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[
"Four-Family N = 1 Supersymmetric Pati-Salam Models from Intersecting D6-Branes",
"Four-Family N = 1 Supersymmetric Pati-Salam Models from Intersecting D6-Branes"
]
| [
"Tianjun Li \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingP. R. China\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\nNo.19A Yuquan Road100049BeijingP. R. China\n",
"Rui Sun [email protected] \nKorea Institute for Advanced Study\n85 Hoegiro, Dongdaemun-Gu02455SeoulKorea\n",
"Chi Zhang [email protected] \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingP. R. China\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\nNo.19A Yuquan Road100049BeijingP. R. China\n"
]
| [
"Institute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingP. R. China",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\nNo.19A Yuquan Road100049BeijingP. R. China",
"Korea Institute for Advanced Study\n85 Hoegiro, Dongdaemun-Gu02455SeoulKorea",
"Institute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingP. R. China",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\nNo.19A Yuquan Road100049BeijingP. R. China"
]
| []
| We investigate the construction of four-family N = 1 supersymmetric Pati-Salam models from Type IIA T 6 /Z 2 × Z 2 orientifold with intersecting D6-branes. Utilizing the deterministic algorithm introduced in Ref.[1], we obtain 274 types of models with three rectangular tori and distinct gauge coupling relations at string scale, while 6 types of models with two rectangular tori and one titled torus. In both cases, there exists a class of models with gauge coupling unification at string scale. In particular, for the models with two rectangular tori, one tilted torus and gauge coupling unification, the gaugino condensations are allowed, and thus supersymmetry breaking and moduli stabilization are possible for further phenomenological study.In addition, we define actions Ω and R of 2 × 2 on Ì 6 , by Ω the parity-reversion on the world-sheet, and R the complex conjugate of Ì 6 as a complex manifold. By the very definition of an orientifold, where the world-sheet parity and complex conjugate make no difference, there are exactly four components of the Ì 6 /( 2 × 2 )-orientifold, namely the image of Ì 6 under the action of ΩR, ΩRθ, ΩRω and ΩRθω. These components are objects bearing RR charges, and thus D6-branes are introduced to cancel their RR charges.Generally, Dp-branes are (p + 1)-dimensional objects in the spacetime, where strings start from and land on. In our case, viewed from the internal space Ì 6 / 2 × 2 , the D6-branes are 3-dimensional objects. It is sufficient to identify D6-branes and 3-cycles for physical considerations. The powerful Eilenberg-Zilber theorem tells us thatIn the following discussion, we denote [a i ], [b i ], i = 1, 2, 3 as generators of H 3 (Ì 2 ; ) for the ith torus respectively. Under the orientifold action 2 × 2 and taking world-sheet into consideration, we find that there are only two patterns for the lattice H 1 (Ì 2 ; ) ≃ 2 : the rectangular one and the tilted one[11,33,35,45]. In the basis [a i ], [b i ], i = 1, 2, 3, the former is generated by [a i ], [b i ] over , while the latter is generated by [ã i ], [b i ] over with | 10.1088/1572-9494/ac6747 | [
"https://arxiv.org/pdf/2202.10252v2.pdf"
]
| 247,011,945 | 2202.10252 | caf8b3f2cfc315de2ab7e4f9faff2ada6fc38c0d |
Four-Family N = 1 Supersymmetric Pati-Salam Models from Intersecting D6-Branes
22 Mar 2022
Tianjun Li
Institute of Theoretical Physics
CAS Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
100190BeijingP. R. China
School of Physical Sciences
University of Chinese Academy of Sciences
No.19A Yuquan Road100049BeijingP. R. China
Rui Sun [email protected]
Korea Institute for Advanced Study
85 Hoegiro, Dongdaemun-Gu02455SeoulKorea
Chi Zhang [email protected]
Institute of Theoretical Physics
CAS Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
100190BeijingP. R. China
School of Physical Sciences
University of Chinese Academy of Sciences
No.19A Yuquan Road100049BeijingP. R. China
Four-Family N = 1 Supersymmetric Pati-Salam Models from Intersecting D6-Branes
22 Mar 2022
We investigate the construction of four-family N = 1 supersymmetric Pati-Salam models from Type IIA T 6 /Z 2 × Z 2 orientifold with intersecting D6-branes. Utilizing the deterministic algorithm introduced in Ref.[1], we obtain 274 types of models with three rectangular tori and distinct gauge coupling relations at string scale, while 6 types of models with two rectangular tori and one titled torus. In both cases, there exists a class of models with gauge coupling unification at string scale. In particular, for the models with two rectangular tori, one tilted torus and gauge coupling unification, the gaugino condensations are allowed, and thus supersymmetry breaking and moduli stabilization are possible for further phenomenological study.In addition, we define actions Ω and R of 2 × 2 on Ì 6 , by Ω the parity-reversion on the world-sheet, and R the complex conjugate of Ì 6 as a complex manifold. By the very definition of an orientifold, where the world-sheet parity and complex conjugate make no difference, there are exactly four components of the Ì 6 /( 2 × 2 )-orientifold, namely the image of Ì 6 under the action of ΩR, ΩRθ, ΩRω and ΩRθω. These components are objects bearing RR charges, and thus D6-branes are introduced to cancel their RR charges.Generally, Dp-branes are (p + 1)-dimensional objects in the spacetime, where strings start from and land on. In our case, viewed from the internal space Ì 6 / 2 × 2 , the D6-branes are 3-dimensional objects. It is sufficient to identify D6-branes and 3-cycles for physical considerations. The powerful Eilenberg-Zilber theorem tells us thatIn the following discussion, we denote [a i ], [b i ], i = 1, 2, 3 as generators of H 3 (Ì 2 ; ) for the ith torus respectively. Under the orientifold action 2 × 2 and taking world-sheet into consideration, we find that there are only two patterns for the lattice H 1 (Ì 2 ; ) ≃ 2 : the rectangular one and the tilted one[11,33,35,45]. In the basis [a i ], [b i ], i = 1, 2, 3, the former is generated by [a i ], [b i ] over , while the latter is generated by [ã i ], [b i ] over with
Introduction
One of the main motivations of string phenomenology is to find a unifying N = 1 supersymmetric quantum field theory, a competent framework among various extensions of the four-dimensional Standard Model (SM). D-branes play an important role in constructing interesting models at the phenomenological level, especially in Type I, Type IIA and Type IIB string theories. Chiral fermions appear at • worldvolume singularities of D-branes [2][3][4][5][6][7][8]; and
• intersecting loci of D-branes in the internal space [9].
Intersecting D6-branes on Type IIA orientifolds have been used to construct threefamily non-supersymmetric models and grand unified models . Even though these models satisfy the Ramond-Ramond (RR) tadpole cancellation conditions, there are Neveu-Schwarz-Neveu-Schwarz (NS-NS) tadpoles remaining due to their non-supersymmetric nature. Moreover, the string scale is close to the Planck scale since the intersecting D6-branes are not transversal in the internal space. As a result, there are large Planck scale corrections at the loop level, leading to the gauge hierarchy problem. As a remedy, a large number of supersymmetric standard-like models and grand unified models [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] have been constructed, with the gauge hierarchy problem solved. We refer to [50] for a comprehensive review for such kind of non-supersymmetric and supersymmetric models.
Among these supersymmetric models, Pati-Salam models has been a prominent road to the Standard Model, without adding any extra U(1) symmetry around the electroweak level. In Refs. [40], Cvetič, Liu and one of us (TL) showed how to systematically construct N = 1 supersymmetric Pati-Salam models from intersecting D6-branes on Type IIA Ì 6 / 2 × 2 -orientifold. After D-brane splitting and supersymmetric preserving Higgs mechanism applied, the Pati-Salam gauge symmetry SU(4) C × SU(2) L × SU(2) R eventually breaks down to the SM gauge symmetry. Due to the supersymmetry breaking and moduli stabilization triggered by their two confining groups in the hidden sectors, these models do have realistic and phenomenological consequences, as shown in Refs. [43,51,52]. Until very recent, intriguingly all possible 202752 three-family N = 1 supersymmetric Pati-Salam models on Ì 6 / 2 × 2 have been found and classified with 33 type of independent models according to gauge coupling relations [1].
Most of these above achievements are based on three-family model buildings. In fact, the study of the SM with four-families is also worthy of attention due to flavor democracy hypothesis. In [53], a democratic mass matrix model was introduced to mainly fix the mass gap problem between three families of fermions of the SM, as well as the hierarchy problem of the Yukawa couplings. Allowing three families of fermions, one gets typical SM predictions, such as a low mass of the top quark and the inequality between three neutrino masses [54]. If one allows four families of fermions in the democratic mass matrix model, three families of precisely massless neutrinos and a massive neutrino can be realized without the assumption on a larger hierarchy of the Yukawa couplings. Moreover, via a slight breaking of democracy, the three massless neutrinos obtain small masses. Then the flavor problem of the SM, can be solved naturally by putting the flavor democracy hypothesis due to the important role democratic mass matrix model plays.
As a matter of fact, the SM does not make a theoretical prediction on the number of families. The only restriction for the number of SM families comes from the requirement made by the Quantum Chromodynamics (QCD). The asymptotic freedom of QCD provide an upper bound 8 for the number of families as discussed in [55]. And if we allow four families of fermions for SM, many open issues of SM can be solved in a natural way. For example, introducing a fourth massive generation to SM can alter the cross section and decay channels of the Higgs particles [56]. When the Yukawa couplings of the fourth generation particles are large enough, these particles are natural candidates for electroweak symmetry breaking [57]. Motivated by constructing realistic four-family SM, in this paper, we concentrate on building four-family N = 1 supersymmetric models with gauge symmetry SU(4) C × SU(2) L × SU(2) R on Ì 6 / 2 × 2 orientifold with intersecting D6-branes as an extension to [1,40]. We obtain four-family Pati-Salam models with gauge coupling unification at string scale or near string scale. In particular, there are models without any filler brane required as well.
The paper is organized as follows. In Section 2 we will briefly review the basics of model building from intersecting D6-branes on Ì 6 / 2 × 2 orientifold. Constraints on D6-brane configuration, such as RR tadpole cancellation condition and supersymmetric condition are also reviewed. In Section 3, we present the symmetry breaking mechanism for U(4) C × U(2) L × U(2) R , i.e., D6-brane splitting and the Higgs mechanism. Four-family of chiral fermion condition and various symmetry relations, such as T-dualities are also discussed. Section 4 is devoted to the phenomenological features of the four-family models.
For each class of model, we list its chiral spectrum for the open string sector. In Section 5, we draw conclusions and briefly discuss the limitations of our work. The four-family Pati-Salam models are presented in the Appendix.
2 Basics of Ì 6 /( 2 × 2 )-Orientifolds Model Construction
To construct realistic and four-family supersymmetric models, we recall the basics of model construction from Type IIA string theory compactified on Ì 6 /( 2 × 2 )-orientifold, with D6-branes intersecting at general tilted angles, under similar settings as in [31] and [33].
We begin with the orientifold Ì 6 /( 2 × 2 ), where D6-branes can be naturally viewed as general 3-cycles. Using the canonical isomorphism Ì 6 ≃ Ì 2 ×Ì 2 ×Ì 2 , we can easily write down the coordinate chart of the orientifold. Let z i , i = 1, 2, 3 be the complex coordinates of the ith torus, respectively. Let θ and ω be the two generators of the abelian group 2 × 2 .
In coordinate (z 1 , z 2 , z 3 ), we define orientifold actions θ and ω of 2 × 2 on Ì 6 by
θ(z 1 , z 2 , z 3 ) := (−z 1 , −z 2 , z 3 ), ω(z 1 , z 2 , z 3 ) := (z 1 , −z 2 , −z 3 ). (2.1) [ã i ] := [a i ]+ 1 2 [b i ]. Under the basis [a 1 ], [b 1 ], [a 2 ], [b 2 ], [a 3 ], [b 3 ], we can represent a 3-cycle [Π a ]
of the orientifold Ì 6 /( 2 × 2 ) by the coordinate (n 1
a , 2 −β 1 l 1 a ) × (n 2 a , 2 −β 2 l 2 a ) × (n 3 a , 2 −β 3 l 3 a )
, where n i a , l i a are all integers, β i takes 0 when the ith torus is rectangular and takes value 1 for the tilted case. Under the action of ΩR, a D6-brane
[Π a ] = (n 1 a , l 1 a ) × (n 2 a , l 2 a ) × (n 3 a , l 3 a ) becomes [Π a ′ ] = (n 1 a , −l 1 a ) × (n 2 a , −l 2 a ) × (n 3 a , −l 3 a ), with [Π a ′ ][Π a ] = (n 1 a , 2 −β 1 l 1 a ) × (n 2 a , 2 −β 2 l 2 a ) × (n 3 a , 2 −β 3 l 3 a ) [Π a ′ ] = (n 1 a , −2 −β 1 l 1 a ) × (n 2 a , −2 −β 2 l 2 a ) × (n 3 a , −2 −β 3 l 3 a )
.
(2.2) Denote [Π ΩR ] = 2 3 (1, 0) × (1, 0) × (1, 0), then we have [Π ΩR ] = 2 3 (1, 0) × (1, 0) × (1, 0), [Π ΩRω ] = −2 3−β 2 −β 3 (1, 0) × (0, 1) × (0, 1), [Π ΩRθω ] = −2 3−β 1 −β 3 (0, 1) × (1, 0) × (0, 1), [Π ΩRωθ ] = −2 3−β 1 −β 2 (0, 1) × (0, 1) × (0, 1),(2.I ab = [Π a ] [Π b ] = 2 −k (n 1 a l 1 b − n 1 b l 1 a )(n 2 a l 2 b − n 2 b l 2 a )(n 3 a l 3 b − n 3 b l 3 a ), I ab ′ = [Π a ] [Π b ′ ] = −2 −k (n 1 a l 1 b + n 1 b l 1 a )(n 2 a l 2 b + n 2 b l 2 a )(n 3 a l 3 b + n 3 b l 3 a ), I aa ′ = [Π a ] [Π a ′ ] = −2 3−k n 1
a l 1 a n 2 a l 2 a n 3 a l 3 a ,
I a O6 = [Π a ] [Π O6 ] = 2 3−k (−l 1 a l 2 a l 3
a + l 1 a n 2 a n 3 a + n 1 a l 2 a n 3 a + n 1 a n 2 a l 3 a ).
(2.5)
The spectrum of intersecting D6-branes is given in Table 1.
Sectors Representations
aa U(N a /2) vector multiplets 3 adjoint chiral multiplets ab + ba I ab ( a , b ) fermions ab ′ + b ′ a I ab ′ ( a , b ) fermions aa ′ + a ′ a 1 2 (I aa ′ − 1 2 I a,O6 ) fermions 1 2 (I aa ′ + 1 2 I a,O6 ) fermions
For three-family N = 1 supersymmetric Pati-Salam model building in Type IIA orientifolds on T 6 /(Z 2 ×Z 2 ) with intersecting D6-branes in which the SU (4) C ×SU (2) L ×SU (2) R gauge symmetries arise from U (n) branes. To get four families of fermions, we require I ab + I ab ′ = 4,
I ac = −4, I ac ′ = 0.
(2.6)
The conditions I ab + I ab ′ = 4 and I ac = −4 ensure that there are four families of SM fermions, and the condition I ac ′ = 0 means that stack a of D6-branes are parallel to the orientifold image of stack c of D6-branes. Thus there should be open strings stretching between those two stack of D6-branes. The light scalar from the NS scalar will obtain mass Z 2 ac ′ /4πα ′ , with Z 2 ac ′ the minimal squared length of the stretching string. Similarly, the light fermions from the R sector acquire the same masses [13,14,39]. These light scalars and fermions form the Higgs fields needed to break the Pati-Salam gauge symmetry to the SM gauge symmetry.
In addition, there are two main constraints for D6-brane configurations and O6-plane configurations, namely the RR Tadpole Cancellation Condition and Supersymmetry Condition. As the sources of RR charges, D6-branes and O6-planes should satisfy the Gauss' law, for the flux of RR fields through the compact space Ì 6 /( 2 × 2 ) without boundary should be conserved. This form of Gauss' law is the so-called RR tadpole cancellation condition. To satisfy this condition, stacks of N a , a = 1, 2, 3 D6-branes are often needed to be introduced as the so-called filler brane, with a running through three families of gauge groups. Then the RR tadpole cancellation condition reads
3 a=1 N a [Π a ] + 3 a=1 N a [Π a ′ ] − 4[Π O6 ] = 0,(2.7)
and the coefficient 4 before [Π O 6 ] comes from the −4 RR charges in the D6-brane charge unit. Other than O6-planes in Equation (2.7), we can also introduce D6-branes between the four O6-planes to cancel out the total RR charges, and rewrite Equation (2.7) free of O6-planes. To do this, we note A a = −n 1 a n 2 a n 3 a , B a = n 1 a l 2 a l 3 a , C a = l 1 a n 2 a l 3 a , D a = l 1 a l 2 a n 3 a ; A a = −l 1 a l 2 a l 3 a ,B a = l 1 a n 2 a n 3 a ,C a = n 1 a l 2 a n 3 a ,D a = n 1 a n 2 a l 3 a .
(2.8) Therefore the tadpole cancellation conditions 2.7 can be expressed in the form Orientifold actions O6-planes (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
−2 k N (1) + 3 a=1 N a A a = −16, −2 k N (2) + 3 a=1 N a B a = −16, −2 k N (3) + 3 a=1 N a C a = −16, −2 k N (4) + 3 a=1 N a D a = −16,ΩR 1 (2 β 1 , 0) × (2 β 2 , 0) × (2 β 3 , 0) ΩRω 2 (2 β 1 , 0) × (0, −2 β 2 ) × (0, 2 β 3 ) ΩRθω 3 (0, −2 β 1 ) × (2 β 2 , 0) × (0, 2 β 3 ) ΩRθ 4 (0, −2 β 1 ) × (0, 2 β 2 ) × (2 β 3 , 0)
where 2N (i) , i = 1, 2, 3, 4 are the number of filler branes between the four O6-branes, as shown in Table 2. Furthermore, the four-dimensional N = 1 supersymmetric conditions require 1/4 supercharges conserved under (i) orientation reversion of D6-branes; and (ii) group action of 2 × 2 . As shown in [9], the four-dimensional N = 1 supersymmetry is preserved under the orientation reversion if and only if rotations of D6-branes with respect to O6-planes are elements of SU(3), while their total rotation angles equal to 0. When the four-dimensional N = 1 supersymmetry is preserved under orientation reversion, it will be preserved under the 2 × 2 -action manifestly. The supersymmetric condition can be written as [33] x
AÃa + x BBa + x CCa + x DDa = 0 , A a /x A + B a /x B + C a /x C + D a /x D < 0 , a = 1, 2, 3 , (2.10) with x A = λ, x B = λ2 β 2 +β 3 /χ 2 χ 3 , x D = λ2 β 1 +β 2 /χ 1 χ 2 .
The positive parameter λ is introduced to put Equation (2.10) on an equal scaling, and χ i = R 2 i /R 1 i , i = 1, 2, 3 is the complex structure modulus parameter for the ith torus respectively.
Gauge Symmetry Breaking via Brane Splittings
To obtain SM or standard-like models via the mechanism of intersecting D6-branes, there should be at least two extra U(1) gauge symmetries for either supersymmetric models or non-supersymmetric models, as a result of the constraints on the quantum number of the right handed electron [14,[31][32][33]. Among these two U(1) gauge symmetries, one is lepton number symmetry U(1) L and another U(1) I 3R is an analogy for right-hand weak isospin. We have the hypercharge Q Y expressed in the form
Q Y = Q I 3R + Q B − Q L 2 . (3.1) The baryonic charge Q B arises from U(1) B , via the decomposition U(3) C ≃ SU(3) C ×U(1) B .
On the other hand, since the U(1) I3R gauge field should be massless, the gauge group U(1) I3R must come from the non-abelian component of U(2) R or USp symmetry, otherwise the U(1) I3R will acquire mass from the B ∧ F couplings. To get an anomaly free U(1) B−L , the U(1) L symmetry should come from some non-abelian group for similar reasons. In previous studies of supersymmetric model building [31,32], U(1) I3R comes from USp groups.
These models indeed have two extra anomaly-free U(1) symmetries, and have at least 8 Higgs doublets. One could in principal break their symmetry groups down to the SM symmetry, but cannot do this without violating the D-flatness and F-flatness, thus the supersymmetry.
In this paper, as introduced in [52], we study a generalized version of the four-family MSSM models. In these models the b or c stacks are with twice the numbers of D6-branes. Then the gauge symmetries of these generalized four-family models can be broken to the standard four-family gauge symmetries SU(4) C ×SU(2) L ×SU(2) R via the Higgs mechanism. Taking gauge symmetries U(4) × U(4) L × U(2) R as example, we consider a U(4) gauge theory with a scalar field in the adjoint representation. By choosing appropriate rotations commuting with the generators of the Lie algebra of SU(4), one can break U(4) to U(2) × U(2), and finally to U(2). We choose the rotation for U(4) scalar field acting on the vacuum expectation value Φ 0 as
Φ 1 = V 1 0 0 0 0 V 1 0 0 0 0 −V 1 0 0 0 0 −V 1 .
(3.2)
We find that the U(4) gauge symmetry breaks spontaneously to U(2) × U(2), as the matrix in (3.2) lies in the center of the Lie algebra of U (4). What left to us is to break down
U(2) × U(2) to U(2). For each U(2) component in U(2) × U(2)
, the generators of its Lie algebra are the standard Pauli matrices. If we choose the rotation matrix for U(2) × U(2) as
Φ 2 = 0 0 V 2 0 0 0 0 V 2 V 2 0 0 0 0 V 2 0 0 ,(3.3)
we find that U(2) × U(2) breaks down to U(2). We also note that a mass of
m 2 = 8g 2 V 2 2 (3.4)
is acquired through the above process. For models with gauge symmetry U(4) × U(2) L × U(4) R and U(4) × U(4) L × U(4) R , one can similarly breaks the symmetry down to U(4) × U(2) L ×U(2) R following the above procedure. The anomalies of the overall U(1) symmetries are canceled by the generalized Green-Schwarz mechanism [13,14,31], while their fields get massive from the linear B ∧ F couplings. The gauge symmetry SU(4) C × SU(2) L × SU(2) R can be further broken down to SM gauge symmetry by D6-brane splitting and Higgs mechanism. Firstly, one can split stack a of N a = 8 D6-branes into stack a 1 of N a 1 = 6 D6-branes and stack a 2 N a 2 = 2 D6-branes. Then the U(4) C symmetry breaks down to U(3) × U(1). We denote the numbers of symmetric and anti-symmetric representations for SU(4) C , SU(2) L and SU(2) R by n a and n a , n b , n b and n c , n c . After splitting, the symmetric and anti-symmetric representations of SU(4) C descend to symmetric representations of SU(3) C and U(1) B−L , and antisymmetric representations of SU(3) C . Note that there are I a 1 a ′ 2 new fields, arising from the intersection of a 1 stack and a 2 stack of D6-branes. The anomaly-free gauge symmetry is
SU(3) C × U(1) B−L , a subgroup of SU(4) C .
Similarly, the stack c of N c = 4 D6-branes can be broen into stack c 1 of N a 1 = 2 D6-branes and stack c 2 N a 2 = 2 D6-branes. Then the U(2) R symmetry breaks down to U(1) I3R . The symmetric representations of SU(2) R descend to the symmetric representations of U(1) I3R only. Also, there are I c 1 c ′ 2 new fields, arising from the intersection of c 1 stack and c 2 stack of D6-branes. The anomaly-free gauge symmetry is U(1) I3R , a subgroup of SU(2) R . After splitting, the gauge symmetry of our model breaks down to
SU(3) C × SU(2) L × U(1) B−L × U(1) I3R .
To get just the SM gauge symmetry, we assume the minimal squared distance Z 2
a 2 c ′ 1
between a 2 stack and the orientifold image of c 1 stack of D6-branes to be very small. Then there are I a 2 c ′ 1 chiral multiplets of light fermions, arising from the open string stretching between a 2 stack of D6-branes and the orientifold image of c 1 stack of D6-branes. These particles break down
SU(3) C × SU(2) L × U(1) B−L × U(1) I3R
to the SM gauge symmetry, playing the same role as the right-handed neutrinos and their complex conjugates. Meanwhile, they preserve the D-flatness and F-flatness, thus the supersymmetry. In conclusion, the whole symmetry-breaking chain is
SU(4) C × SU(2) L × SU(2) R a→a 1 +a 2 − −−−−− → SU(3) C × SU(2) L × SU(2) R × U (1) B−L c→c 1 +c 2 −−−−−→ SU(3) C × SU(2) L × U(1) I3R × U(1) B−L Higgs Mechanism −−−−−−−−−−→ SU(3) C × SU(2) L × U(1) Y . (3.5)
The process of dynamical supersymmetry breaking has been studied in [36] for D6-brane models from Type IIA orientifolds. The kinetic function for a stack a of D6-branes is of the form [43] f a = 1 4κ a (n 1 a n 2 a n 3 a s −
n 1 a l 2 a l 3 a u 1 2 β 2 +β 3 − l 1 a n 2 a l 3 a u 2 2 β 1 +β 3 − l 1 a l 2 a n 3 a u 3 2 β 1 +β 2 ) , (3.6)
where κ a is a constant with respect to the gauge groups, for instance κ a = 1 for SU(N a ). We use moduli parameters s and u i , i = 1, 2, 3 in supergravity basis, which are related to four dimensional dilation parameter φ 4 and complex structure moduli parameters U i , i = 1, 2, 3 as following
Re(s) = e −φ 4 2π Im(U 1 ) Im(U 2 ) Im(U 3 ) |U 1 U 2 U 3 | , Re(u 1 ) = e −φ 4 2π Im(U 1 ) Im(U 2 ) Im(U 3 ) | U 2 U 3 U 1 | , Re(u 2 ) = e −φ 4 2π Im(U 2 ) Im(U 1 ) Im(U 3 ) | U 1 U 3 U 2 | , Re(u 3 ) = e −φ 4 2π Im(U 3 ) Im(U 1 ) Im(U 2 ) | U 1 U 2 U 3 | . (3.7)
In our present models, the U i , i = 1, 2, 3 can be computed as in [45]
U 1 = i χ 1 , U 2 = i χ 2 , U 3 = 2 χ 2 3 + 4 i χ 3 4 + χ 2 3 . (3.8)
Moreover, the Kähler potential takes the form of
K = − ln(S + S) − 3 i=1 ln(U i + U i ). (3.9)
Note that the three moduli parameter χ 1 , χ 2 , χ 3 are not independent, as they can be expressed in terms of x A , x B , x C , x D and the latter parameters are related by the supersymmetric condition (2.10). Actually, one can determine χ 1 , χ 2 , χ 3 up to an overall coefficient, namely an action of dilation on these parameters. So one has to stabilize this dilation to determine all the moduli parameters. Previous studies [58][59][60] employ mechanisms like gaugino condensation to stabilize this overall coefficient, dictating that there should be at least two USp groups in the hidden sectors. Moreover, the one-loop beta functions [40] β g i = −3(
N (i) 2 + 1) + 2|I ai | + |I bi | + |I ci | + 3( N (i) 2 − 1) = −6 + 2|I ai | + |I bi | + |I ci | (3.10)
for each USp(N (i) ) arising from 2N (i) filler branes are required to be negative. However, in this paper, to include other potential mechanisms, we do not restrict ourselves only to models with at least two USp groups in the hidden sectors. The gauge coupling constant related to stack a of D6-branes is 11) and the coupling constant of stack b and stack c of D6-branes are determined in the same way. The kinetic function for U(1) Y is a linear combination of those for SU(4) C and SU(2) R , as shown in [11,43]
g −2 a = |Re(f a )| ,(3.f Y = 3 5 ( 2 3 f a + f c ) . (3.12)
The coupling constant g Y is determined by
g −2 Y = |Re(f Y )| . (3.13)
At tree-level, the gauge couplings have the relation
g 2 a = αg 2 b = β 5 3 g 2 Y = γ(πe −φ 4 ) ,(3.14)
where α, β and γ are ratios between the strong coupling and the weak coupling, and hypercharge coupling, respectively.
T-Duality and its Variations
In string theory, if two models are related by T-duality, these models are considered equivalent. By applying T-duality, one can tremendously simplify the process of searching inequivalent models. Before the discussion of T-duality, we first point out two obvious symmetries that relate equivalent models.
(i) Two models are equivalent, if they are related by a permutation of three Ì 2 ; and (ii) two D6-models are equivalent if their wrapping numbers on any two Ì 2 are in opposite signs, while are the same on the third Ì 2 .
The above two symmetries are known as the D6-brane Sign Equivalent Principle (DSEP). Then, follow the convention of [40], we introduce Type I and Type II T-dualities. Type I duality transformation acts on arbitrary two Ì 2 , say the jth and kth Ì 2 . The wrapping numbers on these tori transform as follows As for Type II T-duality, it acts on all three different Ì 2 . For instance, if we pick the ith, jth and kth Ì 2 , the wrapping numbers on these tori transform as
(n j a , l j a ) → (−l j a , n j a ) , (n k a , l k a ) → (l k a , −n k a ) ,(n i a , l i a ) → (−n i a , l i a ) , (n j a , l j a ) → (l j a , n j a ) , (n k a , l k a ) → (l k a , n k a ) .
(3.16)
In [40], Type II T-duality often combines with the interchange between b and c stacks of
D6-branes b ↔ c (3.17)
associated to SU(2) L and SU(2) R gauge groups.
If we composite Type I T-duality and DSEP, we will get a variation of Type II T-duality.
Under this symmetry transformation, the wrapping numbers on all three Ì 2 change as
(n 1 a , l 1 a ) → (n 1 a , −l 1 a ) , (n 2 a , l 2 a ) → (n 2 a , −l 2 a ) , (n 3 a , l 3 a ) → (n 3 a , −l 3 a ) , b ↔ c . (3.18)
It's easy to see that under variation of Type II T-duality, only the signs ofà a ,B a ,C a ,D a in Equation (2.8) change.
However, it is worth mentioning that, the variation (3.18) of Type II T-duality is not an equivalence in our construction of four-family supersymmetric models, if the model is not invariant under SU(2) L and SU(2) R interchange. This observation makes sense at least phenomenologically. For a four-family supersymmetric model, one can obtain a new model by exchanging the b and c stacks of D6-branes associated to the SU(2) L and SU(2) R groups, as the quantum numbers for SU(2) L and SU(2) R in the particle spectrum will exchange, so will the gauge couplings for these two groups at the string level. We will present with examples of this inequivalence in the next section.
Supersymmetric 4-family Models
Employing the deterministic algorithm in [1], we do not restrict the number of USp groups, and consider two cases, one without any titled torus, and the other with the third torus to be tilted, without loss of generality. Note that four-family models are completely different from the three-family models, thus the argument in [40] to exclude which torus is tilted or not cannot be directly applied to our case, due to the even number of generations.
Models without Tilted Torus
We obtain six classes of 274 supersymmetric four-family models with the deterministic algorithm introduced in [1] with representative models presented in Appendix A. The classification is based on the gauge groups, T-equivalences and phenomenological considerations such as gauge coupling relations.
Model 14 is a class of its own, as it is the representative model without tilted torus achieving exact gauge coupling unification at the string scale. The Higgs-like particles in this model arise from the intersection at b and c ′ stacks of D6-branes, while the Higgs doublets arise from the massless open string states in a N = 2 subsector and form vector-like pairs.
The second class of models has no USp group, which means the tadpole cancellation conditions are satisfied without any filler brane as a rare case. These models are represented by Model 15 and Model 16, which are independent models of T-equivalence. These models are first four-family examples having no confining USp groups to achieve approximate gauge coupling unification at the string scale.
The third class of models includes Model 17 and Model 18 with negative β function and positive β function respectively.
The fourth class of models includes Model 19-21 with two USp groups. These models are independent under T-duality.
As discussed in [59] there are at least two confining USp groups needed, with negative β function, and thus allow for gaugino condensations, these models would need alternative mechanisms to break the supersymmetry. Furthermore, we can observe from the spectrum tables that Model
(−1, 0) × (−1, 1) × (1, 1) DSEP (i) −−−−−→ (−1, 0) × (1, 1) × (−1, 1) DSEP (i) −−−−−→ (1, 0) × (1, 1) × (1, −1).(0, 1) × (−1, 1) × (−3, 1) DSEP (i) −−−−−→ (0, 1) × (−3, 1) × (−1, 1) Type I T-duality − −−−−−−−−− → (0, 1) × (−1, −3) × (1, 1) DSEP (ii) and b ↔ c −−−−−−−−−−−−→ (0, 1) × (1, 3) × (−1, −1).
While the c stack of D6-branes in Model 24 and the b stack of D6-branes in Model 25 are related by Type II T-duality, DSEP and the b ↔ c exchange:
(1, 2) × (−1, 0) × (−1, 1) Type II T-duality − −−−−−−−−−− → (−1, 2) × (0, −1) × (1, −1) DSEP (i) −−−−−→ (−1, 2) × (1, −1) × (0, −1) DSEP (ii) and b ↔ c −−−−−−−−−−−−→ (−1, 2) × (−1, 1) × (0, 1) .
Even though these two models are related by the generalized T-duality as above, they are not phenomenologically equivalent. Model 24 achieves U(4) and U(2) R unification, while Model 25 has U(4) and U(2) L unification due to b and c stacks swapping. The Higgs particles of Model 24 come from intersection of b and c ′ stacks of D6-branes. Since all the beta functions of models within this class are negative, we may break the supersymmetry and stabilize the moduli via gaugino condensations.
The sixth class of models are with large wrapping numbers 5, 6,7,8,9,10,11,13,15,17, represented by Models 26-35 which did not appear in the former search. Three-family models with large wrapping number 5 have been found in [51], but it's the first time to find four-family models with wrapping numbers at this scale.
Models with One Tilted Torus
Employing the deterministic algorithm, we obtain in total 6 types of gauge coupling relations with represented models presented in Appendix B.
Model 36 is a class of its own, as it is the only type of models with one tilted torus achieving exact gauge coupling unification at the string scale. The Higgs particles in this model arise from the massless open string states in a N = 2 subsector and form vector-like pairs because the b stack branes for these models are parallel to c stack brane images on the third two-tori. Since all the beta functions are negative in this model, we can break the supersymmetry and stabilize the moduli via gaugino condensations.
The second class of models includes Model 37 and Model 38, and has no USp group. These models are related by type II T-duality. More specifically, we will below show how they are related by T-dualities explicitly. It's easy to find that the a stacks of both models are related by DSEP and Type II T-duality
(1, −1) × (−1, 0) × (−1, −1) DSEP − −−− → (−1, 0) × (1, −1) × (−1, −1) Type II T-duality − −−−−−−−−−− → (1, 0) × (−1, 1) × (−1, −1) .
Note that the b stack of wrapping numbers of Model 38 are obtained by applying the variation of Type II T-duality and DSEP on the b stack of wrapping numbers of Model 37:
(0, 1) × (−1, −2) × (1, 1) DSEP − −−− → (−1, −2) × (0, 1) × (1, 1) variation of Type II T-duality − −−−−−−−−−−−−−−−−− → (−1, 2) × (0, −1) × (1, −1) DSEP − −−− → (1, −2) × (0, −1) × (−1, 1) .
The c stacks of Model 37 and Model 38 are related by applying DSEP twice:
(1, 0) × (1, −2) × (1, 1) DSEP − −−− → (1, −2) × (1, 0) × (1, 1) DSEP − −−− → (1, −2) × (−1, 0) × (−1, −1) .
Phenomenological Analysis
Models without Tilted Torus
We begin with Model 14. The gauge group of Model 14 is U(4) × U(2) L × U(2) R × USp(2) × USp(4). We tabulate the full spectrum of chiral particles of Model 14 in Table 3. Interestingly, both Model 15 and Model 16 do not have any USp group, and then they do not have any exotic particles charged under USp groups as well. We tabulate the full spectrum of chiral particles of Model 15 below as a representative for this class.
Model 17 and Model 18 are constructed with one USp group. We tabulate the full spectrum of chiral particles of Model 17 in Table 5 (12), respectively. We tabulate the full spectrum of chiral particles in Model 20 in Table 6.
2) L ×U(2) R ×USp(2)×USp(4), U(4)×U(4) L ×U(2) R ×USp(2)× USp(4), U(4) × U(2) L × U(2) R × USp(4) 2 and U(4) × U(4) L × U(2) R × USp(4) × USp
Models 22-25 are built with at least three USp groups. Their gauge groups are U(4) × U(2) L ×U(2) R ×USp(2) 2 ×USp(4), U(4)×U(2) L ×U(2) R ×USp(2) 3 ×USp(8), U(4)×U(2) L × U(2) R × USp(2) 3 × USp(4) and U(4) × U(2) L × U(2) R × USp(2) 3 × USp(4), respectively. Note that Model 24 and Model 25 are related by T-duality, but are not phenomenologically equivalent. This can be easily seen from the fact that Model 24 has U(4) and U(2) R gauge Table 3. Spectrum of chiral particles of Model 14 Table 7. Spectrum of chiral particles of Model 24 Table 7. The exotic particles charged by USp groups may form bound states and composite particles at some intermediate energy scale, as the strong coupling dynamics of the USp groups requires. The composite particles are consistent with anomaly cancellation conditions, as in the QCD case. These composite particles thus are charged only under the SM gauge symmetry [34]. There are essentially two kinds of neutral bound states. The first one comes from decomposing the rank 2 anti-symmetric representation of the USp groups into two fundamental representations, and then taking the pseudo inner product of the fundamental representations. The second one comes from the rank 2N anti-symmetric representation of USp(2N ) for N ≥ 2. The first bound state is similar to a meson that is the inner product of a fundamental representation and an anti-fundamental representation of SU(3) C in QCD. The second bound state is a USp(2N ) singlet, which is an analog to a baryon being a rank 3 anti-symmetric representation of SU(3) C . Models 14, 20 and 25 contain the second kind of bound states. Now we take Models 14, 20 and 25 as examples to show explicitly how bound states are formed.
Model 14 SU(4) C × SU(2) L × SU(2) R × USp(2) × USp(4) Q 4C Q 2L Q 2R Qem B − L Field ab 4 × (4, 2, 1, 1, 1) 1 −1 0 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q L , L L ac 4 × (4, 1, 2, 1, 1) −1 0 1 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q R , L R bc ′ 8 × (1, 2, 2, 1, 1) 0 −1 −1 0, ±1 0 H ′ b1 4 × (1, 2, 1, 2, 1) 0 −1 0 ∓ 1 2 0 a4 2 × (4, 1, 1, 1, 4) 1 0 0 1 6 , − 1 2 1 3 , −1 c4 2 × (1, 1, 2, 1, 4) 0 0 −1 ± 1 2 0 a 1 × (10, 1, 1, 1, 1) −2 0 0 − 1 3 , 1 − 2 3 , 2 a 1 × (6, 1, 1, 1, 1) 2 0 0 1 3 , − 1 3 , −1 2 3 , −2 b 3 × (1, 3, 1, 1, 1) 0 2 0 0, ±1 0 b 3 × (1, 1, 1, 1, 1) 0 2 0 0 0 c 1 × (1, 1, 3, 1, 1) 0 0 2 0, ±1 0 c 1 × (1, 1, 1, 1, 1) 0 0 2 0 0 bc 2 × (1, 2, 2, 1, 1) 0 −1 1 0, ±1 0 H i u , H i d 2 × (1, 2, 2, 1, 1) 0 1 −1Model 15 SU(4) C × SU(2) L × SU(2) R Q 4C Q 2L Q 2R Qem B − L Field ab 8 × (4, 2, 1) 1 −1 0 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q L , L L ab ′ 4 × (4, 2, 1) −1 −1 0 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q L , L L ac 4 × (4, 1, 2) −1 0 1 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q R , L R bc 16 × (1, 2, 2) 0 −1 1 0 ± 1 0 H ′ b 5 × (1, 3, 1) 0 2 0 0, ±1 0 b 5 × (1, 1, 1) 0 2 0 0 0 c 7 × (1, 1, 3) 0 0 −2 0, ±1 0 c 9 × (1, 1, 1) 0 0 2 0, ±1 0 bc ′ 8 × (1, 2, 2) 0 −1 −1 0, ±1 0 H ′ 8 × (1, 2, 2) 0 1 1Model 17 SU(4) C × SU(2) L × SU(2) R × USp(2) Q 4C Q 2L Q 2R Qem B − L Field ab 6 × (4, 2, 1, 1) 1 −1 0 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q L , L L ab ′ 2 × (4, 2, 1, 1) −1 −1 0 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q L , L L ac 4 × (4, 1, 2, 1) −1 0 1 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q R , L R bc 8 × (1, 2, 2, 1) 0 −1 1 0, ±1 0 H ′ c2 4 × (1, 1, 2, 2) 0 0 1 ± 1 2 0 b 3 × (1, 3, 1, 1) 0 2 0 0, ±1 0 b 3 × (1, 1, 1, 1) 0 2 0 0 0 c 7 × (1, 1, 3, 1) 0 0 −2 0, ±1 0 c 9 × (1, 1, 1, 1) 0 0 2 0 0 bc ′ 6 × (1, 2, 2, 1) 0 −1 −1 0, ±1 0 H ′ 6 × (1, 2, 2, 1) 0 1 1Model 20 SU(4) C × SU(2) L × SU(2) R × USp(4) 2 Q 4C Q 2L Q 2R Qem B − L Field ab 6 × (4, 2, 1, 1, 1) 1 −1 0 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q L , L L ab ′ 2 × (4, 2, 1, 1, 1) −1 −1 0 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q L , L L ac 4 × (4, 1, 2, 1, 1) −1 0 1 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q R , L R bc 4 × (1, 2, 2, 1, 1) 0 −1 1 0, ±1 0 H ′ c2 1 × (1, 1, 2, 4, 1) 0 0 −1 ∓ 1 2 0 a4 2 × (4, 1, 1, 1, 4) −1 0 0 − 1 6 , 1 2 − 1 3 , 1 b4 3 × (1, 2, 1, 1, 4) 0 1 0 ± 1 2 0 c4 4 × (1, 1, 2, 1, 4) 0 0 1 ± 1 2 0 a 1 × (10, 1, 1, 1, 1) 2 0 0 1 3 , − 1 3 , −1 2 3 , −2 a 1 × (6, 1, 1, 1, 1) −2 0 0 − 1 3 , 1 3 , 1 − 2 3 , 2 b 1 × (1, 3, 1, 1, 1) 0 2 0 0, ±1 0 b 1 × (1, 1, 1, 1, 1) 0 2 0 0 0 c 5 × (1, 1, 3, 1, 1) 0 0 2 0, ±1 0 c 27 × (1, 1, 1, 1, 1) 0 0 2 0 0 bc ′ 7 × (1, 2, 2, 1, 1) 0 −1 −1 0, ±1 0 H ′ 7 × (1, 2, 2, 1, 1) 0 1 1Model 24 SU(4) C × SU(2) L × SU(2) R × USp(2) 3 × USp(4) Q 4C Q 2L Q 2R Qem B − L Field ab ′ 4 × (4, 2, 1, 1, 1, 1, 1) 1 1 0 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q L , L L ac 4 × (4, 1, 2, 1, 1, 1, 1) −1 0 1 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q R , L R bc 2 × (1, 2, 2, 1, 1, 1, 1) 0 1 −1 0, ±1 0 H ′ bc ′ 4 × (1, 2, 2, 1, 1, 1, 1) 0 −1 −1 0, ±1 0 H a3 1 × (4, 1, 1, 1, 1, 2, 1) −1 0 0 − 1 6 , 1 2 − 1 3 , 1 a4 2 × (4,
The composite particle spectrum for Model 14 is listed in Table 8. The confining group is USp(4), with two charged intersection. The mixing of intersection a4 and c4 results in the chiral supermultiplets (4, 1, 2, 1, 1). Moreover, the confined particle spectrum for Model 20 in Table 9. The confining group is USp(4), and has three charged intersection. Besides self-confinement, it is also viable to form mixed-confinement between sections within the same confining group. The chiral supermultiplets (4, 2, 1, 1, 1), (1, 2, 2, 1, 1) and (4, 1, 2, 1, 1) are yielded by the mixed-confinement between intersections a4, b4 and c4. For Model 25, the composite particle spectrum is given in Table 10. There are two confining groups USp(4) and USp (2), each with two charged intersections. The mixedconfinement between intersections b2, c2 yields the chiral supermultiplet (1, 2, 2, 1, 1, 1, 1), while the mixed-confinement between intersections a4, c4 yields the chiral supermultiplet (4, 1, 2, 1, 1, 1, 1). Note that when there is only one charged intersection, mixed-confinement will not be formed, thus only the tensor representations from self-confinement are left. Checking from the composite particle spectra, one finds that no new anomaly is introduced to the remaining gauge symmetry. Thus our models are free of anomalies. The above analysis for composite particles applies for all our models except Model 15 and Model 16 without any confining group. 1, 1, 1, 1, 1, 2) 3 × (6, 1, 1, 1, 1, 1, 1), 3 × (10, 1, 1, 1, 1, 1, 1), 2 × (4, 1, 2, 1, 1, 1, 1) c4
1 × (1, 1, 2, 1, 1, 1, 2) 1 × (1, 1, 1, 1, 1, 1, 1), 1 × (1, 1, 3, 1, 1, 1, 1) USp(2) 1 b1 1 × (1, 2, 1, 2, 1, 1, 1) 1 × (1, 1, 1, 1, 1, 1, 1), 1 × (1, 3, 1, 1, 1, 1, 1) USp(2) 3 a3 1 × (4, 1, 1, 1, 1, 2, 1) 1 × (10, 1, 1, 1, 1, 1, 1), 1 × (6, 1, 1, 1, 1, 1, 1)
Models with One Tilted Torus
In this section, we show basic phenomenological properties of models with one tilted torus. Model 36 represents the models with exact gauge coupling unification at the string level so far. The gauge symmetries therein are U(4) × U(2) L × U(2) R × USp(2) 2 with two confining groups in the hidden sector. The full spectrum of this model is shown in Table 3. Table 11. Spectrum of chiral particles of Model 36
Model 36 SU(4) C × SU(2) L × SU(2) R × USp(2) 2 Q 4C Q 2L Q 2R Qem B − L Field ab ′ 4 × (4, 2, 1, 1, 1) 1 1 0 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q L , L L ac ′ 4 × (4, 1, 2, 1, 1) −1 0 −1 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q R , L R b4 4 × (1, 2, 1, 1, 2) 0 1 0 ± 1 2 0 c2 4 × (1, 1, 2, 2, 1) 0 0 1 ± 1 2 0 b 3 × (1, 3, 1, 1, 1) 0 −2 0 0, ±1 0 b 2 × (1, 1, 1, 1, 1) 0 2 0 0 0 c 3 × (1, 1, 3, 1, 1) 0 0 2 0, ±1 0 c 3 × (1, 1, 1, 1, 1) 0 0 2 0 0 bc 8 × (1, 2, 2, 1, 1) 0 1 −1 0, ±1 0 H i u , H i d 8 × (1, 2, 2, 1, 1) 0 −1 1
Model 37 and Model 38 have no USp group. The gauge group for the two models is U(4) C × U(2) L × U(2) R . Since there is no USp group, the gaugino condensation mechanism will not work. Thus, one need to find other mechanism for supersymmetry breaking. Also, there are no exotic particles in these two models, as exotic particles are charged under USp groups. We show the full chiral spectrum in the open string sectors for Model 37 in Table 12. Table 13.
Model 37 SU(4) C × SU(4) L × SU(4) R Q 4C Q 2L Q 2R Qem B − L Field ab ′ 4 × (4, 4, 1) −1 −1 0 1 3 , − 2 3 , 1, 0 − 1 3 , 1 Q L , L L ac ′ 4 × (4, 1, 4) 1 0 1 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q R , L R b 2 × (1, 10, 1) 0 −2 0 0, ±1 0 b 2 × (1, 6, 1) 0 2 0 0 0 c 2 × (1, 1, 10) 0 0 2 0, ±1 0 c 2 × (1, 1, 6) 0 0 2 0 0 bc 4 × (1, 4, 4) 0 −1 1 0, ±1 0 H i u , H iModel 39 SU(4) C × SU(4) L × SU(2) R × USp(2) Q 4C Q 4L Q 2R Qem B − L Field ab 4 × (4, 4, 1, 1) 1 −1 0 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q L , L L ac 4 × (4, 1, 2, 1) −1 0 1 − 1 3 , 2 3 , −1, 0 1 3 , −1 Q R , L R bc ′ 4 × (1, 4, 2, 1) 0 1 1 0, ±1 0 H ′ b1 4 × (1, 2, 1, 2) 0 1 0 ± 1 2 0 b 2 × (1, 10, 1, 1) 0 2 0 1 3 , − 1 3 , −1 2 3 , −2 b 2 × (1, 6, 1, 1) 0 −2 0 − 1 3 , 1 − 2 3 , 2 c 3 × (1, 1, 3, 1) 0 0 2 0, ±1 0 c 3 × (1, 1, 1, 1) 0 0 2 0 0 bc 6 × (1, 2, 2, 1) 0 1 −1 0, ±1 0 H i u , H i d 6 × (1, 2, 2, 1) 0 −1 1
For the models with one tilted torus, the models represented by Model 36 are the only class with gauge coupling unification and carry two confining USp groups. Thus gaugino condensation can trigger supersymmetry breaking and moduli stabilization [36].
Discussions and Conclusions
Utilizing the deterministic algorithm, we obtain various classes of four-family supersymmetric models from intersecting D6-branes on Ì 6 / 2 × 2 orientifold, with and without tilted torus. In total, there are 274 physical independent four-family supersymmetric models without tilted torus, and 6 physical independent four-family supersymmetric models with the third torus to be the tilted one, without loss of generality.
For models without tilted torus, Model 14 represents the model with gauge coupling unification at the string scale. Models 15 and 16 are the rare models without any USp group in the hidden sectors, with tadpole cancellation conditions satisfied. Models 22-26 are with at least two confining USp groups. Thus gaugino condensation can be triggered to break the supersymmetry and stabilize the moduli. Moreover, there are Models 27-35 with wrapping numbers absolutely larger than 5 which was not reached for three family supersymmetric Pati-Salam models as discussed in [1].
For models with one tilted torus, Models 37 and 38 satisfy the tadpole cancellation conditions without any filler branes. Models such as Models 39 and 40 are related with b and c stacks of branes swapping. Model 36 represents the models with exact gauge coupling unification at the string scale, with two confining USp groups allowing gaugino condensation as well. This class of models would be ideal for further phenomenology model buildings (such as in [61]) as gaugino condensation can be triggered to break the supersymmetry and stabilize the moduli, while gauge coupling unified at string scale.
A Four-Family Standard Models from Intersecting D6-Branes without Tilted Tori
In the appendix, we list all representative four-family models obtained from our random scanning method. In the first columns for each table, a, b, c represent three stacks of D6-branes, respectively. Also in the first columns, 1, 2, 3, 4 is a short-handed notation for the filler branes along the ΩR, ΩRω, ΩRθω and ΩRθ O6-planes, respectively. The second columns for each table list the numbers of D6-branes in every stack, respectively. In the third columns of each table, we record wrapping numbers of each D6-brane configuration, and designate the third Ì 2 to be tilted.
The rest columns of each table record intersection numbers between stacks. For instance, in the b column of Table 37, from top to bottom, the numbers represent intersection numbers I ab , I bb , I cb , etc.. As usual, b ′ and c ′ are the orientifold ΩR image of b and c stacks of D6-branes. We also list the relation between x A , x B , x C , x D , which are determined by the supersymmetry condition Equation (2.10), as well as the relation between the moduli parameter χ 1 , χ 2 , χ 3 . The one loop β functions β g i are also listed. To have a clearer sight of gauge couplings, we list them up in the caption of each table, which makes it easier to check whether they are unified. Table 14. D6-brane configurations and intersection numbers of Model 14, and its gauge coupling relation is g 2 Table 15. D6-brane configurations and intersection numbers of Model 15, and its gauge coupling relation is g 2 a = 7 Table 16. D6-brane configurations and intersection numbers of Model 16, and its gauge coupling relation is g 2 a = 10 9 g 2 b = 10 3 g 2 c = 50 29 ( 5 3 g 2 Y ) = 2 Table 17. D6-brane configurations and intersection numbers of Model 17, and its gauge coupling relation is g 2 a = 5 Table 18. D6-brane configurations and intersection numbers of Model 18, and its gauge coupling relation is g 2 Table 19. D6-brane configurations and intersection numbers of Model 19, and its gauge coupling relation is g 2 Table 20. D6-brane configurations and intersection numbers of Model 20, and its gauge coupling relation is g 2 a = 21 Table 21. D6-brane configurations and intersection numbers of Model 21, and its gauge coupling relation is g 2 Table 22. D6-brane configurations and intersection numbers of Model 22, and its gauge coupling relation is g 2 Table 23. D6-brane configurations and intersection numbers of Model 23, and its gauge coupling relation is g 2 Table 24. D6-brane configurations and intersection numbers of Model 24, and its gauge coupling relation is g 2 Table 25. D6-brane configurations and intersection numbers of Model 25, and its gauge coupling relation is g 2 Table 26. D6-brane configurations and intersection numbers of Model 26, and its gauge coupling relation is g 2 a = 11 Table 27. D6-brane configurations and intersection numbers of Model 27, and its gauge coupling relation is g 2 Table 28. D6-brane configurations and intersection numbers of Model 28, and its gauge coupling relation is g 2 a = 47 Table 29. D6-brane configurations and intersection numbers of Model 29, and its gauge coupling relation is g 2 Table 30. D6-brane configurations and intersection numbers of Model 30, and its gauge coupling relation is g 2 a = 760 Table 31. D6-brane configurations and intersection numbers of Model 31, and its gauge coupling relation is g 2 a = 1309 Table 32. D6-brane configurations and intersection numbers of Model 32, and its gauge coupling relation is g 2 a = 1248 7 11 , χ 2 = 2 √ 77, χ 3 = 8 7 11 Table 33. D6-brane configurations and intersection numbers of Model 33, and its gauge coupling relation is g 2 a = 1680 Table 34. D6-brane configurations and intersection numbers of Model 34, and its gauge coupling relation is g 2 a = 2176 Table 36. D6-brane configurations and intersection numbers of Model 36, and its gauge coupling relation is g 2 Table 37. D6-brane configurations and intersection numbers of Model 37, and its gauge coupling relation is g 2 a = 2g 2 b = 2g 2 c = 10 7 ( 5 3 g 2 Y ) = 4πe φ 4 . Table 38. D6-brane configurations and intersection numbers of Model 38, and its gauge coupling relation is g 2 a = 2g 2 b = 2g 2 c = 10 7 ( 5 3 g 2 Y ) = 4πe φ 4 . Table 39. D6-brane configurations and intersection numbers of Model 39, and its gauge coupling relation is g 2 a = 2g 2 b = g 2 c = ( 5 3 g 2 Y ) = 8 Table 40. D6-brane configurations and intersection numbers of Model 40, and its gauge coupling relation is g 2 a = g 2 b = 2g 2 c = 10 7 ( 5 3 g 2 Y ) = 8 Table 41. D6-brane configurations and intersection numbers of Model 41, and its gauge coupling relation is g 2 Table 42. D6-brane configurations and intersection numbers of Model 42, and its gauge coupling relation is g 2 a = g 2 b = 2g 2 c = 10 7 ( 5 3 g 2 Y ) = 8 (1, 0) × (1, 0) × (−2, 0) Table 43. D6-brane configurations and intersection numbers of Model 43, and its gauge coupling relation is g 2 a = 8 3 g 2 b = g 2 c = ( 5 3 g 2 Y ) = 16 9 2 3/4 πe φ 4 . Table 44. D6-brane configurations and intersection numbers of Model 44, and its gauge coupling relation is g 2 a = 16
a = g 2 b = g 2 c = ( 5 3 g 2 Y ) = 8 3 4 √ 2πe φ 4 . model 14 U (4) × U (2) L × U (2) R × U Sp(2) × U Sp(4) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 4 a 8 (−1, 0) × (−1, 1) × (1, 2) -1 1 4 0 -4 0 0 2 b 4 (0, 1) × (−1, 2) × (−1, 2) 3 -3 - - 0 -8 -4 0 c 4 (1, 1) × (−1, 0) × (−1, 2) 1 -1 - - - - 0 -2 1 2 (1, 0) × (1, 0) × (1, 0) x A = 1 4 x B = 1 4 x C = 1 2 x D 4 4 (0, 1) × (0, 1) × (1, 0) β g 1 = −2, β g 4 = 0 χ 1 = 1 √ 2 , χ 2 = 1 √ 2 , χ 3 = 1 √ 29 g 2 b = 7 3 g 2 c = 35 23 ( 5 3 g 2 Y ) = 4 5 3/4 πe φ 4 3 √ 3 . model 15 U (4) × U (2) L × U (2) R stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ a 8 (−1, 0) × (−1, 1) × (1, 1) 0 0 8 -4 -4 0 b 4 (−1, 2) × (0, 1) × (−1, 3) 5 -5 - - -16 0 c 4 (1, 2) × (−2, 1) × (−1, 1) -7 -9 - - - - x A = 1 5 x B = 1 6 x C = 1 6 x D χ 1 = √ 5 6 , χ 2 = 1 √ 5 , χ 3 = 2 √ 53 2 3 11 3/4 πe φ 4 . model 16 U (4) × U (4) L × U (4) R stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ a 8 (−1, 0) × (1, 1) × (−1, 1) 0 0 8 -4 -4 0 b 8 (−1, 1) × (−1, 3) × (0, 1) 4 -4 - - 0 -8 c 8 (−1, 1) × (1, 2) × (−1, −1) 8 24 - - - - x A = 1 11 x B = 1 3 x C = 1 3 x D χ 1 = √ 11 3 , χ 2 = 1 √ 11 , χ 3 = 2 √ 116 g 2 b = 5 2 g 2 c = 25 16 ( 5 3 g 2 Y ) = 4 9 14 3/4 πe φ 4 . model 17 U (4) × U (2) L × U (2) R × U Sp(2) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 2 a 8 (−1, 0) × (−1, 1) × (1, 1) 0 0 6 -2 -4 0 0 b 4 (−1, 2) × (0, 1) × (−1, 2) 3 -3 - - -8 0 0 c 4 (1, 2) × (−2, 1) × (−1, 1) -7 -9 - - - - 4 2 2 (1, 0) × (0, 1) × (0, 1) x A = 2 7 x B = 1 4 x C = 1 4 x D β g 2 = −2 χ 1 = 7 2 4 , χ 2 = 2 7 , χ 3 = 2 2 7a = g 2 b = 3g 2 c = 5 3 ( 5 3 g 2 Y ) = 8 3 2 3/4 πe φ 4 . model 18 U (4) × U (4) L × U (4) R × U Sp(4) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 3 a 8 (−1, 0) × (−1, 2) × (1, 1) 1 -1 4 0 -4 0 -2 b 8 (−1, 1) × (0, 1) × (−1, 1) 0 0 - - 0 0 2 c 8 (1, 1) × (−1, 1) × (−1, 1) -4 -12 - - - - -2 3 4 (0, 1) × (1, 0) × (0, 1) x A = 1 4 x B = x C = 1 2 x D β g 3 = 2 χ 1 = √ 2, χ 2 = 1 2 √ 2 , χ 3 = √ 2a = 2g 2 b = g 2 c = ( 5 3 g 2 Y ) = 8 3 2 3/4 πe φ 4 . model 19 U (4) × U (4) L × U (2) R × U Sp(2) × U Sp(4) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 4 a 8 (−1, 0) × (−1, 1) × (1, 2) -1 1 0 4 -4 0 0 2 b 8 (0, 1) × (−1, 1) × (−1, 1) 0 0 - - 2 -6 -2 0 c 4 (1, 1) × (−1, 0) × (−1, 2) 1 -1 - - - - 0 -2 1 2 (1, 0) × (1, 0) × (1, 0) x A = x B = x C = 2x D 4 4 (0, 1) × (0, 1) × (1, 0) β g 1 = −4, β g 4 = 0 χ 1 = √ 2, χ 2 = √ 2, χ 3 = √ 222 g 2 b = 7 2 g 2 c = 7 4 ( 5 3 g 2 Y ) = 4 11 √ 310 3/4 πe φ 4 . model 20 U (4) × U (2) L × U (2) R × U Sp(4) 2 stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 2 4 a 8 (−1, 0) × (1, 1) × (−1, 2) 1 -1 6 -2 -4 0 0 -2 b 4 (−3, 2) × (−1, 2) × (0, 1) 1 -1 - - -4 0 0 3 c 4 (−2, 1) × (1, 2) × (−1, −2) 5 27 - - - - -1 4 2 4 (1, 0) × (0, 1) × (0, 1) x A = 1 20 x B = 3 8 x C = 3 4 x D 4 4 (0, 1) × (0, 1) × (1, 0) β g 2 = −5, β g 4 = 5 χ 1 = 3 5 2 2 , χ 2 = 1 √ 10 , χ 3 = 1 √ 10a = g 2 b = 4g 2 c = 20 11 ( 5 3 g 2 Y ) = 32 9 4 √ 2πe φ 4 . model 21 U (4) × U (4) L × U (2) R × U Sp(4) × U Sp(12) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 2 4 a 8 (−1, 0) × (1, 1) × (−1, 2) 1 -1 4 0 -4 0 0 -2 b 8 (−1, 1) × (−1, 1) × (0, 1) 0 0 - - -6 6 0 2 c 4 (−2, 1) × (1, 2) × (−1, −2) 5 27 - - - - -1 4 2 12 (1, 0) × (0, 1) × (0, 1) x A = 1 16 x B = 1 2 x C = x D 4 4 (0, 1) × (0, 1) × (1, 0) β g 2 = −5, β g 4 = 4 χ 1 = 2 √ 2, χ 2 = 1 2 √ 2 , χ 3 = 1 2 √ 2a = g 2 b = 2g 2 c = 10 7 ( 5 3 g 2 Y ) = 2 √ 2 4 √ 3πe φ 4 . model 22 U (4) × U (2) L × U (2) R × U Sp(2) 2 × U Sp(4) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 2 3 4 a 8 (−1, 0) × (−1, 1) × (1, 1) 0 0 4 0 -4 0 0 -1 1 b 4 (0, 1) × (−1, 3) × (−1, 1) 2 -2 - - 0 -12 1 0 0 c 4 (1, 1) × (−1, 0) × (−2, 2) 0 0 - - - - 2 0 -2 2 4 (1, 0) × (0, 1) × (0, 1) x A = 1 3 x B = 1 3 x C = 1 3 x D 3 2 (0, 1) × (1, 0) × (0, 1) β g 2 = −3, β g 3 = −4, β g 4 = −2 4 2 (0, 1) × (0, 1) × (1, 0) χ 1 = 1 √ 3 , χ 2 = 1 √ 3 , χ 3 = 2 √ 3a = g 2 b = 5 2 g 2 c = 25 16 ( 5 3 g 2 Y ) = 2 3 11 3/4 πe φ 4 . model 23 U (4) × U (2) L × U (2) R × U Sp(2) 3 × U Sp(8) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 2 3 4 a 8 (−1, 0) × (1, 1) × (−1, 1) 0 0 4 0 -4 0 0 0 1 -1 b 4 (−1, 2) × (−1, 1) × (0, 1) 1 -1 - - -4 6 -2 0 0 1 c 4 (−1, 1) × (1, 3) × (−1, −1) 0 12 - - - - 3 -1 3 1 1 2 (1, 0) × (1, 0) × (1, 0) x A = 1 11 x B = 1 2 x C = 1 2 x D 2 8 (1, 0) × (0, 1) × (0, 1) β g 1 = −1, β g 2 = −5, β g 3 = −1, β g 4 = −2 3 2 (0, 1) × (1, 0) × (0, 1) χ 1 = √ 11 2 , χ 2 = 1 √ 11 , χ 3 = 2a = 3 2 g 2 b = g 2 c = ( 5 3 g 2 Y ) = 2 3 3/4 πe φ 4 . model 24 U (4) × U (2) L × U (2) R × U Sp(2) 3 × U Sp(4) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 2 3 4 a 8 (−1, 0) × (−1, 1) × (1, 1) 0 0 0 4 -4 0 0 0 -1 1 b 4 (0, 1) × (−1, 1) × (−3, 1) -2 2 - - 2 -4 -1 3 0 0 c 4 (1, 2) × (−1, 0) × (−1, 1) -1 1 - - - - 0 2 0 -1 1 2 (1, 0) × (1, 0) × (1, 0) x A = 3x B = 3 2 x C = 3 2 x D 2 4 (1, 0) × (0, 1) × (0, 1) β g 1 = −5, β g 2 = −1, β g 3 = −4, β g 4 = −3 3 2 (0, 1) × (1, 0) × (0, 1) χ 1 = √ 3 2 , χ 2 = √ 3, χ 3 = 2 √ 3 4 2 (0, 1) × (0, 1) × (1, 0)a = g 2 b = 3 2 g 2 c = 5 4 ( 5 3 g 2 Y ) = 2 3 3/4 πe φ 4 . model 25 U (4) × U (2) L × U (2) R × U Sp(2) 3 × U Sp(4) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 2 3 4 a 8 (1, 0) × (1, 1) × (1, −1) 0 0 4 0 -4 0 0 0 1 -1 b 4 (−1, 2) × (−1, 1) × (0, 1) 1 -1 - - 4 2 -2 0 0 1 c 4 (0, 1) × (1, 3) × (−1, −1) -2 2 - - - - 3 -1 0 0 1 4 (1, 0) × (1, 0) × (1, 0) x A = 1 3 x B = 1 2 x C = 1 2 x D 2 2 (1, 0) × (0, 1) × (0, 1) β g 1 = −1, β g 2 = −5, β g 3 = −4, β g 4 = −3 3 2 (0, 1) × (1, 0) × (0, 1) χ 1 = √ 3 2 , χ 2 = 1 √ 3 , χ 3 = 2 √ 3 4 2 (0, 1) × (0, 1) × (1, 0)10 g 2 b = 11 5 g 2 c = 55 37 ( 5 3 g 2 Y ) = 8 13 3/4 πe φ 4 7 √ 5 . model 26 U (4) × U (2) L × U (2) R × U Sp(2) 3 stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 2 3 4 a 8 (1, −1) × (1, 0) × (1, 1) 0 0 -3 7 -4 0 -1 0 1 b 4 (−5, 2) × (1, 1) × (−1, 0) -3 3 - - -2 0 -2 5 0 c 4 (−3, 1) × (−1, 1) × (−1, 1) 0 12 - - - - 1 3 3 2 2 (1, 0) × (0, 1) × (0, 1) x A = 26 5 x B = 13x C = 26 5 x D 3 2 (0, 1) × (1, 0) × (0, 1) β g 2 = −1, β g 3 = 2, β g 4 = −1 4 2 (0, 1) × (0, 1) × (1, 0) χ 1 = √ 13, χ 2 = 2 √ 13 5 , χ 3 = 2 √ 13a = 6 7 g 2 b = 2 7 g 2 c = 2 5 ( 5 3 g 2 Y ) = 6 7 √ 2 4 √ 3πe φ 4 . model 27 U (4) × U (2) L × U (2) R stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ a 8 (−2, −1) × (1, 1) × (1, 1) 0 -8 12 -8 -4 0 b 4 (−1, 1) × (6, 2) × (−1, 0) 4 -4 - - 8 0 c 4 (1, 1) × (−1, 0) × (−2, 2) 0 0 - - - - x A = 9x B = 3x C = 9x D χ 1 = √ 3, χ 2 = 3 √ 3, χ 3 = 2 √ 3112 g 2 b = 18 7 g 2 c = 30 19 ( 5 3 g 2 Y ) = 1 3 2 7 23 3/4 πe φ 4 . model 28 U (4) × U (2) L × U (2) R × U Sp(2) × U Sp(4) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 2 a 8 (1, −1) × (1, 0) × (2, 1) 1 -1 -5 9 -4 0 0 -2 b 4 (−7, 2) × (1, 1) × (−1, 0) -5 5 - - 9 11 0 -2 c 4 (−2, 1) × (−2, 1) × (−2, 1) 5 27 - - - - -1 4 1 2 (1, 0) × (1, 0) × (1, 0) x A = 92 7 x B = 46x C = 46 7 x D 2 4 (1, 0) × (0, 1) × (0, 1) β g 1 = −5, β g 2 = 4 χ 1 = √ 23, χ 2 = 2 √ 23 7 , χ 3 = 4 √ 23a = 5 24 g 2 b = 19 8 g 2 c = 95 62 ( 5 3 g 2 Y ) = 2 45 86 3/4 πe φ 4 . model 29 U (4) × U (2) L × U (2) R × U Sp(2) 2 stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 2 4 a 8 (2, −1) × (1, 0) × (2, 1) 0 0 -6 10 -4 0 -2 2 b 4 (−8, 1) × (1, 1) × (−1, 0) -7 7 - - 15 11 -1 0 c 4 (−3, 1) × (−2, 1) × (−2, 1) 9 39 - - - - 4 6 2 2 (1, 0) × (0, 1) × (0, 1) x A = 43 4 x B = 86x C = 43 4 x D 4 2 (0, 1) × (0, 1) × (1, 0) β g 2 = 3, β g 4 = 4 χ 1 = √ 86, χ 2 = 43 2 4 , χ 3 = 2 √ 8631 g 2 b = 9g 2 c = 15 7 ( 5 3 g 2 Y ) = 48 31 √ 2 4 √ 37 3/4 πe φ 4 . model 30 U (4) × U (2) L × U (2) R × U Sp(2) 2 stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 3 a 8 (−1, 1) × (−1, 0) × (1, 1) 0 0 -5 9 -4 0 0 0 b 4 (−2, 1) × (−1, 1) × (−4, 1) 3 29 - - 33 -35 -1 8 c 4 (1, 0) × (−9, −2) × (−1, 1) 7 -7 - - - - 0 2 1 2 (1, 0) × (1, 0) × (1, 0) x A = 42x B = 28 3 x C = 42x D 3 2 (0, 1) × (1, 0) × (0, 1) β g 1 = −5, β g 3 = 4 χ 1 = 2 7 3 , χ 2 = 3 √ 21, χ 3 = 4 747 g 2 b = 5g 2 c = 25 13 ( 5 3 g 2 Y ) = 8 47 4 √ 2185 3/4 πe φ 4 . model 31 U (4) × U (2) L × U (2) R stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ a 8 (0, 1) × (−1, −2) × (2, 1) 0 0 10 -6 -4 0 b 4 (1, −2) × (1, −3) × (4, 1) -23 -73 - - 48 -24 c 4 (1, −10) × (0, −1) × (−2, 1) 8 -8 - - - - x A = x B = 1 5 x C = 1 74 x D χ 1 = 1 √ 370 , χ 2 = 5 74 , χ 3 = 2 74 541 g 2 b = 11g 2 c = 11 5 ( 5 3 g 2 Y ) = 64 123 77 3/4 πe φ 4 . model 32 U (4) × U (2) L × U (2) R × U Sp(14) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 a 8 (−1, 1) × (−1, 0) × (1, 1) 0 0 -5 9 -4 0 0 b 4 (−2, 1) × (−2, 1) × (−4, 1) 13 51 - - 45 -35 -1 c 4 (1, 0) × (−11, −2) × (−1, 1) 9 -9 - - - - 0 1 14 (1, 0) × (1, 0) × (1, 0) x A = 56x B = 112 11 x C = 56x D β g 1 = −5 χ 1 = 447 g 2 b = 13g 2 c = 65 29 ( 5 3 g 2 Y ) = 256 141 4 √ 213 3/4 πe φ 4 . model 33 U (4) × U (2) L × U (2) R × U Sp(10) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 a 8 (−1, 1) × (−1, 0) × (1, 1) 0 0 -5 9 -4 0 0 b 4 (−2, 1) × (−2, 1) × (−4, 1) 13 51 - - 51 -45 -1 c 4 (1, 0) × (−13, −2) × (−1, 1) 11 -11 - - - - 0 1 10 (1, 0) × (1, 0) × (1, 0) x A = 64x B = 128 13 x C = 64x D β g 1 = −5 χ 1 = 8 2 13 , χ 2 = 4 √ 26, χ 3 = 16 2 1353 g 2 b = 15g 2 c = 25 11 ( 5 3 g 2 Y ) = 192 53 4 √ 35 3/4 πe φ 4 . model 34 U (4) × U (2) L × U (2) R × U Sp(6) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 ) n n b b ′ c c ′ 1 a 8 (−1, 1) × (−1, 0) × (1, 1) 0 0 -5 9 -4 0 0 b 4 (−2, 1) × (−2, 1) × (−4, 1) 13 51 - - 57 -55 -1 c 4 (1, 0) × (−15, −2) × (−1, 1) 13 -13 - - - - 0 1 6 (1, 0) × (1, 0) × (1, 0) x A = 72x B = 48 5 x C = 72x D β g 1 = −5 χ 1 = 4 3 5 , χ 2 = 6 √ 15, χ 3 = 8 3 5a = g 2 b = g 2 c = ( 5 3 g 2 Y ) = 2 √ 2πe φ 4 .3 4 √ 2πe φ 4 .) x A = 1 2 x B = 2x C = 1 2 x D β g 1 = −2 χ 1 = √ 2, χ 2 = 1 2 √ 2 , χ 3 = 2 √ 23 4 √ 2πe φ 4 .a = 2g 2 b = g 2 c = ( 5 3 g 2 Y ) = 8 3 4 √ 2πe φ 4 .3 4 √ 2πe φ 4 .x A = 2x B = 1 2 x C = 1 2 x D β g 1 = −2 χ 1 = 1 2 √ 2 , χ 2 = √ 2, χ 3 = 2 √ 25 g 2 b = 2g 2 c = 10 7 ( 5 3 g 2 Y ) = 16 5 √ 2πe φ 4 .
I T-duality applies. Recall the definitions in Equation (2.8), Type I T-duality only makes an exchange between (A a ,Ã a ), (B a ,B a ), (C a ,C a ), (D a ,D a ). Moreover, Type I duality transformation is often combined with the trivial two Ì 2 exchange, and we call the combination an extended Type I T-duality.
15, 17 and 20 do not have the proper Higgs doublets with quantum number (1, 2, 2) under U (4) C × U (2) L × U (2) R gauge symmetry, from neither the b and c or c ′ stacks of brane intersection nor the massless open string states in a N = 2 subsector. Thus, we do not have the SM fermion Yukawa couplings at renormalizable level which are invariant under the global U (1) C × U (1) L × U (1) R symmetry in these models. The fifth class of models no less than three USp groups, and includes Models 19-25. Among these models, Model 24 and Model 25 are related by T-dualities with b and c stacks of branes swapping. To see this, we show how Model 24 and Model 25 are related. To begin with, a stacks of D6-branes in Model 24 and Model 25 are related by the DSEP:
And the b stack of D6-branes in Model 24 are related to the c stack of D6-branes in Model 25, by Type I T-duality (3.15), the DSEP, and the interchange (3.17) of b and c stacks:
Models 39-42 with one USp group in the hidden sector are related by T-dualities in a similar way. The Higgs particles in Model 37 again come from the massless open string states in a N = 2 subsector and form vector-like pairs because the b stack branes for these models are parallel to c stack brane on the third two-tori. Since there are no USp groups in the hidden sectors, gaugino condensations do not work in this case. One needs to stabilize the modulus and break the supersymmetry via different mechanism. For Model 39, there are four Higgs doublets arising from N = 2 subsectors due to the parallel of b stack branes and c stack brane on the third two-tori. In addition, there are models 43 and 44 with distinct gauge coupling relations at string scale.
at the string scale, while Model 25 has U(4) and U(2) L gauge coupling unification. We represent the full spectrum of chiral particles inModel 24 in
only one USp(2) group in the hidden sector. Their gauge symmetry is all U(4) × U(4) L × U(2) R × USp(2). We note thatModel 39 and Model 40 are T-dual to each other, as well as Model 41 and Model 42. But they are clearly not equivalent models at phenomenological level due to b and c stacks of brane swapping. In Model 39, we have SU(3) C and U(1) Y gauge coupling unification at the string level, while in Model 40 this gauge unification get swapped to SU(3) C and SU(2) L gauge coupling unification at the string level. Similarly, this b and c stacks of brane swapping appear between Model 41 and Model 42 as well. The whole spectrum of chiral particles of Model 39 in
model 35 U
35(4) × U (2) L × U (2) R × U Sp(2) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 36 U
36(4) × U (2) L × U (2) R × U Sp(2) 2 stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 37 U
37(4) × U (4) L × U (4) R stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 38 U
38(4) × U (4) L × U (4) R stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 39 U
39(4) × U (4) L × U (2) R × U Sp(2) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 40 U
40(4) × U (2) L × U (4) R × U Sp(2) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 41 U
41(4) × U (4) L × U (2) R × U Sp(2) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 42 U
42(4) × U (2) L × U (4) R × U Sp(2) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 43 U
43(4) × U (2) L × U (2) R × U Sp(2) stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
model 44 U
44(4) × U (2) L × U (2) R × stack N (n 1 , l 1 ) × (n 2 , l 2 ) × (n 3 , l 3 )
a short-handed notation for the image of [Π a ]. To sum up, a D6-brane [Π a ] and its orientifold image [Π a ′ ] take the form
3 )
3where [Π ΩRω ], [Π ΩRθ ], [Π ΩRθω ] are the images of [Π ωR ] under the action of ω, θ and θω respectively. The coefficient 2 3 comes from identifying 8 O6-branes (±1, 0) × (±1, 0) × (±1, 0), under the quotient of Ω and R actions. The intersection number of D6-branes can be easily computed as
4 )
4where k = β 1 + β 2 + β 3 . If we denote [Π O6 ] = [Π ΩR ] + [Π ΩRω ] + [Π ΩRθ ] + [Π ΩRθω ], we have
Table 1 .
1Spectrum of intersecting D6-branes.
Table 2 .
2Configuration of four O6-planes
Table 4 .
4Spectrum of chiral particles of Model 15
Table 5 .
5Spectrum of chiral particles of Model 17
Table 6 .
6Spectrum of chiral particles of Model 20
Table 8 .
8The composite particle spectrum for Model 14Model 14
SU(4) C × SU(2) L × SU(2) R × USp(2) × USp(4)
Confining Force
Intersection
Exotic Particle Spectrum
Confined Particle Spectrum
USp(4) 4
a4
2 × (4, 1, 1, 1, 4)
3 × (6, 1, 1, 1, 1), 3 × (10, 1, 1, 1, 1), 4 × (4, 1, 2, 1, 1)
c4
2 × (1, 1, 2, 1, 4)
3 × (1, 1, 1, 1, 1), 3 × (1, 1, 3, 1, 1)
USp(2) 1
b1
4 × (1, 2, 1, 2, 1)
10 × (1, 1, 1, 1, 1), 10 × (1, 1, 3, 1, 1)
Table 9 .
9The composite particle spectrum for Model 20Model 20
SU(4) C × SU(2) L × SU(2) R × USp(4) 2
Confining Force
Intersection
Exotic Particle Spectrum
Confined Particle Spectrum
USp(4) 4
a4
2 × (4, 1, 1, 1, 4)
3 × (6, 1, 1, 1, 1), 3 × (10, 1, 1, 1, 1), 6 × (4, 2, 1, 1, 1)
b4
3 × (1, 2, 1, 1, 4)
6 × (1, 1, 1, 1, 1), 6 × (1, 3, 1, 1, 1), 12 × (1, 2, 2, 1, 1)
c4
4 × (1, 1, 2, 1, 4)
10 × (1, 1, 1, 1, 1), 10 × (1, 1, 3, 1, 1), 8 × (4, 1, 2, 1, 1)
USp(4) 2
c2
1 × (1, 1, 2, 4, 1)
1 × (1, 1, 3, 1, 1), 1 × (1, 1, 1, 1, 1)
Table 10 .
10The composite particle spectrum for Model 25Model 25
SU(4) C × SU(2) L × SU(2) R × USp(2) 3 × USp(4)
Confining Force
Intersection
Exotic Particle Spectrum
Confined Particle Spectrum
USp(4) 2
b2
3 × (1, 2, 1, 1, 4, 1, 1)
6 × (1, 1, 1, 1, 1, 1, 1), 6 × (1, 3, 1, 1, 1, 1, 1), 6 × (1, 2, 2, 1, 1, 1, 1)
c2
2 × (1, 1, 2, 1, 4, 1, 1)
3 × (1, 1, 1, 1, 1, 1, 1, 1), 3 × (1, 1, 3, 1, 1, 1, 1)
USp(2) 4
a4
2 × (4,
Table 12 .
12Spectrum of chiral particles of Model 37
Table 13 .
13Spectrum of chiral particles of Model 39
Table 35 .
35D6-brane configurations and intersection numbers of Model 35, and its gauge coupling relation is g 2a = 2736
59 g 2
b = 17g 2
c = 85
37 ( 5
3 g 2
Y ) = 64
177 170 3/4 πe φ 4 .
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"Ionized Gas in Damped Lyman-α Systems and Its Effects on Elemental Abundance Studies",
"Ionized Gas in Damped Lyman-α Systems and Its Effects on Elemental Abundance Studies"
]
| [
"J Christopher Howk [email protected] \nDepartment of Physics & Astronomy\nThe Johns Hopkins University\n3400 N. Charles St21218BaltimoreMD\n",
"Kenneth R Sembach [email protected] \nDepartment of Physics & Astronomy\nThe Johns Hopkins University\n3400 N. Charles St21218BaltimoreMD\n"
]
| [
"Department of Physics & Astronomy\nThe Johns Hopkins University\n3400 N. Charles St21218BaltimoreMD",
"Department of Physics & Astronomy\nThe Johns Hopkins University\n3400 N. Charles St21218BaltimoreMD"
]
| [
"The Astrophysical Journal Letters"
]
| Recent high-resolution observations of metal absorption lines in high-redshift damped Lyα systems have shown that Al III, a tracer of moderately-ionized gas, very often has a velocity structure indistinguishable from that of low-ionization gas. Regions of ionized and neutral hydrogen in these systems are likely cospatial. The higher-ionization Si IV and C IV absorption shows a much weaker or non-existent correlation with the low ionization material, implying that the regions traced by Al III are photoionized by a soft (stellar) spectrum, by a hard (power law) spectrum with a very low ionization parameter, or a combination of both. We discuss the ionization of the damped Lyα systems and use photoionization equilibrium models to make quantitative estimates of its effects on abundance studies in these systems. We show that ionization effects may be large enough to account for the observed dispersion in absolute metal abundances in damped Lyα systems, causing systematically higher abundances in lower column density systems. The observed Si + /Fe + and Zn + /Cr + ratios may systematically overestimate the intrinsic Si/Fe and Zn/Cr ratios, respectively, if ionized gas is present in these systems, thereby mimicking the effects of α-element enrichment or dust depletion. | 10.1086/312284 | [
"https://arxiv.org/pdf/astro-ph/9907428v1.pdf"
]
| 18,730,150 | astro-ph/9907428 | 68638aabbcbd5b8007a81116882def35ad6755d5 |
Ionized Gas in Damped Lyman-α Systems and Its Effects on Elemental Abundance Studies
Jul 1999
J Christopher Howk [email protected]
Department of Physics & Astronomy
The Johns Hopkins University
3400 N. Charles St21218BaltimoreMD
Kenneth R Sembach [email protected]
Department of Physics & Astronomy
The Johns Hopkins University
3400 N. Charles St21218BaltimoreMD
Ionized Gas in Damped Lyman-α Systems and Its Effects on Elemental Abundance Studies
The Astrophysical Journal Letters
Jul 1999arXiv:astro-ph/9907428v1 29Subject headings: galaxies: abundances -intergalactic medium -quasars: absorption lines - radiative transfer
Recent high-resolution observations of metal absorption lines in high-redshift damped Lyα systems have shown that Al III, a tracer of moderately-ionized gas, very often has a velocity structure indistinguishable from that of low-ionization gas. Regions of ionized and neutral hydrogen in these systems are likely cospatial. The higher-ionization Si IV and C IV absorption shows a much weaker or non-existent correlation with the low ionization material, implying that the regions traced by Al III are photoionized by a soft (stellar) spectrum, by a hard (power law) spectrum with a very low ionization parameter, or a combination of both. We discuss the ionization of the damped Lyα systems and use photoionization equilibrium models to make quantitative estimates of its effects on abundance studies in these systems. We show that ionization effects may be large enough to account for the observed dispersion in absolute metal abundances in damped Lyα systems, causing systematically higher abundances in lower column density systems. The observed Si + /Fe + and Zn + /Cr + ratios may systematically overestimate the intrinsic Si/Fe and Zn/Cr ratios, respectively, if ionized gas is present in these systems, thereby mimicking the effects of α-element enrichment or dust depletion.
surveys of DLAs find a very conspicuous correlation between the velocity structure seen in the absorption lines of low-ionization species (e.g., Si II, Fe II, and Zn II) and the structure observed in Al III, a tracer of moderately (photo)ionized gas [IP(Al + , Al +2 ) = (18.8, 28.4) eV]. Such an obvious correlation is not observed between the low-ionization species and the more highly-ionized ions such as Si IV or C IV. Similar arrangements of low-and intermediate-ions can be found along selected sightlines extending into the halo of the Milky Way. Towards HD 93521 (Spitzer & Fitzpatrick 1993) and ρ Leo ) the tracers of neutral and photoionized gas have relative velocity component distributions resembling those of Al III and low ions in DLAs. The total hydrogen column densities towards these stars are log N (H I) = 20.10 and 20.44, respectively (Diplas & Savage 1994). In the Milky Way, the scale height of Al III is consistent with that of the free electrons, h z ∼ 1 kpc, which is more extended than the H I distribution (Savage, Edgar, & Diplas 1990).
Most singly-ionized metal species that are dominant ionization stages in H I-bearing regions may also be produced in photoionized clouds where H o is a small fraction of the total hydrogen content. The formation of metal absorption lines in both ionized and neutral regions can have a significant effect on elemental abundance determinations. Ionization can be an important issue for high-precision studies of elemental abundances in the Milky Way (Sofia & Jenkins 1998;Howk, Savage, & Fabian 1999;Sembach et al. 1999). The Al III in DLAs, with velocity structure that is often indistinguishable from that of the low ions (Lauroesch et al. 1996), suggests the long-held assumption that ionization effects are neglible in these systems may be unwarranted. In this work we examine the contribution of photoionized gas to the observed metal-line absorption in damped Lyα systems.
THE IONIZING SPECTRUM
The major uncertainty in determining the ionization balance in the DLAs is the unknown shape of the ionizing spectrum. The two most likely origins for ionizing photons in DLAs are: internal stellar and external background sources. Ionization of the DLAs by external sources, e.g., by the integrated light from QSOs, AGNs, starbursts, and normal galaxies (Haardt & Madau 1996;Madau & Shull 1996), requires that the ionizing photons "leak" into the DLAs. This might seem unlikely given the large observed neutral hydrogen column densities, but the multi-component nature of these systems implies that each individual cloud may have a much lower column density than the total. Furthermore, the ionization of the warm ionized medium (WIM) in the Milky Way requires ∼ 15% of the ionizing photon output of Galactic OB stars (Reynolds 1993). This implies that the gaseous structure of a present day disk galaxy is such that ionizing photons can travel very large distances from their origin, and of order 5% may escape the Galaxy completely (Bland-Hawthorn & Maloney 1999). We assume a similar arrangement in the DLAs. For the external ionization case, we adopt an updated version of the Haardt & Madau (1996; hereafter HM) QSO ultraviolet background in our photoionization models. This modified background spectrum (Haardt 1999, priv. comm.) assumes q o = 0.5 (instead of 0.1), a power law index for the QSO emission spectrum of α = 1.8 (rather than 1.5), and a redshift evolution of the QSO number density that follows the trend described by Madau, Haardt, & Rees (1999).
Internal ionization, in this work, refers to photoionization by stellar sources internal to the DLAs. If DLAs represent the early phases of massive disk galaxies (e.g., Wolfe & Prochaska 1998), it is reasonable to expect some star formation in these systems. Searches for Lyα and Hα emission from DLAs imply low star formation rates:Ṁ * ∼ < 5 − 20 M ⊙ yr −1 (Bunker et al. 1999;Lowenthal et al. 1995), with one detection of Lyα emission suggestingṀ * ≈ 1 M ⊙ yr −1 (Warren & Møller 1996). In the Milky Way, where ionizing photons from early-type stars must leak through the neutral ISM to ionize the WIM, the star formation rate is of order 2 − 5 M ⊙ yr −1 (Mezger 1987;McKee 1989;McKee & Williams 1997). The perpendicular column density of ionized hydrogen in the WIM is about 1/4 that of neutral hydrogen at the solar circle, thus demonstrating that a relatively large fraction of interstellar hydrogen can be ionized with a modest level of star formation. For the internal ionization case, we adopt the spectrum of a typical late O star as the ionizing spectrum. We use an ATLAS line-blanketed model atmosphere (Kurucz 1991) with an effective temperature T ef f = 33, 000 K and log(g) = 4.0. Our work on the ionization of the Galactic WIM (Sembach et al. 1999) suggests that such a spectrum is able to match the constraints imposed by emission line observations of the ionized gas (Reynolds & Tufte 1995;Reynolds et al. 1998;Haffner, Reynolds, & Tufte 1999).
We consider only a single temperature stellar source for the internal case, and a QSO-dominated spectrum for the external ionization case. The reader should be aware that the true ionizing spectrum may be a combination of soft (internal) and hard (external) ionizing spectra. The lack of associated Si IV absorption with the low ions favors either the softer stellar spectrum or a very low ionization parameter.
PHOTOIONIZATION MODELS
We use the CLOUDY ionization equilibrium code (Ferland et al. 1998;Ferland 1996) to model the ionization of DLAs. We assume a plane-parallel geometry with the ionizing spectrum incident on one side. Rather than match the total H I column density in our models, we stop the integration at the point where the local ionization fraction of neutral hydrogen climbs above 10%, i.e.,
x(H o ) ≡ N (H o )/N (H tot ) > 0.1.
Our models therefore treat the (almost) fully-ionized regions assumed to envelop the neutral, H I-bearing clouds. The relative mix of neutral and ionized material can be inferred from observations of adjacent ions, e.g., Al II/Al III. Our models assume a base metal abundance of 0.1 solar, with relative heavy element abundances equivalent to those observed in the Galactic warm neutral medium (Sembach et al. 1999;. We include interstellar grains for heating and cooling processes (see Ferland 1996 andBaldwin et al. 1991), with a dust to gas ratio 0.1 of the Galactic value. Our models are only as accurate as the input atomic data for the CLOUDY code, and we refer the reader to Ferland (1996) and Ferland et al. (1998) for discussions of the uncertainties (see also our earlier work with CLOUDY: Sembach et al. 1999;. In particular, the dielectronic recombination coefficients for elements in the third and fourth row of the periodic table are typically not well known, and the radiative recombination coefficients for many of the heavier elements (e.g., Zn and Cr) are often based on somewhat uncertain theoretical considerations.
We have computed CLOUDY models for the z ≈ 2.0 HM spectrum and for the Kurucz model atmosphere over a range of ionization parameters, Γ. In this case Γ is the dimensionless ratio of total hydrogen-ionizing photon density to hydrogen particle density at the face of the cloud. In Figure 1 we present the ionization fractions of several ions, x(X i ), for the HM spectrum as a function of the assumed ionization parameter. The top panel shows x(X i ) for elements with at least two potentially measurable ionization stages: Si, Fe, and Al. The bottom panel shows the effects of ionization on relative metal abundances, tracing values of x(X i )/x(Fe + ) for several commonly measured ions. These plots can be used to correct for ionization effects if one is able to estimate Γ.
For large values of log Γ ( ∼ > −3.0) the predicted strength of the Si IV becomes large, with x(Si + )/x(Si +3 ) ∼ < 2, contrary to observations. At log Γ = −4.0 this ratio is ≈ 100. We note that the behavior of the ratio x(Ni + )/x(Cr + ) in Figure 1 also suggests a low ionization fraction, given that the observed ratio N (Ni + )/N (Cr + ) is typically very near the solar Ni/Cr ratio. 1 The utility of this ratio as an indicator of the ionization parameter would be improved with better atomic data. Figure 1 shows that while Al III is a tracer of ionized gas, it accounts for less than 10% of the total aluminum column density, even in regions of fully-ionized hydrogen (where Al II or Al IV dominate). Unfortunately, this implies that past arguments for a lack of ionized gas in DLAs based upon a relatively large Al II/Al III ratio are possibly erroneous. Figure 2 shows the CLOUDY photoionization calculations performed assuming internal sources of ionizing photons, i.e., star formation. Again the fraction of aluminum in Al III is relatively small. If we assume that the properties of the ionized gas in the DLAs are similar to those of the WIM in the Milky Way, a relatively low ionization parameter is preferred (e.g., log Γ ∼ < −3.7 is adopted by Sembach et al. 1999). The x(Ni + )/x(Cr + ) ratio suggests a low value of Γ, as in the external ionization case. For the adopted stellar spectrum, the fraction of silicon in the form of Si +3 never rises above 0.1% for the range of ionization parameters considered. Note that this is a considerably smaller fraction than found for high-z Lyman limit systems (Steidel & Sargent 1989;Prochaska 1999) and Lyα forest clouds (Songaila & Cowie 1996). Figures 1 and 2 show that even in the case where the Al II/Al III ratio is large, the amount of ionized gas in DLAs can be significant. Comparing certain metal ions to hydrogen may very well systematically overestimate the abundances of DLAs. Figure 3 shows the implied fraction of ionized hydrogen in DLAs, f (H + ), for the stellar and QSO ionizing spectra in the top panel, where we have plotted the results for several different values of log Γ. In the middle panel we show the logarithmic error introduced into measurements of [Zn/H], defined as It should be pointed out that the atomic data for zinc are quite uncertain, with the recombination coefficients being derived from extrapolations of the results for other elements. The atomic data for silicon are more reliable, though the abundance of this element is complicated by its possible inclusion into dust grains. The behavior of ǫ(Zn/H) and ǫ(Si/H) observed in Figure 3 is a common feature for those elements predominantly found in their singly-ionized stage in neutral gas. The large spread in total metal abundances, [Zn/H] (Pettini et al. 1997a(Pettini et al. , 1999, in DLAs at a given redshift could in part be due to differing ionization conditions. The total spread in abundance at a given redshift can be as high as almost 2.0 dex (Pettini et al. 1997a(Pettini et al. , 1999, which is not easily explained by ionization effects. However, the standard deviations of measurements in a given redshift interval are of order 0.3 − 0.4 dex (Pettini et al. 1997a). This degree of variation is consistent with a range of f (H + ) values between ∼ 0.0 and ∼ 0.6 in these systems.
DISCUSSION
ǫ(Zn/H) ≡ log N (Zn II) N (H I) measured − log N (Zn) N (H) intrinsic ,(1)
If ionization is playing a significant role in determining the apparent distribution of metallicity in DLAs, we might expect lower column density systems to show higher inferred abundances, on average. This is consistent with the claim by Pettini et al. (1999) that the "census" of metals in known DLAs is dominated by high column density, low metallicity systems, while those higher apparent metallicity systems tend to be of lower neutral hydrogen column densities (see also Wolfe & Prochaska 1998). However, one should also be wary of the possible selection effects in identifying high metallicity, high column density absorbers (Pei & Fall 1995;Wolfe & Prochaska 1998; see also Pettini et al. 1999).
Systematic errors in the relative metal abundances can also be significant, depending on the ions compared. Unfortunately, systematic errors in excess of 20% can begin to mimic other effects such as nucleosynthetic enrichment or dust depletion. For example, if the internal stellar ionizing spectrum is appropriate, the errors in the [Si/Fe] abundances inferred from N (Si + )/N (Fe + ) can mimic the preferential inclusion of iron into dust grains, or the enhancement of α-elements over iron. (Pettini et al. 1997b).
There are some ionic ratios that are accurate tracers of relative metal abundances even if ionized gas makes a substantial contribution. For f (H + ) < 0.5, the ratios of Mn II and Mg II to Fe II should trace Mn/Fe and Mg/Fe to within ∼ 10% in the case of the external (hard) ionizing spectrum. The ratio of Si II to Al II should be a reasonable proxy for Si/Al. For the softer stellar spectrum, the ratios of Ni II and Mg II to Si II are reliable tracers of Ni/Si and Mg/Si. Fe III is a much better tracer of ionized gas than Al III in the sense that it is the dominant ionization stage of iron in the photoionized gas. The λ1122 transition of Fe III may be lost in the Lyα forest toward high-redshift quasars, but in select cases this important transition may be useful for providing further information on the ionized gas in the DLAs.
Our calculations suggest that ionized regions may make a significant contribution to the total column density of metal ions in DLAs, and that this contribution can lead to systematic errors in the determination of abundances in these systems. Observational studies of abundances in DLAs should take ionization into account whenever possible, or at the very least assess its possible impact on the derived results.
We thank G. Ferland and collaborators for their years of work on the CLOUDY ionization code, and F. Haardt and P. Madau for providing us an electronic version of their updated UV background spectrum. Our thanks also to M. Pettini, J. Lauroesch, and J. Prochaska for helpful comments that have improved the presentation of our work. We acknowledge support from the NASA LTSA grant NAG5-3485. Haardt & Madau (1996) UV background spectrum appropriate for z = 2.0. The top panel shows the ionization fractions of ions of Fe, Al, and Si, all of which have multiple potentially observable ionization states. The bottom panel shows the ionization fractions of several commonly observed singly-ionized species relative to that of Fe + . These ionization fractions are appropriate for the fully-ionized gas in a system. The atomic input data used for Cr and Zn, in particular, are somewhat uncertain (see text). Figure 1 but assuming an internal source of ionization approximated by a Kurucz (1991) model atmosphere with T ef f = 33, 000 K. Si +3 is not seen on this plot because for all ionization parameters considered, log x(Si +3 ) < −3.
for changing mixtures of neutral and ionized gas, as traced by the Al II/Al III ratio, while the bottom panel shows the equivalent ǫ(Si/H). The predicted values of ǫ(Zn/H) and ǫ(Si/H) vary significantly with the adopted ionizing spectrum and ionization parameter. Errors in the derived values of [Zn/H] or [Si/H] of a few tenths of a dex are easily achievable even when N (Al II) ≫ N (Al III).
Figure 3
3shows that DLAs with f (H + ) ≈ (0.5, 0.4, and 0.2) can have errors of ǫ(Si/H) ≈ (0.1, 0.07, and 0.04) dex and ǫ(Zn/H) ≈ (0.3, 0.2, and 0.1) dex in the case of internal ionization for log Γ = −3.0. This error is larger for smaller ionization parameters. For the external ionizing spectrum these values are
For f (H + ) ≈ (0.5, 0.4, and 0.2), the systematic errors in [Si/Fe] are ǫ(Si/Fe) ≈ (+0.4, +0.3, and + 0.2). Similarly, systematic errors in the [Cr/Zn] abundances can mimic the inclusion of chromium into dust: ǫ(Cr/Zn) ≈ (−0.3, −0.2, and − 0.1) for the same f (H + ) values. The values f (H + ) required to explain the dispersion in inferred [Zn/H] metallicities are also sufficient to provide the dispersion in inferred [Cr/Zn] values
Fig. 1 .
1-Ionization fractions as a function of ionization parameter, Γ, for several ions assuming a
Fig. 2.-As Figure 1 but assuming an internal source of ionization approximated by a Kurucz (1991) model atmosphere with T ef f = 33, 000 K. Si +3 is not seen on this plot because for all ionization parameters considered, log x(Si +3 ) < −3.
Fig. 3 .
3-The ratio N (Al II)/N (Al III) as an indicator of the ionized gas content of DLAs. The top panel shows the fraction of ionized material, f (H + ), in DLAs as a function of the measured N (Al II)/N (Al III) assuming the labelled ionization parameters. The bottom two panels show the error (in dex) of the implied [Zn/H] and [Si/H] abundances if no correction for ionization is made (see text). The ǫ(Zn/H) results are mostly schematic given the uncertainties in the atomic data for zinc.
This result relies on new f -value determinations byFedchak & Lawler (1999). Using these new oscillator strengths we find a (weighted) average abundance [ Cr/Ni ] = +0.013 ± 0.023 in the 11 DLAs containing both elements in theProchaska & Wolfe (1999) sample.
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| []
|
[
"Einstein-Cartan cosmology and the CMB anisotropies",
"Einstein-Cartan cosmology and the CMB anisotropies"
]
| [
"Davor Palle [email protected] \nul. Ljudevita Gaja 35\n10000ZagrebCroatia\n"
]
| [
"ul. Ljudevita Gaja 35\n10000ZagrebCroatia"
]
| []
| We derive linear scalar perturbation equations for Einstein-Cartan field equations of Weyssenhoff fluid, as well as for the corresponding perturbations of Bianchi identity and geodesic equations. The equations are given in both conformal Newtonian and synchronous gauges. They are suitable for numerical implementation when precise evolution of torsion and its perturbation will be extracted from N-body cosmic simulations of the large scale structures in the Universe. A rising number of problems of the concordance cosmological model forces us to include the rotational degrees of freedom realized through torsion in the Einstein-Cartan gravity. | null | [
"https://arxiv.org/pdf/2204.10283v1.pdf"
]
| 248,299,705 | 2204.10283 | 801133eab83fcb6c48ca1abb43434a405d549fd1 |
Einstein-Cartan cosmology and the CMB anisotropies
15 Apr 2022 April 22, 2022
Davor Palle [email protected]
ul. Ljudevita Gaja 35
10000ZagrebCroatia
Einstein-Cartan cosmology and the CMB anisotropies
15 Apr 2022 April 22, 2022
We derive linear scalar perturbation equations for Einstein-Cartan field equations of Weyssenhoff fluid, as well as for the corresponding perturbations of Bianchi identity and geodesic equations. The equations are given in both conformal Newtonian and synchronous gauges. They are suitable for numerical implementation when precise evolution of torsion and its perturbation will be extracted from N-body cosmic simulations of the large scale structures in the Universe. A rising number of problems of the concordance cosmological model forces us to include the rotational degrees of freedom realized through torsion in the Einstein-Cartan gravity.
Introduction and motivation
Despite the huge success of the concordance ΛCDM model, cosmologists are faced with new challenges of the theory rooted in the surprising observational results known as the Hubble tension, 21cm EDGES anomalous absorption signal or anomalous high-redshift galaxy halo number densities (see [1] and references therein). It seems that some of the above problems could be resolved within the Einstein-Cartan cosmology [1].
Since there are a large number of cosmic observables defined as small perturbations of various physical quantities, we derive in this paper scalar perturbations for the CMB, neutrinos, baryons and CDM within the Einstein-Cartan cosmology.
A detailed description of the framework and all equations can be found in the next chapter and in Appendix A. The last chapter and the Appendix B contain some comments and suggestions for numerical implementation.
Linear scalar perturbation equations
We follow closely the definitions of ref. [2] of spatially flat cosmology with metric assignment (-+++) and the standard relation between the proper t and conformal τ time dτ = dt/a(τ ), while derivatives are denoted by dots:ȧ ≡ ∂a/∂τ . All equations will be given in the Fourier k-space with the following definition for any G:
G( x, τ ) = d 3 ke ı k· x G( k, τ ).
The Einstein-Cartan cosmological model [3,4,1] is spatially flat with Ω tot = Ω m +Ω Q +Ω Λ = 1, but with very well determinate Ω m = 2, Ω Q = −1 and Ω Λ = 0. The gauge invariant perturbed densities in the spacetimes with vorticity or shear have more complex structure [4] than in Friedmann spacetimes. Fortunately, the observations suggest that we can ignore small deviations from isotropy and homogeneity of Friedmann geometry. Thus, we perform perturbations in the conformal Newtonian and synchronous gauges on the Friedmann background [2] with the effective energy-momentum tensor of the Einstein-Cartan (EC) Weyssenhoff fluid model [5]. It is possible to define an effective energy-momentum tensor for any model in the Einstein-Cartan theory of gravity [6].
Appendix A is dedicated to the detailed definitions and discussion of the EC field equations for Weyssenhoff fluid and the corresponding Bianchi identity.
Acknowledging the relations of Appendix A and flat geometry perturbation theory [7,2] we arrive at the perturbed EC field equations in the conformal Newtonian gauge (equations analogous to eq. (23a)-(23d) of ref. [2]; φ and ψ are metric perturbations):
3ȧ a (φ +ȧ a ψ) + k 2 φ = − a 2 2 κδρ + a 2 QδQ, k 2 (φ +ȧ a ψ) = a 2 2 κ(ρ + p)ı k · v − a 2 Q 2 ı k · v, φ +ȧ a (ψ + 2φ) + (2ä a −ȧ 2 a 2 )ψ + k 2 3 (φ − ψ) = 1 2 κa 2 δp − a 2 QδQ, k 2 (φ − ψ) = 12πG N a 2 (ρ + p)σ.(1)
The effective energy-momentum tensor appears as:
T ef f µν = (p − κS 2 − Λ)g µν + U µ U ν (p + ρ − 2κS 2 ) −2(−g αβ + U α U β )∇ α [U (µ S ν)β ],(2)only Q 12 = −Q 21 = 0, Q 2 = 1 2 Q µν Q µν , κ = 8πG N , Q = κS.
We apply the assumption of isotropy (Friedmann geometry) deriving the above equations neglecting the terms proportional to torsion such as
k 2 v 1 − k 1 v 2 or k 1 k 2 (v 1 − v 2 )
. Perturbations of the EC field equations in the synnchronous gauge contain the same torsion terms as in the conformal Newtonian gauge.
Perturbation of the EC Bianchi identities described in Appendix A leads us to the following equations for CDM and baryon density contrasts and velocity gradients in the conformal Newtonian gauge:
δ c = −Θ c + 3φ, Θ c = [1 + (−7 + 1 3 ) Q 2 κρ c ] −1 {−ȧ a Θ c + k 2 ψ − 1 κρ c [2k 2 QδQ + k 2 (5 − 1 3 )Q 2 ψ − 8QQΘ c − 12Q 2ȧ a Θ c ]}, δ b = −Θ b + 3φ, Θ b = [1 + (−7 + 1 3 ) Q 2 κρ b ] −1 {−ȧ a Θ b + k 2 ψ + c 2 s k 2 δ b − 1 κρ b [2k 2 QδQ + k 2 (5 − 1 3 )Q 2 ψ − 8QQΘ b − 12Q 2ȧ a Θ b ] + 4ρ γ 3ρ b an e x e σ T (Θ γ − Θ b )},(3)Θ ≡ ı k · v.
