ∂F ∂y (x 2 , f (x 2 )) −1 ∂F ∂x (x 2 , f (x 2 ))(x 1 − x 2 ) + Q(x 1 , x 2 ) ,
avendo raggruppato in Q(x 1 , x 2 ) i termini restanti; da (6.15) segue che Dimostrazione. L'idea consiste nell'applicare il teorema delle funzioni implicite a
lim (x1→x2) Q(x 1 , x 2 ) |x 1 − x 2 | = 0 , per cui fè differenziabile. Il teorema dell'inverso locale. Sia A ⊆ R n aperto, Un'applicazione f ∈ C 1 (A, R n ) si dice diffeomorfismo locale in a ∈ A se esiste un intorno V ∋ a tale che f | Vè un diffeomorfismo.F : A × R n → R n , F (x, y) := y − f (x) , x ∈ A , y ∈ R n .
Osserviamo infatti che preso il nostro a ∈ A, allora risulta
F (a, f (a)) = 0 , det ∂F ∂x (a, f (a)) = − det ∂f ∂x (a)
(notare lo scambio del ruolo di x, y rispetto all'enunciato di Teo.6.6). Per cui, il teorema delle funzioni implicite ci assicura che esistono intorni V ⊆ A, a ∈ V , B ⊆ R n , f (a) ∈ B , ed una funzione g : B → V di classe C 1 tale che F (g(y), y) = 0 , y ∈ B ; in altri termini, f • g(y) = y , y ∈ B . Per cui gè l'inversa locale cercata.
Esempio 6.2. Esibiamo un esempio per il quale non vale il teorema dell'inverso locale. Sia
f : R → R , f (x) := x 2 + x 2 sin 1 x , x = 0 0 , x = 0
Si verifica che fè derivabile su R e f ′ (0) = 1/2 = 0 . Tuttavia f ha infinite oscillazioni in ogni intorno di 0 , e quindi non puó essere localmente invertibile. Si noti che f ′è discontinua in 0 .
Moltiplicatori di Lagrange. Sia f ∈ C 1 (A), A ⊆ R n . Consideriamo una varietá Σ ⊂ A di classe C 1 e dimensione m < n, equipaggiata di una rappresentazione parametrica ϕ : U → Σ , U ⊆ R m , ϕ = (ϕ 1 (u), . . . , ϕ n (u)) , u := (u 1 , . . . , u m ) ∈ U .
Si dicono massimi e minimi vincolati di f rispetto a Σ i massimi e minimi di f • ϕ : U → R. In tal caso, diremo che Σè un vincolo per f . E' immediato dimostrare, tramite la formula di derivazione composta, che In altri termini, ∇fè normale a Σ nel punto ϕ(b). Ora, in generale Σ si puó presentare in forma implicita, ovvero in termini di un luogo Σ := {a ∈ A : F (a 1 , . . . , a n ) = 0} , (6.17)
∇(f • ϕ) = n i=1 ∂f ∂x i (ϕ(u)) ∂ϕ i ∂u k (u)
dove F : A → R mè un'applicazione C 1 . In questo caso, abbiamo il seguente
Teorema 6.8 (Moltiplicatori Lagrangiani). Sia f : A → R, A ⊆ R n , di classe C 1 , e sia Σ ⊂ A l'insieme degli zeri di un'applicazione F : A → R, anch'essa di classe C 1 . Allora, i punti di massimo e minimo (vincolato) della funzione f | Σ : Σ → R sono i punti stazionari liberi della funzione H : A × R → R , H(x, λ) := f (x) + λF (x) .
Dimostrazione. Sia a ∈ Σ tale che ∂F ∂u n (a) = 0 .
Allora, per il teorema delle funzioni implicite (Teo.6.6) esiste un intorno V ∋ (a 1 , . . . , a n−1 ) e g : V → R tali che F (u 1 , . . . , u n ) = 0 ⇔ u n = g(u 1 , . . . , u n−1 ) .
In tal modo, in un intorno di a del tipo V × (a n − δ, a n + δ), il versore normale a Σ si scrive
n = 1 1 + |∇g| 2 − ∂g ∂u 1 , . . . , − ∂g ∂u n−1 , 1 ( 6.8 ) = ∇F |∇F | .
Usando (6.16), troviamo che se a ∈ Σè stazionario per f | Σ , allora ∇f deve essere proporzionale ad n in a. In altri termini, esiste λ ∈ R tale che ∇f (a) + λ∇F (a) = 0 . (6.18)
Quest'ultima equazione, accoppiata alla condizione F (a) = 0 ,è equivalente all'annullarsi del gradiente di H in (a, λ).
Esempio 6.3. Siano g, k > 0 fissati. Consideriamo la circonferenza Σ := {(x, y) ∈ R 2 , x 2 + y 2 − 1 = 0} e la funzione f (x, y) := mgy + 1/2k[(x − 1) 2 + y 2 ], x, y ∈ R. Allora i punti di minimo di f | Σ sono le soluzioni del sistema
k(x − 1) + 2λx = 0 mg + ky + 2λy = 0 .
Cenni sulle forme differenziali e loro integrazione.
Formule di Gauss-Green. Sia U ⊆ R n aperto ed F : U → R n un'applicazione di classe C 1 . La divergenza di Fè data dalla funzione
divF : U → R , divF := n i ∂F i ∂x i =: (∇, F ) .
Un compatto K ⊂ U si dice dominio regolare se il bordo ∂Kè costituito da varietá C 1 a tratti (curve regolari a tratti 22 , nel caso n = 2 ). La formula di Gauss-Green stabilisce che
K divF = ∂K (F, n) ,(6.19)
dove n denota il versore normale esterno al bordo ∂K ⊂ R n e (F, n) : U → Rè la funzione definita dal prodotto scalare
(F, n)(x) := i F i (x)n i (x) , x ∈ U .
Osserviamo che nel caso n = 1 e K = [a, b] abbiamo n(a) = −1 e n(b) = 1 ; inoltre la divergenza di Fè la sua derivata, mentre l'integrale su ∂K = {a, b}è semplicemente la somma
(F, n)(b) + (F, n)(a) = F (b)n(b) + F (a)n(a) = F (b) − F (a) .
Dunque, (6.19) si riduce al teorema fondamentale del calcolo. Nelle righe che seguono diamo l'idea della dimostrazione nel caso n = 2 , con bordo dato da curve regolari (per i dettagli si veda [12]):
• Fissiamo r > 0 ed consideriamo un ricoprimento Q := {Q k } di U , dove ogni Q kè un quadrato con centro (t k , a k ) e lato 2r , r > 0 . Osserviamo che, potendo assumere che Uè limitato, possiamo sempre scegliere Q finito;
• Richiediamo che ogni ∂K ∩ Q k , k ∈ N, sia il grafico di una funzione α k : I k → R per qualche intervallo limitato I k ⊂ R. Osserviamo che grazie al Teorema delle funzioni implicite ( §6.3) ∂K si puó sempre interpretare localmente come il grafico di una funzione, dunque il ricoprimento di cui sopra esiste.
In tale scenario possiamo scrivere il versore normale esterno nel seguente modo:
n(t, α k (t)) = (1 + α ′ k (t) 2 ) −1/2 (−α ′ k (t), 1) , t ∈ I k . Sia ora f ∈ C 1 0 (U ∩ Q k ), cosicché fè nulla al bordo di Q k . Assumiamo che K ∩ Q k sia della forma K ∩ Q k = {(x 1 , x 2 ) ∈ R 2 : x 1 ∈ I k := (t k − r, t k + r) , x 2 ∈ (a k − r, α k (x 1 ))} ,(6.20)
il che vuol dire che stiamo applicando il teorema del Dini esprimendo δK ∩ Q k in funzione della coordinata x 1 . Applicando Fubini troviamo, tenendo conto che per ipotesi f (t, a k −r) = 0 , ∀t ∈ I k ,
K∩Q k ∂f ∂x2 dx 1 dx 2 = I k dt α k (t) a k −r ∂f ∂x2 dx 2 = = I k f (t, α k (t)) dt = = I k f (t, α k (t)) n 2 (t, α k (t)) 1 + α ′ k(riguardando l'ultimo integrale otteniamo K∩Q k ∂f ∂x 2 dx 1 dx 2 = ∂K∩Q k f n 2 ds , ds := 1 + α ′ k (t) 2 dt . (6.21) Analogamente, avendosi f (t k − r, x 2 ) = f (t k + r, x 2 ) = 0 , ∀x 2 ∈ (a k − r, a k + r), K∩Q k ∂f ∂t dtdx 2 = I k dt α k (t) a k −r ∂f ∂t dx 2 = = I k dt d dt α k (t) a k −r f (t, x 2 )dx 2 − f (t, α k (t))α ′ k (t) = = α k (t k +r) a k −r f (t k + r, x 2 ) dx 2 − α k (t k −r) a k −r f (t k − r, x 2 ) dx 2 − I k f (t, α k (t))α ′ k (t) dt = = − I k f (t, α k (t))α ′ k (t) dt = = ∂K∩Q k f n 1 ds .
(6.22) Si verifica facilmente che le uguaglianze precedenti rimangono vere anche per quegli indici k per i quali scrivessimo x 1 in funzione di x 2 , cosicché le assumeremo valide per ogni K ∩ Q k , anche non della forma (6.20). Consideriamo ora (6.21,6.22)
sostituendo ad f le funzioni ρ k F i ∈ C 1 0 (U ∩ Q k ), i = 1, 2 , dove {ρ k ∈ C 1 0 (Q k )} kè una partizione dell'unitá di U subordinata a Q 23 , e calcoliamo ∂K (F, n) = k ∂K∩Q k ρ k (F, n) = = k ∂K∩Q k ρ k (F 1 n 1 + F 2 n 2 ) (6.21,6.22) = = k K∩Q k ∂(ρ k F1) ∂x1 + ∂(ρ k F2) ∂x2 dx 1 dx 2 .
Derivando per parti le funzioni integrande ed usando le identitá
k ρ k (x) = 1 ⇒ k ∂ρ k ∂x i (x) = 0 , ∀x ∈ K , i = 1, 2 ,
e immediato verificare che i contributi delle ρ k si elidono, e ció dimostra (6.19).
Forme Differenziali. Dato uno spazio vettoriale reale V di dimensione finita n, denotiamo con ∧ m V la sua potenza tensoriale esterna 24 di ordine m ≤ n; ricordiamo che ∧ 0 V := R,
∧ 1 V = V e ∧ n V = R. Se {e i }è una base di V , denotiamo con e I := e i1 ∧ . . . ∧ e im , I := {i 1 , . . . , i m } , gli elementi della base di ∧ m V . Nel caso V = R n, * , n ∈ N, abbiamo la base {dx i } , cosicché ∧ m R n, * ha base {dx I := dx i1 ∧ . . . ∧ dx im } .
Sia ora U ⊆ R n un aperto connesso. Una m-forma differenziale di classe C kè il dato di un'applicazione di classe C k ω : U → ∧ m R n, * ;
23 Con ció intendiamo che: (1) ogni ρ k : U → [0, 1]è una funzione C ∞ con supporto in Q k ; (2) in un intorno di ogni x ∈ U viè solo un numero finito di ρ k con ρ k (x) = 0 ; (3) k ρ k (x) = 1 , ∀x ∈ K (convergenza puntuale, quando la sommaè infinita). L'esistenza delle partizioni dell'unitá si dimostra con argomenti che fanno uso del Lemma di Uryshon, vedi [
∧ v 2 = −v 2 ∧ v 1 .
considerando, per ogni multiindice I di lunghezza m, i prodotti scalari a I (x) := (ω(x), dx I ),
x ∈ U , otteniamo una funzione a I ∈ C k (U ) e troviamo
ω(x) = |I|=m a I (x) dx I , x ∈ U .
In accordo alle convenzioni precedenti, una 0 -formaè semplicemente una funzione f ∈ C k (U ). df : U → R n, * , a → df a , a ∈ U .
Denotiamo con Ω m k (U ) lo spazio delle m-forme differenziali di classe k , k ∈ Z + ∪ {∞} . La derivata esternaè definita dall'applicazione d : Ω m k (U ) → Ω m+1 k−1 (U ) , ω → dω := n i=1 ∂a I ∂x i dx i ∧ dx I .
Fattori integranti. Come applicazione del concetto di forma differenziale presentiamo un metodo per la soluzione di una classe di problemi di Cauchy.
Sia U ⊂ R 2 , (t 0 , u 0 ) ∈ U , h, g ∈ C(U ) con g(t 0 , u 0 ) = 0 . Consideriamo il problema u ′ = −h(t, u(t)) g(t, u(t)) −1 u(t 0 ) = u 0 . (6.23)
Definiamo la 1 -forma ω(t, u) := h(t, u) dt + g(t, u) du , (t, u) ∈ U , e supponiamo che essa sia esatta, ovvero ω = df , f ∈ C 1 (U ). Sostituendo eventualmente f con f − f (t 0 , u 0 ) possiamo assumere che f (t 0 , u 0 ) = 0 , per cui abbiamo
f (t 0 , u 0 ) = 0 , ∂f ∂y (t 0 , u 0 ) = g(t 0 , u 0 ) = 0 .
Sono dunque soddisfatte le ipotesi di Teo.6.6, per cui esiste A ⊆ R ed u ∈ C 1 (A) tale che f (t, u(t)) = 0 , t ∈ A. Poiché, grazie a (6.8), si ha
u ′ (t) = − ∂f ∂y (t, u(t)) −1 ∂f ∂x (t, u(t)) = −g(t, u(t)) −1 h(t, u(t)) ,(6.24)
concludiamo che (A, u)è soluzione di (6.23). Qualora ω non sia esatta possiamo cercare un fattore integrante, ovvero una funzione φ ∈ C 1 (U ), φ(t 0 , u 0 ) = 0 , tale che
φω(t, u) := φ(t, u)h(t, u) dt + φ(t, u)g(t, u) du , (t, u) ∈ U ,
sia esatta, ovvero φω = df φ , f φ ∈ C 1 (U ). In tal modo, {∂f φ /∂y}(t 0 , u 0 ) = φ(t 0 , u 0 )g(t 0 , u 0 ) = 0 e troviamo la soluzione u φ di (6.23) definita come in (6.24).
Integrazione di forme e Teorema di Stokes. Discutiamo ora la nozione di integrale di una forma differenziale. Iniziamo osservando che avendosi R ≃ ∧ 0 R * ,n , ∧ n R * ,n , l'integrale di 0 -forme ed n-formeè ben definito nella maniera usuale, visto che queste si possono riguardare come funzioni di classe C k ; in particolare, se K ⊂ Uè compatto allora sappiamo che | K ω| < ∞, ω ∈ Ω 0 k (U ) o Ω n k (U ). Piú in generale possiamo integrare m-forme, 1 ≤ m < n, nel modo che segue. Consideriamo l'aperto A ⊆ R m equipaggiato con la restrizione della misura prodotto di Lebesgue ( §6.6) e γ ∈ C 1 (A, U ); data ω ∈ Ω m k (U ), l'integrale di ω su γ si definisce come
γ ω := |I|=m A a I • γ(u) · det ∂ I γ(u) du I , A ⊆ R m , ω ∈ Ω m k (U ) ,(6.∂γ I : A → M m,m (R) , ∂γ I (v) := ∂γ i k ∂u j (v) j,k=1,...,m , v ∈ A ⊆ R m .
L'integrazione delle formeè strettamente connessa con la loro proprietá di esattezza: un importante risultato stabilisce che ω ∈ Ω 1 0 (U )è esatta se e solo se γ ω = 0 per ogni curva chiusa e regolare a tratti γ ([12, Cor.8.
∂ m : C m (M ) → C m−1 (M ) , ∂ m σ := i (−1) i σ • j i m , σ ∈ C ∞ (∆ m , M ) .
Dei semplici (ma piuttosto tediosi) conti mostrano che ∂ m • ∂ m+1 = 0 , per cui Im(∂ m+1 )è un sottospazio vettoriale di ker(∂ m ). Ha quindi senso definire l'omologia singolare di ordine m come lo spazio quoziente H m (M ) := ker(∂ m )/Im(∂ m+1 ) .
Presa ω ∈ Z m dR (M ), definiamo il funzionale lineare
ω * : ker(∂ m ) → R , ω * (v) := i a i σi ω , ∀v := i a i σ i ∈ ker(∂ m ) .
Grazie al teorema di Stokes troviamo immediatamente
ω * (v) = {ω + dϕ} * (v + ∂ m+1 w) , ∀ϕ ∈ Ω m−1 ∞ (M ) , w ∈ C m+1 (M ) ,
per cui concludiamo che: Le varietá considerate nel precedente teorema formano una classe abbastanza vasta (superficie di Riemann, sfere,...). Segnaliamo il fatto fondamentale (e non banale) che se costruissimo gli spazi di omologia a partire da applicazioni continue σ ∈ C(∆ m , M ) piuttosto che C ∞ otterremmo lo stesso spazio vettoriale H m (M ), il quale dipende quindi solo dalla topologia di M . Per dettagli sulla dualitá di de Rham rimandiamo a [4]; ci limitiamo qui ad elencare i casi delle palle unitarie (D n ) e delle sfere (S n ), n ∈ N, di interesse nell'ambito del teorema di Brouwer (Teo.2.13):
(1) ω * = {ω+dϕ} * dipende solo dalla classe di equivalenza [ω] ∈ H m dR (M ); (2) ω * (v) = ω * (v+∂ m+1 w) dipende solo dalla classe di equivalenza di v in H m (M ). Di conseguenza, abbiamo una ben definita applicazione lineare H m dR (M ) → H * m (M ) , [ω] → ω * ,(6.H k (S n ) = R , k = 0, n {0} , k = 0, n , H k (D n ) = R , k = 0 {0} , k > 0 .
(6.28)
Fondamenti di calcolo variazionale.
Oggetto di studio del calcolo variazionaleè la minimizzazione di applicazioni (comunemente dette funzionali) definite su spazi di funzioni. In termini precisi, consideriamo
f : R 3 → R , f = f (t, x, p) di classe C 2 , a < b ∈ R e lo spazio topologico X := {u ∈ C 1 ([t 0 , t 1 ]) : u(a) = 0, u(b) = L} , L ∈ R ; definiamo quindi il funzionale F : X → R , F (u) := b a f (t, u(t), u ′ (t)) dt . (6.29)
In nostro problemaè quello di trovare u ∈ X che minimizzi F . Come vedremo nelle sezioni seguenti, esiste una stretta relazione tra il calcolo variazionale e la teoria delle equazioni alle derivate parziali.
Diamo ora una condizione necessaria all'esistenza di un minimo u ∈ X per F . Consideriamo funzioni ϕ ∈ C 1 0 ([a, b]) (ovvero, ϕ(a) = ϕ(b) = 0 ). Per ogni λ ∈ R osserviamo che v + λϕ ∈ X , v ∈ X , e definiamo g(λ) := F (u + λϕ), λ ∈ R, cosicché
g ′ (λ) = b a ∂f ∂x (·) ϕ(t) + ∂f ∂p (·) ϕ ′ (t) dt . (6.30)
Chiaramente, se uè di minimo per f allora λ = 0è di minimo per g , per cui esplicitando la condizione g ′ (0) = 0 otteniamo il seguente risultato:
Proposizione 6.10. Se u ∈ Xè di minimo per il funzionale F definito da (6.29), allora b a ∂f ∂x (t, u(t), u ′ (t)) ϕ(t) + ∂f ∂p (t, u(t), u ′ (t)) ϕ ′ (t) dt = 0 , ∀ϕ ∈ C 1 0 ([a, b]) . (6.31)
Teorema 6.11 (Teorema di Eulero). Sia u ∈ X di minimo per il funzionale F definito da (6.29), ∈ (a, b) .
u ∈ C 2 ([a, b]). Allora u soddisfa l'equazione di Eulero-Lagrange d dt ∂f ∂p (t, u(t), u ′ (t)) = ∂f ∂x (t, u(t), u ′ (t)) , t(6.32) Dimostrazione. Integrando per parti otteniamo b a ∂f ∂p (t, u(t), u ′ (t)) ϕ ′ (t) dt = − b a d dt ∂f ∂p (t, u(t), u ′ (t)) ϕ(t) dt , (6.33) per cui b a ϕ(t) d dt ∂f ∂p (t, u(t), u ′ (t)) − ∂f ∂x (t, u(t), u ′ (t)) dt = 0 , ϕ ∈ C 1 0 ([a, b]) .
Da quest'ultima espressioneè semplice concludere che il fattore moltiplicato per ϕ nella funzione integranda deve essere nullo, ed il teoremaè dimostrato.
Il teorema di Eulero fornisce una condizione necessaria affinché esista una soluzione u del problema variazionale associato al funzionale (6.29). Un'importante classe di funzionali tali che l'equazione di Eulero-Lagrangeè anche condizione sufficienteè quella dei funzionali convessi, ovvero
F (λu + (1 − λ)v) ≤ λF (u) + (1 − λ)F (v) , λ ∈ [0, 1] , u, v ∈ X .
(6.34) Teorema 6.12. Sia dato il problema variazionale (6.29), con F convesso. Una funzione u ∈ X ∩ C 1 ([a, b])è di minimo per F se e solo se soddisfa l'equazione di Eulero-Lagrange (6.32).
Dimostrazione. Grazie al teorema di Eulero, per dimostrare il teorema dobbiamo soltanto verificare che una funzione u che soddisfi (6.32)è di minimo per F . Ora, per ogni v ∈ X risulta che ϕ := v − u ∈ C 1 0 ([a, b]) (si ricordino le condizioni al bordo per elementi di X ). Per convessitá, troviamo
F (u + λϕ) = F (λv + (1 − λ)u) ≤ λF (v) + (1 − λ)F (u) . Definendo g(λ) := F (u + λϕ) = F (λv + (1 − λ)u) , abbiamo la condizione g(λ) ≤ λg(1) + (1 − λ)g(0) ⇒ g(1) − g(0) ≥ g(λ) − g(0) λ ,
ovvero g(1) ≥ g(0) + g ′ (0). Ora, applicando (6.30,6.33) otteniamo
g ′ (λ) = b a ϕ(t) ∂f ∂x (·) − d dt ∂f ∂p (·) dt .
Dunque, poiché u soddisfa (6.32) otteniamo g ′ (0) = 0 . Concludiamo quindi che g(1) ≥ g(0), ovvero F (v) ≥ F (u).
Misure ed integrali su spazi prodotto.
Torniamo ora nell'ambito della teoria della misura, occupandoci dell'integrazione su prodotti cartesiani di spazi misurabili.
La costruzione di misure su spazi cartesiani. Siano (X, M, µ), (Y, N , ν) spazi misurabili. Vogliamo equipaggiare il prodotto cartesiano X × Y di una misura, ed a tale scopo introduciamo la famiglia dei rettangoli R X,Y := {A × B : A ∈ M, B ∈ N } , la quale 26 soddisfa le seguenti proprietá:
Lemma 6.13. (1) ∅ ∈ R X,Y ; (2) R, R ′ ∈ R X,Y ⇒ R ∩ R ′ ∈ R X,Y ; (3) Per ogni R ∈ R X,Y esistono R 1 , . . . , R n ∈ R X,Y disgiunti tali che R c =∪ n k=1 R k . Dimostrazione. (1) Ovviamente ∅ × ∅ = ∅ ; (2) Posto R = A × B , R ′ = A ′ × B ′ , abbiamo R ∩ R ′ = (A ∩ A ′ ) × (B ∩ B ′ ), con A ∩ A ′ ∈ M , B ∩ B ′ ∈ N . (3) Posto R = A × B , abbiamo R c = (A c × B)∪(A × B c )∪(A c × B c ), con A, A c ∈ M , B, B c ∈ N . Dal Lemma precedente segue che se R, R ′ ∈ R X,Y ed R ⊂ R ′ allora esistono R 1 , . . . , R n ∈ R X,Y disgiunti tali che R ′ = R∪R 1∪ . . .∪R n ; (6.35) infatti, usando (3) abbiamo R c =∪ n k R ′ k con {R ′ k } ⊂ R X,Y ; per cui R c ∩R ′ =∪ k R k , R k := R ′ ∩R ′ k , e quindi R ′ = R∪(R c ∩ R ′ ) = R∪R 1∪ . . .∪R n . Definiamo ora l'applicazione λ : R X,Y → R + , λ(A × B) := µ(A) · ν(B) , A ∈ M , B ∈ N .
Lemma 6.14. L'applicazione λ soddisfa le seguenti proprietá: (i) λ∅ = 0 ; (ii) Se {R n }è una successione di rettangoli disgiunti tale che R :=∪ n R n ∈ R X,Y , allora λR = n λR n ; (iii) Se {R n }è una successione di rettangoli ed R = ∪ n R n ∈ R X,Y , allora λR ≤ n λR n ; (iv) Se R, R ′ ∈ R X,Y ed R ⊆ R ′ allora λR ≤ λR ′ .
Dimostrazione. (i)è ovvia. Riguardo (ii) scriviamo R := A × B , R n := A n × B n ed osserviamo che, pur essendo gli elementi di {R n } a due a due disgiunti, potremmo tranquillamente avere A n ∩ A m = ∅ , B n ′ ∩ B m ′ = ∅ , per opportune coppie di indici n, m e n ′ , m ′ . Consideriamo la funzione f che associa ad x ∈ X la somma (eventualmente numerabile) delle misure di tutti i B n tali che (x, y) ∈ R n per qualche y ∈ Y ; chiaramente, possiamo scrivere f (x) = n χ An (x)νB n , x ∈ X . D'altra parte, fissato x ∈ A abbiamo che ogni y ∈ Bè tale che (x, y) appartiene ad uno, ed un solo, R n , per cui deve essere, per additivitá numerabile di ν ,
f (x) = χ A (x)νB ; concludiamo che χ A (x)νB = n χ An (x)νB n .
Integrando ed usando il Teorema di convergenza monotóna troviamo λR = µA · νB = χ A dµ · νB = n χ An νB n dµ = n µA n νB n = n λR n .
Riguardo (iii), ripetiamo il ragionamento del punto precedente osservando che stavolta, fissato x ∈ A, la generica coppia (x, y), y ∈ Y , apparterrá ad uno o piú rettangoli R n , per cui abbiamo χ A (x)νB ≤ n χ An (x)νB n e quindi, integrando membro a membro,
λR = χ A dµ · νB ≤ n χ An νB n dµ = n λR n .
Infine, per quanto attiene a (iv) osserviamo che usando (6.35) e (ii) troviamo
λR ′ = λR + n i λR i ≥ λR .
Teorema 6.15 (La misura prodotto). Dati gli spazi di misura completi (X, M, µ), (Y, N , ν), esiste edè unica la misura completa µ × ν su X × Y che estende λ alla σ -algebra generata da R X,Y .
Dimostrazione. Come primo passo mostriamo che
λ * : 2 X×Y → R + , λ * A := inf A⊆∪nRn:{Rn}⊂RX,Y n λR n (6.36) e una misura esterna. Chiaramente λ * ∅ = 0 ; inoltre se A ⊆ A ′ allora A ′ ⊆ ∪ n R n , {R n } ⊂ R X,Y , implica A ⊆ ∪ n R n e quindi (passando agli inf) λ * A ≤ λ * A ′ .
Rimane infine da verificare la subadditivitá numerabile. Se {A n } ⊂ 2 X×Y allora possiamo assumere λ * A n < ∞ per ogni n ∈ N, altrimenti non viè nulla da dimostrare. Per definizione di λ * , per ogni ε > 0 ed n ∈ N esiste una successione {R n,k } tale che A n ⊆ ∪ k R n,k , k λR n,k < λ * A n + 2 −n ε .
Poiché ∪ n A n ⊆ ∪ n,k R n,k , usando la definizione di λ * e le diseguaglianze precedenti troviamo
λ * (∪ n A n ) ≤ n,k λR n,k < n λ * A n + n 2 −n ε = n λ * A n + ε ,
e per arbitrarietá di ε segue la subadditivitá numerabile di λ * , la qualeè quindi una misura esterna. Applicando il Lemma 4.7 otteniamo una misura completa µ × ν su X × Y . Per ultimare la dimostrazione rimane da verificare soltanto il fatto che {µ × ν}R = λR per ogni R ∈ R X,Y . A tale scopo osserviamo che, per costruzione, {µ × ν}R = λ * R (vedi Lemma 4.7); d'altra parteè evidente che l'inf attraverso il qualeè definito λ * Rè raggiunto proprio da λR (vedi Lemma 6.14(iii)), per cui il teoremaè dimostrato.
Denotiamo con R σδ ⊆ 2 X×Y la classe dei sottoinsiemi di X × Y del tipo E = ∩ n (∪ m R n m ), dove R n m ∈ R X,Y ∀n, m ∈ N;è chiaro che ogni elemento di R σδè misurabile rispetto a µ × ν .
Lemma 6.16. Sia E ⊆ X × Y misurabile tale che {µ × ν}E < ∞. Allora esiste R ∈ R σδ tale che E ⊆ R e {µ × ν}E = {µ × ν}R .
Dimostrazione. Per definizione di µ × ν (vedi (6.36)) possiamo trovare una successione {R n } , E ⊆ R n , n ∈ N, i cui elementi sono unione numerabile di rettangoli e tale che scelto ε > 0 esiste n ε ∈ N con
{µ × ν}E ≤ {µ × ν}R n ≤ {µ × ν}E + ε , ∀n ≥ n ε ⇒ {µ × ν}E = lim n {µ × ν}R n .
Eventualmente ridefinendo R n → ∩ n i R i possiamo assumere R n ⊆ R n+1 , n ∈ N, con {R n } ⊆ R σδ . Inoltre avendo E misura finita possiamo assumere {µ × ν}R 1 < ∞. Definendo R := ∩ n R n ed applicando il Lemma 4.3 troviamo {µ × ν}R = lim n {µ × ν}R n = {µ × ν}E .
I teoremi di Fubini e Tonelli. I risultati seguenti permettono di approcciare il calcolo di integrali di funzioni su spazi prodotto. Per prima cosa introduciamo il concetto di sezione di un insieme E ⊆ X × Y :
E x := {y ∈ Y : (x, y) ∈ E} ⊆ Y , x ∈ X (ed analogamente si definisce E y ⊆ X , y ∈ Y ); abbiamo le ovvie proprietá (E c ) x = (E x ) c , (∪ i E i ) x = ∪ i (E i ) x , χ Ex (y) = χ E (x, y) = χ Ey (x) . Lemma 6.17. Sia E ∈ R σδ . Allora E xè misurabile per ogni x ∈ X . Dimostrazione. Il Lemmaè banalmente vero se E ∈ R X,Y . Assumendo quindi che E = ∪ n E n , {E n } ⊆ R X,Y , troviamo χ Ex (y) = χ(x, y) = sup n χ En (x, y) = sup n χ (En)x (y) .
Poiché ogni E n ∈ R X,Y , abbiamo che ogni (E n ) x ⊆ Yè misurabile; quindi χ (En)xè misurabile e, grazie al Teo.4.11, χ Ex = sup n χ (En)xé misurabile, il che vuol dire che E xè misurabile. Infine prendiamo E = ∩ i E i , dove ogni E iè unione numerabile di rettangoli (per cui per ogni E i vale l'enunciato del Lemma). Allora
χ Ex (y) = χ E (x, y) = inf i χ Ei (x, y) = inf i χ (Ei)x (y) ,
e ragionendo come in precedenza concludiamo che E xè misurabile.
Lemma 6.18. Sia E ∈ R σδ con {µ × ν}E < ∞. Allora la funzione f E (x) := νE x , x ∈ X ,è misurabile e f E dµ = {µ × ν}E .
Dimostrazione. Il Lemmaè banalmente vero se E ∈ R X,Y . Se Eè unione numerabile di rettangoli osserviamo che possiamo comunque esprimerlo, usando iterativamente (6.35), come un'unione disgiunta:
E =∪ n E n , {E n } ⊆ R X,Y . Consideriamo quindi le funzioni f n (x) := ν(E n ) x ,
x ∈ X , che sono positive e misurabili, ed osserviamo che f E = n f n , cosicché concludiamo che f Eè misurabile. Per convergenza monotóna ed additivitá numerabile di µ × ν abbiamo
f E dµ = n f n dµ = n {µ × ν}E n = {µ × ν}E ,
per cui il Lemmaè vero per unioni numerabili di rettangoli. Sia ora E = ∩ n E n tale che ogni E nè unione numerabile di rettangoli (per cui per ogni E n vale l'enunciato del Lemma). Eventualmente ridefinendo E ′ n := ∩ n i E i possiamo assumere che E n+1 ⊆ E n per ogni n ∈ N. D'altra parte, essendo E di misura finita, possiamo sempre assumere, ricordando la definizione (6.36), che {µ × ν}E 1 < ∞. A questo punto, posto f n (x) := ν(E n ) x , x ∈ X , grazie al Teorema 4.11 concludiamo che f E = lim n f nè misurabile. D'altra parte abbiamo
f E (x) = νE x Lemma 4.3 = lim n ν(E n ) x = lim n f n (x) , q.o. in x ∈ X ; per cui, avendosi f n ≤ f 1 e f 1 dµ = {µ × ν}E 1 < ∞, per il teorema di convergenza dominata di Lebesgue concludiamo f E dµ = lim n f n dµ = lim n {µ × ν}E n Lemma 4.3 = {µ × ν}E . Lemma 6.19. Sia E ⊆ X × Y misurabile e tale che {µ × ν}E = 0 . Allora q.o. in x ∈ X risulta che E xè misurabile e νE x = 0 .
Dimostrazione. Usando il Lemma 6.16 troviamo che esiste R ∈ R σδ con E ⊆ R e {µ × ν}E = {µ × ν}R = 0 , per cui usando il Lemma precedente troviamo
{µ × ν}R = f R dµ = 0 ⇒ f R (x) = νR x = 0 q.o. in x ∈ X .
D'altro canto per costruzione E x ⊆ R x per ogni x ∈ X , ed essendo ν completa concludiamo che E xè misurabile con νE x = 0 per ogni x tale che νR x = 0 .
Lemma 6.20. Sia E ⊆ X × Y misurabile e tale che {µ × ν}E < ∞. Allora la funzione f E (x) := νE x , x ∈ X ,è misurabile e f E dµ = {µ × ν}E .
Dimostrazione. Grazie al Lemma 6.16 sappiamo che esiste R ∈ R σδ tale che Il seguente teorema costituisce lo strumento principale per il calcolo esplicito di integrali su spazi prodotto e fornisce la famosa regola dello scambio dell'ordine di integrazione:
E ⊆ R e {µ × ν}E = {µ × ν}R . Cosicché ∞ > {µ × ν}R = {µ × ν}E + {µ × ν}(R − E) ⇒ {µ × ν}(R − E) = 0 . Grazie al Lemma 6.19 troviamo ν(R − E) x = 0 q.o. in x ∈ X , per cui f E (x) = f R (x) := νR x , q.o. in x ∈ X .Teorema 6.21 (Fubini). Sia f ∈ L 1 µ×ν (X × Y ). Allora q.o. in x ∈ X risulta che f (x, ·) ∈ L 1 ν (Y ) ed F (x) := Y f (x, y)dν , x ∈ X , appartiene a L 1 µ (X). Inoltre l'affermazione analogaè vera scambiando i ruoli di X ed Y , e X×Y f (x, y) d{µ × ν} = X Y f (x, y)dν dµ = Y X f (x, y)dµ dν . (6.37)
Dimostrazione. Grazie alla simmetria tra x ed yè sufficiente dimostrare il teorema senza scambiare i ruoli delle due variabili. Se la conclusione del teoremaè vera per due funzioni alloraè vera anche per la loro differenza, per cui possiamo ridurci a considerare il caso f ≥ 0 . Il Lemma 6.20 afferma che il teoremaè vero se fè la funzione caratteristica di un insieme E ⊆ X × Y di misura finita, per cui essoè vero anche per funzioni semplici che si annullano al di fuori di un insieme di misura finita (denotiamo con S c (X × Y ) l'insieme di tali funzioni). Ora, ogni funzione integrabile non negativà e limite puntuale di una successione {ψ n } ⊂ S c (X × Y ) monotóna crescente (vedi Prop.4.29), per cui applicando il teorema di Beppo Levi
F (x) := Y f (x, y) dν = lim n Y ψ n (x, y) dν ;
essendo il teorema vero per ogni ψ n concludiamo che, essendo F limite puntuale di funzioni misurabili,è essa stessa misurabile (Teorema 4.11). Applicando ancora il teorema di Beppo Levi, ed usando ancora il fatto che il teoremaè vero per ogni ψ n , troviamo
F dµ = X Y f dµdν = lim n X Y ψ n dµdν = lim n X×Y ψ n d{µ × ν} = X×Y f d{µ × ν} .
Ció conclude la dimostrazione. Dimostrazione. L'unico punto della dimostrazione del teorema di Fubini in cui usiamo l'integrabilitá di fè dove affermiamo che fè limite puntuale di funzioni in S c (X × Y ). Del resto, Prop.4.12 afferma che fè limite puntuale di funzioni in S c (X × Y ) con la sola ipotesi di misurabilitá, a patto che X × Y sia σ -finito. Cióè senz'altro vero se X, Y sono σ -finiti, per cui possiamo ripetere con successo l'argomento della dimostrazione del Teorema di Fubini ed ottenere le proprietá desiderate.
Teorema 6.22 (Tonelli). Siano (X, M, µ),(Y, N , ν) spazi di misura σ -finiti ed f : X × Y → R + una funzione misurabile non negativa. Allora: (1) f (x, ·) : Y → R +è una funzione misurabile q.o. in x ∈ X ; (2) F (x) := Y f (x, y) dν , x ∈ X ,
Nel teorema precedente si considerano funzioni f ≥ 0 e non si fa nessuna affermazione sull'integrabilitá di f . Tuttavia se si suppone che f (x, ·) ∈ L 1 µ (Y ) q.o. in x ∈ X ed F ∈ L 1 µ (X), allora il punto (4) permette di concludere che f ∈ L 1 µ×ν (X × Y ).
Convoluzioni.
Un'importante applicazione dei teoremi di Fubini e Tonelliè quella dei prodotti di convoluzione, i quali a loro volta hanno un ruolo importante nell'ambito degli spazi L p , di Sobolev, e nell'analisi di Fourier. In questa sezione consideriamo gli spazi euclidei R d , d ∈ N, equipaggiati con la misura prodotto di Lebesgue (che otteniamo iterando d − 1 volte sulla retta reale la costruzione della sezione precedente).
Teorema 6.23. Siano f ∈ L 1 (R d ), g ∈ L p (R d ), p ∈ [1, +∞]. Allora q.o. in x ∈ R d la funzione
K(x, y) := f (x − y)g(y) , x, y ∈ R d , è integrabile rispetto ad y su R d . Di conseguenzaè ben definita la funzione
f * g(x) := R d f (x − y)g(y) dy , x ∈ R d , (6.38) la quale soddisfa la diseguaglianza f * g p ≤ f 1 g p , (6.39) per cui f * g ∈ L p (R d ).
Dimostrazione. Dimostriamo il teorema distinguendo i vari casi per p ∈ [1, +∞]. (1) p = ∞: l'enunciatoè ovvio.
(2) p = 1 . Q.o. in y ∈ R d si ha R d |K(x, y)| dx = |g(y)| R d |f (x − y)| dx = |g(y)| f 1 , per cui, avendosi g ∈ L 1 (R d ) troviamo K(·, y) ∈ L 1 (R d ) q.o. in y ∈ R d . Inoltre R d dy R d |K(x, y)| dx ≤ f 1 g 1 < +∞ . (6.40) Per il teorema di Tonelli abbiamo K ∈ L 1 (R d × R d ), mentre per Fubini concludiamo che K(x, ·) ∈ L 1 (R d ) q.o. in x ∈ R d , ovvero R d |f (x − y)g(y)|dy < +∞ , q.o. in x ∈ R d ,
il che dimostra (6.38). Per dimostrare che f * g ∈ L 1 (R d ), basta osservare che
R d dx R d f (x − y)g(y) dy ≤ R d dx R d |f (x − y)g(y)| dy F ubini = R d dy R d |f (x − y)g(y)| dx (6.40) ≤ f 1 g 1 .
(3) p ∈ (1, +∞). Sia g ∈ L p (R d ) e K p (x, y) := |f (x − y)||g(y)| p . Grazie a quanto mostrato per p = 1 , abbiamo
K p (x, ·) ∈ L 1 (R d ) ovvero |K p (x, ·)| 1/p ∈ L p (R d ) , q.o. in x ∈ R d . Poniamo q := p. Poiché f ∈ L 1 (R d ), abbiamo |δf (x, ·)| 1/q ∈ L q (R d ) , δf (x, y) := f (x − y) , q.o. in x ∈ R d . Applicando la disuguaglianza di Holder otteniamo, q.o. in x ∈ R d , +∞ Holder > R d |K p (x, ·)| 1/p · |δf (x, ·)| 1/q = R d |f (x − y)| 1/p |g(y)| · |f (x − y)| 1/q dy ≥ |f * g(x)| . Per cui f * g(x)è definito q.o. in x ∈ R d . Ora, |K p (x, ·)| 1/p p p = R d |f (x − y)||g(y)| p dy = (|f | * |g| p (x)) |δf (x, ·)| 1/q q q = R d |f (x − y)| dy = f 1 ,
per cui scrivendo esplicitamente la precedente disuguaglianza di Holder otteniamo
(|f | * |g| p (x)) 1/p · f 1/q 1 ≥ |f * g(x)| ⇒ |f | * |g| p (x) · f p/q 1 ≥ |f * g(x)| p .
Applicando a |g| p ∈ L 1 (R d ) quanto mostrato nel caso p = 1 abbiamo che |f | * |g| p ∈ L 1 (R d ), per cui |f * g| p ≤ |f | * |g| p · f p/q 1è integrabile.
La funzione f * g si dice convoluzione di f e g . Qui di seguito ne elenchiamo alcune proprietá elementari, la cui dimostrazioneè lasciata per esercizio:
1. f * g = g * f , f ∈ L 1 (R d ), g ∈ L p (R d ); 2. f 1 * (f 2 * g) = (f 1 * f 2 ) * g , f 1 , f 2 ∈ L 1 (R d ); 3. f * (g + h) = f * g + f * h, h ∈ L p (R d ); 4. (af ) * g = f * (ag), a ∈ R d .
Per approcciare la questione della derivabilitá di una convoluzione introduciamo alcune nozioni. Per prima cosa consideriamo F ∈ L 1 (R d , R m ) nel senso di Oss.5.3 ed osserviamo che possiamo scrivere, in termini vettoriali, F = (F 1 , . . . , F m ), dove F i ∈ L 1 (R d ), i = 1, . . . , m. Presa quindi f ∈ L p (R d ) definiamo la convoluzione
F * f : R d → R m , (F * f ) i := F i * f , i = 1, . . . , m ,
cosicché grazie al Teorema 6.23 abbiamo F * f ∈ L p (R d , R m ). Ora, se g ∈ C 1 (R d ) allora possiamo considerare il gradiente
∇g : R d → R d , ∇g := ∂g ∂x 1 , . . . , ∂g ∂x d ; se ∇g ∈ L 1 (R d , R d ) ha quindi senso considerare la convoluzione ∇g * f ∈ L p (R d , R d ), ∀f ∈ L p (R d ).
Dalle definizioni precedenti si deduce banalmente il seguente risultato, conseguenza diretta dei teoremi di derivazione sotto il segno di integrale e della commutativitá del prodotto di convoluzione:
Proposizione 6.24. Sia f ∈ L 1 (R d ) ∩ C 1 (R d ) con ∇f ∈ L 1 (R d , R d ). Allora per ogni g ∈ L 1 (R d )
si ha che f * gè derivabile e ∇(f * g) = (∇f ) * g .
Se anche gè derivabile con ∇g ∈ L 1 (R d , R d ), allora ∇(f * g) = (∇f ) * g = f * (∇g). Osservazione 6.2. Sia G un gruppo topologico localmente compatto di Hausdorff, e µ ∈ R(G) la misura di Haar (vedi §4.1). L'argomento della dimostrazione del teorema precedente si basa sui teoremi di Fubini-Tonelli, l'invarianza per traslazione e la diseguaglianza di Holder: tali proprietá sono tutte verificate dalla misura di Haar, per cui possiamo definire la convoluzione Ora,è facilmente verificabile che nessuna funzione g ∈ L 1 (R)è un'identitá rispetto al prodotto di convoluzione, ovvero f * g = f , ∀f ∈ L 1 (R). Tuttaviaè possibile introdurre dei buoni sostituti dell'identitá, che definiamo qui di seguito.
f * g(s) := G f (st −1 )g(t) dµ(t) , s ∈ G , f ∈ L 1 µ (G) , g ∈ L p µ (G) ,Definizione 6.25. Una successione di funzioni {ρ n } ⊂ L 1 (R d ) si dice identitá approssimata se ρ n ≥ 0 , supp(ρ n ) ⊆ ∆(0, 1/n) , ρ n = 1 , ∀n ∈ N .
In particolare, diremo che {ρ n }è una successione di mollificatori se ρ n ∈ C ∞ c (R d ) 28 ∀n ∈ N. Osservazione 6.3. (1) Possiamo definire in modo del tutto analogo identitá approssimate indicizzate da un parametro a valori reali, piuttosto che dai numeri naturali; (2) Le nozioni di identitá approssimata e successione di mollificatori si possono dare senza variazioni nel caso di funzioni in L 1 (R d , C).
(3) Se f ha supporto compatto allora ogni ρ n * f ha supporto compatto. Proposizione 6.26. Sia p ∈ [1, +∞) e {ρ n } un'identitá approssimata. Allora per ogni g ∈ L p (R d ) risulta g − ρ n * g p → 0 ; in particolare, se {ρ n }è una successione di mollificatori allora ogni ρ n * g
e di classe C ∞ e C ∞ c (R d )è denso in L p (R d ) in norma · p .
Dimostrazione. Iniziamo approssimando in norma · p una funzione g ∈ C c (R d ). Per continuitá e compattezza del supporto abbiamo che gè uniformemente continua, per cui per ogni ε > 0 esiste δ > 0 tale che se |y| < δ allora |g(x − y) − g(x)| < ε . Per 1/n < δ troviamo
|(ρ n * g)(x) − g(x)| ≤ R d |g(x − y) − g(x)| ρ n (y) dy = ∆(0,1/n) |g(x − y) − g(x)| ρ n (y) dy ≤ ε R d ρ n = ε .
Ció implica ρ n * g − g ∞ → 0 ; avendo g ed ogni ρ n * g supporto compatto (il quale non si ingrandisce al crescere di n) concludiamo che
ρ n * g − g p ≤ K 1/p ρ n * g − g ∞ → 0 , dove K < ∞è la misura del supporto di ρ 1 * g . Se f ∈ L p (R d ), allora grazie a Cor.5.8 esiste f ε ∈ C c (R d ) tale che f − f ε p < ε e quindi ρ n * f − f p ≤ ρ n * (f − f ε ) p + ρ n * f ε − f ε p + f ε − f p ≤ 2 f − f ε p + ρ n * f ε − f ε p < 3ε
per n abbastanza grande. Infine, se {ρ n }è una successione di mollificatori allora Prop.6.24 implica che ρ n * g ∈ C ∞ (R d ) per ogni n.
Osservazione 6.4. L'argomento della proposizione precedente dimostra anche il seguente risultato: se g ∈ C(R d )è uniformemente continua, allora ρ n * g − g ∞ → 0 .
Esempio 6.6. Si consideri la funzione
ρ(x) := e 1 x 2 −1 , |x| < 1 0 , |x| ≥ 1
Allora ρ ∈ L 1 (R), e definendo ρ n (x) := ρ −1 1 nρ(nx), x ∈ R, si ottiene una successione di mollificatori.
Esempio 6.7. Per ogni n ∈ N, consideriamo la funzione ρ n := nχ (0, 1 n ) . Chiaramente {ρ n }è un'identitá approssimata. Osserviamo che applicando Prop.6.26 ed il Teorema di Fischer-Riesz otteniamo, per ogni f ∈ L 1 (R), Le convoluzioni dal punto di vista dell'analisi funzionale. Nelle righe che seguono faremo uso delle nozioni di norma di un operatore (vedi (7.13)) ed algebra di Banach (Def.7.8). Denotiamo con BL p (R d ), p ∈ [1, +∞], l'algebra di Banach degli operatori lineari limitati da L p (R d ) in sé. Teo.6.23 afferma che l'operatore
f * ρ k (x) = k R f (x − y)χ (0, 1 k ) (y) dy = k x+1/k x f (s) ds = f (x) , q.o. in x ∈ R (dove l'indice k indica una sottosuccessione {n k } ). Il che fornisce una versione dell'uguaglianza f (x) = lim h→0 1 h x+h x f (s) ds , f ∈ L 1 (R) , q.o. in x ∈ R ,(6.C p f ∈ BL p (R d ) , {C p f }(g) := f * g , ∀g ∈ L p (R d ) ,(6.42)
ha norma ≤ f 1 . Considerando una successione di mollificatori {ρ n } ⊂ L 1 (R d ), otteniamo
{C 1 f }(ρ n ) 1 = f * ρ n 1 n → f 1 , per cui C 1 f ha norma esattamente pari a f 1 . Inoltreè ovvio che f * ( f * g) = (f * f ) * g , f, f ∈ L 1 (R d ), g ∈ L p (R d )
, per cui abbiamo dimostrato: Teorema 6.27. Lo spazio L 1 (R d ), equipaggiato del prodotto di convoluzione,è un'algebra di Banach commutativa che denotiamo con (L 1 (R d ), * ), e (6.42) definisce un'applicazione lineare isometrica
C p : (L 1 (R d ), * ) → BL p (R d ) , f → C p f , tale che C p (f * f ) = {C p f } • {C p f } , ∀f, f ∈ L 1 (R d ).
Un ulteriore aspetto interessante della convoluzioneè il suo rapporto con la trasformata di Fourier, come vedremo nel seguito.
= f (x − y) consideriamo ϕ ∈ L 1 (R d × R d ) tale che esista c > 0 con ϕ(x, ·) 1 , ϕ(·, y) 1 ≤ c , q.o. in x, y ∈ R d . Cosicché, per ogni p ∈ [1, +∞] e g ∈ L p (R d ), l'integrale C ϕ g(x) := R d ϕ(x, y)g(y) dy , x ∈ R d , definisce un operatore lineare limitato C ϕ ∈ BL p (R d ) tale che C ϕ ≤ c.
La trasformata di Laplace. Nello stesso ordine di idee dell'osservazione precedente consideriamo lo spazio L p (R + ), R + := [0, +∞), e definiamo la trasformata di Laplace
Lf (x) := R + e −xt f (t) dt , f ∈ L 1 (R + ) , x ∈ R + . (6.43)
Abbiamo le seguenti proprietá:
• L(f + ag) = Lf + aLg , f, g ∈ L 1 (R + ), a ∈ R;
• Lf ∈ C 0 (R + ); Infatti, poiché |e −xt f (t)| ≤ |f (t)|, possiamo applicare il teorema di Lebesgue e concludere che
x n → x ⇒ lim n Lf (x n ) = lim n R + e −xnt f (t) dt = Lf (x) , x n → ∞ ⇒ lim n Lf (x n ) = lim n R + e −xnt f (t) dt = 0 .
• Lf ∞ ≤ f 1 ; (basta osservare che e −xt ≤ 1 , x, t ∈ R + ).
• L(f * g) = Lf · Lg ; Infatti, prolunghiamo f, g ad R ponendo f (x) = g(x) = 0 , x < 0 , e, usando il teorema di Fubini, calcoliamo 6.8 Esercizi.
L{f * g}(x) = R + e −xt f * g(t) dt = = R + R e −x(t−s) f (t − s) · e −xs g(s) dtds = = R + e −xθ f (θ) dθ R + e −xs g(s) ds = = Lf (x) · Lg(x) .
Esercizio 6.1. Sia p ∈ (1, +∞). Si consideri l'identitá approssimata
ρ := 1 2 χ [−1,1] , ρ ε := 1 ε ρ x ε , ε > 0 ,
e, preso α ∈ (0, 1/p), si calcoli la convoluzione ρ ε * f , dove
f ∈ L p (R) , f (x) := x −α , x ∈ (0, 1] 0 , altrimenti . Verificare che ρ ε * f ∈ C c (R) ∩ L p (R) per ogni ε > 0 , e che lim ε→0 f − ρ ε * f ∞ = 0 . Esercizio 6.2. Sia f (x, y) := x 2 − y 2 (x 2 + y 2 ) 2 , x, y ∈ [0, 1] .
Si verifichi che f dx dy = f dy dx, e si dimostri che f / ∈ L 1 ([0, 1] 2 ).
Soluzione. Per calcolare gli integrali di f rispetto ad x, y , ricordiamo che
s (1 + s 2 ) ds = − 1 2 1 1 + s 2 ⇒ 1 (1 + s 2 ) 2 ds = 1 2 arctan s + s 1 + s 2 , per cui s 2 (1 + s 2 ) 2 ds = 1 2 arctan s − s 1 + s 2 .
Possiamo ora calcolare Usando la simmetria di f rispetto ad x, y , otteniamo anche
ω = { 1 − 2x[(x 2 + y 2 ) −2 ] } dx + { 1 − 2y[(x 2 + y 2 ) −2 ] } dy , (x, y) ∈ R 2 − {0} ϕ = z(x + y) −1 (dx + dy) + log(x + y) dz , x, y, z > 0
Inoltre, si trovi la primitiva F di ϕ tale che F (1, 1, 1) = 1 . dove χ Eè la funzione caratteristica di E , e si mostri che λ * λ ′ ∈ Λ 1 β (R); (3) Si consideri l'applicazione
µ : L 1 (R) → Λ 1 β (R) , f → µ f : µ f E := E f , ∀E ∈ L . (6.46)
Si mostri che µè lineare, isometrica, e che µ f * g = µ f * µ g , ∀f, g ∈ L 1 (R). Si verifichi inoltre che µ nonè suriettiva. (4) Si verifichi che la misura di Dirac δ 0 appartiene a Λ 1 β (R), e che λ * δ 0 = λ, ∀λ ∈ Λ 1 β (R).
(Suggerimenti: per il punto (2), riguardo l'additivitá numerabile di λ * λ ′ si osservi che per ogni successione {E n } di insiemi disgiunti la serie n χ Enè positiva, monotóna crescente e puntualmente convergente a χ E , E :=∪ n E n , per cui si puó applicare il Teorema di Beppo Levi; per il punto (3), riguardo la non suriettivitá si considerino le misure di Dirac).
Analisi Funzionale.
In questa sezione esponiamo le basi dell'analisi funzionale. Tale approccio fa uso di nozioni sia topologiche 29 che di algebra lineare ed i risultati che esso permette di conseguire hanno importanti applicazioni in analisi "hard".
Nelle pagine seguenti tratteremo prevalentemente spazi di Banach e di Hilbert a coefficienti sia reali che complessi. Il caso realeè interessante per le applicazioni alla teoria delle equazioni alle derivate parziali (si veda §10 e la relativa bibliografia). Gli spazi di Hilbert complessi hanno un ruolo importante in meccanica quantistica ([22, Vol.I-IV], [9]) ed in teoria delle rappresentazioni dei gruppi topologici ( [14,9]), ivi compresa l'analisi di Fourier ( §8).
Spazi di Banach e di Hilbert.
Spazi di Banach. Sia E uno spazio vettoriale, reale o complesso. Una seminorma su Eè una funzione p : E → R + tale che
p(λv) = |λ|p(v) , p(v + w) ≤ p(v) + p(w) , λ ∈ R(C) , v, w ∈ E .
In particolare, p si dice norma se p(v) = 0 implica v = 0 . In tal caso, useremo la notazione · := p(·) e diremo che Eè uno spazio normato. Nel seguito, indicheremo con E 1 l'insieme degli elementi di E con norma uguale a 1 . Se f : E → Rè un'applicazione lineare (ovvero, un funzionale lineare), allora introduciamo la notazione
f := sup v∈E |f (v)| v = sup v∈E1 |f (v)| ; (7.1)
se f < +∞, allora diciamo che fè limitato o continuo. L'insieme E * dei funzionali lineari limitati da E in R (o C qualora E sia uno spazio complesso), equipaggiato con la norma (7.1),è uno spazio normato, chiamato il duale di E . Nel seguito, utilizzeremo la notazione Viceversa, se Eè tale che ogni serie assolutamente convergenteè convergente, consideriamo una successione di Cauchy {v n } . Per mostrare che questa ammette un limite, scegliamo una sottosuccessione {n k } ⊂ N tale che per n, m > n k risulti v n − v m < 2 −k , e definiamo la sottosuccesione
f, v := f (v) , ∀f ∈ E * , v ∈ E . (7.2) Una successione {v n } ⊂ E si dice di Cauchy se | ≤ v n − v m n,m → 0 ; osserviamo che | v n − v m | ≤ v n − v m n,m → 0 ,{g k := v n k − v n k−1 } (avendo posto v n0 := 0 ). Allora k g k ≤ g 1 + k 2 −k = g 1 + 1 ,
per cui la serie { k g k }è monotóna e limitata, e quindi convergente. Esiste dunque v := k g k , edè semplice verificare che deve essere anche v = lim n v n .
Proposizione 7.2. Sia E uno spazio normato. Allora il duale E * è uno spazio di Banach. Sia E uno spazio di Banach. Un sottospazio vettoriale E ′ ⊆ E si dice di Banach se essoè completo. Preso un sottospazio vettoriale E ′ ⊆ E , la chiusura di E ′ in E si definisce come lo spazio vettoriale degli elementi di E che sono limite di successioni di Cauchy in E ′ , e si denota con E ′ ; ovviamente E ′è -per costruzione -un sottospazio di Banach. Preso un insieme S ⊂ E , il sottospazio di Banach generato da S si definisce come la chiusura dello spazio vettoriale delle combinazioni lineari di elementi di S .
Dimostrazione. Presa una successione di Cauchy
{f n } ⊂ E * , osserviamo che | f n , v − f m , v | ≤ f n − f m v , ∀v ∈ E ,| f n − f m | ≤ f n − f m , per cui f n è di Cauchy in R ed esiste M > 0 tale che f n < M , n ∈ N. Ora, per ogni ε > 0 esiste m ∈ N tale che |f (v) − f n , v | < ε , n ≥ m; per cui, se v ∈ E 1 troviamo |f (v)| ≤ ε + | f n , v | ≤ ε + M .
Dato un sottospazio vettoriale Esempio 7.1. Sia X uno spazio topologico. Lo spazio C b (X) delle funzioni continue e limitate su X a valori realiè uno spazio di Banach, qualora equipaggiato della norma · ∞ . Una famiglia di funzionali limitati su C b (X)è data dalle delta di Dirac
E ′ ⊆ E l'operazione di restrizione f → R E ′ f := f | E ′ , f ∈ E * , induce un'applicazione lineare R E ′ : E * → E ′ * , e chiaramente, R E ′ f ≤ f , ∀f ∈ E * . Il teorema diδ x , f := f (x) , x ∈ X ⇒ | δ x , f | ≤ f ∞ .≤ z p w q , z + z ′ p ≤ z p + z ′ p , 1 p + 1 q = 1 , (7.3)
ed il teorema di Fischer-Riesz implica che ogni l pè uno spazio di Banach.
Esempio 7.3 (Il Teorema di Riesz-Markov).
Se Xè uno spazio compatto allora C(X), equipaggiato con la norma dell'estremo superiore,è uno spazio di Banach. Assumiamo ora che X sia di Hausdorff; per ogni misura di Radon con segno µ ∈ R(X) definiamo il funzionale
F µ , f := X f dµ , f ∈ C(X) .
E' semplice verificare che F µ = |µ|(X), cosicché abbiamo un'applicazione lineare isometrica
R(X) → C(X) * , µ → F µ . (7.4)
Il Teorema di Riesz-Markov ([25, §13.4]) afferma che (7.4)è anche suriettiva, cosicché abbiamo una caratterizzazione delle misure di Radon su X in termini dei funzionali lineari continui su C(X). Per dare un'idea della dimostrazione consideriamo ϕ ∈ C(X) * positivo, ovvero tale che ϕ, f ≥ 0 ∀f ≥ 0 , ed introduciamo l'applicazione
µ * A := sup{ ϕ, f , f ∈ C(X) , supp(f ) ⊂ A , 0 ≤ f ≤ 1} , A ∈ 2 X .
Si dimostra che la "misura interna" µ * induce una misura di Radon µ definita su un'opportuna σ -algebra M ⊃ τ X , e che F µ = ϕ. La tesi del teorema segue osservando che ogni funzionale su C(X) si decompone in una differenza di funzionali positivi ([25, Prop.13.24]). Il teorema si estende senza difficoltá al caso complesso, cosicché abbiamo un isomorfismo di spazi di Banach R(X, C) → C(X, C) * , dove R(X, C)è lo spazio delle misure di Radon complesse su X .
Spazi di Hilbert e basi ortonormali. Uno spazio di Banach H si dice di Hilbert se questoè equipaggiato di un prodotto scalare (u, v) ∈ R , u, v ∈ H , 30 Le stesse definizioni possono essere formulate usando il campo complesso, ed in tal caso scriveremo l p C , p ∈ [1, ∞] . che ne induce la norma, ovvero v 2 = (v, v), ∀v ∈ H . Nel caso complesso il prodotto scalareè per definizione sesquilineare, ovvero (λu, µv) = λµ(u, v) , (v, u) = (u, v) , ∀u, v ∈ H , λ, µ ∈ C (7.5) (Spesso in letteratura si richiede, a differenza di quanto facciamo noi, linearitá nella prima variabile ed antilinearitá nella seconda, ma chiaramente questa nonè una differenza sostanziale). Il prodotto scalare si puó ricostruire dalla norma grazie all'identitá di polarizzazione Diseguaglianza di Cauchy-Schwarz e dualitá di Riesz. Siano u, v ∈ H e λ > 0 ; allora
(u, v) = 1/4 3 k=0 i k u + i k v 2 , ∀u, v ∈ H ,0 ≤ u − λv 2 = u 2 − 2(u, λv) + λ 2 v 2 ⇒ 2(u, v) ≤ λ −1 u 2 + λ v 2 , cosicché per λ = u v −1 otteniamo (u, v) ≤ u v . Nel caso complesso troviamo, con λ ∈ C, 0 ≤ u − λv 2 = u 2 − 2Re{λ(v, u)} + |λ| 2 v 2 , cosicché, valutando per λ = (u, v) v −1 otteniamo la diseguaglianza Cauchy-Schwarz |(u, v)| ≤ u v , ∀u, v ∈ H .(u, v) = k a k b k = k f, e k b k = f, v , ∀v := k b k e k ,
per cui f = f u ed il teoremaè dimostrato.
7.2
Operatori limitati e C * -algebre.
Siano E, F spazi normati. Si dice operatore limitato, o continuo, un'applicazione lineare T : E → F tale che, per qualche c ∈ R + fissato, T v ≤ c v , ∀v ∈ E . Definendo T := inf{c ∈ R : T v ≤ c v ∀v ∈ E} = sup{ T v , v = 1} (7.13) e la struttura di spazio vettoriale
(T + λT ′ )v := T v + λT v ′ , T, T ′ ∈ B(E, F ) , v, v ′ ∈ E , λ ∈ R ,
otteniamo che l'insieme B(E, F ) degli operatori limitati da E in Fè uno spazio normato. per cui, essendo F completo, possiamo definire T v := lim n T n v ∈ F . L'applicazione T : E → F cosí definitaè chiaramente lineare, e per ogni v ∈ E 1 , ε > 0 ed n > n ε troviamo
T v ≤ T v − T n v + T n v < ε + sup n T n v < ε + sup n T n ;
poiché {T n }è di Cauchy abbiamo che c := sup n T n < ∞, per cui T v ≤ ε + c e Tè limitato. Infine abbiamo
T v − T n v = lim m T m v − T n v ≤ lim sup m,n T m − T n v m,n → 0 ,
per cui Tè limite di {T n } e concludiamo che B(E, F )è uno spazio di Banach.
Se F ′è uno spazio normato allora ha senso effettuare la composizione di operatori
T ′ • T : E → F ′ (spesso scriveremo piú brevemente T ′ T ). Visto che T ′ T v ≤ T ′ T v ≤ T ′ T v , v ∈ E , concludiamo che T ′ T ≤ T ′ T , T ∈ B(E, F ) , T ′ ∈ B(F , F ′ ) .T v ≤ T v − v n + T v n = T v − v n → 0 .T (f ⊕ λ) ∞ ≤ |λ| + f 1 ≤ 2 f ⊕ λ .
L'immagine di T coincide con lo spazio AC([0, 1]) delle funzioni assolutamente continue, il qualè e strettamente contenuto e denso in F . Dunque, T nonè chiuso.
Operatori aggiunti. Un'importante operazioneè quella che associa a T ∈ B(E, F ) l'operatore aggiunto
T * : F * → E * , f → T * f : T * f, v := f, T v , ∀v ∈ E .
L'applicazione T → T * è isometrica, e ció si dimostra tramite il Teorema di Hahn-Banach che dimostreremo nel seguito (Teo.7.18):
Proposizione 7.6. Per ogni T ∈ B(E, F ), si ha T * = T . Dimostrazione. Per ogni v ∈ E ed f ∈ E * abbiamo la stima | T * f, v | = | f, T v | ≤ f T v ,
per cui, passando al sup per v ≤ 1 prima e f ≤ 1 poi, otteniamo T * ≤ T . Per dimostrare la diseguaglianza opposta, prendiamo x ∈ E , poniamo y := T x −1 T x ∈ F 1 , e definiamo i funzionali g : Ry → R (o g : Cy → C nel caso complesso), g, λy := λ, λ ∈ R (o C). Per costruzione, i nostri funzionali hanno norma 1: g = 1 . Grazie a Teo.7.18, per ogni g esiste g ∈ F * tale che g = g = 1 e g| Ry = g (o g| Cy = g ). Inoltre, avendosi 1 = g, y , troviamo
T x = | g, T x | = | T * g, x | ≤ T * g x ≤ T * x . Per cui T ≤ T * , il che conclude la dimostrazione.
Quando abbiamo a che fare con un spazio di Hilbert H possiamo adattare la definizione di operatore aggiunto usando la dualitá di Riesz: Per calcolarne l'aggiunto, osserviamo che
(v, F u) = i 1 0 1 0 v(x)χ [0,x] (t)u(t) dtdx = (F * v, u) , ∀u, v ∈ H , cosicché F * v(t) = −i 1 0 v(x)χ [0,x] (t) dx = −i 1 t v(x) dx , ∀t ∈ [0, 1] , v ∈ H .
Ad uso futuro, osserviamo che
{F − F * }f (x) = i x 0 f (t) dt + 1 x f (t) dt = F f (1) , ∀x ∈ [0, 1] , f ∈ H .
(7.15) Esempio 7.9 (Operatori di traslazione). Consideriamo lo spazio di Hilbert H := L 2 (R, C) e, preso t ∈ R, definiamo l'operatore
U t : H → H : U t f (x) := f (x + t) , ∀x ∈ R , f ∈ H .
32 Con il termine antilineare intendiamo che (λT 1 + T 2 ) * = λT * 1 + T * 2 , λ ∈ C , T 1 , T 2 ∈ B(H) .
Chiaramente |f | 2 = |U t f | 2 per ogni f ∈ H , per cui U tè isometrico e quindi limitato. Inoltre
U t ha, ovviamente, inverso U −1 t = U −t , e (f, U t g) = f (s)g(s + t) ds = f (s ′ − t)g(s ′ ) ds ′ = (U −t f, g) , f, g ∈ H .
Comcludiamo che U tè unitario, con U * t = U −t . Osservazione 7.3 (Alcune proprietá elementari dei proiettori). Ogni proiettore P ∈ B(H) definisce il sottospazio H P := {v ∈ H : P v = v} , il qualeè di Hilbert (ovvero,è chiuso nella topologia della norma). Viceversa, ogni sottospazio di Hilbert H ′ ⊆ H con base hilbertiana {f k } definisce l'operatore
P v := k (f k , v)f k , v ∈ H ,
il qualeè un proiettore (lasciamo la verifica di questi fatti come esercizio). Essendo {f k } un insieme ortonormale per ogni proiettore P ∈ B(H) si ha, usando (7.6),
P v 2 = k |(f h , v)| 2 ≤ v 2 ⇒ P ≤ 1 ;
piú precisamente abbiamo P = 1 , visto che P f k = f k = 1 , ∀k ∈ N. Infine, si verifica facilmente che
H P ⊆ H P ′ ⇔ P ′ P = P P ′ = P , H P ∩ H P ′ = {0} ⇔ P ′ P = P P ′ = 0 . (7.16)
Quando P P ′ = P ′ P = 0è ovvio che P + P ′è un proiettore, ed il relativo sottospazioè in effetti la chiusura del sottospazio
H P + H P ′ := {u + v : u ∈ H P , v ∈ H P ′ } .|(v, u)| ≤ v − v n u + |(v n , u)| = v − v n u → 0 ,
per cui v ∈ V ⊥ e V ⊥è chiuso. D'altro canto, un ragionamento analogo mostra che V ⊥ = (V) ⊥ , dove Vè la chiusura di V . Ora, denotiamo con P il proiettore su V , cosicché w = P w , ∀w ∈ V ; preso u ∈ H definiamo v := P u e, posto v ′ := u − v , troviamo
(v ′ , w) = (u − P u, w) = (u, w) − (P u, w) = (u, P w) − (P u, w) = (u, P w) − (u, P w) = 0 , ∀w ∈ V , cosicché v ′ ∈ V ⊥ . Inoltreè ovvio che V ∩ V ⊥ = {0} , dunque se u = z + z ′ , z ∈ V , z ′ ∈ V ⊥ allora u = v + v ′ = z + z ′ ⇔ V ∋ v − z = z ′ − v ′ ∈ V ⊥ ⇔ 0 = v − z = z ′ − v ′ ⇔ v = z , v ′ = z ′ .
In conclusione, abbiamo la decomposizione ortogonale
H = V ⊕ V ⊥ , (7.19)
il che significa che ogni u ∈ H si scrive in modo unico Operatori e forme bilineari. Un'applicazione bilineare A :
come u = v + v ′ , v ∈ V , v ′ ∈ V ⊥ , e u 2 = v 2 + v ′ 2 . Lemma 7.7. Per ogni sottospazio V ⊂ H , si ha V ⊥⊥ = V . Dimostrazione. Applicando (7.19) a V ⊥ otteniamo, essendo V ⊥ chiuso, H = V ⊥ ⊕ V ⊥⊥ ,(7.H × H → R si dice limitata se esiste c > 0 tale che A(u, v) ≤ c u v , u, v ∈ H , e simmetrica se A(u, v) = A(v, u), u, v ∈ H .
Chiaramente, Aè limitata se e solo se essaè continua nella topologia prodotto su H ×H indotta dalla norma.
Ora
, fissato v ∈ H abbiamo che l'applicazione f A,v (u) := A(u, v), u ∈ H ,è in effetti un funzionale lineare limitato tale che f A,v ≤ c v , per cui il teorema di Riesz implica che esiste ed e unico T A v ∈ H tale che f A,v , u = A(u, v) = (u, T A v) .
Per bilinearitá di A troviamo che T A : H → Hè un'aplicazione lineare, mentre la limitatezza di A implica che T A ≤ c. In particolare, Aè simmetrica se e solo se T A = T * A . Viceversa, ogni operatore limitato T ∈ B(H) definisce la forma bilineare limitata A T (u, v) := (u, T v), u, v ∈ H . Analogo risultato vale nel caso complesso considerando forme sesquilineari, ovvero forme antilineari nella prima variabile e lineari nella seconda. C * -algebre. Le seguenti nozioni formalizzano in termini intrinseci alcune proprietá fondamentali degli operatori limitati.
Definizione 7.8. (1) Un' algebra di Banach (reale o complessa)è un'algebra A che soddisfa le seguenti proprietá: (i) Aè uno spazio di Banach; (ii) per ogni a, a ′ ∈ A risulta aa ′ ≤ a a ′ .
(2) Diciamo che A ha un'identitá se esiste 1 ∈ A tale che 1a = a1 = a, a ∈ A, e che Aè commutativa se aa ′ = a ′ a per ogni a, a ′ ∈ A. (3) Un'algebra di Banach complessa Aè una *-algebra di Banach se esiste un'applicazione * : A → A, a → a * antilineare, isometrica ed idempotente (ovvero a * * = a per ogni a ∈ A). (4) Una *-algebra di Banach si dice C*-algebra seè verificata l'identitá C* a * a = a 2 , a ∈ A .
Esempio 7.10. Sia E uno spazio di Banach. Allora B(E)è un'algebra di Banach, reale o complessa qualora E sia reale o complesso (vedi (7.14)).
Esempio 7.11. Sia H uno spazio di Hilbert complesso. Preso T ∈ B(H) allora (7.14) implica T * T ≤ T 2 , ma d'altra parte T u 2 = (T u, T u) = (u, T * T u) ≤ T * T u 2 , ∀u ∈ H , (7.22) per cui concludiamo che T * T = T 2 e quindi B(H)è una C*-algebra. In realtá si dimostra che per ogni C*-algebra A esiste un opportuno spazio di Hilbert complesso H con un'inclusione A ⊆ B(H); questo risultatoè noto come la costruzione di Gel'fand-Naimark-Segal ("GNS", si veda il seguente Teorema 7.9 e [19,).
Esempio 7.12. Sia X compatto. Allora C(X), equipaggiata con l'usuale moltiplicazione e norma delle'estremo superiore,è un'algebra di Banach commutativa con identitá la funzione costante 1 . Se Xè localmente compatto, allora C 0 (X)è un'algebra di Banach priva di identitá. Passando al caso complesso, definendo f * (x) := f (x) , ∀x ∈ X , f ∈ C(X, C), abbiamo che C(X, C) (per X compatto) e C 0 (X, C) (per X localmente compatto) sono C*-algebre.
Esempio 7.13. L 1 (R), equipaggiato con il prodotto di convoluzione e la norma · 1 ,è un'algebra di Banach commutativa (vedi Teo.6.23 e successive osservazioni). Analogamente L 1 (R, C)è un'algebra di Banach, la qualeè di interesse nell'ambito della trasformata di Fourier. Definiamo ora, per
ogni f ∈ L 1 (R, C), f * (t) := f (−t), t ∈ R;é immediato verificare che l'applicazione f → f * è antilineare, isometrica ed idempotente, cosicché L 1 (R, C)è una * -algebra di Banach.
Esempio 7.14. Sia (X, M, µ) uno spazio misurabile. Allora L ∞ µ (X, C) (equipaggiata con le stesse operazioni di C(X, C))è una C*-algebra commutativa, con identitá la funzione costante 1 .
Sia A una C*-algebra. Un funzionale lineare ω ∈ A * si dice positivo se ω, a * a ∈ R + per ogni a ∈ A; nel seguito denoteremo con A * + l'insieme dei funzionali positivi. E' possibile dimostrare che ogni a = a * ∈ A si scrive a = a * 1 a 1 − a * 2 a 2 , a 1 , a 2 ∈ A (vedi [20, Chap.1]), cosicché se ωè positivo allora ω, a ∈ R per ogni a = a * . Ora, per un generico a ∈ A possiamo scrivere a = a + + ia − , a + := 1/2(a + a * ) , a − := −i/2(a − a * ) e, ovviamente, a * + = a + , a * − = a − , a * = a * + − ia * − . Quindi abbiamo ω, a + , ω, a − ∈ R e ω, a * = ω, a * + − i ω, a * − = ω, a + − i ω, a − = ω, a + − i ω, a − = ω, a , da cui l'utile proprietá ω, a * = ω, a , ∀a ∈ A , ω ∈ A * + . (7.23)
Inoltre, applicando la diseguaglianza di Cauchy-Schwarz (7.9) con A(b, a) : Teorema 7.9 (Gel'fand-Naimark). Sia A una C*-algebra con identitá 1 ∈ A ed ω ∈ A * + . Allora esistono uno spazio di Hilbert complesso H ω , una rappresentazione 33 π ω : A → B(H ω ) ed u ω ∈ H ω tali che ω, a = (u ω , π ω (a)u ω ) , ∀a ∈ A .
= ω, b * a troviamo | ω, b * a | 2 ≤ ω, b * b ω, a * a , ∀a, b ∈ A .
Dimostrazione. Consideriamo l'insieme I ω := {z ∈ A : ω, z * z = 0} . Poiché ωè continuo per ogni successione convergente z n → z troviamo z * z = lim n z * n z n e quindi 0 = ω, z * z = lim n ω, z * n z n , dunque I ωè chiuso. Grazie a (7.24), per ogni a ∈ A e z ∈ I ω troviamo | ω, az | 2 ≤ ω, aa * ω, z * z = 0 , dunque az ∈ I ω ; usando tale proprietá concludiamo che, presi z ′ ∈ I ω , λ ∈ C, ω, (z + λz ′ ) * (z + λz ′ ) = 0 , cosicché I ωè anche un sottospazio chiuso e quindi un ideale sinistro di A. Consideriamo ora lo spazio quoziente
V := A/I ω := {[v] := {v + z : z ∈ I ω } : v ∈ A} ; su V definiamo la forma sesquilineare ([v], [v ′ ]) := ω, v * v ′ , v, v ′ ∈ A , (7.25) la qualeè ben posta in quanto se v = v 0 + z , v ′ = v ′ 0 + z ′ , z, z ′ ∈ I ω ,
allora, usando il fatto che I ὼ e un ideale sinistro e (7.23),
ω, v * v ′ = ω, (v * 0 + z * )(v ′ 0 + z ′ ) = ω, v * 0 v ′ 0 + ω, v * 0 z ′ + ω, v ′ 0 * z + ω, z * z ′ = ω, v * 0 v ′ 0 . D') = ω, v * a * av ≤ ω v * a * a v = ω a 2 v 2 ,
per cui π ω (a) ∈ B(H ω ). E' altrettanto ovvio che π ω (a + a ′ ) = π ω (a) + π ω (a ′ ), π ω (aa ′ ) = π ω (a)π ω (a ′ ), ∀a, a ′ ∈ A, nonché
([v], [av ′ ]) = ω, v * av ′ = ω, (a * v) * v ′ = ([a * v], [v ′ ]) ,
per cui π ω (a * ) = π ω (a) * . Ció mostra che π ωè una rappresentazione. Infine, ponendo u ω :
= [1] ∈ H ω troviamo (u ω , π ω (a)u ω ) = ([1], [a1]) = ω, a ,
e ció conclude la dimostrazione.
Il risultato precedenteè un passo fondamentale della giá menzionata costruzione GNS, la quale permette di concludere che ogni C*-algebra ammette una rappresentazione iniettiva. Tale proprietá puó essere dedotta direttamente dal teorema precedente solo nel caso in cui ω sia fedele (ovvero ω, a * a = 0 implica a = 0 ): infatti, se a = 0 allora {π ω (a)}u ω = [a1] = [a] = ω, a * a 1/2 = 0 , per cui π ω (a) non puó essere l'operatore nullo. In generale, per ottenere la rappresentazione iniettiva desiderata occorre passare ad un'opportuna somma diretta di spazi di Hilbert del tipo H ω (vedi [19,Ex.4.3.17]). Quando Aè commutativa il teorema precedente si puó interpretare in termini di misure di Radon (vedi Esempio 7.3 e l'Esercizio 7.8).
Uniforme limitatezza ed applicazioni aperte.
Lemma 7.10 (Lemma di Baire). Sia X uno spazio metrico completo ed {X n } una successione di chiusi tali cheẊ n = ∅ , n ∈ N (ovvero, ogni X nè rado). Allora [∪ n X n ] · = ∅ (L'unione numerabile di insiemi radiè un insieme rado).
Dimostrazione. L'affermazione da dimostrare equivale a verificare che presa la successione di aperti A n := X − X n , n ∈ N, con A n = X , risulta ∩ n A n = X . Consideriamo allora un aperto U ⊂ X .
Il nostro compitoè dimostrare che U ∩ (∩ n A n )è non vuoto. A tale scopo, scegliamo x 0 ∈ U , r 0 > 0 tali che ∆(x 0 , r 0 ) ⊂ U . Consideriamo quindi, induttivamente (x n+1 , r n+1 ) : x n+1 ∈ ∆(x n , r n ) ∩ A n ∩ U , n ∈ N
(notare che la precedente definizioneè ben posta proprio perché ogni A nè denso). Si verifica facilmente che {x n }è di Cauchy, per cui esiste edè unico il limite x. Per costruzione x ∈ U ∩ (∩ n A n ).
Teorema 7.11 (Teorema di Banach-Steinhaus). Siano E, F spazi di Banach e {T i } i∈I ⊂ B(E, F ) una famiglia tale che sup i T i v < ∞ per ogni v ∈ E . Allora esiste una costante c > 0 tale che T i < c, i ∈ I .
Dimostrazione. Poniamo
X n := {v ∈ E : T i v ≤ n , i ∈ I} , n ∈ N .
Poiché per ipotesi sup i T i v < ∞, abbiamo che ∪ n X n = E . Il lemma di Baire implica che deve esistere n 0 ∈ N tale cheẊ n = ∅ . Per cui, esistono v 0 ∈ E , r 0 > 0 tali che ∆(v 0 , r 0 ) ⊂ X n0 . Quindi otteniamo, per ogni w ∈ E , w ≤ 1 , A corollario dei risultati precedenti segnaliamo la seguente terminologia: dato uno spazio topologico X , un sottoinsieme Y ⊆ X si dice di tipo G δ seè intersezione numerabile di insiemi aperti. Osserviamo che l'intersezione di insiemi G δè G δ , e l'unione finita di insiemi G δè G δ . Inoltre, se Xè metrico, allora ogni chiusoè di tipo G δ . Come esempio, segnaliamo X = R, Y = R − Q .
T i (v 0 + r 0 w) ≤ n 0 ⇒ r 0 T i w ≤ n 0 + T i v 0 < 2n 0 .
A seguire alcune notazioni. Per ogni δ, r > 0 scriviamo E ≤r := {v ∈ E : v ≤ r} . Inoltre definiamo δE ≤r := {δw : w ∈ E ≤r } = E ≤δr e v + E ≤r := {v + w : w ∈ E ≤r } , cosicché se T ∈ B(E, F ) allora
T (v + E ≤r ) = T v + T (E ≤r ) := {T v + T v ′ , v ′ ∈ E ≤r } . Lemma 7.14. Siano E, F spazi di Banach e T ∈ B(E, F ) tale che T (E ≤1 )è denso in qualche F ≤r , r > 0 . Allora per ogni ε > 0 risulta F (1−ε)r ⊂ T (E ≤1 ).
Dimostrazione. Presi w ∈ F ≤r ed ε ∈ (0, 1), per ipotesi esiste w 1 ∈ T (E ≤1 ) tale che w − w 1 < εr . Ora, sempre per ipotesi, abbiamo che εT (E ≤1 )é denso in F ≤εr , per cui troviamo che esiste
w 2 ∈ εT (E ≤1 ) tale che w − w 1 − w 2 < ε 2 r . Procedendo per induzione otteniamo una successione {w n ∈ ε n−1 T (E ≤1 )} tale che w − n k w k < ε n r .
Scegliamo ora v n ∈ E ≤1 tali che w n = ε n−1 T v n , ∀n ∈ N. Allora la serie n ε n−1 v nè assolutamente convergente, e quindi convergente a v ∈ E . Per costruzione T v = n ε n−1 T v n = n w n = w e v ≤ n ε n−1 ≤ (1 − ε) −1 , e ció conclude la dimostrazione. Dimostrazione. Per l'ipotesi di suriettivitá l'insieme ∪ n∈N T (E ≤n )è denso in F , e per il Lemma di Baire esiste almeno un m ∈ N tale che T (E ≤m ) contiene un intorno del tipo
w + F ≤ε , w ∈ F , ε > 0 , cosicché T (E ≤1 ) ⊇ m −1 w+F εm −1 . Del resto, se w 0 ∈ F ≤εm −1 allora m −1 w+w 0 ∈ m −1 w+F ≤εm −1 e quindi troviamo successioni {v k }, {v ′ h } ⊂ E ≤1 tali che m −1 w +w 0 = lim k T v k , m −1 w = lim h T v ′ h ; per cui 1/2w 0 = 1/2 lim k T (v k − v ′ k ) , con 1/2(v k − v ′ k ) ∈ E ≤1 ∀k ∈ N . Dunque T (E ≤1 )è denso in F 1/2εm −1 .
Usando il Lemma precedente, e la linearitá di T , concludiamo che per ogni v ∈ E e ρ > 0 esiste un δ > 0 tale che
T v + F ≤δ ⊂ T (v + E ≤ρ ) . (7.26) Ora, preso un aperto U ⊂ E e v ∈ U troviamo v ∈ v + E ≤ρ ⊂ U per qualche ρ > 0 , e da (7.26) concludiamo che T (v + E ≤ρ ) contiene l'intorno T v + F ≤δ di T v .
In altre parole Tè un'applicazione aperta ed il teoremaè dimostrato.
Corollario 7.16 (Teorema dell'inverso continuo). Sia T ∈ B(E, F ) biettivo. Allora T −1 ∈ B(F , E).
Dimostrazione. Il fatto che T −1è lineare segue dalla linearitá di T , mentre la continuitá (ovvero limitatezza) di T −1 segue dal fatto che Tè un'applicazione aperta.
Il teorema di Hahn-Banach.
Il teorema di Hahn-Banachè uno dei risultati cardine dell'analisi funzionale. Oggetto del teoremà e l'esistenza di funzionali lineari limitati su spazi vettoriali equipaggiati con seminorme.
f (v ′ ) ≤ p(v ′ ) , ∀v ′ ∈ E ′ , (7.27) allora esistef : E → R tale chef | E ′ = f ,f (v) ≤ p(v), ∀v ∈ E .
La diseguaglianza (7.27) puó essere riguardata come una condizione di limitatezza per f , ed il teorema di Hahn-Banach ci assicura che l'estensionef soddisfa la medesima minorazione di f rispetto a p. Il caso che abbiamo in menteè quello in cui Eè uno spazio normato ed f ∈ E ′ * , con
f := sup v ′ ∈E ′ 1 | f, v ′ | , p(v) := f v , v ∈ E .
Dimostrazione. Si tratta di applicare il Lemma di Zorn al seguente insieme parzialmente ordinato:
P := {h : E h → R : E ′ ⊆ E h ⊆ E , h| E ′ = f , h(v) ≤ p(v) , v ∈ E h } equipaggiato con la relazione d'ordine h ≤ h ′ ⇔ E h ⊂ E h ′ e h ′ | E h = h .
Sia C ⊂ P totalmente ordinato; allora, esiste un elemento massimale k per C , definito da
E k := h∈C E h , k(v) := h(v) , v ∈ E h .
Dunque, per il Lemma di Zorn otteniamo che P ammette un elemento massimale, che denotiamo conf . Resta da verificare chefè il funzionale che stiamo cercando. Innanzitutto, osserviamo che se il dominio E f dif non coincidesse con E allora potremmo considerare v 0 ∈ E − E f , il sottospazio
proprio E 0 := E f + Rv 0 di E , e definire f ′ (v + λv 0 ) :=f (v) + λδ , v ∈ E f , λ ∈ R ,
dove δ ∈ Rè una costante che sceglieremo in modo tale che
f ′ (v + λv 0 ) =f (v) + λδ ≤ p(v + λv 0 ) ;
ció sarebbe in contraddizione con la massimalitá dif ed il teorema sarebbe dimostrato. Del resto, usando (7.27), si verifica che basta scegliere
sup v∈E ′ {f (v) − p(v − v 0 )} ≤ δ ≤ inf v∈E ′ {p(v + v 0 ) − f (v)} .
Corollario 7.19. Sia E uno spazio normato, ed E ′ ⊆ E un sottospazio vettoriale. Allora per ogni
f ∈ E ′ * esistef ∈ E * conf | E ′ = f e f = f . Corollario 7.20. Per ogni v ∈ E si ha v = max f ∈E * 1 | f, v | . (7.28) Dimostrazione. Assumiamo v = 0 . E' ovvio che v ≥ | f, v |, f ∈ E * 1 .
D'altra parte, considerando il sottospazio E ′ := Rv ed il funzionale f ∈ E ′ * , f, λv := λ v 2 , λ ∈ R, ed applicando il teorema di Hahn-Banach, otteniamo l'uguaglianza cercata.
Operatori compatti ed il Teorema di Fredholm.
Siano E, F spazi di Banach (reali o complessi). Un operatore limitato T ∈ B(E, F ) si dice compatto se T (E ≤1 )è precompatto nella topologia della norma di F (dove E ≤1è la palla unitaria di E ). Denotiamo con K(E, F ) l'insieme degli operatori compatti da E in F . Dimostrazione. L'immagine di un insieme compatto attraverso un'applicazione continuaè compatta, per cui se S ′ ∈ B(F ), S ∈ B(E), T ∈ K(E, F ) allora applicando il Lemma precedente concludiamo che S(E ≤1 )è limitato (per continuitá di S ), T • S(E ≤1 ) precompatto (per compattezza di T ) ed S ′ • T • S(E ≤1 ) precompatto (per continuitá di S ′ ). Ció mostra che K(E, F )è chiuso rispetto a composizioni con elementi di B(F ), B(E). Inoltre, la sfera unitaria di uno spazio a dimensione finitaè compatta e ció mostra che ogni operatore T a rango finito (ovvero, tale che T (E) ha dimensione finita)è compatto. Mostriamo che K(E, F )è uno spazio vettoriale: se T, T ′ ∈ K(E, F ) ed A ⊂ Eè limitato allora T (A) × T ′ (A)è compatto (essendo il prodotto di compatti) e l'applicazione Esempio 7.15. Sia X uno spazio metrico compatto, K ∈ C(X × X) e µ una misura di Radon su X . Definiamo l'operatore
Lemma 7.21. Sia T ∈ K(E, F ) compatto. Allora T (A)è compatto per ogni A ⊂ E limitato. Dimostrazione. Essendo A limitato essoè contenuto in una palla E ≤r := {v ∈ E : v ≤ r} per qualche r ∈ R. Ora, presa una successione {T v n } ⊂ T (E ≤r ) troviamo che {r −1 T v n } ∈ T (E ≤1 ),T (A) × T ′ (A) → F , (T v, T ′ v ′ ) → T v + T ′ v ′T v (n) n − T v (m) m ≤ T v (n) n − T i v (n) n + T i v (n) n − T i v (m) m + T i v (m) m − T v (T K : L 1 µ (X) → C(X) , T K f (y) := X K(x, y)f (x) dx , y ∈ X .
Verifichiamo che T K sia ben definito: essendo K uniformemente continua (Heine-Cantor), scelto ε > 0 troviamo δ > 0 tale che |K(x, y) − K(x, y ′ )| < ε per ogni x, y, y ′ tali che d 2 ((x, y), (x, y ′ )) = d(y, y ′ ) < δ (qui, e di seguito, usiamo la notazione dell'Esercizio 2.4). Poiché δè indipendente da x, y, y ′ ∈ X concludiamo che
|T K f (y) − T K f (y ′ )| < ε f 1 , (7.29) dunque T K f ∈ C(X). Inoltre |T K f (y)| ≤ κ y ∞ f 1 ≤ K ∞ f 1 , dunque T Kè limitato e T K ≤ K ∞ . Infine, se f 1 ≤ 1 allora usando (7.29) troviamo |T K f (y) − T K f (y ′ )| < ε per d(y, y ′ ) < δ ; poiché il nostro δ non dipende dalla scelta di f concludiamo che la famiglia {T K f } f 1≤1è equicontinua e quindi,
per Ascoli-Arzelá, precompatta. Dunque T Kè un operatore compatto. Consideriamo ora, per ogni p ∈ [1, ∞), l'operatore
I p : C(X) → L p µ (X)
che associa ad f ∈ C(X) la sua classe in L p µ (X); allora I p f p ≤ f ∞ µX 1/p e quindi I pè limitato. Concludiamo dalla proposizione precedente che I p • T K : L 1 µ (X) → L p µ (X)è compatto. Esempio 7.16. Sia (X, M, µ) uno spazio misurabile, h i ∈ L r µ (X), g i ∈ L q µ (X), i = 1, . . . , n ∈ N, q := p. Posto K(x, y) := n i h i ⊗ g i (x, y) = i h i (x)g i (y), x, y ∈ X , abbiamo, con la notazione dell'Esempio 7.18,
T K : L p µ (X) → L r µ (X) , T K f := X K(x, y)f (y) dy = i h i · X f g i .
Dunque T K ha rango finito e quindiè compatto. Riguardo la norma di T K , abbiamo l'ovvia maggiorazione Ad esempio, se T = T * ∈ K(H), allora per il teorema di Fredholm 1−Tè un operatore di Fredholm con indice nullo. Denotiamo con F (H) l'insieme degli operatori di Fredholm su H , il quale diventa naturalmente uno spazio topologico se equipaggiato con la topologia della norma. Per dare una vaga idea di come la geometria sia coinvolta nella nozione di indice segnaliamo il teorema di Atiyah-Janich ([1, Appendix]), il quale afferma che, dato lo spazio compatto di Hausdorff X , l'insieme delle classi di omotopia di funzioni continue f : X → F (H)è isomorfo al gruppo K 0 (X) della K-teoria di X . Quest'ultimoè un importante invariante topologico di X , nonché la nozione fondamentale su cui poggia il famoso teorema di Atiyah-Singer.
T K f r ≤ i h i r g i q · f p .T ∈ B(E, F ); presi v ∈ E ed f ∈ F * si ha l'identitá f, T v = T * f, v , per cui f ∈ ker T * se e soltanto se f ∈ {T (E)} ⊥ , ovvero {T (E)} ⊥ = ker T * ( 7.30) ⇔ T (E) = {ker T * } ⊥ .∃w 0 = lim k T (v n k − z n k ) = lim k T v n k ⇒ v n k = (v n k − T v n k ) + T v n k k → w + w 0 , alché w = {1 − T } lim k v n k = {1 − T }(w + w 0 ) ,= d −1 k (v k − z k ) ⊂ E 1 avremmo s k − T s k = d −1 k (v k − T v k ) k → 0 · w =d(s i k , ker(1 − T )) = d −1 k d(v k − z k , ker(1 − T )) = d −1 k d(v k , ker(1 − T )) = 1 ,∈ E k } tale che u k ≡ 1 e d(u k , E k+1 ) ≥ 1/2 , per cui, per ogni h < k , T u h − T u k = u h − {(1 − T )u h − (1 − T )u k + u k } ≥ d(u h , E h+1 ) ≥ 1/2 .P K u ∈ K tale che u − P K u = min K u − v . E ció avviene se e solo se (u − P K u, v − P K u) ≤ 0 per ogni v ∈ K .1. Esiste edè unico u 0 ∈ K tale che A(u 0 , v − u 0 ) ≥ ϕ, v − u 0 , v ∈ K ;
2. Se Aè simmetrica, allora u 0 ∈ K soddisfa la condizione precedente se e soltanto se
1 2 A(u 0 , u 0 ) − ϕ, u 0 = min v∈K 1 2 A(v, v) − ϕ, v . (7.34) Dimostrazione. Innanzitutto osserviamo che grazie al teorema di Riesz esistono unici T ∈ B(H), f ∈ H tali che A(u, v) = (T u, v), ϕ, v = (f, v), ∀v ∈ H . Per cui il nostro compitoè quello di trovare u 0 ∈ H tale che (T u 0 − f, v − u 0 ) ≥ 0 , v ∈ K .
L'equazione precedenteè equivalente a richiedere che preso un δ > 0 si abbia
(δf − δT u 0 + u 0 − u 0 , v − u 0 ) ≤ 0 , v ∈ K ,
il che -ricordando la definizione di proiezione P K su un sottoinsieme chiuso K di uno spazio euclideo -si puó leggere come il fatto che
u 0 = P K (δf − δT u 0 + u 0 ) . (7.35)
Per cui, adesso l'ideaè quella di trovare δ affinché sia verificata (7.35), usando il teorema delle contrazioni. A tale scopo, definiamo
S δ : K → K , v → P K (δf − δT v + v)
osservando che (7.35) equivale a richiedere che u 0 sia un punto fisso per S δ . A questo punto, stimiamo
S δ v − S δ v ′ 2 ≤ v − v ′ 2 − 2δ(v − v ′ , T (v − v ′ )) + δ 2 T (v − v ′ ) ≤ v − v ′ 2 (1 − 2αδ + c 2 δ 2 ) .
Per cui, affinché S δ sia una contrazione (ed abbia quindi un punto fisso),è sufficiente che sia verificata la disequazione c 2 δ 2 − 2αδ < 0 , la quale ammette certamente soluzioni δ > 0 . Ció dimostra il Punto 1 dell'enunciato del teorema. Infine, se Aè simmetrica allora A(·, ·)è un prodotto scalare su H e per il teorema di Riesz esiste edè unica g ∈ H tale che Inoltre, se Aè simmetrica, u 0 soddisfa (7.37) se e soltanto se
ϕ, v = A(g, v) , v ∈ H . (7.36) Il Punto 1 dell'enunciatoè equivalente a A(g − u 0 , v − u 0 ) ≤ 0 , v ∈ H ⇔ u 0 = P K g , il cheè equivalente a minimizzare, al variare di v in K , la quantitá A(g − v, g − v) o, equivalente- mente, minimizzare A(v, v) − 2A(g, v).1 2 A(u 0 , u 0 ) − ϕ, u 0 = min v∈H 1 2 A(v, v) − ϕ, v . (7.38)
Dimostrazione. Da (7.35) otteniamo (essendo K = H ) f = T u 0 , da cui segue (7.37).
Osservazione 7.7. Applicando il teorema di Riesz abbiamo l'operatore autoaggiunto T ∈ B(H) e g ∈ H in luogo della forma A e di ϕ ∈ H * , dunque possiamo esprimere il risultato precedente come segue: se Tè tale che la relativa forma bilineareè coercitiva, allora essoè biettivo (infatti, la condizione (T u 0 , v) = (g, v), ∀v ∈ H ,è equivalente a dire che T u 0 = g ).
Teoria spettrale.
Gli operatori limitati, in particolare quelli compatti, appaiono frequentemente in analisi sotto la forma di operatori integrali, per cuiè importante conoscerne le proprietá spettrali, che si traducono in termini di soluzioni di problemi integro-differenziali.
Spettro e risolvente. Per ogni algebra di Banach (reale o complessa) A con identitá 1 denotiamo con A −1 il gruppo degli elementi invertibili di A, ovvero di quei T ∈ A tali che esiste T −1 ∈ A con T T −1 = T −1 T = 1 . I seguenti due risultati si applicano (chiaramente) al caso particolare in cui A = B(E) per qualche spazio di Banach E .
Lemma 7.29. A −1è aperto nella topologia della norma.
Dimostrazione. Sia T ∈ A −1 . Mostriamo che esiste δ > 0 tale che ogni T ′ ∈ ∆(T, δ)è invertibile. Definiamo S := 1 − T −1 T ′ ed osserviamo che scegliendo δ < T −1 −1 otteniamo S < 1 . Ora, la serie k=0 S k (per convenzione poniamo S 0 := 1 )è assolutamente convergente e quindi convergente ad A ∈ A.
Inoltre (1 − S)A = lim n (1 − S) n k=0 S k = lim n (1 − S n ) = 1 , per cui A = (1 − S) −1 = (T −1 T ′ ) −1 . Ne segue che T ′ ha inverso AT −1 .
Definizione 7.30. Sia A un'algebra di Banach (reale o complessa) con identitá 1 , e T ∈ A. Il risolvente di Tè dato dall'insieme
ρ(T ) := {λ ∈ K : T − λ1 ∈ A −1 } , K = R, C .
Lo spettro di Tè dato da σ(T ) := K − ρ(T ). Supponiamo ora che A = B(E) per qualche spazio di Banach E ; diciamo che λ ∈ Kè un autovalore di T se esiste un autovettore v ∈ E , ovvero T v = λv ; l'insieme degli autovalori di T si denota con σp(T ). Dimostrazione. Innanzitutto dimostriamo che σ(T ) ⊆ ∆(0, T ); assumendo che T > 0 , prendiamo λ tale che |λ| > T (cosicché λ = 0 ) ed osserviamo che λ −1 T < 1 . Ragionando come nel Lemma precedente abbiamo che la serie n=0 λ −n T nè assolutamente convergente, e quindi convergente ad
A ∈ A tale che (1 − λ −1 T )A = A(1 − λ −1 T ) = 1 . Dunque 1 − λ −1 T (e quindi T − λ1 )è invertibile. Verifichiamo ora che σ(T )è chiuso. Supponiamo per assurdo che esista una successione {λ n } ⊂ σ(T ) tale che λ n → λ ∈ ρ(T ). Cio' vuol dire che T − λ1è invertibile, ma del resto (T − λ n 1) − (T − λ1) = |λ − λ n | n → 0 .
Poiché A −1è aperto nella topologia della norma (Lemma precedente) otteniamo una contraddizione, e ció dimostra la proposizione.
Osservazione 7.8. Sia A una * -algebra di Banach con identitá 1 e T ∈ A. Preso λ ∈ C abbiamo che T − λ1 ha inverso B ∈ A se e solo se T * − λ1 ha inverso B * . Dunque ρ(T * ) = ρ(T ) e σ(T * ) = σ(T ). In particolare, se Hè uno spazio di Hilbert e T = T * ∈ B(H) allora T ha spettro reale σ(T ) ⊂ R.
Il risultato seguente permette di ottenere informazioni sullo spettro di un operatore autoaggiunto su uno spazio di Hilbert: Proposizione 7.32 (Il principio del minimax). Sia T = T * ∈ B(H). Posto
λ + := sup u =1 (u, T u) , λ − := inf u =1 (u, T u) , si ha σ(T ) ⊆ [λ − , λ + ] e λ ± ∈ σ(T ).
Dimostrazione. Abbiamo (u, T u) ≤ λ + u 2 per ogni u ∈ H , per cui se λ > λ + allora esiste ε > 0 tale che
(u, (λ1 − T )u) ≥ λ u 2 − λ + u 2 > ε u 2 .
Dunque la forma definita da λ1 − Tè coercitiva e per il teorema di Lax-Milgram λ1 − Tè biettivo, il che vuol dire che λ / ∈ σ(T ). Passiamo ora al secondo enunciato: per mostrare che λ + ∈ σ(T ) mostreremo che λ + 1 − T non puó essere invertibile. A tale scopo osserviamo che
u, v → (u, (λ + 1 − T )v) , u, v ∈ H ,
e una forma sesquilineare simmetrica e definita positiva, per cui la diseguaglianza di Cauchy-Schwarz (7.9) implica che
|(u, (λ + 1 − T )v)| ≤ (u, λ + u − T u) 1/2 (v, λ + v − T v) 1/2 , ∀u, v ∈ H .
Passando al sup al variare di u ∈ H , u = 1 , troviamo la stima
(λ + 1 − T )v ≤ c(v, λ + v − T v) 1/2 , ∀v ∈ H , (7.40) dove c := sup u =1 (u, λ + u − T u) 1/2 . Ora, per definizione di λ + esiste una successione {v n } tale che v n ≡ 1 e (v n , λ + v − T v n ) → 0 , dunque (λ + 1 − T )v n ( 7.40) ≤ c(v n , λ + v n − T v n ) 1/2 → 0 , e se λ + 1 − T avesse inverso B ∈ B(H) troveremmo v n = B(λ + 1 − T )v n → 0 , il che contraddice la condizione v n ≡ 1 . Concludiamo che λ + ∈ σ(T )
. Ripetendo il ragionamento per λ − (con diseguaglianze invertite) otteniamo quanto desiderato. Dimostrazione. Per la proposizione precedente deve essere (u, T u) = 0 per ogni u ∈ H , e quindi 2(u, Allora Tè compatto (infattiè limite degli operatori di rango finito T n x := (0.x 1 , . . . , x n /n, 0, . . .), x ∈ l 2 C ) ma non autoaggiunto (si calcoli infatti T * e si verifichi che T * = T ). Inoltre abbiamo {T x − λx} n+1 = x n /n − λx n+1 , ∀n ∈ N, per cui T − λ1è invertibile per ogni λ = 0 36 , mentre evidentemente T nonè suriettivo. Concludiamo quindi che σ(T ) = {0} , in contrasto con (7.41).
T v) = (u + v, T (u + v)) − (u, T u) − (v, T v) = 0 , ∀u, v ∈ H . Dunque T = 0 .
Esempio 7.18. Sia X uno spazio compatto e di Hausdorff, µ una misura di Radon su X e K ∈ C(X × X); definiamo l'operatore lineare
T K : C(X) → C(X) , T K f (y) := X K(x, y)f (x) dx , y ∈ X .
Osserviamo che T Kè limitato in quanto T K f ∞ ≤ K ∞ µX f ∞ . Preso λ ∈ R, troviamo che essoè un autovalore di T K se e solo se esiste u ∈ C(X) soluzione del problema λu(y) = X K(x, y)u(x) dx , y ∈ X , (7.42) noto come equazione di Volterra omogenea di seconda specie. 36 Lasciamo a chi legge questa verifica, come semplice esercizio di algebra lineare.
Poiché T u 2 2 = x 2 u(x) 2 dx ≤ u(x) 2 dx troviamo subito T ≤ 1 37 . Valutiamo gli operatori T − λ1 al variare di λ ∈ R; per prima cosa, osserviamo che, preso u ∈ L 2 ,
{T − λ1}u(x) = 0 q.o. ⇒ (x − λ)u(x) = 0 q.o. ⇒ u(x) = 0 q.o. , per cui ogni T − λ1è iniettivo al variare di λ ∈ R. Ora,è conveniente definire la funzione f λ (x) := (x − λ) −1 , x ∈ [0, 1], ed osservare che se λ ∈ R − [0, 1] allora l'operatore S λ ∈ BL 2 : S λ u(x) := f λ (x)u(x) , x ∈ [0, 1] , u ∈ L 2 , e limitato e tale che (T − λ1)S λ = S λ (T − λ1) = 1 , dunque R − [0, 1] ⊆ ρ(T ). D'altro canto, se λ ∈ [0, 1] allora definendo u 0 ∈ L 2 , u 0 (x) := 1 , x ∈ [0, 1], e supponendo u 0 = (T − λ1)v per qualche v ∈ L 2 , troviamo la contraddizione v 2 2 = f λ u 0 2 2 = 1 0 dx (x − λ) 2 = ∞ . Dunque T − λ1 nonè suriettivo e λ ∈ σ(T ). Concludiamo che σ(T ) = [0, 1], σp(T ) = ∅ .
Il teorema spettrale per gli operatori compatti. Passiamo ora a studiare lo spettro di un operatore compatto.
v n+1 = n i a i v i , avremmo T v n+1 = n i a i λ i v i = λ n+1 n i a i v i ⇒ a i (λ i − λ n+1 ) = 0 ∀i = 1, . . . , n .
Poiché λ n+1 = λ i per ogni i otteniamo una contraddizione, e dunque v 1 , . . . , v n+1 sono linearmente indipendenti. Ció implica che abbiamo inclusioni proprie V n ⊂ V n+1 per ogni n ∈ N. D'altra parte, per costruzione,
(T − λ n 1)V n ⊂ V n−1 .
Usando il Lemma di Riesz (Esercizio 7.4(1)) troviamo che esiste una successione {u n } ⊂ E 1 tale che u n ∈ V n e d(u n , V n−1 ) ≥ 1/2 . Ora, se m < n − 1 abbiamo V m ⊂ V n−1 e
λ −1 n T u n − λ −1 m T u m = u n − v ≥ 1/2 (7.43) dove v := u m − λ −1 n (T − λ n 1)u n + λ −1 m (T − λ m 1)u m ∈ V n−1 .
Se, per assurdo, λ = lim n λ n fosse non nullo troveremmo che la successione{λ −1 n T u n } sarebbe limitata; per compattezza di T potremmo estrarne una sottosuccessione convergente, e ció contraddice (7.43). Il teorema spettrale per gli operatori autoaggiunti. Esiste una versione del teorema precedente per il caso degli operatori autoaggiunti limitati, la quale si puó esprimere con il linguaggio della teoria della misura. Allo scopo di facilitarne la comprensione gettiamo uno sguardo un pó diverso al caso compatto: preso T = T * ∈ K(H), osserviamo che lo spettro σ(T ) = {λ k } , essendo numerabile, puó essere riguardato in modo naturale come uno spazio di misura (σ(T ), M, µ) con σalgebra M := 2 σ(T ) (vedi Esempio 4.2). Preso λ k ∈ σ(T ) consideriamo il proiettore P (λ k ) ∈ B(H) sul sottospazio di Hilbert ker(T − λ k 1) cosicché, grazie al Teorema 7.36, abbiamo la decomposizione
ortogonale v = k P (λ k )v ⇒ v 2 = k P (λ k )v 2 , ∀v ∈ H .
Per ogni u, v ∈ H definiamo
µ uv : M → C , µ uv E := λ k ∈E (u, P (λ k )v) ;
è immediato verificare che µ uvè una misura complessa su σ(T ) tale che, in particolare,
µ uv {λ k } = (u, P (λ k )v) , ∀λ k ∈ σ(T ) .
Presa una funzione limitata f : σ(T ) → C (che ovviamenteè µ-misurabile), abbiamo che la forma sesquilineare
A f (u, v) := k f (λ k )µ uv {λ k } , u, v ∈ H , e limitata, in quanto |A f (u, v)| 2 ≤ k |(u, f (λ k )P (λ k )v)| 2 ≤ u 2 k |f (λ k )| 2 P (λ k )v 2 ≤ u 2 f 2 ∞ v 2 , per cui esiste edè unico l'operatore f (T ) ∈ B(H) tale che A f (u, v) = (u, f (T )v). In particolare, poiché T P (λ k )v = λ k P (λ k )v , abbiamo la seguente "diagonalizzazione" di T : (u, T v) = k (u, T P (λ k )v) = k λ k µ uv {λ k } , ∀u, v ∈ H . (7.44)
L'idea alla base dei ragionamenti precedentiè quella che, dati lo spettro di T e la famiglia di misure {µ uv } , riusciamo a ricostruire non solo T , bensí anche ogni operatore del tipo f (T ), dove fè una qualsiasi funzione limitata sullo spettro di T .
Consideriamo ora operatori T ∈ B(H) non necessariamente compatti. Chiaramente ora noǹ e piú detto che lo spettro σ(T ) sia numerabile, né che H ammetta una base di autovettori (vedi Esempio 7.19). Tuttavia si puó ancora dimostrare un analogo di (7.44), e qui di seguito ne esponiamo in buon dettaglio la dimostrazione.
Iniziamo assumendo che H sia uno spazio di Hilbert complesso. Ció sia perché in questo ambito le argomentazioni seguenti sono valide anche per operatori piú generici di quelli autoaggiunti 38 , sia perché in alcuni passi sará cruciale usare tecniche di analisi complessa.
Come primo passo denotiamo con P (σ(T ), C) ⊂ C(σ(T ), C) la * -algebra delle funzioni polinomiali f (λ) := n k z k λ k , z k ∈ C, λ ∈ σ(T ) (per k = 0 poniamo λ 0 = 1 ), e definiamo l'applicazione ∀f, g ∈ P (σ(T ), C), z ∈ C. Il teorema di Stone-Weierstrass (Teo.2.7) ci dice che P (σ(T ), C)è denso in C(σ(T ), C), edè possibile dimostrare che 39
f ∞ = f (T ) , ∀f ∈ P (σ(T ), C) .
Dunque estendendo per continuitá otteniamo un operatore isometrico
C(σ(T ), C) → B(H) , f → f (T ) ,(7.47)
noto come il calcolo funzionale continuo di T ; osserviamo che (7.47)è in realtá una rappresentazione, il che vuol dire che continuano a valere le (7.46) per generici elementi di C(σ(T ), C). Ora, per ogni u, v ∈ H definiamo il funzionale lineare
ϕ uv : C(σ(T ), C) → C , ϕ uv , f := (u, f (T )v) , ∀f ∈ C(σ(T ), C) , il qualeè limitato in quanto | ϕ uv , f | ≤ f (T ) u v = f ∞ u v . (7.48)
Applicando il teorema di Riesz-Markov (Esempio 7.3) troviamo che esiste edè unica la misura di Radon complessa µ uv : M uv → C su σ(T ) tale che
ϕ uv , f = (u, f (T )v) = σ(T ) f (λ) dµ uv (λ) , ∀f ∈ C(σ(T ), C) . (7.49)
Osserviamo che in particolare µ uuè una misura di Radon reale e positiva per ogni u ∈ H . Abbiamo dunque un'applicazione H × H → R(σ(T ), C) , u, v → µ uv , (7.50) la qualeè chiaramente sesquilineare, ovvero
µ au+u ′ ,bv+v ′ = abµ uv + aµ uv ′ + bµ u ′ v + µ u ′ v ′ . (7.51)
Ora, le uguaglianze (7.49) ci dicono che l'applicazione u, v → f dµ uv , u, v ∈ H ,è in realtá una forma sesquilineare limitata, e che f (T ) ∈ B(H)è in effetti l'operatore ad essa associato. In particolare, visto che T = I(T ), troviamo
(u, T v) = σ(T ) λ dµ uv (λ) , ∀u, v ∈ H ,(7.52)
ovvero un analogo non numerabile di (7.44). Diciamo che µ := {µ uv }è una misura spettrale di T . Possiamo ora estendere il calcolo funzionale continuo, nel modo che segue. Denotiamo con L ∞ β (σ(T )) la C*-algebra delle funzioni a valori complessi, boreliane e limitate su σ(T ) ⊂ R. Visto che ogni µ uvè una misura boreliana troviamo che L ∞ β (σ(T )) ⊆ L ∞ µuv (σ(T ), C), ∀u, v ∈ H . Per cui, per ogni f ∈ L ∞ β (σ(T ))è ben definita l'applicazione
H × H → C , u, v → σ(T )
f dµ uv , (7.53) la quale, grazie a (7.51) ed all'Esercizio 4.8,è sequilineare. Inoltre essaè limitata: per verificare ció e sufficiente effettuare una stima nel caso u = v ,
σ(T ) f dµ uu ≤ f ∞ σ(T ) dµ uu ≤ f ∞ u 2 ,
e trattare il caso generale usando l'identitá di polarizzazione µ uv = 1/4
3 k=0 i k µ u+i k v,u+i k v .
Denotando con f (T ) ∈ B(H) l'operatore associato a (7.53) otteniamo un'estensione del calcolo funzionale (7.47),
L ∞ β (σ(T )) → B(H) , f → f (T ) ,(7.µ f g E = E f g , ∀E ∈ M , cosicché (f, T g) = [0,1] f (s)sg(s) ds = [0,1] s · f (s)g(s) ds = [0,1] s dµ f g (s) ,
e µ := {µ f g }è la misura spettrale di T (Osservare che µ{λ} = 0 , ∀λ ∈ σ(T )). Chiaramente, l'analogo argomento vale nel caso di uno spazio di Hilbert reale (tuttavia non sempre ci si trova in una situazione cosí comoda).
7.8
Topologie deboli e spettri di algebre di Banach.
Viene spesso richiesto che successioni in spazi funzionali convergano (si pensi ad esempio alla successione di Peano-Picard). Purtroppo la topologia della norma di uno spazio di Banach risulta essere troppo ricca di aperti per avere buone proprietá rispetto alla compattezza, per cui in generaleè impossibile costruire successioni che ammettano sottosuccessioni convergenti. E' quindi conveniente introdurre topologie meno dotate di aperti rispetto a quella della norma, in modo da migliorare le proprietá inerenti la convergenza.
La topologia debole. Sia E uno spazio di Banach. La topologia debole σ(E, E * ), definita su E , e la topologia meno fine che rende continui i funzionali lineari f ∈ E * . Visto che ogni f ∈ E * è anche limitato (ovvero, continuo in norma), abbiamo che σ(E, E * )è meno fine della topologia della norma di E .
Siano v = v ′ ∈ E ; usando il teorema di Hahn-Banach estendiamo il funzionale f 0 , λ(v − v ′ ) := λ v − v ′ , ∀λ ∈ R, ottenendo f ∈ E * tale che f, v = f, v ′ . Cosicché esiste α ∈ R tale che f, v < α < f, v ′ , e gli aperti in σ(E, E * ) A := f −1 (−∞, α) , A ′ := f −1 (α, +∞)
hanno intersezione vuota e contengono rispettivamente v , v ′ . Concludiamo che (E, σ(E, E * ))è uno spazio di Hausdorff. Diciamo che una successione {v n } ⊂ Eè debolmente convergente a v ∈ E se essa converge rispetto alla topologia debole, cosa che accade se e solo se f, v = lim n f, v n , ∀f ∈ E * ; in tal caso, scriviamo
v n σ → v ovvero v = σ lim n v n .
Elenchiamo nel seguito alcune proprietá elementari della topologia debole:
1. v n σ → v ⇔ f, v n → f, v , ∀f ∈ E * ; 2. v n → v ⇒ v n σ → v ; 3. Se v n σ
→ v , allora { v n }è limitata e v ≤ lim inf n v n . Ció segue osservando che per ogni f ∈ E * la successione {| f, v n |} converge a | f, v |), ed applicando Cor.7.12.
4. Se v n σ → v e f m → f , allora f n , v n → f, v . Infatti, si trova | f, v − f n , v n | ≤ | f, v − v n | + f − f n v n . Esempio 7.21. Sia p ∈ [1, +∞), q := p e (X, M, µ) uno spazio misurabile σ -finito. Una succes- sione {f n } ⊂ L p µ (X)è debolmente convergente ad f ∈ L p µ (X) se e solo se X (f n − f )g n → 0 , ∀g ∈ L q µ (X) .
Ed in tal caso, esiste M ∈ R tale che f n p ≤ M , n ∈ N. Ció implica che e k σ → 0 , nonostante il fatto che e k ≡ 1 .
Usando le proprietá precedenti e l'esistenza di una base finita, si puó dimostrare in modo molto semplice che se E ha dimensione finita, allora la topologia debole e quella della norma coincidono. D'altra parte, se E ha dimensione non finita, allora
• La sfera unitaria E 1 := {v ∈ E : v = 1} nonè chiusa in σ(E, E * ) • Il disco unitario E <1 := {v ∈ E : v < 1} nonè aperto in σ(E, E * ) (si veda [5, §III.2])
. Concludiamo questa breve rassegna su σ(E, E * ) richiamando i seguenti due risultati, conseguenze piuttosto semplici del teorema di Hahn-Banach: Proposizione 7.38. Un sottoinsieme C ⊆ E convessoè debolmente chiuso se e solo seè chiuso nella topologia della norma.
Proposizione 7.39. Sia T ∈ B(E, F ). Allora Tè un'applicazione continua da (E, σ(E, E * )) in (F , σ(F , F * )).
La topologia * -debole. Passiamo ora a considerare un ulteriore tipo di topologia debole, stavolta sul duale E * . Consideriamo l'applicazione lineare canonica
ϕ : E → E * * , v → ϕ v : ϕ v , f = f, v , (7.55)
peraltro isometrica grazie a (7.28). La topologia * -debole su E * è la topologia meno fine che rende continua la famiglia di applicazioni {ϕ v , v ∈ E} , e si denota con σ(E * , E). Poiché ϕ(E) ⊆ E * * , abbiamo che σ(E * , E)è meno fine di σ(E * , E * * ). Lo stesso argomento usato per la topologia debole (ma stavolta nonè neanche necessario invocare il teorema di Hahn-Banach) mostra che lo spazio (E * , σ(E * , E))è di Hausdorff. Una successione {f n } ⊂ E * converge ad f ∈ E * nella topologia * -debole se e solo se f, v = lim n f n , v , ∀v ∈ E ; per indicare la convergenza * -debole di {f n } useremo le notazioni
f n * → f ovvero f = * lim n f n .
In maniera analoga al caso della topologia debole, si dimostrano le seguenti proprietá:
1. f n * → f ⇔ f n , v → f, v , v ∈ E ; 2. f n → f ⇒ f n *
Dimostrazione. Consideriamo lo spazio prodotto R E con elementi {ω v ∈ R} v∈E , equipaggiato con le proiezioni (continue) π v : R E → R, v ∈ E , e l'applicazione (manifestatamente iniettiva)
T : E * → R E , T (f ) := { f, v } v∈E .
Vogliamo mostrare che T : E * → T (E * )è un omeomorfismo. Ora, si ha ϕ v = π v • T , v ∈ E ; essendo ϕ v * -debolmente continua concludiamo che ogni π v • Tè * -debolmente continua per cui, per definizione di topologia prodotto, Tè continua. Per dimostrare che T −1 : T (E * ) → E * è continua,è sufficiente verificare che, per ogni v ∈ E , l'applicazione
ϕ v • T −1 : T (E) → R , T (f ) → ϕ v , f é continua. Ma cióè evidente per definizione di topologia prodotto, in quanto ϕ v • T −1 = π v | T (E * ) . Ora, T (E * ≤1 ) = K 1 ∩ K 2 , dove K 1 := {ω ∈ R E : |ω v | ≤ v , v ∈ E} = v [− v , v ] K 2 := {ω ∈ R E : ω v+aw = ω v + aω w , v, w ∈ E, a ∈ R}
E' evidente che K 1è compatto (Tychonoff). D'altra parte,
K 2 = v,w,a (π v+aw − π v − aπ w ) −1 (0)
dunque (per continuitá di π v+aw , π v , aπ w ) essoè chiuso. Essendo quindi T (E * ≤1 ) intersezione di un chiuso con un compatto,è esso stesso compatto. Essendo T −1 continua, concludiamo che E * ≤1 e compatto.
Corollario 7.42. Sia E uno spazio di Banach. Allora la sfera unitaria E * 1è *-debolmente compatta.
Dimostrazione. Infatti E * 1è chiusa edè contenuta nel compatto E * ≤1 .
Definizione 7.43. Uno spazio di Banach E si dice riflessivo se l'iniezione canonica (7.55)è suriettiva, cosicché si ha un isomorfismo E ≃ E * * .
Il seguente teorema, che forniamo senza dimostrazione, fornisce una caratterizzazione topologica degli spazi riflessivi. Parte del teorema di Kakutaniè semplice da dimostrare. Se Eè riflessivo allora (7.55) si restringe ad un omeomorfismo isometrico ϕ : E ≤1 → E * * ≤1 . Il teorema di Alaoglu implica che (E * * ≤1 , σ(E * * , E * )) e compatto, per cuiè sufficiente verificare che
ϕ −1 : (E * * ≤1 , σ(E * * , E * )) → (E ≤1 , σ(E, E * ))
e un'applicazione continua; ovvero, per definizione della topologia debole su E , che per ogni f ∈ E * , l'applicazione
f • ϕ −1 : E * * ≤1 → R sia * -debolmente continua. Poichéè immediato verificare che f • ϕ −1 = ϕ * f (dove ϕ * f ∈ E * * * è definito secondo l'applicazione canonica ϕ * : E * → E * * * ), concludiamo che f • ϕ −1è * -debolmente continua, e quindi ϕ −1è continua.
Passiamo ora ad elencare alcune proprietá concernenti la separabilitá degli spazi di Banach per la topologia debole: Per quanto concerne gli spazi L p , nel caso in cui X sia separabile abbiamo la seguente situazione:
• L p µ (X), p ∈ (1, ∞),è separabile ([5, Teo.IV.13]), e riflessivo (dualitá di Riesz); • L 1 µ (X)è separabile ([5, Teo.IV.13]), ma non riflessivo (vedi Prop.5.14); • L ∞ µ (X) nonè separabile (a meno che X non sia un insieme numerabile, vedi [5, Lemma IV.2]) e neanche riflessivo (vedi Prop.5.14). Tuttavia, essendo L ∞ µ (X) il duale di L 1 µ (X) (dualitá di Riesz), la palla unitaria di L ∞ µ (X)è compatta rispetto alla topologia *-debole (Teorema di Alaoglu).
Lo spettro di una * -algebra di Banach ed il teorema di Gel'fand-Naimark. Il teorema di Alaoglu si dimostra senza alcuna variazione nel caso di spazi di Banach complessi, ed un'importante applicazione di questo risultatoè la possibilitá di rappresentare, in modo piú o meno accurato a seconda dei casi, una generica * -algebra di Banach commutativa come un'algebra di funzioni continue. E' questo l'oggetto del risultato noto come il teorema di Gel'fand-Naimark.
Sia A una * -algebra di Banach commutativa. Denotiamo con A l'insieme dei caratteri di A, ovvero di quegli ω ∈ A * non nulli tali che ω, aa ′ = ω, a ω, a ′ , ω, a * = ω, a , ∀a, a ′ ∈ A ;
(7.56) osserviamo che Aè naturalmente equipaggiato della topologia *-debole, rispetto alla quale essoè di Hausdorff.
Definizione 7.45. Lo spazio di Hausdorff A si dice lo spettro di A.
Introduciamo ora la trasformata di Gel'fand
A → C ( A, C) , a → a : a(ω) := ω, a , ∀ω ∈ A . (7.57)
Le funzioni a, a ∈ A, sono effettivamente continue proprio per definizione di topologia * -debole (infatti a = ϕ a , nella notazione del paragrafo precedente), per cui (7.57)è ben definita come applicazione. Osserviamo che per costruzione (7.57)è lineare e tale che aa ′ = a a ′ , a * (ω) = a(ω) , ∀a, a ′ ∈ A , ω ∈ A . (7.58)
Supponiamo ora che A abbia identitá 1 ∈ A e poniamo λ := ω, 1 ; allora troviamo ω, a = λ ω, a per ogni a ∈ A e quindi, scegliendo a = 1 , λ 2 = λ; essendo ω non nullo troviamo λ = 0 , e concludiamo che deve essere λ = ω, 1 = 1 per ogni ω ∈ A. Nella seguente proposizione stabiliamo una connessione tra A e gli spettri degli elementi di A.
Lemma 7.46. Sia A una * -algebra di Banach commutativa e con identitá 1 ∈ A. Per ogni a ∈ A ed ω ∈ A si ha ω, a ∈ σ(a), cosicché | ω, a | ≤ a .
Dimostrazione. Iniziamo dimostrando che T ω := a − ω, a 1 nonè invertibile. A tale scopo, osserviamo che (avendosi ω, 1 = 1 ) troviamo ω, T ω = 0 , per cui T ω appartiene al nucleo di ω , che denotiamo con ker ω . Ora, ker ωè chiuso rispetto a prodotti con generici elementi di A 40 , infatti ω, T a = 0 · ω, a = 0 per ogni T ∈ ker ω ed a ∈ A. Osserviamo che ker ω non puó avere elementi invertibili, altrimenti avremmo la contraddizione 1 = ω, T T −1 = 0 · ω, T −1 = 0 , T ∈ ker ω . Concludiamo che T ω nonè invertibile e ω, a ∈ σ(a). Il fatto che | ω, a | ≤ a segue da Prop.7.31.
In conseguenza del Lemma precedente troviamo A ⊆ A * ≤1 ; inoltreè immediato verificare che À e chiuso (il limite nella topologia * -debole di una successione di caratteriè un carattere), per cui il teorema di Alaoglu implica che ( A, σ(A * , A))è compatto, essendo A un chiuso contenuto in un compatto. Dunque, ogni a ∈ C( A, C), a ∈ A, ha norma a ∞ := sup ω∈ A | ω, a | ≤ a . Proposizione 7.47 (Gel'fand-Naimark). Per ogni * -algebra di Banach A commutativa e con identitá 1 , la trasformata di Gel'fandè un operatore lineare limitato ed ha immagine densa in C( A, C).
Dimostrazione. La linearitá di (7.45)è ovvia e la limitatezza segue dal Lemma 7.46, il quale implica che a ∞ ≤ a , a ∈ A. Denotiamo ora con B ⊆ C( A, C) l'immagine di A attraverso la trasformata di Gel'fand; per mostrare la densitá di (7.45) l'ideaè quella di utilizzare il teorema di Stone-Weierstrass (Teo.2.7), per cui occorre verificare che Bè chiusa rispetto al passaggio all'aggiunto in C( A, C) (e cióè ovvio da (7.58)), che B contiene le funzioni costanti z ∈ C( A, C), z ∈ C (e cióè ovvio in quanto z1(ω) = z ω, 1 = z , ∀ω ∈ A), e che B separa i punti di A (e anche questoè ovvio, visto che ω = ω ′ se e solo se ω, a = ω ′ , a per qualche a ∈ A, ovvero a(ω) = a(ω ′ )). Dunque concludiamo che Bè densa in C( A, C) come desiderato.
Ora, peculiaritá delle C*-algebre commutativeè che a = sup ω∈ A | ω, a | per ogni a ∈ A (vedi [19, §4.1.10 e Lemma 4.3.11]). In altri termini la trasformata di Gel'fandè isometrica, e quindi chiusa come applicazione da A in C( A, C). Di conseguenza, l'immagine di A attraverso (7.57) e sia chiusa che densa in C( A, C), e quindi coincide con C( A, C). Dunque abbiamo mostrato il seguente risultato: 40 Ovvero, ker ωè un ideale di A . Teorema 7.48. (Gel'fand-Naimark, [19,Thm.4.3.13]). Per ogni C*-algebra A commutativa con identitá 1 , la trasformata di Gel'fandè iniettiva e suriettiva.
Esempio 7.23. (1) Sia X compatto e di Hausdorff, A := C(X, C); allora Aè omeomorfo ad X (vedi Esercizio 7.9); (2) Sia A := L 1 (R, C) + Cδ 0 , dove L 1 (R, C)è equipaggiato del prodotto di convoluzione e δ 0è la delta di Dirac (qui abbiamo immerso L 1 (R, C) in Λ 1 β (R, C), vedi Esercizio 6.7); allora si ha un omeomorfismo θ : A → T, e la trasformata di Gel'fand coincide, in essenza, con la trasformata di Fourier (vedi Esercizio 7.
p i (v) = 0 , ∀i ∈ I ⇒ v = 0 .
Diciamo che Vè localmente convesso se ammette una famiglia separante di seminorme. La topologia naturale di V , che denotiamo con σV ,è la piú debole topologia che rende continue le applicazioni La relativa topologia naturaleè nota con il nome di topologia forte di B(E) (vedi [19, §4.6]). In particolare, preso uno spazio di Hilbert H ed una successione {T n } ⊂ B(H), le disuguaglianze
p i al variare di i in I . Chiaramente, σVè di Hausdorff; per definizione, una successione {v n } ⊂ V converge a v ∈ V nella topologia σV se e soltanto se p i (v n ) n → p i (v) per ogni i ∈ I .|(u, {T n − T m }v)| ≤ u {T n − T m }v ≤ u T n − T m v , ∀u, v ∈ H ,
mostrano che la convergenza in norma di {T n } implica quella in topologia forte, la quale a sua volta implica la convergenza nella topologia debole del punto (4). Dunque, ogni algebra di von Neumann A ⊆ B(H)è chiusa sia rispetto alla topologia forte che quella della norma (dunque Aè anche una C*-algebra).
Un sottoinsieme C di uno spazio vettoriale V si dice: (1) convesso, se per ogni x, y ∈ C , t ∈ [0, 1] risulta tx + (1 − t)y ∈ C ; (2) bilanciato, se per ogni x ∈ C e λ di modulo 1 41 risulta λx ∈ C ; (3) assorbente, se ∪ t≥0 tC = V .
Uno spazio vettoriale V si dice topologico se su essoè definita una topologia τ tale che le operazioni di somma e moltiplicazione scalare sono applicazioni continue Il teorema di Hahn-Banach si applica senza variazioni agli spazi localmente convessi, per cui questi sono ricchi di funzionali lineari continui rispetto alla topologia naturale. Lo spazio vettoriale avente per elementi tali funzionali si denota con V * , edè chiamato il duale topologico di V ; si
V × V → K , K × V → V , dove K = R,puó dimostrare che V * separa i punti di V (si veda [22, §V.1]). Ora, per ogni v ∈ V definiamo il funzionale ϕ v : V * → R , ϕ v , f := f, v , f ∈ V * .
Equipaggiamo quindi V * della topologia debole σ(V * , V ) che rende continua la famiglia di applicazioni {ϕ v , v ∈ V } . Dati due spazi localmente convessi V e V ′ con famiglie di seminorme {p i } , {q j } , possiamo considerare operatori lineari T : V → V ′ continui rispetto alle rispettive topologie naturali. Si verifica facilmente (vedi [22,Teo.V.2]) che un operatore lineare T : V → V ′è continuo se e solo se per ogni seminorma q j esistono seminorme p 1 , . . . , p n su V e c > 0 tali che
q j (T v) ≤ c n k=1 p k (v) , v ∈ V .
(7.60)
Ora, alcuni semplici argomenti topologici mostrano che, dato il nostro spazio localmente convesso V , le seguenti condizioni sono equivalenti: (1) Vè metrizzabile; (2) 0 ∈ V ammette una base numerabile di intorni; (3) σVè indotta da una famiglia numerabile di seminorme. Per dare un'idea della dimostrazione, se assumiamo che σV sia indotta da una famiglia numerabile {p n } allora possiamo definire la metrica
d(v, w) := n 2 −n p n (v − w) 1 + p n (v − w)
, v, w ∈ V . x α k k , x ∈ R n , e l'operatore D α := ∂ |α| ∂x α1 1 . . . ∂x αn n , α ∈ Z +,n .
Una funzione f ∈ C ∞ (R n ) si dice a decrescenza rapida se per ogni α, β ∈ Z +,n risulta
f α.β := sup x |x α D β f (x)| < ∞ . (7.62)
Denotiamo con S(R n ) lo spazio delle funzioni a decrescenza rapida. Osserviamo che (7.62) ci dice che f ∈ S(R n ), ed ogni sua derivata, decrescono piú rapidamente dell'inverso di un qualsiasi polinomio. E' chiaro che { · α,β }è una famiglia numerabile di seminorme, cosicché S(R n ) ha una struttura di spazio localmente convesso metrizzabile (vedi (7.61)). Usando gli usuali teoremi di uniforme convergenza dimostriamo facilmente che S(R n )è completo, per cui essoè uno spazio di Fréchet. Il duale topologico S * (R n )è chiamato lo spazio delle distribuzioni temperate.
Esempio 7.25. Per ogni x ∈ R n consideriamo la delta di Dirac δ x ∈ S * (R n ), δ x , f := f (x). Poiché δ x , f ≤ f 0,0 concludiamo che δ x ∈ S * (R n ). Definiamo ora δ ′ x : S(R) → R, δ x , f := f ′ (x). Poiché δ ′ x , f ≤ f 0,1 concludiamo che δ ′ x ∈ S * (R); tuttavia, a differenza della delta di Dirac, δ ′
x nonè associata a nessuna misura su R.
Esempio 7.26. Definiamo
P 1/x , f := lim ε→0 + |x|≥ε 1 x f (x) dx = lim ε→0 + ∞ ε 1 x (f (x) − f (−x)) dx , ∀f ∈ S(R) . (7.63) Poiché lim x→0 x −1 (f (x) − f (−x)) = 2f ′ (0) troviamo che l'integrale improprio nell'espressione precedente converge a ∞ 0 1 x (f (x) − f (−x)) dx < ∞ .
Dunque P 1/xè ben definita come applicazione lineare. Per mostrarne la continuitá, osserviamo
che |x −1 (f (x) − f (−x))| ≤ 2 f ′ ∞ per cui, spezzando (7.63) tra [0, 1] ed (1, ∞) troviamo | P 1/x , f | ≤ 2 1 0 f ′ ∞ dx + ∞ 1 |xf (x)| 1 x 2 dx ≤ 2 f 0,1 + f 1,0 .
Dunque P 1/x ∈ S * (R), edè noto come la parte principale di Cauchy.
Da (7.62) segue che S(R n ) ⊂ L p (R n ) per ogni p ∈ [1, +∞], dunque abbiamo un'applicazione canonica i p : S(R n ) → L p (R n ) .
Si verifica che i pè continua. Ad esempio, nel caso n = 1 , p = 1 abbiamo Per dualitá di Riesz ogni g ∈ L q (R n ) definisce il funzionale continuo F g ∈ L p, * (R n ) (vedi (5.15)); per cui F g • i p : S(R n ) → Rè un funzionale continuo rispetto alla topologia naturale di S(R n ). Dunque, abbiamo un'immersione
i 1 f 1 = f 1 = R 1 1 + x 2 {(1 + x 2 )|f (x)|} dx ≤ π{ f ∞ + x 2 f ∞ } ,L q (R n ) → S * (R n ) , g → F g • i p : F g • i p , f = f g , ∀f ∈ S(R n ) .
(7.65) Equipaggiando S * (R n ) della topologia debole σ(S * (R n ), S(R n )), ed usando la continuitá di i p , concludiamo che (7.65)è continua. Restringendo (7.65) a funzioni del tipo i q g ∈ L q (R n ), g ∈ S(R n ), otteniamo l'applicazione continua I : S(R n ) → S * (R n ) , g → I(g) : I(g), f = f g , ∀f ∈ S(R n ) . Chiamiamo T * l'applicazione aggiunta di T .
Esempio 7.27 (La derivata debole). Per ogni α ∈ Z +,n , l'operatore di derivazione definisce l'applicazione continua
D α : S(R n ) → S(R n ) , D α f γ,δ ≤ f γ,δ+α .
L'operatore aggiunto D α, * : S * (R n ) → S * (R n )è noto come la derivata debole. Ora, per ogni f, g ∈ S(R n ) troviamo
D α, * • I(f ), g = D α f (x)g(x) dx per parti = (−1) |α| f (x)D α g(x) dx = I(f ), (−1) |α| D α g .
Dunque se ϕ ∈ S * (R n ) per densitá troviamo
D α, * (ϕ), g = ϕ, (−1) |α| D α g , g ∈ S(R n ) .
Diciamo che una distribuzione temperata ϕ ∈ S * (R n ) si annulla in un aperto Ω ⊆ R n qualora ϕ, f = 0 per ogni f ∈ S(R n ) tale che supp(f ) ⊆ Ω. Il supporto di ϕ, che denotiamo con supp(ϕ), si definisce come il complemento del piú grande insieme aperto sul quale ϕ si annulla. Si dimostra che
supp(ϕ) = {0} ⇒ ϕ = α∈I c α D α, * • δ 0 (7.68) dove I ⊂ Z +,nè un insieme finito e c α ∈ R (vedi [22, Thm.V.11]).
Esempio 7.28 (Rinormalizzazione). Consideriamo la funzione x −1 + (t) := χ (0,+∞) (t) · t −1 , t ∈ R. Poiché 1 0 x −1 + (t) dt = +∞ concludiamo che x −1 + non definisce una distribuzione su S(R). Tuttavia, se consideriamo il sottospazio S 0 (R) di S(R) delle funzioni a decrescenza rapida che si annullano nell'origine, troviamo che il funzionale definito dall'integrale improprio
P + 1/x , f := +∞ 0 1 t f (t) dt , f ∈ S 0 (R) ,
e ben definito e continuo. Applicando il teorema di Hahn-Banach, concludiamo che esistono estensioni ad S(R) di P + 1/x , note come rinormalizzazioni. Prese due rinormalizzazioni ϕ 1 , ϕ 2 , da (7.68) concludiamo ϕ 1 − ϕ 2 = α c α D α, * • δ 0 . In effetti, si dimostra facilmente ([22, Ex.V.3.9]) che tutte e sole le rinormalizzazioni di P + 1/x sono quelle del tipo
ϕ c ∈ S * (R) : ϕ c , f := c 0 f (x) − f (0) x dx + ∞ c f (x) x dx , c > 0 , il che implica ϕ c − ϕ c ′ = − log c c ′ δ 0 . La topologia di Fréchet su C ∞ c (Ω), distribuzioni e soluzioni deboli.
Sia Ω ⊆ R n un aperto connesso e C ∞ c (Ω) lo spazio delle funzioni C ∞ aventi supporto compatto e contenuto in Ω. Abbiamo giá mostrato che C ∞ c (Ω) nonè uno spazio di Banach (vedi Esercizio 2.7), tuttaviaè possibile verificare che essoè uno spazio di Fréchet rispetto alla famiglia di seminorme D α f ∞ , α ∈ Z +,n , f ∈ C ∞ c (Ω) . Esempio 7.30 (Ancora sulla derivata debole). Per ogni g ∈ C(R n ), α ∈ Z +,n ,è ben definita l'applicazione lineare
| ϕ, f | ≤ c K |α|≤mK D α f ∞ , ∀f ∈ C ∞ 0 (K) ⊂ D(Ω) .D α, * g, f := (−1) |α| g(x)D α f (x) dx , f ∈ D(Ω) ,
la qualeè continua in quanto, per ogni compatto K ⊂ Ω,
| D α, * g, f | ≤ vol(K) g| K ∞ D α f ∞ , ∀f ∈ C ∞ 0 (K) .
Esempio 7.31. Per ogni g ∈ L 1 loc (R n ),è ben definita la distribuzione
F g , f := f g , f ∈ D(Ω) : | F g , f | ≤ g| K 1 f ∞ , ∀f ∈ C ∞ 0 (K) , K ⊂ Ω .
Introduciamo ora l'operazione di convoluzione tra una distribuzione ed una funzione in D(Ω). Innanzitutto, richiamiamo le operazioni di traslazione ed antipodo: per ogni f ∈ D(Ω), x ∈ R n definiamo
ǫf (y) := f (−y) , f x (y) := f (y + x) , y ∈ R n ⇒ {ǫf } x (y) = f (x − y) ; e del tutto ovvio che ǫf, f x , {ǫf } x ∈ D(Ω), con D α {ǫf } x ∞ = D α f ∞ , α ∈ Z +,n .
Osserviamo quindi che per ogni x ∈ R n e ϕ ∈ D * (Ω) con associate costanti c K , m K , K ⊂ Ω, nel senso di (7.70), risulta
| ϕ, {ǫf } x | ≤ c K |α|≤mK D α {ǫf } x ∞ = c K |α|≤mK D α f ∞ < ∞ , f ∈ C ∞ 0 (K) .
Per cui,è ben definita la funzione
ϕ * f : R n → R , ϕ * f (x) := ϕ, {ǫf } x , x ∈ R n .
Facciamo qualche semplice osservazione sulla regolaritá di ϕ * f . Poiché f ∈ C ∞ 0 (K)è uniformemente continua, troviamo che per ogni ε > 0 esiste δ > 0 tale che
|x − x ′ | ≤ δ ⇒ (εf ) x − (εf ) x ′ ∞ < ε . Ripetendo l'argomento precedente per ogni insieme finito {D α f } |α|≤m ⊂ C ∞ 0 (K), concludiamo che |ϕ * f (x) − ϕ * f (x ′ )| ≤ c K |α|≤mK D α (εf ) x − D α (εf ) x ′ ∞ ≤ c K M K ε ,
dove M Kè la somma dei multiindici α ∈ Z +,n tali che |α| ≤ m K . Dunque ϕ * f ∈ C(R n ), e ripetendo l'argomento precedente sulle derivate di f concludiamo che ϕ * f ∈ C ∞ (R n ).
Esempio 7.32. Data g ∈ L 1 loc (R n ), in accordo all'Esempio 7.31 per ogni f ∈ D(Ω) troviamo
F g * f (x) = g(y)f (x − y) dy , x ∈ R n ,
per cui ϕ * fè una naturale generalizzazione della convoluzione. Questo giustifica la classica
notazione ϕ * f (x) = ϕ(y)f (x − y)dy . Osserviamo che Prop.6.24 implica F g * f ∈ C ∞ (R n ). Esempio 7.33. Consideriamo la delta di Dirac δ 0 ∈ D * (Ω). Allora δ 0 * f (x) = {ǫf } x (0) = f (x), f ∈ D(Ω), e quindi δ 0 * f = f .
In modo analogo a quanto accade con le funzioni a decrescenza rapida, si puó verificare che la dualitá di Riesz induce un'immersione I : Osserviamo che p(D), p(D * ) sono ben definiti anche qualora c α ∈ C ∞ (R).
Definizione 7.51. Sia p un polinomio a coefficienti in C ∞ (R) di grado massimo k ∈ N e g ∈ C(R). Una soluzione debole dell'equazione alle derivate parziali p(D)u = gè una distribuzione ϕ ∈ D * (R n ) tale che
p(D * )ϕ = I g ⇔ p(D * )ϕ, f = g(x)f (x) dx , ∀f ∈ D(R n ) .
Inoltre, ϕ ∈ D * (R n ) si dice soluzione fondamentale se p(D * )ϕ = δ 0 .
Sia ϕ ∈ D * (R n ) una soluzione fondamentale per l'operatore p(D) ed f ∈ D(R n ). Allora u := ϕ * fè C ∞ ed ha senso considerare la funzione p(D)u . Ora, si puó verificare che vale la proprietá di associativitá
= f , f ∈ C ∞ 0 (R n ), si puó scrivere, formalmente, come ϕ = F Φ , dove Φ(x) := 1/2|x| , n = 1 (2π) −1 log |x| , n = 2 −(4π|x|) −1 , n = 3
Osserviamo che in generale Φ / ∈ L 1 loc (R n ), per cui la notazione F Φ vá intesa nel senso formale.
Operatori non limitati.
Nonostante gli approcci che possiamo adottare a livello topologico (distribuzioni, spazi di Sobolev) sta di fatto che gli operatori differenziali, che ovviamente sono di grande importanza in analisi, non sono limitati né in norma · ∞ né in norma · p . In questa sezione presentiamo alcuni risultati sugli operatori non limitati su spazi di Hilbert. Dimostrazione. Se u ⊕ z ∈ G(D, T ) ⊥ allora, per ogni v ∈ D , abbiamo
0 = (u ⊕ z, v ⊕ T v) = (u, v) + (z, T v) ⇔ f T,z (v) = −(u, v) .
Essendo D denso, l'uguaglianza precedenteè equivalente ad affermare che z ∈ D * e T * z = −u . Possiamo dunque definire l'operatore
U : G(D * , T * ) → G(D, T ) ⊥ , U (z ⊕ T * z) := (−T * z ⊕ z) ,(7.|f T,u (v)| − u S v ≤ |f T +S,u (v)| ≤ |f T,u (v)| + u S v , ∀u ∈ H , v ∈ D , per cui {u ∈ B(H) : f T +S,u < ∞} = D * = D .T * T u = −u ′′ , ∀u ∈ D ∩ C 2 ([0, 1], C) ,
per cui il teorema di von Neumann implica che l'operatore di derivata seconda ammette un'estensione autoaggiunta. Sempre invocando il teorema di von Neumann, concludiamo che per ogni f ∈ H esiste un unico u ∈ D tale che u +
T * T u = f . Se, in particolare, u ∈ C 2 ([0, 1], C) allora uè soluzione unica dell'equazione u − u ′′ = f , u ∈ D .
Notare che il precedente problema differenziale presenta delle condizioni al bordo (u(0) = u(1)), "nascoste" nella definizione del dominio D .
Un operatore simmetrico (D, T ) si dice semicoercitivo se esiste α ≥ 0 tale che (T u, u) ≥ α u 2 per ogni u ∈ D ; in particolare diciamo che Tè coercitivo se α > 0 . Ad esempio, ogni operatore positivo, ovvero tale che (T u, u) ≥ 0 , ∀u ∈ D ,è semicoercitivo (con α = 0 ). Per segnalare l'importanza della nozione di coercitivitá nella teoria delle equazioni alle derivate parziali invitiamo a dare un'occhiata alle sezioni 7. (2) L'operatore 1 − CTè iniettivo e con immagine D , in maniera tale che
i(1 + CT )(1 − CT ) −1 = T . (7.75)
Dimostrazione. Le stesse considerazioni fatte appena prima della definzione di CT mostrano che
(T + i1)u 2 = T u 2 + u 2 = (T − i1)u 2 , ∀u ∈ D , cosicché per ogni v = (T +i1)u abbiamo CT v = v e CT si estende per continuitá a {T + i1}(D). Possiamo quindi definire CT v := CT • P v , v ∈ H , dove P ∈ B(H)è il proiettore sul sottospazio {T + i1}(D) ⊆ H ; ció mostra il punto (1) 43 . Riguardo il punto (2) osserviamo che CT v = v ⇔ (T − i1)u = (T + i1)u ⇔ u = −u = 0 ⇔ v = 0 , ∀v = (T + i1)u , per cui 1 − CTè iniettivo. Per valutare l'immagine di 1 − CT poniamo W := {T + i1}(D) e calcoliamo 1| W − CT = (T + i1)(T + i1) −1 − (T − i1)(T + i1) −1 = 2i(T + i1) −1 1| W + CT = 2T (T + i1) −1 , da cui segue che {1 − CT }(T u + iu) = 2iu , ∀u ∈ D ⇒ {1 − CT }(W ) = D , nonché i(1 + CT )(1 − CT ) −1 = 2iT (T + i1) −1 (2i) −1 (T + i1) = T .
Il seguente risultato mostra il vantaggio apportato dalla trasformata di Cayley, consistente nella possibilitá di descrivere gli operatori simmetrici (non limitati) in termini di isometrie parziali: Dimostrazione. Che l'applicazione {(T, D) → CT } sia iniettiva segue da (7.75). D'altra parteè ovvio che se (T ′ , D ′ ) estende (T, D) allora CT ′ estende CT nel senso dell'enunciato, per cui rimane da verificare solo che la trasformata di Cayleyè suriettiva. A tale scopo consideriamo un'isometria parziale U ∈ B(H) tale che (1 − U )| ker U ⊥ ha immagine densa; per definizione di isometria parziale abbiamo
(U v, U v ′ ) = (v, v ′ ) , ∀v, v ′ ∈ ker U ⊥ , cosicché se U w = w per qualche w ∈ ker U ⊥ allora (w, (1 − U )v) = (w, v) − (w, U v) = (U w, U v) − (w, U v) = (U w − w, U v) = 0 ;
poiché (1 − U )| ker U ⊥ ha immagine densa concludiamo che deve essere w = 0 , per cui 1 − Uè iniettivo su ker U ⊥ . Siamo ora in grado di definire
D := {1 − U }(ker U ⊥ ) , T (1 − U )v := i(1 + U )v , v ∈ ker U ⊥ ,
e dei semplici conti mostrano che
(T (1 − U )v, (1 − U )v) ∈ R , ∀v ∈ ker U ⊥((T * + i1)u = (T + i1)v ⇔ {T * + i1}(u − v) = 0 (infatti T * u = T u avendosi u ∈ D ⊆ D * ). Del resto, per ipotesi abbiamo ker(T * + i1) = {T − i1}(D) ⊥ = {0} , e quindi u = v ∈ D , da cui D = D * .
Gli indici di difetto di (D, T ) si definiscono come Dimostrazione. Sia z ∈ C − σ(D, T ) e w ∈ C con |w| R z −1 < 1 . Allora la serie n=0 R n z w ǹ e assolutamente convergente e, come nel caso limitato, troviamo che 1 − wR zè invertibile, con (1 − wR z ) −1 = n R n z w n . Per cui
n + := dim{T + i1}(D) ⊥ , n − := dim{T − i1}(D) ⊥ , n + , n − ∈ N ∪ {∞} .(1 − wR z ) −1 R z = {(z1 − T )(1 − wR z )} −1 = {(z1 − T )(1 − w(z1 − T ) −1 )} −1 = {(z − w)1 − T } −1 , e quindi C − σ(D, T )è aperto.
In generale nonè detto che lo spettro di un operatore simmetrico sia contenuto in R; infatti, mentre da un lato sappiamo che z1 − Tè iniettivo se Im(z) = 0 (vedi paragrafo sulla trasformata di Cayley), nonè detto che z1 − T sia suriettivo. Tuttavia, troviamo: In analogia con il caso limitato ritroviamo il teorema spettrale per operatori autoaggiunti. Denotiamo con M β (σ(D, T )) e L ∞ β (σ(D, T )) rispettivamente lo spazio delle funzioni boreliane e la C*-algebra delle funzioni boreliane limitate su σ(T ), a valori complessi in entrambi i casi.
A f : D f × D f → C , u, v → σ(D,T ) f dµ uv , D f := {u ∈ H : σ(D,T ) |f | 2 dµ uu < ∞} , definisce una forma sesquilineare e quindi un operatore (D f , f (T )) tale che A f (u, v) = (u, f (T )v), ∀u, v ∈ D f . In particolare (D, T ) = (D I , I(T )), dove I(λ) := λ, ∀λ ∈ σ(D, T ); (3) Se f ∈ L ∞ β (σ(D, T )) allora f (T )è limitato con f (T ) ≤ f ∞ , e si ha una rappresentazione L ∞ β (σ(D, T )) → B(H) , f → f (T ) .
Referenze per una dimostrazione dettagliata del teorema precedente sono [22,VIII.3], [19, 5.3]; l'ordine di ideeè essenzialmente quello del caso limitato, con la differenza sostanziale che stavolta σ(D, T )è uno spazio localmente compatto, per cui occorre apportare delle modifiche all'argomento che utilizza il calcolo funzionale continuo ed il teorema di Riesz-Markov. Osserviamo inoltre che, a differenza del caso limitato, funzioni continue su σ(D, T ) possono essere non limitate e quindi tali che la loro immagine rispetto al calcolo funzionale del punto (2) sia un operatore non limitato, come nel caso dello stesso (D, T ). Invece le misure spettrali del punto (1) rimangono comunque finite. Ora, abbiamo (e t ) * = e −t , e t e −t = 1 , e t+s = e t e s , ∀t, s ∈ R , e visto che il calcolo funzionale boreliano conserva le operazioni di moltiplicazione e passaggio all'aggiunto troviamo
U * t = U −t , U t U −t = 1 , U t+s = U t U s , ∀t, s ∈ R ,(7.76)
cosicché in particolare ogni U t , t ∈ R,è unitario. Chiamiamo U := {U t } il gruppo ad un parametro definito da (D, T ). Visto che e t −1 → 0 puntualmente per t → 0 , in conseguenza dell'Esercizio 7.16 e della relazione U t+s = U t U s troviamo che U soddisfa la seguente proprietá di continuitá rispetto alla topologia forte: lim
t→0 U t v − v = lim t→0 U t+s v − U s v = 0 , ∀v ∈ H , s ∈ R . (7.77)
Ora, per ogni λ ∈ σ(D, T ) abbiamo
|e iλt − 1| ≤ |λt| , ∀t ∈ R ,
per cui per ogni u ∈ D troviamo
{t −1 (U t − 1) − iT }u 2 = (u , {t −1 (U t − 1) − iT } * {t −1 (U t − 1) − iT }u) = |{t −1 (e t − 1) − iλ} * {t −1 (e t − 1) − iλ}| dµ uu (λ) = |t −1 (e iλt − 1) − iλ| 2 dµ uu (λ) ≤ {λ 2 + 2λ 2 + λ 2 }dµ uu (λ) ;
grazie al Teorema 7.64 sappiamo che la funzione I 2 (λ) := λ 2 , λ ∈ σ(D, T )è µ uu -integrabile per ogni u ∈ D , per cui possiamo applicare il teorema di convergenza dominata e passare al limite t → 0 sotto il segno di integrale. Ma
lim t→0 t −1 (e iλt − 1) − iλ = 0 , per cui concludiamo che lim t→0 t −1 (U t − 1)u = iT u , ∀u ∈ D ,(7.U t v − v 2 2 = |v(t + s) − v(s)| 2 ds Es. 5 .2 → 0 , ∀v ∈ H .
Dunque U soddisfa le ipotesi del teorema di Stone. Consideriamo ora lo spazio delle funzioni complesse a decrescenza rapida
S(R, C) := {f + ig : f, g ∈ S(R)} .
Allora per ogni u ∈ D := S(R, C) troviamo
lim t→0 {t −1 (U t − 1)u}(s) = lim t→0 t −1 {u(t + s) − u(s)} = u ′ (s) , ∀s ∈ R ;
poiché u ′ ∈ S(R, C) ⊂ H , da (7.78) concludiamo subito che l'operatore associato ad Uè l'estensione autoaggiunta di (D, T ), T u := −iu ′ , u ∈ D .
I gruppi ad un parametro hanno importanti applicazioni in meccanica quantistica. Consideriamo lo spazio di Hilbert H := L 2 (R d , C) ed un operatore autoaggiunto (D, T ) su H . L'equazione di Schroedinger associata a T con condizione iniziale ψ 0 ∈ Dè data dal problema differenziale ∂ψ ∂t = iT ψ , ψ(·, t 0 ) = ψ 0 , (7.79) la cui funzione incognita ψ : R d × R → Cè tale che ψ(·, t) ∈ D per ogni t ∈ R. Denotato con U = {U t } il gruppo a un parametro associato a (D, T ), usando (7.78) concludiamo che (7.79) ammette soluzione ψ(x, t) :
= {U t ψ 0 }(x) , ∀x ∈ R d , t ∈ R . (7.80)
Portiamo ad esempio l'equazione di Schroedinger per la particella libera, che si ottiene per d = 3 e (D, T ) l'estensione autoaggiunta dell'operatore di Laplace −∆ dell'Esempio 7.37; osserviamo che in tal caso la soluzioneè calcolabile usando la trasformata di Fourier ( §8.2),
ψ(x, t) := R 3 ψ 0 (λ)e −i|λ| 2 t+λ·x dλ , x ∈ R 3 , t ∈ R (7.81)
(per una verifica si derivi sotto il segno di integrale). Non sempre peró ci troviamo in situazioni cosí comode, cosicché diventa essenziale sfruttare le proprietá astratte dei gruppi ad un parametro; ed in tal caso, come passo intermedioè necessario stabilire se il nostro operatore (D, T ), presentato tipicamente in forma differenziale, sia (essenzialmente) autoaggiunto. Come esempio menzioniamo l'equazione di Schroedinger per l'atomo con n elettroni, che si ottiene per
D := C ∞ 0 (R 3n , C) , T u := −∆u + f u , u ∈ D ,{e t (H/k)u}(x) = R 3n u(λ)e −i|λ| 2 k −1 t+λ·x dλ , x ∈ R 3n ,
mentre e t (V /k)u = exp(itf /k)u ; in tal modo, calcolando il limite del teorema precedente otteniamo la famosa formula di Feynman-Kac (vedi [23, §X.11]).
Il Teorema di Schauder.
Come rimarcato in §2.3 i teoremi di punto fisso rivestono una grossa importanza in analisi: soluzioni di problemi integro-differenziali si possono presentare come punti fissi di applicazioni continue definite su spazi di funzioni, come accade ad esempio nel caso del problema di Cauchy rispetto all'operatore di Volterra (3.3). In questa sezione effettuiamo una breve rassegna dei teoremi piú noti e delle loro applicazioni. Enfatizziamo il fatto che la convessitá svolge un ruolo di primo piano per la dimostrazione dei risultati che seguono.
Teorema 7.67 (Schauder). Sia E uno spazio di Banach e C ⊂ E un convesso compatto e non vuoto. Allora ogni applicazione continua f : C → C ammette almeno un punto fisso.
Dimostrazione. Preso ε > 0 consideriamo il ricoprimento X ε := {∆(v, ε) : v ∈ C} ed estraiamo un sottoricoprimento finito X ε,n := {∆(v k , ε) : k = 1, . . . , n} . Definiamo quindi le funzioni continue
g 1 , . . . , g n : C → R + , g k (v) := ε − v − v k , se v − v k ≤ ε 0, se v − v k ≥ ε Osserviamo che n k=1 g k (v) > 0 , v ∈ C , per cui definiamo G n : C → C n , G n (v) := n k=1 g k (v)v k n k=1 g k (v) , dove C n ⊆ Cè l'inviluppo convesso di {v 1 , . . . , v n } . Osserviamo che G n (v) − v ≤ ε , v ∈ C . (7.82)
Consideriamo ora la funzione continua f n := G n • f : C → C n e la restrizione f n : C n → C n ; essendo C n compatto, convesso, e contenuto nello spazio a dimensione finita generato da v 1 , . . . , v n , per il teorema di Brouwer troviamo che esiste w ∈ C n tale che
G n • f (w) = f n (w) = f n (w) = w .
Inoltre, grazie a (7.82) troviamo
f (w) − w = f (w) − G n • f (w)) ≤ ε .
Osserviamo che w dipende da ε . Scegliendo ε = 1/m, m ∈ N, scriviamo w ≡ w m ed osserviamo che le considerazioni precedenti implicano che
∀m ∈ N ∃w m ∈ C : f (w m ) − w m ≤ m −1 .
Per compattezza di C , esiste una sottosuccessione w mi tale che {f (w mi )} converge ad un w 0 ∈ C . La stima
w mi − w 0 ≤ w mi − f (w mi ) + f (w mi ) − w 0
mostra che anche {w mi } converge a w 0 ; per cui, essendo f continua troviamo f (w 0 ) = w 0 .
Osservazione 7.11. Usando essenzialmente lo stesso argomento della dimostrazione precedente si ha la seguente versione del teorema di Schauder: se C ⊂ E un convesso, chiuso, limitato e non vuoto, allora ogni applicazione compatta f : C → C ammette un punto fisso.
Tra le applicazioni del teorema di Schauder, segnaliamo:
1. Una dimostrazione alternativa del Teorema di Peano. Con le notazioni di Teo.3.10, poniamo I := [t 0 − r, t 0 + r], consideriamo lo spazio di Banach E := C(I) e definiamo C := {u ∈ E : u − u 0 ∞ ≤ r} . Si verifica facilmente che Cè convesso, chiuso e limitato. A questo punto, si tratta di verificare che l'operatore di Volterra F u(t) := u 0 + t t0 f (s, u(s)) ds , u ∈ C , definisce un'applicazione compatta da C in sé. Il punto fisso di F fornisce la soluzione cercata. Esercizio 7.2. Sia n ∈ N ed X ⊆ R n ,Ẋ = ∅ , equipaggiato della misura di Lebesgue µ. Preso x ∈ X , dimostrare che la delta di Dirac δ x ∈ C 0 (X) * si estende ad un funzionale F ∈ L ∞, * µ (X), e che F non appartiene all'immagine dell'applicazione canonica (5.15).
Sketch della soluzione. Applicando il teorema di Hahn-Banach troviamo che esiste il funzionale F ∈ L ∞, * µ (X) cercato. Supponiamo ora per assurdo che F = F g per qualche g ∈ L 1 µ (X), ovvero
F, f = f g , f ∈ L ∞ µ (X) . Presa una successione {f n } ⊂ C c (X), 0 ≤ f n ≤ 1 , f n (x) ≡ 1 , tale che f n → χ {x} puntualmente, per convergenza di Lebesgue (|f n g| ≤ |g|, n ∈ N) troviamo 1 = f n (x) = F, f n = f n g dµ n → 0 ,
il cheè assurdo. Osserviamo che F = F g implicherebbe che µ x ≺ µ, dove µ xè la misura di Dirac, il cheè assurdo (vedi Esempio 4.9).
Soluzione. (1) Segue dalla compattezza di T . (2) Visto che | f n , T u | ≤ f n T u ≤ T per ogni u ∈ E ≤1 , troviamo ϕ n ∞ ≤ T e |ϕ n (x) − ϕ n (x ′ )| ≤ |x − x ′ | per ogni n ∈ N ed x, x ′ ∈ X .
(3) Usando Ascoli-Arzelá troviamo una sottosuccessione {ϕ n k } uniformemente convergente, per cui 0
h,k ← ϕ n h − ϕ n k ∞ ≥ sup u∈E ≤1 | f n h , T u − f n k , T u | = T * f n h − T * f n k .
(4-5) Si argomenta in modo analogo ai punti precedenti, usando stavolta successioni {v n } ⊂ E ≤1 . (2) Supponiamo per assurdo che E abbia dimensione infinita: allora esiste una successione di sottospazi propri V n ⊂ V n+1 , n ∈ N. Usando il punto precedente riusciamo a costruire una successione {u n ∈ (V n ) 1 } con d(u n+1 , V n ) ≥ 1/2 e quindi u n − u m ≥ 1/2 , m < n. Ció contraddice l'ipotesi di compattezza di E 1 . (4)). (4) Si mostri che, preso n ∈ N, risulta indS n = −n , ind(S * ) n = n , dove ind denota l'indice nel senso di (7.33).
= {x = {x k } ∈ l ∞ : ∃ lim k x k } ed al funzionale F ∈ E * , F, x := lim k x k ).
Esercizio 7.8 (Misure e rappresentazioni). Sia X uno spazio compatto di Hausdorff. Presa una misura di Radon µ su X consideriamo lo spazio di Hilbert H µ := L 2 µ (X) e, per ogni f ∈ C(X), definiamo l'operatore lineare
π µ (f ) : H µ → H µ : π µ (f )v := f v , v ∈ H µ . (7.85) (1) Si mostri che π µ (f ) := sup v∈H µ,≤1 π µ (f )v ≤ f ∞ ; (2) Si verifichi che (7.85) definisce una rappresentazione π µ : C(X) → B(H µ )
di C(X) (ovvero, si mostri che π µè lineare e che π µ (f g) = π µ (f )π µ (g), ∀f, g ∈ C(X)); (3) Si verifichi che, preso v ∈ H µ , l'insieme [v] := {π µ (f )v, f ∈ C(X)}è un sottospazio vettoriale di H µ ; (4) Sia ν una misura di Radon assolutamente continua rispetto a µ; si dimostri che esiste un operatore lineare T ∈ B(H ν , H µ ) tale che T w = w , ∀w ∈ H ν , T • π ν (f ) = π µ (f ) • T , ∀f ∈ C(X) . (Suggerimenti: (4) Il teorema di Radon-Nikodym fornisce una funzione positiva F ∈ L 1 µ (X) tale che νE = E F dµ, E ∈ M ; si osservi quindi che F 1/2 w ∈ H µ per ogni w ∈ H ν ; (6) Si prenda la funzione v 0 (x) := 1 , x ∈ X , e si usi Prop.5.7.) Esercizio 7.9. Sia X uno spazio compatto e di Hausdorff, ed A := C(X, C). Si mostrino i seguenti punti: (1) Data f ∈ C(X), si ha σ(f ) = f (X) (ovvero, lo spettro di f coincide con la sua immagine); (2) per ogni x ∈ X , si mostri che l'applicazione
ω x : A → C , ω x , f := f (x) , ∀f ∈ A ,
e un carattere di A; (3) Usando i punti precedenti, si mostri che l'applicazione X → A , x → ω x , e un omeomorfismo. (1) Per ogni λ ∈ T si mostri che l'applicazione
ω λ : A → C : ω λ , f := f (t)λ −t dt , ∀f ∈ Á e un carattere di A e si verifichi che ω λ ≤ 1 .
(2) Data f ∈ A, verificare che la funzione f (λ) := ω λ , f , λ ∈ T,é continua e limitata. (3) Presa g ∈ L ∞ (R, C) si consideri F g ∈ A * e si mostri che affinché si abbia F g ∈ A , deve essere g(t) = g(−t), g(t + s) = g(t)g(s), per ogni t, s ∈ R. Si concluda che g(t) = λ t per qualche λ ∈ T.
(Suggerimenti: per il punto (1) si usino i Teoremi di Fubini-Tonelli ed il Teorema di convergenza dominata. Per il punto (2) si mostri che l'applicazione {T ∋ λ → ω λ ∈ A}è continua e poi si usi la definizione di topologia * -debole. Per il punto (3) si scriva esplicitamente F g , f := f g e si impongano le condizioni (7.56).) Esercizio 7.11. Sia {f n } ⊂ L 1 (R) un'identitá approssimata (vedi Def.6.25). (1) Si mostri che i funzionali F n , g := f n g , g ∈ C 0 (R) , sono lineari e continui, ovvero {F n } ⊂ C 0 (R) * . (2) Si verifichi che F n * → δ 0 , dove δ 0 ∈ C 0 (R) * è la delta di Dirac. Esercizio 7.12. Si fissi p = 2 nell'Esempio 7.6 e si calcoli lo spettro dell'operatore T a ∈ B(l 2 C ), a ∈ l ∞ C . Si trovino le condizioni su a tali che T a sia compatto. (1) Si verifichi che πè iniettiva, e posto R := π(L ∞ µ (X, C)) si verifichi che Rè una * -sottoalgebra di B(H); (2) Si consideri la funzione u 0 ∈ H , u 0 (x) := 1 , ∀x ∈ X , e preso T ∈ B(H) si ponga ϕ := T u 0 ∈ H . Si dimostri che se T ∈ R ′ (ovvero T π f = π f T , ∀f ∈ L ∞ µ (X, C)), allora
T f = f ϕ , ∀f ∈ L ∞ µ (X, C) ⊂ H .
(3) Si mostri che in realtá ϕ ∈ L ∞ µ (X, C), con ϕ ∞ ≤ T . (4) Si mostri che T = π ϕ , cosicché R ′ ⊆ R. (5) Si verifichi che R ⊆ R ′ , cosicché R = R ′ , e si concluda che Rè un'algebra di von Neumann.
(Suggerimenti: (1) Se π f = 0 allora in particolare π f u 0 = f = 0 ; (2) Imponendo la condizione T π f = π f T si trova π f T u 0 = T π f u 0 = T f ; (3) Scrivendo esplicitamente la disuguaglianza T f 2 2 ≤ T 2 f 2 2 si trova 0 ≤ |ϕ| 2 |f | 2 ≤ T 2 |f | 2 , ∀f ∈ L ∞ µ (X, C) ⊂ H , e da quest'ultima segue facilmente la stima cercata; (4) Si usi il punto (2) e la densitá di L ∞ µ (X, C) in H (vedi Prop.5.6); (5) Per l'inclusione R ⊆ R ′ si usi il fatto che L ∞ µ (X, C)è un'algebra commutativa, e per il fatto che Rè chiusa nella topologia debole si usi l'esercizio 7.13.)
T 2 k = T 2 k , ∀k ∈ N .
(2) Usando il punto precedente, dimostrare che non possono esistere P, Q ∈ B(H) autoaggiunti che soddisfino le relazioni di Heisenberg (Suggerimenti: riguardo (1) si osservi che l'identitá da dimostrareè certamente vera nel caso k = 1 , in quanto T 2 = T * T (vedi (7.22)), e si proceda per induzione. Riguardo (2) si argomenti per assurdo e si osservi che (7.86) implica P Q n − Q n P = −inQ n−1 , ∀n ∈ N, cosicché
P Q − QP = −i1 .(2 k + 1) Q 2 k (1) = (2 k + 1) Q 2 k ≤ 2 P Q 2 k +1 , ∀k ∈ N .
Riguardo (3) (1) Si mostri che F g ∈ D per ogni g ∈ V := {g ∈ H : (u 0 , g) = 0} , e che se f ∈ Hè ortogonale a tutte le funzioni in V allora fè costante;
{f (T ) − f n (T )}u 2 = (u, {f (T ) − f n (T )} * {f (T ) − f n (T )}u) = σ(D,T ) g n dµ uu , dove g n := (f − f n ) * (f − f n ) ∈ L ∞ β (σ(D, T )) , ∀n ∈ N ,
(2) Usando il punto precedente e (7.15), si mostri che (F T * v, g) = (v, g) , ∀v ∈ D * , g ∈ V ;
(3) Si mostri quindi che per ogni v ∈ D * esiste c ∈ C tale che v = F T * v + c, cosicché se v ∈ D * allora vè la primitiva di una funzione in H ; (4) Si mostri che
v(1) − v(0) = i(v ′ , u 0 ) = 0 , ∀v ∈ D * .l'equazione u ′ − iλu = f , u ∈ D , ha soluzione u(x) = e −iλx f λ (1) e iλ − 1 + f λ (x) , x ∈ [0, 1] .
Dunque (D, T −λ1)è suriettivo e quindi biettivo. Infine, osservando che f λè continua concludiamo facilmente che l'operatore
{S λ f }(x) := e −iλx f λ (1) e iλ − 1 + f λ (x) , x ∈ [0, 1] , f ∈ L 2 ([0, 1], C) ,
e limitato edè in effetti l'inverso di (D, T − λ1).) Esercizio 7.19 (Misure e rappresentazioni, II). Sia X uno spazio compatto e di Hausdorff con famiglia di boreliani βX , e L ∞ β (X, C) la C*-algebra delle funzioni limitate e boreliane su X . Si assuma che esiste uno spazio di Hilbert complesso H ed una rappresentazione
π : L ∞ β (X, C) → B(H) , tale che f n n → f puntualmente ⇒ π(f n ) n
→ π(f ) nella topologia forte (vedi Esempio 7.24 (5)). (1) Si mostri che per ogni coppia u, v ∈ H esiste una misura boreliana µ uv : βX → C, reale per u = λv , λ ∈ R, e complessa altrimenti, tale che
X f dµ uv = (u, π(f )v) , ∀f ∈ L ∞ β (X, C) .
(2) Si consideri un compatto del tipo (Suggerimenti: (1) Si osservi che ogni funzione caratteristica χ E , E ∈ βX ,è boreliana e che π(χ E )è un proiettore; si definisca quindi, per ogni u, v ∈ H ,
X := I∪S ⊂ R ,(7.µ uv E := (u, π(χ E )v) , ∀E ∈ βX .
(2) Si definisca la misura
ω : βX → R , ωE := µ(E ∩ I) + ν(E ∩ S) ,
dove µè la misura di Lebesgue e ν quella di enumerazione, e l'operatore In questa sezione esponiamo i fondamenti dell'analisi di Fourier, in particolare sviluppi in serie e trasformate, accennando poi alle sue generalizzazioni nell'ambito dei gruppi topologici, che formano l'oggetto di studio dell'analisi armonica astratta.
T f (x) := xf (x) , ∀x ∈ X , f ∈ H := L 2 ω (X, C) ,
Serie di Fourier.
Consideriamo una successione uniformemente convergente del tipo
f (x) := a 0 2 + +∞ k=1 {a k cos kx + b k sin kx} , x ∈ R . (8.1)
Vista la periodicitá delle funzioni trigonometriche, la funzione fè completamente determinata dai suoi valori nell'intervallo [−π, π] (o -equivalentemente -su [0, 2π]). Ci chiediamo ora come la condizione di uniforme convergenza influisce sulle proprietá dei coefficienti a k , b k . A tale scopo, richiamiamo le relazioni π −π cos kx cos mx dx = δ km π , π −π cos kx sin mx dx = 0 , π −π sin kx sin mx dx = δ km π , (8.2) dove δ kmè il simbolo di Kronecker. Moltiplicando (8.1) prima per cos mx, poi per sin mx, m ∈ N, ed integrando sull'intervallo [−π, π] troviamo
a k = 1 π π −π f (x) cos kx dx , b k = 1 π π −π f (x) sin kx dx , a 0 = 1 π π −π f (x) dx . (8.3)
Con le precedenti espressioni per i coefficienti a k , b k , chiamiamo (8.1) lo sviluppo in serie di Fourier di f . Sorge in modo naturale la questione di quali funzioni ammettano uno sviluppo in serie di Fourier, e di che tipo di convergenza (puntuale, uniforme, ...) questa abbia. Riguardando (8.3), risulta chiaro che una condizione necessariaè f ∈ L 1 ([−π, π]). Ora, (8.2) suggerisce che possiamo riguardare B := {cos mx, sin kx , k, m ∈ N} come un insieme ortogonale di funzioni per il prodotto scalare di L 2 ([−π, π]); in realtá, come sará chiaro alla fine della sezione, Bè in effetti una base per L 2 ([−π, π]). Come primo passo, ci dedicheremo alla dimostrazione di teoremi di convergenza della serie di Fourier per funzioni regolari a tratti. Il seguente lemma, inerente lo "sviluppo di Fourier" della funzione costante 1 , si dimostra per induzione, e ne omettiamo la dimostrazione.
Lemma 8.2. Sia f ∈ L 2 ([−π, π])
. Allora
a 2 0 2 + ∞ k=1 (a 2 k + b 2 k ) ≤ 1 π π −π f (x) 2 dx . (8.6)
Osserviamo che sapendo che Bè una base ortonormale (8.6) diventa un'uguaglianza. Usando (4.48), otteniamo immediatamente la forma classica del Lemma di Riemann-Lebesgue:
Lemma 8.3. Sia f ∈ L 1 ([−π, π]). Allora lim k→∞ π −π f (x) cos kx dx = lim k→∞ π −π f (x) sin kx dx = 0 . (8.7)
Prima di procedere introduciamo le seguenti convenzioni. Data f : R → R definiamo, qualora esistano, i limiti
f + (x) := lim t→x + f (t) , f − (x) := lim t→x − f (t) , f (x) := 1 2 (f + (x) − f − (x)) , x ∈ R .
Una funzione f : R → R sia dice periodica con periodo a > 0 se f (x + a) = f (x), x ∈ R. E' chiaro che se fè una funzione sviluppabile in serie di Fourier, alloraè periodica con periodo 2π . Una proprietá elementare delle funzioni periodicheè la seguente: Dimostrazione. Sia n ∈ N ed S n la somma parziale n-esima della serie di Fourier di f . Allora S n (x) = 1 π π −π f (t) 1 2 + n k=1 cos kt cos kx + sin kt sin kx dt
a −a f = a−x −a−x f , x ∈ R .= 1 π π −π f (t) 1 2 + n k=1 cos k(t − x) dt u:=t−x = 1 π π+x −π−x f (x + u) 1 2 + n k=1
cos ku du Osservazione 8.1. Lo spazio delle funzioni continue e regolari a tratti su [−π, π]è denso in L 2 ([−π, π]) nella norma · 2 e ció implica che B neè effettivamente una base ortonormale. Inoltre, osservando che
1 π π −π f (t) 2 dt = k (a 2 k + b 2 k ) ≤ k (k 2 a 2 k + k 2 b 2 k ) ,
da (8.10) otteniamo una versione della diseguaglianza di Poincaré (10.8).
Esempio 8.1. Consideriamo la funzione f ♯ con periodo 2π che prolunga f (x) := x, x ∈ [−π, π]. Allora abbiamo lo sviluppo b k = − 2 k cos kπ = (−1) k+1 2 k ⇒ f ♯ (x) = x = 2 k=1 (−1) k+1 sin kx k (Poiché f ♯è dispari, otteniamo una serie di soli seni). Definendo invece f (x) := x 2 , otteniamo una funzione pari f ♯ con sviluppo di Fourier f ♯ (x) = x 2 = π 2 3 + 4 ∞ k=1 (−1) k cos kx k 2 .
Le espressioni precedenti possono essere usate per calcolare esplicitamente la somma delle serie numeriche che si ottengono fissando dei valori di x. Si veda ad esempio i casi x = π, π/2 . L'equazione del calore. Siano ω ∈ R ed f ∈ C 2 ([−π, π]); consideriamo il problema alle derivate parziali
∂ t u = ω 2 ∂ xx u u(x, 0) = f (x) , x ∈ [−π, π] , t ∈ [0, +∞) .∂ x u = k ke −k 2 ω 2 t {−a k cos kx + b k sin kx} ⇒ ∂ xx u = − k k 2 e −k 2 ω 2 t {a k cos kx + b k sin kx} ,
e quindi uè una soluzione di (8.11). I termini a k , b k , k ∈ N, si possono determinare imponendo la condizione iniziale:
f (x) = u(x, 0) = a 0 2 + k {a k cos kx + b k sin kx} per cui a k = 1 π π −π f (x) cos kx dx , b k = 1 π π −π f (x) sin kx dx .
Il richiedere la convergenza della serie per ∂ xx u equivale a richiedere f ∈ C 2 ([−π, π]). In maniera analoga, possiamo risolvere prolemi alle derivate parziali con condizioni miste piuttosto che condizioni iniziali.
Osservazione 8.2 (Il nucleo del calore). Consideriamo il problema (8.11) nel caso ad n dimensioni (ovvero, ∂ t u = ω 2 ∆u , u :
R n × R + → R, con u(·, 0) = f , f : R n → R) e la funzione Φ(x, t) := 1 (4πω 2 t) n/2 e −|x| 2 /(4ω 2 t) , (x, t) ∈ R n × R + .
Osserviamo che Φ / ∈ L 1 loc (R n × R + ), tuttavia si puó verificare che integrando formalmente rispetto ad x otteniamo la famiglia di distribuzioni
F t ∈ D * (R n ) , t ∈ R : F t , f := f (y)Φ(y, t) dy , f ∈ D(R n ) .
In particolare per t = 0 abbiamo la delta di Dirac, F 0 = δ 0 , per cui effettuando la convoluzione troviamo F 0 * f = f . Inoltre, per t > 0 abbiamo ∂ t Φ = ω 2 ∆Φ, dunque ponendo u(x, t) := F t * f otteniamo una soluzione di (8.11), come si verifica derivando formalmente l'espressione esplicita di u :
u(x, t) := f (y)Φ(t, x − y) dy , x ∈ R n , t > 0 .
Per questo motivo Φè detta la soluzione fondamentale, o nucleo, dell'equazione del calore. Nel caso n = 1 , sviluppando una soluzione u = u(x, t) in serie di Fourier (si veda (8.12) e la dimostrazione di Teo.8.4), troviamo
u(x, t) = 1 π f (y) 1 2 + k e −k 2 ω 2 t cos k(x − y) dy , (x, t) ∈ R 2 , per cui Φ(x, t) = 1 2π + 1 π k e −k 2 ω 2 t cos kx , (x, t) ∈ R 2 ,
esprime lo "sviluppo di Fourier" della soluzione fondamentale (vedi (8.27) per un significato preciso dell'espressione precedente). Osservare che la serie nell'espressione precedente non converge per t = 0 .
La trasformata di Fourier.
Da un punto di vista intuitivo la trasformata di Fourier puó essere vista come un analogo continuo delle serie di Fourier, oppure come una "continuazione analitica" della trasformata di Laplace (si veda (6.43)). In questa sezione esporremo in buon dettaglio il caso della retta reale, limitandoci ad accennare ai casi piú generali di R d , d > 1 , e dei gruppi localmente compatti abeliani. Iniziamo con il definire, per ogni f ∈ L 1 (R, C)
f (x) := 1 √ 2π R f (t)e ixt dt , x ∈ R . (8.13)
Innanzitutto osserviamo che l'espressione precedenteè ben definita per ogni x ∈ R in quanto
| f (x)| ≤ 1 √ 2π f 1 , x ∈ R . (8.14)
Elenchiamo alcune proprietá elementari della trasformata di Fourier:
• (f + ag) = f + a g , f, g ∈ L 1 (R, C), a ∈ R;
• f ∈ C 0 (R, C); Infatti, sia {x n } ⊂ R con x n → x; poiché |e ixnt f (t)| = |f (t)|, possiamo applicare il teorema di Lebesgue e concludere che lim n f (x n ) = f (x). Il fatto che f svanisce all'infinito segue dal lemma di Riemann-Lebesgue (si veda (4.48)).
• f ∞ ≤ (2π) −1/2 f 1 ; Ció segue da (8.14).
• (2π) −1/2 (f * g) = f · g ; Infatti, basta usare il teorema di Fubini in modo analogo a (6.44), avendosi f, g ∈ L 1 (R, C).
Nelle righe che seguono stabiliremo alcune proprietá di una successione di funzioni che giocherá un ruolo importante per le trasformate di Fourier, costruita a partire dalla cosiddetta misura Gaussiana. Innanzitutto, definiamo ρ(x) := 1 √ 2π e −x 2 /2 , x ∈ R ⇒ ρ ∈ C ∞ 0 (R) ∩ p∈ [1,+∞] L p (R) , e dimostriamo le seguenti proprietá:
ρ(x) = ρ(x) , x ∈ R . (8.15)
ρ n (x) := nρ(nx) , x ∈ R , n ∈ N ⇒ ρ n 1 ≡ 1 .
ρ n * f (x) = 1 √ 2π lim m→∞ m k=0 (−1) k n 2k+1 k! R (x − t) 2k f (t) dt .
Notare che il termine a destraè un polinomio in x di grado 2m;è questa l'essenza della dimostrazione del teorema di densitá di Weierstrass ([12, Teo.2.8.1]).
Con la seguente notazione, introduciamo l'applicazione nota come antipodo:
ǫf (x) := f (−x) , x ∈ R , f ∈ L 1 (R, C) ⇒ ǫf (x) = f (x) , x ∈ R .
Teorema 8.7 (Parseval). Se f ∈ L 1 (R, C) ∩ L 2 (R, C) allora f ∈ L 2 (R, C) e f 2 = f 2 .
Dimostrazione. Ponendo g := f * ǫf , troviamo g ∈ L 1 (R, C), e
g(y) = R f (y − x)f (−x) dx ⇒ g(0) = f 2 2 .
Inoltre,
g(x) = (f * ǫf ) (x) = f (x) f (x) , x ∈ R ⇒ R g = f 2 2 . (8.21)
Essendo f, ǫf ∈ L 2 (R, C), usando l'Esercizio 5.2 troviamo che gè continua, e grazie a Prop.6.24 lo stessoè vero per ogni g * ρ n , n ∈ N. Usando (8.19) troviamo g * ρ n − g 1 n → 0 , per cui l'Esercizio 4.8 implica lim n g * ρ n (x) = g(x) , x ∈ R .
In particolare, lim n g * ρ n (0) = g(0) = f 2 2 .
D'altra parte, usando (8.18) troviamo g * ρ n (x) = R g(t) ρ t n e −ixt dt , x ∈ R ⇒ g * ρ n (0) = Osserviamo ora che -essendo L 1 ∩ L 2 denso in L 2 -la trasformata di Fourier si estende ad un operatore F ∈ BL 2 (R, C) , il quale, grazie al Teorema di Parseval,è isometrico. Con il prossimo teorema dimostriamo che F e in effetti un operatore unitario.
Teorema 8.10 (Teorema di Fourier-Plancherel). L'estensione F della trasformata di Fourierè un operatore unitario di L 2 (R, C) in sé.
Dimostrazione. Visto che giá sappiamo che Fè isometrico l'unica proprietá che occorre verificarè e la suriettivitá. Per ogni g ∈ L 2 (R, C) definiamo g n := gχ [−n,n] , n ∈ N, ed osserviamo che g n ∈ L 1 (R, C) ∩ L 2 (R, C) (Esercizio 5.3); per cui,è ben definita g n ∈ C 0 (R, C). Il primo passo della dimostrazione sará quello di verificare che ponendȏ g(x) := lim n 1 √ 2π g n (t)e −ixt dt = lim n g n (−x) , q.o. in x ∈ R , (8.25) otteniamo una ben definita funzione in L 2 (R, C). Alché, mostreremo che {g →g} fornisce l'inversa di F . A tale scopo, osserviamo che chiaramente lim n g − g n 2 = 0 per cui -per isometria di Ftroviamo lim n F g − g n 2 = 0 . Applicando Fischer-Riesz e (8.25) troviamo F g(x) = lim k g n k (x) =g(−x) , q.o. in x ∈ R , e dunqueg(x)è ben definito q.o. in x ∈ R ed ivi coincidente con F g(−x). Ció implica |g(x)| 2 dx = |F g(−x)| 2 dx ⇒ g 2 = F g 2 = g 2 < +∞ .
Abbiamo quindi costruito un operatore isometrico F ′ ∈ BL 2 (R, C), F ′ g :=g . Mostriamo ora che F ′ Fè l'identitá su L 2 (R, C); a tale scopo, per densitá,è sufficiente verificare solo per g ∈ L 1 (R, C) ∩ L 2 (R, C), e si ha g * ρ n (x) ( 8.18) = R g(t)ρ t n e −itx dt ( 8.17) = R g(t) ρ n (t)e −itx dt = 1 √ 2π (g * ρ n ) (t) e −itx dt .
L'uguaglianza precedente si puó leggere come g * ρ n = F ′ F (g * ρ n ), n ∈ N. Poiché per (8.20) si ha lim n g − g * ρ n 2 = 0 , otteniamo g = F ′ F g , ed il teoremaè dimostrato.
La trasformata di Fourier in R d . La teoria della trasformata di Fourier si generalizza facilmente al caso R d , d ∈ N. Presa f ∈ L 1 (R d , C) (misura prodotto di Lebesgue) definiamo
f (x) := (2π) −n/2 R d f (t)e ix·t dt , x ∈ R d , dove
x · t denota il prodotto scalare. Gli strumenti di lavoro principali della sezione precedente, le convoluzioni e le misure gaussiane, si utilizzano senza problemi in questo caso piú generale, le prime senza variazioni e le seconde definendo ρ n (x) := (2π) −n/2 e −|x| 2 /2 , x ∈ R d .
In modo analogo al caso d = 1 possiamo definire l'antitrasformatȃ f (x) := f (−x) , ∀x ∈ R d , f ∈ L 1 (R d , C) .
I risultati principali della sezione precedente rimangono, ovviamente, veri:
Teorema 8.11 (Parseval, Fourier, Plancherel). La trasformata di Fourier definisce un operatore lineare limitato L 1 (R d , C) → C 0 (R d , C) , f → f , (8.26) il quale ha inverso {g →g} in C 0 (R d , C) ∩ L 1 (R d , C). Inoltre
f 2 = f 2 , ∀f ∈ L 1 (R d , C) ∩ L 2 (R d , C) ,
e (8.26) si estende ad un operatore unitario F ∈ BL 2 (R d , C).
La dimostrazione del teorema precedente si effettua adattando le tecniche del caso unidimensionale con un uso massiccio del teorema di Fubini. Per dettagli in merito segnalamo [23, Chap.IX], dove un approccio leggermente diverso rispetto a quello della sezione precedente viene adottato con l'uso dello spazio S(R d , C) delle funzioni complesse a decrescenza rapida. dove d : T × T → Rè la metrica di T (il qualeè omeomorfo al cerchio). E' possibile dimostrare che G * è a sua volta localmente compatto e di Hausdorff (oltre che, ovviamente, abeliano, ovvero, χχ ′ = χ ′ χ per ogni χ, χ ′ ∈ G * ). Diamo alcuni esempi fondamentali di gruppi di caratteri:
Osserviamo che il teorema precedente si applica solo nel caso in cui G sia abeliano; infatti un generico gruppo topologico potrebbe essere privo di caratteri non banali, come ad esempio il gruppo SU (2) delle matrici complesse 2 × 2 con determinante 1 .
Denotiamo ora con µ la misura di Haar di G (vedi §4.1) e definiamo, per ogni f ∈ L 1 µ (G, C), Analogamente al caso G = R, usando il teorema di convergenza dominata troviamo che fè una funzione continua, cosicché abbiamo un analogo astratto della trasformata di Fourier. Il fatto che (quando G * nonè compatto) f svanisce all'infinitoè piú delicato da dimostrare, ma comunque vero, cosicché f ∈ C 0 (G * , C), cosí come rimangono veri, nel caso G abeliano, i teoremi di Fourier, Parseval e Plancherel, con la modifica che stavolta abbiamo un operatore unitario si ottiene un gruppo ad un parametro. Infine, si mostri che T f u = f u , ∀f ∈ L 1 (R, C) , u ∈ H .
(Suggerimenti: Per (1) si osservi che, essendo ogni U t unitario, troviamo |(u, U t v)| ≤ u v e quindi A f (u, v) ≤ f 1 u v . Per (2), in particolare l'uguaglianza che coinvolge la convoluzione f * g , si confrontino i prodotti scalari (u, T f * g v) e (u, T f T g v) usando (7.76) ed il teorema di Fubini. Per (3) si usino i teoremi di Lebesgue e di Fubini).
Esercizio 8.6. Si mostri che ϕ ∈ C ∞ 0 (R, C) ∩ L 2 (R, C) per ogni ϕ ∈ C ∞ c (R, C). (Suggerimenti: si usi il Teorema di Parseval. Per la differenziabilitá si usi il Teorema 4.37).
Esercizio 8.7 (Trasformata di Fourier e derivate deboli). Si consideri la trasformata di Fourier come un operatore unitario F ∈ BL 2 (R, C), cosicché F f = f , F * f =f , ∀f ∈ L 1 (R, C) ∩ L 2 (R, C). (1) Si determini l'operatore autoaggiunto (D, T ) associato, nel senso di (7.78), al gruppo ad un parametro (8.32); (2) Si mostri che { U t := F U t F * }è , a sua volta, un gruppo ad un parametro e se ne dia un'espressione esplicita sul sottospazio L 1 (R, C) ∩ L 2 (R, C); (2) Per ogni f ∈ L 1 (R, C) ∩ L 2 (R, C) e t, x ∈ R si trova
{ U t f }(x) = {F U tf }(x) = = e itsf (s)e ixs ds = = f (λ)e is(t+x−λ) dλds = = f (x + t) ,
per cui U tè un operatore di traslazione (Esempio 7.9) e l'operatore autoaggiunto associatoè l'estensione autoaggiunta della derivata (Esempio 7.39). (3) Semplici manipolazioni algebriche. (4) Si calcoli la trasformata di Fourier di T ϕ usando il punto (1)).
Analisi Complessa.
Consideriamo un'applicazione f : U → C, con U ⊆ C aperto. Scrivendo, per ogni z ∈ U , z = x + iy , x, y ∈ R, ed f (z) = u + iv , u, v ∈ R, troviamo che f puó essere riguardata come una coppia di funzioni reali di due variabili reali, cosicché scriviamo f (z) ≡ f (x, y) ≡ u(x, y) + iv(x, y) .
Dunque valgono per le funzioni complesse tutti i risultati dimostrati per le funzioni di due variabili reali a valori in R 2 , in particolare quelli inerenti le forme differenziali ( §6.4). Tuttavia l'analisi delle funzioni di variabile complessa presenta delle importanti peculiaritá, e gioca un ruolo importante in svariati ambiti, dalla geometria (si pensi ad esempio al teorema di Riemann-Roch) all'ingegneria elettrica, per non parlare della teoria quantistica dei campi.
Serie di potenze e funzioni analitiche.
Una serie di potenze si presenta nel seguente modo: ∞ n=0 a n z n , a n ∈ C .
(9.5)
La questione della convergenza di una serie di potenzeè completamente risolta dal seguente Teorema 9.3 (Abel). Data una serie di potenze del tipo (9.5), esiste un reale esteso r ∈ [0, +∞], detto raggio di convergenza, avente le seguenti proprietá: per ogni z ∈ C con |z| < r , risulta che n a n z nè assolutamente convergente, e quindi convergente; se ρ ∈ (0, r), allora (9.5) converge uniformemente per ogni z tale che |z| ≤ ρ; se z ∈ C e |z| > r , allora (9.5) non converge.
Dimostrazione. Definiamo r tramite la formula di Hadamard Se |z| < r , allora per ogni ρ ∈ (|z|, r) si ha chiaramente |z| r < ρ r < 1 , |z| ρ < 1 .
Inoltre, per definizione di r , esiste n 0 ∈ N tale che 1 ρ > 1 r > |a n | 1/n , n > n 0 .
Dunque, 1 > |z| ρ n > |a n ||z| n , per cui (9.5)è assolutamente convergente (e quindi convergente).
Definizione 9.4. Sia U ⊆ C un aperto. Una funzione f : U → C si dice analitica in U se per ogni ζ ∈ U esiste un disco ∆ = ∆(ζ, r), ∆ ⊆ U , tale che f (z) = ∞ n=0 a n (z − ζ) n , z ∈ ∆ , (9.7)
per opportuni coefficienti a n ∈ C.
Allo scopo di iniziare a chiarire la natura delle funzioni analitiche, ed in particolare dei coefficienti a n , n ∈ N, diamo il seguente 1 n! f (n) (ζ)(z − ζ) n , z ∈ ∆ . (9.8)
Integrazione complessa.
In realtá le proprietá di analiticitá ed olomorfia sono del tutto equivalenti. Il fatto che ogni funzione olomorfaè analitica si dimostra utilizzando il teorema di Cauchy, il quale stabilisce l'annullarsi dell'integrale di una funzione olomorfa su una curva chiusa. ). Infine, diremo che la curva continua γè regolare a tratti se I puó essere suddiviso in un numero finito I 1 , . . . , I n di sottointervalli chiusi tali che ogni restrizione γ| I k , k = 1, . . . , n, sia regolare. In tal caso γ ′ puó essere comunque definita su tutto I , estendendo per continuitá a sinistra in ogni estremo destro di I k , k = 1, . . . , n, per cui γ ′ risulta essere continua a tratti. Preso un aperto U ⊂ C, denotiamo con C reg (I, U ) l'insieme delle curve regolari a tratti su I con immagine contenuta in U . La funzione integranda f •γ(t)·γ ′ (t)è per definizione continua a tratti, per cui l'integrale precedente esiste giá nel senso di Riemann. Si noti che cambiando l'orientazione di γ , il che corrisponde ad effettuare il cambio di variabile t → γ(t) := γ(b + a − t), t ∈ I , si ottiene l'inversione di segno
γ f dz = − b a f • γ(b + a − t) · γ ′ (b + a − t) dt = − b a f • γ(s) · γ ′ (s) ds = − γ f dz .
Infine, osserviamo che γ f dz si puó esprimere anche in termini di integrali di forme differenziali, definendo la 1 -forma ω f : U → R 2, * ≃ R 2 ≃ C , ω f (x, y) := f (x + iy){dx + i dy} , e procedendo per integrazione di forme differenziali: infatti, usando (6.25) con m = 1 , n = 2 , si verifica immediatamente che (9.12) coincide con l'integrale di ω f . Nel seguito, seguendo una notazione standard, scriveremo ω f ≡ f dz . Allora γ dz z − ζ = 2πi . (9.14)
Per la verifica, osserviamo che l'integrale precedenteè ben definito perché f (z) := (z − ζ) −1è olomorfa per z ∈ C − {ζ} ; dunque f • γ(t) = ε −1 e −2πit e γ ′ (t) = ε2πie 2πit , cosicché (9.14) si riduce all'integrale su I della funzione costante 2πi , il quale evidentementeè uguale proprio a 2πi .
Domini regolari ed il teorema di Cauchy. Un dominio regolareè un aperto limitato U ⊂ C il cui bordo ∂Uè costituito da un numero finito di curve chiuse, semplici, e regolari a tratti. Osserviamo che dall'ipotesi di limitatezza di U segue che C − U possiede una, ed una sola, componente connessa non limitata V ; chiamiamo frontiera esterna di U l'insieme di curve ∂U est := V ∩ U . Chiamiamo invece frontiera interna l'insieme di curve ∂U int := ∂U − ∂U est . Per distinguere la frontiera interna da quella esterna si prende orientazione antioraria per le curve in ∂U est , ed oraria per quelle in ∂U int . Allora ∂U estè la circonferenza di raggio 4 centrata in 0 , con orientazione antioraria, mentre ∂U int e l'unione delle circonferenze di raggio 1 centrate, rispettivamente, in 2 e −2 , orientate entrambe in senso orario.
Ora, per additivitá dell'integrale di Riemann troviamo che se ∂U = γ 1∪ . . .∪γ n , dove ogni γ k , k = 1, . . . , n,è una curva chiusa, semplice e regolare a tratti, allora
∂U f dz = k γ k f dz .
Sottolineiamo il fatto che l'orientazione delle γ k della frontiera internaè opposta rispetto alle rimanenti altre, il che comporta, come abbiamo visto poc'anzi, il cambio di segno dei relativi integrali qualora volessimo scriverle con l'orientazione antioraria.
Dimostrazione. Supponiamo per assurdo che ζ ∈ U sia un punto di massimo per |f |. Allora esiste un disco ∆ di centro ζ e raggio ε tale che ∆ ⊆ U , e grazie alla formula di Cauchy troviamo Dimostrazione. Posto p(z) = n k=0 a k z k , consideriamo un arbitrario ρ > 0 ed osserviamo che, applicando ricorsivamente la diseguaglianza triangolare, per |z| > ρ abbastanza grande troviamo |p(z)| ≥ |z n | |a n | − n k=0 |a k | |z| n−k ≥ c|z| n , dove cè un'opportuna costante positiva. Se p fosse privo di zeri avremmo che p −1 sarebbe olomorfa, ed applicando il principio di massimo ad U = ∆ := ∆(0, ρ) troveremmo
f (ζ) = ∂∆ f (z) z − ζ dz = 1 2π 2π 0 f (ζ + εe iθ ) dθ ,
Funzioni meromorfe ed il teorema dei residui.
Sia U ⊆ C aperto ed f ∈ O(U ). Un punto z 0 ∈ U si dice zero di f di ordine n ∈ N se esiste un intorno V ∋ z tale che f (z) = a n (z − z 0 ) n + a n+1 (z − z 0 ) n+1 + . . . , a k = 0 , z ∈ V ; in termini equivalenti, esiste una funzione olomorfa h ∈ O(V ) tale che h(z 0 ) = 0 e f (z) = (z − z 0 ) n h(z) , z ∈ V . e sono tutti semplici. Scegliamo quindi r > 0 e γ come la curva il cui graficoè il rettangolo con vertici −r , r , r + πi , −r + πi .
In tal modo l'unico polo interno al grafico di γè ξ := 1/2πi ed il relativo residuoè
Res ξ e az e z + e −z dz = lim z→ξ (z − ξ) e az e z + e −z = −i/2e aiπ/2 . Gli spazi di Sobolev sono stati introdotti come strumento per la dimostrazione di teoremi di esistenza ed unicitá della soluzione di problemi differenziali con condizioni al bordo. L'idea di fondoè quella di combinare il concetto di derivata debole con la teoria degli spazi L p . In un certo senso la teoria degli spazi di Sobolev sovverte il punto di vista del classico calcolo variazionale ( §6.5). Se da un lato, classicamente, la minimizzazione di un funzionale veniva effettuata risolvendo un'equazione differenziale (l'equazione di Eulero-Lagrange), ora l'esistenza e l'unicitá della soluzione di un problema alle derivate parziali vengono dimostrate associando ad esso un problema variazionale. Il vantaggio di questo approccio risiede nel fatto che per portare a buon fine il procedimento di minimizzazione abbiamo a disposizione gli strumenti dell'analisi funzionale (in particolare, i Teoremi di Stampacchia-Lax-Milgram).
In questa sezione considereremo intervalli aperti (a, b) non necessariamente limitati, ovvero sono ammesse le possibilitá a, b = ±∞.
Proprietá di base.
Presi a, b ∈ R, consideriamo il problema differenziale Osserviamo che l'equazione precedente, soddisfatta da ogni soluzione di (10.1), ha senso piú in generamente per funzioni u ∈ C 1 (a, b). In questo modo, possiamo pensare di sostituire il problema iniziale (10.1) con la ricerca di una funzione u che soddisfi l'equazione (10.2), con il vantaggio che lo spazio delle possibili soluzioniè a priori molto piú grande (infatti, cerchiamo funzioni C 1 piuttosto che C 2 ). Dimostrazione. Supposto che esista w ∈ L p (a, b) che soddisfi (10.3) troviamo b a zϕ = 0 , ϕ ∈ C 1 0 (a, b), dove z := v − w ∈ L p (a, b). Per densitá di C 1 0 (a, b) in L p (a, b) concludiamo che il funzionale F z ∈ L q, * (a, b), q := p,è nullo, per cui z = 0 q.o. per dualitá di Riesz.
La funzione v del Lemma precedente si dice la derivata debole di u e nel seguito sará denotata con u ′ . Si osservi che u ′è la derivata debole di u anche nel senso delle distribuzioni (Esempio 7.27), con l'ulteriore proprietá di appartenere ad L p (a, b). L'insieme delle funzioni L p debolmente derivabiliè chiaramente uno spazio vettoriale; chiameremo tale spazio spazio di Sobolev, e lo denoteremo con
W 1,p (a, b) := {u ∈ L p (a, b) | ∃u ′ ∈ L p : − b a u ′ ϕ = b a uϕ ′ , ϕ ∈ C 1 0 (a, b)}
(osservare che il suffisso "1" suggerisce che stiamo considerando un analogo di C 1 ). Osserviamo che nella definizione precedente potremmo considerare equivalentemente ϕ ∈ C ∞ 0 (a, b) o ϕ ∈ C ∞ c (a, b).
Osservazione 10.1. Sia C c (a, b) lo spazio delle funzioni continue a supporto compatto in (a, b); alloraè chiaro che C c (a, b) ⊂ L p (a, b), ∀p ∈ [1, +∞]. Se u ∈ C 1 c (a, b) := C 1 (a, b) ∩ C c (a, b) allora la derivata di u (nel senso classico) appartiene a C c (a, b), e chiaramente essa coincide con la derivata debole. Per cui abbiamo applicazioni canoniche
C 1 c (a, b) → W 1,p (a, b) , ∀p ∈ [1, +∞] .
Tornando al nostro problema iniziale, osserviamo che affinché sia ben definito l'integrale del termine sinistro di (10.2)è sufficiente che sia u ∈ W 1,p (a, b) per un qualche p ∈ [1, +∞]. Inoltre, siamo passati da un problema differenziale ad uno integrale. Consideriamo ora la seguente norma su W 1,p : u W,p := u p + u ′ p , u ∈ W 1,p . Sketch della dimostrazione. La completezza segue osservando che se {u n } ⊂ W 1,p (a, b)è di Cauchy allora esistono u := lim n u n ∈ L p e u 1 := lim n u ′ n ∈ L p ; un passaggio al limite per l'uguaglianza b a u n ϕ ′ = − b a u ′ n ϕ , ϕ ∈ C 1 c ([a, b]) , mostra che effettivamente u 1è la derivata debole di u , dunque W 1,p (a, b)è completo. La riflessivitá segue considerando l'isometria canonica
W 1,p (a, b) → L p (a, b) × L p (a, b) , u → (u, u ′ ) ,
il che permette di esibire W 1,p (a, b) come un sottospazio chiuso dello spazio riflessivo L p (a, b) × L p (a, b). Poiché in generale un sottospazio chiuso di uno spazio riflessivoè riflessivo concludiamo che W 1,p (a, b)è riflessivo. Allo stesso modo, la separabilitá di W 1,p (a, b) segue dalla separabilitá di L p (a, b) × L p (a, b).
Osservazione 10.2. Consideriamo l'applicazione canonica
D : W 1,p (a, b) → L p (a, b) , u → u ′ .
Allora Dè un operatore lineare tale che Du p = u ′ p ≤ u W,p . Per cui gli spazi di Sobolev, analogamente a quanto accade nella teoria delle distribuzioni, sono atti a rendere la derivata (debole) un'applicazione continua. Presi a, b ∈ R diciamo che u ∈ W 1,p (a, b) si annulla al bordo se ció accade al suo rappresentante continuo, ed in tal caso scriveremo u ∈ W 1,p 0 (a, b). Nel seguito, identificheremo u col suo rappresentante continuo.
Osservazione 10.3. Siano a, b ∈ R ed u 1 ∈ L p (a, b) con p ∈ [1, +∞]. Poiché (a, b) ha misura finita abbiamo che u 1è integrabile (Cor.5.4) ed u := x a u 1 (t)dt, x ∈ [a, b],è assolutamente continua e quindi limitata. Ció implica u ∈ L p (a, b) e quindi u ∈ W 1,p (a, b) con derivata debole u 1 . Dunque, su intervalli limitati le funzioni in W 1,p (a, b) sono tutte e sole le primitive di funzioni L p . In particolare, Cosicché la seminorma n(u) := u ′ p , u ∈ W 1,p 0 (a, b),è equivalente a · W,p su W 1,p 0 (a, b). Usando i teoremi di densitá, troviamo che la precedente uguaglianzaè verificata per v ∈ H 1 0 (Ω), e quindi u cè soluzione debole. Viceversa, dimostrare che una soluzione debole u dè anche regolare (ovvero u d ∈ C 2 (Ω))è un risultato non banale; una volta dimostrato che u d ∈ C 2 (Ω) una semplice integrazione per parti permette di concludere che u dè una soluzione classica. Sketch della dimostrazione. Sia osserva che Kè un chiuso convesso in H 1 (Ω), non dipendente dã g ma solo da g . A questo punto, si applica il teorema di Stampacchia.
9. 4
4Funzioni meromorfe ed il teorema dei residui. . . . . . . . . . . . . . . . . . . . . . . 191 9.5 Esercizi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10 Cenni sugli Spazi di Sobolev. 198 10.1 Proprietá di base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.2 Immersioni compatte di W 1,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.3 Ordini e dimensioni generali. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.4 Applicazioni alle equazioni alle derivate parziali. . . . . . . . . . . . . . . . . . . . . 204 1 Introduzione.
Proposizione 2. 3 .
3Sia X uno spazio localmente compatto ed {f n } ⊂ C 0 (X) tale chef n − f m ∞
|f (x) − f nε (x)| + sup x∈X−Kε I risultati precedenti si estendono al caso complesso aggiungendo l'ipotesi che A sia chiusa rispetto al passaggio alla funzione coniugataf → f * : f * (x) := f (x) , ∀x ∈ X , f ∈ C(X, C) (2.3)
[a, b] → Rè continua ed f (a)f (b) < 0 , allora esiste x ∈ (a, b) tale che f (x) = 0 : Proposizione 2.11. (Teorema di Brouwer in dimensione uno) Sia f : [0, 1] → [0, 1] continua. Allora esiste x ∈ [0, 1] tale che f (x) = x.Dimostrazione. Se f (0) = 0 oppure f (1) = 1 non viè nulla da dimostrare, per cui assumiamo f (0) = 0 e f (1) = 1 . Applichiamo allora il teorema del valore intermedio a g(x) := f (x) − x, x ∈ [0, 1].Ora, si ha la seguente generalizzazione del risultato precedente: Teorema 2.12. (Teorema di Brouwer) Sia S ⊂ R n , n ∈ N, un insieme convesso, compatto e non vuoto, ed f : S → S continua. Allora esiste x ∈ S tale che f (x) = x.
F
α := {F : [0, 1] → R , F (x) := x 0 f (t) dt , x ∈ [0, 1] : f ∈ L α } e equicontinua e limitata.Esercizio 2.2. Sia α ∈ (0, 1) e τ ∈ C(R) una funzione con costante di Lipschitz α (ovvero: |τ (x) − τ (y)| ≤ α|x − y|, ∀x, y ∈ R). Scelto c > 0 , si mostri che l'applicazione T : C([0, 1]) → C([0, 1]) , f → T f : T f (s) := c + s 0 τ • f (t) dt , ∀s ∈ [0, 1] , e una contrazione. Si dimostri che esiste edè unica f 0 ∈ C([0, 1]) tale che f 0 = T f 0 . Si dimostri inoltre che f 0è derivabile e che f ′ 0 = τ • f 0 . Infine, si calcoli f 0 nel caso τ (x) = αx, ∀x ∈ R.
Esercizio 2. 5 .
5Sia X uno spazio metrico compatto ed {f n } ⊂ C(X) una successione precompatta e convergente puntualmente ad f : X → R. Si mostri che {f n } converge uniformemente ad f (cosicché fè continua). (Suggerimento: si supponga per assurdo che esiste ε > 0 tale che f − f k ∞ ≥ ε per ogni k appartenente ad un sottoinsieme infinito di N. Essendo {f n } precompatta, esiste una sottosuccessione {f ki } di {f k } uniformemente convergente; ma {f ki } deve necessariamente convergere ad f per convergenza puntuale di {f n } , il che fornisce la contraddizione cercata).Esercizio 2.6. Sia {f n : R → R} una successione di funzioni convesse puntualmente convergente ad f : R → R. Presi α < a < b < β , si verifichino le seguenti proprietá: (1) Le successionim n := (f n (a) − f n (α))(a − α) −1 , M n := (f n (β) − f n (b))(β − b) −1 , n ∈ N ,sono limitate; (2) Usando il punto precedente e Prop.4.48(1), si mostri che esiste L > 0 tale che |f n (x) − f n (y)| ≤ L|x − y| , ∀n ∈ N , x, y ∈ [a, b] .
( 3 )( 4 )
34Usando il punto precedente, si verifichi che esiste C > 0 tale che |f n (x)| ≤ C + L(b − a) , ∀n ∈ N , x ∈ [a, b] . Usando l'Esercizio 2.5 ed i punti precedenti, si verifichi che {f n | [a,b] } converge uniformemente. (Suggerimento: per il punto 4 si usi il teorema di Ascoli-Arzelá).Esercizio 2.7. Presa la successione di funzioni f n : R → R , n ∈ N : f n (x) := e −x 2 , |x| < n e −n 2 [1 − 2n(|x| − n)] , |x| ∈ [n, n + (2n) −1 ] 0 , |x| > n + (2n) −1 , si mostri che: (1) f n ∈ C c (R), ∀n ∈ N; (2) {f n } converge uniformemente ad f ∈ C 0 (R) − C c (R), f (x) := e −x 2 , x ∈ R.
(u 0
0, u 1 , . . . , u n−1 ) : I → R n , u 0 := u , u i := u ′ i−1 , i = 1, . . . , n − 1 , per cui in generale tratteremo problemi del tipo u ′ = f (t, u) , (3.1)
Definiamo quindi I 0 := (t 0 − r 0 , t 0 + r 0 ), e X := {u ∈ C(I 0 ) : u − u 0 ∞ < r ′ } .
F
: X → X , F u(t) := u 0 + t t0 f (s, u(s)) ds , t ∈ I 0 . (3.3)
Teorema 3 . 3 .
33Sia dato il problema di Cauchy (3.2) con f localmente Lipschitz nella seconda variabile. Allora ogni soluzione locale di (3.2) ammette un prolungamento massimale.
Teorema 3. 4 (
4Fuga dai compatti). Sia (I, u), I = (a, b), una soluzione massimale di (3.2). Per ogni compatto K ⊂ A esiste δ > 0 tale che per ogni t ∈ I − (a + δ, b − δ) risulta (t, u(t)) ∈ A − K .
Esempio 3. 1 .
1Studiamo il problema di Cauchy
( 3 . 11 )
311Lemma 3.7 (La diseguaglianza di Gronwall). Sia 0 ∈ I ⊆ R un intervallo ed u ∈ C(I, R) tale che esistano β, α ∈ C(I, R), β ≥ 0 , con u(t) ≤ α(t) + t 0 β(s)u(s) ds , t ∈ [0, T ] ⊂ I . τ )dτ ds , t ∈ [0, T ] .(3.13)Dimostrazione. Poniamo φ(t) := exp − t 0 β(s)ds , t ∈ I , cosicché φ(t) −1 φ(s) s)u(s) ds + u(t) ≤ α(t)β(t)φ(t) .Integrando questa diseguaglianza rispetto a t e dividendo per φ(t) sommando α(t) ed applicando (3.12) troviamo (3.13).
Corollario 3. 8 .
8Siano β > 0, α, γ ∈ R. Se u(t) ≤ α + t 0 (βu(s) + γ) ds , t ∈ [0, T ] ,(3.14)
Osservazione 4. 3 .
3Dal Lemma precedente segue che se fè misurabile allora per ogni
Definizione 4 . 14 .
414Sia {f n } ∪ {f } ⊂ M (X). Diciamo che la successione {f n } converge q.o. ad f se lim n f n (x) = f (x), q.o. in x ∈ X .
Teorema 4 . 15 (
415Egoroff). Sia (X, M, µ) uno spazio di misura finita ed {f n } ⊂ M (X) una successione convergente q.o.
I
aa := {(a, b), a < b ∈ R} , I ac := {(a, b], a < b ∈ R} , I ca := {[a, b), a < b ∈ R} , I cc := {[a, b], a < b ∈ R} ; I := I aa ∪ I ac ∪ I ca ∪ I cc .
l(I) := b − a , ∀I ∈ I , ed introduciamo l'applicazioneµ * : 2 R → R + , µ * A := inf n∈N l(I n ) : A ⊂ n∈N I n , {I n } ⊂ I aa .(4.17)I seguenti risultati mostrano che µ * è in effetti una misura esterna invariante per traslazioni.
Lemma 4 . 16 .
416Valgono le seguenti proprietá:
Lemma 4. 17 .
17Per ogni a, b ∈ R, a < b , si ha b−a = µ * (I) = µ * (I) = µ * (J) = µ * (J ′ ) , I := (a, b) , I := [a, b] , J := [a, b) , J ′ := (a, b] . (4.18) Se Iè non limitato (ovvero a = −∞ o b = ∞) allora µ * I = ∞.
Osservazione 4. 4 .
4Dai risultati precedenti seguono banalmente le seguenti proprietá: (1) Se µ * A = 0 allora µ * (A ∪ B) = µ * B per ogni B ∈ 2 R ; (2) Se Aè numerabile allora µ * A = 0 .
Teorema 4 . 20 .
420La classe Lè una σ -algebra e contiene quella dei boreliani su R. Dimostrazione. Il fatto che Lè una σ -algebra segue dal Lemma 4.7. Verifichiamo che L contiene i boreliani; a tale scopoè sufficiente mostrare che ogni intervallo del tipo J := (a, ∞), a ∈ R, appartiene ad L. Grazie alla regolaritá esterna possiamo prendere una successione {I n } di intervalli aperti tali che, preso A ⊆ R, A ⊆ ∪ n I n , µ * A ≥ n l(I n ) − ε . Posto I − n := J c ∩ I n , I + n := J ∩ I n , abbiamo, grazie al Lemma 4.17, l(I n ) = l(I − n ) + l(I + n ), cosicché
Osservazione 4.5. L'inclusione dei boreliani in Lè stretta, per cui Lè una σ -algebra piuttosto "grande" (vedi [10, §2.2], [16, Es.3.5.3] o [2, Es.3.1.8]). Tuttavia esistono insiemi non misurabili secondo Lebesgue, il piú famoso dei quali, l'insieme di Vitali,è il sottoinsieme V ⊂ [0, 1] che si costruisce come segue: innanzitutto consideriamo la proiezione di [0, 1] sull'insieme delle classi di equivalenza modulo i razionali, π : [0, 1] → [0, 1] Q , π(x) := {y ∈ [0, 1] : y − x ∈ Q} , e quindi definiamo V come l'immagine di una sezione 10 di π (dettagli su [25, §3.4] o [2, §1.8]).Definiamo ora la misura di Lebesgue µ : L → R + , µE := µ * E , E ∈ L .
Teorema 4.21. µè una misura completa, boreliana, regolare esterna, invariante per traslazioni ed estende la funzione lunghezza definita sugli intervalli.
Lemma 4 .
423. (Vitali, [25, Lemma 5.1.1]). Sia A ⊂ R un insieme di misura esterna finita ed I una collezione che ricopre A nel senso di Vitali. Allora, preso ε > 0 , esiste un insieme finito {I 1 , . . . , I n } di intervalli mutualmente disgiunti tale che µ * A −∪ n k I k < ε .
I 1 , . . . , I k , definiamo λ(k) := sup {l(I) : I ∈ I , I ∩ ∪ k j=1 I j = ∅} .
11
Osservare che quando la parte internaȦè l'insieme vuoto, abbiamo G(A) = {0} .
Teorema 4 . 26 (
426Lusin). Sia f una funzione misurabile su un intervallo [a, b]. Allora per ogni ε > 0 esiste una funzione continua ϕ ε su [a, b] tale che µ{x : f (x) = ϕ ε (x)} < ε .Sketch della dimostrazione. Usando Prop.4.25 costruiamo una successione {f n } di funzioni continue tali che µ{x : |f (x) − f n (x)| ≥ 1/n} < 1/n. Ció fornisce una convergenza q.o. ed a questo punto usiamo il teorema di Egoroff.
Proposizione 4.27. (Lebesgue-Vitali, [12, Es.6.4.2]). Una funzione limitata f : [a, b] → R e integrabile secondo Riemann se e soltanto se l'insieme dei punti di discontinuitá di f ha misura (di Lebesgue) nulla.
Esempio 4. 6 (
6La funzione di Dirichlet). Sia f : [0, 1] → R la funzione definita come f (t) = 0 , t ∈ Q ∩ [0, 1], f (t) = 1 , t ∈ [0, 1] − {[0, 1] ∩ Q} . Usando Oss. 4.4(2) concludiamo che, essendo Q ∩ [0, 1] numerabile, µ(Q ∩ [0, 1]) = 0 ,per cui f coincide quasi ovunque con la funzione costante 1 . Essendo la misura di Lebesgue completa (Teo.4.21), usando Prop.4.13 concludiamo che fè misurabile. D'altro canto f nonè integrabile secondo Riemann, infatti l'insieme dei punti di discontinuitá di fè [0, 1], come si verifica osservando che per ogni t ∈ [0, 1] possiamo trovare successioni {q n } ⊂ [0, 1] ∩ Q , {i n } ⊂ [0, 1] ∩ (R − Q) con limite t, cosicché lim n f (q n ) = 0 , lim n f (i n ) = 1 . Ovviamente, la non integrabilitá secondo Riemann di f si puó verificare piú direttamente mostrando che non vale la condizione (4.28) nella sezione seguente.
Proposizione 4 . 30 (
430Convergenza limitata). Sia (X, M, µ) uno spazio di misura finita ed {f n } ⊂ M (X) una successione limitata rispetto alla norma dell'estremo superiore. Se q.o. in x ∈ X esiste lim n f n (x) =: f (x), allora fè misurabile e
Teorema 4 .
432 (Teorema di convergenza monotóna, Beppo Levi). Sia {f n } ⊂ M + (X) una successione crescente di funzioni non negative tale che esista il limite f (x) := lim n f n (x), q.o. in x ∈ X . Il Lemma di Fatou ci assicura che f ≤ lim n inf f n . D'altro canto, per monotonía la successione f n ammette limite lim n f n = lim n inf f n , per cui f ≤ lim n f n . Ora, sempre per monotonía troviamo f n ≤ f , n ∈ N, per cui lim n f n ≤ f .
Osservazione 4. 9 .
9Il teorema di Beppo Levi nonè valido nei seguenti casi: (1) nell'ambito dell'integrale di Riemann (si veda l'Esempio 4.7 piú avanti); (2) per successioni decrescenti ([25, Ex.4.7(b)]).
Dimostrazione. ( 1 )
1L'affermazioneè ovvia per funzioni semplici. Del resto f, g sono limite puntuale di successioni monotóne {ψ n } , {ϕ n } di funzioni semplici (Prop.4.12), per cui applicando il teorema di Beppo Levi troviamo (f + g) = lim n (ψ n + ϕ n ) = lim n ψ n + ϕ n = f + g .
( 2 )
2Se f = 0 q.o. allora f = 0 (vedi Oss.4.7(1)); viceversa, se f = 0 allora, presa la successione {E n } ⊂ M di Prop.4.29, abbiamo 0 = f ≥ n −1 µE n . Quindi supp(f ) = ∪ n E n ha misura nulla. (3) Basta applicare il punto (1) e (4.30). (4) Grazie al punto (1) si ha
Esempio 4. 7 .
7(Riemann vs. Lebesgue in convergenza puntuale). Consideriamo la successione di funzionif n : [0, 1] → R , f n (x) := 1 , x ∈ {q 1 , . . . , q n } 0 , altrimenti , n ∈ N , dove {q n , n ∈ N}è una enumerazione dei razionali in [0, 1] (ovvero, una corrispondenza 1-1 tra N e Q ∩ [0, 1]). Ogni f nè continua in [0, 1] − {q 1 , . . . , q n } ,per cuiè integrabile secondo Riemann con R f n = 0 . Ora, {f n } converge puntualmente a f (x) = 1 , x ∈ Q ∩ [0, 1] 0 , altrimenti , la quale nonè integrabile secondo Riemann, come si verifica in maniera analoga all'Esempio 4.6. Equipaggiamo ora [0, 1] con la misura di Lebesgue. Abbiamo che {f n }è monotóna crescente, limitata e definita su uno spazio di misura finita, per cui possiamo applicare sia il teorema di convergenza limitata che quello di Beppo Levi, concludendo che fè integrabile secondo Lebesgue con 0 = lim n f n = lim n f n = f .
Corollario 4. 34 .
34Sia f ∈ L 1 µ (X), f ≥ 0 . Allora per ogni ε > 0 esiste un δ > 0 tale che A f < ε , ∀A : µA < δ .
Teorema 4 .
436 (Teorema di convergenza di Lebesgue). Sia {f n } una successione di funzioni misurabili con limite f (x) := lim n
(x) dx = n λ−2 [1 − (n + 1)e −n ] , per cui, se λ < 2 otteniamo lim n n f n = 0 , il cheè quanto possiamo aspettarci avendosi lim n f n = 0 puntualmente. Tuttavia, per λ ≥ 2 otteniamo lim n f n = 0 . Dal punto di vista dei teoremi di convergenza sotto il segno di integrale, osserviamo i seguenti fatti: (1) Per λ ∈ [0, 1) abbiamo f n → 0 uniformemente, per cui abbiamo convergenza sotto il segno di integrale senza necessitá di usare i teoremi di Lebesgue; (2) Per λ ∈ [1, 2) abbiamo la stimaf n (x) := n λ xe −nx ≤ g(x) := λ λ e −λ x 1−λ ;per cui 14 , essendo g integrabile, possiamo passare al limite sotto il segno di integrale grazie al teorema di Lebesgue, il quale in questo casoè indispensabile non avendosi convergenza uniforme.(3) Per λ ≥ 2 , chiaramente il teorema di Lebesgue nonè valido.
Teorema 4 . 38 (
438Radon-Nikodym). Sia (X, M, µ) uno spazio misurabile e ν ≺ µ, con ν, µ σ -finite. Allora esiste edè unica f ∈ M + (X) tale che νA = A f dµ , A ∈ M .(4.35)
Teorema 4 . 41 .
441Una funzione f : [a, b] → Rè BV se e soltanto seè la differenza di due funzioni monotóne. Per cui, se f ∈ BV ([a, b]) allora la derivata prima f ′ esiste q.o. in [a, b] (Teo.4.39).
Esempio 4 . 11 .
411La funzione caratteristica f := χ [0,1] : [−1, 1] → Rè monotóna crescente e quindi a variazione limitata. La derivata prima di fè -a meno di equivalenza q.o. -la funzione nulla, per cui (4.38) in questo casoè una diseguaglianza stretta:
Concludiamo questa breve rassegna sulle funzioni BV menzionando il seguente risultato, connesso all'integrale di Stieltjes ([17, 10.36], [25, §12.3]):Teorema 4.44. (Helly, [17, §10.36.5]). Sia {f n } ⊆ BV ([a, b]) una successione puntualmente convergente ad una funzione
grazie a Lemma 4.46) F =F . Viceversa, sia F una primitiva. Allora Fè continua e BV (Prop.4.43). Eventualmente sommando una costante, possiamo assumere che f ≥ 0 , per cui concludiamo che Fè AC grazie a (4.33).
45) il che implica che ϕè Lipschitz in [c, d]. Punto 2: Ogni funzione lipschitzianaè AC. Punto 3: Eq. 4.44 implica che ogni rapporto incrementale di ϕ rispetto ad un x 0 fissatoè una funzione monotóna crescente. Punto 4: Si osservi che
Esercizio 4 . 13 (
413Misure di Lebesgue-Stieltjes). Sia ω : R → R una funzione monotóna crescente e continua a destra. (1) Si mostri che definendo l ω (J) := ω(b)−ω(a), ∀J := (a, b] ∈ I ac , e µ * ω A := inf n l ω (J n ) : A ⊂ ∪ n J n , {J n } ⊂ I ac , ∀A ⊆ R , si ottiene una misura esterna.
(
Suggerimenti: per (1) e (2) si proceda in modo analogo alla misura di Lebesgue; per (3) si mostri, in analogia al Lemma 4.17, che µ * ω (J) = l ω (J), ∀J ∈ I ac (si veda anche l'Esercizio 4.11); per (4) si inizi verificando su funzioni caratteristiche di intervalli in I ac contenuti in [α, β]).
L p (R), L p (X) := L p (a, b), X := (a, b), a, b ∈ R . Inoltre, per economia di notazione d'ora in poi scriveremo L p := L p ([0, 1]), p ∈ [1, +∞), G := G([0, 1]) (notare che in quest'ultimo caso stiamo considerando l'intervallo chiuso [0, 1]). Definiamo ora
h , h ∈ N , e si usi il punto (3).). 20 La proprietá che si chiede di dimostrare in questo punto equivale a dire che {τ h }è una base di Schauder per L p nel senso di §7.1. Un esempio esplicito di {τ h }è dato dalle cosiddette funzioni di Haar, vedi I.Singer: Bases in Banach spaces, Ex.2.3.
e una contrazione. A tale scopo, osserviamo cheRè differenziabile e chiaramente soddisfa (6.10), per cui, in un opportuno intorno dell'origine abbiamo |R(x,
Teorema 6. 7 (
7Teorema dell'inverso locale). Sia A ⊆ R n aperto ed f : A → R n un'applicazione di classe C 1 . Se a ∈ Aè tale che det ∂f ∂x (a) = 0 ,allora fè un diffeomorfismo locale in a.
k=1,...,m = ∇f (ϕ(u)) , ∂ϕ ∂u k (u) k=1,...,m . Per cui, se cerchiamo i punti stazionari b ∈ U per f • ϕ, allora la condizione ∇(f • ϕ) = 0 si traduce in ∇f (ϕ(b)) , ∂ϕ ∂u k (b) = 0 , k = 1, . . . , m . (6.16)
Usando l'antisimmetria del prodotto esterno ed il teorema di Schwartz troviamo d 2 := d • d = 0 . Una m-forma si dice chiusa se dω = 0 , ed esatta se ω = dϕ per qualche m − 1 -forma ϕ; poiché d 2 = 0 ,è chiaro che ogni forma esattaè chiusa. In particolare, una 1 -formaè esatta se ω = df per qualche f ∈ C k (U ), dove dfè l'applicazione definita dal differenziale (vedi (6.4))
25) dove, per ogni I := {i 1 , . . . , i k , . . . , i m } , abbiamo definito l'applicazione
2.1]). Osserviamo inoltre che la proprietá dell'esattezza di ω dipende anche dal dominio scelto: se Uè stellato 25 allora ogni 1 -forma chiusaè anche esatta ([12, Teo.8.2.2]).
Esempio 6. 4 .
4Sia A := R 2 − {0} ed ω := (x 2 + y 2 ) −1 [−y dx + x dy]. Allora ωè chiusa ma non esatta (si calcoli infatti γ ω , γ(t) := (cos t, sin t), t ∈ [0, 1]).Sia ora K ⊂ U un dominio regolare con bordo ∂K . Il teorema di Stokes (vedi [12, Eq.K ⊂ U ⊆ R n , ω ∈ Ω m k (U ) .(6.26) Nel caso m = 1 , n = 2 il teorema di Stokesè una riformulazione della formula di Gauss-Green con ω = F 2 dx 1 − F 1 dx 2 . La dualitá di de Rham. Il teorema di Stokesè un tassello importante di una costruzione fondamentale in analisi e geometria, nota come la dualitá di de Rham. Abbiamo visto nei paragrafi precedenti come l'esattezza di una forma differenziale si traduca nella soluzione di un'equazione differenziale, per cui, data una varietá M che per comoditá riguardiamo come un sottoinsieme M ⊆ R n ,è di interesse caratterizzare le forme differenziali esatte su M . Ovviamente, condizione necessaria (e facilmente verificabile) per l'esattezza della forma ω ∈ Ω m ∞ (M )è la proprietá di essere chiusa, ovvero dω = 0 . Per ogni m ∈ N denotiamo con Z m dR (M ) lo spazio vettoriale delle m-forme chiuse e C ∞ ; chiaramente dΩ m−1 ∞ (M )è un sottospazio vettoriale di Z m dR (M ), e definiamo la coomologia di de Rham di ordine m come lo spazio quoziente H m dR (M ) := Z m dR (M )/dΩ m−1 ∞ (M ) . In base alle considerazioni precedenti, il fatto che H m dR (M ) sia non banale si traduce nell'esistenza di m-forme non esatte. Ebbene, H m dR (M ) puó essere calcolato in termini di proprietá prettamente topologiche di M , e nelle righe seguenti accenneremo alla dimostrazione di questo fatto. Iniziamo definendo, per ogni m ∈ N, il simplesso standard ∆ m := {x ∈ R m : x i ≥ 0 ∀i = 1, . . . , n , i x i ≤ 1} ⊂ R m ; osserviamo che ∆ 1è un intervallo chiuso, ∆ 2 un triangolo chiuso, ∆ 3 un tetraedro. Ora, preso un qualsiasi insieme Σ possiamo definire lo spazio vettoriale V (Σ) avente elementi combinazioni lineari formali del tipo i c i a i , c 1 ∈ R, a i ∈ Σ; per ogni m ∈ N consideriamo quindi l'insieme C ∞ (∆ m , M ) e definiamo C m (M ) := V (C ∞ (∆ m , M )). Lo scopoè ora quello di introdurre sugli spazi C m (M ) degli operatori ∂ : C m (M ) → C m−1 (M ) che formalizzino la nozione di bordo di un sottoinsieme di R n (nozione peraltro usata in (6.26)). A tale scopo, per ogni i = 0, . . . , m − 1 definiamo le applicazioni (dette facce) j i m : ∆ m−1 → ∆ m , j i m (x) := (x 1 , . . . , x i−1 , 0, x i , . . . , x m−1 ) , i = 0 (1 − i x i , x 1 , . . . , x m−1 ) , i = 0 ; il contenuto intuitivo della definizione precedenteè che un m − 1 -simplesso standard si presenta m volte come una delle facce dell'm-simplesso. Possiamo ora costruire gli operatori di bordo estendendo per linearitá sugli elementi di C ∞ (∆ m , M ) ⊂ C m (M ):
′ (t)] 2 dt . Poiché f (t, x, p) = p 2è convessa, Fè convesso. L'equazione di Eulero-Lagrange associata ad F e u ′′ = 0 , con condizioni al bordo u(a) = 0 , u(1) = L . La soluzioneè quindi il minimo per Fè u(t) = L(b − a) −1 (t − a).
è misurabile; (3) le stesse proprietá (1,2) sono vere scambiando i ruoli di x, y ; (4)è verificata l'uguaglianza (6.37).
Osservazione 6. 1 .
1Le convoluzioni hanno la notevole proprietá di essere continue anche nel caso in cui né f né g lo siano: a questo proposito si veda l'Esercizio 5.2. Ad esempio, invitiamo a verificare che, presi 0 < b ≤ a e definite χ a := χ (−a,a) , χ b := χ(−b,b), cosicché χ a , χ b ∈ L p (R) ∀p ∈ [1, +∞], allora χ a * χ bè una funzione a supporto compatto, continua e lineare a tratti.
anche in questo caso piú generale 27 . Per dettagli sull'argomento si veda [14, Vol.I, Cap.5].
41) dimostrabile altrimenti usando la derivabilitá q.o. delle funzioni primitive.
Osservazione 6.5. L'argomento della dimostrazione di Teo.6.23 funziona, piú in generale, se in luogo della funzione δf (x, y) :
trasformata di Laplace definisce un morfismo (limitato) di algebre di BanachL : (L 1 (R + ), * ) → C 0 (R + ) , f → Lf .
f
(x, y) dx dy = − π 4 .Dunque non valgono i teoremi di Fubini-Tonelli; ed infatti f / ∈ L 1 ([0, 1] 2 ), come si verifica ponendo f + (x, y) := sup{f (x, y),
Esercizio 6. 3 .F[FF
3Si calcoli il minimo dei funzionali ((u ′ ) 2 + 2tu] dt , G(u) = 1 0 [(u ′ ) 2 − 2tuu ′ + e t u] dt , u(0) = u(1) = 0 .Esercizio 6.4. Si trovi la soluzione u del problema ((x, y) = y 2 + λx 2 − x 3 , G(x, y) = e xy (y 2 + 1) − y(1 + e 2xy ) , al variare del parametro λ ∈ R.
Si dimostri che le seguenti forme differenziali sono esatte sui relativi domini:
(
Suggerimento: si osservi che il dominio di ω nonè stellato, per cui conviene usare [12, Cor.8.2.1]).
Esercizio 6. 7 (
7Convoluzioni di misure) Sia Λ 1 β (R) lo spazio normato delle misure boreliane finite su R (vedi Def.4.3 ed Esercizio 4.7). Per ogni λ : M → R, λ ′ : M ′ → R, λ, λ ′ ∈ Λ 1 β (R), si mostri che: (1) La misura prodotto λ × λ ′è boreliana e finita su R 2 ; (2) Si definisca {λ * λ ′ }E := R 2 χ E (x + y) dλ(x)dλ ′ (y) , ∀E ∈ M ∩ M ′ , (6.45)
per cui la successione delle norme { v n }è di Cauchy in R e quindi convergente. Una serie {s n := n i v i } ⊂ E si dice convergente se esiste il limite lim n s n , ed assolutamente convergente se esiste il limite della serie reale {a n := n i v i } ⊂ R. Uno spazio normato E si dice di Banach se essoè completo rispetto alla topologia della norma (ovvero, ogni successione di Cauchy converge ad un elemento di E ).
Proposizione 7. 1 .29
1Un spazio normato Eè di Banach se e solo se ogni serie assolutamente convergenteè convergente.Dimostrazione. Se Eè di Banach ed {s n Si pensi alle locuzioni spazio localmente convesso od operatore compatto.
per cui la succesione f n , v ⊂ R (o C)è di Cauchy e possiamo definire f (v) := lim n f n , v , v ∈ E .Si tratta quindi di verificare che l'applicazione f cosí definitaè un funzionale lineare continuo, e che lim n f − f n = 0 . Mostriamo che fè limitata: poiché {f n }è di Cauchy, otteniamo per diseguaglianza triangolare che
Che f sia lineareè evidente per linearitá dei limiti, per cui f ∈ E * e concludiamo la dimostrazione osservando
Hahn-Banach, che mostreremo nel seguito, afferma che R E ′è suriettiva. Una base di Schauder dello spazio di Banach Eè una successione {e i ∈ E} che soddisfa la seguente proprietá: per ogni v ∈ E esiste edè unica la successione reale (o complessa) {a i } tale che v = i a i e i . Non tutti gli spazi di Banach posseggono una base di Schauder (vedi i commenti in [5, Cap.V]).
Esempio 7 . 2 (<
72Gli spazi l p ). Gli spazi L p µ (X) sono degli spazi di Banach per ogni p ∈ [1, +∞] (Teorema di Fischer-Riesz). In particolare denotiamo con l p l'insieme delle successioni z := {z n ∈ R} n tali che +∞ .Definiamo inoltre l ∞ come lo spazio delle successioni tali chez ∞ := sup n |z n | < +∞ .Ora 30 , ogni z ∈ l p si puó riguardare come una funzione sullo spazio discreto N equipaggiato della misura di enumerazione. Per cui, con le tecniche usate per dimostrare Prop.5.1 e Prop.5.2, otteniamo delle versioni delle diseguaglianze di Holder e Minkowski, n |z n ||w n |
la quale si dimostra banalmente usando (7.5) e le identitá Re(z) = 1/2(z + z),Im(z) = −i/2(z − z), ∀z ∈ C. Una famiglia {e i } ⊂ H si dice ortogonale se (e i , e j ) = 0 , ∀i = j ,ed ortonormale se, inoltre, e i , e i = 1 per ogni i ∈ N. Una base hilbertiana di Hè una famiglia ortonormale {e i } tale che lo spazio vettoriale da essa generatoè denso in H . Si puó dimostrare che se Hè separabile allora esso ha una base numerabile (ovvero, la famiglia {e i }è una successione, vedi [17, §4.16.3]). L'esistenza delle basi hilbertiane puó essere dimostrata usando il procedimento di ortogonalizzazione di Gram-Schmidt (ancora, si veda [17, §4.16.3]). Chiaramente, una base hilbertianaè una base di Schauder (per verificarlo, si definiscano gli a i nella definizione di base di Schauder come i prodotti scalari v, e i ).
Esempio 7. 4 .
4Come vedremo in §8, una base per lo spazio di Hilbert reale L 2 ([0, 1])è quella delle funzioni trigonometriche B := {sin(2πnx) , cos(2πmx) , x ∈ [0, 1]} n,m∈N . Passando al caso complesso, le stesse argomentazioni del caso reale permettono di concludere che B C := {e 2πinx , x ∈ [0, 1]} n∈Z e una base per L 2 ([0, 1], C) (osservare che, avendosi cos(2πmx) = 1/2(e 2πinx + e −2πinx ) , sin(2πnx) = −i/2(e 2πinx − e −2πinx ) , ∀x ∈ [0, 1] , e possibile ottenere elementi di B come combinazioni lineari di elementi di B C ).
Proposizione 7. 3 (
3Bessel, Parseval). Dato un insieme ortonormale {e i } dello spazio di Hilbert H , per ogni u ∈ H si ha i |(e i , u)| 2 ≤ u 2 . (7.6) Se, in particolare, {e i }è una base hilbertiana allora u = i (e i , u)e i e u 2 = i |(e i , u)| 2 . Dimostrazione. Posto c i := (e i , u) troviamo (nel caso reale) 0 ≤ u − n i a i e i 2 = (u − n i a i e i , u − n i a i e i ) = u 2 − 2 i a i c i + n i a 2 i = u 2 − n i c 2 i + n i (a i − c i ) 2 .
espressione assume il suo minimo proprio per a i ≡ c i , in concomitanza del quale troviamo 0 ≤ u 2 − i c 2 i ovvero (7.6) 31 . Quando {e i }è una base hilbertiana sappiamo che esiste una serie i a i e i avente limite u , per cui imponendo in (7.7) lim n u − n i a i e i 2 = 0 troviamo, valutando nel minimo a i ≡ c i scalari (e i , u), i ∈ N, si dicono i coefficienti di Fourier di u .
( 7 . 8 )
78Osservazione 7.1. L'argomento della dimostrazione della diseguaglianza di Cauchy-Schwarz vale per ogni forma bilineare (sesquilineare), simmetrica e definita positiva A :E × E → R (C), nel senso che |A(u, v)| ≤ A(u, u) 1/2 A(v, v) 1/2 , ∀u, v ∈ E . (7.9)Per verificare (7.9), nei conti che mostrano (7.8) si sostituisca u 2 con A(u, u).Ora, (7.8) implica che per ogni u ∈ H il funzionalef u : H → R (C) : f u , v := (u, v) , v ∈ H , e limitato ed ha norma ≤ u ; d'altra parte, usando il fatto che | f u , v | = u , v := u/ u , otteniamo f u = u , u ∈ H .(7.10) Teorema 7.4 (Riesz). Sia H uno spazio di Hilbert. Allora l'applicazione H → H * , u → f u , (7.11) e lineare (antilineare nel caso complesso), isometrica e suriettiva. 31 D'altra parte, nel caso complesso (i − c i | 2 e possiamo argomentare in modo analogo al caso reale.Dimostrazione (caso separabile). L'applicazione (7.11)è chiaramente lineare e (7.10) implica che essaè isometrica e quindi iniettiva; per cui resta da verificarne soltanto la suriettivitá. Sia {e k } una base ortonormale per H . Poniamo a k := f, e k ed osserviamo che k e k ∈ H . Ora,
Esempio 7. 5 .
5L 2 µ (X)è uno spazio di Hilbert rispetto al prodotto scalare (f, g) := f g . Nel caso complesso abbiamo invece il prodotto scalare (f, g) := f g , f, g ∈ L 2 µ (X, C) .
Proposizione 7. 5 .
5Se Eè uno spazio normato ed F uno spazio di Banach allora B(E, F )è uno spazio di Banach. Dimostrazione. Se {T n } ⊆ B(E, F )è una successione di Cauchy allora per ogni v ∈ E risulta T n v − T m v ≤ T n − T m v n,n → 0 ;
, per E = F usiamo la notazione B(E) := B(E, E).Esempio 7.6. Sia a := {a n } ∈ l ∞ C . Allora, per ogni p ∈ [1, +∞], l'operatore T a ∈ B(l p C ) , T a z := {a n z n } n , z := {z n } ∈ l p C , e limitato ed ha norma T a = a ∞ .Preso T ∈ B(E, F ), consideriamo l'immagine T (E) := {T u : u ∈ E} ⊆ F e diciamo che T e chiuso se T (E)è un sottospazio chiuso di E . Il nucleo di T si definisce come il sottospazio ker T := {u ∈ E : T u = 0} ⊆ E , il qualeè chiuso in quanto se v = lim n v n , {v n } ⊂ ker T , allora
Esempio 7 . 7 .
77Consideriamo gli spazi di Banach L 1 ([0, 1]), F := C([0, 1]), equipaggiati rispettivamente con le norme · 1 ed · ∞ . Allora la somma diretta E := L 1 ([0, 1]) ⊕ Rè uno spazio di Banach rispetto alla norma f ⊕ λ := sup{ f 1 , |λ|} , e l'applicazione T : E → F , {T (f ⊕ λ)}(x) := λ + x 0 f (t) dt , ∀x ∈ [0, 1] , e un operatore limitato tale che
T
* ∈ B(H) : (T * u, v) := (u, T v) , u, v ∈ H , T ∈ B(H) ; dunque abbiamo un'applicazione isometrica * : B(H) → B(H), T → T * , lineare quando Hè reale ed antilineare 32 quando Hè complesso. Un operatore T ∈ B(H) si dice autoaggiunto se T = T * . Diciamo invece che U ∈ B(H)è unitario qualora U * U = U U * = 1 , e lasciamo come esercizio dimostrare che Uè unitario se e solo se essoè suriettivo ed isometrico. Infine, un proiettore P ∈ B(H)è un operatore autoaggiunto ed idempotente, P = P * = P 2 .
Esempio 7.8 (L'operatore di Volterra). Osservando che H := L 2 ([0, 1], C) ⊂ L 1 ([0, 1], C), definiamo l'operatore F ∈ B(H) , F u(x) := i x 0 u(t) dt , ∀x ∈ [0, 1] , u ∈ H .
( 7 . 17 )
717Complementi ortogonali, nuclei e immagini. Preso un sottospazio V ⊂ H definiamo il complemento ortogonale V ⊥ := {v ∈ H : (v, u) = 0, ∀u ∈ V} . (7.18) La disuguaglianza di Cauchy-Schwarz implica che se v = lim n v n , {v n } ⊂ V , allora
20) per cui, per l'unicitá della decomposizione ortogonale, confrontando (7.20) con(7.19) concludiamo che V ⊥⊥ = V . Sia ora T ∈ B(H). Dall'identitá (T * u, v) = (u, T v), ∀u, v ∈ H ,e dal Lemma precedente, segue immediatamente che {T (H)} ⊥ = ker T * ⇔ T (H) = {ker T * } ⊥ , ∀T ∈ B(H) ; (7.21) in particolare, quando Tè chiuso otteniamo T (H) = {ker T * } ⊥ .
abbondanza di funzionali positivi puó essere dimostrata usando la teoria spettrale ed il Teorema di Hahn-Banach (vedi[19,). Il risultato seguente mostra come usando i funzionali positivi sia possibile interpretare una C*-algebra in termini di operatori su uno spazio di Hilbert.
altro canto (7.25)è anche un prodotto scalare, in quanto ([v], [v]) = 0 se e solo se v ∈ I ω , ovvero [v] = 0 . Definiamo H ω come lo spazio di Hilbert ottenuto completando V rispetto a (7.25). Per ogni a ∈ A definiamo l'applicazione π ω (a) : H ω → H ω , {π ω (a)}[v] := [av] , la qualeè ben posta in quanto [av] = [a(v + z)] per ogni z ∈ I ω . E' ovvio che π ω (a)è lineare, e {π ω (a)}[v] 2 = ([av], [av]
Corollario 7 . 12 .
712Sia {T n } ⊂ B(E, F ) una successione tale che {T n v} converge per ogni v ∈ E . Allora: (1) sup n T n < ∞; (2) Posto T v := lim n T n v , risulta che Tè un operatore limitato e T ≤ lim n inf T n . Osservazione 7.4. Nel corollario precedente non si afferma che T n − T → 0 . Tuttavia si puó verificare con un argomento del tipo " 3 -ε " che sup v∈K T n v − T v → 0 per ogni compatto K ⊂ E .Corollario 7.13. Sia B ⊆ E tale che f (B)è limitato per ogni f ∈ E * . Allora Bè limitato.
Teorema 7 .
715 (Teorema dell'applicazione aperta). Siano E, F spazi di Banach e T ∈ B(E, F ) suriettivo. Allora T (A)è aperto in F per ogni aperto A ⊆ E (ovvero Tè un'applicazione aperta).
Corollario 7. 17 (
17Teorema del grafico chiuso). Siano E, F spazi di Banach e T : E → F un'applicazione lineare. Se il grafico G(T ) := {v ⊕ T (v), v ∈ E}é un sottoinsieme chiuso di E ⊕ F , allora Tè un operatore limitato. Dimostrazione. La somma diretta E ⊕ Fè uno spazio vettoriale nella maniera ovvia, e definendo (v, w) := sup{ v , w } possiamo riguardare E ⊕ F come uno spazio di Banach. Osserviamo che abbiamo gli operatori P ∈ B(E ⊕ F , E), P (v, w) := v , P ′ ∈ B(E ⊕ F , F ), P (v, w) := w , i quali chiaramente hanno norma 1 . Ora, per ipotesi G(T )è un sottospazio di Banach (ovvero chiuso) di E ⊕ F , e per costruzione S := P | G(T ) ha inverso S −1 ∈ B(E, G(T )), S −1 v := (v, T v). Per il teorema precedente S −1è limitato, e di conseguenza T = P ′ • S −1è limitato.
per cui possiamo estrarre una sottosuccessione convergente {r −1 T v n k } ; di conseguenza anche {T v n k } e convergente, per cui T (E ≤r )è precompatto. Concludiamo che la chiusura di T (A), essendo contenuta nel precompatto T (E ≤r ),è compatta 34 . Proposizione 7.22. Siano E, F spazi di Banach. Allora K(E, F )è uno spazio di Banach chiuso rispetto a composizioni con elementi di B(F ), B(E) 35 . Inoltre, se T ∈ B(E, F ) ha rango finito alloraè compatto.
e continua. Ció implica che (T + T ′ )(A)è compatto e quindi K(E, F )è uno spazio vettoriale (che λT , λ ∈ R, C, T ∈ K(E, F ), sia compatto non ci dovrebbero essere dubbi). Infine mostriamo che K(E, F )è chiuso in norma. Consideriamo una successione di Cauchy {T i } ⊂ K(E, F ) e mostriamo che il limite Tè compatto. A tale scopoè sufficiente verificare che, data una successione limitata {v n } ⊂ E , esiste una sottosuccessione convergente di {T v n } , e ció si dimostra utilizzando il seguente argomento diagonale. Per compattezza di T 1 , esiste certamente una sottosuccessione {v
di {v n } tale che {T 1 v
convergente; analogamente, esiste una sottosuccessione {v
converge; iterando il procedimento otteniamo la sottosuccessione {v
34
Infatti in generale un insieme chiuso contenuto in un compattoè esso stesso compatto.35 In particolare, se E = F allora K(E) := K(E, E)è un ideale bilatero chiuso di B(E) .
ed il terzo termine si annullano nel limite i → ∞ per convergenza di {T i } e limitatezza di {v (n) n } , mentre il secondo termine si annulla nel limite m, n → ∞ per convergenza di {T i v (n) n } . Ne segue che {T v (n) n }è convergente e dunque Tè compatto.
Osservazione 7. 5 .
5Lo spazio vettoriale degli operatori a rango finitoè denso in K(E, F ) se F possiede una base di Schauder, e quindi in particolare se Fè uno spazio di Hilbert. Per esempi di casi in cui tale risultato non vale si veda [5, Cap.VI, Oss.1]. Complementi ortogonali in spazi di Banach. Diamo ora una naturale generalizzazione della nozione di complemento ortogonale in uno spazio di Hilbert. Sia E uno spazio di Banach ed M ⊂ E , N ⊂ E * sottospazi vettoriali. I complementi ortogonali di M, N si definiscono rispettivamente come M ⊥ := {f ∈ E * : f, v = 0 , ∀v ∈ M} ⊆ E * , N ⊥ := {v ∈ E : f, v = 0 , ∀f ∈ N } ⊆ E . Analogamente al caso di uno spazio di Hilbert, si verifica facilmente che M ⊥⊥ = M ; (7.30) per dettagli sui complementi ortogonali in spazi di Banach rimandiamo a [5, II.5]. Sia ora
Quando
Tè chiuso, chiaramente abbiamo T (E) = {ker T * } ⊥ .L'alternativa di Fredholm. Possiamo ora dimostrare un risultato fondamentale per la soluzione di equazioni agli autovalori per operatori compatti.Lemma 7.23. Sia T ∈ K(E). Allora ker(1 − T ) ha dimensione finita e 1 − T ∈ B(E)è un operatore chiuso. Dimostrazione. Se ker(1 − T ) avesse dimensione infinita allora troveremmo una successione {v n ∈ ker(1 − T ) ∩ E 1 } priva di sottosuccessioni convergenti in norma (vedi Esercizio 7.4). D'altro canto, essendo T compatto deve esistere una sottosuccessione convergente {T v n k } . Poiché T v n k = v n k , k ∈ N,anche {v n k } converge, e questaè una contraddizione. Dimostriamo che 1 − Tè chiuso: se {v n } ⊂ Eè una successione tale che {v n − T v n } converge a w ∈ E , occorre verificare che w = v − T v per qualche v ∈ E . Considerata la distanza d n := d(v n , ker(1 − T )), n ∈ N, osserviamo che avendo ker(1 − T ) dimensione finita deve esistere z n ∈ ker(1 − T ) tale che d n = v n − z n . Supponiamo di aver mostrato che {v n − z n }è limitata; in tal caso potremmo estrarre una sottosuccessione {v n k − z n k } tale che
e w apparterrebbe all'immagine di 1 − T , il quale sarebbe quindi chiuso. Dunque per dimostrare il Lemma rimane da verificare solo che {v n −z n }è limitata. Se per assurdo cosí non fosse, troveremmo infiniti indici k ∈ N tali che d k → ∞, e posto s k :
il che contraddice il fatto che {s i k } ha limite in ker(1 − T ).
Teorema 7 . 24 (
724Fredholm). Per ogni T ∈ K(E) valgono le seguenti proprietá:1. ker(1 − T ) ha dimensione finita; 2. (1 − T )(E) = ker(1 − T * ) ⊥ ; 3. ker(1 − T ) = {0} ⇔ (1 − T )(E) = E .Dimostrazione. (1) Vedi Lemma 7.23. (2) Grazie al Lemma 7.23 sappiamo che 1 − Tè chiuso, e scrivendo (7.31) per 1 − T si trova (1 − T )(E) = ker(1 − T * ) ⊥ come desiderato. (3) Sia ker(1 − T ) = {0} . Supponendo per assurdo che E 1 := (1−T )(E) sia strettamente contenuto in E troviamo subito T | E1 ⊆ E 1 , T | E1 ∈ K(E 1 ) e, grazie al Lemma precedente, abbiamo che E 2 := (1 − T )| E1è chiuso. Del resto 1 − Tè iniettivo, per cui definendo induttivamente E k := (1 − T )(E k−1 ) , k = 2, . . . , otteniamo una sequenza di sottospazi chiusi di E contenuti strettamente l'uno nell'altro. Grazie al Lemma di Riesz (Esercizio 7.4) esiste una successione {u k
Ció contraddice la compattezza di T , per cui 1 − T deve essere suriettivo. Viceversa sia 1 − T suriettivo. Allora (essendo 1 − T chiuso per il Lemma precedente) troviamo ker(1 − T * ) = {(1 − T )(E)} ⊥ = {0} (vedi(7.31)); per cui, essendo T * compatto (Esercizio 7.3), possiamo applicare a quest'ultimo l'argomento precedente, concludendo che 1 − T * è suriettivo. D'altra parte 1 − T * e chiuso (sempre grazie al Lemma precedente), per cui applicando ancora (7.31) concludiamo che{0} = {(1 − T * )(E * )} ⊥ = ker(1 − T ).Osservazione 7.6. (1) Il teorema di Fredholm si puó enunciare nel seguente modo: l'equazione u − T u = v o ammette soluzione unica per ogni v ∈ E , oppure l'equazione omogenea u − T u = 0 ammette un numero finito di soluzioni linearmente indipendenti (l'alternativa, appunto). In tal caso v ∈ ker(1 − T * ) ⊥ , per cui abbiamo una condizione di ortogonalitá di v rispetto alle soluzioni dell'omogenea. (2) L'alternativa di Fredholmè una delle principali motivazioni di una nozione che svolge un ruolo fondamentale in analisi funzionale ed in geometria (!): dato lo spazio di Hilbert H , un operatore S ∈ B(H) si dice di Fredholm se: (i) ker S ha dimensione finita; (ii) S(H) ha codimensione finita (ovvero, S(H) ⊥ ha dimensione finita). In tal casoè ben definito l'indice indS := dim ker S − dimS(H) ⊥ ∈ Z . (7.33)
Teorema 7 . 28 (
728Lax-Milgram). Sia A : H × H → R una forma bilineare, limitata e coercitiva. Allora, per ogni ϕ ∈ H * esiste edè unico u 0 ∈ H tale che A(u 0 , v) = ϕ, v . (7.37)
Osserviamo cheè del tutto evidente che σp(T ) ⊆ σ(T ), ed in effetti se E ha dimensione finita allora σ(T ) = σp(T ); d'altro canto, consideriamo lo shift S ∈ B(l 2 ) : Sx := (0, x 1 , x 2 , . . .) , x := (x 1 , x 2 , . . .) ∈ l 2 ; (7.39) si verifica immediatamente che 0 ∈ σ(T ) − σp(T ) (infatti Sè iniettivo ma non suriettivo). Proposizione 7.31. Sia A un'algebra di Banach (reale o complessa) con identitá 1 , e T ∈ A. Allora σ(T )è compatto e si ha l'inclusione σ(T ) ⊆ ∆(0, T ) = {λ : |λ| ≤ T } .
Corollario 7 . 33 .
733Se T ∈ B(H)è autoaggiunto allora σ(T ) = {0} ⇒ T = 0 . (7.41)
Esempio 7 . 17 .
717Consideriamo lo spazio di Hilbert l 2 C e l'operatore T x := {0, x 1 , . . . , x n /n, . . .} , x := {x n } ∈ l 2 C .
Esempio 7 . 19 .
719Consideriamo lo spazio di Hilbert L 2 ≡ L 2 ([0, 1]) e l'operatore T ∈ BL 2 : T u(x) := xu(x) , x ∈ [0, 1] , u ∈ L 2 .
Lemma 7 . 34 .
734Sia T ∈ K(E) e {λ n } ⊆ σp(T ) una successione di numeri reali (complessi) distinti e non nulli convergente a λ. Allora λ = 0 . Dimostrazione. Per ogni n ∈ N consideriamo un autovettore v n ∈ E , T v n = λ n v n , e definiamo V n := span{v 1 . . . v n } . Mostriamo ora che v 1 , . . . , v n sono linearmente indipendenti. Poiché cióè chiaramente vero per n = 1 , procediamo induttivamente e, assunto che v 1 , . . . , v n siano linearmente indipendenti, mostriamo che v 1 , . . . , v n+1 sono linearmente indipendenti. Se per assurdo fosse
Teorema 7 . 35 .
735Sia E uno spazio di Banach (reale o complesso) a dimensione infinita e T ∈ K(E). Allora: (1) 0 ∈ σ(T ); (2) σ(T ) − {0} = σp(T ) − {0} ; (3) σ(T )è un insieme al piú numerabile, ed in tal caso 0è l'unico punto di accumulazione. Dimostrazione. (1) Supponendo per assurdo che T sia invertibile troviamo che T −1(E 1 )è limitato e quindi E 1 = T (T −1 (E 1 ))è compatto. Ció contraddice il fatto che E ha dimensione infinita. (2) Sia λ ∈ σ(T ) − {0} . Se ker(T − λ1) = {0} allora per l'alternativa di Fredholm troviamo (T − λ1)E = E per cui λ ∈ ρ(T ). Ció contraddice l'ipotesi λ ∈ σ(T ), per cui deve essere λ ∈ σp(T ). (3) Poniamo A n := σ(T ) ∩ {λ : |λ| ≥ n −1 } ; grazie al punto precedente sappiamo che A n ⊂ σp(T ).Inoltre, ogni A n puó essere al piú finito, altrimenti potremmo estrarne una sottosuccessione convergente a λ = 0 , in contraddizione con il lemma precedente. Per cui, avendosi σ(T ) − {0} = ∪ n A n troviamo che σ(T )è numerabile, e grazie al Lemma precedente sappiamo che 0è l'unico eventuale punto di accumulazione.Il risultato seguente caratterizza lo spettro degli operatori compatti autoaggiunti su uno spazio di Hilbert:Teorema 7.36 (Il teorema spettrale per operatori autoaggiunti compatti). Sia H uno spazio di Hilbert separabile (reale o complesso) e T ∈ K(H) autoaggiunto. Allora H ha una base di autovettori di T . Dimostrazione. Poniamo H 0 := ker T ed H n := ker{T − λ n 1} . Per il teorema di Fredholm (vedi anche Oss.7.6) abbiamo che ogni H n ha dimensione finita. Poiché (u, T v) = λ m (u, v) = (T u, v) = λ n (u, v) concludiamo che H n ⊥ H m , n = m. Vogliamo ora verificare che H := ∪ n≥0 H nè denso in H . A tale scopo osserviamo che T ( H) ⊆ H e che, avendosi (T v, u) = (v, T u) = 0 , u ∈ H , v ∈ H ⊥ , ha senso definire l'operatore T 0 := T | H ⊥ . Osserviamo che T 0è autoaggiunto e compatto. Ora, se λ ∈ σ(T 0 ) − {0} allora, per il teorema precedente, esiste un autovettore v ∈ H ⊥ tale che T 0 u = T u = λu ; per cui si avrebbe λ ∈ σp(T ) e u ∈ H ⊥ ∩ H n , il cheè assurdo. Concludiamo che σ(T 0 ) = {0} , per cui T 0 = 0 grazie a (7.41). Ora, T 0 = 0 equivale ad affermare che H ⊥ ⊆ ker T . Concludiamo che H ⊥ ⊆ ker T ⊆ H , per cui H = {0} .
P
(σ(T ), C) → B(H) , f → f (T ) in particolare T = I(T ), dove I(λ) := λ , ∀λ ∈ σ(T ) . Inoltreè evidente che {f + zg}(T ) = f (T ) + zg(T ) , {f g}(T ) = f (T )g(T ) , {f * }(T ) = f (T ) * , (7.46)
il calcolo funzionale boreliano di T . Si verifica che (7.54)è una rappresentazione tale che f (T ) ≤ f ∞ , ∀f ∈ L ∞ β (σ(T )) (vedi[19, Thm.4.5.4]; osserviamo che comunque (7.54)è isometrica su C(σ(T ), C) ⊂ L ∞ β (σ(T ))). Le precedenti considerazioni portano al seguente teorema: Teorema 7.37 (Hilbert). Se T = T * ∈ B(H) allora esiste una misura spettrale µ di T tale chè e verificata (7.52). Si hanno poi le seguenti proprietá:(1) Per ogni f ∈ L ∞ β (σ(T )), l'operatore f (T ) ∈ B(H) definito da (7.54)è limitato e f (T ) ≤ f ∞ . In particolare T = I(T ), dove I(λ) := λ, ∀λ ∈ σ(T ). (2) L'operatore (limitato) L ∞ β (σ(T )) → B(H) , f → f (T ) e una rappresentazione.Per una dimostrazione dettagliata del teorema precedente rimandiamo a [19, §4.5] o [22, §VII.2]. Qui ci limitiamo ad osservare che, mentre nel caso T ∈ K(H) abbiamo che ogni {λ} ⊆ σ(T ) ha misura non nulla, nel caso generale T ∈ B(H) possiamo trovare µ{λ} = 0 ; in effetti, si puó dimostrare che µ{λ} = 0 se e solo se λ ∈ σp(T ). Per proprietá di continuitá della rappresentazione del punto (2) rimandiamo all'Esercizio 7.16, il quale include anche il caso degli operatori non limitati ( §7.10). Esempio 7.20. Consideriamo lo spazio di Hilbert H := L 2 ([0, 1], C) e l'operatore T ∈ B(H) definito come nell'Esempio 7.19. Le stesse considerazioni fatte nell'esempio di cui sopra mostrano che T = T * e σ(T ) = [0, 1], σp(T ) = ∅ . Denotiamo con M la σ -algebra dei boreliani di [0, 1], e per ogni f, g ∈ L 2 definiamo
Esempio 7 . 22 .
722Sia H uno spazio di Hilbert e {e k } ⊂ H 1 := {v ∈ H : v = 1} una successione ortonormale. Usando il teorema di Riesz e Prop.7.3 troviamo f u , e k = (u, e k ) k → 0 , ∀u ∈ H .
Teorema 7 .
744. (Kakutani, [5, Teo.III.16]). Uno spazio di Banach Eè riflessivo se e solo se la palla unitaria E ≤1 := {v ∈ E : v ≤ 1}è debolmente compatta.
1 .
1Sia E uno spazio di Banach tale che E * sia separabile. Allora Eè separabile ([5, Teo.III.23]). Il viceversaè falso (ad esempio, si prenda E = L 1 ); 2. Un spazio di Banach Eè riflessivo e separabile se e solo E * è riflessivo e separabile ([5, Cor.III.24], l'esempio canonicoè E = L p , p = 1, ∞); 3. Uno spazio di Banach Eè separabile se e solo se (E * ≤1 , σ(E * , E))è metrizzabile ([5, Teo.III.25]). Per dare un'idea della dimostrazione, se assumiamo che E sia separabile, allora possiamo definire la metrica d(f, g) := n 2 −n | f − g, v n | , f, g ∈ E * ≤1 , dove {v n } ⊂ Eè denso. 4. (Eberlein-Smulian, [5, Teo.III.27,Teo.III.28]). Uno spazio di Banach Eè riflessivo se e solo se (E ≤1 , σ(E, E * ))è sequenzialmente compatta (ovvero, da ogni successione limitata in Eè possibile estrarre una sottosuccessione debolmente convergente).
10 e [19, §4.2.8]). (3) Sia T = T * ∈ B(H) ed A ⊆ B(H) l'immagine di C(σ(T ), C) rispetto al calcolo funzionale continuo (7.47); allora si ha un omeomorfismo A ≃ σ(T ) (vedi [19, Prop.4.3.15]).
7. 9
9Cenni su spazi localmente convessi e distribuzioni.Sia V uno spazio vettoriale. Diciamo che una famiglia {p i } i∈I di seminorme su Vè separante se
Esempio 7 .
724. (1) Ogni spazio di Banachè localmente convesso (in quanto la norma separa i punti); (2) Gli spazi L p µ,loc (X) sono localmente convessi (vedi Cor.5.11); (3) Sia V uno spazio vettoriale e {f i ∈ V * } una famiglia di funzionali lineari su V ; allora abbiamo la famiglia di seminorme P := {p i : p i (v) := |f i (v)|, ∀v ∈ V } . Se Pè separante allora V viene equipaggiato di una struttura di spazio localmente convesso. Ad esempio, se Eè uno spazio di Banach allora la famiglia di seminorme P := {p f : p f (v) := | f, v |, ∀v ∈ E} f ∈E * e separante ed induce la topologia debole. Analogamente, la famiglia P * := {p v : p v (f ) := | f, v |, ∀f ∈ E * } v∈È e separante ed induce la topologia * -debole su E * . (4) Dato uno spazio di Hilbert complesso H , la C*-algebra B(H)è equipaggiata con la famiglia di seminorme D := {p u,v (T ) := |(u, T v)|, T ∈ B(H)} u,v∈H , la qualeè separante in quanto per ogni T ∈ B(H) risulta p T u,u (T ) = |(T u, T u)| = T u 2 , ∀u ∈ H . La topologia naturale associata a Dè nota con il nome di topologia debole di B(H) edè molto usata nella teoria delle algebre di operatori. Una * -algebra A ⊆ B(H) contenente l'identitá e chiusa rispetto alla topologia debole si dice algebra di von Neumann (vedi [19, §4.6]). (5) Sia E uno spazio di Banach; l'algebra di Banach B(E)è equipaggiata con la famiglia separante di seminorme S := {p v (T ) := T u , T ∈ B(E)} v∈E .
C a seconda del caso reale o complesso. Ad esempio, ogni spazio localmente convessò e uno spazio vettoriale topologico rispetto alla topologia naturale (semplice esercizio!). Il seguente risultato caratterizza gli spazi vettoriali topologici che sono localmente convessi.Teorema 7.49. Sia (V, τ ) uno spazio vettoriale topologico di Hausdorff. Allora Vè localmente convesso (ovvero, τè indotta da una famiglia separante di seminorme) se e solo se 0 ∈ V ammette una base di intorni convessi, bilanciati ed assorbenti. Sketch della dimostrazione. Sia V localmente convesso. E' immediato verificare che se p 1 , . . . , p n sono seminorme su V allora ogni C ε;p1,...,pn := {x ∈ V : |p k (x)| < ε , k = 1, . . . , n} , ε > 0 , e convesso, bilanciato ed assorbente. Viceversa, se (V, τ )è tale che 0 ammette una base C := {C i } di intorni convessi, bilanciati e assorbenti, allora introducendo i funzionali di Minkowski p i (x) := inf{t ∈ R : x ∈ tC i } , x ∈ V , (7.59) otteniamo una famiglia separante di seminorme.
.
Si dice spazio di Fréchet uno spazio localmente convesso, completo e metrizzabile.Funzioni a decrescenza rapida e distribuzioni temperate. Sia n ∈ N e Z +,n l'insieme delle n-ple di interi non negativi. Definiamo |α| := n k=1 α k , α := (α 1 , . . . , α n ) ∈ Z +,n , x α := n k=1
la continuitá di i 1 segue da (7.60). Per p qualsiasi, poniamo q := p e troviamo f p ≤ |f | p/q |f | 0 , dove c := |f | 1/q p < ∞. I casi n > 1 si trattano in maniera analoga.
polinomi di Hermite si dimostra che I(S(R n ))è denso in S * (R n ) ([22, Thm.V.14]): in termini espliciti, per ogni ϕ ∈ S * (R n ) esiste una successione {g n } ⊂ S(R n ) tale cheϕ, f = lim n g n (x)f (x) dx , ∀f ∈ S(R n ) .(7.67) La densitá in S * (R n ) dell'immagine di (7.66) ha un'importante conseguenza: data un'applicazione continua T : S(R n ) → S(R n ) osserviamo che I•T : S(R n ) → S * (R n )è continua, per cui estendendo per continuitá possiamo definire T * : S * (R n ) → S * (R n ) : T * • I := I • T .
( 7 . 69 )[ 22 ,
76922La dimostrazione del fatto che (7.69) induce una topologia di Fréchetè non banale e coinvolge la nozione di limite induttivo rispetto alla rappresentazione di C ∞ c (Ω) come lProblem V.46]). Qui diamo per buono tale risultato e denotiamo lo spazio di Fréchet cosí ottenuto con D(Ω). Per definizione, un funzionale lineare ϕ : D(Ω) → Rè continuo se e soltanto se per ogni compatto K ⊂ Ω esistono c K > 0 , m K ∈ Z +,n tali che
dei funzionali lineari continui su D(Ω), che denotiamo con D * (Ω),è detto lo spazio delle distribuzioni, e viene equipaggiato della topologia debole indotta da D(Ω).Esempio 7.29. Per ogni x ∈ R ed α ∈ Z +,n , abbiamo, ovviamente, che δ α x , f := D α f (x), f ∈ D(Ω),è una distribuzione.
D(Ω) → D * (Ω) avente immagine densa. Per cui, ogni applicazione continua T : D(Ω) → D(Ω) si estende in modo univoco ad un'applicazione continua T * : D * (Ω) → D * (Ω), T * • I = I • T . Un'importante applicazione delle idee precedentiè la seguente: consideriamo un polinomio p(x) = |α|<k c α x α , c α ∈ R. Alloraè definito l'operatore continuo p(D) : D(Ω) → D(Ω) , p(D)f := |α|<k D α (c α f ) , ∀f ∈ D(Ω) , il quale si estende in modo unico a p(D * ) : D * (Ω) → D * (Ω) , p(D * )ϕ, f = |α|≤k (−1) |α| ϕ, D α (c α f ) , ∀f ∈ D(Ω) .
p(D)(ϕ * f ) = (p(D * )ϕ) * f , per cui concludiamo che p(D)u = δ 0 * f = f . In altri termini, uè una soluzione classica dell'equazione p(D)u = f . Il Teorema di Malgrange-Ehrenpreis ([23, Thm.IX.23]) afferma che ogni operatore differenziale p(D) a coefficienti costanti ammette una soluzione fondamentale. La dimostrazione si basa su tecniche di analisi complessa e l'estensione, tramite il teorema di Hahn-Banach, del funzionale ϕ D : p(−D) (D(R n , C)) → C , p(−D)f → f (0) , ad una distribuzione ϕ ∈ D * (R n , C), dove D(R n , C) := {f + ig : f, g ∈ D(R n )} . Per una rassegna su tale risultato si veda [18]. Esempio 7.34. [23, §IX.5]. La soluzione fondamentale dell'equazione di Laplace-Poisson ∆u
Definizione 7 . 52 .
752Un operatore su uno spazio di Hilbert Hè il dato di una coppia (D, T ), dove D (il dominio di T )è un sottospazio vettoriale di H e T : D → Hè un'applicazione lineare. Diremo che (D, T )è densamente definito se Dè denso in H .Ovviamente ogni T ∈ B(H)è un operatore densamente definito nel senso della definizione precedente, con D = H . D'altro canto, come esempio fondamentale portiamo la derivataH := L 2 (R) , D := C 1 c (R) , T f := f ′ , ∀f ∈ D .Osserviamo che in generaleè giá difficoltoso effettuare la somma o il prodotto di operatori, in quanto occorre fare attenzione ai domini. Invece, quando Tè iniettivo possiamo facilmente definire l'inverso di T come l'operatore(T (D), T −1 ) : T −1 (T u) := u , ∀u ∈ D ; rimarchiamo il fatto che potremmo avere T −1 ∈ B(H) anche quando Tè non limitato. Diciamo che (D ′ , T ′ ) estende (D, T ) se D ⊂ D ′ e T ′ | D = T , ed in tal caso scriviamo (D, T ) ≺ (D ′ , T ′ ); osserviamo che puó capitare che (D, T ) abbia come estensione un operatore limitato. Analogamente al caso limitato, diciamo che u ∈ Dè un autovettore di (D, T ) se T u = λu per qualche λ ∈ C. Diciamo che (D, T )è chiuso se il grafico G(D, T ) := {u ⊕ T u ∈ D ⊕ H} e un sottospazio chiuso di H ⊕ H , ed in tal caso, grazie al teorema del grafico chiuso, abbiamo che se D = H allora Tè limitato. Preso un generico operatore (D, T ), diremo che essoè chiudibile se la chiusura di G(D, T ) in H ⊕ Hè il grafico di un operatore (D, T ); chiaramente, in tal caso (D, T )è chiuso e per definizione estende (D, T ). Chiamiamo (D, T ) la chiusura di (D, T ). Operatori simmetrici ed autoaggiunti. Preso (D, T ) densamente definito, per ogni u ∈ H consideriamo l'applicazione lineare f T,u : D → K , f T,u (v) := (u, T v) (dove K = R, C a seconda del caso reale o complesso), la quale non necessariamenteè limitata; quando essa loè, essendo D denso possiamo estendere f T,u per continuitá ad un funzionale di H , per cui il Teorema di Riesz ci assicura che esiste edè unico T * u ∈ H tale che (T * u, v) = f T,u , v , ∀v ∈ H . (7.71) L'operatore aggiunto di (D, T ) si definisce come la coppia (D * , T * ), dove D * := {u ∈ H : f T,u < ∞} e T * uè definito da (7.71); ovviamente in particolare troviamo (T * u, v) = (u, T v), ∀v ∈ D , il che giustifica la nostra terminologia. Osserviamo che in generale nonè detto che (D * , T * ) sia densamente definito. Proposizione 7.53. Sia (D, T ) densamente definito. Allora (D * , T * )è chiuso.
72)il quale chiaramenteè isometrico; del resto le considerazioni precedenti ci dicono che Uè anche suriettivo e quindi invertibile. Ora, l'ortogonale di un sottospazioè sempre chiuso, per cui, essendo U invertibile ed isometrico, G(D * , T * ) = U −1 G(D, T ) ⊥è chiuso.Corollario 7.54. Con le notazioni della proposizione precedente, si ha la decomposizione ortogonale H ⊕ H = G(D, T ) ⊕ U G(D * , T * ) . Dimostrazione. Usando (7.72) giá sappiamo che G(D, T ) ⊥ = U G(D * , T * ). Del resto G(D, T ) ⊥⊥ = G(D, T ) e ció conclude la dimostrazione. Un operatore densamente definito (D, T ) si dice simmetrico se (T u, v) = (u, T v) , ∀u, v ∈ D . L'espressione precedente ci dice che ogni u ∈ Dè tale che f T,uè limitato e T u = T * u . Dunque, (D, T )è simmetrico se e solo se (D, T ) ≺ (D * , T * ); ció implica che anche (D * , T * )è densamente definito. Se (D, T )è simmetrico allora possiede solo autovalori reali, infatti se u ∈ D e T u = λu , λ ∈ C, allora troviamo (u, T u) = λ(u, u) = (T u, u) = λ(u, u) . Definizione 7.55. Un operatore simmetrico (D, T ) si dice: (1) autoaggiunto, se D = D * (il che implica T = T * ); (2) essenzialmente autoaggiunto, se la chiusura (D, T )è un operatore autoaggiunto. Corollario 7.56. Ogni operatore simmetricoè chiudibile, ed ogni operatore autoaggiuntoè chiuso. Dimostrazione. Segue immediatamente da Prop.7.53. Osservazione 7.9. Se (D, T )è autoaggiunto ed S = S * ∈ B(H) allora (D, T + S)è autoaggiunto; infatti
Esempio 7 . 35 (→
735La derivata). Consideriamo lo spazio di Hilbert complesso H := L 2 ([0, 1], C) e l'operatore di VolterraF u(x) := i x 0 u(t) dt , u ∈ H .Se u ∈ Hè tale che F u = 0 , allora per ogni x, y ∈ (0, 1) troviamo 0 = (χ[x,y] , u) = y x u = −i(F u(y) − F u(x)) = 0 , dunque F uè ortogonale ad ogni funzione a gradini; poiché l'insieme delle funzioni a gradiniè denso in H concludiamo che F u = 0 , cosicché Fè iniettivo. Inoltre F ha immagine F (H) densa in H , in quanto essa contiene il sottoinsieme (denso) {f ∈ C 1 ([0, 1], C) : f (0) = 0} ⊂ H . Definiamo dunque l'operatoreD := {F u + z : z ∈ C , u ∈ H , F u(0) = F u(1) = 0} , T (F u + z) := u .Si verifica facilmente (semplice esercizio sugli integrali multipli!) che (D, T )è simmetrico e che D * = {F u + z : z ∈ C, u ∈ H} , cosicché (D, T ) nonè autoaggiunto (per dettagli si veda[19, §5.1.16]). Si osservi che, applicando il teorema fondamentale del calcolo,T g = −ig ′ , ∀g ∈ D ∩ C 1 ([0, 1], C) .Lemma 7.57. Se S ∈ B(H)è autoaggiunto e iniettivo allora (D, S −1 ), D := S(H),è un operatore autoaggiunto.Dimostrazione. Innanzitutto osserviamo che avendosiD Lemma 7.7 = D ⊥⊥ ( 7.21) = (ker S) ⊥ = {0} ⊥ = H , concludiamo che Dè denso in H . Per ogni u := Su 0 , v := Sv 0 ∈ D osserviamo che (u, S −1 v) = (Su 0 , v 0 ) = (u 0 , Sv 0 ) = (S −1 u, v) , cosicché (D, S −1 )è simmetrico e quindi D ⊆ D * , il dominio dell'aggiunto (D * , S −1, * ). D'altro canto, w ∈ H appartiene a D * se e solo se esiste w ′ ∈ H tale che (w ′ , u) = (w, S −1 u) , ∀u ∈ D . Ma (w ′ , u) = (w ′ , Su 0 ) = (Sw ′ , u 0 ) = (Sw ′ , S −1 u) ⇒ (w, S −1 u) = (Sw ′ , S −1 u) , ∀u ∈ D ,per cui avendosi H = {S −1 u, u ∈ D} concludiamo che w = Sw ′ e quindi w ∈ D . Da ció si deduce D = D * e ció conclude la dimostrazione.Teorema 7.58 (von Neumann). Sia (D, T ) densamente definito e chiuso. Allora esiste un sottospazio D + ⊆ D , denso in H , tale che (D + , T * T )è autoaggiunto e 1 + T * T : D + → Hè biettivo, con (1 + T * T ) −1 ∈ B(H). Dimostrazione. Poiché Tè chiuso abbiamo G(D, T ) = G(D, T ) e grazie a Cor.U G(D * , T * ) P ′ U −1 → D * , dove Pè il proiettore su G(D, T ), 1 ∈ B(H ⊕ H)è l'identitá e P ′è la proiezione di H ⊕ H sulla prima componente della somma diretta. Definiamo gli operatori S : H → D , Su := P ′ P (u ⊕ 0) R : H → D * , Ru := {P ′ U −1 (1 − P )}(u ⊕ 0) .Essendo R, S composizioni di proiettori ed operatori isometrici troviamo R , S ≤ 1 . Applicando la decomposizione ortogonale di Cor.7.54 e (7.72), per ogni u ∈ H abbiamou ⊕ 0 = (Su ⊕ T Su) + (T * Ru ⊕ −Ru) = (S + T * R)u ⊕ (T S − R)u , cosicché u = (S + T * R)u , 0 = (T S − R)u ⇒ R = T S , (1 + T * T )S = 1 .(7.73) Posto D + := S(H) ⊆ D , queste ultime uguaglianze ci dicono che: (1) T (D + ) ⊆ R(H) ⊆ D * , cosicché 1 + T * Tè ben definito su D + ; (2) Sè iniettivo, e quindi anche 1 + T * Tè iniettivo su D + ; (3) l'immagine di 1 + T * T coincide con H . Inoltre troviamo (S * u, u) = (S * (1 + T * T )Su, u) = Su 2 + Ru 2 ∈ R , cosicché S = S * . Usando il Lemma precedente otteniamo che (D + , S −1 )è autoaggiunto, e confrontando con l'ultima uguaglianza di (7.73) concludiamo che (D + , S −1 ) = (D + , 1 + T * T ), cosicché (D + , 1 + T * T )è autoaggiunto. Infine, Oss.7.9 implica che anche (D + , T * T )è autoaggiunto.Esempio 7.36. Consideriamo l'operatore simmetrico (D, T ) dell'Esempio 7.35. Abbiamo che (D, T )è densamente definito, edè semplice verificare che essoè anche chiuso. D'altra parte troviamo
6 e 10.4. E' possibile dimostrare che ogni operatore simmetrico semicoercitivo ha un'estensione autoaggiunta; questo risultatoè noto come l'estensione di Friedrichs ([19, §5.1.13]).
Esempio 7.37 (L'operatore di Laplace). Sia Ω ⊆ R d aperto. Sullo spazio di Hilbert L 2 (Ω) definiamo l'operatore D := C ∞ c (Ω) , Lu := −∆u , ∀u ∈ D . Le formule di Green implicano che (u, Lu) = ∇u · ∇u ≥ 0 , ∀u ∈ D , per cui usando il teorema di Friedrichs concludiamo che (D, L) ammette un'estensione autoaggiunta. Il dominio di tale estensioneè noto come lo spazio di Sobolev H 2 (Ω) (vedi §10).La trasformata di Cayley. In questo paragrafo consideriamo spazi di Hilbert complessi. Sia (D, T ) un operatore simmetrico e λ = x + iy ∈ C; allora anche (D, T − x1)è simmetrico e per ogni u ∈ D risulta(T − λ1)u 2 = ((T − x1)u − iyu, (T − x1)u − iyu) = (T − x1)u 2 + y 2 u 2 ≥ y 2 u 2 .Cosicché T − λ1è iniettivo quando y = 0 , e l'operatore (T − λ1) −1 : {T − λ1}(D) → H e limitato con norma ≤ y −1 . Definiamo la trasformata di Cayley di (D, T ) come l'operatore lineare CT : {T + i1}(D) → {T − i1}(D) , CT := (T − i1)(T + i1) −1 . (7.74) Lemma 7.59. Si hanno le seguenti proprietá: (1) CTè isometrico e suriettivo, per cui si estende ad un operatore CT ∈ B(H), isometrico su {T + i1}(D) e nullo sul suo complementare in H 42 ;
Teorema 7 . 60 .
760La trasformata di Cayley definisce una corrispondenza biunivoca dall'insieme degli operatori simmetrici su quello delle isometrie parziali U ∈ B(H) tali che (1−U )| ker U ⊥ ha immagine densa in H . Inoltre, se (D, T ) ≺ (D ′ , T ′ ) allora CT ′ u = CT u per ogni u ∈ {T + i1}(D) ⊆ {T ′ + i1}(D ′ ).
cosicché (D, T )è simmetrico), e che U = CT . 43 Nel seguito identificheremo CT con CT .I seguenti risultati mostrano l'utilitá della trasformata di Cayley nel determinare se l'operatore simmetrico (D, T )è autoaggiunto. Proposizione 7.61. (D, T )è autoaggiunto se e solo se CTè unitario. Dimostrazione. Se (D, T )è autoaggiunto allora i, −i non sono autovalori di T , per cui {0} = ker(T ± i1) = {T ∓ i1}(D) ⊥ , il che implica che {T ± i1}(D) sono densi in H . Ma T , essendo autoaggiunto,è anche chiuso, per cui {T ± i1}(D) sono spazi chiusi e quindi coincidenti con H ; concludiamo che CTè unitario. Viceversa, se CTè unitario allora T + i1è suriettivo e per ogni u ∈ D * esiste v ∈ D tale che
Segue direttamente da (7.74) e dal Lemma 7.59 che CT (o, per essere pignoli, la sua estensione per continuitá ad H )è unitario se e solo se n + = n − = 0 , per cui (D, T )è autoaggiunto se e solo se n + = n − = 0 . Se invece (D, T ) ha indici di difetto non nulli ma uguali, allora esiste un operatore unitario V : {T + i1}(D) ⊥ → {T − i1}(D) ⊥ (costruito nella maniera banale, mettendo in corrispondenza biunivoca gli elementi delle basi), per cui possiamo estendere CT ponendo U (u + v) := CT u + V v , u ∈ {T + i1}(D) , v ∈ {T + i1}(D) ⊥ ; per costruzione Uè unitario e tale che 1 − U ha immagine densa, per cui essoè la traformata di Cayley di un operatore autoaggiunto (D ′ , T ′ ) il quale, grazie al Teorema 7.60, estende (D, T ). Dunque se n + = n − allora (D, T ) ha un'estensione autoaggiunta, non unica in quanto dipendente dall'operatore V definito poc'anzi. Per esempi di calcolo di indici di difetto rimandiamo all'Esercizio 7.17 e [9, Es.13.2.9] (lo shift). Osservazione 7.10. In modo analogo al caso limitato abbiamo ker T * = T (D) ⊥ per ogni operatore (D, T ) densamente definito. Dunque troviamo n + = dim ker(T * − i1) , n − = dim ker(T * + i1) , cosicché per calcolare gli indici di difetto possiamo procedere risolvendo le equazioni T * u = ±iu , u ∈ D * . Teoria spettrale degli operatori autoaggiunti. Sia (D, T ) un operatore sullo spazio di Hilbert complesso H . Il risolvente di (D, T )è per definizione l'insieme dei z ∈ C tali che esiste R z ∈ B(H) con R z (H) ⊆ D e (z1 − T )R z = 1 , R z (z1 − T ) = 1| D (in termini colloquiali, R z = (z1 − T ) −1 ). Lo spettro di (D, T ) si definisce come il complementare del risolvente e si denota con σ(D, T ). Come vedremo, in questo ambito piú generale abbiamo che σ(D, T )è tipicamente non limitato, mentre invece ritroviamo la proprietá di chiusura del caso degli operatori limitati: Proposizione 7.62. Lo spettro σ(D, T ) ⊆ Cè chiuso, per ogni operatore (D, T ).
Proposizione 7 . 63 .
763Se (D, T )è autoaggiunto allora σ(D, T ) ⊆ R. Dimostrazione. Posto z = a + ib , b = 0 , abbiamo che z1 − Tè invertibile se e solo se S + i1è invertibile, dove (D, S), S := b −1 (a1 − T ),è autoaggiunto. Essendo (D, S) simmetrico abbiamo che S + i1è iniettivo (vedi inizio paragrafo sulla trasformata di Cayley), e quanto visto nella dimostrazione di Prop.7.61 implica che S + i1è anche suriettivo e quindi invertibile.
Teorema 7 .
764 (von Neumann). Sia (D, T ) un operatore autoaggiunto su uno spazio di Hilbert H . Allora: (1) Esiste una misura spettrale µ := {µ uv } u,v∈D per T , tale cheè verificata (7.52) per ogni u, v ∈ D ; (2) Per ogni f ∈ M β (σ(D, T )), l'applicazione
Esempio 7 .
738 (La derivata). Poniamo H := L 2 ([0, 1], C), cosicché H ⊂ L 1 ([0, 1], C), e denotiamo con H 1 C ⊂ H lo spazio delle primitive di funzioni in H . Definiamo quindi D := {u ∈ H 1 C : u(0) = u(1)} , T u := −iu ′ , ∀u ∈ D . L'operatore (D, T )è banalmente simmetrico (per verificare si integri per parti) ed autoaggiunto (Esercizio 7.17), per cui lo spettro di (D, T )è reale, ed in effetti σ(D, T ) = 2πZ (Esercizio 7.18). Consideriamo ora la trasformata di Fourier discreta (x)e 2πinx dx , ∀n ∈ Z , u ∈ H . Essendo B C := {e n (x) := e −2πinx , x ∈ [0, 1]} una base di H (Esempio 7.4), abbiamo (u, v) = n u(n) v(n) per ogni u, v ∈ H . Definiamo la famiglia di misure complesse µ uv E := 2πn∈E u(n) v(n) , ∀E ⊆ 2 σ(D,T ) , u, v ∈ H . Poiché B C ⊂ D possiamo calcolare T e n = −ie ′ n = 2πne n , ∀n ∈ Z , per cui, per ogni u, v ∈ D , troviamo (u, T v) = n u(n)(e n , T v) = n u(n)(T e n , v) = n 2πn u(n) v(n) = λ∈σ(D,T ) λµ uv {λ} , e {µ uv }è una misura spettrale per (D, T ).Gruppi ad un parametro. Sia (D, T ) un operatore autoaggiunto. Consideriamo la famiglia di funzioni {e t } t∈R , e t : R → C , e t (λ) := e iλt , ∀λ ∈ R .E' ovvio che ogni e tè continua e limitata, per cui {e t } ⊂ L ∞ β (σ(D, T )). Dunque, grazie al teorema precedente possiamo definire U t := e t (T ) ∈ B(H) , ∀t ∈ R .
dove fè un termine associato al potenziale elettrico, f (r 1 , . . . , r n ) := −n n k=1 |r k | −1 + k<h |r h − r k | −1 , ∀r k ∈ R 3 − {0} , k = 1, . . . , n . Ovviamente il calcolo esplicito di una soluzioneè ora piú difficoltoso, ed il fatto che (D, T )è essenzialmente autoaggiunto, acclarato dal Teorema di Kato-Rellich,è un risultato non banale, per la cui dimostrazione rimandiamo a [23, Thm.X.16]. Una volta mostrata l'essenziale autoaggiuntezza di (D, T ) possiamo calcolare esplicitamente il relativo gruppo ad un parametro attraverso la formula di Trotter-Kato (vedi [22, §VIII.8]): Teorema 7.66. Siano (D, H), (D ′ , V ) autoaggiunti sullo spazio di Hilbert H e tali che (D ∩ D ′ , H + V ) sia essenzialmente autoaggiunto. Allora, per ogni t ∈ R, e t (H + V )u = lim k→∞ {e t (H/k)e t (V /k)} k u , ∀u ∈ H . Osserviamo che la formula precedenteè tutt'altro che banale, causa il fatto che (D, H), (D ′ , V ) non necessariamente commutano. Il teorema precedente si applica all'atomo con n elettroni ponendo (D, H) = (S(R 3n , C), −∆) e definendo (D ′ , V ) come l'operatore V u := f u , ∀u ∈ D ′ := {u ∈ H : f u ∈ H} . L'espressione esplicita di e t (H/k) si ottiene immediatamente da (7.81),
2 .
2Teoremi di esistenza di soluzioni per equazioni integrali. Un'equazione di Uryshon di seconda specieè un problema del tipou(t) = λ b a ϕ(t, s, u(s)) ds , ϕ ∈ C(A) , A := [a, b] × [a, b] × [−r, r] . (7.83) Poniamo E := C([a, b]) e C := {u ∈ E : u ∞ ≤ r} .Al solito, occorre verificare che Cè convesso, chiuso e limitato, e che l'operatoreF u(t) := λ b a ϕ(t, s, u(s)) ds , u ∈ C , sia un'applicazione compatta da C in sé: in effetti, cosíè se λ ≤ r ϕ −1 ∞ (b − a) −1 ([11, §3]).Menzioniamo infine il seguente risultato:Teorema 7.68. (Leray-Schauder-Tychonoff, [22, Thm.V.19]). Sia S un convesso compatto e non vuoto in uno spazio localmente convesso V . Allora ogni applicazione continua f : S → S ammette punti fissi. Esempio 7.40. La compattezzaè una condizione essenziale per i teoremi precedenti; ad esempio, l'applicazione F : l 2 ≤1 → l 2 ≤1 , x := {x n } → F x : (F x) 0 := 1 − x 2 2 (F x) n := x n−1 e continua, ma evidentemente priva di punti fissi.7.12 Esercizi.Esercizio 7.1. Sia E := C([0, 1]). Fissato α ∈ [0, 1], sia T α : D(T α ) ⊂ E → E l'operatore lineare definito da T α f (x) = x 0 y −α f (y) dy , x ∈ [0, 1] , dove D(T α ) := f ∈ E : x −α f (x) ∈ L 1 .Stabilire se le seguenti affermazioni sono vere o false: (a) T 1 nonè continuo; (b) T αè compatto per α ∈ [0, 1). Inoltre, determinare lo spettro di T α . (Suggerimento: si applichino i teoremi di Ascoli-Arzelá e Fredholm).
Esercizio 7. 3 (
3Schauder). Siano E, F spazi di Banach e T ∈ K(E, F ). (1) Si ponga X := T (E ≤1 ) (chiusura nella topologia della norma) e si mostri che Xè uno spazio metrico compatto. (2) Presa una successione {f n } ⊂ F * ≤1 , si mostri che la famiglia {ϕ n } ⊂ C(X) , ϕ n (x) := f n , x , x ∈ X , e equilimitata ed equicontinua. (3) Usando i punti precedenti, si mostri {T * f n } ⊂ E * ammette una sottosuccessione convergente, cosicché T * è compatto. (4) Si prenda ora un generico T ∈ B(E, F ) e si mostri che se T * ∈ K(F * , E * ) allora Tè compatto. (5) Si concluda che T ∈ K(E, F ) se e solo se T * ∈ K(F * , E * ).
Esercizio 7. 4 (
4Riesz). (1) Sia E uno spazio normato ed M ⊂ E chiuso. Allora per ogni ε > 0 esiste u ∈ E 1 tale che d(u, M ) := inf w∈M u − w ≥ 1 − ε . (7.84)(2) Usando il punto precedente, si concluda che se E 1è compatto nella topologia della norma allora E ha dimensione finita.Soluzione.(1) Sia v ∈ E − M . Poiché Mè chiuso (e diverso da E ) abbiamo d := d(v, M ) > 0 . Per costruzione, per ogni ε > 0 esiste w ∈ M tale che d ≤ v − w ≤ (1 − ε) −1 d. Ponendo u := (v−w)/ v−w si trova il vettore cercato.
Esercizio 7. 5 .
5Si consideri la seguente successione {f n } ⊂ p∈[1,∞] L p (R): f n (x) := 2 −n , x ∈ [2 n , 2 n+1 ] 0 , altrimenti.Si mostri che:(1) f n p → 0 per ogni p ∈ (1, ∞]; (2) {f n } non converge debolmente in L 1 (R).
Esercizio 7. 6 ([ 7 ,
67Ex.5.52]). Si consideri la successione {x (n) } ⊂ l ∞ definita da x Si dimostri che {x (n) } converge nella topologia * -debole, ma non in quella debole. (Suggerimento: riguardo la convergenza debole, si applichi il Teorema di Hahn-Banach al sottospazio E :
Esercizio 7. 7 (
7Ulteriori proprietá dello shift). Dato l'operatore S ∈ B(l 2 ) definito in (7.39), si dimostrino le seguenti proprietá:(1) S * x = (x 2 , x 3 , . . .), x := (x 1 , x 2 , . . .) ∈ l 2 ; (2) S * S = 1 , dove 1 ∈ B(l 2 )è l'identitá; (3)Per ogni x, y ∈ l 2 , risulta lim n (x, S n y) = 0 . Si interpreti ció in termini della topologia debole di B(l 2 ) (vedi Esempio 7.24
( 5 )
5Si mostri che H µ ha dimensione 1 nel caso in cui µ sia una misura di Dirac. (6) Supposto che µ sia regolare esterna, si mostri che esiste v 0 ∈ H µ tale che [v 0 ]è denso in H µ .
Esercizio 7 . 10 .
710Consideriamo la * -algebra di Banach A := L 1 (R, C) (vedi Esempio 7.13).
Esercizio 7 . 13 (
713Commutanti e algebre di von Neumann). Sia H uno spazio di Hilbert complesso. Preso un sottoinsieme S ⊆ B(H), definiamo il commutante di S S ′ := {T ∈ B(H) : T A = AT , ∀A ∈ S} .Si dimostri che: (1) S ′è un'algebra; (2) S ′è chiuso rispetto alla topologia debole di B(H) (nel senso dell'Esempio 7.24(4)); (3) se S = S * := {A * : A ∈ S} allora S ′è una * -algebra e quindi, per il punto precedente, un'algebra di von Neumann. (Suggerimenti: riguardo il punto (2), occorre verificare che se T ∈ B(H) appartiene alla chiusura debole di S ′ allora T A = AT per ogni A ∈ S , ovvero (u, AT v) = (u, T Av) per ogni u, v ∈ H . Del resto, per ipotesi esiste una successione {T n } ⊆ S ′ tale che lim n (u, T n v) = (u, T v) per ogni u, v .) Esercizio 7.14 (Algebre di von Neumann commutative). Sia (X, M, µ) uno spazio di misura finita ed H := L 2 µ (X, C). Si consideri la rappresentazione π : L ∞ µ (X, C) → B(H) , f → π f : π f u := f u , ∀u ∈ H .
Esercizio 7. 15 (
15Le relazioni di Heisenberg). Sia H uno spazio di Hilbert complesso. (1) Preso T = T * ∈ B(H), mostrare che
( 3 )
3Esibire uno spazio di Hilbert complesso H con un sottospazio denso D ⊆ H ed operatori simmetrici (D, P ), (D, Q) tali che {P Q − QP }u = −iu , ∀u ∈ D . (7.87)
si prenda H = L 2 (R, C), D = S(R, C) := {f + ig : f, g ∈ S(R)} , e Qu(x) := xu(x) , x ∈ R , P u := −iu ′ , u ∈ D .Esercizio 7.16 Si consideri l'operatore autoaggiunto (D, T ) ed il relativo calcolo funzionale boreliano del Teorema 7.64. Presa una successione {f n } ⊂ L ∞ β (σ(D, T )) limitata in norma · ∞ e convergente puntualmente a f ∈ L ∞ β (σ(D, T )), si dimostri che {f n (T )} converge a f (T ) nella topologia forte di B(H) (vedi Esempio 7.24(5)). (Suggerimenti: occorre verificare che lim n {f (T ) − f n (T )}u = 0 , ∀u ∈ B(H); d'altra parte
tende puntualmente a 0 . Per cui, avendo σ(D, T ) misura µ uu -finita, basta invocare il teorema di convergenza limitata Prop.4.30.) Esercizio 7.17. Si consideri lo spazio di Hilbert H := L 2 ([0, 1], C), l'operatore (D, T ) dell'Esempio 7.38, e l'operatore di Volterra F ∈ B(H) (Esempio 7.8). Si consideri poi la funzione u 0 (x) := 1 , ∀x ∈ [0, 1], e si osservi che (u 0 , u) = u , ∀u ∈ H .
( 5 )
5Si concluda che D = D * , cosicché (D, T )è autoaggiunto; (6) Usando Oss.7.10, si mostri che gli indici di difetto di (D, T ) sono nulli.(Suggerimenti: (1) Si noti che g = F g(1) e che V ⊥⊥ = Cu 0 ; (2) L'equazione(7.15) ed il punto (1) implicano che (F T * v, g) = (F * T * v + {F T * v(1)}u 0 , g) = (T * v, F g) = (v, T F g), e, del resto, T F g = g ; (3) Dalle uguaglianze precedenti segue che F T * v −vè ortogonale ad ogni g ∈ V , dunque per il punto (1) troviamo che F T * v − vè costante; (4) Si ha v(1) − v(0) = 1 0 v ′ ; (6) L'equazione T * u = −iu ′ = ±iuha soluzioni u(x) := ce ±x , x ∈ [0, 1], c ∈ C, le quali non sono in D . ) Esercizio 7.18 Si consideri l'operatore (D, T ) dell'Esempio 7.38, e si mostri che: (1) (D, T ) ha spettro puntuale 2πZ := {2πk, k ∈ Z} ; (2) Per ogni λ ∈ R−2πZ, l'operatore (D, T −λ1)è biettivo; (3) Si concluda che σ(D, T ) = σp(D, T ) = 2πZ. (Suggerimenti: in via preliminare si osservi che se u ∈ D allora u(0) = u(1). Riguardo i quesiti: (1) Occorre risolvere l'equazione −iu ′ − λu = 0 , u ∈ D , che ha soluzione in D se e solo se λ ∈ 2πZ e u(x) := ce iλx , dove c ∈ C. (2) Grazie al punto precedente giá sappiamo che (D, T − λ1)è iniettivo se λ ∈ R − 2πZ. D'altro canto, con il metodo di variazione delle costanti troviamo che, presa f ∈ L 2 ([0, 1], C) ⊂ L 1 ([0, 1], C) e posto f λ (t) := t 0 f (s)e −iλs ds , t ∈ [0, 1] ,
88)dove Iè l'unione di un numero finito di intervalli compatti mutualmente disgiunti ed S := {x n }∪{x 0 } e dato dalla successione {x n } con punto limite x 0 ∈ R. Si mostri che esistono uno spazio di Hilbert H e T = T * ∈ B(H) tale che σ(T ) = X , σp(T ) = S . (3) Si concluda che per ogni compatto X ⊂ R del tipo (7.88) esistono uno spazio di Hilbert H ed una rappresentazione come in(1).
. 4 .
4Sia f : R → R una funzione con periodo 2π e regolare a tratti. Allora la serie di Fourier (8.1) converge puntualmente alla funzione f , e quindi ad f stessa nei suoi punti di continuitá.
f (x + u) sin(n+1/2)u 2 sin u/2 du .
ultimo termineè una serie convergente.
e
una soluzione u di (8.11) debba essere di classe C 2 rispetto ad x, possiamo sviluppare u(·, t) in serie di Fourier. Inoltre osserviamo che le funzioni e −ω 2 t cos x , e −ω 2 t sin x sono delle soluzioni di (8.11), se non si tiene conto della condizione iniziale. L'ideaè quindi quella di scrivere lo sviluppo −k 2 ω 2 t {a k cos kx + b k sin kx} ;(8.12) assumendo le necessarie condizioni di regolaritá, troviamo
f
uguaglianza precedente ci dice che ρè soluzione del problema di Cauchy u ′ (x) = −(2π) −1/2 xu(x) u(0) = 1 , il quale d'altra parte ha soluzione unica u = ρ; per cui, ρ = ρ. La dimostrazione di (8.16) si effettua usando la sostituzione t → nt nell'integrale ρ n 1 . (8.17) si dimostra usando le uguaglianzeρ n (x) = nρ(nx) = n ρ(nx) = n 2π R e itnx−t 2 /2 dt ed applicando la sostituzione t → nt. Infine, (8.18) si dimostra calcolando f * ρ n (x) = R f (x − y)ρ n (y) dy (s)ρ(t/n)e i(x−s)t dtds = − R 1 √ 2π R f (s)e −ist ds ρ(t/n)e ixt dt = − R f (−t)ρ(t/n)e ixt dt = R f (t)ρ(t/n)e −ixt dt ,avendo usato il teorema di Fubini.Richiamiamo ora la notazione f y (x) := f (x + y), x, y ∈ R, e diamo il seguenteLemma 8.6. Si ha lim n f * ρ n − f 1 n → 0 , f ∈ L 1 (R, C) , , f ∈ L 1 (R, C) ∩ L 2 (R, C) . (8.20)Dimostrazione. Per dimostrare(8.19), effettuiamo la stimaf * ρ n − f 1 ≤ |f (x − y) − f (x)|ρ n (y) dxdy = f −y − f 1 ρ n (y) dy .Ora, usando l'Esercizio 5.2 troviamo che g(y):= f −y − f 1è continua, per cui scelto ε > 0 esiste δ > 0 tale che |y| < δ implica f −y − f 1 < ε . Dunque, f * ρ n − f 1 ≤ ε δ −δ ρ n (y) dy + 2 f 1 |y|≥δ ρ n (y) dy ,e poiché il secondo addendo nell'espressione precedente diventa arbitrariamente piccolo nel limite n → ∞, troviamo quanto volevasi dimostrare. Per dimostrare(8.20), osserviamo che dµ(t) := ρ n (t) dtè una misura di probabilitá, per cui possiamo usare la diseguaglianza di Jensen (4.46) e concludere|f * ρ n (x) − f (x)| 2 = |f (x − y) − f (x)|ρ n (y) dy 2 Jensen ≤ |f (x − y) − f (x)| 2 ρ n (y) dy .Dunque, otteniamo la diseguaglianzaf * ρ n − f 2 ≤ f −y − f 2 ρ n (y) dy e l'argomento usato per dimostrare (8.19) implica quanto volevasi dimostrare. Osservazione 8.3. Il lemma precedente stabilisce che {ρ n } si comporta come una successione di mollificatori, nonostante i supporti supp(ρ n ) non soddisfino la proprietá enunciata in Def.6.25. Quando f ∈ C c (R) il risultato puó essere migliorato, ottenendo una convergenza uniforme (vedi Oss.6.4 o [12, Lemma 2.8.1]). Poiché ogni ρ nè analitica, sviluppando in serie di Taylor troviamo
convergente alla costante (2π) −1/2 nonché monotóna crescente; per cui, per convergenza monotóna (Teo.4.32) e grazie a (
Teorema 8. 8 (
8Teorema di inversione di Fourier). Sia f ∈ L 1 (R, C) tale che f ∈ L 1 (R, C). Allora f (x) = 1 √ 2π f (t)e −ixt dt , q.o. in x ∈ R .(8.24) Dimostrazione. Usando (8.18) ed applicando il teorema di convergenza monotóna alla successione (8.23) troviamo f * ρ n (f (t)e −ixt dt . D'altra parte, grazie a (8.19) ed al teorema di Fischer-Riesz (Teo.5.5), concludiamo che esiste una sottosuccessione {n k } tale che f (x) = lim n f * ρ n k (x), q.o. in x ∈ R. Corollario 8.9. Sia f ∈ L 1 (R, C) tale che f (x) = 0 q.o. in x ∈ R. Allora f (x) = 0 q.o. in x ∈ R.