Density contrasts in the synchronous gauge do not contain torsion terms, just like in the Newtonian gauge.Θ c (synch) vanishes, whileΘ b (synch) has the same form asΘ b (conf ), but without terms proportional to ψ. We discard terms that should vanish owing to the isotropy and put ιk 3 v 3 = 1 3 ι k · v using the same argument.
The Boltzmann equations for the phase-space distributions require the resolution of the perturbed geodesic equations in the EC cosmology:
P 0 P µ dτ +Γ µ (νκ) P ν P κ = 0, Γ µ (νκ) = µ νκ + Q µ νκ. + Q µ κν. , (µν) = 1 2 (µν + νµ), Γ µ νκ = µ νκ + Q µ νκ. + Q µ κν. + Q µ .νκ , torsion tensor = Q µ .νκ = 1 2 (Γ µ νκ −Γ µ κν ).
We verify that the torsion terms cancel out in the perturbed geodesic equations to linear order in both gauges. As a consequence, the Boltzmann equations for photons, massless and massive neutrinos retain their forms as in the Einstein cosmology (see ref. [2]). Now we have a complete set of coupled equations for the CDM, baryons, photons and neutrinos in the EC cosmology.
Conclusion and comments
Inspecting the form of the perturbation equations in the EC cosmology, one can notice that we need the knowledge not only of the torsion, but also of its time derivative and its perturbation (in the Zeldovich model is δQ = Q( Ωc Ωm δ c + Ω b
Ωm δ b )). We can achieve this objective only with the extensive N-body numerical simulations within the EC cosmology starting at large redshifts with a primordial vorticity of the Universe that causes the nonvanishing angular momentum of the Universe which is a nonrelativistic limit of torsion. In Appendix B we suggest how to improve the numerical codes for the CMB anisotropy calculations. Recent analysis of a parity violation in polarization data of Planck [8] refers to the right-handed characteristic. We show in ref. [9] that the Universe must have a preference to right-handedness of its vorticity as a consequence of a lefthanded weak interactions and the resulting abundant right-handed helicity light Majorana neutrinos.
The appearance of the primordial cosmic magnetic field is a inevitable consequence of the existence of the primordial vorticity. All these new phenomena have to be studied both theoretically and observationally.
Appendix A
In this appendix we adopt metric assignment (+ ---), as well as all definitions and conventions as in ref. [5] with intention that a reader can verify some corrections.
The Riemann-Cartan connection can be expressed as:
Γ α βµ = Γ α βµ + Q α .βµ + Q α βµ. + Q α µβ.
. The symmetric part of the EC field equations is:
R (µν) − 1 2R g µν = κT (µν) .
The contracted Bianchi identity has the following form (note the wrong sign in eqs. (2.15),(4.13),(4.14) and (4.16) of ref. [5] in front of the Mathisson-Papapetrou force):
(∇ ν − 2Q ν )T ν .µ + 2Q α .µβ T β .α − S ν .αβR αβ .
.µν = 0 . In the equation preceding eq.(5.1) of ref. [5] the last term is missing:
R (µν) − 1 2R g µν = R µν − 1 2 Rg µν + 2κ∇ α [u (µ S α ν). ] +κ 2 S 2 (2u µ u ν − g µν ) − κg µν g λφ ∇ α [u (λ S α φ).
] . However, the effective energy-momentum tensor in eq.(5.2) is correct. The relation (2.13) of ref. [5] is not generally fulfilled:
(∇ α − 2Q α )S α .µν = T [µν] .(4)
This is not an obstacle since we have, in general, a free choice to define the energy-momentum tensor [6]. Anyhow, instead of the tensor in eq. (2.19) of ref. [5], we can choose the following one:
T µα = (u µ + z µ )P α , u µ z µ = 0, z µ z µ = −1, u µ P µ = ρ .
Inserting the above tensor into the eq.(4), we get the algebraic equations for vector z µ . However, our choice is the "minimal" tensor eq.(5.2) in ref. [5].
Appendix B
A line-of-sight method [10] reduces significantly the time to solve the coupled system of equations with photon anisotropies. Let us write the multipole expansion of the temperature anisotropy:
F γ (τ, k, µ) = ∞ l=0 F γ,l (τ, k)(−ı) l (2l + 1)P l (µ) .
It fulfills the following differential equation:
dF γ dτ + (ıkµ + dκ dτ )F γ = K F (φ, ψ, F γ,0 , Θ b , F γ,2 , G γ,0 , G γ,2 ),
K F is well known f unction, G γ is polarization anisotropy.
The coupled equations are solved up to some l γ = O(10) and then the rest of multipoles are evaluated by the line-of-sight integrals [10] up to some l max = O(1000).
To avoid the problems with k-sampling and precision, one can instead separate F γ into the knownF γ (l γ ) and unknown part ∆F γ (l γ ):
F γ (τ, k, µ) =F γ (l γ , τ, k, µ) + ∆F γ (l γ , τ, k, µ), F γ (l γ , τ, k, µ) = lγ l=0 F γ,l (τ, k)(−ı) l (2l + 1)P l (µ), ∆F γ (l γ , τ, k, µ) = ∞ l=lγ +1 F γ,l (τ, k)(−ı) l (2l + 1)P l (µ) .
The unknown part satisfies the differential equation that has an explicit solution in the form of integrals:
d∆F γ dτ + (ıkµ + dκ dτ )∆F γ = K F − dF γ dτ − (ıkµ + dκ dτ )F γ .
Namely, for known P (τ ) and R(τ ), the equation: dµ∆F γ (l γ , τ, k, µ)P l (µ), f or any l > l γ .
dy(τ ) dτ + P (τ )y(τ ) = R(τ ),
Denoting the initial power spectrum with P init (k), the anisotropy spectrum is obtained:
C l (τ ) = N d 3 kP init (k)|F γ,l (τ, k)| 2 .
Thus, instead of performing the double derivatives, as in ref. [10], one has to evaluate double integrals.
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On chirality of the vorticity of the Universe. D Palle, Entropy. 14958D. Palle, "On chirality of the vorticity of the Universe", Entropy 14, 958 (2012).
A line-of-sight integration approach to cosmic microwave background anisotropies. U Seljak, M Zaldarriaga, ApJ. 469437U. Seljak and M. Zaldarriaga, "A line-of-sight integration approach to cosmic microwave background anisotropies", ApJ 469, 437 (1996).
| []
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[
"DETERMINANTAL REPRESENTATIONS OF SEMI-HYPERBOLIC POLYNOMIALS",
"DETERMINANTAL REPRESENTATIONS OF SEMI-HYPERBOLIC POLYNOMIALS"
]
| [
"Greg Knese "
]
| []
| []
| We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation for hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk. | 10.1307/mmj/1472066143 | [
"https://arxiv.org/pdf/1308.6556v2.pdf"
]
| 28,137,668 | 1308.6556 | 8d77ca4230a4c5bf51d319ff54e3587b7baf2915 |
DETERMINANTAL REPRESENTATIONS OF SEMI-HYPERBOLIC POLYNOMIALS
3 Jun 2016
Greg Knese
DETERMINANTAL REPRESENTATIONS OF SEMI-HYPERBOLIC POLYNOMIALS
3 Jun 2016arXiv:1308.6556v2 [math.AG]
We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation for hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.
Introduction
A homogeneous polynomial P ∈ R[x 0 , x 1 , . . . , x n ] is hyperbolic of degree d with respect to e ∈ R n+1 if P (e) = 0 and if for all x ∈ R n+1 the one variable polynomial t → P (x − te) has only real zeros. This concept was originally studied by Gårding for its relation to PDE (see [7], [15]) but it-and the related concept of stable polynomials-has since become important to convex optimization, combinatorics, probability, combinatorics, and analysis. See the papers and surveys [31], [13], [34], [29], [15], [26].
A deep result in the area is a determinantal representation for trivariate hyperbolic polynomials due to Helton-Vinnikov [16], [32] which solved a 1958 conjecture of Lax [17] (see [18]) and, as is mentioned in [15], can be used to develop the full Gårding theory of hyperbolicity.
Theorem A. Let p ∈ R[x 0 , x 1 , x 2 ] be hyperbolic of degree d with respect to e 2 and monic in x 2 . Then, there exist d × d real symmetric matrices A 0 , A 1 such that p(x 0 , x 1 , x 2 ) = det(x 0 A 0 + x 1 A 1 + x 2 I).
If we relax the problem to finding self-adjoint matrices instead of real symmetric matrices, proofs more amenable to computations are possible (see [10], [30], [33]). The resulting theorem is just as useful for most purposes.
Theorem A*. Let p ∈ R[x 0 , x 1 , x 2 ] be hyperbolic of degree d with respect to e 2 and monic in x 2 . Then, there exist d × d self-adjoint matrices A 0 , A 1 such that
p(x 0 , x 1 , x 2 ) = det(x 0 A 0 + x 1 A 1 + x 2 I).
Our immediate goal is to prove a generalization of this result based on a result in Geronimo et al [8] and an extension to four variables based on a result in Bickel and Knese [2], while our larger goal is to advertise the close connection between determinantal representations of hyperbolic polynomials and sums of squares decompositions for multivariable Schur stable polynomials. See [11], [12], [20], [21] for background on the latter topic.
Our main result establishes a determinantal representation with the assumption of hyperbolicity weakened. We shall call a homogeneous polynomial P ∈ R[x 0 , x 1 , . . . , x n ] a semi-hyperbolic polynomial with respect to the direction e ∈ R n+1 \ {0} if for every x ∈ R n+1 the univariate polynomial t → P (x − te) is either identically zero or only has real roots.
The key distinction between hyperbolic and semi-hyperbolic polynomials is that we do not assume P (e) = 0. Some references actually confuse the two, while Renegar [31] is the only reference we have found that emphasizes the distinction. We elaborate on our motivations in Section 6. We do need to allow for t → P (x − te) to be identically zero, because for instance if P (e) = 0 and x = 0, then P (−te) ≡ 0. We give an example of a semi-hyperbolic polynomial that is not hyperbolic in any direction in Section 3.
Here is our main theorem.
Theorem 1. Let p ∈ R[x 0 , x 1 , x 2 ]
of degree d be semi-hyperbolic with respect to e 2 = (0, 0, 1).
Then, there exist d × d self-adjoint matrices A 0 , A 1 , A 2 with A 2 positive semi-definite and a constant c ∈ R such that p(x) = c det(x 0 A 0 + x 1 A 1 + x 2 A 2 ).
Assuming p has no factors depending on x 0 , x 1 alone, the above data can be chosen to additionally satisfy: See Section 2 for the proof. We can recover Theorem A* when p(e 2 ) = 0 since p will then have degree d in x 2 and then A 2 will be positive definite. We can then factor A 1/2 2 from the right and left of 2 j=0 x j A j in order to get a determinantal representation of the form given in Theorem A*, namely with A 2 = I.
• rank A 1 = deg 1 p, rank A 2 = deg 2 p, • A 1 = B + − B − with B ± both
There is nothing special about the vector e 2 ; a linear change of variables could be used to establish a determinantal representation for other semi-hyperbolic polynomials. The assumption of no factors depending on only x 0 , x 1 is there to avoid certain annoyances that such trivial factors introduce. For instance p(x 0 , x 1 , x 2 ) = x 1 is certainly semi-hyperbolic in the direction e 2 but then A 2 = A 0 = 0 and the signature of the 1 × 1 matrix A 1 does not really provide any useful information.
It follows that a trivariate semi-hyperbolic polynomial p can be lifted to a four variable polynomial
P (x 0 , x 1 , y 1 , x 2 ) = c det(x 0 A 0 + x 1 B + + y 1 B − + x 2 A 2 )
which is hyperbolic in the direction (0, 1, 1, 1) and P (x 0 , x 1 , −x 1 , x 2 ) = p(x 0 , x 1 , x 2 ). So, we are projecting a hyperbolic polynomial (possessing a definite determinantal representation) of four variables to a set where it is not necessarily hyperbolic. It also follows that a trivariate semi-hyperbolic polynomial is a limit of hyperbolic polynomials. Indeed, writing p as in Theorem 1, define for ǫ > 0
(1.1) p ǫ (z) = c det(x 0 A 0 + x 1 A 1 + x 2 (A 2 + ǫI)).
Then, p ǫ → p as ǫ ց 0. We do not know if semi-hyperbolic polynomials in more than three variables are the limit of hyperbolic polynomials. The theorem above has an curious asymmetry in its treatment of x 0 and x 1 . This is partly due to idiosyncrasies of our proof but we also think there are some subtleties to resolve. To be specific, one could certainly break up A 0 into a difference of positive semi-definite matrices according to its signature but we have been unable to connect the signature of the A 0 we construct with geometric properties of p. We have no reason to believe this cannot be done, especially because this issue does not arise in the hyperbolic case. Indeed, we can take A 2 = I and the number of zeros of t → p(tx 0 , tx 1 , i) in C + equals the number of negative eigenvalues of x 0 A 0 + x 1 A 1 . Similarly, the number of zeros of t → p(tx 0 , tx 1 , −i) equals the number of positive eigenvalues of x 0 A 0 + x 1 A 1 . Thus, the signature of x 0 A 0 + x 1 A 1 can be derived from properties of p in the hyperbolic case. Notice that we evaluate p on the complex line (0, t, i) to determine the signature of A 1 while in the theorem above we evaluate on the line (1, t, i), which actually seems less natural. The example in Section 3 shows this is actually necessary: using the line (1, t, i) we get a correct count of the negative eigenvalues of A 1 while using the line (0, t, i) we get an incorrect count. The details are recorded in Section 3.
As a nice corollary, we can quickly recover the following variant of Theorem A*. The original proof, while not difficult, requires transforming a real stable polynomial to a hyperbolic polynomial through a linear transformation. Our signature count of A 1 in Theorem 1 makes the proof go smoothly.
Corollary 1 (See Theorem 6.6 of [3]). If p ∈ R[x 0 , x 1 , x 2 ] is homogeneous of degree d and hyperbolic with respect to all vectors in the cone {(0, v 1 , v 2 ) : v 1 , v 2 > 0}, then there exist d × d self-adjoint matrices A 0 , A 1 , A 2 and a constant c ∈ R such that A 1 , A 2 are positive semi-definite, A 1 + A 2 = I, and
p(x) = c det(x 0 A 0 + x 1 A 1 + x 2 A 2 ).
Since [3] uses Theorem A to prove the above result, all of the matrices can be taken to be real but our proof does not yield this. For p as in the corollary, p(1, x 1 , x 2 ) is known as a real stable polynomial. This formula was used in the recent paper regarding the Kadison-Singer problem [26]. See Section 4 for the very short proof of the corollary.
The key tool for the proof of Theorem 1 is a determinantal representation proven in Geronimo-Iliev-Knese [8] for certain polynomials on the bidisk D 2 = D × D (here D is the open unit disk in the complex plane C). Define D(z) = z 1 D 1 + z 2 D 2 where the D 1 , D 2 are (n + m) × (n + m) matrices given by
D 1 = I n 0 0 0 D 2 = 0 0 0 I m .
For n = n 1 + n 2 , define
P + = I n 1 0 0 0 0 0 0 0 0 P − = 0 0 0 0 I n 2 0 0 0 0
where the blocks correspond to the orthogonal decomposition
C n+m = C n 1 ⊕ C n 2 ⊕ C m . Let E = {z ∈ C : |z| > 1}, T = {z ∈ C : |z| = 1}. Theorem B. Suppose p ∈ C[z 1 , z 2 ] has bidegree (n, m), no zeros in (T × D) ∪ (T × E)
, and no factors depending on z 1 alone. Let n 2 be the number of zeros of p(z 1 , 0) in D. Then, there exists an (n + m) × (n + m) unitary U and a constant c ∈ C such that
p(z 1 , z 2 ) = c det((z 1 P − + P + + D 2 ) − U(P − + z 1 P + + z 2 D 2 )).
This is referred to as a determinantal representation for "generalized distinguished varieties" in [8] since it generalizes a determinantal representation for the "distinguished varieties" of Agler and McCarthy [1] which correspond to the case n 2 = 0. Polynomials defining distinguished varieties are essentially a Cayley transform of real stable polynomials and distinguished varieties have their own motivation in terms of operator theory as shown in [1]. Theorem B is based on first proving a sums of squares decomposition for polynomials p ∈ C[z 1 , z 2 ] with no zeros in T × D ("a face of the bidisk") and no factors in common with
p(z) = z n 1 z m 2 p(1/z 1 , 1/z 2 ). Namely, |p(z)| 2 − |p(z)| 2 = (1 − |z 1 | 2 )(|E 1 (z)| 2 − |E 2 (z)| 2 ) + (1 − |z 2 | 2 )|F (z)| 2 where E 1 ∈ C n 1 [z], E 2 ∈ C n 2 [z], F ∈ C m [z], n = n 1 + n 2 where n 2 is the number of zeros of p(z 1 , 0) in D.
This formula generalizes a sums of squares formula of Cole and Wermer [5] related to Andô's inequality from operator theory (see also [9] and [22]). It would be interesting to characterize which polynomials possess such a sums of squares formula where |F (z)| 2 is also given by a difference of squares |F 1 (z)| 2 − |F 2 (z)| 2 , and-going further-it would be interesting to see what sort of determinantal representation for real homogeneous polynomials comes out of the corresponding development from Theorem B to Theorem 1 presented here. Beyond trivariate polynomials, there are many results on the existence or non-existence of determinantal representations. See [33], [19], [27], [28], [4], [25] for recent results and convenient summaries of the state of the art. Vinnikov [33] conjectures that hyperbolic polynomials always divide a hyperbolic polynomial which has a determinantal representation but with additional requirements placed on the set where the determinantal polynomial is positive. Our next theorem offers a step in the right direction for this conjecture albeit in a special situation. A polynomial p is affine with respect to a variable x j if it has degree one in that variable.
Theorem 2. Let p ∈ R[x 0 , x 1 , x 2 , x 3 ] be hyperbolic of degree d with respect to the cone {(0, v 1 , v 2 , v 3 ) : v 1 , v 2 , v 3 > 0}.
Assume p is affine in x 2 and x 3 and of degree n in x 1 . Then, there exists k ≤ 2n + 4 and k × k self-adjoint matrices
A 0 , A 1 , A 2 , A 3 such that p divides det( 3 j=0 x j A j ), A 1 , A 2 , A 3 are positive semi-definite and A 1 + A 2 + A 3 = I.
See Section 5. Theorem 2 seems to be one of the few higher dimensional situations where one gets a determinantal representation from simple hypotheses. The recent article of Kummer [24] proves the interesting result that a hyperbolic polynomial in n variables with no real singularities divides a determinantal polynomial. This article also obtains bounds on the sizes of the matrices involved under the assumption that some power of the polynomial has a determinantal representation. Theorem 2 requires no assumptions of smoothness and obtains general bounds on the sizes of the matrices involved, but Kummer's result has the advantage that it works in n variables and does not assume degree restrictions.
The key tool for Theorem 2 is the following sums of squares decomposition from Bickel-Knese [2].
Theorem C (Theorem 1.12 of [2]). Let p ∈ C[z 1 , z 2 , z 3 ] have multi-degree (n, 1, 1) and no zeros on D 3 . Then, there exist column-vector valued polynomials
E 1 ∈ C n [z 1 , z 2 , z 3 ], E 2 , E 3 ∈ C 2 [z 1 , z 2 , z 3 ] such that for z = (z 1 , z 2 , z 3 ), w = (w 1 , w 2 , w 3 ) p(z)p(w) −p(z)p(w) = 3 j=1 (1 − z jwj )E j (w) * E j (z) wherep(z) = z n 1 z 2 z 3 p(1/z 1 , 1/z 2 , 1/z 3 ). 2. Proof of Theorem 1 from Theorem B Let C + = {z ∈ C : ℑz > 0}, C − = {z ∈ C : ℑz < 0}. Assume P ∈ R[x 0 , x 1 , x 2 ] is homogeneous of degree d and for every x ∈ R 3 t → P (x − te 2 )
is either identically zero or only has real zeros. We will assume P has no factors depending only on x 0 , x 1 which can easily be incorporated into our final determinantal representation by appending diagonal blocks to our matrices.
Consider q(z 1 , z 2 ) = P (1, z 1 , z 2 ) which has no zeros in (R × C + ) ∪ (R × C − ).
To see this, take z = (a 1 , a 2 + ib 2 ) ∈ (R × C + ) ∪ (R × C − ) with q(z) = 0. Then, P ((1, a 1 , a 2 ) + te 2 ) has the imaginary root t = ib 2 , which would imply t → P (1, a 1 , a 2 + t) is identically zero. This means x 1 − a 1 x 0 divides P which we have ruled out. Now, define
p(z 1 , z 2 ) = q i 1 + z 1 1 − z 1 , i 1 + z 2 1 − z 2 1 − z 1 2i n 1 − z 2 2i m
where q has degree n in x 1 and degree m in x 2 . Setting x 0 = 1 in P (x 0 , x 1 , x 2 ) cannot lower the degree in x 1 or x 2 , so n = deg 1 P , m = deg 2 P . Recall that |z| > 1}. We cannot have p(1, z 2 ) = 0 unless p(z 1 , z 2 ) has z 1 − 1 as a factor. This follows by Hurwitz's theorem since the polynomials z 2 → p(z 1 , z 2 ) will have no zeros in C \ T for z 1 ∈ T with z 1 → 1, and then p(1, z 2 ) will either have the same property or will be identically zero. However such factors cannot exist since they imply q has degree less than n in x 1 . In any case, we can safely divide out factors of p that depend only on z 1 since these can easily be incorporated into our final determinantal representation. Having done this, p satisfies the hypotheses of Theorem B and we may write p(z 1 , z 2 ) = c det((z 1 P − + P + + D 2 ) − U(P − + z 1 P + + z 2 D 2 )) for a unitary U. Notice n 2 is the number of roots of z 1 → p(z 1 , 0) in D which is the same as the number of roots of z 1 → q(z 1 , i) = P (1, z 1 , i) in C + .
z → i 1 + z 1 − z is
We convert back to q via z → z−i z+i . So,
q(z 1 , z 2 ) = p z 1 − i z 1 + i , z 2 − i z 2 + i (z 1 + i) n (z 2 + i) m = p z 1 − i z 1 + i , z 2 − i z 2 + i det((z 1 + i)D 1 + (z 2 + i)D 2 ) = c det((z 1 − i)P − + (z 1 + i)P + + (z 2 + i)D 2 − U((z 1 + i)P − + (z 1 − i)P + + (z 2 − i)D 2 )) = c det((I − U)D(z) − i(I + U)(P − − P + − D 2 )) = ±c det((I − U)(−z 1 P − + z 1 P + + z 2 D 2 ) + i(I + U)). (2.1)
The last line comes from multiplying on the right by det(−P − + P + + D 2 ). Letting M(z) = −z 1 P − + z 1 P + + z 2 D 2 , we now form the spectral decomposition U = V u 0 0 I V * ; V is a unitary, u is a k × k diagonal unitary with no 1's on the diagonal, and k is the rank of U − I.
Factoring V and V * out from the left and right of (2.1) leaves
q(z) = ± c det I − u 0 0 0 V * M(z)V + i I + u 0 0 2I = ±c det(I − u) det I 0 0 0 V * M(z)V + a 0 0 2iI = ±c det(I − u) det (V * M(z)V ) kk + a * 0 2iI = C det((V * M(z)V ) kk + a)
where a = i(I + u)(I − u) −1 is a diagonal matrix with real entries, (V * M(z)V ) kk is the upper k × k block of V * M(z)M, and C is a constant. Now, V * M(z)V = −z 1 V * P − V + z 1 V * P + V + z 2 V * D 2 V and if we set A 0 = a, A 1 = (−V * P − V + V * P + V ) kk , and A 2 = (V * D 2 V ) kk we have a determinantal representation for q:
q(z) = C det(A 0 + z 1 A 1 + z 2 A 2 ).
Notice A 0 , A 1 , A 2 are evidently self-adjoint with A 2 positive semi-definite, and since deg q = d we have d ≤ k. Once we show k = d, we can homogenize to get the determinantal representation for P . It helps to first establish some of the additional details listed in Theorem 1.
It is a general fact that for matrices A, B, the degree of det(tA + B) is at most rank A (we leave this as an exercise). So, deg j q ≤ rank A j for j = 1, 2. On the other hand, by construction rank A 1 ≤ rank (−P − + P + ) = deg 1 q and rank A 2 ≤ rank D 2 = deg 2 q, yielding deg j q = rank A j for j = 1, 2. Next, setting B ± = (V * P ± V ) kk we have A 1 = B + − B − . Since rank A 1 = n 1 + n 2 and rank B + ≤ n 1 and rank B − ≤ n 2 , we must have equality in both inequalities. This also shows the ranges of B + , B − have trivial intersection by considering dimensions. Since P + + P − + D 2 = I, we must have B + + B − + A 2 = I. 6 In order to show k = d, it suffices to show Q(t) := tA 1 + A 2 is non-singular for some t. For then, there would be a t 0 such that t → q(t(t 0 , 1)) has degree k, and since q has degree d, we would have k ≤ d and thus k = d.
Note Q(t) = I + (t − 1)B + − (t + 1)B − . By the spectral theorem
B + = n 1 j=1 ν j v j v * j B − = n 2 j=1 µ j w j w * j where V = {v 1 , I + (t − 1)d + 0 0 0 I − (t + 1)d − 0 0 0 I
for diagonal matrices d + , d − containing the eigenvalues ν 1 , . . . , ν n 1 ,µ 1 , . . . , µ n 2 on the diagonal. The determinant of this vanishes for only finitely many t and so Q(t 0 ) is certainly non-singular for some t 0 . Thus, k = d and we homogenize q at degree d to see that
P (x) = C det(x 0 A 0 + x 1 A 1 + x 2 A 2 ).
This concludes the proof of Theorem 1.
Example
Renegar [31] has an example of a polynomial that is semi-hyperbolic but not hyperbolic in any direction (see Section 2 of that paper); however we have constructed an example that is more illustrative for our purposes.
Let
p(x 0 , x 1 , x 2 ) = 2x 2 0 x 1 − (x 2 0 + 3x 2 1 )x 2 .
Then, t → p(x − te 2 ) clearly has only real roots for x ∈ R 3 since this one variable polynomial has degree 1 and real coefficients. Let
A 0 = i 3 0 −3 − √ 3 3 0 √ 3 √ 3 − √ 3 0 A 1 = 0 0 0 0 1 0 0 0 −1 A 2 = 1 0 0 0 0 0 0 0 0 .
We see that
p(x) = 3 det(x 0 A 0 + x 1 A 1 + x 2 A 2 )
. As remarked in the introduction we can lift to P (x 0 , x 1 , y 1 , x 2 ) = 3x 1 y 1 x 2 − (x 2 + x 1 + 3y 1 )x 2 0 which is hyperbolic in the direction (0, 1, 1, 1) and P (x 0 , x 1 , −x 1 , x 2 ) = p(x 0 , x 1 , x 2 ). We now explain why p is not hyperbolic in any direction.
We first show that {x : p(x) = 0} consists of the two connected components P + = {x : p(x) > 0}, P − = {x : p(x) < 0}. I thank the referee for the following simplified explanation. The hypersurface {x : p(x) = 0} is the graph of the continuous function (x 0 , x 1 ) →
2x 2 0 x 1 x 2 0 +3x 2 1
. Thus, {x : p(x) = 0} is divided into exactly two components: the part above the graph and the part below.
Next, neither component P + , P − is convex. For instance, (−1, 0, −1), (1, 0, −1) ∈ P + but (0, 0, −1) / ∈ P + . One can similarly show P − is not convex. This implies that p is not hyperbolic in any direction since it is a fundamental result of Gårding that if p is hyperbolic in some direction e, then the connected component of {x : p(x) = 0} containing e is convex.
This brings up a potential paradox. Since p is a limit of hyperbolic polynomials p ǫ as in equation (1.1), how is it possible that the connected components of {x : p(x) = 0} are non-convex in the above example? An answer is that a convex component of {x : p ǫ (x) = 0} could shrink to an isolated point (in projective space) as ǫ ց 0. This is something we have seen graphically using the above example.
Finally, in connection with our discussion after Theorem 1 regarding the signatures of A 0 , A 1 , let us point out that p(1, t, i) = 2t − (1 + 3t 2 )i has one zero in C + , which agrees with the number of negative eigenvalues of A 1 . On the other hand, p(0, t, i) = −3t 2 i has no zeros in C + . Notice also that p(t, 1, i) = (2 − i)t 2 − 3i has one zero in C + . This matches the number of negative eigenvalues of A 0 , which is what one would like to have more generally in order for Theorem 1 to have a more symmetric statement.
Proof of Corollary 1
Notice that t → p(x − te 2 ) is either identically zero or only has real roots by Hurwitz's theorem since this polynomial can be obtained as the limit as a ց 0 of t → p(x − t(ae 1 + e 2 )).
Any factors depending only on x 0 , x 1 can easily be dealt with separately so we may assume there are no such factors. So, p satisfies the hypotheses of Theorem 1. Also, t → p(1, t, i) can have no zeros in the upper half plane for if it had such a zero z = x + iy where y > 0, then t → p((1, x, 0) + t(0, y, 1)) would have the non-real zero t = i contradicting hyperbolicity in the direction (0, y, 1). This shows that rank B − = 0 in Theorem 1 and therefore A 1 is positive semi-definite as desired.
Proof of Theorem 2 from Theorem C
We largely follow the scheme of [23]. Let P ∈ R[x 0 , x 1 , x 2 , x 3 ] be homogeneous of degree d of degree 1 in x 2 , x 3 and of degree n in x 1 . Assume P is hyperbolic with respect to the cone
{(0, v 1 , v 2 , v 3 ) : v 1 , v 2 , v 3 > 0}. Then, for x = (x 1 , x 2 , x 3 ) q(x) = P (1, x)
has no zeros in C 3 + ∪ C 3 − and q(x) = q(x). Switching to the tridisk, we see that
f (z) = q i 1 + z 1 1 − z 1 , i 1 + z 2 1 − z 2 , i 1 + z 3 1 − z 3 1 − z 1 2i n 1 − z 2 2i 1 − z 3 2i
has no zeros in D 3 ∪ E 3 . Note that we may as well assume f is irreducible since otherwise f will have a factor depending on one or two variables alone, in which case there is no issue with having a determinantal representation. Let 1/z = (1/z 1 , 1/z 2 , 1/z 3 ) for z ∈ C 3 and definẽ
f (z) = z n 1 z 2 z 3 f (1/z) ∂f ∂z j = z n 1 z 2 z 3 z j ∂f ∂z j (1/z) for j = 1, 2, 3.
Since q has real coefficients one can show thatf = f and
nf = z 1 ∂f ∂z 1 + ∂f ∂z 1 f = z j ∂f ∂z j + ∂f ∂z j for j = 2, 3
after some simple computations. Thus, (n + 2)f = p +p where
p(z) = 3 j=1 ∂f ∂z jp (z) = 3 j=1 z j ∂f ∂z j .
Let f t (z) = f (tz) for 0 < t < 1. Then, f t has no zeros in D 3 and if we setf t (z) = t n+2 f (z/t), then |f t | = |f t | on T 3 (sincef = f ) and sof t /f t is analytic and bounded by 1 in modulus for z ∈ D 3 by the maximum principle. Now, for z ∈ D 3 0 ≤ lim tր1 |f (tz)| 2 − |t n+2 f (z/t)| 2 1 − t 2 (n + 2) = (n + 2) 2 |f (z)| 2 − 2Re(p(z)(n + 2)f (z)) = |p(z)| 2 − |p(z)| 2 since (n + 2)f = p +p with some computations omitted (see [23] for more details). This shows that if p vanishes in D 3 , then so doesp and so does f which by assumption does not happen. Hence, p has no zeros in D 3 . Note that if p andp had a common factor then this would be a factor of f which we have already ruled out; we point out that p andp cannot be multiples of one another sincẽ p vanishes at the origin. The conclusion of Theorem C holds for such a p but since we have only stated it for polynomials with no zeros on D 3 (as opposed to D 3 ) we must explain how to address the case at hand. The main point is that for 0 < t < 1, p t (z) = p(tz) will satisfy the hypotheses of Theorem C and therefore there exist vector polynomials E t 1 , E t 2 , E t shows the vector polynomials E t j are locally bounded in D 3 and hence we can choose subsequences of t ր 1 such that that E t 1 ∈ C 2n [z], E t 2 , E t 3 ∈ C 2 [z] converge to vector polynomials E 1 ∈ C 2n [z], E 2 , E 3 ∈ C 2 [z] and hence we will get a sums of squares decomposition as in Theorem C. Note the polynomials in E 1 , E 2 , E 3 necessarily have degree at most (n − 1, 1, 1), (n, 0, 1), (n, 1, 0) (this is proven in [21] for instance) and they will be non-trivial since p andp have no factors in common. On the zero set Z f of f , p = −p and therefore
(5.1) 0 = 3 j=1 (1 − z jwj )E j (w) * E j (z)
for z, w ∈ Z f . This equation ensures that the map
(5.2) z 1 E 1 (z) z 2 E 2 (z) z 3 E 3 (z) → E 1 (z) E 2 (z) E 3 (z)
defined initially for vectors of the above form with z ∈ Z f , extends linearly to a well-defined (2n + 4) × (2n + 4) unitary U. (Some details: If a combination of vectors from the left side of (5.2) sums to zero, (5.1) shows the corresponding combination on the right sums to zero. So, we get a well-defined linear map from the span of the left side of (5.2) to the span of the right side. Now, (5.1) shows this map is an isometry. Since we are in finite dimensions it can be extended to a unitary.) Note that E 1 , E 2 , E 3 cannot vanish identically in Z f without vanishing in all of C 3 since the degrees are lower and f is irreducible. Let P j for j = 1, 2, 3 be the projection onto the j-th component in the orthogonal decomposition of C 2n+4 = C 2n ⊕ C 2 ⊕ C 2 and let M(z) = 3 j=1 z j P j . By (5.2), for z ∈ Z f
(I − UM(z)) E 1 (z) E 2 (z) E 3 (z) = 0
and therefore det(I −UM(z)) = 0 for z ∈ Z f \{z : E 1 , E 2 , E 3 = 0}. Basic results in algebraic geometry (such as in Chapter 4, Section 4 of [6]) can be used to establish that this implies det(I − UM(z)) vanishes for z ∈ Z f (i.e. Z f \ {z : E 1 , E 2 , E 3 = 0} is Zariski dense in Z f ) since f is irreducible and none of E 1 , E 2 , E 3 vanish identically on Z f . Therefore f divides det(I − UM(z)). Write
f (z)g(z) = det(I − UM(z))
for some polynomial g of degree at most (n, 1, 1). As with Section 2, we convert back to q. There is some repetition in what follows but since the situations are slightly different we include the details. Now, q(z)r(z) = det(( Let U = V u 0 0 I V * be the spectral decomposition of U where u is k × k diagonal with unimodular entries, none of which equals 1. Here k is the rank of I − U. As in Section 2, the determinant (5.3) can be converted to
(5.4) q(z)r(z) = (const) det((V * M(z)V ) kk + a)
where again (V * M(z)V ) kk refers to taking the upper k × k block of the given matrix, and a = i(I + u)(I − u) −1 . Finally, if we homogenize (5.4) at degree k-note this is at most 2n + 4-then
P (x)R(x) = (const) det(x 0 a + 3 j=1 x j A j ) with A j = (V * P j V ) kk and A 1 +A 2 +A 3 = (V * IV ) kk = I, and where R(x) = x k−d 0 r((1/x 0 )(x 1 , x 2 , x 3 )
). This concludes the proof.
Concluding questions and remarks
We think it is worthwhile to discuss or rehash some of the motivations and lingering questions of this paper in more detail.
Semi-hyperbolic polynomials have perhaps been overlooked because they lack one of the key features of hyperbolic polynomials. Specifically, if p is hyperbolic in the direction e, then the connected component of {x : p(x) = 0} containing e is convex (see [31]). No such result holds for semi-hyperbolic polynomials (see Renegar [31] Section 2 or Section 3 above). This convexity property ties hyperbolic polynomials to optimization and is "the cornerstone of hyperbolic programming" [31]. This begs the question, why study semihyperbolic polynomials which may lack this property? First, we think it is a good general principle in mathematics to understand the degenerate versions of objects of interest. Notice that the (local uniform) limit of a sequence of homogeneous polynomials of degree d which are semi-hyperbolic with respect to a specific direction e is either semi-hyperbolic or identically zero. This follows from Hurwitz's theorem applied to each polynomial t → P (x − te). Hyperbolic polynomials do not share this property. Somewhat related is the following question mentioned in the introduction. Question 1. Every trivariate semi-hyperbolic polynomial is a limit of hyperbolic polynomials. Is this true more variables?
Second, our main theorem, Theorem 1, shows that trivariate semi-hyperbolic polynomials possess determinantal representations just as in the hyperbolic case. We think this in itself provides good justification for the study of semi-hyperbolic polynomials. This is a good point to formally state a question from the introduction. Question 2. In Theorem 1, can the signature and rank of A 0 be determined directly from properties of p?
Our own personal motivations for studying semi-hyperbolic polynomials came from the natural connection we presented above between semi-hyperbolic polynomials and two variable polynomials with no zeros on T×D. These latter polynomials appeared in [8] essentially because of the realization that some of the theory of polynomials with no zeros in D 2 could be pushed further to the situation of no zeros in T×D. It was realized later that this initially unnatural condition is closely related to hyperbolicity and indeed is essentially equivalent to semi-hyperbolicity.
Finally, we wish to rehash our larger goal of the paper of connecting sums of squares formulas to determinantal representations. This description will be somewhat imprecise. The approach of this paper shows that if p(z 1 , z 2 , . . . , z n ) has no zeros in D n and possesses a hermitian sums of squares formula (6.1) |p| 2 − |p| 2 = n j=1
(1 − |z j | 2 )SOS j (here each SOS j term is a sum of squared moduli of polynomials) then p+p divides a unitary determinantal polynomial det(I − UD(z)). Here U is a unitary matrix and D(z) is a diagonal matrix with coordinate functions on the diagonal. One can then convert p +p via Cayley transform and homogenization to a hyperbolic polynomial (hyperbolic with respect to all vectors with positive entries) and through some linear algebra get a self-adjoint determinantal polynomial. One can reverse engineer some of this: take a hyperbolic polynomial P (again hyperbolic with respect to vectors with positive entries) convert to a polynomial q satisfying q =q. If q can be written as p +p where p satisfies (6.1) then P divides a determinantal representation. If n > 2, not every p ∈ C[z 1 , . . . , n] with no zeros in D n satisfies an equation of the form (6.1); such polynomials are called Agler denominators. A polynomial is an Agler denominator if and only ifp/p satisfies a multivariable von Neumann inequality (see [21]).
We have also presented a modification to hyperbolicity/semi-hyperbolicity with respect to a specific direction. In n variables this would entail, after various conversions, to understanding polynomials satisfying (6.1) where SOS 1 , . . . SOS n−1 are replaced with differences of squares. With the exception of our work in [8], this is relatively uncharted territory.
positive semi-definite where rank B − equals the number of roots of p(1, t, i) in the upper half plane and rank B + + rank B − = rank A 1 , • and B − + B + + A 2 = I.
a conformal map of the unit disk onto the upper half plane sending T to R ∪ {∞} where 1 → ∞. Thus, p has no zeros in (T \ {1}) × D as well as (T \ {1}) × E where E = {z ∈ C :
. . . , v n 1 }, W = {w 1 , . . . , w n 2 } form orthonormal sets of eigenvectors corresponding to the positive eigenvalues {ν 1 , . . . , ν n 1 }, {µ 1 , . . . , µ n 2 } of B + , B − respectively. Let Y = {y 1 , . . . y k−n } be an orthonormal basis for the complement of V and W . Then, B = V ∪ W ∪ Y is a basis for C k . Let C be a basis dual to B. (Two bases {b 1 , . . . b N },{c 1 , . . . , c N } are dual if b * j c k = δ jk .)The matrix for Q(t) obtained by using C as a basis for the domain and B for the range is of the form
corresponding to p t as in Theorem C. Then,sup T 3 |p(z)| 2 ≥ (1 − |z j | 2 )|E t j (z)| 2
j=1 (z j + i)P j ) − U( 3 j=1 (z j − i)P j )) = det((I − U)M(z) + i(I + U)) (5.3) for r(z) = (z 1 + i) n (z 2 + i)(z 3 + i)g z 1 − i z 1 + i , z 2 − i z 2 + i , z 3 − i z 3 + i .10
AcknowledgmentsI would like to sincerely thank my co-authors J. Geronimo and P. Iliev[8], my co-author K. Bickel[2], and the anonymous referee. I would like to also thank Victor Vinnikov for introducing me to hyperbolic polynomials.
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Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem. Adam W Marcus, Daniel A Spielman, Nikhil Srivastava, DOI10.4007/annals.2015.182.1.8.MR3374963Ann. of Math. 2Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing families II: Mixed character- istic polynomials and the Kadison-Singer problem, Ann. of Math. (2) 182 (2015), no. 1, 327-350, DOI 10.4007/annals.2015.182.1.8. MR3374963
Polynomials with and without determinantal representations. Tim Netzer, Andreas Thom, DOI10.1016/j.laa.2012.04.043.MR2946343Linear Algebra Appl. 4377Tim Netzer and Andreas Thom, Polynomials with and without determinantal representations, Linear Algebra Appl. 437 (2012), no. 7, 1579-1595, DOI 10.1016/j.laa.2012.04.043. MR2946343
Determinantal representations and the Hermite matrix. Tim Netzer, Daniel Plaumann, Andreas Thom, DOI10.1307/mmj/1370870379.MR3079270Michigan Math. J. 622Tim Netzer, Daniel Plaumann, and Andreas Thom, Determinantal representations and the Hermite matrix, Michigan Math. J. 62 (2013), no. 2, 407-420, DOI 10.1307/mmj/1370870379. MR3079270
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Determinantal representations of hyperbolic plane curves: An elementary approach. Daniel Plaumann, Cynthia Vinzant, DOI10.1016/j.jsc.2013.05.004.MR3066450J. Symbolic Comput. 57Daniel Plaumann and Cynthia Vinzant, Determinantal representations of hyperbolic plane curves: An el- ementary approach, J. Symbolic Comput. 57 (2013), 48-60, DOI 10.1016/j.jsc.2013.05.004. MR3066450
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E-mail address: geknese@wustl. E-mail address: [email protected]
| []
|
[
"Probing BFKL dynamics in Mueller-Navelet jet production at the LHC",
"Probing BFKL dynamics in Mueller-Navelet jet production at the LHC"
]
| [
"B Ducloué \nDepartment of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland\n",
"L Szymanowski \nNational Centre for Nuclear Research\nHoża 6900-681WarsawPoland\n",
"S Wallon \nLaboratoire de Physique Théorique\nUMR 8627\nCNRS\nUniv. Paris Sud\nUniversité Paris-Saclay\n91405OrsayFrance\n\nFaculté de Physique\nUPMC Univ\nParis 06, 4 place Jussieu75252, Cedex 05ParisFrance\n"
]
| [
"Department of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland",
"Helsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland",
"National Centre for Nuclear Research\nHoża 6900-681WarsawPoland",
"Laboratoire de Physique Théorique\nUMR 8627\nCNRS\nUniv. Paris Sud\nUniversité Paris-Saclay\n91405OrsayFrance",
"Faculté de Physique\nUPMC Univ\nParis 06, 4 place Jussieu75252, Cedex 05ParisFrance"
]
| []
| We review the results of our studies on the production of two jets with a large interval of rapidity at hadron colliders, which was proposed by Mueller and Navelet as a possible test of the high energy dynamics of QCD, within the next-to-leading logarithm framework. The application of the Brodsky-Lepage-Mackenzie procedure to fix the renormalization scale leads to a very good description of the available CMS data at the LHC for the azimuthal correlations of the jets. We show that the inclusion of next-to-leading order 1 arXiv:1610.03697v1 [hep-ph] | null | [
"https://arxiv.org/pdf/1610.03697v1.pdf"
]
| 118,363,533 | 1610.03697 | 62f249a3d98ad97f4a092634441bcb91c5812ce7 |
Probing BFKL dynamics in Mueller-Navelet jet production at the LHC
June 3, 2022 12 Oct 2016
B Ducloué
Department of Physics
University of Jyväskylä
University of Jyväskylä
P.O. Box 3540014Finland
Helsinki Institute of Physics
University of Helsinki
P.O. Box 6400014Finland
L Szymanowski
National Centre for Nuclear Research
Hoża 6900-681WarsawPoland
S Wallon
Laboratoire de Physique Théorique
UMR 8627
CNRS
Univ. Paris Sud
Université Paris-Saclay
91405OrsayFrance
Faculté de Physique
UPMC Univ
Paris 06, 4 place Jussieu75252, Cedex 05ParisFrance
Probing BFKL dynamics in Mueller-Navelet jet production at the LHC
June 3, 2022 12 Oct 2016
We review the results of our studies on the production of two jets with a large interval of rapidity at hadron colliders, which was proposed by Mueller and Navelet as a possible test of the high energy dynamics of QCD, within the next-to-leading logarithm framework. The application of the Brodsky-Lepage-Mackenzie procedure to fix the renormalization scale leads to a very good description of the available CMS data at the LHC for the azimuthal correlations of the jets. We show that the inclusion of next-to-leading order 1 arXiv:1610.03697v1 [hep-ph]
corrections to the jet vertex significantly reduces the importance of energymomentum non-conservation which is inherent to the BFKL approach, for an asymmetric jet configuration.
One of the most famous testing grounds for BFKL physics [1] are the Mueller Navelet jets [2], illustrated in Fig. 1. Besides the cross section also a more exclusive observable within this process drew the attention, namely the azimuthal correlation between these jets. Considering hadron-hadron scattering in the common parton model to describe two jet production at LO, one deals with a back-to-back reaction and expects the azimuthal angles of the two jets always to be π and hence completely correlated. This corresponds in Fig. 1 to φ J,1 = φ J,2 − π. But when we increase the rapidity difference between these jets, the phase space allows for more and more emissions leading to an angular decorrelation between the jets. The production of two jets of transverse momenta k J,1 , k J,2 and rapidities y J,1 , y J,2 is described by the differential cross-section
dσ d|k J,1 | d|k J,2 | dy J,1 dy J,2 = (1) a,b 1 0 dx 1 1 0 dx 2 f a (x 1 )f b (x 2 ) dσ ab d|k J,1 | d|k J,2 | dy J,1 dy J,2 ,
where f a,b are the usual collinear partonic distributions (PDF). In the BFKL framework, the partonic cross-section reads dσ ab d|k J,1 | d|k J,2 | dy J,1 dy J,2 = dφ J,1 dφ J,2
d 2 k 1 d 2 k 2 V a (−k 1 , x 1 ) G(k 1 , k 2 ,ŝ) V b (k 2 , x 2 ),(2)
where V a,b and G are respectively the jet vertices and the BFKL Green's function. At present, they are known with the next-to-leading logarithm accuracy [3,4,5,6,7]. The cross sections (1,2) are the basic blocks of the calculations presented in [9,10,11] of the decorrelation coefficients cos m(π − ∆φ) , ∆φ = φ J,1 − φ J,2 , m ∈ N , which are observables which can be measured at experiments performed at the LHC. At present the measurements of the CMS collaboration are done for the so called the symmetric configuration of produced jets, i.e. jets in which the lower limit on transverse momentum is the same for both jets.The theoretical estimates obtained in this case for cos m(π − ∆φ) with the use of the Brodsky-Lepage-Mackenzie method to fix the renormalization scale [12], turns out to be in good agreement with the measurement reported recently by the CMS collaboration [8]. This fact is [11] for cos(π − ∆φ) (left panel) and cos 2(π − ∆φ) (right panel), with the measurements by CMS@LHC presented in [8].
y ∆ 0 2 4 6 8 〉 ) φ ∆ - π cos(
clearly illustrated in Fig. 2 and the left panel of Fig. 3 shown in Ref. [8], which also shows the comparison of measurements with various Monte Carlo simulations. The observables which are more robust against theoretical uncertainties, in particular which are more stable against a choice of renormalization and factorization scales, are the ratios of decorrelation coefficients. Fig. 4 shows a good agreeement of results of calculation with the CMS data. The CMS collaboration also measured the azimuthal distribution of the jets, defined as
1 σ dσ dϕ = 1 2π 1 + 2 ∞ n=1 cos (nϕ) cos (nϕ) , ϕ = ∆φ − π .(3)
The good agreement between theoretical estimates of [11] and measurements of this observable is shown in the right panel of Fig. 3.
Up to now we discussed production of jets in the symmetric configuration. From theoretical point of view the Monte Carlo simulations suffer in this case from insta- [8].
y ∆ 0 2 4 6 8 〉 )) φ ∆ - π cos(3( 〈 0 0.
bilities which makes difficult the comparison of theoretical results based on BFKL method with the fixed order calculation. Such comparison of different theoretical predictions can be made in the case of jet production in the asymmetric configuration, in which two jets have very different transverse momenta. In the Fig. 5 and in the left panel of Fig. 6 we present our theoretical predictions for decorrelation coefficients and their ratio confronted with the result of the fixed order calculation of Ref. [13]. It seems that specially in the case of the ratio cos 2ϕ cos ϕ a measurement of this observable could discriminate between two different mechanisms. Unfortunately, for now experimental measurements in such asymmetric configurations, although very desirable, are not available.
The important drawback of the BFKL method is the fact that it does not respect exact energy-momentum conservation. This fact can lead to sizable numerically effects, although formally it represents a non-leading correction. In the Ref. [14] we studied the violation of energy-momentum conservation for asymmetric configuration using the method proposed by Del Duca nd Schmidt in [15]. In consist in introduction of the effective rapidity Y eff defined as
Y eff ≡ Y C 2→3 0 C BFKL,O(α 3 s ) 0 (4)
in [14], where C 2→3 0 is the amplitude for 2 → 3 partonic process (contributing to the cross section) calculated up to O(α 3 s ) accuracy without any approximations and C BFKL,O(α 3 s ) 0 is the amplitude of the same process obtained within BFKL method. If the violation of energy-momentum conservation is not numerically important the ratio Y eff Y should take values close to one. In the right panel of Fig. 6 we show our result for the ratio Y eff Y estimated by taking into account NLO BFKL corrections to the jet production vertex . We see that for very asymmetric jet momenta this ratio takes values close to 1, which justifies our conclusion that the predictions y ∆ cos ϕ (left panel) and the ratio cos 3ϕ cos 2ϕ (right panel), with the measurements by CMS@LHC presented in [8].
obtained for production of jets in asymmetric configuration should not be affected by violation of energy-momentum conservation. as a function of jet momentum k J,2 for fixed k J,1 = 35 GeV for Y = 8 and s = 7 TeV at leading logarithmic (blue) and next-to-leading logarithmic (brown) accuracy
Figure 1 :
1Mueller Navelet jets production.
Figure 2 :
2The comparison of the results of the theoretical calculation of Ref.
Figure 3 :
3The comparison of theoretical calculation of Ref.[11] for cos 3(π − ∆φ) (left panel) and 1 σ dσ dϕ (right panel), with the measurements by CMS@LHC presented in
Figure 4 :
4The comparison of theoretical predictions of Ref.[11] for the ratio cos 2ϕ
Figure 5 :
5Asymmetric configuration. Variation of cos ϕ and cos 2ϕ as a function of rapidity difference Y at NLL accuracy compared with a fixed order treatment.
Figure 6 :
6Left panel: Asymmetric configuration. Variation of the ratio cos 2ϕ cos ϕ as a function of rapidity difference Y at NLL accuracy compared with a fixed order treatment. Right panel: Variation of the ratio Y ef f Y
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| []
|
[
"Efficient joint noise removal and multi exposure fusion",
"Efficient joint noise removal and multi exposure fusion"
]
| [
"Antoni Buades \nInstitute of Applied Computing and Community Code (IAC3)\nDept. of Mathematics and Computer Science\nUniversitat de les Illes Balears\nPalmaSpain\n",
"Jose Luis Lisaniid \nInstitute of Applied Computing and Community Code (IAC3)\nDept. of Mathematics and Computer Science\nUniversitat de les Illes Balears\nPalmaSpain\n",
"Onofre Martorellid *[email protected] \nInstitute of Applied Computing and Community Code (IAC3)\nDept. of Mathematics and Computer Science\nUniversitat de les Illes Balears\nPalmaSpain\n"
]
| [
"Institute of Applied Computing and Community Code (IAC3)\nDept. of Mathematics and Computer Science\nUniversitat de les Illes Balears\nPalmaSpain",
"Institute of Applied Computing and Community Code (IAC3)\nDept. of Mathematics and Computer Science\nUniversitat de les Illes Balears\nPalmaSpain",
"Institute of Applied Computing and Community Code (IAC3)\nDept. of Mathematics and Computer Science\nUniversitat de les Illes Balears\nPalmaSpain"
]
| []
| Multi-exposure fusion (MEF) is a technique that combines different snapshots of the same scene, captured with different exposure times, into a single image. This combination process (also known as fusion) is performed in such a way that the parts with better exposure of each input image have a stronger influence. Therefore, in the result image all areas are well exposed. In this paper, we propose a new method that performs MEF and noise removal. Rather than denoising each input image individually and then fusing the obtained results, the proposed strategy jointly performs fusion and denoising in the Discrete Cosinus Transform (DCT) domain, which leads to a very efficient algorithm. The method takes advantage of spatio-temporal patch selection and collaborative 3D thresholding. Several experiments show that the obtained results are significantly superior to the existing state of the art. OPEN ACCESS Citation: Buades A, Lisani JL, Martorell O (2022) Efficient joint noise removal and multi exposure fusion. PLoS ONE 17(3): e0265464. https://doi.org/ 10. | 10.1371/journal.pone.0265464 | [
"https://arxiv.org/pdf/2112.03701v1.pdf"
]
| 244,921,081 | 2112.03701 | cadff4a42462f30ba589af5a3b78afe49df879a5 |
Efficient joint noise removal and multi exposure fusion
Antoni Buades
Institute of Applied Computing and Community Code (IAC3)
Dept. of Mathematics and Computer Science
Universitat de les Illes Balears
PalmaSpain
Jose Luis Lisaniid
Institute of Applied Computing and Community Code (IAC3)
Dept. of Mathematics and Computer Science
Universitat de les Illes Balears
PalmaSpain
Onofre Martorellid *[email protected]
Institute of Applied Computing and Community Code (IAC3)
Dept. of Mathematics and Computer Science
Universitat de les Illes Balears
PalmaSpain
Efficient joint noise removal and multi exposure fusion
RESEARCH ARTICLE ¶ The authors acknowledge the Ministerio de Ciencia, Innovació n y Universidades (MCIU), the Agencia Estatal de Investigació n (AEI) and the European Regional Development Funds (ERDF) for its support to the project TIN2017-85572-P.
Multi-exposure fusion (MEF) is a technique that combines different snapshots of the same scene, captured with different exposure times, into a single image. This combination process (also known as fusion) is performed in such a way that the parts with better exposure of each input image have a stronger influence. Therefore, in the result image all areas are well exposed. In this paper, we propose a new method that performs MEF and noise removal. Rather than denoising each input image individually and then fusing the obtained results, the proposed strategy jointly performs fusion and denoising in the Discrete Cosinus Transform (DCT) domain, which leads to a very efficient algorithm. The method takes advantage of spatio-temporal patch selection and collaborative 3D thresholding. Several experiments show that the obtained results are significantly superior to the existing state of the art. OPEN ACCESS Citation: Buades A, Lisani JL, Martorell O (2022) Efficient joint noise removal and multi exposure fusion. PLoS ONE 17(3): e0265464. https://doi.org/ 10.
Introduction
Multi-exposure fusion (MEF) methods combine different pictures of the same scene, captured with different exposure times, into a single image. By keeping the best exposed parts of each image, it is possible to reconstruct a result where all the details of the scene are well rendered. Compared to High Dynamic Range (HDR), MEF doesn't require the estimation of the camera response function and the tone mapping of the processed values to the standard 8-bit [0,255] range.
The differences between the existing MEF algorithms in the literature lie in the way the information of the input images is blended. The classical method by Mertens et al. [1] computes weighted averages of the values of the input images at the same pixel location, the weights depending on exposure, saturation and contrast. These averages can also be computed in the Discrete Cosinus Transform (DCT) domain [2]. Other authors [3,4] use gradient information, and obtain the final result by solving a Poisson partial differential equation. Moreover, to prevent the appearance of visual artifacts in the resulting images, the blending is not applied on a per-pixel basis. Instead, a patch-based approach or a multiresolution blending strategy are used [5][6][7]. The use of image fusion techniques is not limited to multi-exposure images. Indeed, the combination of several images permits to improve their quality, removing for example noise [8], compression artifacts [9], haze [10][11][12], blur [13] or shaking blur from hand held video [14,15].
In this paper we tackle the problem of fusing noisy multi-exposed images. A naive approach would consist in independently denoising the multi-exposed images and then applying a fusion technique to the result. We address both problems jointly for the first time in the literature, to the best of our knowledge. We develop a technique that draws inspiration from the well known BM3D denoising method [16] and the multi-exposure fusion method described in [2]. BM3D denoising takes advantage of the redundancy of similar image patches (i.e. small image square blocks). Each image patch is denoised by grouping together similar patches in a local neighborhood and stacking them in a 3D structure to which a 3D transform is applied. In practice, the 3D transform is applied in a separable way, i.e. 2D DCT transforms followed by a 1D DCT or Walsh-Hadamard transform, as detailed in [16]. Denoising is achieved by applying a shrinkage operator to the coefficients in the transformed domain. This denoising technique is known as collaborative filtering.
We adapt BM3D to denoise the whole set of multi-exposure images by using the spatiotemporal patch selection strategy proposed in [8]. The fusion method in [2], on the other hand, merges the 2D DCT coefficients of differently exposed images. This method is not able to merge the coefficients describing the average of the patch and copies this value from the Mertens et al. [6] solution. In this paper we modify the algorithm to get rid of the dependence on Mertens et al.'s result, and include an additional step to make it robust to noise. The fact that both BM3D and the proposed fusion method work on the 2D DCT domain permits to combine them to obtain an efficient algorithm that simultaneously denoises and fuses the multi-exposed images.
The performed experiments in the sequences of Fig 1 show the superior performance of the proposed technique over the naive approach, both in terms of quality of the results and computational efficiency. The paper is organized as follows: Section 2 describes the existing literature on MEF methods. The fusion method is described in Section 3. Section 4 presents the complete proposed technique, including the BM3D-inspired denoising stage and the fusion method. In Section 5, we discuss the implementation of the method and compare with state of the art algorithms. Finally we draw some conclusions in Section 6.
Related work
The literature on MEF is very extensive. Mertens et al. [1] proposed to combine the images by averaging, choosing for each pixel a different weight depending on saturation, contrast and well-exposedness. Since such an average produces ghosting effect if the images are not well aligned, several methods were proposed to take motion into account, such as An et al. [17], Liu et al. [18], Hessel et al. [19] Li et al. [20], Ocampo et al. [21] and Hayat et al. [22].
Instead of averaging directly pixel values, several methods prefer to fuse gradient information, and then obtain the final image by solving a Poisson equation [3]. Some authors [4,[23][24][25] propose to merge all gradient values, while others (Kuk et al. [26]) choose the gradient corresponding to the better exposed image.
Deciding independently for each pixel which is the correct combination is not a robust strategy and leads to visual artifacts. Common approaches to improve the results involve the use of pyramidal image representations or patches. Several methods [6,7,27,28] adopted the Laplacian pyramid [29]. Patch-based methods make the fusion more robust by involving all the pixels in a small window [5,[30][31][32]. The method in [2] uses a DCT transform. The 2D DCT coefficients of the patches at the same spatial location and different exposure are combined depending on its magnitude. This combination is not valid to set the patch illumination, which is obtained using the Mertens et al. [6] algorithm.
The use of alternative color systems to the standard RGB is often proposed, specially YCbCR [33] which separates the luminance from the chromatic components. Since the chromatic components contain few high frequency information, a simpler strategy can be used to fuse them. Published methods differ depending on which one of the previously exposed techniques is applied to the Y component [2,34,35].
For non static sequences, the fusion creates ghosting effects near the boundary of moving objects. A possible strategy to address this problem is to weight the fusion considering patch correlation, thus discarding dissimilar patches which might belong to a different object. Another common choice is to register the whole sequence into a reference exposure [2,[36][37][38]. A straightforward strategy is to select the better exposed image as reference and generate a new sequence where the color of each image is equalized with respect to this reference. This new sequence can be fused using a MEF method, as proposed in [39][40][41][42]. Such a strategy avoids the creation of ghosting effects due to motion, but the fusion does not permit to get rid of noise.
Several methods (Z. G. Li et al. [43,44], Singh et al. [45], Raman et al. [46], Li et al. [47]) divide the images into low and high frequency components using the Bilateral [48] or the Guided filter [49]. This separation permits an additional enhancement of the image details.
Multi-exposure fusion might be accomplished by the use of variational techniques [50][51][52]. The proposed methods favor the geometry of the short exposure views and the chromaticity of the ones with longer exposure. Other methods are defined in order to maximize particular quality measures as [53][54][55][56].
Recently, neural networks have been proposed for multi exposure fusion. They can be divided into two categories, depending on whether they require the ground truth to be correctly exposed in all image areas to define the loss function (supervised methods) or not (unsupervised or self-supervised method). For HDR Kalantari et al. [57] and Wu et al. [58] proposed supervised methods. For MEF, Xu et al. [59] proposed a Generative Adversarial Network (GAN) strategy to fuse pairs of images with different exposure time. In the group of the selfsupervised methods, Prabhakar et al. [60,61] designs a metric which evaluates the quality of the multi-exposure fusion image. Xu et al. [62] trained a self-supervised neural network to preserve the similarity between the fusion result and the source exposures.
Li et al. (CNNFEAT) [63] combines the several exposures depending on a series of descriptors learnt by a neural network. Zhang et al. [64] and Zhang et al. [65] propose unified deep learning frameworks for several fusion tasks, including MEF.
There is very few literature dealing with noise removal during multi-exposure image fusion, and most published papers are focused on HDR. Akyuz et al. [66] denoise each frame before fusion, but this is performed in the radiance domain. Tico et al. [39] combine an initial fusion result with the image in the sequence with the shortest exposure. This combination is performed in the wavelet domain and coefficient attenuation is applied to the coefficients of the difference. Min et al. [67] filter the set of images with spatio-temporal motion compensated anisotropic filters prior to HDR reconstruction. Lee et al. [68] use sub-band architecture for fusion, with a weighted combination using a motion indicator function to avoid ghosting effects. The low frequency bands are filtered with a multi-resolution bilateral filter while the high frequency bands are filtered by soft thresholding. Ahmad et al. [69] identify noisy pixels and reduce their weight during image fusion.
MEF algorithm
We propose a novel algorithm for multi-exposure fusion which will serve as basis for the joint fusion and denoising method. It operates in the DCT domain and it is inspired by the fusion method in [2]. Compared to [2], we introduce a new strategy for fusing the average coefficient of the DCT and a completely different color management. It also includes a novel noise removal step, allowing for its application with moderately noisy images.
The basic algorithm, applied to single-channel images, is described first. Its extension to color images is presented in Section 3.2. Finally, a modification that confers denoising capability to the method is proposed in Section 3.3.
Single channel image fusion
Let's denote by Y k , k = 1, 2, . . ., K a sequence of luminance images acquired with different exposure taking values in the range [0, 1]. This sequence is supposed to be static. We split the images Y k into n b partially-overlapped patches of b × b pixels, fB l k g; l ¼ 1; . . . ; n b , and the 2D DCT transform of each patch is computed.
At a given pixel location, some of these patches may belong to under or over exposed parts of the images, while others may be well exposed. Both under and over exposed patches will have non-zero DCT coefficients of small magnitude, due to the lack of high frequency information. Conversely, the coefficients of well exposed patches will be large. These considerations lead to the following equation that aggregates the non-zero DCT coefficients of all the patches corresponding to the same spatial location:
B l ðxÞ ¼ X K k¼1 w l k ðxÞB l k ðxÞ; x 6 ¼ 0; l ¼ 1; 2; . . . ; n b ;ð1Þ
whereB l k denotes the DCT transform of patch l in image k, and the weights w l k ðxÞ are defined depending on the frequency ξ,
w l k ðxÞ ¼ jB l k ðxÞj p P K n¼1 jB l n ðxÞj p x 6 ¼ 0 :ð2Þ
where p > 0 is a parameter of the method. We observe that, for a given frequency, patches having higher coefficient magnitudes (i.e. well exposed patches) contribute more than the others to the weighted sum in Eq (1). This strategy does not apply to the zero frequency DCT coefficient, ξ = 0, i.e. the average of the patch values. Since large zero frequency coefficients correspond to over-exposed images, applying the same weighted combination would simply overexpose the fused image. We weight these coefficients depending on the average value of the patch and on the average value of the image to which it belongs:
B l ð0Þ ¼ X K k¼1 w l k ð0ÞB l k ð0Þ; l ¼ 1; 2; . . . ; n b ;ð3Þ
where w l k ð0Þ ¼
1 C e À ðB l k ð0ÞÀ 0:5Þ 2 =s 2 l � e À ðm k À 0:5Þ 2 =s 2 g ;ð4Þ
with μ k the average of the values of image k, C a normalizing constant, and σ l and σ g parameters of the method. Observe that 0.5 corresponds to the center of the [0, 1] range. Well exposed images are supposed to have an average value close to 0.5. The weighting factor (4) favors patches whose average value is close to this central value, and that belong to well exposed images. For all the experiments in this paper we fixed σ g = 0.3 and σ l = 0.5. Finally, the fused patches are obtained after applying the inverse DCT transform to the result of the aggregation
B l ðxÞ ¼ F À 1 ðB l ðxÞÞ; l ¼ 1; . . . ; n b :ð5Þ
Since patches are partially overlapped, the pixels in overlapping areas are averaged to produce the final image.
Color image fusion
We use an orthonormal version of the well known YUV color space, described by the following linear transformation.
Y U V 0 B B B @ 1 C C C A ¼ 1 ffi ffi ffi 3 p 1 ffi ffi ffi 3 p 1 ffi ffi ffi 3 p ffi ffi ffi 2 p 2 0 À ffi ffi ffi 2 p 2 1 ffi ffi ffi 6 p À 2 ffi ffi ffi 6 p 1 ffi ffi ffi 6 p 0 B B B B B B B B B @ 1 C C C C C C C C C A R G B 0 B B B @ 1 C C C A
The Y channel, given by the normalized average of the RGB values, represents the luminance, while U and V contain the chromatic information. The use of an orthonormal transform is motivated by the denoising stage described below. The matrix rows are mutually orthogonal which guarantees that noise is not color correlated by the transform. Each row has an Euclidean norm equal to one which permits to maintain the same noise standard deviation of the original image. This transformation can be obtained from the classical YUV decomposition by applying an orthogonalization method [70]. This color transformation was used for example in [16]. The Y channel is processed as described in the previous section. However, for the U and V components, we apply the weighted average defined by Eqs (1) and (2) to all the coefficients, including ξ = 0. The reason is that aggregating the average values of the chromaticity components (i.e. the values of the zero coefficients in the transformed domain) does not increase the risk of over-exposure, but enhances the patch average chromaticity, making the result more colorful than when applying the single channel method to each of the R, G and B components. This is noticeable in Fig 2, in which both strategies are compared.
Noise removal
The most common noise model is the additive white Gaussian noise (AWGN). The observed noisy image v is related to the underlying clean image u by v ¼ u þ n;
being n a noise image, independently and identically distributed at each pixel as a zero-mean Gaussian random variable with standard deviation σ. For other types of noise, the initial data can be modified by using variance stabilization transforms or whitening strategies, or the designed algorithm might be modified by adapting locally the parameters or applying multiscale methods. For this reason, AWGN is the most commonly assumed model in order to design general noise removal algorithms.
Denoising can be achieved by using a thresholding estimator that projects the noisy image to an orthonormal basis and reconstructs the denoised result with the transform coefficients larger than a given threshold [71].
Following this principle, the fusion method proposed in Sections 3.1 can naturally incorporate noise removal by modifying the weight definition (2) to w l k ðxÞ ¼ Thr s ðjB l k ðxÞjÞ p P K n¼1 Thr s ðjB l n ðxÞjÞ p ;
x 6 ¼ 0ð6Þ
where Thr s ðjB l k ðxÞjÞ ¼
0 jB l k ðxÞj < T � s jB l k ðxÞj otherwise 8 > > < > > :
Since the modified YUV color transformation is orthonormal, the noise standard deviation is not modified by the linear transformation proposed in Section 3.2, converting from RGB to the mentioned space. Thus, the same thresholding can be applied to each channel. The threshold parameter T is set to 2.7 as usual when denoising by thresholding in an orthonormal basis [16]. Fig 3 compares the application of the fusion chain with and without this DCT thresholding stage.
When dealing with high levels of noise, the DCT thresholding method described above is not enough to provide good denoising results. The next section describes how the fusion method can be combined with a collaborative denoising technique to obtain much better results.
Joint noise removal and fusion procedure
We assume, as in the previous sections, an AWGN noise model with standard deviation σ and that the input images are co-registered. Many variants have been proposed to deal with the noise removal of image sequences having the same exposure and noise conditions. Such methods cannot be directly used for multi-exposed images.
We propose a joint noise removal and fusion procedure. The noise removal stage is an adaptation of the BM3D collaborative DCT thresholding technique proposed in [16]. The use of such technique permits a natural integration with the fusion method proposed in Section 3, since both denoising and fusion are performed in the DCT domain.
BM3D is a patch-based image denoising method, which means that first the image is split into overlapping patches which are processed independently. Each patch is denoised by finding groups of similar patches in a local neighborhood and stacking them in a 3D structure. A separable 3D transform is applied to this structure (that is, 2D DCT transforms of each patch followed by a 1D transform in the third dimension of the stack) and denoising is achieved by setting to zero all the coefficients below a fixed threshold that depends on the standard deviation of the noise (which is assumed to be known). After computing the inverse 3D transform of the thresholded coefficients a denoised stack of patches is obtained. Since patches are partially overlapped the final denoised image is obtained after averaging, at each pixel position, the contributions of each denoised patch, in a process known as aggregation. This process should not be confused with the aggregation in the DCT domain described in Section 3.1 and modeled by Eqs (1) and (3).
In the original BM3D implementation this process is repeated two times, first applied to the noisy image, and then using as input both the noisy image and the first denoising result (known as oracle). In this second step the thresholding operation in the DCT domain is replaced by a Wiener filtering guided by the oracle image. The authors show that this two steps process restores more details and improves the denoising performance.
A naive solution to our problem would consist in denoising each multi-exposed image with the previous algorithm and then apply the fusion method described in Section 3 to the denoised set. However, we propose a more efficient method, which produces better results.
Instead of processing each image independently, we use information of the whole multiexposed set to denoise each image. In particular, the group of similar patches used to create each 3D stack is searched using the patch-selection procedure proposed in [8], which has proved to reduce the dependence on noise in the patch comparison, improving the robustness of the denoising method and reducing the usual artifacts of collaborative filtering.
Given the multi-exposed set of images and an initial patch location x, we associate to x a 3D block composed by the 2D patches from the set located at the same spatial position. Then, we search for similar 3D blocks associated to other spatial locations y. The distance between different 3D blocks is computed as
d 3D ðx; yÞ ¼ X image i in set jjP i ðxÞ À P i ðyÞjjð7Þ
where P i (x) and P i (y) denote the 2D patches referenced by x and y in image i (each 2D patch is referenced by its top-left vertex). We extract from the selected 3D blocks, the set of 2D patches belonging to each one of the images. Each set of 2D patches is denoised independently applying the collaborative strategy as proposed in [16]. Since the fusion method described in Section 3 operates in the DCT domain, it can be integrated with the BM3D algorithm. For each group of 2D patches belonging to a particular exposure, a collaborative 2D+1D transform and threshold is applied. After applying the inverse 1D transform, and prior to the computation of the inverse 2D-DCT transforms, we group the 2D DCT coefficients of the patches belonging to the same selected spatial location and having different exposures. The multi-exposure fusion described by Eqs (1) and (3) is performed on each of these groups. After this fusion step, only one DCT transformed patch is obtained at each selected location, to which the inverse DCT is applied. Fig 4 illustrates the process.
By repeating the operation on each patch location and applying aggregation in the image domain, we obtain the final denoised and fused result. In our implementation no second iteration of the algorithm is performed.
Discussion and experimental results
In this section we compare the proposed method with state of the art algorithms for exposure fusion. We compare with Mertens et al. [1], Ma et al. [5], Li et al. [44], Kou [22] and Martorell et al. [2]. The results from Ma et al. [5] and Ma et al. [55] were computed with the software downloaded from the corresponding author's webpage. The results of Mertens et al. [1] were obtained from the dataset provided in [56,72] [22,77] were obtained from the corresponding GitHub webpages. The code by Xu et al., originally proposed to fuse only pairs of images, has been adapted to fuse any sequence. We fuse the first two images of the set, this output is then combined with the third input image, and so on until all the images in the sequence have been fused. The results from Li et al. [44], Kou et al. [28] and Martorell et al. [2] were computed with the code provided by the authors. In all cases, default parameter settings are adopted.
Our results were computed using the same parameters for all the tests in this section. We use patches of size 8 × 8 pixels which is the standard size used by patch based denoising algorithms, as for example the BM3D [16]. The threshold for the collaborative filtering is set to the standard value 2.7σ. For the fusion stage, we fixed p = 7 as the power exponent of the coefficient magnitudes in Eq (2), σ g = 0.3 and σ l = 0.5 for the combination of the ξ = 0 coefficients. These latter parameters were set experimentally.
A sliding window approach is applied for the DCT based denoising/fusion. Once it is processed, the window is moved along both directions with a displacement step of N step = 2. The fact that the whole window is fused permits the processing of all the pixels in the image.
Noise-free sequences
In this case, we just compare the ability of fusing the different exposure images. We use for our method just the fusion algorithm described in section 3, but without applying any thresholding of the DCT. Note that none of the compared methods include this noise filtering step. Fig 5 displays the results of all the methods on the "Belgium House" data set. Most methods produce a result with good global illumination. However, looking closer at the images in Fig 6, we observe that many outdoor details in Mertens et al. [1] and Ma et al. [5] are overexposed. Li et al. [44] and Kou et al. [28] are not able to maintain the letters on the blackboard on the right side of Fig 6 and Ma et al. [55] is not able to preserve the details on the tree at the top of the image. Li et al. [63] the fusion result is over-smoothed, while the result by Hayat et al. [22] is over-saturated at the bright parts. The results of [2], Hessel et al [19], Zhang et al. [64] and ours are quite similar.
Noisy sequences with white uniform Gaussian noise
We compare in Fig 7 the results of all the algorithms on a noisy multi exposure sequence obtained after adding noise with standard deviation σ = 15 to the clean images (see Fig 8). We apply all the algorithms with their default parameters. It is clear from this figure, that none of the methods (except ours) is well adapted to the presence of noise. See Fig 9 for details of the results.
In the next experiment we add noise of standard deviation 25 to the clean multi-exposed images (see Fig 8) and apply the original BM3D algorithm [16] to denoise each one of them. We then apply the different multi exposure fusion methods to the denoised data (except our method, which is applied directly to the noisy sequence) and display the results in Fig 10. An excerpt of the results is zoomed in and displayed in Fig 11. We observe that our method is the only one able to denoise and fuse the multi exposure sequence without producing noticeable artifacts.
Numerical evaluation.
We propose a quantitative evaluation of the results by adapting the MEF-SSIM index proposed by Ma et al. [56] to take into account that the initial images are noisy. We begin by briefly describing this measure. Ma et al. decompose each patch x k , k = 1, 2, . . ., K of the set of input images as
x k ¼ jjx k À m x k jj � x k À m x k jjx k À m x k jj þ m x k ¼ c k � s k þ l kð8Þ
where c k ¼ jjx k À m x k jj, l k ¼ m x k and s k ¼ ðx k À m x k Þ=jjx k À m x k jj roughly represent the contrast, luminance and structure components of a patch x k . Using the previous decomposition, they compute the desired contrast and structure of the output image, respectively, aŝ
c ¼ max k¼1;���;K c k ¼ max k¼1;���;K jjx k À m x k jj:ð9Þ
and � s ¼ where w(�) is a weighting function that determines the contribution of each source image patch in the structure of the fused image patch (see Ma et al. [56] for more details on these weights.) With that, the desired output patch result iŝ
P K k¼1 wðx k À m x k Þs k P K k¼1 wðx k À m x k Þ ;ŝ ¼ � s jj� sjj ;ð10Þx ¼ĉ �ŝ:ð11Þ
Finally, the value that measures the structural similarity between a set of input patches {x k }, k = 1, 2, . . ., K from a sequence of multi-exposure images and the corresponding patch of the fused image y is given by The final measure is given by the mean of S({x k }, y) with {x k } centered at each pixel of the image
Sðfx k g; yÞ ¼ 2sx y þ C s 2 x þ s y 2 þ C :ð12ÞQðYÞ ¼ 1 M X M j¼1
Sðfx k gðjÞ; yðjÞÞ:
The defined strength of the signalĉ needs to be modified in order to subtract the noise energy. Indeed, the strength of the noisy patches writes as the sum of the signal and noise energies. In order to evaluate the noise removal in the fused image y, we redefinê c ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi maxðĉ 2 À s 2 ; 0Þ
p :ð14Þ
We denote this modified measure as MEF-SSIM n . Larger values indicate a better performance of the method. The values of MEF-SSIM n for the examples of Figs 7 and 10 are displayed in Table 1. As it can be seen, our method has the best MEF-SSIM n score.
Realistic multi exposure noisy images
The AWGN model does not hold in practice for real photographs. The noise is approximately white and additive at the camera sensor. However, it is signal dependent, meaning that the noise standard deviation at each pixel depends on its noise-free value. The noise characteristics are then modified by the camera processing pipeline, consisting of demosaicking, color processing, gamma correction and compression [78]. In order to test our fusion and noise removal method with a 'realistic' noise case, we use RAW images which contain the acquired data at the sensor of the camera. This data can be obtained by selecting the RAW as output format when using professional reflex cameras. We add signal dependent white noise to a set of multi exposure RAW images and simulate a typical camera processing pipeline, obtaining their final color version in a common graphics format (PNG, JPEG, etc.).
The results in Fig 12 show that, even in this situation, the proposed method is able to denoise and fuse the images without introducing artifacts.
Computational analysis
The time complexity of the proposed algorithm is OðjYjÞ, where |Y| denotes the size of each image in the multi-exposure sequence.
Assuming that the 3D transforms used for the collaborative filtering are performed in a separable way (i.e. 2D transforms followed by 1D transforms, as detailed in [16]), the overall number of operations of the algorithm, per pixel, is approximately
C T 2D þ 2Kb 2 N 2 S þ 2Kb 2 C T 1D þ 2Kkb 2 þ kC 0 T 2D þ kb 2ð15Þ
where:
• C T 2D denotes the number of operations required to compute the 2D DCTof the stack of patches similar to the one centered at the considered pixel. If we consider a neighborhood of size N S × N S around the pixel, this implies the computation of KN 2 S 2D DCTs, where K is the number of images in the sequence. The time complexity can be reduced by pre-computing the transforms in each block of size K × N S × N S and reusing them in overlapping blocks, similarly to what is proposed in [16].
• The second term accounts for the 3D block matching step. This implies the exhaustive search, in a N S × N S neighborhood of the pixel, of 3D blocks of size K × b × b.
• The third term counts the number of operations for the computation of the 1D DCT transforms (direct and inverse) of the k nearest neighbors of each patch, in each frame. C T 1D denotes the cost of computing a 1D DCT transform (direct or inverse) of a vector of size k.
• The fourth term accounts for the fusion step, which involves k 3D blocks of patches of size K × b × b.
• The fifth term accounts for the number of operations needed to compute the k inverse 2D DCT transforms of the fused patches. C 0 T 2D denotes the cost of computing the inverse 2D DCT transform of a patch of b × b pixels.
• Finally, the last term counts the number of operations involved in the image aggregation step.
Observe that the number of denoising operations per pixel, for each image, is smaller than that for the original BM3D algorithm, since only one step of the collaborative filtering is applied. In addition, the inverse DCT is applied only to the fused patches, since it is not necessary to denoise each individual image of the multi-exposure set.
Moreover, the previous estimation assumes that an exhaustive-search algorithm has been used for block matching. The costs C T of the DCT transforms depends on the availability of fast algorithms. By using predictive search techniques and fast separable transforms the complexity of the algorithm could be significantly reduced. Moreover, the overall number of operations can be further reduced by processing only one out of each N step < b pixels in both the horizontal and vertical directions. Due to the overlapping of the patches, the aggregation step used in the final step of the algorithm guarantees that all the pixels are correctly processed. In this case, the overall complexity of the method is reduced by a factor N 2 step .
Conclusions
In this paper we propose a patch-based method for the simultaneous denoising and fusion of a sequence of multi-exposed images. Both tasks are performed in the DCT domain and take advantage of a collaborative 3D thesholding approach similar to BM3D [16] for denoising, and the proposed fusion technique. For the collaborative denoising, a spatio-temporal criterion is used to select similar patches along the sequence, following the approach in [8]. The overall strategy permits to denoise and fuse the set of images without the need of recovering each denoised image image in the multi-exposure set, leading to a very efficient procedure. Several experiments show that the proposed method permits to obtain state-of-the-art fusion results even when the input images are noisy. As future work, we plan to extend the current approach to multi-exposed video sequences.
Fig 1 .
1Multi exposure data sets used for comparison. https://doi.org/10.1371/journal.pone.0265464.g001
Fig 2 .
2Comparison between applying the fusion to each RGB channel independently and using the YUV luminance and chromatic components. The luminance of the two results is identical, while the color is enhanced by the latter strategy.https://doi.org/10.1371/journal.pone.0265464.g002
Fig 3 .
3Noisy example with sequence office and noise with standard deviation σ = 15. Top, from left to right: one image from the multi-exposure set, fusion without DCT thresholding, fusion with DCT thresholding. Bottom: detail of the images. https://doi.org/10.1371/journal.pone.0265464.g003
Fig 4 .
4Processing scheme of a specific reference patch. https://doi.org/10.1371/journal.pone.0265464.g004
. The code for Hessel et al. (EEF) [19, 73], Xu et al. (FusionDN) [35, 74], Zhang et al.(IFCNN) [64, 75], Li et al. (CNNFEAT) [63, 76] and Hayat et al. (MEF-Sift)
Fig 5 .
5Results of fusion of noise-free multi-exposure images with different methods. https://doi.org/10.1371/journal.pone.0265464.g005 Fig 6. Excerpt of the results shown in Fig 5.
https://doi.org/10.1371/journal.pone.0265464.g006
Fig 7 .
7Results of fusion of noisy multi-exposure images with different methods. The noise standard deviation of each input image is 15. https://doi.org/10.1371/journal.pone.0265464.g007 Fig 8. Noisy multi-exposure data sets used for comparison. On the first row, the noise standard deviation of each input image is 15. On the second row the standard deviation is 25. https://doi.org/10.1371/journal.pone.0265464.g008
Fig 9 .
9Excerpt of the results shown in Fig 8. It is clear from this figure, that our method is the only one that takes noise into account. https://doi.org/10.1371/journal.pone.0265464.g009 Fig 10. Results of fusion and denoising. Our method is the only one applied directly to the noisymulti-exposure images. The rest of methods fuse denoised versions of the images obtained using the BM3D algorithm. The noise standard deviation of each input image is 25. https://doi.org/10.1371/journal.pone.0265464.g010
Fig 11 .
11Excerpt of the results shown in Fig 10. https://doi.org/10.1371/journal.pone.0265464.g011
Antoni Buades, Jose Luis Lisani, Onofre Martorell. Writing -original draft: Antoni Buades, Jose Luis Lisani. Writing -review & editing: Antoni Buades, Jose Luis Lisani, Onofre Martorell.
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et al. [28], Ma et al. [55], Hessel et al, (EEF) [19], Xu et al. [35], Zhang et al. (IFCNN) [64], Li et al. (CNNFEAT) [63], Hayat et al. (MEF-Sift)
Table 1 .
1Values of MEF-SSIM n for the examples of Figs 7 and 10. https://doi.org/10.1371/journal.pone.0265464.t001Mertens et al.
[1]
Ma et al.
[5]
Li et al.
[44]
Xu et al.
[35]
Kou et al.
[28]
Ma et al.
[55]
Martorell et al.
[2]
Li et al.
[63]
Hessel et al
[19]
Zhang et al.
[64]
Hayat et al.
[22]
Ours
Fig 7
0.607
0.349
0.611
0.567
0.666
0.599
0.570
0.695
0.553
0.572
0.634
0.841
Fig
10
0.728
0.673
0.727
0.689
0.740
0.730
0.738
0.755
0.709
0.709
0.742
0.758
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AcknowledgmentsThe authors thank Zhengguo Li and Fei Kou for kindly providing the implementation of[28,44], respectively.Author ContributionsMethodology: Antoni Buades, Jose Luis Lisani.Resources: Antoni Buades.
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| [
"Leonard S Kisslinger \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPA\n"
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"Department of Physics\nCarnegie Mellon University\n15213PittsburghPA"
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| We use our mixed hybrid model for the Ψ(2S) state to estimate Ψ(2S) to J/Ψ(1S) suppression in p-Pb collisions, and the Υ(3S) state to estimate Υ(3S) to Υ(1S) suppression in Pb-Pb collisions, and compare to recent experimental measurements. | 10.1007/s10773-015-2824-7 | [
"https://arxiv.org/pdf/1412.4747v6.pdf"
]
| 119,284,481 | 1412.4747 | 2555804818b9e147bca570f292f563c1da66569b |
3 Mar 2015
Leonard S Kisslinger
Department of Physics
Carnegie Mellon University
15213PittsburghPA
3 Mar 2015Ψ(2S), Υ(3S) Suppression in p-Pb, Pb-Pb Collisions and Mixed Hybrid Theory PACS Numbers:12.38.Aw,13.60.Le,14.40.Lb,14.40.Nd
We use our mixed hybrid model for the Ψ(2S) state to estimate Ψ(2S) to J/Ψ(1S) suppression in p-Pb collisions, and the Υ(3S) state to estimate Υ(3S) to Υ(1S) suppression in Pb-Pb collisions, and compare to recent experimental measurements.
Introduction
The production of Ψ and Υ mesons via p − p collisions has been of interest for many years as a test of QCD (Quantum Chromodynamics). More than a decade ago it was shown that the relative production of Ψ(2S) to J/Ψ(1S) in p =p collisions was not consistent with standard QCD models [1]. Similarly, in experiments on Υ(nS) production via p − p collisions it was found [2,3] that Υ(3S) to Υ(1S) production is also not consistent with standard QCD models. In a theoretical study of Ψ and Υ production via p − p or p =p collisions [4] it was shown that the relative probabilities of Ψ(2S) to J/Ψ(1S) and Υ(3S) to Υ(1S) are consistent with experiment if the Ψ(2S) and Υ(3S) are mixed heavy hybrids, discussed below. The fact that Ψ(2S) is a mixed charmonium hybrid meson and Υ(3S) is a mixed bottomonium hybrid meson, while J/Ψ(1S) and Υ(1S) are standard charmonium and bottomonium mesons is the basis for the present work.
Recent experiments using d − Au collisions [5,6] and p − P b collisions [7,8] have shown a strong suppression, S A , of Ψ(2S) relative to J/Ψ(1S). As stated in these articles, this suppression cannot be explained by current theoretical models [9], [10], [11], [12], [13]. In an earlier study of J/Ψ production and absorbtion [14] two scenarios were used for charmonium production: 1. charmonium states are produced with the cc a color octet (|cc(8)g >; and 2. charmonium states are produced as a color singlet |cc >, which is the standard model.
In the present work on Ψ(2S) suppression scenario 1. of Ref [14] is used, as is discussed in Ref [4]. We estimate S A for both J/Ψ(1S) and Ψ(2S) for p − P b collisions using the mixed heavy hybrid theory, and show that the ratio of S A for Ψ(2S) to J/Ψ(1S) is consistent with experiments. CMS experiments have measured Υ states suppression in Pb-Pb collisions [15,16], and estimated the yields of Υ(3S)/Υ(1S) relative to those in p-p collisions [16]. We estimate this ratio using our mixed hybrid theory.
Next we briefly discuss the method of QCD Sum Rules, and how this was used to show that the Ψ(2S) and Υ(3S) are mixed heavy hybrids, defined in the next section.
Mixed Heavy Hybrid States via QCD Sum Rules
The starting point of the method of QCD sum rules [17] for finding the mass of the state referred to as A is the correlator,
Π A (x) = |T [J A (x)J A (0)]| ,(1)
with | the vacuum state and the current J A (x) creating the states with quantum numbers A. The QCD sum rule is obtained by equating a dispersion relation of Π A in momentum space to an operator product expansion of Π A using QCD diagrams with quarks and gluons.
After taking a Borel transform [17], B, in which the momentum variable is replaced by the Borel mass, M B , the QCD sum rule has the form
1 π e −M 2 A /M 2 B + B ∞ so Im[Π A (s)] π(s − q 2 ) ds = B k c A k (q) < 0|O k |0 > ,(2)
where M A is the lowest mass of a state with the properties of A and the right-hand side is the Borel transform of the operator product expansion of Π A . The operator that produces the mixed charmonium and hybrid charmonium states is using Eq(2) with the quantities derived in Ref. [18]. The solution for M C−HC is given by the minimum in the plot. Note that M C−HC ≃ M 2 B for a solution satisfying the method of QCD Sum Rules. This plot is shown in the figure below for (Eq(3)) b 2 = 0.5. From this figure one sees that the minimum in M 2 C−HC (M 2 B ) corresponds to the Ψ ′ (2S) state, with a mass [19] of 3.686 GeV. Therefore the Ψ ′ (2S) meson is 50% normal Charmonium and 50% hybrid Charmonium, while the J/Ψ(1S) is a normal Charmonium meson. The analysis for Upsilon states was similar, with the Υ(3S) being 50% normal Bottomonium and 50% hybrid Bottomonium, while the Υ(1S) and Υ(2S) states are standard Bottomonium mesons. We shall use this to estimate the ratio of suppression of Ψ(2S) to J/Ψ(1S) in p-Pb collisions and Υ(3S) to Υ(3S) in Pb-Pb collisions.
J C−HC = bJ H + √ 1 − b 2 J HH ,(3)
Nuclear Modification and Suppression of Ψ(2S)/ J/Ψ(1S) in p-Pb Cossisions
In this section we derive the relative suppression of Ψ(2S) to J/Ψ(1S) and compare this result to experiment. First the definition of nuclear suppression and experimental data for the relative Ψ(2S) to J/Ψ(1S) suppression is given, and then the theoretical derivation and comparison to experiment is presented. The mixed Charmonium hybrid theory, with the Ψ ′ (2S) meson being 50% normal Charmonium and 50% hybrid Charmonium is directly used in calculating the relative suppression.
Experimental Ψ(2S) to J/Ψ(1S) suppression in p-Pb collisions
The nuclear modification for Φ = J/P si(1S) or Ψ(2S) produced in A-B collisions is defined as [5,7]
R Φ = dN A−B Φ /dy N coll dN pp Φ /dy ,(4)
where dN A−B Φ /dy and dN pp Φ /dy are the invariant yields of Φ in A-B and pp collisions. In this work we consider p-Pb collisions (A=p, B=Pb).
The relative suppression of Ψ(2S) to J/Ψ(1S) is defined as
R Ψ(2S)−J/Ψ(1S) = R Ψ(2S) R J/Ψ(1S) .(5)
The experimental resuls for rapidity 0 ≤ y ≤ 3, as shown in the figure blow is
R Ψ(2S)−J/Ψ(1S) | exp ≃ 0.65 ± 0.1(6)
As stated in Refs. [5,6,7,8], the observed suppression of Ψ(2S) compared to J/Ψ(1S) cannot be explained in standard charmonium models. As stated by J. Matthew Durham [6], "the difference in suppression is too strong to be explained by breakup effects in the nucleus...these observations raise interesting questions about the mechanism of Ψ(2S) suppression when it is produced in a nuclear target."
Recently there was an attempt to explain the Ψ(2S) versus J/Ψ(1S) suppression using a comover interaction approach [20]. In the present work we show that the mixed hybrid theory for the Ψ(2S) state, which has been successful in predicting ratios of Ψ(2S) to J/Ψ(1S) production cross sections in p-p [4] and A-A [21] collisions, can explain the mystery of the he Ψ(2S) versus J/Ψ(1S) suppression.
The experimental results for p-Pb collisions are shown in Figure 2.
TheoreticalΨ(2S) to J/Ψ(1S) suppression in p-Pb collisiona
The suppression, S A , of charmonium states is given by the interaction with nucleons as it traverses the nucleus. For a standard charmonium meson state |cc > or hybrid meson state |ccg >, with the cc having octet color, the equation for suppression is given by [22] S A = e −noσ ΦN L ,
where Φ is a cc or ccg meson, L is the length of the path of Φ in nuclear matter ≃ 8 to 10 fm for p-Pb collisions, with nuclear matter density n o = .017f m −3 , and σ ΦN is the cross section for Φ-nucleon collisions. The cross section for standard charmonium cc meson via strong QCD interactions with nucleons is given by [22] σ ccN = 2.4α s πr 2 cc ,
where the strong coupling constant α s ≃ 0.118 [19], and the charmonium meson radius r cc ≃ h/(2M c c), with M c the charm quark mass. Using 2M c ≃ M J/Ψ ≃ 3 GeV,
r cc ≃ h/(3GeV c) ≃ 6 × 10 −17 m = 0.06f m(9)
From Eqs (8,9) σ ccN ≃ 3.2 × 10 −3 f m 2 = 3.2 × 10 −2 mb .
Taking L ≃ 8-10 fm and n o = .017f m −3 , from Eq (10),
n o σ ccN L ≃ 0.0022 S cc A = e −noσ ccN L ≃ 1.0 .(11)
On the other hand, the cross section for hybrid charmonium ccg meson via strong QCD interactions with nucleons has been estimated in Ref [22] as σ ccgN ≃ 6-7 mb. In the present work we use σ ccgN ≃ 6.5mb .
From this, using L ≃ 8-10 fm and n o = .017f m −3 , from Eq(7 we obtain
n o σ ccgN L ≃ 0.88 to 1.1 S ccg A ≃ 0.4 to 0.33 .(13)
Using our mixed hybrid model, with 50% |cc > and 50% |ccg >, From Eqs (4,11,19), we find
R Ψ(2S)−J/Ψ(1S) | theory ≃ 1 + 0.4 to 0.33 2 = 0.7 to 0.66 .(14)
Comparing Eqn (14) to Eqn(6), one finds that the mixed hybrid theory for the state Ψ(2S) solves the mystery of the large suppression of Ψ(2S) vs J/P si in p-Pb collisions, and therefore in other A-B collisions.
Nuclear Modification and Suppression of Υ(3S)/Υ(1S) in Pb-Pb Collisions
This section is similar to the previous one, with the main difference being that we use the experimental results of Ref [16] for the ratios of the standard Υ(3S) to Υ(1S) rather than the theoretical estimate for Ψ(2S) to Ψ(1S)) used in the previous section.
Experimental Υ(3S) to Υ(1S) suppression in Pb-Pb collisions
As stated in Ref [16],
We also use the result from Ref [16] for the Υ(2S):
Υ(2S)/Υ(1S)| P bP b Υ(2S)/Υ(1S)| pp = 0.21 ± 0.07(stat) ± 0.02(syst); .(16)
Since the Υ(2S) state in the theory of Ref [18], upon which the present work is based, is a standard bb state, we shall use this modified by the reltive bottomium to charmonium nucleation time [22] to estimate the suppression ratio for the standard component of the Υ(3S) state in the next subsection.
Theoretical Υ(3S) to Υ(1S) suppression in Pb-Pb collisions
In deriving S cc A , the suppression for a standard model cc state we used Eq(8) to obtain the cross section for standard charmonium-nucleon cross section. Since the Υ(2S) is a standard bb state, we can a more accurate result for standard bottomium supression Pb-Pb to pp for the bb component of the Υ(3S) from Eq(16) modified by the relative neutralization time [22] of bb vs cc= M c /M b ≃ 0.55
S bb A = Υ(3S)/Υ(1S)| P bP b Υ(3S)/Υ(1S)| pp | sm ≃ 0.11 .(17)
For the cross section for hybrid bottomonium bbg meson via strong QCD interactions with nucleons we use σ bcgN ≃ σ ccgN , therefore from Eq (12) σ bbgN ≃ 6.5mb .
Using L ≃ 15 fm for Pb-Pb collisions and n o = .017f m −3 , from Eq(7 we obtain
n o σ bbgN L ≃ 1.65 S ccg A ≃ 0.19 .(19)
From Eqs (17,19) one obtains R Υ(3S)−Υ(1S) | theory ≃ .11 + .19 2 ≃ 0.15 (20) Although this is somewhat larger than the eperimental ratio shown in Eq(15), the results are in agreement within experimental and theoretical errors.
Conclusions
Using our mixed hybrid theory for the Ψ(2S) and Υ(3S) states we have found approximate agreement with experiment for the Ψ(2S) to Ψ(1S) cross section ratio for p-Pb vs p-p collisions, and the Υ(3S) to Υ(1S) cross section ratio for Pb-Pb vs p-p collisions.
with J H |0 >= |cc(0) >, J HH |0 >= |[cc(8)g](0) >, where |cc(0) > is a standard Charmonium state, while a hybrid Charmonium state |[cc(8)g](0) > has cc(8) with color=8 and a gluon with color=8. For the mixed hybrid Charmonium state produced by J C−HC mass M A of Eq(2) is called M C−HC . To find the mass M C−HC one plots the value of M 2 C−HC vs M 2 B
Figure 1 :
1Mixed Charmonium-hybrid charmonium mass ≃ 3.65 GeV
Figure 2 :
2The relative suppression of Ψ(2S) to J/Ψ(1S) for √ s pp =5.02 TeV (ALICE) with rapidity ≃ -4 and 3; and √ s pp =200 GeV (PHENIX) with rapidity ≃ 0
although the ratios of observed yields of [Υ(2S)/Υ(1S)] pp , [Υ(2S)/Υ(1S)] P bP b , [Υ(3S)/Υ(1S)] pp , and [Υ(3S)/Υ(1S)] P bP b must be corrected for difference in acceptance and efficiency of the Υ(2S) and Υ(3S) states to the Υ(1S) state, by taking ratio of ratios these corrections are not needed. The results for the ratio of ratios needed for the present work is Υ(3S)/Υ(1S)| P bP b Υ(3S)/Υ(1S)| pp = 0.06 ± 0.06(stat) ± 0.06(syst); .
AcknowledgementsThe author thanks Dr. Debasish Das, Saha Institute of Nuclear Physics, for the suggestion to consider the mixed hybrid heavy quark theory as an alternative to the theory used in Ref[20]to explain the results in Refs[5],[6],[7],[8].
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| []
|
[
"A substantial increase of Curie temperature in a new type of diluted magnetic semiconductors via effects of chemical pressure",
"A substantial increase of Curie temperature in a new type of diluted magnetic semiconductors via effects of chemical pressure"
]
| [
"Shuang Yu \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n",
"Guoqiang Zhao \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n",
"Yi Peng \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nDepartment of Materials Science & Engineering\nSichuan University\nChengduChina\n",
"Xiaohong Zhu \nDepartment of Materials Science & Engineering\nSichuan University\nChengduChina\n",
"Xiancheng Wang \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n",
"Jianfa Zhao \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n",
"Lipeng Cao \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n",
"Wenmin Li ",
"Zhi Li [email protected] \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n\nSchool of Materials Science and Engineering\nNanjing University of Science and Technology\n210094NanjingChina\n",
"Zheng Deng [email protected]:[email protected] \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n\nElectronic\n",
"Changqing Jin \nBeijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n\nMaterials Research Lab at Songshan Lake\n523808DongguanChina a) Electronic\n"
]
| [
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"Department of Materials Science & Engineering\nSichuan University\nChengduChina",
"Department of Materials Science & Engineering\nSichuan University\nChengduChina",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"School of Materials Science and Engineering\nNanjing University of Science and Technology\n210094NanjingChina",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Electronic",
"Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Materials Research Lab at Songshan Lake\n523808DongguanChina a) Electronic"
]
| []
| Chemical pressure is an effective method to tune physical properties, particularly for diluted magnetic semiconductors (DMS) of which ferromagnetic ordering is mediated by charge carriers. Via substitution of smaller Ca for larger Sr, we introduce chemical pressure on (Sr,Na)(Cd,Mn) 2 As 2 to fabricate a new DMS material (Ca,Na)(Cd,Mn) 2 As 2 . Carriers and spins are introduced by substitutions of (Ca,Na) and (Cd,Mn) respectively. The unit cell volume reduces by 6.2% after complete substitution of Ca for Sr, suggesting a subsistent chemical pressure. Importantly the local geometry of [Cd/MnAs 4 ] tetrahedron is optimized via chemical compression that increases the Mn-As hybridization leading to enhanced ferromagnetic interactions. As a result, the maximum Curie temperature (T C ) is increased by about 50% while the the maximum saturation moment increases by over 100% from (Sr,Na)(Cd,Mn) 2 As 2 to (Ca,Na)(Cd,Mn) 2 As 2 . The chemical pressure estimated from the equation of state is equal to an external physical pressure of 3.6 GPa. | 10.1063/1.5120719 | [
"https://arxiv.org/pdf/1910.08305v2.pdf"
]
| 204,788,637 | 1910.08305 | 228090fc3484f1f2e29c882b00cd748150116cec |
A substantial increase of Curie temperature in a new type of diluted magnetic semiconductors via effects of chemical pressure
Shuang Yu
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
Guoqiang Zhao
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
Yi Peng
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
Department of Materials Science & Engineering
Sichuan University
ChengduChina
Xiaohong Zhu
Department of Materials Science & Engineering
Sichuan University
ChengduChina
Xiancheng Wang
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
Jianfa Zhao
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
Lipeng Cao
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
Wenmin Li
Zhi Li [email protected]
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
School of Materials Science and Engineering
Nanjing University of Science and Technology
210094NanjingChina
Zheng Deng [email protected]:[email protected]
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
Electronic
Changqing Jin
Beijing National Laboratory for Condensed Matter Physics, and Institute ofPhysics
Chinese Academy of Sciences
100190BeijingChina
School of Physics
University of Chinese Academy of Sciences
100190BeijingChina
Materials Research Lab at Songshan Lake
523808DongguanChina a) Electronic
A substantial increase of Curie temperature in a new type of diluted magnetic semiconductors via effects of chemical pressure
1/ 19
Chemical pressure is an effective method to tune physical properties, particularly for diluted magnetic semiconductors (DMS) of which ferromagnetic ordering is mediated by charge carriers. Via substitution of smaller Ca for larger Sr, we introduce chemical pressure on (Sr,Na)(Cd,Mn) 2 As 2 to fabricate a new DMS material (Ca,Na)(Cd,Mn) 2 As 2 . Carriers and spins are introduced by substitutions of (Ca,Na) and (Cd,Mn) respectively. The unit cell volume reduces by 6.2% after complete substitution of Ca for Sr, suggesting a subsistent chemical pressure. Importantly the local geometry of [Cd/MnAs 4 ] tetrahedron is optimized via chemical compression that increases the Mn-As hybridization leading to enhanced ferromagnetic interactions. As a result, the maximum Curie temperature (T C ) is increased by about 50% while the the maximum saturation moment increases by over 100% from (Sr,Na)(Cd,Mn) 2 As 2 to (Ca,Na)(Cd,Mn) 2 As 2 . The chemical pressure estimated from the equation of state is equal to an external physical pressure of 3.6 GPa.
Chemical pressure is an effective method to tune physical properties, particularly for diluted magnetic semiconductors (DMS) of which ferromagnetic ordering is mediated by charge carriers. Via substitution of smaller Ca for larger Sr, we introduce chemical pressure on (Sr,Na)(Cd,Mn) 2 As 2 to fabricate a new DMS material (Ca,Na)(Cd,Mn) 2 As 2 . Carriers and spins are introduced by substitutions of (Ca,Na) and (Cd,Mn) respectively. The unit cell volume reduces by 6.2% after complete substitution of Ca for Sr, suggesting a subsistent chemical pressure. Importantly the local geometry of [Cd/MnAs 4 ] tetrahedron is optimized via chemical compression that increases the Mn-As hybridization leading to enhanced ferromagnetic interactions. As a result, the maximum Curie temperature (T C ) is increased by about 50% while the the maximum saturation moment increases by over 100% from (Sr,Na)(Cd,Mn) 2 As 2 to (Ca,Na)(Cd,Mn) 2 As 2 . The chemical pressure estimated from the equation of state is equal to an external physical pressure of 3.6 GPa.
2/ 19
The diluted magnetic semiconductors (DMS) have been investigated extensively as they offer an opportunity to control the ferromagnetic properties by changing carrier density. The advantage leads to potential applications in spintronic devices. [1][2][3] Specifically, recently couples of Fe-doped III-V DMS reached relatively high Curie temperature (T C ~ 320-340 K), 4-6 which challenges existing concepts and motivates further understanding of ferromagnetism in DMS. The spin & charge doping are induced by one element doping such as Mn doping into (Ga,Mn)As leading to difficulty in tuning either conducting or magnetic properties. 7 Consequently a series of new type of DMS materials with independent carrier and spin doping have been discovered to overcome aforementioned difficulty, e.g. Li 1+x (Zn,Mn)As termed "111" type or (Ba,K)(Zn,Mn) 2 As 2 (BZA) termed "122" type. BZA holds the record of Curie temperature (230 K) among the "111" and "122" -type DMS. [7][8][9][10][11] Given a DMS material, effective ways to modify T C can be achieved by increasing the carrier density using an applied electric field, photo excitations or pressure. 7,12 Particularly, pressure is expected to increase both carrier concentration and Mn-As hybridization which result in an enhancement of ferromagnetic interactions in DMS materials. 13 On the other hand, internal chemical pressure which plays a comparable role as external physical pressure, is widely used to modify physical properties in many functional materials. For instance, an equivalent increase in superconducting critical temperature in cuprate superconductors has been reported via relatively low pressures (4-6 GPa) induced by chemical pressure. 14,15 Superconductivity in iron-based compound BaFe 2 As 2 can be induced by moderate pressure (<6 GPa) and iso-valent chemical doping (BaFe 2 As 2-x P x ) respectively. 16,17 Comparing to external physical pressure, internal chemical pressure which can be applied by iso-valent substitutions, does not require any specific devices (e.g. diamond
3/ 19
anvil cell or piston cylinder cell). Nevertheless, chemical pressure-effects in DMS materials are rarely reported.