→ 0 ,
0Trasformata di Fourier e distribuzioni temperate. L'utilizzo di S(R d , C) di cui abbiamo appena accennatoè motivato dal fatto che la trasformata di Fourier si restringe ad un operatore lineareT : S(R d , C) → S(R d , C) , T f := f , ∀f ∈ S(R d , C) ⊂ L 1 (R d , C) , il qualeè continuo nella topologia naturale di S(R d , C) (vedi [23, Thm.IX.1]); ció implica cheè ben definita la trasformata di Fourier sulle distribuzioni temperate, come l'aggiunto di T :T * : S * (R d , C) → S * (R d , C) : T * • I = I • T ,(8.27) dove I : S(R d , C) → S * (R d , C)è l'immersione canonica nel senso di (7.66). La trasformata di Fourier dei gruppi localmente compatti abeliani. Sia G un gruppo topologico localmente compatto e di Hausdorff. Il gruppo dei caratteri di Gè l'insieme G * delle funzioni continue del tipo χ : G → T : χ(ts) = χ(s)χ(s) , ∀t, s ∈ G , equipaggiato con il prodotto χχ ′ (s) := χ(s)χ ′ (s), inverso χ −1 (s) := χ(s) , ed identitá e(s) := 1 , s ∈ G. Introduciamo su G * la topologia della convergenza uniforme sui compatti: ∀K ⊆ G compatto ,
f
(s)χ(s) dµ(s) , ∀χ ∈ G * .(8.29)
( 3 )
3Si mostri che l'operatore autoaggiunto ( D, T ) associato ad { U t }è dato da D = F (D), T = F T F * ; (4) Si verifichi (usando l'Esercizio precedente) che −i d ds ϕ(s) = F T ϕ(s) , ∀ϕ ∈ C ∞ c (R, C) ,cosicché, per definizione di T , si ha T F = −id/ds • F sul sottospazio C ∞ c (R, C) ⊂ L 2 (R, C) 45 ; (Suggerimenti: (1) Calcolando il limite (7.78) si trova D = u ∈ L 2 (R, C) : s 2 |u(s)| 2 < ∞ , T u(s) := su(s) , ∀s ∈ R , u ∈ D .
45
In realtá si verifica che ( D, T )è la derivata deboleD = {u ∈ L 2 (R, C) | ∃u ′ ∈ L 2 (R, C) : uϕ ′ = − u ′ ϕ , ∀ϕ ∈ C ∞ c (R, C)} , T u = −iu ′ ;il dominio Dè noto come lo spazio di Sobolev (complesso) H 1 (R, C) (vedi §10); per dettagli si veda[15, §III.3].
Lemma 9 . 5 .
95Ogni funzione analitica f : U → Cè olomorfa. Inoltre, ogni derivata n-esima f (n) e analitica in U (e quindi olomorfa), ed f si sviluppa nella serie di Taylorf (z) = ∞ n=0
a valori complessi. Iniziamo richiamando le seguenti terminologie: una curva γ : I := [a, b] → C , γ(t) = x(t) + iy(t) , t ∈ I (dove x, y : I → R, k = 1, 2 , al solito sono le componenti di γ ) si dice chiusa se γ(a) = γ(b), e semplice se γ| [a,b) , γ| (a,b] sono iniettive. Diremo inoltre che γè regolare se essaè continua ed ha derivata γ ′ : I → C , γ ′ (t) := x ′ (t) + iy ′ (t) continua 47 inİ := (a, b
Esempio 9. 2 .
2La curva seguente γ : [0, 4] → C descrive il perimetro di un quadrato in C edè regolare a tratti,Integrali curvilinei. Sia ora f ∈ C(U, C); per ogni γ ∈ C reg (I, γ(t) · γ ′ (t) dt .(9.12)
Esempio 9. 3 . ( 1 )1( 2 )
312Siano ζ, ζ ′ ∈ C e γ : I := [0, 1] → C definita da γ(t) := (1 − t)ζ ′ + tζ . Allora γ ′ (t) = ζ − ζ ′ e · (ζ − ζ ′ ) dt = ζ − ζ ′ .(9.13) Sia ζ ∈ C, ε > 0 e γ : I → C − {ζ} la curva (chiusa e semplice) γ(t) := ζ + εe 2πit , t ∈ I := [0, 1] .
Esempio 9. 4 . ( 1 )
41Consideriamo la corona circolare aperta U ⊂ C, U := {z ∈ C : 1 < |z| < 2} . Allora Uè un dominio regolare, il cui bordoè l'unione disgiunta ∂U = ∂U int∪ ∂U est delle curve ∂U int : [0, 1] → C , ∂U int (t) := e 2πi(1−t) , ∂U est : [0, 1] → C , ∂U est (t) := 2e 2πit . Osservare l'orientazione inversa di ∂U int rispetto a ∂U est . (2) Come variazione del tema precedente consideriamo ∆(ζ, r) := {z ∈ C : |z − ζ| < r} e definiamo U := ∆(0, 4) − {∆(2, 1) ∪ ∆(−2, 1)} .
ζ + εe iθ )| dθ .(9.23) Supponiamo che esista z ′ := ζ + εe iθ ∈ ∂∆ con |f (z ′ )| < M . Allora troviamo |f (z)| < M in un intorno V ⊆ ∂∆ di z ′ , e quindi (visto che in ∂∆ − V abbiamo |f (z)| ≤ M ) ζ + εe iθ )| dθ ,il che contraddice (9.23). Dunque |f (z)| = M , z ∈ ∂∆. Applicando questo ragionamento per ogni disco di raggio minore di ε , concludiamo che |f | ∆ = M . Applicando il teorema precedente, troviamo che fè costante in ∆ e, per continuazione analitica, concludiamo che fè costante, il che fornisce una contraddizione.Osservazione 9.2. Si hanno versioni del teorema precedente per funzioni armoniche (vedi [30, IV.2.3]), e per soluzioni del problema di Dirichlet non omogeneo (vedi [5, VIII.5,IX.7]).
Corollario 9 . 15 (
915Teorema fondamentale dell'algebra.). Sia p ∈ C[z] non costante. Allora p ammette almeno uno zero in C.
e −r+iy + e −r−iy dy .Effettuando una stima al limite r → ∞ otteniamo che gli integrali tra 0, π tendono a zero; e x+πi + e −x+πi dx = K r + e aπi K r , e x + e −x dx e , passando al lim r→∞ , l'integrale reale che vogliamo calcolare. DunqueI = −i/2e aiπ/2 = K r + e aπi K rIl calcolo di J si effettua in maniera analoga.) Esercizio 9.7. Sia {x} := x−[x] ∈ (0, 1) la parte frazionaria di x ∈ R. Si dimostri che la funzione g(z) := ∞ 1 {x} x z+1 dx , Re(z) > 0 (integrale reale)è olomorfa. (Suggerimento: La funzione x, z → {x}/x z+1è di classe L 1 su [1, ∞) come funzione di x ed olomorfa come funzione di z nel semipiano Re(z) > 0 , infatti x −z−1 = e −(z+1) log x . Applicando il Teorema 4.37 possiamo derivare rispetto a z sotto il segno d'integrale, concludendo che gè olomorfa).
Esercizio 9.8. (La zeta di Riemann). Si ponga U >λ := {z ∈ C : Re(z) > λ} , ∀λ ∈ R, e si consideri un aperto U tale che U ⊂ U >1 . Si prenda quindi la successione f n (z) := n k=1 1 k z , z ∈ U .
problema (10.1) nonè particolarmente interessante, visto che siamo in grado di risolverlo con metodi classici. Tuttavia ce ne serviremo per introdurre alcuni importanti concetti.Consideriamo una soluzione u di (10.1). Allora per ogni ϕ ∈ C 1 0 (a, b) risulta, integrando per parti,
Lemma 10. 1 .
1Siano a, b ∈ R, p ∈ [1, +∞] ed u ∈ L p (a, b). Se esiste v ∈ L p (a, b)
Esempio 10 . 1 .
101Posto a = −1 , b = 1 , allora u(x) := 1/2(|x| + x), x ∈ [−1, 1], appartiene a W 1,p (a, b) per ogni p ∈ [1, +∞]. La derivata debole di uè la funzione di Heaviside H(x) := 0 , x ∈ (−1, 0) 1 , x ∈ [0, 1)D'altra parte, H non appartiene a W 1,p per nessun p ∈ [1, +∞] (lasciamo la verifica di questo fatto come semplice esercizio).
nel caso p = 2 allora possiamo vedere u W,2 come la norma associata al prodotto scalare (u, u)Introduciamo la notazione H 1 (a, b) := W 1,2 (a, b), cosicché H 1 (a, b)è uno spazio pre-Hilbertiano.Proposizione 10.2 (Completezza degli spazi di Sobolev). Per ogni p ∈ [1, +∞], lo spazio di Sobolev W 1,p (a, b)è completo rispetto alla norma (10.4). Inoltre, W 1,p (a, b)è riflessivo per p ∈ (1, +∞), e separabile per p ∈ [1, +∞).
Teorema 10. 3 (
3Esistenza del rappresentante continuo). Sia p ∈ [1, +∞] ed a, b ∈ R. Per ogni u ∈ W 1,p (a, b), esiste edè unicaũ ∈ C([a, b]) tale che u =ũ q.o. in (a, b), in maniera tale chẽ u(x) −ũ(y) = x y u ′ (t) dt , x, y ∈ (a, b) . (10.5) Sketch della dimostrazione. Fissato x 0 ∈ (a, b), l'ideaè quella di considerare la funzione continua u 0 (x) := x x0 u ′ (t) dt , x ∈ (a, b) .Osserviamo che u 0è ben definita in quanto u ′ | (x0,x) ∈ L p ([x 0 , x]) ⊂ L 1 ([x 0 , x]) (vedi Cor.5.4). Cosicché, per ogni ϕ ∈ C 1 0 (a,
0 − u)ϕ ′ = 0 , ϕ ∈ C 1 0 (a, b) .Da quest'ultima uguaglianza si puó dedurre (in modo non banale, vedi [5, Lemma VIII.1,Cor.IV.24]) che u − u 0 coincide q.o. con una costante c ∈ R. Poniamo alloraũ := u 0 + c.
W 1,1 (a, b) = AC([a, b]) , a, b ∈ R .
Proposizione 10. 4 (
4Diseguaglianza di Poincaré). Siano a, b ∈ R, a < b , p ∈ [1, +∞]. Allora esiste una costante c = c(a, b) tale che u W,p ≤ c u ′ p , u ∈ W 1,p0 (a, b).(10.6)
Teorema 10.13 (Il problema di Dirichlet non omogeneo). Sia dato il problema−∆u + u = f u| ∂Ω = g (10.16) dove f ∈ L 2 (Ω), g ∈ C(∂Ω). Se g =g| ∂Ω per qualcheg ∈ H 1 (Ω) ∩ C(Ω),allora esiste edè unica la soluzione debole u ∈ H 1 (Ω) di (10.16), come soluzione del problema variazionale K := {v ∈ H 1 (Ω) : v −g ∈ H 1 0 (Ω)} .
Proposizione 10 . 14 (′ ) 2 + qu 2 ]
101422Il problema di Sturm-Liouville). Sia dato il problema −(pu ′ ) ′ + qu = f u(0) = u(1) = 0 (10.17) dove p ∈ C 1 ([0, 1]), p ≥ α con α ∈ R + − {0} , q ∈ C([0, 1]), f ∈ L 2 . Se q ≥ 0 ,allora esiste edè unica la soluzione debole di (10.17), come soluzione del problema variazionale ≥ ′ v ′ + quv) , u, v ∈ H 1 0 (0, 1) , e coercitiva. Possiamo quindi applicare il teorema di Lax-Milgram. Con metodi analoghi (e qualche piccola variante), siamo in grado di dimostrare il seguente Proposizione 10.15 (Il problema di Neumann). Sia dato il problema −u ′′ + u = f u ′ (0) = u ′ (1) = 0 (10.18)con f ∈ L 2 . Allora esiste edè unica la soluzione debole u ∈ H 2 (0, 1) di(10.17)
Xè una σ -algebra, e µ : M → R + := R + ∪ {+∞} e una funzione, detta misura, tale che:che vengono lasciate indeterminate. Invece, a differenza
di quanto accade nella teoria dei limiti definiamo 0 · ±∞ := 0 . Riguardo la relazione d'ordine,
poniamo −∞ < a < ∞ per ogni a ∈ R.
Definizione 4.2. Uno spazio misurabile (o di misura)è il dato di una terna (X, M, µ), dove
Xè un insieme, M ⊆ 2
2 .
2Misure di Radon e regolaritá interna. Sia (X, M, µ) uno spazio misurabile di Borel con X localmente compatto e di Hausdorff. Denotata con K la famiglia dei compatti su X osserviamo che, poiché Xè di Hausdorff, ogni C ∈ Kè anche chiuso (vedi [6, Prop.10.6] o [19, 1.6.5]), per cui K ⊂ M .Definizione 4.8. Sia X uno spazio localmente compatto di Hausdorff e µ : M → R + una misura di Borel su X . Diciamo che µè una misura di Radon se valgono le seguenti proprietá:
10 )
10dove Ag := {ag, a ∈ A} , gA := {ga, a ∈ A} . Sull'esistenza ed unicitá delle misure di Haar, si veda [25, §14.4- §14.6]; oltre, sempre su questa strada, c'è l'analisi armonica astratta ([14]).Esempio 4.4. Consideriamo il gruppo additivo degli interi (Z, +), sul quale definiamo la topologia discreta. Definiamo su Z la misura di enumerazione µ : 2 Z → R + , costruita come nell'Esempio 4.2. Poiché Zè discreto,è evidente che µè di Radon (in particolare, i compatti di Z sono i sottoinsiemi finiti). Inoltreè chiaro che µA = µ(A + k) , ∀k ∈ Z , dove A + k := {h + k, h ∈ A} . Dunque µè invariante per traslazioni e quindi una misura di Haar.4. Misure di Hausdorff. Sia ora (X, d) uno spazio metrico e δ > 0 . Consideriamo ε > 0 e definiamo
Osservazione 5.4. Il teorema di dualitá di Riesz rimane valido nel caso in cui X abbia misura qualsiasi, a condizione che p sia strettamente maggiore di 1 (vedi [10, Teo.8.5.12] per i dettagli; per un controesempio al caso p = 1 vedi [2, Oss.9.2.3])
19, Prop.1.7.12] per il caso di spazi normali e [12] per R 2 .24 Detta anche antisimmetrica, in quanto si hanno le relazioni v p(1) ∧ . . . ∧ vp m = sgn(p)v 1 ∧ . . . ∧ vm , ∀v i ∈ V , dove pè una qualsiasi permutazione di ordine m . Ad esempio, nel caso m = 2 abbiamo v 1
Sia M una varietá di dimensione d ed avente un ricoprimento finito {U i } tale che ogni intersezione finita ∩ k U i k sia diffeomorfa ad un aperto stellato di R d . Allora27)
dove H *
m (M )è lo spazio duale di H m (M ).
Teorema 6.9 (de Rham). H m (M ) ha dimensione finita per ogni m ∈ N e (6.27) stabilisce un isomorfismo di spazi vettoriali.
Di conseguenza si ha un isomorfismo H m
dR (M ) ≃ H m (M ).
Del resto f Rè misurabile per il Lemma 6.18, e grazie a Prop 4.13 concludiamo che f Eè misurabile. Infine, sempre per il Lemma 6.18 concludiamo che f E dµ = {µ×ν}R = {µ×ν}E .
Osservazione 7.2. (1) Dal teorema precedente segue immediatamente che uno spazio di Hilbert e riflessivo nel senso di Def.7.43; (2) Il teorema di Rieszè valido anche nel caso non separabile: la dimostrazione si basa su un accorto uso delle proiezioni ortogonali sui sottospazi di H (vedi [19, §3.1.6 -3.1.9]).
Teorema 7.18 (Hahn-Banach). Sia E uno spazio vettoriale e p : E → R una seminorma. Se E ′ e un sottospazio di E ed f : E ′ → Rè un'applicazione lineare che soddisfa
Sia H uno spazio di Hilbert, K ⊆ H un convesso chiuso e non vuoto. Presa una forma bilineare, limitata e coercitiva A, per ogni ϕ ∈ H * risulta quanto segue:Definizione 7.26. Sia H uno spazio di Hilbert. Una forma bilineare A : H × H → R si dice
coercitiva se esiste α > 0 tale che A(u, u) ≥ α u 2 , u ∈ H .
Teorema 7.27 (Stampacchia).
78 )
78il che vuol dire che possiamo esprimere T in termini del suo gruppo ad un parametro. In realtá, si dimostra il seguente risultato:Teorema 7.65 (Stone). Ogni famiglia di operatori unitari {U t } t∈R ⊂ B(H) che soddisfi (7.76) e (7.77)è il gruppo ad un parametro di un operatore autoaggiunto (D, T ). della dimostrazione. Occorre, prima di tutto, mostrare che l'insieme D := {u ∈ H : ∃ lim t→0 t −1 (U t − 1)u ∈ H} e un sottospazio denso di H ; si tratta poi di verificare che definendo T u , u ∈ D , a partire da (7.78) si ottiene effettivamente un operatore autoaggiunto. Per dettagli rimandiamo a [22, Thm.VIII.8].Esempio 7.39 (La derivata). Consideriamo la famiglia U = {U t } dell'Esempio 7.9; alloraè immediato verificare che U soddisfa (7.76), e del restoSketch
tenendo a mente l'Esempio 7.19. (3) Si usi l'Esercizio 7.16.)8 Analisi di Fourier. sviluppo di una funzione in serie di Fourier contiene in sé il germe di un concetto molto importante in analisi, quello di base di uno spazio di Hilbert. L'idea ulteriore che gli elementi di tale base (le funzioni trigonometriche) forniscano, per separazione di variabili, le soluzioni di un'equazione alle derivate parziali (l'equazione del calore)è un secondo concetto fondamentale, usato poi anche in altri ambiti (l'equazione dell'oscillatore armonico con i polinomi di Hermite ([30, Compl.II.III.1-3]), l'equazione dell'atomo di idrogeno con i polinomi di Laguerre [30, Compl.II.III.[2][3], . . .). Altrettanto importanteè la trasformata di Fourier, la quale trova applicazioni in svariati campi tra cui le equazioni alle derivate parziali e la teoria della trasmissione dei segnali.Lo
Lemma 8.1. Per ogni n ∈ N, vale la relazione La seguente diseguaglianza di Bessel segue immediatamente dalla considerazione che Bè un insieme ortogonale in L 2 ([−π, π]) (vedi Prop.7.3):1
2
+
n
k=0
cos ky =
sin(n + 1/2)y
2 sin y/2
;
(8.4)
per cui,
1
π
π
0
sin(n + 1/2)y
2 sin y/2
dy =
1
2
.
(8.5)
Va da sé che le citazioni relative a pagine web sono suscettibili di diventare obsolete.
Con il termine algebra intendiamo uno spazio vettoriale equipaggiato con un prodotto (associativo e distributivo). E' ben noto che le funzioni continue formano un'algebra rispetto alle operazioni di combinazione lineare e moltiplicazione.
Per definizione, l'algebra generata da un sottoinsieme S di C(X)è lo spazio vettoriale generato da prodotti del tipo f 1 · · · fn , dove n ∈ N , f 1 , . . . , fn ∈ S .
A volte diremo, piú semplicemente, che Fè limitato.
Qui usiamo la metrica euclidea d 2 ((x, y), (x ′ , y ′ )) := d(x, x ′ ) 2 + d(y, y ′ ) 2 , x, x ′ , y, y ′ ∈ X .
Quando, al contrario, lim t→b − |u(t)| = ∞ allora diciamo che u esplode. In tal caso troviamo necessariamente lim t→b − |u ′ (t)| = ∞ .
Osservare che, sfruttando le proprietá elementari delle σ -algebre, nonè restrittivo supporre che An ∩ Am = ∅ , n = m .
Una sezione dell'applicazione suriettiva π : Y → Xè un'applicazione iniettiva s : X → Y tale che π • sè l'identitá su X . L'esistenza di una sezione in generale nonè assicurata, a meno che non si invochi l'assioma della scelta.
Sul motivo di questa notazione rimandiamo a §5.
La stima precedente si ottiene con il seguente trucco: si definisca F (t) := t λ e −tx , t ∈ R , e si osservi che: (1) max F = F (λ/x) ; (2) fn(x) = xF (n) ≤ xF (λ/x) = g(x) .
In termini piú precisi, passiamo da L p µ (X) al suo spazio quoziente rispetto al sottospazio delle funzioni nulle q.o.. Il nostro abuso di terminologia consiste allora nell'indicare tale quoziente ancora con L p µ (X) .
La funzione segno sgnè definita nell'Esercizio 5.7, che invitiamo a risolvere per meglio maneggiare i conti della presente dimostrazione.
Qui abbiamo scritto, per semplicitá di notazione, µ ≡ µ An .
Ovvero esiste x ∈ U tale che per ogni y ∈ U risulta che i segmento che unisce x ad yè contenuto in U .
Osservare che, nel caso X = Y = R equipaggiato con la misura di Lebesgue, R X,Y ha elementi prodotti cartesiani di generici insiemi misurabili e non solo di intervalli, per cui il termine rettangoloè da intendere in senso lato.
Con la notazione dµ(t) intendiamo il fatto che stiamo integrando rispetto alla variabile t ∈ G .28 Qui con C ∞ c (R d ) intendiamo lo spazio vettoriale delle funzioni C ∞ su R d a supporto compatto.
Con il termine rappresentazione intendiamo che πωè un operatore lineare limitato tale che πω(a * ) = πω(a) * , πω(aa ′ ) = πω(a)πω (a ′ ) , ∀a, a ′ ∈ A .
In effetti T = 1 ; lasciamo la verifica di questa uguaglianza come esercizio.
Ovvero i cosiddetti operatori normali T ∈ B(H) tali che T T * = T * T . Osserviamo che nel caso reale operatori non autoaggiunti non sono in genere diagonalizzabili giá in dimensione finita (quando il polinomio caratteristico ha redici complesse).
Per l'uguaglianza seguente vedi [19, Prop.4.3.15] o [22, Thm.VII.1];è in questo punto cheè importante considerare spazi di Hilbert complessi.
Qui potremmo avere λ ∈ R o λ ∈ C a seconda del caso in cui Vè reale o complesso.
Operatori di questo tipo sono detti isometrie parziali. Si osservi che il nucleo di un'isometria parzialeè per costruzione il complementare del sottospazio sul quale essaè isometrica.
Vedi Def.4.3 ed Esercizio 4.7.
Cióè sempre garantito quando Gè uno spazio metrico (e quindi normale), vedi §5.2. In particolare, Cc(G, C) e denso in H quando Gè un gruppo di Lie.
Qui definiamo γ ′ (a) e γ ′ (b) rispettivamente come i limiti per t → a + e t → b − .
Del resto, anche in Teo.10.9 si richiede p > 1 per avere la compattezza.
Possiamo allora procedere a stimaref (x + u) − f + (x) 2 sin u/2 sin(n+1/2)u du .A questo punto,è conveniente definire e studiare la funzione (|a k | + |b k |) , n ∈ N , e convergente. A tale scopo, osserviamo che poiché fè regolare a tratti la derivata f ′è ben definita e continua in [−π, π] tranne che in un numero finito di punti, nei quali poniamo f ′ := 0 . In tal modo abbiamo che f ′ ∈ L 1 ([−π, π]) ∩ L 2 ([−π, π]), ed integrando per parti possiamo calcolarne i coefficienti di FourierApplicando la diseguaglianza di Bessel (8.6), otteniamoil che implica che la serie k k 2 (a 2 k + b 2 k ) converge. Ora, applicando la diseguaglianza xy ≤ 1/2(x 2 + y 2 ) adove µ * è la misura di Haar su G * . Per dettagli si veda[14,Vol.1, Cap.6],[9, §12].Esempio 8.2. Nel caso G = R d ritroviamo (a meno di radici di 2π ) la classica trasformata di Fourier in R d ; nel caso G = T (G * ≃ Z) abbiamo invece la cosiddetta trasformata di Fourier discreta f (k) := T f (z)z k dµ(z) , ∀k ∈ Z , f ∈ L 1 µ (T, C) ; l'integrale precedenteè effettuato sulla misura di Haar µ di T, la quale coincide essenzialmente con la misura di Lebesgue sull'intervallo [0, 1] una volta usato il cambiamento di variabileInfine, nel caso G = Z (G * ≃ T) la misura di Haar coincide con la misura di enumerazione (Esempio 4.4), cosicché lo spazio delle le funzioni integrabili su Zè dato daDi conseguenza, ogni f ∈ l 1 (Z, C) si puó riguardare come la successione dei coefficienti di Fourier della sua trasformata:Esercizi.Esercizio 8.1. Scrivere gli sviluppi in serie di Fourier delle seguenti funzioni: (1) e αx , α ∈ R;Esercizio 8.2. Calcolare la trasformata di Fourier di f n := nχ [0,1/n] , n ∈ N, e studiare la convergenza della successione { f n } ⊂ C 0 (R, C).Dimostrare che: (1) Valgono le uguaglianzeallora lim n→±∞ n f (n) = 0 .(Suggerimento: per (1) si usi lo stesso metodo usato per dimostrare il Lemma di Riemann-Lebesgue).Si considerino le misure di Dirac µ a , a ∈ R d , e si mostri che µ a (x) = e ix·a , x ∈ R d (cosicché, in particolare, per a = 0 otteniamo la funzione costante 1 ).(Suggerimenti: per il punto (1) si usi il teorema di convergenza dominata, mentre per il punto (2) si osservi che g dµ f = gf per ogni g ∈ L 1 µ f (R d )).Esercizio 8.5 (Trasformata di Fourier e gruppi ad un parametro). Sia U := {U t } un gruppo ad un parametro sullo spazio di Hilbert complesso H (vedi (7.76) e (7.77)). Per ogni f ∈ L 1 (R, C) si definisca(1) Si mostri che A fè una forma bilineare continua, per cui esiste edè unico l'operatore(3) Si prenda H := L 2 (R, C) e si verifichi che definendo U t u(s) := e its u(s) , ∀u ∈ H , t ∈ R , (8.32)Funzioni olomorfe.In analogia al caso reale (in piú variabili, vedi (6.1)), diremo che fè derivabile in ζ ∈ U lungo la direzione v ∈ T := {z ∈ C : |z| = 1} , se esiste finito il limite (con t ∈ R)Al solito, parleremo di funzioni di classe C k (U, C), k = 1, . . . , +∞.Ora, vogliamo introdurre una nozione diversa di derivata, intesa stavolta come limite rispetto alla variabile z ,2) mettendo in evidenza il fatto che z tende a ζ a prescindere dalla direzione. Diremo che una funzione f : U → Cè olomorfa in U se per ogni ζ ∈ U esiste in C il limite (9.2), che chiameremo la derivata complessa di f in ζ . Come si esprime la nozione di olomorfia in termini delle usuali derivate parziali? Per rispondere a questa domanda osserviamo (9.1) e notiamo che a denominatore del rapporto incrementale in effetti non appare la differenza z − ζ = tv , z := ζ + tv , bensí il parametro t. Dunque se vogliamo parlare di derivata nel limite z → ζ occorre innanzitutto dividere ∂f /∂v per v ; l'indipendenza dalla direzione si traduce invece col fatto che la funzionenon dipende da v ∈ T e coincide proprio con la derivata complessa di f .Applicando i consueti risultati di derivazione di combinazioni lineari e prodotti di funzioni otteniamo che l'insieme delle funzioni olomorfe in Uè un'algebra, la quale sará denotata con O(U ); l'algebra delle funzioni olomorfe in ogni intorno di U sará denotata invece con O(U ). Lemma 9.1. Sia f ∈ C ∞ (U, C), f (z) = u(x, y) + iv(x, y). Allora fè olomorfa in U se e solo se valgono le equazioni di Cauchy-Riemann.Scrivendo esplicitamente f (z) = u(x, y) + iv(x, y), otteniamo le relative equazioni in u, v . Supponiamo ora che f soddisfi (9.4). Introduciamo le notazionidove H x , H y , H xy ∈ C ∞ (U ′ , C). Applicando le equazioni di Cauchy-Riemann troviamoOra, per diseguaglianza triangolare abbiamo ξ, η ≤ |z − ζ| = |ξ + iη|; per cui, posto M := max U ′ {|H x |, |H y |, |H xy |} , concludiamo cheDunque, f ′ (ζ)è ben definito come limite.non sono olomorfe. D'altra parte, ogni funzione del tipo f (z) := z k , k ∈ N,è olomorfa.Dimostrazione. Per mostrare che fè olomorfa verifichiamo che soddisfa le equazioni di Cauchy-Riemann. A tale scopo, consideriamo al solito ζ ∈ ∆ ⊆ U tale che f (z)è della forma (9.7), e ne denotiamo conha lo stesso raggio di convergenza di (9.9), e quindi converge uniformemente in ∆ ρ . Per gli usuali teoremi di derivazione di serie uniformemente convergenti, fè derivabile, ePer cui, poiché ogni g k soddisfa (9.4), abbiamo che f soddisfa (9.4) e quindiè olomorfa. L'argomento precedente mostra anche che f ′ = lim k g ′ kè analitica e quindi olomorfa. Infine, confrontando i termini di (9.9) e (9.10) otteniamo f (n) (ζ) = n!a n , da cui (9.8).Un esempio fondamentale di funzione analitica (e quindi olomorfa)è dato daInfatti, per ogni ζ ∈ U e z : |z − ζ| < |ζ| troviamo che la serie ∞ n=0 (−1) n z − ζ ζ ń e assolutamente convergente, e quindi uniformemente convergente in ogni disco chiuso ∆ ⊂ {z : |z − ζ| < |ζ|} . La sommaè chiaramente data (per la regola di divisione tra polinomi) daE' disponibile in letteratura una serie di dimostrazioni del teorema di Cauchy, di natura sia analitica che geometrica. La tecnica piú classicaè quella di approssimare U con un insieme di rettangoli sui qualiè piú semplice verificare la validitá di (9.15), si veda[3]. Nelle righe seguenti mostriamo come il teorema di Cauchy sia conseguenza del teorema di Stokes 48 : se ω :Dunque, se f ∈ O(U ), allora la 1 -forma f dzè chiusa. Applicando il teorema di Stokes a ω = f dz otteniamo immediatamente (9.15). Possiamo ora dimostrare il seguente fondamentale teorema:Dimostrazione. La funzione {z → (z − ζ) −1 }è analitica in U − {ζ} (si veda (9.11)), e quindi olomorfa (Lemma 9.5). Per cui,Definendo γ come nell'Esempio 9.3(2) per il teorema di Cauchy si trova, tenendo conto delle orientazioni 49 ,Per cui, per dimostrare (9.17)è sufficiente valutare l'integrale su γ nell'uguaglianza precedente,Ora, (9.18) implica che l'integrale precedente in realtá non dipende dalla scelta di ε > 0 , per cui passando al limite ε → 0 otteniamo (9.17).48 Visto che C ha dimensione reale 2 il teorema di Stokes si riduce alla formula di Gauss-Green trattata in §6.4. 49In termini espliciti abbiamo ∂(U − ∆) = ∂U∪γ . Per cui, visto che l'integrale cambia segno passando da γ a γ ,Corollario 9.8. Ogni funzione olomorfa su un dominio regolare U ⊂ Cè analitica. Per cui, tutte e sole le funzioni analitiche su U sono le funzioni olomorfe.Dimostrazione. Dopo Lemma 9.5,è sufficiente dimostrare la prima affermazione. A tale scopo, presa f ∈ O(U ), applicando la formula integrale di Cauchy, i teoremi di passaggio al limite sotto il segno di integrale, e (9.11), troviamo, per ζ ∈ ∆(z 0 , ε) e γ definita alla solita maniera,Alcune conseguenze della formula di Cauchy. La formula integrale di Cauchy ha una serie di implicazioni, importanti sia dal punto di vista analitico che algebrico-geometrico. L'esposizione di queste prenderá il resto della sezione.Osservazione 9.1. Da (9.19) e (9.8) segue anche la formula(9.20)la quale implica, nel caso in cui f sia costante,Dimostrazione. Per ipotesi esiste M ∈ R tale che f ∞ < M . Usando z − ζ = εe 2πit e dz = ε2πie 2πit dt otteniamo, da (9.17),Potendo scegliere ε > 0 arbitrariamente grande, troviamo f n (ζ) = 0 e quindi, esprimendo f in serie di Taylor, otteniamo che fè costante.Il seguente risultato mostra che in effetti (9.15) caratterizza le funzioni olomorfe:per cui F (ζ) non dipende dalla scelta di γ . Mostriamo che Fè olomorfa e che f = F ′ , il che implica che fè olomorfa (Lemma 9.5). Scelto ζ ∈ U osserviamo che, per continuitá di f , per ogniavendo usato(9.13). Per cui f = F ′ ed il teoremaè dimostrato. Dimostrazione. Supponiamo che esista ζ ∈ U ed una successione {ζ n } (con ζ n = ζ , n ∈ N) tali che ζ = lim n ζ n , f (ζ) = f (ζ n ) = 0 , n ∈ N. Allora, esiste un disco ∆ = ∆(ζ, ε) contenente ogni ζ n per n maggiore o uguale di un opportuno k , e possiamo scrivere f (z) = k a k (z − ζ) k , z ∈ ∆. Se jè il primo indice tale che a j = 0 , alloraPer cui, f = 0 .Il teorema precedente consente di estendere ai complessi le funzioni reali classiche, considerando gli sviluppi in serieed osservando che per z ∈ R otteniamo esattamente le usuali funzioni reali sin , cos, exp, log . Il principio di continuazione analitica assicura che le funzioni sopra definite sono le uniche funzioni analitiche che estendono quelle reali date: ad esempio, se f ∈ O(C) e f (t) = sin t, ∀t ∈ R, allora f (z) − sin z ha insieme degli zeri contenente R e quindi avente punti di accumulazione, per cui deve essere f (z) − sin z ≡ 0 . Enfatizziamo il fatto che l'unicitá dell'estensione sussiste soltanto se ci restringiamo a considerare funzioni analitiche, infatti possiamo trovare sempre un'infinitá di estensioni a C continue o C ∞ .Esempio 9.5 (Il catino di Cauchy). Consideriamo la funzione realeUn semplice studio mostra che f ∈ C ∞ 0 (R). L'insieme degli zeri di fè dato da {x ≤ 0} ed ha chiaramente punti di accumulazione, per cui non esiste un'estensione analitica di f definita su un aperto U che contenga R. Dimostrazione. Definendo gli operatoritroviamo che l'equazione di Cauchy-Riemann per f si scriveil che implica f ′ = 0 .Teorema 9.14 (Principio del massimo). Sia U ⊂ C un dominio regolare connesso, f ∈ O(U ). Allora il massimo di |f | in Uè assunto in un punto del bordo ∂U .Un punto singolare (o singolaritá) per fè un punto z 0 ∈ C sul quale f nonè definita; diremo che z 0è isolato se esiste un intorno V ∋ z 0 tale che f sia definita in V − {z 0 } . Una singolaritá isolata z 0è un polo di ordine n ∈ N se esiste un intorno V ∋ z 0 tale chesi estende per continuitá ad una funzione olomorfa e non nulla in V . Usando lo sviluppo di Taylor di h, concludiamo che f ammette uno sviluppo di LaurentViceversa, ogni funzione f che ammetta uno sviluppo in serie del tipo precedente ha in ζ un polo di ordine n. Sia ora ζ ∈ C una singolaritá isolata per f ed U ∋ ζ un dominio regolare tale che U − {ζ} sia contenuto nel dominio di olomorfia di f . Dal teorema di Cauchy segue chenon dipende dalla scelta di U . Chiameremo la quantitá definita dall'equazione precedente il residuo in ζ della forma differenziale f dz . Osserviamo che (9.25) fornisce un interpretazione del residuo come un'ostruzione per la forma f dz ad essere chiusa. Chiaramente,Sia ora f del tipodove i coefficienti a −k sono nulli per k ∈ N abbastanza grande. AlloraInfatti, sfruttando il passaggio al limite sotto il segno di integrale, e calcolando il residuo sul bordo di un opportuno disco ∆ di centro ζ , otteniamoed i termimi con n ≥ 0 svaniscono per olomorfia della funzione (z − ζ) n , mentre i termini con n ≤ −2 svaniscono grazie all'uguaglianza (9.21).Il teorema dei residuiè di estrema utilitá nel calcolo di integrali complessi (ed anche reali). Infatti, il calcolo del residuo di una funzione meromorfa (ovvero, il termine destro di (9.27))è un'operazione piuttosto semplice: posto n uguale all'ordine del polo di f in ζ , grazie a (9.24) e (9.26), otteniamo immediatamente(9.28) 9.5 Esercizi.Soluzione. Si consideri la funzione meromorfa f (z) := (1 + z 6 ) −1 , definita su C privato dell'insieme P := {e (2k+1)πi/6 , k = 0, . . . , 5} . Chiaramente, gli elementi di P sono tutti poli semplici per f . Preso R > 0 , consideriamo il dominio regolare U delimitato dal segmento {|Re(z)| ≤ R} e dalla semicirconferenza S = {z = Re iθ , θ ∈ (0, π)} . Per R abbastanza grande, saranno contenuti in U tutti e soli i poli di f contenuti nel semipiano {|Im(z)| ≥ 0} , che sono ζ 1 := e iπ/6 , ζ 2 := e iπ/2 , ζ 3 := e iπ5/6 .Applicando (9.28) abbiamo Resed analogamentePer cui,Valutiamo ora, con R abbastanza grande, l'integrale su S :Esercizio 9.2. Calcolare l'integraleSoluzione. Effettuando la sostituzione (Suggerimento: presi ζ, ζ ′ ∈ U ed un dominio regolare V ⊂ U tale che ζ, ζ ′ ∈ V , si verifichi che per ogni f ∈ F si ha la stimaquindi si usi l'equilimitatezza).dove zeri e poli di f sono contati con la relativa molteplicitá.Soluzione. Preso ξ ∈ U consideriamo un disco ∆ con centro ξ tale che ξè l'unico zero o polo di f in ∆. Per ipotesi esistono n ∈ Z ed una funzione h olomorfa e non nulla in ∆ tale che fè della forma f (z) = (z − ξ) n h(z) , z ∈ ∆ .Se n > 0 allora f ha uno zero in ξ , mentre per n < 0 abbiamo un polo; in entrambi i casi nè, a meno del segno, la relativa molteplicitá. DunqueRipetendo il ragionamento per ogni ξ ∈ U troviamo la formula desiderata.Esercizio 9.6. Si calcoli R e ax + e −ax e x + e −x dx , |a| < 1 . (3) Si usi il principio di continuazione analitica. (4) Si ha Ulteriore aspetto interessante inerente la zeta di Riemannè la formula di EuleroDiseguaglianza di Sobolev e conseguenze. Abbiamo visto in precedenza che se (a, b)è limitato allora vale la stima (10.7), la quale implicaUna versione della diseguaglianza precedente vale anche per intervalli non limitati e generiche u ∈ W 1,p (a, b). Ció ha importanti conseguenze sulla struttura degli spazi di Sobolev: I seguenti interessanti risultati si dimostrano con l'uso di stime derivanti da (10.9) e Teo.10.5:cosicché W 1,p (a, b)è un'algebra.L'immersione continua in L ∞ . Il Lemma 10.6 ci dice cheè ben definito e limitato l'operatore lineare canonico Sketch della dimostrazione. Verifichiamo la compattezza di(10.12). A tale scopo denotiamo con F l'immagine attraverso I cont p della palla unitaria di W 1,p (a, b) ed osserviamo cheDunque Fè equilimitato ed equicontinuo, ed il Teorema di Ascoli-Arzelá (Teo.2.16) implica che Fè precompatto. Infine, per quanto riguarda la compattezza di I q 1 , l'ideaè quella di ripetere il ragionamento precedente applicando il Teorema di Riesz-Fréchet-Kolmogorov. Usando esattamente la stessa tecnica del caso unidimensionale, otteniamo Proposizione 10.11. Per ogni aperto Ω ⊆ R n , lo spazio W 1,p (Ω)è riflessivo per p ∈ (1, +∞) e separabile per p ∈ [1, +∞). In particolare, H 1 (Ω)è uno spazio di Hilbert separabile.Altri risultati, come l'approssimazione con funzioni C ∞ c (Ω) (Teorema di Friedrichs, [5, Teo.IX.2]), la diseguaglianza di Poincaré ([5,Cor.IX.19]), e l'immersione continua di W 1,p (Ω) in L ∞ (Ω), rimangono veri anche in piú dimensioni, seppure in quest'ultimo caso con alcune ipotesi aggiuntive sulla coppia (p, n). In particolare, se p > n ed Ω ⊂ R nè limitato e di classe C 1 allora si ha un'immersione compatta W 1,p (Ω) → C(Ω) (Teorema di Rellich-Kondrachov, [5, Teo.IX.16]) 51 ; al solito, utilizzeremo le notazioni W 1,p 0 (Ω), H 1 0 (Ω) per denotare i sottospazi delle funzioni nulle al bordo di Ω.Applicazioni alle equazioni alle derivate parziali.In questa sezione mostriamo come l'uso combinato degli spazi di Sobolev e del teorema di Lax-Milgram permette di dimostrare risultati di esistenza ed unicitá per problemi differenziali di tipo ellittico con condizioni al bordo.Nel seguito, indicheremo con Ω ⊂ R n un aperto limitato. Allo scopo di avere una notazione piú agile, denotiamo con ∇u · ∇v ∈ L 2 (Ω) , ∇u := ∂u ∂x 1 , . . . , ∂u ∂x 1 , la funzione ottenuta effettuando il prodotto scalare dei gradienti di u, v ∈ H 1 (Ω).
come gruppo additivo) allora G * ≃ R d ; infatti tutti e soli i caratteri di R d sono quelli del tipo χ x : R d → T , χ x (t) := e ix·t. G Se, = R D , D ∈ N, ∀t ∈ R d , x ∈ R dSe G = R d , d ∈ N (come gruppo additivo) allora G * ≃ R d ; infatti tutti e soli i caratteri di R d sono quelli del tipo χ x : R d → T , χ x (t) := e ix·t , ∀t ∈ R d , x ∈ R d .
Infatti i caratteri di T sono tutti e soli quelli del tipo χ k : T → T , χ k (z) := z k. G Se, = T (gruppo Moltiplicativo) Allora G * ≃ Z ; ∀z ∈ T , K ∈ Z, Se G = T (gruppo moltiplicativo) allora G * ≃ Z. Infatti i caratteri di T sono tutti e soli quelli del tipo χ k : T → T , χ k (z) := z k , ∀z ∈ T , k ∈ Z .
. G Se, = Z (gruppo Additivo) Allora G * ≃ T, Z → T , χ z (k) := z k , ∀k ∈ Z , z ∈ TSe G = Z (gruppo additivo) allora G * ≃ T, con caratteri χ z : Z → T , χ z (k) := z k , ∀k ∈ Z , z ∈ T .
Se Gè un gruppo localmente compatto abeliano allora si ha un isomorfismo G ≃ G * * . Sketch della dimostrazione. Si tratta di verificare che l'applicazione G → G * * , s → ϕ s : ϕ s (χ) := χ(s). ' Un, T * * ≃ T {r D } * * ≃ R D, Z * * ≃ Z; Questo Nonè Un Fatto Casuale, quanto si ha il seguente teorema: Teorema 8.12 (Pontryagin-van Kampen). ∀χ ∈ G * , s ∈ G , (8.28Un'occhiata agli esempi precedenti mostra che si hanno isomorfismi {R d } * * ≃ R d , T * * ≃ T, Z * * ≃ Z; questo nonè un fatto casuale, in quanto si ha il seguente teorema: Teorema 8.12 (Pontryagin-van Kampen). Se Gè un gruppo localmente compatto abeliano allora si ha un isomorfismo G ≃ G * * . Sketch della dimostrazione. Si tratta di verificare che l'applicazione G → G * * , s → ϕ s : ϕ s (χ) := χ(s) , ∀χ ∈ G * , s ∈ G , (8.28)
Le verifiche che (8.28)è iniettiva e conserva il prodotto sono semplici e vengono lasciate come esercizio. La continuitá e la suriettivitá sono i punti piú delicati. 14§24.8e un isomorfismo. Le verifiche che (8.28)è iniettiva e conserva il prodotto sono semplici e vengono lasciate come esercizio. La continuitá e la suriettivitá sono i punti piú delicati, e per essi rimandiamo a [14, Vol.1, §24.8].
Si consideri lo spazio di Hilbert H := L 2 µ (G, C) e si denoti con U H il gruppo degli operatori unitari. (1) Preso h ∈ G si definisca u h (g) := u(hg). ; M → R + Di Hausdorff, M ⊂ 2, G Di Haar ; ∀u ∈ H , G ∈ G, ovvero µ(gE) = µE , ∀E ∈ M ). h + v h , u 2 = u h 2 , ∀α ∈ C , u, v ∈ H) Si mostri che U hh ′ = U h U h ′ , ∀h, h ′ ∈ G. (3) Assumendo che lo spazio delle funzioni continue a supporto compatto C c (G, C) sia denso in H 46. si mostri che lim h→h ′ U h u − U h ′ u 2 → 0 , ∀u ∈ HEsercizio 8.8 (La rappresentazione regolare). Sia G un gruppo topologico (localmente com- patto, di Hausdorff ) e µ : M → R + , M ⊂ 2 G , la sua misura di Haar (invariante a sinistra, ovvero µ(gE) = µE , ∀E ∈ M ). Si consideri lo spazio di Hilbert H := L 2 µ (G, C) e si denoti con U H il gruppo degli operatori unitari. (1) Preso h ∈ G si definisca u h (g) := u(hg), ∀u ∈ H , g ∈ G, e si mostrino le seguenti uguaglianze: {αu + v} h = αu h + v h , u 2 = u h 2 , ∀α ∈ C , u, v ∈ H , cosicché l'applicazione U h u := u h , u ∈ H , definisce un operatore unitario U h ∈ U H . (2) Si mostri che U hh ′ = U h U h ′ , ∀h, h ′ ∈ G. (3) Assumendo che lo spazio delle funzioni continue a supporto compatto C c (G, C) sia denso in H 46 , si mostri che lim h→h ′ U h u − U h ′ u 2 → 0 , ∀u ∈ H .
C) e denotato con (·, ·) il prodotto scalare di H , l'applicazione A f. Si Mostri Che, Presa F ∈ L ; H × H → C, A f (u, v) := G f (h)(u, U h v) dµ(hSi mostri che, presa f ∈ L 1 µ (G, C) e denotato con (·, ·) il prodotto scalare di H , l'applicazione A f : H × H → C , A f (u, v) := G f (h)(u, U h v) dµ(h) ,
∈ H Un Operatore Limitato T F ∈ B ; V, Si, T f * l = T f T l , T αf +l = αT f + T l , T f ≤ f 1 , ∀f, l ∈ L 1 µ. G, C) , α ∈ CH) tale che (u, T f v) = A f (u, v), ∀ue una forma sesquilineare limitata, che definisce quindi un operatore limitato T f ∈ B(H) tale che (u, T f v) = A f (u, v), ∀u, v ∈ H . Si verifichino le seguenti relazioni: T f * l = T f T l , T αf +l = αT f + T l , T f ≤ f 1 , ∀f, l ∈ L 1 µ (G, C) , α ∈ C .
Si verifichino i punti precedenti nei casi:(i) G = R (gruppo additivo), dove µè la misura di Lebesgue; (ii) G = Z (sempre gruppo additivo), dove µè la misura di enumerazione. vedi EsempioSi verifichino i punti precedenti nei casi:(i) G = R (gruppo additivo), dove µè la misura di Lebesgue; (ii) G = Z (sempre gruppo additivo), dove µè la misura di enumerazione (vedi Esempio
Si verifichi che {U h χ}(g) = χ(h)χ(g), ∀χ ∈ G * ⊂ H , h, g ∈ G, e che {T f χ}(g) = f (χ)χ(g), ∀f ∈ L 1 µ. G * ⊂ H Sia, G Di, G, C), χ ∈ G * , g ∈ GSia assuma che G sia compatto e si verifichi che vale l'inclusione G * ⊂ H , dove G * è il gruppo dei caratteri di G. (7) Si verifichi che {U h χ}(g) = χ(h)χ(g), ∀χ ∈ G * ⊂ H , h, g ∈ G, e che {T f χ}(g) = f (χ)χ(g), ∀f ∈ L 1 µ (G, C), χ ∈ G * , g ∈ G.
Suggerimenti: (1) La linearitáè ovvia, mentre per l'isometria si usi l'invarianza per traslazioni di µ; (3) Si verifichi per u ∈ C c (G, C) e poi si argomenti per densitá; (4) Si proceda come nell'Esercizio 8.5; (6) Si osservi che χ ∈ G * è continua e che µè una misura finita. Si osservi che, grazie al teorema di Riesz, basta verificare che A f (u, χ) = f (χ)(u, χ), ∀u ∈ H , e si usi il teorema di Fubini sull. integrale doppio A f (u, χ)(Suggerimenti: (1) La linearitáè ovvia, mentre per l'isometria si usi l'invarianza per traslazioni di µ; (3) Si verifichi per u ∈ C c (G, C) e poi si argomenti per densitá; (4) Si proceda come nell'Esercizio 8.5; (6) Si osservi che χ ∈ G * è continua e che µè una misura finita. (7) Si osservi che, grazie al teorema di Riesz, basta verificare che A f (u, χ) = f (χ)(u, χ), ∀u ∈ H , e si usi il teorema di Fubini sull'integrale doppio A f (u, χ)).
Come vedremo negli esempi successivi. Diamo Ora Un'utile Caratterizzazione Delle Funzioni Olomorfe, moltissime" funzioni in C ∞ (U, C) non sono olomorfe. Diamo ora un'utile caratterizzazione delle funzioni olomorfe. Come vedremo negli esempi suc- cessivi, "moltissime" funzioni in C ∞ (U, C) non sono olomorfe, e lo strumento piú comodo per
= (e z − 1) −1 , z ∈ C,è meromorfa ed ha poli ζ k := 2πik , k ∈ Z. Esempio 9.7. La funzione f1Esempio 9.7. La funzione f (z) := (e z − 1) −1 , z ∈ C,è meromorfa ed ha poli ζ k := 2πik , k ∈ Z. Ogni polo ha ordine 1 , avendosi
Si verifichi che {f n } converge uniformemente ad una funzione f , la qualeè quindi olomorfa in U ; (3) Si mostri che fè prolungabile analiticamente ad una funzione F >1 ∈ O(U >1 ). Si mostri che ogni f nè olomorfa in U. esercizio precedente e le identitáSi mostri che ogni f nè olomorfa in U ; (2) Si verifichi che {f n } converge uniformemente ad una funzione f , la qualeè quindi olomorfa in U ; (3) Si mostri che fè prolungabile analiticamente ad una funzione F >1 ∈ O(U >1 ). (4) Usando l'esercizio precedente e le identitá
(a, b)è una successione limitata in norma · W,p . (1) Se p ∈ (1, +∞] allora esiste una sottosuccessione {f kn } convergente in norma · ∞ , e quindi convergente in norma · q per ogni q ∈ [1, +∞]; (2) Se p = 1 allora esiste una sottosuccessione {f kn } convergente. Corollario 10.10. Siano a, b ∈ R ed {f n } ⊂ W 1,pin norma · q per ogni q ∈ [1, +∞Corollario 10.10. Siano a, b ∈ R ed {f n } ⊂ W 1,p (a, b)è una successione limitata in norma · W,p . (1) Se p ∈ (1, +∞] allora esiste una sottosuccessione {f kn } convergente in norma · ∞ , e quindi convergente in norma · q per ogni q ∈ [1, +∞]; (2) Se p = 1 allora esiste una sottosuccessione {f kn } convergente in norma · q per ogni q ∈ [1, +∞).
] e {f n } ⊂ W 1,p (a, b)è una successione debolmente convergente allora quanto visto in §7.8 implica che {f n }è limitata, e quindi si applicano i risultati del corollario precedente. Osservazione 10.5. Se p ∈ (1, +∞Osservazione 10.5. Se p ∈ (1, +∞] e {f n } ⊂ W 1,p (a, b)è una successione debolmente conver- gente allora quanto visto in §7.8 implica che {f n }è limitata, e quindi si applicano i risultati del corollario precedente.
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Introduciamo Ora, modo ricorsivo) gli spazi di Sobolev di ordine superiore al primo: W m,p (a, b) := {u ∈ W m−1,p (a, b) : u ′ ∈ W m−. 1p (a, b)} , m = 2, 3, . .Introduciamo ora (in modo ricorsivo) gli spazi di Sobolev di ordine superiore al primo: W m,p (a, b) := {u ∈ W m−1,p (a, b) : u ′ ∈ W m−1,p (a, b)} , m = 2, 3, . . .
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| []
|
[
"Soft-Output Joint Channel Estimation and Data Detection using Deep Unfolding",
"Soft-Output Joint Channel Estimation and Data Detection using Deep Unfolding"
]
| [
"Haochuan Song [email protected] \nNational Mobile Communications Research Laboratory\nSoutheast University\nNanjingChina\n\nPurple Mountain Laboratories\nNanjingChina\n",
"Xiaohu You \nNational Mobile Communications Research Laboratory\nSoutheast University\nNanjingChina\n\nPurple Mountain Laboratories\nNanjingChina\n",
"Chuan Zhang [email protected] \nNational Mobile Communications Research Laboratory\nSoutheast University\nNanjingChina\n\nPurple Mountain Laboratories\nNanjingChina\n",
"Christoph Studer [email protected] \nDepartment of Information Technology and Electrical Engineering\nETH Zürich\nZürichSwitzerland\n"
]
| [
"National Mobile Communications Research Laboratory\nSoutheast University\nNanjingChina",
"Purple Mountain Laboratories\nNanjingChina",
"National Mobile Communications Research Laboratory\nSoutheast University\nNanjingChina",
"Purple Mountain Laboratories\nNanjingChina",
"National Mobile Communications Research Laboratory\nSoutheast University\nNanjingChina",
"Purple Mountain Laboratories\nNanjingChina",
"Department of Information Technology and Electrical Engineering\nETH Zürich\nZürichSwitzerland"
]
| []
| We propose a novel soft-output joint channel estimation and data detection (JED) algorithm for multiuser (MU) multiple-input multiple-output (MIMO) wireless communication systems. Our algorithm approximately solves a maximum aposteriori JED optimization problem using deep unfolding and generates soft-output information for the transmitted bits in every iteration. The parameters of the unfolded algorithm are computed by a hyper-network that is trained with a binary cross entropy (BCE) loss. We evaluate the performance of our algorithm in a coded MU-MIMO system with 8 basestation antennas and 4 user equipments and compare it to state-ofthe-art algorithms separate channel estimation from soft-output data detection. Our results demonstrate that our JED algorithm outperforms such data detectors with as few as 10 iterations.I. INTRODUCTIONIn the uplink of multiuser multiple-input multiple-output (MU-MIMO) systems, where user equipments (UEs) transmit pilots and data to a base station (BS), deploying optimal joint channel estimation and data detection (JED) is an elusive goal, and has been practicable only for small-scale MIMO systems due to the combinatorial nature and the high-dimensionality of the JED problem [1]-[3]. To overcome the high complexity of such methods, references [4],[5]proposed gradient-based algorithms that efficiently compute approximate solutions to the single-UE JED problem for large BS antenna arrays. The multi-UE JED case has been tackled recently in [6] for cellfree massive MU-MIMO systems, also using a gradient-based algorithm. All of these methods compute hard-output estimates and are, thus, unable to realize the full potential of coded data transmission. A soft-output algorithm that approximates JED using iterative estimation and detection (IED) has been proposed recently in[7], which alternates between channel estimation and data detection. Similar alternating optimization methods have been used before for IED in [8],[9].A. ContributionsWith the recent progress in deep neural networks, optimal JED [1]-[3] is suddenly within reach by leveraging deep | 10.1109/itw48936.2021.9611404 | [
"https://arxiv.org/pdf/2112.00330v1.pdf"
]
| 244,531,580 | 2112.00330 | 39ea6ded11ba209959d76df1992b45d14dc3efd1 |
Soft-Output Joint Channel Estimation and Data Detection using Deep Unfolding
Haochuan Song [email protected]
National Mobile Communications Research Laboratory
Southeast University
NanjingChina
Purple Mountain Laboratories
NanjingChina
Xiaohu You
National Mobile Communications Research Laboratory
Southeast University
NanjingChina
Purple Mountain Laboratories
NanjingChina
Chuan Zhang [email protected]
National Mobile Communications Research Laboratory
Southeast University
NanjingChina
Purple Mountain Laboratories
NanjingChina
Christoph Studer [email protected]
Department of Information Technology and Electrical Engineering
ETH Zürich
ZürichSwitzerland
Soft-Output Joint Channel Estimation and Data Detection using Deep Unfolding
We propose a novel soft-output joint channel estimation and data detection (JED) algorithm for multiuser (MU) multiple-input multiple-output (MIMO) wireless communication systems. Our algorithm approximately solves a maximum aposteriori JED optimization problem using deep unfolding and generates soft-output information for the transmitted bits in every iteration. The parameters of the unfolded algorithm are computed by a hyper-network that is trained with a binary cross entropy (BCE) loss. We evaluate the performance of our algorithm in a coded MU-MIMO system with 8 basestation antennas and 4 user equipments and compare it to state-ofthe-art algorithms separate channel estimation from soft-output data detection. Our results demonstrate that our JED algorithm outperforms such data detectors with as few as 10 iterations.I. INTRODUCTIONIn the uplink of multiuser multiple-input multiple-output (MU-MIMO) systems, where user equipments (UEs) transmit pilots and data to a base station (BS), deploying optimal joint channel estimation and data detection (JED) is an elusive goal, and has been practicable only for small-scale MIMO systems due to the combinatorial nature and the high-dimensionality of the JED problem [1]-[3]. To overcome the high complexity of such methods, references [4],[5]proposed gradient-based algorithms that efficiently compute approximate solutions to the single-UE JED problem for large BS antenna arrays. The multi-UE JED case has been tackled recently in [6] for cellfree massive MU-MIMO systems, also using a gradient-based algorithm. All of these methods compute hard-output estimates and are, thus, unable to realize the full potential of coded data transmission. A soft-output algorithm that approximates JED using iterative estimation and detection (IED) has been proposed recently in[7], which alternates between channel estimation and data detection. Similar alternating optimization methods have been used before for IED in [8],[9].A. ContributionsWith the recent progress in deep neural networks, optimal JED [1]-[3] is suddenly within reach by leveraging deep
unfolding of iterative algorithms [7], [10]- [13]. In this paper, we propose a novel soft-output JED algorithm that builds upon a maximum a-posteriori (MAP) JED problem formulation which we solve approximately using deep unfolding. We derive an iterative algorithm with soft-output capabilities by utilizing the approximate posterior mean estimator (PME) put forward in [14], [15]. All algorithm parameters are generated by a hyper-network that processes estimated channel state information (CSI). The hyper-network is trained with a binary cross entropy (BCE) loss that exploits the soft-output capabilities of our JED algorithm. We provide simulation results for an 8 BS antenna, 4 UE MU-MIMO system and compare our algorithm to state-of-the-art methods that separate channel estimation from soft-output data detection.
B. Notation
Lower case letters denote matrices and upper case boldface letters denote vectors. We use A b,u , a u , and b k to represent the entry in the bth row and uth column of the matrix A, the uth column of matrix A, and the kth entry in the vector b, respectively. The superscripts T and H denote the transpose and Hermitian transpose, respectively. The Frobenius norm and trace of a matrix A is A F and Tr(A). The U × U identity matrix is I U . Sets are denoted by calligraphic letters and the cardinality of Q is |Q|. The operator E denotes expectation.
II. PREREQUISITES
We now introduce the system model and MAP-JED optimization problem from which we derive a computationally efficient soft-output JED algorithm in Section III.
A. System Model
We focus on the uplink of a MU-MIMO communication system in which U single-antenna UEs transmit pilots and data to a BS equipped with B antennas. We assume a block-fading scenario with a coherence time of K = T + D time slots; T time slots are reserved for pilots and D time slots are used for payload data. The transmitted data matrix S = [S T , S D ] contains the pilots S T ∈ C U ×T and the transmit symbols S D ∈ Q U ×D of all UEs, where Q is the constellation set (e.g., QPSK). In what follows, we consider a frequency-flat channel 1 with the following input-output relation [16]:
Y = HS + N.
(1)
Here, Y ∈ C B×K is the receive matrix containing all received symbols at the B BS antennas over the K time slots, H ∈ C B×U is the (unknown) channel matrix, and N ∈ C B×K models thermal noise with i.i.d. circularly-symmetric complex Gaussian entries and variance N 0 per complex dimension.
B. MAP-JED Optimization Problem
We start by formulating the MAP-JED problem. Our goal is to jointly estimate the channel matrix and recover the most likely transmit symbols which requires us to assume priors for the channel matrix and the transmit symbols. For simplicity, we assume i.i.d. circularly-symmetric Gaussian entries in H with entry-wise variance E h and equally likely transmit symbols. These assumptions result in the MAP-JED problem 2 For simplicity of exposition, we will directly work with the transmit symbol matrix S instead of the pilot and data matrices S T and S D , respectively.
H, S D = arg min H∈C B×U S D ∈Q U ×D Y − HS 2 F + λ H 2 F (2) with λ = N 0 /E h .
Since (2) can be written as two nested optimization problems in the variables S and H, we can first determine the optimal channel estimate H given the transmit symbol matrix S and then find the optimal transmit symbol matrix S. Since the optimization problem is quadratic in H, the optimal channel estimate H has the following closed-form solution:
H = YS H M −1 ,(3)
where we use the auxiliary matrix M = SS H + λI U . We can now substitute H into the objective of (2) and perform algebraic simplifications, which leads to an equivalent MAP-JED problem that only depends on the transmit symbol matrix:
S = arg max S∈Q U ×K Tr Y H YS H M −1 S .(4)
After solving the MAP-JED problem in (4), one can determine the optimal channel estimate by plugging S into (3). Note that for λ = 0, the MAP-JED problem in (4) reduces to the well-known maximum likelihood JED problem in [3, Eq. 6].
III. S-JED: SOFT-OUTPUT JOINT CHANNEL ESTIMATION AND DATA DETECTION The combinatorial nature of solving (4) exactly would quickly result in prohibitive complexity which necessitates approximate algorithms. Furthermore, even solving (4) approximately would lead to a hard-output JED method that is unable to realize the full potential of coded data transmission. We now derive a computationally efficient algorithm to approximately solve (4) while being able to compute soft-output information.
A. Smoothening the MAP-JED Problem
The discrete nature of the constellation Q is the culprit of preventing gradient-descent-like methods (and deep unfolding) to approximately solve (4). We therefore propose to first relax the set Q to its convex hull, which is defined as [5]
C = |Q| i=1 α i s i | (α i ∈ R + , ∀i) ∧ |Q| i=1 α i = 1) . (5)
Here, s i is the ith symbol in the constellation Q. We can now replace the discrete constellation Q in (4) by the convex set C, which leads to the smoothened optimization problem
S sm = arg max S∈C U ×K Tr Y H YS H M −1 S .(6)
The resulting smoothened optimization problem has a differentiable objective and a convex constraint, which permits the use of gradient-based methods and deep unfolding.
B. Smoothened MAP-JED via Forward-Backward Splitting
We now show how to use forward-backward splitting (FBS) [17] to approximately solve (6). FBS iteratively solves convex optimization problems of the form
s = arg min s f (s) + g(s),(7)
where the function f is differentiable and convex, and g is convex but not necessarily smooth or bounded. By initializing FBS with s (1) , it solves the problem in (7) for the iterations t = 1, 2, . . . until convergence by computing
s (t+1) = prox g (s (t) − τ (t) ∇f (s (t) ); τ (t) ),(8)
where ∇f (s) is the gradient of f (s) and τ (t) is a carefullychosen step size at iteration t. The proximal operator for g(s) is defined as follows [18]:
prox g (s; τ ) = arg min z τ g(z) + 1 2 z − s 2 2 .(9)
While FBS is able to exactly solve convex optimization problems, it can be used to approximately solve many nonconvex problems [17], which will be described next. For our MAP-JED problem, we define f and g in (7) as
f (S) = Tr Y H YS H M −1 S and g(S) = χ C (S),(10)
where the indicator function χ C (S) implements the convex constraint S ∈ C U ×K in (6), and is zero if S ∈ C and infinity otherwise. The gradient of the function f in S is given by
∇f (S) = M −1 SY H Y(I U − S H M −1 S).(11)
Due to space constraints, the derivation of the gradient will be shown in the future journal version of the paper [19]. The proximal operator in (9) for S is given by
prox g (S; τ (t) ) = arg min X∈C U ×K 1 2 X − S 2 F ,(12)
where we move the indicator function in g(S) back to the constraint. This problem has closed-form expressions for most constellation sets, e.g., for QPSK the projection operator is
proj C ( {S i,j }) = min{max{| {S i,j }| , −α}, α}(13)proj C ( {S i,j }) = min{max{| {S i,j }| , −α}, α},(14)
where α = 1 √ 2 for the set Q = ± 1 √ 2 ± 1 √ 2 j . This FBS algorithm efficiently (and approximately) solves the smoothened MAP-JED problem in (6), but ignores the discrete nature of the constellation set and is unable to compute soft-outputs. Sections III-C and III-D address these issues.
C. Exploiting the Constellation with the PME
In order to exploit the constellation set Q, we model the iterations of FBS after evaluating the gradient step
X (t) = S (t) − τ (t) ∇f (S (t) )(15)
as follows:
X (t) = S + E (t) .(16)
Here, S is the (unknown) true transmitted data matrix and E (t) models estimation errors on these per-iteration estimates. By assuming that the distribution of E (t) is known, one can replace the projection onto the convex hull C in (12) by the entry-wise posterior mean estimate (PME)
S (t+1) u,k = E[S u,k |X (t) u,k ](17)
with u = 1, . . . , U and k = 1, . . . , K. We emphasize that the PME depends on the prior distribution (which is given by the constellation set Q) and the statistics of E (t) . We have the following prior distribution on the transmitted data:
p(S u,k ) = 1 |Q| |Q| i=1 δ(S u,k − s i ).(18)
Here, s i is the ith symbol in the constellation Q and δ is the Dirac delta function. By assuming that the estimation errors E (t) u,k are circularly-symmetric complex Gaussian with variance ν (t) u,k , the PME in (17) has a closed form (see, e.g., [20]) and one can replace the proximal operator in (12) with the PME in (17). This step requires knowledge of the per-iteration estimation error variances ν (t) u,k , ∀u, k, which are difficult to obtain in practice. However, as shown in Section IV, we can use a hyper-network to determine these variances.
D. Approximate PME and Soft-Output Generation
In order to extract soft-outputs, we build upon the approximate PME put forward in [14], [15]. The key idea is to replace (17) by an approximate three-step procedure that (i) converts the per-iteration estimates in (15) into soft-outputs in the form of log-likelihood ratios (LLRs) for every transmitted bit, (ii) transforms these LLR values into probabilities, and (iii) converts these probabilities back into the soft-symbol estimates. We now summarize this approach for QPSK modulation.
Step (i): Assume that the per-iteration estimation errors are circularly-symmetric complex Gaussian with variances ν (t) u,k , ∀u, k. Then, the LLRs for the two bits that map to QPSK symbols are given by [21,Tbl. 4.3]
L (t) 1,u,k = 4 {X (t) u,k } ν (t) u,k and L (t) 2,u,k = 4 {X (t) u,k } ν (t) u,k .(19)
Step (ii): We convert the LLRs in (19) into probabilities as follows [21,Eq. 3.6]:
P (t) b,u,k = 1 2 1 + tanh L (t) b,u,k 2 , b ∈ {1, 2},(20)
which express the probabilities of the bth bit that map to the symbol S u,k (e.g., using Gray mapping) being 1.
Step (iii): We use the probabilities in (20) to compute symbol estimates as follows [21,App. A.4]:
{S (t+1) u,k } = 1 √ 2 (2P (t) 1,u,k − 1) (21) {S (t+1) u,k } = 1 √ 2 (2P (t) 2,u,k − 1),(22)
where S (t+1) u,k approximates the PME output in (17). We emphasize that we are using this three-step procedure instead of the PME in (17) for three reasons: First, in
Step (i) we calculate LLR values for every transmitted bit in each iteration, which provides our algorithm with soft-output data detection capabilities. Second, in Step (ii) we obtain probabilities for every transmitted bit in each iteration, which will be key in learning the parameters of our soft-output JED algorithm (see Section IV-B). Third, this procedure was shown in [14], [15] to be less complex and numerically more stable than evaluating the exact PME in (17).
IV. DEEP UNFOLDING WITH A HYPER-NETWORK
We now explain our deep unfolding strategy for the soft-output JED algorithm and how to train the algorithm parameters. Due to space constraints, we focus on QPSK only-the general case will be presented in [19].
A. Deep-Unfolding Architecture
In order to determine the algorithm parameters, we use an emerging paradigm known as deep unfolding [10]- [12] which we combine with a hyper-network that provides these parameters based on estimated CSI [13]. The idea of deep unfolding is to unfold an iterative algorithm into T max layers (one for every iteration) and use tools of deep learning to determine an optimal set of the algorithm's parameters in every iteration (layer) t = 1, . . . , T max . Instead of hard-coding these parameters after training, we train a hyper-network that generates these algorithm parameters dependent on CSI.
The hyper-network, the unfolded algorithm (which consists of T max layers, each representing a JED iteration), and the parameters are shown in Fig. 1. At layer t, the input S (t) is first updated by a gradient descent step in (15) to obtain the symbol estimates X (t) . Then, the three-step procedure to approximate the PME as described in Section III-D is performed to obtain the next iterate S (t+1) as in (21) and (22). We note that the last layer t = T max only requires the LLR outputs in (19).
Our unfolded architecture requires several algorithm parameters, which are generated by a hyper-network. Specifically, for each iteration (layer) t = 1, . . . , T max , we require the periteration step size τ (t) and the estimation error variances ν (1) (1)
Hyper network
Gradient Calculation
∇ (1) S (1) (1) Soft symbols to LLRs (1) LLRs to probabilities Probabilities to symbols (2) (1) − 1 ∇ (1) (1) of parameters per iteration, we assume that the estimation error variances are fixed with respect to the time slot k, i.e., we only require ν (t) u and use the same variance for all time slots. We note that the hyper-network does not generate ν (t) u , but rather a normalized version η (t)
Gradient Calculation
u = N 0 /ν (t)
u to account for large variations in N 0 . We also require the parameter λ in (2); instead of using the same parameter λ for all iterations t = 1, . . . , T max , each layer uses a different parameter λ (t) .
The inputs to the hyper-network are the vectorized leastsquared channel estimate H LS = Y T S −1 T of the pilot phase (Y T contains the first T columns of the matrix Y) and the noise variance N 0 . The hyper-network itself consists of five dense layers with rectified linear unit (ReLU) activations in each layer except for the last one, which uses an absolute value activation to generate non-negative parameters.
B. Hyper-Network Training
In order to train the hyper-network, we leverage the softoutput capabilities of our algorithm. Specifically, since our JED algorithm computes probabilities for the transmitted bits (20), we can train the hyper-network using the outputs in the last iteration t = T max using the widely-used binary cross entropy (BCE) loss, which is defined as follows:
H(b i , p(b i )) = b i log(p(b i )) + (1 − b i ) log(1 − p(b i )). (23)
Here, b i ∈ {0, 1} is the label of the ith bit and p(b i ) is the predicted probability of this bit being 1. In our case, we utilize the probabilities P (Tmax) b ,u,k in (20) for every transmitted bit b b ,u,k , where b = 1, 2 is the bit index, u = 1, . . . , U the UE index, and k = 1, . . . , K the time slot index, calculated in the last iteration t = T max . Hence, we define the following average BCE loss over all of these probabilities
L = 1 2U K 2 b =1 U u=1 K k=1 H b b ,u,k , P (Tmax) b ,u,k ,(24)
which we use to train the hyper-network parameters. We learn only a single hyper-network for all signal-to-noise-ratio (SNR) values, which is in stark contrast to the common approach of using a different hyper-network for every SNR.
V. SIMULATION RESULTS
We now demonstrate the efficacy our soft-output JED algorithm and compare it to baseline algorithms. We first detail the system setup and then show simulation results.
A. System Setup
We simulate a MU-MIMO system as described in Section II-A with B = 8 BS antennas and U = 4 single-antenna UEs transmitting QPSK symbols for K = 244 time slots. The UEs transmit orthogonal pilots in S T from a 4 × 4 Hadamard matrix. The channel matrices are modelled as Rayleigh fading with i.i.d. complex standard Gaussian entries. We consider per-UE coding with a rate-1/2 low-density parity-check (LDPC) code as in IEEE 802.11n [22] with a block-length of 480 bits; for LDPC decoding, we use a sum-product and layered decoding algorithm with 10 iterations. The hyper-network is trained using an NVIDIA GTX1080 with 1 M transmissions and batch size of 1 k. We use Monte-Carlo simulations to extract the coded packet error rate (PER), uncoded bit error rate (BER), and BCE as in (24). We run T max = 10 iterations of our soft-output JED algorithm (called "S-JED").
B. Baseline Algorithms
In order to evaluate the effectiveness of our S-JED algorithm, we simulate the SIMO lower bound, which cancels MU interference with perfect CSI in a genie-aided fashion [23]. We also compare our algorithm to conventional methods that separate channel estimation from soft-output data detection. For such methods, we simulate a SIMO lower bound with estimated CSI (called "SIMO (est. CSI)"), where we use a leastsquares channel estimator to compute H LS . We also compare S-JED to the widely used soft-output linear minimum meansquare error (L-MMSE) equalizer [21], [24] and the max-log optimal single-tree-search sphere decoder (STS-SD) [25]. Figure 2 shows our simulation results. In Fig. 2(a), we see that S-JED approaches the SIMO lower bound by less than 3 dB at a coded PER of 0.1% and outperforms the SIMO lower bound that uses estimated CSI. S-JED significantly outperforms the max-log optimal soft-output STS-SD algorithm and the widely used L-MMSE equalizer, which both separate channel estimation from detection, by 2 dB and 4 dB, respectively. In Fig. 2(b), we see that the uncoded BER results behave similarly. The results in Fig. 2(c) demonstrate that the BCE accurately characterizes the performance of all methods, as the order between algorithms is preserved with respect to coded PER and uncoded BER-this implies that the BCE loss in (24) is well suited to train soft-output data detectors.
C. Simulation Results
VI. CONCLUSIONS
We have proposed a novel soft-output joint channel estimation and data detection (S-JED) algorithm for MU-MIMO systems. Our method formulates a maximum a-posteriori (MAP) optimization problem and computes approximate LLR values in every iteration. The algorithm parameters are generated by a hyper-network, which is trained using deep unfolding and a BCE cost function. Simulation results have shown that the proposed S-JED algorithm with only 10 iterations significantly outperforms the max-log optimal STS-SD and L-MMSE equalizer, which both separate channel estimation from soft-output data detection.
There are many avenues for future work. Our future journal paper in [19] will include missing derivations, a complexity comparison, and apply S-JED to higher-order modulation schemes as well as to other channel models.
u = 1, . . . , U and k = 1, . . . , K. To reduce the amount last Iteration 1st Iteration ...
Fig. 1 .
1Block diagram of the deep unfolded soft-output JED and the hyper-network. The unfolded algorithm consists of Tmax layers. Each layer takes in soft-symbols from the preceding layer and outputs new soft-symbols; the last layer outputs only LLR values. The hyper-network takes in the LS channel estimate H LS and noise variance N 0 in order to produce the parameters step sizes τ (t) , regularization parameters λ (t) , and normalized error variances η
Fig. 2 .
2Coded PER (a), uncoded BER (b), and BCE (c) performance for a B = 8 BS antenna, U = 4 UE MU-MIMO system with transmitting QPSK for K = 240 time slots. The proposed soft-output JED (S-JED) algorithm approaches the SIMO lower bound and outperforms the SIMO bound with estimated CSI as well as the max-log optimal soft-output STS-SD and the widely used L-MMSE equalizer which separate channel estimation from data detection.
Frequency selective channels can be transformed into parallel frequencyflat subcarriers by means of orthogonal frequency-division multiplexing.2 In Section IV, we let the hyper-network determine an appropriate choice of the parameter λ (t) for every iteration t, which enables us to apply our JED algorithm to channel matrices that are not necessarily i.i.d. Gaussian.
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A 354 Mb/s 0.37 mm2 151 mW 32-User 256-QAM Near-MAP Soft-Input Soft-Output Massive MU-MIMO Data Detector in 28nm CMOS. C Jeon, O Castañeda, C Studer, IEEE Solid-State Circuits Lett. 29C. Jeon, O. Castañeda, and C. Studer, "A 354 Mb/s 0.37 mm2 151 mW 32-User 256-QAM Near-MAP Soft-Input Soft-Output Massive MU-MIMO Data Detector in 28nm CMOS," IEEE Solid-State Circuits Lett., vol. 2, no. 9, pp. 127-130, Sep. 2019.
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| []
|
[
"Domain walls and gauge field localization in strongly-coupled pure Yang-Mills theories",
"Domain walls and gauge field localization in strongly-coupled pure Yang-Mills theories"
]
| [
"Archil Kobakhidze [email protected] \nSchool of Physics\nThe University of Melbourne\n3010VictoriaAustralia\n"
]
| [
"School of Physics\nThe University of Melbourne\n3010VictoriaAustralia"
]
| []
| We present a mechanism of gauge field localization on a domain wall within the framework of strongly coupled pure Yang-Mills theory. arXiv:0807.4578v1 [hep-th] | 10.1007/s10773-010-0640-7 | [
"https://arxiv.org/pdf/0807.4578v2.pdf"
]
| 17,141,745 | 0807.4578 | 2624705bdea02b052f4b5acdfc2c6a15f4354e3d |
Domain walls and gauge field localization in strongly-coupled pure Yang-Mills theories
July 2008
Archil Kobakhidze [email protected]
School of Physics
The University of Melbourne
3010VictoriaAustralia
Domain walls and gauge field localization in strongly-coupled pure Yang-Mills theories
July 2008
We present a mechanism of gauge field localization on a domain wall within the framework of strongly coupled pure Yang-Mills theory. arXiv:0807.4578v1 [hep-th]
Introduction
Some time ago, Dvali and Shifman proposed a field-theoretic mechanism for the localization of a gauge field on a hypersurface (brane) embedded in higher dimensional spacetime (bulk) [1]. The physical picture behind this mechanism is as follows. A gauge field is assumed to be in the confining phase in the bulk and in the Coloumb (or less confining) phase on the brane. Then the field is prevented from spreading out into extra dimensions because of the mass gap generated by the bulk confinement. Although this idea is physically rather transparent, its explicit implementation requires an adequate treatment of quantum effects in non-perturbative regime. An attempt to describe Dvali-Shifman mechanism based on some models of dual superconductivity has been made in [2]. In [3] more involved mechanism has been proposed within N=2 supersymmetric Yang-Mills theories which operates in weak coupling regime.
Most of the explicit models for the Dvali-Shifman localization of gauge fields has been discussed in supersymmetric setting. Also, besides the Yang-Mills fields, the models in [1] and [3], involve many additional fields and somewhat ad hoc mass parameters. Thus, although these models represent the interesting testing ground for theoretical ideas, they are far more complicated compared to a pure Yang-Mills theory where one may think that the non-perturbative mechanism of [1] could also be realized.
In this paper we aim to demonstrate explicitly the phenomenon of gauge field localization in strongly coupled non-supersymmetric pure Yang-Mills theory. In field-theoretic approach brane is usually viewed as a topological defect (domain wall, string, ...) formed upon the condensation of a scalar field. In supersymmetric theories [1], [3] such scalar fields are naturally present as the constituents of supermultiplets. But in pure Yang-Mills theories there are no elementary scalar fields that could condense. However, as has been argued in [4], [5], effective Higgs-like fields can emerge in strongly coupled Yang-Mills theories at low energies. The physical picture in strongly coupled Yang-Mills theories is analogous to the electron spin-charge separation [6] occurring in strongly correlated systems, such as in higher temperature superconducting cuprates. In what follows we will argue that condensation of the Higgs-like effective scalars support a domain wall structure on which massless Abelian gauge field gets localized.
In the present paper we explicitly consider (3+1)-dimensional strongly-coupled SU(2) gauge theory with an Abelian gauge field localized in (2+1)-dimensions. We hope to present generalization to higher dimensions and localized non-Abelian fields in future publications. Having such an explicit field-theoretic mechanism would greatly advance more detailed studies of various interesting domain wall based higher-dimensional models [7]. Aside the higher-dimensional theories, the physics discussed in this paper might be relevant for QCD in the infrared domain.
Spin-charge separation in SU(2) Yang-Mills
We start with a brief review of the spin-charge separation mechanism in SU(2) gauge theory [4]. SU(2)-valued gauge potential can be written as:
A µ = A µ τ 3 + W − µ τ − + W + µ τ + ,(1)
where,
W ± µ = (A 1 µ ∓ iA 2 µ )/ √ 2, τ ± = (τ 1 ± iτ 2 )/ √ 2,
and τ i are the Pauli matrices. SU(2)Yang-Mills Lagrangian then takes the form
L Y M = − 1 4 F µν F µν − W − µ [η µν D α D α − D µ D ν − 2igF µν ] W + ν − 1 4 g 2 W + µ W − ν − W + ν W − µ 2 ,(2)
where F µν = ∂ µ A ν − ∂ ν A µ and D µ = ∂ µ − igA µ is the covariant derivative relative to the Abelian field A µ gauging the Cartan subgroup U (1) C of SU(2). Next, the off-diagonal gauge fields W + µ = (W − µ ) * we decompose as
W + µ = iφ 1 e µ − iφ * 2 e * µ .(3)
In the above decomposition φ 1,2 are the complex scalar fields which carry U (1) C charge degrees of freedom of the W-bosons, while
e µ = 1 √ 2 (e 1 µ + ie 2 µ ) ≡ e iσê µ ,(4)
is a complex vector with the orthonormality condition
e i µ e jµ = δ ij .(5)
This vector describes the spin degrees of freedom of the W fields. It is believed that in the nonperturbative regime at low energies precisely φ i and e i µ (along with the remaining Abelian (Cartan subalgebra) field A µ ) degrees of freedom, rather than the original gauge bosons (A µ , W ± µ ) of SU(2), more adequately describe the dynamics, including the anticipated spontaneous generation of the mass gap.
Some important remarks on the above formalism are in order. First note that φ i and e i µ have the same number of degrees of freedom as the original off-diagonal gauge fields, i.e. 8. Indeed, two complex scalars φ i have 4 and a complex vector e µ with 3 constraints (5) has 5 degrees of freedom, so 9 in total. However, one of these 9 degrees of freedom is redundant because of the local phase invariance,
φ 1 → e iα(x) φ 1 , φ 2 → e iα(x) φ 2 , e µ → e −iα(x) e µ ,(6)
induced by the decomposition, and hence the degrees of freedom match as they should. The composite field,
B µ = ie * α ∂ µ e α ,(7)
plays the role of the gauge boson of this U (1) α invariance. In [4] the field (7) is referred to as the magnetic gauge field and U (1) α as the magnetic gauge invariance. To remove the redundant degree of freedom, we instead consider U (1) α -gauge invariant transverse magnetic field,
B α ≡ iê * µ ∂ αê µ = B α − ∂ α σ , ∂ αB α = 0 ,(8)whereê µ = e iσ(x) e µ , and also, U (1) α -gauge invariant fields,φ 1 = e −iσ(x) φ 1 andφ 1 = e −iσ(x) φ 1 .
With hatted fields we have right number degrees of freedom, namely 8, and no U (1) α local gauge invariance. However, there remains still the global phase invariance, U (1) glob α ,
φ 1 → e iαφ 1 ,φ 2 → e iαφ 2 ,ê µ → e −iαê µ .(9)
This will be important in what follows. Another important point is that the decomposition in (3) is not gauge invariant in general and thus physical relevence of various quantities requires special care. We are interested in gauge invariant observables, and in particular we would like to have a gauge invariant order parameter,
v 2 = |φ 1 | 2 + |φ 2 | 2(10)
This is not invariant under the full SU(2) gauge transformations unless we impose the gauge fixing condition
D µ W + µ = 0 ,(11)
which we assume to hold in what follows. By fixing the gauge (11) SU(2) symmetry is reduced and the remaining local gauge invariance becomes U (1) C , where U (1) C is the diagonal Cartan subgroup of the original SU (2), under which the fields transform aŝ
φ 1 → e iθ(x)/2φ 1 ,φ 2 → e −iθ(x)/2φ 2 ,ê µ →ê µ .(12)
Thus the full symmetry group is U (1) C × U (1) glob α . Finally, we stress non-canonical behavior of some composite objects introduced through the spin-charge decomposition (3) under the Lorentz transformations. Namely, as is discussed in detail in [5], while the object |φ 1 | 2 −|φ 2 | 2 |φ 1 | 2 +|φ 2 | 2 ≡ cos 2β transforms as a Lorentz scalar (it is also U (1) C ×U (1) αinvariant), the objects 2φ 1φ2 |φ 1 | 2 +|φ 2 | 2 and 2φ * 1φ * 2 |φ 1 | 2 +|φ 2 | 2 are realized as projective representations. 1 This will be important in the next section, where we will discuss the ground state. Indeed, due to the constraint,
cos 2 2β + 4|φ| 2 1 |φ 2 | 2 (|φ 1 | 2 + |φ 2 | 2 ) 2 = 1,(13)
non-trivial Lorentz invariant vacuum state can be realized if, and only if,
cos 2β = ±1 .(14)
Summarizing this section, we have expressed original SU(2) Yang-Mills theory with gauge bosons (A µ , W + µ , W − µ ) in terms of new variables (A µ ,B µ ,φ 1 ,φ 2 ). In the gauge (11) the Lagrangian (2) takes the form:
L Y M = − 1 4 F µν F µν + (∂ µ − igA µ − iB µ )φ 1 2 + (∂ µ + igA µ − iB µ )φ 2 2 +(|φ 1 | 2 + |φ 2 | 2 )(∂ αê * µ ∂ αêµ −B αB α ) + (φ 1φ2 ∂ αêµ ∂ αêµ + c.c.) + 4igF µνê µê * ν (|φ 1 | 2 − |φ 2 | 2 ) − 1 8 g 2 |φ 1 | 2 − |φ 2 | 2 2 .(15)
The ground state
With the aim to determine the ground state of the theory given by (15), let us concentrate on the last non-derivative term in (15), This potential is shown in Figure 1 (the left graph). The Lorentz-invariant ground state is realized by homogeneous field configurations |φ 1,2 | = v 1,2 subject to the constraint (14). It is easy to see that at tree level the ground state is given by the trivial configuration, v 1 = v 2 = 0. In order to locate the true ground state one must inspect radiative corrections to (16). This has been done in [8] (see also [9]) and we just borrow the 1-loop effective potential obtained there:
V 0 (φ 1 , φ 2 ) = 1 8 g 2 |φ 1 | 2 − |φ 2 | 2 2(16)V 1 = V 0 (v 1 , v 2 ) − β SU(2) (g 2 ) (v 2 1 − v 2 2 ) 2 4 log |v 2 1 − v 2 2 | µ − 25 12 . (17) β SU(2) (g 2 ) = − 22 3 g 4
16π 2 is the 1-loop SU(2) β-function for the running gauge coupling g(µ): dg d log µ = β SU (2) g . The potential (17) is drawn in Figure 1 (the right graph). The renormalization scale is convenient to associate with the gauge invariant order parameter (10), µ = v The minima now satisfy the equation,
|v 2 1 − v 2 2 | v 2 1 + v 2 2 ≡ | cos 2β| = exp − 6π 11α + 11 3 ,(18)
where α = g 2 4π is the SU(2) fine structure constant. We observe from (18) that nontrivial Lorentzinvariant solutions occur for
α(v) = 18π 121 ≈ 0.47.(19)v 1 ≡ v cos β = v, v 2 ≡ v sin β = 0 ,(20)v 1 ≡ v cos β = 0, v 2 ≡ v sin β = v .(21)
Thus a solution for fieldsφ 1 = f 1 (z),φ 2 = f 2 (z) interpolating, e.g., between vacua (20) at z = −∞ and (21) at z = +∞ will describe topologically stable domain wall 3 (see also [10] for a similar domain-wall configuration). This, the so-called "standard" domain wall configuration, is obtained numerically and is depicted in Figure 2.
Gauge field on the wall
The above domain-wall configuration is not the most general one. In fact a family of solutions can be obtained by multiplying the "standard" domain wall configuration by a phase,
F 1,2 = f 1,2 (z)e iσ/2 .(22)
Indeed, one of the twoφ fields, always can be taken to be real due to the U (1) C gauge invariance (this is just the gauge fixing condition). However, the (asymptotic) solutions on either side of the standard domain wall are continuously degenerated due to the U (1) glob α global invariance which is spontaneously broken only on the domain wall. Thus σ in (22) represents a collective coordinate of the wall which is realized as the Goldstone boson of spontaneously broken U (1) glob α , σ = σ(t, x, y). 4 The low-energy Lagrangian for this mode looks as [3]:
L σ = 1 2 g 2 ∂ m σ∂ m σ ,(23)
where m = 0, 1, 2 and ∼ (gv) −1 . On the other hand, compact scalar field σ is dual to the (2+1)-dimensional gauge field,
F mn = i mnk ∂ k σ .(24)
Then if we identify (2+1)-dimensional gauge coupling g 2 (2+1) = g 2 we obtain the following Lagrangian,
L = − 1 4g 2 (2+1) F mn F mn ,(25)
which describes (2+1)-dimensional Abelian gauge field localized on the domain wall. One can envisage that the above mechanism for gauge field localization is in accord with the heuristic argument suggested in [1] and represents field-theoretic analogue ( see [3] ) of the D-brane picture in string theory. Far away the domain wall v = 0, and the magnetic fieldB µ ≡ (B µ − ∂ µ σ) becomes massive. The spontaneous breaking of the local U (1) α in the bulk must also result in the formation of magnetic Abrikosov-Nielsen-Olesen strings. The fluctuation of the string endpoints attached to the wall gives the mode σ(t, x, y) which in turn is dual to the electric field F mn according to the relation (24). It would be interesting to explore this picture in more quantitative details.
Conclusion
In this paper we have found the domain wall solution in strongly coupled pure SU(2) Yang-Mills theory based on the spin-charge decomposition formalism developed in [4], [5]. The domain wall supports localized collective coordinate of the wall which is dual to the Abelian gauge field. Thus we have found that the mechanism for gauge field localization discussed in [1], [3] can be naturally realized in pure Yang-Mills theories.
Figure 1 :
13D plots of the tree level V 0 (on the left) and one-loop V 1 (on the right) potentials as a functions of (φ 1 ,φ 2 ). Mass is given in units of v. The Lorentz-invariant vacuum for V 0 corresponds to trivial (0, 0) configuration, while at one-loop level one has non-trivial vacua (±1, 0) and (0, ±1). The non-trivial vacua, e.g. (1, 0) and (0, 1), are separated by the potential barrier.
Figure 2 :
2Plot of the numerical solution for the "standard" domain wall corresponding to the field configuration interpolating between vacua (1, 0) and (0, 1). Mass is given in v units.This value is on the edge of perturbation theory where the 1-loop results still can be trusted.2 The equation (19) in priciple determines the vacuum expectation value v in terms of dimensionless SU(2) gauge coupling defined at certain energy scale, the phenomenon known as the dimensional transmutation.Notice the constraint (13) induces global O(3) symmetry. However the Lorentz-invariant ground state has no O(3) degeneracy, but is only doubly degenerate under discrete exchange symmetry, v 1 ↔ v 2 , since the Lorentz invariance strictly implies (14). Therefore, we have
The complex vector fieldê µ also transforms as a projective representation, so that the Lorentz invariance of the total Lagrangian is maintained.
Obviously, the simple 1-loop approximation cannot give quantitatively accurate predictions. Nevertheless, we believe that the qualitative picture of the vacuum structure is adequetely described at 1-loop level.3 We assume for definiteness that the wall lies in the xy plane and is centered at z = 0
The physical picture here fully parallels the one discussed in[3].
Acknowledgments. I am indebted to Ray Volkas for useful discussion, and to Nadine Pesor for her suggestion on numerical calculations. This work was supported by the Australian Research Council.
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| []
|
[
"Impact of Variable Photospheric Radius on Exoplanet Atmospheric Retrievals",
"Impact of Variable Photospheric Radius on Exoplanet Atmospheric Retrievals"
]
| [
"Jake Taylor \nInstitute for Research on Exoplanets\nDepartment of Physics\nUniversity of Montréal\nH2V 0B3MontréalCanada\n"
]
| [
"Institute for Research on Exoplanets\nDepartment of Physics\nUniversity of Montréal\nH2V 0B3MontréalCanada"
]
| [
"MNRAS"
]
| Inverse techniques are used to extract information about an exoplanet's atmosphere. These techniques are prone to biased results if the appropriate forward model is not used. One assumption used in a forward model is to assume that the radius of the planet is constant with wavelength, however a more realistic assumption is that the photospheric radius varies with each wavelength. We explore the bias induced when attempting to extract the molecular abundance from an emission spectrum which was generated with a variable radius. We find that for low gravity planets, the retrieval model is not able to fit the data if a constant radius model is used. We find that biased results are obtained when studying a typical hot Jupiter in the MIRI LRS wavelength range. Finally, we show that high gravity planets do not suffer a bias. We recommend that future spectral retrievals that interpret exoplanet emission spectra should take into account a variable radius. | null | [
"https://arxiv.org/pdf/2203.01839v2.pdf"
]
| 247,222,624 | 2203.01839 | b9a1a008399f0d317348250c31343a700259ed30 |
Impact of Variable Photospheric Radius on Exoplanet Atmospheric Retrievals
2022
Jake Taylor
Institute for Research on Exoplanets
Department of Physics
University of Montréal
H2V 0B3MontréalCanada
Impact of Variable Photospheric Radius on Exoplanet Atmospheric Retrievals
MNRAS
0002022Accepted XXX. Received YYY; in original form ZZZPreprint 8 March 2022 Compiled using MNRAS L A T E X style file v3.0methods: data analysis -techniques: spectroscopic
Inverse techniques are used to extract information about an exoplanet's atmosphere. These techniques are prone to biased results if the appropriate forward model is not used. One assumption used in a forward model is to assume that the radius of the planet is constant with wavelength, however a more realistic assumption is that the photospheric radius varies with each wavelength. We explore the bias induced when attempting to extract the molecular abundance from an emission spectrum which was generated with a variable radius. We find that for low gravity planets, the retrieval model is not able to fit the data if a constant radius model is used. We find that biased results are obtained when studying a typical hot Jupiter in the MIRI LRS wavelength range. Finally, we show that high gravity planets do not suffer a bias. We recommend that future spectral retrievals that interpret exoplanet emission spectra should take into account a variable radius.
INTRODUCTION
Studying the emission spectra of exoplanets using spectral retrievals has been the pioneering technique over the past decade to infer planetary properties (Lee et al. 2012;Line et al. 2014;Stevenson et al. 2014;Line et al. 2016;Evans et al. 2017;Edwards et al. 2020;Changeat 2021). Current work has utilised both the Hubble Space Telescope (HST) and the Spitzer Space Telescope. With such limited spectral coverage, assumptions about the forward model used in the retrieval framework need to be made. These assumptions include, the parameterization of the temperature-pressure structure, the molecules in the atmosphere, and the geometry and the cloud composition.
With the successful launch of the James Webb Space Telescope (JWST) on the 25th of December 2021 the exoplanet community will have access to spectral observations of exoplanets from 0.6 − 12 microns at unprecedented spectral resolution. The higher signal to noise observations will require more complex forward models such as geometries that extend past the current 1D assumption (Feng et al. 2016;Taylor et al. 2020;Feng et al. 2020;Irwin et al. 2020;Changeat & Al-Refaie 2020) and the parameterisation of clouds (Taylor et al. 2021).
A common model assumption is that the radius of the planet is constant with wavelength. Fortney et al. (2019) compared the effect of computing the radius at each wavelength has on the emission spectra. They found that while the effect is not important for current HST observations, it can cause a 10 − 20 % difference in the wavelength dependent eclipse depth for JWST quality data. Therefore in this study we explore how the variable radius assumption can bias our retrieved results when studying JWST quality observations. ★ E-mail: [email protected]
METHODS
In this section we outline the model set up and retrieval framework used.
The outline of the methodology of this study are as follows:
• Generate a model of a hot Jupiter which has a radius that varies with wavelength
• Consider 3 different observing scenarios: NIRSpec PRISM, MIRI LRS and NIRSpec PRISM + MIRI LRS
• For each of the observing scenarios, considering 3 different error envelopes: 100ppm, 60ppm and 30ppm.
• Retrieve on these simulated datasets with a retrieval set up which assumes that the radius is fixed at each wavelength
Model Setup
It was found by Fortney et al. (2019) that the impact of the variable radius is dependent on the surface gravity of the planet. Hence, we explore 3 atmospheric scenarios:
(i) A cloud free model based on the physical properties of WASP-43b.
(ii) Same as (1) however with 10% of the mass. We call this the low gravity scenario. This is analogous to a low gravity planet such as WASP-107b.
(iii) Same as (1) however with 2× the mass. We call this the high gravity scenario. This is analogous to a high gravity planet such as WASP-18b.
These scenarios let us explore a range of planetary scenarios observable with JWST. To generate these model spectra we use the CHIMERA code (see Line et al. (2013) show the main system parameters used in this study in Table 1. We use a parameterised temperature-pressure profile developed by Parmentier & Guillot (2014) which balances incoming shortwave radiation with the outgoing longwave radiation. For simplicity we have used a blackbody spectrum calculated at T = 4400K for the stellar model. We present the models in Figure 1, it can see seen that when comparing the difference between a model with a variable radius and one with a constant radius, that there is a difference which increases with wavelength. We see that the difference is larger for lower gravity cases, which is consistent with Fortney et al. (2019). We note that our study only considers hot Jupiters. Fortney et al. (2019) points out that the impact may be different for ultra hot Jupiters.
Variable Radius Implementation
We upgraded the CHIMERA code to be able to calculate the flux when considering a change in the photospheric radius (defined as the radius at which optical depth ( ) = 2/3) at each wavelength.
The planet-to-star flux is given by the equation:
= ( ) ( ) 2 ( ) ( )(1)
where ( ) is the planetary radius, which is shown to have a wavelength dependence. The planetary radius is often assumed to be constant with wavelength.
To account for the wavelength dependence we rewrite Equation 1 to be:
= + ( ) ( ) 2 ( ) ( )(2)
where ( ) is the change in radius at each wavelength. We can compute this ( ) by first extracting the altitude levels in the atmosphere ( ( )) and dividing this by the planetary radius ( 0 ), ( )/ 0 . By calculating the cumulative optical depth ( ( )) as a function of ( ), it is possible to determine the altitude at ( ) = 2/3, hence this would provide the ( ) at each wavelength. We note that in this study we are considering this effect in 1D and the ( ) = 2/3 should vary for different emission angles.
Sampling Method
To perform the retrievals we used the dynesty package (Speagle 2020) which implements a nested sampling algorithm (Skilling 2004;Skilling 2006). This technique has been effective in extracting atmospheric information from spectra (Rathcke et al. 2021;Lustig-Yaeger et al. 2021;Ahrer et al. 2022).
RESULTS
In this section we will outline the results from the three different planetary scenarios. We retrieve on the variable radius models shown in Figure 1 using a fixed radius assumption in our retrieval model. We perform this for three different wavelength ranges obtainable with JWST. These are: NIRSpec PRISM, MIRI LRS and then combining the two instrument modes. For each of these wavelength ranges a 100ppm, 60ppm and 30ppm error envelope is assumed to assess the limits in which a bias may arise.
WASP-43b
Figure 1 shows that the difference between the variable and constant radius models do not deviate until mid-IR, with around 100ppm difference at 4.5 microns. Figure 2 shows the best fitting spectra for all the observing modes for the 100ppm case. Table 1 shows the retrieved abundances for each modelling set up, accompanied by the reduced 2 .
Observations with NIRSpec PRISM show that the variable radius does not bias the retrieved chemical abundances. For both the 100ppm and 60ppm cases, the abundance of CO 2 is unconstrained, however at the 30ppm level it is possible to constrain the abundance of CO 2 . For observations with MIRI LRS, the abundance of H 2 O is retrieved to the correct value within 2 for the 100ppm and 60ppm case. A biased result is retrieved for the 30ppm case. When combining instrument modes H 2 O and CO are retrieved to be greater than 2 away from the input value. It is interesting that the combining NIRSpec PRISM with MIRI LRS results in a biased abundance of CO, as there is no CO information in this wavelength region. When combining instrument modes we begin to detect CO 2 at the 60ppm level and then constrain the abundance to the correct value at the 30ppm level. Figure 1 shows that the difference between the variable and constant radius models is large throughout the whole infrared. Although the initial 100ppm simulation produced the incorrect chemistry, this is not the interesting takeaway point. The constant radius model is not able to fit the variable radius simulated data, we show this in Figure 3. Therefore low gravity planets will require a variable radius model to be able to fit the observations. After 17 days the MIRI LRS still had not converged, compared to 0.477 days and 0.282 days from the WASP-43b and High Gravity cases. Each simulation was run on 12 central processing units (CPU) cores. The forward model, which assumes a constant radius, is too simple. The sampling algorithm cannot converge to produce an adequate reduced 2 . As a result of the biases being observed at the 100ppm level, we do not explore the 60ppm or 30ppm level. Figure 1 shows that the difference in the models is only around 50ppm at 4.5 microns. Figure 4 shows the best fitting models from the 100ppm case and present the retrieved abundances in Table 3.
Low Gravity Scenario
High Gravity Scenario
For each of the NIRSpec PRISM and NIRSpec PRISM + MIRI LRS observing modes, the abundance of H 2 O and CO were constrained within 2 of the input value. For the MIRI LRS observing mode the H 2 O abundance was constrained within 2 and the rest of the chemistry is unconstrained, which was to be expected. We can summarise that a constant radius model is sufficient to interpret the atmosphere of a high gravity exoplanet.
CONCLUSION
In this study we explored the impact of a variable photospheric radius on a retrieval analysis of emission spectra. We explored three model scenarios which span a range of gravities. Our conclusions can be summarised as follows:
(i) For a typical hot Jupiter, the variable radius assumption is not important when interpreting observations in the NIRSpec PRISM wavelength range. However, the impact of a variable radius becomes important when observing over the MIRI LRS wavelength region. If the variable radius is not assumed in the retrieval model then biased chemistry will be found.
(ii) For a low gravity hot Jupiter it is not possible to fit the emission spectrum with a constant radius model. Hence for all observing scenarios of a low gravity hot Jupiter, a variable radius model needs to be used for accurate interpretation.
(iii) For a high gravity hot Jupiter no biases were found when retrieving on the variable radius model with a constant radius model.
(iv) It is possible to constrain the abundance of CO 2 unambiguously in the WASP-43b case with an error envelope of 30ppm. A similar constraint is placed on the abundance of CO 2 in the high gravity case. The upper limit is well constrained, however a large tail is seen in the lower limit.
From this study we recommend that the community include a variable radius model in their retrieval frameworks as it will be important in the JWST era. This can be implemented in a similar way as discussed in Section 2.2. Given that the optical depths are already computed in the radiative transfer calculation, determining and then adding ( ) does not increase computation time.
ACKNOWLEDGEMENTS
Jake Taylor thanks the Canadian Space Agency for financially supporting this work. He thanks Dr Peter McGill for proof reading the manuscript. Jake thanks Dr Michael Line for discussions about the photospheric variable radius and the use of CHIMERA. Jake thanks Dr Joanna Barstow for the review of this letter which greatly improved its clarity.
DATA AVAILABILITY
Any data generated in this study is available on request. Table 2. The retrieved chemical abundances for the Low Gravity model. We show each observing mode and error envelope. We also present the reduced 2 where n = 7. The error ranges are the 2 ranges as outputted from dynesty.We present the true model values in brackets next to the parameters name. We have colour coordinated the results so they are easier for the reader to interpret. We present results within 2 in purple, outside of 2 in red, results with an upper limit inside 2 in blue and unconstrained results in black. 3.29 Table 3. The retrieved chemical abundances for the High Gravity model. We show each observing mode and error envelope. We also present the reduced 2 where n = 7. The error ranges are the 2 ranges as outputted from dynesty. We present the true model values in brackets next to the parameters name. We have colour coordinated the results so they are easier for the reader to interpret. We present results within 2 in purple, outside of 2 in red, results with an upper limit inside 2 in blue and unconstrained results in black.
865.62
Figure 2 .
2The best fitting model and simulated data for WASP-43b. The top, middle, and bottom panel show the NIRSpec observing mode, the NIRSpec + MIRI LRS observing mode, the MIRI LRS observing mode, respectively. The reduced 2 of the fit is shown inset.
Figure 3 .
3The best fitting model and simulated data for the low gravity case. The top and bottom panel show the NIRSpec observing mode and the NIRSpec + MIRI LRS observing mode, respectively. The reduced 2 of the fit is shown inset.
Figure 4 .
4The best fitting model and simulated data for the high gravity case. The top, middle, and bottom panel show the NIRSpec observing mode, the NIRSpec + MIRI LRS observing mode, the MIRI LRS observing mode, respectively. The reduced 2 of the fit is shown inset.
for more detailed explanation of the capabilities of the framework, the open source code can be found here: https://github.com/mrline/CHIMERA), weFigure 1. Left: Forward models used in this study with variable radius shown by the dashed line and constant radius with the solid line.In purple we present the model based on WASP-43b. In red we present the low gravity scenario. In blue we present the high gravity scenario. Right: The residuals when taking the difference between the variable and constant radius models.10 0
10 1
Wavelength [microns]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
Fp/F
star + Offset
WASP-43b
Low Grav WASP-43b
High Grav WASP-43b
10 0
10 1
0
100
Residual (ppm)
0
2000
10 0
10 1
Wavelength [microns]
0
50
Parameter
Value Parameter
Value
log(H 2 O) 1
-3
log(CO 2 ) 4
-8
log(CO) 2
-3
log(CH 4 ) 3
-8
Teq [K]
1400
Rs [R sun ] 5
0.667
log(
)
-1.5
Tstar [K]
4400
log( )
-0.7
SMA [AU] 5
0.015
Mp [M ] 5
2.034
Rp [R ] 5
1.036
Table 1. Atmospheric and planetary properties used to create the model atmo-
spheres. 1 Polyansky et al. (2018), 2 Li et al. (2015), 3 Yurchenko & Tennyson
(2014), 4 Yurchenko et al. (2020), 5 Gillon et al. (2012).
The error ranges are the 2 ranges as outputted from dynesty. We present the true model values in brackets next to the parameters name. We have colour coordinated the results so they are easier for the reader to interpret. We present results within 2 in purple, outside of 2 in red, results with an upper limit inside 2 in blue and unconstrained results in black.NIRSpec PRISM
Model
logH 2 O (-3) logCO (-3) logCO 2 (-8) logCH 4 (-8)
2 /n
100ppm
-2.99 +0.40
−0.24
-3.05 +0.63
−0.43
-9.67 +2.10
−2.20
-9.35 +2.08
−1.99
0.18
60ppm
-3.02 +0.19
−0.14
-3.10 +0.32
−0.26
-9.68 +1.91
−2.21
-9.46 +2.56
−2.42
0.51
30ppm
-3.03 +0.09
−0.07
-3.12 +0.14
−0.13
-8.74 +0.85
−3.10
-9.59 +2.35
−2.29
1.89
MIRI LRS
Model
logH 2 O (-3) logCO (-3) logCO 2 (-8) logCH 4 (-8)
2 /n
100ppm
-2.24 +1.01
−1.56
-6.65 +4.80
−5.05
-8.25 +3.49
−3.53
-8.68 +3.16
−3.10
0.15
60ppm
-2.24 +0.80
−0.99
-6.75 +4.90
−4.88
-8.60 +3.67
−3.23
-8.81 +2.97
−3.01
0.39
30ppm
-2.21 +0.37
−0.38
-6.75 +4.76
−4.92
-8.61 +3.16
−3.08
-9.09 +2.87
−2.69
1.58
NIRSpec PRISM + MIRI LRS
Model
logH 2 O (-3) logCO (-3) logCO 2 (-8) logCH 4 (-8)
2 /n
100ppm
-2.68 +0.38
−0.33
-2.52 +0.44
−0.42
-9.20 +2.05
−2.65
-9.03 +2.67
−2.81
1.25
60ppm
-2.68 +0.22
−0.21
-2.53 +0.26
−0.27
-8.29 +1.03
−3.51
-9.19 +2.61
−2.67
3.43
30ppm
-2.68 +0.11
−0.11
-2.54 +0.14
−0.14
-7.60 +0.22
−0.41
-9.13 +2.41
−2.71
12.24
Table 1. The retrieved chemical abundances for the WASP-43b model. We show each observing mode and error envelope. We also present the reduced 2 where
n = 7. NIRSpec PRISM
Model
logH 2 O (-3) logCO (-3) logCO 2 (-8) logCH 4 (-8)
2 /n
100ppm
-2.00 +0.18
−0.25
-8.16 +3.81
−3.66
-9.93 +2.28
−1.96
-8.77 +3.18
−3.05
81.96
NIRSpec PRISM + MIRI LRS
Model
logH 2 O (-3) logCO (-3) logCO 2 (-8) logCH 4 (-8)
2 /n
100ppm
-9.26 +2.65
−2.57
-1.76 +0.55
−2.14
-7.03 +1.54
−4.61
-1.83 +0.28
−0.27
MNRAS 000, 1-4 (2022)
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| [
"https://github.com/mrline/CHIMERA),"
]
|
[
"SKEW DYCK PATHS WITH CATASTROPHES",
"SKEW DYCK PATHS WITH CATASTROPHES"
]
| [
"Helmut Prodinger "
]
| []
| []
| Skew Dyck paths are like Dyck paths, but an additional south-west step (−1, −1) is allowed, provided that the path does not intersect itself. Lattice paths with catastrophes can drop from any level to the origin in just one step. We combine these two ideas. The analysis is strictly based on generating functions, and the kernel method is used. | 10.47443/dml.2022.008 | [
"https://arxiv.org/pdf/2201.02518v2.pdf"
]
| 245,827,935 | 2201.02518 | d631355abdae903c7bfe90ae13c94aaf98ddc88f |
SKEW DYCK PATHS WITH CATASTROPHES
10 Jan 2022
Helmut Prodinger
SKEW DYCK PATHS WITH CATASTROPHES
10 Jan 2022arXiv:2201.02518v2 [math.CO]
Skew Dyck paths are like Dyck paths, but an additional south-west step (−1, −1) is allowed, provided that the path does not intersect itself. Lattice paths with catastrophes can drop from any level to the origin in just one step. We combine these two ideas. The analysis is strictly based on generating functions, and the kernel method is used.
Introduction
The standard random walk on the non-negative integers may be visualized by the following graph (only the first 8 states are shown): The Figure 2 shows the graph underlying the catastrophes, drawn in purple. The standard reference for lattice paths with catastrophes is the paper [2]; the very recent paper [3] contains some bijective aspects.
The present note will combine skew Dyck paths (with an extra step (−1, −1)), as analyzed in [5], with the concept of catastrophes.
Date: January 11, 2022. 2020 Mathematics Subject Classification. 05A15. 1 Philippe Flajolet [1] liked the names 'excursion' resp. 'meander, which we will not use.
A warm-up: Dyck paths and catastrophes
To offer a gentle path for our readers, we analyze the paths as in Fig. 2 (which is also contained in the earlier papers [2,3]), since we will approach the skew Dyck paths with catastrophes in a similar style in the next section.
We introduce generating functions f i = f i (z), where the coefficient of z n counts the number of paths starting at the origin (=the big circle) and end after n steps at state i (=level i). The following recursions are easy to see:
f 0 = 1 + z(f 1 + f 2 + f 3 + f 4 + · · · ), f i = zf i−1 + zf i+1 , i ≥ 1.
Since f 0 is somewhat special, we leave it out for the moment and compute the other ones, f i , i ≥ 1. Eventually we will solve the equation for f 0 , which will turn out to be just linear. Therefore we introduce the bivariate generating function
F (u) = F (u, z) = i≥1 u i−1 f i
and we treat f 0 as a parameter. Summing the recursions,
F (u) = zf 0 + zuF (u) + z u [F (u) − f 1 ], or F (u) = zf 0 − z u f 1 1 − zu − z u = zf 1 − zuf 0 zu 2 − u + z = zf 1 − zuf 0 z(u − r 1 )(u − r 2 ) , with r 1 = 1 + √ 1 − 4z 2 2z , r 2 = 1 − √ 1 − 4z 2 2z .
An essential step of the kernel method is that the 'bad' factor (u − r 2 ) must cancel. The reciprocal of this factor does not have a power series expansion in (u, z). Dividing this factor out in both, denominator and numerator, leads to
F (u) = −zf 0 z(u − r 1 ) = zf 0 zr 1 (1 − u/r 1 ) ,
and from this
f i = [u i−1 ]F (u) = zf 0 zr i 1 = f 0 r i 2 .
This formula holds for all i ≥ 0. It is interesting to note that all these functions are just a multiple of the parameter f 0 . Now we can go to the first recursion:
f 0 = 1 + z(f 0 r 1 2 + f 0 r 2 2 + f 0 r 3 2 + · · · ) = 1 + zf 0 r 2 1 − r 2
and solve the equation,
f 0 = 2 − 3z − 2z 2 + z √ 1 − 4z 2 2(1 − z − 2z 2 − z 3 )
.
It has the power series expansion f 0 = 1+z 2 +z 3 +3z 4 +5z 5 +12z 6 +23z 7 +52z 8 +105z 9 +232z 10 +480z 11 +1049z 12 +2199z 13 +· · · , and the list of coefficients is sequence A224747 in the OEIS [6]. The papers [2,3] have this already. We can also compute the number of Dyck paths with catastrophes and open (unspecified) end, via the generating function
f 0 + f 1 + f 2 + f 3 + f 4 + . . . = 1 − z + (1 + z) √ 1 − 4z 2 2(1 − z − 2z 2 − z 3 ) = 1 + z + 2z 2 + 4z 3 + 8z 4 + 17z 5 + 35z 6 + 75z 7 + 157z 8 + 337z 9 + 712z 10 + · · · .
This is almost sequence A274115 in the OEIS [6]; 1 + z(f 0 + f 1 + f 2 + · · · ) has this sequence as coefficients. Thanks go to Michel Marcus to point out the connection with the sequences in the OEIS.
Skew Dyck paths with catastrophes
Skew Dyck are a variation of Dyck paths, where additionally to steps (1, 1) and (1, −1) a south-west step (−1, −1) is also allowed, provided that the path does not intersect itself.
Here is a list of the 10 skew paths consisting of 6 steps: We prefer to work with the equivalent model (resembling more traditional Dyck paths) where we replace each step (−1, −1) by (1, −1) but label it red. Here is the list of the 10 paths again (Figure 4): The rules to generate such decorated Dyck paths are: each edge (1, −1) may be black or red, but and are forbidden. These figures are taken from our earlier paper [5]. As in the motivating example from the previous section, we introduce generating functions. But since we need 3 layers to control everything, we need to introduce f j , g j , h j . The purple edges all end in the origin and represent the catastrophes. If they are ignored, one models skew Dyck paths in this way, in particular f 0 + g 0 + h 0 is the generating function of skew Dyck paths coming back to level 0. However, now, we will deal with the purple arrows as well. The following recursions can be read off immediately from the diagram 5:
f 0 = 1 + z i≥2 f i + z i≥2 g i + z i≥2 h i , f i+1 = zf i + zg i , i ≥ 0, g i = zf i+1 + zg i+1 + zh i+1 , i ≥ 0, h i = zh i+1 + zg i+1 , i ≥ 0.
As before, we first ignore f 0 and treat is as a parameter. We introduce
F (u) = i≥1 u i−1 f i , G(u) = i≥0 u i g i , H(u) = i≥0 u i h i .
Now we sum the recursions and get
i≥0
u i f i+1 = i≥0 u i zf i + i≥0 u i zg i or F (u) = f 0 + uzF (u) + zG(u). Further, i≥0 u i g i = i≥0 u i zf i+1 + i≥0 u i zg i+1 + i≥0 u i zh i+1 which translates into G(u) = zF (u) + z u [G(u) − G(0)] + z u [H(u) − H(0)];
similarly
H(u) = z u [G(u) − G(0)] + z u [H(u) − H(0)].
We will eliminate the functions G and H:
G = − zf 0 + zuF − F z and H = G − zF. Therefore we end up with just one equation for F −uzf 0 − zu 2 F + uF = −z 2 F u − 2z 2 f 0 + 2zF − 2g 0 z 2 − z 3 F + z 3 f 1 which can be solved, F (u) = − (−2zf 0 − 2zg 0 + z 2 f 1 + f 0 u) z −z 2 u + 2z − z 3 + zu 2 − u = − (−2zf 0 − 2zg 0 + z 2 f 1 + f 0 u) z z(u − r 1 )(u − r 2 ) with r 1 = 1 + z 2 + √ 1 − 6z 2 + 5z 4 2z , r 2 = 1 + z 2 − √ 1 − 6z 2 + 5z 4 2z .
The fact that u − r 2 must divide the numerator of F leads to
g 0 = −2zf 0 + z 2 f 1 + r 2 f 0 2z ,
is not in OEIS [6]. To compute the number of all skew Dyck paths coming back to the x-axis, we need
f 0 + g 0 + h 0 = (1 − z 2 )(3 − z)(1 − z − 2z 2 − z 3 ) − (1 − 4z − 3z 2 + z 3 + z 4 ) √ 1 − 6z 2 + 5z 4 2(1 − 2z 2 − 6z 3 − 3z 4 + z 5 + z 6 )
.
The series expansion is 1 + z 2 + z 3 + 4z 4 + 5z 5 + 17z 6 + 25z 7 + 76z 8 + 125z 9 + 353z 10 + 625z 11 + 1681z 12 + · · · .
One can also sum everything to enumerate the open ended paths in this model:
j≥0 f j + j≥0 g j + j≥0 h j = (1 + z)(1 − 4z + 4z 2 + 4z 3 − z 5 ) + (1 + 5z + 3z 2 − z 3 − z 4 ) √ 1 − 6z 2 + 5z 4 2(1 − 2z 2 − 6z 3 − 3z 4 + z 5 + z 6 ) ;
the series expansion is 1 + z + 2z 2 + 4z 3 + 9z 4 + 18z 5 + 41z 6 + 85z 7 + 193z 8 + 410z 9 + 929z 10 + 2004z 11 + · · · . Theorem 1. The generating functions related to the model of skew Dyck paths with catastrophes are
f j = f 0 r j 1 , g j = (r 2 − z) f 0 r j+1 1 , h j = (r 2 − 2z) f 0 r j+1 1 , with r 1 = 1 + z 2 + √ 1 − 6z 2 + 5z 4 2z , r 2 = 1 + z 2 − √ 1 − 6z 2 + 5z 4 2z , and f 0 + g 0 + h 0 = (1 − z 2 )(3 − z)(1 − z − 2z 2 − z 3 ) − (1 − 4z − 3z 2 + z 3 + z 4 ) √ 1 − 6z 2 + 5z 4 2(1 − 2z 2 − 6z 3 − 3z 4 + z 5 + z 6 )
.
The enumeration of all paths, regardless of the final level, is j≥0 f j + j≥0 g j + j≥0 h j = (1 + z)(1 − 4z + 4z 2 + 4z 3 − z 5 ) + (1 + 5z + 3z 2 − z 3 − z 4 ) √ 1 − 6z 2 + 5z 4 2(1 − 2z 2 − 6z 3 − 3z 4 + z 5 + z 6 ) .
Conclusion
From our analysis, other types of catastrophes (example: only from an even-numbered level one can drop to the ground) can be computed without much further effort. This changes only the equation for f 0 ; the other quantities stay as they are.
Figure 1 .Figure 2 .
12Standard symmetric random walk on the non-negative integers Such walks are also known as Dyck paths, going up or down one step at the time. Most of the time, they need to come back to the x-axis (end at state 0), but one also considers open-ended paths where the level of the end of the path is not specified. Catastrophes can lead from a state i ≥ 2 back to the x-axis in one step.
Figure 3 .
3All 10 skew Dyck paths of length 6 (consisting of 6 steps).
Figure 4 .
4The 10 paths redrawn, with red south-east edges instead of south-west edges.
Figure 5 .
5Three layers of states according to the type of steps leading to them (up, down-black, down-red). Purple arrows all end in the origin.
so that g 0 can be eliminated and only f 1 remains as unknown; F = zf 0 1 − zu + z 2 − zr 2 plugging in u = 0 allows to compute it:Therefore the function F is known:From this, the other functions can be computed as well:as well asNow we can finally deal with f 0 . The evaluations that are needed are simple:of course this is better done with a computer. From this, the equation (just linear) for f 0 can be solved:.The series expansion 1 + z 3 + z 4 + 4z 5 + 6z 6 + 18z 7 + 31z 8 + 85z 9 + 157z 10 + 410z 11 + 792z 12 + 2004z 13 + · · ·
Basic analytic combinatorics of directed lattice paths. C Banderier, P Flajolet, Theor. Comput. Sci. 281C. Banderier and P. Flajolet. Basic analytic combinatorics of directed lattice paths, Theor. Com- put. Sci., 281 (2002),37-80.
Lattice paths with catastrophes. C Banderier, M Wallner, Discrete Math. Theor. Comput. Sci. 192332C. Banderier and M. Wallner. Lattice paths with catastrophes, Discrete Math. Theor. Comput. Sci., 19 (2017), Paper No. 23, 32.
Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths. J.-L Baril, S Kirgizov, Discrete Mathematics Letters. 7J.-L. Baril, S. Kirgizov. Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths, Discrete Mathematics Letters, 7 (2021), 5-10.
The Kernel Method: A Collection of Examples. H Prodinger, Séminaire Lotharingien de Combinatoire. 50ppH. Prodinger. The Kernel Method: A Collection of Examples, Séminaire Lotharingien de Combi- natoire, B50f (2004), 19 pp.
H Prodinger, arXiv:2108.09785Partial skew Dyck paths-a kernel method approach. H. Prodinger, Partial skew Dyck paths-a kernel method approach, arXiv:2108.09785 (2021).
Sloane and The OEIS Foundation Inc. The on-line encyclopedia of integer sequences. J A Neil, Neil J. A. Sloane and The OEIS Foundation Inc. The on-line encyclopedia of integer sequences, 2022.
address: [email protected] Stellenbosch, South Africa, and NITheCS (National Institute for Theoretical and Computational Sciences). South Africa EmailHelmut Prodinger, Department of Mathematical Sciences, Stellenbosch UniversityHelmut Prodinger, Department of Mathematical Sciences, Stellenbosch Univer- sity, 7602 Stellenbosch, South Africa, and NITheCS (National Institute for Theo- retical and Computational Sciences), South Africa Email address: [email protected]
| []
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"Gelfand-Kirillov dimension of differential difference algebras",
"Gelfand-Kirillov dimension of differential difference algebras"
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"Yang Zhang [email protected] \nDepartment of Mathematics\nUniversity of Manitoba Winnipeg\nR3T 2N2MBCanada\n",
"Xiangui Zhao \nDepartment of Mathematics\nUniversity of Manitoba Winnipeg\nR3T 2N2MBCanada\n"
]
| [
"Department of Mathematics\nUniversity of Manitoba Winnipeg\nR3T 2N2MBCanada",
"Department of Mathematics\nUniversity of Manitoba Winnipeg\nR3T 2N2MBCanada"
]
| []
| Differential difference algebras were introduced by Mansfield and Szanto, which arose naturally from differential difference equations. In this paper, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand-Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand-Kirillov dimension under some specific conditions and construct an example to show that this upper bound can not be sharpened any more.Assume that R is a unital associative k-algebra. Let σ be an automorphism of R, δ be a σ-derivation of R and A = R[x; σ, δ] be an Ore extension over R. It was shown in [3] that GKdim(A) ≥ GKdim(R) + 1;(1) the inequality becomes an equality provided that the following condition holds:In this paper we investigate the Gelfand-Kirillov dimension of differential difference algebras. We show that Inequality (1) of the Gelfand-Kirillov dimension of an Ore extension can be extended to differential difference algebras, that is, we get a lower bound of the Gelfand-Kirillov dimensions of differential difference algebras. However, owing to the noncommutativity of indeterminates of differential difference algebras, even under conditions similar to ( * ), the equality does not hold for differential difference algebras in general. We find an upper bound for the Gelfand-Kirillov dimension of a differential difference algebra satisfying (an analogues of) condition ( * ), and give a sufficient condition under which the lower bound is reached. We also construct an example to show that the upper bound we obtain cannot be sharpened any more.This paper is organized as follows.Definition and examples of differential difference algebras are given in Section 2. The Gelfand-Kirillov dimension of differential difference algebras is investigated in Section 3. | 10.1112/s1461157014000102 | [
"https://arxiv.org/pdf/1310.2583v2.pdf"
]
| 119,147,877 | 1310.2583 | fcb219f4895a86696267142ecfe74d1f020da283 |
Gelfand-Kirillov dimension of differential difference algebras
7 Dec 2013
Yang Zhang [email protected]
Department of Mathematics
University of Manitoba Winnipeg
R3T 2N2MBCanada
Xiangui Zhao
Department of Mathematics
University of Manitoba Winnipeg
R3T 2N2MBCanada
Gelfand-Kirillov dimension of differential difference algebras
7 Dec 2013differential difference algebraGelfand-Kirillov dimensionOre extension MSC 2010: 16P9016S36
Differential difference algebras were introduced by Mansfield and Szanto, which arose naturally from differential difference equations. In this paper, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand-Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand-Kirillov dimension under some specific conditions and construct an example to show that this upper bound can not be sharpened any more.Assume that R is a unital associative k-algebra. Let σ be an automorphism of R, δ be a σ-derivation of R and A = R[x; σ, δ] be an Ore extension over R. It was shown in [3] that GKdim(A) ≥ GKdim(R) + 1;(1) the inequality becomes an equality provided that the following condition holds:In this paper we investigate the Gelfand-Kirillov dimension of differential difference algebras. We show that Inequality (1) of the Gelfand-Kirillov dimension of an Ore extension can be extended to differential difference algebras, that is, we get a lower bound of the Gelfand-Kirillov dimensions of differential difference algebras. However, owing to the noncommutativity of indeterminates of differential difference algebras, even under conditions similar to ( * ), the equality does not hold for differential difference algebras in general. We find an upper bound for the Gelfand-Kirillov dimension of a differential difference algebra satisfying (an analogues of) condition ( * ), and give a sufficient condition under which the lower bound is reached. We also construct an example to show that the upper bound we obtain cannot be sharpened any more.This paper is organized as follows.Definition and examples of differential difference algebras are given in Section 2. The Gelfand-Kirillov dimension of differential difference algebras is investigated in Section 3.
Introduction
Differential difference algebras (Definition 2.1) arose naturally from differential difference equations [2,10]. The class of differential difference algebras contains several well-known classes of noncommutative algebras, for example, commutative polynomial algebras, quantum planes, and skew polynomial algebras of derivation (or automorphism) type. Rougly speaking, a differential difference algebra is a noncommutative polynomial ring (over an algebra) with two sets D and S of indeterminates, where D originally stands for differential operators and S originally stands for shift operators (the difference operators can be derived from S). Operators in D (S, respectively) commute with each other, but not with those in S (D, respectively). The exact definition is given in Section 2.
Let k be a field and A be a unital associative k-algebra. The Gelfand-Kirillov dimension of A is defined as
GKdim(A) = sup V lim n→∞ log n dim k (V n )
where the supremum is taken over all finite dimensional subspaces V of A. The Gelfand-Kirillov dimension is a very useful and powerful tool for investigating noncommutative algebras. Basic properties of Gelfand-Kirillov dimension can be found in [7]. There have been a number of results concerning Gelfand-Kirillov dimensions of algebras with derivations and/or automorphisms, for example, the Gelfand-Kirillov dimension of Ore extensions of derivation type [9], of Ore extensions of automorphism type [3], of PBW-extensions [11], and of skew polynomial extensions [13].
Preliminaries
Throughout this paper, we assume that k is a field and all algebras are unital associative kalgebras. Denote the set of k-algebra homomorphisms of algebra A by Aut(A). If σ ∈ Aut(A), then a mapping δ on A is called a σ-derivation provided that, for any a, b ∈ A and c ∈ k, δ(ca + b) = cδ(a) + δ(b) and δ(ab) = aδ(b) + δ(a)b. Particularly, if σ = id then δ is called a derivation on A.
First we recall the definition of differential difference algebras, which was introduced by Mansfield and Szanto [10] with some discussions of Gröbner bases. Definition 2.1 (cf., [10]) An algebra A is called a differential difference algebra of type (m, n), m, n ≥ 1, over a subalgebra R ⊆ A if there exist elements S 1 , . . . , S m , D 1 , . . . , D n in A such that (i) the set {S α D β : α ∈ N m , β ∈ N n } forms a basis for A as a free left R-module.
(ii) D i r = rD i + δ i (r) for any 1 ≤ i ≤ n and r ∈ R, where δ i is a derivation on R. (iii) S i r = σ i (r)S i for any 1 ≤ i ≤ m and r ∈ R, where σ i is a k-algebra automorphism on the subalgebra R[D 1 , . . . , D n ] ⊆ A such that σ i | R ∈ Aut(R) and σ i (D j ) = n l=1 a ijl D l , a ijl ∈ R. (iv) S i S j = S j S i , 1 ≤ i, j ≤ m; D i ′ D j ′ = D j ′ D i ′ , 1 ≤ i ′ , j ′ ≤ n. (v) D i S j = S j σ j (D i ), 1 ≤ i ≤ n, 1 ≤ j ≤ m. (vi) For any 1 ≤ i, j ≤ n and 1 ≤ i ′ , j ′ ≤ m, δ i • δ j = δ j • δ i , σ i ′ • σ j ′ = σ j ′ • σ i ′ .
Remark 2.2 In the above definition, both subalgebras R[D 1 , . . . , D n ] and R[S 1 , . . . , S m ] of A are iterated Ore extensions over R. But, in general A is not an iterated Ore extension over R.
The class of differential difference algebras contains several other known classes of algebras, for example, commutative polynomial algebras, quantum planes, and skew polynomial algebras of derivation (or automorphism) type. Example 2.3 Let 0 = q ∈ k and I q be the two-sided ideal of the free associative algebra k x, y generated by the element yx − qxy. Then the quotient algebra k q [x, y] = k x, y /I q is called a quantum plane ( [6], Chapter IV). It is easy to see that k q [x, y] is a differential difference algebra of type (1, 1) over k.
The following example distinguishes differential difference algebras from algebras of solvable type [5], PBW extensions [1], and G-algebras [8].
Example 2.4 Let A be the k-algebra generated by {D 1 , D 2 , S} with defining relations R = {D 2 D 1 = D 1 D 2 , D 1 S 1 = S 1 D 2 , D 2 S 1 = S 1 D 1 }.
Then it is easy to see that A is a differential difference algebra of type (1, 2) over k. However, by the defining relations, A is not an algebra of solvable type [5], PBW extensions [1], and G-algebras [8].
Let D = {D 1 , . . . , D n } and S = {S 1 , . . . , S m }. If A is a differential difference algebra over R defined as Definition 2.1, we denote A = R[S, D; σ, δ]. For α = (α 1 , . . . , α m ) ∈ N m and r ∈ R, we simply write σ α (r) = σ α 1 1 · · · σ αm m (r), D α = D α 1 1 · · · D αm m and |α| = α 1 + · · · + α m . In particular, D 0 i = 1, the identity of R. Similarly, we use notations δ α (r), S β (β ∈ N n ), and so on. Then every element in A can be written uniquely in the form: α,β r α,β S α D β , where r α,β ∈ R and only finitely many r α,β are nonzero.
The following example is taken from [10] with some modifications. This example shows where the differential difference algebras come from.
Example 2.5 Let M, n, p ∈ N and p ≥ 1. Consider the following system, which arises from the calculation of symmetries of discrete systems (cf., [4]),
u n+M +1 = ω(n, u n , u n+1 , . . . , u n+M ); D j F (n, u n , u n+1 , . . . , u n+M ) = 0, 1 ≤ j ≤ p,
where F is the unknown function and w is a given function in the field Q(n, u n , . . . , u n+M ) of rational functions over the rational numbers Q in indeterminates n, u n , u n+1 , . . . , u n+M , such that ∂ω ∂un = 0, and D j : T → T is a linear operator of the form
D j = α=(α 0 ,...,α M )∈N M , β∈N c α,β • s β • ∂ α 0 +···+α M ∂u α 0 n · · · ∂u α M n+M ,
where T = Q(n, u n+t : t ∈ Z), c α,β ∈ Q(n, u n+t : t ∈ Z) are multiplication operators and only finitely many c α,β are nonzero, and s is the shift operator defined by s(n) = n + 1 and s(u n ) = u n+1 . A natural approach to deal with this system is to consider those operators D j and s as elements of the noncommutative algebra A over R generated by operator variables {S, D n , . . . , D n+M }, where S denotes the shift operator s and D n+t denotes the differential operator ∂ ∂u n+t for 0 ≤ t ≤ M, subject to the following commutation rules:
D n+t • S = S • D n+t−1 + ∂ω ∂u n+t • S • D n+M , 1 ≤ t ≤ M; D n • S = ∂ω ∂u n • S • D n+M ; D n+t • D n+t ′ = D n+t ′ • D n+t , 0 ≤ t, t ′ ≤ M; S • r = s(r) • S, r ∈ R; D n+t • r = r • D n+t + ∂r ∂u n+t , r ∈ R, 0 ≤ t ≤ M. Then A = Q(n, u n+t : t ∈ Z)[S, D; σ, δ] is a differential difference algebra of type (1, M + 1) over Q(n, u n+t : t ∈ Z), where δ i = ∂ ∂u i for n ≤ i ≤ n + M and σ| R = s, σ(D n ) = s −1 ( ∂ω ∂un )D n+M , σ(D n+t ) = D n+t−1 + s −1 ( ∂ω ∂u n+t )D n+M for 1 ≤ t ≤ M.
Note that the difference operator ∆, defined by ∆(u i ) = u i+1 − u i for i ∈ Z, can be derived from the shift operator: ∆ = S − id. So the differential difference algebra in the above example actually involves both differential and difference operators.
Gelfand-Kirillov dimension of differential difference algebras
In this section, we consider the Gelfand-Kirillov dimension of differential difference algebras. We first fix some notations. Let GKdim(A) denote the Gelfand-Kirillov dimension of an algebra A, dim(V ) denote the dimension of a k-vector space V , and card(T ) denote the cardinality of a set T . Recall that, for r, s ∈ N, the binomial coefficient r s = r(r−1)···(r−s+1)
s(s−1)···1
if 0 ≤ s ≤ r; and r s = 0 if s < 0 or s > r. The Gelfand-Kirillov dimension of an Ore extension has been discussed in [3].
V of R such that U ⊆ V , σ(V ) ⊆ V and δ(V ) ⊆ V p for some p ≥ 1. Then GKdim(A) = GKdim(R) + 1.
We want to consider the Gelfand-Kirillov dimension of differential difference algebras satisfying "similar" conditions as in Proposition 3.1. Our goal is to find a lower bound and an upper bound of the Gelfand-Kirillov dimension of such a differential difference algebra.
The following proposition gives a general lower bound of the Gelfand-Kirillov dimension of a differential difference algebra.
Proposition 3.2 Let R be a k-algebra and A = R[S, D; σ, δ] be a differential difference algebra of type (m, n). Then GKdim(A) ≥ GKdim(R) + m + n.
Proof. Suppose that V is a finite dimensional generating subspace of R and 1 ∈ V . Then
W = V + n i=1 kD i + m j=1
kS j is a finite dimensional generating subspace of A. For any r ∈ N,
W 3r = (V + n i=1 kD i + m j=1 kS j ) 3r ⊇ α∈N m ,β∈N n 0≤|α|, |β|≤r V r S α D β .
For any k-basis U of V r , the set
{uS α D β : u ∈ U, 0 ≤ |α|, |β| ≤ r, α ∈ N m , β ∈ N n } is a k-basis of 0≤|α|,|β|≤r V r S α D β . Hence, we have that dim(W 3r ) ≥ dim 0≤|α|,|β|≤r V r S α D β = dim(V r ) · card({α : 0 ≤ α 1 + . . . + α m ≤ r}) · card({β : 0 ≤ β 1 + . . . + β n ≤ r}) = dim(V r ) · r + m − 1 m · r + n − 1 n ,
where r+m−1 m and r+n−1 n are polynomials in r of degree m and n respectively. Hence,
GKdim(A) ≥ lim r→∞ log r dim(W r ) = lim r→∞ log r dim(W 3r ) ≥ lim r→∞ log r dim(V r ) · r + m − 1 m · r + n − 1 n = GKdim(R) + m + n.
In the special case R = k, the equality in the above proposition holds, i.e., we have the following proposition, which indicates that the lower bound of GKdim(A) obtained in Proposition 3.2 can not be sharpened any more.
V r = (k + n i=1 kD i + m j=1 kS j ) r ⊆ 0≤|α|,|β|≤r kS α D β ,
where the last inclusion holds since
D β S α ∈ β ′ ∈N n ,|β ′ |=|β| kS α D β ′ , α ∈ N m , β ∈ N n .
So, dim(V r ) ≤ r+m−1 m · r+n−1 n . Hence, GKdim(A) ≤ m + n and thus by Proposition 3.2 GKdim(A) = m + n. Now let us turn to upper bounds for GKdim(A). First we consider the case when R is finitely generated. Lemma 3.4 Let R be a k-algebra with a finite dimensional generating subspace V , and let A = R[S, D; σ, δ] be a differential difference algebra of type (m, n). Suppose that σ i (V ) ⊆ V for 1 ≤ i ≤ m. Then GKdim(A) ≤ 2 GKdim(R) + m + n.
Furthermore, if, for all 1 ≤ i ≤ m and 1 ≤ j ≤ n, σ i (D j ) is contained in the vector space over k generated by {D 1 , . . . , D n }, then GKdim(A) = GKdim(R) + m + n.
Proof. Since V is a generating subspace, there exists p ≥ 1 such that
a ijl ∈ V p , δ i (V ) ⊆ V p , 1 ≤ i, l ≤ n, 1 ≤ j ≤ m,
where the a ijl are the coefficients that appear in Definition 2.1. Then
δ i (V t ) ⊆ V p+t , σ j (V t ) ⊆ V t , 1 ≤ i ≤ n, 1 ≤ j ≤ m, t ≥ 1.
So, eventually replacing V by V p if necessary, we may assume that
1 ∈ V, δ i (V ) ⊆ V 2 , σ j (V ) ⊆ V, 1 ≤ i ≤ n, 1 ≤ j ≤ m. Let X = n i=1 kD i , Y = m j=1
kS j and W = V + X + Y . Then W is a generating subspace of A.
In order to finish the first statement of the this lemma, we have to prove the following three lemmas first.
Lemma 3.5 For any integer s ≥ 1,
(i). XY ⊆ V Y X, XV ⊆ V X + V 2 , Y V = V Y . (ii). X s V ⊆ s i=0 V i+1 X s−i . (iii). X s Y ⊆ s−1 i=0 V s+i Y X s−i .
Proof. (i). It follows easily by definition. (ii). (By induction on s.) If s = 1, then we have XV ⊆ V X + V 2 by the commutation rules of differential difference algebras. Suppose that
X r V ⊆ r i=0 V i+1 X r−i for 1 ≤ r ≤ s. Then X s+1 V ⊆ X s (V X + V 2 ) ⊆ s i=0 V i+1 X s−i+1 + s i=0 V i+1 X s−i V ⊆ s i=0 V i+1 X s−i+1 + s i=0 V i+1 s−i j=0 V j+1 X s−i−j = s i=0 V i+1 X s−i+1 + s i=0 s−i j=0 V i+j+2 X s−i−j = s i=0 V i+1 X s−i+1 + s i=0 s+1 l=i+1 V l+1 X s−l+1 (l := i + j + 1) = s i=0 V i+1 X s−i+1 + s+1 l=1 V l+1 X s−l+1 = s+1 i=0 V i+1 X s−i+1 .
Thus (ii) holds for all s ≥ 1.
(iii). (By induction on s.) If s = 1, then we have V 1 Y X 1 ⊇ XY by (i), and thus (iii) holds.
Suppose that X r Y ⊆ r−1 i=0 V r+i Y X r−i for all 1 ≤ r ≤ s. Then, X s+1 Y ⊆ X s (V Y X) ⊆ s i=0 V i+1 X s−i Y X (by (ii)) = s−1 i=0 V i+1 X s−i Y X + V s+1 Y X ⊆ s−1 i=0 V i+1 s−i−1 j=0 V s−i+j Y X s−i−j+1 + V s+1 Y X = s−1 i=0 s−i−1 j=0 V s+j+1 Y X s−i−j+1 + V s+1 Y X = s−1 i=0 s−1 l=i V s+l−i+1 Y X s−l+1 + V s+1 Y X (l := i + j) ⊆ s−1 i=0 s−1 l=0 V s+l+1 Y X s−l+1 + V s+1 Y X = s−1 l=0 V s+l+1 Y X s−l+1 + V s+1 Y X ⊆ s l=0 V s+l+1 Y X s−l+1 .
Hence (iii) holds.
Lemma 3.6 For all r ≥ 1, W r ⊆ r i=0 r−i j=0 V 2r 2 Y i X j .
Proof. (By induction on r.) If r = 1, then the right hand side of the inclusion is
V 2 + V 2 X + V 2 Y ⊇ W .
Suppose the statement is true for r ≥ 1. Then, by induction hypothesis,
W r+1 ⊆ r i=0 r−i j=0 V 2r 2 Y i X j (V + X + Y ) (induction hypothesis) = r i=0 r−i j=0 V 2r 2 Y i X j V + r i=0 r−i j=0 V 2r 2 Y i X j+1 + r i=0 r−i j=0 V 2r 2 Y i X j Y ⊆ r i=0 r−i j=0 V 2r 2 Y i j l=0
V l+1 X j−l (by Lemma 3.5(ii)) Hence, for any ε > 0, f (n) < n d+ε for almost all n. So f (pn 2 ) < (pn 2 ) d+ε = p d+ε n 2d+2ε < n 2d+3ε for almost all n.
+ r i=0 r−i j=0 V 2r 2 Y i X j+1 + r i=0 r−i j=0 V 2r 2 Y i j−1 l=0 V j+l Y X j−l (by Lemma 3.5(iii)) = r i=0 r−i j=0 j l=0 V 2r 2 +l+1 Y i X j−l + r i=0 r−i j=0 V 2r 2 Y i X j+1 + r i=0 r−i j=0 j−1 l=0 V 2r 2 +j+l Y i+1 X j−l = r i=0 r−i j=0 j p=0 V 2r 2 +j−p+1 Y i X p (p := j − l) + r i=0 r−i+1 j=1 V 2r 2 Y i X j (shift index j) + r+1 i=1 r−i+1 j=0 j p=1 V 2r 2 +2j−p Y i X p (shift i and p := j − l) ⊆ r i=0 r−i j=0 r−i p=0 V 2r 2 +r−p+1 Y i X p + r i=0 r−i+1 j=1 V 2r 2 Y i X j + r+1 i=1 r−i+1 j=0 r−i+1 p=1 V 2r 2 +2r−p Y i X p ⊆ r i=0 r−i p=0 V 2r 2 +r−p+1 Y i X p + r i=0 r−i+1 j=1 V 2r 2 Y i X j + r+1 i=1 r−i+1 p=1 V 2r 2 +2r−p Y i X p ⊆ r+1 i=0 r−i+1 j=0 V 2(r+1) 2 Y i X j . Therefore, W r ⊆ r i=0 r−i j=0 V 2r 2 Y i X j for any r ≥ 1.
Therefore, lim For the second statement of Lemma 3.4, we first note that, for all 1 ≤ i ≤ m and 1 ≤ j ≤ n, if σ i (D j ) ∈ X then XY = Y X. Then, under the given assumptions, we have the following lemma.
V 2r 2 Y i X j = lim r→∞ log r dim(V 2r 2 ) r i=0 r−i j=0 dim(Y i ) dim(X j ) = lim r→∞ log r f (2r 2 ) r i=0 r−i j=0 i + m − 1 m − 1 j + n − 1 n − 1 ≤ lim r→∞ log r f (2rLemma 3.8 For all r ≥ 1, W r ⊆ r i=0 r−i j=0 V 2r−i−j Y i X j .
Proof. (By induction on r.) It is easy to check that the inclusion is true for r = 1. Suppose
W r ⊆ r i=0 r−i j=0 V 2r−i−j Y i X j for r ≥ 1. Then W r+1 ⊆ r i=0 r−i j=0 V 2r−i−j Y i X j (V + X + Y ) (by induction hypothesis) = r i=0 r−i j=0 V 2r−i−j Y i X j V + V 2r−i−j Y i X j+1 + V 2r−i−j Y i X j Y ⊆ r i=0 r−i j=0 V 2r−i−j Y i j l=0 V l+1 X j−l (by Lemma 3.5(ii)) + r i=0 r−i j=0 V 2r−i−j Y i X j+1 + r i=0 r−i j=0 V 2r−i−j Y i+1 X j = r i=0 r−i j=0 j t=0 V 2r−i−t+1 Y i X t (t := j − l) + r i=0 r+1−i j=1 V 2r−i−j+1 Y i X j (shift index j ) + r+1 i=1 r−i j=0 V 2r−i−j+1 Y i X j (shift index i ) ⊆ r+1 i=0 r+1−i j=0 V 2(r+1)−i−j Y i X j
That proves our lemma.
By the above lemma,
W r ⊆ r i=0 r−i j=0 V 2r−i−j Y i X j ⊆ r i=0 r−i j=0 V 2r Y i X j .
Hence, by a similar argument as we used in the proof of the first statement of Lemma 3.4, we have that GKdim(A) ≤ GKdim(R) + m + n. Thus, by Proposition 3.2, GKdim(A) = GKdim(R) + m + n.
Now we are in a position to state our main theorem.
Theorem 3.9 Let R be a k-algebra and A = R[S, D; σ, δ] be a differential difference algebra of type (m, n). Suppose that for any finite dimensional subspace U of R there exist a finite dimensional subspace V of R and an integer p ≥ 1 such that
U ⊆ V, σ i (V ) ⊆ V, δ j (V ) ⊆ V p , 1 ≤ i ≤ m, 1 ≤ j ≤ n.
Then Proof. Let W be a finite dimensional subspace of A with a k-basis w 1 , . . . , w q , q ∈ N. Note that each Thus GKdim(A) ≤ 2 GKdim(R) + m + n since W is arbitrary. Therefore, by Lemma 3.2, GKdim(R) + m + n ≤ GKdim(A) ≤ 2 GKdim(R) + m + n. That completes our proof of the first statement. The second statement follows similarly by using the second part of Lemma 3.4.
GKdim(R) + m + n ≤ GKdim(A) ≤ 2 GKdim(R) + m + n.w i , 1 ≤ i ≤ q,
Immediately from Theorem 3.9, we have the following corollaries. Recall that an algebra A is called locally finite dimensional if every finitely generated subalgebra of A is finite dimensional.
Corollary 3.11 Let R be a k-algebra and A = R[S, D; σ, δ] be a differential difference algebra of type (m, n) satisfying the conditions of the first statement of Theorem 3.9.
(i). If R is locally finite dimensional, then GKdim(A) = m + n.
(ii). If GKdim(R) < ∞, then GKdim(A) < ∞.
Thus, GKdim(A) ≥ lim r→∞ log r dim(W r ) ≥ 4. Therefore, GKdim(A) = 4.
Proposition 3.1 ([3], Corollary 2.4, cf.,[7], Proposition 3.5.) Let R be a k-algebra and A = R[D; σ, δ] be an Ore extension. Suppose that, for each finite dimensional subspace U of R, there exists a finite dimensional subspace
Proposition 3. 3
3Let A = k[S, D; σ, δ] be a differential difference algebra of type (m, n) over k. Then GKdim(A) = m + n.Proof. Let V = k + j . Then V is a finite dimensional generating subspace of A. For any r ∈ N,
Lemma 3. 7
7Let f : N → R be an increasing and positive valued function, and p > 1. Then lim n→∞ log n f (pn 2 ) ≤ 2 lim n→∞ log n f (n).Proof. Let d = limn→∞ log n f (n). By Lemma 2.1 of[7], d = inf{ρ ∈ R : f (n) ≤ n ρ for almost all n ∈ N}.
log n f (pn 2 ) ≤ 2d = 2 lim n→∞ log n f (n). Now let us return to the proof of Lemma 3.4. Let f (r) = dim(V r ) for r ∈ N. Then GKdim(A) = lim r→∞ log r dim(W r ) ≤ lim r→∞ log r dim
(R) + m + n where the last inequality holds because of Lemma 3.7 and the fact that if p(i) is a polynomial in i of degree s then r i=0 p(i) is a polynomial in r of degree s + 1.
Furthermore
, if, for all 1 ≤ i ≤ m and 1 ≤ j ≤ n, σ i (D j ) is contained in the vector space over k generated by {D 1 , . . . , D n }, then GKdim(A) = GKdim(R) + m + n.
is a polynomial in D 1 , . . . , D n , S 1 , . . . , S m with coefficients in R. Let U be the subspace of R spanned by all the coefficients (in R) of w 1 , . . . , w q and all a ijl (defined in Definition 2.1), 1 ≤ i, l ≤ n, 1 ≤ j ≤ m. Then U is finite dimensional and hence there exist a finite dimensional subspace V of R and an integer p ≥ 1 such thatU ⊆ V, σ i (V ) ⊆ V, δ j (V ) ⊆ V p for 1 ≤ i ≤ m, 1 ≤ j ≤ n. LetB be the subalgebra of R generated by V . Then σ i (B) ⊆ B, σ i (D l ) ∈ B[D; δ] and δ j (B) ⊆ B for 1 ≤ i ≤ m, 1 ≤ j, l ≤ n. That is, A ′ = B[S, D; σ, δ] is a differential difference algebra satisfying the conditions of Lemma 3.4. Note that W ⊆ A ′ . So, by Lemma 3.4, we have lim r→∞ log r dim(W r ) ≤ GKdim(A ′ ) ≤ 2 GKdim(B) + m + n ≤ 2 GKdim(R) + m + n.
Corollary 3. 10
10The quantum plane k q [x, y] has Gelfand-Kirillov dimension 2.
Acknowledgements. The authors would like to thank Prof. Günter Krause for his careful reading of the manuscript and valuable comments. This work is supported in part by the National Sciences and Engineering Research Council of Canada.Proof.(i). It follows from the fact that GKdim(R) = 0 if and only if R is locally finite dimensional.(ii). This is clear.Note that if we set R = k in Theorem 3.9, then the conditions of the theorem are satisfied. Thus Theorem 3.9 implies Proposition 3.3.The following example shows that the upper bound of GKdim(A) stated in Theorem 3.9 is the "best" one under the given conditions. Example 3.12 Let A be the k-algebra generated by {z, z −1 , D, S} with defining relationsLet R = k[z, z −1 ] be the algebra of Laurent polynomials over k, and let σ be the automorphism of the algebraThen A = R[S, D; σ, 0] is a differential difference algebra of type (1, 1) andProof.It is easy to see that A can be thought of as an iterated Ore extension over R:Hence {S i D j : i, j ∈ N} forms an R-basis of A. Thus A is a differential difference algebra.Note that the restriction of σ on R is the identity automorphism of R. It is clear that A satisfies all conditions of Theorem 3.9. So, by Theorem 3.9, GKdim(A) ≤ 2 GKdim(R) + 2. Since GKdim(R) = 1 (see, for example, Corollary 8.2.15 of[12]), GKdim(A) ≤ 4.Note that DS = Sσ(D) = SzD = zSD. Then one can prove that D j S = z j SD j by induction on j, and then that D j S i = z ij S i D j by induction on i. Now we claim thatSince (i − q − 1) + p + 1 + (j − p) + q = i + j ≤ r, z l S i D j = S i−q−1 D p SD j−p S q ∈ W r . Thus our claim holds.It is clear that B is k-linearly independent. Then by our claim, dim(W r ) ≥ card(B) = r i=0 r j=0 (ij + 1) = 1 4 r 4 + 1 2 r 3 + 5 4 r 2 + 2r + 1.
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Symmetries and first integrals of ordinary difference equations. P E Hydon, Proceedings of the Royal Society of London (series A). 456P. E. Hydon, Symmetries and first integrals of ordinary difference equations, Proceedings of the Royal Society of London (series A) 456 (2000), 2835-2855.
Non-commutative Gröbner bases in algebras of solvable type. A Kandri-Rody, V Weispfenning, Journal of Symbolic Computation. 91A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type, Journal of Symbolic Computation 9 (1990), no. 1, 1-26.
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Growth of algebras and Gelfand-Kirillov dimension. G Krause, T Lenagan, Graduate Studies in Mathematics. 22AMSG. Krause and T. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Graduate Studies in Mathematics, vol. 22, AMS, 2000.
V Levandovskyy, H Schönemann, Plural: a computer algebra system for noncommutative polynomial algebras, Proceedings of the 2003 international symposium on Symbolic and algebraic computation. ACMV. Levandovskyy and H. Schönemann, Plural: a computer algebra system for noncommu- tative polynomial algebras, Proceedings of the 2003 international symposium on Symbolic and algebraic computation, ACM, 2003, pp. 176-183.
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Elimination theory for differential difference polynomials. E L Mansfield, A Szanto, Proceedings of the 2003 international symposium on symbolic and algebraic computation. the 2003 international symposium on symbolic and algebraic computationACME. L. Mansfield and A. Szanto, Elimination theory for differential difference polynomials, Proceedings of the 2003 international symposium on symbolic and algebraic computation, ACM, 2003, pp. 191-198.
The Gelfand-Kirillov dimension of Poincaré-Birkhoff-Witt extensions, Perspectives in Rings Theory. J Matczuk, F. Van Oystaeyen and L. Le BruynKluwer Academic PublishersJ. Matczuk, The Gelfand-Kirillov dimension of Poincaré-Birkhoff-Witt extensions, Per- spectives in Rings Theory (F. Van Oystaeyen and L. Le Bruyn, eds.), Kluwer Academic Publishers (1988), 221-226.
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[
"arXiv:quant-ph/0610166v3 5 May 2010 epl draft Dynamical Realization of Macroscopic Superposition States of Cold Bosons in a Tilted Double Well",
"arXiv:quant-ph/0610166v3 5 May 2010 epl draft Dynamical Realization of Macroscopic Superposition States of Cold Bosons in a Tilted Double Well"
]
| [
"L D Carr \nDepartment of Physics\nColorado School of Mines\n80401GoldenCO\n",
"D R Dounas-Frazer \nDepartment of Physics\nColorado School of Mines\n80401GoldenCO\n",
"M A Garcia-March \nDepartment of Physics\nColorado School of Mines\n80401GoldenCO\n"
]
| [
"Department of Physics\nColorado School of Mines\n80401GoldenCO",
"Department of Physics\nColorado School of Mines\n80401GoldenCO",
"Department of Physics\nColorado School of Mines\n80401GoldenCO"
]
| []
| We present exact expressions for the quantum sloshing of Bose-Einstein condensates in a tilted two-well potential. Tunneling is suppressed by a small potential difference between wells, or tilt. However, tunneling resonances occur for critical values of the tilt when the barrier is high. At resonance, tunneling times on the order of 10-100 ms are possible. Furthermore, such tilted resonances lead to a dynamical scheme for creating few-body NOON-like macroscopic superposition states which are protected by the many body wavefunction against potential fluctuations. | 10.1209/0295-5075/90/10005 | [
"https://arxiv.org/pdf/quant-ph/0610166v3.pdf"
]
| 119,451,548 | quant-ph/0610166 | 2ea6eec54641ecfc78c516abdad121401b6ecbcc |
arXiv:quant-ph/0610166v3 5 May 2010 epl draft Dynamical Realization of Macroscopic Superposition States of Cold Bosons in a Tilted Double Well
L D Carr
Department of Physics
Colorado School of Mines
80401GoldenCO
D R Dounas-Frazer
Department of Physics
Colorado School of Mines
80401GoldenCO
M A Garcia-March
Department of Physics
Colorado School of Mines
80401GoldenCO
arXiv:quant-ph/0610166v3 5 May 2010 epl draft Dynamical Realization of Macroscopic Superposition States of Cold Bosons in a Tilted Double Well
PACS 03.75.Lm -Tunneling, Bose-Einstein condensates in periodic potentials
We present exact expressions for the quantum sloshing of Bose-Einstein condensates in a tilted two-well potential. Tunneling is suppressed by a small potential difference between wells, or tilt. However, tunneling resonances occur for critical values of the tilt when the barrier is high. At resonance, tunneling times on the order of 10-100 ms are possible. Furthermore, such tilted resonances lead to a dynamical scheme for creating few-body NOON-like macroscopic superposition states which are protected by the many body wavefunction against potential fluctuations.
Bose-Einstein condensates (BECs) in optical lattices are an ideal medium for studying a vast range of quantum many-body phenomena [1,2], including macroscopic quantum tunneling. A two-well potential is a simple limiting case which nevertheless exhibits rich quantum behavior. Spatially separate BECs in a two-well potential have been created in experiments [3] and tunneling times on the order of 50 ms have been observed [4]. The lifetime of a typical experiment is 1-100 s [5]. This system, in certain limits (see Eq. (1) below), maps onto the Lipkin-Meshkov-Glick (LMG) model [6,7], 1 the N -body generalization of the two-state problem, in which N particles can occupy two single-particle modes. Dynamical instability at the level of mean field theory (MFT) is associated with generation of strongly correlated states on a microscopic exact diagonalization (ED) level [8]; the LMG model can be solved by both MFT and ED. BECs offer the exciting possibility of realizing macroscopic superposition (MS) states (NOON states) with tens to thousands of particles, and thereby pushing the limits of quantum mechanics [9]. However, the physical context of a BEC in a double well has quite different regimes. Thus different approaches are required, including MFT [10][11][12], the multiconfigurational time-dependent Hartree (MCTDH) theory [13,14], and our own [15] and others' [16,17] work on ED of the LMG model with additional terms, including tilt or bias. Without the active use of tilt, MS states are destroyed by potential fluctuations [10,15].
In this Letter, we use a biased LMG model to investigate the quantum sloshing of many bosons in a tilted doublewell, with the goal of guiding dynamical creation of MS states in BEC double-well experiments. Quantum sloshing is the tunneling dynamics of a system in which all atoms are initially localized in one well. At the quarter and threequarter periods of the ensuing cyclical dynamics, one finds an MS state. The basic concept of quantum sloshing is similar to what occurs in an rf SQUID, in which the many body wavefunction oscillates between two macroscopically distinct states [18].
The LMG model is applicable for
χ ≡ [(N 2 − 1)U ]/(2 ω) 1 ,(1)
where N is the number of atoms, U is the interaction energy, and ω is the local trap frequency in each of the two wells [15,19]. Criterion (1) means that only the lowest single-particle state in either well is occupied. Additionally, it is required that all dynamical perturbation have an energy much less than ω. Under these restrictions, there are two regimes: ζ/N ≡ J/N |U | ≫ 1, where J is the tunneling energy, we denote the Josephson regime; ζ ≪ 1 we denote the Fock regime [9]. It is in the latter that MS states occur. The experimentally observed phenomenon of self-trapping, in which the system becomes stuck in one well, can be attributed to long tunneling times [20]. These oscillations with long tunneling times are associated with the presence of Schrödinger-cat-like, or NOON-like, MS states. Such states collapse to a narrow distribution of Fock states in the presence of a very small tilt [15], and tunneling is exponentially suppressed. Past studies p-1 on the suppression of tunneling have focused on environmental effects such as finite temperature or coupling to a reservoir [21][22][23]. Therefore, in the presence of a small tilt, finite temperature, or coupling to a reservoir, the MS states are dynamically unaccessible. Tilt displays radically different behavior than these other forms of suppression of the tunneling. Namely, tunneling resonances occur for critical values of the tilt when the barrier is high. At resonance, MS states states reappear [15], and therefore tunneling is again observed. We show that at tunneling resonances the oscillation time between wells is hundreds of orders of magnitude faster and less sensitive to deviations in the tilt than in the symmetric case. This speed-up permits us to propose a simple scheme for the creation of MS states both for few-body and manybody systems. Whereas past proposals involved ramping the barrier height [24] or continuous variation of atomatom interactions via Feshbach resonance [25], MS states are realized periodically in our scheme when all parameters are fixed. These MS states take the form of protected NOON states. Using the tunneling times of quantum sloshing to positively identify NOON or NOON-like states has been a significant tool in the study of SQUIDbased NOON states, where the double well consists of right-and left-circulating states on a ring [18]. In cold atom experiments, quantum tunneling effects have so far been restricted to one or two atoms [26,27], and MS states of NOON-like form have yet to be clearly identified.
We briefly mention methods for regimes other than that of the biased LMG. For χ ≫ 1, MFT is applicable for ζ/N ≫ 1; in this regime, called the "linear" or "Josephson" regime [9], 2 the double-well system can be described by a pendulum in a 2D phase-number phase space [10-12, 24, 28, 29]. In contrast, for ζ/N 1, called the "nonlinear" or "self-trapping" regime, MFT methods find macroscopic self trapping in one well [10][11][12]29]. The concept of phase is well-defined in the Josephson regime, together with a clear semiclassical limit; it is not welldefined in the "nonlinear" regime [12,30], and then requires a more strongly quantum approach [31,32].
MCTDH theory is such an approach. It does indeed find different dynamics from mean-field predictions, particularly in the Fock regime [33]. This purely numerical method is more exact [14] than MFT or LMG approaches [15,20,24,25,34,35] and in principle superior, as it can span all regimes. However, our modified LMG approach has the advantage that it is simpler and leads to exact and perturbative analytical expressions together with straightforward simulations; a mix of analytical and numerical methods at different levels of approximation is useful. Moreover, because our methods are not computationally intensive they can be easily extended to two and three dimensions [19], unlike the more exact but computationally demanding MCTDH theory. Under condition (1), MCTDH and LMG methods should converge.
Thus we work with the biased LMG model. Although mean-field theory for the asymmetric trap has been considered before [11], the MFT approximation is, as we have said, a poor one in the Fock regime, ζ 1. Experimentally, tilt appears both as a systematic error and deliberately in device applications [26,27,[36][37][38]. Tilted optical lattices are especially relevant to applications in gravitometry [36], quantum computing [26,39], and atomtronics [38].
The two-mode Hamiltonian for N weakly interacting bosons in a tilted two-well potential, or biased LMG model, iŝ
H = −J j =j ′b † jb j ′ + U jn j (n j − 1) + ∆Vn L ,(2)
where the subscript j ∈ {L, R} is the well or site index, J is the hopping strength, U is the interaction potential, and ∆V is the tilt. Hereb j andb † j satisfy the usual bosonic annihilation and creation commutation relations andn j ≡b † jb j . Eq. (2) can be derived from first-principles quantum field theory for weakly interacting bosons at zero temperature. An arbitrary state vector in Fock space is given by
|ψ = N nL=0 c nL |n L , N − n L ,(3)
where n L is the number of particles in the left well and c nL ∈ C. We require the total number of particles N to be constant. Under this restriction, the Hamiltonian reduces to an (N + 1) × (N + 1) tridiagonal matrix [24]. We consider the dynamics of a system in which all particles initially occupy the right well, i.e., |ψ(t = 0) = |0, N . In the Schrödinger picture, the time evolved ket is |ψ(t) ≡ exp(−iĤt/ )|ψ . The probability of finding n L particles in the left well at some time t > 0 is P nL (t) ≡ | n L , N − n L |ψ(t) | 2 , the average occupation of the left well is n L (t) ≡ ψ(t)|n L |ψ(t) , and the average variance is σ 2 nL (t) ≡ ψ(t)|n 2 L |ψ(t) − n 2 L . We first consider the simple case of noninteracting particles, U = 0, in a symmetric potential, ∆V = 0, to illustrate the problem. This case is exactly solvable. The probability of finding all particles in the right well, i.e.,
n L = 0, is P 0 (t) = cos 2N (Jt/ ).(4)
The tunneling period is T ≡ π /J, which is independent of N . When t = T /2, the system is in state |N, 0 and all particles have tunneled into the left well. The average occupation and variance of the left well are
n L (t) = N sin 2 (Jt/ ),(5)σ 2 nL (t) = (N/4) sin 2 (2Jt/ ).(6)
The particles therefore tunnel sinusoidally between wells with a frequency 2J/ . The variance is greatest when t = T /4. At this time, the probability of finding n L particles in the left well is
P nL (T /4) = 2 −N N !/[n L !(N − n L )!].(7)
The system is in a truncated coherent state, i.e., a binomial superposition of all number-states. The tilted case is also straightforward [19], although to our knowledge the following expressions have not yet appeared in the literature. The dynamics are slightly different when ∆V = 0, since the occupation of the left well now is
n L (t) = A sin 2 (ωt/2),(8)
where the amplitude and frequency of oscillation are
A ≡ N/[1 + (∆V /2J) 2 ],(9)ω ≡ (2J/ ) 1 + (∆V /2J) 2 .(10)
When ∆V = 2J, only N/2 particles tunnel between wells. Fig. 1(a) shows the probability densities P nL (t) in this case. In Figs. 1(b) and 1(c), Eqs. (9) and (10) are plotted as a function of ∆V . From the expression for A we observe that tunneling between wells is completely suppressed when |∆V | > 2J √ N − 1. Because the hopping strength J is much smaller than the barrier height, tunneling is highly sensitive to small tilt.
We proceed to consider how a small interaction term changes this scenario in the Josephson regime, ζ ≫ 1, in a symmetric potential, ∆V = 0. For the non-interacting system a single frequency 2J/ characterizes n L (t); in contrast, N dominant frequencies emerge in the interacting
to lowest order in perturbation theory [19] in N/ζ. We have also verified this result through simulations. In Eq. (11) the high frequency carrier depends only on the hopping strength J while the low frequency envelope depends on both the interaction potential U and the total number of particles N . The envelope reaches half its maximum value when
t = T 1/2 ≡ ( /U ) cos −1 [2 −1/(N −1) ].(12)
At times t ≪ T 1/2 , all particles tunnel between wells with period T , as in Fig. 2(a). At times near T 1/2 , on the other hand, only half the particles tunnel between wells with period T . When t ≃ 2T 1/2 , there is essentially no tunneling (see Fig. 2(b)). Small interactions thus damp the oscillations between wells [20,40]. However, tunneling revivals occur periodically with period T r ≡ π /U . The first tunneling revival occurs when |t − T r | < T 1/2 , as shown in Fig. 2(c). The separation of time scales, T 1/2 ≪ T r , occurs only for N ≫ 1, as evident in Eq. (12). For the remainder of our discussion, we turn to the high barrier limit, ζ ≪ 1, as it is key to the dynamic production of MS states. We assume U > 0 without loss of generality with respect to the dynamics. Using perturbation theory, it can be shown [15,19] that the eigenstates are MS states of the form |φ ± ≡ (|N − n L , n L ± |n L , N − n L ) / √ 2 to lowest order in ζ. The degenerate number states in p-3 the ζ = 0 limit split into symmetric and antisymmetric MS states for small ζ, with an energy difference of ∆E N −nL . The two eigenstates with the highest eigenvalue are nearly-degenerate MS states of the form |φ ± ≡ (|N, 0 ± |0, N ) / √ 2. The energy difference between |φ ± is
∆E N = 4U (ζ/2) N N/[(N − 1)!].(13)
The characteristic frequency is ω N = ∆E N / . Notice that since ∆E N is a very small number (ζ ≪ 1), ω N is also very small, and decreases rapidly with increasing N . All particles occupy the right well with probability
P 0 (t) = 1 − P N (t) = cos 2 (ω N t/2),(14)
In Fig. 3(a), we plot the probability densities P nL (t) and the average occupation n L (t) as a function of time. The tunneling period is T N ≡ 2π/ω N . The average occupation and variance are
n L (t) = N sin 2 (ω N t/2),(15)σ 2 nL (t) = (N 2 /4) sin 2 (ω N t).(16)
In this regime, as in the noninteracting case, all N particles oscillate sinusoidally between wells. There are two important differences. The first is that the period of oscillation depends on N and can become quite large for large values of N . Note that, for large N , self-trapping is observed for exponentially long times, in agreement with mean-field approaches. Second, at time t = T N /4, we find that P N = P 0 = 1/2. At this time, all particles simultaneously occupy both wells and the system is described by a NOON-like MS state, a new prediction that cannot be achieved in the mean-field limit, and which is intriguing in the framework of quantum information. Guided by this interest, let us characterize the entanglement at t = T N /4, by utilizing four standard entanglement measures: the average local impurity, or "Qmeasure" [41], the local entropy of entanglement [42], the Schmidt rank k [43], and a "macroscopic superposition size" (MSS) based on physical measurement [44]. The Qmeasure is given by
Q = [(N +1)/N ][1−(Trρ 2 L +Trρ 2 R )/2], where ρ L(R) = Tr R(L) |ψ(t) ψ(t)|. The entropy is S = − N nL=0 P nL (t) log N +1 P nL (t)
, and the Schmidt rank k is given by the number of non-zero eigenvalues of the reduced density matrix ρ L . Finally, the MSS measure is C δ = N/n min , where n min is the minimum number of particles one must to measure to distinguish both branches, i.e., |N − n L , n L and |n L , N − n L . These measures take the value Q = S = 0, k = 1, and C δ = 0 if and only if |ψ(t) is a pure state. This occurs at t = T N /2 and T N . At time T N /4, we find that each measure reaches a maximum value of Q = N/[2(N + 1)], S = log N +1 (2), k = 2, and C δ = N/(n L + 1).
We proceed to consider the effects of tilt or bias in the LMG model. For ζ ≪ 1 the tunneling between wells is extremely sensitive to tilt ∆V . Using perturbation theory, it was shown that when ∆V > 2∆E N /N the eigenstates are nearly perfect number-states of the form |n L , N − n L and |N −n L , n L . In this case, since the number states are near-eigenstates of the Hamiltonian, the initial condition |ψ(0) is nearly stationary and tunneling between wells is strongly suppressed. This is quite different from suppression of tunneling due to thermal effects or coupling to a reservoir [21][22][23]; our system is closed and suppression is due to an internal parameter, namely, imperfections in the trapping potential. However, for stronger tilts quasi-degenerate MS eigenstates of the form |φ ± ; p ≡ (|N − p, p ± |0, N ) / √ 2 reappear [15] for ∆V = ∆V p , with a splitting in the energies between the states |φ ± ; p equal to ∆E p N , where
∆E p N = 4U (ζ/2) N −p (N − p) (N − p − 1)! N ! p!(N − p)! ,(17)
In this case, the potential difference is exactly compensated by the repulsive interaction of p particles in the lower well. Then, at resonance, the tunneling frequency is ω p N = ∆E p N / . The average occupation of the left well is
n L (t) = (N − p) sin 2 (ω p N t/2)(18)
to lowest order in ζ. Here, N − p particles tunnel between wells with period T p N = 2π/ω p N . At time t = T p N /2, N − p particles are in the left well. To compensate the tilt, p particles remain in the right well at all times. When t = T p N /4, the system is described by an MS state such that P p = P 0 = 1/2 and the entanglement measures Q, S, and k reach the same values as in the symmetric case. The entanglement measured by C δ is again C δ = N , provided that the measure is taken in the higher well. Interestingly enough, if the measure is taken in the lower one, the entanglement measured is smaller, C δ = N/(p + 1), since p + 1 particles must be measured to distinguish between both branches. In Fig. 3(b) the tunneling dynamics for the second resonance, i.e., p = 2, are illustrated for a system of N = 7 particles. Near a resonance, tunneling is suppressed when
|∆V − ∆V p | > 2∆E p N /(N − p).(19)
This is due to the fact that, near the resonance, the eigenstates are again near-perfect number states when the deviations of the tilt with respect to the resonant one exceed this quantity [15]. Then, the condition for the suppression of the tunneling in the symmetric case is that the tilt exceeds 2∆E N /N while in the case of the asymmetric potential, when the tilt coincides with a resonance, the condition for the suppression of the tunneling is that the difference between the tilt and that of the resonance exceeds 2∆E p N /(N − p). Moreover, tunneling near resonance is much faster than tunneling in a symmetric potential since both tunneling frequencies, ω p N and ω N , are proportional to ∆E p N / and ∆E N / respectively, and we have shown that ∆E p N is greater than ∆E N by many orders of magnitude. In Fig. 4(a), we show the symmetric tunneling period T N versus N when ζ = 0.1. Clearly, T N becomes very long as N becomes large. For instance, in a typical symmetric double-well used in experiments [26], 200 87 Rb atoms tunnel between wells with period T 200 = 1.15 × 10 635 ms when ζ = 0.0964. Furthermore, tunneling is completely suppressed for deviations in the tilt greater than 4.16 × 10 −636 nK · k B . Obviously, one does not expect to observe many-body tunneling in this regime. Notice that, under the same conditions, systems with as few as N = 1, 2, and 3 87 Rb atoms yield tunneling times as long as T 1 = 466 ms, T 2 = 4840 ms, and T 3 = 134000 ms, respectively. Even in a few-particle system, tunneling times can be prohibitively long, demonstrating self-trapping behavior [4].
However, tunneling at resonance can be hundreds of orders of magnitude faster than the symmetric case, as in Fig. 4(b). For the 200-atom system discussed above, when p = 197, we find that N − p = 3 particles tunnel between wells with period T 197,198, and 199, respectively. Thus, the observation of the tunneling of a few 87 Rb atoms is made possible by tunneling resonances in a many-body system. Moreover, embedding NOON-like MS states in the many body wavefunction leads to the possibility of larger MS states of more than three particles; the scaling of the advantage in tunneling time gained via tilt can be calculated as τ ≡ ∆E N /∆E p N , from Eqs. (13) and (17). Taking p ′ ≡ N − p as the number of atoms in the embedded MS state, and using Stirling's approximation ln n! ≃ n ln n − n, we find ln τ ≃ (− ln n + 1 + ln ζ 2 )n + ( p ′ 4n + p ′2 12n 2 − 1 2 ln n)p ′ +( 3 2 ln p ′ − 3 2 + 1 2 ln 4 − ln ζ)p ′ ; (20) the second set of cross terms is vital. This expansion matches the exact expression very well, as illustrated in Fig 5. In conclusion, we used the two-mode approximation to develop a Fock space picture of a system of ultracold bosons in a tilted two-well potential, covering all regimes of barrier height, from the Josephson regime to the regime in which one can find NOON states. In the latter regime, which occurs when the barrier is high, a small tilt causes the complete suppression of tunneling, leading to selftrapping. Long tunneling times prevent the observation of many-body tunneling even in a symmetric potential and MS states are too sensitive to fluctuations in the trapping potential to be realistic. However, in this regime, tunneling resonances occur when the tilt can be compensated by atom-atom interactions. At resonance, tunneling is much faster and less sensitive to tilt than in a symmetric potential. Furthermore, tunneling resonances can be used to create NOON-like MS states embedded in and protected by a larger many-body system.
We thank Ignacio Cirac, Ann Hermundstad, William Phillips, and Trey Porto for useful discussions.
Fig. 1 :
1(Color online) Suppression of tunneling for noninteracting atoms. (a) Shown are the probability densities Pn L (t) (colorbar) for all number states when N = 100, U = 0, and ∆V = 2J. Only N/2 = 50 particles tunnel between wells. (b) The tunneling amplitude and (c) the frequency of oscillation as a function of tilt. When ∆V > 2J√ N − 1, tunneling is completely suppressed. Particles tunnel between wells faster in a tilted potential than in a symmetric potential.
Fig. 2 :
2(Color online) Damped tunneling in the Josephson regime. (a) Probability densities Pn L (t) for all number states when N = 10 and ζ/N ≡ J/N U = 10 for t ≪ T 1/2 . All particles tunnel between wells with period T = π /J. (b),(c) Average occupation (top panel) and number variance (bottom panel) of the left well for longer times. (b) Oscillations between wells are damped by atom-atom interactions. (c) The first tunneling revival occurs when t = Tr ≡ π /U . The colorbar is the same as in Fig. 1. system in the Josephson regime. The average occupation of the left well is given by the modulated signal n L (t) = (N/2) 1 − cos(2Jt/ ) cos N −1 (U t/ ) ,
Fig. 3 :
3(Color online) Tunneling resonances in a few-atom system. Shown are the probability densities Pn L (t) when N = 7 and ζ = 0.1, for (a) ∆V = 0 and (b) ∆V = 4U . (a) All particles tunnel between wells with period TN . At time t = TN /4, the system is described by an MS state. (b) Only N − 2 = 5 particles tunnel between wells. The oscillation frequency is 5 orders of magnitude faster than the symmetric case. (c) Tunneling amplitude as a function of tilt ∆V for N = 5 and ζ = 0.1. Tunneling resonances occur when ∆V = ∆Vp ≡ 2pU . At resonance, N − p particles tunnel between wells. The insert is a zoom around ∆V /2U = 2.
Fig. 4 :
4(Color online) Tunneling periods in a many-body system. (a) Tunneling period TN versus the total number of particles N when ζ = 0.1 and ∆V = 0. For large N , tunneling becomes very slow. (b) At resonance, ∆V = ∆Vp ≡ 2pU , only N − p particles tunnel between wells. Shown are the tunneling periods T p N versus p for N = 40 to 100 with ζ = 0.1. At resonance, the oscillations can be hundreds of orders of magnitude faster than in the symmetric case.
Fig. 5 :
5Embedding the MS state in the many body wavefunction. Relative increase τ in tunneling time and robustness against potential fluctuations, for an MS state of N − p = 10, 50, 100 particles (left to right); plotted as a function of the number of particles in the full state, N , for exact (solid black curves) and approximate (dashed red curves) expressions, all for ζ = 0.1. Grant PHY-0547845 as part of the NSF CAREER program, by the Fulbright Commission, by Spain's Ministerio de Educación y Ciencia (MEC), and by the Fundación Española de Ciencia y Tecnología (FECYT).
197 200 = 117 ms. This resonance occurs when ∆V = ∆V 197 = 210 nK · k B . An MS state of the form |ψ = (|3, 197 − i|0, 200 )/ √ 2 will be realized at T 197 200 /4 = 29.25 ms. Likewise, we find that T 198 200 = 34.3 ms and T 199 200 = 33.0 ms when ∆V = ∆V 198 = 211 nK · k B and ∆V 199 = 212 nK · k B , respectively. Finally, at resonance, this system is sensitive to deviations in the tilt on the order of 0.273 nK·k B , 1.40 nK ·k B , and 2.90 nK·k B for p =
Sometimes called "two-mode Bose Hubbard" or "two-site Bose Hubbard" in cold quantum gases literature, although the LMG model precedes the Bose-Hubbard model by 24 years.
Sols and Leggett subdivide this regime into "Rabi" and "Josephson"; we use only the term "Josephson" for simplicity.
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[
"On Holographic Weyl Anomaly",
"On Holographic Weyl Anomaly"
]
| [
"Mozhgan Mir [email protected] \nPerimeter Institute for Theoretical Physics\nDepartment of Physics\nFerdowsi University of Mashhad\nP.O. Box 1436N2L 2Y5Waterloo, MashhadOntarioCanada, Iran\n"
]
| [
"Perimeter Institute for Theoretical Physics\nDepartment of Physics\nFerdowsi University of Mashhad\nP.O. Box 1436N2L 2Y5Waterloo, MashhadOntarioCanada, Iran"
]
| []
| We use the relation between certain diffeomorphisms in the bulk and Weyl transformations on the boundary to build the conformal structure of the metric in the presence of matter in the bulk. We explicitly obtain the conformal anomaly in any spacetime dimension d in a holographic frame work in the case of gravity coupled to the scalar fields. This way we also provide a holographic construction of the bulk spacetime metric and of the matter fields in the bulk of this spacetime out of sources that are related to conformal field theory data. With the latter, we produce an asymptotic expansion of the bulk fields and scalar fields near the boundary in terms of spacetime dimension in the context of the AdS/CFT correspondence. We work out both the gravitational and matter conformal anomalies of the boundary theory as a coefficient of the infrared logarithmic divergence that appears in the on-shell action. | 10.1007/jhep10(2013)084 | [
"https://arxiv.org/pdf/1307.5514v3.pdf"
]
| 118,428,261 | 1307.5514 | f8408d13057ace62ad8969cec7ae7b47801e8bc0 |
On Holographic Weyl Anomaly
26 Sep 2013
Mozhgan Mir [email protected]
Perimeter Institute for Theoretical Physics
Department of Physics
Ferdowsi University of Mashhad
P.O. Box 1436N2L 2Y5Waterloo, MashhadOntarioCanada, Iran
On Holographic Weyl Anomaly
26 Sep 20130Holography, Weyl anomaly
We use the relation between certain diffeomorphisms in the bulk and Weyl transformations on the boundary to build the conformal structure of the metric in the presence of matter in the bulk. We explicitly obtain the conformal anomaly in any spacetime dimension d in a holographic frame work in the case of gravity coupled to the scalar fields. This way we also provide a holographic construction of the bulk spacetime metric and of the matter fields in the bulk of this spacetime out of sources that are related to conformal field theory data. With the latter, we produce an asymptotic expansion of the bulk fields and scalar fields near the boundary in terms of spacetime dimension in the context of the AdS/CFT correspondence. We work out both the gravitational and matter conformal anomalies of the boundary theory as a coefficient of the infrared logarithmic divergence that appears in the on-shell action.
Introduction
Holography prescribes a correspondence between a (d+1)-dimensional gravitational theory and a d-dimensional field theory referred to bulk/boundary duality [1,2]. The AdS/CFT correspondence provides a precise dictionary between the bulk and boundary physics [3,4] with a rather exact computational frame work [3,5]. It is proposed in [3,5] that the partition function of string theory on AdS spaces with determined boundary conditions for the bulk fields corresponds to a generating functional of cor-relation functions in the conformal field theory (CFT). String theory at low energies is understood as a supergravity theory that corresponds to the large N and strong coupling regime of the CFT [6]. The boundary value of fields is playing the role of sources for operators of the dual CFT 1 . For instance, the boundary metric couples to the boundary stress-energy tensor. Also there is a correspondence between the infrared divergences on the supergravity side and the ultraviolet divergences on the CFT side.
One may construct an asymptotically AdS space through solving the gravitational equations coupled to scalars and the scalar field equation on this manifold, by using the conformal structure at infinity and the boundary value of the scalars as initial data. However, a boundary conformal structure does not determine the bulk metric completely and one needs more CFT data to specify the solution to the higher order terms in the asymptotic expansion of the solution for instance the expectation value of the corresponding operators [7]. Although under certain assumption for example conformally flat bulk metrics does yield the bulk metric up to diffeomorphisms [8]. Let us emphasize that we do not have any assumption about the regularity of the solution, since we only obtain the near boundary solution, therefore we simply do not confront possible singularities in the interior of the bulk.
Any asymptotically AdS metric near the boundary in the Graham-Fefferman coordinate systems [9] can be rewritten as [10,11,12] ds 2 = G µν dx µ dx ν = L 2 dρ 2 4ρ 2 + 1 ρ g ij (x, ρ)dx i dx j , (1.1)
where 2 g(x, ρ) = g (0) + · · · + ρ d 2 g ( d 2 ) + ρ d 2 h ( d 2 ) log ρ + · · · .
(1.
2)
The logarithmic term exists only in even dimensions. L is related to the cosmological constant as Λ = − d(d−1) 2L 2 . For simplicity throughout this paper we set L = 1. Metric G µν in (1.1) has singularity at infinity. However performing a (singular) conformal transformation induces a well-defined boundary metric g (0) at infinity.
The AdS d+1 space has the curvature of G as
R µνρσ = (G µρ G νσ − G µσ G νρ ) + O(ρ).
(1.3) 1 We only assume that the dual CFT exists and do not have any further assumption since our results only depend on the spacetime dimension of the boundary theory. 2 The index in the brackets indicates the half of the number of derivatives involved in that term
The action of Weyl transformations of the boundary metric is realized as a subgroup of the bulk diffeomorphisms for the bulk metric. These conformal transformations on the boundary is called PBH transformations [13]. The PBH transformation does not depend on the details of the solution of interest and therefore it gives information for any such solutions, e.g., even including stringy corrections.
Infrared divergences in the gravitational sector can be seen as the cause of the conformal anomaly [14]. Invariance of AdS under rescaling the radial coordinate that corresponds to dilatation in the CFT, is broken after regulating these infrared divergences. It means that a conformal transformation in the boundary theory is related to a special diffeomorphism in the bulk that preserves the form of the coordinate system (1.1) [13].
In the asymptotic solution (1.2) g (n) with n < d 2 are determined uniquely and locally by g (0) . g ( d 2 ) is a nonlocal quantity which is not obtained only from the boundary condition for the metric. Rather only its trace and covariant divergence are determined.
However, it is determined by the vacuum expectation value of the dual stress-energy tensor. In the case of pure gravity, when the boundary dimension d is odd the coefficient
g ( d 2 ) in (1.
2) is conserved and traceless and there is no quantum correction for the stress-energy tensor. Furthermore it is well-known that h ( d 2 ) is related to the conformal anomaly through the metric variation of the conformal anomaly [15]. It was shown in [8] that, given a conformally flat boundary metric in which the bulk Weyl tensor vanishes, the bulk metric can be determined to all orders. Also in this case h ( d 2 ) in (1.2) vanishes because it is the metric variation of a topological invariant. In addition for the standard Euclidean AdS, g (n) = 0 for 0 < n < d 2 [15].
In the case of pure gravity, the holographic conformal anomalies for the spacetime dimensions up to six were obtained [15] and the holographic results agree with the known field theory calculations [14]. It is well-known that in odd (boundary) dimensions, there are no conformal anomalies. We would like to build holographically the bulk metric coupled to the matter out of the CFT data and extract the associated conformal anomaly as a quantum feature of CFT by applying the holographic prescription [15]. The CFT data are the sources for the operators that are turned on and the corresponding expectation values.
The same discussion is applicable when one adds matter to the system. By con-sidering scalars that correspond to marginal and relevant operators one can compute perturbatively the back reaction of the scalars to the gravitational background. The solution of the massive scalar field has the mass
m 2 = ∆(∆ − d), (1.4)
where ∆ is the conformal dimension of the dual operator [16]. The asymptotic expansion for the scalar field near the boundary in the coordinate system (1.1) is in the form [15] Φ(x, ρ) = ρ
(d−∆) 2 φ(x, ρ), φ(x, ρ) = φ (0) + φ (1) ρ + . . . . (1.5)
Because we want to use the probe approximation, we consider a relevant operator in the boundary theory. It requires in (1.4), m 2 < 0 for the bulk scalar and this approach is valid in the regime ∆ < d 3 . The leading order in the small-ρ expansion of the scalar field is ρ α 2 with α = d − ∆ which appears in the mass formula (1.4) and
0 < α < d 2 .
(1.6)
Although in the case pure gravity for odd boundary dimensions there are no gravitational conformal anomalies, adding relevant deformations there is a possibility of having conformal anomalies in both odd and even dimensions, providing ∆ − d 2 to be an integer, a logarithmic term appears. The logarithmic term ψ (∆− d 2 ) appearing in this situation in (1.5), is related to the matter conformal anomalies [15]. φ (n) with n < ∆ − d 2 and ψ (∆− d 2 ) are determined uniquely in terms of g (0) and φ (0) from the scalar field equation and gravitational equations coupled to scalars that are presented in the
next section. The coefficient φ (∆− d 2 )
is not obtained by solving the field equations. However it is the leading finite part determined by the vacuum expectation value 4 of the dual operator [17] O
(x) = (∆ − d 2 )φ (∆− d 2 ) (x).
In this paper we follow two parallel processes to construct the holographic conformal anomaly in the presence of matter. In section 2 we discuss how to find the anomaly by solving equations of motion. In section 3 we use the PBH transformations that give 3 For a marginal operator with ∆ = d (or α = 0) we will see in (3.11) that the matter part of the anomaly vanishes and we left with the gravitational part. 4 Global condition on the bulk metric corresponds to the choice of a particular vacuum in the conformal field theory.
us the covariant form of the anomaly in the cases we are interested but the overall factors are still undetermined. In section 4 first we obtain an asymptotic solution in the case of gravity coupled to a free scalar by choosing specific dimensions d = 4, 6 for a given boundary metric and Dirichlet value of the scalar field. We use these solutions and their covariant derivatives. In the last three subsections, we exhibit results for the insertion of four scalar fields.
Asymptotic AdS gravity coupled to scalar fields
The Einstein-Hilbert action for a gravity theory with d + 1 dimensions on a manifold M and the d-dimensional boundary 5 ∂M is
S gr = 1 16πG d+1 N [ M d d+1 x √ G(R[G] + 2Λ) − ∂M d d x √ γ2K], (2.1)
where K is the trace of the second fundamental form and γ is the induced metric on the boundary. The boundary term makes the variational problem with Dirichlet boundary conditions be well-defined [18]. Here we work with Euclidean signature of the metric.
The on-shell gravitational action diverges because of the spacetime volume is infinite and also the singularity in the metric at the boundary as was mentioned previously. To regulate the theory, we cut off the spacetime at ρ = for some small parameter near the boundary. In the action (2.1) the bulk integral is calculated in the region ρ ≥ and the boundary term at ρ = . The divergences in the action are in the form of the 1/ k poles and in the case of even (boundary) dimensions there is also a logarithmic divergence [14] S gr,reg = 1 16πG d+1
N d d x √ g (0) − d 2 a (0) + − d 2 +1 a (1) + · · · + −1 a ( d 2 −1) − log a ( d 2 )
+O( 0 ), (2.2) 5 In the following, d+1-dimensional indices are shown with Greek indices µ, ν, · · · , d-dimensional ones with Latin ones i, j, · · · . R ij [G] indicates the Ricci tensor of the metric G, etc.
where the coefficients a (n) 6 are given by local covariant combinations of the g (0) and its curvature tensor.
The renormalized action is obtained by subtracting the divergent terms and then removing the regulator, i.e., → 0. The terms proportional to the negative powers of in the regularized action are invariant under the scale transformations
δg (0) = 2δσg (0) , δ = 2δσ .
The log term transforms as log → log + log(2δσ). The variation from the latter does not depend on and should be canceled by the variation of the finite terms such that the total action (2.2) remains invariant under the transformation. Hence, the log term gives the conformal anomaly A of the renormalized theory on the boundary
S lg,g = − 16πG d+1 N 2 d d x √ g (0) A. (2.3)
Generally the boundary conditions for the metric and scalar fields are arbitrary fields because they are regarded as sources for the boundary operators and we take functional derivatives with respect to them in the corresponding conformal field theory. Also solving the equations of motion and applying the boundary conditions will lead us to the information in the boundary theory such as anomaly. The gravitational field equation in the presence of matter reads
R µν − 1 2 (R + 2Λ)G µν = −8πG N T µν ,(2.4)
with appropriate Dirichlet boundary conditions.
In order to solve (2.4) with a given conformal structure at infinity, we work in the coordinate system (1.1) introduced by Fefferman and Graham [9]. In the case of relevant deformation on the boundary theory, the metric expansion becomes [19]
g ij (x, ρ) = N −1 n=0 ρ n g (n) ij (x) + ∞ m=2 ρ n+m α 2 g (n+m α 2 ) ij (x) +ρ d 2 N −1 n=0 ρ n g ( d 2 +n) ij (x) + ∞ m=2 ρ n+m α 2 g ( d 2 +n+m α 2 ) ij (x) ,(2.5)
where we assume α 2 = N M which M, N are relatively prime.
For even dimensions, a logarithmic term appearers in the expansion as follows [19]:
g ij (x, ρ) = N −1 n=0 ρ n g (n) ij (x) + ∞ m=2 ρ n+m α 2 g (n+m α 2 ) ij (x) +ρ d 2 log ρ N −1 n=0 ρ n f ( d 2 +n) ij (x) + ∞ m=2 ρ n+m α 2 f ( d 2 +n+m α 2 ) ij (x) . (2.6)
We wish to study how the bulk spacetime with a relevant deformation in the boundary theory is constructed holographically . Here we discuss a bulk scalar field but a similar approach is applicable to other kinds of matter. The action for a free massive scalar is given by
S M = 1 2 d d+1 x √ G(G µν ∂ µ Φ∂ ν Φ + m 2 Φ 2 ), (2.7)
where the metric G µν has an expansion of the form (2.6).
Similar to the case of pure gravity, the theory is regularized by integrating the bulk ρ ≥ in the action (2.7) [20,21]
S M,reg = d d x det g (0) α− d 2 a M (0) + α− d 2 +1 a M (1) + · · · + a M (α− d 2 +1) − log a M (α− d 2 )
+O( 0 ), (2.8) where the matter conformal anomaly is given by the coefficient of the logarithmic term and is added to the formula (2.3) to obtain the whole contribution. The form of the expansion of the scalar field Φ is [19] Φ
(x, ρ) = ρ α 2 φ(x, ρ) = ρ α/2 N −1 n=0 ∞ m=0 ρ n+m α 2 φ (n+m α 2 ) (x) +ρ (d−α) 2 N −1 n=0 ∞ m=0 ρ n+m α 2 φ ( d 2 −α+n+m α 2 ) (x). (2.9)
By applying this constraint, we take operator with a particular conformal dimension ∆ to get logarithmic term, i.e.,
∆ = d 2 + k, k = 0, 1, · · · .
The scalar field equation (− G + m 2 )Φ = 0 by using (1.5) becomes
[−α∂ ρ log gφ + 2(d − 2α − 2)∂ ρ φ − 1 √ g ∂ i ( √ gg −1 ij ∂ j φ)] + ρ[−2∂ ρ log g∂ ρ φ − 4∂ 2 ρ φ] = 0.
(2.10)
where we used the relation m 2 = α(α − d) for the mass and
G Φ = 1 √ G ∂ µ ( √ GG µν ∂ ν Φ).
Through solving the equations of motion, one can determine recursively φ (n) , n < d 2 − k. For given φ (0) , when d 2 − k is integer, at this order the coefficient of φ (d/2−k) becomes zero and it is necessary to introduce a logarithmic term at this order, now the expansion of Φ reads
Φ(x, ρ) = ρ α 2 φ(x, ρ) = ρ α/2 N −1 n=0 ∞ m=0 ρ n+m α 2 φ (n+m α 2 ) (x) +ρ (d−α) 2 log ρ N −1 n=0 ∞ m=0 ρ n+m α 2 ψ ( d 2 −α+n+m α 2 ) (x). (2.11)
The coefficient of the logarithmic term ψ (∆− d 2 ) is obtained but φ (∆− d 2 ) remains undetermined. However, it is known [22,17] that the latter is specified by the expectation value of the dual operator. Since the action does not include interactions for the scalar,
we can expect φ (∆− d 2 ) to be linear in φ (0) .
Now we proceed to obtain solutions to Einstein's equations coupled to scalar fields.
The total action is given by the summation of (2.1) and (2.7)
S = S gr + S M . (2.12)
The gravitational equations of motion 7 (2.4) in the coordinate system (1.1) by making use of the expansion of the scalar field (1.5) read [15] ρ
2g ij − 2(g g −1 g ) ij + T r(g −1 g )g ij − R ij (g) − (d − 2)g ij − Tr(g −1 g )g ij = = −8πG d+1 N ρ α−1 α(α − d) d − 1 φ 2 g ij + ρ ∂ i φ∂ j φ (2.13) ∇ i Tr(g −1 g ) − ∇ j g ij = −16πG d+1 N ρ α−1 α 2 φ ∂ i φ + ρ ∂ ρ φ∂ i φ Tr(g −1 g ) − 1 2 Tr(g −1 g g −1 g ) = −16πG d+1 N ρ α−2 αd(α − 1) 4(d − 1) φ 2 + αρ φ∂ ρ φ + ρ 2 (∂ ρ φ) 2 7
Notice that if α < 0 (or ∆ > d) the right hand side of equations diverges near the boundary, because for α < 0 operators are irrelevant. We only consider α ≥ 0 (or ∆ ≤ d) that gives marginal or relevant operators and equations can be calculated perturbatively.
where prime presents differentiation with respect to ρ (with ρ considered an extra parameter, rather than a coordinate), ∇ i is the covariant derivative built from the metric g, and R ij (g) is the Ricci tensor of g.
These equations are solved order by order in ρ. First from the equation in (2.13) the coefficients g (n) for n = d 2 in terms of g (0) are obtained. The expression for g (n) becomes singular when n = d 2 . The trace and divergence of g (n) for any n are determined from the last two equations in (2.13). Since the trace of g ( d 2 ) is related to the vacuum expectation value of the trace stress tensor, if the former is zero we conclude that the stress energy tensor does not receive quantum effects, otherwise it detects the appearance of the conformal anomaly that is obtainable from both coefficients of the logarithmic terms,
h ( d 2 ) and ψ ( d 2 −α)
. From the regulated on-shell action,the value of the bulk integral in (2.12) gives a logarithmic divergence that is where the anomaly emerges [15] S reg (bulk) =
ρ≥ dρ d d x √ G[ d 8πG (d+1) N − m 2 d − 1 Φ 2 ] (2.14) = ρ≥ dρ d d x 1 ρ g(x, ρ)[ d 16πG (d+1) N ρ −d/2 − m 2 2(d − 1) φ 2 (x, ρ)ρ −k ].
where k = d 2 −α, from above formula it is concluded even if d is odd there is a possibility of having conformal anomaly as k is a positive integer.
Anomaly by the PBH transformation approach
AdS/CFT duality states that the conformal transformations in the boundary are emerged from the specific diffeomorphism that preserves the form of the metric (1.1) [23]. Consider a coordinate transformation that keeps the form of the metric in FG
gauge (1.1) ρ = ρ (1 − 2σ(x )) x i = x i + a i (x , ρ ).
The a i (x , ρ ) are defined providing of the form invariance of the metric. To leading order in σ, with boundary condition a i (x, ρ = 0), it is given by
a i (x, ρ) = L 2 2 ρ 0 dρ g ij (x, ρ )∂ j σ(x). (3.1)
The infinitesimal diffeomorphism transformation of g ij (x, ρ) under the PBH transformation [13] is
δg ij (x, ρ) = 2σ(1 − ρ∂ ρ )g ij (x, ρ) + ∇ i a j (x, ρ) + ∇ j a i (x, ρ), (3.2)
where the covariant derivatives are constructed from the metric g ij (x, ρ). The first term is a homogenous term and its coefficient determines the conformal dimension. To see this, regard the power series expansion of the metric in the gravity theory coupled to scalars in the vicinity of ρ = 0
g(x, ρ) = g (0) + g (1) ρ + . . . + g (n) ρ n + ρ α g (α) + g (α+1) ρ + g (α+2) ρ 2 + . . . , (3.3)
then one gets a homogenous scaling term for each coefficient as
δg (n) = −2σ(x)(n − 1)g (n) + · · · ,
it implies that g (n) ij has the conformal dimension 2(n − 1). In particular, g (0) ij has the conformal dimension −2. To find how different components of the metric transform,
we expand a i (x, ρ) in powers of ρ a = a (1) ρ + a (2) ρ 2 + . . . + a (n+1) ρ n+1 + ρ α a (α+1) ρ + a (α+2) ρ 2 + . . . , (3.4)
where pertaining to the boundary condition we set a i (0) (x) to be zero. We substitute above expansions into eq. (3.2) and find 8
δg (0) ij = 2σ g (0) ij , (3.5) δg (1) ij = a k (1) ∂ k g (0) ij + ∂ (i a k (1) g (0) j)k , δg (α) ij = −2σ(α − 1)g (α) ij , δg (α+1) ij = −2σαg (α+1) ij + a k (1) ∂ k g (α) ij + a k (α+1) ∂ k g (0) ij + 2∂ (i a k (1) g (α) j)k + 2∂ (i a k (α+1) g (0) j)k , δg (α+2) ij = −2σ(α + 1)g (α+2) ij + a k (α+2) ∂ k g (0) ij + a k (α+1) ∂ k g (1) ij + a k (1) ∂ k g (α+1) ij + a k (2) ∂ k g (α) ij +2∂ (i a k (α+2) g (0) j)k + 2∂ (i a k (α+1) g (1) j)k + 2∂ (i a k (1) g (α+1) j)k + 2∂ (i a k (2) g (α) j)k , where a i (1) = L 2 2 (g (0) −1 ) ij ∂ j σ, a i (2) = − L 2 4 g −1 (1) ij ∂ j σ, a i (β) = − L 2 2β (g (0) −1 g (β−1) g (0) −1 ) ij ∂ j σ. (3.6)
where β = α + 1, α + 2, . . . 9 . Also for utilization at some point subsequently we have
δR = −2σR − 2(d − 1) σ, δR ij = −(d − 2)∇ i ∇ j σ − g ij σ, (3.7)
where exhibits the Laplacian. Furthermore we acquire these relations by taking advantage of the PBH transformations by noticing the transformation of components of the scalar field under transformations. The series expansion of the scalar field is in the following form:
Φ(x, ρ) = ρ α 2 φ (0) + φ (1) ρ + φ (2) ρ 2 + . . . + ρ (d−α) 2 φ ( d 2 −α) + . . . , (3.8)
therefore, we get
δφ (0) = −ασφ (0) , δφ (1) = −(α + 2)σφ (1) + a k (1) ∂ k φ (0) (3.9) δφ (2) = −(α + 4)σφ (2) + a k (1) ∂ k φ (1) + a k (2) ∂ k φ (0) , also δφ (α) = −3ασφ (α) , δφ (α+1) = −(3α + 1)σφ (α+1) + a k (α+1) ∂ k φ (0) + a k (1) ∂ k φ (α) ,
where the definitions for a k (M ) , M = 1, 2, β are given in (3.6). From the last two terms we see in general the conformal dimension of φ (n) is α + 2n. Now we want to take advantage of the PBH transformations and determine the explicit form of the different components of the metric and the scalar field expansions.
For this purpose, we write down the most covariant combination using the curvature tensor, the scalar field and their covariant derivatives such that each term in this combination with an arbitrary coefficient has the desired conformal dimension. Therefore, by understanding the PBH transformations of any component given in (3.5) and (3.9), we make a comparison with the conformal transformation of the mentioned combination. The equivalence of inhomogeneous parts of transformations in two sides specifies the unknown coefficients. For example the covariant expressions for g (1) and g (2) under 9 Note that by referring to (3.1), β starts from α + 1.
the PBH transformations become
10 g (1) ij = − L 2 d − 2 R (0) ij − R g (0) ij 2(d − 1) , (3.10) g (2) ij = c 1 L 4 C klmn C klmn g (0) ij + c 2 L 4 C iklm C j klm + L 4 d − 4 1 8(d − 1) ∇ i ∇ j R − 1 4(d − 2) R ij + 1 8(d − 1)(d − 2) Rg (0) ij − 1 2(d − 2) R kl R ikjl + d − 4 2(d − 2) 2 R i k R jk + 1 (d − 1)(d − 2) 2 RR ij + 1 4(d − 2) 2 R kl R kl g (0) ij − 3d 16(d − 1) 2 (d − 2) 2 R 2 g (0) ij ,
we see later there are possible contributions coming from the scalar field (see(4.5) and (4.13)).
Lets define a Lagrangian by using the regulated bulk action (2.14) that gives only the logarithmic contribution to the action with one dimension higher
L = ρ≥ dρ 1 ρ g(x, ρ)[ d 16πG (d+1) N ρ − d 2 + α(d − α) 2(d − 1) φ 2 (x, ρ)ρ −k ], (3.11) where k = d 2 − α and we have inserted the definition (1.4).
The result of the integration has a − log part that based on the discussion above (2.3), its coefficient with multiplication of a factor of −2 gives us the anomaly From the regulated action for the metric G and the scalar field Φ. Referring to the logarithmic divergences in (2.14) and the discussion above (2.3), one finds
A = −2( d 16πG (d+1) N a ( d 2 ) + a M (α− d 2 )
).
(3.12)
The gravitational part of the anomaly appears at order of d 2 in the expansion of g(x, ρ) and the matter anomaly occurs at order of k in the expansion of g(x, ρ)(φ(x, ρ)) 2 .
In these cases the integration in (3.11) gives us a logarithmic divergence. In order to expand the square root, we use the general formula for expansion of the square root of 10 Notice that our convention for the curvature tensor differs from one given in [15] in a minus sign.
determinant of a typical matrix A = 1 + B det(1 + B) = 1 + 1 2 TrB − 1 4 TrB 2 + 1 8 (TrB) 2 − 1 8 TrB TrB 2 + 1 6 TrB 3 + 1 48 (TrB) 3 + . . .
where B is obtained by
B = ρg (−1) (0) g (1) + . . . + ρ n g (−1) (0) g (n) + ρ α g (−1) (0) g (α) + ρ α+1 g (−1) (0) g (α+1) + . . . . (3.14)
where n is an integer. The number of derivatives and scalars determine that in which order a logarithmic term appears, as we will see in the next sections. However, in above expansion we do not consider a logarithmic term since in the integrand (3.11), only polynomial terms produce the contributions to the anomaly, even if the definition of the metric and scalar components inside (3.14) at the certain orders includes the logarithmic term.
k=1, general d
Here we want to attain the explicit forms of g (α) and g (α+1) by considering the most general covariant combinations with the conformal dimensions 2(α − 1) and 2α, respectively, we need to have two scalar fields in each term, as result one has
g (α) ij = C 1 (φ (0) ) 2 g (0) ij , g (α+1) ij = C 1 L 2 b 1 (φ (0) ) 2 R ij + b 2 g (0) ij (φ (0) ) 2 R + b 3 ∂ i φ (0) ∂ j φ (0) + b 4 ∇ i ∇ j (φ (0) ) 2 + b 5 g (0) ij (φ (0) ) 2 + b 6 g (0) ij ∇ k φ (0) ∇ k φ (0) . (3.15)
Notice that from the holographic principle, the Ricci tensor and the Ricci scalar are defined in terms of boundary value of the metric g (0) . The value of C 1 is specified by solving the equations of motion (see (4.24)). For α = d 2 − 1 the coefficients b 1 , . . . , b 4 are characterized in terms of b 5 and b 6 Because of the existence or the factor of ρ − d 2 in the first term of the lagrangian (3.11), it turns out in order to find anomaly in the case of k = 1 only the order α + 1 in the expansion of the square root of the determinant of the metric g contributes. In this case the explicit form for the expansion of the metric, according to (2.6), is
b 1 = − 1 2 − 1 d + (2b 5 + b 6 )(1 − d 2 ), b 2 = 2 + (1 + b 6 (d − 2))d 4(d − 1)d , b 3 = −2(1 + 1 d ) + 2(1 − d)(2b 5 + b 6 ), b 4 = 1 2 (d − 2)(2b 5 + b 6 ) + 1 .g(x, ρ) = g (0) (x) + g (1) (x)ρ + . . . + g (n) (x)ρ n +ρ α g (α) (x) + g (α+1) (x)ρ + h (α+1) (x)ρ log ρ + . . . , (3.17)
where n is an integer number and α = d 2 − k. Notice that the logarithmic term only appears when α + 1 = d 2 is an integer. The expansion of the scalar field (2.11) that is applicable here reads
Φ(x, ρ) = ρ α/2 φ (0) (x) + φ (1) (x)ρ + φ (2) (x)ρ 2 + . . . +ρ (d−α)/2 φ (d/2−α) (x) + ψ (d/2−α) (x) log ρ + . . . , (3.18)
the logarithmic term is involved only when d − α is even. For the specific value of α given here, it is concluded that we have the logarithmic term at order ρ.
To obtain the general form for the anomaly in the presence of scalar field, as a covariant expression of terms with two derivatives, one requires to find only the order α+1 in the expansion of the square root which contributes in the anomaly, (see (3.11)).
The result for the expansion of the square root of the determinant of the metric (3.3) by using (3.13), that includes the power α + 1 of ρ, is
detg = detg (0) ρ α+1 1 2 (g −1 (0) ) kl g (α+1) kl − 1 2 (g −1 (0) ) kl g (1) jl (g −1 (0) ) jm g (α) km + 1 4 (g −1 (0) ) kl g (1) kl (g −1 (0) ) jm g (α) jm . (3.19)
expression (3.12) and the gravitational contribution to the anomaly is found as
A g = −2 C 1 d 16πG (d+1) N (2 − 4b 1 + d(−1 + 4b 1 + 4(d − 1)b 2 )) 8(d − 1) R(φ (0) ) 2 + 1 2 (b 4 + d b 5 ) (φ (0) ) 2 + 1 2 (b 3 + d b 6 )∇ k φ (0) ∇ k φ (0) ,(3.20)
where indicates the covariant derivatives with respect to g (0) . The second term is a total derivative and is scheme dependent, i.e. it can be removed by adding a covariant counterterm to the on-shell action [15]. The overall factor is ascertained by solving equations of motion directly and is presented in (4.27). Via substitution values b's in (3.16), the ratio of non vanishing terms is
A g = C( d − 2 4(d − 1) R(φ (0) ) 2 + ∇ k φ (0) ∇ k φ (0) ), (3.21)
where C is given by
C = − C 1 d 16πG (d+1) N (b 3 + d b 6 ).
The formula (3.21) is in exact agreement with the result obtained in [24] for the co-
singular for α = d 2 − 1.
The matter contribution to the anomaly, coming from the second term in (3.12), involves φ (1) that can be obtained in the similar fashion by the PBH transformations.
The following expression satisfies the transformation given in (3.9)
φ (1) = L 2 2(−2α − 2 + d) α 2(1 − d) Rφ (0) + φ (0) . (3.22)
The ratio of two coefficients in the brackets for α = d 2 − 1 and after using by part integration to the last terms is the same as the gravitational conformal anomaly in (3.21). However, the above expression is again singular at order which ρ k = ρ. It means the PBH approach is only useful in order to determine the covariant form of the anomaly although it fails to determine the overall factor. We come back to this issue in more details in section 4.3.
Using the fact that the anomaly is conformally covariant [25], i.e., it transforms up to a total derivative, we check that the variation of the expression (3.19) under the PBH transformations must be at most a total derivative. For this purpose, first notice that only the variation of g (1) and g (α+1) are required to be considered while the variation of other elements only produces homogenous terms. The first term in the PBH transformation of g (α+1) in (3.5) is homogenous, also since the covariant derivative of g (0) vanishes the last term is a total derivative, and we do not consider it as it is scheme dependence. After substituting from (3.6) and taking trace, we find
Tr(δg (α+1) ij (x)) = C 1 2 L 2 (− d 2 + 1) σ(φ (0) ) 2 + Homogenous term + Total derivative. (3.23)
Similarly, from the transformation of g (1) in (3.5), we have
Tr(δg (1) ij (x)) = Tr(2∂ (i a k (1) g (α) j)k (x)) = L 2 σ.
k=2, general d
In this case to procure the anomaly for k = 2, i .e., four derivatives, it is necessitated considering the order α + 2 in the expansion of the square root in (3.13). It implies to take terms terms at order α + 2 in (3.11). The explicit expansion of the metric in this case is
g(x, ρ) = g (0) (x) + g (1) (x)ρ + . . . + g (n) (x)ρ n +ρ α g (α) (x) + g (α+1) (x)ρ + g (α+2) (x)ρ 2 + h (α+2) (x)ρ 2 logρ + . . . . (3.25)
To acquire g (α+2) , it is essential to take into consideration all possible covariant terms with the conformal dimension 2(α + 1) and with two free indices; As a result the feasible contributions come from the terms with two scalar fields having the conformal dimension α and four derivatives which is equal to the number of derivatives that emerge in the variation of (3.25). Then by applying the BPH transformations to the covariant expression and comparison with the last term in (3.5), one could find the unknown coefficients in this combination. However, due to the appearance of the trace of g (α+2) in the anomaly which is achieved from the equations (2.13), we do not present the explicit form of it here. Refereing to (3.13) and (3.14) at order α + 2, for finding the formula for the gravitational conformal anomaly with four derivatives, we use the following expression:
detg = detg (0) ρ α+2 1 2 Tr(g −1 (0) g (α+2) ) − 2 4 Tr(g −1 (0) g (1) g −1 (0) g (α+1) ) − 2 4 Tr(g −1 (0) g (2) g −1 (0) g (α) ) + 2 8 Tr(g −1 (0) g (1) )Tr(g −1 (0) g (α+1) ) + 2 8 Tr(g −1 (0) g (2) )Tr(g −1 (0) g (α) ) − 1 8 Tr(g −1 (0) g (1) g −1 (0) g (1) )Tr(g −1 (0) g (α) ) − 2 8 Tr(g −1 (0) g (1) g −1 (0) g (α) )Tr(g −1 (0) g (1) ) + 3 6 Tr(g −1 (0) g (1) g −1 (0) g (1) g −1 (0) g (α) ) + 3 48 Tr(g −1 (0) g (1) )Tr(g −1 (0) g (1) )Tr(g −1 (0) g (α) ) . (3.26)
In the previous section, we saw the conformally invariant form of the scalar field at order that the logarithmic term exists, is the same as one for the conformal anomaly built of the scalar field. We use this fact to find the covariant form for total anomaly.
In the considering case, φ (2) appears in the matter part for computing the anomaly.
To obtain the form of it by applying the PBH transformations, we write down the most general covariant expression consist of four derivatives and one scalar field with undetermined coefficients which are identified by comparing the PBH transformation of φ (2) with (3.9).
φ (2) P BH = p 1 C klmn C klmn φ (0) + L 4 φ (0) − α 2(d − 1) Rφ (0) − α(−2 + d − 2α) (d − 2) 2 R ij R ij φ (0) + α (d 2 (5 + α) + 4(4 + 3α) − 2d(9 + 5α)) 4(d − 2) 2 (d − 1) 2 R 2 φ (0) − (−4(1 + α) + d(2 + α)) (d − 2)(d − 1) R φ (0) + (−2 + d − 4α) 2(d − 1) ∂ i R∂ i φ (0) + 2(−2 + d − 2α) (d − 2) R ij ∇ i ∇ j φ (0) / 8(−4 + d − 2α)(−2 + d − 2α) ,(3.27)
where α is the conformal dimension of φ (0) and the first term involving the Weyl tensor transforms to itself apart from a constant factor. We will see in (4.34) that the Weyl tensor is not required in (3.27), however, as commented in the discussion section, in general there is feasibility to induct it. We expect that above expression is also proportional to the corresponding anomaly (3.12) (see section 4.4).
In fact if we use the Panetiz operator with four derivatives introduced in [24] acting on one scalar field we have the following expression that from the property of this operator is covariant under the PBH transformations
P 2 φ (0) ∼ φ (0) − (d − 4) 1 4(d − 1) Rφ (0) + 1 (d − 2) 2 R ij R ij φ (0) − R 2 φ (0) × (d − 4) 4(d − 1) 2 4 − 3d (d − 2) 2 − d 4 − (8 + (d − 4)d) 2(d − 2)(d − 1) R φ (0) + 4 R ij ∂ i ∂ j φ (0) d − 2 − (d − 6) 2(d − 1) ∂ i R∂ i φ (0) ,(3.28)
one can see that is proportional to the formula (3.27) for α = d 2 − 2.
Since the anomaly is a covariant object under the transformations, we expect (3.28)
is actually part of the anomaly that includes terms with two scalars and four derivatives.
However, it is not the most practicable covariant form. For instance a combination in terms of Weyl tensor and/or the terms with more than two scalar fields may be added, depending on the form of the action (see the discussion section for further illustration).
To show above expression is covariant under transformations given in the formulas (3.29) where the variation of the Christoffel symbol is given by
δ[∇ k 1 ∇ k 2 . . . ∇ kn A i 1 i 2 ...im ] = −δΓ k k 1 k 2 ∇ k ∇ k 3 . . . ∇ kn A i 1 i 2 ...im − . . . −δΓ k k 1 i 1 ∇ k 2 ∇ k 3 . . . ∇ kn A ki 2 ...im − . . . −∇ k 1 [δΓ k k 2 k 3 ∇ k ∇ k 4 . . . ∇ kn A i 1 i 2 ...im + . . .] −∇ k 1 ∇ k 2 . . . ∇ k n−1 [δΓ k kni 1 A ki 2 ...im + . . .] +∇ k 1 ∇ k 2 . . . ∇ kn [δA i 1 i 2 ...im ],δΓ k ij = ∂ i σδ k j + ∂ j σδ k i − ∂ k σg (0) ij . (3.30)
To corroborate that (3.28)is covariant under the transformation, since the transformation of the inverse of the boundary metric and the boundary value of the scalar field is homogeneous therefore we simply do not need to consider them. For instance more with regarding (3.29) and (3.30) for a scalar field φ we have
δ φ → δφ − (2 − d)∇ i σ∇ i φ.
However, When we encounter the variation of derivatives acting on a tensor, one must consider contributions of transformations of the homogenous parts or the tensor, in particular the homogenous part of transformation for the Ricci scalar (3.7).
Nonetheless it is realized that the PBH transformations do not give us all participation in the anomaly such as contributions with different number of scalars as we encounter in section 4.5. Furthermore these transformations may not fix all the coefficients in the covariant expression. We demonstrate the formula for the anomaly for specific and general dimensions through solving the equations of motion in the next section, that is, we reproduce all the coefficients of the covariant expression for the anomaly that was given by the PBH transformations as well as the overall factor.
Anomaly through solving the back reacted gravitational equations of motion
The conformal dimension of the dual operator is determined by considering that the following expression is covariant under the conformal transformations [24] √ gφ n P k φ n .
For instance, for k = 1,
P 1 = − + d − 2 4(d − 1) R,(4.1)
and P 2 after acting on a scalar was given in (3.28).
Where P k and n scalar fields transform as 11
P k → e (− n 2 −k)σ P k e ( n 2 −k)σ , φ n → e −αnσ φ n ,(4.2)
and by considering the transformation √ g, i.e., e dσ , it turns out that for the dimension d and 2n scalar fields, one has
α = d − 2k 2n . (4.3) 11
Remember that through turning to account conformal invariance to the action we obtain specific values of α to be d 2 − k where k indicates the number of square derivatives.
Therefore, the result for the anomaly for a general dimension is presented in different categories that are characterized by the choice of the number of scalar fields 2n and of derivatives k. However, the anomaly formula for a specific dimension may involve all possible number of scalar fields and their derivatives as well as the Ricci tensor such that each individual term has the desired conformal dimensions. In this section, first we follow the procedure given in section 2 to find the conformal anomaly in the case of n = 1, first for the specific spacetime dimensions d = 4, 6 and the number of derivatives specified by k = 1, 2, we generalize this approach to achieve the anomaly for a general dimension and k = 1, 2. Then in the last three sections we explore the result in the case of n = 2.
k=2, d=4
For a dynamical AdS background, i.e., the gravity coupled to scalar, the gravitational field equations are given in (2.13). The approach is the same as the pure gravity, it means we solve the equations of motion by applying the CFT data. For illustration, we reveal the formula of the conformal anomaly for d = 4 and the massless scalar field, therefore ∆ = 4 and α = 0. The explicit form of the expansions of the scalar field and metric that construct the anomaly with four derivatives and with two scalar fields only involves integer powers of ρ that are as follows:
φ(x, ρ) = φ (0) + ρφ (1) + ρ 2 φ (2) + ρ 2 log ρψ (2) , g(x, ρ) = g (0) + ρg (1) + ρ 2 g (2) + ρ 2 log ρh (2) . (4.4)
The recursion relation for g (1) is determined by zero order of the perturbative expansion of the first equation in (2.13)
g (1) ij = 1 d − 2 −Tr(g (1) )g (0) ij − R ij + 8πG N ∂ i φ (0) ∂ j φ (0) ,(4.5)
where
Trg (1) = −R + 8πG N ∂ k φ (0) ∂ k φ (0) 2(d − 1) . (4.6)
From the third equation in the leading order, one gets
Trg (2) = 1 4 Tr(g (−1) (0) g (1) g (−1) (0) g (1) ) − 8πG N (φ (1) ) 2 . (4.7)
scalar field because m 2 is zero only the gravitational constituent of the anomaly in (3.12) prevails. The anomaly is achievable from (3.13) in which only the first two terms of B exist in (3.14) 12
A (α=0) = − 2d 16πG N 1 2 Trg (2) − 1 4 Tr(g (1) g (1) ) + 1 8 (Trg (1) ) 2 ,(4.8)
where we have used the following expression from the solution of the scalar equation (2.10) that is linear in the scalar field at the leading order
φ (1) = 1 4 φ (0) ,(4.9)
and g (1) is defined according to (4.5).
The second equation in (2.13) provides nonlinear relations between different components of the metric and the scalar field, e.g., the trace of g (1) gives a relation between its divergence and φ (1) as
∇ i (T rg (1) ) − ∇ j g (1) ij = −16πG N φ (1) ∂ i φ (1) . (4.10)
In conclusion, the formula for the anomaly after substitution (4.7) in (4.8) is
A (α=0) = − −2 384Gπ − R 2 + 3R ij R ij + 2(8πG N ∂ i φ (0) ∂ i φ (0) ) 2 +8πG N − 6R ij ∂ i φ (0) ∂ j φ (0) + 2 R ∂ i φ (0) ∂ i φ (0) + 3 (φ (0) ) 2 , (4.11)
where one can see the pure gravitational part has the same ratio of coefficients given in [14]. In other words, when φ (0) is constant, our result for anomaly is the same as the conclusion in [14] for the coefficient of the log in the on-shell action
a (2) = − 1 8πG N (E (4) + I (4) ),
where E (4) and I (4) are 4 dimensional Euler density and a conformal invariant, respectively that are introduced in [14]. For the rest of terms, it is easy to see that the above formula for the anomaly turns into (3.28) by virtue of Bianchi identity and suppressing total derivatives by choice of the scheme through adding local finite counterterms, that is implied the following relations:
R ij ∂ i φ (0) ∂ j φ (0) → 1 4 R (φ (0) ) 2 − R ij φ (0) ∇ i ∇ j φ (0) , (4.12) R∂ i φ (0) ∂ i φ (0) → 1 2 R (φ (0) ) 2 − Rφ (0) φ (0) .
Also the anomaly receives a contribution of four scalar fields with derivatives anting on in the last term of the first line in (4.11) that is covariant under the transformations, since in the four dimensional spacetime, the scalar field does not transform. Therefore, the formula (3.28) should be modified for specific values of dimension. Though it is matched with the remaining terms in (4.11), in particular, it does not involve R 2 (φ (0) ) 2 .
Note that we only get the terms with derivatives acting on the scalars and not some power of the scalar field.
k=1, d=6
In this section, we express an obstacle for finding the anomaly and present a way to handle it. In general the form of the expansion of the metric depends on dimension, however, for the scalar field for a specific k, for any dimension, it remains unchanged.
In the six dimensional spacetime, the expansions of the metric and the scalar field are as follows:
g(x, ρ) = g (0) + ρ g (1) + ρ 2 g (2) + ρ 3 g (3) + ρ 3 log ρh (3) + . . . , φ(x, ρ) = φ (0) + ρ φ (1) + ρ log ρψ (1) + ρ 2 φ (2) + . . . .
In this case, by considering the first equation at the leading order we get the same result for g (1) in (3.10). However, the expression for g (2) resulting from the first order, must be modified as follows for d = 6 and α = 2:
g (2) ij = − 1 2(−4 + d) R (1) ij + 2g (1) ik g (1)k j − Tr(g (1) g (1) ) − 2Trg (2) g (0) ij + 16(d − 2)Gπ d − 1 φ (0) 2 g (0) ij , (4.13)
where R (1) ij is the first order term in the expansion of the Ricci tensor
R (1) ij = − 1 2 g (1) ij + 1 2 ∂ k ∂ i g (1) jk + 1 2 ∂ k ∂ j g (1) ik − 1 2 ∂ i ∂ j Trg (1) = L 2 4(d − 2)(d − 1) 2(d − 1)∂ k ∂ k R ij − (d − 2)∂ i ∂ j R + 4(d − 1)(R ik R j k −R kl R likj ) − ∂ k ∂ k Rg (0) ij .
(4.14)
In fact, one can see the result in (4.13) apart from the last term, by virtue Bianchi identity, is the same as the second formula in (3.10). The Ricci scalar is zero at this order. Also Trg (2) that is another element for constructing of the anomaly is procured directly from the third equation at the leading order as
Trg (2) = 1 4 Tr(g (1) g (1) ) − 4πdG N (d − 1) φ (0) 2 . (4.15)
By referring to the formula (3.13) one needs to find Trg (3) as well that is achieved from the third equation at order ρ as
Trg (3) = 1 6 − Tr(g (1) g (1) g (1) ) + 4Tr(g (1) g (2) ) − 5Trh (3) + 8πG N − 4φ (0) ψ (1) + 2(2 − 3d) d − 1 φ (0) φ (1) , (4.16)
however, the explicit form of g (3) is not granted by means of solving equations of motion.
In the absence of the scalar field, h ( d 2 ) is traceless and covariantly conserved. However, the existence of relevant deformations at the same order, gives us
Trh (3) = − (−2 + 3d)8πG N 3(d − 1) φ (0) ψ (1) . (4.17)
The form of ψ (1) is acquired by taking the scalar equation into consideration
ψ (1) = − 1 4 αTrg (1) φ (0) + φ (0) . (4.18)
According to (3.13), the formula for the gravitational part of the anomaly is given by
A (α=2) g = − 6 16πG N 1 2 Trg (3) − 1 2 Tr(g (1) g (2) ) + 1 4 Trg (1) Trg (2) − 1 8 Trg (1) Tr(g (1) g (1) ) + 1 6 Tr(g (1) g (1) g (1) ) + 1 48 Trg (1) 3 . (4.19)
Now after exerting (4.16) with usage relations (4.13) and (4.15), we obtain
A (α=2) g = −2d 768(d − 4)(d − 2)(d − 1)πG N − R Trg (1) + (d − 2)(d − 1) (d − 4)
(Trg (1) ) 3 + (10 − 3d)Trg (1) Tr(g (1) g (1) ) + 4(d − 4)Tr(g (1) g (1) g (1) ) + g ij
(1) 2(d − 1) R ij − (d − 2)∂ i ∂ j R + 4(d − 1) 2(d − 2)g (1) k i g (1) kj − R k i R kj +R kilj R kl − 5d 192πG N Trh (3) − d 6 φ (0) ψ (1) − d(−2 + 3d) 48(d − 1) Trg (1) (φ (0) ) 2 +4φ (0) φ (1) . (4.20)
However, the relation (4.20) is not a combination resembling (3.21) at the part involving the scalar field. In particular, the explicit form of φ (1) is not determined from the equations because the coefficient of this term at the first order of the scalar equation (2.10) vanishes and at the next, order there is a relation between φ (1) and higher order field, φ (2) . In other words, φ (1) as a nonlocal field is not ascertained via information at the boundary, i.e., g (0) and φ (0) . However, there is another contribution to the anomaly that comes from the mass term in the action [15] that was not involved in the previous section due to the massless scalar. In the expansion of √ g (φ(x, ρ)) 2 in (3.11), only the terms at order one take part in the matter anomaly, and the result is
A M = −2α(d − α) 2(d − 1) 2φ (0) φ (1) + 1 2 Tr(g (1) )(φ (0) ) 2 . (4.21)
In fact, the total anomaly is given by adding (4.20) and (4.21) involve terms with scalars that are proportional to ψ (1) which is covariant under transformations, so the covariance under transformations is recovered. The total anomaly for d = 6 is given by
A (α=2) t = −2 6 × 2 10 (d − 4)(d − 1) 2 (d − 2) 2 πG N 4d (d − 1) d R R + (2 + 3d)R ij R ij R − 4(d − 1) R ij R ij + 2R ij R kl R ikjl − (2 + d)d 2 R 3 − 1 288(d − 1) 2 × − 4 36 + d(−19 + 3d) R(φ (0) ) 2 + d(d − 1)(2 + 3d)φ (0) φ (0) . (4.22)
The first two lines reproduce the results given in [14] 13 in the absence of the scalar field. Also the ratio of terms with two scalars is exactly the same as what we expect in (3.21) for the six dimensional spacetime. 13 beside a minus sign in our convention for the definition of the Ricci tensor and by virtue Bianchi identity to the first term.
k=1, general d
We now proceed to obtain the solution for a general dimension while the relation (4.3) is satisfied, it means we set a constraint on the mass of the scalar field, on other words, for other values of the mass basically there is no anomaly involving the scalar field, therefore we are only interested in the solution in which the anomaly appears. In order to find the anomaly in the presence of scalar field for a general dimension and of only terms with two derivatives, we follow the procedure similar to the previous section. The formula given in (3.15) by using the PBH transformations is singular for α = d 2 − 1, so we employ the equations of motion to attain the anomaly. Here we suppose that α > 1 (or d > 4), therefore the formula for g (1) given in (3.10) is applicable to this case. 14 Via making approach to the formula (3.13), one demands to solve the third equation at order α − 1 to find
Trg (α+1) = − 1 α(1 + α)(d − 1) (d − 1) (1 + 2α)Trh (α+1) − C 1 α 2 Trg (1) (φ (0) ) 2 +8πG d+1 N α (−2 + d + dα)φ (0) φ (1) + 2(d − 1)φ (0) ψ (1) ,(4.23)
where C 1 is obtained by setting (3.15) in the third equation and considering the order
α − 2 C 1 = − 4πG d+1 N d − 1 ,(4.24)
also the coefficient of the logarithmic term at this order is
Trh (α+1) = − 8πG d+1 N (−2 + d + dα)φ (0) ψ (1) (1 + α)(d − 1) ,(4.25)
where ψ (1) is the same as before (4.18). Indeed, this form of ψ (1) is universal and is applicable to any specific dimension with the same number of derivatives and of scalar fields in the anomaly. However, g (α+1) is not obtainable through solving the equations of motion since at this order its coefficient vanishes. We find this component for a different value of α in the next section.
The gravitational part of the anomaly by noticing the formula (3.19) and (4.23) is 14 Although in the next section we refer to the formula in this section for α > 2.
specified as 26) and the matter anomaly is the same as (4.21). In conclusion, only a combination proportional to ψ (1) remains in the total anomaly, see (4.25). After substitution of the trace of g (1) in (3.10) and the expressions (4.23), (4.25) and removing the total derivative, for α = d 2 − 1, we attain
A (α= d 2 −1) g = − −2d 16πG d+1 N (1 + 2α) 2α(1 + α) Trh (α+1) + πG N (−2 + d + dα) (d − 1)(1 + α) Trg (1) (φ (0) ) 2 +4φ (0) φ (1) + 8πG d+1 N 1 + α φ (0) ψ (1) ,(4.A (α= d 2 −1) t = −2 2d (d − 2) 4(d − 1) R(φ (0) ) 2 + ∂ k φ (0) ∂ k φ (0) ,(4.27)
and therefore the anomaly maintains the covariant description in (3.21).
k=2, general d
In this section, we aim to find the anomaly which has the structure of four derivatives and of two scalar fields, for a general dimension, through solving the equations of motion. For this purpose similar to before we solve the coupled system of the first and third equations in (2.13) by inserting the expansions (3.25) and the first term in (4.4) into the equations that is straightforward but somewhat tedious. Again note that this progress breaks down at α = d 2 − 2 and the coefficient of φ 2 vanishes, therefore a logarithmic term is required to get the full solution of the back-reaction of the scalars to the bulk geometry. From the formula (3.26), we need to pursue different components in order to construct the anomaly. The explicit form of g (α+1) is determined via the first equation in (2.13) at order α
g (α+1) ij = 1 (d − 1)(1 + α)(2 − d + 2α) (d − 1) R (α) ij + (1 + α)Trg (α+1) g (0) ij +8πG d+1 N (−2 + d − α)αg (1) ij (φ (0) ) 2 + αg (0) ij Trg (1) (φ (0) ) 2 + 2(d − α)φ (0) φ (1) −(d − 1)∂ i φ (0) ∂ j φ (0) ,(4.28)
where the expansion of the Ricci tensor at this order is
R (α) ij = − 1 2 ∂ k ∂ k g (α) ij + 1 2 ∂ k ∂ i g (α) jk + 1 2 ∂ k ∂ j g (α) ik − 1 2 ∂ i ∂ j Trg (α) = C 1 2 − g (0) ij ∂ k ∂ k (φ (0) ) 2 + (2 − d)∂ i ∂ j (φ (0) ) 2 ,(4.29)
where it is related to the second derivative of the scalar fields, as expected because only g (α) gets contribution at this order. The Ricci scalar reads
R (α) = (1 − d)C 1 ∂ k ∂ k (φ (0) ) 2 . (4.30)
The trace of g (α+1) is the same as (4.23) if we remove the first and last terms. At the same order of the third equation, for given α, by considering the cyclic property of the trace, we have a factor of two for the terms including four components of the metric, as a result, we have
Trg (α+2) = 1 (d − 1)(1 + α)(2 + α) (d − 1) Tr(g (1) g (α+1) )(1 + α) 2 − Trh (α+2) (3 + 2α) +4πG d+1 N (1 + α + α 2 )Tr(g (1) g (1) )(φ (0) ) 2 − (2 + α + α 2 )Trg (2) (φ (0) ) 2 +4(1 + α) − d 4 + α(3 + α) (φ (1) ) 2 − 2α − 4 + d(3 + α) φ (0) φ (2) +2(d − 1)φ (0) ψ (2) , (4.31)
where we have inserted the components of the inverse of the metric, for instance, we
have G (−1)ij (α) = −g ik (0) g (α) kl g lj (0) . (4.32)
In general the explicit forms of g (1) ij and g (2) ij that are given by the first equation in (2.13), depending on the value of α, can receive extra terms coming from the first term in rhs of this equation, however, for α > 2 the formulas resulted from the PBH transformation given in (3.10) are applicable.
Similar to before, from the third equation (2.13), the coefficient of ρ α log ρ indicates
Trh (α+2) = − 8πG d+1 N α(−4 + 3d + dα) (d − 1)(1 + α)(2 + α) φ (0) ψ (2) . (4.33)
Also at order one of the scalar equation (2.10), one gets
φ (2) = 1 8(d − 2α − 4) − 2g kl (1) ∂ k ∂ l φ (0) + 2 φ (1) + (2α + 4)Trg (1) φ (1) + ∂ k Trg (1) ∂ k φ (0) −2∂ k g (1)
kl ∂ l φ (0) − 2αφ (0) Tr(g (1) g (1) ) − 2Trg (2) . (4.34)
Albeit above expression is reaped without considering the logarithmic term and it is not valid for the chosen value of α in this section, because the expansion at order two is truncated. However, it turns out that this expression is proportional to our result given by the PBH transformations (3.27) Since φ (2) for α = d 2 − 2 is singular, it is essential to add a logarithmic term at the corresponding order, and the corresponding coefficient reads as
ψ (2) = − (d − 2α − 4) 2 φ (2) . (4.35)
In fact we will see this covariant combination is also the possible combination that constructs the anomaly. The explicit form of ψ (2) in terms of curvature and scalar field and its derivatives after substitution of the expressions for g (1) ij and g (2) ij in (4.35), is revealed as
ψ (2) = − 1 32 φ (0) − 8 + (d − 4)d 2(d − 2)(d − 1) R φ (0) − (d − 4) 4(d − 1) Rφ (0) + R 2 φ (0) × (d − 4) − 16 + d 16 + (d − 4)d 16(d − 2) 2 (d − 1) 2 − (d − 4) (d − 2) 2 R kl R kl φ (0) + 4 d − 2 R kl ∂ k ∂ l φ (0) − (d − 6) 2(d − 1) ∂ k R∂ k φ (0) (4.36)
Notice that φ (2) was canceled in above expression.
The anomaly pertaining to the gravitational part of the action is given by considering the first part of (3.12) and the formula (3.26)
A (α= d 2 −2) g = −2d 16πG d+1 N 1 2
Trg (α+2) − Tr(g (1) g (α+1) ) − Tr(g (2) g (α) ) + Tr(g (1) g (1) g (α) )
+ 1 4 Trg (1) Trg (α+1) + Trg (2) Trg (α) − Tr(g (1) g (α) )Trg (1) − 1 8 Tr(g (1) g (1) )Trg (α) + 3 48 Trg (1) 2 Trg (α) ,
where we have applied the relation (4.28).
The matter anomaly in this case by referring to the second part of (3.11) becomes
A (α= d 2 −2) M = −2α(d − α) 2(d − 1) 2φ (0) φ (2) + φ (1) φ (1) + 1 2 Trg (2) (φ (0) ) 2 + Trg (1) φ (0) φ (1) − 1 4 Tr(g (1) g (1) )(φ (0) ) 2 + 1 8 Trg (1) 2 (φ (0) ) 2 .
Adding these two parts gives us the total anomaly
A (α= d 2 −2) t = − −2 8(d − 2) 8(d − 4)φ (0) ψ (2) + (d − 1) πG d+1 N Trh (α+2) + ( φ (0) ) 2 + 2(1 + α) Trg (1) φ (0) φ (0) + α(2 + α) Trg (1) 2 (φ (0) ) 2 + 4g (1) ij ∂ i φ (0) ∂ j φ (0) − R (α) ij g ij (1) 2πG d+1 N − (d − 2) 2 d − 1 Tr(g (1) g (1) )(φ (0) ) 2 + 4d d − 1 Trg (2) (φ (0) ) 2 − g (1) ij ∂ i φ (0) ∂ j φ (0) 2(d − 2) .
(4.37)
Similar to the previous section the contribution of φ (2) in the gravitational part of the anomaly, which is not determined through solving the equations of motion, is canceled by the corresponding term in the matter anomaly for α = d/2 − 2. Since Trh (α+2) is proportional to ψ (see (4.33)) and is already in the covariant form. We only need to clear that the rest of the terms in above expression are in the same covariant combination. We call the the collection of terms in the square brackets apart from the first two terms as T . After substitution different components and using by part integration, we have
T = − −2 8(d − 2) ( φ (0) ) 2 − (1 + α) d − 1 Rφ (0) φ (0) − (d − 4) (d − 2) 2 R ij R ij (φ (0) ) 2 + R 2 (φ (0) ) 2 d 2 3 + α(2 + α) + 4 4 + α(2 + α) (1 − d) 4(d − 2) 2 (d − 1) 2 + 2 (d − 2)(d − 1) R ∂ k φ (0) ∂ k φ (0) + 1 d − 1 R ij ∂ i ∂ j (φ (0) ) 2 − 4 d − 2 R ij ∂ i φ (0) ∂ j φ (0) ,
where it is manifestly covariant under the PBH transformations, because it is proportional to the definition of ψ (2) for α = d/2 − 2, i.e.,
T = − 8 (d − 2) φ (0) ψ (2) .
Finally from the formula (4.37), the total conformal anomaly becomes
A t = − 16 d φ (0) ψ (2) ,
where ψ (2) is given in (4.36). This is the anomaly formula for the scalar field which depends on the coordinates of the boundary spacetime.
n=2, k=1, general d
In this section, we aim to find the anomaly for four scalars with two derivatives. In the case of k = 1 and n = 2, the value of α is specified by considering (4.3) as α = (d−2)/4.
The applicable expansions of the scalar field and of the metric are as follows:
φ (α= d 4 − 1 2 ) = φ (0) + ρ φ (1) + . . . + ρ α φ (α) + ρφ (α+1) + ρ log ρψ (α+1) + . . . + . . . , g (α= d 4 − 1 2 ) = g (0) ij + ρ g (1) ij + . . . + ρ α g (α) ij + ρg (α+1) ij + . . . + ρ 2α g (2α) + ρg (2α+1) + ρ log ρh (2α+1) ij + . . . . (4.38)
Since the action (2.12) is invariant under Φ → −Φ, the components at orders with odd coefficients of α 2 vanish in above expansions.
Here one needs to explore the different elements to build the anomaly (3.13). From the third equation at orders α − 2, α − 1 and 2α − 2, we get, respectively (4.39) and finally at order 2α − 1, we obtain
Trg (α) = − 4dπG d+1 N d − 1 (φ (0) ) 2 ,Trg (α+1) = α (1 + α) Tr(g (1) g (α) ) − 8πG d+1 N (−2 + d + dα) (d − 1)(1 + α) φ (0) φ (1) ,Trg (2α) = (−2 + 3α) 4(−1 + 2α) Tr(g (α) g (α) ) + 4πG d+1 N (d + 2α − 3dα) (d − 1)(−1 + 2α) φ (0) φ (α) ,Trg (2α+1) = 1 2α(d − 1)(1 + 2α) (d − 1) − (1 + 4α)Trh (2α+1) + 4α 2 Tr(g (1) g (2α) ) +α − 3αTr(g (1) g (α) g (α) ) + (1 + 3α)Tr(g (α) g (α+1) ) − 8πG d+1 N α − 2(3 + α) + d(5 + 3α) φ (1) φ (α) + (−2 + d − 2α + 3dα)φ (0) φ (α+1) +2(d − 1)φ (0) ψ (α+1) ,(4.40)
where we used the component of the inverse metric given in (4.32) and also
G (−1)ij (α+1) = −g (0)ik g (α+1) kl g (0)lj + g (0)im g (1) mn g (0)nk g (α) kl g (0)lj + g (0)ik g (α)
kl g (0)lm g (1) mn g (0)nj .
Furthermore at this order, the coefficient of log ρ gives us
Trh (2α+1) = − 4πG d+1 N d + 3dα − 2(1 + α) (d − 1)(1 + 2α) φ (0) ψ (α+1) . (4.41)
From the scalar equation at orders α − 1 and α, one gets
φ (α) = α 2(d − 4α) Trg (α) φ (0) , (4.42) ψ (α+1) = − 1 16(2 + d) 16 φ (α) + (d 2 − 4) Trg (α+1) φ (0) − Tr(g (1) g (α) )φ (0) + (d − 2)(d + 6)Trg (α) φ (1) + 12d Trg (1) φ (α) − 8 3Trg (1) φ (α) − ∂ i φ (0) ∂ i Trg (α) + 2∂ i g ij (α) ∂ j φ (0) + 2∂ i ∂ j φ (0) g ij (α)
, and after simplification, we find
ψ (α+1) = 1 16(d − 1) 2 (d + 2) − 16(d − 1) 2 ∂ k ∂ k φ (α) + πG d+1 N (d − 2) 4(d − 1)(16 + 3d) φ (0) (φ (0) ) 2 − (d − 2)(8 + 3d)R(φ (0) ) 3 + 32(d − 1)∂ k (φ (0) ) 2 ∂ k φ (0) , (4.43)
where the first term of the above expression by using the the first formula in (4.42) is a total derivative, so it is removed by adding the appropriate contact term to the action.
By using by part integration, we get
ψ (α+1) = (d − 2)(8 + 3d)πG d+1 N 16(d − 1) 2 (d + 2) (φ (0) ) 2 4(d − 1) φ (0) − (d − 2)R φ (0) . (4.44)
It is easy to see that the ratio of the coefficients is the same as what we expect from (3.21). As a next step, from the first equation at orders α − 1 and α, one has
g (α) ij = − 1 d − 2α g (0) ij Trg (α) + 8πG d+1 N (−d + α) (d − 1)(d − 2α) g (0) ij (φ (0) ) 2 , (4.45) g (α+1) ij = 1 (d − 1)(1 + α)(2 − d + 2α) (d − 1) R (α) ij + 2αg (1) k j g (α) ik + 2αg (1) k i g (α) jk + (1 − α)g (α)
ij Tr g (1) + (d − 1)(1 + α) Tr g (α+1) − Tr(g (1) g (α) ) g
(0) ij + 8πG d+1 N (d − α)αφ (0) (g (1) ij φ (0) + 2g (0) ij φ (1) ) − (d − 1)∂ i φ (0) ∂ j φ (0) .(4.46)
In above the formulas for g (1) and φ (1) The last required element, coming from the order 2α − 1, is
g (2α) ij = 1 2(d − 1)(d − 4α) (d − 1) − 2αg (α)k i g (α) jk + (−1 + α)g (α) ij Trg (α) + g (0) ij − 2Trg (2α) +Tr(g (α) g (α) ) + 8πG d+1 N (−d + α) (φ (0) ) 2 g (α) ij + 2g (0) ij φ (0) φ (α) ,(4.47)
where we have used the fact that the Ricci tensor does not receive any component at this order.
The expressions for two parts of the anomaly are
A (α= d 4 − 1 2 ) g = −2d 16πG d+1 N 1 2 Trg (2α+1) − 1 4 2Tr(g (1) g (2α) ) + 2Tr(g (α) g (α+1) ) + 1 8 2Trg (1) Trg (2α) + 2Trg (α) Trg (α+1) − 1 8 2Trg (α) Tr(g (1) g (α) ) + Trg (1) Tr(g (α) g (α) ) + 1 6 3Tr(g (1) g (α) g (α) ) + 1 48 3Trg (1) Trg (α) 2 , (4.48) A (α= d 4 − 1 2 ) M = −2α(d − α) 2(d − 1) 2φ (1) φ (α) + 2φ (0) φ (α+1) + 1 2 2Trg (1) φ (0) φ (α) + 2φ (0) φ (1) Trg (α) + Trg (α+1) (φ (0) ) 2 − 1 4 2Tr(g (1) g (α) )(φ (0) ) 2 + 1 8 2(φ (0) ) 2 Trg (1) Trg (α) . (4.49)
After inserting all the corresponding metric and scalar components, the value of the total anomaly with one derivative and four scalar fields becomes
A (α= d 4 − 1 2 ) t = − −2d 4(1 + 2α) φ (0) ψ (α+1) + (1 + 4α) 16πG d+1 N α Trh (2α+1) = −2(d + 2) 4d φ (0) ψ (α+1) .
Note that in the first equality, the contributions only come from Trh (2α+1) and ψ (α+1) and other terms have canceled each other. In the last equality, we have used the relation (4.41). The formula for ψ (α+1) is given in (4.44).
n=2, k=0, general d
Here we wish to determine the anomaly in the case of four powers of the scalar field.
Therefore, from the formula (4.3), one has α = d/4. The related expansions are
g (α= d 4 ) = g (0) + ρ g (1) + . . . + ρ α g (α) + ρg (α+1) + . . . + ρ 2α g (2α) + log ρh (2α) + . . . + . . . φ (α= d 4 ) = φ (0) + ρ φ (1) + . . . + ρ α φ (α) + log ρψ (α) + . . . + . . . (4.50)
Considering the third equation, the formula for g (α) remains the same as (4.45). At order α − 2, it is manifested that
Trg (2α) = 1 4(d − 1)α(−1 + 2α) (d − 1) α(−2 + 3α)Tr(g (α) g (α) ) − 2(−1 + 4α)Trh (2α) +16πG d+1 N α (d + 2α − 3dα)φ (0) φ (α) − 2(d − 1)φ (0) ψ (α) ,(4.51)
and the coefficient of the logarithmic term induces
Trh ( d 2 ) = 4πG d+1 N (d + 2α − 3dα) (d − 1)(−1 + 2α) φ (0) ψ (α) ,(4.52)
where ψ (α) is obtained from the scalar equation at order α − 1
ψ (α) = − d 16 Trg (α) φ (0) . (4.53)
The results for the contributions to the anomaly are
A (α= d 4 ) g = −2d 16πG d+1 N 1 2 Trg (2α) − 1 4 Tr(g (α) g (α) ) + 1 8 Trg (α) 2 , A (α= d 4 ) M = −2α(d − α) 2(d − 1) 2φ (0) φ (α) + 1 2 Trg (α) (φ (0) ) 2 ,(4.54)
and the total anomaly has the following structure:
A (α= d 4 ) t = − −2 16(d − 2)(d − 1)πG d+1 N 2(d − 1) 2 Trh (2α) + 8d(d − 1)πG d+1 N φ (0) ψ (α) +(πG d+1 N ) 2 d 3 (φ (0) ) 4 . (4.55)
However, by using the formulas given in (4.52) and (4.53), in this case, the total anomaly becomes zero. That is, there is not a possibility of receiving a contribution of the four powers of the scalar field in the anomaly.
n=3, k=0, general d
We reiterate the discussion in the previous section, in the case of six powers of the scalar field to realize whether there is a contribution to the anomaly for α = d/6. The corresponding expansions are g (α= d 6 ) = g (0) + ρ g (1) + . . . + ρ α g (α) + ρ 2α g (2α) + ρ 3α g (3α) + ρ 3α log ρh (3α) + . . . , φ (α= d 6 ) = φ (0) + ρ φ (1) + . . . + ρ α φ (α) + ρ 2α φ (2α) + ρ 2α log ρψ (2α) + . . . . Trg (3α) = 1 3(d − 1)α(−1 + 3α) (d − 1)α(−3 + 7α)Tr(g (2α) g (α) ) + α(−1 + d + 2α − 2dα)
Tr(g (α) g (α) g (α) ) + (−1 + d + 6α − 6dα)Trh (3α) + 4πG d+1 N α (d + 8α − 9dα)
(φ (α) ) 2 + 2 (d + 4α − 5dα)φ (0) φ (2α) − 2(d − 1)φ (0) ψ (2α) .
(4.57)
Also at the same order, by considering the logarithmic term, we have
Trh (3α) = 8πG d+1 N (d + 4α − 5dα) 3(d − 1)(−1 + 3α) φ (0) ψ (2α) ,(4.58)
where ψ (2α) according to the order 2α − 1 of the scalar equation is
ψ (2α) = d 48 − 2Trg (2α) φ (0) + Tr(g (α) g (α) )φ (0) − 3Trg (α) φ (α) . (4.59)
The contributions to the anomaly are coming from
A (α= d 6 ) g = −2d 16πG d+1 N 1 2 Trg (3α) − 1 4 2Tr(g (α) g (2α) ) + 1 8 2Trg (α) Trg (2α) − 1 8
Trg (α) Tr(g (α) g (α) ) + 1 6 Tr(g (α) g (α) g (α) ) + 1 48 Trg (α) 3 ,
A (α= d 6 ) M = −2α(d − α) 2(d − 1) 2φ (0) φ (2α) + (φ (α) ) 2 + 1 2 Trg (2α) (φ (0) ) 2 + 2Trg (α) φ (0) φ (α) − 1 4
Tr(g (α) g (α) )(φ (0) ) 2 + 1 8 Trg (α) 2 (φ (0) ) 2 , (4.60) and the total anomaly becomes
A (α= d 6 ) t = − −2d 3(d − 2) φ (0) ψ (2α) + 3(d − 1) 8πG d+1 N d Trh (3α) − (πG d+1 N ) 2 d 3 3(d − 1) 2 (φ (0) ) 6 . (4.61)
The same as before, we see that the anomaly after inserting of the formulas given in (4.58) and (4.59) vanishes.
Discussion
We have used the AdS/CFT correspondence that prescribes a duality between bulk/boundary observables in order to construct iteratively the bulk solutions from CFT data. We did not have any assumption except that the CFT has an AdS dual.
We then proceeded to investigate logarithmically divergent part of the asymptotic solution to find the conformal anomaly. All infrared divergences of the bulk on-shell action can be obtained in terms of boundary value of fields. Our discussion for scalar fields can be straightforwardly generalized to other kinds of matter [26,27].
However, in general, a knowledge of sources in the CFT is not sufficient to construct the complete bulk metric-scalar solution. By additional information from the expectation value of the dual operators, the undetermined non-local coefficients in each case are obtainable. On the other hand we expect by using the information from the trace and divergence of g (d) one can find the explicit form which satisfies these constraints.
Therefore, this way one could obtain a quantity that is directly related to expectation value of the corresponding boundary stress-energy tensor in any dimension d. Also the solution to the deep interior region is accessible by extra CFT data (for instance non-local observables such as Wilson loop, Wilson surfaces, etc).
In our discussion we supposed certain covariant terms appeared in the anomaly depending on the relevant deformation exist in the boundary theory. To solve the equations of motion we only discussed a specific action combining of (2.1) and (2.7).
For a general gravitational action which involves a function of the curvature and its covariant derivatives with the possible coupling to any number of matter fields and their derivatives such that d + 1 diffeomorphism leaves it unchanged, a Weyl transformation of the on-shell action gives the anomaly. There are two independent types of trace anomalies that are related to the terms that are present in the effective action. The type A anomaly which has the Euler density structure in the desired dimension and the type B anomalies that are an Weyl invariant expression constructed of the contraction of the Weyl tensors or covariant derivatives of the Weyl tensor [13]. In our calculation we get the form of the type A anomaly for a specific choice of dimensions related to the pure gravity part. From the PBH transformation point of view the type B anomalies are not universal in the sense they appear with arbitrary coefficients. However, these coefficients are fixed by using the equations of motion pertaining to the chosen form of the action.
Throughout this paper we only take the Einstein gravity action into consideration.
If higher derivative corrections are included in the action such as contractions of Weyl Tensors and combinations of curvature tensors like R n [25,28,29], we anticipate that the result for the anomaly will be modified. That is, while the general structure will be the same, the coefficients will have new values depending on the extra gravitational couplings. However, with existence higher derivative corrections in the action, in order to figure out all the coefficients in the expansions of the metric and scalar field, one needs information of the higher point functions of the stress-energy tensor as well.
In addition in our calculations we found that with a free bulk scalar, the anomaly vanishes for some power of the scalar field with n > 2. It is expected by introducing a potential with higher order interactions for the bulk scalar in the action, we will get non-zero contributions of this form to the anomaly. Another extension of the present work would be studying renormalization group flows that are produced by adding a potential for the bulk scalar. One can use the present approach to extend the related discussion to any dimension d. However, we are not yet able to relate our conclusions to the properties of the conformal field theories in various dimensions.
to figure out the infrared divergences of both gravitational and matter sectors. Then in subsections 4.3 and 4.4 we generalize this approach to a general dimension and compute the conformal anomaly, in terms of d with given boundary conditions, as local polynomial functions of two and four derivative terms built of the Ricci tensor and the Ricci scalar of the metric g (0) ij and the scalar field in the boundary value φ (0)
, the expression (3.15) for any value of b 5 and b 6 is conformally invariant. It turns out for particular dimensions they may not be determined by equations of motion The BPH transformation approach is accurate only for polynomial expansions, but b 5 and b 6 are singular at order the anomaly appears and we need the logarithmic term in the expansion. However, there is another method to attain them at this order of equations of motion through performing the PBH transformation for d = 2n + and finding residual values by considering log ρ in the equations of motion as the integration of ρ −1− . Nevertheless we do not follow this approach in view of the fact that we get rid of b 5 and b 6 in the normalized form of the anomaly as it is demonstrated shortly in (3.21).
variant operator with two derivatives apart from a minus sign due to our convention for the definition of the curvature tensor. However, this result only determines the covariant form of the anomaly since from the equations of motion b 5 and b 6 become
From ( 3 .
315) the variation of g (α) is homogenous we only need to consider the transformations (3.23) and (3.24) in (3.19) that manifests the variation of the trace anomaly vanishes under the PBH transformations.
(3. 5 )
5, (3.7) and (3.9) we use the fact that in general the variation of an expression with any number of covariant derivatives and tensor indices is
are given in (3.10) 15 and (3.22). The formula for R (α) ij is the same as (4.29) with the corresponding value of α.
respectively, for the given value of α in this section. From the third equation at order 3α, one obtains
We will see later in the presence of scalar fields, the coefficients a (n) also depend on the scalar field and its covariant derivatives as well.
The symmetrization bracket is defined as A (i B j) = 1/2(A i B j + B i A j ).
From now on we suppress factors of g (0) . Unless we mean the contrary, indices are raised and lowered with metric g (0) also all contractions are built with this metric.
With the same reason given in section 4.3, we suppose α > 1.
Acknowledgments : I would like to thank Robert Myers for introducing me to this problem and for his continued support throughout its completion. I particularly thank Janet Hung for clarifying many aspects and for useful discussions.
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"A HIGH-RESOLUTION COMPTON SCATTERING STUDY OF THE ELECTRON MOMENTUM DENSITY IN Al Typeset using REVT E X 3"
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| [
"T Ohata \nDepartment of Synchrotron Radiation Science\nGraduate University for Advanced Studies\nSynchrotron Radiation Research Institute\nInstitute of Materials Structure Science\nJapan Synchrotron Radiation Research Institute\nAkougun, Hyogo 678-12678-12, 305Kamigouri, Tsukuba, KamigouriAkougun, HyogoJapan, Japan, Japan, Japan\n",
"M Itou \nDepartment of Synchrotron Radiation Science\nGraduate University for Advanced Studies\nSynchrotron Radiation Research Institute\nInstitute of Materials Structure Science\nJapan Synchrotron Radiation Research Institute\nAkougun, Hyogo 678-12678-12, 305Kamigouri, Tsukuba, KamigouriAkougun, HyogoJapan, Japan, Japan, Japan\n",
"I Matsumoto \nHigh Energy Accelerator Research Organization\n305TsukubaIbarakiJapan\n",
"Y Sakurai \nAcademy of Mining and Metallurgy\nDepartment of Physics\nTokyo University of Fisheries\nKounan, Minato, Al.Mickiewicza 30108Tokyo, KrakowJapan, Poland\n",
"H Kawata \nDepartment of Physics\nNortheastern University\n02115BostonMAUSA\n",
"N Shiotani \nand Interfaculty Reactor Institute\nNortheastern University\n02115BostonMAUSA\n",
"S Kaprzyk \nDepartment of Physics\nDelft University of Technology\nThe Netherlands 1 A. Bansil2629 JBDelft\n",
"P E Mijnarends \nNortheastern University\n02115BostonMAUSA\n"
]
| [
"Department of Synchrotron Radiation Science\nGraduate University for Advanced Studies\nSynchrotron Radiation Research Institute\nInstitute of Materials Structure Science\nJapan Synchrotron Radiation Research Institute\nAkougun, Hyogo 678-12678-12, 305Kamigouri, Tsukuba, KamigouriAkougun, HyogoJapan, Japan, Japan, Japan",
"Department of Synchrotron Radiation Science\nGraduate University for Advanced Studies\nSynchrotron Radiation Research Institute\nInstitute of Materials Structure Science\nJapan Synchrotron Radiation Research Institute\nAkougun, Hyogo 678-12678-12, 305Kamigouri, Tsukuba, KamigouriAkougun, HyogoJapan, Japan, Japan, Japan",
"High Energy Accelerator Research Organization\n305TsukubaIbarakiJapan",
"Academy of Mining and Metallurgy\nDepartment of Physics\nTokyo University of Fisheries\nKounan, Minato, Al.Mickiewicza 30108Tokyo, KrakowJapan, Poland",
"Department of Physics\nNortheastern University\n02115BostonMAUSA",
"and Interfaculty Reactor Institute\nNortheastern University\n02115BostonMAUSA",
"Department of Physics\nDelft University of Technology\nThe Netherlands 1 A. Bansil2629 JBDelft",
"Northeastern University\n02115BostonMAUSA"
]
| []
| We report high-resolution Compton profiles (CP's) of Al along the three principal symmetry directions at a photon energy of 59.38 keV, together with corresponding highly accurate theoretical profiles obtained within the localdensity approximation (LDA) based band-theory framework. A good accord between theory and experiment is found with respect to the overall shapes of the CP's, their first and second derivatives, as well as the anisotropies in the CP's defined as differences between pairs of various CP's. There are however discrepancies in that, in comparison to the LDA predictions, the measured profiles are lower at low momenta, show a Fermi cutoff which is broader, and display a tail which is higher at momenta above the Fermi momentum. A number of simple model calculations are carried out in order to gain insight into the nature of the underlying 3D momentum density in Al, and the role of the Fermi surface in inducing fine structure in the CP's. The present results when compared with those on Li show clearly that the size of discrepancies between theoretical and experimental CP's is markedly smaller in Al than in Li. This indicates that, with increasing electron density, the conventional picture of the electron gas becomes more representative of the momentum density and that shortcomings of the LDA framework in describing the electron correlation effects become less important. | 10.1103/physrevb.62.16528 | [
"https://arxiv.org/pdf/cond-mat/0009123v1.pdf"
]
| 119,374,377 | cond-mat/0009123 | c5db23e4e20271cce43cb7d002d20f3e1aa30386 |
A HIGH-RESOLUTION COMPTON SCATTERING STUDY OF THE ELECTRON MOMENTUM DENSITY IN Al Typeset using REVT E X 3
8 Sep 2000
T Ohata
Department of Synchrotron Radiation Science
Graduate University for Advanced Studies
Synchrotron Radiation Research Institute
Institute of Materials Structure Science
Japan Synchrotron Radiation Research Institute
Akougun, Hyogo 678-12678-12, 305Kamigouri, Tsukuba, KamigouriAkougun, HyogoJapan, Japan, Japan, Japan
M Itou
Department of Synchrotron Radiation Science
Graduate University for Advanced Studies
Synchrotron Radiation Research Institute
Institute of Materials Structure Science
Japan Synchrotron Radiation Research Institute
Akougun, Hyogo 678-12678-12, 305Kamigouri, Tsukuba, KamigouriAkougun, HyogoJapan, Japan, Japan, Japan
I Matsumoto
High Energy Accelerator Research Organization
305TsukubaIbarakiJapan
Y Sakurai
Academy of Mining and Metallurgy
Department of Physics
Tokyo University of Fisheries
Kounan, Minato, Al.Mickiewicza 30108Tokyo, KrakowJapan, Poland
H Kawata
Department of Physics
Northeastern University
02115BostonMAUSA
N Shiotani
and Interfaculty Reactor Institute
Northeastern University
02115BostonMAUSA
S Kaprzyk
Department of Physics
Delft University of Technology
The Netherlands 1 A. Bansil2629 JBDelft
P E Mijnarends
Northeastern University
02115BostonMAUSA
A HIGH-RESOLUTION COMPTON SCATTERING STUDY OF THE ELECTRON MOMENTUM DENSITY IN Al Typeset using REVT E X 3
8 Sep 2000(November 3, 2018)2
We report high-resolution Compton profiles (CP's) of Al along the three principal symmetry directions at a photon energy of 59.38 keV, together with corresponding highly accurate theoretical profiles obtained within the localdensity approximation (LDA) based band-theory framework. A good accord between theory and experiment is found with respect to the overall shapes of the CP's, their first and second derivatives, as well as the anisotropies in the CP's defined as differences between pairs of various CP's. There are however discrepancies in that, in comparison to the LDA predictions, the measured profiles are lower at low momenta, show a Fermi cutoff which is broader, and display a tail which is higher at momenta above the Fermi momentum. A number of simple model calculations are carried out in order to gain insight into the nature of the underlying 3D momentum density in Al, and the role of the Fermi surface in inducing fine structure in the CP's. The present results when compared with those on Li show clearly that the size of discrepancies between theoretical and experimental CP's is markedly smaller in Al than in Li. This indicates that, with increasing electron density, the conventional picture of the electron gas becomes more representative of the momentum density and that shortcomings of the LDA framework in describing the electron correlation effects become less important.
In a Compton scattering experiment one measures the so-called Compton profile (CP), J(p z ) = ρ(p)dp x dp y ,
where ρ(p) is the ground state electron momentum density. In an independent particle model the momentum density is given by
ρ(p) = (2π) −3 | ψ(r) exp(ip · r)dr| 2 ,(2)
where ψ(r) denotes the electron wave function. [1][2][3][4] The summation in Eq. (2) extends over all occupied states. The Compton profile, J(p z ), thus contains signatures of the Fermi surface breaks and correlation effects in the underlying three dimensional momentum distribution ρ(p). Since Fermi momenta p f are typically ∼ 1 a.u., a high momentum resolution of ∼0.1 a.u. is essential in the experiment for delineating Fermi surface related fine structure in the CP.
High-resolution Compton studies have recently been reported on Li, 5,6 Be, 7-9 V, 10 and
Cu. 11 In all these cases, careful comparisons of the shapes of the absolute valence electron CP's, and the structure in the first and second derivatives as well as the directional anisotropies of the CP's, have been made with the corresponding parameter-free theoretical predictions based on the use of the LDA. A similar investigation of Li-rich LiMg disordered alloys where disorder effects were treated using the mean field KKR-CPA approach has also been carried out. 12 In this way, the band-theory based LDA approach has been shown to provide a remarkably accurate description of many aspects of the momentum density associated with the quantum mechanical electronic ground state, including the characteristic fine structure induced by the Fermi surface. More exciting however is the fact that the aforementioned comparisons have for the first time clearly established the presence of systematic deviations between theoretical and experimental momentum densities. In Li, the experimental break Z k in the momentum density at p f appears to be very small, nearly zero 6 ;
if so, this is very far from the results of electron gas calculations stretching over the last several decades. 13 In Be, the latest Compton data 9 indicates anisotropic electron correlation effects outside the scope of much of the existing theoretical work which is based on treating properties of the homogeneous electron gas. [14][15][16][17][18][19] For these reasons, a renewed interest in the problem of correlation effects on the momentum density beyond the LDA is natural, [20][21][22][23][24] although much further work is necessary for developing an approach of wide applicability in metals and alloys.
Bearing these considerations in mind, there is strong motivation for undertaking high-
(i) = 1 2 {[y(i + 1) − y(i)]/[x(i + 1) − x(i)] + [y(i) − y(i − 1)]/[x(i) − x(i − 1)]}.
No further smoothing or filtering was applied. The interpolation and differentiation cause some statistical correlation between the data points.
III. COMPUTATIONS
The band structure problem was solved within the all-electron charge self-consistent KKR framework without any free parameter. Exchange-correlation effects were incorporated using the von Barth-Hedin local spin density (LSD) approximation. 35 The lattice constant was computed to be 7.6534 a.u. by minimizing the total energy; for comparison, the experimental lattice constant at room temperature is 7.6559 a.u. The self-consistent crystal potential was obtained by iterating the KKR cycles using an elliptic contour with 48 points in the complex energy plane. The final charge density is self-consistent to an accuracy of about In examining the overall shape of the CP's in Fig. 1, one notes that the experimental points are lower at low momenta compared to the calculated values. We emphasize that this does not imply that the measured 3D momentum density is lower than the theoretical one at all momenta. To see this, recall that 44
ρ(0) = − 1 2π d 2 J av. (p) dp 2 p=0 ,(3)
where J av. (p) denotes the directionally averaged CP which in a cubic crystal may be reasonably approximated by 45
J av. (p) = (1/35)[10J 100 (p) + 16J 110 (p) + 9J 111 (p)].(4)
The bottom row in Fig. 1 shows that the differences between the experimental and theoretical second derivatives at p z = 0 are well within the error bars. In view of Eq. (3), this indicates that the underlying 3D distributions are not significantly different at p z = 0. In fact, this result implies that the measured momentum density must be smaller than the theoretical one at momenta approaching the Fermi momentum p f . This is also borne out by the first derivatives shown in the central row of Fig. 1 which begin to show differences between experiment and theory only above p z ∼ 0.3 a.u.
Further insight is provided by Fig. 2 which shows the spherically averaged 3D momentum density defined by
ρ av. (p) = −(1/2πp)(dJ av. /dp),(5)
where J av. The theoretical curves in Fig. 3 include the LP-correction, while those in Fig. 1 do not. A comparison of Fig. 3 with the last column of Fig. 1 If the momentum density within the Fermi sphere were flat and smooth, the first derivative dJ(p z )/dp z shown in the middle row of Fig. 1 derivative has been observed in positron annihilation 1D-ACAR measurements by Okada
∆ρ(p) = d 3 rρ(r) ρ IN T (p, r s (r)) − ρ N I (p, r s (r)) ,(6)
10 − 4
4electrons and the Fermi energy to 10 −4 Ry. An angular momentum cutoff l max = 2 was employed. A free-electron-like Fermi surface was found with Fermi radii k 100 = 0.9246, k 110 = 0.9255, and k 111 = 0.9292 a.u.; the free-electron value would be 0.9254 a.u. The CP's were obtained by first evaluating the three-dimensional momentum density ρ(p) in terms of the momentum matrix element of the KKR Green's function 36-39 over a fine mesh of 48 × 4851 × 177 p-points, covering momenta up to p max ∼ 5 a.u. This mesh involves 4851 k-points in the 1/48-th irreducible part of the Brillouin zone with each k-point translated into 177 p-points by adding reciprocal lattice vectors; the factor of 48 takes into account the symmetry operations of the cubic point group. The CP's can then be computed accurately by evaluating the two-dimensional integral of Eq. (1) using a generalized linear tetrahedron method. 40 The final CP's have been calculated over a momentum mesh containing 151 p z points in the range 0-3 a.u. along each of the three measured directions. The accuracy of the computed profiles is about 1 part in 10 4 . A similar integration technique has been used in our earlier studies of high-resolution CP's of various metals and alloys. 5,7-9,11,26,41,42 . The Lam-Platzman correction 43 to the CP's was computed using the occupation number density of the uniform electron gas. IV. RESULTS AND DISCUSSION Figure 1 shows the measured and computed CP's of the valence electrons along the [100], [110] and [111] directions; the theoretical CP's are convoluted with a gaussian which represents the experimental resolution of 0.12 a.u. FWHM. The experimental valence CP's have been obtained by subtracting the theoretical core CP's from the measured profiles. In this connection, we used the solid-state core wavefunctions which reflect the slight overlap of the 2p core states in Al. The first and second derivatives of the valence profiles have been obtained by numerical differentiation.
49 Figure 3
493(p) is obtained via Eq.(4). The oscillations in ρ av. (p) at small momenta reflect partly the large (correlated) error bars due to the division of the small derivative by small values of p, and partly the (spherically averaged) effect of Brillouin zone-face interactions to be discussed below. In any event,Fig. 2makes it clear that the experimental 3D momentum density lies below the theoretical predictions as one approaches p f , and that the situation reverses itself above p f . The 2nd derivatives J ′′ inFig. 1all show a peak at p f . There is good agreement between theory and experiment as to the position of the peaks, i.e., the value of p f , but the measured peaks are all lower and broader than the theoretical predictions. Although the shapes of the J ′′ peaks reflect the complex interplay between the effects of experimental resolution, electron correlations and lattice potential on the Fermi cut-off in the momentum density, it is evident from Figs. 1 and 2 that the measured distribution possesses a tail higher than the theory beyond p f . These discrepancies between theory and experiment are similar to those reported earlier in Li 5,6 and other metals[7][8][9]11 and have their origin in the electron correlation effects beyond the LDA which are not treated properly within our theoretical framework. Such correlations are expected to cause (relative to the independent particle model) a decrease of the momentum density as one approaches p f , and a tail at momenta greater than p f ; as indicated above, both features are qualitatively visible in our comparison between theory and experiment.Notably, the deviations from LDA theory are smaller in Al than in Li. For example, the difference between the theoretical and experimental valence profiles at p z = 0 is approximately 16 % for Li and 4.5 % for Al, 46 and the width of the peak at p f in the second derivatives is 0.23 a.u. in Li and 0.15 a.u. in Al; thus, the "blurring" of the Fermi cutoff is more severe in Li than in Al. These characteristic differences between Li and Al are partly related to the difference in the electron density of the two metals. The electron density in terms of r s (the standard parameter for the volume per electron of valence electrons), is 3.21 for Li and 2.12 for Al. Therefore, the bare Coulomb interaction is more effectively screened in Al than in Li. As shown by a variety of treatments of the homogenous interacting electron gas, as the electron density increases, the kinetic energy dominates, and the momentum density is described more closely by the free-electron rectangular distribution with a step-wise cutoff at p f .14-19,47-considers the effect of the isotropic Lam-Platzman (LP) correction on the [111] CP; results along other directions are similar and are not shown in the interest of brevity.
where the integral extends over the Wigner-Seitz cell. The expression within the square brackets gives the difference between the momentum densities of the interacting and noninteracting homogeneous electron gas (denoted by the superscripts 'INT' and 'NI') evaluated at the local density ρ(r) of the physical system and r s (r) is the corresponding electron density parameter. Equation (6) thus attempts to take into account inhomogeneities in the electron gas, whereas the semi-empirical model of Ref.26 replaces the integrand by its value at the average electron density in Al. The matter is quite subtle, and further work is necessary in order to develop a satisfactory treatment of correlation effects on the momentum density in solids.
et al.50 and 2D-ACAR measurements by Mader et al.51 who also ascribed it to zone-face interactions around the W-points.The directional differences, shown inFig. 4, are a measure of the anisotropy. Although the maximum difference is about 1 % of the peak value of the profile itself, they show definite structures which can have several origins. Firstly, for different crystal orientations the plane of integration in Eq. (1) sweeps differently through the Umklapp Fermi spheres centered at the reciprocal lattice points in the higher Brillouin zones. Secondly, the Fermi surface is slightly distorted from a sphere, as witnessed by the different Fermi radii given above, while, thirdly, band-structure effects such as a p dependence of the momentum density within the Fermi spheres and interactions of the electron bands with the Brillouin zone faces with consequent distortion of the wavefunctions will also contribute to the anisotropy. The importance of the first point can be readily studied using a simple model of a spherical freeelectron-like Fermi surface surrounded by seven shells of similar Umklapp Fermi surfaces.The momentum density within each Fermi surface is assumed to be constant and given by the square of the corresponding Fourier component of the electron wave function at Γ 1 . 3 The CP for a given direction then consists of a superposition of inverted parabolas, centered at the projections on that direction of the reciprocal lattice points. The height of each parabola is proportional to the momentum density within the corresponding Fermi sphere, while its cut-off points are found by adding or subtracting p f from the projected center.Figure 5shows the directional differences thus obtained. The positions of the cut-off points havebeen indicated by the arrows at the bottom of the graph, together with a symbol which denotes the direction of projection (a = [100], b = [110], c = [111]) and the coordinates of the center of the Fermi sphere. It should be noted that many Umklapp Fermi spheres coincide in projection and therefore these coordinates are not unique; the simplest set has been noted. The analysis in Fig. 5 shows that much of the important structure in the directional differences stems from the < 111 > Umklapp contributions; the other Umklapps play a less important role. A comparison of Fig. 5 with the calculated differences in Fig. 4 shows an overall qualitative correspondence in the succession of positive and negative peaks. On a more detailed scale, however, there are significant differences which have their origin in the other factors mentioned above. Notable examples are the peaks around 0.85 a.u. in the J [111] − J [100] and J [111] − J [110] directional differences in Fig. 4 which have no clear counterpart in Fig. 5. Kubo et al. 52 have ascribed these features to the fact that in the [111] direction the actual Fermi surface bulges out beyond the free-electron Fermi sphere in the second Brillouin zone while there is a contraction in the third zone. This will strongly affect the [111] profile but not so much the other two. Our simple free-electron model of course does not contain this Fermi surface distortion effect. The calculated curves in Fig. 4, on the other hand, include all of these factors and reproduce the essential characteristics of the measured differences, although some discrepancies remain. It may be noted that non-locality of the exchange and correlation potential in Li reduces the Fermi surface anisotropy 53,54 and thus would affect the anisotropy of the CP's. In this vein, lattice vibrations would reduce the Umklapp contributions and hence the anisotropy of the momentum density. In how far such effects can explain the residual discrepancies in Fig. 4 remains unclear.In principle, the anisotropy in the momentum density may be obtained approximately by expanding both the momentum density and the CP's into lattice harmonics, and establishing the relation between the expansion coefficients for the momentum density and those for the CP's.55 Actually, Eq. (5) represents the l = 0 term in such a scheme. However, we have not attempted to analyze our data along these lines since the number of measured profiles is not large enough.Additional information may be gained from a comparison of CP's with the correspond-ing results of positron annihilation measurements. Both experiments probe the momentum density -in positron annihilation one measures the momentum density of the annihilating electron-positron pair whereas in Compton only the electron momentum density is involved. In Fig. 6 the first derivative of the one-dimensional angular correlation of positron annihilation radiation (1D-ACAR) profile measured by Okada et al. 50 for the [111] orientation is compared with the corresponding CP. The momentum resolution of the 1D-ACAR is 0.11 a.u. which is almost the same as that of the present CP's. Since the 1D-ACAR was area normalized to the theoretical 1D-ACAR calculated by Kubo et al., 52 the peak height at p z = 0 is almost the same as that of the present CP. The slope at the Fermi momentum is steeper in the 1D-ACAR than in the CP, which is direct evidence for enhancement of the annihilation of positrons with the s − p electrons near the Fermi energy predicted first by Kahana 56 on the basis of an interacting electron gas model. Also, the correlation tail for p > p f in the 1D-ACAR is weaker than its counterpart in the CP as a result of the partial cancellation of electron-electron and positron-electron correlation effects. 57 Finally, the fine structure at 0.2 a.u. and 0.7 a.u. is more pronounced in the 1D-ACAR than in the CP. This points to less correlation-induced smearing in positron annihilation compared to Compton scattering. V. SUMMARY AND CONCLUSIONS We have measured the Compton profiles (CP's) of Al along [100], [110] and [111] directions at a photon energy of 59.38 keV and a momentum resolution of 0.12 au. Parallel, highly accurate all-electron computations have been carried out within the LDA-based band-theory framework. Comparisons between theory and experiment at the level of the shapes of the CP's, structure in the 1st and 2nd derivatives of the CP's, and anisotropies obtained by taking differences between three pairs of CP's, all show a good level of accord. However, there are discrepancies as well. In comparison to the LDA predictions, the measured profiles are lower at low momenta, show a Fermi cutoff which is broader, and display a tail which is higher at momenta above the Fermi momentum. The inclusion of correlation effects in the LDA via the standard isotropic Lam-Platzman correction improves things slightly, but the essential discrepancies remain. A model analysis in terms of directionally averaged CP's allows us to get a handle on the 3D momentum density of Al; in this way, we adduce that the experimental 3D density near p=0 does not differ significantly from LDA predictions even though the CP's do. In this vein, CP's are computed using a model 3D distribution in which free electron spheres with appropriate weights are placed on reciprocal lattice points (extending to seven shells around a central sphere) to represent the higher momentum components in the electronic wave functions; the results show that a significant amount of fine structure in the CP's is induced by these higher momentum components and by k-states near the W-points in the Brillouin zone where the free electron spheres overlap. The present results when compared with those reported earlier on Li show clearly that the size of discrepancies between theoretical and experimental CP's is markedly smaller in Al than in Li; in particular, theoretical and experimental profiles at p z = 0 differ by 16% in Li but only by 4.5% in Al, and the peak width in the 2nd derivative at p f is 0.23 au in Li but 0.15 au in Al. It is thus clear that, with increasing electron density, the conventional picture of the electron gas becomes more representative of the momentum density and that shortcomings of the LDA framework in describing the electron correlation effects become less important. Finally, we compare briefly our [111] CP with the positron-annihilation (1D-ACAR) measurements of Okada et al., and show that in the case of positron-annihilation the Fermi cut-off is sharper and that there is less correlation induced smearing of structures in the ACAR spectrum. ACKNOWLEDGMENTS It is a pleasure to acknowledge important conversations with Bernardo Barbiellini. The Compton profile measurements were performed with the approval of the Photon Factory Advisory Committee, Proposal Nos. 92-G257, 94-G351 and 97-G288. This work is supported by the US Department of Energy under contract W-31-109-ENG-38, by the Polish Committee for Scientific Research through grant number 2P03B02814, and benefited from a travel grant from NATO, and the allocation of supercomputer time at NERSC and the Northeastern University Advanced Scientific Computation Center (NU-ASCC).FIGURES FIG. 1. Top: Measured and computed Compton profiles of Al along the [100], [110] and [111] directions. Theoretical profiles (solid lines) have been broadened to reflect experimental resolution. Middle: First derivatives of the measured and computed profiles. Bottom: Second derivatives of the measured and the computed profiles. FIG. 2. Theoretical (solid curve) and experimental (dashed curve) directionally averaged 3D electron momentum density obtained via Eqs. (4) and (5). FIG. 3. Same as the last column of Fig. 1, except that here the theoretical curves in all cases include the Lam-Platzman correction. FIG. 4. Measured and computed directional difference profiles for three different pairs of directions. FIG. 5. Directional difference profiles calculated for a simple quasi-free-electron model of Al in which the CP is given by a superposition of parabolic contributions centered at various reciprocal lattice points (see text). The arrows at the bottom indicate the positions of the cut-off points (each parabola has two cut-off points; the other one lies outside the graph). The directions of projection are indicated by a (=[100]), b (=[110]), and c (=[111]), while the subscripts denote the coordinates of the reciprocal lattice points involved. 000 denotes the cut-off of the central Fermi surface, i.e., the Fermi radius p f . FIG. 6. First derivative of the 1D-ACAR spectrum (open circles) along the [111] direction read off from Ref. 50 is compared with the derivative of the [111] Compton profile shown in Fig. 1.
78.70.Ck, 71.20.Gj, 71.15.Mb, 71.18.+y Typeset using REVT E XI. INTRODUCTION
resolution Compton studies of other systems. Our choice of Al in this connection is an especially good one because Al is trivalent and, therefore, it extends the range of electron densities investigated so far via high-resolution Compton. Correlation effects are of course expected to become less important with increasing electron density as the kinetic energy dominates. Also, Al has been the traditional touchstone of a free-electron-like metal with a nearly spherical Fermi surface (viewed in the extended zone). Neither a high-resolution, high-statistics Compton measurement, nor a band theory computation of high accuracy in order to identify Fermi surface related fine structure in the CP's of Al is currently available in the literature.25,26 The goal of the present work is to fill this gap and determine the extent to which the LDA describes the momentum density in Al. The existing Compton data on Al consists essentially of a number of measured CP's using γ-ray sources and solid state detectors at low momentum resolution.27-30 Quite some time ago, Shiotani et al. 31 obtained the [111] CP of Al at a momentum resolution of 0.08 au, but did not investigate the anisotropy of the CP or the Fermi surface signatures therein. An outline of this article is as follows. In the next section we describe the experimental procedures. Section III gives pertinent details of computations. In section IV the experimental CP's are analyzed in the light of the band theory predictions as well as a number of other model computations. The Compton results are also compared briefly with closely related positron annihilation spectra. Section V summarizes our main conclusions. Single crystals of Al with surface normals oriented along the [100], [110] and [111] directions were used. The thickness of the crystals was about 2 mm. The reader is referred to Sakai 34 ; the integrated intensity of the double scattering events was found to be 10% of the single scattering events.II. EXPERIMENT
Sakurai et al. 32 for details of our Compton spectrometer, and to Tanaka et al. 33 and Itou et
al. 8 for our data processing procedures. Briefly, the spectrometer consists of a Cauchois-type
bent-crystal analyzer of Si(422) with an image plate serving as a position sensitive detector.
The scattering angle is 160 • . The synchrotron radiation X-rays from a multipole wiggler
installed in the 6.5 GeV Accumulation Ring at the National Laboratory for High Energy
Physics are monochromatized by a quasi-doubly bent monochromator to 59.38 keV with an
energy resolution of about 80 eV. The overall momentum resolution is estimated to be 0.12
a.u. The double Compton scattering events were simulated via the Monte Carlo program of
The statistical error of each datum point, given by σ =
√
N + 0.003N, is estimated to be
less than 0.3 %. Since the data points are not measured equidistantly they are interpolated
onto an equidistant mesh of 0.02 a.u. using simple linear interpolation. The data were
numerically differentiated according to y ′
shows that, although the inclusion of the LP-correction improves things, much of the discrepancy between theory and experiment still remains. Interestingly, Ref.26 has recently analyzed the correlation correction to the CP's of Al in terms of a model which involves the break Z k in the momentum density at p fas the only free parameter. By adjusting Z k , Ref. 26 finds that the discrepancy between the
LDA predictions and the measurements can be essentially removed for a Z k value between
0.7 and 0.8, in reasonable accord with the corresponding theoretical values from various
authors which are scattered between 0.76 and 0.85. 14-19 There is no inconsistency between
the present results and those of Ref. 26. To see this relationship, recall that the standard
LP-correction is defined via 43
would be a straight line up to the cutoff at the Fermi momentum. However, at momenta less than the Fermi radius some structure is visible notably in the [111] and the [100] derivatives. In this connection, we note that theFermi sphere overlaps with Umklapp Fermi spheres centered on the (111) reciprocal lattice
points around the W-points in the Brillouin zone. For example, the hexagonal zone face
contains six W-points which all project at ( 1
2 , 1
2 , 1
2 ), i.e., the point p z = 0.71 a.u. on the [111]
axis. Similarly, four W points in the first Brillouin zone project at 0.41 a.u. and another four
at 0.82 a.u. on the [100] direction. Interestingly, the experimental as well as the theoretical
derivatives contain structure around 0.7 a.u. in the [111] and 0.4 a.u. in the [100] CP. This
indicates the importance of the k states near the W points with respect to the fine structure
in the Al CP's. Incidentally, a structure similar to the wiggle around 0.4 a.u. in the [100]
Y. Sakurai, M. Ito, T. Urai, Y. Tanaka, N. Sakai, T. Iwazumi, H. Kawata, M. Ando, and
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. S Manninen, J. Phys. Chem. Solids. 61335S. Manninen, J. Phys. Chem. Solids 61, 335 (2000).
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[
"Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor Gross-Neveu Model *",
"Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor Gross-Neveu Model *"
]
| [
"Zhou Bang-Rong \nCollege of Physical Sciences\nGraduate School\nCCAST (World Laboratory)\nChinese Academy of Sciences\nP.O.Box 8730100049, 100080Beijing, BeijingChina, China\n"
]
| [
"College of Physical Sciences\nGraduate School\nCCAST (World Laboratory)\nChinese Academy of Sciences\nP.O.Box 8730100049, 100080Beijing, BeijingChina, China"
]
| []
| The effective potential analysis indicates that, in a 3D two-flavor Gross-Neveu model in vacuum, depending on less or bigger than the critical value 2/3 of GS/HP , where GS and HP are respectively the coupling constants of scalar quark-antiquark channel and pseudoscalar diquark channel, the system will have the ground state with pure diquark condensates or with pure quark-antiquark condensates, but no the one with coexistence of the two forms of condensates. The similarities and differences in the interplay between the quark-antiquark and the diquark condensates in vacuum in the 2D, 3D and 4D two-flavor four-fermion interaction models are summarized. | 10.1088/0253-6102/47/4/027 | [
"https://arxiv.org/pdf/0704.0829v2.pdf"
]
| 119,571,447 | 0704.0829 | f2c494b2f1a0103b219383044e34231cf292bd45 |
Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor Gross-Neveu Model *
23 Jun 2007
Zhou Bang-Rong
College of Physical Sciences
Graduate School
CCAST (World Laboratory)
Chinese Academy of Sciences
P.O.Box 8730100049, 100080Beijing, BeijingChina, China
Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor Gross-Neveu Model *
23 Jun 2007(Dated:)numbers: 1238Aw1238Lg1210Dm1115Pg Keywords: 3D Gross-Neveu model, quark-antiquark and diquark condensates, effective potential
The effective potential analysis indicates that, in a 3D two-flavor Gross-Neveu model in vacuum, depending on less or bigger than the critical value 2/3 of GS/HP , where GS and HP are respectively the coupling constants of scalar quark-antiquark channel and pseudoscalar diquark channel, the system will have the ground state with pure diquark condensates or with pure quark-antiquark condensates, but no the one with coexistence of the two forms of condensates. The similarities and differences in the interplay between the quark-antiquark and the diquark condensates in vacuum in the 2D, 3D and 4D two-flavor four-fermion interaction models are summarized.
I. INTRODUCTION
It has been shown by effective potential approach that in a two-flavor 4D Nambu-Jona-Lasinio (NJL) model [1], even when temperature T = 0 and quark chemical potential µ = 0, i.e. in vacuum, there could exist mutual competition between the quark-antiquark condensates and the diquark condensates [2]. Similar situation has also emerged from a 2D two-flavor Gross-Neveu (GN) model [3] except some difference in the details of the results [4]. An interesting question is that if such mutual competition between the two forms of condensates is a general characteristic of this kind of two-flavor four-fermion interaction models? For answer to this question, on the basis of research on the 4D NJL model and the 2D GN model, we will continue to examine a 3D two-flavor GN model in similar way. The results will certainly deepen our understanding of the feature of the four-fermion interaction models.
We will use the effective potential in the mean field approximation which is equivalent to the leading order of 1/N expansion. It is indicated that a 3D GN model is renormalizable in 1/N expansion [5].
II. MODEL AND ITS SYMMETRIES
The Lagrangian of the model will be expressed by
L =qiγ µ ∂ µ q + G S [(qq) 2 + (q τ q) 2 ] +H P A=2,5,7 (qτ 2 λ A q C )(q C τ 2 λ A q).(1)
All the denotations used in Eq.(1) are the same as the ones in the 2D GN model given in Ref. [4], except that * The project supported by the National Natural Science Foundation of China under Grant No.10475113.
the dimension of space-time is changed from 2 to 3 and the coupling constant H S of scalar diquark interaction channel is replaced by the coupling constant H P of pseudoscalar diquark interaction channel. Now the matrices γ µ (µ = 0, 1, 2) and the charge conjugate matrix C are taken to be 2 × 2 ones and have the explicit forms
γ 0 = 1 0 0 −1 , γ 1 = 0 i i 0 , γ 2 = 0 1 −1 0 = C.
(2) It is emphasized that, in 3D case, no "γ 5 " matrix can be defined, hence the third term in the right-handed side of Eq.(1) will be the only possible color-anti-triplet diquark interaction channel which could lead to Lorentzinvariant diquark condensates, where we note that the matrix Cτ 2 λ A is antisymmetric. Without "γ 5 ", the Lagrangian (1) will have no chiral symmetry. Except this, it is not difficult to verify that the symmetries of L include:
continuous flavor and color symmetries
SU f (2) ⊗ SU c (3) ⊗ U f (1);
2. discrete symmetry R: q → −q;
3. parity P:
q(t, x) → γ 0 q(t, − x) and q C (t, x) → −γ 0 q C (t, − x); 4. time reversal T : q(t, x) → γ 2 q(−t, x) and q C (t, x) → −γ 2 q C (−t, x); 5. charge conjugate C: q ↔ q C ; 6. special parity P 1 : q(t, x 1 , x 2 ) → γ 1 q(t, −x 1 , x 2 ) and q C (t, x 1 , x 2 ) → −γ 1 q(t, −x 1 , x 2 ); 7. special parity P 2 : q(t, x 1 , x 2 ) → γ 2 q(t, x 1 , −x 2 ) and q C (t, x 1 , x 2 ) → −γ 2 q C (t, x 1 , −x 2 ).
If the quark-antiquark condensates qq could be formed, then the time reversal T , the special parities P 1 and P 2 will be spontaneously broken [6]. If the diquark condensates q C τ 2 λ 2 q could be formed, then the color symmetry SU c (3) will be spontaneously broken down to SU c (2) and the flavor number U f (1) will be spontaneously broken but a "rotated" electric charge UQ(1) and a "rotated" quark number U ′ q (1) leave unbroken [7]. In addition, the parity P will be spontaneously broken, though all the other discrete symmetries survive. This implies that the diquark condensates q C τ 2 λ 2 q will be a pseudoscalar. In this paper we will neglect discussions of the Goldstone bosons induced by breakdown of the continuous symmetries and pay our main attention to the problem of interplay between the above two forms of condensates.
III. EFFECTIVE POTENTIAL IN MEAN FIELD APPROXIMATION
Define the order parameters in the 3D GN model by
σ = −2G S qq and ∆ = −2H P q C τ 2 λ 2 q ,(3)
then in the mean field approximation, the Lagrangian (1) can be rewritten by
L =Ψ(x)S −1 (x)Ψ(x) − σ 2 4G S − |∆| 2 4H P ,(4)
where
Ψ(x) = 1 √ 2 q(x) q C (x) andΨ(x) = 1 √ 2 q(x)q C (x)
are the expressions of the quark fields in the Nambu-Gorkov basis [8]. In the momentum space, the inverse propagator S −1 (x) for the quark fields may be expressed by
S −1 (p) = p − σ −τ 2 λ 2 ∆ −τ 2 λ 2 ∆ * p − σ , p = γ µ p µ .(5)
The effective potential corresponding to L given by Eq. (4) becomes
V (σ, |∆|) = σ 2 4G S + |∆| 2 4H P + i d 3 p (2π) 3 1 2 Tr ln S −1 (p)S 0 (p).
(6) Similar to the case of the 2D NG model [4], the calculations of Tr for (red, green) and blue color degrees of freedom can be made separately thus Eq.(6) will be reduced to
V (σ, |∆|) = σ 2 4G S + |∆| 2 4H P + 2i d 3 p (2π) 3 ln p 2 − (σ − |∆|) 2 + iε p 2 + iε + ln p 2 − (σ + |∆|) 2 + iε p 2 + iε + ln p 2 − σ 2 + iε p 2 + iε(7)
After the Wick rotation, we may define and calculate in 3D Euclidean momentum space
I(a 2 ) = d 3p (2π) 3 lnp 2 + a 2 p 2 = 1 2π 2 a 2 Λ − 2 3 a 3 arctan Λ a ≃ 1 2π 2 a 2 Λ − π 3 |a| 3 , if Λ ≫ |a|,(8)
where Λ is the 3D Euclidean momentum cut-off. Assume that Λ ≫ |σ − |∆||, Λ ≫ σ + |∆| and Λ ≫ σ, then by means of Eq.(8) we will obtain the final expression of the effective potential in the 3D GN model
V (σ, |∆|) = σ 2 4G S + |∆| 2 4H P − 1 π 2 (3σ 2 + 2|∆| 2 )Λ + 1 3π 6σ 2 |∆| + 2|∆| 3 + σ 3 +2θ(σ − |∆|)(σ − |∆|) 3 .(9)
IV. GROUND STATES Equation (9) provide the possibility to discuss the ground states of the model analytically. The extreme value conditions ∂V (σ, |∆|)/∂σ = 0 and ∂V (σ, |∆|)/∂|∆| = 0 will lead to the equations
σ 1 2G S − 6Λ π 2 + 4|∆| π + σ π + 2 π θ(σ − |∆|)(σ − |∆|) 2 = 0,(10)|∆| 1 2H P − 4Λ π 2 + 2|∆| π + 2 π [σ 2 − θ(σ − |∆|)(σ − |∆|) 2 ] = 0.(11)
Define the expressions
K = A B B C = AC − B 2 ,
where A, B and C represent the second order derivatives of V (σ, |∆|) with the explicit expressions
A = ∂ 2 V ∂σ 2 = 1 2G S − 6Λ π 2 + 4|∆| π + 2σ π + 4 π θ(σ − |∆|)(σ − |∆|), B = ∂ 2 V ∂σ∂|∆| = ∂ 2 V ∂|∆|∂σ = 4 π [σ − θ(σ − |∆|)(σ − |∆|)], C = ∂ 2 V ∂|∆| 2 = 1 2H P − 4Λ π 2 + 4|∆| π + 4 π θ(σ − |∆|)(σ − |∆|).(12)
Equations (10) and (11) have the four different solutions which will be discussed in proper order as follows.
(i) (σ, |∆|)=(0,0). It is a maximum point of V (σ, |∆|), since in this case we have
A = 1 2G S − 6Λ π 2 < 0 and K = A 1 2H P − 4Λ π 2 > 0,
assuming Eqs. (10) and (11) have solutions of non-zero σ and |∆|.
(ii) (σ, |∆|)=(σ 1 ,0), where the non-zero σ 1 satisfies the equation
1 2G S − 6Λ π 2 + 3σ 1 π = 0.(13)
When Eq. (13) is used, we obtain A = 3σ 1 π ,
K = = A 1 2H P − 1 3G S + 2σ 1 π > 0, if G S H P > 2 3 .
Hence (σ 1 ,0) will be a minimum point of V (σ, |∆|) when G S /H P > 2/3. (iii) (σ, |∆|)= (0, ∆ 1 ), where non-zero ∆ 1 obeys the equation
1 2H P − 4Λ π 2 + 2∆ 1 π = 0.(14)
By using Eq.(14) we may get
A = 1 2G S − 3 4H P + ∆ 1 π , K = A 2∆ 1 π .
Obviously, (0,∆ 1 ) will be a minimum point of V (σ, |∆|) when G S /H P < 2/3. (iv) (σ, |∆|)=(σ 2 , ∆ 2 ). In view of existence of the function θ(σ − |∆|) in Eqs.(10) and (11), we have to consider the case of σ 2 > ∆ 2 and σ 2 < ∆ 2 respectively. (a) σ 2 > ∆ 2 . In this case, Eqs.(10) and (11) will become
σ 2 1 2G S − 6Λ π 2 + 1 π (3σ 2 2 + 2∆ 2 2 ) = 0, 1 2H P − 4Λ π 2 + 4σ 2 π = 0.
From them we can get
A = 3σ 2 2 − 2∆ 2 2 πσ 2 > 0, K = − 16∆ 2 2 π 2 < 0.
Thus it is turned out that (σ 2 , ∆ 2 ) will be neither a maximum nor a minimum point of (10) and (11) are changed into
V (σ, |∆|) if σ 2 > ∆ 2 . (b) σ 2 < ∆ 2 . Now Eqs.1 2G S − 6Λ π 2 + 4∆ 2 + σ 2 π = 0,(15)∆ 2 1 2H P − 4Λ π 2 + 2 σ 2 2 + ∆ 2 2 π = 0.(16)
Hence we will have the results that
A = σ 2 π , K = 2σ 2 π 2 ∆ 2 (∆ 2 2 − σ 2 2 − 8σ 2 ∆ 2 ),
from which it may be deduced that only if
σ 2 < ( √ 17 − 4)∆ 2 ,(17)
(σ 2 , ∆ 2 ) is just a minimum point of V (σ, |∆|). On the other hand, from Eqs. (15) and (16) obeyed by σ 2 and ∆ 2 we may get
1 2H P − 1 3G S = 2 π∆ 2 √ 13 − 1 6 ∆ 2 + σ 2 × √ 13 + 1 6 ∆ 2 − σ 2 .(18)
Equation (18) indicates that for the minimum point (σ 2 , ∆ 2 ) satisfying Eq.(17) one will certainly have G S /H P > 2/3. Taking this and the result obtained in case (ii) into account we see that if G S /H P > 2/3 the effective potential V (σ, |∆|) will have two possible minimum points (σ 1 , 0) and (σ 2 , ∆ 2 ). To determine which one of the two minimum points is the least value point of V , we must make a comparison between V (σ 1 , 0) and V (σ 2 , ∆ 2 ) with the constraint given by Eq. (17). In fact, it is easy to find out that when Eq.(13) is used,
V (σ 1 , 0) = − σ 3 1 2π ,(19)
and that when Eqs. (15) and (16) are used,
V (σ 2 , ∆ 2 ) = − 1 3π ∆ 3 2 + 3σ 2 2 ∆ 2 + σ 3 2 2 .(20)
By comparing Eq.(13) with Eq.(15) we may obtain the relation
3σ 1 = σ 2 + 4∆ 2 .(21)
By means of Eqs.(19)-(21) it is easy to verify that
V (σ 1 , 0) − V (σ 2 , ∆ 2 ) = − 1 3π 1 9 (23∆ 3 2 − 4σ 3 2 ) + σ 2 ∆ 2 3 (8∆ 2 − 7σ 2 ) < 0,
when Eq.(17) is satisfied. This result indicates that when G S /H P > 2/3, the least value point of V (σ, |∆|) will be (σ 1 , 0) but not (σ 2 , ∆ 2 ).
In summary, if the necessary conditions G S Λ > π 2 /12 and H P Λ > π 2 /8 for non-zero σ and ∆ are satisfied, then the least value points of the effective potential V (σ, |∆|) will be at
(σ, |∆|) = (0, ∆ 1 ) (σ 1 , 0) if 0 ≤ G S /H P < 2/3 G S /H P > 2/3 . (22)
As a result, in the ground state of the 3D two-flavor GN model, depending on that the ratio G S /H P is either bigger or less than 2/3, one will have either pure quarkantiquark condensates or pure diquark condensates, but no coexistence of the two forms of condensates could happen.
V. CONCLUDING REMARKS
The result (22) in the 3D GN model can be compared with the ones in the 4D NJL model and in the 2D GN model. The minimal points of the effective potential V (σ, |∆|) for the latter models have been obtained and are located respectively at
(σ, |∆|) = (0, ∆ 1 ) (σ 2 , ∆ 2 ) (σ 1 , 0) if 0 ≤ G S /H S < 2/[3(1 + C)] 2/[3(1 + C)] < G S /H S < 2/3 G S /H S > 2/3(23)
with C = (2H S Λ 2 4 /π 2 − 1)/3 and Λ 4 denoting the 4D Euclidean momentum cutoff in the 4D two-flavor NJL model, if the necessary conditions G S Λ 2 4 > π 2 /3 and H S Λ 2 4 > π 2 /2 for non-zero σ and ∆ are satisfied [2], and
(σ, |∆|) = (0, ∆ 1 ) (σ 2 , ∆ 2 ) (σ 1 , 0) if G S /H S = 0 0 < G S /H S < 2/3 G S /H S > 2/3(24)
in the 2D two-flavor GN model [4]. In Eqs.(23) and (24), G S and H S always represent the coupling constants in scalar quark-antiquark channel and scalar diquark channel separately. By a comparison among Eqs.(22)-(24) it may be found that the three models lead to very similar results. In all the three models, the interplay between the quarkantiquark and the diquark condensates in vacuum depends on the ratio G S /H D (D = S for the 4D and 2D model and D = P for the 3D model). In particular, the diquark condensates could emerge (in separate or coexistent pattern) only if G S /H D < 2/3. This is probably a general characteristic of the considered two-flavor fourfermion models, since in these models the color number of the quarks participating in the diquark condensates and in the quark-antiquark condensates is just 2 and 3 respectively. However, there are also some differences in the pattern realizing the diquark condensates among the three models, though the pure quark-antiquark condensates arise only if G S /H D > 2/3 in all of them. In the 2D GN model, the pure diquark condensates emerge only if G S /H S = 0 and this is different from the 4D NJL model where the pure diquark condensates may arise if G S /H S is in a finite region below 2/3. Another difference is that in the 3D GN model, there is no coexistence of the quark-antuquark condensates and the diquark condensates but such coexistence is clearly displayed in the 4D and 2D model. This implies that in the 3D GN model, G S /H P = 2/3 becomes the critical value which distinguishes between the ground states with the pure diquark condensates and with the pure quark-antiquark condensates.