Previous studies of physical pressure-effects on "122" BZA only presented negative pressure-effect on T C . The proposed reason is that physical pressure distorts [MnAs 4 ] tetrahedra and then reduces effective Mn-As hybridization which in turn damages ferromagnetic ordering. [18][19][20] In this work we generated chemical pressure by changing atom size on another group of DMS (Sr,Na)(Cd,Mn) 2 As 2 . 21 By replacement of Sr for Ca, (Ca,Na)(Cd,Mn) 2 As 2 was synthesized as a new DMS material. From Sr-to Ca-compound, the unit cell volume decreases by 6.2% suggesting positive chemical pressure effect. It is found that local geometry of [MnAs 4 ] tetrahedron in (Ca,Na)(Cd,Mn) 2 As 2 is optimized by chemical pressure. Consequently, a successful improvement of ferromagnetic ordering by chemical pressure has been observed: comparing to (Sr,Na)(Cd,Mn) 2 As 2 , both maximum Curie temperature and saturation moment in (Ca,Na)(Cd,Mn) 2 As 2 are significantly enhanced.
Polycrystalline samples of (Ca,Na)(Cd,Mn) 2 As 2 were synthesized by solid state reaction with high purity elements. The stoichiometric ratios of starting materials were well mixed and pressed into pellets. All the processes were conducted under the protection of high-purity Argon due to the air-sensitive starting materials. The pellets were sealed in tantalum-tubes with 1 bar of Argon, and then the Ta-tubes were enclosed into evacuated quartz tubes. The samples were firstly heated at 600 °C for 12 hours.
Then the products were reground & pelleted, and sintered at 650 °C for another 12 hours. The recovered samples were characterized by powder X-ray diffraction (PXRD) with a Rigaku diffractometer using Cu-Kα radiation at room-temperature. Scanning Electron Microscope (SEM) was used to investigate the morphology and particle size. alongc-axis. Besides, two more principal deviations between CaCd 2 As 2 and SrCd 2 As 2 is the Cd/Mn-As bond lengths and As-Cd/Mn-As bond angles in Cd 2 As 2 layers which will be discussed in more details. The maximum T C of (Ca,Na)(Cd,Mn) 2 As 2 is about 50%
higher than that of (Sr,Na)(Cd,Mn) 2 As 2 (the maximum T C ~ 13 K). 21 In Figure 2(c) Tc decreases slightly with higher Na-doping level when x>0.05, is presumable due to more defects induced by Na doping in specimens. After reaching maximum T C, ferromagnetic ordering is also weakened by over-doped Mn, similar to analogues (Sr,Na)(Zn,Mn) 2 As 2 and (Sr,Na)(Cd,Mn) 2 As 2 . 21 Figure 2(b).
Coercive fields are smaller than 100 Oe. Saturation moments (M sat ) decrease with increasing Mn (Figure 2(d)), due to increased antiferromagnetic interactions as proposed to explain the decrease of T C . Nevertheless, maximum M sat of (Ca,Na)(Cd,Mn) 2 As 2 is significant larger than that of (Sr,Na)(Cd,Mn) 2 As 2 (MaximumM sat < 1μ B /Mn). The larger M sat indicates that more local spins on Mn are ferromagnetic ordered, consistent with higher T C in (Ca,Na)(Cd,Mn) 2 As 2 . 21 Electrical transport measurements are shown in Figure 3. The temperature dependent resistivity (ρ(T)) for parent compound CaCd 2 As 2 shows semiconducting behavior within temperature range of 2 -300 K (Figure 3(a)). It is worth noting that the resistivity of CaCd 2 As 2 is much smaller than SrCd 2 As 2 (ρ 300K ~ 1*10 4 Ω•mm and ρ 120K ~ 1*10 7 Ω•mm). 21 It is consistent with the aforementioned scenario that shortened Cd/Mn-As bond lengths and optimized As-Zn/Mn-As bond angle within sub-layers enhance intra-sub-layer Cd/Mn-As hybridization and in turn benefit conduction. On the other hand, ρ 2K of CaCd 2 As 2 is 3 orders magnitude larger than all the Na-doped (Ca,Na)(Cd,Mn) 2 As 2 , indicating significantly increased carrier concentrations via Na doping. The scheme is further supported as shown in Figure 3(a) by the decrease of resistivity of (Ca 1-x Na x )(Cd 0.85 Mn 0.15 ) 2 As 2 with increasing Na-doping level. In contrast, as shown in Figure 3(b), resistivity of (Ca 0.95 Na 0.05 )(Cd 1-y Mn y ) 2 As 2 gradual increases with increasing Mn concentrations.
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1/2 2 1/2 2 0 / / ( ) / (2 ) Hv kB n e C eB ħ ħ ,(1)
Where C 0 ≈ 0.605, e is the elemental charge, ħ is the reduced Planck constant, respectively, and 1/2 ≤ n v ≤ 2 depending on the number of hole sub-bands contributing to the charge transport. The best fitting to Eq. (1) gives n v = 0.62, close to that of (Sr,Na)(Cd,Mn) 2 As 2 . The maximum MR is ~15% at T = 2 K and H = 7 T. It is larger than analogues (Sr,Na)(Zn,Mn) 2 As 2 and (Ca,Na)(Zn,Mn) 2 As 2 as well as (Ba,K)(Zn,Mn) 2 As 2 which has a much higher Tc. 10,[27][28] Carrier type of the parent phase CaCd 2 As 2 and doped phases (Ca,Na)(Cd,Mn) 2 As 2 is p-type. The hole concentration of these samples are about 10 19 -10 20 cm -3 . Figure 4(c) shows Hall resistivity (ρ xy (H)) below and above T C for (Ca 0.9 Na 0.1 )(Cd 0.85 Mn 0.15 ) 2 As 2 as a typical example. At T = 2 K, the clear anomalous Hall effect (AHE) is a strong evidence for intrinsic ferromagnetism in a DMS material.
Carrier concentration calculated with linear ρ xy (H) at high-field range is n p = 2.98×10 19 cm -3 . At 300 K ρ xy is proportional to field and we obtain n p = 5.38×10 19 cm -3 .
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Considering key roles of local geometry of [Zn/MnAs] 4 tetrahedra to ferromagnetic interaction in BZA, we compare Cd/Mn-As bond lengths and
As-Cd/Mn-As bond angles of CaCd 2 As 2 and SrCd 2 As 2 to seek microscopic insight into the origin of improved ferromagnetic ordering in CaCd 2 As 2 . For carrier-mediated ferromagnetism in DMS, itinerant carriers play an important role in ferromagnetic interaction. [29][30][31][32][33][34][35][36][37][38] Given the quasi 2D structural of CaCd 2 As 2 and SrCd 2 As 2 , one can expect carriers are more itinerant along ab-plane than c-axis. If one takes a close look at Cd 2 As 2 planes, it is easy to find two sub-layers within one CdAs plane (Figure 1 (a)). It is reasonable to assume that intra-sub-layer component is more important than inter-sub-layer one to modify carriers mobility within the Cd 2 As 2 plane. With the same doping levels, sub-layer of CaCd 2 As 2 has shorter Cd/Mn-As bond length and more optimal As-Cd/Mn-As bond angles than that of SrCd 2 As 2 . As shown in Figure 1 (d), the (Ca 0.95 Na 0.05 )(Cd 0.95 Mn 0.05 )As 2 has the average Cd/Mn-As bond length of 2.700 Å and the average As-Cd/Mn-As bond angle within sub-layers of 108.9° that is close to the ~109.47° for a nondistorted ideal tetrahedron. 18 On the other hand in (Sr 0.95 Na 0.05 )(Cd 0.95 Mn 0.05 )As 2 the average Cd/Mn-As bond length is 2.712 Å and the average As-Cd/Mn-As bond angle is 113.6° that is apparently deviated from the ~109.47°. The shortened Cd/Mn-As bond length will definitely increase Mn-As hybridization. Additionally the ideal As-Cd/Mn-As bond angle will increase the overlap of Mn-As planar orbitals and guarantee the maximum strength of Mn-As hybridization, hence increasing the ferromagnetic interactions. Previous studies of physical pressure-effects on "122" BZA indicated that shortened Zn/Mn-As bond length and optimized As-Zn/Mn-As bond angle (~109.47° for a regular tetrahedron)
will enhance Cd/Mn-As hybridization. 18 In short (Ca,Na)(Cd,Mn) 2 As 2 have stronger intra-sub-layer Cd/Mn-As hybridization than that for (Sr,Na)(Cd,Mn) 2 As 2 . As a result,
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we found improved ferromagnetic ordering in (Ca,Na)(Cd,Mn) 2 As 2 . Consequently, it is reasonable to assume more chemical pressure could further improve Tc within this system. We suggest replacing Ca with Mg for future investigation, due to smaller cation size of Mg.
We calculated the equation of state (EoS) equation with first-principles calculations with plane augmented-wave (PW) pseudopotential method implemented in VASP code 39 to build up relationship between cell volume and pressure (P(V)) of SrCd 2 As 2 ( Figure S2). Based on P(V) curve, we estimate that an external pressure of 3.6 GPa can reduce cell volume of SrCd 2 As 2 to 120.0 Å 3 (volume of CaCd 2 As 2 at ambient pressure).
In summary, we successfully synthesized a new type of DMS, (Ca,Na)(Cd,Mn) 2 As 2 . The carriers and spins are introduced via (Ca,Na) and (Cd, Mn) substitutions independently. The Curie temperature of (Ca,Na)(Cd,Mn) 2 As 2 is 50% higher than that for (Sr,Na)(Cd,Mn) 2 As 2 due to the effects of chemical pressure, and the saturation moments is also enhanced dramatically. The significant improvement of ferromagnetism in (Ca,Na)(Cd,Mn) 2 As 2 indicate the prospect to search for high temperature diluted magnetic semiconductors via proper chemical pressure.
Supplementary Material
See supplementary material for the PXRD pattern of (Ca 0.95 Na 0.05 )(Cd 1-y Mn y ) 2
a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]
(
Ca 0.9546 Na 0.0454 )(Cd 0.9396 Mn 0.0604 ) 2 As 2 . Consequently we use normal composition of each sample in this manuscript, for the sake of simplification. The dc magnetic properties were measured with a Superconductivity Quantum Interference Device (SQUID, Quantum Design), and transport properties were examined by Physical Property Measurement System (PPMS, Quantum design). We calculated the equation of state (EoS) equation by first-principles calculations with plane augmented-wave (PW) pseudopotential and generalized gradient approximation implemented in VASP code with 16168 k-point grid and 500 eV energy cutoff to build up relationship between cell volume and pressure (P(V)) of SrCd 2 As 2 .Both CaCd 2 As 2 and SrCd 2 As 2 crystallize into hexagonal structure with P-3m1 space group (No. 164) as shown inFigure 1(a). Powder X-ray diffraction patterns for samples show that all of the peaks can be well indexed into P-3m1 space group (Figure S1). For all the samples, crystal grain has sharp boundaries indicating good crystallization as shown inFigure 1(b). The lattice constants were calculated by Rietveld refinement. Both of a-and c-axis shrink linearly with increasing Mn doping level in Figure 1(c) because Mn 2+ (0.66 Å) is smaller than Cd 2+ (0.78 Å), well following the Vegard Law, an evidence of successful (Cd,Mn) substitution.CaCd 2 As 2 and SrCd 2 As 2 are quasi 2D-materials where Ca/Sr ions layers and honeycomb-like Cd 2 As 2 layers stack alternately along c axis. 22 Given lattice constants for SrCd 2 As 2 (a ~ 4.4516 Å, c ~ 7.4221 Å, V ~ 127.4Å 3 ) and CaCd 2 As 2 (a ~ 4.3909 Å, c ~ 7.1870 Å, V ~ 120.0 Å 3 ), chemical compression effect is visible in the latter, particularly 5/ 19
Figure 2 H
2= 500 Oe. There is no obvious difference between zero field cooling (ZFC) and field cooling (FC) but clear ferromagnetic signatures are observed for all samples, i.e.sharp upturns with decreasing temperature. T C were determined from valleys of dM/dT curves. Above T C , susceptibility is fitted with Curie-Weiss law (inset ofFigure 2(a)), (χ-χ 0 ) -1 = (T -θ)/C, where χ 0 stands for a temperature-independent term and θ for paramagnetic temperature. Neither T C nor θ monotonically increases with increasing Mn or Na doping level(Figure 2(c)). Maximum T C~ 19 K and θ ~ 22 K are obtained for x = 0.05, y = 0.15.
Figure 4
4Negative magnetroresistance (MR = Δρ/ρ 0 = (ρ H -ρ 0 )/ρ 0 ) is found below ~ 18 K consistent with T C from magnetization data. Above 18 K, positive MR emerges. The consistency indicates that the negative MR is related to ferromagnetic ordering. InFigure 4(b), MR doesn't saturate at H = 7 T and T = 2 K, where the spins are almost fully aligned according to M(H) curve. In (Ga,Mn)As, and analogue (Sr,Na)(Cd,Mn) 2 As 2 the unsaturated MR are explained with giant splitting of the valence band.In order to understand such behavior, the negative magnetroresistance dates at 2 K are fitted with following equation,[24][25][26]
Figure Caption
Figure 1 .
1(0.05 )As 2 . Marked bond length and bond angle are the ones within the CdAs sub-layers.
Figure 2 .
2(a) M(T) measured under H =500 Oe of (Ca 1-x Na x )andθ versus Na-and Mn doping level. (d) M sat versus Mn doping level.
Figure 3
3Figure 3. Temperature dependent resistivity curves of (a) (Ca 1-x Na x )(Cd 0.85 Mn 0.15 ) 2 As 2 , (x = 0.025, 0.05, 0.1) and CaCd 2 As 2 in the insert. (b) (Ca 0.95 Na 0.05 )(Cd 1-y Mn y ) 2 As 2 (y = 0.05, 0.15, 0.2).
Figure 4 .
4(a) ρ(T) curves of (Ca 0.95 Na 0.05 )(Cd 0.85 Mn 0.15 ) 2 As 2 under various field. (b) Magnetoresistance curves of (Ca 0.95 Na 0.05 )(Cd 0.85 Mn 0.15 ) 2 As 2 measured in an external field up to 7 T at T = 2, 10, 20, and 50 K, respectively. The pink circles show the fitting result according to Eq. (1). (c) Hall effect measurement results for
AcknowledgmentsThis work was financially supported by National Key R&D Program of China (No.
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| []
|
[
"Minimizing Travel in the Uniform Dispersal Problem for Robotic Sensors",
"Minimizing Travel in the Uniform Dispersal Problem for Robotic Sensors"
]
| [
"Michael Amir ",
"Alfred M Bruckstein ",
"Michael Amir ",
"Alfred M ",
"\nTechnion -Israel Institute of Technology Haifa\nIsrael\n",
"\nACM Reference Format\nTechnion -Israel Institute of Technology Haifa\nIsrael\n"
]
| [
"Technion -Israel Institute of Technology Haifa\nIsrael",
"ACM Reference Format\nTechnion -Israel Institute of Technology Haifa\nIsrael"
]
| [
"Proc. of the 18th Interna-tional Conference on Autonomous Agents and Multiagent Systems (AAMAS 2019)"
]
| The limited energy capacity of individual robotic agents in a swarm often limits the possible cooperative tasks they can perform. In this work, we investigate the problem of covering an unknown connected grid environment (e.g. a maze or connected corridors) with a robotic swarm so as to minimize the maximal number of steps that each member of the swarm makes and their activity time before their work is finished, thereby minimizing the energy requirements. The robots are autonomous, anonymous and identical, with local sensors and finite memory, and possess no communication capabilities. They are assumed to disperse over time from a fixed location, and to move synchronously. The robots are tasked with occupying every cell of the environment, while avoiding collisions.In the literature such topics are known as uniform dispersal problems. The goal of minimizing the number of steps traveled by the robots has previously been studied in this context. Our contribution is a local robotic strategy for simply connected grid environments that, by exploiting their topology, achieves optimal makespan (the amount of time it takes to cover the environment) and minimizes the maximal number of steps taken by the individual robots before their deactivation. The robots succeed in discovering optimal paths to their eventual destinations, and finish the covering process in 2V − 1 time steps, where V is the number of cells in the environment. | null | [
"https://arxiv.org/pdf/1903.03259v1.pdf"
]
| 72,940,896 | 1903.03259 | 2e2f9cc0d76fe49f16f52d3331abe66bc932918c |
Minimizing Travel in the Uniform Dispersal Problem for Robotic Sensors
IFAAMASCopyright IFAAMAS2019. May 13-17, 2019
Michael Amir
Alfred M Bruckstein
Michael Amir
Alfred M
Technion -Israel Institute of Technology Haifa
Israel
ACM Reference Format
Technion -Israel Institute of Technology Haifa
Israel
Minimizing Travel in the Uniform Dispersal Problem for Robotic Sensors
Proc. of the 18th Interna-tional Conference on Autonomous Agents and Multiagent Systems (AAMAS 2019)
. of the 18th Interna-tional Conference on Autonomous Agents and Multiagent Systems (AAMAS 2019)Montreal, CanadaIFAAMAS2019. May 13-17, 20199 pages.Mobile robotMinimizing movementUnknown environmentUni- form dispersalGrid environmentArea coverage
The limited energy capacity of individual robotic agents in a swarm often limits the possible cooperative tasks they can perform. In this work, we investigate the problem of covering an unknown connected grid environment (e.g. a maze or connected corridors) with a robotic swarm so as to minimize the maximal number of steps that each member of the swarm makes and their activity time before their work is finished, thereby minimizing the energy requirements. The robots are autonomous, anonymous and identical, with local sensors and finite memory, and possess no communication capabilities. They are assumed to disperse over time from a fixed location, and to move synchronously. The robots are tasked with occupying every cell of the environment, while avoiding collisions.In the literature such topics are known as uniform dispersal problems. The goal of minimizing the number of steps traveled by the robots has previously been studied in this context. Our contribution is a local robotic strategy for simply connected grid environments that, by exploiting their topology, achieves optimal makespan (the amount of time it takes to cover the environment) and minimizes the maximal number of steps taken by the individual robots before their deactivation. The robots succeed in discovering optimal paths to their eventual destinations, and finish the covering process in 2V − 1 time steps, where V is the number of cells in the environment.
INTRODUCTION AND RELATED WORK
The objective of swarm robotics is to enable a large group of simple and autonomous mobile robots to work cooperatively towards complex goals. It is often the case, e.g. when the robots are traveling large distances or are airborne, that a lot of energy is required for the sustained activity of robots in the swarm. In this work, we are interested in solving the uniform dispersal problem for simply connected grid environments while minimizing the movement and active time of each individual robot, in order to minimize the energy requirements.
In many real life scenarios, e.g. mapping or hazard detection, one is interested in deploying agents over an unknown area and covering it for the purposes of sensing or reacting [17]. The use of swarm robotics to solve such problems has many inherent advantages, such as scalability, greater coverage, and autonomy in mission execution. In uniform dispersal, a large number of mobile robots emerge over time from a source or several source locations (called "doors" in the literature), and are tasked to completely cover an unknown environment R by occupying every location and to terminate their work in finite time [3]. The robots must not collide (i.e. two robots must never occupy the same location), nor step outside the boundaries of the environment.
Hsiang et al. [19] [18] introduced the problem of uniform dispersal in discrete planar domains by mobile robots endowed only with finite memory, local sensors, and local communication. Their DFS-esque "follow the leader" strategy enables robots to cover the environment in optimal time, assuming a synchronous time scheme. Much follow-up work has focused on achieving dispersal with weaker models of robots, e.g. disallowing communication, reducing memory, or assuming asynchronous time [3] [15] [4]. Barrameda et al. [3] have shown that the dispersal problem is intractable under the usual assumptions if the robots are assumed to be oblivious (that is, to possess no persistent states), though there have been attempts to get around this limitation using randomization [16]. It is standard to assume that the robots are moving in a connected grid environment, as any 2D space can be approximated well by pixelation into tiny grid cells of uniform size.
From a theoretical perspective, the problem of dispersing and coordinating mobile robotic agents while minimizing movement or energy has been studied extensively both as a centralized motion planning problem and in distributed sensor networks [9] [21] [11], and various computational hardness results have been proven in the case of general graph environments [8]. More broadly, multi-agent scheduling problems have been studied in the presence of energy constraints [13]. Specifically in the context of uniform dispersal for robotic sensors, the question of minimizing travel for orthogonal areas that we here concern ourselves with was discussed in the original paper by Hsiang et. al [19] and soon after in Stainzberg's doctoral dissertation [25], and more recently in [14] and [16].
In recent decades there has been considerable effort dedicated to the algorithmic problems of agent coverage or exploration, wherein a robot or team of robots must completely explore, occupy, or map an area. Attention has been given to the case of a single robot tasked with visiting every vertex of a graph or grid environment [5] [28], to single-and multi-robot path planning [1], to natural or pheromonebased computation models [23] [26] [27], to related formation or dispersal problems [22] [7], and to a multitude of other topics. We refer the reader to [12] or [2] for recent surveys. The problem of uniform dispersal distinguishes itself from many of these by its distinctly online nature. The robots emerge onto the environment at different times and must successfully embed themselves into the ongoing exploration effort, without colliding with other robots, and without interrupting the constant outflow of new robots. They must do this under stringent computational, sensory, and communication restrictions-in most recent models, the robots, modelled as finite automata, are not allowed to talk to each other, and cannot even tell the difference between environmental obstacles and the presence of robots active in the formation. We find it fairly surprising that under these restrictions, robots are capable of exploring an entirely unknown environment in theoretically optimal time, as well as (we shall see) walk only in shortest paths to their destinations while doing so.
Much attention has been given to the problem of deployment and coverage in GPS-denied environments, as this may enable the deployment of robotic fleets outside laboratory conditions and their utilization in real world scenarios. Dispersal strategies that operate under stringent restrictions on communication and sensing may be especially relevant to future investigations in this domain. Implementation, however, forms a technical barrier, as when looking at the problem of generating robust uniform coverage from a systems perspective the issues of relative visual localization -range, angular coverage and persistence -become important. There has been progress towards overcoming these barriers in a number of different settings. In [6] the authors discuss a visual relative localization method suited for autonomous navigation and obstacle avoidance in indoor environments, for mobile robots with limited computational power. In [24] the authors present a visual localization method based on an image processing algorithm suitable for use on small quadcopters. The algorithm assumes all the quadcopters have identical but specific markings that ease the localization. These are but examples of the sensors an agent might use when implementing strategies that operate under such restrictions.
Our contribution: Working in a synchronous time setting, Hsiang et al. [19] pose the problem of minimizing the total and individual number of steps the robots take (the "total travel" and "individual travel"), while achieving optimal makespan-the time before complete coverage of the environment. They describe several algorithms for general grid environments that consecutively improve on each other in this respect, but these algorithms do not achieve a global optimum.
We describe a local uniform dispersal strategy that, for simply connected grid environments, achieves optimal makespan and minimizes the total travel and maximal individual travel. The strategy's goal is to enable a robot to settle in place as soon as possible, thereby minimizing the energy consumption. It exploits the ability to decompose simply connected environments into a tree of simply connected sub-environments via "halls"-defined as corners of the environment that also have an obstacle located diagonally opposite to them. We work in a setting similar to [19], where time is synchronous and robots have local sensors and finite memory. Specifically, the robots require 5 bits of persistent memory (2 5 persistent states), and a visibility span of Manhattan distance 2. As is sometimes assumed, e.g. in [15], they are initialized with a common notion of up, down, left and right. Unlike [19], our algorithm works without assuming any inter-robot communication capabilities: the robots are only capable of seeing environmental obstacles (including other robots that block them), and are unable to distinguish between kinds of obstacles.
By attempting to restrict their movement to as few directions as possible, our strategy enables the robots to travel in shortest paths from their arrival point to their eventual, a-priori unknown, settling point. The robots finish dispersing in 2V − 1 time steps, where V is the number of cells in the environment.
We show further that no local strategy can minimize total travel in the general case, i.e. for general grid environments.
MODEL
Consider the integer grid Z 2 = Z × Z, whose vertices are points (x, y) where x and y are both integers, and (x 1 , y 1 ) is connected to (x 2 , y 2 ) if and only if the Manhattan distance |x 1 − x 2 | + |y 1 − y 2 | is exactly 1. A grid environment or region R is defined as a connected sub-graph of Z 2 . The complement of R, denoted R c , is defined as the sub-graph Z 2 − R of Z 2 . We call the vertices of R c walls. Definition 2.1. A region R is said to be simply connected if and only if any path v 1 v 2 . . . v 1 of vertices in R that forms a closed curve does not surround any vertices of R c .
In particular, a region R is simply connected if R c is connected. A robot is a mobile point in R with limited vision and small finite memory. No two robots may occupy the same location. The visibility range of all robots is assumed to be 2, meaning at every time step, a robot is aware of unoccupied vertices in R that are at a Manhattan distance of 2 or less from it. It infers from this the positions of local obstacles (walls or other robots), but cannot distinguish between types of obstacles. All robots have a shared notion of up, down, left and right upon emergence from s.
Time is discretized to steps of t = 1, 2, . . .. At every time step, all robots perform a Look-Compute-Move operation sequence, in which they examine their environment and move to a new location based on a computation they perform (a robot may also choose to stay in place -this counts as a move). This occurs synchronously, meaning that all robots move to their computed next location at the same time. The "beginning" of a time step refers to the configuration of the robots at that time step before the robots move. The "end" of a time step is the configuration at that time step after the robots move.
We denote by prev(A) the position of a robot A at the beginning of the previous time step, and by next(A) its position at the beginning of the next time step.
A given robot is either active or settled. All robots are initially active, and eventually become settled at the end of some time step. Settled robots never move from their current position.
A unique vertex s in R is designated as the source or "door" vertex. If at the beginning of a time step there is no mobile robot at s, a new robot emerges at s at the end of that time step.
Energy and total travel. The "travel" T i of the ith robot is the number of time steps t that begin and end with the robot still active. This definition includes steps where the robot does not change location, since we wish to relate travel to energy expenditure (e.g., a quadcopter floating or circling in place is still traveling, and consumes just as much energy). The total travel of the robots is then the sum T i over all robots, and can be seen as the total amount of energy the robots consume before they settle.
FIND-CORNER DEPTH-FIRST SEARCH
We describe a local rule, "Find-Corner Depth-First Search" (Algorithm 1), that enables the robots to disperse over a simply-connected region R. As in [19], the algorithm has a makespan of 2V − 1 (where V is the number of cells in R, or equivalently, the total area of R when setting every cell to be a unit square). We note that since at best, robots arrive at s once per two time steps, this is the lowest possible makespan.
The purpose of FCDFS is to minimize the individual travel and total travel of the robots. It does this by ensuring that the path of a robot from s to its eventual destination (the vertex at which it settles) is a shortest path in R.
The idea of the algorithm lies in the distinction between a corner and a hall (see Figure 1 and Figure 2):
Definition 3.1. A vertex v of a grid environment R is called a corner if either:
(a) v has one or zero neighbours in R, or (b) v has precisely two neighbours u and u ′ in R, and u and u ′ have a common neighbour w that is distinct from v.
Definition 3.2. A vertex v of R is called a hall if
it has precisely two neighbours u and u ′ , and u and u ′ are both adjacent to the same vertex w in R c . Essentially, halls are vertices in R that are blocked by walls on two sides, and have an additional wall w diagonal to them. Corners are either dead-ends, or vertices in R that are blocked by walls on two sides, and have a vertex w of R diagonal to them. If v is either a hall or a corner, w is called the "diagonal" of v, and is denoted diaд(v). We observe that diagonals are uniquely specified.
Robots executing FCDFS attempt to move only in 'primary' and 'secondary' directions, where the secondary direction is always a 90-degree clockwise rotation of the primary direction (for example "up and right", "right and down", or "down and left"). They may only change their primary direction once they arrive at a hall, and they become settled once both their primary and secondary directions are blocked and they are at a corner.
For the rest of this section, let R(t) be the environment R at time t, i.e. the initial environment R where we have removed from R every vertex that is occupied by a settled robot at the beginning of time step t.
A robot at time t is searching for the corners and halls of R(t). However, robots executing FCDFS are unable to distinguish between active robots, and walls or settled robots. Hence, it is important to design the algorithm so that a robot never misidentifies a corner of R(t) as a hall, or vice-versa, due to an active robot (rather than a wall or a settled robot) occupying the diagonal and being identified as an obstacle. For this purpose we enable our robots to remember their two previous locations. We will show that an active robot can occupy the diagonal of a corner at time t if and only if its predecessor occupied this diagonal at time t − 2, thereby allowing the predecessor to distinguish between 'real' and 'fake' halls.
Algorithm 1 Find-Corner Depth-First Search
Let v be the current location of A. if every neighbouring vertex of v is occupied then Settle. else if A has never moved then ▷ Initialization
Search clockwise, starting from the "up" direction, for an unoccupied vertex, and set primary direction to point to that vertex. end if if A can move in its primary direction then
Step in the primary direction. else if A can step in secondary direction then
Step in the secondary direction. else ▷ We are at a corner or a hall.
if prev(prev(A)) = diaд(v) ∨ diaд(v) is unoccupied then Settle. else
▷ We think we are at a hall.
Set primary direction to point to the neighbour of v different from prev(A).
Move in the primary direction. end if end if
Analysis
In this section we give an analysis of the FCDFS algorithm. To start, we require some lemmas about corners and halls. Proof. Removing c does not affect connectedness, nor does it affect the distance from u to v, as any path going through c can instead go through diaд(c). Further, as c is adjacent to two walls, no path in R − c can surround it, so R − c also remains simply connected. □ An articulation point (also known as a separation or cut vertex) is a vertex of a graph whose deletion increases the number of connected components of the graph (i.e. disconnects the graph) [10]. Lemma 3.4. The halls of a simply connected region are articulation points.
Proof. Let h be a hall of a simply connected region R. Suppose for contradiction that h is not an articulation point, and let u and u ′ be the neighbours of h. Then there is a path from u to u ′ that does not pass through h. Let P be this path, and let P ′ be the path from u to u ′ that goes through h.
When embedded in the plane in the usual way, R is in particular a simply connected topological space. The hall h is embedded onto a unit square, whose four corners each touch a wall: three touch the two walls adjacent to h, and the fourth touches diaд(h). Joined together to form a closed curve, the paths P and P ′ form a rectilinear polygon that must contain at least one corner of h in its interior. Hence, the curve PP ′ contains a part of R c -and we get a contradiction to the simply connected assumption. (See Figure 3). □ Lemma 3.4 indicates that R can be decomposed into a tree structure T (R) as follows: first, delete all halls of R to form separated connected components. Let C 1 , C 2 , . . . , C n be these components, where C i also includes its adjacent halls. Letting the vertices of T (R) be these components, connect C i and C j by an edge if they share a hall. We set C 1 to be the root of the tree, and the connected component containing the door vertex s.
By Lemma 3.3, assuming our robots correctly stop only at corners, R(t) can in the same manner be decomposed into a tree T (R(t)) whose connected components are C 1 (t), C 2 (t), . . .. These components are each a sub-graph of a connected component of T (R).
Let A 1 , A 2 , . . . denote the robots that emerge from s in the order of arrival. In the next several propositions, we make the no fake halls at time t assumption: this is the assumption that for any t ′ < t, at the end of time step t ′ : robots can only become settled at corners of R(t ′ ), and can only change primary directions at halls of R(t ′ ). We do not include the initialization of a primary direction when a robot arrives at s. We will later show that the "no fake halls" assumption is always true, so the propositions below hold unconditionally. Proposition 3.5. Assuming no fake halls at time t, a robot A i active at the beginning of time step t has traveled an optimal path in R from s to its current position.
Proof. By the assumption, the only robots that became settled did so at corners. Consequently, by Lemma 3.3, R(t) is a connected graph, and there is a path in R(t) from s to A i . The path A i took might not be in R(t), but whatever articulation points (and in particular halls) A i passed through must still exist, by definition.
Since A i is active at the beginning of time t, by the algorithm, it has taken a step every unit of time up to t. Until A i enters its first hall, and between any two halls A i passes through, it only moves in its primary and secondary directions. This implies that the path A i takes between the halls of R(t) must be optimal (since it is optimal when embedded onto the integer grid Z 2 ). We note also that A i never returns to a hall h it entered a connected component of R(t) from, since the (possibly updated) primary direction pulls it away from h.
We conclude that A i 's path consists of taking locally optimal paths to traverse the connected components of the tree T (R(t)) in order of increasing depth. Since in a tree there is only one path between the root and any vertex, this implies that A i 's path to its current location is at least as good as the optimal path in R(t). By Lemma 3.3, b, this implies that A i 's path is optimal in R. □ Corollary 3.6. Assuming no fake halls at time t, (a) For all i < j, the distance between the robots A i and A j , if they are both active at the beginning of t, is at least 2(j − i) (b) No collisions (two robots occupying the same vertex) have occurred.
Proof. For proof of (a), note that at least two units of time pass between every arrival of a new robot (since in the first time step after its arrival, a newly-arrived robot blocks s). Hence, when A j arrives, A i will have walked an optimal path towards its eventual location at time t, and it will be at a distance of 2(j − i) from s. This distance is never shortened up to time t, as A i will keep taking a shortest path.
(b) follows immediately from (a). □ From Corollary 3.6 and determinism, we get: Lemma 3.7. Suppose A i is active at the beginning of time step t. Assuming no fake halls at time t, next(A i+1 ) = prev(A i ).
We note that Lemma 3.7 also indicates that if at the beginning of time step t, A i is active, then A i+1 will be active at the beginning of time step t + 1.
We can now show that the "no fake halls" assumption is true, and consequently, the propositions above hold unconditionally. Proposition 3.8. For any t, at the end of time step t: robots only become settled at corners of R(t), and only change primary directions halls of R(t) (not including the primary direction decided at initialization).
Proof. The proof of the proposition is by induction. The base case for t = 1 is trivially true.
Suppose that up to time t − 1, the proposition holds. Note that this means the "no fake halls" assumption holds up to time t, so we can apply the lemmas and propositions above to the algorithm's configuration at the beginning of time t.
We will show that the proposition statement also holds at time t. Let A i be an active robot whose location at the beginning of t is v.
First, consider the case where v = s. The algorithm only enables A i to settle at s if it is surrounded by obstacles at all directions.
Any obstacle adjacent to A i must be a wall of R(t) (as any active robot must be at a distance at least 2 from A i , due to Corollary 3.6). Hence, if A i settles at s, s is necessarily a corner, as claimed.
We now assume that v s. We separate the proof into two cases: Case 1: Suppose A i becomes settled at the end of time step t. Then by the algorithm, at the beginning of t, A i detects obstacles in its primary and secondary directions. These must be walls of R(t) due to Corollary 3.6, so v is either a corner or a hall of R(t).
Since A i settled, we further know that either diaд(v) is empty, or prev(prev(A i )) = diaд(v). In the former case, v is a corner of R(t).
In the latter case, we know from Lemma 3.7 and from the fact that no collisions occur that the only obstacle detected at diaд(v) is A i+1 , which is an active robot, so v is again a corner of R(t). In either case a corner is detected and the agent is settled.
Case 2: Suppose A i changed directions at the end of time step t. Then it sees two adjacent obstacles, and an obstacle at diaд(v). As in case 1, we infer that v is either a corner or a hall. If it is a corner, then diaд(v) is an active agent. By Corollary 3.6, it is either A i+1 or A i−1 . It cannot be A i+1 , as then A i 's position two time steps ago would have been diaд(v), so it would become settled instead of changing directions. It cannot be A i−1 , as diaд(v) is closer to s than v, and A i−1 has arrived earlier than A i , and has been taking a shortest path to its destination. Hence, diaд(v) cannot be an active agent, and v must be a hall as claimed. □
We have shown that the no fake-hall assumption is justified at all times t, hence we can assume that the propositions introduced in this section hold unconditionally. Proof. Propositions 3.5 and 3.8 imply that robots take a shortest path in R to their destination. That means that as long as the destination of a robot is not s itself, robots will step away from s one unit of time after they arrive. Until then, this means that robots arrive at s at rate one per two time steps.
Every robot's end-destination is a corner, and by the initialization phase of the algorithm, the destination is never s unless s is completely surrounded. Since there are no collisions, there can be at most V robots in R at any given time. By Lemma 3.3, robots that stop at corners keep R connected. Furthermore, every R(t) is a rectilinear polygon, so unless it has exactly one vertex, it necessarily has at least two corners. This means that the destination of every robot is different from s unless s is the only unoccupied vertex. Hence, a robot whose destination is s will only arrive when s is the only unoccupied vertex, and this will happen when V robots have arrived, so after at most 2V − 1 time steps. This is exact, since it is impossible to do better than 2V − 1. □ Propositions 3.9 and 3.5, alongside the "no fake halls" proof, complete our analysis. They show that FCDFS has a makespan of 2V − 1, and also that the durations of activity of the individual robots are optimal, since every robot travels a shortest path to its destination without stopping.
As every vertex must be occupied for the dispersal to end, a trivial lower bound on the total travel for any dispersal algorithm is v ∈R dist(s, v). Since this is achieved by our algorithm, total travel is also minimized.
In practice, the energy savings of our algorithm are dependent on the shape of the environment R. We take as a point of comparison the Depth-First Leader-Follower algorithm of Hsiang et al. [19]. On a 1-dimensional line of length n, both FCDFS and DFLF require the same total travel, O(n 2 ), so no improvement is attained. In contrast, on an n-by-n square grid, DFLF requires total travel O(n 4 ) in the worst case, and FCDFS requires O(n 3 ) -significantly less. This is because the DFLF strategy starting from a corner might cause the leader, A 1 , to "spiral" inwards into the grid, covering every one of its n 2 vertices in n 2 − 1 moves; the subsequent robot A i will make n 2 − i moves, for a sum total of O(n 4 ). FCDFS, on the other hand, distributes the path lengths more uniformly. Note that both algorithms take the exact same amount of time to finish.
Where is it best to place s? If we want to minimize the total travel, by the formula given above, the best place to place s is the vertex of R that minimizes the sum of distances v ∈R dist(s, v) (there may be several). This is the discrete analogue of the so-called Fermat-Toricelli point, or the "geometric median" [20].
The number of persistent states
As in previous work on uniform dispersal, our robots are finitestate automatons with O(1) persistent memory bits or states that carry over between time steps. The requirement of finite memory is important, as it allows for scalability: the robots' memory need not scale with the size or complexity of the environment.
There has been some interest in the question of just how little memory one can get away with. It has been shown that oblivious robots -robots with just one persistent state -are incapable of solving the dispersal problem, even with infinite visibility [3]. Consequently, any dispersal algorithm requires some number of persistent states, and we are interested in implementing our algorithm with as few as possible -i.e. bringing the robots as close as possible to "obliviousness" of their prior history and to center their decisions, as much as possible, on their current position and frame of reference.
Moreover, Algorithm 1 required the robots to remember their previous locations relative to their current location and to be able to use them as points of comparison. The 5-bit implementation shows how this could be done through remembering only the previous two relative directions of motion. A robot is then required only to know whether there are obstacles at the four cardinal directions (up, down, left, right), and at its diagonal, which is always at a 135°degree rotation from the primary direction. This simplifies the localization computations.
We implemented a 5-bit or 2 5 -state version of our algorithm on a simulator (see Algorithm 2). A robot's state is described by bits b 1 b 2 b 3 b 4 b 5 . All bits are initially 0. b 1 b 2 describe the primary direction (one of four), and b 3 tells us whether the previous step was taken in the primary direction (if b 3 = 0) or in the secondary direction (if b 3 = 1). b 4 b 5 is a counter that is reset to 10 upon entering a hall or one step after initialization, and thereafter is equal to * 1, where * is a bit that tells us whether we walked in the primary or secondary direction two steps ago (by copying b 3 ). A robot that detects an obstacle at its diagonal interprets its position
Algorithm 2 5-bit FCDFS
Let v be the current location of A.
if v has no unoccupied neighbours then
Settle. b 3 b 4 b 5 ← 011 else if b 4 b 5 = 00 then
Search clockwise, starting from the "up" direction, for an unoccupied vertex, and set primary direction to point to that vertex. b 4 b 5 ← 10 end if if A cannot move in primary or secondary directions then if v has just one neighbour then
Settle. b 3 b 4 b 5 ← 011 else if (b 5 = 1 ∧ b 3 + b 4 = 1) ∨ diaд(v) is unoccupied then Settle. b 3 b 4 b 5 ← 011 else
Set primary direction to obstacle-less direction not equal to 180°rotation of previous direction stepped in (i.e. the neighbour of v we haven't visited yet; this can be inferred from b 1 b 2 and b 3 ). Step in the primary direction. b 3 ← 0 else if A can step in secondary direction then
Step in the secondary direction.
b 3 ← 1 else Settle. b 3 b 4 b 5 ← 011 end if
as a fake hall (i.e. a corner) as long as b 5 = 1 and b 3 + b 4 = 1, that is, as long as at least one time step passed since the last hall, and our previous position was diagonal to us. In order to conserve memory, our robots do not strictly speaking have a "settled" state. Instead, once a robot determines it is in a corner (and so needs to settle), it sets b 3 b 4 b 5 to 011, indicating that it visited its diagonal-this causes it to never move again.
The impossibility of minimizing total travel for general grid environments
We saw that there is a local rule that minimizes total travel for simply connected grid environments. In this section we show that, for robots with finite visibility, there is no local rule that universally minimizes total travel for all connected grid environments.
Let r be the visibility range of the robots. Consider the grid environment in Figure 4 (not drawn to scale). It connects a set of 10r columns of width 1 spaced 2r cells apart. The bottom row has total length 20r 2 . Most of the columns are dead-ends and have a height of 30r 2 . The first column and an additional column connect to the top row, and have height 30r 2 + 1. Label the grid environment where this additional column is the kth column G(k). The door s is at the bottom left. It is readily seen that the total travel required by an optimal solution for any environment G(k) is v ∈G(k ) dist(s, v), where s is the door of G(k) (let a line of robots going up the first column fill the top row, and let robots going to the right fill the other columns). Proposition 3.10. Let ALG be a local rule for uniform dispersal of robots with visibility range r . There is an environment G(k) for which the total travel of ALG is at least v ∈R dist(s, v) + 1.
Proof. (Sketch) We consider the actions of rule ALG on the grid environment G(k). We do not specify the value of k yet.
As before, label the robots emerging at s A 1 , A 2 , . . . in their order of arrival. Since A 1 cannot distinguish between the up and right directions upon arrival at s (any distinct feature of the environment is at distance at least r + 1 and hence is invisible), we can assume without loss of generality that it steps up (if it steps right, simply rotate and reflect G(k)).
Assume for contradiction that the total travel of ALG is T = v ∈G(k ) dist(s, v). This assumption implies that every robot travels a shortest path to its settlement destination. In particular, A 1 must have precisely dist(A 1 , v 1 ) travel, where v 1 is the destination at which A 1 chooses to settle.
We note the following facts:
(1) Once A 1 stepped up, it has committed to stepping up and right until reaching v 1 , as circling in place or going in a third direction increases its travel past dist(A 1 , v 1 ), causing the total travel of ALG to be greater than T -a contradiction. (2) v 1 cannot be a vertex in the first column or in the top row except the top vertex of column k or one vertex to its left, as should v 1 not equal those, settling there would block off the path to the top row going through the first column, and force other robots to travel to the top row through column k. This is sub-optimal, and causes the total travel to increase
beyond T -a contradiction. (3) v 1 cannot be any vertex in the kth column other than the top of the kth column, as this would require A 1 to step downwards.
(*) From (1)-(3) we conclude that v 1 must equal precisely the top vertex of the kth column or one vertex to its left.
Up to the time when A 1 reaches the top row, none of the ends of the other columns have been seen, so ALG will run the same regardless of the value of k. Since total travel is assumed to be optimal, no robot can block s for more than one time step, so by the time A 1 reaches the top row, there will have been created at least 4r robots. Each of these 4r robots must have already entered one of the columns or settled, since they travel optimal paths to their destination, and the total length of the bottom row is 20r 2 , whereas 30r 2 time must have passed for A 1 to reach the top.
As there are 10r columns, there must exist a column that none of the robots A 1 , . . . , A 4r have entered. Set the value of k to equal this column.
When A 1 reaches v 1 , the above indicates that any other robot currently present in the kth column (if there are any) arrived at least 2 · 4r time steps after A 1 . Therefore it is at distance at least 8r from A 1 , meaning that there is a space of 6r vertices in column k that no robot has seen yet. This indicates that ALG must make the same decision for A 1 whether these vertices exist or not. However, if any one of these vertices does not exist, then column k is not connected to the top row, indicating that A 1 cannot settle at the top of the kth column or to its left, else it will block off part of the environment. We arrived at a contradiction to (*).
We conclude that there is an environment G(k) where the total travel of ALG is greater than the optimum, so ALG is sub-optimal. □ By adding more columns to the G(k) construction and increasing the height of the columns, we can force A 1 to go down more and more steps, causing the difference between the optimal total travel and the total travel of ALG to be arbitrarily large. Proposition 3.10 only makes the assumption of limited visibility. It holds even assuming the agents have global communication, infinite memory, and are aware of each others' positions at all times.
We note that we did not exclude the possibility of a local rule that minimizes the maximal individual travel. Furthermore, we did not exclude the possibility of a rule that minimizes total travel when pauses are not counted.
SIMULATIONS, COMPARISONS, AND ALTERNATIVE STRATEGIES
We verified and animated our algorithm by simulating it on our robot simulator. Figures 5 and 6 show four stills from a run of the algorithm on two different environments. Figure 7 shows a FCDFS deadlock scenario in an environment that is not simply connected: the halls constantly redirect the robots, forming a cycle. The door vertex has mistakenly blocked itself off, due to the robots exiting from it mistaking the robots in a cycle for obstacles. The arrows indicate the location and primary direction of the robots, and the diamonds are settled robots. Rather than block active robots, the settled robots form halls to enable the swarm to explore more of the environment. Figure 6: A simulation on a different environment. Note how the trail of robots always forms a shortest path to its current front.
We experimented with two variants of FCDFS that are similarly optimal. FCDFS assumes robots are initialized with a common notion of up, down, left and right, but this assumption is unnecessary if we let robots settle in place as soon as they reach a corner (in FCDFS they keep moving if they can). This modified strategy is illustrated in Figure 8, where robots randomly choose their initial direction. This creates a more "symmetric"-looking dispersal. The strategy shown in Figure 9 is more significantly different: in it, rather than stick to their secondary and primary directions, robots attempt to scale the boundary of the environment with a "left hand on wall" clockwise orientation, until they hit a corner or a wall. Both of these variants achieved the same makespan and total travel as FCDFS, though they are visually distinct.
Empirically, we compared the performance of FCDFS to the performance of our implementation of the DFLF and BFLF algorithms of [19] (adapted to our slightly different model) over a number of simply-connected environments, measuring the total travel and maximal individual travel ( Table 1). Note that though all algorithms are deterministic, some local decisions are not fully specified in [19], hence different implementations may result in slightly different performance, though asymptotically every implementation will perform the same. We let our robots decide between arbitrary local decisions at random, averaging performance over several re-runs.
Only for the sake of this comparison, we elected to exclude time steps where robots are active but do not change location, as such intermediate pauses are not counted in [19]. FCDFS is optimal regardless, and factoring these in leaves the DFLF and FCDFS columns unchanged, since such pauses never occur during their execution. However, including pauses causes the maximal travel of BFLF to become extremely large. Hence, Table 1 shows that BFLF is good at reducing the number of location changes of a robot, but in many applications (e.g. when robots are quadcopters) its energy consumption is very high compared to FCDFS.
DISCUSSION
A robotic swarm must take into account the energy capacity of the individual. We discussed the problem of minimizing travel, hence energy expenditure, in the uniform dispersal problem for simply connected grid regions. We showed the existence of a strategy that minimizes total and individual travel for the case of a single source vertex. We showed also a non-existence result for such strategies in the case of general grid environments. Several extensions of our work can readily be considered. First, as our algorithm deals only with the single door case, it is desirable to find an energy-efficient dispersal algorithm for the case of multiple doors from which robots arrive independently.
Next, synchronicity is a strong assumption, enabling every robot to proceed to its destination without ever being blocked by another robot. To extend our work to less controlled settings, we may assume an asynchronous time scheme-for example, allow a probability q that an agent fails to activate at a given time step. We cannot expect a makespan-and total travel-optimal algorithm to exist in such settings, but we anticipate relatively effective strategies might exist. As a way to proceed, though our algorithm makes the powerful assumption of synchronicity, the strategy of finding corners and not stopping at halls seems general, and could possibly be adopted for the asynchronous case as well.
Finally, our algorithm requires the environments to be simplyconnected orthogonal environments: it would be interesting to see an algorithm that works for broader scenarios, or in the opposite direction, results regarding the non-existence of efficient algorithms for such scenarios under the stringent computational assumptions we made.
Proc. of the 18th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2019), N. Agmon, M. E. Taylor, E. Elkind, M. Veloso (eds.), May 13-17, 2019, Montreal, Canada. © 2019 International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved.
Figure 1 :
1Corners. (Blue vertices are walls; vertices in R c ).
Figure 2 :
2A hall.
Lemma 3 . 3 .
33Let c be a corner of a simply connected region R. Then: (a) R − c is simply connected. (b) For any two vertices u, v in R − c, the distance between u and v is the same as in R.
Figure 3 :
3The two possibilities for PP ′ .
Proposition 3 . 9 .
39Let V be the number of vertices of R. At the end of time-step 2V − 1, every cell is occupied by a robot.
b 4 b 5
5← 10 end if end if if b 4 b 5 was not updated at this time step then ▷ i.e. b 5 = 1 or time to update b 5 b 4 b 5 ← b 3 1 end if if A can move in its primary direction then
Figure 4 :
4The construction G(k).
Figure 5 :
5A simulation of FCDFS. The blue blocks are walls.
Figure 7 :
7A deadlock scenario in environments that are not simply connected.
Figure 8 :
8Multi-directional dispersal strategy.
Figure 9 :
9"Left hand on wall" strategy.
Table 1 :
1A comparison of total travel and maximal individual travel over different environments (excluding pauses). Entries are in the form total travel (maximal travel). See Figures 5, 6, 9 for the specific environments used.
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| []
|
[
"Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics",
"Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics"
]
| [
"V Prat [email protected] \nMax-Planck Institut für Astrophysik\nKarl-Schwarzschild-Str. 185748Garching bei MünchenGermany\n\nUPS-OMP; IRAP; Toulouse\nUniversité de Toulouse\nFrance\n\nCNRS\nIRAP\n14 avenue Édouard Belin; F31400ToulouseFrance\n",
"F Lignières \nUPS-OMP; IRAP; Toulouse\nUniversité de Toulouse\nFrance\n\nCNRS\nIRAP\n14 avenue Édouard Belin; F31400ToulouseFrance\n",
"J Ballot \nUPS-OMP; IRAP; Toulouse\nUniversité de Toulouse\nFrance\n\nCNRS\nIRAP\n14 avenue Édouard Belin; F31400ToulouseFrance\n"
]
| [
"Max-Planck Institut für Astrophysik\nKarl-Schwarzschild-Str. 185748Garching bei MünchenGermany",
"UPS-OMP; IRAP; Toulouse\nUniversité de Toulouse\nFrance",
"CNRS\nIRAP\n14 avenue Édouard Belin; F31400ToulouseFrance",
"UPS-OMP; IRAP; Toulouse\nUniversité de Toulouse\nFrance",
"CNRS\nIRAP\n14 avenue Édouard Belin; F31400ToulouseFrance",
"UPS-OMP; IRAP; Toulouse\nUniversité de Toulouse\nFrance",
"CNRS\nIRAP\n14 avenue Édouard Belin; F31400ToulouseFrance"
]
| []
| Context. The seismology of early-type stars is limited by our incomplete understanding of gravito-inertial modes. Aims. We develop a short-wavelength asymptotic analysis for gravito-inertial modes in rotating stars. Methods. The Wentzel-Kramers-Brillouin approximation was applied to the equations governing adiabatic small perturbations about a model of a uniformly rotating barotropic star. Results. A general eikonal equation, including the effect of the centrifugal deformation, is derived. The dynamics of axisymmetric gravito-inertial rays is solved numerically for polytropic stellar models of increasing rotation and analysed by describing the structure of the phase space. Three different types of phase-space structures are distinguished. The first type results from the continuous evolution of structures of the non-rotating integrable phase space. It is predominant in the low-frequency region of the phase space. The second type of structures are island chains associated with stable periodic rays. The third type of structures are large chaotic regions that can be related to the envelope minimum of the Brunt-Väisälä frequency.Conclusions. Gravito-inertial modes are expected to follow this classification, in which the frequency spectrum is a superposition of sub-spectra associated with these different types of phase-space structures. The detailed confrontation between the predictions of this ray-based asymptotic theory and numerically computed modes will be presented in a companion paper. | 10.1051/0004-6361/201527737 | [
"https://arxiv.org/pdf/1512.08907v2.pdf"
]
| 119,121,271 | 1512.08907 | 2979bec76ab26d9383093c2f9452e14b60f78019 |
Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics
January 18, 2016
V Prat [email protected]
Max-Planck Institut für Astrophysik
Karl-Schwarzschild-Str. 185748Garching bei MünchenGermany
UPS-OMP; IRAP; Toulouse
Université de Toulouse
France
CNRS
IRAP
14 avenue Édouard Belin; F31400ToulouseFrance
F Lignières
UPS-OMP; IRAP; Toulouse
Université de Toulouse
France
CNRS
IRAP
14 avenue Édouard Belin; F31400ToulouseFrance
J Ballot
UPS-OMP; IRAP; Toulouse
Université de Toulouse
France
CNRS
IRAP
14 avenue Édouard Belin; F31400ToulouseFrance
Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics
January 18, 2016Astronomy & Astrophysics manuscript no. Grav1.9.4 c ESO 2016Asteroseismology -Waves -Chaos -Stars: oscillations -Stars: rotation
Context. The seismology of early-type stars is limited by our incomplete understanding of gravito-inertial modes. Aims. We develop a short-wavelength asymptotic analysis for gravito-inertial modes in rotating stars. Methods. The Wentzel-Kramers-Brillouin approximation was applied to the equations governing adiabatic small perturbations about a model of a uniformly rotating barotropic star. Results. A general eikonal equation, including the effect of the centrifugal deformation, is derived. The dynamics of axisymmetric gravito-inertial rays is solved numerically for polytropic stellar models of increasing rotation and analysed by describing the structure of the phase space. Three different types of phase-space structures are distinguished. The first type results from the continuous evolution of structures of the non-rotating integrable phase space. It is predominant in the low-frequency region of the phase space. The second type of structures are island chains associated with stable periodic rays. The third type of structures are large chaotic regions that can be related to the envelope minimum of the Brunt-Väisälä frequency.Conclusions. Gravito-inertial modes are expected to follow this classification, in which the frequency spectrum is a superposition of sub-spectra associated with these different types of phase-space structures. The detailed confrontation between the predictions of this ray-based asymptotic theory and numerically computed modes will be presented in a companion paper.
Introduction
As compared to the wealth of new information obtained from the seismology of solar-type and red-giant stars (Miglio 2015), reliable seismic diagnostics on non-evolved, intermediate-mass and massive pulsators (γ Doradus, δ Scuti, β Cephei, SPB or Be stars) are scarce and, in any case, restricted to atypical slowly rotating stars (Kurtz et al. 2014;Saio et al. 2015;Triana et al. 2015). This stems from our limited understanding of the effects of rotation on oscillation modes (Lignières 2013;Townsend 2014;Reese 2015), at least for the high rotation rates of typical non-evolved, intermediate-mass and massive stars (Royer et al. 2007;Lignières 2013). The two classical approximations to treat rotational effects have been the perturbative expansion in Ω/ω, where Ω is the rotation rate and ω is the pulsation frequency (Saio 1981), and the so-called traditional approximation of the Coriolis acceleration (Eckart 1960). The perturbative expansion happens to be valid for small rotation rates but not for those of most early-type stars (Reese et al. 2006;Ballot et al. 2010Ballot et al. , 2013. The traditional approximation greatly simplifies the understanding of Coriolis effects on low-frequency g modes (e.g. Townsend 2003;Bouabid et al. 2013) but the range of validity and accuracy of this approximation are not known. The centrifugal deformation in particular is not taken into account in the traditional approximation.
Send offprint requests to: F. Lignières More recently, the development of dedicated numerical codes has made it possible to compute modes and frequencies accurately without approximating the Coriolis and centrifugal effects on the modes (Reese et al. 2006(Reese et al. , 2009Ouazzani et al. 2012). This is a crucial step towards a confrontation with observational data. However, in addition to accurate theoretical frequencies, mode identification of observed frequencies requires a priori knowledge of the mode properties. In the case of slowly rotating stars, our knowledge of the expected frequency patterns is important for the interpretation of observed frequency spectra. It comes from the short-wavelength asymptotic theory of oscillation modes, which in the case of spherically symmetric nonrotating stars provides analytical expressions of the asymptotic frequency patterns. When applied to rapidly rotating stars, the same short-wavelength asymptotic analysis is not as straightforward. As in geometrical optics, it first leads to a ray model that describes wave propagation. Then, the conditions to produce a mode from the positive interference between rays that need to be established. Methods and concepts to infer the properties of the modes from those of the rays have been mostly developed in the field of quantum physics. The basic idea is to study rays as trajectories of a Hamiltonian system and then to relate the modes to the phase-space structures of the ray dynamics. For acoustic modes in rotating stars, such a ray-based asymptotic theory provided a physical classification of the modes as well as quantative predictions that have been successfully confronted with numeri-
Eikonal equation for gravito-inertial waves
In this section, we derive an eikonal equation for shortwavelength, adiabatic gravito-inertial waves in a uniformly rotating polytropic model of star. The stellar model is specified in Sect. 2.1, the equations for small perturbations about this model are put in a convenient form in Sect. 2.2, the eikonal equation obtained by applying the WKB approximation to the perturbation equations is presented in Sect. 2.3, and the domains of propagation of gravito-inertial waves are discussed in Sect. 2.4. Only the main equations are presented in this section because their derivation are detailed in Appendix A.
Polytropic model of rotating star
The stellar model is a self-gravitating uniformly rotating monatomic gas that verifies a polytropic relation, which is assumed to give a reasonably good approximation of the relation between pressure and density in real stars (Hansen & Kawaler 1994). The governing equations can be written as
P 0 = Kρ 1+1/µ 0 ,(1)0 = −∇P 0 − ρ 0 ∇ ψ 0 − Ω 2 w 2 /2 ,(2)∆ψ 0 = 4πGρ 0 ,(3)
where P 0 is the pressure, ρ 0 the density, K the polytropic constant, µ the polytropic index, ψ 0 the gravitational potential, Ω the rotation rate, w the distance to the rotation axis, and G the gravitational constant.
The uniform rotation ensures that the flow is barotropic. A pseudo-enthalpy h 0 = dP 0 /ρ 0 = (1 + µ)P 0 /ρ 0 can then be introduced and quantities describing the equilibrium model, such as the sound speed c s = Γ 1 P 0 /ρ 0 , where Γ 1 = 5/3 is the first adiabatic exponent of the gas; the effective gravity g 0 = −∇ ψ 0 − Ω 2 w 2 /2 ; and the Brunt-Väisälä frequency, given by the relation N 2 0 = g 0 · ∇ρ 0 ρ 0 − 1 Γ 1 ∇P 0 P 0 are expressed in terms of h 0 :
g 0 = ∇h 0 , c 2 s = Γ 1 µ + 1 h 0 , N 2 0 = µΓ 1 µ + 1 − 1 g 2 0 c 2 s .(4)
In polytropic models of stars, the density ρ 0 vanishes at the surface. Moreover, the polytropic relation Eq.
(1) implies that the sound speed c s also vanishes, and the last of Eqs. (4) shows that the Brunt-Väisälä frequency N 0 goes to infinity. These properties are not necessarily valid in more realistic stellar models. Specifying the mass and rotation rate of the star is not sufficient to determine the polytropic model in physical units. This requires fixing an additional parameter, for example the stellar radius (suitable parameter choices are discussed in Hansen & Kawaler (1994), for the non-rotating case, and in Christensen- Dalsgaard & Thompson (1999), for the rotating case). In the following, however, we only present dimensionless quantities that do not depend on the choice of this additional parameter. The rotation rate Ω is compared to Ω K = GM/R 3 e 1/2 , the limiting rotation rate for which the centrifugal acceleration equals the gravity at the equator, where M is the stellar mass and R e the equatorial radius. Other frequency units important for our problem are (i) the inverse of the dynamical timescale associated with hydrostatic equilibrium ω 0 = GM/R 3 p 1/2 , which provides a lower bound for acoustic wave frequencies, where R p is the polar radius; (ii) the Brunt-Väisälä frequency N 0 , an upper bound for gravity wave frequencies; and (iii) twice the rotation rate f = 2Ω, the upper bound for pure inertial wave frequencies.
The spatial distribution of the Brunt-Väisälä frequency is shown in Fig. 1. In the non-rotating spherically symmetric case (see the black curve on the upper figure), N 0 vanishes at the centre, has a local maximum, N 0,max , around r = 0.28R and then shows a local minimum, N 0,min , in the stellar envelope near r = 0.74R. A simple analytical expression of N 0 , which is valid for envelope models, (see Appendix E) shows that a local minimum is present at r = 3/4R for all polytropic indices, suggesting that such a minimum of N 0 is a generic feature of stars with radiative envelopes. The Brunt-Väisälä frequency distribution becomes anisotropic in centrifugally deformed stars. In particular, the amplitudes and positions of the local minima are clearly different along the polar and the equatorial axes. This is apparent in Fig. 1, which shows iso-contours of N 0 (bottom) as well as the polar and equatorial radial profiles of N 0 (top) for a Ω/Ω K = 0.84 model. In the following, the local minima along the polar and equatorial axes are denoted N p 0,min and N e 0,min , respectively. By comparison, the variation of the local maximum of the Brunt-Väisälä frequency between its values along the pole (N p 0,max ) and along the equatoral radius (N e 0,max ) remains small. This is because the deviation from sphericity is weaker in the central layers.
Perturbation equations and boundary conditions
Time-harmonic, small-amplitude perturbations of the stellar model are studied under two approximations. The first approximation is to neglect viscous and thermal dissipations. Both processes have a very small effect on the value of oscillation frequencies, at least in the frequency range of observed modes. Non-adiabatic processes play an essential role in the oscillation amplitude, in particular, through excitation mechanisms such as the κ mechanism, but determining mode amplitudes is outside the scope of the present study. The second approximation, known as the Cowling approximation, is to neglect perturbations of the gravitational potential induced by density perturbations. It is justified for oscillations of small wavelengths and, there-fore, fully compatible with the small-wavelength regime considered here. Under these two approximations, the linear equations governing the evolution of small-amplitude perturbations can be written as
∂ t ρ + ∇ · (ρ 0 v) = 0, (5) ρ 0 ∂ t v + ρ 0 f ∧ v = −∇P + ρg 0 ,(6)∂ t P + v · ∇P 0 = c 2 s (∂ t ρ + v · ∇ρ 0 ) ,(7)
where v, ρ, and P, are respectively the Eulerian perturbations of velocity, density, and pressure, and f = 2Ω is the rotation vector. Before applying the short-wavelength approximation, the perturbation equations for v, ρ, and P are reduced to a single equation forP, the complex amplitude of the time-harmonic pressure perturbation P = {P(x) exp(−iωt)}. According to the calculations detailed in Appendix A.1, we obtain the equation
∆P = f 2 ω 2 ∇ 2 zP + N 2 0 ω 2 ∆ ⊥P + (V + iV m ) · ∇P + M P ,(8)
where the operators ∇ 2 z and ∆ ⊥ are related to the unit vectors parallel to the rotation axis, e z = f / f , and the effective gravity, e = − g 0 /g 0 , by
∇ 2 z ≡ ∇ z (∇ z ) with ∇ z ≡ e z · ∇, (9) ∆ ⊥ ≡ ∇ · (∇ ⊥ ) with ∇ ⊥ ≡ ∇ − e ∇ and ∇ ≡ e · ∇.(10)
The expressions of V , V m , and M are given by Eqs. (A.39, A.40, and A.41), respectively. As a linear combination of f and g 0 , the vector V is contained in a meridional plane, whereas V m is colinear to f ∧ g 0 and thus perpendicular to this plane. First order derivatives in the meridional plane can be suppressed from this equation by introducing a variableΨ =P/a and by choosing the function a adequately (see Appendix A.2). We then obtain a wave equation for acoustic and gravito-inertial waves in a uniformly rotating, centrifugally deformed, barotropic star as follows:
∆Ψ = − ω 2 c 2 sΨ + f 2 ω 2 ∇ 2 zΨ + N 2 0 ω 2 ∆ ⊥Ψ + i f ∧ g 0 ωc 2 s T e φ · ∇Ψ +C Ψ ,(11)
where e φ is the unit vector in the azimuthal direction and T is given by Eq. (A.53). The first term on the right-hand side (RHS) of (11) is associated with sound waves whereas the next two terms on the RHS produce gravito-inertial waves. The fourth term on the RHS is non-zero only for non-axisymmetric perturbations in rotating stars. It contains the latitudinal derivative of the Coriolis parameter f cos θ, which gives rise to Rossby and Kelvin waves. Such a simple form for the wave equation comes at the cost of very complicated expressions for a and C as functions of c s , N 0 , g 0 , f , and ω (see Eqs. (A.48) and (A.54)).
The short-wavelength approximation of the perturbation equations
The ray model results from the WKB approximation of the wave equation (11). It consists in looking for wave-like solutions of the form
Ψ = {A(x) exp i[Φ(x) − ωt]} = {Ψ(x) exp(−iωt)},(12)
assuming that the associated wavelength λ w ∼ ∇Φ −1 is much shorter than the typical length scale on which the background medium varies, hereafter denoted L b .
Article number, page 3 of 24 A&A proofs: manuscript no. Grav1.9.4 The phase Φ and the amplitude A ofΨ are expanded into power series of 1/Λ, where Λ = L b /λ w is assumed to be large, yielding
Φ = Λ Φ 0 + 1 Λ Φ 1 + . . . and A = A 0 + 1 Λ A 1 + . . .(13)
Introducing these expansions into the wave equation (11) and considering only the dominant O(Λ) terms, we obtain an equation for Φ 0 , the so-called eikonal equation. The amplitude A 0 is determined at the next order. The eikonal equation obtained with this procedure depends on the frequency range considered.
If ω is of the order of Λ, the dominant terms of the wave equation (11) are the left-hand side (LHS) and the first term on the RHS, leading to an eikonal equation for sound waves ω = c s k, where k = ∇Φ 0 and k = k . If ω = O(1), the dominant terms are those that contain second derivatives ofΨ, leading to an eikonal equation for gravito-inertial waves. The fourth term on the RHS of the wave equation (11) is a first-order azimuthal derivative ofΨ and is therefore negligible when ω = O(1). Thus, Rossby waves are not included in the eikonal equation if one considers the ω = O(1) regime or if one restricts to axisymmetric perturbations. Before writing down the eikonal equation for gravitoinertial waves, the role of the so-called constant term C Ψ must be clarified. Indeed, if a wave, acoustic or gravito-inertial, of vanishingly small wavelength approaches the stellar surface, it will cross the surface without being refracted back into the star. This regime corresponds to the limit λ w → 0, where the constant term C Ψ is negligible and the resulting eikonal equation contains no surface refraction effect. We are interested, however, in waves that are refracted back because oscillations modes are the result of wave interferences within the star. As known from previous studies on acoustic and gravity waves (Aerts et al. 2010), this back-refraction occurs for outwards-travelling waves that encounter atmospheric layers whose pressure scale heights are smaller that the wavelength of the wave. We also know that the pressure scale height strongly decreases as one approaches the surface in such a way that if L b is of the order of the pressure scale height in the interior and H s is the surface pressure scale height, we obtain H s /L b 1. Thus, there exists a wavelength range for which we have at the same time λ w /L b = 1/Λ 1 inside the star and λ w of the order of or larger than H s . In the wave equation, this surface refraction effect is accounted for by the constant term C Ψ because C is inversely proportional to the square of the pressure scale height (see Eq. (16) below). Thus, an eikonal equation containing this term is simply obtained by assuming λ w = O(H s ) in the expansion given by Eq. (13). This yields
k 2 = f 2 ω 2 k 2 z + N 2 0 ω 2 k 2 ⊥ + k 2 φ − C with C = C − ω 2 c 2 s ,(14)
where k z = e z · k, k φ = e φ · k and k ⊥ = e ⊥ · k, where the unit vector e ⊥ is defined as e ⊥ = e φ ∧ e . In Eq. (14), the term in k 2 φ results from the fact that ∇ ⊥ is not just the derivative along e ⊥ . It also contains derivatives in φ, and so does ∆ ⊥ . Although the full expression of C is a very complex function of the equilibrium quantities c s , N 0 , g 0 , f , and of ω (see Eq. (A.54)), we only need to estimate C where it becomes of the same order of magnitude as the other terms in the dispersion relation, that is near the surface. The increase of C towards the surface is due to the sharp increase of N 0 and 1/c s , while g 0 does not vary much in the stellar envelope. In a polytropic model of star, we also have N 0 ∝ 1/c s near the surface. In order to only retain the dominant term, we look for an expansion of C in powers of c s and find that C can be written as the ratio between two polynomials of c 2 s , namely
C = C 0 + C 1 c 2 s + O(c 4 s ) αK + Gc 2 s 2 c 4 s ,(15)
where the terms α = Γ 1 µ µ+1 − 1, K = ( f · g 0 ) 2 − ω 2 g 0 2 , G = ω 2 (ω 2 − f 2 ) and the terms C 0 and C 1 are O(1) at the surface.
In the super-inertial 1 regime ω > f , K does not vanish. The dominant term of C is thus C 0 (αK) 2 c 4 s and we obtain
C = 1 − f 2 ω 2 cos 2 Θ Γ 2 1 4 µ − 1 µ + 1 g 2 0 c 4 s + O(1/c 2 s ),(16)
where cos Θ = − f . g 0 / f g 0 . In the sub-inertial regime ω < f , both K and C 0 vanish at the critical angles Θ c and π − Θ c such that cos 2 Θ c = ω 2 / f 2 . In this regime and in the regions close to the stellar surface and the critical angles, C is given by
C = C 1 (Θ c ) G 2 1 c 6 s (Θ c ) + O(1/c 4 s ).(17)
The detail of the asymptotic expression of C is presented in Appendices A.3 and A.4. In the following, we always use the expression (16) for C. As we see in the next sections, inspection of the eikonal and ray equations near the critical angle indicates that rays tend to avoid the surface layers close to the critical angle. This is further confirmed by our computations that do not show rays reaching the surface close to the critical angle. Using (16), the eikonal equation can be written as
ω 2 = f 2 k 2 z k 2 + k 2 c + N 2 0 k 2 ⊥ + k 2 φ k 2 + k 2 c + f 2 cos 2 Θ k 2 c k 2 + k 2 c ,(18)
with
k 2 c = Γ 2 1 4 µ − 1 µ + 1 g 2 0 c 4 s .(19)
This equation is similar to the local dispersion relations generally used to study gravito-inertial waves in the Earth's atmosphere (Fritts & Alexander 2003) and to construct associated ray models (Marks & Eckermann 1995). The present relation is nevertheless more general in that it takes the centrifugal deformation of the star into account and does not make the traditional approximation. In the following we shall restrict our study to axisymmetric perturbations, that is to the case k φ = 0.
Domains of propagation
The eikonal equation (18) allows us to discuss the domain of propagation of gravito-inertial waves in a more general context than previous works. In particular, when compared to the work of Dintrans & Rieutord (2000), the following discussion includes the effects of the centrifugal deformation and, through the k c term, those of the background compressibility.
According to (12), non-evanescent wave solutions are possible only when both components of the wave vector k are real numbers. The region of space where the eikonal equation admits real solutions thus determines the domain where rays can propagate. After projecting k onto an orthogonal basis, the eikonal equation takes the general form of a conic. Using (e , e ⊥ ) as a basis and the relations k 2 = k 2 + k 2 ⊥ and k z = k cos Θ − k ⊥ sin Θ, the eikonal equation (18) is rewritten as
(N 2 0 + f 2 sin 2 Θ − ω 2 )k 2 ⊥ − 2 f 2 cos Θ sin Θk k ⊥ −(ω 2 − f 2 cos 2 Θ)k 2 − (ω 2 − f 2 cos 2 Θ)k 2 c = 0.(20)
To discuss the constraints on the domain of propagation that can be derived from this relation, it is useful to first consider the non-rotating case. The eikonal equation then becomes
N 2 0 (r) − ω 2 L 2 r 2 − ω 2 k 2 r = ω 2 k 2 c (r),(21)
with k ⊥ = k θ = ±L/r and L is the norm of the vector x ∧ k, which is equivalent to the angular momentum. This equation corresponds to the high-sound-speed limit of eikonal equations already derived in the non-rotating case, for example by Gough (1993). The condition that k 2 r is positive translates into ω 2 < N 2 0 /(1 + r 2 k 2 c /L 2 ) and this inequality fully specifies the spatial limit of the resonant cavity for a wave of given ω and L. In internal regions where r 2 k 2 c /L 2 1, it further simplifies into ω < N 0 , a condition that no longer depends on L or k θ . Closer to the surface, ω N 0 and the condition for propagation is ω < c 1 S L , where S L = Lc s /r is the Lamb frequency and c 1 a constant that can be expressed in terms of Γ 1 and µ. If this last condition is fullfilled at the surface, the gravity ray escapes the star. Otherwise, the ray is refracted back into the star and the relation ω = c 1 S L determines the outer radius of the domain of propagation. An important point is that the condition for the back-refraction in the outer layers of the star depends on L or equivalently on k θ .
Coming back to the rotating case, the condition that k is real can be expressed as
δ = Γk 2 ⊥ − (ω 2 − f 2 cos 2 Θ) 2 k 2 c ≥ 0,(22)
where δ is the reduced discriminant of the eikonal equation viewed as a quadratic equation for the variable k , and
Γ = ω 2 ( f 2 − ω 2 ) + N 2 0 (ω 2 − f 2 cos 2 Θ).(23)
The sign of Γ determines whether gravito-inertial waves can propagate in the inner region where k c can be neglected. In contrast, the outer limit of the domain of propagation is determined by the condition δ ≥ 0 and, as expected from the condition of propagation in the non-rotating case, it depends on the value of k 2 ⊥ . However, contrary to the non-rotating case, k 2 ⊥ is not known a priori because rotation breaks the spherical symmetry and the associated conservation of the norm of the angular momentum. The consequence is that in the general case of gravito-inertial waves in a rotating star, we are not able to predict the outer limit of the resonant cavity from the eikonal equation alone. It is thus necessary to solve the ray dynamics to find out whether a ray of a given frequency remains confined within the star and, if this is the case, the shape of the outer limit of its resonant cavity. In the ray dynamics calculation presented in this paper, all rays stay inside the stars because, at the surface of polytropic models, k c is infinite and δ is thus necessarily negative. In real stars, k c remains finite at the surface, and this implies that rays with sufficently high values of k ⊥ near the surface can propagate outside the stars. Waves associated with such rays are expected to have very high spatial frequencies at the surface, and are thus unlikely to be visible. However, they can play a role in the transport of angular momentum out of the stars.
In the following of the section, we thus only discuss the domain of propagation in the internal region where the k 2 c term can be neglected, since this approximation is valid from the centre up to a certain radius that depends on the given ray. The condition of propagation Γ ≥ 0 is equivalent to
ω 2 − < ω 2 < ω 2 + ,(24)
with
ω 2 ± = f 2 + N 2 0 ± f 2 + N 2 0 2 − 4 f 2 N 2 0 cos 2 Θ 2 .(25)
If the centrifugal deformation is neglected, that is if we replace Θ by the colatitude θ, this condition of propagation is equivalent to that found by Dintrans & Rieutord (2000). Two limit cases, f N 0 and N 0 f , are relevant since the former occurs in the bulk of typical moderately rotating stellar radiative zones, whereas the latter is verified in convective zones or close to the centre of the star where N 0 vanishes. If f N 0 , the conditions (24) simplify to
f 2 cos 2 Θ < ω 2 < N 2 0 + f 2 sin 2 Θ.(26)
On the other hand, if N 0 f , the conditions (24) become
N 2 0 cos 2 Θ < ω 2 < f 2 + N 2 0 sin 2 Θ,(27)
where in the limit of vanishing N 0 , the ω < f rule for pure inertial waves is recovered. We now consider the particular case of uniformly rotating µ = 3 polytropic models. Close to the centre of the star, the approximate conditions (27) hold because N 0 vanishes there. At some distance from the centre, the Brunt-Väisälä frequency always becomes larger than f . Moreover, for moderate rotators, the approximate condition (26) is valid from a small radius up to the surface. Indeed, as long as Ω ≤ 0.38Ω K and r > 0.1R e , the ratio f 2 /N 2 0 is always smaller than 0.10. Two typical domains of propagation are shown in panels (a) and (b) of Fig. 2 in the super-inertial and sub-inertial frequency ranges, respectively. The rotation of the stellar model is Ω = 0.38Ω K . For the super-inertial case, the condition of propagation reduces to ω 2 < N 2 0 + f 2 sin 2 Θ, which leads to the domain shown in panel (a), including the weakly prolate avoided central region. By contrast, in the sub-inertial regime the centre is never forbidden because conditions (27) are automatically fulfilled for ω < f . Away from this central region, the conditions of propagation (26) reduce to f 2 cos 2 Θ < ω 2 for these waves. The resulting domain of propagation is shown in panel (b). A noticeable effect of the centrifugal deformation is that the limit of the domain is no longer strictly conical away from the central region. As the centrifugal deformation of the equipotentials becomes significant towards the surface, the boundary of the domain of propagation is bent towards lower latitudes to keep Θ c = arccos(ω/ f ), the angle between the direction of the effective gravity and the rotation axis, constant.
The local minimum of the Brunt-Väisälä frequency present in the stellar envelope (see Fig. 1) induces an additional forbidden region for frequencies between N e 0,min and N p 0,max . As illustrated on panel (c) for the Ω = 0.38Ω K model, the domains of progagation below and above the position of N e 0,min are disjoint when N 0,min < ω < N 0,max in non-rotating stars or when N p 0,min < ω < N e 0,max in rotating stars. In centrifugally deformed stars, waves with frequencies in the range N e 0,min < ω < N p 0,min also show a forbidden region, but this time it is limited to equatorial regions so that the inner and outer domains communicate
(a) (b) (c) (d) 2¡ 0 N 0,min e N 0,min p N 0,max e (b) (a) (d) (c) ω Fig. 2.
Typical domains of propagation (grey areas) given by the Γ(ω) ≥ 0 condition as a function of the gravito-inertial wave frequency ω in a Ω = 0.38Ω K polytropic model of star. In general, the outer limit of the domain, here loosely represented by the stellar surface, cannot be predicted from ω alone and is obtained by solving the ray dynamics. Panel (a) corresponds to a super-inertial frequency (ω > f ) and panel (b) to a sub-inertial frequency (ω < f ). The two bottom figures show domains with an intermediate forbidden zone when frequencies are between the local minimum and maximum of the Brunt-Väisälä frequency. Panel (c) corresponds to frequencies in the N p 0,min < ω < N e 0,max interval and also exists in non-rotating stars, whereas panel (d) corresponds to frequencies in the N e 0,min < ω < N p 0,min interval and is due to the centrifugal deformation of rotating stars. through polar regions. An example is shown in panel (d) for the Ω = 0.38Ω K model also. We observe that the outer limit of the forbidden region is oblate whereas the inner limit is prolate. As can be inferred from the distribution of N 0 shown in Fig. 1, this property is related to the oblate and prolate shapes of the iso-contours of the Brunt-Väisälä frequency taken in the N e 0,min , N p 0,min interval. These peculiar resonant cavities appear in a frequency range that increases with rotation as the ratio N p 0,min /N e 0,min goes from 1 in the non-rotating case to 1.70 at Ω = 0.84Ω K .
The ray model
In this section, the ray model for progressive gravito-inertial waves is derived from the eikonal equation. Its Hamiltonian structure is emphasised in Sect. 3.1 as it enables us to investigate ray paths using tools and concepts developed for Hamiltonian dynamical systems. The geometrical structure of the phase space in particular is the key to understand the nature of the dynamics. Despite its high dimensionality, the phase space can be explored numerically through cuts known as Poincaré surfaces of section (hereafter PSS). The numerical method and PSS are presented in Sects. 3.2 and 3.3, respectively.
3.1. The Hamiltonian equations describing the ray path and the wave vector evolution along the path
The eikonal equation (18) is a partial differential equation (PDE) for the phase function Φ 0 (x). Instead of solving this equation as a PDE, the gravito-inertial ray model consists in searching for solutions along some path x(t) called the ray path. One then has to solve the coupled differential equations that govern the ray path and the evolution of k along the path. It has been shown that this problem can be written under very general circumstances in a Hamiltonian form (Lighthill 1978;Ott 1993). In the present case, the Hamiltonian is obtained by writing the eikonal equation in the form ω = H(x, k) and the ray path is defined by the group velocity. For a given coordinate system [x i ], the Hamiltonian equations of the ray dynamics are (Lignières 2011)
dx i dt = ∂H ∂k i ,(28)dk i dt = − ∂H ∂x i ,(29)
where the k i = ∂Φ 0 ∂x i are the covariant components of k on the natural basis e i = ∂x ∂x i associated with the coordinate system [x i ]. In Appendix B, these equations are written with spherical coordinates and the k components on the usual unit vectors e r and e θ .
A classical property of gravito-inertial waves is the orthogonality of the group velocity v g = dx/dt and the phase velocity v p = ωk/k 2 . Here, this property is valid in the interior but broken when k c is not negligible. This is apparent in the following formula:
v g · v p = (ω 2 − f 2 cos 2 Θ)k 2 c k 2 (k 2 + k 2 c ) ,(30)
whose derivation is detailed in Appendix C. Another important characteristic is the behaviour of the rays whenever they reach the limits of the domain of propagation. This can be discussed from the relations
dx dt · e = ± √ δ ω(k 2 + k 2 c ) ,(31)dx dt · e ⊥ = Γk ⊥ ± f 2 cos Θ sin Θ √ δ ω(ω 2 − f 2 cos 2 Θ)(k 2 + k 2 c ) ,(32)
also derived in Appendix C. Equation (31) shows that the component of the group velocity parallel to e vanishes at the limits of the domain of propagation, that is when δ = 0. If this occurs when k c is negligible, Eq. (22) finds that Γ also vanishes there. Thus, according to Eq. (32), both components of the group velocity vanish at the limit of the domain of propagation and the turning point is an ordinary cusp point. However, if δ = 0 is reached closer to the stellar surface, where k c cannot be neglected, Eqs. (22) and (32) imply that Γ and dx/dt · e ⊥ do not vanish. The trajectory is then locally parallel to e ⊥ , that is tangent to an equipotential, before it is redirected towards the stellar interior. Figure 3 illustrates the two types of turning points for a given trajectory in the super-inertial regime. The capacity of the numerical method to accurately compute the trajectories near cusp points has been successfully tested against an analytical solution (see Appendix D).
Numerical method for the ray dynamics
The gravito-inertial ray dynamics has been investigated by integrating numerically Eqs. (B.1)-(B.4) using a 5th-order Runge-Kutta method. The step size of the integration is chosen automatically to keep the local error estimate smaller than a given tolerance. The background polytropic model of star is solved numer- ically with an iterative scheme, which is described in Rieutord et al. (2005).
We checked that the PSS shown in Sect. 4 are not significantly modified by decreasing the maximum allowed local error of the Runge-Kutta scheme. We also checked the influence of the resolution of the background polytropic stellar model. Finally, the conservation of the Hamiltonian is used as an independent accuracy test of the computations.
Phase-space visualisation: the Poincaré surface of section
The gravito-inertial ray dynamics is governed by a Hamiltonian with two degrees of freedom, where the phase space is fourdimensional. The fact that ω = H is time independent is equivalent to energy conservation. It implies that phase-space trajectories actually stay on a three-dimensional surface. A PSS is the intersection of all phase-space trajectories with a given threedimensional surface, defined for example by fixing one coordinate. A PSS at a given frequency is therefore a two-dimensional surface.
Different choices are possible for the PSS, but to provide a complete view of phase space, it must be intersected by most phase-space trajectories. Here we used a PSS defined by fixing the colatitude θ to π/2, which corresponds to half of the equatorial plane. Furthermore, to ensure that two trajectories do not intersect on the same point on the PSS, the intersection with θ = π/2 is taken into account only when trajectories cross it from one particular side (here from the θ < π/2 side). For a given frequency ω, the PSS is computed by following many different trajectories over long time intervals. Then to scan the phase space at a given rotation, the PSS is computed for frequencies spanning the 0, N p 0,max interval. For non-integrable systems, the phasespace structure is complex and the features observed in a PSS are expected to depend on the resolution (in k r , r or ω) by which the dynamics is investigated. However, modes occupy a finite phase-space volume because Fourier analysis shows that their wave-vector localisation is inversely proportional to their spatial localisation (see Lignières & Georgeot 2009). Investigating phase space with a finite resolution is therefore sufficient to infer mode properties.
Gravito-inertial ray dynamics in rotating stars
In this section, we study the evolution of the gravito-inertial ray dynamics with rotation. We first describe the case of pure gravity rays in a non-rotating polytropic model (Sect. 4.1) and then consider gravito-inertial rays under the traditional approximation (Sect. 4.2). These are two simple integrable systems that serve as a reference to study the general case of gravito-inertial rays in a rotating star (Sect. 4.3).
4.1. The non-rotating case Ω = 0
In the non-rotating case, the eikonal equation simplifies into Eq. (21), where L is the second invariant that makes the Hamiltonian system with two degrees of freedom integrable. The phase space of integrable systems is made of nested invariant tori. These structures are said to be invariant because any trajectory that starts on one of the structures stays on it; they are called tori because they have the topology of a two-dimensional torus. Any torus is specified by a value of the doublet (ω, L) and its intersection with the θ = π/2 hypersurface is the following f L 2 ,ω (k r , r) = 0 curve:
k 2 r = N 2 0 (r) ω 2 − 1 L 2 r 2 − k 2 c (r),(33)
trivially derived from the eikonal equation (21). As for any integrable system, two types of invariant tori exist. Irrational tori, on which any ray covers the whole torus, or equivalently fills the f L 2 ,ω (k r , r) = 0 curve on the PSS, and rational (or resonant) tori, on which any ray closes on itself before covering the torus. Rays of resonant tori are thus periodic orbits that imprint the PSS on a finite number of points. Figure 4 presents three PSS computed at three different frequencies. For integrable systems, modes can be related to the dynamics through the Einstein-Brillouin-Keller (EBK) quantisation procedure that has been applied to non-rotating spherical stars by Gough (1986) (see also Gough 1993;Lignières & Georgeot 2009). Quantisation conditions actually select tori that support modes in the sense that rays belonging to these tori interfere positively to construct standing waves, i.e. modes. Accordingly, the degree of the spherical harmonic of the mode, , is related to the invariant L by the relation L = + 1/2. We therefore chose to represent in Fig. 4 tori with between 1 and 30. Two of the three PSS have been computed for frequencies corresponding to gravity modes: the (n, ) = (3, 3) and the (n, ) = (50, 3) modes. On these PSS, the torus associated with the mode is highlighted in green. These examples are useful to gain insight into the relation between the position on the PSS and the modes. In the same spirit, the k r normalisation has been chosen in such a way that the value of L of a given k r = f L 2 ,ω (r) curve can be easily retrieved. Indeed, from the eikonal equation (21) and provided that ω N 0,max , we obtain Rωk r /N 0,max LR/r max , where r max = 0.28R is the radius of the inner maximum of N 0 . Thus, the ordinate of the intersection of the k r = f L 2 ,ω (r) curve with the r = 0.25R vertical line is approximatively equal to 4L.
The traditional approximation
The so-called traditional approximation (Eckart 1960;Lee & Saio 1987) The f L 2 ,ω (k r , r) = 0 curves shown are the imprint on the PSS of the invariant tori of the integrable gravity ray dynamics. They are drawn here for L = 1/2 + with taking integer values from 1 to 30 corresponding to degrees of spherical harmonics. The value of L of a given tori can be easily estimated from the intersection with the blue vertical dashed line, since the ordinate of the intersection point is approximately equal to 4L. The EBK quantisation enables us to specify the tori where positive interferences of rays produce modes. For example, the (n, ) = (3, 3) gravity mode on panel (b) and the (n, ) = (50, 3) gravity mode on panel (c) are constructed on the tori highlighted in green. The red curve on panel (a) is a separatrix also called a homoclinic orbit because it connects the hyperbolic point at (k r , r/R) = (0, 0.7968) to itself. As explained in Sect. 4.3.1, it is at the origin of a large chaotic phase-space region observed in the rotating case. are predominantly horizontal, the effect of the Coriolis force in the radial direction is negligible, and the centrifugal deformation is neglected.
Assuming a spherically symmetric stellar model (thus e = e r and e ⊥ = e θ ) and neglecting the f sin θ terms, the eikonal equation (20) simplifies into
N 2 0 (r) − ω 2 k 2 θ − (ω 2 − f 2 cos 2 θ) k 2 r + k 2 c (r) = 0.(34)
From this equation and the associated dynamical equations, we found a second invariant, namely
λ = ω 2 r 2 k 2 θ ω 2 − f 2 cos 2 θ = ω 2 r 2 (k 2 r + k 2 c ) N 2 0 − ω 2 ,(35)
which implies that the associated ray dynamics is integrable. Integrability is expected since the perturbation equation in the traditional approximation is separable. Furthermore, neglecting the k c term, the condition of propagation derived from (34) is
(N 2 0 − ω 2 )(ω 2 − f 2 cos 2 θ) > 0.(36)
For super-inertial waves, the domain of propagation is simply given by ω < N 0 . This frequency range is more restricted than that given by Γ > 0, but both are qualitatively similar and even identical in the limit f /N 0 → 0 if the centrifugal deformation is neglected (see Eq. 26). For sub-inertial waves, the domain of propagation of the traditional approximation consists of two unconnected domains above and below the radius where ω = N 0 . Above this radius, the equatorial region defined by f cos θ < ω and below, the polar cone defined by ω < f cos θ. Thus, the propagation of sub-inertial waves from the N 0 > f layers towards the central region N 0 < f is not allowed by the traditional approximation (for a graphical account of these differences in the domain of propagation, see Fig. 11 of Gerkema et al. 2008). This is a clear limitation of the traditional approximation since, as we have seen from the discussion on the domain of propagation, sub-inertial waves are indeed allowed to go through the centre of the star (and more generally through regions where N 0 < f ). Near the surface, the condition is ω < c 1 S √ λ with S √ λ = √ λc s /r, which is a condition formally similar to the non-rotating case, with λ playing the role of L 2 .
The second expression in (35) yields the imprint of the invariant tori on the PSS. We observe that it has the form f λ,ω (k r , r) = 0 that is the same as in the non-rotating case using λ instead of L 2 . This relation between λ and L 2 is the raydynamics version of the known relation between the separation constant of the perturbation equation in the traditional approximation and the product ( + 1) (Bouabid et al. 2013). The PSS of the traditional approximation has thus the same structure as the non-rotating PSS. The effect of rotation comes from the increase of λ with f /ω, which produces the dilation effect observed between the two PSS shown in Fig. 5. As for the nonrotating case, gravito-inertial modes computed under the traditional approximation can be associated with specific tori. This is the case in Fig. 5, where two gravito-inertial modes labelled by their quantum numbers at zero rotation, namely (n, ) = (3, 3) and (n, ) = (50, 3), are represented in green on the PSS.
Phase-space structure of rotating stars
As rotation increases the gravito-inertial ray dynamics is no longer integrable and undergoes a transition towards a mixed state with coexistence of chaotic regions and invariant tori. This is apparent on Fig. 6, which shows the phase-space structure at Article number, page 8 of 24 V. Prat et al.: Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics Fig. 6. PSS computed at ω/ f = 3.1 for a rotation Ω/Ω K = 0.38 and four gravito-inertial rays in the physical space belonging to the three different types of phase-space structures: a surviving KAM torus (blue), a period-5 island chain (green/light green), and a chaotic zone (red). The light green ray on the middle panel is the stable periodic orbit at the centre of the period-5 island chain. The period of an island chain is defined as the number of corresponding structures on the PSS (see Sect. 4.3.2). These rays cross the PSS on points of corresponding colours. On all figures showing rays in the physical space, the limits of the domain of propagation given by Γ(ω) = 0 are drawn in magenta. Ω = 0.38Ω K and ω/ f = 3.1 together with a selection of rays in the physical space. As compared to the non-rotating PSS, which only exhibits f L 2 ,ω (k r , r) = 0 one-dimensional curves, we now observe three different types of structures. First of all, there are one-dimensional curves similar in shape to the non-rotating curves. The blue ray provides an example of ray associated with this type of tori. A second and new type of phase-space structures forms around stable periodic orbits and is called island chains. An orbit is said to be stable when trajectories induced by small perturbations of the orbit remain close to it. The green ray highlights one of the invariant tori formed around a central periodic orbit represented by the light green ray shown in the physical space. Finally, the third type of structure is the chaotic region, which can be easily identified on the PSS by the fact that the associated rays do no stay on a one-dimensional curve but fill a two-dimensional surface instead. The chaotic zone of Fig. 6 is highlighted by the imprint of a ray also shown in the physical space. .567] of the ray dynamics in the traditional approximation in a spherical star rotating at Ω = 0.38Ω K . The f λ,ω (k r , r) = 0 curves shown are the imprint on the PSS of the invariant tori of the integrable dynamics. They are drawn for the quantised values of λ( , ω/ f ), where is the degree of the spherical harmonics of the corresponding mode at Ω = 0 and is varied from 1 to 30. The increase of the gap between the f λ,ω (k r , r) = 0 curves as ω decreases is due to the increase of λ with f /ω. Tori in green correspond to gravito-inertial modes labelled by their quantum numbers at zero rotation, that is the (n, ) = (3, 3) mode (top) and the (n, ) = (50, 3) mode (bottom).
The apparition of these new structures in the gravito-inertial dynamics is typical of the KAM-type (in reference to the Kolmogorov-Arnold-Moser theorem) transition of Hamiltonian systems from integrability towards chaos (Ott 1993). Accordingly, in systems that experience infinitesimal departures from integrability, all the resonant tori of the integrable system are destroyed and replaced by island chains around stable periodic orbits and small chaotic regions developing around unstable periodic orbits. Meanwhile, irrational tori are continuously perturbed but not destroyed. The evolution towards larger values of the parameter controlling the departure from integrability has been studied in many cases. The common phenomenology emerging from these studies is that, when the control parameter increases, the invariant tori, either the surviving irrational tori or the island chains, are progressively destroyed while chaotic regions occupy larger phase-space volumes. As demonstrated in the simple case of the kicked rotor (Ott 1993), for each irrational torus there is a critical value of the control parameter above which the torus is destroyed and these critical values serve to quantify the progression towards chaos.
The present gravito-inertial ray dynamics clearly undergoes a KAM-type transition towards a mixed state including surviving irrational tori, island chains, and chaotic regions. Neverthe- Fig. 7. PSS computed at ω/ f = 2.4 for a rotation Ω/Ω K = 0.38 and two gravito-inertial rays belonging to a period-1 island chain including the central stable periodic orbit (light blue). less, we find that the actual departure from integrability strongly depends on the frequency ω. Indeed, the surface occupied by the island chains and chaotic regions strongly decreases towards low frequencies. This is obvious when comparing the phasespace structure of a Ω = 0.38Ω K star at three frequencies, ω/ f = {3.1, 2.4, 0.8}, as shown in Figs. 6, 7, and 8, respectively. This observation does not mean that island chains and chaotic regions are absent below some frequency but rather that they are too small to be visible when the PSS are shown on large k r and r scales. The same phenomenology also holds at higher rotation as shown by the PSS of a Ω = 0.84Ω K star computed for three decreasing frequencies, ω/ f = {2, 1.5, 0.5} and shown in Figs. 9, 10, and 11, respectively. We therefore conclude that the departure from integrability as measured by the presence of large island chains and chaotic regions strongly diminishes in the low part of the 0, N p 0,max interval and is weakly dependent on the centrifugal deformation.
Overall we may say that the tori that results from the continuous evolution of the non-rotating tori still shape the largescale organisation of the phase space even at large rotation rates. This is in contrast with the axisymmetric acoustic ray dynamics where, as rotation increases, the phase space is progressively dominated by large islands and a large chaotic region, while irra-Article number, page 10 of 24 V. Prat et al.: Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics Fig. 9. PSS computed at ω/ f = 2.0 for a rotation Ω/Ω K = 0.84 and two gravito-inertial rays belonging to a period-9 island chain (blue) and to a chaotic zone (red), respectively. tional tori only survive for large values of k θ /ω corresponding to whispering gallery modes with low disc-integrated visibilities.
In the following, the properties of each of the three types of phase-space structures are described in more detail. Considering them separately is relevant for the study of gravitoinertial modes because, according to Percival's conjecture (Percival 1973;Berry & Robnik 1984) and following the results obtained for acoustic modes (Lignières & Georgeot 2009), we expect that each type of phase-space structure can be asymptotically related to families of modes with specific properties.
Chaotic regions
PSS computed at different rotation rates show that a large chaotic region is systematically present in the upper part of the 0, N p 0,max frequency interval. In the following we argue that this chaotic region is related to the existence of unstable periodic orbits, which are themselves a direct consequence of the presence of a local minimum of the Brunt-Väisälä frequency in the stellar envelope. This interpretation is supported by our ability to predict the frequency range for which large chaotic regions are seen on the PSS.
In smoothly perturbed integrable Hamiltonian systems, chaotic motions appear near separatrices that connect hyperbolic points, that is unstable periodic orbits or unstable equilibrium points. In the framework of the KAM transition theory, these hyperbolic points naturally originate from the destruction of the resonant tori and chaos first appears around these hyperbolic points and along the structures that connect them (Ott 1993). Chaotic motions, however, can also originate from hyperbolic points already present in the unperturbed system. In our case, the unperturbed system does possess hyperbolic points as can be inferred from the PSS at Ω = 0 shown in panel (a) of Fig. 4. The vicinity of the point (k r , r) = (0, 0.8) is indeed typical of the phase portrait around a hyperbolic point with nearby orbits forming branches of hyperbolas. The separatrix (the red curve in panel (a) of Fig. 4) is also called a homoclinic orbit because it connects the hyperbolic point to itself. The apex of the motion of a pendulum provides a classical example of a hyperbolic point in a system with one degree of freedom, which is in this case a hyperbolic fixed point or equivalently an unstable equilibrium point. In our system with two degrees of freedom, the hyperbolic point is rather an unstable periodic orbit because it corresponds to a circular trajectory at a fixed radius (r ∼ 0.8 on panel (a) of Fig. 4). Since the ray dynamics in a spherical star is separable in the radial and latitudinal coordinates, the radius of the hyperbolic Article number, page 11 of 24 A&A proofs: manuscript no. Grav1.9.4 Fig. 8. PSS computed at ω/ f = 0.8 for a rotation Ω/Ω K = 0.38 and two gravito-inertial rays belonging to a period-11 island chain (blue) and to a surviving KAM torus (green), respectively. point is actually a fixed point of the radial ray dynamics. As in the case of the pendulum, the radial Hamiltonian takes the usual form H r = 1/2k 2 r + V r and the unstable fixed point corresponds to the maximum of the potential V r .
In Appendix E, analytical solutions of this problem are found via the envelope approximation, which involves the assumption that the mass enclosed in a sphere of given r does not depend on r, and we show that these solutions provide a very good approximation to the numerical solutions obtained without the envelope approximation. Accordingly, the gravity ray dynamics possesses a hyperbolic point for any frequency ω within the [0.93N 0,min , N 0,min ] interval. The radius of the hyperbolic point varies monotonically from r = 0.84R for ω = 0.93N 0,min to r = 0.75R for ω = N 0,min , while the value of L characterising the separatrix varies also monotonically from L = 16.92 for ω = 0.93N 0,min to L = +∞ for ω = N 0,min .
To confirm that chaos first occurs around the separatrix, we computed gravito-inertial rays close to this orbit for a slowly rotating model Ω/Ω K = 0.02. As illustrated by Fig. 12, a small chaotic region is found in the expected frequency range and at the expected location. We further make the hypothesis that the large chaotic regions observed at Ω/Ω K = 0.38 and Ω/Ω K = 0.84 are due to the same mechanism. To test this hypothesis, PSS have been computed for frequencies inside and outside 0.93 (N e 0,min ) 2 + f 2 , N p 0,min . To define the lower and upper bounds of this interval, rotational effects have been taken into account on phenomenological grounds. First, rotation changes the frequency range for propagation, and following the conditions (26), a f 2 sin 2 Θ term must be added to N 2 0,min . Second, we included the N e 0,min , N trifugal force. Indeed, we expect that unstable periodic orbits exist for these frequencies because there is always a range of latitudes where the condition ω/ N 2 0,min + f 2 sin 2 Θ ∈ [0.93, 1] is fulfilled. Considering frequencies 5% apart from the interval limits, we found that the large chaotic region is present inside and absent outside the 0.93 (N e 0,min ) 2 + f 2 , N p 0,min interval. At a given frequency, the extent of the chaotic surface seen on the PSS is not simple to predict quantitatively. Comparing Figs. 12, 6, and 9 we nevertheless observe that, as expected, the surface of the chaotic region increases with rotation.
Island chains
As they originate from the destruction of given rational tori, island chains can be associated with the rational number p/q of their parent torus. Then, the central periodic orbit of the p/q island imprints q times the PSS (Ott 1993); in this study we call the number q the period. The number p can be found by constructing a different PSS (for instance by fixing the radius r instead of the colatitude). For super-inertial rays, p corresponds to the number of loops visible in real space. On the six PSS displayed in Figs. 6-11, we highlighted island chains of various periods: one period-1 at ω/ f = 2.4, two of period-5 (but with different p values) at ω/ f = 3.1 and ω/ f = 1.5, respectively, one period-9
at ω/ f = 2, one period-11 at ω/ f = 0.8, and one with a period larger than 26 at ω/ f = 0.5. As expected from theory, the width of the island decreases with their period. Moreover, there is a clear tendency to find only small and large-period island chains in the sub-inertial regime, although there is no obvious change of behaviour at ω/ f = 1.
In a typical KAM transition, island chains form immediately for a non-zero pertubation, grow with the amplitude of the perturbation, and are finally destroyed. We followed the evolution of some low-period island chains with rotation and found that their width slightly increase but no destruction of island chains above a certain rotation rate has been observed. For example, the period-1 island chain is still clearly visible at 99% of the critical velocity.
In the physical space, rays belonging to island chains do not fill their resonant cavity as they remain concentrated around the stable periodic trajectory. From Ballot et al. (2012), we already know that numerically computed modes can be associated with island chains. A detailed confrontation of the island properties with numerically computed modes will be performed in a future paper.
Surviving tori
The irrational tori of the non-rotating case are continuously deformed by the effect of rotation but may nevertheless have survived destruction even at high rotation rates. This is the case, in particular, for high-L tori, since for most frequencies no large island chain or chaotic region is observed in the PSS in this domain, which corresponds to high values of the normalised k r . Some irrational tori with lower L also survive the perturbation induced by rotation. This is obvious at low frequencies where phase space is very much structured by one-dimensional curves broadly similar to the non-rotating curves (see Figs. 8 and 11 and panel (c) of Fig. 4).
This nearly integrable behaviour of the low-frequency, gravito-inertial, ray dynamics might be attributed to the proximity of a hidden integrable system. To test whether the ray dynamics in the traditional approximation could be this inte-grable system, we computed the "traditional" invariant λ along the path of gravito-inertial rays. Considering a sub-inertial ray in a Ω/Ω K = 0.38 star, we found that λ varies by several orders of magnitude along a given ray. As expected, the largest variations occur in regions of space that are outside the "traditional" domain of propagation, but even within the resonant cavity of the traditional approximation, λ still varies by a factor of two. In the super-inertial regime, the variations of λ are of the order of 1.5, which is of the same order of magnitude as the variations of L; the parameter L is the invariant of the non-rotating case, which is not supposed to provide a valid approximation. We conclude that the ray dynamics computed in the traditional approximation does not provide an accurate approximation even in the low-frequency regime, where the full gravito-inertial ray dynamics is close to integrable.
From rays to modes
In this section, we summarise the properties of gravito-inertial modes that can be inferred from the present study of the ray dynamics. Methods to construct modes from the superposition of positively interfering rays have been first proposed for integrable systems. The EBK procedure has been established in the field of quantum physics as a generalisation of the Bohr-Sommerfeld method to quantise systems from their classical trajectories. This method only works in the presence of invariant tori and leads to a regular frequency spectrum in the sense that the spectrum can be described by a function of N integers, where N is the number of degrees of freedom of the system. For chaotic dynamical systems, the EBK quantisation does not work and must be replaced by the Gutzwiller trace formula, which is difficult to apply in practice (Gutzwiller 1990). Nevertheless, generic statistical properties of spectra associated with chaotic dynamical systems, such as the distribution of spacings between consecutive frequencies, have been discovered in quantum systems (Bohigas et al. 1984). For mixed dynamical systems, it has been conjectured that chaotic regions and phase-space regions structured by invariant tori, such as island chains or regions of surviving irrational tori, can be quantised separately. Consequently, the spectrum of mixed systems is expected to be a superposition of frequency subsets, each of which are associated with different phase-space structures. Frequency subsets associated with regions structured by invariant tori should be regular while those associated with chaotic regions should possess generic statistical properties (Percival 1973;Berry & Robnik 1984). A more detailed description of the concepts presented above together with the full references is available in Lignières & Georgeot (2009). Percival's predictions have been applied to acoustic rays in rotating stars and successfully confronted with numerically computed high-frequency acoustic modes (Lignières & Georgeot 2009).
In our case, Percival's conjecture implies that the frequency spectrum of short-wavelength gravito-inertial modes is a superposition of frequency subsets associated with the phase-space structures of the gravito-inertial ray dynamics, notably a regular sub-spectrum associated with the surviving irrational tori, regular sub-spectra associated with island chains, and a subspectrum with generic statistical properties associated with the large chaotic region. In principle, quantitative predictions on the regular sub-spectra can be obtained by conducting the EBK quantisation of the invariant tori. This has been carried out in Pasek et al. (2012) for the period-2 island chain of the acoustic ray dynamics and could be also attempted for the gravito-inertial island chains and surviving tori.
Previous numerical explorations of gravito-inertial modes already provide some insights into the relation between the asymptotic predictions of the gravito-inertial dynamics and the numerically computed modes. Using µ = 3 polytropic models, Ballot et al. (2010) followed low-degree ( ≤ 3) modes in both low-and high-order ranges from Ω = 0 to Ω = 0.7Ω K . The vast majority of these modes undergo a smooth evolution of their spatial distribution with rotation, in which the main change is the equatorial mode confinement in the sub-inertial regime. These modes most probably correspond to the phase-space region structured by the surviving irrational tori. On the other hand, some modes experience a striking modification of their spatial distribution, leading to a concentration of their energy. The connection between these modes called rosette modes and the central periodic orbits of island chains has been clearly established in Ballot et al. (2012). Since then, Takata & Saio (2013) proposed a different approach in which rosette modes result from the rotationally induced coupling of modes, which were nearly degenerate gravity modes at Ω = 0 (see also Takata 2014). The link between these two apparently very different approaches to model rosette modes remains to be understood. In our study, the number K introduced by Takata & Saio (2013) to characterise families of rosette modes corresponds to the ratio 2q/p. 2 The authors found only integer values of K, which they justified by the fact that the only effect of rotation they considered was the Coriolis force. We also present island chains with non-integer values. This is probably because we took the centrifugal deformation into account. Modes that could be related to the large chaotic region that we describe have not been identified yet. By providing an estimate of both the frequencies and wave vectors of chaotic rays, ray dynamics will help to find these modes. More generally, a dedicated numerical study will be necessary to test the predictions of the ray-based asymptotic theory. In this context, tools allowing us to construct phase-space representations for numerically computed modes, such as Husimi distributions, are available. As illustrated by Lignières & Georgeot (2009), those can be used to show the imprint of modes on the PSS.
Discussion and conclusions
The main results presented in the previous sections are (i) the derivation of an eikonal equation for gravito-inertial rays in polytropic uniformly rotating stars, (ii) the analysis of the properties of this equation, including constraints on the domains of propagation for k φ = 0 rays, (iii) the numerical computation of the k φ = 0 ray dynamics for µ = 3 polytropic models with increasing rotation, and (iv) the interpretation of the numerical results using tools and concepts of Hamiltonian dynamics. In particular, the exploration of the phase space provides us with a global view of the properties of gravito-inertial waves in rotating stars. Qualitative predictions on the frequency spectrum of gravito-inertial modes have been made and should be a powerful tool to interpret the properties of numerically computed gravito-inertial modes. A quantitative result already provided by the ray theory is the domain of propagation. Other quantitative predictions are expected from the EBK quantisation of the surviving irrational tori and of the island chains.
A first restriction of the present work is that we did not consider k φ 0 rays. This would be necessary for an asymptotic study of non-axisymmetric gravito-inertial modes. Another re-striction is the use of µ = 3 polytropic models although the formalism can be extended to more realistic rotating stellar models. In particular, the effects of a convective core and/or a convective envelope would be interesting to consider.
The present ray dynamics formalism can be improved by taking the effect of background mean motions and, in particular, differential rotation into account. For example, self-consistent models including baroclinic effects computed by the ESTER code (Rieutord & Espinosa Lara 2013) could be used. According to atmospheric or oceanic studies, the eikonal equation including mean horizontal motions is identical to (18), except that the frequency is changed into the Doppler-shifted frequency, which is the frequency that would be observed in a frame of reference moving with the mean horizontal motions (Fritts & Alexander 2003). Analagous with the effect of wind shear, differential rotation should cause some back-refraction and thus will modify the resonant cavity. Another improvement to the eikonal equation would be to include non-axisymmetric Rossby waves.
Ray models might also be used to study the transport of angular momentum by waves. This would require us to extend the ray formalism to treat the evolution of the wave amplitude A along the ray path (the next order of the WKB approximation). Such amplitude equations are already used to investigate the driving of the atmospheric circulation by waves (Marks & Eckermann 1995). To conduct similar investigations in stars, it will be necessary to model energy deposition processes such as thermal dissipation and critical layers (Fritts et al. 1998;Alvan et al. 2013) and to take phase changes at caustics into account (Broutman et al. 2004).
Angular momentum transport by gravito-inertial waves that occasionally escape the star has been invoked as a possible explanation of the Be phenomenon (Ishimatsu & Shibahashi 2013). Specific properties of chaotic gravito-inertial waves such as those observed in our calculations could be interesting in this context. At any location and in particular at the surface, the horizontal wavelength k ⊥ of chaotic waves can indeed take many different values. Thus a chaotic wave can be back-refracted many times at the surface before it escapes the star because k ⊥ has reached a high enough value.
When compared to the ray theory of Dintrans & Rieutord (2000), the present work includes the effects of centrifugal deformation as well as a realistic back-refraction of rays in the stellar envelope. This last point has a crucial effect on the comparison between the two approaches. In Dintrans & Rieutord (2000), the rays of characteristic are confined into the star by a rigid and stress-free surface and this boundary condition destroys the Hamiltonian character of the dynamics of characteristics. As a consequence, the volume of phase-space elements is not conserved, which in turns enables rays to focus on attractors. This focusing effect is at the origin of the singular gravitoinertial modes that are effectively observed in the presence of rigid boundaries (Maas et al. 1997;Dintrans et al. 1999). However, there are no such rigid boundaries in real stars and singular gravito-inertial modes focused on attractors of characteristics by unrealistic rigid boundaries should not exist in stars. As we have seen in the present paper, the ray dynamics ought to be Hamiltonian in stars as long as dissipative processes are neglected.
Appendix A: Derivation of the perturbation equation (11)
In this appendix, our goal is to transform the perturbation equations (5)-(7) into Eq. (11) forΨ. In Sect. A.1, density and velocity perturbations appearing in Eqs. (5)-(7) are eliminated to obtain an equation for pressure perturbations only. Then in Sect. A.2, the new variableΨ =P/a is introduced to eliminate the first-order derivatives in the meridional plane from the perturbation equation on P. Finally, in Sects. A.3 and A.4, the constant term C is expanded in powers of c 2 s to determine the dominant O(1/H 2 s ) approximation of C.
Appendix A.1: Derivation of an equation for pressure perturbations only
We first rewrite the perturbation equations (5)-(7), using u = ρ 0 v and the definition of the Brunt-Väisälä frequency N 2
0 = g 0 · ∇ρ 0 ρ 0 − 1 Γ 1 ∇P 0
P 0 as follows:
∂ t ρ + ∇ · u = 0, (A.1) ∂ t u + f ∧ u = −∇P + ρ g 0 , (A.2) ∂ t P = c 2 s ∂ t ρ + N 2 0 g 2 0 g 0 · u . (A.3)
In order to eliminate u from these equations, we first calculate f ∧ (A.2) as
f ∧ ∂ t u + ( f · u) f − f 2 u = − f ∧ ∇P + ρ f ∧ g 0 . (A.4)
The first term on the LHS is then developed using ∂ t (A.2) to give
∂ 2 tt u + f 2 u − ( f · u) f = −∇∂ t P + ∂ t ρg 0 + f ∧ ∇P − ρ f ∧ g 0 . (A.5)
We use a similar method to eliminate the scalar product f · u. We compute f · (A.2)
f · ∂ t u = − f · ∇P + ρ f · g 0 , (A.6)
and introduce it into ∂ t (A.5) to get
∂ 3 ttt u + f 2 ∂ t u = −( f · ∇P) f + ρ( f · g 0 ) f − ∇∂ 2 tt P + ∂ 2 tt ρg 0 + f ∧ ∇∂ t P − ∂ t ρ f ∧ g 0 . (A.7)
Defining the operator
L ≡ ∂ 3 ttt + f 2 ∂ t (A∂ t L(P) + α ( f · g 0 )( f · ∇P) + ( f ∧ g 0 ) · ∇∂ t P + g 0 · ∇∂ 2 tt P = c s 2 ∂ t L(ρ) + α ( f · g 0 ) 2 ρ + g 0 2 ∂ 2 tt ρ = c s 2 D(ρ), (A.12) where D ≡ ∂ 4 tttt + ( f 2 + N 0 2 )∂ 2 tt + N 0 2 ( f · g 0 ) 2 g 0 2 . (A.13)
To simplify calculations, we assume that P and ρ are harmonic in time, i.e. P = {Pe −iωt } and ρ = {ρe −iωt }. Then the operator D is simply a multiplication by A.14) and ∂ t L a multiplication by From Eq. (A.12),ρ is expressed as a function ofP as follows:
D = ω 4 − ω 2 ( f 2 + N 0 2 ) + N 0 2 ( f · g 0 ) 2 g 0 2 ,(ρ = GP + H · ∇P c s 2 D , (A.16) with H = α ( f · g 0 ) f − iω f ∧ g 0 − ω 2 g 0 . (A.17)
We then compute ∇ · L(u) from Eq. (A.7), using the general identity ∇ · (ba) = a · ∇b + b∇ · a and the fact that f is uniform
∇ · L(u) = − f · ∇( f · ∇P) − iω∇ · ( f ∧ ∇P) + ω 2 ∆P + ∇ · (ρF), (A.18) with F = ( f · g 0 ) f + iω f ∧ g 0 − ω 2 g 0 . (A.19)
The expression (A.18) can be simplified using
∇ · ( f ∧ ∇P) = ∇P · (∇ ∧ f ) − f · (∇ ∧ ∇P) = 0, (A.20)
which finally yields
∇ · L(u) = ω 2 ∆P − f · ∇( f · ∇P) + ∇ · (ρF). (A.21)
Replacing the expression (A.21) in Eq. (A.9) and using Eq. (A.16) to expressρ, we obtain the following equation forP:
G(GP + H · ∇P) c s 2 D + ω 2 ∆P − f · ∇( f · ∇P) + ∇ · (GP + H · ∇P)F c s 2 D = 0. (A.22) We use the identity ∇ · (a ∧ b) = b · (∇ ∧ a) − a · (∇ ∧ b)
, which implies that ∇ · ( f ∧ g 0 ) = 0, as f is uniform and g 0 derives from a potential. Defining the vector F 0 = ( f · g 0 ) f −ω 2 g 0 , the vectors F and H are written as F = F 0 +iω f ∧ g 0 and H = α(F 0 −iω f ∧ g 0 ). Thus, Eq. (A.22) becomes
G(GP + αF 0 · ∇P) c s 2 D + ω 2 ∆P − f · ∇( f · ∇P)+∇ · (GP + αF 0 · ∇P)F 0 c s 2 D + i(1 − α) ωG c 2 s D ( f ∧ g 0 ) · ∇P − iαω∇ · ( f ∧ g 0 ) · ∇P F c 2 s D + iω α c 2 s D ( f ∧ g 0 ) · ∇(F 0 · ∇P) = 0, (A.23)
where we have used the fact that α is uniform and f ∧ g 0 is parallel to e φ . We then use the relation
F 0 · ∇ F 0 c s 2 D · ∇P = 1 c s 2 D F 0 · ∇(F 0 · ∇P) + F 0 · ∇ 1 c s 2 D F 0 · ∇P (A.24)
to get
Dω 2 ∆P − D f · ∇( f · ∇P) + N 0 2 g 0 2 F 0 · ∇(F 0 · ∇P) + N c s 2 F 0 · ∇P + M c s 2P +i(1 − α) ωG c 2 s ( f ∧ g 0 ) · ∇P − iαωD∇ · ( f ∧ g 0 ) · ∇P F c 2 s D + iω α c 2 s ( f ∧ g 0 ) · ∇(F 0 · ∇P) = 0, (A.25) where M = G G + c s 2 D∇ · F 0 c s 2 D , (A.26) N = (1 + α)G + c s 2 D∇ αF 0 c s 2 D . (A.27)
From the definition of F 0 , we have
F 0 · ∇(F 0 · ∇P) = ( f · g 0 ) 2 f · ∇( f · ∇P) + ω 4 g 0 · ∇(g 0 · ∇P) − ω 2 ( f · g 0 ) f · ∇( g 0 · ∇P) + g 0 · ∇( f · ∇P) + ( f · g 0 ) f · ∇( f · g 0 ) − ω 2 g 0 · ∇( f · g 0 ) f · ∇P, (A.28)
where, thanks to the fact that f is uniform, the last term can be also written 1 2 ( f · ∇K) f · ∇P with
K = ( f · g 0 ) 2 − ω 2 g 0 2 = F 0 · g 0 . (A.29)
where the operators ∇ and ∇ z are defined by Eqs. (9) and (10). Equation (A.38) becomes
a∆Ψ = a f 2 ω 2 ∇ 2 zΨ + a N 2 0 ω 2 ∆ ⊥Ψ + V · ∇Ψ + iaV m · ∇Ψ + M Ψ , (A.45) where V = −2∇a + 2 f 2 ω 2 (∇ z a)e z + 2 N 2 0 ω 2 ∇a − 2 N 2 0 ω 2 (∇ a)e + aV , (A.46) M = −∆a + f 2 ω 2 ∇ 2 z a + N 2 0 ω 2 ∆a − N 2 0 ω 2 ∇ 2 a − N 2 0 ω 2 (∇ · e )∇ a + V · ∇a + M a. (A.47)
The condition V = 0 translates into a first-order linear differential equation for a. The projection of this equation on e and e z leads to two equations for ∇ a/a and ∇ z a/a. These equations can be solved as a linear system for these two unknowns provided that the determinant of the system, which is equal to D, does not vanish. We then use these expressions of ∇ a/a and ∇ z a/a to write where the V m term is now expressed as a function of the dimensionless quantity T , defined by
T = (1 − α) − α c 2 s D G ∇ · F 0 c 2 s D − α r sin θ Gg 0 sin Θ F 0 · ∇ g 0 sin Θ r sin θ . (A.53)
The term C equals M /a and can be expressed in terms of W to be written as
C = − ∇ · W − W 2 + N 2 0 ω 2 ∇ · W − ∇ (W · e ) + N 2 0 ω 2 W 2 − (W · e ) 2 −
N 2 0 ω 2 (∇ · e )(W · e ) + f 2 ω 2 ∇ z (W · e z ) + (W · e z ) 2 + V · W + M .
(A.54)
From the definitions of M and M, it is easy to see that C can be written as C = −ω 2 /c 2 s + C to recover Eq. (11).
Appendix A.3: The constant term
To take into account the back-refraction of outgoing waves in the short-wavelength approximation, C is expanded in powers of H = RT/g 0 or equivalently c 2 s and we only retain the dominant term. Among the quantities involved in C, the inverse of the sound speed and the Brunt-Väisälä frequency tend to become very large towards the stellar surface, whereas f , g 0 , and Γ 1 remain finite. To derive the dominant term of C approaching the stellar surface, we use the following expression for W: and e s is the classical cylindrical unit vector pointing away from the z-axis. The dynamical equations (B.1)-(B.4) in spherical coordinates are dr dt = 1 ω(k 2 + k 2 c ) k r N 0 2 sin 2 (θ − Θ) + f 2 cos 2 θ − ω 2 + k θ N 0 2 sin(θ − Θ) cos(θ − Θ) − f 2 sin θ cos θ , (B.12) dθ dt = 1 ωr(k 2 + k 2 c ) k r N 0 2 sin(θ − Θ) cos(θ − Θ) − f 2 sin θ cos θ + k θ N 0 2 cos 2 (θ − Θ) + f 2 sin 2 θ − ω 2 , (B.13) dk r dt = 1 ω(k 2 + k 2 c ) k r k θ r N 0 2 sin(θ − Θ) cos(θ − Θ) − f 2 sin θ cos θ + k θ 2 r N 0 2 cos 2 (θ − Θ) + f 2 sin 2 θ − ω 2 + ∂ r (k c 2 ) 2
(ω 2 − f 2 cos 2 Θ) − ∂ r (N 0 2 ) 2 [k r sin(θ − Θ) + k θ cos(θ − Θ)] 2 + ∂ r Θ N 0 2 sin(θ − Θ) cos(θ − Θ)(k r 2 − k θ 2 ) + k r k θ 2 cos 2 (θ − Θ) − 1 + f 2 sin Θ cos Θk c 2 , (B.14) dk θ dt = 1 rω(k 2 + k 2 c )
f 2 sin θ cos θ(k r 2 − k θ 2 ) + k r k θ (2 cos 2 θ − 1) + ∂ θ (k c 2 ) 2
(ω 2 − f 2 cos 2 Θ) − ∂ θ (N 0 2 ) 2 [(k r sin(θ − Θ) + k θ cos(θ − Θ)] 2 + f 2 sin Θ cos Θk c 2 ∂ θ Θ + (∂ θ Θ − 1) sin(θ − Θ) cos(θ − Θ)(k r 2 − k θ 2 ) + k r k θ 2 cos 2 (θ − Θ) − 1 N 0 2 − k θ k r N 0 2 sin 2 (θ − Θ) + f 2 cos 2 θ − ω 2 − k 2 θ N 0 2 sin(θ − Θ) cos(θ − Θ) − f 2 sin θ cos θ , (B.15) These equations have been used for the numerical computation of the gravito-inertial dynamics in barotropic models of uniformly rotating stars.
Appendix C: Derivation of relations (30), (31), and (32)
It is also useful to derive other relations to understand ray dynamics. Below, the relation (30) between the group velocity and the phase velocity is derived. Using the first two equations of the Hamiltonian system (B.1) and (B.2), we have dx dt · k = k r ∂H ∂k r + k θ ∂H ∂k θ .
(C.1)
Then, taking the derivatives of the eikonal equation (B.5) with respect to k r and k θ , we obtain (k 2 + k 2 c )ω ∂H ∂k r + ω 2 k r = f 2 k z ∂k z ∂k r + N 2 0 k ⊥ ∂k ⊥ ∂k r , (C.2) (k 2 + k 2 c )ω ∂H ∂k θ + ω 2 k θ = f 2 k z ∂k z ∂k θ + N 2 0 k ⊥ ∂k ⊥ ∂k θ . (C.
3)
It follows that ω(k 2 + k 2 c ) dx dt · k = −ω 2 k 2 + f 2 k z k r ∂k z ∂k r + k θ ∂k z ∂k θ + N 2 0 k ⊥ k r ∂k ⊥ ∂k r + k θ ∂k ⊥ ∂k θ . ∂k ⊥ ∂k θ = cos(θ − Θ), (C.8) and from there we deduce that ω(k 2 + k 2 c ) dx dt · k = −ω 2 k 2 + f 2 k 2 z + N 2 0 k 2 ⊥ (C.9) = (ω 2 − f 2 cos 2 Θ)k 2 c , (C.10) which corresponds to relation (30).
where we have introduced the dimensionless quantitiesω = ω/ GM/R 3 , x = r/R and the notations a = α(µ + 1)/Γ 1 and b = (µ + 1)(µ − 1)/4. In our one-dimensional system, k r and the radial derivative of V r must vanish at fixed points. As H r = 0, the first condition implies that V r also vanishes at the fixed point. The equilibrium will be unstable if V r is maximum there. Asking that both V r and its radial derivative vanish is equivalent to solve the following system of equations for the variable x excluding the x = 0 case:
ã ω 2 (1 − x) − b L 2 x 3 − x 3 (1 − x) 2 = 0, ã ω 2 (1 − x)(4x − 3) − 2 b L 2 x 4 = 0. (E.3)
Using two new variables u = 1− x and v = x 3 , the system above is equivalent to solving a cubic equation for u, a second equation giving the relation betweenω and L as follows:
u 3 − u 2 4 + b 2L 2 u + b 4L 2 = 0, ω 2 = a L 2 2b u(1 − 4u) (1 − u) 4 . (E.4)
The cubic equation has a unique real solution if L is smaller than a minimum value L min , and this solution is not physical as it gives a point outside the star. Above L min , there are in addition two real solutions among which the smaller solution corresponds to the unstable point and the larger solution to the outer stable point. Simple analytical solutions are obtained for the two extreme cases L = L min and L = +∞.
In conclusion, an unstable hyperbolic point is present in the frequency range 0.9348N 0,min ≤ ω ≤ N 0,min . As ω increases in this interval, the position of the unstable point decreases from r = 0.8406R to r = 0.75R, while the L value of the separatrix increases from L min = 16(µ+1)(µ−1) 19 √ 57−143 to L = +∞. It is interesting that some properties of the unstable hyperbolic point, namely the radius and the frequency interval normalised by N 0,min , do not depend on the polytropic index. For a µ = 3 polytropic model, L min = 16.92.
We compared these analytical solutions to numerical determinations of the hyperbolic point obtained for the full µ = 3 polytropic model, that is without the envelope approximation. The agreement is fairly good since the numerical solution gives c 2 N 0,min ≤ ω ≤ N 0,min with 0.9371 < c 2 < 0.9372 and 16.80 < L min < 16.81.
Fig. 3 .
3Gravito-inertial ray showing the two types of turning points, a cusp point on the Γ = 0 boundary (in magenta) and a smooth backrefraction in the surface layers where the decreasing density scale height becomes of the order of the wavelength in that direction.
Fig. 4 .
4consists in neglecting the contribution of the latitudinal component of the rotation vector in the Coriolis force. The solutions of the perturbation equation are then separable into the coordinates θ and r. This approximation is justified if motions Article number, From panel (a) to panel (c), three PSS at ω/N 0,max = [0.66676, 0.50200, 0.05522] in a non-rotating polytropic model of star.
Fig. 5 .
5Two PSS at ω/ f = [2.21, 0
Fig. 10 .
10PSS computed at ω/ f = 1.5 for a rotation Ω/Ω K = 0.84 and two gravito-inertial rays belonging to a period-5 island chain including the central stable periodic orbit (light blue).
Fig. 11 .
11PSS computed at ω/ f = 0.5 for a rotation Ω/Ω K = 0.84 and a gravito-inertial ray belonging to an island chain of large (> 26) period.
Fig. 12 .
12PSS computed at ω/ f = 30.5 for a rotation Ω/Ω K = 0.02 showing the development of a small chaotic zone in the vicinity of the separatrix present in the non-rotating case (red curve on panel (a) of Fig. 4).
to rewrite Eqs. (A.1) and (A.3) as ∂ t L(ρ) = −∇ · L(u), (A.9) ∂ t L(P) = c s 2 ∂ t L(ρ) + αg 0 · L(u), (A.10) where L(u) is given by Eq. (A.7) and α, in a polytropic model. Replacing L(u) in Eq. (A.10) by the RHS of Eq. (A.7), we get
G
= ω 2 (ω 2 − f 2 ). (A.15) Article number, page 16 of 24 V. Prat et al.: Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics
2 cos ΘV + ω 2 V z , (A.51) where V and V z are defined by V = V e + V z e z .With this choice for the function a, the wave equation Eq. (A.φ · ∇Ψ + CΨ, (A.52)
U
= c 2 s D = αK + Gc 2 s , (A.56) and w 0 = w 0 e and w 1 are both O(1) with respect to the c 2 s expansion.
Article number, page 2 of 24 V. Prat et al.: Asymptotic theory of gravity modes in rotating stars. I. Ray dynamicsFig. 1. (Top) Solid black line shows the profile of the Brunt-Väisälä frequency N 0 normalised by ω 0 for the non-rotating stellar model. The red lines show N 0 for the Ω = 0.84Ω K model along the polar (solid line) and equatorial (dashed line) radii. A thick red tick on the x-axis indicates the polar radius R p . (Top) Map in a meridional plane of N 0 normalised by ω 0 for the Ω = 0.84Ω K model. The black line is the star surface.0.0
0.2
0.4
0.6
0.8
1.0
r/R e
0
1
2
3
4
N
o /ω
o
0
4ω ο
N ο
The terms super-inertial and sub-inertial are also used in the geophysical literature but with different meanings as they refer to frequencies higher or smaller than 2Ω cos θ.Article number, page 4 of 24 V. Prat et al.: Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics
When comparing modes and rays, one must be careful about the fact that all modes are either symmetric or anti-symmetric with respect to the equator, which is not the case for ray trajectories.Article number, page 14 of 24 V. Prat et al.: Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics
Acknowledgements. The authors thank Michel Rieutord for his careful reading of the manuscript, and an anonymous referee for contributing to the improvement of this paper. This work was supported by the Centre National de la Recherche Scientifique (CNRS) through the Programme National de Physique Stellaire (PNPS).A&A proofs: manuscript no.Grav1.9.4In the end, we get D∆P − (ω 2 − f 2 − N 0 2 ) f · ∇( f · ∇P) + N 0 2 ω 2 g 0 2 g 0 · ∇(g 0 · ∇P)− N 0 2 g 0 2 ( f · g 0 )[ f · ∇( g 0 · ∇P) + g 0 · ∇( f · ∇P)]which is an equation for a single unknown, as we were looking for.Using the relation D∆P = ω 2 (ω 2 − f 2 )∆P − N 2 0 ∆P + N 2 0 ( f · g 0 ) 2 ω 2 g 2 0 ∆P , from the definition of D, Eq. (A.30) is rewritten asTo greatly simplify the terms involving second-order derivatives, one can express ∇P in the (non-orthonormal) basis defined by f , g 0 , and e φ and take the divergence of the resulting equation. It giveswhere we define the following operators and expressions:with p = f · g 0 , q = g 2 0 , n = f 2 and σ = ( f · g 0 ) 2 − f 2 g 2 0 . If in addition, the operators ∇ z and ∆ ⊥ are defined as in Eqs. (9) and (10), the wave equation (A.31) becomeswhere, using the fact that f ∧ g 0 = − f g 0 sin Θe φ ,Here, Eq. (A.41) for pressure perturbations is written in a normal form by eliminating first-order derivatives in the meridional plane. This can be done by introducing a new variableΨ defined byP = aΨ, where the axisymmetric function a is chosen to eliminate the first-order term V · ∇P. To carry out this substitution, we use the following relations: ∆P = a∆Ψ + 2∇a · ∇Ψ +Ψ∆a, (A.42) ∇ 2 P = a∇ 2 Ψ + 2(∇ a)(∇ Ψ ) +Ψ∇ 2 a, (A.43) ∇ 2 zP = a∇ 2 zΨ + 2(∇ z a)(∇ zΨ ) +Ψ∇ 2 z a, (A.44) A&A proofs: manuscript no.Grav1.9.4The dominant terms in the expression (A.54) of C areSumming these terms we obtain, in the K 0 case,As shown below in Sect. A.4, the calculation of w 0 leads toIn the sub-inertial regime (ω < f ), the dominant term of C vanishes at the critical angles Θ c and π − Θ c such that cos Θ c = ω/ f . Coming back to the full expression of C, Eq. (A.54), it can be shown thatand that C 1 does not vanish when K = 0. Thus, as one approaches the surface along the critical angles, the dominant term in the expression of C is given byIn practice, we found that rays do not reach the surface layers near the critical angle. This can be understood as the dispersion relation and the ray equations show that near the critical angle, k ⊥ vanishes, and this implies that dr(θ)/dθ vanishes as well.Appendix A.4: The w 0 term According to Eqs. (A.55) and (A.49), w 0 is the dominant term of c 2 s UW . From the definitions of W and V , we obtainThis expression is then developed usingThe expression of w 0 becomes 2G αω 2 g 2(A.78)Developing the last expression, we show that(A.81)Appendix B: The ray dynamics equationsIn this section, the ray dynamics equations for axisymmetric rays are written using the spherical coordinates [r, θ] and the wavevector components [k r , k θ ] on the usual orthonormal basis e r , e θ . To derive them, it is convenient to start from the general Hamiltonian equations(28)and(29)for spherical coordinates[r, θ]. These equations are expressed in terms of the covariant components of k on the natural basis (k nat r , k nat θ ), and a change of variables from [r, θ, k nat r , k nat θ ] to [r, θ, k r , k θ ] is necessary. From the relations k r = k nat r and k θ = k nat θ /r, Eqs.(28)and(29)take the following form:where the Hamiltonian H(r, θ, k r , k θ ) is given by the eikonal equation written in the form ω = H(r, θ, k r , k θ ).To develop the previous equations, one possible way is to start from the original eikonal equation(18)in the case k φ = 0 written as(B.5)and to calculate the partial derivatives of this expression with respect to the coordinates [r, θ, k r , k θ ]. To do so, one needs to express k z and k ⊥ as functions of [r, θ, k r , k θ ], namelywhere the last relation has been obtained from e = cos Θ e z + sin Θ e s , (B.8) e ⊥ = − sin Θ e z + cos Θ e s , (B.9) e ⊥ · e r = sin(θ − Θ), (B.10) e ⊥ · e θ = cos(θ − Θ), (B.11) V. Prat et al.: Asymptotic theory of gravity modes in rotating stars. I. Ray dynamics Similar calculations enable us to derive the expression (31). Indeed, using the first two equations of the Hamiltonian system (B.1) and (B.2) and the expression of e in Eq. (B.8), we haveThen, from Eqs. (C.2), (C.3), and (C.5)-(C.8), we findSolving the eikonal equation(20)as a quadratic equation for the variable k , we findwhere δ has been already defined by Eq.(22). Then, the relation (31) is obtained replacing k z by k z = k cos Θ−k ⊥ sin Θ in Eq. (C.12). From the difference of Eq. (30) and Eq. (31), an expression for dx/dt · e ⊥ is obtained. Using the above expression of k ± and Eq.(22), we derive Eq. (32).Appendix D: Test of the numerical method on a regular cusp pointWe considered the simple case of gravity rays in a plane-parallel atmosphere characterised by an exponential variation of the Brunt-Väisälä frequency in the z direction. The ray dynamics equations derived from the dispersion relation ωwhere k x is constant, k z = k z0 − γωt, and γ = d ln N 0 /dz. An analytical solution of these equations going through a regular cusp point of coordinates (x r , z r ) iswhere the time variable t has been replaced by u = k z /k x . The cusp point is reached at u = 0. The code used in the present paper was able to reproduce this analytical solution with a controlled level of accuracy.Appendix E: Unstable fixed point of the non-rotating radial ray dynamicsIn non-rotating stars, the ray dynamics equations can be solved separately in the radial and latitudinal coordinates. The radial dynamics is governed by Eq. (21). For some values of the frequency, the phase portrait of the radial dynamics is typical of doublewell-potential systems with two elliptic regions around the two fixed points at the minima of the potential separated from high-energy motions by a separatrix that goes through the unstable hyperbolic fixed point at the maximum of the potential. This is illustrated by panel (a) ofFig. 4. Equation(21)can be seen as a classical Hamiltonian H r = 1/2k 2 r + V r with H r = 0 and V r a potential that depends on two parameters, ω and L. In our case, the separatrix separates low-L from high-L motions. When such a system is moved away from integrability, chaos appears first near the unstable hyperbolic point. Here, we show that using an envelope model we can analytically determine the frequency range where the non-rotating radial dynamics shows a double-well-potential behaviour as well as the separatrix and the position of the unstable point.In envelope models of stars, the radial dependence of the mass inside a given radius is neglected. Within this approximation, the hydrostatic equation can be integrated to provide simple expressions for the profile of the thermodynamic quantities(Cox 1968). In particular, the Brunt-Väisälä frequency can be written as, (E.1) and can be shown to possess a local minimum N 0,min at r/R = 0.75. This local minimum is expected to be a generic feature of radiative stellar envelopes. It is at the origin of the existence of an unstable fixed point of the ray dynamics. Using Eq. (E.1), Eq. (21) governing the radial dynamics becomes
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| []
|
[
"The lifetime in the Standard Model The lifetime in the Standard Model",
"The lifetime in the Standard Model The lifetime in the Standard Model"
]
| [
"Jason Aebischer [email protected] ",
"Benjamín Grinstein [email protected] ",
"Jason Aebischer ",
"\nPhysik-Institut\nDepartment of Physics\nUniversität Zürich\nCH-8057ZürichSwitzerland\n",
"\nUniversity of California at San Diego\nLa Jolla92093CAUSA\n"
]
| [
"Physik-Institut\nDepartment of Physics\nUniversität Zürich\nCH-8057ZürichSwitzerland",
"University of California at San Diego\nLa Jolla92093CAUSA"
]
| []
| Using an operator product expansion (OPE) approach an updated Standard Model prediction of the lifetime is presented. The computation in three different mass schemes for the heavy quarks leads to three different values consistent with each other and with experiment. Furthermore a novel way to compute the lifetime is presented, taking differences of , and meson decay rates. In this approach the leading contributions from free-quark decays cancel out, leading to a reduction of scale and scheme dependence. | 10.22323/1.380.0145 | [
"https://arxiv.org/pdf/2111.07076v1.pdf"
]
| 244,117,535 | 2111.07076 | 1944819c7bd392d7a3dafef5a5f9e8a2d6e8847d |
The lifetime in the Standard Model The lifetime in the Standard Model
13 Nov 2021
Jason Aebischer [email protected]
Benjamín Grinstein [email protected]
Jason Aebischer
Physik-Institut
Department of Physics
Universität Zürich
CH-8057ZürichSwitzerland
University of California at San Diego
La Jolla92093CAUSA
The lifetime in the Standard Model The lifetime in the Standard Model
13 Nov 2021*** Particles and Nuclei International Conference -PANIC2021 *** *** 5 -10 September, 2021 *** *** Online *** * Speaker Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/
Using an operator product expansion (OPE) approach an updated Standard Model prediction of the lifetime is presented. The computation in three different mass schemes for the heavy quarks leads to three different values consistent with each other and with experiment. Furthermore a novel way to compute the lifetime is presented, taking differences of , and meson decay rates. In this approach the leading contributions from free-quark decays cancel out, leading to a reduction of scale and scheme dependence.
Introduction
The lifetime of the = ( ) meson is an interesting observable, as it allows to put strong constraints on New Physics models such as for instance scalar Leptoquarks or Two-Higgs-Doublet models [1][2][3]. Such models are particularly interesting regarding the charged-current -anomalies seen in and * . The lifetime of the is measured with a small uncertainty by the LHCb [4,5] and CMS [6] collaborations and averages to exp = 0.510 (9)ps .
(1)
From the theory side, different approaches can be employed to determine the lifetime, such as QCD Sum Rules [7], Potential models [8] or Operator Product Expansion (OPE) methods [9][10][11]. The three approaches all predict to be in the ballpark of the experimental value in Eq. (1). In the following we will adopt the approach introduced by Beneke and Buchalla in [9] to compute using Effective Field Theory methods involving the OPE, the effective Hamiltonian as well as Non-Relativistic QCD (NRQCD). It can be summarized as follows: In a first step the heavy degrees of the Standard Model (SM) are integrated out at the electroweak scale to obtain the effective Hamiltonian H eff contributing to the lifetime. In a next step, the Wilson coefficients of the effective operators in H eff are run down to the low scale low by solving their renormalization group equations. This low scale is chosen to be of the order of the -quark mass for -decays and of order for -quarks decays. At these scales an OPE of the transition operator T relevant for the decay is performed. The transition operator is related via the optical theorem to the decay rate of the in the following way:
Γ = 1 2 |T | ,(2)
where the transition operator is the imaginary part of the time-ordered product of two insertions of the effective Hamiltonian
T = Im ∫ 4 H eff ( )H eff (0) .(3)
The products of two operators from H eff at different space-time points are then expanded in the OPE, i.e. written in terms of a series of local operators. The relevant contributions for T can be split into contributions resulting from the -and -quark decays, as well as from Weak Annihilation (WA) and Pauli interference (PI) diagrams:
T = T + T + T WA + T PI .(4)
The first two contributions result from spectator decays of the corresponding quarks inside the meson, generating the free-quark decay contribution (at dimension three) as well as dimensionfive chromomagnetic dipole operators. The WA and PI contributions correspond to dimension-six operators in the OPE. In spite of their higher mass suppression in the expansion their contributions are retained, since they are generated through one-loop diagrams and hence have an enhancement factor of 16 2 compensating the suppression.
The lifetime in the Standard Model
Jason Aebischer
After performing the OPE at the low scale low the QCD fields are expressed in terms of NRQCD fields. Subsequently, a velocity expansion in the small quark velocities of the nonrelativistic quarks is performed. The matrix elements of the resulting operators can be estimated for the leading operators like the kinetic term, the Fermi-and the Darwin term using potential models [12]. In estimating the matrix elements of the four-quark operators spin-symmetry has been used, which relates all matrix elements of the four-fermi operators from WA and PI to a single reduced matrix element.
Result
In this section we present the results for the final decay rates of the , which are discussed in detail in [13]. Assuming a non-zero strange quark mass and computing in three different mass schemes, namely the MSbar scheme, the meson scheme and the Uspilon scheme we find for the decay rates
Γ MS = (1.51 ± 0.38| ± 0.08| n.p. ± 0.02| ± 0.01| ± 0.01| ) ps −1 , Γ meson = (1.70 ± 0.24| ± 0.20| n.p. ± 0.01| ± 0.01| ) ps −1 , Γ Upsilon = (2.40 ± 0.19| ± 0.21| n.p. ± 0.01| ± 0.01| ) ps −1 .(5)
The largest uncertainties in all three schemes stem from residual dependence on the renormalization scale , which can be reduced by taking into account higher-order QCD corrections. Further uncertainties result from non-perturbative (n.p.) corrections. These can be reduced by taking into account higher order corrections in the velocity expansion as well as by having better estimates for the matrix elements, preferably from Lattice calculations. The smallest uncertainties are due to parametric uncertainties of the strange-quark mass and the CKM element . When the strange quark mass is neglected, the central values of the decay rates are enhanced by ∼ 7%.
When combining all the different uncertainties the results in the three schemes in Eq. (5) are compatible with each other and also with the experimental value, which is derived from Eq. (1) to be Γ exp = 1.961(35) ps −1 .
The large spread in the central values obtained using different mass schemes calls however for a computation which includes higher-order QCD-as well as non-perturbative corrections in order to decrease the apparent differences.
New method to determine Γ
In this section we describe a novel way [14] on how to determine the decay rate, using differences of heavy meson decay rates. Generally, the decay rate of a heavy meson with heavy quark can be written as
Γ( ) = Γ (0) + Γ . . ( ) + Γ WA+PI ( ) + O ( 1 4 ) ,(7)
The lifetime in the Standard Model Jason Aebischer 0 , 0 + , 0 0 , + + , + Γ meson 3.03 ± 0.51 3.03 ± 0.53 3.33 ± 1.29 3.33 ± 1.32 where the leading contribution Γ (0) denotes the free quark decay rate, Γ . . results from nonperturbative corrections and Γ WA,PI denote the WA and PI contributions. Taking now the difference of decay rates for the , and mesons, and applying the formula in Eq. (7) one finds
Γ( ) + Γ( ) − Γ( ) = Γ . . ( ) + Γ . . ( ) − Γ . . ( ) + Γ WA+PI ( ) + Γ WA+PI ( ) − Γ WA+PI ( ) .(8)
The right-hand side of Eq. (8) can be computed in a similar way as discussed in the introduction with the only difference that for the and mesons Heavy Quark Effective Theory instead of NRQCD is employed. On the left-hand side of Eq. (8) the decay rates Γ( ) and Γ( ) can be taken from experiment, which then allows to express Γ( ) in terms of known/calculable quantities.
The main advantage of Eq. (8) is that the free-quark decay contribution drops out in the difference together with it's uncertainties. Furthermore, one can use this relation for either charged or neutral mesons, leading to four possible ways to predict the decay rate. For the meson scheme and the four different combinations of mesons used in the relation we report the results in Tab. 1. The obtained results again show some deviation compared to the experimental value in Eq. (6). Several possibilities might serve as a solution to resolve this discrepancy: A first explanation would be the underestimation of the uncertainties from NLO corrections to the Wilson coefficients as well as non-perturbative corrections in our computation. Secondly eye-graph contributions, which are generally neglected in lattice computations of the used matrix elements, might have a sizable impact on the result. Furthermore, as pointed out in [15] neglected dimension-seven contributions in the charm decays can have a large impact. Finally quark-hadron duality might be violated, in which case a completely new method to compute the decay rates of heavy mesons might be in order.
Summary
We present an updated computation of the lifetime in the SM using an EFT approach involving the effective Hamiltonian, the OPE as well as NRQCD. The obtained values in three different mass schemes are compatible with each other and with the experimental value within the given uncertainties. The large scheme dependence of the result, manifesting itself in a large spread of the central values ranging from 1.51 ps −1 to 2.40 ps −1 , calls however for a determination including higher order QCD corrections.
Furthermore a novel way on how to determine the lifetime is presented, taking differences of , and decay rates. The method leads to values that exceed the experimental value, which might have several reasons, like the underestimation of uncertainties, the neglected eye-graph and dimension-seven terms or even the violation of quark-hadron duality.
Table 1 :
1Results obtained for the decay rate Γ( ) in ps −1 in the meson scheme, using the different combinations of and mesons in Eq.(8).
Lifetime of − Constrains Explanations for Anomalies in → ( * ). Rodrigo Alonso, Benjamín Grinstein, Jorge Martin Camalich, Phys. Rev. Lett. 118881802Rodrigo Alonso, Benjamín Grinstein, and Jorge Martin Camalich. Lifetime of − Constrains Explana- tions for Anomalies in → ( * ) . Phys. Rev. Lett., 118(8):081802, 2017.
Impact of polarization observables and → on new physics explanations of the → anomaly. Monika Blanke, Andreas Crivellin, Stefan De Boer, Teppei Kitahara, Marta Moscati, Ulrich Nierste, Ivan Nišandžić, Phys. Rev. D. 99775006Monika Blanke, Andreas Crivellin, Stefan de Boer, Teppei Kitahara, Marta Moscati, Ulrich Nierste, and Ivan Nišandžić. Impact of polarization observables and → on new physics explanations of the → anomaly. Phys. Rev. D, 99(7):075006, 2019.
Addendum to "Impact of polarization observables and → on new physics explanations of the → anomaly. Monika Blanke, Andreas Crivellin, Teppei Kitahara, Marta Moscati, Ulrich Nierste, Ivan Nišandžić, Phys.Rev.D. 100535035Monika Blanke, Andreas Crivellin, Teppei Kitahara, Marta Moscati, Ulrich Nierste, and Ivan Nišandžić. Addendum to "Impact of polarization observables and → on new physics explanations of the → anomaly". 5 2019. [Addendum: Phys.Rev.D 100, 035035 (2019)].
Measurement of the + meson lifetime using + → / + decays. Roel Aaij, Eur. Phys. J. Roel Aaij et al. Measurement of the + meson lifetime using + → / + decays. Eur. Phys. J.
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Measurement of the lifetime of the + meson using the + → / + decay mode. Roel Aaij, Phys. Lett. B. 742Roel Aaij et al. Measurement of the lifetime of the + meson using the + → / + decay mode. Phys. Lett. B, 742:29-37, 2015.
Measurement of b hadron lifetimes in pp collisions at √ = 8 TeV. Albert M Sirunyan, Eur. Phys. Albert M Sirunyan et al. Measurement of b hadron lifetimes in pp collisions at √ = 8 TeV. Eur. Phys.
. Eur.Phys.J.C. 786561CC, 78(6):457, 2018. [Erratum: Eur.Phys.J.C 78, 561 (2018)].
Likhoded. decays and lifetime in QCD sum rules. V V Kiselev, A E Kovalsky, A K , Nucl. Phys. B. 585V.V. Kiselev, A.E. Kovalsky, and A.K. Likhoded. decays and lifetime in QCD sum rules. Nucl. Phys. B, 585:353-382, 2000.
. S S Gershtein, V V Kiselev, A K Likhoded, A V Tkabladze, Physics of B(c) mesons. Phys. Usp. 38S.S. Gershtein, V.V. Kiselev, A.K. Likhoded, and A.V. Tkabladze. Physics of B(c) mesons. Phys. Usp., 38:1-37, 1995.
The Meson Lifetime. Martin Beneke, Gerhard Buchalla, Phys. Rev. D. 53Martin Beneke and Gerhard Buchalla. The Meson Lifetime. Phys. Rev. D, 53:4991-5000, 1996.
Inclusive B(c) decays as a QCD lab. I Y Ikaros, Bigi, Phys. Lett. B. 371Ikaros I.Y. Bigi. Inclusive B(c) decays as a QCD lab. Phys. Lett. B, 371:105-110, 1996.
The Lifetime of meson and some relevant problems. Chao-Hsi Chang, Shao-Long Chen, Tai-Fu Feng, Xue-Qian Li, Phys. Rev. D. 6414003Chao-Hsi Chang, Shao-Long Chen, Tai-Fu Feng, and Xue-Qian Li. The Lifetime of meson and some relevant problems. Phys. Rev. D, 64:014003, 2001.
B(c) spectroscopy. S S Gershtein, V V Kiselev, A K Likhoded, A V Tkabladze, Phys. Rev. D. 51S. S. Gershtein, V. V. Kiselev, A. K. Likhoded, and A. V. Tkabladze. B(c) spectroscopy. Phys. Rev. D, 51:3613-3627, 1995.
Standard Model prediction of the lifetime. Jason Aebischer, Benjamín Grinstein, 52021Jason Aebischer and Benjamín Grinstein. Standard Model prediction of the lifetime. 5 2021.
A novel determination of the lifetime. Jason Aebischer, Benjamín Grinstein, 82021Jason Aebischer and Benjamín Grinstein. A novel determination of the lifetime. 8 2021.
Revisiting Inclusive Decay Widths of Charmed Mesons. Daniel King, Alexander Lenz, Maria Laura Piscopo, Thomas Rauh, Aleksey V Rusov, Christos Vlahos, 92021Daniel King, Alexander Lenz, Maria Laura Piscopo, Thomas Rauh, Aleksey V. Rusov, and Christos Vlahos. Revisiting Inclusive Decay Widths of Charmed Mesons. 9 2021.
| []
|
[
"Spin dependent photoelectron tunnelling from GaAs into magnetic Cobalt",
"Spin dependent photoelectron tunnelling from GaAs into magnetic Cobalt",
"Spin dependent photoelectron tunnelling from GaAs into magnetic Cobalt",
"Spin dependent photoelectron tunnelling from GaAs into magnetic Cobalt"
]
| [
"D Vu ",
"H F Jurca ",
"F Maroun ",
"P Allongue ",
"N Tournerie ",
"A C H Rowe ",
"D Paget ",
"S Arscott ",
"E Peytavit ",
"\nPhysique de la matière condensée\nEcole Polytechnique\nCNRS\n91128PalaiseauFrance\n",
"\nInstitut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN), CNRS UMR8520, Avenue Poincaré, Cité Scientifique\n59652Villeneuve d'AscqFrance\n",
"D Vu ",
"H F Jurca ",
"F Maroun ",
"P Allongue ",
"N Tournerie ",
"A C H Rowe ",
"D Paget ",
"S Arscott ",
"E Peytavit ",
"\nPhysique de la matière condensée\nEcole Polytechnique\nCNRS\n91128PalaiseauFrance\n",
"\nInstitut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN), CNRS UMR8520, Avenue Poincaré, Cité Scientifique\n59652Villeneuve d'AscqFrance\n"
]
| [
"Physique de la matière condensée\nEcole Polytechnique\nCNRS\n91128PalaiseauFrance",
"Institut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN), CNRS UMR8520, Avenue Poincaré, Cité Scientifique\n59652Villeneuve d'AscqFrance",
"Physique de la matière condensée\nEcole Polytechnique\nCNRS\n91128PalaiseauFrance",
"Institut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN), CNRS UMR8520, Avenue Poincaré, Cité Scientifique\n59652Villeneuve d'AscqFrance"
]
| []
| The spin dependence of the photoelectron tunnel current from free standing GaAs films into outof-plane magnetized Cobalt films is demonstrated. The measured spin asymmetry (A) resulting from a change in light helicity, reaches ±6% around zero applied tunnel bias and drops to ±2% at a bias of -1.6 V applied to the GaAs. This decrease is a result of the drop in the photoelectron spin polarization that results from a reduction in the GaAs surface recombination velocity. The sign of A changes with that of the Cobalt magnetization direction. In contrast, on a (nonmagnetic) Gold film A ≈ 0%. | 10.1103/physrevb.83.121304 | [
"https://arxiv.org/pdf/1008.1403v2.pdf"
]
| 119,226,105 | 1008.1403 | 72779e6592c9533107c65910f64321c958546a8b |
Spin dependent photoelectron tunnelling from GaAs into magnetic Cobalt
16 Aug 2010
D Vu
H F Jurca
F Maroun
P Allongue
N Tournerie
A C H Rowe
D Paget
S Arscott
E Peytavit
Physique de la matière condensée
Ecole Polytechnique
CNRS
91128PalaiseauFrance
Institut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN), CNRS UMR8520, Avenue Poincaré, Cité Scientifique
59652Villeneuve d'AscqFrance
Spin dependent photoelectron tunnelling from GaAs into magnetic Cobalt
16 Aug 2010
The spin dependence of the photoelectron tunnel current from free standing GaAs films into outof-plane magnetized Cobalt films is demonstrated. The measured spin asymmetry (A) resulting from a change in light helicity, reaches ±6% around zero applied tunnel bias and drops to ±2% at a bias of -1.6 V applied to the GaAs. This decrease is a result of the drop in the photoelectron spin polarization that results from a reduction in the GaAs surface recombination velocity. The sign of A changes with that of the Cobalt magnetization direction. In contrast, on a (nonmagnetic) Gold film A ≈ 0%.
Since the initial discovery of spin dependent tunnelling between a magnetic metal and a superconductor [1] and subsequently between two magnetic metals, [2,3] spin dependent tunnelling has been extensively studied in fixed, all-solid junctions. This is because such studies reveal details of surface magnetism and also because tunnel junctions, in particular metallic magnetic tunnel junctions, [4] are technologically important. [5] Tunnelling from ferromagnetic and anti-ferromagnetic tips has been successfully employed to observe magnetic ordering in metals down to the atomic scale. [6] Similarly, spin polarized tunnelling from ferromagnetic metals and ferromagnetic semiconductors into nonmagnetic semiconductors has also been reported in both all-solid junctions [7] and from a ferromagnetic tip. [8] In these cases the transient spin polarization of the post-tunnel electrons is measured via the circular polarization of the resulting luminescence. In principle the reverse process should also be possible. The tunnel current of spin polarized photoelectrons into a ferromagnetic surface should depend on the relative orientations of the photoelectron spin to the surface magnetization. This phenomenon was the basis of Pierce's proposal for GaAs tip spin polarized scanning tunnelling microscopy (SPSTM). [9] However, despite significant experimental work, [10,11] the effect has never been convincingly demonstrated, with experimental difficulties attributed to parasitic optical effects yielding apparent spin dependent tunnelling, even on nonmagnetic surfaces. [11,12] Here we demonstrate the spin dependence of the tunnel photocurrent, I ph t (σ ± ), from p + GaAs under circularlypolarized light excitation into ultra-thin Cobalt films magnetized out-of-plane.
In constrast to previous works [10][11][12] spin-polarized electron injection is performed from epitaxial lift-off thin GaAs films deposited using an original microfluidic method on pre-metallized quartz, with an overhanging cantilever of 65µm length (see bottom inset, Fig. 1). [13] As shown in the upper inset of Fig. 1, the photocarriers are generated at the rear (non tunnel) surface, and then diffuse across the film before tunnelling (the film thickness, of 3µm, is comparable with the charge and spin diffusion lengths for a doping level of N A ≈ 10 18 cm −3 and larger than the absorption depth, 1µm, for the hν = 1.59 eV pump light used here). The cantilevers are pressed into mechanical contact with the metal surface, as detected using the reflected part of the incident laser beam with a quadrant photodiode, so that tunnelling of photoelectrons occurs over a relatively large contact area through an interfacial oxide layer of homogeneous thickness. This simple, onedimensional geometry i) avoids poorly controlled direct light excitation at the tip apex, [12] ii) results in a photocurrent which, unlike front surface excitation, [14] does not directly depend on tunnel bias, iii) reduces instabilities due to changes of interfacial chemistry observed for tunnelling from tips, [15] and provides a stable tunnel interface for up to 30 minutes in air at room temperature.
Tunnel injection was performed into an ultrathin Co(0001) layer (thickness ≈ 5 monolayers) epitaxially grown by electrodeposition on an atomically flat Au(111) buffer layer on Si(111). [17] The Co surface was passivated by chemisorbing CO which renders the surface resistant to oxidation in dry air and quenches empty surface states. [18] As shown ( Fig. 2A) by the square magnetization loop measured with the field applied perpendicular to the surface (using the polar magneto optical Kerr effect) averaged over 1 mm 2 , these passivated Co/Au(111) ultra-thin films present a strong perpendicular anisotropy with a coercive field smaller than 200 Oe. The full zero field remanence of the magnetization after application of a magnetic field larger than the coercive field, indicates that the sample is essentially composed of a single domain whose lateral extent is larger than the contact area through which tunnelling occurs. The photoelectron polarization in the cantilevers has been analyzed using polarized luminescence (PL). The σ ± polarized PL spectra [I P L (σ ± )] of the cantilever at a low light intensity of 50 W/cm 2 (hν = 1.59 eV) are shown in curves a and b of Fig. 2B, respectively. As known for p + GaAs, the structure near 1.39 eV is due to acceptor-related recombination [19] and that the above bandgap luminescence degree of circular polarization, [I P L (σ + ) − I P L (σ − )]/[I P L (σ + ) + I P L (σ − )] is equal to 8% as seen from curve c. This polarization corresponds to an average over all photo-electrons in the cantilever. Using this value and by numerically solving the spin diffusion equation, a spin polarization of tunnelling electrons of the order of 16% can be inferred [13] as well as a spinlattice relaxation time for conduction electrons of 0.16 ns, in good agreement with independent measurements on doped GaAs. [20] For the investigation of spin dependent tunnelling, the circular polarization of the pump light excitation (5 mW focussed to a spot of about 10µm diameter) is switched by a Pockels' cell. A measurement cycle consists of the following phases: i) The tunnel current is stabilized at 11 nA in the dark by the feedback loop for a GaAs bias of -1.5 V. ii) The feedback loop is opened and two bias scans of duration 12 ms are performed. One scan is performed in the dark and the other one under σ + illumination. The tunnel photocurrent I ph t (σ + ) is obtained by difference. iii) After a new stabilization sequence, two bias scans are again taken, one in the dark and the other one with a σ − polarized laser. This procedure, lasting about 0.25 s, gives the bias dependence of the spin asymmetry factor A, defined by
A = [I ph t (σ + ) − I ph t (σ − )]/[I ph t (σ + ) + I ph t (σ − )]. A may also be written [2] A = δρ m ρ m δn s n s ,(1)
where δX symbolizes the difference of the quantity X between + and -spins, quantized along the direction of light excitation. ρ m and n s are respectively the total metallic density of states at the tunnel energy and the concentration of the tunnelling electrons. Using, as shown above, δn s /n s ≈ 16%, and δρ m /ρ m ≈ 70% about 1 eV above the Fermi energy [18], an asymmetry of the order of 10 % is anticipated using Eq. 1.
The results averaged over 100 measurement cycles are shown in Fig. 1 and Fig. 3. Curve a of Fig. 1 shows the dependence of the dark current as a function of reverse bias applied to the GaAs cantilever and curve b shows that of the additional current, I ph t , induced by the light excitation. This current increases nonexponentially up to about 100 nA. Curve a of Fig. 3 shows that A varies from 6% at zero bias to 2% at a reverse bias of -1.6 V. The non zero value of A is due to a spin dependence of the tunnelling current since i) reversal of the magnetization of the Cobalt layer by transient application of a magnetic field larger than the corecive field induces a change of sign of the asymmetry without any significant modification of either the absolute value or the bias dependence (curve b in Fig. 2), and ii) measurements on (nonmagnetic) Gold films result in an asymmetry that is always smaller than 1% (curve c) and approximately 0% for zero bias. Moreover the measured asymmetry is similar to the above rough estimate.
A more quantitative interpretation of these results uses a general model recently developed for tunnel injection of photoelectrons into metals. [16] The excellent agreement between the calculated (red lines, Fig. 3) and measured bias dependences indicates that the dominant contribution to the tunnel photocurrent comes from con-duction electrons. The injection energy is almost biasindependent and close to that of the bottom of the conduction band in the bulk since the energy loss, (1 − f )ϕ b , in the depletion layer (see Fig. 4) is smaller than 150 meV. (The surface barrier ϕ b ≈ 0.3eV under light excitation and the numerical factor f is larger than about 0.5 because of surface quantization.) The energy dependence of the total Cobalt density of empty states at this injection energy cannot explain the nonexponential bias dependence of I ph t . [18] In the same way, as shown in curve d of Fig. 3, δρ m /ρ m calculated using the known spin dependent density of empty states, only decreases by 25% which, using Eq. 1, cannot explain the measured bias dependence of A. The decrease of A must therefore be dominated by δn s /n s .
The decrease of δn s /n s and the nonexponential increase of the tunnel photocurrent are caused by the same effect, namely unpinning of the surface Fermi level. [16] As seen in Fig. 4, the application of a bias changes the semiconductor surface charge and shifts the electron quasi-Fermi level away from midgap by a quantity ∆ϕ which is obtained by charge neutrality. [16] The surface recombination velocity is S = S 0 exp(−∆ϕ/k B T )/D(∆ϕ) where S 0 is the value of S for ∆ϕ=0 and D(∆ϕ) is the relative decrease of the density of surface states. [21] The resulting bias-induced decrease of S results in an increase of the effective lifetime of the tunnelling electrons which increases their concentration and increases the spin polarization losses by spin-lattice relaxation.
The tunnel photocurrent is proportional to n s and to the tunnel probability, for which the expressions are found in Ref. 16. Calculation of δn s /n s is performed by solving the equations for spin and charge diffusion [22] from the rear surface to the plane of injection. For a cantilever of thickness l, in the limit of large recombination at the rear surface and of small absorption length, one finds δn s n s = ±0.5 τ s τ sinh(l/L) sinh(l/L s )
1 + S/v d 1 + aS/v d(2)
for σ ∓ polarized light excitation, respectively. Here τ and τ s are the bulk electron lifetime and spin lifetime, L and L s are the charge and spin diffusion lengths, v d = (D/L) coth(l/L) is the effective charge diffusion velocity, D is the diffusion constant and a = (L s /L) coth(l/L)/ coth(l/L s ) is the ratio of v d to the equivalent spin diffusion velocity (here a < 1 since L s < L).
The bias dependences of the tunnel photocurrent, of the dark current and of A are calculated using the model of Ref. 16. The work function for passivated Cobalt is 6 eV, [23] and the dielectric constant of the tunnel gap is equal to 10, close to that of both Gallium Oxide [24] and Cobalt Oxide. [25] The spin diffusion length is 0.6µm, i.e. close to independent estimates. [26] As in Ref. 16, other parameters for non polarized tunnelling have values taken from the literature. Good agreement with the data is obtained for 0.6 < f < 1 when the tunnel distance is adjusted between 0.6 nm and 0.75 nm. The calculated curves in Fig. 1 and Fig. 3 correspond to f ≈ 0.9 and d = 0.74 nm. As seen in Fig. 1, the calculation correctly predicts the bias dependence of the tunnel dark current and photocurrent. Note that these dependences appear to be quite similar since both are determined by the degree of unpinning of the semiconductor surface Fermi level. Curve e of Fig.3 shows the calculated decrease of the polarization of injected electrons. The bias dependence of A calculated using Eq. 2 is shown in curve f and agrees very well with the measured dependence. The zero bias asymmetry is also well accounted for, and is smaller than the rough estimate made above because ∆ϕ is non negligible for the high excitation intensities used here.
We have neglected here the spin dependence of the photovoltage and therefore of ∆ϕ, [10] caused by spin injection into the subsurface depletion layer. This should induce a spin dependence of the surface recombination velocity which, as for bulk spin dependent recombination, [27] increases δn s /n s . Conservation of spin currents shows that the relative change of δn s /n s depends on the balance between the spin lattice relaxation (time T 1s ) and the lifetime of electrons trapped at surface centers. An upper limit for this effect, found by taking for T 1s equal to the spin relaxation time of conduction electrons (0.16 ns), and a hole capture cross section σ p = 2 × 10 −18 m 2 equal to the maximum value obtained for a large variety of midgap centers, [28] indicates that the relative modification of the spin asymmetry is less than 10 −3 . Finally, a possible spin dependence of the tunnel matrix element has also been neglected. While such a dependence is unknown, the good agreement between the model and the experimental results of Fig. 2 indicates that it does not play a crucial role.
In conclusion, the spin dependence of the tunnel current of conduction photoelectrons into a magnetic metal has been clearly demonstrated. In mechanical contact, the bias dependence of A is caused by the decrease of the electron spin polarization due to the decrease of the surface recombination velocity resulting from the unpinning of the quasi electron Fermi level. Spin injection concerns electrons of well-defined energy (comparable with k B T ) and this observation may finally, for larger tunnel distances where the surface recombination velocity is nearly constant, [16] open the way to spin-dependent tunnelling spectroscopy (SPSTS) and SPSTM of magnetic metals as proposed by Pierce more than 20 years ago. [9]
FIG. 1 :
1The top right schematic describes the principle of the experiment in which photoelectrons injected from the rear face of a free standing GaAs layer diffuse to the front face before tunnelling. Also shown, bottom left, is an optical microscope image of the overhanging GaAs layer deposited onto a metallized quartz substrate. Curves a and b correspond to the tunnel dark and photocurrent bias dependences for tunnelling into Cobalt, repectively. The solid, red lines correspond to the calculations of the tunnel currents using a model (Ref.16) describing tunnelling of photoelectrons into metals.
FIG. 2 :
2(A) The magnetic field dependence of the magnetization perpendicular to the surface of the Cobalt film as measured using the polar magneto-optical Kerr effect. (B) Curves a and b show the spectra of the σ ± polarized components of the cantilever luminescence under circularly-polarized excitation. Curve c shows the polarization of the spectrum, about 8% for band-to-band emission.
FIG. 3 :
3Curves a and b show the measured bias dependence of the spin asymmetry of the tunnel photocurrent into magnetized Cobalt before and after reversal of the magnetization by the transient application of a magnetic field. Curve c is the asymmetry measured on a nonmagnetic Gold surface. The calculated spin dependence of the metallic density of states and of the photoelectron spin polarization, after division by factors of 10 and 1.3 respectively, are shown in curves d and e. The calculated asymmetry, shown in curve f, is in excellent agreement the measured dependence.
FIG. 4 :
4Energy band structure for spin polarized tunnelling into Cobalt. The injection energy Eg − (1 − f )ϕ b is shown along with realistic representations of the densities of states of the majority (black) and minority (red) spins. The black (gray) lines represent the case where the applied bias is 0V (-1 V) and indicates that the spin-polarized conduction electrons are injected into the top half of the 3d minority spin band. The top left (right) inset shows the semiconductor surface density of states at an applied bias of 0V (-1 V).
ACKNOWLEDGEMENTSThis work was partially supported by the Agence National de la Recherche (ANR), SPINJECT 06-BLAN-0253.
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| []
|
[
"Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula",
"Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula"
]
| [
"Yohsuke T Fukai \nRIKEN Center for Biosystems Dynamics Research\n\n\nDepartment of Physics\nthe University of Tokyo\n\n",
"Kazumasa A Takeuchi \nDepartment of Physics\nthe University of Tokyo\n\n"
]
| [
"RIKEN Center for Biosystems Dynamics Research\n",
"Department of Physics\nthe University of Tokyo\n",
"Department of Physics\nthe University of Tokyo\n"
]
| []
| We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments of liquid-crystal turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions. arXiv:1909.11920v3 [cond-mat.stat-mech] | 10.1103/physrevlett.124.060601 | [
"https://arxiv.org/pdf/1909.11920v3.pdf"
]
| 202,888,880 | 1909.11920 | f8093709c8d83f4e25ff00f53d29db7611e61588 |
Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
(Dated: February 13, 2020)
Yohsuke T Fukai
RIKEN Center for Biosystems Dynamics Research
Department of Physics
the University of Tokyo
Kazumasa A Takeuchi
Department of Physics
the University of Tokyo
Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
(Dated: February 13, 2020)
We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments of liquid-crystal turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions. arXiv:1909.11920v3 [cond-mat.stat-mech]
Efforts on universal behavior associated with scale invariance, which have established important concepts such as the renormalization group and the universality class, now shed light on novel aspects of nonequilibrium fluctuations. In this respect, the Kardar-Parisi-Zhang (KPZ) universality class [1][2][3][4] plays a distinguished role, because of the existence of exact solutions and experimental realizations. The KPZ class is also known to arise in a variety of problems: besides growing interfaces and directed polymers as originally proposed [1], it also turned out to be relevant for stochastic particle transport, quantum integrable systems [3,4], and fluctuating hydrodynamics [5], to name but a few.
In the following, let us focus on the one-dimensional case, for which exact studies have been developed, and consider growing interfaces described by the height h(x, t) at position x ∈ R and time t ∈ R. The KPZ class describes scale-invariant fluctuations of growing interfaces in the long-time limit, in general situations without particular symmetries and conservation laws. The hallmark of the KPZ class is the scaling laws for the fluctuation amplitude ∼ t β and the correlation length ∼ t 1/z , with universal exponents β and z that take the values β = 1/3 and z = 3/2 for the one-dimensional case [1,2,4]. The height h(x, t) is then generally written, for large t, as
h(x, t) v ∞ t + (Γt) 1/3 χ(X, t)(1)
where χ(X, t) is a stochastic variable, X := x/ξ(t) denotes the coordinate rescaled by the correlation length ξ(t) := 2 A (Γt) 2/3 , and v ∞ , Γ, A are system-dependent parameters. The variable χ(X, t) is expected to be universal, in the sense that its statistical properties do not depend on microscopic details of the systems. The scaling exponents of the KPZ class have been found in various experimental systems [6], including colonies of living cells [7,8], combusting paper [9], and liquid-crystal turbulence [4,[10][11][12].
Recently, remarkable developments triggered by exact studies [3,4] have unveiled novel aspects on the KPZ class. A particularly important outcome is the geometry dependence, which we describe below. If an interface grows on top of a flat substrate, as usually assumed in simulations, the interface roughens but maintains the globally flat profile. In contrast, if an interface in a plane starts to grow from a point nucleus, say, at x = 0, it takes a circular shape with a growing radius. Although this interface becomes flatter and flatter as the radius increases, statistical properties of χ(X, t) remain distinct from the flat case. Specifically, χ(X, t) has different asymptotic behavior as follows
χ(X, t) d → A 1 (X), (flat) A 2 (X) − X 2 , (circular)(2)
where d → denotes convergence in distribution ( d = and d will be used analogously). A 1 (X) and A 2 (X) are called the Airy 1 [13,14] and Airy 2 [15] processes, respectively, and well studied analytically [16]. Due to their translational invariance, as long as one-point properties are concerned, A i (X) can be replaced by a single stochastic variable χ i . Remarkably, the one-point distribution of χ 1 and χ 2 was shown [17][18][19][20] to coincide respectively with the GOE and GUE Tracy-Widom distribution [21], known from random matrix theory [22], which describes the distribution of the largest eigenvalue of random matrices in the Gaussian orthogonal and unitary ensembles (GOE and GUE). This geometry dependence, as well as the emergence of the Tracy-Widom distribution, turned out to be experimentally relevant too, as shown by experiments on liquid-crystal turbulence [4,10,11]. Correlation properties were also shown to be different between the flat and circular cases, even though the scaling exponents β and z take the same values. On the basis of those results, one may state that the flat and circular interfaces constitute different universality subclasses within the single KPZ class, characterized by different yet universal distribution and correlation properties. Those universality subclasses have been, however, mostly studied for a few "canonical" cases including the flat and circular ones. A natural and important question is then what happens for more general initial conditions. Theoretically, the KPZ fixed-point variational formula [16,[23][24][25][26] can be used to predict the asymptotic properties of χ(X, t → ∞) for general initial conditions. On the other hand, experimental and numerical studies have focused on finite-time behavior emerging from intermediate initial conditions. For example, the present authors [12] studied growth from a ring of finite radius R 0 , which then produces two curved interfaces, one growing outward and the other one inward. Focusing on the ingrowing interfaces, we found that finite-time properties of χ(X, t) for different R 0 are controlled solely by the rescaled time τ := v ∞ t/R 0 , as follows: statistical properties of χ(X, t) agree with those for the flat subclass initially (τ 1), until the interfaces nearly collapse at τ ≈ 1 and therefore do not behave as KPZ anymore. Analogous behavior was also observed numerically by Carrasco and Oliveira [27], who used lattice models with system size set to decrease in time (mimicking the shrinking circumference of the ingrowing interfaces). The case of enlarging substrates, which would correspond to the outgrowing case, has also been studied and crossover from the flat to circular subclasses was suggested in this case [27][28][29], which is also expected to be described by τ . However, it remains unclear how universal such finite-time behavior is, why τ is the right parameter to describe it, and above all, how such crossover can be described theoretically.
Those problems are addressed and answered in this Letter. We study outgrowing interfaces from ring initial conditions both numerically and experimentally, using an off-lattice version of the Eden model [30] and the liquidcrystal turbulence [4,[10][11][12]. Scaling functions for the flat-to-circular crossover are determined, and shown to be the same for both of the studied systems. Moreover, we describe this crossover theoretically, by adapting the variational formula [16,[23][24][25][26] for curved initial conditions. The formula is numerically evaluated and shown to reproduce our numerical and experimental results quantitatively, without adjustable parameters. This also implies that the flat-to-circular crossover is indeed universal and, furthermore, should generally appear for any curved interfaces with locally parabolic initial conditions.
We first study the off-lattice Eden model [30], in which a cluster of round particles (with unit diameter) grows by stochastic addition of new particles. The initial condition is set to be a ring of N particles [ Fig. 1(a)]. The evolution rule is as follows (see Ref. [30] for details): at each time step, we randomly choose a particle at the interface, attempt to put a new particle next to it in a random direction and do so if there is no overlapping particle. Time is then increased by 1/(the number of the interfacial particles) whether the new particle was put or not. Particles that cannot contribute further growth were checked and removed from the list of the interfacial particles every time unit. To characterize the height fluctuations, we measure the local radius increment R(θ, t), which is the radial distance between the initial ring and the interface at each angular position θ [ Fig. 1(a)]. Thanks to the rotational symmetry, we have
R(θ, t) d = h(0, t) v ∞ t + (Γt) 1/3 χ(0, t),(3)
but statistical precision can be improved by averaging over θ. In our simulations, we varied the initial size N from 100 to 40000 and obtained 4320 to 14400 realizations for each case (summarized in Table SI [31]). For comparison, we also simulated flat interfaces, for which the initial condition was a line formed by 75000 particles and the periodic boundary condition in the spanwise direction was used, and obtained 14400 realizations.
To characterize statistical properties of the stochastic variable χ(X, t), we first estimated the non-universal parameters v ∞ , Γ and A, from the data for the flat interfaces. v ∞ and Γ were obtained by the standard procedure [4], specifically by using
∂ t h v ∞ + const. × t −2/3 and h 2 c / t 2/3 χ 2 1 c Γ 2/3 , where · · · k
c denotes the kth-order cumulant and here we used the fact that the asymptotic fluctuations of the flat interfaces are given by the GOE Tracy-Widom distribution. We obtained v ∞ = 0.51370(5) and Γ = 0.980(3). The parameter A was obtained by A = 2Γ/v ∞ , the relationship valid for isotropic growth [11].
With those parameter values, we define the rescaled height
q(θ, t) := R(θ, t) − v ∞ t (Γt) 1/3 d χ(0, t)(4)
and measure its mean and variance as functions of time, for different initial particle number N (Fig. 2 left). Fig-ure 2 also shows the rescaled mean velocity [4,12] p(θ, t) :=
3t 2/3 Γ 1/3 [∂ t R(θ, t) − v ∞ ] χ(0, t) + 3t∂ t χ(0, t) ,(5)
which asymptotically goes to χ(0, t) if χ(0, t) converges sufficiently fast. For the flat case (gray circles), q → χ 1 , p → χ 1 and q 2 c → χ 2 1 c as expected. In the case of the ring initial conditions, for large N the data first behave similarly to the flat case, then deviate and approach the values for the circular subclass, χ 2 and χ 2 2 c [32]. This crossover takes place earlier for smaller N . Indeed, when the data are plotted against the rescaled time τ = v ∞ t/R 0 (R 0 = N/2π), all data collapse onto a single curve except for the nonuniversal short-time regime (Fig. 2 right). This suggests that the distribution of χ(0, t) for different R 0 , denoted by χ(0, t; R 0 ), is described by a single stochastic variable χ c (0, τ ), parametrized by τ , as follows:
χ(0, t; R 0 ) d → χ c (0, τ ), (R 0 , t → ∞)(6)
where the double limit is taken with fixed τ = v ∞ t/R 0 . Then the flat-to-circular crossover we found indicates
χ c (0, τ ) d → χ 1 for τ → 0 and χ c (0, τ ) d → χ 2 for τ → ∞. The skewness Sk[R(θ, t)] := R 3 c / R 2 3/2 c → Sk[χ c (0, τ )] and the kurtosis Ku[R(θ, t)] := R 4 c / R 2 2 c → Ku[χ c (0, τ )] show consistent behavior (Fig. S1 [31]).
We also study this crossover in the spatial correlation. In the case of the point initial condition, suppose θ = 0 corresponds to x = 0, then using R(θ, t) = h(x, t) 2 + x 2 h + x 2 2h and Eq. (2), we can show q(θ, t) d → A 2 (X). Therefore, the rescaled spatial covariance C s (∆X, t) := q(θ + ∆θ, t)q(θ, t) − q(θ, t) 2 with ∆X := R(θ, t) ∆θ/ξ(t) can be directly compared with the covariance of the Airy 1 and Airy 2 processes. Our numerical results for the ring initial conditions (Fig. S2 filled symbols) indeed show crossover from the Airy 1 covariance (τ 1) to the Airy 2 covariance (τ 1), consistently to the results on the one-point distribution.
To test universality of our finding, in particular the function forms of χ c (0, τ ) and χ c (0, τ ) 2 c , we conducted experiments on liquid-crystal turbulence [4,[10][11][12]. As in the previous studies, we applied an AC voltage (here, 22 V at 300 Hz) to nematic liquid crystal filling a thin gap between transparent electrodes, and observed growth of a turbulent state called the dynamic scattering mode 2 (DSM2), expanding in a metastable turbulent state, DSM1 (see Supplemental Text [31] for detailed methods). DSM2 was generated by emitting a few ultraviolet laser pulses [4]. Using the holographic technique we previously adopted for the DSM2 growth experiments [12], we formed the laser intensity profile in the shape of The mean and variance of the rescaled height, q(θ, t) and q(θ, t) 2 c , and the rescaled mean velocity p(θ, t) for the Eden model in the outgrowing case. The data are shown against the raw time t (left) and the rescaled time τ = v∞t/R0 (right). The theoretical curves evaluated numerically from the variational formula for the outgrowing interfaces (=var., blue solid line) are shown in the right panels. The values of χ1 and χ2 are shown by the dashed and dotted lines, respectively. The inset of the right-top figure shows the difference between the data and the excepted longtime limit value, χ2 . The black solid line indicates slope −1/3. a ring of a given radius R 0 , which sets the initial condition of the DSM2 interface [ Fig. 1(b)]. We also generated circular interfaces with a point initial condition, and flat interfaces with a linear initial condition. We obtained 941 to 1936 realizations for each case (Table SII [ 31]), recorded by a charge-coupled device camera. The radius R(θ, t) of the DSM2 interfaces (or the height h(x, t) for the flat case) was determined from each image, with the time t defined as the elapsed time after shooting the laser pulses. Then the non-universal parameters v ∞ , Γ, A were evaluated in the same way as for the Eden model, here for the flat and point initial conditions (Table SII [ 31]).
Although the values of v ∞ , Γ, A are expected to be independent of the initial condition, in practice one needs to evaluate for each set of experiments, because of unavoidable slight changes in experimental conditions [11]. For the ring initial conditions, however, the parameter values could not be obtained in the same way because of the time dependence (i.e., crossover) of χ(X, t). We therefore used the values obtained from the flat case for the outgrowing cases, unless otherwise stipulated. Possible shifts in the parameter values were taken into account in the uncertainty estimates for the outgrowing cases, evaluated from the differences in the parameter values between the flat and circular cases. Now we compare the experimental results with those for the Eden model. Figure 3 left panel shows the variance of the rescaled height, q(θ, t) 2 c , against τ = v ∞ t/R 0 , which overlaps on the Eden data within statistical errors and parameter uncertainty (error bars and shades, respectively) apart from the non-universal shorttime behavior. For the rescaled mean velocity p(θ, t) (right panel), the uncertainty of v ∞ was too large to make a meaningful comparison (inset). However, if we instead choose the value of v ∞ in such a way that p(θ, t) at the largest t falls onto the curve for the Eden model (obtained values of v ∞ are given in Table SII), p(θ, t) overlaps for all t (main panel). Those results of q(θ, t) 2 c and p(θ, t) suggest universality of the one-point distribution of χ c (0, τ ). Moreover, the spatial covariance C s (∆X, t) is also found to overlap with the results of the Eden model if the value of τ is close enough (Fig. S2). This suggests that not only the one-point distribution of χ c (0, τ ) but the spatial covariance of χ c (X, τ ) is also universal.
So far we have characterized the flat-to-circular crossover and found it to be controlled by a single parameter τ = v ∞ t/R 0 , but why so and how can this crossover be theoretically described? To answer these questions, we employ the variational formula [16,[23][24][25][26] and apply it to a general, curved initial condition.
The variational formula describes the height h(x, t) for a general initial condition h(x, 0) =: h 0 (x) as follows
h(x, t) d sup y∈R [h circ (x, t; y) + h 0 (y)] ,(7)
where h circ (x, t; y) denotes the height for the point initial condition nucleating at position y, growing with the same realization of noise for different y [23]. Intuitively, this means that the initial condition h(x, 0) can be regarded as a collection of point sources and h(x, t) is then given by the envelope of the circular interfaces from those point sources, a bit analogously to Huygens' principle [33]. The formula (7) involves a mathematical object called the Airy sheet [23,25], but if the interest is only in the one-point distribution, it can be simply expressed by the Airy 2 process, as follows [16,24]:
χ(X, t) d sup Y ∈R A 2 (X − Y ) − (X − Y ) 2 + h 0 (ξ(t)Y ) (Γt) 1/3 .(8)h 0 (x) = R 0 g x R 0(9)
where g(w) is a locally parabolic function, i.e., g(w) = −c 2 w 2 + O w 2 for small |w|. Substituting Eq. (9) into Eq. (8), taking the limit R 0 , t → ∞ with fixed τ = v ∞ t/R 0 , and setting x = 0 yields
χ(0, t) d → sup Y ∈R A 2 (Y ) − (1 + cτ ) Y 2 =:χ (cτ ) (10) with c := (4c 2 Γ)/(A 2 v ∞ )
. This shows that the asymptotic height distribution is parameterized only by cτ , and only the local functional form of g (w) at small |w| is relevant. The characteristic time is τ = 1/c and therefore t = A 2 R 0 /4c 2 Γ, and this is the time at which the initial height difference |h 0 (0) − h 0 (ξ (t))| becomes comparable to the fluctuation amplitude, (Γt) 1/3 . For isotropic growth, the relationship A = 2Γ/v ∞ [11] further yields c = 2c 2 .
For the ring initial conditions, g(w) is given by g(w) = σ √ 1 − w 2 1 |w|<1 − 1 with σ = +1 (−1) for the outgrowing (ingrowing) case. Then we obtain χ(0, t) d χ (στ ), which we have expressed by χ c (0, τ ) for the outgrowing case σ = +1 [Eq. (6)]. Note that, mathemati- [34,35]. In the other limit τ → ∞, clearly,χ(τ ) → A 2 (0) d = χ 2 , i.e., GUE Tracy-Widom distribution. Therefore, χ c (0, τ ) = χ(τ ) indeed has the expected limits on both sides of the flat-to-circular crossover.
cally, it is known thatχ(0) = sup Y ∈R (A 2 (Y )−Y 2 ) d = χ 1 , i.e., GOE Tracy-Widom distribution
To compare the variational formula with the experimental and numerical data for finite τ , we employ a Monte Carlo method to evaluate Eq. (10). The Airy 2 process A 2 (Y ) is in fact known to be equivalent to the largest eigenvalue of large GUE random matrices undergoing Dyson's Brownian motion [16,34]. We therefore implement Dyson's Brownian motion numerically, in the form of the Ornstein-Uhlenbeck process of Hermitian random matrices and obtained approximated realizations of A 2 (Y ) (see Supplemental Text [31] for details). Then we evaluated the supremum of Eq. (10), interpolating the values of A 2 (Y ) between the discrete steps by using the Brownian bridge [31]. The results for the outgrowing case (σ = +1) are shown in Figs. 2 and 3, where the data of the mean q , variance q 2 c , and the rescaled mean velocity p are compared with the corresponding expressions ofχ(τ ), specifically, χ(τ ) , χ(τ ) 2 c [Eq. (4)], and χ(τ ) + 3τ ∂ τ χ(τ ) [Eq. (5)], respectively. The results of the variational formula precisely agree, without any adjustable parameter, with the numerical and experimental data. We also inspected the ingrowing case σ = −1 and confirmed the validity of the variational formula (Fig. S3). The agreement was also underpinned for the skewness and kurtosis (Fig. S4).
In summary, we found KPZ crossover functions that govern height fluctuations of interfaces growing outward from ring initial conditions, parameterized only by the rescaled time τ = v ∞ t/R 0 , and evidenced their universality both experimentally and numerically. We then presented a theoretical description of this crossover, on the basis of the KPZ variational formula for general curved initial conditions. We numerically evaluated the formula and found remarkable agreement with the experimental and numerical data. Our results constitute the first example where the KPZ variational formula was successfully used to describe experimental observations, showing the ability of this formula to explain, or even predict, real data from general initial conditions. We hope our work will trigger further studies to elucidate geometrydependent universality of the KPZ class and beyond.
We thank P. Le Doussal for useful discussions on the variational formula, and F. Bornemann for the theoretical curves of the Airy 1 and Airy 2 covariance [36]. We thank Supercomputer Center of the Institute for Solid State Physics (the University of Tokyo) and Meiji Institute for Advanced Study of Mathematical Sciences (Meiji University) for computational facilities. We acknowledge financial support by KAKENHI from Japan Society for the Promotion of Science (Grant Nos. JP25103004, JP16H04033, JP19H05800, JP19H05144, JP17J05559), by Yamada Science Foundation, and by the National Science Foundation (Grant No. NSF PHY11-25915).
Similarly to the past studies [1][2][3][4][5], we prepared an electroconvection cell by assembling two glass plates with transparent electrodes, sandwiching 12 µm-thickness spacers. A liquid-crystal sample, N -(4-Methoxybenzylidene)-4-butylaniline doped with 0.01wt.% of tetra-n-butylammonium bromide, was filled in a 1.5 cm × 1.5 cm region enclosed by the spacers. The homeotropic alignment was achieved by spin-coating N,N -dimethyl-N -octadecyl-3aminopropyltrimethoxysilyl chloride on the electrodes. The cell was contained in a temperature controller, whose temperature was maintained at 25 • C, with the temporal fluctuation in the order of ±0.01 • C. Under this temperature, the electroconvection was observed by applying an AC voltage to the cell. The cutoff frequency, which separates the conductive and dielectric regimes of the electroconvection [6], was roughly 1.7 × 10 3 Hz. At the frequency we used in the main experiments, 300 Hz, compact DSM2 clusters grew at the amplitude 17.5 V. The voltage amplitude for the main experiments, 22 V, was chosen to be sufficiently higher than that threshold.
To generate a growing DSM2 interface, we first applied an AC voltage of amplitude 22 V and frequency 300 Hz to the cell and the system was set entirely in the metastable DSM1 state. After 5 s, we emitted three ultraviolet laser pulses to the cell (New Wave Research MiniLaseII, wavelength 355 nm, pulse width 4-6 ns, repetition frequency 20 Hz, energy 0.4 mJ after attenuation) to nucleate DSM2 [4]. The laser pulse was reflected by a spatial light modulator (Hamamatsu Photonics, LCOS-SLM X10468-05) and a hologram of the given shape was made, by using the experimental setup reported in Ref. [5]. For the linear initial condition, the line length was approximately 8 mm. Then the growing DSM2 interface was recorded by a charge-coupled device camera for a given time (for the flat interfaces, the region of width ≈ 4 mm near the center of the line was observed). Then we turned off the applied voltage, waited for 30 s, and started the next run.
To obtain the radius R(θ, t) of the DSM2 interfaces (or the height h(x, t) for the flat case), we binarized each image by thresholding. For the ring initial conditions, first we determined the center of the ring by using the ensemble average of the image intensity fields taken at t = 1 s, and used it to define the radius R(θ, t).
SUPPLEMENTAL TEXT 2: EVALUATION OF THE VARIATIONAL FORMULA
Detailed description of the numerical method
We evaluated the variational formula (10) by using Dyson's Brownian motion to approximate the Airy 2 process, as follows:
1. We first prepared an initial Hermitian random matrix drawn from the Gaussian unitary ensemble (GUE), or equivalently, the stationary distribution of Eq. (S2) (below):
H jk (0) = 1 2 N (1) jk (j = k) 1 2 N (1) jk + iN (2) jk (j > k) ,(S1)
where N (m) jk are i.i.d. random variables drawn from the normal distribution with the mean 0 and the variance 1. 2. We simulated the following Ornstein-Uhlenbeck process:
dH jk (u) du = −H jk + η (1) jk (u) (j = k) −H jk + 1 2 η(1)
jk (u) + iη jk (u) = δ j j δ k k δ m m δ(u − u). We employed Gillespie's exact algorithm for the Ornstein-Uhlenbeck process [7] for each element, with time step ∆u.
3. We computed the largest eigenvalue λ (N ) j := λ (N ) (j∆u) for each time step j∆u (j = 0, . . . , j max ). Then, since
√ 2N 1/6 λ (N ) (N −1/3 Y ) − √ 2N d → A 2 (Y ) (N → ∞),(S3)
we rescaled it as follows:λ (N ) j
:= √ 2N 1/6 λ (N ) j − √ 2N . ∆ũ := N 1/3 ∆u.(S4)
4. Using the fact that the Airy 2 process is locally equivalent to the Brownian motion with unit diffusion constant (the standard Brownian motion) [8,9], we approximated the variational formula (10) bỹ
χ (cτ ) = sup Y ∈R A 2 (Y ) − (1 + cτ )Y 2 ≈ max k=−L,...,L−1 r (j) k =:χ N (cτ ),(S5)
where L was taken sufficiently large (see below), j = L, . . . , j max − L, and r (j) k is the maximum of the Brownian bridge (with unit diffusion constant) connecting z
(j) k :=λ (N ) j+k − (1 + cτ ) (k∆ũ) 2 and z (j) k+1 . Specifically, r (j)
k is a random variable whose cumulative distribution function is given by [10] P r (j)
k < z = 1 − exp − (z−z (j) k )(z−z (j) k+1 ) ∆ũ (z ≥ max z (j) k , z (j) k+1 ) 0 (otherwise). (S6)
The range of L was chosen so that it satisfies maxλ Step size
To find an appropriate step size ∆u, we estimated the range of Y that is relevant to the value of the supremum in Eq. (S5). Since the Airy 2 process is locally equivalent to the standard Brownian motion, a value of Y such that (1 + cτ )Y 2 becomes as large as A 2 (Y ), denoted by Y 0 , is given approximately by (1 + cτ )Y 2 0 = √ 2Y 0 . With this Y 0 , ∆ũ should be chosen so that ∆ũ Y 0 . For large τ , Y 0 ≈ τ −2/3 . In the present work, we chose ∆ũ = 10 −3 (∆u = 10 −3 N −1/3 ) so that ∆ũ Y 0 is satisfied for all τ ≤ 3 × 10 3 .
Matrix size N
To quantify the effect of finite matrix size N , we first evaluated the cumulants of the rescaled largest eigenvalue, The results in Fig. S5(a) show that (λ being consistent with theoretical expectation [11]. Sinceχ N (∞) =λ (N ) j , the finite-N corrections shown in Fig. S5(a) are equivalent to those ofχ(τ ) in the limit τ → ∞. Similarly, we evaluated the variational formula (S5) with cτ = 0, for which it is known thatχ(0) = sup Y ∈R (A 2 (Y ) − Y 2 ) d = χ 1 , i.e., GOE Tracy-Widom distribution [12,13]. As displayed in Fig. S5(b), the data obtained from the variational formula indeed show the cumulant values approaching those of χ 1 , again with the finite-size corrections proportional to N −2/3 . From those results, we decided to use N = 512 (largest size) for the main part of the work; at N = 512, the amplitude of the finite-N corrections shown in Fig. S5 is small enough to compare with our numerical and experimental data. The uncertainty of the estimated cumulants displayed in Figs. 3, S3, and S4 is the summation of the statistical uncertainty and the expected amplitude of finite-N correction, the latter being evaluated by the larger value of χ N (∞) Table SI for the number of realizations for the simulations. The experimental data for the variance (left) is omitted because of large finite-time effect [5]. The values for χ1 (flat) are shown by the dashed lines.
FIG. 1 .
1Typical snapshots from the Eden simulations and the liquid-crystal experiments. (a) An Eden interface growing outward from a ring with N = 1000 (dotted line). Time is indicated by the color. (b) A DSM2 cluster (black) growing from a ring with R0 = 366 µm (dotted lines). The elapsed time after shooting laser is indicated above each image. The scale bar corresponds to 1 mm.
FIG. 2 .
2FIG. 2. The mean and variance of the rescaled height, q(θ, t) and q(θ, t) 2 c , and the rescaled mean velocity p(θ, t) for the Eden model in the outgrowing case. The data are shown against the raw time t (left) and the rescaled time τ = v∞t/R0 (right). The theoretical curves evaluated numerically from the variational formula for the outgrowing interfaces (=var., blue solid line) are shown in the right panels. The values of χ1 and χ2 are shown by the dashed and dotted lines, respectively. The inset of the right-top figure shows the difference between the data and the excepted longtime limit value, χ2 . The black solid line indicates slope −1/3.
FIG. 3 .
3Comparison of the results from the experiments (color filled symbols), the Eden simulations (gray open symbols), and the variational formula (=var., blue solid line), for the outgrowing interfaces. The variance of the rescaled height, q(θ, t) 2 c , and the rescaled mean velocity p(θ, t) are shown in the left and right panels, respectively, against τ = v∞t/R0. For the numerical results, data with t > 10 3 are shown by the same symbols as those inFig. 2. For the experimental results, statistical errors are indicated by the error bars on the first and last data points, and uncertainty associated with the parameter estimation is shown by the shaded areas. The values for χ1 (flat) and χ2 (circular) are shown by the dashed and dotted lines, respectively. The inset of the right panel shows the experimental results obtained with v∞ from the flat case, while it was adjusted in the main panel to fit the Eden data at the largest t (see text).
We use Eq. (8) and consider a class of curved initial conditions in the following form
( 2 )
2jk (u) (j > k) (S2) arXiv:1909.11920v3 [cond-mat.stat-mech]
<
(1 + cτ )[(L/1.1)∆ũ] 2 . Then the cumulants χ N (cτ )k c were evaluated by taking the average of the right-hand side of Eq. (S5) over varying j and independent realizations of the Ornstein-Uhlenbeck process. In our simulations for the main results presented in Figs. 3, S3, and S4, we used 5360 realizations of the Ornstein-Uhlenbeck process with N = 512, ∆ũ = 10 −3 (see below), and j max = 500000.
varying N , and compared with the known values for the GUE Tracy-Widom distribution, χ 2 k c .
with the difference decreasing as ∼ N −2/3 ,
FIG
. S1. The skewness and kurtosis of the rescaled height q(θ, t) for the Eden model. The data are plotted against the raw time t (left column) and the rescaled time τ = v∞t/R0 (right column). The values for χ1 and χ2 are shown by the dashed and dotted lines, respectively.FIG. S2. Spatial covariance Cs(∆X, t), plotted against the normalized length ∆X, for the outgrowing interfaces. The solid and dashed lines indicate the Airy 1 (flat) and Airy 2 (circular) covariance, respectively, Ai(X + ∆X)Ai(X) − Ai(X) 2 . FIG. S3. The ingrowing counterpart of Fig. 3. Results from the liquid-crystal experiments (color filled symbols; data adopted from Ref. [5]) and those from Eden simulations [gray open symbols; N = 100000( ), 40000( ), 20000( ), 10000( )] are compared with the curves obtained from the variational formula (=var., red solid line). The shaded area behind the variational formula curves indicates uncertainty of the Monte-Carlo evaluation. See
FIG. S4. The skewness and kurtosis of the rescaled height q(θ, t), plotted against the rescaled time τ = v∞t/R0 for the (a) outgrowing and (b) ingrowing interfaces. The numerical results with t > 10 3 are shown by gray open symbols, and the curves obtained by numerical evaluation of the variational (=var.) formula are drawn with the red solid line. The symbols for the numerical results are for the samples with N = 40000( ), 20000( ), 10000( ), 4000( ), 1000( ), 500( ), 100( ), 50( ) for the outgrowing interfaces, and for N = 100000( ), 40000( ), 20000( ), 10000( ) for the ingrowing interfaces, respectively. The shaded area for the variational formula indicates uncertainty of the Monte-Carlo evaluation. The values for χ1 (flat) and χ2 (circular) are shown by the dashed and dotted lines, respectively. FIG. S5. Finite-N corrections in the cumulants ofχN (∞) = λ (N ) j (a) andχN (0) (b). They are compared with the cumulants of χ2 and χ1, respectively. The errorbars indicate statistical uncertainty. The black solid lines are guides for the eyes showing the exponent −2/3.
TABLE SI .
SIParameters for the Eden simulations.TABLE SII. Experimental conditions and non-universal parameters. initial condition # of samples v∞(µm/s) Γ(µm 3 /s) Values obtained by fitting the last data point of p to the results of the Eden model (see main text).type
outgrowing
N
50
100 500 1000 4000 10000
# of samples 14400 14400 14400 14400 11520 4320
type
outgrowing
ingrowing
N
20000 40000 10000 20000 40000 100000
# of samples 4320 4320 4320 4320 4320 1440
type
flat
length
75000
# of samples 14400
line
1417
30.84(2) 1.25(2) × 10 3
point
941
29.68(3) 1.31(4) × 10 3
R0 = 366 µm
1936
30.84(2) *
-
R0 = 219 µm
1521
30.60(2) *
-
* SUPPLEMENTAL FIGURES
0.22
0.24
0.26
0.28
0.30
Sk [q]
Sk [χ 1 ]
Sk [χ 2 ]
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| []
|
[
"The quest for electronic ferroelectricity in organic charge-transfer crystals",
"The quest for electronic ferroelectricity in organic charge-transfer crystals"
]
| [
"Alberto Girlando \nGroup on Molecular Materials for Advanced Applications (MMAA)\n\n\nDept. of Chemistry\nLife Sciences and Environmental Sustainability\nParco Area delle Scienze 1743124ParmaItaly\n"
]
| [
"Group on Molecular Materials for Advanced Applications (MMAA)\n",
"Dept. of Chemistry\nLife Sciences and Environmental Sustainability\nParco Area delle Scienze 1743124ParmaItaly"
]
| []
| Organic ferroelectric materials are in demand in the growing field of environmentally friendly, lightweight electronics. Donor-Acceptor charge transfer crystals have been recently proposed as a new class of organic ferroelectrics, which may possess a new kind of ferroelectricity, the so-called electronic ferroelectricity, larger and with faster polarity switching in comparison with conventional, inorganic or organic, ferroelectrics. The current research aimed at achieving ambient conditions electronic ferroelectricity in organic charge transfer crystals is shortly reviewed, in such a way to evidence the emerging criteria that have to be fulfilled to reach this challenging goal. arXiv:1904.07836v1 [cond-mat.mtrl-sci] | null | [
"https://arxiv.org/pdf/1904.07836v1.pdf"
]
| 119,307,401 | 1904.07836 | c29ff2c4000b37e309aff78945fdc20483449264 |
The quest for electronic ferroelectricity in organic charge-transfer crystals
Alberto Girlando
Group on Molecular Materials for Advanced Applications (MMAA)
Dept. of Chemistry
Life Sciences and Environmental Sustainability
Parco Area delle Scienze 1743124ParmaItaly
The quest for electronic ferroelectricity in organic charge-transfer crystals
Organic ferroelectric materials are in demand in the growing field of environmentally friendly, lightweight electronics. Donor-Acceptor charge transfer crystals have been recently proposed as a new class of organic ferroelectrics, which may possess a new kind of ferroelectricity, the so-called electronic ferroelectricity, larger and with faster polarity switching in comparison with conventional, inorganic or organic, ferroelectrics. The current research aimed at achieving ambient conditions electronic ferroelectricity in organic charge transfer crystals is shortly reviewed, in such a way to evidence the emerging criteria that have to be fulfilled to reach this challenging goal. arXiv:1904.07836v1 [cond-mat.mtrl-sci]
I. INTRODUCTION
Ferroelectricity, i.e., spontaneous electrical polarization, is a phenomenon analogous to ferromagnetism, yet its discovery in actual materials is relatively recent, dating back to the last century [1]. On the other hand, in the emerging field of lightweight and environmentally friendly materials for electronics, the quest for organic ferroelectrics operating at room temperature has proven to be more promising than that of ferromagnetic solids [2]. The polar nature of ferroelectricity of course requires that the crystal structure does not have an inversion center, and in general the ferroelectrics are divided into different classes on the basis on how this symmetry loss is obtained. In recent years, attention has focused on organic mixed stack charge-transfer (ms-CT) co-crystals, made up by planar π-electron donor (D) and acceptor (A) molecules alternating along the stack direction. These materials are characterized by ρ, the degree of CT, ranging from 0 to 1, with ρ ∼ 0.5 separating the neutral (N) from the ionic (I) ground state [3,4]. Increase of the Madelung energy, following lattice contraction by lowering temperature, may induce a peculiar phase transition, the N to I one, with ρ crossing the N-I borderline. Ionic systems are subject to the Peierls instability, yielding to dimerization of the stack, hence satisfying the above mentioned criterion for ferroelectricity. In addition, in ms-CT crystals the ferroelectricity may occur through a new mechanism, involving the π-electronic cloud along the stack, yielding high values of polarization for an organic system. In this paper, the quest for electronic ferroelectricity in ms-CT crystals will be shortly reviewed.
II. BASIC CHARACTERISTICS AND CLASSIFICATION OF FERROELECTRICS
A crystal possessing spontaneous electrical polarization, i.e., electric dipole, in general is not able to orient nearby electric dipoles since the polarization is canceled out by the so-called leakage currents, due to electrons moving through the crystal, ions moving through the air etc. A net dipole can be temporarily generated by chang- ing temperature, which changes the polarization due to a change in the equilibrium position of the atoms in the crystal. We then have the pyroelectric effect, which lasts until the moving electrons adapt to the new equilibrium. A polar, pyroelectric crystal is not necessarily a ferroelectric crystal, because ferroelectricity implies that the direction of the dipoles can be changed by an external electrical field, giving rise to the typical hysteresis loop of the polarization P vs. electric field E shown on the left side of Figure 1. The ferroelectric loop indicates the alignment of the electric dipole domains inside the crystal, and also implies that above a certain temperature, called Curie temperature T c , there is a phase transition to a paraelectric, non-polar (centrosymmetric) phase.
The phase transition between paraelectric and ferroelectric phase is related to the microscopic origin of the ferroelectricity. In general, it is a disorder-order type if the paralectric phase is characterized by thermally disordered electric dipoles which on the average cancel out the net polarization, or it is a displacitive transition if the electric dipoles in the ferroelectric phase are originated by a collective shift of the barycenter of the opposite charges. Displacitive transitions are characterized by a soft phonon mode, that is a phonon connected to the charges motion, whose frequency goes to zero at the transition. In the proximity of the phase transition the dielectric constant κ, i.e., the response of the system to the electric field, increases up to several orders of magnitude, and peaks at T c , as shown in the right side of Figure 1. Table I reports the above described basic parameters (the Curie temperature T c , the spontaneous polarization P s at T close to T c , and the maximum value of the dielectric constant κ) for some representative ferroelectrics. They have been chosen to represent different classes, both from the point of view of chemical composition (inorganic, organic, organic-inorganic, polymeric, single or double molecular component) and from the microscopic origin of ferroelectricity.
The Rochelle salt, the firstly discovered ferroelectric, is a mixed Na-K tetrahydrated salt of tartaric acid, and the para-to ferro-electric transition is a disorder to order type, implying the position of the ions and possibly of the hydrogen bonds. The shift of the H atom in hydrogen bonded framework is at work also in thiourea, croconic acid, and the co-crystals involving chloranilic acid. On the other hand, the para-to ferro-electric transition of BaTiO 3 is a typical example of displacitive transition, as it is for the ms-CT crystals TTF-BA and TTF-CA, with the difference that in the latter cases the displacement involves molecules rather than atoms.
III. ELECTRONIC FERROELECTRICITY
As Table I shows, progress in organic ferroelectrics has been noticeable in the last few years, reaching performance comparable to that of the prototype BaTiO 3 . What makes ms-CT crystals particularly interesting is that they may display a new type of mechanism for ferroelectricity, which has been named "electronic" ferroelectricity to distinguish it from the conventional, or "ionic", ferroelectricity [13]. In fact, in conventional ferroelectricity the dipole moments arise from the displacement of oppositely charged atoms/molecules in the crystal, or from the displacement of hydrogens in the hydrogen-bond framework. Also in the fully ionic (ρ ∼1) TTF-BA the ferroelectricity arise from the displacement of the molec-ular ions TTF and BA when the stack dimerizes due to the Peierls transition. For TTF-CA, on the other hand, the direction of the polarization is opposite to that due to the molecular ions displacement [12]. This unexpected result has been theoretically understood on the basis of modern polarizability theory, showing that the net polarization is the difference between the polarization due to the ion displacement, P ion , and the larger and opposite polarization due to the shift of the π-electron cloud within the DA dimer in the chain, P el [14]. The electronic response is estimated to be about 20 times as large than the ionic one, and, most importantly, also much faster. All this has been demonstrated for TTF-CA, below the N-I and dimerization transition temperature, namely, below 80 K. After the TTF-CA discovery [12], the research has been addressed to find ms-CT crystals which exhibit electronic ferroelectricity at higher temperatures, possibly at room temperature. The large amount of experimental data and theoretical modeling available for ms-CT crystals and their peculiar N-I phase transition help to establish the main conditions that have to be fulfilled to attain electronic ferrolectricity.
According to the general requirement for ferroelectricity, inversion center symmetry must be lacking, that for 1:1 ms-CT crystals means that the stack must be dimerized. The dimerization is induced by the Peierls mechanism, namely by response of the system to the electron-phonon coupling. The response is effective for systems with ρ > 0.3, and is maximum in the proximity of the N-I boundary, so in general fully neutral crystals have regular stacks also at low temperatures, intermediate ionicity crystals can have dimerized stacks also at room temperature or in the proximity of it, whereas fully ionic (ρ > 0.8) stacks dimerize at low temperatures. The response of the electronic system to external perturbations is also maximum for systems close to the N-I boundary, so one has to look for such systems to achieve electronic ferroelectricity. As a further requirement, either the unit cell has to contain only a DA pair, or two pair have to present in-phase dimerization. This is the case for TTF-CA, whereas a slightly different system, Dimethyl-tetrathiafulvalene-Chloranil (DMeTTF-CA) presents dimerization and intermediate ionicity, but the two DA dimeric units in the unit cell are arranged out-of-phase, so the system is antiferroelectric [15].
A. Lock-arm Supramolecular Ordering (LASO) systems
The goal of room temperature electronic ferroelectricity seemed to have been achieved when Tayi et al. [16] reported ferroelectric hysteresis cycles for three ms-CT crystals obtained on the basis of a novel supramolecular design concept, the Lock-Arm Supramolecular Ordering (LASO), that synergistically combines intermolecular CT and hydrogen bonds. Ferroelectric behavior in three complexes formed by the same electron acceptor 1 and three different donors, 2, 3 and 4. (Figure 2) was ascribed to a sizeable charge ρ transferred from D to A molecules arranged in non-centrosymmetric structure characterized by dimerized stacks, as TTF-CA below 80 K [3]. A degree of CT of 0.67, 0.89 and 0.43 was estimated through infrared (IR) spectroscopy for co-crystals 1 · 2, 1 · 3 and 1 · 4, respectively. The spontaneous polarization P s at room temperature was reported to be about 2 µC cm −2 for all the three co-crystals [16]. However, a theoretical calculation by D'Avino and Verstraete [17] suggested that the LASO systems have a much lower degree of CT, hence supporting a very small polarization. At the same time, a re-examination of the X-ray diffraction data of LASO complexes by the proposing team showed that all the three crystal structure are actually better resolved in terms of centro-symmetric space groups, apparently incompatible with ferroelectricity [18]. At this point an international collaboration [19] set up to independently reproduce the data of Ref. [16]. The co-crystal 1 · 2 was selected and synthesized following the original recipe. The X-ray analysis confirmed that the crystal belongs to the triclinic P 1 space group, with two DA pairs per unit cell. Density functional theory (DFT) calculations confirmed that the P 1 phase of 1 · 2 is the most stable, and that geometry optimization of 1 · 3 and 1 · 4 also leads to non-polar structures [19].
Further characterization was performed by vibrational spectroscopy. Figure 3 compares the polarized IR and Raman spectra of the co-crystal 1 · 2 obtained in Ref. [16] (left side) and in Ref. [19] (right side). Vibrational spectra enable to ascertain the presence or absence of an inversion center (regular or dimerized stack), offering complementary information with respect to X-rays. In fact, the latter probes only the long-range order, while vibrational spectroscopy is also sensitive to local dipolar fluctuations and disorder, which might be at the origin of ferroelectricity. Totally symmetric molecular vibrations show up with large or huge intensity in IR spectra only upon dimerization, where they modulate asymmetric flows of electronic charge [3]. The simultaneous presence of these modes in IR and Raman spectra polarized along the DA stack axis is the typical signature of a (possibly local) symmetry breaking. Tayi et al. [16] claimed to have found such a coincidence (asterisks and dashed lines in the left side of Figure 3), but the frequency coincidences are probably accidental, due to the presence of many bands in their limited quality IR spectra. In fact, IR spectra have been obtained by applying the Kubelka-Munk transformation to "single point reflectance" spectra of single crystals, whereas this transformation is only appropriate for the diffuse reflectance spectra of powders. Indeed IR absorbance and Raman spectra by D'Avino et al. [19] do not present coincident peaks (Figure 3, right side), therefore excluding a possible dimerization of the stack.