It is also indicated that if the two-flavor four-fermion interaction models are assumed to be simulations of QCD (of course, only the 4D NJL model is just the true one) and the four-fermion interactions are supposed to come from the heavy color gluon exchange interactions −g(qγ µ λ a q) 2 (a = 1, · · · , 8; µ = 0, · · · , D − 1) via the Fierz transformation [7], then one will find that in all the three models, for the case of two flavors and three colors the ratio G S /H D are always equal to 4/3 which is larger than the above critical value 2/3. From this we can con-
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| []
|
[
"Resource Efficient 3D Convolutional Neural Networks",
"Resource Efficient 3D Convolutional Neural Networks"
]
| [
"Okan Köpüklü \nInstitute for Human-Machine Communication, TU Munich\nGermany\n",
"Neslihan Kose \nDependability Research Lab\nIntel Labs Europe\nIntel Deutschland GmbH\nGermany\n",
"Ahmet Gunduz \nInstitute for Human-Machine Communication, TU Munich\nGermany\n",
"Gerhard Rigoll \nInstitute for Human-Machine Communication, TU Munich\nGermany\n"
]
| [
"Institute for Human-Machine Communication, TU Munich\nGermany",
"Dependability Research Lab\nIntel Labs Europe\nIntel Deutschland GmbH\nGermany",
"Institute for Human-Machine Communication, TU Munich\nGermany",
"Institute for Human-Machine Communication, TU Munich\nGermany"
]
| []
| Recently, convolutional neural networks with 3D kernels (3D CNNs) have been very popular in computer vision community as a result of their superior ability of extracting spatio-temporal features within video frames compared to 2D CNNs. Although there has been great advances recently to build resource efficient 2D CNN architectures considering memory and power budget, there is hardly any similar resource efficient architectures for 3D CNNs. In this paper, we have converted various well-known resource efficient 2D CNNs to 3D CNNs and evaluated their performance on three major benchmarks in terms of classification accuracy for different complexity levels. We have experimented on (1) Kinetics-600 dataset to inspect their capacity to learn, (2) Jester dataset to inspect their ability to capture motion patterns, and (3) UCF-101 to inspect the applicability of transfer learning. We have evaluated the run-time performance of each model on a single Titan XP GPU and a Jetson TX2 embedded system. The results of this study show that these models can be utilized for different types of real-world applications since they provide real-time performance with considerable accuracies and memory usage. Our analysis on different complexity levels shows that the resource efficient 3D CNNs should not be designed too shallow or narrow in order to save complexity. The codes and pretrained models used in this work are publicly available 1 . | 10.1109/iccvw.2019.00240 | [
"https://arxiv.org/pdf/1904.02422v5.pdf"
]
| 102,353,321 | 1904.02422 | 205f53c5fd7c1629367f6feedf3fd80edcf4ead4 |
Resource Efficient 3D Convolutional Neural Networks
Okan Köpüklü
Institute for Human-Machine Communication, TU Munich
Germany
Neslihan Kose
Dependability Research Lab
Intel Labs Europe
Intel Deutschland GmbH
Germany
Ahmet Gunduz
Institute for Human-Machine Communication, TU Munich
Germany
Gerhard Rigoll
Institute for Human-Machine Communication, TU Munich
Germany
Resource Efficient 3D Convolutional Neural Networks
Recently, convolutional neural networks with 3D kernels (3D CNNs) have been very popular in computer vision community as a result of their superior ability of extracting spatio-temporal features within video frames compared to 2D CNNs. Although there has been great advances recently to build resource efficient 2D CNN architectures considering memory and power budget, there is hardly any similar resource efficient architectures for 3D CNNs. In this paper, we have converted various well-known resource efficient 2D CNNs to 3D CNNs and evaluated their performance on three major benchmarks in terms of classification accuracy for different complexity levels. We have experimented on (1) Kinetics-600 dataset to inspect their capacity to learn, (2) Jester dataset to inspect their ability to capture motion patterns, and (3) UCF-101 to inspect the applicability of transfer learning. We have evaluated the run-time performance of each model on a single Titan XP GPU and a Jetson TX2 embedded system. The results of this study show that these models can be utilized for different types of real-world applications since they provide real-time performance with considerable accuracies and memory usage. Our analysis on different complexity levels shows that the resource efficient 3D CNNs should not be designed too shallow or narrow in order to save complexity. The codes and pretrained models used in this work are publicly available 1 .
Introduction
Ever since AlexNet [18] won the ImageNet Challenge (ILSVRC 2012 [24]), convolutional neural networks (CNNs) have dominated the majority of the computer vision tasks. Then the primary trend has been more on creating deeper and wider CNN architectures to achieve higher accuracies [10,26,29]. However, in real world computer vision applications such as face recognition, robot navigation and augmented reality, the tasks need to be carried out under runtime constraints on a computationally 1 https://github.com/okankop/Efficient-3DCNNs limited platform. Only recently, there has been a rising interest in building resource efficient convolutional neural networks but it is limited with 2-dimensional kernels (2D) [13,11,37,20,25].
The same history is repeating for CNNs with 3dimensional (3D) kernels [9]. Since the large video datasets became available, the primary trend for video recognition tasks is again to achieve higher accuracies by building deeper and wider architectures [31,22,32,9,6]. Considering the fact that 3D CNNs achieve better performance for video recognition tasks compared to 2D CNNs [3], it is very likely that this 3D CNN architecture search will continue until the achieved accuracies saturate. However, real-world applications still require resource efficient 3D CNN architectures taking runtime, memory and power budget into account. This work aims to fill this research gap.
In this paper, we first have created the 3D versions of the well-known 2D resource efficient architectures: SqueezeNet, MobileNet, ShuffleNet, MobileNetV2 and ShuffleNetV2. We have evaluated t-he performance of these architectures on three publicly available benchmarks:
(1) Kinetics-600 dataset [3] to learn models' capacities.
(2) Jester dataset [1] to learn how well the models capture the motion.
(3) UCF-101 dataset [27] to evaluate the applicability of transfer learning for each model.
The computational complexity of the implemented architectures are measured in terms of floating point operations (FLOPs), which is widely used metric among resource efficient architectures. In this paper, the number of FLOPs refers to the number of multiply-adds. However, as highlighted by [20], the number of FLOPs is an indirect metric which does not give an actual performance indication like speed or latency. Therefore, for all the implemented architectures we have also evaluated their run-time performance on two different platforms, which are Nvidia Titan XP GPU and Jetson TX2 embedded system-on-module (SoM) with integrated 256-core Pascal GPU.
Related Work
Lately, there is a rising interest in building small and efficient neural networks [13,11,20,23,34,7]. The common approaches used for this objective can be categorized under two categories: (i) Accelerating the pretrained networks, or (ii) directly constructing small networks by manipulating kernels. For the first one, [7,8,33,21] proposes to prune either network connections or channels without reducing the performance of pretrained models. Additionally, many other methods apply quantization [23,28,34] or factorization [19,14,15] for the same objective. However, our focus is on the second one for directly designing small and resource efficient 3D CNN architectures.
Current well-known resource efficient CNN architectures are all constructed with 2D convolutional kernels and benchmarked at ImageNet. SqueezeNet [13] reduced the number of parameters and computation while maintaining the classification performance. MobileNet [11] makes use of depthwise separable convolutions to construct lightweight deep neural networks. The depthwise separable convolutions factorize the standard convolutions into a depthwise convolution followed by a 1x1 pointwise convolution. Compared to standard convolutions, depthwise separable convolutions use between 8 to 9 times less parameters and computations. ShuffleNet [37] proposes to use pointwise group convolutions and channel shuffle in order to reduce computational cost. MobileNetv2 [25] makes use of the inverted residual structure where the intermediate expansion layer uses depthwise convolutions. ShuffleNetV2 [20] builds on top of ShuffleNet [37] using channel split together with channel shuffle which realizes a feature reuse pattern.
These architectures intensively make use of group convolutions and depthwise separable convolutions. Group convolutions are first introduced in AlexNet [18] and efficiently utilized in ResNeXt [35]. Depthwise separable convolutions are introduced in Xception [5] and they are the main building blocks for majority of lightweight architectures.
All of the above-mentioned resource efficient architectures are 2D CNNs. They are designed to operate on static images and evaluated on a very large benchmark (i.e., Ima-geNet). To the best of our knowledge, this is the first work that proposes and evaluates resource efficient 3D CNNs on large scale video benchmarks.
3D CNNs such as well-known C3D [30] require significantly more parameters and computations compared to their 2D counterparts which make them harder to train and prone to overfitting. With the availability of large scale video datasets such as Sports-1M [16], Kinetics-400 [3], this problem is solved. Moreover, [3] proved that 3D CNNs achieve better accuracies compared to 2D CNNs for video classification task. Consequently, 3D CNN architecture search is an active area in research community to achieve higher accuracies.
Several 3D CNN architectures have been proposed recently. Carreira et al. propose Inflated 3D CNN (I3D) [3], where the filters and pooling kernels of a deep CNN are expanded to 3D, making it possible to leverage successful ImageNet architecture designs and their pretrained models. P3D [22] and (2+1)D [32] propose to decompose 3D convolutions into 2D and 1D convolutions operating on spatial and depth dimensions, respectively. In [9], 3D versions of famous ImageNet architectures such as ResNet [10], Wide ResNet [36], ResNeXt [35] and DenseNet [12] are evaluated and it has been shown that ResNeXt achieves better results compared to others. Recently, Feichtenhofer et al. propose a novel architecture named SlowFast [6], which uses a Slow pathway, operating at low frame rate, to capture static content of a video, and a Fast pathway, operating at high frame rate, to capture the dynamic content of a video.
Up to now, nearly all the 3D CNN architectures in the literature are heavyweight, requiring 10s and even 100s billions of floating point operations (FLOPs). Moreover, majority of these architectures also use optical flow modality, which increases the complexity even further. Our focus in this work is to evaluate 3D CNNs having less than 1 GFLOPs. Consequently, we have implemented the 3D version of SqueezeNet [13], MobileNet [11], MobileNetV2 [25], ShuffleNet [37] and ShuffleNetV2 [20] for 4 different complexity levels and then evaluated them on 3 different video benchmarks. We have evaluated our architectures only using RGB modality without computing costly optical flow modality.
Resource Efficient 3D CNN Architectures
In this section, we explain the details of the resource efficient 3D CNN architectures that have been proposed and evaluated within the scope of this work. We initially introduce the 3D versions of the well-know resource efficient 2D CNN architectures by explaining their building blocks and networks structures. Then we compare these models in terms of number of layers, nonlinearities, and skip connections. We conclude with training details of the models.
3D Versions of Well-known Architectures
In this section, we give the implementation details of our resource efficient architectures with 3-dimensional kernels, which are converted from well-know resource efficient 2D CNN architectures. Main building blocks of each architecture are depicted in Fig. 1. The input is always considered as a clip of 16 frames with spatial resolution of 112 pixels. For all of the 3D CNN architectures, first convolutions always apply stride of (1,2,2). For the rest of the architectures, depth dimension is reduced together with spatial dimensions.
3D-SqueezeNet
SqueezeNet [13] is considered as one of very first resource efficient CNN architectures with notable accuracy performance. It achieves the AlexNet [18]-level accuracy with 50 times fewer parameters and less than 0.5 MB model size.
The main building block of SqueezeNet is Fire block whose 3D version is depicted in Fig. 1 (a). As illustrated in Table 1, 3D-SqueezeNet begins with a convolution layer (Conv1), followed by 8 Fire blocks (Fire-2-9), ending with a final convolutional layer (Conv10).
In our experiments, we use SqueezeNet with simple bypass since it achieves the best result in its 2D version for ImageNet. SqueezeNet does not apply depthwise convolutions which is the main building block for majority of re-source efficient architectures. Instead, it uses three strategies to reduce the number of parameters while maintaining accuracy: (i) Replacing 3x3 filters with 1x1 filters, (ii) decreasing the number of input channels to 3x3 filters, and (iii) downsampling late in the network so that convolution layers have large activation maps. Moreover, compared to other resource efficient architectures, SqueezeNet cannot be modified with width multiplier parameter resulting in different complexities. Therefore, it is only experimented with its default configuration as shown in Table 8.
3D-MobileNetV1
MobileNets [11] apply depthwise separable convolutions which have a form that factorize a standard convolution
Layer / Stride
Filter size Output size
Input clip 3x16x112x112 Conv1/s(1,2,2) 3x3x3 64x16x56x56 MaxPool/s(2,2,2) 3x3x3 64x8x28x28 Fire2 128x8x28x28 Fire3 128x8x28x28 MaxPool/s(2,2,2) 3x3x3 128x4x14x14 Fire4 256x4x14x14 Fire5 256x4x14x14 MaxPool/s(2,2,2) 3x3x3 256x2x7x7 Fire6 384x2x7x7 Fire7 384x2x7x7 MaxPool/s(2,2,2) 3x3x3 384x1x4x4 Fire8 512x1x4x4 Fire9 512x1x4x4 Conv10/s(1,1,1) 1x1x1 NumClsx1x4x4 AvgPool/s(1,1,1) 1x4x4
NumCls Table 1: 3D-SqueezeNet architecture. Details of Fire block is given in Fig. 1
(a).
into a depthwise convolution and 1 × 1 convolution, which is called as pointwise convolution. In MobileNet architectures, the depthwise convolution applies a single filter to each input channel and then the pointwise convolution applies a 1 × 1 convolution to combine the outputs of the depthwise convolution. Different from the standard convolution, the depthwise separable convolution involves two layers which separates filtering and combining operations as illustrated in Fig. 1 (b). This process helps to decrease computation time and model size significantly. Unlike all recent popular CNN architectures, MobileNet does not contain skip connections. Therefore, depth of the network cannot be increased too much which hinders gradient flow. Table 2 shows the details of the 3D-MobileNet architecture. 3D-MobileNet begins with a convolutional layer, followed by 13 MobileNet blocks, ending with a linear layer. MobileNet has 28 layers in case the depthwise and pointwise convolutions in each MobileNet block are counted as separate layers.
3D-MobileNetV2
MobileNetV2 [25] is another 2D resource efficient architecture. It builds upon the main idea of MobileNetV1 by using depthwise separable convolutions; however, it introduces two new components: 1) linear bottlenecks between the layers, and 2) shortcut connections between the bottlenecks. The idea behind 1) is both keeping the size of model low by decreasing number of channels and extracting as much as information by applying depthwise convolution after decompressing the data. This convolutional module allows to
Layer / Stride Repeat Output size Input clip 3x16x112x112 Conv(3x3x3)/s(1,2,2) 1 32x16x56x56 Block/s(2x2x2) 1 64x8x28x28 Block/s(2x2x2) 1 128x4x14x14 Block/s(1x1x1) 1 128x4x14x14 Block/s(2x2x2) 1 256x2x7x7 Block/s(1x1x1) 1 256x2x7x7 Block/s(2x2x2) 1 512x1x4x4 Block/s(1x1x1) 5 512x1x4x4 Block/s(1x1x1) 1 1024x1x4x4 Block/s(1x1x1) 1 1024x1x4x4 AvgPool(1x4x4)/s(1,1,1) 1 1024x1x1x1 Linear(1024xNumCls)
1 NumCls Table 2: 3D-MobileNet architecture. Details of Block is given in Fig. 1
(b).
reduce memory usage during inference. On the other hand, 2) allows training faster and construct deeper models like ResNet architectures [10]. Fig. 1 (c) shows the MobileNetV2 block. Table 3 shows the layers of 3D-MobileNetV2 architecture. 3D-MobileNetV2 begins with a convolutional layer, followed by 17 MobileNetV2 blocks, and then a convolutional layer and finally ending with a linear layer.
3D-ShuffleNetV1
According to [37], ShuffleNet provides superior performance compared to MobileNet [11] by a significant margin, which is reported as absolute 7.8% lower ImageNet top-1 error at level of 40 MFLOPs. The model is also reported to achieve 13× actual speedup over AlexNet while maintain-
Layer / Stride
Repeat Output size ing comparable accuracy. The architecture uses two new operations, which are pointwise group convolution and channel shuffle which is depicted in Fig. 1 (d).
As illustrated in Table 4, 3D-ShuffleNet begins with a convolutional layer followed by 16 ShuffleNet blocks, which are grouped into three stages. In each stage, the number of output channels are kept same with the applied Shuf-fleNet blocks. For the next stage, the output channels are doubled and the spatial and depth dimensions are reduced to half. ShuffleNet architecture ends with a final linear layer. In ShuffleNet units, group number g controls the connection sparsity of pointwise convolutions. In this study, the group number is selected as 3.
3D-ShuffleNetV2
In ShuffleNetV2 [20] architecture, channel split operator is introduced different from V1. As illustrated in Fig. 1 (e), at the beginning of each block, the input of c feature channels are split into two branches with c-c and c channels, respectively. One branch remains as identity, and the other branch includes three convolutions with the same input and output channels. Different from ShuffleNet, the two 1×1 convolutions are not groupwise. After the convolutions, the two branches are concatenated and the number of channels keeps the same. At the end of the block, channel shuffle operation is applied to enable information communication between the two branches. Table 5 shows the layers of 3D-ShuffleNetV2 architecture. 3D-ShuffleNetV2 architecture begins with a convolutional layer, followed by 16 ShuffleNetV2 blocks, and then a convolutional layer and finally ending with a linear layer. Similar to 3D-ShuffleNet, the stack of blocks are grouped into three stages, and at each stage the number of output Conv(1x1x1)/s(1,1,1) 1 c 4 x1x4x4 AvgPool(1x4x4)/s(1,1,1) 1 c 4 x1x1x1 Linear 1 NumCls Table 5: 3D-ShuffleNetV2 architecture. Its' main building block is given in Fig. 1 (e) with stride 1 (left) and spatio temporal 2x downsampling (right). The number of channels (c1, c2, c3, c4) for different complexities are given in Table 6.
Output channels 0.25x 0.5x 1.0x 1.5x 2.0x channels are kept same while with the next stage, they are doubled. Different from the 3D-ShuffleNet, the number of channels in each stage are not fixed. Table 6 shows the number of channels (c 1 , c 2 , c 3 , c 4 ) for different levels of complexities. Also, in 3D-ShuffleNet, the number of output channels in the final layer (c 4 ) is same after the third stage, whereas in 3D-ShuffleNetV2, different number of output channels are selected for different levels of complexities (Table 6).
Comperative Analysis
In this section, we compare the experimented architectures according to the number of layers, nonlinearities and skip connections. These design criteria plays an important role for the performance of the architectures. Comparison of the architectures are given in Table 7. For the number of layers, we counted the convolutional and linear layers. For the skip-connections, we have counted the addition or concatenation operations in the architectures. Finally, for the number of non-linearity, we have counted the ReLU operations in one inference time since it is the only non-linearity used for all the architectures.
It is noticeable that comparatively earlier architectures
Training Details
Learning: For the training of the architectures, Stochastic Gradient Descent (SGD) with standard categorical crossentropy loss is applied. For mini-batch size of SGD, largest fitting batch size is selected, which is usually in the order of 128 videos. The momentum, dampening and weight decay are set to 0.9, 0.9 and 1x10 −3 , respectively. When the networks are trained from scratch, learning rate is initialized with 0.1 and reduced 3 times with a factor of 10 −1 when the validation loss converges. For the training of UCF-101 benchmark, we have used the pretrained models of Kinetics-600. We have frozen the network parameters and fine-tuned only the last layer. For fine-tuning, we start with a learning rate of 0.01 and reduced it two times after 30 th and 45 th epochs with a factor of 10 −1 and optimization is completed after 15 more epochs.
Regularization: Although Kinetics-600 and Jester are very large benchmarks and immune to over-fitting, UCF-101 still requires intensive regularization. Weight decay of 1x10 −3 is applied for all the parameters of the network. A dropout layer is applied before the final conv/linear layer of the networks. While dropout ratio is kept at 0.2 for Kinetics-600 and Jester, it is increased to 0.9 for UCF-101.
Augmentation: For temporal augmentation, input clips are selected from a random temporal position in the video clip. If the video contains smaller number of frames than the input size, loop padding is applied. For the input to the networks, always 16-frame clips are used. For Jester benchmark, it is critical to capture the full content of the gesture video in the selected input clip. Therefore, we have applied downsampling of 2 by selected 16 frames from 32 frames for Jester benchmark [17].
For spatial augmentation, we have selected a random spatial position from the input video. Moreover, we have selected a scale randomly from {1, 1 2 1/4 , 1 2 3/4 , 1 2 } in order to perform multi-scale cropping as in [9]. For Kinetics-600 and UCF-101, input clips are flipped with 50% probability. After the augmentations, input clip to the network has the size of 3 x 16 x 112 x 112 referring to number of input channels, frames, width and height pixels, respectively.
Recognition: For Kinetics-600 and UCF-101, we select non-overlapping 16-frame clips from each video sample. Then center cropping with scale 1 is applied to each clip. Using the pretrained models, class scores for each clip is calculated. For each video, we average the scores of all clips. The class with the highest score indicates the class label of the video.
Implementation: Network architectures are implemented in PyTorch and trained with a single Titan Xp GPU.
Experiments
In this section, we first explain the experimented datasets. Then, we discuss about the achieved results for the experimented network architectures together with their run-time performance on both NVIDIA Titan Xp and Jetson TX2 embedded system.
Datasets
• Kinetics-600 dataset is an extension of Kinetics-400 dataset, which contains 600 human action classes, with at least 600 video clips for each action. Each clip is approximately 10 seconds long and is taken from a different YouTube video. There are in total 392,622 training videos. For each class, there are also 50 and 100 validation and test videos, respectively. Since the labels for the test set is not publicly available, we have conducted our experiments on the validation set.
We selected Kinetics-600 benchmark in order to evaluate the capacity of the experimented networks. It is very rare that a real-life application tries to classify 600 different classes. However, these kind of very large-scale datasets are very useful to evaluate the capacity of the networks to learn. Although it is still necessary to capture the motion patterns in the video, the network should especially capture the spatial content in order to identify the correct class label of the video. For example, there are 9 different "eating something" classes where "something" is one of "burger, cake, carrot, chips, doughnut, hotdog, ice cream, spaghetti, watermelon". Although "eating" action is same for all these, the true label can only be identified when the network captures discriminative features of what is being eaten.
• Jester dataset is currently the largest available hand gesture dataset. In each video sample of the dataset, a person performs pre-defined hand gestures in front of a laptop camera or webcam. There are in total 148,092 gesture videos under 27 classes. The dataset is divided into three subsets: Unlike Kinetics-600 benchmark, in Jester dataset, spatial content of the all video samples are same: A person sitting in front of a camera performs a hand gesture from almost the same distance. Moreover, the selection of classes are more focused on the movement of the hand. That is why, Jester benchmark is suitable to inspect the ability of the networks in capturing motion patterns.
• UCF101 dataset is an action recognition dataset of realistic action videos, collected from YouTube. It consists of 101 action classes, over 13k clips and 27 hours of video data. Compared to Kinetics-600 and Jester datasets, UCF-101 contains very little amount of training videos, hence prone to over-fitting. For the evaluation of UCF-101 dataset, we have used only split-1. We selected UCF-101 benchmark in order to inspect the applicability of transfer learning for the experimented network architectures.
Results
In this section, we elaborate on our findings in the experiments that we have conducted for 5 different network architectures, 4 levels of complexity (except for SqueezeNet) on 3 different benchmarks. Moreover, runtime performance of the models are evaluated on 2 different platforms, namely Titan XP and Jetson TX2 embedded system. According to the results in Table 8, the following conclusions can be inferred:
Accuracy: (i) The deeper architectures (3D-ShuffleNet, 3D-ShuffleNetV2, 3D-MobileNetV2) achieve better results compared to shallower architectures (3D-SqueezeNet, 3D-MobileNetV1). Accordingly, resource efficient 3D CNNs should not be designed too shallow in order to save complexity.
(ii) Motion patterns are better captured with depthwise convolutions. Since depthwise convolutions have kernels of 3x3x3, they can capture relations in depth dimension together with spatial dimension. The main building block of 3D-MobileNetV2 is the inverted residual block, which expands the number of channels to the input of depthwise convolution layers with an expansion ratio. Therefore, it contains more depthwise convolution filters compared to other architectures. Consequently, it achieves by far best performance in Jester benchmark, although it has inferior results in Kinetics-600 and UCF-101 benchmarks.
(iii) All models showed comparatively similar performance on both Kinetics-600 and UCF-101 datasets. This shows transfer learning is a valid approach for resource efficient 3D CNNs since there is a direct correlation between model performances on these two datasets.
Complexity level: (iv) There is a severe performance degradation if the networks are scaled with very small width multiplier in order to satisfy the required computational complexity. For example, in the first block of the Table 8, we can see that 3D-MobileNetV2 0.2x and 3D-ShuffleNetV2 0.25x achieve 5-9% worse than 3D-ShuffleNetV1 0.5x and 3D-MobileNetV1 0.5x in Kinetics-600 benchmark. Capacity of the models degrades severely as the width multiplier gets smaller, especially when it is less than 0.5. We can see the same pattern on all three benchmarks that we have experimented.
(v) The main design criteria of the 3D-SqueezeNet is to save number of parameters, not computations. Therefore it has the smallest number of parameters at the highest complexity level. However, it also has around 300 million more FLOPs compared to other architectures since it does not make use of depthwise convolutions.
Runtime performance: (vi) Although the network architectures contain similar FLOPs, some architectures are much faster than others. As highlighted by [20], this is due to several other factors affecting speed such as memory access cost (MAC) and degree of parallelism, which are not taken into account by FLOPs.
(vii) 3D-SqueezeNet is the only architecture that does not make use of depthwise convolutions, hence contains highest FLOPs. However, surprisingly it has the highest runtime performance. This is due to the latest CUDNN [4] library which is specifically optimized for standard convolutions. Similar results can also be observed with ResNet and ResNeXt architectures.
(viii) Runtime performance heavily depends on the hardware that the network architecture is running. For example, for the highest two complexity levels, 3D-ShuffleNetV1 is the faster than 3D-ShuffleNetV2 on GPU, whereas 3D-ShuffleNetV2 achieves higher runtime than 3D-ShuffleNetV1 on Jetson TX2.
State-of-the-art comparison: (ix) Architectures with more parameters and FLOPs like ResNets, ResNeXt-101 and I3D achieve generally better results for datasets measuring the capacity of the tested architectures like Kinetics dataset as evaluated and shown in Table 8. However, network design makes a huge difference. For example, 3D-ShuffleNetV1 2.0x achieves similar performance with ResNet-18, although ResNet-18 requires 7 times more parameters and 15 times FLOPs .
(x) The architecture design should be done according to the given task. As inverted residual block excels at capturing dynamic motions, 3D-MobileNetV2 1.0x achieves better results than much wider and deeper ResNet-101 (around 20 times more parameters and FLOPs) at Jester benchmark.
Conclusion
In recent years, the research in action recognition has mostly focused on obtaining the best accuracy by generating deep and wide CNN architectures. However, real-world applications require resource efficient architectures that take runtime, memory and power budget into account. Recently, several resource efficient 2D CNN architectures have been proposed. However, there is a lack of architectures for 3D counterparts. This work aims to fill this research gap.
The proposed architectures are generated by implementing the 3D versions of Squeezenet, MobileNet, MobileNetV2, ShuffleNet, ShuffleNetV2 architectures for 4 different complexity levels. The performance of these architectures have been evaluated using 3 different benchmarks, which are selected according to analyze models' capacities, how well the models capture the motion and the applicability of transfer learning for each model.
According to the analysis for 4 different complexity levels, the results show that these resource efficient 3D CNN architectures provide considerable classification performances. Using the width multiplier, the capacity of the architectures can be modified flexibly. The results on Jester benchmark show that depthwise convolutions are very good at capturing motion patterns. Moreover, nearly all models run in real-time both at Titan XP and Jetson TX2. As the results proved the applicability of transfer learning, these architectures can be used for other real-world applications by using pretrained models.
Figure 1 :
1Main building block for each resource efficient 3D CNN architecture. F is the number of feature maps and D × H × W stands for Depth × Height × Width for the input and output volumes. DWConv and GConv stand for depthwise and group convolution, respectively. BN and ReLU(6) stand for Batch Normalization and Rectified Linear Unit (capped at 6), respectively. (a) SqueezeNet's Fire block; (b) MobileNet block; (c) left: MobileNetv2 block, right: MobileNetv2 block with spatiotemporal downsampling (2x); (d) left: ShuffleNet block, right: ShuffleNet block with spatiotemporal downsampling (2x); (e) left: ShuffleNetv2 block, right: ShuffleNetv2 block with spatiotemporal downsampling (2x).
Table 3 :
33D-MobileNetV2 architecture. Block is inverted residual
block whose details are given in Fig. 1 (c) with stride 1 (left) and
spatio temporal 2x downsampling (right).
Table 4 :
43D-ShuffleNet architecture. Its' main building block is
given in Fig. 1 (d) with stride 1 (left) and spatio temporal 2x down-
sampling (right).
Table 6 :
6The number of channels used in 3D-ShuffleNetv2 architecture for different levels of complexities.
Table 7 :
7Comparison of resource efficient 3D architectures according to the number of layers, non-linearity and skip-connections.(i.e. SqueezeNet and MobileNetV1) have smaller num-
ber of layers, non-linearity and skip-connections. On the
other hand, recent resource efficient architectures (i.e. Shuf-
fleNetV1, ShuffleNetV2 and MobileNetV2) are deeper, in
the order of 50 layers and 30 non-linearity. Corollary, they
require more skip connections in order to facilitate better
gradient update mechanism.
Table 8 :
8Comparison of resource efficient 3D architectures over video classification accuracy, number of parameters and speed on two
different platforms and four levels of computation complexity. The calculations of MFLOPs, parameters and speeds are done for Kinetics-
600 benchmark. For speed calculations (clips per second (cps)), the used platforms are Titan Xp and Jetson TX2; and the batch size is set
to 8. All models takes 16 frames input with 112 x 112 spatial resolution except for I3D, which takes 64 frames input with 224 x 224 spatial
resolution.
training set (118,562 videos), validation set (14,787 videos),
and test set (14,743 videos). Since the labels for test set is
not publicly available, we have conducted our experiments
on the validation set.
AcknowledgementsWe give our special thanks to Stefan Hörmann for his assistance to this work. We also gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU and Jetson TX2 Development Kit used for this research.
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| Experimentalists seeking to improve the coherent lifetimes of quantum bits have generally focused on mitigating decoherence mechanisms through, for example, improvements to qubit designs and materials, and system isolation from environmental perturbations. In the case of the phase degree of freedom in a quantum superposition, however, the coherence that must be preserved is not solely internal to the qubit, but rather necessarily includes that of the qubit relative to the "master clock" (e.g. a local oscillator) that governs its control system. In this manuscript we articulate the impact of instabilities in the master clock on qubit phase coherence, and provide tools to calculate the contributions to qubit error arising from these processes. We first connect standard oscillator phase-noise metrics to their corresponding qubit dephasing spectral densities. We then use representative lab-grade and performance-grade oscillator specifications to calculate operational fidelity bounds on trapped-ion and superconducting qubits with relatively slow and fast operation times. We discuss the relevance of these bounds for quantum error correction in contemporary experiments and future large-scale quantum information systems, and discuss potential means to improve master clock stability. arXiv:1602.04551v2 [quant-ph] | null | [
"https://arxiv.org/pdf/1602.04551v2.pdf"
]
| 119,272,882 | 1602.04551 | a34a72d357f7b4dc5d4cd2b20747cebe9ff105dc |
The role of master clock stability in scalable quantum information processing
(Dated: August 17, 2016)
Harrison Ball
School of Physics
ARC Centre for Engineered Quantum Systems
The University of Sydney
2006NSWAustralia
William D Oliver
Department of Physics
Massachusetts Institute of Technology
02139CambridgeMAUSA
MIT Lincoln Laboratory
02420LexingtonMAUSA
Michael J Biercuk
School of Physics
ARC Centre for Engineered Quantum Systems
The University of Sydney
2006NSWAustralia
The role of master clock stability in scalable quantum information processing
(Dated: August 17, 2016)
Experimentalists seeking to improve the coherent lifetimes of quantum bits have generally focused on mitigating decoherence mechanisms through, for example, improvements to qubit designs and materials, and system isolation from environmental perturbations. In the case of the phase degree of freedom in a quantum superposition, however, the coherence that must be preserved is not solely internal to the qubit, but rather necessarily includes that of the qubit relative to the "master clock" (e.g. a local oscillator) that governs its control system. In this manuscript we articulate the impact of instabilities in the master clock on qubit phase coherence, and provide tools to calculate the contributions to qubit error arising from these processes. We first connect standard oscillator phase-noise metrics to their corresponding qubit dephasing spectral densities. We then use representative lab-grade and performance-grade oscillator specifications to calculate operational fidelity bounds on trapped-ion and superconducting qubits with relatively slow and fast operation times. We discuss the relevance of these bounds for quantum error correction in contemporary experiments and future large-scale quantum information systems, and discuss potential means to improve master clock stability. arXiv:1602.04551v2 [quant-ph]
I. INTRODUCTION
A fundamental challenge to the broader application of quantum information science is the management of error in fragile quantum hardware [1,2]. The need for higher-fidelity performance motivates research at all architectural levels [3], from theoretical studies of fault tolerance and analyses of quantum error correction (QEC) implementations down to experimental improvements in the operational fidelity of elemental devices. A familiar aspect of this challenge is decoherence, a process by which even idle qubits undergoing free evolution (i.e., the identity operator) will gradually lose their stored quantum information, rendering them useless in subsequent computation.
A prevalent component of decoherence is dephasing, a randomization of the relative phase between the basis states that form a coherent superposition state. Dephasing in qubit systems is commonly attributed to environmental fluctuations of a qubit bias or control parameter, e.g., an external magnetic field, which modulates the qubit-state energy splitting and hence its dynamic phase evolution. In turn, experimentalists have primarily focused their efforts on reducing both the level of environmental noise and the qubit sensitivity to that which remains (c.f. [4][5][6]). As a result, many qubit technologies [7] have witnessed remarkable performance improvements, with dephasing times approaching milliseconds or even seconds (depending on the qubit modality) [6,[8][9][10][11][12][13], and operational fidelities reaching ∼ 99.999% [14][15][16]. Achieving targets such as these -once thought all but impossible -has led many to recognize that scalable, error-corrected quantum computation may become a reality. In essence, by improving qubit coherence, error rates may be reduced below fault-tolerant thresholds, and in principle QEC may be employed to suppress errors at arbitrary system scales.
This prescription, however, belies issues that only arise as qubit coherence and operational fidelity improve. A fundamental example, one widely studied in the context of precision metrology, is that the control and measurement of highly coherent systems must be compared to a suitably coherent reference [17]. Qubit coherence, likewise, is inferred relative to a reference, generally in the form of a local oscillator (LO) used to control and interrogate the system, for instance by inducing Ramsey or Rabi oscillations. Here, the evolution of the qubit's phase degree of freedom in a quantum superposition is effectively being compared against the accumulated ticks of a clock, defined by the LO [17]. While the importance of phase coherence is known to any experimentalist who has failed to lock an oscillator to a stable reference, in the past, qubit coherence times and operational fidelities have generally been limited by environmental noise. At today's performance levels, however, noise in the master clock is beginning to emerge as a contributor to the overall error rate. And, as qubits continue to improve, mitigating master clock instabilities will become material to high fidelity operation.
This manuscript addresses the topic of master clock instability and its impact on qubit coherence and operational fidelities for quantum information applications. We begin by accounting for the master clock in a Hamiltonian treatment of qubit dephasing. From this starting point, we engage in an analysis of the relevance of master-clock instabilities to qubit coherence and dephasing-induced error rates. We identify master-clock phase fluctuations as an emerging source of error in today's qubit systems and one that will certainly become more prominent as qubit error rates continue to improve. Our presentation serves to unify concepts familiar to quantum information [1], quantum control [9,18,19], engineering [20,21], and precision frequency metrology [22] in order to allow the translation of LO phase noise specifications -as presented in experimental data sheets -into gate error probabilities for a variety of canonical single-qubit operations. We consider gates applied to superconducting and trappedion qubits, prominent qubit modalities that represent systems with relatively fast (10 ns) and slow (100 µs) gate times yielding sensitivity to different frequency regions of the oscillator noise spectrum. Calculations presented in this analysis highlight the fact that qubit dephasing errors arising from the phase fluctuations in commercial precision LOs, while not explicitly limiting experiments today, will likely become a significant consideration in the context of large-scale quantum information. They also reveal that many lab-grade oscillators are beginning to limit achievable fidelities in contemporary systems. We address the relevance of different frequency regimes of LO phase noise, and highlight "far-from-carrier" phase noise as contributing to important upper-bounds on qubit operational fidelities. The materials we present both demonstrate a path to mesoscale quantum information systems using existing master clocks and provide new motivation for investment in LO hardware and precision frequency metrology research in order to underpin large-scale quantum information processing. We augment this review with detailed supplementary material, aggregating a comprehensive theoretical foundation to understand clock-induced errors in quantum systems.
II. QUBIT DEPHASING INDUCED BY THE MASTER CLOCK
In a semiclassical picture, qubit phase coherence corresponds to maintaining the dynamical phase of the qubit relative to a reference (master) clock for the system. Implicit in the Bloch sphere representation of a qubit state is, in general, a transformation to a frame co-rotating with the nominal qubit Larmor frequency, ω 0 . In a dephasing process, the relative phase difference between the qubit state and its reference frame evolves stochastically in time, introducing a degree of randomness to the qubit state. Typical formulations assign this stocastic evolution to instability in qubit frequency, induced, for instance, by fluctuations in external fields. Over time, these fluctuations cause the qubit to randomly advance or retreat relative to the co-rotating frame. However, what about the stability of the rotating frame being used as a phase reference? The simple observation that the qubit phase is defined relative to the rotating frame indicates that an experimentalist must consider not only the stability of the qubit, but also the stability of the rotating frame itself, realized through the use of a local oscillator in the experiment. Accordingly, the master clock has a role that is more fundamental than the synchronization of scheduled operations; the master clock determines, in part, the coherence of the underlying qubits.
The impact of generic qubit dephasing may be formally captured through a Hamiltonian formulation, written as the sum of an ideal control component and a randomly fluctuating noise component in the three-dimensional Pauli basis [23].
H(t) = h(t)σ + δh z (t)σ z ,(1)δh z (t) = δh (env) z (t) + δh (LO) z (t)(2)
where we have restricted attention to longitudinal dephasing noise by setting δh x,y (t) = 0. Terms from environmental dephasing δh of these two noise terms accounts for qubit dephasing at all times t.
Within this framework, we can build a connection between the LO-induced dephasing term and the noise properties of local oscillators. A local oscillator with amplitude A(t) and carrier frequency ω LO possesses a phase that is easily partitioned into a "control" component (desired, deterministic phase evolution), φ C (t), and a "noise" component (unwanted, stochastic phase evolution), φ N (t), in the expression for the oscillator signal:
A(t) cos ω LO t + [φ C (t) + φ N (t)]
. Setting the LO frequency to be on resonance with the qubit, ω LO = ω 0 , the stability of the rotating frame is determined by the noisy phase evolution of the LO, φ N (t), as it produces a time-dependent frequency detuning of the LO relative to the qubit through its time derivative, δω LO (t) ≡φ N (t). With this qualitiative understanding in hand, we are able to associate the Hamiltonian term δh (LO)
z (t) = − 1 2φ N (t)
, explicitly linking LO phase fluctuations to a dephasing Hamiltonian with an impact that is indistiguishable from other environmental dephasing sources.
In Fig. 1 the equivalence of the two dephasing terms described above is represented on the Bloch sphere in the frame co-rotating with the resonant driving field. The term δh Hereafter, without loss of generality, we will ignore environmental dephasing by setting δh
III. CALCULATING DEPHASING DUE TO PHASE NOISE IN THE LOCAL OSCILLATOR
We employ the filter transfer function formalism to incorporate noise power spectral densities for the Hamiltonian noise terms into calculations of operational fidelity and qubit coherence [21]. This approach treats the problem of calculating the impact of noise on a quantum system in terms of the overlap of the noise and an effective frequency-domain filter describing the action of the control [24,25]. It adopts concepts that are well known in the engineering literature [26] and has recently been developed and applied to the control of qubits [27][28][29][30].
In essence, the filter transfer function shapes the spectrum of the underlying noise as it couples to the qubit, passing certain bands and rejecting others. We see this quantitatively for a system Hamiltonian in the form of Eq. 1 by writing the fidelity of an arbitrary qubit operation in terms of the overlap integral of S (1) z (ω), the unilateral (one-sided) power spectral density (PSD) [31] of the dephasing field δh z (t), and the filter transfer function G z,l (ω), which captures the action of the control Hamiltonian h(t)σ [21,32]. We express the average fidelity
F av (τ ) ≈ 1 2 1 + exp[−χ(τ )] ,(3)χ(τ ) = 1 π ∞ 0 dω ω 2 S (1) z (ω) l∈ x,y,z G z,l (ω).(4)
The fact that the error integral, χ(τ ), is expressed as a product of the noise power spectral density and filter transfer functions demonstrates how the frequency-domain shaping of the noise determines the ultimate contribution of the noise to operational infidelity. The transfer functions for the control, G z,l (ω), may be calculated analytically and have contributions along all Cartesian directions (indexed by l), as dephasing noise present during a non-commuting control operation (e.g. ∝σ x ) induces both dephasing and amplitudedamping [21,32]. Alternatively, the G z,l (ω) corresponding to a Ramsey pulse sequence would determine dephasing during free evolution. Our objective is to link noisy fluctuations in the LO phase to the quantity S For this we reference a large base of research from the field of precision metrology, where the characterization of time and frequency signals is a key objective [17]. Conveniently, in this discipline the stability of a signal at a notionally fixed frequency is quantified by characterizing temporal fluctuations in the phase of that signal, represented in the Fourier domain. Similar analyses are found in the context of communications engineering and stochastic signal processing.
The quantity of metrological significance [22] in LO characterization is S
(1) φ N (ω) ≡ lim T →∞ 2 T |Φ T (ω)| 2 , the uni-
lateral power spectral density of the LO phase fluctuations over a measurement time T . In this expression, Φ T (ω) =
T /2 −T /2 φ N (t)e −iωt dt
is the complex amplitude of the harmonic Fourier component at frequency ω for the time-gated signal φ N (t), defined for times |t| < T /2 and zero otherwise. The quantity |Φ T (ω)| 2 is the energy density of this harmonic with units of energy per Hertz; angle brackets indicate an expectation value; and stationarity has been assumed to take the limit T → ∞. The phase fluctuations generate sidebands on the carrier -the unadulterated LO signal -at frequencies ω LO ± ω, effectively broadening the observed linewidth of the LO in frequency space (see Fig. 1). The Fourier harmonic ω of the phase instability is therefore measured as an offset from the carrier, and the power of these sidebands is measured over a 1 Hz bandwidth.
The time-derivative of the phase fluctuations is equivalent to a time-dependent detuning of the LO from resonance, as described above, and we may therefore relate their noise spectra to the dephasing power spectrum used in filter-function calculations as 1 4
ω 2 S (1) φ N (ω) = S(1)
z (ω). The factor 1/4 arises from omitting the counter-rotating term in the rotating wave approximation. While S (1) φ N (ω) is used in the metrological community [22], most LO manufacturers use a metric for the single-sideband phase noise,L(ω) = 10 log 10
[ 1 2 S (1) φ N (ω)]
, expressed in logarithmic units of dBc/Hz. The ultimate relationship therefore allows tabulated single-sideband phase-noise specifications to be converted to unilateral dephasing power spectral densities as
S (1) z (ω) = 1 2 ω 2 10L (ω) 10 .(5)
This correspondence enables a connection between the metrics provided by oscillator manufacturers and those used for both precision oscillator characterization and for qubit dephasing calcuations [22], and it is represented graphically in the difference between curves in Figs. 2a-b. A full accounting of all relevant factors and the formal derivation of these quantities is presented in the Supplementary Information in a single consistent notation.
IV. CLOCK ERRORS INDUCED BY STATE-OF-THE-ART LOCAL OSCILLATORS
With Eq 5 in hand, it is possible to use LO phase-noise specifications found on instrumentation data sheets for calculations of qubit coherence and operational error rates. We calculate the operational infidelity 1−F av (τ ) using published L(ω) for representative synthesizers and analytic filter transfer functions as inputs for Eq. 4. We consider two standard operations: the identity operation,Î, and theX gate, defined as a driven rotation through angle π about thex-axis of the Bloch sphere. These choices are representative of the singlequbit operations comprising a universal gate set for gate-based quantum computation. The former allows insight into the coherent lifetime of the qubit under free evolution (T 2 ), while the latter informs how gate fidelity is reduced during driven evo-
z (ω) via Eq. 5. Data extracted from specification sheets and manufacturer measurements. For the "lab-grade" synthesizer, phase noise at frequencies ω/2π < 10 Hz and ω/2π > 10 MHz has been extrapolated following the trending behaviour of manufacturer-supplied data forL(ω). c-d) Calculated infidelity for different classes of quantum logic operations (see main text). We have numerically confirmed that, for evolution times τ < 100 ms, the presence of a sharp low-frequency cutoff below ω/2π < 1 Hz contributes a negligible correction to our calculated results based on extrapolatingL(ω) as described above.
lution by the presence of LO-induced dephasing noise. For each of these "primitive" operations, the corresponding filter transfer function G z,l (ω) is calculated following the techniques presented in Refs. 19, 21, 30, 32-35. In addition to the primitive forms of these operations, we also employ specific dynamic error suppression (DES) strategies [25,[36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] designed to reduce errors due to dephasing noise. Such controls generally constitute time-dependent modulation of the system dynamics with the aim of coherently averaging out slow fluctuations. While these protocols have typically been associated with the mitigation of environmental decoherence, as the reader might expect based on the discussion in Section II, a growing body of literature has shown that errors induced by imperfect control -including LO phase noise -can also be mitigated using these same techniques [9,27,30]. Here, for the identity operation (free evolution), we use the simple example of a spin echo, while for the drivenX gate, we employ a dynamically corrected version of this gate [46][47][48], also described as a Walsh amplitude modulated filter [30,51]. A description and analytic expressions for the filter functions used here may be found in the Supplementary Information.
For this analysis, we have selected two distinct grades of LO to demonstrate the significance of LO choice in quantum control experiments. Figure 2a shows the singlesideband phase noise at 10 GHz for both a "lab grade" synthesizer (Vaunix LMS-123) and a "precision" synthesizer (Keysight/Agilent 8267D OPT-UNY). Across much of the band, as expected, the phase noise for the precision synthesizer is lower -by approximately 40 dB -than the lab-grade unit. For the LOs studied hereL(ω) declines rapidly with offset frequency from the carrier, exhibiting various powerlaw-dependences over the entire frequency range.
Generically, these dependences are captured analytically by the formL(ω) ∝ ω p (i.e., S z (ω) ∝ ω p+2 ). Moving away from the carrier, dominant processes common in LOs include random-walk frequency noise (p = −4), flicker frequency noise (p = −3), white frequency noise (p = −2), flicker phase noise (p = −1), and white phase noise (p = 0). Note that as expected, the dephasing noise mechanisms carry an exponent p + 2 with respect to the corresponding phase noise mechanisms with exponent p (Fig. 2b). Details of the origins of the underlying processes may be found in [20].
Calculations of infidelity for the four operations outlined above are presented in Fig. 2c-d and summarized in Table I. For the identity operator, we find an approximate improvement of 10 4 in residual error rate due to the use of a highprecision frequency source, and the infidelity remains below 10 −6 out to 100 ms evolution time. In contrast the lab-grade synthesizer induces an error exceeding ∼ 0.1% beyond a few microseconds of evolution time. The plateau-like behavior in infidelity with increasing evolution time is due to the interplay of the dephasing power spectrum and filter-function [52], and is similar to phase-error saturation phenomenology observed in precision oscillator characterization [20].
For both free evolution and driven operations, the deleterious impact of using a LO increases as the duration of the operation grows, with relevance for all major qubit modalities, including trapped-ion [53], superconducting [54], and semiconductor [55] qubits. For instance, for driven operations as long as 100 µs, particularly relevant for atomic qubits, the precision synthesizer only induces an error ∼ 10 −7 while the lab-grade synthesizer induces an error more than 20, 000× larger, reaching ∼ 0.2%. The primary reason for this behavior is the superior far-from-carrier phase-noise performance in the precision synthesizer, especially over the offfset frequency range 1 − 1000 kHz. We note that the selected extrapolation of the high-frequency phase noise beyond ω/2π = 10 MHz (see caption Fig. 2) is particularly favorable to the lab-grade LO, meaning that calculated error rates at the shortest timesτ < 100 ns -may underestimate the actual infidelities.
An interesting observation is that in the presence of these noise power spectral densities and for the realistic evolution times selected for our calculations, DES protocols have a minimal impact on gate performance. In Fig. 2c-d , over the entire range of times -10 ns . . . 100 ms -the use of these protocols offers only a small, sporadic improvement and, over several time spans, can even degrade the operational fidelity (note: these protocols may still give a substantial net improvement in the presence of more dominant environmental noise). This performance is explained by considering that, whileL(ω) declines rapidly with offset frequency, the ω 2 factor that transforms this spectrum to S z (ω) reveals a high-frequency dominance of the resulting dephasing noise, in particular, for the precision LO, which exhibits an approximately Ohmic spectrum (S z (ω) ∝ ω, see Fig. 2b) [56,57]. DES is known to perform poorly for spectra with strong high-frequency content, where the noise evolves rapidly compared to the control and the physics of coherent averaging fails: a violation of the so-called decoupling limit.
These results make clear that reducing far-from-carrier phase noise has the potential to provide augmented fidelities for quantum operations. However it remains a question how well one might do by improving the quality of the LO. We consider achievable gains in operational fidelity by calculating a lower-bound to the error imparted by the LO's thermal noise floor; even if LO hardware were improved we could do no better than saturating the thermal noise floor across the control bandwidth. This is generically quoted as -174 dBm/Hz for a matched load at 290K. This is an absolute noise power, meaning that for LOs with power 0 dBm the single-sideband phase noise floor will take a valueL min = −174 dBc/Hz, and it will worsen (improve) with decreasing (increasing) LO power. We assume an otherwise ideal LO subject to this minimum noise floor [58] by setting the phase noise to the constant valueL(ω) =L min over the effective control bandwidth ω ∈ [0, ω c ]. Using this spectrum we calculate the resulting upper-bound on operational fidelity in the presence of this noise using χ min ≈ (κω c /2π)10L min /10 (see Supplementary Information). In this expression κ is a characteristic scaling factor depending on the control protocol, and χ min is approximately independent of τ in the limit ω c /2π τ −1 . This straightforward calculation reveals that broadband LO phase noise due to thermal effects at room temperature imposes a non-negligible upper bound on gate fidelity, as shown in Fig. 3. For instance, considering a typical bandwidth ∼ 20 GHz the thermal noise floor induces qubit infidelity in excess of 10 −8 . Reducing the thermal noise floor to that associated with a 4K bath (-192 dBc/Hz) improves the fidelity by two orders of magnitude. Similarly, restricting the control bandwidth to 10 MHz sets the infidelity < 10 −11 .
V. DISCUSSION
The calculations we have presented demonstrate that master clock phase fluctuations are an emerging consideration for quantum information applications. Indeed, lab-grade oscillators may already limit the performance of today's qubits with gate times on microsecond scales. On the other hand, using high-performance precision LOs, the calculated error rates for I andX operations with both the fast and slow gates considered here are at several orders of magnitude smaller than the current state-of-art (Table 1). Therefore, although significantly more expensive, the use of such precision LOs is adequate for near-term, proof-of-concept experimental demonstrations with contemporary qubits.
Our results also enhance existing arguments about the merits of qubit modalities that accommodate fast control pulses. These arguments are typically based on a practical "clockspeed" assessment, wherein a technology with faster gates will simply execute an algorithm more quickly. Here, in considering LO phase noise, we find that reducing operation times from 10 µs to 10 ns generally also reduces infidelity (Fig. 2 c-d). The improvement can be substantial (3-4 orders of magnitude) for lab-grade LOs, and it remains about an order of magnitude for the precision LO considered here. The intuition is that shorter pulses are subject to phase fluctuations for less time and, therefore, suffer less dephasing-induced errors.
However, arguments in favor of shorter control operations must be made with a cognizance of the actual LO noise spectrum and the control bandwidth. First, shorter operations ac-cess higher frequencies in the LO spectrum, and, as illustrated in Fig. 2, far-from-carrier phase noise dominates residual errors for commercial sources. Therefore, there may in principle be cases where decreasing the pulse duration actually increases the integrated noise level. Second, bandwidthdependent thermal-noise floors will eventually pose upper bounds on operational fidelities that are more strict when using short control pulses possessing higher bandwidths. And, when they do, suppressing the effective thermal noise floor by further engineering the performance of room-temperature LOs and/or developing cryo-compatible LOs [59] embedded in cryogenic control architectures [60] may play an important role in the development of future, high-fidelity quantum information systems.
Similarly, whereas DES protocols are reasonably good at suppressing dephasing error due to low-frequency environmental fluctuations, they are less efficient at suppressing the impact of LO phase noise (Fig. 2a) due to the high-frequency weight in the dephasing power spectrum (Fig. 2b). In fact, over certain frequency bands in Figs. 2c-d, the DES protocols (dashed lines) can actually enhance error due to LO phase noise. Noise at higher frequencies evolves too rapidly to be coherently averaged by the control, resulting in the filter transfer functions for the DES protocols passing noise in this band. The detailed spectrum ultimately determines whether shorter pulses and DES protocols yield a net win despite the higher noise far from the carrier.
Moving to mesoscale quantum information, such as quantum simulations and/or prototype logical-qubit demonstrations with a few hundred qubits [61][62][63], we believe these results will serve to inform error budgets and motivate careful selection of the precision LO in use. Looking forward, however, the bounds on error rates calculated here for both free and driven evolution with precision oscillators are, in the view of these authors, remarkably high and will likely limit performance in larger-scale systems, particularly in the context of quantum error correction.
It is widely viewed that quantum error correction, QEC, will be necessary to achieve sustained operation in faulty qubit systems. Error bounds due to LO phase instability (of the type calculated here) place architectural constraints on the error correction protocol: the QEC cycle must be completed within a time corresponding to a fixed phase-noise-induced error rate. The tabulated maximum QEC cycle times appearing in Table II show differences of more than 10 5 in the permissible QEC cycle times for the two oscillators considered here.
The use of a precision LO in this context readily supports phase noise infidelities at the level 10 −7 , an order of magnitude below the most pessimistic fault-tolerance thresholds, over a time period 600 µs. This provides strong evidence that, in principle, we should be able to achieve fault-tolerance for QEC with existing oscillator technology. However, scalability requires consideration of corrected fault-tolerant error rates that are sufficiently low to permit algorithmic execution with manageable resource levels [1], a target that grows increasingly challenging for larger-scale applications [2]. Broadly, one would aim to have a logical error rate that scales as the inverse problem size, with only a small chance of a single logical error during the algorithmic execution. This results in calculated logical qubit error rates reaching O(10 −15 ) in the context of Shor's algorithm for factorization of medium-sized keys [64].
Achieving such extraordinary logical-error-rate targets for large-scale machines will involve a consideration of both the physical-qubit error rate and the resource levels required to implement QEC. As error rates approach the fault-tolerance threshold, the encoding resource requirements diverge in time and qubit numbers. Stated in an alternate way, the efficiency of QEC grows as the error rate is suppressed relative to the fault-tolerance threshold. Accordingly, the error floors imposed by LO phase instability must be traded off at a systemlevel against the overheads associated with QEC encoding, for instance in the number of required physical qubits encoding a single logical bit. Calculations suggest that for the Bacon-Shor code, the presence of clock-induced errors near ∼ 10 −7 would require a factor 100 − 1000 times more resources relative to physical-qubit error-rates near 10 −10 . While the LO is only part of the story -errors due to environmental qubit dephasing must also be reduced -one will also need to improve LOs to achieve these levels [64]. Based on current understanding of the relative challenge associated with QEC encoding in either large surface codes or concatenated schemes, and the continuing, anticipated improvements in physical qubit error rates, we believe that the clock-induced error rates we have identified here motivate investment in ultra-low-phase noise LO development.
Another major consideration in the context of QEC is that clock-induced errors are highly correlated in space and time, posing a challenge to existing QEC analyses which have focused primarily on independent and identically distributed stochastic error models. As experimental teams incorporate clock distribution in quantum information architectures [60,65,66], the complexity of QEC analyses will need to be augmented in order to handle the effects of such correlated errors.
While we have focused on phase fluctuations far from the carrier -i.e., on short time scales such as the physical qubit gate time or the QEC cycle period -attention to long-term LO stability is also required. Slow phase diffusion of the LO i.e., close to carrier phase noise -causes substantial error Fig. 2b. Driven-operation error rates (see Fig. 2c) yield similar results. accumulation over long times. In the context of a single LO, such errors appear adiabatic and are generally correctable provided the timescale of the phase diffusion is much longer than the QEC cycle period. By contrast, in systems with multiple control generators, long-time instabilities may be detrimental if the generators exhibit phase diffusion with respect to oneanother. In such cases, the diffusion represents a temporal, stochastic analog to the spatial, deterministic "clock skew" observed in classical semiconductor chips, and it is not generally accommodated by standard quantum error correction protocols. To give a concrete example, whereas generators A and B may individually exhibit correctable slow phase diffusion, the axis for anX gate on generator A may adiabatically skew to be (in an extreme case) in quadrature with the equivalentX gate on generator B. A logical controller directs generator A to apply anX gate to a qubit A, but the rotation is aŶ rotation relative to generator B and the qubits it drives.
Temporally correlated, low-frequency LO phase fluctuations can also introduce challenges in evaluating relevant gate errors for QEC, due to the presence of potential biases in common quantum verification processes such as randomized benchmarking [67]. In evaluating the impact of such errors, it is important to distinguish the contribution of slow LO instabilities to single-operation error rates from their contribution to empirical estimates of qubit fidelities drawn from tomographic measurements over repeated trials. The process of data acquisition and averaging over many individual experiments can contribute error due to slow drifts that dominate the actual error rate experienced in any individual operation or evolution period (see [35] for details).
In summary, we have presented a comprehensive review of critical issues relating to master-clock-induced errors for quantum information applications. We have unified established concepts from frequency metrology and quantum information, permitting experimentalists to translate common LO hardware specifications into estimates of qubit coherence and operational fidelities. Analyses employing information available on representative LO datasheets have revealed that farfrom-carrier phase noise poses a performance limiting upperbound on operational fidelities. As a result we expect to see a growing emphasis on high-performance LO synthesis chains across all technology platforms, and we foresee a growing importance of LO-induced errors in the design of large-scale quantum information systems. We encourage future studies to examine not only clock synthesis, but clock distribution, with an eye towards architectural impacts of skewed clock distribution in quantum systems with single and multiple LOs.
VI. METHODS
In the appropriate interaction picture, environmental dephasing processes and those induced by LO phase fluctuations are formally equivalent. We begin with the physical system Hamiltonian ( = 1) in the laboratory frame for a qubit possessing a nominal transition (angular) frequency ω0, driven by a local oscillator with carrier fre-quency ωLO,
HS = 1 2 ω0σz + 1 2 δω0(t)σz
+ Ω(t) cos ωLOt + [φC (t) + φN (t)] σx.
Appearing in this expression are the operatorsσi representing the Pauli matrices in the Cartesian coordinate basis. The first term corresponds to the Hamiltonian of the qubit under free evolution, while the second term captures environmental noise (e.g. magnetic field fluctuations), which effectively change the nominal qubit transition frequency by an amount δω(t) as ω0 → ω0 + δω0(t).
The third term corresponds to the qubit-field interaction with amplitude Ω(t) driven by a local oscillator with nominal carrier frequency ωLO. The time-dependent phase of the LO is partitioned into a "control" component (desired phase evolution), φC (t), and a "noise" component (unwanted phase fluctuations), φN (t).
Moving to the interaction picture co-rotating with the carrier frequency and making the rotating-wave approximation we obtain
H (ω LO ) I = 1 2 (ω0 − ωLO)σz + 1 2 δω0(t)σz + 1 4 Ω(t) e −i[φ C (t)+φ N (t)]σ + + e i[φ C (t)+φ N (t)]σ −(8)
where we define the qubit ladder operatorsσ± =σx ± iσy. The LO phase fluctuation produces a time-dependent frequency detuning through its time derivative, δωLO(t) ≡φN (t), effectively transforming the LO frequency as ωLO → ωLO + δωLO(t).
In order to make this phenomenology explicitly comparable with environmental dephasing, we perform a second interaction-picture transformation
H (ω LO ,φ N ) I ≡ U † φ N H (ω LO ) I Uφ N − Hφ N(10)
where Uφ N (t) = exp[−i φ N (t) 2σ z ] is the evolution operator under the dephasing Hamiltonian Hφ N ≡ 1 2φ N (t)σz induced by phase fluctuations in the LO. Setting the static LO detuning to zero (ω0 − ωLO = 0) the transformed system Hamiltonian subject to LO phase fluctuations thereby takes the form
H (ω LO ,φ N ) I = 1 2 δω0(t)σz − 1 2φ N (t)σz
The first two terms in this expression represent the two dephasing terms introduced in the main text, δh
ACKNOWLEDGMENTS
The authors acknowledge J. J. Bollinger for motivating careful consideration of the role of LO phase noise on qubit coherence and for discussions on phase noise. We acknowledge useful conversations with S. Shankar and R. J. Schoelkopf, who provided motivation for the choice of lab-grade synthesizer. Work partially supported
(FIG. 1 .
1env) z (t) and LO-induced dephasing, δh (LO) z (t), appear on an equal footing in this formulation (for full derivation, see Methods and Supplementary Information). Once a relative phase relationship between the qubit and LO is established via an initial interaction (control pulse), the presence Schematic representation of the correspondence between phase fluctuations and environmental dephasing. A stable qubit interacting with a LO experiencing frequency instability (captured by the phase instability S φ (ω) -the power spectral density of phase fluctuations), induces random phase accumulation in a freely evolving qubit, represented in the Bloch sphere picture. Phase instability is frequently represented in the fourier domain as a broadened lineshape of the LO's carrier. The same phenomenology arises for a LO outputting a perfectly stable sinusoidal signal in the presence of environmental Hamiltonian terms that produce fluctuations in the frequency of the qubit transition.
) induces rotations of the qubit Bloch vector about theẑ-axis of the Bloch sphere, relative to a fixed coordinate frame set by an ideal LO. Conversely, LO instability captured by δh (LO) z (t) induces rotations of the coordinate frame relative to a fixed Bloch vector representing an ideal qubit. From the perspective of the qubit, both terms are sources of dephasing.
) = 0 in order to restrict attention to LO-induced dephasing.
z
(ω) relevant to the Hamiltonian noise terms. Since we are restricting attention to the case δh z (t) = δh (LO) z (t) (omitting environmental noise for convenience), the PSD S
z
(ω) will solely describe fluctuations in the LO phase.
FIG
. 2. a) Local oscillator phase noise expressed asL(ω) for two grades of synthesizer, b) converted to S
FIG. 3 .
3Calculated infidelity floor (fidelity ceiling) imposed by thermal noise at 4K and 290K with variable LO noise bandwidths. This calculation is independent of operation length, τ under the conditions outlined in the main text. Results hold for Ramsey and both primitive and WAMF π rotations. Infidelity floor for spin echo is three times larger. SeeSupplementary Information.
6 × 10 −9 (1 × 10 −10 )
t) cos[φC (t)]σx + sin[φC (t)]σy .
TABLE I .
IComparison of typical error rates for superconduct-
ing (30 ns) vs trapped-ion (10 µs) driven-gate operations, includ-
ing Walsh Amplitude Modulated Filters (WAMF). Main results
calculated assuming an integration bandwidth of 10 MHz. The
temperature-dependent noise-floor for different cutoff frequencies is
also shown -these data are largely independent of operation time and
type. Noise floor calculations strictly valid only for cutoff frequen-
cies high compared with the inverse operation time.
Time to Reach LO-Induced Error Rate pPrecision LO > 100 ms > 100 ms > 100 ms 80 ms 600 µs TABLE II. Time until a qubit physical error rate p is reached due solely to phase fluctuations in the LO. These times may be viewed as an upper-bound on the allowable QEC cycle period. Achievable cycle periods will be reduced due to other error sources. Error rates are derived from free-evolution calculations presented inp
10 −3
10 −4
10 −5
10 −6 10 −7
Lab-Grade LO 4.0 µs
900 ns
200 ns 30 ns <10 ns
W911NF-12-R-0012the ARC Centre of Excellence for Engineered Quantum Systems CE110001013, the Intelligence Advanced Research Projects Activity (IARPA) through the ARO, the US Army Research Office under Contract. H. & A. Harleythe ARC Centre of Excellence for Engineered Quantum Systems CE110001013, the Intelligence Advanced Research Projects Activ- ity (IARPA) through the ARO, the US Army Research Office un- der Contract W911NF-12-R-0012, and a private grant from H. & A. Harley.
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| []
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[
"ON CONJUGATE PSEUDO-HARMONIC FUNCTIONS",
"ON CONJUGATE PSEUDO-HARMONIC FUNCTIONS"
]
| [
"Yevgen Polulyakh [email protected]. \nNAS of Ukraine\n\n"
]
| [
"NAS of Ukraine\n"
]
| []
| We prove the following theorem. Let U be a pseudoharmonic function on a surface M 2 . For a real valued continuous function V : M 2 → R to be a conjugate pseudo-harmonic function of U on M 2 it is necessary and sufficient that V is open on level sets of U .Let M 2 be a surface, i.e. a 2-dimensional and separable manifold, U : M 2 → R be a real-valued function on M 2 . Denote also bythe open unit disk in the plane.of p on M 2 and a homeomorphism T : D → N such that T (0, 0) = p and a function u = U • T : D → R 2 is harmonic and not identically constant.A neighbourhood N is called simple neighbourhood of p.We can even choose N and T from previous definition to comply with the equalityfor a certain n = n(p) ∈ N (see[2]).Let U : M 2 → R be a pseudo-harmonic function on M 2 and V : M 2 → R be a real valued function. Definition 3 (see [1]). A function V is called a conjugate pseudoharmonic function of U in a point p ∈ M 2 if there exist a neighbourhood N of p on M 2 and a homeomorphism T : D → N such that T (0, 0) = p and u = U • T : D → R 2 and v = V • T : D → R 2 | null | [
"https://arxiv.org/pdf/0905.3184v1.pdf"
]
| 17,865,372 | 0905.3184 | 41323eed069af2c403a2f3b1682c3b6c5e31b0e4 |
ON CONJUGATE PSEUDO-HARMONIC FUNCTIONS
19 May 2009
Yevgen Polulyakh [email protected].
NAS of Ukraine
ON CONJUGATE PSEUDO-HARMONIC FUNCTIONS
19 May 2009a pseudo-harmonic functiona conjugatea surfacean interior transformation
We prove the following theorem. Let U be a pseudoharmonic function on a surface M 2 . For a real valued continuous function V : M 2 → R to be a conjugate pseudo-harmonic function of U on M 2 it is necessary and sufficient that V is open on level sets of U .Let M 2 be a surface, i.e. a 2-dimensional and separable manifold, U : M 2 → R be a real-valued function on M 2 . Denote also bythe open unit disk in the plane.of p on M 2 and a homeomorphism T : D → N such that T (0, 0) = p and a function u = U • T : D → R 2 is harmonic and not identically constant.A neighbourhood N is called simple neighbourhood of p.We can even choose N and T from previous definition to comply with the equalityfor a certain n = n(p) ∈ N (see[2]).Let U : M 2 → R be a pseudo-harmonic function on M 2 and V : M 2 → R be a real valued function. Definition 3 (see [1]). A function V is called a conjugate pseudoharmonic function of U in a point p ∈ M 2 if there exist a neighbourhood N of p on M 2 and a homeomorphism T : D → N such that T (0, 0) = p and u = U • T : D → R 2 and v = V • T : D → R 2
are conjugate harmonic functions.
We can choose N and T from previous definition in such way that u(z) = U • T (z) = Re z n + U(p) , v(z) = V • T (z) = Im z n + V (p) , z = x + iy ∈ D , for a certain n = n(p) ∈ N (see [2]).
Definition 4 (see [1]). A function V is called a conjugate pseudoharmonic function of U on M 2 if it is a conjugate pseudo-harmonic function of U in every p ∈ M 2 .
Let us remind following
Definition 6 (see [3] Then the mapping F : M 2 → C,
F (p) = U(p) + iV (p) , p ∈ M 2
is interior. Proof. Suppose that contrary to the statement of Proposition the equality V • γ(τ 1 ) = V • γ(τ 2 ) is valid for certain τ 1 , τ 2 ∈ I, τ 1 < τ 2 .
Since the function V • γ is continuous and a set [τ 1 , τ 2 ] is compact, then following values
d 1 = min t∈[τ 1 ,τ 2 ] V • γ(t) , d 2 = max t∈[τ 1 ,τ 2 ] V • γ(t) , are well defined. Let us fix s 1 , s 2 ∈ [τ 1 , τ 2 ] such that d i = V • γ(s i ), i = 1, 2.
We designate W = (τ 1 , τ 2 ). It is obviously the open subset ofI. Let us consider first the case Assume now that d 1 = d 2 . Since V • γ(τ 1 ) = V • γ(τ 2 ) due to our previous supposition, then either s 1 or s 2 is contained in W .
d 1 = d 2 . It is clear that [τ 1 , τ 2 ] ⊆ (V • γ) −1 (d 1 ) in
Let s 1 ∈ W (the case s 2 ∈ W is considered similarly). Then V The contradiction obtained shows that our initial supposition is false and the function V • γ is strictly monotone on I. We are going to show that the set F (Q) containes a neigbourhood of F (p). At the same time we shall show that p is an isolated point of a level set F −1 (F (p)).
Without loss of generality we can assume that U(p) = V (p) = 0. Let N be a simple neighbourhood of p and T : D → N be a homeomorphism such that for a certain n ∈ N the folloving equality holds true u(z) = U • T (z) = Re z n , z ∈ D (see Definition 1 and the subsequent remark). It is clear that without losing generality we can regard that N is small enough to be contained in Q.
Observe that for an arbitrary level set Γ of U an intersection Γ ∩
T (D) = Γ ∩ N is open in Γ. Consequently, since T is homeomorphism then a mapping v = V • T : D → R is open on level sets of u = U • T : D → R (see Definition 5).
Let us consider two possibilities.
Case 1. Zero is a regular point of the smooth function u = U • T , i. e. n = 1 and u(z) = Re z, z ∈ D.
In this case u −1 (u(0)) = u −1 (U(p)) = T −1 (U −1 (U(p))) = {0} × (−1, 1). According to Proposition 1 the function v is strictly monotone on every segment which is contained in this interval, so it is strictly monotone on {0} × (−1, 1). Consequently, for points z 1 = 0 − i/2 and z 2 = 0 + i/2 the following inequality holds true v(z 1 ) · v(z 2 ) < 0.
Let us note that from previous it follows that V is monotone on the arc β = T ({0}×(−1, 1)) = U −1 (U(p))∩N. And since F −1 (F (p))∩N ⊂ β then F −1 (F (p)) ∩ N = {p} and p is an isolated point of its level set F −1 (F (p)).
Let
d 1 = v(z 1 ) < 0 and d 2 = v(z 2 ) > 0 (The case d 1 > 0 and d 2 < 0 is considered similarly). Denote ε = 1 2 min(|d 1 |, |d 2 |) > 0 .
Function v is continuous, so there exists δ > 0 such that following implications are fulfilled Let us examine a neighbourhood W = (−δ, δ) × (−1/2, 1/2) of 0, which is depicted on Figure 1. It can be easily seen that for every x ∈ (−δ, δ) following relations are valid
|z − z 1 | < δ ⇒ |v(z) − d 1 | < ε , |z − z 2 | < δ ⇒ |v(z) − d 2 | < ε .u(x + iy) = x , y ∈ (−ε, ε) , v(x − i/2) < v(z 1 ) + ε < −2ε + ε = −ε , v(x + i/2) > v(z 2 ) − ε > 2ε − ε = ε .
From two last lines and from the continuity of v on a segment {x} ×
[−1/2, 1/2] it follows that v({x} × [−1/2, 1/2]) ⊇ (−ε, ε). Therefore F • T ({x} × [−1/2, 1/2]) ⊇ {x} × (−ε, ε) , x ∈ (−δ, δ) . Since T (W ) ⊆ N ⊆ Q by the choise of N, then 0 = F (p) ∈ (−δ, δ) × (−ε, ε) ⊆ F • T (W ) ⊆ F (Q) . Case 2.
Zero is a saddle point of u = U • T , i. e. u(z) = Re z n , z ∈ D for a certain n > 1.
In this case
u −1 (u(0)) = T −1 (U −1 (U(p))) = {0} ∪ 2n−1 k=0 γ k , where γ k = {z ∈ D | z = a · exp(πi(k − 1/2)/n), a ∈ (0, 1)}, k = 1, . . . , 2n − 1.
As above, applying Proposition 1 we conclude that function v = V •T is strictly monotone on each arc γ k , k = 1, . . . , 2n − 1. Since v is continuous and 0 is a boundary point for each
γ k , then v(z) = v(0) for all z ∈ k γ k . Therefore, 0 = (F • T ) −1 (F • T (0)) and F −1 (F (p)) ∩N = {p}, i. e. p is the isolated point if its level set F −1 (F (p).
Let us designate by
R k = z ∈ D z = ae iϕ , a ∈ [0, 1), ϕ ∈ π(k−1/2) 2 , π(k+1/2) 2 , k = 0, . . . , 2n − 1
sectors on which disk D is divided by the level set u −1 (u(0)). We also denote
D l = {z ∈ D | Re z ≤ 0} , D r = {z ∈ D | Re z ≥ 0} .
Consider map Φ : D → D given by the formula Φ(z) = z n , z ∈ D. It is easy to see that for every k ∈ {0, . . . , 2n − 1} depending on its parity sector R k is mapped homeomorphically by Φ either onto D l or onto D r . Let a mapping Φ k : R k → D r is given by relation
Φ k = Φ| R k , if k = 2m , Inv • Φ| R k , if k = 2m + 1 , k = 0, . . . , 2n − 1 ,
where Inv : D → D is defined by formula Inv(z) = −z, z ∈ D.
Evidently, all Φ k are homeomorphisms.
We consider now inverse mappings ϕ k = Φ −1 k : D r → D, k = 0, . . . , 2n − 1. By construction all of these mappings are embeddings. Moreover, it is easy to see that
u k (z) = u • ϕ k (z) = Re z , when k = 2m , − Re z , when k = 2m + 1 .
Let us fix k ∈ {0, . . . , 2n − 1}. It is clear that ϕ k homeomorphically maps a domainD r = {z ∈ D | Re z > 0} onto a domain R k = z ∈ D z = ae iϕ , a ∈ (0, 1), ϕ ∈ π(k−1/2) 2 , π(k+1/2) 2 , so with the help of argument similar to the observation preceding to case 1 we conclude that the mappingv k = v•ϕ k |D r :D r → R is open on level sets of the functionů k = u • ϕ k |D r :D r → R. As above, applying Proposition 1 we conclude that functionv k is strictly monotone on each arc α c =ů −1 k (ů k (c + 0i)) = {z ∈D r | Re z = c} , c ∈ (0, 1) . We already know that the function v is strictly monotone on the arcs γ k and γ s , where s ≡ k + 1 (mod 2n). Therefore the function v k = v • ϕ k : D r → R is strictly monotone on the arcs α − = ϕ −1 k (γ k ) = {z ∈ D r | Re z = 0 and Im z < 0} , α + = ϕ −1 k (γ s ) = {z ∈ D r | Re z = 0 and Im z > 0} . Let us verify that v k is strictly monotone on the arc
α 0 = α − ∪ {0} ∪ α + = u −1 k (u k (0)) = {z ∈ D r | Re z = 0} .
Since v k (0) = v(0) = V (p) = 0 according to our initial assumptions and 0 is the boundary point both for α − and α + , then v k is of fixed sign on each of these two arcs.
So we have two possibilities:
• either v k has the same sign on α − and α + , then v k | α 0 has a local extremum in 0; • or v k has different signs on α − and α + , then v k is strictly monotone on α 0 . Suppose that v k has the same sign on α − and α + . We will assume that v k is negative both on α − and α + . The case when v k is positive on α − and α + is considered similarly.
Denote z 1 = 0 − i/2 ∈ α − , z 2 = 0 + i/2 ∈ α + . Let ε = 1 2 min(|v k (z 1 )|, |v k (z 2 )|) > 0 . From the continuity of v k it follows that there existsδ > 0 to comply with the following implications
(1) |z − z 1 | <δ ⇒ |v k (z) − v k (z 1 )| <ε , |z − z 2 | <δ ⇒ |v k (z) − v k (z 2 )| <ε , |z| = |z − 0| <δ ⇒ |v k (z) − v k (0)| = |v k (z)| <ε .
Let c ∈ (0,δ). Then the point w 0 = c + i0 is situated on the curve α c between points w 1 = c − i/2 and w 2 = c + i/2. It follows from (1) that v k (w 1 ) < −ε, v k (w 2 ) < −ε and v k (w 0 ) ∈ (−ε, 0). But these three correlations can not hold true simultaneously since v k is strictly monotone on α c as we already know.
The contradiction obtained shows us that v k has different signs on α − and α + . So, v k is strictly monotone on α 0 . Now, repeating argument from case 1 we find such ε k > 0 and δ k > 0 that the setŴ It is easy to show that W is an open neighbourhood of 0 in D. From (2) and from our initial assumptions it follows that
k = [0, δ k ) × − 1 2 , 1 2 meets the relations (2) F • T • ϕ k (Ŵ k ) ⊇ [0, δ k ) × (−ε k , ε k ) , if k = 2m , F • T • ϕ k (Ŵ k ) ⊇ (−δ k , 0] × (−ε k , ε k ) , if k = 2m + 1 . Let us denote W k = ϕ k (Ŵ k ), W = 2n−1 k=0 W k , δ = min k=0,...,2n−1 δ k > 0 , ε = min k=0,...,2n−1 ε k > 0 .F (Q) ⊇ F (N) ⊇ F • T (W ) ⊇ (−δ, δ) × (−ε, ε) .
So, we have proved that for an arbitrary point p ∈ M 2 and its open neighbourhood Q a set F (Q) contains a neigbourhood of F (p). Hence the mapping F :
M 2 → C is open.
At the same time we have shown that an arbitrary p ∈ M 2 is an isolated point of its level set F −1 (F (p)). It is easy to see now that any level set F −1 (F (p)) can not contain a nondegenerate continuum.
Consequently, the map F is interior. Let us verify that V c has a local extremum in some p ∈ Γ c . Note that the space Γ c is locally arcwise connected, i. e. for every point a ∈ Γ c and its open neighbourhood Q there exists a neighbour-hoodQ ⊆ Q of a such that every two points b 1 , b 2 ∈Q can be connected by a continuous curve in Q. This is a straightforward corollary of the remark subsequent to Definition 1.
Since It is clear that without loss of generality we can choose N so small that either
V c (b 1 ), V c (b 2 ) ⊂ V c (I) if V c (b 1 ) < V c (b 2 ) , V c (b 2 ), V c (b 1 ) ⊂ V c (I) if V c (b 2 ) > V c (b 1 ) .b 1 , b 2 ∈Ô such that V c (b 1 ) < V c (p) < V c (b 2 ) and either V (b) ≤ V (p) for all b ∈Ô or V (b) ≥ V (p) for all b ∈Ô, i. e. p isV (b) = V c (b) ≤ V c (p) = V (p) for every b ∈ N ∩ Γ c or V (b) ≥ V (p) for all b ∈ N ∩ Γ c .
Let for definiteness p is the local maximum of V c and V (b) ≤ V (p) for every b ∈ N ∩ Γ c . The case when p is the local minimum of V c is considered similarly.
On one hand it follows from what we said above that The contradiction obtained shows that our initial assumption is false and V is open on level sets of U.
{U(p)} × (V (p), +∞) ∩ f (D) = ∅ since u −1 (U(p)) = T −1 (Γ c ∩ N) and v(z) = V (T (z)) ≤ V (p) for all z ∈ T −1 (Γ c ∩ N)
Sufficiency. Let U be a pseudo-harmonic function on M 2 and a continuous function V : M 2 → R be open on level sets of U.
From Lemma 1 it follows that the mapping F : M 2 → C, F (p) = U(p) + iV (p), p ∈ M 2 is interior.
Let p ∈ M 2 and N is a simple neighbourhood of p in M 2 . Then there exists a homeomorphism T : D → N. It is straightforward that for the open set N a mapping F N = F | N : N → C is interior and its composition F N • T = F • T : D → C with the homeomorfism T is also an interior mapping. Now from Stoilov theorem it follows that there exists a complex structure on D such that the mapping F •T is holomorphic in this complex structure (see [3]). But from the uniformization theorem (see [4]) it follows that a simple-connected domain has a unique complex structure. So the mapping F • T is holomorphic on D in the standard complex structure. Thus the functions u = Re(F • T ) = U • T and v = Im(F • T ) = V • T are conjugate harmonic functions on D. Consequently, V is a conjugate pseudo-harmonic function of U in the point p.
From arbitrariness in the choise of p ∈ M 2 it follows that V is a conjugate pseudo-harmonic function of U on M 2 . Proof. This statement follows from Theorem 1, Lemma 1 and the Stoilov theorem which says that there exists a complex structure on M 2 such that the interior mapping F (p) = U(p) + iV (p), p ∈ M 2 is holomorphic in this complex structure (see [3]).
Definition 5 .
5Let U and V be continuous real valued functions on a surface M 2 . We say that V is open on level sets of U if for everyc ∈ U(M 2 ) a mapping V | U −1 (c) : U −1 (c) → R is open on the space U −1 (c) in the topology induced from M 2 .Theorem 1. Let U be a pseudo-harmonic function on M 2 . For a real valued continuous function V : M 2 → R to be a conjugate pseudoharmonic function of U on M 2 it is necessary and sufficient that V is open on level sets of U.
First
we will verify one auxiliary statement. Denote I = [0, 1],I = (0, 1) = I \ {0, 1}.
Proposition 1 .
1In the condition of Lemma 1 the following statement holds true. Let γ : I → M 2 be a simple continuous curve and γ(I) ⊆ U −1 (c) for a certain c ∈ R. If the set γ(I) is open in U −1 (c) in the topology induced from M 2 , then the function V • γ : I → R is strictly monotone.
this case. So the open subset γ(W ) of the level set U −1 (c) is mapped by V onto a one-point set {d 1 } which is not open in R and V is not open on level sets of U.
• γ(W ) ⊆ [d 1 , +∞) and the open subset γ(W ) of the level set U −1 (c) can not be mapped by V to an open subset of R since its image containes the frontier point d 1 = V • γ(s 1 ). So, in this case V is not open on level sets of U.
Proof of Lemma 1 .
1Let p ∈ M 2 and Q be an open neighbourhood of p.
Figure 1 .
1Figure 1.
Figure 2 .
2Figure 2.
Proof of Theorem 1 .
1Necessity. Let U, V : M 2 → R be conjugate pseudoharmonic functions on M 2 (see Definitions 3 and 4). Obviously, V is continuous on M 2 . Suppose that contrary to the statement of Theorem there exists such c ∈ R that V is not open on the level set Γ c = U −1 (c) ⊂ M 2 , i. e. a map V c = V | Γc : Γ c → R is not open on Γ c in the topology induced from M 2 .
the map V c is not open by our supposition, then there exists an open subsetO of Γ c such that its image R = V c (O) is not open in R. Therefore there is a point d ∈ R \ Int R. Fix p ∈ V −1 c (d) ∩ O.Let us show that p is a point of local extremum of V c . Fix a neigh-bourhoodÔ ⊆ O of p such that every two points b 1 , b 2 ∈Ô can be connected by a continuous curve β b 1 ,b 2 : I → Γ c which meets relations β(0) = b 1 , β(1) = b 2 and β(I) ⊆ O. It is clear that an image of a pathconnected set under a continuous mapping is path-connected, therefore following inclusions are valid
Evidently, p is not an interior point of V c (Ô) since it is not the interior point of V c (O) by construction and V c (Ô) ⊆ V c (O). Then there does not exist a pair of points
the point of local extremum of V c . Now, since V is the conjugate pseudo-harmonic function of U in the point p (see Definition 3), we can take by definition a neighbourhood N of p in M 2 and a homeomorphism T : D → N such that a map f : D → C f (z) = u(z) + iv(z) , z ∈ D is holomorphic on D. Here u = U • T : D → R and v = V • T : D → R.
by construction. Therefore a point U(p) + iV (p) = f (T −1 (p)) is not the interior point of a set f (D). On the other hand it is known that the holomorphic map f is open, so the point f (T −1 (p)) must be the interior point of the domain f (D).
Corollary 1 .
1Let U, V : M 2 → R be conjugate pseudoharmonic functions on M 2 . Then there exists a complex structure on M 2 with respect to which U and V are conjugate harmonic functions on M 2 .
). A mapping G : M 2 In order to prove theorem 1 we need following Lemma 1. Let U be a pseudo-harmonic function on M 2 . Let a real valued continuous function V be open on level sets of U.1 → M 2
2 of a surface M 2
1 to
a surface M 2
2 is called interior if it complies with conditions:
1) G is open, i. e. an image of any open subset of M 2
1 is open in
M 2
2 ;
2) for every p ∈ M 2
2 its full preimage G −1 (p) does not contain any
nondegenerate continuum (closed connected subset of M 2
1 ).
A topological characterization of pseudo-harmonic functions. Y Tôki, Osaka Math. Journ. 31Y. Tôki, A topological characterization of pseudo-harmonic functions, Osaka Math. Journ., vol. 3, No 1, 1951, P. 101-122.
Topological methods in the theory of functions of a complex variable. M Morse, PrincetonM. Morse, Topological methods in the theory of functions of a complex variable, Princeton, 1947.
S Stoilow, Leçons sur les principes topologiques de la théorie des fonctions analytiques. Paris; ParisGauthier-Villars2nd. editionStoilow S., Leçons sur les principes topologiques de la théorie des fonctions analytiques, 2nd. edition, Paris, Gauthier-Villars, Paris, 1938.
O Forster, Lectures on Riemann Surfaces. Springer Graduate Texts in Math81O. Forster, Lectures on Riemann Surfaces, Springer Graduate Texts in Math, V. 81, 1981.
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"Lessons learnt from the recent EURADOS intercomparisons in computational dosimetry"
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"Hans Rabus \nPhysikalisch-Technische Bundesanstalt (PTB)\nAbbestrasse 2-1210587BerlinGermany\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n",
"Maria Zankl \nHelmholtz Zentrum München German Research Center for Environmental Health (HMGU)\nNeuherbergGermany\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n",
"José Maria Gómez-Ros \nCentro de Investigaciones Energéticas\nMedioambientales y Tecnológicas (CIEMAT)\nMadridSpain\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n",
"Carmen Villagrasa \nInstitut de Radioprotection et de Sûreté Nucléaire (IRSN)\nFontenay-aux-RosesFrance\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n",
"Jonathan Eakins \nUK Health Security Agency (UKHSA)\nDidcotUnited Kingdom\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n",
"Christelle Huet \nInstitut de Radioprotection et de Sûreté Nucléaire (IRSN)\nFontenay-aux-RosesFrance\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n",
"Hrvoje Brkić \nJ. J. Strossmayer\nUniversity of Osijek (MEFOS)\nOsijekCroatia\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n",
"Rick Tanner \nUK Health Security Agency (UKHSA)\nDidcotUnited Kingdom\n\nEuropean Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany\n"
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"European Radiation Dosimetry Group (EURADOS) e.V\nNeuherbergGermany",
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| Organized by Working Group 6 "Computational Dosimetry" of the European Radiation Dosimetry Group (EURADOS), a group of intercomparison exercises was conducted in which participants were asked to solve predefined problems in computational dosimetry. The results of these comparisons were published in a series of articles in this virtual special issue of Radiation Measurements. This paper reviews the experience gained from the various exercises and highlights the resulting conclusions for future exercises, as well as regarding the state of the art and the need for development in terms of quality assurance for computational dosimetry techniques. | 10.1016/j.radmeas.2022.106822 | [
"https://arxiv.org/pdf/2205.07321v1.pdf"
]
| 248,810,784 | 2205.07321 | fd51942bdfb02c6e3e19803674dd4eb1077188ef |
Lessons learnt from the recent EURADOS intercomparisons in computational dosimetry
Hans Rabus
Physikalisch-Technische Bundesanstalt (PTB)
Abbestrasse 2-1210587BerlinGermany
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
Maria Zankl
Helmholtz Zentrum München German Research Center for Environmental Health (HMGU)
NeuherbergGermany
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
José Maria Gómez-Ros
Centro de Investigaciones Energéticas
Medioambientales y Tecnológicas (CIEMAT)
MadridSpain
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
Carmen Villagrasa
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)
Fontenay-aux-RosesFrance
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
Jonathan Eakins
UK Health Security Agency (UKHSA)
DidcotUnited Kingdom
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
Christelle Huet
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)
Fontenay-aux-RosesFrance
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
Hrvoje Brkić
J. J. Strossmayer
University of Osijek (MEFOS)
OsijekCroatia
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
Rick Tanner
UK Health Security Agency (UKHSA)
DidcotUnited Kingdom
European Radiation Dosimetry Group (EURADOS) e.V
NeuherbergGermany
Lessons learnt from the recent EURADOS intercomparisons in computational dosimetry
1/17
Organized by Working Group 6 "Computational Dosimetry" of the European Radiation Dosimetry Group (EURADOS), a group of intercomparison exercises was conducted in which participants were asked to solve predefined problems in computational dosimetry. The results of these comparisons were published in a series of articles in this virtual special issue of Radiation Measurements. This paper reviews the experience gained from the various exercises and highlights the resulting conclusions for future exercises, as well as regarding the state of the art and the need for development in terms of quality assurance for computational dosimetry techniques.
Introduction
The European Radiation Dosimetry Group (EURADOS) is an association of 80 institutions and more than 600 individual members that promotes harmonization and good practice in dosimetry (Rühm et al., 2018(Rühm et al., , 2020Harrison et al., 2021). EURADOS has eight working groups dealing with different aspects and application areas of radiation dosimetry. EURADOS Working Group 6 on "Computational Dosimetry" has a cross-cutting character. Its main activities include the organization of comparison exercises (Tanner et al., 2004;Gualdrini et al., 2005;Siebert et al., 2006;Price et al., 2006;Gualdrini et al., 2008;Broggio et al., 2012;Vrba et al., 2014Vrba et al., , 2015Caccia et al., 2017) and training courses (Rabus et al., 2021a) as well as studies on fundamental aspects of computational dosimetry.
Recently, several computational dosimetry exercises have been completed, the results of which are compiled in this virtual special issue of Radiation Measurements (De Saint-Hubert et al., 2021Eakins et al., 2021;Gómez-Ros et al., 2021Huet et al., 2022;Rabus et al., 2021b;Villagrasa et al., 2022;Zankl et al., 2021aZankl et al., , 2021bZankl et al., , 2021c. This current article is meant as a synopsis and reflection on the common issues found in those different exercises and the lessons learnt on the state of the art in applied computational dosimetry, as well as conclusions for future exercises.
Overview of the exercises
The exercises can be divided roughly into two classes depending on the nature of their solutions. The first class comprised six exercises on the use of ICRP computational reference phantoms (ICRP, 2009;Zankl et al., 2021b) and one on unfolding of neutron spectra from Bonner sphere measurements . The former will be referred to as "voxel-phantom exercises" throughout this article, the latter as the "Bonner sphere exercise". Despite the quite different nature of the problems to be solved, these exercises had in common that they required the application of well-known methodologies and established computational tools. This allowed the organizers to establish prior reference solutions that could be used to validate the results subsequently submitted by the participants in the exercise.
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The second class are exercises where no reference solutions could be established, since one of the objectives of the exercises was evaluating the possible influence of different cross-section models in the codes used by the participants. Two of these exercises were code intercomparisons, one dealing with the calculation of microdosimetric and nanodosimetric quantities (Villagrasa et al., 2019(Villagrasa et al., , 2022, and the other with the effects of gold nanoparticles on dose deposition at the microscopic scale (Li et al., 2020a(Li et al., , 2020bRabus et al., 2021bRabus et al., , 2021c; the former is called the "uncertainty exercise" in this article and the latter the "nanoparticle exercise". The other two exercises in this class dealt with out-of-field dose calculations. One was about calculating the dose to the foetus during maternal proton therapy treatment and the other was about calculating the secondary neutron fluence. (De Saint-Hubert et al., 2021. These two exercises are referred to as "foetus dose exercises".
It is important to note that none of the exercises was intended to be a code competition. Rather, the aim was to investigate the dispersion of results when the same problem was solved by different people using different approaches and different codes or the same code with different options. The first class of exercises focused on identifying the state of the art in the application of common methods in computational dosimetry. The second class was more exploratory in nature and aimed to assess the state of the art in terms of the capabilities of codes and approaches. All classes contained tasks of different complexity, and thus different demands on the participants' skills.
The exercises were organized and run by ad-hoc teams composed of EURADOS WG 6 members. In general, the preparation of the exercises involved independent simulations by several team members, with their respective results then cross-referenced to identify potential pitfalls in the proposed exercise definitions and to check whether the tasks were solvable based on the information to be provided. For the exercises with reference solutions, the results of these test simulations were also used to establish those values, (e.g. by taking the mean), as well as gain a handle on the typical levels of uncertainty that may be considered acceptable for them.
In some exercises, templates for reporting results were also provided to the participants.
ICRP reference voxel phantom exercises
Of the voxel-phantom exercises, two involved exposure to an external point source, emitting either 60 Co gamma photons or 10 keV neutrons respectively (Huet et al., 2022). The task to be solved was to calculate the organ absorbed doses and the effective dose for a given exposure duration and activity of the source.
A third voxel-phantom exercise also dealt with point source geometries, but for cases of typical Xray examinations (Huet et al., 2022). Here the task was more complex, as participants were required to determine the position of the radiation point source in relation to the phantom. In addition, the results were to be presented as conversion coefficients to organ absorbed doses, both from air kerma and kerma area product. This exercise was thus linked to a potential practical application in which the latter quantities are determined as part of the quality assurance of radiological equipment, and the conversion coefficients sought would enable an assessment of the dose absorbed by the patient during the X-ray examination.
The fourth voxel-phantom exercise considered a uniform planar source of 60 keV photons beneath the phantom (mimicking ground contamination by 241 Am) and required participants to calculate organ absorbed dose rates and the effective dose rate for a given area density of the emission rate (Eakins et al., 2021). In the fifth voxel-phantom exercise, a mixed radiation field of gaseous 16 N was considered, emitting beta and high energy gamma radiation from both inside (lung) and outside the human body (Gómez-Ros et al., 2021); the ratios of organ equivalent dose rates to activity concentration were to be determined.
The most extensive voxel-phantom exercise involved an idealized case of internal dosimetry (Zankl et al., 2021c). For the sake of simplicity, hypothetical radionuclides were considered that were uniformly distributed in specified organs and emitted monoenergetic photon or electrons. Here, absorbed fractions and specific absorbed fractions of energy in the "source" organ and in specified "target" organs were to be determined as well as S-values for the resulting source and target organ combinations for two specific radionuclides.
Bonner sphere spectra unfolding exercise
The tasks were defined by the counts measured by a set of twelve Bonner spheres of different diameters and known sensitivities (as determined by the organizers with radiation transport simulations), located at a measurement point in one of four known environments: inside the bunker of a medical linac; near a radioactive source; in a simulated workplace field within a neutron calibration facility; or outside a nuclear power plant. The count rates measured by the Bonner spheres were determined by the organizers through Monte Carlo radiation transport simulations of the respective complete measurement setup for each Bonner sphere within its environment. In addition, and to recreate a realistic situation, the count rate from one of the Bonner spheres in one of the scenarios was intentionally given an incorrect value in order to test the participants' ability to detect an erroneous measurement and exclude it when applying the deconvolution procedure (Gómez-Ros et al., 2018
Micro-and nanodosimetric uncertainty exercise
In its first phase, this exercise included a microdosimetric and a nanodosimetric intercomparison (Villagrasa et al., 2019). In the frame of the former, frequency distributions of specific energy were to be determined within a microscopic water sphere for different distributions of a lowenergy electron emitter with an energy spectrum derived from the internal-conversion Auger emitter 125 I. In the nanodosimetry part, ionization cluster size distributions were to be determined in target spheres of different sizes located at different distances from a point source with the same energy spectrum as in the microdosimetry part. A sensitivity analysis was also performed on the variation of inelastic cross-sections and its consequences for the calculated ionisation cluster size distributions (Villagrasa et al., 2022). In the second phase of the exercise (in preparation), the focus will be on a comparison of the cross-section datasets for low-energy electron transport and a consideration of their impact on the dispersion of nanodosimetric results.
Nanoparticle exercise
In the nanoparticle exercise, the dose enhancement from a gold nanoparticle, as well as the energy spectrum of electrons emitted from it, were to be determined when irradiated with two low-energy X-ray spectra. The geometry was simply a gold sphere in water irradiated with a parallel beam from a plane photon source, the cross-sectional area of which was slightly larger than that of the nanoparticle. Two different nanoparticle diameters were considered, and the dose enhancement was to be determined in spherical water shells around the nanoparticle. (Li et al., 2020a(Li et al., , 2020bRabus et al., 2021bRabus et al., , 2021c
Foetal dose during maternal proton therapy
This exercise consisted of two parts. The first part dealt with the effects of different calculation phantoms for pregnant women, and different code versions of MCNP, on the predicted dose to the foetus during maternal brain proton therapy. The second part dealt with the dependence of the secondary neutron spectra on the Monte Carlo radiation transport codes and nuclear models that were used, and their effects on the calculated and measured neutron doses during proton therapy. (De Saint-Hubert et al., 2021
Experiences from the exercises
Observations on participants' results
In general, the ensemble of participants' results that were submitted initially showed a large scatter. In the exercises for which a reference solution was available, excellent agreement within the expected statistical fluctuations was found in some cases, while others showed significantly larger deviations, which in some individual cases ranged by up to several orders of magnitude. For the tasks without a reference solution, a subset of the reported results also agreed with each other to some extent, while others deviated significantly from this group. In both classes of exercise, the occurrence of extreme outliers was not correlated with the complexity of the problem.
Some of the deviations were attributable to simple errors, such as copy-and-paste mistakes or incorrect arrangement of the results in the given template. Others resulted from misunderstanding how the final results should be normalised (e.g. normalising to the correct quantity but at a different distance from the source than was required). In the voxel-phantom exercise for the case of X-ray examinations, some participants normalized to the value of air kerma free in air at a specific distance from the source or to the entrance surface dose (which includes backscatter) instead of to air kerma free in air at the skin as was requested. In the microdosimetric and nanodosimetric intercomparisons, the normalisation to 'one decay of the electron source' was not always understood by the participants and was also sometimes difficult to implement for some Monte Carlo codes. The use of a logarithmic scale for the microdosimetric quantity (specific energy distribution) also caused problems with proper normalisation.
Many major deviations were caused by the fact that the participants' simulations deviated from the specifications in terms of geometrical dimensions or the quantities to be determined. One example of this was the "nanoparticle" exercise, where only two out of eleven participants implemented the requested geometry correctly, which consisted of a gold sphere irradiated in water by a collimated parallel photon beam of given dimensions. Another example was the voxel phantom exercise on internal dosimetry, where some participants used organ masses that included blood instead of those given in ICRP Publication 110, as was stated in the exercise definition. In some of the voxel-phantom exercises, the choice of the location of the source was also sometimes a problem due to deviation from the correct reference point (e.g. the edge of the phantom array instead of the phantom's skin).
Other causes of major deviations were that some participants were not familiar with certain concepts, such as the normalization quantity "kerma area product" (in the voxel-phantom X-ray exercise) or effective dose; mistakes for the latter included not applying tissue weighting factors correctly, not averaging and summing over the defined set of organs, neglecting to sex-average, or neglecting to apply the correct energy-dependent radiation weighting factor for neutron exposures.
In the voxel-phantom exercises, many participants had problems applying the method recommended for bone marrow dosimetry in (ICRP, 2010). This finding stimulated writing an article to better explain this approach, which is also part of this Special Issue .
As already mentioned, participants whose results differed from the reference solution (class 1) or from the majority of other participants' solutions (class 2) were informed of this fact and asked to revise their solutions. Not all contacted participants responded to this invitation or provided the requested information on details about their simulations.
Of those participants who submitted a revised solution, some did not indicate what they had changed in their computational procedure to arrive at their revised results. This therefore does not give any additional insight into possible similar errors to be expected in future similar exercises, or hints that could have been communicated to the other participants.
In the nanoparticle exercise, where some inconsistencies became evident after the first publication of the results (Li et al., 2020a) and required a thorough re-evaluation (Li et al., 2020b), consistency checks provided clear indications of the causes of the discrepancies for some results. Nevertheless, some of the participants concerned did not provide revised solutions (Rabus et al., 2021b). However, it must be also stressed that the majority of participants were very supportive of the re-analysis of the results and were eager to clarify the origin of the discrepancies found initially.
Issues with omitted quality assurance of results
Many of the anomalies found in the reported data could have been detected by the participants themselves, e.g. through simple plausibility checks of their results. Examples are briefly discussed in the following.
For example, a very general plausibility consideration is that if the irradiation conditions are quite homogeneous, it may be expected that all organ doses will be of broadly similar magnitudes; a single organ dose result differing by several orders of magnitude from the rest of a given participant's dataset ought therefore to be immediately apparent to them as being potentially erroneous. Similarly, if multiple energies are considered, it is unlikely that the value for a single intermediate energy will be entirely outside the range of values for all other energies.
In the Bonner sphere exercise, there were cases of reported results with negative values for the neutron fluence. These physically impossible values, as well as anomalous spectral shapes, could have been identified by simply plotting the results. Some of the reported spectra differed from the reference solutions by several orders of magnitude; such anomalies could have been easily detected if the unfolded spectra had been convolved with the given sensitivities of the Bonner spheres, to verify that the given count rate was then achieved.
In the voxel phantom exercise for internal dosimetry, a simple plausibility check would have been that the absorbed fraction for electrons and lowenergy photons must be close to unity in a source 5/17 organ and quite small for other organs, since these radiations have a short range in condensed matter and therefore deposit their energy close to the point of release. Moreover, some participants in this exercise reported results for absorbed fractions and specific absorbed fractions for which the ratio of these two quantities varied between different energies of the particles emitted from the (monoenergetic) source. However, since this ratio is simply the mass of the organ, it cannot depend on the energy.
In addition, for many of the tasks in the voxel phantom exercises, literature values are available for fairly similar exposure conditions that could have been used for comparison, at least as a first approximation to indicate the expected magnitudes of the results.
When dealing with voxel phantom simulation, one of the simplest checks might be to visualise the problem in order to ensure the proper positioning of the beam, though it is noted that some software packages struggle due to the sizes of these input files. If one is using any variance reduction it should also be ensured that simulations with and without application of these techniques reproduce the same results, albeit with differing statistical uncertainties.
Issues with exercise definitions
In some cases, inadequacies in exercise definitions became apparent while they were already running. In the nanoparticle exercise, for example, one of the quantities to be reported by the participants was the energy spectrum of electrons "in spherical shells" around the nanoparticle, with the radii of the bounding spherical surfaces given.
Most of the participants interpreted this physically undefined quantity as the energy distribution of the electrons entering the respective volume. However, one participant determined the energy distribution of the balance of the number of electrons traversing the surfaces of the respective volume and withdrew her results on the assumption that the observed negative frequencies indicated an error that she could not locate.
Another problem with this part of the nanoparticle exercise was that there was no default energy binning, so participants chose very different values for the bin size, with some using logarithmic binning and others using linear binning. The large statistical variations in the results obtained with small energy bin sizes masked the variations between the different results when plotted together (Li et al., 2020a).
In the exercise on Bonner sphere spectrum unfolding, it was found during the analysis that in one of the scenarios considered, there was an interference of the Bonner sphere response due to backscattering of neutrons from a nearby concrete wall (Gómez-Ros et al., 2018).
In the voxel phantom exercises featuring 60 Co photons and 10 keV neutrons, the instruction given to participants for the location of the point source was to place it '100 cm from the surface of the chest', which could be interpreted differently. In response, the organizers performed small sensitivity analyses to quantify the impact from the ambiguity of this parameter, the outcomes from which were used to imply appropriate 'tolerances' that could be applied to the submitted results (Huet et al., 2022).
The voxel phantom exercise on internal dosimetry was not wisely designed in several respects. The tasks to be solved were too extensive, which also made evaluation challenging and led to delays in feedback to the participants. For electrons and low-energy photons, the source and target organs were sometimes too far apart, which led to very large statistical uncertainties even in the reference solution. Therefore, the degree of deviation between participant and master solutions could not be reliably quantified in some situations.
In the uncertainty exercise, the use of a multienergy electron source that was similar to the 125 I decay but did not take into account the variability of the actual decay, complicated the understanding of the problem on one hand and, on the other hand, did not favour the analysis of the sensitivity study on the variation of the cross sections. Indeed, the use of monoenergetic electrons would have helped in both aspects.
In the foetal dose exercise, atomic numbers of the elements, mass numbers of the nuclides, and crosssection identifiers had not been fixed for all materials used in the simulations, so participants made their own (different) choices, which caused some of the discrepancies initially noted.
Issues with the timeable of the exercises
Most exercises were planned with a timetable, which in almost all cases proved to be too ambitious and optimistic. This was partly because for many exercises the initial number of participants was lower than was expected and considered adequate for the purpose, so submission deadlines were postponed several times to increase participation after further Manuscript: RADMEAS_VSI_Common_Pitfalls_final.docx.
6/17 publicity for the exercises. Further deadline extensions became necessary at the request of the participants.
After an initial analysis of the submitted solutions, participants whose results differed by more than expected from either the reference solution or from most other participants' solutions (as appropriate) were informed on this fact and invited to revise their solutions. Deadlines for the submission of revised results also had to be postponed several times.
As a result, the total duration of the exercises exceeded the typical length of stay of junior researchers at a given institute, making it difficult, if not impossible, to follow-up on abnormal results in some cases.
Lessons learnt
This section discusses the insights gained from the exercises from the perspective of the organizers.
Problem specification
The participants, as well as the organisers of the exercises, are committed to EURADOS and the intercomparisons in addition to their daily work. Therefore, the topics of the intercomparison exercises must be relevant to the participants' fields of work, and the tasks to be solved should not be overly demanding in terms of setup time. CPU resource requirements may also need to be considered but should generally be less of an issue.
To meet the workload requirements, some of the exercises presented in this Special Issue were designed with simplified idealistic geometric setups and irradiation conditions. Examples include the voxel phantom exercises for monoenergetic point sources, the uncertainty exercise in micro-and nanodosimetry (idealised energy spectrum), and the nanoparticle exercise (simplistic geometry). These simplifications have sometimes raised concerns among reviewers about the usefulness of the respective comparisons but seem justified given the aforementioned time constraints. Regardless of the complexity of a task, a complete description of the problem to be solved with all relevant information must always be given. For Monte Carlo simulation exercises, this means that the radiation source, simulation geometry and materials must be comprehensively specified. On the other hand, it should generally not be specified exactly how the Monte Carlo simulation or the unfolding are to be carried out. The path to the solution, as well as the tools to be used, must be decided at the discretion of the participant. The participant must determine, for example, whether and which variance reduction techniques can or should be used, whether the transport of secondary charged particles should be simulated or how the thermal neutron transport should be performed.
However, depending on the aim of the exercise, a more detailed specification of intermediate steps or procedures may be advisable. For instance, whenever the performance of codes or their differences is to be assessed, it may be wise also to specify some of the aforementioned aspects of the simulations to ensure that the differences between results from different participants only reflect the differences in the codes that one is interested in.
Reporting of results
The task definition should include very precise instructions for reporting results, and templates should be provided where possible. Providing such a template, where the participants were requested to fill in their results in a pre-defined format, not only helps clarify exactly what output is required from them in each case, but also greatly facilitated the evaluation of the results in the respective exercises. This is especially important when spectral information is to be reported, where a lack of specification of bin division can make synopsis quite cumbersome when different participants use different bin sizes and/or linear and logarithmic equidistant bins.
In addition, asking for redundant information can help to identify potential problems with participants' data. Examples of this were: the voxel phantom exercise for internal dosimetry, where absorbed fractions and specific absorbed fractions were to be reported (differing by only one factor, i.e. organ mass); or the voxel phantom exercise for the X-ray examinations, where the results were to be reported normalised to both kerma free-in-air and kerma area product, which again differ by only one factor. In the case of the nanoparticle exercise, only reporting of results normalised to the number of primary particles was required. If normalisation to the area density of the emitted primary photons from the source had also been reported, the incorrect implementations of the simulation geometry would have been detected much earlier during the exercise.
Timing of the exercise
Regarding the problems encountered with nonresponding participants at the revision of results stage, it is planned to set up rules in future exercises to get a more formal commitment from participants. The rules to be established concern deadlines, participation in the feedback loop, and requirements for co-authorship to potential manuscripts (see Supplementary Fig. S1 and Supplementary Fig. S2).
In addition, more timely feedback to the participants might improve their preparedness to disclose details of their computational procedures and improvements. Long feedback intermissions make it difficult for the participants to recall exactly what was done and even what the exercise was about. It should be kept in mind that the organizers, as well as all participants, are performing these exercises alongside their daily duties. In the reanalysis of the nanoparticle exercise, a set of hierarchical MS Excel templates were used that allowed a fast assessment of the internal consistency of the results reported by participants (in an Exceltemplate provided to them) as well as a 'live' synopsis via hyperlinks. As an illustration of this approach, Supplementary Fig. S3 to Supplementary Fig. S6 show screenshots of the "Synopsis" worksheets with easy-to-assess graphs and calculated figures of merit (integral quantities normalised so that their expected values are close to unity).
Using these templates to assess the consistency of a participant's results took only a few minutes and required only copying and pasting the results from the Excel templates completed by the participant into the template used for the analysis. In many cases, this enabled feedback in less than an hour. Of course, it was more time-consuming to identify the more sophisticated deviations from the exercise specifications.
Another issue with timing is that calculations with voxel phantoms may require large amounts of time. With some codes, even the visualisation of them can take up to several hours, and the production calculations even weeks. Some codes have ability to skip the geometry check at the beginning of production calculations (e.g. DBCN card in MCNP) and in this way speed up the calculations significantly.
Quality assurance of results
As indicated in Section 3.2, in all exercises some of the participants seemed to have submitted their results without first carrying out adequate quality control of their solutions, e.g. by simple plausibility checks or by comparison with literature data, if available. Approaches such as the one mentioned in Section 4.2 could allow for faster identification of outliers and more timely feedback to participants. This could alleviate some of the problems, such as where there was a lack of response from participants following feedback regarding abnormal results.
However, a significant degradation in the quality of results may persist, as the evolution of many computational tools towards greater ease of use also allows their use without a certain level of expertise, which may still be required for meaningful results. Interaction with some exercise participants who reported unreasonable results revealed a lack of understanding of several fundamental aspects. For example, that the results of calculations are not just numbers, but physical quantities (which have dimensions).
Some participants were also unaware that there are different ways of specifying the categorical variable of a histogram (lower or upper limit of the bin or the bin centre), which can vary between different codes and affect the comparison of results, such as when reported with different definitions of the meaning of the values on the x-axis. This could be countered by requiring both the lower and upper bin limits to be reported.
There were also cases where participants determined ratios with a finer bin size than specified in the task, and then re-binned their results for reporting by averaging the ratios over the larger bins instead of calculating the ratio between the sum of the numerators and the sum of the denominators. Detection of such elementary mistakes requires access to the original results, so their origins were not always immediately apparent to the organizers.
Considering that most exercises will lead to publications in the form of EURADOS reports or journal articles, compliance with the principles of FAIR data (an acronym for findability, accessibility, interoperability and reusability, (Wilkinson et al., 2016)) is also a matter that should be given more emphasis in the future. This is, of course, the responsibility of each participant, but appropriate commitments can be included in the application forms ( Supplementary Fig. S1 and Supplementary Fig. S2). As a minimum requirement, participants should provide documentation on where the following files are stored and backed up:
data files containing the reported results data files with the raw simulation output files used for their production (material data or other input files, code, etc.) -log files and other supplementary output files of the simulations. This information is essential when participants need to review their work for possible errors. However, in the exercises, delayed responses from participants were sometimes explained by difficulties in finding data, uncertainty about which version of a code was used, and similar such problems. Therefore, it may be useful for the organisers to collect this information -or even the files containing the metadata of the simulations -as part of the reporting of the results. This might be the case especially for participants who are inexperienced users of simulation codes, who may not be aware that the log files etc. generated by their codes contain information that complements their simulation results and is important for their quality.
It was not uncommon for files of original simulation results, which were shared by participants with the organisers during the feedback loops, to contain only columns of numbers. A header indicating which quantities are listed and which units have been used, and ideally also containing information on the code used and its version, as well as the date when the file was written, would be minimum requirements for the useability of these data files. In addition, the aforementioned auxiliary information is also needed.
Conclusions
Beyond doubt, the reported EURADOS exercises are beneficial to the field of computational dosimetry. They directly contribute to the training of the participants by improving their computational procedures through feedback with the task organisers. They lead also to the availability of representative dose values for various exposure conditions that may aid future novice users in the quality assurance of their methods. In addition, they also provide a snapshot of how well (or otherwise) the computational techniques are being applied within the community in general, and how well some of the concepts recommended by organizations such as ICRP are understood; the observed difficulties in correctly defining and evaluating effective doses (Eakins et al., 2021;Huet et al., 2022), or in determining bone marrow doses , are clear examples of the latter.
The obvious question of what could be done better in future exercises has been partly addressed in Section 4. A general answer to this question is not easy, since it depends on the objective of the exercises. The question will therefore continue to be the subject of discussion within EURADOS Working Group 6 when new exercises are prepared.
To avoid participants wasting their time on the tasks of an exercise in cases where they are prone to give incorrect results, due to a wrong idea of the task or ignorance of the dosimetric quantities to be determined, several modifications of the exercises can be considered. One could be to define explicitly the dosimetric quantities and normalisation quantities to be used. Another possibility would be to include in the definition of the task a list of checks to be made by the participants on the results, or even provide them with templates like those used in the reanalysis of the nanoparticle exercise.
Such changes to the exercises would mitigate the risk of potential errors by the participants. However, while this closer guidance may lead to improved results, this better agreement will only reflect the participants' capability to follow detailed instructions and not their actual state of expertise or their performance in real-world applications. Moreover, concepts like effective dose are widely used and already well-defined elsewhere; arguably, it should not therefore be the role of EURADOS WG6, which focusses on computational dosimetry, to coach radiation professionals on the basic concepts that underpin radiological protection.
Most of the exercises were the types of task that the participants may be confronted with in their professional activities or in research. Generally, they then would have to perform the simulation or unfolding, and calculate the dosimetric quantities of interest, without specific guidance. A supervisor of an early-stage researcher, or a reviewer of potential papers arising from such work, can be a source of feedback that hopefully reveals major nonplausibilities in the results, if any. However, as discussed by (Rabus et al., 2021c), these potential quality filters often seem to have failed for simulation studies on nanoparticles so far.
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A better option could therefore be to create a questionnaire to assess the knowledge of the participants, and then give more detailed instructions to the less experienced participants. One could also start the exercise with a webinar explaining what is expected, what participants should do, and explaining the importance of quality assurance. A recording of the webinar could also be made available on the EURADOS website so that participants who join the exercise later can refer to it.
Another potential improvement for some of the exercises, if repeated, could be to include several reporting steps in the exercise. For example, in the voxel phantom exercise for the X-ray examinations, participants could first need to report their results for the position of the radiation source in relation to the phantom; they could then get feedback if it is outside the uncertainty band of the reference value and be invited to report a revised value. On request they could also get the correct positions as feedback and run their simulations for these. Alternatively, such future exercises could have a first step with as few specifications as possible, a second with plausibility checks to be performed by the participants, and a third with feedback on the submitted results and a request for revision. Supplementary Fig. S3: Screenshot of the section with the diagrams in the "Synopsis" worksheet of the Microsoft Excel template used in the nanoparticle exercise to test the internal consistency of the results reported by a participant for the energy spectrum of electrons emitted from the gold nanoparticle for the four combinations of nanoparticle size and X-ray spectrum. The plots show comparisons of the energy distributions of the number of emitted electrons (normalised to the average number of photon interactions in the gold nanoparticle for the simulation geometry, as defined in the exercise). The two plots in the upper row compare different nanoparticle sizes for the same radiation quality, where similar values are expected for high-energy electrons and higher frequencies for lowenergy electrons for the smaller nanoparticle. The bottom row shows the comparison for the same nanoparticle size and different energy spectra, where one expects similar values for low electron energies because the range of these electrons is smaller than the size of the nanoparticle. The data shown have been calculated from the participants data by rebinning and normalization to the expected number of photon interactions in the nanoparticle. They are plotted in "microdosimetry style" such that the area under the curves is proportional to the number of electrons emitted per photon interaction in the respective energy interval.
The data for emitted electrons (and the corresponding data for energy deposition shown in Supplementary Supplementary Fig. S4) correspond to one of the cases where the participant correctly applied the required geometry. Examples of how different the respective graphs look for the case of deviating geometry can be found in (Rabus et al., 2021c).
AcknowledgementsThe support of all members of EURADOS WG 6 who contributed to the organization of the exercises and the efforts of the participants to solve the tasks are gratefully acknowledged. The authors also thank the Editor-in-Chief of Radiation Measurements, Adrie de Bos, for making this special issue possible, and the EURADOS Council for providing the necessary funding.Supplementary Fig. S4: Screenshot of the section with the diagrams in the "Synopsis" worksheet of the Microsoft Excel template used in the nanoparticle exercise to test the internal consistency of the results reported by a participant for the energy deposition in spherical shells around a gold nanoparticle. The four plots in the left and middle columns show comparisons between the results for the different cases (two radiation qualities, two nanoparticle diameters) after conversion to easily understood quantities (number of ionizations in the spherical shells around the nanoparticle per photon interaction in the nanoparticle). The diagram on the upper right (blue frame) shows a comparison of all four data sets of the absorbed dose convergence for water as the size of the scoring region increases. The bottom right plot (yellow frame) shows the results for the quantity of interest (dose enhancement ratio) for the four cases studied. For details see(Rabus et al., 2021b(Rabus et al., , 2021c. For details see(Rabus et al., 2021b(Rabus et al., , 2021c.
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| []
|
[
"One-time learning in a biologically-inspired Salience-affected Artificial Neural Network (SANN)",
"One-time learning in a biologically-inspired Salience-affected Artificial Neural Network (SANN)"
]
| [
"Leendert A Remmelzwaal \nDepartment of Electrical Engineering\nDepartment of Mathematics and Applied Mathematics\nUniversity of Cape Town\n7700Cape TownSouth Africa\n",
"George F R Ellis [email protected] \nMARCS Institute for Brain, Behaviour and Development\nUniversity of Cape Town\n7700, Cape TownSouth Africa\n",
"Jonathan Tapson [email protected]*correspondingauthor:[email protected] \nWestern Sydney University\nSydneyAustralia\n"
]
| [
"Department of Electrical Engineering\nDepartment of Mathematics and Applied Mathematics\nUniversity of Cape Town\n7700Cape TownSouth Africa",
"MARCS Institute for Brain, Behaviour and Development\nUniversity of Cape Town\n7700, Cape TownSouth Africa",
"Western Sydney University\nSydneyAustralia"
]
| []
| Standard artificial neural networks (ANNs), loosely based on the structure of columns in the cortex, model key cognitive aspects of brain function such as learning and classification, but do not model the affective (emotional) aspects of brain function. However these are a key feature of the brain (the associated 'ascending systems' have been hard-wired into the brain by evolutionary processes). These emotions are associated with memories when neuromodulators such as dopamine and noradrenaline affect entire patterns of synaptically activated neurons. Here we present a bio-inspired ANN architecture which we call a Salience-Affected Neural Network (SANN), which, at the same time as local network processing of task-specific information, includes non-local salience (significance) effects responding to an input salience signal. During pattern recognition, inputs similar to the salience-affected inputs in the training data will produce reverse salience signals corresponding to those experienced when the memories were laid down. In addition, training with salience affects the weights of connections between nodes, and improves the overall accuracy of a classification of images similar to the salience-tagged input after just a single iteration of training. Note that we are not aiming to present an accurate model of the biological salience system; rather we present an artificial neural network inspired by those biological systems in the human brain, that has unique strengths.GlossaryAI = artificial intelligence arXiv:1908.03532v3 [cs.NE] 23 Aug 2019 Biologically-inspired Salience Affected Artificial Neural Network (SANN) 2 AND = affective neural darwinism [28] ANN = artificial neural network [28] BBD = brain-based device [22] MNF = non-negative matrix factorization [49] MSE = mean squared error NN = neural network RNN = recurrent neural network SANN = salience-affected neural network SNN = spiking neural networkEach memory has associated with it an emotional tag that is recalled when the memory is recalled, which triggers feelings associated with the relevant context or event. According to Edelman [20][24][21], emotional tags are a dimension of salience that is present in addition to standard neural network training (e.g. pattern recognition) that takes place in the cortex.In this paper we clearly distinguish between 'neurotransmitters' and 'neuromodulators'. Neurotransmitters are chemicals responsible for a targeted, rapid signaling between one synapse and another post-synaptic neuron. Neurotransmitters are only released at the synaptic cleft, and are only received by specific receptors at the post-synaptic terminal. Neurotransmitters are said to have an impact on the 'channel' [69] [55] [42] [56].By comparison, neuromodulators are messenger molecules that are diffusely dispersed across the cortex via the 'ascending systems'(Fig.1). Neuromodulators can impact many neurons simultaneously, which may result in post-synaptic gain. Some neuromodulators have a gradual impact over time, while others (e.g. those responsible for registering fear) can be responsible for single-exposure learning. The impact of neuromodulators depend on the chemical, the quantity and the location in the cortex that it is dispersed. In contrast to neurotransmitters, neuromodulators are described as having an impact on the 'state' of a neuron -post-synaptic gain -rather than an impact on the 'channel' [56] [33] [55] [6].Neuromodulators not only impact the synapses in the cortex, but they also impact the rest of the body (e.g. adrenaline). As a result, the diffuse projections from the arousal system can be considered as both ascending and descending (seeFigure 1). In addition to neuromodulators, other chemicals produced in the human body can impact cognition (such as hormones from the pituitary, and peptides from the gut) [42] [9]. However our focus in this paper is on the 'ascending systems' and their function. It is worth remembering here that the hippocampus is a key target of the arousal system.[63] | 10.1101/726331 | [
"https://arxiv.org/pdf/1908.03532v3.pdf"
]
| 201,615,807 | 1908.03532 | 6df50ac7c99354a5497ad56a2704093fcd834a03 |
One-time learning in a biologically-inspired Salience-affected Artificial Neural Network (SANN)
Leendert A Remmelzwaal
Department of Electrical Engineering
Department of Mathematics and Applied Mathematics
University of Cape Town
7700Cape TownSouth Africa
George F R Ellis [email protected]
MARCS Institute for Brain, Behaviour and Development
University of Cape Town
7700, Cape TownSouth Africa
Jonathan Tapson [email protected]*correspondingauthor:[email protected]
Western Sydney University
SydneyAustralia
One-time learning in a biologically-inspired Salience-affected Artificial Neural Network (SANN)
artificial neural networkssalience effectsone time learning
Standard artificial neural networks (ANNs), loosely based on the structure of columns in the cortex, model key cognitive aspects of brain function such as learning and classification, but do not model the affective (emotional) aspects of brain function. However these are a key feature of the brain (the associated 'ascending systems' have been hard-wired into the brain by evolutionary processes). These emotions are associated with memories when neuromodulators such as dopamine and noradrenaline affect entire patterns of synaptically activated neurons. Here we present a bio-inspired ANN architecture which we call a Salience-Affected Neural Network (SANN), which, at the same time as local network processing of task-specific information, includes non-local salience (significance) effects responding to an input salience signal. During pattern recognition, inputs similar to the salience-affected inputs in the training data will produce reverse salience signals corresponding to those experienced when the memories were laid down. In addition, training with salience affects the weights of connections between nodes, and improves the overall accuracy of a classification of images similar to the salience-tagged input after just a single iteration of training. Note that we are not aiming to present an accurate model of the biological salience system; rather we present an artificial neural network inspired by those biological systems in the human brain, that has unique strengths.GlossaryAI = artificial intelligence arXiv:1908.03532v3 [cs.NE] 23 Aug 2019 Biologically-inspired Salience Affected Artificial Neural Network (SANN) 2 AND = affective neural darwinism [28] ANN = artificial neural network [28] BBD = brain-based device [22] MNF = non-negative matrix factorization [49] MSE = mean squared error NN = neural network RNN = recurrent neural network SANN = salience-affected neural network SNN = spiking neural networkEach memory has associated with it an emotional tag that is recalled when the memory is recalled, which triggers feelings associated with the relevant context or event. According to Edelman [20][24][21], emotional tags are a dimension of salience that is present in addition to standard neural network training (e.g. pattern recognition) that takes place in the cortex.In this paper we clearly distinguish between 'neurotransmitters' and 'neuromodulators'. Neurotransmitters are chemicals responsible for a targeted, rapid signaling between one synapse and another post-synaptic neuron. Neurotransmitters are only released at the synaptic cleft, and are only received by specific receptors at the post-synaptic terminal. Neurotransmitters are said to have an impact on the 'channel' [69] [55] [42] [56].By comparison, neuromodulators are messenger molecules that are diffusely dispersed across the cortex via the 'ascending systems'(Fig.1). Neuromodulators can impact many neurons simultaneously, which may result in post-synaptic gain. Some neuromodulators have a gradual impact over time, while others (e.g. those responsible for registering fear) can be responsible for single-exposure learning. The impact of neuromodulators depend on the chemical, the quantity and the location in the cortex that it is dispersed. In contrast to neurotransmitters, neuromodulators are described as having an impact on the 'state' of a neuron -post-synaptic gain -rather than an impact on the 'channel' [56] [33] [55] [6].Neuromodulators not only impact the synapses in the cortex, but they also impact the rest of the body (e.g. adrenaline). As a result, the diffuse projections from the arousal system can be considered as both ascending and descending (seeFigure 1). In addition to neuromodulators, other chemicals produced in the human body can impact cognition (such as hormones from the pituitary, and peptides from the gut) [42] [9]. However our focus in this paper is on the 'ascending systems' and their function. It is worth remembering here that the hippocampus is a key target of the arousal system.[63]
Introduction
Some aspects of human cognition, such as the ability to learn and classify input signals, can be modelled with artificial neural networks (ANN). However, standard ANNs do not incorporate the non-specific, diffuse dispersion of neuromodulators in the cortex. The presence of neuromodulators in the cortex, originating from the arousal system, allow memories to be tagged with emotion and salience. In this paper we present a Salience Affected Neural Network (SANN), an artificial neural network (ANN) with an additional trainable dimension of salience or significance compared to standard models. This is a novel ANN architecture model inspired by the effect that neuromodulators have on memories -a crucial feature of the way the human brain functions [28]. It enables powerful single-exposure learning in ANNs, affecting memories proportional to the strength of a global salience signal, and retrieval of that salience signal when the same image is encountered later (a Reverse Salience Signal ). It is important to note that we are not presenting an accurate model of the biological salience system, rather we present an artificial neural network inspired by the salience systems in the human brain; this allows it to have unique attributes.
Our paper provides a proof of principle that this proposal can be made to work, and could be employed in far more complex reinforcement learning dynamics in Deep Neural Networks. A key feature of the paper is the distinction made between Generic and Specific Semantic Memory ( §2.2), and our use of the SANN to determine both.
Affective systems, neurotransmitters, and neuromodulators
Human cognition does not just consist of cortical processing, but also includes sub-cortical processing such as the primary and secondary emotions of the affective system [55] [17] [72] [35] [15] [16]. Primary emotions, for example, are critical to guiding the cortex in the context of ongoing interaction with the physical, ecological, and social environment [61] [27]. Emotions are associated with memories when neuromodulators such as dopamine and noradrenaline affect patterns of synaptically activated neurons. Neuromodulators are dispersed in the cortex by diffuse projections originating from the arousal system (see Figure 1). The release of neuromodulators in the cortex are the source of primary emotions [61], and they have the effect of altering neural network weights as described in Edelman's concept of "Neural Darwinism" [20] [21]. Thus emotions affect the cortex via Affective Neural Darwinism (AND) [28].
Like most discussion of brain function [13] [36], most artificial neural network structures focus on information processing tasks such as pattern recognition, classification, information storage, and sequence prediction. Inter alia this involves them storing information ("memories") and retrieving them ("remembering"). However, standard artificial neural networks omit representing a crucial aspect of cognitive function: namely "affect" [61] [16] [29], i.e. the effect of primary and secondary emotions. In this paper we focus on the impact of neuromodulators in the cortex, and it is this diffuse effect of post-synaptic gain that we model in a SANN.
Salience Affected Neural Network
In this paper we present a Salience Affected Artificial Neural Network that represents the non-local signaling in the cortex via neuromodulators associated with rewards and reinforcement signals broadcast from the arousal system, and demonstrates that this makes single-exposure learning possible. The SANN model we present is not trying to be an accurate model of the biological salience system, rather it is an architecture for an artificial neural network inspired by the salience systems in the human brain. This work is based on the relation between these important structural and functional aspects of the human brain, and thereby opens up a new class of ANN that may be useful in computational applications.
In particular a SANN demonstrates that these kinds of networks not only enable single-exposure learning by laying down memory patterns associated with strong salience signals, but enables recall of those salience signals when the relevant stimulus is encountered at later times. The SANN models both the way memories associated with emotionally-laden events [17] may be embodied in neural networks, and the way those emotional associations can be recalled when the events are remembered and so alter immediate responses appropriately.
The architecture we propose in this paper is a key breakthrough in ANN structuring, because our model adds salience as an additional dimension to the network, and allows for salience training by single exposure. By doing so, we demonstrate how a neural network can be modified to include a salience signal without needing to re-train it on the original data set. While salience can be applied post-training by single exposure, the SANN also allows for salience to be applied during the initial training. Single exposure training has practical application for large neural networks which have already been trained on extensive data sets, and would not wish to repeat the training for specific new additions.
We present a one-dimensional affective system (just positive/negative affect). In reality the innate primary emotional systems are multi-dimensional [61] [79] because this enables innate responses to a variety of circumstances. Extensions of the model proposed here should be able to handle this higher dimensional emotional structure.
Significance: Robotics
The emotional systems in humans have been developed by evolutionary processes over millions of years because they produce mental processes that significantly enhance survival rates [27]. It might therefore be expected that using simulated emotional systems in robots would also improve their abilities to learn and survive in a variety of different contexts, where the 'affective' experiences they have guide them as to how to behave, even though no qualia are associated with them. As in the human case, they would be provided with a set of primary pre-primed systems giving guidance to the main ANNs as to what kind of behaviour is likely to enhance survival in the context of particular experiences. This would then shape both immediate behaviour and alter neural connection strengths so that the pattern of active connections as a whole at that time would be enhanced, perhaps even being learned at one go. Thus what is demonstrated here as a matter of principle could be incorporated as aspects of far more complex neural networks or robotic applications. Thus while this paper uses a simple SANN to demonstrate the effect, it is a proof in principle of the possibility of existence of applications where inclusion of the architecture we lay out here in deep learning reinforcement neural networks [30] could enhance their capabilities considerably. This is a subject for future research.
There is now a substantial literature on 'emotional robots' that simulate emotions in order to have a more effective robot-human interaction [3] [7]. However they do not simulate the affect-neocortical relations of the kind we discuss here, where emotions have an important effect on cognition and memory [17] [61] [27]. Such 'affective' systems as proposed here could potentially further enhance robot-human interactions by inclusion in such robots, allowing them to relate to humans more meaningfully on an emotional level (for example, retrieved memories would have an associated emotional tone that would affect behaviour).
The Big Picture
The big picture here is how genuine complexity emerges from the underlying physics, the two key cases (apart from life in general) being the brain on the one hand and digital computers on the other.
The way the functioning of the mind/brain occurs at the micro level is discussed in [26], where the physical nature of bio-molecules such as voltage gated ion channels in axons and dendrites is identified as the key enabler allowing branching logic to emerge from the underlying unitary physics. However this only leads to our ability to think when billions of neurons are linked by synapses to form immensely complex hierarchically structured neural nets such as occur in neocortical columns [42]. Given the neural and synaptic structure, that architecture is the key to brain function.
The way digital computers are able to perform the logical operations on the basis of the underlying physics is discussed in [25], where it is the detailed physical structure of transistors in integrated circuits that is the key enabler allowing digital logic to emerge from the underlying unitary physics. These again have to be built into a dense network of interconnected transistors forming a hierarchically structured whole [77] in order to perform the magic of turning the logic of an algorithm into its representation in terms of digitally coded electron flows in transistors. Given the functioning of transistors, it is that detailed architecture that is the key to computer function. Now one can model biological neural networks through suitable computer programs that code for Artificial Neural Networks (ANNs) (see Figure 6), and this gives remarkable properties such as pattern recognition and classification. In essence they are a simplified representation of the structure of cortical columns in the brain. These models can be spiking or non-spiking; the majority in broad use are non-spiking, although increasingly Spiking Neural Networks (SNNs) are being used in artificial intelligence applications [64].
The key point is that neither ANNs or SNNs in use at present represent the architecture embodied in the "ascending systems" in the brain, with diffuse projections that we model here. The burden of this paper is that doing so has the potential to greatly increase the power of both ANNs and SNNs. We demonstrate this in the case of an ANN by implementing the architecture whose essence is shown in Figure 7. That is, we show that this highly simplified version of the arousal system in the brain -which is there for very good functional reasons, leading to such enhanced survival prospects that its existence has been hardwired into our genes [27] -can in the case of the simple ANN we implement be demonstrated to give greatly improved learning performance. It can be expected that the same will be true if this architecture is implemented in the much more complex deep learning networks in use in the AI community.
Structure of this paper
Using a cognitive architecture approach [52], we consider the effect of emotions on memory in the neocortex (Section 2) and associated cortical structures of diffuse projections in the human brain. We conduct an investigation of existing comparable frameworks (Section 3) and then present the expected characteristics of the SANN model and its design (Section 4), and its implementation (Section 5). Thereafter we present the results from simulations (Section 6) and discuss conclusions and future work (Section 7).
The effect of emotions on memory
This section considers the way emotional effects affect cortical function, and how this relates to semantic memory (generic and specific).
Edelman's value system
In this paper, we build on Edelman's framework of the arousal system (which he calls the "value system") [20]. There are four stages of emotional effect in the cortex. Firstly, external experiences and internal thoughts 5 give rise to an image in the cortex. External experiences are first received by the senses and are then sent via the thalamus (A) to both the amygdala (B), generating an unconscious response, and to the cortex (D), responsible for conscious response. Secondly, the experience can then cause a conscious (E), and/or a sub-conscious (C) emotional response. At the same time, top-down corticothalamic connections send information from the cortex back to the thalamus (G) [67]. Thirdly, the arousal system releases neuromodulators into the neocortex via diffuse ascending connections (F), having the effect of associating an emotional tag with a memory [22]. Fourthly and finally, if the same or similar events contexts/places/people are encountered later, the memory is recalled with the same emotional tag that was stored.
According to Edelman [24] [21], there are four stages of emotional influence in the cortex (shown in Figure 2):
(i) External experiences (seeing your dog) and internal thoughts (remembering something good or bad) activates an image in the cortex. External experiences are first received by the senses and are then sent via the thalamus [73] (label "A" in Figure 2) to both the the amygdala (generating a unconscious response, label "B" in Figure 2), and to the cortex (responsible for conscious response, label "D" in Figure 2) [48] (ii) The experience can then cause a conscious emotional response (label "E" in Figure 2), and/or a sub-conscious emotional response (label "C" in Figure 2). This emotional response takes the form of the release of neuromodulators (such as dopamine, noradrenaline and serotonin) from nuclei in the arousal system. This process of triggering is modulated by experience, and can be altered through learning (for example, learning to identify your mother's face in a crowd). At the same time, top-down corticothalamic connections send information from the cortex back to the thalamus (label "G" in Figure 2), which allow for contextual prediction to take place in the thalamus [67].
(iii) The associated spread of neuromodulators from these nuclei to the neocortex via diffuse ascending connections [44] (labelled as "F" in Figure 2) has three effects:
(a) It induces emotional feelings (qualia), and can also possibly initiate immediate action in response in a sub-cortical (fast) way; (b) It alters neural connection strengths [20] so that memories of the event and associated contexts/places/people are stored together with the appropriate associated emotional tag (positive or negative) [22]; this alteration of cortical connection weights is the AND affect-shaped mechanism of neural plasticity [28]; In addition, once-off learning takes place if a strong enough emotional tag is attached to the event. (c) It results in a cognitive response, as the conscious brain responds to what has happened [17]. This can then lead to resulting considered action.
(iv) If the same or similar events contexts/places/people are encountered later, the memory is recalled with the same emotional tag that was stored.
In this paper we focus on the effect that neuromodulators have on memories (Item 3b), the storing of emotional tags associated with memories, and their recall (Item 4). We demonstrate how once-off learning takes place if a strong enough emotional tag is attached to the event. We do not model the immediate effects of the emotions (Item 3a), nor do we attempt to model how the emotional system is triggered (Item 2).
Types of memory
Long term memory is commonly classified as either explicit (declarative) or implicit (non-declarative) memory [5] (see Figure 3). Non-declarative memory is considered to act unconsciously, and covers activities like tying a shoelace, driving a car, or riding a bike. By comparison, declarative memory is available to the conscious mind, and can be broadly categorized as either semantic or episodic memory [75]. Semantic memory on the one hand describes facts, ideas, meaning, and concepts [65], and on the other specific members of classes identified in semantic memory, with an emotional tag attached to each. Episodic memory places specific items in a specific context, connecting specific semantic memories in a sequence, with a location, time, and with emotions [80].
To illustrate the difference between semantic and episodic memory, we provide an example. Semantic memory described unique classes of objects (e.g. a woman, your mother, a book, a bench). Concepts can be structured in a hierarchy of complexity, and (as with objects inheritance in computer science) semantic memories can inherit from parent classes. Semantic memories can be further classified as either Generic (classes of concepts) or Specific (instances of a concept); this corresponds to the distinction between a class and an instance of a class in object oriented programming [11]. For example, your mother's face could be a specific instance of the generic concept of a woman's face, which in turn could be a sub-class of the generic concept of a human face. An example of a computational model of semantic memory would be an Artificial Neural Network (ANN) that is designed to classify objects.
By comparison an episodic memory places a memory in a context (e.g. a location, with other objects, and in a particular instance in time). For example, a memory of "Mother reading while sitting on a bench in the park, at 2pm on 14th June." would be an episodic memory. This concept can be expressed in computer science as an instance of a class. An example of a computational model of episodic memory would be a Recurrent Artificial Neural Network (RNN), that is able to classify a sequence of objects.
In this paper we focus on modelling the value system associated with semantic memories, with the aim to extend this to episodic memory in the future. In the following section we take a closer look at the structure of diffuse projections, and the effects of salience in the cortex. Types of long term memory include Explicit and Implicit memories. Explicit memories can be further divided into Semantic and Episodic memories. Semantic memory be divided into Generic (classes) and Specific (instances of those classes). While emotions can be associated with both Explicit and Implicit memories, in this paper we focus on Semantic memories, and specifically how neuromodulators associated with emotions impact Generic and Specific memories.
Cortical structures of diffuse projections
Within the motor, sensory and cognitive systems, the human brain has at least three classes of connections relating to the cortex [21]. These are all crucial to brain function, which is why they have been built into our brain through a variety of evolutionary processes [27].
The first class consists of localized inter-neuronal connections forming layered neural networks, which are modeled by standard artificial neural networks (ANNs). Standard ANNs [10] model the way signals flow locally in the columns in the cortex through local connections in those columns, allowing complex processes such as pattern recognition [10], image classification [14] [46] and naming [36] to occur.
The second class are recurrent connections between different cortical areas [21]. These are important in terms of cortical functioning, but are not the concern of the present paper.
A third class consists of diffuse projections from the arousal system [54] to the cortex, known as 'modulatory' systems [44] or 'ascending' systems [24]. From their nuclei of origin they send axons up and down the nervous system in a diffused spreading pattern (see Figure 1). The effect of the modulatory systems projecting profusely is that each associated neuromodulator (for example norepinephrine and dopamine) affects large populations of neurons, allowing non-local interactions to occur in the brain. The release of the neuromodulators affects both the neurons, and the strength of the connections between neurons (see Figure 4). Specifically, neuromodulators affect the probability that neurons in the neighborhood of value-system axons [20] [21] will fire after receiving excitatory input, thus they are an important mechanism effecting neural plasticity in the long term. These systems bias neuronal responses affecting both learning and memory by guiding neuronal group selection, and for this reason that they are sometimes termed value systems [23] [24]. Standard ANNs do not model this non-local flow of information represented by the ascending systems, which are a significant feature of the structure of the brain. The aim of this paper is to model the effects of such connections in a new class of ANNs, which we refer to as Salience-affected neural networks (SANNs) because they allow representation of salience effects. The release of neuromodulators by the arousal system affects a whole region in the cortex simultaneously via the ascending systems, impacting those neurons that are active at the time the neuromodulators are released in proportion to their level of activation. A released neuromodulator affects both the neurons themselves (their behaviour) as well as the strength of the connections between neurons at synapses. Specifically, neuromodulators affect the probability that neurons will fire after receiving excitatory input, and they also affect the quantity of neurotransmitters released across the chemical synapses between neurons.
The salience of an entity refers to its state or quality of standing out relative to neighboring entities. For example, a salient memory might be one that significantly stands out among others because of its emotional content. Beebe and Lachmann defined the three principles of salience that describe the interaction structures in the first year of life [8]. These are the principles of ongoing regulations, disruption and repair, and heightened affective moments. Ongoing regulation describes the characteristic pattern of repeated interactions, such as a child interacting with their mother or father. This is well represented by standard ANNs, which allow associational learning with multiple-trial (i.e. ongoing regulations). However heightened affective moments are dramatic moments standing out among other events [8], often leading to single-exposure learning, which is an established psychological phenomenon [70], and has been the subject of discussion in the psychology literature for over 50 years.
While the details of how single-trial training works in the cortex are in dispute, the fact that single-trial training sometimes occurs is well established [4] [51]. When it occurs, single-trial learning is often associated with the neuromodulator dopamine [81]. This is one of the neuromodulators released throughout the cortex by the ascending systems, and is associated both with the neural coding of basic rewards, and in reinforcement signals defining an agent's goals [72] [18] [57]; hence it is known to play an important role in brain functioning. Standard ANNs do not currently model the salience effects of neuromodulators such as dopamine, and are unable to implement single-exposure learning; but SANNs do both.
Characteristics of neuromodulators
A key characteristic of a neuromodulator is that it is present or absent in a region of the cortex, rather than being associated with specific synaptic connections. It can occur with various strengths in such a region, therefore we model the input salience signal of a neuromodulator associated with positive salience in the SANN model as a positive signal with value between 0 and 1, and the input salience signal of a neuromodulator associated with negative salience as a negative signal with value between −1 and 0. ‡ A second key characteristic of the effect of neuromodulators is that they affect the the entire pattern of activated nodes according to the level of activation of each synapse. Therefore, in the presence of a salience signal, both the connection weights and the sigmoidal activation function were adjusted depending on the current level of activation of each neuron/synapse (see Equation 1 and Equation 6 below); hence there is no change if the synapse is not active. The key feature of the SANN is that it is able to strengthen an entire pattern of activated nodes proportional to the strength of the salience signal, rather than just individual connections.
In the human brain, neuromodulators only influence those neurons that possess the relevant receptors for each modulator. Furthermore, neuromodulators receptors come in different sub-types. However, in modelling terms for the purposes of demonstrating the concept, we assume that neuromodulators are global in nature, affecting all nodes and weights in a network. The salience signal is therefore modelled as a single value, and associated with each node and weight in the network (see Figure 7).
Comparison with other Frameworks
In this section we compare SANNs with other neural network models.
Standard ANNs
Most neural network learning processes take many repetitions before a memory is stored. But in real life, some memories are stored after just one experience ('once-off' learning). In this paper we demonstrate that once-off learning takes place if a strong enough emotional tag is attached to the event.
Husbands' model of gas diffusion
The only other class of ANNs of which we are aware that represent non-local effects are models of the effects of Nitric Oxide (NO), which is a freely diffusing neuromodulator [39]. A model of gas diffusion is used in a class of ANNs in which nodes are capable of non-locally modulating the behaviour of other nodes. These models are called GasNets [39], as they simulate the presence of the gas NO in the environment surrounding the neurons. Like SANNs, this form of modulation allows a kind of plasticity in the network in which the intrinsic properties of nodes are changing as the network operates [39]. However to the best of our knowledge, GasNets have not been modified to train and test an ANN with specific salience, nor do they have the potential for single-trial learning in ANNs.
Our model differs from the research on GasNets by Husbands [39] as it models both the way memories associated with emotionally-laden events may be embodied in neural networks, and the way those emotional associations can be recalled when the events are remembered. Furthermore, this research introduces an additional input salience signal during training, and each node produces a nodal reverse salience signal during testing. This research also demonstrates that these kinds of networks enable single-exposure learning by associating strong salience signals with input combinations. ‡ In future work we will model a multidimensional salience system, with different neuromodulators associated with each dimension [61] [79]; however for this paper it will be adequate to think of a single neuromodulator that can have positive or negative valence.
Juvina's model of valuation and arousal
Juvina [41] presents an approach to adding primitive evaluative capabilities to a cognitive architecture in which two sub-symbolic quantities called valuation and arousal are learned for each declarative memory element, based on usage statistics and the reward it generates. Consequently each memory element can be characterized as positive or negative and having a certain degree of affective intensity. In turn, these characteristics affect the latency and probability of retrieval for that memory element. This is similar to what we do, but using a very different cognitive architecture. Ours is based closely on the way the diffuse systems interact with cognitive functioning in the human brain [20] [21], which is an affective process because these systems are those implicated in the primary emotional systems identified by Panksepp [61].
Edelman's Robots
Gerald Edelman created a range of brain-based devices (BBDs) including Darwin VII and Darwin X. [22] [45]. Edelman created 'Darwin VII' to capture a holistic picture of its environment, incorporating 3 different senses (vision, auditory, conductance or "taste"), a motor system, and a "value system" (attempting to model an ascending neuromodulatory value or reward system). Edelman then attempted to model long-term episodic memory (behaviour attributed to the hippocampus) in his model 'Darwin X'.
Edelman's robot 'Darwin VII' contained a computational nervous system of 20,000 neuronal units, and 450,000 synaptic connections. After some training 'Darwin VII' successfully learned to associate the taste value of the blocks with their visual patterns [22]. The sensory systems interpreted incoming signals (such as conductance of a block) as a 'value signal". In Edelman's model of the value system, the strength of connections between two neural units could change based on the value signal (i.e. a positive value signal would strengthen a connection).
Key difference relative to our paper: Edelman allowed the 'salience' signal in 'Darwin VII' to directly impact the strength of synaptic connections between neurons. By doing so, Edelman's model incorporates 'salience' as just another input signal (along with the visual, auditory and sensory signals) that requires extensive training. As a result, Edelman's BBD models required many iterations of training for the salience to be embedded in the model ,and associated with sensory input patterns. Due to the design of his model, Edelman's model was not able to demonstrate single exposure learning on an previously trained dataset.
By contrast, in this paper we demonstrate how the 'salience' signal from the arousal system (which Edelman called the 'value' system) can be modelled as an additional dimension to the neural network model. We demonstrate how the 'salience' signal can be either part of the training, or applied afterwards as a single exposure. By modelling the 'salience' signal as an additional dimension to the neural network model, we are able to demonstrate how the arousal system can affect neural activation patterns (e.g. memories) during single exposure, without affecting synaptic weights of the neural network.
Salience-affected neural network (SANN)
To model the effect that neuromodulators have on semantic memories we present a neural network that allows for feature extraction training on a standard data set, while in addition also allowing for the network to be affected globally by a salience signal related to a specific individual item.
Definition of terms We define the salience of an entity as its state or quality of standing out relative to neighboring entities, representing a positive or negative evaluation of that entity relative to individual welfare, and a salience signal (S) as an additional input signal to a SANN representing such a quality that affects each node during training. We also define the reverse salience signal (S ) as the sum of the reverse salience signals produced by each node during testing. In practice this means that the SANN will allow for a salience signal to be associated with a specific image (i.e. a weighted combination of features), and for that same image (for example, a specific face) to product a reverse salience signal if that specific combination of features is experienced again in the future.
Expected performance characteristics
A Salience-affected neural network (SANN) that models diffuse projections in the cortex must be able to demonstrate the following characteristics:
(i) The salience signal should affect entire patterns of activated nodes, not just individual nodes, and not all the nodes in the network equally: rather the entire current pattern of activation of nodes is either strengthened or weakened.
(ii) The network should be able to be trained with the standard training data set, as well as an additional dimension of an incoming salience signal associated with each image.
(iii) The network should be able to produce a reverse salience signal (as an output) for a specific input and the associated pattern of activation.
(iv) Training with salience should have an overall positive impact on the classification accuracy of the network as a whole, and for images in the same class.
(v) Training with salience should have a positive impact on the classification confidence of the specific image tagged with salience during training.
It is important to note that in this paper we do not model how salience activation occurs; rather we model its outcomes.
Design: SANN
The Salience-affected Neural Network (SANN) presented in this paper is designed to classify handwritten digits (0-9). As part of the pre-processing, non-negative matrix factorization [49] (NMF) was applied to extract features from the dataset, reducing the size of the input layer.
In addition to a standard ANN, each node of the neural network is adapted to accept a global salience signal (S) during training, and to respond with a reverse salience signal (S ) during classification (see Figure 5). Furthermore, when exposed to salience signal, the SANN also globally strengthens the connection weights in the network proportional to the activation levels of the parent nodes.
Dataset
We use the MNIST database of handwritten digits [47], consisting of 70,000 labelled, 28x28 grayscale images of the 10 handwritten digits (0-9). The training set consists of 60,000 images while the testing set consists of 10,000 images (see Figure 9). This dataset has been widely used in conjunction with supervised and unsupervised neural networks models [12] [43] [59], to provide a realistic dataset for simulations of biologically-inspired models, similar to the one we present in this paper. We present a salience-affected neural network (SANN). The architecture of the model starts with a feature extraction algorithm, to extract features from the dataset. Feature extraction also has the impact of reducing the size of the input layer required for the neural network. Each image is then reduced to a weighted set of features, and this input is fed to the input layer of the neural network. The neural network is a standard feed-forward neural network, and is trained as a classifier. In addition to the input values, we add an additional dimension: a global salience signal. This salience signal (S) consists of a single value, and affects all nodes in the neural network. Also, the neural network is designed to produce a reverse salience signal (S') when an image is presented to the model, so that each image produces both an output (classification) as well as a reverse salience signal (S').
Feature extraction (NMF)
As part of the pre-processing, non-negative matrix factorization [49] (NMF) was applied to extract features from the dataset. A NMF component layer size of 49 features was chosen. The MNIST dataset was therefore reduced from a set of images of 784 pixels each, to a set of images with with 49 features per images. The input layer of the neural network was reduced from 784 to 49 nodes (see Figure 6).
NMF pre-processing was chosen for a two reasons. Firstly, it is known that 2-layer feed forward networks can achieve 98.4 -99.3 % classification accuracy [74]. We require a wider range of accuracy to demonstrate the impact of an SANN on classification accuracy, hence we degrade the maximum accuracy by reducing the dimensionality from 784 (28x28) input values, to 49 input values using NMF. Secondly, NMF is known to produce "human like" features from raw data, so it is a reasonable technique to use for the purpose of demonstrating the SANN model.
Neural network model
The model we use is a fully-connected artificial neural network (ANN), configured with 49 input nodes, a hidden layer of 100 nodes, and an output layer of 10 nodes (see Figure 6). Each node in the output layer represents a specific digit (0-9).
We chose an ANN because it is a well-characterized and common neural network from which many other neural networks are derived [2]. For this research, the neural network was adapted from an open-source neural network model [71]. The neural network model could be extended by addition of many more layers, or even as a deep neural network [60].
In this model we chose to use the sigmoidal activation function. We felt that the sigmoidal activation function was better suited than other activation functions (e.g. maxout functions [34], ReLu functions [58], PLUs [78]) to the SANN model for two reasons. Firstly, because sigmoidal activation functions are more biologically realistic: most biological systems saturate at some level of stimulation (where activation functions like ReLu do not). And secondly, sigmoidal functions allow for bipolar activations, whereas functions like ReLu are effectively monopolar and hence not useful for a reverse salience signal with both polarities of activation.
Salience signal
To incorporate salience, an additional signal was embedded in a standard ANN [2] (the salience signal), such that it directly affected every individual node during training, and collectively produced a reverse salience signal during testing (see Figure 7). Each node in the network then produces a reverse salience signal during testing, which signals are summed to produce a single reverse salience signal for the network. During training, the salience signal also has the impact of altering the weights (blue arrows) of the connections between all nodes in the network simultaneously by an amount proportional to the activation level of the subsequent node and the salience signal strength.
Similar ANN input combinations would be expected to produce similar reverse salience signals, because the reverse salience signal is defined as the summation of the nodal reverse salience signals observed at each node.
Implementing a salience signal in the SANN
The addition of a global salience signal in the SANN introduces an additional dimension to the standard neural network (ANN). The global salience signal (S) was designed to have two distinct effects.
First, to accept a global salience signal (S) which would affect each node in the network simultaneously, and produce a reverse salience signal for future inputs. To achieve this we enable the global salience signal to affect the sigmoidal activation function of each node in the ANN. In this section we discuss and evaluate 3 different ways in which we allow the salience signal to impact the activation function.
Secondly, the global salience signal (S) was also designed to to alter the strength (weights) of all connections between nodes, by an amount proportional to the activation level and the salience signal strength (Equation 6).
The variations of the activation function proposed in this research for use with the SANN closely mirror the variations described by Husbands [39]. This research focuses firstly on the threshold offset variable, referred to as b i by Husbands [39], and secondly on the alteration of node weight, corresponding to Edelman's idea of Neural Darwinism [20] [28].
In this section we discuss the two effects the input salience signal has in the SANN, firstly the effect on the activation function at each node, and secondly, the effect on the weight of each connection in the network. We also discuss the reverse salience signal, which can occur because of the alteration to the activation function.
Effect 1: Activation Function
Each node was assigned a salience value. This salience value (S i ) was defined as a value between −1 and 1, calculated relative to the input salience signal (S global ), and the current activation level of the node (A i ). A constant (β) was introduced to allow for different learning rates, which might differ from neuron type to neuron type, or with the kind or neuromodulator modelled.
S i (new) = S i (old) + (1 − S i (old)) × A i × S global × β(1)
In the SANN, the input salience signal (S global ) affects the network such that the network produces a reverse salience signal (S ) for future input values. As previously mentioned, neuromodulators affect the probability that neurons will fire after receiving excitatory input. To maintain a link to biological behaviour of neurons, salience was implemented such that it affected the activation function. The activation function chosen in this paper is a sigmoid activation function (see Equation 2).
y(x) = 1 1 + e −x(2)
Modifications We investigate three possible modifications to the activation function, namely (a) a horizontal offset, (b) a change in the gradient, and (c) a change in the amplitude (see Figure 8). Effect 1a: Horizontal Offset The first of three effects was to introduce a horizontal shift of the activation function along the x-axis (see Figure 8A). A positive salience signal would shift the activation function to the left, resulting in a higher output from the activation the next time. The activation function incorporated the horizontal offset as shown in Equation 3.
y(x) = 1 1 + e −(x+Si)(3)
Effect 1b: Gradient Change The second of three effects was to introduce a change in the gradient of the activation function (see Figure 8B). A positive salience signal would reduce the gradient of the activation function, resulting in a higher output from the activation the next time. The activation function incorporated the gradient change as shown in Equation 4.
y(x) = 1 1 + e −(0.5 −S i ×x)(4)
Effect 1c: Amplitude Change The third of three effects was to introduce a change in the amplitude of the activation function (see Figure 8C). A positive salience signal would increase the amplitude of the activation function, resulting in a higher output from the activation the next time. The activation function incorporated the amplitude change as shown in Equation 5.
y(x) = 0.5 −Si 1 + e −x(5)
Effect 2: Strengthening of connection weights
In addition to producing a reverse salience signal (S ), the global salience signal (S global ) also has the impact of altering the weights of the connection between all nodes (W ij ) in the network simultaneously, by an amount proportional to the current weight of the connection (W ij ), the strength of the input salience signal (S global ), and the activation level of the destination node (A j ).
W ij (new) = W ij (old) × (1 + α × |S global × A j |)(6)
Here α is a constant which might differ from neuron type to neuron type, or with the kind or neuromodulator modelled. Note that because the updating equation for each synapse weight depends on the current state of activation A i of that synapse, just as in the case of Neural Darwinism [20], it is the entire current pattern of activation that gets reinforced by the salience signal.
Classification accuracy
To measure the impact of salience on the neural network, we measured the classification accuracy (τ ) of the class, and the classification confidence (φ) of the individual image.
Classification accuracy (class): The classification accuracy (τ ) of a class was calculated as the number of correct classifications, divided by the total number of classifications performed. The inverse of the classification accuracy is often referred to as the error rate.
Classification confidence (individual image): The classification confidence (φ) of the individual image (i) described how confident the classification algorithm is to assigning an image to a class. The classification confidence (φ) was calculated as the highest output value the neural network (max(o)). The amplitude change was normalized when calculating the classification accuracy by a factor of 0.5 (−S global ) , because it was the only function to change the amplitude of the signal propagating through the network.
Reverse Salience Signal
The changes made to the activation function allow the value of salience associated with a specific image to be read off when that image is presented. This is the reverse salience signal (S network ). If an image (or a member of its class) had high salience signal (S global ) associated with it during training, then we would expect a high reverse salience signal (S network ) during testing. The reverse salience signal for each node (S i ) was calculated as the product of the current activation level A i , and the reverse salience signals of each node (S i ) (see Equation 7). The reverse salience signal for the network (S network ) was calculated as the sum of the reverse salience signal of each node in the network (see Equation 8).
S i = |A i | × S i (7) S network = S i(8)
It was observed that, due to the nature of the definition of reverse salience signal value, once the SANN was trained with salience, the reverse salience signal values would have a global offset. We refer to this offset as the residual reverse salience value. The reverse salience signal was defined as the average reverse salience signal over a set of test images. The reverse salience signal was then adjusted to exclude the residual reverse salience signal.
Results
In this section we describe the results obtained from training the SANN with salience. We are concerned with the effect of a salience signal applied to a particular single representative image within a class of images both on (a) recognition of that class of images (Semantic memory: Generic, see Figure 3), and (b) recognition of that particular image (Semantic memory: Specific, see Figure 3), so we clearly separate these out in what follows.
Neural network training
In this section, we review the impact of the changes to the activation function indicated in Fig. 8 (Eqns. (3)-(5)), as well as the change in weights (Eqn. (7)). We clearly separate effects on class recognition and on individual image recognition in our results below.
Classification training: The neural network was designed with 49 inputs, a single hidden layer of 100 nodes and an output layer of 10 nodes (see Figure 7). The neural network was trained on 1,000 images, over a period of 25 epochs, with a learning rate of 0.5. The trained network was then tested on 1000 test images, and correctly classified 776 of the 1000 test image (categorization accuracy of 77.6% for the MNIST dataset). Our aim in this paper is not to optimize the performance of the neural network by changing the size of the neural network layers, the size of the training set, the number of epochs, or the learning rate. Rather we establish the baseline performance of a sufficiently accurate neural network classifier, and document the impact of training the network with a global salience signal (S).
Single exposure salience training: After the SANN had been trained to classify the images in the MNIST dataset, the network was exposed to a salience signal as discussed above in previous sections. To achieve this, a single image in the training set was tagged with salience, and then the network underwent a single-exposure training event (see Figure 9). We repeated this process for every individual image in the training set.
Effect 1: Activation Function
In this section we will discuss the results observed for (a) class recognition, and for (b) individual image recognition. Specific measures we will discuss include reverse salience signal value, and classification accuracy. We will also discuss the results of the three possible modifications to the activation function, namely (a) the horizontal offset, (b) the gradient, and (c) the amplitude.
Reverse salience signal value (class):
The results show that after a single exposure training iteration, the SANN associates a strong reverse salience value (S ) to all inputs that are similar to the salience-affected value. The reverse salience signal (S ) was notably grouped by similarity and by class (see Figure 10). The more similar an input image was to the image trained with salience, the higher the reverse salience signal. Similarity was calculated using MSE. This result supports the principle that salience applied to the member of a class, will have an impact on similar inputs, having the highest impact on the other members of the same class. After salience training, the neural network produced a reverse salience signal (S ) for every input image. Input images most similar to the salience-tagged image (based on an MSE calculation) presented with the highest reverse salience signal (S ). The reverse salience signal was also notably grouped by classification (i.e. digit). In this diagram image #7 in the training set was tagged with salience. Image #7 was correctly classified as belonging to class: Digit #1.
Reverse salience signal value (individual image):
The results shows that the image trained with salience produced the highest reverse salience signal (S ) (see Figure 10). This result supports the principle that if an input is accompanied with salience, then that same input will produce a strong reverse salience signal if it is seen again.
Classification accuracy (class): At the class (generic) level, we observed that each of the three modifications to the activation function had varying effects on the network's classification accuracy (φ). The horizontal offset and amplitude change of the activation function both had a negative impact on the average classification accuracy, however, the gradient change had a positive impact on the classification accuracy (see Figure 11). It was observed that the higher the salience magnitude (for gradient change), the higher the overall improvement in network classification. The highest average improvement in network classification accuracy was observed with a gradient change only and a salience factor of 1.0, resulting in an average improvement of classification accuracy (φ) of 0.61%, from the benchmark performance of 77.60% to 78.21%. For each modifications to the activation function, the input salience factor (S) was varied from 0 -1, and the network performance was recorded. It was observed that the horizontal offset and amplitude change of the activation function both had a negative impact on the average classification accuracy, however, the gradient change had a positive impact on the classification accuracy. The highest average improvement in network classification accuracy was observed with a gradient change only and a salience factor of 1.0, resulting in an average improvement of 0.61%, from the benchmark performance of 77.60% to 78.21%.
Classification accuracy (individual image): At the individual image (instance) level, we observed that each of the three modifications to the activation function had varying effects on the network's classification confidence (τ ) for the salience-tagged image. The horizontal offset and amplitude change of the activation function both had a negative impact on the average classification confidence. However, the gradient change had a positive impact on the classification accuracy (see Figure 12). It was observed that the higher the salience magnitude (for gradient change), the higher the overall improvement in the classification confidence for the salience-tagged image. The highest average improvement in network classification confidence for the salience-tagged image was observed with a gradient change only and a salience factor of 1.0, resulting in an average improvement in classification confidence (τ ) of 4.89%, from the benchmark performance of 90.41% to 95.20%. In this section we have observed the effect of a global salience signal on the activation function of each node and shown how a salience signal can improve the average classification accuracy (φ) by 0.61%, and the average classification confidence (τ ) by 4.89%. In the next section we discuss the effects of the global salience signal on the connection weights.
Effect 2: Strengthening of connection weights
In addition to impacting the activation function of each node, the global salience signal (S) also affects each of the connection weights. It is important to note that this process is very different from back-propagation because it affects all weights globally at the same time, only factoring in the global salience signal and the current activation level of each node.
Classification accuracy (class): During a single exposure of global salience (S), the salience signal affects the connection weights proportional to the current activation level of the nodes. The impact of this single iteration of training on the overall network was a clear improvement in the classification accuracy (φ) of images. The highest average improvement in network classification accuracy (φ) was observed with a gradient change only and a salience factor of 1.0, resulting in an average improvement of classification accuracy (φ) of 0.61%, from the benchmark performance of 77.60% to 78.21%.
Classification confidence (individual images): At the individual image (instance) level, we also observed an improvement in the classification confidence (τ ) of images in response to a change in weights (see Figure 14). The highest average improvement in network classification confidence (τ ) for the salience-tagged image was observed with a gradient change only and a salience factor of 1.0, resulting in an Fig 13. In addition to affecting the activation function of each node in the network, the salience signal (S) also strengthens the weights of connections between nodes, proportional to the activation level of the post-synaptic node. The input salience factor (S) was varied from 0 -1, and the network performance was recorded. It was observed that for a salience magnitude of 0.8 resulted in the greatest overall improvement in network classification: 2.49%. After salience training, the SANN network was able to accurately categorize 80.06% for the MNIST dataset, compared to the benchmark of 77.6% accuracy before the salience training.
average improvement in classification confidence (τ ) of 5.56%, from the benchmark performance of 90.41% to 95.97%. At a salience factor of 0.8, the average improvement in classification confidence (τ ) is still significant: 5.48%, from the benchmark performance of 90.41% to 95.89%. At the individual image (instance) level, we also observed an improvement in the classification confidence (τ ) of images in response to a change in weights. The highest average improvement in network classification confidence (τ ) for the salience-tagged image was observed with a gradient change only and a salience factor of 1.0, resulting in an average improvement in classification confidence (τ ) of 5.56%, from the benchmark performance of 90.41% to 95.97%.
In this section we have observed the effect of a global salience signal on the activation function on the connection weights in the network. With a salience factor of 0.8, this effect improves the average classification accuracy (φ) by 2.49%, and the average classification confidence (τ ) by 5.48%.
Combining Effect 1 and Effect 2
Having reviewed the results of Effect 1 and Effect 2 separately, we then combined Effect 1 (combined gradient and amplitude change) and Effect 2 (changing connection weights). By combining these effect we were able to test the impact of single-iteration training of a global salience signal on both the activation function of each node (Effect 1) and the connection weight of each connection in the network (Effect 2).
Benchmark performance: The benchmark performance of the neural network before salience training was a classification accuracy (φ) of 77.6%, and a classification confidence (τ ) of 90.41%. The best performance for Effect 1 alone took place with a salience magnitude of 1.0: an improvement in network classification accuracy (φ): 0.61%, and an improvement in network classification confidence (τ ): +4.89%. The best performance for Effect 2 alone took place with a salience magnitude of 0.8: an improvement in network classification accuracy (φ) of 2.49%, and an improvement in network classification confidence (τ ) of 5.56%.
Classification accuracy (class): We combined Effect 1 and Effect 2, and varied the salience factor for each effect to create a performance heat map. We noticed an improved overall higher classification accuracy (φ) across the entire training set, compared to either effect alone (see Figure 15A). The best performance for Effect 1 and Effect 2 combined was an improvement in network classification accuracy (φ) 2.56% (an improvement from 77.6% to 80.16%). This result is 0.07% higher than Effect 2 alone, and 1.95% higher than Effect 1 alone.
In addition to calculating average performance, we also recorded the standard deviation of the results (see Figure 15C). The standard deviation of the dataset describes how similar the results were within the result set. The smaller the standard deviation, the closer to the average the results in the dataset were. While we would like to maximize the average performance improvement, we will seek to minimize the standard deviation of the performance improvement. The lowest standard deviation of classification accuracy (φ) observed for Effect 1 and Effect 2 combined was 0.00% (when both salience factors were set to 0). The higher the salience factors, the higher the standard deviation for the result set.
Classification confidence (individual images): After combining Effect 1 and Effect 2, we also noticed an improvement in average classification confidence (τ ) across each individual image in the training set (see Figure 15B). The best performance for Effect 1 and Effect 2 combined was an improvement in classification confidence (τ ) from 90.41% to 96.0% (an improvement of 5.59%). This was 0.03% higher than Effect 2 alone, and 0.7% higher than Effect 1 alone.
We also performed a standard deviation analysis on the classification confidence (τ ) (see Figure 15D). The lowest standard deviation of classification accuracy (φ) observed for Effect 1 and Effect 2 combined was 19.54% (when salience factor of Effect 1 was 0.0 and the salience factor for Effect 2 was 1.0). However, there were several values of salience factors that produced this result.
Optimization (overall performance): To optimize the salience factors for the SANN model as a whole, an optimization value (γ) was calculated to maximize the average values of classification accuracy (φ) and classification confidence (τ ), and to minimize the standard deviations of classification accuracy (φ) and classification confidence (τ ) (see Equation 9).
γ = φ × τ stdev(φ) × stdev(τ )(9)
The optimization value was calculated for varying salience factors, and a heatmap of this optimization value was created to visualize the optimal salience factors for the SANN model as a whole (see Figure 15E). The optimal salience factors to maximize the average values of classification accuracy (φ) and classification confidence (τ ), and to minimize the standard deviations of classification accuracy (φ) and classification confidence (τ ) were a salience factor of 0.7 for Effect 1, and 0.4 for Effect 2. At these salience factor values, the network demonstrates an improvement in average classification accuracy (φ) of 80.1%, and an improvement in average classification confidence (τ ) of 96.0%.
Benchmark comparison: Compared to the benchmark performance of the neural network before salience training, a salience factor of 0.7 for Effect 1, and 0.4 for Effect 2 resulted in the following performance improvement: average classification accuracy (φ) improvement of 2.51%, and an improvement in average classification confidence (τ ) of 5.60%.
Discussion
We made notable observations at various levels, namely at the level of (a) the entire network, (b) each classification category, and (c) individual images.
Level A: entire network: After salience training, the SANN model as a whole experienced an additional dimension: salience. A global salience signal (S) was added as a single input during training, and a reverse salience signal (S ) was added as an additional output. The results shows that the image trained with salience produced the highest reverse salience signal (S ) (see Figure 10) during testing, and that images similar to the salience-tagged image also produce a relatively high reverse salience signal (S ) during testing.
Level B: classification category (generic memories): An individual image tagged with salience would experience, on average, an improvement in classification accuracy (φ) improvement of 2.51% (see Figure 15A).
Level C: individual images (specific memories): Crucially, the specific image tagged with salience was positively affected by the salience training. An individual image tagged with salience would experience, on average, an improvement in average classification confidence (τ ) of 5.60% (see Figure 15B).
A classification accuracy (φ) improvement of 2.51% at the class level may not seem significant at first, but this result is in fact quite significant. The classification accuracy benchmark for the neural network before salience training was 77.6%, which was achieved after 50 epochs of training with a training set of 1000 images per epoch. To achieve a 2.51% improvement in accuracy, the same network would require an additional 38 epochs of back propagation training, with 1000 input images per epoch: a total total exposure of 38,000 images (see Figure 17). The results show that the SANN model is able to achieve this significant improvement in overall classification accuracy with just a single iteration of global salience training.
Having demonstrated that a classification accuracy (φ) improvement of 2.51% at the class level is significant, it is then clear that an average classification confidence (τ ) Fig 15. Having tested the independent impact of salience on the node's activation function (Effect 1) and impact on the connection weights (Effect 2), these two effects were then combined. Simulations were run varying the salience factor for both Effect 1 and Effect 2, to observe the effects of the average (A) and standard deviation (B) of the classification accuracy (φ) of the entire network, as well as the average (A) and standard deviation (B) of the classification confidence (τ ) of each individual images. These results were combined as described in Equation 9 to calculate the optimal salience factors to ensure optimal performance of the SANN. The optimal salience factors to maximize the average values of classification accuracy (φ) and classification confidence (τ ), and to minimize the standard deviations of classification accuracy (φ) and classification confidence (τ ) were a salience factor of 0.4 for Effect 1, and 0.7 for Effect 2. The values with a darker border indicate the maximum values in each grid.
improvement at the individual level of 5.60% is even more significant. This means that a strong salience signal attached to an individual image makes its recognition much more likely. This demonstrates the efficacy of the mechanism whereby a strong salience signal attached to an individual image strengthens the entire activation pattern associated with that specific image, which is the central message of this paper. After only a single iteration of salience training, the SANN produces a reverse salience signal (S ). This reverse salience signal is present after Effect 1 only, but is amplified with addition of Effect 2. The reverse salience signal (S ) is the highest for the handwriting digits in the same classification group as the salience-tagged image. In this figure, we demonstrate the reverse salience signal for the scenario where Image #7 (classified as a handwriting digit #1) in the training set was tagged with salience during training.
Conclusion
In this paper we have presented a ANN that incorporates some fundamental features and behaviors that we associate with salience and learning in the cortex, as indicated in Figure 7. We do not claim that the SANN accurately models the biological salience system, but rather that the SANN is an artificial neural network model with architecture inspired by the salience systems in the human brain, leading to important performance enhancement. Specifically, the SANN incorporates a global salience signal, modelled on the diffuse projections from the arousal system observed in the cortex.
A SANN accepts a global salience signal in addition to input and output values representing incoming sensory data. We have not modelled what leads to affect, or its effect on behaviour, but rather two distinct effects in the cortical network related to memory. Firstly, it affects each node by altering the activation function. Secondly, it affects the weights of every connection in the network. In both cases, the impact on the network is proportional to the activation level of the nodes in the network, which closely models the effect of the neuromodulators associated with the ascending systems on the cortex. This is a powerful mechanism underlying learning via neural plasticity, because it reinforces an entire neural activation pattern in one go.
After the neural network had undergone standard supervised training (on the MNIST dataset), the SANN model was trained with a single iteration of salience. Following this training the SANN model produced a reverse salience signal for similar images to the salience-affected input image during training. In addition, training with salience strengthened the weights of connection during the single iteration of salience training, which further increased the reverse salience signal and the classification accuracy for images similar to the salience-tagged image. Combining Effect 1 and Effect 2, the SANN model was able to demonstrate a 2.51% improvement in accuracy for the class after just a single iteration of training (a jump from 77.6% to 80.11% accuracy). To highlight the significance of this achievement, we measure how many additional training epochs using backpropagation the SANN would have required to achieve the same improvement. To achieve the same improvement with only back-propagation training, the same network would require an additional 38 epochs of training, with 1000 input images per epoch: a total additional exposure of 38,000 images.
Notably, we demonstrated that the SANN can be trained with just a single training iteration. It was found that the reverse salience signal was proportional to the magnitude of the input salience signal. We associate this result with the expectation that single events of high salience can produce a learned response; to the best of our knowledge, this is the first demonstration of this phenomenon in an ANN.
Measured performance characteristics
We demonstrated the following characteristics:
(i) The salience signal affect patterns of activated nodes, not just individual nodes, and not all the nodes in the network equally: rather the entire current pattern of activation of nodes is either strengthened or weakened. (ii) The network was able to be trained with the standard training data set, as well as an additional dimension of an incoming salience signal associated with each image.
(iii) The network produced a reverse salience signal (as an output) for a specific input and the associated pattern of activation.
(iv) Training with salience had a overall positive impact (2.6% improvement) on the classification accuracy of the network as a whole.
(v) Training with salience had a positive impact on the classification confidence of the specific image tagged with salience during training, raising the confidence from 99.85% to 100%.
Specific and generic images: Results
The models and results published here link back into the distinction between generic and specific images (Figure 3). While salience affects the generic class (e.g. women), it has a heightened effect on the specific memory (e.g. your mother) [75] [65]. This also links back to the differentiation between a class and an instance of a class in object oriented programming [11]. The SANN has shown that single iteration salience training can affect both the generic class of memories (e.g. the class: Digit 1), as well as the memory of a specific memory (e.g. Digit 1: Image 7 / 1000). The effect on the specific image that has been tagged with salience is significantly greater than the effect for the class to which it belongs. This is a key feature of the way affect (salience) affects cognitive functioning: we recognize specific contexts or individuals as safe and friendly or threatening and dangerous on the basis of previous experience, and act accordingly. Enhanced recognition and the reverse salience signal are both key to this dynamic.
Research Implications
The research explored here has many implications, especially at the computational, biological and psychological levels.
At a computation level, the modeling of ongoing regulations in neural networks will allow us to extract more information (in the form of salience) from a standard ANN, without significantly adding to the complexity of the ANN structure. Furthermore, exploring the effect of heightened affective moments on a neural network will enable the future training of neural networks with a single training iteration. A major point is that salience signals can strengthen an entire pattern of neural activation at one go. This is the key feature that makes this architecture so powerful.
At the biological level, one of the puzzling issues about the brain is why evolutionary selection lead to the very complex "wet" structure of chemical synapses, involving transformation of the signal from electrical to chemical form and then back again, as opposed to the much more direct gap junction (electrial) synapses, which are much faster [42] [9]. Our analysis, following on the work of Edelman [20] [24] [21], provides an answer: signal transduction via neuromodulators at synapses allows the non-local effects of the ascending systems to implement the very effective one-time learning process we have demonstrated in this paper, rather than the customary laborious and computationally intensive gradient descent algorithms like back-propagation involving thousands of training events in conventional ANNs. It also allows activation of the reverse salience signals, a key aspect of how these systems are related to Panksepp's primary emotional systems [61] [27], as elucidated in [28].
At a behavioural level, a SANN allows the modeling of the effect of affective states on memory (for example, adding an emotional tag to significant memories), because the ascending systems originate in the arousal system which is the seat of affective states. Thus a SANN potentially represents a key effect of emotions on cortical activity, which is known to be significant. It does not however model how affect alters perception and action, which is a different topic [17]; nor does it model what effects trigger salience signals.
Future Work
Future work to be done includes:
(i) Extending the SANN to include episodic memory, and not just semantic memory. This could be achieved by feeding the output back into the input to create a recurrent neural network. Salience signals would then apply to a sequence of events, rather than a single event.
(ii) Investigating alternative variations to the sigmoidal activation function at each node in response to the input salience signal, for example activation function gradient.
(iii) Extending the SANN to multi-dimensional salience signals [79].
(iv) Extending the hidden layer size, and the number of hidden layers.
(v) Extending the salience signal to a biologically-accurate spiking-neuron SNN.
(vi) Extension of deep learning reinforcement neural networks [30] to include the principles laid out here This research will also eventually be incorporated in a model two of the authors of this paper (LR and GE) are proposing jointly with A. Mishra of cortico-thalamic interactions [67]. This models how perception works [31] via top-down predictive connections from the cortex to the thalamus [1]. Conjoining this with the results of this paper would open the way to modelling how affect alters perception.
Supporting Material
The source code for the SANN, as well as records of the tests conducted in this paper, are publicly available online [66]. For additional information, please contact the authors.
Fig 1 .
1The release of neuromodulators by the arousal system in the cortex are the source of primary emotions. The arousal system sends neuromodulators (e.g. dopamine and noradrenaline) into the cortex by means of diffuse projections. These neuromodulators simultaneously affect patterns of activated neurons at the time the neuromodulators are delivered, acting on the memory of the object currently being observed.
Fig 2 .
2Fig 2. There are four stages of emotional effect in the cortex. Firstly, external experiences and internal thoughts 5 give rise to an image in the cortex. External experiences are first received by the senses and are then sent via the thalamus (A) to both the amygdala (B), generating an unconscious response, and to the cortex (D), responsible for conscious response. Secondly, the experience can then cause a conscious (E), and/or a sub-conscious (C) emotional response. At the same time, top-down corticothalamic connections send information from the cortex back to the thalamus (G) [67]. Thirdly, the arousal system releases neuromodulators into the neocortex via diffuse ascending connections (F), having the effect of associating an emotional tag with a memory [22]. Fourthly and finally, if the same or similar events contexts/places/people are encountered later, the memory is recalled with the same emotional tag that was stored.
Fig 3 .
3Fig 3. Types of long term memory include Explicit and Implicit memories. Explicit memories can be further divided into Semantic and Episodic memories. Semantic memory be divided into Generic (classes) and Specific (instances of those classes). While emotions can be associated with both Explicit and Implicit memories, in this paper we focus on Semantic memories, and specifically how neuromodulators associated with emotions impact Generic and Specific memories.
Fig 4 .
4Fig 4. The release of neuromodulators by the arousal system affects a whole region in the cortex simultaneously via the ascending systems, impacting those neurons that are active at the time the neuromodulators are released in proportion to their level of activation. A released neuromodulator affects both the neurons themselves (their behaviour) as well as the strength of the connections between neurons at synapses. Specifically, neuromodulators affect the probability that neurons will fire after receiving excitatory input, and they also affect the quantity of neurotransmitters released across the chemical synapses between neurons.
Fig 5 .
5Fig 5. We present a salience-affected neural network (SANN). The architecture of the model starts with a feature extraction algorithm, to extract features from the dataset. Feature extraction also has the impact of reducing the size of the input layer required for the neural network. Each image is then reduced to a weighted set of features, and this input is fed to the input layer of the neural network. The neural network is a standard feed-forward neural network, and is trained as a classifier. In addition to the input values, we add an additional dimension: a global salience signal. This salience signal (S) consists of a single value, and affects all nodes in the neural network. Also, the neural network is designed to produce a reverse salience signal (S') when an image is presented to the model, so that each image produces both an output (classification) as well as a reverse salience signal (S').
Fig 6 .
6Each layer on the SANN model has different dimensions. The input layer consists of 784 values per image (28 x 28 pixels). Each image is passed through a feature extraction algorithm, reducing its dimension to a weighted set of 49 features. The input layer of the neural network is 49 nodes wide, to accept the set of 49 features. The hidden layer of the neural network is 100 nodes, and the output layer of the neural network is 10 nodes, each node in the output layer representing a specific digit (0-9).
Fig 7 .
7The Salience-affected neural network (SANN) model is an artificial neural network (ANN) with an additional signal dimension added: the salience signal. The salience signal affects all nodes (green arrows) in the network directly during training.
Fig 8 .
8Three changes to the activation function were explored: (A) Change in the horizontal offset of the activation function: A positive salience signal would shift the activation function to the left, resulting in a higher output from the activation the next time. (B) Change in the gradient of the activation function: A positive salience signal would reduce the gradient of the activation function, resulting in a higher output from the activation the next time. (C) Change in the amplitude of the activation function: A positive salience signal would increase the amplitude of the activation function, resulting in a higher output from the activation the next time.
Fig 9 .
9The SANN model was trained using the MNIST database of handwritten digits[47], consisting of 70,000 labelled, 28x28 grayscale images of the 10 handwritten digits (0-9). The training set consists of 60,000 images while the testing set consists of 10,000 images. A single image in the training set chosen for salience tagging in one particular run is shown here.
Fig 10 .
10Fig 10. After salience training, the neural network produced a reverse salience signal (S ) for every input image. Input images most similar to the salience-tagged image (based on an MSE calculation) presented with the highest reverse salience signal (S ). The reverse salience signal was also notably grouped by classification (i.e. digit). In this diagram image #7 in the training set was tagged with salience. Image #7 was correctly classified as belonging to class: Digit #1.
Fig 11 .
11Each of the three modifications to the activation function had varying effects on the network's classification accuracy (φ) at an individual image level.
Fig 12 .
12Each of the three modifications to the activation function had varying effects on the network's classification confidence (τ ) for the salience-tagged image. The horizontal offset and amplitude change of the activation function both had a negative impact on the average classification confidence. However, the gradient change had a positive impact on the classification accuracy. The highest average improvement in network classification confidence for the salience-tagged image was observed with a gradient change only and a salience factor of 1.0, resulting in an average improvement in classification confidence (τ ) of 4.89%, from the benchmark performance of 90.41% to 95.20%.
Fig 14 .
14Fig 14. At the individual image (instance) level, we also observed an improvement in the classification confidence (τ ) of images in response to a change in weights. The highest average improvement in network classification confidence (τ ) for the salience-tagged image was observed with a gradient change only and a salience factor of 1.0, resulting in an average improvement in classification confidence (τ ) of 5.56%, from the benchmark performance of 90.41% to 95.97%.
Fig 16 .
16Fig 16. After only a single iteration of salience training, the SANN produces a reverse salience signal (S ). This reverse salience signal is present after Effect 1 only, but is amplified with addition of Effect 2. The reverse salience signal (S ) is the highest for the handwriting digits in the same classification group as the salience-tagged image. In this figure, we demonstrate the reverse salience signal for the scenario where Image #7 (classified as a handwriting digit #1) in the training set was tagged with salience during training.
Fig 17 .
17Fig 17. Combining Effect 1 and Effect 2, the SANN model was able to demonstrate a 2.51% improvement in accuracy for the class after just a single iteration of training (a jump from 77.6% to 80.11% accuracy). To highlight the significance of this achievement, we measure how many additional training epochs using backpropagation the SANN would have required to achieve the same improvement. To achieve the same improvement with only back-propagation training, the same network would require an additional 38 epochs of training, with 1000 input images per epoch: a total additional exposure of 38,000 images.
AcknowledgementsA preliminary version of this work was the subject of an MSc thesis of one of us (LR), supervised by JT and GE, which was put on the internet as a preprint a while ago[68]. The present version is so improved and updated that it is essentially a new paper, in particular because, apart from a greatly improved presentation of the logic of the project and its relation to brain structure and function, it has added the dynamics of a shift of weight in proportion to the salience signal and two further effects on the activation function, as well as extensive testing of how this works out in practice.We are grateful to Mark Solms for very helpful comments.This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. We declare no conflicting interests.
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"Learning Dynamics of Linear Denoising Autoencoders",
"Learning Dynamics of Linear Denoising Autoencoders"
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| Denoising autoencoders (DAEs) have proven useful for unsupervised representation learning, but a thorough theoretical understanding is still lacking of how the input noise influences learning. Here we develop theory for how noise influences learning in DAEs. By focusing on linear DAEs, we are able to derive analytic expressions that exactly describe their learning dynamics. We verify our theoretical predictions with simulations as well as experiments on MNIST and CIFAR-10. The theory illustrates how, when tuned correctly, noise allows DAEs to ignore low variance directions in the inputs while learning to reconstruct them. Furthermore, in a comparison of the learning dynamics of DAEs to standard regularised autoencoders, we show that noise has a similar regularisation effect to weight decay, but with faster training dynamics. We also show that our theoretical predictions approximate learning dynamics on real-world data and qualitatively match observed dynamics in nonlinear DAEs. | null | [
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| 49,207,733 | 1806.05413 | 7444b97174930beccc6df4295b8d776bfab89e9d |
Learning Dynamics of Linear Denoising Autoencoders
Arnu Pretorius
Steve Kroon
Herman Kamper
Learning Dynamics of Linear Denoising Autoencoders
Denoising autoencoders (DAEs) have proven useful for unsupervised representation learning, but a thorough theoretical understanding is still lacking of how the input noise influences learning. Here we develop theory for how noise influences learning in DAEs. By focusing on linear DAEs, we are able to derive analytic expressions that exactly describe their learning dynamics. We verify our theoretical predictions with simulations as well as experiments on MNIST and CIFAR-10. The theory illustrates how, when tuned correctly, noise allows DAEs to ignore low variance directions in the inputs while learning to reconstruct them. Furthermore, in a comparison of the learning dynamics of DAEs to standard regularised autoencoders, we show that noise has a similar regularisation effect to weight decay, but with faster training dynamics. We also show that our theoretical predictions approximate learning dynamics on real-world data and qualitatively match observed dynamics in nonlinear DAEs.
Introduction
The goal of unsupervised learning is to uncover hidden structure in unlabelled data, often in the form of latent feature representations. One popular type of model, an autoencoder, does this by trying to reconstruct its input (Bengio et al., 2007). Autoencoders have been used in various forms to address problems in machine translation (Chandar et al., 2014;Tu et al., 2017), speech processing (Elman & Zipser, 1987;Zeiler et al., 2013), and computer vision (Rifai et al., 2011;Larsson, 2017), to name just a few areas. Denoising autoencoders (DAEs) are an extension of autoencoders which learn latent features by reconstructing data from corrupted versions of the inputs (Vincent et al., 2008). Although this corruption step typically leads to improved performance over standard autoencoders, a theoretical understanding of its effects remains incomplete. In this paper, we provide new insights into the inner workings of DAEs by analysing the learning dynamics of linear DAEs.
We specifically build on the work of Saxe et al. (2013a;, who studied the learning dynamics of deep linear networks in a supervised regression setting. By analysing the gradient descent weight update steps as time-dependent differential equations (in the limit as the learning rate approaches a small value), Saxe et al. (2013a) were able to derive exact solutions for the learning trajectory of these networks as a function of training time. Here we extend their approach to linear DAEs. To do this, we use the expected reconstruction loss over the noise distribution as an objective (requiring a different decomposition of the input covariance) as a tractable way to incorporate noise into our analytic solutions. This approach yields exact equations which can predict the learning trajectory of a linear DAE.
Our work here shares the motivation of many recent studies (Advani & Saxe, 2017;Pennington & Worah, 2017;Pennington & Bahri, 2017;Nguyen & Hein, 2017;Dinh et al., 2017;Louart et al., 2017;Swirszcz et al., 2017;Lin et al., 2017;Neyshabur et al., 2017;Soudry & Hoffer, 2017; working towards a better theoretical understanding of neural networks and their behaviour. Although we focus here on a theory for linear networks, such networks have learning dynamics that are in fact nonlinear. Furthermore, analyses of linear networks have also proven useful in understanding the behaviour of nonlinear neural networks (Saxe et al., 2013a;Advani & Saxe, 2017).
First we introduce linear DAEs ( §2). We then derive analytic expressions for their nonlinear learning dynamics ( §3), and verify our solutions in simulations ( §4) which show how noise can influence the shape of the loss surface and change the rate of convergence for gradient descent optimisation. We also find that an appropriate amount of noise can help DAEs ignore low variance directions in the input while learning the reconstruction mapping. In the remainder of arXiv:1806.05413v2 [stat.ML] 29 Jul 2018 the paper, we compare DAEs to standard regularised autoencoders and show that our theoretical predictions match both simulations ( §5) and experimental results on MNIST and CIFAR-10 ( §6). We specifically find that while the noise in a DAE has an equivalent effect to standard weight decay, the DAE exhibits faster learning dynamics. We also show that our observations hold qualitatively for nonlinear DAEs.
Linear Denoising Autoencoders
We first give the background of linear DAEs. Given training data consisting of pairs {(x i , x i ), i = 1, ..., N }, wherex represents a corrupted version of the training data x ∈ R D , the reconstruction loss for a single hidden layer DAE with activation function φ is given by
L = 1 2N N i=1 ||x i − W 2 φ(W 1xi )|| 2 .
Here, W 1 ∈ R H×D and W 2 ∈ R D×H are the weights of the network with hidden dimensionality H. The learned feature representations correspond to the latent variable z = φ(W 1x ).
To corrupt an input x, we sample a noise vector , where each component is drawn i.i.d. from a pre-specified noise distribution with mean zero and variance s 2 . We define the corrupted version of the input asx = x + . This ensures that the expectation over the noise remains unbiased, i.e. E (x) = x.
Restricting our scope to linear neural networks, with φ(a) = a, the loss in expectation over the noise distribution is
E [L] = 1 2N N i=1 ||x i − W 2 W 1 x i || 2 whitece + s 2 2 tr(W 2 W 1 W T 1 W T 2 ),(1)
See the supplementary material for the full derivation.
Learning Dynamics of Linear DAEs
Here we derive the learning dynamics of linear DAEs, beginning with a brief outline to build some intuition.
The weight update equations for a linear DAE can be formulated as time-dependent differential equations in the limit as the gradient descent learning rate becomes small (Saxe et al., 2013a). The task of an ordinary (undercomplete) linear autoencoder is to learn the identity mapping that reconstructs the original input data. The matrix corresponding to this learned map will essentially be an approximation of the full identity matrix that is of rank equal to the input dimension. It turns out that tracking the temporal updates of this mapping represents a difficult problem that involves dealing with coupled differential equations, since both the on-diagonal and off-diagonal elements of the weight matrices need to be considered in the approximation dynamics at each time step.
To circumvent this issue and make the analysis tractable, we follow the methodology introduced in Saxe et al. (2013a), which is to: (1) decompose the input covariance matrix using an eigenvalue decomposition;
(2) rotate the weight matrices to align with these computed directions of variation; and (3) use an orthogonal initialisation strategy to diagonalise the composite weight matrix W = W 2 W 1 . The important difference in our setting, is that additional constraints are brought about through the injection of noise.
The remainder of this section outlines this derivation for the exact solutions to the learning dynamics of linear DAEs.
Gradient descent update
Consider a continuous time limit approach to studying the learning dynamics of linear DAEs. This is achieved by choosing a sufficiently small learning rate α for optimising the loss in (1) using gradient descent. The update for W 1 in a single gradient descent step then takes the form of a time-dependent differential equation
τ d dt W 1 = N i=1 W T 2 x i x T i − W 2 W 1 x i x T i whitesp − εW T 2 W 2 W 1 = W T 2 (Σ xx − W 2 W 1 Σ xx ) − εW T 2 W 2 W 1 .
Here t is the time measured in epochs, τ = N α , ε = N s 2 and Σ xx = N i=1 x i x T i , represents the input covariance matrix. Let the eigenvalue decomposition of the input covariance be Σ xx = V ΛV T , where V is an orthogonal matrix and denote the eigenvalues λ j = [Λ] jj , with λ 1 ≥ λ 2 ≥ · · · ≥ λ D . The update can then be rewritten as
τ d dt W 1 = W T 2 V Λ − V T W 2 W 1 V Λ V T morewhitespace − εW T 2 W 2 W 1 .
The weight matrices can be rotated to align with the directions of variation in the input by performing the rotations W 1 = W 1 V and W 2 = V T W 2 . Following a similar derivation for W 2 , the weight updates become
τ d dt W 1 = W T 2 Λ − W 2 W 1 Λ − εW T 2 W 2 W 1 τ d dt W 2 = Λ − W 2 W 1 Λ W T 1 − εW 2 W 1 W T 1 .
Orthogonal initialisation and scalar dynamics
To decouple the dynamics, we can set W 2 = V D 2 R T and W 1 = RD 1 V T , where R is an arbitrary orthogonal matrix and D 2 and D 1 are diagonal matrices. This results in the product of the realigned weight matrices
W 2 W 1 = V T V D 2 R T RD 1 V T V = D 2 D 1
to become diagonal. The updates now reduce to the following scalar dynamics that apply independently to each pair of diagonal elements w 1j and w 2j of D 1 and D 2 respectively:
τ d dt w 1j = w 2j λ j (1 − w 2j w 1j ) − εw 2 2j w 1j (2) τ d dt w 2j = w 1j λ j (1 − w 2j w 1j ) − εw 2j w 2 1j .(3)
Note that the same dynamics stem from gradient descent on the loss given by
= D j=1 λ j 2τ (1 − w 2j w 1j ) 2 + D j=1 ε 2τ (w 2j w 1j ) 2 . (4)
By examining (4), it is evident that the degree to which the first term will be reduced will depend on the magnitude of the associated eigenvalue λ j . However, for directions in the input covariance Σ xx with relatively little variation the decrease in the loss from learning the identity map will be negligible and is likely to result in overfitting (since little to no signal is being captured by these eigenvalues). The second term in (4) is the result of the input corruption and acts as a suppressant on the magnitude of the weights in the learned mapping. Our interest is to better understand the interplay between these two terms during learning by studying their scalar learning dynamics.
Exact solutions to the dynamics of learning
As noted above, the dynamics of learning are dictated by the value of w = w 2 w 1 over time. An expression can be derived for w(t) by using a hyperbolic change of coordinates in (2) and (3), letting θ parameterise points along a dynamics trajectory represented by the conserved quantity w 2 2 − w 2 1 = ±c 0 . This relies on the fact that is invariant under a scaling of the weights such that w = (w 1 /c)(cw 2 ) = w 2 w 1 for any constant c (Saxe et al., 2013a). Starting at any initial point (w 1 , w 2 ) the dynamics are
w(t) = c 0 2 sinh (θ t ) ,(5)
with
θ t = 2tanh −1 (1 − E) ζ 2 − β 2 − 2βδ − 2(1 + E)ζδ (1 − E) (2β + 4δ) − 2(1 + E)ζ
where β = c 0 1 + ε λ , ζ = β 2 + 4, δ = tanh θ0 2 and E = e ζλt/τ . Here θ 0 depends on the initial weights w 1 and w 2 through the relationship θ 0 = sinh −1 (2w/c 0 ). The derivation for θ t involves rewriting τ d dt w in terms of θ, integrating over the interval θ 0 to θ t , and finally rearranging terms to get an expression for θ(t) ≡ θ t (see the supplementary material for full details). To derive the learning dynamics for different noise distributions, the corresponding ε must be computed and used to determine β and ζ. For example, sampling noise from a Gaussian distribution such that ∼ N (0, σ 2 I), gives ε = N σ 2 . Alternatively, if is distributed according to a zero-mean Laplace distribution with scale parameter b, then ε = 2N b 2 .
The Effects of Noise: a Simulation Study
Since the expression for the learning dynamics of a linear DAE in (5) evolve independently for each direction of variation in the input, it is enough to study the effect that noise has on learning for a single eigenvalue λ. To do this we trained a scalar linear DAE to minimise the loss λ = λ 2 (1−w 2 w 1 ) 2 + ε 2 (w 2 w 1 ) 2 with λ = 1 using gradient descent. Starting from several different randomly initialised weights w 1 and w 2 , we compare the simulated dynamics with those predicted by equation (5). The top row in Figure 1 shows the exact fit between the predictions and numerical simulations for different noise levels, ε = 0, 1, 5.
The trajectories in the top row of Figure 1 converge to the optimal solution at different rates depending on the amount of injected noise. Specifically, adding more noise results in faster convergence. However, the trade-off in (4) ensures that the fixed point solution also diminishes in magnitude.
To gain further insight, we also visualise the associated loss surfaces for each experiment in the bottom row of Figure 1. Note that even though the scalar product w 2 w 1 defines a linear mapping, the minimisation of λ with respect to w 1 and w 2 is a non-convex optimisation problem. The loss surfaces in Figure 1 each have an unstable saddle point at w 2 = w 1 = 0 (red star) with all remaining fixed points lying on a minimum loss manifold (cyan curve). This manifold corresponds to the different possible combinations of w 2 and w 1 that minimise λ . The paths that gradient descent follow from various initial starting weights down to points situated on the manifold are represented by dashed orange lines.
For a fixed value of λ, adding noise warps the loss surface making steeper slopes and pulling the minimum loss manifold in towards the saddle point. Therefore, steeper descent directions cause learning to converge at a faster rate to fixed points that are smaller in magnitude. This is the result of a sharper curving loss surface and the minimum loss manifold lying closer to the origin.
We can compute the fixed point solution for any pair of initial starting weights (not on the saddle point) by taking
λ = λ 2 (1 − w2w1) 2 + ε 2 (w2w1) 2 for λ = 1
, as well as the gradient descent paths (dashed orange lines) for randomly initialised weights. The cyan hyperbolas represent the global minimum loss manifold that corresponds to all possible combinations of w2 and w1 that minimise λ . Left: ε = 0, w * = 1. Middle: ε = 1, w * = 0.5. Right: ε = 5, w * = 1/6. the derivative
d λ dw = − λ τ (1 − w) + ε τ w,
and setting it equal to zero to find w * = λ λ+ε . This solution reveals the interaction between the input variance associated with λ and the noise ε. For large eigenvalues for which λ ε, the fixed point will remain relatively unaffected by adding noise, i.e., w * ≈ 1. In contrast, if λ ε, the noise will result in w * ≈ 0. This means that over a distribution of eigenvalues, an appropriate amount of noise can help a DAE to ignore low variance directions in the input data while learning the reconstruction. In a practical setting, this motivates the tuning of noise levels on a development set to prevent overfitting.
The Relationship Between Noise and Weight Decay
It is well known that adding noise to the inputs of a neural network is equivalent to a form of regularisation (Bishop, 1995). Therefore, to further understand the role of noise in linear DAEs we compare the dynamics of noise to those of explicit regularisation in the form of weight decay (Krogh & Hertz, 1992). The reconstruction loss for a linear weight decayed autoencoder (WDAE) is given by
1 2N N i=1 ||x i − W 2 W 1 x i || 2 + γ 2 ||W 1 || 2 + ||W 2 || 2 (6)
where γ is the penalty parameter that controls the amount of regularisation applied during learning. Provided that the weights of the network are initialised to be small, it is also possible (see supplementary material) to derive scalar dynamics of learning from (6) as
w γ (t) = ξE γ E γ − 1 + ξ/w 0 ,(7)
where ξ = (1 − N γ/λ) and E γ = e 2ξt/τ . Figure 2 compares the learning trajectories of linear DAEs and WDAEs over time (as measured in training epochs) for λ = 2.5, 1, 0.5 and 0.1. The dynamics for both noise and weight decay exhibit a sigmoidal shape with an initial period of inactivity followed by rapid learning, finally reaching a plateau at the fixed point solution. Figure 2 illustrates that the learning time associated with an eigenvalue is negatively correlated with its magnitude. Thus, the eigenvalue corresponding to the largest amount of variation explained is the quickest to escape inactivity during learning.
The colour intensity of the lines in Figure 2 correspond to the amount of noise or regularisation applied in each run, with darker lines indicating larger amounts. In the continuous time limit with equal learning rates, when compared with noise dynamics, weight decay experiences a delay in learning such that the initial inactive period becomes extended for every eigenvalue, whereas adding noise has no effect on learning time. In other words, starting from small weights, noise injected learning is capable of providing an equivalent regularisation mechanism to that of weight decay in terms of a constrained fixed point mapping, but with zero time delay.
However, this analysis does not take into account the practice of using well-tuned stable learning rates for discrete optimisation steps. We therefore consider the impact on training time when using optimised learning rates for each approach. By using second order information from the Hessian as in Saxe et al. (2013a), (here of the expected reconstruction loss with respect to the scalar weights), we relate the optimal learning rates for linear DAEs and WDAEs, where each optimal rate is inversely related to the amount of noise/regularisation applied during training (see supplementary material). The ratio of the optimal DAE rate to that for the WDAE is
R = 2λ + γ 2λ + 3ε .(8)
Note that the ratio in (8) will essentially be equal to one for eigenvalues that are significantly larger than both ε and γ, with deviations from unity only manifesting for smaller values of λ.
Furthermore, weight decay and noise injected learning result in equivalent scalar solutions when their parameters are related by γ = λε λ+ε (see supplementary material). This leads to the following two observations. First, it shows that adding noise during learning can be interpreted as a form of weight decay where the penalty parameter γ adapts to each direction of variation in the data. In other words, noise essentially makes use of the statistical structure of
λ = λ 2 (1 − w2w1) 2 + ε 2 (w2w1) 2 + γ 2 (w 2 2 + w 2 1 )
. Gradient descent paths (orange/magenta dashed lines), minimum loss manifold (cyan curves), saddle point (red star). Middle: Simulated learning dynamics. Right: Norm of the weights over time for each simulated run. Top: Noise with λ = 1, ε = 0.1 and γ = 0. Bottom: Weight decay with λ = 1, ε = 0 and γ = λ(0.1)/(λ + 0.1) = 0.091. The magenta line in each plot corresponds to a simulated run with small initialised weights.
the input data to influence the amount of shrinkage that is being applied in various directions during learning. Second, together with (8), we can theoretically compare the learning dynamics of DAEs and WDAEs, when both equivalent regularisation and the relative differences in optimal learning rates are taken into account.
The effects of optimal learning rates (for λ = 1), are shown in Figure 3. DAEs still exhibit faster dynamics (left panel), even when taking into account the difference in the learning rate as a function of noise, or equivalent weight decay (middle panel). In addition, for equivalent regularisation effects, the ratio of the optimal rates R can be shown to be a monotonically decreasing function of the noise level, where the rate of decay depends on the size of λ. This means that for any amount of added noise, the DAE will require a slower learning rate than that of the WDAE. Even so, a faster rate for the WDAE does not seem to compensate for its slower dynamics and the difference in learning time is also shown to grow as more noise (regularisation) is applied during training (right panel).
Exploiting invariance in the loss function
A primary motivation for weight decay as a regulariser is that it provides solutions with smaller weight norms, producing smoother models that have better generalisation performance. Figure 4 shows the effect of noise (top row) compared to weight decay (bottom row) on the norm of the weights during learning. Looking at the loss surface for weight decay (bottom left panel), the penalty on the size of the weights acts by shrinking the minimum loss manifold down from a long curving valley to a single point (associ-ated with a small norm solution). Interestingly, this results in gradient descent following a trajectory towards an "invisible" minimum loss manifold similar to the one associated with noise. However, once on this manifold, weight decay begins to exploit invariances in the loss function to changes in the weights, so as to move along the manifold down towards smaller norm solutions. This means that even when the two approaches learn the exact same mapping over time (as shown by the learning dynamics in the middle column of Figure 4), additional epochs will cause weight decay to further reduce the size of the weights (bottom right panel). This happens in a stage-like manner where the optimisation first focuses on reducing the reconstruction loss by learning the optimal mapping and then reduces the regularisation loss through invariance.
Small weight initialisation and early stopping
It is common practice to initialise the weights of a network with small values. In fact, this strategy has recently been theoretically shown to help, along with early stopping, to ensure good generalisation performance for neural networks in certain high-dimensional settings (Advani & Saxe, 2017). In our analysis however, what we find interesting about small weight initialisation is that it removes some of the differences in the learning behaviour of DAEs compared to regularised autoencoders that use weight decay.
To see this, the magenta lines in Figure 4 show the learning dynamics for the two approaches where the weights of both the networks were initialised to small random starting values. The learning dynamics are almost identical in terms of their temporal trajectories and have equal fixed points. However, what is interesting is the implicit regularisation that is brought about through the small initialisation. By starting small and making incremental updates to the weights, the scalar solution in both cases end up being equal to the minimum norm solution. In other words, the path that gradient descent takes from the initialisation to the minimum loss manifold, reaches the manifold where the norm of the weights happen to also be small. This means that the second phase of weight decay (where the invariance of the loss function would be exploited to reduce the regularisation penalty), is not only no longer necessary, but also does not result in a norm that is appreciably smaller than that obtained by learning with added noise. Therefore in this case, learning with explicit regularisation provides no additional benefit over that of learning with noise in terms of reducing the norm of the weights during training.
When initialising small, early stopping can also serve as a form of implicit regularisation by ensuring that the weights do not change past the point where the validation loss starts to increase (Bengio et al., 2007). In the context of learning dynamics, early stopping for DAEs can be viewed as a method that effectively selects only the directions of variation deemed useful for generalisation during reconstruction, considering the remaining eigenvalues to carry no additional signal.
Experimental Results
To verify the dynamics of learning on real-world data sets we compared theoretical predictions with actual learning on MNIST and CIFAR-10. In our experiments we considered the following linear autoencoder networks: a regular AE, a WDAE and a DAE.
For MNIST, we trained each autoencoder with small randomly initialised weights, using N = 50000 training samples for 5000 epochs, with a learning rate α = 0.01 and a hidden layer width of H = 256. For the WDAE, the penalty parameter was set at γ = 0.5 and for the DAE, σ 2 = 0.5. The results are shown in Figure 5 (left column).
The theoretical predictions (solid lines) in Figure 5 show good agreement with the actual learning dynamics (points). As predicted, both regularisation (orange) and noise (green) suppress the fixed point value associated with the different eigenvalues and, whereas regularisation delays learning (fewer fixed points are reached by the WDAE during training when compared to the DAE), the use of noise has no effect on training time.
Similar agreement is shown for CIFAR-10 in the right column of Figure 5. Here, we trained each network with small randomly initialised weights using N = 30000 training samples for 5000 epochs, with a learning rate α = 0.001 and a hidden dimension H = 512. For the WDAE, the penalty parameter was set at γ = 0.5 and for the DAE, σ 2 = 0.5.
Next, we investigated whether these dynamics are at least also qualitatively present in nonlinear autoencoder networks. Figure 6 shows the dynamics of learning for nonlinear AEs, WDAEs and DAEs, using ReLU activations, trained on MNIST (N = 50000) and CIFAR-10 (N = 30000) with equal learning rates. For the DAE, the input was corrupted using sampled Gaussian noise with mean zero and σ 2 = 3. For the WDAE, the amount of weight decay was manually tuned to γ = 0.0045, to ensure that both autoencoders displayed roughly the same degree of regularisation in terms of the fixed points reached. During the course of training, the identity mapping associated with each eigenvalue was estimated (see supplementary material), at equally spaced intervals of size 10 epochs.
The learning dynamics are qualitatively similar to the dy-namics observed in the linear case. Both noise and weight decay result in a shrinkage of the identity mapping associated with each eigenvalue. Furthermore, in terms of the number of training epochs, the DAE is seen to learn as quickly as a regular AE, whereas the WDAE incurs a delay in learning time. Although these experimental results stem from a single training run for each autoencoder, we note that wall-clock times for training may still differ because DAEs require some additional time for sampling noise. Similar results were observed when using a tanh nonlinearity and are provided in the supplementary material.
Related Work
There have been many studies aiming to provide a better theoretical understanding of DAEs. Vincent et al. (2008) analysed DAEs from several different perspectives, including manifold learning and information filtering, by establishing an equivalence between different criteria for learning and the original training criterion that seeks to minimise the reconstruction loss. Subsequently, Vincent (2011) showed that under a particular set of conditions, the training of DAEs can also be interpreted as a type of score matching. This connection provided a probabilistic basis for DAEs. Following this, a more in-depth analysis of DAEs as a possible generative model suitable for arbitrary loss functions and multiple types of data was given by Bengio et al. (2013).
In contrast to a probabilistic understanding of DAEs, we present here an analysis of the learning process. Specifically inspired by Saxe et al. (2013a), as well as by earlier work on supervised neural networks (Opper, 1988;Sanger, 1989;Baldi & Hornik, 1989;Saad & Solla, 1995), we provide a theoretical investigation of the temporal behaviour of linear DAEs using derived equations that exactly describe their dynamics of learning. Specifically for the linear case, the squared error loss for the reconstruction contractive autoencoder (RCAE) introduced in Alain & Bengio (2014) is equivalent to the expected loss (over the noise) for the DAE. Therefore, the learning dynamics described in this paper also apply to linear RCAEs.
For our analysis to be tractable we used a marginalised reconstruction loss where the gradient descent dynamics are viewed in expectation over the noise distribution. Whereas our motivation is analytical in nature, marginalising the reconstruction loss tends to be more commonly motivated from the point of view of learning useful and robust feature representations at a significantly lower computational cost (Chen et al., 2014;2015). This approach has also been investigated in the context of supervised learning (van der Maaten et al., 2013;Wang & Manning, 2013;Wager et al., 2013). Also related to our work is the analysis by Poole et al. (2014), who showed that training autoencoders with noise (added at different levels of the network architecture), is closely connected to training with explicit regularisation and proposed a marginalised noise framework for noisy autoencoders.
Conclusion and Future Work
This paper analysed the learning dynamics of linear denoising autoencoders (DAEs) with the aim of providing a better understanding of the role of noise during training. By deriving exact time-dependent equations for learning, we showed how noise influences the shape of the loss surface as well as the rate of convergence to fixed point solutions. We also compared the learning behaviour of added input noise to that of weight decay, an explicit form of regularisation. We found that while the two have similar regularisation effects, the use of noise for regularisation results in faster training. We compared our theoretical predictions with actual learning dynamics on real-world data sets, observing good agreement. In addition, we also provided evidence (on both MNIST and CIFAR-10) that our predictions hold qualitatively for nonlinear DAEs.
This work provides a solid basis for further investigation. Our analysis could be extended to nonlinear DAEs, potentially using the recent work on nonlinear random matrix theory for neural networks (Pennington & Worah, 2017;Louart et al., 2017). Our findings indicate that appropriate noise levels help DAEs ignore low variance directions in the input; we also obtained new insights into the training time of DAEs. Therefore, future work might consider how these insights could actually be used for tuning noise levels and predicting the training time of DAEs. This would require further validation and empirical experiments, also on other datasets. Finally, our analysis only considers the training dynamics, while a better understanding of generalisation and what influences the quality of feature representations during testing, are also of prime importance.
Supplementary material
The following section provides detail omitted in the paper regarding the derivation of certain equations as well as additional comments.
A. Expected loss for linear DAEs
We derive the expected reconstruction loss over the noise distribution as presented in (1) in the paper. The expected loss can be written as
E [L] = 1 2N N i=1 E ||x i − W 2 W 1xi || 2 . wherex i = x i + i , with sampled from an isotropic noise distribution with component variance s 2 . Let SE(x i ) = ||x i − W 2 W 1xi || 2 and M = W 2 W 1 . Then E [SE(x i )] = E ||(I − M )x i + M (x i −x i )|| 2 = SE(x i ) + E ||M (x i −x i )|| 2 because the cross product terms vanish, since E [x i ] = x i : 0 = E x T i (I − M ) T M (x i −x i ) = E (x i −x i ) T M T (I − M )x i .
We also have that
||M (x i −x i )|| 2 = (x i −x i ) T M T M (x i −x i ) = tr (x i −x i ) T M T M (x i −x i ) = tr M (x i −x i )(x i −x i ) T M T = tr M i T i M T dueE [L] = 1 2N N i=1 ||x i − W 2 W 1 x i || 2 whitece + s 2 2 tr W 2 W 1 W T 1 W T 2 .
Now, if we assume w 2 = w 1 , and let a = ∂ 2 ε ∂w 2 1 = ∂ 2 ε ∂w 2 2 and b = ∂ 2 ε ∂w2w1 , the eigenvalues for the Hessian can be shown to be λ H = a − b or λ H = a + b. The second order update for a single weight w at time t is then given by
w t+1 = w t − ∂ ε ∂w t /λ H ,
where the maximum λ H , is when w 2 = w 1 = 1, such that
λ H = 1 τ (λ + ε) + 2 τ (λ + ε) − λ τ = 2λ + 3ε τ .
Therefore, the optimal learning rate is
α ε = 1/λ H = τ 2λ + 3ε .
For WDAEs with penalty parameter γ, a very similar derivation gives
α γ = τ 2λ + γ .
Taking the ratio of the optimal DAE rate to that for the WDAE gives R = α ε α γ = 2λ + γ 2λ + 3ε .
E. Equivalent scalar solutions
In Section 4 of the paper, the DAE fixed point solution is shown to be w * ε = λ λ + ε .
Now if w = w 2 w 1 and w 2 = w 1 , then for WDAE we have that the scalar loss is given by
γ = λ 2τ (1 − w) 2 + γ τ w, and ∂ γ ∂w = − λ τ (1 − w) + γ τ .
Setting the above equal to zero and solving gives w * γ = 1 − γ/λ.
To obtain the value of γ for which the two fixed points are equal, we set w * γ = w * ε and solve for γ to find γ = λε λ + ε .
F. Estimated dynamics for nonlinear networks
The dynamics for the nonlinear networks trained in Figure 6 in the paper were estimated using the following approach. First, compute
Σ xx = N i=1 x i x T i = V ΛV T ,
using an eigen-decomposition giving eigenvalues λ j , j = 1, ..., D. Then at regular intervals computê
Σ xx (t) = N i=1
x ixi (t) T , wherex(t) is the estimated reconstruction of input at time t generated by the autoencoder network. Finally, using the following rotation to obtain the diagonal matrix
Λ(t) = V TΣ xx (t)V,
where the diagonal contains the estimated eigenvaluesλ j (t), we can compute an estimate for the identity mapping associated with each eigenvalue asλ j (t)/λ j ∈ [0, 1].
G. Learning dynamics for tanh autoencoder networks
We investigated the dynamics of learning for nonlinear AEs, WDAEs and DAEs, using tanh activations. Figure 7 shows the dynamics for these networks trained on MNIST (N = 50000) and CIFAR-10 (N = 30000) with equal learning rates. For the DAE, the input was corrupted using sampled Gaussian noise with mean zero and σ 2 = 2. For the WDAE, the amount of weight decay was set to γ = 0.0045. During the course of training, the identity mapping associated with each eigenvalue was estimated using the approach described in Section F, at equally spaced intervals of size 100 epochs.
Figure 1 .
1Learning dynamics, loss surface and gradient descent paths for linear denoising autoencoders. Top: Learning dynamics for each simulated run (dashed orange lines) together with the theoretically predicted learning dynamics (solid green lines). The red line in each plot indicates the final value of the resulting fixed point solution w * . Bottom: The loss surface corresponding to the loss
Figure 2 .
2Theoretically predicted learning dynamics for noise compared to weight decay for linear autoencoders. Top: Noise dynamics (green), darker line colours correspond to larger amounts of added noise. Bottom: Weight decay dynamics (orange), darker line colours correspond to larger amounts of regularisation. Left to right: Eigenvalues λ = 2.5, 1 and 0.5 associated with high to low variance.
Figure 3 .
3Learning dynamics for optimal discrete time learning rates (λ = 1). Left: Dynamics of DAEs (green) vs. WDAEs (orange), where darker line colours correspond to larger amounts noise or weigh decay. Middle: Optimal learning rate as a function of noise ε for DAEs, and for WDAEs using an equivalent amount of regularisation γ = λε/(λ + ε). Right: Difference in mapping over time.
Figure 4 .
4The effect of noise versus weight decay on the norm of the weights during learning. Left: Two-dimensional loss surface
Figure 5 .
5Learning dynamics for MNIST and CIFAR-10. Solid lines represent theoretical dynamics and 'x' markers simulated dynamics. Shown are the mappings associated with the set of eigenvalues {λi, i = 1, 4, 8, 16, 32}, where the remaining eigenvalues were excluded to improve readability. Top: Noise: AE (blue) vs. DAE with σ 2 = 0.5 (green). Bottom: Weight decay: AE (blue) vs. WDAE with γ = 0.5 (orange). Left: MNIST. Right: CIFAR-10.
Figure 6 .
6Learning dynamics for nonlinear networks using ReLU activation. AE (blue), WDAE (orange) and DAE (green). Shown are the mappings associated with the first four eigenvalues, i.e. {λi, i = 1, 2, 3, 4}. Left: MNIST Right: CIFAR-10.
to the invariance of the trace under cycle permutation of products. Therefore, in expectation over the noise we haveE ||M (x i −x i )|| 2 = tr M (s 2 I)M T ,and as a result
Figure 7 .
7Learning dynamics for nonlinear networks using tanh activation. AE (blue), WDAE (orange) and DAE (green). Left: MNIST Right: CIFAR-10.
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AcknowledgementsWe would like to thank Andrew Saxe for early discussions that got us interested in this work, as well as the reviewers for insightful comments and suggestions. We would like to thank the CSIR/SU Centre for Artificial Intelligence Research (CAIR), South Africa, for financial support. AP would also like to thank the MIH Media Lab at Stellenbosch University and Praelexis (Pty) Ltd for providing stimulating working environments for a portion of this work.B. Learning dynamics for linear DAEsWe derive the expression for the learning dynamics of a linear DAE as presented in (5) in the paper. As departure point, we start by examining the expected scalar update equations over the noise model for a small learning rate α, which can be written aswhere τ = N α , with N representing the number of training samples. Define w = w 2 w 1 and using the product rule the update for w then becomesNext we make the following hyperbolic change of coordinateswhere θ parameterises points along the dynamics trajectory represented by w 2 2 − w 2 1 = ±c 0(Saxe et al., 2013a). Note that with this change of coordinates we obtainSimilarly,Plugging these results into the update for w given in(9), yieldsand as a result,(where θ t = θ f (t)) to track the weight trajectory gives equation (5) in the paper.C. Learning dynamics for linear WDAEsWe derive the expression for the learning dynamics of a linear WDAE as presented in(7)in the paper. Reconstruction loss with weight decay gives the scalar loss associated with an eigenvalue λ aswhere γ is the penalty parameter that controls the amount of regularisation that is being applied. The update equations for the weights then follow asassuming the initial w 2 = w 1 (which holds approximately for small initial values), we have for w = w 2 w 1 thatThen solving for w f giveswhere E γ = e 2ξt/τ and ξ = (1 − N γ/λ).D. Optimal learning ratesWe derive expressions for the optimal learning rates for linear DAEs and WDAEs as presented in(8)in the paper. First, consider the expected scalar DAE lossThe Hessian of ε is given by H =
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Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. J Pennington, S Schoenholz, S Ganguli, Advances in Neural Information Processing Systems. Pennington, J., Schoenholz, S., and Ganguli, S. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. In Advances in Neural Information Processing Systems, pp. 4788-4798, 2017.
B Poole, J Sohl-Dickstein, S Ganguli, arXiv:1406.1831Analyzing noise in autoencoders and deep networks. Poole, B., Sohl-Dickstein, J., and Ganguli, S. An- alyzing noise in autoencoders and deep networks. arXiv:1406.1831, 2014.
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Exact solution for on-line learning in multilayer neural networks. D Saad, S A Solla, Physical Review Letters. 74214337Saad, D. and Solla, S. A. Exact solution for on-line learning in multilayer neural networks. Physical Review Letters, 74(21):4337, 1995.
Optimal unsupervised learning in a singlelayer linear feedforward neural network. T D Sanger, Neural Networks. 2Sanger, T. D. Optimal unsupervised learning in a single- layer linear feedforward neural network. Neural Net- works, 2(6):459-473, 1989.
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Learning hierarchical category structure in deep neural networks. A M Saxe, J L Mcclelland, S Ganguli, Proceedings of the Cognitive Science Society. the Cognitive Science SocietySaxe, A. M., McClelland, J. L., and Ganguli, S. Learning hierarchical category structure in deep neural networks. In Proceedings of the Cognitive Science Society, pp. 1271- 1276, 2013b.
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| []
|
[
"Topological Spin Texture in a Quantum Anomalous Hall Insulator",
"Topological Spin Texture in a Quantum Anomalous Hall Insulator"
]
| [
"Jiansheng Wu \nDepartment of Physics\nHong Kong University of Science and Technology\nClear Water BayHong KongChina\n",
"Jie Liu \nDepartment of Physics\nHong Kong University of Science and Technology\nClear Water BayHong KongChina\n\nInstitute for Advanced Study\nHong Kong University of Science and Technology\nClear Water BayHong KongChina\n",
"Xiong-Jun Liu \nDepartment of Physics\nHong Kong University of Science and Technology\nClear Water BayHong KongChina\n\nInstitute for Advanced Study\nHong Kong University of Science and Technology\nClear Water BayHong KongChina\n"
]
| [
"Department of Physics\nHong Kong University of Science and Technology\nClear Water BayHong KongChina",
"Department of Physics\nHong Kong University of Science and Technology\nClear Water BayHong KongChina",
"Institute for Advanced Study\nHong Kong University of Science and Technology\nClear Water BayHong KongChina",
"Department of Physics\nHong Kong University of Science and Technology\nClear Water BayHong KongChina",
"Institute for Advanced Study\nHong Kong University of Science and Technology\nClear Water BayHong KongChina"
]
| []
| The quantum anomalous Hall (QAH) insulator, also called Chern insulator, is a two-dimensional (2D) topological state of quantum matter which exhibits a gap in the bulk and chiral gapless edge states in the boundary [1]. Different from the quantum Hall effect driven by external magnetic fields [2, 3], the QAH effect applies no magnetic field, and in the recent experimental discovery the spin-orbit coupling and ferromagnetic ordering are essential to reach this state [4]. In this work, we find a novel phenomenon in the Chern insulator that the edge states are chiral in both the orbital and spin degrees of freedom when a chiral-like symmetry is present, and exhibit topologically stable spin texture in the position space. The chiral edge spin texture has no correspondence in the conventional trivial materials, and may have potential applications in designing topological-state-based spin devices which might be applicable to future spintronic technologies.Discovery of the quantum Hall effect in 1980s brought about a fundamental concept, topological state of quantum matter, to condensed matter physics [2, 3]. In the quantum Hall effect an external magnetic field drives the electrons to fill in discrete Landau levels. This leads to a gap in the 2D bulk, while the boundary exhibits 1D chiral gapless edge states, namely, the edge modes propagate one-way in each edge. In the integer quantum Hall effect the Hall conductance is quantized by Chern numbers, which are integers and characterize the global properties (topology) of the system [5]. The topological interpretation of the quantized Hall conductance implies that to obtain a quantum Hall state does not necessitate a magnetic field. The theoretical idea for quantum Hall effect without Landau levels, i.e. the QAH effect, was first introduced by Haldane in a honeycomb lattice over two decades ago [1]. The recent interests in this topological state have been revived due to the great development in the field of time-reversal invariant topological insulators [6, 7], with numerous new theoretical proposals having been introduced [8-13]. Importantly, following the proposal in Ref. [11], the QAH state has been detected in a recent experiment using ferromagnetic thin-film topological insulator, with the quantized Hall conductance having been observed [4].The minimal model for the Chern insulator is described * email: [email protected] by a two-band Hamiltonian H = k ψ † k H(k)ψ k , where the spin basis ψ k = (c k↑ , c k↓ ) T . Around the Γ point H(k) takes the simple (2 + 1)D Dirac formwhere for the realization with thin-film ferromagnetic topological insulators [4, 11] m z depends on the Zeeman term induced by the ferromagnetic order, the B-term characterizes the hybridization between top and bottom thin-film surfaces, and 2A 1,2 equal Fermi velocities of the surface Dirac cones of the parent topological insulator [14, 15]. The QAH phase is obtained when m z B < 0, with the Chern number C 1 = sgn(m z A 1 A 2 ) [11, 16]. In the solid-state experiment, the Cr-doped (Bi 1−x Sb x ) 2 Te 3 was used to achieve the above Hamiltonian and the QAH phase [4]. On the other hand, this model can also be realized in a square optical lattice [17] with spin-orbit coupling generated based on cold atom experiments [18][19][20].In the isotropic case |A 1 | = |A 2 | the Hamiltonian (1) respects a chiral-like symmetry defined by S = σ n1 M n2 , with n 1 ⊥ n 2 being arbitrary 2D orthogonal unit vectors in the x-y plane. Here σ n1 and M n2 represent the n 1component of the Pauli matrix acting on the spin space and the spatial reflection along the n 2 direction, respectively. It can be seen that SH(k)S † = −H(k). Then for two edges normal to n 1 axis, the edge states with k n2 = 0 must be eigenstates of σ n1 , with the spin oppositely polarized in the opposite edges [21]. Furthermore, using the k · p theory one can expand the 1D edge Hamiltonian up to the leading order of momentum and confirm that the edge states with nonzero k n2 are also eigenstates of σ n1 . This result is shown numerically inFig. 1. We see that the spin of edge states is in-plane polarized, and varies one cycle following the 1D closed path of the boundary. This spin texture shows that the edge states are chiral in both the orbital and spin degrees of freedom. Interestingly, the spin chirality gives a nontrivial quantized Berry phase for the edge modes after evolving one cycle along the boundary: γ C = ±π, and this defines a 1D winding number N 1d = ±1 which can be verified to correspond to the bulk Chern invariant via C 1 = N 1d sgn(m z ). With fixed sgn(m z ), changing the edge spin chirality reverses Chern number of the QAH insulator, while varying both the spin chirality and sgn(m z ) gives the phases with the same C 1 (see e.g.Fig. 1 a and f). InFig. 1we have shown the chiral spin texture and edge currents of the 2D insulator in different parameter regimes, and with square (a-d) and circular (e-h) geometries, respectively. arXiv:1401.0415v2 [cond-mat.mes-hall]FIG. 1: Topological spin texture of the edge states. a-d, The edge spin texture for square geometry of the boundary, with the Zeeman term mz > 0 and B < 0. e-h, The edge spin texture for circular geometry of the boundary, with mz < 0 and B > 0. For the other parameters, we take that A1, A2 > 0 (a,e); A1 > 0, A2 < 0 (b,f); A1, A2 < 0 (c,g); and A1 < 0, A2 > 0 (d,h). The topological spin textures give rise to quantized Berry phases if evolving the edge spin one circle along the boundary, which define 1D nontrivial topological states in the position space of the boundary and characterized by the 1D winding number N 1d . The bulk Chern number C1 corresponds to N 1d with an additional sign factor sgn(mz).The above study demonstrates a correspondence between the nontrivial topologies exhibited in the bulk and the boundary. The bulk Chern number is a topological invariant of the first Brillouin zone, which is a 2D closed manifold in the momentum space due to the band gap of the insulator. However, the edge modes are gapless and generically the 1D boundary is a closed manifold not in the momentum space, but in the position space. Therefore the 1D edge invariant N 1d is obtained in the real space rather than in the k-space. Both the bulk and edge topological states are classified by integers Z. Note that a 1D topological state necessitates the symmetry protection [22]. The correspondence between the bulk and edge topological phases relies on the chiral-like symmetry as introduced above, albeit the Chern insulator is an intrinsic topological state not depending on symmetry. These properties implies that from an intrinsic bulk topological phase we obtain a real-space topological state in the reduced dimension of the boundary when the additional S symmetry is present.While the 1D topological state relies on the symmetry protection, the chirality enables the topological spin texture to be insensitive to local perturbations which explicitly break the S symmetry. We have verified that the topological spin texture is stable against in-plane Zeeman fields without driving bulk phase transition, and insensitive to any type of local magnetic and nonmagnetic disorder scatterings, even when the disorder strength is comparable with the bulk gap. Actually, like that the orbital chirality of the edge modes prohibits the back scattering, the spin chirality ensures that no scattering occurs between two edge modes with opposite local spin polarizations. On the other hand, in the Methods section we show that the chiral spin texture may be scattered by the discrete lattice anisotropy in the high-energy regime.The edge channel of the Chern insulator can be described by 1D chiral Luttinger liquid [23, 24]. Furthermore, the above study shows that the edge modes are chiral in both orbital and spin degrees of freedom. As the topological spin texture leads to quantized Berry phases, which can be integrated by a Berry's connection, the edge states are governed by the following effective HamiltonianHere ψ s denotes the orbital part of the edge states,x is the position parameter along the edge, and the Berry's connection A s = i χ s (x|∂ x |χ s (x) , with |χ s (x) representing the polarized spin degree of freedom. The integral of A s along the 1D boundary gives¸dxA s (x) = N 1d π. The π-Berry phase is equivalent to a half magnetic fluxquanta threading through the 2D sample and encircled by the edge. According to the study by Wilczek in 1982 [25], a half quantum flux can lead to 1/2-fractionalization of the orbital angular momentum. As a result, for the 2D sample with circular geometry, the orbital angular momentum of the edge modes should be fractionalized as l z = m + N 1d /2, with m being integers. The fractionalization of the orbital angular momentum has an observable in the edge spectrum E lz = v edge l z R −1 lz , with R lz the effective radius of edge state wave function. Due to the 1/2-fractionalization no zero-energy (mid-gap) edge state exists, and therefore the total number of edge states is even. However, threading an additional magnetic 1/2flux-quanta can exactly push one original state to zero energy, changing the total number of edge modes to be odd, which provides an observable for the 1/2-fractionalization of orbital angular momentum. 3 FM quantum anomalous Hall insulator Normal lead a FM FM FM c d e b / ( ) (0) − plane y− plane −y plane = 0.5 = 0.25 = 0.5 = 1.0 FM lead FIG. 2: Angle-dependence of tunneling conductance.a, A normal metallic lead is strongly coupled to the left-hand edge and a ferromagnetic lead is weakly coupled to the righthand edge of the Chern insulator. Here the parameters for the quantum anomalous Hall phase satisfy mz > 0, A1,2 > 0, and B < 0. Then spin of the edge states in the right-hand edge points to the +y direction. b, The tunneling conductance σ(φ) is plotted versus azimuthal angle φ of the magnetization in the ferromagnetic lead, with the magnetization varying in the x−y plane (c), y−z plane (d), and x−z plane (e), respectively. The numerical results are presented for different polarization ratios P in the ferromagnetic lead. The maximum tunneling conductance is obtained when the magnetization aligns with the edge spin-polarization direction.The spin and orbital chirality make the edge of the Chern insulator be an exotic 1D metal which has no correspondence in conventional 1D materials. The topological spin texture of the edge modes may lead to strong spin-dependent effects as presented below, which on one hand can provide new unambiguous verification of the QAH state in the experiment, on the other hand, are applicable to spintronics by designing topological spin devices [26]. As illustrated inFig. 2a, we attach two metallic leads to the QAH sample, with a normal lead strongly coupled to the left-hand edge and a ferromagnetic lead weakly coupled to the right-hand edge. Due to the spin texture, the couplings between the sample edge and leads are fully spin selective, which can lead to strong anisotropic effects in the tunneling conductance when changing the direction of magnetization M FM in the ferromagnetic lead, as shown inFig. 2b-e. The tunneling conductance σ(φ)(Fig. 2 b)exhibits a clear angle-dependence when M FM varies in x − y and y − z planes (c and d), while it is a constant when M FM varies along x − z plane (e). This measures that the edge spin polarizes to the y direction. The angle-dependence implies a strong magnetoresistive effect by setting M FM along ±y directions. The magnetoresistance, given by MR = [σ(0) − σ(π)]/σ(π) × 100%, is plotted inFig. 3as a function of chemical potential µ in the Chern in-sulator and the polarization ratio P of the ferromagnetic lead. Due to the full spin-polarization, the edge of the sample can be regarded as an ideal dissipationless half-metal. This gives that MR ≈ 2P/(1 − P ) × 100%, which is significantly larger than the corresponding tunneling magnetoresistance obtained in conventional ferromagnet/insulator/ferromagnet devices with the same polarization ratio (the inserted panel ofFig. 3)[27]. The strong magnetoresistive effect has been widely applied to spintronics, especially to designing read heads [26]. 0.0 0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 0.5 P 0 100 200 300 400 500 600 700 800 MR (%) 0.0 0.2 0.4 0.6 0.8 0 200 400 600 800 O O MR (%) P =0 =0.6 2P/(1-P) 2P 2 /(1-P 2 ) O FIG. 3: Magnetoresistance for the setup in Fig. 2a by setting magnetization of the ferromagnetic lead along ±y directions.The magnetoresistance (MR) is plotted numerically as a function of the polarization ratio P in the ferromagnetic lead and the chemical potential µ (in unit of B) in the QAH insulator. The parameters for the QAH phase are taken as mz = −0.3A1,2 = −0.3B, which gives the bulk gap Eg = 2|mz| = 0.6B. The MR is uniform versus µ when the chemical potential is within the bulk gap (|µ| < 0.3B) and decreases when µ lies out of the gap. The inserted panel shows that the MR coincides with 2P/(1 − P ) for |µ| < 0.3B (black solid and red circled curves), which is significantly larger than the corresponding tunneling MR, given by 2P 2 /(1 − P 2 ) (green curve), in the conventional ferromagnet/insulator/ferromagnet devices with polarization ratio P in both ferromagnets [27].Another interesting application of the topological spin texture is to design controllable spin-filtering devices, as illustrated inFig. 4. The edge spin texture ensures that the output current is fully spin-polarized, with the polarization direction depending on which edge the drain lead is attached to. From the numerical results we see that when the voltage of the output lead lies in the sample bulk gap, the output current is 100% polarized to the same direction, reflecting that the spin texture is identical for all edge states. We note that the band gap of the currently realized QAH effect is small, while for realistic applications a larger topological gap is necessary. The new physics unveiled in this work and the proposed potential applications will motivate the search for new novel 4 materials of Chern insulator with sizable band gaps in the future. | 10.1103/physrevlett.113.136403 | [
"https://arxiv.org/pdf/1401.0415v3.pdf"
]
| 7,037,183 | 1401.0415 | a562dbb0da1314c2814d12ed488d7976daa038c3 |
Topological Spin Texture in a Quantum Anomalous Hall Insulator
6 Jan 2014
Jiansheng Wu
Department of Physics
Hong Kong University of Science and Technology
Clear Water BayHong KongChina
Jie Liu
Department of Physics
Hong Kong University of Science and Technology
Clear Water BayHong KongChina
Institute for Advanced Study
Hong Kong University of Science and Technology
Clear Water BayHong KongChina
Xiong-Jun Liu
Department of Physics
Hong Kong University of Science and Technology
Clear Water BayHong KongChina
Institute for Advanced Study
Hong Kong University of Science and Technology
Clear Water BayHong KongChina
Topological Spin Texture in a Quantum Anomalous Hall Insulator
6 Jan 2014(Dated: January 7, 2014)
The quantum anomalous Hall (QAH) insulator, also called Chern insulator, is a two-dimensional (2D) topological state of quantum matter which exhibits a gap in the bulk and chiral gapless edge states in the boundary [1]. Different from the quantum Hall effect driven by external magnetic fields [2, 3], the QAH effect applies no magnetic field, and in the recent experimental discovery the spin-orbit coupling and ferromagnetic ordering are essential to reach this state [4]. In this work, we find a novel phenomenon in the Chern insulator that the edge states are chiral in both the orbital and spin degrees of freedom when a chiral-like symmetry is present, and exhibit topologically stable spin texture in the position space. The chiral edge spin texture has no correspondence in the conventional trivial materials, and may have potential applications in designing topological-state-based spin devices which might be applicable to future spintronic technologies.Discovery of the quantum Hall effect in 1980s brought about a fundamental concept, topological state of quantum matter, to condensed matter physics [2, 3]. In the quantum Hall effect an external magnetic field drives the electrons to fill in discrete Landau levels. This leads to a gap in the 2D bulk, while the boundary exhibits 1D chiral gapless edge states, namely, the edge modes propagate one-way in each edge. In the integer quantum Hall effect the Hall conductance is quantized by Chern numbers, which are integers and characterize the global properties (topology) of the system [5]. The topological interpretation of the quantized Hall conductance implies that to obtain a quantum Hall state does not necessitate a magnetic field. The theoretical idea for quantum Hall effect without Landau levels, i.e. the QAH effect, was first introduced by Haldane in a honeycomb lattice over two decades ago [1]. The recent interests in this topological state have been revived due to the great development in the field of time-reversal invariant topological insulators [6, 7], with numerous new theoretical proposals having been introduced [8-13]. Importantly, following the proposal in Ref. [11], the QAH state has been detected in a recent experiment using ferromagnetic thin-film topological insulator, with the quantized Hall conductance having been observed [4].The minimal model for the Chern insulator is described * email: [email protected] by a two-band Hamiltonian H = k ψ † k H(k)ψ k , where the spin basis ψ k = (c k↑ , c k↓ ) T . Around the Γ point H(k) takes the simple (2 + 1)D Dirac formwhere for the realization with thin-film ferromagnetic topological insulators [4, 11] m z depends on the Zeeman term induced by the ferromagnetic order, the B-term characterizes the hybridization between top and bottom thin-film surfaces, and 2A 1,2 equal Fermi velocities of the surface Dirac cones of the parent topological insulator [14, 15]. The QAH phase is obtained when m z B < 0, with the Chern number C 1 = sgn(m z A 1 A 2 ) [11, 16]. In the solid-state experiment, the Cr-doped (Bi 1−x Sb x ) 2 Te 3 was used to achieve the above Hamiltonian and the QAH phase [4]. On the other hand, this model can also be realized in a square optical lattice [17] with spin-orbit coupling generated based on cold atom experiments [18][19][20].In the isotropic case |A 1 | = |A 2 | the Hamiltonian (1) respects a chiral-like symmetry defined by S = σ n1 M n2 , with n 1 ⊥ n 2 being arbitrary 2D orthogonal unit vectors in the x-y plane. Here σ n1 and M n2 represent the n 1component of the Pauli matrix acting on the spin space and the spatial reflection along the n 2 direction, respectively. It can be seen that SH(k)S † = −H(k). Then for two edges normal to n 1 axis, the edge states with k n2 = 0 must be eigenstates of σ n1 , with the spin oppositely polarized in the opposite edges [21]. Furthermore, using the k · p theory one can expand the 1D edge Hamiltonian up to the leading order of momentum and confirm that the edge states with nonzero k n2 are also eigenstates of σ n1 . This result is shown numerically inFig. 1. We see that the spin of edge states is in-plane polarized, and varies one cycle following the 1D closed path of the boundary. This spin texture shows that the edge states are chiral in both the orbital and spin degrees of freedom. Interestingly, the spin chirality gives a nontrivial quantized Berry phase for the edge modes after evolving one cycle along the boundary: γ C = ±π, and this defines a 1D winding number N 1d = ±1 which can be verified to correspond to the bulk Chern invariant via C 1 = N 1d sgn(m z ). With fixed sgn(m z ), changing the edge spin chirality reverses Chern number of the QAH insulator, while varying both the spin chirality and sgn(m z ) gives the phases with the same C 1 (see e.g.Fig. 1 a and f). InFig. 1we have shown the chiral spin texture and edge currents of the 2D insulator in different parameter regimes, and with square (a-d) and circular (e-h) geometries, respectively. arXiv:1401.0415v2 [cond-mat.mes-hall]FIG. 1: Topological spin texture of the edge states. a-d, The edge spin texture for square geometry of the boundary, with the Zeeman term mz > 0 and B < 0. e-h, The edge spin texture for circular geometry of the boundary, with mz < 0 and B > 0. For the other parameters, we take that A1, A2 > 0 (a,e); A1 > 0, A2 < 0 (b,f); A1, A2 < 0 (c,g); and A1 < 0, A2 > 0 (d,h). The topological spin textures give rise to quantized Berry phases if evolving the edge spin one circle along the boundary, which define 1D nontrivial topological states in the position space of the boundary and characterized by the 1D winding number N 1d . The bulk Chern number C1 corresponds to N 1d with an additional sign factor sgn(mz).The above study demonstrates a correspondence between the nontrivial topologies exhibited in the bulk and the boundary. The bulk Chern number is a topological invariant of the first Brillouin zone, which is a 2D closed manifold in the momentum space due to the band gap of the insulator. However, the edge modes are gapless and generically the 1D boundary is a closed manifold not in the momentum space, but in the position space. Therefore the 1D edge invariant N 1d is obtained in the real space rather than in the k-space. Both the bulk and edge topological states are classified by integers Z. Note that a 1D topological state necessitates the symmetry protection [22]. The correspondence between the bulk and edge topological phases relies on the chiral-like symmetry as introduced above, albeit the Chern insulator is an intrinsic topological state not depending on symmetry. These properties implies that from an intrinsic bulk topological phase we obtain a real-space topological state in the reduced dimension of the boundary when the additional S symmetry is present.While the 1D topological state relies on the symmetry protection, the chirality enables the topological spin texture to be insensitive to local perturbations which explicitly break the S symmetry. We have verified that the topological spin texture is stable against in-plane Zeeman fields without driving bulk phase transition, and insensitive to any type of local magnetic and nonmagnetic disorder scatterings, even when the disorder strength is comparable with the bulk gap. Actually, like that the orbital chirality of the edge modes prohibits the back scattering, the spin chirality ensures that no scattering occurs between two edge modes with opposite local spin polarizations. On the other hand, in the Methods section we show that the chiral spin texture may be scattered by the discrete lattice anisotropy in the high-energy regime.The edge channel of the Chern insulator can be described by 1D chiral Luttinger liquid [23, 24]. Furthermore, the above study shows that the edge modes are chiral in both orbital and spin degrees of freedom. As the topological spin texture leads to quantized Berry phases, which can be integrated by a Berry's connection, the edge states are governed by the following effective HamiltonianHere ψ s denotes the orbital part of the edge states,x is the position parameter along the edge, and the Berry's connection A s = i χ s (x|∂ x |χ s (x) , with |χ s (x) representing the polarized spin degree of freedom. The integral of A s along the 1D boundary gives¸dxA s (x) = N 1d π. The π-Berry phase is equivalent to a half magnetic fluxquanta threading through the 2D sample and encircled by the edge. According to the study by Wilczek in 1982 [25], a half quantum flux can lead to 1/2-fractionalization of the orbital angular momentum. As a result, for the 2D sample with circular geometry, the orbital angular momentum of the edge modes should be fractionalized as l z = m + N 1d /2, with m being integers. The fractionalization of the orbital angular momentum has an observable in the edge spectrum E lz = v edge l z R −1 lz , with R lz the effective radius of edge state wave function. Due to the 1/2-fractionalization no zero-energy (mid-gap) edge state exists, and therefore the total number of edge states is even. However, threading an additional magnetic 1/2flux-quanta can exactly push one original state to zero energy, changing the total number of edge modes to be odd, which provides an observable for the 1/2-fractionalization of orbital angular momentum. 3 FM quantum anomalous Hall insulator Normal lead a FM FM FM c d e b / ( ) (0) − plane y− plane −y plane = 0.5 = 0.25 = 0.5 = 1.0 FM lead FIG. 2: Angle-dependence of tunneling conductance.a, A normal metallic lead is strongly coupled to the left-hand edge and a ferromagnetic lead is weakly coupled to the righthand edge of the Chern insulator. Here the parameters for the quantum anomalous Hall phase satisfy mz > 0, A1,2 > 0, and B < 0. Then spin of the edge states in the right-hand edge points to the +y direction. b, The tunneling conductance σ(φ) is plotted versus azimuthal angle φ of the magnetization in the ferromagnetic lead, with the magnetization varying in the x−y plane (c), y−z plane (d), and x−z plane (e), respectively. The numerical results are presented for different polarization ratios P in the ferromagnetic lead. The maximum tunneling conductance is obtained when the magnetization aligns with the edge spin-polarization direction.The spin and orbital chirality make the edge of the Chern insulator be an exotic 1D metal which has no correspondence in conventional 1D materials. The topological spin texture of the edge modes may lead to strong spin-dependent effects as presented below, which on one hand can provide new unambiguous verification of the QAH state in the experiment, on the other hand, are applicable to spintronics by designing topological spin devices [26]. As illustrated inFig. 2a, we attach two metallic leads to the QAH sample, with a normal lead strongly coupled to the left-hand edge and a ferromagnetic lead weakly coupled to the right-hand edge. Due to the spin texture, the couplings between the sample edge and leads are fully spin selective, which can lead to strong anisotropic effects in the tunneling conductance when changing the direction of magnetization M FM in the ferromagnetic lead, as shown inFig. 2b-e. The tunneling conductance σ(φ)(Fig. 2 b)exhibits a clear angle-dependence when M FM varies in x − y and y − z planes (c and d), while it is a constant when M FM varies along x − z plane (e). This measures that the edge spin polarizes to the y direction. The angle-dependence implies a strong magnetoresistive effect by setting M FM along ±y directions. The magnetoresistance, given by MR = [σ(0) − σ(π)]/σ(π) × 100%, is plotted inFig. 3as a function of chemical potential µ in the Chern in-sulator and the polarization ratio P of the ferromagnetic lead. Due to the full spin-polarization, the edge of the sample can be regarded as an ideal dissipationless half-metal. This gives that MR ≈ 2P/(1 − P ) × 100%, which is significantly larger than the corresponding tunneling magnetoresistance obtained in conventional ferromagnet/insulator/ferromagnet devices with the same polarization ratio (the inserted panel ofFig. 3)[27]. The strong magnetoresistive effect has been widely applied to spintronics, especially to designing read heads [26]. 0.0 0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 0.5 P 0 100 200 300 400 500 600 700 800 MR (%) 0.0 0.2 0.4 0.6 0.8 0 200 400 600 800 O O MR (%) P =0 =0.6 2P/(1-P) 2P 2 /(1-P 2 ) O FIG. 3: Magnetoresistance for the setup in Fig. 2a by setting magnetization of the ferromagnetic lead along ±y directions.The magnetoresistance (MR) is plotted numerically as a function of the polarization ratio P in the ferromagnetic lead and the chemical potential µ (in unit of B) in the QAH insulator. The parameters for the QAH phase are taken as mz = −0.3A1,2 = −0.3B, which gives the bulk gap Eg = 2|mz| = 0.6B. The MR is uniform versus µ when the chemical potential is within the bulk gap (|µ| < 0.3B) and decreases when µ lies out of the gap. The inserted panel shows that the MR coincides with 2P/(1 − P ) for |µ| < 0.3B (black solid and red circled curves), which is significantly larger than the corresponding tunneling MR, given by 2P 2 /(1 − P 2 ) (green curve), in the conventional ferromagnet/insulator/ferromagnet devices with polarization ratio P in both ferromagnets [27].Another interesting application of the topological spin texture is to design controllable spin-filtering devices, as illustrated inFig. 4. The edge spin texture ensures that the output current is fully spin-polarized, with the polarization direction depending on which edge the drain lead is attached to. From the numerical results we see that when the voltage of the output lead lies in the sample bulk gap, the output current is 100% polarized to the same direction, reflecting that the spin texture is identical for all edge states. We note that the band gap of the currently realized QAH effect is small, while for realistic applications a larger topological gap is necessary. The new physics unveiled in this work and the proposed potential applications will motivate the search for new novel 4 materials of Chern insulator with sizable band gaps in the future.
The quantum anomalous Hall (QAH) insulator, also called Chern insulator, is a two-dimensional (2D) topological state of quantum matter which exhibits a gap in the bulk and chiral gapless edge states in the boundary [1]. Different from the quantum Hall effect driven by external magnetic fields [2,3], the QAH effect applies no magnetic field, and in the recent experimental discovery the spin-orbit coupling and ferromagnetic ordering are essential to reach this state [4]. In this work, we find a novel phenomenon in the Chern insulator that the edge states are chiral in both the orbital and spin degrees of freedom when a chiral-like symmetry is present, and exhibit topologically stable spin texture in the position space. The chiral edge spin texture has no correspondence in the conventional trivial materials, and may have potential applications in designing topological-state-based spin devices which might be applicable to future spintronic technologies.
Discovery of the quantum Hall effect in 1980s brought about a fundamental concept, topological state of quantum matter, to condensed matter physics [2,3]. In the quantum Hall effect an external magnetic field drives the electrons to fill in discrete Landau levels. This leads to a gap in the 2D bulk, while the boundary exhibits 1D chiral gapless edge states, namely, the edge modes propagate one-way in each edge. In the integer quantum Hall effect the Hall conductance is quantized by Chern numbers, which are integers and characterize the global properties (topology) of the system [5]. The topological interpretation of the quantized Hall conductance implies that to obtain a quantum Hall state does not necessitate a magnetic field. The theoretical idea for quantum Hall effect without Landau levels, i.e. the QAH effect, was first introduced by Haldane in a honeycomb lattice over two decades ago [1]. The recent interests in this topological state have been revived due to the great development in the field of time-reversal invariant topological insulators [6,7], with numerous new theoretical proposals having been introduced [8][9][10][11][12][13]. Importantly, following the proposal in Ref. [11], the QAH state has been detected in a recent experiment using ferromagnetic thin-film topological insulator, with the quantized Hall conductance having been observed [4].
The minimal model for the Chern insulator is described * email: [email protected] by a two-band Hamiltonian H = k ψ † k H(k)ψ k , where the spin basis ψ k = (c k↑ , c k↓ ) T . Around the Γ point H(k) takes the simple (2 + 1)D Dirac form
H(k) = m z + 2B(k 2 x + k 2 y ) 2A 1 k y + i2A 2 k x 2A 1 k y − i2A 2 k x −m z − 2B(k 2 x + k 2 y ) ,(1)
where for the realization with thin-film ferromagnetic topological insulators [4,11] m z depends on the Zeeman term induced by the ferromagnetic order, the B-term characterizes the hybridization between top and bottom thin-film surfaces, and 2A 1,2 equal Fermi velocities of the surface Dirac cones of the parent topological insulator [14,15]. The QAH phase is obtained when m z B < 0, with the Chern number C 1 = sgn(m z A 1 A 2 ) [11,16]. In the solid-state experiment, the Cr-doped (Bi 1−x Sb x ) 2 Te 3 was used to achieve the above Hamiltonian and the QAH phase [4]. On the other hand, this model can also be realized in a square optical lattice [17] with spin-orbit coupling generated based on cold atom experiments [18][19][20].
In the isotropic case |A 1 | = |A 2 | the Hamiltonian (1) respects a chiral-like symmetry defined by S = σ n1 M n2 , with n 1 ⊥ n 2 being arbitrary 2D orthogonal unit vectors in the x-y plane. Here σ n1 and M n2 represent the n 1component of the Pauli matrix acting on the spin space and the spatial reflection along the n 2 direction, respectively. It can be seen that SH(k)S † = −H(k). Then for two edges normal to n 1 axis, the edge states with k n2 = 0 must be eigenstates of σ n1 , with the spin oppositely polarized in the opposite edges [21]. Furthermore, using the k · p theory one can expand the 1D edge Hamiltonian up to the leading order of momentum and confirm that the edge states with nonzero k n2 are also eigenstates of σ n1 . This result is shown numerically in Fig. 1. We see that the spin of edge states is in-plane polarized, and varies one cycle following the 1D closed path of the boundary. This spin texture shows that the edge states are chiral in both the orbital and spin degrees of freedom. Interestingly, the spin chirality gives a nontrivial quantized Berry phase for the edge modes after evolving one cycle along the boundary: γ C = ±π, and this defines a 1D winding number N 1d = ±1 which can be verified to correspond to the bulk Chern invariant via C 1 = N 1d sgn(m z ). With fixed sgn(m z ), changing the edge spin chirality reverses Chern number of the QAH insulator, while varying both the spin chirality and sgn(m z ) gives the phases with the same C 1 (see e.g. Fig. 1 a and f). In Fig. 1 we have shown the chiral spin texture and edge currents of the 2D insulator in different parameter regimes, and with square (a-d) and circular (e-h) geometries, respectively.
1 = 1 1 = 1 1 = −1 a b c d e g h f 1 = −1 1 = −1 1 = 1 1 = 1 1 = −1 1 = 1 1 = −1 1 = −1 1 = 1 FIG. 1:
Topological spin texture of the edge states. a-d, The edge spin texture for square geometry of the boundary, with the Zeeman term mz > 0 and B < 0. e-h, The edge spin texture for circular geometry of the boundary, with mz < 0 and B > 0. For the other parameters, we take that A1, A2 > 0 (a,e); A1 > 0, A2 < 0 (b,f); A1, A2 < 0 (c,g); and A1 < 0, A2 > 0 (d,h). The topological spin textures give rise to quantized Berry phases if evolving the edge spin one circle along the boundary, which define 1D nontrivial topological states in the position space of the boundary and characterized by the 1D winding number N 1d . The bulk Chern number C1 corresponds to N 1d with an additional sign factor sgn(mz).
The above study demonstrates a correspondence between the nontrivial topologies exhibited in the bulk and the boundary. The bulk Chern number is a topological invariant of the first Brillouin zone, which is a 2D closed manifold in the momentum space due to the band gap of the insulator. However, the edge modes are gapless and generically the 1D boundary is a closed manifold not in the momentum space, but in the position space. Therefore the 1D edge invariant N 1d is obtained in the real space rather than in the k-space. Both the bulk and edge topological states are classified by integers Z. Note that a 1D topological state necessitates the symmetry protection [22]. The correspondence between the bulk and edge topological phases relies on the chiral-like symmetry as introduced above, albeit the Chern insulator is an intrinsic topological state not depending on symmetry. These properties implies that from an intrinsic bulk topological phase we obtain a real-space topological state in the reduced dimension of the boundary when the additional S symmetry is present.
While the 1D topological state relies on the symmetry protection, the chirality enables the topological spin texture to be insensitive to local perturbations which explicitly break the S symmetry. We have verified that the topological spin texture is stable against in-plane Zeeman fields without driving bulk phase transition, and insensitive to any type of local magnetic and nonmagnetic disorder scatterings, even when the disorder strength is comparable with the bulk gap. Actually, like that the orbital chirality of the edge modes prohibits the back scattering, the spin chirality ensures that no scattering occurs between two edge modes with opposite local spin polarizations. On the other hand, in the Methods section we show that the chiral spin texture may be scattered by the discrete lattice anisotropy in the high-energy regime.
The edge channel of the Chern insulator can be described by 1D chiral Luttinger liquid [23,24]. Furthermore, the above study shows that the edge modes are chiral in both orbital and spin degrees of freedom. As the topological spin texture leads to quantized Berry phases, which can be integrated by a Berry's connection, the edge states are governed by the following effective Hamiltonian
H edge = iv edgeˆdx ψ * s (x) ∂x − iA s (x) ψ s (x).(2)
Here ψ s denotes the orbital part of the edge states,x is the position parameter along the edge, and the Berry's connection A s = i χ s (x|∂ x |χ s (x) , with |χ s (x) representing the polarized spin degree of freedom. The integral of A s along the 1D boundary gives¸dxA s (x) = N 1d π.
The π-Berry phase is equivalent to a half magnetic fluxquanta threading through the 2D sample and encircled by the edge. According to the study by Wilczek in 1982 [25], a half quantum flux can lead to 1/2-fractionalization of the orbital angular momentum. As a result, for the 2D sample with circular geometry, the orbital angular momentum of the edge modes should be fractionalized as l z = m + N 1d /2, with m being integers. The fractionalization of the orbital angular momentum has an observable in the edge spectrum E lz = v edge l z R −1 lz , with R lz the effective radius of edge state wave function. Due to the 1/2-fractionalization no zero-energy (mid-gap) edge state exists, and therefore the total number of edge states is even. However, threading an additional magnetic 1/2flux-quanta can exactly push one original state to zero energy, changing the total number of edge modes to be odd, which provides an observable for the 1/2-fractionalization of orbital angular momentum. a, A normal metallic lead is strongly coupled to the left-hand edge and a ferromagnetic lead is weakly coupled to the righthand edge of the Chern insulator. Here the parameters for the quantum anomalous Hall phase satisfy mz > 0, A1,2 > 0, and B < 0. Then spin of the edge states in the right-hand edge points to the +y direction. b, The tunneling conductance σ(φ) is plotted versus azimuthal angle φ of the magnetization in the ferromagnetic lead, with the magnetization varying in the x−y plane (c), y−z plane (d), and x−z plane (e), respectively. The numerical results are presented for different polarization ratios P in the ferromagnetic lead. The maximum tunneling conductance is obtained when the magnetization aligns with the edge spin-polarization direction.
The spin and orbital chirality make the edge of the Chern insulator be an exotic 1D metal which has no correspondence in conventional 1D materials. The topological spin texture of the edge modes may lead to strong spin-dependent effects as presented below, which on one hand can provide new unambiguous verification of the QAH state in the experiment, on the other hand, are applicable to spintronics by designing topological spin devices [26]. As illustrated in Fig. 2 a, we attach two metallic leads to the QAH sample, with a normal lead strongly coupled to the left-hand edge and a ferromagnetic lead weakly coupled to the right-hand edge. Due to the spin texture, the couplings between the sample edge and leads are fully spin selective, which can lead to strong anisotropic effects in the tunneling conductance when changing the direction of magnetization M FM in the ferromagnetic lead, as shown in Fig. 2 b-e. The tunneling conductance σ(φ) (Fig. 2 b) exhibits a clear angle-dependence when M FM varies in x − y and y − z planes (c and d), while it is a constant when M FM varies along x − z plane (e). This measures that the edge spin polarizes to the y direction. The angle-dependence implies a strong magnetoresistive effect by setting M FM along ±y directions. The magnetoresistance, given by MR = [σ(0) − σ(π)]/σ(π) × 100%, is plotted in Fig. 3 as a function of chemical potential µ in the Chern in-sulator and the polarization ratio P of the ferromagnetic lead. Due to the full spin-polarization, the edge of the sample can be regarded as an ideal dissipationless half-metal. This gives that MR ≈ 2P/(1 − P ) × 100%, which is significantly larger than the corresponding tunneling magnetoresistance obtained in conventional ferromagnet/insulator/ferromagnet devices with the same polarization ratio (the inserted panel of Fig. 3) [27]. The strong magnetoresistive effect has been widely applied to spintronics, especially to designing read heads [26]. The magnetoresistance (MR) is plotted numerically as a function of the polarization ratio P in the ferromagnetic lead and the chemical potential µ (in unit of B) in the QAH insulator. The parameters for the QAH phase are taken as mz = −0.3A1,2 = −0.3B, which gives the bulk gap Eg = 2|mz| = 0.6B. The MR is uniform versus µ when the chemical potential is within the bulk gap (|µ| < 0.3B) and decreases when µ lies out of the gap. The inserted panel shows that the MR coincides with 2P/(1 − P ) for |µ| < 0.3B (black solid and red circled curves), which is significantly larger than the corresponding tunneling MR, given by 2P 2 /(1 − P 2 ) (green curve), in the conventional ferromagnet/insulator/ferromagnet devices with polarization ratio P in both ferromagnets [27].
Another interesting application of the topological spin texture is to design controllable spin-filtering devices, as illustrated in Fig. 4. The edge spin texture ensures that the output current is fully spin-polarized, with the polarization direction depending on which edge the drain lead is attached to. From the numerical results we see that when the voltage of the output lead lies in the sample bulk gap, the output current is 100% polarized to the same direction, reflecting that the spin texture is identical for all edge states. We note that the band gap of the currently realized QAH effect is small, while for realistic applications a larger topological gap is necessary. The new physics unveiled in this work and the proposed potential applications will motivate the search for new novel materials of Chern insulator with sizable band gaps in the future.
B = A1 = A2 (a); B = −A1 = −A2 (b); mz = A1 = A2 (c); and mz = −A1 = −A2 (d).
The sign change of pout in d and e from the topological region with B < 0 to the region B > 0 implies that the edge spin reverses direction.
Methods
Lattice model. We consider the lattice model for the present QAH insulator, and require that the lattice Hamiltonian reduces to Eq. (1) around Γ point. The main results of this work are not lattice configuration dependent. Here we examine the following square lattice
Hamiltonian H(k) = m z + 2Ba −2 (2 − cos k x a − cos k y a) σ z + 2A 1 a −1 sin k y aσ x − 2A 2 a −1 sin k x aσ y ,(3)
where a is the lattice constant and can be set as a = 1, and m z is assumed to be |m z | < 4|B| for convenience. The above Hamiltonian describes a topological (trivial) phase for m z B < 0 (m z B > 0), with the Chern number
C 1 = sgn(A 1 A 2 )[sgn(m z ) − sgn(B)]/2. For |m z | 4|B|
, the low-energy continuous Hamiltonian (1) can capture the physics of the Chern insulating phases.
Symmetry-breaking perturbations. Under lowenergy continuous approximation the Hamiltonian H(k) anticommutes with the symmetry operator S = σ n1 M n2 . The local perturbation which breaks this symmetry includes the in-plane Zeeman fields V pert = m x σ x + m y σ y , the nonmagnetic and magnetic disorders rj ,s V non dis ( r j )n rj ,s + rj ,α V mag dis,α ( r j )ψ † rj σ α ψ rj , with the particle number operator n rj ,s = c † rj ,s c rj ,s , s =↑, ↓, and α = x, y, z. Here V non dis ( r j ) and V mag dis,α ( r j ) represent nonmagnetic and magnetic random disorder potentials, respectively. In the Supplementary Information we show that the edge spin texture is not affected by the in-plane Zeeman fields without driving phase transition in the bulk, and also insensitive to the local disorder perturbations.
In the high-energy regime, due to discrete lattice anisotropy generically the Hamiltonian H(k) only anticommutes with S = M y σ x and S = M x σ y . This ensures that the edge spin aligns along x (y) axis in the edges normal toê y (ê x ) direction and far away from sample corners. On the other hand, around the corners of the square sample the spin polarization of the high-energy edge modes can be verified to have a sizable tilt to the perpendicular direction.
Quantum tunneling transport. The tunneling transport is studied with Landauer formalism in the 2D tight-binding model of the Chern insulator. With the coupling to normal and ferromagnetic leads, we determine the retarded Green's function of the Chern insula-
tor by G R (ω) = (ω − H − Σ R ) −1 ,
where H is the tight binding Hamiltonian of the Chern insulator, and Σ R is the self-energy due to the couplings to the leads. Using the Fisher-Lee relation we can obtain the scattering matrix based on the Green's function and self-energies [28]
S ss p,q = −δ p,q δ αβ + i[Γ s p ] 1/2 G R ss [Γ s q ] 1/2 .(4)
Here, the matrix element S ss p,q (s, s =↑, ↓) denotes the scattering amplitude of the process that an electron is scattered from the spin state s in lead q to the spin state s in lead p, with p = q = L, R representing the left and right-hand lead, respectively. Γ s
p = i[(Σ ss p ) R − (Σ ss p ) A ],
where (Σ ss p ) R/A is the s-spin component retarded/advanced self-energy due to the coupling to lead p. By determining the scattering matrix one can obtain the transmission coefficients regarding different spin channels. Then we obtain the tunneling conductances under different configurations of the ferromagnetic and normal-metal leads, and the output spin-polarized current in the spin-filtering process.
Supplementary Information
S-1. TWO-BAND MODEL FOR QUANTUM ANOMALOUS HALL INSULATOR
The minimal realization for the quantum anomalous Hall effect is to consider a two-band model with spin-orbit coupling and magnetization. The main results in this work are not lattice configuration dependent. Here, for convenience we consider the square lattice model with the Hamiltonian given by
H = k ψ † k H(k)ψ k ,(S1)
where the spin basis is defined by ψ k = (c k↑ , c k↓ ) T with c k,s the electron annihilation operator in spin state s =↑, ↓, and the Bloch Hamiltonian H(k) takes the form
H(k) = 2A 1 sin k y σ x − 2A 2 sin k x σ y + m z + 2B(2 − cos k x − cos k y ) σ z .(S2)
In the above form we have taken the lattice constant to be a = 1. In the case |m z | 4|B|, the physics of the system can be fully governed by the low-energy Hamiltonian
H(k) = 2A 1 k y σ x − 2A 2 k x σ y + m z + 2B(k 2 x + k 2 y ) σ z .(S3)
The Bloch Hamiltonian can be rewritten in the form H
(k) = d(k) · σ, where the d-vector is defined through d x = 2A 1 k y , d y = −2A 2 k x , and d z = m z + 2B(k 2 x + k 2 y )
. The topology of the system is determined by the first Chern number, which can be calculated by
C 1 = 1 4πˆd 2 kn · ∂n ∂k x × ∂n ∂k y , n = 1 | d(k)| (d x , d y , d z ).(S4)
The integrand in the right hand side of the above equation describes a mapping between the momentum k-space to the spherical surface S 2 formed by the unit vector n(k). Therefore the first Chern invariant is given by the number of times that this mapping can cover the whole spherical surface. For each fixed momentum k = (k 2 x + k 2 y ) 1/2 , the n-vector can cover one circle in the x − y plane. Moreover, it is easy to see that for both k = 0 and k = ∞, the unit vector n(k) points along the z or −z axis. Therefore the mapping must cover integer times of the whole spherical surface. In particular, when m z B > 0, the z-component of the n-vector is always positive (for m z > 0) or negative (for m z < 0), and the number of coverage is zero. This gives that C 1 = 0. On the other hand, for m z B < 0 and m z > 0, the unit vector n points to +z and −z directions when k = 0 and k = ∞, respectively. We then get a single coverage of the spherical surface in the mapping, and obtain the Chern number C 1 = 1. Finally, from the Eq. (S4) we can see that the Chern number is odd with respect to each component of n(k), and it changes sign if reversing the sign of any component n j . With these results in mind we conclude that
C 1 = sgn(A 1 A 2 )[sgn(m z ) − sgn(B)]/2.(S5)
The Hall conductance of the quantum anomalous Hall insulator is given by σ xy = C 1 e 2 /h, where e and h are the electron charge and Plank constant, respectively.
S-2. TOPOLOGICAL EDGE SPIN TEXTURE
We consider the isotropic low-energy Dirac Hamiltonian (S3) with A 1 = ηA 2 , where η = ±1. In this case the low-energy Hamiltonian is rotationally invariant and can be written down in the following generic form
H(k) = 2A 1 k n2 σ n1 − 2A 2 k n1 σ n2 + (m z + 2Bk 2 )σ z ,(S6)
where the arbitrary in-plane orthogonal unit vectors are n 1 = un x + vn y and n 2 = −vn x + un y , where the coefficients satisfy u 2 + v 2 = 1. The components of the momenta and Pauli matrices are given by σ n1 = n 1 · σ, σ n2 = n 2 · σ, k n1 = uk x + vηk y , and k n2 = −vηk x + uk y . It can be verified that the Hamiltonian (S6) respects the chiral-like symmetry defined by S = σ n2 M n1 (also valid for σ n1 M n2 ), with M n1 representing the spatial reflection with respect to the n 1 direction. It follows that
SH(k)S † = −H(k).(S7)
We show below that due to this symmetry the edge states of the quantum anomalous Hall insulator are in-plane spinpolarized, and exhibit topological spin texture in the boundary. In particular, we consider three different situations as illustrated in Fig. S1: the open boundary (a), the circular close boundary (b), and the generic closed boundary (c). Furthermore, we should emphasize that while we use here the Hamiltonian (S6) as an example for the study, which is relevant for the realistic experiment. the edge spin polarizations shown below are generic results for quantum anomalous Hall insulators satisfying the chiral-like symmetry given in Eq. (S7). In the illustration we consider the parameter regime that mz > 0, B < 0, and A1,2 > 0.
A. Open boundary
We first consider the simplest situation that the system has two edges normal to n 2 direction, while it is infinite (or periodic) along n 1 axis [ Fig. S1 (a)]. This study can be applied to the system with smooth and slowly varying (closed) boundaries. In this case the momenta k n1 are good quantum numbers, and the Hamiltonian can be described by
H = kn 1 H 1D (k n1 ),(S8)
where H 1D (k n1 ) is a k n1 -parameterized 1D Hamiltonian. The end states of H 1D (k n1 ) are edge states of the original 2D quantum anomalous Hall system with momentum k n1 . The 1D Hamiltonian can be expressed as
H 1D (k n1 ) = −ˆdx n2 ψ † kn 1 (x n2 ) 2B∂ 2 xn 2 σ z + iA 1 ∂ xn 2 σ n1 ψ kn 1 (x n2 ) +ˆdx n2 ψ † kn 1 (x n2 ) (m z + 2Bk 2 n1 )σ z − A 2 k n1 σ n2 ψ kn 1 (x n2 ) .(S9)
The transformation in Eq. (S7) is defined for the first quantization Hamiltonian. Accordingly, for the second quantization Hamiltonian, the symmetry operator S transforms the basis according to S(c kn 1 ↑ , c † kn 1 ↓ ) T = (c † −kn 1 ↓ , c −kn 1 ↑ ) T . This is followed by
SH 1D (k n1 )S † = H 1D (−k n1 ),(S10)
which gives that SHS † = H. It is trivial to know that at k n1 = 0, which is a reflection invariant momentum, the 1D Hamiltonian H 1D (0) is invariant under the S-transformation. This implies that the system H 1D (0) belongs to the 1D chiral unitary (AIII) class [21,22]. Indeed, this can be more transparent if rotating the spin operators (σ n1 , σ n2 ) → (σ y , −σ x ) in H 1D (0). In this case, one can verify that the time-reversal and charge conjugation symmetries, defined respectively by T = iKσ y with K the complex conjugation, and C : (ĉ kn 1 σ ,ĉ † kn 1 σ ) → (σ z ) σσ (ĉ † −kn 1 σ ,ĉ −kn 1 σ ), are generically broken for H 1D , while at k n1 = 0 the chiral symmetry S = T C is preserved. Therefore H 1D (0) describes 1D topological insulating phase belonging to the chiral unitary (AIII) class, whose end states are eigenstates of the chiral operator σ n2 , with the spin oppositely polarized in the opposite edges [21].
The spin-polarization of the edge states with k n1 = 0 can be obtained using the k · p theory. By expanding up to the leading order of the momentum k n1 the effective edge Hamiltonian for the two edges normal to n 2 direction, under the restriction of the S symmetry, should take the generic form v edge k n1 σ n2 . Therefore the edge states with nonzero k n1 are also eigenstates of σ n2 . The edge spin polarizations are illustrated in Fig. S1 (a) under the condition with m z > 0, B < 0, and A 1,2 > 0. We note that the Eqs. (S8) and (S10) are directly derived from the symmetry given in Eq. (S7), not dependent on the specific Hamiltonian (S6) or (S9). Therefore the edge spin polarizations are generic results for any 2D Hamiltonian satisfying the S symmetry.
B. Circular closed boundary
We turn to the edge spin texture for the finite system with closed boundary. For convenience we study the quantum anomalous Hall sample with circular geometry, and show below that the edge modes exhibit topological spin texture as illustrated in Fig. S1 (b). Noting that the boundary is rotationally invariant, it is convenient to reexpress the Hamiltonian in the polar coordinate (r, ϕ) system
H(r, ϕ) = 2B( 1 r ∂ r r∂ r + 1 r 2 ∂ 2 ϕ ) + m z σ z + i 2A 1 r σ r ∂ ϕ − i2A 2 σ ϕ ∂ r ,(S11)
where σ r = cos ϕσ x + sin ϕσ y and σ ϕ = cos ϕσ y − sin ϕσ x . The eigenstates of H(r, ϕ) can generically be described by
|Φ m (r, ϕ) = |φ m (r, ϕ) e imϕ ,(S12)
where m are integers. For bulk states, each m corresponds to two solutions |φ (±) m (r, ϕ) , with eigenvalues ±E m (E m > 0) respectively. On the other hand, for edge modes each m corresponds to only a single eigenstate denoted by |φ edge m (r, ϕ) . These states can be solved by
2B( 1 r ∂ r r∂ r − m 2 − i2m∂ ϕ − ∂ 2 ϕ r 2 ) + m z σ z − i2A 2 σ ϕ ∂ r − 2A 1 r σ r (m − i∂ ϕ ) |φ (±) m (r, ϕ) = ±E m |φ (±) m (r, ϕ) ,(S13) and 2B( 1 r ∂ r r∂ r − m 2 − i2m∂ ϕ − ∂ 2 ϕ r 2 ) + m z σ z − i2A 2 σ ϕ ∂ r − 2A 1 r σ r (m − i∂ ϕ ) |φ edge m (r, ϕ) = E m |φ edge m (r, ϕ) .(S14)
For the present circular boundary, the chiral-like symmetry is given by S = σ r M ϕ , where M ϕ transforms ϕ to −ϕ. Using S to act on both sides of the eigen-equations for the bulk and edge states yields
2B( 1 r ∂ r r∂ r − m 2 + i2m∂ ϕ − ∂ 2 ϕ r 2 ) + m z σ z − i2A 2 σ ϕ ∂ r + 2A 1 r σ r (m + i∂ ϕ ) |φ (±) −m (r, −ϕ) = ±E m |φ (±) −m (r, −ϕ) ,(S15)
and
2B( 1 r ∂ r r∂ r − m 2 + i2m∂ ϕ − ∂ 2 ϕ r 2 ) + m z σ z − i2A 2 σ ϕ ∂ r + 2A 1 r σ r (m + i∂ ϕ ) |φ edge −m (r, −ϕ) = −E m |φ edge −m (r, −ϕ) .(S16)
Taking that ϕ → −ϕ, we can rewrite the above equations in the form
2B( 1 r ∂ r r∂ r − m 2 − i2m∂ ϕ − ∂ 2 ϕ r 2 ) + m z σ z − i2A 2 σ ϕ ∂ r + 2A 1 r σ r (m − i∂ ϕ ) |φ (±) −m (r, ϕ) = ±E m |φ (±) −m (r, ϕ) , (S17) and 2B( 1 r ∂ r r∂ r − m 2 − i2m∂ ϕ − ∂ 2 ϕ r 2 ) + m z σ z − i2A 2 σ ϕ ∂ r + 2A 1 r σ r (m − i∂ ϕ ) |φ edge −m (r, ϕ) = −E m |φ edge −m (r, ϕ) .(S18)
From the last four formulas we find that the bulk states |φ (S17) and (S18) the left-hand side is not the original Hamiltonian). This implies that for the present circular boundary, in the edge state spectrum E 0 = 0. From Eqs. (S15) and (S16) we have that the corresponding eigenstates transform according to
S|φ (±) m (r, ϕ) = |φ (∓) −m (r, −ϕ) , S|φ edge m (r, ϕ) = |φ edge −m (r, −ϕ) . (S19)
Furthermore, we note that the reflection operator M ϕ transforms the bulk and edge states via
M ϕ |φ (±) m (r, ϕ) = |φ (±) −m (r, −ϕ) , M ϕ |φ edge m (r, ϕ) = |φ edge −m (r, −ϕ) . (S20)
Together with the results in Eqs. (S19) and (S20) we can deduce for the bulk and edge states that (up to ± signs)
σ r |φ (±) m (r, ϕ) = |φ (∓) m (r, ϕ) , σ r |φ edge m (r, ϕ) = |φ edge m (r, ϕ) .(S21)
This shows that the edge modes are eigenstates of σ r and then the edge spin polarization takes the spatial configuration illustrated in Fig. S1 (b) (with the parameter regime that m z > 0, B < 0, and A 1,2 > 0), while the bulk states are generically not eigenstates of σ r and do not exhibit topological spin texture in the position space.
C. Generic boundary
Now we generalize the above results to the situation with generic boundary geometries [ Fig. S1 (c)]. We require that in the sample there are no narrow areas where the edge modes localized in different edges may couple to each other, leading to the finite size effects. Let the boundary be characterized by a curve C l : f (r, l) = 0, with (r, l) consisting of a local orthogonal coordinate system. Around the boundary, one can always locally describe the Hamiltonian by
H(r, l) = 2B(p 2 r + p 2 l ) + m z σ z + 2A 1 p l σ r − 2A 2 p r σ l ,(S22)
where p l and p r are momentum operators with respect to the local tangent ( n l ) and normal ( n r ) directions of the boundary, respectively. The Pauli matrices σ r = n r · σ and σ l = n l · σ. For the above Hamiltonian the symmetry operator is defined by S = σ r M l . Similarly, the edge states can be described by |Φ edge k l (r, l) , with k l the quasimomentum along the boundary. Through the same procedure as done in the above subsection, we can verify that the eigenvalues for |Φ edge k l (r, l) and |Φ edge −k l (r, −l) are E k l and −E k l , respectively. The transformations on these states satisfy S|Φ edge k l (r, l) = |Φ edge −k l (r, −l) and M l |Φ edge k l (r, l) = |Φ edge −k l (r, −l) . Therefore, we again have σ r |Φ edge k l (r, l) = |Φ edge k l (r, l) up to a ± sign. These results conclude that the edge states exhibit topological spin texture in the boundary.
The topological edge spin texture leads to a nontrivial quantized Berry phase in the boundary, which can be calculated by
γ C =˛dxA s (x), A s = i χ s (x|∂ x |χ s (x) ,(S23)
where we have denoted by |χ s (x) the spin part of the edge state wave function, withx the coordinate along the boundary. Due to the topological spin texture, the Berry phase reads γ C = ±π, which defines a 1D topological state characterized by the winding number N 1d = γ C /π in the edge. Note that the Berry phase is determined by the two in-plane spin components, it should be odd under the transformation by M n2 or M n1 , and be even under the transformation σ z → −σ z . Therefore the 1D winding number is given by N 1d = sgn(A 1 A 2 ), and it has a correspondence to the bulk Chern number via
C 1 = sgn(m z )N 1d .(S24)
It is worthwhile to note that there is an additional sign factor in the correspondence between the 2D bulk topological state and the 1D topological state in the boundary. This is reasonable, since the first Chern invariant is a 2D winding number, and it necessitates the inclusion of one more dimension relative to that for N 1d . The additional sign factor accounts for such difference in the dimension.
S-3. SYMMETRY-BREAKING PERTURBATIONS
In the low-energy regime, the nonperturbed Hamiltonian is rotationally invariant and respect the chiral-like S symmetry. The local perturbation which breaks this symmetry includes the in-plane Zeeman fields, the nonmagnetic and magnetic disorders. In the high-energy regime, due to discrete lattice anisotropy the chiral-like symmetry is generically broken in the Bloch Hamiltonian, unless for several special directions.
A. In-plane magnetic fields
The in-plane Zeeman couplings V pert = m x σ x + m y σ y can break the S symmetry. For the sake of generality, we write down the Hamiltonian in the bases of momenta and Pauli matrices with respect to the generic orthogonal unit vectors n 1 and n 2
H(k) = (2A 1 k n2 + m 1 )σ n1 − (2A 2 k n1 + m 2 )σ n2 + (m z + 2Bk 2 )σ z ,(S25)
where m 1 = um x + vm y and m 2 = vm x − um y . The in-plane Zeeman fields shift the bulk band edge from the Γ point to the one with finite momenta. In the parameter regime m z B < 0, increasing the Zeeman field can lead to a topological phase transition at the critical point
m = m c = 2 1/2 A |m z | 1/2 |B| 1/2 ,(S26)
where m = (m 2 x + m 2 y ) 1/2 and A = |A 1,2 |. The condition with m < m c corresponds to the topological phase. Similar as the study in the subsection II (A), we consider the boundary modes localized in the two edges normal to n 2 direction. It is straightforward to verify that for S = σ σn 2 M n1 we have
SH(k)S † = −H(k) − 2m 2 σ n2 .(S27)
Therefore, in the presence of in-plane Zeeman fields, generically no symmetry is preserved for the system. However, it is interesting that the Hamiltonian also satisfies
σ n2 H(k)σ −1 n2 = −H(k) − 2(m 2 + 2A 2 k n1 )σ n2 .(S28)
This implies that the Hamiltonian H 1D (k 0 n1 , x n2 ) with k 0 n1 = −m 2 /(2A 2 ) again defines a 1D chiral unitary (AIII) class insulator [21,22]. Then in the topological phase the edge states localized in the left-hand side |Φ edge Furthermore, for the generic 1D momentum k n1 = k 0 n1 +k n1 , we shift the zero point of the momentum to k 0 n1 , and can then rewrite the Hamiltonian in the form
H 1D (k n1 , x n2 ) = −ˆdx n2 ψ † kn 1 (x n2 ) 2B∂ 2 xn 2 σ z + iA 1 ∂ xn 2 σ n1 ψ kn 1 (x n2 ) +ˆdx n2 ψ † kn 1 (x n2 ) (m z + 2Bk 2 n1 )σ z − A 2kn1 σ n2 ψ kn 1 (x n2 ) ,(S29)
wherem z = m z + 4Bk 0 n1kn1 + 2Bk 2 n1 . It can be seen that the above 1D Hamiltonian is formally equivalent to the one in Eq. (S9). Therefore, according to the previous results, all the boundary states localized in the edges normal to n 2 direction are eigenstates of σ n2 . With this result we confirm that the edge spin texture cannot be affected by the in-plane Zeeman fields without driving the phase transition in the bulk.
B. Magnetic and non-magnetic disorders
The nonmagnetic and magnetic disorders can also break the chiral-like symmetry S of the system. For convenience we adopt the lattice model to study the disorder effects. The local on-site magnetic and nonmagnetic disorders can be described by
V dis = rj ,s V non dis ( r j )n rj ,s + rj ,α V mag dis,α ( r j )ψ † rj σ α ψ rj ,(S30)
where ψ rj = (c rj ,↑ , c rj ,↓ ), the particle number operator n rj ,s = c † rj ,s c rj ,s with r j the 2D coordinate for lattice sites, s =↑, ↓, and α = x, y, z. Here V non dis ( r j ) ∈ [−V 0 /2, V 0 /2] and V mag dis,α ( r j ) ∈ [−V α /2, V α /2] represent nonmagnetic and magnetic random disorder potentials, respectively, and for magnetic disorder we take that V 2
x + V 2 y + V 2 z = V 2 0 . The configuration averaging of the disorder potentials vanishes
V non dis ( r j ) = V mag dis,α ( r j ) = 0.(S31)
In the regime that the disorder strength is weak relative to the bulk gap of the quantum anomalous Hall insulator, we expect that the chiral edge spin texture cannot be scattered. The reason is because the spin chirality of the edge states ensures that edge modes with opposite spin polarizations are spatially localized far away from each other, and weak disorder, while breaking the symmetry of the bulk Hamiltonian, cannot lead to the scattering between such two edge modes. If we consider only the nonmagnetic disorder, the commutation relation between S = σ n2 M n1 and the second quantization Hamiltonian satisfies
SHS † = H − rj ,s V non dis ( r j )n rj ,s ,(S32)
where the condition V non dis ( r j ) = 0 has been applied. On the other hand, if there is only magnetic disorder, we have
SHS † = H − rj V mag dis,n2 ( r j )ψ † rj σ n2 ψ rj .(S33)
Comparing with the formulas (S32) and (S33) we can see that under the same strength of disorder potentials, the topological spin texture should be more insensitive to the magnetic disorders. This is because, for example, for the edges normal to n 2 direction, only the magnetic disorder component V mag n2
breaks the S symmetry and may affect the edge spin polarizations. In particular, the magnetic disorder with polarization along z direction has no effect on the edge spin texture. The numerical results for nonmagnetic and magnetic disorder effects are shown in Fig. S2 and Fig. S3, respectively. The effects of disorder scattering on the spin textures are shown in (a) to (f) in the two figures, from which one can see that the topological spin texture is nearly unaffected even when the disorder strength equals the bulk gap E gap . In Fig. S2 (g) and Fig. S3 (g) we show the Root Mean Square of x, y, and z components of the edge spin
γ Sα = S 2 α 1/2 , α = x, y, z,(S34)
with these spin components satisfying
S 2 x + S 2 y + S 2 z = 1.(S35)
The magnitude of γ Sz quantitatively reflects the deviation of the spatial spin configuration from the in-plane topological spin texture. It can be seen that γ Sz exhibits a very weak dependence on the disorder scattering, especially in weak (magnetic and nonmagnetic) disorder regime with V 0 < 0.2E gap [ Fig. S2 (g) and Fig. S3 (g)]. Even in the strongest disorder regime with V 0 = 4.0E gap , the relative Root Mean Square p = γ Sz /(γ Sx + γ Sy + γ Sz ) is less than 12% for nonmagnetic disorder and less than 9% for magnetic disorder. Under the same disorder strength, the edge spin texture is clearly more insensitive to magnetic disorder scattering, consistent with the results in Eqs. (S32) and (S33). The weak dependence of the edge spin texture on disorder scattering implies that the proposed topological spin devices in the main text are also insensitive to the local disorder perturbations. x + V 2 y + V 2 z ) 1/2 ]; (g) Root Mean Square for Sx,y,z versus disorder strength V0. Relative to the nonmagnetic disorder regime, the magnetic disorder has a stronger effect in randomizing the wave-function distribution of edge states, but clearly has a weaker scattering effect on the spin texture.
C. Effect of discrete lattice structure anisotropy
The topological spin texture relies on the continuous approximation of the Bloch Hamiltonian, which is valid in the low-energy regime. For the realistic materials [4,11], the bulk gap of the quantum anomalous Hall insulator is much less than the bandwidth of the system. In this case, all physics, including the topology and the edge states, can be captured by the low-energy Bloch Hamiltonian. Therefore the continuous approximation is typically well justified.
The topological spin texture can in principle be scattered by the discrete lattice anisotropy in the high-energy regime. Indeed, if we consider the parameter regime that |m z | is close to or larger than |A 1,2 | and |B|, the bulk gap of the system is in the order of the band width. Then the high-energy edge states exist and to study them one has to go beyond the low-energy Bloch Hamiltonian. From Eq. (S2) one can see that in the high-energy regime, due to discrete lattice anisotropy generically the Bloch Hamiltonian H(k) only anticommutes with S = M y σ x and S = M x σ y . This ensures that for a square sample with boundaries along the directions of Bravais lattice vectors, the edge spin aligns along x (y) axis in the edges normal to y (x) direction and far away from sample corners. On the other hand, around the corners of the square sample the spin polarization of the high-energy edge modes may have a finite tilt to the perpendicular direction due to the broken down of the S symmetry.
The numerical results are shown in Fig. S4, where we take the parameters which are rescaled to be dimensionless that A 1 = A 2 = −B = 1.0 and m z = 1.5. In this regime the bulk band gap of the system E gap = 3.0, which is close to the bandwidth. It can be seen from Fig. S4 (a) that the edge states with energies |E| < 0.5 exhibit topological spin texture, with spin polarization well within x − y plane and having negligible spin tilting to z direction at the corners. However, when energy increases, the edge spins at corners exhibit a more and more pronounced tile to the perpendicular direction. On the other hand, once going away from the corners by only few sites, we can find that the edge spins exactly point to x or y direction for all edge modes, reflecting that the chiral-like S symmetry is recovered in these directions even in the high-energy regime.
S-4. ORBITAL ANGULAR MOMENTUM FRACTIONALIZATION
The edge channel of the quantum anomalous Hall insulator can be described by 1D chiral Luttinger liquid. Furthermore, the topological spin texture leads to quantized Berry phases, which define nontrivial topological states in the boundary. Taking into account the Berry phase effect, the chiral edge states can be governed by the following effective Hamiltonian
H edge = iv edgeˆdx ψ * s (x) ∂x − iA s (x) ψ s (x).(S36)
Here ψ s denotes the orbital wave-function of the edge states. The integral of A s along the 1D boundary giveş dxA s (x) = N 1d π. The π-Berry phase is equivalent to a half magnetic flux-quanta threading through the 2D sample 1,2 = − = 1, = 1.5 = 3.0 FIG. S4: Edge spin texture byond low-energy limit in quantum anomalous Hall insulator with square boundary. The parameters are rescaled to be dimensionless that A1 = A2 = −B = 1.0 and mz = 1.5, which leads to the bulk gap Egap = 3.0. (a) The spin texture for edge modes of different energies, with only part of the states shown here. It can be seen that the edge states with energies |E| < 0.5 exhibit topological spin texture, with spin polarization well within x − y plane. Beyond such energy scale the edge spins exhibit a more and more pronounced tile to the perpendicular direction at square corners; (b) Wave function profiles of the edge states (for energy E < 1.5) and bulk states (for E > 1.5). and encircled by the edge. According to the study by Wilczek in 1982 [25], a half quantum flux can lead to 1/2fractionalization of the orbital angular momentum. As a result, for the 2D sample with circular geometry, the orbital angular momentum of the edge modes should be fractionalized as l z = m + N 1d /2, with m being integers. The energy spectrum of the edge states is corresponding to fractionalization of the orbital angular momentum and is given by
E lz = m + 1 2 N 1d v edge R lz , m = 0, ±1, ±2, ...(S37)
where R lz is the effective radius of edge state wave function. Due to the 1/2-fractionalization no zero-energy edge state exists. Then the number of edge states is N = even. This result can also be derived from the Eq. (S14). If separating the edge state wave-function by spin and orbital parts |φ edge m (r, ϕ) = φ m (r)|χ s (ϕ) , from Eq. (S14) one can obtain the energy by
E m = 2A 1 φ m (r)| 1 r |φ m (r) χ s (ϕ)|σ r (m − i∂ ϕ )|χ s (ϕ) = m + 1 2 2A 1 R m , m = 0, ±1, ±2, ...(S38)
where R m is the expectation value of 1/r. Now let us thread an additional magnetic flux Φ through the sample, which is described by a gauge A = A ϕêϕ , with A ϕ = (Φ/Φ 0 ) 1 r , where Φ 0 represents the flux quanta. In the presence of the external magnetic flux, the equation for |φ edge m (r, ϕ) reads
2B( 1 r ∂ r r∂ r − (m − Φ/Φ 0 ) 2 − i2(m − Φ/Φ 0 )∂ ϕ − ∂ 2 ϕ r 2 ) + m z σ z − i2A 2 σ ϕ ∂ r − 2A 1 r σ r (m − Φ Φ 0 − i∂ ϕ ) |φ edge m (r, ϕ)
= E m |φ edge m (r, ϕ) . (S39)
From the above equation one can find that the energy spectrum of edge states is shifted to be 2A1 Rm (m + 1/2 − Φ/Φ 0 ). Therefore, when an additional magnetic 1/2-flux-quanta Φ = Φ 0 /2 is threaded through the sample, the edge state |φ edge 0 (r, ϕ) is exactly pushed to zero energy, and the number of edge states becomes N = odd. The change in the number of edge modes between even and odd by threading a half flux-quanta provides an observable for the 1/2-fractionalization of orbital angular momentum.
FIG. 2 :
2Angle-dependence of tunneling conductance.
FIG. 3 :
3Magnetoresistance for the setup in Fig. 2a by setting magnetization of the ferromagnetic lead along ±y directions.
FIG. 4 :
4Spin filtering effect and output spin-polarized current. a, For a quantum anomalous Hall insulator with circular boundary, the edge spin-polarization depends on the direction of the 1D edge. This provides a controllable way to generate spin-polarized current by attaching normal-metal leads to different directions of the sample edge. b-e, The polarization ratio pout of the output spin current is plotted as a function of voltage eV in a drain lead and the Zeeman term mz (b,c) or B (d,e), with the magnitudes rescaled by the spin-orbit coefficient A (= |A1,2|). Other parameters are taken as
FIG. S1 :
S1Edge spin polarizations for the quantum anomalous Hall insulator with different boundary geometries. (a) The open boundary which has two edges normal to n2 direction, and is infinite (or periodic) along n1 axis; (b) The circular closed boundary which is rotationally invariant; (c) The generic boundary.
m
(r, ϕ) and |φ (−) −m (r, −ϕ) have opposite energies E m and −E m , respectively. For edge states we have opposite energies E m and −E m for |φ edge m (r, ϕ) and |φ edge −m (r, −ϕ) , respectively. On the other hand, generically we have E m = E −m and E m = −E −m (in Eqs.
2 ) R are eigenstates of σ n2 , with the spin oppositely polarized in the opposite edges[21].
FIG. S2 :
S2Effect of nonmagnetic disorder scattering on the edge spin texture. The parameters for the numerical calculation are rescaled to be dimensionless and are taken as A1 = A2 = −B = 1.0 and mz = 0.3. (a-f) The edge spin configuration with different magnitudes of the nonmagnetic disorder potential V0. The colors represent the wave-function distribution of the edge states; (g) Root Mean Square for Sx, Sy and Sz versus V0.
S3: Effect of magnetic disorder scattering on the edge spin texture. The parameters for the numerical calculation are the same as those in Fig. S2 A1 = A2 = −B = 1.0 and mz = 0.3. (a-f) The edge spin configuration with different magnitudes of the total magnetic disorder potential V0 [= (V 2
[ 1 ]
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AcknowledgementWe appreciate the helpful discussions with K. T. Law, Yayu Wang, J. Sinova, Z. -X. Liu, M. Cheng, T. K. Ng, and Y. Zhou. The authors thank the support of HKRGC through Grant 605512 and through Grant HKUST3/CRF09.Author ContributionsAll authors contributed to the studies of this work, and contributed to the writing of the manuscript. X.J.L. has planed this project.Additional InformationThe authors declare no competing financial interests. Supplementary Information accompanies this paper. Materials and correspondence can be addressed to X.J.Liu. | []
|
[
"THE EFFECTIVE MASS PROBLEM FOR THE LANDAU-PEKAR EQUATIONS",
"THE EFFECTIVE MASS PROBLEM FOR THE LANDAU-PEKAR EQUATIONS"
]
| [
"Dario Feliciangeli ",
"Simone Rademacher ",
"Robert Seiringer "
]
| []
| []
| We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar[15]. | 10.1088/1751-8121/ac3947 | [
"https://arxiv.org/pdf/2107.03720v2.pdf"
]
| 235,765,686 | 2107.03720 | ffb827925f8ac97078d3bca12462d56037356a3b |
THE EFFECTIVE MASS PROBLEM FOR THE LANDAU-PEKAR EQUATIONS
25 Jan 2022
Dario Feliciangeli
Simone Rademacher
Robert Seiringer
THE EFFECTIVE MASS PROBLEM FOR THE LANDAU-PEKAR EQUATIONS
25 Jan 2022arXiv:2107.03720v2 [math-ph]
We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar[15].
INTRODUCTION
The polaron is a model of an electron interacting with its self-induced polarization field of the underlying crystal. The description of the polarization as a quantum field corresponds to the Fröhlich model [8]. In the classical approximation, on the other hand, the dynamics of a polaron is described by the Landau-Pekar (LP) equations [14,24,15]. For (ψ t , ϕ t ) ∈ H 1 (R 3 ) × L 2 (R 3 ), where ψ t is the electron wave function and ϕ t denotes the phonon field, these equations read in suitable units (see [8] or [1])
i∂ t ψ t = h ϕt ψ t , iα 2 ∂ t ϕ t = ϕ t + σ ψt ,(1.1)
where h ϕ is the Schrödinger operator
h ϕ = −∆ + V ϕ (1.2)
with potential V ϕ (x) = 2 Re [(−∆) −1/2 ϕ](x) = π −2 |x| −2 * Re ϕ, (1.3) and
σ ψ (x) = (−∆) −1/2 |ψ| 2 (x) = (2π 2 ) −1 |x| −2 * |ψ| 2 , (1.4) where * denotes convolution. The parameter α > 0 quantifies the strength of the coupling of the electron's charge to the polarization field.
Despite its long history, the polaron model continues being actively investigated. For recent experimental and numerical work, we refer to [5,21,25,27,28] and references there.
The LP equations can be derived from the dynamics generated by the (quantum) Fröhlich Hamiltonian for suitable initial states in the strong coupling limit α → ∞ [17] (see also [6,7,13,18,22] for earlier results on this problem). One of the outstanding open problems concerns the polaron's effective mass [19,26,29]: due to the interaction with the polarization field, the electron effectively becomes heavier and behaves like a particle with a larger mass. This mass increases with the coupling α, and is expected to diverge as α 4 as α → ∞. A precise asymptotic formula was obtained by Landau and Pekar [15] based on the classical approximation, and hence it is natural to ask to what extent the derivation of the LP equations in [17] allows to draw conclusions on the effective mass problem.
It is, however, far from obvious how to rigorously obtain the effective mass even on the classical level, i.e., from the LP equations (1.1). A heuristic derivation, reviewed in Section Date: January 26, 2022. 1 4.1 below, considers traveling wave solutions of (1.1) for non-zero velocity v ∈ R 3 , and expands the corresponding energy for small v. The existence of such solutions remains unclear, however, and we in fact conjecture that no such solutions exist for non-zero v. This is related to the fact the energy functional corresponding to (1.1) (given in Eq. (2.1) below) does not dominate the total momentum, and a computation of the ground state energy as a function of the (conserved) total momentum would simply yield a constant function (corresponding to an infinite effective mass). Due to the vanishing of the sound velocity in the medium, a moving electron can be expected to be slowed down to zero speed by emitting radiation. (See [9,10,2,11,12] for the study of a similar effect in a model of a classical particle coupled to a field.)
In this paper, we provide a novel definition of the effective mass for the LP equations. We shall argue that all low energy states have a well-defined notion of (initial) velocity, and hence we can minimize the energy functional among states with given velocity. Expanding the resulting energy-velocity relation for small velocity gives a definition of the effective mass, which coincides with the prediction by Landau and Pekar [15].
1.1. Structure of the paper. In Section 2, we explain our rigorous approach to derive the energy-velocity relation of the system, allowing for a precise definition and computation of the effective mass. After introducing some notation and recalling fundamental properties of the Pekar energy functional in Section 2.1, we identify in Section 2.2 a set of initial data for the LP equations for which it is possible to define the position, and consequently the velocity, at any time. We then arrive at an energy-velocity relation by defining E(v) in Section 2.3 as the minimal energy among all admissible initial states of fixed initial velocity v. Finally, in Section 2.4 we state our main result, an expansion of E(v) for small velocities v, allowing for the computation the effective mass of the system. Section 3 contains the proof of our main result, Theorem 2.1.
In Section 4 we discuss the formal approach to the effective mass via traveling waves. Moreover, we investigate an alternative definition of the effective mass, through an alternative notion of velocity of low-energy states.
MAIN RESULTS
2.1. Preliminaries. We start by introducing further notation and recalling some known results. The classical energy functional corresponding to the Landau-Pekar equations (1.1)
is defined on H 1 (R 3 ) × L 2 (R 3 ) as G(ψ, ϕ) = ψ, h ϕ ψ + ϕ 2 2 for ψ 2 = 1. (2.1)
Equipped with the symplectic form 1 2i´d ψ ∧ dψ + α 2 2i´d ϕ ∧ dφ, it defines a dynamical system leading to the LP equations (1.1). Moreover, G is conserved along solutions of (1.1).
Let e P denote the Pekar ground state energy e P := min G(ψ, ϕ) .
(2.2)
(For an estimation of its numerical value, see [23].) It was proved in [20] that the minimum in 2.2 is attained for the Pekar minimizers (ψ P , ϕ P ), which are radial smooth functions in C ∞ (R 3 ) satisfying ψ P > 0, ϕ P = −σ ψ P and ψ P = ψ ϕ P , where ψ ϕ denotes the ground state of h ϕ whenever it exists. Moreover, this minimizer is unique up to the symmetries of the problem, i.e., translation-invariance and multiplication of ψ by a phase. We shall denote
H P = h ϕ P − µ P with µ P = inf spec h ϕ P . (2.3)
Associated to G, there are the two functionals
E(ψ) := inf ϕ∈L 2 (R 3 ) G(ψ, ϕ) =ˆR 3 |∇ψ(x)| 2 dx − 1 4πˆR6 |ψ(x)| 2 |ψ(y)| 2 |x − y| dx dy (2.4) and F (ϕ) := inf ψ∈H 1 (R 3 ) ψ 2 =1 G(ψ, ϕ) = inf spec h ϕ + ϕ 2 2 (2.5)
and clearly e P = min G(ψ, ϕ) = min E(ψ) = min F (ϕ). We also define the manifolds of minimizers
M G := {(ψ, ϕ) | G(ψ, ϕ) = e P }, M E := {ψ | E(ψ) = e P } , M F := {ϕ | F (ϕ) = e P } . (2.6)
The results in [20] imply that we can write these in terms of the Pekar minimizers (ψ P , ϕ P ) as
M G = {(e iθ ψ y P , ϕ y P ) | θ ∈ [0, 2π), y ∈ R 3 }, M E = {e iθ ψ y P | θ ∈ [0, 2π), y ∈ R 3 }, M F = {ϕ y P | y ∈ R 3 } (2.7)
where f y := f ( · − y) for any function f . Furthermore, it can be deduced from the results in [16] that the energy functionals F and E are both coercive (see [3, Lemmas 2.6 and 2.7]), i.e., there exists C > 0 such that
F (ϕ) ≥ e P + C dist 2 L 2 (ϕ, M F ), E(ψ) ≥ e P + C dist 2 H 1 (ψ, M E ). (2.8)
The following Lemma on properties of the projection onto the manifold M F will be important for our analysis below. Its proof will be given in Appendix A.
Lemma 2.1. There exists δ > 0 such that the L 2 -projection onto M F , is well-defined (i.e., unique) on
(M F ) δ := {ϕ ∈ L 2 (R 3 ) | dist L 2 (ϕ, M F ) ≤ δ} .
(2.9)
For any ϕ ∈ (M F ) δ , we define z ϕ ∈ R 3 via
P M F L 2 (ϕ) = ϕ zϕ P . (2.10)
Then z ϕ is a differentiable function from (M F ) δ to R 3 and its partial derivative in the direction η ∈ L 2 (R 3 ) is given by
∂ t z ϕ+tη ↾ t=0 = A −1 ϕ Re η|∇ϕ zϕ P ,(2.
11)
where A is the invertible matrix defined for any ϕ ∈ (M F ) δ by A i,j := − Re ϕ|∂ i ∂ j ϕ zϕ P . Remark 2.1. Likewise, it can be shown that the H 1 -(resp. L 2 -) projection onto M E have similar properties: There exists δ > 0 such that the H 1 -(resp. the L 2 -) projection onto M E
P M E H 1 (ψ) = e iθ ψ ψ y ψ P , resp. P M E L 2 (ψ) = e iθ ′ ψ ψ y ′ ψ P (2.12) is well-defined on the set (M E ) H 1 δ := {ψ ∈ L 2 (R 3 ) | dist H 1 (ψ, M E ) ≤ δ} (resp. (M E ) L 2 δ := {ψ ∈ L 2 (R 3 ) | dist L 2 (ψ, M E )
≤ δ}) and the functions y ψ , θ ψ (resp. y ′ ψ , θ ′ ψ ) defined through (2.12) are differentiable functions from (M E ) H 1 δ (resp. (M E ) L 2 δ ) to R 3 and R/(2πZ).
Position and velocity of solutions.
In this section, we give a meaning to the notion of position, and therefore velocity, for solutions of the Landau-Pekar equations (at least for a class of initial data). There is a natural way of defining, given ψ t , the position of the electron at time t, which is simply given by X el (t) := ψ t |x|ψ t .
(2.13)
This yields, by straightforward computations using (1.1), that
V el (t) := d dt X el (t) = 2 ψ t |−i∇|ψ t .
(2.14)
Note that (2.14) is always well-defined for ψ ∈ H 1 (R 3 ), even although (2.13) not necessarily is.
For the phonon field, the situation is more complicated as ϕ cannot be interpreted as a probability distribution over positions. This calls for a different approach. By (2.8), Lemma 2.1 and the conservation of G along solutions of (1.1), we know that there exists δ * such that for any initial condition (ψ 0 , ϕ 0 ) such that
G(ψ 0 , ϕ 0 ) ≤ e P + δ * , (2.15) ϕ t admits a unique L 2 -projection ϕ z(t) P onto M F for all times. We use this to define X ph (t) := z(t), V ph := d dt X ph (t) =ż(t). (2.16)
Note that X ph (t) is indeed differentiable by Lemma 2.1 and the differentiability of the LP dynamics. At this point, for any initial data satisfying (2.15), we have a well-defined notion of position and velocity for all times, admittedly in a much less explicit form for the phonon field.