Infrared spectroscopy allows also the estimation of the ionicity ρ through the frequency shift of properly chosen "charge sensitive" vibrational modes. Tayi et al. [16] attributed a sizeable CT to the three LASO compounds on the basis of analysis conducted on the carbonyl modes of compound 1, which appear with polarization perpendicular to the stack (bottom spectrum of the left side of Figure 3). Apart from the already mentioned problem of IR spectra quality, Tayi et al. [16] used a rather incomprehensible calibration procedure employing TCNQ complexes, which lead to a frequency increase upon the addition of an electron. As a matter of fact, according to chemical intuition and DFT calculations, the frequency of carbonyl modes is expected to largely decrease upon negatively charging 1. Independently from calibration procedures, the bottom panel of the right side of Fig-ure 3 compares the IR spectra of neutral 1 and 2 with the one of 1 · 2 (polarization perpendicular to the stack). The latter is clearly the superposition of those of its single neutral components in the whole spectral range and specifically in the carbonyl region. This definitely proves that the 1·2 complex is essentially neutral (ρ ∼ 0), as the other two LASO compounds are, owing to the minimal shifts of the carbonyl modes. The absence of intermolecular charge transfer in the ground state of the three LASO compound has been also confirmed by DFT calculations that did not evidence any sizeable difference among the electronic structure of the three systems [19]. As a matter of fact, largely neutral systems, more akin to van der Waals crystals rather than CT salts, are not susceptible to dimerization on the basis of the Peierls mechanism [3,4].
The crystallographic and spectroscopic data presented by D'Avino et al. [19] exclude any indirect signature of electronic ferroelectricity. In addition, electrical polarization hysteresis measurements at various temperatures between 7 and 400 K, with different fields and frequencies, failed to obtain a hysteresis loop, casting doubts on the claimed ferroelectricity of 1·2, and indirectly on that of the other two co-crystals as well [19]. In response to the criticism, Tayi et al. [20] reproduced the hysteresis loop for 1 · 3, and reported ferroelectricity for another LASO crystal, made up by 1 and a variant of 2 (one NH 2 group substituted by OH) [21]. This latter co-crystal, having a 2:1 ratio of A and D and crystallizing in P 1 space group, has been reported to display second harmonic generation (SHG), indicative the lack of inversion center. The authors attribute the discrepancy between crystallographic structure and SHG to hydrogen atoms, not detected by X-ray.
B. Tetramethylbenzidine-Acceptor series
Collective proton transfer phenomena are a known source of ferroelectricity in molecular systems characterized by the presence of hydrogen bonds [6,8,9]. Apart from the question of reproducibility, this fact may provide an alternative explanation for the ferroelectricity of LASO systems and its elusiveness to X-rays or vibrational spectroscopy. Therefore the quest for room temperature electronic ferroelectricity cannot yet be considered as successful. Another series of ms-CT co-crystal presently being investigated concerns the strong electron donor, 3,3,5,5-Tetramethylbenzidine (TMB) coupled with a series of π-electron molecules of increasing acceptor strength [22]. The co-crystal of TMB with Tetracyanoquinodimethane (TCNQ) at room temperature has a degree of CT of about 0.3. Around 200 K it undergoes a first-order valence instability transition, with a small increase of ρ to 0.4, as measured by the decrease of the C=C frequency from 1532 to 1528 cm −1 (left side of Figure 7). The simultaneous appearance of strong bands in the IR spectra polarized along the stack (right side of Figure 7) at the same frequency of totally symmetric Raman modes unambiguously signals the dimerization of the stack [23].
The X-ray structure of the low-temperature phase confirms the loss of the inversion center [24]. The unit cell contains two DA pairs, arranged ferroelectrically (right side of Figure 5). If we compare the valence instability of TMB-TCNQ with the transition of TTF-CA, we see strong analogies. The change of space group at the transition is the same, from P 2 1 /n to P n, and in both cases the ionicity is close to the N-I interface (0.6 vs 0.4). Therefore also TMB-TCNQ might exhibit electronic ferroelectricity like TTF-CA, but at higher temperature (below 200 K rather than below 80 K). Polarization measurements are underway, but without many perspectives of detecting ferroelectricity. First of all, the crystal is damaged at the transition, since it contracts along one crystallographic axis and expands along the other two [24], likely spoiling the polarization measurements. In addition, the transition implies a strong dimerization, with molecular reorientation ( Figure 5) so it is likely that the polarization cannot be easily changed by the electric field: We could have a pyroelectric, polar crystal, but no ferroelectricity.
By replacing TCNQ with a stronger electron acceptor, 2,5-Difluoro-TCNQ, the degree of CT of the co-crystal, TMB-TCNQF 2 , increases from about 0.3 to about 0.7 at room temperature [22]. The stack is dimerized, as shown by IR spectra (not reported here) and by the Xray analysis. However, the space group (P 2 1 ) is different from that of the low-temperature phase of TMB-TCNQ, and the DA dimers are arranged anti-ferroelectrically, as shown in Figure 6.
Higher degree of CT can be attained by combining TMB with Tetrafluoro-TCNQ, TCNQF 4 [22]. The identification of TCNQF 4 charge sensitive modes, whose frequencies are reported in the upper panel of Figure 7, yield ρ = 0.9. TMB-TCNQF 4 crystallizes in the centrosymmetric monoclinic space group C2/m. However, IR spectra polarized along the stack axis compared to the Raman spectrum (Figure 7, lower panel) show several frequency coincidences, indicating that the inversion center is locally lost. One might suppose that we have a ferroelectric system in the paralectric phase. Measurements are underway, but structural and spectroscopic measurements down to 100 K do not show evidence of phase transitions, retaining the contradiction between Xray and spectroscopic measurements.
IV. CONCLUSIONS
After about five years, the challenge to find room temperature electronic ferroelectrics is still on the table. As a matter of fact, no ms-CT crystal has been found to exhibit unquestionable electronic ferroelectricity besides TTF-CA, even at low temperature. Some hints about the conditions which have to be met come from the studies reported in this paper. First of all, the ms-CT crystal should have intermediate ionicity, which also implies a strong tendency towards stack dimerization and high response to the electric field, i.e., high polarization. If there are two DA dimers per unit cell, they have to be arranged ferroelectrically (cf. Figure 5, right side, and Figure 6). Finally, the dimerization should not be too strong or imply molecular distortions, otherwise it might be difficult to reverse, and the system would be pyroelectric, but not ferroelectric.
FIG. 1 :
1Ferroelectrics: Typical hysteresis loop (P vs. E) and temperature dependence of the dielectric constant κ.
FIG. 2 :
2Structural formulas of A and D molecules used in Ref.[16].
FIG. 3 :
3Raman and IR spectra of crystal 1·2. Left side, traces from top to bottom: Raman spectrum, IR spectrum polarized along the stack axis and perpendicular to it (adapted from Ref.[16]); Right side: Top panel, Raman and IR absorption spectrum polarized along the stack axis; bottom panel, IR absorption spectrum polarized perpendicular to the stack axis compared with the spectra of the neutral components 1 and 2 (adapted from Ref.[19]).
FIG. 4 :
4IR spectra of TMB-TCNQ above (HT) and below (LT) the phase transition. Left panel: Spectra polarized perpendicular to the stack. Right panel: Spectra polarized parallel to the stack. From Ref.[23]
FIG. 5 :
5TMB-TCNQ structure at room temperature and 130 K viewed along the [001] direction. Donor and Acceptor molecules are drawn in red and light blue, respectively. Hydrogens are omitted for clarity. The arrows in the 130 K structure evidence the ferroelectric arrangements of the DA dimers. FIG. 6: Room temperature crystal structure of TMB-TCNQF2, viewed along the [001] direction. Donor and Acceptor molecules are drawn in red and light blue, respectively. Hydrogens are omitted for clarity. The arrows evidence the anti-ferroelectric arrangements of the DA dimers. FIG. 7: Room temperature polarized IR and Raman spectra of TMB-TCNQF4. From Ref.[22].
TABLE I :
IDiiso-propylammonium; TTF: Tetrathiafulvalene; BA: Bromanil; CA: Chloranil.Basic parameters of selected ferroelectrics
Material a
Curie temp. Tc (K) Ps (µC cm −2 @ Tc) Max. diel. const. κmax Ref.
Rochelle salt
297
0.25 @ 276
4 ·10 3
[5]
BaTiO3
381
26 @ 300
10 4
[5]
Thiourea
169
3.2 @ 120
10 4
[6]
VDF0.65-TrFE0.35
363
8 @ 300
50
[7]
Phenazine-H2ca
253
1.8 @ 160
3·10 3
[8]
Croconic acid
>400
21 @ 300
-
[9]
DIPA-Br
420
10 @ 400
-
[10]
TTF-BA
50
0.12 @ 19
20
[11]
TTF-CA
81
6.3 @ 59
500
[12]
a VDF0.65-TrFE0.35: Vinylidene fluoride-Trifluoroethylene copolymer; H2ca: Chloranilic acid;
DIPA:
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| []
|
[
"Discontinuous energy shaping control of the Chaplygin sleigh",
"Discontinuous energy shaping control of the Chaplygin sleigh"
]
| [
"Joel Ferguson [email protected] ",
"Alejandro Donaire [email protected] ",
"Richard H Middleton [email protected]. ",
"\nSchool of Electrical Engineering and Computing and PRC CDSC\nDepartment of Electrical Engineering and Information\nTheory and PRISMA Lab\nThe University of Newcastle\n2308CallaghanNSWAustralia\n",
"\nItaly, and with the School of Electrical Eng. and Comp. Sc.\nUniversity of Naples Federico II\n80125Napoli\n",
"\nQueensland University of Technology\nBrisbaneQLDAustralia\n"
]
| [
"School of Electrical Engineering and Computing and PRC CDSC\nDepartment of Electrical Engineering and Information\nTheory and PRISMA Lab\nThe University of Newcastle\n2308CallaghanNSWAustralia",
"Italy, and with the School of Electrical Eng. and Comp. Sc.\nUniversity of Naples Federico II\n80125Napoli",
"Queensland University of Technology\nBrisbaneQLDAustralia"
]
| []
| In this paper we present an energy shaping control law for set-point regulation of the Chaplygin sleigh. It is well known that nonholonomic mechanical systems cannot be asymptotically stabilised using smooth control laws as they do no satisfy Brockett's necessary condition for smooth stabilisation. Here, we propose a discontinuous control law that can be seen as a potential energy shaping and damping injection controller. The proposed controller is shown to be robust against the parameters of both the inertia matrix and the damping structure of the open-loop system. | 10.1016/j.ifacol.2018.06.056 | [
"https://arxiv.org/pdf/1801.06278v1.pdf"
]
| 39,957,711 | 1801.06278 | 58a3f4a27632b977e9401493d0f05d27cba71cdc |
Discontinuous energy shaping control of the Chaplygin sleigh
Joel Ferguson [email protected]
Alejandro Donaire [email protected]
Richard H Middleton [email protected].
School of Electrical Engineering and Computing and PRC CDSC
Department of Electrical Engineering and Information
Theory and PRISMA Lab
The University of Newcastle
2308CallaghanNSWAustralia
Italy, and with the School of Electrical Eng. and Comp. Sc.
University of Naples Federico II
80125Napoli
Queensland University of Technology
BrisbaneQLDAustralia
Discontinuous energy shaping control of the Chaplygin sleigh
Nonholonomic systemsport-Hamiltonian systemsdiscontinuous controlrobust control
In this paper we present an energy shaping control law for set-point regulation of the Chaplygin sleigh. It is well known that nonholonomic mechanical systems cannot be asymptotically stabilised using smooth control laws as they do no satisfy Brockett's necessary condition for smooth stabilisation. Here, we propose a discontinuous control law that can be seen as a potential energy shaping and damping injection controller. The proposed controller is shown to be robust against the parameters of both the inertia matrix and the damping structure of the open-loop system.
INTRODUCTION
Mechanical systems are often subject to constraints which restrict the motion of the system. These constraints are often categorised as begin either holonomic or nonholonomic. Holonomic constraints refer to static relationships between configuration variables which, in effect, restricts the configuration space of a system. Nonholonomic constraints, however, refers to all constraints that cannot be described in this manner (Goldstein, 1980). Of particular interest to this work, the constraints that arise from nonslip condition of wheels are necessarily described as a relationship between the configuration and velocity of a system (Bloch et al., 2003). As such, these constraints restrict in what directions the system can move and therefore, they are nonholonomic. In this work, we consider the Chaplygin sleigh which is a benchmark system widely used for nonholonomic control design (van der Schaft and Maschke, 1994;Astolfi, 1996;Lee, 2007;Fujimoto et al., 2012;Tian and Li, 2002;Bloch and Reyhanoglu, 1992).
Nonholonomic systems with constraints that are linear in velocities can be represented as port-Hamiltonian (pH) system with Lagrange multipliers that enforce the constraints. In the work of van der Schaft and Maschke (1994) it was shown that by reducing the dimension of the momentum space, these systems have an equivalent representation without Lagrange multipliers. The reduced representation is essentially 'constraint free' insofar as any state in the reduced state-space is permissible. Important to this work, the Chaplygin sleigh admits such a representation (Astolfi et al., 2010).
Here, we utilise this 'constraint free' representation of the Chaplygin system to develop a control law to achieve setpoint regulation of the system. Unfortunately, as it is well known, Brockett's necessary condition for asymptotic stabilisation using smooth feedback control is not satisfied by nonholonomic mechanical systems. As a consequence, this class of system cannot be stabilised using continuously differentiable control laws (Brockett, 1983). This restriction does not rule out the possibility of asymptotic stabilisation using non-smooth controllers, which has been achieved in (Astolfi, 1996;Fujimoto et al., 2012). In this work, we propose a discontinuous energy shaping control law for the Chaplygin system. While control of the Chaplygin system (and nonholonomic systems generally) has been extensively studied, control methods that exploit the natural passivity of the system are quite limited. Similar to the method proposed here, a discontinuous energy shaping control law was proposed by Fujimoto et al. (2012) for the rolling coin system-which is encompassed in the Chaplygin system used here-to asymptotically stabilise the system. A different approach was taken by Lee (2007) where a switching strategy was used to drive a mobile robot-which again is encompassed in the Chaplygin system used here-to a compact set containing the origin.
Previously, we studied the control of the Chaplygin system in (Ferguson et al., 2016) by switching between two manifold regulating control laws where each law could be considered to be energy shaping controllers. Here, we extend this previous work by proposing a single energy shaping control law that drives the configuration of the system to the origin. By exploiting the passivity properties y x (x, y) θ l u Fig. 1. The Chaplygin sleigh is fixed to the ground at the point (x, y). It is able to pivot about this point and move forwards in the u direction. The point (x, y) is constrained from moving in a direction perpendicular to u. The centre of mass is indicated by the red spot and is a distance l from the point (x, y)
.
of the open-loop system, the controller is robust against both the inertia and damping matrices.
The remained of the paper is structured as follows: The Chaplygin sleigh model is presented and the problem formulated in section 2. In Section 3 the discontinuous, potential energy shaping controller is presented and the stability properties of the closed-loop are analysed in section 4. A numerical simulation of the closed-loop is presented in Section 5 and conclusions drawn in section 6.
Notation: For a differentiable function H(x), ∇H denotes the column vector of partial derivatives ∂ H ∂x . Given a differentiable function f (x) ∈ R n , ∂f ∂x denotes the standard Jacobian matrix. 0 n is a matrix of dimension n × n with all elements equal to zero whereas 0 n×m is a n × m matrix of all zeros. I n denotes a n × n dimension identity matrix.
PROBLEM FORMULATION
Chaplygin sleigh model
This paper is focused on control design for the Chaplygin sleigh system (Figure 1). This system can be modelled as a pH system of the form (Astolfi et al., 2010):
qṗ = 0 3 Q(q) −Q (q) J(p) − D(q, p) ∇ q H ∇ p H + 0 3×2 I 2 u y = ∇ p H H = 1 2 p M −1 p,(1)
with generalised coordinates q = (x, y, θ), where x and y denote the position at which the sleigh is fixed to the ground in the x − y plane and θ describes the sleigh's heading angle, p = (p 1 , p 2 ) is the momentum and D(q, p) is the damping matrix satisfying D = D ≥ 0. The system matrices are given by
Q(q) = cos θ 0 sin θ 0 0 1 J(p) = 0 ml J + ml 2 p 2 − ml J + ml 2 p 2 0 M = m 0 0 J + ml 2 ,(2)
where m is the mass of the sleigh and J is the rotational inertia about the centre of mass.
In the special case where l = 0, this system coincides with the knife edge system (Bloch et al., 2003). The knife edge system is closely related to the rolling disk system, studied in (Fujimoto et al., 2012), which has an extra state associated with the roll angle of the disk. If the roll angle from the rolling disk is ignored, it coincides with the knife edge system. Due to the relationship between these systems, the regulating controller developed in this paper can be applied to either of these examples.
Problem statement
Considering the Chaplygin sleigh system (1), our objecting is to design a control law u = u(q, p)
(3) such that lim t→∞ q(t) = 0 n .
DISCONTINUOUS CONTROL LAW
The approach taken to control the Chaplygin system is to first perform two consecutive coordinate transformations, q → z → w. The control law is then designed in the w coordinates where the control objective can be achieved by potential energy shaping using a quadratic potential function. The stability analysis relies heavily on the relationship between the z and w coordinates.
Transformation q → z
The first of our coordinate transformations q → z is defined by the mapping
z = z 1 z 2 z 3 = f z (q) 0 0 1 cos θ sin θ 0 sin θ − cos θ 0 x y θ .
(4)
The Chaplygin system (1) can be expressed in the coordinates (z, p) by the equations
żṗ = 0 3 Q z (z) −Q z (z) J(p) − D z (z, p) ∇ z H ∇ p H + 0 3×2 I 2 u y = ∇ p H H = 1 2 p M −1 p,(5)
with Q z and D z defined by
Q z (z) = ∂f z ∂q (q)Q(q)| q=f −1 z (z) = 0 1 1 −z 3 0 z 2 D z (z, p) = D(p, q)| q=f −1 z (z) .(6)
The reason for expressing the system as a function of z is the structure of Q z in (6). Importantly, Q z has a full-rank left annihilator
Q ⊥ z (z) = [−z 2 0 1] .(7)
3.2 Transformation z → w Similar to the transformation to z in the previous subsection, we now consider the coordinate transformation z → w. The proposed transformation is given by
w = w 1 w 2 w 3 = f w (z) z 1 z 2 z −1 1 − 2z 3 z −2 1 2z 3 z −2 1 .(8)
The inverse transformation z = f −1 w (z) is given by
z = z 1 z 2 z 3 = f −1 w (w) w 1 w 1 w 2 + w 1 w 3 1 2 w 2 1 w 3 .(9)
This are two important properties that have motivated this choice for f w : Firstly, the mapping f −1 w : w → z is smooth. This means that if a solution w(t) is bounded, then z(t) is be bounded also. Secondly, the entire set {w ∈ R 3 |w 1 = 0} corresponds to z = 0 3×1 . This means that in the w coordinates, the control objective can be addressed simply by regulating the variable w 1 to zero whilst keeping w 2 and w 3 bounded.
The Chaplygin system (5) can be expressed as a function of (w, p) by the equations
ẇ p = 0 3 Q w (w) −Q w (w) J(p) − D w (w, p) ∇ w H ∇ p H + 0 3×2 I 2 u y = ∇ p H H = 1 2 p M −1 p,(10)
with Q w and D w defined by
Q w (w) = ∂f w ∂z (z)Q z (z)| z=f −1 w (w) = 0 1 1 − 2w 2 w 1 − w 3 − 1 2 w 2 1 w 3 0 2w 2 w 2 1 D w (w, p) = D z (p, z)| z=f −1 w (w) .(11)
Importantly, the matrix Q w is ill-defined at w 1 = 0. This has the consequence of the dynamics (10) being undefined at this point. As such we define the set on which the dynamics (10) are defined:
U = {(w, p) ∈ R 5 |w 1 = 0}.(12)
The dynamics (10) are well defined on the set U .
Regulation control law
Consider the following control law as a solution to the problem statement:
u(w, p) = −Q w Lw −D∇ p H − 0 0 0 k w 2 1 Di(w) ∇ p H(13)
whereD ∈ R 2×2 is positive definite, L = diag(l 1 , l 2 , l 3 ) where each l i ∈ R is positive and k > 0 is a constant. Remark 1. Considering the Chaplygin system (5), the term ∇ p H can be expressed as a function of z,ż as
∇ p H = z 3 1 1 0 ż 1 z 2 .(14)
Thus, the control law (13) can be written independent of the mass matrix M . Remark 2. The control law (13) is independent of the open-loop damping D w and is, thus, robust against this parameter. Remark 3. The control law (13) has been given as a function of (w, p) but can be equivalently expressed as a function of (z, p) or (q, p) using the mappings f w (8) and f z (4).
Now we show that the closed-loop system admits a Hamiltonian representation. Proposition 4. Consider the Chaplygin system (10) in closed-loop with the control law (13). The closed-loop dynamics are given by
ẇ p = 0 n×n Q w (w) −Q w (w) J(p) − D d (w, p) ∇ w H d ∇ p H d H d = 1 2 p M −1 p + 1 2 w Lw,(15)
where
D d (w, p) = D w (w, p) +D + D i (w).(16)
Proof. The proof follows from direct matching. 2
The control law (13) can be interpreted as potential energy shaping plus damping injection. To see this, first notice that the role of the term −Q w Lw is to add the term 1 2 w Lw to the closed-loop Hamiltonian. This term can be considered a potential function in w. Secondly, the term −D∇ p H − D i ∇ p H is to increase the damping from D w to D d in closed-loop.
STABILITY ANALYSIS
We now analyse the asymptotic behaviour of the closedloop system (15). Note, however, that the analysis is not straightforward as the right-hand side of the dynamic equation is discontinuous. In fact, considering the form of Q w in (11), the closed-loop dynamics are not defined at w 1 = 0. This is even more troublesome when we consider that we wish to regulate the system to a configuration satisfying w 1 = 0.
With this in mind, we will determine the asymptotic behaviour of the system in two steps: Firstly, it is shown that the choice of D i in (13) has the consequence that, provided that w 1 (0) = 0, then w 1 (t) cannot reach zero in finite time. This means that the closed-loop dynamics are well defined for all finite time. The second step is to show that the system cannot be positively invariant on the set U . As a consequence, we show that w 1 tends towards zero asymptotically but will not reach this configuration in finite time.
Our result requires the following Lemma:
Lemma 5. Any real valued function f (x) satisfies the inequality, (17) where x 2 > x 1 are in the domain of f .
− 1 x 2 − x 1 x2 x1 f (x)dx 2 ≥ − x2 x1 f 2 (x)dx
Proof. The proof is provided in the appendix. 2
It will now be shown that any solution to the closed-loop dynamics (15) cannot satisfy w 1 (T ) = 0 for any finite time T < ∞. Lemma 6. The set U is positively invariant. That is, if (w(0), p(0)) ∈ U , (w(t), p(t)) ∈ U for all time t ≥ 0.
Proof. First note that the time derivative of H d satisfieṡ
H d = −∇ p H d D d ∇ p H d < ∇ p H d D i ∇ p H d ≤ 0.(18)
As H d is quadratic in p, w, (18) this implies that for any solution with initial conditions in U , p(t) and w(t) will be bounded over any time interval in which the solution is contained within U . We denote such an interval as ∆t = [0, T ). Considering Q w in (11), as p(t) is bounded ∆t, w 1 (t) is bounded on the same time interval. Boundedness ofẇ 1 implies that lim t→T w 1 (t) exists for all T . Now, for the sake of contradiction, assume that lim t→T w 1 (t) = 0 for some finite T ∈ [t 0 , ∞). Taking any interval [t 1 , T ], such that t 1 ≥ 0, pick t such that
w 1 (t ) = max{w 1 (t)} ∀t ∈ [t 1 , T ]. From (18) it can be verified that time derivative of H d satisfieṡ H d (t) ≤ − k w 2 1 (t) ∇ p2 H 2 d (t) = − k w 2 1 (t)ẇ 2 1 (t)(19)
Integrating with respect to time from t to T
H d (T ) − H d (t ) ≤ − T t k w 2 1 (t)ẇ 2 1 (t)dt(20)
As w 1 (t ) = max{w 1 (t)}∀t ∈ [t 0 , T ],
H d (T ) − H d (t ) ≤ − k w 2 1 (t ) T t ẇ 2 1 (t)dt(21)
Applying Lemma 5 to this inequality yields
H d (T ) − H d (t ) ≤ − k w 2 1 (t ) 1 T − t T t ẇ 1 (t)dt 2 ≤ − k w 2 1 (t ) 1 T − t (w 1 (T ) − w 1 (t )) 2 ≤ − k w 2 1 (t ) 1 T − t w 2 1 (t ) ≤ − k T − t .
(22) As T −t ≤ T −t 1 is arbitrarily small, the right hand side of this inequality can be made arbitrarily large by choosing a small enough time interval. However, H d is lower bounded, thus we have a contradiction. Thus, we conclude that there is no finite T such that lim t→T w 1 (t) = 0. As a consequence, U is positively invariant. 2
As the set U is positively invariant, the closed-loop dynamics (15) are well defined for all time. We now show that w 1 (t) tends to zero asymptotically. This will be done by considering two properties. Firstly, asḢ d ≤ 0, the trajectories (w(t), p(t)) are confined to a compact set. Secondly, it is shown that there is no subset of U that satisfieṡ H d = 0 identically. Combining these two properties, it can be deduced that w 1 → 0. Lemma 7. Consider the closed-loop dynamics (15). On the set U there is no solution to (w(t), p(t)) satisfyingḢ d = 0 identically.
Proof. From (18), it can be seen that time derivative of
H d satisfiesḢ d ≤ −p M −1D M −1 p.(23)
AsD, M > 0, for (23) to be identically equal to zero, p must be identically equal to zero. This means thatṗ = 0 along such a solution.
Evaluating theṗ dynamics of (15) at p =ṗ = 0 results in
−Q w (w)Lw = − Q (z) ∂ f w ∂z z=f −1 w (w) Lw = 0 2 . (24)
Recalling that Q z has a left annihilator given by (7), (24) is
satisfied if ∂ f w ∂z z=f −1 w (w) Lw = Q ⊥ z z=f −1 w (w) a(w),(25)
where Q ⊥ z is defined by (24) and a ∈ R is an unknown, possibly state dependant, function. Rearranging (25) results in
Lw = ∂ f −1 w ∂w Q ⊥ z z=f −1 w (w) a.(26)
Using the definition of f −1 w in (9) and Q ⊥ z in (7), (26) can be evaluated to find
L w 1 w 2 w 3 = 1 w 2 + w 3 w 1 w 3 0 w 1 0 0 w 1 1 2 w 2 1 −w 1 w 2 − w 1 w 3 0 1 a. (27)
The second row of (27) implies that w 2 = 0. Substituting w 2 into the first row of (27) implies that w 1 = 0. However, such a solution is not contained in U , thus, there is no trajectory in U such thatḢ d = 0 identically. 2
We are now in a position to determine the asymptotic behaviour of the closed-loop system (15). The typical approach to verifying asymptotic properties of pH systems is to first show that the system is stable as H d (t) ≤ H d (0). Then asymptotic stability is shown by application of LaSalle's theorem together with some detectability requirements. Here, there are two problems with this approach. Firstly, the dynamics are ill-defined at w 1 = 0, thus, the point (w, p) = (0 3×1 , 0 2×1 ) cannot be an equilibrium-although it behaves just like one in the sense that if (w, p) starts small, it stays small. Secondly, as the system dynamics are not defined for w 1 = 0, LaSalle's theorem does not apply. The following Proposition provides an argument which is similar in nature to LaSalle's theorem to show that w 1 (t) tends towards 0. Considering the transformation (9), this means that z(t) tends towards 0 3×1 , satisfying the control objective. Proposition 8. Consider the closed-loop dynamics (15) with initial conditions (w(0), p(0)) ∈ U . The system verifies:
(i) For each > 0 there exists a δ( ) > 0 such that ||(w(0), p(0))|| < δ =⇒ ||(w(t), p(t))|| < . (ii) lim t→∞ q(t) = 0 3×1 .
Proof. For this proof, we let x = (p, w). Noting that aṡ H d ≤ 0 for all time, H d (t) ≤ H d (0). As H d is quadratic in w and p, claim (i) can be verified to be true. Furthermore, this means that the set
{x|H d (x) ≤ H d (0)}(28)
is both bounded and positively invariant.
The proof of claim (ii) follows from similar argument to LaSalle's invariance principle. The proof is as follows:
First note that lim t→∞ H d = H L exists and is in the set [0, H d (0)] as H d (t) is monotonic and bounded below by zero. Now define the set
V = U ∩ {x|H d (x) ≤ H d (0)}(29)
which is bounded. By Lemma 6, together with claim (i),
the set V is positively invariant. Let x(x 0 , t) denote the solution such the x(x 0 , 0) = x 0 ∈ V .
Consider a solution x(x 0 , t) to the system. As the right hand side of (15) is smooth on V , it is locally Lipschitz. Thus, the solution x(x 0 , t) exists and is unique for all time.
By the Bolzano-Weierstrass theorem, the solution admits an accumulation point as t → ∞. The set of all accumulation points is denoted L + . Furthermore, L + is compact and x(t) → L + as t → ∞. (See Section C.3 Khalil for details).
Now suppose that W = L + ∩ V = ∅. By definition, for each y ∈ W , there exists a sequence t n such that lim n→∞ x(t n ) = y. As H d is continuous and lim t→∞ H d = H L , H d (W ) = H L .
By the continuity of solutions on V and claim (i), a solution x(y, t) is contained in W . Thus, such a solution satisfiesḢ d (t) = 0.
But by Proposition 7, there is no solution in the set U satisfyingḢ = 0 identically. Thus we conclude that W = ∅ L + is contained in the set
V \ V = {x|H(x) ≤ H(0), w 1 = 0}.(30)
As x(t) → L + , w 1 → 0.
Considering the transformation f −1 w in (9) and the fact that w(t) is bounded, w 1 (t) → 0 implies that z(t) → 0 3×1 . Then, considering the transformation f z in (4), z(t) → 0 3×1 implies that q(t) → 0 3×1 as desired. 2
Notice that although q tends towards the origin, the asymptotic behaviour of p has not been established. Clearly p(t) ∈ L ∞ as 1 2 p M −1 p < H d (t) ≤ H d (0) for all time. Further analysis is considered beyond the scope of this paper and left as future work.
SIMULATION RESULTS
In order to demonstrate the effectiveness of the control strategy, a numerical simulation was preformed. The parameters used for the open-loop Chaplygin system were m = 2, J = 1, l = 1 and Fig. 2. Chaplygin sleigh initialised from several initial conditions. The system is controlled by a discontinuous, potential energy shaping controller which drives the configuration to the origin.
D = 1 0.1 + p 2 1 0 0 1 0.1 + p 2 2 .(31)
The expression
1 √ 0.1+p 2 i
is an approximation of Coulomb friction (Gómez-Estern and van der Schaft, 2004) and assumed to be unknown for control purposes. The inertial parameters, m and J, are also assumed to be unknown.
The control law (13) was utilised for control with the following parameters:
L = diag(2, 0.5, 0.8) k = 0.1 D = diag(4, 8).(32)
The Chaplygin sleigh was initialised from an assortment of positions and the simulation was run for 100 seconds. The resulting path of each simulation is shown in Figure 2 where the ghosted images of the sleigh represent the initial positions and the solid image of the sleigh is the target final position. The time histories of the configuration, momentum and control signals for each run can be found in Figures 3, 4 and 5, respectively. Notice that although we have not proved convergence of the functions p(t) or u(t), they appear well behaved in the numerical simulation.
Notice that the simulation plotted in blue, which was initialised at (x, y, θ) = (−3, −2, 1 8 π), takes a rather inefficient path to the origin. This is because it was initialised close to the set parametrised by w 1 = z 1 = θ = 0, on which the control law and closed-loop are not defined. As a result of this singularity, initial conditions close to θ = 0 will have a large initial closed-loop Hamiltonian H d . To dissipate this energy from the closed-loop, the system traverses a long path before converging. Simulations with initial conditions away from this singularity take more 'natural' paths. In this paper we presented a discontinuous control law for the Chaplygin sleigh system which is robust against both the inertial parameters and damping structure of the open-loop system. The proposed control law is successful in driving the configuration of the system to the origin. The results were demonstrated by performing numerical simulations to verify the theoretical claims. In future work, we aim to both extend the analysis to characterise the behaviour of the momentum and control signals as well as extend the controller to apply to a wider class of nonholonomic systems.
Fig. 3 .Fig. 4 .
34Time history of the configuration variables of the closed-loop Chaplygin sleigh system from several simulation scenarios. Time history of the momentum variables of the closed-loop Chaplygin sleigh system from several simulation scenarios.6. CONCLUSION
Fig. 5 .
5Time history of the control signals of the closedloop Chaplygin sleigh system from several simulation scenarios.
Appendix A. PROOFSProof of Lemma 5 By the Schwarz inequality(Lieb and Loss, 2001), any two real valued functions f (x), g(x) satisfyTaking the negative of this inequality results inas desired. 2
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A globally exponentially convergent immersion and invariance speed observer for mechanical systems with nonholonomic constraints. A Astolfi, R Ortega, A Venkatraman, Automatica. 461Astolfi, A., Ortega, R., and Venkatraman, A. (2010). A globally exponentially convergent immersion and invari- ance speed observer for mechanical systems with non- holonomic constraints. Automatica, 46(1), 182-189.
Nonholonomic Mechanics and Control. A Bloch, J Baillieul, P Crouch, J Marsden, Springer-VerlagNew YorkBloch, A., Baillieul, J., Crouch, P., and Marsden, J. (2003). Nonholonomic Mechanics and Control. Springer-Verlag, New York.
Control and stabilization of nonholonomic dynamic systems. A Bloch, M Reyhanoglu, IEEE Transactions on Automatic Control. 3711Bloch, A. and Reyhanoglu, M. (1992). Control and stabilization of nonholonomic dynamic systems. IEEE Transactions on Automatic Control, 37(11), 1746-1757.
Asymptotic stability and feedback stabilization. R W Brockett, Differential Geometric Control Theory. BostonBirkhauserBrockett, R.W. (1983). Asymptotic stability and feedback stabilization. In Differential Geometric Control Theory, 181-191. Birkhauser, Boston.
Switched PassivityBased Control of the Chaplygin Sleigh. J Ferguson, A Donaire, R H Middleton, Proc. IFAC Symposium on Nonlinear Control Systems. IFAC Symposium on Nonlinear Control SystemsMontereyElsevier B.VFerguson, J., Donaire, A., and Middleton, R.H. (2016). Switched PassivityBased Control of the Chaplygin Sleigh. In Proc. IFAC Symposium on Nonlinear Control Systems, 1012-1017. Elsevier B.V., Monterey, Califor- nia.
Passivity based control of a class of Hamiltonian systems with nonholonomic constraints. K Fujimoto, S Sakai, T Sugie, Automatica. 4812Fujimoto, K., Sakai, S., and Sugie, T. (2012). Passivity based control of a class of Hamiltonian systems with nonholonomic constraints. Automatica, 48(12), 3054- 3063.
Classical Mechanics. H Goldstein, Addison-WesleyReading, MA2 editionGoldstein, H. (1980). Classical Mechanics. Addison- Wesley, Reading, MA, 2 edition.
Physical damping in IDA-PBC controlled underactuated mechanical Systems. F Gómez-Estern, A Van Der Schaft, European Journal of Control. 105Gómez-Estern, F. and van der Schaft, A. (2004). Physi- cal damping in IDA-PBC controlled underactuated me- chanical Systems. European Journal of Control, 10(5), 451-468.
Passivity-Based Switching Control for Stabilization of Wheeled Mobile Robots. D Lee, Proc. Robotics: Science and Systems. Robotics: Science and Systems8Lee, D. (2007). Passivity-Based Switching Control for Stabilization of Wheeled Mobile Robots. In Proc. Robotics: Science and Systems, 8.
Analysis. E H Lieb, M Loss, American Mathematical SocietyProvidence, RI2 editionLieb, E.H. and Loss, M. (2001). Analysis. American Mathematical Society, Providence, RI, 2 edition.
Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control. Y P Tian, S Li, Reports on Mathematical Physics. Automatica. van der Schaft, A. and Maschke, B.M.342On the Hamiltonian formulation of nonholonomic mechanical systemsTian, Y.P. and Li, S. (2002). Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control. Automatica. van der Schaft, A. and Maschke, B.M. (1994). On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics, 34(2), 225- 233.
| []
|
[
"Luminosity Functions of XMM-LSS C1 Galaxy Clusters Accepted XXXX Xxxxx XX. Received XXXX Xxxxx XX; in original from XXXX Xxxxx XX",
"Luminosity Functions of XMM-LSS C1 Galaxy Clusters Accepted XXXX Xxxxx XX. Received XXXX Xxxxx XX; in original from XXXX Xxxxx XX"
]
| [
"Abdulmonem Alshino \nSchool of Physics and Astronomy\nThe University of Birmingham\nB15 2TTBirminghamUK\n",
"Habib Khosroshahi \nSchool of Astronomy\nInstitute for Research in Fundamental Sciences (IPM)\nP. O. Box19395-5531TehranIran\n",
"Trevor Ponman \nSchool of Physics and Astronomy\nThe University of Birmingham\nB15 2TTBirminghamUK\n",
"Jon Willis \nDepartment of Physics and Astronomy\nUniversity of Victoria\nElliot Building, 3800 Finnerty RoadV8P 1A1VictoriaBCCanada\n",
"Marguerite Pierre \nDAPNIA/SAp\nCEA Saclay\n91191Gif sur YvetteFrance\n",
"Florian Pacaud \nArgelander-Institut für Astronomie\nUniversity of Bonn\nAuf dem Hügel 7153121BonnGermany\n",
"Graham P Smith \nSchool of Physics and Astronomy\nThe University of Birmingham\nB15 2TTBirminghamUK\n"
]
| [
"School of Physics and Astronomy\nThe University of Birmingham\nB15 2TTBirminghamUK",
"School of Astronomy\nInstitute for Research in Fundamental Sciences (IPM)\nP. O. Box19395-5531TehranIran",
"School of Physics and Astronomy\nThe University of Birmingham\nB15 2TTBirminghamUK",
"Department of Physics and Astronomy\nUniversity of Victoria\nElliot Building, 3800 Finnerty RoadV8P 1A1VictoriaBCCanada",
"DAPNIA/SAp\nCEA Saclay\n91191Gif sur YvetteFrance",
"Argelander-Institut für Astronomie\nUniversity of Bonn\nAuf dem Hügel 7153121BonnGermany",
"School of Physics and Astronomy\nThe University of Birmingham\nB15 2TTBirminghamUK"
]
| [
"Mon. Not. R. Astron. Soc"
]
| CFHTLS optical photometry has been used to study the galaxy luminosity functions of 14 X-ray selected clusters from the XMM-LSS survey. These are mostly groups and poor clusters, with masses (M 500 ) in the range 0.6 to 19 ×10 13 M ⊙ and redshifts 0.05 ≤ z ≤ 0.61. Hence these are some of the highest redshift X-ray selected groups to have been studied. Lower and upper colour cuts were used to determine cluster members. We derive individual luminosity functions (LFs) for all clusters as well as redshift-stacked and temperature-stacked LFs in three filters, g ′ , r ′ and z ′ , down to M = −14.5. All LFs were fitted by Schechter functions which constrained the faint-end slope, α, but did not always fit well to the bright end. Derived values of α ranged from −1.03 to as steep as −2.1. We find no evidence for upturns at faint magnitudes. Evolution in α was apparent in all bands: it becomes shallower with increasing redshift; for example, in the z ′ band it flattened from -1.75 at low redshift to -1.22 in the redshift range z =0.43-0.61. Eight of our systems lie at z ∼ 0.3, and we combine these to generate a galaxy LF in three colours for X-ray selected groups and poor clusters at redshift 0.3. We find that at z ∼ 0.3 α is steeper (-1.67) in the green (g ′ ) band than it is (-1.30) in the red (z ′ ) band. This colour trend disappears at low redshift, which we attribute to reddening of faint blue galaxies from z ∼ 0.3 to z ∼ 0. We also calculated the total optical luminosity and found it to correlate strongly with X-ray luminosity (L X ∝ L 2.1 OP T ), and also with ICM temperature (L OP T ∝ T 1.62 ), consistent with expectations for self-similar clusters with constant mass-to-light ratio. We did not find any convincing correlation of Schechter parameters with mean cluster temperature. | 10.1111/j.1365-2966.2009.15734.x | [
"https://arxiv.org/pdf/0909.3810v2.pdf"
]
| 1,336,533 | 0909.3810 | bc63a5c730c8d2e1fdc00d6130eef2180b3df12b |
Luminosity Functions of XMM-LSS C1 Galaxy Clusters Accepted XXXX Xxxxx XX. Received XXXX Xxxxx XX; in original from XXXX Xxxxx XX
2009
Abdulmonem Alshino
School of Physics and Astronomy
The University of Birmingham
B15 2TTBirminghamUK
Habib Khosroshahi
School of Astronomy
Institute for Research in Fundamental Sciences (IPM)
P. O. Box19395-5531TehranIran
Trevor Ponman
School of Physics and Astronomy
The University of Birmingham
B15 2TTBirminghamUK
Jon Willis
Department of Physics and Astronomy
University of Victoria
Elliot Building, 3800 Finnerty RoadV8P 1A1VictoriaBCCanada
Marguerite Pierre
DAPNIA/SAp
CEA Saclay
91191Gif sur YvetteFrance
Florian Pacaud
Argelander-Institut für Astronomie
University of Bonn
Auf dem Hügel 7153121BonnGermany
Graham P Smith
School of Physics and Astronomy
The University of Birmingham
B15 2TTBirminghamUK
Luminosity Functions of XMM-LSS C1 Galaxy Clusters Accepted XXXX Xxxxx XX. Received XXXX Xxxxx XX; in original from XXXX Xxxxx XX
Mon. Not. R. Astron. Soc
0002009Printed (MN L A T E X style file v2.2)galaxies: clusters: general -galaxies: evolution -galaxies: luminosity function - galaxies: structure
CFHTLS optical photometry has been used to study the galaxy luminosity functions of 14 X-ray selected clusters from the XMM-LSS survey. These are mostly groups and poor clusters, with masses (M 500 ) in the range 0.6 to 19 ×10 13 M ⊙ and redshifts 0.05 ≤ z ≤ 0.61. Hence these are some of the highest redshift X-ray selected groups to have been studied. Lower and upper colour cuts were used to determine cluster members. We derive individual luminosity functions (LFs) for all clusters as well as redshift-stacked and temperature-stacked LFs in three filters, g ′ , r ′ and z ′ , down to M = −14.5. All LFs were fitted by Schechter functions which constrained the faint-end slope, α, but did not always fit well to the bright end. Derived values of α ranged from −1.03 to as steep as −2.1. We find no evidence for upturns at faint magnitudes. Evolution in α was apparent in all bands: it becomes shallower with increasing redshift; for example, in the z ′ band it flattened from -1.75 at low redshift to -1.22 in the redshift range z =0.43-0.61. Eight of our systems lie at z ∼ 0.3, and we combine these to generate a galaxy LF in three colours for X-ray selected groups and poor clusters at redshift 0.3. We find that at z ∼ 0.3 α is steeper (-1.67) in the green (g ′ ) band than it is (-1.30) in the red (z ′ ) band. This colour trend disappears at low redshift, which we attribute to reddening of faint blue galaxies from z ∼ 0.3 to z ∼ 0. We also calculated the total optical luminosity and found it to correlate strongly with X-ray luminosity (L X ∝ L 2.1 OP T ), and also with ICM temperature (L OP T ∝ T 1.62 ), consistent with expectations for self-similar clusters with constant mass-to-light ratio. We did not find any convincing correlation of Schechter parameters with mean cluster temperature.
INTRODUCTION
Most of our knowledge of galaxies is based on observations of the local universe, although distant universe observations have also provided a wealth of information. Statistical studies of galaxies at high redshift are mostly limited to rich galaxy clusters mainly due to observational limitations. Galaxy clusters are important cosmological environments where key galaxy transformation such as stripping and strangulation occur. However, in the hierarchical formation of structure rich clusters are the latest structures to be formed. Lower mass systems or galaxy groups may have been the place where galaxies experience a substantial degree of evolution through processes such as mergers and tidal interaction, as a re-⋆ E-mail: [email protected] sult of the higher efficiency of these processes in the lower velocity dispersion environment of groups.
The Galaxy luminosity function (LF) -the number of galaxies per unit volume in the luminosity interval L to L + dL -has been widely used to study the formation of galaxies and the evolution of galaxy populations with redshift. It is also an excellent statistical tool for describing how different environments influence the properties of galaxies.
Both the bright end (Bower et al. (2006), Naab et al. (2007)) and the faint end (Marzke et al. (1994), Khochfar et al. (2007)) of the LF have been the subject of in-depth studies, as they offer strong observational constraints for models of galaxy formation and evolution. While the bright end of the LF is affected by AGN feedback (Bower et al. (2006)), the faint-end slope is predominantly influenced by feedback from supernovae (Dekel et al. (1986)), and provide a direct indicator of the significance of dwarf galaxies, which are expected to behave differently in rich and poor clusters. Multi-colour LFs, in particular, probe the history of the faint galaxy population, including its star formation history -see for example, Adami et al. (2007).
The vast majority of studies of the galaxy LF give faint-end slopes in the range ∼ −1 to ∼ −2. Most of these have limited magnitude depth (M > −16) and recent deep studies are mostly confined to rich local clusters (See Table 1 in Popesso et al. (2005a) and Table A.1 in Boué et al. (2008) and references therein). These studies not only disagree on the value of the faint-end slope, but they also disagree on the exact form of it, as some studies (e.g. Gonzàlez et al. (2006)) found upturns; a single Schechter function was not an adequate fit to the faint end, and a double Schechter function was required to give a reasonable fit. The existence of these upturns is very sensitive to the method used to determine galaxy membership, with some approaches including spurious galaxies or excluding genuine cluster members due to their low surface brightness.
The evolution of the faint-end slope is hard to study, mainly because the number of faint galaxies detected decreases sharply with increasing redshift. Liu et al. (2008) found that the faint-end slope of a field galaxy population became shallower with increasing redshift (up to z = 0.5) for all galaxy spectral types. However, to account for the photometric redshift errors of the galaxies, they weighted the galaxies as probability-smoothed luminosity distribution at the redshift at which they were measured. This places an important caveat on the interpretation of their data, and hence on their results. On the other hand, simulations by Khochfar et al. (2007) show a measurable dependence of the faint-end slope of the galaxy luminosity function on redshift. However, most of this dependence is seen over a relatively large redshift range, ∆z ≥ 2. Furthermore, it is hard to discriminate galaxy environments in such studies.
X-ray surveys remain one of the most popular methods of finding galaxy systems. Due to the strong density dependence of X-ray emissivity, X-ray cluster selection is much less vulnerable to contamination along the line-of-sight than optical methods. The XMM-Large Scale Survey (XMM-LSS) (Pierre et al. (2004)), a contiguous X-ray survey, has a well-defined selection function which is used to produce a sample of galaxy groups to study their intracluster medium and galaxy properties at medium to high redshift. Pacaud et al. (2007) have presented a study of a sample of 29 galaxy systems from the XMM-LSS survey, drawn from an area of 5 deg 2 out to a redshift of z = 1.05. The cluster distribution peaks around z = 0.3 and T=1.5 keV, half of the objects being groups with a temperature below 2 keV.
In this paper, we use the XMM-LSS optical follow-up observations to study the evolution of the galaxy luminosity function in galaxy groups and poor clusters since z ≈ 0.6. Given the observational biases -distant groups are more massive and hotterwe study whether the redshift dependencies are weaker or stronger when the intrinsic properties of the systems, for instance, intracluster medium temperature, are taken into account.
By using a deep (m g ′ = 24) optical survey of X-ray selected galaxy clusters up to redshift of z = 0.61, we aim in this paper to clarify the debate on the faint-end slope of the LFs of low-mass (M500 ≤ 20×10 13 M⊙) galaxy clusters (or groups), and to explore the existence of any dips, or upturns at the faint end, and to establish whether the slope shows trends with redshift or intracluster medium temperature. Comparison with previous results can help to elucidate the universality of galaxy cluster LFs. Furthermore, the scaling relation of total optical luminosity with temperature and X-ray luminosity for our cluster sample can shed light on the massto-light ratios of low-mass systems when compared to rich clusters.
The paper is constructed as follows: In section 2, we describe the optical catalogue used to calculate the LFs. Then, we describe the data reduction and the method used to construct the colourmagnitude diagrams (CMD) and the subsequent LFs, and the technique adopted for the background subtraction. In section 3, we describe our results, starting with the individual cluster LFs, and then the redshift-stacked clusters and temperature-stacked clusters. In section 4, we discuss our results and compare them with other studies. Finally, in section 5, we summarise our conclusions.
Throughout this article, we adopt the cosmological parameters estimated by Spergel et al. (2007), namely: H0 = 73 km s −1 Mpc −1 , Ωm = 0.24, ΩΛ = 0.76.
DATA
Observations
Optical photometry of the XMM-LSS survey was obtained from the Canada-France-Hawaii Telescope Wide Synoptic Legacy Survey 1 , referred to as the CFHTLS Wide survey. Data were obtained in five passbands (u * , g ′ , r ′ ,i ′ , z ′ ) down to a nominal magnitude limit of i ′ = 24.5. Of the 19 deg 2 of CFHTLS Wide data available in the W1 survey area, 4 deg 2 overlap with the X-ray selected cluster catalogue presented by Pacaud et al. (2007). Hence our photometric data are drawn from four 1 • × 1 • catalogues derived from the survey data.
The data used in this paper are based upon the reduction procedure outlined in Hoekstra et al. (2006). Source extraction and photometry were performed using SExtractor v2.5.0 (Bertin and Arnouts 1996). Zero point information for sources detected in the CFHTLS Wide field survey W1 area was extrapolated from common sources detected in the Sloan Digital Sky Survey equatorial patch which overlaps the southern edge of the W1 area.
XMM-LSS Class 1 (C1) clusters are a well-controlled X-ray selected and spectroscopically confirmed cluster sample. The criteria used to construct the sample guarantee negligible contamination by point-like sources. The observations of the clusters were performed in a homogeneous way (10-20 ks exposures). For full details of the C1 sample, see Pacaud et al. (2007). The main properties of the sample are shown in Table 1. Detailed information on the C1 selection process can be found in Pacaud et al. (2006). 17 out of the 29 XMM-LSS C1 clusters are covered by the CFHTLS Wide field survey. In this paper, we study the luminosity functions of 14 of these 17 clusters -dropping the three with the highest redshifts (clusters with XLSSC numbers 2,29, and 1) because their photometric data is too poor to allow useful constraints to be obtained.
Analysis
Galaxies were detected by SExtractor (Bertin & Arnouts (1996)). Luminosity functions (LFs) were produced in three of the five CFHTLS (u * , g ′ , r ′ ,i ′ , z ′ ) filter bands, namely, g ′ , r ′ and z ′ . To determine the completeness of the LFs, we took into account the limiting apparent magnitude in each field. The completeness limits for each filter was determined using the apparent magnitude LFs of all data (down to the faintest magnitudes available) for each C1 cluster individually. Variations in seeing and exposure time across the CFHTLS fields used here are small, and it was found that for each filter there was a common completeness limit at which all LFs started to drop below the faint end power law slope. Note that the LF turn-up reported by some authors (see section 4.2), which could potentially introduce an error into this method for estimating completeness, falls beyond the faint limit of our LFs (e.g. at -16 in g ′ band), except in our three closest clusters, and hence cannot seriously bias our estimates of the completeness limits. The completeness threshold magnitudes for the three filters g ′ , r ′ and z ′ were found to be 24, 23.5 and 23, respectively. These values are also consistent with results based on comparison of the number counts per field to deeper data from the CFHTLS Deep Field and CCCP Megacam observations (Urquhart et al. (2009)).
Each entry in the catalogues is associated with a FLAG value which indicates the degree of reliability of the data. Flag is a short integer, and a value of 0 denotes good data. The more unreliable the data is, the higher the FLAG value becomes. We included all catalogue entries with FLAG ≤ 3, which includes sources with very close and bright neighbours or some bad pixels and sources which are originally blended with other sources. This may admit some problematic galaxies but this is better than excluding many genuine cluster members, because many clusters contain significant number of blended sources. Factors that may raise the FLAG to > 3 include sources with saturated pixels, truncated sources, incomplete or corrupted data and sources with memory overflow during deblending or extraction. Catalogue entries with FLAG > 3 constitute only ≃ 5% of the total number of entries, and were all removed. Many of the removed entries are fainter than the threshold magnitude and hence would have been removed anyway.
Each entry in the catalogue also includes a stellarity class value, STAR, with values ranging from 0 to 1. The lower its value, the more likely the detected object is a galaxy. Data points with different STAR values were checked by IRAF and their radial profiles were examined to see if they matched the typical profile of a star or a galaxy. Typically, objects with STAR > 0.85 were found to be stars, whilst those with < 0.85 were galaxies. Therefore, only catalogue entries with STAR values of 0.85 or less were included when constructing the LFs.
Spectral temperatures of the XMM-LSS clusters, and the resulting values of R500 (which allow for the evolution in critical density with redshift), were measured by the Saclay team (Pacaud et al. (2007)). To construct colour-magnitude diagrams, we selected all galaxies within a circle of radius, R * = 1.5 × R500 of the clusters. This radius limit, R * , represents an estimate of R200. Colourmagnitude diagrams were produced for all 14 clusters for colour bands: g ′ , r ′ and z ′ . The factor, 1.5 does not have a large effect on the fitted parameters of the Schechter function; we compared the results of 1.0 × R500 to 1.8 × R500 and found that the faint-end slope, α, was only changed within its 1 σ errors.
The CMD were used to colour select galaxies which might be cluster members, hence reducing the background due to interlopers. The colours used for this were u * 2 − g ′ 2 versus g ′ kron , g ′ 2 − r ′ 2 versus r ′ kron , and i ′ 2 − z ′ 2 versus z ′ kron for the three filters respectively, where the subscript 2 refers to the 2 arcsec aperture used in the magnitude measurements. To define and select cluster members in the CMD, we defined upper and lower colour cuts and only galaxies between these two lines were used to produce the LF, as galaxies outside these limits were most likely not cluster members. To define these two colour cuts, we first defined the red sequence line in the CMD and then pushed this line up and down to allow for statistical errors, and for the likely range of galaxy colours.
To define the red sequence line, we first defined the slope and then its Y-intercept. We checked that the BCGs lay at the centre of the X-ray emission in all clusters, and then calculated the red sequence slope in each case by fitting a straight line to the bright galaxies in the CMD. Bright galaxies are defined as those with magnitude ranging from that of the brightest cluster galaxy, mBCG, to a magnitude of mBCG + 3, inclusive. We found that the slope of the red sequence line for the 14 C1 clusters showed a mild trend with redshift: it was steeper for high-redshift clusters. A similar trend was observed by Gilbank et al. (2008) and attributed to a deficit of faint red galaxies at high redshifts, consistent with the galactic downsizing picture.
We divided our cluster sample into two redshift ranges: lowredshift clusters (z < 0.2) and intermediate-redshift clusters (0.2 ≤ z ≤ 0.61). Clusters from each group share a common red sequence line slope with a small variation. The common slope for the low-redshift range was -0.007 and -0.025 for the second range. Instead of using a different red sequence slope for each cluster, we used the common slope of the redshift groups for all clusters belonging to that redshift group.
The Y-intercepts of the red sequence lines were different for each galaxy cluster and depended on the average colour of the bright galaxies as defined above. To fix the value of the intercept for each cluster, the red sequence line was normalised so as to pass through the point in the CMD which has a magnitude of mBCG + 1.5 and colour equal to the average colours of the bright galaxies. This point and the value of the slope completes the definition of the red sequence line.
Both upper and lower colour cuts have the same slope as the red sequence line. In order to define the upper colour cut, we have to determine the upper (red) limit to the cluster red sequence. We took into account the statistical scatter of the colours of the faintest galaxies on the red sequence. These galaxies are defined as those inside a 1.0 × 0.1 (magnitude by colour units) box in the CMD centred on the faint end of the red sequence line (see figure 1 ). The size of this box was chosen to include the faintest galaxies most probably belonging to the red sequence after studying the CMD of the C1 sample. The expected scatter of these galaxies, σ, is calculated by averaging their colour errors, that is the Y-axis errors in the CMD. The upper colour cut, is then taken to be the red sequence line pushed upward by 2σ. By taking into account this scatter, we ensure that almost all genuine cluster red sequence galaxies should fall beneath the red cut, since the statistical error on the brighter galaxies will be smaller.
Similarly, the lower (blue) colour cut is the red sequence line pushed downward in the CMD. In this case, the shift has to account for both statistical scatter, and for the fact that late-type cluster galaxies are intrinsically bluer than red sequence galaxies. The shift was therefore taken to be −(2σ + △). Where △ is the theoretical colour difference between ellipticals and spirals. This was estimated using a simple model which calculates what colour latetype galaxies would have when redshifted by different amounts, as described in King & Ellis (1985). △ is a function of redshift only and the redshift of the galaxy cluster was used to determine its value. This method of estimating △ ignores any intrinsic evolution in the colour offset between red and blue cluster galaxies. However, the detailed COMBO-17 study of Bell et al. (2004) (see their Figure 1) shows that the colour difference between blue and red sequence cluster galaxies changes little over the redshift range (0-0.7) spanned by our clusters. Figure 1 shows an example of our use of colour cuts for selection of cluster galaxies.
Of course, background and foreground galaxies will still con- taminate the sample after the colour cut has been applied, and this contamination must be estimated and removed statistically. For this purpose we used all data in the catalogue to which a given galaxy cluster belonged. In addition to simple Poisson fluctuations, uncertainties in removing background and foreground galaxies arise from large scale structure. To quantify the extra fluctuations arising from this, we proceeded as follows. The whole catalogue 1 • × 1 • area was divided into smaller blocks with areas comparable to that of the cluster in question. Any of these blocks covered mostly (60% or more, by area) by a galaxy cluster, were considered to be dominated by a cluster and hence were discarded from the background calculation. Blocks covered by clusters to an extent less than 60%, were not discarded but the portion covered by the R * circle of any galaxy cluster was removed, so the final blocks used have somewhat different areas. For each background block, an LF was produced in just the same way as for the cluster itself. The same values of the upper and lower red sequence limits of the galaxy cluster in question, were applied to all its background block areas, so galaxies beyond those limits were removed. The application of colour cuts to both source and background fields reduces the noise level in both of them, and hence in the final background-subtracted LF.
We then divided each background block LF by its area, added them and normalised the resulting single LF to the area of the galaxy cluster in question. The error bars on the averaged background LF were calculated from the scatter of the individual block LFs contributing to it. This method of estimating the background has some advantages over the more conventional background estimation method using an outer annulus around the galaxy cluster, since it uses a large background region, and the error estimate allows for the variance arising from the large scale structure. Finally, for each cluster we subtracted its composite background LF from the cluster LF, and propagated the errors. Apparent magnitudes were converted to absolute magnitudes, using the distance for each cluster, and applying K-corrections calculated from Table 3 (for Hubble type E) in Frei & Gunn (1994). The use of early-type K-corrections is common in cluster studies, and justified by the dominance of early-type galaxies in clusters. However, if there were a systematic trend in early-type fraction with magnitude, then this could lead to some distortion of the LF slope. To quantify the maximum possible effect, we note that, using the tables in Frei & Gunn (1994), the K-correction for Hubble type E at z=0.6 for z ′ is 0.37 while it is 0.05 for Hubble type Im). Assuming (very conservatively) a systematic change from a 100% early-type to 100% late-type population across the faint end slope of our LFs, the impact of a differential error of 0.3 magnitudes on our determination of α would still only amount to ∆α ≈ 0.04, which is small compared to the trends in alpha which represent some of our main results. The tables in Frei & Gunn (1994) apply to SDSS filters, which differ slightly from the corresponding Mega-Cam filters. The resulting differences in K-corrections are much smaller than the differences between early-type and late-type galaxies (about 0.03 at z=0.1 and 0.06 at z=0.6), and will have negligible effect on our derived LF slopes.
Finally, the data were binned into bins of width 0.5 magnitude (experiments showed that this bin size was a good choice in terms of fit quality and parameter confidence regions), and the resulting LFs were then fitted by a Schechter function model (Schechter (1976)
), φ(M )dM = 0.4ln(10)φ * e −X X 1+α dM,
where X = 10 −0.4(M −M * ) , M * is the characteristic magnitude, φ * is the characteristic number density and α is the faint-end slope, Lin et al. (1996). Contour plots of the 1σ, 2σ and 3σ confidence levels of α and M * were also produced. The errors in the text and tables refer to the 1σ errors. We also calculated the total optical luminosity LOP T of each cluster by integrating the fitted Schechter function from 5 × M * to -16.
In addition to single LFs for each galaxy cluster in our sample, we produced stacked luminosity functions. The radius used to determine the volume is the R * of the cluster. Before stacking different clusters together, to correct for the evolution in the critical density of the universe, we multiply the LF by
ρc(z = 0) ρc(z = z cl ) ,
where z is the redshift, ρc is the critical density of the universe, a function of z, and z cl is the redshift of the cluster. This correction is necessary for high-redshift clusters if stacked with low-redshift clusters to scale the galaxy density in each cluster to the density at redshift=0. The faintest magnitude bin is not necessarily the same for each cluster and to account for this, we divided the total number of galaxies in each magnitude bin by the summed volume of galaxy clusters that contributed to that bin only. The stacked LFs should enable us to study the evolution of the LF with redshift and to explore any differences between clusters of different temperature.
RESULTS
Individual cluster luminosity functions
The fitted values of the three Schechter parameters α and M * and LOP T of the individual C1 clusters are presented in (2007)). The three highest redshift clusters (2,29 and 2) though covered by the survey, were not included in our analysis because their data were too poor to yield useful fits.
plane for passbands r ′ and z ′ are shown in Figures A1 and B1 respectively. For some of the C1 clusters, the fitting failed to constrain some of the parameters, M * in particular, due to poor statistics or the lack of any well-defined turnover in the LF at the bright end. For these clusters the LF and best fit are presented without any accompanying confidence contour plot. These LFs are placed at the bottom of the figures. For clusters with unconstrained M * , LOP T was also not constrained, because its value depends on both α and M * . Therefore, we excluded these clusters in the part of the analysis related to LOP T .
The average values of α for our sample of clusters are −1.70 ± 0.10, −1.64 ± 0.04 and − 1.43 ± 0.03
for the g ′ , r ′ and z ′ passbands respectively. The correlations between LOP T , α and M * and redshift, temperature (T ) and the Xray luminosity (LX ) taken from Pacaud et al. (2007), were tested using Pearson's correlation coefficient. These coefficients are computed from the ratio of the covariance of the tested variables, X and Y, to the square root of the product of the variances of these variables, i.e.
r = COV (X, Y ) V AR(X) * V AR(Y ) .
This correlation coefficient measures the linear correlation, if it is 1 or -1 then the two variables are perfectly positively or negatively linearly correlated, respectively. To compute the upper and lower 1 σ errors on the correlation coefficient r, we used Fisher's Z transformation: Z = tan −1 r. The strongest correlations found are those between LOP T and T and between LOP T and LX , both of which are expected from the scaling relations of galaxy clusters. In our sample, they both have a correlation coefficient of at least 0.9, see Table 3.
Because higher redshift clusters are more difficult to detect than nearby ones, they will tend to be more massive and hence hotter than typical nearby clusters, see Figure 3 in Pacaud et al. (2007). This (Malmquist) selection effect is present in any deep cluster survey. To account for the T − z correlation arising from this selection effect in the C1 sample, for each correlation coefficient between a quantity and T or z, we have also calculated the partial correlation coefficient between the same two quantities, which attempts to remove any part of the correlation which arises due to the intrinsic trend in T with z within our sample. For this we used an Interactive Data Language (IDL) routine, p correlate.pro to compute the partial correlation coefficient. This uses the following method, which to be concrete we explain using the example of the correlation between α and redshift. Let α and redshift z are the variables of primary interest, whilst temperature T is a third variable whose effects we wish to remove. First, the routine calculates the residuals after regressing α on T ; these are the parts of α that cannot be predicted by T . Likewise, it calculates the residuals after regressing z on T . Finally, the partial correlation coefficient between α and z, adjusted for T , is the correlation between these two sets of residuals.
The results of our correlation analysis for the unstacked clusters, are tabulated in the top section of Table 3. The correlation coefficients between the faint-end slope, α of the individual clusters and redshift are 0.44±0.27 for the r ′ band and 0.54±0.25 for the z ′ band. These coefficients, including the coefficient for the g ′ band, get stronger after the application of the partial analysis and the errors on the coefficients become smaller. This strongly suggests evolution of α with redshift in our sample. We will further scrutinise this possibility in the section of redshift-stacked clusters, because stacking LFs of clusters with similar redshifts should lower scatter in the data and provide a means to probe possible trends. M * also shows a negative correlation with redshift and with temperature but these correlations become insignificant in a partial correlation analysis.
Global scaling relations
The relationships between the global cluster properties, LOP T , LX and T provide a probe of cluster self-similarity. LOP T is strongly correlated to the temperature of our clusters -the correlation coefficients between LOP T and T are 0.95±0.06, 0.96±0.04 and 0.97±0.03 for the g ′ , r ′ and z ′ bands respectively, whilst the partial Figure 2. Correlation diagrams of L OP T versus X-ray gas temperature, T (top panel) and L OP T versus X-ray luminosity, L X (bottom panel) of C1 clusters for passbands g ′ (stars),r ′ (triangles) and z ′ (squares). Clusters with unconstrained M * and hence unconstrained L OP T were excluded. correlation coefficients for the same quantities, factoring out the effects of z, are 0.87±0.16, 0.89±0.11 and 0.92±0.06, see third row in Table 3. The removal of the z effects has lowered the values of the coefficients but they are still high and significant. Correlation between LOP T and LX is also quite strong: 0.92±0.11 (g ′ band), 0.93±0.07 (r ′ band) and 0.90±0.08 (z ′ band).
In Figure 2, we plot LOP T versus T (top panel) and LOP T versus LX (bottom panel). We calculate the slopes for these plots using the Fortran package ODRPACK (Akritas et al. (1996)), which uses numerical orthogonal distance regression method to minimise perpendicular distances between points and the fitted line. One advantage of this is that the slope value will not change if the quantities in question switch axes. In addition, ODRPACK takes into account errors on both X-values and Y-values which are available for LOP T , T and LX .
The logarithmic slopes for the LOP T − T relation for the three filters g ′ , r ′ and z ′ , respectively are 1.57 ± 0.17, 1.51 ± 0.17 and 1.79 ± 0.12, giving an average value of 1.62 ± 0.11. For the LOP T − LX relation, the slopes are 0.47 ± 0.07, 0.43 ± 0.08 and 0.50 ± 0.07, and the average value is 0.47 ± 0.05. Note that the slopes do not differ significantly for the three filters, except the slope of LOP T versus T in the z ′ filter. Such relations between LOP T on one hand, and LX and the gas temperature on the other, are expected because richer and hence more luminous clusters have deeper gravitational potential wells which in turn raise the ICM temperature and its X-ray output by adiabatic compression and shocks generated by supersonic motion. We will discuss this further in Sec. 4.7.
The correlation coefficients between LOP T and redshift are high (all above 0.8), but when the effects of temperature are removed they become insignificant in at least two of the filter set, therefore, this correlation is most likely due to selection effects, Malmquist effect, and does not reflect any genuine relationship between LOP T and z. Correlations between α and M * and T , z and LX were also computed, but none of those showed significantly high values.
Following the above analysis of trends in the properties of individual clusters, we now perform a stacking analysis, grouping clusters first by redshift, and then by temperature. This provides LFs of higher statistical quality, enabling the behaviour to be examined in greater detail.
Redshift-stacked clusters
The 14 C1 clusters span a redshift range 0.05 to 0.61. This range was divided into five redshift bins: 0.05-0.14, 0.26-0.26, 0.29-0.29, 0.30-0.32 and 0.43-0.61. The number of clusters in each bin ranges from two to four. The redshift ranges of these bins were chosen according to two criteria: first, the redshift range of the combined clusters was not too large, and second we required adequate data quality in each bin, to allow a well-constrained Schechter function fit. We kept the number of bins to at least five because a smaller number of bins increases the errors on the correlation coefficients. Plots of the redshift-stacked data with fitted Schechter functions for the three photometric bands are shown in Figures 10, 11 and 12, and results of the fits are given in Table 4.
The faint-end slope, α of the Schechter function of the stacked data shows an evolutionary trend, becoming less steep with increasing redshift. Three of the redshift bins (0.26-0.26, 0.29-0.29, 0.30-0.32) have very similar redshifts and in general the α values for these three bins agree within their errors.
The Pearson and partial correlation coefficients were calculated for α and z, see Table 3. The coefficients are high (≥ 0.88) but with relatively large errors, mainly due to the small number of bins. The partial correlation analysis lowered the values of the coefficients and enlarged the errors. Evidence for evolution in α is seen in all three bands, arising primarily from the fact that the faint-end slope is steeper (α = −1.75 to -1.8) in the low z bin than in the higher redshift bins.
One obvious concern in probing evolutionary trends in the Schechter function fits is that the fitted magnitude range decreases systematically with redshift, due to the apparent magnitude limit of our data. A second effect which might bias α is that within a given redshift bin, the contributing clusters are probed down to different absolute magnitudes, according to their distance. Hence at the faint end, clusters may progressively drop out of the stacked LF. This is especially the case for the lowest and highest redshift bin, which are both much broader that the three bins at z ≈ 0.3. To show the scale of this latter effect, we have drawn a vertical dotted line on each of the stacked LF plots (9, 10, 11, 12, 13, 14 and 15) to show the faintest magnitude to which all clusters in the bin contribute. To the right (fainter side) of this line, one or more of the clusters in the redshift bin drop out of the stacked data.
To check whether the trend of α with redshift is robust against these two effects, we carried out tests on the stacked data, by progressively removing the faintest magnitude bin in the stacked LFs and re-fitting. In general, we found no significant change in the fitted values of α (which changed only within their errors), or in the Table 3. Pearson's correlation coefficients (P.C. Coeff.) of individual C1 clusters, redshift-stacked clusters and temperature-stacked clusters for the three-filter set (g ′ , r ′ ,z ′ ). 'X,Y,Z (Partial)' denotes partial correlation coefficient of quantities X and Y with effects of quantity Z removed, to be compared with the line directly above it, where correlation coefficient of the same quantities X and Y is presented without partial analysis.
α-z correlation when the LFs were truncated at the vertical dashed line, or when the LFs for all redshift bins were fitted to the same limiting absolute magnitude (which is set by the most distant systems). There was one exception to this. The three clusters in the highest redshift band (clusters 1, 6 and 49) all have α values (albeit with large errors) steeper that the shallow slope of -1.31 which fits to the stacked data in the g ′ band for this high redshift bin. As the faintest bins, to the right of the dashed line in the bottom plot of Figure 10, are progressively removed, the fitted slope steepens. Hence the flat slope of -1.31 must be regarded as unsafe, and the very high α-z correlation in g ′ band, given in Table 3, is probably overestimated. Rather, we have a situation in all three photometric bands, where the faint-end slope is steeper at z < 0.2 than it is at higher redshift.
To visualise the behaviour of the faint-end slope in terms of both redshift and colour, we plot the faint end of the fitted luminosity functions for the three bands in Figure 3 using green for g ′ band, red for r ′ band and black for z ′ band. For this plot, we have divided the sample into three redshift bins: low (0.05-0.14), intermediate (0.26-0.32) and high (0.43-0.61), denoted by different line styles. All LFs have been renormalised to have φ = 1 at M=-19.5.
The Figure shows how the faint end slope steepens towards low z. It also illustrates colour trends in α. At low redshift (solid lines), the slopes are very similar (though the curves are separated The values of α in Table 4 also show a trend with colour. The faint-end slope of z-stacked clusters becomes steeper as we move from z ′ (red side) band to g ′ (blue side). This trend is very obvious in the second, third and fourth redshift bins (0.29 ≤ z ≤ 0.32) and much less obvious and maybe absent (within the errors) in the first bin(z ≤ 0.14), see Figure 4 in which we plotted the values of α for the three bands for the lower-and intermediate redshift bins. The increase in the faint-end slope of the Schechter function in the bluer bands means that at the faint side of the colour-magnitude diagram the blue galaxies outnumber red ones.
To explore this we produced K-corrected colour-magnitude diagram ( Figure 5) of g ′ 2 − z ′ 2 versus absolute r kron magnitude for 0.29 ≤ z ≤ 0.32 (six clusters: 8,13,18,22,27 and 40) in which this trend is most obvious, and the same plot for the first redshift bin, 0.05 ≤ z ≤ 0.14 in which no such trend is apparent. Figure 5 clearly demonstrate how the distribution of cluster galaxy colours changes from z ∼ 0 to z ∼ 0.3. In the g ′ band the evolution of α is much stronger, especially after removing the effects of the temperature (partial correlation). These trends in α show that the fraction of blue faint galaxies at z ∼ 0.3 was larger than it is now, and suggests that these galaxies have reddened and moved The Schechter function characteristic magnitude M * in the redshift-stacked clusters showed a negative correlation with redshift. The correlation coefficients between M * and redshift are high but less significant than those between α and redshift. However, when the partial calculations were carried out, these coefficients dropped and became consistent with zero. Hence the trend in M * with z appears to be due to a selection effect: hotter clusters are more luminous, and so are more easily detected at high redshift, and these brighter clusters also tend to have brighter M * (Zandivarez (2006)).
Luminosity functions of z ∼ 0.3 clusters
Eight amongst the 14 C1 clusters, more than half of our sample, lie within the narrow redshift range 0.26 to 0.32. These clusters are representative of low-mass clusters at intermediate redshifts -a population which dominates the XMM-LSS cluster dataset. Stacking these clusters together provides the best available composite LF for X-ray selected poor clusters at z ∼ 0.3, which should be valuable for future comparative studies. The LFs and their associated error contours are shown in Figures 6, 7 and 8.
These clusters range in temperature from 1.3 to 2.8 keV. Schechter fits give α values -1.66±0.11, -1.50±0.05 and −1.36 ± 0.05, and M * values -21.07±0.38, -22.21±0.22 and −22.83 ± 0.17, for the g ′ , r ′ and z ′ bands respectively. Their faint-end slopes are shallower than the local clusters (z ≤ 0.14) but steeper than higher-redshift (z ≥ 0.43) ones. The colour trend of α is very obvious and seems to be a characteristic of z ∼ 0.3 clusters compared to other clusters in other redshift bins, as mentioned above. Wilman et al. (2005a) studied a sample of poor clusters at redshift z ∼ 0.4 selected from the CNOC2 galaxy redshift survey. Comparing this optically selected sample with nearby clusters, they found that the fraction of passive galaxies declines with redshift, which is consistent with our finding of larger population of faint blue galaxies at z ∼ 0.3. However, these authors did not study the LF of their intermediate redshift groups.
In a recent study, Harsono & Propris (2009) presented composite LFs of six rich (T ∼ 7-9 keV) clusters with redshifts ranging from 0.14 to 0.40 (averaging to 0.246) in the B,g,V,r,i and z bands. The LFs were well fitted by a single Schechter function with α values for g, r and z bands as follows: −1.31 ± 0.04, −1.33 ± 0.03 and −1.45 ± 0.02 and the corresponding M * values were −20.94±0.17, −21.95±0.29 and −22.26±0.30. Their M * values are in reasonable agreement with ours, but their slopes are shallower, and show no trend with colour. However, their data were limited to 20-40% of the area within r200, and they suggest that the lack of any upturn in the slope at faint magnitudes may be related to this -the extra faint galaxies responsible for the upturn being associated with a population infalling into clusters. In contrast, our data extend to 1.5r500, which is approximately equal to r200.
Temperature-stacked clusters
The 14 C1 clusters span a temperature range of 0.64 to 4.80 keV This was divided into five subranges: 0.64-1.00, 1.30-1.34, 1.60-2.20, 2.80-3.20 and 4.80-4.80, using the same criteria, discussed above, which were used for stacking into redshift bins. The highest temperature bin consists of only one cluster because after removing the three high-redshift clusters (2,5,29), the temperature difference between the two highest temperature clusters was too large to stack them together, and the LF of the highest temperature cluster, cluster 6 (T=4.80 keV) was of sufficient quality that it provides useful constraints on its own. The second highest temperature bin consists of two clusters and the rest have at least three clusters.
The correlation coefficients of α with temperature are not high enough to establish any trend, especially when we take into consideration the reverse in sign of the coefficients after the partial correlation calculation, see Table 3. But in Table 4 the highest temperature bin contains only one cluster (cluster 6) and the other bins show some indication that α increases (slope decreases) with temperature, especially in bands r ′ and z ′ . Further investigation is needed to arrive at more conclusive results about the α tend with temperature. As to M * , the stacked results do not show any trend with temperature. The LFs of the temperature stacked data are shown in Figures 13, 14 and 15.
Some previous studies (see for example, Miles et al. (2004)) found that galaxy clusters with low X-ray luminosity (comparable to the coolest clusters in our C1 sample) exhibit dips in their LFs. In our data, some of the temperature stacked LF plots (13, 14 and 15), especially those with high temperature (≥ 2.8 keV) showed signs of dips in the faint end of the LF. It can be hard to distinguish between scatter of the data points and a genuine dip in the LF.
To test the genuineness of these dips we fitted a Schechter function minus a Gaussian function defined by three parameters (central magnitude, width and depth) to these temperature stacked LFs. The two fits, with and without the Gaussian were statistically compared using their χ 2 values, and an F-test applied to assess the significance of the improvement resulting from inclusion of the dip. In some cases the dip improved the fit at a confidence level of more than 90%. See, for example Figure 9.
However, careful examination of the stacked LF and the individual clusters LFs in these cases suggested that the dip is produced by the stacking of clusters with different faintest magnitude limits, rather than lying within the magnitude range shared by all the combined clusters. This was found to be true for all stacked LFs that showed a statistical improvement in fit on inclusion of a Gaussian dip. We therefore conclude that our data show no evidence for real dips in the optical LFs of the C1 clusters.
DISCUSSION
Faint-end slope of the luminosity function
In this work we have studied the LFs of the individual clusters in the C1 sample from XMM-LSS. A Schechter function provided a reasonable fit across most of the LF for most clusters, especially in the z ′ band. But the bright end was poorly-fitted for nearly half of the sample (6 out of 14) and M * values were often not well constrained. The faint-end slope ranges are −1.03 ≤ α ≤ −2.1, −1.19 ≤ α ≤ −1.89 and −1.06 ≤ α ≤ −1.77 with averages −1.70 ± 0.10, −1.64 ± 0.04 and −1.43 ± 0.03 for the g ′ , r ′ and z ′
Redshift-Stacked z-Range
Average T Clusters- Table 4. Results of the Schechter function fitting of the redshift-stacked and temperature-stacked clusters for the three-filter set (g ′ , r ′ ,z ′ ).
α (g ′ Band) α (r ′ Band) α (z ′ Band) M * (g ′ Band) M * (r ′ Band) M * (z ′ Band)(keV)T-Range Average z Clusters- α (g ′ Band) α (r ′ Band) α (z ′ Band) M * (g ′ Band) M * (r ′ Band) M * (z ′ Band)(keV)
passbands respectively. The mean faint-end slope, averaging over all the three filters, is αavg = −1.59 ± 0.05.
Comparison of fitted Schechter parameters from different studies should take into account the passband, cluster redshift, and the procedure used in constructing the LF, including the methods used to determine the cluster membership and the background subtraction, since all of these factors may affect the results and therefore the accuracy of comparison.
Previous studies of galaxy cluster LFs have found a wide range for α, from α ∼ −1 (Paolillo et al. (2001)) to α ∼ −2 (Popesso et al. (2006)), but generally, LFs of clusters (both highmass and low-mass systems) are found to have steeper slopes than field galaxy LFs, which usually span values α ∼ −0.7 (Lin et al. (1996)) to -1 (Loveday et al. (1995)). The values we obtain for α fall into the cluster LF range. The mass (M500) range of the C1 clusters is 0.6-19 ×10 13 M ⊙ (Pacaud et al. (2007)) and this puts the C1 sample in the lower-mass class of galaxy clusters (poor clusters and groups). This indicates that low-mass systems have almost the same range of faint-end LF slopes as more massive systems.
Moreover, Gonzàlez et al. (2006) studied LFs of galaxy clusters with a virial mass range 0.01 − 20 × 10 13 M ⊙ and redshift 0.03 < z < 0.06 and found slopes of −1.9 < α < −1.6 at the faint end (Mr ≥ −18). This is consistent with our result for clusters with comparable redshift; clusters 11 and 21, which have estimated masses of 0.6 and 0.9 ×10 13 M ⊙ and redshifts 0.05 and 0.08, show faint-end slopes of −1.80 ± 0.05 and −1.89 ± 0.06 in the r ′ band, with magnitude limits of -14.5 and -15 respectively.
The study of Popesso et al. (2005a) on X-ray selected rich clusters with z ≤ 0.25 also gave a steep faint-end (Mg ≥ −16) slope: −2.1 ≤ α ≤ −1.6 in the SDSS g band. C1 clusters with redshifts ≤ 0.26, namely clusters 11,21,25,41 and 44, have a g ′ band slope range of −2.1 ≤ α ≤ −1.67, which agrees well with Popesso et al. (2005a). The redshift-stacked clusters with redshift z ≤ 0.32 (the first four redshift bins) gave a slope range of −1.79 ≤ α ≤ −1.59 in the g ′ band which is also consistent with Popesso et al. (2005a). The C1 clusters are low-mass systems, whilst the Popesso et al. (2005a) systems are rich clusters. The steep faint end slopes seen in both indicate a larger fraction of dwarf galaxies in both groups and clusters, compared with the shallower LF slopes usually found in the field.
However, as is the case with richer clusters, the results from different studies of the luminosity function of group galaxies arrive at different results. For instance, Miles et al. (2004) derived a very flat (α ∼ −1) Schechter slope for X-ray bright groups -though they found a faint upturn in X-ray dim systems -and Zandivarez (2006) derived similarly low faint end slopes for SDSS groups. Miles et al. (2004) used photometric data of X-ray selected systems and used all galaxies with B-R < 1.7 from the regions outside a radius of R500 from the centre of the group as the background for subtraction, whilst Zandivarez (2006) used spectroscopic data for membership determination for their friends-of-friends selected clusters. Robotham et al. (2006) extracted LFs for 2PIGG groups, derived from the 2dF galaxy redshift survey, and obtained good fits with Schechter functions, with faint end slopes which increased from α ∼ −1 for red galaxies to α ∼ −1.5 for blue galaxies. These discrepancies in the faint-end slope from different studies could arise from a variety of causes: different cluster selection methods (X-ray selected clusters in our case), spectroscopic or photometric selection of cluster galaxies, different galaxy background subtraction techniques (see discussion in section 4.3), and possibly because different clusters have different faint-end slopes depending on their large-scale environment, which will affect the incidence of infalling galaxies.
The absence of the upturn in the faint end of LFs
Both Popesso et al. (2005a) and Gonzàlez et al. (2006) reported an upturn at the faint end of their stacked LF, and required a sum of two Schechter functions, rather than a single Schechter, to obtain reasonable fits. Popesso et al. (2005a) located the upturn at -16 in the g ′ band, and -18.5 in z ′ ; the upturn of Gonzàlez et al. (2006) was found at a similar magnitude: -18 in the r ′ band. In our sample, only the LFs for clusters 11,21 and 41 extend to the faint magnitudes in which Popesso et al. (2005a) and Gonzàlez et al. (2006) found their upturns. The composite LF for these systems is the first in the redshift-stacked LFs, see Figures regarding the steep values of the faint-end slope, we do not find any evidence for a departure from a simple power law at the faint end.
Other studies gave steep slopes at the faint end of cluster LFs but without evidence of sudden upturns, see for example Durret et al. (2002). Garilli et al. (1999) studied composite LFs of 65 clusters ranging in redshift from 0.05 to 0.25 and did not find upturns in their composite LFs. Popesso et al. (2005a) argued that Garilli et al. (1999) did not see this upturn in their stacked LF because they used a weighting for the individual LFs which depends strongly on the cluster magnitude limit, such that clusters with fainter magnitude limits, which contribute to the faint magnitude bins of the stacked LF, were heavily down-weighted. In our sample, only the LFs for clusters 11,21 and 41 extend to the faint magnitude region in which Popesso et al. (2005a) and Gonzàlez et al. (2006) found their upturn. The composite LF for these systems is the first in the redshift-stacked LFs, see Figures 10, 11 and 12. We did not apply any weighting method that depends on the magnitude limit and although faint-end slopes are steep in all three bands for the stacked LF of clusters 11,21 and 41, they lack any upturn at the locations found by Popesso et al. (2005a) and Gonzàlez et al. (2006). Furthermore, individual LFs of these three clusters do not show any obvious upturn in the faint-end part of the LF that can be distinguished from the scatter of the data relative to the fitted Schechter function. Popesso et al. (2006) decomposed their LF by galaxy type and showed that the late-type galaxies LF was well fitted by a single Schechter function with a steep slope (α = 2.0 ), while the earlytype galaxies LF could not be fitted by a single Schechter function, and a composite of two Schechter functions was needed, such that the faint-end upturn of the global cluster LF was due to the earlytype cluster galaxies. This suggests one way of reconciling our results with those of Popesso et al. (2006). If in our poorer clusters late-type galaxies outnumber early-types in the intermediate and faint magnitude ranges then the LF would be steep and without any upturns. This needs to be further investigated by studying the earlytype and the late-type LF separately. Another possibility for the difference between our results and those of Popesso et al. (2006) lies in the techniques used to remove non-cluster galaxies, as we discuss in the next section.
Membership determination methods: Effects on α
The steepness of the faint end of the luminosity function reflects the number of dwarf galaxies within a cluster. Estimates of this number are very sensitive to the method used to estimate and remove the contribution of background and foreground galaxies before constructing the cluster luminosity function.
Rines & Geller (2008) compared methods of membership determination based on spectroscopic data and on photometric data (which we used) with regard to the resulting LF. They highlighted the advantages of spectroscopic identification of cluster members. Where automated photometric methods are used, they found, for example, that many large galaxies, especially those with low surface brightnesses, may be detected as many small separate objects, and warned that if these pieces of galaxies are not removed, they can produce an artificial excess of faint galaxies in cluster fields.
However, we have to emphasise that although spectroscopic data can give precise information on the cluster membership, their use to study cluster LFs is limited to relatively nearby clusters, since for higher-redshift clusters, spectroscopy is feasible only for the bright cluster galaxies. Boué et al. (2008) used deep multicolour photometry to study the LF of A496, using colour selection to reduce contamination by red background galaxies, and did not find the large fraction of dwarf galaxies (α = 2.0) inferred by some other authors, including Popesso et al. (2006). They suggested that this excess of dwarf galaxies in some studies might arise from inadequate removal of background, due to use of inadequate (or no) colour cuts. They claimed that the red sequence used by Popesso et al. (2006) was polluted by field galaxies because they used u * − r ′ vs i ′ in their CMD which Boué et al. (2008) showed was not efficient in rejecting background galaxies. In our study, we did not use u * − r ′ vs i ′ to define the colour cuts. Instead, we used u * − g ′ vs g ′ for the g ′ band, g ′ − r ′ vs r ′ for the r ′ band and i ′ − z ′ vs z ′ for the z ′ band. Moreover, our method of field LF subtraction is based on global background LF constructed by using the whole 1 • × 1 • field of the cluster. Therefore, we don't see any obvious reason why we might have contaminated the red sequence with field galaxies in such away as to give a false steep faint-end slope.
Origin of the faint galaxies
Our results indicate that larger numbers of faint galaxies exist in cluster environments than in the field. It is not straightforward to understand this result, since various dynamical processes that can destroy dwarfs act more effectively in dense environments. Several ideas have been proposed to explain the excess of dwarfs in clusters. Babul & Rees (1992) argued that a primordial population of dwarf galaxies is preserved in high-pressure environments, whilst it fades away in low-pressure regions. Alternatively, dwarfs could be formed by galaxies that fell into clusters from the surrounding field and were morphologically transformed. The transformation mechanism could be tidal fragmentation or so-called harassment of infalling late-type spiral galaxies by the cluster potential or by close encounters (Moore et al. (1996)) or ram pressure stripping of dwarf irregular galaxies (e.g., van Zee et al. (2004)). Boselli et al. (2008) showed that both simulations and observations are consistent regarding the scenario of recent accretion and transformation of low-luminosity star-forming galaxies in the Virgo cluster into quiescent dwarfs due to ram pressure gas stripping and galaxy starvation. They also showed that this process of transformation results in galaxies with structural and spectrophotometric properties similar to those of dwarf ellipticals. If the whole star-forming dwarf galaxy population dominating the faint end of the field luminosity function were accreted, it could be totally transformed by the cluster environment into dwarf ellipticals on timescales as short as 2 Gyr. These vigorous forces acting in cluster environments may explain the steepness of LFs faint-end slopes in nearby clusters.
The evolution of α
Our results show an evolutionary trend of the faint-end slope, α, in all bands used: g ′ , r ′ and z ′ . Liu et al. (2008) examined the faintend slope of the V-band LF of field galaxies with redshifts z < 0.5 and found that it becomes shallower with increasing redshift: their α changed from −1.24 for the lowest redshift bin 0.02 ≤ z < 0.1 to −1.12 for the highest redshift bin 0.4 ≤ z < 0.5. In clusters, a recent study by Lu et al. (2009) of an optically selected cluster sample found steepening of the faint end with decreasing redshift since z ∼ 0.2, and that the relative number of red-sequence dwarf galaxies had increased by a factor of ∼ 3.
It is possible that this LF slope trend with redshift is linked to the finding of that the 'upturn' in the LF faint end (i.e. the excess of galaxies above a single Schechter function) is found only in low redshift clusters. They attributed this to the recent infall of star-forming field galaxies or the whittling down of formerly more massive objects. The impact of recent infall of galaxies into clusters is also supported by the work of Lisker et al. (2007), who showed that dEs in the Virgo cluster fall into two major morphological subclasses: a) dEs with blue centres, thick disks or features reminsiscent of late-type galaxies, such as spiral arms or bars; this class showed no central clustering, suggesting that they are an unrelaxed population formed from infalling galaxies. The second subclass is b) nucleated dEs -a fairly relaxed population of spheroidal galaxies indicating that they have resided in the cluster for a long time, or were formed along with it. Lisker et al. (2007) also pointed to other studies deriving similar results (see references therein), indicating that this subclassification is not specific to the Virgo cluster.
Colour trends
The faint-end slopes, α, of the redshift-stacked groups are steeper in bluer bands in almost all redshift bins. However, this trend is significant (> 1σ) only for the redshift range 0.29 ≤ z ≤ 0.32. The redshift bin in which this effect seems to be absent is the first bin: 0.05 ≤ z ≤ 0.14. This suggests that the fraction of faint blue galaxies in clusters of redshift z ∼ 0.3 are higher than in local systems. Figure 5 further illustrates this and it also shows that these blue faint galaxies appear to have reddened and moved upwards in the colour-magnitude diagram. This is consistent with the findings of Wilman et al. (2005a) who compared the fractions of passive (red and quiescent) and blue star-forming galaxies in cluster at 0.3 ≤ z ≤ 0.55 with nearby (z ≃ 0) clusters. They found that the fraction of passive galaxies declined strongly with redshift to at least z ≃ 0.45. These results are also consistent with the wellknown Butcher-Oemler effect in clusters and support the idea that dense environments are responsible for galaxy transformation from blue to red because these trends are less obvious in field environments, see Wilman et al. (2005b).
Our result is also consistent with Yee et al. (2005) who studied the colours of galaxies as a function of luminosity and environment using the Red Sequence Cluster Survey and the SDSS. They found a higher incidence of faint to moderate luminosity galaxies in high density environments at z > 0.2 compared to lower redshifts and lower density environments. They interpreted this as arising from the shut-down of star formation in low mass galaxies within clusters at z < 0.3, in contrast to the situation in the field (c.f. Balogh et al. (2004)).
The fact that such transformations are observed in low-mass clusters like our C1 sample, as well as in richer clusters, favours mechanisms for suppression of star formation which operate in shallower potential wells, such as strangulation, tidal interactions and galaxy mergers, rather than ram pressure stripping, which is effective mostly in rich environments with high velocity dispersions.
Correlation between global properties of clusters
The optical luminosity is a good indicator of cluster richness, and hence should be closely related to cluster mass, velocity dispersion and temperature (Popesso et al. (2005b)). Assuming that cluster mass is directly proportional to the optical light (i.e. M/LOP T is constant), that the ICM is in hydrostatic equilibrium and that Xray luminosity LX scales with gas temperature T as LX ∝ T 3 (Xue & Wu (2000)), it is expected that LOP T ∝ T 1.5 , and that LOP T ∝ L 0.5 X . Our scaling results for LOP T with LX and T mostly agree well with these expectations. For the LOP T − LX relation, the logarithmic slopes are 0.47 ± 0.07 (g ′ band), 0.43 ± 0.08 (r ′ band) and 0.50 ± 0.07 (z ′ band). While for the LOP T − T relation, we have 1.57 ± 0.17 (g ′ band), 1.51 ± 0.17 (r ′ band) and 1.79 ± 0.12 (z ′ band). Popesso et al. (2004) found 0.38 ± 0.02 for the LOP T − LX relation and 1.12 ± 0.08 for the LOP T − T relation in the z SDSS band within a cluster radius of 0.5 Mpc (chosen to minimise the scatter in their scaling relations). The systems they used for their analysis, the RASS-SDSS sample, were X-ray selected, and ranged from low-mass systems of 10 12.5 M⊙ to massive clusters of 10 15 M⊙, over a redshift range from 0.002 to 0.45. Their logarithmic slope value for the LOP T − LX relation is not inconsistent with our value (within the errors), however, their LOP T − T value is lower than ours. They attributed the departure of their re-sults from the expected values to a breakdown in the assumption of constant mass-to-light ratio. More precisely, they argued that if the assumption of hydrostatic equilibrium was retained, their results would be in a good agreement with M/L ∝ L 0.33±0.03 , as found by Girardi et al. (2002). However, we note that extracting LOP T within a fixed metric radius, will include a smaller fraction of the virial radius for higher mass clusters. Hence it should be no surprise if the LOP T − T relation is flattened below the expected slope of 1.5 for self-similar clusters. Using clusters from the RASS-SDSS sample, Popesso et al. (2005b) calculated LOP T within r500 and r200. Their r200 results were (we used 1.5 × r500): 0.57 ± 0.03 (g band), 0.58 ± 0.03 (r band) and 0.58 ± 0.03 (z band) for the LOP T − LX relation and 1.62 ± 0.10 (g ′ band), 1.64 ± 0.09 (r ′ band) and 1.62 ± 0.10 (z ′ band) for LOP T − T relation. These are in better agreement with our results than Popesso et al. (2004), and demonstrate the importance of the radius used to estimate the LOP T .
CONCLUSIONS
We have studied the luminosity functions of 14 Class 1 (C1) XMM-LSS galaxy clusters in three CFHTLS MegaCam bands: g ′ , r ′ and z ′ . The X-ray selected clusters have masses ranging from 0.6 to 19 ×10 13 M⊙, a redshift range of 0.05 to 0.61, and ICM temperature range of 0.64 to 4.80 keV. We used colour-magnitude lower and upper cuts to reduce contamination by cluster non-members, and performed background subtraction using the 1 • × 1 • field of view in which the cluster lies. K-corrected luminosity functions of galaxies within 1.5 × r200 were constructed for each cluster and fitted with a Schechter function. Total optical luminosities of the individual clusters were also computed by integrating over the fitted Schechter functions. The individual LFs were also stacked together into 5 different redshift and temperature bins. The main findings are:
1.57 ± 0.17, 1.51 ± 0.17 and 1.79 ± 0.12 for the g ′ , r ′ and z ′ bands respectively.
• The slopes of the LOP T − LX and LOP T − T relations are consistent with the established, non-self-similar, cluster LX −T relation and constant mass-to-light ratio, except for the z ′ band value of the LOP T − T relation which is higher than the expected value (1.5) by ∼ 0.3.
• Some of our stacked LFs show dips, but these appear to be artefacts arising where clusters with different faintest magnitude limits are stacked together. We therefore we conclude there is no evidence for real dips in the optical LFs of the C1 clusters. Figure 10. LFs of the stacked clusters for 5 redshift ranges and the associated 1σ, 2σ and 3σ contours for g ′ band. All stacked clusters contributed to all magnitude bins are at the left side (brighter side) of the vertical dotted line which is at the faintest common magnitude bin of the clusters. Whereas at the right side (fainter side) of it, some clusters did not have data in some magnitude bins because they already reached their faintest magnitude limit. Figure 11. LFs of the stacked clusters for 5 redshift ranges and the associated 1σ, 2σ and 3σ contours for the r ′ band. The vertical dotted line is at the faintest common magnitude value of all stacked clusters. Figure 12. LFs of the stacked clusters for 5 redshift ranges and the associated 1σ, 2σ and 3σ contours for the z ′ band. The vertical dotted line is at the faintest common magnitude value of all stacked clusters. Figure 13. LFs of the stacked clusters for 5 temperature ranges and the associated 1σ, 2σ and 3σ contours for the g ′ band. The vertical dotted line is at the faintest common magnitude value of all stacked clusters. Figure 14. LFs of the stacked clusters for 5 temperature ranges and the associated 1σ, 2σ and 3σ contours for the r ′ band. The vertical dotted line is at the faintest common magnitude value of all stacked clusters. Figure 15. LFs of the stacked clusters for 5 temperature ranges and the associated 1σ, 2σ and 3σ contours for the z ′ zand. The vertical dotted line is at the faintest common magnitude value of all stacked clusters.
Figure 1 .
1Colour-magnitude diagram of cluster 25 (redshift=0.26). All galaxies (crosses) on the left of the vertical dotted line are the bright galaxies with magnitude ≤ m BCG + 3. The red sequence line (the solid line) is defined by the point (diamond) with magnitude of m BCG + 1.5 and colour equal to the average colours of all bright galaxies and by the slope of -0.025. The statistical scatter, σ, is estimated by the average colour errors of the galaxies within the 1.0 × 0.1 box on the faint end of the red sequence line. In the case of cluster 25, σ = 0.07. The dashed lines are the upper and lower colour cuts. The upper colour cut is the red sequence line pushed upward by 2σ and the lower colour cut is the red sequence line pushed downward by 2σ + △, where △ = 0.175 for redshift of 0.26. See text for definition of △.
Figure 3 .
3The faint end of the fitted LFs of the C1 sample grouped into three redshift bins: low (0.05-0.14, solid lines), intermediate (0.26-0.32, dashed lines) and high (0.43-0.61, dotted lines). Colours represent the filter bands: green for g ′ , red for r ′ and black for z ′ . All LFs were normalised to have φ = 1 at M=-19.5 for easy comparison. The faint end slopes become shallower with increasing redshifts. Also, at intermediate redshift (dashed lines), the slope shows a trend with colour, becoming steeper towards the blue. This colour trend largely vanishes at low redshifts (solid lines).
Figure 4 .
4The faint-end slope of the fitted Schechter function in g ′ , r ′ and z ′ bands for local clusters with redshift 0.05 to 0.14 and for intermediate redshift (0.29 ≤ z ≤ 0.32) clusters. due to their different values of M * ), whilst at intermediate redshift (dashed lines), the slope shows a strong trend with colour.
Figure 5 .
5Colour-magnitude diagram: g ′ − z ′ versus r ′ (K-corrected) for low-redshift (z=0.05-0.14) clusters and intermediate-redshift (z=0.29-0.32) clusters.
Figure 6 .
6LFs of the 8 stacked C1 clusters with redshift 0.2 to 0.4 and their associated 1σ, 2σ and 3σ contours of confidence levels for α and M * in the g ′ band.
Figure 7 .
710, 11 and 12. Although our results agree with Popesso et al. (2005a) and Gonzàlez et al. (2006) LFs of the 8 stacked C1 clusters with redshift 0.2 to 0.4 and their associated 1σ, 2σ and 3σ contours of confidence levels for α and M * in the r ′ band.
Figure 8 .
8LFs of the 8 stacked C1 clusters with redshift 0.2 to 0.4 and their associated 1σ, 2σ and 3σ contours of confidence levels for α and M * in the z ′ band.
Figure 9 .
9LF of the stacked clusters 1 and 27 (fourth temperature bin: 2.80 ≤ T ≤ 3.20) with plot of the 1σ, 2σ and 3σ confidence contours in the α-M * plane. The line is the fit of a Schechter function plus a Gaussian dip. The fitted dip position is −19.9±0.1 (z ′ filter). The vertical dotted line marks the faintest magnitude at which both stacked clusters contribute.
Figure A1 .Figure B1 .
A1B1LFs of the 14 individual C1 clusters and contours of the well-fitted clusters for the r ′ band. Contours plots of the 1σ, 2σ and 3σ confidence levels of α and M * are placed next to their associated LF. Clusters with failed constrained M * (and no contours) were placed at the bottom. APPENDIX A: INDIVIDUAL LUMINOSITY FUNCTIONS OF C1 CLUSTERS IN r ′ BAND APPENDIX B: INDIVIDUAL LUMINOSITY FUNCTIONS OF C1 CLUSTERS IN z ′ BAND LFs of the 14 individual C1 clusters and the associated 1σ, 2σ and 3σ contours of confidence levels of α and M * for the well-fitted clusters for the z ′ band.
Table 2 .
2The LF plots with the associated 1σ, 2σ and 3σ contours in the M * -αTable 1. List of the 17 C1 galaxy clusters covered by CFHTLS optical survey and their properties sorted according to their redshifts(Pacaud et al. XLSSC
R.A.
Dec
Redshift
T
L X
r 500
number
(J2000)
(J2000)
(keV )
10 43 ergs 2
(Mpc)
11
36.54
-4.97
0.05
0.64
0.11
0.29
21
36.23
-5.13
0.08
0.68
0.11
0.3
41
36.38
-4.24
0.14
1.34
2.4
0.44
25
36.35
-4.68
0.26
2.0
4.6
0.53
44
36.14
-4.23
0.26
1.3
1.2
0.4
22
36.92
-4.86
0.29
1.7
6.2
0.47
27
37.01
-4.85
0.29
2.8
4.8
0.65
8
36.34
-3.8
0.3
1.3
1.2
0.4
13
36.86
-4.54
0.31
1.0
1.3
0.34
40
35.52
-4.55
0.32
1.6
1.6
0.44
18
36.01
-5.09
0.32
2.0
1.3
0.52
6
35.44
-3.77
0.43
4.8
60.3
0.84
49
35.99
-4.59
0.49
2.2
4.3
0.49
1
36.24
-3.82
0.61
3.2
33.2
0.58
2
36.38
-3.92
0.77
2.8
19.6
0.49
29
36.02
-4.23
1.05
4.1
48.3
0.52
5
36.79
-4.3
1.05
3.7
17.1
0.49
Table 2 .
2Results of the Schechter function fitting of the LFs of the 14 C1 galaxy clusters. For some clusters, the M * values were not constrained by the fitting program and the errors of these unconstrained M * are starred. Also, the errors of the corresponding L OP T values are starred, since the computation of L OP T depends on both α and M * .XLSSC g Band
r Band
z Band
number
α
1
-1.94±0.23
-1.59±0.2
-1.06±0.17
6
-1.61±0.16
-1.7±0.09
-1.31±0.09
8
-1.53±0.37
-1.39±0.2
-1.15±0.16
11
-1.67±0.09
-1.8±0.05
-1.71±0.04
13
-1.63±0.63
-1.5±0.07
-1.51±0.08
18
-1.21±0.88
-1.76±0.13
-1.53±0.12
21
-2.01±0.11
-1.89±0.06
-1.77±0.06
22
-1.62±0.26
-1.19±0.19
-1.16±0.15
25
-2.1±0.12
-1.73±0.09
-1.57±0.08
27
-1.78±0.14
-1.85±0.12
-1.56±0.1
40
-1.03±0.3
-1.55±0.13
-1.27±0.09
41
-1.84±0.07
-1.86±0.09
-1.67±0.08
44
-1.75±0.12
-1.47±0.07
-1.44±0.09
49
-1.99±0.38
-1.65±0.16
-1.36±0.12
M *
1
-34.65±***
-23.7±0.91
-23.47±0.32
6
-20.96±0.43
-23.24±0.5
-22.98±0.23
8
-21.42±2.11
-21.79±0.83
-22.55±0.69
11
-20.61±1.5
-21.81±1.84
-21.13±0.73
13
-19.78±1.22
-22.19±0.37
-24.31±0.73
18
-19.66±1.43
-31.09±***
-22.23±0.41
21
-30.21±***
-29.02±***
-21.16±0.8
22
-20.26±0.84
-20.62±0.39
-22.15±0.5
25
-29.29±***
-22.66±0.78
-22.98±0.52
27
-22.22±1.32
-33.02±***
-23.52±0.76
40
-21.3±0.78
-22.95±0.96
-23.21±0.5
41
-30.57±***
-32.2±***
-23.38±1.5
44
-33.23±***
-22.79±0.53
-23.42±0.62
49
-31.66±***
-23.06±0.82
-23.72±0.46
L OP T 10 11 L ⊙
1
57.51±***
24.66±17.2
37.27±11.98
6
21.64±10.25
41.53±20.52
56.29±14.72
8
3.98±3.63
3.07±1.92
7.09±3.74
11
1±0.82
1.11±0.97
1.51±0.85
13
1.86±1.53
12.93±4.75
23.17±12.84
18
1.27±0.99
107.34±***
14.61±6.25
21
0.94±***
4.19±***
1.82±1.11
22
3.58±2.45
4.22±1.64
7.27±3.13
25
10.95±***
10.81±6.65
14.78±6.95
27
7.19±5.75
42.62±***
14.81±8.72
40
5.2±2.95
10.1±6.79
12.87±5.47
41
12.99±***
12.21±***
5.06±4.05
44
95.01±***
9.45±4.25
13.28±6.7
49
17.41±***
13.86±9.14
25.97±11.23
See http://www.cfht.hawaii.edu/Science/CFHLS/
ACKNOWLEDGEMENTSThe results presented in this paper are based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.