2.3. Initial conditions of velocity v. For any v ∈ R 3 (or at least for |v| sufficiently small), we are now interested in considering all initial conditions (ψ 0 , ϕ 0 ) whose solutions have instantaneous velocity v at t = 0 (both in the electron and in the phonon coordinate) and to then minimize the functional G over such states. This will give us an explicit relation between the energy and the velocity of the system, allowing us to define the effective mass of the polaron in the classical setting defined by the Landau-Pekar equations. Note that by radial symmetry of the problem only the absolute value of the velocity, and not its direction, affects our analysis. Hence, for v ∈ R, we consider initial conditions
(ψ 0 , ϕ 0 ) such that (i) (ψ 0 , ϕ 0 ) ∈ H 1 (R 3 ) × L 2 (R 3 ) with ψ 0 2 = 1 and such that (2.15) is satisfied, (ii) V el (0) = V ph (0) = v(1, 0, 0).
The set of admissible initial conditions of velocity v ∈ R can hence be compactly written as
I v := {(ψ 0 , ϕ 0 ) | (i), (ii) are satisfied}.
(2.17)
We will show below that it is non-empty for small enough v.
2.4.
Expansion of the energy. In order to compute the effective mass of the polaron, we now minimize the energy G over the set I v . To this end, we define the energy
E(v) := inf (ψ 0 ,ϕ 0 )∈Iv G(ψ 0 , ϕ 0 ). (2.18)
The following theorem gives an expansion of E(v) for sufficiently small velocities v. Its proof will be given in Section 3.
Theorem 2.1. As v → 0 we have E(v) = e P + v 2 1 4 + α 4 3 ∇ϕ P 2 2 + O(v 3 ). (2.19)
Since the kinetic energy of a particle of mass m and velocity v equals mv 2 /2, (2.19) identifies the effective mass of the system as
m eff = lim v→0 E(v) − e P v 2 /2 = 1 2 + 2α 4 3 ∇ϕ P 2 2 .
(2.20)
The first term 1/2 is simply the bare mass of the electron in our units, while the second term 2α 4 3 ∇ϕ P 2 2 corresponds to the additional mass acquired through the interaction with the phonon field. It agrees with the prediction in [15], and is conjectured to coincide with the effective mass in the Fröhlich model in the limit α → ∞. Note that since (−∆) 1/2 ϕ P = −|ψ P | 2 , ∇ϕ P 2 = ψ P 2 4 , which can be evaluated numerically [23].
Remark 2.2 (Traveling waves). The heuristic computations contained in the physics literature concerning m eff [1,15] all rely, in one way or another, on the existence of traveling wave solutions of the LP equations of velocity v (at least for sufficiently small velocity), i.e. solutions with initial data (ψ v , ϕ v ) such that
(ψ t (x), ϕ t (x)) = (e −ievt ψ v (x − vt), ϕ v (x − vt)) (2.21)
for suitable e v ∈ R. Such solutions would allow to define the energy of the system at velocity v as E TW (v) = G(ψ v , ϕ v ), and a perturbative calculation (discussed in Section 4.1 below) yields indeed
lim v→0 E TW (v) − e P v 2 /2 = 1 2 + 2α 4 3 ∇ϕ P 2 ,(2.22)
in agreement with (2.20). Unfortunately, this approach turns out to be only formal, and we conjecture traveling wave solutions to not exist for any α > 0, v > 0, as explained in more detail in Section 4.1.
Remark 2.3. In Section 2.2, we used the standard approach from quantum mechanics to define the electron's position (2.13) and velocity (2.14). We could, instead, use also for the electron a similar approach to the one we use for the phonon field (i.e. (2.16)) through the projection onto the manifold of minimizers M E . A natural question is whether one obtains the same effective mass using this different notion of position. In Section 4. [15]), but differs in the O(1) term. In fact, as we discuss in Section 4.2, one has m eff < m eff .
PROOF OF THEOREM 2.1
Let us denote δ 1 = ψ 0 − ψ P and δ 2 = ϕ 0 − ϕ P . Expanding G in (2.1) and using that ϕ P = −σ ψ P we find
G(ψ 0 , ϕ 0 ) = G(ψ P + δ 1 , ϕ P + δ 2 ) = e P + 2 ψ P | h ϕ P |Re δ 1 + δ 1 |h ϕ P |δ 1 + 2 Re δ 1 | V δ 2 |ψ P + δ 2 2 2 + δ 1 |V δ 2 |δ 1 . (3.1)
Since ψ 0 is normalized, we have
1 = ψ 0 2 2 = ψ P + δ 1 2 2 = 1 + δ 1 2 2 + 2 ψ P |Re δ 1 ⇐⇒ 2 ψ P |Re δ 1 = − δ 1 2 2 . (3.2) Hence 2 ψ P | h ϕ P |Re δ 1 = 2µ P ψ P |Re δ 1 = −µ P δ 1 2 2 , (3.3) and using V δ 2 δ 1 2 ≤ C δ 2 2 δ 1 H 1 (see, e.g., [18, Lemma III.2]) we arrive at G(ψ 0 , ϕ 0 ) = e P + δ 1 |H P |δ 1 + 2 Re δ 1 | V δ 2 |ψ P + δ 2 2 2 + O( δ 2 2 δ 1 2 H 1 ). (3.4) By completing the square, we have Re δ 2 2 2 + 2 Re δ 1 | V δ 2 |ψ P = Re δ 2 + 2(−∆) −1/2 (ψ P Re δ 1 ) 2 2 − 4 Re δ 1 |ψ P (−∆) −1 ψ P |Re δ 1 (3.5)
and therefore G(ψ 0 , ϕ 0 ) = e P + Im ψ 0 |H P |Im ψ 0 + Im ϕ 0 2 2
+ Re δ 2 + 2(−∆) −1/2 (ψ P Re δ 1 ) 2 2 + Re δ 1 |H P − 4X P |Re δ 1 + O( δ 2 2 δ 1 2 H 1 ),(3.6)
where X P is the operator with integral kernel X P (x, y) := ψ P (x)(−∆) −1 (x, y)ψ P (y). Since X P is bounded, and P ψ P Re δ 1 = δ 1 2 2 /2 by (3.2) (with P ψ P = |ψ P ψ P | the rank one projection onto ψ P ), we also have G(ψ 0 , ϕ 0 ) = e P + Im ψ 0 |H P |Im ψ 0 + Im ϕ 0 2 2
+ Re δ 2 + 2(−∆) −1/2 (ψ P Re δ 1 ) 2 2 + Re δ 1 |Q(H P − 4X)Q|Re δ 1 + O( δ 2 2 δ 1 2 H 1 ) + O( δ 1 3 L 2 ), (3.7) where Q = 1 1 − P ψ P .
Upper Bound: For sufficiently small v, we use as a trial state
(ψ 0 ,φ 0 ) = f v ψ P + ig v H −1 P ∂ 1 ψ P , ϕ P + ivα 2 ∂ 1 ϕ P (3.8) with f v , g v > 0 given by f 2 v := 2v 2 H −1 P ∂ 1 ψ P 2 2 1 − 1 − 4v 2 H −1 P ∂ 1 ψ P 2 2 , g 2 v := 1 − 1 − 4v 2 H −1 P ∂ 1 ψ P 2 2 2 H −1 P ∂ 1 ψ P 2 2 .
(3.9)
Note that ∂ 1 ψ P is orthogonal to ψ P , hence H −1 P ∂ 1 ψ P is well-defined. We begin by showing that (3.8) is an element of the set of admissible initial data I v in (2.17). To prove that (ψ 0 ,φ 0 ) satisfies (i), we only need to check thatψ 0 is normalized (which follows easily from (3.9)) as the condition (2.15) will follow a posteriori from the energy bound we shall derive. We now proceed to show that (ψ 0 ,φ 0 ) satisfies (ii). For the electron, using that H −1 P ∂ j ψ P = −x j ψ P /2 (which can be checked by applying H P and using that [H P , x 1 ] = −2∂ 1 ) and consequently that
∂ i ψ P | H −1 P |∂ j ψ P = δ ij /4 (3.10)
since ψ P is radial, we can conclude that
−2 ψ 0 i∂ j ψ 0 = 4f v g v H −1 P ∂ 1 ψ P |∂ j ψ P = vδ j1 ,(3.11)
i.e., that V el (0) = v (1, 0, 0), as required.
For the phonons, we first note that X ph (0) = 0, since Reφ 0 = ϕ P . Next, we derive a relation for the velocity of the phonons V ph (t) =ż(t) in terms of their position X ph (t) = z(t) for general time t. Since
min z ϕ t − ϕ z P 2 2 = ϕ t − ϕ z(t) P 2 2 ,(3.12)
the position z(t) necessarily has to satisfy
Re ϕ t (u · ∇)ϕ z(t) P = 0 for all u ∈ S 2 ⇐⇒ Re ϕ t ⊥ span{∇ϕ z(t) P }. (3.13)
Differentiating (3.13) w.r.t. time, using (1.1) and evaluating the resulting expression at t = 0, we arrive at
0 = Re −iα −2 (u · ∇)(φ 0 + σψ 0 ) ϕ P − Re (ż(0) · ∇)φ 0 |(u · ∇)ϕ P = −α −2 Imφ 0 (u · ∇)ϕ P − (ż(0) · ∇) Reφ 0 |(u · ∇)ϕ P = − v∂ 1 ϕ P |(u · ∇)ϕ P − (ż(0) · ∇)ϕ P |(u · ∇)ϕ P ,(3.14)
which the velocityż(0) has to satisfy for all u ∈ S 2 , given its position X ph (0) = z(0) = 0. By invertibility of the coefficient matrix, (3.14) has the unique solutionż(0) = v(1, 0, 0), and we indeed conclude that V ph (0) = v (1, 0, 0). We now evaluate G(ψ 0 ,φ 0 ).
Since f v = 1 + O(v 2 ), g v = v + O(v 3 )
, using (3.7) and (3.10) we find
E(v) ≤ G(ψ 0 ,φ 0 ) = e P + v 2 1 4 + α 4 ∂ 1 ϕ P 2 2 + O(v 3 ) (3.15)
verifying on the one hand (2.15) for sufficiently small v, and on the other hand the r.h.s. of (2.19) as an upper bound on E(v) (using that ϕ P is radial). Lower Bound: We first observe that to derive a lower bound on E(v) we can w.l.o.g. restrict to initial conditions (ψ 0 , ϕ 0 ) satisfying additionally
P M E L 2 (ψ 0 ) > 0,(3.16)
X ph (0) = 0.
I v such that dist H 1 (ψ 0 , M E ) = O(v) = dist L 2 (ϕ 0 , M F ) for small v.
Since the L 2 -projection of ϕ 0 is ϕ P by (3.17), it immediately follows that δ 2 2 = O(v). We now proceed to show that necessarily also δ 1 H 1 = O(v). Let y ′ , y ∈ R 3 and θ ∈ [0, 2π) be such that
P M E L 2 (ψ 0 ) = ψ y ′ P , P M E H 1 (ψ 0 ) = e iθ ψ y P ,(3.18)
where we recall that the L 2 -projection is assumed to be positive by (3.16). Combining the upper bound derived in the first step with [3, Eq. (53)], we get
ϕ 0 − ϕ y P 2 2 ≤ C (G(ψ 0 , ϕ 0 ) − e P ) ≤ Cv 2 . (3.19)
There exist δ, C 1 , C 2 > 0 such that (3.20) and this allows to conclude that |y| = O(v). In other words, assuming centering w.r.t. to translations in the phonon coordinate (i.e. (3.17)) forces, at low energies, also the centering w.r.t. translations in the electron coordinate, at least approximately. At this point, it is also easy to verify that θ = O(v) (and, as an aside, that |y ′ | = O(v)), since, by the upper bound derived in the first step and the coercivity of E, we have
ϕ P − ϕ y P 2 ≥ C 1 |y| ∇ϕ P 2 , |y| ≤ δ C 2 |y| > δ ,ψ y ′ P − e iθ ψ y P 2 ≤ ψ y ′ P − ψ 0 2 + e iθ ψ y P − ψ 0 2 = O(v). (3.21)
In particular, we conclude that
δ 1 H 1 ≤ e iθ ψ y P − ψ 0 H 1 + ψ P − e iθ ψ y P H 1 = O(v). (3.22)
Using again (3.7) and that Q(H P − 4X P )Q ≥ 0, we conclude that for any (ψ 0 , ϕ 0 ) ∈ I v satisfying (3.16) and (3.17), as well as G(ψ 0 , ϕ 0 ) ≤ e P + O(v 2 ), we have G(ψ 0 , ϕ 0 ) ≥ e P + Im ψ 0 |H P |Im ψ 0 + Im ϕ 0
2 2 + O(v 3 ).
(3.23)
By arguing as in (3.14), we see that the conditions X ph (0) = 0 and V ph (0) = v imply that
P ∇ϕ P (Im ϕ 0 + vα 2 ∂ 1 Re ϕ 0 ) = 0,(3.24)
where P ∇ϕ P denotes the projection onto the span of ∂ j ϕ P , 1 ≤ j ≤ 3. Since P ∇ϕ P ∂ 1 is a bounded operator, and δ 2 2 = O(v), we find Completing the square, we find
Im ϕ 0 2 2 ≥ P ∇ϕ P Im ϕ 0 2 2 = v 2 α 4 ∂ 1 ϕ P + P ∇ϕ P ∂ 1 Re δ 2 2 2 ≥ v 2 α 4 ∂ 1 ϕ P 2 2 − O(v 3 ).Im ψ 0 |H P |Im ψ 0 = H P Im ψ 0 − v∂ 1 ψ P |H −1 P |H P Im ψ 0 − v∂ 1 ψ P + 2v Im ψ 0 |∂ 1 ψ P − v 2 ∂ 1 ψ P |H −1 P |∂ 1 ψ P ≥ 2v Im ψ 0 |∂ 1 ψ P − v 2 ∂ 1 ψ P |H −1 P |∂ 1 ψ P . (3.28)
From the constraint (3.27) and (3.10), it thus follows that
Im ψ 0 |H P |Im ψ 0 ≥ v 2 4 + O(v 3 ). (3.29)
By combining (3.23), (3.25) and (3.29), we arrive at the final lower bound
E(v) ≥ e P + v 2 1 4 + α 4 ∂ 1 ϕ P 2 2 + O(v 3 ). (3.30)
Again, since ϕ P is radial, this is of the desired form, and hence the proof is complete.
FURTHER CONSIDERATIONS
In this section, we carry out the details related to Remarks 2.2 and 2.3.
4.1.
Effective mass through traveling wave solutions. A possible way of formalizing the derivation of the effective mass in [1,15] relies on traveling wave solutions of the Landau-Pekar equations. A traveling wave of velocity v ∈ R 3 is a solution (ψ t , ϕ t ) of (1.1) of the form
(ψ t , ϕ t ) = (e −ievt ψ TW v (· − vt), ϕ TW v (· − vt)) (4.1)
for all t ∈ R, with e v ∈ R defining a suitable phase factor. As before, by rotation invariance we can restrict our attention to velocities of the form v(1, 0, 0) with v ∈ R in the following. Note that in the case α = 0, where ϕ t = −σ ψt for all t ∈ R, the LP equations simplify to a non-linear Schrödinger equation (also known as Choquard equation). In this case, a traveling wave is given by
ψ TW v (x) = e ix 1 v/2 ψ P (x) with e v = µ P + v 2 4
, and its energy can be computed to be
G ψ TW v , −σ ψ TW v = e P + v 2 4 , (4.2)
yielding an effective mass m = 1/2 at α = 0. For the case α > 0, on the other hand, we conjecture that there are no traveling wave solutions of the form (4.1). The motivation for this conjecture comes from the vanishing of the sound velocity in the medium. An initially moving electron can be expected to be slowed down to zero speed by emitting radiation. Establishing this effect rigorously for the LP equations remains an open problem, however.
If one assumes the existence of traveling wave solutions, at least for small v, one can predict an effective mass that agrees with our formula (2.20), as we shall now demonstrate. From the LP equations (1.1) one easily sees that a traveling wave solution needs to satisfy
−iv∂ 1 ψ TW v = h ϕ TW v + e v ψ TW v −iα 2 v∂ 1 ϕ TW v = ϕ TW v + σ ψ TW v . (4.3)
We shall denote by E TW (v) the energy of the traveling wave as a function of the velocity v ∈ R, i.e.
E TW (v) := G(ψ TW v , ϕ TW v ). (4.4)
In the following, we assume that e v = µ P + O(v 2 ) and that the traveling wave is of the form
(ψ TW v , ϕ TW v ) = ψ P + vξ v ψ P + vξ v 2 , ϕ P + vη v ,(4.5)
with both ξ v and η v bounded in v and converging to some (ξ, η) as v → 0. In other words, we assume that the traveling waves have a suitable differentiability in v, at least for small v, and converge to the standing wave solution (e −iµ P t ψ P , ϕ P ) for v = 0. W.l.o.g. we may also assume that ξ v is orthogonal to ψ P . We can then use that
1 ψ P + vξ v 2 2 = 1 − v 2 ξ v 2 2 ψ P + vξ v 2 2 = 1 − v 2 ξ 2 2 + o(v 2 ) (4.6)
in order to linearize the traveling wave equations (4.1), obtaining that (ξ, η) solves
i∂ 1 ψ P iα 2 ∂ 1 ϕ P = H P 2ψ P (−∆) −1/2 Re 2(−∆) −1/2 ψ P Re 1 ξ η , (4.7)
where H P = h ϕ P − µ P , as defined in (2.3). Splitting into real and imaginary parts, we equivalently find
H P Im ξ = ∂ 1 ψ P (4.8) Im η = α 2 ∂ 1 ϕ P (4.9)
H P Re ξ + 2ψ P (−∆) −1/2 Re η = 0 (4.10)
2 (−∆) −1/2 ψ P Re ξ + Re η = 0. Combining (4.10) and (4.11) gives (H P − 4X P ) Re ξ = 0, with X P defined after (3.6).
It was shown in [16] that the kernel of H P − 4X P is spanned by ∇ψ P , hence Re ξ ∈ span{∇ψ P }. Eq, (4.11) then implies that Re η ∈ span{∇ϕ P }.
Using these equations and (4.6) in the expansion (3.7), it is straightforward to obtain
E TW (v) = e P + v 2 1 4 + α 4 ∂ 1 ϕ P 2 2 + o(v 2 ),(4.12)
which agrees with (4.2) for the case α = 0, and also with (2.19) to leading order in v. In particular, (2.22) holds.
4.2.
Effective mass with alternative definition for the electron's velocity. In this Section, we discuss a different approach to the definition of the effective mass. This approach is based on an alternative way of defining the electron's position and velocity. While in Section 2.2 we use the standard definition from quantum mechanics, here we use a definition similar to the one of the phonons' position and velocity (2.16). For this purpose, we recall Remark 2.1 and that δ * has been chosen such that the condition E(ψ 0 ) ≤ G(ψ 0 , ϕ 0 ) ≤ e P + δ * ensures that for all times there exists a unique L 2 -projection e iθ(t) ψ y(t) P of ψ t onto the manifold M E . Then, we define the electron's position and velocity by X el (t) = y(t), V el (t) =ẏ(t).
(4.13)
Similarly to the conditions (i) and (ii) in Section 2.2, we define the set of admissible initial data as
I v = {(ψ 0 , ϕ 0 ) | (i),(ii') are satisfied} (4.14) where (ii') V el (0) = V ph (0) = v(1, 0, 0).
Note that we are leaving the parameterθ(0) free, which in this case is also relevant. In other words, we have
I v = ∪ κ∈R I v,κ ,(4.15)
where leads to an energy expansion in v that differs from the one of Theorem 2.1 in its second order.
I v,κ = {(ψ 0 , ϕ 0 ) | (i),(ii') are satisfied andθ(0) = κ}.Proposition 4.1. As v → 0, we have E(v) = e P + v 2 ∇ψ P 4 2 3 ∇ϕ P 2 2 + α 4 3 ∇ϕ P 2 2 + O(v 3 ). (4.18)
The energy expansion in (4.18) leads to the effective mass
m eff = lim v→0 E(v) − e P v 2 /2 = 2 ∇ψ P 4 2 3 ∇ϕ P 2 2 + 2α 4 3 ∇ϕ P 2 2 (4.19)
which agrees with (2.20) and (2.22) in leading order for large α only (and thus still confirms the Landau-Pekar prediction [15]), but differs in the O(1) term. In fact, it turns out that m eff < m eff with m eff defined in (2.22). This follows from the observation that the trial state
( ψ 0 , ϕ 0 ) = f v ψ P + ivH −1 P ∂ 1 ψ P f v ψ P + ivH −1 P ∂ 1 ψ P , ϕ P + ivα 2 ∂ 1 ϕ P (4.20) with f v = 1 2 1 + 1 − v 2 /(4 ∂ 1 ψ P 2
2 ) (which coincides up to terms of order v 2 with the trial state (3.8)) is an element of I v,κ forκ = −µ P + 4 ∂ 1 ψ P 2 2 (f v − 1) and is such that
G( ψ 0 , ϕ 0 ) = e P + m eff v 2 /2 + O(v 3 )
. Thus, m eff ≤ m eff and equality holds if and only if equality (up to terms o(v 2 )) holds in (4.36). This is the case if and only if i.e. , recalling the radiality of ψ P , if and only if ψ P is a Gaussian with variance σ 2 = 1/(2c).
Q ψ P Im ψ 0 − cv∂ 1 ψ P = o(v).
Since ψ P satisfies the Euler-Lagrange equation
H P ψ P = 0 ⇐⇒ V ϕ P ψ P = (−∆ + µ P )ψ P ,(4.23)
it cannot be a Gaussian and thereforem eff < m eff . We present only a sketch of the proof of Proposition 4.1, since it uses very similar arguments as the proof of Theorem 2.1.
Sketch of Proof of Proposition 4.1. Upper bound:
We use the alternative trial state
( ψ 0 , ϕ 0 ) = f v ψ P + ivc∂ 1 ψ P f v ψ P + ivc∂ 1 ψ P , ϕ P + ivα 2 ∂ 1 ϕ P ,(4.24)
with
f v := 1 + 1 + 4c 2 v 2 ∂ 1 ψ P 2 2 , c := ∂ 1 ψ P 2 ∂ 1 ϕ P 2 . (4.25)
With similar arguments as in the previous section, one can verify that ( ψ 0 , ϕ 0 ) ∈ I v , in
particular ( ψ 0 , ϕ 0 ) ∈ I v,κ with κ = −µ P + −1+ √ 1+4c 2 v 2 ∂ 1 ψ P 2 2c
. Note that, similarly to (3.14), one can derive necessary conditions for the velocitieṡ y(0),θ(0) (using X el (0) = 0, θ(0) = 0), namely [h Re ϕ 0 +θ(0)] Im ψ 0 −ẏ(0) · ∇ Re ψ 0 (u · ∇)ψ P = 0 for all u ∈ S 2 (4.26) and ψ P (h Re ϕ 0 +θ(0)) Re ψ 0 +ẏ(0) · ∇ Im ψ 0 = 0. (4.27)
Straightforward computations then show that
E(v) ≤ G( ψ 0 , ϕ 0 ) = e P + v 2 ∂ 1 ψ P 4 2 ∂ 1 ϕ P 2 2 + α 4 ∂ 1 ϕ P 2 2 + O(v 3 ). (4.28)
Lower bound: We proceed similarly to the lower bound in the previous section. First, we assume w.l.o.g. that
P M E L 2 (ψ 0 ) = ψ y(0) P , P M F L 2 (ϕ 0 ) = ϕ P ,(4.29)
i.e., centering with respect to translations and changes of phase. We can then substitute the two conditions of (ii') and the conditions for ψ y(0) P (resp. ϕ P ) to be the L 2 -projection of ψ 0 (resp. ϕ 0 ) onto M E (resp. M F ) with their analogue necessary conditions (whose computations proceed along the lines of (4.26) and (4.27)). With this discussion, we are left with the task of minimizing G over the set As in the previous section, one can argue by coercivity of E and F and the upper bound that it is possible to restrict to initial conditions such that δ 2 2 , δ 1 H 1 , y(0) are all O(v). Moreover, the second constraint of the r.h.s. of (4.31) shows that κ = −µ P + O(v). Thus, we are left with minimizing G over the set
I ′ v := κ∈R I ′ v,κ (4.30) with I ′ v,κ := (ψ 0 , ϕ 0 ) ∈ I * |P ∇ψ y(0) P [(h Re ϕ 0 + κ) Im ψ 0 − v∂ 1 Re ψ 0 ] = 0, P ψ y(0) P [(h Re ϕ 0 + κ) Re ψ 0 + v∂ 1 Im ψ 0 ] = 0, P ∇ϕ P (Im ϕ 0 − vα 2 ∂ 1 Re ϕ 0 ) = 0 .I ′′ v := I ′ v ∩ {κ + µ P = O(v), δ 1 H 1 = O(v), δ 2 2 = O(v)} . (4.33)
The lower bound is proven in the same way as before. But instead of the constraint (3.27), this time we need to minimize w.r.t.
P ∇ψ y(0) P [(h Re ϕ 0 + κ) Im ψ 0 − v∂ 1 Re ψ 0 ] = 0. (4.34)
Since κ + µ P , y 0 , δ 1 H 1 and δ 2 2 are all order v and ψ P ∈ C ∞ 0 (R 3 ) (and these facts also allow to infer that ψ y(0) P = ψ P + O(v)), the constraint (4.34) can be written as
∇ψ P |H P | Im ψ 0 = v ∂ 1 ψ P 2 2 (1, 0, 0) + O(v 2 ).
(4.35)
Denoting c = ∂ 1 ψ P 2 2 / ∂ 1 ϕ P 2 2 , we complete the square Im ψ 0 |H P |Im ψ 0 = Im ψ 0 − vc∂ 1 ψ P |H P |Im ψ 0 − vc∂ 1 ψ P + 2vc Im ψ 0 |H P |∂ 1 ψ P − c 2 v 2 ∂ 1 ψ P |H P |∂ 1 ψ P ≥ 2cv Im ψ 0 |H P ∂ 1 ψ P − c 2 v 2 ∂ 1 ψ P |H P |∂ 1 ψ P .
(4.36)
With the constraint (4.34) and ∂ i ψ P |H P |∂ j ψ P = δ i,j ∂ j ϕ P 2 2 , we arrive at (4.18).
CONCLUSIONS
While a rigorous determination of the effective mass of a polaron described by the Fröhlich model remains an outstanding open problem, we solve here the classical analog of this problem, where the polaron is described by the Landau-Pekar equations. Even though these equations are often invoked in heuristic derivations of the effective polaron mass, it is not at all obvious how to make such derivations rigorous since they rely, in one form or another, on the assumption of the existence of traveling waves. As argued above, the latter can not be expected to exist, however. We overcome this problem by introducing a novel variational principle, minimizing the Pekar energy functional over states of given initial velocity v, which can be defined in a natural way for all low-energy states. We hope that this novel point of view may in the future also shed some light on the corresponding problem for the Fröhlich polaron, in particular in view of the recent derivation [17] of the Landau-Pekar equations from the Fröhlich model in the strong coupling limit.
APPENDIX A. WELL-POSEDNESS AND REGULARITY OF THE PROJECTIONS ONTO M F
Similar arguments to the ones used in the following proof are contained in [4], where the functional F is investigated in the case of a torus in place of R 3 . Remark 2.1 on the properties M E can be shown with a similar approach, but we omit its proof.
Proof of Lemma 2.1. We need to prove that there exists δ > 0 such that for any ϕ ∈ (M F ) δ there exists a unique z ϕ identifying the projection of ϕ onto M F , and such that z ϕ is differentiable at any ϕ ∈ (M F ) δ . As the problem is invariant w.r.t. translations, we can w.l.o.g. restrict to show differentiability at ϕ 0 ∈ (M F ) δ such that z ϕ 0 = 0.
We define the function F : L 2 (R 3 ) × R 3 → R 3 given, component-wise, by By definition of z ϕ , we have F (ϕ 0 , 0) = 0 and F (ϕ, z ϕ ) = 0, for any ϕ in a sufficiently small neighborhood of ϕ 0 . Hence, we set out to use the implicit function theorem to determine properties of z ϕ . Observe that, for any η ∈ L 2 (R 3 ), z ∈ R 3 and i, j ∈ {1, 2, 3}, we have ∂ t F i (ϕ + tη, z) = Re η|∂ i ϕ z P and ∂ z j F i (ϕ, z) = − Re ϕ|∂ i ∂ j ϕ z P .
(A.2)
Since ϕ P ∈ C ∞ (R 3 ), the map (M F ) δ ∋ ϕ → det ∂F i ∂z j (ϕ, z)
i,j=1,...,3 is continuous w.r.t the L 2 -norm and, by radiality of ϕ P , det ∂F i ∂z j (ϕ P , 0) i,j=1,...,3
= 1 9 ∇ϕ P 2 2 > 0 . (A.3)
Thus, it follows that det ∂F i ∂z j (ϕ 0 , 0) i,j=1,...,3 > 0, uniformly in ϕ 0 for sufficiently small δ > 0. By the implicit function theorem, there exists a unique differentiable z ϕ : (M F ) δ → R 3 whose partial derivative in the direction η ∈ L 2 (R 3 ) at ϕ 0 is given by
∂ t z ϕ 0 +tη ↾ t=0 = ∂F i ∂z j (ϕ 0 , z ϕ 0 ) i,
follows from the invariance of G under translations (of both ψ and ϕ) and under changes of phase of ψ. Moreover, by the upper bound derived in the first step of this proof and the coercivity of E and F in (2.8), we conclude that it is sufficient to minimize over elements of
( 3 .
325) We are left with giving a lower bound on Im ψ 0 |H P |Im ψ 0 , under the condition that2 ψ 0 |−i∇|ψ 0 = 4 Im ψ 0 |∇ Re ψ 0 = v(1, 0, 0).(3.26) We already argued in (3.22) that ψ 0 − ψ P H 1 = O(v), and therefore 4 Im ψ 0 |∇ψ P = v(1, 0, 0) + O(v 2 ). (3.27)
Conjecture 4. 1 .
1For α > 0, there are no solutions to the LP equations (1.1) of the form (4.1) with v = 0.
the energy over all states of the set I v E(v) := inf (ψ 0 ,ϕ 0 )∈ Iv G(ψ 0 , ϕ 0 ), (4.17)
4.20), equality holds if and only if0 = H −1 P ∂ 1 ψ P − c∂ 1 ψ P = − (x 1 /2 + c∂ 1 ) ψ P (4.22)
I,
* := (ψ 0 , ϕ 0 ) | G(ψ 0 , ϕ 0 ) ≤ e P + δ * Re ϕ 0 ⊥ ∇ϕ P . (4.32)
F
i (ϕ, z) = Re ϕ|∂ i ϕ
No. 694227 (D.F. and R.S.) and under the Marie Skłodowska-Curie Grant Agreement No. 754411 (S.R.) is gratefully acknowledged.
2, we show that, in fact, this alternate definition yields a different effective mass equal tom eff =
2 ∇ψ P
4
2
3 ∇ϕ P 2
2
+
2α 4
3
∇ϕ P
2
2 .
(2.23)
This coincides with (2.20) and (2.22) for large α (hence still confirming the prediction in
Acknowledgments:We thank Herbert Spohn for helpful comments. Funding from the European Union's Horizon 2020 research and innovation programme under the ERC
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| []
|
[
"On the Simplicity of the Eigenvalues of the Non-self-adjoint Mathieu-Hill Operators",
"On the Simplicity of the Eigenvalues of the Non-self-adjoint Mathieu-Hill Operators"
]
| [
"O A Veliev [email protected] \nDepart. of Math\nDogus University\nAcıbadem, IstanbulKadiköyTurkey\n"
]
| [
"Depart. of Math\nDogus University\nAcıbadem, IstanbulKadiköyTurkey"
]
| []
| We find conditions on the potential of the non-self-adjoint Mathieu-Hill operator such that the all eigenvalues of the periodic, antiperiodic, Dirichlet and Neumann boundary value problems are simple. | null | [
"https://arxiv.org/pdf/1301.1011v2.pdf"
]
| 54,784,061 | 1301.1011 | e033f5a15ab2618a3bd5d77a045536c692fa7f4f |
On the Simplicity of the Eigenvalues of the Non-self-adjoint Mathieu-Hill Operators
9 Jan 2013
O A Veliev [email protected]
Depart. of Math
Dogus University
Acıbadem, IstanbulKadiköyTurkey
On the Simplicity of the Eigenvalues of the Non-self-adjoint Mathieu-Hill Operators
9 Jan 2013Mathieu-Hill operatorSimple eigenvalues AMS Mathematics Subject Classification: 34L0534L20
We find conditions on the potential of the non-self-adjoint Mathieu-Hill operator such that the all eigenvalues of the periodic, antiperiodic, Dirichlet and Neumann boundary value problems are simple.
Introduction and Preliminary Facts
Let P (q), A(q), D(q), N (q) be the operators in L 2 [0, π] associated with the equation − y ′′ (x) + q(x)y(x) = λy(x) (1) and the periodic y(π) = y(0), y ′ (π) = y ′ (0),
antiperiodic y(π) = −y(0), y ′ (π) = −y ′ (0),
Dirichlet y(π) = y(0) = 0,
Neumann y ′ (π) = y ′ (0) = 0 (5) boundary conditions respectively. It is well known that the spectra of the operators P (q) and A(q) consist of the eigenvalues λ 2n and λ 2n+1 , called as periodic and antiperiodic eigenvalues, that are the roots of
F (λ) = 2 & F (λ) = −2,(6)
where n = 0, 1, ..., F (λ) =: ϕ ′ (π, λ) + θ(π, λ) is the Hill discriminant and ϕ(x, λ), θ(x, λ) are the solutions of the equation (1) satisfying the initial conditions θ(0, λ) = ϕ ′ (0, λ) = 1, θ ′ (0, λ) = ϕ(0, λ) = 0.
The eigenvalues of the operators D(q) and N (q), called as Dirichlet and Neumann eigenvalues, are the roots of ϕ(π, λ) = 0 & θ ′ (π, λ) = 0 (8) respectively. The spectrum of the operator L(q) associated with (1) and the boundary conditions y(2π) = y(0), y ′ (2π) = y ′ (0) (9) is the union of the periodic and antiperiodic eigenvalues. In other words, the spectrum of L(q) consist of the eigenvalues λ n for n = 0, 1, .... that are the roots of the equation
(F (λ) − 2)(F (λ) + 2) = 0.(10)
The operators P (q), A(q), D(q) and N (q) are denoted respectively by P (a, b), A(a, b), D(a, b) and N (a, b) if
q(x) = ae −i2x + be i2x ,(11)
where a and b are complex numbers. If b = a then, for simplicity of the notations, these operators are redenoted by P (a), A(a), D(a) and N (a). The eigenvalues of P (a) and A(a) are denoted by λ 2n (a) and λ 2n+1 (a) for n = 0, 1, .... We use the following two classical theorems (see p.8-9 of [8] and p.34-35 of [6]).
Theorem 1 If q(x) is an even function, then ϕ(x, λ) is an odd function and θ(x, λ) is an even function. Periodic solutions are either ϕ(x, λ) or θ(x, λ) unless all solutions are periodic (with period π or 2π). Moreover, the following equality holds ϕ ′ (π, λ) = θ(π, λ).
Theorem 2 For all n and for any nonzero a the geometric multiplicity of the eigenvalue λ n (a) of the operators P (a) and A(a) is 1 ( that is, there exists one eigenfunction corresponding to λ n (a)) and the corresponding eigenfunction is either ϕ(x, λ n (a)) or θ(x, λ n (a)), where, for simplicity of the notations, the solutions of the equation
− y ′′ (x) + (2a cos 2x)y(x) = λy(x)(13)
satisfying (7) are denoted also by ϕ(x, λ) and θ(x, λ).
In [8,6] these theorems were proved for the real-valued potentials. However, the proofs pass through for the complex-valued potentials without any change.
The We use also the following result of [11].
Theorem 3 If ab = cd, then the Hill discriminants F (λ, a, b) and F (λ, c, d) (see (6)) for the operators P (a, b) and P (c, d) are the same.
By Theorem 2 the geometric multiplicity of the eigenvalues of P (a) and A(a) for any nonzero complex number a is 1. However, in the non-self-adjoint case a ∈ C\R, the multiplicity (algebraic multiplicity) of these eigenvalues, in general, is not equal to their geometric multiplicity, since the operators P (a) and A(a) may have associated functions (generalized eigenfunctions). Thus in the non-self-adjoint case the multiplicity (algebraic multiplicity) of the eigenvalues may be any finite number when the geometric multiplicity is 1 or 2. Therefore the investigation of the multiplicity of the eigenvalues for complex-valued potential is more complicated.
In this paper we find the conditions on a such that the all eigenvalues of the operators P (a), A(a), D(a) and N (a) are simple, namely we prove the following Note that there are a lot of papers about the asymptotic analyses and about the basis property of the root functions of the operators P (a, b) and A(a, b) (see [1-5, 7, 10] and the references in them). We do not discuss those papers, since in this paper we consider the another aspects of these operators and use only Theorems 1-3.
On the Even Potentials
In this section we analyze, in general, the even potentials. In the paper [9] the following statements about the connections of the spectra of the operators P (q), A(q), D(q) and N (q), where q is an even potential, were proved. Lemma 1 of [9]. If q is an even potential and λ is an eigenvalue of both operators D(q) and N (q), then
F (λ) = ±2, dF dλ = 0,(14)
that is, λ is a multiple eigenvalue of L(q). Proposition 1 of [9]. Let q be an even potential. Then λ is an eigenvalue of L(q) if and only if λ is an eigenvalue of D(q) or N (q).
First using (12) and the Wronskian equality
θ(π, λ)ϕ ′ (π, λ) − ϕ(π, λ)θ ′ (π, λ) = 1(15)
we prove the following improvements of these statements.
Theorem 6 Let q be an even complex-valued function. A complex number λ is both a Neumann and Dirichled eigenvalue if and only if it is an eigenvalue of the operator L(q)
with geometric multiplicity 2.
Proof. Suppose λ is both a Neumann and Dirichled eigenvalue, that is, both equality in (8) hold. On the other hand, it follows from (12), (8) and (15) that
θ(π, λ) = ϕ ′ (π, λ) = ±1.(16)
Now using (8), (16) and (7) one can easily verify that both θ(x, λ) and ϕ(x, λ) satisfy either periodic or anti-periodic boundary condition, that is, λ is an eigenvalue of the operator L(q) with geometric multiplicity 2. Conversely, if λ is an eigenvalue of L(q) with geometric multiplicity 2, then both θ(x, λ) and ϕ(x, λ) satisfy either periodic or anti-periodic boundary condition. Therefore by (7) the equalities in (8) hold, that is, λ is both Neumann and Dirichled eigenvalue. Theorem 7 Let q be an even complex-valued function. A complex number λ is an eigenvalue of multiplicity s of the operator L(q) if and only if it is an eigenvalue of multiplicities u and v of the operators D(q) and N (q) respectively, where u + v = s and u = 0 (v = 0) means that λ is not an eigenvalue of D(q) (N (q)).
Proof. It is well-known and clear that λ 0 is an eigenvalue of multiplicities u, v and s of the operator D(q), N (q) and L(q) respectively if and only if
ϕ(π, λ) = (λ 0 − λ) u f (λ), θ ′ (π, λ) = (λ 0 − λ) v g(λ)(17)
and
(F (λ) − 2)(F (λ) + 2) = (λ 0 − λ) s h(λ),(18)
where f (λ 0 ) = 0, g(λ 0 ) = 0 and h(λ 0 ) = 0. On the other hand by (12) and (15) we have
(F (λ) − 2)(F (λ) + 2) = 4θ 2 (π, λ) − 4 = 4(θ(π, λ)ϕ ′ (π, λ) − 1) = 4ϕ(π, λ)θ ′ (π, λ). (19)
Thus the proof of the theorem follows from (17)-(19) To analyze the periodic and antiperiodic eigenvalues in detail let us introduce the following notations and definitions.
Definition 1 Let σ(T ) denotes the spectrum of the operator T. A number λ is called P DN (q) (periodic, Dirichled and Neumann) eigenvalue if λ ∈ σ(P (q)) ∩ σ(D(q)) ∩ σ(N (q)). A num- ber λ ∈ σ(P (q)) ∩ σ(D(q)) is called P D(q) (periodic and Dirichled) eigenvalue if it is not P DN (q) eigenvalue. A number λ ∈ σ(P (q)) ∩ σ(N (q)) is called P N (q) (periodic and Neu- mann) eigenvalue if it is not P DN (q) eigenvalue. Everywhere replacing P (q) by A(q) we get the definition of ADN (q), AD(q) and AN (q) eigenvalues.
Using Theorems 6, 7, Definition 1 and the equality σ(P (q)) ∩ σ(A(q)) = ∅ we obtain (c) A complex number λ is an eigenvalue of geometric multiplicity 1 of the operator P (q) if and only if it is either P D(q) or P N (q) eigenvalue.
The theorem continues to hold if P (q), P DN (q), P D(q) and P N (q) are replaced by A(q), ADN (q), AD(q) and AN (q) respectively. Now we prove the main theorem of this section.
Theorem 9 Let q be an even complex-valued function and λ be an eigenvalue of geometric multiplicity 1 of the operator P (q). Then the number λ is an eigenvalue of multiplicity s of P (q) if and only if it is an eigenvalue of multiplicity s either of the operator D(q) (first case) or of the operator N (q) (second case). In the first case the system of the root functions of the operators P (q) and D(q) consists of the same eigenfunction ϕ(x, λ) and associated functions ∂ϕ(x, λ) ∂λ ,
1 2! ∂ 2 ϕ(x, λ) ∂λ 2 , ..., 1 (s − 1)! ∂ s−1 ϕ(x, λ) ∂λ s−1 .(20)
In the second case the system of the root function of the operators P (q) and N (q) consists of the same eigenfunction θ(x, λ) and associated functions
∂θ(x, λ) ∂λ , 1 2! ∂ 2 θ(x, λ) ∂λ 2 , ..., 1 (s − 1)! ∂ s−1 θ(x, λ) ∂λ s−1 .(21)
The theorem continues to hold if P (q) is replaced by A(q).
Proof. Let λ be an eigenvalue of geometric multiplicity 1 and multiplicity s of the operator P (q). By Theorem 1 there are two cases. Case 1. The corresponding eigenfunction is ϕ(x, λ). Case 2. The corresponding eigenfunction is θ(x, λ). We consider Case 1. In the same way one can consider Case 2. In Case 1, θ(x, λ) is not a periodic solution, that is, it does not satisfy the periodic boundary condition (2). On the other hand, the first equality of (6) with (12) and (7) implies that
θ(π, λ) = 1 = θ(0, λ),(22)
that is, θ(x, λ) satisfies the first equality in (2). Therefore θ(x, λ) does not satisfies the second equality of (2), that is,
θ ′ (π, λ) = 0.(23)
This inequality means that v = 0, where v is defined in Theorem 7. Therefore, by Theorem 7 we have u = s, that is, λ is an eigenvalue of multiplicity s of the operator D(q). Now suppose that λ is an eigenvalue of multiplicity s of D(q). Then by (8) and (7) ϕ(π, λ) = 0 = ϕ(0, λ).
On the other hand, using the first equality of (6), (12) and (7) we get
ϕ ′ (π, λ) = 1 = ϕ ′ (0, λ).(25)
Therefore ϕ(x, λ) is an eigenfunction of P (q) corresponding to the eigenvalue λ. Then, by Theorem 1, θ(x, λ) is not a periodic solution. This, as we noted above, implies (23) and the equality u = s. Thus, by Theorem 7, λ is an eigenvalue of multiplicity s of P (q). If λ is an eigenvalue of multiplicity s of the operators P (q) and D(q), then
F (λ) = 2, dF dλ = 0, d 2 F dλ 2 = 0, ..., d s−1 F dλ s−1 = 0 (26) and ϕ(π, λ) = 0, dϕ(π, λ) dλ = 0, d 2 ϕ(π, λ) dλ 2 = 0, ..., d s−1 ϕ(π, λ) dλ s−1 = 0.(27)
Since ϕ(0, λ) = 0 and ϕ ′ (0, λ) = 1 for all λ, we have
ϕ(0, λ) = 0, dϕ(0, λ) dλ = 0, d 2 ϕ(0, λ) dλ 2 = 0, ..., d s−1 ϕ(0, λ) dλ s−1 = 0 (28) and ϕ ′ (0, λ) = 1, dϕ ′ (0, λ) dλ = 0, d 2 ϕ ′ (0, λ) dλ 2 = 0, ..., d s−1 ϕ ′ (0, λ) dλ s−1 = 0.(29)
Moreover, using (26) and (12) we obtain
ϕ ′ (π, λ) = 1, dϕ ′ (π, λ) dλ = 0, d 2 ϕ ′ (π, λ) dλ 2 = 0, ..., d s−1 ϕ ′ (π, λ) dλ s−1 = 0.(30)
Thus, by (27)
we obtain
−( 1 k! ∂ k ϕ(x, λ) ∂λ k ) ′′ + (q(x) − λ) 1 k! ∂ k ϕ(x, λ) ∂λ k = 1 (k − 1)! ∂ k−1 ϕ(x, λ) ∂λ k−1
for k = 1, 2, ..., (s − 1). Therefore ϕ(x, λ) and the functions in (20) are the root functions of the operators P (q) and D(q). Thus the first case is proved. In the same way we prove the second case. The proof of this results for A(q) are similar
Main Results
In this section we consider the operators P (a), A(a), D(a) and N (a) with potential
q(x) = 2a cos 2x,(32)
where a is a nonzero complex number. By Theorem 2 the geometric multiplicity of the eigenvalues of P (a) and A(a) is 1. Therefore it follows from Theorem 8 that Clearly, the eigenfunctions corresponding to P N (a) eigenvalues, P D(a) eigenvalues, AD(a) eigenvalues and AN (a) eigenvalues have the forms
σ(P (a)) = {P D(a) eigenvalues} ∪ {P N (a) eigenvalues},(33)σ(A(a)) = {AD(a) eigenvalues} ∪ {AN (a) eigenvalues},(34)Ψ P N (x) = a 0 √ 2 + ∞ k=1 a k cos 2kx,(37)Ψ P D (x) = ∞ k=1 b k sin 2kx,(38)Ψ AD (x) = ∞ k=1 c k sin(2k − 1)x,(39)
and
Ψ AN (x) = ∞ k=1 d k cos(2k − 1)x(40)
respectively. For simplicity of the calculating we normalize these eigenfunctions as follows
∞ k=0 |a k | 2 = 1, ∞ k=1 |b k | 2 = 1, ∞ k=1 |c k | 2 = 1, ∞ k=1 |d k | 2 = 1.(41)
Substituting the functions (37)-(40) into (13) we obtain the following equalities
λa 0 = √ 2aa 1 , (λ − 4)a 1 = a √ 2a 0 + aa 2 , (λ − (2k) 2 )a k , = aa k−1 + aa k+1 ,(42)(λ − 4)b 1 = ab 2 , (λ − (2k) 2 )b k , = ab k−1 + ab k+1 ,(43)(λ − 1)c 1 = ac 1 + ac 2 , (λ − (2k − 1) 2 )c k , = ac k−1 + ac k+1 ,(44)(λ − 1)d 1 = −ad 1 + ad 2 , (λ − (2k − 1) 2 )d k , = ad k−1 + ad k+1(45)
for k = 2, 3, .... Here a k , b k , c k , d k depend on λ and a 0 , b 1 , c 1 , d 1 are nonzero constants (see [6] p. 34-35). By Theorem 10, if the eigenvalue λ corresponding to one of the eigenfunctions (37)-(40), denoted by Ψ(x), is multiple then there exists associated function Φ satisfying
− (Φ(x, λ)) ′′ + (q(x) − λ)Φ(x, λ) = Ψ(x).(46)
Since the boundary conditions (2)-(5) are self-adjoint λ and Ψ(x) are eigenvalue and eigenfunction of the adjoint operator. Therefore multiplying both sides of (46) by Ψ we get (Ψ, Ψ) = 0, where (., .) is the inner product in L 2 [0, π]. Thus, if the eigenvalues corresponding to the eigenfunctions (37)-(40), are multiple, then we have
∞ k=0 a 2 k = 0, ∞ k=1 b 2 k = 0, ∞ k=1 c 2 k = 0, ∞ k=1 d 2 k = 0.(47)
To prove the simplicity of the eigenvalue λ corresponding, say, to (40) we show that there is not a sequence {d k } satisfying the above 3 equalities: (45), (41) and (47), since these equalities hold if λ is a multiple eigenvalue. For this we use following proposition which readily follows from (41) and (47).
|d n (λ)| 2 > 1 2 ,(48)
then λ is a simple AN (a) eigenvalue, where a = 0. The statement continues to hold for AD(a), P D(a) and P N (a) eigenvalues if d n is replaced by c n , b n and a n respectively.
To apply the Proposition 1, we use following lemmas.
|d k | 2 ≤ 1 2 ,(49)|d k ± d m | 2 ≤ 1,(50)|d k | 2 ≤ |a| 2 |λ − (2k − 1) 2 | 2 .(51)(b) If Re λ < (2p − 1) 2 − 2 |a| for some p ∈ N, then |d k−1 | > |d k | > 0 and |d k+s | < |2a| s+1 |d k−1 | |λ − (2k − 1) 2 | |λ − (2(k + 1) − 1) 2 | ... |λ − (2(k + s) − 1) 2 |(52)
for all k > p and s = 0, 1, .... Proof. (a) If (49) does not hold for some k, then by Proposition 1 λ is a simple eigenvalue that contradicts the assumption of the lemma.
Using the last equalities of (47) and (41), we obtain
(d k ± d m ) 2 = − n =k,m d 2 n ± 2d k d m ≤ n =k,m |d n | 2 + |d k | 2 + |d m | 2 = 1,
that is, (50) holds. Now (51) follows from (45) and (50).
(b) Suppose that |d k | ≥ |d k−1 | for some k > p > 0. By (45) λ − (2k − 1) 2 |d k | ≤ |a| |d k−1 | + |a| |d k+1 | .
On the other hand, using the condition on λ we get λ − (2k − 1) 2 > 2 |a| . Therefore
|d k+1 | ≥ 2 |d k | − |d k−1 | ≥ |d k | .
Repeating this process s times we obtain |d k+s | ≥ |d k+s−1 | for all s ∈ N. It means that {|d k+s | : s ∈ N} is a nondecreasing sequence. On the other hand, |d k | + |d k+1 | = 0, since if both d k and d k+1 are zero, then using (45) we obtain that d j = 0 for all j ∈ N, that is, the solutions (40) is identically zero. Therefore d k does not converge to zero being the Fourier coefficient of the square integrable function Ψ AN (x). This contradiction shows that {|d k+s | : s ∈ N} is a decreasing sequence. Thus |d k | > 0 for all k > p. Now let us prove (52). Using (45) and the inequality |d k−1 | > |d k | > 0, we get
|d k+s | < |2a| |d k+s−1 | |λ − (2(k + s) − 1) 2 |(54)
for all s = 0, 1, .... Iterating (54) s times we obtain (52). (c) By (45) we have
k∈I |d k | 2 ≤ k∈I |a| 2 (|d k−1 | + |d k+1 |) 2 (d(λ, I)) 2 ≤ k∈I 2 |a| 2 (|d k−1 | 2 + |d k+1 | 2 ) (d(λ, I)) 2 .
Note that in case k = 1 instead of d k−1 we take d 1 (see the first equality of (45)). Now (53) follows from (41). (d) Everywhere replacing d k by c k we get the proof of the last statement In the similar way we prove the following lemma for P (a).
Lemma 2 Suppose that λ is a multiple P D(a) eigenvalue corresponding to the eigenfunction (38)
, where a = 0. Then (a) For all k ∈ N, m ∈ N, n ∈ N, n = m the following inequalities hold
|b m | 2 ≤ 1 2 , |b n ± b m | 2 ≤ 1, |b k | 2 ≤ |a| 2 |λ − (2k) 2 | 2 . (55) (b) If Re λ < (2p) 2 − 2 |a| for some p ∈ N, then |b k−1 | > |b k | > 0 and |b k+s | < |2a| s+1 |b k−1 | |λ − (2k) 2 | |λ − (2(k + 1)) 2 | ... |λ − (2(k + s)) 2 |(56)
for all k > p and s = 0, 1, ...
(c) Let I ⊂ N and b(λ, I) = min k∈I λ − (2k) 2 = 0. Then k∈I |b k | 2 ≤ 4 |a| 2 (b(λ, I)) 2 .(57)
(d) If λ is a multiple P N (a) eigenvalue corresponding to (37) then the statements (a) and (b) continue to hold for k > 1, m ≥ 0 and the statement (c) continues to hold for
I ⊂ {0, 1, ...} if b j is replaced by a j .
Introduce the notation D n = {λ ∈ C : λ − (2n − 1) 2 ≤ 2 |a| }. Proof. By (34) if λ is an eigenvalues of the operator A(a), then the corresponding eigenfunction is either Ψ AN (x) or Ψ AD (x) (see (39) or (40)). Without loss of generality, we assume that the corresponding eigenfunction is Ψ AN (x).
(a) Since d k → 0 as k → ∞, there exists n ∈ N such that
|d n | = max k∈N |d k | .
Therefore (a) follows from (45) for k = n.
(b) Suppose that λ ∈ D n is a multiple eigenvalue corresponding to the eigenfunction Ψ AN (x). By definition of D n for k = n we have
λ − (2k − 1) 2 ≥| (2n − 1) 2 − (2k − 1) 2 | − |2a| ≥ (2n − 3) 2 − (2n − 1) 2 − |2a| .
This together with the condition on n and the definition of d(λ, I) (see Lemma 1(c)) gives d(λ, N\{n}) > 2 √ 2 |a| . Thus, using (53) and (41) we get
k =n |d k | 2 < 1 2 & |d n | 2 > 1 2
which contradicts Proposition 1. Instead of Lemma 1 using Lemma 2 in the same way we prove the following Proof. Since 8 > 8 √ 6 (1 + √ 2), by Theorem 11(b) the ball D n for n > 2 does not contain the multiple eigenvalues of the operator A(a). Therefore we need to prove that the ball D n for n = 1, 2 also does not contain the multiple eigenvalues. Since the balls D 1 and D 2 are contained in the half plane {λ ∈ C : Re λ < 16 } we consider the following two strips {λ ∈ C : 9 < Re λ < 16 }, {λ ∈ C : 6 < Re λ ≤ 9 } and half plane {λ ∈ C : Re λ ≤ 6 } separately. We consider the AN (a) eigenvalues, that is, the eigenvalues corresponding to the eigenfunction (40). Consideration of the AD(a) eigenvalues are the same.
To prove the simplicity of the eigenvalues lying in the above strips, we assume that λ is a multiple eigenvalue. Using Lemma 1 by direct calculating (see Estimation 1 and Estimation 2 in Appendix) we show that (48) for n = 2 holds that contradicts Proposition 1.
Investigation the half plane Re λ ≤ 6 is more complicated. Here we use the first two equalities of (45)
(λ − 1)d 1 = −ad 1 + ad 2 , (λ − 9)d 2 , = ad 1 + ad 3 .(58)
By direct calculating we get (see Estimation 3 and Estimation 4 in the Appendix)
∞ k=3 |d k | 2 < 0.03 415, |d 3 | |d 2 | < 0.174 32(59)
Then by (41) we have
|d 1 | 2 + |d 2 | 2 > 1 − ε,(60)
where ε = 0.03 415. On the other hand, by (49), |d 1 | 2 ≤ 1 2 , |d 2 | 2 ≤ 1 2 . These inequalities and (47) imply that
|d 1 | 2 = 1 2 − ε 1 , |d 2 | 2 = 1 2 − ε 2 , d 2 2 = − d 2 1 + ε 3 ,
where ε 1 ≥ 0, ε 2 ≥ 0, ε 1 + ε 2 = ε, |ε 3 | < 0.03 415. Now, one can easily see that
( d 2 d 1 ) 2 = −1 + α, d 2 d 1 = ±(i + δ),
where |α| < 0.03 415 0.5−0.03 415 < 0.074, |δ| < 1 2 |0.074| + 1 7 |0.074| 2 < 0.0 4. Therefore we have
d 2 d 1 − d 1 d 2 = ± (i + δ) 2 − 1 i + δ = ± 2i(i + δ) + δ 2 i + δ = ±2i + γ,(61)
where |γ| < (0.04) 2 1−0.04 < 0.002. On the other hand, dividing the first equality of (58) by d 1 and the second by d 2 and then subtracting second from the first and taking into account (61) we get 8
a = ±2i − 1 + γ − d 3 d 2 ,(62)
where by assumption 8 a ≥ √ 6. Therefore using the second estimation of (59) in (62) we get the contradiction 2. 449 5 < √ 6 ≤ 8 a < √ 5 + 0.174 32 + 0.002 < 2. 412 5
In the same way we consider the simplicity of the eigenvalues of the operators P (a), D(a) and N (a). First let us investigate the eigenvalues of D(a). Since the eigenvalues of D(a) is the union of P D(a) and AD(a) eigenvalues and the AD(a) eigenvalues are investigated in Theorem 13, we investigate the P D(a) eigenvalue. Proof. The second statement follows from the first statement and Theorem 13. Therefore we need to prove the first statement by using (43). Since 14 > 5(1 + √ 2), by Theorem 12, the P D(a) eigenvalues lying in the ball B n for n > 3 are simple.
If λ ∈ B 3 , then 26 ≤ Re λ ≤ 46. Using Lemma 2 and (41) we obtain the estimations (see
Estimation 5 in Appendix) k =3 |b k | 2 < 1 2 , |b 3 | 2 > 1 2
which, by Proposition 1, proves the simplicity of the P D(a) eigenvalues lying in B 3 . Now we need to prove that the balls B and B 2 does not contain the multiple P D(a) eigenvalues. Since these balls are contained in the strip {λ ∈ C : Re λ ≤ 26 } we consider the following cases: 16 < Re λ ≤ 26, 12 < Re λ ≤ 16 and Re λ ≤ 12.
In the first two cases using Lemma 2 we get the inequality (see Estimation 6 and Estimation 7) obtained from (48) for n = 2 by replacing d n with b n which proves, by Proposition 1, the simplicity of the eigenvalues. Now consider the third case Re λ ≤ 12. Using Lemma 2 we obtain (see Estimation 8 and Estimation 9 in Appendix)
∞ k=3 |b k | 2 < 1 15 , |b 3 | |b 2 | < 0.213 1(63)
The first inequality of (63) with (41) implies that
|b 1 | 2 + |b 2 | 2 > 1 − β,(64)
where β < 1 15 . Instead of (60) using (64) and repeating the proof of (61) we obtain
b 2 b 1 − b 1 b 2 = (i + δ) 2 − 1 i + δ = 2i(i + δ) + δ 2 i + δ = ±2i + γ 1 ,(65)
where |γ 1 | < 0.01. Now dividing the first equality of (43) by b 1 and the second equality of (43) for k = 2 by b 2 and then subtracting second from the first and using (65) we get
12 a = ±2i + γ 1 − b 3 b 2 ,(66)
where by assumption 12 a ≥ 2.4. Thus, using (63) in (66) we get the contradiction 2. 4 ≤ 12 a < 2 + 0.213 1 + 0.01 = 2. 223 1
Theorem 15 If 0 < |a| ≤ 4 3 , then the all eigenvalues of the operators P (a) and N (a) are simple.
Proof. By Theorem 13 and Theorem 14 we need to prove that if |a| ≤ 4 3 , then all P N (a) eigenvalues are simple. Since 6 > (1 + √ 2) 4 3 , by Theorem 12, the P N (a) eigenvalues lying in the ball B n for n > 1 are simple. Now we prove that the balls A 0 and A 1 does not contain the multiple P N (a) eigenvalues. Since these balls are contained in {λ ∈ C : Re λ < 8 } we consider the following cases: Case 1: 3 ≤ Re λ < 8. Using (42) and Lemma 2 (see Estimation 10 in Appendix) we obtain |a 1 | 2 > 1 2 which, by Proposition 1, proves the simplicity of the eigenvalues. Case 2: Re λ < 3. Using Lemma 2 we obtain ( see Estimations 11 and 12 in Appendix) ∞ k=2 |a k | 2 < 1 58 , |a 2 | |a 1 | < 0.103 01 (67)
The first inequality of (67) with (41) implies that
|a 0 | 2 + |a 1 | 2 > 1 − ρ,(68)
where ρ < 1 58 . Instead of (60) using (68) and repeating the proof of (61) we obtain a 1 a 0 − a 0 a 1 = ±2i + γ,
where |γ| < 0.0006. Now dividing the first equality of (42) by a 0 and the second by a 1 and then subtracting second from the first and taking into account (69) we get
4 a = ±2 √ 2i + √ 2γ − a 2 a 1 ,(70)
where by assumption 4 a ≥ 3. Therefore using (67)
spectrum of P (a), A(a), D(a), N (a) for a = 0 are {(2k) 2 : k = 0, 1, ...}, {(2k + 1) 2 : k = 0, 1, ...}, {k 2 : k = 1, 2, ...}, {k 2 : k = 0, 1, ...} respectively. All eigenvalues of P (0), except 0, and A(0) are double, while the eigenvalues of D(0) and N (0) are simple.
Theorem 4 (
4Main results for the operators P (a), A(a), D(a) and N (a)): (a) If 0 < |a| ≤ 8 √ 6 , then the all eigenvalues of the operators A(a) and D(a) are simple. (b) If 0 < |a| ≤ 4 3 , then the all eigenvalues of the operators P (a) and N (a) are simple. This theorem with Theorem 3 implies Theorem 5 (Main results for the operators A(a, b) and P (a, b)): (a) If 0 < |ab| ≤ 64 6 , then the all eigenvalues of the operator A(a, b) are simple. (b) If 0 < |ab| ≤ 16 9 , then the all eigenvalues of the operator P (a, b) are simple.
Theorem 8
8Let q be an even complex-valued function. Then (a) The spectrum of P (q) is the union of the following three pairwise disjoint sets: {P DN (q) eigenvalues}, {P D(q) eigenvalues} and {P N (q) eigenvalues}. (b) A complex number λ is an eigenvalue of geometric multiplicity 2 of the operator P (q) if and only if it is P DN (q) eigenvalue.
-(30), ϕ(x, λ) and the functions in (20) satisfy both the periodic and Dirichlet boundary conditions. On the other hand, differentiating s − 1 times, with respect to λ, the equation − ϕ ′′ (x, λ) + q(x)ϕ(x, λ) = λϕ(x, λ)
where P D(q), P N (q), AD(q) and AN (q) (see Definition 1) are denoted by P D(a), P D(a), P D(a) and P D(a) when the potential q is defined by (32). Moreover, Theorem 7, Theorem 2 and Theorem 9 yield the equalitiesσ(D(a)) = {P D(a) eigenvalues} ∪ {AD(a) eigenvalues}, (35) σ(N (a)) = {P N (a) eigenvalues} ∪ {AN (a) eigenvalues} (36) and the following theorem. Theorem 10 For any a = 0 the eigenvalue λ of the operator P (a) or A(a) is multiple if and only if it is a multiple eigenvalue either of D(a) or N (a). Moreover, the operators P (a), A(a), D(a) and N (a) have associated functions corresponding to any multiple eigenvalues.
Proposition 1
1If there exists n ∈ N = : {1, 2, ..., } such that
Lemma 1
1Suppose that λ is a multiple AN (a) eigenvalue corresponding to the eigenfunction (40), where a = 0. Then (a) For all k ∈ N, m ∈ N, k = m the following inequalities hold
(c) Let I ⊂ N and d(λ, I) =: min k∈I λ − (2k − 1) If λ is a multiple AD(a) eigenvalue corresponding to the eigenfunction (39), then the inequalities (49)-(53) continues to hold if d j is replaced by c j .
Theorem 11 (a) All eigenvalues of the operator A(a) lie on the unions of D n for n ∈ N. (b) If 4n − 4 > (1 + √ 2) |a|, where a = 0, then the eigenvalues of A(a) lying in D n are simple.
Theorem 12 (a) All P D(a) eigenvalues lie in the unions of B =: {λ : |λ − 4| ≤ |a| } and B n =: {λ : λ − (2n) 2 ≤ 2 |a| } for n = 2, 3, ..... All P N (a) eigenvalues lie in the unions of A 0 = {λ : |λ| ≤ √ 2 |a| }, A 1 = {λ : |λ − 4| ≤ (1 + √ 2) |a| } and B n for n = 2, 3, .... (b) If 4n − 2 > (1 + √ 2) |a| and n > 1, where a = 0, then the eigenvalues of P (a) lying in B n are simple. Now we prove the main result for A(a). Theorem 13 If 0 < |a| ≤ 8 √ 6 , then the all eigenvalues of the operator A(a) are simple.
Theorem 14
14If 0 < |a| ≤ 5, then all P D(a) eigenvalues are simple. Moreover, if 0 < |a| ≤ 8 √ 6 , then the all eigenvalues of the operator D(a) are simple.
Estimation 1: Let 9 < Re λ < 16. By (51) we have Since d(λ, {4, 5, ...}) < 33 using (53) we get Estimation 2. Let 6 < Re λ ≤ 9. By (51) Estimation 12. Here we estimate a2 a1 for Re λ < 3. Iterating (42) for k = 2, we get (λ − 16)(λ − 36) . |61| |33| 2 |13| 2 < 0.103 01we get the contradiction
3 ≤ 4
a <
√
2(2 + 0.0006) + 0.103 01 = 2. 932 3
4 Appendix
|d 1 |
2 ≤
|a|
2
|λ − 1|
2 ≤
8
√
6
2
|8|
2 =
1
6
, |d 3 |
2 ≤
|a|
2
|λ − 25|
2 ≤
8
√
6
2
|9|
2 =
32
243
.
∞
k=4
|d k |
2 <
4 8
√
6
2
|33|
2 =
128
3267
.
These inequalities imply that
k =2
|d k | 2 <
128
3267
+
32
243
+
1
6
=
19 849
58 806
<
1
2
.
|d 1 |
2 ≤
8
√
6
2
|5|
2 =
32
75
, |d 3 |
2 ≤
8
√
6
2
|16|
2 =
1
24
.
∞
k=2
|a k |
2 ≤
16
1521
+
64
9801
<
1
58
.
a 2 =
aa 1 + aa 3
λ − 16
=
aa 1
λ − 16
+
a
λ − 16
(
aa 2 + aa 4
λ − 36
)
(74)
=
aa 1
λ − 16
+
a 3 a 1
(λ − 16) 2 (λ − 36)
+
a 3 a 3
(λ − 16) 2 (λ − 36)
+
a 2 a 4
Now dividing both sides of (74) by a 1 and using Lemma 2(d), (56) we obtain
|a 2 |
|a 1 |
≤
4
3
13
+
4
3
3
|33| |13| 2 +
4 4
3
5
|33| 2 |13| 3 +
8 4
3
5
Estimation 3. Let Re λ ≤ 6. By (52) and (49) we haveNow using (51) and (53) and taking into account d(λ, {6, 7, ...}) ≤ 115 we obtainThese inequalities imply thatEstimation 4. Now we estimate |d3| |d2| for Re λ ≤ 6. Iterating (45) for k = 3, we get.Therefore, dividing both sides of (72) by d 2 and using (52) we obtainEstimation 5. Let 26 ≤ Re λ ≤ 46. Using (56) and (58) we obtain.Now dividing both sides of (73) by b 2 and using (56) we obtain
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The asymptotics of the gap in the Mathieu equation. J Avron, B Simon, Ann. Phys. 134J. Avron, B. Simon, The asymptotics of the gap in the Mathieu equation, Ann. Phys., 134 (1981) 76-84.
Asymptotics of instability zones of the Hill operator with a two term potential. P Djakov, B Mityagin, J. Funct. Anal. 2421P. Djakov and B.Mityagin, Asymptotics of instability zones of the Hill operator with a two term potential. J. Funct. Anal. 242 (2007), No. 1, 157-194.
Instability zones of periodic 1D Schr¨odinger and Dirac operators. P Djakov, B Mityagin, Russian Math. Surveys. 614P. Djakov and B. Mityagin, Instability zones of periodic 1D Schr¨odinger and Dirac operators, Russian Math. Surveys 61 (2006), No. 4, 663-766.
Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials. P Djakov, B Mityagin, Math. Ann. 351P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann. 351 (2011), 509-540.
The spectral theory of periodic differential operators. M S P Eastham, Hafner, New YorkM. S. P. Eastham, The spectral theory of periodic differential operators, Hafner, New York 1974.
On the effect of the boundary conditions on the eigenvalues of ordinary differential equations. E Harrell, Amer. J. Math. John Hopkins Presssupplement 1981, dedicated to P. HartmanE. Harrell, On the effect of the boundary conditions on the eigenvalues of ordinary differential equations, Amer. J. Math., supplement 1981, dedicated to P. Hartman, Baltimore, John Hopkins Press.
Hill's Equation. W Magnus, S Winkler, Interscience Publishers, John WileyW. Magnus and S. Winkler, Hill's Equation, Interscience Publishers, John Wiley, 1966.
Determining the potential of a Sturm-Liouville operator from its Dirichlet and Neumann spectra, Pacific journal of math. V Pierce, 204V. Pierce, Determining the potential of a Sturm-Liouville operator from its Dirichlet and Neumann spectra, Pacific journal of math., 204 (2002), No.2, 497-509.
O A Veliev, arXiv:1202.4735Spectral Analysis of the Non-self-adjoint Mathieu-Hill Operator. O. A. Veliev, Spectral Analysis of the Non-self-adjoint Mathieu-Hill Operator, arXiv:1202.4735 (2012).
O A Veliev, arXiv:1202.6048Isospectral Mathieu-Hill Operators. O.A. Veliev, Isospectral Mathieu-Hill Operators, arXiv:1202.6048 (2012).
| []
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[
"Angular momentum and topological dependence of Kepler's Third Law in the Broucke-Hadjidemetriou-Hénon family of periodic three-body orbits",
"Angular momentum and topological dependence of Kepler's Third Law in the Broucke-Hadjidemetriou-Hénon family of periodic three-body orbits"
]
| [
"Marija R Janković \nFaculty of Physics\nInstitute of Physics\nBelgrade University\nStudentski Trg 1211000BelgradeSerbia\n",
"V Dmitrašinović \nBelgrade University\nPregrevica 118P.O.Box 5711080Zemun, BelgradeSerbia\n"
]
| [
"Faculty of Physics\nInstitute of Physics\nBelgrade University\nStudentski Trg 1211000BelgradeSerbia",
"Belgrade University\nPregrevica 118P.O.Box 5711080Zemun, BelgradeSerbia"
]
| []
| We use 57 recently found topological satellites of Broucke-Hadjidemetriou-Hénon's periodic orbits with values of the topological exponent k ranging from k = 3 to k = 58 to plot the angular momentum L as a function of the period T , with both L and T rescaled to energy E = − 1 2 . Upon plotting L(T /k) we find that all our solutions fall on a curve that is virtually indiscernible by naked eye from the L(T ) curve for non-satellite solutions. The standard deviation of the satellite data from the sixth-order polynomial fit to the progenitor data is σ = 0.13. This regularity supports Hénon's 1976 conjecture that the linearly stable Broucke-Hadjidemetriou-Hénon orbits are also perpetually, or Kolmogorov-Arnold-Moser stable. | 10.1103/physrevlett.116.064301 | [
"https://arxiv.org/pdf/1604.08358v1.pdf"
]
| 1,164,764 | 1604.08358 | 44fe2dbf4fbd95ceb10e13ca825ad3d44adcce00 |
Angular momentum and topological dependence of Kepler's Third Law in the Broucke-Hadjidemetriou-Hénon family of periodic three-body orbits
28 Apr 2016
Marija R Janković
Faculty of Physics
Institute of Physics
Belgrade University
Studentski Trg 1211000BelgradeSerbia
V Dmitrašinović
Belgrade University
Pregrevica 118P.O.Box 5711080Zemun, BelgradeSerbia
Angular momentum and topological dependence of Kepler's Third Law in the Broucke-Hadjidemetriou-Hénon family of periodic three-body orbits
28 Apr 2016(Dated: April 29, 2016)arXiv:1604.08358v1 [physics.class-ph]numbers: 4550Jf, 0545-a, 9510Ce Keywords: celestial mechanicsthree-body systems in Newtonian gravitynonlinear dynamics
We use 57 recently found topological satellites of Broucke-Hadjidemetriou-Hénon's periodic orbits with values of the topological exponent k ranging from k = 3 to k = 58 to plot the angular momentum L as a function of the period T , with both L and T rescaled to energy E = − 1 2 . Upon plotting L(T /k) we find that all our solutions fall on a curve that is virtually indiscernible by naked eye from the L(T ) curve for non-satellite solutions. The standard deviation of the satellite data from the sixth-order polynomial fit to the progenitor data is σ = 0.13. This regularity supports Hénon's 1976 conjecture that the linearly stable Broucke-Hadjidemetriou-Hénon orbits are also perpetually, or Kolmogorov-Arnold-Moser stable.
Introduction
Numerical studies of periodic three-body orbits have increased their output over the past few years -more than 40 new orbits, and their "satellites" have been discovered, Refs. [1][2][3][4]. Unlike periodic two-body orbits, which are all ellipses, and thus are all topologically equivalent, the non-colliding three-body periodic orbits have one of infinitely many different topologies. Montgomery, Ref. [5], had devised an algebraic method to associate a free-group element ("word") w with a three-body orbit's topology, and thus to label and classify such periodic orbits; for an elementary introduction to this method, see Ref. [6]. That classification method has recently acquired practical importance in the identification of new three-body orbits, Refs. [1,3,4].
A number of newly discovered orbits, Refs. [1][2][3][4], were of the so-called topological satellite type. Such satellite orbits, are also known as "bifurcation" in the older literature, Refs. [2,7], where they were only loosely defined in terms of their presumed origin. It was only in Ref. [3] that a precise definition of a topological satellite was given. When this definition was applied to the figure-8 satellites [25], reported in Ref. [3], it led to the discovery of a remarkable "topological Kepler's third law"-like regularity for arbitrary orbits with vanishing angular momenta, Ref. [8]. The immediate question is whether this regularity persists when the angular momentum does not vanish?
The present Letter is an attempt to answer that question, albeit in a single, specific family of three-body orbits, viz. in the Broucke-Hadjidemetriou-Hénon (BHH) family [9][10][11][12][13][14][15], that has the simplest non-trivial topology (free group element w=a). The main reason for selecting only this family of orbits is that it is the most thoroughly studied family thus far: it is the only family of orbits with a previously determined dependence of the period T on the angular momentum L of (non-satellite, or progenitor) periodic orbits, Refs. [9][10][11][12][13][14][15]. No such, or comparable, study of any of the remaining known families exists to our knowledge at this moment. Moreover, the BHH family is one of only two families [26] of periodic threebody orbits that have been observed in astronomy: all known "hierarchical" triple star systems belong to BHH orbits. Moreover, the Sun-Earth-Moon system may be viewed as a BHH solution, albeit with highly asymmetrical mass ratios.
The first step towards this goal, the one of finding as many different BHH satellite orbits as possible, has already been accomplished in Ref. [16]. Previously, Davoust and Broucke, Ref. [7], had found one (the first k=3) satellite of one retrograde BHH orbit. Ref. [16] extended the search for retrograde BHH satellite orbits systematically up to values k ≤ 19 of the topological exponent k, and more haphazardly up to k = 58; thus several different types of BHH satellites with identical values of k were discovered, [27], as were a few prograde BHH satellites, see the Supplemental Material [17] and the Web site [18]. Prograde BHH satellites have not been studied systematically, as yet, mostly due to their paucity at the values of the angular momentum covered in the searches in Ref. [16]. Presently it is not known how many satellites ought to exist, and under which conditions. It is interesting, however, that the observed satellites correspond only to linearly stable BHH progenitor orbits. This is in line with Hénon's 1976 conjecture [14,17] about Kolmogorov-Arnold-Moser (KAM) stability of linearly stable BHH orbits. Then, motivated by the findings reported in Ref. [8], we checked for similar regularities of satellite BHH orbits with non-zero angular momentum. Firstly, we formulated the topological dependence of Kepler's third law for three-body orbits with non-zero angular momenta, and secondly we tested it on the presently known satellites of the retrograde BHH family. Secondly we found a striking result: all of our retrograde BHH satellites fall on a single (continuous) curve L(T /k), Fig. 3 that is practically indiscernible by naked eye from the L(T ) curve, Fig. 1, for non-satellite (progenitor) retrograde BHH solutions, whereas the "topologically uncorrected" curve L(T ) looks very differently, see Fig. 2. A quantitative measure of this (dis)agreement is shown in terms of corresponding standard deviations.
Preliminaries
Broucke [7,9,10], Hadjidemetriou [11][12][13] and Hénon [14,15] (BHH) explored a set of periodic planar threebody orbits with equal mass bodies. These orbits form two continuous curves in the L-T plane whose lower (retrograde) terminus ("end") is the collinear collision (Schubart) orbit, and both the retrograde and the direct L(T ) curves approach the same high-L limit at their upper termini, Fig. 1.
Although BHH write of two families of orbits -direct, or prograde, and retrograde -all of these orbits belong to a single topological family: during one period the orbit completes a single loop around one of the poles on the shape sphere. This loop can be described by the conjugacy class of the fundamental group/free group element a, according to the topological classification used in Refs. [1,6]. It turns out, however, that there are numerous relative periodic orbits with topology a k , with k = 2, 3... Such orbits are sometimes called satellites [2,3], whereas other authors call them "bifurcation orbits" [7].
Scaling laws for three bodies
It is well known that Kepler's third law (for two bodies) follows from the spatio-temporal scaling laws, which, in turn, follow from the homogeneity of the Newtonian gravity's static potential, Ref. [19]. These scaling laws read r → λr, t → λ 3/2 t, and consequently v → v/ √ λ. The (total) energy scales as E → λ −1 E, the period T as T → λ 3/2 T and angular momentum as L → λ 1/2 L, i.e., differently than either the period T , or "size" R, which is the reason why only the vanishing angular momentum L = 0 is a "fixed point" under scaling. For this reason, we use scale-invariant angular momentum L r = L|E| 1/2 , scale-invariant period T r = T |E| 3/2 and, for simplicity's sake, equal masses. Thus, we may replace the "mean size"R of the three-body system in Kepler's third law T ∝R 3/2 with the inverse absolute value of energy |E| −1 , i.e., T ∝ |E| −3/2 , or equivalently T |E| 3/2 = T r = const. .
The "constant" on the right-hand-side of this equation is not a universal one in the three-body case, as it is in the two-body case (where it depends only on the masses and the Newtonian coupling G): it may depend both on the family w of the three-body orbit, described by the free-group word w, and on the scale-invariant angular momentum L r = L|E| 1/2 of the orbit, see Refs. [14,15], as follows
T (w) |E| 3/2 = T (w) r = f (L (w) |E| 1/2 ) = f (L (w) r ),
or as an inverse function:
L (w) r = L (w) |E| 1/2 = f −1 (T (w) |E| 3/2 ) = f −1 (T (w) r ).
Thus, the curve L (w)
r (T (w) r ) = L (w) |E| 1/2 (T (w) |E| 3/2 ) as a function of T (w) r = T (w) |E| 3/2
is a fundamental property of any family w of periodic orbits. For the BHH family the L(T) curve, for fixed energy E = −0.5 orbits, based on the data from Refs. [9][10][11][12][13][14][15] is shown in Fig. 1.
We wish to see if the zero-angular-momentum relation T r (w k ) = kT r (w), Ref. [8], or some similar statement holds also at non-zero angular momentum? The analogon of this relation for orbits with non-zero angular momenta would be a simple relation between L(T) curves for the progenitor orbit L r (T r ) and its k-th satel-
lite L (w k ) r (T (w k ) r ): L (w) r (T (w) r ) = L (w k ) r (T (w k ) r /k).(1)
We shall test this relation in the BHH family of solutions, and in order to do so, we use the BHH satellite orbits from Ref. [16].
L(T) curves for BHH satellites
The L-T plot of different-k satellite orbits are scattered over a large region and do not intersect the BHH progenitor family of orbits' L(T) curve when plotted as a function of the (un-divided) period T, see Fig. 2. Note the large span of periods T in the data, Table I, and in Fig. 2, as well as two large "gaps" in the data. These gaps are due to the exigencies of the search reported in Ref. [16], which was not conducted with the intention of testing the hypothetical topological Kepler's third law. The values in Table I have been rounded off to five sig- Table I: Properties of satellite orbits in the retrograde branch of the BHH family. Here k is the topological power of the orbit, T is its period, and L its angular momentum. All orbits have the same energy E = − 1 2 . For the raw data and a discussion of numerical errors, see the Supplemental Material [17]. nificant decimal places. So, the numerical error is less than one part in 10,000. Such an error would be invisible in the Figs. 2,3,4 meaning that the "size of the points" in these figures is larger than the expected error. After dividing the period T (at fixed energy) by the topological exponent k, T ′ = T/k, we can see in Fig. 3 that the satellite orbits' L(T/k) curve (the angular momentum L as a function of topologically-rescaled period T/k) approximately coincides with the L(T) curve of BHH retrograde orbits. It seems that such an appearance of order out of apparent disorder cannot be an accident.
Next, in Fig. 3 we look more closely at the section of the L(T) curve of progenitor BHH retrograde orbits Table I. in which we have found all but one of our satellites. We have interpolated Hénon's, [14], 18 stable retrograde data points with a piecewise polynomial fit in this part of the L(T) curve. The standard deviations from this interpolated curve were calculated for: 1) Broucke's 10 progenitor retrograde orbits, [9,10], and 2) the 56 out of 57 new satellite orbits from Table I (excluding one orbit that lies near the "shoulder" at T=14 in Fig. 3), with the following results. 1) σ = 0.0034 for Broucke's orbits; and 2) σ = 0.1269 for satellite orbits. This difference of two orders of magnitude between these two numbers clearly indicates that the rescaled satellites' periods do not coincide exactly with the progenitor ones, but only approximately. Moreover, when one assembles Hénon's and Broucke's, [9,10], retrograde orbits in one set and fits the aggregate data by a polynomial of the sixth degree, Fig. 3, the standard deviation of the fit is σ = 0.0313, whereas the standard deviation of all satellite orbits from this polynomial curve is σ = 0.1315, roughly four times bigger. It is (statistically) clear that the satellites do not follow exactly the same L(T) curve as the progenitors, but the deviation is not large. This constitutes the evidence for the analogon of the topological dependence of Kepler's third law for the L = 0 case, Ref. [8].
Finally, we note that all of our newly found satellite orbits fall into a region of the progenitor L(T) curve that corresponds to stable progenitor BHH orbits, with one possible exception (the red point near the "shoulder" at T=14 in Fig. 3), that "sits" on the border point between stable and unstable regions. We have not found any other satellites in this, the second stable region of BHH retrograde orbits. In Fig. 4 we show the fine structure in the satellites' L(T/k) curve, that remains to be studied in finer detail and be better understood.
We have not studied the direct/prograde (sub)family of BHH orbits, as Ref. [16] did not search for their satellites, but found four almost inadvertently. Certainly, that task ought to be completed in the future.
Summary, Conclusions and Outlook
We have used 57 new satellite orbits from Ref. [16], in the family of Broucke-Hadjidemetriou-Hénon, Refs. [9][10][11][12][13][14][15], relative periodic solutions to the planar three body problem. Thence followed a striking relation between their kinematic and topological properties.
BHH orbits constitute a family with a simple topology, described by the free group element a according to the classification on the shape sphere, and their satellites are orbits of the topology a k . The BHH orbits' angular momenta L and periods T form a continuous curve L(T), at fixed energy. Our satellite orbits form a scattered set of points on the same L(T) plot, but all of them exhibit the property that after their period T is divided by their topological order k, they approximately fall on the L(T) curve of the original (k = 1) BHH orbits. This study was motivated by the discovery, Ref. [8], of a relation between the topology and periods among the satellites of the figure-eight orbit, Ref. [3], and one other type ("moth I" -"yarn" in Ref. [1]), of three-body orbits at vanishing angular momentum. This Letter shows that Kepler's third law's topological dependence also holds for orbits with L = 0, albeit only approximately. It remains to be seen just precisely what this discrepancy depends on?
These results are even more striking if one remembers that among our results there are several distinct types of satellite orbits of the same topological power k, some with quite different values of L and T, which all display this property. A closer look at the L(T/k) curve revealed a fine structure, which should be investigated in higher detail in the future. An extension of the search conducted in Ref. [16], into hitherto unexplored regions of the L-T plane ought to provide (new) data that will further test our result.
Our results indirectly confirm Hénon's 1976 conjecture, see page 282 in Ref. [14], reproduced in the Supplemental Material [17], that the linearly stable BHH orbits are also nonlinearly, or perpetually, or KAM stable. Such KAM stability implies the existence of quasiperiodic orbits with periods that conform to the quasiperiodicity condition (i.e. with periods that are "almost commensurate" with the BHH progenitor's period), as predicted by the KAM theorem, Refs. [22][23][24].
Our study opens several new questions: 1) The most commonly observed hierarchical triple star systems belong to the BHH family. Are there BHH topological satellites among astronomically observed three-body systems? It is important to extend the present study to the realistic case of three different masses: some early work has already been done in this direction by Broucke and Boggs, Ref. [9], and by Hadjidemetriou and Christides, Ref. [12]. 2) In recent years there have been formal "proofs of existence" given for at least some BHH orbits, Refs. [20,21]. This begs the question: can one "prove existence" of their satellite orbits, and, if yes, of how many satellites, and under which conditions? Acknowledgments M. R. J. was a recipient of the "Prof Dr Djordjě Zivanović" scholarship awarded jointly by the Faculty of Physics and the Institute of Physics, Belgrade University, and was also supported by a City of Belgrade studentship (Gradska stipendija grada Beograda). The work of V. D. was supported by the Serbian Ministry of Science and Technological Development under grant numbers OI 171037 and III 41011. The computing cluster Zefram (zefram.ipb.ac.rs) at the Institute of Physics Belgrade has been extensively used for numerical calculations.
Figure 1 :
1L(T ) curves for direct, or prograde (green, upper set of points) and retrograde (blue, lower set of points) BHH orbits, all at fixed energy E = −0.5.
Figure 2 :
2L(T) dependence of retrograde BHH orbits' (blue dots of different hues) and their satellites' (red), with various values of k, all at fixed energy E = −0.5. The data are fromTable I.
Figure 3 :
3L(T'=T/k) dependence at fixed energy E = −0.5 for the aggregate set of retrograde BHH orbits (blue dots of different hues) and their satellites (red dots) with various values of k, together with the fitted interpolating function (blue solid). The data are from
Figure 4 :
4Enlargement of the L∈[1.5, 3] region of the retrograde BHH orbits (blue dots) and their satellites (red dots) with various values of k L(T'=T/k) dependence at fixed energy E = −0.5. Note that the size of the dots on the diagram exceeds the corresponding numerical uncertainties ("error bars").
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The search method is described, tables with the data regarding the new BHH satellites, together with numerical uncertainties, are shown. We briefly discuss Hénon's conjecture about KAM stability of BHH orbits. 10.1103/PhysRevLett.000.000000forSee the Supplemental Material atSee the Supplemental Mate- rial at http://link.aps.org/ supplemental/10.1103/PhysRevLett.000.000000 for [The search method is described, tables with the data regarding the new BHH satellites, together with numerical uncertainties, are shown. We briefly discuss Hénon's conjecture about KAM stability of BHH orbits.].
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Satellite orbits of the figure-eight were first observed in Ref. [2] and further investigated in Refs. 3, 4Satellite orbits of the figure-eight were first observed in Ref. [2] and further investigated in Refs. [3, 4]
The other one being the Lagrange family of orbits. The other one being the Lagrange family of orbits.
The presence of multiple satellites with the same topology is not the first known instance of its kind: there are (many) different satellites of the figure-eight orbit with identical values of k, see Ref. 3, 4], albeit with zero angular momentumThe presence of multiple satellites with the same topol- ogy is not the first known instance of its kind: there are (many) different satellites of the figure-eight orbit with identical values of k, see Ref. [3, 4], albeit with zero an- gular momentum.
| []
|
[
"Information Accessibility and Cryptic Processes",
"Information Accessibility and Cryptic Processes"
]
| [
"John R Mahoney \nComplexity Sciences Center and Physics Department\nUniversity of California at Davis\nOne Shields Avenue95616DavisCA\n",
"Christopher J Ellison \nComplexity Sciences Center and Physics Department\nUniversity of California at Davis\nOne Shields Avenue95616DavisCA\n",
"James P Crutchfield \nComplexity Sciences Center and Physics Department\nUniversity of California at Davis\nOne Shields Avenue95616DavisCA\n\nSanta Fe Institute\n1399 Hyde Park Road, Santa Fe87501NM\n"
]
| [
"Complexity Sciences Center and Physics Department\nUniversity of California at Davis\nOne Shields Avenue95616DavisCA",
"Complexity Sciences Center and Physics Department\nUniversity of California at Davis\nOne Shields Avenue95616DavisCA",
"Complexity Sciences Center and Physics Department\nUniversity of California at Davis\nOne Shields Avenue95616DavisCA",
"Santa Fe Institute\n1399 Hyde Park Road, Santa Fe87501NM"
]
| []
| We give a systematic expansion of the crypticity-a recently introduced measure of the inaccessibility of a stationary process's internal state information. This leads to a hierarchy of k-cryptic processes and allows us to identify finite-state processes that have infinite crypticity-the internal state information is present across arbitrarily long, observed sequences. The crypticity expansion is exact in both the finite-and infinite-order cases. It turns out that k-crypticity is complementary to the Markovian finite-order property that describes state information in processes. One application of these results is an efficient expansion of the excess entropy-the mutual information between a process's infinite past and infinite future-that is finite and exact for finite-order cryptic processes. PACS numbers: 02.50.-r 89.70.+c 05.45.Tp 02.50.Ey | 10.1088/1751-8113/42/36/362002 | [
"https://arxiv.org/pdf/0905.4787v1.pdf"
]
| 16,385,728 | 0905.4787 | 6a7abd5d7c0bbe976d8b7cad6c6958ec500e6613 |
Information Accessibility and Cryptic Processes
29 May 2009 (Dated: May 29, 2009)
John R Mahoney
Complexity Sciences Center and Physics Department
University of California at Davis
One Shields Avenue95616DavisCA
Christopher J Ellison
Complexity Sciences Center and Physics Department
University of California at Davis
One Shields Avenue95616DavisCA
James P Crutchfield
Complexity Sciences Center and Physics Department
University of California at Davis
One Shields Avenue95616DavisCA
Santa Fe Institute
1399 Hyde Park Road, Santa Fe87501NM
Information Accessibility and Cryptic Processes
29 May 2009 (Dated: May 29, 2009)Santa Fe Institute Working Paper 09-05-XXX
We give a systematic expansion of the crypticity-a recently introduced measure of the inaccessibility of a stationary process's internal state information. This leads to a hierarchy of k-cryptic processes and allows us to identify finite-state processes that have infinite crypticity-the internal state information is present across arbitrarily long, observed sequences. The crypticity expansion is exact in both the finite-and infinite-order cases. It turns out that k-crypticity is complementary to the Markovian finite-order property that describes state information in processes. One application of these results is an efficient expansion of the excess entropy-the mutual information between a process's infinite past and infinite future-that is finite and exact for finite-order cryptic processes. PACS numbers: 02.50.-r 89.70.+c 05.45.Tp 02.50.Ey
INTRODUCTION
The data of phenomena come to us through observation. A large fraction of the theoretical activity of model building, though, focuses on internal mechanism. How are observation and modeling related? A first step is to frame the problem in terms of hidden processes-internal mechanisms probed via instruments that, in particular, need not accurately report a process's internal state. A practical second step is to measure the difference between internal structure and the information in observations.
We recently established that the amount of observed information a process communicates from the past to the future-the excess entropy-is the mutual information between its forward-and reverse-time minimal causal representations [1,2]. This closed-form expression gives a concrete connection between the observed information and a process's internal structure.
Excess entropy, and related mutual information quantities, are widely used diagnostics for complex systems. They have been applied to detect the presence of organization in dynamical systems [3,4,5,6], in spin systems [7,8,9], in neurobiological systems [10,11], and even in language [12,13], to mention only a very few uses. Thus, understanding how much internal state structure is reflected in the excess entropy is critical to whether or not these and other studies of complex systems can draw structural inferences about the internal mechanisms that produce observed behavior.
Unfortunately, there is a fundamental problem. The excess entropy is not the internal state information the process stores-rather, the latter is the process's statistical complexity [1,2]. On the positive side, there is a diagnostic. The difference between, if you will, experiment and theory (between observed information and internal structure) is controlled by the difference between a process's excess entropy and its statistical complexity. This difference is called the crypticity-how much internal state information is inaccessible [1,2]. Here we introduce a classification of processes using a systematic expansion of crypticity.
The starting point is computational mechanics's minimal causal representation of a stochastic process Pthe ǫ-machine [14,15]. There, a process is viewed as a channel that communicates information from the past, ← − X = . . . X −3 X −2 X −1 , to the future, − → X = X 0 X 1 X 2 . . ..
(X t takes values in a finite measurement alphabet A.)
The excess entropy is the shared (or mutual) information between the past and the future:
E = I[ ← − X ; − → X ]
. The amount of historical information that a process stores in the present is different. It is given by the Shannon information C µ = H[S] of the distribution over the ǫ-machine's causal states S. C µ is called the statistical complexity and the causal states are sets of pasts ← −
x that are equivalent for prediction [14]:
ǫ( ← − x ) = { ← − x ′ : Pr( − → X | ← − x ) = Pr( − → X | ← − x ′ )} .(1)
Causal states have a Markovian property that they render the past and future statistically independent; they shield the future from the past [15]:
Pr( ← − X , − → X |S) = Pr( ← − X |S) Pr( − → X |S) .(2)
ǫ-Machines are also unifilar [14,16]: From the start state, each observed sequence . . . x −3 x −2 x −1 . . . corresponds to one and only one sequence of causal states. The signature of unifilarity is that on knowing the current state and measurement, the uncertainty in the next state vanishes:
H[S t+1 |S t , X t ] = 0.
Although they are not the same, the basic relationship between these quantities is clear: E is the process's effective channel utilization and C µ is the sophistication of that channel. Their difference, one of our main concerns in the following, indicates how a process stores, manipulates, and hides internal state information.
Until recently, E could not be as directly calculated from the ǫ-machine as the process's entropy rate h µ and its statistical complexity. Ref. [1] and Ref. [2] solved this problem, giving a closed-form expression for the excess entropy:
E = I[S + ; S − ] ,(3)
where S + are the causal states of the process scanned in the "forward" direction and S − are the causal states of the process scanned in the "reverse" time direction. This result comes in a historical context. Some time ago, an explicit expression for the excess entropy had been developed from the Hamiltonian for onedimensional spin chains with range-R interactions [8]:
E = C µ − R h µ .(4)
A similar, but slightly less compact form is known for order-R Markov processes:
E = H[X R 0 ] − R h µ ,(5)
where X R 0 = X 0 , . . . , X R−1 . It has also been known for some time that the statistical complexity is an upper bound on the excess entropy [16]:
E ≤ C µ ,
which follows from the equality derived there:
E = C µ − H[S + | − → X ] .
Using forward and reverse ǫ-machines, Ref. [1] extended this, deriving the closed-form expression for E in Eq. (3) and two new bounds on E: E ≤ C − µ and E ≤ C + µ . It also showed that:
H[S + | − → X ] = H[S + |S − ](6)
and identified this quantity as controlling how a process hides its internal state information. For this reason, it is called the process's crypticity:
χ + = H[S + | − → X ] .(7)
In the context of forward and reverse ǫ-machines, one must distinguish two crypticities; depending on the scan direction one has:
χ + = H[S + |S − ] or χ − = H[S + |S − ] .
In the following we will not concern ourselves with reverse representations and so can simplify the notation, using C µ for C + µ and χ for χ + . Here we show that, for a restricted class of processes, the crypticity in Eq. (6) can be systematically expanded to give an alternative closed-form to the excess entropy in Eq. (3). One ancillary benefit is a new and, we argue, natural hierarchy of processes in terms of information accessibility.
K-CRYPTICITY
The process classifications based on spin-block length and order-R Markov are useful. They give some insight into the nature of the kinds of process we can encounter and, concretely, they allow for closed-form expressions for the excess entropy (and other system properties). In a similar vein, we wish to carve the space of processes with a new blade. We define the class of k-cryptic processes and develop their properties and closed-form expressions for their excess entropies.
For convenience, we need to introduce several shorthands. First, to denote a symbol sequence that begins at time t and is L symbols long, we write X L t . Note that X L t includes X t+L−1 , but not X t+L . Second, to denote a symbol sequence that begins at time t and continues on to infinity, we write − → X t .
Definition. The k-crypticity criterion is satisfied when
H[S k | − → X 0 ] = 0 .(8)
Definition. A k-cryptic process is one for which the process's ǫ-machine satisfies the k-crypticity criterion.
Definition. An ∞-cryptic process is one for which the process's ǫ-machine does not satisfy the k-crypticity criterion for any finite k.
Lemma 1. H[S k | − → X 0 ] is a nonincreasing function of k.
Proof. This follows directly from stationarity and the fact that conditioning on more random variables cannot increase entropy:
H[S k+1 | − → X 0 ] = [S k | − → X −1 ] ≤ H[S k | − → X 0 ] .
Lemma 2.
If P is k-cryptic, then P is also j-cryptic for all j > k.
Proof. Being k-cryptic implies H[S k | − → X 0 ] = 0. Applying Lem. 1, H[S j | − → X 0 ] ≤ H[S k | − → X 0 ] = 0.
By positivity of entropy, we conclude that P is also j-cryptic.
This provides us with a new way of partitioning the space of processes. We create a parametrized class of sets {χ k : k = 0, 1, 2, . . .}, where χ k = {P : k-cryptic and not (k − 1)-cryptic}.
The following result provides a connection to a very familiar class of processes. Proposition 1. If a process P is order-k Markov, then it is k-cryptic.
Proof. If P is order-k Markov, then H[S k |X k 0 ] = 0. Conditioning on more variables does not increase uncertainty, so:
H[S k |X k 0 , − → X k ] = 0 .
But the lefthand side is H[S k | − → X 0 ]. Therefore, P is k-cryptic.
Note that the converse of Prop. 1 is not true. For example, the Even Process (EP), the Random Noisy Copy Process (RnC), and the Random Insertion Process (RIP) (see Ref. [1] and Ref. [2]), are all 1-cryptic, but are not order-R Markov for any finite R.
Note also that Prop. 1 does not preclude an order-k Markov process from being j-cryptic, where j < k. Later we will show an example demonstrating this.
Given a process, in general one will not know its crypticity order. One way to investigate this is to study the sequence of estimates of χ at different orders. To this end, we define the k-cryptic approximation.
Definition. The k-cryptic approximation is defined as
χ(k) = H[S 0 |X k 0 , S k ] .
The k-Cryptic Expansion
We will now develop a systematic expansion of χ to order k in which χ(k) appears directly and the k-crypticity criterion plays the role of an error term. Theorem 1. The process crypticity is given by
χ = χ(k) + H[S k | − → X 0 ] .(9)
Proof. We calculate directly, starting from the definition, adding and subtracting the k-crypticity criterion term from χ's definition, Eq. (7):
χ = H[S 0 | − → X 0 ] − H[S k | − → X 0 ] + H[S k | − → X 0 ] .
We claim that the first two terms are χ(k). Expanding the conditionals in the purported χ(k) terms and then canceling, we get joint distributions:
H[S 0 | − → X 0 ] − H[S k | − → X 0 ] = H[S 0 , − → X 0 ] − H[S k , − → X 0 ] .
Now, splitting the future into two pieces and using this to write conditionals, the righthand side becomes:
H[ − → X k |S 0 , X k 0 ] + H[S 0 , X k 0 ] − H[ − → X k |S k , X k 0 ] − H[S k , X k 0 ] .
Appealing to the ǫ-machine's unifilarity, we then have:
H[ − → X k |S k ] + H[S 0 , X k 0 ] − H[ − → X k |S k , X k 0 ] − H[S k , X k 0 ] .
Now, applying causal shielding gives:
H[ − → X k |S k ] + H[S 0 , X k 0 ] − H[ − → X k |S k ] − H[S k , X k 0 ] .
Canceling terms, this simplifies to:
H[S 0 , X k 0 ] − H[S k , X k 0 ] .
We now re-expand, using unifilarity to give:
H[S 0 , X k 0 , S k ] − H[S k , X k 0 ] .
Finally, we combine these, using the definition of conditional entropy, to simplify again:
H[S 0 |X k 0 , S k ] .
Note that this is our definition of χ(k). This establishes our original claim:
χ = χ(k) + H[S k | − → X 0 ] ,
with the k-crypticity criterion playing the role of an approximation error.
Corollary 1. A process P is k-cryptic if and only if χ = χ(k) .
Proof. Given the order-k expansion of χ just developed, we now assume the k-crypticity criterion is satisfied; viz.,
H[S k | − → X 0 ] = 0.
Thus, we have from Eq. (9):
χ = χ(k) .
Likewise, assuming χ = χ(k) requires, by Eq. (9) that H[S k | − → X 0 ] = 0 and thus the process is k-cryptic.
Corollary 2. For any process, χ(0) = 0.
Proof.
χ(0) = H[S 0 |X 0 0 , S 0 ] = H[S 0 |S 0 ] = 0 . Convergence Proposition 2. The approximation χ(k) is a nonde- creasing function of k. Proof. Lem. 1 showed that H[S k | − → X 0 ]
is a nonincreasing function of k. By Thm. 1, χ(k) must be a nondecreasing function of k.
Corollary 3. Once χ(k) reaches the value χ, χ(j) = χ for all j > k.
Proof. If there exists such a k, then by Thm. 1 the process is k-cryptic. By Lem. 2, the process is j-cryptic for all j > k. Again, by Thm. 1, χ(j) = χ. Proof. Applying stationarity,
χ(1) = H[S 0 |X 0 , S 1 ] = H[S k |X k , S k+1 ].
We are given χ(1) = 0 and so H[S k |X k , S k+1 ] = 0. We use this below. Expanding χ(k + 1),
χ(k + 1) = H[S 0 |X k+1 0 , S k+1 ] = H[S 0 |X k 0 , X k , S k+1 ] = H[S 0 |X k 0 , S k , X k , S k+1 ] ≤ H[S 0 |X k 0 , S k ] = χ(k) .
The third line follows from χ(1) = 0. By Prop. 2, χ(k + 1) ≥ χ(k). Therefore, χ(k + 1) = χ(k). Finally, using χ(1) = 0, we have by induction that χ(k) = 0 for all k.
Corollary 6. If there is a k ≥ 1 for which χ(k) = 0, then χ(j) = 0 for all j ≥ 1.
Proof. This follows by composing Cor. 4 with Cor. 5.
Together, the proposition and its corollaries show that χ(k) is a nondecreasing function of k which, if it reaches χ at a finite k, remains at that value for all larger k. Proposition 3. The cryptic approximation χ(k) converges to χ as k → ∞.
Proof. Note that χ = lim k→∞ H[S 0 |X k 0 ] and recall that
χ(k) = H[S 0 |X k 0 , S k ].
We show that the difference approaches zero:
H[S 0 |X k 0 ] − H[S 0 |X k 0 , S k ] = H[S 0 , X k 0 ] − H[X k 0 ] − H[S 0 , X k 0 , S k ] + H[X k 0 , S k ] = H[S 0 , X k 0 ] − H[X k 0 ] − H[S 0 , X k 0 ] + H[X k 0 , S k ] = H[X k 0 , S k ] − H[X k 0 ] = H[S k |X k 0 ] . Moreover, lim k→∞ H[S k |X k 0 ]
= 0 by the ǫ map from pasts to causal states of Eq. (1). Therefore, as k → ∞, χ(k) → χ.
Excess Entropy for k-Cryptic Processes
Given a k-cryptic process, we can calculate its excess entropy in a form that involves a sum of ∝ |A k | terms, where each term involves products of k matrices. Specifically, we have the following.
Corollary 7. A process P is k-cryptic if and only if E = C µ − χ(k).
Proof. From Ref. [1], we have E = C µ − χ, and by Cor. 1, χ = χ(k). Together, these complete the proof.
The following proposition is a simple and useful consequence of the class of k-cryptic processes.
Corollary 8. A process P is 0-cryptic if and only if E = C µ .
Proof. If P is 0-cryptic, our general expression then reads
E = C µ − H[S 0 |X 0 0 , S 0 ] = C µ .
To establish the opposite direction, E = C µ and Cor. 7 imply that χ(k) = 0 for all k. In particular, χ(0) and the process is 0-cryptic.
Crypticity versus Markovity
Equation (4) and Equation (5) give expressions for E in the cases when the process is order-R Markov and when it is an order-R spin chain. These results hinge on whether or not H[X R 0 ] = C µ . Reference [8] stated a condition under which equality holds in terms of transfer matrices. Here we state a simpler condition by equating two chain rule expansions of
H[X R 0 , S R ]: H[X R 0 |S R ] + H[S R ] = H[S R |X R 0 ] + H[X R 0 ] . H[S R |X R 0 ]
= 0 by virtue of the fact that each such (history) word maps to exactly one causal state by Eq. (1). Thus, we conclude that for order-R Markov processes:
H[X R 0 ] = H[S R ] ⇐⇒ H[X R 0 |S R ] = 0 .
So, an order-R Markov process is also a spin chain if and only if H[X R 0 |S R ] = 0. This means that there is a 1 − 1 correspondence between the R-blocks and causal states, confirming the interpretation specified in Ref. [8].
We can also extend the condition for H[X R 0 ] = C µ to the results presented here in the following way.
Proposition 4. H[X R 0 |S R ] = 0 ⇐⇒ χ(R) = R h µ ,(10)
where h µ is the process's entropy rate.
Proof. The proof is a direct calculation:
χ(R) = H[S 0 |X R 0 , S R ] = H[S 0 , X R 0 ] − H[X R 0 , S R ] = H[S 0 , X R 0 ] − H[X R 0 |S R ] − H[S R ] = H[S 0 , X R 0 ] − H[X R 0 |S R ] − H[S 0 ] = H[X R 0 |S 0 ] − H[X R 0 |S R ] = Rh µ − H[X R 0 |S R ] .
Proposition 5. Periodic processes can be arbitrary order-R Markov, but are all 0-cryptic.
Proof. According to Ref. [17], we have E = C µ . By Cor. 8 the process is 0-cryptic.
Proposition 6.
A positive entropy-rate process that is an order-R Markov spin chain is not (R − 1)-cryptic.
Proof. Assume that the order-R Markov spin chain is (R − 1)-cryptic. For R ≥ 1, If the process is (R − 1)-cryptic, then by Cor. 1 χ(R − 1) = χ. Combining this with the above Prop. 4, we have
χ(R − 1) = (R − 1)h µ − H[X R−1 0 |S R−1 ].
If it is an order-R Markov spin chain, then we also have from Eq. (4) that χ = Rh µ . Combining this with the previous equation, we find that H[X R−1
0 |S R−1 ] = −h µ .
By positivity of conditional entropies, we have reached a contradiction. Therefore an order-R Markov spin chain must not be (R − 1)-cryptic.
For R = 0, the proof also holds since negative cryptic orders are not defined.
Proposition 7.
A positive entropy-rate process that is an order-R Markov spin chain is not (R − n)-cryptic for any 1 ≥ n ≥ R.
Proof. For R ≥ 1, By Lem. 2, if the process were (R − n)-cryptic for some 1 ≥ n ≥ R, then it would be (R − 1)-cryptic. By Prop. 6, this is not true. Therefore, the primitive orders of Markovity and crypticity are the same. Similarly, for R = 0, the proof also holds since negative cryptic orders are not defined.
EXAMPLES
It is helpful to see crypticity in action. We now turn to a number of examples to illustrate how various orders of crypticity manifest themselves in ǫ-machine structure and what kinds of processes are cryptic and so hide internal state information from an observer. For details (transition matrices, notation, and the like) not included in the following and for complementary discussions and analyses of them, see Refs. [1,2,17].
We start at the bottom of the crypticity hierarchy with a 0-cryptic process and then show examples of 1-cryptic and 2-cryptic processes. Continuing up the hierarchy, we generalize and give a parametrized family of processes that are k-cryptic. Finally, we demonstrate an example that is ∞-cryptic.
Even Process: 0-Cryptic Figure 1 gives the ǫ-machine for the Even Process. The Even Process produces binary sequences in which all blocks of uninterrupted 1s are even in length, bounded by zeros. Further, after each even length is reached, there is a probability p of breaking the block of 1s by inserting one or more 0s. Reference [2] showed that the Even Process is 0-cryptic with a statistical complexity of C µ = H (1/(2 − p)), an entropy rate of h µ = H(p)/(2 − p), and crypticity of χ = 0. If p = 1 2 , then C µ = log 2 (3)− 2 3 bits and E = log 2 (3)− 2 3 bits. (As Ref. [2] notes, these closed-form expressions for C µ and E have been known for some time.) To see why the Even Process is 0-cryptic, note that if X = 0, then S 0 = A; and if X = 1, then S 0 = B. Therefore, the 0-crypticity criterion of Eq. (8) is satisfied.
It is important to note that this process is not order-R Markov for any finite R [17]. Nonetheless, our new expression for E is valid. This shows the broadening of our ability to calculate E even for low complexity processes that are, in effect, infinite-order Markov.
Golden Mean Process: 1-Cryptic Figure 2 shows the ǫ-machine for the Golden Mean Process [17]. The Golden Mean Process is one in which no two 0s occur consecutively. After each 1, there is a probability p of generating a 0. As sequence length grows, the ratio of the number of allowed words of length L to the number of allowed words at length L − 1 approaches the golden ratio; hence, its name. The Golden Mean Process ǫ-machine looks remarkably similar to that for the Even Process. The informational analysis, however, shows that they have markedly different properties. Reference [2] showed that the Golden Mean Process has the same statistical complexity and entropy rate as the Even Process: C µ = H (1/(2 − p)) and h µ = H(p)/(2 − p). However, the crypticity is not zero (for 0 < p < 1). From Cor. 1 we calculate:
χ = χ(1) = H[S 0 |X 1 0 , S 1 ] = H[S 0 |X 1 0 ] = P r(0)H[S 0 |X 0 = 0] + P r(1)H[S 0 |X 0 = 1] = H(p)/(2 − p) . If p = 1 2 , C µ = log 2 (3) − 2 3
bits, an excess entropy of E = log 2 (3) − 4 3 bits, and a crypticity of χ = 2 3 . Thus, the excess entropy differs from that of the Even Process. (As with the Even Process, these closed-form expressions for C µ and E have been known for some time.)
The Golden Mean Process is 1-cryptic. To see why, it is enough to note that it is order-1 Markov. By Prop. 1, it is 1-cryptic. We know it is not 0-cryptic since any future beginning with 1 could have originated in either state A or B. In addition, the spin-block expression for excess entropy of Ref. [17], Eq. (4) here, applies for an R = 1 Markov chain.
Butterfly Process: 2-Cryptic
The next example, the Butterfly Process of Fig. 3, illustrates in a more explicit way than possible with the previous processes the role that crypticity plays and how it can be understood in terms of an ǫ-machine's structure. Much of the explanation does not require calculating much, if anything. It is first instructive to see why the Butterfly Process is not 1-cryptic.
A B C D E 1 2 |2 1 2 |0 1 2 |1 1 2 |3 1 2 |0 1 2 |1 1 2 |4 1 2 |6 1 2 |5
If we can find a family { − → x 0 } such that H[S 1 | − → X 0 = − → x 0 ] = 0, then the total conditional entropy will be positive and, thus, the machine will not be 1-cryptic. To show that this can happen, consider the future − → x 0 = (0, 1, 2, 4, 4, 4, . . .). It is clear that the state following 1 must be A. Thus, in order to generate 0 or 1 before arriving at A, the state pair (S 0 , S 1 ) can be either (B, C) or (D, E). This uncertainty in S 1 is enough to break the criterion. And this occurs for the family
{ − → x 0 } = {0, 1, . . .}.
To see that the process is 2-cryptic, notice that the two paths (B, C) and (D, E) converge on A. Therefore, there is no uncertainty in S 2 given this future. It is reasonably straightforward to see that indeed any (X 0 , X 1 ) will lead to a unique causal state. This is because the Butterfly Process is a very limited version of an 8-symbol order-2 Markov process.
Note that the transition matrix is doubly-stochastic and so the stationary distribution is uniform. The statistical complexity is rather direct in this case: C µ = log 2 (5). We now can calculate χ using Cor. 1: (01) For comparison, if we had assumed the Butterfly Process was 1-cryptic, then we would have:
χ = χ(2) = H[S 0 |X 2 0 , S 2 ] = H[S 0 |X 2 0 ] = PrE = C µ − χ(1) = C µ − (H[S 0 , X 0 ] − H[S 1 , X 0 ]) ≈ log 2(5) − (3.3219 − 2.5062) = log 2(5) − 0.8156 ≈ 1.5063 bits.
We can see that this is substantially below the true value: a 25% error.
Restricted Golden Mean: k-Cryptic
Now we turn to illustrate a crypticity-parametrized family of processes, giving examples of k-cryptic processes for any k. We call this family the Restricted Golden Mean as its support is a restriction of the Golden Mean support. (See Fig. 4 for its ǫ-machines.) The k = 1 member of the family is exactly the Golden Mean.
It is straightforward to see that this process is order-k Markov. Proposition 1 then implies it is (at most) k-cryptic. In order to show that it is not (k − 1)-cryptic, consider the case − → x 0 = 1 k , 0, . . .. The first (k − 1) 1s will induce a mixture over states k and 0. The following future − → x k = 1, 0, . . . is consistent with both states k and 0. Therefore, the (k − 1)-crypticity criterion is not satisfied. Therefore, it is k-cryptic. For arbitrary k, there are k + 1 causal states and the stationary distribution is:
π = 2 k + 2 , 1 k + 2 , 1 k + 2 , . . . , 1 k + 2 .
The statistical complexity is
C µ = log 2 (k + 2) − 2 k + 2 .
For the k-th member of the family, we have for the crypticity:
χ = χ(k) = 2k k + 2 .
And the excess entropy follows directly from Cor. 7:
E = C µ − χ = log 2 (k + 2) − 2(k + 1) k + 2 ,
which diverges with k. (Calculational details will be provided elsewhere.)
Stretched Golden Mean
The Stretched Golden Mean is a family of processes that does not occupy the same support as the Golden Mean. Instead of requiring that blocks of 0s are of length 1, we require that they are of length k. Here, the Markov order (k) grows, but the cryptic order remains 1 for all k.
Again, it is straightforward to see that this process is order-k Markov. To see that it is 1-cryptic, first note that if X 0 = 1, then S 1 = 0. Next consider the case when X 0 = 0. If the future − → x 1 = 1, . . ., then S 1 = k. Similarly, if the future − → x 1 = 0 n , 1, . . ., then S 1 = k − n. This family exhibits arbitrary separation between its Markov order and its cryptic order and so demonstrates that these properties are not redundant. The stationary distribution is the same as for the Restricted Golden Mean and so, then, is the statistical complexity. In addition, we have:
χ = χ(1) = H[S 0 |X 0 , S 1 ] = h µ .
Consequently,
E = C µ − χ = C µ − h µ .
The Nemo Process: ∞-Cryptic
We close our cryptic process bestiary with a (very) finite-state process that has infinite crypticity: The three-state Nemo Process. Over no finite-length sequence will all of the internal state information be present in the observations. The Nemo Process ǫ-machine is shown in Fig. 6.
Its stationary state distribution is
Pr(S) ≡ π = 1 3 − 2p A B C 1 1 − p 1 − p ,
from which one calculates the statistical complexity:
C µ = log 2 (3 − 2p) − 2(1 − p) 3 − 2p log 2 (1 − p) . A B C p|1 1 − p|0 1|0 1 − q|0 q|1 FIG[S k | − → X 0 = − → x ] > 0.
The family of futures we use begins with all 0s and then has a 1. Intuitively, the 1 is chosen because it is a synchronizing word for the process-after observing a 1, the ǫ-machine is always in state A. Then, causal shielding will decouple the infinite future from the first few symbols, thereby allowing us to compute the conditional entropies for the entire family of futures.
First, recall the shorthand:
Pr(S k | − → X 0 ) = lim L→∞ Pr(S k |X L 0 ) .
Without loss of generality, assume k < L. Then,
Pr(S k |X L 0 ) = Pr(X k 0 , S k , X L k ) Pr(X L 0 ) = Pr(X L k |X k 0 , S k ) Pr(X k 0 , S k ) Pr(X L 0 ) = Pr(X L k |S k ) Pr(X k 0 , S k ) Pr(X L 0 )
, where the last step is possible since the causal states are Markovian [15], shielding the past from the future. Each of these quantities is given by:
Pr(X L k = w|S k = σ) = [T (w) 1] σ Pr(X k 0 = w, S k = σ) = [πT (w) ] σ Pr(X L 0 = w) = πT (w) 1 .
where T (w) ≡ T (x0) T (x1) · · · T (xL−1) , 1 is a column vector of 1s, and T (x) σσ ′ = Pr(S ′ = σ ′ , X = x|S = σ). To establish H[S k | − → X 0 ] > 0 for any k, we rely on using values of k that are multiples of three. So, we concentrate on the following for n = 0, 1, 2, . . .:
H[S 3n |X 3n+1 0 = 0 3n 1, − → X 3n+1 ] > 0 .
Since 1 is a synchronizing word, we can greatly simplify the conditional probability distribution. First, we freely include the synchronized causal state A and rewrite the conditional distribution as fraction:
Pr(S 3n |X 3n+1 0 = 0 3n 1, − → X 3n+1 ) = Pr(S 3n |X 3n+1 0 = 0 3n 1, S 3n+1 = A, − → X 3n+1 ) = Pr(S 3n , X 3n+1 0 = 0 3n 1, S 3n+1 = A, − → X 3n+1 ) Pr(X 3n+1 0 = 0 3n 1, S 3n+1 = A, − → X 3n+1 ) .
Then, we factor everything except − → X 3n+1 out of the numerator and make use of causal shielding to simplify the conditional. For example, the numerator becomes:
Pr(S 3n , X 3n+1 0 = 0 3n 1, S 3n+1 = A, − → X 3n+1 ) = Pr( − → X 3n+1 |S 3n , X 3n+1 0 = 0 3n 1, S 3n+1 = A) × Pr(S 3n , X 3n+1 0 = 0 3n 1, S 3n+1 = A) = Pr( − → X 3n+1 |S 3n+1 = A) × Pr(S 3n , X 3n+1 0 = 0 3n 1, S 3n+1 = A) = Pr( − → X 3n+1 |S 3n+1 = A) Pr(S 3n , X 3n+1 0 = 0 3n 1) .
Similarly, the denominator becomes:
Pr(X 3n+1
0 = 0 3n 1, S 3n+1 = A, − → X 3n+1 ) = Pr( − → X 3n+1 |S 3n+1 = A) Pr(X 3n+1 0 = 0 3n 1) .
Combining these results, we obtain a finite form for the entropy of S 3n conditioned on a family of infinite futures, first noting:
Pr(S 3n |X 3n+1 0 = 0 3n 1, − → X 3n+1 ) = Pr(S 3n |X 3n+1 0 = 0 3n 1) .
Thus, for all − → x 3n+1 , we have:
H[S 3n |X 3n+1 0 = 0 3n 1, − → X 3n+1 = − → x 3n+1 ] = H[S 3n |X 3n+1 0 = 0 3n 1] .
Now, we are ready to compute the conditional entropy for the entire family. First, note that T (0) raised to the third power is a diagonal matrix with each element equal to (1 − p)(1 − q). Thus, for j = 1, 2, 3 . . .:
T (0) 3j σσ = (1 − p) j (1 − q) j .
Using all of the above relations, we can easily calculate:
Pr(S 3n |X 3n+1 0 = 0 3n+1 1) = 1 3 − 2p A B C p 0 q(1 − p) .
Thus, for p, q ∈ (0, 1), we have:
H[S 3n | − → X 0 ] ≥ H[S 3n |X 3n+1 0 = 0 3n 1, − → X 3n+1 ] = − → x 3n+1 Pr X 3n+1 0 = 0 3n 1, − → X 3n+1 = − → x 3n+1 × H[S 3n |X 3n+1 0 = 0 3n 1, − → X 3n+1 = − → x 3n+1 ] = H[S 3n |X 3n+1 0 = 0 3n 1] × − → x 3n+1 Pr X 3n+1 0 = 0 3n 1, − → X 3n+1 = − → x 3n+1 = H[S 3n |X 3n+1 0 = 0 3n 1] Pr(X 3n+1 0 = 0 3n 1) = p 3 − 2 log 2 3 − 2p p + q(1 − p) 3 − 2p log 2 q(1 − p) 3 − 2p × [(1 − p)(1 − q)] 3n > 0 . So, any time k is a multiple of three, H[S k | − → X 0 ] > 0. Finally, suppose (k mod 3) = i, where i = 0. That is, suppose k is not a multiple of three. By Lem. 1, H[S k | − → X 0 ] ≥ H[S k+i | − → X 0 ]
and, since we just showed that the latter quantity is always strictly greater than zero, we conclude that H[S k | − → X 0 ] > 0 for every value of k. The above establishes that the Nemo Process does not satisfy the k-crypticity criterion for any finite k. Thus, the Nemo process is ∞-cryptic. This means that we cannot make use of the k-cryptic approximation to calculate χ or E.
Fortunately, the techniques introduced in Ref. [1] and Ref. [2] do not rely on an approximation method. To avoid ambiguity denote the statistical complexity we just computed as C + µ . When the techniques are applied to the Nemo Process, we find that the process is causally reversible (C + µ = C − µ ) and has the following forwardreverse causal-state conditional distribution:
Pr(S + |S − ) = 1 p + q − pq A B C D p 0 q(1 − p) E 0 q p(1 − q) F q p(1 − q) 0 .
With this, one can calculate E, in closed-form, via:
E = C + µ − H[S + |S − ] .
(Again, calculational details will be provided elsewhere.)
CONCLUSION
Calculating the excess entropy I[ ← − X ; − → X ] is, at first blush, a daunting task. We are asking for a mutual information between two infinite sets of random variables.
Appealing to E = I[S;
− → X ], we use the compact representation of the ǫ-machine to reduce one infinite set (the past) to a (usually) finite set. A process's k-crypticity captures something similar about the infinite set of future variables and allows us to further compact our form for excess entropy, reducing an infinite variable set to a finite one. The resulting stratification of process space is a novel way of thinking about its structure and, as long as we know which stratum we lie in, we can rapidly calculate many quantities of interest.
Unfortunately, in the general case, one will not know a priori a process's crypticity order. Worse, as far as we are aware, there is no known finite method for calculating the crypticity order. This strikes us as an interesting open problem and challenge.
If, by construction or by some other means, one does know it, then, as we showed, crypticity and E can be calculated using the crypticity expansion. Failing this, though, one might consider using the expansion to search for the order. There is no known stopping criterion, so this search may not find k in finite time. Moreover, the expansion is a calculation that grows exponentially in computational complexity with crypticity order, as we noted. Devising a stopping criterion would be very useful to such a search.
Even without knowing the k-crypticity, the expansion is often still useful. For use in estimating E, it provides us with a bound from above. This is complementary to the bound below one finds using the typical expansion E(L) = H[X L 0 ] − h µ L [17]. Using these upper and lower bounds, one may determine that for a given purpose, the estimate of χ or E is within an acceptable tolerance.
The crypticity hierarchy is a revealing way to carve the space of processes in that it concerns how they hide internal state information from an observer. The examples were chosen to illustrate several features of this new view. The Even Process, a canonical example of order-∞ Markov, resides instead at the very bottom of this ladder. The two example families show us how k-cryptic is neither a parallel nor independent concept to order-R Markov. Finally, we see in the last example an apparently simple process with ∞-crypticity.
The general lesson is that internal state information need not be immediately available in measurement values, but instead may be spread over long measurement sequences. If a process is k-cryptic and k is finite, then internal state information is accessible over sequences of length k. The existence, as we demonstrated, of processes that are ∞-cryptic is rather sobering. (The Appendix comments on what happens when one fails to appreciate this.) Interpreted as a statement of the impossibility of extracting state information, it reminds us of earlier work on hidden spatial dynamical systems that exhibit a similar encrypting of internal structure in observed spacetime patterns [18].
Due to the exponentially growing computational effort to search for the crypticity order and, concretely, the existence of ∞-cryptic processes, the general theory introduced in Ref. [1] and Ref. [2] is seen to be necessary. It allows one to directly calculate E and crypticity and to do so efficiently.
Corollary 4 .
4If there is a k ≥ 1 for which χ(k) = 0, then χ(1) = 0.Proof. By positivity of the conditional entropy H[S 0 |X 0 , S 1 ],
FIG. 1 :
1A 0-cryptic process: Even Process. The transitions denote the probability p of generating symbol x as p|x.
FIG. 2: A 1-cryptic process: Golden Mean Process.
FIG. 3: A 2-cryptic process: Butterfly Process over a 6symbol alphabet.
FIG
. 4: k-cryptic processes: Restricted Golden Mean Family.
FIG
. 5: k-cryptic processes: Stretched Golden Mean Family.
. 6: The ∞-cryptic Nemo Process.The Nemo Process is not a finite-cryptic process. That
is, there exists no finite k for which H[S k |
− →
X 0 ] = 0. To
show this, we must demonstrate that there exists a family
of futures such that for each future H
AcknowledgmentsChris Ellison was partially supported on a GAANN fellowship. The Network Dynamics Program funded by Intel Corporation also partially supported this work.APPENDIX: CRYPTICITY UNTAMEDRecently, Ref.[19]asserted that a process's E can be obtained from its ǫ-machine using the following expression:. Though renamed, I erased is the crypticity of Ref.[1]. However, as we showed in the main development, it is χ + (1) and so the above expression is valid only for 0-cryptic and 1-cryptic processes.Ref.[19]considered only the Even and Golden Mean Processes. These, as we saw, are 0-cryptic and 1-cryptic and so it is no surprise that the expression worked. Indeed, their low-order crypticity is why closed-form expressions for their excess entropies have been known for quite some time, prior to the recent developments.In short, the claims in Ref.[19]are incorrect. The implication there that all ǫ-machines are 1-cryptic is also. The examples we gave show how wrong such an approximation can be. We showed how large the errors can grow. The full theory of Ref.[1]and Ref.[2]is required. The richness of the space of processes leads us to conjecture that it will suffer no shortcuts.
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| []
|
[
"The number of triple systems without even cycles",
"The number of triple systems without even cycles"
]
| [
"Dhruv Mubayi ",
"Lujia Wang "
]
| []
| []
| For k 4, a loose k-cycle C k is a hypergraph with distinct edges e 1 , e 2 , . . . , e k such that consecutive edges (modulo k) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that for every even integer k 4, there exists c > 0 such that the number of triple systems with vertex set [n] containing no C k is at most 2 cn 2 .An easy construction shows that the exponent is sharp in order of magnitude. This may be viewed as a hypergraph extension of the work of Morris and Saxton, who proved the analogous result for graphs which was a longstanding problem. For r-uniform hypergraphs with r > 3, we improve the trivial upper bound but fall short of obtaining the order of magnitude in the exponent, which we conjecture is n r−1 .Our proof method is different than that used for most recent results of a similar flavor about enumerating discrete structures, since it does not use hypergraph containers. One novel ingredient is the use of some (new) quantitative estimates for an asymmetric version of the bipartite canonical Ramsey theorem.F as a (not necessarily induced) subgraph. Henceforth we will call G an F -free r-graph. Write Forb r (n, F ) for the set of F -free r-graphs with vertex set [n]. Since each subgraph of an F -free r-graph is also F -free, we trivially obtain |Forb r (n, F )| 2 ex r (n,F ) by taking subgraphs of an F -free r-graph on [n] with the maximum number of edges. On the other hand for fixed r and F ,so the issue at hand is the factor log n in the exponent. The work of Erdős-Kleitman-Rothschild [25] and Erdős-Frankl-Rödl [26] for graphs, and Nagle-Rödl-Schacht [45] for hypergraphs (see also[44]for the case r = 3) improves the upper bound above to obtain |Forb r (n, F )| = 2 ex r (n,F )+o(n r ) .Although much work has been done to improve the exponent above (see[1,6,7,8,31,34,48]for graphs and[10,11,21,47,13,50]for hypergraphs), this is a somewhat satisfactory state of affairs when ex r (n, F ) = Ω(n r ) or F is not r-partite. | 10.1007/s00493-018-3765-6 | [
"https://arxiv.org/pdf/1701.00269v2.pdf"
]
| 119,325,194 | 1701.00269 | 02c0d6d64382d185e301168d29a7a6645fa995dd |
The number of triple systems without even cycles
8 Feb 2017 November 10, 2018
Dhruv Mubayi
Lujia Wang
The number of triple systems without even cycles
8 Feb 2017 November 10, 2018
For k 4, a loose k-cycle C k is a hypergraph with distinct edges e 1 , e 2 , . . . , e k such that consecutive edges (modulo k) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that for every even integer k 4, there exists c > 0 such that the number of triple systems with vertex set [n] containing no C k is at most 2 cn 2 .An easy construction shows that the exponent is sharp in order of magnitude. This may be viewed as a hypergraph extension of the work of Morris and Saxton, who proved the analogous result for graphs which was a longstanding problem. For r-uniform hypergraphs with r > 3, we improve the trivial upper bound but fall short of obtaining the order of magnitude in the exponent, which we conjecture is n r−1 .Our proof method is different than that used for most recent results of a similar flavor about enumerating discrete structures, since it does not use hypergraph containers. One novel ingredient is the use of some (new) quantitative estimates for an asymmetric version of the bipartite canonical Ramsey theorem.F as a (not necessarily induced) subgraph. Henceforth we will call G an F -free r-graph. Write Forb r (n, F ) for the set of F -free r-graphs with vertex set [n]. Since each subgraph of an F -free r-graph is also F -free, we trivially obtain |Forb r (n, F )| 2 ex r (n,F ) by taking subgraphs of an F -free r-graph on [n] with the maximum number of edges. On the other hand for fixed r and F ,so the issue at hand is the factor log n in the exponent. The work of Erdős-Kleitman-Rothschild [25] and Erdős-Frankl-Rödl [26] for graphs, and Nagle-Rödl-Schacht [45] for hypergraphs (see also[44]for the case r = 3) improves the upper bound above to obtain |Forb r (n, F )| = 2 ex r (n,F )+o(n r ) .Although much work has been done to improve the exponent above (see[1,6,7,8,31,34,48]for graphs and[10,11,21,47,13,50]for hypergraphs), this is a somewhat satisfactory state of affairs when ex r (n, F ) = Ω(n r ) or F is not r-partite.
Introduction
An important theme in combinatorics is the enumeration of discrete structures that have certain properties. Within extremal combinatorics, one of the first influential results of this type is the Erdős-Kleitman-Rothschild theorem [25], which implies that the number of triangle-free graphs with vertex set [n] is 2 n 2 /4+o(n 2 ) . This has resulted in a great deal of work on problems about counting the number of graphs with other forbidden subgraphs [6,7,8,14,15,26,31,40,48] as well as similar question for other discrete structures [10,11,17,18,35,46,47,49,51]. In extremal graph theory, these results show that such problems are closely related to the corresponding extremal problems. More precisely, the Turán problem asks for the maximum number of edges in a (hyper)graph that does not contain a specific subgraph. For a given r-uniform hypergraph (henceforth r-graph) F , let ex r (n, F ) be the maximum number of edges among all r-graphs G on n vertices that contain no copy of In the case of r-partite r-graphs, the corresponding questions appear to be more challenging since the tools used to address the case ex r (n, F ) = Ω(n r ) like the regularity lemma are not applicable. The major open problem here when r = 2 is to prove that |Forb r (n, F )| = 2 O(exr(n,F )) .
The two cases that have received the most attention are for r = 2 (graphs) and F = C 2l or F = K s,t . Classical results of Bondy-Simonovits [16] and Kovári-Sós-Turán [36] yield ex 2 (n, C 2l ) = O(n 1+1/l ) and ex 2 (n, K s,t ) = O(n 2−1/s ) for 2 s t, respectively. Although it is widely believed that these upper bounds give the correct order of magnitude, this is not known in all cases. Hence the enumerative results sought in these two cases were |Forb 2 (n, C 2l )| = 2 O(n 1+1/l ) and |Forb 2 (n, K s,t )| = 2 O(n 2−1/s ) .
In 1982, Kleitman and Winston [32] proved that |Forb 2 (n, C 4 )| = 2 O(n 3/2 ) which initiated a 30-year effort on searching for generalizations of the result to complete bipartite graphs and even cycles. Kleitman and Wilson [33] proved similar results for C 6 and C 8 in 1996 by reducing to the C 4 case. Finally, Morris and Saxton [42] recently proved that |Forb 2 (n, C 2l )| = 2 O(n 1+1/l ) and Balogh and Samotij [14,15] proved that |Forb 2 (n, K s,t )| = 2 O(n 2−1/s ) for 2 s t. Both these results used the hypergraph container method (developed independently by Saxton and Thomason [50], and by Balogh-Morris-Samotij [13]) which has proved to be a very powerful technique in extremal combinatorics. For example, [13] and [50] reproved |Forb r (n, F )| = 2 exr(n,F )+o(n r ) using containers.
There are very few results in this area when r > 2 and ex r (n, F ) = o(n r ). The only cases solved so far are when F consists of just two edges that intersect in at least t vertices [9], or when F consists of three edges such that the union of the first two is equal to the third [12] (see also [4,5,22,23] for some related results). These are natural points to begin these investigations since the corresponding extremal problems have been studied deeply.
Recently, Kostochka, the first author and Verstraëte [37,38,39], and independently, Füredi and Jiang [29] (see also [30]) determined the Turán number for several other families of r-graphs including paths, cycles, trees, and expansions of graphs. These hypergraph extremal problems have proved to be quite difficult, and include some longstanding conjectures. Guided and motivated by these recent developments on the extremal number of hypergraphs, we consider the corresponding enumeration problems focusing on the case of cycles.
Definition 1 For each integer k 3, a k-cycle C k is a hypergraph with distinct edges e 1 , . . . , e k and distinct vertices v 1 , . . . , v k such that e i ∩ e i+1 = {v i } for all 1 i k − 1, e 1 ∩ e k = {v k } and e i ∩ e j = ∅ for all other pairs i, j.
Sometimes we refer to C k as a loose or linear cycle. To simplify notation, we will omit the parameter r when the cycle C k is a subgraph of an r-graph.
Since ex r (n, C k ) = O(n r−1 ), we obtain the upper bound |Forb r (n, C k )| = 2 O(n r−1 log n) when r and k are fixed and n → ∞. Our main result is the following theorem, which improves this upper bound and generalizes the Morris-Saxton theorem [42] to 3-graphs.
Theorem 2 (Main Result) For integers r, k 3, there exists c = c(r, k), such that |Forb r (n, C k )| < 2 c n 2 if r = 3 and k is even, 2 c n r−1 (log n) (r−3)/(r−2) if r > 3.
Since trivially ex r (n, C k ) = Ω(n r−1 ) for all r 3, we obtain |Forb 3 (n, C k )| = 2 Θ(n 2 ) when k is even. We conjecture that a similar result holds for r > 3 and cycles of odd length.
Conjecture 3 For fixed r 3 and k 3 we have |Forb r (n, C k )| = 2 Θ(n r−1 ) .
Almost all recent developments in this area have relied on the method of hypergraph containers that we mentioned above. What is perhaps surprising about the current work is that the proofs do not use hypergraph containers. Instead, our methods employ old and new tools in extremal (hyper)graph theory. The old tools include the extremal numbers for cycles modulo h and results about decomposing complete r-graphs into r-partite ones, and the new tools include the analysis of the shadow for extremal hypergraph problems and quantitative estimates for the bipartite canonical Ramsey problem. An r-partite r-graph H is an r-graph with vertex set r i=1 V i (the V i s are pairwise disjoint), and every e ∈ H satisfies |e ∩ V i | = 1 for all i ∈ [r]. When all such edges e are present, H is called a complete r-partite r-graph. When |V i | = s for all i ∈ [r], a complete r-partite r-graph H is said to be balanced, and denoted K s:r .
Definitions and notations
For each integer k 1, a (loose, or linear) path of length k denoted by P k , is a collection of k edges e 1 , e 2 , . . . , e k such that |e i ∩ e j | = 1 if i = j + 1, and e i ∩ e j = ∅ otherwise.
We will often omit floors and ceilings in our calculations for ease of notation and all logs will have base 2.
Proof of the main result
The proof of Theorem 2 proceeds by counting edge-colored (r − 1)-graphs with certain restrictions; the details differ quite substantially for the cases r = 3 and r > 3. In this section we state the main technical statement (Theorem 5) about these edge-colorings that will be needed, as well as some other tools, and then prove Theorem 2 using these results.
Main technical statement
Given an (r − 1)-graph G with V (G) ⊂ [n], a coloring function is a function χ : G → [n] such that χ(e) = z e ∈ [n] \ e for every e ∈ G. We call z e the color of e. The vector of colors N G = (z e ) e∈G is called an edge-coloring of G. The pair (G, N G ) is an edge-colored (r − 1)-graph. A color class is the set of all edges that receive the same color.
Given G, each edge-coloring N G defines an r-graph H(N G ) = {e ∪ {z e } : e ∈ G}, called the extension of G by N G . When there is only one coloring that has been defined, we also use the notation G * = H(N G ) for the extension. Observe that any subgraph G ′ ⊂ G also admits an extension by N G , namely, G ′ * = {e ∪ {z e } : e ∈ G ′ } ⊂ G * . If G ′ ⊂ G and χ| G ′ is one-to-one, then G ′ is called rainbow colored. If a rainbow colored G ′ further satisfies that z e / ∈ V (G ′ ) for all e ∈ G ′ , then G ′ is said to be strongly rainbow colored. Note that a strongly rainbow colored graph C k ⊂ G ′ gives rise to 3-graph C k in G ′ * ⊂ G * . Definition 4 Given r 3, k 3, s 1, let f r (n, k, s) be the number of edge-colored balanced complete (r−1)-partite (r−1)-graphs G = K s:r−1 with V (G) ⊂ [n], whose extension G * is C k -free.
The function f r (n, k, s) allows us to encode r-graphs, and our main technical theorem gives an upper bound for this function.
Theorem 5 Given r 3, k 3 there exist D = D k/2 , c 2 = c 2 (r, k), such that f r (n, k, s) 2 (5/2)ks log n+4s 2 log D if r = 3, k is even, 2 (c 2 +2r)s r−2 log n+s r−1 (log(c 2 +r)+(r−2) log s) if r > 3.
For r and k fixed, the bounds above can be written as
f r (n, k, s) = 2 O(s log n+s 2 ) if r = 3, k is even, 2 O(s r−2 log n+s r−1 log s) if r > 3.
Note that the trivial upper bound is f r (n, k, s) n (r−1)s+s r−1 ∼ 2 s r−1 log n (first choose (r − 1)s vertices, then color each of its s r−1 edges) so Theorem 5 is nontrivial only if s = o(n). The proof of Theorem 5 will be given in Sections 3-6.
Decomposing r-graphs into balanced complete r-partite r-graphs
Chung-Erdős-Spencer [19] and Bublitz [3] proved that the complete graph K n can be decomposed into balanced complete bipartite graphs such that the sum of the sizes of the vertex sets in these bipartite graphs is at most O(n 2 / log n). See also [55,43] for some generalizations and algorithmic consequences. We need the following generalization of this result to r-graphs.
Theorem 6 Let n r 2.
There exists a constant c ′ 1 = c ′ 1 (r), such that any n-vertex r-graph H can be decomposed into balanced complete r-partite r-graphs K s i :r , i = 1, . . . , m, with s i (log n) 1/(r−1) and m i=1 s r−1 i c ′ 1 n r /(log n) 1/(r−1) .
Proof. An old result of Erdős [27] states that for any integers r, s 2 and n rs, we have ex r (n, K s:r ) < n r−1/s r−1 .
Note that for r = 2, this was proved much earlier by Kővári-Sós-Turán [36].
We first assume that n 2r. Taking the derivative, one can show that for each r 2, n −→ n/r − (log n) 1/(r−1) is an increasing function in n, hence its minimum is achieved at n = 2r. So for all n 2r, we have
n r − (log n) 1/(r−1) 2r r − (log 2r) 1/(r−1) = 2 − (log 2r) 1/(r−1) 0.
Thus, for any s (log n) 1/(r−1) n/r, the Turán number for K s:r is ex r (n, K s:r ) < n r−1/s r−1 .
Next, we give an algorithm of decomposing H into K s:r s. Let H 1 = H. For i 1, repeatedly find a K s i :r ⊂ H i with maximum s i subject to s i (log n) 1/(r−1) and delete it from H i to form H i+1 . The loop terminates at step i if |H i | n r /(log n) 1/(r−1) . Then let the remaining graph be decomposed into single edges (K 1:r s).
By the algorithm, the vertex size of each K s i :r satisfies that s i (log n) 1/(r−1) , 1 i m automatically. So we are left to show the upper bound for m i=1 s r−1 i .
We divide the iterations of the above algorithm into phases, where the kth phase consists of those iterations where the number of edges in the input r-graph of the algorithm lies in the interval (n r /(k + 1), n r /k]. In other words, in phase k, each K s i :r to be found is in an r-graph H i with |H i | > n r /(k + 1). Define s(k) = (log n/ log(k + 1)) 1/(r−1) . Then it is easy to see s(k) (log n) 1/(r−1) n/r. So by Erdős' result n r k + 1 = n r−1/s(k) r−1 > ex(n, K s(k):r ).
Hence, K s(k):r ⊂ H i . So in phase k, the minimum s i of a K s i :r we are able to find has the lower bound
s i s(k) = log n log(k + 1) 1/(r−1) . Now notice that m i=1 s r−1 i = m i=1 s r i · 1 s i .
Dividing up the terms in the summation according to phases, we observe that this is a sum of the number of edges deleted in the kth phase times a weight of 1/s i for each edge. Also notice that there are at most n r /(log n) 1/(r−1) single edges, we have
m i=1 s r−1 i n r (log n) 1/(r−1) + ∞ k=1 n r 1 k − 1 k + 1 log(k + 1) log n 1/(r−1) = n r (log n) 1/(r−1) + ∞ k=1 n r (log(k + 1)) 1/(r−1) k(k + 1)(log n) 1/(r−1) = c ′ 1 n r (log n) 1/(r−1) , where c ′ 1 = 1 + ∞ k=1 (log(k + 1)) 1/(r−1) k(k + 1) .
Finally, for n < 2r, we just let H be decomposed into K 1:r s. Then we obtain the following bound for
m i=1 s r−1 i . m i=1 s r−1 i = m 2r r c ′ 1 n r (log n) 1/(r−1)
with appropriately chosen c ′ 1 = c ′ 1 (r). This completes the proof.
A corollary of Theorem 5
Theorem 5 is about the number of ways to edge-color complete (r − 1)-partite (r − 1)-graphs with parts of size s and vertex set in [n]. In this section, we use Theorems 5 and 6 to prove a related statement where we do not require the (r − 1)-partite condition and the restriction to s vertices.
Definition 7 For r 3 and k 3, let g r (n, k) be the number of edge-colored (r − 1)-graphs G with V (G) ⊂ [n] such that the extension G * is C k -free.
Lemma 8 Let r 3, k 3, and n be large enough. Then there exist c 1 = c 1 (r), c 2 = c 2 (r, k) and D = D k/2 , such that
g r (n, k) 2 (3kc 1 +4 log D)n 2 if r = 3, k is even, 2 2(c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) if r > 3.
Note that if r and k are fixed, for both cases we have
g r (n, k) = 2 O(n r−1 (log n) (r−3)/(r−2) ) .
Proof. Given any (r − 1)-graph G, by applying Theorem 6 with parameter r − 1 instead of r, we may decompose G into balanced complete (r − 1)-partite (r − 1)-graphs K s 1 :r−1 , . . . , K sm:
r−1 , with s i (log n) 1/(r−2) and m i=1 s r−2 i c 1 n r−1 /(log n) 1/(r−2) , where c 1 = c 1 (r) = c ′ 1 (r − 1)
. Then we trivially deduce the following two facts.
• From the second inequality, we have m c 1 n r−1 /(log n) 1/(r−2) .
• Using the fact that these copies of K s i :r−1 are edge disjoint, we have
m i=1 s r−1 i n r − 1 < n r−1 .
Therefore, to construct an edge-colored G, we need to first choose a sequence of positive integers (m, s 1 , . . . , s m ) such that m c 1 n r−1 /(log n) 1/(r−2) , and s i (log n) 1/(r−2) for all i. More formally, let
S n,r = {(m, s 1 , s 2 , . . . , s m ) : m c 1 n r−1 /(log n) 1/(r−2) , 1 s i (log n) 1/(r−2) , 1 i m}. Then |S n,r | c 1 n r−1 (log n) 1/(r−2) (log n) 1/(r−2) c 1 n r−1 (log n) 1/(r−2) = 2 log c 1 n r−1 (log n) 1/(r−2) + c 1 n r−1 log(log n) 1/(r−2) (log n) 1/(r−2) 2 c 1 n r−1 .
Then, we sequentially construct edge-colored K s i :r−1 for each i ∈ [m]. To make sure G * is C k -free, K * s i :r−1 has to be made C k -free in the first place. Applying Theorem 5, we get the following upper bounds. For r = 3 and even k,
g 3 (n, k) (m,s 1 ,...,sm)∈S n,3 m i=1 f 3 (n, k, s i ) (m,s 1 ,...,sm)∈S n,3 m i=1 2 (5/2)ks i log n+4s 2 i log D (m,s 1 ,...,sm)∈S n,3 2 m i=1 (5/2)ks i log n+4s 2 i log D (m,s 1 ,...,sm)∈S n,3 2 (5/2)k log n(c 1 n 2 / log n)+4n 2 log D 2 c 1 n 2 · 2 (5/2)kc 1 n 2 +4n 2 log D 2 (3kc 1 +4 log D)n 2 .
For r > 3, and (m, s 1 , . . . , s m ) ∈ S n,r , the number of ways to construct these copies of
K s i :r−1 is at most m i=1 f r (n, k, s i ) m i=1 2 (c 2 +2r)s r−2 i log n+s r−1 i (log(c 2 +r−1)+(r−2) log s i ) = 2 m i=1 (c 2 +2r)s r−2 i log n+s r−1 i (log(c 2 +r−1)+(r−2) log s i ) 2 (c 2 +2r) c 1 n r−1 (log n) 1/(r−2) log n+n r−1 (log(c 2 +r−1)+(r−2) log(log n) 1/(r−2) ) 2 (c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) +n r−1 (log(c 2 +r−1)+(r−2) log(log n) 1/(r−2) ) 2 (3/2)(c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) .
Note that this is the only place in the proof where we use s i (log n) 1/(r−2) . Therefore,
g r (n, k) (m,s 1 ,...,sm)∈Sn,r m i=1 f r (n, k, s i ) 2 c 1 n r−1 · 2 (3/2)(c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) 2 2(c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) ,
and the proof is complete.
Finding a cycle if codegrees are high
A crucial statement that we use in our proof is that any r-graph such that every sub-edge has high codegree contains rich structures, including cycles. This was explicitly proved in [37] and we reproduce the proof here for completeness.
Lemma 9 (Lemma 3.2 in [37]) For r, k
3, if all sub-edges of an r-graph H have codegree greater than rk, then C k ⊂ H.
Proof. Let F = ∂ r−2 H be the (2-)graph that consists of pairs that are contained in some edge of H. Note that each edge of H induces a K r in F , so all edges of F are contained in some triangle (C 3 ). Furthermore, since all sub-edges of H have codegree greater than rk, each edge of F is in more than rk triangles. We will first find a k-cycle in F as follows. Starting with a triangle C 3 , for i = 3, . . . , k − 1 pick an edge e ∈ C i , form C i+1 by replacing e by the other two edges of one of at least rk − i + 2 triangles containing e and excluding other vertices of C i .
Next, let C k ⊂ F be a k-cycle with edges f 1 , . . . , f k . Find in H a subgraph C = {e i : e i = f i ∪ g i , i ∈ [k]} such that V (C) = ∪ k i=1 e i is of maximum size. Suppose C is not a k-cycle in H. Then there are distinct i, j such that g i ∩ g j = ∅. Pick v ∈ g i ∩ g j and consider the sub-edge e i \{v} = f i ∪g i \{v}. The codegree d H (e i \{v}) > rk by assumption. On the other hand, |V (C)| < rk since C is not a k-cycle, so there exists a vertex v ′ ∈ N H (e i \ {v}) \ V (C). Replacing e i by e i \ {v} ∪ {v ′ },
we obtain a C ′ with a larger vertex set, a contradiction. So H contains a C k .
Proof of Theorem 2
Now we have all the ingredients to complete the proof of our main result.
Proof of Theorem 2. Starting with any r-graph H on [n] with C k ⊂ H, we claim that there exists a sub-edge with codegree at most rk. Indeed, otherwise all sub-edges of H will have codegree more than rk, and then by Lemma 9 we obtain a C k ⊂ H. Let e ′ be the sub-edge of H with 0 < d H (e ′ ) rk such that it has smallest lexicographic order among all such sub-edges. Delete all edges of H containing e ′ from H (i.e. delete {e ∈ H : e ′ ⊂ e}). Repeat this process of "searching and deleting" in the remaining r-graph until there are no such sub-edges. We claim that the remaining r-graph must have no edges at all. Indeed, otherwise we get a nonempty subgraph all of whose sub-edges have codegree greater than rk, and again by Lemma 9, we obtain a C k ⊂ H.
Given any C k -free r-graph H on [n], the algorithm above sequentially decomposes H into a collection of sets of at most rk edges who share a sub-edge (an (r − 1)-set) in common. We regard the collection of these (r − 1)-sets as an (r − 1)-graph G. Moreover, for each edge e ∈ G, let N e be the set of vertices v ∈ V (H) such that e ∪ {v} is an edge of H at the time e was chosen. So N e ∈ [n]\e rk , for all e ∈ G. Thus, we get a map
φ : Forb r (n, C k ) −→ (G, N G ) : G ⊂ [n] r − 1 , N G = N e ∈ [n] \ e rk : e ∈ G .
We observe that φ is injective. Indeed,
φ −1 ((G, N G )) = H(N G ) = {e ∪ {z e } : e ∈ G, z e ∈ N e },
therefore |Forb r (n, C k )| = |φ(Forb r (n, C k ))|. Let P = φ(Forb r (n, C k )) which is the set of all pairs (G, N G ) such that H(N G ) is C k -free. Next we describe our strategy for upper bounding |P |.
For each pair (G, N G ) ∈ P and e ∈ G, we pick exactly one z 1 e ∈ N e . Thus we get a pair
(G 1 , N G 1 ), where G 1 = G, and N G 1 = (z 1 e : e ∈ G 1 ). Then, delete z 1 e from each N e , let G 2 = {e ∈ G 1 : N e \ {z 1 e } = ∅} and pick z 2 e ∈ N e \ {z 1 e } to get the pair (G 2 , N G 2 ). For 2
i < rk, we repeat this process for G i to obtain G i+1 . Since each N G i contains only singletons, the pair (G i , N G i ) can be regarded as an edge-colored (r − 1)-graph. Note that we may get some empty G i s. This gives us a map
ψ : P −→ (G 1 , . . . , G rk ) : G i ⊂ [n] r − 1 is edge-colored for all i ∈ [rk] .
Moreover, it is almost trivial to observe that ψ is injective, since if y = y ′ , then either the underlying (r − 1)-graphs of y and y ′ differ, or the (r − 1)-graphs are the same but the color sets differ. In both cases one can easily see that ψ(y) = ψ(y ′ ). Again, we let Q = ψ(P ).
By Lemma 8, we have
|Forb r (n, C k )| = |P | = |Q| rk i=1 g r (n, k) 3k i=1 2 (3kc 1 +4 log D)n 2 if r = 3, k is even rk i=1 2 2(c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) if r > 3 2 3k i=1 (3kc 1 +4 log D)n 2 if r = 3, k is even 2 rk i=1 2(c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) if r > 3 2 (9k 2 c 1 +12k log D)n 2 if r = 3, k is even 2 2rk(c 2 +2r)c 1 n r−1 (log n) (r−3)/(r−2) if r > 3 = 2 c n r−1 (log n) (r−3)/(r−2) .
where c = 9k 2 c 1 + 12k log D if r = 3 and k is even, and c = 2rk(c 2 + 2r)c 1 if r > 3. The constants c 1 = c 1 (r), c 2 = c 2 (r, k) and D = D k/2 are from Theorem 6 and Theorem 5, respectively.
Proof of Theorem 5 for r > 3
In the remaining part of the paper, we prove Theorem 5. The cases r = 3 and r > 3 have quite different proofs. In this short section we prove the case r > 3.
We need one more ingredient to prove Theorem 5, namely a partite version of the extremal result for C 2l . This is a corollary of the main result of [37] although it can also be proved directly by analyzing the shadow with a much better bound.
Lemma 10 Let r 3, k 3 and G be an r-partite r-graph on vertex sets
r i=1 V i with |V i | = s, for all i. There exists c ′ 2 = c ′ 2 (r, k), such that if |G| > c ′ 2 s r−1 then G contains a cycle of length k.
Proof. By Theorem 1.1 of [37], we have
ex r (n, C k ) ∼ n r − n − ⌊(k − 1)/2⌋ r ∼ ⌊(k − 1)/2⌋ (r − 1)! n r−1 .
So, we may take c ′ 2 large enough such that
c ′ 2 s r−1 = c ′ 2 r r−1 n r−1 ⌊(k − 1)/2⌋ (r − 1)! n r−1 ,
to guarantee the existence of a copy of C k in G. We remark that
c ′ 2 = ⌊k/2⌋r r−1 (r−1)! suffices. Proof of Theorem 5 for r > 3. Let G = K s:r−1 with V (G) ⊂ [n] and C k ⊂ G * . For any edge-coloring N G = (z e : e ∈ G) of G, let Z = {z e : e ∈ G} ⊂ [n]
be the set of all its colors. We first argue that |Z| < (
c 2 + r − 1)s r−2 , where c 2 = c 2 (r, k) = c ′ 2 (r − 1, k) = ⌊k/2⌋(r − 1) r−2 /(r − 2)!, the constant from Lemma 10. Indeed, if |Z| (c 2 + r − 1)s r−2 , then |Z \ V (G)| (c 2 + r − 1)s r−2 − s(r − 1) > (c 2 + r − 1 − (r − 1))s r−2 = c 2 s r−2 . For each color v ∈ Z \ V (G) pick an edge in G with color v. We get a subgraph G ′ ⊂ G that is strongly rainbow with |G ′ | = |Z \V (G)| > c 2 s r−2 .
By Lemma 10, we find an (r − 1)-uniform C k in G that is strongly rainbow, which contradicts the fact that C k ⊂ G * .
We now count the number of edge-colored K s:r−1 as follows: first choose s(r − 1) vertices from [n] as the vertex set, then choose at most (c 2 + r − 1)s r−2 colors, finally color each edge of the K s:r−1 . As |K s:r−1 | = s r−1 , this yields f r (n, k, s) n s(r−1)+(c 2 +r−1)s r−2 ((c 2 + r − 1)s r−2 ) s r−1 = 2 (s(r−1)+(c 2 +r−1)s r−2 ) log n+s r−1 (log(c 2 +r−1)+(r−2) log s) 2 (c 2 +2r)s r−2 log n+s r−1 (log(c 2 +r−1)+(r−2) log s) .
Proof of Theorem 5 for r = 3 and even k
The rest of the paper is devoted to the proof of Theorem 5 for r = 3 and even k. For simplicity of presentation, we write k = 2l where l 2. Our two main tools are the following lemmas about edge-coloring bipartite graphs.
Lemma 11
Let l 2, s, t 1, G = K s,t be an edge-colored complete bipartite graph with V (G) ⊂ [n] and Z = {z e : e ∈ G} ⊂ [n] be the set of all colors. If G contains no strongly rainbow colored C 2l , then |Z| < 2l(s + t).
Lemma 12 For each l 2, there exists a constant D = D l > 0, such that the following holds. Let s, t 1, G = K s,t be a complete bipartite graph with vertex set V (G) ⊂ [n], and Z ⊂ [n] be a set of colors. Then the number of ways to edge-color G with Z such that the extension G * contains no C 2l , is at most D (s+t) 2 .
The proofs of these lemmas require several new ideas which will be presented in the rest of the paper. Here we quickly show that they imply Theorem 5 for r = 3 and even k.
Proof of Theorem 5 for r = 3 and k = 2l. Recall that r 3, l 2, and that f 3 (n, 2l, s) is the number of edge-colored copies of K s,s whose vertex set lies in [n] and whose (3-uniform) extension is C 2l -free. To obtain such a copy of K s,s , we first choose from [n] its 2s vertices, then its at most 4ls colors by Lemma 11 and finally we color this K s,s by Lemma 12. This yields f 3 (n, 2l, s) n 2s+4ls D (2s) 2 2 5ls log n+4s 2 log D = 2 (5/2)ks log n+4s 2 log D ,
where the last inequality holds since l 2. Note that D = D l = D k/2 is the desired constant.
Proof of Lemma 11
In this section we prove Lemma 11. Our main tool is an extremal result about cycles modulo h in a graph. This problem has a long history, beginning with a Conjecture of Burr and Erdős that was solved by Bollobás [2] in 1976 via the following result: for each integer m and odd positive integer h, every graph G with minimum degree δ(G) 2((h + 1) h − 1)/h contains a cycle of length congruent to m modulo h. The lower bound on δ(G) was improved by Thomassen [53] who also generalized it to the case with all integers h. It was conjectured by Thomassen that graphs with minimum degree at least h + 1 contain a cycle of length 2m modulo h, for any h, m 1. The conjecture received new attention recently. In particular Liu-Ma [41] settled the case when h is even, and Diwan [24] proved it for m = 2. To date, Sudakov and Verstraëte [52] hold the best known bound for the general case on this problem.
We need the very special case m = 1 of Thomassen's conjecture and in order to be self contained, we give a proof below. The idea behind this proof can be found in Diwan [24].
Lemma 13
If G is an n-vertex graph with at least (h + 1)n edges, then G contains a cycle of length 2 modulo h.
Proof. By removing vertices of degree at most h, we may assume that G has minimum degree at least h + 1. Let P be a longest path in G. Assume that P is of length l. Let V (P ) = {x 0 , x 1 , . . . , x l }, where x 0 , x l are the two end-vertices of P , and x i−1 x i ∈ P for all i ∈ [l]. Then we observe that N (x 0 ) ⊂ V (P ). Otherwise we can extend P to a longer path by x 0 y with some vertex y ∈ N (x 0 ) \ V (P ). So N (x 0 ) = {x i : i ∈ I} for some I ⊂ [l]. Note that 1 ∈ I, |I| = |N (x 0 )| δ(G) h + 1, and the distance dist P (x i , x j ) = |j − i|. Consider the set J = {i − 1 : i ∈ I, i = 1}, note that this is the set of all distances dist P (x 1 , x i ) with x i ∈ N (x 0 ) and i = 1. Clearly, |J| = |I| − 1 h. If there exists i − 1 ∈ J with i − 1 ≡ 0 (mod h), then we are done, since the sub-path of P from x 1 to x i together with x 0 x 1 , x 0 x i form a cycle of length 2 modulo h. So none of the numbers in J are multiples of h. By the pigeonhole principle, there are at least two elements i − 1, j − 1 ∈ J such that i − 1 ≡ j − 1 (mod h), thus dist P (x i , x j ) = |j − i| ≡ 0 (mod h). Again, we can find a cycle of length 2 modulo h by taking the sub-path of P connecting x i , x j and edges x 0 x i , x 0 x j .
Lemma 14 Let integers l 2, s, t 1, G = K s,t with V (G) ⊂ [n]
be edge-colored. If G contains a strongly rainbow colored cycle of length 2 (mod 2l − 2), then G contains a strongly rainbow colored C 2l .
Proof. Let us assume that C is the shortest strongly rainbow colored cycle of length 2 modulo 2l − 2 in G. Then C has at least 2l edges. We claim that C is a C 2l . Suppose not, let e be a chord of C (such a chord exists as G is complete bipartite), such that C is cut up into two paths P 1 and P 2 by the two endpoints of e, and |P 1 | = 2l − 1. Let Z 1 , Z 2 be the set of their colors respectively. If the color z e / ∈ Z 1 ∪ V (P 1 ) \ e, then P 1 ∪ e is a strongly rainbow colored cycle of length 2l, a contradiction. Therefore z e ∈ Z 1 ∪ V (P 1 ) \ e, but then z e / ∈ Z 2 ∪ V (P 2 ) \ e, yielding a shorter strongly rainbow colored cycle P 2 ∪ e of length 2 modulo 2l − 2, a contradiction.
We now have all the necessary ingredients to prove Lemma 11.
Proof of Lemma 11. Suppose that |Z| 2l(s + t). Then |Z \ V (G)| (2l − 1)(s + t). For each color v in Z \ V (G), pick an edge e of G with color v. We obtain a strongly rainbow colored subgraph G ′ of G with at least (2l − 1)(s + t) edges. Lemma 13 guarantees the existence of a rainbow colored cycle of length 2 modulo 2l − 2 in G ′ . By construction, this cycle is strongly rainbow. Lemma 14 then implies that there is a strongly rainbow colored C 2l in G.
Proof of Lemma 12
Our proof of Lemma 12 is inspired by the methods developed in [37]. The main idea is to use the bipartite canonical Ramsey theorem. In order to use this approach we need to develop some new quantitative estimates for an asymmetric version of the bipartite canonical Ramsey theorem.
Canonical Ramsey theory
In this section we state and prove the main result in Ramsey theory that we will use to prove Lemma 12. We are interested in counting the number of edge-colorings of a bipartite graph, such that the (3-uniform) extension contains no copy of C 2l . The canonical Ramsey theorem allows us to find nice colored structures that are easier to work with. However, the quantitative aspects are important for our application and consequently we need to prove various bounds for bipartite canonical Ramsey numbers. We begin with some definitions.
Let G be a bipartite graph on vertex set with bipartition X ⊔ Y . For any subsets X ′ ⊂ X,
Y ′ ⊂ Y , let E G (X ′ , Y ′ ) = G[X ′ ⊔ Y ′ ] = {xy ∈ G : x ∈ X ′ , y ∈ Y ′ }, and e G (X ′ , Y ′ ) = |E G (X ′ , Y ′ )|.
If X ′ contains a single vertex x, then E G ({x}, Y ′ ) will be simply written as E G (x, Y ′ ). The subscript G may be omitted if it is obvious from context.
Definition 15
Let G be an edge-colored bipartite graph with V (G) = X ⊔ Y .
• G is monochromatic if all edges in E(X, Y ) are colored by the same color.
• G is weakly X-canonical if E(x, Y ) is monochromatic for each x ∈ X.
• G is X-canonical if it is weakly X-canonical and for all distinct x, x ′ ∈ X the colors used on E(x, Y ) and E(x ′ , Y ) are all different.
In all these cases, the color z x of the edges in E(x, Y ) is called a canonical color.
Lemma 16 Let G = K a,b be an edge-colored complete bipartite graph with bipartition A⊔B,
with |A| = a, |B| = b. If G is weakly A-canonical, then there exists a subset A ′ ⊂ A with |A ′ | = √ a such that G[A ′ ⊔ B] = K √ a,b is A ′ -canonical or monochromatic.
Proof. Take a maximal subset A ′ of A such that the coloring on Our next lemma guarantees that in an "almost" rainbow colored complete bipartite graph, there exists a rainbow complete bipartite graph.
E(A ′ , B) is A ′ -canonical. If |A ′ | √ a,
Lemma 17
For any integer c 2, and p > c 4 , if G = K p,p is an edge-colored complete bipartite graph, in which each color class is a matching, then G contains a rainbow colored K c,c .
Proof. Let A ⊔ B be the vertex set of G. Pick two c-sets X, Y from A and B respectively at random with uniform probability. For any pair of monochromatic edges e, e ′ , the probability that they both appear in the induced subgraph E(X, Y ) is
p−2 c−2 p c 2 = c(c − 1) p(p − 1) 2 .
On the other hand, the total number of pairs of monochromatic edges is at most p 3 /2, since every color class is a matching. Therefore the union bound shows that, when p > c 4 , the probability that there exists a monochromatic pair of edges in E(X, Y ) is at most
p 3 2 c(c − 1) p(p − 1) 2 = pc 4 2(p − 1) 2 < 1.
Consequently, there exists a choice of X and Y such that the E(X, Y ) contains no pair of monochromatic edges. Such an E(X, Y ) is a rainbow colored K c,c . Now we are ready to prove the main result of this section which is a quantitative version of a result from [38]. Note that the edge-coloring in this result uses an arbitrary set of colors. Since the conclusion is about "rainbow" instead of "strongly rainbow", it is not essential to have the set of colors disjoint from the vertex set of the graph.
Theorem 18 (Asymmetric bipartite canonical Ramsey theorem) For any integer l 2, there exists real numbers ǫ = ǫ(l) > 0, s 0 = s 0 (l) > 0, such that if G = K s,t is an edgecolored complete bipartite graph on vertex set X ⊔ Y with |X| = s, |Y | = t with s > s 0 and s/ log s < t s, then one of the following holds:
• G contains a rainbow colored K 4l,4l ,
• G contains a K q,2l on vertex set Q ⊔ R, with |Q| = q, |R| = 2l that is Q-canonical,
where q = s ǫ ,
• G contains a monochromatic K q,2l on vertex set Q ⊔ R, with |Q| = q, |R| = 2l, where q = s ǫ .
Note that in the last two cases, it could be Q ⊂ X, R ⊂ Y or the other way around.
Proof. We will show that ǫ = 1/18l. First, fix a subset Y ′ of Y with |Y ′ | = t 1/4l and let
W = x ∈ X : there exists a Y ′′ ∈ Y ′ 2l such that E G (x, Y ′′ ) is monochromatic . If |W | > s/2l, then the number of Y ′′ ∈ Y ′ 2l such that E G (x, Y ′′ ) is monochromatic for some x (with repetition) is greater than s/2l. On the other hand, | Y ′ 2l | < |Y ′ | 2l = √ t. By the pigeonhole principle, there exists a Y ′′ ∈ Y ′ 2l such that at least s 2l √ t s 2l √ s s 1/3
vertices x have the property that E G (x, Y ′′ ) is monochromatic. Let Q 1 be a set of s 1/3 such x. Then we obtain a weakly Q 1 -canonical K s 1/3 ,2l on Q 1 ⊔Y ′′ which, by Lemma 16, contains a canonical or monochromatic K s 1/6 ,2l . Since ǫ < 1/6, this contains a K s ǫ ,2l as desired.
We may now assume that |W | s/2l. By definition of W and the pigeonhole principle, E G (x, Y ′ ) contains at least |Y ′ |/2l (distinct) colors for every x ∈ X \ W . Hence, for each x ∈ X \ W we can take |Y ′ |/2l distinctly colored edges from E(x, Y ′ ) to obtain a subgraph
G ′ of G on (X \ W ) ⊔ Y ′ with |X \ W ||Y ′ |/2l edges.
Pick a subset X ′ ⊂ X \W with |X ′ | = s 1/16l 2 and e G ′ (X ′ , Y ′ ) |X ′ ||Y ′ |/2l. This is possible by an easy averaging argument. Let
Z = y ∈ Y ′ : there exists an X ′′ ∈ X ′ 2l such that E G ′ (X ′′ , y) is monochromatic .
If |Z| > |Y ′ |/20l, then the number of X ′′ ∈ X ′ 2l such that E G ′ (X ′′ , y) is monochromatic for some y (with repetition) is greater than |Y ′ |/20l. On the other hand, | X ′ 2l | < |X ′ | 2l = s 1/8l . By the pigeonhole principle, there exists a X ′′ ∈ X ′ 2l such that at least
|Y ′ | 20ls 1/8l = t 1/4l 20ls 1/8l s 1/4l (log s) 1/4l 20ls 1/8l s 1/9l = s 2ǫ
vertices y have the property that E G ′ (X ′′ , y) is monochromatic. Let Q 2 be a set of s 2ǫ such y. We find a weakly Q 2 -canonical K 2l,s 2ǫ on X ′′ ⊔ Q 2 . Again, by Lemma 16, a copy of K 2l,s ǫ that is monochromatic or canonical is obtained.
Finally, we may assume that |Z| |Y ′ |/20l. Then
e G ′ (X ′ , Y ′ \ Z) e G ′ (X ′ , Y ′ ) − |X ′ ||Z| 1 2l |X ′ ||Y ′ | − 1 20l |X ′ ||Y ′ | = 9 20l |X ′ ||Y ′ | 9 20l |X ′ ||Y ′ \ Z|.
Since each vertex y ∈ Y ′ \ Z has the property that E G ′ (X ′ , y) sees each color at most 2l − 1 times, for each y ∈ Y ′ \ Z we may remove all edges from E G ′ (X ′ , y) with duplicated colors (keep one for each color). We end up getting a bipartite graph G ′′ on X ′ ⊔ (Y ′ \ Z) with at least 9|X ′ ||Y ′ \ Z|/40l 2 edges. By the Kővári-Sós-Turán theorem [36], there is a c > 0 such that G ′′ contains a copy K of K p,p where p > c log s. Let V (K) = A ⊔ B. For each x ∈ A, the edges set E(x, B) is rainbow colored, and for each y ∈ B, the edge set E(A, y) is rainbow colored. Therefore each color class in K is a matching. By Lemma 17 and s > s 0 > 2 (4l) 4 /c , we can find a rainbow colored K 4l,4l in K as desired.
The Induction argument for Lemma 12
We are now ready to prove Lemma 12. Let us recall the statement.
Lemma 12
For each l 2, there exists a constant D = D l > 0, such that the following holds. Let s, t 1, G = K s,t be a complete bipartite graph with vertex set V (G) ⊂ [n], and Z ⊂ [n] be a set of colors. Then the number of ways to edge-color G with Z such that the extension G * contains no C 2l , is at most D (s+t) 2 .
Proof of Lemma 12. Let the vertex set of G be S ⊔ T with |S| = s and |T | = t. We apply induction on s + t. By Lemma 11, |Z| := σ < 2l(s + t). The number of ways to color G is at most σ st . As long as s + t D/2l, we have
σ st D st D (s+t) 2
and this concludes the base case(s).
For the induction step, we may henceforth assume s + t > D/2l, and the statement holds for all smaller values of s + t. Let us also assume without loss of generality that t s.
Next, we deal with the case t s/ log s. Let D > 16l 2 . Then s > (s + t)/2 > D/4l > 4l and the number of ways to color G is at most
σ st (2l(s + t)) st 2 s 2 log(2l(s+t)) log s 2 s 2 log(4ls) log s 2 2s 2 2 (s+t) 2 log D = D (s+t) 2 .
Therefore, we may assume that s/ log s < t s, and s > D/4l > s 0 (l) so the conditions of Theorem 18 hold. Let N G = (z e ) e∈G be an edge-coloring of G using colors in Z. By Theorem 18, such an edge-colored G will contain a subgraph G ′ that is either • a rainbow colored K 4l,4l , or • a Q-canonical K q,2l , or
• a monochromatic K q,2l , where |Q| = q = s ǫ .
Claim 19 G ′ cannot be a rainbow colored K 4l,4l .
Proof of Claim 19. Suppose for a contradiction that G ′ = K 4l,4l is rainbow colored and Z ′ is the set of colors used on G ′ . Then |Z ′ \ V (G ′ )| 16l 2 − 8l. Pick an edge of each color in Z ′ \ V (G ′ ) to obtain a strongly rainbow colored subgraph G ′′ of G ′ with |G ′′ | = 16l 2 − 8l (2l − 1)8l. By Lemma 13, G ′′ contains a strongly rainbow colored cycle of length 2 mod 2l − 2. Lemma 14 now implies the existence of a strongly rainbow colored C 2l in G ′′ , which forms a linear C 2l in G * , a contradiction.
Let q = s ǫ and α be the number of edge-colorings of G that contain a Q-canonical subgraph G ′ which is a copy of K q,2l and let β be the number of edge-colorings of G that contain a monochromatic subgraph G ′ which is a copy of K q,2l . We will prove that both α and β are at most (1/2)D (s+t) 2 and conclude by Claim 19 that the total number of colorings is at most α + β D (s+t) 2 as desired.
Let the vertex set of
G ′ = K q,2l be Q ⊔ R, where Q ∈ X q , R ∈ Y 2l and {X, Y } = {S, T }.
6.2.1
The canonical case
Our goal is to show that α (1/2)D (s+t) 2 . Recall that for each x ∈ Q, the edges in E(x, R) all have the same color z x which is called a canonical color. Let Z c = {z x : x ∈ Q} be the set of all canonical colors. For each edge xy with x ∈ Q, y ∈ Y \ (R ∪ Z c ), a color z xy = z x is called a free color. We will count the number of colorings of E(Q, Y ), and then remove Q to apply the induction hypothesis. For each coloring N G , consider the following partition of Y \ (R ∪ Z c ) into two parts:
Y 0 = {y ∈ Y \ (R ∪ Z c )
: E(y, Q) sees at most 11l − 1 free colors},
Y 1 = {y ∈ Y \ (R ∪ Z c )
: E(y, Q) sees at least 11l free colors}.
We claim that the length of strongly rainbow colored paths that lie between Q and Y 1 is bounded.
Claim 20
If there exists a strongly rainbow colored path P = P 2l−2 ⊂ E(Q, Y 1 ) with both end-vertices u, v ∈ Q, then there exists a C 2l in G * .
Proof of Claim 20. Clearly, P extends to a linear P 2l−2 in G * . We may assume both z u , z v / ∈ V (P * ), where P * = {e ∪ {z e } : e ∈ P } is the extension of P . Otherwise, suppose w.l.o.g. z u ∈ V (P * ), let y be the vertex next to u in P , let S y be of maximum size among sets {x ∈ Q : xy all colored by distinct free colors}.
Since y ∈ Y 1 , |S y | 11l. Note that |V (P * )| = 4l − 3 and |V (P * ) ∩ Y 1 | l − 1, we have |S y \ V (P * )| 11l − (4l − 3 − (l − 1)) 8l. Since |V (P * )| < 4l, E(y, S y ) is rainbow, and G ′ is Q-canonical, there must be at least 4l vertices in S y \ V (P * ) whose canonical color is not in V (P * ). Among these 4l vertices there is at least one u ′ with z u ′ y / ∈ V (P * ). Replacing u by u ′ , we get a strongly rainbow colored path of length 2l − 2 with z u / ∈ V (P * ). Now, Since |R| = 2l, we can find a vertex y ∈ R such that y / ∈ {z e : e ∈ P }. Further, since both z u , z v / ∈ V (P * ) and z u = z v , the set of edges
P * ∪ {uyz u , vyz v }
forms a copy of C 2l in G * .
Thanks to this observation about strongly rainbow paths, we can bound the number of colorings on E(Q, Y 1 ) as follows. It is convenient to use the following notation.
Definition 21 Given X ′ ⊂ X and Y ′ ⊂ Y , let #E(X ′ , Y ′ ) be the number of ways to color the edges in E(X ′ , Y ′ ).
Claim 22 #E(Q, Y 1 ) (2l) q · (32l 2 ) bq · (qb) 2lq · σ 6lq+8l 2 b .
Proof of Claim 22. By Claim 20, according to the length of the longest strongly rainbow colored path starting at a vertex, Q can be partitioned into 2l − 3 parts 2l−3 i=1 Q i , where Q i = {x ∈ Q : the longest strongly rainbow colored path starting at x and contained in E(Q, Y 1 ) has length i}.
For each i, let q i = |Q i |. We now bound the number of colorings of the edges in E(Q i , Y 1 ).
Firstly, for each x ∈ Q i , choose an i-path P x ⊂ E(Q, Y 1 ) starting at x and color it strongly rainbow. The number of ways to choose and color these paths for all the vertices x ∈ Q i is at most
((qb) ⌈(i+1)/2⌉ σ i ) q i (qbσ) iq i .
Fix an x ∈ Q i . Partition Y 1 into 3 parts depending on whether y is on the extension P * x of the path starting at x, or the color of xy is on P * x or else, i.e.
Y 1 = 3 j=1 Y (j) i,x , where Y (1) i,x = Y 1 ∩ V (P * x ), Y (2) i,x = {y ∈ Y 1 \ Y (1) i,x : z xy ∈ V (P * x )}, Y (3) i,x = Y 1 \ (Y (1) i,x ∪ Y (2) i,x ).
Depending on the part of Y 1 that a vertex y lies in, we can get different restrictions on the coloring of the edges in E(y, Q i ).
• If y ∈ Y (1) i,x , then z xy has as many as σ choices. Note that |P * x | = 2i + 1, and |Y (1) i,x | i + ⌈i/2⌉ 2i. This gives #E(x, Y (1) i,x ) σ 2i .
• If y ∈ Y (2) i,x , then z xy ∈ V (P * x ), so there are at most 2i + 1 choices for this color and #E(x, Y (2) i,x ) (2i + 1) b .
• Lastly, let |Y
(3) i,x | = b i,x . If y ∈ Y (3)
i,x , then xy extends P x into a strongly rainbow colored path P ′ x = P x ∪ {xy} of length i + 1, which forces the edges x ′ y to be colored by V (P ′
x * ) for each x ′ ∈ Q i \ V (P ′
x * ). Otherwise, the path P ′
x ∪ {x ′ y} is a strongly rainbow colored path of length i + 2 starting at a vertex x ′ ∈ Q i , contradicting the definition of Q i . Therefore, z x ′ y has at most 2i
+ 3 choices if x ′ ∈ Q i \ V (P ′ x * ). Putting this together, for each y ∈ Y (3) i,x , we have #E(Q i \ V (P ′ x * ), y) (2i + 3) q i . Noticing that |Q i ∩ V (P ′ x * )| i + 1 + ⌈(i + 1)/2⌉ 2i + 1, we have #E(Q i , y) #E(Q i ∩ V (P ′ x * ), y) · #E(Q i \ V (P ′ x * ), y) σ 2i+1 (2i + 3) q i .
Hence the number of ways to color
E(x, Y 1 ) ∪ E(Q i , Y (3) i,x ) is at most 2 b · σ 2i · (2i + 1) b · σ (2i+1)b i,x (2i + 3) q i b i,x .
The term 2 b arises above since Y (1) i,x has already been fixed before this step, so we just need to partition Y 1 \ Y (1) i,x to get Y (2) i,x and Y (3) i,x .
Now we remove x from Q i , Y(3)
i,x from Y 1 and repeat the above steps until we have the entire E(Q i , Y 1 ) colored. Note that x∈Q i b i,x b, and that i 2l − 3 which implies 2i + 3 < 4l. We obtain
#E(Q i , Y 1 ) (qbσ) iq i x∈Q i 2 b · σ 2i · (2i + 1) b · σ (2i+1)b i,x (2i + 3) q i b i,x (qbσ) 2lq i x∈Q i 2 b · σ 4l+4lb i,x · (4l) b+q i b i,x (qbσ) 2lq i · 2 bq i · σ 4lq i +4lb · (4l) bq i +bq i = (32l 2 ) bq i · (qb) 2lq i · σ 6lq i +4lb . Because 2l−3 i=1 q i = q, taking the product over i ∈ [2l − 3], we obtain #E(Q, Y 1 ) (2l − 3) q 2l−3 i=1 #E(Q i , Y 1 ) (2l − 3) q 2l−3 i=1 (32l 2 ) bq i · (qb) 2lq i · σ 6lq i +4lb (2l) q · (32l 2 ) bq · (qb) 2lq · σ 6lq+8l 2 b ,
where (2l − 3) q counts the number of partitions of Q into the Q i .
Since G ′ = E(Q, R) is Q-canonical, #E(Q, R) σ q .
As |Z c | q,
#E(Q, Y ∩ Z c ) σ q 2 .
By definition of Y 0 ,
#E(Q, Y 0 ) (σ 11l (11l + 1) q ) b (σ 11l (12l) q ) b .
Therefore to color E(Q, Y ), we need to first choose the subsets R and Z c ∩ Y of Y and then take a partition to get Y 0 and Y 1 . We color each of E(Q, R), E(Q, Y ∩ Z c ), E(Q, Y 0 ) and E(Q, Y 1 ). This gives
#E(Q, Y ) b 2l b q 2 b · #E(Q, R) · #E(Q, Y ∩ Z c ) · #E(Q, Y 0 ) · #E(Q, Y 1 ) b 2l b q 2 b · σ q · σ q 2 · (σ 11l (12l) q ) b · [(2l) q · (32l 2 ) bq · (qb) 2lq · σ 6lq+8l 2 b ] = b 2l 2 b · (2lb) q · (384l 3 ) qb · (qb) 2lq · σ q+q 2 +11lb+6lq+8l 2 b .
Finally, we apply the induction hypothesis to count the number ways to color G = K s,t .
Recall that q = s ǫ < s/ log s < t s, σ 2l(s + t) 4ls. There are two cases.
• (X, Y ) = (S, T ) and (a, b) = (s, t)
Recall that we must first choose Q ⊂ X.
#E(X, Y ) s q · #E(Q, Y ) · #E(X \ Q, Y ) s q · t 2l 2 t · (2lt) q · (384l 3 ) qt · (qt) 2lq · σ q+q 2 +11lt+6lq+8l 2 t · D (s+t−q) 2 s q t 2l · 2 t · (2lt) q · (384l 3 ) qt · (qt) 2lq · (4ls) q+q 2 +11lt+6lq+8l 2 t · D (s+t−q) 2 s q t 2l · t q · (2 1/q (2l) 1/t 384l 3 ) qt · (qt) 2lq · (4ls) 9l 2 t · D −2qt · D −2qs+q 2 · D (s+t) 2 2 1/q (2l) 1/t 384l 3 (4l) 9l 2 /q D 2 qt · t 2l (st) q (qt) 2lq s 9l 2 t · D −qs · D (s+t) 2 2 1/q (2l) 1/t 384l 3 (4l) 9l 2 /q D 2 qt · q 2lq t 2l+q+2lq s q+9l 2 t D qs · D (s+t) 2 1 4 D (s+t) 2 .
To show the last inequality above, it is obvious that 2 1/q (2l) 1/t 384l 3 (4l) 9l 2 /q < D 2 for large enough D, so we are left to show that 4q 2lq t 2l+q+2lq s q+9l 2 t < D qs for large D. Taking logarithms, we have log 4q 2lq t 2l+q+2lq s q+9l 2 t = 2 + 2lq log q + (2l + q + 2lq) log t + (q + 9l 2 t) log s qs log D.
This holds for large D because qs has the dominating growth rate among all the above terms as q = s ǫ and s is large.
• (X, Y ) = (T, S) and (a, b) = (t, s)
#E(X, Y ) t q · #E(Q, X) · #E(Y \ Q, X) t q · s 2l 2 s · (2ls) q · (384l 3 ) qs · (qs) 2lq · σ q+q 2 +11ls+6lq+8l 2 s · D (s+t−q) 2 t q s 2l · 2 s · (2ls) q · (384l 3 ) qs · (qs) 2lq · (4ls) q+q 2 +11ls+6lq+8l 2 s · D (s+t−q) 2 t q s 2l · s q · (2 1/q (2l) 1/s 384l 3 ) qs · (qs) 2lq · (4ls) 9l 2 s · D −2qs · D −2qt+q 2 · D (s+t) 2 2 1/q (2l) 1/s 384l 3 (4l) 9l 2 /q D 2 qs · s 2l (st) q (qs) 2lq s 9l 2 s · D −qt · D (s+t) 2 2 1/q (2l) 1/s 384l 3 (4l) 9l 2 /q D 2 qs · q 2lq t q s 2l+q+2lq+9l 2 t D qt · D (s+t) 2 1 4 D (s+t) 2 .
Again, to show the last inequality above, it is clear that 2 1/q (2l) 1/s 384l 3 (4l) 9l 2 /q < D 2 for large D, so we are left to show that 4q 2lq t q s 2l+q+2lq+9l 2 s < D qs for large D. Taking logarithms, we have log 4q 2lq t q s 2l+q+2lq+9l 2 s = 2 + 2lq log q + q log t + (2l + q + 2lq + 9l 2 s) log s qt log D.
This holds for large D because qt has the dominating growth rate among all the above terms.
In summary, the number of colorings of G such that there exists a G ′ ⊂ G that is a Qcanonical K q,2l is
α 1 4 D (s+t) 2 + 1 4 D (s+t) 2 = 1 2 D (s+t) 2 .
The monochromatic case
Our goal is to show that β (1/2)D (s+t) 2 . Recall that the vertex set of
G ′ = K q,2l is Q ⊔ R,
where Q ∈ X q and R ∈ Y 2l . The term canonical color now refers to the only color z c that is used to color all edges of G ′ , and Z c = {z c } still means the set of canonical colors. A free color is a color that is not z c . As before we will count the number of colorings of E(Q, Y ), and then remove Q to apply the induction hypothesis.
Let Y 1 = Y \ (R ∪ Z c )
. Similar to Claim 20, we claim that the length of a strongly rainbow colored path between Q and Y 1 is bounded.
Claim 23
If there exists a strongly rainbow colored path P = P 4l−2 ⊂ E(Q, Y 1 ) with both end-vertices u, v ∈ Q, then there exists a C 2l in G * .
Proof of Claim 23. We observe that z c appears in the path or the color of the path at most once, as P is strongly rainbow. Hence, by the pigeonhole principle, there exists a sub-path P ′ of length 2l − 2 such that z c / ∈ V (P ′ * ) and both end-vertices u, v of P ′ are in Q.
Now, Since |R| = 2l, we can find two vertices y, y ′ ∈ R such that y, y ′ / ∈ {z e : e ∈ P ′ }. Thus, the edges P ′ * ∪ {uyz c , vy ′ z c } yield a copy of C 2l in G * .
Again, we first use this claim to color E(Q, Y 1 ).
Claim 24 #E(Q, Y 1 ) (4l) q · (128l 2 ) qb · (qb) 4lq · σ 12lq+32l 2 b .
Proof of Claim 24. The proof proceeds exactly the same as that of Claim 22, except that Q is partitioned into 4l − 3 parts 4l−3 i=1 Q i . So in the calculation at the end, we have i 4l − 3 which gives 2i + 3 < 8l and
#E(Q i , Y 1 ) (qbσ) iq i x∈Q i 2 b · σ 2i · (2i + 1) b · σ (2i+1)b i,x (2i + 3) q i b i,x (qbσ) 4lq i x∈Q i 2 b · σ 8l+8lb i,x · (8l) b+q i b i,x (qbσ) 4lq i · 2 bq i · σ 8lq i +8lb · (8l) bq i +bq i (128l 2 ) bq i · (qb) 4lq i · σ 12lq i +8lb .
Again, note that 4l−3 i=1 q i = q. Taking the product over i ∈ [4l − 3], we obtain
#E(Q, Y 1 ) (4l − 3) q 4l−3 i=1 #E(Q i , Y 1 ) (4l − 3) q 4l−3 i=1 (128l 2 ) bq i · (qb) 4lq i · σ 12lq i +8lb (4l) q · (128l 2 ) qb · (qb) 4lq · σ 12lq+32l 2 b ,
where (4l − 3) q counts the number of partitions of Q into the Q i .
Similarly, to color E(Q, Y ), we need to choose the subsets R and Y ∩ Z c , and what remains is Y 1 . Consequently,
#E(Q, Y ) b 2l b · #E(Q, R) · #E(Q, Y ∩ Z c ) · #E(Q, Y 1 ) b 2l b · σ · σ q · [(4l) q · (128l 2 ) qb · (qb) 4lq · σ 12lq+32l 2 b ] = b 2l+1 (4l) q · (128l 2 ) qb · (qb) 4lq · σ 1+q+12lq+32l 2 b .
We apply the induction hypothesis to count the number ways to color G = K s,t . Recall that q = s ǫ < s/ log s < t s, σ 2l(s + t) 4ls. We split the calculation into two cases.
• (X, Y ) = (S, T ) and (a, b) = (s, t)
Recall that we must first choose Q ⊂ X.
#E(X, Y ) s q · #E(Q, Y ) · #E(X \ Q, Y )
s q · t 2l+1 (4l) q · (128l 2 ) qt · (qt) 4lq · σ 1+q+12lq+32l 2 t · D (s+t−q) 2 s q t 2l+1 · (4l) q · (128l 2 ) qt · (qt) 4lq · (4ls) 1+q+12lq+32l 2 t · D (s+t−q) 2 s q t 2l+1 · ((4l) 1/t 128l 2 ) qt · (qt) 4lq · (4ls) 33l 2 t · D −2qt · D −2qs+q 2 · D (s+t) 2 (4l) 1/t 128l 2 (4l) 33l 2 /q D 2 qt · t 2l+1 s q (qt) 4lq s 33l 2 t · D −qs · D (s+t) 2 (4l) 1/t 128l 2 (4l) 33l 2 /q D 2 qt · q 4lq t 2l+1+4lq s q+33l 2 t D qs · D (s+t) 2
1 4 D (s+t) 2 .
To show the last inequality above, it is obvious to see that (4l) 1/t 128l 2 (4l) 33l 2 /q < D 2 for large D, so we are left to show that 4q 4lq t 2l+1+4lq s q+33l 2 t < D qs for large D. Taking logarithms, we have log 4q 4lq t 2l+1+4lq s q+33l 2 t = 2 + 4lq log q + (2l + 1 + 4lq) log t + (q + 33l 2 t) log s qs log D.
This holds for large D because qs has dominating growth rate among all the terms above.
• (X, Y ) = (T, S) and (a, b) = (t, s) #E(X, Y ) t q · #E(Q, Y ) · #E(X \ Q, Y ) t q · s 2l+1 (4l) q · (128l 2 ) qs · (qs) 4lq · σ 1+q+12lq+32l 2 s · D (s+t−q) 2 t q s 2l+1 · (4l) q · (128l 2 ) qs · (qs) 4lq · (4ls) 1+q+12lq+32l 2 s · D (s+t−q) 2 t q s 2l+1 · ((4l) 1/s 128l 2 ) qs · (qs) 4lq · (4ls) 33l 2 s · D −2qs · D −2qt+q 2 · D (s+t) 2 (4l) 1/s 128l 2 (4l) 33l 2 /q D 2 qs · s 2l+1 t q (qs) 4lq s 33l 2 s · D −qt · D (s+t) 2 (4l) 1/s 128l 2 (4l) 33l 2 /q D 2 qs · q 4lq t q s 2l+1+4lq+33l 2 t D qt · D (s+t) 2
1 4 D (s+t) 2 .
To show the last inequality above, it is obvious to see that (4l) 1/s 128l 2 (4l) 33l 2 /q < D 2 for large D, so we are left to show that 4q 4lq t q s 2l+1+4lq+33l 2 t < D qt for large D. Taking logarithms, we have log 4q 4lq t q s 2l+1+4lq+33l 2 t = 2 + 4lq log q + q log t + (2l + 1 + 4lq + 33l 2 t) log s qt log D.
This holds for large D because qt has the dominating growth rate among all the terms above.
Finally, we have
β 1 4 D (s+t) 2 + 1 4 D (s+t) 2 = 1 2 D (s+t) 2 ,
and the proof is complete.
Concluding remarks
• A straightforward corollary of Theorem 2 is the very same result for hypergraph paths P k . Indeed, for the upper bound on Forb r (n, P k ) one has to just observe that P k ⊂ C 2⌈(k+1)/2⌉ , while the lower bound is trivial.
• Although we were unable to use hypergraph containers to prove Theorem 2 it would be very interesting to give a new proof using containers. In particular, this would entail proving some supersaturation type results for this problem which may be of independent interest. It would also likely yield some further results in the random setting which we have not addressed.
• The main open problem raised by our work is to solve the analogous question for larger r and for odd cycles (Conjecture 3).
For r = 3, our method will not work for odd cycles as it relies on finding a bipartite structure from which it is difficult to extract odd 3-uniform cycles (although this technical hurdle could be overcome to solve the corresponding extremal problem in [37]).
For larger r, our method does not work because the cost of decomposing a complete r-graph into complete r-partite subgraphs is too large to remain an error term. More precisely, for r = 3, we implicitly applied Lemma 12 (in the proof of Lemma 8) to reduce the number of ways to color a graph to at most 2 O(n 2 ) instead of the trivial 2 O(n 2 log log n) . But for r > 3 the main term in the calculation turns out to be 2 O(n r−1 (log n) (r−3)/(r−2) ) which comes from choosing the colors for the copies of K s i :r−1 (see Section 3). This cannot be improved due to Theorem 6 and Lemma 10 each of which gives a bound that is sharp in order of magnitude. Consequently, even if we proved a version of Lemma 12 for r > 3 (and the tools we have developed should suffice to provide such a proof) this would not improve Theorem 2 for r > 3.
• Another way to generalize the result of Morris-Saxton to hypergraphs is to consider similar enumeration questions when the underlying r-graph is linear, meaning that every two edges share at most one vertex. Here the extremal results have recently been proved in [20] and the formulas are similar to the case of graphs. The special case of this question for linear triple systems without a C 3 is related to the Ruzsa-Szeméredi (6, 3) theorem and sets without 3-term arithmetic progressions.
that the method of hypergraph containers would not apply easily to prove Theorem 2 and we therefore developed new ideas. We are also grateful to Jozsef Balogh for providing us with some pertinent references, and to Jie Han for pointing out that our proof of Theorem 2 applies for r > 3 and odd k.
Define a = |X| and b = |Y | so {a, b} = {s, t}.
Throughout this paper, we let [n] denote the set {1, 2, . . . , n}. Write X r = {S ⊂ X : |S| = r} and X r = {S ⊂ X : |S| r}. For X ⊂ [n], an r-uniform hypergraph or r-graph H on vertex set X is a collection of r-element subsets of X, i.e. H ⊂ X r . The vertex set X is denoted by V (H). The r-sets contained in H are edges. The size of H is |H|. Given S ⊂ V (H), the neighborhood N H (S) of S is the set of all T ⊂ V (H) \ S such that S ∪ T ∈ H. The codegree of S is d H (S) = |N H (S)|. When the underlying hypergraph is clear from context, we may omit the subscripts in these definitions and write N (S) and d(S) for simplicity. The sub-edges of H are the (r − 1)-subsets of [n] with positive codegree in H. The set of all sub-edges of H is called the shadow of H, and is denoted ∂H.
then we are done. So, we may assume that |A ′ | < √ a. By maximality of A ′ , there are less then √ a canonical colors. By the pigeonhole principle, there are at least |A|/|A ′ | a/ √ a = √ a vertices of A sharing the same canonical color, which gives a monochromatic K √ a,b .
Acknowledgment. We are very grateful to Rob Morris for clarifying some technical parts of the proof in[42]at the early stages of this project. After those discussions, we realized
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[
"An affine scaling method using a class of differential barrier functions",
"An affine scaling method using a class of differential barrier functions"
]
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"Abdessamad Barbara "
]
| []
| []
| In this paper we address a practical aspect of differential barrier penalty functions in linear programming. In this respect we propose an affine scaling interior point algorithm based on a large classe of differential barrier functions. The comparison of the algorithm with a vesion of the classical affine scaling algorithm shows that the algorithm is robust and efficient. We thus show that differential barrier functions open up new perspectives in linear optimization. | 10.1080/02331934.2020.1812606 | [
"https://arxiv.org/pdf/1705.07667v1.pdf"
]
| 119,141,220 | 1705.07667 | debc8dfa5d006c34b11f64c2cbdaa5b7d2855499 |
An affine scaling method using a class of differential barrier functions
Abdessamad Barbara
An affine scaling method using a class of differential barrier functions
arXiv:1705.07667v1 [math.OC] 22 May 2017 manuscript No. (will be inserted by the editor)Barrierconcave gaugedifferential barrierinterior point meth- odslinear programsprimal algorithm AMS Subject Classifications: 90C0590C5149M3049N15
In this paper we address a practical aspect of differential barrier penalty functions in linear programming. In this respect we propose an affine scaling interior point algorithm based on a large classe of differential barrier functions. The comparison of the algorithm with a vesion of the classical affine scaling algorithm shows that the algorithm is robust and efficient. We thus show that differential barrier functions open up new perspectives in linear optimization.
Introduction
In this paper we present an algorithm based on a family of penalty functions introduced in [1]. Contrary to the classical logarithmic barrier function, these functions are not necessarily barriers, since they can be well defined on the positive orthant including its boundary. But they are differentially barriers. In fact, these functions generalize the notion of barrier functions since (Proposition 17 of [1]) a barrier function is in particular a differential barrier one. We recall that (Definition 1 of [1]) a function F is said to be a differential barrier on the positive orthant P = [0, +∞) n if F is differentiable on (0, +∞) n and lim sup x→x ′ x>0 ||∇F (x)|| = +∞, for every x ′ being on the boundary of P. So ∇F plays the role of a barrier. Also, the fact that a method based on the minimization of Abdessamad BARBARA Institut de Mathématiques de Bourgogne(IMB)-UMR 5584 CNRS Université de Bourgogne 9 avenue Alain Savary BP 47870, 21078 Dijon cedex, France E-mail: [email protected] a penalty function is of interior points type is closely related to the following property. Proposition 1.1 (Proposition 18 of [1]) Let F be a convex, lower semi-continuous and differential barrier function on P. Then every optimal solution x to the problem
min{F (x) : Ax = b, x ≥ 0}
is an interior point of the positive orthant.
Through the example of the concave gauge functions 1 we will consider, we will show the important role that penalty functions of the differential barrier type can play as an alternative to the classical logarithmic barrier function. In this respect we consider the familiy of differential barrier functions builded from the following concave gauge functions:
ξ r : x → x r i 1 r if x ∈ [0, +∞) n , −∞ elsewhere,
where r is taken arbitrary in (0, 1). To be more precise, let us consider the linear program given by min{ c, x : Ax = b, x ∈ [0, +∞) n } (LP ) where A is an m × n matrix of rank m, c, x ∈ R n and b ∈ R m . By definition of a concave gauge function the positive cone can be expressed as [0, +∞) n = {x ∈ R n : ξ r (x) ≥ 0} .
Hence the original linear program can be equivalently rewritten as min{ c, x : Ax = b, ξ r (x) ≥ 0}.
Applying the approach developed in [1], we propose to penalize the constraint ξ r (x) ≥ 0 by the functions
g r : x → − 1 r ξ r (x) r if x ∈ [0, +∞) n , +∞ elsewhere,
So the nonlinear optimization problem approximating the linear program 2 is as follow min{F r,µ (x) : Ax = b} (P µ,r ) µ>0 1 A background on concave gauge functions is given in [1] and a complete description is done in [2] 2 We recall that the idea to approximate a linear program by a nonlinear optimization problem is du originally to Courant [3] in 1941 with a penalty function of exterior type and later to Frisch [4], in 1955, when he introduced the logarithmic barrier function which is an interior penalty one. We recall also that the notion of interior penalty operators were introduced by Auslender [5] in 1976 to generalize the concept of barrier functions.
where
F r,µ (x) = c, x + µg r (x) if x ∈ [0, +∞) n , +∞ elsewhere.
It is easy to see that F r,µ is a differential barrier function and then Proposition 1.1 the optimal solution of (P µ,r ) belongs to the interior of the positive orthant. The algorithm we build, called galpv4, is of primal type and uses an affine scaling approach 3 . It consists of two combined phases. The first one improves the feasibility of the current point and the second brings the point closer to an optimal solution. At each iteration, this requires the computation of two directions. The direction d k , bringing a current point x k closer to the optimal solutions' set is obtained as follows. We compute at x k the Newton direction d k (µ) for the problem (P µ,r ). Vector d k is then the part of the expression of d k (µ), independent of parameter µ and satisfies
δ µ d k (µ) = d k + O(µ), where δ µ is a positive real function of µ. That is d k = lim µ↓0 δ µ d k (µ). The direction d ′k
that improves the feasibility of the current point is obtained by using the same process with the linear program
min{λ : Ax + λ b − Ax k = b, x ∈ [0, +∞) n and λ ∈ [0, +∞)}.
We show that the sequence (x k ) converges. Its limit is an interior point of the optimal solutions' face of the linear program when β ∈ 0, 2 3 , where β is the factor of the maximal step size with respect to x k and d k . Moreover, by calculating d k , the algorithm generates a sequence (y k , s k ) that converges to the ξ ⊕ r -analytic center of the dual optimal solutions' face, where ξ ⊕ r is the polar concave gauge function (see [2]) of ξ r . In Proposition 2.1 of Section 2, we show that galpv4 includs the classical affine scaling approach by setting r = 0. In this respect, we compare the algorithm's performances between different values of r ∈ [0, 1) through numerical experiments using the familiar netlib test set [10]. The paper is organized as follows. In section 2 we present the algorithm, the computation of an affine scaling direction and how to find approximately a relative interior feasible solution. Section 3 deals with the convergence results and a stopping criteria, followed by numerical results and comments in Section 4. Finally, we close the paper by some concluding remarks in Section 5.
Presentation of a primal affine scaling method
In order to take account of possible bound constraints, we consider in all the following, the linear program
min{ c, x : Ax = b, x ≥ 0, x i ≤ u i i ∈ I} (LP B)
where I is a subset to {1, 2, · · · , n} and u ∈ R n is given such that u i > 0 if i ∈ I and u i = +∞ if not. It is easy to see that the dual problem of (LP B) is max{ b, y − u I , w I : A t y + s − w = c, w I = 0, s, w ≥ 0} (LDB) where I = {1, 2, · · · , n} − I. Moreover if (x, y, s, w) is a primal-dual optimal solution then it is easy to see from the KKT optimality conditions that
s = c − A t y + w, w I = −U −1 I X I (c − A t y) I and w I = 0(1)
where U I = diag (u I ) and X I = diag (x I ). We assume that there exists a relative interior feasible solution to (LP B) and that the minimum is finite. Hence the optimal solutions' set of (LP B) and of (LDB) are non empty, and there is no duality gap. Moreover the set of optimal solutions to (LDB) is compact. Now taking account of the slack variables u i − x i , we adapt the definition of ξ r as
ξ r : x → x r i + i∈U (u i − x i ) r 1 r if x ≥ 0 and u i − x i ≥ 0, ∀i ∈ I, −∞ elsewhere.
The following algorithm, called galpv4, uses an approach based on a version of the classical affine scaling algorithm presented in [7,8,11]. Algorithm galpv4 Initialization Construct a starting point x as described just bellow and choose r ∈ [0, 1). Compute y, w, s according to (8), (9) and (10) respectively. Compute the expected relative duality gap Rgap according to (11) Set the feasibility measure Rf ← Ax − b ∞ b ∞ + 1 Choose ǫ > 0 a stopping rule parameter.
While min(Rf, Rgap) > ǫ do
Compute d x the feasible direction according to (7) Compute d the descent direction according to (6) Set t max ← min min
dx i <0 − x i d xi , min dx i >0,i∈I − u i − x i d xi , 1 If Rf > ǫ then t ← 0.95t max else set t ← 0.65t max Set x ← x + td x Set t max ← min min di<0 − x i d i , min di>0,i∈I − u i − x i d i If Rf > ǫ then t ← 0.65t max else set t ← 0.95t max Update x ← x + td
Update y, w, s according to (8), (9) and (10) respectively. Update the expected relative duality gap Rgap according to (11)
Update Rf ← Ax − b ∞ b ∞ + 1 End while
Let us describe now how to construct, empirically, a starting point. In fact we construct two starting points x 1 and x 2 . The first one is defined as follow.
For j = 1, 2, .., n, x 1 j = min n A j. , 0.9u j if c j < 0 and x 1 j = min n A j. , 0.1u j otherwise, where A j. is the j th column of matrix A. The second one is defined as in the routine pcinit.f of the software HOPDM of Gondzio [12]. Our starting point x is chosing as follow. If min x 2 i > min j x 1 j or min j x 1 j < 1 then we set x = x 2 else we set x = x 1 . Note that the algorithm can be extended to the case r = 0. It is justified by the following proposition.
Proposition 2.1 Set n I = card(I) and define ξ 0 as
ξ 0 (x) = i∈{1,···,n} x i i∈I (u i − x i ) 1 n+n I if x ∈ [0, +∞) n −∞ elsewhere. For any r ∈ (−∞, 0) ∪ (0, 1) we setξ r = 1 (n+nI ) 1 r ξ r . Then for every x ∈ (0, +∞) n , (i) lim r↑0ξ r (x) = lim r↓0ξ r (x) = 1 n+nI ξ 0 (x), (ii) lim r↑0 ∇ξ r (x) = lim r↓0 ∇ξ r (x) = 1 n+nI ∇ξ 0 (x), (iii) lim r↑0 ∇ 2ξ r (x) = lim r↓0 ∇ 2ξ r (x) = 1 n+nI ∇ 2 ξ 0 (x).
Proof. (i). Without loss of generality we can assume that I = ∅. Let x ∈ (0, +∞) n and set ψ 1 (r) = ln( 1 n Σx r i ) and ψ 2 (r) = r. We have lim
ψ 1 (r) ψ 2 (r) = lim r→0 ψ ′ 1 (r) ψ ′ 2 (r) , but ψ ′ 1 (r) = n i=1 x r i ln x i n i=1 x r i . The result
follows.
(ii) and (iii). Using (i) and the expressions of ∇ξ r and ∇ 2ξ r , it is easy to see that lim r→0 ∇ξ r (x) = ∇ 1 n ξ 0 (x) and lim
r→0 ∇ 2ξ r (x) = ∇ 2 1 n ξ 0 (x). ⊓ ⊔
Finding a descent direction
Let x be a relative interior feasible point to (LP B), µ > 0 and r ∈ (0, 1). The Newton direction at x to the penalized problem (P µ,r ) is obtained by solving the minimization problem
min 1 2 ∇ 2 F r,µ (x)d, d + ∇F r,µ (x), d : Ad = 0 .
Using the KKT optimality conditions, the problem amounts to finding d(µ) ∈ R n and y ∈ R m solutions to the system of linear equations
∇ 2 F r,µ (x)d(µ) + ∇F r,µ (x) + A t y = 0 Ad(µ) = 0.(2)
We have ∇F r,µ (x) = c−µGe and ∇ 2 F r,µ (x) = µ(1−r)H where e t = (1, 1, .., 1) ∈ R n , G and H are diagonal matrices defined respectively by
G ii = x r−1 i if i / ∈ I, x r−1 i − (u i − x i ) r−1 otherwise(3)
and
H ii = x r−2 i if i / ∈ I, x r−2 i + (u i − x i ) r−2 otherwise.(4)
Then setting
P = I − H − 1 2 A t AH −1 A t −1 AH − 1 2 ,(5)
the projection matrix on the kernel of AH − 1
2 , system (2) reduces to P H − 1 2 ∇F r,µ (x)+ µ(1 − r)H 1 2 d(µ) = 0 and then µ(1 − r)d(µ) = −H − 1 2 P H − 1 2 ∇F r,µ (x) = −H − 1 2 P H − 1 2 c + µH − 1 2 P H − 1 2 Ge.
Since µ(1 − r) > 0 we can so take as an affine scaling direction at x to the linear program vector d given by
d = lim µ↓0 µ(1 − r)d(µ) = −H − 1 2 P H − 1 2 c(6)
Observe that since c, d = − P H − 1 2 c 2 2 < 0, d is a descent direction to the linear program at every point to R n . Remark: To improve the quality of the direction d, in order to maintain a good feasibility to the current point, we can compute in addition the direction H − 1 2 P H 1 2 d which can be used instead of d. Which in fact amounts to projecting a second time the direction H 1 2 d onto the vector subspace ker AH − 1 2 . Of course, theoretically the two directions are identical, but numerically there is a significant difference. However the computation of H − 1 2 P H 1 2 d has some extra cost in number of operations. Therefore we use the technic only when the relative duality gap is less than 0.001 or the current number of iterations exceeds 20.
Finding a feasible solution
It is well known that an approximate relative interior feasible solution to (LP B) can be obtained by solving a linear problem of the form min λ :
Ax + λ(b − Ax 0 ) = b, x ≥ 0, x i ≤ u i , ∀i ∈ I, λ ≥ 0 , (F LP ) where x 0 is a point arbitrarily chosen in (0, +∞) n . Write (F LP ) as min c,x :Ãx = b,x ≥ 0,x i ≤ u i , ∀i ∈ I , wherex = x λ ,c = 0 R n 1 andà = ( A b − Ax 0 )
. Then using (6), the affine scaling directiond with respect tox is given byd
= −H − 1 2PH − 1 2c whereH = H 0 0 λ r−2 andP = I −H − 1 2Ã t ÃH −1Ãt −1ÃH − 1 2 .
But the matrixÃH −1Ãt will be generally dense when there is one dense column inÃ. Column b − Ax 0 , in most cases, is dense. So for large-scale applications, we split such column from the others. We proceed as follow.
Set v = b − Ax 0 . ThenÃH −1Ãt = AH −1 A t + λ 2−r vv t . Using the Sherman- Morrison formula we have ÃH −1Ãt −1 = AH −1 A t −1 + δλ 2−r ww t , where w = AH −1 A t −1 v and δ = −1 1 + λ 2−r w, v . Sõ P = P − δλ 2−r H −1 2 A t ww t AH −1 2 δλ 1− r 2 H −1 2 A t w δλ 1− r 2 w t AH −1 2 −δ = P 0 0 0 − δ λ 1− r 2 H −1 2 A t w −1 λ 1− r 2 w t AH −1 2 −1 . It follows thatd = −δλ 2−r H −1 A t w −1 . Since −δλ 2−r > 0 the search
directions with respect to x and λ can be expressed respectively as d
x = −H −1 A t AH −1 A t −1 (b − Ax 0 ) and d λ = −1.
But numerical experiments show that as iterations go, the constraint Ax + λ b − Ax 0 = b is less and less satisfied. This is due to the rounding off errors generated by the projection onto kerà at each iteration and thus creating a snowball effect. To work around this problem, we proceed as follows: Let x k be a current point in the feasibility searching phase. Then x k 1 is a feasible point of problem
min λ : Ax + λ b − Ax k = b, x ≥ 0, x i ≤ u i , ∀i ∈ I, λ ≥ 0 .
In this case the search direction with respect to x k is
d k x = −H −1 A t AH −1 A t −1 (b − Ax k )(7)
It follows that the point
x k+1 λ k+1 = x k 1 + t k d k x −1 for a step size t k
suitably chosen, does not suffer from the snowball effect mentioned above.
Remark: To compute AH −1 A t −1 (b − Ax)
and P H − 1 2 c we solve for w and ∆ by Cholesky factorization the linear systems
AH −1 A t w = b − Ax and AH −1 A t ∆ = H − 1 2 c and then we compute P H − 1 2 c = H − 1 2 c − H − 1 2 A t ∆.
Convergence, dual solution and stopping criteria
Without loss of generality we can assume in this section that I = ∅. In this case
H = X r−2 , P = I − X 1− r 2 A t (AX 2−r A t ) −1 AX 1− r 2 , where X = diag(x) and x ∈ (0, +∞) n .
To simplify we assume that the starting point x 0 is a relative interior feasible solution to the linear program. So we consider (x k ) k∈N the sequence defined by
x k+1 = x k + βt k max d k , where β ∈ (0, 1) and t k max is the max- imum step length with respect to x k and d k = −X 1− r 2 k P X 1− r 2 k c = −X 2−r k c + X 2−r k A t (AX 2−r k A t ) −1 AX 2−r k c. Set y k = AX 2−r k A t −1 AX 2−r k c and s k = c − A t y k .
Here is our main result.
Theorem 3.1 Assume that 0 < β < 2 3
. Then x k , y k , s k k∈N converges to (x, y, s), where (y, s) is the ξ t -analytic center to the dual optimal face of the linear program, t is such that 1 t + 1 r = 1 and x belongs to the relative interior of the primal optimal face of the linear program.
Before giving the proof of the theorem, we first establish some preliminary results.
Proposition 3.1 ∞ k=0 βt k max P X 1− r 2 k c 2 2
is a converging series.
Proof. We have c, x k+1 − c, x k = βt k max c, d k = −βt k max P X 1− r 2 k c 2 2 .
The sequence c, x k k∈N is then decreasing. Since we assumed that the optimal value of the linear program is finite, the sequence is bounded and then converges. Set c its limit. Then we have
∞ k=0 βt k max P X 1− r 2 k c 2 2 = c, x 0 − c <
+∞. The result then follows. ⊓ ⊔ Now let us recall an important result. It was proved by Monteiro et al. [13], Saigal [11], Tseng and Luo [14] and Tsuchiya [15]. Corollary 3.1 Let x ∈ (0, +∞) n . Then X 1− r 2 P X 1− r 2 c satisfies
X 1− r 2 P X 1− r 2 c 2 ≤ L(A, c) c, X 1− r 2 P X 1− r 2 c .
Proof. First observe that X 1− r 2 P X 1− r 2 c P X 1− r 2 c 2 can be viewed as the optimal solution to the following ellipsoidal problem max c, w :
Aw = 0, X r 2 −1 w 2 2 ≤ 1 (EP r )
Hence using Theorem 3.2 by consideringX = X 1− r 2 instead of X, the result follows.
⊓ ⊔ Proposition 3.2 (x k ) k∈N is a converging sequence, say to x. Furthermore,
for each k ∈ N, x k − x 2 ≤ h c, x k − c , where h = 1 L(A, c) . Proof. By Corollary 3.1 we have c, x k − c, x k+1 = −βt k max c, d k ≥ L(A, c) βt k max d k 2 = L(A, c) x k+1 − x k 2 . It follows that +∞ > c, x 0 −c = 0≤k<+∞ c, x k − x k+1 ≥ L(A, c) 0≤k<+∞ x k+1 − x k 2 .
Thus (x k ) k∈N converges. Now using again Corol-
lary 3.1 we have c, x k −c = ∞ j=0 c, x k+j − x k+j+1 ≥ 1 h ∞ j=0 x k+j − x k+j+1 2 ≥ 1 h ∞ j=0 x k+j − x k+j+1 2 = 1 h x k − x 2 . The result follows. ⊓ ⊔
Now we recall the next theorem proved by Dikin [16]. A proof can also be found in [11,17,18,19].
Theorem 3.3
For every x > 0 and for every p ∈ R n , we have
AX 2 A t −1 AX 2 p 2 ≤ q(A) p 2 ,
where q(A) is a constant only function of A.
Proposition 3.3
The sequences (y k ) and (s k ) are bounded.
Proof According to Theorem 3.3, for every x > 0 and for every p ∈ R n , we have
y k 2 = AX 2−r k A t −1 AX 2−r k c 2 = A X 1− r 2 k 2 A t −1 A X 1− r 2 k 2 c 2 ≤ q(A) c 2 and then s k 2 = c − A t y k 2 ≤ (1 + q(A) A 2 ) c 2 .
The result then follows. ⊓ ⊔ Let us now consider the following notation. Given x ∈ (0, +∞) n and s ∈ R n we set I r (x, s) = {i :
x 1−r i |s i | = X 1−r s ∞ }.
Lemma 3.1 Let (x, s) ∈ (0, +∞) n × R n be such that Xs = 0. One has for every (r, r ′ ) ∈ [0, 1] 2 , if r < r ′ then x ir ≥ x i r ′ and |s ir | ≤ |s i r ′ |, ∀(i r , i r ′ ) ∈ I r (x, s) × I r ′ (x, s).
Proof. We have
0 < x 1−r i r ′ |s i r ′ | ≤ X 1−r s ∞ = x 1−r ir |s ir |(1)
and
0 < x 1−r ′ ir |s ir | ≤ X 1−r ′ s ∞ = x 1−r ′ i r ′ |s i r ′ |(2)
Multiplying side by side (1) and (2)
one has 0 < x 1−r i r ′ x 1−r ′ ir |s ir ||s i r ′ | ≤ x 1−r ir x 1−r ′ i r ′ |s i r ′ ||s ir |. That is x r ′ −r i r ′ ≤ x r ′ −r ir
and then x i r ′ ≤ x ir . Now using (2) one has 0 <
x 1−r ′ i r ′ |s ir | ≤ x 1−r ′ ir |s ir | ≤ X 1−r ′ s ∞ = x 1−r ′ i r ′ |s i r ′ |.
The result then follows. ⊓ ⊔ Define I = {i : x i = 0}, J = {i : x i > 0} and n I = card(I).
Lemma 3.2 There ish > 0 such that x k J − x J 2 ≤h x k I 2 , ∀k ∈ N.
Proof. Let (y, s) be an accumulation point of (y k , s k ). The existence of (y, s) is ensured by Proposition 3.3. Using Proposition 3.2 we have
x k −x 2 2 = x k I 2 2 + x k J − x J 2 2 ≤ h 2 c, x k − x 2 = h 2 s, x k − x 2 = h 2 s I , x k I 2 ≤ h 2 s I 2 2 x k I 2 2
and then x k J − x J Proposition 3.4 Let β ∈ (0, 1). Then there exists K ∈ N such that i) ∀k ≥ K, ∀r ∈ (0, 1), ∀i r ∈ I r (x k , s k ),
x k ir = O( x k I ∞ ) and s k ir = O(1). Furthermore X 1−r k s k ∞ = X 1−r k,I s k I ∞ = O x I 1−r ∞ and there exists a constantĈ such that s k J 2 ≤Ĉ x I 2−r ∞ . ii) ∀k ≥ K, c, x k+1 −c ≤ L( c, x k −c), where L = 1−β L(A, c) 6−r 2−r 2 3−r 2−r g 6−2r 2−r n 7 2 − r(1−r) 2(2−r) I . iii) ∞ k=0 x k I a ∞ < +∞, ∀a > 0. iv) x k I , s k I = O( x k I ∞ ). v) c, x k − c = O( x k I ∞ ) and x k − x 2 = O( x k I ∞ ). Proof. i) and ii) We have c, x k+1 − c = c, x k − c − βt max P X 1− r 2 k c 2 2 = c, x k −c−βt max X 1− r 2 k s k 2 2 ≤ c, x k −c−βt max X 1− r 2 k s k 2 ∞ . Now t k max = min − x k i d k i : d k i < 0 ≥ 1 X −1 k d k ∞ = 1 X 1−r k s k ∞ . Then c, x k+1 − c ≤ c, x k −c−β X 1− r 2 k s k 2 ∞ X 1−r k s k ∞ . Let i r 2 , i r ∈ I r 2 (x k , s k )×I r (x k , s k ). From Lemma 3.1 one has x k i r 2 ≥ x k ir and then c, x k+1 − c ≤ c, x k − c − βx k i r 2 s k i r 2 2 s k ir(1)
Using Proposition 3.2 , the fact that X
1− r 2 k P X 1− r 2 k c P X 1− r 2 k c 2
is the optimal solution to (EP r ) and
x k − x X r 2 −1 k x k − x 2 is a feasible solution of (EP r ), L(A, c) x k − x 2 X r 2 −1 k (x k − x) c, x k − x X r 2 −1 k (x k − x) 2 ≤ c, X 1− r 2 k P X 1− r 2 k c P X 1− r 2 k c 2 = P X 1− r 2 k c 2 = X 1− r 2 k s k 2 . It follows that L(A, c) 2 x k I 2 2 + x k J − x J 2 2 X k r 2 I e I 2 2 + X k r 2 −1 J (x k J − x J ) 2 2 ≤ X 1− r 2 k s k 2 2 . Now
using Lemma 3.2, the fact that r ∈ (0, 1) and the fact that lim k→∞ x k I = 0, for k being large enough one has X k
r 2 −1 J (x k J − x J ) 2 2 ≤ M r−2h2 x k I 2 2 = M r−2h2 X k 1− r 2 I X k r 2 I e I 2 2 ≤ M r−2h2 x k I 2−r ∞ X k r 2 I e I 2 2 ≤ X k r 2 I e I 2 2
, where e I is the vector of R nI whose components are equal to 1. According to iii)
(z) = i∈I z r 2 i 2 r if z ∈ [0, +∞) nI , −∞ elsewhere and ξ r r−2 ,I (z) = i∈I z r r−2 i r−2 r if z ∈ (0, +∞) nI , 0 if z ∈ ∂[0, +∞) nI , −∞ elsewhere.
Here . Then using (29bis)
∂[0, +∞) nI de- notes the boundary of [0, +∞) nI . That is i∈I x k i r = X k r 2 I e I 2 2 = i∈I x k i r ≤ n 1− r 2 I x k I r 2 . Hence L(A, c) 2 2n 1− r 2 I x I 2−r 2 ≤ L(A, c) 2 x k I 2 2 + x k J − x J 2 2 X k r 2 I e I 2 2 + X k r 2 −1 J (x k J − x J )M 2−r s k J 2 ≤ M 1− r 2 X 1− r 2 k,J s k J 2 ≤ min j∈J x k j 1− r 2 X 1− r 2 k,J s k J 2 ≤ X 1− r 2 k P X 1− r 2 k c J 2 ≤ X 1− r 2 k P X 1−we get X 1−r k,J s k J 2 ≤ L(A, c) √ n I s I ∞ M x k I ∞ X 1−r k,I s k I 2 .
Hence using again the fact that lim
k↑∞ x k I = 0 we get X 1−r k s k ∞ = X 1−r k,I s k I ∞ for k large enough.
Turn back now to (2). Then when k is large enough we have
L(A, c) 2 n 1− r 2 I x k I 2−r 2 ≤ 2 X 1− r 2 k,I s k I 2 2 ≤ 2n I X 1− r 2 k,I s k I 2 ∞ = 2n I x k i r 2 1− r 2 s k i r 2 2 ≤ 2n I x k I 2−r 2 s k i r 2 2(3)
and
L(A, c) 2 n 1− r 2 I x k I 2−r 2 ≤ 2n I x k i r 2 1− r 2 s k i r 2 2 ≤ 2n I x k i r 2 2−r s k I 2 ∞ .(4)
Using (3) and Lemma 3.1 it follows that
s k ir ≥ s k i r 2 ≥ L(A, c) √ 2n 1− r 4 I(5)
and then s k ir = O(1). Now using (4), (5) and the fact that +∞ > g = sup k∈N s k ∞ , we get iii) By ii) and Proposition 3.
x k I 2 ≥ x k i r 2 ≥ L(A, c)I x k I 2 . But c, x k −x = s, x k −x = s I , x k I ≤ s I 2 x k I 2 ≤ √ n I g x k I 2 . It follows that x k i r 2 s k i r2, x k I ∞ ≤ x k − x 2 ≤ H c, x k − c ≤ HL k−K c, x K − c , ∀k ≥ K. The result follows since 0 < L a = 1 − L(A, c) 6−r 2−r 2 3−r 2−r g 6−r 2−r n 7 2 − r(1−r) 2(2−r) I a < 1, ∀a > 0. iv) We have L(A, c) x k I 2 ≤ L(A, c) x k − x 2 ≤ c, x k − c = s k , x k − x = s k I , x k I + s k J , x k J − x J = s I , x k I ≤ s I 2 x k I 2 .
The result then follows from i). v) Let (y, s) be an accumulation point to (y k , s k ). We have c, x k − c = A t y + s, x k − x = s I , x k I . Using Proposition 3.2 and the Cauchy-Schwartz inequality we get
s 2 x k I 2 ≥ s, x k = c, x k − c ≥ 1 h x k − x 2 ≥ 1 h x k I 2 .
The result then follows. ⊓ ⊔ Now we establish some technical results given by Saigal [11] in the classical case.
Proposition 3.5 Let (u k ) the sequence defined by u k = X r 2 k P X 1− r 2 k c c, x k − c = X k s k c, x k − c .
Then we have:
i) The sequence (u k ) is bounded. ii) ∞ k=0 u k J 2 < +∞.
iii) ∞ k=0 |δ k | < +∞, where δ k = e I , u k I − 1.
Proof.
i) and ii) By Proposition 3.2, there is h > 0 such that
x k − x ≤ h c, x k − c .
Then u I 2 = X k,I s k
I 2 c, x k − c ≤ h X k,I s k I 2 x k − x 2 ≤ h x k I ∞ s k I 2
x k I ∞ = h s k I 2 Hence u k I is bounded according to Proposition 3.3. According to Proposition 3.4
u k J 2 = X k,J s k J 2 c, x k − c ≤ h x k J ∞ s k J 2 x k − x 2 ≤ hM s k J 2 x k I 2 ≤ hĈM x k I 1−r .
Then i) and ii) follows by using Proposition 3.4. iii) Set SD = {(y, s) : A t y + s = c, s J = 0} the expected dual optimal solutions' set. Let ŷ k ,ŝ k a solution to the problem min s k − s 2 : (y, s) ∈ SD .
We have c,
x k − c = c, x k − x = ŝ k + A tŷk , x k − x = ŝ k I , x k I . Then e I , u k I − 1 = x k I , s k I − ŝ k I , x k I c, x k − c ≤ s k I −ŝ k I 2 x k I 2 c, x k − c . By Proposition 3.2 we have x k I 2 ≤ x k − x 2 ≤ h c, x k − c . Hence e I , u k I − 1 2 ≤ h s k I −ŝ k I 2 .
From Theorem 7 of [11], there isM such that ŝ k − s k 2 ≤M s k J 2 . Using Proposition 3.4 we get s k I −ŝ k I 2 ≤M s k J 2 ≤ĈM x I 2−r ∞ . The result then follows by using iii) of Proposition 3.4.
⊓ ⊔ Now let us introduce the potential function F defined as follow:
F (x) = ln c, x − c ξ r (x) , whereξ r : x → i∈I x r i 1 r if x ≥ 0, −∞ elsewhere.
The following Proposition holds.
Proposition 3.6 There is ∆ ∈ R such that for every k ≥ 0, F (x k ) ≥ ∆ > −∞.
Proof. By Theorem 3.1 and Proposition 4.3 of [2],ξ r (x k )ξ t (e) ≤ x k I , e I , where t is such that 1 t Proof. Let us proof at first that given β ∈ (0, 1), there is K ∈ N such that
+ 1 r = 1. Henceξ r (x k ) ≤ 1 n 1 t I i∈I x k i ≤ n 1 2 n 1 t I x k I 2 = n 1 r − 1 2 x k I 2 = n 1 r − 1 2 x k − x 2 ≤ n 1 r − 1 2 h c, x k − c and then −∞ < ln 1 n 1 r − 1 2 h ≤ F (x k ). The result then follows. ⊓ ⊔ Proposition 3.7 Let β ∈ 0, 2 3 . There exists K ∈ N such that ∀k ≥ K F (x k+1 )−F (x k ) ≤ −θ k Υ k + (1 − Υ k ) 2 − 3β 3(1 − β) X r 2 k,I w k I 2 2 −(1+Υ k ) θ k j∈I x k r δ k −θ k γ k where Υ k = 1 if max i∈I w k i ≤ 0, 0 if max i∈I w k i > 0, θ k =t k 1 − 1 i∈I x k i rt k ,t k = t k c, x k − c , t k = βt k max , γ k = X∀k ≥ K, F x k+1 − F x k ≤ ln 1 − θ k X r 2 k,I w k I − 2 θ k i∈I x k i r δ k − θ k γ k − i∈I x k i r i∈I x k i r ln 1 − θ k w k i . We have F (x k+1 ) − F (x k ) = ln c, x k+1 − c c, x k − c − 1 r ln x k+1 r i x k r i , c, x k+1 − c = c, x k − c − t k X −r k X k S k , X k s k , u k = X k s k c, x k − c andt k = ( c, x k − c)t k . Then c, x k+1 − c c, x k − c = 1 −t k X −r k,I u k , u k = 1 −t k X −r k,I u k I , u k I −t k γ k . Now u k I = X r k,I w k I + 1 i∈I
x k r i X r k,I e I , X −r k,I u k I = w k I + 1 i∈I x k r i e I and δ k = u k I , e I −1 = X r k,I w k I , e I . Then X −r k,I u k I , u k I = X r k,I w k
I , w k I + 2 i∈I x k r i δ k + 1 i∈I x k r i . It follows that c, x k+1 − c c, x k − c = 1 −t k X r k,I w k I , w k I − 2t k i∈I x k r i δ k −t k i∈I x k r i −t k γ k = 1 −t k i∈I x k r i 1 − θ k X r k,I w k I , w k I − 2θ k i∈I x k r i δ k − θ k γ k .
Let us show now, for β ∈ (0, 1), θ k =t
k 1 −t k i∈I x k r i = i∈I x k r it k i∈I x k r i −t k > 0, for k large enough. We have c, x k − c = s k , x k − x = s k I , x k I + s k J , x k J − x J . Then i∈I x k r i −t k = i∈I x k r i − β c, x k − c max i∈I (X 1−r k s k ) i = i∈I x k r i max i∈I (X 1−r k s k ) i − β( s k I , x k I + s k J , x k J − x J ) max i∈I (X 1−r k s k ) i ≥ i∈I x k r i (x k 1−r i s k i ) − β s k I , x k I − β s k J , x k J − x J ) max i∈I (X 1−r k s k ) i = x k I , s k I max i∈I (X 1−r k s k ) i 1 − β s k J , x k J − x J x k I , s k I − β ≥ x k I , s k I max i∈I (X 1−r k s k ) i 1 − β s k J 2 x k J − x J 2 x k I , s k I − β
Using the fact that lim k↑∞ x k −x = 0, from Proposition 3.4 we get
s k J 2 x k J − x J 2 x k I , s k I = o( x k I 1−r ). Therefore θ k > 0. Now 1 r ln i∈I x k r+1 i i∈I x k r i = 1 r ln i∈I x k r i j∈I x k r j (1 −t k x k −r i u k i ) r .
Since the function t → ln t, t > 0, is concave, one has
1 r ln i∈I x k r+1 i i∈I x k r i r ≥ i∈I x k r i j∈I x k r j ln 1 −t k x k −r i u k i = i∈I x k r i j∈I x k r j ln 1 −t k j∈I x k r j −t k w k i = ln 1 −t k j∈I x k r j + i∈I x k r i j∈I x k r j ln 1 − θ k w k i . Hence F (x k+1 )−F (x k ) ≤ ln 1 − θ k X r k,I w k I , w k I − 2θ k i∈I x k r i δ k − θ k γ k − i∈I x k r i j∈I x k r j ln 1 − θ k w k i .
Assume now that max i∈I w k i > 0. Then using Lemma 8 of [11] or its proof we can easily see
i∈I x k r i j∈I x k r j ln 1 − θ k w k i ≥ − θ k j∈I x k r j δ k − θ k 2 2 j∈I x k r j X r 2 k,I w k I 2 1 − θ k max i∈I w k i .
Using in addition the fact that ln(1 − a) ≤ −a, ∀a < 1 we get
F (x k+1 ) − F (x k ) ≤ −θ k 1 − θ k 2 j∈I x k r j 1 1 − θ k max i∈I w k i X r 2 k,I w k I 2 2 − θ k j∈I x k r j δ k − θ k γ k . Now 1 − θ k 2 j∈I x k r j 1 1 − θ k max i∈I w k i = 1 − θ k 2 j∈I x k r j 1 1 + θ k j∈I x k r j − θ k max i∈I (X −r k u k ) i = 1 − 1 2 j∈I x k r i θ k 1 + θ k j∈I x k r j 1 1 − θ k 1 + θ k j∈I x k r j max i∈I (X −r k u k ) i .
We have on the one hand θ k =t
k 1 −t k j∈I x k r j and thent k = θ k 1 + θ k j∈I x k r j . On the other hand,t k = β c, x k − c max i∈I (X 1−r k s k ) i = β max i∈I (X −r k u k ) i . It follows that 1 − θ k 2 j∈I x k r j 1 1 − θ k max i∈I w k i = 1 −t k 2(1 − β) i∈I x k r i = 1 − β 2(1 − β) c, x k − c i∈I x k r i max j∈I (X 1−r k s k ) i .
if t ∈ (0, 1) and
ξ t,I (s I ) = i∈I s i t 1 t if s I ∈ (0, +∞) nI 0 if s I ∈ ∂([0, +∞) nI ) −∞ elsewhere,
if t ∈ (−∞, 0). The unicity of the solution is ensured by the strict quasiconcavity of ξ t and Lemma 1 of [1]. The KKT optimality conditions are then expressed as follow. There exist (y, s) ∈ R m × R n and v ∈ R n satisfying the following conditions, ∇ξ t,I (s I ) = v I , Av = 0, A t y + s = c, s J = 0, s ≥ 0.
k,I i∈I x k i r i∈I x k i r c, x k − c X 1−r k,I s k I − e = 0 (1) For all k ∈ N we set I(k) = {i : x k i ≥ x k I 2 ∞ }.
We shall prove that for some K chosen large enough, I(k) ⊂ I(k + 1), ∀k ≥ K. For k ∈ N and i ∈ {1, · · · , n} we set ǫ k i = j∈I x k j r c, x k − c x k i 1−r s k i − 1. Let ǫ > 0 be small enough. Then by (1) there exists K ∈ N large enough such that ∀k ≥ K, |ǫ k i | ≤ ǫ, ∀i ∈ I(k). Let k ≥ K and i ∈ I(k). Since K is assumed to be large we have necessarily from (1) s k i > 0. Using in addition the fact that s k ∞ ≤ g and Proposition 3. Table 1 ⋆⋆ : Number of iterations exceeds 300. * : Best optimal value obtained with Rgap ∈ (10 −8 , 10 −10 ).
At first we can observe that for every problem there is at least one r value for which the problem is solved. Also, we can see that most problems are solved for r between 0 and 0.5. The maximum number of problems solved is reached for r = 0.2, as shown in the graph below. The results are conclusive and show that differentially barriers penalty functions offer effective alternative to conventional logarithmic barrier function in linear programming.
As we point out in the introduction, we chose an algorithm of affine scaling type for the simplicity of its implementation. But the "Predictor-corrector" method of Mehrotra [20] has proved highly efficient in the classical case (r = 0). Our immediate goal is to adapt this method to these new penalty functions.
Theorem 3. 2
2There exists a constant L(A, c) > 0 such that every optimal solution w to the following ellipsoidal problem max c, w : Aw = 0, w 2 ≤ L(A, c) c, w .
1 ≥ 0 also means that s 2 ≥ L(A, c), for every s be an accumulation point of (s k ). In all the following we set g = sup k∈N s k ∞ , x J > 0, M > 0 and that according to Proposition 3.3 g < +∞.
where ξ r 2 ,I and ξ r r−2 ,I are the concave gauge functions respectively defined by ξ r 2 ,I
∞
) and x k ir = O x k I 2 , witch implies that i r ∈ I. Now using (5),(6) and the fact that |s k ir | ≤ s k ∞ ≤ g we get x
(
c, x k − c) and then by (1), c, x k+1 − c ≤ L( c, x k − c).
ǫ k i ) ≥ L(A, c) x k − x 2 (1 + ǫ k i ) ≥ L(A, c) x k I ∞ (1 − ǫ). Hence x s I , x I . It follows that X then i ∈ I(k +1). Set nowÎ = ∪ k∈N I(k) and let us prove that in factÎ = I. Assume for contradiction that there is i ∈ I−Î. Then ∀k ≥ K
An affine scaling algorithm was originally proposed by Dikin in 1967[6]. It was rediscovered by several researchers such as barnes[7] and Vanderbei et al[8] after Karmarkar[9] proposed his famous projective scaling algorithm.
t if x I ≥ 0 −∞ elsewhere,
. R.J. Vanderbei, M.J. Meketon and B.A. Freedman,A Modification of Karmarkar's linear programming algorithm, Algorithmica 1(1986)395-407.
Using the fact that i∈I x k r i maxx k i s k i and the fact that c,x k r δ k −θ k γ k . Consider now the case max i∈I w k i ≤ 0.Then i∈I x k i r i∈I x k i r ln 1 − θ k w k i ≥ 0 and the result follows by using the fact thatThen using again Proposition 3.4 we gettBut β ∈ (0, 1). The result then follows.⊓ ⊔ Now as is mentioned in Theorem 1 of[1], given t in (−∞, 0) ∪ (0, 1), we say the ξ t -dual-analytic center the unique optimal solution to the problem max y,s. Using in addition the fact that s k is bounded it follows thatwhich is absurde. Henceand then there is necessarily τ > 0 such that τ e I < s k I for k being large enough. Let now (ỹ,s) be an accumulation point to (y k , s k ). Then we have Xs = 0, Ax = b, Aỹ +s = c, x ≥ 0 ands ≥ 0. The KKT optimality conditions of (LP ) are then satisfied and then x is an (LP ) optimal solution and (ỹ,s) is a dual optimal solution. Moreover since x I = 0,s I > 0, x J > 0 and s J = 0 the strict complementary slackness condition holds. Let now k be large enough. Then it is easy to see with the help of Proposition 3.4 that ( c,have ξ t,I (∇ξ r,I (x k I )) = 1 and then, by continuity of ξ t,I on (0, +∞) nI , we getis bounded. Let then z be a limit of a convergent subsequence of (z k ). Then s I ξ t,I (s I ) = ∇ξ r,I (z I ). Using again The result then follows ⊓ ⊔ Turn back now to the case where I = ∅. Then using (1) of Section 2 and adapting results of this section, the expected dual approximate optimal solution vectors y, s and w, associated to a current point x arewhere U = diag(u). Hence the expected relative duality gap isNumerical resultsThe porpose of the following tests is to compare the algorithm performance according to different values of r between 0 and 1. We have opted to consider the following values r = 0 (the classical case), r = 0.1, r = 0.2, r = 0.3, r = 0.4, r = 0.5, r = 0.6 and r = 0.7. We solved a large set of testing problems, taken from the familiar Netlib test set (GAY[10]czprob 80 64 61 57 59 61 77 ⋆⋆ d2q06c 71 69 69 72 75 103 134 ⋆⋆ d6cube 117 87 85 72 68 70 74 82 degen2 44 43 43 37 42 44 49 62 degen3 53 51 50 47 51 49 64 91 dfl001 ⋆⋆ ⋆⋆ 144 166 174 ⋆⋆ ⋆⋆ ⋆⋆ e226 53 53 52 51 69 74 88 ⋆⋆ etamacro 49 50 55 65 79 100 ⋆⋆ ⋆⋆ fffff800 61 56 55 58 68 62 140 212 finnis 78 82 80 85 ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ fit1d 41 40 66 65 52 76 ⋆⋆ ⋆⋆ fit1p 31 32 32 41 48 50 73 ⋆⋆ fit2d 55 45 43 42 44 58 ⋆⋆ ⋆⋆ fit2p 44 47 48 51 45 56 73 110 forplan 48 46 46 50 52 55 64 ⋆⋆ ganges 27 26 34 29 32 36 44 56 gfrd-pnc 29 30 37 49 65 89grow7 97 87 83 82 81 79 72 67 grow15 113 97 101 94 88 81 71 66 112 109 103 89 82 72 63 60 israel 54 55 67 82 91 118 153 ⋆⋆ kb2 38 42 36 40 45 61 65 ⋆⋆ lotfi 46 46 48 53 58 66 88 106 maros 58 61 63 66 77 10168 recipe 38 38 37 38 38 33 31 48 sc105 26 27 27 28 26 29 29 32 sc205 35 37 31 34 35 40 47 63 sc50a 33 32 23 23 23 22 23 25 sc50b 23 23 22 21 21 21 20 21 scagr25 31 32 32 34 36 39 45 63 scagr7 37 29 31 31 32 42 39 52 scfxm1 44 44 45 48 53 61 83 118 scfxm2 46 47 51 54 62 74 95 148 scfxm3 45 45 50 54 64 81 102 146 scorpion 45 44 46 44 46 ⋆⋆ ⋆⋆ ⋆⋆ scrs8 ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ 107 134 190 ⋆⋆ scsd1 47 45 44 41 41 37 40 41 scsd6 53 47 46 43 41 41 32 34 scsd8 42 41 39 38 37 37 29 30 sctap1 49 49 51 51 55 52 62 73 sctap2 52 51 48 48 49 46 54 67 sctap3 56 49 47 46 48 50 59 73 seba 47 48 47 43 45 56 62 99 share1b 59 66 73 113 119 145 141 ⋆⋆ share2b 37 39 38 38 29 33 33 50 shell 51 48 47 45 46 48 54 67 ship04l 52 50 49 47 46 37 37 42 ship04s 52 50 48 46 36 45 35 39 ship08l 49 48 48 45 36 37 34 38 ship08s 52 50 49 48 46 39 37 50 ship12l 51 49 48 46 44 45 49 ⋆⋆ ship12s 49 48 46 46 40 40 49 53 sierra 52 51 51 54 68 86 141 ⋆⋆ stair 46 42 34 45 41 46 59 99 standata 56 63 57 57 63 63 75 89 standgub 56 54 53 57 55 58 73 88 standmps 71 66 64 65 68 69 81 101 stocfor1 43 43 43 45 49 48 56 73 stocfor2 70 72 78 89 99 116
the symbolic Cholesky factorization we use rdmps1, rdmps2, rdspec, prepro and all dependencies written by Gondzio. Our numerical experiments were performed on a laptop hp ZBook (Processor: Intel core i7-4810 MQ, CPU 2.804 × 8HZ -Operating system: Ubuntu linux). 12We use no presolvingthe symbolic Cholesky factorization we use rdmps1, rdmps2, rdspec, prepro and all dependencies written by Gondzio[12]. We use no presolving. Our numerical experiments were performed on a laptop hp ZBook (Processor: Intel core i7-4810 MQ, CPU 2.804 × 8HZ -Operating system: Ubuntu linux).
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Hopdm -Modular Solver for LP Problems, User's Guide to Version 2.12, Working Paper WP-95-50, International Institute for Applied Systems Analysis. J Gondzio, M Makowski, Laxenburg, AustriaGondzio J. and M. Makowski, Hopdm -Modular Solver for LP Problems, User's Guide to Version 2.12, Working Paper WP-95-50, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1995.
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| [
"P A Skordos \nCenter for Nonlinear Studies\nLos Alamos National Lab\nB258, 87545Los AlamosNM\n\nMassachusetts Institute of Technology\n545 Technology SquareNE43-432, 02139CambridgeMA\n"
]
| [
"Center for Nonlinear Studies\nLos Alamos National Lab\nB258, 87545Los AlamosNM",
"Massachusetts Institute of Technology\n545 Technology SquareNE43-432, 02139CambridgeMA"
]
| []
| A tantalizing version of Maxwell's demon is presented which appears to operate reversibly. A container of hard core disks is separated into two chambers of equal volume by a membrane that selects which disk can penetrate depending on the disk's angle of incidence. It is shown that the second law of thermodynamics requires the incompressibility of microscopic dynamics or an appropriate energy cost for compressible microscopic dynamics. | 10.1103/physreve.48.777 | [
"https://arxiv.org/pdf/chao-dyn/9305006v1.pdf"
]
| 45,432,644 | chao-dyn/9305006 | 63315603e4c72ac1c393b326fdf51330aa198197 |
May 14, 1993
P A Skordos
Center for Nonlinear Studies
Los Alamos National Lab
B258, 87545Los AlamosNM
Massachusetts Institute of Technology
545 Technology SquareNE43-432, 02139CambridgeMA
May 14, 19931 Submitted August 92, Accepted May 93, Phys. Rev. E 2
A tantalizing version of Maxwell's demon is presented which appears to operate reversibly. A container of hard core disks is separated into two chambers of equal volume by a membrane that selects which disk can penetrate depending on the disk's angle of incidence. It is shown that the second law of thermodynamics requires the incompressibility of microscopic dynamics or an appropriate energy cost for compressible microscopic dynamics.
Introduction
The second law of thermodynamics is usually attributed to the fact that states of maximumdisorder in a statistical system have the largest probability of occurrence among all possible states. The microscopic dynamics of a statistical system are assumed to be of minor importance. However the microscopic dynamics can not be arbitrary, and they must satisfy certain conditions for the second law to hold. In particular they must conserve phase space volume. Hamiltonian dynamics conserve phase space volume according to Liouville's theorem. However the condition of incompressibility is more general than the condition of Hamiltonian dynamics, and it is important to discuss incompressibility and the second law directly without the aid of Hamiltonians.
This paper examines the relation between compressible microscopic dynamics and the second law of thermodynamics using a time-reversible system of hard core disks and a membrane. The possibility that a Maxwell's demon could imitate the membrane of our system in a reversible manner is also explored, and it is shown that a Maxwell's demon can only approximate the membrane using irreversible operations. The membrane selects which disk can penetrate depending on the disk's angle of incidence, and creates a density di erence between two chambers initialized to have equal density.
The analysis of our system con rms that the second law of thermodynam-ics requires the incompressibility of microscopic dynamics or an appropriate energy cost for compressible microscopic dynamics 1].
The next section describes our system and explains how the membrane creates a density di erence between the two chambers. Section 3 estimates the density di erence theoretically, and section 4 con rms the theoretical estimate using a computer simulation. Section 5 explains how the membrane challenges the second law of thermodynamics, and discusses the evolution of the system of disks in phase space. Section 6 shows that the membrane compresses the phase space of the disks, and section 7 demonstrates that if the membrane interaction obeyed incompressible dynamics, then it would not create a density di erence (for piecewise di erentiable maps). Section 8 reviews another system that obeys compressible dynamics: a pump of disks consisting of a microscopic trapdoor and a cooling mechanism.
Section 9 analyzes the membrane as a Maxwell's demon that interacts with the disks based on information about the disks. The new demon uses a tennis racket to bounce the disks and to select which disk can penetrate the membrane depending on the disk's angle of incidence.
Description of the Model
Our system is shown in gure 1. It consists of a box containing hard core disks, and a membrane that separates the box into two chambers of equal volume. The hard core disks move and collide with each other elastically.
The membrane interacts with each incident disk according to the following equation, where V 1 ; V 2 are the velocity components of the disk before the interaction and V 0 1 ; V 0 2 are the velocity components of the disk after the interaction.
+45 :
8 > > < > > : V 0 1 = V 2 V 0 2 = V 1 9 > > = > > ; if 8 > > < > > : (V 1 < 0 V 2 < 0 jV 1 j > jV 2 j) or (V 1 > 0 V 2 > 0 jV 1 j < jV 2 j) 9 > > = > > ; 45 : 8 > > < > > : V 0 1 = V 2 V 0 2 = V 1 9 > > = > > ; if 8 > > < > > : (V 1 < 0 V 2 > 0 jV 1 j > jV 2 j) or (V 1 > 0 V 2 < 0 jV 1 j < jV 2 j) 9 > > = > > ; +90 : 8 > > < > > : V 0 1 = V 1 V 0 2 = V 2 9 > > = > > ; otherwise (1)
The labels +45, 45, and +90 are motivated by the discussion of section 9 that views the membrane as a Maxwell's demon playing tennis with the disks.
The above equation says that an incident disk in octants 2,4,5, and 7 (refer to gure 2) reverses the x-component of its velocity when it hits the membrane and is not allowed to penetrate. It also says that an incident disk in the remaining octants penetrates the membrane by being de ected toward the x-axis when coming from the left, and away from the x-axis when coming from the right. The speed and the kinetic energy of the disk are conserved. Assuming that the velocities of the disks are distributed isotropically inside the container, it follows from geometrical considerations that the impact rate of disks hitting the membrane is a cosine of the impact angle in absolute value, impact-rate / j cos j : (2) The membrane exploits this cosine distribution of impact angles by allowing disks with \high rate" impact angle to penetrate from the right, and allowing disks with \low rate" impact angle to penetrate from the left. To achieve reversibility, the remaining impact angles (\low rate" from the right and \high rate" from the left) are blocked and do not penetrate the membrane.
They are simply re ected back. Furthermore, each \high rate" angle from the right is rotated onto a \low rate" angle when the disk penetrates the membrane, and vice versa.
The membrane interaction of equation 1 makes the membrane more permeable from the right side than from the left. This leads to an excess ux of disks from the right side, and creates a density di erence between the two chambers. When the density di erence reaches an equilibrium value (which is calculated in the next section), the uxes of disks between the two sides of the membrane become equal.
Estimate of the Density Di erence
At equilibrium the uxes of disks from the left and from the right side of the membrane are equal to each other. In other words if N L is the normalized density in the left chamber, and P (L!R) is the probability of an individual disk to penetrate the membrane coming from the left, we require that N L P (L!R) = N R P (R!L) :
(
We can estimate P (L!R) using the fact that among all the disks that strike the membrane from the left only those with trajectories in the third and sixth octants of gure 2 are allowed to penetrate. In particular, these disks have impact angles in the intervals ( =4; =2) and ( ; 5 =4). In addition, we assume that the probability of an individual disk to strike the membrane varies as the cosine of the impact angle and is independent of the density in each chamber. Although the total impact rate depends linearly on the density of the disks, the probability of an individual disk to reach the membrane is independent of the density to a rst approximation. Thus we get,
P (L!R) ' 2 C Z =2 =4 cos d ' 2 C 0:3 ;(4)
for some normalization constant C; and similarly,
P (R!L) ' 2 C Z 5 =4 4 =4
cos d ' 2 C 0:7 ; (5) which gives N L ' 0:7 and N R ' 0:3 :
In other words the system of membrane and disks reaches equilibrium when the uxes of disks from the left and from the right side of the membrane are balanced, and this happens when the normalized density is approximately 0:7 in the left chamber and approximately 0:3 in the right chamber.
Simulation Results
To check the theoretical results of the previous section, a two-dimensional system of hard core disks with a membrane has been simulated. The computer program used in these simulations is the same program as the one described in detail in reference 2] with a few modi cations to simulate the membrane. In particular, when the center of an incident disk reaches the membrane, the disk's velocity is transformed according to equation 1.
In the experiments reported below forty disks are used. The size of each chamber is 24:3 10 13 cm 2 (equal size chambers), and the disk radius is
The Loss of Ergodicity
Our membrane appears to create a density di erence without doing any work.
In particular, it appears to establish the equivalent of a lower potential energy in the left chamber that creates a density di erence in order to equalize the potential di erence between the two sides. This can not be correct however.
If we cut a hole in the membrane, a return ux of disks will result through the hole which can be used to convert thermal energy into useful work in violation of the second law of thermodynamics.
In order to rescue the second law of thermodynamics we can follow a number of di erent approaches. The rst approach is to assume that the membrane can not be cut, that it must be a closed surface. In nature for example a chemical potential di erence exists at the interface between two di erent materials such p-n semiconductors (see 3]). There is nothing in our model of the membrane however that implies that the membrane is an interface between two di erent materials and that it can not be cut. So we have to look elsewhere for an explanation of why the membrane can not violate the second law of thermodynamics. We start by examining the evolution of the disks in phase space. In general there are many ways in which a system can fail to be ergodic.
A trivial way to lose ergodicity in the context of billiard balls is to remove all interactions between the disks. Then the disks can not see each other and bounce between the walls of the container undisturbed, and the system does not attain a Maxwellian velocity distribution. This is a trivial example that shows that collisions between disks are necessary for modeling ideal gas.
Binary collisions give rise to chaotic dynamics and allow a system of hard disks to explore fully its phase space.
Binary collisions are not enough to guarantee ergodicity however. A system based on binary collisions and some other dynamics can fail to be ergodic if the additional dynamics introduce an attractor in phase space. In particular this can occur when the evolution map M : ! does not conserve measure in phase space; that is the Jacobian determinant of the evolution map is not equal to one (absolute value). The simplest example of a map that does not conserve measure is a two-to-one map, which means that two distinct representative points in are mapped onto the same point. In physical space it means that two distinct trajectories of disks are mapped onto the same trajectory. An example is the one-way valve Maxwell's demon described in section 9.
It turns out that the membrane and disks system also compresses phase space volume. The following three sections analyze this compression and demonstrate that the second of thermodynamics requires the incompressibility of microscopic dynamics or an appropriate energy cost for compressible microscopic dynamics.
Compression of Phase Space Volume
To examine the compressibility of dynamics, we consider a membrane system that contains only one disk for simplicity. In other words, we have the same container and membrane as shown in gure 2 and we have a single disk inside.
The phase space is three dimensional X; Y; where X,Y is the position of the disk and is the velocity angle. We examine the compressibility of dynamics using this one disk system.
It is easy to see that during free motion and collisions with the wall, the evolution of the system conserves phase space volume. The question is what happens during the membrane-disk interaction. Referring to gure 3 we see that the transformation of the velocity angle of an incident disk is a linear map that has slope minus one, which suggests that phase space volume is conserved. This is incorrect however. In order to calculate the compressibility of dynamics we must consider the evolution of all three X; Y; dimensions of the phase space together, and not only the dimension that is shown in gure 3.
For concreteness we look at the time evolution of an in nitesimal phase space volume !(X 1 ; Y 1 ; 1 ) centered at the representative point X 1 ; Y 1 ; 1 .
We assume that the membrane is located at X = 0, and that X 1 is near the membrane with X 1 > 0 , and that 1 is inside the interval 3 =4 < 1 < .
We assume that after a time interval of 1:0 (in appropriate units) every point X; Y; in the volume !(X 1 ; Y 1 ; 1 ) has penetrated the membrane from the right and moved to the left of the membrane. We denote by X 0 ; Y 0 ; 0 the image of X; Y; under the evolution map, and we have the following equation, X 0 = X + cos t c sin (1 t c )
where t c is the time it takes for point X; Y; to move to the membrane and is equal to X= cos . Therefore, X 0 = sin Xsin =cos Y 0 = Y X(1 + sin =cos ) cos 0 = 3 =2
To check whether the evolution map compresses the volume !(X 1 ; Y 1 ; 1 )
we evaluate the Jacobian determinant of the above equations. We nd, sin =cos 0 cos X=cos 2 1 sin =cos 1 sin X=cos 2 0 0 1 = sin =cos = tan (9) For points X; Y; inside the volume !(X 1 ; Y 1 ; 1 ), the angle is inside the interval 3 =4 < < . Thus, the Jacobian determinant is always less than one. In other words the evolution map compresses phase space volume when a disk penetrates the membrane from right to left. Repeating the above steps, equations 8 and 9 using f( ), we nd f( ) = sin 1 ( sin + C) (10) where C is an arbitrary constant. If C is zero, f( ) corresponds to a transparent membrane (i.e. no membrane at all) or a completely impenetrable membrane (i.e. a wall). If C is non-zero, f( ) corresponds to a membrane that maps velocity angles in a non-linear fashion. In analogy with the mem-
Microscopic Recti ers
Another system that obeys compressible microscopic dynamics with an energy cost is a pump of disks consisting of a microscopic trapdoor and external cooling. Such a pump is often used to explain thermalization 2, 8,9]. In this section we examine the pump in terms of phase space compression.
A microscopic trapdoor without external cooling is an example of a system that is designed to extract work from the thermal motion of disks by rectifying spontaneous variations in density in a system of disks. Such a trapdoor is also called a microscopic recti er, and it does not succeed as explained in references 2, 8,9] because the rectifying mechanism becomes thermalized and starts moving randomly in every possible way. In order to succeed the rectifying mechanism must be kept at a lower temperature than the system of disks.
Interestingly the prevention of thermalization by a cooling process compresses phase space volume. To see how compression of phase space volume occurs when a microscopic recti er is cooled, we examine the trapdoor system of reference 2]. For simplicity we consider only one disk at rst. We denote by x; y; u; v the coordinates and velocity of the disk, and by X; U the position and velocity of the trapdoor. We assume that the cooling operation of reference 2] occurs at time T c for 0 < T c < 1 (in appropriate units of time), and we consider the evolution of the system between times 0 and 1.
The new state is given by the following equations,
x 0 = x + u T c + u (1 T c )
y 0 = y + v T c + v (1 T c ) u 0 = u v 0 = v X 0 = X + U T c + U (1 T c ) U 0 = U(14)
where 0 < < 1 is the cooling parameter (a xed number), and is chosen so that the total energy of the system is conserved. If m; M are the masses of the disk and the trapdoor, the parameter is given by the formula,
= v u u t 1 + MU 2 m(u 2 + v 2 ) (1 2 ) :(15)
After some algebra we nd that the Jacobian determinant of the evolution equation 14 is equal to . In other words, phase space volume is compressed by a factor of during a cooling operation. A similar calculation for a system containing a large number of disks, for example N disks, gives the following Jacobian determinant,
Jacobian = 1 + (1 2 ) MU 2 P N i=1 m(u 2 i + v 2 i ) ! (N 1) ;(16)
which reduces to (1 + (1 2 )) if we assume that the total energy of the disks is much larger than the energy of the trapdoor, and that is the ratio of the energy of the trapdoor to the average energy of the disks. Further if the trapdoor is colder than the disks, the energy ratio is a small number, and the Jacobian determinant is approximately equal to . Hence, cooling the trapdoor is accompanied by compression of phase space volume.
A cooled recti er is simply a pump and is not a threat to second law of thermodynamics 2]. The external cooling does work on the system and is responsible for compressing the phase space of the trapdoor and disks. When the cooled recti er is enclosed in a larger system that is closed and isolated (for example if hot and cold reservoirs of nite heat capacity are used to perform the cooling), then the extended phase space evolves incompressibly and heat can only be converted to work while the system is approaching equilibrium. The membrane and disks system can also be viewed as a pump if we assume that the membrane does work in order to compress the phase space of the disks. 9 Maxwell's demon An alternative way of bringing the membrane in accordance with the second law of thermodynamics is to view the membrane as a Maxwell's demon that interacts with the disks according to information about each incoming disk.
For this purpose we review rst the traditional one-way valve Maxwell's demon.
Maxwell's demon is an imaginary being (or device) that operates a microscopic door between two chambers containing disks 5, 2, 6, 7]. The simplest version of the demon opens the door when a disk is coming from the right, and closes the door when a disk is coming from the left. The demon's operations lead to a density di erence between two chambers, and the density di erence can be used to extract work from the thermal motion of the disks.
If the demon could operate in a complete cycle dissipating less energy than the work that can be extracted after the demon has nished its operations,
Figure 2 .
2The vertical solid line in gure 2 denotes the membrane, and the other solid lines denote a division of the plane into octants. The two dashed lines inside the rst and sixth octants (counting counterclockwise) denote trajectories that are de ected and penetrate the membrane according to the case +45 of equation 1. The transformation of the velocity takes place instantaneously when the center of an incident disk reaches the membrane. Another way of examining equation 1 is to rewrite the equation as a map of the velocity angle (impact angle). This is shown in gure 3 where is the impact velocity angle and 0 is the transformed velocity angle after the membrane interaction has occurred. Both and 0 range from to . The map consists of eight line segments which can be interpreted as follows: A completely transparent membrane is a straight line of slope \1" passing through the origin, and a completely impenetrable membrane (mirror-re ecting) is a line of slope \ 1" moved upwards radians away from the origin and wrapped around periodically. The eight line segments of gure 3 can be interpreted as breaking a line of slope \ 1" (mirror-re ecting) and shifting some of the broken pieces up and down. The broken pieces that are shown dashed in gure 3 correspond to impact angles that penetrate the membrane. The broken pieces that are shown solid in gure 3 correspond to impact angles that are mirror-re ected back. The membrane of equation 1 leads to a large density di erence between the two chambers. The success of the membrane depends on the thermal motion of the disks and the impact rate of disks hitting the membrane.
3
10 8 cm. These numbers give a mean free path of the order of 10 6 cm which is approximately the length of each chamber. The average speed of each disk is 3:56 10 4 cm=sec. In gure 4 we can see that the time average of the number of disks in the left chamber increases from an initial value of 0:5 (normalized) to a value of 0:7 as a result of the membrane interaction. The smooth curve of gure 4 plots the cumulative time average of the number of disks in the left chamber, which approaches a steady value as the averaging time increases. The noisy curve of gure 4 plots a sequence of running averages of the number of disks in the left chamber (each one taken over 1:25 10 10 sec), and provides an indication of the density uctuations for the chosen system parameters.
Figure 5
5shows the same quantities as gure 4, but examines them on a much longer time scale. The running averages of gure 5 are based on longer time intervals (25 10 10 sec), so the size of uctuations is accordingly reduced. The cumulative time average of the number of disks in the left chamber is a straight line that intersects the y-axis at the value of 0:68 (normalized). This corresponds to 0:69 density (normalized) if we take into account that the unequal number of disks in each chamber changes the available area. The simulation results are in good agreement with the theoretical estimate of 0:7 normalized density di erence of section 3.
A
system of N disks in two dimensions can be represented as a 4N vector (: : :; X i ; Y i ; U i ; V i ; : : :) of real numbers in R 4N phase space. This 4N vector is the representative point of the system, and it speci es exactly the positions and velocities of the particles in the system at any given time (all particles have equal mass). As the system evolves in time, the representative point moves inside a constant energy subsurface of the R 4N phase space because the total energy is conserved. The total linear momentum is not conserved because wall collisions re ect a disk'because membrane interactions permute and/or re ect a disk's momentum according to equation 1. The total linear momentum is only conserved in a time average sense, and this has been checked by computer simulations. The membrane and disks system is not ergodic because it does not spend equal times in equal regions of the accessible phase space (for example see 4, p.68]). If the representative point of the system visited regions of that have more disks in the left chamber as often as regions of that have more disks in the right chamber, then the time average density would be equal in the two chambers. Instead, the membrane and disks system approaches irreversibly a state of 0:7=0:3 density di erence independent of initial conditions.
Figure 6
6shows geometrically how the compression of phase space volume occurs. The gure is con ned to two dimensions for graphical reasons. The two rectangles shown in solid lines correspond to phase space volumes that are mapped onto each other under the evolution map | they correspond to !(X; Y; ) with the angle chosen constant for all points. The vertical line X = 0 of gure 6 corresponds to the membrane. It is easy to see that the edges of the two rectangles (the original rectangle and its image under the evolution map) are equal between the two rectangles, but the corresponding angles are not equal, and hence the areas of the two rectangles are not equal. Phase space volume is compressed when penetrating the membrane from right to left and expanded when penetrating from left to right.7 Other Membrane MapsThe signi cance of compressible dynamics can be appreciated if we attempt to nd a new membrane map that would result in a density di erence while preserving phase space volume. It turns out that this is not possible, at least for piecewise di erentiable maps. To see this, we seek a map f( ) mapping the impact velocity angle to a new velocity angle f( ) so that the condition of incompressibility (Jacobian determinant unity) is satis ed.
brane of equation 1, we apply the non-linear map f( ) of equation 10 to a selected region of velocity angles and block the remaining angles. In this way we hope that the probabilities of penetrating the membrane from the left and from the right will be unequal (seesection 2). A simple calculation however shows that the probabilities of penetrating the membrane from the left and from the right will be equal for all possible choices of the constant C in equation 10.For example let us suppose that ( ; d) is an interval of velocity angles that penetrate the membrane from the right side, and (f( d); f( )) is the image of ( ; d) under the membrane map f( ) = sin 1 (sin + C)where C is a positive constant C <= 1 sin d. Also let us assume that the remaining velocity angles are blocked and do not penetrate the membrane.Then the probability of an individual molecule to penetrate the membrane from the right (see section 3) is, probabilities of penetrating the membrane from the right and from the left are equal to each other. Therefore a membrane map (piecewise di erentiable) that obeys incompressible dynamics can not create a density di erence.The above discussion con rms that the membrane of equation 1 achieves a density di erence by compressing the phase space of the disks. The density di erence achieved by the membrane can be brought in accordance with the second law of thermodynamics by assuming that the membrane does a certain amount of work in order to compress the phase space of the disks. In other words, the second law of thermodynamics requires the incompressibility of microscopic dynamics or an appropriate energy cost for compressible microscopic dynamics.
then the second law of thermodynamics would be violated. This conundrum has inspired a large volume of literature aimed at exorcising Maxwell's de-mon 6].A popular way of exorcising Maxwell's demon (see 10]) is to assume that the demon operates its trapdoor according to information about each incoming disk, and to observe that the demon must erase the information that it has about the present disk in order to process the next disk. The erasure of information is irreversible because the demon maps two distinct disk trajectories onto the same trajectory every time it interacts with a disk by opening and closing its trapdoor. In particular after an interaction it is impossible to distinguish whether the disk came from the opposite chamber or whether the disk bounced o the demon's door.It is postulated that the irreversible erasure of information must be accompanied by a minimum amount of entropy production which is ln(2) in the case of the one-way valve Maxwell's demon 10]. Thus the demon is brought into accordance with the second law because the reduction of entropy achieved by the demon is counterbalanced by an equal amount of entropy production that is necessary to implement the demon's irreversible operations. The membrane of equation 1 can also be viewed as a Maxwell's demon by imagining that the membrane-disk interactions are the result of a demon playing tennis with the disks. The demon moves a tiny racket up or down so as to intercept each incoming disk at the membrane line. In addition, the demon orients the racket in one of three possible orientations +45, 45, and +90 degrees, so as to re ect the incoming disk according to equation 1 (also see gure 2). The tennis demon di ers from the one-way valve Maxwell's demon in that the racket of the tennis demon must be positioned along a continuum of locations depending on where the disk intersects the membrane line. In contrast, the door of the one-way valve Maxwell's demon must be positioned in one of two locations, closed or open. Another di erence between the tennis demon and the one-way valve Maxwell's demon is that the tennis demon discards no information about the disks: The disk trajectories evolve bijectively so that the state of a disk (position, velocity) after a racket collision determines completely the state of the disk before the racket collision. In contrast, the one-way valve Maxwell's demon maps two disk trajectories onto the same trajectory every time it interacts with a disk. How can we then exorcise the tennis demon using information ideas? In section 6 we saw that the evolution of the disks compresses phase space volume. Now we may notice that the compression of phase space volume occurs because the demon chooses the location of the racket as a function of the incoming disk's position. In contrast, a racket that is positioned at a xed location does not cause any compression of phase space volume. Therefore the problem must lie in choosing the racket locations. Suppose we consider a tennis demon that can only position the racket at a discrete number of xed locations along the membrane line. In particular we assume that the possible racket locations are uniformly spaced, and that the distance between successive locations is equal to x. If a disk trajectory does not fall exactly on one of the racket locations, the demon picks the nearest location possible. With the help of a diagram (see gure 7) we can see that no matter how small (but non-zero) the spacing x is, there are always disk trajectories that are mapped on top of each other. The discrete tennis demon does not compress the phase space of the disks in the continuous manner of section 6, but it does compress the phase space of the disks because of the many-to-one map of disk trajectories.The discrete tennis demon operates with nite information about each incoming disk, but it erases information irreversibly. The irreversible manyto-one map of disk trajectories disappears only in the limit of x going to zero (the spacing between racket locations). In this limit the discrete compression of phase space volume is replaced by the continuous compression of phase space volume of section 6. Moreover the reversibility of trajectories that is achieved in the limit comes at the expense of requiring the demon to operate with in nite information. Because Maxwell's demon can only operate with nite information (we can think of it as a microscopic computer), it follows that the tennis demon can not imitate the membrane of equation 1 reversibly.A tennis demon can only approximate the membrane of equation 1 using irreversible operations.
Figure 1 :
1A system of disks and a membrane can be viewed as a Maxwell's demon (see text). The membrane, displayed as a dashed line, interacts with the disks according to a time-reversible and energy conserving rule that creates a density di erence between the two chambers.
Figure 2 :
2The membrane interaction is shown graphically. The vertical solid line denotes the membrane, and the other solid lines denote a division of the plane into octants. The dashed lines correspond to disk trajectories that penetrate the membrane.
Figure 3 :
3The membrane interaction is shown as a map of the velocity angle.The impact velocity angle is mapped to the new velocity angle 0 . Both angles range from to . The dashed line segments correspond to trajectories that penetrate the membrane, while the solid line segments correspond to trajectories that are mirror-re ected back.
Figure 4 :
4The membrane creates a density di erence between two chambers initialized to have equal density. The smooth curve plots the cumulative time average of the number of disks in the left chamber, which increases from an initial value of 0:5 to approximately 0:7. The noisy curve plots a sequence of running averages of the number of disks in the left chamber, each one taken over 1:25 10 10 sec.
Figure 5 :
5The same quantities as in gure 4 are examined on a much longer time scale. The running averages are based on time intervals of 25 10 10 sec.
Figure 6 :
6The membrane compresses the phase space of the disks. A region of phase space that corresponds to disks with identical velocity angle is compressed when the points penetrate the membrane from right to left.
Figure 7 :
7A nite demon can position its racket at a discrete number of locations that are uniformly spaced along the membrane line. The demon picks the nearest location possible to position its racket, but there are always trajectories that are mapped on top of each other. The y-axis corresponds to the membrane and the solid lines are disk trajectories.
An interesting discussion of compressible dynamics has recently appeared in a paper that arrives at similar conclusions as ours, Kechen Zhang and Kezhao Zhang, \Mechanical models of Maxwell's demon with noninvariant phase volume. Phys. Rev. A. 468An interesting discussion of compressible dynamics has recently appeared in a paper that arrives at similar conclusions as ours, Kechen Zhang and Kezhao Zhang, \Mechanical models of Maxwell's demon with noninvariant phase volume", Phys. Rev. A 46 (8) 4598|4605 (October 1992).
\Maxwell's demon, recti ers, and the second law: Computer simulation of Smoluchowski's trapdoor. P A Skordos, W H Zurek, Am. J. Phys. 6010P.A. Skordos and W.H. Zurek, \Maxwell's demon, recti ers, and the second law: Computer simulation of Smoluchowski's trapdoor", Am. J. Phys. 60 (10) 876|882 (October 1992).
R P Feynman, R B Leighton, M Sands, The Feynman Lectures on physics. Reading, MassachusettsAddison-Wesley Publication3R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on physics -Vol.3, (Addison-Wesley Publication, Reading, Massachusetts 1963, 1965), chapter III-14, pp 14-9 | 14-10.
Measure and Integration for Use. H R Pitt, Clarendon PressNew YorkH.R. Pitt, Measure and Integration for Use, (Clarendon Press, New York 1985), pp. 63|69.
\Maxwell's Demon. W Ehrenberg, Sci. Am. 217103W. Ehrenberg, \Maxwell's Demon", Sci. Am. 217 103|110, (November 1967).
H S Le, A F Rex, \Resource Letter MD-1: Maxwell's demon. 58H.S. Le and A.F. Rex, \Resource Letter MD-1: Maxwell's demon", Am. J. Phys. 58 (3) 201|209, (March 1990) ;
Demon: Entropy, Information. ' Maxwell, H.S. Le and A.F. RexHilger/Princeton U.P.; PrincetonMaxwell's Demon: Entropy, Infor- mation, edited by H.S. Le and A.F. Rex, (Hilger/Princeton U.P., Princeton 1990).
J C Maxwell, Theory of Heat. Longmans, GreenJ.C. Maxwell, Theory of Heat, (Longmans, Green, London 1871), pp. 308| 309.
R P Feynman, R B Leighton, M Sands, The Feynman Lectures on physics. Reading, MassachusettsAddison-Wesley Publication146R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on physics -Vol.1, (Addison-Wesley Publication, Reading, Massachusetts 1963, 1965), chapter I-46.
\Experimentell nachweisbare der ublichen Thermodynamik widersprechende Molekularphanomene. M Smoluchowski, Phys. Z. 13M. Smoluchowski, \Experimentell nachweisbare der ublichen Thermody- namik widersprechende Molekularphanomene", Phys. Z. 13, 1069|1080 (1912).
C H Bennett, \Demons, Engines, and the Second Law. 257C.H. Bennett, \Demons, Engines, and the Second Law", Sci. Am. 257, 108| 116, (November 1987).
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| [
"Stefan Kolb \nQUANTUM SYMMETRIC KAC-MOODY PAIRS\n\n"
]
| [
"QUANTUM SYMMETRIC KAC-MOODY PAIRS\n"
]
| [
"Mathematics Subject Classification. 17B37, 17B67"
]
| The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba. | 10.1016/j.aim.2014.08.010 | [
"https://arxiv.org/pdf/1207.6036v3.pdf"
]
| 119,608,860 | 1207.6036 | fe6d44978cf12c0862df62828e844371d94caea1 |
2010
Stefan Kolb
QUANTUM SYMMETRIC KAC-MOODY PAIRS
Mathematics Subject Classification. 17B37, 17B67
2010Contentsand phrases Kac-Moody algebrasinvolutionssymmetric pairsquantum groupscoideal subalgebras
The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.
Let g be a symmetrizable Kac-Moody algebra defined over an algebraically closed field K of characteristic 0. The quantized enveloping algebra U q (g) of g, discovered by Drinfeld [Dri87] and Jimbo [Jim85] nearly thirty years ago, is an integral part of representation theory with many deep applications. Let θ : g → g be an involutive Lie algebra automorphism and let g = k ⊕ p be the decomposition of g into the (+1)-and the (−1)-eigenspace of θ. The present paper is concerned with the construction and the structure theory of quantum group analogs of U (k) as right coideal subalgebras B = B(θ) of U q (g). We call such analogs quantum symmetric pair coideal subalgebras and refer to (B, U q (g)) as a quantum symmetric pair.
If g is finite dimensional then there exist two approaches to the construction of quantum symmetric pairs. In the early nineties, Noumi, Sugitani, and Dijkhuizen constructed quantum group analogs of U (k) as coideal subalgebras of U q (g) for all g of classical type [Nou96], [NS95], [NDS97]. Their approach is based on explicit solutions of the reflection equation [Che84], [KS92] and hence follows in spirit the methods developed by the (then) Leningrad school of mathematical physics [FRT88]. Independently, G. Letzter developed a comprehensive theory of quantum symmetric pairs for all semisimple symmetric Lie algebras [Let97], [Let99], [Let00], [Let02]. [Let03], [Let04], [Let08]. Her theory is based on the Drinfeld-Jimbo presentation of quantized enveloping algebras. It is well understood that the two approaches to quantum symmetric pairs provide essentially the same coideal subalgebras of U q (g), see [Let99,Section 6].
Over the last decade, numerous examples of quantum symmetric pairs for infinite dimensional symmetrizable Kac-Moody algebras have appeared in the literature.
Here we group these examples in three classes.
(1) q-Onsager algebras: The q-Onsager algebra is a quantum symmetric pair coideal subalgebra for the Chevalley involution of the affine Lie algebraŝl 2 (K). It derives its name from the fact that the Lie subalgebra ofŝl 2 (C) fixed under the Chevalley involution appeared in Onsager's investigation of the planar Ising model [Ons44], see also [Ter06,Remark 9.1] for historical comments. The q-Onsager algebra appeared, as an algebra, in Terwilliger's investigation of tridiagonal pairs and polynomial association schemes [Ter93,Lemma 5.4]. The name q-Onsager algebra, however, goes back to Baseilhac and Koizumi [BK05] who observed its role as a symmetry algebra for a class of quantum integrable models. An embedding of the q-Onsager algebra as a coideal subalgebra of U q (ŝl 2 (K)) was established in [IT10,Proposition 1.13], see also [Bas05,Section 2]. Recently, it was proposed to study quantum symmetric pair coideal subalgebras corresponding the Chevalley involution for arbitrary affine Kac-Moody algebras [BB10] under the name generalized q-Onsager algebras. These algebras previously appeared in [DG02, Section 3] and [DM03,3.4].
(2) Twisted quantum loop algebras of the second kind: Assume that g is finite dimensional and let L(g) = g ⊗ K[t, t −1 ] denote the corresponding loop algebra. The involutive automorphism θ of g lifts to an involutive automorphism θ L of L(g) defined byθ L (x ⊗ t n ) = θ(x) ⊗ t −n for all x ∈ g, n ∈ Z. The same formula also extends θ to an involutive automorphismθ of the untwisted affine Lie algebraĝ of g. The fixed Lie subalgebras of L(g) andĝ are isomorphic and we call them twisted loop algebras of the second kind. Correspondingly, any quantum symmetric pair coideal subalgebra for the involutionθ will be called a twisted quantum loop algebra of the second kind. The twisted quantum loop algebras (of the second kind) corresponding to the Chevalley involution coincide with the generalized q-Onsager algebras discussed in (1). The most prominent examples of twisted quantum loop algebras, however, were introduced by Molev, Ragoucy, and Sorba [MRS03] for the involutions corresponding to the symmetric pairs (sl N (K), so N (K)) and (sl 2m (K), sp 2m (K)). Molev, Ragoucy, and Sorba call their algebras twisted q-Yangians. A twisted quantum loop algebra corresponding to the symmetric pair (sl N (K), sl N (K) ∩ (gl r (K) ⊕ gl N −r (K))) was constructed in [GM]. Both [MRS03] and [GM] use an FRT approach for their constructions and work with gl N (K) rather than sl N (K). Examples of quantum symmetric pair coideal subalgebras of U q (ŝl 2 (K)) which are not twisted quantum loop algebras were recently constructed in [Reg12].
(3) Quantized GIM Lie algebras: Generalized intersection matrix (GIM) Lie algebras, originally introduced by Slodowy [Slo84], [Slo86], form a class of Lie algebras which generalize Kac-Moody algebras by allowing Cartan matrices with positive off-diagonal entries. It was realized by Berman [Ber89] that GIM Lie algebras are subalgebras fixed under an involution of certain symmetrizable Kac-Moody algebras. Recently, quantum group analogs of GIM Lie algebras associated to two-fold affinizations of Cartan matrices of type ADE were constructed by Tan [Tan05] in an attempt to better understand the quantum toroidal algebras defined in [GKV95]. In [LT12] the resulting quantized GIM Lie algebras were realized as coideal subalgebras of quantized enveloping algebras. Quantized GIM Lie algebras provide examples of quantum symmetric pairs for non-affine symmetrizable Kac-Moody algebras.
In the present paper a comprehensive theory of quantum symmetric pairs for symmetrizable Kac-Moody algebras is developed. This theory comprises all of the above example classes and reduces to Letzter's theory if g is finite dimensional. The paper develops algebraic properties of the resulting quantum symmetric Kac-Moody pairs. This includes triangular decompositions, in particular an analog of the Iwasawa decomposition. It also includes explicit presentations of quantum symmetric pair coideal subalgebras in terms of generators and relations and a description of their centers. Moreover, we investigate the transformation behavior of quantum symmetric pairs under Hopf algebra automorphisms of U q (g) and the behavior of quantum symmetric pairs in the limit q → 1, commonly referred to a specialization. It has to be stressed that for finite dimensional g most of these results were obtained by Letzter and that the proofs in the Kac-Moody case are for the most part straightforward translations of her arguments. Nevertheless, we feel that a detailed presentation of the construction of quantum symmetric pairs and of the proofs of their properties is appropriate. Firstly, this guarantees that everything does indeed translate to the Kac-Moody setting. This fact was not used by any of the papers referred to in (1), (2), and (3) above (except [Reg12]), although it significantly conceptualizes and generalizes those constructions. This will become apparent in the final two sections of the present paper where the theory is applied to define quantum loop algebras and quantized GIM algebras in full generality. Delius and MacKay asked in [DM03,p. 189] for an affine generalization of Noumi's and Letzter's theories. In [BB10, Section 4] Baseilhac and Belliard asked for a Drinfeld-Jimbo realization of the twisted q-Yangians in [MRS03]. The present paper answers both these questions. Secondly, a detailed presentation allows for some minor changes to Letzter's original approach. This will hopefully help to convince the reader that quantum symmetric pairs appear very naturally and are technically not more involved than quantized enveloping algebras themselves.
An important ingredient in Letzter's construction is the classification of involutive automorphisms of simple complex Lie algebras up to conjugation. Being equivalent to the classification of real simple Lie algebras, this classical problem was essentially solved by E. Cartan [Car14]. It was revisited by Araki [Ara62] using Cartan subalgebras h of g, invariant under θ, such that h ∩ p is maximally abelian in p. Dijkhuizen observed in [Dij96] that this condition is crucial for the general construction of quantum symmetric pairs. In [Let99] Letzter specifies the choice of θ even further. Let g = n + ⊕ h ⊕ n − be the triangular decomposition of g with respect to a fixed choice Π = {α i | i ∈ I} of simple roots and let e i , f i , h i for i ∈ I denote the Chevalley generators of g. Moreover, for any subset X ⊂ I let g X denote the Lie subalgebra of g generated by all e i , f i , h i for i ∈ X. To construct quantum symmetric pairs for finite dimensional g, Letzter assumes that
θ(h) = h (1.1)
and that there exists a subset X ⊂ I such that θ| gX = id gX (1.2) and θ(e i ) ∈ n − and θ(f i ) ∈ n + if i ∈ I \ X. (1.3) For finite dimensional g the above conditions can be achieved for any involutive automorphism θ via conjugation. It turns out that Letzter's theory of quantum symmetric pairs can be extended to symmetrizable Kac-Moody algebras and involutions θ which satisfy (1.1)-(1.3) for a subset X ⊂ I of finite type. More precisely, recall that a Lie algebra automorphism σ : g → g is said to be of the second kind if the standard Borel subalgebras b + and b − of g satisfy [b + : σ(b − ) ∩ b + ] < ∞, [Lev88, III.1]. Any automorphism θ satisfying (1.1)-(1.3) for a subset X ⊂ I of finite type is of the second kind. It is one of the main observations of the present paper that Letzter's theory extends to involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras.
Involutive automorphisms of the second kind were essentially classified by Kac and Wang [KW92]. It follows from their results that any involution of the second kind is conjugate to an involution satisfying (1.1)-(1.3). This involution can be written explicitly in terms of certain pairs (X, τ ) where X ⊂ I as above, and τ is a diagram automorphism of I. The pairs (X, τ ) corresponding to involutive automorphisms of the second kind can be described solely in terms of the root system of g and will be called admissible pairs. They are natural generalizations of Satake diagrams [Ara62, p. 32/33] to the Kac-Moody setting. The explicit form of the involution θ(X, τ ) corresponding to the admissible pair (X, τ ) immediately suggests the definition of a quantum group analog θ q (X, τ ). Similarly, a set of generators of the Lie subalgebra k fixed under θ(X, τ ) suggests a set of generators and hence the definition of the corresponding quantum symmetric pair coideal subalgebra.
In the following the content of the present paper is described in some more detail. The classification of involutions of the second kind of symmetrizable Kac-Moody algebras in terms of admissible pairs is recalled in Section 2. This section contains all facts about Kac-Moody algebras used subsequently for the construction of quantum symmetric pairs. Only the proof of the main result, Theorem 2.7, is moved to Appendix A. Section 3 briefly recalls the definition and properties of quantized enveloping algebras defined over the field K(q) of rational functions in q. Here the focus is on the relation between Lusztig's braid group action on U q (g) and the adjoint action of U q (g) on itself. The aim is to find q-analogs of all the constituents of the involution θ(X, τ ) of g. With the preparations provided by Sections 2 and 3 at hand, it is straightforward to define the quantum group analog θ q (X, τ ) of θ(X, τ ) as an K(q)-algebra automorphism of U q (g) in Section 4. This definition is one of the more essential differences between the present paper and Letzter's constructions for finite dimensional g where the quantum group analog of θ maps q to q −1 , see Subsection 4.5. In Section 5 quantum symmetric pair coideal subalgebras B c,s are defined depending on two families of additional parameters c, s ∈ K(q) I\X . The next aim is to describe B c,s explicitly in terms of generators and relations. To this end, three crucial triangular decompositions are presented in Section 6. The first two decompositions provide bases of U q (g) and B c,s in terms of certain monomials in generators of B c,s . The third decomposition is a quantum analog of the Iwasawa decomposition. These tools make it possible to give the desired presentation of B c,s in Section 7. The relations are given explicitly for all generalized Cartan matrices with entries ≥ −2, but the method works in full generality. Here the q-Onsager algebra appears in Example 7.6. In Section 8 the center of B c,s is determined. It is shown that if B c,s corresponds to an indecomposable Cartan matrix of infinite type then the center Z(B c,s ) is trivial. The center of B c,s in the finite case was previously determined in [KL08] and is here briefly recalled for completeness. Section 9 analyzes the behavior of quantum symmetric pairs under the action of the group Aut Hopf (U q (g)) of Hopf algebra automorphisms of U q (g). The aim is in particular to find a class of representatives of the Aut Hopf (U q (g))-orbits of quantum symmetric pairs. In Section 10 we discuss non-restricted specialization, that is, the behavior of θ q (X, τ ) and B c,s in the limit q → 1. Using specialization, it is in particular shown in Theorem 10.11 that B c,s satisfies a maximality condition. In these specialization arguments it is crucial that the triangular decompositions involving B c,s remain true over the localization A = K[q, q −1 ] (q−1) , see Theorem 10.7. The maximality property of B c,s is used to identify the twisted q-Yangians from [MRS03] with quantum symmetric pair coideal subalgebras in Section 11. It should be noted that twisted q-Yangians contain an additional infinite dimensional central subalgebra due to the fact that they are constructed from gl n (K) and not from sl n (K). The final Section 12 provides the construction of quantized GIM Lie algebras in full generality, going beyond the two-fold affinizations considered by Lv and Tan in [Tan05], [LT12].
At the end of each section that emulates Letzter's constructions for finite dimensional g, we provide detailed references to her work and point out any differences between her constructions and those presented here in the Kac-Moody case. This applies to Sections 4, 5, 6, 7, 9, 10, and 11. April 2012. The author is grateful to the organizers, Loek Helminck and Ralf Koehl, for this occasion which significantly furthered the completion of the paper.
Involutive automorphisms of the second kind
Automorphisms of infinite-dimensional, symmetrizable Kac-Moody algebras belong to either of two disjoint classes, automorphisms of the first and automorphisms of the second kind. In principle, involutive automorphisms of the second kind were classified in [KW92]. In the very final remark of their paper, Kac and Wang announce that a combinatorial description of involutions of the second kind in terms of Dynkin diagrams is possible. This section is the author's take on what Kac and Wang might have had in mind. This will, moreover, allow us to fix notations for Kac-Moody algebras and their automorphisms. The main result of this section, Theorem 2.7, should be attributed to [KW92], however, the author was unable to locate the given explicit formulation in the literature. In Appendix A we will reduce Theorem 2.7 to results stated in [KW92]. A similar result is contained in [BBBR95] where almost split real forms of symmetrizable Kac-Moody algebras are classified.
All through this paper K denotes an algebraically closed field of characteristic 0.
2.1. Kac-Moody algebras and groups. Let I be a finite set and let A = (a ij ) i,j∈I be a generalized Cartan matrix. Recall that A is a square matrix with integer entries such that a ii = 2 for all i ∈ I, a ij ≤ 0 if i = j, and (a ij = 0 ⇔ a ji = 0). We always assume A to be symmetrizable, that is, there exists a diagonal matrix D = diag(ǫ i |i ∈ I) with coprime entries ǫ i ∈ N such that DA is symmetric. Moreover, we assume A to be indecomposable. Let P ∨ be the free abelian group of rank 2|I| − rank(A) with Z-basis {h i | i ∈ I} ∪ {d s | s = 1, . . . , |I| − rank(A)}, set h = K ⊗ Z P ∨ , and define P = {λ ∈ h * | λ(P ∨ ) ⊆ Z}. As usual, we call P the weight lattice and P ∨ the dual weight lattice associated to A. Moreover, define Π ∨ = {h i | i ∈ I} and choose a linearly independent subset Π = {α i | i ∈ I} of h * such that
α j (h i ) = a ij and α j (d s ) ∈ {0, 1} (2.1)
for all i, j ∈ I and all s = 1, . . . , |I| − rank(A). In this case (h, Π, Π ∨ ) is a minimal realization of A. Let Q = ZΠ and Q ∨ = ZΠ ∨ denote the root lattice and the dual root lattice, respectively, and set Q + = i∈I N 0 α i . For any µ, ν ∈ h * we write
µ ≥ ν if µ − ν ∈ Q + .
The Kac-Moody algebra g = g(A) is the Lie algebra over K generated by h and elements e i , f i for i ∈ I and the relations given in [Kac90,1.3]. The derived algebra g ′ = [g, g] is the Lie subalgebra of g generated by the elements e i , f i for i ∈ I. Let n + and n − denote the Lie subalgebras of g generated by the elements of the sets {e i | i ∈ I} and {f i | i ∈ I}, respectively. One has triangular decompositions
g = n + ⊕ h ⊕ n − g ′ = n + ⊕ h ′ ⊕ n −
where h ′ = i∈I Kh i . As a h-module g decomposes into root spaces
g = β∈h * g β where g β = {x ∈ g | [h, x] = β(h)x, for all h ∈ h}. Let Φ = {β ∈ h * | g β = 0}
denote the set of roots and set Φ + = Φ ∩ Q + .
For any i ∈ I the fundamental reflection r i ∈ GL(h) is defined by
r i (h) = h − α i (h)h i for all h ∈ h.
The Weyl group W is the subgroup of GL(h) generated by the fundamental reflections r i . It is a Coxeter group given by defining relations
r 2 i = 1, (r i r j ) mij = 1 (2.2)
for all i, j ∈ I where m ij = 2, 3, 4, 6, and ∞ if a ij a ji = 0, 1, 2, 3, and ≥ 4, respectively, [Kac90,Proposition 3.13]. Via duality W also acts on h * by
r i (α) = α − α(h i )α i for all α ∈ h(h i , h) = α i (h)/ǫ i ∀h ∈ h, i ∈ I, (d m , d n ) = 0 ∀n, m ∈ {1, . . . , s}.
We denote the induced bilinear form on h * by the same symbol and observe that
(α i , α j ) = ǫ i a ij for all i, j ∈ I.
The action of W on h can be interpreted in terms of the Kac-Moody group G associated to g ′ . We briefly recall the construction of G along the lines of [KW92, 1.3]. Let G * denote the free product of the additive groups g α for α ∈ Φ re and let i α : g α → G * be the canonical inclusion. Recall that a g ′ -module (V, π) is called integrable if π(e i ) and π(f i ) act locally nilpotently on V for all i ∈ I. For any integrable g ′ -module (V, π) define a homomorphism π * : G * → Aut(V ) by π * (i α (x)) = exp(π(x)) for x ∈ g α , α ∈ Φ re . Let N * denote the intersection of all ker(π * ). By definition G := G * /N * is the Kac-Moody group associated to g ′ . We write π * : G * → G to denote the canonical projection. To shorten notation we write exp(x) := π * (i α (x)) for x ∈ g α , α ∈ Φ re . Let Aut(g) denote the group of Lie algebra automorphisms of g. There is a group homomorphism Ad : G → Aut(g) corresponding to the adjoint action of g ′ on g. More explicitly, we have Ad(exp(x))) = exp(ad(x)) for all x ∈ g α , α ∈ Φ re . Consult [KW92, 1.3] for more details.
We are now ready to lift the action of r i on h to an element in Aut(g). For any i ∈ I define an element m i ∈ G by
m i = exp(e i ) exp(−f i ) exp(e i ).
By [Kac90, Lemma 3.8] one has Ad(m i )(g α ) = g ri(α) for all α ∈ h * and Ad(m i )| h = r i for all i ∈ I. Moreover, by [Kac90, Remark 3.8] the elements m i satisfy the relations
m i m j m i . . . mij factors = m j m i m j . . . mij factors (2.3)
with m ij as in (2.2).
2.2.
Automorphisms of the first and second kind. Let b + = n + ⊕ h and b − = h ⊕ n − denote the standard Borel subalgebras of g. One says that σ ∈ Aut(g) is of the first kind if σ(b + ) = Ad(g)(b + ) for some g ∈ G. By [KW92,4.6] this is equivalent to
dim(σ(b + ) ∩ b − ) < ∞.
Similarly, σ is of the second kind if σ(b + ) = Ad(g)(b − ) for some g in G, or equivalently,
dim(σ(b + ) ∩ b + ) < ∞.
If g not of finite type then any automorphism of g is either of the first or of the second kind but never both. In this case the group Aut(g) is Z 2 -graded with automorphisms of the first kind in degree 0 and automorphisms of the second kind in degree 1.
Apart from Ad(G), four other subgroups of Aut(g) are relevant in the following. The setH = Hom(Q, K × ) of group homomorphism from Q to the multiplicative group K × is a group under multiplication. For any x ∈H define Ad(x) ∈ Aut(g) by Ad(x)| h = id h and Ad(x)(v) = x(α)v for all α ∈ Φ, v ∈ g α . Then Ad(H) is a subgroup of Aut(g). The automorphisms in Ad(H) are of the first kind.
Let Aut(A) denote the group of all permutations σ of the set I such that a i,j = a σ(i),σ(j) . View Aut(A) as a subgroup of Aut(g ′ ) by requiring σ(e i ) = e σ(i) , σ(f i ) = f σ(i) . The action of Aut(A) on g ′ can be extended to an action on g following [KW92,4.19]. Then the induced map on h * satisfies σ(α i ) = α σ(i) . Viewed as a subgroup of Aut(g), the group Aut(A) consists of automorphisms of the first kind.
Recall the Chevalley involution ω defined by As ω commutes with all elements in Aut(A), one obtains that Out(A) is a subgroup of Aut(g).
ω(e i ) = −f i ω(f i ) = −e i ω(h) = −h
Let Aut(g, g ′ ) denote the group of automorphisms of g which restrict to the identity on g ′ . The following decomposition of Aut(g) is established in [KW92,4.23].
Proposition 2.1. Aut(g) = Out(A) ⋉ Aut(g, g ′ ) × (Ad(H) ⋉ Ad(G)) .
The above proposition implies that any σ ∈ Aut(g) of the second kind can be written in the form
σ = Ad(s) • f • τ • ω • Ad(g) (2.5)
for some s ∈H, f ∈ Aut(g, g ′ ), τ ∈ Aut(A), and g ∈ G.
2.3. Longest elements in parabolic subgroups of W . Special elements in G play a major role in the classification of involutive automorphism of the second kind in Theorem 2.7. Let X ⊂ I be a subset of finite type, let W X ⊂ W be the corresponding parabolic subgroup with longest element w X , and let Φ X be the corresponding root system considered as a subset of Φ. Fix a reduced decomposition w X = r i1 . . . r i k and define m X = m i1 . . . m i k . By the word property for Coxeter groups [BB06] and Relation (2.3), the element m X ∈ G is independent of the chosen reduced decomposition of w X .
Let g X ⊆ g denote the semisimple Lie algebra generated by {e i , f i | i ∈ X} with Chevalley automorphism ω X mapping e i to f i . Let moreover τ X denote the diagram automorphism of g X corresponding to the longest element w X . The following proposition can be found in [BBBR95,4.9,4.10]. We use the notation Aut(A, X) = {σ ∈ Aut(A) | σ(X) = X}.
Proposition 2.2.
(1) The automorphism Ad(m X ) of g leaves g X invariant and satisfies the relation
Ad(m X )| gX = τ X ω X .
(2) In Aut(g) the relation
Ad(m 2 X ) = Ad(exp(iπ2ρ ∨ X )) holds, where 2ρ ∨ X denotes the sum of the positive coroots of Φ X . (3)
The automorphism Ad(m X ) of g commutes with all elements in Aut(A, X) and with the Chevalley involution ω.
Proof. Parts (1) and (2)
σ(Ad(m i )x) = Ad(m σ(i) )σ(x) (2.6)
holds for any σ ∈ Aut(A) and any x ∈ g. If σ ∈ Aut(A, X) and w X = r i1 . . . r i k is a reduced decomposition then w X = r σ(i1) . . . r σ(i k ) . Hence
m X = m σ(i1) . . . m σ(i k )
because m X is independent of the choice of reduced decomposition for w X . Now (2.6) implies σ(Ad(m X )(x)) = Ad(m X )(σ(x)) and thus Ad(m X ) and σ commute.
An sl 2 -argument shows the relation
m i = exp(−f i ) exp(e i ) exp(−f i ) in G.
Hence
ω(Ad(m i )x) = Ad(exp(−f i ) exp(e i ) exp(−f i ))(ω(x)) = Ad(m i )(ω(x))
holds for any x ∈ g. Thus Ad(m X ) commutes with ω.
2.4. Admissible pairs. We introduce the notion of an admissible pair which generalizes Satake diagrams as given in [Ara62] in the finite case to symmetrizable Kac-Moody algebras. It is slightly more explicit than a similar notion used in [BBBR95,Def. 4.10 b), Cor. 4.10.4] in so far as we also include property (3), below, in the definition.
Definition 2.3. A pair (X, τ ) consisting of a subset X ⊆ I and an element τ ∈ Aut(A, X) is called admissible if the following conditions are satisfied:
(1) τ 2 = id I .
(2) The action of τ on X coincides with the action of −w X .
(3) If j ∈ I \ X and τ (j) = j then α j (ρ ∨ X ) ∈ Z. It is an instructive exercise to verify that the Satake diagrams in [Ara62, p. 32/33] describe all admissible pairs for finite dimensional simple g.
Example 2.4. Consider g = sl 4 (K) with I = {1, 2, 3} and the standard choice of simple roots and coroots. Then the pair ({1, 3}, id I ) is admissible because −w X acts as the identity on X and ρ ∨ We now associate an involutive automorphism of the second kind to any admissible pair. Let > be a fixed total order on the set I. Let (X, τ ) be an admissible pair. We define an element s(X, τ ) ∈H by
X = (h 2 + h 3 )/2 satisfies α 1 (ρ ∨ X ) = α 3 (ρ ∨ X ) = 1, α 2 (ρ ∨ X ) = −1. The pair ({1}, id I ), however, is not admissible because in this case ρ ∨ X = h 1 /2 satisfies α 1 (ρ ∨ X ) = 1, α 2 (ρ ∨ X ) = −1/2, α 3 (ρ ∨ X ) =s(X, τ )(α j ) = 1 if j ∈ X or τ (j) = j, i αj(2ρ ∨ X ) if j / ∈ X and τ (j) > j, (−i) αj(2ρ ∨ X ) if j / ∈ X and τ (j) < j, (2.7)
where i ∈ K denotes a square-root of −1.
Theorem 2.5. Let (X, τ ) be an admissible pair. Then
θ(X, τ ) = Ad(s(X, τ )) • τ • ω • Ad(m X ) (2.8)
defines an involutive automorphism of g of the second kind which commutes with Ad(s(X, τ )).
Proof. Note first that Ad(s(X, τ )) as defined by (2.7) commutes with Ad(m X ) and with τ • ω and hence it commutes also with θ(X, τ ) as defined by (2.8). Note also that θ(X, τ ) is an automorphism of second kind because Ad(s(X, τ )), τ , and Ad(m X ) are automorphisms of the first kind, and ω is an automorphism of the second kind. By Proposition 2.2.(3) the map Ad(m X ) commutes with τ and ω and hence one obtains
θ(X, τ ) 2 = Ad(s(X, τ )) 2 • Ad(m X ) 2 .
Both Ad(s(X, τ )) 2 and Ad(m X ) 2 act trivially on h and hence it remains to show that
Ad(s(X, τ )) 2 • Ad(m X ) 2 (x) = x (2.9)
for all x ∈ g αj , j ∈ I. By Proposition 2.2.(2) on has Ad(m X ) 2 (x) = (−1) αj (2ρ ∨ X ) x for all x ∈ g αj . Relation (2.9) now follows immediately from Definition (2.7) and condition (3) in Definition 2.3.
Remark 2.6. The definition of s(X, τ ) in (2.7) depends on the chosen total order on the set I and hence so does θ q (X, τ ). More explicitly, for a different total order on I the map s(X, τ ) may change by a factor −1 on both α j and α τ (j) if j = τ (j). The corresponding map θ(X, τ ) for the new total order is conjugate to the original θ(X, τ ) by an element in Ad(H).
The following theorem provides the main result of this section, namely the classification of involutive automorphisms of the second kind of g in terms of admissible pairs. Note that the group Aut(A) acts on the set of all admissible pairs by
σ((X, τ )) = (σ(X), σ • τ • σ −1 ) for σ ∈ Aut(A).
Theorem 2.7. The map (X, τ ) → θ(X, τ ) gives a bijection between the set of Aut(A)-orbits of admissible pairs for g and the set of Aut(g)-conjugacy classes of involutive automorphisms of the second kind.
We will prove Theorem 2.7 in Appendix A by reducing it to results in [KW92]. By construction one has θ(X, τ )(h) = −w X τ (h) for all h ∈ h. Hence, by duality, θ = θ(X, τ ) induces the map
Θ : h * → h * , Θ(α) = −w X τ (α). (2.10)
As the bilinear form (·, ·) on h * is W -invariant one obtains (α, β) = (Θ(α), Θ(β)) (2.11) for all α, β ∈ Q.
2.5. The invariant Lie subalgebras k and k ′ . Let (X, τ ) be an admissible pair and let θ = θ(X, τ ) be the corresponding involution of the second kind defined by (2.8). Let k = {x ∈ g | θ(x) = x} and k ′ = {x ∈ g ′ | θ(x) = x} denote the invariant Lie subalgebras of g and g ′ , respectively. To define quantum group analogs of k and k ′ we determine a set of generators.
Lemma 2.8. The Lie algebra k is generated by the elements
e i , f i for i ∈ X, (2.12) h ∈ h with θ(h) = h, (2.13) f i + θ(f i ) for i ∈ I \ X. (2.14)
Similarly, the Lie algebra k ′ is generated by the elements (2.12), (2.14), and all h ∈ h ′ with θ(h) = h.
Proof. Letk denote the Lie subalgebra of g generated by the elements (2.12), (2.13), and (2.14). Observe thatk ⊆ k because all generators ofk are invariant under θ. Conversely, assume that x ∈ k. Write x = x + + x 0 + x − with x + ∈ n + , x 0 ∈ h, and x − ∈ n − . As θ(f i ) ∈ n + for i ∈ I \ X there exists u ∈k such that u − x − ∈ n + + h. Hence we may assume that x − = 0. Similarly, x 0 ∈k and hence we may assume that x = x + ∈ n + . Write x as a sum of weight vectors x = β∈Q + x β . As θ(g β ) = g −wX τ (β) the relation x β = 0 implies β ∈ i∈X N 0 α i . Hence x ∈k.
The corresponding generators of the universal enveloping algebra can be modified by constant terms. The nonstandard quantum symmetric pair coideal subalgebras which will be introduced in Definition 5.6 contain quantum analogs of such modified generators.
Corollary 2.9. Let s = (s i ) i∈I\X ∈ K I\X . The universal enveloping algebra U (k) is generated by the elements (2.12), (2.13), and
f i + θ(f i ) + s i for i ∈ I \ X. (2.15)
as a unital algebra.
Remark 2.10. A complete set of defining relations of U (k) or U (k ′ ) in terms of the generators (2.12)-(2.14) can be written down. For the GIM Lie algebras described in the introduction and in Section 12 this was achieved by Berman in [Ber89]. The author is not aware of any publication describing defining relations of U (k) or U (k ′ ) for general admissible pairs. A rather indirect way to obtain such relations is provided by specialization of the defining relations of their quantum analogs using the results of Sections 7 and 10 of the present paper.
2.6. Compatible minimal realizations. For τ ∈ Aut(A) the corresponding Lie algebra automorphism τ : g → g defined in [KW92,4.19] does not necessarily leave P ∨ invariant. If, however, there exists a permutation σ of the set {1, . . . , s} such that
α τ (i) (d σ(j) ) = α i (d j )
for all i ∈ I, j ∈ {1, . . . , s} (2.16) then one has τ (P ∨ ) = P ∨ with τ (d j ) = d σ(j) . In this case we say that the minimal realization is compatible with τ . In the next subsection we will need to choose a compatible minimal realization in order to lift τ to an automorphism of the quantized enveloping algebra of g.
Example 2.11. Let A = 2 −2 −2 2 be the generalized Cartan matrix of affine sl 2 with I = {0, 1} and P ∨ = Zα 0 ⊕ Zα 1 ⊕ Zd. Then the minimal realization defined by α 0 (d) = 1, α 1 (d) = 0 is not compatible with the transposition (01) ∈ Aut(A). However, the minimal realization defined by α 0 (d) = 1 = α 1 (d) is compatible with (01).
Proposition 2.12. Let A be of affine type and τ ∈ Aut(A). Then there exists a minimal realization for A which is compatible with τ .
Proof. Assume that A = (a ij ) i,j∈I is given by one of the Dynkin diagrams in [Kac90, p. 54/55] and I = {0, 1, . . . , n}, where α 0 denotes the additional affine root. There exist positive integers b j , uniquely determined by b 0 = 1, such that n j=0 b j a ij = 0. If τ (0) = 0 then define α i ∈ h * by (2.1) and α i (d 1 ) = δ i,0 . If τ (0) = 0 then define α i by (2.1) and α i (d 1 ) = δ i,0 + δ i,τ (0) . In both cases it follows from the positivity of the coefficients b j that {α i | i ∈ I} is a linearly independent subset of h * . Relation (2.16) holds by construction with σ = id {1} .
Remark 2.13. The above proof does not work in the general symmetrizable Kac-Moody case. It would be good to know if any automorphism of a symmetrizable generalized Cartan matrix has a compatible minimal realization.
Lusztig automorphisms
This section provides notation and background on quantum groups. It is well known that quantized enveloping algebras have only very few Hopf algebra automorphisms, see Theorem 3.2. A quantum analog of the involutive automorphism θ(X, τ ) corresponding to an admissible pair can therefore be defined only as an algebra automorphism of the quantized enveloping algebra of g. It is straightforward to define quantum group analogs of the Chevalley involution and of elements of Aut(A). To define a quantum group analog of the automorphism Ad(m X ) we briefly recall the definition and properties of the Lusztig automorphisms. In particular, in Sections 3.3, 3.4 we use results by Kébé [Kéb99] to describe the action of certain Lusztig automorphisms on Chevalley generators of quantized enveloping algebras.