LF for the majority of clusters in our sample. The value of α range from −1.03 to −2.1, but no evidence is found for upturns at the faint end of the Schechter function, even in the lowest redshift systems, for which our LFs extend well into the dwarf regime. • M * ranges from −19.66 to −24.31. However, for many (nearly a third) of the clusters' M * values are not well-constrained. • The redshift-stacked LFs confirm that α becomes shallower low redshift. • A Schechter, 0.05-0.14flattening to −1.22 ± 0.06 at high redshift (0.43-0.61) in the z ′ band. Similar trends are present in the other two bands• A Schechter function provides a good fit across most of the LF for the majority of clusters in our sample. The value of α range from −1.03 to −2.1, but no evidence is found for upturns at the faint end of the Schechter function, even in the lowest redshift sys- tems, for which our LFs extend well into the dwarf regime. • M * ranges from −19.66 to −24.31. However, for many (nearly a third) of the clusters' M * values are not well-constrained. • The redshift-stacked LFs confirm that α becomes shallower low redshift (0.05-0.14), flattening to −1.22 ± 0.06 at high redshift (0.43-0.61) in the z ′ band. Similar trends are present in the other two bands.
also steepens significantly from the red (z ′ ) to the blue (g ′ ) band for clusters at redshift ∼ 0.3. This effect is not present in our local clusters (z ∼ 0), suggesting reddening of the faint blue. • , • α, also steepens significantly from the red (z ′ ) to the blue (g ′ ) band for clusters at redshift ∼ 0.3. This effect is not present in our local clusters (z ∼ 0), suggesting reddening of the faint blue
the temperature-stacked LFs do not exhibit any strong evidence for trends of the Schechter parameters with ICM temperature. • Total optical luminosities for our sample range from 1.0 to 56.3 ×10 11 L⊙, and correlate strongly with X-ray luminosity. The logarithmic slopes of the LOP T − LX relation are 0. • After removing the effects of redshift (correcting for the Malmquist effect. 47 ± 0.07, 0.43±0.08 and 0.50±0.07 for the g ′ , r ′ and z ′ bands respectively. • Also, LOP T correlate strongly with the X-ray gas temperature, T. The logarithmic slopes of the LOP T − T relation are REFERENCES• After removing the effects of redshift (correcting for the Malmquist effect), the temperature-stacked LFs do not exhibit any strong evidence for trends of the Schechter parameters with ICM temperature. • Total optical luminosities for our sample range from 1.0 to 56.3 ×10 11 L⊙, and correlate strongly with X-ray luminosity. The logarithmic slopes of the LOP T − LX relation are 0.47 ± 0.07, 0.43±0.08 and 0.50±0.07 for the g ′ , r ′ and z ′ bands respectively. • Also, LOP T correlate strongly with the X-ray gas temper- ature, T. The logarithmic slopes of the LOP T − T relation are REFERENCES
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| []
|
[
"Diversity from the Topology of Citation Networks",
"Diversity from the Topology of Citation Networks"
]
| [
"V Vasiliauskaite \nCentre for Complexity Science\nTheoretical Physics Group\nImperial College London\nSW7 2AZU.K\n",
"T S Evans \nCentre for Complexity Science\nTheoretical Physics Group\nImperial College London\nSW7 2AZU.K\n"
]
| [
"Centre for Complexity Science\nTheoretical Physics Group\nImperial College London\nSW7 2AZU.K",
"Centre for Complexity Science\nTheoretical Physics Group\nImperial College London\nSW7 2AZU.K"
]
| []
| We study transitivity in directed acyclic graphs and its usefulness in capturing nodes that act as bridges between more densely interconnected parts in such type of network. In transitively reduced citation networks degree centrality could be used as a measure of interdisciplinarity or diversity. We study the measure's ability to capture "diverse" nodes in random directed acyclic graphs and citation networks. We show that transitively reduced degree centrality is capable of capturing "diverse" nodes, thus this measure could be a timely alternative to text analysis techniques for retrieving papers, influential in a variety of research fields. | null | [
"https://arxiv.org/pdf/1802.06015v1.pdf"
]
| 3,370,422 | 1802.06015 | 3ce99a8286f20aca4d959fdf76d6aab9a23b39a5 |
Diversity from the Topology of Citation Networks
V Vasiliauskaite
Centre for Complexity Science
Theoretical Physics Group
Imperial College London
SW7 2AZU.K
T S Evans
Centre for Complexity Science
Theoretical Physics Group
Imperial College London
SW7 2AZU.K
Diversity from the Topology of Citation Networks
Directed Acyclic GraphCentralityCitation NetworkTransitive Reduction
We study transitivity in directed acyclic graphs and its usefulness in capturing nodes that act as bridges between more densely interconnected parts in such type of network. In transitively reduced citation networks degree centrality could be used as a measure of interdisciplinarity or diversity. We study the measure's ability to capture "diverse" nodes in random directed acyclic graphs and citation networks. We show that transitively reduced degree centrality is capable of capturing "diverse" nodes, thus this measure could be a timely alternative to text analysis techniques for retrieving papers, influential in a variety of research fields.
Introduction
A Directed acyclic graph (DAG) is a network with a distinct topological structure. Foremost, DAGs do not contain cycles, reflecting some intrinsic constraint present in the system. This lack of cycles ensures that the process of transitive reduction (TR) [1,2] results in a unique network. During the TR process as many links as possible are eliminated from the network, with a requirement that the connectivity of a network remains unchanged. An example of a DAG and its transitively reduced version is shown in Fig. 1a. Since the result of transitive reduction is unique, it hinders that only structurally essential links are kept after the process.
DAGs are a common representation of various data. For instance, a food chain can be represented as a DAG where the nodes are species and directed edges are linking predators to prey. Trophic levels are unveiled by an analysis of paths in this DAG [3]. Another example is the dependencies (the edges) between different parts (typically each class is a node) of a software package.
In this paper we will analyse data from another type of DAG: a citation network. Citations between documents such as scientific papers, court judgements or patents are also well represented by DAGs [4,5]. In such citation networks, nodes represent documents and a directed edge, pointing from node v to node u indicates citation of a document u by a document v. The constraint here is time; all edges point backward in time since one paper cannot cite another if the latter has not yet been published. Network analysis has long been an important part of bibliometric analysis [4]. Centrality measures may be used to indicate the most important scientific papers. The simplest centrality measure is degree centrality -a count of number of connections a node has. In this bibliometric context degree is citation count: the more a paper is cited, the more important it is [6].
Researchers' motivations for citing are ambiguous and varied among research disciplines [7]. Thus it is unclear whether the number of citations acquired by a paper is due to its significance, specificity of the research field it is published in, or the fact that high degree nodes in growing networks tend to acquire ever more edges due to some preferential attachment mechanisms [8]. It poses a question whether importance, captured by degree in such networks, is affected by node's age and whether any of the edges, adjacent to the node are better indicators of importance.
In terms of network analysis, degree is a tool used across all types of networks. The question we ask here is if there is another simple network centrality measure that is sensitive to the special topological properties of a DAG. A previous study [5] showed that TR eliminates as many as 80% of edges in citation networks of scientific papers and significantly reduces degree centrality of many highly cited publications. Clough et al. [5] suggested that the ideas of the papers that retained high degree count after TR are those with high impact among multiple research areas. Consequently, they are essential for linking ideas across research fields. We explore this idea in this work and prove it by using text analysis and clustering tools.
We will test and validate following propositions.
Proposition 1. If two nodes u and v in a DAG have the same high in-or out-degree centrality (or, simply, degree k u , k v ) maintains high degree centrality after transitive reduction (thereafter called reduced degree and defined k TR u ), this node u is in between more densely interconnected parts of a network. Proposition 2. Consider citation network in which an edge (v, u) indicates citation of an older paper u by a newer paper v. In this network high in-degree of a node u, k in u indicates high citation count of a paper u. If after TR u maintains high in-degree centrality, ideas in that paper must have influenced papers across a variety of disciplines. In other words, interdisciplinarity of a paper is depicted in its reduced in-degree.
Research methodology Notation
A DAG G is a set of nodes and edges: G = (V, E), that contains N = |V| nodes and |E| directed edges, none of which induce cycles in the graph. Here a directed edge from node v to node u is denoted (v, u).
We define the predecessors of a node u to be N + u = {v|(v, u) ∈ E}. These will be the immediate neighbours which are in the future of document u in the context of our citation networks.
(a) An example of a DAG. Vertical position of nodes is such that all edges are pointing downwards. The grey edges are transitively reducible. In case of a citation network, the time flow is opposite to edge direction: the closer a node is to the top of the figure, the later its publishing time.
(b) The same network as in Fig. 1a with potential clusters highlighted as cells, bounded by red-dotted lines. All nodes in the same cluster have the same cluster label. Blue node's diversity is defined as a number of unique cluster labels of its predecessors, the green nodes, belong to. In this example, its diversity score is 0.75, as its predecessors belong to 3 unique clusters out of 4 possible. Our nodes will also carry a label taken from a set L which we use to label different clusters. The label for node u is denoted c u ∈ L. It is also useful to refer to the set cluster labels carried by neighbours of a node u and this set of labels is denoted as
L + u = {c v |v ∈ N + u }.
Our assumption is that while each node u is a member of a single cluster, the cluster labels of the predecessor nodes will give us a more nuanced indication of the relationship between one node u and the different clusters. For example many academic papers may be assigned to a single academic field for convenience, they are published in journals associated with one field or are uploaded first to one particular section of a repository such as arXiv. So we calculate a measure of the diversity of cluster membership for each node, what we call the interdisciplinarity I u of node u. Our interdisciplinarity index is a measure of diversity, in this case it is a normalised version of what is called "richness" in ecology (we also consider another measure based on entropy in Appendix A which is referred to as "diversity" in ecology). This diversity score is equal to the fraction of unique clusters to which node's predecessors belong to, that is
I u = |L + u | |L| .(1)
An example of how we calculate diversity is illustrated in Fig. 1b.
To test propositions 1 and 2, we study the correlations between the node measures: interdisciplinarity I u , k in u the indegree, and the corresponding in-degree in the transitively reduced network k in,TR u . For instance, the Pearson correlation between two vectors defined over the nodes, X u and Y u , would be
ρ(X, Y) = 1 |V| v∈V (X v −X)(Y v −Ȳ ) σ X σ Y ,X = 1 |V| v∈V X v , (σ X ) 2 = 1 |V| v∈V (X v −X) 2 ,(2)
with similar definitions of mean and standard deviation for the Y vector. For propositions 1 and 2 to be correct we are looking for ρ(I, k TR ) ρ(I, k).
Stochastic model
One way we test our propositions is to use a stochastic model to create a DAG with an appropriate planted structure. In our model we have two main types of nodes: interdisciplinary and intradisciplinary nodes. The interdisciplinary nodes connect to all other nodes with equal likelihood, while the intradisciplinary nodes prefer to connect to nodes with the same cluster label. We keep the average degree the same for all nodes and the size of each cluster the same. We impose an order on the nodes to ensure we have a directed acyclic graph and we will do this with an integer index i = 1, . . . , N . Our convention will be that if i and j are allowed indices with i < j then v i and v j are vertices while the only edge possible is from the larger index vertex v j to the smaller index vertex v i , so an edge (v j , v i ) is possible but (v i , v j ) is never present. Our aim is to see if the in-degree of nodes in the transitively reduced graph is positively correlated with indicators of interdisciplinarity of networks based on the metadata of the nodes in our model, the cluster labels.
There are many ways to define a suitable model but for our purposes we implement the simplest version. This is not intended to model any particular real world context. Instead our model is used to illustrate the principle behind our propositions and the methods we use to demonstrate them.
In more detail, we first choose how many clusters we will work with, |L|. We then select one cluster, say the one labelled by 0, to be the subset of interdisciplinary nodes. The remaining (|L| − 1) clusters contain intradisciplinary nodes which we will label with integers 1 to (|L| − 1) for convenience. We assign the cluster labels to vertices sequentially in terms of the vertex index i, that is the cluster label of v i is the remainder of i divided by |L|, c i = i mod |L|. We then connect an edge from j to i with probability p ji where
p ji = 0 if j ≤ i λ if j > i , c i or c j = 0 γ if j > i , c j = c i = 0 ϕ if j > i , c j = c i = 0 .(3)
Here c i ∈ L is the cluster label of the node v i so then v i ∈ C ci . In order to keep the average degree the same for all nodes, we will choose
λ = (|L| − 2) · ϕ + γ |L| − 1 .(4)
The average total degree (in-degree plus out-degree) is then k ≈ λN for large N . This leaves us with two free parameters controlling the number of and distribution of edges and we choose these to satisfy 1 > γ > ϕ > 0 so that intradisciplinary nodes prefer to connect to nodes of the same label. For the two types of nodes we have k intra,in ≈ N 2|L| (|L|−2)·ϕ+γ +λ for intradisciplinary nodes while k inter,in ≈ N λ 2 for interdisciplinary nodes, which are identical when relation 4 is enforced. The expressions for the average out-degrees are identical as the probabilities defined in Eq. 3 are symmetric in terms of the cluster labels, c i and c j , and the clusters are spread evenly throughout the DAG.
Citation data
To test our approach on actual data, we use the example of a citation network in which documents are nodes and we will draw an edge from a newer document when it cites an older document. In the context of citation networks, our clusters of nodes represent collections of documents all on a single subtopic. Our assumption that each document has a single label corresponds to the common assignment of a single label for every document, perhaps as part of some hierarchical classification system. The reality is that many documents deal with several topics and we are looking to measure that Figure 2: An example of a random DAG with planted partition. The network is composed of 3 communities, which are labelled in different node colours. Blue nodes are the interdisciplinary nodes, whereas green and red communities are composed of intradisciplinary nodes. Each edge is labelled with the appropriate connection probability used to create that edge.
"diversity", the "interdisciplinary" nature of documents and to show that it is related to the high value for the degree in the transitively reduced network, a purely topological graph measure.
We used two datasets: citations between papers in the arXiv High Energy Physics-Theory repository, for papers published between 1992 and 2003 (referred to here as hep-th) [9] and Web of Science Astronomy and Astrophysics papers, published between 2003 and 2010 (referred to below as astro) [10,11]. The hep-th dataset contains a set of 27,770 papers and 351,500 edges, that represent citations between the papers. The astro dataset contains a set of 103,526 papers and 921,880 edges. 1 Note that in this paper we use a "conventional" edge direction: edge (v, u) indicates citation of u by v, k in u is the citation count of document u, whereas k out v is the length of the bibliography of document v.
Text analysis for node clustering
One way to group documents is to look for similarities in their texts. We will use standard text analysis methods to find such groups of similar texts and these will be our "clusters". We will assume that the clusters we find in our academic paper datasets represent different research subfields. First, we create a list of the stems of important words and their frequency from the abstract of each paper [12], see Appendix B for details. Important words are those which occur in many documents, but do not occur too often. For instance, the values could be 80% and 20% respectively. This would mean that a term is considered important if it does not occur in more than 80% of documents, as then it probably is generic and carries little meaning. On the other hand, if it occurs in less than 20% of documents, it is probably insignificant. We found that in this study of research abstracts, values of 90 − 97% and 1 − 5% respectively were sufficient.
The frequency vectors for each paper were used to create a term frequency-inverse document frequency matrix tf-idf matrix [12]. Clusters of the abstracts and, consequently, of papers themselves were obtained by using K-means algorithm [13].This algorithm creates a pre-determined number of clusters; we chose values of 10, 20 and 30.
Data Availability
The data that was generated to support the findings of this study is available from the corresponding author on reasonable request. The hep-th dataset is available from the KDD Cup 2003 website [9]. Access to the astro dataset is on request as described in the paper of Gläser et al. [10] and the official website of the Topic Extraction Challenge [14].
Data Analysis Stochastic Model DAGs
Using our stochastic model we generated DAGs ranging in size from 1,000 to 15,000 nodes in increments of 1,000 nodes. For each size of network, 100 random graphs were generated. We varied the predetermined number of clusters, |L|, 10, 20 and 30. Lastly, we used edge formation probabilities ϕ = 0.00001 and γ = 0.01. Values of the variables are summarised in Table 1 The results are shown in Fig. 3. The figure illustrates that the importance of a node, which is depicted by in-degree centrality, is reduced by a significantly smaller amount for nodes that are interconnected. In contrast, the nodes with a high intraconnectivity and small interconnectivity seem to lose a substantial amount of edges. This is expected, as within a cluster which is, by definition, highly connected, most of the edges are redundant. Since all redundant edges are removed, for nodes which degree centrality is acquired mainly from the nodes in the same cluster, their centrality is reduced significantly after TR. This effect is also notably more prominent for larger networks. Since in all cases, edge loss is significantly larger for intradisciplinary nodes, proposition 1 is proven.
Citation networks
To test proposition 2, we clustered papers using the methodology described in section and used these clusters to calculate our interdisciplinary measure I of (1) for each paper. We correlated diversity score for individual papers with their actual degrees and reduced degrees. We also varied a number of clusters, which again was selected to be either 10, 20 or 30.
The results are provided in Fig. 4 for the astro dataset and in Fig. 5 for the hep-th dataset. These figures show that the diversity of topics covered by a paper, as given by our measure of interdisciplinarity I of (1), does appear to be well correlated with the reduced in-degree in both datasets. This relationship is much less visible in the plots of interdisciplinarity and the full in-degree. Correlation coefficients, given in Table 2, show that there is indeed a very strong non-linear correlation between the reduced degree k in,TR and our diversity measure I regardless of the number of clusters. At the same time we see a significantly lower correlation between diversity and the in-degree k in in the citation network. This tendency is seen using both Pearson correlation measure and Spearman correlation measure. Table 2: Correlation between diversity (interdisciplinarity) I and degree k before and after transitive reduction in hep-th and astro datasets. Using both, Pearson correlation coefficient and Spearman correlation coefficient results in higher correlation between degree and diversity after transitive reduction. This result also seems to scale with the increase of the number of clusters.
Discussion and conclusions
In this paper we validated the proposition, initially suggested by Clough et al. [5], that for citation networks, in-degree centrality remains high after transitive reduction for nodes (papers) that have a large cross-disciplinary influence. We further validated this result using a simple model which produced random DAGs with planted partitions including nodes which connected homogeneously to others (interdisciplinary nodes) while most nodes preferred to connect within their community (intradisciplinary nodes). This study has shown several important results. It seems that cross-disciplinary nodes are highlighted by a large reduced degree in networks that contain a large number of clusters. This result is expected as in a network with only a few clusters, edges between clusters are more frequent, and the edges adjacent to interdisciplinary nodes will play a less important role. However, when the number of clusters increases, interdisciplinary nodes start to play a key role in sustaining these connections. Such connections are essential for the causal structure of the DAG and so a large proportion will be retained under transitive reduction. These edges contribute to k in,TR so a high value will flag interdisciplinary nodes.
In a citation network, highly cross-disciplinary nodes represent papers with a wide range of influence. For example, consider two papers from the hep-th dataset: "Large N Field Theories, String Theory and Gravity" by Aharony et al. (hep-th/9905111) and "The Large N Limit of Superconformal Field Theories and Supergravity" by Malcadena (hep-th/9711200). The first paper has a relatively large in-degree: k in = 806 and the largest reduced degree of all papers: k in,TR = 77. This paper also has a very large diversity score: 1.0, 0.9 and 0.73(3) for clusterings of size 10, 20, 30 Figure 4: Correlation between diversity of papers and their citation count (in-degree) before (Fig. 4a) and after (Fig. 4b) TR in the astro dataset. Different colours indicate results when nodes are clustered into a different number of groups: |C| = 10, 20, 30 for green squares, blue triangles and red circles respectively. There seems to be little correlation between degree and interdisciplinarity, however, diversity score increases with reduced degree -points in Fig. 4b are more compactly aligned along a curve. For clarity, in this and following figures linear-logarithmic scale for axes is used.
respectively. By investigating it's arXiv frontpage, one can find that it is listed under a number of different fields: High Energy Physics -Theory, General Relativity and Quantum Cosmology, High Energy Physics -Lattice and High Energy Physics -Phenomenology. Thus the paper seems highly interdisciplinary. In contrast, Malcadena's paper has a smaller interdisciplinarity score of 0.8, 0.5 and 0.43(3) for clusterings of size 10, 20 and 30 respectively and it is also listed only under the hep-th section of arXiv which suggests this paper is much more field specific. Indeed, its degree is reduced from 2414 to 15 and it is not highlighted by the reduced degree as a paper that is cross-disciplinary.
The measure of a reduced in-degree centrality has a potential to be a useful bibliometric metric. We have proved here that it highlights the interdisciplinarity of scientific papers, an important topic in bibliometrics [15,16]. Furthermore, it has also been demonstrated that transitive reduction of a growing directed acyclic network is a fast calculation, taking O(N M ) for N nodes and M vertices [1,2], so it would be a computationally cheap metric to implement. It could be used to produce recommendations of papers which might appeal to researchers from outside a given topic. For instance, if a physicist searches for a paper on transitive reduction algorithms, a search engine could return a list ordered by reduced degree k in,TR . That would highlight papers written by computer scientists specifically for non-specialists which are likely to be more digestible by researchers in other fields looking to use the latest algorithms on their data, or examples of scientists using transitive reduction algorithms but in specialised contexts that is more readily appreciated by the physicist searching for information.
Lastly, our successful demonstration of reduced degree as a measure of interdisciplinarity in a generative and structureless random DAG gives us an inkling that the result may give important insights in other networks with such structure. For instance with food webs, degree quantifies species' participation in the food web. Let us choose such edge direction that in-degree of a species is equivalent to the number of species dependent on it as a food source. However, some of these dependencies are non-essential, i.e. if this species becomes extinct, alternative prey could be found. On the contrary, species with high reduced degree are essential in the ecosystem represented by the food web, as alternative food source is impossible for many predators if this prey becomes extinct. Another application could be to improve software. Representing software dependencies as a DAG, a high k in,TR shows that a piece of software plays a critical role across a wide range of software packages.
Our work also highlights the importance of using centrality measures which are adapted to their context. Although there are a vast number of centrality measures available, simple measures as degree if appropriately adapted can give us important insights without unnecessary over-complications. Figure 5: Correlation between diversity of papers and their citation count (in-degree) before (Fig. 5a) and after (Fig. 5b) TR in the hep-th dataset. Correlation between diversity score and degree is higher after TR, which is also visible in the figures: points in Fig. 5b are more compactly aligned along a curve.
Supplementary Information
A Interdisciplinarity using alternative diversity measures
There are various ways to evaluate diversity. Indices of diversity are commonly used in ecosystem analysis: they provide information about composition of communities and abundances of species in those communities. In our context, a node's "ecosystem" is a set of its predecessors. 2 What we called interdisciplinarity is closely related to species richness -number of species present in an ecosystem. Alternatively, one could use measures of diversity that could give more information about the composition and infrastructure of ecosystems in contrast to measures of richness, as they also account for commonness of species. One example of such measures is Shannon's entropy [17]. For a node u it is defines as
H u = − v∈N + u p v lnp v (A1)
where p v is proportion of the system, composed of species v. Then interdisciplinarity, defined in terms of (normalised) diversity isĨ
u = e Hu S ,(A2)
where S is the number of unique species. In our context, S is a number of cluster of semantically similar papers. p v is equal to a number neighbours of u that belong to the same community, as a fraction of all neighbours of u. This value is then summed over all communities that constitute the "ecosystem" of node u. Fig. A6 shows results of this measure of interdisciplinarity, applied to our previously studied hep-th and astro datasets. It is clear that the correlation between interdisciplinarityĨ and reduced degree is higher than the real degree. Results are summarised in Table A3. Table A3: Correlation between Shannon diversity (interdisciplinarity)Ĩ and degree k before and after transitive reduction in the hep-th and astro datasets. Using both, Pearson correlation coefficient and Spearman correlation coefficient result in higher correlation between degree and diversity after transitive reduction. This result also seems to scale with the increase of the number of clusters.
hep-th |C| PearsonR(Ĩ, k TR,in ) PearsonR(Ĩ, k in ) SpearmanR(Ĩ, k TR,
(a) Full in-degree Figure A6: Correlation between Shannon diversity of papers and their citation count (in-degree) before (Fig. A6a, Fig. A6c) and after (Fig. A6b, Fig. A6d) TR in the hep-th and astro datasets respectively. Different colours indicate results when nodes are clustered into a different number of groups, see legends. Shannon diversity measure further highlights the ability of reduced degree capture diversity of papers in a citation network.
(b) TR in-degree (c) Full in-degree (d) TR in-degree
B Terms
Producing the terms
We used the abstract to characterise each paper. There are many reasons for doing this. The primary reason is that we had easy access to the abstracts. We also believe that in the context of the two data sets we used here, that the prominence and brevity of abstracts leads to assumption that authors choose their words carefully, making an abstract an effective list of keywords. By way of comparison, the full text contains too many generic terms, too many terms used in generic discussions of wider contexts to reflect the specific content of a paper. Abstracts also have few equations and abbreviations, and they do not contain other complicated structures such as figures and tables. Equations we will discuss in more detail below. As for abbreviations, these are often avoided in an abstract or, if not a widely used and standard abbreviation, the full text will be given as well and that text will be correctly processed in our approach. Since abstracts are read by both experts and to a large extent by non-experts, they ought to be geared toward non-experts. However we also note that for a study of much smaller number of papers in bibliometrics [18], researchers found that clustering based on abstracts and titles alone gave a poorer match with an expert based classification than when clustering using full texts. For text analysis we used corresponding NLTK (Python natural language toolkit) tokenize package functions. We also used the list of English stopwords provided in the NLTK. These words, such as "the", "are" were removed from the list of words that would be considered for being important words -terms. Furthermore, purely numeric stems are not considered either. We decided not to alter the list of stopwords manually in order not to introduce any subjectivity to our analysis. We also considered terms, composed of more than one stem. For instance, in Astronomy, "dark" could be related to "dark matter" or "dark energy" which are essentially distinct topics. Such word combinations, composed of n tokens are called n-grams. We used n-grams composed of maximum 3 stemmed tokens.
Punctuation is used to detect sentences, which are further split into words ("tokens") by spaces. A token is then stemmed -broken down into the word's root. Apostrophes are not dealt with very well; for instance "McDonald's" would be split into two tokens "mcdonald" and "'s". Brackets, however, are removed so that a term such as "SU(2)" becomes two tokens, "su" and "2". This also means that the symbol "$" is removed so that the characters in L A T E Xequations (which are surrounded by $ symbols) are then processed like other text. For instance "$\lambda = \sqrt(2)$" would leave us with tokens "\\lambda, \\sqrt". Backslashes signify regular expressions, in this case special mathematical symbols.
Many of the resulting tokens are not useful nor reliable as terms; they often have multiple meanings or are used only in exactly the same form by a tiny number of authors. In many cases these will be excluded by the the requirement that a term must appear in a minimum number of documents but not more than a maximum number, thresholds frequencies which we discuss below. So for instance a stem "lambda" is too frequent to be considered a term, whereas the stems of h µν are too rare. The role of equations in these academic documents is another reason why abstracts are a better source to use than the full texts. It is hard to find examples of abstracts with sufficiently large numbers of large equations.
Geared with terms for all abstracts, we formed a term frequency -inverse document frequency (tfi-df) matrix by using tf-idf vectoriser. It creates a matrix in which rows correspond to terms and columns correspond to documents: an entry i, j is equal to the number of times i th term occurred in j th document. Two thresholds define whether a stem is considered a term and is included in the matrix, these are the minimum and maximum frequency of the feature (stem) within documents. The texts are then clustered based on the frequency vectors of these terms (the tfi-df matrix) using K-means algorithm.
Stems of Important words in the hep-th data yang-mil, expans, boundari, classic, equival, way, argu, limit, class, hole, associ, group, descript, exist, discuss, dual, black, effect, n=2, obtain, potenti, invari, manifold, analysi, field theori, arbitrari, previous, coupl, interact, interpret, studi, correspond, number, topolog, quantize, vacuum, matter, differ, dynam, boson, propos, onli, point, valu, supergrav, structur, graviti, term, represent, depend, symmetri, singular, defin, phase, properti, u, conform, deriv, close, massless, consist, perturb, gaug, provid, paramet, appear, result, explicit, space-tim, generat, formal, set, dimension, determin, shown, equat, n, correct, given, complet, known, calcul, consid, contain, constant, possibl, supersymmetr, natur, transform, problem, string theori, hamiltonian, work, magnet, exact, introduc, paper, extend, describ, base, construct, certain, cosmolog, general, includ, loop, charg, oper, finit, local, standard, spectrum, present, larg, quantum, vector, new, 's, model, formul, gaug theori, chiral, relat, use, matrix, geometri, scalar, theori, dualiti, string, investig, particular, space, free, black hole, lead, brane, massiv, form, order, connect, fix, approach, condit, supersymmetri, action, mechan, appli, mass, physic, metric, level, recent, exampl, background, simpl, energi, spin, ani, direct, allow, method, spacetim, configur,
Figure 1 :
1Visualisation of notation used.
Figure 3 :
3Edge loss for interdisciplinary nodes (green points) and intradisciplinary nodes (red points) in various sizes of random DAGs, generated using a variation of SBM with planted partition and injection of homogeneous nodes, as described in section . Results with various sizes of clustering sets are illustrated in the figures. In all cases, interdisciplinary nodes seem to lose a significantly smaller fraction of their in-degree, regardless of the size of a network and the size of clustering. However, the difference between the loss of the degree in inter-and intra-disciplinary nodes seems to increase with both the size of a network and increased number of clusters. For ϕ = 0.00001 and γ = 0.01 with λ set using Eq. 4.hep-th |C| PearsonR(I, k TR,in ) PearsonR(I, k in ) SpearmanR(I, k TR,in ) SpearmanR(I, k in ) |C| PearsonR(I, k TR,in ) PearsonR(I, k in ) SpearmanR(I, k TR,in ) SpearmanR(I, k in )
|C| PearsonR(Ĩ, k TR,in ) PearsonR(Ĩ, k in ) SpearmanR(Ĩ, k TR,in ) SpearmanR(Ĩ, k in )
.Variable
Values
|L|
10, 20, 30
|N |
1, 000 − 15, 000 with increment of 1,000
|E|
∼ 177 − 126, 000
k in
∼ 0.3545 − 8.4
λ
0.00112, 0.0005358, 0.0003545 for |L| = 10, 20, 30
ϕ
0.00001
γ
0.01
Table 1: Values of parameters and network metrics of stochastic model DAGs
In practice there are often "bad" links which are in the "wrong" direction, from an older to a newer document. This is because documents are published in different versions and the text available may not have been created at the time associated with the document in the data set. For instance, a revised version of an arXiv paper carry the same index as the first version. A journal article has several associated dates: first submitted, date accepted, published online, formal publication date and so forth. Such bad links can introduce cycles and these must be dealt with. For our arXiv data we simply dropped these bad links as they account for less than 1% of the data. For the astro data we use ISI identifiers and assume that the larger the value, the older the paper, and only add links when there is a citation from one paper of higher ISI identifier than the paper being. This eliminates all cycles but 5973 citations are not encoded as edges in our network as a result, which is 0.64% of the total in this data.
(a) Full in-degree(b) TR in-degree
For instance, in a citation network, a node's "ecosystem" is composed of papers that cited that paper.
AcknowledgementThis work was funded by Engineering and Physical Sciences Research Council grant EP-R512540-1.Author contributions statement V.V. conducted the data analysis. The rest of the work was done jointly by both authors. Both authors reviewed the manuscript.Competing interestsThe authors declare no competing interest.
An algorithm for finding a minimum equivalent graph of a digraph. M Dennis, Gerald L Moyles, Thompson, Journal of the ACM (JACM). 163Dennis M Moyles and Gerald L Thompson. An algorithm for finding a minimum equivalent graph of a digraph. Journal of the ACM (JACM), 16(3):455-460, 1969.
The transitive reduction of a directed graph. A V Aho, M R Garey, J D Ullman, SIAM Journal on Computing. 12A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131-137, jun 1972.
Stratigraphy of a random acyclic directed graph: The size of trophic levels in the cascade model. Tomasz Luczak, Joel E Cohen, The Annals of Applied Probability. 32Tomasz Luczak and Joel E. Cohen. Stratigraphy of a random acyclic directed graph: The size of trophic levels in the cascade model. The Annals of Applied Probability, 3(2):403-420, 1993.
Networks of scientific papers. Derek J De Solla Price, Science. 1493683Derek J. de Solla Price. Networks of scientific papers. Science, 149(3683):510-515, 1965.
Transitive reduction of citation networks. J R Clough, J Gollings, T V Loach, T S Evans, Journal of Complex Networks. 32J. R. Clough, J. Gollings, T. V. Loach, and T. S. Evans. Transitive reduction of citation networks. Journal of Complex Networks, 3(2):189-203, sep 2014.
Citation frequency as a measure of research activity and performance. Eugene Garfield, Essays of an Information Scientist. 12Eugene Garfield. Citation frequency as a measure of research activity and performance. Essays of an Information Scientist, 1(2):406-408, 1973.
What do citation counts measure? a review of studies on citing behavior. Lutz Bornmann, Hans-Dieter Daniel, Journal of Documentation. 641Lutz Bornmann and Hans-Dieter Daniel. What do citation counts measure? a review of studies on citing behavior. Journal of Documentation, 64(1):45-80, jan 2008.
Measuring the preferential attachment mechanism in citation networks. Mingyang Wang, Guang Yu, Daren Yu, Physica A: Statistical Mechanics and its Applications. 38718Mingyang Wang, Guang Yu, and Daren Yu. Measuring the preferential attachment mechanism in citation networks. Physica A: Statistical Mechanics and its Applications, 387(18):4692 -4698, 2008.
Same data-different results? towards a comparative approach to the identification of thematic structures in science. Jochen Gläser, Wolfgang Glänzel, Andrea Scharnhorst, Scientometrics. 1112Jochen Gläser, Wolfgang Glänzel, and Andrea Scharnhorst. Same data-different results? towards a comparative approach to the identification of thematic structures in science. Scientometrics, 111(2):981-998, mar 2017.
Nees Jan van Eck, Theresa Velden, and Ludo Waltmann. Topic identification challenge. Kevin Boyack, Wolfgang Glänzel, Jochen Gläser, Frank Havemann, Andrea Scharnhorst, Scientometrics. 1112Kevin Boyack, Wolfgang Glänzel, Jochen Gläser, Frank Havemann, Andrea Scharnhorst, Bart Thijs, Nees Jan van Eck, Theresa Velden, and Ludo Waltmann. Topic identification challenge. Scientometrics, 111(2):1223-1224, mar 2017.
Understanding inverse document frequency: on theoretical arguments for idf. Stephen Robertson, Journal of documentation. 605Stephen Robertson. Understanding inverse document frequency: on theoretical arguments for idf. Journal of docu- mentation, 60(5):503-520, 2004.
An efficient k-means clustering algorithm: analysis and implementation. T Kanungo, D M Mount, N S Netanyahu, C D Piatko, R Silverman, A Y Wu, IEEE Transactions on Pattern Analysis and Machine Intelligence. 247T. Kanungo, D.M. Mount, N.S. Netanyahu, C.D. Piatko, R. Silverman, and A.Y. Wu. An efficient k-means clustering algorithm: analysis and implementation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(7):881- 892, jul 2002.
Diversity and network coherence as indicators of interdisciplinarity: case studies in bionanoscience. Ismael Rafols, Martin Meyer, Scientometrics. 822Ismael Rafols and Martin Meyer. Diversity and network coherence as indicators of interdisciplinarity: case studies in bionanoscience. Scientometrics, 82(2):263-287, 2010.
Indicators of the interdisciplinarity of journals: Diversity, centrality, and citations. Loet Leydesdorff, Ismael Rafols, Journal of Informetrics. 51Loet Leydesdorff and Ismael Rafols. Indicators of the interdisciplinarity of journals: Diversity, centrality, and citations. Journal of Informetrics, 5(1):87-100, Jan 2011.
Entropy and diversity. Lou Jost, Oikos. 1132Lou Jost. Entropy and diversity. Oikos, 113(2):363-375, 2006.
Stems of Important words in the astro data general galaxi, object, galaxi cluster, techniqu photometr, supernova general, star format, sun flare, star variabl, gamma ray burst, shock, starburst, radiat transfer, large-scal, galaxi individu, star, oscil, abund star, survey, high-redshift, galaxi high-redshift, plasma, evolut galaxi, radiat mechan, galaxi structur, galaxi nuclei, extinct, hole, kinemat, quasar general, observ, open cluster, associ, mhd, galaxi kinemat, nuclei, evolut star, rotat, general star, kinemat dynam galaxi, sun magnet, cluster general, supernova remnant, black, x-ray binari, photometr, stellar content, cluster general galaxi, kinemat dynam, galaxi ism, analysi, fundament, continuum, sun corona sun, radiat, format star, agb, lens, globular cluster, activ, interact, radio continuum, intergalact, galaxi halo, matter, dynam, sun magnet field, abund, na, instabl, data, gamma, structur, magnet field, circumstellar, statist, magnetospher, galaxi starburst, sun, galaxi evolut galaxi, medium, large-scal structur, supernova, neutron, shock wave, hydrodynam, ngc, close, wind, wave, planet, paramet, x-ray, stellar, jet, x-ray star, jet outflow, galaxi evolut, burst, ionospher, disk, variabl, numer, intergalact medium, accret accret disk, individu ngc, method data analysi, format galaxi, star oscil, polar, method statist, acceler, eclips, quasar, cluster associ, process, ism individu, flare, corona, format, radio, techniqu, data analysi, solar, dust extinct, dust, cosmic ray, magnet, ray burst, black hole physic, techniqu spectroscop, accret disk, molecul, atmospher, cosmolog theori, cosmolog, general, infrar star, galaxi activ galaxi, region, ism jet outflow, circumstellar matter, ellipt, dynam galaxi, evolut, disc, infrar, corona sun, galaxi individu ngc, spectroscop, cluster, galaxi, large-scal structur univers, dark matter, star format star, gravit lens, dark, activ galaxi, x-ray galaxi, relat, individu, star individu, line, ism, open cluster associ, binari, theori, radio line, ism cloud, ism molecul, space, fundament paramet, dwarf, black hole, distanc, hole physic, sun corona, star abund, mechan, infrar galaxi, turbul, mass, cluster individu, star evolut, ism jet, galaxi kinemat dynam, evolut galaxi format, galaxi stellar content, physic, globular, method numer, halo, structur univers, gravit, gamma ray, content, galaxi stellar, outflow, galaxi format, background. Patrick Glenisson, Wolfgang Glänzel, Frizo Janssens, Bart De Moor, ; , Combining full text and bibliometric information in mapping scientific disciplines. Information Processing & Management. 41star fundament paramet, satellit, pulsar, simul, cosmolog observ, method, planetari, star neutron, star agb, star fundament, univers, particl, cloud, accret accret, remnant, gamma-ray, galaxi format galaxi, field, open, method data, instrument, transfer, galaxi activ, binari close, ray, cosmic, accret, galaxi cluster generalPatrick Glenisson, Wolfgang Glänzel, Frizo Janssens, and Bart De Moor. Combining full text and bibliometric information in mapping scientific disciplines. Information Processing & Management, 41(6):1548-1572, 2005. express, particl, time, su, univers, type, integr, fermion, function, comput, field, zero, case, aris, tensor, algebra, dimens, solut, state, scale. Stems of Important words in the astro data general galaxi, object, galaxi cluster, techniqu photometr, supernova general, star format, sun flare, star variabl, gamma ray burst, shock, starburst, radiat transfer, large-scal, galaxi individu, star, oscil, abund star, survey, high-redshift, galaxi high-redshift, plasma, evolut galaxi, radiat mechan, galaxi structur, galaxi nuclei, extinct, hole, kinemat, quasar general, observ, open cluster, associ, mhd, galaxi kinemat, nuclei, evolut star, rotat, general star, kinemat dynam galaxi, sun magnet, cluster general, supernova remnant, black, x-ray binari, photometr, stellar content, cluster general galaxi, kinemat dynam, galaxi ism, analysi, fundament, continuum, sun corona sun, radiat, format star, agb, lens, globular cluster, activ, interact, radio continuum, intergalact, galaxi halo, matter, dynam, sun magnet field, abund, na, instabl, data, gamma, structur, magnet field, circumstellar, statist, magnetospher, galaxi starburst, sun, galaxi evolut galaxi, medium, large-scal structur, supernova, neutron, shock wave, hydrodynam, ngc, close, wind, wave, planet, paramet, x-ray, stellar, jet, x-ray star, jet outflow, galaxi evolut, burst, ionospher, disk, variabl, numer, intergalact medium, accret accret disk, individu ngc, method data analysi, format galaxi, star oscil, polar, method statist, acceler, eclips, quasar, cluster associ, process, ism individu, flare, corona, format, radio, techniqu, data analysi, solar, dust extinct, dust, cosmic ray, magnet, ray burst, black hole physic, techniqu spectroscop, accret disk, molecul, atmospher, cosmolog theori, cosmolog, general, infrar star, galaxi activ galaxi, region, ism jet outflow, circumstellar matter, ellipt, dynam galaxi, evolut, disc, infrar, corona sun, galaxi individu ngc, spectroscop, cluster, galaxi, large-scal structur univers, dark matter, star format star, gravit lens, dark, activ galaxi, x-ray galaxi, relat, individu, star individu, line, ism, open cluster associ, binari, theori, radio line, ism cloud, ism molecul, space, fundament paramet, dwarf, black hole, distanc, hole physic, sun corona, star abund, mechan, infrar galaxi, turbul, mass, cluster individu, star evolut, ism jet, galaxi kinemat dynam, evolut galaxi format, galaxi stellar content, physic, globular, method numer, halo, structur univers, gravit, gamma ray, content, galaxi stellar, outflow, galaxi format, background, star fundament paramet, satellit, pulsar, simul, cosmolog observ, method, planetari, star neutron, star agb, star fundament, univers, particl, cloud, accret accret, remnant, gamma-ray, galaxi format galaxi, field, open, method data, instrument, transfer, galaxi activ, binari close, ray, cosmic, accret, galaxi cluster general.
| []
|
[
"Excitonic fine structure of epitaxial Cd(Se,Te) on ZnTe type-II quantum dots",
"Excitonic fine structure of epitaxial Cd(Se,Te) on ZnTe type-II quantum dots"
]
| [
"Petr Klenovský \nDepartment of Condensed Matter Physics\nFaculty of Science\nMasaryk University\nKotlářská 267/261137BrnoCzech Republic\n\nCzech Metrology Institute\nOkružní 3163800BrnoCzech Republic\n",
"Piotr Baranowski \nInstitute of Physics\nPolish Academy of Sciences\nAl Lotników 32/46PL-02-668WarsawPoland\n",
"Piotr Wojnar \nInstitute of Physics\nPolish Academy of Sciences\nAl Lotników 32/46PL-02-668WarsawPoland\n"
]
| [
"Department of Condensed Matter Physics\nFaculty of Science\nMasaryk University\nKotlářská 267/261137BrnoCzech Republic",
"Czech Metrology Institute\nOkružní 3163800BrnoCzech Republic",
"Institute of Physics\nPolish Academy of Sciences\nAl Lotników 32/46PL-02-668WarsawPoland",
"Institute of Physics\nPolish Academy of Sciences\nAl Lotników 32/46PL-02-668WarsawPoland"
]
| []
| The structure of the ground state exciton of Cd(Se,Te) quantum dots embedded in ZnTe matrix is studied experimentally using photoluminescence spectroscopy and theoretically using k · p and configuration interaction methods. The experiments reveal a considerable reduction of fine-structure splitting energy of the exciton with increase of Se content in the dots. That effect is interpreted by theoretical calculations to originate due to the transition from spatially direct (type-I) to indirect (type-II) transition between electrons and holes in the dot induced by increase of Se. The trends predicted by the theory match those of the experimental results very well. The theory identifies that the main mechanism causing elevated fine-structure energy in particular in type-I dots is due to the multipole expansion of the exchange interaction. Moreover, the theory reveals that for Se contents in the dot > 0.3, there exist also a peculiar type of confinement showing signatures of both type I and type II and which exhibits extraordinary properties, such as almost purely light hole character of exciton and toroidal shape of hole states. | 10.1103/physrevb.105.195403 | [
"https://arxiv.org/pdf/2112.06798v2.pdf"
]
| 245,124,428 | 2112.06798 | cf1e5f7d6934753122466bcd0be94b888e6daa5a |
Excitonic fine structure of epitaxial Cd(Se,Te) on ZnTe type-II quantum dots
Petr Klenovský
Department of Condensed Matter Physics
Faculty of Science
Masaryk University
Kotlářská 267/261137BrnoCzech Republic
Czech Metrology Institute
Okružní 3163800BrnoCzech Republic
Piotr Baranowski
Institute of Physics
Polish Academy of Sciences
Al Lotników 32/46PL-02-668WarsawPoland
Piotr Wojnar
Institute of Physics
Polish Academy of Sciences
Al Lotników 32/46PL-02-668WarsawPoland
Excitonic fine structure of epitaxial Cd(Se,Te) on ZnTe type-II quantum dots
(Dated: March 2, 2022)
The structure of the ground state exciton of Cd(Se,Te) quantum dots embedded in ZnTe matrix is studied experimentally using photoluminescence spectroscopy and theoretically using k · p and configuration interaction methods. The experiments reveal a considerable reduction of fine-structure splitting energy of the exciton with increase of Se content in the dots. That effect is interpreted by theoretical calculations to originate due to the transition from spatially direct (type-I) to indirect (type-II) transition between electrons and holes in the dot induced by increase of Se. The trends predicted by the theory match those of the experimental results very well. The theory identifies that the main mechanism causing elevated fine-structure energy in particular in type-I dots is due to the multipole expansion of the exchange interaction. Moreover, the theory reveals that for Se contents in the dot > 0.3, there exist also a peculiar type of confinement showing signatures of both type I and type II and which exhibits extraordinary properties, such as almost purely light hole character of exciton and toroidal shape of hole states.
INTRODUCTION
Key components of the future quantum devices for usage in information technology will be on demand sources of single photons and entangled photon-pairs. A prominent candidate systems in this respect are currently the quantum dots (QDs). Mainly semiconductor type-I QDs where both electron and hole wavefunctions are bound inside QD body, show excellent optical properties combined with their compatibility with current semiconductor processing technology and, moreover, they offer the potential for scalability [1][2][3][4][5][6][7][8]. QDs currently cover a rather wide range of topics, from quantum cryptography protocols [9,10], sources of polarization-entangled photon pairs [11][12][13], quantum key distribution [14,15], quantum gates [16][17][18][19][20], or as nanomemories [18,[21][22][23][24][25][26].
One of the key challenges for turning QDs into sources of entangled photons is to zero the tiny energy separation of the bright doublet of the ground state exciton (X 0 ), dubbed the fine-structure splitting (FSS). That can be achieved, e.g., by externally applying elastic strain [27][28][29][30][31][32], electric [13], or magnetic [33][34][35] fields. Further option to have QDs with negligible FSS is provided by growing QDs on lattice matched materials [36,37].
Another route of obtaining small FSS is utilizing type-II QDs where one of the quasiparticles, electron or hole, is bound outside of QD body, while the other resides inside [7,[38][39][40][41][42][43]. The aforementioned type-II QDs were mostly realized on group III-V materials, InAs, GaAs, and GaSb. However, there exist another class of type-II QD structures based on II-VI materials, like the Cd(Se,Te)/ZnTe dots. Note, however, that our purpose in this work is not to study the feasibility of Cd(Se,Te)/ZnTe dots for generation of entangled photons but rather to study the reduction of FSS with increased electron-hole spatial separation.
CdSe/ZnTe is a semiconductor system, which is wellknown for its type-II band alignment, in which the spatially indirect optical emission appears in the near infrared spectral region, i.e., at 1.0 eV-1.1 eV. This fact has been demonstrated experimentally in CdSe/ZnTe quantum wells [44], CdSe/ZnTe core/shell nanowires [45] and colloidal core/shell nanocrystals [46]. However, the lattice mismatch which is the driving force for the formation of self assembled QDs is very small and amounts to 0.003 in this material system, which prevents the formation of type-II CdSe/ZnTe QDs.
On the other hand, a sufficiently large lattice mismatch of 0.07 is present in CdTe/ZnTe semiconductor system which has enabled the growth of CdTe/ZnTe QDs by molecular beam epitaxy [47,48]. Subsequently, their optical properties have been subject of extensive investigations [49][50][51]. In particular, it was found that this semiconductor system is characterized by the type-I confinement which is manifested by quite short excitonic lifetimes, i.e., below 500 ps [52]. However, the valence band offset in these structures is quite small [53] whereas only strain effects are responsible for its type-I character. That is why the addition of a certain amount of selenium into Cd(Se,Te) QD-layer, leading to the shift the valence band towards lower energies, results in the transition from type-I to type-II confinement [54]. At the same time the Cd(Se,Te)/ZnTe lattice mismatch is sufficiently large to induce the QD formation and the optical emission from these structures is intense enough to enable the observation of the emission from individual QDs, which enables the unique study of FSS in those type-II QDs that we discuss in this work. Cd(Se,Te)/ZnTe QDs and measurements of their optical emission, revealing FSS of that system as function of Se content. That is followed by theory discussion of electronic structure of Cd(Se,Te)/ZnTe QDs, starting from analysis of the single-particle states and carrying on to computations of correlated excitons, finally showing that the trends predicted by theory match those of the experimental results very well. Furthermore, theory shows a rather unusual behavior of Cd(Se,Te)/ZnTe QDs related to light hole exciton and Aharonov-Bohm effect.
EXPERIMENT
The samples containing self assembled Cd(Se,Te)/ZnTe QDs are grown by the molecular beam epitaxy. The details of the growth procedure are described in Ref. [54]. The optically active part of the samples consists of a layer of Cd(Se,Te) QDs embedded in ZnTe matrix. Three samples with a different average Se concentrations within the dots equal to 0.002, 0.03, and 0.1 are investigated for the purposes of the present work. Se content within QDs can be effectively changed by varying the growth parameters during the deposition of the QD-layer. That layer consists consecutively of three CdTe monolayers, one CdSe sub-monolayer, and two CdTe monolayers. Depending on the coverage of the central CdSe monolayer, which is controlled by the exposure time of Se-flux, the average Se concentration can be varied from 0 up to 0.17. The largest Se-concentration corresponds to the deposition of a complete central CdSe monolayer. After the deposition of the QDs-layer the QD-formation process does not take place spontaneously despite of the large lattice mismatch, as is common for II-VI semiconductor system. It has to be additionally induced by tellurium deposition at low substrate temperature and its subsequent thermal desorption [47,48]. In the final step, the dots are capped with a 50 nm thick ZnTe layer.
Photoluminescence (PL) measurements performed at low temperature reveal that the emission energy strongly depends on Se concentration within the Cd(Se,Te) QDs which is induced, most likely, by the change of confinement from type-I to type-II [54]. In particular, it is found that the maximum emission energy amounts to 1.98 eV, 1.83 eV, and 1.69 eV for the investigated mean Se concentrations of 0.002, 0.03, and 0.1, respectively. This choice of the samples along with the inhomogeneous broadening of the emission bands, which amounts typically to 80 meV, ensures that the emission lines from individual QDs can be found in the entire spectral range, from 1.5 eV up to 1.9 eV.
In order to assess the emission from individual QDs, µ-PL measurements in which the excitation laser spot is of panel (a) shows the measured (points) and fitted (lines) dependencies of X (black) and XX (red) with slopes a = 0.9 and a = 1.7, respectively. The dots are taken from three samples with a different average Se concentrations of 0.002, 0.03, and 0.1 which is marked in (c) with different colors: blue, red, and dark grey, respectively. Temperature during these measurements was kept at 7 K, the excitation laser wavelength was 405 nm and the laser spot diameter was ∼ 3µm. reduced to 3 µm are performed. Further reduction of the excitation area is obtained using apertures with diameter of 400 nm within a 150 nm thick gold layer deposited on top of the structures. For the measurements, the samples are placed inside a continuous flow cryostat in which the temperature is kept at 7 K.
Several emission lines with the spectral width in the range of 500 µeV -1 meV originating from individual QDs are observed. In order to determine the corresponding excitonic FSS values, linear polarization of the optical emission spectrum has been measured. This study is performed in geometry in which light propagates perpendicular to the sample surface and the linear polarization vector is always parallel to the sample plane. It is found that the emission energy slightly depends on the linear polarization angle for all measured bands. In Figure 1 (a), the emission lines are measured in two orthogonal linear polarizations corresponding to the largest change of the emission energy. In such configuration, the energy difference is given by the value of FSS. In order to determine the best FSS-values, the spectral position is plotted as a function of the polarization angle for both emission lines, see Fig. 1 (b). It is found that this dependence can be well fitted with a sine square function, whereas its amplitude gives us directly the FSS value [49,55]. FSS values of the two emission bands presented in Fig. 1 (a) and (b) are found to be very similar. However, both polarization angle dependencies are shifted by 90 • with respect to each other, see Fig. 1 (b). This feature is characteristic for biexciton and single exciton emission and indicates that both bands originate from the same QD. In order to identify whether the particular band corresponds to the single exciton or to biexciton emission, excitation power dependence of the optical emission spectrum has been measured, inset. The intensity of the high energy line at 1.798 eV increases almost linearly with increasing excitation power whereas the intensity of the low energy line at 1.796 eV increases superlinearly which leads us to associate them to the single exciton and biexciton emission, respectively.
In Figure 1 (c), FSS values from over 40 individual QDs are plotted as a function of the single exciton emission energy. A large distribution of FSS values decreasing from ∼ 300 µeV to almost zero is found among the investigated dots. Most importantly, they depend conspicuously on the exciton emission energy. Significantly, smaller FSS-values are observed, in average, for the dots emitting at lower energies compared to the dots emitting at higher energies. The large variation of these values at a fixed energy results, most likely, from the anisotropy of the potential localizing charge carriers which is induced by the shape and/or strain anisotropy of the dots similar to CdTe/ZnTe QDs without Se [50]. On the other hand, the maximum emission energy depends primarily on the Se concentration within the dots [54]. At the same time, it is found that the sizes and shapes of the dots do not change significantly as a function of Se-content within the investigated concentration range as demonstrated previously by atomic force microscopy [54]. Thus, it is reasonable to conclude that the increase of the average Se concentration within Cd(Se,Te) QDs leads to the overall decrease of FSS values. A possible explanation of this effect relies on the change of the confinement of the dot/matrix interface character from type-I to type-II, leading to the increase of the electron-hole spatial separation. Furthermore, the effect of mutual compensation of electron and hole wavefunction anisotropies may result in the observed decrease of FSS-values in type-II QDs, as predicted theoretically in Ref. [56].
Based on the growth procedure and the optical measurements presented above we cannot draw any definite conclusion about the Se composition profile within the dots. In our considerations an uniform Se-distribution is assumed for simplicity reasons. In fact, the presence of Seand Te-rich regions within the dot inducing additional electron-hole separation within the dots cannot be excluded. Such effects have already been studied in entirely type-I QD systems in which Cd(Se,Te) were embedded into ZnSe matrix [57]. One of the conclusions of that work was that the FSS values were even slightly increased in the presence of Se-atoms as compared to pure CdSe/ZnSe QDs and CdTe/ZnTe QDs. Since in the presently described Cd(Se,Te)/ZnTe QDs a decrease of the average FSS values takes place with an increasing Se-content, we do not expect that the local variation of Se and Te within the dots impacts significantly our results.
The most distinct experimental trends concerning the optical emission from Cd(Se,Te)/ZnTe QDs which appear as a function of increasing Se concentration within the dots are presented in Figure 2. First of all, a considerable redshift of the emission energy from 1.98 eV down to 1.6 eV is observed, Figure 2 (a). That is accompanied by a decrease of the decay rate by one order of magnitude, Figure 2 (b). Since the PL-decays can be well described by biexponential functions [54] for all samples, the fast and slow decay rates are determined and plotted in blue and red in Fig. 2 (b), respectively. Finally, those results are compared to the dependence of FSS values as a function of Se concentration, which are the main subject of this work, see the optical emission was too weak to perform a detailed FSS investigation like in the other samples with lower Se-content. Note that the distinct decrease of the decay rates strongly indicates the type-II character of the QDs and is caused directly by the electron-hole wavefuncion spatial separation. The huge emission energy redshift of 350 meV is also consistent with the type-II confinement in CdSeTe/ZnTe QDs with relatively large Se-content. Cd(Se,Te) bandgap reduction cannot explain this effect since it is expected to amount to only 135 meV at maximum (for Se concentration of 0.4) due to the bowing effect [58].
THEORY
Based on the aforementioned experimental results we will now provide the theoretical reason for the reduction of FSS values with increasing Se content. In order to do that we calculate the correlated electronic structure of the ground state exciton (X 0 ) using a combination of the eight-band k · p method [59][60][61], providing singleparticle (SP) basis states for the configuration interaction [62] (CI) algorithm which we developed earlier, see Ref. [43]. During the CI calculation our CI code evaluates also the emission radiative rate [18,43] utilizing the Fermi's golden rule [63].
More specifically, we consider [35,43,61] the SP states as linear combination of s orbital like and x, y, z p orbital like Bloch waves at Γ point of the Brillouin zone, i.e. Ψ ai (r) = ν∈{s,x,y,z}⊗{↑,↓}
χ ai,ν (r)u Γ ν .(1)
Here u Γ ν is the Bloch wave-function of an s-like conduction band or a p-like valence band at Γ point, ↑/↓ mark the spin, and χ ai,ν is the envelope function for
a i ∈ {e i , h i }.
On the other hand, in CI we consider the excitonic wavefunction as a linear combination of the Slater determinants (SDs)
ψ X i (r) = nSD m=1 η i,m D X m (r),(2)
where n SD is the number of SDs D X m (r), and η i,m is the i-th CI coefficient which is found along with the eigenenergy using the variational method by solving the Schrödinger equation
H X ψ X i (r) = E X i ψ X i (r),(3)
where E X i is the i-th eigenenergy of excitonic state ψ X i (r), andĤ X is the CI Hamiltonian which readŝ
H X =Ĥ SP 0 +V X ,(4)
whereĤ SP 0 andV X represent the Hamiltonian of the noninteracting SP states and the Coulomb interaction between them, respectively. The matrix element ofV X is [18,35,43]
D X n |V X |D X m = − 1 4π 0 ijkl drdr e 2 (r, r )|r − r | × {Ψ * i (r)Ψ * j (r )Ψ k (r)Ψ l (r ) − Ψ * i (r)Ψ * j (r )Ψ l (r)Ψ k (r )}.(5)
where e labels the elementary charge and (r, r ) is the spatially dependent dielectric function. Note that minus sign in front of the integral in Eq. (5) results from different sign of the charge of the electron and hole from which exciton is composed. The sixfold integral in Eq. (5) is evaluated using the Green's function method [35,43,62,64]. Note, that for (r, r ) in Eq. (5) we use the positionally dependent bulk dielectric constant in our CI calculations. Further, the multipole expansion of the exchange interaction is included in our CI for CI basis consisting of two electron and two hole SP ground states following the theory outlined in Refs. [56,65]. In (b) ve show the SP energies (blue curve) and that computed using CI without (orange curve) and with (green curve) the inclusion of the effect of Coulomb correlation. In (b) we also give the binding energy of X 0 with respect to SP transition (black broken curve) with energy axis on the right. The insets in (b) show side cuts of our QD (green object) and the SP electron (blue object) and hole (red object) probability densities. Panel (c) gives the conduction (CB, black), heavy-hole (HH, red), light-hole (LH, green), and spin-orbit split off (SO, blue) Bloch band content of X 0 as a function of Se concentration computed [35] [100]
[100]
[100]
[001] bellow
[100]
[100]
[100]
[001] 4. Cuts of the single-particle (SP) probability densities of Cd(Se,Te)/ZnTe QDs for (a) zero Se content, (b) Se content of 0.2, and (c) Se content of 0.4, corresponding to type-I, type-II, and type-I/II confinement, respectively. The letters QD in top row mark that the first column showing the cuts of the simulated QD body, the numbers in the first row enumerate SP states, starting from the ground state marked by zero. The last column gives the Miller indices of the planes where the cut was performed in each row of the figure. The abbreviations "el." and "hl." mark the electrons and holes, respectively. In (b) the designations "[001] above" and "[001] bellow" identify that the cuts of the hole densities were performed above and bellow QD body, respectively, and correspond to the side cut given in the last row of panel (b).
plane (a) (b) (c) { { { { { { FIG.
RESULTS AND DISCUSSIONS
The electronic states of Cd(Se,Te)/ZnTe QDs computed using the aforementioned k · p+CI method are shown in Fig. 3. Motivated by typical structure pinpointed in Ref. [54], the computed shape of the Cd(Se,Te) QD was truncated cone with lower and upper basis diameters of 36 nm and 22 nm, respectively, and with height of 4 nm. Except of QD body, the rest of the simulation space consisted of ZnTe. After definition of the structure, the elastic strain tensor was obtained in the whole simulated structure by grid-point-wise minimization of elastic energy. Thereafter, the Coulomb potential energy, including the effects of the piezoelectricity, was obtained by solving the Poisson's equation in the whole structure. The resulting Hamiltonian matrix was then diagonalized using Nextnano++ [60,66] simulation tool, which was also used for the aforementioned computation steps. Note, that all the material parameters including the effective masses were taken from the library of the Nextnano++ software [66].
The resulting eigenenergies and eigenfunctions are shown in Fig. 3 (a) and Fig. 4, respectively, for twelve electron and twelve hole SP states. Depending on Se content, we have identified three types of confinement in our QDs, i.e., type-I for Se=0-0.15, type-II for Se=0.15-0.3, and so-called type I/II for Se=0.3-0.4. Note, that the type-II confinement was identified by the spatial location of hole wavefunctions {inset in Fig. 3 (b) and Fig. 4 (b)} being outside of QD body, while electrons are firmly bound inside QD for all Se contents. The type I/II confinement is peculiar and shows features of the remaining two types of confinement, see also in the following.
While the energy of electron SP states reduces with similar rate for all Se contents, including that for SP excited states, the holes are affected by Se content and the associated type of confinement considerably more. While for type I and associated Se contents < 0.15, hole SP energy decreases (i.e., the absolute value of hole energy increases), for type II that increases only slightly, and, finally, for type I/II the increase of hole SP energy (decrease of the absolute value of that) with increasing Se content is observed.
As a result of the aforementioned discussion, the SP electron-hole transition energy {see blue curve in Fig. 3 (b)} remains almost constant for increasing Se content in type I, while its magnitude is reduced with increase of Se for type II and type I/II.
The Se-content-dependent energies of X 0 computed by CI without and with considering the effect of correlation are shown by orange and green curves, respectively, in Fig. 3 (b). Note, that by the "effect of correlation" we mean specifically the expansion of CI complexes into the basis consisting not only from ground but also excited SP states. Now, the binding energy of X 0 {Fig. 3 (b)} compared to electron-hole transition SP energy decreases in type I from 70 meV for Se content of zero to 10 meV for content of 0.15. The large binding energies in type I are due to the attractive Coulomb interaction between tightly quantum confined electrons and holes in QD [67]. For Se content between 0.15 and 0.3 (type II) the binding energy remains constant at 10 meV and further increases with Se content to 30 meV in type-I/II. Correlation further increases the binding energy by 20 meV for Se content of zero up to by almost 30 meV for Se content of 0.4. In type II the additional binding energy due to correlation is ∼ 2 meV. Finally, note that the magnitude of the reduction of X 0 energy with increasing Se content matches that observed from spectral shift of the maximum of photoluminescence spectra in Ref. [54].
In Fig. 3 (c) we show the Se-dependent Bloch state content of X 0 , obtained from the Bloch state composition of electron and hole SP states, utilizing the squares of CI wavefuntion coefficients η i,m from Eq. (2). The method was previously developed in Refs. [13,35]. Note, that the studied Bloch state composition in this work is that of the conduction band (CB), heavy-hole (HH), light-hole (LH), and spin-orbit (SO) valence bands.
The content of CB in X 0 is ∼ 50 % for all studied Se contents. While HH content is also ∼ 50 % for smaller Se concentrations, the increase of the amount of Se causes considerable progressive admixing of LH states in X 0 , reaching values of ∼ 40 % for Se contents > 0.35. At the same time, we observe increase of admixing of SO state for Se contents > 0.3, reaching values as high as ∼ 10 %. We note that for Se < 0.3 the SO content of X 0 is negligibly small.
Thus, both the type-II regime and in particular type-I/II show unusual composition of X 0 . Strikingly, in type I/II X 0 has almost LH like character with small addition of SO states and negligible HH content. Such LH X 0 would be advantageous in quantum information technology, such as, e.g., enabling coherent conversion of photons into electron spins [68], and was first experimentally reported in Ref. [69] for GaAs/AlGaAs QD system where the LH character was obtained by externally applied tensile strain. However, we predict that in Cd(Se,Te)/ZnTe QDs such an LH exciton is present for Se contents > 0.35 without the necessity of any external tuning.
We now turn our attention to the theory analysis of the fine-structure of X 0 and show the results of that in Fig. 5. We computed the fine structure employing three levels of approximation in CI, i.e., (i) without considering the effect of correlation and with monopole-monopole term of the exchange interaction only, (ii) with correlation and monopole-monopole term of exchange, and (iii) without correlation but with assuming the monopolemonopole, monopole-dipole, and dipole-dipole terms of the exchange interaction what we call multipole expansion. [56,70] The results for FSS of X 0 are given in Fig. 5 (a) by circles and full curves. Firstly, we see that FSS values computed using approximations (i) and (ii) defined in previ-ous paragraph are < 10 µeV for all studied Se concentrations. However, the multipole expansion of exchange, i.e., point (iii) above, gives more realistic estimation of FSS, when compared to experimental data {Figs. 1 and 2}, in particular for type-I confinement. We, thus, conclude that FSS in type I and type II confinement of our system is dominated by multipole expansion of the exchange interaction, predominantly by the dipole-dipole term [67]. However, in the regime of type I/II FSS is increased up to 20 µeV due to the effect of Coulomb correlation.
In Fig. 5 (b) we show the energy difference between optically active (bright), and inactive (dark) X 0 doublets and we mark that as X 0 BD in that panel. We see that similarly to FSS, X 0 BD is larger in type-I regime. Interestingly, contrary to FSS, BD is dominated in type I by the approximation (ii), i.e., the correlation, which causes FSS to be more than twice larger than that found by approximations (i) and (iii). On the other hand, in type II, BD is caused predominantly by approximation (iii), i.e., multipole expansion. Finally, BD for type I/II is again caused mainly by correlation {approximation (ii)}.
Furthermore, we see the theory prediction of X 0 radiative rate, computed by our k · p+CI computational complex, in Fig. 5 (c). We observe that the computed rates for all three approximations (full curves) are close to experimental results in Fig. 2 (b). As expected, the emission rate of our dots in type II (∼ 10 −3 1/ns) is reduced by two orders of magnitude compared to type I (∼ 10 −1 1/ns). Moreover, for the type-I/II regime that is reduced by another two orders of magnitude (to ∼ 10 −5 1/ns). However, we note that in our CI calculations we omit the short-range interaction within the unit crystal cell. Thus, we neglect also the effect of that on the emission rate, resulting in the difference to the experimental data.
The aforementioned behavior can be understood with inspection of the probability densities given in Fig. 4. We see that in type I {Fig. 4 (a)} both electron and hole probability densities reside inside QD body, hence, the overlap is largest of the studied types of confinement. On the other hand, in type II {Fig. 4 (b)} while the electrons still reside inside QD, holes are pushed to the surrounding ZnTe material above and below QD, what also explains the reduced oscillator strength of X 0 . The confinement for holes is provided by the bandedge changes caused by the elastic strain around QD and piezoelectricity originating in lattice mismatch between dot and buffer materials [7,42]. Finally, for type I/II in Fig. 4 (c) we find the holes to be confined again inside QD body. However, holes, in particular those lowest in energy, are confined on the outskirts of QD and their overlap with electrons is very faint as the wavefunctions almost exactly "miss" each other. It is that kind of behavior, combining localization of holes inside QD with very small overlap with electrons, that led us to the nomenclature of this confinement as type-I/II. However, the aforementioned analysis of the radiative emission rate of X 0 from Cd(Se,Te)/ZnTe QDs indicates that both type II and type I/II regimes are very hard to be accessed by optical means, as the dots would emit one photon in ∼ 1 µs or even in ∼ 100 µs for the former and latter confinements, respectively.
Finally, we would like to comment on the topology of the hole wavefunctions in Cd(Se,Te)/ZnTe QDs. From the probability densities shown in Fig. 4 we can observe, that the holes for type-II and type-I/II {Fig. 4 (b) and (c)} have toroidal shape for both ground and excited SP states. Thus, the hole states in Cd(Se,Te)/ZnTe QDs with type-II or type-I/II might be utilized for the realization of the Aharonov-Bohm effect [71], similarly as was proposed, e.g., in Ref. [72].
CONCLUSIONS
We have studied the excitonic structure of Cd(Se,Te) QDs embedded in ZnTe matrix. The photoluminescence spectroscopy analysis revealed reduction of FSS of ground state exciton with increasing Se content in dots. This was found to be associated with type-II character of the dots for larger Se contents. We confirmed that by detailed k · p and CI calculations, the results of which explained the experimentally observed trends very well. The theory identified the main mechanism causing larger FSS, in particular in type-I dots, to be due to the multipole expansion of the exchange interaction. Furthermore, using our theory we found that for Se contents in the dot larger than ∼ 0.3 a peculiar type I/II confinement occurs, causing an almost purely light hole character of exciton and toroidal shape of hole states. ACKNOWLEDGMENTS P.W. acknowledges the financial supported from the National Centre of Science (Poland) through grant 2017/26/E/ST3/00253. P.K. was financed by the project CUSPIDOR, which has received funding from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme. In addition, this project has received national funding from the Ministry of Education, Youth and Sports of the Czech Republic and funding from European Union's Horizon 2020 (2014-2020) research and innovation framework programme under Grant agreement No. 731473. The work reported in this paper also was partially funded by projects 20IND05 QADeT, 20FUN05 SE-QUME, 17FUN06 SIQUST that received funding from the EMPIR programme co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation programme.
FIG. 1 .
1Fine structure splitting (FSS) of individual Cd(Se,Te)/ZnTe quantum dots (QDs) for (a) exciton (X) and biexciton (XX) emission from a single Cd(Se,Te)/ZnTe QD measured in two orthogonal linear polarizations corresponding to the anizotropy axes of the dot (b) spectral position of the exciton and biexciton emission from the same dot as a function of the polarization angle. Solid lines represent fits with sine square functions from which the FSS value of 275 µeV is determined (c) FSS values for various individual QDs as a function of the exciton emission energy. The inset
FIG. 2 .
2Fig. 2 (c). The values presented inFig. 2(c) are obtained from the arithmetic average from all QD excitons observed on a given sample. The three experimental points correspond to the three samples with different average Se concentrations investigated in this work. Despite of the fact that there is a large distribution of FSS values a clear decrease of average FSS values with increasing Se concentration is observed. In the case of Cd(Se,Te)/ZnTe QDs with Se-content of 0Dependence of selected optical properties on the Se concentration within Cd(Se,Te)/ZnTe quantum dots (a) emission energy (b) decay rates, whereas fast and slow decay are shown in blue and red, respectively, (c) average FSS determined from the data presented inFigure 1 (c). Temperature of the measurements was 7 K and the excitation laser wavelength 405 nm.
FIG. 3 .
3Electronic states of Cd(Se,Te)/ZnTe QDs. In panel (a) we show twelve single-particle (SP) energies of electrons (blue) and holes (red). The inset in (a) gives markings of different types of confinement in Cd(Se, Te)/ZnTe QDs, i.e., type-I (mark T-I) for Se=0-0.15, type-II (mark T-II) for Se=0.15-0.3, and a type I/II (mark T-I/II) for Se=0.3-0.4.
FIG. 5 .
5Fine energy structure of ground state exciton (X 0 ) in Cd(Se,Te)/ZnTe QDs as a function of Se content. In (a) we show the theory values of FSS obtained using CI with different level of approximation, i.e., without including the effect of correlation as well as without considering the multipole expansion of exchange interaction {"CI (i)", red curve}, that for data with correlation included and without multipole {"CI (ii)", blue curve}, and without the effect of correlation but with multipole expansion included {"CI (iii)", green curve}. Panel (b) shows the calculated values of energy splitting between bright and dark exciton (X 0 BD) and in (c) we give the computed values of corresponding X 0 emission rates. Note that colors of theory curves in (b) and (c) correspond to the same CI approximations as was described for (a). The transitions between different confinements are marked in all panels by black dotted vertical lines.
The paper is organized as follows.We start with description of experiments, i.e., the growth of arXiv:2112.06798v2 [cond-mat.mes-hall] 1 Mar 2022
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[
"A Mixed-ADC Receiver Architecture for Massive MIMO Systems",
"A Mixed-ADC Receiver Architecture for Massive MIMO Systems"
]
| [
"Ning Liang [email protected] \nDepartment of EEIS\nKey Laboratory of Wireless-Optical Communications\nChinese Academy of Sciences\nUniversity of Science and Technology\nChina Emails\n",
"Wenyi Zhang [email protected] \nDepartment of EEIS\nKey Laboratory of Wireless-Optical Communications\nChinese Academy of Sciences\nUniversity of Science and Technology\nChina Emails\n"
]
| [
"Department of EEIS\nKey Laboratory of Wireless-Optical Communications\nChinese Academy of Sciences\nUniversity of Science and Technology\nChina Emails",
"Department of EEIS\nKey Laboratory of Wireless-Optical Communications\nChinese Academy of Sciences\nUniversity of Science and Technology\nChina Emails"
]
| []
| Motivated by the demand for energy-efficient communication solutions in the next generation cellular network, a mixed-ADC receiver architecture for massive multiple input multiple output (MIMO) systems is proposed, which differs from previous works in that herein one-bit analog-to-digital converters (ADCs) partially replace the conventionally assumed high-resolution ADCs. The information-theoretic tool of generalized mutual information (GMI) is exploited to analyze the achievable data rates of the proposed system architecture and an array of analytical results of engineering interest are obtained. For deterministic single input multiple output (SIMO) channels, a closed-form expression of the GMI is derived, based on which the linear combiner is optimized. Then, the asymptotic behaviors of the GMI in both low and high SNR regimes are explored, and the analytical results suggest a plausible ADC assignment scheme. Finally, the analytical framework is applied to the multi-user access scenario, and the corresponding numerical results demonstrate that the mixed system architecture with a relatively small number of high-resolution ADCs is able to achieve a large fraction of the channel capacity without output quantization.Index Terms-Analog-to-digital converter, generalized mutual information, massive MIMO, mixed architecture, multi-user access. | 10.1109/itwf.2015.7360769 | [
"https://arxiv.org/pdf/1507.07290v1.pdf"
]
| 17,310,680 | 1507.07290 | 3c8d7591804d0c692dfff06f8446c199051f01a3 |
A Mixed-ADC Receiver Architecture for Massive MIMO Systems
27 Jul 2015
Ning Liang [email protected]
Department of EEIS
Key Laboratory of Wireless-Optical Communications
Chinese Academy of Sciences
University of Science and Technology
China Emails
Wenyi Zhang [email protected]
Department of EEIS
Key Laboratory of Wireless-Optical Communications
Chinese Academy of Sciences
University of Science and Technology
China Emails
A Mixed-ADC Receiver Architecture for Massive MIMO Systems
27 Jul 2015
Motivated by the demand for energy-efficient communication solutions in the next generation cellular network, a mixed-ADC receiver architecture for massive multiple input multiple output (MIMO) systems is proposed, which differs from previous works in that herein one-bit analog-to-digital converters (ADCs) partially replace the conventionally assumed high-resolution ADCs. The information-theoretic tool of generalized mutual information (GMI) is exploited to analyze the achievable data rates of the proposed system architecture and an array of analytical results of engineering interest are obtained. For deterministic single input multiple output (SIMO) channels, a closed-form expression of the GMI is derived, based on which the linear combiner is optimized. Then, the asymptotic behaviors of the GMI in both low and high SNR regimes are explored, and the analytical results suggest a plausible ADC assignment scheme. Finally, the analytical framework is applied to the multi-user access scenario, and the corresponding numerical results demonstrate that the mixed system architecture with a relatively small number of high-resolution ADCs is able to achieve a large fraction of the channel capacity without output quantization.Index Terms-Analog-to-digital converter, generalized mutual information, massive MIMO, mixed architecture, multi-user access.
I. INTRODUCTION
The prosperity of mobile Internet calls for new technologies to meet the exponential increase in demand for mobile data traffic. In recent years, a heightened attention has been focused on massive multiple input multiple output (MIMO) systems, which achieves significant gains in both energy efficiency and spectral efficiency, and thus are envisioned as a promising key enabler for the next generation cellular network [1] [2].
Thus far, most of the works on massive MIMO assume perfect hardware implementation. However, this assumption is not well justified, since the hardware cost and circuit power consumption scale linearly with the number of BS antennas and thus soon become economically unbearable unless lowcost, energy-efficient hardware is deployed which however easily suffers from impairments. Among various sources of hardware impairment, low-resolution analog-to-digital converters (ADCs) have attracted ubiquitous attention due to their favorable property of low cost, low power consumption and feasibility of implementation [3]. For Nyquist-sampled real Gaussian channel, the authors of [4] established some general results regarding low-resolution output quantization. The authors of [5] designed a modified minimum mean square error (MMSE) receiver for MIMO systems with output quantization. In [6], the authors investigated a practical monobit digital receiver paradigm for impulse radio ultra-wideband (UWB) systems. Recently, the authors of [7] examined the impact of one-bit quantization on achievable rates of massive MIMO systems with both perfect and estimated channel state information (CSI). The authors of [8] addressed the high signal-to-noise (SNR) capacities of both single input multiple output (SIMO) and MIMO channels with one-bit quantization.
Despite of its great superiority in deployment cost and energy efficiency, one-bit quantization generally has to tolerate large rate loss, especially in the high SNR regime [8], thus highlighting the indispensability of high-resolution ADC for digital receiver. Motivated by such consideration, in this paper we propose a mixed receiver architecture for massive MIMO systems in which one-bit ADCs partially, but not completely, replace conventionally assumed high-resolution ADCs. This architecture has the potential of allowing us to remarkably reduce the hardware cost and power consumption while still maintain a large fraction of the performance gains promised by massive MIMO.
Recognizing the challenge in working with the channel capacity directly, we take an alternative route and seek to characterize the achievable data rates specified to certain encoding/decoding scheme. To this end, we exploit the information-theoretic tool of generalized mutual information (GMI) [9] to address the achievable data rates of our proposed system architecture. As a performance measure for mismatched decoding, GMI has proved convenient and useful in several important scenarios such as fading channels with imperfect CSI at the receiver [9] and channels with transceiver distortion [10], [11].
Exploiting a general analytical framework developed in [10], we obtain a series of analytical results. First, we consider a deterministic SIMO channel where the BS is equipped with N antennas but only has access to K pairs of high-resolution ADCs and (N − K) pairs of one-bit ADCs, and derive a closed-form expression of the GMI for general ADC assignment and linear combiner. This enables us to optimize the linear combiner design and further explore the asymptotic behaviors of the GMI in both low and high SNR regimes that in turn help us suggest a plausible ADC assignment scheme. The corresponding numerical results indicate that even with a small number of high-resolution ADCs, our system architecture already achieves a substantial fraction of the channel capacity without output quantization, thus verifying the effectiveness of the mixed architecture. Further issues such as ergodic fading channels with or without channel estimation error are included in an extended work [12].
Then, we apply our analysis to the multi-user access scenario. Numerical results reveal that the mixed system architecture with a small number of high-resolution ADCs achieves a large fraction of the channel capacity without output quantization, provided that the multi-user system is properly loaded. In summary, the proposed mixed architecture strikes a reasonable and attractive balance between cost and spectral efficiency, for both single-user and multi-user scenarios. Thus we envision it as a promising receiver paradigm for energy-efficient massive MIMO systems.
The remaining part of this paper is organized as follows. Section II outlines the system model. Adopting GMI as the performance metric, Section III establishes the theoretical framework for deterministic SIMO channels, based on which the optimal linear combiner design and the asymptotic behaviors of the GMI in both low and high SNR regimes are explored. Section IV extends the theoretical framework to the multi-user access scenario. Numerical results are given in Section V to corroborate the analysis. Finally, Section VI concludes the paper.
Notation: Throughout this paper, we use x to represent the 2-norm of vector x, and let X * , X T and X H denote the conjugate, transpose and conjugate transpose of X, respectively. Complex Gaussian distribution with mean µ and variance σ 2 is denoted by CN(µ, σ 2 ), while CN(µ, C) stands for the distribution of a circularly symmetric complex Gaussian random vector with mean µ and covariance matrix C. We use log(x) to denote the natural logarithm of positive real number x.
II. SYSTEM MODEL
We start by focusing on a single-user system where a singleantenna user communicates with an N -antenna BS. In this paper, we consider a narrow-band channel model, for which the frequency-flat fading channel h is chosen according to CN(0, I) and is fixed throughout the transmission of the codeword. Moreover, the realization of the channel is assumed to be perfectly known by the BS and thus is deemed as deterministic in the subsequent analysis. The received signal at the BS can be expressed as
y l = hx l + z l , for l = 1, 2, ..., L,(1)
where x l is the complex signal transmitted at the l-th symbol time, z l ∼ CN(0, σ 2 I) models the independent and identically distributed (i.i.d.) complex Gaussian noise vector, and L is the codeword length. We consider a mixed architecture, as illustrated in Figure 1, in which only 2K high-resolution ADCs are available and all the other 2(N − K) ADCs are with only one-bit resolution. We further let the I/Q outputs at each antenna be quantized by two ADCs of the same kind. Thus the quantized output can be expressed as
r l n = δ n · (h n x l + z l n ) +δ n · sgn(h n x l + z l n ),(2)
for l = 1, ..., L, n = 1, ..., N . Hereδ n 1 − δ n , and δ n ∈ {0, 1} is the binding indicator: δ n = 1 means that the ADCs corresponding to the n-th antenna are high-resolution, whereas δ n = 0 indicates that they are with one-bit resolution. Note that here we assume sufficiently high resolution, so that the residual quantization noise is negligible, for δ n = 1. For notation simplicity, we further define a binding vector δ [δ 1 , ..., δ N ] T , which should be optimized according to the channel state h so that the limited number of high-resolution ADCs will be well utilized to enhance the system performance.
For transmission of rate R, the user selects a message m from M = {1, 2, ..., ⌊2 LR ⌋} uniformly randomly, and maps the selected message to a transmitted codeword, i.e., a length-
L complex sequence, {x l (m)} L l=1 .
In this paper, we restrict the codebook to be drawn from a Gaussian ensemble; that is, each codeword is a sequence of L i.i.d. CN(0, E s ) random variables, and all the codewords are mutually independent. Such a choice of codebook satisfies the average power constraint
1 L L l=1 E[|x l (m)| 2 ] ≤ E s .
We thus define the SNR as SNR d = E s /σ 2 , and let σ 2 = 1 thereafter for convenience.
For an N -antenna SIMO channel, we let C(N, K 1 , K 2 ) denote its capacity when equipped with K 1 pairs of high-resolution ADCs and K 2 pairs of one-bit ADCs, where 0 ≤ K 1 , K 2 , K 1 + K 2 ≤ N . As discussed in the introduction, the evaluation of C(N, K, N − K) is a formidable task. Therefore in the following, we adopt the nearest-neighbor decoding rule at the decoder, and leverage the general framework developed in [10] to investigate the GMI of our proposed system architecture. The GMI acts as an achievable rate and thus also a lower bound of C(N, K, N − K). To this end, we introduce a linear combiner to process the channel output vector, as illustrated in Figure 1. Thus the processed channel output iŝ
x l = w H r l ,(3)
for l = 1, ..., L, where w is designed according to the channel state h and the binding vector δ.
With nearest-neighbor decoding, upon observing {x l } L l=1 , the decoder computes, for all possible messages, the distance metrics
D(m) = 1 L L l=1 |x l − ax l (m)| 2 , m ∈ M,(4)
and decides the received message as the one that minimizes (4). The scaling parameter a is selected appropriately for optimizing the decoding performance.
III. GMI AND OPTIMIZATION OF COMBINER
A. GMI of the Proposed System Architecture
From now on, we suppress the time index l for notational simplicity. To facilitate the exposition, we summarize (2) and
(3) asx = w H r f (x, h, z),(5)
where f (·) is a memoryless nonlinear distortion function that incorporates the effects of output quantization as well as linear combining. Although δ and w are made invisible in the function f (·) since they are both determined by h, we need to keep in mind that f (·) implicitly includes δ and w.
We apply the general framework developed in [10] to derive the GMI of our proposed system architecture. Particularly, employing the similar procedure as [10, Appendix C], we have the conditional GMI given as follows.
Proposition 1: With Gaussian codebook ensemble and nearest-neighbor decoding, the GMI conditioned on w and δ is given as
I GMI (w, δ) = log 1 + κ(w, δ) 1 − κ(w, δ) ,(6)
where the parameter κ(w, δ) is
κ(w, δ) = |E[f * (x, h, z) · x]| 2 E s E[|f (x, h, z)| 2 ] .(7)
The corresponding optimal choice of the scaling parameter a is
a opt (w, δ) = E[f (x, h, z) · x * ] E s .(8)
Note that κ(w, δ) is the squared correlation coefficient of channel input x and the processed output f (x, h, z), and thus is upper bounded by one, from Cauchy-Schwartz's inequality. Moreover, I GMI (w, δ) is a strictly increasing function of κ(w, δ) for κ(w, δ) ∈ (0, 1). Therefore in the following, we will seek to maximize κ(w, δ) by choosing well designed linear combiner w and binding vector δ. To start with, we derive a closed-form expression of κ(w, δ), given as follows.
Proposition 2: Given w and δ, for (7) in Proposition 1, we have
κ(w, δ) = w H R rx R H rx w E s w H R rr w ,(9)
where R rx E[rx * ] is the correlation vector between r and x, with its n-th element being (R rx ) n = h n E s δ n +δ n · 4 π(|h n | 2 E s + 1)
,
and R rr E[rr H ] is the covariance matrix of r, with its (n, m)th entry being (R rr ) n,m =
1 + δ n · |h n | 2 E s +δ n , if n = m, h n h * m E s δ n δ m + δ nδm · 4 π(|hm| 2 Es+1) + δ n δ m · 4 π(|hn| 2 Es+1) + δ nδm · 4 π arcsin (hnh * m ) R Es √ |hn| 2 Es+1 √ |hm| 2 Es+1 + i·arcsin (hnh * m ) I Es √ |hn| 2 Es+1 √ |hm| 2 Es+1
, if n = m.
(11) The corresponding optimal choice of the scaling parameter a in
(8) is a opt (w, δ) = 1 E s w H R rx .(12)
Proof: See [12, Appendix A].
B. Optimization of Linear Combiner
In this subsection, we turn to optimize w such that the GMI is maximized for given h and δ. The subsequent proposition summarizes our result.
Proposition 3: For given h and δ, the optimal linear combiner w takes the following form
w opt = R −1 rr R rx ,(13)
which is in fact a minimum mean square error (MMSE) combiner that minimizes the mean squared estimation error of x upon observing r. The corresponding κ(w, δ) is
κ(w opt , δ) = a opt (w opt , δ) = 1 E s R H rx R −1 rr R rx .(14)
Proof: See [12, Section III-B] for the proof.
C. Asymptotic Behaviors of I GMI (w opt , δ)
In the previous subsection, the optimal linear combiner for our proposed framework is derived. Thus we are ready to examine its asymptotic performance in both low and high SNR regimes. Letting SNR tend to zero, we have the following corollary.
Corollary 1: As E s → 0, for given δ we have
I GMI (w opt , δ) = N n=1 δ n +δ n · 2 π |h n | 2 E s + o(E s ). (15)
See [12, Appendix B] for its proof. Comparing with C (N, N, 0) in the low SNR regime, i.e., C(N, N, 0) = N n=1 |h n | 2 E s + o(E s ), we conclude that part of the achievable rate is degraded by a factor of 2 π due to one-bit quantization. For the high SNR case, the subsequent corollary collects our results.
Corollary 2: As E s → ∞, for given δ we have the effective SNR in (6) as
κ(w opt , δ) 1 − κ(w opt , δ) = p 2 E s + [4 + O(1/E s )]q H B −1 q π − [4 + O(1/E s )]q H B −1 q ,(16)
with p, q, and B given by
p [h 1 , ..., h K ] T ,(17)q h K+1 /|h K+1 |, ..., h N /|h N | T ,(18)(B) n,m 4 π arcsin (h n+K h * m+K ) R |h n+K h * m+K | + i · arcsin (h n+K h * m+K ) I |h n+K h * m+K | +O(1/E s ),(19)
where h is a rearrangement of h, by stacking the channel coefficients of the antennas equipped with high-resolution ADCs in the first K positions of h. See [12, Appendix C] for the proof. An observation from (16) indicates that the contributions of high-resolution ADCs and one-bit ADCs in the high SNR regime are separate, as the first term corresponding to high-resolution ADCs increases linearly with E s , whereas the second term coming from one-bit ADCs tends to be a positive constant independent of E s . Comparing with Corollary 1, we infer that one-bit ADCs are getting less beneficial as the SNR grows large, as will be validated by numerical study in Section V. Moreover, both (15) and (16) suggest that high-resolution ADCs should be assigned to the antennas with the strongest K link magnitude gains, at least in the low and high SNR regimes.
IV. EXTENSION TO MULTI-USER SCENARIO
In this section, the BS serves M single-antenna users simultaneously. For simplicity, we focus on the deterministic channel case, and the channel matrix between the users and the BS is denoted by H [h 1 , ..., h N ] ∈ C M×N , whose elements are i.i. d. CN(0, 1), i.e., h n [h 1n , ..., h Mn ] T collecting the channel coefficients related to the n-th BS antenna. There are still only K pairs of high-resolution ADCs available at the BS. Thus we rewrite the quantized output at the n-th antenna, with user j considered, as
r mu n = δ n · M ι=1
h ιn x ι +z n +δ n ·sgn M ι=1 h ιn x ι +z n , (20) where x ι ∼ CN(0, E s ) denotes the i.i.d. coded signal dedicated to the ι-th user, and M ι =j h ιn x ι + z n summarizes the cochannel interference and noise for the considered user j. For a fair comparison, the SNR in this situation is defined as SNR d = M E s , reflecting the total transmit power from all users.
Following a similar derivation procedure as that constructed in Section III, we get the GMI of the considered user.
Proposition 4: For given H and δ, when treating other users' signals as noise, the GMI of user j is
I mu GMI = log 1 + κ mu 1 − κ mu ,(21)
where the parameter κ mu is
κ mu = 1 E s (R mu rx ) H (R mu rr ) −1 R mu rx .(22)
R mu rx is the correlation vector between r mu and x j , with its n-th entry given as
(R mu rx ) n = h jn E s δ n +δ n · 4 π( h n 2 E s + 1) ,(23)
and R mu rr is the covariance matrix of r mu , with its (n, m)-th entry being (R mu
rr ) n,m = 1 + δ n · h n 2 E s +δ n , if n = m, h T n h * m E s δ n δ m + δ nδm · 4 π( hm 2 Es+1) + δ n δ m · 4 π( hn 2 Es+1) + δ nδm · 4 π arcsin (h T n h * m ) R Es √ hn 2 Es+1 √ hm 2 Es+1 + i·arcsin (h T n h * m ) I Es √ hn 2 Es+1 √ hm 2 Es+1
, if n = m.
(24) In the multi-user scenario, there is no clear clue about how to assign the high-resolution ADCs. For this reason, we consider two heuristic ADC assignment schemes in the numerical study. That is • Scheme #1: high-resolution ADCs are assigned to the antennas with the maximum M j=1 |h jn | 2 . • Scheme #2: high-resolution ADCs are assigned randomly. Scheme #1 follows from the asymptotic property of I mu GMI in the low SNR regime, and the proof is omitted due to space limitation.
V. NUMERICAL RESULTS
In this section, we validate our previous analysis with numerical results. The GMI and the channel capacity are both averaged over a large number of channel realizations, for both single-user and multi-user scenarios.
A. GMI for Single-User Scenario
We assign the high-resolution ADCs to the antennas with the strongest K link magnitude gains, as suggested by Corollary 1 and Corollary 2. In Figure 2 and Figure 3, the solid curves represent I GMI (w opt , δ), the dashed lines correspond to the channel capacity without output quantization, i.e., C(N, N, 0) = log(1 + h 2 SNR d ), and the dash-dot curves account for the channel capacity under antenna selection, i.e., C(N, K, 0) = log(1 + N n=1 δ n · |h n | 2 SNR d ). Several observations are in order. First, both figures clearly show that the gain of deploying more high-resolution ADCs decreases rapidly, and thus a small number of high-resolution ADCs actually already achieves a large fraction of C (N, N, 0). For example, our proposed system architecture with 10 pairs of high-resolution ADCs attains 84% of C(100, 100, 0) when SNR d = 0dB, and it achieves 91% of C(100, 100, 0) when K increases to 20. Besides, both figures indicate that, compared to the antenna selection paradigm, one-bit ADCs in our architecture are less beneficial when the SNR grows large, but still significantly improve the system performance in the low to intermediate SNR regime.
B. GMI for Multi-user Scenario
Now, we examine the feasibility of our proposed system architecture in the multi-user scenario. Figure 4 collects the result, where the solid curves correspond to Scheme #1 suggested by asymptotic analysis in the low SNR regime, the dash-dot curves are obtained by random assignment per Scheme #2, and the dashed lines refer to the per-user capacity with N pairs of high-resolution ADCs, i.e., 1 M log det(I + E s HH H ). We notice that though Scheme #1 is only analytically validated for the low SNR case, it does achieve better performance than Scheme #2. For the special case of K = N , it is well known that the linear MMSE receiver is suboptimal for MIMO channel [13], and thus we observe a distinguishable gap between the per-user capacity and the GMI. Most importantly, here a small number of high-resolution ADCs also attain a large fraction of the channel capacity with N pairs of high-resolution ADCs. For example, when SNR d = 0dB and N = 100, Scheme #1 with K = 10 achieves 76% of the per-user capacity with N pairs of high-resolution ADCs, and this number rises to 80% when K = 20. Figure 5 accounts for the impact of increasing number of users on the achievable sum rates, focusing on K = 20 and SNR d = 0dB. For a fair comparison, we introduce another performance curve corresponding to MMSE receiver with N pairs of high-resolution ADCs. We conclude that our proposed architecture achieves satisfactory performance even when the system is heavily loaded, by noticing that, even for M = N = 100, our proposed architecture still attains 70% of that achieved by MMSE receiver without quantization.
VI. CONCLUSION
In this paper, we propose a mixed-ADC receiver architecture, and leverage the GMI to analytically evaluate its achievable data rates under two scenarios. The corresponding numerical study concludes that the proposed architecture with a small number of high-resolution ADCs suffices to achieve a significant fraction of the channel capacity without output quantization, for both single-user and multi-user scenarios. Thus our approach provides a systematic way of designing energy-efficient massive MIMO receivers.
A number of interesting and important problems remain unsolved beyond this paper, such as designing the optimal ADC assignment scheme for any SNR, especially for the multi-user scenario; extending the analysis to more comprehensive hardware impairment models beyond ADC; among others. Besides, in order to make this approach effective for wideband channels which are more prevailing in the future communication systems, it is particularly crucial to extend the analysis to frequencyselective fading channels. We note that this is feasible but beyond the scope of this paper, and will be treated in a future work.
Fig. 1 .
1Illustration of the system architecture.
Fig. 2 .
2GMI of the proposed system architecture for different number of highresolution ADCs, N = 100.
Fig. 3 .
3GMI of the proposed system architecture for different number of highresolution ADCs, N = 400.
Fig. 4 .
4GMI of the considered user j in multi-user scenario, N = 100, M = 10.
Scheme #1 Scheme #2 Capacity w/o quantization MMSE w/o quantization
Fig. 5 .
5Sum GMI for users in multi-user scenario: increasing number of users, N = 100, K = 20, SNR d =0dB.
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| []
|
[
"Persuading Voters in District-based Elections",
"Persuading Voters in District-based Elections"
]
| [
"Matteo Castiglioni [email protected] \nPolitecnico di Milano\nPiazza Leonardo da Vinci 32I-20133MilanItaly\n",
"Nicola Gatti [email protected] \nPolitecnico di Milano\nPiazza Leonardo da Vinci 32I-20133MilanItaly\n"
]
| [
"Politecnico di Milano\nPiazza Leonardo da Vinci 32I-20133MilanItaly",
"Politecnico di Milano\nPiazza Leonardo da Vinci 32I-20133MilanItaly"
]
| []
| We focus on the scenario in which an agent can exploit his information advantage to manipulate the outcome of an election. In particular, we study district-based elections with two candidates, in which the winner of the election is the candidate that wins in the majority of the districts. District-based elections are adopted worldwide (e.g., UK and USA) and are a natural extension of widely studied voting mechanisms (e.g., k-voting and plurality voting). We resort to the Bayesian persuasion framework, where the manipulator (sender) strategically discloses information to the voters (receivers) that update their beliefs rationally. We study both private signaling, in which the sender can use a private communication channel per receiver, and public signaling, in which the sender can use a single communication channel for all the receivers. Furthermore, for the first time, we introduce semi-public signaling in which the sender can use a single communication channel per district. We show that there is a sharp distinction between private and (semi-)public signaling. In particular, optimal private signaling schemes can provide an arbitrarily better probability of victory than (semi-)public ones and can be computed efficiently, while optimal (semi-)public signaling schemes cannot be approximated to within any factor in polynomial time unless P = NP. However, we show that reasonable relaxations allow the design of multi-criteria PTASs for optimal (semi-)public signaling schemes. In doing so, we introduce a novel property, namely comparative stability, and we design a bi-criteria PTAS for public signaling in general Bayesian persuasion problems beyond elections when the sender's utility function is state-dependent. arXiv:2012.05002v2 [cs.GT] 10 Dec 2020 1 In k-voting, a candidate wins if he collects at least k votes. 2 In majority voting, the candidate with the most votes wins. | null | [
"https://arxiv.org/pdf/2012.05002v2.pdf"
]
| 228,064,124 | 2012.05002 | 097638237894cbf9a56675f746a2eb207aeba883 |
Persuading Voters in District-based Elections
Matteo Castiglioni [email protected]
Politecnico di Milano
Piazza Leonardo da Vinci 32I-20133MilanItaly
Nicola Gatti [email protected]
Politecnico di Milano
Piazza Leonardo da Vinci 32I-20133MilanItaly
Persuading Voters in District-based Elections
We focus on the scenario in which an agent can exploit his information advantage to manipulate the outcome of an election. In particular, we study district-based elections with two candidates, in which the winner of the election is the candidate that wins in the majority of the districts. District-based elections are adopted worldwide (e.g., UK and USA) and are a natural extension of widely studied voting mechanisms (e.g., k-voting and plurality voting). We resort to the Bayesian persuasion framework, where the manipulator (sender) strategically discloses information to the voters (receivers) that update their beliefs rationally. We study both private signaling, in which the sender can use a private communication channel per receiver, and public signaling, in which the sender can use a single communication channel for all the receivers. Furthermore, for the first time, we introduce semi-public signaling in which the sender can use a single communication channel per district. We show that there is a sharp distinction between private and (semi-)public signaling. In particular, optimal private signaling schemes can provide an arbitrarily better probability of victory than (semi-)public ones and can be computed efficiently, while optimal (semi-)public signaling schemes cannot be approximated to within any factor in polynomial time unless P = NP. However, we show that reasonable relaxations allow the design of multi-criteria PTASs for optimal (semi-)public signaling schemes. In doing so, we introduce a novel property, namely comparative stability, and we design a bi-criteria PTAS for public signaling in general Bayesian persuasion problems beyond elections when the sender's utility function is state-dependent. arXiv:2012.05002v2 [cs.GT] 10 Dec 2020 1 In k-voting, a candidate wins if he collects at least k votes. 2 In majority voting, the candidate with the most votes wins.
Introduction
The fairness and efficiency of democratic elections largely depend on the news provided by the media. Indeed, often, citizens are called to express opinions on complex choices they do not know deeply enough to express informed judgments. Therefore, multiple and reliable sources of information providing fair and in-depth coverage of the public debate are crucial to guarantee the democratic process. However, most of the information shaping the voters' opinions is not disclosed to inform in a disinterested way but instead aims to direct voters' political orientation, thus persuading them to prefer one specific candidate over another.
As recently showed by Allcott and Gentzkow (2017) and Guess, Nyhan, and Reifler (2018) for the 2016 US presiden-tial election, the spread of fake news has become a major public concern for democracy. The problem of assessing the extent to which it is possible to manipulate an election has received considerable attention under the general framework of election control and has been investigated according to several perspectives, such as control by bribery (Faliszewski et al. 2009;Erdélyi, Reger, and Yang 2020) or by adding and deleting candidates and voters (Loreggia et al. 2015;Faliszewski, Hemaspaandra, and Hemaspaandra 2011;Liu et al. 2009;Chen et al. 2017). More recently, Sina et al. (2015), Faliszewski et al. (2018), Wilder and Vorobeychik (2018), Wilder and Vorobeychik (2019), and Castiglioni, Ferraioli, and Gatti (2020) studied social influence as a means of election control. In this paper, we pose the following question: can an informed agent use his information advantage to influence an election's outcome by the partial disclosure of information to rational voters?
According to the classical Bayesian persuasion framework by Kamenica and Gentzkow (2011), the above problem can be formulated as a game with asymmetric information, where a sender can influence the behavior of the receiver(s) through the strategic provision of payoff-relevant information. In particular, the sender can strategically reveal information by means of a signaling scheme that determines "who knows what" about the parameters that govern the payoff functions. Alonso and Câmara (2016), Chan et al. (2019), and Bardhi and Guo (2018) provide the seminal attempts to apply the Bayesian persuasion framework to voting. More recently, Castiglioni, Celli, and Gatti (2020a) and Castiglioni, Celli, and Gatti (2020b) investigated its computational issues. All the aforementioned works focus on kvoting or plurality-voting elections. Differently, in this paper, we study for the first time how Bayesian persuasion can be adopted in more challenging settings such as districtbased elections with two candidates, in which the winner of the election is the candidate winning in the majority of the districts. We focus on the setting with no inter-agent externalities where each receiver's utility depends only on his action and the realized state of nature, but not on the other receivers' actions. This assumption is well-motivated, as voting for the most preferred candidate is a weakly dominant strategy in two-candidate elections.
Two forms of signals are customarily investigated in the literature. With private signals, the sender can target differ-ent information to different receivers. Instead, with public signals, the sender can only communicate the same information to every receiver. Even if private persuasion may be more beneficial for the sender, sometimes, as in election settings where there are too many receivers, privately communicating to each receiver may be impracticable. At the same time, public communication is much easier to implement, e.g., through TVs or newspapers. We introduce a new form of signaling, called semi-public, to model situations between private and public settings in district-based elections, where all the receivers of the same district observe the same signal, but the sender can target different information to different districts. Indeed, the voters are often reached by local communication shared with the voters in the same location, e.g., local newspapers, electoral posters and rallies.
Original Contributions We study the efficiency and complexity of signaling in district-based elections with two candidates. First, we compare private, public, and semi-public signaling schemes in terms of efficiency when used to manipulate elections, showing that private signaling schemes perform arbitrarily better than (semi-)public schemes. Then, we show that optimal private signaling schemes can be computed efficiently, while the direct use of the results provided by Castiglioni, Celli, and Gatti (2020a) shows that the problem is inapproximable with (semi-)public signaling. However, we prove that multi-criteria Polynomial-Time Approximation Schemes (PTASs) for public and semi-public signaling schemes are possible when some relaxations are made. In particular, in the case of semi-public persuasion, we allow -persuasiveness and lower the number of districts to control needed to win the election by an arbitrary constant factor w.r.t. the majority. Instead, in the case of public persuasion, we also need to lower the number of votes needed to win in a district by an arbitrary constant factor w.r.t. the majority. In doing so, we introduce a novel property, namely comparative stability, and we design a bi-criteria PTAS to compute public signaling schemes in general Bayesian persuasion problems beyond district-based elections. Our result extends that by Xu (2020), allowing state-dependent sender's utility functions and generalizing from stable to comparative stable sender's utility functions.
Related Works The seminal model of Bayesian persuasion with a single receiver is introduced by Kamenica and Gentzkow (2011). This model is extended, allowing multiple receivers, by Bergemann and Morris (2016a), Bergemann and Morris (2016b), Wang (2013), and Taneva (2015). Furthermore, Alonso and Câmara (2016), Bardhi and Guo (2018) and Chan et al. (2019) provide the first attempts of applying the Bayesian persuasion framework to voting. In particular, Bardhi and Guo (2018) and Chan et al. (2019) study unanimity voting and k-voting rules, respectively, in settings with binary actions and state spaces. Instead, Alonso and Câmara (2016) employ a novel geometric tool to characterize an optimal public signaling scheme in voting.
In addition to the works mentioned above, which provide the economic groundings of Bayesian persuasion in (sim-ple) voting settings, other works study election problems from a computational perspective. In particular, Arieli and Babichenko (2019) study the problem of private Bayesian persuasion with no inter-agent externalities. In the case of binary state spaces and k-voting rule, they provide a characterization of the optimal private signaling schemes. 1 Cheng et al. (2015) study the same k-voting problem with public persuasion, providing a polynomial-time approximation algorithm for a relaxed version of the problem in which the number of votes needed to win the election is reduced by an arbitrary constant factor and -persuasiveness is adopted. Castiglioni, Celli, and Gatti (2020a) extend the previous models to settings with an arbitrary number of states of nature and candidates. In particular, they prove that a private signaling scheme for k-voting can be computed in polynomial time, while the optimal public signaling scheme is NPhard to approximate within any factor. Castiglioni, Celli, and Gatti (2020b) strengthen this hardness result, showing that finding a public signaling scheme that is approximately optimal and -persuasive requires quasi-polynomial time, assuming the exponential time hypothesis.
Our work is also closely related to Bayesian persuasion in general settings beyond elections and the relation between stability of the sender's utility function and the computation of approximations. In particular, Cheng et al. (2015) introduce the notion of stability and show that this is a sufficient property to compute approximately optimal -persuasive signaling schemes in polynomial time. Xu (2020) extends this framework to incorporate α-approximable sender's utility functions and shows that approximately optimal andpersuasive signaling schemes can be computed in polynomial time when the sender's utility function is stable and independent of the state of nature.
Problem Formulation
In this section, we introduce the two frameworks we use in our work: district-based elections and Bayesian persuasion.