3.1. Quantum groups. Let K(q) be the field of rational functions in an indeterminate q. Recall that D = diag(ǫ i |i ∈ I) and define q i = q ǫi for any i ∈ I. Up to minor notational changes we follow the presentation in [Lus94] and define for g = g(A) the quantized enveloping algebra U q (g) to be the associative K(q)-algebra with generators E i , F i , and K µ for all i ∈ I, µ ∈ P ∨ and relations For later use we make the quantum Serre relations more explicit. For any i, j ∈ I let F ij (x, y) denote the noncommutative polynomial in two variables defined by
(1) K 0 = 1 and K h K h ′ = K h+h ′ for all h, h ′ ∈ P ∨ . (2) K h E i = q αi(h) E i K h for all i ∈ I, h ∈ P ∨ . (3) K h F i = q −αi(h) F i K h for all i ∈ I, h ∈ P ∨ . (4) E i F i − F i E i = δ ij K i − K −1 i q i − q −1 i for all i ∈ I where K i = K ǫi hi .F ij (x, y) = 1−aij n=0 (−1) n 1 − a ij n qi x 1−aij −n yx n .F ij (E i , E j ) = F ij (F i , F j ) = 0 for all i, j ∈ I. (3.2)
The algebra U q (g) is a Hopf algebra with coproduct ∆, counit ε, and antipode S given by
∆(E i ) = E i ⊗ 1 + K i ⊗ E i , ε(E i ) = 0, S(E i ) = −K −1 i E i , (3.3) ∆(F i ) = F i ⊗ K −1 i + 1 ⊗ F i , ε(F i ) = 0, S(F i ) = −F i K i , (3.4) ∆(K h ) = K h ⊗ K h , ε(K h ) = 1, S(K h ) = K −h (3.5)
for all i ∈ I, h ∈ P ∨ . We denote by U q (g ′ ) the Hopf subalgebra of U q (g) generated by the elements E i , F i , and K ±1 i for all i ∈ I.
Remarks 3.1. 1) The Hopf algebra U q (g ′ ) coincides with the Hopf algebra U q (g C ) for C = DA defined in [Jos95, 3.2.9]. 2) In later sections we will sometimes need to work with U q (g) defined over a field extension of K(q). In particular, in Subsections 4.4 and 9 we will work over K(q 1/2 ) and K(q), respectively. In this case U q (g) is defined by the same relations (1)-(4) and (3.2).
As usual let U + , U − , and U 0 denote the subalgebra of U q (g) generated by the elements {E i | i ∈ I}, {F i | i ∈ I}, and {K h | h ∈ P ∨ } respectively. Moreover define U 0′ to be the subalgebra of U 0 generated by the elements {K ±1 i | i ∈ I}. By [Lus94, 3.2] the multiplication maps give isomorphisms of vector spaces
U + ⊗ U 0 ⊗ U − ∼ = U q (g), U + ⊗ U 0′ ⊗ U − ∼ = U q (g ′ ). (3.6) Let K(q)[Q]
be the group algebra of the root lattice. There is an algebra isomorphism K(q)[Q] → U 0′ such that α i → K i . For any β ∈ Q we hence write
K β = i∈I K ni i if β = i∈I n i α i . (3.7)
The commutation relations (2), (3) then take the form
K β E i = q (β,αi) E i K β , K β F i = q −(β,αi) F i K β for all β ∈ Q, i ∈ I.
For any U q (g ′ )-module M and any µ ∈ P let
M µ = {m ∈ M | K i m = q µ(hi) i m for all i ∈ I}
denote the corresponding weight space. In particular, with respect to the left adjoint action of U q (g ′ ) on itself or on U q (g) one obtains a Q-grading of both U q (g ′ ) and U q (g). More precisely, we use Sweedler notation to define the left adjoint action of U q (g) on itself by
ad(x)(u) = x (1) uS(x (2) )
for all x, u ∈ U q (g).
Then one has ad(K i )(u) = K i uK −1 i and hence
U q (g) β = {u ∈ U q (g) | K i uK −1 i = q (αi,β) u}
for all β ∈ Q, and U q (g ′ ) β is defined analogously.
3.2. Automorphisms of U q (g) and U q (g ′ ). Define ω to be the algebra automorphism of U q (g) determined by
ω(E i ) = −F i , ω(F i ) = −E i , ω(K h ) = K −h . (3.8)
Observe that our definition of ω differs by a sign from the definition given in [Lus94, 3.1.3]. We choose to add the sign in order to make ω a quantum analog of the Chevalley involution given by (2.4). The map ω is a coalgebra antiautomorphism of U q (g).
Using the notation introduced in Subsection 2.2 also in the quantum case, we set H = Hom(Q, K(q) × ) where K(q) × denotes the multiplicative group of K(q). For any x inH define a Hopf algebra automorphism Ad(x) of U q (g) by
Ad(x)(u) = x(β)u
for all u ∈ U q (g) β and β ∈ Q.
Note that Ad(x) leaves U q (g ′ ) invariant. Moreover, one has
Ad(x) • ω = ω • Ad(x −1 ) for all x ∈H. (3.9)
By [KW92,4.19] the group Aut(A) acts on g(A) by Lie algebra automorphisms. It seems not straightforward to find an analogue action of Aut(A) on U q (g) because it is unclear how to define the action on the generators K ds . However, for any τ ∈ Aut(A) there exists an algebra automorphism of U q (g ′ ), denoted by the same symbol τ , such that
τ (E i ) = E τ (i) , τ (F i ) = F τ (i) , τ (K ±1 i ) = K ±1 τ (i) . (3.10)
Moreover, if the minimal realization of A is compatible with τ in the sense of Subsection 2.6 with corresponding permutation σ of {1, . . . , s}, then one can extend τ to an algebra automorphism of U q (g) given by (3.10) and τ (d j ) = d σ(j) . The action of Aut(A) on Q allows us to form the semidirect productH ⋊ Aut(A) which acts faithfully on U q (g ′ ) by Hopf algebra automorphisms. Let Aut Hopf (U q (g ′ )) denote the group of Hopf algebra automorphisms of U q (g ′ ). The following theorem is given in [Twi92, Theorem 2.1].
Theorem 3.2. The map
H ⋊ Aut(A) → Aut Hopf (U q (g ′ )), (x, τ ) → Ad(x) • τ
is an isomorphism of groups.
3.3. Parabolic decompositions. We recall here some results of [Kéb99] which will be used to make the quantum group analogue of the involution θ(X, τ ) explicit in Theorem 4.4. Let X ⊂ I be a subset of finite type. Let M X denote the subalgebra of U q (g) generated by
{E i , F i , K ±1 i | i ∈ X}. Write M − X , M + X , and M 0 X for the subalgebras generated by {F i , | i ∈ X}, {E i | i ∈ X}, and {K ±1 i | i ∈ X}, respectively. For any i ∈ I \ X the subspace ad(M X )(E i ) of U is finite-dimensional and contained in U + .
This follows from the triangular decomposition of M X , the fact that ad(F j )(E i ) = 0 for j ∈ X, the quantum Serre relations, and the fact that
ad(E j )u ∈ U + for all u ∈ U + . Moreover, the ad(M X )-module ad(M X )(E i ) is irreducible. Now define V +
X to be the subalgebra generated by the elements of all the finite dimensional subspaces ad(M X )(E i ) for i / ∈ X. It is proved in [Kéb99] that multiplication gives an isomorphisms of vector spaces
U + ∼ = V + X ⊗ M + X . (3.11)
To write explicitly the action of Lusztig's isomorphisms corresponding to the longest element in W X we provide the following lemma. Recall from (3.3)-(3.5) that S denotes the antipode of U q (g).
Lemma 3.3. Let i ∈ I \ X. (1) If v ∈ U + is a highest weight vector for the adjoint action of M X U 0 of weight w X α i then v ∈ ad(M X )(E i ). (2) If v ∈ S(U − ) is a lowest weight vector for the adjoint action of M X U 0 of weight −w X α i then v ∈ ad(M X )(F i K i ).
Proof. To prove (1) note that the assumption on the weight of v and (3.11) imply that
v ∈ ad(M + X )(E i ) ⊗ M + X . With respect to this decomposition write v = l u l ⊗ m l with linearly independent weight vectors u l ∈ ad(M + X )(E i ). The ad(M + X )-invariance of v implies l ad(E i )(u l ) ⊗ m l = − l ad(K i )(u l ) ⊗ ad(E i )(m l ) (3.12)
for all i ∈ X and hence the space M v spanned by the elements m l is ad
(M + X M 0 X )- invariant. It follows that M v is contained in the locally finite part F l (M X ) = {m ∈ M X | dim(ad(M X )(m)) < ∞}, see for example [FM98, Proof of Lemma 3.1.1]. Choose m l of maximal weight (w.r.t. ad(M X )). Then (3.12) implies that m l ∈ F l (M X ) ∩ M + X is a highest weight vector (w.r.t. ad(M X )) and hence m l ∈ K(q)1. Hence v ∈ ad(M + X )(E i ) ⊗ K(q)1 which proves (1). Property (2) is proved analogously using the decomposition S(U − ) ∼ = V − X ⊗ S(M − X )
where V − X denotes the subalgebra of S(U − ) generated by the elements of all the finite dimensional subspaces ad(M X )(F j K j ) for j / ∈ X.
In the following subsection we will give the highest and the lowest weight vector from the previous lemma explicitly in terms of Lusztig's automorphisms.
3.4. Definition and properties of Lusztig automorphisms. For any i ∈ I let T i be the algebra automorphism of U q (g) denoted by T ′′ i,1 in [Lus94, 37.1]. In particular, T i satisfies the relations
T i (K h ) = K ri(h) , T i (E i ) = −F i K i , T i (F i ) = −K −1 i E i (3.13)
for any h ∈ P ∨ . Moreover, by [Lus94,37.24] the inverse of T i satisfies
T −1 i = σ • T i • σ (3.14) where σ : U q (g) → U q (g) is the unique K(q)-algebra antiautomorphism of U q (g) determined by σ(E i ) = E i , σ(F i ) = F i , and σ(K h ) = K −h for all i ∈ I, h ∈ P ∨ .
In particular one has [Lus94,39.4.3] the automorphisms T i satisfy braid relations. Hence, for any w ∈ W one may define an algebra automorphism T w : U q (g) → U q (g) by
T −1 i (K h ) = K ri(h) , T −1 i (E i ) = −K −1 i F i , T −1 i (F i ) = −E i K i . ByT w = T i1 . . . T i k where w = r i1 . . . r i k is a reduced expression for w. One hence has T w (K h ) = K w(h)
for all h ∈ P ∨ and, with the notation (3.7),
T w (K β ) = K w(β) for all β ∈ Q.
Let X be a subset of I of finite type. We would like to replace the automorphism Ad(m X ) in (2.8) by the Lusztig automorphism T wX where w X denotes the longest element in the parabolic subgroup W X . We first determine the action of T wX on the generators corresponding to the subset X.
Lemma 3.4. Let X be a subset of I of finite type. Define a permutation τ of X by w X (α i ) = −α τ (i) for all i ∈ X. For all i ∈ X one has
T wX (E i ) = −F τ (i) K τ (i) , T wX (F i ) = −K −1 τ (i) E τ (i) , T wX (K i ) = K −1 τ (i) , T −1 wX (E i ) = −K −1 τ (i) F τ (i) , T −1 wX (F i ) = −E τ (i) K τ (i) , T −1 wX (K i ) = K −1 τ (i) , Proof. Write w X = w ′ r i for some w ′ ∈ W X . Then w ′ (α i ) = α τ (i) because w X (α i ) = −α τ (i) . By [Jan96, Proposition 8.20] one obtains T w ′ (E i ) = E τ (i) and hence T wX (F i ) = T w ′ (−K −1 i E i ) = −K −1 τ (i) E τ (i)
. Similarly one obtains the other two expressions. The formulas for T −1 wX follow by conjugation with the algebra antiautomorphism σ, see (3.14), using w X = w −1 X . Now we determine the action of T wX and T −1 wX on the remaining generators. This will allow us to identify the ad(M X )-highest weight vector in ad(M X )(E i ) and the ad(M X )-lowest weight vector in ad(M X )(F i K i ) for i ∈ I \X. The following lemma is a generalization of [CK90, Remark 1.6].
Lemma 3.5. Let X be a subset of I of finite type and i ∈ I \ X.
(
1) The subspace ad(M X )(E i ) of U + is a finite dimensional, irreducible ad(M X )- submodule of U q (g) with highest weight vector T wX (E i ) and lowest weight vector E i . (2) The subspace ad(M X )(F i K i ) of S(U − ) is a finite dimensional, irreducible ad(M X )-submodule of U q (g) with highest weight vector F i K i and lowest weight vector T −1 wX (F i K i ).
Proof. For any i ∈ I \ X, j ∈ X one has by Lemma 3.4 the relation
0 = T wX (ad(F j )(E i )) = T wX (F j E i K j − E i F j K j ) = −K −1 τ (j) E τ (j) T wX (E i ) − K τ (j) T wX (E i )K −1 τ (j) E τ (j) K −1 τ (j) = −K −1 τ (j) ad(E τ (j) )(T wX (E i )) K −1 τ (j)
and hence T wX (E i ) is a highest weight vector for the adjoint action of M X of weight w X α i . By Lemma 3.3 one obtains that T wX (E i ) is a highest weight vector of the irreducible ad(M X )-module generated by E i . Similarly, for any i ∈ I \ X, j ∈ X one obtains
0 = T −1 wX (ad(E j )(F i K i )) = −K −1 τ (j) (adF τ (j) )(T −1 wX (F i K i ))K −1 τ (j) . Moreover, T −1 wX (F i K i ) = σ • T wX (F i )K wX αi ∈ S(U − ).
Hence we may apply Lemma 3.3 and obtain that
T −1 wX (F i K i ) ∈ ad(M X )(F i K i ) is a lowest weight vector.
Quantum involutions
From now on until the end of Section 10 fix an admissible pair (X, τ ). In this section the automorphism θ(X, τ ) defined by (2.8) is deformed to an automorphism of U q (g). We first consider the case X = ∅. Then we use the Lusztig automorphism corresponding to the longest word w X ∈ W X to also deform the automorphism Ad(m X ).
4.1. The case X = ∅. In this case θ(∅, τ ) = τ • ω by (2.8). Recall that we use the same symbols τ and ω in the quantum case. The automorphism τ • ω : U q (g ′ ) → U q (g ′ ) is a natural quantum analog of θ(∅, τ ), but to make the definition in this case compatible with the definition for general X ⊂ I we consider a slightly different map. Define an algebra automorphism ψ :
U q (g) → U q (g) by ψ(E i ) = E i K i , ψ(F i ) = K −1 i F i , ψ(K h ) = K h (4.1) for all i ∈ I, h ∈ P ∨ . Now define θ q (∅, τ ) = ψ • τ • ω : U q (g ′ ) → U q (g ′ )
as the q-deformation of θ(∅, τ ). If the minimal realization of A is compatible with τ then θ q (∅, τ ) also defines an automorphism of U q (g).
4.2.
A q-analog of Ad(m X ). Assume now that X = ∅. Define
T X = T wX • ψ : U q (g) → U q (g)
where ψ is given by (4.1). The following lemma is a q-analogue of statement (1) and the first part of (3) in Proposition 2.2. The first statement in the lemma is the reason for introducing the additional isomorphism ψ.
Lemma 4.1. (1) T X • τ • ω| MX = id| MX .
(2) T X commutes with τ as an automorphism of U q (g ′ ).
Proof. The first claim follows immediately from Lemma 3.4 and the definition (4.1) of ψ. To verify (2), observe that by definition of T i one has
τ (T i (E j )) = T τ (i) (E τ (j) ) (4.2)
for any i, j ∈ I, see [Lus94, 37.1.3]. Let w X = r i1 . . . r i k be a reduced expression. Then the relation w X = r τ (i1) . . . r τ (i k ) and (4.2) imply τ (T wX (E j )) = T wX (τ (E j )). Similar relations hold for F j and K j and hence T wX commutes with τ . As ψ commutes with τ so does T wX • ψ.
Remark 4.2. If the minimal realization of A is compatible with τ then T X and τ commute as automorphisms of U q (g).
4.3. The quantum involution θ q (X, τ ). We are now in a position to define a quantum analog of the involutive automorphism θ(X, τ ) defined by (2.8). Recall the definition of the element s(X, τ ) ∈H from (2.7) and of the subgroup Ad(H) of Aut(U q (g)) from Subsection 3.2.
Definition 4.3. We call the automorphism θ q (X, τ ) :
U q (g ′ ) → U q (g ′ ) defined by θ q (X, τ ) = Ad(s(X, τ )) • T X • τ • ω the quantum involution corresponding to (X, τ ).
By Lemma 3.5 for any i ∈ I \ X there exist r ∈ N 0 , monomials
Z − i (X) = F i1 . . . F ir ∈ M − X , Z + i (X) = E j1 . . . E jr ∈ M + X , (4.3) and coefficients a + i , a − i ∈ K(q) such that T −1 wX (F i K i ) = a − i ad(Z − i (X))(F i K i ), T wX (E i ) = a + i ad(Z + i (X))(E i ). (4.4)
By construction the quantum involution θ q (X, τ ) is a K(q)-algebra automorphism of U q (g ′ ). It is not involutive, but by the following theorem it retains crucial properties of θ(X, τ ).
Theorem 4.4. The quantum involution θ q (X, τ ) has the following properties:
(
1) θ q (X, τ )| MX = id| MX . (2) For all β ∈ Q one has θ q (X, τ )(K β ) = K Θ(β) . (3) For any i ∈ I \ X there exist u i , v i ∈ K(q) × such that θ q (X, τ )(E i ) = −u i σ ad(Z − τ (i) (X))(F τ (i) K τ (i) ) , θ q (X, τ )(F i K i ) = −v i ad(Z + τ (i) (X))(E τ (i) ).
Proof. Claim (1) follows from Lemma 4.1 and the fact that Ad(s(X, τ )) restricts to the identity on M X . One obtains Claim (2) by the following calculation
θ q (X, τ )(K β ) = T wX (K −1 τ (β) ) = K −1 wX (τ (β)) = K Θ(β) . To verify Claim (3) observe for i ∈ I \ X the relation θ q (X, τ )(F i K i ) = Ad(s(X, τ )) • T X • τ • ω(F i K i ) = Ad(s(X, τ )) • T X (−E τ (i) K −1 τ (i) ) = Ad(s(X, τ )) • T wX (−E τ (i) ) (4.4) = −a + i Ad(s(X, τ )) • ad(Z + τ (i) (X))(E τ (i) )
. The map Ad(s(X, τ )) multiplies by a nonzero scalar. This proves the second formula in (3). The first formula follows analogously from the first formula in (4.4) and relation (3.14).
Remark 4.5. If the minimal realization of A is compatible with τ then θ q (X, τ ) extends to an algebra automorphism of U q (g). 4.4. Commutation with ω. By Proposition 2.2.(3) the automorphism Ad(m X ) of g commutes with the Chevalley involution ω of g. As shown below, however, the automorphism T X of U q (g) does not commute with the Chevalley involution ω of U q (g). Commutation with ω can be achieved by a slight modification of T X if on works with U q (g) defined over the field K(q 1/2 ). In the following we make this explicit although commutation of ω and T X is not necessary for the construction of quantum symmetric pairs. The content of this subsection will not be used in the rest of the paper.
For any i ∈ I one has
ω • ψ(E i ) = q −2 i ψ • ω(E i ), ω • ψ(F i ) = q 2 i ψ • ω(F i ). Hence one obtains the relation ψ • ω = ω • ψ • Ad(ν 2 ) (4.5) where ν ∈H is defined by ν(α i ) = q i . By [Lus94, 37.2.4] one has T i (ω(u)) = q −(αi,β) ω(T i (u)) for any i ∈ I, u ∈ U q (g) β (4.6)
where the additional sign appearing in [Lus94, 37.2.4] is conveniently hidden in our definition of ω. Relation (4.6) implies in particular that
T w (ω(u)) = i∈I q mi i ω(T w (u)) (4.7)
for all u ∈ U q (g) β and wβ − β = i∈I m i α i , see [Jan96,8.18(5)]. We abbreviate the above expression in the special case w = w X . For any i ∈ I define
Q X,i = j∈I q mj /2 j ∈ K(q 1/2 ) (4.8)
where w X α i − α i = αj ∈π m j α j . For the rest of this subsection assume that U q (g) is defined over the field K(q 1/2 ). Formula (4.7) implies that
T wX • ω = ω • T wX • Ad(η 2 X ) (4.9)
where η X ∈H is defined by η X (α i ) = Q X,i . Combining (4.5) and (4.9) one obtains
ω • T wX • ψ • Ad(η X ν) = T wX • ω • ψ • Ad(η −1 X ν) = T wX • ψ • ω • Ad(η −1 X ν −1 ) (3.9) = T wX • ψ • Ad(η X ν) • ω. (4.10) Define T ′ X = T wX • ψ • Ad(η X ν).
One can now formulate a version of Lemma 4.1 which includes commutation with ω.
Lemma 4.6. (1) T ′ X • τ • ω| MX = id| MX .
(2) T ′ X commutes with τ and ω as an automorphism of U q (g ′ ). Proof. Property (1) follows from the fact that Ad(η X ν)| MX = id| MX and from Lemma 4.1.(1). Commutation of τ and T ′ X follows from q i Q X,i = q τ (i) Q X,τ (i) and from Lemma 4.1.
(2). Commutation of ω and T ′ X was verified in (4.10).
4.5.
References to Letzter's constructions. For finite dimensional g, an automorphismθ 2 very similar to θ q (X, τ ) was constructed in [Let99, Theorem 3.1]. The definition ofθ 2 contains an additional Hopf algebra automorphism χ which produces an additional parameter. In our setting this parameter will be introduced in the next subsection. More importantly, the automorphismθ 2 constructed in [Let99, Theorem 3.1] is only a K-algebra automorphism which maps q to q −1 and
K β to K −Θ(β) for all β ∈ Q.
Letzter developed the main body of her theory in [Let99] in terms of right coideal subalgebras of U q (g). In the subsequent papers [Let00] and [Let02] she changed conventions to left coideal subalgebras. This is convenient for the study of quantum Harish-Chandra modules as defined in [Let00, Definition 3.1] but it comes at the price that the definition of the quantum involutions given in [Let02,Theorem 7.1] involves the right adjoint action. Moreover, the connection with Lusztig's automorphisms, which was evident in [Let99], only appears implicitly in the later papers, for example in the proof of [Let02, Theorem 7.1].
Quantum symmetric pairs
To shorten notation we just write θ q instead of θ q (X, τ ) to denote the quantum involution associated to the fixed admissible pair (X, τ ) by Definition 4.3. Recall that Corollary 2.9 provides a set of generators of the enveloping algebra U (k) of the invariant Lie algebra k. Replacing these generators by suitable elements in U q (g ′ ) we now define a quantum analog of U (k ′ ) as a right coideal subalgebra of U q (g ′ ).
For the elements (2.12) and (2.13) this is straightforward. Indeed, the generators
e i , f i for i ∈ X are replaced by M X while the generators h ∈ h ′ with θ(h) = h are replaced by K h if h ∈ Q ∨ .
For the generators f i + θ(f i ) + s i for i ∈ I \ X from (2.15), however, there exist families of possible q-analogs which are given by
B i = F i + c i θ q (F i K i )K −1 i + s i K −1 i for all i ∈ I \ X (5.1) for suitable (c i ) i∈I\X ∈ (K(q) × ) I\X and (s i ) i∈I\X ∈ K(q) I\X . Following [Let02,
Section 7] we will call a q-analog of U (k ′ ) standard if s i = 0 for all i ∈ I \ X. If there are also generators of the form (5.1) with s i = 0 then the quantum analog of U (k) will be called nonstandard. In the following subsection we give rigorous definitions of quantum symmetric pairs coideal subalgebras. We then collect properties of their generators which will eventually, in Section 7, lead to a presentation of quantum symmetric pair coideal subalgebras in terms of generators and relations.
Definition of quantum symmetric pair coideal subalgebras
B c,s . De- fine Q Θ = {β ∈ Q | Θ(β) = β} and let U 0 Θ ′ denote the subalgebra of U 0′ generated by the elements K β for all β ∈ Q Θ . Definition 5.1. For any c = (c i ) i∈I\X ∈ (K(q) × ) I\X and s = (s i ) i∈I\X ∈ K(q) I\X define B c,s = B c,s (X, τ ) to be the subalgebra of U q (g ′ ) generated by M X , U 0 Θ ′ , and the elements (5.1) for all i ∈ I \ X.
The algebra B c,s is a quantum analog of U (k ′ ) only for suitable parameters c and s. In the remainder of this subsection the necessary restrictions on the parameters will be elaborated. Independently of these restrictions, however, B c,s is always a coideal of U q (g ′ ).
Proposition 5.2. Let c ∈ (K(q) × ) I\X and s ∈ K(q) I\X . Then B c,s is a right coideal subalgebra of U q (g ′ ).
Proof. Clearly, M X and U 0 Θ ′ are Hopf subalgebras of U q (g ′ ). It remains to show that
∆(B i ) ∈ B c,s ⊗ U q (g ′ ) (5.2) for all i ∈ I \ X. Theorem 4.4 implies that there exists v i ∈ K(q) × such that B i = F i − c i v i ad(Z + τ (i) (X))(E τ (i) )K −1 i + s i K −1 i . (5.3)
If one applies the relation
∆(ad(x)(u)) = x (1) u (1) S(u (3) ) ⊗ ad(x (2) )(u (2) ) to x = Z + τ (i) (X) and u = E τ (i) then one obtains in view of Equation (3.3) the relation ∆ ad(Z + τ (i) (X))(E τ (i) ) − ad(Z + τ (i) (X))(E τ (i) ) ⊗ 1 ∈ M + X K τ (i) ⊗ U q (g ′ ). (5.4)
Relations (5.3) and (5.4) together imply that
∆(B i ) − B i ⊗ K −1 i ∈ M + X U 0 Θ ′ ⊗ U q (g ′ ) (5.5)
and hence Relation (5.2) holds for all i ∈ I \ X.
A desirable condition for B c,s to qualify as a quantum analog of
U (k ′ ) is that B c,s ∩ U 0′ = U 0 Θ ′ .
Lemmas 5.4 and 5.5 below impose restrictions on the parameters c and s for which this condition is satisfied.
Lemma 5.3. Let i ∈ I \ X such that τ (i) = i and (α i , Θ(α i )) = 0. Then (α i , α τ (i) ) = 0 and Θ(α i ) = −α τ (i) . Proof. As τ (i) = i and (α i , α j ) ≤ 0 for all j = i the relation (α i , −w X α τ (i) ) = (α i , Θ(α i )) = 0 implies that (α i , α τ (i) ) = 0. (5.6) Moreover, α i + Θ(α τ (i) ) = α i − w X α i ∈ ZX and hence α i + Θ(α τ (i) ) = Θ(α i + Θ(α τ (i) )) = α τ (i) + Θ(α i ). (5.7)
Equations (5.6), (5.7), and the assumption (α τ (i) , Θ(α τ (i) )) = (α i , Θ(α i )) = 0 imply
(α τ (i) + Θ(α i ), α τ (i) + Θ(α i )) = (α τ (i) + Θ(α i ), α i + Θ(α τ (i) )) = 0 and hence Θ(α i ) = −α τ (i) . Lemma 5.4. Let B be a right coideal subalgebra of U q (g). Let i, j ∈ I such that (α i , α j ) = 0, (α i , α i ) = (α j , α j ), and F i − c i E j K −1 i ∈ B, F j − c j E i K −1 j ∈ B for some c i , c j ∈ K(q) × . If c i = c j then (K i K j ) −1 ∈ B.
Proof. The claim follows from the relation
[F i − c i E j K −1 i , F j − c j E i K −1 j ] = −c i K j − K −1 j q j − q −1 j K −1 i + c j K i − K −1 i q i − q −1 i K −1 j (5.8)
by applying the coproduct to the right hand side.
By the above lemma and Lemma 5.3 the condition B c,s ∩ U 0′ = U 0 Θ ′ can only be satisfied if the parameter c is contained in the set
C = {c ∈ (K(q) × ) I\X | c i = c τ (i) if τ (i) = i and (α i , Θ(α i )) = 0}. (5.9)
The parameters s are also subject to restrictions. Define
I ns = {i ∈ I \ X | τ (i) = i and α i (h j ) = 0 ∀j ∈ X}.
(5.10) By the following lemma it is only reasonable to allow s i = 0 for i ∈ I ns .
Lemma 5.5. Let B ⊆ U q (g ′ ) be a subalgebra which contains U 0 Θ ′ and the element B i defined by (5.1) for some c i , s i ∈ K(q) × , i ∈ I \ X. If i / ∈ I ns then K −1 i ∈ B. Proof. If τ (i) = i then K i K −1 τ (i) ∈ U 0 Θ ′ .
Conjugating B i by this element one obtains
K −1 i ∈ B. Similarly, if α i (h j ) = 0 for some j ∈ X, then one conjugates by K j ∈ U 0 Θ ′ to obtain K −1 i ∈ B.
It will turn out that the condition
s i = 0 if i / ∈ I ns is not sufficient to ensure B c,s ∩ U 0′ = U 0 Θ ′ . Consider the set S = {s ∈ K(q) I\X | s i = 0 ⇒ (i ∈ I ns and a ij ∈ −2N 0 ∀j ∈ I ns \ {i})}. (5.11)
The relevance of the additional condition a ij ∈ −2N 0 in the definition of S will become apparent in the proof of Relation (5.21), see also Remark 5.12.
Definition 5.6. We call B c,s for c ∈ C and s ∈ S a quantum symmetric pair coideal subalgebra of U q (g ′ ). If s = 0 = (0, 0, . . . , 0) then B c = B c,0 is called standard. If s i = 0 for some i ∈ I \ X then B c,s is called nonstandard.
Remark 5.7. The construction of the quantum symmetric pair coideal subalgebras B c,s may seem rather ad hoc, since they are defined by giving explicit quantum analogs of the generators of U (k ′ ). It will be shown in Section 10 that B c,s specializes to U (k ′ ) and that B c,s is maximal with this property. For finite dimensional g, Letzter showed that any coideal subalgebra of U q (g) with these two properties has to be of the form B c,s , see [Let99,Theorem 5.8], [Let02,Theorem 7.5]. An analog of Letzter's classification result in the Kac-Moody case would provide a stronger justification for the definition of quantum symmetric pair coideal subalgebras. This problem is left for future work.
Remark 5.8. The algebra B c,s for c ∈ C, s ∈ S is a quantum analog of U (k ′ ), see Theorem 10.8. If the minimal realization of A is compatible with τ then one may
replace U 0 Θ ′ by the algebra U 0 θ = K(q) K h | h ∈ P ∨ , θ(h) = h to obtain a quantum analog of U (k).
5.2.
Decompositions and projections for U q (g). The triangular decomposition (3.6) for U q (g) implies that the multiplication map gives an isomorphism of vector spaces
U + ⊗ U 0 ⊗ S(U − ) ∼ = U q (g). (5.12)
This leads to a direct sum decomposition
U q (g) = ⊕ λ∈P ∨ U + K λ S(U − ) (5.13)
For any λ ∈ P ∨ let P λ : U q (g) → U + K λ S(U − ) denote the projection with respect to this decomposition. It follows from the formulas for the coproduct of U q (g) that the map P λ is a homomorphism of left U q (g)-comodules, that is
∆ • P λ (x) = (id ⊗ P λ )∆(x)
for all x ∈ U q (g). (5.14)
This relation implies the following grading of right coideal subalgebras of U q (g).
Lemma 5.9. Let B be a right coideal subalgebra of U q (g).
Then B = ⊕ λ∈P ∨ P λ (B). Proof. Let b ∈ B and λ ∈ P ∨ . Relation (5.14) implies that ∆(P λ (b)) = b (1) ⊗ P λ (b (2) ) ∈ B ⊗ U q (g). Application of id ⊗ ε implies P λ (b) ∈ B.
We may also consider the direct sum decomposition
U q (g) = ⊕ α,β∈Q + U + α U 0 U − −β . (5.15) Let π α,β : U q (g ′ ) → U + α U 0 U − −β denote
the projection with respect to this decomposition.
5.3. Quantum Serre relations for B c,s . All through this subsection fix c ∈ C and s ∈ S and consider the corresponding quantum symmetric pair coideal subalgebra B c,s . Recall the definition of the noncommutative polynomials F ij given for any i, j ∈ I by (3.1). The next lemmas collect properties of F ij evaluated on the elements relevant for the construction B c,s .
Lemma 5.10. The following relations hold for all i, j ∈ I:
F ij (F i K i , F j K j ) = 0, (5.16) F ij (θ q (F i K i )K −1 i , θ q (F j K j )K −1 j ) = 0, (5.17) F ij (F i , K −1 j ) = 0. (5.18)
Proof. Property (5.16) follows from (3.2) because
(F i K i ) 1−aij −n F j K j (F i K i ) n = F 1−aij −n i F j F n i K 1−aij i K j (5.19) if 0 ≤ n ≤ 1 − a ij .
As θ q is an algebra automorphism one obtains
F ij (θ q (F i K i ), θ q (F j K j )) = 0 for all i, j ∈ I.
This implies (5.17) by a calculation analog to (5.19). Finally, to verify (5.18) note that the relation F
1−aij −n i K −1 j F n i = q aij n i F 1−aij j K −1 i implies F ij (F i , K −1 j ) = 1−aij n=0 (−1) n q aij n i 1 − a ij n qi F 1−aij i K −1 j = 0 by [Jan96, 0.2.(4)].
We extend the definition of B i given in (5.1) for i ∈ I \ X to all elements of I by defining B i := F i for i ∈ X.
Lemma 5.11. The following relations hold in U q (g):
F ij (B i , B j ) = 0
for all i ∈ X, j ∈ I, (5.20)
π 0,0 (F ij (B i , B j )) ∈ U 0 Θ ′ for all i, j ∈ I. (5.21) Proof. For i ∈ X, j ∈ I one has θ q (F i K i ) = F i K i and hence F ij (B i , B j ) = F ij (F i , F j + c j θ q (F j K j )K −1 j + s j K −1 j ) (5.18) = F ij (F i , F j + c j θ q (F j K j )K −1 j ) (3.2) = c j F ij (F i , θ q (F j K j )K −1 j ) = c j F ij (θ q (F i K i )K −1 i , θ q (F j K j )K −1 j ) (5.17) = 0.
This proves (5.20). We will now verify relation (5.21) in several steps. By (5.20) we may assume that i / ∈ X.
Step 1:
π 0,0 (F ij (B i , B j )) = 0 if i ∈ I, j ∈ X. Indeed, assume that π 0,0 (F ij (B i , B j )) = 0 for some i ∈ I \ X, j ∈ X. For weight reasons this is only possible if τ (i) = i, a ij = −1, and w X (α i ) = α i + α j . The latter relation, however, implies that (α i + α j )(h k ) = 0 for all k ∈ X \ {j}. Hence α i (h k ) = α j (h k ) = 0 for all k ∈ X \ {j}. This, in turn, implies that w X (α i ) = s j (α i ) = α i − a ji α j and hence a ji = −1. But then α i (ρ ∨ X ) = α i (h j /2) = a ji /2 = −1/2 which contradicts the fact that (X, τ ) is an admissible pair.
By
Step 1 we may from now on assume that neither i nor j are contained in X.
Step 2:
π 0,0 (F ij (B i , B j )) ∈ U 0 Θ ′ if i, j / ∈ I ns . Again, in this case π 0,0 (F ij (B i , B j )) = 0 implies that F ij (B i , B j ) has a zero weight summand. By definition of F ij and B i , B j , however, this is only possible if a ij = 0 and Θ(α i ) = −α j . In this case c i = c j by definition (5.9) of C and hence B i = F i − c i E j K −1 i and B j = F j − c j E i K −1 j . Now relation (5.8) implies that F ij (B i , B j ) ∈ U 0 Θ ′ .
Step 3: π 0,0 (F ij (B i , B j )) = 0 if (i / ∈ I ns and j ∈ I ns ) or (i ∈ I ns and j / ∈ I ns ). This holds for weight reasons.
Step 4: π 0,0 (F ij (B i , B j )) = 0 if i, j ∈ I ns and −a ij ∈ 2N 0 . Observe first that in this case
π 0,0 (F ij (B i , B j )) = s j π 0,0 F ij (B i , K −1 j ) (5.22)
for weight reasons. Moreover, for −a ij ∈ 2N 0 the non-commutative polynomial (3.1) can be written as
π 0,0 B m i K −1 j B n i − B n i K −1 j B m i = 0 (5.24) if n + m is odd.
Let M denote the free monoid generated by symbols E, F, K −1 and let ι : M → M denote the monoid anti-automorphism defined by ι(E) = F , ι(F ) = E, and ι(K −1 ) = K −1 . Let, moreover, ℓ : M → N 0 denote the length function and let π : M → U q (g ′ ) denote the monoid homomorphisms defined by π(E) = E i K −1 i , π(F ) = F i , and π(K −1 ) = K −1 i . To verify (5.24) it suffices to show that π 0,0 π(u)K −1 j π(v) − π(ι(v))K −1 j π(ι(u)) = 0 (5.25) for all u, v ∈ M with ℓ(u) = m and ℓ(v) = n. The above relation follows from π 0,0 (π(w)) = π 0,0 (π(ι(w))) which holds for all w ∈ M. This proves Step 4, and hence completes the proof of Equation (5.21).
Remark 5.12. Assume, contrary to the definition of S given by (5.11), that s j = 0 for some i, j ∈ I ns with −a ij odd. For λ ij = (1 − a ij )α i + α j one then obtains
P −λij • π 0,0 F i,j (B i , K −1 j ) = 0 in general.
It is straightforward to verify this for −a ij = 1 or −a ij = 3 by direct computation. Hence in this case
P −λij • π 0,0 (F i,j (B i , B j )) = 0.
As will be seen in the proof of Proposition 5.16, the above relation would imply that B c,s ∩ U 0′ = U 0 Θ ′ . This is the reason why we restrict to parameters s in the set S given by (5.11)
Remark 5.13. By the above proof, Relation (5.21) can be refined. Indeed, on has π 0,0 (F i,j (B i , B j )) = 0 unless Θ(α i ) = −α j and a ij = 0 in which case c i = c j , s i = s j = 0, and
F ij (B i , B j ) = c i K i K −1 j − K j K −1 i q i − q −1 i
as calculated in the proof of Lemma 5.4.
The following technical lemma will be used in the proof of Proposition 5.16.
Lemma 5.14. Let α, β ∈ Q + . If π α,β (F ij (B i , B j )) = 0 then λ ij − α / ∈ Q Θ and λ ij − β / ∈ Q Θ .
Proof. By (5.20) there is nothing to show if i ∈ X. Hence we may assume that i / ∈ X. Consider first the case that j ∈ X. Then (5.17) implies
0 = F ij (θ q (F i K i )K −1 i , θ q (F j K j )K −1 j ) = F ij (θ q (F i K i )K −1 i , F j ). Hence, if π α,β (F ij (B i , B j )) = 0 for some α, β ∈ Q + then 0 ≤ β ≤ λ ij − α i and 0 ≤ α ≤ −Θ(λ ij − α i ). This implies that λ ij − β ≥ α i , −Θ(λ ij ) − α ≥ −Θ(α i ).
As Θ(α i ) ∈ −Q + one gets λ ij − β / ∈ Q Θ and −Θ(λ ij ) − α / ∈ Q Θ . As λ ij + Θ(λ ij ) ∈ Q Θ the second relation implies that λ ij − α / ∈ Q Θ . Finally, we turn to the case that j / ∈ X. By (3.2) and (5.17) the relation π α,β (F ij (B i , B j )) = 0 implies 0 ≤ β < λ ij and α = α ′ + α ′′ with 0 ≤ α ′ < τ (λ ij ) and α ′′ ∈ i∈X N 0 α i . Hence λ ij − β and τ (λ ij ) − α ′ are nonzero elements in N 0 α i + N 0 α j and N 0 α τ (i) + N 0 α τ (j) , respectively. Now the relations Θ(α k ) ∈ −Q + for k ∈ I \ X and α ′′ , λ ij − τ (λ ij ) ∈ Q Θ imply the claim.
Lemma 5.15. The following relations hold in B c,s :
K λ B i = q −(αi,λ) B i K λ for all i ∈ I, λ ∈ Q Θ , (5.26) E j B i − B i E j = δ ij K i − K −1 i q i − q −1 i for all i ∈ I, j ∈ X. (5.27)
Proof. Recall from (2.11) that (α, β) = (Θ(α), Θ(β)) holds for all α, β ∈ Q. In particular one has (α i , λ) = (Θ(α i ), λ) if λ ∈ Q Θ . This implies (5.26).
Relation (5.27) holds for i ∈ X by the defining relation 3.1.(4) of U q (g). For i / ∈ X Equation (5.27) follows from
[E j , θ q (F i K i )K −1 i ] = θ q [E j , F i K αi−Θ(αi) ]
= 0 which holds as (α j , α i ) = (α j , Θ(α i )) by (2.11).
For any J = (j 1 , . . . , j n ) ∈ I n define wt(J) = n i=1 α ji and
E J = E j1 . . . E jn , F J = F j1 . . . F jn , B J = B j1 . . . B jn . (5.28)
Fix i, j ∈ I and recall that λ ij = (1 − a ij )α i + α j . We want to further investigate properties of Y := F ij (B i , B j ). By Relation (5.5) and Lemma 5.15 one has
∆(Y ) ∈ Y ⊗ K −λij + {J | wt(J)<λij } M + X U 0 Θ ′ B J ⊗ U q (g ′ ). (5.29)
For later use we observe the following properties of the second term
S(U − )U 0′ ∩ {J | wt(J)<λij } M + X U 0 Θ ′ B J = S(U − )U 0′ ∩ M X U 0 Θ ′ = S(M − X )U 0 Θ ′ , (5.30) U + U 0′ ∩ {J | wt(J)<λij } M + X U 0 Θ ′ B J = U + U 0′ ∩ M X U 0 Θ ′ = M + X U 0 Θ ′ (5.31)
which follow from the linear independence of the terms F J for all J with wt(J) < λ ij . The following proposition provides the main tool to write the algebra B c,s efficiently in terms of generators and relations.
Proposition 5.16. In U q (g) one has P −λij (F ij (B i , B j )) = 0 for all i, j ∈ I.
Proof. To abbreviate notation set Z := P −λij (F ij (B i , B j )). Relations (5.14) and (5.29) imply
∆(Z) ∈ Y ⊗ K −λij + {J | wt(J)<λij } M + X U 0 Θ ′ B J ⊗ U + K −λij S(U − ). (5.32)
Assume now, that Z = 0. Choose α ∈ Q + maximal with respect to the partial order such that π α,β (Z) = 0 for some β ∈ Q + . In this case by (3.3), (3.4). If α = 0 then Relations (5.32) and (5.30) imply K −λij +α ∈ U 0 Θ ′ in contradiction to Lemma 5.14. Hence α = 0 and Z ∈ S(U − )K −λij . Now choose β ∈ Q + maximal such that π 0,β (Z) = 0. In this case 0 = (id ⊗ π 0,β )∆(Z) ∈ K −λij +β ⊗ S(U − β )K −λij As before, Relations (5.32) and (5.31) together with Lemma 5.14 imply β = 0. Hence Z = π 00 (Z). But then 0 = π 0,0 (Z) ∈ K(q)K −λij U 0 Θ ′ in contradiction to Equation (5.21). Hence Z = 0.
0 = (id ⊗ π α,0 )∆(Z) ∈ S(U − )K −λij +α ⊗ U + α K −λij
Corollary 5.17. In B c,s one has the relation
F ij (B i , B j ) ∈ {J∈J | wt(J)<λij } M + X U 0 Θ ′ B J for all i, j ∈ I. (5.33)
Proof. This follows from Proposition 5.16 by applying the counit to the second tensor factor in (5.32) where Z = P −λij (F ij (B i , B j )) = 0.
Remark 5.18. For i ∈ I ns one has
∆(B i ) = B i ⊗ K −1 i + 1 ⊗ (F i − c i E i K −1 i ) (5
Triangular decompositions
From now on until the end of Section 8 we fix parameters c ∈ C, s ∈ S and hence a corresponding quantum symmetric pair coideal subalgebra B c,s of U q (g ′ ). By Corollary 5.17 there exist relations between the generators B i , i ∈ I, of B c,s which are similar to the quantum Serre relations for U q (g ′ ). To make the lower order terms in these relations explicit, and to show that these relations together with those from Lemma 5.15 form a complete set of relations for B c,s , one uses triangular decompositions of U q (g ′ ) involving B c,s . The triangular decompositions in this section will also be used in Section 10 to describe the behavior of B c,s under specialization.
6.1. A one-sided U + U 0′ -module basis of U q (g ′ ). Recall from (5.28) that for any multi-index J = (j 1 , . . . , j n ) ∈ I n we defined wt(J) = n i=1 α ji and F J = F j1 . . . F jn and B J = B j1 . . . B jn . In this case we also define |J| = n. Let J be a fixed subset of n∈N0 I n such that {F J | J ∈ J } is a basis of U − . By the triangular decomposition (3.6) of U q (g ′ ) the set {F J | J ∈ J } forms a basis U q (g ′ ) as a left U + U 0′ -module. By the following proposition this basis can be replaced by the set
{B J | J ∈ J }.
Define a filtration F * of U − by F n (U − ) = span{F J | J ∈ I m , m ≤ n} for all n ∈ N 0 . As the quantum Serre relations are homogeneous, the set {F J | J ∈ J , |J| ≤ n} forms a basis of F n (U − ).
Proposition 6.1. The set {B J | J ∈ J } is a basis of the left (or right) U + U 0′module U q (g ′ ).
Proof. We prove the result for the left U + U 0′ -module U q (g ′ ). Let J ∈ J and |J| = n. We first show by induction on n that F J is contained in the left U + U 0′ -module generated by the set
{B J | J ∈ J }. Indeed, F J − B J ∈ U + U 0′ F n−1 (U − ). Using the quantum Serre relations for U − one hence obtains that F J − B J is contained in the left U + U 0′ -submodule of U q (g ′ ) generated by the set {F L | L ∈ J , |L| ≤ n − 1}.
The induction hypothesis implies the desired result.
It remains to show that the set {B J | J ∈ J } is linearly independent over U + U 0′ . To this end assume that there exists a non-empty finite subset J ′ ⊂ J such that J∈J ′ a J B J = 0 (6.1) for some a J ∈ U + U 0′ . Let m ∈ N be maximal such that there exists J ∈ J ′ with |J| = m. In view of the specific form (5.1) of the generators B i , Equation (6.1) implies that
J∈J ′ ,|J|=m a J F J = 0. (6.2)
By the triangular decomposition (3.6) of U q (g ′ ) and the fact that {F J | J ∈ J } is linearly independent, Equation (6.2) implies that a J = 0 for all J ∈ J ′ with |J| = m. Induction on m gives the desired result. Again, one can reformulate the above proposition by saying that the multiplication map
M + X ⊗ U 0 Θ ′ ⊗ B c,s,J → B c,s
is an isomorphism of vector spaces.
6.3. The quantum Iwasawa decomposition. Fix a subset I * of I \X containing exactly one element of each τ -orbit of I \ X. In particular, I * contains all i ∈ I \ X with τ (i) = i and precisely one of j, τ (j) if τ (j) = j. Let U ′ Θ denote the subalgebra of U 0′ generated by all elements in the set {K i , K −1 i | i ∈ I * }. The multiplication map gives an isomorphism of algebras
U ′ Θ ⊗ U 0 Θ ′ ∼ = U 0′ . (6.5)
Recall from Section 3.3 that V + X denotes the subalgebra of U + generated by the elements in ad(M X )(E i ) for all i ∈ I \ X. By (3.11) and (6.5) the multiplication map gives an isomorphism
V + X ⊗ U ′ Θ ⊗ M + X ⊗ U 0 Θ ′ ∼ = U + U 0′
of vector spaces. By (6.5) and Proposition 6.2 the above decomposition of U + U 0′ implies the following result.
Proposition 6.3. The multiplication map gives an isomorphism of vector spaces
V + X ⊗ U ′ Θ ⊗ B c,s ∼ = U q (g ′ ).
Remark 6.4. The Iwasawa decomposition of g ′ is given by g ′ = n + X ⊕a ′ ⊕k ′ where a ′ = {h ∈ h ′ | θ(h) = −h} and n + X denotes the g + X -module generated by {e i | i ∈ I \ X} under the adjoint action. In the above decomposition, V + X is a q-analog of U (n + X ) and B c,s is a q-analog of U (k ′ ). The algebra U ′ Θ , however, is a somewhat coarse analog of U (a ′ ). A better q-analog of U (a ′ ) is given by A Θ = K(q) K β | Θ(β) = −β . However, relation (6.5) does not hold with U ′ Θ replaced by A Θ . This problem can be can be circumvented by adjoining certain roots of K ±1 i to U 0′ , as done for instance in [Let99], but this is not necessary for our purposes. Theorem 7.1. LetB be the algebra freely generated over M + X U 0 Θ ′ by elementsB i for all i ∈ I and let Π denote the canonical algebra homomorphism Π :B → B c,s given byB
i → B i , m → m for all i ∈ I, m ∈ M + U 0 Θ ′ . (7.1)
WriteB J =B j1 · · ·B jn for any J = (j 1 , . . . , j n ) ∈ I n . Then there exist elements
C ij (c) ∈ wt(J)<λij M + X U 0 Θ ′B J
for all i, j ∈ I, i = j such that the kernel ker(Π) is the ideal ofB generated by the following elements:
K βBi − q −(β,αi)B K β for all β ∈ Q Θ , i ∈ I, (7.2) E iBj −B j E i − δ ij K i − K −1 i q i − q −1 i for all i ∈ X, j ∈ I, (7.3) F ij (B i ,B j ) −C ij (c)
for all i, j ∈ I, i = j. Proof. It follows from Corollary 5.17 that there exist elements
C ij (c) ∈ wt(J)<λij M + X U 0 Θ ′B J such that F ij (B i ,B j )−C ij (c)
lies in ker(Π) for any i, j ∈ I, i = j. Let L be the ideal ofB generated by the elements in (7.2), (7.3), and (7.4) for this choice ofC ij (c). It follows from Lemma 5.15 and Corollary 5.17 that L is contained in ker(Π). Hence there is a well defined surjective mapB/L → B c,s given by (7.1). As in the proof of Proposition 6.2 one shows thatB/L is spanned as a left M + X U 0 Θ ′ -module by the elementsB J for J ∈ J . Now Proposition 6.2 implies that the surjective map B/L → B c,s is also injective. The formal independence ofC ij (c) of s was already noted in Remark 5.18.
In the following we use the notation C ij (c) = Π(C ij (c)). The results from Subsection 5.3 can be efficiently used to find explicit formulas for the elements C ij (c). This allows us to obtain a complete presentation of the algebras B c,s in terms of generators and relations without the need to embark on explicit calculations. The method used in Theorem 7.4 below was developed in [Let03] for finite dimensional g, but it also works in the symmetrizable Kac-Moody case. To apply it we need the following first order calculation of ∆(B i ). Recall Theorem 4.4.(3) and define
Z i = −v i ad(Z + τ (i) )(K 2 τ (i) )K −1 τ (i) K −1 i for any i ∈ I \ X where we abbreviate Z + τ (i) = Z + τ (i) (X).
Lemma 7.2. For any i ∈ I \ X one has
∆(B i ) = B i ⊗ K −1 i + 1 ⊗ F i + c i Z i ⊗ E τ (i) K −1 i + Υ (7.5) for some Υ ∈ M X U 0 Θ ′ ⊗ γ>α τ (i) U + γ K −1 i .
Proof. We apply the formula ∆(ad(x)(u)) = x (1) u (1) S(x (3) ) ⊗ ad(x (2) )(u (2) ) (7.6) to x = Z + τ (i) and u = E τ (i) . By (3.3) one obtains
∆(ad(Z + τ (i) )(E τ (i) )) = ad(Z + τ (i) )(E τ (i) ) ⊗ 1 + ad(Z + τ (i) )(K 2 τ (i) )K −1 τ (i) ⊗ E τ (i) + Ψ for some Ψ ∈ M + X K τ (i) ⊗ γ>α τ (i) U + γ . By definition of B i and Z i one obtains the claim of the Lemma with Υ = c i v i Ψ(K −1 i ⊗ K −1 i ).
By Theorem 7.1 one has
Z i B j = q (αi−α τ (i) ,αj ) B j Z i for all i, j ∈ I \ X. (7.7)
As before fix distinct i, j ∈ I and write Y = F ij (B i , B j ). By Proposition 5.16 one has Z = P −λij (Y ) = 0. Using (5.32) as in the proof of Corollary 5.17 one obtains
C ij (c) = −(id ⊗ ε)(∆(Z) − Y ⊗ K −λij ) (5.14) = −(id ⊗ ε)(id ⊗ (P −λij • π 0,0 ))(∆(Y ) − Y ⊗ K −λij ). (7.8) The expression (id ⊗ (P −λij • π 0,0 ))(∆(Y ) − Y ⊗ K −λij )
can be evaluated in general because many terms in (id ⊗ π 0,0 )∆(Y ) vanish. More explicitly, we use the fact that summands of ∆(Y ) need to contain the same number of F i 's and E i 's in the second tensor factor in order to contribute nontrivially. By Lemma 7.2, however, this can only occur if i ∈ {τ (i), τ (j)}. One obtains the following result. Recall from Equation (5.20) that C ij (c) = 0 if i ∈ X. Hence we only need to consider the case that i ∈ I \ X. In the following two subsections we distinguish the two cases j / ∈ X and j ∈ X.
7.2.
Determining C ij (c) in the case i, j ∈ I \X. The next theorem gives explicit expressions for C ij (c) for a ij ∈ {0, −1, −2} if j ∈ I \ X. In this case, summands of ∆(Y ) involving the term Υ in Equation (7.5) will never survive under id ⊗ π 0,0 . Recall the definition of the q-number [n] q for n ∈ N.
Theorem 7.4. Assume that i, j ∈ I \ X. The elements C ij (c) = Π(C ij (c)) from Theorem 7.1 are given by the following formulas. Case 1: a ij = 0.
C ij (c) = δ i,τ (j) (q i − q i ) −1 c i Z i − c j Z j . (7.9)
Case 2: a ij = −1.
C ij (c) = δ i,τ (i) q i c i Z i B j − δ i,τ (j) (q i + q −1 i ) q i c j Z j + q −2 i c i Z i B i . (7.10) Case 3: a ij = −2. C ij (c) =δ i,τ (i) q i (q i + q −1 i ) 2 c i Z i (B i B j − B j B i ) (7.11) + δ i,τ (j) [3] qi q i (q 4 i − 1) q −8 i c i Z i B 2 i − c j Z j B 2 i .
Proof. To abbreviate notation we write Q −λij = id ⊗ (P −λij • π 0,0 ) for any i, j ∈ I. Case 1: By (7.8) one has
C ij (d) = −(id ⊗ ε) • Q −λij ∆(B i B j − B j B i ) − (B i B j − B j B i ) ⊗ K −λij . (7.12) For j ∈ I \ X one obtains Q −λij (∆(B i B j ) − B i B j ⊗ K −λij ) = Q −λij c j Z j ⊗ F i E τ (j) K −1 j = Q −λij −δ i,τ (j) c j Z j ⊗ K i − K −1 i q i − q −1 i K −1 j = δ i,τ (j) (q i − q −1 i ) −1 c j Z j ⊗ K −λij
. By (7.12) this implies the desired formula (7.9) for C ij (d).
Case 2: For a ij = −1 one has Y = B 2 i B j − (q i + q −1 i )B i B j B i + B j B 2 i .
Performing a calculation analogous to Case 1, one obtains
Q −λij (∆(Y ) − Y ⊗ K −λij ) = δ i,τ (i) C 1 ⊗ K −λij + δ i,τ (j) C 2 ⊗ K λij (7.13) where C 1 ⊗ K −λij =Q −λij c i Z i B j ⊗ F i E i K −1 i K −1 j − (q i + q −1 i )B j c i Z i ⊗ F i K −1 j E i K −1 i + B j c i Z i ⊗ K −1 j F i E i K −1 i = − q i c i Z i B j ⊗ K −λij (7.14)
and similarly, for the term C 2 where τ (i) = j one obtains
C 2 ⊗ K −λij = Q −λij B i c j Z j ⊗ K −1 i F i E i K −1 j + B i c j Z j ⊗ F i K −1 i E i K −1 j −(q i + q −1 i )c j Z j B i ⊗ F i E i K −1 j K −1 i − (q i + q −1 i )B i c i Z i ⊗ K −1 i F j E j K −1 i +c i Z i B i ⊗ F j E j K −2 i + B i c i Z i ⊗ F j K −1 i E j K −1 i = (q i + q −1 i ) q i c j Z j B i + q −2 i c i Z i B i ⊗ K −λij . (7.15)
Equations (7.14) and (7.15) imply Formula (7.10).
Case 3: For a ij = −2 one has Y = B 3 i B j − [3] qi B 2 i B j B i + [3] qi B i B j B 2 i − B j B 3 i where [3] qi = q −2 i + 1 + q 2 i .
Again Relation (7.13) holds with λ ij = 3α i + α j . If τ (i) = i then (7.7) gives
Z i B i = B i Z i , Z i B j = B j Z i .
With these relations one determines
C 1 = q i (q i + q −1 i ) 2 c i Z i (B j B i − B i B j ) (7.16)
by a calculation analogous to Case 2. Similarly, if τ (i) = j one uses the relations
Z i B i = q 4 i B i Z i , Z i B j = q −4 i B j Z i , Z j B i = q −4 i B i Z j . (7.17) to obtain C 2 = [3] qi q i (1 − q 4 i )B 2 i c i Z i − q −8 i c j Z j . (7.18)
One obtains Relation (7.11) by inserting (7.16) and (7.18) into Equation (7.13).
Remark 7.5. The method of the above proof can in principle by used to determine the lower order terms C ij (c) for a ij = −3, −4, . . . . This becomes very tedious. In the case where g is of finite type G 2 and a ij = −3 the term C ij (c) was calculated in [Let03, Theorem 7.1]. For c = (1, 1) it is also given in [KP11, Proposition 3.1 ] in the precise conventions of the present paper. In this case C = (K(q) × ) 2 and S = K(q) 2 . The quantum symmetric pair coideal subalgebra B c,s corresponding to c = (c 0 , c 1 ) ∈ C and s = (s 0 , s 1 ) ∈ S is generated by two elements
B i = F i − c i E i K −1 i + s i K −1 i for i = 0, 1.
One has Z 0 = Z 1 = −1. By Theorem 7.4 the algebra B c,s is given by the defining relations
B 3 0 B 1 −[3] q B 2 0 B 1 B 0 +[3] q B 0 B 1 B 2 0 −B 1 B 3 0 = q(q+q −1 ) 2 c 0 (B 1 B 0 −B 0 B 1 ), (7.19) B 3 1 B 0 −[3] q B 2 1 B 0 B 1 +[3] q B 1 B 0 B 2 1 −B 0 B 3 1 = q(q+q −1 ) 2 c 1 (B 0 B 1 −B 1 B 0 ). (7.20)
This algebra is the q-Onsager algebra discussed in the introduction.
7.3.
Determining C ij (c) in the case i ∈ I \ X and j ∈ X. In this case Lemma 7.2 implies that C ij (c) = 0 unless i = τ (i). Moreover, the expansion (7.5) needs to be extended to include terms of weight α i + α j in the second tensor factor.
Lemma 7.7. Assume i ∈ I \ X, j ∈ X, and τ (i) = i. Then there exists W ij ∈ M + X which is independent of c, such that
∆(B i ) = B i ⊗ K −1 i + 1 ⊗ F i + c i Z i ⊗ E i K −1 i + c i W ij K j ⊗ ad(E j )(E i )K −1 i + Υ for some Υ ∈ M X U 0 Θ ′ ⊗ γ>α i γ =α i +α j U + γ K −1 i .
Proof. For β ∈ Q let π β : U q (g) → U q (g) β denote the projection onto the corresponding weight space. Relation (7.6) implies that
(id ⊗ π αi+αj )∆(ad(Z i )(E i )) ∈ U + K j K i ⊗ ad(E j )(E i )
With this observation the claim follows in the same way as Lemma 7.2.
Theorem 7.8. Assume that i ∈ I \X and j ∈ X. The elements C ij (c) = Π(C ij (c)) from Theorem 7.1 are given by the following formulas. Case 1: a ij = 0.
C ij (c) = 0. (7.21)
Case 2: a ij = −1.
C ij (c) = δ i,τ (i) c i 1 q i − q −1 i q 2 i B j Z i − Z i B j + q i + q −1 i q j − q −1 j W ij K j . (7.22) Case 3: a ij = −2. C ij (c) = δ i,τ (i) c i q i − q −1 i q 2 i [3] qi B i B j − (q 2 i + 2)B j B i Z i (7.23) − Z i (2 + q −2 i )B i B j − [3] qi B j B i −δ i,τ (i) q i − q −1 i q j − q −1 j (q i + q i ) 2 [3] qi B i c i W ij K i .
Proof. If a ij = 0 then Y = B i B j − B j B i and hence Y = 0 if j ∈ X by (5.20). Cases 2 and 3 follow from Lemma 7.7 by direct computation in a similar way as the corresponding cases in Theorem 7.4 follow from Lemma 7.2. One has to keep in mind, however, that in the present case Z i does in general not q-commute with B j . As i = τ (i) the elements Z i and B i still commute.
Example 7.9. Consider g = sl 4 (K) with X = {1, 3} and τ = id. In this case I ns = ∅ and hence only the standard quantum symmetric pair coideal subalgebra B c exists. It is generated by
M X = K(q) E 1 , F 1 , K ±1 1 , E 3 , F 3 , K ±3
3 and the element
B 2 = F 2 − c 2 T 1 T 3 (E 2 )K −1 2 = F 2 − c 2 ad(E 3 E 1 )(E 2 )K −1 2 . To determine C 21 (c) one calculates ∆(B 2 ) =B 2 ⊗ K −1 2 + 1 ⊗ F 2 − c 2 (1 − q −2 ) 2 E 1 E 3 ⊗ E 2 K −1 2 − c 2 (1−q −2 )E 3 K 1 ⊗ ad(E 1 )(E 2 )K −1 2 − c 2 (1−q −2 )E 1 K 3 ⊗ ad(E 3 )(E 2 )K −1 2 − c 2 K 1 K 3 ⊗ ad(E 1 E 3 )(E 2 )K −1 2 which implies Z 2 = −(1 − q −2 ) 2 E 1 E 3 , W 21 = −(1 − q −2 )E 3 .
By (7.22) one obtains
C 21 (c) = −q −1 (q − q −1 ) 2 c 2 F 1 E 1 E 3 − q −2 c 2 K −1 1 E 3 − c 2 K 1 E 3 . 7.4.
References to Letzter's constructions. For finite dimensional g, the above method to determine the coefficients C ij (c) was devised in [Let03, Section 7] and the result corresponding to Theorem 7.4 is stated in [Let03, Theorem 7.1]. By reference to the finite list of cases in [Ara62, p. 32/33], Letzter's formulas become simpler than those given here, even for a ij = −1, −2. The necessity to include terms of weight α i + α j in the second tensor factor of ∆(B i ) in the calculation of C ij (c) for j ∈ X seems to have been overlooked in [Let03]. As Example 7.9 above shows, the formulas in [Let03, Theorem 7.1(iv)] are only valid for j / ∈ X.
The center of quantum symmetric pair coideal subalgebras
The center Z(B c,s ) of the algebra B c,s can be described in full generality. There are two distinct cases. Recall that the Cartan matrix A is always assumed to be indecomposable. If g = g(A) is infinite dimensional then the center Z(B c,s ) is trivial, as will be shown in Subsection 8.2. If g is finite dimensional then the center of a slight extension of B c,s was determined in [KL08]. For completeness, and as the conventions of [KL08] differ from those of the present paper, we summarize the structure of the center for the finite case in the brief Subsection 8.3. In both cases, the description of the integrable part of U q (g ′ ) due to [JL92], [JL94] provides an essential ingredient in the proof. 8.1. Integrability of B c,s -invariant elements. For any left U q (g ′ )-module M define its integrable part I(M ) by
I(M ) = {m ∈ M | ∀i ∈ I ∃k ∈ N such that E k i m = 0 = F k i m}.
Observe that I(M ) is a U q (g ′ )-submodule of M . We will be in particular interested in the integrable part of U q (g ′ ) considered as a module over itself via the left adjoint action. In this case one has by [JL92, 6.4], [Jos95, 7.1.6] the relation
I(U q (g ′ )) = λ∈−R + ad(U q (g ′ ))(K −λ ) (8.1) where R + = Q ∩ 2P + with P + = {λ ∈ P | λ(h i ) ≥ 0 ∀i ∈ I}. Recall that a U q (g ′ )- module M is called a weight module if there exists a direct sum decomposition M = µ∈P M µ
such that for all m ∈ M µ one has
K i m = q (αi,µ) m, E i m ∈ M µ+αi , F i m ∈ M µ−αi .
Finally, for any U q (g ′ )-module M define the subspace of B c,s -invariant elements by
M Bc,s = {m ∈ M | bm = ε(b)m ∀b ∈ B c,s }.
To obtain the following result we adapt the proof of [Let97,Lemma 4.4] to the present setting.
Proposition 8.1. Let M be a U q (g ′ )-weight module.
(1) If M Bc,s = {0} then there exists v ∈ M such that F i v = 0 for all i ∈ I.
(2) M Bc,s ⊆ I(M ).
Proof. Assume m ∈ M Bc,s and write m = µ∈P m µ with m µ ∈ M µ . Choose µ ∈ P minimal such that m µ = 0. Then B i m = ε(B i )m implies that F i m µ = 0 for all i ∈ I. This proves (1). Next we show that for all µ ∈ P there exists k ∈ N such that F k i m µ = 0. Indeed, otherwise choose λ ∈ P minimal such that F k i m λ = 0 for all k ∈ N. By minimality, for all µ < λ there exists k µ ∈ N such that F kµ i m µ = 0. Hence B n i m µ ∈ ν>µ−kµαi M ν if µ < λ. For large n this implies that B n i m has M λ−nαicomponent F n i m λ = 0. This is a contradiction to B n i m = ε(B i ) n m. In the same way one shows that for all µ ∈ P there exists k ∈ N such that E k i m µ = 0. To this end one has to replace the generators B i in the above argument by elements C i = E i + C i with C i ∈ U − U 0′ . Such elements exist in B c,s as can be seen via the adjoint action of M − X on B i K i for i ∈ I \ X. This completes the proof of (2). 8.2. The center of B c,s for A of infinite type. We now want to determine the center Z(B c,s ) of the algebra B c,s . Observe first that In view of Equation (8.1), to determine Z(B c,s ) one hence has to investigate the subspaces (ad(U q (g ′ ))(K −λ )) Bc,s of U q (g ′ ) for λ ∈ R + .
Z(B c,s ) = {z ∈ B c,s | ad(b)(z) = ε(b)z ∀b ∈ B c,s },Lemma 8.2. Assume that g = g(A) is infinite dimensional. If λ ∈ R + satisfies λ(h i ) = 0 for some i ∈ I then (ad(U q (g ′ ))(K −λ )) Bc,s = {0}.
Proof. Let V (λ/2) denote the integrable left U q (g ′ )-module of highest weight λ/2. Its dual space V (λ/2) * is a lowest weight U q (g ′ )-module of lowest weight −λ/2 with left action given by
(uf )(v) = f (S(u)v) for all f ∈ V (λ) * , v ∈ V (λ), u ∈ U q (g ′ ).
By [Jos95, Proof of Lemma 7.1.15 (iii)] one has an isomorphism of left U q (g ′ )modules With the above lemma one can show that the center of B c,s is trivial unless A is of finite type.
ad(U q (g ′ ))(K −λ ) ∼ = V (λ/2) ⊗ V (λ/2) * .
Theorem 8.3. Assume that g is infinite dimensional. Then Z(B c,s ) = K(q)1.
Proof. Assume that z ∈ Z(B c,s ). By relations (8.1) and (8.2) one can write z = λ∈R + z λ with z λ ∈ (ad(U q (g ′ ))(K −λ )) Bc,s .
By Lemma 8.2 the relation z λ = 0 implies that λ(h i ) = 0 for all i ∈ I. But in this case ad(U q (g ′ ))(K −λ ) = K(q)K −λ and hence z is a linear combination of the elements K −λ for λ ∈ R + with λ(h i ) = 0 for all i ∈ I. Applying the coproduct to z one may assume that z = K −λ for some λ ∈ R + with λ(h i ) = 0 for all i ∈ I. For each such λ, however, one has Θ(λ) = −τ (λ) and hence K λ ∈ U 0 Θ ′ implies that λ = 0. 8.3. The center of B c,s for A of finite type. If g is of finite dimensional then it is convenient to augment U 0′ to the group algebra of the weight lattice and to work over the field K(q 1/2 ). The resulting extensionǓ q (g) of U q (g ′ ) hence has generators E i , F i and K λ for all i ∈ I and λ ∈ P . It is sometimes called the simply connected quantized enveloping algebra of g. Relation (8.1) simplifies to
I(Ǔ q (g)) = λ∈P + ad(U q (g))(K −2λ )
Similarly, one extends B c,s to the algebraB c,s generated by U 0 Θ = K(q 1/2 ) K λ | λ ∈ P, Θ(λ) = λ and by M X and the elements B i defined by (5.1) for all i ∈ I \ X as before. Define a subset P + Θ of the set of dominant integral weights by P + Θ = {λ ∈ P + | Θ(λ) = λ + w 0 λ − w X λ} where w 0 denotes the longest element of the Weyl group W . The following theorem summarizes the main results of [KL08] in the conventions of the present paper.
Theorem 8.4. Assume that g is finite dimensional.
(1) Z(B c,s ) = λ∈P + Θ Z(B c,s ) ∩ ad(U q (g))(K −2λ ). (2) dim Z(B c,s ) ∩ ad(U q (g))(K −2λ ) = 1 if λ ∈ P + Θ , 0 else.
(3) The center Z(B c,s ) is polynomial ring in rank(k) variables.
The projection P −2λ from Section 5.2 satisfies P −2λ (I(Ǔ q (g))) = ad(U q (g))(K −2λ ). By Lemma 5.9 this proves (1), see also [KL08,footnote p. 318]. The proof of Property (2) is more involved and takes up most of [KL08]. Property (3) follows from (1) and (2) and an analysis of the set P + Θ , see [KL08, Section 9].
Equivalence of coideal subalgebras
In this section we investigate equivalence of quantum symmetric pair coideal subalgebras in the sense of the following definition.
Definition 9.1. Let C and D be right coideal subalgebras of a Hopf algebra H. We say that C is equivalent to D if there exists a Hopf algebra automorphism ϕ of H such that ϕ(C) = D. In this case we write C ∼ D.
To obtain more automorphisms, during this section, we work with U q (g ′ ) defined over the algebraic closure K(q). Accordingly, we extend the set of parameters and define
C = {c ∈ (K(q) × ) I\X | c i = c τ (i) if τ (i) = i and (α i , Θ(α i )) = 0} S = {s ∈ K(q) I\X | s i = 0 ⇒ (i ∈ I ns and a ij ∈ −2N 0 ∀j ∈ I ns \ {i})}.
Recall from Subsection 6.3 that we have fixed a subset I * ⊂ I \X containing exactly one element of each τ -orbit in I \ X. As we will see, any B c,s is equivalent to a B d,s ′ for some s ′ ∈ S and d in the subset
D = {d ∈ (K(q) × ) I\X | d i = 1 if τ (i) = i or (α i , Θ(α i )) = 0 or i / ∈ I * } (9.1) of (K(q) × ) I\X .
9.1. Equivalence under the action ofH. Let B and B ′ be coideal subalgebras of U q (g ′ ). Refining the notation introduced in Definition 9.1 we write BH ∼ B ′ if there exists x ∈H such that Ad(x)(B) = B ′ . Moreover, define an equivalence relation on S by
s S ∼ s ′ ⇐⇒ s i = ±s ′ i for all i ∈ I ns . if s = (s i ) i∈I\X and s ′ = (s ′ i ) i∈I\X .
For the following result the fact that U q (g ′ ) is defined over K(q) is necessary because one needs to take square roots.
i ∈ I \ X such that τ (i) = i choose d i ∈ K(q) such that d 2 i = c i . Define x ∈H by x(α i ) = 1 if τ (i) = i or i ∈ X and x(α i ) = d −1 i else. Then Ad(x)(B i ) = Ad(x)(F i + c i θ q (F i K i )K −1 i + s i K −1 i ) = d i (F i + θ q (F i K i )K −1 i + s i d i K −1 i )
for all i with τ (i) = i, and Ad(x) leaves M X , U 0 Θ ′ , and B j with τ (j) = j invariant.
Hence B c,sH ∼ B c ′ ,s ′ where c ′ ∈ C and s ′ ∈ S are given by
c ′ i = 1 if τ (i) = i, c i else, and s ′ i = s i d i ∀ i ∈ I ns .
Assume now that τ (j) = j and (α j , Θ(α j )) = 0. By Lemma 5.3 we have (α j , α τ (j) ) = 0 and Θ(α j ) = −α τ (j) and therefore α j (h l ) = 0 = α j (ρ ∨ X ) for all l ∈ X. Hence
B j = F j − c j E τ (j) K −1 j and B τ (j) = F τ (j) − c τ (j) E j K −1 τ (j) . As c ∈ C one ob- tains c j = c τ (j) . Define x ∈H by x(α j ) = c −1 j and x(α i ) = 1 if i = j. Then Ad(x)(B j ) = c j (F j − E τ (j) K −1 j ), Ad(B τ (j) ) = F τ (j) − E j K −1 τ (j) and Ad(x)(B i ) = B i if i = j.
(2) Assume that s, s ′ ∈ S satisfy s ∼ s ′ . Define x ∈H by
x(α i ) = 1 if i / ∈ I ns or (i ∈ I ns and s i = s ′ i ), −1 else.
One sees immediately that Ad(x)(B d,s ) = B d,s ′ holds for all d ∈ D.
For the converse implication we make use of the following fact which is a consequence of Proposition 6.1:
( * ) Let d = (d i ) i∈I\X ∈ D, s = (s i ) i∈Ins ∈ S, and c i ∈ K(q) × , t i ∈ K(q) for some i ∈ I * . If F i +c i θ q (F i K i )K −1 i +t i K −1 i ∈ B d,d i = d ′ i = 1 and θ q (F i K i ) = −E i and hence Ad(x)(B ′ i ) = x(−α i ) F i − x(2α i )E i K −1 i + x(α i )s ′ i . Property ( * ) implies that x(α i ) 2 = 1 and x(α i )s ′ i = s i . Hence s i = ±s ′ i . Next assume that i ∈ I * with τ (i) = i. By definition of D we have in particular d τ (i) = 1 = d ′ τ (i) . It suffices to show that B i = B ′ i .
To this end one calculates
Ad(x)(B ′ i ) = x(−α i )F i + x(−Θ(α i ))d ′ i θ q (F i K i )K −1 i = x(−α i )[F i + x(α i + w X α τ (i) )d ′ i θ q (F i K i )K −1 i ]
. By ( * ) the above relation implies that
d i = x(α i + w X α τ (i) )d ′ i . (9.2) Analogously one obtains with d τ (i) = 1 = d ′ τ (i) the relation Ad(x)(B ′ τ (i) ) = x(−α τ (i) )[F τ (i) + x(α τ (i) + w X α i )θ q (F i K i )K −1 i ] and hence x(α τ (i) + w X α i ) = 1. The relation α i − α τ (i) ∈ Q Θ implies that α i + w X α τ (i) = α τ (i) + w X α i . The desired relation d i = d ′ i now follows from (9.2).
Remark 9.3. The definition of D suggests the investigation of the set
I D = {i ∈ I * | τ (i) = i, (α i , Θ(α i )) = 0}.
For simple g of finite type it was observed in [Let02, Section 7, Variation 1] that I D is empty unless the restricted root system corresponding to the involution θ is of nonreduced type BC. In the latter case I D contains precisely one element. For g = g(A) of affine type with indecomposable A the set I D contains at most two elements. This can be seen by direct inspection of the affine Dynkin diagrams in [Kac90,pp. 54,55]. As an example with |I D | = 2 consider g = sl 4 (C) with X = ∅ and τ = (01)(23).
9.2.
Equivalence under the action of Aut Hopf (U q (g ′ )). It is possible that B d ∼ B d ′ for d, d ′ ∈ D with d = d ′ , even for finite dimensional, simple g. Let τ ′ ∈ Aut(A) be a diagram automorphism which commutes with τ and which leaves X invariant.
By Proposition 9.2.(1) one has τ ′ (B d )H ∼ B d ′ for some d ′ ∈ D. As the following example shows it is possible that d ′ = d.
Example 9.4. For g = sl 3 (C), X = ∅, and τ = (12) with I * = {1} the standard quantum symmetric pair coideal subalgebra B d is generated by the elements
F 1 − dE 2 K −1 1 , F 2 − E 1 K −1 2 if d = (d, 1). Hence τ (B d ) is generated by the elements F 1 − E 2 K −1 1 and F 2 − dE 1 K −1 2 . Now apply Ad(x) where x(α 1 ) = d −1 and x(α 2 ) = 1. The subalgebra Ad(x) • τ (B d ) is generated by the elements F 1 − d −1 E 2 K −1 1 , F 2 − E 1 K −1 2
and hence coincides with B d ′ where d ′ = (d −1 , 1).
The above example can immediately be generalized to the following statement.
Proposition 9.5. Assume that g is finite dimensional and simple. Let d, d ′ ∈ D and denote their only nontrivial entry (if any) by d i and d ′ i , respectively. Then
B d ∼ B d ′ if and only if d i = d ′ i or d −1 i = d ′ i .
In the symmetrizable Kac-Moody case there may be more diagram automorphisms which commute with τ and leave X invariant. Hence one obtains additional Hopf algebra automorphisms which identify B d and B d ′ for different d, d ′ ∈ D.
Example 9.6. As in Remark 9.3 consider g = sl 4 (C) with X = ∅ and τ = (01)(23). By definition, the corresponding quantum symmetric pair coideal subalgebra B d for d = (d 0 , 1, d 2 , 1) is generated by elements
B 0 = F 0 −d 0 E 1 K −1 0 , B 1 = F 1 −E 0 K −1 1 , B 2 = F 2 −d 2 E 3 K −1 2 , B 3 = F 3 −E 2 K −1 3 . Consider the diagram automorphism τ ′ = (02)(13). One has τ ′ (B d ) = B d ′ where d ′ = (d 2 , 1, d 0 , 1).
To describe equivalence classes of quantum symmetric pair coideal subalgebras in general, define
Aut(A, X) τ = {σ ∈ Aut(A, X) | σ • τ = τ • σ}.
Via Proposition 9.2 the action of Aut(A, X) τ on the set of quantum symmetric pair coideal subalgebras induces an action of Aut(A, X) τ on the set D × S/ S ∼. By Theorem 3.2 any automorphism of U q (g ′ ) which maps a quantum symmetric pair coideal subalgebra corresponding to (X, τ ) to another such coideal subalgebra, is of the form Ad(x) • σ form some x ∈H and σ ∈ Aut(A, X) τ . This implies the following parametrization of equivalence classes of quantum symmetric pair coideal subalgebras corresponding to the admissible pair (X, τ ).
Theorem 9.7. There is a one-to-one correspondence between the equivalence classes of quantum symmetric pair coideal subalgebras of U q (g ′ ) corresponding to the admissible pair (X, τ ) and the Aut(A, X) τ -orbits in D × S/ S ∼.
9.3. References to Letzter's constructions. For finite dimensional g, the parameter sets D and S appear implicitly in [Let02, Variants 1 and 2] and explicitly in [Let03, Section 2]. It was asked at the end of [Let02, Variant 1] if B d and B d ′ for d, d ′ ∈ D can be isomorphic as algebras. Example 9.4 shows that they can even be isomorphic via a Hopf algebra automorphism of U q (g). It remains an interesting question, however, if they can be isomorphic as algebras if d and d ′ belong to different Aut(A, X) τ -orbits. The main interest in equivalence of quantum symmetric pair coideal subalgebras stems for the classification result [Let99, Theorem 5.8], [Let02,Theorem 7.5] which states that all right coideal subalgebras of U q (g) which specialize to U (k) and satisfy a maximality condition (explained in 10.5) are equivalent to a quantum symmetric pair coideal subalgebra. As indicated in Remark 5.7, it would be interesting to extend this result to the setting of symmetrizable Kac-Moody algebras. Then Theorem 9.7 would provide a parametrization up to Hopf algebra automorphism of all coideal subalgebras of U q (g ′ ) which can reasonably be considered as quantum analogs of U (k ′ ).
Specialization
In the present section we consider the limit of quantum symmetric pair coideal subalgebras as q tends to 1. This is done using non-restricted specialization as outlined in [CK90,1.5]. We follow the presentation in [HK96]. All through this section we abbreviate k = K(q).
10.1. Specialization of θ q (X, τ ). We recall specialization of the k-algebra U q (g ′ ) following [HK96, 3.3, 3.4]. Let A = K[q] (q−1) denote the localization of the polynomial ring K[q] with respect to the prime ideal generated by q − 1. For any i ∈ I define
(K i ; 0) q = K i − 1 q − 1 . The A-form U ′ A of U q (g ′ ) is defined to be the A-subalgebra of U q (g ′ ) generated by the elements E i , F i , K ±1
i , and (K i ; 0) q for all i ∈ I. Consider K as an A-module via evaluation at 1. The algebra
U ′ 1 = K ⊗ A U ′ A
is called the specialization of U q (g ′ ) at q = 1. For any x ∈ U ′ A we denote its image in U ′ 1 by x. The following fact is well known and a minor variation on [HK96, Theorem 3.4.9].
Theorem 10.1. There exists an isomorphism of algebras
U ′ 1 → U (g ′ ) such that E i → e i , F i → f i , and (K i ; 0) q → ǫ i h i . Let now φ : U q (g ′ ) → U q (g ′ ) be a k-linear map. We say that φ is specializable if φ(U ′ A ) ⊆ U ′ A . In this case φ induces a K-linear map id ⊗ φ| U ′ A : U ′ 1 → U ′ 1 .
Via Theorem 10.1 one thus obtains a K-linear map φ : U (g ′ ) → U (g ′ ). We say that φ specializes to φ. Observe that if k-linear maps f, g : U q (g ′ ) → U q (g ′ ) are both specializable then so is f • g and f • g = f • g. We now apply specialization to the quantum involution θ q (X, τ ).
Proposition 10.2. For any admissible pair (X, τ ) the quantum involution θ q (X, τ ) specializes to θ(X, τ ).
Proof. Clearly both τ and ω are specializable and specialize to the corresponding automorphisms of U (g ′ ). By the explicit formulas given in [Lus94, 37.1.3] the Lusztig automorphism T i is also specializable for any i ∈ I. It specializes to the automorphism T i : U (g ′ ) → U (g ′ ) given by
e i → −f i , f i → −e i , h → r i (h), e j → ad(e i ) −aij (−a ij )! (e j ), f j → ad(f i ) −aij (−a ij )! (f j ), for j = i.
One checks by an sl 2 -argument that T i = Ad(m i )| U(g ′ ) . Hence T X specializes to Ad(m X )| U(g ′ ) . Now the theorem follows from the fact observed above that specialization is compatible with composition of maps.
We give an immediate application of the above proposition. We call a pair of multi-indices (c, s) ∈ (k × ) I\X × k I\X specializable if c i , s i ∈ A and c i (1) = 1 for all i ∈ I \ X.
Corollary 10.3. Let (c, s) ∈ (k × ) I\X × k I\X be specializable. The generators B i of B c,s belong to U ′ A and satisfy B i = f i + θ(f i ) + s i for all i ∈ I \ X.
Proof. Let i ∈ I \X. As F i K i ∈ U ′ A the above proposition gives θ q (F i K i )K −1 i ∈ U ′ A and θ q (F i K i )K −1 i = θ(f i ).
Together with the assumptions on c i and s i this implies
that B i ∈ (B c,s ) A and B i = f i + θ(f i ) + s i .
10.2. Properties of A-modules. The ring A is a principal ideal domain. We begin by recalling two facts about principal ideal domains which will be repeatedly used in the sequel. The first statement of the following proposition is a consequence of the fundamental theorem of finitely generated modules over principle ideal domains. The second statement can be found in [Eis95, Corollary 6.3].
Proposition 10.4. Let R be principal ideal domain.
(1) Any finitely generated, torsion-free module over R is free.
(2) An R-module is torsion-free if and only if it is flat.
As a first application one obtains the following result.
Lemma 10.5. Let x ∈ U ′ A . Then x = 0 if and only if x ∈ (q − 1)U ′ A . Proof. Consider the short exact sequence 0 → (q − 1)A → A → A/(q − 1)A ∼ =K → 0 of A-modules. By the above proposition U ′ A is a flat A-module. Hence, tensoring by U ′ A , one obtains an exact sequence 0 → (q − 1)A ⊗ A U ′ A ∼ =(q−1)U ′ A → U ′ A → K ⊗ A U ′ A → 0.
This proves the lemma.
Let W be a k-vector space. All A-submodules of W are torsion-free. Hence, for any A-submodule M of W the map M → M ⊗ A k is injective. As a consequence of the above proposition one obtains the following Lemma which will be used in the Subsection 10.3 to verify triangular decompositions over A analog to those in Section 6.
Lemma 10.6. Let W be a k-vector space and let M and M ′ be A-submodules of W . Let M k and M ′ k denote the k-vector subspace of W generated by M and M ′ , respectively. Then the following hold:
(1) The map ι M :
M ⊗ A k → M k is an isomorphism. (2) The map ι M,M ′ : M ⊗ A M ′ → M k ⊗ k M ′ k is injective. Proof. (1) Assume that m = N i=1 m i ⊗ a i ∈ ker(ι M ). LetM denote the A-N ′ i=1 b i ⊗ a ′ i for some a ′ i ∈ k. There exists n ∈ N such that (q − 1) n a ′ i ∈ A for all i = 1, . . . , N ′ . Hence m ∈ ker(ι M ) implies that 0 = ι M (m(q − 1) n ) = N ′ i=1 b i a ′ i (q − 1) n .
By linear independence of the set {b i | i = 1, . . . , N ′ } one obtains a ′ i (q − 1) n = 0 and hence m = 0. This proves that ι M is injective, and surjectivity holds by construction.
(2) Part (1) implies that
M k ⊗ k M ′ k ∼ = M ⊗ A ⊗k ⊗ k k ⊗ A M ′ ∼ = M ⊗ A k ⊗ A M ′ ∼ = (M ⊗ A M ′ ) ⊗ A k.
The A-module M is torsion free and hence flat by Proposition 10.4. Therefore the
map M ⊗ A M ′ → M ⊗ A (M ′ ⊗ A k) ∼ = (M ⊗ A M ′ ) ⊗ A k is injective. This proves that ι M,M ′ is injective.
10.3. Specialization and triangular decompositions. For any subspace W ⊂ U q (g ′ ) we define W A = W ∩ U ′ A and W = K ⊗ A W A ⊂ U ′ 1 . By [HK96, Proposition 3.3.3] the multiplication map yields an isomorphism
U + A ⊗ A U 0 A ′ ⊗ A U − A ∼ = U ′ A (10.1)
analogous to the triangular decomposition (3.6). By the following theorem all triangular decompositions from Section 6 also hold true over A. Set A × = A ∩ k × and recall the definition of the subspace B c,s,J given in (6.3).
Theorem 10.7. Let (c, s) ∈ C × S be specializable. The multiplication maps give the following isomorphisms of A-modules.
(
1) (U 0 Θ ′ ) A ⊗ A (U ′ Θ ) A ∼ = U 0 A ′ , (2) (V + X ) A ⊗ A (M + X ) A ∼ = U + A , (3) U + A ⊗ A U 0 A ′ ⊗ A (B c,s,J ) A ∼ = U ′ A , (4) (M + X ) A ⊗ A (U 0 Θ ′ ) A ⊗ A (B c,s,J ) A ∼ = (B c,s ) A , (5) (V + X ) A ⊗ A (U ′ Θ ) A ⊗ A (B c,s ) A ∼ = U ′ A . Proof.
Injectivity follows in all five cases from Lemma 10.6 and from the triangular decompositions in Section 6. It remains to prove surjectivity.
(1) U 0 A ′ is the A-subalgebra of U 0′ generated by the elements
K i , K −1 i , (K i ; 0) q (10.2)
for all i ∈ I. If i ∈ X or i ∈ I * then the elements (10.2) belong to (U 0 Θ ′ ) A and (U ′ Θ ) A , respectively. Hence in this case, the elements (10.2) are in the image of the multiplication map. If i ∈ I \ (I * ∪ X) then
K i K −1 τ (i) , K −1 i K τ (i) ∈ (U 0 Θ ′ ) A and K ±1 τ (i) ∈ (U ′ Θ ) A .
Hence K i and K −1 i lie in the image of the multiplication map. Finally, the relation
(K i ; 0) q = (K i K −1 τ (i) ; 0) q − K i (K −1 τ (i)
; 0) q shows that in this case (K i ; 0) q also lies in the image of the multiplication map.
(2) U + A is the A-subalgebra of U q (g ′ ) generated by all E i for i ∈ I. If i ∈ X then
E i ∈ (M + X ) A . If i / ∈ X then E i ∈ (V + X ) A .
Hence, all E i lie in the image of the multiplication map.
(3) Recall the filtration F * of U − defined at the beginning of Subsection 6.1. Via the triangular decomposition (3.6) the filtration F * extends to a filtration of U q (g ′ ) by
F n (U q (g ′ )) = U + ⊗ U 0′ ⊗ F n (U − ).
We show by induction on n that F n (U q (g ′ )) A lies in the image of the multiplication map. For n = 0 this holds true because (10.1) implies that
F 0 (U q (g ′ )) A = U + A ⊗ A U 0 A ′ ⊗ A A.
For y ∈ F n (U q (g ′ )) A , again by (10.1), there exist
u J ∈ U + A ⊗ A U 0 A ′ such that y − J∈I n ∩J u J F J ∈ F n−1 (U q (g ′ )) A . Hence y − J∈I n ∩J u J B J ∈ F n−1 (U q (g ′ )) A .
By induction hypothesis y hence lies in the image of the multiplication map (3).
(4) Let b ∈ (B c,s ) A . By Proposition 6.2 one has b = J∈J a J B J for some a J ∈ M + X U 0 Θ ′ . Let m ∈ N 0 be maximal such that a J = 0 for some J ∈ J with |J| = m. By the triangular decomposition (10.1) one can write
b = J∈J ,|J|≤m b J F J for some b J ∈ (U + U 0′ ) A . As B i − F i ∈ (U + U 0′ ) A one obtains a J = b J if |J| = m. Hence a J ∈ (M + X U 0 Θ ′ ) A for all J ∈ J with |J| = m. By induction on m one obtains a J ∈ (M + X U 0 Θ ′ ) A for all J ∈ J .
(5) Surjectivity follows by inserting (4) into the left hand side of (5) and using (3) together with (1) and (2).
10.4. Specialization of B c,s . We now want to show that for specializable (c, s) ∈ C × S the quantum symmetric pair coideal subalgebra B c,s specializes to U (k ′ ). To this end, let g + X and h ′ θ denote the Lie subalgebras of g ′ generated by the sets {e i | i ∈ X} and {h i + θ(h i ) | i ∈ I}, respectively. One verifies that
M + X = U (g + X ) U 0 Θ ′ = U (h ′ θ ). (10.3)
With this observation, Theorem 10.7.(4) gives the following result.
Theorem 10.8. Let (c, s) ∈ C × S be specializable. Then B c,s = U (k ′ ).
Proof. As (c, s) is specializable, Corollary 10.3 implies that f i + θ(f i ) + s i ∈ B c,s for all i ∈ I \ X. As F i ∈ B c,s one has f i ∈ B c s for all i ∈ X. Moreover, g + X ⊂ B c,s and h ′ θ ⊂ B c,s by (10.3). By Corollary 2.9 this proves that U (k ′ ) ⊆ B c,s . Conversely, Theorem 10.7.(4) and (10.3) imply that B c,s ⊆ U (k ′ ).
10.5. A maximality condition. Observe that U + A is generated by all ordered monomials in the generators E i , i ∈ I, as an A-module. There are only finitely many such ordered monomials of any given weight. Hence, for any finite dimensional subspace V ⊂ U + one obtains that V A is a submodule of a finitely generated Amodule. As A is Noetherian this implies that V A is finitely generated. Similarly one shows that if V is a finite dimensional subspace of U 0′ then V A is finitely generated. Combining these observations with the triangular decomposition 10.1 one obtains the following result.
Lemma 10.9. Let V ⊂ U q (g ′ ) be a finite dimensional subspace. Then V A is a finitely generated, free A-submodule of U ′ A . The above lemma and Theorem 10.7.(5) imply the following result.
Lemma 10.10. Let (c, s) ∈ C × S be specializable and u ∈ U q (g ′ ). Then ku + B c,s A (B c,s ) A is a finitely generated, free A-module.
Proof. If u ∈ B c,s then there is nothing to show. Otherwise, choose finite dimensional subspaces V ⊂ V + X , T ⊂ U ′ Θ , and D ⊂ B c,s such that u ∈ V ⊗ T ⊗ D with respect to the quantum Iwasawa decomposition. We may assume that the unit 1 of U q (g ′ ) is contained in both V and T . Set V + = V ∩ ker ε and T + = T ∩ ker ε. Then the subspace ku + B c,s is contained in the direct sum
(V + ⊗ T ⊗ D) (k ⊗ T + ⊗ D) B c,s .
Observe that V A = (V + ) A ⊕ A and T A = (T + ) A ⊕ A. By Theorem 10.7.(5) this implies that (ku + B c,s ) A is contained in
((V + ) A ⊗ A T A ⊗ A D A ) (A ⊗ A (T + ) A ⊗ A D A ) (B c,s ) A .
Hence one obtains an injective A-module homomorphism
ku + B c,s A (B c,s ) A → ((V + ) A ⊗ A T A ⊗ A D A ) (A ⊗ A (T + ) A ⊗ A D A ).
By Lemma 10.9 the A-module on the right hand side is finitely generated and free. Hence, by Proposition 10.4, so is ku + B c,s A (B c,s ) A as A is Noetherian.
Theorem 10.11. Let W be a vector subspace of U q (g ′ ) which contains B c,s for some specializable (c, s) ∈ C × S.
If W = U (k ′ ) then W = B c,s .
Proof. Assume that there exists u ∈ W \ B c,s . By Lemma 10.10 the nonzero A- 10.6. References to Letzter's constructions. Specialization is a major theme in both [Let99] and [Let02]. For finite dimensional g, Theorems 10.8 and 10.11 are stated as [Let99,Theorem 4.8] and [Let02,(7.24), (7.25)]. The additional assumption that C is an algebra made in [Let99,Theorem 4.8] and [Let02, (7.25)] seems not to be necessary for the proof. The fact that the triangular decompositions from Section 6 also hold for the A-forms, Theorem 10.7, is not made explicit in Letzter's papers. Instead she uses a process of 'rescaling' and 'subtracting', see [Let99,proof of Theorem 4.8] and [Let02,before (7.23)]. The fact that this process terminates seems to boil down to the properties of the principal ideal domain A stated in Proposition 10.4. A proof of Lemma 10.5 in a similar spirit can be found in [Let99, Theorem 2.5]. As already indicated in Subsection 9.3, Letzter's analysis proceeds one step beyond the present paper with the classification theorem [Let99, Theorem 5.8], [Let02,Theorem 7.5]. It is to be expected that a rigorous proof of this result in the Kac-Moody case will involve the triangular decompositions of A-forms given in Theorem 10.7.
module N A = ku+B c,s A (B c,s ) A is free. Let v ∈ ku+B c,
Twisted quantum loop algebras of the second kind
In this section and the next we will apply the theory developed so far to construct two classes of quantum symmetric pair coideal subalgebras which in special cases have been considered previously in the literature. The present section is devoted to quantum versions of twisted loop algebras of the second kind which are defined in general in Subsection 11.1. It is then shown, in Subsection 11.2, how the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba in [MRS03] appear as examples of such twisted quantum loop algebras. 11.1. Symmetric loop algebras and their quantization. Let g be a finitedimensional simple Lie algebra over K and let θ : g → g be an involutive automorphism. Assume that rank(g) = n and that I = {1, 2, . . . , n}. As before write A = (a ij ) i,j∈I to denote the Cartan matrix of g. By Theorem 2.7 the involution θ is determined up to conjugation by an admissible pair (X, τ ). Letĝ = g ⊗ K[t, t −1 ] ⊕ Kc ⊕ Kd be the corresponding untwisted affine Kac-Moody algebra as defined in [Kac90,7.2]. SetÎ = {0, 1, . . . , n} and recall that the Cartan matrix ofĝ is given by = (a ij ) i,j∈Î . Here a 00 = 2 and a i0 = a 0i = −α max (h i ) for all i ∈ I where α max denotes the highest root in the finite root system corresponding to g. Letĥ andb denote the standard Cartan and Borel subalgebra ofĝ, respectively. Consider the involutive automorphismθ :ĝ →ĝ given byθ
(x ⊗ t n ) = θ(x) ⊗ t −n ,θ(c) = −c,θ(d) = −d.
(11.1) Letτ :Î →Î be the map given byτ | I = τ andτ (0) = 0. It follows from the explicit description ofθ that the induced involutionΘ :ĥ * →ĥ * is given byΘ = −w X •τ . Hence (ĥ,b) is a split pair forθ in the sense of Definition A.2 andθ is an involutive automorphism of the second kind. The preceding discussion implies the following lemma.
Lemma 11.1. Let (X, τ ) be an admissible pair for g and θ = θ(X, τ ).
(1) There existsτ ∈ Aut(Î, X) such thatτ (0) = 0 andτ (i) = τ (i) for all i ∈ I.
(2) The pair (X,τ ) is admissible forĝ.
(3) The automorphismθ :ĝ →ĝ given by (11.1) is the involutive automorphism of the second kind corresponding to the admissible pair (X,τ ).
By the above lemma, all the results of this paper can be applied to the involutive automorphismθ corresponding to the admissible pair (X,τ ). In particular, for any c ∈ C and s ∈ S one obtains the corresponding quantum symmetric pair coideal subalgebra B c,s of U q (g ′ ).
The Lie algebraĝ ′ = g ⊗ K[t, t −1 ] ⊕ Kc has a one-dimensional center spanned by c. The Cartan matrix ofĝ is of rank n. There exist uniquely determined integers b j ∈ N, j = 0, 1, . . . , n, with b 0 = 1, given in [Kac90,p. 54], such that n j=0 b j a ij = 0 for all i ∈Î.
The above relation implies that α i ( n j=0 b j ǫ j h j ) = 0 for all i ∈Î and therefore n i=0 b j ǫ j h j spans the center of g ′ . The quotient
L(g) =ĝ ′ /Kc ∼ = g ⊗ K[t, t −1 ]
is the loop algebra corresponding to g. The involutionθ ofĝ ′ induces an involutive automorphismθ L : L(g) → L(g) and the invariant Lie subalgebra k ′ L = {x ∈ L(g) |θ L (x) = x} is isomorphic to k ′ becauseθ(c) = −c.
Consider the quantum analog of the central element c given by K c = n j=0 K bj j . The Hopf algebra U q (L(g)) defined by
U q (L(g)) = U q (g ′ )/(K c − 1)U q (g ′ )
is a q-analog of the universal enveloping algebra of the loop algebra L(g). Let π : U q (g ′ ) → U q (L(g)) denote the canonical projection. We call the image π(B c,s ) a twisted quantum loop algebra (of the second kind). It is a right coideal subalgebra of U q (L(g)). By the following Lemma the twisted quantum loop algebra π(B c,s ) is isomorphic to B c,s as an algebra.
Lemma 11.2. For any c ∈ C and s ∈ S one has (K c − 1)U q (g ′ ) ∩ B c,s = ∅.
Proof. By the quantum Iwasawa decomposition (Proposition 6.3) and the fact that
K c − 1 is central in U q (g ′ ), it suffices to show that (K c − 1)U ′ Θ ∩ U 0 Θ ′ = ∅.
Any element of U ′ Θ can be written as a Laurent polynomial in K 0 with coefficients in
K(q) K ±1 i | i ∈ I * \ {0} .
The product of any such a Laurent polynomial with (K c − 1) will always depend on K 0 and therefore does not belong to U 0 Θ ′ .
Remark 11.3. A remark on terminology is in order. Besides (11.1) there is another way to extend the involution θ of g toĝ. Indeed, consider the involutive automorphismθ 1 :ĝ →ĝ given bŷ
θ 1 (x ⊗ t n ) = θ(x) ⊗ (−t n ),θ 1 (c) = c,θ 1 (d) = d. (11.2)
The automorphismθ 1 is of the first kind. Formula (11.2) also defines an involution of the loop algebra L(g). The existing literature on twisted loop algebras is concerned with the Lie subalgebra of L(g) fixed underθ 1 , mostly in the case when θ is a diagram automorphism. It seems natural to say that the subalgebra of L(g) fixed byθ 1 is a twisted loop algebra of the first kind. The terminology for twisted loop algebras then reflects the terminology for involutions of Kac-Moody algebras.
In [MRS03] Molev, Ragoucy, and Sorba use the name twisted q-Yangians for quantum analogs of U (k ′ L ). As pointed out in the introduction of [MRS03], however, Ol'shanskiȋ's original definition of twisted Yangians involves involutions of the first kind [Ol'92]. For this reason we prefer to call the quantum analogs of U (k ′ L ) twisted quantum loop algebras of the second kind.
The structure theory of U q (L(g)) is very similar to the structure theory of U q (g ′ ).
In particular, one can formulate specialization for U q (L(g)). The results of Section 10 literally translate to U q (L(g)) and π(B c,s ) and one obtains the following result in the same way as Theorems 10.8 and 10.11.
Theorem 11.4. Let c ∈ C and s ∈ S be specializable.
(1) The subalgebra π(B c,s ) of U q (L(g)) specializes to U (k ′ L ).
(2) Let W be a vector subspace of U q (L(g)) which contains π(B c,s ). If W specializes to U (k ′ L ) then W = π(B c,s ). 11.2. FRT-realizations of twisted quantum loop algebras of the second kind. In this subsection we restrict to the case g = sl N (K) for n = N − 1 ∈ N. We assume that the Cartan matrix A = (a ij ) 1≤i,j≤n is given in the standard form
a ij = 2 if i = j, −1 if |i − j| = 1, 0 else.
Let σ : I → I denote the only non-trivial element in Aut(A). There exist three types of admissible pairs for g:
AI: (X, τ ) = (∅, id I ),
AII: (X, τ ) = ((1, 3, . . . , 2m − 1), id I ) if n = 2m − 1, AIII/IV: (X, τ ) = ((r + 1, r + 2, . . . , n − r), σ) for any 1 ≤ r ≤ (n + 1)/2.
The corresponding fixed Lie subalgebras of g coincide with so N (K), sp 2m (K), and sl N (K) ∩ (gl r (K) ⊕ gl N −r (K)), respectively. In [MRS03] Molev, Ragoucy, and Sorba construct q-analogs of U (k ′ L ) in the cases AI and AII. Here we recall their construction and relate it to the coideal subalgebras π(B c,s ) of U q (L(g)) constructed in the last subsection. The construction in [MRS03] uses the FRT realization of quantized enveloping algebras [FRT88] as opposed to the Drinfeld-Jimbo realization used in the present paper. The translation between the different realizations of quantized enveloping algebras is given in detail by Frenkel and Mukhin in [FM02]. Moreover, [MRS03] work with gl N (K) instead of sl N (K). Let Id N denote the identity matrix in gl N (K). The involution θ = θ(X, τ ) : sl N (K) → sl N (K) corresponding to each of the three admissible pairs described above can be extended to an involutive automorphism of gl N (K) by setting θ(λId N ) = −λId N for any λ ∈ K. Just as in the previous subsection one obtains an involutive automorphism of the loop algebra
L(gl N (K)) = gl N (K) ⊗ K[t, t −1 ] bŷ θ : L(gl N (K)) → L(gl N (K)),θ(x ⊗ t k ) = θ(x) ⊗ t −k .
Let L(gl N (K))θ denote the Lie subalgebra of L(gl N (K)) fixed underθ. For λ = 1 the Lie subalgebra of gl N (K) fixed under θ coincides with the fixed Lie subalgebra in sl N (K). However, L(gl N (K))θ contains an infinite dimensional central subspace spanned by the elements C k = Id N ⊗ t k − Id N ⊗ t −k for all k ∈ N which are not contained in L(sl N (K)).
Following [FM02,2.3] and [MRS03, Section 3] we now recall the definition of the Hopf algebra U R q (ĝl N ) which is a q-analog of the universal enveloping algebra of L(gl N (K)). For 1 ≤ i, j ≤ N let E ij denote the N × N matrix with only nonzero entry 1 in the (i, j)-th position and set
R(z, w) =(z − w) i =j E ii ⊗ E jj + (q −1 z − qw) i E ii ⊗ E ii + (q −1 − q)z i>j E ij ⊗ E ji + (q −1 − q)w i<j E ij ⊗ E ji .
By definition, the algebra U R q (ĝl N ) is generated by the coefficients L ± ij [±k] of the formal series
L ± ij (z) = ∞ k=0 L ± ij [±k]z ±k 1 ≤ i, j ≤ N subject to the relations L + ji [0] = L − ij [0] = 0 for 1 ≤ i < j ≤ N, (11.3) L − ii [0]L + ii [0] = L + ii [0]L − ii [0] = 1 for 1 ≤ i ≤ N,
R(z, w)L ± 1 (z)L ± 2 (w) = L ± 2 (w)L ± 1 (z)R(z, w), (11.4) R(z, w)L + 1 (z)L − 2 (w) = L − 2 (w)L + 1 (z)R(z, w). (11.5) Here (11.4) and (11.5) are equations in U R q (ĝl N )[[z, z −1 , w, w −1 ]] ⊗ End(K(q) N ) ⊗2 and
L ± 1 (z) = N i,j=1 L ± ij (z) ⊗ E ij ⊗ Id N , L ± 2 (z) = N i,j=1 L ± ij (z) ⊗ Id N ⊗ E ij .
The relations (11.4) and (11.5) can be written more explicitly as
(q −δij z − q δij w)L ± ia (z)L ± jb (w) + (q −1 − q)(zδ i>j + wδ i<j )L ± ja (z)L ± ib (w) (11.6) = (q −δ ab z − q δ ab w)L ± jb (w)L ± ia (z) + (q −1 − q)(zδ a<b + wδ a>b )L ± ja (w)L ± ib (z) and (q −δij z − q δij w)L + ia (z)L − jb (w) + (q −1 − q)(zδ i>j + wδ i<j )L + ja (z)L − ib (w) = (q −δ ab z − q δ ab w)L − jb (w)L + ia (z) + (q −1 − q)(zδ a<b + wδ a>b )L − ja (w)L + ib (z)
, see also [GM10, (2.42)-(2.44)]. In particular, if one collects coefficients of z 0 w 1 then one obtains the relations
q δij L ± ia [0]L ± jb [0] − q δ ab L ± jb [0]L ± ia [0] = (q − q −1 )(δ b<a − δ i<j )L ± ja [0]L ± ib [0] (11.7) and q δij L + ia [0]L − jb [0] − q δ ab L − jb [0]L + ia [0] (11.8) = (q − q −1 ) δ a>b L − ja [0]L + ib [0] − δ i<j L + ja [0]L − ib [0] . The algebra U R q (ĝl N )
is a Hopf algebra with coproduct given by
∆(L ± ij (z)) = N k=1
L ± ik (z) ⊗ L ± kj (z) (11.9) and antipode S(L ± ij (z)) = L ± (z) −1 . In [MRS03] the authors define Y tw q (o N ) to be the subalgebra of U R q (ĝl N ) generated by the coefficients s ij [−k] of the entries of the matrix S(z) = (s ij (z)) = L − (z)L + (z −1 ) t .
More precisely, one considers the power series in z −1 given by
s ij (z) = N a=1 L − ia (z)L + ja (z −1 ) for 1 ≤ i, j ≤ N and defines s ij [−k] by s ij (z) = ∞ k=0 s ij [−k]z −k .
Similarly, in the case N = 2m, consider
S(z) = L − (z)GL + (z −1 ) t
where G denotes the N × N -matrix given by
G = q m k=1 E 2k−1,2k − m k=1 E 2k,2k−1 .
Molev, Ragoucy, and Sorba define Y tw q (sp 2m ) to be the subalgebra of U R q (ĝl N ) generated by the coefficients s ij [−k] of the entries of the matrix S(z) and the elements s i,i+1 [0] −1 for i = 1, 3, . . . , 2m − 1. It follows from (11.9) that Y tw q (o N ) and Y tw q (sp 2m ) are left coideal subalgebras of U R q (ĝl N ). One can perform specialization for U R q (ĝl N ) in the same way as for U q (ĝ) and U q (L(g)), see [MRS03, Section 3]. The following statement is made in [MRS03, Corollaries 3.5, 3.12].
Lemma 11.5. The subalgebras Y tw q (o N ) and Y tw q (sp 2m ) of U R q (ĝl N ) specialize to U (L(gl N (K))θ) for the involution θ corresponding to the admissible pairs of type AI and AII, respectively.
To relate the algebras Y tw q (o N ) and Y tw q (sp 2m ) to twisted quantum loop algebras (of the second kind) as defined in the previous subsection we use the fact that the Hopf algebra U q (L(sl N (K))) can be embedded into U R q (ĝl N ).
Proposition 11.6 ([FM02, Lemma 3.8]). The following formulas define an embedding of Hopf algebrasÎ : U q (L(sl N (K))) → U R q (ĝl N ):
I(E 0 ) = (−q) N (q −1 − q) −1 L − 1N [−1]L + N N [0], I(F 0 ) = (−q) −N (q − q −1 ) −1 L − N N [0]L + N 1 [1], I(E i ) = (q −1 − q) −1 L − i+1,i [0]L + ii [0], I(F i ) = (q − q −1 ) −1 L − ii [0]L + i,i+1 [0], I(K i ) = L + ii [0]L − i+1,i+1 [0] (i = 1, . . . , N − 1).
In the following we will suppress the symbolÎ and consider U q (L(sl N (K))) as a Hopf subalgebra of U R q (ĝl N ). For the admissible pairs of type AI and AII one has S = {0} and hence the corresponding twisted quantum loop algebras are parametrized by elements c ∈ C.
We are now in a position to establish the desired relation between the algebras Y tw q (o N ) and Y tw q (sp 2m ) defined above and the twisted quantum loop algebras π(B c ) in the cases AI and AII, respectively. As Y tw q (o N ) and Y tw q (sp 2m ) are left coideal subalgebras we use the antipode S to turn the right coideal subalgebra π(B c ) into a left coideal subalgebra. The following theorem is the main result of this section.
Theorem 11.7. (1) Let (X, τ ) be of type AI, set c = (c 0 , c 1 , c 3 , . . . , c N −1 ) = (q −2(N −1) , q 2 , q 2 , . . . , q 2 ), and let π(B c ) ⊂ U q (L(sl N (K))) be the corresponding twisted quantum loop algebra. Then S(π(B c )) = Y tw q (o N ) ∩ U q (L(sl N (K))).
(2) Let (X, τ ) be of type AII with N = 2m, set c = (c 0 , c 2 , c 4 , . . . , c N −2 ) = (q −2N +7 , q 7 , q 7 , . . . , q 7 )
and let π(B c ) ⊂ U q (L(sl N (K))) be the corresponding twisted quantum loop algebra. Then S(π(B c )) = Y tw q (sp 2m ) ∩ U q (L(sl N (K))).
Proof.
(1) It follows from the definition of the A-form U R q (ĝl N ) A of U R q (ĝl N ) given by [MRS03,(3.8), (3.9)] that U q (L(sl N (K))) A ⊂ U R q (ĝl N ) A . Hence Lemma 11.5 implies that Y tw q (o N ) ∩ U q (L(sl N (K))) specializes to a subalgebra of
U (L(sl N (K))) ∩ U (L(gl N (K))θ) = U (k ′ L )
where as in Subsection 11.1 we write k ′ L = {x ∈ L(sl N (K)) |θ(x) = x}. In the following we will show that
S(π(B c )) ⊆ Y tw q (o N ) ∩ U q (L(sl N (K))) (11.10)
Specialization is compatible with the antipode S. Hence S(π(B c )) specializes to U (k ′ L ). By (11.10) this implies that Y tw q (o N ) ∩ U q (L(sl N (K))) also specializes to
U (k ′ L ). Hence W = S −1 (Y tw q (o N ) ∩ U q (L(sl N (K)))
) is a subspace of U q (L(sl N (K))) which contains π(B c ) and specializes to U (k ′ L ). Theorem 11.4 implies that W = π(B c ).
It remains to verify the inclusion (11.10). To this end one uses Proposition 11.6 and calculates
s 1N [−1] = N a=1 L − 1a [0]L + N a [1] + L − 1a [−1]L + N a [0] (11.3) = L − 11 [0]L + N 1 [1] + L − 1N [−1]L + N N [0] = (q − q −1 )(−q) N (K 0 F 0 − q −2N E 0 ) = −(q − q −1 )(−q) N −2 S(F 0 − q −2(N −1) E 0 K −1 0 ) (11.11)
Similarly, for i = 1, . . . , N −1, one calculates
s i+1,i [0] = N a=1 L − i+1,a [0]L + ia [0] (11.3) = L − i+1,i [0]L + ii [0] + L − i+1,i+1 [0]L + i,i+1 [0] = −(q − q −1 )(E i − K i F i ) = −(q − q −1 )q −2 S(F i − q 2 E i K −1 i ). (11.12)
Relations (11.11) and (11.12) show that S(π(B i )) ∈ Y tw q (o N ) for all generators B i of B c . Hence (11.10) is verified which completes the proof of part (1) of the theorem.
(2) By the same argument as in the proof of part (1) it suffices to show that S(π(B c )) ⊆ Y tw q (sp 2m ) ∩ U q (L(sl N (K))). We first show that M X is contained in Y tw q (sp 2m ). Indeed, for i = 1, 3, . . . , 2m − 1 one has
s i,i+1 [0] = qL − ii [0]L + i+1,i+1 [0] = qK −1 i s ii [0] = qL − ii [0]L + i,i+1 [0] = q(q − q −1 )F i , s i+1,i+1 [0] = qL − i+1,i [0]L + i+1,i+1 [0] = −q(q − q −1 )E i K −1 i . As the inverse of s i,i+1 [0]
is by definition also contained in Y tw q (sp 2m ) one obtains M X ⊂ Y tw q (sp 2m ). Without loss of generality we now restrict to the case m = 2.
Recall that
B 2 = F 2 − c 2 ad(E 3 E 1 )(E 2 )K −1 2 , B 0 = F 0 − c 0 ad(E 1 E 3 )(E 0 )K −1 0 , and define B ′ 2 = ad(F 3 )(B 2 K 2 )K −1 2 , B ′ 0 = ad(F 1 )(B 0 K 0 )K −1 0 . The algebra π(B c ) is generated by M X and B ′ 0 , B ′ 2 . Hence, it suffices to show that S(B ′ i ) ∈ Y tw q (sp 4 ) for i = 0, 2. One calculates S(B ′ 2 ) = q 2 [F 2 , F 3 ] q K 2 − c 2 q −2 [E 2 , E 1 ] q −1 K −1 1 , (11.13) S(B ′ 0 ) = q 2 [F 0 , F 1 ] q K 0 − c 0 q −2 [E 0 , E 3 ] q −1 K −1 3 (11.14)
where [a, b] v = ab − v ba. Proposition 11.6 and Relations (11.7), (11.8) imply that
[F 2 , F 3 ] q = −q(q − q −1 ) −1 L − 22 [0]L + 24 [0], [E 2 , E 1 ] q −1 = q −1 (q − q −1 ) −1 L − 31 [0]L + 11 [0]
. Entering these relations into (11.13) one obtains
S(B ′ 2 ) = − q 4 q − q −1 L − 33 [0]L + 24 [0] + c 2 q −7 L − 31 [0]L + 22 [0] .
The assumption c 2 = q 7 gives S(B ′ 2 ) = − q 3 q−q −1 s 32 [0] and hence S(B ′ 2 ) ∈ Y tw q (sp 4 ). We next determine S(B ′ 0 ). Collecting coefficients of z −1 w in (11.6) with two minus signs as superscripts one obtains the relations Using the above relations, the relation L +
44 [0]L − 43 [0] = q −1 L − 43 [0]L + 44 [0]
, Proposition 11.6, and the fact that N = 4 is even, one calculates
[E 0 , E 3 ] q −1 = q N −1 (q − q −1 ) −1 L − 13 [−1]L + 33 [0]. (11.15)
Similarly, collecting coefficients of z 0 w 2 in (11.6) with two plus signs as superscripts one obtains the relations
L + 12 [0]L + 41 [1] − L + 41 [1]L + 12 [0] = (q − q −1 )L +[F 0 , F 1 ] q = −q −N +1 (q − q −1 ) −1 L − 44 [0]L + 42 [1]. (11.16)
Entering (11.15) and (11.16) into (11.14) one obtains
S(B ′ 0 ) = − q −N +4 q − q −1 L − 11 [0]L + 42 [1] + c 0 q 2N −7 L − 13 [−1]L + 44 [0] .
The assumption c 0 = q −2N +7 gives S(B ′ 2 ) = − q −N +3 q−q −1 s 14 [−1] and hence S(B ′ 0 ) ∈ Y tw q (sp 4 ). Hence all generators of π(B c ) are contained in Y tw q (sp 2m ) which completes the proof of the theorem.
Remark 11.8. An FRT-construction of a family of twisted quantum loop algebras of type AIII/IV was given in [GM] depending on one parameter apart from q. It is to be expected that the analog of the above theorem holds for this family and the corresponding twisted quantum loop algebras π(B c,s ) considered in Subsection 11.1. Moreover, for type AIII/IV Theorem 9.7 provides a family of nonequivalent quantum symmetric pair coideal subalgebras π(B c,s ) depending on two parameters apart from q. It would be interesting to discover the second parameter in the constructions of [GM].
Remark 11.9. By [MRS03, Section 4.1] both Y tw q (o N ) and Y tw q (sp 2m ) contain central subalgebras which are polynomial rings in infinitely many variables. To treat both cases simultaneously we write Y tw q to denote either of Y tw q (o N ) and Y tw q (sp 2m ), we denote the central polynomial subalgebra in each case by K(q)[C N ], and, as before, we let π(B c ) be the corresponding twisted quantum loop algebra. By Theorem 8.3 the algebra S(π(B c )) has trivial center. It would be interesting to know if the multiplication map
S(π(B c )) ⊗ K(q)[C N ] → Y tw q
is bijective. This seems natural as
L(gl N (K))θ ∼ = L(sl N (K))θ ⊕ C N where C N denotes the infinite dimensional central subspace of L(gl N (K))θ spanned by the elements C k = Id N ⊗ t k − Id N ⊗ t −k for all k ∈ N.
11.3. References to Letzter's constructions. For finite dimensional g, the strategy of the proof of Theorem 11.7 was employed in [Let99, Section 6] to show that the coideal subalgebras constructed by Noumi and Sugitani in [Nou96], [NS95] coincide with the quantum symmetric pair coideal subalgebras constructed in [Let99].
Quantized GIM Lie algebras
As a second application of the theory developed in this paper we construct quantum group analogs of generalized intersection matrix (GIM) Lie algebras. These Lie algebras are generalizations of Kac-Moody Lie algebras for Cartan matrices which also allow positive off-diagonal entries. They were originally introduced by Slodowy in his study of the deformation theory of singularities [Slo84], [Slo86]. In Section 12.1 we define GIM Lie algebras. In Section 12.2 their quantum group analogs are constructed in full generality. Special examples of quantized GIM Lie algebras were previously constructed by Y. Tan in [Tan05]. The motivation of that paper was the apparent similarity of their defining relations to those of the quantum toroidal Lie algebras considered by Ginzburg, Kapranov, and Vasserot in [GKV95, Section 3].
12.1. Generalized intersection matrix algebras.
Definition 12.1. A generalized intersection matrix (GIM) is an integral matrix A = (a ij ) i,j∈I for some finite set I such that
a ii = 2, a ij > 0 ⇔ a ji > 0, a ij < 0 ⇔ a ji < 0
for all i, j ∈ I. A generalized intersection matrix A is called symmetrizable if there exists a diagonal matrix D = diag(ǫ i | i ∈ I) with coprime entries ǫ i ∈ N such that DA is symmetric.
Associated to any generalized intersection matrix A is a GIM Lie algebra L(A) which is given explicitly in terms of generators an relations in [Slo84], [Slo86]. Here, however, L(A) will be defined using an observation by Berman [Ber89, Proposition 2.1] which identifies L(A) with the Lie subalgebra of a Kac-Moody algebra fixed under an involution. To this end one associates to A a generalized Cartan matrix C(A) as follows. Let I be an identical copy of I with elements denoted by i for all i ∈ I and defineÎ = I ∪ I. Then C(A) = (c ij ) i,j∈Î where c ii = c i i = 2 for all i ∈ I and
c ij = c i j = a ij if a ij ≤ 0, c ij = c i j = 0 if a ij > 0 and i = j, c ij = 0 if a ij ≤ 0 or i = j, c ij = c i j = −a ij if a ij > 0 and i = j.
The definition of C(A) is most easily understood in terms of Dynkin diagrams as explained in [Ber89, Section 2]. If A is symmetrizable with diagonal matrix D, then so is C(A) with diagonal matrix D ⊕ D. The generalized intersection matrix A is called unoriented if C(A) is indecomposable. By construction, there exists σ ∈ Aut(Î) given by σ(i) = i and σ(i) = i for all i ∈ I. As σ 2 = idÎ , the map θ = θ(∅, σ) = σ • ω defines an involutive Lie algebra automorphism of g(C(A)).
Definition 12.2. Let A be an unoriented GIM and g = g(C(A)). The Lie subalgebra L(A) of g(C(A)) fixed under the involution θ is called the GIM Lie algebra corresponding to A.
12.2. Quantized GIM algebras. Quantum symmetric pairs provide an immediate quantum analog of the above notion of a GIM Lie algebra. Retain the notations of the previous subsection. As the entries of C(A) satisfy c ii = c ii = 0 for all i ∈ I, the parameter set C defined by (5.9) forÎ and (X, τ ) = (∅, σ) becomes C = {c ∈ K(q)Î | c i = c i for all i ∈ I} = K(q) I .
Similarly, the set S defined by (5.11) is trivial in this case. Hence there exist only standard quantum symmetric pair coideal subalgebras corresponding to the involution θ = θ(∅, σ) of g(C(A)).
Definition 12.3. Let A be an unoriented, symmetrizable GIM, and θ = θ(∅, σ) the corresponding involution of g(C(A)) and c ∈ C = K(q) I . The corresponding quantum symmetric pair coideal subalgebra B c of U q (g(C(A)) ′ ) is called a quantum GIM algebra of A and is denoted by U q (L(A)) c .
By definition, U q (L(A)) c is hence the subalgebra of U q (g(C(A)) ′ ) generated by the elements
B i = F i − c i E i K −1 i , B i = F i − c i E i K −1 i , K i K −1 i , K i K −1 i
for all i ∈ I. By Theorem 7.1 and Theorem 7.4 it is straightforward to obtain a presentation of U q (L(A)) c in terms of generators and relations. Theorem 7.3 implies that if c ij = 0 for i, j ∈Î then σ(i) / ∈ {i, j} and therefore C ij (c) = 0. Hence the quantum Serre relations for U q (L(A)) c can have nonzero lower order terms only if c ij = 0.
The noncommutative polynomial F ij (x, y) was defined in (3.1) only for a specified generalized Cartan matrix. To express the quantum Serre relations for U q (L(A)) c in terms of A, introduce noncommutative polynomials −k F i in two variables x, y defined for any i ∈ I and any k ∈ N 0 by −k F i (x, y) = 1+k n=0 (−1) n 1 + k n qi x 1+k−n yx n .
In the case that A itself is a generalized Cartan matrix one hence has aij F i (x, y) = F ij (x, y). For c ∈ C let G q (A) c denote the unital, associative K(q)-algebra with generators G i , G i , L i , L i for all i ∈ I and the following defining relations:
(1) L i L i = 1 = L i L i for i ∈ I,
(2) L i G j = q −aij i G j L i , L i G j = q aij i G j L i for all i, j ∈ I, (3) If a ij = 0 then [G i , G j ] = [G i , G j ] = 0 and [G i , G j ] = δ ij c i L i − L i q i − q −1 i
(4) If a ij < 0 then aij F i (G i , G j ) = 0 = aij F i (G i , G j ) and [G i , G j ] = 0.
(5) If a ij > 0 then −aij F i (G i , G j ) = 0 = −aij F i (G i , G j ) and [G i , G j ] = 0 = [G i , G j ]. As indicated above, the following theorem is a direct consequence of Theorems 7.1 and 7.4.
Theorem 12.4. Let A be an unoriented, symmetrizable GIM and c ∈ C. There is a uniquely determined algebra isomorphism ϕ : G q (A) c → U q (L(A)) c such that ϕ(L i ) = K i K −1 i , ϕ(G i ) = B i , ϕ(G i ) = B i for all i ∈ I.
Remark 12.5. For two-fold affinizations of Cartan matrices of type ADE a quantized GIM algebra was introduced in [Tan05]. In this case the algebra defined in [Tan05, Definition 2.1] coincides with the algebra G q (A) c for c = (1, 1, . . . , 1) up to extension of U 0 Θ ′ via a compatible minimal realization of C(A) as in Remark 5.8. In [LT12, Theorem 3.1] the above theorem is verified in this special case by explicit computation. Moreover, in [Tan05] an action of a braid group on quantized GIM algebras associated to two-fold ADE affinizations is constructed. This braid group action can be interpreted within a general framework of braid group actions on quantum symmetric pair coideal subalgebras outlined in the finite case in [KP11].
Appendix A. Classification of involutions of the second kind
In this appendix the statement of Theorem 2.7 is reduced to results explicitly stated in [KW92].
A.1. Cartan and Borel subalgebras. Let g denote an indecomposable, symmetrizable Kac-Moody algebra as in Section 2.1 with standard Cartan subalgebra h and standard Borel subalgebras b + = h ⊕ n + and b − = h ⊕ n − . More generally, any maximal ad-diagonalizable Lie subalgebra of g is called a split Cartan subalgebra of g, and any maximal completely solvable Lie subalgebra of g is called a Borel subalgebra of g [KW92, 1.25, 1.26]. As in the finite case, all Cartan subalgebra of g are conjugate under the action of the Kac-Moody group G. Borel subalgebras, however, are generally either conjugate to b + or to b − . More precisely, one has the following result. . Let b be a Borel subalgebra of g and let t be a Cartan subalgebra of g such that t ⊂ b. There exists g ∈ G such that (Ad(g)(b), Ad(g)(t)) = (b + , h) or (Ad(g)(b), Ad(g)(t)) = (b − , h).
A.2. Split pairs. As in Section 2, let θ : g → g be an involutive automorphism of the second kind. By [KW92, Lemma 5.7, Corollary 5.8 (ii) ⇒ (i)] every Borel subalgebra of g contains a θ-stable Cartan subalgebra t. As in (2.10) we denote the induced map on t * by Θ.
Definition A.2 ([KW92, 5.15]). Let t be a θ-stable Cartan subalgebra of g and let b be a Borel subalgebra of g containing t. Let ∆ and ∆ + denote the set of roots of t in g and b, respectively, and let Π denote the set of simple roots of t in b.
The pair (t, b) is called a split pair for θ if there exists a subset X of Π satisfying the following conditions:
(1) ∆ + ∩ Θ(∆ + ) = ∆ + ∩ ZX.
(2) θ| gα = id gα for all α ∈ ∆ + ∩ ZX.
Assume that (t, b) is a split pair for the involutive automorphism θ of g, and that X ⊂ Π is as in the above definition. Let, moreover, g X denote the Lie subalgebra of g generated by g ±α for all α ∈ X. By [KW92,5.16] the Lie algebra g X is finitedimensional and semisimple. Hence, by Property (2) of the above definition one has θ| gX = id gX .
The involution θ of g induces an involution of G. Let G θ denote the fix point subgroup and observe that Ad(g) • θ = θ • Ad(g) for all g ∈ G θ . The following result is the main ingredient in the proof of Theorem 2.7. (1) There exists a split pair (t, b) for θ.
(2) Assume that (t 1 , b 1 ) and (t 2 , b 2 ) are split pairs for θ such that Ad(g)(b 1 ) = b 2 for some g ∈ G. Then there exists g ′ ∈ G θ such that Ad(g ′ )(t 1 ) = t 2 and Ad(g ′ )(b 1 ) = b 2 .
A.3. Square roots. In the following we will encounter involutive automorphisms of g which we would like to commute with elements of Ad(H) or Aut(g, g ′ ). The following lemma will provide a useful tool for this purpose. We will say that a group H allows square roots if for any h ∈ H there exists an element g ∈ H such that g 2 = h.
Lemma A.4. Let θ be an involutive automorphism of g and H ⊂ Aut(g) a commutative subgroup which allows square roots and satisfies θ • H • θ = H. Then for any h ∈ H we can write
h • θ = h θ • θ h
where θ h ∈ Aut(g) is H-conjugate to θ and h θ ∈ H commutes with θ h .
Proof. For any k ∈ H define θ(k) = θ • k • θ ∈ H. Given g, h ∈ H with g 2 = h define θ h := θ(g −1 ) • θ • θ(g) and h θ := g • θ(g) = θ(g) • g. Then
h θ • θ h = g • θ(g) • θ(g −1 ) • θ • θ(g) = g • θ • θ(g) = h • θ
and on the other hand θ h • h θ = θ(g −1 ) • θ • θ(g) • θ(g) • g = θ(g −1 ) • h • θ(g) • θ = h • θ which proves the lemma.
A.4. Proof of Theorem 2.7. Let θ : g → g be an involutive automorphism of the second kind. By Theorem A.3.(1) there exists a split pair (t, b) for θ. Conjugating θ by an element of Ad(G) and using Proposition A.1 we may assume that (t, b) = (h, b + ) or (t, b) = (h, b − ). Note that (h, b + ) is a split pair for θ if and only if (h, b − ) is a split pair for θ. Hence, we may assume that (t, b) = (h, b + ).
Remark A.5. Note that the subset X ⊆ Π obtained in this way is uniquely determined by the original involution θ. Indeed, given g ∈ G with Ad(g)(h) = h and Ad(g)(b) = b, then g ∈ N G (h) represents an element w in the Weyl group W which satisfies w(∆ + ) = ∆ + and hence w(Π) = Π. But this implies that w is the unit element in W [Kac90, 3.11 b)].
As (t, b) = (h, b + ) we may identify Π with the index set I and consider X as a subset of I. As observed before Theorem A.3 the subset X thus obtained is of finite type. Hence we may use notation and results from Subsection 2.3. The automorphism Ad(m X ) commutes with θ because Ad(e i ) and Ad(f i ) do so for all i ∈ X. Hence the map −w X • Θ on h * , which is induced by ω • Ad(m X ) • θ, satisfies (−w X • Θ) 2 = id. (A.1) By Proposition 2.2 the automorphism ω • Ad(m X ) • θ ∈ Aut(g) leaves (h, b + ) and g X invariant. Hence there exists τ ∈ Aut(A, X) such that τ •ω •Ad(m X )•θ induces the trivial action on ∆. Hence −τ • w X = Θ and (A.1) implies
τ 2 = id. (A.2)
As τ • ω • Ad(m X ) • θ induces the trivial action on ∆ we can find s ∈H such that Ad(s) • τ • ω • Ad(m X ) • θ| g ′ = id g ′ . This relation, together with θ| gX = id gX from Subsection A.2, implies that the restriction s| QX ≡ 1 is the constant function. Hence Ad(s) commutes with Ad(m X ). Moreover, as Ad(H) is a commutative subgroup of Aut(g) which is invariant under conjugation by θ, we may apply Lemma A.4 and assume that Ad(s) and θ commute. By (A.3) this assumption implies that Ad(s) and τ • ω • Ad(m X ) commute and therefore Ad(s) also commutes with τ • ω. Hence we obtain s(α i ) = s(−α τ (αi) ) = s(α τ (αi) ) −1 (A.5) for all i ∈ I. This implies in particular that s(α i ) 2 = 1 if τ (i) = i. Now we use θ 2 = id g as well as the commutativity from above and from Proposition 2.2.(3) to obtain the relation (Ad(s)) 2 • (Ad(m X )) 2 | g ′ = (Ad(s) • τ • ω • Ad(m X )) 2 | g ′ = id g ′ (A.6) Using Proposition 2.2.(2) one obtains from (A.5) and (A.6) the relation 3) implies that there exists φ ∈ Aut(g; g ′ ) such that θ = φ • Ad(s) • τ • ω • Ad(m X ).
α i (ρ ∨ X ) ∈ Z if τ (i) = i.
As Aut(g; g ′ ) ⊂ Aut(g) is a commutative subgroup which allows square-roots (compare [KW92,4.20] for an explicit description of Aut(g; g ′ )) we may apply Lemma A.4 and assume that φ commutes with θ. But then relation (A.8) implies φ 2 = id which by [KW92,4.20] is impossible if char(K) = 0 unless φ = id g . The following proposition collects the result of the discussion up to this point.
Proposition A.6. Let θ : g → g be an involutive automorphism of the second kind. Then there exists an admissible pair (X, τ ) such that θ is Aut(g)-conjugate to θ(X, τ, s) := Ad(s) • τ • ω • Ad(m X ) for some s ∈H such that Ad(s) commutes with θ(X, τ, s).
By the discussion preceding Proposition A.6 we know that s| QX ≡ 1 and that the relations s(α j ) = s(α τ (j) ) −1 and s(α j ) 2 = (−1) αj (2ρ ∨ X ) hold for all j ∈ I \ X. Note that these relations imply s(α j ) = ±s(X, τ )(α j ) for all j ∈ I. Define u ∈H by u(α j ) := 1 if s(α j ) = s(X, τ )(α j ), i if s(α j ) = −s(X, τ )(α j ).
Then θ(X, τ ) = Ad(u) • θ(X, τ, x) • (Ad(u)) −1 which proves that θ is conjugate to an automorphism of the form θ(X, τ ) as given by (2.8).
It is straightforward to check that if two admissible pairs belong to the same Aut(A)-orbit, then the corresponding involutive automorphisms are Aut(g) conjugate. Hence it remains to show that θ(X, τ ) = φ • θ(X ′ , τ ′ ) • φ −1 (A.9) for some φ ∈ Aut(g ′ ) implies that (X, τ ) and (X ′ , τ ′ ) belong to the same Aut(A)orbit. Indeed, note that in this case both (h, b) and (φ(h), φ(b)) are split pairs for θ := θ(X, τ ). Moreover, replacing φ by φ • ω if necessary, on may assume that φ(b) and b are Ad(G)-conjugate. By Theorem A.3.(2) there exists g ∈ G θ such that φ(h) = Ad(g)(h) and φ(b) = Ad(g)(b). Hence, replacing φ by Ad(g −1 ) • φ, we may assume that φ(h) = h and φ(b) = b. (A.10) Let Θ, Θ ′ , and Φ denote the automorphisms of h * induced by θ, θ ′ := θ(X ′ , τ ′ ), and φ, respectively. Relation (A.10) implies Φ ∈ Aut(A) and by relation (A.9) one has Θ = Φ • Θ ′ • Φ −1 . Thus Φ(X ′ ) = X and
τ ′ = −w X ′ • Θ ′ = −Φ −1 • w X • Φ • Φ −1 • Θ • Φ = Φ −1 • τ • Φ
which shows that (X, τ ) and (X ′ , τ ′ ) belong to the same Aut(A)-orbit. This completes the proof of Theorem 2.7.
i ∈ I and h ∈ h. Define Out(A) = Aut(A) if A is of finite type and Out(A) = Aut(A) ∪ ωAut(A) else [KW92, 1.32].
0 and hence condition (3) is not satisfied. The pair ({2}, id I ) is not admissible for the same reason while the pair ({1, 2}, id I ) violates (2). Finally, the pair ({2}, (13)) is admissible because now condition (3) is empty.
( 5 )
5The quantum Serre relations given in [Lus94, 3.1.1.(e)].
the q i -binomial coefficient defined for instance in [Lus94, 1.3.3]. By [Lus94, Corollary 33.1.5], the quantum Serre relations in (5) can be written in the form
F ij (x,
aij −n yx n − x n yx 1−aij −n . (5.23) By (5.22) and (5.23) it suffices to show that
∆(B i ) does not explicitly depend on the parameter s i but only on c i . By the proof of the above theorem, this implies that the expression of F ij (B i , B j ) as an element of {J∈J | wt(J)<λij } M + X U 0 Θ ′ B J does not explicitly depend on s. 5.4. References to Letzter's constructions. For finite dimensional g, standard and nonstandard quantum symmetric pairs were introduced in full generality in [Let02, Variants 1 and 2, after (7.25)]. Previously, the parameters c ∈ C were implicitly included via a Hopf algebra automorphism χ which entered the definition of the automorphismθ 2 in [Let99, Theorem 3.1]. The nonstandard quantum symmetric pair coideal subalgebras seem not to be defined explicitly in that paper, however, their existence is already observed in [Let99, (5.14) and Remark 5.10]. In [Let00, Section 2] nonstandard analogs were defined for a parameter set larger than S but this was corrected at the end of [Let02, Variant 2]. The discussion in the present subsection partly follows [Let02, Section 7]. Proposition 5.2 is [Let02, Theorem 7.2] or [Let99, Corollary 4.2]. Subsection 5.3 follows the procedure outlined in [Let02, after Theorem 7.2] to describe quantum symmetric pair coideal subalgebras in terms of generators and relations. In particular, Lemma 5.14 corresponds to [Let02, Lemma 7.3] and Lemma 5.15 and Corollary 5.17 are subsumed in [Let02, Theorem 7.4] in the finite case. The proof of Proposition 5.16 follows the proof of [Let02, Theorem 7.4]. A discussion similar to Lemma 5.11 is contained in [Let02, before Lemma 7.3]. However, this discussion is significantly simpler than the proof of Lemma 5.11 because Letzter excludes nonstandard quantum symmetric pairs and uses casework which is possible in the finite case due to the classification of admissible pairs in [Ara62, p.32/33].
. 2 .
2Define a subspace of B c,s by B c,s,J = ⊕ J∈J K(q)B J . (6.3) Proposition 6.1 can be reformulated by saying that the multiplication mapU + ⊗ U 0′ ⊗ B c,s,J → U q (g ′ ) (6.4)is an isomorphism of vector spaces.6.2. A one-sided M + X U 0 Θ ′ -module basis of B c,s . By Proposition 6.1 any element in B c,s can be written as a linear combination of elements in {B J | J ∈ J } with coefficients in U + U 0′ . Actually, it is sufficient to allow coefficients from M + The set {B J | J ∈ J } is a basis of the left (or right) M + X U 0 Θ ′module B c,s . Proof. We prove the result for the left M + X U 0 Θ ′ -module B c,s . Let L ∈ I n . One can apply the quantum Serre relations (3.2) repeatedly to write F L = J∈J ,|J|=n a J F J for some a J ∈ K(q). Hence, using Corollary 5.17 and the relations in Lemma 5.15, one obtains B L − J∈J ,|J|=n a J B J ∈ m<n J∈I m M + X U 0 Θ ′ B J .One obtains B L ∈ J∈J M + X U 0 Θ ′ B J by induction on n = wt(L). This proves that the set {B J | J ∈ J } spans B c,s as a left M + X U 0 Θ ′ -module. On the other hand, the set {B J | J ∈ J } is linearly independent over U + U 0′ by Proposition 6.1. Hence it is also linearly independent over M +
6. 4 .
4References to Letzter's constructions. For finite dimensional g, Proposition 6.2 appears in the proof of [Let02, Theorem 7.4]. The quantum Iwasawa decomposition is stated along the lines of Remark 6.4 in [Let99, Theorem 4.5]. Four variants of Proposition 6.3 which interchange the order of the factors and exchange V + X with V − X appear in [Let04, Theorem 2.2]. In an earlier paper [Let97, Theorem 2.4] the quantum Iwasawa decomposition was established in the case X = ∅. 7. Generators and relations 7.1. Generators and relations for B c,s . The following theorem summarizes the results of Subsection 5.3 and is identical to [Let02, Theorem 7.4] if g is finite dimensional.
Moreover, the formal expression of the elementsCij (c) in wt(J)<λij M + X U 0 Θ ′B Jis independent of the parameter s ∈ S.
Theorem 7 . 3 .
73For any i, j ∈ I such that i / ∈ {j, τ (i), τ (j)} one has C ij (c) = F ij (B i , B j ) = 0 in B c,s .
Example 7. 6 .
6Consider the affine Lie algebraŝl 2 (K) with generalized Cartan ma-I = {0, 1}. Choose the admissible pair (X, τ ) = (∅, id).
c,s ) ⊂ I(U q (g ′ )) Bc,s . (8.2)
by [Jos95, 7.1.15 (ii)] the simple highest weight U q (g ′ )-module V (λ/2) is infinite dimensional and therefore does not contain a nonzero element annihilated by all F i for i ∈ I. Hence V (λ) ⊗ V (λ/2) * does not contain a nonzero element annihilated by all F i for i ∈ I. By Proposition 8.1.(1) one obtains (V (λ/2) ⊗ V (λ/2) * ) Bc,s = {0} which by (8.3) completes the proof of the Lemma.
Proposition 9 . 2 .
92(1) Let c ∈ C and s ∈ S. Then there exist d ∈ D and s ′ ∈ S such that B c,sH ∼ B d,s ′ .(2) Let d, d ′ ∈ D and s, s ′ ∈ S. Then B d,sH ∼ B d ′ ,s ′ if and only if d = d ′ and s S ∼ s ′ . Proof. (1) For all
submodule of M generated by the set {m i | i = 1, . . . , N }. Being finitely generated, M is a free A-module by Proposition 10.4.(1). We can choose a basis {b i | i = 1, . . . , N ′ } and write m =
s A be a representative of a basis element of N A . The assumption W = U (k ′ ) together with Theorem 10.8 imply that v = b for some b ∈ (B c,s ) A . By Lemma 10.5 one obtains v − b ∈ (q − 1)U ′ A ∩ ku + B c,s = (q − 1) ku + B c,s A . This contradicts the assumption that v represents a basis element of N A .
Theorem A.3 ([KW92, 5.19, 5.32]). Let θ : g → g be an involutive automorphism of the second kind.
τ
• ω • Ad(m X )| gX = id gX . (A.4)
and (A.7) imply that (X, τ ) is an admissible pair. Moreover, (Ad(s) • τ • ω • Ad(m X )) 2 | h = id h and hence (A.6) implies Ad(s) • τ • ω • Ad(m X ) 2 = id g (A.8) Relation (A.
* and the root system Φ is stable under W . Let Φ re = W Φ and Φ im = Φ\ Φ re denote the sets of real and imaginary roots, respectively.By [Kac90, 2.1] there exists a nondegenerate, symmetric, bilinear form (·, ·) on
h such that
s then c i = d i and t i = s i .Now write d = (d i ) i∈I\X , s = (s i ) i∈I\X and d ′ = (d ′ i ) i∈I\X , s ′ = (s ′ i ) i∈I\Xand assume that B d,s = Ad(x)(B d ′ ,s ′ ) for some x ∈H. We denote the generators of B d,s and B d ′ ,s ′ by B i and B ′ i , respectively. Assume first that i ∈ I ns . Then
Consult [FRT88, Section 3], [FM02, 2.3], [MRS03, Section 3], or [GM10, 2.3] for more details.
With these relations and Proposition 11.6 one gets42 [1]L +
11 [0]
L +
11 [0]L +
41 [1] − qL +
41 [1]L +
11 [0] = 0.
Acknowledgments. The author is grateful to N. Guay for pointing out the preprint [GM], to Y. Tan for providing the paper[LT12]ahead of print, and to J. Stokman for valuable comments. The initial work on this project was done in 2009 at the Universities of Edinburgh and Amsterdam. The results were announced at a miniworkshop on 'Generalizations of Symmetric Spaces' in Oberwolfach in
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[
"Analysis of theDΣ pentaquark molecular states with QCD sum rules",
"Analysis of theDΣ pentaquark molecular states with QCD sum rules",
"Analysis of theDΣ pentaquark molecular states with QCD sum rules",
"Analysis of theDΣ pentaquark molecular states with QCD sum rules"
]
| [
"Zhi-Gang Wang \nDepartment of Physics\nNorth China Electric Power University\n071003BaodingP. R. China\n",
"Zhi-Gang Wang \nDepartment of Physics\nNorth China Electric Power University\n071003BaodingP. R. China\n"
]
| [
"Department of Physics\nNorth China Electric Power University\n071003BaodingP. R. China",
"Department of Physics\nNorth China Electric Power University\n071003BaodingP. R. China"
]
| []
| In this article, we study theDΣc,DΣ * c ,D * Σc andD * Σ * c pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 13 in a consistent way. The present calculations support assigning the Pc(4312) to be theDΣc pentaquark molecular state with J P = 1 2 − , assigning the Pc(4380)to be theDΣ * c pentaquark molecular state with J P = 3 2 − , assigning the Pc(4440/4457) to be theD * Σc pentaquark molecular state with J P = 3 2 − or theD * Σ * c pentaquark molecular state with J P = 5 2 − . Special attentions are payed to the operator product expansion. | 10.1142/s0217751x19500970 | [
"https://arxiv.org/pdf/1806.10384v3.pdf"
]
| 247,582,902 | 1806.10384 | 01f097721d3e84591affa6df38f815d104fbc475 |
Analysis of theDΣ pentaquark molecular states with QCD sum rules
25 May 2019
Zhi-Gang Wang
Department of Physics
North China Electric Power University
071003BaodingP. R. China
Analysis of theDΣ pentaquark molecular states with QCD sum rules
25 May 2019number: 1239Mk1420Lq1238Lg Key words: Pentaquark molecular statesQCD sum rules
In this article, we study theDΣc,DΣ * c ,D * Σc andD * Σ * c pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 13 in a consistent way. The present calculations support assigning the Pc(4312) to be theDΣc pentaquark molecular state with J P = 1 2 − , assigning the Pc(4380)to be theDΣ * c pentaquark molecular state with J P = 3 2 − , assigning the Pc(4440/4457) to be theD * Σc pentaquark molecular state with J P = 3 2 − or theD * Σ * c pentaquark molecular state with J P = 5 2 − . Special attentions are payed to the operator product expansion.
Introduction
In 2015, the LHCb collaboration studied the Λ 0 b → J/ψK − p decays and observed two pentaquark candidates P c (4380) and P c (4450) in the J/ψp mass spectrum with the significances of more than 9 standard deviations [1]. acceptable solutions. More experimental data are still needed to determine the quantum numbers unambiguously. In 2016, the LHCb collaboration inspected the Λ 0 b → J/ψK − p decays for the presence of J/ψp or J/ψK − contributions with minimal assumptions about K − p contributions and obtained model-independent support for the evidences of the P + c (4380/4500) [2]. Also in 2016, the LHCb collaboration obtained additional support for the existences of the two P + c (4380/4450) in the Λ 0 b → J/ψπ − p decays [3]. There have been several possible assignments since the observations of the P c (4380) and P c (4450), such as the pentaquark molecular states [4,5,6,7,8,9,10,11] (or not the molecular pentaquark states [12]), the diquark-triquark type pentaquark states [13], the diquark-diquarkantiquark type pentaquark states [14,15,16,17], re-scattering effects [18], etc. In Table 1, we present some typical assignments in the scenario of pentaquark molecular states, in this article, we will focus on this scenario, and examine the possible molecule assignments based on the QCD sum rules.
The QCD sum rules is a powerful theoretical tool in studying the ground state hadrons [19,20,21,22]. The diquark-diquark-antiquark type hidden-charm pentaquark states have been studied in details with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 10 in a consistent way [16,17]. In Ref. [5], Chen et al study theD * Σ c and DΣ * c −D * Λ c pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 8. In Ref. [10], Chen et al construct many interpolating currents to study the meson-baryon type pentaquark molecular states with the spin J = 1 2 , 3 2 and 5 2 extensively. In Ref. [23], Azizi, Sarac and Sundu study theD * Σ c and DΣ * c −D * Λ c pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 6. In Refs. [5,10,23], also in the QCD sum rules for the tetraquark states [24], the QCD spectral densities have two energy scales, Table 1: Some typical pentaquark molecule assignments. µ = m c for the M S mass m c (m c ) and µ = 1 GeV for other input parameters. In Refs. [5,10], m c (m c ) = 1.23 GeV, while in Ref. [23], m c (m c ) = 1.27 GeV.
In Refs. [25,26], we study the diquark-antidiquark type tetraquark states and meson-meson type molecular states with the QCD sum rules by calculating the vacuum condensates up to dimension-10 in the operator product expansion in a systematic way, and explore the energy scale dependence of the hidden-charm (hidden-bottom) tetraquark states and molecular states in details for the first time, and suggest a formula
µ = M 2 X/Y /Z − (2M Q ) 2 ,(1)
with the effective heavy quark masses M Q to determine the optimal energy scales of the QCD spectral densities, which works very well for the hidden-charm (hidden-bottom) tetraquark states and molecular states [25,26,27], and hidden-charm pentaquark states [16,17]. In calculations, we take the M S masses m Q (m Q ) from the Particle Data Group [28]. In the QCD sum rules for the multiquark states, it is difficult to satisfy the pole dominance or ground state dominance, the energy scale formula µ = M 2 X/Y /Z − (2M Q ) 2 can enhance the pole contributions remarkably, and improve the convergent behaviors of the operator product expansion considerably.
In this article, we extend our previous works [16,17,25,26,27] to study the masses and pole residues of theDΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 13, and revisit the assignments of the P c (4380) and P c (4450). In calculations, we separate the contributions of the negative parity and positive parity pentaquark molecular states unambiguously, and study the hidden-charm pentaquark molecular states in three cases in details.
After the present work was finished and submitted to https://arxiv.org/, and appeared as arXiv:1806.10384, the LHCb collaboration observed a narrow pentaquark candidate P c (4312) in the J/ψp mass spectrum with the statistical significance of 7.3σ, and confirmed the P c (4450) pentaquark structure, and observed that it consists of two narrow overlapping peaks P c (4440) and P c (4457) with the statistical significance of 5.4σ [29]. The measured masses and widths are
P c (4312) : M = 4311.9 ± 0.7 +6.8 −0.6 MeV , Γ = 9.8 ± 2.7 +3.7 −4.5 MeV , P c (4440) : M = 4440.3 ± 1.3 +4.1 −4.7 MeV , Γ = 20.6 ± 4.9 +8.7 −10.1 MeV , P c (4457) : M = 4457.3 ± 0.6 +4.1 −1.7 MeV , Γ = 6.4 ± 2.0 +5.7 −1.9 MeV .(2)
The P c (4312) may be aDΣ c pentaquark molecule candidate [29,30]. We modify the assignments according to the new experimental data and add the QCD sum rules for theDΣ c pentaquark molecular state. The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of theDΣ c ,DΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion.
2 QCD sum rules for theDΣ c ,DΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states
In the following, we write down the two-point correlation functions Π(p), Π µν (p) and Π µναβ (p) in the QCD sum rules, listed in Table 2.
In general, we expect to solve the eigenequation of the QCD Hamiltonian and obtain the eigenstates and eigenvalues for the five-quark systems. By analyzing the eigenvalues and substructures of the eigenstates, we can distinguish the diquark-diquark-antiquark type pentaquark states and meson-baryon type pentaquark molecular states. However, at the present time, it is a very difficult work to solve eigenequation of the QCD Hamiltonian for the five-quark systems.
At the phenomenological side, we insert a complete set of intermediate pentaquark molecular states with the same quantum numbers as the current operators J(x), iγ 5 J(x), J µ (x), iγ 5 J µ (x), J µν (x) and iγ 5 J µν (x) into the correlation functions Π(p), Π µν (p) and Π µναβ (p) to obtain the hadronic representation [19,20], because the scattering meson-baryon states can only contribute a finite width to the pentaquark molecular states to modify the dispersion relation. After isolating the pole terms of the lowest hidden-charm pentaquark molecular states, we obtain the following Schwinger spinors [35],
s U U = ( p + M ± ) ,(16)s U µ U ν = ( p + M ± ) −g µν + γ µ γ ν 3 + 2p µ p ν 3p 2 − p µ γ ν − p ν γ µ 3 p 2 ,(17)s U µν U αβ = ( p + M ± ) g µα g νβ + g µβ g να 2 − g µν g αβ 5 − 1 10 γ µ γ α + γ µ p α − γ α p µ p 2 − p µ p α p 2 g νβ − 1 10 γ ν γ α + γ ν p α − γ α p ν p 2 − p ν p α p 2 g µβ − 1 10 γ µ γ β + γ µ p β − γ β p µ p 2 − p µ p β p 2 g να − 1 10 γ ν γ β + γ ν p β − γ β p ν p 2 − p ν p β p 2 g µα ,(18)
and p 2 = M 2 ± on the mass-shell. In this article, we choose the structures p, 1, pg µν , g µν and p (g µα g νβ + g µβ g να ), g µα g νβ +g µβ g να for the correlation functions Π(p), Π µν (p) and Π µναβ (p) respectively to study the J P = 1 where the s 0 are the continuum threshold parameters and the T 2 are the Borel parameters. We separate the contributions of the negative parity pentaquark molecular states from that of the positive parity pentaquark molecular states unambiguously.
In the following, we briefly outline the operator product expansion for the correlation functions Π(p), Π µν (p) and Π µναβ (p) in perturbative QCD. We contract the u, d and c quark fields in the correlation functions Π(p), Π µν (p) and Π µναβ (p) with Wick theorem, and obtain the results:
ΠD Σc (p) = −i ε ijk ε i ′ j ′ k ′ d 4 xe ip·x γ α γ 5 C k ′ k (x)γ 5 γ β − T r [iγ 5 C m ′ m (−x)iγ 5 U mm ′ (x)] T r γ α D jj ′ (x)γ β CU T ii ′ (x)C +T r iγ 5 C m ′ m (−x)iγ 5 U mi ′ (x)γ β CD T jj ′ (x)Cγ α U im ′ (x) ,(24)ΠD Σ * c µν (p) = i ε ijk ε i ′ j ′ k ′ d 4 xe ip·x C k ′ k (x) − T r [iγ 5 C m ′ m (−x)iγ 5 U mm ′ (x)] T r γ µ D jj ′ (x)γ ν CU T ii ′ (x)C +T r iγ 5 C m ′ m (−x)iγ 5 U mi ′ (x)γ ν CD T jj ′ (x)Cγ µ U im ′ (x) ,(25)ΠD * Σc µν (p) = −i ε ijk ε i ′ j ′ k ′ d 4 xe ip·x γ α γ 5 C k ′ k (x)γ 5 γ β − T r [γ ν C m ′ m (−x)γ µ U mm ′ (x)] T r γ α D jj ′ (x)γ β CU T ii ′ (x)C +T r γ ν C m ′ m (−x)γ µ U mi ′ (x)γ β CD T jj ′ (x)Cγ α U im ′ (x) ,(26)ΠD * Σ * c µναβ (p) = i ε ijk ε i ′ j ′ k ′ d 4 xe ip·x C k ′ k (x) − T r [γ α C m ′ m (−x)γ µ U mm ′ (x)] T r γ ν D jj ′ (x)γ β CU T ii ′ (x)C −T r [γ β C m ′ m (−x)γ µ U mm ′ (x)] T r γ ν D jj ′ (x)γ α CU T ii ′ (x)C −T r [γ α C m ′ m (−x)γ ν U mm ′ (x)] T r γ µ D jj ′ (x)γ β CU T ii ′ (x)C −T r [γ β C m ′ m (−x)γ ν U mm ′ (x)] T r γ µ D jj ′ (x)γ α CU T ii ′ (x)C +T r γ α C m ′ m (−x)γ µ U mi ′ (x)γ β CD T jj ′ (x)Cγ ν U im ′ (x) +T r γ β C m ′ m (−x)γ µ U mi ′ (x)γ α CD T jj ′ (x)Cγ ν U im ′ (x) +T r γ α C m ′ m (−x)γ ν U mi ′ (x)γ β CD T jj ′ (x)Cγ µ U im ′ (x) +T r γ β C m ′ m (−x)γ ν U mi ′ (x)γ α CD T jj ′ (x)Cγ µ U im ′ (x) ,(27)
where the U ij (x), D ij (x) and C ij (x) are the full u, d and c quark propagators respectively (S ij (x) = U ij (x), D ij (x)),
S ij (x) = iδ ij x 2π 2 x 4 − δ ij qq 12 − δ ij x 2 qg s σGq 192 − ig s G a αβ t a ij ( xσ αβ + σ αβ x) 32π 2 x 2 − 1 8 q j σ µν q i σ µν + · · · ,(28)C ij (x) = i (2π) 4 d 4 ke −ik·x δ ij k − m c − g s G n αβ t n ij 4 σ αβ ( k + m c ) + ( k + m c )σ αβ (k 2 − m 2 c ) 2 − g 2 s (t a t b ) ij G a αβ G b µν (f αβµν + f αµβν + f αµνβ ) 4(k 2 − m 2 c ) 5 + · · · , f αβµν = ( k + m c )γ α ( k + m c )γ β ( k + m c )γ µ ( k + m c )γ ν ( k + m c ) ,(29)
and t n = λ n 2 , the λ n is the Gell-Mann matrix [20,36], then compute the integrals both in the coordinate and momentum spaces to obtain the correlation functions Π(p), Π µν (p) and Π µναβ (p) therefore the QCD spectral densities ρ 1
Π 1 2 (p) = − p + M − p 2 − M 2 − + i p 2 Γ − (p 2 ) λ − 1 2 2 − p − M + p 2 − M 2 + + i p 2 Γ + (p 2 ) λ + 1 2 2 + · · · , Π 3 2 (p) = − p + M − p 2 − M 2 − + i p 2 Γ − (p 2 ) λ − 3 2 2 − p − M + p 2 − M 2 + + i p 2 Γ + (p 2 ) λ + 3 2 2 + · · · , Π 5 2 (p) = − p + M − p 2 − M 2 − + i p 2 Γ − (p 2 ) λ − 5 2 2 − p − M + p 2 − M 2 + + i p 2 Γ + (p 2 ) λ + 5 2 2 + · · · .(30)
In calculations. we observe that the zero width approximation will not impair the predictive ability significantly even for large widths [37], the scattering baryon-meson states can be neglected safely. Furthermore, from Eqs.(24)- (27), we can see that there are two heavy quark propagators and three light quark propagators in the correlation functions, if each heavy line emits a gluon and each light quark line contributes a quark pair, we obtain a operator GGūuūudd, which is of dimension 13, we should take into account the vacuum condensates at least up to dimension 13. In this article, we carry out the operator product expansion to the vacuum condensates up to dimension-13 and assume vacuum saturation for the higher dimensional vacuum condensates. We take the truncations n ≤ 13 and k ≤ 1 in a consistent way, the operators of the orders O(α k s ) with k > 1 are discarded. In previous QCD sum rules for the pentaquark molecular states, the operator product expansion was carried out up to the vacuum condensates of the dimension 8 or 6 [5,10,23], the vacuum condensates qq 3 , qg s σGq 2 , qq 2 qg s σGq and qq qg s σGq 2 were discarded. The vacuum condensates qq 2 qg s σGq and qq qg s σGq 2 , which come from the Feynman diagrams shown in Figs.1-2, play an important role in determining the Borel windows, as there appear terms of the orders O 1
T 2 , O 1 T 4 , O 1 T 6
in the QCD spectral densities, which manifest themselves at small values of the Borel parameter T 2 , we have to choose large values of the T 2 to warrant convergence of the operator product expansion and appearance of the Borel platforms. In the Borel windows, the vacuum condensates qq 2 qg s σGq and qq qg s σGq 2 play a less important role. Although the vacuum condensates qq αs π GG , qq 2 αs π GG and qq 3 αs π GG are the vacuum expectations of the operators of the order O(α s ), and they are neglected due to the small contributions of the gluon condensates in the QCD sum rules for the multiquark states [25,26,27].
Once the analytical QCD spectral densities ρ 1 j,QCD (s) and ρ 0 j,QCD (s) are obtained, we can take the quark-hadron duality below the continuum thresholds s 0 and introduce the weight functions s exp − s T 2 and exp − s T 2 to obtain the following QCD sum rules:
2M − λ − j 2 exp − M 2 − T 2 = s0 4m 2 c ds √ sρ 1 j,QCD (s) + ρ 0 j,QCD (s) exp − s T 2 ,(31)2M + λ + j 2 exp − M 2 + T 2 = s0 4m 2 c ds √ sρ 1 j,QCD (s) − ρ 0 j,QCD (s) exp − s T 2 ,(32)where j = 1 2 , 3 2 , 5 2 , ρ 0 j,QCD (s) = m c ρ 0 j,QCD (s), ρ 1 j,QCD (s) = ρ 1 0 (s) + ρ 1 3 (s) + ρ 1 4 (s) + ρ 1 5 (s) + ρ 1 6 (s) + ρ 1 8 (s) + ρ 1 9 (s) + ρ 1 10 (s) +ρ 1 11 (s) + ρ 1 13 (s) , ρ 0 j,QCD (s) = ρ 0 0 (s) + ρ 0 3 (s) + ρ 0 4 (s) + ρ 0 5 (s) + ρ 0 6 (s) + ρ 0 8 (s) + ρ 0 9 (s) + ρ 0 10 (s) +ρ 0 11 (s) + ρ 0 13 (s) ,(33)
the explicit expressions of the QCD spectral densities ρ 1/0 i (s) with i = 0, 3, 4, 5, 6, 8, 9, 10, 11 and 13 are given in the appendix.
We differentiate Eqs.(31)-(32) with respect to τ = 1 T 2 , then eliminate the pole residues λ ± 1 2 / 3 2 / 5 2 and obtain the QCD sum rules for the masses of the pentaquark molecular states,
M 2 − = − d dτ s0 4m 2 c ds √ s ρ 1 QCD (s) + ρ 0 QCD (s) exp (−τ s) s0 4m 2 c ds √ s ρ 1 QCD (s) + ρ 0 QCD (s) exp (−τ s) ,(34)M 2 + = − d dτ s0 4m 2 c ds √ s ρ 1 QCD (s) − ρ 0 QCD (s) exp (−τ s) s0 4m 2 c ds √ s ρ 1 QCD (s) − ρ 0 QCD (s) exp (−τ s) ,(35)
where ρ 1 QCD (s) = ρ 1 j,QCD (s) and ρ 0 QCD (s) = ρ 0 j,QCD (s). In numerical calculations, we observe that the masses M + of theDΣ c ,DΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states with the J P = 1 which are much larger than the correspondingD + Σ c ,D + Σ * c ,D * + Σ c andD * + Σ * c threshold holds 4.318 GeV, 4.382 GeV, 4.460 GeV and 4.524 GeV respectively. In this article, we would not pay attention to the pentaquark molecular states with positive parity, as they may be resonance states or virtual states.
Numerical results and discussions
We take the standard values of the vacuum condensates qq = −(0.24 ± 0.01 GeV) 3 , qg s σGq = m 2 0 qq , m 2 0 = (0.8 ± 0.1) GeV 2 , αsGG π = (0.33 GeV) 4 at the energy scale µ = 1 GeV [19,20,21], and choose the M S mass m c (m c ) = (1.28 ± 0.03) GeV from the Particle Data Group [28]. Furthermore, we take into account the energy-scale dependence of the input parameters, 12 25 ,
qq (µ) = qq (1GeV) α s (1GeV) α s (µ)qg s σGq (µ) = qg s σGq (1GeV) α s (1GeV) α s (µ) 2 25 , m c (µ) = m c (m c ) α s (µ) α s (m c ) 12 25 , α s (µ) = 1 b 0 t 1 − b 1 b 2 0 log t t + b 2 1 (log 2 t − log t − 1) + b 0 b 2 b 4 0 t 2 ,(36)where t = log µ 2 Λ 2 , b 0 = 33−2n f 12π , b 1 = 153−19n f 24π 2 , b 2 = 2857− 5033 9 n f + 325 27 n 2 f 128π 3
, Λ = 210 MeV, 292 MeV and 332 MeV for the flavors n f = 5, 4 and 3, respectively [28,38,39].
In this article, we study the pentaquark molecular states in three cases, (I). We evolve the input parameters to the energy scale µ = M 2 P − (2M c ) 2 to extract the masses M P with the truncation of the operator product expansion D = 13; (II). We evolve the input parameters except for m c (m c ) to the energy scale µ = 1 GeV to extract the masses M P with the truncation of the operator product expansion D = 10; (III). We evolve the input parameters except for m c (m c ) to the energy scale µ = 1 GeV to extract the masses M P with the truncation of the operator product expansion D = 13. Now we take a short digression to discuss the energy scale formula, µ = M 2 P − (2M Q ) 2 . In the heavy quark limit, the Q-quark serves as a static well potential and can combine with a light quark q to form a heavy diquark in color antitriplet, or combine with a light diquark in color antitriplet to form a baryon state in color singlet. The Q-quark serves as another static well potential, and can combine with a light diquark ε ijk q i q ′j to form a heavy triquark in color triplet, or combine with a light quark q to form a heavy meson in color singlet,
q j + Q k → ε ijk q j Q k , ε ijk q i q ′j + Q k → ε ijk q i q ′j Q k , ε ijl q i q ′j + Q k → ε lkm ε ijl q i q ′j Q k , q j + Q k → δ jk Q q ,(37)
where the i, j, k, l, m are color indexes. Then
ε ijk q j Q k + ε imnq′m Q n → compact tetraquark states , ε lkm ε ijl q i q ′j Q k + ε mnb q ′′n Q b → compact pentaquark states , Qq + q ′ Q → tetraquark molecular states , Qq + ε ijk q ′i q ′′j Q k → pentaquark molecular states .(38)
The two heavy quarks Q andQ stabilize the four-quark systems qq ′ QQ or the five quark systems qq ′ q ′′ QQ , just as in the case of the (µ − e + )(µ + e − ) molecule in QED [40]. The tetraquark (molecular) states qq ′ QQ (X, Y, Z) and pentaquark (molecular) states qq ′ q ′′ QQ (P ) are characterized by the effective heavy quark masses M Q (or constituent quark masses not as robust) and the vir-
tuality V = M 2 X/Y /Z/P − (2M Q ) 2
(or bound energy not as robust). The QCD sum rules have three typical energy scales µ 2 , T 2 , V 2 . It is natural to take the energy scales of the QCD spectral densities to be µ = V .
The effective Q-quark masses M Q embody the net effects of the complex dynamics, appear as parameters and their values are fitted by the QCD sum rules. The M Q have uncertainties, the optimal values in the diquark-antidiquark (diquark-diquark-antiquark) systems are not necessary the optimal values in the meson-meson (meson-baryon) systems. In the multiquark states consist of color singlet constituents, irrespective of the meson-meson type or meson-baryon type multiquark states, or in the multiquark states consist of color (anti)triplet constituents, irrespective of the diquark-antidiquark type or diquark-diquark-antiquark type multiquark states, the effective Qquark masses M Q should have universal values.
We fit the effective Q-quark masses M Q to reproduce the experimental masses of the Z c (3900) and Z b (10610) in the scenario of tetraquark states or molecular states [25,26], as there are controversies concerning the tetraquark and molecule assignments, then use the energy scale formula µ = M 2 X/Y /Z/P − (2M Q ) 2 to study the hidden-charm (hidden-bottom) tetraquark states and hidden-charm (hidden-bottom) pentaquark states or hidden-charm (hidden-bottom) tetraquark molecular states and hidden-charm (hidden-bottom) pentaquark molecular states.
In Ref. [26], we obtain the optimal value M c = 1.84 GeV for the tetraquark molecular states. Later, we re-checked the numerical calculations and corrected a small error involving the mixed condensates and obtained the updated value M c = 1.85 GeV [41].
In the case (I), we choose the Borel parameters T 2 and continuum threshold parameters s 0 to satisfy the following four criteria:
C1. Pole dominance at the phenomenological side; C2. Convergence of the operator product expansion; C3. Appearance of the Borel platforms; C4. Satisfying the energy scale formula, by try and error.
In the cases (II) and (III), we choose the Borel parameters T 2 and continuum threshold parameters s 0 to satisfy the three criteria, C1, C2 and C3. Now we write down the contributions of the different terms in the operator product expansion,
D(n) = s0 4m 2 c ds ρ n (s) exp − s T 2 s0 4m 2 c ds ρ(s) exp − s T 2 ,(39)
where the ρ n (s) are the QCD spectral densities for the vacuum condensates of dimension n, and the total spectral densities ρ(s) = √ sρ 1 QCD (s) + ρ 0 QCD (s). There is another definition for the D(n),
D(n) = ∞ 4m 2 c ds ρ n (s) exp − s T 2 ∞ 4m 2 c ds ρ(s) exp − s T 2 ,(40)
which enhance the contributions of the vacuum condensates of low dimension and lead to smaller Borel parameters. Such a definition only warrants the operator product expansion is convergent if all the hadron states are taken into account on the phenomenological side. In this article, we prefer the definition shown in Eq.(39) as we only take into account the ground state contributions. The contributions of the perturbative terms D(0) are usually small for the multiquark states, we approximate the continuum contributions as ρ(s)Θ(s − s 0 ), and define the pole contributions (PC) or ground state contributions as
PC = s0 4m 2 c ds √ sρ 1 QCD (s) + ρ 0 QCD (s) exp − s T 2 ∞ 4m 2 c ds √ sρ 1 QCD (s) + ρ 0 QCD (s) exp − s T 2 .(41)
In Ref. [34], we separate the contributions of the positive parity and negative parity baryon states explicitly, and study the heavy, doubly-heavy and triply-heavy baryon states with the QCD sum rules in a systematic way. In calculations, we observe that the continuum threshold parameters √ s 0 = M gr + (0.6 − 0.8) GeV can reproduce the masses of the observed heavy and doubly-heavy baryon states [28], where the subscript gr denotes the ground baryon states. The pentaquark states and pentaquark molecular states are another type baryon states considering the fractional spins 1 2 , 3 2 , 5 2 , we can take the continuum threshold parameters as √ s 0 < M P + 0.8 GeV. The resulting Borel parameters or Borel windows T 2 , continuum threshold parameters s 0 , optimal energy scales of the QCD spectral densities and pole contributions of the ground state pentaquark molecular states are shown explicitly in Table 3. From the table, we can see that the pole dominance or the C1 is satisfied in the cases (I) and (II), while in the case (III) the pole contributions are very small, less than 25%.
In the QCD sum rules for the multiquark states, we usually choose the same pole contributions as (40−60)% [16,17,25,26,27], which satisfy the pole dominance, the resulting Borel windows are small, T 2 max − T 2 min ≈ 0.4 GeV 2 . If we enlarge or narrow the pole contributions, the Borel windows are changed, the corresponding predictions are also changed. In Refs. [16,17,25,26,27], we study the tetraquark states, tetraquark molecular states and pentaquark states with the QCD sum rules in a consistent way by choosing the pole contributions (40 − 60)%, and obtain satisfactory results in assigning the exotic states. In the present work, we choose the pole contributions (40 − 60)% in the case (I) and expect to obtain reliable predictions.
In Figs.3-5, we plot the contributions of the vacuum condensates of dimension n with variations of the Borel parameters T 2 for the central values of other input parameters shown in Table 3 in the cases (I), (II) and (III), respectively. From the figures, we can see that the contributions D(n) change quickly with variations of the Borel parameters at the regions T 2 ≤ 3.0 GeV 2 , 2.6 GeV 2 and 3.3 GeV 2 in the cases (I), (II) and (III), respectively, which cannot lead to stable QCD sum rules, and the operator product expansion is not convergent, we should choose (much) larger Borel parameters T 2 . In Fig.6, we plot the absolute contributions of the vacuum condensates of dimension n for the central values of the input parameters shown in Table 3 In calculations, we observe that in the case (II), we take into account the vacuum condensates up to dimension 10, not to dimension 13, there are no terms associated with 1 T 2 , 1 T 4 , 1 T 6 , which warrant those terms manifest themselves at low T 2 and appearance of the Borel platforms, the predicted masses increase monotonously with increase of the Borel parameters. We choose small Borel windows T 2 max − T 2 min = 0.4 GeV 2 , and obtain the Borel platforms by requiring the uncertainties δMP MP induced by the Borel parameters are about 1%.
We take into account all uncertainties of the input parameters, and obtain the masses and pole residues of the J P = 1 2 − , 3 2 − and 5 2 − hidden-charm pentaquark molecular states, which are shown explicitly in Table 4 and Figs.7-12. From Table 4, we can see that the criterion C4 is satisfied in the case (I).
In Figs.7-12, we plot the masses and pole residues at much larger ranges of the Borel parameters than the Borel windows. From Figs.7-8, we can see that the predicted masses and pole residues in the case (I) decrease monotonously and quickly with increase of the Borel parameters at the region T 2 ≤ 2.0 GeV 2 , then reach small platforms and increase slowly with increase of the Borel parameters. From Figs.9-10, we can see that the predicted masses and pole residues in the case (II) increase monotonously and quickly with increase of the Borel parameters at the region T 2 < 2.6 GeV 2 , then increase slowly with increase of the Borel parameters. From Figs.11-12, we can see that the predicted masses and pole residues in the case (III) decrease monotonously and quickly with increase of the Borel parameters at the region T 2 < 3.0 GeV 2 , then decrease very slowly with increase of the Borel parameters. In all the three cases, we can define Borel platforms by requiring the uncertainties δMP MP induced by the Borel parameters are about 1%, the criterion C3 can be satisfied. The flatness of the platforms have relation (III) > (I) > (II).
In summary, in the case (I), the criteria C1, C2, C3, C4 can be satisfied; in the case (II), the criteria C1, C2, C3 can be satisfied; in the case (III), the criteria C2, C3 can be satisfied. While the convergent behaviors have relation (I) > (II) > (III) and the flatness of the platforms have relation (III) > (I) > (II).
In the case (III), if we choose small Borel parameters, the pole contributions can be enhanced, however, the convergence of the operator product expansion breaks down. On the other hand, if we choose larger continuum threshold parameters to enhance the pole contributions, we can obtain much larger masses than the total masses of the two constituents, which correspond to virtual states or resonances, not meson-baryon bound states. The masses extracted from the continuum state dominated QCD sum rules are not robust, the case (III) are not preferred.
Compared to the QCD sum rules in the case (II), the QCD sum rules in the case (I) have better convergent behaviors in the operator product expansion and more flat Borel platforms. We do not prefer the case (II) as they lead to two energy scales, µ = m c and µ = 1 GeV, in the QCD spectral densities, just like in the case of the ssqqc pentaquark states [42].
In this article, we prefer the QCD sum rules in the case (I), which support assigning the P c (4312) to be theDΣ c pentaquark molecular state with J P = 1 2 − , assigning the P c (4380) to be theDΣ * c pentaquark molecular state with J P = 3 2 − , assigning the P c (4440/4457) to be thē D * Σ c pentaquark molecular state with J P = 3 2 − or theD * Σ * c pentaquark molecular state with J P = 5 2 − , see Table 4. As the mass alone cannot identify a hadron, more experimental data are still needed to determine the P c (4312), P c (4380), P c (4440) and P c (4457) unambiguously. In other words, the QCD sum rules indicate that there maybe exist theDΣ c ,DΣ * c ,D * Σ c andD * Σ * Table 4. We have to study the two-body strong decays of the pentaquark molecular statesDΣ c ,DΣ * c ,D * Σ c ,D * Σ * c → J/ψp with the three-point QCD sum rules to assign the P c (4312), P c (4380), P c (4440) and P c (4457) in a more robust way, as we need more parameters beyond the masses to assign the P c (4312), P c (4380), P c (4440) and P c (4457) unambiguously. However, it is a difficult work to deal with the tensor (or spinor) structures in the three-point QCD sum rules for the hadronic coupling constants involving the pentaquark molecular states with the spin ≥ 3 2 . It is our next work. In Fig.13, we plot the masses of the pentaquark molecular states with variations of the Borel parameter T 2 for the central values of other input parameters in Table 3 in the case (I) with truncations of the operator product expansion D = 8, 9, 10, 11 and 13, respectively. From the figure, we can see that the predicted masses change significantly outside of the Borel windows, the higher dimensional vacuum condensates play an important role in determining the Borel windows; the regions between the two perpendicular lines are the Borel windows. Even in the Borel windows, Table 3: The truncations of the operator product expansion D, optimal energy scales µ, Borel parameters T 2 , continuum threshold parameters s 0 and pole contributions (pole) for the hiddencharm pentaquark molecular states, the energy scale µ = 1 GeV denotes the input parameters except for the m c (m c ) are taken at 1 GeV.
J P D µ(GeV) T 2 (GeV 2 ) s 0 (GeV 2 ) polē D 0 Σ + c (2455)
the predicted masses change considerably with the truncations of the operator product expansion, we should truncate the operator product expansion in a consistent way. Now we discuss the possible uncertainties originate from the energy scales in the case (I). In calculations, we observe that the predicted masses M decrease monotonously and slowly with the increase of the energy scales µ. If we choose the same continuum threshold parameters s 0 as that shown in Table 3, and take the uncertainties δµ = ±0.2 GeV and vary the Borel parameters T 2 to retain the same pole contributions as that shown in Table 3, we obtain the uncertainties δM = +0.00 −0.02 GeV, +0.00 −0.02 GeV, +0.00 −0.01 GeV and +0.00 −0.01 GeV for theDΣ c ,DΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states, respectively. In fact, if we take the uncertainties δµ = ±0.2 GeV and vary both the Borel parameters T 2 and continuum threshold parameters s 0 to retain the same pole contributions as that shown in Table 3, we can obtain the tiny uncertainties δM ≈ 0, so the uncertainties δM originate from the δµ near the optimal energy scales shown in Table 3 can be neglected.
We can define the QCD side of the QCD sum rules as
Π(µ) = s0 4m 2 c (µ) ds √ sρ 1 QCD (s, µ) + ρ 0 QCD (s, µ) exp − s T 2 .(42)
The Π(µ) evolves with the renormalization group equation, we can take into account the energyscale dependence according to the following equation,
Π(µ) = α s (µ 0 ) α s (µ) γ Π(µ 0 ) ,(43)
where the γ is the anomalous dimension of the correlation function, and we expect that the energy scale dependence can be factorized out and absorbed into the pole residue, the predicted mass M is energy scale independent, see Eq. (34). The anomalous dimensions γ for the QCD sum rules involving the massive quarks are unknown up to now [43]. We have to perform the following routine to take into account the energy scale dependence. 12 25 , qq (µ 0 ) α s (µ 0 ) α s (µ) 12 25 , qg s σGq (µ 0 ) α s (µ 0 ) α s (µ) 2 25 , (44) Table 4: The predicted masses and pole residues of the hidden-charm pentaquark molecular states. and evolve the c-quark mass and vacuum condensates to the optimal energy scales µ = 2.2, GeV, 2.4 GeV, 2.5 GeV and 2.6 GeV, respectively. In the operator product expansion, the energy scale µ separates the regions of short and long distances, the interactions at momenta p 2 > µ 2 are included in the Wilson's coefficients, while the effects at p 2 < µ 2 are absorbed into the vacuum condensates, which are energy scale dependent and can be evolved to arbitrary energy scales according to the renormalization group equation. The scale µ (normalization point) should be large enough in order to justify the calculations of the Wilson's coefficients in QCD perturbation theory. In this article, the energy scales µ = 2.2, GeV, 2.4 GeV, 2.5 GeV and 2.6 GeV are suitable. We obtain the masses M of the pentaquark molecular states through a fraction, the energy scale dependence in the numerator and denominator are canceled out to some extent, the δM induced by the δµ near the optimal energy scales are very small.
Π(µ) = Π (m c (µ), qq (µ), qg s σGq (µ)) = Π m c (µ 0 ) α s (µ) α s (µ 0 )
Conclusion
In this article, we study theDΣ c ,DΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 13 in a consistent way. In calculations, we separate the contributions of the negative parity and positive parity pentaquark molecular states unambiguously, and study the masses and pole residues of the hidden-charm pentaquark molecular states with the QCD sum rules in details. Special attentions are payed to the operator product expansion, as the predicted masses change remarkably with the truncations of the operator product expansion, we should truncate the operator product expansion in a consistent way. The present calculations support assigning the P c (4312) to be theDΣ c pentaquark molecular state with J P = 1
The Breit-Wigner masses and widths are M Pc(4380) = 4380 ± 8 ± 29 MeV, M Pc(4450) = 4449.8 ± 1.7 ± 2.5 MeV, Γ Pc(4380) = 205 ± 18 ± 86 MeV, and Γ Pc(4450) = 39 ± 5 ± 19 MeV, respectively. The preferred quantum numbers of the (P c (4380), P c (4450)) are J
QCD (s) through the dispersion relation. In Eq.(28), we retain the term q j σ µν q i comes from the Fierz re-arrangement of the q iqj to absorb the gluons emitted from other quark lines to extract the mixed condensate qg s σGq .From Eqs.(24)-(27), we can see that there are two type contributions, one contains two Tr's, one contains one Tr. The terms with two Tr's have both factorizable contributions and nonfactorizable contributions, while the terms with one Tr have only non-factorizable contributions. At the leading order, the perturbative terms have only factorizable contributions. The non-factorizable contributions play a important role in determining the pentaquark molecular states. If there are only factorizable contributions of the terms in the two Tr's, the intermediate scattering baryonmeson states will dominate the QCD sum rules. On the other hand, if we take into account both the factorizable contributions and non-factorizable contributions, the intermediate baryon-meson loops only contribute a finite imaginary part to modify the dispersion relation at the hadron side,
Figure 1 :
1The diagrams contribute to the mixed condensate qq 2 qg s σGq of dimension 11. Other diagrams obtained by interchanging of the heavy quark lines (dashed lines) or light quark lines (solid lines) are implied.
Figure 2 :
2The diagrams contribute to the mixed condensate qq qg s σGq 2 of dimension 13. Other diagrams obtained by interchanging of the heavy quark lines (dashed lines) or light quark lines (solid lines) are implied.
√
in the cases (I), (II) and (III), respectively. From the figure, we can see that the contributions of the perturbative terms D(0) are not the dominant contributions, the contributions of the vacuum condensates of dimensions 6 and 8 are very large. If we take the contributions of the vacuum condensates of dimension 6 as milestones, the contributions of the vacuum condensates |D(n)| decrease quickly with increase of the dimensions n, the operator product expansion is well convergent. The convergent behaviors have relation (I) > (II) > (III).
−
, respectively, which lie in the correspondingDΣ c ,DΣ * c ,D * Σ c andD * Σ * c thresholds, respectively, see
J
P D µ(GeV) M (GeV) λ(10 −3 GeV 6 ) Thresholds (MeV)
Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :
34567The contributions of the vacuum condensates of dimension n = 0, 3, 4, · · · with variations of the Borel parameter T 2 for central values of other input parameters in the case (I), where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c ,D * Σ c andD * Σ * c , respectively. The contributions of the vacuum condensates of dimension n = 0, 3, 4, · · · with variations of the Borel parameter T 2 for central values of other input parameters in the case (II), where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c ,D * Σ c andD * Σ The contributions of the vacuum condensates of dimension n = 0, 3, 4, · · · with variations of the Borel parameter T 2 for central values of other input parameters in the case (III), where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c ,D * Σ c andD * Σ The absolute contributions of the vacuum condensates of dimension n for central values of the input parameters in the Borel windows in the cases (I), (II) and (III), where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c ,D * Σ c andD * Σ * c , respectively. The masses of the pentaquark molecular states with variations of the Borel parameter T 2 in the case (I), where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c , D * Σ c andD * Σ * c , respectively.
Figure 8 :
8The pole residues of the pentaquark molecular states with variations of the Borel parameter T 2 in the case (I), where the A, B, C and D denote the pentaquark molecular states DΣ c ,DΣ * c ,D * Σ c andD * Σ * c , respectively.
Figure 9 :Figure 10 :
910The masses of the pentaquark molecular states with variations of the Borel parameter T 2 in the case (II), where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c , D * Σ c andD * Σ * c , respectively. The pole residues of the pentaquark molecular states with variations of the Borel parameter T 2 in the case (II), where the A, B, C and D denote the pentaquark molecular states DΣ c ,DΣ * c ,D * Σ c andD * Σ * c , respectively.
Figure 11 :
11The masses of the pentaquark molecular states with variations of the Borel parameter T 2 in the case (III), where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c , D * Σ c andD * Σ * c , respectively.
Figure 12 :Figure 13 :
1213The pole residues of the pentaquark molecular states with variations of the Borel parameter T 2 in the case (III), where the A, B, C and D denote the pentaquark molecular states DΣ c ,DΣ * c ,D * Σ c andD * Σ * c , respectively. The masses of the pentaquark molecular states with variations of the Borel parameter T 2 in the case (I) with truncations of the operator product expansion D = 8, 9, 10, 11 and 13, where the A, B, C and D denote the pentaquark molecular statesDΣ c ,DΣ * c ,D * Σ c andD * Σ * c , respectively.
2 −
2, assigning the P c (4380) to be theDΣ * c pentaquark molecular state with J P = 3 2 − , assigning the P c (4440/4457) to be thē D * Σ c pentaquark molecular state with J P = 3 2 − or theD * Σ * c pentaquark molecular state with J P = 5 2 − . The QCD sum rules indicate that there maybe exist theDΣ c ,DΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states with the J P
AppendixThe explicit expressions of the QCD spectral densities:For theDΣ c pentaquark molecular states,For theDΣ * c pentaquark molecular states,For theD * Σ c pentaquark molecular states,For theD * Σ * c pentaquark molecular states,where dydz =
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| []
|
[
"MMF: A loss extension for feature learning in open set recognition",
"MMF: A loss extension for feature learning in open set recognition"
]
| [
"Jingyun Jia \nFlorida Institute of Technology\n0000−0002, 3878−4205, 32901−, MelbourneFLUSA\n",
"− 049x \nFlorida Institute of Technology\n0000−0002, 3878−4205, 32901−, MelbourneFLUSA\n",
"Philip K Chan \nFlorida Institute of Technology\n0000−0002, 3878−4205, 32901−, MelbourneFLUSA\n"
]
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"Florida Institute of Technology\n0000−0002, 3878−4205, 32901−, MelbourneFLUSA",
"Florida Institute of Technology\n0000−0002, 3878−4205, 32901−, MelbourneFLUSA",
"Florida Institute of Technology\n0000−0002, 3878−4205, 32901−, MelbourneFLUSA"
]
| []
| The objective of open set recognition (OSR) is to classify the known classes as well as the unknown classes when the collected samples cannot exhaust all the classes. For example, the frequently emerged new malware classes require a system to classify the known classes and identify the unknown malware classes. This paper proposes a loss extension that leverages the neural network to find representations for the known classes so that the representations of the known and the unknown classes become more effectively separable. Our contributions include: First, we introduce an extension that can be incorporated into different loss functions to find more discriminative representations. Second, we show that the proposed extension can significantly improve the performances of two different types of loss functions on datasets from two different domains. Third, we show that with the proposed extension, one loss function outperforms the others in training time and model accuracy. | 10.1007/978-3-030-86340-1_26 | [
"https://arxiv.org/pdf/2006.15117v2.pdf"
]
| 220,128,140 | 2006.15117 | 11f6bc43b74be9558aad5a0c596dee7223f569e5 |
MMF: A loss extension for feature learning in open set recognition
Jingyun Jia
Florida Institute of Technology
0000−0002, 3878−4205, 32901−, MelbourneFLUSA
− 049x
Florida Institute of Technology
0000−0002, 3878−4205, 32901−, MelbourneFLUSA
Philip K Chan
Florida Institute of Technology
0000−0002, 3878−4205, 32901−, MelbourneFLUSA
MMF: A loss extension for feature learning in open set recognition
Open set recognition · Feature learning · Loss extensions
The objective of open set recognition (OSR) is to classify the known classes as well as the unknown classes when the collected samples cannot exhaust all the classes. For example, the frequently emerged new malware classes require a system to classify the known classes and identify the unknown malware classes. This paper proposes a loss extension that leverages the neural network to find representations for the known classes so that the representations of the known and the unknown classes become more effectively separable. Our contributions include: First, we introduce an extension that can be incorporated into different loss functions to find more discriminative representations. Second, we show that the proposed extension can significantly improve the performances of two different types of loss functions on datasets from two different domains. Third, we show that with the proposed extension, one loss function outperforms the others in training time and model accuracy.
Introduction
For a multinomial classification problem, the OSR problem's objective is to classify the multiple known classes while identifying the unknown classes. The OSR problem defines a more realistic scenario and has drawn significant attention in application areas such as face recognition [12], malware classification [5] and medical diagnoses [15].
In this paper, we introduce an MMF loss extension to help the existing loss functions better handle the open set scenario. The MMF extension is inspired by Extreme Value Signatures (EVS) in [17]. Borrowing from a pre-trained neural network for regular classification, EVS uses only the top K activations at one layer for calculating the distance between an instance and a class. The EVS distance function can help identify the unknown class. Instead of using a pretrained network, we directly learn more discriminative features for the known and unknown classes. We name the approach Min Max Feature (MMF) loss extension because we emphasize the features with the smallest and largest magnitudes during network training. Although the MMF extension is not a standalone loss function, it can be incorporated into different loss functions. Our contribution in Partially supported by grants from Amazon and Rockwell Collins to Philip Chan. arXiv:2006.15117v2 [cs.LG] 3 May 2021 this paper is threefold: First, we propose MMF as an extension to different types of loss functions for the OSR problem. Second, we show that MMF achieves statistically significant AUC ROC improvement when applied to two types of loss functions (classification and representation loss functions) on four datasets from two different domains (images and malware). Third, our results indicate that the combination of MMF and the ii loss function [5] outperforms the other combinations in both training time and overall F1 score.
We organize the paper as follows. In section 2, we give an overview of related work. Section 3 presents the MMF loss extension. Section 4 shows that the MMF extension can improve different types of loss functions significantly.
Related Work
The OSR problem is related to PU (Positive and Unlabeled) learning [10], which can be regarded as a binary classification problem with the absence of negative samples. The OSR problem extends the binary classification problem to a multiclass classification problem, with some classes missing from the training set, and will be recognized as an unknown class during testing. We can divide OSR techniques into three categories based on the training set compositions. The first category includes the techniques that borrow additional data in the training set. To better discriminate between known class and unknown class, unlabeled data is introduced during training in [18]. Dhamija et al. [2] utilize the differences of feature magnitudes between known and borrowed unknown samples as part of the objective function. Hendrycks et al. [6] propose Outlier Exposure(OE) to distinguish between anomalous (unknown) and in-distribution (known) examples. In general, although borrowing and annotating additional data turns OSR into a common classification problem, the retrieval and selection of additional datasets remain an issue.
The research works that generate additional data in training data fall in the second category of open set recognition techniques. Most data generation methods are based on GANs. Neal et al. [11] add another encoder network to traditional GANs to map from images to a latent space. Lee et al. [9] generate "boundary" samples in the low-density area of in-distribution acting as unknown samples. While generating unknown samples for the OSR problem has achieved great performance, it requires more complex network architectures.
The third category of open set recognition does not require additional data. Most of the research works require outlier detection for the unknown class. Pidhorskyi et al. [13] propose manifold learning based on training an Adversarial Autoencoder (AAE) to capture the underlying structure of the distributions of known classes. Hassen and Chan [5] propose ii loss for open set recognition. It first finds the representations for the known classes during training and then recognizes an instance as unknown if it does not belong to any known classes. In EVS, Schultheiss et al. [17] investigate class-specific representations for novelty detection tasks. The research work shows that each class's mean representation can capture discriminative information of both known and unknown classes. EVS focuses on the top K activations via binarizing the activations; however, choosing an appropriate K can be challenging. Also, EVS assumes that all the activation values are positive and only looks at the larger ones. We address both limitations in our proposed approach.
While our proposed approach can be incorporated into different loss functions, we focus on two types of loss functions in this paper: the classification loss functions and the representation loss functions. The objective of classification loss is to lower the classification error of the training data, cross-entropy loss is widely used. The representation loss functions are normally applied to the representation layers, such as triplet loss in [16] and ii loss in [5]. Triplet loss intends to find an embedding space where the distance between an anchor instance and another instance from the same class is smaller by a user-specified margin than the distance between the anchor instance and another instance from a different class. Ii loss aims to maximize the distance between different classes (inter-class separation) and minimize the distance of an instance from its class mean (intra-class spread).
Approach
We propose the MMF extension to learn more discriminative representations through known classes, thus better separating known and unknown classes. The proposed MMF extension does not borrow or generate additional data for the unknown class, and it can be incorporated into different loss functions. We focus on classification loss functions such as cross-entropy loss and representation loss functions, such as triplet loss and ii loss (Section 2).
A typical classification neural network consists of an input layer, hidden layers, and classification layer. We can consider the hidden layers as different levels of representations of the input. We call the values of the last hidden layer activation vector (AV), and each activation is a learned feature. The mean activation The network in Figure 1a contains one convolutional layer, one fully connected layer, one representation layer (representation layer Z), and one classification layer (softmax layer). In some scenarios, a neural network only consists of the input layer and hidden layers as in Figure 1b, where we use learned representations instead of a classification layer for classification tasks. Figure 3a shows the learned MAV values from the representation layer using standalone crossentropy loss. The red boxes are the features with the highest absolute values (magnitudes), the yellow boxes are the features with the lowest absolute values (magnitudes), and the green column is the MAV of the unknown class.
To improve the accuracy of detecting open set samples from unknown classes, we can increase the distances (we use Euclidean distance here) between the learned features of known and unknown samples, summarized by the MAVs of the known and unknown classes. Squared differences are the components of Euclidean distance. Thus we can increase the distance by increasing squared differences. Figure 2 depicts the relationship between squared differences with the absolute feature values (feature magnitudes) of the six known classes. The x-axis is the absolute feature values in six features, and the y-axis is their corresponding squared differences to the unknown class. We consider a feature with a larger magnitude is more significant than that with a smaller magnitude. We observe that a more significant feature leads to a higher squared difference to the unknown class. The reason is the MAV of the unknown class has a relatively small magnitude as we observe in Figure 3a. The small magnitude is due to the unknown class being absent from training and hence its features are not learned. More importantly, the squared difference increases faster with more significant features, which indicates a slight improvement in a more significant feature will increase squared difference more. Thus, we want the features with larger magnitudes to become even more significant to increase the distance between the unknown and known classes.
However, based on the preliminary experiments, we found that after enlarging the magnitudes of the most significant features for the known classes, the unknown class's MAV became further away from the origin, which reduces the increase in the distance between the known and unknown classes. As shown in Figure 3b, the MAV of the unknown class (green column) has significantly increased compared to the one only using standalone cross-entropy loss in Figure 3a. To further improve accuracy and increase the magnitudes of the most significant feature, we also decrease the magnitudes of the least significant features to mitigate the increase of the MAV of the unknown class. Comparing Figure 3c and Figure 3a, we can see that after reducing the magnitude of the least significant features, the feature values of unknown classes indeed get smaller.
Therefore, our MMF extension has two properties. Property A maximizes the most significant feature, i.e., the feature with the largest magnitude, for all the known classes. Property B minimizes the least significant feature; i.e., the feature with the smallest magnitude, for all the known classes. As a result, the learned representations for known classes should be more discriminative, while the unknown classes should be less affected.
Learning objectives
Let x ∈ X be an instance and y ∈ Y be its label. The hidden layers in a neural network can be considered as different levels of representations of input x. Suppose that there are C known classes in training data, and C + 1 classes in test data with the additional class as unknown class. We denote the MAV of class i as µ i , and µ ij represents the j th feature of the MAV of class i. Assume the AVs and MAVs have F dimensions, representing F features, we stack the MAVs for all the classes to form a representation matrix U C×F . To satisfy Property A, we first select the most significant features for each class to form the "max feature" vector. The i th element in "max feature" is for class i:
max f eature i = max 1≤j≤F |µ ij |,(1)
In the example of Figure 3a, the "max feature" would be (1.8, 1.2, 1.3, 1.4, 1.6, 1.4) (the absolute values of the red boxes). Likewise, for Property B, we measure the vector of the "min feature" as the least significant feature for each class. The i th element is for class i:
min f eature i = min 1≤j≤F |µ ij |(2)
The "min feature" in the example of Figure 3a would be (0.13, 0.45, 0.27, 0.34, 0.25, 0.32) (the absolute values of the yellow boxes). Then, we maximize the smallest value in "max feature" directly. In this way, all the values in the "max feature" would be maximized, thus the most significant features for all the known classes would be maximized as Property A. Meanwhile, we minimize the largest value in the "min feature" to implicitly minimize all the values in the "min feature", therefore the least significant features for all the known classes would be minimized as Property B. As a result, the proposed MMF extension satisfies both properties:
M M F = max 1≤i≤C (min f eature i ) − min 1≤i≤C (max f eature i )(3)
In the example of Figure 3a, we would like to maximize the "1.2" in the "max feature" and minimize the "0.45" in the "min feature". There are alternative methods to generate the "max feature" and "min feature", for example, instead of selecting the highest absolute values for "max feature", we experimented with the highest values (max f eature 1i = max 1≤j≤F (µ ij )) and the lowest values (max f eature 2i = max 1≤j≤F (−µ ij )) to form two "max feature" vectors and later to be maximized at the same time. However, our experiments indicate that using the single "max feauture" vector can achieve better performances. There are also other methods to implicitly maximize the most significant features and minimize the least significant values for all the classes, such as maximizing the average value of the "max feature", or minimizing the average value of the "min feature", i.e.
C i=1
1 C (min f eature i − max f eature i ). However, the results of using average value are weaker than using the extreme values across all classes, hence we choose to use the extreme values in our extension function and in our experiments.
Training with MMF and Open Set Recognition
In addition to Properties A and B, the MMF extension can be incorporated into different loss functions. We focus on two types of loss functions: a) loss functions designed for decision layers such as cross-entropy loss; b) loss functions designed for representation layers such as triplet loss and ii loss. Notably, we combine the MMF extension with these two types of loss functions differently, as Figure 1.
We use the network architecture in Figure 1a to simultaneously train the network with classification loss functions and the MMF extension. During each iteration, first, we extract AVs and generate the representation matrix; second, we construct the MMF extension function from the "max feature" vector and "min feature" vector; third, the weights of each layer of the network are first updated to minimize the MMF extension then updated to minimize classification loss functions using stochastic gradient descent.
The MMF extension can also be incorporated into representation loss functions such as triplet loss and ii loss. As both representation loss functions and the MMF extension should be applied to the layer learning representations, their combination gives us:
L = L rep + λM M F,(4)
L rep is a representation loss function, and λ is a hyperparameter that strikes a balance between the representation loss function and the MMF extension. Figure 1b shows the network architecture using a representation loss function with an MMF extension. The combination serves on the Z-layer of the network. Moreover, the network weights get updated using stochastic gradient descent during each iteration. After the training process, we obtain the representation centroids for each class. Then during the inference, we use the same strategy as used in ii loss [5]. First, we calculate the outlier score as the distance of learned representation to the nearest representation centroid. Then we sort the outlier score of the training data in descending order and pick the 99 percentile outlier score value as the outlier threshold. If the outlier score of a test sample is beyond the threshold, it will be recognized as the unknown class, otherwise, it will be classified as the known class with the nearest representation centroid.
Experimental Evaluation
We evaluate the MMF extension with simulated open-set datasets from the following four datasets. MNIST [14] contains 70,000 handwritten digits from 0 to 9, which is 10 classes in total.
To simulate an open-set dataset, we randomly pick six digits as the known classes participant in the training, while the rest are treated as the unknown class only existing in the test set. CIFAR-10 [7] contains 60,000 32x32 color images in 10 classes, with 6,000 images per class. There are 50,000 training images and 10,000 test images. We first convert the color images to grayscale and randomly pick six classes out of the ten classes as the known classes, while instances from the remaining classes are treated as the known class only existing in the test set. Microsoft Challenge (MC) [8] contains disassembled malware samples from 9 families. We use 10260 samples that can be correctly parsed then extract their function call graphs (FCG) as in [4] for the experiment. The dimensionality of the FCG is 63x63. Again, to simulate an open-set dataset, we randomly pick six classes as the known classes, while the rest are considered unknowns. Android Genome (AG) [19] consists of malicious android apps from many families in different sizes. We use nine families (986 samples) with a relatively larger size for the experiment to be fairly split into the training set, the test set, and the validation set. we first use [3] to extract the function instructions and then extract 1453 raw FCG features as in [4]. Like the MNIST and the MC dataset, we randomly pick six classes as the known classes in the training set and consider the rest as the unknown class, which are only used in the test phase.
Network Architectures and Evaluation Criteria
We evaluate the MMF extension associated with two types of loss functions: classification loss functions and representation loss functions. Specifically, we use the cross-entropy loss as the example of classification loss functions, and use ii loss [5] and triplet loss [16] as the examples of representation loss functions. Moreover, we compare compare these pairs with OpenMax [1].
For the MNIST dataset, the padded input layer is of size (32, 32), followed by two non-linear convolutional layers with 32 and 64 nodes. We also use the max-polling layers with kernel size (3,3) and strides (2, 2) after each convolutional layer. We use two fully connected non-linear layers with 256 and 128 hidden units after the convolutional component. Furthermore, the linear layer Z, where we extract the representation matrix, is six dimensions in our experiment.
We use the Relu activation function for all the non-linear layers and set the Dropout's keep probability as 0.2 for the fully connected layers. We use Adam optimizer with a learning rate of 0.001. The network architecture of the CIFAR-10 experiment is similar to the MNIST dataset, except the padded input layer is of size (36, 36). The experiment for the MS Challenge dataset also implements two convolutional layers. The padded input layer is of size (67, 67). However, we only use one fully connected layer instead of two after the convolutional layers. Also, we make the keep probability of Dropout as 0.9. The Android Genome dataset does not use the convolutional component. We use a network with one fully connected layer of 64 units before the linear layer Z. We also used Dropout with a keep probability of 0.9 for the fully connected layers. We set the learning rate of Adam optimizer as 0.1. Besides, we use batch normalization in all the layers to prevent features from getting excessively large. And as mentioned in section 3.2, we use contamination ratio of 0.01 for the threshold selection.
As we discussed in Equation 4, we use a hyperparameter λ combine the MMF extension with the representation loss functions (i.e. ii loss and triplet loss in the experiments) as: L = L rep + λM M F . While the range of λ is (0, 1], we set λ as 0.2 and 0.5 for ii loss and triplet loss for the MNIST and CIFAR-10 datasets. For the MC dataset, we set λ as 0.5 and 0.3 for ii loss and triplet loss. We set λ as 0.4 for both ii loss and triplet loss in the AG dataset's experiments.
For each dataset, we simulate three different groups of open sets then repeat each group 10 runs, so each dataset has 30 runs in total. When measuring the model performance, we use the average AUC scores under 10% and 100% FPR (False Positive Rate) for recognizing the unknown class, as lower FPR is desirable in the real world for cases like malware detection. We measure the F1 scores for known and unknown classes (C + 1 classes) separately as one of the OSR tasks is to classify the known classes. Moreover, we perform t-tests with 95% confidence in both the AUC scores and F1 scores to see if the proposed MMF extension can significantly improve different loss functions.
Experimental Results
We compare the model performances of OpenMax as well as three loss function quadruples: cross-entropy loss, ii loss, and triplet loss. Table 1 shows the AUC scores of the models in the four datasets; mainly, we focus on comparing the "Standalone" with the corresponding "+MMF" subcolumns. We observe that the quadruples in general achieve better AUC scores than OpenMax. Moreover, with the MMF extension, the AUC scores of the loss functions have achieved statistically significant improvements in 16 out of 24 cases (3 loss functions×4 datasets×2 FPR values). Table 2 shows the average F1 scores for the four datasets. We first calculate the F1 scores for each of the C known classes and the unknown class, then average the C + 1 classes as the Overall F1 scores. We can see the loss functions with the MMF extension have better results than their corresponding standalone versions for both the known and the unknown classes. We observe that ii loss with the MMF extension is more accurate than the other five methods in six out of twelve F1 scores. Particularly, it achieves the highest Overall F1 scores for three out of four datasets. Table 3 shows the comparison of the average training time of the 30 runs for the MNIST dataset with 5000 iterations via NVIDIA Tesla K80 GPU on AWS. We find that adding the MMF extension almost doubles the training time of using standalone cross-entropy. While for ii loss and triplet loss, adding the extension increases the training time by around 1%. The reason is that the MMF extension needs to create the representation matrix from scratch for the network with ce loss, which needs an extra backpropagation step, both of which take more time. We also observe that with our MMF extension, ii loss has the fastest training time among three loss functions. Overall F1 scores and training time indicate that "ii+MMF" is the most accurate and efficient combination. Figure 3c shows the heatmap of MAV values of the simulated open MNIST dataset trained by cross-entropy loss with the MMF extension. We take digits "0", "2", "3", "4", "6", "9" as the known classes and the remaining digits as the unknown class. Comparing with the MAV values from the network with triplet+MMF. The left subplots of (a) and (b) are the representations of the unknown class (a mixture of digits "1", "5", "7" and "8"), and the right plots are the representations of the known classes. standalone cross-entropy loss (Figure 3a), we can find that the MAVs of the known classes become more discriminative from each other, and each of the known classes has its representative feature. (e.g. Z1 for class "0", Z2 for class "2"). Whereas the MMF extension has less effect on the unknown class, and its MAV values are relatively evenly distributed.
Analysis
Since we recognize the unknown class based on the outlier score described in section 3.3, we analyze both the test samples' outlier scores from the known classes and the unknown class from the MNIST experiment. Figure 4 shows the histogram of the distributions of the outlier scores in triplet loss experiments and triplet loss with the MMF extension. Compared with using standalone triplet loss, adding an MMF extension increases the outlier scores of the unknown class, which pushes the score distributions further away from those of the known classes and results in fewer overlaps between the known classes the unknown class. It is the reduced overlaps that make the known classes and the unknown classes more separable than before. Figure 5 shows the t-SNE (perplexity: 50) plots of the Z-layer representations of the MNIST dataset from the same experiments. We can see that with the MMF extension, not only the known classes and the unknown class are more separate from each other, the known classes become more disparate than before. We also perform an ablation analysis for the MMF loss extension to understand the importance of the MMF extension's two properties. As shown in Table 1, our baselines include (1) standalone loss functions; (2) loss functions with an extension that maximize the most significant feature as Property A (MaxF); (3) loss functions with an extension that minimizes the least significant feature as Property B (MinF). In general, the MMF extension with both properties outperforms the baselines. This result is consistent with our motivation for the two properties at the beginning of Section 3. Moreover, we find that MaxF and MinF extensions can also achieve better performance than standalone loss functions. While both properties improve AUC scores, Property A (MaxF) has a larger improvement. Hence, Property A plays a more critical role in AUC improvement than Property B.
To investigate why MinF also helps improve AUC performance, we show the heatmap of the MAV for the unknown class in the experiment of ce on the MC dataset in Figure 6. Comparing Figure 6a and Figure 6b, we observe that MinF reduced the feature magnitudes for the unknown class, thus increased the distance between the known and unknown classes. Similarly, from Figure 6c and Figure 6d, we observe that the feature magnitudes of the unknown class in MMF (MaxF+MinF) are much smaller than the ones in MaxF. The second observation is consistent with the earlier discussion on adding MinF to help MaxF in MMF at the beginning of Section 3.
Conclusion
We introduced an add-on loss function extension for the OSR problem. The extension maximizes the feature with the largest magnitude meanwhile minimizes the one with the smallest magnitude for all the known classes during training so that the learned presentations are more discriminative from each other. We have shown that while the known classes are more discriminative from each other, the feature values of unknown classes are less affected by the extension, hence simplifying the open set recognition. We incorporated the proposed extension into both classification and representation loss functions and evaluated them in images and malware samples. The results show that the proposed approach has achieved statistically significant improvements for different loss functions.
Fig. 1 :Fig. 2 :
12An overview of the network architectures of different types of loss functions. The convolutional layers are optional. The MMF module in red is our proposed loss extension. Squared differences of MAV values between the known and unknown classes inFigure 3a.
Fig. 3 :
3The heatmap of MAVs (Mean Activation Vectors) of the classes from the MNIST dataset using cross entropy loss with different extensions. vectors (MAV) of a class is the average of the activation vectors of the class.
Fig. 4 :
4The distributions of outlier scores in MNIST.
Fig. 5 :
5The t-SNE plots of the MNIST dataset in the experiments of triplet vs.
Fig. 6 :
6The heatmap of the unknown class's MAV in the experiment of cross entropy loss (ce) on the Microsoft Challenge dataset (MC).
Table 1 :
1The average AUC scores of 30 runs at 100% and 10% FPR of Open-Max and three loss functions quadruples. The underlined values are statistical significant better than the standalone loss functions via t-test with 95% confidence. The values in bold are the highest values in each quadruple. The values in brackets are the highest values in each row.OpenMax
ce
ii
triplet
FPR
Standalone +MMF +MaxF +MinF Standalone +MMF +MaxF +MinF Standalone +MMF +MaxF +MinF
MNIST
100% 0.9138
0.9255
0.9479 0.9515 0.9393
0.9578 [0.9649] 0.9579 0.9607
0.9496 0.9585 0.9480 0.9404
10%
0.0590
0.0765
0.0744 0.0761 0.0751
0.0821 [0.0842] 0.0826 0.0830
0.0750 0.0796 0.0777 0.0739
CIFAR-10
100% [0.6757]
0.5803
0.5982 0.6103 0.5807
0.6392
0.6419 0.6437 0.6439
0.6106 0.6248 0.6131 0.6127
10%
0.0065
0.0070
0.0089 0.0090 0.0077 [0.0103] 0.0096 0.0100 0.0100
0.0089 0.0102 0.0092 0.0093
MC
100% 0.8739
0.9148 [0.9500] 0.9387 0.9352
0.9385
0.9461 0.9407 0.9397
0.9240 0.9430 0.9317 0.9178
10%
0.0405
0.0530
0.0635 0.0600 0.0588
0.0627 [0.0656] 0.0629 0.0619
0.0565 0.0622 0.0563 0.0546
AG
100% 0.4150
0.7506
0.8205 0.8152 0.7501
0.8427
0.8694 0.8763 [0.8831] 0.8271 0.8379 0.8203 0.8256
10%
0.0010
0.0058
0.0148 0.0163 0.0036
0.0285
0.0305 [0.0368] 0.0366
0.0229 0.0275 0.0260 0.0235
Table 2 :
2The average F1 scores of 30 runs of OpenMax and three loss functions
pairs. The underlined values show statistically significant improvements (t-test
with 95% confidence) comparing with the standalone loss functions. The values
in bold are the highest values in each column.
Table 3 :
3The comparison of training time for the MNIST dataset.Regular +MMF delta
ce
119.33 230.43 +111.1
ii
122.17 123.30 +1.14
triplet 223.27 225.70 +2.43
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| []
|
[
"Continuous and Discontinuous Phase Transitions in the evolution of a polygenic trait under stabilizing selective pressure",
"Continuous and Discontinuous Phase Transitions in the evolution of a polygenic trait under stabilizing selective pressure"
]
| [
"Annalisa Fierro [email protected] \nDipartimento di Medicina Molecolare e Biotecnologie Mediche\nCNR-SPIN Complesso Univ. Monte S. Angelo\nVia CinthiaI-80126NaplesItaly\n",
"Sergio Cocozza \nIstituto di Endocrinologia ed Oncologia Sperimentale\nUniversità degli Studi di Napoli \"Federico II\"\nNaplesItaly\n",
"Antonella Monticelli \nINFN -Sez. Napoli\nCNR Napoli\nNaplesItaly\n",
"Giovanni Scala \nDipartimento di Fisica \"Ettore Pancini\", Università degli Studi di Napoli \"Federico II\", and INFN -Sez. Napoli\nComplesso Univ. Monte S. Angelo\nVia Cinthia, I80126NaplesItaly\n",
"Gennaro Miele \nComplesso Univ. Monte S. Angelo\nVia Cinthia, I80126NaplesItaly\n"
]
| [
"Dipartimento di Medicina Molecolare e Biotecnologie Mediche\nCNR-SPIN Complesso Univ. Monte S. Angelo\nVia CinthiaI-80126NaplesItaly",
"Istituto di Endocrinologia ed Oncologia Sperimentale\nUniversità degli Studi di Napoli \"Federico II\"\nNaplesItaly",
"INFN -Sez. Napoli\nCNR Napoli\nNaplesItaly",
"Dipartimento di Fisica \"Ettore Pancini\", Università degli Studi di Napoli \"Federico II\", and INFN -Sez. Napoli\nComplesso Univ. Monte S. Angelo\nVia Cinthia, I80126NaplesItaly",
"Complesso Univ. Monte S. Angelo\nVia Cinthia, I80126NaplesItaly"
]
| []
| The presence of phenomena analogous to phase transition in Statistical Mechanics, has been suggested in the evolution of a polygenic trait under stabilizing selection, mutation and genetic drift.By using numerical simulations of a model system, we analyze the evolution of a population of N diploid hermaphrodites in random mating regime. The population evolves under the effect of drift, selective pressure in form of viability on an additive polygenic trait, and mutation. The analysis allows to determine a phase diagram in the plane of mutation rate and strength of selection. The involved pattern of phase transitions is characterized by a line of critical points for weak selective pressure (smaller than a threshold), whereas discontinuous phase transitions, characterized by metastable hysteresis, are observed for strong selective pressure.A finite size scaling analysis suggests the analogy between our system and the mean field Ising model for selective pressure approaching the threshold from weaker values. In this framework, the mutation rate, which allows the system to explore the accessible microscopic states, is the parameter controlling the transition from large heterozygosity (disordered phase) to small heterozygosity (ordered one). | 10.1140/epjp/i2017-11566-9 | [
"https://arxiv.org/pdf/1409.0654v3.pdf"
]
| 14,992,247 | 1409.0654 | 27d9ee74c525a30f4e1e0818db99f446d3c3e75c |
Continuous and Discontinuous Phase Transitions in the evolution of a polygenic trait under stabilizing selective pressure
Annalisa Fierro [email protected]
Dipartimento di Medicina Molecolare e Biotecnologie Mediche
CNR-SPIN Complesso Univ. Monte S. Angelo
Via CinthiaI-80126NaplesItaly
Sergio Cocozza
Istituto di Endocrinologia ed Oncologia Sperimentale
Università degli Studi di Napoli "Federico II"
NaplesItaly
Antonella Monticelli
INFN -Sez. Napoli
CNR Napoli
NaplesItaly
Giovanni Scala
Dipartimento di Fisica "Ettore Pancini", Università degli Studi di Napoli "Federico II", and INFN -Sez. Napoli
Complesso Univ. Monte S. Angelo
Via Cinthia, I80126NaplesItaly
Gennaro Miele
Complesso Univ. Monte S. Angelo
Via Cinthia, I80126NaplesItaly
Continuous and Discontinuous Phase Transitions in the evolution of a polygenic trait under stabilizing selective pressure
The presence of phenomena analogous to phase transition in Statistical Mechanics, has been suggested in the evolution of a polygenic trait under stabilizing selection, mutation and genetic drift.By using numerical simulations of a model system, we analyze the evolution of a population of N diploid hermaphrodites in random mating regime. The population evolves under the effect of drift, selective pressure in form of viability on an additive polygenic trait, and mutation. The analysis allows to determine a phase diagram in the plane of mutation rate and strength of selection. The involved pattern of phase transitions is characterized by a line of critical points for weak selective pressure (smaller than a threshold), whereas discontinuous phase transitions, characterized by metastable hysteresis, are observed for strong selective pressure.A finite size scaling analysis suggests the analogy between our system and the mean field Ising model for selective pressure approaching the threshold from weaker values. In this framework, the mutation rate, which allows the system to explore the accessible microscopic states, is the parameter controlling the transition from large heterozygosity (disordered phase) to small heterozygosity (ordered one).
Introduction
The need of applying a statistical approach arises in physics when the global properties of ensembles containing a huge number of elementary constituents are studied. In this case, the behavior of the single element is irrelevant in favor of more informative averages on the whole ensemble. In this sense, the situation is strictly analogous in both Population and Quantitative Genetics, where the focus is on the analysis of allele frequencies and phenotype distribution parameters rather than the genetic/phenotypic description of each individual. The analogy between Quantitative Genetics and thermodynamics was noted from the very beginning by R. A. Fisher himself [1]. The development of such analogy allowed Iwasa to introduce a concept of entropy for Population Genetics, which satisfies the analogous of H-theorem [2]. Such information entropy measure ensures an exact solution at statistical equilibrium [2], [3], [4], [5]. This point of view, can be alternatively seen by defining a free fitness, namely the entropy divided by population size plus the mean fitness. The free fitness is maximized at equilibrium, when natural selection and drift (random sampling) are at work [2], [5], and provides an analogous of free energy in thermodynamics.
Many analogies between biological evolution and statistical physics are present in literature (for a review see [6], [7], and reference therein). The presence of phenomena analogous to phase transition has been also suggested [8], [9], [10].
In this paper, we analyze in details such phase transition phenomena in a stochastic model, extensively analyzed in the past years (see for instance Ref. [3], and references therein). It consists in a population of N diploid hermaphrodite individuals, reproducing in pairs in a random mating regime, evolving under the effect of drift, selective pressure in form of viability, and mutation (we assume "substitution" mutations). Following Wright's seminal paper [17], we consider M different bi-allelic genes additively combining on the character, and the individual viability following a Gaussian profile in the trait. Using numerical simulations of such a model, we determine a phase diagram in the plane of mutation rate and strength of selection. The involved pattern of phase transitions is characterized by a transition from a state, where the alleles of individuals are roughly randomly distributed, to a state of clones, where individuals display a unique genome. This transition presents feature of a second order transition for weak selective pressure (smaller than a threshold), whereas discontinuous phase transitions, characterized by metastable hysteresis, are observed for strong selective pressure.
The model
Using numerical simulations, we study a model for N diploid individuals, sexually reproducing with random mating between any pairs of individuals. Each individual i (with i = 1, . . . , N ) is represented by two sequences of M variables, σ ikj (where k = 1, 2 stands for the two genome replicas, and j = 1, . . . , M runs on different loci). We refer to σ ikj as alleles, and assume that each allele can take two values, ±1. A mutation rate, µ, is introduced, as the probability of an allele to mutate at each generation (σ ikj → −σ ikj ). Similar models were studied in Refs. [18], [19].
A stabilizing selective pressure on such a phenotype can be implemented via the survival probability of an individual i, typically known as viability,
S(p i ) ≡ N exp −(p i − p m ) 2 ω 2 /2 ,(1)
where N is the suitable normalization constant, ω measures the strength of selection, p i = M j=1 2 k=1 σ ikj is the additive polygenic phenotype random variable, and p m stands for the optimum phenotype.
For ω → 0, there is no selective pressure, and all the microscopic states are equivalent. Whereas, for ω → ∞, the selective pressure is at maximum, and the only surviving individuals are those with p i = p m . In general, the effect of the selective pressure is to reduce the accessible phase space to those microscopic states better conform to the constraint on the phenotype.
Numerical simulations of the model are performed for different set of the parameters N , M , ω, and µ. Starting from a common initial state, where the M variants are chosen equal to ±1 with equal probability, 30 independent populations evolve with different random noise. Our choice of the initial state corresponds to an initial frequency of the allele "1" in locus j, ρ in (j) = 0.5, ∀j = 1, .., M . During the evolution 1) two individuals, i 1 and i 2 , are randomly chosen and an off-spring i is generated, such that σ ikj of the off-spring is equal to σ i1kj or σ i2kj with equal probability; 2) the alleles are mutated (i.e., σ ikj → −σ ikj ) with probability µ (called mutation rate); 3) the newborn individual survives with probability S(p i ), given by Eq. (1); 4) the point 1-3 are iterated until N newborn individuals are generated. Then, the old generation is replaced by a new generation of same size N , formed by off-springs of the previous individuals. Note that the population size is fixed and not allowed to fluctuate.
The equilibrium results are independent of the initial assumption about ρ in (j). Further analysis is necessary to evaluate the effect of the initial state on the out-of-equilibrium behavior. Hereafter, we choose p m = 0, however preliminary simulations show the model with a different optimum (p m = 0) displays qualitatively similar behavior. The connection between the present model and a usual system of Statistical Mechanics with Ising spins is rather natural (one can speculate that the random mating is similar to a long range interaction between pairs of spins in the same locus). The biological model in absence of selective pressure remembers M independent systems of 2N spins, the introduction of a selective pressure instead corresponding to a coupling between different systems. With this analogy in mind, we introduce the following quantities in order to describe the macroscopic state of the biological model:
q ≡ 1 M M j=1 |q(j)| ,(2)
with
q(j) ≡ 1 2N N i=1 2 k=1 σ ikj ,(3)
where . . . stands for the average over the independently evolving populations (hereafter simply denoted by ensemble of populations). In our Statistical Mechanics analogue, the quantity q(j), Eq. (3), should correspond to the magnetization per spin in a system of 2N Ising spins, and q, Eq. (2), to the average of the magnetization modulus over different systems. Following the same analogy, we also introduce the susceptibility, as
χ ≡ 1 M M j=1 χ(j),(4)with χ(j) ≡ 2N q(j) 2 − |q(j)| 2 .
It is interesting to note that the magnetization q(j) of j-th locus is related to the expected heterozygosity (fraction of heterozygous individuals expected on the basis of Hardy-Weinberg equilibrium condition) in the same locus, denoted by h s (j), which is a more familiar quantity in the Population Genetics context. Indeed, one can easily prove that
h s (j) ≡ 1 4N 2 N i,l=1 2 k,n=1 1 − δ σ ikj σ lnj = = 1 2 1 − q(j) 2 .(5)
In our case, h s (j) essentially coincides with the observed heterozygosity (observed fraction of heterozygous individuals). From Eq. (5), we see that the minimum of magnetization corresponds to the maximum of heterozygosity, and vice-versa. Denoting with H s the average of h s (j) over different loci and on the ensemble, one can easily prove that
H s = 1 2 1 − χ 2N − q 2(6)
for a very large number of realizations. This occurs since, in this limit, |q(j)| is independent of j.
Results and Discussion
For any fixed set of the parameters, we follow the evolution of a given population till it asymptotically reaches a stationary state, which we refer to as steady state, where we evaluate q and χ. Let us start by focusing our attention on the role played by mutation rate and selection strength only, and to this aim we fix the values of N = 1000 and M = 50. In general, the system reaches the steady state for values of generation number, which depend on ω and µ. Two different behaviors are observed in the regime of small and large selective pressure strength, respectively. In Fig. 1, we report q as a function of µ for different values of ω. As shown in figure, by decreasing the mutation rate µ, q goes from small-q (which vanishes in the limit of large-N ) to q ∼ 1. Note that, for large µ, the alleles ±1 have roughly equal probability (namely, h s (j) ∼ 0.5 for each locus), and hence the steady state does not significantly differ from the initial one. On the contrary, for small µ, the system reaches fixation (i.e., h s (j) ∼ 0 for each locus). In this case, the individuals are just clones, namely, for each realization, the population is represented by a unique genome that is a generic combination of ±1 in a neighborhood of the phenotype optimum (exactly in the optimum for ω → ∞). The crossover from the state with small q (large heterozygosity) to the state with q ∼ 1 (small heterozygosity) is characterized by a maximum in the susceptibility, χ. The value of µ corresponding to such a maximum is a monotonic increasing function of ω. Moreover, concerning its dependence on M , it is interesting to observe that it simply scales as 1/M , as one can expect since 2µM , representing the mutation rate per individual, is the relevant quantity, ruling the mutations during evolution. From Fig. 1, it can be easily observed that for weak selection strength, roughly ω < 0.4 (with blue circles in figure corresponding to ω = 0.4), one has a smooth crossover that becomes abrupt for larger ω.
To better analyze the nature of these steady states, and the crossover from small-q states to large-q ones, we perform the following numerical experiment. For any value of ω, starting from a configuration at high mutation rate, we decrease µ at a given rateμ ≡ ∆µ/∆n (n denoting the generation number). In other words, the system is kept at a given value of the mutation rate for an interval ∆n, and, at the end of it, q and χ are measured. Afterward, the value of µ is decreased of ∆µ and the procedure is iterated till µ reaches zero. At this point the procedure is inverted and µ is increased at the same rate, in analogy to a physical system, first cooled and then heated at given rate. As usual in thermodynamics, for any µ, two states are considered macroscopically equivalent if the measured values of q and χ coincide. As we will show in the following sections, this procedure confirms the presence of two different regimes, for ω < ω c and ω ≥ ω c , respectively, where the threshold ω c 0.4 + O(N −1 ).
Small selective pressure
In Fig. 2, q and χ vs µ are plotted for ω = 0.1 at two different values of the cooling rate. As we see in figure, for small enough cooling rate, the curves do not depend on the cooling rate,μ, and the two branches, obtained by decreasing and increasing µ respectively, always coincide. These behaviors are observed for each value of ω < ω c (small selective pressure). Moreover, the states obtained with this procedure in the limit of small cooling rate coincide with the above defined steady states (pink stars in Fig. 2), and in some sense, these states can be considered equilibrium states of the system. The crossover here observed from large to small heterozygosity, ruled by the mutation rate and characterized by a maximum in the susceptibility, strongly resembles a continuous (second order) transition in Figure 2. Main frame: Order parameter, q vs µ, for ω = 0.1 and N = 1000. The full lines are obtained first cooling the system at fixedμ (the red line corresponds to 10 −8 and the blue line to 10 −9 ), and then heating it at the same rate. The pink stars correspond to the steady states. Inset: Susceptibility, χ vs µ, with the same symbols as in the main frame.
a physical system. This observation suggests to study this transition as a usual critical phenomenon.
From the intersection of the fourth order cumulant, evaluated for different sizes of the system, the critical mutation rate, µ c ≡ lim N →∞ µ c (N ), is estimated, and a finite size scaling analysis is performed in order to evaluate the critical exponents of the transition.
For an Ising model with vanishing magnetic field 1 , the reduced fourth order cumulant of the order parameter [20] is given by
U 4 = 1 − m 4 3 m 2 2 ,(7)
where m is the magnetization. Following Ref. [20], as the system size N → ∞, U 4 → 0 for T > T c and U 4 → 2/3 for T < T c . For large enough values of the size N , all curves representing U 4 as a function of temperature cross in a point whose location gives the critical point. A natural extension of Eq. (7) to our biological system is
U 4 ≡ 1 M M j=1 U 4 (j),(8)
where
U 4 (j) = 1 − q(j) 4 3 q(j) 2 2(9)
is the fourth order cumulant of j-th locus, and U 4 denotes the average over the M loci.
1. For vanishing magnetic field (H = 0) the Ising model undergoes a second order phase transition from a disordered paramagnetic phase (vanishing magnetization) to an ordered ferromagnetic one (not vanishing magnetization), at temperature Tc. In the ferromagnetic phase (T ≤ Tc), the magnetization → 0 at the critical temperature as a power law with exponent β. For fixed temperature T < Tc, a first order transition controlled by the magnetic field, is found for vanishing H. In the present case, increasing the mutation rate, in each locus j it is observed a transition from a disordered phase to an ordered one. This is in perfect analogy with the Ising model. Hence, we expect that varying the size N of the system, all curves for U 4 as a function of µ cross in a point which provides the critical mutation rate µ c ≡ lim N →∞ µ c (N ). Note that µ c keeps a dependence on ω.
In Fig. 3, U 4 is plotted as a function of µ for different values of the size N , having fixed ω = 0.2. As expected, in the limit of large N we find that U 4 tends to 2/3 and to zero for small and large mutation rate, respectively. The crossing point, µ c , represented by the red circles in Fig. 4, increases by increasing ω and vanishes in the limit ω → 0. Blue stars in the same figure correspond to the maximum points of the susceptibility, namely µ c (N ) for N = 1000. We find that µ c (N ) is smaller than µ c for each non vanishing ω, whereas this behavior reverses for ω = 0.
Next, the critical behavior of the order parameter and of the susceptibility are studied. Extending the predictions from finite size scaling analysis in the Ising model [20] to the present system, we expect that near the critical point
q = N −a q 0 ( N c ) , χ = N b χ 0 ( N c ) ,(10)
where = (µ−µ c )/µ c , and q 0 and χ 0 are scaling functions. A finite size scaling analysis allows to evaluate the exponents a, b, and c. For each value of ω, N a q and N −b χ are plotted as a function of N c , where a, b, and c are chosen in order to rescale the curves for different N onto a unique one (data not shown).
In a d-dimensional physical system, a, b and c are related to the critical exponents, ν, β and γ, by the following relations [20]: a = β/νd, b = γ/νd, and c = 1/νd, where ν, β and γ depend on the euclidean dimension, and tend to the mean field exponents in the limit of high dimension d. Although the space dimension is here not defined at all, we can evaluate β and γ, from a, b and c, as β = a/c and γ = b/c. In Table 1, µ c and the scaling exponents obtained for different ω are listed 2 . Although the errors are rather large (of the order of the 10%) and further analysis is necessary to confirm these findings, the critical exponents seem to change along the critical line, and to tend to the mean field Ising critical exponents (i.e., β = 0.5 and γ = 1) by approaching ω c . This result can be interpreted in the following way. The correspondence between micro-states with optimum phenotype and energy minima in physical systems is rather natural. Following this analogy, the growth of ω would correspond to increase the barriers between two minima. This suggests the possibility that the selective pressure in some sense plays the role of the euclidean dimension, controlling the energy landscape of the system, and the maximum selective pressure corresponds to the mean field limit, where energy barriers between different minima become infinite.
For ω = 0, no crossing point is observed in the fourth order cumulant. Consistently, the maximum point of the susceptibility µ c (N ) goes to zero as 1/N (data not shown) in the limit N → ∞, and the following trivial finite size scaling is found (data not shown):
q = q 0 (µN ) , χ = N χ 0 (µN ) .(11)
It is worth observing, that due to this scaling behavior of q and χ, the heterozygosity, H s , defined in Eq. (6) results scale free for ω = 0. Moreover, in absence of selective pressure, since the loci are independent, there is no dependence on M at all.
Hysteresis cycles at large selective pressure
Interestingly, in the region of large selective pressure, for ω ≥ ω c , a metastable hysteresis appears between small 2. Note that the scaling relation 2β + γ = νd, which in terms of a and b becomes 2a + b = 1, is almost everywhere verified. Figure 5. Order parameter, q vs µ, for N = 1000 and ω → ∞. The different cycles have been obtained by using different cooling rates.The black stars reproduce data plotted in Fig. 1 for ω → ∞, obtained by following the system up to 2 · 10 6 generation numbers. and close to 1 values of q, as shown in Fig. 5. Again, for large mutation rate, the system is at equilibrium in states with small-q, and, for small mutation rate, is at equilibrium in states with q ∼ 1. However, the two branches, at small−q and q ∼ 1 respectively, are both observed for intermediate values of the mutation rate, depending on the pattern of µ-variation. Although by decreasing the cooling rate the hysteresis cycle shrinks, we always see two well distinct branches on our observation time scales. This behavior is reminiscent of a discontinuous (first order) transition, where metastable hysteresis is usually observed. In this case, the distinction of long-lived metastable states from equilibrium states is rather difficult, since the lifetime of the metastable states may be longer than the observation time. As it can be seen in Fig. 5, the states, obtained decreasing µ at smallμ, coincide with the steady states, reached by the system for very large generation numbers. Hence, in this case the so-called steady states, which are stationary on our observation time scales, are likely metastable. Note that in Fig. 5 the hysteresis curves are plotted for ω → ∞, where this phenomenon is more evident. Our findings are efficaciously summarized in Fig. 6, where the phase diagram for a system of N = 1000 individuals is shown in the plane (µ, 1/ω). For ω < ω c , we plot 1/ω as function of the maximum point of χ, µ c (ω, N ) (data already shown in Fig. 4). The blue line should give, in the thermodynamic limit, a line of critical points, where the continuous transition from small-q to q = 1 phase should be observed. In the region at large µ, the system is found in the Disordered Phase (DP), and in the region at small µ it is found in the Ordered Phase (OP). Above ω c , data depend on the cooling rate and the susceptibility displays two maxima, depending on the pattern of µ-variation. Red circles in Fig. 6 correspond to the maximum points of χ, along the hysteresis loop obtained at the smallest cooling rate,μ = 10 −9 . In this region, the system behaves as a physical system undergoing a discontinuous transition controlled by the mutation rate. Between the two red lines, the two phases coexist.
Heterozygosities
The analysis on the dependence of the order parameter, q, on the mutation rate, µ, is also carried out for the expected heterozygosity of the single population, H s , and for the expected heterozygosity measured on the set of populations as a whole, H t . As for q, hysteresis cycles are observed for ω ≥ ω c (data not shown). Interestingly, H s and H t display different behaviors for small mutation rate. Figs. 7 shows that H s ∼ 0.5 and H t ∼ 0.5, for large mutation rate, whereas H s ∼ 0 and H t ∼ 0.5 (we expect H t = 0.5 for N p → ∞), for small mutation rate, where the independently evolving populations (although initially identical) reach fixation in generally different (but macroscopically equivalent) micro-states, developing a genetic diversity. This phenomenon is in some sense analogous of the spontaneously symmetry breaking in physical system.
Comparison with literature and Conclusions
In summary, we have analyzed the evolution of a population of N diploid individuals, sexually reproducing with random mating, evolving under the effect of a Gaussian viability depending on an additive polygenic trait. Using the standard tools of Statistical Mechanics, we show that the system displays a complex phase diagram with a transition from a disordered to an ordered phase, controlled by the mutation rate. We provide the phase diagram in the (µ, ω) plane, showing that the order of the transition changes depending on the strength of selection, being continuous for weak selective pressures and discontinuous for strong ones. Similar findings are expected for a population of 2N haploid individuals.
Many analogies are found in literature between evolution and Statistical Mechanics, and they are not all equivalent. In our picture, the mutation rate plays the role of temperature in statistical physics (and N plays the same role of the finite dimension in physics systems), in agreement with Leuthausser's analogy between the Eigen model and an Ising system [12]. In other formulations (see for instance [4], [21]), temperature is instead related to population size. In Ref. [4], small mutation rates are considered, and populations are always in our fixation limit, i.e. they are made by clones. Then, the system state is a point in the genome space (which here is a 2M dimensional space), and not a point in the 2M N configurational space of individuals. In this limit, the mutation rate does not affect the steady state, and can merely influence the dynamics of the system. These two divergent points of view can be reconciled if one thinks about the main source of stochasticity that for large µ and N is dominated by the mutation rate (present analysis), and that on the contrary, for small µ and N is dominated by the random drift (see for instance [4]). Since the quantity representing the main source of stochasticity is the natural candidate to play the role of temperature, this would explain the different approaches present in literature.
The presence of phenomena analogous to phase transitions is also not new in biological evolution, in particular in the quasi-specie context [11], [12], [13], [14], [15], [16], in strict analogy with the critical mutation rate here found, the error threshold is the critical value of the mutation rate, below that the population is closely centered around the fitness peak, and above that it is roughly distributed over all the accessible space, losing the favorable sequence. We observe that µ c is a monotonic increasing function of ω, with a fix point in µ c = 0 for ω = 0 (obviously, in absence of selective pressure, the system is always in the disordered phase). Increasing the selective pressure, values of µ c roughly between 6 · 10 −4 and 2 · 10 −3 are observed. Since the relevant quantity is the mutation rate per individual, 2µM , we expect that µ c simply scales as 1/M (similar behaviors are observed for the error threshold in the singlepeaked landscape [6]). Thus, we can speculate that, in viral populations, where mutation rate is estimated between 10 −4 and 10 −5 , systems near these transitions can exist.
For future, we intend to study the effect of a different choice for the optimum of the viability, p m . It is interesting to explore how our findings change considering a less degenerate case (the case here considered p m = 0 is the most degenerate one), or even a non-reachable optimum value. In particular, we intend to investigate how the hysteresis cycles found at large selective pressure depend on this particular choice. Preliminary simulations show that the model with a different optimum (p m = 0) displays qualitatively similar behavior, with ω c decreasing as |p m | increases. However, further work is necessary to confirm this behavior and to understand its meaning. Finally, the effect of a different form for the viability will be also investigated. In particular, the model can be easily extended to include multiple optimal phenotypes and, thus, be used to study speciation.
Figure 1 .
1Order parameter, q vs µ, in the steady states for N = 1000 and ω = 0, 0.1, 0.2, 0.4, 0.5, 1, ∞ (from left to right). The continuous lines are guides for eyes.
Figure 3 .
3U 4 vs µ, for ω = 0.2 and different values of N . The crossing point is µc ∼ 0.0021. The continuous lines are guides for eyes.
Figure 4 .
4Critical mutation rate, µc (red circles) vs ω, compared with µc(N ), for N = 1000 (blue stars).
Figure 6 .
6Phase diagram in the plane (µ, 1/ω) for a system of size N = 1000 (see text for explanations). DS indicates the Disordered Phase, OS the Ordered Phase and CR the Coexistence Region. The continuous lines are guides for eyes.
Figure 7 .
7Main frame: Ht vs µ, in the steady states for N = 1000 and ω = 0, 0.1, 0.2 (from left to right). Inset: Hs vs µ, in the steady states for N = 1000 and ω = 0, 0.1, 0.2 (from left to right). The continuous lines are guides for eyes.
AcknowledgmentsThe authors would like to thank L. Peliti and A. Coniglio for valuable discussions.Author's contributions AF conceived the model, implemented the software and drafted the manuscript; SC and GM conceived the study, participated in its design and coordination, and helped to draft the manuscript. AM and GS contributed to the discussion of the results, and helped to draft the manuscript. All authors have read and approved the final manuscript.
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"Decoupling Graphene from SiC(0001) via Oxidation",
"Decoupling Graphene from SiC(0001) via Oxidation"
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"R M Tromp \nIBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY\n",
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"Z Chen \nIBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY\n",
"Y Sun \nIBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY\n",
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"IBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY",
"IBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY",
"IBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY",
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"IBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY",
"IBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY",
"IBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY",
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"IBM Research Division\nT.J. Watson Research Center\n10598Yorktown HeightsNY"
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| When epitaxial graphene layers are formed on SiC(0001), the first carbon layer (known as the "buffer layer"), while relatively easy to synthesize, does not have the desirable electrical properties of graphene. The conductivity is poor due to a disruption of the graphene -bands by covalent bonding to the SiC substrate. Here we show that it is possible to restore the graphene -bands by inserting a thin oxide layer between the buffer layer and SiC substrate using a low temperature, CMOS-compatible process that does not damage the graphene layer. | 10.1103/physrevb.82.041411 | [
"https://arxiv.org/pdf/1003.5702v1.pdf"
]
| 95,531,756 | 1003.5702 | 0025c22e78c34447ecf1a6d2539c61829ae005a2 |
Decoupling Graphene from SiC(0001) via Oxidation
S Oida
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
F R Mcfeely
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
J B Hannon
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
R M Tromp
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
M Copel
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
Z Chen
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
Y Sun
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
D B Farmer
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
J Yurkas
IBM Research Division
T.J. Watson Research Center
10598Yorktown HeightsNY
Decoupling Graphene from SiC(0001) via Oxidation
number: 6835-p6837Nq6172Nn6835Md
When epitaxial graphene layers are formed on SiC(0001), the first carbon layer (known as the "buffer layer"), while relatively easy to synthesize, does not have the desirable electrical properties of graphene. The conductivity is poor due to a disruption of the graphene -bands by covalent bonding to the SiC substrate. Here we show that it is possible to restore the graphene -bands by inserting a thin oxide layer between the buffer layer and SiC substrate using a low temperature, CMOS-compatible process that does not damage the graphene layer.
Following its experimental realization by Novoselov et al. 1 , graphene, one, or a very few, layers of carbon in hexagonal sp 2 -hybridized sheets, has been the subject of intensive investigation. Its unique electronic properties 2,3 have attracted great interest owing to potential applications in nanoelectronics 4,5 . Graphene films can be produced in a variety of ways, e.g., exfoliation of samples from pyrolytic graphite 1 , chemical vapor deposition (CVD) [6][7][8] , or sublimation of Si from SiC(0001) substrates [9][10][11][12] . From the standpoint of compatibility with current device fabrication processes, Si sublimation from SiC(0001) is particularly appealing: Wafer-sized graphene films of controlled thickness can be grown directly on a semi-insulating substrates. However, graphene growth on SiC(0001) 13 , either by sublimation of Si or by carbon CVD, has a serious drawback. The first graphene layer, while easy to grow uniformly, is non-conductive 14,15 . Thus from an electronic point of view, this layer is not graphene at all, but rather a "buffer layer" on which additional, electrically-active graphene must be grown. Band structure measurements by angle-resolved photoemission spectroscopy 16 and first-principles calculations 14,15 show disruption of the buffer layer -bands by strong covalent bonding to the SiC substrate.
Recently, it was shown that annealing in hydrogen at temperatures above 600 C can decouple the buffer layer from the the SiC substrate, resulting in the appearance of the graphene band structure 17 . Here we describe a lowtemperature oxidation process that accomplishes the same decoupling. When the buffer layer is exposed to oxygen at 250 C, an oxide layer of 3 Å is formed between the buffer layer and the SiC(0001) substrate. Surprisingly, this ultra-thin layer is sufficient to decouple the buffer layer from the substrate, restoring the -band structure characteristic of free-standing graphene. We correlate the existence of graphene-like bands with the appearance of the plasmon in electron energy loss spectroscopy (EELS).
Although it perhaps seems counter intuitive to attempt to improve the conductivity of a structure by oxidation, the formation of a SiO 2 decoupling layer between the graphene and the SiC substrate has a fair amount of a priori thermodynamic and kinetic plausibility. The free energy of formation of SiO 2 is more negative than that of CO 2 by approximately 100 kcal/mole at 500 K. Thus if a buffer layer / SiC structure were oxidized under such conditions so as to achieve thermodynamic equilibrium, essentially all of the oxygen reacted would be in the form of SiO 2 . Of course, this offers no guarantee that the desired structure can be synthesized, since equilibrium is not achievable under practical oxidation conditions. In fact, the equilibrium products of SiC oxidation are undesirable, as graphitic carbon, presumably highly disordered, would be produced in equimolar amounts to the SiO 2 . Instead the ideal to be sought is a kinetic regime in which graphene remains inert to the oxidant (e.g. O 2 ), SiC is oxidized to produce sufficient SiO 2 , and the carbon liberated from the oxidation of SiC is oxidized and carried away. While these requirements appear quite stringent, graphene is well known for its chemical inertness, and, as we show, no more than about a monolayer of SiO 2 is required to decouple the film from the substrate. In addition there is precedent for oxidation selectivity such as we desire between the graphene and the nascent carbon formed by SiC oxidation: when carbon nanotubes are grown from alcohol precursors it is believed that one of the roles of the oxygen is to scavenge any amorphous carbon formed in the pyrolyitc process 18 . In what follows we demonstrate all three of these requirements can be met by a process of low temperature, high pressure oxidation. By this means, the buffer layer can be electronically decoupled from the SiC substrate, restoring the -bands to a substantially unperturbed condition.
The synthesis of epitaxial graphene layers on SiC(0001) was carried out in an ultra-high vacuum system equipped with low-energy electron microscopy (LEEM) [11][12][13] . The graphene layers were formed at elevated temperature while the surface was imaged with LEEM. Details on the preparation of clean, flat SiC(0001) samples are given elsewhere 11,12 . Two different synthesis approaches were used, yielding essentially identical results. In the first process, the sample is annealed above 1300 C in a background pressure of disilane until the SiC decomposes, creating 1-3 ML of carbon in a controlled manner. In the second process, a small amount of ethylene (e.g. 1 × 10 −7 Torr) is added to the disilane background below the temperature at which SiC decomposes. A buffer layer film limited to a single carbon layer can be formed with this CVD approach. This latter method has the advantage that the CVD process is self limiting, yielding a single graphene buffer layer, with no possibility of producing additional graphene, which would complicate the analysis of the experiments. However, the nucleation rate of the graphene buffer layer during CVD growth is difficult to control, and the domain size of the CVD films can be significantly smaller than that of films grown by thermal decomposition. After synthesis, the graphene layer thickness was verified using the LEEM reflectivity method developed by Hibino et al. 19 . EELS was used to monitor the integrity of the -bands. The LEEM instrument employed for these experiments includes an energy filter, enabling us to obtain EELS spectra in situ from the same area of the surface that is imaged 20 . A focused ion beam (FIB) system was used to mill out alignment marks on the SiC substrate before graphene synthesis, which allowed us to obtain EELS spectra and images from a specific area of the surface, remove the sample from the LEEM chamber for oxidation, return it to the LEEM chamber and collect LEEM images and EELS spectra from exactly the same area of the sample. For reference, an EELS spectrum from a thick exfoliated graphene flake placed on SiC(0001) is shown in Fig. 1.
FIG. 1: Electron energy loss spectra recorded from (a) a thick graphite flake placed on SiC(0001) (black), (b) a graphene buffer layer on SiC(0001) before oxidation (blue), and (c) the same buffer layer after oxidation (red). All spectra were recorded using 33 eV electrons at near-normal incidence (q ~ 0). The incident electron energy was 33 eV and the scattering geometry was such that both the incident and scattered beams were approximately normal to the surface (q~0). The feature near 6.2 eV loss energy corresponds to the surface -plasmon of graphite [21][22][23] . A spectrum obtained under identical scattering conditions from a graphene buffer layer synthesized on SiC via CVD is also shown. As expected, owing to the disruption of the graphene -bands, no plasmon loss features are observed, confirming that the electronic structure of the covalently-bonded buffer layer is different from that of graphene.
Following LEEM image collection and selected area EELS characterization, the sample was removed from the LEEM and atomic force microscopy (AFM) images were obtained from the same area in which the EELS spectra were obtained. It was then introduced into an oxidation chamber connected to an x-ray photoemission (XPS) spectrometer. The sample was oxidized in 1 atm of O 2 at 250 C for 5 s. The effect of the oxidation on the sample was characterized using XPS, as illustrated in Fig. 2.
FIG. 2: XPS spectra of the (a) Si 2p and (b) C 1s core levels from a buffer layer grown on SiC(0001). The bottom spectrum in each panel is from the buffer layer before oxidation. The middle spectra are from the buffer layer after oxidation. The top spectra are from a thick graphene film grown on SiC(0001) via high-temperature sublimation.
In each panel, the bottom spectra show the Si 2p and C 1s core levels of the buffer layer sample after growth but prior to oxidation. The Si 2p region shows a single peak corresponding to Si in SiC. (Some slight tailing to higher binding energy is observed due to a small amount of oxide present on this airexposed sample.) The C 1s region shows a peak at 283.42 eV binding energy corresponding to carbidic carbon and a second peak shifted to higher binding energy by 1.81 eV, corresponding to buffer layer carbon. After oxidation, the spectra in the middle rows are obtained. The Si 2p spectrum shows a weak satellite feature at higher binding energy, which we attribute to the formation of oxidized Si. (The O 1s spectrum, not shown, confirms the uptake of oxygen by the system.) From the area of the weak satellite feature, we estimate the thickness of this oxidized layer, analyzed as SiO 2 , to be surprisingly thin: no more than about 3 Å. Quantitative ion scattering measurements to determine the oxide thickness more precisely are described below. For these oxidation conditions, the growth of the oxide saturates within seconds and there is little qualitative difference between samples oxidized for 5 s or 1 hr. In the C 1s spectrum, the graphene buffer layer peak has shifted by approximately 0.26 eV towards lower binding energy with respect to the carbidic carbon peak. This is hardly surprising, since, as we shall show below, the valence electronic structure of the graphene layer has undergone a dramatic change. However we note that the graphenic carbon intensity is unchanged from the unoxidized sample, indicating that within sensitivity of the XPS measurements (about 5%) the graphene layer is not chemically attacked by the oxygen, and any carbon liberated from the SiC via oxidation is removed from the sample by the oxygen, either as CO or as CO 2 . The latter conclusion is reinforced by experiments on the oxidation of clean SiC(0001), which show that under the above oxidation conditions, the formation of SiO 2 proceeds without buildup of graphitic or amorphous carbon on the surface.
In addition to the changes in shape in the Si 2p and C 1s spectra upon oxidation, both spectra shift rigidly to lower binding energy. This is indicative of a band bending effect caused by the introduction of negative charge in the surface region. We compare this oxidationinduced band bending of the middle row of Fig. 2 with the band bending resulting from growing graphene multilayers on SiC made via SiC decomposition, shown in the top panel. We note that the band bending induced by adding electrically active graphene onto the surface, shown in the top panel, is similar to the band bending produced by low temperature oxidation. An obvious interpretation of this coincidence is that following oxidation the graphene has become electronically decoupled from the substrate, and has become electrically active, exhibiting roughly the same local chemical environment (e.g. doping level) as few-layer graphene. After oxidation, subsidiary experiments were performed which showed that flash heating to 1200 C causes the oxide to decompose. Analysis of the C 1s spectra of these samples before and after flashing shows no significant difference in the intensities of the graphenic carbon peaks before and after flashing, indicating that the graphene layer is unaffected by this process, an observation we shall exploit below.
Selected-area low-energy electron diffraction (LEED) and ex situ AFM images, recorded from the same area of the surface before and after oxidation, are shown in Fig. 3.
FIG. 3: LEED patterns (a,c) and AFM images (b,d) recorded from a graphene buffer layer on SiC(0001) before (a,b) and after (c,d) oxidation of the SiC substrate. The AFM images are nearly identical, while the diffraction pattern show that strong coupling to the SiC lattice is lifted by the oxidation.
The AFM images are virtually identical, demonstrating that the graphene is not consumed during the oxidation of the substrate. Note that the defect features, such as the small holes in the graphene layer (arising from the high nucleation rate of the graphene during CVD), have identical shapes and sizes before and after oxidation. This "edge graphene" would certainly be the most reactive feature of the buffer layer, and even it is apparently unaffected. In addition, we observe no features which could be attributed to silicon oxide on the surface (e.g. hillocks or protrusions), which suggests that the oxide formed is sub-surface, as desired. While AFM suggests no significant change in the surface morphology, the LEED patterns are quite different, suggesting a decoupling of the buffer layer from the substrate. Before oxidation, the expected 63 × 63 diffraction pattern of the buffer layer is observed. The fractional-order spots arise from double diffraction from the SiC(0001) and graphene lattices 9 , consistent with a strong coupling of the graphene buffer layer to the substrate. However, following oxidation, the fractional-order spots are extinguished. The pattern corresponds to a superposition of diffraction from graphene and from SiC(0001), indicative of a weaker coupling to the substrate, e.g. due to the formation of a thin amorphous silicon oxide layer between the graphene layer and the substrate. Similar changes in the LEED pattern are observed during H intercalation 17 .
EELS spectra recorded after oxidation suggest that the buffer layer has adopted the electronic structure of graphene. The EELS spectrum (Fig. 1c)) exhibits a loss feature at 6.2 eV, where none was present before (Fig. 1b). This feature occurs at the same energy as the plasmon feature seen on the multilayer graphene flake (Fig. 1a). From this observation we conclude that the oxidation process has indeed decoupled the buffer layer from the substrate and restored the graphene-like bands.
The XPS and AFM data suggest that the oxidation process results in a very thin oxide layer under the buffer layer. In order to directly determine both the oxygen content and the location of the oxygen relative to the graphene, we performed structural measurements using medium energy ion scattering (MEIS) 24 . Using this highresolution form of Rutherford backscattering, we measured the depth profiles for oxygen, carbon, and silicon, verifying that the oxygen accumulates in a thin SiO 2 layer underneath the graphene. For these experiments SiC(0001) samples with a 1-2 graphene layers were prepared via sublimation and characterized using XPS. In Fig. 4 we show typical data for a sample before and after oxidation, taken using a normally incident beam of 100 keV protons. The data for each element have been replotted on a depth scale, and the intensities have been nornmalized to the cross sections.
FIG. 4: Medium energy ion scattering spectra for graphene/SiC(0001) before (a) and after oxidation (b). Contributions from carbon, silicon and oxygen have been shifted and plotted on the same depth scale. After oxidation, the oxygen leading edge occurs deeper than the carbon edge, showing that the oxygen is subsurface.
Before oxidation (Fig. 4a), a large surface carbon peak is seen, caused by the graphene layer. For this sample, the graphene film thickness corresponded to roughly 2 ML (the buffer layer and one additional carbon layer). Deeper into the sample, the carbon intensity drops to the same level as the subsurface silicon peak, representing the contribution from the outermost layers of the SiC substrate. Only a small oxygen peak is observed, due to ambient exposure during transfer to the MEIS system. After oxidation (Fig. 4b), a more pronounced oxygen peak is observed. Note that the leading edge of the oxygen signal occurs deeper than the carbon edge, demonstrating that the oxygen located below the surface. The carbon peak has a slightly smaller intensity in the oxidized spectrum, which we attribute to non-uniformity in the graphene film thickness, consistent with XPS measurements which suggest that the thickness varies from 1-2 ML over this sample. Quantitative modeling of the spectrum supports the idea of an oxide-supported graphene film; we were able to accurately fit the results with of the unoxidzed sample as 1.9 layers of Gr/SiC(001) and after oxidation as 1.9 layers of Gr/ 3.4 Å SiO 2 / SiC(0001).
In summary, we have shown that the covalent bonding of the graphene buffer layer to the SiC(0001) substrate can be lifted by the insertion of an ultra-thin (~3 Å) oxide layer between the graphene and the substrate. The activated buffer layer exhibits the -plasmon characteristic of graphene, showing that the band structure of graphene has been largely recovered. The lowtemperature oxidation method offers potential advantages for the device fabrication. It is simple to implement, can be carried out on pre-fabricated devices (i.e. with metal contacts in place), and is compatible with conventional CMOS processes. This work was supported by DARPA under contract FA8650-08-C-7838 through the CERA program.
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| []
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[
"Constraining dark energy with Hubble parameter measurements: an analysis including future redshift-drift observations",
"Constraining dark energy with Hubble parameter measurements: an analysis including future redshift-drift observations"
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| [
"Rui-Yun Guo \nDepartment of Physics\nCollege of Sciences\nNortheastern University\n110004ShenyangChina\n",
"Xin Zhang \nDepartment of Physics\nCollege of Sciences\nNortheastern University\n110004ShenyangChina\n\nCenter for High Energy Physics\nPeking University\n100080BeijingChina\n"
]
| [
"Department of Physics\nCollege of Sciences\nNortheastern University\n110004ShenyangChina",
"Department of Physics\nCollege of Sciences\nNortheastern University\n110004ShenyangChina",
"Center for High Energy Physics\nPeking University\n100080BeijingChina"
]
| []
| The nature of dark energy affects the Hubble expansion rate (namely, the expansion history) H(z) by an integral over w(z). However, the usual observables are the luminosity distances or the angular diameter distances, which measure the distance-redshift relation. Actually, the property of dark energy affects the distances (and the growth factor) by a further integration over functions of H(z).Thus the direct measurements of the Hubble parameter H(z) at different redshifts are of great importance for constraining the properties of dark energy. In this paper, we show how the typical dark energy models, for example, the ΛCDM, wCDM, CPL, and holographic dark energy models, can be constrained by the current direct measurements of H(z) (31 data used in total in this paper, covering the redshift range of z ∈ [0.07, 2.34]). In fact, the future redshift-drift observations (also referred to as the Sandage-Loeb test) can also directly measure H(z) at higher redshifts, covering the range of z ∈ [2, 5]. We thus discuss what role the redshift-drift observations can play in constraining dark energy with the Hubble parameter measurements. We show that the constraints on dark energy can be improved greatly with the H(z) data from only a 10-year observation of redshift drift. 2 and modified gravity (MG) is a major mission in modern cosmology. A basic strategy is to accurately measure the both histories of cosmic expansion and growth of structure and to compare them for a consistency check.The main property of dark energy is characterized by its equation-of-state parameter (EoS) w(z). In fact, dark energy affects the expansion history and growth of structure of the universe in a subtle way. To measure the history of the cosmic expansion, the most important way is to measure the distance-redshift relation.For example, through the observations of type Ia supernovae, one measures the luminosity distances at different redshifts, and through the observations of baryon acoustic oscillations (BAO), one measures the angular diameter distances at different redshifts. The cosmic distance, whether the luminosity distance or the angular diameter distance, is linked to the Hubble expansion rate H(z) through an integration, namely, D c (z) = z 0 dz /H(z ), where D c (z) is the comoving line-of-sight distance to an object at redshift z in a flat universe. The luminosity distance D L (z) and the angular diameter distance D A (z) can be expressed as | 10.1140/epjc/s10052-016-4016-x | [
"https://arxiv.org/pdf/1512.07703v4.pdf"
]
| 118,843,944 | 1512.07703 | ad07936828e007aecfd26d0fc9b78a8916f5123e |
Constraining dark energy with Hubble parameter measurements: an analysis including future redshift-drift observations
Rui-Yun Guo
Department of Physics
College of Sciences
Northeastern University
110004ShenyangChina
Xin Zhang
Department of Physics
College of Sciences
Northeastern University
110004ShenyangChina
Center for High Energy Physics
Peking University
100080BeijingChina
Constraining dark energy with Hubble parameter measurements: an analysis including future redshift-drift observations
The nature of dark energy affects the Hubble expansion rate (namely, the expansion history) H(z) by an integral over w(z). However, the usual observables are the luminosity distances or the angular diameter distances, which measure the distance-redshift relation. Actually, the property of dark energy affects the distances (and the growth factor) by a further integration over functions of H(z).Thus the direct measurements of the Hubble parameter H(z) at different redshifts are of great importance for constraining the properties of dark energy. In this paper, we show how the typical dark energy models, for example, the ΛCDM, wCDM, CPL, and holographic dark energy models, can be constrained by the current direct measurements of H(z) (31 data used in total in this paper, covering the redshift range of z ∈ [0.07, 2.34]). In fact, the future redshift-drift observations (also referred to as the Sandage-Loeb test) can also directly measure H(z) at higher redshifts, covering the range of z ∈ [2, 5]. We thus discuss what role the redshift-drift observations can play in constraining dark energy with the Hubble parameter measurements. We show that the constraints on dark energy can be improved greatly with the H(z) data from only a 10-year observation of redshift drift. 2 and modified gravity (MG) is a major mission in modern cosmology. A basic strategy is to accurately measure the both histories of cosmic expansion and growth of structure and to compare them for a consistency check.The main property of dark energy is characterized by its equation-of-state parameter (EoS) w(z). In fact, dark energy affects the expansion history and growth of structure of the universe in a subtle way. To measure the history of the cosmic expansion, the most important way is to measure the distance-redshift relation.For example, through the observations of type Ia supernovae, one measures the luminosity distances at different redshifts, and through the observations of baryon acoustic oscillations (BAO), one measures the angular diameter distances at different redshifts. The cosmic distance, whether the luminosity distance or the angular diameter distance, is linked to the Hubble expansion rate H(z) through an integration, namely, D c (z) = z 0 dz /H(z ), where D c (z) is the comoving line-of-sight distance to an object at redshift z in a flat universe. The luminosity distance D L (z) and the angular diameter distance D A (z) can be expressed as
I. INTRODUCTION
In 1998, two observation teams independently found that the universe is currently undergoing an accelerating expansion, through the observations of type Ia supernovae [1,2]. Though the statistical significance was not high enough, the supernovae evidence for cosmic acceleration was quickly accepted by the community at large because the subsequent observations of cosmic microwave background (CMB) [3,4] and large-scale structure (LSS) [5,6] soon provided substantial independent evidence supporting the conclusion of supernovae observations. If the theory of general relativity (GR) is valid on all scales of the universe, the fact of cosmic acceleration implies that a new energy component with negative pressure, referred to as "dark energy" [7][8][9][10][11][12][13][14][15][16], is needed in the universe. However, there still exists another possibility: that the cosmic acceleration arises from a breakdown of GR on cosmological scales. To distinguish between dark energy D L = (1 + z)D c and D A = (1 + z) −1 D c , respectively. In fact, the linear growth factor also involves a further integration over a function of H(z).
Furthermore, the property of dark energy affects the Hubble expansion rate H(z) also through an integral, namely, in a flat universe, we have
H 2 (z) H 2 0 = Ω r (1 + z) 4 + Ω m (1 + z) 3 + (1 − Ω r − Ω m )X(z),(1)
where Ω r and Ω m are the current density parameters of radiation and matter, respectively, and X(z) describes how dark energy density evolves with redshift, X(z) ≡ ρ de (z)/ρ de (0) = exp 3 z 0 1 + w(z ) 1 + z dz .
Therefore, it is extremely difficult to constrain the property of dark energy using the measurements of cosmic distances and growth rate of structure, because there are two integrals between these observables and w(z). Obviously, to accurately constrain the history of dark energy evolution, a more important way is to directly measure the Hubble parameter H(z), owing to the fact that between H(z) and w(z) there is only one integral. While difficult, a number of measurement data of H(z) have been accumulated and studied in recent years .
Through two astrophysical methods, namely, the measurement of differential age of galaxies and the measurement of clustering of galaxies or quasars, more than 30 observational data of H(z) have been obtained [18, 21, 23, 29, 30, 32, 33, 38-41, 45, 46]. One of the major aims of this paper is to have a look at how these H(z) data can constrain dark energy. We perform such an analysis by taking several typical dark energy models as examples. We only focus on the expansion history of the universe, thus we do not consider MG models in this paper. Since the current observations show that the spatial curvature of the universe is very small, |Ω k | O(10 −3 ) [49], we only consider a flat universe in the analysis of this paper.
The current data of H(z) are all in the range of z 2. Obviously, measuring H(z) at higher redshifts could provide additional accurate information as regards Ω m h 2 , thus helping break the low-redshift parameter degeneracies, which is of great importance to constrain the property of dark energy. Recently, there have been a number of works discussing the observations of redshift drift [50][51][52][53][54][55][56][57][58][59][60][61][62][63], which probe the expansion history of the universe in the "redshift desert" of 2 z 5. Through monitoring the shift of Lyman-α forest absorption line of a distant quasar over a period of a few decades, one can detect the time variation of its redshift, namely, the redshift drift. This is equivalent to measure the Hubble parameter at a high redshift.
This method is also referred to as the "Sandage-Loeb test" (SL test) [64,65]. The highly accurate COsmic Dynamics EXperiment (CODEX) spectrograph on the 39m Extremely Large Telescope (ELT) being built is expected to perform such a task [53]. The forecast analyses of using the redshift-drift observations to constrain dark energy have been recently done in a number of work [50][51][52][53][54][55][56][57][58][59][60][61][62][63]. The combination of SL test data and current Hubble parameter data was also preliminarily discussed in [61]. In this paper, we wish to perform an uniform analysis for several popular, typical dark energy models, by combining the current H(z) data with the future high-redshift H(z) data from the redshift-drift observations.
The simplest candidate for dark energy is the "cosmological constant" Λ proposed by Einstein, of which the corresponding cosmological model is the Λ cold dark matter (ΛCDM) model. The ΛCDM model is very simple and is favored by the current cosmological observations, in particular, the observation of the Planck satellite mission [49], thus it is widely viewed as a prototype of the standard cosmological model.
However, actually, current observations have not excluded the dynamical dark energy models, and in fact the ΛCDM model needs to be tested further in a more accurate manner. Thus it is extremely important to probe the dynamics of dark energy. The simplest extension to Λ is the dark energy with a constant w, of which the corresponding cosmological model is the so-called wCDM model. The shortcoming of this model is that the constant w is usually viewed unphysical or unreal. To consider a model with time-varying w, the most popular way is to parametrize w(a) in the form of w(a) = w 0 + w a (1 − a), which is often called the Chevallier-Polarski-Linder (CPL) model [66,67]. However, the CPL model has an evident shortcoming that it has two more additional parameters than ΛCDM, which adds enormous complexities leading to the fact that w 0 and w a (in particular w a ) are very difficult to be well constrained. To remain the same number of parameters with wCDM and to simultaneously consider the evolution of w, we take the holographic dark energy (HDE) model [68][69][70] into account. The HDE model originates from the consideration of the holographic principle of quantum gravity, and it can fit the observational data fairly well [71][72][73][74][75][76][77][78][79][80][81], thus it is a rather competitive model among the many dark energy models [75,76,81]. Therefore, in this paper, in order to make a comprehensive analysis, we take the ΛCDM, wCDM, CPL, and HDE models as typical examples.
The organization of this paper is as follows. In Sect. II, we describe the current measurements of the The Hubble parameter H(z) is defined to be the rate of the relative expansion of the universe,
H(z) =ȧ a = − 1 1 + z dz dt ,(3)
where a is the cosmic scale factor andȧ is its rate of change with respect to the cosmic time t. H(z) is usually expressed in the unit of km s −1 Mpc −1 . Directly measuring H(z) is always a major challenge in modern cosmology.
In recent years, enormous efforts have been made in the measurements of H(z). Currently, more than 30 H(z) data have been accumulated, from two kinds of different measurement methods. The first method was proposed by Jimenez and Loeb [17] in 2002. One could take the passively evolving galaxies as standard cosmic chronometers whose differential age evolution as a function of the redshift can directly probe H(z), as is given by the second equal sign of Eq. (3). This method is usually called differential age method, abbreviated as "DA" method in this paper. We use 25 data obtained from the DA method through more than 10 years' effort, as listed in Table I. These data include eight new measurements of H(z) in 2012 [29] with smaller error bars compared to the earlier data [28]. In the current literature [28,32,35], it has been shown that the constraints from them on cosmological models are almost equal to those from current type Ia supernova apparent magnitude versus redshift data. Besides, we add two latest H(z) data obtained in 2015 [46], up to z ∼ 2 (z = 1.363 and z = 1.965). It has been shown [46] that there is a detectable improvement (∼ 5%) on Ω m and w compared to previous measurements when they are used to estimate the accuracy on cosmological parameters in the ΛCDM and wCDM models.
The second popular way to directly measure H(z) is through the clustering of galaxies or quasars.
Hereafter, this approach is called "Clustering" for convenience. One could get a direct measurement of H(z) by using the BAO peak position as a standard ruler in the radial direction [21]. Through the BAO detection, methods will not be discussed in detail in this paper; for more details, see Refs. [21,30,32,41,45].
Importantly, we use the latest BAO measurement H(z) = 96.8 ± 3.4 km s −1 Mpc −1 at z = 0.57 [41] and H(z) = 222 ± 7 km s −1 Mpc −1 at z = 2.34 [45] instead of the previous measurements at the same redshifts. The total six "Clustering" measurements of H(z) are also listed in Table I.
Note here that in this paper we adopt most of the compilation of the current H(z) data from Ref. [44].
In Ref. [44], the sources of these H(z) data are clearly given, and the statistical and systematical errors are discussed in detail. The other updated H(z) data are also discussed in Refs. [41,45,46] in detail. For the utilization of the data from BAO, some authors thought that they are not totally model-independent and thus may not be used in the cosmological parameter constraints [20,[82][83][84][85][86][87][88]. We admit that there are indeed some problems in the utilization of the H(z) data, but these are not the focus of this paper. The main aim of this paper is to have a look at how the future redshift-drift measurements can improve the constraints on cosmological parameters with the H(z) data alone. In order not to deviate from the main aim of this paper, we do not address these issues in this paper.
We plot these H(z) data points in the left panel of Fig. 1. The 25 data points from the "DA" measurement are in the range of 0.07 ≤ z ≤ 1.965 and the six data points from the "Clustering" measurement are in the range of 0.35 ≤ z ≤ 2.34. For these data points, the highest redshift is z = 2.34 [45], corresponding to the point obtained from the BAO measurement in the Ly-α forest of BOSS quasars, in the "Clustering"
dataset. (Note that using this high-redshift measurement, the evidence of evolving dark energy has been demonstrated in Ref. [43].) The other five points in the "Clustering" dataset are all in the range of z ∈ [0.35, 0.73]. Comparing these data points in Fig. 1, we apparently find that the error bars of points from the "Clustering" dataset are much less than those from the "DA" dataset.
In order to constrain the cosmological models with these H(z) data points, we need to perform a χ 2 statistical analysis. The χ 2 function of this analysis is given by
χ 2 H (p) = N i=1 [H th (z i ; p) − H obs (z i )] 2 σ 2 H,i ,(4)
where N denotes the number of data points, z i is the redshift at which H(z i ) has been measured, p represents This shows that solely using the current H(z) measurements could provide rather tight constraints on the cosmological parameters.
In this paper, we study how accurate high-redshift H(z) data could be provided by the future redshiftdrift observation and how these data would impact on constraining dark energy with the H(z) measurements alone.
The redshift-drift observation, sometimes called the "SL test", is not only conceptually simple, but also is a direct probe of cosmic dynamic expansion, although being observationally challenging. We adopt an experiment like CODEX [53] to perform a forecast analysis for the predicted accuracy of observations. The major observation facilities, e.g. ELT, aim at directly measuring the accelerating expansion of the universe by detecting the cosmological redshift drift of the Lyman-α forest from QSOs lying in 2 z 5.
The main observation of SL test is the redshift variation, expressed as a spectroscopic velocity shift [65],
∆v = ∆z 1 + z = H 0 ∆t o 1 − E(z) 1 + z ,(5)
where ∆t o is the time interval of observation, and E(z) = H(z)/H 0 is decided by specific cosmological models. According to the performance of the Monte Carlo simulations of Lyman-α absorption lines, the uncertainty on ∆v can be written as [53] σ ∆v = 1.35 S /N 2370 show the accuracy of the 10-year H(z) data from the SL test, we plot the forecast data points in Fig. 2; for a convenient display, we show the H(z)/(1 + z) plots. The fiducial models for simulating the forecast data are chosen to be the ΛCDM, wCDM, and CPL models in this example, as shown in the three panels of 10-year SL H(z) point has. This implies that the SL test would play a more important role for constraining the models with more parameters.
−1 N QSO 30 −1/2 1 + z QSO 5 x cm s −1 ,(6)
In the mock data simulation, we adopt the scheme accordant with our previous papers [59,60,62,63],
i.e., we choose the best-fitting specific dark energy model in study as the fiducial model to produce the simulated H(z) data. The best fit of the dark energy model is given by the current H(z) data. This aims to avoid the potential tension between the current H(z) data and the simulated future H(z) data. In most papers on the redshift-drift observation [50-54, 56-58, 61], the fiducial model for simulating data is chosen to be the ΛCDM model no matter what dark energy model is in study, which sometimes leads to the evident tension between the current data and the simulated data in the combined analysis. Our scheme can efficiently avoid such a problem. In the following, we use the current and future H(z) data to uniformly constrain the typical dark energy models and study what role the high-redshift H(z) measurement from the 10-year SL test would play in constraining dark energy with the H(z) data alone.
III. CONSTRAINTS ON DARK ENERGY MODELS FROM HUBBLE PARAMETER MEASUREMENTS INCLUDING REDSHIFT-DRIFT OBSERVATIONS
In the section, we study the capability of the H(z) measurements in constraining dark energy models.
First, we study how the current 31 H(z) data can be used to constrain the typical dark energy models. Then we use the each best-fitting dark energy model itself as the fiducial model to produce the simulated mock We choose four specific dark energy models as representatives of cosmological models to make the analysis. They are the ΛCDM, wCDM, CPL, and HDE models. In the ΛCDM model, the EoS of dark energy is fixed to be w = −1. In the wCDM model, the EoS of dark energy, w, is a constant. In the CPL model, the EoS of dark energy is parametrized as w(z) = w 0 + w a z 1+z [66,67]. In the HDE model, the EoS of dark energy is given by w(z) = −1/3 − (2/3c)
√ Ω de (z) [68], where c is a dimensionless parameter and the function Ω de (z) is the solution to the differential equation [68], with the prime denoting the derivative with respect to ln a.
Ω de = Ω de (1 − Ω de )[1 + (2/c) √ Ω de ]
We constrain the four dark energy models by using the current H(z) data and the combination of the current and SL 10-year H(z) data. The fit results are given in Table II. We find that, using the current H(z) precisions of all the parameters, in particular the parameter Ω m .
σ(w 0 ) − 0.2541 0.3842 − − 0.2152 0.2671 − σ(w a ) − − 1.7619 − − − 0.4923 − σ(c) − − − 1.0145 − − − 0.4482 σ(Ω m )
In order to quantify the improvements, we list the errors and constraint precisions of parameters in the four models for the fits to the current H(z) data and the current + SL 10-year H(z) data, in Table III. Based on the best-fit value and the error of the parameter in the fit, we can evaluate the constraint precision of the parameter. For a parameter ξ, one can define the constraint precision as ε(ξ) = σ(ξ)/ξ bf , where ξ bf denotes the best-fit value of ξ.
We find that the precision of Ω m can be enhanced by nearly one order of magnitude when the SL 10-year H(z) data are combined. Concretely, the precision of Ω m is improved from 11.57% to 2.11% for ΛCDM, from 13.75% to 3.12% for wCDM, from 100.20% to 5.41% for CPL, from 13.88% to 5.20% for HDE. The constraint precision of the parameter h is also evidently enhanced for all the four models; for details, see √ Ω de (z) [68]. Clearly, in the early times (z → ∞
and Ω de → 0), one has w(z → ∞) = −1/3, and in the far future (z → −1 and Ω de → 1), one has w(z → −1) = −1/3 − 2/3c; thus the HDE model does not involve the ΛCDM model. In Fig. 5, we show the reconstructed evolutions of w(z) for CPL and HDE with 1σ and 2σ errors obtained from the H(z)+SL
10-year data. We find that it is possible to differentiate dynamical dark energy from ΛCDM by only using the H(z) measurements in the future.
IV. CONCLUSION
The direct measurements of the Hubble parameter at different redshifts are vitally important for constraining the property of dark energy. Usually, the constraints on dark energy are often provided by the distance-redshift relation measurements, but the distance (luminosity distance or angular diameter distance)
is linked to dark energy by an integral over 1/H(z), and H(z) is affected by dark energy via another integral over w(z). Thus using the distance measurements to constrain the history of w(z) is extremely difficult, but using the H(z) measurements to constrain the dark energy is much simpler and more feasible.
Though directly measuring H(z) is a challenging task, in recent years some H(z) data have been accumulated through the great efforts of astronomers. Up to now, we have about 31 H(z) data in total, covering the redshift range of z ∈ [0.07, 2.34]. In these data, about 25 data points come from the "DA" measurement (0.07 ≤ z ≤ 1.965) and about six data points come from the "Clustering" measurement (0.35 ≤ z ≤ 2.34).
We show that the two datasets of H(z) are consistent with each other, and solely using the current H(z) data (the combination of the two datasets) could provide fairly good constraints on the typical dark energy models.
In addition, the future redshift-drift observations (i.e., the SL test) could actually also directly measure H(z) at higher redshifts, covering the redshift range of z ∈ [2,5]. Thus we also discuss what role the redshiftdrift observation can play in constraining dark energy with the Hubble parameter measurements. We choose four specific dark energy models as typical examples to make an analysis. They are the ΛCDM, wCDM, CPL, and HDE models. We consider a 10-year observation of redshift drift and produce 30 simulated H(z) data at the redshift range of z ∈ [2,5]. We show that the constraints on the dark energy models can be improved greatly when the high-redshift H(z) data from only a 10-year observation of redshift drift are combined. We expect that the redshift-drift observation would be successfully implemented and the accurate high-redshift H(z) data could be obtained to make great contribution to the study of dark energy.
FIG. 1 :
1Left: The current Hubble parameter measurements data (31 points referenced in total), where 25 H(z) data points (0.07 ≤ z ≤ 1.965) come from the "DA" measurement and six data points (0.35 ≤ z ≤ 2.34) come from the "Clustering" measurement. In these data points, the highest redshift is z = 2.34, corresponding to the point obtained from the BAO measurement in the Ly-α forest of BOSS quasars [45], in the "Clustering" dataset; whereas all the other five points in the "Clustering" dataset are in the range of z ∈ [0.35, 0.73]. Right: The constraints on the ΛCDM model with the current H(z) measurements (the 68% and 95% CL contours are shown in the Ω m -h plane). We show the constraints from the "DA" and "Clustering" datasets, separately, and we also show the constraints from the combination of the two. a measurement of D V = D 2/3 A (z/H(z)) 1/3 was obtained with a combination of the Hubble parameter H(z) and the angular diameter distance D A (z). Then one can measure H(z) and D A (z) through a variety of scientific methods [21, 30, 32, 41, 45]. For example, Gaztanaga et al. [21] separated the clustering of the LRG sample in the SDSS DR6 and DR7 into the line-of-sight and transverse information, and obtained H(z) = 79.69 ± 2.65 km s −1 Mpc −1 at z = 0.24 and H(z) = 86.45 ± 3.68 km s −1 Mpc −1 at z = 0.43. But the two data are not used in our analysis (consistent with [35-37, 47]) because they have unreasonably small error bars, causing a strong controversy in the existing papers [22, 24, 27]; Blake et al. [30] extracted H(z) and D A (z) by combining the acoustic parameter A(z) ∝ [D 2 A (z)/H(z)] 1/3 and the Alcock-Paczynski distortion parameter F(z) ∝ D A (z)H(z), and obtained H(z) = 82.6 ± 7.8 km s −1 Mpc −1 at z = 0.44, H(z) = 87.9 ± 6.1 km s −1 Mpc −1 at z = 0.6, and H(z) = 97.3 ± 7.0 km s −1 Mpc −1 at z = 0.73; and so on. Detailed separation
model parameters, H th and H obs are the predicted value of H(z) in the cosmological model and the measured value, respectively, and σ H,i is the standard deviation of the ith point.Since the ΛCDM model is widely viewed as a prototype of the standard cosmology, we take this model as a reference model to test the consistency of the two datasets of H(z) measurements. In the right panel ofFig. 1, we plot the two-dimensional posterior contours (68% and 95% confidence level) in the Ω m -h plane of the ΛCDM model using the DA and Clustering data of H(z). We find that the two datasets are rather consistent with each other. Although the number of data points is less, the constraining power of the "Clustering" set is evidently better than the "DA" dataset. The combination of the two datasets provides a much tighter constraint on the cosmological model; see the red contours in this figure. The combined H(z) measurements with 31 data in total give the fit results: Ω m = 0.2654 +0.0325 −0.0287 and h = 0.7043 +0.0241 −0.0246 , constraining the parameter Ω m to the precision of ∼ 11.57% and the parameter h to the precision of ∼ 3.46%.
where S /N = 3000 is defined as the spectral signal-to-noise per 0.00125 nm pixel, N QSO and z QSO are the number and redshift of QSOs, respectively. In addition, the last exponent x = −1.7 for 2 z 4 and x = −0.9 for z > 4. In our simulation, 30 SL test data are chosen to uniformly distribute over six redshift bins of z QSO ∈ [2, 5] (namely, the redshift interval ∆z = 0.5 for each bin), by observing 30 bright QSOs at high redshifts. The observation of the redshift drift is equivalent to the observation of the Hubble parameter, since we have the simple relationships: H(z) = (H 0 − ∆v/∆t o )(1 + z) and σ H = (1 + z)σ ∆v /∆t o . Thus we can use the SL test to simulate 30 mock H(z) data in the redshift range of z ∈ [2, 5]. In this paper, we choose to consider a 10-year observation of redshift drift (∆t o = 10 year) to make the analysis, because in our opinion a 10-year forecast is fairly proper and meaningful for our study of the H(z) constraints on dark energy. To
Fig. 2 .FIG. 2 :
22The black curve with cyan band represents the best fit with 1σ uncertainty, reconstructed from the current H(z) measurements (31 data in total). We find that the higher redshift is, the smaller error bar the The evolutions of H(z)/(1 + z) in the ΛCDM, wCDM, and CPL models. The black curves and cyan bands (best fit with 1σ uncertainty) are reconstructed from the current H(z) measurements. The red error bars (1σ) on the black curves are estimated from the 10-year redshift-drift observation.
redshift H(z) data (z ∈[2,5]) from a 10-year redshift-drift observation and combine the current and future H(z) data to constrain the dark energy model. Our aim is to see how the future high-redshift H(z) measurements from SL test would improve the constraining power in the study of dark energy with the H(z) observations.
PrecisionFIG. 3 :
3ΛCDM wCDM CPL HDE ΛCDM wCDM CPL HDE , the ΛCDM model can be well constrained, but other dark energy models that have one or two more parameters than ΛCDM can only be loosely constrained. However, when the SL 10-year H(z) data are combined, all the constraint results are improved significantly.To see the improvements from the SL 10-year measurement visually, we show the constraint results inFigs. 3 and 4. In Fig. 3, we show the two-dimensional posterior distribution contours (68% and 95% CL) in the Ω m -h plane for the four dark energy models. The pink contours are from the constraints of current H(z) data and the blue contours are from the constraints of the combination of current and SL 10-year H(z) data.For all the cases, we find that the degeneracy directions are evidently changed by adding the SL 10-year data. InFig. 4, we show the two-dimensional marginalized contours in the Ω m -w plane for the wCDM model, in the w 0 -w a plane for the CPL model, and in the Ω m -c plane for the HDE model. From these figures, we clearly see that adding the SL 10-year data leads to significant improvements for the constraint Constraints (1σ and 2σ CL) on the ΛCDM, wCDM, CPL, and HDE models in the Ω m -h plane from the current H(z) measurements (pink contours) and current H(z) + future 10-year redshift-drift measurements (blue contours).
FIG. 4 :
4Constraints (1σ and 2σ CL) on the wCDM, CPL, and HDE models from the current H(z) measurements (pink contours) and current H(z) + future 10-year redshift-drift measurements (blue contours). We show the twodimensional marginalized contours in the Ω m -w plane for the wCDM model, in the w 0 -w a plane for the CPL model, and in the Ω m -c plane for the HDE model. For the models that contain ΛCDM as a sub-model, namely wCDM and CPL, the positions of the cosmological constant are clearly denoted.
FIG. 5 :
5The reconstructed evolutions of w(z) for CPL and HDE with errors (1σ and 2σ) obtained from the H(z)+ SL 10-year data. the H(z) data from an only 10-year observation of redshift drift are included. Among the four dark energy models analyzed in this paper, the CPL and HDE models could describe time-evolving EoS w(z). For the CPL model, there are two parameters, w 0 and w a , describing the property of dark energy, and the ΛCDM model is contained in this model as a sub-model with (w 0 , w a ) = (−1, 0). The HDE model is totally different from the CPL model; it has only one parameter (namely, c) to describe the property of dark energy, w(z) = −1/3 − (2/3c)
TABLE I :
IData of the Hubble parameter H(z) versus the redshift z, where H(z) and σ H are in units of km s −1 Mpc −1 .z
H(z) σ H Reference Method
0.07 69.0 19.6
[38]
DA
0.1 69.0 12.0
[23]
DA
0.12 68.6 26.2
[38]
DA
0.17 83.0 8.0
[23]
DA
0.179 75.0 4.0
[29]
DA
0.199 75.0 5.0
[29]
DA
0.2 72.9 29.6
[38]
DA
0.27 77.0 14.0
[23]
DA
0.28 88.8 36.6
[38]
DA
0.352 83.0 14.0
[29]
DA
0.4 95.0 17.0
[23]
DA
0.48 97.0 62.0
[23]
DA
0.593 104.0 13.0
[29]
DA
0.68 92.0 8.0
[29]
DA
0.781 105.0 12.0
[29]
DA
0.875 125.0 17.0
[29]
DA
0.88 90.0 40.0
[23]
DA
0.9 117.0 23.0
[23]
DA
1.037 154.0 20.0
[29]
DA
1.3 168.0 17.0
[23]
DA
1.363 160.0 33.6
[46]
DA
1.43 177.0 18.0
[23]
DA
1.53 140.0 14.0
[23]
DA
1.75 202.0 40.0
[23]
DA
1.965 186.5 50.4
[46]
DA
0.35 82.7 8.4
[32]
Clustering
0.44 82.6 7.8
[30]
Clustering
0.57 96.8 3.4
[41]
Clustering
0.60 87.9 6.1
[30]
Clustering
0.73 97.3 7.0
[30]
Clustering
2.34 222.0 7.0
[45]
Clustering
TABLE II :
IIFit results for the ΛCDM, wCDM, CPL, and HDE models using the current H(z) data and the H(z) + SL 10-year data.H(z)
H(z)+ SL 10-year
Parameter
ΛCDM
wCDM
ΛCDM
wCDM
w
−1 (fixed) −0.8174 +0.2519
−0.2563
−1 (fixed) −0.8151 +0.1884
−0.2391
Ω m
0.2654 +0.0325
−0.0287
0.2662 +0.0355
−0.0376
0.2654 +0.0056
−0.0055
0.2663 +0.0093
−0.0071
h
0.7043 +0.0241
−0.0246
0.6742 +0.0489
−0.0451
0.7042 +0.0126
−0.0126
0.6735 +0.0423
−0.0420
Parameter
CPL
HDE
CPL
HDE
w 0
−0.8591 +0.3981
−0.3697
−
−0.8531 +0.2689
−0.2652
−
w a
0.8583 +0.4995
−2.4411
−
0.8366 +0.4849
−0.4996
−
c
−
1.1383 +1.3681
−0.4323
−
1.1642 +0.5430
−0.3271
TABLE III :
IIIConstraint errors and precisions of parameters in the ΛCDM, wCDM, CPL, and HDE models for the fitsto the current H(z) data and the H(z) + SL 10-year data.
H(z)
H(z)+ SL 10-year
Error ΛCDM wCDM CPL HDE ΛCDM wCDM CPL HDE
Table III .
IIIFor the property of dark energy, the constraints are also improved moderately. For example, in the wCDM model, the precision of w is improved from 31.09% to 26.40%; in the CPL model, the precision of w 0 is improved from 44.72% to 31.30%; and in the HDE model, the precision of c is improved from 89.12% to 38.50%. In addition, for the CPL model, w a is loosely constrained by only using the current H(z) data (its precision is 205.28%), but when the SL 10-year observation is considered, w a can be constrained to 58.85%. Hence, we see that the H(z) constraints on dark energy models can be improved greatly when
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| []
|
[
"Inverse Optimal Control Adapted to the Noise Characteristics of the Human Sensorimotor System",
"Inverse Optimal Control Adapted to the Noise Characteristics of the Human Sensorimotor System"
]
| [
"Matthias Schultheis [email protected] ",
"Dominik Straub [email protected] ",
"Constantin A Rothkopf [email protected] ",
"\nCentre for Cognitive Science\nCentre for Cognitive Science Technical\nTechnical University of Darmstadt\nDarmstadtGermany\n",
"\nCentre for Cognitive Science Technical\nUniversity of Darmstadt\nDarmstadtGermany\n",
"\nUniversity of Darmstadt\nDarmstadtGermany\n"
]
| [
"Centre for Cognitive Science\nCentre for Cognitive Science Technical\nTechnical University of Darmstadt\nDarmstadtGermany",
"Centre for Cognitive Science Technical\nUniversity of Darmstadt\nDarmstadtGermany",
"University of Darmstadt\nDarmstadtGermany"
]
| []
| Computational level explanations based on optimal feedback control with signaldependent noise have been able to account for a vast array of phenomena in human sensorimotor behavior. However, commonly a cost function needs to be assumed for a task and the optimality of human behavior is evaluated by comparing observed and predicted trajectories. Here, we introduce inverse optimal control with signaldependent noise, which allows inferring the cost function from observed behavior. To do so, we formalize the problem as a partially observable Markov decision process and distinguish between the agent's and the experimenter's inference problems. Specifically, we derive a probabilistic formulation of the evolution of states and belief states and an approximation to the propagation equation in the linear-quadratic Gaussian problem with signal-dependent noise. We extend the model to the case of partial observability of state variables from the point of view of the experimenter. We show the feasibility of the approach through validation on synthetic data and application to experimental data. Our approach enables recovering the costs and benefits implicit in human sequential sensorimotor behavior, thereby reconciling normative and descriptive approaches in a computational framework. | null | [
"https://arxiv.org/pdf/2110.11130v1.pdf"
]
| 239,050,542 | 2110.11130 | 301491287876a0f9034c763ebe69b2ee436c8215 |
Inverse Optimal Control Adapted to the Noise Characteristics of the Human Sensorimotor System
Matthias Schultheis [email protected]
Dominik Straub [email protected]
Constantin A Rothkopf [email protected]
Centre for Cognitive Science
Centre for Cognitive Science Technical
Technical University of Darmstadt
DarmstadtGermany
Centre for Cognitive Science Technical
University of Darmstadt
DarmstadtGermany
University of Darmstadt
DarmstadtGermany
Inverse Optimal Control Adapted to the Noise Characteristics of the Human Sensorimotor System
Computational level explanations based on optimal feedback control with signaldependent noise have been able to account for a vast array of phenomena in human sensorimotor behavior. However, commonly a cost function needs to be assumed for a task and the optimality of human behavior is evaluated by comparing observed and predicted trajectories. Here, we introduce inverse optimal control with signaldependent noise, which allows inferring the cost function from observed behavior. To do so, we formalize the problem as a partially observable Markov decision process and distinguish between the agent's and the experimenter's inference problems. Specifically, we derive a probabilistic formulation of the evolution of states and belief states and an approximation to the propagation equation in the linear-quadratic Gaussian problem with signal-dependent noise. We extend the model to the case of partial observability of state variables from the point of view of the experimenter. We show the feasibility of the approach through validation on synthetic data and application to experimental data. Our approach enables recovering the costs and benefits implicit in human sequential sensorimotor behavior, thereby reconciling normative and descriptive approaches in a computational framework.
Introduction
Computational level theories of behavior strive to answer the questions, why a system behaves the way it does and what the goal of the system's computations is. Such goals can be formalized based on the reward hypothesis. In the words of Richard Sutton, the reward hypothesis assumes, that "goals and purposes can be well thought of as maximization of the expected value of the cumulative sum of a received scalar signal" [1]. Thus, to understand human sensorimotor behavior, it is essential to characterize its goals and purposes quantitatively in terms of costs and benefits. But particularly in every-day tasks, the cost and benefits underlying behavior are unknown.
Stochastic optimal control allows formulating behavioral goals in terms of a cost function for tasks involving sequential actions under action variability, uncertainty in the internal model, and delayed rewards. The solution to the optimization problem entailed in the cost function is a sequence of actions, which is then compared to human movements. While early models optimized costs related to deterministic kinematics of a movement to a target [2], task goals were subsequently formalized as costs on the variance of stochastic movements' endpoint distances to a goal target [3]. Importantly, although [3] considered open-loop control, it revealed the importance of modeling the specific variability of human movements, which increases linearly with the magnitude of the control signal [4]. As the neuronal control signal increases, so does its variability, leading to optimal movements trading off between achieving task goals and reducing the expected impact of movement variability.
Including sensory feedback in stochastic optimal control leads to a computationally much more intricate problem, which can be formulated as a partially observable Markov decision process (POMDP) and is intractable in general. One of the few tractable cases is the linear-quadratic Gaussian (LQG) setting, where dynamics are linear, costs are quadratic, and variability is additive and Gaussian, leading to sensory inference of the state and control to be decoupled [5]. However, not only is human movement variability signal dependent, but additionally the uncertainty of sensory signals increases linearly with the magnitude of the stimulus, a phenomenon known as Weber's law [6]. Todorov [7] extended the LQG case by introducing stochastic optimal feedback control with signaldependent noise, which allows the specification of noise models in line with what is known about the human sensorimotor system. This model [8] has been able to explain a broad range of phenomena in sensorimotor control [9,10], including linear movement trajectories, smooth velocity profiles, speed-accuracy tradeoffs, and corrections of errors only if they influence attaining the behavioral goal. Particularly incorporating signal-dependent noise has been crucial in explaining experimental data, ranging from how corrections of movements during action execution depend on feedback and task goals [11], that movements consider sensory uncertainty and temporal delays in real-time [12], that movement plans in novel environments are reoptimized based on the learning of internal models to minimize implicit motor costs and maximize rewards [13], and many others [14,15,16,17].
Explaining human behavior in these studies usually starts by hypothesizing the cost function describing a task, obtaining the optimal feedback controller, and comparing simulated trajectories to those observed experimentally. This line of inquiry, therefore, utilizes similarity of trajectories to quantify the degree of optimality in human behaviors. In some cases [14], trajectories are simulated from the model to check for robustness with respect to changes in the model parameters. If our goal is to use optimal control models to infer such quantities, which often cannot be measured independently, from behavior, it would instead be desirable to invert the problem and find those parameter settings which are consistent with the observed trajectories. While such inverse methods have been developed both in the field of reinforcement learning to infer the rewards being optimized by an agent [18,19,20] as well as in optimal control for the LQG case [21,22,23], this is currently not possible for the noise characteristics of the human sensorimotor system.
Here, we introduce a probabilistic formulation of inverse optimal feedback control under signaldependent noise in the tradition of rational analysis [24,25]. Our starting point is the forward problem introduced in [8]. We formulate the inference problem faced by an agent as a POMDP and distinguish it from the inference problem of an experimenter observing the agent. We proceed by deriving the likelihood of a sequence of observed states and provide an approximation to the non-Gaussian uncertainty due to the signal-dependent noise. First, this allows recovering the cost function underlying the agent's behavior from observed behavioral data. Second, we extend the inference of the cost function to the case in which the state variables are only partially observable to the experimenter, e.g., when only measuring the position of the agent's movement. Third, we show through simulated and experimental data that the cost functions can indeed be recovered. Fourth, the probabilistic formulation allows recovering the agent's belief during the experiment as well as the experimenter's uncertainty about the inferred belief.
Related Work
Inferring the cost functions underlying an agent's behavior has long been of interest in different scientific fields ranging from economics [26] and psychology [27] to neuroscience [28] and artificial intelligence, particularly reinforcement learning [18,19,29,20]. Inverse Reinforcement Learning (IRL) specifically addresses the question of inferring the cost function being optimized [18,19,29] or approximately optimized [30] by an agent, with more recent approaches employing deep neural networks [31,32]. Some work has particularly addressed sensorimotor behavior [33,34,35,36] and extensions to the partially observable setting have been developed [37].
More relevant to the present study are computational frameworks that invert Bayesian models of perception and decision making to infer beliefs and costs of the agent. While this literature is extensive, exemplary studies for trial-based actions include cognitive tomography [38] and inference of the cost function in sensorimotor learning [39]. Extensions to sequential tasks have also been proposed, particularly considering active perception [40,41], real-world behavior [42,43], and general formulations [44,45]. Very similarly, work on Bayesian theory of mind uses highly related computational models, which also allow for the subjective beliefs of the agent to be different from the observer's [46,47].
In the context of inverse optimal control, related work has considered different problem settings, e.g., deterministic MDPs without additive noise and full observability [48]. More specifically in the LQR and LQG domain, [21] is concerned with finding the cost matrices and noise covariances given a known system with controller and Kalman gain. Similarly, [49] infers the above-mentioned cost matrices in the LQG setting based on observations but additionally takes constraints into account. The extension by [23] infers the terminal cost and the state cost function together with an exponential discount factor in the LQR setting. A different line of work has been concerned with estimating the dynamics, state sequence, and delay of internal LQG models from neural population activity [22,50] under the assumption that an agent might not know the dynamics, e.g., of a brain-computer interface. Other work [51] is concerned with learning a control policy from states, observations, and controls.
Our approach is different from previous research in two important ways. First, while most other approaches take different quantities such as the filter or controller gains or the agent's observations or controls as given, we consider the setting that is typical in a behavioral experiment: The filter and controller as well as the agent's observations are internal to the experimental subject and cannot be observed. Instead, we assume that only trajectories x 1:T are observed. Second, other approaches to inverse optimal control in the LQG setting do not involve signal-dependent noise, which we address in this paper.
Background: LQG with sensorimorotor noise characterisitics
We model a human subject as an agent in a partially-observable environment as introduced by Todorov [7] and depicted in Fig. 1 A. For this we consider a discrete-time linear dynamical system with state x t ∈ R m and control u t ∈ R p with both control-independent and control-dependent noises
x t+1 = Ax t + Bu t + V ξ t + c i=1 i t C i u t .(1)
The noise terms ξ t ∈ R m and i t ∈ R are standard Gaussian random vectors and variables, respectively, resulting in control-independent noise with covariance V V T and control-dependent noise having covariance i C i u t u T t C T i . The agent receives an observation y t ∈ R k from the observation model
y t = Hx t + W ω t + d i=1 ε i t D i x t .(2)
The noise terms ω t ∈ R k and ε t ∈ R are again standard Gaussian, so that the covariance of the state-independent observation noise is W W T , while for the state-dependent observation noise it is
i D i x t x T t D T i .
All matrices of the linear dynamical system can in principle be time-varying, but we leave out the time indices for notational simplicity. The objective of the agent is to choose u t to minimize a quadratic cost function,
J(u 1:T ) = E x 1:T T t=1 x T t Q t x t + u T t R t u t .(3)
While the original LQG problem without control-and state-dependent noises can be solved exactly by determining an optimal linear filter and controller independently [5], this separation principle is no longer applicable in the case considered here. Todorov [7] introduced an approximate solution method in which the optimal filters K t and controllers L t are iteratively determined in an alternating fashion, leaving the respective other one constant. The resulting optimal filter which minimizes the expected cost, is of the form where η t is a standard Gaussian random vector and represents internal estimation noise. The optimal linear control law can be formulated as
x t+1 = Ax t + Bu t + K t (y t − Hx t ) + Eη t ,(4)u t = −L txt .(5)
The equations for determining the matrices L t and K t are given in Appendix B. For a detailed derivation the reader is referred to [7].
Inverse Optimal Control
In this paper, we consider the inverse problem, i.e., we observe an agent who is acting optimally in an agent-experiment loop ( Fig. 1 B) according to the model of Section 2, and want to infer properties of the agent's perceptual and action processes, which are represented by parameters θ.
In the examples in this paper, we have treated all matrices except the subjective control costs (R) and parameters of the task objective (Q) as given. This choice is motivated by the fact that the cost function is usually the least understood quantity in a behavioral experiment, while sensorimotor researchers often have quite accurate models for the dynamics in the tasks they are studying and for subjects' noise characteristics. In principle, however, our probabilistic formulation of the inverse optimal control problem allows inferring parameters θ of any of the matrices of the system by evaluating the likelihood function w.r.t. those parameters. Given a set of N independent trajectories {x 1:T } i=1:N , each of length T , we can infer θ by maximizing the product of their likelihoods p(x 1:T | θ), each decomposing as
p(x 1:T | θ) = p θ (x 1 ) T −1 t=1 p θ (x t+1 | x 1:t ).(6)
In the following, we drop the explicit dependency of the parameters θ. The graphical model from the agent's point of view ( Fig. 1 A) is structurally identical to that from the experimenter's perspective ( Fig. 1 C). But, since we as experimenter observe the true states x 1:T instead of the agent's noisy observations y 1:T , the usual Markov property does not hold and each x t generally depends on all previous states x 1:t−1 via the agent's estimates and actions. To efficiently compute the likelihood factors p(x t | x 1:t−1 ), we track our belief about the agent's belief p(x t | x 1:t ), which gives a sufficient statistic for the history. This approach allows propagating our uncertainty about the agent's beliefs and actions over time and estimating the agent's belief.
To compute the likelihood function for some value of θ, we first determine the control and filter gains L t and K t using the iterative method introduced by Todorov [7]. We then compute an approximate likelihood factor p(x t | x 1:t−1 ) for each time step in the following way (see Algorithm 1):
First, we determine the distribution p(x t+1 ,x t+1 | x t ,x t ), which describes the joint evolution of x t andx t (Section 3.1). Second, we combine it with the belief distribution p(x t | x 1:t ), yielding p(x t+1 ,x t+1 | x 1:t ) (Section 3.2). As this step cannot be done in closed-form due to the signaldependent noise, we introduce a Gaussian approximation of this quantity. Third, marginalizing over x t+1 gives the desired likelihood factor p(x t+1 | x 1:t ), while conditioning on the observed true states x t+1 gives the statistic of the history p(x t+1 | x 1:t+1 ), which we use for computing the likelihood factor of the following time step.
In Section 3.3, we extend this procedure to the setting where the state x t is only partially observed as a noisy linearly transformed version o t (Fig. 1 D).
Algorithm 1: Approximate Likelihood Computation
Result: Approximate log likelihood of parameters θ Input: Parameters θ, Data {x i 1:T } i=1:N , Model L t , K t ← Approximate optimal controller and filter using the method of Todorov [7]; initialize p(x 0 | x 0 ) as the experimenter's initial belief of the agent's belief; for each trajectory
x 1:T from {x i 1:T } i=1:N do for t ← 0 to T − 1 do Compute p(x t+1 ,x t+1 | x 1:t ) using Eq. (10); Marginalize overx t+1 to get p(x t+1 | x 1:t ); Condition on x t+1 to get p(x t+1 | x 1:t+1 ) using Eq. (11); end end return N i=1 log p(x i 1 ) + T t=2 log p(x i t | x i 1:t−1 )
Joint dynamics of states and estimates
In this section, we derive the joint dynamics of states and estimates, specifying the distribution
p(x t+1 ,x t+1 | x t ,x t ).
To do so, we build on work by Van Den Berg et al. [52], who introduced this idea for the standard LQG case in the context of planning, and extend it to the model with state-and action-dependent noises as considered in Section 2. First, we substitute the control in the state update (1) with its law (5), giving
x t+1 = Ax t − BL txt + V ξ t − d i=1 ε i t C i L txt ,(7)
and rewrite the filter update equation (4) as
x t+1 = (A − BL t )x t + K t (y t − Hx t ) + Eη t = (A − BL t − K t H)x t + K t Hx t + K t W ω t + K t c i=1 i t D i x t + Eη t .(8)
In the last equation, we have again inserted the control law (5), then the observation model (2), and rearranged terms. Equations (7) and (8) give us a representation of x t andx t which only depends on states or estimates from the previous time step. Stacking both equations together specifies the
distribution p(x t+1 ,x t+1 | x t ,x t ), with x t+1 x t+1 = A −(B + d i=1 ε i t C i )L t K t (H + c i=1 i t D i ) A − BL t − K t H x t x t + V 0 0 0 K t W E ξ t ω t η t =: (F t + M t c i=1 i t D i )x t + (F t + d i=1 ε i t C iMt )x t + G t ζ t ,(9)
where ζ t ∼ N (0, I). For a detailed definition of the matrices F t ,F t , M t ,M t see Appendix C.1.
Approximate propagation
We obtain the distribution p(x t+1 ,x t+1 | x 1:t ) by propagating p(x t | x 1:t ) through the joint dynamics model (9). But, since the latter involves a product of Gaussian random variables ε t andx t , the resulting distribution is no longer Gaussian. To make likelihood computation tractable, we approximate it by a Gaussian using moment matching. This allows us to maintain an approximate Gaussian belief about the agent's belief and gives us an approximation of the likelihood function in Eq. (6).
First, we assume that our belief of the agent's belief at time step t is given by a Gaussian distribution p(x t | x 1:t ) = N µx |x , Σx |x . To approximately propagate p(x t | x 1:t ) and the observation x t through Eq. (9), we compute the mean and variance of the resulting distribution via moment matching (see Appendix E) and obtain the approximation
p(x t+1 ,x t+1 | x 1:t ) ≈ N x t+1 x t+1 μ t+1 = μ x µx ,Σ t+1 = Σ xxΣxx Σx xΣxx ,(10)witĥ µ t+1 = F t x t +F t µx |x , Σ t+1 = M t ( c i=1 D i x t x T t D T i )M T t + d i=1 C iMt (Σx |x + µx |x µ T x|x )M T t C T i +F t Σx |xF T t + GG T .
Marginalizing overx t+1 gives an approximation of the likelihood factor of time step t + 1,
p(x t+1 | x 1:t ) ≈ N (μ x ,Σ xx ).
On the other hand, conditioning on observation x t+1 gives the belief of the agent's belief for the following time step, p(
x t+1 | x 1:t+1 ) = N μx |x ,Σx |x , witĥ µx |x =μx +Σx xΣ −1 xx (x t+1 −μ x ),Σx |x =Σxx −Σx xΣ −1 xxΣxx .(11)
We initialize p(x 0 | x 0 ) with the initial belief of the agent.
Partial observability from the observer's point of view
In practice, we often do not have access to the full state x t in the model, e.g., if there are unmeasured quantities of the physical world such as velocity and acceleration when using a tracking system which only provides measurements of position in time, or if we have latent variables in our model representing internal states of the observed agent. Furthermore, measurements might be noisy, e.g., due to the use of imprecise tracking hardware. We therefore consider the case where the state is partially observable for both the agent, i.e., the subject in the experiment, and the observer, i.e., the experimenter. In this case, we assume that the experimenter observes a linear transformation o ∈ R s of the state x t with additive Gaussian noise, i.e.,
o t = Sx t + U ϑ t ,(12)
where ϑ t is a standard Gaussian random vector, resulting in the distribution p(o t | x t ) = N Sx t , U U T . The resulting Bayesian network is shown in Fig. 1 D. We can again formulate a joint dynamical system of x t andx t with additional observations o t , resulting in
x t+1 x t+1 o t+1 = A −(B + d i=1 ε i t C i )L t K t (H + c i=1 i t D i ) A − BL t − K t H SA −S(B + d i=1 ε i t C i )L t x t x t + V 0 0 0 0 K t W E 0 SV 0 0 U ξ t ω t η t ϑ =: (F t + M t c i=1 i t D i )x t + (F t + d i=1 ε i t C iMt )x t + G t ζ t ,(13)
with matrices F t ,F t , M t ,M t defined accordingly (for definitions see Appendix C.2). Note that this equation is structurally the same as for the fully-observable case (Eq. (9)) and we have overloaded the matrix definitions to highlight that both can be treated similarly.
The likelihood of an observed trajectory decomposes as p(o 1:
T | θ) = p θ (o 1 ) T t=2 p θ (o t | o 1:t−1 )
. For computing the factors p(o t | o 1:t−1 ), we follow structurally the same steps as for the fullyobservable case, but now p(x t ,x t | o 1:t ) serves as sufficient statistic of the history: We first assume the distribution p(x t ,x t | o 1:t ) to be Gaussian distributed and approximately propagate it through the joint dynamics model (13)) by computing the mean and variance. We marginalize the resulting Gaussian approximation of p(
p(x t+1 ,x t+1 , o t+1 | x t ,x t ) (Eq.x t+1 ,x t+1 , o t+1 | o 1:t ) over x t+1 ,x t+1 , yielding the likelihood factors p(o t+1 | o 1:t ).
On the other hand, conditioning on the observation o t gives the history statistic p(x t ,x t | o 1:t ) for the following time step. All steps are very similar to the fully-observable case, but a more detailed description is given in Appendix D.
Parameter inference
In the previous sections, we have provided an algorithm for computing an approximate likelihood of the parameters θ given a set of observed trajectories {x i 1:T } i=1:N . To determine the optimal parameters, we maximize the likelihood, giving us a point estimate θ MLE of the true parameters. As one has to solve the control problem (determining L t and K t ) by an iterative procedure [7] for every likelihood evaluation, computing gradients of the likelihood (although possible) is not very efficient. We instead use the robust gradient-free optimizer BOBYQA 2 [53], which minimizes the negative log likelihood based on a quadratic approximation. Our implementation in jax [54] is available on github. 3
Evaluation and applications 4.1 Validation on synthetic reaching data
We apply the introduced method to recover parameters in a single-joint reaching task with controldependent noise and 5-dimensional state space (details in Appendix F.1). The goal is to bring the hand to a target while minimizing control effort. The cost function has three parameters: (i) v, the cost of the velocity at the final time step, (ii) f , the cost of the acceleration at the final timestep, (iii) r, the cost of actions at each timestep. Simulated data for the parameters r = 10 −5 , v = 0.2, f = 0.02 are shown in Fig. 2 A. Visual inspection of the likelihood function (Fig. 2 B) shows that the maximum likelihood estimate (MLE) is very close to the true parameter values.
Once we have obtained the MLE, we can perform belief tracking, i.e., computing our approximate belief of the agent's belief p(x t |o 1:t ). As an example, we simulated N = 20 trajectories {o 1:T } i=1:N with T = 30 from a partially observed version of the reaching task used above in which we only observe the position and treat velocity and acceleration as latent variables. Fig. 2 C shows our approximate belief about the agent's belief for the MLE parameters, together with the true agent's belief. Note that we can recover the agent's belief of state, velocity, and acceleration quite accurately from noisy observations of the position only. For the evaluation of parameter estimates, we compute their root mean squared errors (RMSEs) in logarithmic space. The effect of estimation errors on the resulting trajectories is illustrated in Fig. 2 D, where we simulated trajectories as in Fig. 2 A with different mean parameter errors. To show that our method yields good parameter estimates over a range of different parameter settings, we perform maximum likelihood estimations (MLEs) of all three parameters for different true parameter values of which two were chosen from a pairwise grid while the third one was left as in Fig. 2 A. In this analysis, we used 100 simulated trajectories and 10 repetitions each. In Fig. 2 E, which shows the resulting RMSEs for different combinations of true parameters on pairwise grids, we demonstrate that the RMSEs are small over a wide range of parameter values. The RMSEs (Fig. 2 F) across different values for the respective other two parameters and 10 repetitions were 2.4 × 10 −2 (r), 2.1 × 10 −2 (v) and 3.1 × 10 −2 (f ). We compared the RMSEs obtained by our estimation method to the ones of two baseline approaches. A first baseline is obtained by running a version of the algorithm without signal-dependent noise (i.e., using the basic LQG during inference), for which the likelihood can be evaluated exactly in closed-form. The results on the reaching problem (Section 4.1) in terms of RMSE of the parameter estimates are worse by roughly two orders of magnitude (RMSE of 1.766 vs 0.027). A second, stronger baseline is obtained by setting the additive noise in the standard LQG to the average noise magnitude of simulated trajectories. In this case, the RMSEs are still worse by roughly an order of magnitude (RMSE of 0.702 vs 0.027). Note that the information on the average noise level would not be readily available for real data without knowing the true parameters and therefore constitutes a strong baseline. A plot visualizing the results of the baselines for each parameter is provided in Appendix G.1.1.
Finally, we investigated the influence of the number of samples by evaluating estimates of trajectories as in Fig. 2 A for different numbers of trials (Fig. 2 G). As expected, more trials increase the accuracy of the estimates and lead to lower error. We estimated the convergence rate by fitting a line to the log-log plot and obtained 0.78 for the RMSE. An analysis using the Kullback-Leibler divergence between the empirical trajectory distributions of the true and maximum likelihood parameters as an additional evaluation metric can be found in Appendix G.1.2.
We perform a similar analysis for generated partially observed trajectories in which we only observe the position and treat velocities, accelerations, and forces as latent variables. The results are qualitatively very similar and are therefore presented in Appendix G.1.3. An additional empirical evaluation of the impact of the moment matching approximation is given in Appendix G.1.4.
Application to real reaching movements
To show the applicability to real data, we apply our method to reaching trajectories from a previously published experiment [55], in which a rhesus monkey had to perform center-out reaching movements and hold its hand at the target to receive a reward. Since the data contains only measurements of position (velocity and acceleration are computed using finite differences), we use the partially observable version of the reaching model described in Section 4.1, treating velocity and acceleration as latent variables. The approximate likelihood functions with respective MLEs of the three parameters are shown in Fig. 3 A, indicating that we can determine the parameter set for which the trajectories are most likely. Fig. 3 B shows the given trajectories together with simulations using our model with the MLE parameters. We observe that the inferred parameters produce simulated data that convincingly look like the real data and provides smooth estimates of the latent velocity and acceleration profiles.
Application to eye movements
A Saccadic eye movements B Parameter estimates We also apply our method to a model of saccadic eye movements which was presented by Crevecoeur and Kording [56]. This model captures fixating one's eyes to an initial point and then performing a saccade to fixate another point. A cost parameter (r) is used to trade off the cost of the movement and the deviation from the target. As this model is an LQG model with controldependent noises, it directly allows the application of our method for recovering the parameter. Fig. 4 A shows simulated trajectories representing typical eye movements encountered in the experiments. The MLEs based on simulated data for a range of 20 different parameter values (100 repetitions each) are shown in Fig. 4 B. Except for very few outliers, the estimated parameters are very close to the true parameters.
Application to random problems
To demonstrate that the inference method works on a wide range of problems defined according to the model definition (Section 2), we evaluate it on randomly generated problems with 5-dimensional state-space and two-dimensional action space. Detailed information on the generation procedure is given in Appendix F.3. Each model has two parameters (r 1 and r 2 ) which again represent the cost of control effort in each of the two movement directions. For each model, we sampled a true value of parameter r 2 from a uniform random distribution and estimated both parameters jointly by maximizing the approximate likelihood. In Fig. 5 A we show the errors for a range of parameter values r 1 for different random models. The median and quantiles for the results of 2000 random problems are shown in Fig. 5 B. One can observe that the estimates are generally very close to the true parameters. The results for the other parameter r 2 are basically identical since the problem is symmetric w.r.t. the parameters, but we include the results in Appendix G.2.
Conclusion
In this paper, we investigated the inverse optimal control problem under signal-dependent noise. We formalized the problem as a POMDP and introduced a first method for inferring cost parameters of an agent in a linear-quadratic control problem with signal-dependent noise. Numerical simulations show A Individual random seed B Aggregated that accurate inference of cost parameters given synthetic data is feasible in random control problems, simulated arm movements, and simulated eye movements. Additionally, the method can be used to probabilistically infer the belief of the considered agent. Furthermore, the method was applied to real data from a macaque monkey performing reaching movements. The inferred parameters reproduce reaching data in simulation that convincingly agrees with the original data and provides smooth estimates of the latent velocity and acceleration profiles. More recent and more general methods for optimal control in high-dimensional continuous domains exist, but while some of these may provide interpretable non-linear features, they consider deterministic MDPs without noise and assume full observability [48], others relying on function approximation through neural networks including GANs, are useful in engineering applications but may not provide a computational level explanation of behavior [31,32]. Taken together, our method does not require designing a cost function and testing for similarity of simulated trajectories with experimental data, but allows inferring the cost functions directly from behavior, thereby reconciling normative and descriptive approaches to human sensorimotor behavior.
Limitations and Future work The proposed algorithms are based on stochastic optimal feedback control with linear dynamics, quadratic cost functions, and signal-dependent noise. As such, the first limitation lies in the restriction to problems with linear dynamics. An extension to non-linear dynamics could be achieved by linearizing the dynamics locally or more generally by using a framework that iteratively linearizes the dynamics at each time step, e.g., iLQG [57]. Similarly, while quadratic cost functions allow modeling a wide range of costs and benefits within sensorimotor control, certain cost functions such as exponential discounting may be more cumbersome to accommodate.
While the presented method was able to recover cost functions in the considered problems, higherdimensional parameter spaces will likely pose difficulties in finding unique point estimates of parameters. This problem could be addressed by using appropriate structured prior distributions over parameters. A fully Bayesian treatment could be realized by using Markov chain Monte Carlo involving our likelihood model. Further research should similarly investigate the limits of the Gaussian approximation of the likelihood. In the cases considered here, the approximation of the likelihood function appeared to be unbiased up to an RMSE of approximately 3 × 10 −2 , however, approximating distributions by simpler ones may introduce systematic biases and noise. Possible extensions could resort to using particle filters albeit at a higher computational cost.
Another issue regarding higher-dimensional parameter spaces is that evaluation of the likelihood function becomes quite expensive by determining the optimal controller and optimal filter iteratively. Even if this procedure takes only one second on a common PC for the reaching task, optimization in high-dimensional spaces requires a larger number of function evaluations, which renders inference costly. By relying on the iterative procedure, it also becomes difficult to compute gradients, which may prevent the use of efficient gradient-based solvers. While for our applications a solver based on quadratic approximations was efficient, one could resort to Bayesian optimization.
Future work in the area of robotics may explore applying the inferred cost functions in the training of visuomotor policies [58,59] in humanoid robots with reinforcement learning [60]. Possible applications include utilizing the inferred cost functions in apprenticeship learning [61,62,63], in which a policy is learned from demonstrations of a potentially suboptimal demonstrator or teacher. Similarly, applications may also include transfer learning [64,65,66], in which learned policies or cost functions are transferred to related tasks.
Finally, characterizing individual human subjects by analyzing their behavior may in principle be used with negative societal impact. In the context of scientific investigations of human sensorimotor control within cognitive science and neuroscience, only anonymized behavioral data for the understanding of the human mind and brain are employed. Table 1 provides an overview of the notation used in this paper. We distinguish between variables used in the original model described in Section 2 [7] and quantities necessary for our inference method described in Section 3. scaling of the signal-dependent noises in the joint dynamical system G scaling of the signal-independent noise in the joint dynamical system o t ∈ R s experimenter's observation of the statê µ t ,Σ t mean and covariance of the Gaussian approximation of state and agent's estimatê µx |x ,Σx |x mean and covariance of the experimenter's belief about the agent's estimate
A Notation
B Approximate Optimal Control of LQG systems with sensorimotor noise characteristics
For approximately solving a system as described in Section 2, the optimal filters K t and controllers L t can be iteratively determined in an alternating fashion, leaving the respective other one constant Todorov [7].
Given filter matrices K t , the optimal control matrices L t are computed in form of a backward pass as
L t = R t + B T P x t+1 B + i C T i P x t+1 + P e t+1 C i −1 B T P x t+1 A P x t = Q t + A T P x t+1 (A − BL t ) + i D T i K T t P e t+1 K t D i P e t = A T P x t+1 BL t + (A − K t H) T P e t+1 (A − K t H) s t = tr P x t+1 V V T + P e t+1 V V T + EE T + K t W W T K T t + s t+1 ,
where we initialize P x T = Q T , P e T = 0, s T = 0.
Given optimal control matrices L t , the optimal filter matrices K t are computed in form of a forward pass as
K t = AΣ e t H T HΣ e t H T + W W T + i D i Σ e t + Σx t + Σ ex t + Σ xe t D T i −1 Σ e t+1 = V V T + EE T + (A − K t H) Σ e t A T + i C i L t Σx t L T t C T i Σx t+1 = EE T + K t HΣ e t A T + (A − BL t ) Σx t (A − BL t ) T + (A − BL t ) Σ xe t H T K T t + K t HΣ ex t (A − BL t ) T Σ xe t+1 = (A − BL t ) Σ xe t (A − K t H) T − EE T , with Σ ex t = Σ xe t T and we initialize Σ e 1 = Σ 1 Σx 1 =x 1x T 1 Σ xe 1 = 0.
C Joint update equation derivation C.1 Fully-observable state
Stacking Eq. (7) and Eq. (8) into a vector, gives
x t+1 x t+1 = Ax t − BLx t + V ξ t − d i=1 ε i t C i L txt , (A − BL t − K t H)x t + K t Hx t + K t W ω t + K t c i=1 i t D i x t + Eη t = A K t H x t + −BL t A − BL t − K t H x + 0 K t c i=1 i t D i x t + d i=1 ε i t C i −L t 0 x + V 0 0 0 K t W E ξ t ω t η t = A K t H + 0 K t c i=1 i t D i x t + −BL t A − BL t − K t H + d i=1 ε i t C i −L t 0 x + V 0 0 0 K t W E ξ t ω t η t =: (F t + M t c i=1 i t D i )x t + (F t + d i=1 ε i t C iMt )x + G t ζ t .
C.2 Partially-observable state
Stacking Eq. (7), Eq. (8), and Eq. (12) into a vector, gives
x t+1 x t+1 = Ax t − BLx t + V ξ t − d i=1 ε i t C i L txt (A − BL t − K t H)x t + K t Hx t + K t W ω t + K t c i=1 i t D i x t + Eη t S Ax t − BLx t + V ξ t − d i=1 ε i t C i L txt + U ϑ t = A K t H SA x t + −BL t A − BL t − K t H −SBL t x + 0 K t 0 c i=1 i t D i x t + d i=1 ε i t C i −L t 0 −SL t x + V 0 0 0 0 K t W E 0 SV 0 0 U ξ t ω t η t ϑ = A K t H SA + 0 K t 0 c i=1 i t D i x t + −BL t A − BL t − K t H −SBL t + d i=1 ε i t C i −L t 0 −SL t x + V 0 0 0 0 K t W E 0 SV 0 0 U ξ t ω t η t ϑ =: (F t + M t c i=1 i t D i )x t + (F t + d i=1 ε i t C iMt )x + G t ζ t .
D Approximate likelihood in case of partially-observable state
In the following, we definex t as the true state x t and the agent's beliefx t stacked together, i.e.,
x t = [x t ,x t ].
First, we assume that the belief of the agent's belief at time step t is given by a Gaussian distribution
p(x t ,x t | o 1:t ) = N x t x t µx |o = µ x|o µx |o , Σx |o = Σ x|o Σ xx|o Σx x|o Σx |o .
To approximately propagate p(x t ,x t | o 1:t ) through Eq. (9), we compute the mean and variance of the resulting distribution via moment matching (see Appendix E) and obtain the approximation
p(x t+1 , o t+1 | o 1:t ) ≈ N x t+1 o t+1 μ t+1 = µx µ o ,Σ t+1 = ΣxΣx ô Σ oxΣo ,(14)
withμ t+1 = F t µ x|o +F t µx |o =F µx |o ,
Σ t+1 = M t ( c i=1 D i (Σ x|o + µ x|o µ T x|o )D T i )M T + d i=1 C iM (Σx |o + µx |o µ T x|o )M T C T i +F Σx |oF T + GG T ,
whereF t consists of F t andF t vertically stacked, i.e.,F t :
= F T tF T t T .
Marginalizing over x t+1 andx t+1 gives an approximation of the likelihood factor of time step
t + 1, p(o t+1 | o 1:t ) ≈ N (μ o ,Σ oo ).
On the other hand, conditioning on observation o t+1 gives the belief of the state and the agent's estimate for the following time step, p(x t+1 ,x t+1 | o 1:t+1 ) = N μx |o ,Σx |o , witĥ
µx |o =μx +Σx oΣ −1 oo (o t+1 −μ o ),Σx |o =Σxx −Σx oΣ −1 ooΣox .(15)
We initialize p(x 0 ,x 0 | o 0 ) with our initial belief of the state and of the agent's belief.
E Derivation of approximate propagation
For a fully-observable state, the goal is to derive a closed-form approximation for p(x t+1 ,x t+1 | x 1:t ) when propagating the (approximate) belief p(x t |x 1:t ) and the state x t through the extended dynamics model p(x t+1 ,x t+1 | x t ,x t ) by matching the result with a Gaussian distribution. In this section we will consider first the more general case where the state is partially observable, and derive the equations for a fully-observable state as a special case afterwards. As the fully-and partiallyobservable cases differ in the number of random variables (for partial-observability we have random variables o t in addition to the true state x t ), we will consider a general joint dynamics model p(w t+1 | x t ,x t ) and general observations o t . Then, the goal becomes to derive the approximation for p(w t+1 | o 1:t ) when propagating the belief p(x t ,x t | o 1:t ) through the model p(w t+1 | x t ,x t ).
We restate the update equation for w t+1 , coinciding with both Eq. (9) and Eq. (13),
w t+1 = (F t + M t c i=1 i t D i )x t + (F t + d i=1 ε i t C iMt )x + G t ζ t ,(16)
where the matrices F t , M t ,F t , C t , and G t are partitioned as described in Appendix C.1.
We assume that p(x t ,x t | o 1:t ) follows a Gaussian distribution with
p(x t ,x t | o 1:t ) ∼ N x t x t µ t = µ x µx , Σ t = Σ xx Σ xx Σx x Σxx .(17)
To match p(w t+1 | o 1:t ) with a Gaussian distribution, we compute the meanμ and varianceΣ of Eq. (16) where x andx are distributed according to Eq. (17). In the following, we will drop time indices to enhance readability. For the mean, we obtain
µ = E x, ,x,ε,ζ (F + M c i=1 D i i )x + (F + d i=1 C i ε iM )x + Gζ = E x, (F + M c i=1 D i i )x + Ex ,ε (F + d i=1 C i ε iM )x + E ζ [Gζ] = E x (F + M c i=1 D i E i i )x + Ex (F + d i=1 C i E ε i ε i M )x + G E ζ [ζ] = E x [F x] + Ex Fx + G E ζ [ζ] = F µ x +F µ x =:μ x +μ x .
For the variance, we use that ζ is independent ofx := x T ,x T T , thereforê Σ =Σx +Σ ζ , withΣ ζ = GG T and we define
T x + Tx := (F + M c i=1 D i i )x + (F + d i=1 C i ε iM )x.
To deriveΣx, we first regard the terms E x, T x T T x :
E x, T x T T x = E x, (F + M c i=1 D i i )xx T (F T + c i=1 i D T i M T ) = E (F + M c i=1 D i i ) E x xx T (F T + c i=1 i D T i M T ) = E (F + M c i=1 D i i )Υ xx (F T + c i=1 i D T i M T ) = F Υ xx F T + E F Υ xx c i=1 i D T i M T + E M ( c i=1 D i i )Υ xx F T + E M ( c i=1 D i i )Υ xx ( c i=1 i D T i M T ) = F Υ xx F T + F Υ xx c i=1 E i D T i M T + M ( c i=1 D i E i )Υ xx F T + E M ( c i=1 D i i )Υ xx ( c i=1 i D T i M T ) = F Υ xx F T + E M ( c i=1 D i i )Υ xx ( c i=1 i D T i M T ) = F Υ xx F T + E M ( c i=1 c j=1 D i i Υ xx j D T j )M T = F Υ xx F T + E M ( c i=1 D i i Υ xx i D T i )M T = F Υ xx F T + M ( c i=1 D i E i i i Υ xx D T i )M T = F Υ xx F T + M ( c i=1 D i Υ xx D T i )M T ,
where we defined the (raw) second moments of x as
Υ x := E x xx T = Σ x + µ x µ T x .
With that, we obtain for E
x, T x T T x − µ x µ T x : E x, T x T T x − µ x µ T x = F Υ xx F T + M ( c i=1 D i Υ xx D T i )M T − F µ x µ T x F T = F (Υ xx − µ x µ T x )F T + M ( c i=1 D i Υ xx D T i )M T = F Σ xx F T + M ( c i=1 D i Υ xx D T i )M T A similar derivation follows for Ex ,ε TxT T x − µxµ T x , giving Ex ,ε TxT T x − µxµ T x =F ΣxxF T + d i=1 C iM ΥxxM T C T i .
Using these intermediate results, we can now computeΣx:
Σx = E x, ,x,ε (T x + Tx)(T x + Tx) T − E x, ,x,ε [(T x + Tx)] E x, ,x,ε [(T x + Tx)] T = E x, ,x,ε (T x T T x + T x T T x + TxT T x + TxT T x ) − (µ x + µx)(µ x + µx) T = E x, T x T T x − µ x µ T x + E x, ,x,ε T x T T x − µ x µ T x + E x, ,x,ε TxT T x − µxµ T x + Ex ,ε TxT T x − µxµ T x ( * ) = F Σ xx F T + M ( c i=1 D i Υ xx D T i )M T +F ΣxxF T + d i=1 C iM ΥxxM T C T i + E x, ,x,ε T x T T x − µ x µ T x + E x, ,x,ε TxT T x − µxµ T x = M ( c i=1 D i Υ xx D T i )M T + d i=1 C iM ΥxxM T C T i + F Σ xx F T +F ΣxxF T + F Σ xxF T +F Σx x F T = M ( c i=1 D i Υ xx D T i )M T + d i=1 C iM ΥxxM T C T i +F ΣF T ,
where in ( * ) we used the previously derived results and definedF t as F t andF t vertically stacked, i.e.,F t := F T tF T t T .
By putting everything together, we get w t+1 ∼ N (μ w ,Σ w ) witĥ
µ w = F µ x +F µx =F µx, Σ w = M ( c i=1 D i Υ xx D T i )M T + d i=1 C iM ΥxxM T C T i +F ΣF T + GG T = M ( c i=1 D i (Σ xx + µ x µ T x )D T i )M T + d i=1 C iM (Σxx + µxµ T x )M T C T i +F ΣF T + GG T ,(18)
whereF t consists of F t andF t vertically stacked, i.e.,F t := F T t ,F T t T .
E.1 Partially-observable state
In case of partial observability, we have w t+1 = x T t+1 ,x T t+1 , o T
E.2 Fully-observable state
In case of a fully-observable state, we have w t+1 = x T t+1 ,x T t+1 T , with observation o t = x t , so we are interested in approximating p(x t+1 ,x t+1 | x 1:t ). We assume p(x t | x 1:t ) = N µx |x , Σx |x and x t to be observed and therefore deterministic. We can then informally write
p(x t ,x t | x 1:t ) ∼ N x t x t µ t = x t µx |x , Σ t = 0 0 0 Σx |x .
Plugging this into Eq. (18), we obtain w t+1 ∼ N (μ w ,Σ w ), witĥ
µ w = F µ x +F µx = F x t +F µx |x , Σ w = M ( c i=1 D i Υ xx D T i )M T + d i=1 C iM ΥxxM T C T i + F Σ xx F T +F ΣxxF T + F Σ xxF T +F Σx x F T + GG T = M ( c i=1 D i x t x T t D T i )M T + d i=1 C iM Υx |xM T C T i +F Σx |xF T + GG T = M ( c i=1 D i x t x T t D T i )M T + d i=1 C iM (Σx |x + µx |x µ T x|x )M T C T i +F Σx |xF T + GG T ,
where the time-dependency of the matrices was omitted to enhance readability.
F Further information on Applications
If not stated otherwise, we used 100 trajectories to determine the MLEs. For optimization, we ran the optimizer 10 times with random initial points and took the optimal value to avoid local optima.
F.1 Reaching model
The reaching model was the same one used in the original publication [7], where all details can be found. The cost function which was minimized, is given by
x p (T ) − x * p 2 + (v · x v (T )) 2 + (f · x f (T )) 2 + r M − 1 M −1 k=0 u(k∆) 2 ,
where x p is the position, x v the velocity, x f the force, x * p the target position, and ∆ the time discretization, discretizing T into M time steps, i.e., ∆M = T . Note that there is no explicit parameter for the cost of the end-point position needed because the other parameters are relative to this quantity.
We used the same model for the real reaching data, which were taken from the Database for Reaching Experiments and Models 4 and were previously published [55]. We took the horizontal component of reaching movements towards targets at 0 degrees (right of center) and truncated each trial so that it contained the movement only.
F.2 Saccadic eye movement problem
We used the model by Crevecoeur and Kording [56] with a time discretization of 1.25 ms. The initial angle was set to −10 and the target angle to 10 as shown in Figure 1b of the referenced paper.
F.3 Random models
The random models were inspired by the work of Todorov [7]. For these models, the state space was four-dimensional with an additional dimension for modelling the target that the first dimension of the state should be controlled to. The action space was n = 2-dimensional. The matrices A, B and H of the dynamical system were randomly sampled with A ij , B ij , H ij ∼ N (0, 1).
A and B were normalized to 1 using the Frobenius norm. The additive noises V , W , and E were sampled from LKJ-Cholesky distributions V, W ∼ LKJ-Cholesky (1) and the multiplicative noises were sampled with C ij , D ij ∼ Uniform(0, 0.5).
The state cost matrices Q t were set to Q 1:T −1 = 0 and Q T = dd T with d = [1 0 . . . −1] yielding a state cost at the last time step of (x T − 1)(x T − 1) T . The control cost was parametrized with R = diag [r 1 , . . . , r n ]. We used our maximum likelihood method to infer the parameters r 1 , . . . , r n .
G Additional Results
In this section we will provide additional results for the reaching and random problems.
G.1.2 Kullback-Leibler divergence as evaluation measure
As an alternative evaluation measure, we propose to compare the distributions induced by the true parameters and the maximum likelihood parameters. For this, we estimate empirical distributions of the trajectories by generating 10,000 trajectories and computing the mean and variance for each time step to approximate the distribution for each time step by a Gaussian. The Kullback-Leibler (KL) divergence between two Gaussian distributions p = N (µ p , Σ p ) and q = N (µ q , Σ q ) can be calculated in closed form as
D KL (p q) = 1 2 log |Σ q | |Σ p | − k + (µ p − µ q ) T Σ −1 q (µ p − µ q ) + tr Σ −1 q Σ p .
Instead of using the KL divergence directly, which is not symmetric, we consider instead a symmetric version,
1 2 D KL (p q) + 1 2 D KL (q p),
and compute the mean over time to aggregate the values over time.
The results like in Fig. 2 E and F with KL divergence instead of RMSE as a metric are shown in Fig. 7.
G.1.3 Partially observable state
We evaluate a version of the reaching model in which the experimenter observes only the position and treats velocity and acceleration as latent variables. The results are qualitatively similar to the fully observed case. However, there are regions in the parameter space where estimates are worse (Fig. 8 A). Additionally, estimates of the parameters representing the penalty on velocity (v) and acceleration (f ) are worse by an order of magnitude compared to the fully observed case (Fig. 8 B), which is to be expected when only the position is observed. Specifically, the average RMSEs are 4.0 × 10 −2 (r), 1.1 × 10 −1 (v), and 2.6 × 10 −1 (f ). However, the parameter errors do not result in large differences in the KL divergence of the simulated trajectories (position only) w.r.t. the observed trajectories, so we suspect that the higher RMSEs in estimated parameters are due to ambiguities in the trajectories for this particular model.
G.1.4 Evaluation of moment matching approximation
In Section 3.2 we introduced a moment matching approximation to make computation of the likelihood tractable. An experimentalist comparing an optimal control model to experimental data might be interested in the influence of this approximation on trajectories. For the reaching model from Section 4.1, we therefore compare the empirical distributions over trajectories estimated using Monte Carlo rollouts (using 10,000 trajectory samples) to the approximate distribution over trajectories determined using our method (given the true parameters). The difference in symmetrized KL (see Appendix G.1.2) between the empirically estimated distribution and our approximation is found to be 1.60 × 10 −3 . Additionally, we compare this result to a baseline by replacing the signal-dependent noise by additive noise, for which the trajectory distribution can be calculated in closed form. The additive noise magnitude is chosen as the average of the signal-dependent noise magnitudes for the whole trajectories. Note that this quantity is not directly available and therefore has to be also WLPHVWHS SRVLWLRQ WLPHVWHS YHORFLW\ WLPHVWHS DFFHOHUDWLRQ Figure 10: Empirical evaluation of the moment matching approximation. Trajectory distributions of the reaching task using a Monte-Carlo approximation (mean: black solid line, 2 × STD in gray), our moment-matching approximation (orange, dashed), and a baseline where the signaldependent noise is replaced by additive noise (blue, dotted). The gray and orange areas are hardly distinguishable visually, showing that the moment matching approximation estimates the trajectory distribution very precisely. The baseline overestimates the variance through the signal-dependent noise in early time steps, leading to an overall too high variance of the trajectories.
A Individual random seed B Aggregated estimated, e.g., via Monte-Carlo rollouts. The difference in symmetrized KL between the empirical estimate and the baseline is 6.05. A plot of the resulting distributions is shown in Fig. 10. As expected, the moment matching approximation estimates the trajectory distribution very precisely in comparison to the baseline.
G.2 Random problems
In Fig. 11 A we show the errors for a range of parameter values r 2 for different random models. The median and quantiles for the results of 1000 random problems are shown in Fig. 11 B. One can observe that the estimates are generally very close to the true parameters.
Figure 1 :
1The agent-experimenter loop and its formalization as POMDP together with the inference problems from the agent's point of view and from the experimenter's point of view.
Figure 2 :
2Validation on synthetic reaching data. A Simulated trajectories of the reaching task. B Negative log likelihood for different combinations of the parameters given data and the resp. third parameter. C Belief tracking: During a reaching movement, the agent has a belief about the position, velocity, and acceleration (green dotted lines). The experimenter observes a noisy version of the agent's actual movements (blue) and computes an estimate of the agent's belief (orange dashed lines with shaded region representing 2 × SD). D Mean trajectories with RMSE 0.016 (MLE), 0.1, 0.5, and 1.0. E RMSE of the MLEs for all parameters on pairwise grids (for one value respective third parameter). The colors are the same as in the previous subplot. F Distribution over 10 repetitions of RMSE for each parameter across different values for the other parameters. G RMSE and KL divergences (between empirical distributions of true trajectories and simulated ones based on the MLE) with 0.2 and 0.8 quantiles for different numbers of trajectories with parameters as in (A).
Figure 3 :
3Application to real reaching data. A Negative log likelihoods for the three model parameters. B Reaching trajectories (velocities and accelerations computed with finite differences) and simulated data from the model with MLE parameters.
Figure 4 :
4Saccadic eye movements. A Generated trajectories. B Median and quantiles of MLEs for a range of true parameter values of r.
Figure 5 :
5Random problems. A MLEs for a range of parameter values of r 1 for different random problems and different values of r 2 (color). B Aggregated results (median, percentiles) for 2000 random problems.
t+1 T
t+1with observations o t , so the goal becomes to approximate p(x t+1 ,x t+1 , o t+1 |o 1:t ). If p(x t ,x t |o 1:t ) is distributed as in Eq. (17), Eq. (18) gives directly the formula for the approximation of p(x T t+1 ,x T t+1 , o T t+1 | o 1:t ).
G. 1
1Synthetic reaching data G.1.1 Comparison to a baseline
Fig. 6
6shows the RMSEs of our method in comparison to the two baselines described in Section 4.1. 4 https://crcns.org/data-sets/movements/dream/overview
Figure 6 :Figure 7 :
67Evaluation of the proposed inverse optimal control method in terms of RMSE of the maximum-likelihood parameter estimates. Left: Our method, in which the likelihood is approximated via moment-matching. Middle: Standard LQG (without signal-dependent noise), for which the likelihood of the parameters given the data can be calculated in closed-form. Right: Standard LQG where the additive noise level was set to the average noise magnitude in the trajectories.A KL for pairs B Average KL KL divergences. As a secondary evaluation metric, we show the KL divergence between trajectories simulated using the true parameters and trajectories simulated using the MLEs. The results are qualitatively similar to the RMSEs and show that the inferred parameters generate trajectories very similar to the observed data.
Figure 8 :Figure 9 :
89Evaluation of partially observed model. RMSEs and KLs for partially observed reaching model. Negative log likelihod of the partially observed model.
Figure 11 :
11Random problems (second parameter). A MLEs for a range of parameter values of r 2 for different random problems and different values of r 1 . B Aggregated results (median, percentiles) for 1000 random problems.
A Simulated data B Likelihood function E RMSE for pairs D Mean simulated data C Belief tracking G RMSE # trials F RMSE per parameter
Table 1 :
1Notation
Python implementation under GNU GPL available at PyPI 3 https://github.com/RothkopfLab/inverse-optimal-control
Acknowledgments and Disclosure of FundingWe acknowledge support from the Lichtenberg high performance computing facility of the TU Darmstadt, the projects 'The Adaptive Mind' and 'The Third Wave of AI' funded by the Excellence Program and the project 'WhiteBox' funded by the Priority Program LOEWE of the Hessian Ministry of Higher Education, Science, Research and Art.
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"Paolo Bertozzini ",
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"Chatchai Puttirungroj \nDepartment of Mathematics and Statistics\nFaculty of Science and Technology\nThammasat University\n12121PathumthaniThailand\n"
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"Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza\nUniversità di Roma\nVia A. Scarpa 16, I00161RomaItaly",
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]
| []
| Suitable duals of multimodules are introduced and used to provide transposition contravariant right semiadjunctions (and dualitites under reflexivity). Several additional notions on multimodules are discussed: generalized morphisms and involutions; tensor products and contractions; inner products; first-order differential operators. A construction of an (involutive) colored properad of multimodules is suggested.The study of categories of modules and bimodules over unital associative rings or algebras is one of the most developed subjects of modern algebra and its inception might be traced back to the work of R.Dedekind, E.Noether and B.L.van den Waerden, among many others.Multimodules over unital associative rings and algebras are quite a natural generalization of right/left-modules and bimodules that, as far as we know, have been first described in N.Bourbaki [Bourbaki 1942]. Since, for most purposes, multimodules are equivalently seen as bimodules over tensor products of rings and algebras, it can be claimed that their investigation essentially reduces to the study of special classes of bimodules and not much attention has been paid to them (we have been able to locate only one specific reference on multimodules[Kertész 1962] and some sporadic mentioning of them, for example in [Takeuchi 1987, section 0]).The "substitution" of multimodules with corresponding bimodules over tensor products turns out to be problematic whenever the category of morphisms is extended with the inclusion of maps that have different covariance properties with respect to the several actions involved. One could still substitute multimodules with bimodules over tensor products of rings, as long as such tensor products of rings are simultaneously equipped with different products (all distributive with respect to the same Abelian group structure), but this essentially amounts to define an "hyper-algebra structure" on the tensor product multimodule of the rings (see remark 3.3).The basic algebraic material here presented naturally arose as a byproduct in our study of non-commutative generalizations of contravariant calculus. 1 Since quite surprisingly we have not been able to locate any relevant source dealing with this topic, we thought that the subject deserves an adequate separate treatment. Specifically (anticipating arguments and motivations pertaining to the aforementioned work) in non-commutative (algebraic) geometry, it is a common thread to look for generalizations of the usual notion of "differential operator" to the case of maps between bimodules over a non-commutative algebra A and it often happens (for example whenever one is considering "double derivations" on A) that the spaces of such "non-commutative differential operators" are naturally equipped with a multimodule structure over the original non-commutative algebra A. Although of tangential interest for this work, a general definition of first-order differential operators between multimodules, covering in particular all such cases, will be included in appendix B. Further developments in these directions, including investigations of non-commutative vector fields and non-commutative connections on (multi-)modules, will have to wait subsequent works (see the paper in footnote 1 and references therein for more details).In short, the specific goals of the present work are to: define multimodules based over an arbitrary Z-central bimodule 2 (more generally over a Z-central unital associative ring R Z ) instead of just an Abelian group: this allows to discuss mutually commuting (right/left) actions that are compatible with a certain fixed Z-linear structure, but that can still have alternative R-linearity properties;introduce a notion of involution for multimodules that allows for different covariance/contravariance: since involutions for us are just involutive morphisms, this requires an appropriate definition of category of multimodules, where morphisms (necessarily Z-linear) can have different covariance properties (and even different conjugate-R Z -linearity properties) with respect to the different actions involved;provide a systematic treatment of the several (Z-linear) duals of multimodules, their associated categorical semi-adjunctions and (under saturation conditions for evaluations) establish transposition dualities: 1 P.Bertozzini, R.Conti, C.Puttirungroj, Non-commutative Contravariant Differential Calculus (in preparation). 2 Where Z is a commutative unital associative ring/algebra | 10.1016/j.indag.2022.02.005 | [
"https://arxiv.org/pdf/2111.02184v1.pdf"
]
| 241,035,518 | 2111.02184 | a227190af427690073ead57bb3e272151d6548bb |
Duality for Multimodules
Nov 2021 MCS-2020: 16D10, 16D80, 16D90, 16D99, 18M85
Paolo Bertozzini
Roberto Conti [email protected]
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza
Università di Roma
Via A. Scarpa 16, I00161RomaItaly
Chatchai Puttirungroj
Department of Mathematics and Statistics
Faculty of Science and Technology
Thammasat University
12121PathumthaniThailand
Duality for Multimodules
Nov 2021 MCS-2020: 16D10, 16D80, 16D90, 16D99, 18M85dedicated to N.Bourbaki who initiated the study of multimodulesMultimoduleInvolutionDualitySemi-adjunctionInvolutive Colored Properad
Suitable duals of multimodules are introduced and used to provide transposition contravariant right semiadjunctions (and dualitites under reflexivity). Several additional notions on multimodules are discussed: generalized morphisms and involutions; tensor products and contractions; inner products; first-order differential operators. A construction of an (involutive) colored properad of multimodules is suggested.The study of categories of modules and bimodules over unital associative rings or algebras is one of the most developed subjects of modern algebra and its inception might be traced back to the work of R.Dedekind, E.Noether and B.L.van den Waerden, among many others.Multimodules over unital associative rings and algebras are quite a natural generalization of right/left-modules and bimodules that, as far as we know, have been first described in N.Bourbaki [Bourbaki 1942]. Since, for most purposes, multimodules are equivalently seen as bimodules over tensor products of rings and algebras, it can be claimed that their investigation essentially reduces to the study of special classes of bimodules and not much attention has been paid to them (we have been able to locate only one specific reference on multimodules[Kertész 1962] and some sporadic mentioning of them, for example in [Takeuchi 1987, section 0]).The "substitution" of multimodules with corresponding bimodules over tensor products turns out to be problematic whenever the category of morphisms is extended with the inclusion of maps that have different covariance properties with respect to the several actions involved. One could still substitute multimodules with bimodules over tensor products of rings, as long as such tensor products of rings are simultaneously equipped with different products (all distributive with respect to the same Abelian group structure), but this essentially amounts to define an "hyper-algebra structure" on the tensor product multimodule of the rings (see remark 3.3).The basic algebraic material here presented naturally arose as a byproduct in our study of non-commutative generalizations of contravariant calculus. 1 Since quite surprisingly we have not been able to locate any relevant source dealing with this topic, we thought that the subject deserves an adequate separate treatment. Specifically (anticipating arguments and motivations pertaining to the aforementioned work) in non-commutative (algebraic) geometry, it is a common thread to look for generalizations of the usual notion of "differential operator" to the case of maps between bimodules over a non-commutative algebra A and it often happens (for example whenever one is considering "double derivations" on A) that the spaces of such "non-commutative differential operators" are naturally equipped with a multimodule structure over the original non-commutative algebra A. Although of tangential interest for this work, a general definition of first-order differential operators between multimodules, covering in particular all such cases, will be included in appendix B. Further developments in these directions, including investigations of non-commutative vector fields and non-commutative connections on (multi-)modules, will have to wait subsequent works (see the paper in footnote 1 and references therein for more details).In short, the specific goals of the present work are to: define multimodules based over an arbitrary Z-central bimodule 2 (more generally over a Z-central unital associative ring R Z ) instead of just an Abelian group: this allows to discuss mutually commuting (right/left) actions that are compatible with a certain fixed Z-linear structure, but that can still have alternative R-linearity properties;introduce a notion of involution for multimodules that allows for different covariance/contravariance: since involutions for us are just involutive morphisms, this requires an appropriate definition of category of multimodules, where morphisms (necessarily Z-linear) can have different covariance properties (and even different conjugate-R Z -linearity properties) with respect to the different actions involved;provide a systematic treatment of the several (Z-linear) duals of multimodules, their associated categorical semi-adjunctions and (under saturation conditions for evaluations) establish transposition dualities: 1 P.Bertozzini, R.Conti, C.Puttirungroj, Non-commutative Contravariant Differential Calculus (in preparation). 2 Where Z is a commutative unital associative ring/algebra
it is already known that in the case of bimodules one needs to separately consider right and left duals in place of the usual notion of dual vector space; in the case of multimodules, the situation is a bit more involved and one can construct different (conjugate)-duals for any choice of subfamilies of left/right actions (and corresponding conjugations of R Z ); each dual is defined in this work via a universal factorization property and its elements are concretely realized as Z-multilinear functions that have selective R Z -(conjugate)-linearity properties with respect to the specified actions;
introduce universal traces, and more generally contractions, on multimodules;
traces of linear operators and contractions of tensors are quite standard operations performed in multilinear algebra; we reframe such notions in the more general context of multimodules, providing again a definition via universal factorization properties;
discuss, for multimodules over involutive rings/algebras, suitable notions of "inner products" and (under conditions of non-degeneracy/fullness) establish Riesz isomorphisms:
inner products on multimodules also come in several types, each corresponding to a different dual, and are here realized as certain balanced multi-sesquilinear maps; involutive algebras are necessary in order to give a meaning to Hermitianity conditions on inner products; every inner-product induces a canonical Riesz morphisms of a multimodule into a corresponding dual; non-degeneracy and fullness are required to obtain an isomorphism. describe first order differential operators between multimodules: the first order condition in non-commutative geometry [Connes 1994, sections 4.γ and 4.δ], usually formulated in the case of operators between bimodules, is here expanded to cover the general setting of multimodules; make the first steps toward a study of involutive colored properads using multimodules as a template:
the material here included is mostly intended to provide a usable language for quite practical situations (some of which have been actually originating from work in categorical non-commutative geometry) where multimodules and their duals might be used and manipulated. As a consequence, we have not been looking for maximal generality in the statements and we kept a rather low sophistication level in the discussion of all the category-theoretical aspects of the subject; a more detailed study of these topics is under way, but we can already anticipate that it will fall within the scope of certain variants of involutive colored properads and involutive polycategories. As stated above, we plan to address more properly these points in subsequent works.
Here below is a more detailed description of the content of the paper.
In section 2 we modify the usual setting of bimodules over unital associative rings considering, in place of the initial ring Z a commutative unital associative ring Z and instead of rings acting on Abelian groups (Z-bimodules), Z-central algebras A acting in a Z-bilinear way on Z-central bimodules M. Morphisms are in this case pairs of Z-linear maps (in place of additive maps) that induce a unital Z-linear covariant or contravariant grade-preserving homomorphism on the associated N-graded algebras M := A ⊕ M ⊕ {0} · · · of the bimodules. This kind of environment can immediately describe, as a special case, categories of K-linear covariant or contravariant morphisms of unital bimodules over K-algebras, for a certain field K in place of Z.
The existence of many situations requiring the usage of non-trivial (involutive auto)morphisms for the base field K and the consequent need to deal simultaneously with maps that are not K-linear, imposes a further refinement of the structure: the common base commutative associative unital ring Z is replaced by a Z-central unital associative ring R Z . The family of unital covariant or contravariant Z-linear homomorphisms φ of Z ⊕ R identifies the possible alternative notions of φ-linearity with respect to the base ring R. The paradigmatic situation with R := C and Z := R imposes only R-linearity on morphisms that are further classified as C-linear and C-conjugate-linear depending on the choice of the C-automorphism φ; but the formalism can be used in the case of algebras over arbitrary extensions of fields (or more generally extensions of rings).
Section 3 puts forward our definition of multimodules over families of unital associative algebras over R Z . We stress that taking R = Z = Z, we just reproduce the usual definition of multimodules in [Bourbaki 1942] and taking Z → R an extension of fields we obtain multimodules as Z-vector spaces equipped with Z-bilinear actions of R-algebras, where morphisms can be φ-linear for any Z-linear automorphism φ of R. The unavoidability of multimodules (in every context dealing with bimodules) is witnessed by the construction of Z-central multimodules of Z-linear maps, and Z-tensor products, between Z-central bimodules.
In section 4 we specialize to the treatment of involutive endomorphisms of Z-central multimodules over Z-central R Z -algebras and we examine how involutions on bimodules (and multimodules) propagate to involutions for spaces of Z-linear morphisms and tensor products of multimodules.
The main result of the paper is contained in section 5 where we introduce definitions of duals of multimodules via universal factorization properties and we prove that transpositions functors in the category of multimodules give rise to contravariant right semi-adjunctions (theorem 5.3) that, for multimodules satisfying reflexivity, produce dualities. In general there exist different conjugate-duals for a Z-central multimodule (A α ) A M (B β ) B over R Z -algebras, each one of them (γ i ) I M (γ j ) J specified by certain families (γ i ) i∈I and (γ j ) j∈J of R-conjugations, with arbitrary sets of indexes I ⊂ A and J ⊂ B.
In the first part of section 6 we define universal contractions/traces on multimodules via universal factorization properties and we construct them quotienting the original multimodules with respect to certain commutator sub-multimodules. The remaining part of section 6 discusses tentative generalizations, to the setting of multimodules over involutive algebras, of the familiar notion of inner product for vector spaces or modules and for each such inner product defines its Riesz "natural transformation". 3 Under conditions of non-degeneracy and fullness of the inner products, we also provide a multimodule version of Riesz isomorphism theorem. The inner products here introduced are not necessarily positive: a positivity requirement can be added (at such an abstract level) imposing the existence of positive cones on the algebras.
The final outlook section 7 briefly expands on the already mentioned planned utilization of the categories of multimodules, here developed, as a paradigmatic example in the study of the abstract notion of "involutive colored properad" and their associated involutive "convolution hyper-algebroids" following the lines that some of us have discussed in previous papers [Bertozzini Conti Lewkeeratiyutkul Suthichitranont 2020].
In appendix A, we briefly recall the notion of (contravariant) semi-adjunction [Medvedev 1974], a special case of regular full functorial pairings later defined in [Wisbauer 2013], that will be needed to describe the dualities for contravariant trasposition functors in categories of multimodules. Special attention has been devoted to the explicit characterization of semi-adjunctions for contravariant functors.
As already mentioned, the present paper was motivated by an ongoing effort towards the study of non-commutative vectors fields and contravariant non-commutative differential calculus (see footnote 1); in appendix B we present the generalization, to the case of multimodules, of a definition of first-order differential operator on bimodules over non-commutative algebras, that has been useful in that context. Further extensions in the direction of differential analysis on multimodules (starting with a theory of connections) are briefly mentioned in the outlook section and will be dealt with elsewhere.
Generalities
We start specifying basic settings and definitions; for more details on background material that is not explicitly mentioned, we refer to the texts [Aluffi 2009] and [Bourbaki 1942].
We assume Z to be a commutative unital associative ring. All the rings R here considered will be unital associative (not necessarily commutative) and Z-central rings: they are equipped with a unital homomorphism of rings ι R : Z → Z(R) := {r ∈ R | ∀x ∈ R : r · x = x · r}, where Z(R) denotes the center of the ring R (itself a commutative unital associative ring).
All the R-bimodules M considered in this paper are assumed to be unital (1 R · x = x, for all x ∈ M) and Z-central R-bimodules, meaning that there is a unital homomorphism of rings ι M :
Z → Z(M) 0 , where we define Z(M) 0 := {r ∈ Z(R) | ∀x ∈ M : r · x = x · r} as the center ring of the R-bimodule M (that is itself a Z-central unital sub-ring of R). Similarly, Z(M) 1 := {x ∈ M | ∀r ∈ R : r · x = x · r} denotes the center module of M (itself a Z-central unital R-bimodule).
Here Z-central R-algebras are defined as Z-central R-bimodules A :
= R A R with a distributive multiplication • such that: (r · x) • y = r · (x • y), (x · r) • y = x • (r · y) and x • (y · r) = (x • y) · r,
for all x, y ∈ A and r ∈ R. In this way, multiplication in a Z-central R-algebra is necessarily Z-bilinear and every Z-central ring R becomes an (associative unital) Z-central algebra over itself. We will usually consider Z-central R-algebras A that are unital and associative. We will consider Z-central A-bimodules M that are unital and hence become canonically Z-central R-bimodules with action r · x := (r · 1 A ) · M x, for r ∈ R and x ∈ M.
Since Z is initial in the category of unital associative rings, Z · 1 R ⊂ Z ⊂ Z(R) and the characteristic of R is the minimum n ∈ N such that Ker(ι) = n · Z, where ι : Z → R is the initial unital homomorphism z → z · 1 R . Whenever the characteristic is a prime number, R is actually an F-algebra over the finite field F := Z/ Ker(ι).
Particular attention should be given to the definition of morphisms for bimodules over Z-central R-algebras.
Definition 2.1. Let Z be a commutative unital associative ring and R Z a unital associative Z-central ring.
A map M Φ − → N between two Z-central unital bimodules M := M Z , N := N Z , is said to be Z-linear if: Φ(x + y) = Φ(x) + Φ(y), Φ(ι M (z) · x) = ι N (z) · Φ(x), ∀x, y ∈ M, z ∈ Z. A map A φ − → B between Z-central unital associative rings A := A Z , B := B Z is • covariant if: φ(x • A y) = φ(x) • B φ(y), for all x, y ∈ A, • contravariant if: φ(x • A y) = φ(y) • B φ(x), for all x, y ∈ A, • unital if: φ(1 A ) = 1 B ,
• homomorphism if: it is Z-linear covariant and unital,
• anti-homomorphism if: it is Z-linear contravariant and unital.
A Z-linear map M := A M A Φ − → B N B between Z-central unital bimodules over Z-central unital associa- tive rings A := A Z , B := B Z is said to be φ-linear, for a certain Z-linear unital homomorphism (anti- homomorphism) A φ − → B if: Φ(a 1 · x · a 2 ) = φ(a 1 ) · Φ(x) · φ(a 2 ), ∀x ∈ M, a 1 , a 2 ∈ A, in the φ-covariant case, Φ(a 1 · x · a 2 ) = φ(a 2 ) · Φ(x) · φ(a 1 ), ∀x ∈ M, a 1 , a 2 ∈ A, in the φ-contravariant case. A φ-linear covariant (contravariant) morphism of Z-centralFor Z-central bimodules A M A , B N B over Z-central unital associative algebras R A R , R B R over a Z-central unital associative ring R Z , morphisms are still denoted by (φ, Φ), where Φ := (Φ 0 , Φ 1 ) is a pair of φ-linear unital morphisms Φ 0 : A → B of algebras and Φ 1 : M → N of bimodules, such that Φ 1 is Φ 0 -linear: Z ι R / / R φ ι A / / A Φ 0 M Φ 1 Φ 1 (a 1 · x · a 2 ) = Φ 0 (a 1 ) · Φ 1 (x) · Φ 0 (a 2 ) covariant φ-linear case Z ι R / / R ι B / / B N Φ 1 (a 1 · x · a 2 ) = Φ 0 (a 2 ) · Φ 1 (x) · Φ 0 (a 1 ) contravariant φ-linear case.
Remark 2.2. We have a category of Z-linear maps between Z-central unital bimodules over Z-central unital associative R Z -algebras. Such category is not Z 2 -graded with respect to covariance / contravariance, since the same morphism Φ can be φ-covariant or φ-contravariant depending on the choice of φ.
A better alternative consists, as we did, in defining morphisms as triples
A M A (φ,Φ) − −−− → B N B of Z-linear maps φ : R Z → R Z , Φ 0 : A R → B R and Φ 1 : M A → N B with (φ, Φ 0 ) and (Φ 0 , Φ 1 ) both φ-linear morphisms.
In this case the category is Z 2 -graded (by the covariance of the triple) furthermore it is isomorphic to the Z 2 -graded category of degree zero unital Z-linear (covariant or contravariant) morphisms (φ, Φ 0 , Φ 1 ) between graded unital associative Z-central algebras of the form M :
= R ⊕ A ⊕ M ⊕ {0} · · · . Definition 2.3. A covariant (respectively contravariant) involution on a Z-central unital associative ring R Z is a Z-linear covariant (respectively contravariant) map R ⋆ − → R that is involutive (x ⋆ ) ⋆ = x, for all x ∈ R.
Whenever dealing with Z-central algebras A R over a Z-central unital associative ring R Z , we use the term R Z -conjugation to denote an involution of the Z-central unital associative ring R Z .
A covariant (contravariant) involution ⋆ on R A R is said to be γ-conjugate-linear if it is γ-linear for a certain covariant (contravariant) R Z -conjugation γ, specifically: (r 1 · x · r 2 ) ⋆ = γ(r 1 ) · x ⋆ · γ(r 2 ) in the γ-covariant case; (r 1 · x · r 2 ) ⋆ = γ(r 2 ) · x ⋆ · γ(r 1 )) in the γ-contravariant case, for all r 1 , r 2 ∈ R and x ∈ A.
Remark 2.4. For Z-central algebras A R over non-commutative rings R, covariant (contravariant) involutions can be γ-conjugate-linear only with respect to a covariant (contravariant) conjugation γ. Whenever R is commutative, there is no difference between covariant and contravariant conjugations and hence, for an arbitrary conjugation γ, we can have covariant or contravariant involutions on A R that are γ-conjugate-linear.
Notice that for involutive
Z-central R-algebras A, we necessarily have η A (Z) ⊂ Z(A R ) 0 ∩ {x ∈ A | x ⋆ = x}.
It is of course possible, for a certain Z-central ring R to have involutions γ that do not necessarily leave η R (Z) invariant or that do not necessarily fix all the elements of η R (Z); in this case one can further "restrict" the commutative algebra Z in order to make γ a conjugation: given a certain family Γ of additive (covariant or contravariant) involutions of R, we see that Z Γ := η −1 R γ∈Γ {x ∈ Z(R) | γ(r) = r} is a unital sub-algebra of Z making all the γ ∈ Γ conjugations of R as Z Γ -central ring.
There are universal ways to reformulate γ-conjugate-linear unital morphisms of Z-central R-algebras (and also of Z-central R-bimodules) as covariant R-linear unital morphisms.
Definition 2.5. Given a Z-central R-algebra A R and a conjugation γ in R, a γ-conjugate of A R consists of a γ-conjugate-linear unital morphism of Z-central R-algebras A Remark 2.6. Unicity of γ-conjugates up to a unique isomorphism compatible with the universal property is standard, their existence can be provided as follows.
Given a Z-central unital associative R-algebra A R and a conjugation γ in R, take as a Z-central bimodule A γ := A and define η A : A → A γ as the identity map, here denoted as A ∋ x →x ∈ A γ . If γ is a contravariant conjugation, define r 1·x· r 2 := γ(r 2 ) · x · γ(r 1 ) andx•ŷ := y • x, for all x, y ∈ A and r 1 , r 2 ∈ R. If γ is a covariant conjugation, define r 1·x· r 2 := γ(r 1 ) · x · γ(r 2 ) andx•ŷ := x • y, for all x, y ∈ A and r 1 , r 2 ∈ R. Notice that in both cases A γ becomes a Z-central R-bimodule with the new actions· and it becomes a Z-central R-algebra with the new product•; furthermore the map η A : A → A γ turns out to be a Z-linear γ-conjugatelinear contravariant (respectively covariant) unital homomorphism. For any γ-conjugate-linear unital contravariant (respectively contravariant) homomorphism φ : A → B, we necessarily need to defineφ(x) := φ(x), and we verify thatφ : A γ → B is an R-linear unital covariant homomorphism in both cases.
Multimodules Over Unital Associative Z-central R-algebras
We introduce here multimodules over families of unital associative Z-central R-algebras. 4 In the following, we adapt the general definition of multimodule from [Bourbaki 1942, section II.1.14]:
Definition 3.1. Let Z be a commutative unital associative ring and R be a unital associative Z-central ring. Given two families of unital associative Z-central R-algebras (A α ) α∈A and (B β ) β∈B , an • an injective function f :
(A α )-(B β ) multimodule (A α ) M (B β ) is a Z-central bimodule that is a Z-central unital A α -B β bimodule for every (α, β) ∈ A × BA morphism of multimodules (A α ) A M (B β ) B (φ,η,Φ,ζ,ψ) f − −−−−−−− → (C γ ) C N (D δ ) D , (A + , B + )-covariant inA ⊎ B → C ⊎ D, with A + = A ∩ f −1 (C), B + = B ∩ f −1 (D); • two maps A φ − → End Z (R) ψ ← − B associating to every pair of indexes α ∈ A and β ∈ B two Z-linear unital endomorphisms φ α , ψ β of R Z , covariant for (α, β) ∈ A + × B + and contravariant for (α, β) ∈ A − × B − , • for (α, β) ∈ A + × B + , Z-linear covariant unital homomorphisms A α (φ α ,η α ) − −−−− → C f (α) , B β (ψ β ,ζ β ) − −−−− → D f (β) ; • for (α, β) ∈ A − × B − , Z-linear contravariant unital homomorphisms A α (φ α ,η α ) − −−−− → D f (α) , B β (ψ β ,ζ β ) − −−−− → C f (β) ; • a Z-linear map M Φ − → N such that Φ(a · x · b) = η α (a) · Φ(x) · ζ β (b), for all (α, β) ∈ A + × B + and Φ(a · x · b) = ζ β (b) · Φ(x) · η α (a), for all (α, β) ∈ A − × B − , (a, b) ∈ A × B and x ∈ M.
The signature of the morphism is (φ, η, ζ, ψ) f . The function f is the covariance of signature of the morphism and covariant morphisms are those for which f (A) ⊂ C and f (B) ⊂ D. The pair (φ, ψ) is the R-linearity of the signature of the morphism and R-linear morphisms are those for which both φ and ψ are constant equal to Id R . In some cases we will denote by Φ σ a morphism (φ, η, Φ, ζ, ψ) f with signature σ = (φ, η, ζ, ψ) f .
The composition of morphisms
(A α ) A M (B β ) B (φ 2 ,η 2 ,Φ 2 ,ζ 2 ,ψ 2 ) f 2 − −−−−−−−−−−− → (A ′ α ′ ) A ′ N (B ′ β ′ ) B ′ (φ 1 ,η 1 ,Φ 1 ,ζ 1 ,ψ 1 ) f 1 − −−−−−−−−−−− → (A ′′ α ′′ ) A ′′ P (B ′′ β ′′ ) B
′′ of multimodules is given componentwise:
(φ 1 , η 1 , Φ 1 , ζ 1 , ψ 1 ) f 1 • (φ 2 , η 2 , Φ 2 , ζ 2 , ψ 2 ) f 2 := (φ 1 • φ 2 , η 1 • η 2 , Φ 1 • Φ 2 , ζ 1 • ζ 2 , ψ 1 • ψ 2 ) f 1 • f 2 . The identity of a multimodule (A α ) A M (B β ) B is the morphism (Id R , (Id A α ) A , Id M , (Id B β ) B , Id R ) Id A⊎B .
Remark 3.2. The map Φ : M → N between multimodules does not have an intrinsic covariance: for every left index α ∈ A and for every right index β ∈ B the morphisms (η α , Φ) and (ζ β , Φ) are covariant or contravariant depending on the sign ± indicated in the subsets A ± and B ± .
Similarly, the map Φ : M → N between multimodules is always Z-linear, but it does not have an intrinsic φ-linearity with respect to R for a fixed Z-linear morphism φ: for every left index α ∈ A and right index β ∈ B, the morphism (η α , Φ) is φ α -linear and the morphism (ζ β , Φ) is ψ β -linear.
We have a category M [R Z ] of morphisms of Z-central multimodules over R Z -algebras with composition of morphisms defined componentwise. The subcategories of M [R Z ] consisting of (A α ) α∈A -(B β ) β∈B multimodules, over the same two families of unital associative R Z -algebras, and morphisms given by (φ, η, Φ, ζ, ψ) f , with f := Id A⊎B , φ α := Id R =: ψ β and η α =
Id A α , ζ β = Id B β for all (α, β) ∈ A × B, are denoted by (A α ) A M (B β ) B .
In case of A-bimodules, we use the notation A M A .
This essential remark explains why the study of multimodules cannot be "reduced" to the theory of bimodules.
Remark 3.3. If R = Z, it is common to dismiss the usage of multimodules (A α ) M (B β ) in favor of their "equiv- alent" description as bimodules R α A α M R β B β over tensor product R-algebras R α∈A A α and R β∈B B β since: if R = Z, there is a categorical isomorphism between the sub-category (A α ) A M (B β ) B of covariant R-linear morphisms of (A α ) A -(B β ) B -multimodules and the category R β A α M R β B β of covariant R-linear morphisms of bimodules over the R-balanced tensor product of the R Z -algebras.
As soon as one considers morphisms of multimodules with arbitrary covariance f , it is actually impossible to impose a unique unital associative product on the R-tensor product algebras in order to obtain a similar equivalent treatment via categories of bimodules.
A perfectly possile alternative (that we do not pursue here) would be to work with the category of "bimodules" over hyper-Z-central R-algebras: Z-central bimodules Z α∈A A α equipped with many different R-actions (on each of the tensor-factors) and different Z-bilinear associative unital binary product operations suitably compatible with the R-actions (see for example [Bertozzini Conti Lewkeeratiyutkul Suthichitranont 2020, section 5.3]); but in this case multimodules need anyway to be used in order to define hyper-algebras.
To a certain extent, the usage of general morphisms of multimodules (with arbitrary conjugation and convariance signatures as in definition 3.1) can be avoided, replacing the target multimodule with a suitable "twisted version" (depending on the signatures of the original morphism) and obtaining as a result an R-linear covariant morphism into such "twisted multimodule". The construction follows similar steps as in definition 2.5 and remark 2.6 and it simultaneously extends to multimodules the notions of conjugate-dual, opposite, restriction of rings, pull-back.
Definition 3.4. Let (A α ) A M (B β ) B and (C γ ) C N (D δ ) D be two Z-central multimodules over R Z -algebras, and let σ := (φ, η, ζ, φ) f
be a given signature for multimodule morphisms between M and N.
A σ-twisted multimodule 6 of N consists of a morphism of multimodules (A α
) A N σ (B β ) B Θ σ N − −− → (C γ ) C N (D δ ) D ,
with signature σ, such that the following universal factorization property is satisfied: for any other morphism of
multimodules (A α ) A M (B β ) B Φ − → (C γ ) C N (D δ ) D , with signature σ, there exists a unique covariant R-linear morphism of multimodules (A α ) A M (B β ) B Φ σ − − → Φ (A α ) A N σ (B β ) B in the category (A α ) A M (B β ) B such that Φ = Θ σ N • Φ σ . Remark 3.5.
As any definition via universal factorizations, σ-twisted of a given multimodule are unique, up to a unique isomorphism compatible with the factorization property. A construction can be achieved as follows. Consider N σ := N as a Z-central bimodule and Θ σ N : N σ → N as the identity map. For all x ∈ N we will denote by x σ ∈ N σ its corresponding element, hence Θ σ
N (x σ ) = x, for all x ∈ N. For all σ-covariant indexes (α + , β + ) ∈ A + × B + , and σ-contravariant indexes (α − , β − ) ∈ A − × B − , we define new actions on N σ : a · α + x σ · β + b := η α + (a) · f (a + ) x · f (b + ) ζ b + (b) σ , ∀(a, b) ∈ A α + × B β + , ∀x σ ∈ N σ , a · α − x σ · β − b := ζ b − (b) · f (b − ) x · f (a − ) η α − (a) σ , ∀(a, b) ∈ A α − × B β − , ∀x σ ∈ N σ , obtaining a multimodule (A α ) A N σ (B β ) B such that the map Θ σ N : x σ → x is a morphism of multimodules with signature σ. Finally, given any other morphism (A α ) A M (B β ) B Φ − → (C γ ) C N (D δ ) D of multimodules with signature σ, the function Φ σ : m → (Φ(m)) σ ∈ N σ , (due to the bijectivity of Θ σ N ) is the unique map that satisfies Θ σ N (Φ σ (m)) = Θ σ N ((Φ(m)) σ ) = Φ(m)
, for all m ∈ M, and by direct calculation, we see that it is also a morphism of multimodules with identity signature.
As typical of any category of homomorphisms of algebraic structures, sub-structures can be defined via algebraically closed subsets and quotient-structures via congruences. 6 We might also write Φ-twisted of N, for a morphism M
Φ − → N, instead of σ(Φ)-twisted, where σ(Φ) denotes the signature of Φ. Definition 3.6. Given a multimodule (A α ) A M (B β ) B over Z-central R-algebras,
• a sub-multimodule of M is a subset N ⊂ M that is algebraically closed under all the operations:
0 M ∈ N, x, y ∈ N ⇒ x+y ∈ N, x ∈ N ⇒ a· α x· β b ∈ N, ∀(α, β) ∈ A×B, (a, b) ∈ A α ×B β , x, y ∈ N; • a multimodule congruence on (A α ) A M (B β ) B is an equivalence relation E ⊂ M × M such that: x ∼ E y ⇒ (x+z) ∼ E (y+z), x ∼ E y ⇒ (a· α x· β b) ∼ E (a· α y· β b), ∀(α, β) ∈ A×B, (a, b) ∈ A α ×B β , x, y, z ∈ M. A quotient multimodule of M by the congruence E is the multimodule (A α ) A ( M E ) (B β ) B consisting of the quotient set M/E equipped with the well-defined addition [x] E + [y] E := [x + y] E , for all x, y ∈ M, and the well-defined actions: a · α [x] E · β b := [a · α x · β b] E , for all x ∈ M, (α, β) ∈ A × B and (a, b) ∈ A α × B β .
Remark 3.7. As usual, any multimodule congruence E uniquely determines the M-sub-multimodule [0 M ] E ; reciprocally any M-sub-multimodule N uniquely determines a multimodule congruence x ∼ y :⇔ x − y ∈ N whose equivalence classes, for x ∈ M, are the affine spaces [x] ∼ = x + N := {x + y | y ∈ N}. The notation M N is used to identify the quotient of M by the congruence uniquely determined by the sub-multimodule N.
Inclusions of sub-multimodules
N ι − → M and quotients M π − → M N are morphisms in the category (A α ) A M (B β ) B .
Despite being rarely mentioned, multimodules naturally appear whenever bimodules are around:
Proposition 3.8. Let A M B and A ′ N B ′ be Z-central bimodules over Z-central unital associative R-algebras A, A ′ , B, B ′ . The set Hom Z (M; N) of Z-linear maps φ : M → N is a left-(A ′ , B) right-(A, B ′ ) multimodule with the following actions, for all a ∈ A, a ′ ∈ A ′ , x ∈ M, b ∈ B, b ′ ∈ B ′ : left external action: (a ′ · φ)(x) := a ′ φ(x),
right external action:
(φ · b ′ )(x) := φ(x)b ′ , left internal action: (b ⊙ φ)(x) := φ(xb), right internal action: (φ ⊙ a)(x) := φ(ax). Proof. By direct calculation, x → (a ′ · φ)(x), x → (φ · b ′ )(x), x → (b ⊙ φ)(x), x → (φ ⊙ a)(x)
are all Z-linear and the above defined maps are all Z-bilinear actions. To prove the multimodule structure on Hom Z (M; N) we check that the actions pairwise commute, for all a ∈ A, a ′ ∈ A ′ and all b ∈ B, b ′ ∈ B ′ :
(a ′ · φ) · b ′ = a ′ · (φ · b ′ ), (a ′ · φ) ⊙ a = a ′ · (φ ⊙ a), (b ⊙ φ) ⊙ a = b ⊙ (φ ⊙ a), (b ⊙ φ) · b ′ = b ⊙ (φ · b ′ ). Remark 3.9. More generally, if (A α ) M (B β ) and (C γ ) N (D δ ) are Z-central multimodules over Z-central R-algebras, the Z-central bimodule Hom Z (M; N) becomes a left-(B β , C γ ) β∈B,γ∈C and a right-(A α , D δ ) α∈A,δ∈D Z-central mul- timodule with internal/external actions given by: (c γ · φ · d δ )(x) := c γ · φ(x) · d δ , (b β ⊙ φ ⊙ a α )(x) := φ(a α · x · b β ), for all (α, β, γ, δ) ∈ A × B × C × D, (a α , b β , c γ , d δ ) ∈ A α × B β × C γ × D δ , x ∈ M and φ ∈ Hom Z (M; N).
Tensor products provide other examples of multimodules [Bourbaki 1942, section II.3.4]:
Proposition 3.10. Let (A α ) M (B β ) and (C γ ) N (D δ ) be Z-central multimodules over Z-central R-algebras. Their tensor product M ⊗ Z N over Z is a left-(A α , C γ ) α∈A,γ∈C right-(B β , D δ ) β∈B,δ∈D Z-central multimodule.
Proof. The definition of tensor product (via universal factorization property for Z-balanced bi-homomorphism) and its construction are well-known: see for example [Bourbaki 1942, section II.3, proposition 3]; we only recall here the relevant actions on simple tensors:
a · (x ⊗ Z y) = (a · x) ⊗ Z y, c · (x ⊗ Z y) = x ⊗ Z (c · y), (x ⊗ Z y) · b = (x · b) ⊗ Z y, (x ⊗ Z y) · d = x ⊗ Z (y · d), for all (x, y) ∈ M × N, (a, b, c, d) ∈ A α × B β × C γ × D δ , (α, β, γ, δ) ∈ A × B × C × D.
Remark 3.11. One can actually define tensor products of multimodules in much greater generality.
Instead of taking only the tensor product over the algebra Z of "scalars" and use Z-bilinear maps, we can "contract" over arbitrary families of shared Z-central R-algebras acting on the two multimodules and utilize suitable maps that are "balanced" over the "contracted actions", obtaining multimodules over the remaining "un-contracted" actions, as detailed in the following exposition.
Let (A α ) A M (A β ) B and (A γ ) C N (A δ ) D be a pair of multimodules; consider the relation A ⊎ B Σ − → C ⊎ D, defined by (ξ, ζ) ∈ Σ ⇔ A ξ = A ζ (where ξ ∈ A ⊎ B and ζ ∈ C ⊎ D), and let A ⊎ B ⊃ A ′ ⊎ B ′ Γ − → C ′ ⊎ D ′ ⊂ C ⊎ D be a bijective function between subsets of indexes, such that Γ ⊂ Σ (in practice A ξ = A Γ(ξ) , for all ξ ∈ A ′ ⊎ B ′ ).
A tensor product of multimodules M and N over Γ consists of:
• a left-(A ξ ) ξ∈(A−A ′ )⊎(C−C ′ ) right-(A ζ ) ζ∈(B−B ′ )⊎(D−D ′ ) multimodule (A ξ ) (A−A ′ )⊎(C−C ′ ) (M ⊗ Γ N) (A ζ ) (B−B ′ )⊎(D−D ′ ) , • a Γ-balanced bi-morphism M × N η − → M ⊗ Γ N, that means a Z-bilinear map that satisfies: η(a · ξ x, y) = η(x, a · Γ(ξ) y), ∀a ∈ A ξ , (ξ, Γ(ξ)) ∈ A ′ × C ′ , (x, y) ∈ M × N, η(x · ξ a, y) = η(x, y · Γ(ξ) a), ∀a ∈ A ξ , (ξ, Γ(ξ)) ∈ B ′ × D ′ , (x, y) ∈ M × N, η(x · ξ a, y) = η(x, a · Γ(ξ) y), ∀a ∈ A ξ , (ξ, Γ(ξ)) ∈ B ′ × C ′ , (x, y) ∈ M × N, η(a · ξ x, y) = η(x, y · Γ(ξ) a), ∀a ∈ A ξ , (ξ, Γ(ξ)) ∈ A ′ × D ′ , (x, y) ∈ M × N, (3.1) η(a · ξ x, y) = a · ξ η(x, y), ∀a ∈ A ξ , ξ ∈ A − A ′ , (x, y) ∈ M × N, η(x, c · ξ y) = c · ξ η(x, y), ∀c ∈ A ξ , ξ ∈ C − C ′ , (x, y) ∈ M × N, η(x · ξ b, y) = η(x, y) · ξ b, ∀b ∈ A ξ , ξ ∈ B − B ′ , (x, y) ∈ M × N, η(x, y · ξ d) = η(x, y) · ξ d, ∀d ∈ A ξ , ξ ∈ D − D ′ , (x, y) ∈ M × N,
in such a way that the following universal factorization property holds: for any other Γ-balanced bi-morphism
M×N Φ − → P into a left-(A ξ ) (A−A ′ )⊎(C−C ′ ) right-(A ζ ) ζ∈(B−B ′ )⊎(D−D ′ ) multimodule P, there exists a unique morphism of multimodules M ⊗ Γ NΦ − → P (over the same indexed families of algebras) such that Φ =Φ • η.
Its construction is standard and consists of the quotient of a free multimodule over M × N by the congruence generated by the required axioms of Γ-balanced bi-morphism. More specifically we recall that:
• a free (A α ) α∈A -(B β ) β∈B multimodule, over a set X, is function X η X − − → (A α ) A F(X) (B β ) B , with values into a (A α ) α∈A -(B β ) β∈B multimodule F(X)
, such that the following universal factorization property is satisfied:
for any other map X Φ − → (A α ) A M (B β ) B into an (A α ) α∈A -(B β ) β∈B multimodule M, there exists a unique morphism of multimodules F(X)Φ − → M in the category (A α ) A M (B β ) B such that φ =Φ • η X ;
• a construction of free multimodule over X can be achieved taking
x∈X [(⊗ Z α∈A A α ) ⊗ Z (⊗ Z β∈B B)], the set of finitely supported functions from X into (⊗ Z α∈A A α ) ⊗ Z (⊗ Z β∈B B)
with pointwise addition and pointwise outer target actions as specified in footnote 11, defining η X x (y) :=
(⊗ Z α∈A 1 A α ) ⊗ Z (⊗ Z β∈B 1 B β ), y = x, (⊗ Z α∈A 0 A α ) ⊗ Z (⊗ Z β∈B 0 B β )
, y x, for all x, y ∈ X, and checking the universal factorization property;
• the congruence E Γ generated by the relations in equations (3.1) is just the intersection of the set of congruences of (
A α ) A -(B β ) B multimodule in F(M × N)
, that contain all of the differences between left and right terms in each of the equations 3.1;
• the tensor product consists of the quotient multimodule M ⊗ Γ N :
= F(M×N) E Γ , with the Γ-balanced bi- morphism η := π • η M×N , where F(M × N) π − → M ⊗ Γ N is the quotient morphism.
Involutions in Multimodules
In parallel with the case of morphisms, also the nature of involutions in multimodules is more delicate and an involution is an involutive endomorphism inducing involutions on the algebras and conjugations on R.
Definition 4.1. Let (A α ) A M (A β ) B be a Z-central multimodule over Z-central unital associative R-algebras. A multimodule involution on M is a morphism (A α ) A M (A β ) B (φ,η,⋆,ζ,ψ) f − −−−−−−− → (A α ) A M (A β ) B that is involutive: • A ⊎ B f − → A ⊎ B is an involutive function f • f = Id A⊎B ; 7 • for all (α 1 , α 2 ) ∈ f ∩(A×A), A α 1 = A α 2 , † α 1 := φ α 1 = φ α 2 is a covariant R Z -conjugation, ‡ α 1 := η α 1 = η α 2 is a covariant † α 1 -linear involution; • for all (β 1 , β 2 ) ∈ f ∩ (B × B), A β 1 = A β 2 , † β 1 := ψ β 1 = ψ β 2 is a covariant R Z -conjugation, ‡ β 1 := ζ β 1 = ζ β 2 is a covariant † β 1 -linear involution; • for all (α, β) ∈ f ∩ (A × B), A α = A β , † α := φ α = ψ β =: † β is a contravariant R Z -conjugation, ‡ α := η α = ζ β =: ‡ β is a contravariant † α -linear involution; • A α M A β ⋆ − → A α M A β is an involution such that: ∀(α, β) ∈ A + × B + : (a · α x · β b) ⋆ = a ‡ α · f (α) x ⋆ · f (β) b ‡ β , ∀(a, b) ∈ A α × A β , x ∈ M; ∀(α, β) ∈ A − × B − : (a · α x · β b) ⋆ = b ‡ β · f (β) x ⋆ · f (α) a ‡ α , ∀(a, b) ∈ A α × A β , x ∈ M.
If necessary, we will denote an involution of (
A α ) A M M (A β ) B M by ( † σ M , ‡ σ M , ⋆ M ) σ∈ f M , where: Z ι R / / R † σ M ι A / / A σ 1 ‡ σ M M ⋆ M Z ι R / / R ι B / / A σ 2 M σ := (σ 1 , σ 2 ) ∈ f M ⊂ (A M ⊎ B M ) × (A M ⊎ B M ).
Here we examine involutions for multimodules of morphisms between involutive bimodules.
R † ρ := † A ι A / / A ‡ ρ := ‡ A Hom Z (M; N) ⋆ B ‡ σ := ‡ B R † σ := † B ι B o o R ι A / / A Hom Z (M; N) B R ι B o o f := f * M ⊎ f N , ρ ∈ f * M , σ ∈ f N .
The involution ⋆ has covariance signature and R-linearity signatures that, for inner actions, coincide with those of ⋆ M ; and for outer actions with those of ⋆ N . The involutivity of ⋆ follows from:
If ( C P C , ⋆ P ) is an involutive Z-central bimodule over R Z -algebras, we have (T • S ) ⋆ = T ⋆ • S ⋆ ,(T ⋆ ) ⋆ = ⋆ N • ⋆ N • T • ⋆ M • ⋆ M = T , for all T ∈ Hom Z (M; N).
For the actions, if (A, ‡ A ) and (B, ‡ B ) are contravariantly involutive, we necessarily have:
(c · b ⊙ T ⊙ a · d) ⋆ (x) = (c · T (a · x ⋆ M · b) · d) ⋆ N = (d ‡ B · T ((b ‡ A · x · a ‡ A ) ⋆ M ) ⋆ N · c ‡ B ) = (d ‡ B · a ‡ A ⊙ T ⋆ ⊙ b ‡ A · c ‡ B )(x), ∀a, b ∈ A, c, d ∈ B, x ∈ M.
Whenever (A, ‡ A ) and (B, ‡ B ) are covariantly involutive, we obtain:
(c · b ⊙ T ⊙ a · d) ⋆ (x) = (c · T (a · x ⋆ M · b) · d) ⋆ N = (c ‡ B · T ((a ‡ A · x · b ‡ A ) ⋆ M ) ⋆ N · d ‡ B ) = (c ‡ B · b ‡ A ⊙ T ⋆ ⊙ a ‡ A · d ‡ B )(x), ∀a, b ∈ A, c, d ∈ B, x ∈ M.
The remaining two cases with opposite contravariance between (A, ‡ A ) and (B, ‡ B ) are treated similarly.
Finally (T •S ) ⋆ = ⋆ P •T •S •⋆ M = ⋆ P •T •⋆ N •⋆ N •S •⋆ M = T ⋆ •S ⋆ , ∀(T, S ) ∈ Hom K (N; P)×Hom K (M; N).
Notice that the involution ⋆ : T → T ⋆ has R-linearity signature † M for the inner actions and the R-linearity signature of † N for the outer actions. The following proposition describes involutions in the case of tensor products of involutive multimodules.
Proposition 4.4. Let (A α ) A M (A β ) B , ((⋆ α ) A , ⋆ M , (⋆ β ) B ) f and (B γ ) C N (B δ ) D , ((⋆ γ ) C , ⋆ N , (⋆ δ ) D ) g be involutive Z-central multimodules over R Z -algebras; the tensor product multimodule (A α ,B γ ) A⊎C (M ⊗ Z N) (A β ,B δ ) B⊎D has an involution ((⋆ α , ⋆ β ) A⊎C , ⋆ M ⊗ Z ⋆ N , (⋆ γ , ⋆ δ ) B⊎D ) ( f,g) . Proof. Define A ⊎ B ⊎ C ⊎ D ( f,g) − −− → A ⊎ B ⊎ C ⊎ D as the "disjoint union" of the involutions A ⊎ B f − → A ⊎ B and C ⊎ D g − → C ⊎ D. It follows that ( f, g) is an involution. Furthermore, for all T ∈ {A, B}, for all τ ∈ {α, β, γ, δ} we have (T τ , ⋆ τ ) = (T ( f,g)(τ) ⋆ ( f,g)(τ) ). The Z-linear map M ⊗ Z N ⋆ M ⊗ Z ⋆ N − −−−−−−− → M ⊗ Z N, defined by universal factorization property from the Z-bilinear map M × N ∋ (x, y) → x ⋆ M ⊗ Z y ⋆ N ∈ M ⊗ Z N, for all x ∈ M, y ∈ N, is involutive.
The covariance/contravariance behavior of the involution ⋆ M ⊗ Z ⋆ N with respect to the several actions is described as follows, denoting by τ ± , for τ ∈ {α, β, γ, δ}, the indexes corresponding respectively to covariantly/contravariantly involutive algebras:
∀(α + , β + , γ + , δ + ) ∈ A + × B + × C + × D + , ∀(a, b) ∈ A α + × A β + , (c, d) ∈ B γ + × B δ + , (x, y) ∈ M × N : (a · α + c · γ + (x ⊗ Z y) · β + b · δ + d) ⋆ M ⊗ Z ⋆ N = (a · α + x · β + b) ⋆ M ⊗ Z (c · γ + y · δ + d) ⋆ N = (a ⋆ α+ · f (α + ) x ⋆ M · f (β + ) b ⋆ β+ ) ⊗ Z (c ⋆ γ+ · g(γ + ) y ⋆ N · g(δ + ) d ⋆ δ+ ) = a ⋆ α+ · f (α + ) c ⋆ γ+ · g(γ + ) (x ⊗ Z y) ⋆ M ⊗ Z ⋆ N · f (β + ) b ⋆ β+ · g(δ + ) d ⋆ δ+ , ∀(α − , β − , γ − , δ − ) ∈ A − × B − × C − × D − , ∀(a, b) ∈ A α − × A β − , (c, d) ∈ B γ − × B δ − , (x, y) ∈ M × N : (a · α − c · γ − (x ⊗ Z y) · β − b · δ − d) ⋆ M ⊗ Z ⋆ N = (a · α − x · β − b) ⋆ M ⊗ Z (c · γ − y · δ − d) ⋆ N = (b ⋆ β− · f (β − ) x ⋆ M · f (α − ) a ⋆ α− ) ⊗ Z (d ⋆ δ− · g(δ − ) y ⋆ N · g(γ − ) c ⋆ γ− ) = b ⋆ β− · f (β − ) d ⋆ δ− · g(δ − ) (x ⊗ Z y) ⋆ M ⊗ Z ⋆ N · f (α − ) a ⋆ α− · g(γ − ) c ⋆ γ− , where we used the fact that (T τ ± , ⋆ τ ± ) = (T ( f,g)(τ ± ) , ⋆ ( f,g)(τ ± ) ), for τ ∈ {α, β, γ, δ}.
More generally, we can use the tensor product over subfamilies defined in remark 3.11.
Remark 4.5. Let (A α ) A M (A β ) B , ((⋆ α ) A , ⋆ M , (⋆ β ) B ) f M and (B γ ) C N (B δ ) D , ((⋆ γ ) C , ⋆ N , (⋆ δ ) D ) f N be two involu- tive multimodules. The "internal tensor product" M ⊗ Γ N over an indexed family Γ ⊂ Σ := {(α, β) | A α = B β } of common subalgebras A β = B γ , with (β, γ) ∈ Γ, that is stable under f := ( f M , f N ), the disjoint union of the support involutions f M , f N : (ξ, ζ) ∈ Γ ⇒ ( f (ξ), f (ζ)) ∈ Γ, ∀ξ, ζ ∈ A ⊎ B ⊎ C ⊎ D,
becomes an involutive multimodule with involution ⋆ := ⋆ M ⊗ Γ ⋆ N . The involution ⋆ is well-defined by universal factorization property of tensor products. 10
Pairing Dualities in Z-central Multimodules
Here we provide an extension, to the case of Z-central multimodules, of the notion of duality of vector spaces.
Although tensor products are always introduced via their universal factorization property, and later used to provide examples of monoidal categories, in the literature duals are almost never defined via universal factorization properties and are rather described either with non-categorical definitions or as dual objects inside suitable monoidal categories.
Our main purpose here will be to directly discuss the several pairing dualities for multi-modules.
Let us more generally consider the case of Z-central left-
(A α ) α∈A right-(B β ) β∈B multimodules (A α ) α∈A M (B β ) β∈B .
We can define several notions of duals, one for every subset of indexes I × J ⊂ A × B: 10 Apart from checking directly that, under the stability condition, the involution is well-defined, it is also possible to obtain the same result, considering first the involution already defined in proposition 4.4 and making use of proposition 6.4 together with remark 6.5. 11 With some abuse of notation, will denote by · the "outer actions" on the tensor product multimodule (⊗
Definition 5.1. Given a Z-central multimodule (A α ) α∈A M (B β ) β∈B over R Z -algebras (A α ) α∈A -(B β ) β∈B and a family of indexes I×J ⊂ A×B, an (I, J)-dual of the multimodule M is a pair (N, τ), where (B β ) β∈B N (A α ) α∈A is a Z-central (B β ) β∈B -(A α ) α∈A multimodule over the R Z -algebras (B β ) β∈B -(A α ) α∈A and τ : N × M → (⊗ Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β ) is a Z-multilinear (A − I, B − J)-balanced (I, J)-multilinear map: 11 ∀t, x ∈ M, τ(t, a · α x · β b) = a · α τ(t, x) · β b, τ(b · β t · α a, x) = b · β τ(t, x) · α a, ∀(a, b) ∈ A α × B β , (α, β) ∈ I × J, τ(b · β t · α a, x) = τ(t, a · α x · β b), ∀(a, b) ∈ A α × B β , (α, β) ∈ (A − I) × (B − J),Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β ) given by: a · βo [(⊗ Z α∈I x α ) ⊗ Z (⊗ Z β∈J y β )] · αo b := (⊗ Z α∈I x ′ α ) ⊗ Z (⊗ Z β∈J y ′ β ), where x ′ α := x α , α α o a · x αo , α = α o and y ′ β := y β , β β o y βo · b, β = β o for all (α o , β o ), (α, β) ∈ I × J, (a, b) ∈ A αo × B βo , (x α , y β ) ∈ A α × B β .
satisfying the following universal factorization property: for any (A − I, B − J)-balanced (I, J)-multilinear map
Φ : N × M → (⊗ Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β ) where (B β ) β∈B N (A α ) α∈A is another Z-central (B β ) β∈B -(A α ) α∈A multimodule over R Z -algebras, there exists a unique morphism of multimodulesΦ : N → N such that Φ = τ • (Φ, Id M ). 12
Again, if an (I, J)-dual exists, it is unique up to a unique isomorphism of (B β ) β∈B -(A α ) α∈A multimodules satisfying the previous universal factorization property. The existence is provided in the following result.
* I M * J , τ) of the Z-central multimodule M over R Z -algebras (A α ) α∈A -(B β ) β∈B .
Proof. For every (I, J) with I × J ⊂ A × B, consider the following set:
* I M * J := M φ − → (⊗ Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β ) | ∀(α, β) ∈ I × J, ∀(a, b) ∈ A α × B β : φ(a · α x · β b) = a · α φ(x) · β b . We see that * I M * J is a Z-central (B β ) β∈A -(A α ) α∈A multimodule defining, for all φ, ψ ∈ * I M * J and x ∈ M: 13 φ + ψ : x → φ(x) + ψ(x), b • β φ • α a : x → b • β φ(x) • α a, ∀(α, β) ∈ I × J, (a, b) ∈ A α × B β , b ⊙ β φ ⊙ α a : x → φ(a · α x · β b), ∀(α, β) ∈ (A − I) × (B − J), (a, b) ∈ A α × B β .
The evaluation map τ(φ, x) := φ(x), for all φ ∈ * I M * J and x ∈ M turns out to be an (
A − I, B − J)-balanced (I, J)-multilinear map τ : * I M * J × M → (⊗ Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β ).
To every Z-multilinear (A− I)-(B− J)-balanced and (I, J)-multilinear map
Φ : N×M → (⊗ Z α∈I A α )⊗ Z (⊗ Z β∈J B β ), the usual Curry isomorphism associates the map Φ : N → [(⊗ Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β )] M that to every element t ∈ N associates the map Φ t : M → (⊗ Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β ) given by Φ t (x) := Φ(t, x), for all x ∈ M.
The defining properties of Φ entail that Φ t ∈ * I M * J , for all t ∈ N and that the map Φ : N → * I M * J given by t → Φ t is a morphism of (B β ) β∈B -(A α ) α∈A multimodules. Finally Φ(t, x) = Φ t (x) = τ( Φ t , x), for t ∈ N and x ∈ M.
Φ − → M 2 such that Φ(a · α x · β b) = a · α Φ(x) · β b, for all (α, β) ∈ A × B, (a, b) ∈ A α × B β , x ∈ M.
For every subset I × J ⊂ A × B of indexes, we have a different contravariant right semi-adjoint functorial
pairing I ♭ J | ϑ ⇆ θ | I ♯ J between the transposition functors 14 (A α ) A M (B β ) B I ♭ J , , I ♯ J m m (B β ) B M (A α ) A that
• on objects of the respective categories, are given by duals: 12 We are assuming here that, in the category of Z-central R Z -multimodules, ⊗ Z α∈∅ A α := R =: ⊗ Z β∈∅ B β . 13 To avoid confusion, we will denote by • the "inner actions" on the tensor product multimodule (⊗
I ♭ J : M → * I M * J , ∀M ∈ Ob (Aα ) A M (B β ) B , I ♯ J : N → * J N * I , ∀N ∈ Ob (B β ) B M (Aα ) A ;Z α∈I A α ) ⊗ Z (⊗ Z β∈J B β ) given by: b • βo [(⊗ Z α∈I x α ) ⊗ Z (⊗ Z β∈J y β )] • αo a := (⊗ Z α∈I x ′ α ) ⊗ Z (⊗ Z β∈J y ′ β ), where x ′ α := x α , α α o x αo · a, α = α o and y ′ β := y β , β β o b · y βo , β = β o for all (α o , β o ), (α, β) ∈ I × J, (a, b) ∈ A αo × B βo , (x α , y β ) ∈ A α × B β .
14 The apparent distinction between ♭ and ♯ is purely formal since they interchange by permuting the sets of indexes: I ♭ J = J ♯ I . N 2 ), are respectively given by µ-pull-backs and ν-pull-backs:
(M 2 ) I ♭ J µ I ♭ J − −− → (M 1 ) I ♭ J , µ I ♭ J (φ) := φ • µ, ∀φ ∈ * I M * J 2 , (5.1) (N 2 ) I ♯ J ν I ♯ J − −− → (N 1 ) I ♯ J , ν J ♭ I (ψ) := ψ • ν, ∀ψ ∈ * J N * I 2 ;
where unit and co-unit of the semi-adjunction are given by the following natural evaluation transformations: 15
Id (Aα ) α∈A M (B β ) β∈B θ − → ♯ • ♭, M → θ M , M θ M − − → (M I ♭ J ) J ♯ I , θ M x : φ → φ(x), ∀φ ∈ M I ♭ J , x ∈ M, Id (B β ) β∈B M (Aα ) α∈A ϑ − → ♭ • ♯, N → ϑ N , N ϑ N − − → (N J ♯ I ) I ♭ J , ϑ N y : ψ → ψ(y), ∀ψ ∈ N J ♯ I , y ∈ N.
Restricting the previous contravariant right semi-adjunction I ♭ J | ϑ ⇆ θ | I ♯ J to the full reflective subcategories (whose objects are those multimodules for which the evaluation natural transformations are isomorphisms), we obtain a categorical duality.
Proof. The contravariant functorial nature of I ♭ J and I ♯ J is standard from their definitions.
By direct computation θ M is a morphism in (A α ) α∈A M (B β ) β∈B and ϑ N is a morphism in (B β ) β∈B M (A α ) α∈A furthermore for every pair of morphisms (M 1 ) µ − → (M 2 ) in (A α ) α∈A M (B β ) β∈B and (N 1 ) ν − → (N 2 ) in (B β ) β∈B M (A α ) α∈A : θ M 2 • µ = (µ I ♭ J ) J ♯ I • θ M 1 , ϑ N 2 • ν = (ν J ♯ I ) I ♭ J • ϑ N 1 .
Finally we check the right semi-adjunction condition I ♭ J | ⇆ | I ♯ J using formula (A.2):
( I ♭ J (θ M )) • ϑ (M I ♭ J ) = ι (MI ♭ J ) , ∀M ∈ Ob (Aα ) A M (B β ) B , [( I ♭ J (θ M ) • ϑ M I ♭ J )(φ)](x) = [(θ M ) I ♭ J (ϑ M I ♭ J φ )](x) = [(ϑ M I ♭ J φ ) • θ M ](x) = ϑ M I ♭ J φ (θ M x ) = θ M x (φ) = φ(x) = [ι (MI ♭ J ) (φ)](x), ∀φ ∈ M I ♭ J , x ∈ M.
For the full reflective subcategories of the semi-adjunction we have a categorical duality (see remark A.3).
Remark 5.4. There is of course the possibility to define also I γ J -conjugate duals of (A α ) A M (B β ) B for any family of R Z -conjugations (γ k ) k∈I⊎J , for (i, j) ∈ I × J ⊂ A × B. For this purpose is just enough to repeat the previous construction of duals utilizing maps that are γ k -conjugate-R Z -linear. Whenever γ k = Id R , for all k ∈ I ⊎ J, we re-obtain the previous definition as a special case.
Here we discuss how our definition of duals relates to already available notions in the case of bimodules.
Remark 5.5. The notion of I-J dual of an (A α ) A -(B β ) B multimodule over R Z -algebras that we have just introduced in definition 5.1 is a direct generalization of some much more familiar constructs for bimodules. In the following we study the "inclusion relations" between the different duals of a given multimodule.
Here below, we consider an A-bimodule
A M A as an (A α ) A -(B β ) B -multimodule, with A := {α o }, B := {β o }, A × B = {(α o ,a · (x ⊗ Z y) · b := (ax) ⊗ Z (yb), for all x, y, a, b ∈ A; where M ∨ is an A-bimodule with the actions (b ⊙ φ ⊙ a)(x) := φ(a · x · b), for all a, b ∈ A, x ∈ M, φ ∈ M ∨ . Taking I × J = A × B
Remark 5.6. Consider the auxiliary (I 2 , J 2 )-global (I 1 , J 1 )-dual multimodules, for I 1 × J 1 ⊂ I 2 × J 2 ⊂ A × B:
I 1 I 2 M J 1 J 2 := M φ − → (⊗ Z α∈I 2 A α ) ⊗ Z (⊗ Z β∈J 2 B β ) | ∀(α, β) ∈ I 1 × J 1 , ∀(a α , b β ) ∈ A α × B β : φ(a α xb β ) = a α φ(x)b β ,
equipped with the multimodule actions specified as follows (see footnotes 11 13), for all φ ∈ I 1
I 2 M J 1 J 2 and x ∈ M: b • β φ • α a : x → b • β φ(x) • α a, ∀(α, β) ∈ I 2 × J 2 , (a, b) ∈ A α × B β , b ⊙ β φ ⊙ α a : x → φ(a · α x · β b), ∀(α, β) ∈ (A − I 1 ) × (B − J 1 ), (a, b) ∈ A α × B β , (a · α φ · β b) := x → a · α φ(x) · β b, ∀(α, β) ∈ (I 2 − I 1 ) × (J 2 − J 1 ). (5.2)
Notice that whenever (I 1 , J 1 ) = (I 2 , J 2 ), we have I 1 I 2 M J 1 J 2 = * I 1 M * J 1 as a multimodule and that the extra multimodule actions in line (5.2) appear only when I 1 × J 1 I 2 × J 2 .
If I 1 × J 1 ⊂ I ′ 1 × J ′ 1 we have natural set theoretic inclusions:
I ′ 1 I 2 M J ′ 1 J 2 η M / / I 1 I 2 M J 1 J 2 ,
that are also covariant morphisms of multimodules for all the common actions involved (inner target actions • for indexes in I 2 × J 2 ; internal source actions ⊙ for indexes in (A − I ′ 1 ) × (B − J ′ 1 ) and external target action · for indexes in the set (I 2 − I ′ 1 ) × (J 2 − J ′ 1 )). Keeping (I 1 , J 1 ) fixed, if I 2 × J 2 ⊂ I ′ 2 × J ′ 2 , we define the following embedding map:
I 1 I 2 M J 1 J 2 ζ M :=(⊗ Z α∈I ′ 2 −I 2 1 Aα )⊗ Z −⊗ Z (⊗ Z β∈J ′ 2 −J 2 1 B β ) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ I 1 I ′ 2 M J 1 J ′ 2 , that to every φ ∈ I 1 I 2 M J 1 J 2 associates the map M ∋ x → (⊗ Z α∈I ′ 2 −I 2 1 A α ) ⊗ Z φ(x) ⊗ Z (⊗ Z β∈J ′ 2 −J 2 1 B β ) in I 1 I ′ 2 M J 1 J ′ 2
, that is actually a covariant morphism of multimodules for all the common relevant actions involved (inner target actions • for indexes in I 2 × J 2 ; internal source actions ⊙ for indexes in (A − I 1 ) × (B − J 1 ) and external target action · for indexes in (I 2 − I 1 ) × (J 2 − J 1 )).
Proposition 5.7. Given a multimodule (A α ) α∈A M (B β ) β∈B , for all the inclusions I 1 × J 1 ⊂ I 2 × J 2 ⊂ A × B of indexes, we have the following natural transformations between contravariant functors from the category
(B β ) β∈B M (A α ) α∈A into the category M [R Z ] of Z-central multimodules over R Z -algebras: M → * I 2 M * J 2 I ′ 1 I 2 M J ′ 1 J 2 η M / / I 1 I 2 M J 1 J 2 − −−−−−−−−−−−−−−−− → I 1 I 2 M J 1 J 2 ζ M :=(⊗ Z β∈J−J ′ 1 B β )⊗ Z −⊗ Z (⊗ Z α∈I−I ′ 1 Aα ) ← −−−−−−−−−−−−−−−−−−−−−−−−−−− − * I 1 M * J 1 .
17 For algebras over R := Z := K this is just the usual dual as a K-vector space.
Proof. The passage associating to a multimodule M ∈ Ob (Aα ) A M (B β ) B its (I 2 , J 2 )-global (I 1 , J 1 ))-dual multimodule I 1 I 2 M J 1 J 2 , is a contravariant functor acting on morphisms by transposition as in equation (5.1):
M µ − → N → I 1 I 2 M J 1 J 2 µ • ← − I 1 I 2 N J 1 J 2 , where µ • (φ) := φ • µ,
and from η M • µ I 2 ♭ J 2 = µ • • η N and µ • • ζ N = ζ M • µ I 1 ♭ J 1 we see that η and ζ are natural transformations.
Theorem 5.8. For every inclusion of indexes
I 1 × J 1 ⊂ I 2 × J 2 ⊂ A × B, considering the two contravariant right semi-adjunctions I k ♭ J k | I k ϑ J k ⇆ I k θ J k | I k ♯ J k ,• for all morphisms M µ − → N in (A α ) A M (B β ) B and P ν − → Q in (B β ) B M (A α ) A we have commutative diagrams: (P) I 1 ♯ J 1 ζ P | | ① ① ① ① ① ① ① ① ① (Q) I 1 ♯ J 1 ν I 1 ♯ J 1 o o ζ Q " " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ I 1 I 2 P J 1 J 2 I 1 I 2 Q J 1 J 2 ν • o o (P) I 2 ♯ J 2 η P b b ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ (Q) I 2 ♯ J 2 ν I 2 ♯ J 2 o o η Q < < ① ① ① ① ① ① ① ① (M) I 1 ♭ J 1 ζ M { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ (N) I 1 ♭ J 1 µ I 1 ♭ J 1 o o ζ N # # • • • • • • • • • I 1 I 2 M J 1 J 2 I 1 I 2 N J 1 J 2 µ • o o (M) I 2 ♭ J 2 η M c c • • • • • • • • (N) I 2 ♭ J 2 µ I 2 ♭ J 2 o o η N
In the second pair of commuting diagrams, by the exchange symmetry M ↔ Q, it is enough to prove the first. Consider x ∈ M, φ ∈ (M) I 1 ♭ J 1 and ψ ∈ (M) I 2 ♭ J 2 :
[ζ • M • ev M (x)](φ) = [ζ • M (ev M x )](φ) = ev M x (ζ M (φ)) = (⊗ Z α∈I 2 −I 1 1 A α ) ⊗ Z φ(x) ⊗ Z (⊗ Z β∈J 2 −J 1 1 B β ) = [ζ (M) I 1 ♭ J 1 (( I 2 θ J 2 M ) x )](φ), [η • M • ev M (x)](ψ) = [η • M (ev M x )](ψ) = ev M x (η M (ψ)) = ψ(x) = [η (M) I 1 ♭ J 2 ( I 2 θ J 2 M ) x ](ψ).
We define a poset category I of index pairs via the order relation (I 1 , J 1 ) ≤ (I 2 , J 2 ) ⇔ (I 1 ⊂ I 2 ) ∧ (J 1 ⊂ J 2 ).
We consider S M the category whose objects are contravariant right semi-adjunctions and whose morphisms are specified by the previous commuting diagrams of natural transformations ζ, η.
Traces and Inner Products on Multimodules
In the first part of this section we generalize to the setting of multimodules the well-known multilinear algebraic operations producing contractions of tensors (over pairs of contravariant/covariant indexes) over a vector space and hence the equally familiar notion of trace of linear operators.
We proceed again introducing the relevant universal factorization properties.
T (a · ξ x) = T (a · ζ x), ∀x ∈ M, ∀a ∈ A ξ = A ζ , (ξ, ζ) ∈ Γ ∩ (A × A), T (x · ξ a) = T (x · ζ a), ∀x ∈ M, ∀a ∈ A ξ = A ζ , (ξ, ζ) ∈ Γ ∩ (B × B), T (a · ξ x) = T (x · ζ a), ∀x ∈ M, ∀a ∈ A ξ = A ζ , (ξ, ζ) ∈ Γ ∩ (A × B). A Γ-contraction of the multimodule (A α ) A M (A β ) B , consists of a Γ-tracial morphism of (A α ) α∈A Γ -(A β ) β∈B Γ multi- modules over R Z -algebras (A α ) A Γ M (A β ) B Γ T Γ M − −− → (A α ) A Γ M | Γ (A β ) B Γ such that the following universal factorization property is satisfied: for any Γ-tracial morphism (A α ) A Γ M (A β ) B Γ T − → (A α ) A Γ N (A β ) B Γ of multimodules, there exists a unique morphism (A α ) A Γ M | Γ (A β ) B ΓT − → (A α ) A Γ N (A β ) B Γ of multimodules such that T =T • T Γ M . Remark 6.2.
As usual, Γ-contractions of Z-central multimodules are unique up to a unique isomorphisms compatible with the defining factorization property; their existence is provided by the following construction.
Given the Z-central multimodule (A α ) A M (A β ) B and the injective symmetric relation Γ ⊂ (A ⊎ B) × (A ⊎ B) with A ξ = A ζ whenever (ξ, ζ) ∈ Γ, defining A Γ := A − Dom(Γ) and B Γ := B − Im(Γ), consider the Z-central (A α ) α∈A Γ -(A β ) β∈B Γ sub-multimodule (A α ) A Γ [M, Γ] (A β ) B Γ of (A α ) A Γ M (A β ) B Γ
generated by the elements of the form: We briefly examine how involutions in multimodules descend to their contractions.
a · ξ x − a · ζ x, ∀x ∈ M, ∀a ∈ A ξ = A ζ , (ξ, ζ) ∈ Γ ∩ (A × A), x · ξ a − x · ζ a, ∀x ∈ M, ∀a ∈ A ξ = A ζ , (ξ, ζ) ∈ Γ ∩ (B × B), a · ξ x − x · ζ a, ∀x ∈ M, ∀a ∈ A ξ = A ζ , (ξ, ζ) ∈ Γ ∩ (A × B).Definition 6.3. An involution ( † σ M , ‡ σ M , ⋆ M ) σ∈ f M on a Z-central multimodule (A α ) A M (A β ) B over R Z -algebras is a Γ-compatible involution, where Γ ⊂ (A ⊎ B) × (A ⊎ B) is an injective symmetric relation on A ⊎ B, if: (ξ, ζ) ∈ Γ ⇒ ( f (ξ), f (ζ)) ∈ Γ, ∀ξ, ζ ∈ A ⊎ B.
(6.1)
Proposition 6.4. Suppose that ( (A α ) A M (A β ) B , ⋆ M ) in an involutive Z-central multimodule over R Z -algebras. If the involution is Γ-compatible with a Γ-contraction,(α, β) ∈ A Γ ×B Γ ) and T Γ (x ⋆ M ) = T Γ (x) ⋆ , for all x ∈ M.
Remark 6.5. Looking at the universal contructions of tensor products of multimodules in remark 3.11 and of contrations in remark 6.2, we obtain the following familiar result:
(A α ) α∈A M (B β ) β∈B ⊗ Γ (C γ ) γ∈C M (D δ ) δ∈D ≃ T Γ (A α ) α∈A M (B β ) β∈B ⊗ Z (C γ ) γ∈C M (D δ ) δ∈D ,
tensor products over Γ ⊂ (A ⊎ C) ⊎ (B ⊎ D) are naturally isomorphic to Γ-contracted tensor products over Z.
We pass now to consider the generalization of inner product couplings for multimodules.
Definition 6.6. Suppose that the unital associative R Z -algebras A α and B β , for (α, β) ∈ A × B, are all contravariantly involutive. Given a multimodule (A α ) A M (B β ) B and sub-indexes I × J ⊂ A × B, an (I, J)-right-inner product on M consists of a bi-additive map · | · I−J :
M × M → Z i∈I A i ⊗ Z Z j∈J B j such that: x | a · α y · β b I−J = a · α x | y I−J · β b, ∀x, y ∈ M, (a, b) ∈ A α × B β , (α, β) ∈ I × J, a · α x · β b | y I−J = b * • β x | y I−J • α a * , ∀x, y ∈ M, (a, b) ∈ A α × B β , (α, β) ∈ I × J, a · α x · β b | y I−J = x | a * · α y · β b * I−J , ∀x, y ∈ M, (a, b) ∈ A α × B β , (α, β) ∈ (A − I) × (B − J). A (I, J)-left-inner product on M is a bi-additive map I−J · | · : M × M → Z i∈I A i ⊗ Z Z j∈J B j such that: I−J a · α x · β b | y = a · α I−J x | y · β b, ∀x, y ∈ M, (a, b) ∈ A α × B β , (α, β) ∈ I × J, I−J x | a · α y · β b = b * • β I−J x | y • α a * , ∀x, y ∈ M, (a, b) ∈ A α × B β , (α, β) ∈ I × J, I−J a · α x · β b | y = I−J x | a * · α y · β b * , ∀x, y ∈ M, (a, b) ∈ A α × B β , (α, β) ∈ (A − I) × (B − J).
Proposition 6.7. For every right-(I, J)-inner product (x, y) → x | y I−J we have its:
transpose (x, y) → y | x I−J , * -conjugate (x, y) → x | y * I−J , * -adjoint (x, y) → y | x * I−J .
The transpose and conjugate are left-(I, J)-inner products; the adjoint is a right-(I, J)-inner product on M.
Remark 6.8. Without entering into a detailed discussion of "positivity" for inner products, we simply mention that stating this condition requires an additional compatible "order structure" on the involved rings and algebras. Whenever the commutative unital associative involutive ring Z is equipped with a positive cone Z + (that by definition is a pointed subset 0 Z ∈ Z + ⊂ Z, stable under addition Z + + Z + ⊂ Z + , stable under multiplication
Z + · Z + ⊂ Z + , sharp Z + ∩ (−Z + ) = {0 Z } and involutive Z ⋆ Z + ⊂ Z + )
, any Z-central unital associative algebra R (and hence any Z-central R Z -algebra) canonically inherits a positive cone R + := Z + · 1 R ⊂ R. In this case, positivity of a right-(I, J)-inner product can be imposed requiring
x | x I−J ∈ Z i∈I A i ⊗ Z Z j∈J B j +
, for all x ∈ M. Similar condition can be imposed for left-(I, J)-inner products.
Theorem 6.9. An (I, J)-inner product (right or left) induces canonical Riesz maps Proof. From definition 6.6 we see that
M I − → Λ J − −− → * I M * J , I − → Λ J x : x → I − → Λ J x I − → Λ J x : y → x | y I−J , M I ← − Λ J − −− → * IM * J , I ← − Λ J y : y → I ← − Λ J y I ← − Λ J y : x → x | y I−J , where I − → Λ J isI − → Λ J x ∈ * I M * J , for all x ∈ M. The map I − → Λ J : M → * I M * J is a contravariant morphism of multimodules: I − → Λ J a· α x· β b (y) = a · α x · β b | y I−J = b * • β x | y I−J • α a * = b * • β I − → Λ J x (y) • α a * = (b * • β I − → Λ J x • α a * )(y), ∀(α, β) ∈ I × J, x, y ∈ M, (a α , b β ) ∈ A α × B β ; I − → Λ J a· α x· β b (y) = a · α x · β b | y I−J = x | a * · α y · β b * I−J = I − → Λ J x (a * · α y · β b * ) = (b * ⊙ β I − → Λ J x ⊙ α a * )(y), ∀(α, β) ∈ (A − I) × (B − J), x, y ∈ M, (a α , b β ) ∈ A α × B β .
The proof for the case of I ← − Λ J can be obtained passing to the transpose inner product. The contravariant nature of Riesz maps requires contravariant involutions in the definition of inner products; alternative possibilities can be explored with "inner couplings" on M with more general signatures.
Remark 6.12. Thinking of multimodules in the 1-
category (A α ) A M (B β ) B as "1-arrows" (A α ) A M ← − − (B β ) B , in a 2-category of morphisms (A α ) A (B β ) B M j j N t t ✤ ✤ ✤ ✤ K S Φ ,+ + * I M * J k k ⇓ * I − → Λ * J (B β ) B
following the definition of hybrid 2-category described in [Bertozzini Puttirungroj 2014]. We will pursue such developments elsewhere.
Remark 6.13. In definition 6.6 in order to keep the closest possible resemblance to the usual axioms for inner products in Hilbert spaces and Hilbert-C*-modules, we have imposed covariance, for certain actions, only one of the two variables and contravariance on the other. It is perfectly possible to consider more general cases, where covariance and contravariance are simultaneously present in both variables (on disjoint sets of indexes): let I × J ⊂ A × B with I := I l ∪ I r , J = J l ∪ J r and I l ∩ I r = ∅ = J l ∩ J r , a (I l , J l )-left (I r , J r )-right inner product
on (A α ) A M (B β ) B is a bi-additive map (x, y) → I l −J l x | y I r −J r , for x, y ∈ M, such that: ∀x, y ∈ M, I l −J l x | a · α y · β b I r −J r = a · α x | y I r −J r · β b, ∀(a, b) ∈ A α × B β , (α, β) ∈ I r × J r , I l −J l a · α x · β b | y I r −J r = a · α x | y I r −J r · β b, ∀(a, b) ∈ A α × B β , (α, β) ∈ I l × J l , I l −J l a · α x · β b | y I r −J r = b * • β I l −J l x | y I r −J r • α a * , ∀(a, b) ∈ A α × B β , (α, β) ∈ I r × J r , I l −J l x | a · α y · β b I r −J r = b * • β I l −J l x | y I r −J r • α a * , ∀(a, b) ∈ A α × B β , (α, β) ∈ I l × J l , I l −J l a · α x · β b | y I r −J r = I l −J l x | a * · α y · β b * I r −J r , ∀(a, b) ∈ A α × B β , (α, β) ∈ (A − I) × (B − J).
Riesz maps I l −J l − → Λ I r −J r and I l −J l ← − Λ I r −J r can be similarly defined and a perfect parallel of theorem 6.9 holds.
Outlook
Although we are not going here into specific details, that will be subject of a forthcoming work, we preview some of the categorical features making multimodules a quite intriguing playground.
The family of multimodules, with their several tensor products, constitutes a paradigmatic example of "algebraic structure" consisting of "many inputs / many outputs nodes" that can be "linked" in many different ways: each multimodule (A α ) A M (B β ) B should be interpreted as an arrow with sources (B β ) B and targets (A α ) A ; 19 every tensor product over a subfamily provides a possible "concatenation" of arrows and such compositions will be subject to associativity and unitality axioms typical of category theory.
At the 1-categorical level (when only multimodules as 1-arrows and tensor products as compositions are considered) the structure seems to be describable as a colored properad [Hackney Robertson Yau 2015], a horizontal categorification (i.e. a many objects version) of the notion of properad introduced by [Vallette 2007].
Dualities of multimodules seem to provide the easiest examples of involutions for arrows in a colored properad and can be taken as a paradigmantic template in order to axiomatize a notion of "involutive colored properad". Contractions can be used to introduce "sinks" and "sources", hence more general types of "partial involutions".
As already mentioned in remark 6.12, we plan to further study Riesz dualities as examples of hybrid natural transformations, between functors with different covariance, in the context of hybrid 2-categories introduced in [Bertozzini Puttirungroj 2014].
Covariant morphisms of multimodules should be interpreted (exactly as in the usual case of categories of bimodules) as cubical 2-arrows. In this way, one obtains for multimodules a colored properad analog of the usual double category of covariant morphisms of bimodules.
It is also possible to iterate the construction of multimodules over multimodules creating a vertical categorification ladder that can be used to define "involutive higher colored properads" (possibly requiring the noncommutative exchange property introduced in [Bertozzini Conti Lewkeeratiyutkul Suthichitranont 2020]).
The purely algebraic theory of Z-central multimodules over R Z -algebras here presented can be subject to a functional analytic treatment as soon as topologies/uniformities are introduced and the actions are required to be continuous in the suitable sense. We will explore in the future the more restrictive axioms for a (higher) C*-algebraic version of this material and obtain (infinite-dimensional) functional analytic generalizations of the (essentially finite-dimensional) reflexivity (θ M covariant isomorphism) and self-duality ( J − → Λ I M contravariant isomorphism) conditions on multimodules.
Although the basic definition of first-order differential operator between Z-central multimodules over noncommutative R Z -algebras is included in appendix B, much more needs to be done regarding the full differential theoretic theory of multimodules (and also bimodules!), starting with a theory of connections on multimodules and possibly proceeding in the direction of properadic non-commutative geometry as a natural extension of our current efforts in categorical non-commutative geometry. An exploration of the interplay between duality (for bimodules) and first-order differential operators associated to covariant differential calculi on a non-commutative Z-central algebra is carried on in our forthcoming work (mentioned in footnote 1).
Notes and Acknowledgments
A Functorial Pairings and Semi-adjunctions
In order to properly discuss the categorical features of dual pairing of multimodules, we need to deal with a variant of the well-known notion of adjunction between functors, originally introduced in [Medvedev 1974
Hom B (F(A), B) λ AB ⇆ ρ AB Hom A (A, G(B)), ∀(A, B) ∈ Ob A × Ob B . (A.1)
The full pairing is regular if both ρ, λ are regular maps:
ρ AB • λ AB • ρ AB = ρ AB , λ AB • ρ AB • λ AB = λ AB , ∀(A, B) ∈ Ob A × Ob B .
A covariant full functorial pairing F λ ⇆ ρ | G will be called [Medvedev 1974] a:
right semi-adjunction F λ ⇆ ρ | G if: ρ AB • λ AB = Id Hom A (A,G(B)) , ∀(A, B) ∈ Ob A × Ob B , left semi-adjunction F λ ⇆ ρ | G if: λ AB • ρ AB = Id Hom B (F(A),B) , ∀(A, B) ∈ Ob A × Ob B .
The following remark follows immediately from [Wisbauer 2013, section 2].
Remark A.2. Note that, similarly to adjunction (as indicated by our notation F λ ⇆ ρ | G), full functorial pairing is an "asymmetric" notion, with the functor G right functorially paired to F (equivalently F left paired to G).
The existence of the natural transformation ρ (respectively λ) in formula (A.1) is equivalent to the existence of a unit Id
ρ AB (x) := G(x) • η A ∈ Hom A (A; G(B)), ∀(A, B) ∈ Ob A × Ob B , ∀x ∈ Hom B (F(A); B), λ AB (y) := ǫ B • F(y) ∈ Hom B (F(A); B), ∀(A, B) ∈ Ob A × Ob B , ∀y ∈ Hom A (A; G(B)).
The semi-adjunction conditions, can be equivalently written via composition of units and co-units in the 2-category of natural transformations as: 21
F λ ⇆ ρ | G A η A :=ρ AF(A) (ι F(A) ) − −−−−−−−−−− → G(F(A)), B ǫ B :=λ G(B)B (ι G(B) ) ← −−−−−−−−−− − F(G(B)), F λ ⇆ ρ | G F is left semi-adjoint of G: ǫ F(A) • B F(η A ) = ι F(A) ⇔ λ • ρ = Id Hom B (F(A);B) ; F λ ⇆ ρ | G G is right semi-adjoint of F: G(ǫ B ) • A η G(B) = ι G(B) ⇔ ρ • λ = Id Hom A (A;G(B)) .
In view of the established equivalence description of semi-adjunctions via (λ, ρ) or via (η, ǫ) we will liberally utilize the alternative notations:
F λ ⇆ ρ | G ⇔: F ǫ ⇆ η | G
A semi-adjunction is necessarily a regular full functorial pairing; whenever ρ and λ are inverse of each other, the necessarily regular full functorial pairing reproduces an adjunction F ⊣ G with unit η and co-unit ǫ.
We will need to utilize semi-adjunctions in the case of contravariant functors.
F | λ ⇆ ρ | G, Hom B (B, F(A)) λ AB ⇆ ρ AB Hom A (A, G(B)), ∀(A, B) ∈ Ob A × Ob B ,
and respectively to a left contravariant functorial pairing:
F λ ⇆ ρ G, Hom B (F(A), B) λ AB ⇆ ρ AB Hom A (G(B), A), ∀(A, B) ∈ Ob A × Ob B .
The definitions of contravariant regularity and contravariant semi-adjunction remain the same.
For all possible cases of contravariant semi-adjunction, the equivalent statements in terms of the associated 21 These are actually the original equations used by [Medvedev 1974] to define right and left semi-adjunctions.
unit and co-unit natural transformations can be summarized as follows, for all (A, B) ∈ Ob A × Ob B : Notice, due to the contravariance, the "change of direction" and respectively the order of composition of the natural transformations involved (so that for contravariant right functorial pairings we have in practice two unit, and for contravariant left functorial pairings, we actually have two co-unit natural transformations).
F | ǫ ⇆ η | G A η A :=ρ AF(A) (ι F
Notice also that the "doubling" of the semi-adjointness conditions is just an apparent artifact of notation since:
F | λ ⇆ ρ | G ⇔ G | ρ ⇆ λ | F and similarly F λ ⇆ ρ G ⇔ G ρ ⇆ λ F.
Whenever ρ and λ are inverse of each other, the necessarily regular full contravariant right functorial pairing reproduces a contravariant right adjunction F ⊢⊣ G with two units η and ǫ that, upon restriction to the full subcategories of reflexive objects provides a duality.
Let us exemplify the required semi-adjunction in the case of Z-central bimodules over Z-central R-algebras. Proof. The transposition duality functor associates to every morphism Ω 1 Φ − → Ω 2 of Z-central Z-bimodules the transposed map Ω * 1 Φ * ← − − Ω * 2 defined, for all ψ ∈ Ω * 2 , by Φ * (ψ) := ψ • Φ ∈ Ω * 1 ; the map Φ * is Z-linear and the transposition Φ → Φ * is a contravariant endofunctor: (Φ • Ψ) * = Ψ * • Φ * and (Id Ω ) * = Id Ω * , for all Φ, Ψ ∈ Hom(M Z ) and Ω ∈ Ob(M Z ). The evaluation transform ev : Ob(M Z ) → Hom(M Z ), given by Ω → ev Ω (where ev Ω : Ω → Ω * * is the Z-linear map x → ev Ω x that, for x ∈ Ω, is defined as ev Ω x (φ) := φ(x), for all φ ∈ Ω) is a natural tranformation between the covariant functors Id M Z ev − → * • * , since: for all morphisms Ω 1 Φ − → Ω 2 in M Z , we have ev Ω 2 • Φ = Φ * * • ev Ω 1 .
We observe that the natural transformation Id M Z ev − → * • * satisfies the following "weakened version" of the triangle (right-right contravariant) adjunction identities:
• (ev Ω ) * • ev Ω * = Id Ω * , for all Ω ∈ Ob(M Z ),
• ev Ω * • (ev Ω ) * : Ω * * * → Ω * * * , for all Ω ∈ Ob(M Z ), is an idempotent "projecting" the Z-bimodule Ω * * * onto its Z-submodule ev Ω * (Ω * ) := {ev Ω * φ | φ ∈ Ω * } ⊂ Ω * * * .
Whenever we impose on the objects Ω, the condition of reflexivity, i.e. we ask that Ω ev Ω − − → Ω * * is an isomorphism in M Z , the previous contravariant right semi-adjunction becomes a duality (contravariant right-right adjoint equivalence).
In the case of Z-central multimodules over Z-central R-algebras, proposition A.4 generalizes in the form described in theorem 5.3.
B First-Order Differential Operators on Z-central Multimodules
In this last appendix, we briefly preview a definition of first-order differential operator between multimodules. The first-order condition B.1, is just a reformulation for multimodules of the usual first-order condition for operators acting on A-bimodules, put forward in [Connes 1994, sections 4.γ and 4.δ]. We encountered such notion during our investigation of non-commutative vector fields and derivations of non-commutative algebras (see footnote 1) and although we needed there to consider mostly first-order differential operators defined on A ⊗ Z A, for a non-commutative unital associative R Z -algebra A, here for completeness we present the basic definition in a more general context. A perfectly similar reinterpretation in terms of commutators, provides:
δ(a · α x · β b) + a · α δ(x) · β b = δ(a · α x) · β b + a · α δ(x · β b), ∀(α, β) ∈ A × B, (a, b) ∈ A α × B β , (B.1) δ(a · α b · α ′ x) + a · α b · α ′ δ(x) = b · α ′ δ(a · α x) + a · α δ(b · α ′ x), ∀α α ′ ∈ A, (a, b) ∈ A α × A α ′ , (B.2) δ(x · β a · β ′ b) + δ(x) · β a · β ′ b = δ(x · β a) · β ′ b + δ(x · β ′ b) · β a, ∀β β ′ ∈ B, (a, b) ∈ B β × B β ′ .[[δ, L α a ] − , L α ′ b ] − = 0 Hom Z (M;N) = [[δ, L α ′ b ] − , L α a ] − , ∀α α ′ ∈ A, (a, b) ∈ A α × A α ′ ,
as an equivalent reformulation of equation (B.2) for the left-(A α , A α ′ ) bimodules A α ,A α ′ M, A α ,A α ′ N and
[[δ, R β a ] − , R β ′ b ] − = 0 Hom Z (M;N) = [[δ, R β ′ b ] − , R β a ] − , ∀β β ′ ∈ B, (a, b) ∈ A β × A β ′ ,
as a replacement of equation (B.3) for the right-(B β , B β ′ ) bimodules M B β ,B β ′ , N B β ,B β ′ .
Remark B.3. Definition B.1 is actually a special (R-linear covariant) case of a much more general notion of first-order differential operator that allows to discuss Z-linear differential operators that are possibly contravariant and R-conjugate linear. Making use of exactly the same notations introduced in definition 3.1 for morphisms (zero-order differential operators) between multimodules, we say that a first-order differential
→ A γ that satisfies the universal factorization property: for any γ-conjugate-linear unital morphism ofZ-central R-algebras A φ − → B, there exists a unique covariant R-linear homomorphism A γφ − → B such that φ =φ • η A .In the case of Z-central unital R-bimodules R M R the definition of γ-conjugate M η M − −→ M γ is given via the same universal factorization property diagram of R-bimodules, "forgetting" the multiplication.
the sub-families of indexes A + ⊂ A, B + ⊂ B and (A − , B − )-contravariant in the sub-families of indexes A − := A − A + , B − := B − B + , consists of:
Proposition 4. 2 .
2Suppose that A and B are both Z-central R Z -algebras with involutions ‡ A and ‡ B over the respective R-conjugations † A and † B . If the Z-central bimodules A M A and B N B are both involutive with involutions ( † A , ‡ A , ⋆ M ) f M and ( † B , ‡ B , ⋆ N ) f N , also the Z-central multimodule B,A Hom Z (M; N) A,B , considered in proposition 3.8, is equipped with an involutive map ⋆ : T → T ⋆ := ⋆ N • T • ⋆ M and becomes an involutive multimodule with multimodule involution ( †, ‡, ⋆) f , defined as follows:
for all (T, S ) ∈ Hom K (N; P) × Hom K (M; N). In particular (Hom Z (M; M), •, ⋆) is a unital associative Z-central algebra with a covariant involution. 8 Proof. If T ∈ Hom Z (M; N) with R-linearity signature φ T , the composition T ⋆ := ⋆ N • T • ⋆ M is Z-linear and with R-linearity signature † N • φ T • † M and hence T → T ⋆ is well-defined as an endo-map of Hom Z (M; N).
Remark 4. 3 .
3The previous proposition can easily be further generalized: whenever (A α ) M (A β ) and (B γ ) N (B δ ) are Z-central multimodules over Z-central R Z -algebras, any pair ( † M , ‡ M , ⋆ M ) f M and ( † N , ‡ N , ⋆ N ) f N of involutions, induces an involution ⋆ : T → T ⋆ := ⋆ N • T • ⋆ M of Hom Z (M; N), that is compatible with all the external and internal actions of the multimodule (B γ ,A β ) Hom K (M; N) (A α ,B δ ) defined in remark 3.9 and hence, defining f := f * M ⊎ f N , † := † M ⊎ † N , ‡ := ‡ M ⊎ ‡ N , we see that ( †, ‡, ⋆) f is an involution of the Z-central multimodule (B γ ,A β ) Hom K (M; N) (A α ,B δ ) over R Z -algebras. 9
Theorem 5. 2 .
2For every (I, J) with I × J ⊂ A × B, there exists an (I, J)-dual (
For every pair of families of unital associative Z-central R Z -algebras (A α ) α∈A and (B β ) β∈B and every family of indexes I × J ⊂ A × B, (I, J)-transposition functors (and evaluation natural transformations) give us a contravariant right semi-adjunction according to the definitions fully recalled in appendix A, remark A.3. Theorem 5.3. Let (A α ) A M (B β ) B be the category with objects Z-central (A α ) A -(B β ) B multimodules over unital associative R-algebras and with morphism Z-linear maps M 1
∈
Hom (Aα ) A M (B β ) B (M 1 ; M 2 ) and (N 2 ν ← − N 1 ) ∈ Hom (B β ) B M (Aα ) A (N 1 ;
β o )} singleton sets and with A α o := A =: B β o .The "double dual" 16 M ∨ of an A-bimodule A M A (see for example[Fernández 2017, section 2.1] for more details) is the central Z-bimodule M ∨ := Hom A M A (M; · (A ⊗ Z A) · ) of covariant homomorphisms of bimodules, from A M A , with values into · (A ⊗ Z A) · seen as an A-bimodule with the "exterior actions" given by:
as a singleton (only one right and only one left action) in definition 5.1, we see that M ∨ = * I M * J .The well-known notions (see for example[Borowiec 1997]) of "right dual" M * := Hom A (M A ; A) and "left dual" * M := Hom A ( A M; A) of a bimodule A M A are just the dual of the right A-module M A (respectively the dual of the left A-module A M), as in[Bourbaki 1942, section II.3], equipped with the following actions(a · φ ⊙ b)(x) := aφ(bx), for all a, b ∈ A, x ∈ M and φ ∈ M * (respectively (a ⊙ φ · b)(x) := ψ(xa)b, for all a, b ∈ A, x ∈ Mand ψ ∈ * M). When A × B is a singleton, taking I := ∅, J := B, we recover M * = * I M * J and, when I := B, J := ∅, we get * M = * I M * J . Finally the "scalar dual" of a bimodule 17 A M A , defined as M ′ := Hom R (M; R ⊗ Z R), equipped with the actions (b ⊙ φ ⊙ a)(x) := φ(a · x · b), for all a, b ∈ A, x ∈ M and φ ∈ M ′ , can be obtained from our definition as M ′ = * I M * J , taking A × B to be, as usual, a singleton and I := ∅ =: J.
To every index pair (I, J) ∈ Ob I we associate the contravariant right semi-adjunction ⋆ (I,J)M := I ♭ J | I ϑ J ⇆ I θ J | I ♯ Jand to every morphism (I 1 , J 1 ) ≤ (I 2 , J 2 ) in I we associate the morphism ⋆ (I 2 → S M is a contravariant functor.
Definition 6. 1 .
1Given a Z-central multimodule (A α ) A M (A β ) B over R Z -algebras and Γ ⊂ (A ⊎ B) × (A ⊎ B) an injective symmetric relation 18 such that A ξ = A ζ , for all (ξ, ζ) ∈ Γ, let A Γ := A − Dom(Γ) and B Γ := B − Im(Γ). A Z-linear map (A α ) A M (A β ) B T − → V, of Z-central bimodules, is Γ-tracial if it satisfies the following properties:
The quotient map T Γ M : M → M [M,Γ] =: M | Γ onto the quotient (A α ) A Γ -(A β ) B Γ multimodule, satisfies the universal factorization property, since every Γ-tracial homomorphism M T − → N of (A α ) A Γ -(A β ) B Γ multimodules entails [M, Γ]⊂ Ker(T ) and hence canonically factorizes via T Γ M . Notice that it is possible to have multimodules that only possess trivial Γ-traces (for a certain family Γ) and hence they have trivial universal Γ-contractions.
there exists a unique contracted involution onto M | Γ and the contraction map MT Γ − − → M | Γ is involutive. Proof. Condition (6.1) implies that the involution ⋆ M leaves invariant the submultimodule (A α ) A Γ [M, Γ] (B β ) B Γ and hence, defining (x + [M, Γ]) ⋆ := x ⋆ M + [M, Γ],for all x ∈ M, the involution will pass to the quotient multimodule (with the same covariance properties in
a contravariant morphism of multimodules into * I M * J , the (I, J)-dual of M, and respectively I ← − Λ J is a covariant morphism of multimodules into the I − → Λ J -twisted of * I M * J , here denoted by * IM * J .
Definition 6 . 10 .
610An inner product · | · I−J is * -Hermitian if it coincides with its * -adjoint; non-degenerate if both the Riesz maps I − → Λ J and I ← −Λ J are injective; algebraically full if M | M I−J = Z i∈I A i ⊗ Z Z j∈J B j ; saturated if both I − → Λ J and I ← − Λ J are surjective.Remark 6.11. Notice that the Riesz map I − → Λ J is contravariant and hence, under non-degeneracy and saturation, an (I, J)-inner product always induces an anti-isomorphism between M and its (I, J) dual * I M * J .Under fullness and saturation,* I M * J ( I − → Λ J ) −1 − −−−− → M is a ( I − → Λ J ) −1 -twisted of M as defined in 3.4 and 3.5.
we see that, for all I × J ∈ A × B, (I, J)-duals provide an {0, 1}-contravariant involution (A α ) A * I N * J + + * I M * J 3 3 ✤✤ ✤✤ * I Φ * J (B β ) B , over objects and 1-arrows, in the sense described in [Bertozzini Conti Lewkeeratiyutkul Suthichitranont 2020, section 4]. Riesz maps can be considered as "natural transformations" examples of hybrid 2-arrows (A α ) A M
|
G, one defines, for all (A, B) ∈ Ob A × Ob B ,η A := ρ AF(A) (ι F(A) ), ǫ B := λ G(B)B (ι G(B) );in the reverse direction, given unit Id A η − → G • F and co-unit F • G ǫ − → Id B natural transformations, one defines
Remark A. 3 .
3In the case of contravariant functors A F & & G g g B , the usual right-right and left-left adjunctions, are corresponding to a right contravariant full functorial pairing:
||η
G F is right semi-adjoint of G: F(η A ) • B ǫ F(A) = ι F(A) ⇔ λ AB • ρ AB = Id Hom B (F(A);B) , G G is right semi-adjoint of F: G(ǫ B ) • A η G(B) = ι G(B) ⇔ ρ AB • λ AB = Id Hom A (A;A :=ρ AF(A) (ι F(A) ) ← −−−−−−−−−− − G(F(A)), B ǫ B :=λ G(B)B (ι G(B) ) is left semi-adjoint of G: ǫ F(A) • B F(η A ) = ι F(A) ⇔ λ AB • ρ AB = Id Hom B (F(A);B) is left semi-adjoint of F: η G(B) • A G(ǫ B ) = ι G(B) ⇔ ρ AB • λ AB = Id Hom A (A;G(B)) .
Proposition A. 4 .
4In the category M Z of morphisms of Z-central Z-bimodules, the transposition pairing duality Ω → Ω * , for Ω ∈ Ob(M Z ), is a contravariant endofunctor. The evaluation natural transformation ev induces a right contravariant semi-adjoint endo-funtorial pairing M Z * ) ) * i i M Z , hence the transposition duality endofunctor is right semi-adjoint to itself: Upon restriction to the subcategory of reflexive objects (those objects Ω for which ev Ω is an isomorphism) the functorial pairing above is a duality.
Definition B. 1 .
1Let (A α ) M (B β ) and (A α ) N (B β ) be two multimodules, over the families of unital associative R Z -algebras (A α ) α∈A and (B β ) β∈B . The set Diff 1 (A α )−(B β ) (M; N) of first-order differential operators from M to N consists of those R-linear maps M δ − → N that satisfy the first-order conditions, for all x ∈ M:
. 2 .
2Defining L α a (x) := a · α x and R β b (x) := x · β b, for all (α, β) ∈ A × B, (a, b) ∈ A α × B β and x ∈ M, a direct computation assures that, for all (α, β) ∈ A × B, equation (B.1) above is equivalent to [[δ, L α a ] − , R β b ] − = 0 Hom Z (M;N) = [[δ, R β b ] − , L α a ] − , ∀(a, b) ∈ A α × B β ,that is the familiar Connes' first-order condition for the operator M δ − → N on the bimodules A α M B β , A α N B β .
unital bimodules, over Z-central unital associative rings, consists of a pair (φ, Φ) as above. In the case of Z-central unital associative algebras over Z-centralunital associative rings, the morphism A
Φ
− → B must be unital and covariant (contravariant).
such that every pair of left actions and every pair of right actions commute. 5
: P.Bertozzini thanks Starbucks Coffee (1 st floor of Emporium Tower, Emquartier Sky Garden, Jasmine City) where he spent most of the time dedicated to this research project; he thanks Fiorentino Conte of "The Melting Clock" for the great hospitality during many crucial on-line dinner-time meetings Bangkok-Rome. [Mesablishvili Wisbauer 2013] A Mesablishvili Bachuki, Wisbauer Robert (2013) On Rational Pairings of Functors Appl Categ Structures 21(3):249-290 DOI:10.1007/s10485-011-9264-1 arXiv:1003.3221 Vallette B (2007) A Koszul Duality for Props Trans Amer Math Soc 359:4865-4943 DOI:10.1090/S0002-9947-07-04182-7 arXiv:math/0402213 Regular Pairings of Functors and Weak (Co)Monads Algebra Discrete Math 15(1):127-154 arXiv:1101.1195[Takeuchi 1987]
1
Takeuchi Mitsuhiro (1987)
√
Morita Theory Journal of the Mathematical Society of Japan 39(2):301-36
DOI:10.2969/jmsj/03920301
[Vallette 2007]
7
[Wisbauer 2013]
1, A, A.1, A
Wisbauer R (2013)
], and generalized (in a much wider context) in [Mesablishvili Wisbauer 2013, Wisbauer 2013]. Definition A.1. A full functorial pairing F λ ⇆ ρ | G between the covariant functors A F & & G g g B is a pair of natural transformations between the (left-contravariant, right-covariant) Hom-bifunctors [Wisbauer 2013]:
Due to the different covariance of the functors involved, a categorical discussion of the "naturality" of Riesz morphisms would require the usage of hybrid 2-categories[Bertozzini Puttirungroj 2014] of multimodules.
This generalizes the special case of multimodules over unital associative K-algebras over the field K: in this case one can take R := K and Z a subfield of K consisting of fixed points for all the relevant conjugations of K (in practice it is always possible, in each characteristic p, to take Z as the initial field of that characteristic: Q in characteristic 0 and F p for any p prime).5 We assume the existence of a common Z-central bimodule structure on M compatible with all the Z-bilinear right/left actions.
From the involutivity of f , we have f (A + ) = A + , f (B + ) = B + , f (A − ) = B − and f (B − ) = A − .
Notice that the involution ⋆ is multiplicative independently from the convariace/contravariace of the original involutions on M. 9 Here, given two functions F : A → B and G : C → D with define F ⊎ G : A ⊎ B → C ⊎ D the "disjoint union" of the two maps.
These evaluations maps are just obtained applying Curry isomorphism to the pairing duality τ in definition 5.1.16 To be precise, the double dual is obtained choosing here Z := K; this is a slight generalization that we found particularly useful in our treatment of contravariant non-commutative differential calculus (see footnote 1).
We can also assume that Γ is irreflexive: (ξ, ζ) ∈ Γ ⇒ ξ ζ; since "tracing an action over itself" does not have any effect.
We are using here, for the tensor products, the same "reversed order" notation of the functional compositions.
:Considering the category I (actually the poset) of pairs (I, J), with I × J ⊂ A × B, where for every inclusion I 1 ×J 1 ⊂ I 2 ×J 2 there is a unique morphism of pairs (I 1 , J 1 ) − → (I 2 , J 2 ), we have that every multimodule M has an associated dual functor IProof. In the first pair of diagram, due to the exchange symmetry µ ↔ ν it is sufficient to prove the second. Taking φ ∈ (N) I 1 ♭ J 1 and ψ ∈ (N) I 2 ♭ J 2 we immediately get, for x ∈ M:operator between multimodules consists of the following data and conditions:Whenever f , (φ α ) A , (ψ β ) B , (η α ) A and (ζ β ) B are all identity functions, we recover the initial definition B.1.Given an arbitrary signature σ := (φ, η, ζ, ψ) f , we denote by Diff 1 σ (M; N) the family of first-order differential operators defined by all the conditions above. Making use of remark 3.5 we can obtain a bijective correspondence between Diff 1 σ (M; N) and Diff 1 (A α ) A −(B β ) B (M; N σ ) that is associating to each first-order differential operator δ ∈ Diff 1 σ (M; N), with signature σ, the unique first-order differential operator δ σ ∈ Diff 1The family of first-order differential operators Diff 1 (A α ) A −(B β ) B (M; N) is a central Z-bimodule, but does not usually have other well-defined actions, even of the algebra R. Whenever all the R Z -algebras involved are R-central bimodules (in particular if R = Z) the following immediate result is of interest.For any pair of sub-families of indexes I × J ⊂ A × B, define I I α :=The spaces Diff 1are all Z-central multimodules with respect to the following actions:x ∈ M, (r · α δ · β s)(x) := r · α δ(x) · β s, ∀(α, β) ∈ I × J, (r, s) ∈ I I α × J J β = R, x ∈ M, (s ⊙ β δ ⊙ α r)(x) := δ(r · α x · β s), ∀(α, β) ∈ I × J, (r, s) ∈ I I α × J J β = R, x ∈ M.Whenever (I 1 , J 1 ) ≤ (I 2 , J 2 ), we have inclusions Diff 1 (I
. Aluffi Paolo, American Mathematical Society127Bertozzini Conti Lewkeeratiyutkul Suthichitranont 2020] 1, 3.3, 6.Aluffi Paolo (2009) Algebra 0 American Mathematical Society [Bertozzini Conti Lewkeeratiyutkul Suthichitranont 2020] 1, 3.3, 6.12, 7
P Bertozzini, R Conti, W Lewkeeratiyutkul, N Suthichitranont, arXiv:1709.09339On Strict Higher C*-categories Cahiers de Topologie et Géométrie Différentielle Catégoriques LXI-3:1-110. 127Bertozzini Puttirungroj 2014] 3, 6.Bertozzini P, Conti R, Lewkeeratiyutkul W, Suthichitranont N (2020) On Strict Higher C*-categories Cahiers de Topologie et Géométrie Différentielle Catégoriques LXI-3:1-110 arXiv:1709.09339 [Bertozzini Puttirungroj 2014] 3, 6.12, 7
Paolo Bertozzini, Chatchai Puttirungroj, Hybrid Categories Proceedings of AMM2014 (19 th Annual Meeting in Mathematics. Pattaya, ThailandA-One HotelPaolo Bertozzini, Chatchai Puttirungroj (2014) Hybrid Categories Proceedings of AMM2014 (19 th Annual Meeting in Mathematics, 20-22 March 2014, A-One Hotel, Pattaya, Thailand) 119-128
Vector Fields and Differential Operators: Noncommutative Case Quantum Groups and Integrable Systems I (Prague, 1997). Andrzej Borowiec, 10.1023/A:1021697831180Czechoslovak Journal of Physics. 4711Andrzej Borowiec (1997) Vector Fields and Differential Operators: Noncommutative Case Quantum Groups and Integrable Systems I (Prague, 1997) Czechoslovak Journal of Physics 47(11):1093-1100 DOI:10.1023/A:1021697831180 arXiv:q-alg/9710006
. N Bourbaki, Bourbaki N (1989
. David Fernández, arXiv:1708.02650The Kontsevich-Rosenberg Principle for Bi-Symplectic FormsHackney Robertson Yau 2015Fernández, David (2017) The Kontsevich-Rosenberg Principle for Bi-Symplectic Forms arXiv:1708.02650 [Hackney Robertson Yau 2015] 7
. P Hackney, M Robertson, D Yau, 10.1007/978-3-319-20547-2Infinity Properads and Infinity Wheeled Properads Lecture Notes in Mathematics. Hackney P, Robertson M, Yau D (2015) Infinity Properads and Infinity Wheeled Properads Lecture Notes in Math- ematics 2147 DOI:10.1007/978-3-319-20547-2 arXiv:1410.6716 [Kertész 1962] 1
. Kertész Andor, 10.1007/BF01650073On Multimodules Archiv Der Mathematik. 13121Medvedev 1974] 1, A, A.1Kertész Andor (1962) On Multimodules Archiv Der Mathematik 13(1):267-74 DOI:10.1007/BF01650073 [Medvedev 1974] 1, A, A.1, 21
Semiadjoint Functors and Kan extensions. M Medvedev, Ja, 10.1007/BF00967444Siberian Mathematical Journal. 154Medvedev M Ja (1974) Semiadjoint Functors and Kan extensions Siberian Mathematical Journal 15(4):952-956 DOI:10.1007/BF00967444
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[
"Notes on primordial black hole origin for thermal gamma-ray bursts",
"Notes on primordial black hole origin for thermal gamma-ray bursts"
]
| [
"Tyler Mcmaken \nJILA and Department of Physics\nUniversity of Colorado\n80309BoulderColoradoUSA\n"
]
| [
"JILA and Department of Physics\nUniversity of Colorado\n80309BoulderColoradoUSA"
]
| [
"MNRAS"
]
| Recently, an alleged plausible astrophysical scenario was proposed for the production of observed thermal gamma-ray bursts, via Hawking radiation emitted from a primordial black hole (PBH) freely falling into a more massive black hole. Here the implausibility of that scenario is demonstrated, and the key flaws in that paper's calculations and assumptions are elucidated through a discussion of some common misconceptions concerning black holes and general relativity. In particular, the predicted radiance observed from Earth is found to be orders of magnitude lower than what any instrument could detect, and the PBH-BH merger signature would be completely overwhelmed by the background Hawking signature from free PBHs. | 10.1093/mnras/stac196 | [
"https://arxiv.org/pdf/2111.07045v2.pdf"
]
| 244,117,604 | 2111.07045 | d5c948f113ed7eb6300183520b8dd06e65b10240 |
Notes on primordial black hole origin for thermal gamma-ray bursts
2021
Tyler Mcmaken
JILA and Department of Physics
University of Colorado
80309BoulderColoradoUSA
Notes on primordial black hole origin for thermal gamma-ray bursts
MNRAS
00020218 February 2022Preprint 8 February 2022 Compiled using MNRAS L A T E X style file v3.0black hole physics -gamma-ray bursts -dark matter -black hole mergers
Recently, an alleged plausible astrophysical scenario was proposed for the production of observed thermal gamma-ray bursts, via Hawking radiation emitted from a primordial black hole (PBH) freely falling into a more massive black hole. Here the implausibility of that scenario is demonstrated, and the key flaws in that paper's calculations and assumptions are elucidated through a discussion of some common misconceptions concerning black holes and general relativity. In particular, the predicted radiance observed from Earth is found to be orders of magnitude lower than what any instrument could detect, and the PBH-BH merger signature would be completely overwhelmed by the background Hawking signature from free PBHs.
INTRODUCTION
Gamma-ray bursts (GRBs) are some of the most energetic events observed in the Universe. Such events are typically characterized by non-thermal spectra, and despite the diversity in their light curves and the uncertainty in their exact emission mechanisms, the sources for GRBs are almost unanimously agreed upon to fall under two classes, based on the GRB's duration.
Short GRBs, with a duration of tenths of a second, are associated with kilonovae produced from compact binary mergers. Such a view has been confirmed both computationally and observationally (Nakar 2007;Metzger et al. 2010;Tanvir et al. 2013;Berger et al. 2013), notably with the detection of the short GRB 170817A alongside LIGO's and VIRGO's detection of gravitational wave GW170817 from a neutron star merger event (Abbott et al. 2017). On the other hand, the progenitors of long GRBs, with a duration of tens of seconds, are massive stars undergoing core-collapse (MacFadyen et al. 2001;Woosley & Bloom 2006). This collapsar model has been confirmed with numerous coincident observations, including GRB 980425 & SN 1998bw (Kulkarni et al. 1998), GRB 030329 & SN 2003dh (Mazzali et al. 2003, and GRB 060218 & SN 2006aj (Sollerman et al. 2006).
Because of the diversity in the population of GRBs and the continued speculation surrounding the exact details of the emission mechanisms from the above-mentioned progenitors, some have proposed alternative exotic GRB sources. Most recently, Barco (2021) has claimed that the Hawking radiation from an atom-sized primordial black hole (PBH) falling into a larger black hole would produce a thermal spectrum consistent with that of a GRB. According to his model, as the PBH falls sufficiently close to the horizon, the Hawking temperature undergoes a significant Lorentz boost and leads to an energetic, highly collimated beam of blackbody radiation that reaches Earth.
Aside from the fact that pure thermal spectra are only observed for a rare handful of GRBs that are already well-explained by the fireball ★ E-mail: [email protected] model (Ghirlanda et al. 2013), the work of Barco (2021) contains several misleading claims, erroneous calculations, and nonphysical assumptions, most notably in the use of special relativistic equations to describe Schwarzschild near-horizon behavior and in the choice of 1.1 × 10 17 for the PBH's initial Lorentz factor . Thus, a follow-up analysis of the work is warranted. This work will not consider the (un)likelihood of the existence of PBHs as a source for dark matter, as the subject has already been considered at length in the literature (see, e.g. Villanueva-Domingo et al. (2021) for a review). Instead, this work is concerned solely with the viability of the detection of such objects if they were to fall into a black hole. Section 2 outlines a rectified approach to the problem considered by Barco, yielding a vastly different light curve than what he has presented. Then, Section 3 enumerates specific misconceptions that one may glean from Barco's work, and Section 4 concludes the paper.
CALCULATION
In what follows, geometrized units are used where = = ℏ = = 1. The metric signature is (− + ++), and with the exception of the subscript , all Greek indices are tensorial spacetime indices. Section 2.1 details the setup of the problem considered by Barco (2021), the calculation itself (different from Barco's ad hoc approach) is performed in Section 2.2, and numerical results are presented in Section 2.3.
Problem setup
Consider a Schwarzschild black hole with mass • , described by the line element
2 = −Δ 2 + Δ −1 2 + 2 2 + sin 2 2 ,(1)
where Δ( ) ≡ 1 − 2 • / is the horizon function. To this spacetime add a small, spherical object with mass e ≪ • and radius 2 e , located at = e (where the subscript e stands for "emitter," since it will be emitting Hawking radiation). Without loss of generality, the object will be placed at the pole = 0 to remove any dependence on the azimuthal angle . This object will model a PBH, which will be assumed (as is standard) to be in free fall from rest at infinity (this assumption will be revisited in Section 3.3 and will be found to be approximately true for physical scenarios). Quantitatively, this free fall condition implies that the object's four-velocity e ≡ e / in the static coordinate frame ( , , , ) is (Misner et al. 1973):
e = Δ( e ) , − 2 − 1 + 2 2 e Δ( e ), − 2 e , 0 .(2)
The specific energy and specific angular momentum will be left arbitrary for now, though it should be noted that the condition of free fall from rest at infinity implies that = 1. Further, assume that the PBH emits blackbody radiation isotropically from its surface with a constant Hawking temperature ′ in its own rest frame. Each photon from the PBH begins with an angular frequency ′ ( e ) corresponding to the emitter's rest frame and propagates outward until reaching an observer on Earth, located a distance from the black hole. Here and elsewhere, primed quantities are measured in the free fall frame (i.e. the rest frame of the emitter), and unprimed quantities are measured in the static frame (i.e. the rest frame of the observer on Earth).
The observer on Earth is assumed to be static, with a four-velocity
o = 1 Δ( o ) , 0, 0, 0 .(3)
Any other four-velocity can be inserted here if one wishes to consider a PBH-BH system moving with respect to Earth, but since other galaxies are only redshifting and receding from Earth, such a choice would only act to decrease the amount of observed radiation. The goal is then to calculate the spectral irradiance from the PBH emitter detected by an observer on Earth.
Radiance calculation
Though radiation emitted just above the horizon may be modified by absorption and emission from a black hole's accretion disc, the simplified setup of this problem assumes that no radiative transfer is required-the spectral radiance from the emitter simply needs to be ray-traced to the observer on Earth along a null geodesic bundle.
The spectral radiance ′ (also called the specific intensity) of a photon with angular frequency ′ is defined by:
′ N ≡ ′ cos ′ ′ Ω ′ ′ ′ ,(4)
where N is the number of photons emitted from an area element ′ in the emitter's proper time ′ through the solid angle Ω ′ , and ′ is the angle at which each photon is emitted to the normal of the area element (Rybicki & Lightman 1985;Yoshino et al. 2019).
For a blackbody emitter, the spectral radiance is given by Planck's law:
′ = ′3 2 2 e ′ / ′ − 1 ,(5)
where the Hawking temperature ′ emitted in the PBH's rest frame is assumed to be constant over the duration of the free fall event:
′ = 1 8 e .(6)
As Barco notes, introducing a spin for the PBH (i.e. using Kerr instead of Schwarzschild) will slightly modify this formula, but doing so can only decrease the PBH's Hawking temperature.
To calculate the spectral radiance ( o ) observed from Earth, a key insight from analyzing how equation (4) behaves under transformations is that the occupation number ∝ / 3 is Lorentzinvariant (Misner et al. 1973). Thus, one can write:
( o ) = 3 ′ ( e ),(7)
where ≡ ( o )/ ′ ( e ) is the blueshift factor, greater than 1 for blueshifted photons and less than 1 for redshifted photons (note that is distinct from the astronomer's redshift factor ≡ −1 − 1). Now all that needs to be done is to calculate the frequencies ( o ) and ′ ( e ), representing the time components of the photon fourmomentum measured in the frame of and evaluated at the position of the observer and the emitter, respectively. The angular frequency in either frame is given by (Misner et al. 1973):
≡ − ,(8)
where the four-velocity = is given by equation (2) or (3), and the photon's coordinate frame four-momentum ≡ / (normalized to its frequency at infinity) is
= 1 Δ , 1 − 2 2 Δ, − 2 , 0 ,(9)
for the impact parameter . The choice of will uniquely determine the specific photon path connecting the emitter to the observer. Equivalently, the photon path can be constrained by specifying the position e of the emitter and the angle e at which the photon is emitted with respect toˆ . The relation between and e can be found by transforming into the static orthonormal tetrad frame and solving tan e = .
Combining equations (2), (3), (8), (9), and (10), the blueshift factor becomes
= Δ( e ) Δ( o ) Δ( e ) + cos e 2 Δ( e ) − 1 − 2 2 e − e sin e −1 .(11)
The observed spectral radiance then becomes
( o ) = ( o ) 3 2 2 e ( o )/ eff − 1 ,(12)
with the effective temperature
eff = ′ .(13)
Any other radiometric quantities, such as the spectral irradiance = ∫ Ω or the irradiance = ∫ , can then be derived from equation (12).
Numerical results
The effective temperature obtained in the Section 2.2, equation (13) looks quite different than the analogous temperature calculated by Barco (2021) in his equation (18). Part of the reason for the differences is that the derivation here uses the emission angle e measured in the observer's frame, whereas Barco uses the angle ′ e measured in the emitter's frame. Using ′ e makes the algebra needlessly more complicated; for example, the relationship between and ′ e analogous to Effective temperature eff as a function of the coordinate time as a PBH falls from a distance of ∼20 Gm above the horizon to a distance of ∼1 μm above the horizon. The green shaded region shows eff , calculated from equations (2) and (13), for a realistic set of parameters (see Sec. 3.3) over the allowed PBH mass range (10 −11 ∼ 10 −14 ⊙ ). For comparison, the black curve shows the value of eff assuming the parameters used by Barco (2021) ( • = 5 × 10 5 ⊙ , e = 2.5 × 10 −13 ⊙ , ≈ 2.23 × 10 −13 , and = 1.1 × 10 17 ), and the black squares and blue curve show the analogous values presented by Barco. equation (10) will include extra -dependent terms. Barco's analysis neglects many of these terms, and his assumption that ( ) ≫ ( ) should actually remove any -dependence from his final expression.
What, then, will a radiating PBH actually look like as it falls into a more massive black hole? The spectrum will appear as a blackbody, in accordance with equation (12). The observed temperature of this blackbody will be given by equation (13) and will cool as the PBH approaches the horizon, but at a different rate and a much lower overall magnitude than that predicted by Barco. Fig. 1 shows the comparison between the cooling behavior found here and that found by Barco (2021) (cf. Fig. 4 of that work).
As can be seen from Fig. 1, the observed temperature of the PBH will actually be much lower than that calculated by Barco. The choice of constants used for the black curve in this plot is the same as that chosen by Barco (in particular, = 1.1 × 10 17 ), while a more realistic, physically motivated set of constants are used for the green shaded region; a discussion of the physical validity of those constants is deferred to Sec. 3.3.
The effective temperature depends inversely on both (with the green region approximately at the lower bound for ) and e (the PBH would need a mass of e ∼ 10 −30 ⊙ to match Barco's values in the observable regime). The choice of only changes eff by less than an order of magnitude, and the choice of • only changes the scaling of the Schwarzschild time and distances. Also, note that Barco's equation (20) (used to calculate the blue curve in Fig. 1) contains a typo and should read " / 0 " in the denominator instead of " / ". Barco also excludes a necessary factor of in the exponent when performing his broken power law fit (Ryde 2004).
The dependence of the emission from the infalling PBH on the observer's time is found by integrating e / e from equation (2) along the null geodesic connecting the observer and emitter. The polar coordinate e will also change as the PBH approaches the horizon. Though it is not stated explicitly, if Barco did use the same constant value for this angle for different emitter radii, he would essentially be relocating the position of Earth for every new position e as the emitter falls in, so that Earth is always in the optimal spot to receive maximum radiation.
As a final comment, note that Fig. 1 traces back the PBH much farther from the horizon than what Barco had presented. Doing so reveals that for the given choices of parameters, the observed radiation would have lasted much longer than Barco's reported 22.7 seconds-Barco considers only the final microns of the PBH's descent, though much more radiation will actually be observed when the PBH is farther from the horizon. In fact, the most radiation would actually be observed in the asymptotic regime before the PBH even reaches the black hole. The PBH dark matter scenario would thus predict a free particle Hawking background many orders of magnitude higher than the radiation signature from PBH-BH mergers.
MISCONCEPTIONS
To explain the key differences in these calculations and the assumptions that lead to these differences, the discussion that follows will be structured around three main misconceptions about general relativity that pervade Barco's work.
Special relativity isn't an add-on to general relativity
The first misconception concerns the applicability of formulas from special relativity and general relativity. The equations of special relativity apply to flat spacetimes, whereas general relativity encompasses all spacetimes. Though special relativistic effects like kinematical boosting may be relevant for fast objects in non-flat spacetimes, these effects do not need to be added in separately, since they are already encompassed by the equations of general relativity.
As an example, consider the blueshift factor appearing in equation (7). This factor can be expressed in a form that reveals two independent effects (Yoshino et al. 2019):
≡ ( o ) ′ ( e ) = ( o ) ( e ) ( e ) ′ ( e ) = Δ( e ) Δ( o ) 1 (1 + cos e ) ,(14)
which is valid when = 0. The first term in parentheses on the right hand side of equation (14) is the gravitational redshift between the near-horizon emitter and the observer, whereas the second term is associated with the Doppler effect from the relative motion between the emitter and the observer. Both of these effects are implicitly included in the calculations of Section 2; no additional relativistic formulas are needed.
In contrast, Barco adds in factors from relevant effects ad hoc, and more problematically, uses some equations that are only valid in special relativity. One example, cast into the notation of this paper, is his equation (11):
= ′ (1 − cos e )(15)
(and note that the Lorentz factor given by equation (17) of Barco (2021) contains a typo and should read −2 instead of 2 ). This equation correctly models the deboosting of radiation from a blackbody emitter travelling away from an observer, but it is only applicable in flat spacetimes (Henry et al. 1968). For an emitter mere microns away from the horizon of an intermediate mass black hole, spacetime is certainly not flat. Deboosting effects will still be present, but they will not take the same form as in the simple flat case and instead will naturally appear in the full general relativistic calculation.
Black holes don't enhance/collimate near-horizon light
As an external observer views an emitter approaching the horizon of a black hole, the radiation from that emitter is exponentially dimmed and redshifted until it merges with the extremely cold Hawking radiation noise from the black hole itself. Why, then, does Barco argue that the radiation from the PBH is enhanced as it approaches the horizon until it forms an "extremely collimated beam"?
Part of the confusion comes from the concept of an escape cone. As an isotropic emitter falls into a black hole, the amount of radiation that is able to escape to infinity decreases until the emitter passes the horizon, after which all radiation (even photons travelling directly outwards in the emitter's rest frame) will remain trapped in the black hole. The escape cone is the solid angle of radiation from the nearhorizon emitter that is able to escape to infinity. This cone is not a concentrated enhancement of the radiation; it is simply an everdecreasing selection of a portion of the isotropic radiation.
In fact, the radiation from the escape cone is not collimated in the slightest. As shown in Fig. 2, for any external emitter near the black hole, light from the escape cone will diffusely spread across the entire sky. Only a vanishingly small portion of this radiation will reach Earth. The light may appear collimated in the emitter's local free fall frame (the upper left insets of Fig. 2), but when transforming back to the observer's frame, it is clear that the light does not form a concentrated beam.
Once again, Barco misapplies special relativity here, in the assumption that the least deboosted radiation will come from the edge of the escape cone (at a critical angle ′ ) and travel straight back to Earth. In actuality, photons emitted at the angle ′ will circle around the black hole as they asymptotically approach the photon sphere (this path is shown by the dashed curves in Fig. 2) and eventually veer off to infinity. Such a choice for the emission angle would be an extremely poor choice if one were using the correct equations from general relativity, since a vanishingly small number of the ray-traced photons from this angle would actually reach Earth after their endless traverse along the photon sphere.
Black holes are not cosmic vacuum cleaners
The final and perhaps most widespread misconception about a black hole is that it is a sort of cosmic "vacuum cleaner" that sucks in everything even remotely close to it at near-light speeds. But the truth is that black holes have no special power beyond that of any other gravitating mass. If the Sun were suddenly replaced by a 1 solar mass black hole, the gravitational dynamics of the solar system would remain completely unchanged.
How does this misconception come into play in Barco's model? First, note that it is extremely improbable for an object with a random trajectory to be captured by a black hole. The black hole will not suck in everything nearby; since PBHs are essentially collisionless, any merger must be the result of a direct head-on collision with an extremely compact region of space. Even objects already in a black hole's accretion disc will rarely fall within the innermost stable circular orbit (ISCO) and dive below the horizon.
To make this first note more quantitative, consider the capture rate of PBHs by more massive black holes given the currently observed local dark matter density of DM ∼ 0.5 GeV/cm 3 (Read 2014). The capture rate F can be estimated by integrating a Maxwellian dark matter distribution over the parameter space of and that would lead to capture within a radius , with up to the characteristic scale (1/3)¯ 2 and constrained by equation (18) below: (Kouvaris 2008;Capela et al. 2013), where the velocity dispersion¯ is taken to be 7 km/s. Even in the extremely unlikely scenario that PBH were small enough for radiation to be detectable from the few black holes within a kiloparsec from Earth, and all dark matter were composed of PBHs (i.e. Ω PBH = Ω DM ), and all PBHs that reach within a black hole's ISCO were captured, the estimated capture rate would be F ≈ 10 −9 yr −1 .
F = Ω PBH Ω DM √ 6 PBH PBH 2 • ¯ (1 − 2 • / )(16)
This number is several orders of magnitude lower than the observed GRB rate (Podsiadlowski et al. 2004), and, as has been argued, such a PBH would be visible long before reaching any black hole.
Putting the unlikelihood of capture aside, one other comment is warranted concerning perhaps the most egregious assumption made by Barco. When specifying the constants of motion for the infalling PBH, Barco arbitrarily chooses ≈ 2.23 × 10 −13 and = 1.1 × 10 17 . The choice of is not too important-for a nonrelativistic object to fall within a radius , it must have (Kouvaris 2008), which for the capture radius at the ISCO yields values of centered around 0 with a maximum close to unity. Within this range, different choices for have a negligible effect on the calculation of eff . However, is non-negligible, and Barco's value is much too high. Barco gives the justification that the central black hole will boost the emitter up to ultrarelativistic speeds, but his chosen value is simply absurd, for two reasons.
2 < 2( + 1) 1 − 2 • /(18)
First, the specific energy is not "boosted" to high values as the PBH approaches the black hole (again, black holes don't have that kind of sucking power).
is a constant of motion that does not change as the emitter speeds up close to the horizon. Even though a distant observer may see the emitter speed up in the black hole's gravitational well, the effect is exactly cancelled by gravitational time dilation so that the observed specific energy remains conserved.
The confusion may come from the fact that the local specific energy = e given by equation (2) (or the equivalent equation (1) in Barco (2021)) does diverge as the emitter approaches the horizon. However, measures the energy in the unphysical frame of "shell observers" who are able to remain static arbitrarily close to the horizon. In the physical frame of observers at rest at infinity, no such extreme boosting occurs-the fastest observed black hole inflow is only ∼ 0.3 (Pounds et al. 2018), and an object in radial free fall from rest at infinity in a Schwarzschild spacetime will only reach a maximum observed speed of ≈ 0.385 (this is left as an exercise to the reader).
Second, now that it is established that black holes themselves will not substantially boost the observed energy of an infaller, all that remains is to show that the choice of the PBH's initial specific energy is too high. Disregarding the effects of spacetime curvature, an object with a specific energy (and therefore Lorentz factor) of = 1.1 × 10 17 would be so fast that it would be able to traverse the entire Milky Way galaxy in a mere .029 milliseconds of its own proper time. Its velocity would be
which, given Barco's quoted mass of 2.5 × 10 −13 ⊙ , would correspond to a total energy of 3 × 10 61 GeV. Not only is this number so far above the grand unified scale that no known laws of physics would apply (certainly not the classical general relativistic calculations done here), it also is only a few orders of magnitude away from the total estimated mass-energy content of the entire observable Universe. What would be a more realistic choice for the value of ? The simplest option would be = 1, corresponding to a free-faller initially at rest at infinity (the initial velocity 0 at infinity is related to the specific energy via the formula = [1 − ( 0 / ) 2 ] −1/2 ). Such an assumption was made in the context of PBHs in e.g. Jackson & Ryan (1973).
If one wanted a more precise value for 0 (or equivalently, ), one could follow Hawking's original supposition that 0 lies in the range 50-1000 km/s (or equivalently, 1.00000001 < < 1.000006), similar to that of other bodies like stars and galaxies that move through the Universe (Hawking 1971;Bird et al. 2016;Yalinewich & Caplan 2021). This range of values is not only consistent with cosmological constraints on the baryon-dark matter relative velocity after kinematic decoupling when PBHs would first be formed (Ali-Haïmoud & Kamionkowski 2017;Dvorkin et al. 2014), but it is also consistent with local observational measurements of the current velocity distribution of dark matter (Herzog-Arbeitman et al. 2018). Some authors have supposed PBH velocities up to ∼0.6 ( ∼ 1.25) when studying gravitational interactions with other bodies like neutron stars (Capela et al. 2013), but even this value is close enough to = 1 that the simplified limit can be taken without question. Any other studies that consider higher, ultrarelativistic black hole velocities (e.g. D'Eath 1978) only do so in a highly theoretical context and make no claims of any connections to PBHs or even to any physically realistic observations.
DISCUSSION
Throughout the years since PBHs were first proposed, they have been used to abduce (i.e. back-explain) a variety of exotic phenomena, many of which have later been proven to be manifestly false. Even Hawking's original work on PBHs supposed not only that they could explain Weber's controversial claims of gravitational wave detections from 1970 but also that they could solve the solar neutrino problem that was later resolved with the theory of flavor oscillations (Hawking 1971). It would appear that such a false abduction has occurred once again in the context of thermal-dominated GRBs.
Three key premises of the arguments of Barco (2021) give rise to the misleading assertion that PBH mergers with larger black holes produce observable GRBs: first, Barco makes special relativistic approximations that do not apply in the near-horizon regime; second, Barco assumes that an infalling emitter's radiation will be boosted and enhanced into a collimated beam when in fact it will be dimmed and redshifted from the observer's perspective; and third, Barco chooses a value for the PBH's initial velocity that is wildly, unrealistically high. Regarding the second point, from personal correspondences with Barco it is clear that the intent of that work was not to claim that radiation from PBHs would be Lorentz-boosted or enhanced directly because of the massive black hole; nevertheless, the claims as they are currently written in his paper are potentially misleading in that regard.
In summary, PBH-BH mergers cannot produce observable GRBs. Not only would their radiation be much too low to be detected, but even if it were high enough, the cooldown signature would look nothing like a GRB. Since the central black hole only acts to dim the PBH's radiation, the PBH would have been visible long before nearing another black hole's horizon. Instead of observing a burst, one would detect a constant streak, in the same way that meteors are observed for the entire duration of their descent through the sky, not just in the final milliseconds of their ablation.
As mentioned in Sec. 1, thermal-dominated GRBs already have well-known progenitors to explain their origins. These events are always extra-galactic and are associated with galaxies with rapid star formation (in contrast, a PBH origin would predict a higher occurrence rate in quenched galaxies, which host more black holes). The prevailing view, supported by multi-messenger observations, is that short GRBs are produced from kilonovae, and long GRBs are produced from collapsars. The only questions that remain are associated with how exactly the energy from these powerful events can be converted into the observed radiation through known physical processes.
Figure 1 .
1Figure 1. Effective temperature eff as a function of the coordinate time as a PBH falls from a distance of ∼20 Gm above the horizon to a distance of ∼1 μm above the horizon. The green shaded region shows eff , calculated from equations (2) and (13), for a realistic set of parameters (see Sec. 3.3) over the allowed PBH mass range (10 −11 ∼ 10 −14 ⊙ ). For comparison, the black curve shows the value of eff assuming the parameters used by Barco (2021) ( • = 5 × 10 5 ⊙ , e = 2.5 × 10 −13 ⊙ , ≈ 2.23 × 10 −13 , and = 1.1 × 10 17 ), and the black squares and blue curve show the analogous values presented by Barco.
Figure 2 .
2Null geodesics emanating from an emitter (small, brown circle) at a radius (a) e = 4 • and (b) e = 2.3 • . The geodesics are equally spaced in the free fall frame with a separation of 2 • for (a) and 1 • for (b), and for simplicity, only the right half of the set of geodesics that can reach infinity are shown. The insets in the upper left show a close-up of the geodesics in the emitter's local tetrad frame. The color shows the degree of redshift along each geodesic, and the dashed curve shows the geodesic that asymptotes to the photon sphere. The black disc shows the portion of the black hole within the horizon ( ≤ 2 • ).
© 2021 The Authors
MNRAS 000, 1-6 (2021)
This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-6 (2021)
ACKNOWLEDGEMENTSThe author would like to thank Andrew Hamilton, Lia Hankla, and others in JILA for helpful discussions surrounding this work. The author would also like to thank the peer reviewers of this work for their comments and feedback.DATA AVAILABILITYAll data and numerical results generated or analysed during this study are included in this paper or the references therein. The author is happy to provide the M notebook used to produce these results to any interested readers.
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"Disc instabilities and semi-analytic modelling of galaxy formation",
"Disc instabilities and semi-analytic modelling of galaxy formation"
]
| [
"E Athanassoula \nTechnopole de l'Etoile -Site de Chateau-Gombert\nLaboratoire d'Astrophysique de Marseille\nObservatoire Astronomique de Marseille Provence\n38 rue Frédéric Joliot-Curie13388Marseille Cédex 13France\n"
]
| [
"Technopole de l'Etoile -Site de Chateau-Gombert\nLaboratoire d'Astrophysique de Marseille\nObservatoire Astronomique de Marseille Provence\n38 rue Frédéric Joliot-Curie13388Marseille Cédex 13France"
]
| [
"Mon. Not. R. Astron. Soc"
]
| The Efstathiou, Lake and Negroponte (1982) criterion can not distinguish bar stable from bar unstable discs and thus should not be used in semi-analytic galaxy formation simulations. I discuss the reasons for this, illustrate it with examples and point out shortcomings in the recipes used for spheroid formation. I propose an alternative, although much less straightforward, possibility. | 10.1111/j.1745-3933.2008.00541.x | [
"https://arxiv.org/pdf/0808.0016v1.pdf"
]
| 15,504,785 | 0808.0016 | bd4326f8593e4f7b79d61a255f5206721fc49614 |
Disc instabilities and semi-analytic modelling of galaxy formation
2006. July 31. 2008
E Athanassoula
Technopole de l'Etoile -Site de Chateau-Gombert
Laboratoire d'Astrophysique de Marseille
Observatoire Astronomique de Marseille Provence
38 rue Frédéric Joliot-Curie13388Marseille Cédex 13France
Disc instabilities and semi-analytic modelling of galaxy formation
Mon. Not. R. Astron. Soc
0002006. July 31. 2008Accepted . Received ;(MN L A T E X style file v2.2)galaxies: evolution -galaxies : haloes -galaxies: structure -galaxies: kinematics and dynamics -methods: numerical
The Efstathiou, Lake and Negroponte (1982) criterion can not distinguish bar stable from bar unstable discs and thus should not be used in semi-analytic galaxy formation simulations. I discuss the reasons for this, illustrate it with examples and point out shortcomings in the recipes used for spheroid formation. I propose an alternative, although much less straightforward, possibility.
INTRODUCTION
Technically, it is not yet possible to make full cosmological simulations which include both the dark matter and the baryons, down to the scale of individual galaxies. Yet the dark matter only simulations have reached very high resolutions (Springel et al. 2008, priv. comm.) and it is crucial to find ways of exploiting them fully. A scheme involving resampling has been devised, in which a given object and its surrounding region are singled out and resimulated at high resolution (e.g. Katz & White 1993). Nevertheless, this technique can only give information on one, or at best a few objects. For this reason, semi-analytical models were introduced, which can give information on properties of galaxy populations. These models are tagged on to the dark matter only simulations and use 'recipes' to describe the evolution of the baryons. It is clear that the results will be useful only if the recipes in question are correct and adequately chosen. This is not an easy task, since these recipes need to include in a relatively simple way a fair fraction of the available information on many key astrophysical processes.
In their quest to have more spheroids, either bulges or ellipticals, some semi-analytic modellers are now considering disc instabilities. They use the Efstathiou, Lake and Negroponte (1982, hereafter ELN)) criterion to distinguish between bar stable and bar unstable discs. Using this criterion, this information can be obtained very simply from the maximum rotational velocity and the disc mass and scale-length. Once a disc is found to be bar unstable, it is turned instantaneously in the model into an elliptical (e.g. Bower et al. 2006), or, alternatively, half of its mass is turned into a bulge and this repeatedly until disc stability according to the ELN criterion is achieved (e.g. De Lucia & Helmi 2008). In this way a considerable amount of mass is turned into a spheroid and several problems concerning the K-band luminosity function are alleviated (see Parry et al. 2008, for a discussion of the relative merit of the two approaches). There are, however, a number of shortcomings in this modelling approach.
BAR FORMATION
Efstathiou, Lake and Negroponte (1982) ran a number of twodimensional N -body simulations of a purely stellar disc in a rigid halo and, based on these, proposed a very simple criterion to distinguish bar stable from bar unstable discs. This reads
RELN = VM /(MD G/RD) 1/2 ,
where Vm is the maximum rotational velocity, MD is the mass of the disc, RD is its scale length and G is the gravitational constant. If RELN < 1.1, ELN propose that the disc is bar unstable, while it will be bar stable for values larger than that limit. This criterion was subsequently used by Mo, Mao & White (1998) and is sometimes referred to as the Mo, Mao & White criterion.
The ELN criterion was derived for purely stellar discs. Christodoulou, Shlosman & Tohline (1995) rederived it for the case of purely gaseous discs, and found that the stability threshold is considerably lower. Thus, purely gaseous discs will be stable if RELN > 0.9. So far, however, this criterion has not been extended to the physically relevant case where both stars and gas are available in the same disc, even when no star formation or feedback is present. Our discussion will therefore, by necessity, follow this limitation.
Other variants of this criterion have also been proposed (see e.g. van den Bosch 1998). In all its forms, however, the ELN criterion is intimately related to the question of whether disks are self gravitating in their inner parts, the so-called "maximum disc' problem (see Bosma 2004 for a review).
The ELN criterion was derived more than 25 years ago and since then our understanding of bar formation has advanced considerably. A number of criticisms of this criterion can be and have been made. First the central concentration of the halo has not been fully taken into account. In the linear theory, if this was sufficiently high it would cut the swing amplifier cycle and thus stop any growth of the bar mode (Toomre 1981). In simulations, however, there can still be a tunnelling through this barrier (e.g. Sellwood 1989), so . In all cases the projected density of the disc is given by grey-scale and also by isocontours (spaced logarithmically) and the numerical value of R ELN is given in the upper left corner of the face-on views. that a bar may still grow. Thus this criticism may not be particularly acute.
A more serious one is that the disc velocity dispersion is not taken into account. A few years after the ELN criterion was published, Athanassoula & Sellwood (1986) showed that velocity dispersion has an important influence on bar stability, which is, in fact, a function of both disc random motions and halo mass. They also presented a disc with no halo, which is stable because of its high velocity dispersion. Although these results are based on 2D simulations, i.e. have by necessity rigid haloes, they still have the advantage of stressing the effect of random motions on stability and thus underline an inadequacy of the ELN criterion.
A major problem of the ELN simulations is that they are twodimensional. Such simulations have by definition rigid haloes, i.e. haloes which are represented by an external forcing and can not respond to the evolution of the disc. This approximation was reasonable at the time, due to the limited computer means then available. With the increase of computer power, however, high resolution 3D simulations with adequate self-consistent treatment of both the baryonic and the dark matter component have become the norm. Athanassoula (2002) compared the results of such fully self-consistent 3D simulations to those of simulations with rigid haloes and showed that the results obtained with the latter can be totally unreliable. In particular, she showed that rigid haloes can erroneously claim stability for cases which the live haloes show are unstable. She explained the difference by the fact that angular momentum exchange between the halo and the disc is de facto not existent in simulations with rigid haloes. On the contrary, in fully self-consistent simulations angular momentum is emitted by nearresonant material in the bar region and absorbed by near-resonant material in the halo and in the outer disc (Athanassoula 2003). This leads to a considerable growth of the bar and explains why strong bars can be seen in discs immersed in massive haloes (Athanassoula & Misiriotis 2002). All this complexity is of course not contained in the very simple ELN criterion or in the work by Ostriker & Peebles (1973), which come to the opposite conclusions, since they do not include the interaction between the disc and the halo.
To make the inadequacies of the ELN criterion clearer, I have chosen amongst my N -body simulations (Athanassoula 2003(Athanassoula , 2007 six examples, three with RELN smaller than 1.1 and three with larger. The three first examples, shown in Fig. 1, have identical RELN values but different disc velocity dispersion. Since RELN = 0.89, they should all three be bar unstable by the ELN criterion, but Fig. 1 shows that this is not the case. The example on the left panel has a very hot disc which makes it stable against bar formation, the middle one shows an average-sized bar, while the one in the right panels shows a strong bar. The value of RELN chosen is nearly half way between the minimum allowed value (0.63, for a bare exponential disc with no halo) and the stability threshold of 1.1 and shows that even for RELN values far from the stability threshold the disc velocity dispersion can stabilise the disc for very long times, of the order of, or more than a Hubble time. This is true for even lower values of RELN . For example, as already stated, Athanassoula & Sellwood (1986) produced a model with no halo at all which was stable over a Hubble time. This is further enhanced in the case where strong central concentrations, and/or high velocity dispersions in the halo component add their stabilising effect to that of the disc velocity dispersion. Thus, a galaxy may be bar stable when the ELN criterion predicts instability.
The three examples given in Fig. 2 have values of RELN equal to 1.22 for the simulation on the left panel, and 1.19 for the two others. Thus, the ELN criterion predicts that all three are bar stable, since their RELN values are larger than 1.1. Fig. 2 shows that one of the three cases is indeed bar stable, the second one forms a small inner bar and the third one a fair sized bar. The edge-on views are also widely different, since the first two have only a disc component, while the third one shows a clear peanut bulge. So again the ELN criterion is found faulty. This is due to the fact that the live halo helps bar growth by absorbing at its resonances the angular momentum that the inner disc can emit. Of course, there is a limit beyond which the disc would not be able to emit sufficient angular momentum, e.g. because it is not sufficiently massive and/or because it is too hot. In such cases, the angular momentum exchange within the galaxy would be limited by the emitters (or lack thereof) and no bar could grow within an astronomically relevant time even though the halo has responsive resonances (Athanassoula 2003). This limit, however, is very far from the ELN predictions, and, more important, does not depend only on the mass ratios, but also on the velocity dispersions of the various components.
To summarise, the ELN criterion should not be used in semianalytic simulations. In cases where it predicts instability, the disc can still be stabilised by factors which RELN does not take into account, such as a strong central concentration, strong random motions in the disc and halo, or, even better, a combination of all three. Conversely, in cases where the ELN criterion predicts stability, a bar can still form due to the destabilising influence of the halo resonances. Shifting the threshold up or down will not solve the problem as long all the other stabilising/destabilising influences are not taken into account. Finally, note that in all cases, both bar stable and bar unstable, the disc is preserved, i.e. no elliptical galaxy is formed.
SUMMARY AND DISCUSSION
In the previous section I showed that the ELN criterion is too simplistic to be able to describe a complex phenomenon such as bar formation and to distinguish bar stable from bar unstable discs. It thus cannot be used in semi-analytic calculations.
Further problems concern the subsequent evolution, after the bar has formed. No simulation has ever shown an unstable disc turn into an elliptical, or acquire a very massive classical bulge instantaneously. The bar can form a boxy/peanut bulge (see Figs 1 and 2) or a discy bulge, and the formation of a classical bulge from ei-ther of those is not excluded 1 . This classical bulge, however, would be much smaller than required by the semi-analytic models, and, furthermore, it would not form instantaneously since the bar needs some time to form, more time is necessary for the boxy/peanut, or discy bulge formation and yet more time is necessary for the putative conversion into a classical bulge. Thus the second part of the De Lucia & Helmi (2008) model has shortcomings, but these may turn out to be quantitative rather than qualitative. On the contrary, the Bower et al. (2006) model has shortcomings at the qualitative level, since the disc can not disappear, i.e. an elliptical could not form from a bar unstable disc without a merger. Both models of course have already problems in the first part, i.e. deciding whether a disc is bar unstable or not, since they both use the ELN criterion.
Can we replace the simple ELN recipe for handling disc instabilities by a better one? This task is not easy, since we need to include in this recipe information on disc stability, bar formation and evolution and the subsequent formation of the different types of bulges, all of which are complex, nonlinear processes. I firmly believe that it is not possible to find one, or a few simple formulas, like the ELN criterion, that can describe all the necessary ingredients of these processes. It might thus be preferable to seek a solution intermediate between the full cosmological simulation including both the dark matter and the baryons at sufficient resolution, which will not be available in the foreseeable future, and the equally difficult task of finding appropriate recipes.
Computer simulations allow us today to routinely make Nbody simulations which can describe the evolution of a single galaxy. In fact a very large number are already available. Using their results instead of the recipes is a possibility well worth exploring. For example, for the problem at hand one would need a small library of N -body simulations with and without gas, describing bar formation and evolution. These simulations should cover, albeit very crudely, the necessary parameter space describing the halo, stellar disc and gas components. This would include not only the mass and scale-length ratios of the various components, but also the different amounts of random motion in the stellar disc and in the spheroids. From these simulations one should extract all the necessary parameters, such as the times necessary for bar and peanut formation, the amount of mass in the boxy/peanut bulge or the discy bulge, etc. This information would be assembled in some sort of table. Then at each time step of the semi-analytic simulation, instead of checking the criterion or applying the recipe, one would extract from the above table the relevant information as a function of the properties of the galaxy at the time under consideration.
The above scheme is of course not a substitute for full scale cosmological simulations, starting ab initio. It also has a number of shortcomings, the most important of which is that our knowledge about the velocity dispersions in disc galaxies is very limited. Nevertheless, it is a very important improvement with respect to the presently used scheme.
Figure 1 .
1The disc component of three simulations which, by the ELN criterion, should form a bar. The upper panels show the face-on views the middle ones the edge-on, side-on view (i.e. with the line of sight along the bar minor axis) and the lower ones the edge-on, end-on view (i.e. with the line of sight along the bar major axis)
Figure 2 .
2Same as forFig. 2, but for three simulations which, by the ELN criterion, should not form a bar.
SeeAthanassoula (2005) for the definition and properties of the different types of bulges.
ACKNOWLEDGEMENTSI thank Albert Bosma and Gabriella De Lucia for useful discussions and Simon White and Albert Bosma for comments on the manuscript. I also thank Guinevere Kauffmann and Dimitri Gadotti for inviting me to MPA where this work started. This work was partially supported by grant ANR-06-BLAN-0172.
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| []
|
[
"Giant Thermal Magnetoresistance Driven by Graphene Magnetoplasmon",
"Giant Thermal Magnetoresistance Driven by Graphene Magnetoplasmon"
]
| [
"Ming-Jian He \nSchool of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China\n\nMinistry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China\n",
"Hong Qi \nSchool of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China\n\nMinistry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China\n",
"Yan-Xiong Su \nSchool of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China\n\nMinistry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China\n",
"Ya-Tao Ren \nSchool of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China\n\nMinistry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China\n",
"Yi-Jun Zhao \nSchool of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China\n",
"Mauro Antezza \nLaboratoire Charles Coulomb (L2C)\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance\n\nInstitut Universitaire de France\n1 rue DescartesF-75231ParisFrance\n"
]
| [
"School of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China",
"Ministry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China",
"School of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China",
"Ministry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China",
"School of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China",
"Ministry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China",
"School of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China",
"Ministry of Industry and Information Technology\nKey Laboratory of Aerospace Thermophysics\n150001HarbinP. R. China",
"School of Energy Science and Engineering\nHarbin Institute of Technology\n150001HarbinP. R. China",
"Laboratoire Charles Coulomb (L2C)\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance",
"Institut Universitaire de France\n1 rue DescartesF-75231ParisFrance"
]
| []
| In this work, we have predicted a giant thermal magnetoresistance for the thermal photon transport based on the tunable magnetoplasmon of graphene. By applying an external magnetic field, we find that the heat flux can be modulated by approximately three orders of magnitude. Accordingly, negative and giant relative thermal magnetoresistance ratios are both achieved for magnetic fields with a maximum strength of 4 Tesla. This effect is mainly caused by the suppression and enhancement of scattering interactions mediated by graphene magnetoplasmon. Specifically, it has never been achieved before for nanoparticles, which have no response to magnetic fields. The effect is remarkable at these reasonable strengths of fields, and thus has considerable significance for the real-life applications. It is also expected to enable technological advances for the thermal measurement-based magnetic sensor and magnetically thermal management. | 10.1063/5.0022261 | [
"https://arxiv.org/pdf/2007.09596v1.pdf"
]
| 220,647,439 | 2007.09596 | 8dc1cc841d6f64aef5e72a35de46c15dbcb6e76e |
Giant Thermal Magnetoresistance Driven by Graphene Magnetoplasmon
Ming-Jian He
School of Energy Science and Engineering
Harbin Institute of Technology
150001HarbinP. R. China
Ministry of Industry and Information Technology
Key Laboratory of Aerospace Thermophysics
150001HarbinP. R. China
Hong Qi
School of Energy Science and Engineering
Harbin Institute of Technology
150001HarbinP. R. China
Ministry of Industry and Information Technology
Key Laboratory of Aerospace Thermophysics
150001HarbinP. R. China
Yan-Xiong Su
School of Energy Science and Engineering
Harbin Institute of Technology
150001HarbinP. R. China
Ministry of Industry and Information Technology
Key Laboratory of Aerospace Thermophysics
150001HarbinP. R. China
Ya-Tao Ren
School of Energy Science and Engineering
Harbin Institute of Technology
150001HarbinP. R. China
Ministry of Industry and Information Technology
Key Laboratory of Aerospace Thermophysics
150001HarbinP. R. China
Yi-Jun Zhao
School of Energy Science and Engineering
Harbin Institute of Technology
150001HarbinP. R. China
Mauro Antezza
Laboratoire Charles Coulomb (L2C)
UMR 5221
CNRS
Université de Montpellier
F-34095MontpellierFrance
Institut Universitaire de France
1 rue DescartesF-75231ParisFrance
Giant Thermal Magnetoresistance Driven by Graphene Magnetoplasmon
In this work, we have predicted a giant thermal magnetoresistance for the thermal photon transport based on the tunable magnetoplasmon of graphene. By applying an external magnetic field, we find that the heat flux can be modulated by approximately three orders of magnitude. Accordingly, negative and giant relative thermal magnetoresistance ratios are both achieved for magnetic fields with a maximum strength of 4 Tesla. This effect is mainly caused by the suppression and enhancement of scattering interactions mediated by graphene magnetoplasmon. Specifically, it has never been achieved before for nanoparticles, which have no response to magnetic fields. The effect is remarkable at these reasonable strengths of fields, and thus has considerable significance for the real-life applications. It is also expected to enable technological advances for the thermal measurement-based magnetic sensor and magnetically thermal management.
The giant magnetoresistance effect discovered by Grünberg and Fert in 1988 [1] is considered as one of the most fascinating advances in solid state physics. Since then, extensive applications of magnetoresistance have been developed in electronics, such as magnetic sensors and hard-disk read-heads [2]. Inspired by the unique effect, a thermal analog named giant thermal magnetoresistance (GTM) effect is predicted in magneto-optical plasmonic structures in the context of radiative heat transfer [3]. Nowadays, the radiative heat transfer at the nanoscale is of great current interest [4][5][6][7][8] and has highlighted the possibility of modulating heat flux with entirely novel schemes [9][10][11][12][13]. Among them, the magneto-optical material InSb has showed remarkable performance in modulating the radiative heat transfer due to the magnetically tunable properties [5,[14][15][16][17]. More recently, by investigating the near-field radiative heat transfer between two InSb particles, a huge anisotropic thermal magnetoresistance with values of up to 800% is achieved with a magnetic field of 5 T [18]. The GTM effect has great application significance for the thermal measurement-based magnetic sensing and magnetically thermal management. However, the GTM in plasmonic structures has so far strictly relied on the magneto-optical nanoparticles made of semiconductors, like InSb. The realization of GTM is still demanding for nanoparticle structures made of conventional materials, which has no response to magnetic field.
By locating a substrate near two nanoparticles, Dong et al. [19] have introduced a new channel of propagating surface waves to assist the heat transfer. They have successfully delayed the deterioration of thermal photon transport, especially in long distance. Then the performance of the long-distance energy-exchange was improved by exciting the surface plasmon polaritons of graphene [8,20]. Specifically, in the presence of an external magnetic field, hybridization occurs between cyclotron excitations and plasmons in graphene, originating magnetoplasmon polaritons (MPP) [21]. The MPP effect has been utilized to realize magnetically tunable near-field radiative heat transfer between suspended graphene sheets [22,23] and gratings [24]. In the present work, based on the graphene MPP, we have successfully proposed a scheme to induce a GTM effect between two nanoparticles, made of common materials silicon dioxide (SiO 2 ). It should be mentioned that the giant thermal magnetoresistance effect obtained in the present work is indeed remarkable and much stronger than those in the previous study [3].
Here we consider two SiO 2 [25] nanoparticles located above a graphene sheet with a distance z n , and they are separated with a distance d as illustrated in Fig. 1. The radii of the two nanoparticles are identical and selected as 5 nm, which has been widely used in the previous studies [15,20,26]. We limit the calculations with the 2 particle-surface distance z n = 50 nm and particle-particle distance d ≥ 100 nm to guarantee the validity of the dipolar approximation [15,20]. The two nanoparticles are kept at temperature T 1 and T 2 . A static magnetic field with intensity B is applied perpendicularly to the graphene sheet. It should be mentioned that, the substrate effect is ignored in the present work to avoid the MPP hybridization with other modes. Thus the pure effect of MPP on the heat transfer mechanism can be distinguished clearly. Moreover, suspended graphene sheets are always considered as ideal physical models for the studies of near-field radiative heat transfer [11,22,23,27]. As illustrated in Fig. 1, the thermal photons transfer through two channels, (1) the direct particle-particle channel via vacuum interaction and (2) the particle-graphene-particle channel via scattering interaction.
With the application of an external magnetic field, a characteristic optical quantum Hall effect occurs to graphene electrons [28]. The conductivity of graphene becomes a tensor with nonzero elements in off-diagonal
parts xx xy L H H L yx yy σ σ σ σ σ σ σ σ = − (1)
where σ L and σ H denote the longitudinal and Hall conductivities, respectively. The magneto-optical conductivities are taken from Refs. [28,29] and the chemical potential of graphene is selected as µ = 0.08 eV throughout the letter. The intensity of magnetic field is limited to B = 0~4 T, which has significance for the real-life applications.
Under this circumstance, the effect of magnetic fields on modifying the optical properties of SiO 2 can be ignored [30]. Based on the dipole approximation, the electric polarizabilities of the nanoparticles are given as [20]
( ) ( ) ( ) ( ) 0 3 1 4 2 R ε ω α ω π ε ω − = +(2)
where R and ε(ω) are the radius and dielectric function of the nanoparticles, respectively. The polarizability needs to be modified by fluctuation-dissipation theorem
( ) ( ) ( ) 3 2 0 Im 6 k χ ω α ω α ω π = − (3) where ( ) ( ) ( ) ( ) ( ) ( ) 0 0 3 3 / 1 / 6 i c α ω α ω ω α ω π = −
is the dressed polarizability with the radiation correction and k 0 =ω/c.
Then we introduce the radiative heat transfer in the proposed system. The whole system is assumed to be thermalized at T=T 1 =T 2 =300 K at the initial state, then nanoparticle 1 is heated up to 1 T T T = + ∆ . This causes a 3 heat flux ϕ between the two nanoparticles. A radiative heat transfer conductance is defined to quantitatively evaluate the heat flux ( ) ( )
4 2 * 0 0 0 , lim 4 Tr 2 GG T n T d h k T T ω ϕ ω ω χ π +∞ ∆ → ∂ = = ∆ ∂ ∫ (4) where ( ) 1 B , exp( ) 1 n T k T ω ω − = −
is the Bose-Einstein distribution and * denotes conjugate transpose. G is the dyadic Green tensor composed of two parts, i.e., G = G (0) +G (sc) . G (0) and G (sc) represent the contributions of vacuum and scattering interaction, accounting for direct particle-particle and particle-interface-particle channels, respectively. The two parts read as
( ) 0 0 2 3 0 0 0 0 0 4 0 0 G ik d a e b k d b π = (5-a) ( ) ( ) sc 0 2 2 0 2 2 G S P z ik z s p z dk ike r r k k π +∞ = + ∫ (5-b) where 0 2 2 a ik d = − , 0 2 2 0 1 b k d ik d = + − . k and 2 2 0 z k k k = −
are the parallel and perpendicular wave-vectors.
More details including the matrices S and P can be found in Refs. [20]. We note that due to the Hall conductivities, the cross-polarization reflection coefficients r sp and r ps are involved in the reflection characteristics of magneto-optical graphene [22][23][24]. They do not appear in Eq. (5-b), whereas it does not mean that the Hall conductivities play no role in the scattering interaction. In particular, the r s and r p in Eq. (5-b) is modified to take in consideration of the effects of both σ L and σ H [31] ( ) ( )( )
2 2 2 0 2 2 0 2 2 2 h L L H s h e L L H Z r Z Z σ η σ σ σ σ η σ + + = − + + − (6-a) ( ) ( )( ) 2 2 2 0 2 2 0 2 2 2 e L L H p h e L L H Z r Z Z σ η σ σ σ σ η σ + + = + + + (6-b)
where Z h =iωµ 0 /k, Z e =ik/ωε 0 , and η 0 denotes the free-space impedance.
A recent experimental study has shown that the radiative heat transfer between two coplanar membranes can be modulated by bringing a third planar object into close proximity [9]. The configuration is similar to the present 4 work. However, with the determined geometric parameters in the system, the heat flux is fixed. This work presents a novel scheme for the configuration to dynamically modulate thermal transport with the magnetic method. In Fig. 2(a) are achieved at B = 4 T and 1.09 T for 7734% and -58.5%, respectively. It should be mentioned that the giant R TMR = 7734% is much larger than those of previous studies [3,14,18]. Additionally, the negative R TMR = -83.7% reveals relatively strong enhancement of heat transfer with reasonable strengths of the fields, which are much weaker than those of the existing studies on magnetically tunable radiative heat transfer [17,22]. To make out the heat transfer mechanism accounting for the above GTM, we plot in Fig. 2 dashed line corresponding to η S = 1 is added in Fig. 2(b), which denotes the circumstance when scattering interactions vanish. The results demonstrate that scattering ratios η S in Fig. 2(b) and the modulation factors η in Fig. 2(a) exhibit similar variety trends with B. As discussed above, the higher η S stands for more participation of scattering interaction, which is dominated by the graphene MPP. Despite of the slight oscillation at weak fields, the primary trend of η S is decaying with B. This phenomenon implies that a transition from scattering enhancement to scattering suppression occurs with the enlarging magnetic fields.
To explore the physical mechanism of the GTM, the radiative heat transfer conductance h(B, ω) is demonstrated in Fig. 3(a) for d=1556 nm. As investigated in previous studies, the phonon polaritons of SiO 2 are excited in the frequency ranges 8.67×10 13~9 .47×10 13 rad/s and 2.03×10 14 ~2.35×10 14 rad/s [8,32]. The two branches at the specific frequencies are also observed in Fig. 3(a), whereas they differ from each other sharply in magnetic-dependent spectrum. The low-frequency branch converts to broadband at weak fields due to the interactions with graphene MPP. Then, as B enhances, this branch is divided into two parts. They correspond to the intraband (low-frequency) and the first interband transitions (high-frequency) of graphene MPP [24,28], respectively. The maximum of h(B, ω) occurs at B = 1.17 T and ω = 0.92×10 14 rad/s, where the first interband MPP interacts with low-frequency phonon polaritons of SiO 2 and forms a strong coupling. To take a deep insight into the effect of the strengths of magnetic fields, we plot h(ω) in Fig. 3(b) corresponding to different cases: (1) the vacuum conductance without graphene, and (2) different strengths of fields, which are sliced with dashed lines in Fig. 3(a). The peak value of h(ω) for B = 1.17 T is larger than those of B = 0, 2 T and B = 3, 4 T by approximately one and two orders of magnitude. We can infer that this enhancement of h(ω) leads to the R TMR = -83.7% in Fig. 2(a). In addition, we find that as B enhances and exceeds 3 T, the intraband MPP almost fade out and the interband MPP decouple with phonon polaritons of SiO 2 . This results in the decaying trend of h(ω) with B,
and h(ω) nearly reduces to that of vacuum interaction with B > 3 T. The above results imply that the evolution of the MPP modes with magnetic fields plays a crucial role in GTM. Therefore in Fig. 3(c), we demonstrate the propagating properties of the MPP modes. They are given by the reciprocal of localization length in x direction, defined as l x =q'/q" [21]. q' and q" represent the real and imaginary parts of complex longitudinal wave vector, respectively. A red dashed line is added in Fig. 3(c), indicating the same frequency with that in Fig. 3(b). The different peak values of h(ω) can be well explained by the magnitudes of l x . We can confirm that the MPP modulates the scattering interactions based on the strong dependence of propagating length on the intensities of magnetic fields.
To have an intuitive understanding of the scattering characteristics, in Fig. 4(a), the electric field energy density u e [19] is illustrated in x-y plane (z = z n /2) for ω = 0.92×10 14 rad/s, corresponding to the peak in Fig. 3(b). Fig. 4(b) between the two nanoparticles, and it is concentrated above the graphene with a distance z ≈ 1.5 µm. Interestingly, the strongest enhancement locates away from the graphene surface, and even higher than the nanoparticles. As is known, the surface plasmon polaritons of graphene are 6 always excited and confined near the interface [21], and it is the same with the scattering enhancement by graphene [8]. The unique scattering enhancement at this position has never been observed in graphene plasmons before. Specifically, a remarkable scattering suppression is observed in Fig. 4(c) near the graphene sheet. The B = 4 T field shuts off the energy-exchange channel dominated by graphene, and thus an attenuation works on the heat flux between the particles. Then, we can conclude from the above results that, the negative and giant thermal magnetoresistance effects are attributed to the MPP scattering enhancement and MPP scattering suppression, respectively.
In summary, we have predicted a negative and a giant thermal magnetoresistance effect between two SiO 2 nanoparticles based on the graphene MPP. The relative thermal magnetoresistance ratio can reach values of up to 7734% and low to -83.7% for a magnetic field of 4 T and 1.17 T, respectively. These values are indeed remarkable at these strengths of fields and have never been achieved in nanoparticle structures made of conventional materials, which has no response to magnetic field. We show that this behavior is mainly resulted from the suppression and enhancement of scattering interaction mediated by graphene MPP. The effect in the present work is promising for the thermal measurement-based magnetic sensing and magnetically thermal management. The physics in this work are limited to the fixed chemical potentials of graphene and geometry sizes (R, z n ). We believe that the optimized parameters can result in more considerable GTM effect. Moreover, we expect in the future work, magneto-optical materials like InSb can act as the substrate and assist the GTM.
FIG. 1 Schematic of radiative heat transfer between two nanoparticles (separated with a distance d) located above a graphene sheet with a distance z n . A static magnetic field with intensity B is applied perpendicularly to the graphene sheet. There are two channels for energy-exchange, (1) the direct particle-particle channel via vacuum interaction, and (2) the particle-graphene-particle channel via scattering interaction.
, for different separation distance d between the two particles, the modulation performance of magnetic field is demonstrated by the modulation factors η=h(B)/h(0) as the function of B. We show that by tuning the intensities of the magnetic fields, the heat transfer can be either enhanced or suppressed compared to that of zero field. The factor η can be tuned nearly over three orders of magnitude with the reasonable strength of fields, B = 0-4 T. In addition, we observe a slight oscillation of η emerging at weak fields B = 0-0.6 T. To reveal the GTM in the proposed system, a relative thermal magnetoresistance ratio is defined asR TMR =[R(B)-R(0)]/R(0)=[h(0)/h(B)-1]×100%, where the thermal magnetoresistance is given by R=1/h. As given by the definition, the positive and negative values of R TMR represent the decayed and enhanced heat transport compared to the zero field. It is shown inFig. 2(a) that at d = 1556 nm, R TMR reaches values of up to 635% and low to -83.7% at the field B = 4 T and 1.17 T, respectively. For d = 673 nm, the maximum and minimum of R TMR
(b) the scattering ratios defined as η S = h/h (0,0) , where h (0,0) denotes the contribution of vacuum interactions in the heat transfer conductance. A horizontal
u e of B = 4 T are considerably weakened compared to those of zero field and B = 1.17 T. In addition, compared to zero field, u e has enhanced when a magnetic field B = 1.17 T is applied. In Figs. 4(b) and 4(c), the ratios of u e in B = 1.17 T and B = 4 T to that in zero field are demonstrated in the x-z plane. The two nanoparticles are indicated by red spots near x = ± 0.8 µm. The positive and negative ratios represent the enhancement and suppression of u e . A scattering enhancement occurs in
11 FIG. 2
112(a) Modulation factors η = h(B)/h(0) together with the relative thermal magnetoresistance ratio R TMR = [R(B)-R(0)]/R(0)=[h(0)/h(B)-1]×100%. (b) Scattering ratios defined as η S = h/h (0,0) , where h (0,0) denotes the contribution of vacuum interactions in the heat transfer conductance.12 FIG. 3 (a) Radiative heat transfer conductance h(B, ω) for d=1556 nm. (b) Spectral heat transfer conductance h(ω) for vacuum interaction and different fields with scattering. (c) Propagating properties of the MPP modes as the reciprocal of localization length in x direction.13 FIG. 4 (a) Electric field energy density u e in x-y plane (z = z n /2) for ω = 0.92×10 14 rad/s. Ratio of u e in (b) B = 1.17 T and (c) B = 4 T to that in zero field in the x-z plane. The two nanoparticles are indicated by red spots near x = ± 0.8 µm.
ACKNOWLEDGEMENTSDATA AVAILABILITYThe data that support the findings of this study are available from the corresponding author upon reasonable request.7
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| []
|
[
"Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model",
"Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model"
]
| [
"Konstadinos Sfetsos [email protected] \nDepartment of Nuclear and Particle Physics Faculty of Physics\nUniversity of Athens\n15784AthensGreece\n",
"Konstadinos Siampos [email protected] \nMécanique et Gravitation\nUniversité de Mons\n7000MonsBelgique\n"
]
| [
"Department of Nuclear and Particle Physics Faculty of Physics\nUniversity of Athens\n15784AthensGreece",
"Mécanique et Gravitation\nUniversité de Mons\n7000MonsBelgique"
]
| []
| We study what we call the all-loop anisotropic bosonized Thirring σ-model. This interpolates between the WZW model and the non-Abelian T-dual of the principal chiral model for a simple group. It has an invariance involving the inversion of the matrix parametrizing the coupling constants. We compute the general renormalization group flow equations which assume a remarkably simple form and derive its properties. For symmetric couplings, they consistently truncate to previous results in the literature. One of the examples we provide gives rise to a first order system of differential equations interpolating between the Lagrange and the Darboux-Halphen integrable systems. | 10.1016/j.nuclphysb.2014.06.012 | [
"https://arxiv.org/pdf/1405.7803v4.pdf"
]
| 119,119,273 | 1405.7803 | e2d96656039463bf82410afe42c8f932be89ce6d |
Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model
14 Jul 2014
Konstadinos Sfetsos [email protected]
Department of Nuclear and Particle Physics Faculty of Physics
University of Athens
15784AthensGreece
Konstadinos Siampos [email protected]
Mécanique et Gravitation
Université de Mons
7000MonsBelgique
Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model
14 Jul 2014
We study what we call the all-loop anisotropic bosonized Thirring σ-model. This interpolates between the WZW model and the non-Abelian T-dual of the principal chiral model for a simple group. It has an invariance involving the inversion of the matrix parametrizing the coupling constants. We compute the general renormalization group flow equations which assume a remarkably simple form and derive its properties. For symmetric couplings, they consistently truncate to previous results in the literature. One of the examples we provide gives rise to a first order system of differential equations interpolating between the Lagrange and the Darboux-Halphen integrable systems.
Introduction
The change in the behaviour of a field theoretical system is encoded in the way the coupling constants of the theory alter with the energy scale. This is studied mathematically under the general frame of the renormalization group (RG), a systematization started in the early seventies [1]. These investigations typically give rise to a system of first order coupled non-linear differential equations, the RG flow or β-function equations, for the couplings of the theory (for a thorough introduction and a review of the subject see [2]). Typically one starts from an asymptotically free theory in the UV or from a conformal field theory (CFT) and then flows away by perturbing with relevant operators. In traditional approaches the RG flow equations are determined order by order in perturbation theory. It is a natural question to ask if it is possible to compute these equations exactly in the coupling constants or at least for some of them. This is important from a physics view point since one could discover new fixed point theories towards the IR. Mathematically it is a very difficult task since in doing so one has to take into account irrelevant operators as well.
Given the above comments it is always exciting if we can obtain the exact RG flow equations for (at least some of) the couplings of a theory. Intuitively we expect that this could be feasible if the perturbed theory is highly symmetric. Such cases arise when the starting point is a two-dimensional CFT with infinitely dimensional current algebra symmetries for the left and the right movers. A model where this is possible to a certain extend is the bosonized version [3,4] (and references therein) of the non-Abelian Thirring model [5]. We will call in short this the non-Abelian Thirring model which has an action of the form
S = S 0 + kλ π J a + J a − ,(1.1)
where S 0 describes a CFT containing right and left affine Lie algebras both at level k with currents J a + and J a − , respectively. The β-function for this theory was computed in [6] to all orders in λ and to leading order in 1 k . The generalization of this computation to the anisotropic non-Abelian Thirring model in which the current-current interaction in (1.1) is replaced by an arbitrary symmetric coupling matrix was performed in [7]. In these cases the computations were performed using current-algebra techniques without much reference to the geometrical details of the background in (1.1). It is very interesting to obtain an effective action in which all effects of the parameter λ have been incorporated exactly and where the only perturbative expansion is with respect to 1/k.
In a recent development a large family of σ-models was constructed in [8] by a gauging procedure. It interpolates between the Wess-Zumino-Witten (WZW) model and the non-Abelian T-dual of the principal chiral model (PCM) model for a simple group G. In the simplest case this σ-model action was shown to be integrable [8] by demonstrating that certain algebraic constraints for integrability [9] were satisfied. It incorporates non-trivially a single parameter, for small values of which it coincides with (1.1). Remarkably, the RG flow equation to leading order in the 1/k expansion was computed in [10] and coincides with the one in [6]. Based on that, it was proposed that this action is the all loop effective of the non-Abelian Thirring model (1.1). In further support, both actions share the same global symmetries and roughly speaking both possess an additional symmetry under the inversion of the deformation parameter λ. This is manifest for the action of [8] but also arises implicitly from path integral considerations involving (1.1) and symmetry arguments, in [11].
The initial aim of this work was to compute the RG flows for the anisotropic G = SU(2) case, with diagonal coupling matrix λ ab and then to compare the results with the analogue ones in [12] which were found using [7]. Nevertheless, we managed to compute the RG flow equations for the most general class of the σ-models of [8] containing a general deformation matrix λ ab and for general simple group G, and present the result in a remarkably simple and compact form. In the above SU (2) case the derived RG-flow interpolate, between the Lagrange and Darboux-Halphen integrable systems, naturally explained by the interpolating nature of our σ-models.
We believe, but we do not have a proof, that our general RG-flow equations coincide with those of [7,13] when such a comparison can be made, i.e. when λ ab = λ ba . However, we provide three non-trivial examples, and found agreement with results following by using the expressions of [7,13]. In that sense the general σ-model of [8] can be thought of as the effective action for the most general anisotropic non-Abelian Thirring model where all effects related to the coupling matrix λ ab (which may have an antisymmetric part) have been taken into account. As a byproduct of our work we obtain the RG flow equations of the non-Abelian T-dual of the general PCM and we prove that they match those of the general PCM.
The organization of this paper is as follows: In section 2 we set up the general class of interpolating σ-models as suited for our purposes. In section 3 we study various of its symmetries and limits as well as the resulting constraints for the RG flow equations for the coupling matrix λ ab . In section 4 we compute the generalized spinconnection and Ricci tensor corresponding to the metric and antisymmetric tensor of our σ-models. In section 5 we derive and study the β-function for the couplings λ ab .
In section 6 we present two examples based on the anisotropic SU(2) and on the symmetric coset G/H space. In section 7 we compare with existing literature results. We end up our work with a wrap up and a discussion on future directions in section 8.
Setting the frame
In this section we present the two-dimensional σ-models of interest to us, in a way suitable for studying their behaviour under RG flow in subsequent sections.
We will study the σ-models of [8] which we first briefly review for the reader's convenience. Consider a general compact simple group G and a corresponding group element g parametrized by X µ , µ = 1, 2, . . . , dim(G). The right and left invariant Maurer-Cartan forms, as well as the orthogonal matrix relating them, are defined as
J a + = −i Tr(t a ∂ + gg −1 ) = R a µ ∂ + X µ , J a − = −i Tr(t a g −1 ∂ − g) = L a µ ∂ − X µ , R a = D ab L b , D ab = Tr(t a gt b g −1 ) , (2.1) which obey dL a = 1 2 f abc L b ∧ L c , dR a = − 1 2 f abc R b ∧ R c . (2.2)
The matrices t a obey the commutation relations [t a , t b ] = i f abc t c and are normalized as Tr(t a t b ) = δ ab . Then the form of the general σ-model action is given by [8] S k,E (g) = S WZW,k (g) +
k 2 π J a + (E − k(D T − I)) −1 ab J b − , (2.3)
where E is a real matrix parametrizing the coupling constants of the theory. The first term is the WZW action for a group G which can be explicitly written as 1
S WZW,k (g) = k 2π L a µ L a ν ∂ + X µ ∂ − X ν + k 12π B f abc L a ∧ L b ∧ L c . (2.4) 1
The relative coefficient of the quadratic and cubic terms is completely dictated by the Polyakov-Wiegmann (PW) formula [14] S WZW,k (g 1 g 2 ) = S WZW,k (g 1 ) + S WZW,k (g 2 ) − k π Tr(g −1
1 ∂ − g 1 ∂ + g 2 g −1 2 ) ,
which is also very practical in evaluating the action for specific parametrizations of g ∈ G.
The J a ± are the chirally and antichirally conserved currents of the WZW model.
It is better for our purposes to reparametrize the couplings by introducing the matrix
λ = k(k I + E) −1 . (2.5)
Then the action (2.3) becomes
S k,λ (g) = S WZW,k (g) + k π J a + (λ −1 − D T ) −1 ab J b − . (2.6)
If the matrix λ is proportional to the identity, then the corresponding σ-model is of special interest since it is actually integrable. This was proven in [8] by showing that the corresponding metric and antisymmetric tensor fields satisfy the algebraic constraints for integrability of [9] and [15]. In addition, in [15] an S-matrix for the SU (2) case was put forward and checked successfully against perturbation theory. A form of the action similar to (2.6) has appeared before in [16], along with related to this action discussion.
Symmetries and limits of the RG flow
In this section we study various symmetries, properties and limits of (2.6) as well as the emerging constraints on the RG flow.
The action (2.6) has a remarkable symmetry under the inversion of the matrix λ, of the group element g and a simultaneous flip of the sign of the overall scaling k. This is encoded mathematically in the relation
S −k,λ −1 (g −1 ) = S k,λ (g) . (3.1)
We note that both terms in (2.6) with the precise coefficients are necessary for the proof, which is otherwise quite straightforward.
Consider the limit where all the entries of the coupling matrix λ are small and go to zero at the same rate, i.e. the ratio of any two entries is finite. In this limit the action (2.6) can be approximated by
S k,λ (g) = S WZW,k (g) + k π λ ab J a + J b − + O(λ 2 ) ,(3.2)
corresponding to the WZW theory perturbed by the current bilinear J a + J b − with arbitrary coupling matrix λ ab . The first two terms define what we will call the anisotropic non-Abelian Thirring model in analogy with the non-Abelian Thirring model [5], [4].
It becomes already apparent in this limit that the left-right current algebra symmetry of the WZW model breaks down completely for a generic matrix λ implying that the σ-model (2.6) has no isometries whatsoever. However, if we allow for a simultaneous transformation of the coupling matrix λ then the σ-model action (2.6) (and of course its limit (3.2)) is invariant under
g → g −1 0 gg 0 , λ → D T 0 λD 0 , (3.3)
where g 0 ∈ G is a constant group element and D 0 is defined in (2.1) using g 0 . If λ is invariant under the above transformation for all constant group elements g 0 then, from Schur's lemma, λ is necessarily proportional to the unit matrix, i.e. λ ab = λδ ab .
In that case the action (2.6) becomes
S k,λ (g) = S WZW,k (g) + kλ π J a + (I − λD T ) −1 ab J b − (3.4)
and has a true isometry associated with the transformation g → g −1 0 gg 0 .
We believe that this action can be uniquely determined under certain assumptions and by symmetry considerations. The argument goes as follows: assuming that such an action contains the WZW action term and that any additional term preserves twodimensional Lorentz invariance and contains two world-sheet derivatives, implies a term of the form F(D) ab J a + J b − . The matrix F(D) has to transform covariantly under the global symmetry g → g −1 0 gg 0 (perhaps accompanied by a transformation of the coupling constants that F(D) contains). If we subsequently demand that F(D) contains a single coupling parameter λ and invariance under the symmetry (3.1) then we find no other possibility but the action (3.4). Generalizing to a general coupling matrix λ leads rather straightforwardly to the more general action (3.4).
The symmetry (3.1) is very powerful and restricts the form of the RG flow equations for the λ ab 's. The corresponding β-function at one-loop in the 1/k expansion is clearly of the form
β λ = dλ dt = − f (λ) k ,(3.5)
where t = ln µ, with µ being the energy scale and where f (λ) is a matrix to be determined as we will explicitly do so in section 4. Here we note that due to the symmetry
(3.1) we have the relation λ f (λ −1 )λ = f (λ) , (3.6)
which severely constrains the matrix f (λ).
In fact when λ ab = λδ ab this symmetry together with CFT arguments allowed for the almost complete determination of β λ . In this case, the β-function computed in [10] coincides with the all-loop (and leading order in 1/k) result for the non-Abelian
Thirring model found in [6]. The map λ → 1/λ and k → −k (for large values of k) was also noted in [11] without however an explicit realization for the action such as the one in (3.4). It was rather deduced though path integral and symmetry considerations.
In addition, invariance under the symmetry (3.
3) implies that
f (D T 0 λD 0 ) = D T 0 f (λ)D 0 . (3.7)
Finally, we note that for k ≫ 1 we may show that the action (2.6) becomes the non-Abelian T-dual of the σ-model action
S PCM = 1 π E ab L a + L b − = 1 π E ab L a µ L b ν ∂ + X µ ∂ − X ν , (3.8)
which is the PCM action with general coupling matrix E ab , a fact proven in [8]. We reproduce the proof here for the reader's convenient. Expanding the matrix elements of λ in (2.5) near the identity we have that
λ ab = δ ab − 1 k E ab + O 1 k 2 . (3.9)
To get a finite result in the limit k → ∞ in the action (2.6), one is forced to also expand the group element near the identity as
g = I + i v k + O 1 k 2 , v = v a t a . (3.10)
This effectively introduces a non-compact set of variables v a in place of the original ones X µ . In that limit we have that
J a ± = ∂ ± v k + O 1 k 2 , D ab = δ ab + f ab k + O 1 k 2 , f ab = f abc v c . (3.11)
Then in the limit k → ∞ the action (2.6) becomes
S non−Abel (v) = 1 π ∂ + v a (E + f ) −1 ab ∂ − v b , (3.12)
which is indeed the non-Abelian T-dual of the PCM action (3.8) [17] (for the case with E ab = δ ab this was shown before for SU (2) in [18] and for a general group in [19]).
The generalized spin connection and Ricci tensor
In this section we compute the generalized spin connection that includes the torsion and the associated Ricci tensor for the background corresponding to the σ-model (2.6).
These are necessary in order to determine the β-function equations for the couplings λ ab . Our computation will parallel, in a sense, the one in [10] for the case with λ ab = λδ ab .
From the metric corresponding to the action (2.6) we extract the frame fields
ds 2 = g ab e a e b , g ab = (I − λ T λ) ab , e a = (D − λ) −1 ab R b , g abẽ aẽb ,g ab = (I − λλ T ) ab ,ẽ a = (D T − λ T ) −1 ab L b . (4.1)
Hence, depending on the frame we will use, we will bear in mind the use of the metric in raising and lowering indices. This will be done with the metrics g ab andg ab
and their inverses which will be denoted by g ab = g −1 ab andg ab =g −1 ab . It turns out that the corresponding background is non-singular if and only if these metrics are positivedefinite. We also note that the frame fields transform under (3.1) as (e,ẽ) → (λe, λ Tẽ ) and that the metric picks up an overall minus sign since we have not included k 2π in its definition.
The matrix relating the above frames reads
e a = Λ ab e b , Λ = (I − Dλ T ) −1 (D − λ) = (D T − λ T ) −1 (I − D T λ) . (4.2)
It transforms under (3.1) as Λ → λ T Λλ −1 and satisfies the condition 2
Λ T (I − λλ T )Λ = I − λ T λ . (4.3)
We note that Λ is an orthogonal matrix only when λ is proportional to the identity.
From the action (2.6) we also read off the antisymmetric two-form
B = k π B 0 + R T λ ∧ e = k π B 0 − L T λ T ∧ẽ , (4.4)
where B 0 is the antisymmetric two-tensor corresponding to the WZW model action.
The three-form field strength associated to B 0 is
H 0 = − 1 6 f abc L a ∧ L b ∧ L c = − 1 6 f abc R a ∧ R b ∧ R c . (4.5)
In the following we will use theẽ a frame and we will notationally supply the geometric quantities associated with it with a tilde. The spin-connection ω ab is equal to
ω ab = ω ab|cẽ c , ω ab = − ω ba , ω ab|c = 1 2 C abc − C cab + C bca , dẽ a = 1 2 C a bcẽ b ∧ẽ c , C a bc := − C a cb , C abc :=g ad C d bc .
(4.6)
A simple computation shows that
dẽ a = − 1 2 (D T − λ T ) −1 ab f bcd (D T − λ T ) ce (D T + λ T ) d fẽ e ∧ẽ f . (4.7)
Using f abc D ia D jb D kc = f ijk and the first of the identities
(D T − λ T ) −1 = (1 − λλ T ) −1 (Λ −T + λ) , (D T − λ T ) −1 = (Λ + λ)(I − λ T λ) −1 , (4.8) with notation Λ −T = (Λ −1 ) T , we may further write that dẽ a = − 1 2 f abcẽ b ∧ẽ c − 1 2 (1 − λλ T ) −1 (Λ −T + λ) ad × λ ed f ebc − λ be λ c f f de f ẽ b ∧ẽ c .
(4.9) 2 To prove this we found useful to make use of the identity
I − λλ T = (I − λD T )(I − Dλ T ) + (D − λ)λ T + λ(D T − λ T ) .
After some further manipulations we arrive at
dẽ a = − 1 2g am f mbc − λ me λ bn λ cℓ f enℓ + Λ −T m f λ e f f ebc − λ bn λ cℓ f f nℓ ẽ b ∧ẽ c , (4.10)
from which we compute ω ab|c .
Next we turn to the computation of the field strength of the two-form (4.4). Using the identities 11) this is found to be
(I − λλ T )(Λ + λ)(I − λ T λ) −1 = Λ −T + λ , (D − λ)λ T Λ(D − λ) −1 = Λ −T λ T ,(4.H = − 1 6 f abc − λ ad λ be λ c f f de f + 3Λ −T c f λ m f f abm − λ ad λ be f de f ẽ a ∧ẽ b ∧ẽ c . (4.12)
Then the generalized spin connection ω − ab that includes the torsion
ω − ab = ω − ab|cẽ c , ω − ab|c = ω ab|c − 1 2 H abc ,(4.13)
is found to be
ω − ab|c = Λ −T cd (λ md f mab − λ am λ bn f dmn ) .
(4.14)
To proceed with the computation of the Ricci tensor we require the exterior derivative of the matrix Λ −T . We found that
dΛ −T ab = (I − λ T λ) −1 mb λ dm f ad f − λ ac λ f e f cme − Λ dm f da f − λ dn λ ac λ f e f nce + Λ −T ac λ f e f mce − λ dm λ nc f dn f − Λ −T ac Λ dm λ nc f dn f − λ dn λ f e f nce ẽ f . (4.15)
Finally we compute the generalized Ricci tensor by employing the general formula for an antisymmetric spin-connection
R ± ab = ∂ c ω ±c a|b − ω ± ac|d ω ∓ b d|c − ∇ ± b ω ±c a|c , ω ± ab|c − ω ∓ ac|b = C abc ,(4.16)
where ∂ a = e a µ ∂ µ . We have used this particular form since, as will see, it will make manifest the appearance of diffeomorphism terms in the RG flow of the coupling matrix elements λ ab . Using (4.16) and (4.10), (4.14) and (4.15) after some manipulations we find that the generalized Ricci tensor is
R − ab = − λ ℓi f aℓp − λ aq λ pℓ f qiℓ (λ ce f rme − λ nr λ dm f ndc ) g imgpc Λ −1 rb − ∇ b ω −c a|c . (4.17)
Note that in computing the relevant part of the generalized Ricci tensor for the RG flow it was not necessary to know the precise form of R − ab , let alone to know first the exact expression of the generalized Riemann tensor. These would have required a much more involved computation. The reader will appreciate this remark if he or she gives a glance at eqs. (A.11) and (A.12) of [10] for the simplest case with λ ab = λδ ab .
Single coupling
We specialize to the case with λ ab = λ δ ab . Then the previous expressions simplify drastically and we find that
dẽ a = − 1 2 (c 1 + c 2 Λ) ab f bcdẽ c ∧ẽ d , H = −(1 − λ 2 ) c 1 6 f abcẽ a ∧ẽ b ∧ẽ c + c 2 3 f abd Λ cdẽ a ∧ẽ b ∧ẽ c , ω − ab|c = (1 − λ 2 ) c 2 Λ cd f dab , dΛ ab = c 1 f adc Λ dbẽ c + c 2 ( f abc − Λ ae f ebc + Λ ae Λ db f edc )ẽ c , R − ab = −c G c 2 2 Λ −1 ab − ∇ − b ω −c ac , ω −c ac = −c 2 f abc Λ bc ,(4.18)
where we note that in this case Λ −1 = Λ T and that
c 1 = 1 + λ + λ 2 1 + λ , c 2 = λ 1 + λ . (4.19)
The above equal the corresponding expressions in [8] and [10] (up to an appropriate rescaling of the vielbein and a rewriting of dΛ ab ).
Computation of the RG flow equations
In this section we derive the RG flow equations for (2.6) and then study various properties and limits.
The one-loop β-function equations for a general σ-model are given by [20][21][22] dG µν dt
+ dB µν dt = R − µν + ∇ − ν ξ µ , (5.1)
where the second term corresponds to diffeomorphisms along ξ µ . Passing to the tangent space indices with the frameẽ a =ẽ a µ dX µ and using the definitions
dG µν dt =β g abẽ a µẽ b ν , dB µν dt =β B abẽ a µẽ b ν , (5.2) we have thatβ g ab +β B ab = R − ab + ∇ − bξ a ,(5.3)
Associate to the matrixg ab we also define a two-formb ab from B µν =b abẽ a µẽ b ν .
We next compute the left hand side of (5.3). In this computation we reinsert the parameter k into the definitions of G µν and B µν . Since the WZW model term in (2.6) does not depend on the matrix λ we immediately obtain that
dG µν dt + dB µν dt = 2k d dt R a µ (λ −1 − D T ) −1 ab L b ν = 2kR a µ (1 − λ D T ) −1 dλ dt (1 − λD T ) −1 ab L b ν , = 2k e a µ Λ T dλ dt ab e b ν = 2kẽ a µ dλ dt Λ −1 abẽ b ν ,(5.4)
where in the last steps we used (4.1) and (4.2). For completeness we note that if we had not assumed that k is fixed, we would have obtained the expressioñ
β G ab +β B ab = 2k dλ dt Λ −1 ab + 1 k dk dt g +b ab . (5.5)
Using the latter, (4.17) and (5.3) we conclude that
dλ ab dt = − 1 2k λ ℓi f aℓp − λ aq λ pℓ f qiℓ (λ ce f bme − λ nb λ dm f ndc ) g imgpc ,(5.6)
whereξ a = ω −c a|c and that k does not flow. Thus the topological nature of its quantization, due to the WZW limit (achieved when λ → 0) [3], persists at one-loop.
We can write the above system in terms of matrices N a (λ) with elements (N a (λ)) p m = λ ac λ pd f cdi − f apc λ ci g im =: N (λ) ap m . (5.7)
Then 3 dλ ab dt = 1 2k Tr N a (λ)N b (λ T ) = 1 2k N (λ) ap m N (λ T ) bm p . (5.8)
Comparing with (3.5) we find that the matrix f (λ) has elements
f ab (λ) = − 1 2 Tr N a (λ)N b (λ T ) . (5.9)
Properties of the RG flow equations
There are several properties of the RG flow equations (5.8) which we list below: This is readily seen from (5.8) if we use that λ T = λ. Hence it is consistent to restrict to symmetric couplings as in [7] which we examine closer in sections 6 and 7 below.
A purely antisymmetric coupling matrix is not consistent with the RG flow equa-
tions. Hence, if at some energy scale the matrix λ ab is antisymmetric this will not persist along the flow.
6. In the special case of λ ab = λ δ ab or λ ab = λ (D 0 ) ab (for constant group elements), we find that dλ dt = − c G λ 2 2k(1 + λ) 2 , (5.12) where c G is the quadratic Casimir in the adjoint representation, defined from the relation f acd f bcd = c G δ ab . 7. The system (5.8) is in agreement with general CFT considerations
dλ ab dt = − 1 2k f ace f bd f λ cd λ e f + O(λ 3 ) , (5.13)
where a CFT is perturbed with operators of mass dimension equal to two as in our case.
Finally, we note that (5.8) encodes the RG flow equations for the non-Abelian
T-dual of the PCM given in (3.12). We provide some details below.
RG flows in the general PCM and its non-Abelian T-dual
In the limit (3.9) the system (5.8) becomes (5.14) where G ab = 1 2 (E ab + E ba ) and G ab = G −1 ab . These are the RG flow equations corresponding to the limit action (3.12). Since this action is equivalent to the PCM action by a classical canonical transformation 4 we expect that the physical information contained in the β-function equations will be preserved. That implies that the βfunction equations for the general PCM should be given by (5.14) as well. This is already proven due to the equivalence of RG flow system of equations in Poisson-Lie T-duality related σ-models in [23]. In this context non-Abelian T-duality is a particular case. Nevertheless, we present for completeness an independent proof that (5.14) also follow from the β-function equations of the PCM models action (3.8). For this action we can derive the metric and the two-form
dE ab dt = 1 8 G pc G mi (E pl f ail + E aq f qip − E li f al p )(E nb f nmc + E dm f bdc − E ce f bme ) ,ds 2 = G ab L a L b , B = 1 2 B ab L a ∧ L b , (5.15) with G = 1 2 (E + E T ) and B = 1 2 (E − E T ) and
where we have omitted a factor of 1 π . It can be easily shown that the generalized spin-connections are
ω + ab|c = 1 2 (E da f dbc − E cd f dab + E db f dca ) , ω − ab|c = 1 2 (E ad f dbc − E dc f dab + E bd f dca ) . (5.16)
Using (4.16) and the latter, we can easily find that (5.17) with no appearance of diffeomorphisms. Plugging the latter into the RG flow equa-
R − ab = 1 4 G pc G mi (E pl f ail + E aq f qip − E li f al p )(E nb f nmc + E dm f bdc − E ce f bme ) ,tions 1 π dG µν dt + dB µν dt = 1 2π R − µν =⇒ dE ab dt = 1 2 R − ab , (5.18)
we readily find (5.14). Thus the RG flow equations of the general PCM are the same with its non-Abelian T-dual as stated above.
Applications
In this section we focus on cases of particular interest which involve a truncation of the form of the matrix λ. This has to be done with care since setting arbitrarily entries of the matrix λ to zero will not be preserved by the flow (5.8).
The SU(2) case: Lagrange and Darboux-Halphen systems
Consider the simplest case with G = SU (2) and
λ = diag(λ 1 , λ 2 , λ 3 ) . (6.1)
Using for representation matrices t a = σ a √ 2 , where σ a , a = 1, 2, 3 are the Pauli matrices leads, due to our normalization conventions, for the structure constants to f abc = √ 2ε abc . Then from (5.8) we find the system of differential equations
dλ 1 dt = − 2 k (λ 2 − λ 1 λ 3 )(λ 3 − λ 1 λ 2 ) (1 − λ 2 2 )(1 − λ 2 3 ) (6.2)
and cyclic in 1, 2, 3. It turns out that dλ ij dt = 0, ∀i = j, so that the restriction (6.1) to a diagonal matrix is a consistent one. Note the symmetry of the system under the
transformation k → −k , λ i → 1 λ i , i = 1, 2, 3 , (6.3)
which follows from (3.1).
For λ i ≪ 1 this system behaves as
dλ 1 dt = − 2 k λ 2 λ 3 + O(λ 3 ) (6.4)
and cyclic in 1, 2, 3, which is the Lagrange system.
In the opposite limit, when λ a → 1, let
λ a = 1 − x a k + O 1 k 2 , a = 1, 2, 3 . (6.5)
Then in the limit k → ∞ we obtain
dx 1 dt = 1 + 1 2x 2 x 3 (x 2 1 − x 2 2 − x 2 3 ) (6.6)
and cyclic in 1, 2, 3, which is the Darboux-Halphen system. This system also follows from (5.14) with E ab = diag(x 1 , x 2 , x 3 ). It first appeared in RG flows for the PCM for SU(2) with the above diagonal matrix E ab in [24]. The Lagrange and the Darboux-Halphen systems arose before in general relativity by imposing the self-dual condition on the Bianchi IX with SU(2) isometry four-dimensional Euclidean metrics [25].
In conclusion the system (6.2) interpolates between the Lagrange and the Darboux-Halphen systems. These have a Lax pair formulation, i.e. [26]. It is very interesting to investigate if this is the case for the interpolating system (6.2) as well.
The two coupling case using a symmetric coset
Let's split the group index into a part corresponding to a subgroup H of G and the rest belongings to the coset G/H. We will keep denoting by Latin letters the subgroup indices and by Greek letters the coset indices. Consider the case in which the matrix λ has elements
λ ab = λ H δ ab , λ αβ = λ G/H δ αβ . (6.7)
It turns out that the above restriction is consistent with the system (5.8) only for symmetric coset spaces G/H, for which the structure constants f αβγ = 0. Using that for symmetric spaces
f abc f abd = c H δ cd , f αβc f αβd = (c G − c H )δ cd , f cβγ f cβδ = c G 2 δ γδ ,(6.8)
we find the system of equations
dλ H dt = − c G λ 2 G/H (1 − λ 2 H ) 2 + c H (λ 2 H − λ 2 G/H )(1 − λ 2 H λ 2 G/H ) 2k(1 + λ H ) 2 (1 − λ 2 G/H ) 2 , dλ G/H dt = − c G λ G/H (λ H − λ 2 G/H ) 2k(1 + λ H )(1 − λ 2 G/H ) . (6.9)
In the limit of λ G/H = 0 they consistently truncate to the RG flow equation (5.12) with (λ, c G ) replaced by (λ H , c H ). 5 For small couplings these read
dλ H dt = − 1 2k c H λ 2 H + (c G − c H )λ 2 G/H + O(λ 3 ) , dλ G/H dt = − c G 2k λ H λ G/H + O(λ 3 ) ,(6.
10) 5 In fact this is the case for general cosets and for coupling matrices of the block diagonal form
λ = λ H ⊕ O G/H , where λ H is a general dim H square matrix.
To prove this we observe that the nonvanishing components of the matrices N m are
(N a ) b c = λ ad λ be f de f − f abe λ e f g f c , (N α ) β c = − f αβd λ de g ec .
Then clearly the RG flow equations (5.8) become that for a subgroup H ∈ G with λ replaced by λ H . Consistent to these is the fact that the action (2.6) retains its form but with λ replaced by λ H and with the indices a, b taking values in H. The interpretation of this new more general than (2.6) action is of the all-loop action of the anisotropic non-Abelian Thirring model for a group G where the perturbation is by a general current bilinears in H ∈ G.
which is in agreement with general CFT expectations, i.e. with (5.13). Note that for groups for which their rank can be taken arbitrarily large the exact expression for the running of couplings can be simply obtained from the perturbative result. This can be seen by noting that when c G → ∞ and c H → ∞ so that their ratio remains finite we may define x H = c G λ H and x G/H = c G λ G/H . Then the running of the x's coincides with that we would have obtained from the leading terms in (6.10).
Flows from coset CFTs
Recall that in [8] variables. For more details the interested reader is refereed to [8]. Returning to our case note that in this decoupling limit of the subgroup the parameter λ H = 1. Then we find from (6.9) that dλ G/H dt = − c G λ G/H 4k , (6.11) which again is in agreement with general CFT expectations. It is interesting that the all-loop result is identical to the one-loop in λ G/H perturbative CFT result, at leadingorder in the 1/k expansion. The result in (6.11) is essentially the same as that obtained in [10] for the simplest case with G = SU (2) and H = U (1). In addition the RG flow will be the same no matter what the subgroup H is, as long as the coset G/H is a symmetric space. We note that λ H = 1 is a fixed point, only if the subgroup is Abelian, see also [23]. 6 6 It turns out that λ H = 1 corresponds to a fixed point of the group G since λ G/H = 1 is also enforced. To prove this, we use (6.9) for λ H → 1 − and we find for Abelian subgroups that
dλ G/H dλ H ≃ (1 − λ 2 G/H ) 2 2λ G/H (1 − λ H ) 2 =⇒ λ G/H ≃ 1 − 1 − λ H 2 + O(1 − λ H ) 2 .
Hence, the σ-model flows towards the IR to the non-Abelian T-dual of the PCM for the group G.
Comparison with literature
The purpose of this section is to compare results following from our general formula for the β-functions (5.8) with existing ones in the literature.
The authors in [7,13], considered the anisotropic Thirring model action given by the first two terms in (3.2) with symmetric coupling matrix λ and computed the corresponding β-functions using current algebra CFT techniques. The general formula they obtained is not apparently the same as (5.8) and in fact it looks more complicated. Given our completely different approach it is important to make a comparison.
First we briefly review the results of [7]. One considers a perturbation of the form
S pert = ∑ A h A O A , O A = dim G ∑ a,b=1 d A ab J a + J b − , (7.1)
where d A ab are pure numbers and define the perturbation. In this work the d A ab 's were taken to be symmetric in the lower indices a, b = 1, 2, . . . , dim(G). The upper index A takes as many values as the number of independent coupling constants h A (denoted by g A in [7]). Hence, comparing with our notation we have the identification λ ab =
O A (z,z)O B (0, 0) = 1 |z| 2 C AB C O C (0, 0) + · · · ,(7.2)
one extracts the structure constants C AB C . The following three conditions ensure closeness of this algebra and renormalizability at all orders
d A ab d B cd f ace f bd f = C AB C d C e f , d A ac d B bc = D AB C d C ab , d A cd f aec f ebd = R A B d B ab . (7.3)
These relations define the set of coefficients D AB C and R A B . Note also the consistency
relations C AB C = C BA C , D AB C = D BA C , D AC D D DB E = D AB D D DC E . (7.4)
Finally one defines the vector and the matrix
C A (x, y) = C BC A x B y C , D A B = D AC B h C , (7.5)
for any two vectors x and y, as well as
h A = h B ((I − D 2 ) −1 ) B A . (7.6)
Then the β-function equations are given by [7,13] dh A dt = 1 k
1 2 C B (h,h)(I + D 2 ) B A − C B (hD,hD)D B A +h B (DRD) B A , (7.7) where (hD) A =h B D B A .
Finally we note that it is consistent to truncate to a subgroup H ∈ G, by considering d A ab 's with lower indices only in the Lie algebra of H. This is congruous with the discussion in footnote 5.
In the following we concentrate on two non-trivial examples, namely the SU (2) and the symmetric coset G/H cases, where we will use (7.7) to compute explicitly the RG flow equations for the couplings. We will find perfect agreement with (6.2) and (6.9) we have found using our general formula (5.8). Based on that we believe that the system (7.7) becomes identical to that in (5.8) for the case of a symmetric, but otherwise general, coupling matrix λ. The proof should be a nice mathematical exercise.
The SU(2) case
For the case of SU(2) consider turning on just three couplings, h A , A = 1, 2, 3 a case that has been considered in [12]. Referring to (7.1) and choosing as the non-vanishing
d A ab those with d 1 11 = d 2 22 = d 3 33 = 1 ,(7.8)
implies, by comparing with (6.1), that λ 1 = h 1 , etc. Then the non-vanishing C AB C 's, D AB C 's and R A B 's are
C 12 3 = C 21 3 = −2 , D 11 1 = 1 , R 1 2 = R 2 1 = 2 , C 1 (x, y) = −2(x 2 y 3 + x 3 y 2 ) , D 1 1 = λ 1 (7.9)
and cyclic permutations in 1, 2, 3. Then plugging these into (7.7) we find (6.2), a result consistent with that in [12] (after relabeling and rescaling).
The two coupling case using a symmetric coset
Let us concentrate on the case of symmetric coset spaces for which f αβγ = 0. We turn on two couplings, one for the subgroup h H and one for the coset h G/H . As before let
Latin indices denote the subgroup and Greek ones the coset. Hence, we have that
d 1 ab = δ ab , d 2 αβ = δ αβ ,(7.10)
where 1 and 2 refer to the subgroup and coset, respectively. Then the non-vanishing C AB C 's and D AB C 's are
C 11 1 = c H , C 12 2 = C 21 2 = c G 2 , C 22 1 = c G − c H , D 11 1 = D 22 2 = 1 ,(7.11)
where c G , c H are the quadratic Casimir in the adjoint representation of G and H, respectively. Also
R 1 1 = −c H , R 1 2 = R 2 2 = − c G 2 , R 2 1 = −c G + c H . (7.12)
In deriving the above we used (6.8). Plugging these into (7.7) we find precisely (6.9) with the identification λ H = h H and λ G/H = h G/H .
Non-diagonal case
Finally, the reader might also require a further example, involving a non-diagonal (symmetric) coupling matrix, so to strengthen the declared equivalence between (5.8) and (7.7). We did so for a consistent truncation in the SU(2) case, i.e. λ 12 = λ 21 and λ 33 and the results followed from these two expressions are in perfect agreement.
Conclusion and outlook
The main result of the present paper is the proof of the one-loop renormalizability and the computation of the RG flow equations for the coupling matrix λ ab of the action (2.6). This computation was achieved using the standard expression for the one-loop renormalizability of two-dimensional σ-models of [20][21][22] and the explicit expression for the β-function equations is given in (5.8). We used this result to further claim that the action (2.6) is the effective action encoding all loop effects in the coupling matrix λ ab of the fully anisotropic non-Abelian Thirring model action defined by the first two terms in (3.2). The basic support for the above claimed equivalence is the fact that in some highly non-trivial cases our RG flow equations are the same as the ones we have found using a general formula for the running of couplings of the anisotropic symmetric non-Abelian Thirring model action given by (7.7). This formula was obtained in [7] using current-algebra techniques and is an all order in the couplings result (but leading in 1/k). Even though we have not proven the equivalence of (5.8) and (7.7) in general we believe this to be the case for the case (of course when λ ab is symmetric). Our result (5.8) has the advantage of being more general since it includes cases of non-symmetric λ ab and in addition it has a remarkably much simpler form. It will be interesting to investigate the RG flow equations (5.8) for specific low dimensional groups.
We have found a particularly interesting result for the anisotropic SU(2) case with diagonal coupling matrix λ ab . The RG flow equations interpolate between the Lagrange and Darboux-Halphen integrable systems of differential equations. It will be interesting if the integrability property of the system is maintained in general beyond the these limit cases.
Finally we note the existence of the all-loop RG flow equations for the anisotropic bosonized non-Abelian Thirring model with different left and right levels of the current algebra [27]. In particular, using Eq.(4.6) of this work we have computed the β-function for left and right currents with different levels k L and k R . The result is
dλ 1 dt = − (k L + k R )λ 1 (λ 2 2 + λ 2 3 ) − 2(1 + k L k R λ 2 1 )λ 2 λ 3 (1 − k L k R λ 2 2 )(1 − k L k R λ 2 3 )
and cyclic in 1, 2 and 3. This is the analogue of (6.2) to which reduces for the special case with k L = k R = k and λ i → λ i /k. We have also computed the analogue of (6.9) for the case of symmetric cosets G/H with different levels for the left and right current algebras but will not present here the result. Such models are not captured by our all-loop action Eq.(2.6), since the WZW action provides a left and a right current algebra with equal levels. It is will be interesting to realize the above RG flow systems of equations with specific σ-models.
As mentioned, for the general σ-model action (2.6) only the case with λ ab = λδ ab corresponding to (3.4) has been proven to be integrable. It is likely that other choices for λ ab may correspond to integrable models as well. A necessary condition for a model to be integrable is that the non-Abelian action (3.12) to which it tends in the limit (3.9) is integrable. Since non-Abelian T-duality preserves integrability (see appendix D of [8]) this is equivalent to looking at cases with proven integrability in PCM which have been worked out in the literature. Such a case is the anisotropic with diagonal λ ab coupling matrix SU(2) PCM for which integrability was shown in [28][29][30].
It will be highly non-trivial if this model is integrable.
1 .
1The system (5.8) satisfies the condition (3.6) due to the transformationN (λ) ap m → λ −1 ac λ −1 pd λ mn N (λ) cd n .(5.10) 2. In addition it respects the symmetry (3.3) due to the transformation N (λ) ap m → (D 0 ) ca (D 0 ) qp (D 0 ) nm N (λ) cq n . (5.11) 3. It holds its form under λ → λ T . 4. A symmetric coupling constant matrix λ remains symmetric under the RG flow.
a second class of σ-models was constructed interpolating between coset CFTs realized by gauged WZW models and the non-Abelian T-dual of coset PCM models. This construction is based in a limiting procedure in which the part of the coupling matrix E (taken to have a block diagonal structure in the subgroup H and the coset G/H spaces) in (2.3) with subgroup indices is taken to be zero. Then the action (2.6) depends actually not on all of the dim(G) variables X µ but on dim(G/H)
One assumes that the operators O A form a closed set. Then, from the double pole in the operator product expansionh A d A
ab . Due to that d A
ab = d A
ba the comparison with our results can only be made for
symmetric matrices λ.
In this rewriting of β-equations, it becomes apparent that N (λ) ab c play the rôle of deformed, by the matrix λ, structure constants.
This was first shown for the PCM with E ab = δ ab : For the case of SU(2) in[18] and for a general group in[19]. For general coupling matrix E ab the canonical equivalence was established in[17].
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"A primal-dual algorithm with optimal stepsizes and its application in decentralized consensus optimization",
"A primal-dual algorithm with optimal stepsizes and its application in decentralized consensus optimization"
]
| [
"Zhi Li [email protected] ",
"Ming Yan [email protected] ",
"Z Li ",
"M Yan ",
"Zhi Li ",
"Ming Yan ",
"\nDepartment of Computational Mathematics, Science and Engineering\nDepartment of Computational Mathematics, Science and Engineering Department of Mathematics\nMichigan State University\nEast LansingMIUSA\n",
"\nMichigan State University\nEast LansingMIUSA\n",
"\nIntroduction\n\n"
]
| [
"Department of Computational Mathematics, Science and Engineering\nDepartment of Computational Mathematics, Science and Engineering Department of Mathematics\nMichigan State University\nEast LansingMIUSA",
"Michigan State University\nEast LansingMIUSA",
"Introduction\n"
]
| []
| We consider a primal-dual algorithm for minimizing f (x) + h(Ax) with differentiable f . The primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP 2 O) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we extend it to solve f (x) + h l(Ax) with differentiable l * and prove its convergence under a weak condition (i.e., under a large dual stepsize). With additional assumptions, we show its linear convergence. In addition, we show that this condition is optimal and can not be weaken. This result recovers the recent proposed positive-indefinite linearized augmented Lagrangian method.Then we consider the application of this primal-dual algorithm in decentralized consensus optimization. We show that EXact firsT-ordeR Algorithm (EXTRA) and Proximal Gradient-EXTRA (PG-EXTRA) can be consider as the primal-dual algorithm applied on a problem in the form of h l(Ax). Then, the optimal upper bound of the stepsize for EXTRA/PG-EXTRA is derived. It is larger than the existing work on EXTRA/PG-EXTRA. Furthermore, for the case with strongly convex functions, we proved linear convergence under the same condition for the stepsize. | null | [
"https://arxiv.org/pdf/1711.06785v1.pdf"
]
| 24,896,127 | 1711.06785 | 8795810cb32992e97da49a13e7ec3db56c24de5c |
A primal-dual algorithm with optimal stepsizes and its application in decentralized consensus optimization
18 Nov 2017
Zhi Li [email protected]
Ming Yan [email protected]
Z Li
M Yan
Zhi Li
Ming Yan
Department of Computational Mathematics, Science and Engineering
Department of Computational Mathematics, Science and Engineering Department of Mathematics
Michigan State University
East LansingMIUSA
Michigan State University
East LansingMIUSA
Introduction
A primal-dual algorithm with optimal stepsizes and its application in decentralized consensus optimization
18 Nov 2017arXiv:1711.06785v1 [math.OC] Noname manuscript No. (will be inserted by the editor) the date of receipt and acceptance should be inserted laterlinearized augmented Lagrangian · primal-dual · decentralized consensus optimization
We consider a primal-dual algorithm for minimizing f (x) + h(Ax) with differentiable f . The primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP 2 O) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we extend it to solve f (x) + h l(Ax) with differentiable l * and prove its convergence under a weak condition (i.e., under a large dual stepsize). With additional assumptions, we show its linear convergence. In addition, we show that this condition is optimal and can not be weaken. This result recovers the recent proposed positive-indefinite linearized augmented Lagrangian method.Then we consider the application of this primal-dual algorithm in decentralized consensus optimization. We show that EXact firsT-ordeR Algorithm (EXTRA) and Proximal Gradient-EXTRA (PG-EXTRA) can be consider as the primal-dual algorithm applied on a problem in the form of h l(Ax). Then, the optimal upper bound of the stepsize for EXTRA/PG-EXTRA is derived. It is larger than the existing work on EXTRA/PG-EXTRA. Furthermore, for the case with strongly convex functions, we proved linear convergence under the same condition for the stepsize.
Introduction
Minimizing the sum of two functions has applications in a variety of areas including image processing, machine learning, and decentralized consensus optimization [2,3,12,19]. In this paper, we aim to minimize the sum of two functions in the following form:
x * = arg min x∈X f (x) + h l(Ax),(1)
It is equivalent to the following saddle-point problem
min x∈X max s∈S f (x) + Ax, s − h * (s).(3)
In order to solve (2) (or (3)), a primal-dual algorithm was proposed in different fields under different names [5,9,16]. Loris and Verhoeven [16] focused on a particular smooth function f (x) = 1 2 Kx − y 2 , where K is a linear operator. Chen, Huang, and Zhang [5] considered the general problem (2) and proposed a Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP 2 O). Then the same algorithm was rediscovered under the name Proximal Alternating Predictor-Corrector (PAPC) in [9] to solve (2) and its extension to a finite sum of composite functions when h is separable. One iteration of the algorithm is
s k+1 ∈ γ λ I + ∂h * −1 γ λ (I − λAA ⊤ )s k + A x k − γ∇f (x k ) ,(4a)x k+1 = x k − γ∇f (x k ) − γA ⊤ s k+1 .(4b)
Here λ and γ are two positive parameters, and the convergence of this algorithm is shown when λ ≤ 1/λ max (AA ⊤ ) and γ < 2/L, where L is the Lipschitz constant of ∇f .
There are many other algorithms for solving (2) and its extensions. For example, Condat-Vu [4,7,21] solves a more general problem than (2) with an additional non-differential function. However, the corresponding parameters λ and γ have to satisfy λ · λ max (AA ⊤ ) + 2γ/L ≤ 1, and Condat-Vu converges slower than PAPC in solving (2). When f = 0, Condat-Vu reduces to Chambolle-Pock [2]. There are several other primal-dual algorithms for minimizing the sum of three functions with one differentiable function [6,22]. Interested readers are referred to [13,22] for the comparison of different primal-dual algorithms for minimizing the sum of three functions.
When there is only one function f (x), i.e., h = 0, we let A = 0, and the primal-dual algorithm reduces to the gradient descent with stepsize γ. Therefore, the condition γ < 2/L can not be relaxed. Then whether the condition λ ≤ 1/λ max (AA ⊤ ) can be relaxed? In [5, Fig. 1], the authors numerically showed that a larger stepsize (e.g., λ = 4/(3λ max (AA ⊤ ))) gives a better performance than stepsizes satisfying the condition λ ≤ 1/λ max (AA ⊤ ). However, there is no theoretical result for the convergence under this large stepsize.
For linearized Augmented Lagrangian Method (ALM) [23]-a special case of the primal-dual algorithm (4)-the positive definiteness is relaxed in [10]. Consider the constrained optimization problem
minimize s h * (s), subject to − A ⊤ s = b.
Its dual problem is
minimize x b ⊤ x + h(Ax), which is the problem in (2) with f (x) = b ⊤ x. The linearized ALM is s k+1 = arg min s h * (s) + β 2 s − s k − 1 β A(x k − γ(A ⊤ s k + b)) 2 , (6a) x k+1 = x k − γ(A ⊤ s k+1 + b).(6b)
It is exactly the primal-dual algorithm (4) with β = γ/λ. Note that the step in (6a) can be written as arg min
s h * (s) − x k , A ⊤ s + b + γ 2 A ⊤ s + b 2 2 + β 2 s − s k 2 I−(γ/β)AA ⊤ .
In [23], positive-definiteness of I − (γ/β)AA ⊤ is required for proving the convergence. Then the authors in [10] showed that the matrix I − (γ/β)AA ⊤ can be positive-indefinite. More specifically, (γ/β) ≤ 4/(3λ max (AA ⊤ )) is the necessary and sufficient condition for the convergence of linearized ALM. This result motivates us showing the convergence of (4) under a weak condition. In this paper, we extend the result to the primal-dual algorithm (4) by providing a necessary and sufficient condition on λ for its convergence. The extension is nontrivial because the differentiable function f (x) = b ⊤ x in linearized ALM and the Lipschitz constant of ∇f is 0. Furthermore, we consider the more general problem (1) with infimal convolution instead of (2). There are few applications using infimal convolution [8] in image processing [11] and motion planning for robotics [14]. In this paper, we present another application in decentralized consensus optimization with the sum of smooth and nonsmooth functions. For details about decentralized consensus optimization, please see [17,18,15] and references therein. In decentralized consensus on an undirected network, node i has one part of s (for simplicity, we assume that s i is stored and computed on node i). The final purpose is to make sure that the values on all nodes are consensual, i.e., s 1 = · · · = s n , and to obtain the minimizer of the objective function
h * (s) + l * (s) = n i=1 (h * i (x i ) + l * i (x i ))
. The consensus condition can be enforced by A ⊤ s = 0 with ker(A ⊤ ) spanned by {1}, and the problem can be rewritten in the following form:
minimize s h * (s) + l * (s) + ι {0} (−A ⊤ s).
Its corresponding dual problem is minimize x h l(Ax).
Therefore, the decentralized consensus problem is also a special case of (1) with f (x) = 0. We will show in Section 3 that the famous decentralized algorithm-Proximal Gradient EXact firsT-ordeR Algorithm (PG-EXTRA) [19] is exactly the primal-dual algorithm in (7), which is a generalization of (4) for solving (1). In addition, we relax the parameter λ in the primal-dual algorithm and provide a optimal upper bound for λ that is verified by an example from decentralized optimization. Note the convergence of EXTRA under a large stepsize is demonstrated numerically in [20] without theoretical analysis.
The contributions of this paper can be summarized as follows:
-We extend PDFP 2 O/PAPC to solve the problem (1) with an infimal convolution term. -We relax the parameter for the primal-dual algorithm and provide an optimal bound for the parameters. This results recovers the positive-indefinite ALM in [10]. -For decentralized consensus, we show that PG-EXTRA is equivalent to the primal-dual algorithm applied to the dual problem and provide optimal bounds for its parameters that are larger than those gave in [19]. Then we prove the linear convergence of EXTRA under the same weak condition for the stepsize and an additional assumption for the smooth functions.
The rest of this paper is organized as follows. In Section 2, we present the algorithm to solve (1). We show its convergence for the general case in Section 2.2 and linear convergence rate under additional assumptions in Section 2.3. In Section 3, we build the connection between the proposed algorithm with PG-EXTRA in decentralized consensus optimization and provide an optimal bound for its parameters. Then we end this paper with a short conclusion.
New convergence results
A primal-dual algorithm
In this paper, we extend the existing primal-dual algorithm (4) for solving (1) with an infimal convolution term and show its new convergence results. Given x k and s k , one iteration of the primal-dual algorithm is
s k+1 ∈ γ λ D + ∂h * −1 1 γ Ms k + A x k − γP −1 ∇f (x k ) − ∇l * (s k ) , (7a) x k+1 = x k − γP −1 ∇f (x k ) − γP −1 A ⊤ s k+1 ,(7b)
where M = γ 2 λ (D−λAP −1 A ⊤ ). Here λ and γ are two positive parameters, and P and D are two positive definite operators defined on X and S, respectively.
Let I be the identity operator defined on a Hilbert space. For simplicity, we do not specify the space on which it is defined when it is clear from the context. When ∇l * ≡ 0 and P = D = I, the iteration reduces to (4), which is an existing primal-dual algorithm proposed in [5,9,16]. Its convergence is shown if I − λAA ⊤ is positive semidefinite and γ < 2/L with L being the Lipschitz constant of ∇f .
Except extending this existing primal-dual algorithm to (7) for solving the problem (1) with an infimal convolution, we also show its convergence with a larger λ. Specifically, we show that we can choose λ such that D− 3 4 λAP −1 A ⊤ is positive definite, i.e., the upper bound for λ is increased by 1/3.
For convenience, we introduce two operators as
M 1 := γ 2 λ (D − θλAP −1 A ⊤ ), M 2 := γ 2 (1 − θ)AP −1 A ⊤ .
Here θ ∈ (3/4, 1] is chosen such that M 1 is positive definite and M 2 is positive semidefinite. We can find such θ when λ ≤ 4/(3λ max (D −1/2 AP −1 A ⊤ D −1/2 )).
Then we define
C 1 = λ max (M −1/2 1 M 2 M −1/2 1 ) ≥ 0.
With these two operators, we have M = M 1 − M 2 . In addition, we define a positive definite operator as follows
M := M 1 + M 2 .
We let s, t M := s, Mt and s 2 M = s, Ms for any self-adjoint operator M. Note that s 2 M can be negative if M is not positive semidefinte. When M is positive definite, we further define the induced norm as
s M = s, s M . For (x, s) ∈ X × S, we define (x, s) 2 P,M = x 2 P + s 2 M .
Assumption 1 Let (x * , s * ) be any fixed point of (7). For any x ∈ X and s ∈ S, we have
x − x * , ∇f (x) − ∇f (x * ) ≥β ∇f (x) − ∇f (x * ) 2 P −1 ,(8)s − s * , q h (s) − q h (s * ) ≥0,(9)s − s * , ∇l * (s) − ∇l * (s * ) ≥β ∇l * (s) − ∇l * (s * ) 2 M −1 1 ,(10)for some β > 0. Here q h (s) ∈ ∂h * (s) and q h (s * ) ∈ ∂h * (s * ).
This assumption is satisfied if both f and l * have Lipschitz continuous gradients. For example, (8) is satisfied with P = I if f has a 1/β Lipschitz continuous gradient [1,Theorem 18.15]. Also, if ∇f (or ∇l * ) is fixed for all x (or s), e.g., the linear f in linearized ALM, then (8) (or (10)) is satisfied with any β > 0.
Assumption 2 Let (x * , s * ) be any fixed point of (7). There exist τ f ≥ 0, τ h ≥ 0, and τ l ≥ 0, such that, for any x ∈ X and s ∈ S,
x − x * , ∇f (x) − ∇f (x * ) ≥τ f x − x * 2 P ,(11)s − s * , q h (s) − q h (s * ) ≥τ h s − s * 2 M1 ,(12)s − s * , ∇l * (s) − ∇l * (s * ) ≥τ l s − s * 2 M1 ,(13)
where q h (s) ∈ ∂h * (s) and q h (s * ) ∈ ∂h * (s * ).
The assumption is satisfied if functions f , h * , and l * are convex, and in this case, τ f = τ h = τ l = 0.
Convergence for general convex functions
First of all, we find a subgradient of h * at s k+1 :
q h (s k+1 ) := 1 γ Ms k − 1 γ Ms k+1 + Ax k+1 − ∇l * (s k ) ∈ ∂h * (s k+1 ).(14)
It can be easily obtained from (7), and its proof is omitted here. Let (x * , s * ) be any fixed point of (7), and we have a subgradient of h * at s * :
q h (s * ) :=Ax * − ∇l * (s * ) ∈ ∂h * (s * ).(15)
Lemma 1 (fundamental inequality) Let (x * , s * ) be any fixed point of (7). Then we have
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤ (x k , s k ) − (x * , s * ) 2 P, M − s k − s k+1 2 M1 − 2γ s k+1 − s * , q h (s k+1 ) − q h (s * ) + ∇l * (s k ) − ∇l * (s * ) + 2γ ∇f (x k ) − ∇f (x * ), x * − x k + (4θ − 3)(x k − x k+1 ) − (4θ − 3) x k − x k+1 2 P + 4(1 − θ)γ 2 ∇f (x k ) − ∇f (x * ) 2 P −1 .(16)
Proof The definitions of q h (s k+1 ) and q h (s * ) in (14) and (15), respectively, and the update of
x k+1 in (7b) show 2γ s k+1 − s * , q h (s k+1 ) − q h (s * ) + ∇l * (s k ) − ∇l * (s * ) (14),(15) = 2γ s k+1 − s * , 1 γ Ms k − 1 γ Ms k+1 + Ax k+1 − Ax * =2 s k+1 − s * , s k − s k+1 M + 2γ s k+1 − s * , Ax k+1 − Ax * =2 s k+1 − s * , s k − s k+1 M + 2γ A ⊤ s k+1 − A ⊤ s * , x k+1 − x * (7b) = 2 s k+1 − s * , s k − s k+1 M + 2 x k − x k+1 , x k+1 − x * P − 2γ ∇f (x k ) − ∇f (x * ), x k+1 − x * (17) = s k − s * 2 M − s k+1 − s * 2 M − s k − s k+1 2 M + x k − x * 2 P − x k+1 − x * 2 P − x k − x k+1 2 P + 2γ ∇f (x k ) − ∇f (x * ), x * − x k+1 ,
where we expand the first two terms in (17) with 2 a, b = a+b 2 − a 2 − b 2 to obtain the last equality. Therefore, we have
(x k+1 , s k+1 ) − (x * , s * ) 2 P,M =2γ ∇f (x k ) − ∇f (x * ), x * − x k+1 − 2γ s k+1 − s * , q h (s k+1 ) − q h (s * ) + ∇l * (s k ) − ∇l * (s * ) + (x k , s k ) − (x * , s * ) 2 P,M − x k − x k+1 2 P − s k − s k+1 2 M .(18)
The fact that M = M 1 −M 2 gives us an upper bound for the last term of (18).
− s k − s k+1 2 M = − s k − s k+1 2 M1 + s k − s k+1 2 M2 = − s k − s k+1 2 M1 + s k − s * + s * − s k+1 2 M2 ≤ − s k − s k+1 2 M1 + 2 s k − s * 2 M2 + 2 s k+1 − s * 2 M2 . (19)
Adding 2 s k+1 −s * 2 M2 onto both sides of (18), recalling that M = M 1 +M 2 = M + 2M 2 , and combining (19) and (18), we have
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤2γ ∇f (x k ) − ∇f (x * ), x * − x k+1 − 2γ s k+1 − s * , q h (s k+1 ) − q h (s * ) + ∇l * (s k ) − ∇l * (s * ) + (x k , s k ) − (x * , s * ) 2 P, M − x k − x k+1 2 P − s k − s k+1 2 M1 + 4 s k+1 − s * 2 M2 .(20)
With the definition of M 2 , the last term in (20) can be written as
4 s k+1 − s * 2 M2 =4(1 − θ) γP −1 A ⊤ s k+1 − γP −1 A ⊤ s * 2 P =4(1 − θ) x k − γP −1 ∇f (x k ) − x k+1 + γP −1 ∇f (x * ) 2 P =4(1 − θ) x k − x k+1 2 P + 4(1 − θ)γ 2 ∇f (x k ) − ∇f (x * ) 2 P −1 − 8(1 − θ)γ x k − x k+1 , ∇f (x k ) − ∇f (x * ) ,(21)
where the second equality comes from (7b). Then, we plug (21) into (20) and obtain
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤2γ ∇f (x k ) − ∇f (x * ), x * − x k + (4θ − 3)(x k − x k+1 ) − 2γ s k+1 − s * , q h (s k+1 ) − q h (s * ) + ∇l * (s k ) − ∇l * (s * ) + (x k , s k ) − (x * , s * ) 2 P, M − s k − s k+1 2 M1 − (4θ − 3) x k − x k+1 2 P + 4(1 − θ)γ 2 ∇f (x k ) − ∇f (x * ) 2 P −1 .
The result is proved.
⊓ ⊔ Lemma 2 Let (10) be satisfied, then − s k − s k+1 2 M1 − 2γ s k+1 − s * , ∇l * (s k ) − ∇l * (s * ) ≤ − (1 − γ/(2β)) s k − s k+1 2 M1 . Proof Because M 1 is positive definite, we have − s k − s k+1 2 M1 − 2γ s k+1 − s * , ∇l * (s k ) − ∇l * (s * ) = − s k − s k+1 2 M1 − 2γ s k+1 − s k , ∇l * (s k ) − ∇l * (s * ) − 2γ s k − s * , ∇l * (s k ) − ∇l * (s * ) ≤ − s k − s k+1 2 M1 + γ 2β s k − s k+1 2 M1 + 2γβ ∇l * (s k ) − ∇l * (s * ) 2 M −1 1 − 2γβ ∇l * (s k ) − ∇l * (s * ) 2 M −1 1 = − s k − s k+1 2 M1 + γ 2β s k − s k+1 2 M1 ,
where the inequality comes from the Cauchy-Schwarz inequality and (10). ⊓ ⊔ Theorem 1 Let Assumption 1 hold, θ ∈ (3/4, 1], and γ ∈ (0, 2β). For any fixed point (x * , s * ) of (7), we have
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M − (x k , s k ) − (x * , s * ) 2 P, M ≤ − 1 − γ 2β s k − s k+1 2 M1 − (4θ−3)(2β−γ) 2β−4(1−θ)γ x k − x k+1 2 P .(22)
Proof Applying Lemma 2 and (9) to the inequality (16) in Lemma 1 gives
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤ (x k , s k ) − (x * , s * ) 2 P, M − (1 − γ/(2β)) s k − s k+1 2 M1 + 2γ ∇f (x k ) − ∇f (x * ), x * − x k A +4(1 − θ)γ 2 ∇f (x k ) − ∇f (x * ) 2 P −1 − (4θ − 3) x k − x k+1 2 P + 2γ(4θ − 3) ∇f (x k ) − ∇f (x * ), x k − x k+1 B .(23)
Next we bound terms A and B. For term A, the assumption (8) implies
2γ ∇f (x k ) − ∇f (x * ), x * − x k ≤ −2γβ ∇f (x k ) − ∇f (x * ) 2 P −1 ,(24)
and the Cauchy-Schwarz inequality applied to term B implies
2γ(4θ − 3) ∇f (x k ) − ∇f (x * ), x k − x k+1 ≤(2γβ − 4(1 − θ)γ 2 ) ∇f (x k ) − ∇f (x * ) 2 P −1 + γ(4θ−3) 2 2β−4(1−θ)γ x k − x k+1 2 P ,(25)
where θ ∈ (3/4, 1] and γ ∈ (0, 2β). Plugging (24) and (25) into (23), we have
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤ (x k , s k ) − (x * , s * ) 2 P, M − (1 − γ/(2β)) s k − s k+1 2 M1 − (4θ − 3) x k − x k+1 2 P + γ(4θ−3) 2 2β−4(1−θ)γ x k − x k+1 2 P = (x k , s k ) − (x * , s * ) 2 P, M − (1 − γ/(2β)) s k − s k+1 2 M1 − (4θ−3)(2β−γ) 2β−4(1−θ)γ x k − x k+1 2 P
The inequality (22) is proved. ⊓ ⊔ Remark 1 When β = +∞, i.e., the Lipschitz constant of ∇f and ∇l * is 0, then (22) becomes
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M − (x k , s k ) − (x * , s * ) 2 P, M ≤ − s k − s k+1 2 M1 − (4θ − 3) x k − x k+1 2 P .
This is the key result in [10, Theorem 3.1] for linearized ALM.
Note that λ is the product of the primal stepsize γ and the dual stepsize. Larger λ means larger dual stepsize. So this result shows that we can choose a larger dual stepsize.
Theorem 2 Assume that X and S are finite dimensional. Under the assumptions in Theorem 1, the sequence {(x k , s k )} converges to a fixed point of (7).
Proof
The inequality (22) shows that {(x k , s k )} is a bounded sequence. The finite dimensionality of X and S yields the compactness of X and S. Then there exists a subsequence {(x kn , s kn )} that converges to (x * ,s * ). In addition, we have lim k→∞ (x k , s k ) − (x k+1 , s k+1 ) 2 P,M1 = 0 from the inequality (22), and the subsequence {(x kn+1 , s kn+1 )} converges to the same point (x * ,s * ). Therefore, (x * ,s * ) is a fixed point of (7).
Letting (x * , s * ) in (22) be (x * ,s * ), we have
(x k+1 , s k+1 ) − (x * ,s * ) 2 P, M ≤ (x k , s k ) − (x * ,s * ) 2 P, M .
Therefore, the sequence {(x k , s k )} converges to (x * ,s * ), which is a fixed point of (7). ⊓ ⊔
Linear convergence
In this subsection, we prove the linear convergence of the sequence {(x k , s k )} in Theorem 3 under the additional Assumption 2. This general linear convergence result requires τ f > 0. Then, for the special case when f = 0 and h = 0, we show the linear convergence of the sequence in Theorem 4, and this result will be applied to obtain a stronger result for EXTRA than previous work.
Before showing the linear convergence, we prove the following lemma, which is different from Lemma 2. (10) and (13) be satisfied, then
Lemma 3 Let
− s k − s k+1 2 M1 − 2γ s k+1 − s * , ∇l * (s k ) − ∇l * (s * ) ≤ − M 2 (s k+1 − s k ) + γAx k+1 − γAx * − γq h (s k+1 ) + γq h (s * ) 2 M −1 1 (26) − 2γ − γ 2 /β τ l s k − s * 2 M1 . Proof Because M 1 is positive definite, we have − s k − s k+1 2 M1 − 2γ s k+1 − s * , ∇l * (s k ) − ∇l * (s * ) = − s k − s k+1 2 M1 − 2γ M 1/2 1 (s k+1 − s k ), M −1/2 1 (∇l * (s k ) − ∇l * (s * )) − 2γ s k − s * , ∇l * (s k ) − ∇l * (s * ) = − M 1/2 1 (s k+1 − s k ) + M −1/2 1 γ(∇l * (s k ) − ∇l * (s * )) 2 + γ 2 ∇l * (s k ) − ∇l * (s * ) 2 M −1 1 − 2γ s k − s * , ∇l * (s k ) − ∇l * (s * ) .(27)
The first term on the right hand side of (27) becomes
− M 1/2 1 (s k+1 − s k ) + M −1/2 1 γ(∇l * (s k ) − ∇l * (s * )) 2 = − M 1 (s k+1 − s k ) + γ(∇l * (s k ) − ∇l * (s * )) 2 M −1 1 = − M 2 (s k+1 − s k ) + M(s k+1 − s k ) + γ(∇l * (s k ) − ∇l * (s * )) 2 M −1 1 (14),(15) = − M 2 (s k+1 − s k ) + γAx k+1 − γAx * − γq h (s k+1 ) + γq h (s * ) 2 M −1 1 ,
where the second equality comes from M = M 1 − M 2 . For the other two terms on the right hand side of (27), we have
γ 2 ∇l * (s k ) − ∇l * (s * ) 2 M −1 1 − 2γ s k − s * , ∇l * (s k ) − ∇l * (s * ) =γ 2 ∇l * (s k ) − ∇l * (s * ) 2 M −1 1 − (γ 2 /β) s k − s * , ∇l * (s k ) − ∇l * (s * ) − (2γ − γ 2 /β) s k − s * , ∇l * (s k ) − ∇l * (s * ) (10),(13) ≤ − (2γ − γ 2 /β)τ l s k − s * 2 M1 .
Combining both together with (27)
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤ ρ 1 (x k , s k ) − (x * , s * ) 2 P, M ,(28)
where
ρ 1 = max 1−(2γ−γ 2 /β)τ l +C1 1+2γτ h +C1 , 1 − (2γ − γ 2 /β)τ f . The sequence {(x k , s k )} converges linearly to the fixed point (x * , s * ) with rate ρ 1 < 1 if γ ∈ (0, 2β), τ h + τ l > 0, and τ f > 0.
Proof Applying Lemma 3 to (16) in Lemma 1 gives
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤ (x k , s k ) − (x * , s * ) 2 P, M − 2γ − γ 2 /β τ l s k − s * 2 M1 − 2γ s k+1 − s * , q h (s k+1 ) − q h (s * ) + 2γ ∇f (x k ) − ∇f (x * ), x * − x k + (4θ − 3)(x k − x k+1 ) − (4θ − 3) x k − x k+1 2 P + 4(1 − θ)γ 2 ∇f (x k ) − ∇f (x * ) 2 P −1 = (x k , s k ) − (x * , s * ) 2 P, M − 2γ − γ 2 /β τ l s k − s * 2 M1 − 2γ s k+1 − s * , q h (s k+1 ) − q h (s * ) − 2γ ∇f (x k ) − ∇f (x * ), x k − x * + γ 2 ∇f (x k ) − ∇f (x * ) 2 P −1 − (4θ − 3) x k − x k+1 − γP −1 (∇f (x k ) − ∇f (x * )) 2 P . Note that − 2γ ∇f (x k ) − ∇f (x * ), x k − x * + γ 2 ∇f (x k ) − ∇f (x * ) 2 P −1 (8) ≤ − (2γ − γ 2 /β) ∇f (x k ) − ∇f (x * ), x k − x * (11) ≤ − (2γ − γ 2 /β)τ f x k − x * 2 P .
Then we have, together with (12),
(x k+1 , s k+1 ) − (x * , s * ) 2 P, M ≤ (x k , s k ) − (x * , s * ) 2 P, M − 2γ − γ 2 /β τ l s k − s * 2 M1 − 2γτ h s k+1 − s * 2 M1 − (2γ − γ 2 /β)τ f x k − x * 2 P . That is x k+1 − x * 2 P + s k+1 − s * 2 (1+2γτ h )M1+M2 ≤(1 − (2γ − γ 2 /β)τ f ) x k − x * 2 P + s k − s * 2 (1−(2γ−γ 2 /β)τ l )M1+M2 .
(29) For the last term on the right hand of (29), we have
s k − s * 2 (1−(2γ−γ 2 /β)τ l )M1+M2 = M 1/2 1 (s k − s * ) 2 (1−(2γ−γ 2 /β)τ l )I+M −1/2 1 M2M −1/2 1 ≤ 1−(2γ−γ 2 /β)τ l +C1 1+2γτ h +C1 M 1/2 1 (s k − s * ) 2 (1+2γτ h )I+M −1/2 1 M2M −1/2 1 = 1−(2γ−γ 2 /β)τ l +C1 1+2γτ h +C1 s k − s * 2 (1+2γτ h )M1+M2
. Therefore, the inequality (28) is proved.
⊓ ⊔ This theorem requires both τ f > 0 and τ h + τ l > 0 for the linear convergence. Thus it does not cover EXTRA, which is the case when f = 0 and h = 0. For the case when f = 0 and h = 0, we have linear convergence when τ l > 0. The result is shown in Theorem 4, while the connection to EXTRA will be explained in details in the next section. For simplicity, we let D = I and P = I.
Theorem 4 Let f = 0, h = 0, D = I, and P = I. Let (x * , s * ) be a fixed point of (7) and assumptions (10) and (13) hold. Define
M := M 1 + 2θ−1 2(1−θ) M 2 , when θ < 1, M 1 ,
when θ = 1,
and we have 1 + (4θ−3)C2 4θ−3+4(1−θ)C1 x k+1 − x * 2 + s k+1 − s * 2 M ≤ρ 2 1 + (4θ−3)C2 4θ−3+4(1−θ)C1 x k − x * 2 + s k − s * 2 M ,(30)
where
C 2 = λ·λmin(AA ⊤ ) 1−θλ·λmin(AA ⊤ )
and
ρ 2 = max 4θ−3+4(1−θ)C1 (4θ−3)(C2+1)+4(1−θ)C1 , 1−(2γ−γ 2 /β)τ l + 2θ−1 2(1−θ) C1 1+ 2θ−1 2(1−θ) C1
.
Here λ min (AA ⊤ ) is the smallest nonzero eigenvalue of AA ⊤ . The sequence {(x k , s k )} converges linearly to the fixed point (x * , s * ) with rate ρ 2 < 1 if γ ∈ (0, 2β) and τ l > 0.
Proof Because x k+1 − x * is in the range of A ⊤ , let x k+1 − x * = A ⊤ u for some u. In addition, let AA ⊤ = VΣV ⊤ be its eigendecomposition with orthonormal V and diagonal Σ. Then we have
γA(x k+1 − x * ) 2 M −1 1 = γAA ⊤ u 2 M −1 1 = λ u 2 AA ⊤ (I−θλAA ⊤ ) −1 AA ⊤ =λ V ⊤ u 2 Σ(I−θλΣ) −1 Σ ≥ λ·λmin(AA ⊤ ) 1−θλ·λmin(AA ⊤ ) V ⊤ u 2 Σ = C 2 u 2 AA ⊤ =C 2 x k+1 − x * 2 .(31)
(1) We consider the case with θ = 1 first. The equation (18) becomes
x k+1 − x * 2 + s k+1 − s * 2 M1 = x k − x * 2 + s k − s * 2 M1 − 2γ s k+1 − s * , ∇l * (s k ) − ∇l * (s * ) − x k − x k+1 2 − s k − s k+1 2 M1 (26) ≤ x k − x * 2 + s k − s * 2 M1 − x k − x k+1 2 − 2γ − γ 2 /β τ l s k − s * 2 M1 − γA(x k+1 − x * ) 2 M −1 1 (31) ≤ x k − x * 2 + (1 − 2γ − γ 2 /β τ l ) s k − s * 2 M1 − C 2 x k+1 − x * 2 .
Therefore, we have
(1 + C 2 ) x k+1 − x * 2 + s k+1 − s * 2 M1 ≤ρ 2 (1 + C 2 ) x k − x * 2 + s k − s * 2 M1 with ρ 2 = max 1 1+C2 , 1 − 2γ − γ 2 /β τ l .
(2) Then we consider the case with θ < 1. The definition of M 2 and (7b) give
x k − x k+1 2 = γA ⊤ s k+1 − γA ⊤ s * 2 = 1 1−θ s k+1 − s * 2 M2 .(32)
From (18) and (32), we have
x k+1 − x * 2 + s k+1 − s * 2 M = x k − x * 2 + s k − s * 2 M − 2γ s k+1 − s * , ∇l * (s k ) − ∇l * (s * ) − 1 1−θ s k+1 − s * 2 M2 − s k − s k+1 2 M (26) ≤ x k − x * 2 + s k − s * 2 M − 1 1−θ s k+1 − s * 2 M2 + s k − s k+1 2 M2 − 2γ − γ 2 /β τ l s k − s * 2 M1 − M 2 (s k+1 − s k ) + γA(x k+1 − x * ) 2 M −1 1 .(33)
In addition, we have
s k+1 − s * 2 M2 ≤ 2 s k+1 − s * 2 M2 + s k − s * 2 M2 − 1 2 s k+1 − s k 2 M2 . (34)
Adding (34) multiplied by 1 2(1−θ) onto (33) gives
x k+1 − x * 2 + s k+1 − s * 2 M + 1 2(1−θ) s k+1 − s * 2 M2 ≤ x k − x * 2 + s k − s * 2 M + 1 2(1−θ) s k − s * 2 M2 − 4θ−3 4(1−θ) s k+1 − s k 2 M2 − 2γ − γ 2 /β τ l s k − s * 2 M1 − M 2 (s k+1 − s k ) + γA(x k+1 − x * ) 2 M −1 1 .(35)
We have the following inequality
− 4θ−3 4(1−θ) s k+1 − s k 2 M2 − M 2 (s k+1 − s k ) + γA(x k+1 − x * ) 2 M −1 1 ≤ − 4θ−3 4(1−θ)C1 M 2 (s k+1 − s k ) 2 M −1 1 − M 2 (s k+1 − s k ) 2 M −1 1 − γA(x k+1 − x * ) 2 M −1 1 − 2 M 2 (s k+1 − s k ), γA(x k+1 − x * ) M −1 1 ≤ − γA(x k+1 − x * ) 2 M −1 1 + 4(1−θ)C1 4θ−3+4(1−θ)C1 γA(x k+1 − x * ) 2 M −1 1 = − 4θ−3 4θ−3+4(1−θ)C1 γA(x k+1 − x * ) 2 M −1 1 ≤ − (4θ−3)C2 4θ−3+4(1−θ)C1 x k+1 − x * 2 .
Then, (35) becomes
(1 + (4θ−3)C2 4θ−3+4(1−θ)C1 ) x k+1 − x * 2 + s k+1 − s * 2 M1 + 2θ−1 2(1−θ) s k+1 − s * 2 M2 ≤ x k − x * 2 + s k − s * 2 M1 + 2θ−1 2(1−θ) s k − s * 2 M2 − 2γ − γ 2 /β τ l s k − s * 2 M1
. Therefore, (30) is proved.
⊓ ⊔
Application in decentralized consensus optimization
In this section, we show that the algorithm (7) recovers PG-EXTRA [19] for decentralized consensus optimization. Then we prove its convergence under a weak condition that is more general than that in [19] and provide an optimal bound for the stepsize. We use the same notation as [19]. The decentralized consensus problem is
minimize x∈R p n i=1 s i (x) + r i (x),
where s i : R p → R and r i : R → (−∞, +∞] are propoer lsc convex functions held privately by node i to encode the node's objective function. The objective of decentralized consensus is minimizing the sum of all private objective functions while using information exchange between neighboring nodes in a network. Here s i has a Lipschitz continuous gradient with parameter L > 0 and the proximal mapping of r i is simple. We let x i be one copy of x kept at node i. These {x i } n i=1 are not the same in general, and we say that it is consensual if they are the same. Stacking all the copies together, we define
x := − x ⊤ 1 − − x ⊤ 2 − . . . − x ⊤ n − ∈ R n×p , and s(x) = n i=1 s i (x i ), r(x) = n i=1 r i (x i ).
Then the decentralized consensus problem becomes
minimize x s(x) + r(x), subject to x 1 = x 2 = · · · = x n .
The gradient of s at x is written in the following matrix form:
∇s(x) := − (∇s 1 (x 1 )) ⊤ − − (∇s 2 (x 2 )) ⊤ − . . . − (∇s n (x n )) ⊤ − ∈ R n×p ,
and · F is the Frobenius norm for a matrix in R n×p . One iteration of PG-EXTRA reads as
z k+1 = z k − x k + I + W 2 (2x k − x k−1 ) − α∇s(x k ) + α∇s(x k−1 ),(36a)
x k+1 = arg min
x r(x) + 1 2α x − z k+1 2 F ,(36b)
where α is the stepsize and W is a symmetric doubly stochastic matrix that represents information exchange between neighboring nodes. Thus I − W is positive semidefinite, and we can find A such that I− W = AA ⊤ . In addition, we assume that Null(A ⊤ ) = Null(I − W) = span(1 n×1 ), which means that A ⊤ x = 0 is equivalent to x 1 = x 2 = · · · = x n . Therefore, the decentralized consensus problem becomes
minimize x s(x) + r(x) subject to A ⊤ x = 0.
Its dual problem, in the form of (1), is
minimize y r * s * (Ay),(37)
where r * and s * are convex conjugate functions of r and s, respectively. We apply (7) to (37) (h ⇒ r * , l ⇒ s * , x ⇒ y, s ⇒ x) and arrive at
z k+1 = (I − λAA ⊤ )x k + λ γ Ay k − λ γ ∇s(x k ),(38a)
x k+1 = arg min
x {r(x) + γ 2λ x − z k+1 2 F },(38b)y k+1 = y k − γA ⊤ x k+1 .(38c)
Combining (38a) and (38c), we get
z k+1 = z k − x k − (I − λAA ⊤ )(2x k − x k−1 ) − λ γ ∇s(x k ) + λ γ ∇s(x k−1 ). (39)
We let λ = 1 2 and γ = 1 2α , then (39) is exactly (36a). Because M = 2γ 2 (I − (1/2)AA ⊤ ) = γ 2 (I+W) is positive definite, we can let
M 1 = M. If {∇s i (x)} n i=1
are Lipschitz continuous with constant L > 0, the other condition for convergence is
γ < 2β ≤ 2 L λ min (M 1 ) = 2γ 2 L λ min (I + W),
where the second inequality comes from
∇s(x) − ∇s(x),x −x ≥ 1 L x −x 2 ≥ 1 L λ min (M 1 ) x −x 2 M −1 1 .
Therefore, we obtain the condition on the stepsize
α = 1 2γ < λ min (I + W)/L.
This is exactly the upper bound in [19].
The previous upper bound is obtained with θ = 1. By letting θ = 3/4 + ǫ with an arbitrary small ǫ > 0, we have M 1 = γ 2 (2I − (3/4 + ǫ)AA ⊤ ) and In addition, the condition that W = I − AA ⊤ being doubly stochastic can be relaxed. The new condition is that M 1 = γ 2 (2I − (3/4 + ǫ)AA ⊤ ) = γ 2 ((5/4 − ǫ)I + (3/4 + ǫ)W) is positive definite. That is 5I + 3W is positive definite. The comparison for both convergence conditions is in Table 1. Since EXTRA [18] is a special case of PG-EXTRA when r(x) = 0, then the results in Table 1 also apply to EXTRA. Note that this is also the stepsize for linear convergence of EXTRA when the functions s(x) satisfy (13) with l * being s. λ(W) stepsize liear convergence [18,19] (−1, 1] α < λ min (I + W)/L α < µgλ min (I + W)/L 2 our result (− 5 3 , 1] α < 3 4 λ min (I + W) + 1 2 /L α < 3 4 λ min (I + W) + 1 2 /L Table 1 The comparison of convergence conditions for EXTRA/PG-EXTRA with respect to the eigenvalues of W and the upper bound of stepsize α. µg is the restricted strongly convex constant of s(x) + 1 4α x 2 (I−W)/2 with respect to x * , which is difficult to find and depends on α. Our result is better than that in [18,19], and it is optimal.
Optimal stepsize
In this subsection, we show that the upper bound of the stepsize α in Table 1 is optimal. We consider a special problem with r(x) = 0 and s(x) = n i=1 1 2 x i − y i 2 . Note that the Lipschitz constant L = 1 in this case. Then the iteration reads as
x k+1 = (I + W)x k − I + W 2 x k−1 − αx k + αx k−1 .
It can be formulated as the following fixed point problem The eigenvalues of dI + (1 + 1 2d )(I + W) increase continuously to infinity as d increases, while α(1 + 1 d ) decreases. Therefore, det(M F + dI) > 0 for all d ≥ 1 implies λ min (I + (1 + 1 2 )(I + W)) > 2α. Thus the step-size α < 3 4 λ min (I + W) + 1 2 is also a necessary condition, which shows that the upper bound given in Table 1 is optimal.
x k x k+1 = M F x k−1 x k , where M F = 0 I − I+W 2 + αI I + W − αI .
Conclusion
In this paper, we extend the primal-dual algorithm in [5,9,16] to solve the problem f (x) + h l(x) and show its convergence under an optimal condition.
The condition for the primal stepsize is the same, and the dual stepsize can be increased by 1/3. We provide an example to show that this condition can not be weaken. This result recovers and is more general than the positive-indefinite linear ALM proposed in [10]. Then we apply this result to decentralized consensus optimization and extend the stepsize in PG-EXTRA/EXTRA for both convergence and linear convergence. For EXTRA, the stepsize for the linear convergence is the same as the stepsize for convergence and larger than the previous result.
where X and S are two Hilbert spaces; f : X → (−∞, +∞], h : S → (−∞, +∞], and l : S → (−∞, +∞] are proper lower semi-continuous (lsc) convex functions; h l is the infimal convolution of h and l that is defined as h l(s) = inf t∈S h(t) + l(s − t); the linear operator A : X → S is bounded. In addition, we assume that f and l * (the conjugate function of l) have Lipschitz continuous gradients and the proximal operator of h, which is defined asprox λh (t) = (I + λ∂h) −1 (t) := arg min s∈S h(s) + 1 2λ s − t 2 , has a closed-form solution or can be easily computed. Let l be the indicator function ι {0} that returns 0 if s = 0 and +∞ otherwise. Its conjugate function is l * (s) = 0. Then the infimal convolution h l degenerates to h, and the problem (1) becomes x * = arg min x∈X f (x) + h(Ax).
Let (x * , s * ) be a fixed point of (7) and Assumptions 1 and 2 hold. Define M := (1 + 2γτ h )M 1 + M 2 , and we have
M 2 = (1/4 + ǫ)γ 2 AA ⊤ . Then a larger upper bound for the stepsize α = 1 2γ ≤ λ min (2I − (3/4 + ǫ)AA ⊤ )/L <λ min (2I − (3/4)AA ⊤ )/L = ((3/4)λ min (I + W) + 1/2)/L, is derived.
A necessary condition for the convergence of the iteration is λ(M F ) ∈ (−1, 1], which implies the determinant det(M F + dI) > 0 for all d ≥ 1 because the dimension of M F is 2n × 2n. Since det(M F + dI)
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| []
|
[
"Ab-Initio Theory of Moiré Superlattice Bands in Layered Two-Dimensional Materials",
"Ab-Initio Theory of Moiré Superlattice Bands in Layered Two-Dimensional Materials"
]
| [
"Jeil Jung ",
"Arnaud Raoux ",
"Zhenhua Qiao ",
"A H Macdonald ",
"\nThe University of Texas at Austin\n78712AustinTexasUSA\n",
"\nNational University of Singapore\n117551Singapore\n",
"\nUMR 8502\nCNRS\nUniv. Paris-Sud\nF-91405OrsayFrance\n",
"\nThe University of Texas at Austin\n78712AustinTexasUSA\n"
]
| [
"The University of Texas at Austin\n78712AustinTexasUSA",
"National University of Singapore\n117551Singapore",
"UMR 8502\nCNRS\nUniv. Paris-Sud\nF-91405OrsayFrance",
"The University of Texas at Austin\n78712AustinTexasUSA"
]
| []
| When atomically thin two-dimensional (2D) materials are layered they often form incommensurate noncrystalline structures that exhibit long-period moiré patterns when examined by scanning probes. In this paper we present an approach which uses information obtained from ab initio calculations performed on short-period crystalline structures to derive effective Hamiltonians that are able to efficiently describe the influence of the moiré pattern superlattices on electronic properties. We apply our approach to the cases of graphene on graphene (G/G) and graphene on hexagonal boron nitride (G/BN), deriving explicit effective Hamiltonians that have the periodicity of the moiré pattern and can be used to calculate electronic properties of interest for arbitrary twist angles and lattice constants. | 10.1103/physrevb.89.205414 | [
"https://arxiv.org/pdf/1312.7723v1.pdf"
]
| 118,364,010 | 1312.7723 | 999c50c6fb4525c1764e18bd32d697c67ff85485 |
Ab-Initio Theory of Moiré Superlattice Bands in Layered Two-Dimensional Materials
Jeil Jung
Arnaud Raoux
Zhenhua Qiao
A H Macdonald
The University of Texas at Austin
78712AustinTexasUSA
National University of Singapore
117551Singapore
UMR 8502
CNRS
Univ. Paris-Sud
F-91405OrsayFrance
The University of Texas at Austin
78712AustinTexasUSA
Ab-Initio Theory of Moiré Superlattice Bands in Layered Two-Dimensional Materials
numbers: 7322Pr7120Gj7115Mb3115aq
When atomically thin two-dimensional (2D) materials are layered they often form incommensurate noncrystalline structures that exhibit long-period moiré patterns when examined by scanning probes. In this paper we present an approach which uses information obtained from ab initio calculations performed on short-period crystalline structures to derive effective Hamiltonians that are able to efficiently describe the influence of the moiré pattern superlattices on electronic properties. We apply our approach to the cases of graphene on graphene (G/G) and graphene on hexagonal boron nitride (G/BN), deriving explicit effective Hamiltonians that have the periodicity of the moiré pattern and can be used to calculate electronic properties of interest for arbitrary twist angles and lattice constants.
I. INTRODUCTION
Since shortly after it was first isolated for electronic property studies in 2004, 1 the graphene family of two-dimensional electron systems has attracted great interest. Recently attention has expanded 2 to include other extremely anisotropic materials, including hexagonal boron nitride 3 (hBN) and transition metal dichalcogenides, 4 and to structures in which combinations of these materials are stacked in various different ways. All these materials share hexagonal lattice structures, and have low-energy electronic states located at momenta near the two-dimensional lattice's Brillouin-zone corners.
Because the lattice constants of these 2D materials differ, and because the hexagonal lattice orientations of different layers are not always identical, multilayer systems usually do not form two-dimensional crystals. For example, the lattice constant of hBN is approximately 1.7% larger than that of graphene. Differences in lattice constant or orientation produce moiré patterns 5 that are apparent in scanning probe studies of electronic properties [6][7][8][9] when graphene is placed on a graphite or hBN substrate. The moiré pattern is responsible for Hofstadter 10 gaps 11-14 that occur within Landau levels when samples are placed in a perpendicular magnetic field.
The period of the moiré pattern is unrelated to true twodimensional crystallinity, which for a given lattice constant difference is present only at discrete relative orientations, and appears to have little relevance for observed properties. The absence of crystallinity nevertheless complicates theoretical descriptions of electronic properties because it removes the simplifications which would otherwise be afforded by Bloch's theorem. This obstacle has forced researchers to proceed either by using simplified tight-binding models, [15][16][17][18][19] or by performing ab initio or tight-binding calculations 14,20,21 for longperiod crystal approximants to real structures. An alternative approach, [22][23][24][25][26][27][28][29][30] is based on the assumption that interlayer tun-neling amplitudes in 2D materials vary slowly on an atomic scale with changes in either initial or final two-dimensional position. When this assumption is valid, it is possible to formulate an effective theory of low-energy electronic structure in which the Hamiltonian is periodic with the periodicity of the moiré pattern, and therefore to employ Bloch's theorem. We refer to models of this type, which seek mainly to describe electronic properties in systems with long moiré period structures over a limited energy range, as moiré band models.
In this paper we extend the moiré band approach, explaining how moiré band models can be systematically obtained from ab initio electronic structure calculations performed only on short-period commensurate multilayer structures. The moiré band Hamiltonian is position dependent and acts on real spin and on orbital, sublattice and layer pseudo spin degrees of freedom. A moiré band Hamiltonian does not account for the presence or absence of commensurability between the underlying lattices. Moiré band Hamiltonians are particularly advantageous for theories of electronic properties in the presence of an external magnetic field for which other more direct approaches are usually not practical. We illustrate our method of deriving moiré band Hamiltonians by applying it to the case of two graphene layers, and to the case of graphene on hBN.
Our paper is organized as follows. In Section II we explain our approach, which can be applied to any system of layered 2D materials in which the local inter-layer stacking arrangement varies slowly on an atomic scale. The parameters of the model can be extracted from ab initio electronic structure calculations by examining the dependence of electronic states on relative displacements between layers in crystalline stacked structures. In Section III we discuss the 2D band structures of crystalline graphene on graphene (G/G) [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] and graphene on hexagonal boron nitride (G/BN). [31][32][33][34][35][36][37] We extract the quantitative values of the small number of parameters which characterize the corresponding moiré band models from these calculations. (As experimental results emerge, the model parame-arXiv:1312.7723v1 [cond-mat.mes-hall] 30 Dec 2013 ters could instead be fit to observed properties if deemed more reliable than ab initio electronic structure calculations.) In Sections III B and III C we describe applications to G/G and G/BN. For G/BN, it is possible to derive a two-band moiré band model which describes only the graphene layer π-bands. We expect that this simplified model, which is explained in Section III D, will be widely applicable to evaluate many physical properties of graphene on BN substrates. Finally in Section IV we summarize our results and briefly discuss some issues which may arise in applying our approach to other 2D material stacks, and in accounting many-body effects that are important for theories of some physical properties.
II. MOIRÉ BAND MODEL DERIVATION
Our method applies to stacks of two-dimensional crystals with the same lattice structure, similar lattice constants, and relative orientation angles that are not too large. It is ideally suited to stacks composed of graphene and hBN layers in arbitrary order with arbitrary orientations, or to group VIB transition-metal dichalcogenide semiconductor stacks. The basic idea is that the lattice representation Hamiltonian,
H lat = ls L|H|l s L ,(1)
depends mainly on the local coordination between layers l and l , and that this dependence can be characterized performing calculations for crystalline structures in which the layers are displaced arbitrarily, but share the same lattice constant and orientation. In Eq. (1) l labels the layers, each of which is assumed to form a 2D crystal, s labels sites within the 2D crystal unit cell, and L labels lattice vectors. If more than one atomic orbital were relevant at each lattice site, as would be the case for transition metal dichalcogenides for example, s would label both site and relevant orbitals on that site. For graphene and hBN we will restrict our attention to the π-bands so we will consider only one orbital per atom.
The moiré band model is defined by matrix elements of H lat calculated in the representation of the 2D Bloch states of the individual layers. Below we first explain the approximation we use for H lat , and then explain how we use it to evaluate Bloch state matrix elements. We will focus on the case of 2D honeycomb lattices so that our discussion applies specifically to the graphene and hBN cases of primary interest. In this paper we focus on the two layer case, and comment on the more general case only in the discussion section.
When the individual layer 2D lattices have the same orientation and identical lattice constants, the overall material is crystalline. In that case we can exploit translational symmetry and solve the electronic structure problem using Bloch's theorem. Using Bloch state completeness properties it is easy to show that
ls L|H( d)|l s L = 1 N ∑ k∈BZ exp(i k · ( L + τ s )) H ls,l s ( k : d) exp(−i k · ( L + τ s )).(2)
Here H ls,l s ( k : d) is the Wannier representation Bloch-band Hamiltonian; we have explicitly indicated that it is a non-trivial function of any rigid displacement d of the top layer with respect to the bottom layer. (We include the displacement in the site positions so that i.e. τ s → τ s + d in the top layer when it is displaced. d is defined to be zero for AA stacking.) Note that ls L|H( d)|l s L is a function only of L − L, and not of L and L separately, and that H ls,l s ( k : d) = H ls,l s ( k : d + L). The geometry of two stacked honeycomb lattices is illustrated in Fig. 1. The moiré band model is intended to provide a low-energy effective model of electronic states for the case in which the top layer lattice is expanded by factor α and rotated counterclockwise by rotation rotation angle θ with respect to the bottom layer lattice. Note that rigid displacements of incommensurate layers lead only to a spatial shift in the moiré pattern and otherwise have no effect on the electronic structure. 22 When we wish to retain the dependence on initial translation, we denote it by τ . For simplicity we first discuss the case in which τ = 0, and later restore the matrix-element phase-factor changes introduced by this initial translation. The shift in lattice positions of the top layer with respect to the original positions can then be expressed in terms of α and the rotation operator R(θ ):
d( L) ≡ αR(θ ) L − L = ((α cos(θ ) − 1)L x − α sin(θ )L y , (α cos(θ ) − 1)L y + α sin(θ )L x ) .(3)
We obtain our moiré band model by approximating the lattice matrix elements of the scaled and rotated structure using Eq. (2) with d replaced by d( L) in Eq. (3). In using this approximation we are assuming that d( L) varies slowly on an atomic length scale, i.e. that θ and α − 1 ≡ ε are small. In this limit pattern formed by rotation and scaling is discussed further in Appendix A.) Once this substitution is made ls L|H( d)|l s L depends on both L and L and not just on L − L. It follows that the Hamiltonian is no longer Bloch diagonal in a momentum space representation. Our local displacement approximation is obviously exact for ε = θ = 0, and we believe that it is accurate over useful ranges of α and θ as discussed further below. We defer further comment on the accuracy of the approximation until we discuss the two explicit examples explored in this article, G/G and G/BN.
d( L) = ε L + θẑ × L.(4)
In order to use this approximation conveniently, we note that H ls,l s ( k : d) is a periodic function of d with lattice periodicity, i.e. that H ls,l s ( k : d) = H ls,l s ( k : d + L). It can therefore be expanded in terms of reciprocal lattice vectors
H ls,l s ( k : d) = ∑ G H ls,l s ( k : G) exp(−i G · d) × exp(−i G( τ s − τ s ) δ ll ).(5)
The phase factor exp(−i G( τ s − τ s ) δ ll ) is included in the definition of the Fourier expansion coefficients in order to make their symmetry properties more apparent, and δ ll = (1 − δ ll ) where l, l are layer indices. We show below that for G/G and G/BN only a few terms in this Fourier expansion are large. As we explain there, we expect this to be a general property of 2D material stacks. H ls,l s ( k : d) can be calculated relatively easily by performing ab initio supercell density-functional-theory (DFT). The number of atoms per unit cell in these calculations is modest, four for example in the cases with two crystal layers with two atoms per cell considered explicitly in this paper. The Fourier coefficients which describe the dependence of H ls,l s ( k : d) on d are obtained by evaluating the inverse Fourier transform numerically:
H ls,l s ( k : G) = 1 A 0 A 0 d d H ls,l s ( k : d) exp(i G · d) × exp(i G( τ s − τ s ) δ ll )(6)
where A 0 is the integration area of a commensurate configuration primitive cell shown in Fig. 1.
We are now in a position to derive our low-energy model. First of all we use Eqs. (3) and (5) to construct the momentum space matrix elements of our model. We assume that each layer is still accurately crystalline and for each layer evaluate matrix elements using Bloch states defined using the twodimensional crystal structure of that layer. Summing independently over the lattice vectors L of layer l and L of layer l and using Eqs.
( k : G) exp(i G · ( τ s − τ s ) δ ll ) ∆( k − k −G)(7)
where ∆( k) = 1 when k is a reciprocal lattice vector and is zero otherwise and In applying this formula we wish to describe electronic states derived from Bloch orbitals close to a particular or several particular points in momentum space. For graphene and hBN we wish to describe states close to the Dirac points K and K . (Below we consider the K Dirac point for definiteness.) We further assume that the interlayer coupling processes re-sponsible for l = l and G = 0 terms in Eq. (7) are small compared to the l = l , G = 0 term, but that they vary with momentum k on the same reciprocal lattice vector scale. These assumptions allow us to replace H ls,l s ( k : G) by its value at the Dirac point when l = l or G = 0 to obtain ls k|H|l s k = δ l,l H ls,l s ( k :
G = ε G − θẑ × G.(8)G = 0)δ k, k + ∑ G =0 H ls,l s ( K : G) ∆( k − k −G) + δ l,l ∑ G H ls,l s ( K : G) exp(i G · ( τ s − τ s )) ∆( k − k −G)(9)
where the first and second lines are respectively the intralayer and interlayer terms.
As we see in Eq. (9) the Hamiltonian is constructed as the sum of several contributions: i) an isolated layer twodimensional band structure obtained by averaging over displacements d, ii) a sublattice pseudo-spin dependent term which acts within layers and accounts for the influence of nearby layers on-site-energies and hopping within layers and iii) an inter-layer tunneling term which is also strongly dependent on the local stacking arrangement. The first term in this equation reduces to the isolated layer Hamiltonian when interlayer coupling is absent. In fact, according to our numerical calculations, the difference between this term and the isolated layer Hamiltonian is negligible for both G/G and G/BN cases.
The second and third terms are off diagonal in momentum and therefore account for the moiré pattern which breaks translational symmetry. As we illustrate below, using G/G and G/BN as examples, useful models can be obtained with a small number of independent H ls,l s ( K : G) parameters, partly because of symmetry. The values of the parameters can be evaluated by using ab initio electronic structure calculations (the approach we follow in this paper) to examine the relative displacement d dependence of the electronic structure of two-dimensional layers that have a common Bravais lattice. Since the basic premises of DFT theory are reasonably reliable for graphene and hBN, the resulting model is expected to capture all qualitative electronic structure features associated with the moiré pattern. More accurate simple models might eventually be achievable by using this approach to construct similar phenomenological models with parameters derived from experimental observations. We illustrate the power of this simple formula below by applying it to the G/G and G/BN cases.
III. AB-INITIO MOIRÉ-BAND-MODELS
A. Electronic Structure Calculations
We calculate our moiré band parameters starting from Wannier-function lattice representations of bilayer perfect crystal Hamiltonians. In this section we present a brief summary of the first-principles methods employed to obtain the Wannier-function representation Hamiltonian matrices, and discuss some qualitative aspects of the perfect crystal bands of G/G and G/BN that hint at important moiré band properties. Our microscopic calculations were performed for twolayer systems with four atoms per unit cell. We used the software package Quantum Espresso 38 that is interfaced with the package wannier90. 39,40 The calculations were performed using a 42 × 42 k-point sampling density, an energy cutoff of 60 Ry, vonBarth-Car norm conserving pseudopotentials, and the Perdew-Zunger LDA parametrization. (C,B,N.pzvbc.UPF) The same k-point sampling density was maintained for the Wannier representation construction of the Hamiltonian projected to 10 localized orbitals, 6 corresponding to the σ bonds and 4 to p z orbitals centered on the four atoms. The convergence criteria used for self-consistent total energy in the DFT calculations was 10 −9 eV per unit cell.