District-based Elections
There is a set of candidates C = {c 0 , c 1 } and a set of voters R = {r 1 , . . . , r |R| } divided in a set D of districts. The set of voters of district d ∈ D is denoted with R d . Each voter casts a vote for one of the two candidates. Once the voters expressed their preferences, the election process proceeds in two steps. For the sake of simplicity, we study the basic case in which both steps follow a majority-voting rule. 2 The election works as follows. 1. For each d ∈ D, the votes expressed by all r ∈ R d are locally aggregated, and the candidate with the majority of the votes is elected as the winner of the district.
2. The outcomes of all the districts are aggregated, and the candidate that is the winner in the majority of the districts is chosen as the winner of the district-based election.
We assume that the manipulator prefers c 0 to be the winner of the election. Let c ∈ C be a tuple composed by the votes of all the voters, where C = C |R| . Similarly, c d is the tuple of the votes of the voters in district d. The manipulator's utility W : C → {0, 1} is defined as the composition of a collection of functions W d : C |R d | → C, each representing the majority voting run in district d, and the function W : C |D| → {0, 1}, representing the majority voting that aggregates the outcomes of all the districts. We define K D = |D|/2 and, for each district d,
K d = |R d |/2 . Then, W is defined as W(c) = W (W 1 (c 1 ), . . . , W D (c |D| )), where W d (c d )
assumes value c 0 if at least K d of the voters in district d vote for candidate c 0 , and W assumes value 1 if and only if c 0 wins in at least K D districts.
We introduce some relaxations for the majority-voting rules W d and W . In the first relaxation, we allow the number of votes that the target candidate c 0 needs to win in each district d to be smaller than K d . We denote with W d δ the resulting majority voting rule. Formally, W d δ : C |R d | → C assumes value c 0 if at least (1−δ) K d voters in district d vote for c 0 and c 1 otherwise. The manipulator's utility function of this first relaxed problem, denoted with W δ , is defined as
W δ = W (W 1 δ (c 1 ), . . . , W D δ (c |D| )).
In the second, stronger relaxation, we also allow the number of districts that the target candidate c 0 needs to control to win the election to be smaller than K D . We denote with W δ the resulting majority voting rule aggregating the outcomes of the districts. Formally, W δ : C |D| → {0, 1} assumes value 1 when c 0 wins in at least (1 − δ) K D districts. The manipulator's utility function of this second relaxed problem, denoted with W δδ , is defined as
W δδ (c) = W δ (W 1 δ (c 1 ), . . . , W D δ (c D )).
Bayesian Persuasion Framework Our model includes a sender (the manipulator) and a set R of receivers (voters) that must choose an action (a candidate) from the set C = {c 0 , c 1 }. Each voter r's utility u r depends only on his own action and a state of nature θ ∈ Θ drawn from a prior distribution µ ∈ ∆ Θ , where ∆ Θ is the set of probability distributions supported on Θ. In particular, we define u r : Θ × C → [0, 1], where u r (θ, c) expresses how much receiver r appreciates candidate c when the state of nature is θ. We use u r (θ) = u r (θ, c 0 )−u r (θ, c 1 ) to denote how much voter r prefers candidate c 0 over c 1 , in state of nature θ. In general Bayesian persuasion problems, the sender's utility, usually denoted with f θ , depends on the state of nature θ and maps the receivers' action profiles to values in [0, 1].
In our setting, f θ does not depend on θ and is set equal to W, W δ , W δδ depending on the specific problem we tackle. The interaction among the sender and the receivers goes as follows (see Fig. 1). The sender commits to a randomized publicly known signaling scheme φ that maps states of nature to signals for the receivers. The signal set of a receiver r is denoted with S r , while s r ∈ S r is a signal for receiver r. The set of possible signals is then S = × r∈R S r , while a profile of signals is denoted with s = (s 1 , . . . , s |R| ). The sender observes the state of nature sampled from µ and computes s ∈ S according to φ. After observing the signal s r , each receiver r performs a Bayesian update, and infers a posterior belief p r ∈ P (where P = ∆ Θ ) as follows: the realized state of nature is θ with probability p r θ = µ θ φ(θ,sr) θ ∈Θ µ θ φ(θ ,sr) . Then, each receiver plays an action maximizing his expected utility according to posterior p r .
We introduce three forms of signaling schemes. A private signaling scheme exploits a private communication channel toward each receiver. Sometimes this assumption is not realistic, and the sender has only a single communication channel observed by all the receivers, i.e., s r = s r for all r, r ∈ R. We call these signaling schemes public. Finally, we introduce a novel form of communication that suits our election model, where the sender has a communication channel toward each district d, and all the receivers in the same district receive the same signal, i.e., s r = s r for all r, r ∈ R d . We call these signaling schemes semi-public.
In all these settings, a revelation-principle style argument shows that there always exists a signaling scheme that is direct and persuasive. More precisely, a signaling scheme is direct if the signals are action recommendations, while it is persuasive if each receiver has the interest to follow the recommendations. Thus, a direct signaling scheme is a mapping φ : Θ → ∆ C , and φ(θ, c) is the probability whereby the sender recommends c in state θ. In order for the signaling scheme to be persuasive, the receivers must have an incentive to follow the recommendation. This is customarily assured by forcing constraints on φ depending on the specific form of signaling. In particular, the incentive constraints associated with a receiver r in a district d are:
• θ,c:cr=c φ(θ, c)(u r (θ, c) − u r (θ, c )) ≥ 0 ∀c, c ∈ C (private signaling); • θ φ(θ, c)(u r (θ, c r ) − u r (θ, c )) ≥ 0 ∀c ∈ C, c ∈ C (public signaling); • θ,c:c d =c φ(θ, c)(u r (θ,c r ) − u r (θ, c )) ≥ 0 ∀c ∈ C |R d | , c ∈ C (semi-public signaling).
Similarly, a direct signaling scheme is -persuasive if the incentive constraints are violated by at most .
Finally, we state the optimization problems we study in this paper. PRIVATE-DBE is the problem of designing a private signaling scheme maximizing the probability of having candidate c 0 elected in district-based elections. PUBLIC-DBE and SEMIPUBLIC-DBE refer to the same problem with public and semi-public signaling, respectively.
An Example of Inefficiency of (Semi-)Public Persuasion
To clarify better the Bayesian persuasion framework, we provide an example of its application to majority voting without districts. This example is also useful to show that the restriction to (semi-)public signaling can decrease the sender's utility by an arbitrarily large factor. Example 1. Consider a (non-relaxed) majority-voting election with seven voters R = {r 1 , r 2 , r 3 , r 4 , r 5 , r 6 , r 7 } and two candidates C = {c 0 , c 1 }. The objective of the sender is to maximize the probability with which candidate c 0 is elected. Therefore, he needs to persuade at least half of the voters (i.e., |R|/2 = 4) to make candidate c 0 be the winner. There are three states of nature, namely, Θ = {θ A , θ B , θ C }, and each state is equally probable. Tab. 1 provides the parameters u r (θ) of the voters, defined as u r (θ) = u r (θ, c 0 ) − u r (θ, c 1 ) and capturing the net payoff of voter r from having candidate c 0 elected, in state of nature θ. The sender can design a direct and persuasive private signaling scheme such that at least four voters prefer candidate c 0 over c 1 for every signal profile s. Hence, this scheme ensures that candidate c 0 is elected with a probability of 1. Specifically, in each state θ the scheme recommends candidate c 0 to every voter r with utility u r (θ) ≥ 0 and to one voter among those with u r (θ) < 0 chosen randomly with uniform probability. It is easy to see that this private signaling scheme satisfies the incentive constraints. Consider, for example, voter r 1 . The marginal probabilities with which he is recommended to vote for candidate c 0 are: φ 1 (θ A , c 0 ) = 1, φ 1 (θ B , c 0 ) = 1/4 and φ 1 (θ C , c 0 ) = 1/4. Therefore, when he receives the recommendation to vote for c 0 , he has a posterior distribution p with p θ A = µ θ A · φ1(θ A ,c0) θ∈Θ µ θ · φ1(θ,c0) = 1/3 1/3+1/3·1/4+1/3·1/4 = 2/3 and p θ B = p θ C = 1/6. Thus, the voter has expected utility u(θ A )p θ A + u(θ B )p θ B + u(θ C )p θ C = 0 and will follow the recommendation. Similarly, we can show that the incentive constraints associated with the other voters are satisfied.
State θ A State θ B State θ C Voters r 1 ,r 2 +1/2 −1 −1 r 3 ,r 4 −1 +1/2 −1 r 5 ,r 6 −1 −1 +1/2 r 7 +1/2 +1/2 +1/2
We switch to public signals and we show that we cannot design a public signaling scheme that guarantees candidate c 0 to be elected with positive probability. Any public signaling scheme making candidate c 0 win the election with positive probability must assign a strictly positive probability to at least one signal that makes at least four voters prefer candidate c 0 over c 1 . We show that we cannot design such a public signal. In particular, we show that there is no posterior p ∈ P that provides an expected utility larger than or equal to zero to at least four voters. 3 Since receiver r 7 prefers candidate c 0 in every state of nature, he votes for c 0 independently from the posterior induced by the signal. Therefore, it is sufficient to persuade three voters among the first six. Suppose that voters r 1 and r 2 vote for c 0 . This im-
plies that p θ A /2 − p θ B − p θ C = p θ A /2 − (1 − p θ A ) ≥ 0 and p θ A ≥ 2/3.
Suppose, by contradiction, that also voters r 3 and r 4 vote for c 0 . This requires that −p θ A +p θ B /2−p θ C ≥ 0 and p θ B ≥ 2/3, reaching a contradiction with p ∈ P. It is easy to see that, by the symmetry of the instance, all the other sets of four voters cannot vote for c 0 at the same time.
From the previous example, we can state the following:
Proposition 1. There is an instance of majority-voting election in which the optimal private signaling scheme guarantees that candidate c 0 wins the election with a probability of 1, while the optimal public signaling scheme cannot guarantee a winning probability strictly larger than 0. This inefficiency result can be easily generalized to the case of public and semi-public signaling scheme in districtbased elections. Indeed, with only a single district, semipublic signals correspond to public signals and a districtbased election reduces to a simple majority-voting election as the one presented above.
Private Persuasion in District-based Elections
In this section, we show that an optimal private signaling scheme for district-based elections can be found in polynomial time. Our result is built upon the previous works by Arieli and Babichenko (2019) and Castiglioni, Celli, and Gatti (2020a) on k-voting. Let a d,θ be the probability with which K d voters vote for c 0 in district d when the state of nature is θ. Similarly, let α θ be the probability that c 0 wins in at least K D districts with state of nature θ. Finally, given a direct private signaling scheme φ, we denote with φ r (θ, c) = c:cr=c φ(θ, c) the marginal probabilities of φ whereby c is recommended to r with state of nature θ. We can compute an optimal private signaling scheme by LP (1) (all the proofs are in the Supplemental Material). Proof sketch. Constraints (1b) force the marginal probabilities φ r (θ, c 0 ) of the signaling scheme φ to satisfy the incentive constraints. For each state of nature θ, the maximum probability a d,θ with which at least K d receivers in R d vote for c 0 given marginal probabilities φ r (θ, c 0 ) is:
a d,θ = min min m∈{0,...,K d −1} 1 K d − m v θ,m ; 1 ,
where v θ,m is the sum of the lowest |R d | − m elements in the set {φ r (θ, c)} r∈R d ; for further details, see Arieli and Babichenko (2019). The above equation is enforced via Constraints (1f). Constraints (1g) and (1h) ensure that the values of v θ,m are consistent with the values of the other variables. The computation of the maximum probability α θ with which at least K D districts elect c 0 given probabilities a d,θ is similar to the computation of a d,θ given φ r (θ, c 0 ). This is enforced by Constraints (1c), (1d), and (1e). Finally, Objective (1a) maximizes the sum over all θ ∈ Θ of the prior probability multiplied by α θ , i.e., the probability that that c 0 wins when the state of nature is θ.
µ θ α θ (1a) s.t. θ∈Θ µ θ φ r (θ, c o ) u r (θ) ≥ 0 ∀r ∈ R (1b) α θ ≤ 1 K D − m i θ,m (1c) ∀θ ∈ Θ, ∀m ∈ {0, . . . , K D − 1} i θ,m ≤ (|D| − m)l θ,m + d∈D o d,θ,m (1d) ∀θ ∈ Θ, ∀m ∈ {0, . . . , K D − 1} a d,θ ≥ l θ,m + o d,θ,m (1e) ∀d ∈ D, ∀θ ∈ Θ, ∀m ∈ {0, . . . , K D − 1} a d,θ ≤ 1 K d − m v d,θ,m (1f) ∀d ∈ D, ∀θ ∈ Θ, ∀m ∈ {0, . . . , K d − 1} v d,θ,m ≤ (|R d | − m)t d,θ,m + r∈R d z d,θ,r,m (1g) ∀d ∈ D, ∀θ ∈ Θ, ∀m ∈ {0, . . . , K d − 1} φ r (θ, c 0 ) ≥ t d,θ,m + z d,θ,r,m (1h) ∀d ∈ D, ∀r ∈ R d , ∀θ ∈ Θ, ∀m ∈ {0, . . . , K d − 1}
Public and Semi-public Persuasion in District-based Elections
We turn our attention to the design of optimal public and semi-public signaling schemes. There is a sharp distinction between the nature of these problems and that one of private signaling. Indeed, in addition to being inefficient w.r.t. private signals (see Proposition 1), optimal (semi-)public signaling schemes are also inapproximable. The hardness follows from previous results with public signaling. Specifically, Castiglioni, Celli, and Gatti (2020a) prove that it is NP-hard to approximate the optimal public signaling scheme within any factor in elections with majority voting. The extension of this hardness result to public and semi-public signaling in district-base elections is direct as a district-based election reduces to majority-voting when there is only a single district. Thus, we focus on possible relaxations that make the problem computationally tractable. Motivated by the fact that voters are somewhat biased to follow the sender's recommendations, several works relax the incentive constraints allowing the receivers to vote for the target candidate even if other candidates give them a slightly better expected utility ( -persuasiveness). Recently, Castiglioni, Celli, and Gatti (2020b) prove that even allowing this relaxation the problem of designing an approximate public signaling scheme remains intractable with majority voting. Therefore, we focus on other different relaxations. In particular, Cheng et al. (2015) employ two forms of relaxation, adopting -persuasiveness and lowering the number of votes needed to win the election by an arbitrary constant factor. With these two relaxations, they prove that an approximate public signaling scheme with majorityvoting can be computed efficiently. We prove that, adapting these two relaxations to our settings, both PUBLIC-DBE and SEMIPUBLIC-DBE admit a multi-criteria PTAS. As a preliminary step, we prove some results on the relation between the notion of stability and the design of approximately optimal signaling schemes that are of general interest in Bayesian persuasion beyond elections.
Comparative Stability and Public Signaling Schemes
We refer to the notion of stability of a function introduced by Xu (2020). In particular, a function is said stable if, for every action profile, the introduction of small perturbations leads to small changes in the value of the function. Here, we extend the notion of stability to pairs of functions, and we call it comparative. Our extension is such that comparative stability corresponds to (simple) stability in the degenerate case in which the two functions of the pair are the same. Furthermore, if function g satisfies the comparative stability property w.r.t. function h, we also say that g is β-stable compared with h. Initially, we introduce the notion of perturbation by the concept of α-noisy distribution.
Definition 1. Let c ∈ C be an action profile and y be a probability distribution supported on ∆ C . For any α ∈ (0, 1], we say that y is an α-noisy distribution around c if for all i ∈ {1, . . . , n} : Prỹ ∼y [ỹ i = c i ] ≤ α.
Hence, an α-noisy distribution bounds the marginal probability of any single element of {1, . . . , n} to be corrupted. However, no assumption is made on how the corruptions of the elements correlate with each other. Now, we define our notion of comparative stability.
Definition 2. Given two functions g, h : C → [0, 1] and a real number β ≥ 0, we say that g is β-stable compared with h if and only if the following holds for all action profiles c ∈ C, α ∈ (0, 1], and α-noisy distributions y around c:
Ẽ y∼y [g(ỹ)] ≥ h(c)(1 − αβ).
Intuitively, if g satisfies the comparative stability property w.r.t. h, then, for every action profile, the value of h in that action profile is close to the value of g in the corresponding perturbed action profile.
We exploit the notion of comparative stability to design an efficient algorithm that computes approximate public signaling schemes. More precisely, we study a generic multi-agent Bayesian persuasion problem, where the sender faces a set of receivers R, and each receiver needs to choose an action between a couple of alternatives. Let g, h be two sets of arbitrary functions depending on the state of nature θ and denoted with g θ : C → [0, 1] and h θ : C → [0, 1], respectively. According to Definition 2, we say that g is β-stable compared with h if g θ is β-stable with respect to h θ for all the states of nature θ ∈ Θ.
For the sake of clarity, in the following, we use indirect signaling schemes, and we express a signaling scheme as a weighted set of posteriors to which the receivers respond at best. Now, we describe the optimal behavior of the receivers.
Definition 3 (Receivers' behavior with persuasiveness).
Given a set of functions {f θ } θ∈Θ such that f θ : C → [0, 1], the receivers' optimal behavior b p ∈ C with persuasiveness given posterior p ∈ P is as follows. Let: • A = {r ∈ R : θ p θ u r (θ) > 0} the set of receivers whose unique best response is action c 0 , • B = {r ∈ R : θ p θ u r (θ) < 0} the set of receivers whose unique best response is action c 1 , • E = {r ∈ R : θ p θ u r (θ) = 0} the set of receivers who are indifferent between action c 0 and c 1 .
Then, we have: b p = arg max c∈C:cr=c0∀r∈A, cr=c1∀r∈B θ p θ f θ (c).
Similarly, we define the notion of -best response.
Definition 4 (Receivers' behavior with -persuasiveness).
Given a set of functions {f θ } θ∈Θ such that f θ : C → [0, 1], the receivers' optimal behavior b p, ∈ C withpersuasiveness given posterior p ∈ P is as follows. Let:
• A = {r ∈ R : θ p θ u r (θ) > } the set of receivers whose unique best response is action c 0 , • B = {r ∈ R : θ p θ u r (θ) < − } the set of receivers whose unique best response is action c 1 , Now, we show that computing a direct public signaling scheme is equivalent to derive a Bayes plausible distribution of posteriors γ ∈ ∆ P that maximizes the sender's utility. Let supp(γ) denote the set of posteriors induced with strictly positive probability. Similarly, let supp(φ) denote the set of posteriors induced by φ with strictly positive probability. Finding a public signaling scheme is equivalent to finding a probability distribution γ ∈ ∆ P on the set of posteriors P such that p∈supp(γ) γ p p θ = µ θ for every θ ∈ Θ. Given a well-defined distribution over posteriors γ, we can recover a direct signaling schemes φ that induces such a probability distribution by setting φ θ (c) = p∈supp(γ):c=b p γ p p θ . For this reason, in the following, we represent signaling schemes as probability distributions on the posteriors. We introduce some further notation. For every p ∈ P and set of functions f = {f θ } θ∈Θ , we define the sender's expected utility with persuasiveness as f (p) = θ p θ f θ (b p ), and withpersuasiveness as f (p) = θ p θ f θ (b p, ). Finally, we define q-uniform probability distributions as follows.
• E = {r ∈ R : θ p θ u r (θ) ∈ [− , ]}
Definition 5. A probability distribution x ∈ ∆ X is quniform if and only if it is the average of a multiset of q basis vectors in |X|-dimensional space.
Therefore, we say that a probability distribution p ∈ P is q-uniform if each of its entry p θ is a multiple of 1/q. Moreover, we use the notation Q ⊂ ∆ Θ to denote the set of all q-uniform distributions over Θ. Our first result shows that we can decompose each posterior in a convex combination γ ∈ ∆ Q of q-uniform posteriors (with q constant), such that p∈Q γ p g (p) closely approximates h(p * ). This is a generalization of the result by Xu (2020) to state-dependent utility functions (and couples of functions), and it is crucial to prove the following results.
Lemma 1. Let β, > 0, η ∈ (0, 1] and set q = 32 log 4 η min{1; 1/β} / 2 . Then, given a posterior p * ∈ P and two sets of functions g, h with g β-stable compared with h, there exists a γ ∈ ∆ Q with p∈Q γ p p = p * and p∈Q γ p
θ p θ g θ (b p, ) ≥ (1 − η) θ p * θ h θ (b p * ). (2)
Now, we can prove the main result of this section. Consider a couple of sets of functions g, h where g is β-stable compared with h. With abuse of notation, we define g(φ) and h(φ) as the functions which evaluate the expected sender's utility of a public signaling scheme φ with h and g, respectively. We can resort to Lemma 1 to state the following result. The proof is based on solving a linear program that works only with q-uniform posteriors.
Theorem 2. Let β, > 0 and η ∈ (0, 1]. Consider two arbitrary state-dependent sets of functions g, h such that g θ :
C → [0, 1] is β-stable compared with h θ : C → [0, 1] for all θ ∈ Θ.
Then there exists a poly |R| |Θ| log( 1 η min{1;1/β} )/ 2 time algorithm that returns an -persuasive public signaling scheme φ such that:
g(φ ) ≥ (1 − η) max φ∈Φ h(φ),
where Φ is the set of persuasive signaling schemes. By setting h = g, we obtain a generalization of the result by Xu (2020) to state-dependent functions.
Comparative Stability of Voting Functions We apply this novel concept of stability to voting problems. Our first result proves that the two relaxed majority-voting functions previously introduced satisfy the comparative stability property. This result is similar to that by Cheng et al. (2015). However, we use multiplicative factors (in place of additive factors) and prove a slightly stronger result than stability. In particular, we prove that the decrease in utility is small even if only the perturbations from action c 0 to c 1 are bounded.
Lemma 2. W δ is 1/δ-stable compared with W . Moreover, for all c ∈ C, r ∈ R, α ∈ (0, 1], and y ∈ ∆ C such that Pr y (ỹ r = c 1 ∧ c r = c 0 ) ≤ α, it holds:
Eỹ ∼y [W δ (ỹ)] ≥ W (c) 1 − α δ .
We can use the result above to prove that W δδ satisfies the property of comparative stability with respect to W. Intuitively, the result follows from the observation that W is the composition of two majority-voting steps.
Lemma 3. W δδ is 1 δ 2 -stable with respect to W. Finally, we derive a stronger decomposition lemma for majority-voting. Specifically, Lemma 1 shows that the decrease in the expected sender's utility when decomposing a posterior in q-uniform posteriors can be bounded. However, in generic settings, the sender's expected utility in a given state of nature can change arbitrarily. This is not the case in majority voting, where, instead, this decrease is bounded. In particular, we can show the following, that is crucial when addressing the SEMIPUBLIC-DBE problem.
Lemma 4. Let
> 0, η ∈ (0, 1] and set q = 32 log 4 ηδ / 2 . Then, given a posterior p * ∈ P, there exists a γ ∈ ∆ Q with p∈Q γ p p = p * and
p∈Q γ p p θ W δ (b p, ) ≥ (1 − η) p * θ W (b p * ) ∀θ ∈ Θ.
Computing Public and Semi-public Signaling Schemes in District-based Elections We present two multi-criteria PTASs for the SEMIPUBLIC-DBE and PUBLIC-DBE problems, respectively, when our relaxations are adopted. First, we focus on the problem of designing public signaling schemes. We assume -persuasive signaling schemes, and we replace function W with W δδ (this corresponds to relaxing both the majority voting within every single district and the majority voting aggregating the outcomes of all the districts). Let W(φ) and W δδ (φ) denote the functions returning the sender's expected utility provided by a public signaling scheme φ with voting rules W and W δδ , respectively. We show that it is possible to compute efficiently anpersuasive public signaling scheme φ that approximates the optimal persuasive signaling scheme with an approximation factor arbitrarily close to 1. Since the relaxed function W δδ is 1/δ 2 -stable compared to the non-relaxed function W by Theorem 3, we can immediately apply Theorem 2 to these functions and then derive the following.
Corollary 1. Let > 0, δ ∈ (0, 1) and η ∈ (0, 1], then there exists a poly |R| |Θ| log 1 η δ 2 / 2 time algorithm that returns an -persuasive public signaling scheme φ such that:
W δδ (φ ) ≥ (1 − η) max φ∈Φ W(φ),(3)
where Φ is the set of persuasive signaling schemes. Then, we focus on the SEMIPUBLIC-DBE problem. As highlighted above, to overcome the intractability result, also in this setting, it is necessary to relax the problem. Specifically, we use -persuasive signaling schemes and we replace function W with W δ (this corresponds to relaxing the majority voting aggregating the outcomes of all the districts). We show that it is possible to compute efficiently an -persuasive semi-public signaling scheme φ that approximates the optimal persuasive signaling scheme with an approximation factor arbitrarily close to 1. Computing a semi-public signaling scheme φ amounts to determining a collection {φ d } d∈D of |D| public signaling schemes, one for each district, and correlate them. The crucial point concerns the computation of good marginal probabilities of the signaling scheme. Indeed, their aggregation is equivalent to computing a private signaling scheme in majority-voting elections, and this can be done efficiently (see LP (1) and Theorem 1). The main idea of our proof is that there are approximately optimal marginal probabilities of the signaling scheme that use only q-uniform posteriors (with q constant). Let α θ be the probability that c 0 wins in at least K D districts with state of nature θ, a δ d,θ be the probability that candidate c 0 receives at least (1 − δ) K d votes in district d with state of nature θ, and γ d be a probability distribution over posteriors for the receivers in district d. Finally, let I[E] denote the indicator function for the event E. Then, the following formulation computes an approximately optimal signaling scheme in polynomial time.
max α∈[0,1] |Θ| , a δ ∈[0,1] |D|×|Θ| i,l∈R |Θ|×K D , o∈R |D|×|Θ|×K D γ d ∈∆ Q ∀d∈D θ∈Θ µ θ α θ (4a) s.t. α θ ≤ 1 K D − m i θ,m (4b) ∀θ ∈ Θ, ∀m ∈ {0, . . . , K D − 1} i θ,m ≤ (|D| − m)l θ,m + d∈D o d,θ,m (4c) ∀θ ∈ Θ, ∀m ∈ {0, . . . , K D − 1} a δ d,θ ≥ l θ,m + o d,θ,m (4d) ∀d ∈ D, ∀θ ∈ Θ, ∀m ∈ {0, . . . , K D − 1} a δ d,θ ≤ p∈Q γ d p p θ µ θ I W d δ (b p, ) = c 0 (4e) ∀d ∈ D, ∀θ ∈ Θ p∈Q γ d p p θ = µ θ ∀d ∈ D, ∀θ ∈ Θ (4f)
Theorem 3. Let > 0, δ ∈ (0, 1) and η ∈ (0, 1], then there exists a poly |R| |Θ| log( 1 η δ )/ 2 time algorithm that outputs an -persuasive semi-public signaling scheme φ such that:
W δ (φ ) ≥ (1 − η) max φ∈Φ W(φ),(5)
where Φ is the set of persuasive signaling schemes.
Conclusions and Future Works
In this paper, we study how a manipulator can exploit his information advantage to manipulate a district-based election through the strategic provision of information to rational voters. We show that private signaling schemes can be computed efficiently while computing optimal (semi-)public signaling schemes is intractable. However, we show that reasonable relaxations allow the design of multi-criteria PTASs for (semi-)public persuasion. An interpretation of these relaxations is that (semi-)public signaling is often tractable, except when the target candidate wins in at least half of the districts, but it is impossible to make slightly more than half of them elect such a candidate. In most cases, the receivers are slightly biased to follow the sender recommendations, and the manipulator's preferred candidate can either win or not win by at least a small, but not negligible, margin. With these assumptions, our algorithm approximates arbitrarily well the optimal signaling scheme in polynomial time.
In the future, we will study classes of instances in which optimal (semi-)public signaling schemes can be computed efficiently. We are also interested in settings in which the sender is uncertain about the voters' preferences. Omitted Proofs on "Private Persuasion" Theorem 1. LP (1) computes an optimal solution of PRIVATE-DBE in polynomial time.
Proof. LP (1) has a polynomial number of variables and constraints and, therefore, it can be solved in polynomial time. Thus, we just need to prove that LP (1) actually computes an optimal solution to PRIVATE-DBE. First, we remark that all the marginal probabilities φ r (θ, c 0 ) of the signaling scheme φ must satisfy the incentive Constraints (1b). a d,θ represents the probability of having at least K d votes in district d, given state of nature θ. We need to show a d,θ is computed correctly given the other variables of LP (1). In particular, for every state of nature θ, the maximum probability with which at least K d of the receivers in R d vote for c 0 given marginals probabilities φ r (θ, c 0 ) is:
a d,θ = min min m∈{0,...,K d −1} 1 K d − m v θ,m ; 1 ,
where v θ,m is the sum of the lowest |R d |−m elements in the set {φ r (θ, c)} r∈R d ; further details are provided by (Arieli and Babichenko 2019). This definition is encoded by Constraints (1f). Constraints (1g) and (1h) ensure the values v θ,m are well defined and derived from the dual of a simple LP of this kind: min y∈R n x y 1 y = w 0 ≤ y ≤ 1 where x ∈ R n is the vector from which we want to extract the sum of the smallest w entries. Finally, we prove that α θ is computed correctly. The computation of the maximum probability α θ with which at least K D districts elect c 0 given probabilities a d,θ is similar to the computation of a d,θ given φ r (θ, c 0 ). For a similar argument as above, Constraints (1c), (1d), and (1e) correctly compute α θ aggregating the marginal probabilities {a d,θ } d∈D,θ∈Θ . Objective (1a) is given by the sum over all θ ∈ Θ of the prior of state θ, multiplied by α θ . Thus, by definition of α θ , we are maximizing the probability of having c 0 locally elected in more than K D districts.
Finally, we prove how to construct a signaling scheme φ with the same objective function of LP (1). In particular, we find marginal signaling schemes φ r such that the incentive constraints relative to c 0 and c 1 are satisfied and φ r (θ, c 0 ) ≥ φ r (θ, c 0 ) for all r and θ. Since we do not introduce the incentive constraint relative to action c 1 , they could not be satisfied by φ. However, from the optimal marginal probabilities φ r (θ, c 0 ), it is straightforward to compute the marginal probabilities { φ r (θ, c 0 ), φ r (θ, c 1 ) } r∈R,θ∈Θ . For each state of nature θ, let φ r (θ, c 0 ) = 1 if u r (θ) ≥ 0 and φ r (θ, c 0 ) = φ r (θ) otherwise. Then, φ r (θ, c 1 ) = 1 − φ r (θ, c 0 ). The marginal signaling scheme φ r is persuasive as c 1 is recommended only when it is the optimal action, while φ r (θ, c 0 ) ≥ φ r (θ, c 0 ) if and only if u θ ≥ 0. Formally, θ∈Θ µ θ φ r (θ, c 0 ) u r (θ) ≥ θ∈Θ µ θ φ r (θ, c 0 ) u r (θ) ≥ 0 by constraints (1b). Finally, we can aggregate the marginal probabilities of the signaling scheme by using the same approach proposed by Arieli and Babichenko (2019).
Omitted Proofs on "Comparative Stability and
Public Signaling Schemes" Lemma 1. Let β, > 0, η ∈ (0, 1] and set q = 32 log 4 η min{1; 1/β} / 2 . Then, given a posterior p * ∈ P and two sets of functions g, h with g β-stable compared with h, there exists a γ ∈ ∆ Q with p∈Q γ p p = p * and p∈Q γ p
θ p θ g θ (b p, ) ≥ (1 − η) θ p * θ h θ (b p * ). (2)
Proof. Letγ ∈ Q be the empirical distribution of q i.i.d. samples drawn from p * , where each θ has probability p * θ of being sampled. Therefore, the vectorγ is a random variable supported on q-uniform posteriors with expectation p * . Moreover, let γ ∈ ∆ Q be a probability distribution such as, for every p ∈ Q, it holds γ p = Pr(γ = p). It is easy to see that p * = p∈Q γ p p. We need to prove that Equation (2) holds. For every p ∈ Q, we define with γ (θ,i) p the conditional probability of having observed posterior p given that the posterior assigns a probability of i/q to state θ. Formally, for every p ∈ Q, we have:
γ (θ,i) p = γ p p ∈Q:p θ =i/q γ p if p θ = i/q 0 otherwise .
Then, the random variableγ (θ,i) ∈ Q is such that, for every p ∈ Q, it holds Pr(γ (θ,i) = p) = γ (θ,i) p . For each r ∈ R, we define P r ⊆ Q as the set of posteriors that do not change the expected utility of r by more than with respect to p * . Formally, p ∈ P r if and only if | θ p θ u r (θ)− θ p * θ u r (θ)| ≤ . Finally, let α = η min{1; 1/β}.
To complete the proof, we introduce the following three lemmas. First, given a probability distribution p * and a state of nature θ ∈ Θ, the following lemma bounds the maximum probability mass that γ assigns to posteriors p ∈ Q in which the probability assigned to state of nature θ deviates from the one prescribed by p * by at least /4. Lemma 5. Given p * ∈ P, for each θ ∈ Θ, it holds: i:|i/q−p * θ |≥ /4 p∈Q:p θ =i/q
γ p ≤ α 2 p * θ ,
where γ is the probability distribution of q i.i.d samples drawn from p * .
Proof. We observe that the random variableγ θ is drawn from a Binomial probability distribution. We consider two possible cases. If p * θ ≥ 1/8, then by Hoeffding's inequality we can write the following:
Pr |γ θ − p * θ | ≥ 4 ≤ 2 e −2 q ( /4) 2 = (6a)
= 2 e −4 log(4/α) ≤ (6b)
≤ α/16 ≤ α 2 p * θ .(6c)
Figure 1 :
1Interaction between the sender and a receiver.
Theorem 1 .
1LP (1) computes an optimal solution of PRIVATE-DBE in polynomial time.
the set of receivers who are indifferent between action c 0 and c 1 . Then, we have: b p, = arg max c∈C:cr=c0∀r∈A cr=c1∀r∈B θ p θ f θ (c).
Table 1 :
1Payoffs of the voters in Example 1.
|D|×|Θ| i,l∈R |Θ|×K D , o∈R |D|×|Θ|×K D t d,θ,m , v d,θ,m ∈R ∀d∈D,θ∈Θ,m∈{1,...,K d } z d,θ,r,m ∈R ∀d∈D,θ∈Θ,r∈R,m∈{1,...,K d }max
α∈[0,1] |Θ| , a∈[0,1] φr(·,c0)∈[0,1] |Θ| ∀r∈R
θ∈Θ
Liu, H.; Feng, H.; Zhu, D.; and Luan, J. 2009. Parameterized computational complexity of control problems in voting systems. THEOR COMPUT SCI 410(27-29): 2746-2753. Loreggia, A.; Narodytska, N.; Rossi, F.; Venable, K. B.; and Walsh, T. 2015. Controlling elections by replacing candidates or votes. In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, 1737-1738. Sina, S.; Hazon, N.; Hassidim, A.; and Kraus, S. 2015. Adapting the social network to affect elections. In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, 705-713. Taneva, I. A. 2015. Information design . Wang, Y. 2013. Bayesian persuasion with multiple receivers. Available at SSRN 2625399 . Wilder, B.; and Vorobeychik, Y. 2018. Controlling elections through social influence. In Proceedings of the 17th international conference on autonomous agents and multiagent systems, 265-273. International Foundation for Autonomous Agents and Multiagent Systems. Wilder, B.; and Vorobeychik, Y. 2019. Defending elections against malicious spread of misinformation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, 2213-2220. Xu, H. 2020. On the Tractability of Public Persuasion with No Externalities. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2708-2727. SIAM.
Recall that in a public signaling scheme, all the receivers observe the same signal, perform the same update of the belief, and have the same posterior belief.
AcknowledgmentsThis work has been partially supported by the Italian MIUR PRIN 2017 Project ALGADIMAR "Algorithms, Games, and Digital Market".Instead, if p * θ ≤ 1/8, then by Chernoff's bound we can write the following:≤ e −2 log(4/α) log( 7 8p * θ ) = (7b) = ( 8 7 p * θ ) 2 log(4/α) = (7c)= 1 e 8 7 e p * θ 2 log(4/α) ≤ (7d)≤ (e) −2 log(4/α) 8 7 e p * θ ≤ (7e)Moreover, we can write:where in Equations (8d) and (8e) we use that e x ≥ e x and log(α/4) < −1 as α ∈ (0, 1]. Hence, we obtain the following inequality:i:|i/q−p * θ |> /4 p∈Q:p θ =i/qwhich concludes the proof.The second lemma we introduce proves that, when p θ is close to p * , then the utility of every receiver is close to the utility in p * with high probability.Lemma 6. Given p * ∈ P, for each receiver r ∈ R, each state θ ∈ Θ and each i : |i/q − p * θ | ≤ /4, it holds:where γ is the distribution of q i.i.d samples from p * .Proof. Fixθ ∈ Θ, r ∈ R and i with |i/q − p * θ | ≤ /4. Then, lett = θγ (θ,i) θ u r (θ) and t = θ p * θ u r (θ), where the notationγ (θ,i) θ is employed to denote the value of p θ given that the random variableγ (θ,i) ∈ Q assumes value p. First, we show that | E[t ] − t | ≤ /2. This is equivalent to prove the following:| ≥ /2) by the triangular inequality. Then, we use the Hoeffding's inequality to bound the last term:Before introducing the last lemma, we need some further notation. More precisely, given a posterior, we partition the receivers in three sets, depending on their possible bestresponses. We define the partition on the set of receivers induced by p * ∈ P as follows:Similarly, any q-uniform posterior p ∈ Q induces the following partition to the set of receivers whenpersuasiveness is adopted:Then, we define an auxiliary variable y p ∈ C as follows:Note that, by construction, y p is a valid action profile under -persuasiveness. Moreover, by the optimality of thepersuasive best-response, the following holds for every posterior p:Finally, letỹ (θ,i) ∈ C be the random variable such that:Now, we introduce the last lemma we use to complete the proof. This lemma proves thatỹ θ,i are α 2 -noisy probability distributions around b p * .Lemma 7. Given p * ∈ P, for each θ ∈ Θ and i :Proof. We need to prove that for every receiver r, it holds. This concludes the proof. Now, we are ready to prove Equation (2). θ p∈Q(By restricting the set of posteriors.)(By stability of g compared to h and Lemma 7.)This concludes the proof.Theorem 2. Let β, > 0 and η ∈ (0, 1]. Consider two arbitrary state-dependent sets of functions g, h such that g θ :Then there exists a poly |R| |Θ| log( 1 η min{1;1/β} )/ 2 time algorithm that returns an -persuasive public signaling scheme φ such that:where Φ is the set of persuasive signaling schemes.Proof. For every constant β, > 0, η ∈ (0, 1], by Theorem 1, we know that any posterior p * ∈ P guaranteeing a value h(p * ) can be expressed as a convex combination of q-uniform posteriors such that p∈Q γ p g (p) ≥(1 − η) h(p * ). Therefore, given the optimal persuasive public signaling scheme φ * optimizing h, we can decompose each posterior probability distribution p ∈ supp(φ * ) into a convex combination of q-uniform posteriors and obtain an -persuasive public signaling scheme φ maximizing g that satisfies the inequalities stated in the theorem. Let q = 32 log 4 η min{1; 1/β} / 2 . Since, for a fixed number of samples q, the number of q-uniform probability distributions is at most |Θ| q , we can search for the -persuasive public signaling scheme maximizing g over probability distributions p ∈ Q, by solving the following Linear Program composed of O(|Q|) variables and constraints:Finally, given the probability distribution on the q-uniform posteriors γ ∈ ∆ Q , it is easy to derive the corresponding public signaling scheme φ by setting the following for every θ ∈ Θ and c ∈ C:Omitted Proofs on "Comparative Stability and Voting Functions" Lemma 2. W δ is 1/δ-stable compared with W . Moreover, for all c ∈ C, r ∈ R, α ∈ (0, 1], and y ∈ ∆ C such that Pr y (ỹ r = c 1 ∧ c r = c 0 ) ≤ α, it holds:Proof. To prove the first part of the lemma, we need to show that for every voting profilec ∈ C and α-noisy probability distribution y aroundc with α ∈ (0, 1], the following inequality holds:Given that W and W δ assume values exclusively in {0, 1}, Inequality(12)This implies that, where the last inequality follows from |R|/2 |V c0 (c)| ≤ |R|/2 |R|/2 ≤ 1 and from W (c) = 1 by assumption. Finally, to prove the second part of the lemma, we can employ Algorithm 1 to show that for allc ∈ C and for all probability distributions y aroundc such that Prỹ ∼y [ỹ r = c 1 ∧c r = c 0 ] ≤ α, there is an α-noisy probability distribution y guaranteeing. It is easy to see that y is α-noise: y has null probability on all the voting profiles c with a r ∈ R such that c r = c 0 ∧c r = c 1 , i.e., V c0 (c) V c0 (c), while, Prỹ ∼y [ỹ r = c 1 ∧c r = c 0 ] = Prỹ ∼y [ỹ r = c 1 ∧c r = c 0 ] ≤ α. Moreover, since Algorithm 1 moves probability mass from an action profile c to an action profile c with V c0 (c ) ⊆ V c0 (c), it does not increase the expected value of W δ . This concludes the proof.Lemma 3. W δδ is 1 δ 2 -stable with respect to W. Proof. We need to prove that the following inequality holds for all c ∈ C |R| and α-noisy distribution y around c with α ∈ (0, 1].The value of function W δδ depends on the values of all the district functions W d δ . Indeed, given a voting profile c ∈ C, the function W δδ assumes value W δδ (c) = W δ ( W 1 δ ( c 1 ), . . . , W D δ (c D ) ). Therefore, when it is perturbed by an α-noisy probability distribution y, its expected value can be expressed as:Eỹ ∼y [W δδ (ỹ)] = Eỹ ∼y W δ ( W 1 δ (ỹ 1 ), . . . , W D δ (ỹ D ) ) . Lemma 2 can be applied to all the couples of functions W d , W d δ , deriving the following inequality for every d ∈ D, c ∈ C |R| , α ∈ (0, 1]:We can use the above inequality and the fact thatW is a majority-voting function to apply Lemma 2 to the couple of functionsW andW δ , thus showing the following:≥W W 1 (c 1 ), . . . , W |D| (c |D| ) 1 − α δ 2 . This implies that W δδ is 1/δ 2 stable compared to W.Lemma 4. Let > 0, η ∈ (0, 1] and set q = 32 log 4 ηδ / 2 . Then, given a posterior p * ∈ P, there exists a γ ∈ ∆ Q with p∈Q γ p p = p * andProof. The proof follows the same steps of the proof of Lemma 1. In the following, we just highlight the differences between the two proofs. In the steps from Equation(10a)to Equation (10j), we remove the summation over the states of nature. All the other steps hold, except for Equation (10d). Indeed, since -best response is computed maximizing the expected utility of the sender, there are no guarantees that for each state of nature θ it holds g θ (b p, ) ≥ g θ (y p ). However, since W δ is state-independent and monotone non-decreasing in the number of receivers that vote for c 0 , the best response b p, is given by b p, r = c 0 for all the voters with utility u r (θ) ≥ − . Thus, we are guaranteed that, for every y p ∈ C, it holds W δ (y p ) ≤ W δ (b p, ) independently from the state of nature θ. Taking into account Lemma 2, the derivation is straightforward.Omitted Proofs on "Computing a Semi-PublicSignaling Scheme"Theorem 3. Let > 0, δ ∈ (0, 1) and η ∈ (0, 1], then there exists a poly |R| |Θ| log( 1 η δ )/ 2 time algorithm that outputs an -persuasive semi-public signaling scheme φ such that:where Φ is the set of persuasive signaling schemes.Proof. Let q = 32 log 4 η δ / 2 and Q ⊂ ∆ Θ be the set of q-uniform probability distributions on Θ. We show that, given the optimal semi-public signaling scheme φ * , there is a solution φ to LP (4) with W δ (φ ) ≥ (1−η)W(φ * ). Given the signaling scheme φ * , let:• a * d,θ be the probability that c 0 wins in district d when the state of nature is θ and • α * θ be the probability that c 0 wins in at least K d when the state of nature is θ.Then, as showed in Theorem 1, the probability such that c 0 wins in at least K D districts with state of nature θ is:where v θ,m is the sum of the lowest |R d | − m elements in the set {a * d,θ } d∈D . We show that there is a solution to LP (4) with a δ d,θ ≥ (1 − η)a * d,θ for every d and θ. Since the value of each a d,θ is reduced by a multiplicative factor (1 − η), Equation (17) implies that α θ ≥ (1 − η)α * θ and θ µ θ α θ ≥ (1 − η) θ µ θ α * θ . 4 4 See Theorem 1 for details on how LP (4) computes α θ from β δ . Hence, we conclude the proof showing that a δ d,θ ≥ (1 − η)a * d,θ for every d and θ. Let: • φ * d be the marginal probabilities of the signaling scheme φ restricted to the receivers in district d, • γ * ∈ ∆ P be the probability distribution on posteriors induced by φ * d , • γ p ∈ ∆ P be the probability distribution on q-uniform posteriors obtained decomposing a posterior p as prescribed by Lemma 4, and • γ d ∈ ∆ Q be the distribution on q-uniform posteriors obtained by decomposing each posterior induced by φ * d as in Lemma 4, i.e., γ d p = p ∈supp(φ * ) γ * p γ p p for every p. We conclude proving that γ d is a q-uniform distribution that induces a a δ d,θ ≥ (1 − η)a * d,θ for every θ.= a δ d,θ . This concludes the proof.
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The complexity of bribery and control in group identification. G Erdélyi, C Reger, Yang , Y , Autonomous Agents and Multi-Agent Systems. 3418Erdélyi, G.; Reger, C.; and Yang, Y. 2020. The complexity of bribery and control in group identification. Autonomous Agents and Multi-Agent Systems 34(1): 8.
Opinion Diffusion and Campaigning on Society Graphs. P Faliszewski, R Gonen, M Kouteckỳ, N Talmon, IJCAI. Faliszewski, P.; Gonen, R.; Kouteckỳ, M.; and Talmon, N. 2018. Opinion Diffusion and Campaigning on Society Graphs. In IJCAI, 219-225.
Multimode control attacks on elections. P Faliszewski, E Hemaspaandra, L A Hemaspaandra, Journal of Artificial Intelligence Research. 40Faliszewski, P.; Hemaspaandra, E.; and Hemaspaandra, L. A. 2011. Multimode control attacks on elections. Journal of Artificial Intelligence Research 40: 305-351.
Llull and Copeland voting computationally resist bribery and constructive control. P Faliszewski, E Hemaspaandra, L A Hemaspaandra, J Rothe, Journal of Artificial Intelligence Research. 35Faliszewski, P.; Hemaspaandra, E.; Hemaspaandra, L. A.; and Rothe, J. 2009. Llull and Copeland voting computa- tionally resist bribery and constructive control. Journal of Artificial Intelligence Research 35: 275-341.
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| []
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[
"Imaging a 1-electron InAs quantum dot in an InAs/InP nanowire",
"Imaging a 1-electron InAs quantum dot in an InAs/InP nanowire"
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"A C Bleszynski \nDepartment of Physics\nDivision of Engineering and Applied Sciences\nHarvard Univ\n02138CambridgeMassachusettsUSA\n",
"L E Fröberg \nSolid State Physics/the Nanometer Structure Consortium\nLund Univ\nBox 118S-221 00LundSweden\n",
"M T Björk \nSolid State Physics/the Nanometer Structure Consortium\nLund Univ\nBox 118S-221 00LundSweden\n",
"H J Trodahl \nDepartment of Physics\nDivision of Engineering and Applied Sciences\nHarvard Univ\n02138CambridgeMassachusettsUSA\n",
"L Samuelson \nSolid State Physics/the Nanometer Structure Consortium\nLund Univ\nBox 118S-221 00LundSweden\n",
"R M Westervelt \nDepartment of Physics\nDivision of Engineering and Applied Sciences\nHarvard Univ\n02138CambridgeMassachusettsUSA\n"
]
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"Department of Physics\nDivision of Engineering and Applied Sciences\nHarvard Univ\n02138CambridgeMassachusettsUSA",
"Solid State Physics/the Nanometer Structure Consortium\nLund Univ\nBox 118S-221 00LundSweden",
"Solid State Physics/the Nanometer Structure Consortium\nLund Univ\nBox 118S-221 00LundSweden",
"Department of Physics\nDivision of Engineering and Applied Sciences\nHarvard Univ\n02138CambridgeMassachusettsUSA",
"Solid State Physics/the Nanometer Structure Consortium\nLund Univ\nBox 118S-221 00LundSweden",
"Department of Physics\nDivision of Engineering and Applied Sciences\nHarvard Univ\n02138CambridgeMassachusettsUSA"
]
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| PACS indexing codes: 73.63. Nm, Nanowire heterostructures define high-quality few-electron quantum dots for nanoelectronics, spintronics and quantum information processing. We use a cooled scanning probe microscope (SPM) to image and control an InAs quantum dot in an InAs/InP nanowire, using the tip as a movable gate. Images of dot conductance vs. tip position at T = 4.2 K show concentric rings as electrons are added, starting with the first electron. The SPM can locate a dot along a nanowire and individually tune its charge, abilities that will be very useful for the control of coupled nanowire dots. | 10.1103/physrevb.77.245327 | [
"https://arxiv.org/pdf/0802.3236v1.pdf"
]
| 119,284,584 | 0802.3236 | 204a0bde99b474d1467aea49e66c6efdaa87e3db |
Imaging a 1-electron InAs quantum dot in an InAs/InP nanowire
A C Bleszynski
Department of Physics
Division of Engineering and Applied Sciences
Harvard Univ
02138CambridgeMassachusettsUSA
L E Fröberg
Solid State Physics/the Nanometer Structure Consortium
Lund Univ
Box 118S-221 00LundSweden
M T Björk
Solid State Physics/the Nanometer Structure Consortium
Lund Univ
Box 118S-221 00LundSweden
H J Trodahl
Department of Physics
Division of Engineering and Applied Sciences
Harvard Univ
02138CambridgeMassachusettsUSA
L Samuelson
Solid State Physics/the Nanometer Structure Consortium
Lund Univ
Box 118S-221 00LundSweden
R M Westervelt
Department of Physics
Division of Engineering and Applied Sciences
Harvard Univ
02138CambridgeMassachusettsUSA
Imaging a 1-electron InAs quantum dot in an InAs/InP nanowire
(receipt date: )
PACS indexing codes: 73.63. Nm, Nanowire heterostructures define high-quality few-electron quantum dots for nanoelectronics, spintronics and quantum information processing. We use a cooled scanning probe microscope (SPM) to image and control an InAs quantum dot in an InAs/InP nanowire, using the tip as a movable gate. Images of dot conductance vs. tip position at T = 4.2 K show concentric rings as electrons are added, starting with the first electron. The SPM can locate a dot along a nanowire and individually tune its charge, abilities that will be very useful for the control of coupled nanowire dots.
Semiconducting nanowire heterostructures [1] provide an excellent system in which to make high-quality ultra-small quantum dots for applications in nanoelectronics, spintronics and quantum information processing (QIP) [2,3]. The bottom up nature of the construction of quantum dots in nanowire heterostructures [1,4,5] results in atomically sharp interfaces and highly controllable dot size, shape, and composition. Few-electron nanowire quantum dots have been recently reported [6,7] that exhibit a well-defined atom-like electronic shell structure down to the last electron [8,9]. The ability to operate nanowire quantum dots in the one-electron regime makes them attractive candidates for QIP.
InAs is a particularly attractive material for several reasons. InAs has a large g-factor, making it useful for spintronic and QIP devices [2,3]. Furthermore, the g-factor of an InAs nanowire quantum dot can be varied from 2 to the bulk value 14, by varying the dot size [10], a consequence of quantum confinement [11,12]. Lastly, whereas some semiconductors have surface depletion, the surface of InAs is known to have a charge accumulation layer [13]. This potentially allows for very small diameter nanowires and ultrasmall dots that are not depleted of electrons, as well as for Schottky-barrier-free contacts to metallic leads.
In order to control the charge in an individual nanowire quantum dot in a dot circuit, new gating techniques will be needed. A conventional back gate couples to the entire nanowire and all of the dots in it. A lithographically defined gate for an individual dot has to be small and aligned with high precision, which is particularly difficult for heterostructure nanowire quantum dots, because of their small size and the uncertainty in their location. A scanned probe microscope (SPM) can overcome both of these obstacles by using the conducting tip as a movable gate. Cooled SPMs have proven to be powerful tools for imaging the electronic properties of nanoscale systems including nanotubes, nanowires and two-dimensional electron gas structures [14][15][16][17][18][19], and they can image the presence of a single electronic charge [17][18][20][21][22][23]. In previous work, a cooled SPM was used to image quantum dots unintentionally formed in carbon nanotubes [17,18] and in semiconducting nanowires [24].
In this letter, we present scanning gate images at 4.2 K of a one-electron InAs quantum dot formed in an InAs/InP nanowire heterostructure by two InP barriers. We use the conducting tip of a cooled SPM as movable gate to locate the quantum dot and tune its charge, starting with the first electron. The images show concentric rings of high conductance about the dot corresponding to Coulomb blockade peaks as an electron is added. In this way, the SPM can locate a dot along the nanowire and individually tune its charge. This ability will be very useful for the manipulation of multiple coupled quantum dots grown along a nanowire heterostructure. technique [17][18][20][21][22][23], the charged SPM tip is scanned in a plane above the nanowire, and the resulting change in nanowire conductance G is recorded to form a twodimensional image at 4.2 K. Modeling the dot as a small metal sphere, the charge induced by the tip is
d t t d t t t dot V r C r V q = * ) ( ) , ( , where C t d (r t )
is the tip-dot capacitance at a distance r t away and V t d = V tip + V cont is the voltage difference between the tip and the dot for tip voltage V tip including the contact potential V cont . We
assume C t d (r t ) << C
where C is total dot capacitance to ground. Scanning the tip with fixed V t changes C t d (r t ) , which varies q dot V t ,r t ( ) and causes oscillations in dot conductance G each time an electron is added to the dot. In the images, this behavior manifests itself as concentric rings of peaked conductance surrounding the quantum dot with each ring corresponding to a Coulomb blockade peak. The rings thus locate the quantum dot. If the tip is scanned along one of these rings, the induced charge q dot V t ,r t ( ) remains constant -the rings are contours of constant tip-dot coupling.
The nanowires used in this experiment were catalytically grown from Au nanoparticles on an InAs <111> B substrate using chemical beam epitaxy [25]. A TEM image of a typical nanowire quantum dot is shown in Fig. 1 ; the dot is emptied of electrons for V bg < 0.4V due to quantum confinement in the growth direction [8]. The plot displays regions of zero conductance, Coulomb diamonds, when V sd is smaller than the energy required to add another electron to the dot. The diamond size is seen to vary with electron number, revealing a shell structure of electronic states in the quantum dot [9].
A similar set of Coulomb diamonds shown in Fig. 2 For different fixed backgate voltages, we obtained differently sized Coulomb diamonds in tip-voltage plots similar to Fig. 2(b).
SPM images of the last electron on the quantum dot are shown in Figs. 3(a-b). We adjust V bg and V tip so the dot is tuned to the zero-one electron transition when the tip is nearby. The images of G vs. tip position in Figs. 3(a-b), recorded as the tip is scanned in a plane above the nanowire with fixed tip voltage V tip , display a ring centered on the InAs dot that corresponds to the Coulomb-blockade conductance peak as the first electron is added the dot. When the tip is inside the ring in Fig. 3(a), the dot is empty, and when the tip is just outside the ring, the dot holds one electron. The Coulomb blockade ring for the addition of the second electron is visible at the corners of Fig. 3(a). As V tip is made less negative in Fig. 3(b), the first Coulomb blockade ring shrinks to a point and the ring for the second electron also shrinks in size. [24] comparable to the length between the grown-in dot and the contact as shown.
One-electron double dots, grown inside an InAs/InP nanowire, are attractive for spin manipulation, because the dots can be very small and closely spaced. However, it is difficult to gate each dot individually, because the dot size and spacing are often smaller than the spatial resolution of e-beam lithography. Fuhrer et al. [26] used an array of many gates to characterize nanowire double dots, by carefully measuring the dot-gate couplings and tuning the gate voltages accordingly.
Using a conducting SPM tip, we should be able to individually tune the charge in of each dot in an InAs double dot grown in an InAs/InP nanowire. Figures 5(a-b)
Figure 1 (
1a) shows the SPM imaging setup. Using a Coulomb-blockade imaging
(b): the dark sections are InAs and the light sections are InP. The InAs/InP heterostructure is formed by alternating the gas precursors during the growth process. The diameter of the wires is ~ 50 nm and their lengths are ~ 3 μm. An 18 nm long InAs quantum dot is formed between two 8 nm thick InP barriers. The 600 meV conduction band offset between InAs and InP produces electron confinement inside the InAs quantum dot. After growth, the nanowires are deposited onto a degenerately doped Si substrate capped with a 100 nm SiO 2 layer. This conducting substrate acts as a non-local back gate that, through an applied voltage V bg , can tune the Fermi level in the nanowire. Ni/Au electrode contacts, spaced by 2 μm, are defined with e-beam lithography as indicated in Fig. 1(a). The thickness of the InP barriers and the InAs dot are tuned such that the few-electron Coulomb blockade regime can be reached for small V bg . The number of electrons on the dot and the energy of the first few electron states can be determined from the Coulomb blockade diamonds, plots of G vs. gate voltage and source-to-drain voltage V sd shown in Fig. 2. Figure 2(a) was recorded by sweeping V sd and the backgate voltage V bg while the tip position was fixed 50 nm above the dot with constant tip voltage V tip = 2.0V
(b) was obtained by fixing the tip position 70 nm directly over the dot and sweeping V tip and V sd with fixed V bg = 1.2V . The dot is emptied of electrons for V tip < 2.0V . From the change in tip voltage required to add one electron we find C t d r t ( )~0 .4aF for tip distance r t~7 0nm. Like the back gate, the tip can tune the dot's charge to zero electrons. Unlike the back gate, the tip offers the extra advantage of movability: the tip's coupling to the dot can be varied through positioning. The sizes of the Coulomb diamonds change with different combinations of tip and backgate voltages V tip and V bg , as shown in Figs. 2(a) and 2(b).This occurs because V tip and V bg induce charge in the nanowire dot by shifting the electron density profile sideways, toward or away from the substrate; this compresses the wavefunction against the sides of the nanowire and changes the energy of quantum states.
Figures 3 (
3c-d) show SPM Coulomb blockade conductance images of the InAs dot when it is tuned to hold a larger number of electrons. Again, rings of peaked conductance surround the quantum dot. Moving radially outwards, each new ring corresponds to the addition of an electron to the dot, as indicated by the integers. As V bg is changed to a more positive value in Fig. 3(d), the rings shrink in radius, because the back gate pulls more electrons onto the dot. To obtain the same number of electrons, the induced charge q dot V tip ,r t ( ) must be made more negative by moving the tip closer. For certain combinations of V tip and V bg , the SPM images, such as Fig. 4, show an additional set of narrowly spaced Coulomb conductance rings centered about a section of the nanowire to the left of the grown-in InAs quantum dot, clear evidence of the formation of a second quantum dot. By comparing the spacing of the rings surrounding the two dots, the length of the extra dot is found to be ~ 200 nm,
show simulated SPM conductance images of two InAs quantum dots, each with length 25 nm and diameter 50 nm, defined by 5 nm InP barriers. The nanowire lies on its side on a substrate, and an SPM tip is scanned in a plane 25 nm above its diameter with constant tip voltage. The number of electrons on a given dot increases in integer steps with tip-dot radius to form a bullseye-shaped diagram, as described above; rings of high conductance occur at radii where the electron number changes. For a double dot, the centers of the two bullseyes are at different tip positions, as shown in Fig. 5(b). The SPM conductance image locates the double dot along the nanowire, and by simply moving the tip, one can individually tune the charge on each dot. Conductance through two dots in series occurs when both dots conduct, so conductance in the image of Figs. 5(a-b) only occurs at the intersections of the conductance rings for each dot. Conductance images of nanowire double dots vs. like Fig. 5 are equivalent to traditional two-dimensional plots of double-dot conductance vs. gate voltage for two lithographically defined gates. Using an SPM tip as a gate should allow a full range of experiments to manipulate charges and spins on nanowire double dots, that take advantage of the small dot size, large g-factor, and relatively high operating temperature.FIGURES FIG. 1 (a) Experimental setup. A metallized scanning probe microscope (SPM) tip is scanned at a fixed height above a nanowire. Conductance between source and drain electrodes is recorded vs. tip position to obtain an image. The tip is scanned in a plane typically 20 nm to 100 nm above the nanowire with tip voltages -3V to 3V. (b) TEM image of an InAs/InP heterostructure nanowire similar to the ones used in this experiment. Individual atomic layers are clearly visible, indicating the high quality of the epitaxial growth. An InAs quantum dot (dark) with a well defined disc geometry is confined between two InP barriers (light) with InAs leads (dark). FIG. 2 Coulomb blockade diamonds for an InAs quantum dot in an InAs/InP nanowire heterostructure, plotting differential conductance vs. V sd and either (a) back gate voltage V bg or (b) tip voltage V tip . These data show that either the back gate or the SPM tip can be used to tune the dot's charge. The differing size of the diamonds indicates the atomic-like shell structure of the quantum dot. Both the tip and the back gate can reduce the number of electrons on the dot to 0 or 1 as indicated by the large area of zero conductance at the left of the two diamond plots. FIG. 3 SPM Coulomb blockade conductance images (a)-(b) of the last electron on the InAs nanowire quantum dot, and (c)-(d) of a higher number of electrons vs. tip position, as the tip is scanned in a plane 100 nm above the dot. The rings of high conductance correspond to Coulomb blockade conductance peaks. The integers indicate the number of electrons on the dot when the tip lies inside or between the rings. The tip and backgate voltages are (a) V tip = -2.5 V, (b) V tip = -1.5 V. (c-d) V tip = -2.0 V. The back gate voltage for (a-b) is 0.43 V. As the back gate voltage is increased from (c) V bg = 2.5 V to (d) 2.7 V, more electrons are added to the dot; the yellow triangle tracks the conductance peak corresponding to addition of the eleventh electron. The scale bar lengths are (a) 100 nm and (c) 200 nm. FIG. 4 SPM Coulomb conductance image obtained with tip voltage V t = 0.25 V and gate voltage V bg = 1.8 V. In addition to the rings surrounding the intentionally defined quantum dot, a second set of rings is seen in the lower left of the image. The extra set is centered another section of the nanowire, indicating that an extra dot has formed. The closer spacing of the second set of rings as well as their elongated shape indicate that the extra dot is much longer than the 18 nm defined dot. The scale bar is 200 nm. FIG. 5 SPM conductance image simulations of a double quantum dot formed in a 50nm diameter InAs/InP nanowire (outlined in blue). Dark (light) regions correspond to low (high) conductance. (a) The two InAs quantum dots (shaded in blue) are each 25 nm long and are defined by 5 nm thick InP tunnel barriers. The tip is scanned in a plane 25 nm above the nanowire. (b) Zoom-in of the boxed area in (a). The number of electrons on the left and right dot when the tip lies in that region are indicated.
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| []
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[
"Poissonian bursts in e-mail correspondence",
"Poissonian bursts in e-mail correspondence"
]
| [
"C Anteneodo ",
"R Dean Malmgren \nDepartment of Chemical and Biological Engineering\nNorthwestern University\n60208EvanstonILUSA (\n",
"D R Chialvo \nDepartment of Physiology\nFeinberg Medical School\nNorthwestern University\n303 East Chicago Ave. Chicago60611ILUSA\n",
"\nDepartamento de Física\nPUC-Rio and National Institute of Science and Technology for Complex Systems\nRua Marquês de São Vicente 22522453-900 RJRio de JaneiroCEPBrazil (\n"
]
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"Department of Chemical and Biological Engineering\nNorthwestern University\n60208EvanstonILUSA (",
"Department of Physiology\nFeinberg Medical School\nNorthwestern University\n303 East Chicago Ave. Chicago60611ILUSA",
"Departamento de Física\nPUC-Rio and National Institute of Science and Technology for Complex Systems\nRua Marquês de São Vicente 22522453-900 RJRio de JaneiroCEPBrazil ("
]
| []
| Recent work has shown that the distribution of inter-event times for e-mail communication exhibits a heavy tail which is statistically consistent with a cascading Poisson process. In this work we extend the analysis to higher-order statistics, using the Fano and Allan factors to quantify the extent to which the empirical data depart from the known correlations of Poissonian statistics.The analysis shows that the higher-order statistics from the empirical data is indistinguishable from that of randomly reordered time series, thus demonstrating that e-mail correspondence is no more bursty or correlated than a Poisson process. Furthermore synthetic data sets generated by a cascading Poisson process replicate the burstiness and correlations observed in the empirical data. Finally, a simple rescaling analysis using the best-estimate rate of activity, confirms that the empirically observed correlations arise from a non-homogeneus Poisson process. PACS numbers: 02.50.-r 89.20.-a 89.75.-k, | 10.1140/epjb/e2010-00139-9 | [
"https://arxiv.org/pdf/0907.1263v1.pdf"
]
| 55,085,982 | 0907.1263 | 23d944200c5bf3b7b04b98865b72fa698ca6aa7b |
Poissonian bursts in e-mail correspondence
7 Jul 2009
C Anteneodo
R Dean Malmgren
Department of Chemical and Biological Engineering
Northwestern University
60208EvanstonILUSA (
D R Chialvo
Department of Physiology
Feinberg Medical School
Northwestern University
303 East Chicago Ave. Chicago60611ILUSA
Departamento de Física
PUC-Rio and National Institute of Science and Technology for Complex Systems
Rua Marquês de São Vicente 22522453-900 RJRio de JaneiroCEPBrazil (
Poissonian bursts in e-mail correspondence
7 Jul 2009(Dated: July 7, 2009)
Recent work has shown that the distribution of inter-event times for e-mail communication exhibits a heavy tail which is statistically consistent with a cascading Poisson process. In this work we extend the analysis to higher-order statistics, using the Fano and Allan factors to quantify the extent to which the empirical data depart from the known correlations of Poissonian statistics.The analysis shows that the higher-order statistics from the empirical data is indistinguishable from that of randomly reordered time series, thus demonstrating that e-mail correspondence is no more bursty or correlated than a Poisson process. Furthermore synthetic data sets generated by a cascading Poisson process replicate the burstiness and correlations observed in the empirical data. Finally, a simple rescaling analysis using the best-estimate rate of activity, confirms that the empirically observed correlations arise from a non-homogeneus Poisson process. PACS numbers: 02.50.-r 89.20.-a 89.75.-k,
1
The assessment of human activity patterns is crucial for many applications, such as optimization of information traffic, service scheduling and human resource planning. In particular, the temporal dynamics of e-mail correspondence sparked recent interest [1,2,3,5,6,7,8] because of its importance as a communication medium and the availability of very large databases. Recent research [2,3,4,5,7] has shown that the probability distribution of the time elapsed between consecutively sent e-mails by a single user exhibits heavy tails.
The origin of such heavy-tailed statistics is controversial and the focus of much debate.
One explanation for the existence of the inter e-mail times distribution is a priority queuing model [2]. Another contrasting view is a cascading Poisson process [7,9]. In this latter model, there is a primary non-homogeneous Poisson process, which explicitly incorporates daily and weekly modulations, each of whose events triggers a secondary process which is also Poissonian but with a much larger characteristic rate. According to this model, "bursts" of e-mail activity occur in non-overlapping homogeneous Poisson cascades (as opposed to the overlapping cascades of Ref. [10], for instance) separated by long periods of inactivity defined by the primary process. The resulting inter-event time distribution predicted by the model is therefore heavy-tailed due to the mixture of several different scales of rates of activity.
While the cascading Poisson process has been shown to be statistically consistent with empirical inter-event time distributions of several individuals [7], it is unclear whether higherorder statistical patterns are present in the data and whether the cascading Poisson process adequately captures these patterns. Here, we quantify the burstiness and correlations in the empirical and synthetically generated point process data sets using standard statistical measures. We show that the burstiness and correlations in e-mail communication patterns are Poissonian, which are in fact reproduced by the cascading Poisson process.
We study here a database considered previously [1,2,3,5,7] comprised of 3,188 e-mail accounts over an 83-day period at a European University. From this database, we restrict our analysis to the 394 "typical" users that send at least 40 messages over 83 days [5,7], referred here as "empirical" time series. An example is presented in Figure 1 (for User 467) which depicts some of the typical features, namely that there are long pauses between e-mails of the order of 16 and 48 hours and that the short intervals are uncorrelated as the inset shows. The second set of time series to be analyzed, was generated by the cascading Poisson model in Ref [7], one per user, referred to here as "synthetic" time series. For a given user, the sequence of e-mails in time can be seen as a point process [11]. Typically, point processes are characterized with inter-event times or counting statistics, so one might naturally characterize higher-order statistical patterns of a point process with multivariate distributions of these quantities. Multivariate distributions, however, are difficult to assess for time series with few events, as in the present case. We therefore use the Fano and Allan factors, two standard metrics for point processes that provide reliable results for time series with few events [10,11,12,13], to gain insight into the higher-order statistical structure of e-mail correspondence.
The scaling. This can be also seen in the calculation of the local slope of the log-log plot, which changes continuously denoting absence of scaling (Fig. 2, lower panels). This finding is relevant at the light of alternative models [2] of this type of processes.
To further evaluate the empirical time series' departure from Poissonian statistics, we analyzed the surrogate time series obtained by randomly reordering the sequence of interevent times. If the empirical time series exhibits the same behavior in F F and AF as the shuffled time series, then the observed departure from Poissonian statistics is only due to the distribution of inter-event times and not due to their particular ordering; that is, the inter-event times are independent, as is illustrated in Fig. 2 (upper panel, white circles).
Thus, the observed "departure" from Poissonian statistics seen here is merely an artifact of the heavy-tailed inter-event time distributions, and not of some higher-order statistical structure.
Now, we proceed to compare the Fano and Allan factors of the empirical time series of this typical user with a synthetic time series generated from the cascading Poisson process from Ref. [7]. The model uses the best-estimate parameters for this specific user and the same number of events as the empirical time series. As Fig. 3 shows, the agreement between the empirical and synthetic F F and AF curves is remarkable, indicating that the cascading Poisson process is capturing not only the density distribution (already discussed at length in Ref. [7]) but also the higher-order statistical features of e-mail communication. This agreement is also consistent with (and anticipated by) the similarities between the empirical and shuffled time series (Fig. 2).
The analysis up to now indicates that the origin of the observed higher order correlations is related with the non-homogeneity in the rate of e-mail activity. If this is the case, we should be able to rescale time such that the resulting point process appears to have originated from a homogeneous Poisson process. Specifically, any non-homogeneous Poisson process with occurrence rate ρ(t), can be mapped onto a homogeneous Poisson process through a simple transformation of the timescale, namelyt = t 0 ρ(t) [11]. In this new time scale, the Poisson process has unit rate,ρ(t) = 1. In the particular case of a cascading Poisson process with known best-estimate parameters ρ(t) and ρ a and where we know which events are associated with which process [7], we rescale the times between consecutive events accordingly. The the process exhibits correlations, they are trivial as they are found to be originated by daily and weekly cycles of activity. The trivial origin of these correlations is further confirmed by a rescaling transformation which leads to a homogeneous Poisson process. There was no evidence of scale free process at any of the levels of e-mail activity analyzed. To conclude, the present findings are consistent with the cascading non-homogeneous Poisson model as the best description for the human activity patterns behind the e-mail sequences.
FIG. 1 :
1Time series of inter e-mail intervals for a representative user, and its autocorrelation function (inset).
Fano and Allan factors are calculated by dividing the whole observation time interval into W non-overlapping time windows of equal length T and counting the number of events N k in each time window, indexed by k. The Fano factor (F F ) is the ratio of the variance to the mean of the number of events in each time window, F F = ( N 2 k − N k 2 )/ N k , and it represents a measure of the dispersion-burstiness-of the resulting time series relative to a homogeneous Poisson process with the same rate. The Allan factor (AF ) quantifies the difference in variance of counts of adjacent time windows, AF = ( (N k+1 − N k ) 2 )/(2 N k ),and it is a measure of the correlation of counts between successive time windows relative to
FIG. 2 :FIG. 3 :
23Fano and Allan factors as a function of the time window T for User 467, a typical e-mail user (top panels). For comparison, we plot the mean and standard error of the Fano and Allan factors for 30 realizations of shuffled time series as well as the mean and standard error of the Fano and Allan factors for 30 realizations of the homogeneous Poisson process with the same rate. The bottom panels show the respective logarithmic local slopes of the empirical Fano and Allan factor curves.the expectation from a homogeneous Poisson process with the same rate.If the time series were generated by a homogeneous Poisson process, then the number of counts N k in each time window would be independent and identically distributed random variables drawn from a Poisson distribution. In such a case, F F (T ) = AF (T ) = 1, regardless of the time window length T(Fig. 2, top panels, grey circles). Deviations from unity therefore quantify departures from Poissonian statistics. For example, F F (T ) > 1 would indicate that the time series is more bursty than expected from a homogeneous Poisson process at a particular time-scale T . Indeed, oftentimes researchers identify scale-free features in point processes with a power-law increase in the Fano and Allan factors[12].We begin by analyzing the Fano and Allan factors as a function of the length of the time window for a representative e-mail user, to be complemented later in the manuscript with the averages for the entire database. The results for this user's activity are plotted Fano and Allan factors as a function of the length of the counting time window T , for the empirical and synthetic time series for User 467. The synthetic time series is constructed with the same number of events as the empirical time series from a cascading Poisson process with best-estimate parameters obtained in Ref.[5].
Fig. 2 ,
2(top panels, black circles, User 467 ). Notice that for time windows shorter than a few minutes the point process of e-mails is essentially Poisson, denoted by the fact that both indices remain close to unity. For longer times the F F and AF curves noticeably depart from unity, which might suggest that there are some non-Poissonian effects in e-mail communication. Nevertheless, notice that the non-unitary region exhibits no power-law
FIG. 4 :FIG. 5 :
45results presented in Fig. 4 confirm our hypothesis: the rescaled inter-event time sequence exhibit F F and AF values close to unity for time window lengths up to about ten times the characteristic time. By comparison, the results corresponding to thirty realizations of a homogeneous Poisson process with unit rate are presented as well. We know move beyond the analysis of User 467 to present the results of our analysis of all 394 users. The results presented in Fig. 5 show the same analysis presented in Figs. 2Fano and Allan factors as a function of the time window T for the rescaled time series of User 467. For comparison, we compute the Fano and Allan factors for 30 realizations of a homogeneous Poisson process with unit rate and the same number of events. The large fluctuations observed at long time-scales are due to poor statistics (e.g. W = 8 in the longest time window). now computed for all 394 e-mails users in the database. In Fig. 5 (left top panel) it can be seen that the mean F F curve for all users replicates well the mean F F curve for the shuffled time series for each of the 394 users. Thus, at the population level, higher order correlations of the process exhibit the same basic features: for short time windows (up to a few minutes), the F F is unitary for all users, a fingerprint of a Poissonian process; and, for longer time windows, both the empirical and shuffled data sets exhibit a slight increase. As discussed before, this is indicative of correlations in the empirical data which are trivially related to the distribution of inter-event times. In that figure the mean logarithmic difference is also plotted, this is quantified, for each user and time window size with log-distances, Summary of agreement between the empirical time series, shuffled time series (left panels) and cascading Poisson model (right panels). The top panels show the mean and standard deviation (whiskers) Fano factor as a function of the time window length for the time series averaged over all 394 users. The bottom panels show the mean logarithmic distance between the F F results for the empirical and shuffled time series (left) or cascading Poissonian model (right) for all users.
RFIG. 6 :
6i (T ) = log 10 [F F e,i (T )/F F s,i (T )],where subscript e indicates the empirical time series and s the series compared, either shuffled, synthetic or homogeneous Poisson. We do not find any significant deviations between the empirical and shuffled F F curves. The analysis of the entire database also confirms the similarities between the empirical and synthetic time series. This is presented in right top panel in Fig. 5, demonstrating that the mean behavior of the synthetic F F curves is consistent with the empirical mean F F curve with no significant systematic deviations between them. Similar conclusions can be reached from the results of the F F curves of rescaled data compared to a homogeneous Poisson process for all 394 users considered here. This is summarized in Fig. 6. The statistical distance indicates that after properly rescaling time, a homogeneous Poisson process is in fact recovered. Summarizing, the analysis described here shows that the sequences of inter e-mail times are uncorrelated for time scales shorter than a few minutes. At longer time scales, Top panel: Mean and standard deviation (whiskers) of Fano factor as a function of the time window length for the rescaled time series averaged over all 394 users. Bottom panel: shows the mean distance between the F F of the rescaled and that of a homogeneous Poisson process.
Acknowledgements:We are grateful to J-P Eckmann for providing the data. CA 9 acknowledges Northwestern University for the kind hospitality and Brazilian agencies CNPq and Faperj for partial financial support, DRC acknowledges support by NIH NINDS of USA (Grants NS58661).
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| []
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"ALOHA With Collision Resolution(ALOHA-CR): Theory and Software Defined Radio Implementation",
"ALOHA With Collision Resolution(ALOHA-CR): Theory and Software Defined Radio Implementation"
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"Xin Liu \nElectrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA\n",
"John Kountouriotis \nElectrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA\n",
"Athina P Petropulu \nElectrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA\n",
"Kapil R Dandekar \nElectrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA\n"
]
| [
"Electrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA",
"Electrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA",
"Electrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA",
"Electrical & Computer Engineering Department\nDrexel University\nPhiladelphiaPA"
]
| []
| A cross-layer scheme, namely ALOHA With Collision Resolution (ALOHA-CR), is proposed for high throughput wireless communications in a cellular scenario. Transmissions occur in a time-slotted ALOHA-type fashion but with an important difference: simultaneous transmissions of two users can be successful. If more than two users transmit in the same slot the collision cannot be resolved and retransmission is required. If only one user transmits, the transmitted packet is recovered with some probability, depending on the state of the channel. If two users transmit the collision is resolved and the packets are recovered by first over-sampling the collision signal and then exploiting independent information about the two users that is contained in the signal polyphase components. The ALOHA-CR throughput is derived under the infinite backlog assumption and also under the assumption of finite backlog. The contention probability is determined under these two assumptions in order to maximize the network throughput and maintain stability. Queuing delay analysis for network users is also conducted.The performance of ALOHA-CR is demonstrated on the Wireless Open Access Research Platform (WARP) test-bed containing five software defined radio nodes. Analysis and test-bed results indicate that ALOHA-CR leads to significant increase in throughput and reduction of service delays.keywords-multi-user system, blind source separation, MIMO systems, collision resolution, software defined radio | 10.1109/tsp.2010.2048315 | [
"https://arxiv.org/pdf/0909.3091v1.pdf"
]
| 15,387,518 | 0909.3091 | 5f603b9203a49768deceba49391ce3e0b6c069fb |
ALOHA With Collision Resolution(ALOHA-CR): Theory and Software Defined Radio Implementation
16 Sep 2009
Xin Liu
Electrical & Computer Engineering Department
Drexel University
PhiladelphiaPA
John Kountouriotis
Electrical & Computer Engineering Department
Drexel University
PhiladelphiaPA
Athina P Petropulu
Electrical & Computer Engineering Department
Drexel University
PhiladelphiaPA
Kapil R Dandekar
Electrical & Computer Engineering Department
Drexel University
PhiladelphiaPA
ALOHA With Collision Resolution(ALOHA-CR): Theory and Software Defined Radio Implementation
16 Sep 20090
A cross-layer scheme, namely ALOHA With Collision Resolution (ALOHA-CR), is proposed for high throughput wireless communications in a cellular scenario. Transmissions occur in a time-slotted ALOHA-type fashion but with an important difference: simultaneous transmissions of two users can be successful. If more than two users transmit in the same slot the collision cannot be resolved and retransmission is required. If only one user transmits, the transmitted packet is recovered with some probability, depending on the state of the channel. If two users transmit the collision is resolved and the packets are recovered by first over-sampling the collision signal and then exploiting independent information about the two users that is contained in the signal polyphase components. The ALOHA-CR throughput is derived under the infinite backlog assumption and also under the assumption of finite backlog. The contention probability is determined under these two assumptions in order to maximize the network throughput and maintain stability. Queuing delay analysis for network users is also conducted.The performance of ALOHA-CR is demonstrated on the Wireless Open Access Research Platform (WARP) test-bed containing five software defined radio nodes. Analysis and test-bed results indicate that ALOHA-CR leads to significant increase in throughput and reduction of service delays.keywords-multi-user system, blind source separation, MIMO systems, collision resolution, software defined radio
I. INTRODUCTION
In a wireless uplink scenario, collisions occur when two or more users transmit at the same time over the same channel. Traditionally, once a collision occurs, retransmissions are requested. Retransmissions lower throughput and waste power and bandwidth. Well studied schemes for avoiding collisions include Carrier Sensing Multiple Access with Collision Avoidance (CSMA/CA) (e.g., see IEEE 802.11 [1]). In order to overcome the hidden terminal problem, IEEE 802.11 incorporates a positive acknowledgment scheme, i.e., Request To Send (RTS) followed by Clear To Send (CTS). However, in most protocols, collisions occur more frequently as the traffic load increases, in which case the RTS/CTS scheme becomes less effective due to collisions of the RTS reservation packets.
Collision resolution can be viewed as multiuser separation. However, well known approaches that allow for multiuser separation, such as TDMA, FDMA, OFDMA, CDMA or use of multiple antennas, might not be a good fit for wireless networks. Wireless network traffic can be bursty, users operate on limited battery power, and in certain cases, wireless receivers have physical size limitations. TDMA, FDMA and OFDMA approaches are fixed resource allocation schemes and thus are not efficient for bursty traffic.
The CDMA approach requires bandwidth expansion, which results in increased power consumption for each wireless network user. The use of multiple antennas, might not be feasible for compact wireless receivers. Wireless network-friendly approaches to achieve diversity include the NDMA protocol [6,19], ALLIANCES [10,17] and ZigZag decoding [7]. In these protocols, collisions are resolved by combining collided packets and several retransmissions. In these schemes it is assumed that nodes transmit with the same power, and that there is no significant power decrease due to propagation in small-scale networks.
For cases in which users transmit using different power levels, user separation could be achieved via successive interference cancellation (SIC) [14]. However, it might not be a good approach to assign different power to different users. Collisions only happen with some probability, therefore, it would not be good for a user to transmit at low power all the time just to be separable in the event of a collision. A potentially network-friendly approach that does not require retransmissions and allows users to transmit at the same power was recently proposed in [12,11,18]. According to [12,11,18], by upsampling the received signal and viewing its polyphase components as independent linear mixtures of the collided packets, under certain conditions, the collided packets can be recovered in a blind fashion based on a single collision. In [18], user separation was enabled by different carrier frequency offsets (CFO) and user delays. In [12,11], pulse-shape diversity was investigated as source of additional diversity in case user delays and CFOs are small.
In this paper, a novel cross-layer scheme is proposed for high throughput wireless communications in a cellular scenario. Transmissions occur in a time-slotted ALOHA-type fashion but with an important difference: simultaneous transmissions of two users can be successful. The wireless channel is assumed to be flat fading and constant over the duration of one time slot. A user i with a non-empty queue transmits a packet with some probability p in the beginning of each time slot, after waiting for a random time interval τ i . Each user embeds orthogonal pilots in its packet. In each slot, the base station (BS) determines the number of transmitting users. If there are more than two users, the packets are discarded and the users are asked to retransmit at a later time. If there is only one user present, its packet is recovered with some probability, depending on the state of the channel. If there are two users present, the users are separated and their packets are recovered by first oversampling and exploiting independent information about the two users that is contained in the polyphase components of the received signal. The properties of the user delays τ i 's are determined so that the probability of user separation is maximized. The system throughput is derived under the infinite backlog assumption, i.e., the network users always have data in their queues, and also under the assumption of finite backlog. Queuing delay analysis for network users is also conducted.
The performance of the proposed approach is demonstrated via simulations, and also via experiments conducted on a software defined radio (SDR) [2] testbed. This experimental wireless network consists of five nodes, i.e., one base station (BS) or access point, and four users, and was deployed in an indoor laboratory environment. The experimental results suggest that 70% of two order collisions can be resolved by the BS under realistic conditions, which results in higher throughput and lower service delays.
1) Relation to other published work:
Collision resolution is based on the ideas of [18], where naturally occurring user delays and carrier frequency offsets were shown to provide diversity that enables blind user separation, i.e., separation without knowledge of the channel. However, in our experimental setup, we observed that naturally occurring delays and CFOs are rather small. Thus, in this work, we ignore CFOs and introduce intentional delays in addition to the naturally occurring ones. The statistical characteristics of the intentional delays are chosen to enhance the separability of the users. In order to keep the complexity low and the probability of user separation high, resolution of only second order collisions is considered here. The work of [18] was concerned with the physical layer only. Here, we propose a cross-layer approach, and study throughput and queuing performance as well as physical layer issues. Further, a host of physical layer issues motivated by the real implementation are studied.
Multiuser separation based on user delays was also considered in [4]. The approach of [4] considers transmission of isolated frames; it exploits the edges of a frame over which users do not overlap, and assumes knowledge of the channel. However, noise can be a problem when exploiting edge effects as samples are taken at points where the pulse is quite low.
In relation to the collision resolution approaches NDMA [6,19], ALLIANCES [10,17] and ZigZag decoding [7], the proposed approach resolves collisions of order two without retransmissions. Thus, no storage of the collision signals is needed, and network users not involved in the collision do not need to wait until the collision is resolved.
2) Organization: The rest of the paper is organized as follows: The physical layer of ALOHA-CR is introduced in Section II and several implementation considerations are discussed in Section III. Network performance quantities for the Aloha-CR, like throughput and packet delays, are analytically derived in Section IV. A brief description of the SDR platform used to implement Aloha-CR appears in Section V, while specifics of our implementation of the physical and MAC layers on the SDR platform appear in section VI. The obtained experimental results are presented in Section VII and are compared with analytical and simulation results. Finally, conclusions are drawn in Section VIII. The channel between transmitter and receiver is assumed to be flat fading. Moreover, the channel is quasi-static, i.e., the channel remains unchanged over the duration of a packet.
3) Notation
If within a given time slot K users transmit, the baseband signal received by the BS equals
y(t) = K k=1 a k x k (t − τ k ) + w(t),(1)
where a k denotes the channel coefficient between the k − th user and the BS; τ k is a random delay associated with the user k; w(t) represents noise; and x k (t) is the k-th user signal, i.e., x k (t) = i s k (i)p(t − iT s ) where s k (i) is the i−th symbol of user k; T s is the symbol interval; and p(t) is a pulse shaping function with main lobe support [−T s , T s ]. The mainlobe of neighboring pulses overlap by 50%.
In each symbol interval, the received signal is upsampled by a factor of P , with sampling locations at t = iT s + mT s /P , m = 1, 2, · · · , P . The m−th polyphase component of the sampled output is
y m (i) = K k=1 a k h mk (i) * s k (i) + w m (i)(2)
where "*" denotes convolution, and h mk (i) equals h mk (i) = p(iT s + mTs P −τ k ) for i = ...−2, −1, 0, 1, 2, .... Using a pulse with low sidelobe, at the sampling points over the i−th symbol interval, the only interference is from the (i + 1)-th symbol. Therefore, the channel h mk (i) can be approximated as of length 2. The IOTA pulse is a good choice for maintaining low intersymbol interference [9]. September 16, 2009 DRAFT The convolutional MIMO problem can be transformed to a scalar one as
y m (i) = h m Ds(i) + w m (i)(3)
where
h m = [[h m1 (0) h m1 (−1)], ..., [h mK (0) h mK (−1)]] , D = diag([a 1 , a 1 , · · · a K , a K ])
and
s(i) = [s 1 (i) s 1 (i + 1)] , · · · , [s K (i) s K (i + 1)] T .
Let us form the vector y(i) by appending y m (i), m = 1, ..., P . We have
y(i) = As(i) + w(i);(4)
where A = HD and H = [h T 1 , h T 2 , · · · , h T P , ] T . If pilot data are available, the matrix A can be estimated based on the pilot symbols, and then used for the recovery of the information bearing symbols. If no pilots are available, estimating A and then recovering s(i) is still possible by viewing (4) as a P ×2K instantaneous blind MIMO problem. Assuming that P ≥ 2K, and under certain conditions on A, the system is identifiable [18]. Any blind source separation algorithm (e.g., the JADE algorithm [5]) can be applied at this point to obtain an estimate of A, i.e.,Â, within a column permutation ambiguity and a a constant diagonal matrix, Λ, with complex nonzero diagonal elements, which represents phase ambiguity. These ambiguities are trivial, and are inherent in blind estimation problems. Based on and using a least-squares equalizer we can get the de-coupled signalsŝ(i), within permutation and phase ambiguities aŝ
s(i) = (Â HÂ ) −1Â H y(i) = e jArg{−Λ} |Λ| −1 P T s(i)(5)
where Λ is a diagonal matrix and P is a permutation matrix. Denoting by θ k the k−th diagonal element of Arg{Λ}, the k−th input signal can be recovered within a phase ambiguity asŝ k (i) = s k (i)e −jθk .
Although uniform sampling was described above, non-uniform sampling can also be done [18].
A. About users' delays
For fixed sampling locations and a fixed pulse-shape function, the condition number of A can be controlled by the user delays τ k , where k = 1, 2, ..., K. If τ k 's are close to each other, the columns of A will be highly correlated, which results in high condition number.
Since naturally occurring delays are too small to guarantee a well conditioned A, we propose that, before transmission, each node introduces an intentional delay. Let τ k be the sum of the naturally occurring delay and the intentional delay. In this work we try to resolve collision of order two only. Let us express the delay difference between the two users is τ = α+δ, where α is the difference between the intentional delays between users i and j, and δ is the difference between the naturally occurring delays. Let f δ (x) be the pdf of the natural delays differences, and further assume that f δ (x) is symmetric around the origin. The proof is given in Appendix A.
III. PHYSICAL LAYER IMPLEMENTATION ISSUES
Several issues need to be addressed in a practical implementation of the proposed approach.
A. Frequency offsets and phase tracking
In a practical system there are always CFOs between transmitters and receiver, resulting from mismatch between transmitters and receiver oscillators, and also from Doppler shifts due to relative movement between transmitters and receiver. In this case the continuous-time base-band received signal y(t) is of the form:
y(t) = K k=1 a k x k (t − τ k )e j2πFkt + w(t)(6)
where F k is the CFO for user k. In [18] the CFOs were used as source of diversity that enables use separation. In the implementation that we consider here the CFOs are too small to provide diversity, and thus are ignored in the problem formulation. However, their effect is still present in the separated symbols, i.e.,ŝ
k (i) = s k (i)e j(−θk+2πFkTsi)(7)
The effect of the CFO on the separated signal can be mitigated using a phase locked loop (PLL) device [16]. The input output relationship for the PLL is S O (i) = S I (i)e jφ(i) , thus, the CFO estimate can be obtained asF k = 1 2π dφ/dt. The phase ambiguity, θ k can be compensated for through use of pilot symbols, or by using differential phase offset keying.
B. Successive interference cancellation
Successive interference cancellation applied on a mixture of signals treats one of the components of the mixture as the signal of the interest and the rest as interference. The approach of Section II can be combined with SIC to further improve packet recovery performance. In particular, after blind source separation, the contribution of the strongest user signal can be reconstructed and deflated from the received signal. This usually provides a better estimate for the weak user.
One way to determine which is the strongest user is to look for the signal that has the smallest variance around the known constellation. Let the strong user be user k. Reconstruction of the contribution of the k-th user to the received signal requires knowledge of the pulse shape waveform, and estimates for CFO (F k ), channel coefficient (â k ), and delay (τ i ). The reconstructed signal is:
y k (t) =â k iŝ k (i)p(t − iT s −τ i )e j2πFkt .(8)
whereŝ k (i) are the estimated symbols.
The channel coefficient estimates can be obtained by cross-correlating the known pulse-shape waveform with the signal returned by the JADE algorithm. The user delays can be estimated at the synchronization step (see Section III-C), and the CFOs can be estimated as described in Section III-A.
Due to the delay between users, the peaks of different user pulses do not overlap. One could naturally wonder whether applying successive interference cancellation would be sufficient instead of upsampling the signal and performing packet separation as in Section II. As will be shown in Section VII via both simulation and testbed measurements, using SIC directly results in inferior results.
C. Determining sampling points
In order to determine the beginning of the packet, some form of synchronization is required. For synchronization purposes, users are assigned distinct pseudo random sequences (pilots). The base station keeps a record of all pilot sequences in use in the network.
When the packet arrives, the base station uses the beginning part of the received signal to perform correlation with every entry of the code book. A peak in the correlation of the received signal with code i indicates the presence of user i. The peak location provides the beginning of the packet of user i, while the peak value provides the corresponding channel coefficient.
This can be repeated for all possible users, however, in practice the following approach works better.
The strongest user is identified as the one that produces the largest peak in the correlation. Then, the user pilot signal is reconstructed based on estimated channel coefficients and delays, and is subsequently deflated from the pilot portion of the received signal along the lines of Section III-B. The CFO effect is ignored here because of the short duration of the pilot segment.
For synchronization purposes, the best pulse shaping waveform for the pilots is the raised root cosine (RRC) function, as this function maximizes the SNR at the output of the matched filter [13] while it eliminates ISI at the sampling points.
We should note that the part of the packet containing the actual information will need to be oversampled in order for the method described in Section II to be applied. As explained in that section, the best pulse shaping for that purpose is the IOTA pulse [9]. However, we could not have used IOTA for the pilots, because the convolution of IOTA with itself introduces ISI at t = nT s , thus the matched filter would not work well.
D. Blind versus pilot-based user separation
Since a real communication system always uses pilots for synchronization purposes, one would think that these pilots could be used to estimate the matrix A in (4), which then could be used to recover the information bearing symbols. However, the fact that different pulse shape waveforms are used for pilots and information bearing symbols renders that approach impossible. As was already mentioned, in order to maximize the matched filter performance, RRC pulse shaping is used for the pilot symbols to be used in synchronization. Also, in order to minimize intersymbol interference between neighboring symbols of a user, IOTA pulse shaping is used for the payload symbols. Thus, the estimate of matrix A based on the pilots would be different than that corresponding to the payload (based on (3), A depends on the pulse shape function).
However, we can first estimate channel coefficients a k and user delay τ k based on the pilot symbols, and subsequently combine them with the IOTA pulse shape function and sampling points (see (3)) to get the estimate of channel matrixÂ. Following the estimation of the symbols can be recovered via least-squares. We term this approach as training method. In VII we compared the training method to the blind approach, in which the matrix A is considered to be unknown. As it will be seen in that section, the estimation errors in channel coefficients and user delays render the training method inferior to the blind one.
E. Dealing with collisions of any order and packet recovery
In theory, one could use the pilots to determine the number of users present in a collision. This is a detection problem and in low signal-to-noise ratio cases can lead to erroneous conclusions. Instead of attempting to estimate the collision order, we propose the following procedure.
The received signal is always treated as if it contained two users. The users are first separated as explained in Section II, and the strongest user is deflated from the received collision as described in
Section III-B. For the remaining signal one of the following possibilities holds: (i) the signal is just noise; this is when there is only one user signal, or there are more than two users in the received signal.
(ii) the remaining signal corresponds to the signal of the second user that was involved in the collision.
Although the above approach treats even the case of a single user as a potential collision, our experience indicates that this is a more robust approach than detecting the number of users first and then acting accordingly.
Following the synchronization step, regardless of the collision order, the incoming packet would be over-sampled by 4. By applying the blind separation method of Section II we would get 4 sequences. Each sequence would be passed through a PLL. Since the output of the PLL would be scattered around the nominal constellation the sequence with the smallest variance would be chosen. We refer to the strongest signal as s u (.). The decision on whether this is s u (i) or s u (i + 1) can be resolved using the user ID (i.e., the MAC address). Next, the strong user would be deflated from the received signal as discussed in Section III-B. If there was only one user in the received signal, or if there were more than two users, the remaining signal after deflation would not have a meaningful structure; this could be determined using the user ID. Otherwise, the deflation would yield the signal of the second user.
IV. THROUGHPUT ANALYSIS OF ALOHA-CR
Consider a cellular network of K users who communicate with a base station (BS). Users transmit their packets in a time slotted fashion with probability p. Each packet contains multiple symbols, and the time slot duration is equal to the packet duration plus two symbols.
The proposed ALOHA-CR schemes follows the slotted ALOHA protocol, expect that second-order collisions can be resolved. Since collision of packets can be resolved, it is expected that ALOHA-CR will have higher throughput than slotted ALOHA. In this section we first analyze the throughput of ALOHA-CR for the simple case of a network with infinite backlog, i.e. the case in which the queue of the nodes can never be empty, and each node always contends with some probability. In this case we analyze the throughput of the network. Then, we consider the case in which the nodes have finite backlog and analyze the throughput and service delay of ALOHA-CR.
The throughput is defined here as the number of successfully delivered packets per slot. We consider a network of J users with J > 2, and each user contends with probability p. The following possibilities exist for each slot.
• No transmissions are attempted (empty slot).
• A single transmission is attempted. In this case, let depending on the channel state the probability of successful reception be P 0 .
September 16, 2009 DRAFT • Two transmissions are attempted. Let the probability of receiving correctly both of the transmissions be P 1 and the probability of successfully receiving only one of the two transmitting messages be P 2 (i.e., the probability of failing to receive any of the messages is 1 − P 1 − P 2 ).
• More than two transmissions are attempted. In this case none of the transmitted messages can be successfully received and users have to retransmit at some later time.
A. Network with infinite backlog and infinite number of users
For slotted ALOHA, the throughput is well established as C(J) = Jp(1 − p) J−1 , which is maximized
for p * = 1/J with maximum throughput C(p * ) → e −1 as J → ∞.
For ALOHA-CR, the maximum throughput and optimum contention probability are given in the following proposition.
Proposition 2:
The maximum network throughput is:
C = 2P 2 0 P 0 − 2P ′ + P 2 0 + 4P ′2 1 + 2P ′ P 0 − 2P ′ + P 2 0 + 4P ′2 e −2P 0 P 0 −2P ′ + √ P 2 0 +4P ′2 ,(9)
where P ′ = P 1 + P2 2 , and is achieved for contention probability equal to:
p = 2P 0 2P 0 + a + √ a 2 + 4P 0 b(10)
where a = (P 0 − 2P ′ )(J − 1) and b = P ′ (J − 1)(J − 2).
Proof : see Appendix II.
B. Network with finite backlog and finite number of users
In this case the nodes with empty queue will not contend for medium access. Throughput analysis for this case is carried our by extending the approach of [15] to the case in which the receiver can resolve second order collisions with a certain probability. The beauty of the method in [15] is that it approximates the performance of J coupled queues with J uncoupled geom/geom/1 queues, an approximation that simplifies the analysis greatly.
The assumptions in this section follows those in [15], i.e.,
• The arrival rate for each queue in the system is Bernoulli with rate r, i.e., the total arrival rate for a system with J users is rJ.
• A queue k is active in a time slot if it has one or more packets eligible for transmission, else it is inactive.
• Each active queue k = 1, ..., J contends with a common fixed contention probability p.
We further assume that there is an acknowledgment feedback loop, so that the transmitter knows whether the packet that was transmitted was successfully received or has to be re-transmitted. Assuming that the probability that a queue is active in a typical time slot in steady state is q, the probability of success for a queue becomes:
s(q) = P 0 p (1 − qp) J−1 + P 1 + P 2 2 (J − 1) p 2 (1 − qp) J−2(11)
where we assumed that in the case of two message transmissions where only one message is successfully received, it could be any of the two messages with equal probability, i.e., we assume all the links to be equivalent.
1) Active Probability q:
Applying Little's Law to the server, we find that q = r s , where r is the arrival rate and s is given by (11). Following the steps from [15], let us define f (z) = P 0 z (1 − z) J−1 +
P 1 + P2 2 (J − 1) z 2 (1 − z) and f max = f (p * ), where p * is the maximizer for f (z). Based on Appendix I p * = 2P0 2P0+a+ √ a 2 +4P0b
. The function f (z) corresponds to the success probability of a queue in the system when the queues are unstable and thus always active. In other words, f (z) corresponds to the maximum possible success probability. Since a queue cannot output more packets than the ones that arrive in the queue, we can distinguish between two different modes of operation of the queue as a function of the arrival rate:
For r : r > f max the arrival rate in the queue is larger than the maximum possible rate that the packets can exit the queue. In this case the queue is always active (i.e., q = 1), and it's success probability is simply f (p), with p the contention probability.
For r : r < f max the stability of the queue depends on the contention probability, since the physical layer can support a departure rate greater than the arrival rate. But the queue is not stable for all possible contention probabilities. The equation f (p) = r in this case has two real solutions, let these be p min and p max . For p < p min and p > p max , the queue is unstable and the active probability q = 1. This instability is due to either a very conservative choice of contention probability (for the p < p min case) or a very aggressive one (for the case of p > p max ). On the other hand, for p ∈ p min , p max , the queue becomes stable (active probability q < 1), as in this region of operation the physical layer can support a greater departure rate than r. Since a queue cannot output more packets than the ones arriving in the queue, we conclude that the departure rate in this region equals the arrival rate in the queue. In order to calculate the active probability in this region of operation we can simply solve the equation f (qp) = r.
It is straight forward to see that the two solutions of the equation are qp = p min and gp = p max . Solving for q, we get that q = p min p and q = p max p . Since p ∈ p min , p max and q ∈ [0, 1], the only possible solution is q = p min p . Summarizing, the active probability of each node equals q = p min p , r < f max and p ∈ p min , p max 1, otherwise
The equations f (p) = r and f (qp) = r can efficiently be solved for p and q using any numerical method, for example Newton's method.
2) Approximate Throughput: The throughput of J independent queues, using q, p min , p max and f max is τ = Jr, r < f max and p ∈ p min , p max
JP 0 p (1 − p) J−1 + J P 1 + P2 2 (J − 1) p 2 (1 − p) J−2 , otherwise(13)
The system throughput (average number of successful transmissions per slot), when queues are stable is limited by the rate at which messages arrive at each of the queues, while in the region where the queues are unstable (q = 1), the throughput is limited by the maximum achievable throughput of the physical layer, as in the case of infinite backlog.
3) Average Total Delay (Queue+Service delays):
For the regions of operation where the active probability q is less than 1 and thus the queue is stable, we can further calculate the total delay a packet will experience from the time it enters the queue, until it is successfully transmitted. As is shown in section IV-B.1, the queue is stable when r < f max and p ∈ p min , p max . Using the well-known results from queuing theory for the geom/geom/1 queue, the total delay (queuing plus service delay) equals [8] D tot = 1
s 1 − r(1−s) s(1−r)
, r < f max and p ∈ p min , p max
where r (1 − s) is the "birth probability", s (1 − r) is the "death probability" of the queue and s comes from (11), after we calculate the active probability q from eq. 12.
4) Average Delay in Server:
Since for an geom/geom/1 queue with service rate s the average service delay is δ = s −1 , the average service delay of the K independent queues, using q, p min , p max and f max is δ = q r , r < f max and p ∈ p min , p max
P 0 p (1 − p) J−1 + P 1 + P2 2 (J − 1) p 2 (1 − p) J−2 −1 , otherwise(15)
V. TESTBED SETTING
The proposed approach was implemented on the WARP testbed [3,2]. WARP hardware consists of two components: digital baseband processing and analog RF processing. Baseband processing is all done on the main system board which houses a Xilinx Virtex II Pro FPGA for all digital baseband PHY and MAC layer functionality. The main board contains four sets of connectors which provide a set of digital connections to four possible daughtercards that fit onto the main board. These daughtercards perform ADC and DAC functions along with up-and down-conversion to and from the ISM and UNII frequency bands.
In this study we used the non-real time stage of the WARP testbed, which makes use of an API called
WARPLab. WARPLab allows all processing and modulation to be done in Matlab, turning the FPGA of WARP into a simple buffer. Matlab can be used to create a set of data, modulate it, apply the designed pulse shaping function, and transfer the data to the radio card. On the receive side, WARPLab allows for data to be processed in Matlab immediately after it has been downconverted by the RFIC on the radio card.
VI. DETAILS ON THE SDR IMPLEMENTATION
The user packet is structured as shown in Fig. 1. The SDR implementation was carried out in the following steps.
At the transmitter:
• Paylaod -The payload contained 414 bits (32 bits for the user ID and 382 random bits). Convolutional coding with rate 1/2 was applied to get 828 bits. The coded bits were then interleaved. Specifically, the interleaver writes the input sequence in a matrix in row-wise fashion and then reads it in columnwise fashion. Differential quadrature phase shift keying (DQPSK) was used to modulate the data.
The IOTA pulse shape waveform was used for transmission.
• Pilots -A 32 bit m-sequence was added at the beginning as a sequence of pilots; it was BPSK modulated and RRC pulse shape waveform was used for the transmission of the pilot symbols. A code book of four m-sequences was generated. The code book was kept at the BS and linked with to the user IDs.
• Sampling rate -The sampling rate of the board was 40 Msamples/second and 32 samples/symbol were taken, yielding data rate of 1.25 Msymbols/second. At the receiver:
• Synchronization -The signal was read from the receiver buffer where it was already down-converted to 5 MHz. Subsequently it was down-converted to baseband. All entries of the code book were used to perform correlations with the header of the received signal. The entry which gave the largest correlation peak was chosen to indicate who the corresponding user was. For the following discussion, suppose that this is user u 1 (u 1 could be any of the users present in the system).
The delay and channel coefficient of user u 1 was estimated based on the location and value of the peak, respectively. The chosen m-sequence was deflated from the pilot portion of the received signal. All entries of the code book were used to perform correlation with the pilot portion of the deflated signal. The entry that gave the largest correlation peak indicated the second user, u 2 , and the corresponding delay and channel coefficient was estimated.
Note that if there was only one user, the remaining signal after deflation would be just noise. The same would hold in the case that the received signal contained more that 2 users.
• Symbol recovery -The received signal was up-sampled by 4, with sampling points occurred at Let s u (.) be the strongest signal, i.e., the sequence that has the smallest variance around the known constellation. The symbols corresponding to s u (.) were demodulated. The use of DQPSK modulation allowed for removal of phase ambiguity. The demodulated output was passed through a de-interleaver and decoder, to get 414 decoded bits. The result could be either s u (i) or s u (i + 1). If we incorrectly misinterpreted s u (i) for s u (i + 1), the de-interleaver would give a meaningless output. We used the user ID part in the beginning of the decoded output to do correlation with the corresponding entry of the user ID book in order to determine whether the recovered signal was s u (i) or s u (i + 1), and also whether the recorded signal corresponded to user u 1 or user u 2 .