Although mirror symmetry is broken in bilayers for general d, the coupling between π and σ bands is always weak because their energy separation near the Dirac point is ∼ 10 eV, and large compared to coupling matrix elements that are always smaller than 0.1 eV. 42 We therefore retain only the π-electron degrees of freedom in our moiré band models. Because there is only one p z orbital per carbon atom, the Wannier-representation Hamiltonians discussed below are 4 × 4 matrices with row and column indices that can be labelled by the four sites in a two-layer crystal. We characterized the dependence on the relative displacement between the layers by performing calculations on a d-sampling grid with 21 × 36 points in the a × √ 3a area plotted in Fig. 2. We have chosen a coordinate system in which graphene's triangular Bravais lattice has primitive lattice vectors
a 1 = a(1, 0), a 2 = a − 1 2 , √ 3 2 ,(10)
where a = 2.46Å is the lattice constant of graphene. The corresponding primitive reciprocal lattice vectors are
b 1 = 2π a (1, 1 √ 3 ) , b 2 = 2π a 0, 2 √ 3 .(11)
The A and B sublattice positions in the bottom layer are
τ A = (0, 0, 0), τ B = (0, a √ 3 , 0).(12)τ A = (d x , d y , c), τ B = (d x , a √ 3 + d y , c).(13)
In Eq. (13) c is the layer separation that we assume to be constant. Results for geometries where the out of plane z direction coordinate is relaxed as function of d are discussed in Appendix B.
In Fig. 2 we plot results for the dependence of total energy and the band gap at the Dirac point on displacement d for both G/G and G/BN. In the G/BN case, the difference in lattice constant between graphene and hBN layers will play an essential role. The bulk lattice constant of graphite is a G = 2.461Å whereas a hBN = 2.504Å, implying a difference of about 1.7%. For the commensurate calculations summarized in Fig. 2 we used the self-consistent LDA lattice constant of single layer graphene a G = 2.439Å for both graphene and boron nitride sheets. Notice that for G/BN there is a gap at the Dirac point at any value of d.
B. Graphene on Graphene
Moiré Band Model
In Fig. 3 we illustrate the dependence on d of both interlayer and intra-layer values of H ls,l s ( K : d) for the case of two-coupled graphene layers. The intra-layer parameters are typically ∼ 5 meV in the graphene case and do not play an essential role in the moiré bands; we will see later that their role is much more essential in the graphene on boron nitride case. The inter-layer coupling at the Dirac point is larger and more strongly dependent on d. As we now explain, a single real parameter is sufficient to accurately describe the full d dependence of the four complex inter-layer coupling matrix elements. The vast simplification is related to the smooth variation of inter-layer coupling on d, which is related 22 in turn to the fact that the distance between layers of these van der Waals coupled two-dimensional materials is substantially larger than the separation between atoms within a layer. We find that for graphene on graphene the only large corrections to the isolated layer Hamiltonian are for inter-layer tunneling, and that these are large at G = 0 and at two non-zero values of G, that they are real, and that they are identical in the three cases. The interlayer parameters do not have the symmetry of the reciprocal lattice because they are evaluated at the Brillouin-zone corner Dirac point, rather than at the zone center. The three G's which yield large parameters share the minimum value of | K + G|. The entire inter-layer coupling part of the Hamiltonian is accurately captured by a single real parameter with the value 0.113 ± 0.001 eV. We now explain the physics behind this seemingly surprising simplification. Our low energy model is naturally employed in combination with a continuum model in which wave vectors are measured from the Dirac point. The condition that k = k +G then translates into the condition where j = 0, ± and the indices correspond to the three G's for which H ls,l s ( K : G) is large: G = (0, 0) and
Gy (2⇡/a) (eV ) Gx (2⇡/a) Re[H AA 0 ,G ] (eV ) Re[H AA 0 ,G ] Gy (2⇡/a) −2 0 2 −2 0 2 0 0.02 0.04 0.06 0.08 0.1 −6 −4 −2 0 2 4 6 −0.05 0 0.05 0.1 0.15 0.2 G x =−2 G x =−1 G x =0 G x =1 G x =2q = q + K − K +G = q + Q j (14) 1 2 3 4 5 6 G 0 G + G K 0 K +K Q Q + Q 0 A B C 1 3 5 2 4 6 0 G e G = ✓ẑ ⇥GG ± = 4π √ 3a (− √ 3 2 , ± 1 2 ) = K(− 3 2 , ± √ 3 2 ).(15)
Here K is the magnitude of the Dirac wave vector. Taking account of the difference between the rotated and unrotated system reciprocal lattices we find that to first order in ε and θ
Q j = ε K j − θẑ × K j(16)
where K j = K + G j . Note that, independent of the values of θ and ε, the three vectors Q j have the same magnitude K √ ε 2 + θ 2 and that they are related by 120 • rotations. In the graphene case the parameter ε that accounts for the difference in lattice constant between the layers is equal to zero, but we retain it here because of the close similarity between the G/G interlayer hopping terms and the G/BN cases discussed below. When momenta are measured from the Dirac point, a state in one-layer is coupled to states in the same layer separated in momentum space by moiré pattern reciprocal lattice vectors, and to states in the opposite layers separated by moiré pattern reciprocal lattice vectors ± Q j . (See Fig. 5.) For G/G the intra-layer contribution to the Hamiltonian is negligible for G = (0, 0), and for G = (0, 0) and k = K its dependence on site labels is proportional to a unit matrix. It follows that the G = (0, 0), k = K Hamiltonian can be set to zero by choosing the zero of energy appropriately. The dependence of the G = (0, 0) interlayer Hamiltonian on k satisfies the same symmetry requirements as the isolated layer Hamiltonian. We have found that for both G/G and G/BN cases, the difference between the G = (0, 0) interlayer Hamiltonian and the isolated layer Hamiltonian is negligible. x 10 When only the largest non-zero interlayer coupling terms are retained, the Hamiltonian is the sum of three terms, each a product of a coupling constant t bt = 113meV, a momentum boost factor δ q , q+ Q j and a sublattice dependent factor
−4 G x =−2 G x =−1 G x =0 G x =1 G x =2 −6 −4 −2 0 2 4 6 −2 −1 0 1 2 3 4 x 10 −3 G x =−2 G x =−1 G x =0 G x =1 G x =2 Re[H AA, G ] Im[H AA, G ] Gy (T j s,s = t bt exp(i G j · ( τ s − τ s − τ)),(17)
where we have restored the phase change due to the translation τ prior to rotation. When the momentum boost operator is written in real space it it local and has a plane-wave spatial dependence. We therefore obtain a Hamiltonian with a space-dependent inter-layer coupling Hamiltonian that has a sub lattice pseudo-spin dependence:
H bt ( r) = ∑ j exp(−i Q j · r) T j s,s(18)
where
T j = t bt exp(−i G j τ) 1 exp(−i jφ ) exp(i jφ ) 1(19)
and φ = 2π/3. A similar formula was derived previously starting from ad hoc π-band tight-binding models 15,22 . (Note that in Ref. [22] the initial displacement τ was defined relative to AB stacking.) This position-dependent inter-layer tunneling can be understood in terms of local interlayer coordination which varies with the moiré periodicity between AA, AB and intermediate arrangements.
Here we demonstrate by explicit first-principles calculations that this model for twisted layer electronic structure is quite accurate. Our ab initio calculations give rise to a coupling constant of t bt = 113meV, nearly identical to the value t bt = 110meV estimated previously by fitting tight-binding models to the experimentally known Dirac point spectrum of bilayer graphene. In the phenomenological tight-binding model context, the applicability of this model was justified on the basis of the argument 19 that any reasonable inter-layer tunneling ansatz yields a dependence on two-dimensional position that is smooth at atomic scale. Our microscopic calculations free us from an ad hoc tight-binding model and confirm the expected smoothness. The success of the ad hoc tight-binding model in describing interlayer tunneling effects may be traced to the property that only one number, namely t bt , is important for the lowenergy electronic structure. Any microscopic model which is adjusted so that it gives an appropriate value for t bt will yield similar predictions. The explicit form of the d-dependent inter-layer Hamiltonian which retains only the single strong moiré band model parameters follows from Eq. (5):
H AA ( K : d) = H BB ( K : d) = (20) = t bt 1 + exp(−i G + · d) + exp(−i G − · d) H AB ( K : d) = t bt 1 + exp(−iφ ) exp(−i G + · d) + exp(iφ ) exp(−i G − · d) H BA ( K : d) = t bt 1 + exp(iφ ) exp(−i G + · d) + exp(−iφ ) exp(−i G − · d)
Although they are relatively weak in the G/G case, for completeness we specify the leading intralayer terms as well. The Fourier transforms of the intra-layer matrix elements have the same magnitude within a shell of reciprocal lattice vectors. Including the first shell only, we obtain the following parametrization of the d-dependent intra-layer Hamiltonian matrix elements:
H ii ( K : d) = C 0ii + 2C ii Re[ f ( d) exp[iϕ ii ]],(21)H i j ( K : d) = g(C i j , ϕ i j ), (i = j) where f ( d) = exp[−iG 1 d y ] + 2 exp[i G 1 d y 2 ] cos( √ 3 2 G 1 d x ),(22)G 1 = 4π/ √ 3a.
The matrix elements labelled by ii = AA, BB, A A , B B , are the d-dependent site energies. The matrix elements labelled by AB and A B describe inter-sublattice tunneling at the Dirac point within the layers. In Eq. (21)
g(C, ϕ) = 2C cos √ 3G 1 2 d x cos G 1 2 d y − ϕ (23) − 2C cos (G 1 d y + ϕ) − i2 √ 3C sin √ 3G 1 2 d x sin G 1 2 d y − ϕ .
Using the numerical labels for the G vectors in Fig. 5 H ii,
G 1 = H ii, G 3 = H ii, G 5 = C ii exp(iϕ ii ),(24)H ii, G 2 = H ii, G 4 = H ii, G 6 = C ii exp(−iϕ ii )
for the diagonal terms, and
H A ( ) B ( ) , G 1 = H * A ( ) B ( ) , G 4 = C A ( ) B ( ) exp(i(−ϕ AB − π)),(25)H A ( ) B ( ) , G 3 = H * A ( ) B ( ) , G 2 = C A ( ) B ( ) exp(i(−ϕ AB + π/3)), H A ( ) B ( ) , G 5 = H * A ( ) B ( ) , G 6 = C A ( ) B ( ) exp(i(−ϕ AB − π/3))
for the off diagonal terms. For graphene on graphene, the expansion coefficients satisfy the symmetry properties
H AA, G = H * BB, G = H * A A , G = H B B
, G for the diagonal terms and H AB, G = H A B , G for the off diagonal terms, C 0AA = C 0BB , and ϕ AB = ϕ A B = 0. The numerical values of the nonzero parameters that define the model for G/G are
C AA = C B B = 1.10 meV, ϕ AA = ϕ B B = 82.54 • , (26) C BB = C A A = C AA , ϕ BB = ϕ A A = −ϕ AA , C AB = 2.235 meV . The sublattice site-energy difference 2H z ( K : d) = H AA ( K : d) − H BB ( K : d)
vanishes for AA stacking and reaches its maximum value ∼12 meV for the AB stacking configuration. This value is in reasonable agreement with the ∼15 meV site-energy difference estimated elsewhere for AB stacked bilayer graphene. 42 It will be interesting to see if these relatively small terms which are normally neglected in two-layer graphene systems have any observable consequences. Eq. (26) also implies spatial variations of the average site-energy H 0 ( K : d) = (H AA ( K : d)+H BB ( K : d))/2 that are smaller than 1 meV. These variations will tend to drive small charge transfers between different parts of the moiré pattern, but their role is not especially important because of their small value.
First shell approximation for commensurate AB, AA limits
In the following we test the single-parameter moiré band model in which the inter-layer Hamiltonian is truncated at the first shell of its Fourier expansion by applying it to the crystalline AA and AB stacking limits. In the crystalline limit, d is independent of position and Q j = (0, 0) for j = 0, ±. In the AA stacking configuration, interlayer coupling is maximized because carbon atoms in different layers sit exactly on top of each other. Mirror symmetry leads to layer-symmetric and layer-antisymmetric copies of the single-layer Dirac spectrum. With our conventions, AA stacking corresponds to (0, a/ √ 3)) = 354 meV, with all other interlayer coupling elements vanishing. (The small difference of 7 meV with respect to the calculation for AB bilayer graphene presented in Ref. [42] is due to the slightly smaller in-plane lattice constants used here.) These comparisons demonstrate that the truncated Fourier expansion single-parameter model yields matrix elements that are are typically inaccurate by ∼ 15 meV, or by around 5% in relative terms. We emphasize that the truncation at the first shell in the Fourier expansions is not essential to our approach, but is attractive because it yields a model that is specified by a single parameter. The approximate band structures obtained using the above interlayer coupling matrices are compared against the first principles LDA bands in Fig. 8.
G x =−2 G x =−1 G x =0 G x =1 G x =2G x =−2 G x =−1 G x =0 G x =1 G x =2
Application to twisted bilayer graphene
For G/G we have ε = 0 so that the moiré pattern reciprocal lattice vectors are related to the honeycomb reciprocal lattice vectors by
G = −θẑ × G.(27)
The Hamiltonian matrix for a given wave vector k in the moiré Brillouin-zone (MBZ) can be constructed using Eq. (9). The momentum boost operators in the inter-layer Hamiltonian terms connect states whose momenta differ by Q j , while those in the intra-layer Hamiltonian terms connect states whose momenta differ by G. From Eq. (16) the explicit expression for 3) bridge stacking. The electronic structure for BA stacking is identical to AB stacking. We find excellent agreement between the direct and approximate calculations, demonstrating the accuracy of the first shell approximation for interlayer coupling. The intralayer tight-binding Hamiltonian uses the models in Refs. [41,42] with the experimental lattice constants of a = 2.46Å whereas the interlayer coupling is given by the first shell approximation as parametrized in Eq. (20). The small differences can be attributed mainly to the approximations involved in the first shell approximation for describing the interlayer coupling. FIG. 9: Moiré band structure and density of states of two graphene layers for four different relative orientation angles. Our results are similar to those obtained in Refs. [22,23]. We plot the band structure as a function of momentum along the straight lines in k-space connecting points A, B, C and A in Fig. 5. The accompanying densityof-states plots demonstrate the complex influence of interlayer coupling, which is responsible for many van Hove singularities.
A B C A E (eV ) E (eV ) A B C A A B C A A B C A D(E) D(E) D(E) D(E) ✓ = 1 ✓ = 2 ✓ = 5 ✓ = 10
the Q j 's is
Q 0 = θ K(0, −1)(28)Q + = θ K(− √ 3 2 , 1 2 ) Q − = θ K( √ 3 2 ,1 2 )
.
For every k in the MBZ we can construct matrices with 2 × 2 sub-lattice blocks. The isolated layer Dirac Hamiltonian contributes blocks that are diagonal in wave vector and layer. The blocks that account for tunneling from bottom to top layers involve momentum boosts by Q j whereas those that connect the same layer involve momentum boosts by a moiré pattern reciprocal lattice vector. Since the Q j 's change sign with tunneling direction, and the difference between any pair of Q j 's is a moiré pattern reciprocal lattice vector (see Fig. 5), the crystal momentum defined by the moiré pattern periodicity is a good quantum number. For every k in the moiré pattern Brillouin-zone, a finite matrix can be constructed by cutting off the plane-wave expansion.
The moiré bands obtained by diagonalizing the matrix constructed in this way are plotted in Fig. 9. We find bands that are similar to those described in Refs. (22,23) in which moiré bands were derived from phenomenological tight-binding models rather than form ab initio DFT calculations. The close agreement is expected since both models are accurately approximated by a model with a single interlayer tunneling parameter, as explained above. We now turn to the G/BN case in which the layer coupling effects are more complex. There we will see that our approach, which provides a route to build an effective model based on DFT bands, has distinct advantages over a purely phenomenological approach.
C. Graphene on Boron Nitride
Moiré Band Model
The crystalline lattices we used to derive the moiré band model parameters for G/BN were identical to those used for the G/G case, except that the bottom layer was changed from graphene to hBN. Because hBN has a slightly larger lattice constant than graphene, the moiré pattern reciprocal lattice vectors are in this case given by the more general expression (Eq. (8)) which accounts for both dilation and twist. The moiré pattern Brillouin zone therefore continuously changes its orientation as a function of twist angle θ as we illustrate in Fig. 10.
In order to capture the local coordination dependence of the electronic structure we have evaluated Wannier-representation bands over the complete range of inter-layer displacement d values. The dependence on d of Dirac point matrix elements is summarized in Fig. 11. As in the G/G case, these ab initio results provide the chemical information that we use to construct a moiré-band model that can account for the lattice constant difference between graphene and hBN, and for the layer orientation difference of particular bilayers. Because these tightbinding model parameters are smooth functions of d, we can represent them in the moiré band model by a small number of parameters. In Figs. 12-14 we plot moiré band parameters obtained from the information in Fig. 11 using Eq. (6). We find that the k-dependence of the Hamiltonian (retained only in the G = (0, 0) moiré band Hamiltonian term calculated by averaging the Wannier representation Hamiltonian over d) is accurately captured by the Dirac form in both graphene and BN layers.
The interlayer hopping physics is quite similar in the G/G and G/BN cases. By examining Fig. 11 and comparing with the previous G/G discussion, we conclude that as for G/BN three Fourier coefficients dominate and yield a simple transparent model. Including only these coefficients, we obtain for G/BN
where
T j = exp(−i G j τ) t BC t BC exp(−i jφ ) t NC exp(i jφ ) t NC .(30)
In this case there are two distinct interlayer tunneling parameters which have the values t BC = 144 meV and t NC = 97 meV. The notation is suggested by comparing these moiré band matrix elements with those constructed from ad hoc microscopic tight-binding models. 35 The difference between the boron to carbon and nitrogen to carbon hopping parameters, t BC and t NC , is not unexpected since p z orbitals centered on the boron sites should have larger atomic radii than p z orbitals centered on the larger Z nitrogen sites. The remaining large contributions to the G/BN moiré band model are absent for G/G and are discussed below.
In the G/BN case, coupling between layers is responsible not only for interlayer tunneling but also for substantial changes within the individual layers. From our microscopic calculations we find that the carbon site energy parameter is large for the first shell of reciprocal lattice vectors. The momentum space pattern is clearly quite different from that of the inter-layer hopping processes. The intralayer Hamiltonian matrix elements can be presented using the same formulas as in Eq. (21) used earlier for the G/G case. What is different in the G/BN case is that our moiré band model is specified by two interlayer tunneling parameters and by 12 intralayer parameters. The values of the 12 intralayer coefficients are listed below: The Fourier expansion coefficients of the Hamiltonian can be related with the above set of parameters through the same Eqs. (24,25) used in the graphene on graphene case.
C 0 AA = 3.332 eV, C 0 BB = −1.493 eV,(31)
First shell approximation for commensurate AA, AB and BA limits
In the following we apply our model Hamiltonian to crystalline AA, AB and BA stacking limits using the same lattice constant for both graphene and hBN. We proceed with our analysis in a manner similar to the G/G case. The first shell approximation for the commensurate interlayer coupling Hamiltonian closely follows the G/G case, except that it is necessary to distinguish the parameters for tunneling from the boron site and the nitrogen atom site. From Eq. (30) for the AA ( τ AA = (0, 0)), AB ( τ AB = (0, a/ √ 3)) and BA ( τ AB = (0, 2a/ √ 3)) stacking configurations we obtain: which imply relative differences smaller than 2% for the main tunneling terms.
H bt ( K : τ AA ) = 3 t BC 0 0 t NC ,(32)H bt ( K : τ AB ) = 3 0 0 t NC 0 , H bt ( K : τ BA ) = 3 0 t BC 0 0 .G x =−1 G x =0 G x =1 G x =2 −6 −4 −2 0 2 4 6 −0.05 0 0.05 0.1 0.15 0.2 G x =−2 G x =−1 G x =0 G x =1 G x =2
D. Effective low energy model for G/BN
Our model for the electronic structure of graphene on hBN can be further simplified by formulating a version which acts only on the low-energy degrees of freedom within the carbon layers. We expect that this version of our model will be broadly applicable to describe electronic properties of graphene sheets that are weakly influenced by a hBN substrate. As we see, the influence will tend to be stronger when the orientation angle difference between graphene and hBN layers is small. In this approach we integrate out the boron nitride layer degrees-of-freedom to obtain a two-band model for graphene.
When written in terms of 2 × 2 blocks, the four-band model is given for each d by
H f ull = H BN T BN,G T G,BN H G(33)
where the entries in this matrix are 2×2 matrices that map sub lattices to sub lattices. We choose the zero of energy at the carbon site energies of the graphene layer. The effective Hamiltonian for graphene obtained by integrating out the boron nitride orbitals is
H = H G − T G,BN H −1 BN T BN,G .(34)
This expression is valid to leading order in an expansion in powers of the ratio of interlayer tunneling amplitudes to the hBN gap ∼ t BN /(C 0AA −C 0BB ).
In this two-band model we can identify four different physical effects of the hBN substrate: i) There is a ddependent difference between the two carbon site energies in while the imaginary part is proportional to the coefficient of σ y . iv) The final contribution to the effective model is due to virtual occupation of hBN sites and captured by the second term on the right hand side of Eq. (34). The full effective Hamiltonian can be expanded in terms of Pauli matrices to yield an intuitive representation of the Hamiltonian's sublattice dependence. The term proportional to the identity matrix can be viewed as a potential term, the term proportional to σ z as a mass term, and the terms proportional to σ x and σ y as gauge potentials which account for substrate-induced bonding distortions. Virtual processes contribute to all the effectivemodel matrix elements discussed above.
G x =−2 G x =−1 G x =0 G x =1 G x =2G x =−1 G x =0 G x =1 G x =2 −2 0 2 −2 0 2 −5 0 5 x 10 −3 Im[H BB,G ] Re[H BB,G ] −6 −4 −2 0 2 4 6 −1 0 1 2 3 4 x 10 −3 G x =−2 G x =−1 G x =0 G x =1 G x =2G x =−1 G x =0 G x =1 G x =2 −2 0 2 −2 0 2 −5 0 5 x 10 −3 Re[H A 0 A 0 ,G ] Im[H A 0 A 0 ,G ]G x =−1 G x =0 G x =1 G x =2 −2 0 2 −2 0 2 −3 −2 −1 0 1 2 3 x 10 −3 Im[H B 0 B 0 ,G ] Re[H B 0 B 0 ,G ]G x =−1 G x =0 G x =1 G x =2 Im[H A 0 B 0 ,G ] Re[H A 0 B 0 ,G ]G x =−2 G x =−1 G x =0 G x =1 G x =2G x =−2 G x =−1 G x =0 G x =1 G x =2
The microscopic origin of the mass term can be traced to the difference in electronegativity between nitrogen and boron which both leads to differences in charging, and modifies the in-plane sigma bonds. Both effects lead to a mass term in the Hamiltonian that is proportional to σ z . The nitrogen (boron) is negatively (positively) charged. Because the interlayer distance is large, one can crudely approximate the resulting Hartree potential by a Coulomb potential with an effective charge Ze (−Ze) with 0, Z < 1 acting on the carbon atom just on top of it. This picture has been explored from a phenomenological point of view 35 and gives rise to a mass contribution to the Hamiltonian which is qualitatively similar to the one derived here from first principles.
We construct the moiré band Hamiltonian by letting d →
G x =−2 G x =−1 G x =0 G x =1 G x =2 Re[H 0 ] Im[H 0 ]H ss = H 0 ss + H MB ss(35)
where H 0 ss is the non-local Hamiltonian which describes the Dirac cones,
H 0 ss = H s,s ( k : G = 0)δ k, k(36)
and H MB ss is the term which captures the moiré band modulation:
H MB ss = ∑ G =0 H s,s ( K : G) ∆( k − k −G).(37)
This model can be viewed as the Hamiltonian of graphene subject to external periodic pseudospin-dependent potentials represented in a Fourier expanded form as a sum in the G lattice vectors of the moiré reciprocal lattice. The form of the Hamiltonian is informed by first principles calculations that account not only for the variation in carbon layer site-energies with local coordination, but also for variations in inter-carbon hopping and for virtual hopping between graphene and boron nitride layers. As we will show shortly, thanks to the smooth displacement dependence of the d-dependent effective Hamiltonian, the moiré patterns of the pseudospin fields are accurately captured by three pairs of parameters, one pair for each pseudospin effective field component. These generalized superlattice potentials determine the quasiparticle velocity and gaps in the moiré superlattice band structure. 43 Our model provides a simple and accurate starting point from which we can calculate the electronic structure of graphene superlattices subject to moiré patterns of the pseudospin fields shown in Figs. 15 and 16.
G x =−1 G x =0 G x =1 G x =2 −2 0 2 −2 0 2 −1 −0.5 0 0.5 1 x 10 −3 −6 −4 −2 0 2 4 6 −2 −1 0 1 2 3 4 x 10 −3 G x =−2 G x =−1 G x =0 G x =1 G x =2 Re[H z ] Im[H z ]
Our numerical results for the Fourier expansion coefficients of the effective model matrix elements are summarized in Figs. [17][18][19]. Once again the expansion coefficients are dominated by the first shell of G's, and the number of independent coefficients is reduced by symmetry. We find that
H 0 ( K : d) = 2C 0 Re[ f ( d) exp[iϕ 0 ]], H z ( K : d) = 2C z Re[ f ( d) exp[iϕ z ]](38)H AB ( K : d) = 2C AB cos( √ 3 2 G 1 d x ) cos G 1 d y 2 − ϕ AB + sin G 1 d y 2 − ϕ AB − π 6 + 2C AB sin G 1 d y + ϕ AB − π 6 + i 2C AB sin( √ 3 2 G 1 d x ) cos G 1 d y 2 − ϕ AB − sin G 1 d y 2 − ϕ AB − π 6 .
The site-independent term H 0 gives rise to an overall potential shift in the graphene layer depending on the local stacking order. The pseudospin in-plane terms H x and H y together with and the mass term H z , are the coefficients of σ x , σ y , and σ z in the local 2 × 2 moiré band Hamiltonian. The in-plane pseudospin terms H x and H y , can be viewed as a gauge fields A, 44 that shift the Dirac cone band edges away from the original position. Together these coefficients determine the local Dirac point gap through the relation
∆( r) = 2 H 2 x ( r) + H 2 y ( r) + H 2 z ( r).(39)
This local Dirac point gap is not directly related to the overall gap of the moiré pattern because of the non-locality of the momentum-dependent isolated layer Dirac Hamiltonian. (We also expect that the gap will be strongly influenced by manybody effects.) We see in Fig. 16 that the Dirac point gap is everywhere at least 30 meV because H x , H y and H z do not vanish simultaneously. Our effective model is completely specified by six numbers:
C 0 = −10.13 meV, ϕ 0 = 86.53 • ,(40)
C z = −9.01 meV, ϕ z = 8.43 • , C AB = 11.34 meV, ϕ AB = 19.60 • .
As described in detail for the G/G case, wave vector reduced to the moiré Brillouin-zone is a good quantum number for this model, and band eigenstates may be obtained by making plane-wave expansions. The graphene layer Dirac Hamiltonian contributes to diagonal blocks in the plane-wave representation of the moiré band Hamiltonian. The Fourier expansion of the Hamiltonian in G vectors can be related to the above parameters through Eqs. (24) for the diagonal terms, either in the pseudospin or sublattice basis, and for the offdiagonal terms shown in Fig. 19 we have the following form
H AB, G 1 = H * AB, G 4 = C AB exp(i(2π/3 − ϕ AB )),(41)H AB, G 3 = H * AB, G 2 = C AB exp(−iϕ AB ), H AB, G 5 = H * AB, G 6 = C AB exp(i(−2π/3 − ϕ AB )).
The applicability of the effective model is evidenced by its accuracy in describing the band structure for the commensurate stacking arrangements shown in Fig. 20. In these plots local potential fluctuations due to H 0 are manifested by a small offset between the graphene and hBN bands. In the presence of a finite twist angle, the H 0 term leads to an effective potential that varies in space as shown in Fig. 21. These potentials variation on the Moire pattern scale leads to the local densityof-state variations seen experimentally. 9 Even when a graphene sheet on a hBN substrate is globally neutral the charge density will vary locally. The regions within the moiré pattern in which positive and negative charge densities are expected can be identified by neglecting the non-local Dirac Hamiltonian (which vanishes at the Dirac point) and the H x and H y sublattice coupling terms. In this limit charge puddles should be expected wherever the chemical potential, set by imposing global charge neutrality, lies below the lower sublattice site energy or above the upper sub lattice site energy. Since the chemical potential at neutrality is very close to the average site energy, which we have chosen as the energy zero, this condition for the formation of charge puddles is equivalent to |H 0 ( r)| > |H z ( r)|, with the carrier type being electrons if H 0 > H z and holes if H 0 < H z . In Fig. 22 we apply this criterion to obtain a map of electron and hole puddles from the parametrization for H 0 and H z presented in Eq. (38). To obtain a more quantitatively accurate map it will be necessary to restore the H x and H y and to take into account other effects such as the electrostatic screening and many-body corrections. Lattice relaxations, whose influence is discussed in the appendix, will also play a role.
IV. SUMMARY AND DISCUSSION
We have presented a method which can be used to derive approximate electronic structure models for layered semiconductors, semimetals, and gapless semiconductors containing a finite number of two-dimensional crystals with different lattice constants and/or different crystal orientations. The method is intended to be useful for multilayer graphene systems, multilayer transition metal dichalcogenide systems, and for multi-layer systems containing both graphene and boron nitride. When several layers are present simultaneously, structures of this type are not in general two-dimensional crystals, and electronic structure theory can therefore be awkward to apply directly because Bloch's theorem is not valid.
Our approach focuses on the influence on the electronic structure of slowly varying relative displacements d( L) between individual crystalline layers due to a small difference in lattice constants or crystal orientations. The dependence of electronic structure on d can be calculated without experimental input using density-functional-theory. Our analysis produces a moiré band model which is periodic under translations R for which
d( L + R) = d( L) + L(42)
for some two-dimensional lattice vector L . The system is microscopically crystalline only if the vectors R for which Eq. (42) is satisfied are lattice vectors of the two-dimensional crystal. The vectors R are the lattice vectors of the moiré pattern. Like the moiré pattern itself, 7 our moiré band models have a periodicity defined by spatially varying layer alignment, and can therefore be analyzed using Bloch's theorem for a superlattice Hamiltonian that has the periodicity of the moiré pattern. The models consists of massless or massive Dirac models for each two-dimensional layer, combined with a spatially local effective potential which acts on sublattice degrees of freedom.
Our approach to coupled bilayer systems has three main limitations. First of all it does not apply to cases in which differences in lattice constants or rotation angles between adjacent layers are large. This limitation can be overcome, however, by building a theory that is based on larger unit cells with more sub lattice sites and lattice constant ratios between neighboring layers closer to one. For example, for G/G one could for example build models that are similar to the ones discussed here which would apply at rotation angles close to the short-period commensurate rotation angles. Secondly, because it attempts to describe bands over a relatively small part of the Brillouin-zone, it is valid over a limited energy range. Finally, it assumes that the individual layers are indeed crystalline whereas we should in fact expect that the moiré pattern will induce small structural distortions within each layer. For the van der Waals epitaxial systems of interest, however, it seems reasonable to expect these distortions to be small and to neglect them, at least as a first approximation.
We have applied our moiré band method to two different two-layer systems, one with two graphene layers and one with a graphene layer and a hexagonal boron nitride layer. For the case of graphene on boron nitride, which has a large energy gap, we have also derived a simpler model, specified by Eqs. (35,40), in which the boron nitride degrees of freedom are treated perturbatively to obtain an explicit model for graphene on a boron nitride substrate which retains only the graphene π-electron degrees of freedom. In the case of graphene on graphene our calculations explain why the dependence on relative orientation angle of G/G electronic structure is accurately described by a model with a single-interlayer tunneling parameter. For the case of graphene on boron nitride, the models we produce are more complicated because of the need to account for the dependence on d of graphene-layer site energies and interlayer tunneling amplitudes, but still have a small number of parameters. Nevertheless, the graphene only model for G/BN is able to accurately describe the dependence of low-energy bands on rotation angle using six parameters which we have calculated from the d-dependence of ab initio bands.
The models derived in this paper can be used as a starting point to account for the influence of either graphene or hBN substrates on the electronic structure of a graphene layer. We expect that they will be applicable to examine a wide variety of electronic properties. Because our models are derived from local-density-approximation band-structures, they do not account for the non-local exchange and correlation effects which are known to be responsible for large Fermi velocity enhancements in isolated graphene systems. 45 The same effects are likely to be important in multi-layer systems, possibly enhancing band gaps produced by the moiré pattern potentials. 46 Our electronic structure models are sufficiently simple that important many-body physics effects can be addressed separately where they play an essential role. In this appendix we present another set of model parameters obtained allowing lattice relaxation for the interlayer distance in the self-consistent LDA calculations, instead of fixing the vertical atomic separations at the experimental interlayer spacing c = 3.35Å of graphite. Even though the LDA approximation does not accurately captures the non-local van der Waals type interlayer interactions that are important in these systems, its tendency to over bind covalent bonds allows it to hold the weakly interacting layers together. and describe the interlayer lattice constants and the forces between van der Waals layered materials reasonably well. LDA results tend to have reasonable agreement with sophisticated RPA and beyond total energy calculations for thin jellium metal slabs, 47 hexagonal boron nitride, 48 and graphite 49 and other layered materials. 50 We allowed relaxation of the atomic positions in the out of plane z direction using the same 42×42 k-points grid and using a slightly coarser threshold of total energy convergence in the geometry relaxation of 10 −8 a.u. per unit cell and total force of 10 −7 a.u. In both G/G and G/BN cases the overall effect of the relaxation is to increase the interlayer separation by ∼ 0.2Å with respect to the closest interlayer separation. Theis changes leads to a weakening of the interlayer coupling strength in the first shell approximation by about 7% . which implies that farther G vectors in the Fourier expansion become more important. The changes in the position of the Fermi energy with respect to the unrelaxed geometry are in the order of ∼ 10 meV for G/G and ∼ 30 meV for G/BN providing a measure of changes in the shifts in the site potential offsets between the layers near the AA stacking configurations that are incorporated in the relaxed parameter set. Because the pseudopotentials are referenced to vacuum this information gives an estimate of the modulation in the work function of the graphene sheet due to its coupling with the hBN layer. The above observations suggest that geometry relaxation can introduce small but non-negligible changes in the potential map and details of valence-conduction bands overlap in the G/BN case whose band structure near the Fermi energy is determined simultaneously by in-plane xy and z pseudospin terms of comparable magnitudes. This and other details of the electronic structure in a G/BN hetrostructure will be presented elsewhere.
Relaxed geometry parameters for G/G
The d-vector dependent maps of the Hamiltonian matrix elements are changed only quantitatively relative to the unrelaxed case. The numerical values of the parameters that define the intralayer model for G/G in Eq. whereas the interlayer tunneling constants are t bt = 98 meV. The average interlayer separation distance lies between the minimum 3.347Å and the maximum of 3.563Å for AA stacking.
Relaxed geometry parameters for G/BN
For G/BN we have kept the coordinates fixed for the BN sheet while we allowed the carbon atoms to relax in the out of plane direction as a function of d. T The d-vector dependent map of the Hamiltonian matrix elements for G/BN for relaxed geometries also has some quantitative changes relative to the unrelaxed calculations. The numerical values of the parameters that define the intralayer model for G/BN together with the Eqs. The parameters in the sublattice basis and in the pseudospin basis can be related through
C AA = − C 2 0 +C 2 z − 2C 0 C z cos(ϕ 0 − ϕ z )(B4)
ϕ AA = tan −1 C 0 sin(ϕ 0 ) −C z sin(ϕ z ) C 0 cos(ϕ 0 ) −C z cos(ϕ z )
C BB = C 2 0 +C 2 z + 2C 0 C z cos(ϕ 0 − ϕ z ) ϕ BB = tan −1 C 0 sin(ϕ 0 ) +C z sin(ϕ z ) (C 0 cos(ϕ 0 ) +C z cos(ϕ z ) .
FIG. 2 :
2(Color online) Left panel: Total energy per unit cell relative to AA stacking as a function of displacement d. These results are for G/G with constant in-plane lattice constant and vertical separation c = 3.35Å. The highest energy configuration corresponds to AA stacking and the lowest to AB or BA stacking. Gap refers to the separation between conduction and valence bands at the Dirac point, which vanishes at AB and BA points. Right panel: The same plots for G/BN with both sheets constrained to have the in-plane self-consistent LDA lattice constant of graphene and vertical separation c = 3.35Å. The lowest energy stacking configuration corresponds to BA stacking with one of the two carbon atoms in the graphene unit cell sitting on top of boron. The highest energy stacking configuration corresponds to the AA arrangement in which the two carbon atoms in the unit cell sit on top of B and N. The AB configuration which has C on top of N has an intermediate energy. The scale of the dependence of total energy on d is similar in the graphene and hBN cases. Note that the Dirac point gap in the G/hBN case does not vanish at any value of d, but that the typical gap scale is larger in the G/G case. This later property reflects stronger inter-layer coupling.
FIG. 3 :
3(Color online) Dirac-point π-band Wannier-representation Hamiltonian matrix elements as a function of sliding vector d for graphene/graphene. Top panel: Left to right real and then imaginary parts of the AA and then AB interlayer matrix elements as a function of position d in the rectangular cell of Fig.1. The BA matrix element is closely related to the AB matrix element as shown in Eq. (20) and the BB matrix element is identical to the AA matrix element. Interlayer coupling matrix elements have a typical magnitude ∼ 300 meV. We show later that the dependence of these four complex numbers on d is accurately described by a single real number. The color scales show energies in units of eV. Bottom panel: Intralayer Wannier-representation Hamiltonian matrix elements. Left to right the real parts of the AA and BB matrix elements followed by the real then imaginary parts of the AB matrix element. Typical matrix element values are ∼ 5 meV. For graphene on graphene the spatial variation of intra-layer matrix elements has a negligible influence on electronic properties. The color scale shows energy in units of meV. and the A and B positions in the second layer are
FIG. 4 :
4(Color online) Real part of the Fourier transform of H AA ( K : d) evaluated at K = (4π/3a, 0). In Fourier space, interlayer coupling is strong only for three reciprocal lattice vectors, G = 0 and the two non-zero reciprocal lattice vectors for which | K + G| = | K|. The imaginary part of H AA ( K : d) vanishes. At these values of G, H AA ( K : G) is real with identical values of 0.113 ± 0.001 eV.
Fig. 4
4illustrates typical results of a moiré band parameter calculation performed by using Eq. (6) and integrating over d.
0 FIG. 5 :
05Left Panel: Representation of the first shell of G reciprocal lattice vectors with their corresponding numeral labels used in the main text. The three circles in red correspond to the G 0 , G ± vectors with large inter-layer tunneling coefficients. The three vectors K j = K + G j j = 0, ± have the same magnitude. Right Panel: First shell moiré reciprocal lattice vectors G = −θẑ × G for graphene on graphene result differ from the honeycomb reciprocal lattice vectors by a clockwise 90 • rotation and a reduction in size by a factor proportional to θ . The solid black arrows represent the Q j vectors that connect a q vector in the bottom layer to one in the top layer and the red hexagon encloses the moiré pattern Brillouin-zone.
=
online) Real and imaginary parts of the Fourier transforms of the intra-layer AA site diagonal Hamiltonian matrix element. Because of the symmetries of the honeycomb lattice the Gdependent site potentials satisfy H AA, G = H * H B B , G .These contributions to the twisted layer Hamiltonian for G/G are small and can often be neglected.
d = (0, 0) independent of L, and to Direct evaluation of the AA Wannier matrix elements yields H AA ( K : d = (0, 0)) = H BB ( K : d = (0, 0)) = 355 meV and H AB ( K : d = (0, 0)) = H BA ( K : d = (0, 0)) = 0. The matrix element from the Fourier expansion model truncated at the first shell is 3t bt = 339 meV. For the case of AB stacking ( d = (0, a/ √ 3), a direct evaluation of the Wannier matrix elements yields H BA ( K : d =
FIG. 7 :
7(Color online) Real and imaginary parts of the Fourier transform of the intralayer, intersubband Hamiltonian matrix element H AB, G . It follows from symmetry that H AB, G = H A B , G . These matrix elements are also small.
FIG. 8 :
8(Color online) Comparison of LDA G/G band structures (solid lines) and the one-parameter moiré band model (blue dashed lines), which retains only the first shell Fourier-expansion of interlayer coupling. Results are shown for commensurate G/G with AA, AB, and τ br = (0, a/2 √
. 10: (Color online) Schematic Brillouin zones of graphene in black lines, of hBN in blue, and the moiré Brillouin zone (MBZ) in red for incommensurate G/BN. The difference between the lattice constants has been exaggerated to aid visualization. Both the size and orientation of the MBZ change continuously with twist angle due to the lattice constant mismatch. The three Q j vectors given in Eq.(16) connect the K-points of graphene to those of hBN. The moiré pattern reciprocal lattice vectors can be constructed by summing pairs of Q j vectors.
C 0 A A = 0, C 0 B B = 0, C AA = 5.733 meV, ϕ AA = 90 • , C BB = 4.826 meV, ϕ BB = 65.49 • , C A A = −5.703 meV, ϕ A A = 87.51 • , C B B = −3.596 meV, ϕ B B = 65.06 • , C AB = 4.418 meV, ϕ AB = 26.10 • , C A B = 1.987 meV, ϕ A B = 3.50 • .
In the first shell approximation the tunneling amplitudes are 3t BC = 432 meV and 3t NC = 291 meV respectively. In comparison, direct calculations for these stacking configurations give H AA ( K : τ AA ) = 437 meV, H BB ( K : τ AA ) = 294 meV, H BA ( K : τ AB ) = 296 meV, and H AB ( K : τ BA ) = 439 meV. The deviations from 3t BC and 3t NC are in the order of a few meVFIG. 11: (Color online) Displacement vector d dependence of Wannier representation interlayer Hamiltonian matrix elements for G/BN. Upper panel: Matrix elements AA , AB connecting boron with carbon, and matrix elements BA , BB connecting nitrogen with carbon. The interlayer coupling matrix elements vary over a large range ∼ 600 meV. Lower panel: Displacement vector d dependence of intra-layer Wannier representation Hamiltonian matrix elements for G/BN. On-site energies in the graphene layers vary by ∼ 60 meV. In these plots the site energies are plotted relative to their spatial averages C 0ii . (See Eq. (31).) The carbon layer AB inter-sublattice terms vary over a range of ∼ 35 meV whereas the BN layer A B terms vary by ∼ 15 meV. Re[H AB 0 ,G ] Re[H AA 0 ,G
FIG. 12 :
12(Color online) G/BN interlayer moiré band model parameters obtained by evaluating Fourier expansion coefficients for the layer separation dependence of Dirac point Wannier representation Hamiltonian matrix elements. As in the G/G case three Fourier coefficients dominate interlayer coupling.
FIG. 13 :
13Fourier expansion of the Wannier-representation matrix-elements H AA ( K : d), H BB ( K : d), H A A ( K : d) and H B B ( K : d) that describe the interlayer displacement dependence of site-energies in the hBN and graphene layers. These numerical results demonstrate that the site energies are accurately approximated by the model that includes only the first shell of reciprocal lattice vectors. The parameters of this model are listed in the main text. We have chosen the average energy on the carbon sites as the zero of energy. With this choice, the elements corresponding to G = 0 are H AA ( K : G = 0) = 3.332 eV for boron, H BB ( K : G = 0) = −1.493 eV for nitrogen, H A A ( K : G = 0) = 0 eV and H A A ( K : G = 0) = 0 eV. the honeycomb unit cell that is absent for an isolated layer. When viewed as a substrate contribution to graphene's twodimensional Dirac equation H z = (H A A − H B B )/2 can be viewed as a d-dependent mass. Note that the effective mass is sometimes positive and sometimes negative. The A site energy is maximized at AB points, where the carbon A site is on top of a boron atom and far from nitrogen atoms. Similarly the B site energy is minimized at BA points, where the carbon B site is on top of a nitrogen atom and far from boron atoms.ii) H 0 = (H A A + H B B )/2 can be viewed as a d-dependent potential term. iii) H A B = H * B A captures the influence of the substrate on hopping between carbon sublattices. This quantity vanishes by symmetry at the Dirac point for an isolated sheet. Our calculations demonstrate that the reduction in symmetry due to the substrate yields a d-dependent contribution to the Hamiltonian that is roughly of the same size as the mass and potential terms. When the operators that act on sub lattice degrees of freedom are described using Pauli spin matrices, the real part of H A B is proportional to the coefficient of σ x
14: (Color online) Fourier expansion of the Wannier-representation matrix-element H AB ( K : d), which describes inter-sublattice tunneling within the hBN layer, and H A B ( K : d), which describes inter-sublattice tunneling within the graphene layer. These numerical results demonstrate that the local coordination dependence of interlayer hopping processes is accurately approximated by the model that includes only the first shell of reciprocal lattice vectors. The parameters of this model are listed in the main text.FIG. 15: (Color online) Relative displacement d-dependence of the matrix elements of the two-band low-energy effective model for graphene on a hBN substrate. In the effective model map the H 0 = (H AA + H BB )/2 term represents a sublattice independent potential and H z = (H AA − H BB )/2 acts as a mass term in the Dirac equation. The off diagonal matrix-elements H AB accounts for changes in the bonding pattern within the graphene layer. Here A, B refer to the sublattice sites of graphene.
FIG. 16 :
16Mass and pseudospin field terms in the effective Hamiltonian as a function of displacement d. The H z term is due to sublattice potential difference and vanishes along lines of this two-dimensional plot. The simultaneous presence of finite H x and H y and H z terms in the effective Hamiltonian implies that the Dirac-point gap (Eq. (39)) does not vanish at any relative displacement.
FIG. 17 :
17(Color online) Fourier expansion coefficients for the effective model matrix element H 0 ( K : d). This term captures the variation of the site-averaged potential across the moiré pattern. The first shell of reciprocal lattice vectors dominates. d( L) as explained in section II. The moiré band model is particularly simple when constructed from the two-band effective model:
FIG. 18 :
18(Color online) Fourier expansion coefficients for the mass term in the effective model, H z ( K : d) = (H AA ( K : d)−H BB ( K : d))/2 The first shell of reciprocal lattice vectors dominates.
FIG
. 19: (Color online) Fourier expansion coefficients for the off diagonal effective model matrix element H AB ( K : d). The first shell of reciprocal lattice vectors dominates.
20: (Color online) Comparison of the LDA band structure (solid black), the four-band moiré band model (dashed blue lines) and the low energy two-band model (dashed red lines) with the first shell used for the superlattice potentials. The commensurate G/BN arrangements plotted are AA, AB, BA and a bridge stacking with τ br = (0, a/2 √ 3). Note that he electronic structures of AB and BA stacking are different in the G/B case. For the intermediate bridge stacking we see a substantial reduction of the band gap due to a shifting in the Dirac cone momentum space location caused by inplane pseudospin terms. The intralayer Hamiltonian of graphene is approximated using the massless Dirac model with the LDA Fermi velocity 41 while the boron nitride Hamiltonian is modeled with the same Dirac model with a mass term compatible with the LDA gaps. The interlayer coupling is given by the first shell approximation using the parametrizations of Eq. (20), with tunneling from boron and carbon sites distinguished as in Eq. (30) and the parameter set in Eq. (31). In these plots the energy origin of the represented bands has been adjusted so that zero is in the middle of the band gap. FIG. 21: Modulation of the local potential fluctuations H 0 ( r) in real space for different twist angles. These plots illustrate the rotation of the moire pattern when |ε| ∼ θ , and the property that the Moire periodicity L M ∼ a/ √ ε 2 + θ 2 becomes shorter with increasing twist angle. The other pseudospin components of the local Hamiltonian illustrated Fig. 15 produce similar spatial superlattice patterns.
22: (Color online) Left Panel: Schematic illustration of potential variations and local band edges as a function of d y for fixed d x = 0. The approximate conditions for local electron/hole charging discussed in the main text are satisfied over the segments identified by bold black lines. The difference between the Dirac point gap ∆ and the absolute value of the mass |H z | reflects the influence of the in-plane pseudospin fields terms. Right Panel: Sliding vector d dependent map of electron and hole puddles.
(21) for relaxed geometries areC AA = 2.3 meV, ϕ AA = 27.5 • ,(B1) C BB = C AA , ϕ BB = −ϕ AA , C AB = 2.08 meV.
FIG. 23 :
23(Color online) Changes in total energy, Dirac point gaps, average layer separation distance and Fermi energy resulting from allowing self-consistent LDA relaxation in the out of plane z-axis for G/G.FIG. 24: (Color online) Changes in total energy, Dirac point gaps, average layer separation distance measured from the minimum separation distance and Fermi energy resulting from allowing selfconsistent LDA relaxation in the out of plane z-axis for G/BN.
(21) for relaxed geometries areC 0 AA = 3.334 eV, C 0 BB = −1.494 eV, (B2) C 0 A A = 0, C 0 B B = 0, C AA = 5.643 meV, ϕ AA = 56.37 • , C BB = 4.216 meV, ϕ BB = 59.98 • , C A A = −7.402 meV, ϕ A A = 77.71 • , C B B = −4.574 meV, ϕ B B = 85.78 • , C AB = 4.01 meV, ϕ AB = 22.2 • , C A B = 1.90 meV, ϕ A B = 1.30 • . whereas the interlayer tunneling constants are t BC = 130 meV and t NC = 87 meV. The average interlayer separation distance lies between 3.256Å and the maximum of 3.466Å for AA stacking. The parameters of the effective model of G/BN for the relaxed geometries in the sublattice basis are given by C AA = −13.3 meV, ϕ AA = 63.63 • , (B3) C BB = 14.0 meV, ϕ BB = −51.27 • , C AB = 9.53 meV, ϕ AB = 21.82 • .
(The distinction between d( L) and d( L ) is second order in the small parameters ε, θ and therefore neglected. The moiréFIG. 1: (Color online) Left: Schematic representation of two commensurate honeycomb layers with bottom layer sites indicated by light grey circles and top layer sites indicated by dark grey circles. The unit cell of the bilayer contains four sites, A and B for bottom layer and A and B for the top layer. The shaded region represents the primitive cell area A 0 used for the Fourier integrals described in the text. Middle: The relative displacement between the honeycombs is specified by the displacement vector d. We choose d = 0 for AA stacking in which the two honeycombs have no lateral displacement. For d = (0, a/ √ 3) we have the AB stacking where the top layer A site is directly above the bottom layer B site. The bilayer lattice is a periodic function of d and the primitive cell for this periodic dependence is shaded grey in the left figure. It is convenient to use the rectangular a × √ 3a area enclosed by a dotted line in the left figure to illustrate the dependence of the bilayer Bloch bands on d. Right: This panel specifically indicates the points within the rectangular area at which the high symmetry AA, BA, and AB stacking arrangements occur and is helpful for the interpretation of later figures.
Appendix B: Influence of vertical lattice constant relaxation
AcknowledgmentsThis work was supported by the Department of Energy, Office of Basic Energy Sciences under contract DE-FG02-ER45118, and by Welch Foundation grant TBF1473. JJ was partially supported by the National Research Foundation of Singapore under its Fellowship program (NRF-NRFF2012-01). Helpful conversations with Rafi Bistritzer and Byounghak Lee are gratefully acknowledged. We appreciate assistance and computer time provided by the Texas Advanced Computing Center.Appendix A: Exact form of the displacement vector and moire reciprocal latticesIn the main text we used the small angle approximation for the displacement vectors and the moire reciprocal lattices. These approximations are accurate for rotations angles
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"Crystallization Inhibitors: Explaining Experimental Data through Mathematical Models",
"Crystallization Inhibitors: Explaining Experimental Data through Mathematical Models"
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"M P Bracciale \nDipartimento di Ingegneria Chimica Materiali Ambiente\nSapienza Università di Roma\nVia Eudossiana 1800184RomeItaly\n",
"G Bretti \nIstituto per le Applicazioni del Calcolo \"Mauro Picone\"\nvia dei Taurini 1900185RomaItaly\n",
"A Broggi \nDipartimento di Ingegneria Chimica Materiali Ambiente\nSapienza Università di Roma\nVia Eudossiana 1800184RomeItaly\n",
"M Ceseri \nIstituto per le Applicazioni del Calcolo \"Mauro Picone\"\nvia dei Taurini 1900185RomaItaly\n",
"A Marrocchi \nDipartimento di Chimica, Biologia e Biotecnologie\nUniversità degli Studi di Perugia\nVia Elce di Sotto 806123PerugiaItaly\n",
"R Natalini \nIstituto per le Applicazioni del Calcolo \"Mauro Picone\"\nvia dei Taurini 1900185RomaItaly\n",
"C Russo \nDipartimento di Chimica, Biologia e Biotecnologie\nUniversità degli Studi di Perugia\nVia Elce di Sotto 806123PerugiaItaly\n"
]
| [
"Dipartimento di Ingegneria Chimica Materiali Ambiente\nSapienza Università di Roma\nVia Eudossiana 1800184RomeItaly",
"Istituto per le Applicazioni del Calcolo \"Mauro Picone\"\nvia dei Taurini 1900185RomaItaly",
"Dipartimento di Ingegneria Chimica Materiali Ambiente\nSapienza Università di Roma\nVia Eudossiana 1800184RomeItaly",
"Istituto per le Applicazioni del Calcolo \"Mauro Picone\"\nvia dei Taurini 1900185RomaItaly",
"Dipartimento di Chimica, Biologia e Biotecnologie\nUniversità degli Studi di Perugia\nVia Elce di Sotto 806123PerugiaItaly",
"Istituto per le Applicazioni del Calcolo \"Mauro Picone\"\nvia dei Taurini 1900185RomaItaly",
"Dipartimento di Chimica, Biologia e Biotecnologie\nUniversità degli Studi di Perugia\nVia Elce di Sotto 806123PerugiaItaly"
]
| []
| In this paper we propose a new mathematical model describing the effect of phosphocitrate (PC) on sodium sulphate crystallization inside bricks. This model describes salt and water transport, and crystal formation in a one dimensional symmetry. This is the first study that takes into account mathematically the effects of inhibitors inside a porous stone. To this aim, we introduce two model parameters: the crystallization rate, which depends on the nucleation rate, and the specific volume of precipitated salt. These two parameters are determined by numerical calibration of our system model for both the treated and non treated case. | null | [
"https://arxiv.org/pdf/1501.05835v1.pdf"
]
| 118,365,008 | 1501.05835 | e3e05bcdfe5b98a63860ec2647371b968f848af9 |
Crystallization Inhibitors: Explaining Experimental Data through Mathematical Models
January 26, 2015
M P Bracciale
Dipartimento di Ingegneria Chimica Materiali Ambiente
Sapienza Università di Roma
Via Eudossiana 1800184RomeItaly
G Bretti
Istituto per le Applicazioni del Calcolo "Mauro Picone"
via dei Taurini 1900185RomaItaly
A Broggi
Dipartimento di Ingegneria Chimica Materiali Ambiente
Sapienza Università di Roma
Via Eudossiana 1800184RomeItaly
M Ceseri
Istituto per le Applicazioni del Calcolo "Mauro Picone"
via dei Taurini 1900185RomaItaly
A Marrocchi
Dipartimento di Chimica, Biologia e Biotecnologie
Università degli Studi di Perugia
Via Elce di Sotto 806123PerugiaItaly
R Natalini
Istituto per le Applicazioni del Calcolo "Mauro Picone"
via dei Taurini 1900185RomaItaly
C Russo
Dipartimento di Chimica, Biologia e Biotecnologie
Università degli Studi di Perugia
Via Elce di Sotto 806123PerugiaItaly
Crystallization Inhibitors: Explaining Experimental Data through Mathematical Models
January 26, 2015Index terms-mathematical modellingporous mediasalt crystalscrystallization in- hibitors
In this paper we propose a new mathematical model describing the effect of phosphocitrate (PC) on sodium sulphate crystallization inside bricks. This model describes salt and water transport, and crystal formation in a one dimensional symmetry. This is the first study that takes into account mathematically the effects of inhibitors inside a porous stone. To this aim, we introduce two model parameters: the crystallization rate, which depends on the nucleation rate, and the specific volume of precipitated salt. These two parameters are determined by numerical calibration of our system model for both the treated and non treated case.
Introduction
It is well known that one of the major causes of building degradation is the crystallization of salts into the porous matrix [1,2,3,4]. Salt is present inside building stones as free ions: it can be a natural element of the material, created by reaction with atmospheric pollutants or introduced by water solutions penetrating into the porous matrix by capillarity [5]. The latter is the main mechanism leading to buildig damage and has received much attention from the scientific investigation [6,7,8,9,10] but remains not yet fully understood. Salt decay requires the simultaneous presence of soluble salts and water in the porous material, as well as appropriate environmental conditions. Indeed, it originates from salt-ions (e.g. chloride, nitrate, sulphate) that migrate while dissolved in liquid water which flows in the pore network of building materials. Liquid water may penetrate these materials by different processes, including hygroscopic moisture, penetration of rainwater (through, e.g. construction joints, damaged roofs, and cracks), dew point condensation, and rising damp. The latter is probably the most frequent and perhaps one of the most difficult sources of water to remove, when dealing with old buildings. Consider an initially dry porous stone (such as a masonry brick) that is wetted by a salt water solution. During the wetting phase, water fills up the stone bringing the dissolved salt present in the outside environment. If the stone is in contact with ambient air, water molecules are exchanged with the environment by evaporation thus starting a drying phase; the rate of dehydration depends on the relative humidity of the atmosphere. At this point, salt content in water increases and solution may become supersaturated. Once a high degree of supersaturation is reached, salt starts crystallizing: if crystals are formed inside the porous matrix we talk of subflorescence or cryptoflorescence; if crystallization takes place on the exterior boundaries of the stone we talk of efflorescence. Subflorescence causes the formation of large crystals into the pores: once the pres1e exerted by these crystals exceeds the tensile strength of the porous matrix, it can lead to widespread loss of surface, e.g. exfoliation, detachments. The occurrence of efflorescence or subflorescence (cryptoflorescence) depends on several factors including salt type and concentration, microclimate, evaporation rate [6], substrate porosity characteristics [11,12,13,14] and surface tension and viscosity of the solution [15,16,17]. The in-pore crystallization causes a reduction of the pore volume, breaking the liquid network and delaying water transport. Since pore clogging affects the location and quantity of crystals, it might have implications for stress development and deterioration of the material [18]. Common constructions contain different kind of salts such as chlorides, sulphates, nitrates, and carbonates, with their own solubility, crystalline structure and crystallization properties. Among these, sodium sulphate is probably one of the most complex and damaging salt types involved in salt decay processes. Indeed, this salt has three different phases of crystallization at various microclimate conditions, can easily supersaturate and has a solubility which is highly temperature dependent [19,20]. Both crystallization and hydration transformations in sodium sulphate, resulting in significant volumetric changes, have been blamed for the destructive mode of action of this salt [21].
One way to prevent the stone breakage is to treat the porous material with a substance that inhibit subflorescence: these crystallization inhibitors reduce the pressure associated with the growing crystals trying to keep it below the breakage modulus of the substrate. The organic as well as inorganic ion and molecule additives alter the surface properties of the crystals which lead to changes in nucleation, growth, and thereby changes in the shape of the crystals as well as in their agglomeration/dispersion behaviour. Examples of well-known additives with extended technological and industrial uses are the families of (poly)phosphates, carboxylates, polyacrylic acid derivatives, and benzotriazoles [22,23,24]. These additives are widely used as scale-inhibitors to prevent undesired effects associated with sparingly soluble salts (e.g. sulphates, carbonates) precipitating in oil extraction pipelines [25] industrial boilers, heat exchangers, house appliances or water pipes [26,27] and others. The effectiveness of a given inhibitor depends on many variables: salt type, pore structure properties of the substrate, application methodology, the composition of the inhibiting solution to cite a few. Hence, a given modifier has to be evaluated for each stone and for each salt [28]. On the other hand, adding a crystal inhibitor does not affect surface tension nor contact angle of the wetting liquid, since there have not been observed any significant effect on solution transport [29]. Although the effectiveness of some salt crystallization inhibitors in bulk solution has been proved, the possibility of using these products for the prevention of salt decay in building materials is still controversial because it is not clear how these inhibitors act. However, experiments suggest two possible mechanisms [30,22,31,32]: nucleation delay enhances salt transport toward the surface, thus increasing efflorescence; crystal habit modification by absorption on specific faces of a growing crystal that decreases crystal growth rate. Another matter of discussion is the fact that crystal reduction would result in higher supersaturated solutions. It has been speculated, but not actually observed, that in this case the inhibitor may promote salt precipitation at higher supersaturation levels and, hence, the quick formation of large crystals. Therefore, a modifier would eventually increase the crystal pressure and the risk of damage instead of reducing it [33]. Our group has undertaken a broad research project [34,35,36] focusing on the effects of environment-friendly, non-invasive inhibitor systems on saline solutions percolating and crystallizing in a porous media following evaporation, in order to develop a sound methodology suitable for addressing the conservation needs of different salt-weathered sites. Our attention has been particularly focused on the crystallization inhibition properties of functionalized polycarboxylates (i.e. maleate, citrate, phosphocitrate, tartrate), with an emphasis on the phosphorylated family members. Indeed we have demonstrated that phosphocitrate (PC) has been revealed to be one of the most promising inhibitors, because of its effectiveness in controlling the crystallization of different salts (i.e. sodium sulphate, sodium chloride, sodium nitrate, calcium carbonate) and salt mixtures in a wide range of porous materials and in various ambient conditions.
In this work we developed a mathematical model describing the effect of phosphocitrate (PC) on sodium sulphate crystallization inside a brick's porous matrix. There are plenty of mathematical models describing salt crystallization in porous stone. They consists of 3D multiphase systems of equations for heat and mass transport with various degree of complexity. Some models might also couple the governing equations with other effects: osmosis, stress tensor deformations and latent heat release due to salt crystals formation [37,38,39,40]. For the present study we have developed a simple mathematical model of salt and water transport and crystal formation. In fact, we limit our research to the considerations on few available data, which can be obtained using simple laboratory equipments, and so it would not have made sense to include further effects. Moreover, since the experiments were carried out in laboratory at constant temperature, we did not consider directly temperature variations; we just included evaporation rate into the porous stone simply by defining an appropriate sink term in the water balance equation. Actually, this work is a preliminary study to describe mathematically the effects of inhibitors inside a porous stone: to our knowledge, this is the first attempt to develop a mathematical model for the effects of crystallization modifiers. As we shall see, we identified two model parameters that will be crucial for the appropriate description of an inhibitor: K s the crystallization rate taking into account the nucleation rate;
γ the specific volume of precipitated salt, describing the crystal habit modification.
These two parameters will be determined by the numerical calibration of our model -i.e. by comparing our numerical results with the available experimental data -for both the treated and non treated case. The remain of our paper is organized as follows: the second section will describe the materials considered and the experiments performed; the third and fourth sections will introduce the mathematical model and describe the numerical scheme applied to solve the system equations; in section five we will describe our results. The paper ends up with few conclusions.
Materials and Methods
In this section, we will introduce the experimental settings we are going to consider [41]. Commercially produced brick is tested. Bulk density ρ v was determined by weighing and measuring of dimensions of dry prismatic samples. The matrix density ρ mat was measured by helium pycnometer. The porosity n 0 [%] was calculated according to the equation
n 0 = 100 · (1 − ρ v /ρ mat ) .(1)
The porosity determined in this way is 28.51% ± 0.04%. Pore size distribution was determined by mercury intrusion porosimetry (MIP) by Carlo Erba instrument on a about 1g of material. All experiments were performed in air conditioned laboratory at 25±2 • C and 30±5% RH. This set of experiments were conducted, according to standard UNI EN 1925 (Determination of water absorption coefficient by capillarity) and NORMAL 29/88 (Drying Behaviour), without the presence of salt. It will serve as a control sample to test transport properties of the materials under study. The brick specimen has the form of a cube of side 5 cm, is positioned in a bucket containing water and immersed for 3 mm in height. The water absorption for capillarity, expressed in g/cm 2 , is defined as the quantity of water absorbed by the specimen having the base surface in contact with water as a function of time t, with room temperature and pressure. At different time intervals the specimen is taken and tamponed only on the wet surface and then weighted until the variation in the quantity of absorbed water between two consecutive measurements, for a 24 hours interval, is less than 1% of the water mass. The determination of the quantity of water absorbed by the specimen per time unit is given by
W = (m i −m 0 ) S expressed in g/cm 2 ,
where W is the quantity of water absorbed (expressed in g) and S = 25 cm 2 is the surface of the specimen in contact with the porous frame. The experiment is applied to a number of specimen and then the average of the time dependent values W obtained for the different specimens is computed. Finally we get the averaged quantity Q(t k ), with t k the time instants expressed in s 1/2 .
Experiment 2: brick's capillary absorption and drying test in a salt saturated water solution.
Both in untreated and treated brick's samples with PC the water and salt concentration profiles were determined experimentally using prismatic specimen 2 × 2.5 × 12 cm positioned vertically in a bucket containing a salt water solution of N a 2 SO 4 (99.5 g/L) (see Fig. 3). In order to determine the concentration profiles the specimens were cut into 4 pieces with similar dimension and re-assembled sealing the lateral sides with epoxy resin; in this way only the top side of the brick is in contact with ambient air. On the other hand, the immersed part of the specimen is pervious and liquid can flow through the lateral side. The insulated specimens were dried at 65 ± 2 • C to the constant mass. When the solution in the bucket is totally absorbed by the specimens, the water content was obtained as difference of the mass of the saturated specimens and of the sample's mass after drying at 110 ± 2 • C to the constant weight. The concentration of sulphates in the dried samples was determined as follows: the samples were placed in plastic container, 200 mL of boiling water was added and the container was sealed. This procedure was repeated every day for 1 week. Then the dry samples were weighted and the concentration of sulphates was calcutated.
The mathematical model
Here we want to introduce a model of coupled water and sulphate transport taking into account not only the influence of water flow on salt transport but also the effect of bound sulphates on pore walls, and the effects of porosity changes (due to the salt bonding) on moisture transport. Regarding the mathematical domain, a reasonable assumption is to consider a one dimensional geometry since the domain is sealed on its lateral side; hence, flow is predominantly vertical. We denote by n the porosity, i.e. the fraction of volume occupied by voids, and we denote the fraction of volume occupied by the liquid and by the gas (composing the fluid) within the representative element of volume, respectively by θ l and θ g . The following relation holds:
n = θ l + θ g .(2)
The mass balance equation for a liquid of density ρ l reads as:
∂ ∂t (ρ l θ l ) + ∂ ∂z (ρ l q) = f (θ l )(3)
where q is the water flux into the porous matrix and f (θ l ) is the evaporation rate inside the specimen. Both q and f (θ l ) will be specified later on. Let us denote by c i the concentration of free ions in water and with c s the density of bound salt, the mass balance equation for salt dissolved in water is given by:
∂ t (θ l c i ) + ∂ z (c i q) = D∂ z (θ l ∂ z c i ) − ∂c s ∂t ,(4)
where D is the salt diffusion coefficient, while the sink term on the right hand side takes into account the crystal formation into the porous matrix. In this work, we assume that crystal growth depends on the following properties: the concentration of salt dissolved in the liquid, the fraction θ g and the degree of supersaturation. If we indicate the supersaturation level with c we have:
∂c s ∂t = K s c i θ 2 g + K(c i −c) + θ l .(5)
with K s and K two crystallization coefficient and (·) + is the positive part function (or the second term is active only when salt saturation into the liquid exceeds the supersaturation level). The term θ g on the right hand side is raised to power two in order to capture the following fact: the higher the water content, the smaller the crystallization into the pores. The power two simply slows down the crystal formation in saturated regions. The second term on the right hand side has been defined for the sake of completeness; in fact, in our experiments and in the subsequent simulations, salt supersaturation has never been exceeded and term K has not been determined.
Since the overall porostity changes as the salts growth into the porous material, the following equation holds:
n(t) = n 0 − γc s ,(6)
with γ the specific volume of sulphate crystal.
Darcy's law
Water flow into a porous medium is given by the well known Darcy's law [42,43]:
q = − k(s) µ l n n 0 2 (∂ z P c (s) − ρ l g)(7)
with P c = P c (θ l /n) the capillary pressure, k the permeability of the porous matrix, µ l the viscosity of the fluid, the term (n/n 0 ) 2 is a shape factor for the influence of the porosity variation to the water flux and s = θ l /n. Capillary pressure is usually given as a function of water saturation and is defined through a state equation. In literature, one can find capillary pressure state functions for several applications; in building materials, however, despite the number of experimental study, there is not a relation correlating capillary pressure with moisture content into the porous matrix. To overcome this problem, we will approximate Darcy's law through a polynomial function with some free parameters that will be found through model calibration. Thus, we proceed as in [44]. First of all, since the dimensions of the brick are small, gravity effects can be safely disregarded from (7). Then we introduce function B such that
∂ z B = − k(·) µ l ∂ z P c (·).
We know that P c (s) is a decreasing function of s = θ l /n < 1 and vanishes whenever the medium is completely saturated, i.e. θ l = n. On the other hand, permeability k = k(s) is a non-negative increasing function of s and it is bounded from above by its value at saturation. Taking into account these observations, the first derivative of function B with respect to s = θ l /n can be given by the ansatz
B (s) = max 4c (1 − a) 2 (a − s)(s − 1), 0(8)
with a such that k(a) = 0. Constants a and c are physical properties of the porous material involved and will be determined later on. The quantity a · n is the minimum value for saturation ensuring the hydraulic continuity -i.e. water transport through the porous medium. On the other hand c has the dimensions of a diffusivity. The term 4c/(1 − a) 2 is chosen so that max{B (s)} = c. Integrating B (s) we obtain the following expression (see Fig. 1):
B(s) =
Water evaporation
Once water content decreases below the quantity an, the hydraulic continuity is broken and fluid trasport is no longer ensured. Since drying experiments end up with a completely dry stone, we added a sink term in the water balance equation (3) to take into account the effect of evaporation inside the porous matrix. In our mathematical model, we made the simplifying assumption that evaporation is maximum when moisture content is below the value an and decreases quickly as the porous medium becomes saturated: thus liquid flow and evaporation acts at almost separated stages (one is strong while the other is weak and viceversa). This is reasonable since in our controlled experimental setting temperature is constant and does not play a significant role.
We defined the evaporation rate as follows with K T a (temperature dependent) constant and H ε is defined as follows:
f (θ l ) = −ρ l K T θ l H ε (θ l )(9)H ε (θ) = 1 if 0 < θ < an, an+ε−1 ε θ + (an+ε)(1−an) ε if an ≤ θ ≤ an + ε an+ε n(a−1)+ε θ − n(ε+an) n(a−1)+ε if x > an + ε.(10)
see Fig. 2. In our simulations, we took ε = 0.25 an.
The Complete Mathematical model
Summing up, the mathematical model we are going to consider is the following: Table 2 shows the known parameters of the problem. Since some coefficients are unknown, we will calibrate the model versus experimental data. The obtained values will give some insight about the action of the inhibitor in the crystallization process. Table 3 lists the coefficients to be determined.
∂ t θ l = ∂ z n n 0 2 ∂ z B(θ l /n) − K T θ l H ε (θ l ), ∂ t (θ l c i ) = ∂ z c i n n 0 2 ∂ z B(θ l /n) + θ l D∂ z c i − ∂cs ∂t , n(t) = n 0 − γc s , ∂cs ∂t = K s c i (n − θ l ) 2 + K(c i −c) + θ l .(11)
Boundary Conditions
For each experiment we will describe the initial and boundary conditions to apply to model (11). In some cases, we are even able to simplify the model equations.
Experiment 1: pure water
The immersed part of the brick (for −h 2 ≤ z ≤ 0) is pervious to later water flow and we assume that it is initially saturated. In this way we can simply confine ourselves to mathematically describe the domain 0 ≤ z ≤ h 1 . Moreover, since there is no salt, our mathematical model reduces considerably; indeed, we can only retain the water continuity equation (3), that in this setting, is given by:
∂ t θ l = ∂ zz B − K T θ l H ε (θ l ).(12)
Given the absence of salt, porosity will remain constant and, thus, will not affect water flow. Equation (12) has to be coupled with reasonable initial and boundary conditions. For the experiment of imbibition, we assume the conditions
θ l (z, 0) = 0, θ l (0, t) = n 0 ,(13)
that is, the sample is initially dry while its botton side is always saturated. To reproduce the loss of water at the upper boundary z = h 1 due to evaporation, we derive θ l (h 1 , t) from the following relations:
∂ z B = K l |θ l − θ l | α−1 (θ l − θ l ), if θ l >θ l , θ l =θ l , otherwise.(14)
In the above conditions,θ l is the moisture content of the ambient air (assumed constant) while K l is the exchange coefficient with the environment. The exponent α > 1 takes into account that water evaporation from the top of the doamin depends non-linearly on the difference between the quantity of water within the specimen and the valueθ l . Once the imbibition stage is terminated, we stop the simulation and switch to another settings to deal with drying. In this case we consider the whole domain [−h 2 , h 1 ], since we do not add water at the bottom of the specimen. The other changes regard the initial and boundary conditions. If we denote by t s the final time of imbibition and with θ f in (z) = θ l (z, t s ) the value of θ l after imbibition, the initial condition for the new setting is given by
θ l (z, 0) = θ f in (z), for z ∈ [0, h 1 ], θ l (z, 0) = n 0 , for z ∈ [−h 2 , 0](15)
meaning that the initial water content is the final value obtained for the imbibition test. Moreover, at z = −h 2 we impose a no-flux boundary condition:
∂ z θ l (−h 2 , t) = 0,(16)
while at z = h 1 we retain condition (14) again.
Experiment 2: salt saturated water solution
Experiments with salt solution were performed on bricks with height 12 cm. As above, we will consider during imbibition that the first three millimiters are submerged with water, thus we confine ourselves to the domain [0, h 3 ], while during evaporation, to the domain [−h 2 , h 3 ] (see table 2). For imbibition, we assume the initial conditions for the system (11): Figure 3: Setup of experiment 2 as described in Section 2.2.
c s (z, 0) = 0, c i (z, 0) = 0, θ l (z, 0) =θ l , n(z, 0) = n 0 .(17)
As boundary conditions for t ∈ [0, t s ], we impose for the ion content, at z = 0, the salt concentrationc i of the solution used in the experiment:
c i (0, t) =c i(18)
withc i the actual concentration of sodium sulphate in water and a saturation condition for the water content θ l (0, t) = n(0, t).
At the top boundary z = h 3 , we impose:
∂ z c i (h 3 , t) = 0,(20)
i.e. zero ion flux through the upper brick boundary and condition (14). For the drying phase, we assume the initial conditions:
c s (z, 0) = c s (z, t s ), c i (z, 0) = c i (z, t s ), θ l (z, 0) = θ l (z, t s ), n(z, 0) = n(z, t s ),(21)
with z ∈ [0, h 3 ] and for the immersed part, corresponding to z ∈ [−h 2 , 0], of the specimen we set:
c s (z, 0) = 0, c i (z, 0) =c i , θ l (z, 0) = n 0 , n(z, 0) = n 0 .(22)
From now on we consider separately the four bricks composing the specimen. To this aim we define as h b i the height of the broken brick and the points
b i = h b i , with a i = 0 for i = 1, 2, 3, 4.
The i-th brick is then parametrized as the interval [a i , b i ] for i = 1, 2, 3, 4. Then as boundary conditions we impose at the bottom z = a i , zero ion flux through the lower brick boundary
∂ z c i (a i , t) = 0, i = 1, 2, 3, 4(23)
and as a boundary condition for θ l reproducing the loss of water at the lower boundary we assume θ l (a i , t) =θ l , i = 1, 2, 3, 4.
At the upper boundary z = b i we assume the conditions
∂ z c i (b i , t) = 0, i = 1, 2, 3, 4(25)
and
θ l (b i , t) =θ l , i = 1, 2, 3, 4(26)
and we putθ l = 0 in both conditions (24) and (26), in order to reproduce the situation inside the oven.
Calculation of parameterθ l
Since we do not have any measurements of the relative humidity of the ambient air surrounding the sample, we set the value of the moisture content in the environment using the value of the average quantity of water within the brick measured in the imbibition-drying experiment with the sole water. In particular, using the measured average value Q s (quantity of water at saturation of the specimen) and the average value Q d = Q f in − Q s (loss of water at the end of the drying the experiment) we compute the final quantity of water Q f in = Q s + Q d = 0.31274 g/cm 2 and then we get:θ
l = Q f in ρ l h 1 = 0.06254.(27)
Numerical approximation
Here we propose a numerical scheme for the model (11). We mesh the interval [0, h] with a step ∆z = h N and we denote λ = ∆t ∆z , z j = j∆z, j = 1, ..., N.
We also set w k j = w(z j , t k ) the approximation of the function w at the height z j and at the time t k . As showed in [47] The simplest and consistent approximation of ∂ z (r(z)∂ z w) by means of Taylor expansions is the following first order approximation:
∆ j (r, w) := (r j + r j+1 )(w j+1 − w j ) − (r j−1 + r j )(w j − w j−1 ) 2∆z 2 .(28)
From now on, we will omit for simplicity the subscript l of θ. Then, the discretization in explicit form the first equation of the model (11) is:
θ k+1 j − θ k j ∆t = ∆ j ((n k /n 0 ) 2 , B k ) − ∆tK T H ε (θ k j )θ k+1 j ,(29)
Now, if we consider the velocity field computed in the equation (29) and we set it as V = (n/n 0 ) 2 ∂ z B(θ/n), we can rewrite the second equation of the system (11) as:
∂ t (θc i ) − ∂ z (c i V ) = ∂ z (Dθ∂ z c i ) − K s c i (n − θ) − K(c i −c) + θ.(30)
We can assume:
V k j = n k j n 0 2 B θ k j+1 n k j+1 − B θ k j−1 n k j−1 2∆z , for j = 1, . . . , N − 1,(31)
with the boundary values set as follows:
V k 0 = 0, for the imbibition phase, − n k j n 0 2 K l (θ l − θ k j ), for the drying phase,(32)
and
V k N = n k j n 0 2 K l (θ l − θ k j ), for both phases.(33)
Therefore, an explicit and monotonic scheme for (30) reads as:
(θc i ) k+1 j − (θc i ) k j ∆t = V k j+1 c k i,j+1 − V k j−1 c k i,j−1 2∆z + |V k j+1 |c k i,j+1 − 2|V k j |c k i,j + |V k j−1 |c k i,j−1 2∆z + ∆ j (Dθ k , c k i ) − K s c k i,j (n k j − θ k j ) − K(c k i,j −c) + θ k j ,(34)
which is convergent under the CFL condition ∆t ≤ inf θ j ∆z 2 Dn 0 + sup|V |∆z + (K s +K)n 0 ∆z 2 .
We observe that the CFL may become very restrictive during the drying phase, since θ j tends to zero. For this reason we simulated separately the two phases (imbibition and drying) using two different lower bounds for the CFL taking into account the evolution of θ j in the two cases. Then, using the Euler's method for approximation of the third equation in (11), we can write the discretized problem as:
θ k+1 j = θ k j + ∆t ∆ j ((n k /n 0 ) 2 , B k ) − ∆tK T H ε (θ k j )θ k+1 j , j = 1, . . . , N − 1 c k+1 s,j = c k s,j + ∆t[K s c k i,j (n k j − θ k j ) + K(c k i,j −c) + θ k j ], j = 0, . . . , N n k+1 j = n 0 − γc k+1 s,j , j = 0, . . . , N c k+1 i,j = 1 θ k+1 j θ k j c k i,j + λ |V k j+1 |c k i,j+1 −2|V k j |c k i,j +|V k j−1 |c k i,j−1 2 +∆t∆ j (Dθ k , c k i ) + λ V k j+1 c k i,j+1 −V k j−1 c k i,j−1 2 − ∆t[K s c k i,j (n k j − θ k j ) + K(c k i,j −c) + θ k j ] , j = 1, . . . , N − 1,(35)
with suitable boundary conditions described in the next subsections. In particular, for the first equation of the scheme we have:
θ k+1 j = C(θ k j + ∆t ∆ j ((n k /n 0 ) 2 , B k ))(36)
with
C = 1 1 + ∆tK T H ε (θ k j )
.
Note that the scheme in the last equation of (35) may become degenerate if θ k+1 j is null, thus we put into the numerical algorithm a threshold for θ in order to avoid this possibility.
Boundary conditions for the imbibition phase
At the bottom boundary of the brick, we assume the condition for the ion content according to the concentration value of the experiment (18), which reads as
c k+1 i,0 =c i(37)
and the condition (47). At the top boundary of the brick, we impose the zero ion flux condition (20) for the ion content, discretized with a second order approximation:
c k+1 i,N = 4 3 c k+1 i,N −1 − 1 3 c k+1 i,N −2 .(38)
Let us now consider the discretization of the condition (14), reproducing the exchange with the environment. Note that in the case of the experiment 1 with sole water in the condition (14) we have to replace n k j with the constant value n 0 . At the node z N we need to solve the equation
3 2∆z B(θ k+1 N /n k+1 N ) + K l |θ l − θ k+1 N | α−1 (θ l − θ k+1 N ) = 4B(θ k+1 N −1 /n k+1 N −1 ) − B(θ k+1 N −2 /n k+1 N −2 ) 2∆z ,(39)
with the function to be inverted
g 1 (θ) = 3 2∆z B(θ/n) − K l |θ l − θ| α−1 (θ l − θ).
The invertibility condition is
g 1 = 3 2n∆z ∂ θ B(θ/n) + K l α|θ l − θ| α−1 > 0(40)
on a compact set, with
∂ θ B(s = θ/n) = d a+1 n 0 θ n 0 − θ n 0 2 1 n 0 − a n 0 , if s ∈ [a, 1], 0, elsewhere .(41)
Note that the condition ∂ θ B(θ/n) > 0 is always satisfied for θ ∈ [a · n, n], so that (40) holds. Therefore, at the upper boundary of the brick we need to solve, using for example with Newton's method:
θ k+1 N = g −1 1 4B(θ k+1 N −1 /n k+1 N −1 ) − B(θ k+1 N −2 /n k+1 N −2 ) 2∆z .(42)
13
Boundary conditions for the drying phase
In order to model the loss of water, we use the zero ion flux at the bottom of the brick, discretized with a second order approximation as
c k+1 i,0 = 4 3 c k+1 i,1 − 1 3 c k+1 i,2 ,(43)
and condition (38) at the top boundary. Let us now consider the discretization of the conditions (24) and (26), reproducing the situation of the specimen inside the oven we set at the lower boundary:
θ k+1 0 =θ l ,(44)
and analogously at the upper boundary:
θ k+1 N =θ l ,(45)
withθ l = 0.
5 Numerical Results and comparison with experimental data 5.1 Calibration of parameters a, c, K l , K T , α.
Now we describe the calibration procedure to determine a, c, K l , K T and α for both the phases of imbibition and evaporation of water in the brick using the experimental data of experiment 1.
We need to compute the total quantity of water absorbed and lost by the brick at time t k given by:
h 1 0 ρ l θ(z, t k )dz,(46)
thus we need to solve problem (12). We compute θ(z, t k ) numerically with the forward-central approximation scheme
and Neumann condition θ z (0, t) = 0 of null flux, only for the drying phase, that numerically results to be θ k+1
0 = 4 3 θ k+1 1 − 1 3 θ k+1 2 .(48)
Let us define t s the saturation time at the end of the imbibition phase and Q s the corresponding value. Then we compute the approximated values of the quantity of water in the brick Q num k at time t k as follows. With the trapezoidal rule we compute the integral (46): in order to compare the numerical quantity of water to experimental data Q k at time t k . The error to be minimized is then defined as E(a, c, K l , K T , α) = 1 N meas
Q num k = ρ ∆z 2 θ k 0 + 2 N −1 j=1 θ k j + θ k N , Quantities Value Dimensions a 0.21904 - c 9.8073 × 10 −4 cm 2 s −1 K l 3 × 10 −5 s −1 K T 3.2 × 10 −7 s −1 α 0.9 -Nmeas k=1 |Q num k − Q k | |Q| ,
withQ the average value among data. The calibration procedure has been carried out in MATLAB c applying the simulated annealing method. The computational time for a single simulation with fixed parameters takes 900 seconds on an Intel(R) Core(TM) i7-3630QM CPU 2.4 GHz. Table 4 lists the results obtained within an error of about 7%. Figure 4 shows the comparison between measured data and numerical simulations after calibration.
Calibration of constants K s and γ
As described in Section 2.2 for experiment 2, the bricks were first broken in four pieces with similar dimensions, both for the treated and non treated cases; for any brick, we measured its salt content.
In order to determine constant K s and γ we need to define an appropriate functional to be minimized. We proceed as follows. First we define the average quantity of salt in i-th brick as:
A i B i b i a i c s (z,t)dz = 1 h b i b i a i c s (z,t)dz for i = 1, 2, 3, 4.(49)
wheret is a sufficiently long time when we can assume that the water is completely evaporated.
Here A i and B i represent the cross section and the volume of brick i, respectively. If we denote by q num i the average quantity of salt in i-th brick obtained discretizing formula (49) with the trapezoidal quadrature rule, the values of K s and γ can then be found solving the following minimization problem min Ks,γ
1 4 4 i=1 |q num i − q i | |q| ,(50)
withq the average salt content among the four bricks. For the case without inhibitor, the profiles of the quantities obtained numerically at the end of the imbibition experiment (47 days) for the not treated bricks, indicated by NTi, i = 1, 2, 3, 4, are depicted in Figg. 5 and 6. As expected, the quantity of water in the brick θ l is a decreasing function of the height of the brick, since the top of the brick is interested by water exchange with the exterior. The graphs of the same quantitites at the end of the drying phase are depicted in Figg. 7 and 8. We observe that the amount of bound salts is, as expected, an increasing function of the height of the brick, since crystals mostly form where the quantity of water is lower. The calibration procedure gives the following result: we obtain an error of about 11.6% for the values K s = 4.1 · 10 −5 s −1 and γ = 0.6 cm 3 g −1 . In Table 5 we report the comparison between measured data and numerical values obtained using the parameters deriving from the calibration procedure.
For the experiment of the bricks treated with PC-10 −6 M , at the end of the calibration procedure we obtain an error of about 13.7% for the values K s = 6 · 10 −5 s −1 and γ = 0.53 cm 3 g −1 .
In Table 6 we report the comparison between measured data and numerical values obtained using the parameters deriving from the calibration procedure for the four bricks, indicated by PCi, i = 1, 2, 3, 4. In Fig. 9 we depicted the profile of c s for the not treated case (NT) and in presence of PC-10 −6 M (PC). As observed experimentally, the amount of salt crystals is higher in the case of the treatment with the crystallization modifier. Table 5: Salt content in any small brick in the not treated case (NT). We reported the measured salt content q i and the numerical values q num i expressed in mg/cm 3 . Table 6: Salt content in any small brick in the treated case (PC). We reported the salt content q i and the numerical values q num i expressed in mg/cm 3 .
Let us define the average porosity in i-th brick as:
1 h b i b i a i
n(z,t)dz for i = 1, 2, 3, 4,
then for completeness, we report in the next Table 7 the average porosity obtained numerically n num i , i = 1, . . . , 4 for the four bricks obtained discretizing (51) with the trapezoidal quadrature rule, both in the not treated (NT) and treated (PC) case. Table 7: Porosity in any small brick. We reported the average value for the porosity n num i , i = 1, . . . , 4 for the not treated and the treated bricks.
Conclusions
We developed a mathematical model to describe the action of crystallization inhibitors into a porous stone. This simple model is able to capture the main features of the inhibitor from experiments carried out on a set of commercially available bricks. According to the current knowledge, the model describes the action of inhibitors through two coefficients: crystallization rate, K s , taking into account nucleation, and the specific volume γ, taking into account the crystal habit modification. From the calibration of the mathematical model described in section 3, we found out that the action of phosphocitrate (PC) increases the crystallization rate and decreases the crystal specific volume. This means that, although crystals form faster in the presence of the inhibitor, nevertheless they occupy a smaller volume, thus lowering the development of tensile stresses, and, on the other hand, ensuring the hydraulic continuity into the porous stones. In the future, we will repeat the same study varying the materials and with more detailed experiments in order to test and improve our mathematical model. Our aim is to end up with a sound simulation tool to investigate crystallization modifier.
− 1 − 2s) + (1 − a) , if s ∈ [a, 1], 0, if s ∈ [0, a), B(1) = 2 3 c(1 − a), if s > 1. Summing up, Darcy's law can be expressed as follows:
Figure 1 :
1Graph of the functions B a,c (θ) (above) and B a,c (θ) (below) for s ∈ [a, 1] for the choice a = 0.219 and c = 9.87 · 10 −4 .
Figure 2 :
2Profile of the function H ε (θ), with ε = 0.25 an.
2 (B a,c (θ k j+1 /n 0 ) − 2B a,c (θ k j /n 0 ) + B a,c (θ k j−1 /n 0 )) with the boundary condition at the top boundary (42) under the CFL condition ∆t ∆z 2 ≤ n 0 2∂ z B a,c = n 0 2c , with θ k j = θ(z j , t k ), z j = j∆z, j = 0, ..., N = h 1 ∆z , {t k } k=1,...,Nmeas . At the bottom boundary we use the imbibition condition
Figure 4 :
4Data fitting result: comparison between data points and fitting values obtained for a = 0.21904, c = 9.8073 × 10 −4 , K l = 10 −5 , K T = 3.2 × 10 −7 , α = 0.9.
Figure 5 :
5Experiment 2. Imbibition phase in the salty solution: profile of θ l and c i depicted at the final time of the experiment T = 1128 h, with K s = 4.1 · 10 −5 s −1 and γ = 0.6.
Figure 6 :
6Experiment 2. Imbibition phase in the salty solution: profile of c s and n depicted at the final time of the experiment T = 1128 h, with K s = 4.1 · 10 −5 s −1 and γ = 0.6.
Figure 7 :
7Experiment 2. Drying phase: profile of θ l and c i , with K s = 4.1·10 −5 s −1 and γ = 0
Figure 8 :
8Experiment 2. Drying phase: profile of c s and n, with K s = 4.1·10 −5 s −1 and γ = 0.6. The computational time for a single simulation with fixed parameters both for the treated and the not-treated case takes 2240 seconds on an Intel(R) Core(TM) i7-3630QM CPU 2.
Figure 9 :
9Experiment 2. Comparison between the profile of c s without and in presence of PC-10 −6 M .
Table 1
1shows the pore size distribution of the considered brick.Pore Radius
Interval (µm)
0.001-0.01 0.01-0.1 0.1-1 1-10
Distribution (%)
2.8
9.3
42.2 45.7
Table 1: Pore size distribution in the brick under consideration.
2.1 Experiment 1: brick's capillary absorption and drying test in pure water.
Table 2 :
2Parameters of the problem.Description
Units
a
physical property of the porous matrix
-
c
physical property of the porous matrix cm 2 /s
γ
Specific volume of crystal
cm 3 /g
K l
Exchange coefficient
cm/s
K s
Crystallization rate coefficient
s −1
K
Growth rate of hydrated crystals
s −1
K T
Evaporation rate
s −1
α
Evaporation exponent
-
Table 3 :
3Model coefficient to be calibrated
Table 4 :
4Results of the calibration for the imbibition and drying stages without salts. The overall error is about 7%.
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| []
|
[
"Multirefringence phenomena in nonlinear electrodynamics",
"Multirefringence phenomena in nonlinear electrodynamics"
]
| [
"Vitorio A De Lorenci ",
"Renato Klippert ",
"Shi-Yuan Li ",
"Jonas P Pereira ",
"\nInstituto de Física e Química\nInstitute of Cosmology\nDepartment of Physics and Astronomy\nUniversidade Federal de Itajubá\n37500-903ItajubáMGBrazil\n",
"\nInstituto de Matemática e Computação\nTufts University\n02155MedfordMassachusettsUSA\n",
"\nSchool of Physics\nUniversidade Federal de Itajubá\n37500-903ItajubáMGBrazil\n",
"\nShandong University\n250100JinanP. R. China\n",
"\nDipartamento di Fisica and ICRA\nUniversité de Nice Sophia Antipolis\nSapienza Università di Roma, I06103, Cedex 2, 00185Nice, RomeFrance, Italy\n"
]
| [
"Instituto de Física e Química\nInstitute of Cosmology\nDepartment of Physics and Astronomy\nUniversidade Federal de Itajubá\n37500-903ItajubáMGBrazil",
"Instituto de Matemática e Computação\nTufts University\n02155MedfordMassachusettsUSA",
"School of Physics\nUniversidade Federal de Itajubá\n37500-903ItajubáMGBrazil",
"Shandong University\n250100JinanP. R. China",
"Dipartamento di Fisica and ICRA\nUniversité de Nice Sophia Antipolis\nSapienza Università di Roma, I06103, Cedex 2, 00185Nice, RomeFrance, Italy"
]
| []
| Wave propagation in nonlinear theories of the electromagnetism described by Lagrangian densities dependent upon its two local invariants L(F, G) is revisited. On the light of the recent findings in metamaterials, it is here shown that trirefringence is also a possible phenomenon to occur in the realm of such nonlinear theories. A specific model exhibiting this effect is investigated both in terms of phase and group velocities. It is claimed that wave propagation in some well known nonlinear models for spin-one fields, like QED and QCD in certain regimes, may exhibit trirefringence. 42.25.Lc, 42.25.Bs, 42.15.Dp | 10.1103/physrevd.88.065015 | [
"https://arxiv.org/pdf/1502.05663v1.pdf"
]
| 54,545,936 | 1502.05663 | 44be8b2b294b88122f4aa5971f3cd53466ef1aa0 |
Multirefringence phenomena in nonlinear electrodynamics
19 Feb 2015
Vitorio A De Lorenci
Renato Klippert
Shi-Yuan Li
Jonas P Pereira
Instituto de Física e Química
Institute of Cosmology
Department of Physics and Astronomy
Universidade Federal de Itajubá
37500-903ItajubáMGBrazil
Instituto de Matemática e Computação
Tufts University
02155MedfordMassachusettsUSA
School of Physics
Universidade Federal de Itajubá
37500-903ItajubáMGBrazil
Shandong University
250100JinanP. R. China
Dipartamento di Fisica and ICRA
Université de Nice Sophia Antipolis
Sapienza Università di Roma, I06103, Cedex 2, 00185Nice, RomeFrance, Italy
Multirefringence phenomena in nonlinear electrodynamics
19 Feb 2015arXiv:1502.05663v1 [physics.optics]numbers: 4215-i4225Lc4225Bs4215Dp
Wave propagation in nonlinear theories of the electromagnetism described by Lagrangian densities dependent upon its two local invariants L(F, G) is revisited. On the light of the recent findings in metamaterials, it is here shown that trirefringence is also a possible phenomenon to occur in the realm of such nonlinear theories. A specific model exhibiting this effect is investigated both in terms of phase and group velocities. It is claimed that wave propagation in some well known nonlinear models for spin-one fields, like QED and QCD in certain regimes, may exhibit trirefringence. 42.25.Lc, 42.25.Bs, 42.15.Dp
I. INTRODUCTION
As it is well known, nonlinear theories of electromagnetism exhibit birefringence phenomenon. The most popular example appears in the quantum electrodynamics (QED) where polarization effects are activated in the limit of large fields (B cr ∼ E cr = m 2 c 3 /e ), inducing an effective optical axis in the vacuum. In such situation, a light ray is expected to split in two rays propagating with different velocities [1][2][3][4]. An experimental setup designed to measure the birefringent properties of the QED vacuum was long ago proposed [5]. However, direct measurements of this effect are not yet conclusive and are still under consideration [6]. The influence of the nontrivial vacua on the propagation of electromagnetic waves was discussed in several distinct physical configurations [7][8][9][10][11][12]. Conditions for the occurrence of birefringence of gluon fields was also studied [13].
In the context of material media, birefringence effects are expect to occur in several distinct situations. It occurs naturally, for instance in certain crystals presenting optical axes [14,15], or artificially when optical axes are induced by means of external applied electromagnetic fields [16,17]. Nowadays, birefringent materials and methods including this effect have been incorporated in * [email protected] † [email protected] ‡ [email protected] § [email protected] several technological devices [18]. Birefringence is also a powerful optical tool to investigate properties of new materials, biological systems and others [19][20][21].
On the other hand, trirefringence was only recently considered as a possible phenomenon in material media. It was measured [22] in tailored photonic crystals [23], and the theoretical description of this effect in media characterized by effective dielectric coefficients was proposed [24]. In this case [24], only when some of the dielectric coefficients are negative, could trirefringence take place. Metamaterials [25][26][27][28] seem to be good candidates for supporting this effect, due to the controllability of their dielectric tensors. With the present day technology of producing such new media, it is expected that trirefringence will play some important role in technology of optical systems, as birefringence has done.
Usually, effects occurring in the realm of Maxwell electromagnetism in material media are expected to occur in the realm of nonlinear electromagnetic theories. It is possible to build up analogue models between these two domains where the coefficients describing a specific dielectric medium are mapped as derivatives of the Lagrangian density describing a nonlinear theory. In this way, trirefringence should also be a possible effect in nonlinear electromagnetism. By deriving and using the general description for wave propagation in the limit of geometrical optics, in this paper we show that trirefringence is in fact a possible effect in nonlinear electrodynamics. It is not our purpose to set the general conditions a model must fulfill in order to present trirefringence, but only to show the effect as a possible one in the domain of nonlin-ear electromagnetism. A particular model is thus investigated where such phenomenon is shown to occur provided that convenient external fields are set. The phase and group velocities of the waves are derived, as well as the corresponding polarization vectors. A numerical example is graphically studied.
The model examined in the paper corresponds to the effective Lagrangian density for QED in the regime of large fields. The trirefringence phenomenon is shown to occur whenever the model applies, although its measurability requires the control of very large fields. A possible arena to search for this effect could be the special fluids recently produced by high energy collisions, as addressed later in the concluding section. Further, this model is also useful in the context of analogue models in material media, where the predicted effect could be tested on optical systems, for instance in metamaterials.
In the next section, nonlinear electromagnetism is briefly revisited and the field equations are presented in terms of the general two parameters Lagrangian density. In Sec. III, the corresponding wave propagation is examined. The eigenvalue problem is stated and solved, resulting in the general fourth degree equation for the phase velocities. This equation is solved for a specific nonlinear model in Sec. IV. The corresponding polarization states and the description of the effect in terms of group velocities are also discussed. Conclusions and final remarks are presented in Sec. V.
Throughout this paper we employ the Minkowski metric η µν = diag(+1, −1, −1, −1). The completely skewsymmetric tensor η αβµν is defined by η 0123 = 1. We set the units such that the velocity of light in empty space is c = 1.
II. NONLINEAR ELECTRODYNAMICS: FIELD EQUATIONS
Nonlinear Abelian theories for electromagnetism can be formulated by means of the general Lagrangian density L = L(F, G), where F and G are the two local gauge invariants of the electromagnetic field. These invariants are defined in terms of the electromagnetic tensor field F µν , and its dual *
F αβ = 1 2 η αβ στ F στ ,(1)
as
F = F µν F µν (2) G = F µν * F µν .(3)
In terms of the electric E and magnetic B field strengths we have F = −2(E 2 − B 2 ) and G = −4 E · B.
The field equation can be obtained from the least action principle and it can be presented as [3],
2N µναβ F αβ,ν + L F F µν ,ν = 0,(4)
where N µναβ is defined by
N µναβ . = L F F F µν F αβ + L GG * F µν * F αβ +L F G F µν * F αβ + * F µν F αβ .(5)
Use is being made here of the notation L X 1 X 2 ···X n = ∂ n L/∂X 1 ∂X 2 · · · ∂X n previously introduced [3], where each X i is one of the two invariants F or G upon which the Lagrangian L arbitrarily depends. We notice that the above defined rank-4 tensor presents the following symmetries:
N µναβ = −N νµαβ , N µναβ = −N µνβα and N µναβ = N αβµν .
In addition to Eq. (4), F µν satisfies the Bianchi identity * F µν ,ν = 0, which implies in the existence of a potential vector A µ as
F µν = A µ,ν − A ν,µ .(6)
III. NONLINEAR ELECTRODYNAMICS: WAVE PROPAGATION
Let us now discuss the propagation of electromagnetic waves in the general formulation of nonlinear electrodynamics. We restrict ourselves to the propagation of monochromatic waves in the limit imposed by geometrical optics [14,15]. The method of field discontinuities will be used, which can be briefly stated as follows [3,29].
Consider a differentiable inextendible oriented borderless hypersurface Σ, defined locally by φ(x µ ) = 0, where φ is a real differentiable scalar field which locally is a function of the spacetime coordinates x µ = (t, x). Let U + be the spacetime points whose coordinates satisfy φ(x µ ) > 0, and similarly U − be such that φ(x µ ) < 0. Let P be any given point of Σ. For each sufficiently small r > 0, let V r (P ) be a neighborhood of P which consists of the spacetime points Q whose Euclidean distance from P is [(t Q − t P ) 2 + || x Q − x P || 2 ] (1/2) smaller than r. Let P + ∈ U + ∩ V r (P ) and P − ∈ U − ∩ V r (P ) be any two neighbor points from P arbitrarily chosen at opposite sides of Σ. Let f be any given tensor field defined at V r (P ). The Hadamard discontinuity at P of f across Σ is defined as
[f ] Σ (P ) . = lim r→0 + f (P + ) − f (P − ) .(7)
Suppose f such that [f ] Σ = 0 for each P ∈ Σ. Following Hadamard [29], we have [f ,λ ] Σ (P ) = k λf (P ), where k λ = φ ,λ | P is the normal vector to Σ at P andf is a tensor field defined at Σ with the same rank and the same algebraic symmetries as those of f .
We assume the electromagnetic tensor field F µν to be smooth in each U ± , but merely continuous at Σ (that is to say, the F µν are continuous functions at Σ but their derivatives may present discontinuities at Σ). The Hadamard discontinuities at Σ of Eq. (4) and of the derivative of Eq. (6) lead to [3]
f βλ k λ + 2 L F N β µνρ f νρ k µ = 0(8)
and
f αβ = ǫ α k β − ǫ β k α ,(9)
where the quantities f αβ are related to the derivatives of F αβ on Σ by [F αβ,λ ] Σ = f αβ k λ and ǫ µ is the polarization vector [A µ,αβ ] Σ = e µ k α k β . We set k λ = ωV λ + q λ as the wave 4-vector, where V λ = δ 0 λ is the 4-velocity of the observer which decomposes F µν into electric and magnetic fields. The components of this 4-vector k λ are thus the frequency ω and the wave vector q = qq, where q 2 = −q λq λ = 1. Taking together Eqs. (8) and (9), we obtain the general eigenvalue equation [1,3]
Z µ ν ǫ ν = 0,(10)
where
Z µ ν . = k 2 δ µ ν + 4 L F N µα νβ k α k β ,(11)
with k 2 . = k λ k λ . Nontrivial solutions of Eq. (10) can be found only if det | Z µν | = 0, the well known generalized Fresnel equation, and yields
α(k 2 ) 2 + βf 2 k 2 + γ(f 2 ) 2 = 0,(12)
where
f 2 . = F αµ F α ν k µ k ν , and α = L 2 F + 2L F (GL F G − F L GG ) −(L F F L GG − L 2 F G )G 2 ,(13)β = 4L F (L F F + L GG ) − 8(L F F L GG − L 2 F G )F, (14) γ = 16(L F F L GG − L 2 F G ).(15)
One can also recast the quantity f 2 that appears in Eq.
(12) as
f 2 = ( q · E) 2 − ω 2 E 2 + (q · B) 2 − q 2 B 2 + 2ω q · E × B. (16)
The phase velocity v .
= ω/q of the electromagnetic waves can be obtained from Eq. (12). In fact, it is straightforward to show that this equation can be presented as a fourth-degree equation for the phase velocity v as
a 4 v 4 + a 3 v 3 + a 2 v 2 + a 1 v + a 0 = 0,(17)
where we have defined
a 4 . = α − βE 2 + γE 4(18)a 3 . = 2(β − 2γE 2 )q · E × B(19)a 2 . = −2α + β[E 2 − B 2 + (q · E) 2 + (q · B) 2 ] +2γ{2(q · E × B) 2 − [(q · E) 2 + (q · B) 2 −B 2 ]E 2 }(20)a 1 . = −2{β − 2γ[(q · E) 2 + (q · B) 2 −B 2 ]}q · E × B(21)a 0 . = α + β[B 2 − (q · E) 2 − (q · B) 2 ] +γ[B 2 − (q · E) 2 − (q · B) 2 ] 2 .(22)
As stated by Eq. (17), we can find up to four solutions for the phase velocity in the same wave direction. In the next section we will analyze some special cases where multirefringence phenomena may occur. Dispersion relations for light propagation in nonlinear electrodynamics can also be investigated by means of the photon mass operator [30,31]. In such context the propagation of photons in homogeneous magnetic field was investigated long ago [32] and birefringence phenomena was described for some field configurations.
IV. A MODEL FOR TRIREFRINGENCE
In what follows we shall study a particular model for nonlinear electromagnetism which presents interesting multirefringence features. Let the one-parameter nonlinear Lagrangian density
L N L = − 1 4 b 0 F log F λ 2 ,(23)
where b 0 and λ are constants. Particularly these constants can be chosen in order to split the above model in a Maxwellian part plus a nonlinear contribution. This model appears in different contexts in the literature, for instance as the effective Lagrangian density for quantum electrodynamics (QED) [33] in the regime of large fields [34][35][36][37]. In this case the constants in Eq. (23) are related to the fine structure constant α and to the critical electromagnetic fields for which vacuum polarization effects begin to become important. Furthermore, with an appropriate choice of dielectric coefficients this model can describe several kinds of magnetic materials which can be used as analogue systems to investigate properties of the vacuum of the non-Abelian gauge field [38]. A more detailed discussion about this issue is presented in the concluding section.
A. Phase velocity
The study of wave propagation in this special case can be done by following the lines presented in Sec. III. We will investigate multirefringence phenomena using L N L in the Abelian case only.
To study a simplified case, let us assume constant external electric E = Ex and magnetic B = Bŷ fields, much larger than their wave counterparts. The wave vector q is assumed to lie in the xz-plane so that,q ·ŷ = 0, q ·x = sin θ, andq ·ẑ = cos θ. Thus, θ is the angle betweenq andẑ directions. From the above notation, q · E × B = EB cos θ. Taking L N L in Eq. (17) we obtain the following results v 0 = 1,
v ± = 2EB cos θ ± (χ − 2B 2 )(χ − 2E 2 cos 2 θ) χ ,(24)
where it was introduced the shortcut
χ . = 2E 2 − F 2 1 + log F λ 2 .(26)
The quantity v 0 is isotropic and does not depend on any choice for the configuration of the fields nor direction of propagation. However, the other solutions v ± will be different for different configurations of fields andq direction, set by the angle θ. As one can see, three different velocities in the same direction occur provided the square-root is smaller than 2BE cos θ. This naturally imposes conditions on the fields. We observe that trirefringence will occur in a region defined by −θ c < θ < θ c , where
θ c = arccos − F 4E 2 3 + log F λ 2 .(27)
Therefore, it does exist iff
0 < F < F c , F c λ 2 . = e −3 ≃ 0.0498(28)
and
E > E m , E 2 m . = − F 4 3 + log F λ 2 .(29)
Birefringence takes place in the regions θ c < θ < π − θ c and −(π − θ c ) < θ < −θ c . For the remaining angles, just the ordinary wave exists. When F = F c , trirefringence is not present for any direction. If one keeps F fixed and satisfying Eq. (28) and decreases the electric field [satisfying Eq. (29)], then the region where trirefringence takes place decreases and the difference between the moduli of the extraordinary solutions increases. In Fig. 1, the normal surfaces [14,15] associated with Eqs. (24) and (25) are plotted for a selected set of the fields. It can easily be seen there the region exhibiting trirefringence, as anticipated, for fields satisfying Eqs. (28) and (29).
B. Polarization
Let us now return to the Fresnel-like eigenvalue Eq. (10). The matrix Z µ ν given by Eq. (11) reduces, for the Lagrangian in Eq. (23), to
Z µ ν q 2 = (v 2 − 1)δ µ ν + 4 F 1 + log F λ 2 vE µ + (q · E)V µ +(q × B) µ vE ν + (q · E)V ν + (q × B) ν . (30)
This suggests that the polarization vector ǫ ν should conveniently be decomposed as a linear combination of the three vectors which appear in Z µ ν as (29); hence trirefringence is present. Since this is the case, also birefringence and one refraction must take place. The regions where these effects take place are related to the critical angle, θc, as given by Eq. (27). The trirefringent region lies between the dotted straight lines, while the birefringent regions are constituted by the angles limited by the dotted and dot-dashed straight lines. The region presenting one refraction lies between the dot-dashed straight lines.
ǫ ν = avE ν + b(q · E)V ν + c(q × B) ν + dk ν ,(31)
where a, b, c are arbitrary constants with the same physical dimension. The fourth term, with an arbitrary constant d, was introduced because Eq. (9) remains unchanged by it. Equation (10) then reads
[a(v 2 − 1) − Y ]vE µ + [b(v 2 − 1) − Y ](q · E)V µ +[c(v 2 − 1) − Y ](q × B) µ = 0,(32)
where Y is a shortcut for
Y . = 4 F 1 + log F λ 2 av 2 E 2 − (a + c)vq · ( E × B) +c(q × B) 2 − b(q · E) 2 .(33)
For the v 2 = 1 case, Eq. (32) yields Y = 0, from which the polarization state is given by a = (q · E) 2 , b = E 2 − (q × B) 2 , c = −(q · E) 2 up to a global multiplicative factor. For the v 2 = 1 case, Eq. (32) yields a = b = c = 1 up to a global multiplicative factor. Assuming E · B = 0 andq · B = 0, then the phase velocities for this case are given by Eq. (25). Once the physical parameters a, b, c were found in either case, then the particular gauge choice d = −b(q · E)/(qv) ensures ǫ ν to lie in the space orthogonal to V ν .
C. Group velocity
It is well known that, in the geometrical optics approximation the wave equation is a linear equation for the perturbed fields, even in the context of nonlinear electrodynamics [1]. In this case wave packets can be build up by superposing plane wave solutions, whose phase velocities were obtained above. Thus, it is important to deal with the group velocities u . = dω/d q for the propagation analysis [14]. For the particular case of the nonlinear Lagrangian density L N L , with the same configuration of fields and wave vectors we assumed above, the associated group velocities are
u = u xx + u zẑ ,(34)
where u x = sin θ and u z = cos θ for the ordinary v = 1 mode, thus stating that the group and phase velocities coincide for this case. For the extraordinary modes, we obtain
u ± x = χv 2 ± − 4EBv ± cos θ + 2E 2 cos 2 θ χv ± − 2EB cos θ sin θ,(35)
and
u ± z = χv 2 ± cos θ − 2EBv ± cos 2θ − 2E 2 cos θ sin 2 θ χv ± − 2EB cos θ ,(36)
where we are considering v ± as given by Eq. (25). Hence, two extraordinary rays and one ordinary ray can be found in the xz-plane. But their dependence on direction must be investigated more carefully, as we shall do in what follows. If one defines ϕ as the angle between the group velocity and the z-axis, then ϕ = θ for the v = 1 case. For the extraordinary modes, it follows from Eqs. (35)-(36) that
tan ϕ = (χv 2 ± − 4EBv ± cos θ + 2E 2 cos 2 θ) sin θ χv 2 ± cos θ − 2EBv ± cos 2θ − 2E 2 cos θ sin 2 θ ,(37)
which gives us ϕ as a function of θ. For the case of the extraordinary solutions, the analytical inversion of Eq. (37) to give θ as a function of ϕ is very involved. Hence, numerical analyses turns out to be more clarifying. Fig. 2 summarizes such a numerical analysis for the same parameters assumed in Fig. 1. One notices that the region of the xz plane where the ordinary and extraordinary group velocities can be found constitutes a trirefringent region. For the complementary region of the plane of propagation, there exists just the ordinary group velocity.
Comparing Figs. 1 and 2 we find angular sectors for which plane waves associated with the extraordinary polarization modes are supposed to propagate, but with no propagation of the corresponding wave packets. This feature relies on the fact that, in our analysis, the wave packets for the extraordinary modes do propagate along directions which are not generally equal to the directions of the corresponding plane waves components, as it is explicitly shown in Eq. (37). This is due to the fact that the phase velocity is dependent upon the direction of the wave vector; hence ω = v(q)q. This dependence leads to a term in the group velocity that is perpendicular to the phase velocity. The magnitude of such a term may be comparable to the magnitude of the phase velocity itself, yielding therefore to a possibly different behavior of the two aforementioned velocities.
V. CONCLUSION AND DISCUSSION
Trirefringence phenomena is not an effect exclusively occurring in nonlinear metamaterials [22,24]. As shown here, it is possible to formulate a nonlinear model describing electromagnetism where this effect is also expected to occur. Possible extensions of the presented model can be sought by adding the dual invariant G, or else trying other nonlinear Lagrangian models.
As it is well known, in the regime of small fields QED is governed by the Euler-Heisenberg effective Lagrangian density [33]. In this situation birefringence effect is predicted to occur [1]. Experiments are still under consideration in order to confirm such prediction [6,39]. However, when the regime of large fields is considered, the effec-tive Lagrangian governing QED presents the same form as the nonlinear model stated in Eq. (23), leading to the conclusion that trirefringence phenomenon is expected to occur in this regime.
Multirefringence phenomena could also be found for systems described by nonlinear Lagrangian densities which depend on non-Abelian gauge fields. In such cases, the field strength tensors would not be gauge-invariant. Nevertheless, it can be easily shown that the propagation of the field disturbances would be described by the same equations presented above, since only second-order derivatives of the gauge vector field may present non-zero Hadamard discontinuities.
The nonlinear model discussed in the text was proposed long ago [38,40,41] as the effective Lagrangian of quantum-chromodynamics (QCD) or other Yang-Mills theories with non-trivial vacuum properties. When considered for this purpose, the Lorentz invariant parameter F in L(F ) is extrapolated to be F µν(a) F µν(a) , where the index (a) runs in the inner Non-Abelian group space. Considering the regime of small coupling and taking the limit of large mean fields (F/λ 2 ≫ 1) a Lagrangian density with the same functional form as appearing in Eq. (23) is obtained [38]. In this context b 0 may be identified as a β-function coefficient at leading order and λ a constant related to the mass scale. It was claimed [42] that the form for L N L may also result in the leading terms in the limit of weak fields (F/λ 2 ≪ 1), as in both cases | log(F/λ 2 )| ≫ 1. Hence trirefringence of non-Abelian gauge fields can also occur and in principle it can be observed, provided the required field configuration is approached. The practical arena for such kind of study is a quark-gluon plasma (QGP), recently observed in highenergy heavy ion collision experiments where the gluon field is deconfined and can propagate in the bulk of the QGP. Some symmetric field configuration has been investigated, and a possible observable has been proposed [13]. With the progress of measurements on asymmetries in the experiments, which is now a hot topic in RHIC and LHC, other kind of field configurations can be further investigated.
FIG. 1 .
1(color online). Normal surfaces for the nonlinear theory described by Eq. (23). The ordinary wave is represented by the circular thick line and the extraordinary waves are represented by the dashed and dot-dashed curves. The symbols +, and − and o indicate the solutions presented in Eqs. (25) and (24). The chosen values of the electric and magnetic fields are B = 0.1λ and E = 0.09λ. These values satisfy Eqs. (28) and
FIG. 2 .
2(color online). Ray velocities for the nonlinear model described by Eq.(23). The fields are the same as inFig. 1. The symbols +, − and o indicate the solutions presented in Eqs.(25) and(24)when substituted in Eqs. (34)-(36). As it can be seen, trirefringence occurs in the region lying between the dotted straight lines. In the complementary region only the isotropic ray solution propagates.
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| []
|
[
"Model Independent Extraction of S-Matrix Poles from Experimental Data",
"Model Independent Extraction of S-Matrix Poles from Experimental Data"
]
| [
"S Ceci \nRudjer Bošković Institute\nBijenička 54HR-10000ZagrebCroatia\n",
"M Korolija \nRudjer Bošković Institute\nBijenička 54HR-10000ZagrebCroatia\n",
"B Zauner \nRudjer Bošković Institute\nBijenička 54HR-10000ZagrebCroatia\n"
]
| [
"Rudjer Bošković Institute\nBijenička 54HR-10000ZagrebCroatia",
"Rudjer Bošković Institute\nBijenička 54HR-10000ZagrebCroatia",
"Rudjer Bošković Institute\nBijenička 54HR-10000ZagrebCroatia"
]
| []
| By separating data points close to a resonance into intervals, and fitting all possible intervals to a simple pole with constant coherently added background, we obtained a substantial number of convergent fits. After a carefully chosen set of statistical constraints was imposed, we calculated the average of a resonance pole position from the statistically acceptable results. We used this method to find pole positions of Z and N(1440) resonances, and to show that the strong discrepancy between the old and new measurements of the Υ(11020) mass stems from specious comparison of the Υ(11020) pole with its Breit-Wigner mass.Breit-Wigner (BW) parameters are often used for the description of unstable particles (see e.g. Review of Particle Physics [1]), although shortcomings of such choice have been pointed out on numerous occasions. For example, Sirlin showed that the BW parameters of the Z boson were gauge dependent[2]. To resolve this issue he redefined BW parameters, but also suggested usage of the S-matrix poles as an alternative, since poles are fundamental properties of the S-matrix and therefore gauge independent by definition. In a somewhat different study, Höhler advocated using S-matrix poles for characterization of nucleon resonances [3] in order to reduce confusion that arises when different definitions of BW parameters are used[4]. However, loosely defined [5] BW parameters of mesons and baryons are still being extracted from experimental analyses, compared among themselves [1], and used as input to QCD-inspired quark models [6] and as experiment-to-theory matching points for lattice QCD[7].The main motivation for this research are strong discrepancies between the old and newly obtained parameters of some well known resonances. In particular, BaBar collaboration recently reported that the mass of Υ(11020) is 10996±2 MeV [8], significantly different from the old value of 11019±2 MeV [1]. Furthermore, the width turned out to be less than a half of the expected, namely 37±3 MeV instead of 79±16 MeV.In this paper, we developed a reliable method for model-independent extraction of S-matrix pole positions directly from the data, and connected them to the Breit-Wigner parameters in order to understand the observed discrepancies. We showed that the both results, BaBar and PDG, are consistent: the old ones should be interpreted as BW mass and width, while the new ones are pole parameters.The first step in devising a method for extraction of the pole parameters from the experimental data is to set up * Email: [email protected] an appropriate parameterization. The parameterization presented here is based on the assumption that close to a resonance, the T matrix will be well described with a simple pole and a constant background. The similar assumption was used in Höhler's speed plot technique[3]. The speed plot is a method used for the pole parameter extraction from the known scattering amplitudes. It is based on calculating the first order energy derivative of the scattering amplitude, with the key assumption that the first derivative of the background is negligible.The T matrix with a single pole and constant background term is given bywhere W is center-of-mass energy, r b and b b are complex, while M p and Γ p are real numbers. Total cross section is then given by σ ≈ |T | 2 /q 2 , where q is the initial center-ofmass momentum. Equation(1), as well as other similar forms (see e.g.[1]), are standardly called Breit-Wigner parameterizations, which can be somewhat misleading since M p and Γ p are generally not Breit-Wigner, but pole parameters (hence the index p). The square of the T matrix defined in Eq. (1) is given bywhere, for convenience, we simplified the numerator by combining the old parameters into three new real-valued ones: T ∞ , M z , and Γ z . Pole parameters M p and Γ p are retained in the denominator. With such a simple parameterization, it is crucial to use only data points close to the resonance peak. To avoid picking and choosing the appropriate data points by ourselves, we analyzed the data from a wider range around the resonance peak, and fitted localy the parameterization (2) to each set of seven successive data points (seven data points is minimum for our five-parameter fit). Then we increased the number of data points in the sets to eight and fitted again. We continued increasing the number of data points in sets until we fitted the | null | [
"https://arxiv.org/pdf/1007.4207v1.pdf"
]
| 118,169,879 | 1007.4207 | 9728fc3c21f92242f5b637ca1ccb574aea024162 |
Model Independent Extraction of S-Matrix Poles from Experimental Data
23 Jul 2010
S Ceci
Rudjer Bošković Institute
Bijenička 54HR-10000ZagrebCroatia
M Korolija
Rudjer Bošković Institute
Bijenička 54HR-10000ZagrebCroatia
B Zauner
Rudjer Bošković Institute
Bijenička 54HR-10000ZagrebCroatia
Model Independent Extraction of S-Matrix Poles from Experimental Data
23 Jul 2010(Dated: July 27, 2010)
By separating data points close to a resonance into intervals, and fitting all possible intervals to a simple pole with constant coherently added background, we obtained a substantial number of convergent fits. After a carefully chosen set of statistical constraints was imposed, we calculated the average of a resonance pole position from the statistically acceptable results. We used this method to find pole positions of Z and N(1440) resonances, and to show that the strong discrepancy between the old and new measurements of the Υ(11020) mass stems from specious comparison of the Υ(11020) pole with its Breit-Wigner mass.Breit-Wigner (BW) parameters are often used for the description of unstable particles (see e.g. Review of Particle Physics [1]), although shortcomings of such choice have been pointed out on numerous occasions. For example, Sirlin showed that the BW parameters of the Z boson were gauge dependent[2]. To resolve this issue he redefined BW parameters, but also suggested usage of the S-matrix poles as an alternative, since poles are fundamental properties of the S-matrix and therefore gauge independent by definition. In a somewhat different study, Höhler advocated using S-matrix poles for characterization of nucleon resonances [3] in order to reduce confusion that arises when different definitions of BW parameters are used[4]. However, loosely defined [5] BW parameters of mesons and baryons are still being extracted from experimental analyses, compared among themselves [1], and used as input to QCD-inspired quark models [6] and as experiment-to-theory matching points for lattice QCD[7].The main motivation for this research are strong discrepancies between the old and newly obtained parameters of some well known resonances. In particular, BaBar collaboration recently reported that the mass of Υ(11020) is 10996±2 MeV [8], significantly different from the old value of 11019±2 MeV [1]. Furthermore, the width turned out to be less than a half of the expected, namely 37±3 MeV instead of 79±16 MeV.In this paper, we developed a reliable method for model-independent extraction of S-matrix pole positions directly from the data, and connected them to the Breit-Wigner parameters in order to understand the observed discrepancies. We showed that the both results, BaBar and PDG, are consistent: the old ones should be interpreted as BW mass and width, while the new ones are pole parameters.The first step in devising a method for extraction of the pole parameters from the experimental data is to set up * Email: [email protected] an appropriate parameterization. The parameterization presented here is based on the assumption that close to a resonance, the T matrix will be well described with a simple pole and a constant background. The similar assumption was used in Höhler's speed plot technique[3]. The speed plot is a method used for the pole parameter extraction from the known scattering amplitudes. It is based on calculating the first order energy derivative of the scattering amplitude, with the key assumption that the first derivative of the background is negligible.The T matrix with a single pole and constant background term is given bywhere W is center-of-mass energy, r b and b b are complex, while M p and Γ p are real numbers. Total cross section is then given by σ ≈ |T | 2 /q 2 , where q is the initial center-ofmass momentum. Equation(1), as well as other similar forms (see e.g.[1]), are standardly called Breit-Wigner parameterizations, which can be somewhat misleading since M p and Γ p are generally not Breit-Wigner, but pole parameters (hence the index p). The square of the T matrix defined in Eq. (1) is given bywhere, for convenience, we simplified the numerator by combining the old parameters into three new real-valued ones: T ∞ , M z , and Γ z . Pole parameters M p and Γ p are retained in the denominator. With such a simple parameterization, it is crucial to use only data points close to the resonance peak. To avoid picking and choosing the appropriate data points by ourselves, we analyzed the data from a wider range around the resonance peak, and fitted localy the parameterization (2) to each set of seven successive data points (seven data points is minimum for our five-parameter fit). Then we increased the number of data points in the sets to eight and fitted again. We continued increasing the number of data points in sets until we fitted the
By separating data points close to a resonance into intervals, and fitting all possible intervals to a simple pole with constant coherently added background, we obtained a substantial number of convergent fits. After a carefully chosen set of statistical constraints was imposed, we calculated the average of a resonance pole position from the statistically acceptable results. We used this method to find pole positions of Z and N(1440) resonances, and to show that the strong discrepancy between the old and new measurements of the Υ(11020) mass stems from specious comparison of the Υ(11020) pole with its Breit-Wigner mass. Breit-Wigner (BW) parameters are often used for the description of unstable particles (see e.g. Review of Particle Physics [1]), although shortcomings of such choice have been pointed out on numerous occasions. For example, Sirlin showed that the BW parameters of the Z boson were gauge dependent [2]. To resolve this issue he redefined BW parameters, but also suggested usage of the S-matrix poles as an alternative, since poles are fundamental properties of the S-matrix and therefore gauge independent by definition. In a somewhat different study, Höhler advocated using S-matrix poles for characterization of nucleon resonances [3] in order to reduce confusion that arises when different definitions of BW parameters are used [4]. However, loosely defined [5] BW parameters of mesons and baryons are still being extracted from experimental analyses, compared among themselves [1], and used as input to QCD-inspired quark models [6] and as experiment-to-theory matching points for lattice QCD [7].
The main motivation for this research are strong discrepancies between the old and newly obtained parameters of some well known resonances. In particular, BaBar collaboration recently reported that the mass of Υ(11020) is 10996±2 MeV [8], significantly different from the old value of 11019±2 MeV [1]. Furthermore, the width turned out to be less than a half of the expected, namely 37±3 MeV instead of 79±16 MeV.
In this paper, we developed a reliable method for model-independent extraction of S-matrix pole positions directly from the data, and connected them to the Breit-Wigner parameters in order to understand the observed discrepancies. We showed that the both results, BaBar and PDG, are consistent: the old ones should be interpreted as BW mass and width, while the new ones are pole parameters.
The first step in devising a method for extraction of the pole parameters from the experimental data is to set up * Email: [email protected] an appropriate parameterization. The parameterization presented here is based on the assumption that close to a resonance, the T matrix will be well described with a simple pole and a constant background. The similar assumption was used in Höhler's speed plot technique [3]. The speed plot is a method used for the pole parameter extraction from the known scattering amplitudes. It is based on calculating the first order energy derivative of the scattering amplitude, with the key assumption that the first derivative of the background is negligible.
The T matrix with a single pole and constant background term is given by
T (W ) = r p Γ p /2 M p − W − i Γ p /2 + b p ,(1)
where W is center-of-mass energy, r b and b b are complex, while M p and Γ p are real numbers. Total cross section is then given by σ ≈ |T | 2 /q 2 , where q is the initial center-ofmass momentum. Equation (1), as well as other similar forms (see e.g. [1]), are standardly called Breit-Wigner parameterizations, which can be somewhat misleading since M p and Γ p are generally not Breit-Wigner, but pole parameters (hence the index p). The square of the T matrix defined in Eq. (1) is given by
|T (W )| 2 = T 2 ∞ (W − M z ) 2 + Γ 2 z /4 (W − M p ) 2 + Γ 2 p /4 ,(2)
where, for convenience, we simplified the numerator by combining the old parameters into three new real-valued ones: T ∞ , M z , and Γ z . Pole parameters M p and Γ p are retained in the denominator. With such a simple parameterization, it is crucial to use only data points close to the resonance peak. To avoid picking and choosing the appropriate data points by ourselves, we analyzed the data from a wider range around the resonance peak, and fitted localy the parameterization (2) to each set of seven successive data points (seven data points is minimum for our five-parameter fit). Then we increased the number of data points in the sets to eight and fitted again. We continued increasing the number of data points in sets until we fitted the whole chosen range. Such procedure allowed different background term for each fit, which is much closer to reality than assuming a single constant background term for the whole chosen data set (see e.g. discussion on the problems with speed plot in Ref. [9]). In the end, we imposed a series of statistical constraints to all fits to distinguish the good ones.
In order to pinpoint the statistical strategy to be used, we did a substantial number of simulations with the data sets that had known poles and zeros. It turned out that the most successful strategy was to make an ordered list of all fit results, from best to worst, and then to drop the worst three quarters using the following goodness-offit measures: Akaike information criterion [11], Schwartz (Bayesian information) criterion [12], and P-values of the extracted fit parameters (in particular, M p and Γ p ). Eventually, we kept the intersection of the fits that satisfied all criteria. Results closest to the original poles were produced by averaging the obtained pole positions of all good fits. The standard deviation turned out to be a good estimate for errors of obtained parameters. All other approaches we tested, such as keeping only a handful of the best fits, or keeping just those whose values of reduced χ 2 were close to one, failed to accurately reproduce the originalpole parameters. The whole analysis was done in Wolfram Mathematica 7 using Nonlinear-ModelFit routine [13].
Having defined the fitting strategy, we tested the method by applying it to the case of the Z boson. The data set we used is from the PDG compilation [1], and shown in Fig. 1. Extracted pole masses are shown in the same figure: filled histogram bins show pole masses from the good fits, while the empty histogram bins are stacked to the solid ones to show masses obtained in the discarded fits. Height of the pole-mass histogram in Fig. 1 is scaled for convenience.
Extracted S-matrix pole mass and width of Z boson are given in Table I. The pole masses are in excelent agreement, while the pole widths are reasonably close. It is important to stress that the difference between the pole and BW mass of the Z boson is fundamental and statistically significant. Distribution of discarded and good results is shown in the lower part of Fig. 1. Next, we turn to the data from BaBar collaboration [8] to determine whether the PDG averages for Υ(11020), or the newly reported resonance parameters obtained in Ref. [8] are correct. Our local fit of the Υ(11020) pole parameters is shown in Fig. 2. As in the case of Z boson, the full and empty histograms show how many of the extracted pole mass fits were accepted or discarded in the analysis.
Average values obtained for the resonance mass and the width are given in Table II, together with the same parameters obtained in the BaBar analysis, and those quoted by PDG. Extracted pole parameters of Υ(11020) are practically the same as those reported in [8], even though our parameterization is much simpler (single pole plus constant background vs. two constant background, and two pole terms). The original results cited in PDG (from CUSB [15], and CLEO [16]) were obtained by fitting Gaussians to the resonance peaks in the data, and peak positions are usually closer to the BW mass.
To investigate this case further, we analyze another resonance with strong difference between the pole and BW mass, the Roper resonance N(1440). We extracted Roper resonance pole parameters from the πN elastic P 11 partial wave obtained in the GWU analysis [10]. According to PDG, this wave has a very rich structure: there is a four-star Roper resonance, a three-star N(1710) resonance, and a one-star N(2100) resonance. However, the GWU analysis reports only one resonance in this partial wave, the Roper N(1440) resonance. In a preliminary analysis, we could see some indication for all resonances mentioned by PDG but, for this study, we focus on N(1440) because of its unusually strong shift between the pole and BW mass (roughly 75 MeV).
Our results for N(1440) are given in Table III, where we see that the pole parameters are in an excellent agreement with the PDG estimates. Unlike the pole mass, BW masses are situated closer to the positions of the peak (see Figs. 2 and 3).
The field-theory reason for the resonance pole shift is the energy dependence of the imaginary part of resonance self energy, which is commonly modeled by the energy dependent width [18,19]. Equation (1) would be exact if the self energy was constant. In more realistic cases, the T-matrix denominator D(W ) is given by
D(W ) = M b − W − i Γ(W )/2,(3)
where we introduce the BW mass and M b , which is generally not the real part of the pole position. Keeping only the first two terms in Taylor expansion of Γ(W ) about W = M b (as done in Ref. [19]) the width becomes
Γ(W )/2 = Γ b /2 + tan θ (W − M b ),(4)
where the Γ b is a shorthand for Γ(M b ), and tan θ is the slope of Γ/2 at W = M b . The pole position M p − iΓ p /2 is obtained by solving D(W ) = 0, which yields
M p = M b + sin θ cos θ Γ b /2,(5)Γ p = cos 2 θ Γ b .(6)
Relations (5) and (6), originally introduced in Ref. [19], may be used to cross check estimates for pole and BW parameters. Table IV shows angles θ for all resonances analyzed in this paper, calculated from PDG estimates for pole and BW widths using Eq. (6). Since M p is smaller than M b , we chose negative θ solution (cf. Eq. (5). It turns out that the BaBar value of Υ(11020) mass is accurately reproduced. Does this θ carry any physical meaning? For resonances with one dominant decay channel, such as the ∆(1232), we can impose a unitarity condition (Im T = |T | 2 ) to Eq. (1) and learn that r p = e 2iθ , and b p = e iθ sin θ. It is the same θ and represents a half of the complex residue phase. Indeed, from Ref. [1] we read that ∆(1232) has (−47 ± 1) • for pole residue phase, quite consistent with -46 • , a double value of the θ from Table IV. However, this simple relation is lost when important inelastic channels are open, e.g. in the N(1440) case, where 2 θ ≈ -75 • , which is significantly larger than its residue phase -100 • [1]. The difference comes from different Γ(W ) in the denominator and numerator of T matrix: total decay width is in the denominator, while the partial widths are in the numerator. Since the energy dependence of the two is in general different, their slopes (i.e. tan θ) will be different as well.
Since our pole extraction method confirmed BaBar result, the successful cross check is the last piece of the puzzle. PDG estimates of Υ(11020) are consistent with BW parameters.
In conclusion, we have developed a model-independent method for extraction of resonance pole parameters from total cross sections and partial waves. Very good estimates for Z boson, Υ(11020), and N(1440) pole positions were obtained. Furthermore, we showed that the strong discrepancy between PDG estimates and BaBar result for Υ(11020) comes from specious comparison of the pole and BW mass.
We are today witnessing the dawn of ab-initio calculations in low-energy QCD. In order to compare theoretical predictions with experimentally determined resonance states, we need first to establish proper point of comparison. The case of Υ(11020) is a vivid example how particularly careful we must be when choosing this point. Therefore, we would like to express our concern about other potentially problematic comparisons between the pole and BW parameters in the literature, in particular in the Review of Particle Physics, and recommend drawing a clear distinction between the two in future publications.
PACS numbers: 11.55.Bq, 12.40.Yx, 13.25.Gv, 14.20.Gk, 14.40.Pq, 14.70.Hp.
figure] PDG compilation of Z data [1] and histogram of obtained pole masses. Line is the fit result with the lowest reduced χ 2 (just for illustration). Dark (red online) colored histogram bins are filled with statistically preferred results. [Lower figure] Pole masses vs. pole widths. Dark (red online) circles show statistically preferred results we use for averages.
FIG
. 2: Υ(11020) resonance pole obtained by our method. Grey (green online) rectangles represents PDG range for Υ(11020) mass (both figures) and width (only lowerfigure).
FIG. 3 :
3N(1440) resonance pole obtained by our method. Grey (green online) rectangles represents PDG range for N(1440) Breit-Wigner mass (both figures) and width (only lower figure).
S.C. owes a debt of gratitude to A.Švarc, S. Krewald, M. Manley, H. Haberzettl, M. Döring, A. Sibirtsev, V. Brigljević, S. Szilner, and N. Tepić for their support and will to participate in discussions about the topics addressed herein. This work is supported in part by the DAAD (Deutscher Akademischer Austauschdienst) grant No. D/08/00215 and the DFG (Deutsche Forschungsgemeinschaft, Gz.: DO 1302/1-2).
TABLE I :
IPole parameters of Z obtained in this work. PDG values of pole and BW parameters are given for comparison.Z
Pole
Pole PDG [1]
BW PDG [1]
M /MeV
91159 ± 8
91162 ± 2
91188 ± 2
Γ/MeV
2484 ± 10
2494 ± 2
2495 ± 2
TABLE II :
IIParameters of Υ(11020) meson. Pole parameters are results of this work.Υ(11020)
Pole
BaBar [1, 8]
PDG [1]
M /MeV
10999 ± 1
10996 ± 2
11019 ± 8
Γ/MeV
38 ± 1
37 ± 3
79 ± 16
TABLE III
III: N(1440) resonance parameters.
N (1440)
Pole
Pole PDG [1]
BW PDG [1]
M /MeV
1370 ± 6
1365 ± 15
1440 ± 30
20
Γ/MeV
197 ± 6
190 ± 30
300 ± 150
100
1100 1200 1300 1400 1500 1600 1700
0.
TABLE IV :
IVThe connection between S-matrix pole and Breit-Wigner parameters using only the PDG values.θ/ •
Mp/MeV PDG[1]
Mp/MeV Eq. (5)
∆(1232)
-23.0
1210 ± 1
1210
N (1440)
-37.3
1365 ± 15
1368
Υ(11020)
-46.8
10996 a ± 2
10999
Z
-1.26
91162 ± 2
91161
a BaBar value.
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. D M Manley, Phys. Rev. D. 514837D. M. Manley, Phys. Rev. D 51, 4837 (1995);
Lichtenberg. D B , Phys. Rev. D. 103865D. B. Licht- enberg, Phys. Rev. D 10, 3865 (1974).
| []
|
[
"Electromagnons in multiferroic RMn 2 O 5 compounds and their microscopic origin",
"Electromagnons in multiferroic RMn 2 O 5 compounds and their microscopic origin",
"Electromagnons in multiferroic RMn 2 O 5 compounds and their microscopic origin",
"Electromagnons in multiferroic RMn 2 O 5 compounds and their microscopic origin"
]
| [
"A B Sushkov [email protected] \nMaterials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland\n",
"M Mostovoy \nZernike Institute for Advanced Materials\nUniversity of Groningen\n9747 AGGroningenThe Netherlands\n",
"R Valdés Aguilar \nMaterials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland\n",
"S-W Cheong \nRutgers Center for Emergent materials and Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew Jersey\n",
"H D Drew \nMaterials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland\n",
"A B Sushkov [email protected] \nMaterials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland\n",
"M Mostovoy \nZernike Institute for Advanced Materials\nUniversity of Groningen\n9747 AGGroningenThe Netherlands\n",
"R Valdés Aguilar \nMaterials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland\n",
"S-W Cheong \nRutgers Center for Emergent materials and Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew Jersey\n",
"H D Drew \nMaterials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland\n"
]
| [
"Materials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland",
"Zernike Institute for Advanced Materials\nUniversity of Groningen\n9747 AGGroningenThe Netherlands",
"Materials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland",
"Rutgers Center for Emergent materials and Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew Jersey",
"Materials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland",
"Materials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland",
"Zernike Institute for Advanced Materials\nUniversity of Groningen\n9747 AGGroningenThe Netherlands",
"Materials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland",
"Rutgers Center for Emergent materials and Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew Jersey",
"Materials Research Science and Engineering Center\nUniversity of Maryland\n20742College Park, Maryland"
]
| []
| We summarize the existing experimental data on electromagnons in multiferroic RMn 2 O 5 compounds, where R denotes a rare earth ion, Y or Bi, and discuss a realistic microscopic model of these materials based on assumption that the microscopic mechanism of magnetically-induced ferroelectricity and electromagnon absorption relies entirely on the isotropic Heisenberg exchange and magnetostrictive coupling of spins to a polar lattice mode and does not involve relativistic effects. This model explains many magnetic and optical properties of RMn 2 O 5 manganites, such as the spin re-orientation transition, magneticallyinduced polarisation, appearance of the electromagnon peak in the non-collinear spin state and the polarisation of light for which this peak is observed. We compare experimental and theoretical results on electromagnons in RMn 2 O 5 and RMnO 3 compounds. | 10.1088/0953-8984/20/43/434210 | [
"https://arxiv.org/pdf/0806.1207v1.pdf"
]
| 204,978,360 | 0806.1207 | bf39df5f1ccb7fec72ab6ea2b0064695925fb30a |
Electromagnons in multiferroic RMn 2 O 5 compounds and their microscopic origin
6 Jun 2008
A B Sushkov [email protected]
Materials Research Science and Engineering Center
University of Maryland
20742College Park, Maryland
M Mostovoy
Zernike Institute for Advanced Materials
University of Groningen
9747 AGGroningenThe Netherlands
R Valdés Aguilar
Materials Research Science and Engineering Center
University of Maryland
20742College Park, Maryland
S-W Cheong
Rutgers Center for Emergent materials and Department of Physics and Astronomy
Rutgers University
08854PiscatawayNew Jersey
H D Drew
Materials Research Science and Engineering Center
University of Maryland
20742College Park, Maryland
Electromagnons in multiferroic RMn 2 O 5 compounds and their microscopic origin
6 Jun 2008Focus Issue Articlenumbers: 6320Ls 7550Ee7830Hv7530Et
We summarize the existing experimental data on electromagnons in multiferroic RMn 2 O 5 compounds, where R denotes a rare earth ion, Y or Bi, and discuss a realistic microscopic model of these materials based on assumption that the microscopic mechanism of magnetically-induced ferroelectricity and electromagnon absorption relies entirely on the isotropic Heisenberg exchange and magnetostrictive coupling of spins to a polar lattice mode and does not involve relativistic effects. This model explains many magnetic and optical properties of RMn 2 O 5 manganites, such as the spin re-orientation transition, magneticallyinduced polarisation, appearance of the electromagnon peak in the non-collinear spin state and the polarisation of light for which this peak is observed. We compare experimental and theoretical results on electromagnons in RMn 2 O 5 and RMnO 3 compounds.
Introduction
Multiferroic materials that exhibit simultaneous magnetic and ferroelectric order have attracted much attention recently because of the fundamental interest of systems with coupled order parameters and because of their potential for cross electric and magnetic functionality [1,2,3,4,5,6,7]. Many recently discovered multiferroics, e.g. TbMnO 3 [8], TbMn 2 O 5 [9], and Ni 3 V 2 O 8 [10], are improper ferroelectrics in which the electric polarisation is induced by spin ordering. These materials show striking cross-coupling effects such as magnetic field induced polarisation switching [8,9] and giant magnetocapacitance [11] and a strong coupling between spin and lattice excitations that leads to electric dipole excitation of magnons, or electromagnons [1,12,13,14], which is the subject of this paper. The fundamental interest in multiferrocity also derives from the strong interplay between magnetic frustration, ferroelectric order and unusual symmetry breaking in phase transformations that characterize these materials [6,15].
The known microscopic mechanisms of magnetically-induced ferroelectricity include lattice distortion (exchange striction) and redistribution of electron density in response to spin ordering. Such processes occur locally in all magnetic materials. However, only when a spin ordering breaks inversion symmetry do these local electric dipoles add into a macroscopic electric polarisation. Spin orders that break inversion symmetry are rare and the best systems to look for them are frustrated magnets, where competing interactions and the geometry of spin lattice favor unconventional magnetic states. In most of the recently discovered multiferroic materials, such as TbMnO 3 , Ni 3 V 2 O 8 , MnWO 4 and CuO, competing interactions force spins to form a cycloidal spiral. This non-collinear spin order breaks inversion symmetry and activates antisymmetric Dzyaloshinskii-Moriya interaction proportional to the crossproduct of spins, S 1 × S 2 [16]. The concomitant lattice and electronic distortion induces electric polarisation [17,18,19].
When inversion symmetry is broken by a collinear magnetic ordering the strongest spin interaction that can shift ions and polarise electronic clouds is the symmetric Heisenberg exchange, proportional to the scalar product of spins, S 1 · S 2 . This mechanism was proposed to explain multiferroic properties of RMn 2 O 5 , where R denotes a rare earth ion, Y or Bi, and orthorombic manganites showing the E-type antiferromagnetic ordering [20,21,22].
These two microscopic mechanisms of magnetically-induced ferroelectricity give rise to two different forms of phenomenological magnetoelectric coupling: electric polarisation induced by a low-pitch spiral is described by the third-order coupling term P (L 1 ∂ L 2 − L 2 ∂ L 1 ), where P is electric polarisation and L 1,2 are magnetic order parameters describing the sinusoidal and cosinusoidal components of the spiral [23,24], while the coupling working in collinear spin states has the form P L 2 1 − L 2 2 , where L 1 and L 2 are components of a two-dimensional irreducible representation describing magnetic states with opposite electric polarisations [25,20,22].
The magnetoelectric interactions that induce electric polarisation in magnets can also couple oscillations of magnetisation to polar lattice vibrations. The oscillations of polarisation at magnon frequency and vice versa give rise to dynamic magnetoelectric effects, such as , (b) Excitation of the two-magnon continuum via the third-order magnetoelectric coupling ('charged magnons') and (c) Photoexcitation of a single magnon (electromagnon) due to the third-order magnetoelectric coupling, where photon with the frequency ω and zero wave vector scatters off the static spin modulation with the wave vector Q producing a magnon with the same wave vector and the frequency ω. electromagnon excitations. In usual magnets, an oscillating electric field of photons can excite a three-particle continuum consisting of two magnons and one phonon [26]. This process results from the fourth-order spin-lattice coupling (see figure 1a). The third-order couplings in multiferroics, discussed above, make possible photo-excitation of two-magnon continuum without a phonon ('charged magnons' [27]) shown in figure 1b). Replacing one of the magnons by the static modulation of spin density appearing in the ordered spin state, we obtain a process that converts photon into a single magnon, which is the electromagnon (see figure 1c). This process is usually mediated by a polar phonon linearly coupled to both magnons and light, which for low-frequency phonons can lead to a resonant enhancement of the photo-excitation of electromagnons.
As there are two possible contributions to the polarisation -from ionic displacements and from electronic density redistribution -one can think of two corresponding electric contributions to electromagnon -from phonons and from electronic excitations. This means a transfer of the electric dipole spectral weight from phonons (hω ∼ 10 − 100 meV) and/or electronic excitations (hω ∼ 2 eV) down to magnons (hω ∼ 1 − 5 meV). Such a transfer, in turn, leads to a step-like anomaly in the temperature dependence of the dielectric constant. Current research on electromagnons, including this paper, is focused mainly on magnonphonon coupling while magnon-electron aspect is much less explored as it is generally expected to be weak because of the large energy difference between the electronic excitations and the magnetic excitations.
The possibility of electromagnon excitations in multiferroics has long been anticipated theoretically [1], but only recently were they observed in experiment. Pimenov et al [12] reported observation of electric active modes at magnon frequencies in GdMnO 3 and TbMnO 3 which exist only in some magnetically ordered phases and can be suppressed by magnetic field. Further, they found in GdMnO 3 the spectral weight transfer from the lowest frequency phonon down to the electromagnon mode [28]. Sushkov et al [13] [14] and its continuum version was given by de Sousa et al [32,33] who applied it to describe low-energy excitations in BiFeO 3 . Fang and Hu [34] discussed electromagnon absorption in RMn 2 O 5 compounds assuming Heisenberg interactions between spins. The outstanding fundamental questions for electromagnon are the microscopic origin of these novel excitations (Heisenberg or Dzyaloshinskii-Moriya type exchange) in the different classes of compounds (RMnO 3 and RMn 2 O 5 ), the explanation of the observed selection rules, and whether electromagnons may occur in a wider range of materials. The practical issues are enhancing the magneto-capacitance effect and its temperature range and possibly applying these new excitations to metamaterials and/or achieving negative index of refraction in the magnon range of frequencies.
The remainder of this paper is organized as follows. In section 2, we discuss optical absorption spectra of multiferroic YMn 2 O 5 and TbMn 2 O 5 compounds taken at different temperatures. We present experimental evidence allowing us to identify the low-frequency peaks appearing in the non-collinear phase of these materials as electromagnons. For comparison, in section 3 we present data on electromagnons peaks in the spiral multiferroic material Eu 0.75 Y 0.25 MnO 3 . In section 4, we discuss a simple microscopic model of YMn 2 O 5 manganites, based on isotropic Heisenberg exchange interactions between spins, with which we describe magnetic, multiferroic and optical properties of these compounds. We also briefly discuss phenomenological description of magnetic orders and magnetoelectric coupling in RMn 2 O 5 compounds based on their symmetry. In section 5, we discuss the microscopic origin of the electromagnon peaks in both RMn 2 O 5 and RMnO 3 and how the electromagnon relates to the spontaneous polarisation. Finally, we summarize our experimental and theoretical results.
Far infrared spectroscopy of RMn 2 O 5 compounds
The family of RMn 2 O 5 compounds has long been studied [35]. Small variations of the exchange integrals with temperature lead to the complex phase diagram for this multiferroic family [36]. Earlier spectroscopic works revealed far infrared absorption modes activated at low temperatures in EuMn 2 O 5 [37], YMn 2 O 5 [38] and GdMn 2 O 5 [39]. Recently, we have shown that in YMn 2 O 5 and TbMn 2 O 5 strong electric dipole active modes emerge at magnon frequencies in the lowest in temperature ferroelectric phase [13]. As in the work by Pimenov et al [12] on RMnO 3 compounds, we assign this modes as electromagnons.
In figure 2, we show the optical conductivity spectra of YMn 2 O 5 and TbMn 2 O 5 at three different temperatures corresponding to three magnetic/ferroelectric phases. These spectra were obtained by fitting measured transmission spectra [13] with the model dielectric function [40], assuming all modes to be of the electric dipole nature. The spectra measured at 7 K (blue curves) show the characteristic strong and sharp (electromagnon) absorption peaks at lowest frequencies. In this phase, the spontaneous electric polarisation is relatively small and the angles between neighbouring spins along the b axis are large. The spectra measured at 25 K (green curves) have one broad absorption peak near 20 cm −1 . In this phase spins are almost collinear and electric polarisation is large. Red curves, taken just above the Néel temperature, show a single broad absorption band below the phonon frequencies.
Identifying the low-frequency excitations as electromagnons requires addressing several questions. First, to avoid confusion with possible transitions between f -levels of rare earth ions, we have studied YMn 2 O 5 . The second issue is electric versus magnetic dipole (antiferromagnetic resonance) activity. We measured transmission spectra for various mutual orientations of the electric and magnetic field of light (respectively, e and h) with respect to the crystal axes and we found the absorption only for e b P , where P is the spontaneous polarisation vector, independent of the orientation of h, which implies that the excitations are electric dipole active. Can these resonances be new phonons, activated in the low temperature phase? We have performed shell model calculations that put the lowest phonon near 100 cm −1 for any reasonable parameters. Thus, we believe that we can reliably identify these lowfrequency peaks as electromagnons.
Another check of the electromagnon origin of this modes is comparison of their contribution to the step-like anomaly in the temperature dependence of ε 1 , calculated using the Kramers-Kronig relation, with the measured one. Figure 3 shows such a comparison. We have chosen TbMn 2 O 5 for this purpose because of the larger sample size and higher frequency of the electromagnon peaks. It is clear from figure 3 that the whole step-like anomaly in ε 1 (T ) comes entirely from the sharp electromagnon peak.
The frequency and temperature behaviour, presented in figures 2 and 3 is typical for a set of RMn 2 O 5 compounds: R = Er, Ho, Y, Dy, Tb, Gd and Eu [41,42,13,39,37,43]. The appearance of electromagnon peaks as well as the ε step-like anomaly seem to be correlated with the transition into the non-collinear spin state. Notably, BiMn 2 O 5 whose spin ordering is nearly collinear at all temperatures, shows neither the ε step-like anomaly nor electromagnon absorption [43].
An inelastic neutron scattering study of YMn 2 O 5 is in progress. Preliminary data by S.-H. Lee et al [44] show several scattering peaks in energy scans at the wave vector of the static spin structure. A strong neutron feature is observed at 1 meV in good agreement with the sharp low frequency feature in the low temperature infrared spectrum (figure 2a).
Katsura et al [14] predict that the electromagnon originating from antisymmetric exchange has a 'transversal' polarisation with respect to the spontaneous polarisation: e ⊥ P . The observed polarisation selection rule for electromagnons in RMn 2 O 5 compounds is 'longitudinal': e P . We will show that such 'longitudinal' electromagnon can be obtained in a model based on the isotropic Heisenberg exchange.
Far infrared spectroscopy of RMnO 3
It is interesting to compare electromagnons in the two families of multiferroic manganites. We begin with Eu 0.75 Y 0.25 MnO 3 -a compound that mimics lattice parameters of TbMnO 3 but does not have f -f transitions in the far infrared [29]. To extract the parameters of the oscillators, we fit the transmission spectra with a Lorentzian model of the dielectric constant ε(ω) for electric dipole transitions:
ε(ω) = ε ∞ + ∑ j S j ω 2 j − ω 2 − ıωγ j (1)
where ε ∞ is the high frequency dielectric constant, j enumerates the oscillators, S j is the spectral weight, ω j is the resonance frequency, and γ j is the damping rate. Figure 4 shows optical conductivity spectra for three phases of Eu 0.75 Y 0.25 MnO 3 . Figure 5 shows temperature dependence of the fit parameter S j (spectral weight of the peaks in figure 4). A broad absorption (electromagnon) band exists well above the Néel temperature 47 K (red curve in figure 4 and curve 3 in figure 5). Upon cooling, the spectral weight S of this broad band is growing down to T FE =30 K and stays at this level at lower temperatures. Such a behaviour of the background can be seen from the spectra -the absorption at 40 cm −1 , the frequency least affected by two absorption peaks, stays constant at all T < T FE . This absorption produces smooth growth of the ε a (T ) at T FE < T . T N =47 K is the inflection point of the curve 3 which is an evidence for the magnetic origin of this low frequency broad electric absorption. Two electromagnon peaks emerge sharply at the T FE (blue curve in figure 4 and curves 1 and 4 in figure 5). They produce all further growth of the ε a (T ) at T < T FE . The electromagnon spectrum of TbMnO 3 is very similar to that one of Eu 0.75 Y 0.25 MnO 3 , except the large electromagnon peak is at 60 instead of 80 cm −1 [43].
It is also interesting to follow the temperature dependence of the redistribution of the electric dipole spectral weight S. Since electromagnons result from a small admixture of phonons to magnons, the total spectral weight should be conserved. Comparing the curves 2 and 5 in figure 5, one can estimate how much spectral weight the 120 cm −1 phonon is loosing and how much electromagnons acquire. As electromagnons gain more spectral weight than the phonon is loosing, we checked the rest of the phonon peaks for the same polarisation of light. The inset in figure 5 shows that, indeed, phonons lose just enough of the spectral weight to conserve total spectral weight.
In their recent inelastic neutron scattering work, Senff et al [45] reported a set of modes at the incommensurate zone center of TbMnO 3 . The frequency of the lowest mode is equal to the frequency of the lowest infrared peak (24 cm −1 ). We agree with Senff et al in assignment of this mode as electromagnon. However, despite the satisfied polarisation selection rule e ⊥ P , this is not the electromagnon predicted by Katsura et al [14], as we discuss below.
The model of RMn
2 O 5
The presence of two different types of magnetic ions and geometric frustration of spin interactions give rise to rather complex magnetic structures RMn 2 O 5 compounds. Nonetheless, a number of salient properties of these materials, such as the magneticallyinduced electric polarisation, photo-excitation of magnons as well as the spin re-orientation transition, can be understood within a simplified microscopic model, which we discuss in this section. Our starting point is the assumption that multiferroic and optical properties of these materials are governed by the isotropic Heisenberg exchange, although we do include magnetic anisotropy to explain the spin re-orientation transition that has a strong effect on the low-frequency absorption spectrum. We discuss ordered spin states of the model, the mechanism of magnetoelectric coupling and calculate the optical absorption spectrum at magnon frequencies for different magnetic states.
We start by considering a single magnetic ab-layer including Mn 3+ and Mn 4+ ions. The model describes interactions between the spins and their coupling to a polar phonon mode:
H = 1 2 ∑ i, j J i j (P) S i · S j − 1 2 ∑ iα K iα S i ·k iα 2 − − ∑ i µ i (S i · H) +V P 2 2χ (0) 1 − P E − χ 2 E 2 2 .(2)
Here the first term is the isotropic Heisenberg spin exchange, while the second term is the single-ion anisotropy. The antisymmetric Dzyaloshinskii-Moriya exchange as well as other types of anisotropic exchange interactions are not included, as in the scenario discussed below they play no role. We assume for simplicity that in the ordered state all spins lie in the ab plane [21] and neglect the small out-of-plane components found in recent neutron diffraction experiments on single crystals [46,47]. Thus, the easy and intermediate magnetic axes on each Mn site (α = 1, 2) lie in the ab plane, while the hard axisk i3 ĉ. The third term in (2) is the interaction of spins with an applied magnetic field and the last term describes the dielectric response of the system, where χ (0) 1 is the 'bare' dielectric susceptibility related to the polar lattice mode (not including the magnetic contribution calculated below) and χ 2 is the remaining part of the dielectric susceptibility of non-magnetic origin. Finally, V is the system volume.
The coupling between the spins and the polar phonon mode results from the dependence of the exchange coupling on the electric polarisation, which in RMn 2 O 5 is parallel to the b axis:
J i j (P b ) = J i j (0) + J ′ i j (0)P b + 1 2 J ′′ i j (0)P 2 b + . . .(3)
The last two terms give rise to the cubic and the quartic magneto-electric couplings.
Magnetic ordering and spin re-orientation transition
Here, we adopt the model of Chapon et al [21] with 5 exchange constants between pairs of nearest-neighbour Mn ions: J 1 and J 2 couple Mn 4+ ions along the c direction, J 3 and J 5 couple the spins of neighbouring Mn 3+ and Mn 4+ ions and J 4 is the coupling between two neighbouring Mn 3+ ions (see figure 6). Figure 6 shows the minimal energy spin configurations obtained by the numerical minimization of the spin energy (2) on the subspace of the commensurate magnetic states with the wave vector Q = (1/2, 0, 0) for two different sets of exchange constants. The exchange constants J 4 and J 5 were chosen to be positive and large compared to other exchange constants, which gives rise to antiferromagnetic zig-zag chains along the a axis with nearly collinear spins (marked by dashed lines) observed in neutron experiments [21,47].
The angle between spins in neighbouring chains sensitively depends on the ratio between the interchain coupling J 3 and the magnetic anisotropy parameters K i . We assume that the easy magnetic axis is parallel to the a axis. The interplay between magnetic anisotropy and interchain interaction determines the angle between spins in neighbouring antiferromagnetic chains. We first note that if spins in each antiferromagnetic chain would be perfectly collinear than the interchain interactions would cancel as a result of geometric frustration. Conversely, the interchain coupling J 3 results in spin rotations, which destroy the collinearity of spins in each chain. Consider, for example, the spin of the Mn 4+ ion, marked in figure 6(a) by an arrow. Because of a nonzero angle between spins in the neighbouring a-chains, the interaction of the marked spin with the spin of the Mn 3+ ion from the neighbouring chain will give rise to a rotation of the marked spin. These small spin rotations lift the frustration and lead to some energy gain due to interchain interactions. It easy to show that this energy gain is maximal when spins in neighbouring antiferromagnetic chains are orthogonal to each other. Thus, while the magnetic anisotropy favours an almost collinear spin configuration, interchain interactions favour the 90 • angle between spins of neighbouring chains.
This competition gives rise to a very strong sensitivity of the angle between spins in neighbouring chains to the interchain coupling J 3 . For weak interchain coupling J 3 = −2 K, this angle is relatively small and minimal-energy spin configuration shown in figure 6(a) is similar to the one observed in the high-temperature 'collinear' phase of YMn 2 O 5 by Chapon et al [21]. A small change in J 3 from −2 K to −4 K transforms the configuration shown in figure 6(a) into the one shown in figure 6(b), which may explain the spin re-orientation observed in RMn 2 O 5 with R =Tb, Ho, Dy and Y, provided that the interchain coupling is temperature-dependent. Although the rotations that make spins in each chain non-collinear are barely visible, they are sufficient to produce the large changes in the spin configuration, since the magnetic anisotropy is relatively weak. In RMn 2 O 5 , the spin re-orientation transition is accompanied by the loss of commensurability of the spin structure in the a and c directions (which is also a consequence of magnetic frustration). This latter aspect of the transition is, however, less important for the photo-excitation of magnons, discussed below, than the re-orientation of spins.
Magnetically-induced polarisation
Minimizing (2) with respect to P y , we obtain expression for the magnetically-induced electric polarisation,
P b ≈ − χ (0) 1 2V ∑ i, j J ′ i j (0) S i S j ,(4)
which only involves scalar products of spins. Figure 7 gives a simplified view of the Mn layer in the ab plane, in which the spins inside squares depict the spins of Mn 4+ ions located inside oxygen octahedra, while the spins inside triangles are the spins of Mn 3+ ions in oxygen pyramids. The nearly collinear magnetic ordering in the high-temperature ferroelectric phase consists of antiferromagnetic chains along the a direction.
The mechanism responsible for electric polarisation in this magnetic state involves, however, the ↑↑↓ and ↓↓↑ spin chains along the b axis, such as shown in figure 8. These chains contain the polar Mn 4+ -Mn 3+ bonds, connecting parallel spins, and the Mn 3+ -Mn 4+ bonds with opposite polarity, connecting antiparallel spins. Importantly, the charge and This mechanism also works in the low-temperature (incommensurate) ferroelectric phase, which has the same periodicity in the b-direction. The amplitude of the exchange striction is, however, the largest for collinear spins [see (4)], which explains the drop of the polarisation at the transition to the low-temperature phase. For example, if the value of the magnetoelectric coupling ∝ J ′ 3 − J ′ 4 − J ′ 5 , is chosen such that the electric polarisation induced by the 'high-temperature' configuration shown in figure 6(a) is 1000 µC m −2 , then for the 'low-temperature' configuration shown in figure 6(b) it equals 500 µC m −2 .
Phenomenological approach
In this subsection we discuss the phenomenological description of spin states and the magnetoelectric coupling mechanism discussed above, which will clarify the similarities between RMn 2 O 5 and other multiferroic materials. The positions of Mn 3+ and Mn 4+ in the paramagnetic unit cell are shown in table 1 and the eight vector order parameters can be found in table 2 (we use the notations of Bertaut et al [35]).
r 2 = (−x, −y, 1/2) r 6 = (1/2, 0, −z)r 3 = (1/2 − x, 1/2 + y, 1/2) r 7 = (0, 1/2, z)r 4 = (1/2 + x, 1/2 − y, 1/2) r 8 = (0, 1/2, −z)
For discussion of phenomenological description of the magnetoelectric coupling, we have chosen the relatively simple case of BiMn 2 O 5 , which shows the commensurate spin ordering with Q = (1/2, 0, 1/2). In this case the components of the order parameters belong Table 3. Irreducible representations of the space group Pbam for Q = (1/2, 0, 1/2).
F = S 1 + S 2 + S 3 + S 4 F ′ = S 5 + S 6 + S 7 + S 8 C = S 1 + S 2 − S 3 − S 4 C ′ = S 5 + S 6 − S 7 − S 8 G = S 1 − S 2 + S 3 − S 4 G ′ = S 5 − S 6 + S 7 − S 8 A = S 1 − S 2 − S 3 + S 4 A ′ = S 5 − S 6 − S 7 + S 82 x 2 y 2 z m x m y m z I Γ 1 0 1 −1 0 1 0 0 −1 0 −1 −1 0 1 0 0 −1 0 1 −1 0 1 0 0 1 0 −1 −1 0 Γ 2 0 −1 1 0 1 0 0 −1 0 1 1 0 −1 0 0 1 0 1 −1 0 −1 0 0 −1 0 −1 −1 0Γ 1 F x C x C y F y G x −A x −A y G y C ′ z F ′ z G ′ x −A ′ x −A ′ y G ′ y Γ 2 F z C z G z −A z C ′ x F ′ x F ′ y C ′ y G ′ z −A ′ z
to one of the two two-dimensional representations, Γ 1 or Γ 2 , of the Pbam group [35,48]. BiMn 2 O 5 only shows the 'collinear' state with the a-components of the Mn 3+ and Mn 4+ spins described, respectively, by the order parameters F x = −3.1µ B and G ′ x = 2.4µ B with small b components C y = −0.8µ B and A ′ y = 0.6µ B [48] corresponding to a small rotation between spins in neighbouring antiferromagnetic chains.
Since F x is the part two-dimensional representation F x C x ∈ Γ 1 , the state described by the order parameter C x is another ground state of the system. These two states are related by inversion, which transforms F x into −C x and vice versa. It is easy to check that the magnetoelectric coupling of the form −λ x P y F 2 x −C 2 x is invariant upon all symmetry transformations of the paramagnetic phase, so that the order parameters F x and C x describe two ferroelectric states with opposite directions of electric polarisation. It also easy to check that the couplings −λ y P y F 2 y −C 2 y and −λ z P z F 2 z −C 2 z are also allowed by symmetry, which is a strong indication that the mechanism inducing electric polarisation in magnetically ordered state is invariant upon the global spin rotation and the coupling can be written in the form λ P y F 2 − C 2 [20].
Due to the exchange coupling between the Mn 3+ and Mn 4+ ions, the order parameters G ′ x and F x are strongly coupled. Since
G ′ x −A ′
x also belongs to Γ 1 representation, the coupling between the two spin subsystems is phenomenologically described by [20]. Thus, more generally, the magnetoelectric coupling should be written in the form
−g (F x G ′ x −C x A ′ x )Φ me = −λ P y η 2 1 − η 2 2 ,(5)
where η 1 η 2 belongs to Γ 1 representation. This form of the third-order magnetoelectric coupling was discussed previously in the context of the orthorombic manganites with the E-type magnetic ordering [22] and is typical for improper ferroelectrics [25].
Static and dynamic dielectric susceptibility
The contribution of the coupled spin-lattice degrees of freedom to static dielectric susceptibility is given by
χ −1 1 ≈ χ (0) 1 −1 − 1 V ∑ i, j I i A −1 i j I j + 1 2V ∑ i, j J ′′ i j (0) S i S j ,(6)
where
A i j = J i j S i S j + δ i j ∑ α=1,2 K iα 2 S ikiα 2 − S 2 i − ∑ k J ik (S i S k )(7)
and
I i = ∑ j J ′ i j S i × S j c ,(8)
The second term in (6) is the spin contribution to the dielectric constant due to virtual excitations of magnons by electric field (this will become more apparent in the discussion of the dynamic susceptibility). The last term in (6) describes the shift of the phonon frequency due to a change of the spring constants in magnetically ordered states. This effect is known in condensed matter spectroscopy as spin-phonon coupling [49,50]. Phenomenologically, this effect is described by the fourth-order magnetoelectric coupling of the type P 2 L 2 , where L is a magnetic order parameter. In most cases, magnon and phonon branches coupled through this term experience 'repulsion' and phonon hardens. However, in the magnetically frustrated compounds, this contribution to the dielectric constant can have either sign and can result in either hardening or softening of phonons in the magnetic phase (see, e. g., [51] on CdCr 2 S 4 spinel).
Equations of motion describing the coupled spin-lattice dynamics have the form,
P b = − χ 0 ω 2 0 V ∂ H ∂ P b , S i = ∂ H ∂ S i × S i ,(9)
where χ 0 = χ (0) 1 + χ 2 and ω 0 is the bare frequency of the polar phonon. Omitting the fourth order coupling term and solving linearized equations of motion, we obtain the dynamic dielectric susceptibility:
χ −1 (ω) ≈ χ 0 −1 1 − ω 2 ω 2 0 − 1 V ∑ i, j I i BA − ω 2 −1 B i j I j ,(10)
where
B i j = J i j + δ i j S 2 i ∑ α=1,2 K iα S ikiα 2 − K i3 S 2 i − δ i j S 2 i ∑ k J ik (S i S k ) .(11)
The second term in (10) describes the transfer of a part of electric dipole spectral weight from phonon to magnon frequencies, which turns magnons coupled to phonons into electromagnons. If such an electromagnon has lower frequency than the phonon, the static dielectric constant ε(0) = 1 + 4π χ(0) increases as a result of the coupling showing a steplike anomaly. Furthermore, frequencies of the mixed spin-lattice excitations (poles of the dielectric susceptibility (10)) are shifted down with respect to the 'bare' magnon frequencies, found from det BA − ω 2 = 0. (12) Note that the electromagnon term in (10) disappears for collinear spin states, as I i , defined by (8), is zero in this case. This can be understood as follows. Classically, magnons correspond to spin oscillations that are orthogonal to ordered spin vectors. The change of the scalar product of a pair of collinear spins is then proportional to the second power of the amplitude of the oscillations. Since the magnetoelectric coupling in our model originates solely from Heisenberg exchange and only involves scalar products of spins, the linear coupling of electric field to magnons is absent for collinear spins and the lowest-order process is the photoexcitation of a pair of magnons.
In figures 9(a) and (b), we plot the real and imaginary parts of the dielectric function (red and blue lines, respectively) for the 'high-temperature' and 'low-temperature' states shown in, respectively, figures 6(a) and (b) (the imaginary part was obtained by the shift ω → ω + i γ 2 with γ = 1 K). As was explained above, the coupling of magnons to the electric component of light and significant electric dipole absorption at magnon frequencies is only present in the non-collinear 'low-temperature' phase, in agreement with experiment. This result may seem somewhat counterintuitive: while the spontaneous electric polarisation induced by the Heisenberg exchange striction is maximum for the collinear state, the excitation of magnons by the electric component of light (electromagnons) requires non-collinear spins and is only observed below the spin re-orientation transition.
The red points in figure 9 mark the bare frequencies of the softest magnons with zero wave vector, found from (12). Since the magnetic unit cell in this calculation contains 16 mangetic ions, the total number of such magnons is also 16. However, only one of them is strongly coupled to the electric component of light and significantly contributes to the dielectric constant. This electromagnon corresponds to relative rotation of spins of the neighbouring antiferromagnetic chains, which gives rise to oscillations of the induced electric Figure 9. (Colour online) The model calculation results: Frequency dependence of the real (ε 1 ) and imaginary (ε 2 ) parts of the dielectric function (red and blue lines, respectively) for the 'high-temperature' collinear state (panel a) and the 'low-temperature' non-collinear state (panel b) shown, respectively, in panels (a) and (b) of figure 6. Red points are selected magnon frequencies of spins decoupled from the lattice found as the roots of (12). A magnon at 11.8 cm −1 couples to a polar phonon at 100 cm −1 and becomes an electromagnon observable as the peak of the ε 2 . polarisation in the b direction. The frequency of this uncoupled magnon for the non-collinear spin configuration shown in figure 6(b) is 11.8 cm −1 . As the position of the absorption peak in figure 9(b) is clearly lower that this magnon frequency, our parameters correspond to the strong magnetoelectric coupling case. The strong coupling is apparently necessary, if the large difference between dielectric constants of the 'high'-and 'low'-temperature phases ∆ε ′ (0) (= 3.25 in our calculation) is associated solely with the absorption peak emerging in the noncollinear state.
The results of our model calculations presented in figure 9(b) show that the symmetry properties of one magnon and the lattice allow an electromagnon in this system. Next, we discuss the strength of coupling. In our model, the first derivatives of the exchange integrals J i j are the coupling constants. Both magnetically induced polarisation and electromagnon were calculated using the same set of J ′ values. Characteristic value of J ′ used in our calculation was dJ 3 /dy = 4 meV/Å which is much less than, for example, 40 meV/Å for ZnCr 2 O 4 [52].
Discussion
Our model calculations show that the observed values of the polarisation and electromagnon strength are obtained with realistic symmetric exchange constants. Also, the electromagnon coupling has been shown to be of Heisenberg type by the observed selection rules. These are strong arguments in favour of the Heisenberg exchange origin of the polarisation and electromagnons in RMn 2 O 5 family. However, in comparing the experimental and theoretical results, we should keep in mind that in the compounds of interest both symmetric and antisymmetric exchange mechanisms are operative, and they both may produce polarisation and electromagnons. Therefore, it is important to compare both the spontaneous polarisation and the electromagnon selection rules in both RMnO 3 and RMn 2 O 5 with these two mechanisms.
In particular, the origin of ferroelectricity in RMn 2 O 5 remains controversial. An alternative to the magnetostriction scenario discussed here is electric polarisation induced by the bc spiral, as observed in recent experiments ( [46,47,53]) via the inverse Dzyaloshinskii-Moriya mechanism of relativistic origin [17,18,24]. One natural question is whether the electromagnon study can clarify this controversy.
Electromagnon excitations for the cycloidal spiral state were studied by Katsura et al [14]. According to their theory, the electric field of light that excites electromagnon has to be orthogonal to the direction of the electric polarisation induced by the spiral, which can be easily understood as follows. The spin spiral induces electric polarisation that lies in the spiral plane and is orthogonal to the propagation wave vector of the spiral [24]. Thus the bc-spiral in RMn 2 O 5 with the wave vector along the c axis would induce polarisation along the b axis, in agreement with experiment. An electric field applied in the direction perpendicular to the spiral plane will result in a small rotation of this plane, as a result of which the spontaneous polarisation vector acquires a component parallel to the applied field. An oscillating electric field orthogonal to the spiral plane will then excite oscillations of this plane, which is precisely the electromagnon of Katsura et al [14]. Since the polarisation lies in the spiral plane, the polarisation of light should be orthogonal to the spontaneous polarisation. Thus, to excite the oscillations of the bc spiral would require e a, whereas in experiment electric field is parallel to the b axis and the direction of electric polarisation. Indeed, this can be an argument against the spiral scenario of multiferroicity of RMn 2 O 5 .
However, the situation is complicated by the fact that even in RMnO 3 compounds, where the spiral origin of the magnetically-induced electric polarisation is well established, the selection rule e ⊥ P is not obeyed. This was demonstrated by optical measurements in magnetic field. Magnetic field applied to DyMnO 3 [31] and TbMnO 3 [43] gives rise to the magnetic flop transition at which the bc-spiral is replaced by the ab-spiral. The spiral flop, however, has no effect on the polarisation of electromagnons, which are always observed for e a. Also in Eu 0.75 Y 0.25 MnO 3 , where the spin spiral lies in the ab plane already for zero magnetic field and the induced electric polarisation is parallel to a, the selection rule for electromagnons remains the same: e a [29]. Therefore, the selection rules in both RMnO 3 and RMn 2 O 5 are tied to the lattice, not the spin plane. The selection rule for RMnO 3 appears to originate from the GdFeO 3 distortions of the perovskite structure of orthorombic manganites, which generates new types of magnetoelectric interactions and couples light to magnetic excitations that are different from the oscillations of the spiral plane [54]. Furthermore, these distortions may also couple to the zone edge magnons and account for the broad peak observed in RMnO 3 at high frequencies. However, an alternative mechanism for the broad peak is the photoexcitation of bi-magnons and so the precise origin of this high frequency peak in RMnO 3 remains an open question.
In any case, it appears that electromagnons in RMnO 3 are also induced by symmetric exchange even though the spontaneous polarisation is produced by the DM-type antisymmetric exchange mechanism. This is understood as a consequence of the vanishing of any static polarisation induced by symmetric exchange due to the symmetry of the lattice [18]. Another important question then is why the antisymmetric exchange does not produce electromagnon excitations? We suspect that the relativistic interactions are just too weak to produce observable signals. The observation of these weak DM induced electromagnons would help clarify the overall picture of of spin-lattice interactions in multiferroics.
Consequently, while our experimental and theoretical results strongly suggest that the magnetoelectric coupling in RMn 2 O 5 is governed by Heisenberg exchange, we cannot safely rule out the spiral scenario for the spontaneous polarisation in these materials on the basis of optical data. However, the results of recent neutron scattering experiments [53] suggest rather that the spiral components appear as a consequence of the existence of spontaneous polarisation.
Summary
We have presented experimental evidence and theoretical analysis that demonstrate that electromagnon excitations are present in both RMn 2 O 5 and RMnO 3 multiferroics. These electric dipole mixed magnon-phonon modes are observed in the far infrared and match excitations observed in inelastic neutron spectra in both classes of materials. The polarisation selection rules observed in the infrared experiments show that the electromagnon excitations are generated only by symmetric exchange in both classes of materials. The observed selection rules are tied to the lattice, not the spin plane, in contrast to the predictions of the antisymmetric exchange model.
To theoretically account for electromagnons, we have considered third and fourth order coupling between the lattice and spins. The fourth order terms produce spin-phonon interactions that lead to shifts of the magnon and phonon frequencies near magnetic phase transitions observed in many magnetic materials. The third order coupling terms can produce mixed magnon-phonon excitations -the electromagnons -only for non-collinear spin orders. The third order terms are also responsible for the spontaneous polarisation in multiferroics [6]. We also developed a simple microscopic model of RMn 2 O 5 that explains the transition between collinear and non-collinear spin states, the magnetically-induced electric polarisation in the collinear state and the appearance of the electromagnon peak in the non-collinear state as well the polarisation of light for which this peak is observed. The mechanism of magnetoelectric coupling in this model relies entirely on isotropic Heisenberg exchange and magnetostrictive coupling of spins to a polar phonon mode.
Figure 1 .
1(Colour online) Feynman diagrams describing photo-excitation of magnons by the electric field of light (here photon and phonon are represented, respectively, by dashed and solid lines, while the wavy line corresponds to magnon): (a) Photoexcitation of two magnons and one phonon via the fourth-order magnetoelectric coupling (the Lorenzana-Sawatzky mechanism)
Figure 2 .
2(Colour online) Optical conductivity of YMn 2 O 5 and TbMn 2 O 5 for the electric field of light e b in three phases. Strong peaks at 113 and 97 cm −1 are the lowest phonons, other peaks are electromagnons.
Figure 3 .
3(Colour online) Dielectric constant of TbMn 2 O 5 from fits of infrared spectra (lower curve) in comparison with kHz measurements. The whole step-like anomaly is due to electromagnons (figure 2a, blue curve).
Figure 4 .
4(Colour online) Optical conductivity of Eu 0.75 Y 0.25 MnO 3 for e a in the ferroelectric (blue), the spin density wave (dashed), and the paramagnetic (red) phases. Electromagnon band consists of a broad background and two peaks.
Figure 5 .
5(Colour online) Spectral weight of the absorption peaks below 140 cm −1 in figure 4. The numbers from 1 to 5 enumerate the curves. Frequencies in the legend are the lowest temperature resonance frequencies of each mode. Inset: Total spectral weight of the eight phonons above 140 cm −1 .
Figure 6 .
6(Colour online) Minimal energy spin configurations for J 4 = J 5 = 40 K and the anisotropy parameter K a (Mn 3+ ) = 0.6 K for all Mn 3+ ions (red) and K a (Mn 4+ ) = 0.1 K for all Mn 4+ ions (blue). The value of the interchain coupling J 3 is −2 K for the structure in panel (a) and −4 K for the one panel (b).
Figure 7 .
7A cartoon of the magnetic ordering in the high-temperature collinear phase.Figures 7 and 8show why in the high-temperature ferroelectric phase the polarisation vector is oriented along the b axis.
Figure 8 .
8Electric polarisation induced by magnetostriction along the b-chains. The two order parameters η 1 and η 2 describe degenerate ferroelectric states with opposite directions of electric polarisation. spin modulations in the chains have the same period, in which case the conventional exchange striction destroys the cancellation of electric dipoles of the polar bonds and induces electric polarisation along the chains, as illustrated in figure 8.
found out that the electromagnon polarisation in DyMnO 3 stays with the lattice when the spin plane is rotated by the external magnetic field. Electromagnons, observed so far only in non-collinear spin phases of RMnO 3 and RMn 2 O 5 compounds, have common features for both families: they are active only in one polarisation e a axis for RMnO 3 and e b axis for RMn 2 O 5 , where e is electric field of light; well defined peaks exist only in the low-temperature magnetic ferroelectric phase. A theory of electromagnons for the circular magnetic spiral was developed by Katsura et alreported observation
of electromagnons in YMn 2 O 5 and TbMn 2 O 5 . Electromagnons in both these RMn 2 O 5
compounds have very similar spectrum and exist only in magnetic ferroelectric phases that
proves their magnetic origin. Valdés Aguilar et al [29] showed that electromagnon absorption
in Eu 0.75 Y 0.25 MnO 3 occurs over a broad band with two peaks, both of which exist only in the
magnetic ferroelectric phase below 30 K. Pimenov et al [30] explored a composition set
Eu 1−x Y x MnO 3 for 0 ≤ x ≤ 0.5 and confirmed main features of electromagnons in RMnO 3 .
Kida et al [31]
Table 1 .
1The coordinates of Mn ions, where x ≈ 0.41, y ≈ 0.35 and z ≈ 0.26 (for BiMn 2 O 5 ).Mn 3+
Mn 4+
r 1 = (x, y, 1/2)
r 5 = (1/2, 0, z)
Table 2 .
2Magnetic order parameters.Mn 3+
Mn 4+
Table 4 .
4Basis vectors of the space group Pbam for Q = (1/2, 0, 1/2).
AcknowledgmentsThis work was supported in part by the National Science Foundation MRSEC under Grant No. DMR-0520471. MM gratefully acknowledges the support by DFG (Mercator fellowship) and the hospitality of Cologne University. We acknowledge S. Park, C.L. Zhang and Y.J. Choi for the growth and characterization of the single crystals.
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| []
|
[
"First-principles study of lattice dynamical properties of the room-temperature P 2 1 /n and ground-state P 2 1 /c phases of WO 3",
"First-principles study of lattice dynamical properties of the room-temperature P 2 1 /n and ground-state P 2 1 /c phases of WO 3"
]
| [
"Hamideh Hassani \nPhysique Théorique des Matériaux\nQMAT\nCESAM\nUniversité de Lìege\nB-4000Sart-TilmanBelgium\n\nDepartment of Physics\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium\n",
"Bart Partoens \nDepartment of Physics\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium\n",
"Eric Bousquet \nPhysique Théorique des Matériaux\nQMAT\nCESAM\nUniversité de Lìege\nB-4000Sart-TilmanBelgium\n",
"Philippe Ghosez \nPhysique Théorique des Matériaux\nQMAT\nCESAM\nUniversité de Lìege\nB-4000Sart-TilmanBelgium\n"
]
| [
"Physique Théorique des Matériaux\nQMAT\nCESAM\nUniversité de Lìege\nB-4000Sart-TilmanBelgium",
"Department of Physics\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium",
"Department of Physics\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium",
"Physique Théorique des Matériaux\nQMAT\nCESAM\nUniversité de Lìege\nB-4000Sart-TilmanBelgium",
"Physique Théorique des Matériaux\nQMAT\nCESAM\nUniversité de Lìege\nB-4000Sart-TilmanBelgium"
]
| []
| Using first-principles density functional theory, we investigate the dynamical properties of the room-temperature P 21/n and ground-state P 21/c phases of WO3. As a preliminary step, we assess the validity of various standard and hybrid functionals, concluding that the best description is achieved with the B1-WC hybrid functional while a reliable description can also be provided using the standard LDA functional. We also carefully rediscuss the structure and energetics of all experimentally observed and few hypothetical metastable phases in order to provide deeper insight into the unusual phase diagram of WO3. Then, we provide a comprehensive theoretical study of the lattice dynamical properties of the P 21/n and P 21/c phases, reporting zone-center phonons, infrared and Raman spectra as well as the full phonon dispersion curves, which attest for the dynamical stability of both phases. We carefully discuss the spectra, explaining the physical origin of their main features and evolution from one phase to another. We reveal a systematic connection between the dynamical and structural properties of WO3, highlighting that the number of peaks in the high-frequency range of the Raman spectrum appears as a fingerprint of the number of antipolar distortions that are present in the structure and a practical way to discriminate between the different phases.arXiv:2111.11573v1 [cond-mat.mtrl-sci] 22 Nov 2021 | 10.1103/physrevb.105.014107 | [
"https://arxiv.org/pdf/2111.11573v1.pdf"
]
| 244,488,530 | 2111.11573 | 18acfe0cf670d1e04e763a5efb8a06035d45213a |
First-principles study of lattice dynamical properties of the room-temperature P 2 1 /n and ground-state P 2 1 /c phases of WO 3
Hamideh Hassani
Physique Théorique des Matériaux
QMAT
CESAM
Université de Lìege
B-4000Sart-TilmanBelgium
Department of Physics
University of Antwerp
Groenenborgerlaan 1712020AntwerpBelgium
Bart Partoens
Department of Physics
University of Antwerp
Groenenborgerlaan 1712020AntwerpBelgium
Eric Bousquet
Physique Théorique des Matériaux
QMAT
CESAM
Université de Lìege
B-4000Sart-TilmanBelgium
Philippe Ghosez
Physique Théorique des Matériaux
QMAT
CESAM
Université de Lìege
B-4000Sart-TilmanBelgium
First-principles study of lattice dynamical properties of the room-temperature P 2 1 /n and ground-state P 2 1 /c phases of WO 3
(Dated: November 24, 2021)
Using first-principles density functional theory, we investigate the dynamical properties of the room-temperature P 21/n and ground-state P 21/c phases of WO3. As a preliminary step, we assess the validity of various standard and hybrid functionals, concluding that the best description is achieved with the B1-WC hybrid functional while a reliable description can also be provided using the standard LDA functional. We also carefully rediscuss the structure and energetics of all experimentally observed and few hypothetical metastable phases in order to provide deeper insight into the unusual phase diagram of WO3. Then, we provide a comprehensive theoretical study of the lattice dynamical properties of the P 21/n and P 21/c phases, reporting zone-center phonons, infrared and Raman spectra as well as the full phonon dispersion curves, which attest for the dynamical stability of both phases. We carefully discuss the spectra, explaining the physical origin of their main features and evolution from one phase to another. We reveal a systematic connection between the dynamical and structural properties of WO3, highlighting that the number of peaks in the high-frequency range of the Raman spectrum appears as a fingerprint of the number of antipolar distortions that are present in the structure and a practical way to discriminate between the different phases.arXiv:2111.11573v1 [cond-mat.mtrl-sci] 22 Nov 2021
I. INTRODUCTION
Tungsten trioxide, WO 3 , is a wide bandgap semiconductor transition metal oxide with peculiar electronic, chromic and optical properties that makes it appealing for photocatalytic, electrochromic and optoelectrical devices applications [1,2]. WO 3 can be classified as a perovskite-like ABO 3 compound but with an empty A site. As such, it possesses an aristotype P m3m cubic structure, in which W atoms sit at the center of oxygen octahedra sharing a corner and forming a continuous network. However, WO 3 is typically not observed in that reference cubic structure and experimental measurements at various temperatures reveal a very complex phase diagram, not fully elucidated yet [3]. At about 1300 K, WO 3 is reported to crystallize in a P 4/nmm tetragonal phase [3][4][5]. Then, on cooling, it shows consecutive phase transitions to (i) a P 4/ncc tetragonal phase at about 1150 K [3][4][5][6], (ii) an eventual P 2 1 /c monoclinic phase at about 1050 K [3], (ii) a P bcn orthorhombic phase at about 1000 K [3,4,6,7], (iii) a P 2 1 /n monoclinic phase at about 600 K [3,4,6,8], (iv) a P 1 triclinic phase at 273 K [4,9,10] and, ultimately, (v) a P c monoclinic polar phase at around 200 K [4,[11][12][13] that was recently proposed to be instead a related non-polar P 2 1 /c monoclinic phase [14]. All these phases appear as small distortions of the cubic artistotype structure but including distinct rotations and tilts of the oxygen octahedra and different shifts of W atoms within these octahedra.
Experimentally, WO 3 appears also in practice as a substoichiometric (WO 3−x ) compound. This, on the one hand, confers it some unique properties exploited in a variety of technological applications [15] but, on the other hand, makes its study significantly more complex. In fact, the off-stoichiometry makes it a n-doped semiconductor and the excess electrons interact with the polarizable crystal lattice in order to form polarons and bipolarons [16] that strongly influence its properties.
Although extensive studies have been reported concerning the electronic and optical properties of WO 3 [15,[17][18][19], there is a much more limited amount of works dedicated to the investigation of its lattice dynamical properties. The phonon dispersion curves of cubic WO 3 and its elastic properties have been studied by Fan et al. [20] or together with the electron-phonon coupling by Mascello et al. [21]. Some lattice dynamical properties of distinct WO 3 phases have also been reported by Yang et al. [22]. Furthermore, Hamdi et al. [14] studied the phonon instabilities of the cubic phase together with the various potential metastable phases originating from these instabilities. This latter study has shown that the B1 Wu-Cohen (B1-WC) hybrid exchange-correlation functional [23] can accurately reproduce the experimental results regarding both the electronic and structural properties of WO 3 .
In the present work, we first carefully re-discuss the phase diagram of WO 3 , assessing the validity of distinct functionals. We then provide a comprehensive theoretical study of the lattice dynamical properties of the P 2 1 /n room temperature phase and P 2 1 /c expected groundstate phase, reporting infrared and Raman spectra together with the full phonon dispersion curves. On the one hand, we carefully discuss the spectra, explaining the physical origin of their main features and evolution from one phase to another and providing meaningful benchmark results for the interpretation of experimental measurements. On the other hand, polaron formation is directly linked to electron-phonon interaction so that the present understanding of the phonon properties appears as a useful step toward the understanding of polarons in these two phases.
The paper is organized as follows. In the first part, we compare the structural parameters and energetics of various phases of WO 3 , as obtained using standard (LDA and GGA) and more advanced (B1-WC, HSE06) hybrid functionals. We also take this opportunity to re-discuss briefly the phase diagram of WO 3 . In the second part, we carefully discuss the lattice dynamical properties of the P 2 1 /n room temperature and P 2 1 /c ground state structures, comparing our theoretical results with experimental data when available and interpreting the shape of the spectra. From this analysis, Raman measurements emerges as a promising tool to identify the various phases of WO 3 .
II. COMPUTATIONAL DETAILS
Our calculations are performed in the framework of density functional theory (DFT) as implemented in CRYSTAL17 [24] based on a linear combination of localized basis functions and ABINIT [25,26] making use a plane-wave basis set. Calculations are reported using distinct approximations of the exchange-correlation energy including the local density approximation (LDA), the generalized gradient approximation (GGA) and different hybrid schemes. Integrations over the Brillouin zone are approximated by sums over a mesh of 8 × 8 × 8 k-points for the cubic phase or meshes providing a comparable sampling for other phases (meshes of 6 × 6 × 8, 6 × 6 × 4 and 4 × 4 × 4 are used for P4/nmm, P4/ncc and Pbcn phases respectively).
ABINIT is used with the LDA with Perdew-Wang's parametrization [27] and the GGA with both the PBEsol [28] and Wu and Cohen (WC) functionals [29]. These calculations make use of norm-conserving pseudopotentials from the Pseudo Dojo Table v0.4 [30], considering the 2s and 2p orbitals of O and the 5s, 5p, 5d and 6s orbitals of W as valence states. The electronic wave functions are expanded in plane-waves up to an energy cutoff of 60 Ha. Full relaxations of the lattice parameters and atomic positions are performed with each functional using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [31] until the maximal forces and stresses are less than 10 −6 Ha/Bohr and 10 −4 GPa, respectively. The electronic self-consistent calculations were converged until the difference of the total energy is smaller than 10 −9 Ha. Dynamical matrices, Born effective charges and dielectric tensors are obtained within a variational approach [32] to density functional perturbation theory (DFPT) [33].
CRYSTAL is used for the hybrid functional calculations with both the B1-WC functional [34] and the HSE06 functional [35]. Benchmark calculations in LDA have also been performed for comparison with ABINIT. We use all-electron double-ζ basis sets for oxy- gen whereas, for the heavy tungsten atom, we use the effective core pseudopotential (ECP) technique as implemented in the small core Detlev-Figgen pseudopotentials associated with correlation consistent polarized Valence Triple-ζ Basis Set (cc-pVDZ) [36] for explicit treatment of valence electrons. The truncation thresholds in the evaluation of Coulomb and exchange series appearing in the SCF equation for periodic systems are adjusted to 10 −7 for Coulomb overlap tolerance, 10 −7 for Coulomb penetration tolerance, 10 −7 for exchange overlap tolerance, 10 −7 for exchange pseudo-overlap in the direct space, and 10 −14 for exchange pseudo-overlap in the reciprocal space. The tolerance on change in total energy for the SCF convergence is adjusted to 10 −10 Ha. Full structural relaxations are carried out by using a quasi-Newton algorithm with a BFGS Hessian updating scheme so that a convergence of less than 5 × 10 −5 Ha/Bohr and 10 −3 Bohr is reached respectively on the root mean square of the gradient and displacements. The phonon frequencies are computed using the frozen phonon numerical differences [37,38] (for detailed information check Fig. A1a in the appendix). The Born effective charges and the optical dielectric tensors are evaluated via the Berry phase technique [39] and the Coupled-Perturbed Kohn-Sham approach [40,41], respectively.
III. STRUCTURAL PROPERTIES
We begin our study by considering the high-symmetric parent cubic structure (P m3m) of WO 3 . Fig. 1 shows its phonon dispersion curves calculated using the B1-WC hybrid functional. The high-frequency branches show slightly distinct dispersion than those previously reported in [14]. The previous results were unfortunately affected by a bug in the implementation of the Fourier interpolations technique in the CRYSTAL package while the new curves reported here rely on a proper treatment of the dipole-dipole interactions as implemented in ABINIT (for further information we refer to Fig. A1a in the appendix).
Imaginary frequencies appear as negative numbers in Fig. 1 and are associated to phonon structural instabilities of the parent cubic phase. In fact, the distinct phases of WO 3 , observed at different temperatures, arise from the individual or combined condensation of the cooperative atomic motions related to these unstable phonon modes. The latter include (i) polar modes (PM) at Γ (corresponding to irreducible representation Γ − 4 ), which are related to the motion of W against the O atoms, (ii) anti-polar modes (AP) at the M and X points (M − 3 , X − 5 ), which correspond to opposite displacements W against the O atoms from unit cell to unit cell along the [110] and [100] directions, respectively, leading to the displacement of the W atoms from the center of the octahedral and forming chains with alternating short and long W-O bond lengths, and finally (iii) antiferrodistortive modes (AFD) at the M and R points (M + 3 and R + 4 ), which consist in rotation and tilts of the oxygen cages about the central W atom.
Phonon dispersion curves obtained with different functionals are compared in the appendix. Although they appear very similar, we point out that the M + 3 AFD mode is slightly unstable using hybrid and LDA functionals while it is slightly stable using GGA functionals.
In what follows, we will consider all the experimentally observed phases as well as six additional metastable phases arising all from distinct combinations of the unstable modes to gain more insight into the energy landscape of WO 3 . Full structural optimizations of the lattice parameters and atomic positions have then been performed for each phase using different exchange-correlation functionals.
A. Lattice parameters
In Table I, we compare to experiment the calculated lattice parameters obtained with the HSE06 and B1-WC hybrid functionals, the WC and PBEsol GGA functionals, the LDA functional as well as the results relying on the PBE GGA functional as reported recently by Yang et al. [22]. In line with what was previously reported by Hamdi et al. [14], starting from the experimental structure with the P c space group and performing full structural relaxations, the system relaxes back to a P 2 1 /c phase of higher symmetry using all considered functionals. Although a P 2 1 /c phase has only been experimentally observed at about 1000 K [3], its structural parameters are very similar to those of the P c phase observed at about 200 K [11] (the only difference is a small polar distortion as further discussed in the next Section). In Table I, the lattice parameters of the relaxed P 2 1 /c phase are therefore compared both to those of the high-temperature P 2 1 /c phase (P 2 1 /c-HT) and low-temperature P c phase to which we also refer to as P 2 1 /c-LT hereafter.
For the high-temperature P 4/nmm phase, all functionals provide a rather fair agreement with experimental data although PBE results of Yang et al. show a significant overestimation (also for other phases). Both GGA-WC and PBEsol provide also a good prediction of the lattice constants for the P 4/ncc phase. It should be noticed however that the experiments on these high-T phases are performed above 1000 K [3] while the calculations correspond to 0 K. It is therefore possible that this good agreement arises from a cancellation of errors coming, on the one hand, from neglecting the thermal expansion and, on the other hand, from a natural propensity of these GGA functionals to overestimate the lattice parameters of WO 3 . In line with that suggestion, the GGA-WC and PBEsol are indeed in better agreement with the P 2 1 /c-HT phase and their overestimate of lattice parameters tends to increase for the low-temperature phases. On the contrary, the LDA tends to systematically underestimate the lattice constants as expected and provides a good agreement for the P 2 1 /c-LT phase. HSE06 provides good estimates although with a tendency to slightly overestimate the volume of the low-temperature phases. B1-WC is confirmed to be the one providing the best overall estimates for all phases, and gives good description of the P 2 1 /c-LT and P 2 1 /n phases under investigation here. The computed lattice parameters of few additional metastable phases, not observed experimentally, are reported in Table II.
B. Atomic distortion
In this section, we analyze further the structural distortions of each phase with respect to the reference cubic phase by projecting the internal atomic displacements on symmetry adapted modes of the parent cubic phase using the AMPLIMODE software package [43]. The results are presented in Fig. 2 for the experimentally observed phases and in Fig. 3 for the few other phases. For the latter, missing phases in GGAs correspond to cases for which the phase is not metastable and the system relaxes back to a higher symmetry.
As highlighted in Fig. 2, the P 4/nmm phase arises from the condensation of the only anti-polar M − 3 mode along the z direction. This mode will remain present in all experimentally observed phases with a rather constant amplitude. The P 4/ncc phase includes the additional condensation of the AFD R + 4 mode along the same direction yielding a tilt pattern (a 0 a 0 c − ) in Glazer's notations. The P 2 1 /c phase arises from the addition of AFD R + 4 contributions, yielding a tilt pattern (a − a − c − ). A secondary antipolar X − 5 mode in the xy-plane also appears thanks to an anharmonic coupling between the R + 4 and M − 3 modes [44][45][46]. In the P bcn phase, a M + 3 mode emerges giving rise to a distinct tilt pattern a 0 b + c − . The secondary antipolar X − 5 mode in this phase is only along the y direction. Also, another secondary antipolar X + 5 mode along y appears through a coupling between the an out-of-phase octahedral tilting around the second axis yielding to tilt pattern a − b − c − . This phase has the same antipolar M − 3 and X + 5 modes as in the P 2 1 /n phase, but the antipolar X − 5 mode has components in all directions. Finally, the experimental P c phase appears similar to the P 2 1 /c phase except for the additional appearance of a polar Γ − 4 mode. As previously discussed, our calculations in P c symmetry always relax back to P 2 1 /c to which the P c phase is therefore compared. We insist that all the phases except P c are non-polar. Yang et al. [22] report a net dipole moment for the P 2 1 /n phase which is however inconsistent with the symmetry assignment. Fig. 3 also compares the results for few additional metastable phases not observed experimentally.
From Fig. 2 and Fig. 3, it appears that, on the one hand, all functionals predict rather consistently the appearance and amplitudes of polar and antipolar distor-
tions (M − 3 , X − 5 , X + 5 , Γ − 4 )
while, on the other hand, GGA functionals strongly underestimate the amplitudes of rotations and tilts (R + 4 , M + 3 ), as further discussed in the next Section. In line with that, GGAs do not reproduce some metastable phases and also produce the worst description of the P 2 1 /n and and P 2 1 /c phases. Hybrid functionals typically provide a very good description of all phases while LDA gives also reasonable results, significantly better than GGAs. Fig. 4 compares the energies of the various phases under study in comparison to that of the parent cubic phase taken as zero energy reference. The results are shown for distinct DFT functionals. The energy is reported in meV per four-atoms formula unit (meV/f.u.).
C. Energetics
A first observation is that, globally, the gains of energy are slightly larger using the hybrid functionals than the LDA/GGA functionals. This is in line with the slightly larger amplitudes of phonon unstabilities predicted with the hybrid functionals (see the appendix). It is also consistent with the slightly larger amplitudes of distortions obtained with the hybrid functionals, although LDA shows energies more comparable to GGA but amplitudes of distortion closer to hybrid functionals.
Comparing the lowerings of energy produced by the individual condensation of distinct kinds of instabilities (AFD, PM, AP), we observe that the smallest ones are related to the AFD motions (P 4/mbm, P nma, R3c phases). This is compatible with the fact that AFD modes are less unstable than FE or AP modes. The energy lowerings produced by AFD modes are also significantly smaller with GGA functionals that also show slightly smaller AFD instabilities (the M + 3 mode is even stable in GGA-WC). Consequently, some AFD phases are not even appearing as metastable in GGA. We further notice that the energy lowerings produced by AFD modes are also smaller with HSE06 than B1-WC although in this case the phonon dispersion curves and the amplitudes of distortion are relatively close. Only LDA shows an energetics of AFD comparable to B1-WC.
Regarding the individual condensation of PM and AP modes, although the strongest instability is observed in the dispersion curves for the Γ − 4 mode with all functionals, the largest gain of energy arises from the condensation of the AP X − 5 (P 2 1 /m phase) except with the B1-WC for which the R3m phase resulting from the condensation of Γ − 4 is lower in energy. We notice also that while condensation of the AP M − 3 mode (P 4/nmm phase) produces a lowering of energy smaller than the polar Γ − 4 (R3m phase) with hybrid functionals, the opposite is true for LDA and GGA functionals.
As previously discussed by Hamdi et al. [14], the lowest energy phases of WO 3 result from the combination of distinct instabilities and anharmonic couplings play an important role in the formation of P bcn, P 2 1 /n, P 1 and P 2 1 /c phases. For this reason it is not straightforward to clarify the origin of differences in the energy landscape obtained with different functionals. As illus- trated in Fig. 4, all functionals locate consistently these four phases very close in energy (except for B1-WC that locates the P bcn phase at slightly higher energy) but nevertheless in a different order for each of them. We notice that only B1-WC and LDA predict the P 2 1 /c phase as the ground state. Moreover, although not a requirement for internal energies, the energetics of the different phases in B1-WC is in line with the sequence of phase transitions observed experimentally.
All this highlights that WO 3 exhibits a very delicate energy landscape. Choosing the most appropriate functional remains a delicate issue that goes beyond trying to best reproduce cell parameters and atomic distortions. Our study confirms the B1-WC hybrid functional as the most appropriate choice and points the LDA as a reasonable alternative.
D. Phase diagram
In line with its complex energy landscape, WO 3 presents a complicated phase diagram which, as mentioned before, is still partly debated. One question is the potential polar nature of its ground state phase: as previously discussed and confirmed by the present study, the P 2 1 /c phase of stoichiometric WO 3 is predicted as the theoretical ground state and does not show any tendency to polar instability so that the experimental P c assignment, deviating only marginally from the P 2 1 /c prediction, might be related to extrinsic effects [16]. Then, particularly puzzling is the intermediate P 2 1 /c phase appearing in a narrow temperature range at high-temperature and re-emerging as the ground state at low temperature.
In Fig. 5, we compare the symmetry-mode analysis of all experimental phases from high to low temperatures. The AP M − 3 is common to all phases and remains roughly constant in amplitude. Then, on cooling, the AFD R + 4 mode appears from the P 4/ncc phase and is preserved down to the ground state. We notice that the Glazer tilt pattern associated to the different phases (i.e. number and orientation of R + 4 modes) is evolving from one phase to another. In the P bcn and P 2 1 /n phases there is the emergence of an additional AFD M + 3 distortion which however compete with the R + 4 one [14] and disappears in the P1 and P 2 1 /c-LT phases.
The main difference between the high-and lowtemperature P 2 1 /c phases is the amplitude AFD R + 4 mode. It might happen that at high-temperature at which the R + 4 distortion are reduced, the M + 3 distortion can appear to produce intermediate phases while, due to mode competition, the latter disappear when the R + 4 distortion becomes larger at low temperatures.
Our calculations show that the P bcn, P 2 1 /n, P 1 and P 2 1 /c-LT phases are very close in energy. This might contribute to explain the origin of the very complex and unusual phase diagram of WO 3 . It suggests also that distinct phases could coexist and/or could be stabilized under slight variation of the experimental conditions.
IV. BORN EFFECTIVE CHARGES AND DIELECTRIC CONSTANT
We consider now the Born effective charges (Z * ) and dielectric tensor ( ∞ ). All results in the main text are obtained using the LDA functional. Since the optimized LDA lattice parameters slightly underestimate the experimental values, for better comparison with experiment, we decided to work with phases for which (i) lattice parameters are fixed at the experimental values for P 2 1 /n phase and at the optimized B1-WC values for P 2 1 /c phase and (ii) atomic positions are fully relaxed under that constraint. In practice, the deviations of atomic positions remain negligible compared to the structures fully relaxed with B1-WC (see Tab. A1). Further comparison of the Z * and ∞ computed in LDA and B1-WC functionals are also provided in the appendix (Tables A2 and A3), showing close agreement.
In Tab. III the Born effective charge tensors (Z * ) of WO 3 are presented for the eight nonequivalent atoms in the P 2 1 /n phase, and the four nonequivalent atoms in the P 2 1 /c phase. Some Z * are anomalously large in comparison to the nominal atomic charges (+6e, −2e for W and O atoms, respectively) and these values can be linked to the dynamical changes in the hybridization between the 2p orbitals of the oxygen and the 5d orbitals of the W atoms. The main values of the symmetric part of these tensors have also been reported for easier comparison of the different phases and atoms.
In the cubic structure of WO 3 , the WO 6 -octahedra are undistorted and no symmetry breaking is observed. This implies the existence of diagonal, isotropic Born effective charge tensors: +13.47e for W, −9.79e for O || , and −1.8e for O ⊥ consistent with those previously reported in Ref. [47]. By distorting the cubic environment, the off-diagonal values of Z * increase and the main values of Z * decrease with respect to the cubic phase, as previously observed in perovskites [48,49]. In both the P 2 1 /c and P 2 1 /n phases the off-diagonal terms of Z * are small, with the only exceptions the O 1 and O 2 atoms in the basal plane of the P 2 1 /c crystal. This is due to the fact that Z * is presented in the cartesian basis while the unit cell basis of P 2 1 /c is rotated by 45 degrees around the z axis.
We believe that the significant differences between the diagonal components of Z * (W 1 ) (and Z * (W 2 )) for the P 2 1 /n phase can be associated with the distance between two consecutive tungsten atoms in W-O-W chains, which are 3.70, 3.80 and 3.84 Å along the x, y and z directions, respectively. Therefore, by decreasing the interatomic distance the diagonal components of Z * (W 1 ) and Z * (W 2 ) will increase. Furthermore, the same trend exists when the long-short bond alternation of W-O is decreased. Although the same pattern is observed for the P 2 1 /c phase, the differences between the diagonal components of Z * (W 1 ) are smaller than for the P 2 1 /n phase.
The optical dielectric tensor (ε ∞ ij ), and related refractive indices (n i ) are reported in Tab. IV and Tab. V. The experimental values [50] of the refractive indices measured at room temperature are also reported for comparison. The results for the room temperature P 2 1 /n phase show a good agreement with the experimental data. Our calculations indicate a high index of refraction for both the P 2 1 /n and P 2 1 /c phases. This is due to the large local field in these two phases of WO 3 . Also, the birefringence (∆n) is larger in the P 2 1 /n phase (0.23, 0.34 and 0.1 for n 1 -n 2 , n 1 -n 3 and n 2 -n 3 , respectively) than in the P 2 1 /c phase [51].
For comparison, we also computed the optical dielectric tensor ε ∞ ij , and Born effective charges Z * of the ground state P 2 1 /c phase using the B1-WC and HSE06 hybrid functional. The results are presented in the ap- pendix Tab. A2, A3. We can conclude that LDA results are in reasonable agreement with the B1-WC results. However, LDA usually overestimates the absolute value of ε ∞ , which is due to the underestimation of the electronic bandgap in line with the lack of polarization dependence of this local exchange-correlation functional [52].
V. DYNAMICAL PROPERTIES
We focus now on the dynamical properties of the roomtemperature P 2 1 /n phase and ground-state P 2 1 /c phase of WO 3 . In each case, we first discuss the phonons at the zone center (Γ point), reporting both Raman and infrared spectra, and also the full phonon dispersion curves. As in the previous Section, all results in the main text have been obtained in LDA. Further comparison of LDA and B1-WC frequencies are reported in the appendix (Table A4), showing very close agreement regarding both phonon frequencies and eigenvectors.
A. The room temperature P 21/n phase
Irreducible representations at Γ
For the non-polar P 2 1 /n structure, which belongs to the C 5 2h point group, there are 32 atoms in the primitive cell and so 96 phonon modes at each k-point. The zone-center phonons can be classified according to the irreducible representations of this point group as
Γ phonons = 24A g ⊕ 24A u ⊕ 24B g ⊕ 24B u .
These include: (i) three acoustic modes (Γ acoustic = A u + 2B u ), (ii) 48 Raman active modes (A g and B g ) and (iii) 45 infrared (IR) active modes (A u and B u ).
Infrared active modes
TO modes -The calculated TO frequencies of the IR active phonons are listed in Tab. VI, besides the measured experimental data (in brackets) from Ref. [53] and (23) 1015 -their possible assignments. In fact, the assignment of the IR active modes remains still experimentally unexplored. Thus, we assigned each experimental frequency to the mode with largest mode effective charge in the same frequency range (see Tab. A6 in the appendix). It can be observed that the experimental values and our calculated frequencies are overall in good agreement. LO modes -The IR active LO modes take different frequencies when approaching Γ from different directions due to the low symmetry of the P 2 1 /n structure. In Tab. A6 in the appendix, the LO frequencies of the IR active modes along different directions are reported together with related TO frequencies and mode effective charges. Since there is no one-to-one correspondence, the mapping between LO and TO modes is done by comparing the eigenvectors and looking for the maximum overlap.
The polarities of the A u phonon modes are along the y direction, thus, they do not show any LO (17) and B u (1) modes are 18.1e and 19.5e, respectively, along the x direction, and almost zero along z, confirming that these modes are mostly polarized along the x direction. Note that this point is in line with the considerably large (xx) components of the Z * (W 1 ), Z * (W 2 ), Z * (O 1 ) and Z * (O 2 ) tensors of the P 2 1 /n phase. IR reflectivity spectra -The infrared reflectivity spectra are also calculated at normal incidence for all three Cartesian directions, showing contributions of B u (along x), A u (along y) and of B u (along z) modes respectively. The experimental reflectivity spectrum of the [001] crystal surface of WO 3 measured by Gabrusenoksis et al. [54] is also presented in panel (c) of Fig. 6. In our calculations, the reflectivity spectra saturate to unity because our formalism neglects damping. In spite of that, a reasonably good agreement between the theoretical and experimental reflectivity spectrum is observed. However it seems that the small feature at 850 cm −1 corresponding to the VIII: The comparison between calculated Raman modes of WO 3 P 2 1 /n phase within the LDA and available experimental data. The bold fonts represent the frequencies of Raman lines in the calculated Raman spectrum of Fig. 8.
Irrep LDA Expt. [55] Expt. [56] Expt. [57] Expt. [53] Expt. [ FIG. 6: The calculated infrared reflectivity of the P 2 1 /n phase: (a) B u with the polarity along x, (b) A u , (c) B u with the polarity along z, also the experimental reflectivity spectrum of the [001] crystal surface of WO 3 measured at room temperature in the spectral range from 50 to 1200 cm −1 , taken from Ref. [54]. 20) mode is missed in the theoretical spectrum. This mode is correctly predicted by the calculation but is predicted with a negligible mode charge.
B u (
Raman active modes
Raman modes -Tab. VII shows the calculated frequencies of Raman active phonons classified according to their symmetry. To the best of our knowledge, the only theoretical study of the Raman spectrum of the P 2 1 /n phase of WO 3 so far is done by Yang et al. [22]. In contrast to both experimental data and our calculations, they re-FIG. 7: The polarized Raman spectra of the P 2 1 /n phase, calculated within the LDA (for visualization purposes the low frequency range between 100-550 cm −1 is magnified).
port three dominant Raman peaks in the high-frequency range, located around 610, 650, 850 cm −1 . The comparison between calculated Raman active mode frequencies, and available experimental data is given in Tab. VIII where a reasonable agreement is observed. In Raman measurements of Ref [53], it has been mentioned that the crystal samples were so small that the Raman tensors could not be determined except for the 'c' element corresponding to A g symmetry and these modes are in line with the assignments from our calculations. The peak located at 393 cm −1 has only been experimentally measured by Garcia-Sanchez et al. [57] at a frequency of 376 cm −1 . The reason that this peak is absent in most of the other experimental studies is that it corresponds to irreducible representation B g and the Raman polarizability tensor for B g has only non-zero off-diagonal components (see next Section), resulting in a very low Raman inten-sity (the concentration of the active species is very low in off-diagonal directions) [58][59][60][61][62].
Raman spectra -Based on group theory, the Raman susceptibility tensors for the A g and the B g modes are
given by: A g = a d 0 d b 0 0 0 c , B g = 0 0 e 0 0 f e f 0
so that tensor elements could be determined by the polarized Raman scattering measurement on single crystals in different experimental configurations. As illustrated in Fig. 7, the calculated polarized Raman spectra show different intensities depending on the geometrical orientation. Unfortunately, no experimental polarized Raman spectra have been reported so far for WO 3 single crystals so that our calculations can be used as benchmark results for the interpretation of future measurements. We also determined the powder spectrum by averaging over different orientations of the polarized spectrum. The experimental Raman spectrum measured on nanoparticle powder of WO 3 reported in a recent work by Thummavichai et al. [56], and the calculated powder Raman spectrum are compared in Fig. 8. It appears there that our calculations provide an overall good description for the powder Raman spectrum. In the high-frequency range, the spectrum has two peaks at the correct frequency position and with a relatively good prediction of the intensities. In the low-frequency range peaks are at the right frequency position, but the intensities are strongly underestimated.
The two intense peaks in the high-frequency range are dominated by antipolar motions of the oxygen atoms (heavy W atoms are not moving significantly in this frequency range) along z and y orientations (see III B). The A g (22) mode at 766 cm −1 with the maximum intensity is dominated by the similar antipolar mode at M in the reference cubic phase (antipolar M − 3 mode related to oxygen motions along z). The second high-intensity peak at 689 cm −1 is associated to the A g (20) mode which is dominated by the similar antipolar mode at X in the reference cubic phase (antipolar X − 5 mode related to oxygen motions along y). Also, in the low-frequency range, we observe peaks associated to modes with dominant antipolar motions, but involving this time both W and O atoms (A g (6) and A g (11)). Other peaks are mostly associated with W-O-W bending modes of the bridging oxygen. Fig. 9 illustrates the phonon dispersion curves and the phonon density of states of the P 2 1 /n phase. Red and blue colors distinguish the involvement of W and O atoms. There are no imaginary frequencies in the phonon bands, indicating that this phase, although not the ground state, is dynamically stable. The phonon bands are spread into three regions separated by two gaps of 100-150 cm −1 . The low-frequency region is associated to modes involving motions of W and O atoms while the medium-frequency and high-frequency regions concern nearly pure oxygen motions.
Phonon dispersion curves
B. The ground state P 21/c phase In this section, we report the dynamical properties of the P 2 1 /c phase with the LDA functional and further assess the dynamical stability of the P 2 1 /c phase with respect to a potential P c ground state (discussed in Sec. III A).
Irreducible representations at Γ
The non-polar P 2 1 /c structure is similar to the P 2 1 /n phase and it belongs to the C 5 2h point group. However, the primitive unit cell is rotated by 45 • and only contains 16 atoms so that there are 48 Γ-phonon modes at each k-point. The zone-center phonons can be classified according to the irreducible representations of this point group as
Γ phonons = 12A g ⊕ 12A u ⊕ 12B g ⊕ 12B u .
These include: (i) three acoustic modes (Γ acoustic = A u + 2B u ), (ii) 24 Raman active modes (A g and B g ) and (iii) 21 infrared (IR) active modes (A u and B u ).
Infrared active modes
TO modes -In the first part of Tab. IX we report the calculated TO frequencies of the infrared active modes. According to this Table, there is no unstable mode at Γ: the lowest polar mode has a frequency of 139 cm −1 so that it is far from being unstable and the result appears robust (in agreement with Hamdi et al. [14]).
LO modes -The investigation of the LO modes along the different high-symmetry directions of the Brillouin zone, their corresponding irreducible representations, and mode effective charges Z * of the infrared active modes along the x, y and z direction are presented in Tab. A5 of the appendix. This IR reflectivity spectra -The infrared reflectivity spectra are also calculated at normal incidence for all the Cartesian directions. Comparing to the reflectivity spectra of the P 2 1 /n phase, here, between 750 to 800 cm −1 two peaks of A u and B u (z) are absent. This point can be exploited for distinguishing these two phases in future infrared measurements.
FIG. 8:
The unpolarized Raman spectra of the P 2 1 /n phase, the room temperature structure of WO 3 , the calculated within the LDA (for the visualization purposes the low frequency range (between 100-550 cm −1 ) are magnified), and the experimental Raman spectrum measured on nanoparticle powder of WO 3 . The spectrum has been acquired using a 532 nm laser at room temperature [56].
Raman active modes
Raman modes -In the second part of Tab. IX we report the calculated TO frequencies of the Raman active modes classified according to their irreducible representations. We are not aware of any experimental data nor other theoretical values for comparison.
Raman spectra -The calculated polarized, and unpolarized Raman spectra of the P 2 1 /c phase are presented in Fig. 11 and Fig. 12, respectively. The Raman susceptibility tensors for the A g and B g modes are the same as in the P 2 1 /n phase. Although the number of Raman active modes of the P 2 1 /c phase are half the number of Raman active modes of the P 2 1 /n phase, more peaks are observed in the Raman spectrum of the P 2 1 /c phase, because the Raman intensities are higher in the P 2 1 /c phase. This might be because in this phase the density of identical bond characters are more pronounced. In the high-frequency range, the powder Raman spectrum of the P 2 1 /n phase shows only two intense peaks, whereas there are three intense and one small peak in P 2 1 /c phase spectrum. The small peak at 716 cm −1 corresponds mainly to the Jahn-Teller distortion. The mode with the highest intensity at 766 cm −1 is dominated by the antipolar oxygen motions along the z direction (related to M − 3 from the reference cubic phase, see III B). Other two peaks with noticeable intensities are also dominated by the antipolar oxygen motions but this time in the xy-plane (X − 5 antipolar mode from the reference cubic phase).
As a result, the observation of the high-frequency range of the spectrum can provide a clear way to distinguish the P 2 1 /n and P 2 1 /c phases. More generally, it is worth noticing that the number of intense peaks in the highfrequency range reflects the number of antipolar distortions that have been condensed in the crystal structure. Indeed, condensing an anti-polar distortion transforms the related anti-polar mode originally at the zone boundary into a Raman active mode at Γ. In the P 2 1 /n (resp. P 2 1 /c) phase, we have antipolar distortions along z and y directions (resp. z, y and x directions) and two (resp. three) intense peaks. In order to further validate this statement, we calculated also the Raman spectrum of P 4/ncc and P 4/nmm phases, both containing only one antipolar distortion along z (M − 3 ) and, in both cases, only one intense peak appears in the high-frequency range.
Comparing the Raman spectrum of the P 2 1 /n and P 2 1 /c phases, we further observe a small shift of the
Phonon dispersion curves
The phonon dispersion curves and phonon projected density of states of the P 2 1 /c phase are shown in Fig. 13. Similarly to the P 2 1 /n phase, there is no imaginary frequency, confirming the dynamical stability of this phase. Again, the frequencies are spread into 3 well-separated regions. The main differences between the two phases appear in the intermediate region, involving the stretching vibration of O-W-O. Also the DOS of the low-energy region is shifted to slightly higher frequencies in the P 2 1 /c phase.
VI. CONCLUSIONS
In this work we have verified the ability of various DFT functionals (B1-WC, HSE06, LDA, GGA-WC, PBEsol) to describe the structural energetics of all the experimentally observed phases as well as six additional metastable phases of WO 3 . We have analyzed the atomic distortions of these phases in terms of symmetry adapted modes of the hypothetical cubic phase and quantified the amplitude of the distortions. We have analyzed their energetics based on the instabilities presented in the phonon dispersion curve of the parent cubic phase. It can be concluded that (i) the B1-WC hybrid functional is the most appropriate to describe WO 3 while (ii) among the investigated standard functionals, LDA is the most suitable to reproduce the structural distortions of the different phases of WO 3 and to predict the correct ground state. We have also re-discussed briefly the unusual phase diagram of WO 3 .
We have exhaustively described the dielectric and lattice dynamical properties of the ground state P 2 1 c phase and room temperature P 2 1 /n phase. The Born effective charges, dielectric tensors and the refractive indices were obtained. The phonon dispersion curves were also reported; no imaginary frequency was observed, indicating that both phases are dynamically stable. The Γ-phonon modes of both phases were carefully studied and compared to available data. The infrared reflectivity and Raman spectra were also calculated and reasonable agreement between experimental and calculated spectra was found for the room temperature phase. Also, we have carefully discussed the Raman spectra, explaining the physical origin of their main features and evolution from one phase to another. We pointed out that the alternation of long-short bonds along O − W − O chains can lead to important consequences on the dynamical properties of different phases of WO 3 . We also revealed that the number of peaks in the high-frequency range of the Raman spectrum appears as a fingerprint of the number of antipolar distortions that are present in the structure so that measurement of the Raman spectrum of WO 3 appears as an efficient way to distinguish between its different phases.
Our calculations appear as benchmark results for the interpretation of future experimental measurements. Furthermore, many physical properties of WO 3 are affected by polarons, which arise from the coupling of excess charge carriers with specific phonon modes. Our work provides all the ingredients for future more systematic analysis of which phonon modes are involved in polaron formation and key to localize the charge. dence between the phonon eigenmodes, and thus also a match between the corresponding phonon frequencies, which are shown in Tab. A4. The comparison of the calculated phonon frequencies by both LDA and B1-WC indicates that they are in good agreement.
Appendix C: LO-TO modes
In Tab. A5 and Tab. A6, the LO frequencies of the IR active modes for the P 2 1 /c and P 2 1 /n phase, respectively, are reported along different directions together with related TO frequencies and mode effective charges. The TO phonon frequencies of the ground state P 2 1 /c phase at Γ within the B1-WC and LDA approaches, in cm −1 , and the overlap matrix between the phonon eigenmodes computed from the diagonalization of the dynamical matrix by ABINIT/LDA (η A ) and those computed by CRYSTAL/B1-WC (η C ). The LO frequencies (in cm −1 ) of the infrared active modes along the different high-symmetry directions of the Brillouin zone, their corresponding irreducible representations, and mode effective charges (Z * ) of the TO infrared active modes along the x, y and z is presented for the P 2 1 /n phase calculated by LDA. The frequencies shown by light gray in the LO columns are equal to the TO frequencies. The frequencies shown in bold font present the strong shifts in LO-TO splitting. The coordinates of the high-symmetry points are as follows: Y(1/2,0,0), Z(0,1/2,0), B(0,0,1/2), C(1/2,1/2,0), D(0,1/2,1/2), A(1/2,0,1/2) and E(1/2,1/2,1/2).
FIG. 1 :
1The phonon dispersion curve of the cubic phase of WO 3 , computed with the B1-WC hybrid functional and proper treatment of the dipole-dipole interactions.Lines are colored according to the involvement of W (red) and O (blues) atoms in each mode.
FIG. 2 :
2Symmetry adapted mode decomposition of distorted WO 3 phases that are observed experimentally. Comparison between B1-WC, HSE06, GGA-WC, PBEsol, LDA and experimental results[3,6,11]. The modes with an amplitude lower than 0.04 Angstrom are not shown.R + 4 and M + 3 modes. Besides, the antipolar M − 3 mode is along the z direction but with a very slight tilt toward the x direction. This small x component of the M − 3 mode appears through a coupling between the secondary X + 5 and X − 5 modes and the primary R + 4 mode. The P 2 1 /n phase involves the same modes but combined differently within a tilt pattern a − b + c − . Again, the antipolar M − 3 mode is along z with a very small x component. Also, the antipolar X − 5 and X + 5 modes are along y direction but with a very small z component. The P1 phase shows FIG. 3: The symmetry adapted mode decomposition of distorted WO 3 phases which are not observed experimentally. Comparison of B1-WC, HSE06, GGA-WC, PBEsol and LDA. The modes with an amplitude lower than 0.04 Angstrom are not shown. The black lines correspond to the cases where the phases are not condensed.
FIG. 4 :
4Calculated energy gain with respect to the cubic phase of different phases of WO 3 . Comparison between B1-WC, HSE06, GGA-WC, PBEsol and LDA.
FIG. 5 :
5The experimental symmetry adapted mode decomposition of observed phases presented in the order they appear with decreasing temperatures.
III: Calculated (LDA) Born effective charge tensors (Z * ) and their main values (between parentheses) of the atoms of the asymmetric units (see notations in Tab. A1 in the appendix) of the P 2 1 /n and P 2 1 /c phases of WO 3 .
FIG. 9 :
9The calculated full phonon dispersion curve and phonon density of states of the room temperature structure P 21/n of WO 3 within LDA. The blue and red colors distinguish the contributions of oxygen and tungsten atoms, respectively.FIG. 10: The calculated infrared reflectivity of the P 2 1 /c phase: (a) B u with the polarity along x, (b) A u , (c) B u with the polarity along z. FIG. 11: The polarized Raman spectra of the P 2 1 /c phase of WO 3 calculated within the LDA (for visualization purposes the low frequency range between 450-150 cm −1 is magnified).
FIG. 12 :
12The unpolarized Raman spectra of the P 2 1 /c phase, the ground state of WO 3 , calculated within LDA (for the visualization purposes the low frequency range (between 450-150 cm −1 ) is magnified).FIG. 13: The calculated full phonon dispersion curves and phonon density of states of the ground state P 2 1 /c phase of WO 3 within the LDA. The blue and red colors represent the contribution of oxygen and tungsten atoms, respectively. peaks. The larger the alternation of long-short O−W −O bonds, the higher is the frequency of the peaks. The 766 cm −1 peak is an exact match. This is expected because P 2 1 /c and P 2 1 /n phases have the same long-short O − W − O bonds length equal to 1.76-2.09 Å along z. The 689 cm −1 peak in the Raman spectrum of the P 2 1 /n is attributed to the long-short O − W − O bonds equal to 1.78-2.04 Å along y, while in the P 2 1 /c phase, the second peak is shifted to the lower frequency (644 cm −1 ) due to the smaller alternation of long-short O − W − O bonds
ACKNOWLEDGEMENTS
This work has been funded by the Communauté Française de Belgique (ARC AIMED G.A. 15/19-09) and a Methusalem project of the University of Antwerp . EB thanks the FRS-FNRS for support. The authors acknowledge the CECI supercomputer facilities funded by the F.R.S-FNRS (Grant No. 2.5020.1), the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles funded by the Walloon Region (Grant No. 1117545), and the computing facilities of the Flemish Supercomputer Center. We acknowledge that the results of this research have been achieved using the DECI resource BEM based in Poland at Wrocław with support from the PRACE OFFSPRING project.
FIG. A1: The phonon dispersion curves of the cubic phase of WO 3 , obtained by (a) the CRYSTAL17 code, and the improper implementation of the mixed-space approach, and (b) the ABINIT code, and the proper employment of Fourier interpolation technique.
FIG. A2: The comparison of phonon dispersion curves of the cubic phase obtained by B1-WC and (a) HSE06, (b) LDA, (c) PBEsol, (d) GGA-WC.
A5: The LO frequencies (in cm −1 ) of the infrared active modes along the different high-symmetry directions of the Brillouin zone, their corresponding irreducible representations, and mode effective charges (Z * ) of the TO infrared active modes along the x, y and z is presented for the P 2 1 /c phase calculated by LDA. The frequencies shown by light gray in the LO columns are equal to the TO frequencies. The frequencies shown in bold font present the strong shifts in LO-TO splitting. The coordinates of the high-symmetry points are as follows:
TABLE I :
IExperimental and calculated lattice parameters (Å) of different distorted phases of WO 3 , calculated by LDA, GGA-WC, PBEsol, HSE06 and B1-WC hybrid functionals. Also the PBE results of Yang et al.[22] are included. Data in parentheses represent relative deviations from the experimental values in percent.Phase
Exp
HSE06 (∆%)
B1-WC (∆%)
GGA-WC (∆%)
PBEsol (∆%)
PBE [22] (∆%)
LDA (∆%)
5.29 a
5.29 (+0.0)
5.29 (+0.0)
5.31 (+0.3)
5.30 (+0.1)
5.37 (+1.5)
5.27 (-0.3)
P4/nmm
3.92
3.96 (+1.0)
3.92 (+0.0)
3.92 (+0.0)
3.92 (+0.0)
4.05 (+3.3)
3.87 (-1.1)
5.27 a
5.20 (-1.3)
5.16 (-2.0)
5.26 (-0.1)
5.24 (-0.5)
-
5.17 (-1.8)
P4/ncc
7.84
7.95 (+1.4)
7.87 (+0.3)
7.85 (+0.1)
7.85 (+0.1)
-
7.74 (-1.2)
5.27 a
5.28 (+0.1)
5.26 (-0.1)
5.33 (+1.1)
5.27 (+0.0)
-
5.23 (-0.7)
P 21/c-HT
5.26
5.21 (-0.9)
5.15 (-2.0)
5.31 (+0.9)
5.27 (+0.1)
-a Ref [3]
b Ref [6]
c Ref [42]
d Ref [11]
TABLE II :
IILattice parameters (Å) of metastable distorted phases of WO 3 not observed experimentally, calculated using the LDA, GGA-WC, PBEsol, HSE06 and B1-WC hybrid functionals. Dash symbols correspond to cases for which the phase is not metastable and the system relaxes back to a higher symmetry.Phase
HSE06 B1-WC GGA-WC PBEsol LDA
P m3m
3.78
3.78
3.79
3.79
3.76
5.30
5.26
-
5.35
5.25
P4/mbm
3.79
3.78
-
3.79
3.76
3.88
3.85
3.85
3.85
3.81
P 21/m
7.43
7.44
7.46
7.45
7.41
3.88
3.85
3.85
3.85
3.81
R3c
5.31
5.35
5.36
5.35
5.29
R3m
3.82
3.81
3.81
3.81
3.78
R3c
5.39
5.35
-
-
5.32
5.32
5.32
-
-
5.30
Pnma
7.44
7.42
-
-
7.39
5.23
5.22
-
-
5.20
TABLE
TABLE IV :
IVOptical dielectric tensor (ε ∞ ij ) of the P 2 1 /n and P 2 1 /c phases of WO 3 within LDA.
TABLE V :
VCalculated (LDA) and experimental refractive indices (n i ) of the P 2 1 /n and P 2 1 /c phases of WO 3Phase
n1
n2
n3
P 21/n (EXP) [50] 2.70±0.035 2.37±0.035 2.28±0.035
P 21/n (LDA)
2.65
2.41
2.31
P 21/c (LDA)
2.63
2.54
2.33
TABLE VI :
VIThe TO phonon frequencies of the infrared active modes of the P 2 1 /n room temperature phase within LDA in cm −1 . The values in brackets correspond to the experimental measurements reported in[53] Irrep
LDA [Exp]
Irrep
LDA [Exp]
Au(1)
46
Bu(1)
126
Au(2)
57
Bu(2)
194
Au(3)
74
Bu(3)
203
Au(4)
108
Bu(4)
221
Au(5)
134
Bu(5)
235
Au(6)
206
Bu(6)
265
Au(7)
226 [230]
Bu(7)
280
Au(8)
257
Bu(8)
283 [285]
Au(9)
266
Bu(9)
303
Au(10)
272
Bu(10)
310 [310]
Au(11)
308
Bu(11)
315
Au(12)
326
Bu(12)
332 [335]
Au(13)
333
Bu(13)
344
Au(14)
346
Bu(14)
362
Au(15)
362 [370]
Bu(15)
379
Au(16)
425
Bu(16)
407
Au(17)
446
Bu(17)
625 [665]
Au(18)
609
Bu(18)
756
Au(19)
695
Bu(19)
776
Au(20)
699
Bu(20)
847 [825]
Au(21)
766
Bu(21)
1042
Au(22)
1010 [920]
Bu(22)
1048
Au
-TO splitting along the Γ − Y and Γ − B directions. On the other hand, some A u modes experience large shifts of their LO frequencies along the [010] direction (with a width of 45 cm −1 for A u (8), 48 cm −1 for A u (15) and 262 cm −1 for A u (19)). Along the [011] direction, although there is a component parallel to the polarity, no significant shift of LO frequencies is found for A u modes. The A u (15) and A u (19) modes experience a strong LO-TO shifting along [110] (44 cm −1 and 263 cm −1 , respectively). Along
TABLE VII :
VIIPhonon frequencies of Raman active modes of the P 2 1 /n room temperature phase within LDA in cm −1 . direction. As illustrated inTable A6, the B u modes have their polarity in the xz-plane. The two largest LO-TO splittings of 334 and 275 cm −1 are observed for the B u (17) and B u (1) modes respectively and are related to the opposite motion of W and O atoms along x. The mode effective charges of the B uIrrep
LDA
Irrep
LDA
Ag(1)
48
Bg(1)
46
Ag(2)
53
Bg(2)
75
Ag(3)
59
Bg(3)
81
Ag(4)
72
Bg(4)
178
Ag(5)
79
Bg(5)
186
Ag(6)
117
Bg(6)
202
Ag(7)
159
Bg(7)
218
Ag(8)
206
Bg(8)
249
Ag(9)
214
Bg(9)
255
Ag(10)
269
Bg(10)
291
Ag(11)
271
Bg(11)
312
Ag(12)
304
Bg(12)
336
Ag(13)
321
Bg(13)
354
Ag(14)
329
Bg(14)
359
Ag(15)
342
Bg(15)
363
Ag(16)
419
Bg(16)
393
Ag(17)
434
Bg(17)
430
Ag(18)
443
Bg(18)
433
Ag(19)
589
Bg(19)
608
Ag(20)
689
Bg(20)
738
Ag(21)
704
Bg(21)
756
Ag(22)
766
Bg(22)
842
Ag(23)
809
Bg(23)
1003
Ag(24)
1051
Bg(24)
1079
the [111] direction the A u (7) mode and again the A u (19)
mode show giant shifts of 171 cm −1 and 268 cm −1 , re-
spectively. In fact, all these giant shifts correspond to
modes with very large mode effective charges that are
associated with the opposite motion of W and O atoms
along the y
TABLE
Table shows
showsthat along
all the high-symmetry directions of the Brillouin zone,
there are only three large LO-TO splittings in the same
LO frequency ranges: at 260 ± 20 cm −1 , 415 ± 10 cm −1 ,
and 960 ± 20 cm −1 . In addition, the strongest LO-TO
splitting is located at 962 cm −1 with a width of 322 cm −1
and is associated to the B u (8) mode in the [101] direc-
tion. This mode is associated with the opposite motions
of W and O along the x direction.
TABLE IX :
IXThe TO phonon frequencies of the ground state P 2 1 /c phase at Γ within LDA.Irrep
Freq(cm −1 )
Irrep
Freq(cm −1 )
Au
139
Bu
166
Au
174
Bu
241
Au
221
Bu
272
Au
255
Bu
318
Au
284
Bu
345
Au
302
Bu
360
Au
339
Bu
453
Au
365
Bu
640
Au
659
Bu
766
Au
737
Bu
1041
Au
1020
Ag
50
Bg
53
Ag
79
Bg
76
Ag
136
Bg
178
Ag
179
Bg
196
Ag
195
Bg
272
Ag
252
Bg
327
Ag
294
Bg
410
Ag
372
Bg
423
Ag
427
Bg
450
Ag
644
Bg
628
Ag
716
Bg
839
Ag
765
Bg
1076
TABLE A1 :
A1Atomic positions of P 2 1 /n and P 2 1 /c phases of W O 3 in reduced coordinates. Only atomic positions are relaxed by LDA with the fixed lattice constants of the experimental values, and calculated within B1-WC for P 2 1 /n(7.30, 7.53, 7.69 Å), and P 2 1 /c (5.26, 5.15, 7.61 Å), respectively.Phase
atom
Wyckoff
Position
P 21/n
W1
4e
0.2487 0.0274 0.2203
W2
4e
0.2507 0.0286 0.7211
O1
4e
0.5011 0.0338 0.2713
O2
4e
0.4993 0.4657 0.2707
O3
4e
0.2158 0.2611 0.2380
O4
4e
0.2851 0.2611 0.7509
O5
4e
0.2269 0.0199 0.4927
O6
4e
0.2281 0.5067 0.5067
P 21/c
W1
4e
0.2541 -0.2338 -0.2180
O1
4e
0.0343 0.0383 -0.2209
O2
4e
0.4580 0.4637 -0.2794
O3
4e
0.2521 -0.3077 0.0078
TABLE A2 :
A2The Born effective charge tensors (Z * ) of the P 2 1 /c phase of WO 3 , calculated by both LDA and B1-WC approaches (see notations of Tab. A1).
TABLE A3 :
A3Comparison between optical dielectric tensor (ε ∞ ij ) of the P 2 1 /c phase of WO 3 , calculated by HSE06, B1-WC hybrid functional and LDA.
TABLE A4 :
A4
TABLE
TABLE A6 :
A6
Appendix A: Phonon dispersion calculationsIn the CRYSTAL17 package, the phonon frequencies are computed using the frozen phonon numerical differences approach.Fig. A1ashows the phonon bands of the parent cubic phase of WO 3 generated by CRYSTAL17. The unphysical oscillations and intersections observed in the high frequency range of the phonon bands are due to the improper implementation of the mixed-space approach [A63] (the cancellation of the artificial macroscopic electric field is not applied) which causes problems in Fourier interpolations at arbitrary, non high symmetry q-points. As a result, the correct phonon bands has been obtained just after a proper use of the Fourier interpolation technique (shown inFig. A1b).Appendix B: Comparison of LDA and B1-WC for lattice dynamical properties of P 21/c phase In order to be able to accurately match the Γ-phonon modes obtained by LDA and B1-WC, we have evaluated the overlap matrix between the phonon eigenmodes computed from the diagonalization of the dynamical matrix by ABINIT/LDA (η A ) and those computed by CRYSTAL/B1-WC (η C ), namely η C | M | η A , where M is the mass matrix. In case of two corresponding eigenmodes, this overlap η C | M | η A should be approximately 1. We could realize such a one-to-one correspon-
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| []
|
[
"Microscopic Conductivity of Lattice Fermions at Equilibrium -Part I: Non-Interacting Particles",
"Microscopic Conductivity of Lattice Fermions at Equilibrium -Part I: Non-Interacting Particles"
]
| [
"J.-B Bru ",
"W De Siqueira ",
"Pedra C Hertling "
]
| []
| []
| We consider free lattice fermions subjected to a static bounded potential and a time-and space-dependent electric field. For any bounded convex region R ⊂ R d (d ≥ 1) of space, electric fields E within R drive currents. At leading order, uniformly with respect to the volume |R| of R and the particular choice of the static potential, the dependency on E of the current is linear and described by a conductivity (tempered, operator-valued) distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of R, in accordance with Ohm's law in Fourier space. This finite measure is the Fourier transform of a time-correlation function of current fluctuations, i.e., the conductivity distribution satisfies Green-Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace-Fourier transform of a so-called quantum current viscosity. The real and imaginary parts of conductivity distributions are related to each other via the Hilbert transform, i.e., they satisfy Kramers-Kronig relations. At leading order, uniformly with respect to parameters, the heat production is the classical work performed by electric fields on the system in presence of currents. The conductivity measure is uniformly bounded with respect to parameters of the system and it is never the trivial measure 0 dν. Therefore, electric fields generally produce heat in such systems. In fact, the conductivity measure defines a quadratic form in the space of Schwartz functions, the Legendre-Fenchel transform of which describes the resistivity of the system. This leads to Joule's law, i.e., the heat produced by currents is proportional to the resistivity and the square of currents.From (86) it follows that αρ Λ l (J ) ⊂ ρ Λ l (αJ ). If α = 0 then, by replacing (J , α) with (αJ , α −1 ), one gets that ρ Λ l (αJ ) ⊂ αρ Λ l (J ). (i.c) Let J 1 , J 2 ∈ Dom(ρ Λ l ) and take any E which together with (86) yields the converse inclusion ρ Λ l (J 1 + J 2 ) ⊂ ρ Λ l (J 1 ) + ρ Λ l (J 2 ) .(ii) Take any J ∈ Dom(ρ Λ l ) and E J ∈ ρ Λ l (J ). We infer from (75) and Lemma 4.3 that J , E J = J E J , E J = Q Λ l (E J ) . | 10.1063/1.4919967 | [
"https://arxiv.org/pdf/1611.07730v1.pdf"
]
| 119,317,432 | 1611.07730 | bd88f9dd041e4fa9d58613301e0288e198c51998 |
Microscopic Conductivity of Lattice Fermions at Equilibrium -Part I: Non-Interacting Particles
23 Nov 2016 November 24, 2016
J.-B Bru
W De Siqueira
Pedra C Hertling
Microscopic Conductivity of Lattice Fermions at Equilibrium -Part I: Non-Interacting Particles
23 Nov 2016 November 24, 2016
We consider free lattice fermions subjected to a static bounded potential and a time-and space-dependent electric field. For any bounded convex region R ⊂ R d (d ≥ 1) of space, electric fields E within R drive currents. At leading order, uniformly with respect to the volume |R| of R and the particular choice of the static potential, the dependency on E of the current is linear and described by a conductivity (tempered, operator-valued) distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of R, in accordance with Ohm's law in Fourier space. This finite measure is the Fourier transform of a time-correlation function of current fluctuations, i.e., the conductivity distribution satisfies Green-Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace-Fourier transform of a so-called quantum current viscosity. The real and imaginary parts of conductivity distributions are related to each other via the Hilbert transform, i.e., they satisfy Kramers-Kronig relations. At leading order, uniformly with respect to parameters, the heat production is the classical work performed by electric fields on the system in presence of currents. The conductivity measure is uniformly bounded with respect to parameters of the system and it is never the trivial measure 0 dν. Therefore, electric fields generally produce heat in such systems. In fact, the conductivity measure defines a quadratic form in the space of Schwartz functions, the Legendre-Fenchel transform of which describes the resistivity of the system. This leads to Joule's law, i.e., the heat produced by currents is proportional to the resistivity and the square of currents.From (86) it follows that αρ Λ l (J ) ⊂ ρ Λ l (αJ ). If α = 0 then, by replacing (J , α) with (αJ , α −1 ), one gets that ρ Λ l (αJ ) ⊂ αρ Λ l (J ). (i.c) Let J 1 , J 2 ∈ Dom(ρ Λ l ) and take any E which together with (86) yields the converse inclusion ρ Λ l (J 1 + J 2 ) ⊂ ρ Λ l (J 1 ) + ρ Λ l (J 2 ) .(ii) Take any J ∈ Dom(ρ Λ l ) and E J ∈ ρ Λ l (J ). We infer from (75) and Lemma 4.3 that J , E J = J E J , E J = Q Λ l (E J ) .
Introduction
The present paper belongs to a succession of works on Ohm and Joule's laws starting with [BPH1], where heat production of free lattice fermions subjected to a static bounded potential and a time-and space-dependent electric field has been analyzed in detail.
Note that there are mathematical results, previous to [BPH1], on transport properties of different models that yield Ohm's law in some form. The closest results to ours are [KM1,KM2,KLM], where the concept of a "conductivity measure" is introduced for a system of non-interacting fermions subjected to a random potential. [BC] proves Ohm's law for free fermions in graphene-like materials subjected to space-homogeneous timeperiodic electric fields. In [FMU], Ohm's law in the DC-regime is stated for contact interactions between two quasi-free reservoirs with the steady current being a function of the chemical potential difference between the reservoirs. This corresponds to an open quantum system approach to transport properties as in [JOP1,JOP2,JOP3,CMP]. In particular, in contrast to our approach, the conductivity derived in [FMU] is not a bulk property. We rather consider the current response of a closed infinite system of fermions to time-dependent electric fields so that properties of bulk coefficients can be studied in the AC-regime [BPH2]. For previous results on heat production in infinite non-autonomous (closed) quantum systems, see, e.g., [FMSU].
Ohms law is also valid at microscopic scales. Indeed, in a recent work [W] the authors experimentally verified the validity of Ohm's law at the atomic scale for a purely quantum system. Such a behavior was unexpected [F]:
... In the 1920s and1930s, it was expected that classical behavior would operate at macroscopic scales but would break down at the microscopic scale, where it would be replaced by the new quantum mechanics. The pointlike electron motion of the classical world would be replaced by the spread out quantum waves. These quantum waves would lead to very different behavior. ... Ohm's law remains valid, even at very low temperatures, a surprising result that reveals classical behavior in the quantum regime.
[D. K. Ferry, 2012] One aim of the present paper is to establish a form of Ohm and Joule's laws at microscopic scales, by introducing the concept of microscopic conductivity distributions for bounded regions R ⊂ R d of space, whose existence and basic properties follow from rather general properties of fermion systems at equilibrium.
More precisely, consider any arbitrary smooth compactly supported function E : R → R which yields a space-homogeneous electric field 1[x ∈ R] E t w at time t ∈ R oriented along the normalized vector w := (w 1 , . . . , w d ) ∈ R d in some open convex domain R ⊂ R d . For free lattice fermions at thermal equilibrium subjected to a static bounded potential, we show the existence of finite symmetric measures {µ R } R⊂R d on R taking values in the set B + (R d ) of positive linear operators on R d such that, uniformly with respect to (w.r.t.) the volume |R| and the choice of the static potential, the induced mean current response J
R (t) = 1 2 RÊ (t) ν µ R (dν) w + i 2 R H(Ê (t) ) (ν) µ R (dν) w + O E 2 ∞ ,
withÊ being the Fourier transform of E,Ê
R (t) = µ R (Ê (t) ) + iµ ⊥ R (Ê (t) ) w + O E 2 ∞ ,
see Equations (54)-(55). By B (R d )-valued tempered distributions, we mean a map from the space S (R; C) of Schwartz functions to the space B (R d ) of linear operators on R d where each entry w.r.t. the canonical orthonormal basis of R d is a (tempered) distribution. µ R is the linear response in-phase component of the total conductivity in Fourier space and µ R + iµ ⊥ R is named the (microscopic, B(R d )-valued) conductivity distribution of the region R, while µ R is the (in-phase) conductivity measure, similar to [KLM].
We show four important properties of µ R :
• It is the Fourier transform of a time-correlation function of current fluctuations, i.e., the microscopic conductivity measures satisfy Green-Kubo relations. See Theorem 3.1 and Equation (46).
• µ R (R) op is uniformly bounded w.r.t. R and µ R (R\{0}) > 0. See Theorem 3.1.
• If a cyclic representation of the equilibrium state of the system is denoted by (H, π, Ψ), then µ R is the spectral measure of the Liouvillean L of the system w.r.t. a vector Ψ R ∈ H. We show that µ R (R\{0}) = 0 if and only if Ψ R ∈ ker L. Thus, µ R (R\{0}) > 0 is equivalent to the geometric condition Ψ R / ∈ ker L which is easily verified in the present case. See Equation (111), Theorem 5.6 and Corollary 5.7.
• µ R can also be constructed on R\{0} as the boundary value of the Laplace-Fourier transform of a so-called quantum current viscosity. See Equations (32) and (40) as well as Theorem 5.9.
If the first law of thermodynamics holds true for the system under consideration, then the existence and basic properties of the microscopic conductivity measures are, roughly speaking, consequences of very general properties of KMS (Kubo-Martin-Schwinger) states and decay bounds of space-time correlation functions of the equilibrium state.
Indeed, the existence of the (in-phase) conductivity measure is related to the positivity of the heat production induced by the electric field on the fermion system at thermal equilibrium. When the so-called AC-condition
R E t dt = 0(1)
holds, the total heat production per unit of volume (of R) as the electric field is switched off turns out to be equal to
RÊ ν w, µ R (dν) w + O E 3 ∞ = R E t w, µ R (Ê (t) ) w dt + O E 3 ∞ ,
uniformly w.r.t. |R| and the choice of the static potential. Since
R E t w, µ ⊥ R (Ê (t) ) w dt = 0 ,
this expression is the classical work performed by the electric field on the fermion system in the presence of currents J
R :
R E t w, J(E)R (t) dt + O E 3 ∞ .(2)
As µ R (R\{0}) > 0, this implies that electric fields generally produce heat in such systems and heat production is directly related to the electric conductivity. Note that the elements of the dual S * 0 of the space S 0 of Schwartz functions R → R satisfying the AC-condition (1) are restrictions to S 0 of tempered distributions. S * 0 is interpreted here as a space of AC-currents and (S 0 , S * 0 ) is a dual pair. To obtain Joule's law in its original formulation, which relates the heat production with currents rather than with electric fields, we consider the Legendre-Fenchel transform Q * R of the positive quadratic form
Q R (E) := R E t w, µ R (Ê (t) ) dt .
Let ∂Q R (E) ⊂ S * 0 be the subdifferential of Q R at the point E ∈ S 0 .The multifunction
E → σ R (E) = 1 2 ∂Q R (E)
from S 0 to S * 0 (i.e., the set-valued map from S 0 to 2 S * 0 ) is single-valued with domain Dom(σ R ) = S 0 . It is interpreted as the AC-conductivity of the region R. Similarly, the multifunction J → ρ R (J ) = 1 2 ∂Q * R (J ) from S * 0 to S 0 (i.e., the set-valued map from S * 0 to 2 S 0 ) is the AC-resistivity of the region R. Indeed, for all J ∈ Dom(ρ R ) = ∅ and E ∈ Dom(σ R ) = S 0 ,
σ R (ρ R (J )) = {J } and ρ R (σ R (E)) ⊃ {E} .
Moreover, the multifunction ρ R is linear, in the sense described in Section 4.5, and, for any J ∈ Dom(ρ R ),
{Q * R (J )} = J , ρ R (J ) = Q R (ρ R (J )) .(3)
Thus, J , ρ R (J ) is the heat production (per unit of volume) in presence of the current J ∈ Dom(ρ R ). In other words, (3) is an expression of Joule's law in its original formulation, that is, the heat produced by currents is proportional to the resistivity and the square of currents. Remark that we use the Weyl gauge for which E is minus the time derivative of the potential A. Thus, the quantity R E t dt is the total shift of the electromagnetic potential A between the times where the field E is turned on and off. For this reason, we impose the AC-condition (1) to identify the total electromagnetic work with the total internal energy change of the system, which turns out to be the heat production, by [BPH1,Theorem 3.2]. This condition is however not used in our proofs and a general expression of the heat production as a function of the applied electric field at any time is obtained.
Indeed, based on Araki's notion of relative entropy, [BPH1] proves for the fermion system under consideration that the first law of thermodynamics holds at any time: We identify the heat production with an internal energy increment and define an electromagnetic potential energy as being the difference between the total and the internal energy increments. Both energies are studied in detail here to get the heat production at microscopic scales for all times.
Besides the internal energy increment we introduce the paramagnetic and diamagnetic energy increments. The first one is the part of electromagnetic work implying a change of the internal state of the system, whereas the diamagnetic energy is the raw electromagnetic energy given to the system at thermal equilibrium. The paramagnetic energy increment is associated to the presence of paramagnetic currents, whereas the second one is caused by thermal and diamagnetic currents. We show that these currents have different physical origins:
• Thermal currents are currents coming from the space inhomogeneity of the system. They exist, in general, even at thermal equilibrium.
• Diamagnetic currents correspond to the raw ballistic flow of charged particles due to the electric field, starting at thermal equilibrium.
• Diamagnetic currents produced by the electric field create a kind of "propagating wave front" that destabilizes the whole system by changing its internal state. In presence of inhomogeneities the system opposes itself to the propagation of that front by progressively creating so-called paramagnetic currents. Such induced currents act as a sort of friction (cf. current viscosity) to the diamagnetic current and produce heat as well as a modification of the electromagnetic potential energy.
We thus analyze the linear response in terms of diamagnetic and paramagnetic currents, which form altogether the total current of the system and yield the conductivity distribution. For more details on the features of such currents, see Sections 3.5 and 4.4.
For the sake of technical simplicity and without loss of generality, note that we only consider in the sequel an increasing sequence {Λ l } ∞ l=1 of boxes instead of general convex regions R where the electric field is non-vanishing. We obtain uniform bounds permitting to control the behavior of µ Λ l at large size l ≫ 1 of the boxes {Λ l } ∞ l=1 . The uniformity of our results w.r.t. l and the choice of the static potential is a consequence of tree-decay bounds of the n-point, n ∈ 2N, correlations of the many-fermion system [BPH1,Section 4]. Such uniform bounds are crucial in our next paper [BPH2] on Ohm's law to construct the macroscopic conductivity distribution in the case of free fermions subjected to random static potentials (i.e., in the presence of disorder).
The validity of Ohm's law at atomic scales mentioned in [W, F] suggests a fast convergence of µ Λ l , as l → ∞. Hence, we expect that the family {µ Λ l } ∞ l=1 of measures on R obeys a large deviation principle, for some relevant class of interactions between lattice fermions. This question is, however, not addressed here.
To conclude, our main assertions are Theorems 3.1 (existence of the conductivity measure), 3.3 (cf. Ohm's law) and 4.1, 4.7 (cf. Joule's law). This paper is organized as follows:
• In Section 2 we briefly describe the non-autonomous C * -dynamical system for (free) fermions associated to a discrete Schrödinger operator with bounded static potential in presence of an electric field that is time-and space-dependent. For more details, see also [BPH1,Section 2].
• Section 3 introduces Ohm's law at microscopic scales via paramagnetic and diamagnetic currents. Mathematical properties of the corresponding conductivities are explained in detail and a notion of current viscosity is discussed.
• Section 4 is devoted to the derivation of Joule's law at microscopic scales. In particular, we introduce there four kinds of energy increments: the internal energy increment or heat production, the electromagnetic potential energy, the paramagnetic energy increment and the diamagnetic energy. The AC-resistivity is also described.
• All technical proofs are postponed to Section 5. Additional properties on the conductivity measure are also proven, see Section 5.1.2.
• Finally, Section A is an appendix on the Duhamel two-point function. It is indeed an important mathematical tool used here which frequently appears in the context of linear response theory.
Notation 1.1 (Generic constants)
To simplify notation, we denote by D any generic positive and finite constant. These constants do not need to be the same from one statement to another.
Setup of the Problem
The aim of this section is to describe the non-autonomous C * -dynamical system under consideration. Since almost everything is already described in detail in [BPH1, Section 2], we only focus on the specific concepts or definitions that are important in the sequel.
Free Fermion Systems on Lattices
Algebraic Formulation of Fermion Systems on Lattices
The d-dimensional lattice L := Z d (d ∈ N) represents the (cubic) crystal and we define P f (L) ⊂ 2 L to be the set of all finite subsets of L. We denote by U the CAR C * -algebra of the infinite system and define annihilation and creation operators of (spinless) fermions with wave functions ψ ∈ ℓ 2 (L) by
a(ψ) := x∈L ψ(x)a x ∈ U , a * (ψ) := x∈L ψ(x)a * x ∈ U .
Here, a x , a * x , x ∈ L, and the identity 1 are generators of U and satisfy the canonical anti-commutation relations: For any x, y ∈ L, a x a y + a y a x = 0 , a x a * y + a * y a x = δ x,y 1 .
Static External Potentials
Let Ω := [−1, 1] L . For any ω ∈ Ω, V ω ∈ B(ℓ 2 (L)) is defined to be the self-adjoint multiplication operator with the function ω : L → [−1, 1]. The static external potential V ω is of order O(1) and we rescale below its strength by an additional parameter λ ∈ R + 0 (i.e., λ ≥ 0).
Dynamics on the One-Particle Hilbert Space
Let ∆ d ∈ B(ℓ 2 (L)) be (up to a minus sign) the usual d-dimensional discrete Laplacian defined by
[∆ d (ψ)](x) := 2dψ(x) − z∈L, |z|=1 ψ(x + z) , x ∈ L, ψ ∈ ℓ 2 (L) .(5)
Then, for ω ∈ Ω and λ ∈ R + 0 , the dynamics in the one-particle Hilbert space ℓ 2 (L) is implemented by the unitary group {U
(ω,λ) t } t∈R generated by the (anti-self-adjoint) operator −i(∆ d + λV ω ): U (ω,λ) t := exp(−it(∆ d + λV ω )) ∈ B(ℓ 2 (L)) , t ∈ R .(6)
Dynamics on the CAR C * -Algebra
For all ω ∈ Ω and λ ∈ R + 0 , the condition
τ (ω,λ) t (a(ψ)) = a((U (ω,λ) t ) * (ψ)) , t ∈ R , ψ ∈ ℓ 2 (L) ,(7)
uniquely defines a family τ (ω,λ) := {τ (ω,λ) t } t∈R of (Bogoliubov) * -automorphisms of U, see [BR2,Theorem 5.2.5]. The one-parameter group τ (ω,λ) is strongly continuous and we denote its generator by δ (ω,λ) . Clearly,
τ (ω,λ) t (B 1 B 2 ) = τ (ω,λ) t (B 1 )τ (ω,λ) t (B 2 ) , B 1 , B 2 ∈ U , t ∈ R .(8)
In the following, we will need the time-reversal operation Θ. It is the unique map Θ : U → U satisfying the following properties:
• Θ is antilinear and continuous.
• Θ (1) = 1 and Θ (a x ) = a x for all x ∈ L.
• Θ (B 1 B 2 ) = Θ (B 1 ) Θ (B 2 ) for all B 1 , B 2 ∈ U. • Θ (B * ) = Θ (B) * for all B ∈ U.
In particular, Θ is involutive, i.e., Θ • Θ = Id U . This operation can be explicitly defined by using the Fock representation of U. It is called time-reversal of the dynamics τ (ω,λ) t because of the following identity
Θ • τ (ω,λ) t = τ (ω,λ) −t • Θ ,
which is valid for all ω ∈ Ω, λ ∈ R + 0 and t ∈ R, see Lemma 5.1. This feature is important to obtain a symmetric conductivity measure.
Thermal Equilibrium State
For any realization ω ∈ Ω and strength λ ∈ R + 0 of the static external potential, the thermal equilibrium state of the system at inverse temperature β ∈ R + (i.e., β > 0) is by definition the unique (τ (ω,λ) , β)-KMS state ̺ (β,ω,λ) , see [BR2,Example 5.3.2.] or [P,Theorem 5.9]. It is well-known that such a state is stationary with respect to (w.r.t.) the dynamics, that is,
̺ (β,ω,λ) • τ (ω,λ) t = ̺ (β,ω,λ) , β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R .(9)
The state ̺ (β,ω,λ) is gauge-invariant and quasi-free. Such states are uniquely characterized by bounded positive operators d ∈ B(ℓ 2 (L)) obeying 0 ≤ d ≤ 1. These operators are named symbols of the corresponding states. The symbol of ̺ (β,ω,λ) is given by
d (β,ω,λ) fermi := 1 1 + e β(∆ d +λVω) ∈ B(ℓ 2 (L)) .(10)
Let us remark here that ̺ (β,ω,λ) is time-reversal invariant, i.e., for all parameters
β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , ̺ (β,ω,λ) • Θ (B) = ̺ (β,ω,λ) (B) , B ∈ U .
See Lemma 5.1.
Fermion Systems in Presence of Electromagnetic Fields
Electric Fields
Using the Weyl gauge (also named temporal gauge), the electric field is defined from a compactly supported potential
A ∈ C ∞ 0 = l∈R + C ∞ 0 (R × [−l, l] d ; (R d ) * ) by E A (t, x) := −∂ t A(t, x) , t ∈ R, x ∈ R d .(11)
Here, (R d ) * is the set of one-forms 1 on R d that take values in R and
A(t, x) ≡ 0 whenever x / ∈ [−l, l] d and A ∈ C ∞ 0 (R ×[−l, l] d ; (R d ) * ). Since A ∈ C ∞ 0 , A(t, x) = 0 for all t ≤ t 0 , where t 0 ∈ R is some initial time.
We also define the integrated electric field between x (2) ∈ L and x (1) ∈ L at time t ∈ R by
E A t (x) := 1 0 E A (t, αx (2) + (1 − α)x (1) ) (x (2) − x (1) )dα ,(12)where x := (x (1) , x (2) ) ∈ L 2 .
1 In a strict sense, one should take the dual space of the tangent spaces T (R d ) x , x ∈ R d .
Discrete Magnetic Laplacian
We consider without loss of generality negatively charged fermions. Thus, using the (minimal) coupling of A ∈ C ∞ 0 to the discrete Laplacian −∆ d , the discrete time-dependent magnetic Laplacian is (up to a minus sign) the self-adjoint operator
∆ (A) d ≡ ∆ (A(t,·)) d ∈ B(ℓ 2 (L)) , t ∈ R , defined 2 by e x , ∆ (A) d e y = exp i 1 0 [A(t, αy + (1 − α)x)] (y − x)dα e x , ∆ d e y(13)
for all t ∈ R and x, y ∈ L. Here, ·, · is the scalar product in ℓ 2 (L) and {e x } x∈L is the canonical orthonormal basis e x (y) ≡ δ x,y of ℓ 2 (L). In (13), αy + (1 − α)x and y − x are seen as vectors in R d .
Perturbed Dynamics on the One-Particle Hilbert Space
The dynamics of the system under the influence of an electromagnetic potential is defined via the two-parameter group {U
(ω,λ,A) t,s } t≥s of unitary operators on ℓ 2 (L) generated by the
(time-dependent anti-self-adjoint) operator −i(∆ (A) d + λV ω ) for any ω ∈ Ω, λ ∈ R + 0 and A ∈ C ∞ 0 : ∀s, t ∈ R, t ≥ s : ∂ t U (ω,λ,A) t,s = −i(∆ (A(t,·)) d + λV ω )U (ω,λ,A) t,s , U (ω,λ) s,s := 1 . (14)
The dynamics is well-defined because the map
t → (∆ (A(t,·)) d + λV ω ) ∈ B(ℓ 2 (L))
from R to the set B(ℓ 2 (L)) of bounded operators acting on ℓ 2 (L) is continuously differentiable for every A ∈ C ∞ 0 . Note that, as explained in [BPH1,Section 2.3], the interaction between magnetic fields and electron spins is here neglected because such a term will become negligible for electromagnetic potentials slowly varying in space, see Section 2.3.1. This justifies the assumption of fermions with zero-spin.
Perturbed Dynamics on the CAR C * -Algebra
For all ω ∈ Ω, λ ∈ R + 0 and A ∈ C ∞ 0 , the condition
τ (ω,λ,A) t,s (a(ψ)) = a((U (ω,λ,A) t,s ) * (ψ)) , t ≥ s, ψ ∈ ℓ 2 (L) ,(15)
uniquely defines a family of Bogoliubov automorphisms of the C * -algebra U, see [BR2,Theorem 5.2.5]. The family {τ (ω,λ,A) t,s } t≥s is itself the solution of a non-autonomous evolution equation, see [BPH1,].
2 Observe that the sign of the coupling between the electromagnetic potential A ∈ C ∞ 0 and the laplacian is wrong in [BPH1,Eq. (2.8)].
Time-Dependent State
Since ̺ (β,ω,λ) is stationary (cf. (9)) and A(t, x) = 0 for all t ≤ t 0 , the time evolution of the state of the system equals
ρ (β,ω,λ,A) t := ̺ (β,ω,λ) , t ≤ t 0 , ̺ (β,ω,λ) • τ (ω,λ,A) t,t 0 , t ≥ t 0 .(16)
This state is gauge-invariant and quasi-free for all times, by construction.
Space-Scale of Fields, Linear Response Theory and Scanning Gate Microscopy
From Microscopic to Macroscopic Electromagnetic Fields
For space scales large compared to 10 −14 m, electron and nuclei are usually treated as point systems and electromagnetic phenomena are governed by microscopic Maxwell equations. However, the electromagnetic fields produced by these point charges fluctuate very much in space and time and macroscopic devices generally measure averages over intervals in space and time much larger than the scale of these fluctuations. This implies relatively smooth and slowly varying macroscopic quantities. As explained in [Ja, Section 6.6], "only a spatial averaging is necessary." The macroscopic electromagnetic fields are thus coarse-grainings of microscopic ones and satisfy the so-called macroscopic Maxwell equations. In particular, their spacial variations become negligible on the atomic scale. Similarly, we consider that the infinite bulk containing conducting fermions only experiences mesoscopic electromagnetic fields, which are produced by mesoscopic devices. In other words, the heat production or the conductivity is measured in a local region which is very small w.r.t. the size of the bulk, but very large w.r.t. the lattice spacing of the crystal. We implement this hierarchy of space scales by rescaling vector potentials. That means, for any l ∈ R + and A ∈ C ∞ 0 , we consider the space-rescaled vector potential A l defined by
A l (t, x) := A(t, l −1 x) , t ∈ R, x ∈ R d .(17)
Then, to ensure that an infinite number of lattice sites is involved, we eventually perform the limit l → ∞. See [BPH2] for more details. Indeed, the scaling factor l −1 used in (17) means, at fixed l, that the space scale of the electric field (11) is infinitesimal w.r.t. the macroscopic bulk (which is the whole space), whereas the lattice spacing gets infinitesimal w.r.t. the space scale of the electric field when l → ∞.
Linear Response Theory
Linear response theory refers here to linearized non-equilibrium statistical mechanics and has been initiated by Kubo [K] and Mori [M]. Ohm's law is one of the first and certainly one of the most important examples thereof. It is indeed a linear response to electric fields. Therefore, we also rescale the strength of the electromagnetic potential A l by a real parameter η ∈ R and eventually take the limit η → 0.
When |η| ≪ 1 and l ≫ 1, it turns out that, uniformly w.r.t. l, the mean currents J (ω,ηĀ l ) p and J (ω,ηĀ l ) d , defined below by (42)-(43), are of order O (η). Similarly, the energy increments S (ω,ηA l ) , P (ω,ηA l ) , I (ω,ηA l ) p and I (ω,ηA l ) d , respectively defined by (58), (59), (62) and (63), are all of order O η 2 l d . Such results are derived in the next sections by using tree-decay bounds of the n-point, n ∈ 2N, correlations of the many-fermion system [BPH1, Section 4].
Experimental Setting of Scanning Gate Microscopy
Our setting is reminiscent of the so-called scanning gate microscopy used to perform imaging of electron transport in two-dimensional semiconductor quantum structures. See, e.g., [S]. In this experimental situation, the two-dimensional electron system on a lattice experiences a time-periodic space-homogeneous electromagnetic potential perturbed by a mesoscopic or microscopic time-independent electric potential. Physically speaking, this situation is, mutatis mutandis, analogous to the one considered here. Therefore, we expect that our setting can also be implemented in experiments by similar technics combined with calorimetry to measure the heat production.
Microscopic Ohm's Law
In his original work [O] G.S. Ohm states that the current in the steady regime is proportional to the voltage applied to the conducting material. The proportionality coefficient is the conductivity of the physical system. Ohm's laws is among the most resilient laws of (classical) electricity theory and is usually justified from a microscopic point of view by the Drude model or some of its improvements that take into account quantum corrections.
[Cf. the Landau theory of Fermi liquids.] As in the Drude model we do not consider here interactions between charge carriers, but our approach will be also applied to interacting fermions in subsequent papers.
In this section, we study, among other things, (microscopic) Ohm's law in Fourier space for the system of free fermions described in Section 2. Without loss of generality, we only consider space-homogeneous (though time-dependent) electric fields in the box
Λ l := {(x 1 , . . . , x d ) ∈ L : |x 1 |, . . . , |x d | ≤ l} ∈ P f (L)(18)
with l ∈ R + . More precisely, let w := (w 1 , . . . , w d ) ∈ R d be any (normalized) vector, A ∈ C ∞ 0 (R; R) and set E t := −∂ t A t for all t ∈ R. Then,Ā ∈ C ∞ 0 is defined to be the electromagnetic potential such that the value of the electric field equals E t w at time t ∈ R for all x ∈ [−1, 1] d and (0, 0, . . . , 0) for t ∈ R and x / ∈ [−1, 1] d . This choice yields rescaled electromagnetic potentials ηĀ l as defined by (17) for l ∈ R + and η ∈ R.
Before stating Ohm's law for the system under consideration we first need some definitions.
Current Observables
For any pair x := (x (1) , x (2) ) ∈ L 2 , we define the paramagnetic and diamagnetic current observables I x = I * x and I A x = (I A x ) * for A ∈ C ∞ 0 at time t ∈ R by
I x := −2Im(a * x (2) a x (1) ) = i(a * x (2) a x (1) − a * x (1) a x (2) )(19)
and
I A x := −2Im e i 1 0 [A(t,αx (2) +(1−α)x (1) )](x (2) −x (1) )dα − 1 a * x (2) a x (1) .(20)
These are seen as currents because, by (14)-(15), they satisfy the discrete continuity equation
∂ t n x (t) = −τ (ω,λ,A) t,t 0 z∈L 1 [|z| = 1] I (x,x+z) + I A (x,x+z)(21)
for x ∈ L and t ≥ t 0 , where
n x (t) := τ (ω,λ,A) t,t 0 (a * x a x )(22)
is the density observable at lattice site x ∈ L and time t ≥ t 0 . The notions of paramagnetic and diamagnetic current observables come from the physics literature, see, e.g., [GV,Eq. (A2.14)]. The paramagnetic current observable 1 [|z| = 1] I (x,x+z) is intrinsic to the system whereas the diamagnetic one I A x is only non-vanishing in presence of electromagnetic potentials.
Observe that the minus sign in the right hand side of (21) comes from the fact that the particles are negatively charged, I (x,y) being the observable related to the flow of particles from the lattice site x to the lattice site y or the current from y to x without external electromagnetic potential. [Positively charged particles can of course be treated in the same way.] As one can see from (21), current observables on bonds of nearest neighbors are especially important. Thus, we define the subset
K := x := (x (1) , x (2) ) ∈ L 2 : |x (1) − x (2) | = 1(23)
of bonds of nearest neighbors. In fact, by using the canonical orthonormal basis {e k } d k=1 of the Euclidian space R d , we define the current sums in the box Λ l (18) for any l ∈ R + , A ∈ C ∞ 0 , t ∈ R and k ∈ {1, . . . , d} by
I k,l := x∈Λ l I (x+e k ,x) − ̺ (β,ω,λ) I (x+e k ,x) 1 and I A k,l := x∈Λ l I A (x+e k ,x) .(24)
In particular, ̺ (β,ω,λ) (I k,l ) = 0, while I A k,l = 0 when A(t, ·) = 0.
Adjacency Observables
Let P x , x = (x (1) , x (2) ), be the second-quantization of the adjacency matrix of the oriented graph containing exactly the pairs (x (2) , x (1) ) and (x (1) , x (2) ), i.e.,
P x := −a * x (2) a x (1) − a * x (1) a x (2) , x := (x (1) , x (2) ) ∈ L 2 .(25)
The observable P x is related to the current observable I x in the following way: For any
x := (x (1) , x (2) ) ∈ L 2 , 2a * x (1) a x (2) = −P x + iI x , [P x , I x ] = 2i a * x (2) a x (2) − a * x (1) a x (1) .(26)
The importance of the adjacency observable P x in the linear response regime results from the fact that
I ηA x = ηP x 1 0 [A(t, αx (2) + (1 − α)x (1) )](x (2) − x (1) )dα + O η 2 .(27)
Then, similar to the diamagnetic current sum I A k,l (24), we define the observables
P k,l := x∈Λ l P (x+e k ,x) ∈ U , l ∈ R + , k ∈ {1, . . . , d} .(28)
Microscopic Transport Coefficients
Now, for any β ∈ R + , ω ∈ Ω and λ ∈ R + 0 we define two important functions associated with the observables I x and P x :
(p) The paramagnetic transport coefficient σ (ω) p ≡ σ (β,ω,λ) p is defined by σ (ω) p (x, y, t) := t 0 ̺ (β,ω,λ) i[I y , τ (ω,λ) s (I x )] ds , x, y ∈ L 2 , t ∈ R . (29) (d) The diamagnetic transport coefficient σ (ω) d ≡ σ (β,ω,λ) d is defined by σ (ω) d (x) := ̺ (β,ω,λ) (P x ) , x ∈ L 2 .(30)At x ∈ L 2 , σ (ω) d (x)
is obviously the expectation value of the adjacency observable P x in the thermal state ̺ (β,ω,λ) of the fermion system. This coefficient is diamagnetic because of (27). For any bond x ∈ K, it can be interpreted as being the kinetic energy in x: The total kinetic energy observable in the box Λ l equals
2d x∈Λ l a * x a x − x=(x (1) ,x (2) )∈K∩Λ 2 l a * x (2) a x (1) = 2d x∈Λ l a * x a x + 1 2 x∈K∩Λ 2 l P x .
The particle number observables a * x a x , x ∈ Λ l , are rather related to the (kinetic) energy in the lattice sites.
The physical meaning of σ (ω) p is less obvious. We motivate in the following that it is a linear coupling between the diamagnetic current in the bond y and the paramagnetic current in the bond x: Indeed, define by δ (ω,λ) the generator of the group τ (ω,λ) , see (7). Then, for any fixed β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , η ∈ R and y ∈ K, let the symmetric derivationδ (η,y)
:= δ (ω,λ) + iη [I y , · ](31)
be the generator of the (perturbed) group {τ (η,y) t } t∈R of automorphisms of the C * -algebra U. Note that this perturbation corresponds at leading order in η to an electromagnetic potential ηA (y) of order η along the bond y. See, e.g., Lemma 5.11. This small electromagnetic potential yields a diamagnetic current observable of the order ηP y on the same bond y, cf. (27). Since I y ∈ U (cf. (19)), we may use a Dyson-Phillips series to obtain for small |η| ≪ 1 that
τ (η,y) t (B) = τ (ω,λ) t (B) + η t 0 τ (ω,λ) t−s i[I y , τ (ω,λ) s (B)] ds + O η 2
for any B ∈ U. If |η| ≪ 1, then the diamagnetic current behaves as (25) and (27). On the other hand, by (9) and (29), the so-called paramagnetic current
J (η,y) d := ̺ (β,ω,λ) (τ (η,y) t (I ηA (y) y )) = η̺ (β,ω,λ) (P y ) + O η 2 |t| with ̺ (β,ω,λ) (P y ) = O (1), seeJ (η,y) p (x, t) := ̺ (β,ω,λ) (τ (η,y) t (I x )) − ̺ (β,ω,λ) (I x ) satisfies ∂ t J (η,y) p (x, t) = J (η,y) d v (y) (x, t) + O |J (η,y) d | 2 |t| for any x, y ∈ K and t ∈ R, where v (y) (x, t) := 1 ̺ (β,ω,λ) (P y ) ̺ (β,ω,λ) i[I y , τ (ω,λ) t (I x )] = ∂ t σ (ω) p (x, y, t) σ (ω) d (y) .(32)
In other words, v can be interpreted as a (time-dependent) quantum current viscosity. For any l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 we define two further important functions, the analogues of σ (ω) p and σ (ω) d , associated with the observables I k,l and P k,l :
(p) The space-averaged paramagnetic transport coefficient t → Ξ (ω) p,l (t) ≡ Ξ (β,ω,λ) p,l (t) ∈ B(R d ) is defined, w.r.t. the canonical orthonormal basis of R d , by Ξ (ω) p,l (t) k,q := 1 |Λ l | t 0 ̺ (β,ω,λ) i[I k,l , τ (ω,λ) s (I q,l )] ds(33)
for any k, q ∈ {1, . . . , d} and t ∈ R.
(d) The space-averaged diamagnetic transport coefficient Ξ (ω) d,l ≡ Ξ (β,ω,λ) d,l ∈ B(R d )
corresponds to the diagonal matrix defined by
Ξ (ω) d,l k,q := δ k,q |Λ l | ̺ (β,ω,λ) (P k,l ) , k, q ∈ {1, . . . , d} .(34)
Of course, by (24) and (29)- (30),
Ξ (ω) p,l (t) k,q = 1 |Λ l | x,y∈Λ l σ (ω) p (x + e q , x, y + e k , y, t) (35) for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k, q ∈ {1, . . . , d} and t ∈ R, while Ξ (ω) d,l k,k = 1 |Λ l | x∈Λ l σ (ω) d (x + e k , x) .(36)
Both coefficients are typically the paramagnetic and diamagnetic conductivity one experimentally measures for large samples, i.e., large enough boxes Λ l . Indeed, we show in [BPH2] that the limits l → ∞ of Ξ (ω) p,l and Ξ
(ω) d,l generally exist and define so-called macroscopic paramagnetic and diamagnetic conductivities. Before going further, we first discuss some important mathematical properties of Ξ (ω) p,l and Ξ (ω) d,l . By using the scalar product ·, · in ℓ 2 (L), the canonical orthonormal basis {e x } x∈L of ℓ 2 (L) and the symbol d (β,ω,λ) fermi defined by (10), we observe from (36) that
Ξ (ω) d,l k,k = 2 |Λ l | x∈Λ l Re e x+e k , d (β,ω,λ) fermi e x ∈ [−2, 2](37)
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and k ∈ {1, . . . , d}. The main property of the paramagnetic transport coefficient Ξ (ω) p,l is proven in Section 5.1.2 and given in the next theorem. To present it, we introduce the notation B + (R d ) ⊂ B (R d ) for the set of positive linear operators on R d . For any B(R d )-valued measure µ on R, we additionally denote by µ op the measure on R taking values in R + 0 that is defined, for any Borel set X , by
µ op (X ) := sup i∈I µ (X i ) op : {X i } i∈I is a finite Borel partition of X . (38)
We, moreover, say that µ is symmetric if µ(X ) = µ(−X ) for any Borel set X ⊂ R. With these definitions we have the following assertion:
Theorem 3.1 (Microscopic paramagnetic conductivity measures)
For any l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 , there exists a non-zero symmetric
B + (R d )-valued measure µ (ω) p,l ≡ µ (β,ω,λ) p,l on R such that R (1 + |ν|) µ (ω) p,l op (dν) < ∞ ,(39)uniformly w.r.t. l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , and Ξ (ω) p,l (t) = R (cos (tν) − 1) µ (ω) p,l (dν) , t ∈ R .
Proof: The assertions follow from Theorems 5.4 and 5.5 combined with Corollary 5.7 and Lemma 5.10.
Corollary 3.2 (Properties of the microscopic paramagnetic conductivity)
For l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 , Ξ
p,l has the following properties:
(i) Time-reversal symmetry: Ξ (ω) p,l (0) = 0 and Ξ (ω) p,l (−t) = Ξ (ω) p,l (t) , t ∈ R . (ii) Negativity of Ξ (ω) p,l : Ξ (ω) p,l (t) ≤ 0 , t ∈ R . (iii) Cesàro mean of Ξ (ω) p,l : lim t→∞ 1 t t 0 Ξ (ω) p,l (s) ds = −µ (ω) p,l (R\ {0}) ≤ 0 . (iv) Equicontinuity: The family {Ξ (β,ω,λ) p,l } l,β∈R + ,ω∈Ω,λ∈R + 0 of maps from R to B(R d ) is equicontinuous. (v) Macroscopic paramagnetic conductivity measures: The family {µ (ω) p,l } l∈R + has weak * - accumulation points.
Proof:
(i)-(iii) are direct consequences of Theorem 3.1 and Lebesgue's dominated convergence theorem. To prove (iv), observe that the uniform bound (39) implies that, for
any ν 0 ∈ R + 0 , µ (ω) p,l (R\ [−ν 0 , ν 0 ]) = O ν −1 0 uniformly w.r.t. l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 .
(v) follows from Theorem 3.1 and the weak * -compactness of the unit ball in the set of measures on R taking values in the set of positive elements of B(R d ).
The B + (R d )-valued measures µ (ω)
p,l can be represented in terms of the spectral measure of an explicit self-adjoint operator w.r.t. explicitly given vectors, see Equation (111). From this representation, one concludes for instance that, if the operator (∆ d + λV ω ) has purely (absolutely) continuous spectrum (as for λ = 0) then, for any k, q ∈ {1, . . . , d},
µ (ω) p,l (R\ {0}) k,q = 1 |Λ l | (I k,l , I q,l ) (ω) ∼ .
Here, (·, ·) (ω) ∼ is the Duhamel two-point function (·, ·) (ω) ∼ , which is studied in detail in Section A. In fact, the constant µ (ω) p,l (R\ {0}) is the so-called static admittance of linear response theory, see Theorem 5.8. Moreover, Theorem 5.9 explains how µ (ω) p,l can also be constructed from the space-averaged quantum current viscosity
V (ω) l (t) := Ξ (ω) d,l −1 ∂ t Ξ (ω) p,l (t) ∈ B(R d )(40)
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ∈ R. Compare with (32). More precisely, it is the boundary value of the (imaginary part of the) Laplace-Fourier transform of Ξ
(ω) d,l V (ω) l .
Recall that, as asserted in Theorem 3.1, the measure µ (ω) p,l is never the zero-measure. Nevertheless, it is a priori not clear whether the weak * -accumulation points of the family {µ (ω) p,l } l∈R + also have this property. We show in a companion paper that, as l → ∞, the measure µ (ω) p,l converges to the zero-measure if λ = 0 but, for λ ∈ R + , there is generally a unique weak * -accumulation point of {µ (ω) p,l } l∈R + , which is not the zero-measure.
Paramagnetic and Diamagnetic Currents
Recall that we assume in this section that the current results from a space-homogeneous electric field ηE t w at time t ∈ R in the box Λ l , where w :
= (w 1 , . . . , w d ) ∈ R d , E t := −∂ t A t for all t ∈ R, and A ∈ C ∞ 0 (R; R)
. This electric field corresponds to the (rescaled) electromagnetic potential ηĀ l . We also remind that {e k } d k=1 is the canonical orthonormal basis of the Euclidian space R d .
Generally, even in the absence of electromagnetic fields, i.e., if η = 0, there exist (thermal) currents coming from the inhomogeneity of the fermion system for λ ∈ R + . For any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and k ∈ {1, . . . , d},
J (ω) k,l ≡ J (β,ω,λ) k,l := |Λ l | −1 x∈Λ l ̺ (β,ω,λ) (I (x+e k ,x) )(41)
is the density of current along the direction e k in the box Λ l . In the space-homogeneous case, by symmetry, J
(ω) k,l = 0 but in general, J (ω) k,l = 0. We prove in [BPH2] that lim l→∞ J (ω) k,l = 0
almost surely if ω ∈ Ω is the realization of some ergodic random potential. Then, for any l,
β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , η ∈ R, w ∈ R d , A ∈ C ∞
The paramagnetic current density is only related to the change of internal state ρ (β,ω,λ,A) t produced by the electromagnetic field. We will show below that these currents carry the paramagnetic energy increment defined in Section 4.3. The diamagnetic current density corresponds to a raw ballistic flow of charged particles caused by the electric field, at thermal equilibrium. It directly comes from the change of the electromagnetic potential expressed in terms of the observable (57) defined below. We will show that it yields the diamagnetic energy defined in Section 4.3. With this, diamagnetic and paramagnetic currents are respectively "first order" and "second order" with respect to changes of the electromagnetic potentials and thus have different physical properties. See for instance Theorems 3.3 and 4.1.
Current Linear Response
We are now in position to derive a microscopic version of Ohm's law. We use the spaceaveraged paramagnetic and diamagnetic transport coefficients Ξ
(ω) p,l (33) and Ξ (ω) d,l (34) to define the R d -valued functions J (ω,A) p,l ≡ J (β,ω,λ, w,A) p,l and J (ω,A) d,l ≡ J (β,ω,λ, w,A) d,l by J (ω,A) p,l (t) := for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , w ∈ R d and A ∈ C ∞ 0 (R; R)
. They are the linear responses of the paramagnetic and diamagnetic current densities, respectively:
Theorem 3.3 (Microscopic Ohm's law)
For any w ∈ R d and A ∈ C ∞ 0 (R; R), there is η 0 ∈ R + such that, for |η| ∈ [0, η 0 ], J (ω,ηĀ l ) p (t) = ηJ (ω,A) p,l (t) + O η 2 and J (ω,ηĀ l ) d (t) = ηJ (ω,A) d,l (t) + O η 2 , uniformly for l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 .
Proof: See Lemmata 5.14-5.15.
The fact that the asymptotics obtained are uniform w.r.t. l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 is a crucial property to get macroscopic Ohm's law in [BPH2]. Note also that Theorem 3.3 can easily be extended to macroscopically space-inhomogeneous electromagnetic fields, that is, for all space-rescaled vector potentials A l (17) with A ∈ C ∞ 0 , by exactly the same methods as in the proof of Theorem 4.1. We refrain from doing it at this point, for technical simplicity. The result above can indeed be deduced from Theorem 4.1, see Equations (65)-(66).
As a consequence, Ξ
p,l and Ξ (ω) d,l can be interpreted as charge transport coefficients. Observe that Ξ (ω) p,l (0) = 0, by Corollary 3.2 (i). Therefore, when the electric field is switched on, it accelerates the charged particles and first induces diamagnetic currents, cf. (45). This creates a kind of "wave front" that destabilizes the whole system by changing its internal state. By the phenomenon of current viscosity discussed in Section 3.3, the presence of such diamagnetic currents leads to the progressive appearance of paramagnetic currents. We prove in Section 4 that these paramagnetic currents are responsible for heat production and modify as well the electromagnetic potential energy of charge carriers. Indeed, the positive measures of Theorem 3.1 are directly related to heat production (cf. Section 4.4) and are the boundary values of the (imaginary part of the) Laplace-Fourier transforms of the current viscosities as discussed in the previous section.
Note that Theorem 3.3 also leads to (finite-volume) Green-Kubo relations, by (33) and (44). Indeed, by (24), |Λ l | − 1 2 I k,l is a current fluctuation and (33) gives:
Ξ (ω) p,l (t) k,q = t 0 ̺ (β,ω,λ) i |Λ l | − 1 2 I k,l , |Λ l | − 1 2 τ (ω,λ) s (I q,l ) ds(46)
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R and k, q ∈ {1, . . . , d}. In the limit l → ∞ we show in [BPH2] that Ξ (ω) p,l is related to a quasi-free dynamics on the CCR algebra of (current) fluctuations.
Theorem 3.3 together with (44)-(45) gives a natural notion of linear conductivity of the fermion system in the box Λ l : It is the map
t → Σ (ω) l ≡ Σ (β,ω,λ) l (t) ∈ B(R d ) defined by Σ (ω) l (t) := 0 , t ≤ 0 , Ξ (ω) d,l + Ξ (ω) p,l (t) , t ≥ 0 ,(47)for l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 . The total current J (ω,A) l (t) := J (ω,A) p,l (t) + J (ω,A) d,l (t) , t ≥ t 0 ,
which as in [GV,Eq. (A2.14)] is the sum of paramagnetic and diamagnetic current densities, has the following linear response:
J (ω,A) l (t) = R Σ (ω) l (t − s) w E s ds = {Σ (ω) l w} 1 * E . . . {Σ (ω) l w} d * E .(48)
In particular, if the electric field stays constant for sufficiently large times, i.e., E t = D for arbitrary large times t ∈ [T, ∞) with T > t 0 , then in the situation where t ≫ T , i.e., in the DC-regime, we deduce from Corollary 3.2 (iii) and (47)-(48) that
|t| −1 J (ω,A) l (t) = D(Ξ (ω) d,l − µ (ω) p,l (R\ {0})) + o (1) .(49)
It is not a priori clear whether µ
(ω) p,l (R\ {0}) = Ξ (ω)
d,l or not. We prove in [BPH2] that this last equality actually holds in the limit l → ∞. [Recall that A ∈ C ∞ 0 is compactly supported in space and time, but it can be switched off at arbitrary large times.]
In order to express the in-phase current from (48), we define by Σ
(ω) l,+ the symmetriza- tion of Σ (ω) l , that is, Σ (ω) l,+ (t) := Σ (ω) l (|t|) = Ξ (ω) d,l + Ξ (ω) p,l (t) , t ∈ R ,(50)
see Corollary 3.2 (i). Similarly, the anti-symmetrization Σ
(ω) l,− of Σ (ω) l is given by Σ (ω) l,− := sign(t)Σ (ω) l (|t|) , t ∈ R .(51)
With these definitions the current linear response (48) equals
J (ω,A) l (t) = 1 2 R Σ (ω) l,+ (t − s) w E s ds + 1 2 R Σ (ω) l,− (t − s) w E s ds .(52)
The first part in the right hand side of this equality is by definition the in-phase current. This last equation is directly related to Ohm's law in Fourier space: Similar to [KLM], it is indeed natural to define the conductivity measure µ
(ω) Λ l ≡ µ (β,ω,λ) Λ l as being the Fourier transform of Σ (ω)
l,+ (t). By Theorem 3.1 and (50),
µ (ω) Λ l (X ) = µ (ω) p,l (X ) + (Ξ (ω) d,l − µ (ω) p,l (R))1 [0 ∈ X ]
with X ⊂ R being any Borel set. Therefore, we can rewrite the current linear response (52) as
J (ω,A) l (t) = 1 2 RÊ (t) ν µ (ω) Λ l (dν) w + i 2 R H(Ê (t) ) (ν) µ (ω) Λ l (dν) w(53)
withÊ being the Fourier transform of E,Ê (t) ν := e iνtÊ ν , and where H is the Hilbert transform, i.e.,
H (f ) (ν) := − 1 π lim ε→0 + [−ε −1 ,−ε]∪[ε,ε −1 ] f (ν − x) x dx , ν ∈ R .
Here, f : R → C belongs to the space Υ of functions which are the Fourier transforms of compactly supported and piece-wise smooth functions R → R. Equation (53) corresponds to Ohm's law in Fourier space at microscopic scales, in accordance with experimental results of [F, W]. Moreover, by Corollary 3.2 (v) together with Equation (37), Theorem 3.1 and the Bolzano-Weierstrass theorem, the family {µ (ω) Λ l } l∈R + has weak * -accumulation points. As a consequence, the current linear response converges pointwise along a subsequence to
J (ω,A) ∞ (t) = 1 2 RÊ (t) ν µ (ω) R d (dν) w + i 2 R H(Ê (t) ) (ν) µ (ω) R d (dν) w with µ (ω) R d being some weak * -accumulation point of {µ (ω) Λ l } l∈R + . µ (ω)
R d can be interpreted as a macroscopic conductivity measure and is under reasonable circumstances unique. In fact, we give in [BPH2] a detailed analysis of such limits by considering random static external potentials.
Observe that iH (Υ) ⊂ Υ and H • H = −1 on Υ. In particular, the two functionals
µ Λ l : Υ → R , µ Λ l (f ) := 1 2 R f (ν)µ (ω) Λ l (dν) , µ ⊥ Λ l : Υ → R , µ ⊥ Λ l (f ) := 1 2 R H (f ) (ν)µ (ω) Λ l (dν) ,
satisfy Kramers-Kronig relations:
µ Λ l • H =µ ⊥ Λ l and µ ⊥ Λ l • H = −µ Λ l .(54)
Note that, w.r.t. the usual topology of the space
S (R; C) of Schwartz functions, Υ ∩ S (R; C) is dense in S (R; C) and µ Λ l , µ ⊥ Λ l are continuous on Υ ∩ S (R; C). Hence, each entry of µ Λ l , µ ⊥ Λ l w.r.t.
the canonical orthonormal basis of R d can be seen as a tempered distribution. Moreover, (53) yields
J (ω,A) l (t) = µ Λ l (Ê (t) ) + iµ ⊥ Λ l (Ê (t) ) w . (55) Therefore, the B(R d )-valued distribution µ Λ l is the linear response in-phase component of the total conductivity in Fourier space. For this reason, µ Λ l + iµ ⊥ Λ l is named here the (microscopic, B(R d )-valued) conductivity distribution of the box Λ l . Similarly, the limit J (ω,A) ∞ obeys (55) with µ (ω) R d replacing µ (ω) Λ l .
Microscopic Joule's Law
...the calorific effects of equal quantities of transmitted electricity are proportional to the resistances opposed to its passage, whatever may be the length, thickness, shape, or kind of metal which closes the circuit : and also that, coeteris paribus, these effects are in the duplicate ratio of the quantities of transmitted electricity ; and consequently also in the duplicate ratio of the velocity of transmission. [Joule, 1840] In other words, as originally observed [J] by the physicist J. P. Joule, the heat (per second) produced within an electric circuit is proportional to the electric resistance and the square of the current. The aim of this section is to prove such a phenomenology for the fermion system under consideration. Before studying Joule's effect we need to define energy observables and increments:
Energy Observables
For any L ∈ R + , the internal energy observable in the box Λ L (18) is defined by
H (ω,λ) L := x,y∈Λ L e x , (∆ d + λV ω )e y a * x a y ∈ U .(56)
It is the second quantization of the one-particle operator ∆ d + λV ω restricted to the subspace ℓ 2 (Λ L ) ⊂ ℓ 2 (L). When the electromagnetic field is switched on, i.e., for t ≥ t 0 , the (time-dependent) total energy observable in the box Λ L is then equal to H
(ω,λ) L + W A t , where, for any A ∈ C ∞ 0 and t ∈ R, W A t := x,y∈Λ L e x , (∆ (A) d − ∆ d )e y a * x a y ∈ U(57)
is the electromagnetic potential energy observable. We define below four types of energies because we have the two above energy observables as well as two relevant states, the thermal equilibrium state ̺ (β,ω,λ) and its time evolution ρ
(β,ω,λ,A) t .
Time-dependent Thermodynamic View Point
In [BPH1], we investigate the heat production of the (non-autonomous) C * [BPH1,Theorem 3.2] that the fermion system under consideration obeys the first law of thermodynamics. It means that the heat production due to the electromagnetic field is equal to an internal energy increment. The latter is directly related to the family {H (ω,λ) L } L∈R + of internal energy observables. We also consider an electromagnetic potential energy defined from the observable W A t . Hence, we define the following energy increments:
-dynamical system (U, τ (ω,λ,A) t,s ) for any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and A ∈ C ∞ 0 . We show in(Q) The internal energy increment S (ω,A) ≡ S (β,ω,λ,A) is a map from R to R + 0 defined by S (ω,A) (t) := lim L→∞ ρ (β,ω,λ,A) t (H (ω,λ) L ) − ̺ (β,ω,λ) (H (ω,λ) L ) .(58)
It takes positive finite values because of [BPH1, Theorem 3.2].
(P) The electromagnetic potential energy (increment) P (ω,A) ≡ P (β,ω,λ,A) is a map from R to R defined by
P (ω,A) (t) := ρ (β,ω,λ,A) t (W A t ) = ρ (β,ω,λ,A) t (W A t ) − ̺ (β,ω,λ) (W A t 0 ) .(59)
In other words, S (ω,A) is the increase of internal energy of the fermion system due to the change of its internal state, whereas P (ω,A) is the electromagnetic potential energy of the fermion system in the state ρ (β,ω,λ,A) t . By [BPH1, Theorem 3.2], S (ω,A) equals the heat production of the fermion system. Moreover, by [BPH1,Eq. (24)], the increase of total energy of the infinite system
lim L→∞ ρ (β,ω,λ,A) t (H (ω,λ) L + W A t ) − ̺ (β,ω,λ) (H (ω,λ) L ) = S (ω,A) (t) + P (ω,A) (t) (60)
is exactly the work performed by the electromagnetic field at time t ≥ t 0 :
S (ω,A) (t) + P (ω,A) (t) = t t 0 ρ (β,ω,λ,A) s ∂ s W A s ds .(61)
Electromagnetic View Point
In the previous subsection the total energy increment is decomposed into two components (60) that can be identified with heat production and potential energy. This total energy increment can also be decomposed in two other components which have interesting features in terms of currents. Indeed, for any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and A ∈ C ∞ 0 , we define:
(p) The paramagnetic energy increment J (ω,A) p ≡ I (β,ω,λ,A) p is the map from R to R defined by I (ω,A) p (t) := lim L→∞ ρ (β,ω,λ,A) t (H (ω,λ) L + W A t ) − ̺ (β,ω,λ) (H (ω,λ) L + W A t ) . (62) (d) The diamagnetic energy (increment) I (ω,A) d ≡ I (β,ω,λ,A) d is the map from R to R defined by I (ω,A) d (t) := ̺ (β,ω,λ) (W A t ) = ̺ (β,ω,λ) (W A t ) − ̺ (β,ω,λ) (W A t 0 ) .(63)
Note that the limit (62) exists at all times because of (60)-(61). In particular,
I (ω,A) p (t) + I (ω,A) d (t) = t t 0 ρ (β,ω,λ,A) s ∂ s W A s ds(64)
for any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , A ∈ C ∞ 0 and times t ≥ t 0 . The term J (ω,A) p is the part of electromagnetic work implying a change of the internal state of the system, whereas the diamagnetic energy is the raw electromagnetic energy given to the system at thermal equilibrium. Indeed, because of the second law of thermodynamics, in presence of non-zero electromagnetic fields the system constantly tends to minimize the (instantaneous) free-energy associated with H (ω,λ) L + W A t and it is thus forced to change its state as time evolves.
We show below that J (ω,A) p and I
(ω,A) d cannot be identified with either P (ω,A) or S (ω,A) but are directly related to paramagnetic and diamagnetic currents, respectively.
Joule's Effect and Energy Increments
By Theorem 3.3, for each l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and any electromagnetic potential A ∈ C ∞ 0 , the electric field in its integrated form E ηA l t (cf. (11)- (12) and (17)) implies paramagnetic and diamagnetic currents with linear coefficients being respectively equal to
J (ω,A) p,l (t, x) := 1 2 t t 0 y∈K σ (ω) p (x, y,t − s) E A l s (y)ds ,(65)J (ω,A) d,l (t, x) := t t 0 σ (ω) d (x) E A l s (x)ds ,(66)
at any bond x ∈ K (see (23)) and time t ≥ t 0 . Recall that σ Provided |η| ≪ 1, the electric work produced at any time t ≥ t 0 by paramagnetic currents is then equal to
η 2 2 t t 0 x∈K J (ω,A) p,l (s, x)E A l s (x)ds ,(67)
whereas the diamagnetic work equals
η 2 2 t t 0 x∈K J (ω,A) d,l (s, x)E A l s (x)ds = η 2 4 x∈K J (ω,A) d,l (t, x) t t 0 E A l s (x)ds .(68)
Remark that the factor η 2 /2 (instead of η 2 ) in (67)-(68) is due to the fact that K is a set of oriented bonds and thus each bond is counted twice. As explained in Section 3.4, there exist also thermal currents
̺ (β,ω,λ) (I x ) , x ∈ K ,(69)
coming from the inhomogeneity of the fermion system for λ ∈ R + . Thermal currents imply an additional raw electromagnetic work
− η 2 x∈K ̺ (β,ω,λ) (I x ) t t 0 E A l s (x)ds(70)
at any time t ≥ t 0 . Since A is by assumption compactly supported in time, the corresponding electric field satisfies the AC-condition
t t 0 E A (s, x)ds = 0 , x ∈ R d ,(71)
for times t ≥ t 1 ≥ t 0 . Here,
t 1 := min t ≥ t 0 : t ′ t 0 E A (s, x)ds = 0 for all x ∈ R d and t ′ ≥ t
is the time at which the electric field is definitively turned off. In this case, the electric works (68) and (70) We prove this heuristics in Section 5.2.1 and obtain the following theorem:
Theorem 4.1 (Microscopic Joule's law -I)
For any A ∈ C ∞ 0 , there is η 0 ∈ R + such that, for all |η| ∈ (0, η 0 ], l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 , the following assertions hold true: (p) Paramagnetic energy increment:
I (ω,ηA l ) p (t) = η 2 2 t t 0 x∈K J (ω,A) p,l (s, x)E A l s (x)ds + O(η 3 l d ) .
(d) Diamagnetic energy:
I (ω,ηA l ) d (t) = − η 2 x∈K ̺ (β,ω,λ) (I x ) t t 0 E A l s (x)ds + η 2 4 x∈K J (ω,A) d,l (t, x) t t 0 E A l s (x)ds + O(η 3 l d ) .
(Q) Heat production -Internal energy increment:
S (ω,ηA l ) (t) = − η 2 2 x∈K J (ω,A) p,l (t, x) t t 0 E A l s (x)ds +I (ω,ηA l ) p (t) + O(η 3 l d )
(P) Electromagnetic potential energy:
P (ω,ηA l ) (t) = η 2 2 x∈K J (ω,A) p,l (t, x) t t 0 E A l s (x)ds +I (ω,ηA l ) d (t) + O(η 3 l d ) .
The correction terms of order O(η 3 l d ) in assertions (p), (d), (Q) and (P) are uniformly bounded in β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 .
Proof: The first two assertions are Theorem 5.12, whereas (Q) and (P) are direct consequences of (58)-(59), (62)-(63), Theorem 5.12 and Lemma 5.13.
We emphasize the fact that the asymptotics obtained are uniform w.r.t. l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 . This is a crucial property to get macroscopic Joule's law when l → ∞. See [BPH2].
Remark 4.2 (Total energy)
One can easily deduce from Lemma 5.11 the asymptotics of the total work performed by the electric field, which is equal to
t t 0 ρ (β,ω,λ,A) s ∂ s W A s ds ,
similar to what is done in Theorem 4.1.
Theorem 4.1 describes, among other things, how resistance in the fermion system converts electric energy into heat. Indeed, by [BPH1,Theorem 3.2], for any A ∈ C ∞ 0 , there is η 0 ∈ R + such that, for all |η| ∈ (0, η 0 ], l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 ,
η 2 2 t t 0 x∈K J (ω,A) p,l (s, x)E A l s (x)ds − η 2 2 x∈K J (ω,A) p,l (t, x) t t 0 E A l s (x)ds ≥ O(η 3 l d ) .
The latter is the positivity of the heat production, i.e., S (ω,ηA l ) (t) ∈ R + 0 , which for times t ≥ t 1 ≥ t 0 equals, at leading order, the work of paramagnetic currents (67), that is,
η 2 4 t t 0 ds 1 s 1 t 0 ds 2 x,y∈K σ (ω) p (x, y,s 1 − s 2 ) E A l s 2 (x)E A l s 1 (y) ≥ O(η 3 l d ) .(72)
This is nothing but Joule's law expressed w.r.t. electric fields and conductivity (instead of currents and resistance). Indeed, Joule's law in its original form describes a quadratic relation between heat production and currents. The last result gives a quadratic relation between heat production and electric fields, instead (see also (73) and (76)). Joule's law for currents follows from its version for electric fields above, by taking the Legendre-Fenchel transform. For more details, see Section 4.5.
In fact, for any space-homogeneous electric field E ∈ C ∞ 0 (R; R) in the box Λ l for l ∈ R + (as described at the beginning of Section 3), the left hand side of Equation (72) can be rewritten by using (35) and Theorem 3.1 as
η 2 |Λ l | t t 0 ds 1 s 1 t 0 ds 2 w, Ξ (ω) p,l (s 1 − s 2 ) w E s 2 E s 1 = η 2 |Λ l | 2 R |Ê ν | 2 w, µ (ω) p,l (dν) w ≥ 0(73)
for all t ≥ t 1 , withÊ ν being the Fourier transform of E t . In particular,
η 2 2 |Ê ν | 2 w, µ(ω)
p,l (dν) w is, at leading order, the heat production per unit volume due to the component of frequency ν of the electric field, in accordance with Joule's law in the AC-regime. In presence of electromagnetic fields, i.e., at times t ∈ [t 0 , t 1 ] for which the ACcondition (71) does not hold, the situation is more complex. Indeed, at these times, J is generated by paramagnetic currents, see (65). By contrast, the raw electromagnetic energy I (ω,A) d is carried by diamagnetic and thermal currents, see (66) and (69) and compare Theorem 4.1 (d) with (68) and (70). These currents are physically different: Diamagnetic currents correspond to the raw ballistic flow of charged particles due to the electric field, whereas only paramagnetic currents partially participates to the heat production S (ω,A) , a portion of paramagnetic currents being also responsible for the modification of the electromagnetic potential energy:
• Part of the electric work performed by paramagnetic currents participates to the electromagnetic potential energy as explained in Theorem 4.1 (P). The same phenomenon appears for thermal currents defined by (69). Indeed, observe that any current J(t, x) on the bound x at time t yields a contribution
J(t, x) t t 0 E A l s (x)ds
to the electromagnetic potential energy. Compare (70) and P (ω,ηA l ) − I (ω,ηA l ) d via Theorem 4.1 (P) . This potential energy disappears as soon as the electromagnetic potential is switched off.
• Then, the remaining energy coming from the whole paramagnetic energy I (ω,ηA l ) p is a heat energy or quantity of heat, by Theorem 4.1 (Q) and [BPH1,Theorem 3.2]. It survives even after turning off the electromagnetic potential.
Resistivity and Joule's Law
Joule's observation in [J] associates heat production in electric circuits with currents and resistance, rather than electric fields and conductivity. We thus explain in this subsection how to get such a relation between heat production and currents from (72)-(73), which express the total heat production as a function of electric fields and conductivity.
Note that the concept of resistivity is less natural as the one of conductivity. Indeed, the current is an effect of the imposed electric field (and not the other way around). Moreover, from the mathematical point of view, the resistivity is a kind of inverse of the conductivity, which is a measure, as shown above. See Theorem 3.1. To give a precise mathematical meaning to such an inverse of the conductivity measure we use the following observation: Take the function q : e → ae 2 /2 from R to R with a > 0. Its Legendre-Fenchel transform is the function q * from R to R defined by
q * (j) := sup e∈R {je − q (e)} = je j − q (e j ) = j 2 2a , j ∈ R .
Similarly, q (e) := sup j∈R {ej − q * (j)} = ej e − q * (j e ) , e ∈ R .
Their derivatives are respectively equal to ∂ e q (e) = ae = j e and ∂ j q * (j) = j a = e j .
Hence, for any j, e ∈ R, ∂ e q (∂ j q * (j)) = ∂ e q (e j ) = j e j = j , ∂ j q * (∂ e q (e)) = ∂ j q * (j e ) = e je = e . (74) In our construction below, j corresponds to a current J , whereas e refers to an electric field E. Thus, the derivative ∂ e q (e) can be seen as a function that maps each electric field e in the current j e produced by it, i.e., ∂ e q is the conductivity (map) of the system. By (74), ∂ j q * gives thus the corresponding resistivity (map). Below, the function q is replaced by the heat production Q Λ l , which is a quadratic functional of the electric field E, see (75)-(76). The derivatives ∂ e q and ∂ j q * define usual functions. In the case of the Legendre-Fenchel transform Q * Λ l of Q Λ l , we do not have usual derivatives, but only subdifferentials ∂Q * Λ l . Hence, in general, the resistivity is a set-valued map (i.e., a multifunction), see (84). This makes the mathematical statement of Joule's law in its original formulation more abstract, see Theorem 4.7.
For the sake of simplicity, we restrict our analysis to space-homogeneous electric fields E t w in the box Λ l for l ∈ R + , as described at the beginning of Section 3. Here, E ∈ C ∞ 0 (R; R) and w := (w 1 , . . . , w d ) ∈ R d . In this subsection, we fix l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 . Now, we are position to perform the construction heuristically presented above.
By Corollary 3.2 (i), observe that, for times t ≥ t 1 ≥ t 0 ,
t t 0 ds 1 s 1 t 0 ds 2 w, Ξ (ω) p,l (s 1 − s 2 ) w E s 2 E s 1 = 1 2 R ds 1 R ds 2 w, Ξ (ω) p,l (s 1 − s 2 ) w E s 2 E s 1 ds 2 ds 1 .
Therefore, we define the subspace
S 0 := E ∈ S (R; R) : R E s ds = 0
of R-valued Schwartz functions satisfying the AC-condition as well as the functional Q Λ l ≡ Q (β,ω,λ) Λ l on S 0 , the total heat production per unit of volume, by
Q Λ l (E) := 1 2 R ds 1 R ds 2 w, Ξ (ω) p,l (s 1 − s 2 ) w E s 2 E s 1 ds 2 ds 1 , E ∈ S 0 .(75)
It is a finite, positive quadratic form on S 0 . Indeed, by Theorem 3.1,
Q Λ l (E) = 1 2 R |Ê ν | 2 w, µ (ω) p,l (dν) w ∈ R + 0 , E ∈ S 0 ,(76)
and w, µ
p,l w is a positive measure. It thus defines a semi-norm · Λ l ≡ · (β,ω,λ) Λ l on S 0 by
E Λ l := Q Λ l (E) , E ∈ S 0 .(77)
Note that S 0 is a closed subspace of the locally convex (Fréchet) space S (R; R). Let S * 0 be the dual space of S 0 , i.e., the set of all continuous linear functionals on S 0 . S * 0 is equipped with the weak * -topology. By the Hahn-Banach theorem, the elements of the dual S * 0 are restrictions to S 0 of tempered distributions. S * 0 is in fact a space of in-phase AC-currents.
Let
∂Q Λ l (E) ⊂ S * 0 be the subdifferential of Q Λ l at the point E ∈ S 0 . The multi- function σ Λ l ≡ σ (β,ω,λ) Λ l from S 0 to S * 0 (i.e., the set-valued map from S 0 to 2 S * 0 ) is defined by E → σ Λ l (E) = 1 2 ∂Q Λ l (E) .
It is single-valued with domain Dom(σ Λ l ) = S 0 :
Lemma 4.3 (Properties of the AC-conductivity)
The multifunction σ Λ l has domain
Dom(σ Λ l ) := {E ∈ S 0 : ∂Q Λ l (E) = ∅} = S 0 and, for all E ∈ S 0 , σ Λ l (E) = {J E } with J E ,Ẽ = 1 2 R R w, Ξ (ω) p,l (s 1 − s 2 ) w Ẽ s 1 E s 2 ds 2 ds 1 ,Ẽ ∈ S 0 .(78)
[We use the standard notation for distributions:
J E ,Ẽ ≡ J E (Ẽ).]
Proof: We prove that, for all E ∈ S 0 , 2J E is the unique tangent functional of Q Λ l at the point E. Indeed,
Q Λ l (E + E 1 ) − Q Λ l (E) = 2 J E , E 1 + Q Λ l (E 1 )(79)
for all E 1 ∈ S 0 . Since Q Λ l (E 1 ) ≥ 0, the functional 2J E is tangent to Q Λ l at E ∈ S 0 . In particular, Dom(σ Λ l ) = S 0 . The uniqueness of the tangent functional follows from the fact that 2J E is the Gâteaux derivative of Q Λ l at E ∈ S 0 . To see this, replace E 1 with ǫE 1 in (79) and take the limit ǫ → 0.
Equation (78) is directly related to Ohm's law in Fourier space. For this reason, σ Λ l is named here the AC-conductivity of the region Λ l . By Ohm and Joule's laws, a more resistive system produces less heat at fixed electric field. We thus define a AC-resistivity order from the total heat production Q Λ l ≡ Q (β,ω,λ) Λ l (per unit of volume) on the space S 0 of electric fields:
Definition 4.4 (AC-Resistivity order)
For all l ∈ R + , we define the partial order relation ≺ for the system parameters (β, ω, λ) ∈ R + × Ω × R + 0 by
(β 1 , ω 1 , λ 1 ) ≺ (β 2 , ω 2 , λ 2 ) iff Q (β 1 ,ω 1 ,λ 1 ) Λ l ≥ Q (β 2 ,ω 2 ,λ 2 ) Λ l .
This definition is reminiscent of the approach of [LY] to the entropy. Observe also that
(β 1 , ω 1 , λ 1 ) ≺ (β 2 , ω 2 , λ 2 ) iff µ (β 1 ,ω 1 ,λ 1 ) p,l | R\{0} ≥ µ (β 2 ,ω 2 ,λ 2 ) p,l | R\{0} .
Furthermore, this partial order can be rewritten in terms of a quadratic function of currents, in accordance with Joule's law in its original form. To see this, observe that (S 0 , S * 0 ) is a dual pair, by [R,Theorem 3.10]. Therefore, Q Λ l : S 0 → [0, ∞) has a well-defined Legendre-Fenchel transform Q * Λ l ≡ (Q (β,ω,λ) Λ l ) * which is the convex lower semi-continuous functional from S * 0 to (−∞, ∞] defined in our setting by
Q * Λ l (J ) := 2 sup E∈S 0 J , E − 1 2 Q Λ l (E) , J ∈ S * 0 .(80)
The square root of Q * Λ l (J ) can be seen as the norm of the linear map J : (S 0 , · Λ l ) → R:
Lemma 4.5 (Q * Λ l as a semi-norm on S * 0 ) Assume that Q Λ l is not identically zero. Then,
Q * Λ l (J ) = sup | J , E | : E ∈ S 0 , E Λ l = 1 2 .
If Q Λ l is identically zero, Q * Λ l (J ) = ∞ for all J ∈ S * 0 \{0} and Q * Λ l (0) = 0.
Proof: The assertion for Q Λ l ≡ 0 is a direct consequence of (80). Assume that Q Λ l is not identically zero. For any J ∈ S * 0 , define the map
x → f J (x) := sup E∈S 0 : E Λ l =x | J , E | − x 2 2 from R + 0 to R.
By rescaling, observe that, for any x ∈ R + ,
f J (x) = sup E∈S 0 : E Λ l =1 x | J , E | − x 2 2 .(81)
In particular, for any J ∈ S * 0 , f J is clearly continuous. Therefore, we infer from (80)
that Q * Λ l (J ) = 2 sup x∈R + 0 f J (x) = 2 sup x∈R + f J (x) ,(82)
which, combined with (81) and straightforward computations, leads to the assertion.
The above lemma implies that the domain
Dom Q * Λ l = J ∈ S * 0 : Q * Λ l (J ) < ∞ of the functional Q * Λ l is a subspace of S * 0 . Similar to (77), we define the semi-norm · ( * ) Λ l ≡ · ( * ,β,ω,λ) Λ l by J ( * ) Λ l := Q * Λ l (J ) = sup | J , E | : E ∈ S 0 , E Λ l = 1(83)
for any J ∈ S * 0 . Let ∂Q * Λ l (J ) ⊂ S 0 be the subdifferential of Q * Λ l at the point J ∈ S * 0 . The multifunction ρ Λ l ≡ ρ (β,ω,λ) Λ l from S * 0 to S 0 (i.e., the set-valued map from S 0 to 2 S * 0 ) is defined by
J → ρ Λ l (J ) = 1 2 ∂Q * Λ l (J ) .(84)
It is named here the AC-resistivity of the region Λ l because it is a sort of inverse of the AC-conductivity:
Lemma 4.6 (Properties of the AC-resistivity)
The multifunction ρ Λ l has non-empty domain equal to
Dom(ρ Λ l ) := J ∈ S * 0 : ∂Q * Λ l (J ) = ∅ = E∈S 0 σ Λ l (E) .
Furthermore, for all J ∈ Dom(ρ Λ l ) and E ∈ Dom(σ Λ l ) = S 0 ,
σ Λ l (ρ Λ l (J )) = {J } and ρ Λ l (σ Λ l (E)) ⊃ {E} .(85)
Proof: Young's inequality asserts that
1 2 Q * Λ l (J ) + 1 2 Q Λ l (E) ≥ J , E with equality iff 2J ∈ ∂Q Λ l (E). As Q Λ l = Q * * Λ l , 1 2 Q * Λ l (J ) + 1 2 Q Λ l (E) = J , E iff 2E ∈ ∂Q * Λ l (J )
. In other words,
E ∈ ρ Λ l (J ) ⇐⇒ J ∈ σ Λ l (E) .(86)
As a consequence,
J E ∈ σ Λ l (E) (cf. Lemma 4.3) yields E ∈ ρ Λ l (J E ). It follows that E∈S 0 σ Λ l (E) ⊂ Dom(ρ Λ l ) and ρ Λ l (σ Λ l (E)) ⊃ {E} .
Now, let J ∈ Dom(ρ Λ l ) and E ∈ ρ Λ l (J ). Then, by (86), J ∈ σ Λ l (E) and we infer from the uniqueness of the tangent functional (Lemma 4.3) that J = J E . Therefore,
σ Λ l (ρ Λ l (J )) = {J } and Dom(ρ Λ l ) ⊂ E∈S 0 σ Λ l (E) .
Note that Q Λ l : S 0 → [0, ∞) is a convex continuous functional, by positivity of the conductivity measure, see Theorem 3.1 and (76). In particular,
Q Λ l (E) := 2 sup J ∈S * 0 J , E − 1 2 Q * Λ l (J ) .(87)
Therefore, we deduce from (80) and (87) that
(β 1 , ω 1 , λ 1 ) ≺ (β 2 , ω 2 , λ 2 ) iff (Q (β 1 ,ω 1 ,λ 1 ) Λ l ) * ≤ (Q (β 2 ,ω 2 ,λ 2 ) Λ l ) * .
Furthermore, by using (77) and similar arguments as in Lemma 4.5, if Q Λ l is not identically zero, then:
E Λ l = sup | J , E | : J ∈ S * 0 , J ( * ) Λ l = 1 .
We are now in position to obtain Joule's law in its original form. To this end, we say that a multifunction ρ from S * 0 to S 0 is linear if:
(a) Its domain Dom(ρ) is a subspace of S * 0 .
(b) For α ∈ R\{0} and J ∈ Dom(ρ), ρ (αJ ) = αρ (J ) and 0 ∈ ρ (0).
(c) For J 1 , J 2 ∈ Dom(ρ), ρ (J 1 + J 2 ) = ρ (J 1 ) + ρ (J 2 ).
Then, one gets that the heat produced by currents is proportional to the resistivity and the square of currents:
Theorem 4.7 (Microscopic Joule's law -II) (i) ρ Λ l is a linear multifunction and σ Λ l (ρ Λ l (J )) = {J } for all J ∈ Dom(ρ Λ l ).
(ii) For any J ∈ Dom(ρ Λ l ),
{Q * Λ l (J )} = J , ρ Λ l (J ) = Q Λ l (ρ Λ l (J )) .
(iii) There is a bilinear symmetric positive map (·, ·) ( * )
Λ l on Dom(ρ Λ l ) such that Q * Λ l (J 1 ) = (J 1 , J 1 ) ( * ) Λ l and J 1 , ρ Λ l (J 2 ) = {(J 1 , J 2 ) ( * ) Λ l } for all J 1 , J 2 ∈ Dom(ρ Λ l ).
Proof: (i.a) The fact that Dom(ρ Λ l ) is a subspace of S * 0 is a direct consequence of Lemmata 4.3 and 4.6. (i.b) Let α ∈ R and J ∈ Dom(ρ Λ l ). Take any E J ∈ ρ Λ l (J ) and observe that J = J E J , by using Lemmata 4.3 and 4.6. Then,
Since 1 2 Q * Λ l (J ) + 1 2 Q Λ l (E J ) = J , E J , we also deduce that Q * Λ l (J ) = Q Λ l (E J ). (iii) For all J 1 , J 2 ∈ Dom(Q * Λ l ), define (J 1 , J 2 ) ( * ) Λ l := 1 4 Q * Λ l (J 1 + J 2 ) − Q * Λ l (J 1 − J 2 ) .(88)
This quantity is clearly symmetric w.r.t. J 1 , J 2 and
(J , J ) ( * ) Λ l = Q * Λ l (J ) ≥ 0 , J ∈ Dom(Q * Λ l ) ,
by Lemma 4.5. Using the linearity of ρ Λ l and the fact that J , ρ Λ l (J ) ⊂ R + 0 contains exactly one element for all J ∈ Dom(ρ Λ l ), we compute that, for any J 1 , J 2 ∈ Dom(ρ Λ l ),
1 2 {Q * Λ l (J 1 + J 2 ) − Q * Λ l (J 1 − J 2 )} = J 2 , ρ Λ l (J 1 ) + J 1 , ρ Λ l (J 2 ) .
Again by linearity of ρ Λ l , this implies that (88) defines a bilinear form on Dom(ρ Λ l ). We also infer from the above equation that the set
J 2 , ρ Λ l (J 1 ) ⊂ R contains exactly one element. Let E J 1 ∈ ρ Λ l (J 1 ) and E J 2 ∈ ρ Λ l (J 2 ) with J 1 = J E J 1 and J 2 = J E J 2 . Then, by Lemma 4.3, J 2 , ρ Λ l (J 1 ) = J E J 2 , E J 1 = J E J 1 , E J 2 = J 1 , ρ Λ l (J 2 ) .
Technical Proofs
This section is divided in two parts: Section 5.1 gives a detailed proof of Theorem 3.1 as well as additional properties of paramagnetic transport coefficients defined in Section 3.3. In Section 5.2 we prove Theorems 3.3 and 4.1. Note that we start in this second subsection with the proof of Theorem 4.1 because the other one is simpler and uses similar arguments.
Paramagnetic Transport Coefficients
Microscopic Paramagnetic Transport Coefficients
We study in this subsection the microscopic paramagnetic transport coefficient σ (ω) p which is defined by (29), that is,
σ (ω) p (x, y, t) := t 0 ̺ (β,ω,λ) i[I y , τ (ω,λ) s (I x )] ds , x, y ∈ L 2 , t ∈ R .
Recall that I x is the paramagnetic current observable defined by (19), that is,
I x := i(a * x (2) a x (1) − a * x (1) a x (2) ) , x := (x (1) , x (2) ) ∈ L 2 .(89)
The coefficient σ (ω) p can explicitly be written in terms of a scalar product involving current observables. To show this, we introduce the Duhamel two-point function (·, ·) (ω) ∼ defined by
(B 1 , B 2 ) ∼ ≡ (B 1 , B 2 ) (β,ω,λ) ∼ := β 0 ̺ (β,ω,λ) B * 1 τ (ω,λ) iα (B 2 ) dα(90)
for any B 1 , B 2 ∈ U. The properties of this sesquilinear form are described in detail in Appendix A. In particular, by Theorem A.1 for X = U, τ = τ (ω,λ) and ̺ = ̺ (β,ω,λ) , (B 1 , B 2 ) → (B 1 , B 2 ) ∼ is a positive sesquilinear form on U. We then infer from Lemma A.14 that
σ (ω) p (x, y, t) = (I y , τ (ω,λ) t (I x )) ∼ − (I y , I x ) ∼ ,(91)
for all β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , x, y ∈ L 2 and t ∈ R. By Theorem A.16, it follows that σ (ω) p is symmetric w.r.t. time-reversal and permutation of bonds.
Indeed, by using the time-reversal operation Θ : U → U defined in Section 2.1.4, one proves:
Lemma 5.1 (Time-reversal symmetry of the fermion system)
Let β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . Then,
Θ • τ (ω,λ) t = τ (ω,λ) −t • Θ , t ∈ R ,(92)
and
̺ (β,ω,λ) (B) = ̺ (β,ω,λ) • Θ (B) , B ∈ U .(93)
Proof: By continuity of the maps Θ and τ (ω,λ) t as well as the density of polynomials in the creation and annihilation operators in U, it suffices to prove the first assertion for monomials in a x , a *
x , x ∈ L. Now, since Θ(H (see (56)), by [BPH1,Theorem A.3 (i)
], Θ • τ (ω,λ) t (B) = τ (ω,λ) −t • Θ (B) , B ∈ U 0 , t ∈ R ,
which implies (92). The second assertion is a consequence of the uniqueness of the (τ (ω,λ) , β)-KMS state ̺ (β,ω,λ) together with Lemma A.12.
Since Θ (I x ) = −I x for any x ∈ L 2 , we deduce from Lemma 5.1 and Theorem A.16 for X = U, τ = τ (ω,λ) and ̺ = ̺ (β,ω,λ) that the function σ (ω) p from L 4 × R to R is symmetric w.r.t. time-reversal and permutation of bonds:
σ (ω) p (x, y, t) = σ (ω) p (x, y, −t) = σ (ω) p (y, x, t) , x, y ∈ L 2 , t ∈ R .
Thermal equilibrium states ̺ (β,ω,λ) are by construction quasi-free and gauge-invariant. This fact implies that σ (ω) p can be expressed in terms of complex-time two-point correla-
tion functions C (ω) t+iα ≡ C (β,ω,λ) t+iα defined by C (ω) t+iα (x) := ̺ (β,ω,λ) (a * x (1) τ (ω,λ) t+iα (a x (2) )) , x := (x (1) , x (2) ) ∈ L 2 ,(94)
for all β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R and α ∈ [0, β]. This is shown in the following assertion:
Lemma 5.2 (σ (ω) p in terms of two-point correlation functions)
Let β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . Then, for all x, y ∈ L 2 and t ∈ R,
σ (ω) p (x, y, t) = β 0 C (ω) t+iα (x, y) − C (ω) iα (x, y) dα ∈ R , where C (ω) t+iα ≡ C (β,ω,λ) t+iα is the map from L 4 to C defined by C (ω) t+iα (x, y) := π,π ′ ∈S 2 ε π ε π ′ C (ω) t+iα (y π ′ (1) , x π(1) )C (ω) −t+i(β−α) (x π(2) , y π ′ (2) )(95)
for any x := (x (1) , x (2) ) ∈ L 2 and y := (y (1) , y (2) ) ∈ L 2 . Here, π, π ′ ∈ S 2 are by definition permutations of {1, 2} with signatures ε π , ε π ′ ∈ {−1, 1}. 2) ) ∈ L 2 and y := (y (1) , y (2) ) ∈ L 2 . From Equation (91) together with (166),
Proof: Fix β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R, α ∈ [0, β], x := (x (1) , x (σ (ω) p (x, y, t) = β 0 ̺ (β,ω,λ) I y τ (ω,λ) t+iα (I x ) − ̺ (β,ω,λ) I y τ (ω,λ) iα (I x ) dα .(96)
Direct computations using (8) and (19) yield
I y τ (ω,λ) t+iα (I x ) = − a * y (1) a y (2) − a * y (2) a y (1) τ (ω,λ) t+iα (a * x (1) )τ (ω,λ) t+iα (a x (2) ) (97) + a * y (1) a y (2) − a * y (2) a y (1) τ (ω,λ) t+iα (a * x (2) )τ (ω,λ) t+iα (a x (1) ) .
Note that, for all x ∈ L 2 and x ∈ L, the maps
z → τ (ω,λ) z (I x ) , z → τ (ω,λ) z (a * x ) , z → τ (ω,λ) z (a x ) ,(98)
defined on R have unique analytic continuations for z ∈ C and (97) makes sense.
Recall that e x (y) ≡ δ x,y is the canonical orthonormal basis of ℓ 2 (L) and, as usual,
{B 1 , B 2 } := B 1 B 2 + B 2 B 1 , B 1 , B 2 ∈ U .
Therefore, using the anti-commutator relation (4) and (7), we get the equality
{a y (2) , τ (ω,λ) t+iα (a * x (1) )} = e y (2) , (U (ω,λ) t+iα ) * e x (1) 1 , see̺ (β,ω,λ) a * y (1) a y (2) τ (ω,λ) t+iα (a * x (1) )τ (ω,λ) t+iα (a x (2) ) = −̺ (β,ω,λ) a * y (1) τ (ω,λ) t+iα (a * x (1) )a y (2) τ (ω,λ) t+iα (a x (2) ) +̺ (β,ω,λ) {a y (2) , τ (ω,λ) t+iα (a * x (1) )} ̺ (β,ω,λ) a * y (1) τ (ω,λ) t+iα (a x (2) ) .(99)
Since ̺ (β,ω,λ) is by construction a quasi-free state, we use [BR2,p. 48], that is here, ̺ (β,ω,λ) (a * (f 1 ) a * (f 2 ) a (g 1 ) a (g 2 )) = ̺ (β,ω,λ) (a * (f 1 ) a (g 2 ))̺ (β,ω,λ) (a * (f 2 ) a (g 1 )) −̺ (β,ω,λ) (a * (f 1 ) a (g 1 ))̺ (β,ω,λ) (a * (f 2 ) a (g 2 )) , to infer from Equation (99) that
̺ (β,ω,λ) a * y (1) a y (2) τ (ω,λ) t+iα (a * x (1) )τ (ω,λ) t+iα (a x (2) ) = ̺ (β,ω,λ) (a * y (1) a y (2) )̺ (β,ω,λ) (τ (ω,λ) t+iα (a * x (1) )τ (ω,λ) t+iα (a x (2) )) +̺ (β,ω,λ) a * y (1) τ (ω,λ) t+iα (a x (2) ) ̺ (β,ω,λ) a y (2) τ (ω,λ) t+iα (a * x (1) ) .(100)
Remark that the KMS property (164) together with (9) and the Phragmén-Lindelöf theorem [BR2,Proposition 5.3.5] yields
̺ (β,ω,λ) (τ (ω,λ) t+iα (B)) = ̺ (β,ω,λ) (B) , B ∈ U .(101)
See also [BR2,Proposition 5.3.7]. We thus combine (101) and (164) with Equation (8) and the analyticity of the maps (98) to deduce from (94) that
C (ω) −t+i(β−α) (x) = ̺ (β,ω,λ) (a x (2) τ (ω,λ)
t+iα (a * x (1) )) . Using this together with (94), (101) and again the analyticity of the maps (98), we get from Equation (100) that
̺ (β,ω,λ) a * y (1) a y (2) τ (ω,λ) t+iα (a * x (1) )τ (ω,λ) t+iα (a x (2) ) = C (ω) 0 (y (1) , y (2) )C (ω) 0 (x (1) , x (2) ) + C (ω) t+iα (y (1) , x (2) )C (ω) −t+i(β−α) (x (1) , y (2) ) .
Then we use this last equality together with (97) to get
̺ (β,ω,λ) I y τ (ω,λ) t+iα (I x ) = − π,π ′ ∈S 2 ε π ε π ′ C (ω) t+iα (y π ′ (1) , x π(2) )C (ω)
−t+i(β−α) (x π(1) , y π ′ (2) ) +C (ω) 0 (y π ′ (1) , y π ′ (2) )C (ω) 0 (x π(1) , x π(2) ) .
(102) Therefore, the assertion follows by combining (96) with (102) for any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R, α ∈ [0, β], x := (x (1) , x (2) ) ∈ L 2 and y := (y (1) , y (2) ) ∈ L 2 .
Lemma 5.2 is a useful technical result because the complex-time two-point correlation functions C (ω) t+iα can be expressed in terms of the one-particle bounded self-adjoint operator (∆ d + λV ω ) ∈ B(ℓ 2 (L)) to which the spectral theorem can be applied. Indeed, for all β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R and α ∈ [0, β], one gets from (7), (10) and (94)
that C (ω) t+iα (x) = e x (2) , e −it(∆ d +λVω) F β α (∆ d + λV ω ) e x (1) ,(103)
where F β α is the real function defined, for every β ∈ R + and α ∈ R, by t+iα , see [BPH2]. An important consequence of (103) is the fact that the coefficient C (ω) t+iα defined by (95) can be seen as the kernel (w.r.t. the canonical basis {e x ⊗ e x ′ } x,x ′ ∈L ) of a bounded operator on ℓ 2 (L) ⊗ ℓ 2 (L). This operator is again denoted by C (ω) t+iα :
Lemma 5.3 (C (ω)
t+iα as a bounded operator) Let β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R and α ∈ [0, β]. Then, there is a unique bounded operator C (ω)
t+iα on ℓ 2 (L) ⊗ ℓ 2 (L) with e x (1) ⊗ e x (2) , C (ω) t+iα (e y (1) ⊗ e y (2) ) ℓ 2 (L)⊗ℓ 2 (L) = C (ω) t+iα ((x (1) , x (2) ), (y (1) , y (2) ))
for all (x (1) , x (2) ), (y (1) , y (2) ) ∈ L 2 , and
C (ω) t+iα op ≤ 4 and lim α→0 + C (ω) iα − C (ω) 0 op = 0 ,
where · op is the operator norm.
Proof: By (95) and (103), the bounded operator C (ω) t+iα exists, is unique, and one directly gets
1 4 C (ω) t+iα op ≤ e (−it+α)(∆ d +λVω) 1 + e β(∆ d +λVω) op e (it+β−α)(∆ d +λVω) 1 + e β(∆ d +λVω) op ≤ 1
for any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , t ∈ R and α ∈ [0, β]. Moreover, in the same way, (95) and (103) also lead to
1 4 C (ω) iα − C (ω) 0 op ≤ e α(∆ d +λVω) − 1 op + e −α(∆ d +λVω) − 1 op(104)
for any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , and α ∈ [0, β]. Recall that the self-adjoint operator ∆ d + λV ω is bounded, i.e., ∆ d + λV ω ∈ B(ℓ 2 (L)). It follows that the one-parameter group {e α(∆ d +λVω) } α∈R is uniformly continuous (norm continuous). Therefore, the second assertion is deduced from (104) in the limit α → 0 + .
Space-Averaged Paramagnetic Transport Coefficients
Equation (33) and Lemma A.14 for X = U, τ = τ (ω,λ) and ̺ = ̺ (β,ω,λ) yield
Ξ (ω) p,l (t) k,q = 1 |Λ l | (I k,l , τ (ω,λ) t (I q,l )) ∼ − (I k,l , I q,l ) ∼(105)
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k, q ∈ {1, . . . , d} and t ∈ R. Since Θ (I x ) = −I x for any x ∈ L 2 , by Theorem A.16, the operator Ξ
(ω) p,l (t) is symmetric at any fixed time t ∈ R while the B(R d )-valued function Ξ (ω)
p,l is symmetric w.r.t. time-reversal. In other words,
Ξ (ω) p,l (t) k,q = Ξ (ω) p,l (−t) k,q = Ξ (ω) p,l (t) q,k ∈ R(106)
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k, q ∈ {1, . . . , d} and t ∈ R. Because of (105) it is convenient to use the Duhamel GNS (Gelfand-Naimark-Segal)) representation (H,π,Ψ) ≡ (H (β,ω,λ) ,π (β,ω,λ) ,Ψ (β,ω,λ) ) of the (τ (ω,λ) , β)-KMS state ̺ (β,ω,λ) for any β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . See Definition A.8 with X = U and ̺ = ̺ (β,ω,λ) . Note that we identify here the Duhamel two-point function defined by (90) on the CAR algebra U with the scalar product (·, ·) ∼ of the Hilbert spaceH, see Remark A.11. Other cyclic representations could be used instead, but the Duhamel one makes the proofs involving the representation of the paramagnetic conductivity as a spectral measure more transparent via the results of [NVW].
The CAR C * -algebra U is the inductive limit of (finite dimensional) simple C *algebras {U Λ } Λ∈P f (L) , see [Si,Lemma IV.1.2]. By [BR1,Corollary 2.6.19.], U is thus simple. This property has some important consequences: The (τ (ω,λ) , β)-KMS state ̺ (β,ω,λ) is faithful. In particular,π is injective. Remark thatΨ ≡ 1 ∈ U and U is a dense set ofH, butπ (B)Ψ is generally not equal to B ∈ U, in contrast to the usual GNS representation. For this reason, we do not identifyπ (U) with U. Moreover, by Theorem A.9 for X = U and ̺ = ̺ (β,ω,λ) , the * -automorphism group τ = τ (ω,λ) can be extended to a unitary group on the whole Hilbert spaceH:
τ (ω,λ) t (B) = e itL B , t ∈ R , B ∈ U ⊂H ,(107)
withL ≡L (β,ω,λ) being a self-adjoint operator acting onH. The domain ofL includes the domain of the generator δ (ω,λ) of the one-parameter group τ (ω,λ) , i.e., Dom(L) ⊃ Dom(δ (ω,λ) ), whilẽ
L (B) = −iδ (ω,λ) (B) , B ∈ Dom(δ (ω,λ) ) ⊂ U ⊂H .(108)
Equation (107) is an important representation of the dynamics because we can deduce from (105) the existence of the paramagnetic conductivity measure from the spectral theorem.
To present this result, recall that B + (R d ) ⊂ B (R d ) denotes the set of positive linear operators on R d and any B(R d )-valued measure µ on R is symmetric iff µ(X ) = µ(−X ) for any Borel set X ⊂ R. Then, we derive the paramagnetic conductivity measure:
Theorem 5.4 (Conductivity measures as spectral measures)
For any l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 , there exists a finite symmetric
B + (R d )-valued measure µ (ω) p,l ≡ µ (β,ω,λ) p,l on R such that Ξ (ω) p,l (t) = R (cos (tν) − 1) µ (ω) p,l (dν) , t ∈ R .(109)
Proof: Fix l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . LetẼ ≡Ẽ (β,ω,λ) be the (projection-valued) spectral measure of the self-adjoint operatorL. Then, by combining (105)-(106) with (107), we directly arrive at the equality
Ξ (ω) p,l (t) k,q = 1 4 |Λ l | R e itν − 1 (I k,l ,Ẽ(dν)I q,l ) ∼ + 1 4 |Λ l | R e itν − 1 (I q,l ,Ẽ(dν)I k,l ) ∼ + 1 4 |Λ l | R e −itν − 1 (I k,l ,Ẽ(dν)I q,l ) ∼ + 1 4 |Λ l | R e −itν − 1 (I q,l ,Ẽ(dν)I k,l ) ∼(110)
for any k, q ∈ {1, . . . , d} and t ∈ R. Note that, for any Borel set X ⊂ R and all k, q ∈ {1, . . . , d}, (I k,l ,Ẽ (X ) I q,l ) ∼ + (I q,l ,Ẽ (X ) I k,l ) ∼ ∈ R .
Thus, define the B(R d )-valued measure µ (ω) p,l by u, µ (ω) p,l (X ) w = 1 4 |Λ l | k,q∈{1,...,d} u k w q (I k,l ,Ẽ (X ) I q,l ) ∼ + 1 4 |Λ l | k,q∈{1,...,d} u k w q (I q,l ,Ẽ (X ) I k,l ) ∼ + 1 4 |Λ l | k,q∈{1,...,d} u k w q (I k,l ,Ẽ (−X ) I q,l ) ∼ + 1 4 |Λ l | k,q∈{1,...,d} u k w q (I q,l ,Ẽ (−X ) I k,l ) ∼(111)
for any u := (u 1 , . . . , u d ) ∈ R d , w := (w 1 , . . . , w d ) ∈ R d and all Borel sets X ⊂ R.
Here, ·, · denotes the usual scalar product of R d . Obviously, by construction,
u, µ (ω) p,l (X ) w = w, µ (ω) p,l (X ) u and w, µ (ω) p,l (X ) w ≥ 0 ,
for any u, w ∈ R d and all Borel sets X ⊂ R. Moreover, µ
p,l is a symmetric measure and, by (110), we obtain Equation (109).
For any β ∈ R + , ω ∈ Ω and λ ∈ R + 0 , it is useful at this point to also consider the GNS representation (H, π, Ψ) ≡ (H (β,ω,λ) , π (β,ω,λ) , Ψ (β,ω,λ) ) of the (τ (ω,λ) , β)-KMS state ̺ (β,ω,λ) and to describe its relation to the Duhamel GNS representation. To this end, we denote by L ≡ L (β,ω,λ) the standard Liouvillean of the system under consideration, i.e., the self-adjoint operator acting on H which implements the dynamics as
π (τ t (B)) = e itL π (B) e −itL , t ∈ R, B ∈ U ,(112)
with LΨ = Ψ. Let E ≡ E (β,ω,λ) be the (projection-valued) spectral measure of L. We also use the (Tomita-Takesaki) modular objects
∆ ≡ ∆ (β,ω,λ) := e −βL , J ≡ J (β,ω,λ) ,
of the pair (π (U) ′′ , Ψ). Theorem A.1 says that
(B 1 , B 2 ) ∼ = Tπ (B 1 ) Ψ, Tπ (B 2 ) Ψ H , B 1 , B 2 ∈ U ,(113)
where T ≡ T (β,ω,λ) is the operator defined by (160) for τ = τ (ω,λ) and ̺ = ̺ (β,ω,λ) , that is,
T := β 1/2 1 − e −βL βL 1/2 .(114)
Note that T is unbounded, but
π (U) Ψ ⊂ Dom(∆ 1/2 ) ⊂ Dom(T) .(115)
The
B + (R d )-valued measure µ (ω)
p,l of Theorem 5.4, which is defined by (111), can also be studied via (113). Indeed, (113) and (115) together with Theorem A.7 and (163) imply that (I k,l ,Ẽ (X ) I q,l ) ∼ = TE (X ) π (I k,l ) Ψ, TE (X ) π (I q,l ) Ψ H
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k, q ∈ {1, . . . , d} and any Borel set X ⊂ R. The existence of the first moment of µ (ω) p,l is a direct consequence of the above equation. To see this, recall that µ (ω) p,l op is the measure on R taking values in R + 0 that is defined, for any Borel set X ⊂ R and µ = µ (ω) p,l , by (38). Then, one gets the following assertions:
Theorem 5.5 (Existence of the first moment of µ (ω) p,l ) For any l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 , the B + (R d )-valued measure µ (ω) p,l of Theorem 5.4 satisfies the following bounds:
R µ (ω) p,l op (dν) ≤ 1 |Λ l | d k=1 ̺ (β,ω,λ) I 2 k,l , R |ν| µ (ω) p,l op (dν) ≤ 2 |Λ l | d k=1 ̺ (β,ω,λ) I 2 k,l , R |ν| µ (ω) p,l op (dν) ≤ 2 |Λ l | d k=1 ̺ (β,ω,λ) I 2 k,l ̺ (β,ω,λ) (δ (ω,λ) (I k,l )) 2 .
Proof:
Fix l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . By positivity of the measure µ (ω) p,l and linearity of the trace,
µ (ω) p,l op (X ) ≤ Trace B(R d ) µ (ω) p,l (X )
for any Borel set X ⊂ R. This implies that
R µ (ω) p,l op (dν) ≤ Trace B(R d ) R µ (ω) p,l (dν) and R |ν| µ (ω) p,l op (dν) ≤ Trace B(R d ) R |ν| µ (ω) p,l (dν) .
Hence, by (111), it suffices to prove that
R (I k,l ,Ẽ(dν)I k,l ) ∼ ≤ ̺ (β,ω,λ) I 2 k,l ,(117)R |ν| (I k,l ,Ẽ(dν)I k,l ) ∼ ≤ 2̺ (β,ω,λ) I 2 k,l ,(118)
R |ν| (I k,l ,Ẽ(dν)I k,l ) ∼ ≤ 2 ̺ (β,ω,λ) I 2 k,l ̺ (β,ω,λ) (δ (ω,λ) (I k,l )) 2 ,
for any k ∈ {1, . . . , d}.
Inequality (117) is a direct consequence of Theorem A.4. The second upper bound is derived as follows: Fix k ∈ {1, . . . , d}. We infer from (114) and (116)
that R |ν| (I k,l ,Ẽ(dν)I k,l ) ∼ = 1 − e −βL 1/2 E R + 0 π (I k,l ) Ψ 2 H (120) + e −βL − 1 1/2 E R − π (I k,l ) Ψ 2 H .
Clearly, one has the upper bound
1 − e −βL 1/2 E R + 0 π (I k,l ) Ψ 2 H ≤ ̺ (β,ω,λ) I 2 k,l ,(121)
while
e −βL − 1 1/2 E R − π (I k,l ) Ψ 2 H ≤ ∆ 1/2 π (I k,l ) Ψ 2 H ,(122)
with ∆ := e −βL being the modular operator. Using now the anti-unitarity of J, J 2 = 1 and J∆ 1/2 π (I k,l ) Ψ = π (I k,l ) * Ψ = π (I k,l ) Ψ ,
one gets that ∆ 1/2 π (I k,l ) Ψ 2 H = π (I k,l ) Ψ 2 H = ̺ (β,ω,λ) I 2 k,l .(123)
Therefore, by combining Equation (120) Since I k,l ∈ U 0 ⊂ Dom(δ (ω,λ) ),
Lπ (I k,l ) Ψ = −iπ δ (ω,λ) (I k,l ) Ψ ,(125)
see (112). Therefore, by additionally using the Cauchy-Schwarz inequality of (·, ·) ∼ and Theorem A.4, one gets (119) similarly as above.
Equation (116) also leads to a characterization of the non-triviality of the conductivity measure at non-zero frequencies via a geometric condition:
Theorem 5.6 (Geometric interpretation of the AC-conductivity measure) Let l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . Then,
lin {π (I k,l ) Ψ : k ∈ {1, . . . , d}} ⊂ ker (L) iff µ (ω) p,l (R\{0}) = 0 .
Here, lin stands for the linear hull of some set.
Proof: Fix l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . If lin {π (I k,l ) Ψ : k ∈ {1, . . . , d}} ⊂ ker (L) , then we infer from (111) and (116) Proof: By explicit computations, for any k ∈ {1, . . . , d},
δ (ω,λ) (I k,l ) = λA (ω) k,l + B k,l ,(126)
where A
(ω) k,l , B k,l ∈ U are defined, for ω ∈ Ω and l ∈ R + , by
A (ω) k,l := x∈Λ l (V ω (x + e k ) − V ω (x)) P (x,x+e k ) and B k,l := x,z∈L,|z|=1,z =±e k (1 [x ∈ (Λ l + z) \Λ l ] − 1 [x ∈ Λ l \ (Λ l + z)]) P (x,x+e k +z) + x∈L (1 [x ∈ (Λ l + e k ) \Λ l ] − 1 [x ∈ Λ l \ (Λ l + e k )]) 2a * x a x − P (x+e k ,x−e k )
with P (x,y) being defined by (25) for any x, y ∈ L. In particular, δ (ω,λ) (I k,l ) is not zero and hence π (I k,l ) Ψ / ∈ ker (L), because π is injective and the cyclic vector Ψ is separating which, by Corollary 3.2 (iii), implies that
µ (ω) p,l (R\{0}) = 1 |Λ l | (I k,l ,Ẽ (R\{0}) I q,l ) ∼ k,q∈{1,...,d} .
Note that the quantity
Ξ (ω) d,l lim ǫ↓0 L[V (ω) l ](ǫ) ∈ B(R d )
is the so-called static admittance of linear response theory, which equals, in our case, the measure of R\{0} w.r.t. the AC-conductivity measure. In fact, the quantum current viscosity uniquely defines the AC-conductivity measure:
Theorem 5.9 (Reconstruction of µ (ω) p,l from the quantum current viscosity) Let l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . Then, for all w := (w 1 , . . . , w d ) ∈ R d and any continuous and compactly supported real-valued functionÊ withÊ 0 = 0,
RÊ ν w, µ (ω) p,l (dν) w = lim ǫ↓0 1 π R dν ∞ 0 ds (ǫ cos (νs) − ν sin (νs)) e −ǫs ν 2 + ǫ 2 ×Ê ν w, Ξ (ω) d,l V (ω) l (s) w . Proof: Fix l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . For any w ∈ R d , define the complex-valued function F w (z) := R 1 ν − z w, µ (ω) p,l (dν) w , z ∈ C + ,
where C + is the set of complex numbers with strictly positive imaginary part. F
By (111), observe that
F w (z) = 1 4 |Λ l | k,q∈{1,...,d} w k w q I k,l , ((L − z) −1 + (−L − z) −1 )I q,l ∼ + 1 4 |Λ l | k,q∈{1,...,d} w k w q I q,l , ((L − z) −1 + (−L − z) −1 )I k,l ∼
for any z ∈ C + and w := (w 1 , . . . , w d ) ∈ R d . Using
(±L − z) −1 = i ∞ 0 e izs e ∓isL ds , z ∈ C + ,
as well as Theorem A.16 for X = U, τ = τ (ω,λ) and ̺ = ̺ (β,ω,λ) , we obtain F w (z) = i |Λ l | k,q∈{1,...,d} w k w q ∞ 0 e izs (I k,l , τ (ω,λ) t (I q,l )) ∼ ds for every z ∈ C + and w := (w 1 , . . . , w d ) ∈ R d . Using (33) and (105), we now integrate by parts the r.h.s of the above equation to get
F w (z) = − 1 |Λ l | k,q∈{1,...,d} w k w q z −1 ∞ 0 e izs ̺ (β,ω,λ) i[I k,l , τ (ω,λ) s (I q,l )] ds − 1 |Λ l | k,q∈{1,...,d} w k w q z −1 (I k,l , I q,l ) ∼(131)
for any z ∈ C + and w := (w 1 , . . . , w d ) ∈ R d . The function ImF w is the Poisson transform of the positive measure (130). Hence, we invoke [Jak,Theorem 3.7] to conclude that, for any real-valued continuous compactly supported functionÊ :
R → R, lim ǫ↓0 RÊ ν ImF w (ν + iǫ) dν = RÊ ν w, µ (ω) p,l (dν) w .
In particular, by (131) and under the condition thatÊ 0 = 0, we arrive at the assertion.
To conclude, we show the uniformity of the upper bounds of Theorem 5.5 w.r.t. to the parameters l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . These upper bounds all depend on the observable |Λ l | − 1 2 I k,l , which is a current fluctuation, by (24). With this aim we define the linear subspace
I := lin Im(a * (ψ 1 ) a (ψ 2 )) : ψ 1 , ψ 2 ∈ ℓ 1 (L) ⊂ ℓ 2 (L) ⊂ U ,(132)
which is the linear hull (lin) of short range bond currents. It is an invariant subspace of the one-parameter group τ (ω,λ) = {τ (ω,λ) t } t∈R for any ω ∈ Ω and λ ∈ R + 0 . Indeed, the unitary group {(U (ω,λ) t ) * } t∈R (see (6) and (7)) defines a strongly continuous group on (ℓ 1 (L) ⊂ ℓ 2 (L), · 1 ).
Let the positive sesquilinear form ·, ·
(ω) I,l ≡ ·, · (β,ω,λ) I,l in I be defined by
I, I ′ (ω) I,l := ̺ (β,ω,λ) F (l) (I) * F (l) (I ′ ) , I, I ′ ∈ I ,(133)
for any l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . Here, F (l) is the fluctuation observable defined by
F (l) (I) = 1 |Λ l | 1/2 x∈Λ l χ x (I) − ̺ (β,ω,λ) (I) 1 , I ∈ I ,(134)
for each l ∈ R + , where χ x , x ∈ L, are the (space) translation automorphisms. Compare (24) with (134). For instance, the first upper bound of Theorem 5.5 can be rewritten as Therefore, we show that the fermion system has uniformly bounded fluctuations, i.e., the quantity I, I ′ (ω) I,l , I, I ′ ∈ I, is uniformly bounded w.r.t. l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 :
Lemma 5.10 (Uniform boundedness of ·, ·
(ω) I,l ) There is a constant D ∈ R + such that, for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and all ψ 1 , ψ 2 , ψ ′ 1 , ψ ′ 2 ∈ ℓ 1 (L),
Im(a * (ψ 1 ) a (ψ 2 )), Im(a * (ψ ′ 1 ) a (ψ ′ 2 )) (ω) I,l ≤ D ψ 1 1 ψ 2 1 ψ ′ 1 1 ψ ′ 2 1 .
Proof: Let ψ 1 , ψ 2 , ψ ′ 1 , ψ ′ 2 ∈ ℓ 1 (L) ⊂ ℓ 2 (L) and without loss of generality assume that the functions ψ 1 , ψ 2 , ψ ′ 1 , ψ ′ 2 are real-valued. Then, by definition,
Im(a * (ψ 1 ) a (ψ 2 )), Im(a * (ψ ′ 1 ) a (ψ ′ 2 )) (ω) I,l = x:=(x (1) ,x (2) ),y:=(y (1) ,y (2) )∈L 2 ψ 1 (y (1) )ψ 2 (y (2) )ψ ′ 1 (x (1) )ψ ′ 2 (x (2) ) × 1 4 |Λ l | z 1 ,z 2 ∈Λ l ̺ (β,ω,λ) I fl y+(z 2 ,z 2 ) I fl x+(z 1 ,z 1 ) , where I fl x := I x − ̺ (β,ω,λ) (I x ) 1 , x ∈ L 2 .
Recall that I x is the paramagnetic current observable defined by (19). Hence, it suffices to prove the existence of a finite constant D ∈ R + such that, for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and all x, y ∈ L 2 , 1 4 |Λ l | z 1 ,z 2 ∈Λ l ̺ (β,ω,λ) I fl y+(z 2 ,z 2 ) I fl
x+(z 1 ,z 1 ) ≤ D .(135)
This can be shown by using Lemma 5.3. Indeed, we infer from (102) at t = α = 0 that, for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , x, y ∈ L 2 and all z 1 , z 2 ∈ Λ l , ̺ (β,ω,λ) I fl y+(z 2 ,z 2 ) I fl x+(z 1 ,z 1 ) = ̺ (β,ω,λ) I y+(z 2 ,z 2 ) I x+(z 1 ,z 1 ) −̺ (β,ω,λ) I y+(z 2 ,z 2 ) ̺ (β,ω,λ) I x+(z 1 ,z 1 ) = C
(ω) 0 (x + (z 1 , z 1 ) , y + (z 2 , z 2 )) ,
where C
(ω) t+iα is the map from L 4 to C defined at t ∈ R and α ∈ [0, β] by (95). Now, take the canonical orthonormal basis {e x } x∈L 2 of ℓ 2 (L) ⊗ ℓ 2 (L) defined by
e x := e x (1) ⊗ e x (2) , x := (x (1) , x (2) ) ∈ L 2 .
Recall that e x (y) ≡ δ x,y ∈ ℓ 2 (L). Then, the coefficient C (ω) t+iα can be seen as a kernelw.r.t. the canonical basis {e x } x∈L 2 -of an operator on ℓ 2 (L) ⊗ ℓ 2 (L), again denoted by C (ω) t+iα . Then, we observe from (136) that 1 4 |Λ l | z 1 ,z 2 ∈Λ l ̺ (β,ω,λ) I fl y+(z 2 ,z 2 ) I fl x+(z 1 ,z 1 ) = 1 4 |Λ l | z 1 ,z 2 ∈Λ l e x+(z 1 ,z 1 ) , C
0 (e y+(z 2 ,z 2 ) )
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and x, y ∈ L 2 . By Lemma 5.3, the operator C (ω) t+iα always satisfies C (ω) t+iα op ≤ 4 and hence,
1 4 |Λ l | z 1 ,z 2 ∈Λ l e x+(z 1 ,z 1 ) , C (ω) 0 e y+(z 2 ,z 2 ) ≤ 1(138)
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and x, y ∈ L 2 . By (137), it follows that
1 4 |Λ l | z 1 ,z 2 ∈Λ l ̺ (β,ω,λ) I fl y+(z 2 ,z 2 ) I fl x+(z 1 ,z 1 ) ≤ 1
for any l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and all x, y ∈ L 2 .
Tree-Decay Bounds and Uniformity of Responses
Uniformity of Energy Increment Responses
For the reader's convenience we start by reminding a few definitions and some standard mathematical facts used in our proofs. First of all, we recall that in [BPH1, Section 5.2] we give an explicit expression of the automorphisms τ (ω,λ,A) t,s of U in terms of series involving multi-commutators, see [BPH1,Eqs. (3.14)-(3.15)]. Indeed, in [BPH1,Eq. (5.15)] we represent the automorphisms τ for any B ∈ U and t ≥ s. Here, for any t, s ∈ R,
W A t,s := τ (ω,λ) t (W A s ) ∈ U(140)
is the time-evolution of the electromagnetic potential energy observable W A s defined by (57), that is,
W A s := x,y∈L exp i 1 0 [A(s, αy + (1 − α)x)] (y − x)dα − 1 × e x , ∆ d e y a * x a y ,(141)
for any A ∈ C ∞ 0 and s ∈ R. The expression (139) is useful because we can apply tree-decay bounds on multicommutators. These bounds, derived in [BPH1,Section 4], are useful to analyze multicommutators of products of annihilation and creation operators. Using them, we show for instance in [BPH1,Lemma 5.10] that, for any A ∈ C ∞ 0 , there is η 0 ∈ R + such that, for l, ε ∈ R + , there is a ball B(0, R) := {x ∈ L : |x| ≤ R} of radius R ∈ R + centered at 0 such that, for all |η| ∈ (0, η 0 ], β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , and t 0 ≤ s 1 , . . . , s k ≤ t,
x∈Λ L \B R z∈L,|z|≤1 k∈N (t − t 0 ) k k! ̺ (β,ω,λ) [W ηA l s k −t 0 ,s k , . . . , W ηA l s 1 −t 0 ,s 1 , τ (ω,λ) t−t 0 (a * x a x+z )] (k+1) ≤ ε .
This property together with (58) and (139) implies that, for all |η| ∈ (0, η 0 ], l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 ,
S (ω,ηA l ) (t) = k∈N x,z∈L,|z|≤1 e x , (∆ d + λV ω ) e x+z i k t t 0 ds 1 · · · s k−1 t 0 ds k ̺ (β,ω,λ) [W ηA l s k −t 0 ,s k , . . . , W ηA l s 1 −t 0 ,s 1 , τ (ω,λ) t−t 0 (a * x a x+z )] (k+1) .(142)
See [BPH1,Section 5.5] for more details. These assertions are important to get uniform bounds as explained in Theorems 3.3 and 4.1. Indeed, it is relatively straightforward to get the asymptotics of the elements W ηA l t and ∂ t W A t when (η, l −1 ) → (0, 0) by using the integrated electric field
E A t (x) := 1 0 E A (t, αx (2) + (1 − α)x (1) ) (x (2) − x (1) )dα(143)
between x (2) ∈ L and x (1) ∈ L at time t ∈ R (cf. (12)) and the subset
K := x := (x (1) , x (2) ) ∈ L 2 : |x (1) − x (2) | = 1(144)
of bonds of nearest neighbors (cf. (23)).
Lemma 5.11 (Asymptotics of the potential energy observable)
For any η, l ∈ R + , A ∈ C ∞ 0 and t ≥ t 0 , there are complex numbers
D ηA l x,y (t) x,y∈L ⊂ C and a (η, t)-independent subset Λ l ∈ P f (L) of diameter of order O(l) such that W ηA l t = − 1 2 x∈K η t t 0 E A l s (x)ds I x + η 2 2 t t 0 E A l s (x)ds 2 P x +η 3 x∈ Λ l z∈L,|z|≤1D
ηA l
x,x+z (t)a * x a x+z withD ηA l x,x+z (t) and ∂ tD ηA l
x,x+z (t) being uniformly bounded for all η in compact sets, all x, z ∈ L such that |z| ≤ 1, and all ω ∈ Ω, λ ∈ R + 0 and l ∈ R + .
Lemma 5.11 and Equation (145) that, for any A ∈ C ∞ 0 , there is η 0 ∈ R + such that, for all |η| ∈ (0, η 0 ], l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 ,
Lemma 5.13 (Potential energy difference)
For any A ∈ C ∞ 0 , there is η 0 ∈ R + such that, for all |η| ∈ (0, η 0 ] and l ∈ R + ,
P (ω,ηA l ) (t) − I (ω,ηA l ) d (t) = η 2 4 x,y∈K t t 0 E A l s (x)ds t t 0 σ (ω) p (x, y, t − s) E A l s (y)ds +O(η 3 l d ) ,
uniformly for β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 .
Proof: The proof is very similar to the one of Theorem 5.12. In particular, to get the asymptotics, it suffices to observe that, for any A ∈ C ∞ 0 , there is η 0 ∈ R + such that, for all |η| ∈ (0, η 0 ], l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 ,
ρ (β,ω,λ,ηA l ) t (W ηA l t ) − ̺ (β,ω,λ) (W ηA l t ) = t t 0 ̺ (β,ω,λ) i[τ (ω,λ) s (W ηA l s ), τ (ω,λ) t (W ηA l t )] ds + O(η 3 l d ) ,(151)
by (8)- (9), the Dyson-Phillips expansions (139), Lemma 5.11 and tree-decay bounds on multi-commutators [BPH1,Corollary 4.3]. Note that the correction term of order O(η 3 l d ) in (151) is again uniformly bounded in β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 . Then, we use Lemma 5.11 in (151) to obtain ρ (β,ω,λ,ηA l ) t
(W ηA l t ) − ̺ (β,ω,λ) (W ηA l t ) = η 2 4 x,y∈K t t 0 E A l s (x)ds t t 0 ds 1 s 1 t 0 E A l s 2 (y)ds 2 ×̺ (β,ω,λ) i[τ (ω,λ) s 1 (I y ) , τ (ω,λ) t (I x )] + O(η 3 l d ) ,
uniformly for β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 . We then obtain
ρ (β,ω,λ,ηA l ) t (W ηA l t ) − ̺ (β,ω,λ) (W ηA l t ) (152) = η 2 4 x,y∈K t t 0 E A l s (x)ds t t 0 ζ (ω) y,x (t, s) E A l s (y)ds + O(η 3 l d ) ,
by using (148), (150) and an integration by parts. We now invoke Equation (149) in (152) to arrive at the assertion.
Therefore, Theorem 4.1 (Q) and (P) are direct consequences of (58)-(59), (62)-(63), Theorem 5.12 and Lemma 5.13.
Uniformity of Current Linear Response
Following Section 3 we take w := (w 1 , . . . , w d ) ∈ R d , A ∈ C ∞ 0 (R; R) and E t := −∂ t A t for any t ∈ R, with E t w being the space-homogeneous electric field. Then,Ā ∈ C ∞ 0 is defined to be the electromagnetic potential such that the value of the electric field equals for any B 1 , B 2 ∈ U. Its name comes from the clear relation to Duhamel's formula, see [Si,Section IV.4] for more details. This sesquilinear form appears in different contexts. For instance, it has been used by Bogoliubov [B] for finite volume quantum systems in quantum statistical mechanics. It is an useful tool in the first mathematical justification -by Ginibre [G] in 1968 -of the Bogoliubov approximation for the Bose gas. This sesquilinear form is also used in the context of linear response theory, see for instance [BR2,Discussion after Lemma 5.3.16 and Section 5.4]. In fact, it is also named in the literature Bogoliubov or Kubo-Mori scalar product as well as the canonical correlation. A detailed analysis of this sesquilinear form for KMS states has been performed by Naudts, Verbeure and Weder in the paper [NVW]. Their aim was to extend to infinite systems some results of linear response theory initiated by Kubo [K] and Mori [M].
Note that our definition of (·, ·) ∼ is slightly different from the usual one because of the missing normalization factor β −1 in front of the integral in (156). Discussions on Duhamel two-point functions and examples of applications can also be found in [MW, H, FB, NV, R, DLS].
A first way to study this sesquilinear form is to use finite volume systems. This is possible because, by using the Phragmén-Lindelöf theorem [BR2,Proposition 5.3.5] and [BPH1,Theorem A.3], one checks that the formal expression
̺ (β,ω,λ) B * τ (ω,λ) iα (B) = ̺ (β,ω,λ) (τ (ω,λ) iα/2 (B)) * τ (ω,λ) iα/2 (B) ≥ 0 is correct for any B ∈ U and all α ∈ [0, β]. So (B 1 , B 2 ) → (B 1 , B 2 ) ∼ is a positive semi- definite sesquilinear form on U.
It is however important for the study of the conductivity measure to know that this form defines a scalar product. To this end, we invoke the modular theory to have access to functional calculus as it is done in the paper [NVW].
A.2 Duhamel Two-Point Functions on von Neumann Algebras
We consider in all the following subsections an arbitrary strongly continuous one-parameter group τ := {τ t } t∈R of automorphisms of a C * -algebra X as well as an arbitrary (τ, β)-KMS state ̺ ∈ X * for some β > 0. Similar to (156), the Duhamel two-point function (·, ·) ∼ on the C * -algebra X is defined by
(B 1 , B 2 ) ∼ := β 0 ̺ (B * 1 τ iα (B 2 )) dα , B 1 , B 2 ∈ X .(157)
We have in mind the example X = U, τ = τ (ω,λ) and ̺ = ̺ (β,ω,λ) for β ∈ R + , ω ∈ Ω and λ ∈ R + 0 , of course. The GNS representation of ̺ is denoted by (H, π, Ψ). There is a unique normal state of the von Neumann algebra M := π (X ) ′′ , also denoted by ̺ ∈ M * to simplify notation, with ρ = ρ • π on X . By [BR2,Corollary 5.3.4], there is a unique σ-weakly continuous * -automorphism group on M, which is again denoted by τ = {τ t } t∈R , such that τ t • π = π • τ t , t ∈ R, on X . Moreover, the normal state ̺ ∈ M * is a (τ, β)-KMS state on M and it thus satisfies the KMS (or modular) condition, that is, for any b 1 , b 2 ∈ M, the map from R to C extends uniquely to a continuous map
m b 1 ,b 2 on R × [0, β] ⊂ C which is holomorphic on R × (0, β) whereas m b 1 ,b 2 (iβ) = ̺(b 2 b 1 ) , b 1 , b 2 ∈ M .
Here, ·, · H denotes the scalar product of the Hilbert space H. See, e.g., [BR2,Proposition 5.3.7]. Because ̺ is invariant with respect to τ , the * -automorphism group τ has a unique representation by conjugation with unitaries {U t } t∈R ⊂ M, i.e.,
τ t (b) = U t bU * t , t ∈ R , b ∈ M ,
such that U t Ψ = Ψ. As t → τ t is σ-weakly continuous, the map t → U t is strongly continuous. Therefore, the unitary group {U t } t∈R has an anti-self-adjoint operator iL as generator, i.e., U t = e itL . In particular, Ψ ∈ Dom(L) and L annihilates Ψ, i.e., LΨ = 0. The operator L is known in the literature as the standard Liouvillean of τ associated with ̺. The spectral theorem applied to the self-adjoint operator L ensures the existence of a projection-valued measure E on the real line R such that
L = R ν dE(ν) .
We now use the (Tomita-Takesaki) modular objects ∆, J of the pair (M, Ψ). In particular,
J∆ 1/2 (bΨ) = b * Ψ , b ∈ M .(158)
By [P,Proposition 5.11], the modular operator ∆ is equal to
∆ = exp (−βL) = R e −βν dE(ν)(159)
and U t = ∆ −itβ −1 . Now, let the (unbounded) positive operator T acting on H be defined by
T := β 1/2 R 1 − e −βν βν 1/2 dE(ν) .(160)
Here, 1 − e −β·0 β · 0 := 1 .
The Duhamel two-point function (·, ·) ∼ is directly related to this operator:
Theorem A.1 (Duhamel two-point function in the GNS representation)
For any B 1 , B 2 ∈ X ,
(B 1 , B 2 ) ∼ = Tπ (B 1 ) Ψ, Tπ (B 2 ) Ψ H .
In particular, (B 1 , B 1 ) ∼ ≥ 0.
Proof: The proof can be found in [NVW,Theorem II.4]. Since it is short, we give it here for completeness. Note first that, for any b 1 , b 2 ∈ M,
Ψ, b 1 ∆ 1/2 b 2 Ψ H = ∆ 1/2 b * 1 Ψ, b 2 Ψ H = Jb 2 Ψ, b 1 Ψ H = ∆ 1/2 J∆ 1/2 b 2 Ψ, b 1 Ψ H = Ψ, b 2 ∆ 1/2 b 1 Ψ H ,
where we have used ∆ = ∆ * , the anti-unitarity of J, J 2 = 1, and J∆ 1/2 J = ∆ −1/2 . Using this fact and properties of the map m b 1 ,b 2 from R×[0, β] ⊂ C to C together with the Phragmén-Lindelöf theorem [BR2,Proposition 5.3.5] one shows that, for any b 1 , b 2 ∈ M,
m b 1 ,b 2 (iβα) = Ψ, b 1 ∆ α b 2 Ψ H , α ∈ [0, 1/2] , Ψ, b 2 ∆ 1−α b 1 Ψ H , α ∈ [1/2, 1] .
By (157) and (158), it follows that
(B 1 , B 2 ) ∼ = β 1/2 0 π (B 1 ) Ψ, ∆ α π (B 2 ) Ψ H dα (161) +β 1/2 0 J∆ 1/2 π (B 2 ) Ψ, ∆ α J∆ 1/2 π (B 1 ) Ψ H dα . Because J 2 = 1, J∆ α J = ∆ −α and J is anti-unitary, note that J∆ 1/2 π (B 2 ) Ψ, ∆ α J∆ 1/2 π (B 1 ) Ψ H = J∆ α J∆ 1/2 π (B 1 ) Ψ, ∆ 1/2 π (B 2 ) Ψ H = ∆ −α ∆ 1/2 π (B 1 ) Ψ, ∆ 1/2 π (B 2 ) Ψ H
for all α ∈ [0, 1/2]. Therefore, we deduce from (160) and (161) that
(B 1 , B 2 ) ∼ = β π (B 1 ) Ψ, ∆ − 1 ln ∆ π (B 2 ) Ψ H = Tπ (B 1 ) Ψ, Tπ (B 2 ) Ψ H , using that 1/2 0 ∆ α bΨ dα = ∆ 1/2 − 1 ln ∆ bΨ and 1/2 0 ∆ −α bΨ dα = 1 − ∆ −1/2 ln ∆ bΨ for any b ∈ M.
By (160), one checks that Dom(∆ 1/2 ) ⊂ Dom(T) and thus, MΨ ⊂ Dom(T). It is therefore natural to define the Duhamel two-point function, again denoted by (·, ·) ∼ , on the von Neumann algebra M := π (X ) ′′ by
(b 1 , b 2 ) ∼ := Tb 1 Ψ, Tb 2 Ψ H , b 1 , b 2 ∈ M .(162)
This sesquilinear form is a scalar product:
Theorem A.2 (Duhamel two-point function as a scalar product)
The sesquilinear form (·, ·) ∼ is a scalar product of the pre-Hilbert space M.
Proof: The positivity of the sesquilinear form (·, ·) ∼ is clear. Therefore, it only remains to verify that it is non-degenerated. This is proven in [NVW,Lemma II.2.] as follows: First note that 0 is not an eigenvalue of T. This follows from (160). Indeed, for all ν ∈ R,
1 − e −βν βν 1/2 > 0 .
Since ̺ is a (τ, β)-KMS state, the cyclic vector Ψ is also separating for M, by [BR2,Corollary 5.3.9.]. Therefore, (b, b) ∼ = 0 yields TbΨ = 0 which in turn implies that bΨ = 0 and b = 0.
Note that the kernel of π is a closed two-sided ideal. If the C * -algebra X is simple (like U), i.e., when {0} and X are the only closed two-sided ideals, it then follows that ker (π) = {0}.
Using this and Theorem A.2 we deduce that the Duhamel two-point function (157) for B 1 = B 2 ∈ X \{0} is never zero:
Theorem A.3 (Duhamel two-point function -Strict positivity)
If the C * -algebra X is simple then (B, B)
∼ > 0 for all non-zero B ∈ X \{0}.
Finally, we observe that it is a priori not clear that the scalar products (·, ·) ∼ and ·, · H are related to each other via some upper or lower bounds. In fact, a combination of Roepstorff's results [R,Eq. (10)] for finite dimensional systems with those of Naudts and Verbeure on von Neumann Algebras yields the so-called auto-correlation upper bounds [NV,Theorem III.1], also called Roepstorff's inequality. For self-adjoint observables, these upper bounds read:
Theorem A.4 (Auto-correlation upper bounds for observables) For any self-adjoint element
b = b * ∈ M, (b, b) ∼ ≤ bΨ, bΨ H . In particular, for all B = B * ∈ X , (B, B) ∼ ≤ ̺(B 2 ) ≤ B 2 X .
Proof: This theorem is a particular case of [NV,Theorem III.1], by observing in its proof that (u −v) log(u/v) should be replaced by u when u = v. See also [BR2,Theorem 5.3.17].
Note that the authors derive in [R, NV] further upper and lower bounds related the scalar products (·, ·) ∼ and ·, · H . These are however not used in the sequel. For more details, we refer to [NV] or [BR2,Section 5.3.1]. We only conclude this subsection by an important equality for the Duhamel two-point function (·, ·) ∼ which was widely used for finite volume systems. See, e.g., [G,Eq. (2.4)].
This equality does not seem to be proven before for general KMS states. It is a straighforward consequence of Theorem A.1. To this end, denote by δ the generator of the strongly continuous one-parameter group τ := {τ t } t∈R of automorphisms of the C *algebra X .
Theorem A.5 (Commutators and Duhamel two-point function)
For all B 1 ∈ X and B 2 ∈ Dom(δ),
−i(B 1 , δ (B 2 )) ∼ = ̺ ([B * 1 , B 2 ]) .
Proof:
It is a direct consequence of (158)- (160) and (162): For any B 1 ∈ X and B 2 ∈ Dom(δ), −i(B 1 , δ (B 2 )) ∼ = Tπ (B 1 ) Ψ, Tπ (δ (B 2
)) Ψ H = π (B 1 ) Ψ, π (B 2 ) Ψ H − ∆ 1/2 π (B 1 ) Ψ, ∆ 1/2 π (B 2 ) Ψ H = π (B 1 ) Ψ, π (B 2 ) Ψ H − π (B * 2 ) Ψ, π (B * 1 ) Ψ H = ̺ ([B * 1 , B 2 ]) .
See also Theorem A.1.
Corollary A.6 (Duhamel two-point function and generator of dynamics)
For any self-adjoint element B = B * ∈ Dom(δ) ⊂ X , (B, δ (B)) ∼ = 0 and − i̺ ([δ (B) , B]) = (δ (B) , δ (B)) ∼ ≥ 0 .
A.3 Duhamel GNS Representation
In view of Theorem A.2, we denote byH the completion of M w.r.t. the scalar product (·, ·) ∼ . This Hilbert space is related to the GNS Hilbert space of ̺ by a unitary transformation: Proof: Since U ∼ b H = b ∼ , the operator U ∼ defined by U ∼ b = TbΨ for b ∈ M has a continuous isometric extension onH. Then, one checks that the range of T is dense in H and is included in the range of U ∼ . For more details, see [NVW,Theorem II.3.].
A simple consequence of Theorem A.7 is a cyclic representation based on the Duhamel two-point function:
Definition A.8 (Duhamel GNS representation)
The Duhamel GNS representation of the (τ, β)-KMS state ̺ ∈ X * is defined by the triplet (H,π,Ψ) wherẽ
Ψ := U * ∼ Ψ = U * ∼ TΨ ∈H andπ (B) = U * ∼ π (B) U ∼ , B ∈ X .
If X has an identity 1, thenΨ = π(1) ∈ M ⊂H.
This cyclic representation of KMS states does not seem -at least to our knowledge -to have been previously used, even if it is a direct consequence of [NVW,Theorem II.3.].
In particular, the name Duhamel GNS representation is not standard and it could also be called Bogoliubov or Kubo-Mori GNS representation in reference to the scalar product (·, ·) ∼ . As explained in Section A.2, there is a unique σ-weakly continuous * -automorphism groupτ = {τ t } t∈R on the von Neumann algebraM :=π (X ) ′′ , such that τ t =τ t • π, t ∈ R. It has a representation by conjugation with unitaries {e itL } t∈R ⊂ M, the self-adjoint operatorL being equal tõ
L = U * ∼ LU ∼ .(163)
Clearly,Ψ ∈ Dom(L) andLΨ = 0. The normal state̺ ∈M * is a (τ , β)-KMS state. At the end of the previous subsection we explain that if the C * -algebra X is simple, like the CAR algebra U, then π : X → M is injective and one can see the C * -algebra X as a subspace ofH. In particular, if X has an identity 1, theñ Ψ = 1 ∈ X ⊂ M ⊂H .
Note additionally that, in this case, for any element B ∈ X and time t ∈ R, one has τ t (B) ∈ X ⊂H and it is straightforward to check (cf. [NVW, Section III]) that iL is the generator of a unitary group extending τ to the whole Hilbert spaceH: Theorem A.9 (Duhamel GNS representation and dynamics) Assume X is simple. Then, for B ∈ X ⊂H and t ∈ R, τ t (B) = e itL B with (B,LB) ∼ = 0 if B ∈ Dom(L).
Proof:
See [NVW, Section III]: By Theorem A.7, for any B ∈ X ⊂ M ⊂H and t ∈ R, τ t (B) = U * ∼ Tπ (τ t (B)) Ψ = U * ∼ Te itL π (B) Ψ = U * ∼ e itL Tπ (B) Ψ = U * ∼ e itL U ∼ B = e itU * ∼ LU∼ B .
Recall that (H, π, Ψ) is the GNS representation of the (τ, β)-KMS state ̺ and L is the associated standard Liouvillean. See also (163). The equality (B,LB) ∼ = 0 results from Corollary A.6.
Note that Theorem A.9 directly yields the invariance of the norm of B ∈ X ⊂H w.r.t. to the group τ acting on the subspace X ⊂H.
Corollary A.10 (Stationarity of the Duhamel norm)
Assume X is simple. Then, for B ∈ X ⊂H and t ∈ R, τ t (B) ∼ = B ∼ with · ∼ denoting the (Duhamel) norm ofH associated with the scalar product (·, ·) ∼ . Therefore, by Theorem A.9, we can invoke the spectral theorem in order to analyze the dynamics in relation with the scalar product (·, ·) ∼ . This is exploited for instance in Theorem 5.4 to extract the conductivity measure from a spectral measure. from R to C extends uniquely to a continuous map m B 1 ,B 2 on R × [0, β] ⊂ C which is holomorphic on R × (0, β) while, again by stationarity and hermiticity of ρ, m B 1 ,B 2 (t + iβ) = ρ (τ t (Θ (B * 1 )) Θ (B * 2 )) = ρ (Θ (B * 1 ) Θ (τ t (B * 2 ))) = ρ Θ (τ t (B 2 ) B 1 )
for any t ∈ R and B 1 , B 2 ∈ X . As a consequence, ρ Θ is a (τ, β)-KMS state, see [BR2,Proposition 5.3.7].
This lemma implies that, if ̺ is the unique (τ, β)-KMS state with τ being time-reversal invariant, then ̺ is time-reversal invariant. Let These spaces are closed real subspaces of X . Furthermore, they are real pre-Hilbert spaces w.r.t. the Duhamel two-point function (·, ·) ∼ defined by (157).
Lemma A.13 (X ± as real pre-Hilbert spaces)
Assume that τ is time-reversal invariant and ̺ is a time-reversal invariant (τ, β)-KMS state defining the Duhamel two-point function (·, ·) ∼ . Then, for all B 1 , B 2 ∈ X − and all B 3 , B 4 ∈ X + ,
(B 1 , B 2 ) ∼ = (B 2 , B 1 ) ∼ ∈ R and (B 3 , B 4 ) ∼ = (B 4 , B 3 ) ∼ ∈ R .
Proof: For any B 1 , B 2 ∈ X − , one clearly has (B 1 , B 2 ) ∼ = (Θ (B 1 ) , Θ (B 2 )) ∼ .
Thus, we have to prove that (Θ (B 1 ) , Θ (B 2 )) ∼ = (B 2 , B 1 ) ∼ , B 1 , B 2 ∈ X − .
By the Phragmén-Lindelöf theorem [BR2,Proposition 5.3.5], the stationarity of KMS states and Definition (157), it suffices to show that ̺ (Θ (B 1 ) τ t (Θ (B 2 ))) = ̺ (B 2 τ t (B 1 )) for all t ∈ R and every B 1 , B 2 ∈ X − . In fact, by the time-reversal invariance of ̺, the stationarity of KMS states and the hermiticity of states, ̺ (Θ (B 1 ) τ t (Θ (B 2 ))) = ̺ (B 1 τ −t (B 2 )) = ̺ (τ t (B 1 ) B 2 ) = ̺ (B 2 τ t (B 1 )) .
As (·, ·) ∼ is a sesquilinear form, we thus have
(B 1 , B 2 ) ∼ = (B 2 , B 1 ) ∼ = (B 2 , B 1 ) ∼ ∈ R , B 1 , B 2 ∈ X − .
The assertion for X + is proven in the same way.
This lemma can be generalized for time-dependent Duhamel correlation functions. To this end, we show the following assertions:
Lemma A.14 (Commutators and Duhamel correlation functions)
Let ̺ be a (τ, β)-KMS state defining the Duhamel two-point function (·, ·) ∼ . Then, for any B 1 , B 2 ∈ X and all t ∈ R, t 0 ̺ (i [B 1 , τ s (B 2 )]) ds = (B 1 , τ t (B 2 )) ∼ − (B 1 , B 2 ) ∼ .
Proof: It is an obvious consequence of Theorem A.5. The assertion can also be deduced from [NVW,Theorem II.5]. We give here another proof because some of its arguments are used elsewhere in the paper.
By assumption, for any B 1 , B 2 ∈ X , the map from R to C defined by t → ̺ (B 1 τ t (B 2 )) uniquely extends to a continuous map
z → ̺ (B 1 τ z (B 2 ))
on the strip R+i[0, β], which is holomorphic on R+i(0, β). The KMS property of ̺, that is,
̺(B 1 τ t+iβ (B 2 )) = ̺(τ t (B 2 )B 1 ) , B 1 , B 2 ∈ X , t ∈ R ,(164)
implies that, for any B 1 , B 2 ∈ X and t ∈ R,
̺ ([B 1 , τ t (B 2 )]) = ̺ (B 1 τ t (B 2 )) − ̺ (B 1 τ t+iβ (B 2 )) .
As a consequence, by the Cauchy theorem for analytic functions, we obtain that for any B 1 , B 2 ∈ X and t ∈ R. The group property of τ obviously yields ̺ (B 1 τ t+z (B 2 )) = ̺ (B 1 τ z (τ t (B 2 )))
for all z, t ∈ R. On the other hand, the KMS property (164) of ̺ leads to Equation (165) for all z ∈ R + iβ. Therefore, we infer from the Phragmén-Lindelöf theorem [BR2,Proposition 5.3.5] that, for any B 1 , B 2 ∈ X , (165) holds true for all z ∈ R + i[0, β]. In particular, β 0 ̺ (B 1 τ t+iα (B 2 )) dα = (B 1 , τ t (B 2 )) ∼ .
Lemma A.15 (Time-reversal symmetry of commutators)
Assume that τ is time-reversal invariant and ̺ is a time-reversal invariant state. Then, for any B 1 , B 2 ∈ X − (or X + ) and all t ∈ R, Proof: The first equality follows from the following assertions: For any B 1 , B 2 ∈ X − (or X + ) and t ∈ R, for any B 1 , B 2 ∈ X − (or X + ) and t ∈ R.
We are now in position to prove a generalization of Lemma A.13:
Theorem A.16 (Symmetries of Duhamel correlation functions)
Assume that τ is time-reversal invariant and ̺ is a time-reversal invariant (τ, β)-KMS state defining the Duhamel two-point function (·, ·) ∼ . Then, for all B 1 , B 2 ∈ X − (or X + ) and t ∈ R, (B 1 , τ t (B 2 )) ∼ = (B 1 , τ −t (B 2 )) ∼ = (B 2 , τ t (B 1 )) ∼ ∈ R .
Proof: By Lemma A.14, for all B 1 , B 2 ∈ X − (or X + ) and t ∈ R. Observe that
̺ (i[B 1 , τ s (B 2 )]) ∈ R ,
for all B 1 , B 2 ∈ X − (or X + ) and s ∈ R, because B 1 , B 2 are self-adjoint elements of X . From Lemma A.13, it follows that, for any B 1 , B 2 ∈ X − (or X + ) and t ∈ R,
(B 1 , τ t (B 2 )) ∼ ∈ R .
Moreover, by Lemmata A.13 and A.15, (B 1 , τ t (B 2 )) ∼ = (B 1 , τ −t (B 2 )) ∼ = (B 2 , τ t (B 1 )) ∼ for any B 1 , B 2 ∈ X − (or X + ) and t ∈ R.
Trimester Program entitled "Mathematical challenges of materials science and condensed matter physics" for the opportunity to work together on this project at the Hausdorff Research Institute for Mathematics in Bonn. This work has also been supported by the grants MTM2010-16843, MTM2014-53850 (MINECO), the FAPESP grant 2013/13215-5, the grant IT641-13 (Basque Government), the BERC 2014-2017 program (Basque Government) as well as the BCAM Severo Ochoa accreditation SEV-2013-0323 (MINECO). Finally, we thank very much the referee for her/his work and interest in the improvement of the paper.
R
(t) at time t within R obeys:J (E)
ν , and where H is the Hilbert transform. This expression allows us to define B(R d )-valued tempered distributions µ R , µ ⊥ R satisfying Kramers-Kronig relations and such that J (E)
the microscopic charge transport coefficients defined by (29)-(30).
vanish for t ≥ t 1
1and (67) stays constant. Following Joule's effect, for t ≥ t 1 , this energy should correspond to a heat production as defined in[BPH1, Definition 3.1]. The latter equals the energy increment S (ω,ηA l ) , by[BPH1, Theorem 3.2].
identified with either P (ω,A) or S (ω,A) . From Theorem 4.1 (p), the energy J (ω,A) p
βκ , κ ∈ R . Equation (103) provides useful estimates like space-decay properties of complex-time two-point correlation functions C (ω)
with (121)-(123) we arrive at Inequality (118). Finally, to prove (119), observe that R |ν| (I k,l ,Ẽ(dν)I k,l ) ∼ = Tπ (I k,l ) Ψ, E R + 0 TLπ (I k,l ) Ψ H (124) − Tπ (I k,l ) Ψ, E R − TLπ (I k,l ) Ψ H .
= 0 .
0Observe that T acts as the identity on the kernel of L. Assume now that µ by (111) for X = R\{0}, implies that(I k,l ,Ẽ (R\{0}) I k,l ) ∼ = 0 , k ∈ {1, . . . , d} .As a consequence, any linear combination of elements {I k,l } k∈{1,...,d} ∈ U ⊂H belongs to the kernel ofL, i.e.,lin {I k,l : k ∈ {1, . . . , d}} ⊂ ker(L) .By Theorem A.7 and (163), this property in turn yieldslin {π (I k,l ) Ψ : k ∈ {1, . . . , d}} ⊂ ker (L) .
Corollary 5. 7 (
7Non-triviality of the measure µ For any l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 , the B + (R d )-valued measure µ
k [W A s k −s,s k , . . . , W A s 1 −s,s 1 , τ (ω,λ) t−s (B)] (k+1)
Theorem A. 7 (
7Unitary equivalence of H andH) U ∼H = H with U ∼ being the unitary operator defined by U ∼ b = TbΨ for b ∈ M.
X
+ := {B = B * ∈ X : Θ (B) = B} , X − := {B = B * ∈ X : Θ (B) = −B} .
̺
(B 1 τ t+iα (B 2 )) dα − (B 1 , B 2 ) ∼
t 0 ̺
0(i[B 1 , τ s (B 2 )]) ds = −t 0 ̺ (i[B 1 , τ s (B 2 )]) ds = t 0 ̺ (i[B 2 , τ s (B 1 )]) ds .
̺̺
(i[B 1 , τ s (B 2 )]) ds .Furthermore, by stationarity of KMS states, (i[B 1 , τ s (B 2 )]) ds
(B 1 0 ̺
10, τ t (B 2 )) ∼ = t (i[B 1 , τ s (B 2 )]) ds + (B 1 , B 2 ) ∼
(R; R) and t ≥ t 0 , the (increment of) current density resulting from the space-homogeneous electric perturbation E in the box Λ l is the sum of two current densities defined from (24):(p) The paramagnetic current densityJ (ω,ηĀ l ) p (t) ≡ J (β,ω,λ,ηĀ l ) p (t) ∈ R dis defined by the space average of the current increment vector inside the box Λ l , that is for any k ∈ {1, . . . , d},J (ω,ηĀ l ) p (t) k := |Λ l | −1 ρ (β,ω,λ,ηĀ l ) t (I k,l ) .(42)(d) The diamagnetic (or ballistic) current densityJ (ω,ηĀ l ) d (t) ≡ J (β,ω,λ,ηĀ l ) d (t) ∈ R dis defined analogously, for any k ∈ {1, . . . , d}, by J (ω,ηĀ l ) d (t) k := |Λ l | −1 ρ (β,ω,λ,ηĀ l ) t (I ηĀ l k,l ) .(43)
t t 0 Ξ (ω) p,l (t − s) w E s ds , t ≥ t 0 ,(44)J (ω,A) d,l (t) := Ξ (ω) d,l w t t 0 E s ds , t ≥ t 0 ,(45)
t → m b 1 ,b 2 (t) := ̺(b 1 τ t (b 2 )) = Ψ, b 1 τ t (b 2 )Ψ H
Acknowledgments:We would like to thank Volker Bach, Horia Cornean, Abel Klein and Peter Müller for relevant references and interesting discussions as well as important hints. JBB and WdSP thank Mr. and Mrs. Bru for their hospitality, support and interesting discussions. JBB and WdSP are also very grateful to the organizers of the Hausdorfffor π (U) ′′ , see[BR2,Corollary 5.3.9.]. Therefore, the assertion is a direct consequence of Theorem 5.6.We now give another construction of the (AC-conductivity) measure µ(ω)p,l on R\{0} from the diamagnetic transport coefficient Ξ (ω) d,l (34) and the space-averaged quantum current viscositysee (40). W.r.t. the canonical orthonormal basis of R d ,for any k, q ∈ {1, . . . , d} and t ∈ R. Compare(127)l . In fact, one has:Theorem 5.8 (Static admittance) Let l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . Then the limit of L[V(ω)l ](ǫ) exists as ǫ ↓ 0 and satisfies:Note thatẼ (R\{0}) is not the identity becauseL1 = 0.Proof: Fix l, β ∈ R + , ω ∈ Ω and λ ∈ R + 0 . By[NVW,], observe that ΞOn the other hand, by(105)and(107),for any t ∈ R + and k, q ∈ {1, . . . , d}. The von Neumann or mean ergodic theorem (see, e.g.,[P,Theorem 3.13]) implies thatwhereẼ ({0}) is the orthogonal projection on the kernel ofL. By combining (128)-(129) we obviously getProof: Note that (143) yieldsTherefore, the statement is a straightforward consequence of Equations(5),(141)By combining this lemma with (142) one can obtain Theorem 4.1(S). However, by using (64), it is easier to start with the paramagnetic and diamagnetic energies J Theorem 5.12 (Microscopic paramagnetic and diamagnetic energies) For any A ∈ C ∞ 0 , there is η 0 ∈ R + such that, for all |η| ∈ (0, η 0 ], l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 , one has: (p) Paramagnetic energy increment:(d) Diamagnetic energy:.The correction terms of order O(l d η 3 ) in assertions (p) and (d) are uniformly bounded inProof: (p) Using W A t = 0 for any t ≤ t 0 and (9) we note that, for any t ≥ t 0 ,[BPH1,Eqs. (5.41)]. As a consequence, by (62)-(64), the paramagnetic energy increment equalsfor any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , A ∈ C ∞ 0 and t ≥ t 0 . Similar to the proof of[BPH1,Lemma 5.10], one uses Dyson-Phillips expansions (139) and tree-decay bounds on multi-commutators[BPH1,Corollary 4.3] to infer fromThis last correction term of orderfor any B 1 , B 2 ∈ U and all s 1 , s 1 ∈ R. Therefore, we insert this equality and the asymptotics given by Lemma 5.11 in Equation(146)to arrive at the equalityNote that the function ζ (ω)x,y is a map from R 2 to R. By combining (148) with (8)-(9) and (29), we observe thatfor any β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , x, y ∈ L 2 and s 1 , s 2 ∈ R, whileAs a consequence, the assertion follows from(147)and an integration by parts.(d) is a direct consequence of (30), (63) and Lemma 5.11.It remains to study the entropic energy increment S (ω,ηA l ) and the electromagnetic energy P (ω,ηA l ) defined by(58)and(59), respectively. To this end, it suffices to study the potential energy differencefor all times t ≥ t 0 . This is done in the following lemma: E t w at time t ∈ R for all x ∈ [−1, 1] d and (0, 0, . . . , 0) for t ∈ R and x / ∈ [−1, 1] d . This choice yields rescaled electromagnetic potentials ηĀ l as defined by(17)for l ∈ R + and η ∈ R. Recall that A t := 0 for all t ≤ t 0 , where t 0 ∈ R is any fixed starting time. We also recall that {e k } d k=1 is the canonical orthonormal basis of the Euclidian space R d . In this case, the electromagnetic potential energy observable defined by (57) equalsfor any l ∈ R + , η ∈ R, w := (w 1 , . . . , w d ) ∈ R d , A ∈ C ∞ 0 (R; R) and t ∈ R. The full current density is the sum of the paramagnetic and diamagnetic currents J (ω,ηĀ l ) p and J(ω,ηĀ l ) d that are respectively defined by(42)and(43). These currents are directly related to the transport coefficients Ξ(34)). We show this in two lemmata that yield Theorem 3.3:Lemma 5.14 (Linear response of paramagnetic currents) For anyProof:The first assertion is proven by essentially the same arguments as in Section 5.2.1. Indeed, one uses the stationarity (9) of the (τ (ω,λ) , β)-KMS state ̺ (β,ω,λ) , Dyson-Phillips expansions (139) for the non-autonomous dynamics, Lemma 5.11, and treedecay bounds on multi-commutators[BPH1,Corollary 4.3]as in[BPH1,Lemma 5.10]in order to deduce from (42) the existence of η 0 ∈ R + such that, for |η| ∈ [0, η 0 ],uniformly for all l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k ∈ {1, . . . , d} and t ∈ R. Then, for |η| ∈ [0, η 0 ], we employ (153) and derive an assertion similar to Lemma 5.11 in order to getuniformly for all l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k ∈ {1, . . . , d} and t ∈ R. It follows from an integration by parts thatwhich, combined with (33) and(106), yields the assertion.Lemma 5.15 (Linear response of diamagnetic currents)For any w := (w 1 , . . . , w d )uniformly for l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 and t ≥ t 0 .Proof: By (9), for any k ∈ {1, . . . , d}, note thatwhileuniformly for all β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k ∈ {1, . . . , d} and t ∈ R. Therefore, using again for the non-autonomous dynamics, Lemma 5.11, and tree-decay bounds on multi-commutators[BPH1,Corollary 4.3]one deduces the existence of η 0 ∈ R + such that, for |η| ∈ [0, η 0 ], the first term in the right hand side of (154) is of order O (η 2 ), uniformly for l, β ∈ R + , ω ∈ Ω, λ ∈ R + 0 , k ∈ {1, . . . , d} and t ≥ t 0 . Then the assertion follows by combining this property with (34) and (154)-(155).A Duhamel Two-Point FunctionsA.1 Duhamel Two-Point Function on the CAR AlgebraThe Duhamel two-point function (·, ·) (ω) ∼ is defined by (90), that is,A.4 Duhamel Two-Point Function and Time-Reversal SymmetryLet X be a C * -algebra with unity 1 and assume the existence of a map Θ : X → X with the following properties:• Θ is antilinear and continuous.• Θ (1) = 1 and Θ • Θ = Id X .Such a map is called a time-reversal operation of the C * -algebra X . Observe that, for any strongly continuous one-parameter group τ := {τ t } t∈R of automorphisms of X , the family τ Θ := {τ Θ t } t∈R defined byis again a strongly continuous one-parameter group of automorphisms. Similarly, for any state ρ ∈ X * , the linear functional ρ Θ defined byis again a state. We say that τ and ρ are time-reversal invariant if they satisfy τ Θ t = τ −t for all t ∈ R and ρ Θ = ρ.If τ is time-reversal invariant then, for all β > 0, there is at least one time-reversal invariant (τ, β)-KMS state ̺ ∈ X * , provided the set of (τ, β)-KMS states is not empty. This follows from the convexity of the set of KMS states:Lemma A.12 (Existence of time-reversal invariant (τ, β)-KMS states)Assume that τ is time-reversal invariant and ̺ is a (τ, β)-KMS state. Then, ρ Θ is a (τ, β)-KMS state. In particular, 1 2 ρ + 1 2 ρ Θ is a time-reversal invariant (τ, β)-KMS state.Proof: For any t ∈ R and B 1 , B 2 ∈ X , ρ Θ (B 1 τ t (B 2 )) = ρ (Θ (B 1 ) τ −t (Θ (B 2 ))) = ρ (Θ (B * 2 ) τ t (Θ (B * 1 ))) , using the stationarity of KMS-states and hermiticity of states. Since ρ is by assumption a (τ, β)-KMS state, the continuous function t → m B 1 ,B 2 (t) := ρ (Θ (B * 2 ) τ t (Θ (B * 1 )))
all the paper the Duhamel two-point function (·, ·) ∼ defined by (156) on the CAR C * -algebra U with the scalar product (·, ·) ∼ defined by (162) for ̺ = ̺ (β,ω,λ) and τ = τ (ω,λ) on M := π (U) ′′ ⊂H. Note that U ≡ π(U) ⊂ M. is a pre-Hilbert space w.r.t. (·, ·) ∼all the paper the Duhamel two-point function (·, ·) ∼ defined by (156) on the CAR C * -algebra U with the scalar product (·, ·) ∼ defined by (162) for ̺ = ̺ (β,ω,λ) and τ = τ (ω,λ) on M := π (U) ′′ ⊂H. Note that U ≡ π(U) ⊂ M is a pre-Hilbert space w.r.t. (·, ·) ∼ .
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"\nRenormalised Chern-Weil forms associated with families of Dirac operators * Jouko MICKELSSON\nSylvie PAYCHA\n"
]
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"Renormalised Chern-Weil forms associated with families of Dirac operators * Jouko MICKELSSON\nSylvie PAYCHA"
]
| []
| We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in [Sc] and later in[PS1]. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern-Weil forms in as far as they coincide with regularised Chern-Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature formula. We interpret the odd degree Chern-Weil type forms built from superconnections as Wodzicki residues and establish a transgression formula along the lines of known transgression formulae for η-forms. * MSC: 53Z05, 58J40, 58J28, 81T13, 81T50 1 the only two up to linear combinations [LP] | 10.1016/j.geomphys.2007.03.001 | [
"https://arxiv.org/pdf/math/0607148v1.pdf"
]
| 18,824,576 | math/0607148 | edfe4d9ff73ae525f9765dc0979eb4848885dae6 |
6 Jul 2006 September 14, 2018
Renormalised Chern-Weil forms associated with families of Dirac operators * Jouko MICKELSSON
Sylvie PAYCHA
6 Jul 2006 September 14, 2018
We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in [Sc] and later in[PS1]. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern-Weil forms in as far as they coincide with regularised Chern-Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature formula. We interpret the odd degree Chern-Weil type forms built from superconnections as Wodzicki residues and establish a transgression formula along the lines of known transgression formulae for η-forms. * MSC: 53Z05, 58J40, 58J28, 81T13, 81T50 1 the only two up to linear combinations [LP]
Introduction
Chern-Weil formalism in finite dimensions assigns to a connection ∇ on a principal bundle P → B over a manifold B a form f (∇) on B with values in the adjoint bundle Ad P :
f : C(P ) → Ω (B, AdP ) ∇ → f (∇), which is closed with de Rham cohomology class independent of the choice of connection. Here C(P ) is the space of connections on P and Ω(B, W ) the space of differential forms with values in a vector bundle W over B.
In the context of Ψdo−bundles-i.e. bundles with structure group the group of zero order invertible pseudodifferential operators Cℓ * 0 -the trace used in finite dimensions to build maps f j (∇) = tr(∇ 2j ) can be replaced by two 1 natural traces on the algebra Cℓ 0 of zero order pseudodifferential operators, namely the Wodzikci residue and the leading symbol trace. Such constructions were investigated in [PR] and lead to maps which project down to quotient connections∇ on the quotient bundleP with structure group Cℓ * 0 / (1 + Cℓ −∞ ) * where Cℓ −∞ is the algebra of smoothing operators and (1 + Cℓ −∞ ) * the group of invertibles. In other words, they project down to maps:
f : C(P ) → Ω (B, AdP ) ∇ →f (∇).
We call such maps local in as far as they are insensitive "to smoothing perturbations".
In contrast, on a principal bundle with structure group (1 + Cℓ −∞ ) * ⊂ Cℓ * 0 one can mimic the ordinary Chern-Weil construction to build Chern classes using the ordinary trace on Cℓ −∞ . We are concerned in this paper with possible extensions of these Chern forms to Ψdo−bundles. Since the ordinary trace on Cℓ −∞ extends to linear functionals on Cℓ 0 obtained from regularised (or weighted) traces, one might want to try to extend the ordinary Chern-Weil constructions to Ψdo−bundles using these regularised traces. Such issues were addressed in [PR]; the fact that regularised traces do not yield genuine traces gives rise to obstructions to carrying out the Chern-Weil construction since the regularised Chern forms obtained from regularised traces are not closed. However, it is useful to keep in mind that the obstruction to their closedness can be expressed in terms of local maps in the above sense.
In this paper, we discuss ways to "renormalise" the regularised Chern forms by adding to them local maps in order to turn them into closed forms with de Rham classes independent of the connection. To do so, we compare them with Chern forms previously investigated in [Sc] and later [PS1], which are built from superconnections; in some cases they differ by local expressions so that a renormalisation procedure can indeed be carried out adding local counterterms. More precisely, letting (say in the Z 2 -graded case) I A = D + ∇ be a superconnection associated with a Dirac operator D, then the expression tr D 2 (∇ 2j ) − tr I A 2 ( I A 2j ) [2j] -which compares the naive infinite dimensional analog tr D 2 (∇ 2j ) of the finite dimensional Chern form tr(∇ 2j ) and the closed form tr I A 2 ( I A 2j ) [2j] built from the super connection-is local in the above sense.
Here tr D 2 (B) := fp z=0 TR(B(D 2 + π D ) −z ) is the D 2 -weighted (or ζ-regularised) trace of B obtained as the finite part at z = 0 of the meromorphic expansion TR(B(D 2 + π D ) −z ) where B is a form-valued pseudodifferential operator and TR the canonical trace on non integer order pseudodifferential operators [KV]. π D stands for the orthogonal projection onto the kernel of D. This "renormalisation" procedure applies to the geometric setup corresponding to families of Dirac operators associated with a trivial fibration of manifolds (see Theorem 1).
In the case of a family of Dirac operators associated with a general fibration of manifolds, such a straightforward "renormalisation procedure" is not possible due to the presence of an extra curvature term arising from a horizontal distribution on the fibration. Indeed, the Chern-Weil forms associated with a superconnection then differs from a weighted Chern form by (a priori) non local terms involving this extra curvature term. .
On the grounds of the previous discussion, when the fibration is trivial, it differs from renormalised weighted Chern forms by local terms. As it could be expected in analogy with the finite dimensional situation, in the graded case, the first Chern form str I A 2 ( I A 2 ) [2] turns out to be proportional to the curvature of the determinant bundle associated with the family of Dirac operators from which the superconnection is built. But there is also an j-th residue Chern form associated with a superconnection I A (which is new to our knowledge) the 2j − 1-th form part of which which reads (see Theorem 2): .
sres | I A| 2j−1 [2j−1] = √ π (−1) j (2j − 1)!! (2iπ)
In the non graded case, the second residue Chern form (i.e. for j = 2) turns out to be proportional to the curvature of the gerbe associated with the family of Dirac operators from which the superconnection is built, which was investigated by Lott [L]. Following a similar scheme to that of Lott 2 we derive a transgression formula for the j-th residue Chern form (see Theorem 3)
sres | I A λ | 2j−1 [2j−1] = a j · d (η λ ) [2j−2] ,
using the η-invariantη λ (see [BC], [L]) associated with a family of invertible Dirac type operators D(λ) = D − λI. These perturbed operators differ from that of Lott but match physicists' needs.
The relation to gauge anomalies is explained in the last section of the paper. In particular, a new formula for the curvature of the local fermionic vacuum line bundles (see Theorem 5) is derived using a residue correction term added to the naive curvature formula (see Theorem 4), the latter coming by analogy from the geometry of finite-dimensional Grassmann manifolds, replacing the finite-dimensional trace by a weighted trace.
The geometric setup
Let π : E → M be a vector bundle over a closed manifold M . Cℓ 0 (M, E) denotes the Fréchet Lie algebra of 0-order classical pseudo-differential operators (Ψdo−s) acting on smooth sections of E and Cℓ * 0 (M, E) the Fréchet Lie group of invertible 0-order classical pseudo-differential operators. Let P → B be a G = Cℓ * 0 (M, E) principal bundle and AdP = P × G Cℓ 0 (M, E) the adjoint bundle, so that locally, AdP |U ≃ U × Cℓ 0 (M, E). We equip P with a connection 1-form Θ : T P → Cℓ 0 (M, E) which induces a connection ∇ Ad on AdP . In local coordinates we have ∇ Ad = d + [Θ, ·] with Θ the above Cℓ 0 (M, E)-valued one form.
The dual bundle AdP ⋆ to AdP comes equipped with the dual connection ∇ Ad ⋆ defined for any section λ of Ad P ⋆ and any section σ of AdP by
dλ(σ) = ∇ Ad ⋆ λ (σ) + λ(∇ Ad σ).
On the other hand, G acts on the space C ∞ (M, E) of smooth sections of E and the associated vector bundle E = P × G C ∞ (M, E) comes equipped with the connection ∇, locally of the form ∇ = d + Θ. Then, locally ∇ Ad = d+[Θ, ·] and ∇ Ad * = d− [Θ, ·]. It is therefore convenient to write ∇ Ad σ = [∇, σ] for any section σ of AdP and ∇ Ad * λ = [∇, λ] for any section λ of AdP ⋆ . With these notations we have:
d (λ(σ)) = [∇, λ](σ) + λ([∇, σ]).
The group (1 + Cℓ −∞ (M, E)) * , where Cℓ −∞ (M, E) denotes the algebra of smoothing operators, is a normal subgroup of
Cℓ * 0 (M, E). Quotienting Cℓ * 0 (M, E) by (1 + Cℓ −∞ (M, E)) * yields quotient bun- dlesP → B andĒ =P ×Ḡ C ∞ (M, E) with structure group G := Cℓ * 0 (M, E)/ (1 + Cℓ −∞ (M, E)) *
equipped with the induced connection∇. Let C(P ) and C(P ) denote the space of connections on P andP .
Definition 1
We call a map
f : C(P ) → Ω(B, Cℓ(P )) ∇ → f (∇) 2
We derive a complete proof clarifying some steps in Lott's proof. Our proof is carried out for operators D(λ) = D−λ I (which are differential operators) but it easily extands to operators Dα = D + hα(D) (which are pseudo-differential operators) used by Lott where hα is a smooth function with compact support. local whenever it projects down to:f : C(P ) → Ω(B, Cℓ(P )) ∇ →f (∇).
Let Cℓ(E) = P × G Cℓ(M, E) denote the bundle of classical pseudo-differential operators with fibre the whole algebra Cℓ(M, E) of classical pseudo-differential operators acting on sections of E. Clearly, AdP ⊂ Cℓ(E) is a subbundle of Cℓ(E).
In view of the following constructions, it is useful to mention that when M = { * } is a point, then
E = V is a vector space, Cℓ(M, E) = Cℓ 0 (M, E) = End(V ), Cℓ * (M, E) = Cℓ * 0 (M, E) = GL(V )
so that P → B boils down to an ordinary GL(V )-principal bundle and both Cℓ(E) → B and AdP → B boil down to its adjoint bundle AdP = P × G End(V ) for the adjoint action of GL(V ) on End(V ). Thus, Ψdo−bundles can be seen as natural generalisations of ordinary principal bundles.
Q-weighted traces (a short review)
A first atempt to generalise to Ψdo−bundles the construction of Chern-Weil forms on ordinary bundles, is to use regularised (or weighted) traces of powers of the curvature as an Ersatz for ordinary traces of powers of the curvature which provide representatives of Chern-Weil classes in finite dimensions [PR]. We give a brief review of weighted traces of classical pseudo-differential operators. Let Q ∈ Cℓ(M, E) be an invertible admissible elliptic operator of positive order q, where by admissible we mean that its leading symbol admits a spectral cut θ 3 . If Q is not invertible, we replace it by Q + π Q where π Q is the orthogonal projection onto the kernel of Q. An invertible admissible elliptic operator Q has complex powers
Q z θ = 1 2iπ Γ θ λ z (Q − λ) −1 dλ
where Γ θ is a contour around the spectral cut and hence its logarithm log θ Q = d dz Q z | z=0 which is not classical anymore. In applications to follow, Q is non negative self-adjoint so that θ = π can be chosen as a spectral cut. We shall henceforth drop out the explicit mention of the spectral cut writing simply Q −z and log Q. Recall that for any A ∈ Cℓ(M, E) and provided Q has positive order, the map z → TR (A Q −z ) is meromorphic with simple pole at 0 and its finite part at 0
tr Q (A) := fp z→0 TR A Q −z is called the Q-weighted (or ζ-regularised) trace of A.
Here TR is the canonical trace on non integer order classical Ψdo−s [KV]. Even though it is not cyclic and hence not a genuine trace, the Q-weighted trace deserves the name of a trace in as far as it coincides with the ordinary trace on trace-class operators and hence on Cℓ −∞ (M, E) and therefore extends it to a linear map on Cℓ(M, E). In contrast, the Wodzicki residue defined for A ∈ Cℓ(M, E) by
res(A) = 1 (2π) n S * M tr x (σ A (x, ξ)) −n dx d S ξ
vanishes on trace-class operators and hence on Cℓ −∞ (M, E). Here S * M stands for the cotangent unit sphere, d S ξ the canonical volume measure on SM , σ A is the symbol of A, tr x the fibrewise trace and the subscript −n stands for the −n (positively) homogeneous part of the symbol. When A is a differential operator we have [PS2]:
tr Q (A) = − 1 q res (A log Q)(1)
where the residue on the r.h.s is defined by the above formula in spite of A log Q not being classical anymore. The fact that A is differential ensures that the residue is well-defined. In general tr Q (A) depends on Q for a given A ∈ Cℓ(M, E); given two weights Q 1 , Q 2 ∈ Cℓ(M, E) with positive orders q 1 , q 2 and same spectral cut we have:
tr Q1 (A) − tr Q2 (A) = −res A log Q 1 q 1 − log Q 2 q 2 .
(2) Also, tr Q is not cyclic: the obstruction to the cyclicity of tr Q is measured by a Wodzicki residue:
tr Q ([A, B]) = − 1 q res (A [B, log Q]) ,(3)
where now the residue is applied to a genuine classical operator since the bracket [B, log Q] is classical.
We shall need the following technical lemma (see [BGV] Lemma 9.35).
Lemma 1 Let f be a smooth function on ]0, +∞[ with asymptotic behaviour at 0 of the type
f (ε) ∼ ε→0 ∞ j=0 a j ε α−j
for some real number α (depending on f ) and such that for large enough ε,
|f (ε)| ≤ Ce −ε λ for some λ > 0, C > 0. Then its Mellin transform z → M(f )(z) := 1 Γ(z) ∞ 0 ε z−1 f (ε) dt
defines a meromorphic map on the complex plane (which turns ut to be holomorphic at z = 0) and
fp ε=0 f (ε) = fp z=0 M(f )(z) = M(f )(0).
In particular, if f (ε) = √ ε g(ε) then fp ε=0 f (ε) = √ π res z=0 M(g)(z + 1 2 ) .
Proof: The first part of the lemma is well known (see e.g. [BGV] Lemma 9.35). Let us check the formula relating finite parts of f (ε) = √ εg(ε) and its Mellin transform.
fp ε=0 f (ε) = fp z=0 M(f )(z) = fp z=0 1 Γ(z) ∞ 0 √ ε ε z−1 g(ε)dε = fp z=0 1 Γ(z) ∞ 0 ε z+ 1 2 −1 g(ε)dε = fp z=0 Γ(z + 1 2 ) Γ(z) M(g)(z + 1 2 ) = Γ( 1 2 ) fp z=0 z M(g)(z + 1 2 ) = √ π res z=0 M(g)(z + 1 2 ) .
⊔ ⊓
The Mellin transform provides a stepping stone between heat-kernel regularisation and ζ-regularisation methods:
Proposition 1 For any Q ∈ Cℓ(M, E) non negative self-adjoint elliptic and any A ∈ Cℓ(M, E) with vanishing Wodzicki residue: tr Q (A) = fp ε=0 tr A e −εQ .
Proof: This follows from Lemma 1 applied to f (ε) = tr(A e −εQ ). ⊔ ⊓
Q-weighted Chern forms
We define weighted Chern-Weil forms as in [PR] and briefly recall the obstructions to the closedness. Weighted traces extend to Ψdo−valued forms in the following manner. Given a Ψdo−vector bundle E, Q is a section of Cℓ(E) which is elliptic, admissible and has positive constant order q. Note that these properties, ellipticity, admissibility and of constant order q are invariant under the adjoint action of the group Cℓ * (M, E) of invertible classical pseudo-differential operators. The definition of the Qweighted trace and the Wodzicki residue then extend to Ψdo−valued forms setting for b ∈ U ⊂ B and α ⊗ A ∈ Ω (U, Cℓ(E)), with α ∈ Ω(U ), A ∈ C ∞ (U, Cℓ(E)):
tr Q (α ⊗ A)(b) := α(b) ⊗ tr Q b (A(b)); res(α ⊗ A)(b) := α(b) ⊗ res(A(b)).
Properties (1), (2) (3) extend in a straighforward manner to forms:
tr Q (α) = − 1 q res(α log Q) tr Q1 (α) − tr Q2 (α) = − res α log Q 1 q 1 − log Q 2 q 2 tr Q ([α, β]) = − 1 q res(α [β, log Q]),(4)
where the first identity holds whenever α is a differential operator valued form whereas the others hold for any Cℓ (E)-valued forms α, β. The Wodzicki residue commutes with differentiation [∇, res] = 0 whereas weighted traces do not. The obstruction is measured in terms of a Wodzicki residue. Indeed, it follows from (4) that locally, ·] in local coordinates, we have:
[d tr Q ](α) = − 1 q res(α d log Q) as a result of which, writing ∇ Ad = d + [θ,[∇, tr Q ](α) = d tr Q (α) − tr Q ([∇, α]) = d tr Q (α) − tr Q (d α) − tr Q ([θ, α]) = [d tr Q ](α) − 1 q res (α [θ, log Q]) = − 1 q res(α d log Q) − 1 q res (α [θ, log Q]) = − 1 q res (α [∇, log Q]) ∀α ∈ Ω (B, Cℓ(E)) .(5)
The curvature Ω = ∇ Ad 2 of ∇ Ad lies in Ω 2 (P, Cℓ (E)) so that the Q-weighted trace tr Q (Ω i ) defines a 2i-form on B. The following proposition tells us that the obstruction to the closedness is local in the sense of the above definition.
Proposition 2 The exterior differential of the weighted Chern-Weil form tr Q (Ω i ) is local, i.e. of the form dtr Q (Ω i ) = f i (∇) for somef i : C(P ) → Ω(Cℓ(P )).
Proof: Since ∇ Ad (Ω) = [∇, Ω i ] = 0, by (5) we have
dtr Q (Ω i ) = [∇, tr Q ](Ω i ) + tr Q ([∇, Ω i ]) = [∇, tr Q ](Ω i ) + i j=1 tr Q (Ω j [∇, Ω] Ω i−j ) = [∇, tr Q ](Ω i ) = − 1 q res Ω i [∇, log Q] .
Since the Wodzicki residue vanishes on smoothing operators,
d tr Q (Ω i ) = d tr Q (Ω i ) =f (∇),
and hence the locality property of the obstruction to the closedness. ⊔ ⊓ .
From superconnections to Chern-Weil type forms
We review and extend constructions of Chern-Weil type forms carried out in [PS1] using superconnections. Let E be a vector bundle associated with a Ψdo−principal bundle P as before.
Chern forms associated with superconnections
• The Z 2 -graded case: Let us assume that
E = E + ⊕ E − is a Z 2 -graded super bundle on B.
The canonical trace TR for non integer order operators in Cℓ(E) is replaced by the super canonical trace sTR whereas weighted traces tr Q for operators in Cℓ(E) are replaced by weighted supertraces str Q with respect to even weights Q = Q + ⊕ Q − . They vanish on odd Ψdos−and give the difference of weighted traces on even Ψdos−:
sTR(A) := TR(A ++ ) − TR(A −− ); str Q (A) := tr Q + (A ++ ) − tr Q − (A −− )
with obvious notations. This grading combined with the Z 2 -grading on forms Ω
(B, E) = Ω ev (B, E) ⊕ Ω od (B, E) gives rise to: Ω + (B, E) = Ω ev (B, E + ) ⊕ Ω od (B, E − ); Ω − (B, E) = Ω od (B, E + ) ⊕ Ω ev (B, E − ).
Definition 2 [BGV] A superconnection is an odd-parity first order differential operator
I A : Ω + − (B, E) → Ω − + (B, E) which satisfies the (graded) Leibniz rule. If α ∈ Ω(U ) for some open subset U ⊂ B and B ∈ C ∞ (U, Cℓ(E)) then I A(α ⊗ B) = dα ⊗ B + (−1) |α| α ⊗ I A B.
The curvature I A 2 of a superconnection I A on E lies in Ω(B, Cℓ(E)). Following Quillen [Q] we say a superconnection I A on E is associated with a smooth family of elliptic differential operators
{D b , b ∈ B} whenever I A [0] = D.
Example 1 I A = D + ∇ defines a particular super connection associated with D with curvature
I A 2 = D 2 + ∇ Ad D + Ω = Q + [∇, D] + Ω where we have set Q = D 2 . Here [∇, D] = ∇D + D∇ is the anticommutator.
Remark 1 In the following we systematically use graded commutators of operator valued forms: anticommutator for odd-odd forms and usual commutator otherwise.
• The non graded case: Let us assume that E is an ordinary vector bundle on B. Following Quillen, we introduce an extra grading σ such that σ 2 = 1 and build the right C ⊕ C σ-module:
Cℓ σ (E) := Cℓ (E) (C ⊕ C σ) .
Odd degree Ψdos−lie in Cℓ (E) (C σ) whereas Cℓ (E) is identifued with even degree Ψdos−. The ordinary canonical trace TR extends to non integer order operators in Cℓ σ (E) by sTR(α + σβ) = TR(β) so that weighted traces tr Q are replaced by
str Q (α + σβ) := tr Q (β).
These definitions extend to the space Ω σ (B, E) of Cℓ σ (E)-valued forms on B in a straightforward manner.
Definition 3 [BGV] A superconnection is a first order differential operator
I A : Ω σ (B, E) → Ω σ (B, E)
which commutes with σ and satisfies the (graded) Leibniz rule.
As in the even case, it is associated with a family {D b , b ∈ B} of elliptic differential operators whenever I A [0] = D.
Example 2 I A := σ D + ∇ is a particular superconnection associated with D, the curvature of which reads 4
I A 2 = D 2 + ∇ Ad (σD) + Ω = Q + [∇, σ D] + Ω with Q = (σ D) 2 = D 2 as before.
Let us recall from [Sc] (see also [PS1]) that weighted traces can be extended to include weights I A 2 which are Ψdo−valued forms and analogs of Chern-forms can be constructed, which turn out to be closed. Writing
I A 2 = I A 2 [0] + I A 2 [1] + I A 2 [2] = D 2 + I A 2 [>0]
where the subscript [j] stands for the j-th degree part, and [> 0] for non no zero degree part, can be useful to derive explicit expansions in increasing form degree. For example,
(λ − I A 2 ) −1 = λ − D 2 − I A 2 [>0] −1 = K j=0 λ − D 2 −1 I A 2 [>0] λ − D 2 −1 · · · I A 2 [>0] λ − D 2 −1 + S K (D 2 , I A 2 [>0] , λ),(6)
where S K has form degree > K and where I A 2
[>0] λ − D 2 ) −1 arises j times in the j-th term of the sum. By convention the j = 0 term reduces to λ − D 2 ) −1 . Hence, for any positive integer K
(λ − I A 2 ) −1 [K] = K j=0 λ − D 2 −1 I A 2 [>0] λ − D 2 ) −1 · · · I A 2 [>0] λ − D 2 −1
has a finite expansion in increasing form degree. Also, whenever I A [0] = D is invertible, so is I A 2 invertible and its modulus | I A| := I A 2 1 2 can be defined using a contour integration (cfr Section 2):
| I A| = i 2π Γ √ λ ( I A 2 − λ) −1 dλ
where Γ is a contour around the spectrum of D 2 which is a subset of R + . In general, we set
| I A| := I A 2 + π I A
where π I A is the orthogonal projection onto the kernel of I A 2
[0] . This defines a form provided Ker I A 2
[0]
has constant dimension.
Remark 2 Note that for any α ∈ Ω(B, Cℓ (E)),
str I A 2 (α) = fp z=0 sTR α I A 2 + π I A −z = fp ε=0 str α e −ε I A 2
since I A 2 is a differential operator-valued form and hence has vanishing Wodzicki residue.
The following proposition extends results of [PS1]. 4 Here D commutes with σ whereas ∇ anticommutes with σ.
Proposition 3 Let P be a polynomial function. Forms str I A 2 (P ( I A 2 )) and sres (P (| I A)|) associated with a superconnection I A are closed. Their de Rham class is independent of the choice of connection one forms I A [1] .
Remark 3 The residue form sres (P (| I A|)) is in fact insensitive to the projection π I A which is a smoothing operator and hence does not affect the Wodzicki residue.
Proof: We extend the argument used in [PS1] for the closedness of forms str I A 2 ( I A 2i ) to any str I A 2 (P ( I A)).
Equations (5) and (3) extend replacing the connection ∇ by the superconnection I A and the weight Q by the Ψdo−valued form I A 2 of order 2 [PS1] and we have
d str I A 2 P ( I A 2 ) = [ I A, str I A 2 ] P ( I A 2 ) + str I A 2 [ I A, P ( I A 2 )] = − 1 2 sres P ( I A 2 )[ I A, log( I A 2 + π I A )] = − 1 2 d dt t=0 i 2π Γ λ t sres P ( I A 2 )[ I A, ( I A 2 + π A − λ) −1 ] dλ = 0 since [ I A, ( I A 2 + π A − λ) −1 ] is smoothing.( I A 2 ) = I A 2i d dt str I A 2 t I A 2i t = d dt fp ε=0 str I A 2i t e −ε I A 2 t = i j=1 fp ε=0 str I A 2(i−j−1) t [ I A t ,İ A t ] I A 2j t e −ε I A 2 t − ε str I A 2i t [ I A t ,İ A t ] e −ε I A 2 t = i fp ε=0 str [ I A t , I A 2(i−1) tİ A t e −ε I A 2 t ] − ε str [ I A t , I A 2i tİ A t e −ε I A 2 t ] = i fp ε=0 d str I A 2(i−1) tİ A t e −ε I A 2 t − ε d str I A 2i tİ A t e −ε I A 2 t = d i fp ε=0 str I A 2(i−1) tİ A t e −ε I A 2 t − ε str I A 2i tİ A t e −ε I A 2 t(7)
is exact. Here we have used the fact that d dt
I A 2 t =İ A t I A t + I A tİ A t = [ I A t ,İ A t ]
, the graded commutator of I A t with the form Ψdo−valued formİ A t . It follows that the de Rham class of str I A 2 I A 2i and hence of str I A 2 P ( I A 2 ) is independent of the choice of connection.
Similarly, since | I A| j = i 2π Γ λ j 2 ( I A 2 + π A − λ) −1 dλ and since: d dt ( I A 2 t + π A − λ) −1 = −( I A 2 t + π I A − λ) −1 d dt I A 2 t ( I A 2 t + π I A − λ) −1 = −( I A 2 + π I A − λ) −1 [ I A t ,İ A t ]( I A 2 t + π I A − λ) −1 = − I A t , ( I A 2 t + π I A − λ) −1İ A t ( I A 2 t + π I A − λ) −1 ,
it follows that the variation
d dt sres | I A t | j = sres i 2π Γ λ j 2 d dt ( I A 2 t + π A − λ) −1 dλ = − i 2π sres Γ λ j 2 I A t , ( I A 2 t + π I A − λ) −1İ A t ( I A 2 t + π I A − λ) −1 dλ = − i 2π sres Γ λ j 2 I A t + π I A , ( I A 2 t + π I A − λ) −1İ A t ( I A 2 t + π I A − λ) −1 dλ since res[π I A , ·] = 0 = − i 2π sres I A t + π I A , Γ λ j 2İ A t ( I A 2 t + π I A − λ) −2 since [ I A t + π I A , ( I A 2 t + π I A − λ) −1 ] = 0 = − i 2π sres I A t , Γ λ j 2İ A t ( I A 2 t + π I A − λ) −2 since res[π I A , ·] = 0 = j 2 i 2π sres I A t , Γ λ j−2 2İ A t ( I A 2 t + π I A − λ) −1 = j 2 sres I A t ,İ A t ( I A 2 t + π I A ) j−2 2 = j 2 d sres İ A t | I A t | j−1(8)
is also exact, which ends the proof of the proposition. ⊔ ⊓
Chern-forms associated with superconnections I A = ∇ + D
We now specialise to the case I A [2] = 0 and consider a superconnection I A = D + ∇ in the graded setup and I A = σ D + ∇ in the ungraded setup. The following theorem compares the closed Chern-forms str I A 2 ( I A 2i ) with the (non closed in general) weighted Chern forms.
Theorem 1 In the Z 2 -graded set up and provided the superconnection I A = D + ∇, the (closed) Chern-forms str I A 2 ( I A 2j ) [2j] differ from the (non closed) Q-weighted Chern forms str Q (Ω j ) by a local map i.e.
str I A 2 I A 2j [2j] − str Q (Ω j ) =f j (∇). for somef j : C(P ) → Ω(B,P ).
In the ungraded setup and provided the superconnection I A = σ D+∇, (closed) Chern-forms str I A 2 ( I A 2j ) [2j−1] differ from the (non closed) Q-weighted forms str Q (Ω j−1 [∇, σ D]) by a local map i.e.
str I A 2 I A 2j [2j−1] − j str Q (Ω j−1 [∇, σ D]) =ḡ j (∇) for someḡ j : C(P ) → Ω(B,P ).
Remark 4 This does not hold anymore if I A [2] = 0 as can easily be seen from the proof below. When I A [2] = 0, on the grounds of this proposition, str I A 2 I A 2j
[2j] can be interpreted as a renormalised version of str Q (Ω j ).
Proof: Let us observe in the graded case that since I A 2 = Q + [∇, D] + Ω, we have:
str Q I A 2j [2j] = str Q Ω j ,
and similarly in the ungraded case, we have
str Q I A 2j [2j−1] = jstr Q Ω j−1 [∇, σ D] .
Using a Campbell-Hausdorff formula for pseudo-differential operators [O] combined with formula (2) extended to form valued weights, we have
str I A 2 I A 2j [2j] = str Q I A 2j [2j] + str I A 2 I A 2j [2j] − str Q I A 2j [2j] = str Q Ω j − 1 2 sres I A 2j (log I A 2 − log Q) [2j] = str Q (Ω j ) − 1 2 sres I A 2j log(1 + Q −1 ([∇, D] + Ω)) + [log Q, log(1 + Q −1 ([∇, D] + Ω))] + ... [2j] = str Q (Ω j−1 ) + f i (∇),
with f j (∇) the Wodzicki residue of a polynomial expression in D, D −1 and ∇ of total form degree 2j. As a Wodzicki residue, it is insensitive to a perturbation of the connection by a smoothing operator so that f j (∇) =f j (∇). This shows that
str I A 2 I A 2i [2j] − str Q (Ω j ) =f j (∇)
is local. A similar computation shows that
str I A 2 I A 2j [2j−1] − j str Q (Ω j [∇, σ D]) =ḡ j (∇)
is also local. ⊔ ⊓
Residue Chern forms as Wodzicki residues
In order to derive an explicit expression for the residue Chern forms in terms of Wodzicki residues, we borrow the following notations from [CoM] and [H]. For A in Cl(M, E) of order a, a given ∆ ∈ Cℓ(M, E) and any j ∈ N we set:
A (j) := ad j ∆ (A), where ad ∆ (B) = [∆, B], so that A (0) = A, A (j+1) = ad ∆ (A (j) ) = [∆, A (j)
]. When ∆ of order 2 has scalar leading symbol then A (j) has order a + j + 1.
Proposition 4 Let I A be a superconnection associated with an operator D, the square of which has scalar leading symbol. For any positive integer K sres | I A| 2j−1
[K] = K l=0 k1≥0 · · · k l ≥0 2j−1 2 · · · 2j−1 2 − |k| − l (k 1 + · · · + k l + l)! c(k 1 , · · · , k l ) · ·sres I A 2 [>0] (k1) I A 2 [>0] (k2) · · · I A 2 [>0] (k l ) D 2 2j−1 2 −|k|−l [K]
, where we set c(k 1 ) = 1 for any positive integer k and where, for a multi index k = (k 1 , · · · , k l ) for j > 1 we set c(k 1 , · · · , k l ) = (k 1 + · · · + k l + l)! k 1 ! · · · k j !(k 1 + k 2 + 1) · · · (k 1 + · · · + k l−1 + l) .
In particular,
sres (| I A|) [1] = k≥0 1 2 · · · 1 2 − k − 1 (k + 1)! · sres I A 2 [1] (k) D 2 1 2 −k−1 ,
Remark 5 If D is a differential operator then I A 2 is a differential operator and sres | I A| 2j = sres I A 2 j which is why we only consider odd powers.
Remark 6 Since the operator order of I A 2 [>0] is no larger than 1, I A 2
[1]
(k) has order ≤ 1 + k and
I A 2 [>0](k2)
· · · I A 2
[>0]
(k l ) D 2 2j−1 2 −|k|−l has order ≤ 2j − 1 − |k| − l which decreases as |k| or l increases. Thus the Wodzicki residue vanishes for large enough |k| or l and the seemingly infinite series in the proposition is in fact finite.
Proof: We introduce notations borrowed from [H] and [CoM]. Let T ∈ Cℓ(M, E) and T k , k ∈ N be operators in Cℓ(M, E) with decreasing order in k. Then
T ≃ k≥0 T k ⇐⇒ ∀N ∈ N, ∃K(N ) T − K(N ) k=0 T k ∈ Cℓ −N (M, E).
With these notations, for any non negative integer h we have [H] (see the proof of Proposition 4.14)
(λ − D 2 ) −h A ≃ k≥0 (h + k − 1)! (h − 1)!k! A (k) (λ − D 2 ) −h−k .
As a result, the j-th term in (6) reads:
(λ − D 2 ) −1 I A 2 [>0] · · · (λ − D 2 ) −1 I A 2 [>0] (λ − D 2 ) −1 ≃ k1≥0 I A 2 [>0] (k1) (λ − Q) −2−k1 I A 2 [>0] · · · (λ − D 2 ) −1 I A 2 [>0] (λ − D 2 ) −1 ≃ k1≥0 I A 2 [>0] (k1) k2≥0 (−1) k2 (k 1 + k 2 )! k 1 !k 2 ! I A 2 [>0] (k2) (λ − D 2 ) −3−k1−k2 I A 2 [>0] · · · (λ − D 2 ) −1 I A 2 [>0] (λ − D 2 ) −1 ≃ k1≥0 I A 2 [>0] (k1) k2≥0 (k 1 + k 2 )! k 1 !k 2 ! I A 2 [>0] (k2) k3≥0 · (−1) k3 (k 1 + k 2 + 1)! (k 1 + k 2 + 1)!k 3 ! I A 2 [>0] (k3) (λ − D 2 ) −4−k1−k2−k3 I A 2 [>0] · · · · · · (λ − D 2 ) −1 I A 2 [>0] (λ − D 2 ) −1 ≃ |k|≥0 c(k 1 , · · · , k j ) I A 2 [>0] (k1) I A 2 [>0] (k2) · · · I A 2 [>0] (kj ) (λ − D 2 ) −|k|−j−1 .
Letting Γ be a contour around the spectrum spec(D 2 ) ⊂ R + , it follows that for any positive integer K:
I A 2 2i−1 2 [K] = K j=0 1 2iπ Γ λ 2i−1 2 (λ − D 2 ) −1 I A 2 [>0] · · · I A 2 [>0] (λ − D 2 ) −1 I A 2 [>0] (λ − D 2 ) −1 [K] dλ = K j=0 |k|≥0 c(k 1 , · · · , k j ) I A 2 [>0] (k1) I A 2 [>0] (k2) · · · I A 2 [>0] (kj ) 1 2iπ Γ λ 2i−1 2 (λ − Q) −|k|−j−1 dλ [K] = K j=0 |k|≥0 2i−1 2 · · · 2i−1 2 − |k| − j (|k| + j)! c(k 1 , · · · , k j ) I A 2 [>0] (k1) I A 2 [>0] (k2) · · · I A 2 [>0] (kj ) D 2 2i−1−|k|−j [K]
where the last equality follows by integration by parts. Applying the Wodzicki residue yields the result of the proposition. ⊔ ⊓ 5 Getzler's rescaling Let E → B be a Ψdo−vector bundle and let {D b , b ∈ B} be a smooth family of elliptic differential operators parametrised by B acting on the fibres of E. Getzler's rescaling transforms a homogeneous form α [i] of degree i to the expression
δ ε · α [i] · δ −1 ε = α [i] √ ε i ,soI A ε = δ ε · I A · δ −1 ε = I A [0] + I A [1] √ ε + I A [2]
ε .
Here we allow higher forms in the superconnection, keeping in mind later applications involving the Bismut superconnection for families of Dirac operators. Following the usual conventions, for a given superconnection I A = I A [0] + I A [1] + I A [2] we set:
I A ε := √ εĨ A ε = √ ε δ ε · I A · δ −1 ε = √ ε I A [0] + I A [1] + I A [2] √ ε .
Remark 7 Different notations are used in the literature, namely some authors set t := √ ε which leads to (see e.g. [L])Ī
A t := I A t 2 = t I A [0] + I A [1] + I A [2] t ,(9)
a notation which we shall also use in this paper.
The following result shows how the j-th Chern (resp. residue-) Chern form picks up the 2j (resp. 2j − 1-) form degree part of str I A 2 I A 2j .
Proposition 5 Let I A be a superconnection associated with a family of elliptic differential operators parametrised by B. Then, with the notations of (9)
str I A 2 I A 2j [2j] = fp t=0 str Ī A 2j t e −Ī A 2 t [2j]
and
sres | I A| 2j−1 [2j−1] = 1 2 √ π fp t=0 str Ī A 2j t e −Ī A 2 t [2j−1]
.
Proof: Recall that since D is a differential operator, so is I A 2 a differential operator valued form and (see Proposition1)
str I A 2 I A 2j = fp ε→0 str I A 2j e −ε I A 2 .
On the other hand, for any t > 0 we have:
str Ī A 2j t e −Ī A 2 t [2j] = str I A 2j t 2 e − I A 2 t 2 [2j] = str t 2jĨ A 2j t 2 e −t 2Ĩ A 2 t 2 [2j] = str t 2j δ t 2 I A 2j δ −1 t 2 e −t 2 δ t 2 I A 2 δ −1 t 2 [2j] = str t 2j δ t 2 I A 2j e −t 2 I A 2 δ −1 t 2 [2j] = str I A 2j e −t 2 I A 2 [2j]
.
Since the r.h. side is of the type str I A 2j e −ε I A 2 with I A 2 an elliptic differential operator valued form and I A 2j ∈ Ω(B, Cℓ(E)), it has a known asymptotic expansion at 0 and taking finite parts when t → 0 yields the first part of the proposition.
Similarly, since D is a differential operator we have sres | I A| 2j−1 = 1 2 √ π fp t=0 t str I A 2i e −t 2 I A 2 and for any t > 0
str Ī A 2j t e −Ī A 2 t [2j−1] = str I A 2j t 2 e − I A 2 t 2 [2j−1] = str t 2jĨ A 2j t 2 e −t 2Ĩ A 2 t 2 [2j−1] = str t 2j δ t 2 I A 2j δ −1 t 2 e −t 2 δ t 2 I A 2 δ −1 t 2 [2j−1] = str t 2j δ t 2 I A 2j e −t 2 I A 2 δ −1 t 2 [2j−1] = t str I A 2j e −t 2 I A 2 [2j−1]
.
As before, since the r.h.s. is of the type str I A 2j e −ε I A 2 with I A 2 an elliptic differential operator valued form and I A 2j ∈ Ω(B, Cℓ(E)), it has a known asymptotic expansion at 0 and taking finite parts when t → 0 yields the first part of the lemma. Taking finite parts when t → 0 therefore yields the second part of the lemma. ⊔ ⊓
Superconnections associated with Dirac operators
We now specialise to Chern forms built from superconnections associated with families of Dirac operators. Let π : I M → B be a smooth fibration of closed spin manifolds with fibre M and I E → B a Clifford bundle with an associated family of Dirac operators parametrised by B. The vector bundle E := π * I E is an (infinite rank) Ψdo−vector bundle with fibres modelled on C ∞ ( I M/B, I E I M/B ). According to whether the manifolds are even or odd dimensional, E will be Z 2 -graded or not. The vertical Riemannian metric g T M and hermitian metric h I E on I E induce an L 2 -inner product on E. From a connection ∇ I E on I E compatible with h I E , one can build a connection∇ E X σ := ∇ I Ẽ X σ(b) on E = π * I E whereX is the horizontal lift of X ∈ T b B and from there a unitary connection ∇ E on E. The corresponding Bismut superconnection associated with this fibration reads:
I A = D + ∇ E + c(T ) even case, I A = σ D + ∇ E + σ c(T ) odd case,
where T ∈ Ω 2 ( I M, T M ) is the curvature of the horizontal distribution on I M and c the Clifford multiplication. Along the lines of the heat-kernel proof of the index theorem we introduce the kernel k ε ( I A 2 ) of e −ε I A 2 for some ε > 0. Since D is a family of Dirac operators, we have (see e.g. chap. 10 in [BGV] in the even dimensional case and [BC] in the odd dimensional case)
k ε ( I A 2 )(x, x) ∼ ε→0 1 (4πε) n 2 ∞ j=0 ε j k j ( I A 2 )(x, x)(10)
Proposition 6 Let I A be a superconnection adapted to a smooth family of Dirac operators parametrised by B.
1. The j-th Chern form associated with I A is given by an integration along fiber of I M :
str I A 2 I A 2j = (−1) j j! (4π) n 2 I M/B
str(k j+ n 2 ( I A 2 )).
2. If the kernel of D has constant dimension, the j-th residue Chern form associated with I A reads:
√ π sres | I A| 2j−1 = (−1) j (2j − 1)!! (4π) n 2 2 j−1 I M/B
str(k j+ n−1 2 ( I A 2 )).
Here | I A| 2j−1 is defined as before by the contour integral
| I A| 2j−1 = i 2π Γ λ 2j−1 2 ( I A 2 + π I A − λ) −1 dλ.
Remark 8 It follows from this last formula that adding a smoothing Ψdo−valued zero form to I A 2 does not affect the residue form sres | I A| 2j−1 [2j−1] so that one expects formulae for the residue Chern form to be independent of π I A .
Proof: The trace under the integral sign is just the matrix trace for endomorphims of a finite rank vector bundle E whereas on the left-hand-side the trace is computed in the Hilbert space of squareintegrable sections of E over M.
By the above remark, I A being a differential operator, it has vanishing Wodzicki residue and we have:
str I A 2 I A 2j = fp ε=0 str I A 2j e −ε I A 2 = (−1) j fp ε=0 ∂ j ε str e −ε I A 2 |ε=0 .
By equation (10), this yields
str I A 2 I A 2j = (−1) j fp ε=0 ∂ j ε str e −ε I A 2 = (−1) j fp ε=0 ∂ j ε I M/B str k ε ( I A 2 ) = (−1) j (4π) n 2 fp ε=0 ∂ i ε ∞ j=0 ε j− n 2 I M/B str(k j ( I A 2 )) = (−1) i (4π) n 2 fp ε=0 ∞ l=0 (l − n 2 ) · · · (l − n 2 − j + 1)ε l− n 2 −j I M/B str(k l ( I A 2 ) = (−1) j j! (4π) n 2 I M/B
str(k j+ n 2 ( I A 2 )).
This proves the first part of the proposition. On the other hand, again by (10) we have: On the other hand, Lemma 1 applied to g(ε) = str I A 2j e −ε I A 2 then yields:
fp ε=0 str √ ε I A 2j e −ε I A 2 = (−1) j fp ε=0 √ ε ∂ j ε str e −ε I A 2 = (−1) j , fp ε=0 √ ε ∂ j ε I M/B str k ε ( I A 2 ) = (−1) j (4π) n 2 fp ε=0 √ ε ∂ j ε ∞ l=0 ε l− n 2 I M/B str(k l ( I A 2 )) = (−1) j (4π) n 2 fp ε=0 ∞ l=0 (l − n 2 ) · · · (l − n 2 − j + 1)ε l− n−1 2 −j I M/B str(k l ( I A 2 )) = (−1) j (j − 1 2 )(j − 3 2 ) · · · 1 2 (4π)fp ε=0 √ ε str I A 2j e −ε I A 2 = Γ( 1 2 ) res z=0 str I A 2j ( I A 2 ) −z− 1 2 = 2 √ π sres I A 2j ( I A 2 ) − 1 2 so that, √ π sres I A 2j I A 2 − 1 2 = (−1) j (2j − 1)!! (4π) n 2 2 j I M/B str(k j+ n−1 2 ( I A 2 ))
which proves the second part of the the proposition. ⊔ ⊓
The following result relates the j-th (resp. residue) Chern form with the 2j-th (resp. 2j − 1-th) form degree part of the Chern character lim t→0 ch( I A t ).
Theorem 2 In the Z 2 -graded case the j-th Chern form associated with a superconnection I A reads:
str I A 2 I A 2j [2j] = (−1) j j! (2iπ) n 2 I M/BÂ ( I M/B) ∧ ch( I E I M/B ) [2j] = (−1) j j! lim t→0 ch( I A t ) [2j]
.
(11)
In the ungraded case the j-th residue Chern form associated with the superconnection with kernel of D of constant dimension reads:
sres | I A| 2j−1 [2j−1] = (−1) j (2j − 1)!! (2iπ) n+1 2 2 j−1 I M/BÂ ( I M/B) ∧ ch( I E I M/B )) [2j−1] = (−1) j (2j − 1)!! 2 j−1 √ π lim t→0 ch( I A t ) [2j−1] .(12)
Proof: As in [BGV] par. 10.4, using the asymptotic expansion of the kernel k t (x, x) of the heatoperator e −t I A 2 :
k t (x, x) ∼ t→0 1 (4πt) n 2 ∞ j=0 t j k j (x, x)
we have:
ch( I A t ) = δ t str(e −t I A 2 ) ∼ t→0 (4πt) − n 2 j t j I M/B δ t str(k j ( I A 2 )) ∼ t→0 (4π) − n 2 j,p t j−(n+p)/2 I M/B str k j ( I A 2 ) [p]
, so that
fp t=0 ch( I A t ) [p] = (4π) − n 2 I M/B str k p+n 2 ( I A 2 ) [p]
.
(13)
• Z 2 -graded case. The family index theorem [B] (see also Theorem 10.23 in [BGV]) yields the existence of the limit as t → 0 and .
Inserting this in Proposition 6 yields and hence (11).
str I A 2 I A 2j [2j] = (−1) j j!(
• Ungraded case. The family index theorem yields the existence of the limit and [BC]. Combined with (13) this yields: .
lim t→0 ch( I A t ) = √ π (2πi)I M/B str k n+2j−1 2 ( I A 2 ) [2j−1] = √ π (4π)
Inserting this in Proposition 6 gives: and hence (12).
sres | I A| 2j−1 [2j−1] = (−1) j (2j − 1)!! (2iπ)
⊔ ⊓
Corollary 1 Whenever the fibration I M → B is trivial, then
tr Q (Ω j ) − (−1) j j! (2iπ) n 2 I M/BÂ ( I M/B) ∧ ch( I E) [2j] =f j (∇).(14)
is local in the sense of the above definition.
Example 3 In the Z 2 -graded case, corresponds to the curvature on the determinant line bundle associated with a family of Dirac operators [BF]. The formula corresponding to j = 1 in the above theorem
str I A 2 I A 2 [2] = − 1 (2i π) n 2 I M/BÂ ( I M/B) ∧ ch( I E I M/B ) [2]
expresses the curvature on the determinant bundle as − 1 (2i) n 2 times the degree 2 part of the first Chern form associated with the superconnection I A, thereby generalising the relation that holds in finite dimensions (corresponding to the case n = 0 of a zero dimensional fibre M ) relating the first Chern form on a finite rank supervector bundle with minus the curvature on its determinant bundle (see [PR] for a discussion concerning this relation).
Example 4 In the ungraded case, I M/BÂ ∧ ch( I E I M/B )) [2j−1] corresponds to the curvature of a gerbe with connection associated with the family of Dirac operators [CM], [EM], [L]. The formula obtained in the above theorem for j = 2 sres I A 2 | I A| [3] = 3 2 (2iπ)
n+1 2 I M/BÂ ∧ ch( I E I M/B )) [3](15)
where we have set | I A| = ( I A 2 ) 1 2 relates this curvature with the degree 3 part of the residue Chern form sres I A 2 | I A| [3] .
Transgressed residue Chern forms
Let as before π : I M → B be a smooth fibration of closed odd dimensional spin manifolds with fibre M and I E → B a Clifford bundle. The vector bundle E := π * I E is an (infinite rank) Ψdo−vector bundle with fibres modelled on C ∞ ( I M/B, I E I M/B ). Let {D b , b ∈ B} be a smooth family of Dirac operators associated with this fibration. We need to work with invertible operators and introduce for this purpose a covering of B by open sets {U λ } λ∈R with the property that for any b ∈ U λ the (discrete) spectrum of D b does not contain λ. Then D(λ) = D − λI is a family of Dirac type operators which is everywhere invertible on U λ and
I A λ := σ D(λ) + ∇ E + σ c(T ) 4
is a superconnection associated with D(λ).
With the notations of (9) we setĪ
A t := t σ D + ∇ E + σ c(T ) 4t ,
where σ is the grading, which defines a smooth family of superconnections adapted to D.
Let for t > 0Ī
A λ,t := σ t D(λ)
+ ∇ E + σ c(T ) 4t ,
which defines a smooth family of superconnections adapted to D(λ). The following technical result will be useful for what follows.
Lemma 2 Let γ be a differential operator valued form on B.
1. The function t → str γ e −Ī A 2 λ,t decreases faster than any power of t as t → ∞. As t → 0 it behaves as a finite linear combination of expressions ∞ j=0 α j t j−δ for some integer δ and complex numbers α j , β k . 2. If γ is an even form, then = 0 ∀ t > 0.
Remark 9
The lemma easily extends to Ψdo−valued forms if we allow for logarithmic divergences in t in which case δ is a real number.
Proof:
1. Let us first introduce notations similar to notations of [H]. Let T ∈ Cℓ(M, E) and T k , k ∈ N be operators in Cℓ(M, E) with decreasing order in k. Then
T ∼ k≥0 T k (16) ⇐⇒ ∃C ∈ Cℓ(M, E) invertible, s.t. ∀N ∈ N, ∃K(N ) T − K(N ) k=0 T k C ∈ Cℓ −N (M, E).
In the sequel, the operator e −t 2 D 2 plays the role of the invertible operator C.
We also need to extend to Ψdo−valued forms, notations previously used for ordinary classical pseudo-differential operators. For β ∈ Ω ((B, Cℓ (E)) and any j ∈ N we set:
α (j) (β) := ad j D 2 (β), where ad D 2 (β) = [D 2 , β], so that β (0) = β, β (j+1) = ad D 2 (β (j) ) = [D 2 , β (j) ]. SinceĪ A λ,t = t σ D(λ) + Ī A λ,t [>0] we havē I A 2 λ,t = t 2 D(λ) 2 + Ī A 2 λ,t [>0] = t 2 D(λ) 2 + σ t [∇ E , D(λ)] + [∇ E , c(T )] 4t + 1 4 [D(λ), c(T )] + c 2 (T ) 16 t 2 + Ω E = t 2 D(λ) 2 + σ Ī A λ,t [>0,od] + Ī A λ,t [>0,ev] where we have set Ī A λ,t [>0,od] := t [∇ E , D(λ)] + [∇ E ,c(T )] 4t
which only involves odd powers of t and t −1 . Duhamel's formula then yields (see e.g. [H]):
e −Ī A 2 λ,t = (−1) n ∆ l e −u0 t 2 D 2 Ī A 2 λ,t [>0] · · · e −u l−1 t 2 D(λ) 2 Ī A 2 λ,t [>0] e −u l t 2 D(λ) 2 du 1 · · · du l ∼ |k|≥0 (−1) |k| t 2|k| c(k) (|k| + n)! Ī A 2 λ,t [>0] (k1) · · · Ī A 2 λ,t [>0] (k l ) e −t 2 D(λ) 2 .
Here ∆ l := {(u 0 , · · · , u l ), u i ≥ 0, l i=0 u i = 1} is the unit simplex and with the coefficient c(k) as previously defined.
Since Ī A 2 λ,t [>0]
has positive degree, only a finite number of terms of the sum will contribute for a fixed form degree. Now, for a differential operator C of order c, the map t → str C e −t 2 D 2 decreases faster than any power of t at infinity and behaves asymptotically as follows as t → 0:
str C e −t 2 D 2 ∼ t→0 ∞ j=0 α j t j−c−n(17)
where c is the order of C and n the dimension of the manifold. Since expressions of the type Ī A 2 λ,t [>0] are linear combinations of differential operator valued forms with coefficients given by powers of t, for any Ψdo−valued form γ on B
str γ Ī A 2 t [>0](k1)
· · · Ī A has the expected asymptotic behaviour at 0 and at ∞. It follows that so does str γ e −Ī A 2 t have a similar asymptotic behaviour. This ends the proof of the first part of the lemma.
2. The second part of the lemma requires a closer look at the expressions involved. Since ,ev] and since str vanishes on terms of the type σβ, in the expression
Ī A λ,t [>0] = σ Ī A λ,t [>0,od] + Ī A λ,t [>0str σ γ e −Ī A 2 λ,t = |k|≥0 (−1) |k| t 2|k| c(k) (|k| + n)! str σ γ Ī A 2 λ,t [>0] (k1) · · · Ī A 2 λ,t [>0] (k l ) e −t 2 D(λ) 2
(which we recall only contains a finite number of terms for fixed form degree) only those terms will remain that involve an odd number of expressions of the type Ī A λ,t [>0,od] and hence odd powers of t and t −1 . Since γ is assumed to be of even degree, it follows that the total expression str σ γ e −Ī A 2 λ,t is an odd degree form which only involves odd powers of t and t −1 so that
fp t=0 str σ γ e −Ī A 2 λ,t = 0; str σ γ e −Ī A 2 λ,t [2j] = 0 ∀t > 0.
This proves the second part of the lemma.
⊔ ⊓
The following theorem provides a transgression formula for the residue Chern forms sres | I A| 2j−1 [2j−1] .
Theorem 3
On every open subset U λ , the following transgression formula holds:
sres | I A λ | 2j−1 [2j−1] = a j · d (η λ ) [2j−2] ,
where d is the exterior differential, a j := (−1) j (2j−1)!! 2 j−1 and
η λ = fp t=0 ∞ t str d dsĪ A λ,s e −Ī A 2 λ,s
ds is the η invariant associated with D(λ) (see [BC], [L]).
Proof: We first show that sres | I A| 2j−1 [2j−1] = sres | I A λ | 2j−1 [2j−1] and then show a transgression formula for sres | I A λ | 2j−1 [2j−1] . 1. Let us consider a smooth family D(λ)(ε) := D − ε λ I of first order elliptic differential operators interpolating D and D(λ) between 0 and 1 and the corresponding superconnections
I A λ,t (ε) := I A λ,t − ε σ λ I.
Differentiating w.r. to ε we have:
d dε str e −Ī A 2 λ,t (ε) = −str [Ī A λ,t (ε), d dεĪ A λ,t (ε)] e −Ī A 2 λ,t (ε) = − str [Ī A λ,t , d dεĪ A λ,t (ε) e −Ī A 2 λ,t (ε) ] = −d str d dεĪ A λ,t (ε) e −Ī A 2 λ,t (ε) = λ d str σ e −Ī A 2 λ,t (ε) .
By part 2 of Lemma 2 applied to γ = I, we find that for any positive integer j
d dε str e −Ī A 2 λ,t (ε) [2j−1] = λ d str σ e −Ī A 2 λ,t (ε) [2j−2]
= 0 as a consequence of which str e −Ī A 2
λ,t (ε) [2j−1]
is actually independent of ε and
str e −Ī A 2 λ,t [2j−1] = str e −Ī A 2 t [2j−1]
.
By formula (12) in Theorem 2, the limit on either side therefore exists as t → 0 and
lim t→0 str e −Ī A 2 λ,t [2j−1] = lim t→0 ch ( I A t ) [2j−1] = (−1) j 2 j−1 (2j − 1)!! sres | I A| 2j−1 [2j−1] .(18)
2. We now derive a transgression formula for str e −Ī A 2
λ,t [2j−1]
, from which will then follow a transgression formula for sres | I A| 2j−1 [2j−1] as a consequence of (18).
d dt str e −Ī A 2 λ,t = −str [Ī A λ,t , d dtĪ A λ,t ] e −Ī A 2 λ,t = −str [Ī A λ,t , d dtĪ A t e −Ī A 2 λ,t ] = −d str d dtĪ A λ,t e −Ī A 2 λ,t .(19)
The first part of Lemma 2 provides a control as t → 0 and as t → ∞ on the asymptotic behaviour of the last expression str d dsĪ A λ,s e −Ī A 2 λ,s arising in (19). Its primitive in t
η λ (t) := ∞ t str d dsĪ A λ,s e −Ī A 2 λ,s
ds which exists as a consequence of the invertibility of D(λ), has a similar asymptotic behaviour as t → 0. Integrating (19) from t to ∞ and taking the finite part as t → 0 we find that the η invariant (we borrow notations from [L], see his formula (3.19))
η λ := fp t=0ηλ (t) = fp t=0 ∞ t str d dsĪ A λ,s e −Ī A 2 λ,s ds trangresses fp t=0 str e −Ī A 2 λ,t : fp t=0 str e −Ī A 2 λ,t = dη λ .
But by the first part of the theorem (see equation (18)), this leads to
sres | I A λ | 2j−1 [2j−1] = (−1) j (2j − 1)!! 2 j−1 d (η λ ) [2j−2] .
Relation to hamiltonian gauge anomalies
We first review a finite dimensional situation which will serve as a model for infinite dimensional genralisations. We consider the finite-dimensional Grassmann manifold Gr(n, n) consisting of rank n projections in C 2n , which we parametrise by grading operators F = 2P − 1, where P is a finite rank projection.
Lemma 3 The even forms
ω 2j = tr F (dF ) 2j ,(20)
where j = 1, 2, . . . are closed forms on Gr(n, n).
Proof: By the traciality of tr we have
d ω 2j = d tr F (dF ) 2j = tr (dF ) 2j+1 = tr F 2 (dF ) 2j+1 since F 2 = 1 = −tr F (dF ) 2j+1 F since F dF = −dF F = −tr (dF ) 2j+1 F 2 since tr([A, B]) = 0 = −tr (dF ) 2j+1 = 0.(21)
⊔ ⊓
In fact it turns out that the cohomology of Gr(n, n) is generated by even (nonnormalized) forms of the type ω 2j , j = 1, · · · , n [MS].
Let us now consider the infinite dimensional geometric setup described in the previous section up to the fact that π : I M = M × B → B is now a trivial fibration with typical fibre a closed (Riemannian)
spin manifold M . On each open subset U λ := {b ∈ B, λ / ∈ spec(D b )} ⊂ B there is a well defined map 5 F : B → Cℓ 0 (M, E) b → F b := (D b − λI)/|D b − λI|. Since F 2 b = F b , P b := I+F b 2
is a projection, the range Gr(M, E) := ImF of F coincides with the Grassmannian consisting of classical pseudodifferential projections P with kernel and cokernel of infinite rank, acting in the complex Hilbert space H := L(M, E). Here L 2 (M, E) denotes the space of squareintegrable sections of the vector bundle E over the compact manifold M. This map b → F b is generally not contractible and we want to define cohomology classes on B as in (11) up to some modifications required by the specific situation. This problem usually arises in hamiltonian quantization in field theory, when M is an odd dimensional manifold, the physical space. Although here we deal with even forms for odd order operators, there is a relation to the previous discussion on odd forms for odd order operators which is explained in the end of this section. The problem here is similar in spirit to the earlier discussion in as far as we want to modify the naive cohomology classes, imitating the finite-dimensional case, by local corrections arising from the infinite dimensionality of the problem. Indeed, in this infinite dimensional setup traces are generally ill-defined, so that we cannot a priori extend the above computation to Gr(M, E). 6 . However, we can define an analog of (20) at the cost of replacing the trace by a weighted trace.
Proposition 7 Let Q ∈ Cℓ(M, E) be a fixed admissible elliptic operator with positive order. The exterior differential of the form
ω Q 2j (F ) = tr Q F (dF ) 2j(22)
on Gr(M, E):
dω Q 2j = 1 2q res [log Q, F ](dF ) 2k+1 F .
is a local expression which only depends on F modulo smoothing operators.
Proof: The locality and the dependence on F modulo smoothing operators follow from the expression of the exterior differential in terms of a Wodzicki residue. To derive this expression, we mimic the finite dimensional proof, taking into account that this time tr Q is not cyclic:
dω Q 2j = dtr Q F (dF ) 2j = tr Q (dF ) 2j+1 = tr Q F 2 (dF ) 2j+1 = −tr Q F (dF ) 2j+1 F since F dF = −dF F = 1 q res [log Q, F ](dF ) 2j+1 F − tr Q (dF ) 2j+1 F 2 = 1 q res [log Q, F ](dF ) 2j+1 F − tr Q (dF ) 2j+1 ,
where we have used (3) to write To justify this, let us first observe that dicontinuities of F give rise to jumps measured by smoothing operators (see e.g. [Me]); indeed, since D b is a smooth family of self-adjoint elliptic operators on a closed manifold, the discontinuities of F b are measured by differences of projections P b,µ over the finite dimensional space generated by eigenvectors of D − λI with eigenvalues in [0, µ] or [µ, 0] according to whether µ is positive or negative. Finite rank projections being smoothing, it follows that the discontinuities are measured by smoothing operators so that the projected map b →F b turns out to be continuous. Hence if B is contractible, the map σ is contractible.
tr Q [F, (dF ) 2j+1 F ] = − 1 q res F [(dF ) 2j+1 F, log Q] = 1 q res [F, log Q] (dF ) 2j+1 F . Hence tr Q F 2 (dF ) 2j+1 = 1 2q res [log Q, F ](dF ) 2j+1 F from
Example 5 A standard example in physics is the case when
• the base space B is a subset of connections in a (hermitian) finite-rank vector bundle E → M over a closed (Riemannian) spin manifold M . Since the space of connections in a fixed vector bundle is an affine space, if B is the whole space of connections, it is contractible and the map σ is indeed contractible.
• the operators D ∇ (written D A with ∇ = d + A in local coordinates) are (twisted) Dirac operators coupled to a connection ∇ ∈ B.
The following theorem builds "renormalised" forms from the original forms ω Q 2j .
Theorem 4 When σ(B) is contractible, there are even forms θ Q 2j such that
ω ren 2j = ω Q 2j − θ Q 2j
is closed. The forms θ Q 2j vanish when the order of (dF ) 2j+1 is less than -dim M . This holds in particular if the order of (dF ) 2j is less than -dim M in which case ω ren 2j = ω Q 2j = tr(F (dF ) 2j ) is independent of Q.
Remark 10 In the case of Dirac operators parametrised by gauge connections the order of (dF ) 2j+1 is less than -dim M for all j > 0 if the dimension of M is less or equal to −3. This known fact is seen by a simple asymptotic expansion of the differential dF . Using any fixed local trivialization of the underlying vector bundle E, we write D ∇ = D + A and F = (D + A) |D + A| −1 where D is the ordinary Dirac operator on R n so that the infinitesimal variation dF coincides up to order 1 in dA with (D + dA) |D + dA| −1 − D |D| −1 . One can check that the square of the operator |D| −1 (1 − 1 2 (D −1 dA + dAD −1 )) is equal to (D + dA) −2 up to operators of order −3. Hence dF = (D + dA)|D
+ dA| −1 = D/|D| − 1 2 |D| −1 D −1 × [dA, D] + .
. . up to operators of order −2. Here one has to take into account that the commutator of |D| with A is of order zero. It follows that D/|D| differs from (D + dA)/|D + dA| by an operator of order -1 so that dF has order −1. This argument fails for the case of families of metrics because the perturbations of Dirac operators are differential operators of order one. It therefore does not extend to the case of Dirac operators parametrised by metrics since in that case the principal symbol depends on the parameters and the differential dF is a zero order operator.
Proof of Theorem 4: The form dω Q 2j being a Wodzicki residue, by Prop. 7, it only depends on the projectionF and is therefore a pull-back by the projection map p of a form β Q 2j . The pull-back of β Q 2j with respect to σ is a closed form θ Q 2j+1 on B which is exact since σ is contractible. Indeed, selecting a contraction σ t with σ 1 = σ and σ 0 a constant map, we have the standard formula for the potential,
dθ Q 2j = θ Q 2j+1 , with θ 2j = 1 2j + 1 1 0 t 2j ισ t θ Q 2j+1 (σ t )dt.(23)
where ι X is the contraction by a vector field X and the dot means differentiation with respect to the parameter t.
When the order of (dF ) 2j+1 is less than -dim M the correction terms θ Q 2j vanish and if the order of (dF ) 2j is less than -dim M , the weighted trace tr Q coincides with the usual trace so that the naive expression ω Q 2j is a closed form independent of Q. ⊔ ⊓
The 2-form case arises in the quantum field theory gerbe [CMM]. Let B be a contractible parameter space for Dirac operators. For each real number λ, let as before U λ ⊂ B be the set of parameters for which the Dirac operator D(λ) = D − λ I does not have λ as an eigenvalue. Denote U λλ ′ = U λ ∩ U λ ′ and let L λλ ′ (A) be the top exterior power of the spectral subspace E λλ ′ defined by λ <
D A < λ ′ for A ∈ U λλ ′ . The complex lines L λλ ′ (A) form a complex line bundle L λλ ′ over U λλ ′ . For λ < λ ′ < λ ′′ we have the canonical identification L λλ ′ ⊗ L λ ′ λ ′′ = L λλ ′′ .(24)
This family of line bundles defines a gerbe over B. Since B is contractible this gerbe is trivial in the sense that
L λλ ′ = L λ ⊗ L * λ ′(25)
for some line bundles L λ → U λ . The curvature of L λλ ′ is 1/2π times
ω 2 λλ ′ = tr P (λλ ′ )(dP (λλ ′ )) 2(26)
where P (λλ ′ ) is the projection onto E λλ ′ . Denote as before by F (λ) the grading operator (D A − λ)/|D A − λ| on U λ and let P (λ) = 1 2 (F (λ) + 1) be the corresponding spectral projection. In the Hilbert-Schmidt case, when the grading operators are in Gr res (M, E) one proves by a direct computation that
ω 2 λλ ′ = ω 2 λ − ω 2 λ ′(27)
on U λλ ′ with ω 2 λ = 1 8 tr F (λ)(dF (λ)) 2 = tr P (λ)(dP (λ)) 2 .
Remark 11 This last equality follows from the cyclicity of the trace on Hilbert-Schmidt operators. Indeed, since d P (λ) P (λ) + P (λ) d P (λ) = d P (λ) we have tr F (λ)(dF (λ)) 2 = 8 tr P (λ)(dP (λ)) 2 − 4 tr (dP (λ)) 2 = 8 tr P (λ)(dP (λ)) 2 − 4 dtr (P (λ) dP (λ)) = 8 tr P (λ)(dP (λ)) 2 + 4 dtr (dP (λ) P (λ)) = 8 tr P (λ)(dP (λ)) 2 − 4 dtr (dP (λ) P (λ)) = 8 tr P (λ)(dP (λ)) 2 .
In the general case the forms ω 2 λ have to be replaced by the 'renormalised' forms as in Theorem 4. However, we still have Theorem 5 The cocycle of forms ω 2 λλ ′ is trivialized by 1 8 times the forms
ω 2 λ = tr Q F (λ)(dF (λ)) 2 − θ Q 2(29)
or equivalently by 1 8 times the forms
ρ 2 λ = 8 tr Q P (λ)(dP (λ)) 2 − θ Q 2
where θ Q 2 is as in Theorem 4 (with j = 1), restricted to the open set U λ . In particular,
ω 2 λ − ω 2 λ ′ = ρ 2 λ − ρ 2 λ ′ = tr F (λ)(dF (λ)) 2 − F (λ ′ )(dF (λ ′ )) 2 = tr P (λ)(dP (λ)) 2 − P (λ ′ )(dP (λ ′ )) 2 .(30)
Proof: First observe that the correction terms θ Q 2 arising in the differences of the forms ω 2 λ on the intesections B λλ ′ cancel: they do not depend on the parameter λ since a change of λ gives rise to finite rank perturbations of F (λ) and hence to smoothing pertubrations on which the Wodzicki residue vanishes. Let us show first (30). For λ < λ ′ we have tr Q (dP (λ)) 2 − tr Q (dP (λ ′ )) 2 = tr Q (dP (λ)) 2 − (dP (λ ′ )) 2 = tr Q (dP (λλ ′ )) 2 + dP (λ ′ )dP (λλ ′ ) + dP (λλ ′ )dP (λ ′ ) = tr (dP (λ ′ )dP (λλ ′ ) + dP (λλ ′ )dP (λ ′ )) = 0, since tr(dP ) 2 = 0 for any finite rank projector and by cyclicity of the ordinary trace, from which it follows that tr Q F (λ)(dF (λ)) 2 − tr Q F (λ ′ )(dF (λ ′ )) 2 = 8 tr Q P (λ)(dP (λ)) 2 − tr Q P (λ ′ )(dP (λ ′ )) 2 .
To show (29) we expand ω λ 2 in powers of the difference projection P (λλ ′ ) and observe that the zeroth order term is equal to ω λ ′ 2 , the third order term is ω λλ ′ 2 . The mixed terms are ordinary traces, since the operators contain the finite rank projector P (λλ ′ ) as a factor; using the cyclicity of the trace and repeatedly dP P = P dP = dP for any projector and dP P ′ + P dP ′ = 0 for any pair of mutually orthogonal projectors, we get ω λ 2 − ω λ ′ 2 = tr (P (λλ ′ )dP (λλ ′ )dP λλ ′ ) + tr (P (λ ′ )dP (λ ′ )dP (λλ ′ )) + + tr (P (λ ′ )dP (λλ ′ )dP (λ ′ )) + tr (P (λ ′ )dP (λλ ′ )dP (λλ ′ )) + tr (P (λλ ′ )dP (λ ′ )dP (λ ′ )) + tr (P (λλ ′ )dP (λ ′ )dP (λλ ′ )) + tr (P (λλ ′ )dP (λλ ′ )dP (λ ′ ))
= ω λλ ′ 2 − 3tr ((1 − P (λ ′ ) − P (λλ ′ ))dP (λ ′ )dP (λλ ′ )) .
Next for any triple of mutually orthogonal projectors one has tr (P dP ′ dP ′′ ) = 0, again by the above mentioned operator identities; applying this to P = 1 − P (λ ′ ) − P (λλ ′ ), P ′ = P (λ ′ ) and P ′′ = P (λλ ′ ) we see that the mixed terms on the right-hand-side of the above equation vanish and the Theorem follows. ⊔ ⊓ Remark 12 Actually, this is just the degree 2 cohomology part of the statement that the Chern characters for direct summand in vector bundles add up to the Chern character of the sum; for the chosen curvature forms the statement is valid on the level of de Rham forms, not just for de Rham classes.
The forms ω λ 2 are related but not equal to the gerbe eta forms studied in [L]. Let G be the group of smooth based group gauge transformations acting on smooth sections C ∞ (M, E) as unitary operators. On the base B = A/G equipped with the open cover V λ = π(U λ ) we have well-defined eta forms η λ such that dη λ = Ω 3 , where Ω 3 is the Dixmier-Douady 3-cohomology class classifying the gerbe on the base B. (Here we ignore possible torsion in cohomology and work with de Rham representatives.) Let π : A → A/G be the canonical projection where G is the gauge group. The pull-back π * (Ω 3 ) is exact, dΘ = π * (Ω 3 ). These are related to ω λ 2 , as cohomology classes, by
[ω λ 2 ] = [Θ − π * (η λ )].
The above relation holds only in cohomology, not as a relation of forms. This is related to the fact that the difference of the pull-back forms π * (η λ ) must vanish in gauge directions whereas the difference ω λ 2 − ω λ ′ 2 = ω λλ ′ 2 is nonvanishing even in gauge directions. However, we have
Proposition 8 Restricted to gauge orbits, the forms ω λλ ′ 2 vanish as cohomology classes. More precisely, ω λλ ′ 2 = dω λλ ′ 1 on gauge orbits, where ω λλ ′ 1 (X) = −tr ( XP (λλ ′ )) .
Here we can use ordinary trace since P (λλ ′ ) has finite rank; X is an element of the Lie algebra of G acting as multiplication operator in the Hilbert space H.
Proof: The gauge group acts on projections by conjugation P → gP g −1 and so where in the last step we have used the projection property P 2 = P and the cyclicity of trace for finite rank operators; L X is the Lie derivative by vector field along gauge orbits corresponding to the conjugation action of the group G. ⊔ ⊓ Remark 13 A similar modification can be made for the gauge action on the local forms ω λ 2 to show that the action is consistent on overlaps. Again, restricting to gauge orbits, using F 2 = 1 and rearranging terms, one can write where on the right the first term is a trivial cocycle and the rest, being a residue, does not depend on finite rank perturbations and in particular not on the parameter λ.
(dω λλ ′ 1 )(X, Y ) = L X ω λλ ′ 1 (Y ) − L Y ω λλ ′ 1 (X) − ω λλ ′ 1 ([X, Y ]) = −
Here we have only discussed the cocycles of degree 2 because they are the most relevant in gauge theory; it is clear that similar computations can be performed with the higher cocycles. The 2-forms ω λ 2 are directly 'seen' in quantum field theory in the following way. These forms appear as curvature forms of local vacuum line bundles for fermion field in gauge background, [CMM, EM]. The gauge action on gauge connections lifts to an action of an extension of the group of gauge transformations on the local line bundles. On the Lie algebra level, the 2-cocycle describing the Lie algebra extension("hamiltonian anomaly" [Mi]) is just the curvature form evaluated in the gauge directions. In the case of fields in one space dimension, this extension (for a simple compact gauge group) defines an affine Kac-Moody algebra.
For a family of Dirac operators associated with a general fibration of spin manifolds π : I M → B, on the grounds of the family index theorem, we identify Chern forms associated with the superconnection with form components of I M/BÂ ( I M/B) ∧ ch( I E I M/B ) where I E → I M is a vector bundle over I M. The j-th Chern form associated with a superconnection I A introduced in [PS1] (following ideas of[Sc]) has 2j-form part (see Theorem 2
Similarly, d sres (P (| I A|)) = sres ([ I A, P (| I A|)]) = 0 since [ I A, log P (| I A|)] = 0. The forms are therefore closed. Let us check that their de Rham classes are independent of the choice of superconnection. Let I A t be a smooth one parameter family of superconnections then for any monomial P
(
I A 2 )).
lim t→0 ch( I A t ) = (2iπ) − n 2 I M/BÂ ( I M/B) ∧ ch( I E I M/B ).
) ∧ ch( I E I M/B ).
odd powers of t and t −1 . In particular, fp t=0 str σ γ e −Ī
which the result then follows. ⊔ ⊓ Let us consider the map σ : B → Cℓ 0 (M, E)/Cl −∞ (M, E) b →F (b) := p • F (b) where p : Cℓ 0 (M, E) → Cℓ 0 (M, E)/Cl −∞ (M, E) is the canonical projection map. In (quantum field theoretic) applications the map b → σ(b) =F b can be contractible without the map b → F b being contractible, a situation which can occur when B is contractible.
tr (Y [P (λλ ′ ), X]) + tr (X[P (λλ ′ ), Y ]) + tr ([X, Y ]P (λλ ′ )) = −tr ([X, Y ]P (λλ ′ )) = tr (P (λλ ′ )[[P (λλ ′ ), X], [P (λλ ′ ), Y ]]) = ω λλ ′ 2 (X, Y ),
An operator Q ∈ Cℓ(M, E) of positive order is called admissible if there is a proper subsector of C with vertex 0 which contains the spectrum of the leading symbol σ L (Q) of Q. Then there is a half line L θ = {r e iθ , r > 0} (a spectral cut) with vertex 0 and determined by an Agmon angle θ which does not intersect the spectrum of Q. If Q is invertible, thenL θ = {r e iθ , r ≥ 0} does not intersect the spectrum of Q.
t [>0] (kn) e −t 2 D 2
Note here again, the difference in conventions compared to[L]. We follow[CMM],[CM], whereas in[L] the operators D b are perturbed as D b → D b + h(D b ) by a smoothing function h to avoid the zero modes.6 Unless we restrict to the submanifold Grres(M, E) ⊂ Gr(M, E) consisting of points F such that F − ǫ is Hilbert-Schmidt for some fixed point ǫ ∈ Gr(M, E)[PS]
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[
"Trisections on Certain Rational Elliptic Surfaces and Families of Zariski Pairs Degenerating to the same Conic-line Arrangement",
"Trisections on Certain Rational Elliptic Surfaces and Families of Zariski Pairs Degenerating to the same Conic-line Arrangement"
]
| [
"S Bannai ",
"N Kawana ",
"R Masuya ",
"H Tokunaga "
]
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| In this paper, we study the geometry of trisections on certain rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to construct trisections and related plane curves with interesting properties explicitly. As a result we are able to construct new examples of Zariski pairs. Especially, we show the existence of a family of Zariski pairs that degenerate to the same conic-line arrangement. | 10.1007/s10711-021-00672-5 | [
"https://arxiv.org/pdf/2103.07639v1.pdf"
]
| 232,233,040 | 2103.07639 | baf9705360612653573839c1e1d6b098cd84bbfa |
Trisections on Certain Rational Elliptic Surfaces and Families of Zariski Pairs Degenerating to the same Conic-line Arrangement
13 Mar 2021 March 16, 2021
S Bannai
N Kawana
R Masuya
H Tokunaga
Trisections on Certain Rational Elliptic Surfaces and Families of Zariski Pairs Degenerating to the same Conic-line Arrangement
13 Mar 2021 March 16, 2021arXiv:2103.07639v1 [math.AG]
In this paper, we study the geometry of trisections on certain rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to construct trisections and related plane curves with interesting properties explicitly. As a result we are able to construct new examples of Zariski pairs. Especially, we show the existence of a family of Zariski pairs that degenerate to the same conic-line arrangement.
Introduction
Let ϕ : S → C be an elliptic surface over a smooth projective curve C satisfying the conditions as follows:
(i) The morphism ϕ is relatively minimal.
(ii) There exists at least one section O : C → S. We identify O with its image, i.e., a curve on S meeting any fiber of ϕ at one point.
(iii) There exists at least one singular fiber.
Under these assumptions, the Néron-Severi group NS(S) of S is finitely generated and torsionfree by [17,Theorem 1.2]. Let MW(S) be the set of sections of ϕ : S → C. As O ∈ MW(S), MW(S) = ∅. Note that for a section s : C → S, we identify s with its image on S. Let E S be the generic fiber of ϕ : S → C. E S can be regarded as a curve of genus 1 over the rational function field, C(C), of C. Under our setting, S is known as the Kodaira-Néron model of E S and we can identify the set of C(C)-rational points E S (C(C)) of E S with MW(S). Here we define the zero of E S (C(C)) to be the C(C)-rational point given by the restriction O to E S . We also denote it by O for simplicity. For P ∈ E S (C(C)), we denote the corresponding section by s P .
Let D be a divisor on S and let d D be the restriction of D to E S . d D is a divisor on E S defined over C(C). By Abel's theorem, we have P D ∈ E S (C(C)) from d D and the corresponding section s PD which we also denote by s(D). This relates NS(S) with E S (C(C)) and as is shown in [17], we have where T ϕ is a subgroup of NS(S) generated by O and irreducible curves contained in fibers. (A more detailed description for T ϕ can be found in Section 1.)
In [17,Lemma 8.2], Shioda defined a homomorphism φ : E S (C(C)) → NS(S) ⊗ Q by which he defined a pairing •, • on E S (C(C)) called the height pairing and went on to define a lattice structure on the free part of E S (C(C)), namely the Mordell-Weil lattice. In [3,Section 2], the first and the last author defined a homomorphism φ o : Div(S) → NS(S) ⊗ Q such that φ(P D ) = φ o (D). With φ, φ o and Theorem 1, the geometric relation between D and P D can be investigated.
In this article, we consider curves whose classes are mapped to a given P ∈ E S (C(C)) in the case when S is rational, in which case C = P 1 . We need to introduce some terminologies, before we go on to explain our problem.
For a divisor D on S, we say that D is vertical if all irreducible components are contained in fibers. D is said to be horizontal if D contains no vertical components. Note that an 1-section is just an ordinary section. We call D a bisection or trisection if m = 2 or = 3, respectively.
The purpose of this article is to study trisections of S to find plane curves in which we are interested, in particular curves that are interesting from the viewpoint of the embedded topology of plane curve arrangements with irreducible components of low degree. The first and the last authors have been investigating bisections and trisections along this line when S is a certain rational elliptic surface associated to a reduced plane quartic Q and a smooth point z o . In order to make our problem clear we explain how we obtain such a surface more precisely.
Let Q be a reduced quartic which is not the union of 4 concurrent lines. Let z o be a smooth point of Q. We associate a rational elliptic surface, S Q,zo , to Q and z o as follows:
(i) Let f ′ Q : S ′ Q → P 2 be the double cover branched along Q. We denote its canonical resolution by µ : S Q → S ′ Q (see [10] for the canonical resolution). Note that S Q is a rational surface.
(ii) Let Λ zo be the pencil of curves of genus 1 on S Q induced from the pencil of lines through z o . The pencil Λ zo has a unique base point f −1 Q (z o ) of multiplicity 2. Let ν zo : S Q,zo → S Q be the resolution of the indeterminacy for the rational map given by Λ zo . The induced morphism ϕ Q,zo : S Q,zo → P 1 is an elliptic fibration. ϕ Q,zo satisfies the three conditions at the beginning of the Introduction. Thus we have a rational elliptic surface S Q,zo and the diagram below:
S ′ Q µ ← −−− − S Q νz o ← −−− − S Q,zo f ′ Q fQ fQ,z o P 2 ← −−− − q P 2 ← −−− − qz o ( P 2 ) zo ,
where q is a composition of a finite number of blowing-ups so that the branch locus becomes smooth and q zo is a composition of two blowing-ups. The map f Q,zo : S Q → ( P 2 ) zo is the induced double cover which coincides with the quotient morphism determined by the involution [−1] ϕQ,z o on S Q,zo , which is given by the inversion with respect to the group law on the generic fiber.
(iii) The surface ( P 2 ) zo can be blown down to Σ 2 such that the proper transform of Q is mapped to the trisection T SQ,z o as in Section 5.3. Let l zo be the tangent line at z o . By our construction, ϕ Q,zo has a singular fiber F zo containing irreducible componets mapped to l zo . The type of F zo is determined by the residual intersections between l zo and Q. For details on the types of singular fibers, see [15].
Given m-sections D i and the corresponding sections
s i := s PD i (i = 1, . . . , r), we have plane curves C Di := f Q,zo (D i ), C si ( or C PD i ) := f Q,zo (s i ) (i = 1, . . . , r)
. The embedded topology of unions of Q and these curves are of interest. The first and the last authors have previously studied the following cases:
• In [4], they considered the case where Q is a union two general conics. They make use of bisections D i (i = 1, . . . r) such that C Di (i = 1, . . . , r) are conics in order to construct Zariski N -tuples for conic arrangements.
• In [5], they consider the case where Q is an irreducible quartic with two nodes. They make use of bisections D i (i = 1, . . . , r) such that C Di (i = 1, . . . , r) are conics in order to give a Zariski 5-tuple for Q and two conics of this form.
• In [3], they go on to consider trisections of S Q,zo when Q is a union of general four lines. In this case, E SQ,z o (C(t)) ∼ = A * 1 ⊕ (Z/2Z) ⊕2 . In [3], a generator P o for the A * 1 part is chosen and P D for certain trisections D are explicitly computed in terms of P o and the torsion part. As an application, a result given in [7] was refined.
See [1] for definitions of Zariski pairs/triples.
In this article, we continue to study trisections of S Q,zo which are obtained as follows: Let E be a smooth cubic such that the intersection number i p (E, Q) is even for every p ∈ Q ∩ E and also z o ∈ E. In S Q,zo , E gives rise to either an irreducible curveẼ which is a 6-section or two irreducible components E ± , both of which are trisections of ϕ Q,zo . In the former case, PẼ = O and in the latter case, [−1]P E + = P E − and C E ± = E. Note that if there exists p ∈ Q ∩ E with i p (E, Q) being odd, then E gives rise to an an irreducible curve in S Q,zo . We apply the methods introduced in [3] in order to compute P E ± . In fact, we have the following: Proposition 2. Let Q be a nodal quartic curve, (i.e. a possibly reducible quartic curve whose singularities are at most nodes), which is not a union of four lines. Let z o ∈ Q be a general point on a non-linear component of Q. Let E ± be trisections obtained as above. Then the value of the height pairing of P E ± on S Q,zo is given by
P E + , P E + = P E − , P E − = 3 2 − r 2 ,
where r is the number of nodes of Q that E passes through. In particular, r ≤ 3.
We go on to apply this result in the case when Q consists of components of small degree. We will consider the cases where Q is (i) a union of a smooth conic C o and two lines L 1 and L 2 all meeting transversely, or (ii) a union of two smooth conics C 1 and C 2 meeting transversely, and z o is a smooth point on a conic component. We introduce the following notation for each case:
Case (i): L 1 ∩ L 2 = {p 0 }, L 1 ∩ C o = {p 1 , p 2 } and L 2 ∩ C o = {p 3 , p 4 }. Case (ii): C 1 ∩ C 2 = {p 1 , p 2 , p 3 , p 4 }.
In these cases, we have
E SQ,z o (C(t)) ∼ = (A * 1 ) ⊕2 ⊕ (Z/2Z) ⊕2 for (i) (A * 1 ) ⊕3 ⊕ (Z/2Z) for (ii).
As the first and last authors have considered the case where E SQ,z o (C(t)) ∼ = A * 1 ⊕ (Z/2Z) ⊕2 , it seems to be natural to study the above two cases as a next step from the viewpoint of E SQ,z o (C(t)), and our goal of this article is to give new examples that present new phenomena in the study of the embedded topology of plane curves.
In what follows, we assume that Q is one of the above (i) and (ii) unless otherwise is stated. We have the following for C P E ± : Proposition 3. Under the notation given above, we have:
(i) Suppose that Q = L 1 + L 2 + C o . Then r = 1, 2 or 3. More precisely, letting P = {p 0 , p 1 , p 2 , p 3 , p 4 }, we have the following:
(i-1) If r = 1, then E ∩ P = {p 0 } and C P E ± is a line through p 0 and is tangent to C.
(i-2) If r = 2, then E ∩ P = {p i , p j }, i ∈ {1, 2}, j ∈ {3
, 4} and C P E ± is the line through p i and p j .
(i-3) If r = 3, then E ∩ P = {p 0 , p 1 , p 2 } or {p 0 , p 3 , p 4 } and C P E ± = L 1 if E ∩ P = {p 0 , p 1 , p 2 }, L 2 if E ∩ P = {p 0 , p 3 , p 4 }.
(ii) Suppose that Q = C 1 +C 2 . Then r = 0 or 2. More precisely, letting P = {p 1 , p 2 , p 3 , p 4 }, we have the following:
(ii-1) If r = 0, then C P E ± is a bitangent line to Q.
(ii-2) If r = 2, then E ∩ P = {p i , p j } and C P E ± is the line through p i and p j .
As applications of Proposition 3, we construct Zariski pairs/triples for cubic-conic-line arrangements in Section 4.3. In particular, for the following example based on Proposition 3, (ii-1), it may be worthwhile to state it here as a theorem:
Theorem 2. Let Q = C 1 + C 2 as in Proposition 3 (ii). There exist two families of plane curves {B i,s } s∈∆ǫ (i = 1, 2) over ∆ ǫ = {s ∈ C | |s| < ǫ} for sufficiently small ǫ > 0, such that In particular there exists a family of Zariski pairs which degenerate to the same conic-line arrangement.
Remark 0.1. The existence of such families given in Theorem 2 was expected, but the significance of the above Theorem is in giving a simple explicit example.
Our new key tool to construct curves as above is 'the Mumford representation of semireduced divisors' on E SQ,z o . This makes it possible for us to treat multi-sections concretely. The Mumford representation was first considered in [16] in order to describe the Jacobian of hyperelliptic curves explicitly, based on Jacobi's ideas. It has been exploited in the study of hyperelliptic curve cryptography. Our study can be regarded as a new application of Mumford representations.
This article consists of 6 sections. In Section 1 we first summarize previous results which we need later, and prove Proposition 2. In Section 2, we explain two rational elliptic surfaces which play importan roles to prove Proposition 3 and Theorem 2. Propositon 3 will be proven in Section 3. We explain the 'splitting type' introduced in [2] in Section 4. The notion of splitting type plays a key role in order to distinguish the embedded topology of our examples and we prove Theorem 2 (iii) assuming the existence of E i,s (i = 1, 2). In Section 5, we explain the Mumford representation of semi-reduced divisors. In Section 6, we show the existence of plane curves with the desired properties and construct E i,s (s = 0) in Theorem 2 (i) explicitly based on the method given in Section 5.
1 Elliptic surfaces 1.1 Some terminologies and notation for elliptic surfaces We here define some notation and terminologies which are necessary for our later argument. For general references we refer to [12,14,17]. As for bisectios see [4]. Let ϕ : S → C be an elliptic surface over a smooth projective curve C. Throughout this article, we always assume that S satisfies the three conditions in the Introduction. We introduce two subsets Sing(ϕ) and Red(ϕ) concerning singular fibers as follows, where F v = ϕ −1 (v) for v ∈ C:
Sing(ϕ) := {v ∈ C | F v is not a curve of genus 1.}, Red(ϕ) := {v ∈ Sing(ϕ) | F v is reducible.}.
For v ∈ Red(ϕ), m v denotes the number of irreducible components and we denote the irreducible decomposition of F v by
F v = Θ v,0 + mv−1 i=1 a v,i Θ v,i , where Θ v,0 is the unique component satisfying Θ v,0 · O = 1.
We call Θ v,0 the identity component. We use Kodaira's symbol ( [12]) in order to describe the types of singular fibers. Irreducible components of singular fibers are labeled as in [22, pp. 81-82]. We denote the set of sections of ϕ : S → C by MW(S), which is not empty by our assumption. The set MW(S) is endowed with an abelian group structure by considering fiberwise addition with O as the zero element. Let E S be the generic fiber of ϕ : S → C. Then E S can be regarded as a curve of genus 1 over the rational function field, C(C), of C. Under our setting, S is known as the Kodaira-Néron model of E S and we can identify the set of C(C)-rational points
E S (C(C)) of E S with MW(S).
Here the zero of E S (C(C)) is the C(C)-rational point given by the restriction of O to E S . We also denote it by O for simplicity.
We denote the Néron-Severi group of S by NS(S), which is finitely generated and torsionfree by [17,Theorem 1.2]. Following [17], we define the subgroup T ϕ of NS(S) generated by O, F : a general fiber and
Θ v,i (1 ≤ i ≤ m v − 1, v ∈ Red(ϕ))
. For a divisor D on S, the restriction d D of D to E S is a divisor defined over C(C). In [17], Shioda defined a structure on the free part of E S (C(C)) called the height pairing, which we denote by •, • . We use properties of •, • freely. For details, see [17].
For a reducible singular fiber [14, p. 70], which describes at which irreducible component a given section meets F v . This γ coincides with the maps considered in [3] and [21, p. 83].
F v (v ∈ Red(ϕ)), R v denotes the subgroup of T ϕ generated by Θ v,i (1 ≤ i ≤ m v −1). We denote its dual by R ∨ v . The map γ : MW(S) → ⊕ v∈Red(ϕ) R ∨ v /R v is the homomorphism given in
The height pairing and the homomorphism φ o
In this section we review the homomorphism φ o : Div(S) → NS(S) ⊗ Q considered in [3, Section 2] and its relation between φ o (D) and the height P D , P D . Put NS(S) Q := NS(S)⊗Q.
For D ∈ Div(S), we define c(v, D) to be c(v, D) := D · Θ v,1 . . . D · Θ v,mv−1 and put F v = [Θ v,1 , . . . , Θ v,mv−1 ]. By [17, Lemma 5.1], we have ( * ) D ≈ s(D) + (d − 1)O + nF + v∈Red(ϕ) F v A −1 v (c(v, D) − c(v, s(D)),
where ≈ denotes algebraic equivalence between divisors, and d and n are integers defined as follows:
d = D · F, n = (d − 1)χ(O S ) + O · D − s(D) · O. Furthermore, A v is the intersection matrix (Θ v,i · Θ v,j ) 1≤i,j≤mv −1 . Since the entries of A −1 v are not necessarily integers, the condition A −1 v (c(v, D) − c(v, s(D)
)) ∈ Z ⊕mv −1 imposes some restriction at which irreducible components of F v , D and s(D) meet. For example we have the following Lemma: (i) For P ∈ E S (C(C)) and its corresponding section s P , we have φ(P ) = φ o (s P ).
Lemma 1.1. If F v is a singular fiber of type I 2 , c(v, D) − c(v, s(D)) is even (Note that c(v, D) becomes an integer in this case). Let φ o : Div(S) → NS(S) Q be the homomorphism introduced in [3] whose explicit form is ( * * ) φ o (D) = D − dO − (dχ(O S ) + O · D)F − v∈Red(ϕ) F v A −1 v c(v, D). Let φ : E S (C(C)) → NS(S) Q be
(ii) For D ∈ Div(S) and its corresponding point
P D ∈ E S (C(C)), we have φ(P D ) = φ o (D). Corollary 1.3. For divisors D 1 , D 2 , we have P D1 , P D2 = −φ o (D 1 ) · φ o (D 2 ).
Proof. As P D1 , P D2 = −φ(P D1 ) · φ(P D2 ), our statement is immediate from Lemma 1.2.
By Corollary 1.3, P D1 , P D2 can be computed by the geometric data of D 1 and D 2 . Also, by calculating φ o (D 1 ) · φ o (D 2 ) using ( * * ), we have
( * * * ) φ o (D 1 )·φ o (D 2 ) = D 1 ·D 2 −d 2 D 1 ·O−d 1 D 2 ·O−d 1 d 2 χ(O S )− v∈Red(ϕ) t c(v, D 1 )A −1 v c(v, D 2 ), where d i = D i · F (i = 1, 2). In particular, if D 1 = D 2 , we have φ o (D 1 ) · φ o (D 1 ) = D 2 1 − 2d 1 D 1 · O − d 2 1 χ(O S ) − v∈Red(ϕ) t c(v, D 1 )A −1 v c(v, D 1 ).
Furthermore we have the following lemma which we will use later.
P D1 , P D2 = −φ o (D 1 )·φ o (D 2 ) = − D 1 · D 2 − d 1 d 2 χ(O S ) + 1 2 v∈Red(ϕ) t c(v, D 1 )c(v, D 2 ) .
Proof. Since the reducible singular fibers are all of type I2, we have
A −1 v = − 1 2 for all v ∈ Red(ϕ). Also, since D i (i = 1, 2) are integral trisections, we have D i · O = 0 and d i = D i · F = 3 (i = 1, 2)
. Then our result is obtained directly from ( * * * ).
The proof of Proposition 2
In this subsection we prove Proposition 2. Let Q be a nodal quartic curve which is not a union of four lines and let z o ∈ Q be a general point on a non-linear component of Q. By these assumptions, S Q,zo is a rational elliptic surface whose reducible singular fibers are all of type I2 or III. Note that one reducible singular fiber F ∞ corresponds to the tangent line L ∞ of Q at z o and the other singular fibers correspond to the lines passing through z o and a node of Q.
Let E be a smooth cubic curve such that the intersection number i p (E, Q) is even for every p ∈ Q ∩ E and z o ∈ E. We assume that E give rises to two trisections E ± in S Q,zo . Note that since z o ∈ E, E ± become integral trisections. Let r be the number of the nodes of Q that E passes through. Let E ⊂ ( P) zo be the strict transform of E under the compositions of blow-ups q zo • q. Then we have E · E = 9 − r since E is blown-up once at each of the r
nodes. Furthermore, since f * Q,zo ( E) = E + + E − , we have (E + + E − ) 2 = 2 E · E = 2(9 − r) and since (E + ) 2 = (E − ) 2 , we obtain (E + ) 2 + E + · E − = 9 − r.
On the other hand, since
E + · E − = 1 2 Q · E, where Q is the strict transform of Q in S Q,zo , we have E + · E − = 6 − r. Hence we have (E ± ) 2 = 3. Next, we consider t c(v, E ± ). Let F ∞ = Θ ∞,0 + Θ ∞,1 , where Θ ∞,0 is the pre-image under f Q,
zo of the exceptional divisor of the last blow-up in q zo • q, and Θ ∞,1 is the pre-image of the strict transform of the tangent line L ∞ . Then since E · L ∞ = 3, we have c(∞, E ± ) = [3] and t c(∞, E ± )c(∞, E ± ) = 9. Next, let F v be a reducible fiber corresponding to a node v of Q.
Then F v = Θ v,0 + Θ v,1 , where Θ v,0 is the pre-image under f Q,zo of the strict transform L v of the line L v through z o and v, and Θ v,1 is the pre-image E v of the strict transform of the exceptional divisor E v of the blow-up at v. Then since E · E v = 1 (v ∈ E) 0 (v ∈ E)
we have
c(v, E ± ) = 1 (v ∈ E) 0 (v ∈ E).
Now since all of the reducible singular fibers of S Q,zo are of type I2 and E ± are integral trisections, we can apply Lemma 1.4 and obtain
P E + , P E + = P E − , P E − = − (E + ) 2 − 9χ(O SQ,z o ) + 1 2 v∈Red(ϕ) t c(c, E + )c(c, E + ) = − 3 − 9 + 1 2 (9 + r) = 3 2 − 1 2 r
which proves Proposition 2.
Two rational elliptic surfaces for the cases (i) and (ii)
In this section, we give rather detailed descriptions of rational elliptic surfaces for the cases (i) and (ii) in the Introduction. Both of them were considered in [22] and explicit models for both cases were given. As for the translation by a 2-torsion, see [14, Lecture VII] or [21].
The case (i): A rational elliptic surface attached to a conic and two lines
Let Q = C o + L 1 + L 2 be as in the Introduction and we keep our notation there. Take a smooth point of C o as z o and let S Q,zo be the rational elliptic surface associate to Q and z o and we denote its elliptic fibration by ϕ Q,zo : S Q,zo → P 1 . As we have seen in [22], S Q,zo satisfies the following:
(i) The elliptic fibration ϕ Q,zo has 6 singular fibers of type I2 or III and no other singular fiber. They arise from the tangent line at z o and lines connecting z o and p i (i = 0, 1, . . . , 4). We denote them by F ∞ and F i (i = 0, 1, . . . , 4), respectively and their irreducible decompositions by (iv) For each P ij , C [2]Pij is a conic inscribed by Q.
F • = Θ •,0 + Θ •,1 (• = ∞, 0, 1, . . . , 4). (ii) The group E SQ,z o (C(t)) is isomorphic to (A * 1 ) ⊕2 ⊕ (Z/2Z) ⊕2 . (iii) Let L ij (i < j)
2.2
The case (ii): A rational elliptic surface attached to two coincs
General properties for case (ii)
Let Q = C 1 + C 2 be as in the Introduction and we keep the notation in the Introduction. Take a smooth point z o of Q such that l zo is not a bitangnet line to Q. The rational elliptic surface ϕ Q,zo : S Q,zo → P 1 associate to Q and z o satifies the following properties: (ii) The group E SQ,z o (C(t)) is isomorphic to (A * 1 ) ⊕3 ⊕Z/2Z. We denote the unique 2-torsion point by T , which arises from the conic containing z o .
(iii) Let L ij (i < j) denote a line connecting p i and p j . The 6 lines L ij (1 ≤ i < j ≤ 4)
give rise to 12 rational points P ij and
[−1]P ij (1 ≤ i < j ≤ 4) of E SQ,z o (C(t))
, each of which has the height P ij , P ij = 1/2. We may assume
(A * 1 ) ⊕3 ∼ = ZP 12 ⊕ ZP 13 ⊕ ZP 23
and P ij+ T = P kl , where {i, j, k, l} = {1, 2, 3, 4}.
Rational points arising from lines in case (ii)
We consider the structure of S Q,zo in more detail for later use.
P · O = 0, then C P ∩ q • q zo • f Q,zo (F ) = q • q zo • f Q,zo (s P ∩ F ). As ♯(s P ∩ F ) = 1, C P is a line. Lemma 2.2. Let P ∈ E SQ,z o (C(t))
be a point such that C P is a line.
(i) For p ∈ C P ∩ Q, the intersection multiplicity i p (C P , Q) at p is even.
(ii) The height pairing of P is given by
P, P = 3 2 − 1 2 ♯(C P ∩ C 1 ∩ C 2 ).
Conversely, if l is a line such that (a) z o ∈ l and (b) i p (l, Q) is even for ∀p ∈ l ∩ Q, then l gives rise to two points P l and [−1]P l in E SQ,z o (C(t)) such that
P l , P l = 3 2 − 1 2 ♯(l ∩ C 1 ∩ C 2 ).
Proof. (i) If there exists a point p ∈ C P ∩ Q such that i p (C P , Q) is odd, f ′ Q,zo gives a ramified double cover of C P , this is impossible as the image of s [−1]P is also C P .
(ii) Since s P · O = 0, s P · Θ zo,1 = 1 and s P · Θ i,1 = 1 if and only if p i ∈ C P . Hence by [17,Theorem 8.6,(8.12)], our statement follows. The remaining statements follows easily as C P l = l.
Bitangents to Q and triangles
△ ijk = L ij + L ik + L jk in case (ii)
Let s T be the section corresponding to T . Assume that z o ∈ C 1 . Then we infer that C 1 gives rise to a 2-torsion point P C1 , i.e., P C1 = T . Hence we have s T · Θ i,1 = 1 and s T · Θ ∞,0 = 1.
Thus we see P ij+ T = P kl by the property of the map γ (see [14, p. 70]. Also [12, Theorem 9.1] for explicit description).
We now consider rational points in E SQ,z o (C(t)) given by a triangle △ 123 := L 12 + L 13 + L 23 . Since each L ij produces P ij and [−1]P ij ([±1]P ij for short), we have 8 rational points in E SQ,z o (C(t)) from △ 123 :
Q 1 := P 12+ P 13+ P 23 , Q 2 := [−1]P 12+ P 13+ P 23 , Q 3 := P 12+ [−1]P 13+ P 23 , Q 4 := P 12+ P 13+ [−1]P 23 ,
and [−1]Q j (j = 1, 2, 3, 4). As P 12 , P 13 , P 23 is a basis of the free part of E SQ,z o (C(t)), these 8 points are distinct. For these points, we have
• Q i , Q i = [−1]Q i , [−1]Q i = 3/2 and • C Qi = C [−1]Qi .
The proof of Proposition 3
We prove the statements for C E + only, since P E − = [−1]P E + and C P E − = C P E + . As c(i, E + ) = c(i, s P E + ) (i = 0, 1, . . . , 4) and c(∞, s P E + ) = [1], we have:
P E + , P E + = 3 2 − 1 2 r
by Proposition 2. On the other hand, by the explicit formula of the height pairing, we have
P E + , P E + = 3 2 + s P E + · O − 1 2 r,
hence s P E + · O = 0. Now, by Lemma 2.1, C P E + is a line not passing through z o .
Proof for the case (i)
Q = C o + L 1 + L 2
We first show that r = 0. If r = 0, there exists p ∈ L 1 \ P such that i p (E, L 1 ) = i p (E, Q) = 1. This is impossible as i p (E, Q) is even for all p ∈ E ∩ Q.
(i-1) Suppose that r = 1. If C P E + ∩ P = {p 1 }, then E meets L 2 at L 2 \ P. This means that E ∩ L 2 contains a point p with i p (E, Q) = 1, which is impossible as E gives rise to E ± . Similarly p i ∈ C P E + ∩ P. = 1, . . . , 4), we infer that C P E + is a line through p 0 and tangent to C o .
Hence C P E + ∩ P = {p 0 }. Now since c(∞, s P E + ) = c(0, s P E + ) = [1], c(i, s P E + ) = [0] (i
(i-2) Suppose that r = 2. If p 0 ∈ E ∩ Q, then by a similar argument to the case (i-1), we infer that E has a transversal intersection point with either L 1 or L 2 , which is not a node of Q. Hence p 0 ∈ E ∩ Q. If p 0 ∈ E ∩ Q = {p 1 , p 2 } or {p 3 , p 4 }, we infer that E must passes through p 0 . Hence these cases do not occur and C P E + is a line described in the statement.
(i-3) As we see in (i-2), If {p 1 , p 2 } or {p 3 , p 4 } ⊂ E ∩ Q, then p 0 ∈ E ∩ Q. Hence C P E + is a line described in the statement.
Proof for the case (ii)
Q = C 1 + C 2
If r = 1 or 3, we infer that E intersects Q at a smooth point p with i p (E, Q) odd. This is impossible as i p (E, Q) is even for any p ∈ E ∩ Q. Hence r = 0 or 2. Since c(i, E + ) = c(i, s P E + ), we infer that C P E + is a line described in the statement.
Splitting types and the topology of curve arrangements
In this section, we recall the basics on splitting types introduced in [2] which can be used to distinguish the embedded topology of certain plane curves. Furthermore, we calculate the splitting types for certain plane curves derived from sections and trisections of rational elliptic surfaces considered in the previous section.
The definition of splitting types
Let B ⊂ P 2 be a reduced curve of even degree and let π B : B → P 2 be the double cover of P 2 branched along B. An irreducible plane curve C ⊂ P 2 is called a contact curve of B if the local intersection multiplicities i p (B, C) are even for ∀p ∈ B ∩ C. If C is a contact curve, the pre-image π * B (C) of C under π B may be reducible, in which case we call C a splitting curve with respect to B. Now given a pair of splitting curves C 1 , C 2 with respect to B, we define the splitting type as follows. Let π * B (C i ) = C + i + C − i be the irreducible decomposition of π * B (C i ) (i = 1, 2). Definition 4.2. Let m 1 ≤ m 2 be non-negative integers. Suppose that C 1 ∩ C 2 ∩ B = ∅. The splitting type of the triple (C 1 , C 2 ; B) is defined to be (m 1 , m 2 ) if for a suitable choice of labels of C ± 1 , C ± 2 the equalities C + 1 · C + 2 = m 1 and C + 1 · C − 2 = m 2 hold. The following proposition allows us to distinguish the embedded topology of cures of the form B + C 1 + C 2 by studying their splitting types.
Proposition 4.3. [[2], Proposition 2.5]
Let B 1 , B 2 be plane curves of even degree and let C j1 , C j2 be splitting curves with respect to B j (j = 1, 2). Suppose that C j1 ∩ C j2 ∩ B j = ∅, C j1 and C j2 intersect transversally and that (C 11 , C 12 ; B 1 ) and (C 21 , C 22 ; B 2 ) have distinct splitting types. Then a homeomorphism h : P 2 → P 2 such that h(B 1 ) = B 2 and {h(C 11 ), h(C 12 )} = {C 21 , C 22 } does not exist.
Computations of splitting types for some curve arrangements related to Proposition 2
We keep the previous notation given so far.
The case (i)
Let Q = C o + L 1 + L 2 be a quartic as in Proposition 2 and let ϕ Q,zo : S Q,zo → P 1 be the rational elliptic surface in Section 2.1. Let E 1 and E 2 be smooth cubics as in Proposition 3 (i-2) such that E 1 ∩P = {p 1 , p 3 } and E 2 ∩P = {p 1 , p 4 }. Let E ± 1 and E ± 2 be trisections arising from E 1 and E 2 , respectively. By Proposition 3 (i-2) , we may assume that P E + 1 = P 13 and P E + 2 = P 14 . Note that E ± i (i = 1, 2) are all integral trisections and
c(∞, E + 1 ) = [3], c(i, E + 1 ) = [0], (i = 0, 2, 4), c(i, E + 1 ) = [1] (i = 1, 3)
and 1, 4). Let C [2]P13 be the conic as in Section 2.1. We compute the splitting types of (E 1 , C [2]P13 ; Q) and (E 2 , C [2]P13 ; Q). Put f * Q,zo C [2]P13 = C + + C − . By Lemma 1.4, we have
c(∞, E + 2 ) = [3], c(i, E + 2 ) = [0], (i = 0, 2, 3), c(i, E + 2 ) = [1] (i =P E + 1 , P C ± = −(E + 1 · C ± − 3).
As P E + 1 , P C + = 1 and P E + 1 , P C − = −1, the splitting type of (E 1 , C [2]P13 ; Q) is (2,4). A similar computation shows that the splitting type of (E 2 , C [2]P13 ; Q) is (3, 3).
The case (ii)
Let Q = C 1 + C 2 be a quartic as in Proposition 2 and let ϕ Q,zo : S Q,zo → P 1 be the rational elliptic surface in Section 2.2. Let Q 1 and Q 2 be the rational points as in Section 2.2.3. Note that C Qi (i = 1, 2) are bitangents to Q. Choose two smooth cubics E 1 and E 2 such that the trisections E ± i arising from E i satisfies P E + i = Q i for each i. Note that E ± i (i = 1, 2) are all integral trisections and
c(∞, E + 1 ) = [3], c(i, E + 1 ) = [0], (∀i), and c(∞, E + 2 ) = [3], c(i, E + 2 ) = [0], (∀i)
. We now compute the splitting types of (E 1 , C Q1 ; Q) and (E 2 , C Q1 ; Q). Put f * Q,zo (C Q1 ) = s Q1 + s [−1]Q1 . Since Q 1 = P 12+ P 13+ P 23 and Q 2 = [−1]P 12+ P 13+ P 23 , we have
P E + 1 , [±1]Q 1 = Q 1 , [±1]Q 1 = ± 3 2 ,
and
P E + 2 , [±1]Q 1 = Q 2 , [±1]Q 1 = ± 1 2 .
By Lemma 1.4, we have
P E + i , [±1]Q 1 = − E + i · s [±1]Q1 − 3 2 , (i = 1, 2).
Therefore the splitting type of (E 1 ,
C Q1 ; Q) is (0, 3), while that of (E 2 , C Q1 ; Q) is (1, 2).
Some Zariski pairs and the proof of Theorem 2 (iii)
• A Zariski pair from the case (i). Let Q, E i (i = 1, 2) and C [2]P13 be those in the case (i) as above. Put B i = Q + C [2]P13 + E i (i = 1, 2). If B 1 and B 2 have the same combinatorics, then by Proposition 4.3, (B 1 , B 2 ) is a Zariski pair. We give an explicit example in Section 6.1.
• A Zariski pair from the case (ii) and the proof of Theorem 2 (iii). Assume that E 1 and E 2 are as in the case (ii) as above. Put B i = Q + E i + C Q1 (i = 1, 2). If B 1 and B 2 have the same combinatorics, then (B 1 , B 2 ) is a Zariski pair. As for the existence of families of cubics E i,s , s ∈ ∆ ǫ such that (i) for s = 0, E 1,s and E 2,s satisfy the condition on the cubics E 1 and E 2 as above, and (ii) E 1,0 = E 2,0 = L 12 + L 13 + L 23 , it is shown in Section 6.2.
Remark 4.4. So far, we have only considered smooth cubics E such that it gives rise to two components. Another case, i.e., E gives an irreducible componentẼ, actually happens for the cases (i) and (ii). Let E 3 be such a cubic. By a similar argument to that of [6], we see that (Q + E j , Q + E 3 ) (j = 1, 2) are Zariski pairs if Q + E j (j = 1, 2) and Q + E 3 have the same combinatorics. Explicit construction E 3 can be found in [19]. Thus we have a Zariski triple for each case.
Remark 4.5. We also note that the above Zariski pairs can be distinguished by studying the existence/non-existence of certain dihedral covers branched along B i (i = 1, 2), as considered in [22].
Mumford Representations
Semi-reduced divisors on hyperelliptic curves and their representations
For terminologies for curves defined over a field K, we refer to [18]. Let C be a hyperelliptic curve of genus g defined over K (char K = 2) given by an affine equation
y 2 = f (x) = x 2g+1 + . . . + c 2g+1 c i ∈ K (i = 1, . . . , 2g + 1).
We denote the point at infinity by O. By considering π : (x, y) → x, C can be considered as a double cover of P 1 . We denote the covering morphism by π : C → P 1 and the hyperelliptic involution by ι : (x, y) → (x, −y). The branch points of π are the zeros of f (x) = 0 and x = ∞ over which we have O. Let Div 0 (C) and Pic 0 (C) be the group of divisors and the divisor class group of degree 0 on C, respectively. In [16], Mumford gave a description of an element of Pic 0 (C) by two polynomials in K[x], based on Jacobi's idea. It has been used in the study of hyperelliptic curve cryptography in order to compute the addition on Pic 0 (C). We here explain the Mumford representation briefly. For a divisor d = P ∈C m P P on C, we put Supp(d) := {P ∈ C | m P = 0}. A divisor d is said to be an affine divisor if O ∈ Supp(d). For an effective affine divisor d, we can rewrite d in the form
d = d sr + π * d o , d sr = P ∈C n P P (n P ≥ 0),
where d o is a divisor on P 1 and • if P ∈ Supp(d) and P = ι(P ), then ι(P ) ∈ Supp(d) and
• if P ∈ Supp(d) and P = ι(P ), then n P = 1.
Definition 5.1. Under the notation given above, we have the following definitions:
(i) An affine effective divisor d on C is said to be semi-reduced if d = d sr . (ii) A semi-reduced divisor d is said to be h-reduced if deg d ≤ g.
Remark 5.2. In [9,13], an h-reduced divisor defined as above is simply called a reduced divisor. On the other hand, the word 'reduced' is used in a different meaning in standard textbooks of algebraic geometry (e.g., [11]). Hence we use the word 'h-reduced' in order to avoid confusion. We denote the coordinate ring K[x, y]/ y 2 − f (resp. K[x, y]/ y 2 − f ) of the affine part of C over K by K[C] (resp. over K by K[C]). For g ∈ K[x, y], we denote its class in K[C] by [g]. For P ∈ C, O P denotes the local ring at P and ord P means the discrete valuation at P .
Let d = r i=1 n i P i , P i = (x i , y i ) be a semi-reduced divisor on C. The divisor d can be represented by a pair of polynomials in K[x] which is called the Mumford representation. We briefly explain it. We first recall the following lemma:
Lemma 5.3. There exist unique polynomials u(x), v(x) ∈ K[x] such that (i) u(x) := r i=1 (x − x i ) ni , (ii) deg v(x) < deg u(x), ord Pi ([y − v(x)]) ≥ n i , and (iii) v(x) 2 − f is divisible by u.
For a proof, see [9,Lemma 10.3.5]. Conversely if u, v satisfying (i), (ii) and (ii) in Lemma 5.3 are given, then we recover d as the zero set of the ideal generated by [u] and [y −v] in K[C] counted with proper multiplicities (see [9] in detail).
I(d) := {ξ ∈ K[C] | ord P (ξ) ≥ n P for ∀P ∈ Supp(d)}, I(d) := {g ∈ K[x, y] | [g] ∈ K[C]}.
As it can be seen in [20], the Gröbner basis of I(d) with respect to the pure lexicographic
oder > p , y > p x is of the form {u(x), y − v(x)} where (u, v) is the Mumford representation of d and I(d) ∩ K[x] = u(x) .
Semi-reduced divisors of degree 3 on elliptic curves over K
Let E be an elliptic curve given by a Weierstrass equation
y 2 = f (x) = x 3 + c 1 x 2 + c 2 x + c 3 , c i ∈ K.
If a semi-reduced divisor d is defined over K, then the polynomials u, v in the Mumford representation of d are in K[x]. For later use, we consider the case of deg d = 3. Let d = P 1 + P 2 + P 3 be a semi-reduced divisor of degree 3 (P i 's are not necessarily distinct) and put P d := P 1+ P 2+ P 3 .
Lemma 5.6. Assume that P d = O and let (u d , v d ) be the Mumford representation of d. Then we have
(i) P d = P i (i = 1, 2, 3). (ii) deg v d = 2.
Proof. (i) By the definition of the addition on E,
P 1 + P 2 + P 3 − 3O ∼ P d − O.
Hence, if P d = P 1 , then P 2 + P 3 − 2O ∼ 0. This means P 3 = ι * P 2 , but this does not occur as d is semi-reduced. Similarly we have P d = P 2 , P 3 .
(ii) Since P d = O, the points P 1 , P 2 and P 3 are not collinear. As the curve defined by y − v d = 0 passes through P 1 , P 2 , P 3 and deg v d ≤ 2, deg v d = 2.
Lemma 5.7. We keep the notation as before. Assume that d is defined over K. Put P d := (x d , y d ). Then we have the following:
(i) The point P d is a K-rational point of E, i.e., x d , y d ∈ K. (ii) The two polynomials u d , v d satisfy u d , v d ∈ K[x]. In particular, v d is of the form b 0 (x − x d )(x − b 1 ) − y d (b 0 , b 1 ∈ K). Proof. (i) Since P d := P 1+ P 2+ P 3 and d is defined over K, P d is Gal(K/K)-invariant. Hence x d , y d ∈ K.
(ii) The first statement follows from [9,Lemma 10.3.10]. We go on to show the second statement. By Lemma 5.3, the divisor of [y − v d ] is P 1 + P 2 + P 3 + P 4 − 4O for some P 4 ∈ E. Hence P 1 + P 2 + P 3 + P 4 + ι * P 4 − 5O ∼ ι * P 4 − O.
As
P 1 + P 2 + P 3 − 3O ∼ P d − O and P 4 + ι * P 4 − 2O ∼ 0, P d − O ∼ ι * P 4 − O.
Hence ι * P 4 = P d , i.e., P 4 = ι * P d = (x d , −y d ) and v d has the desired form.
Trisections of elliptic surfaces and their Mumford representations
In what follows, we only consider the case when C = P 1 , i.e., S is an elliptic surface over P 1 . In [4,5], we made use of properiies of bisections of certain elliptic surfaces in order to construct Zarisiki N -ples. Our description for bisections there is nothing but the Mumford representation for bisections. One of our purposes of this article is to consider a geometric application of Mumford representations of trisections along the line of [4,5]. To this purpose, in this section, we consider the Mumford representations of trisections. Let D be a trisection and d D be a divisor of degree 3 obtained by the restriction of D to E S . In this article we only consider trisections such that (ii) the coordinates of U 1 and U 2 are (t, x) and (s, x ′ ), respectively with s = 1/t and
x ′ = x/t d .
There exists a unique section ∆ 0 with respect to the ruling Σ d → P 1 such that ∆ 2 0 = −d. With the coordinates as abve, ∆ 0 is given by x = x ′ = ∞. We also have a section given by x = c 0 t d + c 1 t d−1 + . . . + c d , c i ∈ C on U 1 which is linearly equivalent to ∆ 0 + df, f being a fiber of the ruling Σ d → P 1 .
Under these settings, S is obtained in the following way:
(i) There exists a reduced divisor T S on Σ d such that (a) on U 1 , T is given by
T S : f TS (t, x) = x 3 + a d (t)x 2 + a 2d (t)x + a 3d (t) = 0,
where a di (t) ∈ C[t], deg a di ≤ di (i = 1, 2, 3), (b) T S has at most simple singularities (see [8] for simple singularities) and (c) E S is given by the Weierstrass equation y 2 = f T (t, x).
(ii) The affine surface given by y 2 = f TS (t, x) can be extended to a double cover f ′ S : S ′ → Σ d and S is obtained as the minimal resolution of S ′ . We denote the resolution by µ : S → S ′ and put [4,22,14], for detail. With the coordinates as above, C(P 1 ) = C(t).
f S = f ′ S • µ. Note that f S (O) = ∆ 0 . See= (x D , y D ) ∈ E S (C(t)). Since d D is defined over C(t), u D , v D ∈ C(t)[x]. By Lemma 5.6 and 5.7, v D is of the form b 0 (x − x D )(x − b 1 ) − y D , b 0 , b 1 ∈ C(t),
and
(v D ) 2 − f TS = b 2 0 u D (x − x D ). Conversely, assume that P o = (x o , y o ) ∈ E S (C(t)) is given. Then for any polynomial v ∈ C(t)[x] of the form b 0 (x − x o )(x − b 1 ) − y o b 0 , b 1 ∈ C(t), v 2 − f TS has a decomposition b 2 0 (x − x o )u where u ∈ C(t)[x]
is monic and deg u = 3. Hence, u and v gives a divisor In the next section, we make use of this construction for trisections for a given P o with
d(u, v) of degree 3 on E S such that d(u, v) − 3O ∼ P o − O. As u is determined by P o and b 0 (x − x o )(x − b 1 ) − y o ,x o , y o ∈ C[t] and v of the form b o (x − x o )(x − b 1 (t)) − y o , b o ∈ C × , b 1 ∈ C[t].
Examples
We apply our observation in Section 5 to consider explicit examples for trisections given by smooth cubics that appear in Proposition 3. As a result we obtain explicit examples of the Zariski pairs in Section 4.3. We keep the notation as before.
The case
Q = C o + L 1 + L 2
We recall Example 5.2 in [22]. Let [T, X, Z] be homogeneous coordinates of P 2 and let (t, x) = (T /Z, X/Z) be affine coordinates for C 2 = P 2 \ {Z = 0}. Consider Q = C o + L 1 + L 2 given by as z o , we may assume that E Q,zo is given by
y 2 = f TS Q,zo = (x − t 2 )(x + 3t + 2)(x − 3t + 2).
Under these settings, P 13 and P 14 are
P 13 = (1, −3(t + 1)(t − 1)), P 14 = (t + 2, −2 √ 2(t + 1)(t − 2)).
For simplicity we put b 0 = 1. We need to choose b 1 appropriately so that u = 0 gives a smooth cubic and we have the following:
(i) For D(P 13 , 1, c), c ∈ C,
u = x 3 − 2(c + 1)x 2 + (7t 2 + c 2 + 2c − 11)x − (6c − 14)t 2 − (c − 3) 2 .
The curve C D(P13,1,c) is given by u = 0 and it is smooth for general c.
(ii) For D(P 14 , 1, b 1 ) b 1 = −(2 √ 2 + 3)t + c), c ∈ C, u = x 3 + a 1 x 2 + a 2 x + a 3 ,
where a 1 := (4 √ 2 + 5)t − 2c − 3,
a 2 := (12 √ 2 + 12)t 2 − (4 √ 2c + 4c + 12 √ 2 + 13)t + c 2 + 4c − 8 √ 2 − 6,a 3 := −(−6c + 36 √ 2 + 36)t 2 − (c 2 − 12 √ 2c − 12c + 24 √ 2 + 40)t −2c 2 + 8 √ 2c − 16.
The curve C D(P14,1,b1) is given by u = 0 and it is smooth for general c.
As [2]P 14 = 9 8 t 2 , − √ 2 32 t(9t 2 − 16) , C [2]P14 is given by x − 9 8 t 2 = 0. Put B 1 = Q + C [2]P14 + C D(P13,1,c) , B 2 = Q + C [2]P14 + C D(P14,1,b1) .
Since B 1 and B 2 have the same combinatorics for general c, (B 1 , B 2 ) is a Zariski pair by Section 4.3.
6.2 The case Q = C 1 + C 2 : The existence of curves satisfying Theorem 2, (i) and (ii)
By choosing suitable coordinates, we assume that Q is given by f TS Q,zo = 0 and that the generic fiber E Q,zo is given by y 2 = f TS Q,zo . Let P ij and Q j be the points in considered in Section 2.2. Let d j (j = 1, 2, 3, 4) be semi-reduced divisors We give a observation about the Mumford representations of d j (j = 1, 2, 3, 4). Let (u dj , v dj ) be the Mumford representation of d j . Put
[±1]P ij = (x ij (t), ±y ij (t)), [±1]Q j = (x Qj (t), ±y Qj (t)).
As C Pij and C Qj are given by x − x ij (t) = 0 and x − x Qj (t) = 0, respectively, deg x ij = deg x Qj = 1. Hence
u dj = (x − x 12 (t))(x − x 13 (t))(x − x 23 (t)), j = 1, 2, 3, 4,
i.e., u dj = 0 is △ 123 and v dj satisfies
(v 2 dj − f TS Q,zo ) = d j + [−1]Q j − 4O, and v 2 dj − f TS Q,zo = c(x − x Qj (t))u j , c ∈ C(t) × .
We now show the following Lemma which assures that the coefficient c in the above equation is a constant.
Lemma 6.1. The polynomial v dj is of the form v dj = c 0 x 2 + c 1 x + c 2 , c i ∈ C[t], deg c i = i.
In particular, c in the above is a constant.
Proof. We prove the statement for j = 1 only as the remaining cases can be proven in the same way. Put Q Q1 = L 12 + L 13 + L 23 + L Q1 and L Q1 ∩ Q = {q 1 , q 2 }. Consider a pencil Λ 1 of quartics given by |λQ + µQ Q1 | [λ,µ]∈P 1 . Λ 1 satisfies the following properties:
• The base loucs of Λ 1 is 4(p 1 + p 2 + p 3 ) + 2(q 1 + q 2 ), where the coefficients of p i , q j 's mean the multiplicities.
• Since Q, Q Q1 ∈ Λ 1 , general members of Λ 1 are irreducible.
• General members of Λ 1 have nodes at p i (i = 1, 2, 3).
Let C o be the unique conic passing through p i (i = 1, 2, 3) and q j (j = 1, 2). Choose r ∈ C o \ (Q ∪ Q Q1 ). Let Q µr be a member of Λ 1 given by Q + µ r Q Q1 satisfying r ∈ Q µr . Then since the divisor on C o cut out by Q µr is C o | Qµ r = 2(p 1 + p 2 + p 3 ) + q 1 + q 2 + r, C o is an irreducible component of Q µr . Write Q µr = C o + C ′ o . Then C ′ o is irreducible and C ′ o | Qµ r = 2(p 1 + p 2 + p 3 ) + q 1 + q 2 , i.e., C ′ o = C o . Thus 2C o ∈ Λ 1 . As z o ∈ C o , C o is given by an equation of the form
c 0 x 2 + c 1 x + c 2 , c i ∈ C[t], deg c i = i.
and, as 2C o = Q + µ r Q Q1 , we have (c 0 x 2 + c 1 x + c 2 ) 2 = f TS Q,zo + µ r u d1 (x − x Q1 (t)).
By the uniqueness of the Mumford representation, we have v d1 = c 0 x 2 + c 1 x + c 2 and c = µ r . By Lemma 6.1 and −y Qj = c 0 x 2 Qj + c 1 x Qj + c 2 , we may assume v dj is of the form b 0 (x − x Qj )(x + b 10 t + b 11 ) − y Qj , b 0 ∈ C × , b 1j ∈ C, j = 0, 1.
For each Q j , consider the one parameter family of polynomials over s ∈ ∆ ǫ , b 0 (s) = 0 given by v j,s := b 0 (s)(x − x Qj )(x + b 10 (s)t + b 11 (s)) − y Qj , such that v j,0 = v dj . Since v j,s (x Qj (t)) = −y Qj , we have v 2 j,s − f TS Q,zo = b 0 (s) 2 (x − x Qj )u j,s (x).
Hence (u j,s , v j,s ) is a one parameter family of Mumford representations such that d(u j,s , v j,s )− 3O ∼ Q j − O. We also obtain a family of plane curves given by u j,s = 0 for each j, which can be candidates of cubics in Theorem 2. We now go on to give an explicit example, which shows the existence of families of plane curves stated in Theorem 2.
Example 6.2. We use the same coordinates as in the case (i). Let Q = C 1 + C 2 given by By choosing [0, 1, 0] as z o , we may assume that E Q,zo is given by
y 2 = f TS Q,zo = (x − t 2 )(x 2 − 10tx + 25x − 36).
Under these settings, we have P 12 = (5t−6, −5(t−2)(t−3)), P 13 = (9t−18, −3(t−3)(t−6)), P 23 = (8t−12, −4(t−2)(t−6)).
Put Q 1 := P 12+ P 13+ P 23 , Q 2 := [−1]P 12+ P 13+ P 23 . Straightforward computations show Q 1 = (0, −6t), Q 2 = (10t − 25, −6(t − 5)).
Hence Q has bitangents L 1 : x = 0, L 2 : x − 10t + 25 = 0.
Put v 1,s := 1 6
x (x − (11t − 36 + s)) + 6t v 2,s := (x − 10t + 25) (x − (6t − 6 + s)) + 6(t − 5),
where s ∈ C. By computing v 2 j,s − f TS Q,zo (j = 1, 2), we have where u 1,s := x 3 + (−2s − 22t + 36)x 2 + (s 2 + 22st + 157t 2 − 72s − 504t + 396)x −360t 3 + 72st + 1692t 2 − 2592t + 1296 u 2,s := x 3 − (2s + 22t − 36)x 2 − (−s 2 − 32st − 157t 2 + 62s + 504t − 396)x −10s 2 t − 120st 2 − 360t 3 + 25s 2 + 432st + 1692t 2 − 360s − 2592t + 1296.
Thus we have two families of Mumford representations (u j,s , v j,s ) (j = 1, 2) such that (i) the divisor d(u j,0 , v j,0 ) satisfies d(u j,0 , v j,0 ) = P 12 + P 13 + P 23 and (ii) we have linear equivalences d(u j,s , v j,s ) − 3O ∼ Q j − O for j = 1, 2.
Let D 1,s := D(Q 1 , 1 6 , 11t−36+s) and D 2,s := D(Q 2 , 1, 6t−6+s) be trisections determined by d(u 1,s , v 1,s ) and d(u 2,s , v 2,s ), respectively. Put E j,s = C Dj,s (j = 1, 2) be cubics given by u j,s = 0 (j = 1, 2), respectively. Then we can check that there exists ǫ > 0, such that for s ∈ ∆ ǫ , s = 0 the following hold:
(a) The curves E j,s (j = 1, 2) are smooth.
(b) The curves E j,s (j = 1, 2) meet L 1 at three distinct points.
(c) The curves E j,s (j = 1, 2) are tangent to Q at six distinct points.
Let B j,s := Q + E j,s + L 1 (j = 1, 2). From the above facts and Section 3.3, we see that the following statements hold:
(i) The curve B 1,0 is equal to B 2,0 .
(ii) For each s = 0, (B 1,s , B 2,s ) is a Zariski pair.
Theorem 1 .
1([17, Theorem 1.3]) ψ : NS(S)/T ϕ ∼ = E S (C(C)),
Definition 1 .
1A horizontal divisor D is said to be an m-section if D meets in m distinct points with a general fiber. Furthermore if D · O = 0, D is called an integral m-section.
( i )
iFor s = 0, B i,s = Q + E i,s + L, (i = 1, 2), where L is a fixed bitangent line to Q and E i,s (i = 1, 2) are smooth cubics as in Proposition 3 (ii-1) such that C P Both of B i,0 (i = 1, 2) are Q + L 12 + L 13 + L 23 + L, where L ij is the line through p i and p j .
( iii )
iiiThe pairs (B 1,s , B 2,s ) are Zariski pairs for all s = 0.
Lemma 1. 4 .
4If the reducible singular fibers of S are all of type I2 or III and both D 1 and D 2 are integral d 1 -and d 2 -sections respectively, then we have
be the line connecting p i and p j . Let L 1 = L 12 and L 2 = L 34 . The 6 lines L ij give rise to sections of ϕ Q,zo which can be identified with points in E SQ,z o (C(t)). More precisely, both of L 1 and L 2 give 2-torsion points, and as for the other four lines L 13 , L 14 , L 23 and L 24 , each of them give rise to two points, resulting in eight points P ij , [−1]P ij (i, j) = (1, 3), (1, 4), (2, 3),(2,4). We may assume(A * 1 ) ⊕2 ∼ = ZP 13 ⊕ ZP 14and that P 23 , P 24 are obtained by translation by suitable 2-torsion points.
(i) The elliptic fibration ϕ Q,zo has 5
5singular fibers of type I2 or III and no other reducible singular fiber. They arise from the tangent line at z o and lines connecting z o and p i (i = 1, . . . , 4). We denote them by F ∞ and F i (i = 1, . . . , 4), respectively and their irreducible decompositions by F • = Θ •,0 + Θ •,1 (• = ∞, 1, . . . , 4).
Lemma 2. 1 .
1The curve C P is a line if and only if s P ·Θ ∞,1 = 1 and s P ·O = 0. In particular, if C P is a line, then z o ∈ C P .Proof. Choose a general fiber F of ϕ Q,zo . By our construction, q • q zo • f Q,zo (F ) is a line through z o . We may assume F ∩ O ∩ s P = ∅. If C P is a line, C P and q • q zo • f Q,zo (F ) meet one point, which is the image z 1 of s P ∩ F . By our choice of F , z 1 = z o and z o ∈ C P . As both O and Θ ∞,0 are mapped to z o , s P · Θ ∞,1 = 1 and s P · O = 0. Conversely, if s P · Θ ∞,1 = 1 and s
By the property of γ, we infer that sQj ·Θ i,1 = s [−1]Qj ·Θ i,1 = 0 (j = 1, 2, 3, 4) and s Qj ·Θ ∞,1 = s [−1]Qj · Θ ∞,1 = 1 (j = 1, 2, 3, 4). Hence by [17, Theorem 8.6, (8.12)], s Qj · O = s [−1]Qj · O = 0 (j =1, 2, 3, 4). Thus by Lemmas 2.1 and 2.2, C Qj (j = 1, 2, 3, 4) are the four bitangent lines of Q. For other triangles △ ijk (1 ≤ i < j < k ≤ 4), as P ij+ T = P kl for {i, j, k, l} = {1, 2, 3, 4}, we have the same rational points [±1]Q j (j = 1, 2, 3, 4) and the same 4 bitangent lines.
Example 4. 1 .
1Each of the 6 lines L ij in Section 2.1 are splitting curves with respect to Q.
Definition 5 . 4 .
54For a semi-reduced divisor d, (u, v) is said to be the Mumford representation of d. Also, we denote the divisor given by a Mumford representation (u, v) by d(u, v).
Remark 5 . 5 .
55Define ideals I(d) and I(d) in K[C] and K[x, y], respectively as follows:
d
D is a semi-reduced divisor on E S . We denote the rational point determined by d D by P D , i.e, the rational point obtained by Theorerem 1. In order to give the Mumford representation of d D , we recall our setting in [4, Sections 2.2.2, 2.2.3]. Let Σ d (d = χ(O S )) be the Hirzebruch surface of degree d. Take affine open subsets U 1 and U 2 of Σ d so that (i) U i = C 2 (i = 1, 2) and
Remark 5 . 8 .
58For the case of d = 2, S is a rational elliptic surface. If S has a reducible singular fiber, S can be obtained as S Q,zo for some quartic Q and a smooth point z o on Q. If we choose coordinate of P 2 suitably, f TS (t, x) is a defining equation of Q. See [4, Section 2.2.2] for details.We now consider the Mumford representation of d D via the Weiestrass equation of E S as above. Let (u D , v D ) be the Mumford representation of d D and we denote P D
we denote the trisection given by d(u, v) by D(P o , b 0 , b 1 ). Note that ψ(D(P o , b 0 , b 1 )) = P o holds and C D(Po,b0,b1) is given by u = 0.
C o : x − t 2 = 0, L 1 : x + 3t + 2 = 0, L 2 : x − 3t + 2 = 0. In this case, p o = [0, −2, 1] and we label C o ∩ (L 1 ∪ L 2 ) by p 1 = [−1, 1, 1], p 2 = [−2, 4, 1], p 3 = [1, 1, 1] and p 4 = [2, 4, 1] (Note that we label L 1 and L 2 in a different way from [22, Example 5.2]). The lines L 13 and L 14 are given by x − 1 = 0 and x − t − 2 = 0, respectively. By choosing [0, 1, 0]
C 1 :
1x − t 2 = 0, C 2 : x 2 − 10tx + 25x − 36 = 0.In this case, p 1 = [3, 9, 1], p 2 = [2, 4, 1], p 3 = [6, 36, 1], p 4 = [−1, 1, 1] and L 12 : x − 5t + 6 = 0, L 13 : x − 9t + 18, L 23 : x − 8t + 12 = 0.
s − f TS Q,zo = (x − 10t + 25)u 2,s ,
d 1 := P 12 + P 13 + P 23 , d 2 := [−1]P 12 + P 13 + P 23 , d 3 := P 12 + [−1]P 13 + P 23 , d 4 := P 12 + P 13 + [−1]P 23 .
A survey on Zariski pairs. E Artal, J Cogolludo-Agustín, H Tokunaga, Algebraic geometry in East Asia-Hanoi. Tokyo50Math. Soc. JapanE. Artal, J. Cogolludo-Agustín, and H. Tokunaga. A survey on Zariski pairs. In Algebraic geometry in East Asia-Hanoi 2005, volume 50 of Adv. Stud. Pure Math., pages 1-100. Math. Soc. Japan, Tokyo, 2008.
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An elementary introductio to hyperelliptic curves. A J Memezes, Y.-H Wu, R J Zuccherato, N. Koblitz: Algebraic Aspects of Cryptography. BerlinSpringer-VerlagA. J. Memezes, Y.-H. Wu, and R. J. Zuccherato. An elementary introductio to hy- perelliptic curves. In N. Koblitz: Algebraic Aspects of Cryptography, pages 157-178. Springer-Verlag, Berlin, 1998.
The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. R Miranda, ETS Editrice. Doctorate in Mathematical ResearchR. Miranda. The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research]. ETS Editrice, Pisa, 1989.
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Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers. H.-O Tokunaga, J. Math. Soc. Japan. 662H.-o. Tokunaga. Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers. J. Math. Soc. Japan, 66(2):613-640, 2014.
. Hitachinaka-Shi Nakane, 312-8508 JAPAN [email protected], Hitachinaka-shi, Ibaraki-Ken 312-8508 JAPAN [email protected]
| []
|
[
"Asteroseismology of hot pre-white dwarf stars: the case of the DOV stars PG 2131+066 and PG 1707+427, and the PNNV star NGC 1501",
"Asteroseismology of hot pre-white dwarf stars: the case of the DOV stars PG 2131+066 and PG 1707+427, and the PNNV star NGC 1501"
]
| [
"A H Córsico [email protected] \nFacultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque S/N, (1900) La PlataArgentina\n\nInstituto de Astrofísica La Plata\nIALP\nCONICET-UNLP\nArgentina\n\nCarrera del Investigador Científico y Tecnológico\nCONICET\nArgentina\n",
"L G Althaus [email protected] \nFacultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque S/N, (1900) La PlataArgentina\n\nInstituto de Astrofísica La Plata\nIALP\nCONICET-UNLP\nArgentina\n\nCarrera del Investigador Científico y Tecnológico\nCONICET\nArgentina\n",
"M M Miller Bertolami \nFacultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque S/N, (1900) La PlataArgentina\n\nInstituto de Astrofísica La Plata\nIALP\nCONICET-UNLP\nArgentina\n\nFellow of CONICET\nArgentina\n",
"E García-Berro \nDepartament de Física Aplicada\nEscola Politècnica Superior de Castelldefels\nUniversitat Politècnica de Catalunya\nAv. del Canal Olímpic, s/n08860CastelldefelsSpain\n\nInstitute for Space Studies of Catalonia\nc/Gran Capità 2-4, Edif. Nexus 10408034BarcelonaSpain\n"
]
| [
"Facultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque S/N, (1900) La PlataArgentina",
"Instituto de Astrofísica La Plata\nIALP\nCONICET-UNLP\nArgentina",
"Carrera del Investigador Científico y Tecnológico\nCONICET\nArgentina",
"Facultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque S/N, (1900) La PlataArgentina",
"Instituto de Astrofísica La Plata\nIALP\nCONICET-UNLP\nArgentina",
"Carrera del Investigador Científico y Tecnológico\nCONICET\nArgentina",
"Facultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque S/N, (1900) La PlataArgentina",
"Instituto de Astrofísica La Plata\nIALP\nCONICET-UNLP\nArgentina",
"Fellow of CONICET\nArgentina",
"Departament de Física Aplicada\nEscola Politècnica Superior de Castelldefels\nUniversitat Politècnica de Catalunya\nAv. del Canal Olímpic, s/n08860CastelldefelsSpain",
"Institute for Space Studies of Catalonia\nc/Gran Capità 2-4, Edif. Nexus 10408034BarcelonaSpain"
]
| []
| Aims.We present an asteroseismological study on the two high-gravity pulsating PG1159 (GW Vir or DOV) stars, PG 2131+066 and PG 1707+427, and on the pulsating [WCE] star NGC 1501. All of these stars have been intensively scrutinized through multi-site observations, so they have well resolved pulsation spectra. Methods. We compute adiabatic g-mode pulsation periods on PG1159 evolutionary models with stellar masses ranging from 0.530 to 0.741M ⊙ . These models take into account the complete evolution of progenitor stars, through the thermally pulsing AGB phase, and born-again episode. We constrain the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 by comparing the observed period spacing with the asymptotic period spacing and with the average of the computed period spacings. We also employ the individual observed periods in search of representative seismological models for each star. Results. We derive a stellar mass of 0.627 M ⊙ for PG 2131+066, 0.597 M ⊙ for PG 1707+427, and 0.571 M ⊙ for NGC 1501 from a comparison between the observed period spacings and the computed asymptotic period spacings, and a stellar mass of 0.578 M ⊙ for PG 2131+066, 0.566 M ⊙ for PG 1707+427, and 0.576 M ⊙ for NGC 1501 by comparing the observed period spacings with the average of the computed period spacings. We also find, on the basis of a period-fit procedure, asteroseismological models representatives of PG 2131+066 and PG 1707+427. These best-fit models are able to reproduce the observed period patterns of these stars with an average of the period differences of δΠ i = 1.57 s and δΠ i = 1.75 s, respectively. The best-fit model for PG 2131+066 has an effective temperature T eff = 102 100 K, a stellar mass M * = 0.589 M ⊙ , a surface gravity log g = 7.63, a stellar luminosity and radius of log(L * /L ⊙ ) = 1.57 and log(R * /R ⊙ ) = −1.71, respectively, and a He-rich envelope thickness of M env = 1.6 × 10 −2 M ⊙ . We derive a seismic distance d ∼ 830 pc and a parallax π ∼ 1.2 mas. The best-fit model for PG 1707+427, on the other hand, has T eff = 89 500 K, M * = 0.542 M ⊙ , log g = 7.53, log(L * /L ⊙ ) = 1.40, log(R * /R ⊙ ) = −1.68, and M env = 2.5 × 10 −2 M ⊙ , and the seismic distance and parallax are d ∼ 730 pc and π ∼ 1.4 mas. Finally, we have been unable to find an unambiguous best-fit model for NGC 1501 on the basis of a period-fit procedure.Conclusions. This work closes our short series of asteroseismological studies on pulsating pre-white dwarf stars. Our results demonstrate the usefulness of asteroseismology for probing the internal structure and evolutionary status of pre-white dwarf stars. In particular, asteroseismology is able to determine stellar masses of PG1159 stars with an accuracy comparable or even better than spectroscopy. | 10.1051/0004-6361/200810727 | [
"https://arxiv.org/pdf/0903.3628v1.pdf"
]
| 19,008,085 | 0903.3628 | ec76e70d0388692da7ee9572386ab2cd63e02069 |
Asteroseismology of hot pre-white dwarf stars: the case of the DOV stars PG 2131+066 and PG 1707+427, and the PNNV star NGC 1501
20 Mar 2009 March 20, 2009 March 20, 2009
A H Córsico [email protected]
Facultad de Ciencias Astronómicas y Geofísicas
Universidad Nacional de La Plata
Paseo del Bosque S/N, (1900) La PlataArgentina
Instituto de Astrofísica La Plata
IALP
CONICET-UNLP
Argentina
Carrera del Investigador Científico y Tecnológico
CONICET
Argentina
L G Althaus [email protected]
Facultad de Ciencias Astronómicas y Geofísicas
Universidad Nacional de La Plata
Paseo del Bosque S/N, (1900) La PlataArgentina
Instituto de Astrofísica La Plata
IALP
CONICET-UNLP
Argentina
Carrera del Investigador Científico y Tecnológico
CONICET
Argentina
M M Miller Bertolami
Facultad de Ciencias Astronómicas y Geofísicas
Universidad Nacional de La Plata
Paseo del Bosque S/N, (1900) La PlataArgentina
Instituto de Astrofísica La Plata
IALP
CONICET-UNLP
Argentina
Fellow of CONICET
Argentina
E García-Berro
Departament de Física Aplicada
Escola Politècnica Superior de Castelldefels
Universitat Politècnica de Catalunya
Av. del Canal Olímpic, s/n08860CastelldefelsSpain
Institute for Space Studies of Catalonia
c/Gran Capità 2-4, Edif. Nexus 10408034BarcelonaSpain
Asteroseismology of hot pre-white dwarf stars: the case of the DOV stars PG 2131+066 and PG 1707+427, and the PNNV star NGC 1501
20 Mar 2009 March 20, 2009 March 20, 2009Astronomy & Astrophysics manuscript no. article c ESO 2009stars: evolution -stars: interiors -stars: oscillations -stars: variables: other (GW Virginis)-white dwarfs
Aims.We present an asteroseismological study on the two high-gravity pulsating PG1159 (GW Vir or DOV) stars, PG 2131+066 and PG 1707+427, and on the pulsating [WCE] star NGC 1501. All of these stars have been intensively scrutinized through multi-site observations, so they have well resolved pulsation spectra. Methods. We compute adiabatic g-mode pulsation periods on PG1159 evolutionary models with stellar masses ranging from 0.530 to 0.741M ⊙ . These models take into account the complete evolution of progenitor stars, through the thermally pulsing AGB phase, and born-again episode. We constrain the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 by comparing the observed period spacing with the asymptotic period spacing and with the average of the computed period spacings. We also employ the individual observed periods in search of representative seismological models for each star. Results. We derive a stellar mass of 0.627 M ⊙ for PG 2131+066, 0.597 M ⊙ for PG 1707+427, and 0.571 M ⊙ for NGC 1501 from a comparison between the observed period spacings and the computed asymptotic period spacings, and a stellar mass of 0.578 M ⊙ for PG 2131+066, 0.566 M ⊙ for PG 1707+427, and 0.576 M ⊙ for NGC 1501 by comparing the observed period spacings with the average of the computed period spacings. We also find, on the basis of a period-fit procedure, asteroseismological models representatives of PG 2131+066 and PG 1707+427. These best-fit models are able to reproduce the observed period patterns of these stars with an average of the period differences of δΠ i = 1.57 s and δΠ i = 1.75 s, respectively. The best-fit model for PG 2131+066 has an effective temperature T eff = 102 100 K, a stellar mass M * = 0.589 M ⊙ , a surface gravity log g = 7.63, a stellar luminosity and radius of log(L * /L ⊙ ) = 1.57 and log(R * /R ⊙ ) = −1.71, respectively, and a He-rich envelope thickness of M env = 1.6 × 10 −2 M ⊙ . We derive a seismic distance d ∼ 830 pc and a parallax π ∼ 1.2 mas. The best-fit model for PG 1707+427, on the other hand, has T eff = 89 500 K, M * = 0.542 M ⊙ , log g = 7.53, log(L * /L ⊙ ) = 1.40, log(R * /R ⊙ ) = −1.68, and M env = 2.5 × 10 −2 M ⊙ , and the seismic distance and parallax are d ∼ 730 pc and π ∼ 1.4 mas. Finally, we have been unable to find an unambiguous best-fit model for NGC 1501 on the basis of a period-fit procedure.Conclusions. This work closes our short series of asteroseismological studies on pulsating pre-white dwarf stars. Our results demonstrate the usefulness of asteroseismology for probing the internal structure and evolutionary status of pre-white dwarf stars. In particular, asteroseismology is able to determine stellar masses of PG1159 stars with an accuracy comparable or even better than spectroscopy.
Introduction
Pulsating PG1159 stars (also called GW Vir or DOV stars) are very hot hydrogen-deficient post-Asymptotic Giant Branch (AGB) stars with surface layers rich in helium, carbon, and oxygen (Werner & Herwig 2006) that exhibit multiperiodic luminosity variations with periods ranging from 5 to 50 minutes, attributable to non-radial pulsation g-modes (see for a recent review). PG1159 stars are thought to be the evolutionary link between Wolf-Rayet type central stars of planetary nebulae and most of the hydrogen-deficient white dwarfs (Althaus et al. 2005). It is generally accepted that these stars have their origin in a born-again episode induced by a post-AGB Send offprint requests to: A. H. Córsico helium thermal pulse -see Iben et al. (1983), Herwig et al. (1999), Lawlor & MacDonald (2003), and Althaus et al. (2005) for recent references.
Recently, considerable observational effort has been invested to study pulsating PG1159 stars. Particularly noteworthy are the works of Vauclair et al. (2002) on RX J2117.1+3412, Fu et al. (2007) on PG 0122+200, and and on PG 1159−035. These stars have been monitored through long-term observations carried out with the Whole Earth Telescope (Nather et al. 1990). On the theoretical front, recent important progress in the numerical modeling of PG1159 stars (Althaus et al. 2005; has paved the way for unprecedented asteroseismological inferences for the mentioned stars , Córsico et al. 2007a, 2008. The new generation of PG1159 evolutionary models of Miller is derived from the complete evolutionary history of progenitor stars with different stellar masses and an elaborate treatment of the mixing and extra-mixing processes during the core helium burning and born-again phases. The success of these models at explaining the spread in surface chemical composition observed in PG1159 stars , the short born-again times of4334 , and the location of the GW Vir instability strip in the log T eff − log g plane ) renders reliability to the inferences drawn from individual pulsating PG1159 stars.
Besides the mentioned three well-studied pulsating PG1159 stars, there exist two other variable stars of this class that have been also intensively scrutinized through the multi-site observations of the WET: PG 2131+066 and PG 1707+427. In addition, there is a variable central star of planetary nebula (PNNV), NGC 1501, which has been the subject of a nearly continuous photometric coverage from a global observing campaign by Bond et al. (1996). We briefly summarize the properties of these stars below.
PG 2131+066 was discovered as a variable star by Bond et al. (1984) with periods of about 414 and 386 s, along with some other periodicities. On the basis of an augmented set of periods from WET data, Kawaler et al. (1995) obtained a precise mass determination of M * = 0.61 M ⊙ , a luminosity of 10 L ⊙ , and a seismological distance from the Earth of d = 470 pc. Spectroscopic constraints of Dreizler & Heber (1998), on the other hand, gave M * = 0.55 M ⊙ , T eff = 95 000 K, 39.8 L ⊙ , and log g = 7.5 for PG 2131+066. By using this updated determination of the effective temperature, Reed et al. (2000) refined the procedure of Kawaler et al. (1995) and found M * = 0.608 M ⊙ , L * = 26 L ⊙ and d = 668 pc.
PG 1707+427 was discovered to be a pulsator by Bond et al. (1984). Dreizler & Heber (1998) obtained T eff = 85 000 K, log g = 7.5, and then M * = 0.54 M ⊙ and L * = 25 L ⊙ were inferred from their spectroscopic study. Recently, Kawaler et al. (2004) reported the analysis of multi-site observations of PG 1707+427 obtained with WET. Preliminary seismic analysis by using 7 independent ℓ = 1 modes with periods between 334 and 910 s suggest an asteroseismological mass and luminosity of 0.57 M ⊙ and 23 L ⊙ , respectively.
NGC 1501 was classified as a [WCE] star, an early low-mass Wolf Rayet-type PNNV with spectra dominated by strong helium and carbon emission lines (Werner & Herwig 2006). The effective temperature and gravity of this star are T eff = 134 000 K and log g = 6.0 (Werner & Herwig 2006). The variable nature of NGC 1501 was discovered by Bond & Ciardullo (1993). The star shows ten periodicities ranging from 5200 s down to 1154 s, although the largest amplitude pulsations occur between 1154 s and 2000 s. Based on period-spacing data, Bond et al. (1996) found a stellar mass of 0.53 ± 0.03 M ⊙ for NGC 1501.
In this work we complete our small survey of asteroseismological inferences on pulsating PG1159 stars -see Córsico et al. (2007aCórsico et al. ( , 2007bCórsico et al. ( , 2008 for the previous studies of this series -by performing a detailed study of the GW Vir stars PG 2131+066 and PG 1707+427, and the [WCE] star NGC 1501. We employ the same stellar models and numerical tools as in our previous works. In particular, we go beyond the mere use of the observed period-spacing data by performing, in addition, detailed period-to-period fits on the pulsation spectrum of these stars. In our approach, we take full advantage of the state-of-the-art PG1159 evolutionary models devel-oped by Miller . The paper is organized as follows. In Section 2 we briefly describe our PG1159 evolutionary models. In Sect. 3 we derive the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 by using the observed period-spacing data alone. In Sect. 4 we infer structural parameters of these stars by employing the individual observed periods. In this section we derive asteroseismological models representative of PG 2131+066 and PG 1707+427 (4.1), and discuss their main structural and pulsational characteristics (4.2). In Sect. 5 we compare the results of the present paper with those of the asteroseismological study of (hereinafter CA06). Finally, in Sect. 6 we summarize our main results and make some concluding remarks.
Evolutionary models and numerical tools
The pulsation analysis presented in this work relies on a new generation of stellar models that take into account the complete evolution of PG1159 progenitor stars. Specifically, the stellar models were extracted from the evolutionary calculations recently presented by Althaus et al. (2005, and , who computed the complete evolution of model star sequences with initial masses on the ZAMS ranging from 1 to 3.75 M ⊙ . All of the post-AGB evolutionary sequences were computed using the LPCODE evolutionary code (Althaus et al. 2005) and were followed through the very late thermal pulse (VLTP) and the resulting born-again episode that gives rise to the H-deficient, He-, C-, and O-rich composition characteristic of PG1159 stars. The masses of the resulting remnants are 0. 530, 0.542, 0.556, 0.565, 0.589, 0.609, 0.664, and 0.741 M ⊙ . For details about the input physics and evolutionary code used, and the numerical simulations performed to obtain the PG1159 evolutionary sequences employed here, we refer the interested reader to the works by Althaus et al. (2005) and Miller .
We computed ℓ = 1 g-mode adiabatic pulsation periods with the same numerical code and methods we employed in our previous works, see for details. In addition, we performed nonadiabatic computations with the help of the code employed in to evaluate the pulsational stability of the asteroseismological models presented in Sect. 4. We analyzed about 3000 PG1159 models covering a wide range of effective temperatures (5.4 > ∼ log(T eff ) > ∼ 4.8) and luminosities (0 < ∼ log(L * /L ⊙ ) < ∼ 4.2), and a range of stellar masses (0.530 ≤ M * /M ⊙ ≤ 0.741).
Mass determination from the observed period spacing
In this section we constrain the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 by comparing the asymptotic period spacing and the average of the computed period spacings with the observed period spacing. These approaches take full advantage of the fact that the period spacing of PG1159 pulsators depends primarily on the stellar mass, and the dependence on the luminosity and the He-rich envelope mass fraction is negligible (Kawaler & Bradley 1994; Córsico et al. (2007a). c Córsico et al. (2007b). d Córsico et al. (2008). e Reed et al. (2000). f Fu et al. (2007). g Kawaler et al. (2004). h Vauclair et al. (2002). i . j Bond et al. (1996). PG 1159−035. To assess the total mass of NGC 1501 we have considered the high-luminosity regime of the evolutionary sequences, while for PG 2131+066 and PG 1707+427 we have focused on the stages following the "evolutionary knee" for the PG 1159 stars, i.e. the low-luminosity regime.
3.1. First method: comparing the observed period spacing (∆Π O ℓ ) with the asymptotic period spacing (∆Π a ℓ ) Fig. 1 displays the asymptotic period spacing for ℓ = 1 modes as a function of the effective temperature for different stellar masses. Also shown in this diagram is the location of PG 2131+066, with T eff = 95 ± 5 kK (Dreizler & Heber 1998), and ∆Π O ℓ=1 = 21.6 ± 0.4 s (Reed et al. 2000), PG 1707+427, with T eff = 85±5 kK (Dreizler & Heber 1998), and ∆Π O ℓ=1 = 23.0±0.3 s (Kawaler et al. 2004), and NGC 1501, with T eff = 134 ± 5 kK (Werner & Herwig 2006), and ∆Π O ℓ=1 = 22.3 ± 0.3 s (Bond et al. 1996). The asymptotic period spacing is computed as
∆Π a ℓ = Π 0 / √ ℓ(ℓ + 1), where Π 0 = 2π 2 r 2 r 1 (N/r)dr −1(1)
and N is the Brunt-Väisälä frequency (Tassoul et al. 1990). From the comparison between the observed ∆Π O ℓ=1 and ∆Π a ℓ=1 we found a stellar mass of 0.627 M ⊙ for PG 2131+066, 0.597 M ⊙ for PG 1707+427, and 0.571 M ⊙ for NGC 1501 (second column in Table 1).
The method employed here is computationally inexpensive and widely used because it does not involve pulsational calculations. However, we must emphasize that the derivation of the stellar mass using the asymptotic period spacing may not be entirely reliable in pulsating PG1159 stars that pulsate with modes characterized by low and intermediate radial orders (see Althaus et al. 2007). This is particularly true for PG 2131+066 and PG 1707+427. This shortcoming of the method is due to that the asymptotic predictions are strictly valid in the limit of very high radial order (long periods) and for chemically homogeneous stellar models, while PG1159 stars are supposed to be chemically stratified and characterized by strong chemical gradients built up during the progenitor star life. A more realistic approach to infer the stellar mass of PG1159 stars is presented below.
3.2. Second method: comparing the observed period spacing (∆Π O ℓ ) with the average of the computed period spacings (∆Π ℓ )
The average of the computed period spacings is assessed as ∆Π ℓ = (N − 1) −1 k ∆Π k , where the "forward" period spacing is defined as ∆Π k = Π k+1 − Π k (k being the radial order) and N is the number of computed periods laying in the range of the observed periods. For PG 2131+066, Π k ∈ [340, 600] s, according to Kawaler et al. (1995); for PG 1707+427, Π k ∈ [330, 920] s, according to Kawaler et al. (2004); and for NGC 1501, Π k ∈ [1150, 2000] s, according to Bond et al. (1996).
This method is more reliable for the estimation of the stellar mass of PG1159 stars than that described above because, provided that the average of the computed period spacings is evaluated at the appropriate range of periods, the approach is appropriate for the regimes of short, intermediate and long periods (i.e., ∀k) as well. When the average of the computed period spacings is taken over a range of periods characterized by high k values, then the predictions of the present method become closer to those of the asymptotic period spacing approach. On the other hand, the present method requires of detailed period computations, at variance with the method described in the above section. In addition, we note that both methods for assessing the stellar mass rely on the spectroscopic effective temperature, and the results are unavoidably affected by its associated uncertainty.
In Fig. 2 we show the run of average of the computed period spacings (ℓ = 1) for PG 2131+066, PG 1707+427, and NGC 1501 in terms of the effective temperature for all of our PG1159 evolutionary sequences. The run of ∆Π ℓ depends on the range of periods on which the average of the computed period spacing is done. Note that the lines shown in Fig. 2 are very jagged and jumped. This is because that, for a given star, the av-erage of the computed period spacings is evaluated for a fixed period interval, and not for a fixed k-interval 1 .
By adopting the effective temperature of PG 2131+066, PG 1707+427, and NGC 1501 as given by spectroscopy we found a stellar mass of 0.578 M ⊙ , 0.566 M ⊙ , and 0.576 M ⊙ , respectively. Our results are shown in the third column of Table 1. These values are 8.5% (for PG 2131+066) and 5.5% (for PG 1707+427) smaller than those derived through the asymptotic period spacing, showing once again that the asymptotic approach overestimates the stellar mass of PG1159 stars that, like PG 2131+066 and PG 1707+427, exhibit short and intermediate pulsation periods (see Althaus et al. 2008a). On the contrary, there is a very small discrepancy (in the opposite direction) of ∼ 0.9% for the case of NGC 1501, showing that in the long-period regime the results for the stellar mass obtained using the asymptotic period spacing and the average of the computed period spacings nicely agree each other. A similar situation is found in the case of RX J2117.1+3412 (Córsico et al. 2007a).
Third method: using an approximate formula
To compare with previous works, we make an additional estimation of the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 using the approximate expression for the overall structure parameter Π 0 derived by Kawaler & Bradley (1994):
Π 0 ≈ 15.5 M * M ⊙ −1/3 L * 100 L ⊙ −0.035 q y 10 −3 −0.00012 (2)
where q y is the He-rich envelope mass fraction. This expression is derived by considering the dependence of the asymptotic period spacing on the total mass, stellar luminosity, and thickness of the He-rich outer envelope for a large grid of "quasi evolutionary" PG1159 models in the luminosity range 1.6 < ∼ log(L * /L ⊙ ) < ∼ 3.0. Both the present method and the method described in Sect. 3.1 are almost equivalent because they are based on the asymptotic period spacing.
Due to the very weak dependence of Π 0 on q y , we arbitrarily fix it to a value of 10 −2 . Since the luminosity is not known at the outset, we can compute it as L * = 4πσR 2 * T 4 eff , where R 2 * = GM * /g. We use the values of g and T eff inferred through spectroscopy. Assuming that Π 0 is known from the observed period spacing
(Π 0 ∼ √ ℓ(ℓ + 1) ∆Π O ℓ )
, we obtain an estimation of the stellar mass from Eq. (2). Our results are shown in the fourth column of Table 1. For PG 2131+066 and PG 1707+427 the stellar masses obtained in this way are in very good agreement with our values derived from the asymptotic period spacing (first row in Table 1). This is not an unexpected result because, as mentioned, the expression of Kawaler & Bradley (1994) is also based on the asymptotic period spacing. The slight differences found (below ≈ 2.5%) could be due to differences in the modeling of PG 1159 stars. For NGC 1501, instead, there is a substantial difference (≈ 7%) between the prediction of this formula and our value inferred from the asymptotic period spacing. This could be due to the inadequacy of the formula of Kawaler & Bradley (1994) for the high-luminosity regime characterizing the evolutionary status of NGC 1501.
Constraints from the individual observed periods
In this approach we seek pulsation models that best match the individual pulsation periods of PG 2131+066, PG 1707+427, and NGC 1501. For the three stars, we assume that all of the observed periods correspond to ℓ = 1 modes because the observed period spacings and the frequency splittings by rotation are consistent with ℓ = 1 (Kawaler et al. 1995;Kawaler et al. 2004;Bond et al. 1996). To measure the goodness of the match between the theoretical pulsation periods (Π T k ) and the observed individual periods (Π O i ), we follow the same χ 2 procedure as in our previous works -see, e.g, Córsico et al. (2007a) and Townsley et al. (2004). Specifically, we employ the quality function defined as
χ 2 (M * , T eff ) = 1 n n i=1 min (Π O i − Π T k ) 2 (3)
where n is the number of observed periods. The observed periods are shown in the first column of Tables 2 and 3 for PG 2131+066 and PG 1707+427, and in Table 7 of Bond et al. (1996) for NGC 1501. Next, we briefly explain the procedure we follow to found a model representative of a target star. For a given model (characterized by a given T eff ) corresponding to an evolutionary sequence of stellar mass M * , we consider the first observed period of the list, namely Π O 1 , and compute the successive squared dif-
ferences (Π O 1 − Π T k ) 2 ,
where the radial order k varies from 1 to a given maximum value, k max , which corresponds to a theoretical period far longer that the maximum period observed in the star. Next, we retain the minor squared difference, and then we repeat the procedure but this time considering the second observed period, namely Π O 2 . After the minor squared difference associated with this observed period is stored, we proceed with the next observed period, and so until the minor squared difference associated with the last observed period (Π O n ) is stored. The next step is to calculate the sum of these differences and then obtain the value of χ 2 (M * , T eff ) for the model under consideration. It is worth mentioning that with this algorithm, the value of χ 2 (M * , T eff ) does not depend on the particular order in which the observed periods are fitted. The complete algorithm is repeated for all of the models of the sequence, and then, a curve of χ 2 (M * , T eff ) versus T eff is obtained for the complete sequence. This procedure is carried out for all of our sequences. For each star of interest, the PG 1159 model that shows the lowest value of χ 2 is adopted as the "best-fit model".
The search for the best-fit models
We evaluate the function χ 2 = χ 2 (M * , T eff ) for stellar masses of 0.530, 0.542, 0.556, 0.565, 0.589, 0.609, 0.664, and 0.741M ⊙ . For the effective temperature we employed a much finer grid (∆T eff = 10 − 30 K). The quantity (χ 2 ) −1 in terms of the effective temperature for different stellar masses is shown in Fig. 3 for PG 2131+066 (upper panel), PG 1707+427 (middle panel), and NGC 1501 (lower panel), together with the corresponding spectroscopic effective temperatures. We prefer to show in our plots the quantity (χ 2 ) −1 instead χ 2 in order to emphasize the location of models providing good agreements between observed and theoretical periods. As mentioned, the goodness of the match between the observed and theoretical periods is reflected by the value of χ 2 . The lower the value of χ 2 , the better the period match. We will consider -admittedly somewhat arbitrarily-that a peak in the quality function with χ 2 5 (that is, (χ 2 ) −1 0.2) is a good match between the theoretical and the observed periods.
For PG 2131+066 we find one strong maximum of (χ 2 ) −1 for a model with M * = 0.589 M ⊙ and T eff ≈ 102 kK. Such a pronounced maximum in the inverse of χ 2 implies an excellent agreement between the theoretical and observed periods. Another much less pronounced maxima, albeit at effective temperatures closer to the spectroscopic estimation for PG 2131+066, are encountered for M * = 0.609 M ⊙ at T eff ≈ 92.4 kK and M * = 0.565 M ⊙ at T eff ≈ 99.8 kK. However, because the agreement between observed and theoretical periods for these models are substantially poorer than for the one with M * = 0.589 M ⊙ , we adopt this last model as the best-fit asteroseismological model. A detailed comparison of the observed ℓ = 1, m = 0 periods in PG 2131+066 with the theoretical periods of the best-fit model is provided in Table 2. The high quality of our period fit is quantitatively reflected by the average of the absolute period differences
δΠ i = 1 n n i=1 |δΠ i |(4)
where δΠ i = Π O i − Π T k and by the root-mean-square residual
σ δΠ i = ( n i=1 |δΠ i | 2 ) n .(5)
Note that σ δΠ i is simply the function χ 2 evaluated at the best-fit model. For the best-fit model of PG 2131+066 we obtain δΠ i = 1.57 s and σ δΠ i = 2.32 s, which are indeed very small. For PG 1707+427 we find one strong maximum of (χ 2 ) −1 for a model with M * = 0.542 M ⊙ and T eff ≈ 89.5 kK, an effective temperature compatible with the spectroscopic determination. Another somewhat less pronounced maximum is found for a model with M * = 0.556 M ⊙ and T eff ≈ 83.6 kK. Despite the fact that this model has an effective temperature very close to the spectroscopic one, we choose the model with M * = 0.542 M ⊙ as the best-fit model for PG 1707+427, because the period fit is characterized by a better quality. Table 3 shows a comparison between observed ℓ = 1, m = 0 and computed periods of the bestfit model. We found in this case δΠ i = 1.75 s and σ δΠ i = 1.99 s.
The situation for NGC 1501 is markedly different than for PG 2131+066 and PG 1707+427. Indeed, as clearly shown in the lower panel of Fig. 3, for this star the (χ 2 ) −1 function exhibits numerous local maxima at several values of the effective temperature and the stellar mass (M * /M ⊙ = 0.556, 0.565, 0.589, 0.664, and 0.741) that have roughly the same amplitudes, making virtually impossible to isolate a clear and unambiguous seismological solution. Thus, for NGC 1501 we are unable to find a best-fit seismological model. This could be, in part, due to the fact that the periods detected in NGC 1501 can be associated to eigenmodes with radial orders k quite different from each other (with a mean spacing of ∆k > ∼ 6), in such a way that it is easy to find numerous models (characterized by strongly different T eff and M * ) that reproduce to a some extent the observed period spectrum of NGC 1501. It could also be that the impossibility to find a best fit model would be reflecting a different evolutionary history for NGC 1501 than that assumed in this work for our PG1159 sequences.
The fourth column in Tables 2 and 3 shows the rates of period change associated with the fitted modes for PG 2131+066 and PG 1707+427, respectively. Our calculations predict that all of the pulsation periods increase with time (Π > 0), in accordance with the decrease of the Brunt-Väisälä frequency in the core of the models induced by cooling. At the effective temperatures of PG 2131+066 and PG 1707+427, cooling has the largest effect onΠ, while gravitational contraction, which should result in a decrease of periods with time, becomes negligible and no longer affects the pulsation periods, except for the case of modes trapped in the envelope. Until now, the only secure measurements ofΠ in pre-white dwarf stars are those of PG 1159−035, the prototype of the class, by Costa et al. (1999) and more recently by . In this last paper the authors Table 2. Observed and theoretical (ℓ = 1) periods of the bestfit model for PG 2131+066 (M * = 0.589M ⊙ , T eff = 102 179 K, log(L * /L ⊙ ) = 1.57). Periods are in seconds and rates of period change (theoretical) are in units of 10 −12 s/s. The observed periods are taken from Kawaler et al. (1995). obtained a mix of positive and negative values ofΠ, indicating that for that star gravitational contraction is still important. In principle, the needed time interval that the observational data should cover in order to reach a measurement of aΠ in PG 1159 stars like PG 2131+066 and PG 1707+427 is of about ten years. Unfortunately, no future observations in the short term that could allow a determination ofΠ for these stars and thus to check the predictions of our models are foreseen. Finally, the last column in Tables 2 and 3 gives information about the pulsational stability/instability nature of the modes associated with the periods fitted to the observed ones. Full nonadiabatic calculations employing the pulsation code described in indicate that, for the case of PG 2131+066, all the fitted modes except one (that with period 507.9 s) are predicted to be unstable. For the case of PG 1707+427, our nonadiabatic computations are able to explain the existence of periodicities in the range 330 < ∼ Π O < ∼ 680 s only, while they fail to predict pulsational instability for the observed modes with periods at 726, 746, and 909 s.
Π O i Π T k kΠ
Characteristics of the best-fit models for PG 2131+066 and PG 1707+427
The main features of our best-fit model for PG 2131+066 are summarized in Table 4, where we also provide the parameters of the star extracted from other published studies. Specifically, the second column corresponds to spectroscopic results from Werner & Herwig (2006), whereas the third and fourth columns present results from the pulsation studies of Kawaler et al. (1995) and Reed et al. (2000), and from the asteroseismological model of this work, respectively. The number in parenthesis is the spectroscopic estimation of the stellar mass employing the evolutionary tracks of Miller Bertolami & . In the present work, errors in T eff and log(L * /L ⊙ ) are estimated from the width of the maximum in the function χ 2 with respect T eff and log(L * /L ⊙ ), respectively. The error in the stellar mass comes from the grid resolution in M * . Errors in the rest of the quantities are derived from these values. The effective temperature of our best-fit model (T eff = 102 180 K) is somewhat higher than -but still compatible with-the spectroscopic value (T eff = 95 000±5 000 K). On the other hand, the total mass of the best-fit model (M * = 0.589 M ⊙ ) is in agreement with the value derived from the average of the computed period spacings (M * ∼ 0.578 M ⊙ ), but at odds (∼ 6% smaller) with that inferred from the asymptotic period spacing (M * = 0.627 M ⊙ ) (see Table 1). Also, the M * value of our best-fit model is substantially larger than the spectroscopic mass of 0.55 M ⊙ derived by Miller Bertolami & , but very similar to 0.58 M ⊙ according to Werner & Herwig (2006). A discrepancy between the asteroseismological and the spectroscopic values of M * is generally encountered among PG1159 pulsators -see . Until now, the asteroseismological mass of PG 2131+066 has been about 11% larger (∆M * ≈ 0.06 M ⊙ ) than the spectroscopic mass if we consider the early estimation for the seismological mass quoted by Reed et al. (2000) and the derivation of Miller Bertolami & for the spectroscopic mass 2 . In light of the best-fit model derived in this paper, this discrepancy is slightly reduced to about 7% (∆M * ≈ 0.04 M ⊙ ). Finally, our best-fit model for PG 2131+066 is somewhat more luminous and less compact than what is suggested by the results of Reed et al. (2000). The main properties of our best-fit model for PG 1707+427 are shown in Table 5. The second column corresponds to spectroscopic results from Werner & Herwig (2006), whereas the third and fourth columns present results from the pulsation study of Kawaler et al. (2004) and from the asteroseismological model of this work, respectively. As for the case of PG 2131+066, the effective temperature of our best-fit model for PG 1707+427 is slightly larger than the spectroscopic measurement, but even in good agreement with it. Regarding the stellar mass, our best-2 We elect the value of the spectroscopic mass of PG 2131+066 inferred by Miller Bertolami & for this comparison because they use the same post-born again PG1159 evolutionary models we employ here in the determination of the asteroseismological mass, and because the spectroscopic masses quoted by Werner & Herwig (2006) are based on old post-AGB tracks. . Until now, the asteroseismological mass of PG 1707+427 has been more than 7% larger (∆M * ≈ 0.04 M ⊙ ) than the spectroscopic mass if we adopt for the seismological mass the value found by Kawaler et al. (2004) and the derivation of Miller Bertolami & for the spectroscopic mass. In light of our best-fit model, this discrepancy is strongly reduced to about 2% (∆M * ≈ 0.012 M ⊙ ). Finally, our best-fit model for PG 1707+427 is slightly more luminous than what is suggested by Kawaler et al. (2004).
The asteroseismological distance and parallax of PG 2131+066 and PG 1707+427
As in our previous works -see, e.g., Córsico et al. (2007a) -we employ the luminosity of our best-fit models to infer the seismic distance to PG 2131+066 and PG 1707+427. Following Kawaler et al. (1995), we adopt BC = −6.0 ± 0.5 for both stars (Werner et al. 1991). We account for the interstellar absorption, A V , using the interstellar extinction model of Chen et al. (1998). With all these ingredients the seismic distance, d, can be easily computed using the apparent magnitudes, which are m v = 16.6 for PG 2131+066 and m v = 16.7 for PG 1707+427 (Bond et al. 1984). We obtain a distance d ∼ 830 pc and an interstellar extinction A V ∼ 0.18 for PG 2131+066 and d ∼ 730 pc and A V ∼ 0.12 for PG 1707+427.
Our estimation of the distance to PG 2131+066 is ∼ 15% larger than that derived by CA06 (d ∼ 716 pc). This is because our asteroseismological model is somewhat more luminous than that of CA06 (log(L * /L ⊙ ) = 1.57 versus log(L * /L ⊙ ) = 1.37). On the other hand, our distance is 20 − 25% larger than that obtained by Reed et al. (2000) (d ∼ 668 pc) on the basis of their own asteroseismological analysis. This difference can be understood on the basis that Reed et al. (2000) uses a luminosity of log(L * /L ⊙ ) ∼ 1.4, somewhat lower than that of our best-fit model for PG 2131+066, of log(L * /L ⊙ ) ∼ 1.6. On the other hand, our asteroseismological distance for PG 2131+066 is about 1.2 times longer than that quoted by Reed et al. (2000) of ∼ 680 pc obtained on the basis of spectrum fitting, although both estimations are compatible at the 1σ level.
For PG 1707+427, our asteroseismological distance is in agreement with that quoted by CA06, of ∼ 697 pc. Werner et al. (1991) obtain a distance to PG 1707+427 of ∼ 1300 pc, substantially larger than our estimation, but still within the quoted error bars. The different value of Werner et al. (1991) is due to that they use a luminosity of log L * /L ⊙ = 2.15, substantially higher than the luminosity of our best fit model (log L * /L ⊙ = 1.4).
Comparison with the results of CA06
Following the recommendations of an anonymous referee, we include in this section a detailed comparison between the PG1159 models and the asteroseismological results of the present paper and those of the previous study by CA06. These authors performed an asteroseismological analysis of four GW Vir stars (PG 0122+200, PG 1159−035, PG 2131+066, and PG 1707+427) on the basis of a set of twelve PG1159 evolutionary sequences with different stellar masses (M * = 0.53, 0.54, 0.55, · · ·, 0.62, 0.63, 0.64M ⊙ ) artificially derived from the full evolutionary sequence of 0.5895M ⊙ computed by Althaus et al. (2005). That sequence is one of the sequences we use in the present paper. Specifically, the sequences of CA06 were constructed using LPCODE by appropriately scaling the stellar mass of the 0.589M ⊙ sequence before the models reach the low-luminosity, high-gravity stage of the GW Vir domain. Although this procedure leads to a series of unphysical stellar models for which the helium-burning luminosity is not consistent with the thermo-mechanical structure, the transitory stage vanishes shortly before the star reaches the evolutionary "knee" in the HR diagram (see Fig. 2 of CA06). As a consequence, those PG1159 models were not suitable for the highluminosity, low-gravity regime corresponding, for instance, to RX J2117.1+3412, NGC 1501, K 1-16, HE 1429-1209, etc.
Because the sequences of CA06 with different stellar masses were created starting from a single sequence with M * = (2006) clearly illustrate that, when the complete evolution of the PG1159 progenitor stars is taken into account, different stellar masses are associated with different central abundances of C and O, and different sizes of the C-O core. Specifically, the more massive the models, the lower (higher) the central abundance of O (C) and the larger the C-O core. In summary, for a given value of M * , and for stages after the evolutionary knee, the only structural/physical difference between the PG1159 models employed in the present work and those of CA06 is related to the size of the C-O core and the central abundances of O and C. In Fig. 4 we compare some evolutionary tracks (M * = 0.530, 0.542, 0.589, 0.609M ⊙ ) of the PG1159 sequences employed in the present work with the corresponding evolutionary tracks of CA06 (M * = 0.53, 0.54, 0.59, 0.61M ⊙ ). A careful inspection of this figure reveals that both sets of tracks generally differ, but when the models reach the beginning of their white dwarf stage (log T eff 5), they turn be in very close agreement. This agreement is reached earlier in the case of sequences with stellar masses close to M * = 0.589M ⊙ , the value of the sequence from which the remainder sequences of CA06 were generated. Interestingly enough, the regime in which the tracks of CA06 are in agreement with those employed in the present paper embraces the location of PG 2131+066 and PG 1707+427, which have log T eff = 4.97 and log T eff = 4.93, respectively. Because of this, it is expected that the global pulsation properties (i.e., asymp- totic period spacing, average of the computed period spacing) of both sets of models in that regime should nearly agree, and consequently the asteroseismological inferences on PG 2131+066 and PG 1707+427 based on these two different sets of models should not be substantially distinct.
In Table 6 we present a comparison between the stellar masses inferred in CA06 (their Table 1) and in the present study. Note that, not surprisingly, the stellar masses derived in CA06 are in excellent agreement with the values obtained in the present work, irrespective of the particular method employed, being the differences in all of the cases below 2%. This is a clear indication that the asteroseismological results of this paper and that of CA06 for PG 2131+066 and PG 1707+427 are not seriously affected by the differences between both sets of models. This adds credibility and robustness to the asteroseismological results of the present study. This conclusion should change for the case of PG1159 pulsators with stellar masses too departed from the value ∼ 0.589M ⊙ and/or located at earlier evolutionary stages.
Summary and conclusions
In this paper we presented an asteroseismological study of the high-gravity, low-luminosity pulsating PG1159 stars PG 2131+066 and PG 1707+427 and of the high-luminosity PNNV [WCE] star NGC 1501. This is the fourth article of a short series of studies aimed at exploring the internal structure and evolutionary status of pulsating PG1159 stars which have been intensively observed through multi-site campaigns. Our analysis is based on the full PG1159 evolutionary models of Althaus et al. (2005 and . These models represent a solid basis to analyze the evolutionary and pulsational properties of pre-white dwarf stars like PG 2131+066, PG 1707+427, and NGC 1501.
We first used the observed period-spacing data to obtain estimations of the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501. The results are summarized in Table 1, where we provide a summary of results from the present and the previous works by us, and also from other pulsation and spectroscopic studies. We obtained three mass values for each star: the first one by comparing the observed period spacing with the asymptotic period spacing of our models (an inexpensive and widely used approach that does not involve pulsational calculations); the second one by comparing the observed period spacings with the average of the computed period spacing (an approach that requires of detailed period computations); and the third one on the basis of the approximate formula of Kawaler & Bradley (1994), which is based on the behavior of the asymptotic period spacing of a large grid of quasi-evolutionary PG1159 models. The first and the third approaches are almost equivalent, and lead to similar, somewhat overestimated values of the stellar mass for PG 2131+066 and PG 1707+427. The second approach, clearly more realistic, conducts to smaller values of M * (and closer to the spectroscopic inferences) in the case of PG 2131+066 and PG 1707+427, and virtually the same M * value than that obtained from the first approach in the case of NGC 1501.
In the second part of our work, we sought for the models that best reproduce the individual observed periods of each star. The period fits were made on a grid of PG1159 models with a quite fine resolution in effective temperature (∆T eff ∼ 10 − 30 K) although admittedly coarse in stellar mass (∆M * ∼ 0.01 − 0.08 M ⊙ ). We found asteroseismological models only for PG 2131+066 and PG 1707+427. For NGC 1501 we were unable to find a clear and unambiguous seismological solution due to the existence of numerous and equivalent minima characterizing the quality function employed in the period-fit procedure. The pulsational properties of the "best-fit" models for PG 2131+066 and PG 1707+427 are summarized in Tables 2 and 3, respectively. In particular, we predict the values of the rates of period change to be positive and in the range (2 − 5) × 10 −12 s/s. Unfortunately, we have been unable to check the reality of this prediction because the lack of any measurement ofΠ for PG 2131+066 and PG 1707+427 for the moment. The structural characteristics of these best-fit models are shown in Tables 4 and 5. In particular, the seismological masses are closer to the spectroscopic ones in light of our best-fit models, as we found in our previous works. In these tables we also show the seismological distances and parallaxes of PG 2131+066 and PG 1707+427. We found a reasonable agreement between our results and those of Kawaler et al. (1995), Reed et al. (2000), Kawaler et al. (2004), and CA06. We stress that almost all differences between our results (Sections 4.2 and 4.3) and those of earlier works are within the quoted errors.
In summary, in this work we have been able to estimate the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 on the basis of the period-spacing information alone. We have also been successful in finding asteroseismological models for PG 2131+066 and PG 1707+427 from period-to-period comparisons. In particular, the T eff and log g of the best-fit models are in very good agreement with the spectroscopic measurements. Unfortunately, we fail to found an asteroseismological model for NGC 1501. In principle, this shortcoming of our study could be indicating some inadequacy inherent to the stellar modeling. Another possible alternative could be the fact that the period spectrum of NGC 1501 includes periodicities associated with g-modes with radial orders very spaced from each other, in such a way that our χ 2 procedure of period-fit is inefficient to isolate a clear and unambiguous asteroseismological solution. On the other hand, it would be kept in mind that while both PG 2131+066 and PG 1707+427 are classified as PG1159 stars, NGC 1501 is a [WCE] star. Although both classes are suspected to form an evolutionary sequence, this possibility is still under debate (Crowther 2008;Todt et al. 2008) and it could not be the case. Therefore, the failure of our models to fit the period spectrum of NGC 1501 might be indicative that this star has a very different evolutionary history than PG 2131+066 and PG 1707+427.
We have also included a comparison between the models and results of the present work and those of the study by CA06. The models employed in the present work are the result of the complete evolution of the progenitor stars, and as a result, they are characterized by central chemical abundances and a size of the C-O core which are consistent with the value of the stellar mass. This is not the case for the models employed by CA06, which were artificially derived from the full evolutionary sequence of 0.589M ⊙ computed by Althaus et al. (2005). In spite of these differences, our asteroseismological results are in excellent agreement with those of CA06. This adds credibility and robustness to the results of the present study.
As the main conclusions of the present work, we can mention:
-The full evolutionary models of PG1159 stars employed in the present work lead to asteroseismological results for PG 2131+066 and PG 1707+427 that do not differ substantially from those predicted by CA06 on the basis of evolutionary sequences generated artificially. We note, however, that this agreement between both sets of computations is valid for PG1159 stars located at the low-luminosity, highgravity regime after the stars have passed the evolutionary knee in the HR diagram. It should be kept in mind that this conclusion should change for the case of stars located at earlier stages of evolution. -At present, the PG1159 evolutionary models used in this work -and in the previous studies of this series-remain the only suitable for asteroseismological inferences on stars that are located at the high-luminosity, low-gravity regime before the evolutionary knee, such as RX J2117.1+3412 and NGC 1501. -The detailed fitting of the individual periods (Sect. 4) gives somewhat different masses than analysis based on asymptotic period spacing (methods 1 and 3 of Sect. 3), but in very good agreement with the values of M * derived from the average of the computed period spacings (method 2 of Sect.
3). Thus, method 2 is a very appropriate way to estimating stellar masses, and detailed period fits do not significantly improve the mass determinations. We note, however, that the period-fit approach yields an asteroseismological model from which one can infer, in addition to M * , the luminosity, radius, gravity, and distance of the target star. In addition, the period-fit approach does not require -in principle-external constraints such as the spectroscopic values of T eff and g, i.e., the method works "by letting the pulsation modes speak for themselves" (see Metcalfe 2005 for an interesting discussion about this). -The nonadiabatic stability analysis does not at the moment predict instability for all of the fitted modes. This means that, in the frame of the linear nonadiabatic pulsation theory, some pulsation modes detected in PG 2131+066 and PG 1707+427 should not be excited. It is not clear at this stage the origin of this discrepancy. Maybe it could be attributed to the extreme sensitivity of the stability analysis of PG1159 stars to the exact amounts of the main atmospheric constituents (see Quirion et al. 2004 for details). -The main conclusion of this series of papersthe present work and Córsico et al. (2007aCórsico et al. ( , 2007bCórsico et al. ( , 2008)-is that for most well-observed pulsating PG1159 stars (RX J2117.1+3412, PG 0122+200, PG 1159−035, PG 2131+066, and PG 1707+427) it is possible to found a stellar model (the asteroseismological model) with M * and T eff near the spectroscopic measurements to a high internal accuracy. The next step is of course an assessment of the question if the asteroseismological models can provide more accurate masses for these objects. The scatter in the masses derived from the different asteroseismological methods (see Table 1) suggests that it may not be the case. In fact, when all asteroseismological methods are considered, the uncertainty in the determination of the mass amounts to ∼ 0.05M ⊙ , com-parable to the spectroscopic one (∼ 0.05 − 0.1M ⊙ ; Werner et al. 2008). However, it is worth noting that, when results based on asymptotic period spacing (an approach that is not correct for the high-gravity regime of PG1159 stars; see Althaus et al. 2008a) are not taken into account, the scattering in the derived masses is of only ∼ 0.02M ⊙ .
We close the paper by noting that the PG1159 evolutionary models employed in our series of asteroseismological studies are characterized by thick He-rich outer envelopes, as they are predicted by the standard theory for the formation of PG1159 stars. However, Althaus et al. (2008b) have recently demonstrated that the assumption of thinner He-rich envelopes solves the longstanding discrepancy between the measured rates of period change in the prototypical star PG 1159−035 and the predictions of theoretical models. In view of this important result, we are planning future asteroseismological studies for all the pulsating PG1159 stars analyzed in our series of articles, but with non-canonical PG1159 models characterized by thinner He-rich envelopes.
Fig. 1 .
1The dipole asymptotic period spacing in terms of the effective temperature. Numbers along each curve denote the stellar mass (in solar units). Dashed (solid) lines correspond to evolutionary stages before (after) the turning point at the maxima effective temperature of each track. Also shown are the locations of PG 2131+066, PG 1707+427, and NGC 1501. The masses of these stars as derived by comparing ∆Π a ℓ=1 with ∆Π O ℓ=1 are M * = 0.627 M ⊙ , M * = 0.597 M ⊙ , and M * = 0.571 M ⊙ , respectively. Note that to infer the mass of NGC 1501, we have considered the stages before the evolutionary knee (dashed lines).
Fig. 2 .
2Same asFig. 1, but for the average of the computed period spacings. For PG 2131+066 and PG 1707+427 only the stages after the "evolutionary knee" have been plotted. The masses of PG 2131+066, PG 1707+427, and NGC 1501 as derived by comparing ∆Π ℓ=1 with ∆Π O ℓ=1 are M * = 0.578 M ⊙ , M * = 0.566 M ⊙ , and M * = 0.576 M ⊙ , respectively. As inFig. 1, we infer the mass of NGC 1501 by considering the stages before the evolutionary knee (dashed lines).
Fig. 3 .
3The inverse of the quality function of the period fit in terms of the effective temperature (see text for details). The vertical grey strip indicates the spectroscopic T eff and its uncertainties. The curves have been arbitrarily shifted upward (with a step of 0.1). Upper panel corresponds to PG 2131+066, middle panel corresponds to PG 1707+427, and lower panel corresponds to NGC 1501. For NGC 1501, the stages before the evolutionary knee are displayed with dotted lines.
Fig. 4 .
4Comparison between some PG1159 evolutionary tracks employed in the present work (thick lines) and those of CA06 (thin lines). The values are in solar masses. Note that the track corresponding to the sequence of ≈ 0.59M ⊙ is the same for the two sets of computations. 0.589M ⊙ , the central abundances of C and O and the spatial extension of the C-O core are not completely consistent with the value of the stellar mass, except in the case of the models of the 0.589M ⊙ sequence itself. For instance, a model of CA06 with M * = 0.64M ⊙ has a C-O core that is somewhat smaller and the central abundance of O is substantially higher than what would be expected if the complete evolution of the progenitor star were performed, as it is the case for the models employed in the present paper.Fig. 2of Miller Bertolami & Althaus
Tk
unstable
341.45 341.88 14 2.04
yes
-
363.66 15 3.90
yes
384.27 383.94 16 2.60
yes
403.93 403.30 17 3.30
yes
426.36 426.48 18 4.14
yes
450.28 445.31 19 2.89
yes
462.39 465.91 20 4.53
yes
-
488.81 21 4.11
no
507.91 507.91 22 3.79
no
Table 3. Same as Table 2, but for the best-fit model for PG
1707+427 (M * = 0.542 M ⊙ , T eff = 89 504 K, log(L * /L ⊙ ) =
1.40). The observed periods are taken from Kawaler et al.
(2004).
Π O
i
Π T
k
kΠ T
k
unstable
334.62 333.26 12 2.13
yes
-
355.73 13 1.19
yes
-
378.53 14 2.17
yes
-
401.64 15 1.51
yes
-
423.68 16 1.84
yes
448.07 447.31 17 2.46
yes
-
469.26 18 1.49
yes
494.39 492.56 19 2.76
yes
-
516.50 20 2.61
yes
536.41 537.67 21 1.66
yes
-
562.32 22 3.30
yes
-
584.78 23 2.37
yes
-
606.62 24 2.73
yes
-
632.11 25 3.25
yes
-
653.15 26 2.99
yes
677.89 677.38 27 3.43
yes
-
701.20 28 3.03
yes
726.02 722.82 29 3.95
no
745.78 747.58 30 3.42
no
-
770.93 31 3.71
no
-
793.13 32 4.45
no
-
817.86 33 3.61
no
-
841.70 34 4.64
no
-
863.34 35 3.91
no
-
888.59 36 5.03
no
909.05 912.34 37 4.21
no
Table 4 .
4The main characteristics of PG 2131+066.Quantity
Spectroscopy
Seismology
This work
T eff [kK]
95 ± 5
-
102.18 +3.0
−2.8
M * [M ⊙ ]
0.58 ± 0.1
0.608 ± 0.01 0.589 +0.020
−0.024
(0.55 ± 0.1)
log g [cm/s 2 ]
7.5 ± 0.5
-
7.63 +0.12
−0.14
log(L * /L ⊙ )
-
1.41 ± 0.5
1.57 +0.07
−0.06
log(R * /R ⊙ )
-
−1.73
−1.71 +0.06
−0.05
M env [M ⊙ ]
-
3.8 × 10 −3
0.016
M V [mag]
-
7.69 +0.25
−0.18
6.825 +0.655
−0.675
M bol [mag]
-
-
0.825 +0.155
−0.175
A V [mag]
-
-
0.18
d [pc]
-
668 +78
−83
830 +300
−224
π [mas]
-
1.50 ± 0.2
1.2 +0.4
−0.3
Table 5 .
5Same asTable 4, but for PG 1707+427.Quantity
Spectroscopy Seismology
This work
T eff [kK]
85 ± 4.5
-
89.5 +1.7
−1.8
M * [M ⊙ ]
0.59 ± 0.1
0.57 ± 0.02
0.542 +0.014
−0.012
(0.53 ± 0.1)
log g [cm/s 2 ]
7.5 ± 0.3
-
7.53 +0.09
−0.08
log(L * /L ⊙ )
-
1.36
1.40 ± 0.04
log(R * /R ⊙ )
-
-
−1.68 ± 0.04
M env [M ⊙ ]
-
-
0.025
M V [mag]
-
-
7.25 ± 0.6
M bol [mag]
-
-
1.25 ± 0.1
A V [mag]
-
-
0.12
d [pc]
-
-
730 +230
−175
π [mas]
-
-
1.4 ± 0.4
fit model has M * = 0.542 M ⊙ , which is in agreement with the
value derived from the average of the computed period spacings
(M * ∼ 0.566 M ⊙ ), but at odds (∼ 9% lower) with that inferred
from the asymptotic period spacing (M * = 0.597 M ⊙ ) (see Table
1). On the other hand, we note that M * for the best-fit model
is in excellent agreement with the spectroscopic derivation of
Miller Bertolami & Althaus (2006) (0.542 M ⊙ versus 0.53 M ⊙ ),
but is substantially lower than the spectroscopic value quoted by
Werner & Herwig (2006) (0.59 M ⊙ )
Table 6 .
6Comparison between the stellar mass values (in M ⊙ ) obtained in CA06 and in the present work.PG 2131+066
∆Π a
ℓ
∆Π ℓ
Period fit
CA06
0.615 0.575
0.60
This work
0.627 0.578
0.589
∆M * /M *
2 %
0.3 %
2 %
PG 1707+427
∆Π a
ℓ
∆Π ℓ
Period fit
CA06
0.595 0.565
0.55
This work
0.597 0.566
0.542
∆M * /M *
0.3 % 0.1 %
1.5 %
As the star evolves towards higher (lower) effective temperatures, the periods generally decrease (increase) with time. At a given T eff , there are N computed periods laying in the chosen period interval. Later when the model has evolved enough (heated or cooled) it is possible that the accumulated period drift nearly matches the period separation between adjacent modes (|∆k| = 1). In these circumstances, the number of periods laying in the chosen (fixed) period interval is N ± 1, and ∆Π ℓ exhibits a little jump.
Acknowledgements. Part of this work was supported by the MEC grant AYA05-08013-C03-01, by the European Union FEDER funds, by the AGAUR, by AGENCIA through the Programa de Modernización Tecnológica BID 1728/OC-AR, and by the PIP 6521 grant from CONICET. This research has made use of NASA's Astrophysics Data System.
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| []
|
[
"Anomalous translational velocity of vortex ring with finite-amplitude Kelvin waves",
"Anomalous translational velocity of vortex ring with finite-amplitude Kelvin waves"
]
| [
"C F Barenghi \nSchool of Mathematics\nUniversity of Newcastle\nNE1 7RUNewcastleUK\n",
"R Hänninen \nDepartment of Physics\nOsaka City University\nSugimoto 3-3-138558-8585OsakaJapan\n",
"M Tsubota \nDepartment of Physics\nOsaka City University\nSugimoto 3-3-138558-8585OsakaJapan\n"
]
| [
"School of Mathematics\nUniversity of Newcastle\nNE1 7RUNewcastleUK",
"Department of Physics\nOsaka City University\nSugimoto 3-3-138558-8585OsakaJapan",
"Department of Physics\nOsaka City University\nSugimoto 3-3-138558-8585OsakaJapan"
]
| []
| We consider finite-amplitude Kelvin waves on an inviscid vortex assuming that the vortex core has infinitesimal thickness. By numerically solving the governing Biot-Savart equation of motion, we study how the frequency of the Kelvin waves and the velocity of the perturbed ring depend on the Kelvin wave amplitude. In particular, we show that, if the amplitude of the Kelvin waves is sufficiently large, the perturbed vortex ring moves backwards. | 10.1103/physreve.74.046303 | [
"https://arxiv.org/pdf/physics/0604008v3.pdf"
]
| 11,425,808 | physics/0604008 | f963e62aac8eddd458eff6c1f7ae111c0780c902 |
Anomalous translational velocity of vortex ring with finite-amplitude Kelvin waves
16 Oct 2006
C F Barenghi
School of Mathematics
University of Newcastle
NE1 7RUNewcastleUK
R Hänninen
Department of Physics
Osaka City University
Sugimoto 3-3-138558-8585OsakaJapan
M Tsubota
Department of Physics
Osaka City University
Sugimoto 3-3-138558-8585OsakaJapan
Anomalous translational velocity of vortex ring with finite-amplitude Kelvin waves
16 Oct 2006(Dated: November 6, 2018)arXiv:physics/0604008v3 [physics.flu-dyn]numbers: 4732cf6740Vs6757Fg
We consider finite-amplitude Kelvin waves on an inviscid vortex assuming that the vortex core has infinitesimal thickness. By numerically solving the governing Biot-Savart equation of motion, we study how the frequency of the Kelvin waves and the velocity of the perturbed ring depend on the Kelvin wave amplitude. In particular, we show that, if the amplitude of the Kelvin waves is sufficiently large, the perturbed vortex ring moves backwards.
I. INTRODUCTION
Vortex rings are among the most important and most studied objects of fluid mechanics [1,2]. It has been known since the times of Lord Kelvin [3] that a vortex ring is subject to wavy distortions (sinusoidal displacements of the vortex core) called Kelvin waves [4]. In the case of viscous vortex rings, the stability of these waves is a problem with subtle aspects [5] which are still the focus of intense mathematical scrutiny [6]. Our concern is the simpler case in which the fluid is inviscid and the vortex core has infinitesimal thickness. This case refers to the idealized context of classical Euler fluids, but is realistic for superfluids, which have zero viscosity and microscopic vortex core thickness.
Vortex rings have indeed been central to superfluidity [7] since the pioneering experiments on the nucleation of quantized vorticity by moving ions [8], the early investigations into rotons as ghosts of vanished vortex rings [9], and the nature of the superfluid transition [10]. The current interest in superfluid vortex rings extends to the physics of cold atomic gases [11] and the discovery of new nonlinear solutions [12] of the Gross-Pitaevskii's nonlinear Schrödinger equation (NLSE) for a Bose-Einstein condensate. Vortex rings are also important in the study of superfluid turbulence [13]. For example, they have been used as tools to study the Kelvin wave cascade [14] which is responsible for the dissipation of turbulent kinetic energy near absolute zero, and to investigate the effects of vortex reconnections [15], which are the key feature of turbulence; they are also used as simple models of the vortex loops which make up the turbulence [16].
Kelvin waves play a role in all examples listed above. The dispersion relation of Kelvin waves of infinitesimal amplitude A on a circular vortex ring of given radius R, circulation κ, and vortex core radius a is [17]
ω = κ 2πa 2 1 − 1 + ka K 0 (ka) K 1 (ka) ,(1)
where ω is the angular velocity of the wave and k the wave number. Functions K n (x) are modified Bessel functions of order n. The above dispersion relation is also valid for waves on a straight vortex [18]. The properties of small-amplitude Kelvin waves have been already investigated [19], but little is known of what happens at large wave amplitude. The stability problem becomes nonlinear, hence more difficult, and a numerical approach is necessary.
Recently, an astonishing prediction was made by Kiknadze and Mamaladze [20] that, at sufficiently large amplitude, the perturbed vortex ring moves backwards. Unfortunately the prediction arises from numerical analysis based on the local induction approximation (LIA) to the exact equation of motion, which is the Biot-Savart law (BSL). The advantage of the LIA over the BSL is that it is analytically simpler and computationally cheaper. If N is the number of discretization points along a vortex filament, the cost of the computation grows as N under the LIA, whereas under the BSL it grows as N 2 . The use of the LIA was pioneered by Schwarz [22] in his numerical studies of homogeneous isotropic turbulence. His results obtained using the LIA compared reasonably well with results obtained using the BSL, because long-range effects tend to cancel out in the isotropy vortex configurations which he considered. In less isotropic cases, however, for example in rotating turbulence [23], the LIA may not be a good approximation. In particular, the LIA yields wrong predictions about the stability and motion of vortex knots [24], structures which are geometrically similar to (although topological different from) the perturbed vortex rings considered by Kiknadze and Mamaladze [20].
Our first aim is thus to use the exact BSL to investigate the claim of Kiknadze and Mamaladze that the perturbed vortex ring can move backwards [20]. Our second aim is to carry out a more detailed examination of the effects of large-amplitude Kelvin waves on the motion of a vortex ring.
II. MODEL
Our approach is based on the vortex filament model of Schwarz [22] which is appropriate to superfluid helium due to the smallness of the vortex core radius a com-pared to the radius of the vortex ring R. Essentially, a vortex is treated as a topological line defect, that is to say a curve in three-dimensional space. In the absence of dissipation (zero temperature), the vortex at the point r moves with velocity dr/dt = v L where v L is equal to the local superfluid velocity v s that is given by the following Biot-Savart line integral calculated along the entire vortex configuration:
v s (r, t) = κ 4π (s − r) × ds |s − r| 3 .(2)
Here s denotes a variable location along the vortex filament. To implement the BSL, the vortex configuration is discretized into a large number of segments. The technique to handle the singularity that one meets when one tries to evaluate the integral at those discrete points that are used to describe the vortex line can be avoided by splitting the integral into local and nonlocal parts [22]. The velocity of a point s on the vortex is thus
v L = κ 4π s ′ × s ′′ ln 2 l + l − e 1/2 a + κ 4π ′ (s 1 − s) × ds 1 |s 1 − s| 3 .
(3) where ξ is the arc length, the vectors s ′ = ds/dξ, s ′′ = d 2 s/dξ 2 are, respectively, the local tangent and the local normal to the vortex at the point s. The quantities l − and l + are the lengths of the line segments connected to the discretization point s and the prime above the integral symbol means that the line integration now extends only along the remaining vortex segments. One should note that we use a hollow core vortex, which results that the scaling factor in front of a in Eq. (3) is exp(1/2) rather than exp(1/4) which is for a solid rotating core and appears in a paper by Schwarz [22]. The recent progress in using the Gross-Pitaevskii nonlinear Schrödinger equation for quantum fluids suggests that the hollow core model should be more appropriate [25]. The exact value of the core size is not important here. What matters is that it is orders of magnitudes smaller than the radius of the ring or the amplitude of the waves, so that we can use the concept of vortex filament. For example, in a typical helium turbulence experiment the measured vortex line density L is 10 4 or 10 6 cm −2 , which means that the intervortex spacing is 1/ √ L = 0.01 or 0.001 cm, which is a million or hundred thousands times bigger than the vortex core radius (10 −8 cm) in 4 He.
The local induction approximation (LIA) is obtained by neglecting the nonlocal part and is typically written in the form:
v L = βs ′ × s ′′ ,(4)
where β = κ ln(c R /a)/4π, R is some average curvature, and c is of order unit; the last two parameters are adjusted to obtain better agreement with full nonlocal calculations. By choosing c = 8 exp(−1/2) and R to be the local radius of curvature one obtains fairly good results and additionally a limit that gives correctly the velocity for the perfect ring. The calculation of the kinetic energy E of the vortex would not be accurate if carried out on a threedimensional mesh around the vortex due to rapid changes of the velocity field near the vortex core. Fortunately in our case the vortex filament forms a closed loop and the velocity field goes to zero at infinity (the calculation is performed in an infinite box), hence it is appropriate [26] to use Saffman's formula [2]
E = κρ s v s · s × ds,(5)
where the line integration is performed along the vortex filament and ρ s is the superfluid density. The initial condition consists of a vortex ring of radius R with superimposed N Kelvin waves of amplitude A (that is, the wavelength of the perturbation is 2πR/N ). Using cylindrical coordinates r, φ, and z, the Cartesian coordinates of the initial vortex ring are thus
x = R cos φ + A cos(N φ) cos φ, y = R sin φ + A cos(N φ) sin φ, (6) z = −A sin(N φ).
In the absence of Kelvin waves (A = 0) the circular vortex ring moves in the positive z direction with selfinduced translational speed [27] v ring = κ 4πR
[ln (8R/a) − 1/2].
We have tested that, in the case of a circular ring, our numerical method agrees fairly well with this result. All results presented here are obtained using ring radius R = 0.1 cm and values of a and κ which refer to 4 He (κ = h/m 4 = 9.97 × 10 −4 cm 2 /s, where m 4 is the mass of one atom, and a = 1.0 × 10 −8 cm). The dependence of the results on a is small, since a appears only in the slow varying logarithmic term in Eq. (3). The numerical method to evolve the perturbed vortex ring under the BSL is based on a fourth-order Runge-Kutta scheme. The spatial discretization is typically ∆ξ/R = 0.02 and the time step ∆t = 0.5 × 10 −3 s. The time step is well below the one that for a given space resolution provides stable motion of a circular vortex ring without fluctuations and resolves the oscillations of the Kelvin waves. Numerical calculations are also performed using the LIA to compare against the exact BSL.
We are unable to perform a precise stability analysis of large-amplitude Kelvin waves under the Biot-Savart Law or a stability analysis of the Runge-Kutta scheme when applied to the Biot-Savart motion -both problems are practically impossible. We find that for very large times (larger then reported in the following section) the perturbed vortex ring always breaks up at some point (that is, first deforms and later possibly attempts to reconnect with itself). We do not know whether this fate z/R indicates an instability of the vortex for large-amplitude Kelvin waves or a numerical instability. What matters is that the lifetime of the perturbed vortex and the spatial range that it travels are much larger than the time scale of the Kelvin oscillations and the size of the ring itself, because it implies that the results which we describe are physically significant and observable in a real system.
III. RESULTS
The first result of our numerical simulations is that Kiknadze and Mamaladze's prediction [20] obtained using the LIA is indeed correct. Integration of the motion using the exact BSL shows that, provided the amplitude of the Kelvin waves is large enough, the vortex ring moves (on the average) backwards. This result is illustrated in Figs. 1 and 2: the former shows snapshots of the ring at different times as it travels, the latter gives the average translational velocity of the ring along the z direction as a function of the amplitude A of the Kelvin waves. It is apparent that the translational velocity decreases with increasing amplitude of the Kelvin waves and can even become negative.
At some critical value of the amplitude A the translational velocity is zero and the perturbed vortex ring hovers like a stationary helicopter. In the case of N = 10 Kelvin waves this happens when A/R = 0.17 approximately, which is quite close to the LIA prediction, A/R = 0.16. For N = 6 and N = 20 the critical value is, respectively, A/R = 0.32 and A/R = 0.085. This dependence of the critical amplitude on N is in approximate agreement with the LIA prediction [20]. The backward velocity of the perturbed vortex ring depends nonlinearly on the amplitude A of the Kelvin waves. At large enough amplitude A this velocity will slow down. This can be clearly seen in Fig. 2. The Kelvin waves, that can be imagined to behave like small vortex rings, tend to turn backwards, or more precisely, on the direction opposite to the motion of the unperturbed vortex ring. The larger the amplitude the larger fraction of the ring velocity is oriented downwards. This is compensated by the decrease in velocity of the single ring, which is inversely proportional to the amplitude, resulting an optimum value at some amplitude. For N = 20 the optimum amplitude A ≈ 0.25R resulting a downward velocity that is already slightly higher than the velocity upwards of the unperturbed ring.
In addition to Kelvin waves, the translational velocity of the vortex ring can be reduced by having an additional swirl velocity along the vortex core. This was considered by Widnall, Bliss, and Zalay [21]. However, this effect does not matter in our limit of thin-core vortices, which is relevant to superfluids.
The dispersion relation of large-amplitude Kelvin waves can be obtained by tracking the motion of the vortex on the y = 0 plane, for example. If the amplitude A of the Kelvin wave is small, the vortex draws a circle at approximately the same angular frequency that is obtained analytically for small-amplitude Kelvin waves and given by Eq. (1). In the long wavelength limit (k → 0) this relation becomes
ω = − κk 2 4π ln 2 ka − γ ,(8)
where γ = 0.5772 · · · is Euler's constant and the negative sign only indicates that the Kelvin waves rotate opposite to the circulation. Again the above equation differs slightly (−γ in stead of 1/4 − γ) from the form given by Schwarz [22], but this is again only due to the definition of the core type. We find that if we increase the amplitude of the Kelvin waves on the ring then the angular frequency decreases, a result which we also verified in the case of a straight vortex. Some example curves drawn by the vortex on the y = 0 plane are shown in Fig. 3. The average angular frequency is plotted in Fig. 4, which shows also the dispersion relation of waves on a straight vortex for comparison.
It is important to notice that, under the LIA used by Kiknadze and Mamaladze [20] the vortex length remains constant [22], whereas the quantity which is conserved under the exact BSL is the energy. Length and energy are proportional to each other only if the vortex filament is straight, which is not the case in our problem. Indeed, further investigation reveals that the vortex motion contains two characteristic frequencies. The first is the Kelvin frequency and the second is the frequency that is related to the oscillations of the vortex length and illustrated in Fig. 5. If the ratio of the two periods is rational one observes a fully periodic motion (in addition to translational motion along the z axis). At some values of the amplitudes which we calculated, this condition is almost satisfied. At higher values of amplitude one observes that the average radius of the vortex ring oscillates, as shown in Fig. 1. These variations in the total length were observed but not discussed in a recent calculation of the motion of vortex rings using the NLSE model [28].
The accuracy of our numerical method is tested by calculating the energy of the vortex ring. At zero temperature, without any dissipation, the energy (and the momentum) should remain constant. This condition can be quite well satisfied in our calculations. We do get some small oscillations in energy, as seen in Fig. 5, but we have checked that by increasing the space resolution we can reduce them at will, whereas the oscillations in length are independent of the numerical resolution.
IV. CONCLUSION
It is well known that a circular vortex ring has a translational velocity which arises from its own curvature (the smaller the radius R of the ring, the faster the ring travels). Using the exact Biot-Savart law, we have analyzed the motion of a vortex ring perturbed by Kelvin waves of finite amplitude. We have found that the translational velocity of the perturbed ring decreases with increasing amplitude; at some critical amplitude the velocity becomes zero, that is, the vortex ring hovers like a helicopter. A further increase of the amplitude changes the sign of the translational velocity, that is, the vortex ring moves backward. Our finding confirms preliminary results obtained by Kiknadze and Mamaladze using the local induction approximation [20].
This remarkable effect is due to the tilt of the plane of the Kelvin waves which induce motion in the "wrong" direction. The magnitude of the tilt oscillates and what results is a wobbly translational motion in the backward direction. We have also found that the frequency of the Kelvin wave decreases with increasing amplitude and that the total length of the perturbed vortex ring oscillates with time. This oscillation in vortex length is related to the oscillation of the tilt angle.
Time of flight measurements of large, electrically charged, perturbed vortex rings in 4 He could easily detect the decreased translational velocity. Another context in which the effect can be studied is Bose-Einstein condensation in ultra-cold atomic gases, which allow simple visualization of individual vortex structures. For these systems, however, it would be necessary to assess the effect of the nonhomogeneity of the superfluid.
online) Snapshots of the vortex ring of radius R = 0.1 cm perturbed by N = 10 Kelvin waves of various amplitude A taken during the motion of the vortex. In the left panel (a) the amplitude of the Kelvin waves is small, A/R = 0.05, but the perturbed vortex ring (red color) already moves slower than the unperturbed vortex (blue color). In the center panel (b) the Kelvin waves have large amplitude, A/R = 0.35, and the perturbed vortex ring moves backwards (negative z direction) on average. The top right panel (c) shows the top (xy) view of the large amplitude vortex at t = 0 s (blue) and t = 26 s (red, outermost). For comparison, a nondisturbed vortex is shown with dashed line (green). The lower right panel (d) gives the averaged location of the ring as a function of time. From top to bottom the curves correspond to A/R = 0.0, 0.05, 0.10, . . . , 0.35.
FIG. 2 :
2Average translational velocity of the vortex ring as a function of the initial oscillation amplitude A/R. Velocity is scaled by the velocity of the unperturbed ring, vring. The dash-dotted line corresponds to N = 20, solid line to N = 10, and the dashed line to N = 6 in Eq. (6). Critical amplitudes, above which the velocities become negative, are A/R = 0.085, 0.17, and 0.32, respectively.
FIG. 3 :
3Curve drawn by the vortex at y = 0 plane. Here the z coordinate is the coordinate relative to the average location of the vortex and N = 10 in Eq.[6]. In the top left panel (a) the amplitude is A/R = 0.05 and in the top right panel (b) A/R = 0.20. In both panels only the first 30 sec are shown. The thickness of the plotted curve arises from the chaotic motion rather than initial transient. The bottom panel (c) corresponds to A/R = 0.50 and we have drawn the curve for the first 90 sec. The time step between the markers is 2 ms; it is apparent that at large amplitudes the vortex is far from a sinusoidal helix and that the rotational speed at y = 0 plane varies significantly.
FIG. 4 :
4Main figure: Angular frequency of Kelvin waves ω relative to the value ω0 obtained in the small amplitude limit A/R = 0.001 and presented as a function of the wave amplitude A/R. The dashed line is for N = 6, the solid line for N = 10, and the dash-dotted line for N = 20. The inset shows the same when plotted as a function of A/λ, where λ is the wavelength of the Kelvin wave. The additional dotted line is the result obtained for straight vortex when using a wavelength of 0.1 cm together with periodic boundary conditions and using 25 periods above and below to numerically determine the vortex motion.
FIG. 5 :
5(Color online) The observed vortex length compared with the initial length L0 = 2π √ R 2 + N 2 A 2 is illustrated by solid (blue) lines and plotted as function of time in case of N =10. For comparison, the dashed (red) lines show the fluctuations in energy that are due to numerical errors and which can be reduced by increasing the space resolution. With increasing amplitude of oscillations the parameters for A/R shown are: 0.20, 0.30, 0.40, and 0.50.
of C.F.B was supported by EPSRC Grant No. GR/T08876/01 and Grant No. EP/D040892/1. M.T. acknowledges the support of a Grant-in-Aid for Scientific Research from JSPS (Grant No. 18340109) and a Grant-in-Aid for Scientific Research on Priority Areas from MEXT (Grant No. 17071008).
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