Note that although we use correlation with the user IDs to determine the user, we cannot use this information to estimate the channel. This is because the received packet is interleaved and coded, thus the beginning part of the frame is a random sequence until decoding.
Next, we used the detected s u (i) to obtain the corresponding channel estimate using cross-correlation with the IOTA pulse. Note that we already had obtained a channel estimate for that user during the synchronization step. However, the estimate obtained based on the recovered symbols would be more robust as it is based on 414 symbols; the estimate obtained during the synchronization step was based on 32 symbols. Finally, we deflated the corresponding signal from the received mixture.
In the testbed experiment, we assumed that we knew the collision order in each transmission. If the transmission was collision free, we would stop after the first round of detection. In a real case, where the receiver is not aware of the collision order, we would proceed with SIC. The collision free signal would yield a meaningless remainder after SIC and that signal would not pass a checksum verification.
VII. TESTBED MEASUREMENTS
A. A two-user system
In this experiments, nodes 3 and 5 are the two transmitters, and node 2 is the BS. For each time slot both nodes transmit with probability 1.
1) BER comparison:
In this section we show the testbed performance of the ALOHA-CR using blind source separation followed by SIC, described in Section III-B (denoted in the figures as blind), ALOHA-CR using training based source separation followed by SIC, described in Section III-D, (denoted in the figures as training), and ALOHA-CR using SIC only, described in Section III-B, (denoted in the figures as SIC). The blind source separation algorithm used was the JADE method [5], which was downloaded from: http://www.tsi.enst.fr/∼cardoso/guidesepsou.html.
We first consider the raw BER performance (BER before decoding) of the proposed scheme. In this scenario, the location of the transmitter and receiver, and the antenna gains are fixed. By varying the amplitude of the input signal we can look at the BER performance at different SNR levels. For each SNR level, 600 packets were transmitted from the sender to the BS. Since the indoor wireless channel is time varying in both phase and amplitude, the received SNR of the two users varies between transmissions.
The SNR difference of user 1 and user 2 was within 3dB; 94% of them were within 1dB.
For comparison purposes only, in this BER evaluation we only include the delay differences in the
range [T s /2 − T s /8, T s /2 + T s /8].
When the delay differences are smaller, all methods yield high BER and their performance is the same. The BER performance of the blind approach as captured by the testbed is shown in Fig. 3. One can see that the proposed blind separation scheme works very well. The BER approaches 10 −3 at an SNR of about 20dB. The performance of the training method is also shown in Fig. 3. One can see for the testbed measurements results there is about 3dB performance advantage of the blind over the training method when the SNR is smaller than 20dB, and this advantage further increases at higher SNR level. The inferior performance of the training method is due to the sensitivity of least-squares at low SNR. Moreover, in the testbed measurements there is distortion of the pulse shape due to the antenna, drifting of the sampling point, and error in the channel coefficient estimates, which also results in degradation of BER performance. The performance of the SIC method is also included in Fig. 3. We can see that there is an error floor which does not decrease with increasing SNR. This is due to the fact that when we attempt to detect the first user, we treat the other user as interference.
Computer simulations were also conducted to produce the BER for this case. In the simulation the channel coefficients, a k , k = 1, 2, were taken to have amplitude one and random phase. It was assumed that the channel remains the same within each block. The delays and CFOs were set equal to the values observed during the testbed experiments. The estimation results were averaged over 100 independent channels, and 10 Monte-Carlo runs for each channel. In Fig. 3 one can see that there is only 1dB gap between the testbed measurements and the computer simulations.
2) Throughput comparison: The throughput performance of the three methods is given in Fig. 4. In this figure all received packets are taken into account. We assume that any error in the decoding output results in failure of transmission. The throughput was computed as the number of successful delivered packet per time slot. We can see that, as expected, the blind separation method gives the highest throughput.
The throughput of SIC is bounded by 0.3, and does not increase with increasing SNR level.
Comparison of blind and training methods in terms of throughput fairness between the users is given in Fig. 5. We can see that in the high SNR region (SN R ≥ 20dB), the throughput of the two users is almost the same; in the low SNR region, the throughput of user 1 is a little bit lower than user 2, which is caused by the SIC. As we first detect and deflate user 1 from the mixture of received signal, the signal to interference and noise ratio (SINR) of user 1 is lower than that when we detect user 2. This SINR difference will be in favor of user 2 in the low SNR case. In the high SNR region, user 1 can be detected very well even in the presence of user 2. Then the throughput of these two users are almost the same.
3) Throughput vs different shift scenarios:
In this section we demonstrate the advantage of intentional delays. In the previous experiment, we introduce random delay to all users. In this experiment we look at the performance without any intentional delays. The results are shown in Fig. 6. It is clear that if we do not introduce any delays and let nodes transmit freely, the throughput is quite low. This is because the naturally occurring delays are small, thus affecting the condition number of A and resulting in high BER.
B. Buffered Slotted Aloha Measurements
For this set of measurements, we employed 5 SDR nodes, as depicted in Fig In order to gather meaningful data, we had to make sure that the system was at steady state. Since the actual transmission and reception operations were time-consuming, the measurement process was performed in two steps. For each arrival rate and contention probability the system started at the empty state(all queues empty) and for the first 100,000 slots we were not performing any actual transmissions, but rather decided on the outcome of each slot based on the values of P 0 , P 1 and P 2 that were measured off-line for this topology (namely, P 0 was measured to be 0.998, P 1 was 0.965 and P 2 was found to be 0.009) and the number of contenting stations. To be more specific, when no transmitters were trying to transmit, the slot was considered empty. When only one transmitter was trying to access the medium, its queue would decrease by one with probability P 0 . When two transmitters were contenting for the medium there was probability P 1 of both transmissions being successful, i.e., both of the queues would decrease their size by one with probability P 1 , and with probability P 2 only one of the transmitters would decrease its queue. In the later case, where only one of the two nodes was successful, the successful transmission was assigned on either of the two contenting transmitters with equal probability (i.e., probability of 0.5 for each). If more than 2 transmitters were trying to access the medium, a collision was declared and no queue would decrease its size.
After these initial 100,000 slots, there were 3,000 extra slots during which actual transmissions were employed and data for service delay, total delay, throughput and active probability was gathered. For each slot, the outcome was determined by the receiver, depending on how many messages it was able to receive without any errors. Any message that was successfully received, was removed from the corresponding queue, but in case of errors the message had to remain in the queue and be re-transmitted until being successfully received.
The data that was gathered is plotted against the analytically calculated values that were determined as is described in section IV-B. In Fig. 7 the results for the active probability appear, in Fig. 8 we plot the measured and numerically calculated throughput and in Figs. 9 and 10 the measured and analytical results for the total and service delays respectively are plotted. From the plot of the total delay, the lines that correspond to arrival rate of 1/2 do not appear, since in this case the queues are unstable for all possible contention probabilities and thus the total delay goes to infinity. For the active probability and throughput we see that there is almost a perfect match between the measured and the analytically predicted quantities, using the independent queue approach, while for the delay quantities the match is still pretty good, even though it is not as good as for the throughput and active probability. Comparing the system with the conventional buffered slotted Aloha, looking for example at the results from [15], were no collisions can be resolved, we can see that the achieved throughput for ALOHA-CR is more than doubled, the service and total delays are considerably less and further, the system is stable for a much greater span of arrival rates and contention probabilities.
VIII. CONCLUSION
In this paper, we have proposed ALOHA-CR, which is a novel cross-layer scheme for high throughput wireless communications in a cellular scenario. This scheme can resolve second order collisions in the network without requiring retransmissions. We have described in detail the physical and MAC layers of the proposed scheme and derived analytical expressions to predict its performance. Further, the proposed scheme was implemented in a 5 node SDR system and its measured performance showed very good agreement with the analytical derived results. The conducted measurements show that such a system can achieve more than twice the throughput of a conventional slotted Aloha scheme, while maintaining stability for a much wider range of arrival rates and contention probabilities. This indicates that Aloha-CR might be an excellent option for system deployments that can afford some extra complexity on the access point, while requiring low transmitter complexity (compared to other collision resolution schemes) to meet power or pricing requirements.
APPENDIX I PROOF OF THE PROPOSITION 1
Proof :
Let f (x) be the pdf of the relative delay τ between the two users. The probability that the collision is not resolvable is:
P c = ∞ n=−∞ ∆/2+nTs −∆/2+nTs f (x)dx,(16)
where ∆ is some number smaller that T representing the smallest distance between the peaks of the two users that still allows the users to be resolved.
In (16) n runs from −∞ to ∞, because when the relative delay τ is increased by by nT s , n ∈ Z, the channel channel matrix A remains the same.
Because the intentional delays are uniformly distributed in [0, T ], the pdf of α is :
f α (x) = 1/T − |x|/T 2 if |x| ≤ T 0 otherwise(17)
Since τ = α + δ, The PDF of τ is:
f (x) = T −T f α (v)f δ (x − v)dv = T 0 ( 1 T − v T 2 )f δ (x − v)dv + 0 −T ( 1 T + v T 2 )f δ (x − v)dv(18)
Substituting (18) into (16), the probability of collision can be represented as:
P c = C T 0 ( 1 T − v T 2 )f δ (x − v)dvdx + C 0 −T ( 1 T + v T 2 )f δ (x − v)dvdx(19)
where
C dx = ∞ n=−∞ ∆/2+nTs
−∆/2+nTs dx. Now P c is a function of T . Taking the first order derivative of P c with respect to T , we have
dP c dT = 1 T 2 C T 0 ( 2v T − 1)f δ (x − v)dvdx Ξ1 + 1 T 2 C T 0 ( 2v T − 1)f δ (x + v)dvdx Ξ2 .(20)
Next we will show dPc
dT | T =Ts = 0. Defining Φ(x) = x −∞ f δ (v)dv and b a f δ (v)dv = Φ(a) − Φ(b), we get Ξ 1 | T =Ts = 1 T 2 s Ts 0 ( 2v T s − 1) ∞ n=−∞ Φ(∆/2 + nT s − v) − Φ(−∆/2 + nT s − v)dv = 1 T 2 s Ts/2 −Ts/2 2u T s ∞ n=−∞ Φ(∆/2 + nT s − u − T s /2) − Φ(−∆/2 + nT s − u − T s /2) φ(u)
du (21) where u = v − T s /2. As 2u/T s is an odd function in [−T s /2, T s /2], if φ(u) is an even function of u,
then Ξ 1 | T =Ts = 0. Indeed, since Φ(x) = 1 − Φ(−x)
, it can be easily seen that φ(−u) = φ(u). Similarly we can show that Ξ 2 | T =Ts = 0. Thus dPc dT | T =Ts = 0. Next we show that d 2 Pc dT 2 | T =Ts > 0.
d 2 P c dT 2 = dΞ 1 dT + dΞ 2 dT (22) dΞ 1 dT = −6 T 4 C T 0 vf δ (x − v)dvdx + 2 T 3 C T 0 f δ (x − v)dvdx + 1 T 2 C f δ (x − T )dx = − 2 T Ξ 1 + 1 T 2 C f δ (x)dx − 2 T 4 C T 0 vf δ (x − v)dvdx,(23)where C f δ (x)dx = C f δ (x − T )dx. Let us assume that C f δ (x)dx > 1
Ts
Ts 0
C f δ (x − v)
dxdv. This means that the probability for non resolvable collisions when we do not introduce any intentional random delays to any user is larger that that when we only introduce any intentional random delays to one of the two users involved in the collision. The is intuitively correct, and was further confirmed in our testbed.
By applying Ξ 1 | T =Ts = 0, we get
dΞ 1 dT | T =Ts > 1 T 3 s Ts 0 C f δ (x − v)dxdv − 2 T s C Ts 0 vf δ (x − v)dvdx = − Ξ 1 | T =Ts T s = 0.(24)
Similarly, we can prove that dΞ2 dT | T =Ts > 0, which leads to d 2 Pc dT 2 | T =Ts > 0. Thus we has shown that if we assign an intentional delayτ k to each user that is uniformly distributed in [0, T s ], the collision probability achieves a local minimum value.
APPENDIX II
PROOF OF THE PROPOSITION 2
Recall that for a single transmission the probability of successful reception is P 0 ; for order two collision the probability of receiving correctly both of the transmission messages is P 1 and the probability of successfully receiving only one of the two transmitting messages is P 2 . The throughput of this network is :
C(p) = P 0 Jp(1 − p) J−1 + (2P 1 + P 2 ) J 2 p 2 (1 − p) J−2 = P 0 Jp(1 − p) J−1 + (P 1 + P 2 2 ) P ′ J(J − 1)p 2 (1 − p) J−2(25)
Let us find the value of p that maximizes throughput. Taking the derivative of C(p) with respect to p,
we get dC(p) dp = J(1 − p) J−3 [P 0 (1 − p) 2 − (P 0 − 2P ′ )(J − 1) a p(1 − p) − P ′ (J − 1)(J − 2) b p 2 ].(26)
Forcing dC(p)/dp = 0, besides a trivial solution at p = 1, we get two zeros at
p * 1 = 2P0+a− √ a 2 +4P0b 2(P0+a−b) = 2P0 2P0+a+ √ a 2 +4P0b p * 2 = 2P0+a+ √ a 2 +4P0b 2(P0+a−b) = 2P0 2P0+a− √ a 2 +4P0b .(27)
Because b > 0, then a− √ a 2 + 4P 0 b < 0. The possible range for p * 2 is (−∞, 0) or (1, ∞), which violates the requirement that 0 < p < 1. Hence only p * 1 is the valid solution. Moreover it is easy to see that when 0 < p < p * 1 , dC(p)/dp > 0, while p * 1 < p < 1, dC(p)/dp < 0. Thus C(p) is maximized when p = p * 1 . As a and b are related to J, defining η(J) = 2P 0 + a + √ a 2 + 4P 0 b, we get p * 1 = 2P 0 /η(J),
lim J→∞ J η(J) = 1 P 0 − 2P ′ + P 2 0 + 4P ′2 ,(28)
Proposition 1 :
1Let the intentional delays be uniformly distributed over some interval [0, T ]. If T = T s , the probability of the collision being non-resolvable achieves a local minimum, independent of f δ (x).
Fig. 2
2shows an experimental configuration where a single host computer controls five nodes. The host computer acts as the BS and controls all the nodes in order to provide the correct synchronization between the transmitters and the receiver. The separation between nodes 1 and 2 was about 5 meters, and the separation between the node 3 and 4, and nodes 4 and 5 was also about 5 meters. The separation between nodes 1 and 3, and nodes 2 and 5 was about 10 meters. All the nodes transmitted narrowband signals simultaneously, both using the same carrier frequency 2.447 GHz (channel N0.8 of 802.11b WiFi channel).
•
Introducing user delays -A random number of zero samples, chosen uniformly in [0, 32], was added in the beginning of the payload. • Transmission -The signal was first up-converted to 5 MHz and sent to the transmission buffer. The board used channel 8 of the IEEE 802.11 standard to transmit the signal, i.e., the carrier frequency was 2.44GHz.
[ 57 ,
5762, 67, 72] taps of each received pulse. The resulting 4 polyphase components were input to the JADE algorithm for source separation. The output of the JADE algorithm was input to a PLL. The results were 4 sequences, i.e., s u1 (i), s u1 (i+1), s u2 (i), s u2 (i+1), within phase and delay ambiguities.
. 2 .
2Node 5 had the role of the BS and all the other nodes were trying to communicate with it. The transmitted messages consisted of random bits and the length was 414 before coding. Upon reception, the message was decoded and the transmission was considered successful if there were no bits in error. Each node had an independentBernoulli arrival process of rate r, which resulted in a system arrival rate of 4r. For the measurements, so as to measure the system performance in various loads. The contention probability for each of the above arrival rates took values in [0.05, 0.95] with steps of 0.05.
+4P
′2 , (30) which gives us the asymptotical throughput when then number of users increases.
Fig. 3 .
3BER performance of different separation scheme
Fig. 7 .
7Active Probability vs contention probability of Aloha-CR for different arrival rates(4 users)
Fig. 8 .
8Throughput of Aloha-CR vs contention probability for different arrival rates (
Fig. 9 .
9Total delay of Aloha-CR vs contention probability for different arrival rates(4 users)
Fig. 10 .
10Service delay of Aloha-CR vs contention probability for different arrival rates(4 users)
: Bold capitals denote matrices. Bold lower cases denote vectors. H denotes transpose conjugate. The subscript T denotes transpose. The subscript † denotes pseudo-inverse. · F denotes Frobenius-norm. Diag(v) denotes the diagonal matrix with diagonal elements v. ⌈·⌉ denotes rounding up to the nearest integer. II. ALOHA-CR: PHYSICAL LAYER
Throughput Throughput :
ThroughputMeasurements vs Analysisr=1/32 An
r=1/32 Meas
r=1/16 An
r=1/16 Meas
r=1/8 An
r=1/8 Meas
r=1/4 An
r=1/4 Meas
r=1/2 An
r=1/2 Meas
Total Delay : Measurements vs Analysisr=1/32 An
r=1/32 Meas
r=1/16 An
r=1/16 Meas
r=1/8 An
r=1/8 Meas
r=1/4 An
r=1/4 Meas
Service Delay : Measurements vs Analysisr=1/32 An
r=1/32 Meas
r=1/16 An
r=1/16 Meas
r=1/8 An
r=1/8 Meas
r=1/4 An
r=1/4 Meas
r=1/2 An
r=1/2 Meas
This work has been supported by the Office of Naval Research under Grant ONR-N-00014-07-1-0500.September 16, 2009 DRAFT
September 16, 2009 DRAFT
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| []
|
[]
| [
"Luca Sanguinetti \nAlcatel-Lucent Chair\nEcole supérieure d'électricité (Supélec)\nGif-sur-YvetteFrance\n",
"Aris L Moustakas ⋆ \nDepartment of Physics\nNational & Capodistrian University of Athens\nAthensGreece\n\nAlcatel-Lucent Chair\nEcole supérieure d'électricité (Supélec)\nGif-sur-YvetteFrance\n",
"Emil Björnson \nDepartment of Signal Processing\nACCESS Linnaeus Centre\nKTH Royal Institute of Technology\nStockholmSweden\n\nAlcatel-Lucent Chair\nEcole supérieure d'électricité (Supélec)\nGif-sur-YvetteFrance\n",
"♦ ",
"\nMérouane Debbah ‡ † Dipartimento di Ingegneria dell'Informazione\nUniversity of Pisa\nPisaItaly\n"
]
| [
"Alcatel-Lucent Chair\nEcole supérieure d'électricité (Supélec)\nGif-sur-YvetteFrance",
"Department of Physics\nNational & Capodistrian University of Athens\nAthensGreece",
"Alcatel-Lucent Chair\nEcole supérieure d'électricité (Supélec)\nGif-sur-YvetteFrance",
"Department of Signal Processing\nACCESS Linnaeus Centre\nKTH Royal Institute of Technology\nStockholmSweden",
"Alcatel-Lucent Chair\nEcole supérieure d'électricité (Supélec)\nGif-sur-YvetteFrance",
"Mérouane Debbah ‡ † Dipartimento di Ingegneria dell'Informazione\nUniversity of Pisa\nPisaItaly"
]
| []
| In this work, we consider the downlink of a single-cell multi-user multiple-input multiple-output system in which zero-forcing precoding is used at the base station (BS) to serve a certain number of user equipments (UEs). A fixed data rate is guaranteed at each UE. The UEs move around in the cell according to a Brownian motion, thus the path losses change over time and the energy consumption fluctuates accordingly. We aim at determining the distribution of the energy consumption. To this end, we analyze the asymptotic regime where the number of antennas at the BS and the number of UEs grow large with a given ratio. It turns out that the energy consumption is asymptotically a Gaussian random variable whose mean and variance are derived analytically. These results can, for example, be used to approximate the probability that a battery-powered BS runs out of energy within a certain time period. | 10.1109/icassp.2014.6854502 | [
"https://arxiv.org/pdf/1403.1596v2.pdf"
]
| 12,959,529 | 1403.1596 | 9287c3f916011e07711797f153ee875bcd3ff47c |
IMPACT OF USER MOBILITYCopyright IMPACT OF USER MOBILITY19 Mar 2014
Luca Sanguinetti
Alcatel-Lucent Chair
Ecole supérieure d'électricité (Supélec)
Gif-sur-YvetteFrance
Aris L Moustakas ⋆
Department of Physics
National & Capodistrian University of Athens
AthensGreece
Alcatel-Lucent Chair
Ecole supérieure d'électricité (Supélec)
Gif-sur-YvetteFrance
Emil Björnson
Department of Signal Processing
ACCESS Linnaeus Centre
KTH Royal Institute of Technology
StockholmSweden
Alcatel-Lucent Chair
Ecole supérieure d'électricité (Supélec)
Gif-sur-YvetteFrance
♦
Mérouane Debbah ‡ † Dipartimento di Ingegneria dell'Informazione
University of Pisa
PisaItaly
IMPACT OF USER MOBILITY19 Mar 2014arXiv:1403.1596v2 [cs.IT]
In this work, we consider the downlink of a single-cell multi-user multiple-input multiple-output system in which zero-forcing precoding is used at the base station (BS) to serve a certain number of user equipments (UEs). A fixed data rate is guaranteed at each UE. The UEs move around in the cell according to a Brownian motion, thus the path losses change over time and the energy consumption fluctuates accordingly. We aim at determining the distribution of the energy consumption. To this end, we analyze the asymptotic regime where the number of antennas at the BS and the number of UEs grow large with a given ratio. It turns out that the energy consumption is asymptotically a Gaussian random variable whose mean and variance are derived analytically. These results can, for example, be used to approximate the probability that a battery-powered BS runs out of energy within a certain time period.
INTRODUCTION
The data traffic in cellular networks has increased exponentially for a long time and is expected to continue this trend, at least for the next five years [1]. Currently, one of the biggest challenges related to the traffic growth is the increasing energy consumption of the cellular infrastructure equipments [2]. This means that the energy consumption must be taken into account from the very beginning when designing cellular networks of the future. This is particularly important when deploying BSs in new rural regions of the world, where the electrical grid is unreliable or even non-existing. Off-grid deployments rely on combinations of diesel generators, batteries, and local energy harvesting (e.g., from solar panels) [2]. Since the supply of energy is either limited or fluctuates with the harvesting, it is of paramount importance to operate the BS such that it will not run out of energy, also known as power outage.
Most of the existing works dealing with the development of energy-efficient transmission schemes rely on extensive Monte-Carlo simulations that do not provide valuable insights on the interplay between the different system parameters and the impact of the propagation environment. To partially bridging this gap, in [3] the authors make use of stochastic geometry to model the energy consumption of a cellular network where each UE is connected to its closest BS equipped with a single antenna. The energy consumption is expressed as a function of the distance between BSs and UEs, while taking into account the interference from other BSs. In [4], the authors go a step further in this direction when considering a refined energy consumption model that includes the energy of broadcast messages, traffic activity, and user mobility. Instead of relying on simulation results, as was done in [3], closed-form formulas for different statistical properties of the energy consumption are derived. From a network deployment perspective, [5] shows how the number of BS antennas, number of active UEs, and the data rates can be analytically optimized for high energy efficiency. This is achieved using a refined energy consumption model where the three optimization variables appears explicitly.
In this work, we consider the downlink of a single-cell multiuser multiple-input multiple-output (MIMO) system. The BS is equipped with an array of N antennas and serve K UEs simultaneously by using zero-forcing (ZF) precoding. The UEs are assumed to move around in the cell according to a Brownian motion model. Based on this mobility pattern, we aim at determining the statistical distribution of the energy consumption required to guarantee a given data rate at all UEs. By considering the asymptotic regime where K and N grow large with a given ratio, we prove that energy consumption statistics converge in distribution to a Gaussian random variable whose mean and variance can be analytically derived using random matrix theory tools and standard central limit theoretic results. It turns out in the large limit the variance of the energy consumption is dominated by the fluctuations induced by user mobility. The analytical expressions are shown to closely match the numerical results for different settings. Finally, we exemplify how the new statistical characterization can be used to characterize the probability that a battery-powered BS runs out of energy within a certain time period.
The following notation is used throughout this work. Ez[·], COVz[·] and VARz[·] indicate that the expectation, covariance and variance are respectively computed with respect to z. The notation || · || stands for the Euclidean norm whereas Jn(·) denotes the norder Bessel function. We call IK the K × K identity matrix and δ(t) the Dirac delta function. We denote by Q(z) the Gaussian tail function and use Q −1 (z) to indicate its inverse.
SYSTEM AND SIGNAL MODEL
We consider the downlink of a single-cell multi-user MIMO system in which the BS makes use of N antennas to communicate with K single-antenna UEs. The K active UEs change over time and are randomly selected from a large set of UEs that are moving around within the coverage cell A of area A. We assume that the user den-sity ν and the number of UEs K can increase arbitrarily, while the area A is maintained fixed. This amounts to saying that K/ν is constant and equal to A. To simplify the computations, we assume that the UEs are uniformly distributed in a circular cell with radius R such that A = πR 2 and adopt a Brownian motion (or random walk mobility) model with diffusion constant D and constrained in the circular region A [6]. The location of UE k at time t is denoted by x k (t) ∈ R 2 . The BS is located in the centre of the cell and its N transmit antennas are adequately spaced apart such that the channel components to any UE are uncorrelated. We assume that N increases as K becomes larger while the ratio K/N is kept constant and equal to c with 0 < c < 1. Perfect channel state information is assumed to be available at the BS and the same rate is guaranteed to each UE [5].
We call s(t) ∈ C N×1 the signal transmitted at (slotted) time t and denote by G(t) ∈ C N×N its precoding matrix. We assume that s(t) originates from a Gaussian codebook with zero mean and covariance matrix Es[s(t)s H (t)] = IK . Letting y(t) ∈ C K×1 be the vector collecting the samples received at the UEs, we may write
y(t) = H(t)G(t)s(t) + n(t)(1)
where n(t) ∈ C K×1 is a circularly-symmetric complex Gaussian random vector with zero-mean and covariance matrix σ 2 IK and H(t) ∈ C K×N is the channel matrix at time t. The (k, n)th entry [H(t)] k,n accounts for the channel propagation coefficient between the nth antenna at the BS and the kth UE. In particular, we assume
[H(t)] k,n = g(x k (t)) [W(t)] k,n(2)
where g(·) is the path-loss function and the entries [W(t)] k,n account for the fast fading component and are modelled as independent and identically distributed circularly-symmetric complex Gaussian random variables with zero-mean and unit variances, i.e., [W(t)] k,n ∼ CN (0, 1). The temporal correlations of W(t) are modelled according to the Jakes model [7]. We assume that
g(x k (t)) = 1 x k (t) β + r β 0(3)
with β and r0 being the path-loss exponent and some cutoff parameter, respectively. As seen, g(x k (t)) is assumed to be independent over n. This is a reasonable assumption since the distances between UEs and BS are much larger than the distance between the antennas. For analytical convenience, we consider the ZF precoding matrix
G(t) = √ ρ H H (t) H(t)H H (t) −1(4)
where ρ is a design parameter. Substituting G(t) into (1) the achievable data rate of the kth UE is
r k = log 2 1 + ρ σ 2 .(5)
Note that we assume that the same rate is achieved by each UE [5].
The extension to the case in which different rates are required by different UEs can be easily handled by combining ZF with a proper power allocation [8].
The power consumption P (t) = Es[||G(t)s(t)|| 2 ] at time t is given by
P (t) = ρ tr H(t)H H (t) −1(6)
while the energy consumption ET for a given time interval
[0, T ] is ET = T 0 P (t)dt = T 0 ρ tr H(t)H H (t) −1 dt.(7)
MAIN RESULTS
The energy ET is clearly a random quantity, which depends (through H(t)) on the realizations of W(t) as well as on the user positions {x k (t)} throughout the period 0 ≤ t ≤ T . We aim at determining the statistics of ET in the large system limit, i.e., K, N → ∞ with K/N = c. For notational convenience, we denote by ki the ith zero of the first derivative of J1(·) and call
φi = 2 1 0 J0(kit)t β+1 dt.(8)
Observe that the values of {ki} and {φi} can be calculated explicitly as illustrated in [9].
The following theorem summarizes the main results of this work.
Theorem 1 In the large system limit, if ZF precoding is used then the following convergence holds true:
ET − E [ET ] VAR [ET ] −→ K,N→∞ N (0, 1)(9)
where the mean is given by
E [ET ] = T ρcR β 1 − c 2 β + 2 + r β 0 R β(10)
while the variance depends on the user mobility model and takes the form
VAR [ET ] = T R 2 DK ρ 2 c 2 R 2β (1 − c) 2 Θ + O(K −2 ) (11) with Θ = ∞ i=1 2φ 2 i k 2 i J 2 0 (k i ) 1 0 (1 − e − k 2 i DT t R 2 ) 2 dt.
Sketch of proof
The complete proof of Theorem 1 is omitted for space limitations.
In the sequel, we describe the main steps. We begin by observing that when K, N → ∞ with 0 < c < 1, the average of P (t) with respect to the fast fading channel W(t) hardens to a deterministic scalar given by [10,11]
EW [P (t)] − ρc 1 − c 1 K K k=1 1 g(x k (t)) → 0.(12)
Since the path-loss functions g(x k (t)) are independent of each other, from (12) it follows that in the large system limit
Ex,W [P (t)] − ρc 1 − c Ex 1 g(x k (t)) → 0(13)= A1 + A2 with A1 = T 0 T 0 Ex COVW P (t), P (t ′ ) dtdt ′ (14) A2 = T 0 T 0 COVx EW [P (t)] , EW P (t ′ ) dtdt ′ . (15)
Let us start with the computation of A1. For this purpose, we observe that according to the Jakes model [7], the correlation time τ d = 1/f d (with f d being the Doppler frequency) is exceedingly small compared to the typical times related with the UE movements. This means that we can reasonably assume that the power is δ-correlated or, equivalently, that COVW [P (t),
P (t ′ )] = VARW [P (t)] τ d δ(t − t ′ )
. As a result, the fluctuation term A1 in the energy can be approximated as
A1 ≈ T τ d Ex [VARW [P (t)]].
From [12], using (3) we obtain
Ex [VARW [P (t)]] − ρ 2 c 3 (1 − c) 3 T τ d K 2 Ex 1 g 2 (x k (t)) → 0 (16) with Ex 1 g 2 (x k (t)) = R 2β 2β + 2 + 4r β 0 R β β + 2 + r 2β 0(17)
as obtained using (3). Plugging the above results into (14) reveals that A1 decreases proportionally to K −2 when K → ∞.
We are now left with the computation of A2 in (15), which is due to the fluctuations induced by the UE mobility in the cell. In particular, it turns out that
A2 = ρ 2 c 2 K(1 − c) 2 T 0 T 0 COVx x(t) β , x(t ′ ) β dtdt ′ .(18)
According to the Brownian motion model, averaging over the initial positions of the UE (or random walkers in the Brownian motion parlance) yields
COVx x(t) β , x(t ′ ) β = (19) x,x ′ ∈A x β x ′ β F (x, x ′ ; t − t ′ ) − F (x, x ′ ; t + t ′ ) dxdx ′ with F (x, x ′ ; t)
being the probability that a random walker at position x ′ reaches position x at time t. Denote now by ξ the time of each step of the random walker (or, equivalently, the "forgetting time"-the time after which the walker forgets his original direction) and call ℓ the spatial length of each step. For values of t much larger than ξ, F (x, x ′ ; t) can be obtained in the continuum limit by solving a diffusion equation, whose corresponding diffusion constant turns out to be equal to D = ℓ 2 /(4ξ). The diffusion equation can be solved very simply in the circular domain by providing the proper initial condition F (x, x ′ , t = 0) = δ(x − x ′ ). To impose the condition that the random walker does not exit the domain, we need to set the derivative along the radial direction at the boundary equal to zero, i.e.,r T ∇ F (x, x ′ )| x =R = 0 withr being the unit vector along the radial direction. As a result, an eigenfunction expansion for F (x, x ′ ) can be obtained and used to compute (19) from which using (18) the result in (11) follows.
In writing (11) we have taken into account that A1 is ∝ τ d T R 2β /K 2 while A2 is ∝ T R 2β+2 /(DK). This difference in scaling arises from two different effects. Firstly, the extra factor of K −1 in the former is due to the fact that the singular values of W(t) are strongly correlated, while the positions of the UEs are independent. Secondly, the decorrelation time for the fading is τ d , while that for the user mobility is R 2 /D, which is much larger than τ d . Hence, both factors make the variability of the fast fading channel less important for the analysis.
From the results of Theorem 1, it is seen that both the mean and the variance of ET (at least for long enough time interval T such that DT ≫ R 2 ) are proportional to T . This means that the variability of energy consumption will be less important as T becomes larger.
Finally, it can be shown that all higher-order moments of ET asymptotically vanish for large values of K. In particular, tools from random matrix theory [10,11] can be used to control the variations of the fast fading channel while standard central limit theoretic results are needed to prove the convergence of the fluctuations arising from user movements.
Numerical Validation
The accuracy of the results in Theorem 1 are now validated numerically by Monte-Carlo simulations for 10000 different initial positions of the UEs. For illustration purposes, we simply set R = 1, r0 = 0.1, ρ = 1, ℓ = 0.05 and ξ = ℓ 2 = 0.0025. (10) and (11), respectively, are given in Table 1 for completeness. From Figs. 2 and 3, it is seen the numerical results match very well with the theoretical ones for all the investigated scenarios. The small discrepancy observed for T = 2 vanishes if a smaller step is used in the random walk model.
Application: Dimensioning of Cell Batteries
A possible application of the results of Theorem 1 is as follows. Assume that the energy level η of a battery-powered BS has to be designed such that the achievable rate of each UE is r k = log 2 1 + ρ/σ 2 and the probability of running out of energy (before replacement or reloading) is smaller than some given threshold ǫ. Mathematically, this amounts to saying that Pr (ET > η) ≤ ǫ. From the results of Theorem 1, we have that
Pr (ET > η) = Q η − E [ET ] VAR [ET ](20)
from which it follows that
η ≥ VAR [ET ]Q −1 (ǫ) + E [ET ] .(21)
Consider for example a cell operating over a bandwidth B = 10 MHz with a radius R = 1 km and a cut-off parameter r0 = 50 m. Assume K = 32, N = 64 and β = 4. If the noise power is σ 2 = 10 −14 W/Hz and ρ is chosen equal to 3 × 10 −14 W/Hz, then the achievable data rate of each UE is r k = log(1 + ρ/σ 2 ) = 2 bit/s/Hz. Set ℓ = 10 m and ξ = 1 minute such that D = ℓ 2 /(4ξ) = 25 m 2 /minute. Assume that the replacing (or recharging) time T is 1 day. In the above circumstances, from (10) and (11)
Notice that the condition that ǫ = 1% makes the necessary battery level η be substantially higher than E [ET ]. It is also worth observing that the above value accounts only for the energy required to transmit the signal s(t) within the time interval T . The design of the battery level must also take into account the energy required for digital signal processing, channel coding and decoding, channel estimation and pre-coding, and so forth (see [5] for more details). However, all these quantities can be somehow quantified off-line and easily added to η for a correct design.
CONCLUSIONS
In this work, we have studied the energy consumption dynamics in the large limit for a MIMO cellular network in which the UEs move around according to a Brownian motion model. In particular, we have shown that the energy consumption converges in distribution to a Gaussian random variable and we have computed its mean and variance analytically. We have shown that user mobility plays a key role in determining the fluctuations of energy consumption. Numerical results have been used to show that the analytical expressions yield accurate approximations for different settings. As an application of these results, we have dimensioned a battery-powered BS so as to satisfy a certain probability of running out of energy. The results of this work could also be used to get some insights on how designing the number of cells required to cover a given area while taking into account the implementation costs.
It is worth observing that the simplicity of the analytical results comes from the adoption of a ZF precoding technique at the BS. The use of different techniques (such as the regularized ZF) makes the computations much more involved and is currently under investigation. The way different user mobility models impact energy consumption dynamics is also an ongoing research activity.
Fig. 1 .
1Histogram of (ET − E [ET ])/ VAR [ET ] in comparison with the normal distribution N (0, 1) for T = 10, K = 16, N = 32 and β = 4.
Fig. 1
1depicts the histogram of (ET − E [ET ])/ VAR [ET ] when T = 10, K = 16, N = 32 and β = 4. Comparisons are made with the Gaussian distribution N (0, 1). The good match between the two curves validates the results of Theorem 1. Figs. 2 and 3 show the cumulative distribution function (CDF) of ET /T for different values of K, N , T and β. The values of E [ET ] and VAR [ET ] as obtained with
FigFig. 2 .
2Outage probability Pr E T T > α for T = 2 and T = 10 with K = 16, β = 4 and N = 32 or 64.
Fig. 3 .
3Outage probability Pr E T T > α for T = 2 and T = 10 with K = 64, N = 128 and β = 4 or 6.
The variance of the energy ET can be rewritten (using the covariance decomposition formula) as VAR[ET ] from which using (3) the result in (10) easily follows. Observe that
the exceedingly simple form in (10) hold true only for ZF precoding
technique. It becomes more involved if other precoding techniques
(such as the regularized ZF) are considered.
Table 1 .
1Numerical values of E [ET ] and VAR [ET ] for the parameter setting in Figs. 2 and 3
we obtain E [ET ] = 1.73 × 10 3 J and VAR [ET ] = 5.65 × 10 4 J 2 .Plugging these values into (21) and setting ǫ = 1% shows that the battery level must satisfy η ≥ 2.28 × 10 3 J.
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"G P Benham \nLadHyX\nUMR CNRS 7646\nEcole polytechnique91128PalaiseauFrance\n",
"R Gonzalez-Farina \nMathematical Institute\nUniversity of Oxford\nAndrew Wiles Building\n\nRadcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom\n",
"K B Kiradjiev \nMathematical Institute\nUniversity of Oxford\nAndrew Wiles Building\n\nRadcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom\n",
"W T Lee \nSchool of Computing and Engineering\nUniversity of Huddersfield\nHD1 3DHQueensgate, HuddersfieldUnited Kingdom\n\nDepartment of Mathematics and Statistics\nMACSI\nUniversity of Limerick\nLimerickIreland\n",
"S Tibos \nPepsiCo International\n4 Leycroft RoadLE4 1ETLeicesterUnited Kingdom\n",
"T Babb \nMathematical Institute\nUniversity of Oxford\nAndrew Wiles Building\n\nRadcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom\n",
"G P Benham \nLadHyX\nUMR CNRS 7646\nEcole polytechnique91128PalaiseauFrance\n",
"R Gonzalez-Farina \nMathematical Institute\nUniversity of Oxford\nAndrew Wiles Building\n\nRadcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom\n",
"K B Kiradjiev \nMathematical Institute\nUniversity of Oxford\nAndrew Wiles Building\n\nRadcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom\n",
"W T Lee \nSchool of Computing and Engineering\nUniversity of Huddersfield\nHD1 3DHQueensgate, HuddersfieldUnited Kingdom\n\nDepartment of Mathematics and Statistics\nMACSI\nUniversity of Limerick\nLimerickIreland\n",
"S Tibos \nPepsiCo International\n4 Leycroft RoadLE4 1ETLeicesterUnited Kingdom\n"
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"Mathematical Institute\nUniversity of Oxford\nAndrew Wiles Building",
"Radcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom",
"Mathematical Institute\nUniversity of Oxford\nAndrew Wiles Building",
"Radcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom",
"School of Computing and Engineering\nUniversity of Huddersfield\nHD1 3DHQueensgate, HuddersfieldUnited Kingdom",
"Department of Mathematics and Statistics\nMACSI\nUniversity of Limerick\nLimerickIreland",
"PepsiCo International\n4 Leycroft RoadLE4 1ETLeicesterUnited Kingdom",
"Mathematical Institute\nUniversity of Oxford\nAndrew Wiles Building",
"Radcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom",
"LadHyX\nUMR CNRS 7646\nEcole polytechnique91128PalaiseauFrance",
"Mathematical Institute\nUniversity of Oxford\nAndrew Wiles Building",
"Radcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom",
"Mathematical Institute\nUniversity of Oxford\nAndrew Wiles Building",
"Radcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUnited Kingdom",
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| When frying potato snacks, it is typically observed that the dough, which is submerged in hot oil, after some critical time increases its buoyancy and floats to the surface. The lift-off time is a useful metric in ensuring that the snacks are properly cooked. Here we propose a multiphase mathematical model for the frying of potato snacks, where water inside the dough is evaporated from both the top and bottom surfaces of the snack at two receding evaporation fronts. The vapour created at the top of the snack bubbles away to the surface, whereas the vapour released from the bottom surface forms a buoyant blanket layer. By asymptotic analysis, we show that the model simplifies to solving a one-dimensional Stefan problem in the snack coupled to a thin-film equation in the vapour blanket through a non-linear boundary condition. Using our mathematical model, we predict the change in the snack density as a function of time, and investigate how lift-off time depends on the different parameters of the problem. | 10.1137/19m1294253 | [
"https://arxiv.org/pdf/1910.04458v1.pdf"
]
| 204,008,640 | 1910.04458 | a1ac36e779582da6b0ef3ad78fca9f8fb3430091 |
Predicting lift-off time when deep-frying potato dough snacks
T Babb
Mathematical Institute
University of Oxford
Andrew Wiles Building
Radcliffe Observatory Quarter
Woodstock RoadOX2 6GGOxfordUnited Kingdom
G P Benham
LadHyX
UMR CNRS 7646
Ecole polytechnique91128PalaiseauFrance
R Gonzalez-Farina
Mathematical Institute
University of Oxford
Andrew Wiles Building
Radcliffe Observatory Quarter
Woodstock RoadOX2 6GGOxfordUnited Kingdom
K B Kiradjiev
Mathematical Institute
University of Oxford
Andrew Wiles Building
Radcliffe Observatory Quarter
Woodstock RoadOX2 6GGOxfordUnited Kingdom
W T Lee
School of Computing and Engineering
University of Huddersfield
HD1 3DHQueensgate, HuddersfieldUnited Kingdom
Department of Mathematics and Statistics
MACSI
University of Limerick
LimerickIreland
S Tibos
PepsiCo International
4 Leycroft RoadLE4 1ETLeicesterUnited Kingdom
Predicting lift-off time when deep-frying potato dough snacks
When frying potato snacks, it is typically observed that the dough, which is submerged in hot oil, after some critical time increases its buoyancy and floats to the surface. The lift-off time is a useful metric in ensuring that the snacks are properly cooked. Here we propose a multiphase mathematical model for the frying of potato snacks, where water inside the dough is evaporated from both the top and bottom surfaces of the snack at two receding evaporation fronts. The vapour created at the top of the snack bubbles away to the surface, whereas the vapour released from the bottom surface forms a buoyant blanket layer. By asymptotic analysis, we show that the model simplifies to solving a one-dimensional Stefan problem in the snack coupled to a thin-film equation in the vapour blanket through a non-linear boundary condition. Using our mathematical model, we predict the change in the snack density as a function of time, and investigate how lift-off time depends on the different parameters of the problem.
Introduction
Frying is one of the most common and oldest forms of food cooking. Frying has multiple functions including to sterilise, dehydrate and create product texture [1]. Generally there are two types of frying: shallow-fat frying and deep-fat frying. Here we focus on deep fat frying in which the food product being cooked is fully immersed in the oil. During deep-fat frying, some food products undergo density changes that cause them to rise within the oil bath. This process can be exploited in food manufacturing, either as a way of determining the stage of cooking, or as a mechanism to collect the food from the hot oil. For example, in the production of potato snacks, uncooked snacks are submerged in hot oil by a conveyor belt; as the dough cooks they become buoyant and then detach from the conveyor belt. This must happen at precisely the right moment in order to maximise product quality and the productivity of the process. To ensure that the snacks robustly detach at the right time, a better quantitative understanding of the underlying mechanism is needed. In particular, there are two major contributions to buoyancy due to the generation of steam, which either escapes from the snack causing a reduction in density, or becomes trapped underneath the snack in a vapour blanket.
Several different mathematical modelling approaches can be found within the food frying literature. For a comprehensive summary of all relevant types of models, see [13]. Many of these emphasise transport mechanisms of gases and liquids in porous media [6,8,13,14,20]. A commonly used modelling approach is the crust-core model, in which there are two regions: a dry crust where the water has evaporated, and a wet core. In the crust-core approach, mass and energy equations are used to describe the heat and flow in each region, and a moving boundary tracks the evaporation front at the crust-core interface. One major concern in the deep-fat frying literature is oil uptake into the snack, and several experiments have been gathered regarding this issue [5,10,15,16,21]. However, most of these models focus on the oil absorbtion post-frying, since this is when most of the oil (approximately 80%) enters the snack [14,20]. Another important objective of many of these studies is to predict quality changes (puffiness, shrinkage, etc) in the snacks as they fry [10,13,20]. Some models also account for the decrease in the temperature of the oil due to moisture loss from the chip [6,13].
A dominant feature of the frying process is the evaporation of the water, which can be observed both from bubbles rising to the surface, and in a vapour layer surrounding the snack. Despite the formation of a vapour blanket being mentioned in several papers (see for instance [8] where it is stated that the bubbles impede oil inflow through the bottom boundary) this process has not been described in mathematical terms before within the deep-fat frying literature. In other contexts, film boiling has been studied and expressions for the vapour layer thickness have been derived, as well as bubble generation and release frequencies [4,7]. However, none of the above studies address the density changes undergone due to the formation of the vapour blanket, and lift-off is not investigated at all. Furthermore, the effect of the vapour layer, which is a poor conductor, on the heat transfer in the snack is also not discussed.
In this study, we focus on predicting when a snack becomes buoyant, which happens within a few seconds of being introduced into the fryer. Thus, we do not consider structural changes, which occur later on in the frying process; or oil-uptake, which primarily occurs post-frying. We follow the crust-core modelling approach, and we introduce the novel detail of the formation of a vapour layer under the snack. We show that the timescales associated with evaporation indicate that the formation of the vapour blanket is the dominant mechanism for lift-off. We model the growth of the vapour blanket by coupling a thin film equation to the moving-boundary problem in the snack. We show that the insulating features of the vapour blanket play an important role in the dynamics of the evaporation fronts. Whilst all of the models in the above literature are solved numerically by either finite differences or finite volumes, here we combine both numerical and analytical results and compare them together. In particular, we derive closed form solutions for the long-time behaviour of the evaporation fronts and the shape of the vapour blanket, which are useful for the manufacturing process. Furthermore, we show that lift-off times are crucially dependent on the heat transfer properties of the snack.
The remainder of this paper is organised as follows. In Section 2 we introduce the nondimensional mathematical model for the thermal and flow problems within the snack and vapour blanket. By exploiting the small size of some dimensionless groups, the problem simplifies to solving an energy conservation equation for each region and a thin-film equation for the vapour blanket. A formula that relates the density of the snack to the vapour blanket thickness and the position of the evaporation fronts is also given. We first solve our model numerically in Section 3 using the enthalpy method, and we are able to identify several regimes in the frying process: a heating period, the formation of the vapour blanket, and a regime where the bubble volume is constant. Motivated by these numerical results, and considering that the Stefan number of the problem is large, in Section 4 we investigate a further simplification to the model, called the quasi-steady limit. In this limit, where the only time-dependence of the system originates from the motion of the evaporation fronts, we obtain analytical solutions that agree well with the numerical results, and provide insight to the frying behaviour. We discuss our key findings and their relevance to the snack frying process in Section 5.
z = s 2 (x, t) z = s 1 (x, t) z = 1 z = 0 z = −h(x, t)
Vapour bubbles
A Multiphase model for snack frying
In Figure 1, we illustrate the scenario considered. We focus on the two-dimensional case, as shown in the diagram, but we keep the formulation of our mathematical model in three dimensions to be as general as possible. We propose that the snack is divided into four regions, containing different combinations of dough, water and water vapour. Initially, we assume the dough to be entirely composed of a liquid (water) and solid phase (potato), which is defined as region 2 in our diagram. When the snack is introduced into the fryer, the water begins to evaporate, starting from the exterior. This creates two outer layers containing water vapour and solid, which we denote regions 1 and 3. As the water evaporates from the upper evaporation front, it is bubbled away into the surrounding oil. By contrast, water evaporating from the lower front forms a vapour layer beneath the snack, which we denote region 4.
In this section, we present a non-dimensional mathematical model for the frying of a long thin snack, which consists of energy, mass and momentum conservation equations for each of the different regions of the snack. We simplify these equations by exploiting small parameters in the system. For predicting lift-off time, we introduce a relation between the density of the snack, the size of the vapour blanket and the position of the evaporation fronts.
Mathematical model
First, we present the governing equations for each of the regions in Figure 1. We keep all the equations in non-dimensional form for convenience, but later we provide further discussion on the derivation, including a list of how each non-dimensional parameter is defined. As illustrated in the diagram, the domain is long and thin with aspect ratio ε = H/L 1. We model regions 1 and 3
using an advection-diffusion equation for the temperature, and Darcy's law for the fluid 1 St
∂T i ∂t + Pe w i ∂T i ∂z + ε 2 u i · ∇ xy T i = ∂ 2 T i ∂z 2 + ε 2 ∇ 2 xy T i , i = 1, 3,(1)u i = −∇P i , i = 1, 3,(2)0 = ∂ 2 P i ∂z 2 + ε 2 ∇ xy P i , i = 1, 3,(3)
where
T i (x i , t) is the temperature, u i (x, t) = (u i , v i , w i )
is the velocity of the fluid, and P i (x, t) is the pressure. Subscripts are used to denote the different regions and ∇ xy = ( ∂ ∂x , ∂ ∂y , 0) is the gradient in the x-y plane. Our dimensionless parameters are the Péclet number Pe, and the Stefan number St. We assume that the flow in the core region 2 of the snack is negligible, and so there is no need for any mass or momentum equations. The heat equation in this region is C St
∂T 2 ∂t = K 1 ∂ 2 T 2 ∂z 2 + ε 2 ∇ 2 xy T 2 ,(4)
where K 1 and C are the ratios of thermal conductivities and volumetric heat capacities between regions 2 and 1. In region 4, we have an advection-diffusion equation for the temperature and the Navier-Stokes equations for the fluid flow,
Pe K 2 1 τ ∂T 4 ∂t + u 4 · ∇T 4 = ∂ 2 T 4 ∂z 2 + ε 2 ∇ 2 xy T 4 ,(5)Re 1 τ ∂u 4 ∂t + u 4 · ∇u 4 = −β ∂P 4 ∂x , ∂P 4 ∂y , 1 ε 2 ∂P 4 ∂z T + ∂ 2 u 4 ∂z 2 + ε 2 ∇ 2 xy u 4 − Re Fr 2ẑ ,(6)∇ · u 4 = 0,(7)
where K 2 is a ratio of thermal conductivities between regions 4 and 1, τ is the ratio of the timescale of evaporation to the timescale of evolution of the vapour blanket z = −h, Re is the Reynolds number, Fr is the Froude number, and β is a measure of the relative size of the hydrostatic pressure of the oil acting on the gas in region 4 and the pressure drop needed to maintain the Darcy gas flux in regions 1 and 3. On the boundaries at z = 1 and z = −h, we have Newton's law of heating 1 N
∂T 1 ∂z = 1 − T 1 , z = 1(8)K 2 N 1 + ε 2 (∇ xy h) 2 ∂T 4 ∂z + ε 2 ∇ xy h · ∇ xy T 4 = T 4 − 1, z = −h,(9)
where N is the Nusselt number, measuring the ratio between heat transfer at the boundary and heat conduction in the snack. At the boundary, z = −h, we have the kinematic and dynamic boundary conditions
1 τ ∂h ∂t = w 4 − u 4 · ∇ xy h, z = −h,(10)D · n = β ε P 4 − h − κ Bo n, z = −h,(11)
where Bo is the Bond number, κ is the curvature, D is the strain rate tensor, and n is the normal to the vapour blanket, given by
κ = ∇ 2 xy h (1 + ε 2 (∇ xy h) 2 ) 3/2 ,(12)D = 2ε ∂u 4 ∂x ε 2 ( ∂v 4 ∂x + ∂u 4 ∂y ) ε 2 ∂w 4 ∂x + ∂u 4 ∂z ε 2 ( ∂u 4 ∂y + ∂v 4 ∂x ) 2ε ∂v 4 ∂y ε 2 ∂w 4 ∂y + ∂v 4 ∂z ∂u 4 ∂z + ε 2 ∂w 4 ∂x ∂v 4 ∂z + ε 2 ∂w 4 ∂y 2ε ∂w 4 ∂z (13) n = 1 1 + ε 2 (∇h) 2 −ε ∂h ∂x −ε ∂h ∂y 1 .(14)
On the evaporation fronts, z = s i for i = 1, 2, we require that the temperature matches the evaporation temperature of water
T 1 = T 2 = 0, z = s 1 ,(15)T 2 = T 3 = 0, z = s 2 .(16)
We also have a Stefan condition describing the motion of the evaporation fronts. This condition can be derived by balancing the latent energy required to vaporise water with difference in heat flux on either side of the boundary. This gives uṡ
s 1 = K 1 ∂T 2 ∂z − ∂T 1 ∂z + ε 2 ∇ xy s 1 · ∇ xy (T 1 − K 1 T 2 ), z = s 1 ,(17)s 2 = K 1 ∂T 2 ∂z − ∂T 3 ∂z + ε 2 ∇ xy s 2 · ∇ xy (T 3 − K 1 T 2 ), z = s 2 .(18)
The change in density undergone when the water vaporises creates a flow in regions 1 and 3. As discussed by [17], the equations that describe this volume change generated flow are
1 − 1 R ṡ 1 = −w 1 + ε 2 u 1 · ∇ xy s 1 , z = s 1 ,(19)1 − 1 R ṡ 2 = −w 3 + ε 2 u 3 · ∇ xy s 2 , z = s 2 ,(20)
where R is the ratio of the density of water to the density of steam. This signifies the volume change that happens when the water is vaporised, which drives the gas flow. Finally, at the interface between the snack and the vapour blanket, we have continuity of temperature, mass, pressure, and heat flux
T 3 = T 4 , P 3 = ε −2 βΓP 4 , K 2 ∂T 4 ∂z = ∂T 3 ∂z , z = 0,(21)ε 2 u 3 , ε 2 v 3 , w 3 = [u 4 , v 4 , w 4 ] z = 0,(22)
where Γ is the non-dimensional permeability of the snack. The dimensionless initial conditions are given by
T 2 (x, t) = T * ,(23)s 1 (x, y, 0) = 1,(24)s 2 (x, y, 0) = 0, (25) h(x, y, 0) = 0.(26)
In Table 1 we list the dimensionless parameters of the system, their definitions in terms of dimensional parameters, and their approximate values. The dimensional parameters appearing in Table Parameter Definition 1 are: L v , the latent heat of vaporisation of water; α, the porosity of the snack; χ, the permeability of the snack; h c , the heat transfer coefficient; γ, the interfacial tension between water and oil; g, the acceleration due to gravity; H, the height of the snack; L, the length of the snack; µ v , the viscosity of water vapour; T o , T l , T a , the temperature of the oil, the evaporation temperature of water, and the ambient air temperature of the snack before entry into the oil; and ρ o , ρ l , ρ v , the density of oil, water, and vapour, respectively. Finally, we have some compound parameters for regions 1, 2, and 3, each with a subscript denoting the relevant region. These are: ρ j , the compound density; k j the compound thermal conductivity; and c p,j , the compound specific heat capacity, with j = 1 − 4. These compound parameters have been determined by taking a volume-weighted-average of the parameters for the individual phases (solid snack, water, vapour) in each region (see for instance, [11]). For example, ρ 1 = α s ρ s + α v ρ v , where each α represents a mass fraction.
Value St L v αρ l /(ρ 1 c p,1 (T o − T e )) 8.4 C c p,2 ρ 2 /(c p,1 ρ 1 ) 2.1 ε H/L 1.1 × 10 −2 Pe c p,4 (T o − T e )/L v 5.8 × 10 −2 K 1 k 2 /k 1 1.4 K 2 k 4 /k 1 4.2 × 10 −2 τ αρ l /ρ v 5.8 × 10 2 Re k 1 (T o − T e )/(L v µ v ) 9.2 × 10 −1 β L v gρ v ρ o H 5 /(k 1 L 2 µ v (T o − T e )) 6.5 × 10 −1 Γ χ/H 2 2.0 × 10 −4 . N h c H/k 1 1.3 T * (T a − T e )/(T o − T e ) −1.1 Bo ρ o gL 2 /γ 1.1 × 10 3 R ρ l /ρ v 1.7 × 10 3 Fr (k 1 (T o − T e )/ρ v L v H 2 ) L/g 4.3
Model simplifications
Having calculated the non-dimensional parameters in Table 1, we are motivated to consider the asymptotic limit of Pe, ε, Re, Bo −1 , Pe/K 2 , Pe/K 2 τ, Re/τ, ε 2 K 1 , Re/Fr 2 , 1/R → 0.
Note that although a few of these parameter groups associated with the vapour layer (Re, Pe/K 2 ) are marginal in this scaling, we have also carried out a more complex scaling in which the thicknesses of regions 3 and 4 are scaled separately. This scaling confirms that all the dimensionless quantities listed above are small. Under these limits, the only coupling between the flow and thermal problems is through the boundary conditions (10), (11), (19) and (20). The simplified governing equations for the heat problem are 1 St
∂T 1 ∂t = ∂ 2 T 1 ∂z 2 , s 1 ≤ z ≤ 1,(28)
C St
∂T 2 ∂t = K 1 ∂ 2 T 2 ∂z 2 , s 2 ≤ z ≤ s 1 ,(29)
1 St
∂T 3 ∂t = ∂ 2 T 3 ∂z 2 , 0 ≤ z ≤ s 2 .(30)
The only dependence of the heat problem on the thickness of the vapour blanket h is through the lower boundary condition. Hence, the complete set of boundary conditions for the heat problem are 1 N
∂T 1 ∂z = 1 − T 1 , z = 1,(31)T 1 = T 2 = 0, z = s 1 ,(32)s 1 = K 1 ∂T 2 ∂z − ∂T 1 ∂z , z = s 1 ,(33)T 2 = T 3 = 0, z = s 2 ,(34)s 2 = K 1 ∂T 2 ∂z − ∂T 3 ∂z , z = s 2 ,(35)1 N ∂T 3 ∂z hN K 2 + 1 = T 3 − 1, z = 0,(36)
where equation (36) is derived by solving for T 4 and inserting the solution into (21). Specifically, T 4 is given in terms of T 3 and h by
T 4 = 1 N ∂T 3 ∂z z=0 (z + h)N K 2 + 1 + 1.(37)
In order to obtain an equation for h, we need to follow a series of steps. Firstly, taking the third component of the simplified version of (6) together with the reduced form of (11) we obtain P 4 = h throughout region 4. Now, u 4 can be found simply by integrating the reduced form of the first two components of (6), as well as (7). Substituting this into the kinematic condition (10) gives
1 τ ∂h ∂t = β 3 ∇ xy · h 3 ∇ xy h − w 4 | z=0 .(38)
Finally, by considering the fluid problem in region 3, and using the simplified version of (20), we see that w 4 | z=0 =ṡ 2 . Thus, the governing thin-film equation for the vapour blanket becomes
1 τ ∂h ∂t = β 3 ∇ · h 3 ∇h +ṡ 2 .(39)
We would expect that at the edges of the snack, h would take some finite value and the pressure would be equivalent to the hydrostatic pressure of the oil. However, in our thin film equation (39) we cannot impose both conditions, so we choose
h = 0, at δΩ 0 ,(40)
as the lateral boundary condition, where Ω 0 is the cross-section of the snack at z = 0, and δΩ 0 is the boundary of Ω 0 .
Note that whilst the vapour blanket thickness depends spatially on x and y, h = h(x, y, t), the temperature only depends on z, except for the boundary condition (36). Hence, it is convenient to replace h in (36) by an average film thicknessh = ∂Ω 0 h dxdy. Making this substitution, the thermal problem is purely in terms of z, and the vapour blanket problem is in terms of x and y. We can simplify even further by assuming that the snack is uniform in the y direction, giving us a one-dimensional model for the thermal problem in z, and a one-dimensional model for the vapour blanket problem in x. This is the approach that we take for the remaining of the paper.
Density calculation and lift-off time
A necessary condition for the snack to detach from the conveyor belt is that its density is less than that of the surrounding oil. The reduction of the density of the snack is due to two processes. Firstly there is loss of mass as water evaporates into steam and leaves the snack. Secondly the formation of the vapour blanket increases the volume of the snack.
The dimensionless density, ρ, is scaled by the density of oil, ρ o , so that ρ = 1 when the snack is neutrally buoyant. The density is given by
ρ snack (t) = 1 1 + 1 0 h dx ρ v ρ o 1 0 h dx + ρ l ρ o α l (s 1 − s 2 ) + ρ v ρ o α v [1 − (s 1 − s 2 )] + ρ s ρ o α s .(41)
The denominator is the volume of the snack, including the volume of the bubble given by integrating over h. The numerator is the mass of the snack broken into contributions from the gas in the bubble, liquid water in region 2, water vapour in regions 1 and 3 and the solid component of the snack. Therefore, the non-dimensional lift-off time, which we denote t * , is the first time 2 for which ρ snack (t * ) < 1.
Numerical Approach
Our first approach is to solve the problem (28)-(30), (39) numerically using the enthalpy method [3,18]. The non-dimensional temperature is related to the non-dimensional enthalpy in the following way:
T = StK 1 C θ : θ < 0, 0 : 0 ≤ θ ≤ 1, St(θ − 1) : θ > 1.(43)
The enthalpy method conveniently reduces the problem to solving the single partial differential equation
∂θ ∂t = ∂ 2 T ∂z 2 ,(44)
within the entire domain 0 ≤ z ≤ 1, where θ and T are related via (43). We use the method of lines with an explicit forward Euler scheme to solve (39) and (44), where at each time step we update T using the relation (43). We plot the solution in Figure 2, illustrating the evolution of both the temperature, the vapour blanket, and the resultant snack density. We identify several clear regimes in the frying process,
t = 0 t = 0.02 t = 0.3 t = 0.8 a) b) c) d)
Heat diffusion Bubble inflation
Quasi-steady e) f) which we indicate in the density plot. Initially the snack is plunged into the oil at room temperature, and so the first regime consists of a heating period, bringing the temperature within the snack to the evaporation temperature. During this regime the snack is entirely composed of liquid and solid (region 2). Once the temperature is near the boiling point everywhere, and equal to the boiling temperature at the edges of the snack, the latent heat begins to be removed. As the latent heat is removed from the edges of the snack, two evaporation fronts recede into the interior of the snack, bubbling away vapour through the top and bottom. This is the second regime of the process, during which the vapour blanket is formed, and inflates very rapidly, causing a sudden drop in density. The vapour blanket quickly reaches a steady state, bringing us to the final regime. During this regime, the evaporation fronts continue to move inwards (hence it is called the quasisteady regime), and the temperature within each region is approximately linear with z, which is due to the large Stefan number [2,3,12]. Meanwhile the bubble remains at near-constant volume, which can only be explained by a constant growth rate of the evaporation frontṡ 2 in (39). The lift-off time of the snack can be taken as the time at which the density falls below the oil density (42). For the parameters used here, this corresponds to a time of t = 0.1, or in dimensional terms, 1 second, which is in agreement with observations in the frying industry. A key result from our model is that the lift-off time is largely controlled by the inflation of the vapour blanket. In fact, since the bubble inflation is so rapid, one can approximate the lift-off time as the time needed for first evaporation. Hence, as a proxy for the lift-off time, one can simply solve the initial heat diffusion problem (first regime) and find the time at which the temperature in the snack becomes uniformly equal to the evaporation temperature. If we do so, one of the key parameters that determines the lift-off time is the Nusselt number N, which is a measure of the heat conduction at the boundaries. In the literature the Nusselt number for snacks varies between 0.3 and 1.3. Therefore, in Figure 3 we plot the variation of approximate lift-off time with Nusselt number, where we also indicate some lift-off times calculated by solving the full numerical problem for t * such that ρ(t * ) < 1. The lift-off time is a monotonic decreasing function of N, as expected. In dimensional terms lift-off occurs for times between 0.5 and 2.6 seconds.
As a further motivation for our vapour blanket model, suppose instead we were to ignore the vapour blanket, and just solve the classic Stefan problem with Newton heating boundary conditions (i.e. h = 0). In this case, we skip the second regime since there is no bubble inflation, and simply move from a heat diffusion regime to a quasi-steady regime. From Figure 2e) we see that the density decay in the quasi-steady regime is much slower than that caused by bubble inflation. This results in a lift-off time closer to t = 1, which in dimensional terms corresponds to more than 10 seconds, and this is a factor of ten larger than experimental observations. Hence, this serves as a good indication that our vapour blanket model is accurate, and provides the essential ingredients to predict the lift-off time during frying. Motivated by the above simulations, we now consider a further limiting case of the mathematical model called the quasi-steady limit. In this limit, we can further simplify the governing equations and find some analytical results that provide useful insight to the problem.
Quasi-steady limit
The quasi-steady limit corresponds to when the thermal problem (28)-(30) becomes independent of time except through the motion of the evaporation fronts. This limit, which is typical in such phase change problems, is a result of the fact that the Stefan number is large [2,3,12]. To study this limit, we restrict our attention to the second and third regimes of the above simulations. That is to say, we replace the above initial conditions of room temperature with initial conditions at the evaporation temperature T (t = 0) = 0. As before, we restrict our attention to the case where the evaporation fronts move uniformly, such that s 1 , s 2 and T 1 -T 3 are independent of x and y. Hence, the temperature in each region is given by
T i = A i (t)z + B i (t),(45)
for some functions A i , B i , for i = 1 − 4. Applying the boundary conditions (31)-(36), we obtain
T 1 = z − s 1 1 + 1/N − s 1 ,(46)T 2 = 0,(47)T 3 = s 2 − z 1/N +h/K 2 + s 2 ,(48)s 1 = −1 1 + 1/N − s 1 ,(49)s 2 = 1 1/N +h/K 2 + s 2 .(50)
The last equation (50) contains the spatial average of h, which is found by solving the thin-film equation
1 τ ∂h ∂t = β 3 ∂ ∂x h 3 ∂h ∂x + 1 1/N +h/K 2 + s 2 ,(51)
together with the boundary conditions h = 0 at x = 0, 1 from (40). We can solve (49) immediately, finding
s 1 = 1 N 1 + N − 1 + 2N 2 t .(52)
The form of (52) reveals the classic t 1/2 similarity behaviour that is discussed for classic Stefan problems in the literature [9,19]. coupled system (50)-(51). In Figure 4 we display the numerical solution to this system, calculated using the method of lines, as before. We see a fast early-time growth of the evaporation front s 2 , causing a rapid inflation of the bubble over a timescale of around t = 0.01. After this inflation period, the growth rate of s 2 is almost constant, and consequently the bubble shape reaches a steady state, which is consistent with Figure 2.
To understand the apparent steady state, let us consider the evolution equation for the lower evaporation front (50). It is not immediately obvious that (50) yields a constant growth rate solution. However, the non-dimensional conductivity ratio is very small K 2 ≈ 0.04, and s 2 in Figure 4 is also very small, suggesting that perhaps the variables s 2 and h ought to be rescaled by K 2 appropriately. Since, for the steady state solution, we expect h to be independent of time but dependent on space, and we expect the evaporation front to move at a linear growth rate, we seek a rescaling of the form
s 2 = K c 2 (a + bt) + O(K 2c 2 ),(53)h = K d 2 H(x) + O(K 2d 2 ),(54)
for some unknown coefficients a, b, c, d > 0. By inserting the above into (50)-(51), we can see that a steady state is only possible (to leading order) if we choose
d − 1 + c = 0, (55) 4d = c,(56)
which has solution c = 4/5 and d = 1/5. Taking the limit of small K 2 , the resulting system of equations is
b = 1 βH ,(57)1 3 H 3 H x x + 1 βH = 0,(58)
andH is found by integrating (59), which gives
H = Γ(5/4) Γ(3/4) 8π 2 27β 1/4 4/5 ,(60)
where Γ is the Euler Gamma function. Note that the above is only valid for times much smaller than t ≈ K −4/5 2 ≈ 13. However, the snack frying process all takes place within 0 ≤ t ≤ 1, so this is acceptable. Note also that the linear behaviour of s 2 with respect to time is different from the square root behaviour of s 1 observed in (52). Hence, the vapour blanket completely changes evaporation at the lower boundary.
In Figure 5 we display a comparison of the results from the quasi-steady limit, including the steady state, to the original numerical solution from Figure 2. For the comparison, we look at the long-time evolution of the film thickness, the density, the evaporation fronts and the temperature within the snack. We see that in all cases there is close agreement between the numerical solution to the full problem, the quasi-steady solution and the steady state. There is a slight discrepancy (∼ 5%) for the steady state solution to the thin film, and this can be explained by the asymptotic approximation (54). This discrepancy could be mitigated by going to higher order terms in the asymptotic expansion.
There is also a slight disagreement (∼ 5%) between the early-time density predictions of the numerical solution to the full problem and the quasi-steady solution. This can be explained by the way in which we calculate the speed of the lower evaporation frontṡ 2 , which largely controls the density at early times via the inflation of the vapour blanket. In the quasi-steady approximation we calculate the evaporation front s 2 using a numerical discretisation scheme in time to solve (50), with time step δt = 2 × 10 −7 , providing very smooth results. On the other hand, in the numerical solution to the full problem, since we calculate the temperature using the enthalpy method, which does not require tracking the position of the fronts, the evaporation front is calculated by finding the grid point that separates liquid and gas phases. Since the grid spacing is finite, this leads to non-smooth step changes in s 2 and spikes in the time-derivative of s 2 , which we have attempted to smooth using a damping method. Nevertheless, even with a time step δt = 2 × 10 −7 , this produces inevitable error associated with the inflation of the vapour blanket, and this is reflected in the slight disagreement for the density prediction at early times.
Closer agreement can be attained with an even smaller spatial discretisation, but due to the explicit discretisation method, this results in lengthy computation times. Hence, interestingly the quasi-steady solution, though it only applies to an asymptotic limit, is generally more accurate than the numerical solution to the full problem. Since the critical time of interest is the lift-off time, which still shows close agreement between these two approaches, we do not consider this discrepancy to be very important.
Finally, in Figure 5 c,d) we display a comparison of the predictions of the evaporation fronts and the temperature. On a macroscopic level, there is very close agreement, and in particular the steady state solution performs remarkably well. After a time of t = 1, or 10 seconds in dimensional terms, nearly half the liquid in the snack has evaporated and the density has dropped by a factor of around 2. The overall thickness of the vapour blanket is nearly equal to the total width of the snack, which is also consistent with experimental observations.
Conclusions
We considered a mathematical model of potato-snack frying in order to obtain an estimate for the lift-off time of the snack from the conveyor belt. To that end, we modelled the frying process as a Stefan problem with two propagating evaporation fronts where the liquid in the dough turns into vapour and decreases the density of the snack. In addition, a key feature in our model is the presence of a vapour blanket that forms underneath the snack as liquid evaporates. The moving vapour fronts and the vapour blanket were assumed to be the two main mechanisms for density reduction of the snack and, therefore, its eventual lift-off from the belt. Numerical results of the full system, using the enthalpy method, revealed that, indeed, both of these mechanisms were essential to predict a physically realistic lift-off time of the order of a second. Furthermore, we considered a simplified quasi-steady model due to the large Stefan number. Numerical solutions to the reduced problem agreed very well with solutions to the full system and thus allow for a computationally cheaper way to investigate properties of our model and, in particular, the lift-off time.
One of the key dimensionless parameters that emerged as part of our analysis was the Nusselt number N, which is the ratio between heat transfer at the snack boundary and heat conduction in the snack interior. We investigated how changing N affects the lift-off time of the snack. This is important to snack manufacturers since changing the dough, for example, can change the material properties and hence the parameters of the system. Having the dependence of the lift-off time on these parameters is useful in determining the optimal cooking strategies.
To further improve the prediction of lift-off time for the snack there should be a consideration of other forces. These could include interfacial tension between the snack and belt as well as the peeling energy required to overcome the dough elasticity. As a result the orientation of the snack on the belt and indeed the belt design and material could have a further impact on the lift-off time of the snack.
Figure 1 :
1Schematic diagram of the different regions in the snack.
Figure 2 :
2(a,b,c,d) Numerical solution at t = 0, 0.02, 0.1, 0.8, showing a colour plot of the temperature in the snack, a corresponding line plot of the temperature, and the film thickness beneath the snack. (e) Density evolution over time, indicating the critical density for lift-off ρ = ρ oil . (f) Evolution of the evaporation fronts s 1 and s 2 .
Figure 3 :
3Variation of the non-dimensional lift-off time with Nusselt number, showing approximate time calculated by solving the proxy heat diffusion problem until the snack reaches evaporation temperature, and the precise times calculated by solving the full numerical problem until ρ(t * ) < 1.
Figure 4 :
4The remaining unknowns h and s 2 are found by Solution to the quasi-steady approximation. (a) Evolution of the thin film h at various times. (b) Evolution of the lower evaporation front s 2 (t).
Figure 5 :
5dx is the average film thickness, which is a constant in the steady state. Long time results from the quasi-steady approximation. (a) Evolution of the thin film h at various times, compared to the analytical solution for the steady state. (b) Density ρ(t) as a function of time, indicating the lift-off density ρ = ρ oil . (c) Evolution of the lower Stefan boundary s 1 (t), compared to the analytical solution for the steady stateṡ 1 H. (d) Temperature profiles T (z, t) at different times between t = 0 and t = 1, indicating liquid and vapour regions.solve (58) to give H in terms of its average value
Table 1 :
1Dimensionless parameters and their approximate numerical values.
All authors contributed equally to this work, and are placed in alphabetical order.
Note that in reality, there may be some surface tension effects holding the snack down to the solid substrate, therefore delaying lift-off time. However, since these depend on the specific surface properties of the fryer substrate, we do not study such effects here.
AcknowledgmentsThe authors would like to aknowledge the 138th European Study Group with Industry which was held in Bath,[16][17][18][19][20]
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| []
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[
"Guiding center dynamics as motion on a slow manifold in loop space",
"Guiding center dynamics as motion on a slow manifold in loop space"
]
| [
"J W Burby \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n"
]
| [
"Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA"
]
| []
| Since the late 1950's, the dynamics of a charged particle's "guiding center" in a strong, inhomogeneous magnetic field have been understood in terms of near-identity coordinate transformations. The basic idea has been to approximately transform away the coupling between the fast gyration around magnetic fields lines and the remaining slow dynamics. This basic understanding now serves as a foundation for describing the kinetic theory of strongly magnetized plasmas. I present a new way to understand guiding center dynamics that does not involve complicated coordinate transformations. Starting from a dynamical systems formulation of the motion of parameterized loops in a charged particle's phase space, I identify a slow manifold in loop space. Dynamics on this slow manifold are equivalent to guiding center dynamics to all orders in perturbation theory. After demonstrating that loop space dynamics comprises an infinite-dimensional noncanonical Hamiltonian system, I recover the well-known Hamiltonian formulation of guiding center motion by restricting the (pre-) symplectic structure on loop space to the finite-dimensional guiding center slow manifold. | 10.1063/1.5119801 | [
"https://arxiv.org/pdf/1905.04410v2.pdf"
]
| 152,282,856 | 1905.04410 | 829491bea99754d661beaaff08f1223a8d78a50e |
Guiding center dynamics as motion on a slow manifold in loop space
11 May 2019
J W Burby
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
Guiding center dynamics as motion on a slow manifold in loop space
11 May 2019(Dated: 14 May 2019)
Since the late 1950's, the dynamics of a charged particle's "guiding center" in a strong, inhomogeneous magnetic field have been understood in terms of near-identity coordinate transformations. The basic idea has been to approximately transform away the coupling between the fast gyration around magnetic fields lines and the remaining slow dynamics. This basic understanding now serves as a foundation for describing the kinetic theory of strongly magnetized plasmas. I present a new way to understand guiding center dynamics that does not involve complicated coordinate transformations. Starting from a dynamical systems formulation of the motion of parameterized loops in a charged particle's phase space, I identify a slow manifold in loop space. Dynamics on this slow manifold are equivalent to guiding center dynamics to all orders in perturbation theory. After demonstrating that loop space dynamics comprises an infinite-dimensional noncanonical Hamiltonian system, I recover the well-known Hamiltonian formulation of guiding center motion by restricting the (pre-) symplectic structure on loop space to the finite-dimensional guiding center slow manifold.
Since the late 1950's, the dynamics of a charged particle's "guiding center" in a strong, inhomogeneous magnetic field have been understood in terms of near-identity coordinate transformations. The basic idea has been to approximately transform away the coupling between the fast gyration around magnetic fields lines and the remaining slow dynamics. This basic understanding now serves as a foundation for describing the kinetic theory of strongly magnetized plasmas. I present a new way to understand guiding center dynamics that does not involve complicated coordinate transformations. Starting from a dynamical systems formulation of the motion of parameterized loops in a charged particle's phase space, I identify a slow manifold in loop space. Dynamics on this slow manifold are equivalent to guiding center dynamics to all orders in perturbation theory. After demonstrating that loop space dynamics comprises an infinite-dimensional noncanonical Hamiltonian system, I recover the well-known Hamiltonian formulation of guiding center motion by restricting the (pre-) symplectic structure on loop space to the finite-dimensional guiding center slow manifold.
I. INTRODUCTION
Charged particles move in helical trajectories that wind around magnetic field lines.
When the strength of the magnetic field is high this spinning motion is exceedingly fast and the corresponding helix is tightly wound. Thus on timescales large compared with the cyclotron period a charged particle's trajectory resembles the motion of an approximately circular ring that may drift along or across the magnetic lines of force. Modeling these averaged, or "guiding center," dynamics efficiently, especially in non-uniform magnetic fields, is known as the guiding center problem.
While researchers have developed various ingenious strategies [1][2][3][4][5][6][7] for solving the guiding center problem, the technique that has become most widely adopted was developed by Kruskal in Ref. 8 in the context of a broad class of oscillatory dynamical systems. After introducing a special sequence of near-identity coordinate transformations, Kruskal observed that short-timescale oscillations in these systems approximately decouple from the slower drift dynamics. By successively refining the near-identity transformation, the decoupling becomes increasingly complete. When applied to charged particle dynamics, Kruskal's method describes precisely the slow evolution of the fiducial ring swept out by a particle's gyration.
Kruskal's technique owes its popularity to its rigorous mathematical foundation, its tractability at low orders in perturbation theory, and its ability to explain the general phenomenon of adiabatic invariance. However, the technique also possesses several important drawbacks related to its use of complicated near-identity coordinate transformations. Three of these are:
1. While the method is arbitrarily accurate in principle, finding high-order approximations to the averaged dynamics requires a heroic amount of algebra, even by computer algebra standards. 9 2. While Kruskal shows that adiabatic invariance may be understood as a consequence of Noether's theorem, the symmetry that gives rise to the adiabatic invariant in Kruskal's theory is hidden. Uncovering the symmetry is akin to unearthing an infinite fossil; fractional progress requires painstaking effort, and completing the task is impossible.
3. The coordinates introduced by the method break spatial locality. Therefore the region in phase space occupied by an object that may interrupt a particle's motion, e.g. a wall, is rendered extremely complicated.
In this article I will describe an alternative rigorous solution of the guiding center problem that is completely free of drawbacks (2) and (3), and that suffers from drawback (1) less severely. In particular, I will realize guiding center dynamics as the restriction of loop space dynamics to a slow manifold. 10-13 R. S. MacKay 14 refers to such a demonstration as constructing a "slow manifold with internal oscillation." Here loops are periodic parameterized curves in a charged particle's phase space that evolve by being dragged along by Newton's second law.
Using the fact that loop space dynamics and its associated slow manifold are invariant under an obvious relabeling symmetry, I will show that the symmetry underlying adiabatic invariance is rendered obvious from the loop space perspective; it is no longer hidden. By showing that the slow manifold may be calculated without introducing spatially-nonlocal coordinates in phase space, I will demonstrate that the guiding center problem may be solved without breaking spatial locality. Finally, by formulating the guiding center problem as a slow manifold reduction problem, the challenge of capturing high-order effects in the drift dynamics will be recast as a the challenge of selecting initial conditions for loop space dynamics sufficiently close to the slow manifold. This reformulation is interesting in light of the numerical method formulated by Gear, Kaper, Kevrekidis, and Zagaris 15,16 for generating points in phase space arbitrarily close to a given slow manifold.
II. LOOP SPACE DYNAMICS
This section will define and describe loop space dynamics in the context of general dynamical systems. A special feature of loop space dynamics associated with Hamiltonian systems will also be explained. Loop space dynamics governed by the Lorentz force Law will be presented as an example of the general theory.
A. Abstract Background
Recall that a dynamical system on a set P is a one-parameter family of mappings F t : P → P with the properties F 0 = id P and F t+s = F t • F s . Such a family of mappings is also referred to as a flow or flow map. Given a point z ∈ P , the trajectory through z is defined in terms of the flow as the parameterized curve R → P : t → F t (z). When P is a manifold and F t is smooth, the dynamical system may be recovered from its infinitesimal generator, which is the vector field X on P given by
X(z) = d dǫ 0 F ǫ (z).(1)
In other words, X(z) is the initial velocity of the trajectory through z. In many situations it is easier to specify a dynamical system by giving its infinitesimal generator instead of its flow. Nevertheless, it is sometimes more convenient to specify the flow map directly. Given a dynamical system F t on P , it is possible to construct various other induced dynamical systems on spaces constructed out of P . In particular, there is a dynamical system induced on the space ℓP of mapsz 0 : S 1 → P , i.e. the space of parameterized loops in P . (Here the circle S 1 is defined as the set R mod 2π.) The flow map F t : ℓP → ℓP is given by
( F t (z 0 ))(θ) = F t (z 0 (θ)).(2)
It will be convenient to refer to this dynamical system on ℓP as the loop-parallelized dynamics in P . When P is a smooth manifold and F t is smooth, the infinitesimal generator X 0 of loop-parallelized dynamics is given by
( X 0 (z 0 ))(θ) = X(z 0 (θ)).(3)
Here we have identified the tangent space to ℓP atz 0 with the space of smooth vector fields alongz 0 . Thus, loop-parallelized dynamics merely parallelizes the original dynamics on P over the loop parameter θ ∈ S 1 .
From here on, suppose that P is a smooth manifold and that F t is smooth. Starting from loop-parallelized dynamics in P , loop space dynamics in P is constructed as follows. Fix a smooth functional Ω : ℓP → R that is invariant under the phase shiftz 0 →z ψ 0 for each
ψ ∈ S 1 , wherez ψ 0 (θ) =z 0 (θ + ψ).(4)
(Note that the phase shift for fixed ψ defines an invertible mapping on loop space ℓP .)
Lift loop-parallelized dynamics to the dynamical system on on ℓP × S 1 whose infinitesimal generator is given by
X Ω 0 (z 0 , S) = X 0 (z 0 ) + Ω(z 0 )∂ S .(5)
This dynamical system is a lift in the sense that dynamics in first factor ℓP reproduce loopparallelized dynamics; in other words there is a one-way coupling betweenz 0 and S. Next apply the invertible transformation ℓP × S 1 → ℓP × S 1 given by
Φ : (z 0 , S) → (z −S 0 , S).(6)
This transformation may be thought of as "spinning" the loopz 0 by the phase S. Loop space dynamics is then defined as the dynamical system on ℓP × S 1 with infinitesimal generator
X Ω = Φ * X Ω 0 .
Here Φ * denotes the pushforward along Φ. Therefore X Ω is merely X Ω 0 expressed in the new "coordinates" on ℓP × S 1 defined by Φ. An explicit expression for the infinitesimal generator of loop space dynamics is given by
X Ω (z, S) = (Φ * X Ω 0 )(z, S) = T Φ • X Ω 0 • Φ −1 (z, S) = T Φ X Ω 0 (z S ) + Ω(z S )∂ S = X Ω 0 (z) − Ω(z)∂ θz + Ω(z)∂ S ,(7)
where we have used the invariance of Ω under phase shifts. Thus, the trajectory of an element (z, S) ∈ ℓP × S 1 satisfies the system of equations ∂ tz (θ, t) + Ω(z(t))∂ θz (θ, t) = X(z(θ, t)) (8)
S(t) = Ω(z(t)).(9)
The discussion so far has emphasized the generality and geometric origins of loop space dynamics. It is also useful to be aware of the relationship between loop space dynamics and the so-called nonlinear WKB approximation. 17 Suppose z(t) is a solution of the ordinary differential equationż = X(z) comprising a rapid oscillation superimposed on top of a slowly evolving envelope. The nonlinear WKB approach to describing such a solution is to leverage the ansatz z(t) =z(t, S(t)), where S(t) is a rapidly rotating phase and the profilez is periodic in its second argument. Apparently solutions of this form must satisfy ∂ tz (t, S(t))+ S∂ θz (t, S(t)) = X(z(t, S(t))). IfṠ is chosen to approximate the rate of phase oscillations,
say Ω, then the scale-separation assumption implies that the profile approximately satisfies ∂ tz (t, θ) + Ω∂ θz (t, θ) = X(z(t, θ)) for each θ ∈ S 1 . This is of course the governing equation of loop space dynamics. It is interesting to notice that, from the WKB perspective, loop space dynamics would appear to represent an approximation based on scale separation.
However, the geometric picture emphasized in this section shows that there is no need invoke approximations in order to provide loop space dynamics with a useful interpretation.
Namely, loop space dynamics describes the evolution of parameterized loops, "spun" by a phase, entrained in the flow of a given dynamical system. A field-theoretic generalization of this construction is discussed in Ref. 18.
Loop space dynamics associated with a Hamiltonian system enjoys an important special property that will be exploited later when connecting loop space dynamics with guiding center theory. Let (P, −dϑ) be an exact symplectic manifold (not necessarily a cotangent bundle) and fix a function H : P → R that will serve as the Hamiltonian. There is then a unique vector field X H on P that satisfies
ι X H dϑ = −dH.(10)
The vector field X H , which is known as the Hamiltonian vector field, is the infinitesimal generator of a Hamiltonian dynamical system on P . A useful way of characterizing the dynamical system generated by X H is in terms of the so-called phase space variational principle. This variational principle asserts that a parameterized curve [t 1 ,
t 2 ] → P : t → z(t)
is a trajectory of some point z ∈ P under the dynamical system generated by X H if and only if the first fixed-endpoint variation of the action functional
A(z) =ˆt 2 t 1 ϑ(z(t))[ż(t)] − H(z(t)) dt(11)
vanishes at z. The special feature of loop space dynamics induced by X H is that they also obey a phase space variational principle. In particular, if t → (z(t), S(t)) is a trajectory of the loop space dynamics associated with X H , then the first fixed-endpoint variation of the
actionà (z, S) =ˆt 2 t 1 ϑ(z(θ, t))[∂ tz (θ, t) +Ṡ(t)∂ θz (θ, t)] − H(z(θ, t)) dθ dt(12)
vanishes at (z, S). Here ffl = (2π) −1´2 π 0 . Note that this variational principle renders loop space dynamics associated with a Hamiltonian system as a classical field theory on a (1 + 1)dimensional spacetime. Therefore Noether's theorem may be used to extract conservation laws from symmetries. The conserved quantity associated with time translation invariance is the loop energy
H(z) = H(z(θ)) dθ,(13)
while the conserved quantity associated with (time-independent) shifts S → S + ψ is the (normalized) loop action
J(z) = ϑ(z(θ))[∂ θz (θ)] dθ = 1 2πˆz ϑ.(14)
B. Example: The Lorentz Force
When the electric field is zero and the magnetic field is time-independent, Lorentz force dynamics are governed by the ordinary differential equation
on P = Q × R 3 given bẏ v = 1 ǫ v × B(x)(15)
x =v,
where x is contained in Q = R 3 or Q = (S 1 ) 3 , v ∈ R 3 , B is vector field on Q that may be written as the curl of another vector field A, and ǫ is the mass-to-charge ratio. (The mass-to-charge ratio may be negative.) This ordinary differential equation may be identified with the vector field X on P given by
X(x, v) = v · ∂ x + 1 ǫ v × B(x) · ∂ v .(17)
After equipping P with the exact symplectic form −dϑ, where
ϑ(x, v) = 1 ǫ A(x) · dx + v · dx,(18)
and introducing the Hamiltonian
H(x, v) = 1 2 |v| 2 ,(19)
it is straightforward to verify that X = X H is the Hamiltonian vector field associated with H.
Loop space dynamics associated with the Lorentz force is governed by the system of
equations ∂ tṽ (θ, t) + Ω(x(t),ṽ(t))∂ θṽ (θ, t) = 1 ǫṽ (θ, t) × B(x(θ, t)) (20) ∂ tx (θ, t) + Ω(x(t),ṽ(t))∂ θx (θ, t) =ṽ(θ, t)(21)S(t) = Ω(x(t),ṽ(t)),(22)
where the frequency functional Ω can, in principle, be any phase-shift invariant functional on ℓP . The choice for Ω that will be used in this article is
Ω(x,ṽ) = 1 ǫ |B(x)|,(23)
where x = fflx dθ, which clearly satisfies the phase-shift invariance property.
The action (12) that governs loop space dynamics for the Lorentz force is given bỹ
A(x,ṽ, S) =ˆt 2 t 1 1 ǫ A(x(θ, t)) +ṽ(θ, t) · ∂ tx (θ, t) − 1 2 |ṽ(θ, t)| 2 dθ dt +ˆt 2 t 1Ṡ (t) 1 ǫ A(x(θ, t)) +ṽ(θ, t) · ∂ θx (θ, t) dθ dt.(24)
The first fixed-endpoint variation of the action is given by
δà = −ˆt 2 t 1 1 ǫ B(x) × [∂ tx +Ṡ ∂ θx ] + [∂ tṽ +Ṡ ∂ θṽ ] · δx(θ, t) dθ dt −ˆt 2 t 1 ṽ − [∂ tx +Ṡ ∂ θx ] · δṽ(θ, t) dθ dt −ˆt 2 t 1 d dt 1 ǫ A(x) +ṽ · ∂ θx dθ δS(t) dt.(25)
The Euler-Lagrange equations are therefore
1 ǫ B(x) × [∂ tx +Ṡ ∂ θx ] + [∂ tṽ +Ṡ ∂ θṽ ] = 0 (26) v − [∂ tx +Ṡ ∂ θx ] = 0 (27) d dt 1 ǫ A(x) +ṽ · ∂ θx dθ = 0.(28)
Note that the first two equations, Eqs. (26) and (27), imply the third, Eq. (28), because d dt
1 ǫ A(x) +ṽ · ∂ θx dθ = 1 ǫ B × ∂ tx + ∂ tṽ · ∂ θx dθ − ∂ tx · ∂ θṽ dθ = −Ṡ 1 ǫ B × ∂ θx + ∂ θṽ · ∂ θx dθ − (ṽ −Ṡ∂ θx ) · ∂ θṽ = − 1 2 ∂ θ |ṽ| 2 dθ =0.(29)
It follows that the Euler-Lagrange equations will be satisfied if and only if
∂ tṽ (θ, t) +Ṡ(t)∂ θṽ (θ, t) = 1 ǫṽ (θ, t) × B(x(θ, t)) (30) ∂ tx (θ, t) +Ṡ(t)∂ θx (θ, t) =ṽ(θ, t).(31)
Note in particular that the Euler-Lagrange equations are satisfied if (x,ṽ, S) obeys loop space dynamics, regardless of the frequency functional Ω. In other words, the initial value problem for the Euler-Lagrange equations is ill-posed. This ill-posedness, along with the redundancy of the Euler-Lagrange equation (28), is a hallmark of gauge symmetry. There is not necessarily an issue with the fact that the Euler-Lagrange equations are ill-posed. As long as there is some well-posed differential equation whose solutions satisfy the Euler-Lagrange equations, many of the nice tools offered by variational principles are still applicable. In this case, loop space dynamics with a given frequency functional furnish such a differential equation.
III. A SLOW MANIFOLD IN LOOP SPACE
A. Motivating Ideas
An old and intuitive picture of the dynamics of charged particles in a strong magnetic field replaces the particle with a charged, superconducting ring of current. One reason for introducing loop space dynamics in the study of charged particle motion is to make this intuitive picture mathematically precise.
There is an apparent gap between the intuitive picture of moving rigid rings and the loop space description, which involves deformable loops. This is not merely a technical annoyance. The evolution of an arbitrary loop in the Lorentz force phase space will not approximate the motion of a rigid ring in any sense. Indeed, most loops become extremely contorted as time evolves, especially when B is chosen such that Lorentz force dynamics is chaotic.
The way to establish a link between loop space dynamics and the rigid ring picture is to introduce the concept of a slow manifold. Generally speaking, a slow manifold is a special submanifold in the phase space of a dynamical system with multiple timescales. When an initial condition is chosen to lie on the slow manifold, its subsequent time evolution will remain close to the slow manifold for a long period of time. Thus, a slow manifold is an example of an almost invariant set. Moreover, motion on the slow manifold only weakly couples to the fast timescale. This is the sense in which is a slow manifold is "slow."
The remainder of this Section will demonstrate that the phase space for loop space dynamics contains a slow manifold. Moreover, points on this slow manifold may be identified with rigid rings in phase space. Interestingly, these rigid rings are not geometric circles.
Instead their shape is described by a set of non-trivial shape functions whose asymptotic expansion in powers of ǫ may be computed systematically. In light of the intuitive picture of guiding center dynamics as dynamics of rigid charged superconducting rings, these results strongly suggest that loop motion on the slow manifold corresponds in some way to guiding center dynamics. This intuition will be justified in Section III E by proving that dynamics on the slow manifold is equivalent to guiding center dynamics to all orders in perturbation theory.
The purpose of Section IV will be to demonstrate the sense in which the rigid rings that support guiding center motion are "superconducting." According to Eq. (28), the dynamics of a general loop conserves action. In particular, the dynamics of a loop that evolves on the slow manifold conserves action. At leading-order, it will turn out that the expression for the action of a rigid loop is the magnetic flux through the loop, i.e. the usual magnetic moment adiabatic invariant is recovered. Thus, to leading order, the rigid rings conserve flux exactly as superconductors do. At higher orders, this picture has to be distorted slightly because the exact expression for the action of a rigid ring differs from the flux. As a more complete way of describing the all-orders picture, and in order to illuminate the simple origins of the symmetry underlying adiabatic invariance for charged particles, I will show how the Hamiltonian formulation of guiding center dynamics due to Littlejohn 5 may be recovered by restricting the presymplectic form on loop space associated with the action functional (24) to the slow manifold.
B. Fast-Slow Systems and Their Slow Manifolds
A useful class of dynamical systems for the precise study of slow manifolds consists of the fast-slow systems.
Definition 1.
A fast-slow dynamical system is a dynamical system on a cartesian product P = X × Y of Banach spaces X, Y whose infinitesimal generator (ẋ,ẏ) has the form
ǫẏ =f ǫ (x, y) (32) x =g ǫ (x, y),(33)
where f ǫ , g ǫ depend smoothly on ǫ in some open interval containing ǫ = 0 and D y f 0 (x, y) is invertible whenever f 0 (x, y) = 0.
Remark 1. The technical hypothesis on D y f 0 ensures that the limiting differential algebraic
equation (DAE), 0 = f 0 (x, y) (34) x = g 0 (x, y),(35)
has differentiation index 1. In other words, when solving the DAE, Eq. (34) may first be eliminated by solving for y as a function of x, giving a function y * 0 (x). The function y * 0 (x) may then be substituted into Eq. (35) in order to obtain an autonomous ordinary differential equation for x. Fast-slow systems therefore provide a paradigm for studying dimensionality reduction.
In the fast-slow setting, a slow manifold may be defined precisely as follows.
Definition 2. Given a fast-slow system, ǫẏ = f ǫ (x, y),ẋ = g ǫ (x, y), a slow manifold is a formal power series y * ǫ (x) = y * 0 (x) + ǫ y * 1 (x) + ǫ 2 y * 2 (x) + . . .(36)
that satisfies the invariance equation
ǫDy * ǫ (x)[g ǫ (x, y * ǫ (x))] = f ǫ (x, y * ǫ (x)),(37)
to all orders in ǫ.
Remark 2. Note that if y * ǫ is a genuine solution of the invariance equation for each ǫ, then the set Γ ǫ = {(x, y) | y = y * ǫ (x)} is invariant under the dynamics of the fast-slow system for each ǫ. Therefore, when y * ǫ is a slow manifold, it is suggestive, though not rigorous, to think of the "graph" of y * ǫ as an invariant manifold for each ǫ. This is the rationale behind referring to an asymptotic series as a "manifold."
One of the main motivations for studying fast-slow systems is the fact that such systems always contain unique slow manifolds, as described by the following Proposition. Proposition 1. Associated with each fast-slow system is a unique slow manifold. Moreover, the coefficient y * k of the slow manifold may be computed algorithmically for any k ∈ {0, 1, 2, . . . }. In particular, the first two coefficients are determined by the equations
f 0 (x, y * 0 (x)) = 0 (38) y * 1 (x) = [D y f 0 (x, y * 0 (x))] −1 Dy * 0 (x)[g 0 (x, y * 0 (x))] − f 1 (x, y * 0 (x)) ,(39)
where
f 1 = d dǫ 0 f ǫ .
Thus, fast-slow systems always contain formal invariant sets given as graphs of the fast variable y over the slow variable x. Dynamics on such a set are formally prescribed by the infinitesimal generatorẋ = g ǫ (x, y * ǫ (x)) on X. Because g ǫ depends smoothly on ǫ in a neighborhood of 0, dynamics on the slow manifold apparently do not involve the O(ǫ) timescale, and are in this sense slow.
Of course, there is no reason to expect that in general the series defining a slow manifold converges to give a true invariant set on which dynamics is slow. In the normally hyperbolic case in finite dimensions, where the eigenvalues of D y f 0 (x, y * 0 (x)) are purely real, Fenichel 10 effectively established the convergence of the series using transversality arguments. In the normally elliptic case, where the eigenvalues of D y f 0 (x, y * 0 (x)) are purely imaginary, the series are known to diverge in general due to resonance between the fast normal dynamics and the slow dynamics. However, 14 for sufficiently smooth fast-slow systems, truncations of the series y * ǫ often define almost invariant sets, meaning trajectories that begin near the truncated slow manifold remain nearby for long periods of time. In the analytic case, the time for "sticking" to a truncated slow manifold may even be exponentially long. This state of affairs might be summarized by saying the series defining a slow manifold are meaningful even when they don't converge. This point is amplified vividly by Vanneste in Ref. 19, who applies Borel summation to a normally-elliptic slow manifold to define an "optimal" almost invariant set. What emerges from this analysis is a detailed picture of exponentiallysmall high-frequency oscillations that are generated spontaneously by motions along the optimally-invariant set.
In the following two subsections, III C and III D, I will show that loop space dynamics associated with the Lorentz force may be written as a fast-slow system, and then explicitly compute the first two coefficients in the series y * ǫ defining the associated slow manifold.
These low-order terms in the series will strongly suggest that dynamics on the slow manifold in loop space correspond in some way to guiding center dynamics. I will then prove the equivalence of dynamics on the slow manifold with guiding center dynamics to all orders in perturbation theory in subsection III E. In so doing, I will have shown that guiding center dynamics constitutes an example of what MacKay 14 calls a "a slow manifold with internal oscillation."
C. Fast-Slow formulation of Lorentz Loop Dynamics
Consider now the problem of determining wether loop space dynamics associated with the Lorentz force comprise a fast-slow system. The infinitesimal generator for these dynamics is given in Eqs. (20)- (22). The first step in finding a fast-slow split for this dynamical system is to introduce a decomposition of ℓP into mean and fluctuating subspaces,
ℓP = P ⊕ ℓP , where ℓP = ( X, V) ∈ ℓP X(θ) dθ = V(θ) dθ = 0 .(40)
For the remainder of this article, elements in P will be denoted (x, v). Thus, the phase space for loop space dynamics is now expressed as P × ℓP × S 1 , with typical elements denoted (x, v, X, V, S). The relationship between (x, v, X, V, S) and the original loop space variables
(x,ṽ, S) isx =x + X (41) v =v + V.(42)
The second step is to scale the fluctuating particle position X according to X → ǫ −1 X.
The transformed variable will be denoted ρ = ǫ −1 X. The complete expression for the transformation applied in this second step is then (
x, v, X, V, S) → (x, v, ρ, V, S), which
may be regarded as an invertible mapping from P × ℓP × S 1 into itself because ℓP is scale invariant. The relationship between the original loop space variables (x,ṽ, S) and
(x, v, ρ, V, S) is given byx =x + ǫ ρ (43) v =v + V.(44)
The third step is to parameterize the mean velocity variable v as follows. Let e 1 , e 2 be orthogonal unit vector fields on R 3 that are everywhere orthogonal to the vector field B. (As usual when discussing strongly magnetized particles, the field B is assumed to be nowhere vanishing.) Set b = B/|B| and assume b = e 1 × e 2 . Now introduce the mapping
(x, v) → (x, u, v 1 , v 2 ), where u =b(x) · v (45) v 1 =e 1 (x) · v (46) v 2 =e 2 (x) · v.(47)
This mapping amounts to expressing v in a moving orthonormal frame aligned with the magnetic field. The change of variables on discrete loop space in the third step is then given
by (x, v, ρ, V, S) → (x, u, v 1 , v 2 , ρ, V, S). The relationship between the original loop space variables (x,ṽ, S) and (x, u, v 1 , v 2 , ρ, V, S) is given bỹ x =x + ǫ ρ (48) v =ub(x) + v 1 e 1 (x) + v 2 e 2 (x) + V.(49)
We remind those readers familiar with conventional guiding center theory that ( ρ, V) is an arbitrary element of ℓP . In particular, ρ is not required to be orthogonal to b.
The fourth and final step is to parameterize the fluctuating velocity variable V as follows.
First decompose V into the sum of its first Fourier harmonic V 1 and its higher harmonic
content V 2+ , V = V 1 + V 2+ .(50)
Then parameterize the first harmonic V 1 using the vectors V + 1 , V − 1 according to
V 1 (θ) = V + 1 cos θ + V − 1 sin θ.(51)
Next express V + 1 , V − 1 in the moving frame (b, e 1 , e 2 ) as
V + 1 =u + b(x) + v + 1 e 1 (x) + v + 2 e 2 (x) (52) V − 1 =u − b(x) + v − 1 e 1 (x) + v − 2 e 2 (x).(53)
Finally, introduce the components of the adiabatic velocity
w 1 = 1 2 (v + 1 − v − 2 ) (54) w 2 = 1 2 (v + 2 + v − 1 ),(55)
and the components of the non-adiabatic velocity
ω 1 = 1 2 (v + 1 + v − 2 ) (56) ω 2 = 1 2 (v + 2 − v − 1 ).(57)
This sequence of definitions may be interpreted as a mapping (
x, u, v 1 , v 2 , ρ, V, S) → (x, u, v 1 , v 2 , ρ, u + , u − , w 1 , w 2 , ω 1 , ω 2 , V 2+ , S).
The relationship between the original loop space dynamics phase space variables (x,ṽ, S) and the new variables is given explicitly bỹ
x =x + ǫ ρ (58) v =(u + u + cos θ + u − sin θ)b(x) + v ⊥ + (cos θI + sin θb × ) · ω ⊥ + (cos θI − sin θb × ) · w ⊥ + V 2+ ,(59)
where the following useful shorthand notation has been introduced:
v ⊥ =v 1 e 1 (x) + v 2 e 2 (x) (60) w ⊥ =w 1 e 1 (x) + w 2 e 2 (x) = 1 2 (1 − bb) · ( V + 1 + b × · V − 1 ) (61) ω ⊥ =ω 1 e 1 (x) + ω 2 e 2 (x) = 1 2 (1 − bb) · ( V + 1 − b × · V − 1 ),(62)and the tensor b × is defined by b × · a = b × a.
The infinitesimal generator for discrete loop space dynamics expressed in terms of the final set of new variables and the scaled phase S = ǫS is given by
ǫ ∂ t u + = − |B|(x)u − + ǫ v · ∇b · (w ⊥ + ω ⊥ ) + 2b · cos θṽ × δB dθ (63) ǫ ∂ t u − =|B|(x)u + + ǫ v · ∇b · (w ⊥ × b − ω ⊥ × b) + 2b · sin θṽ × δB dθ (64) ǫ ∂ t ω 1 =2|B|(x) ω 2 + ǫ (cos θe 1 + sin θe 2 ) ·Ṽ × δB dθ − ǫ 1 2 v · ∇b · (u + e 1 + u − e 2 ) − v · R ω 2 (65) ǫ ∂ t ω 2 = − 2|B|(x) ω 1 + ǫ (cos θe 2 − sin θe 1 ) ·ṽ × δB dθ − ǫ 1 2 v · ∇b · (u + e 2 − u − e 1 ) + v · R ω 1 (66) ǫ ∂ t v 1 =|B|(x) v 2 + ǫe 1 · ṽ × δB dθ − ǫ u v · ∇b · e 1 − v · R v 2 (67) ǫ ∂ t v 2 = − |B|(x) v 1 + ǫe 2 · ṽ × δB dθ − ǫ u v · ∇b · e 2 + v · R v 1 (68) ǫ ∂ t V 2+ = V 2+ × B(x) − |B|(x) ∂ θ V 2+ + ǫ π 2+ (ṽ × δB) (69) ǫ ∂ t ρ = V − |B|(x) ∂ θ ρ (70) ∂ t w 1 = − 1 2 v · ∇b · (u + e 1 − u − e 2 ) + v · R w 2 + (cos θe 1 − sin θe 2 ) ·ṽ × δB dθ (71) ∂ t w 2 = − 1 2 v · ∇b · (u + e 2 + u − e 1 ) − v · R w 1 + (cos θe 2 + sin θe 1 ) ·ṽ × δB dθ (72) ∂ t u =v · ∇b · v + b · ṽ × δB dθ (73) ∂ t x =ub(x) + v ⊥ (74) S =|B(x)|(75)
where π 2+ is the L 2 -orthogonal projection onto the space of Fourier harmonics greater than or equal to 2, R = (∇e 1 ) · e 2 , and the symbol δB is defined as
δB =ˆ1 0 ρ · ∇B(x + λǫ ρ) dλ.(76)
Note that B(x) = B(x) + ǫ δB. smoothly on ǫ and f 0 is given by
f 0 (x, y) = (˙ ρ 0 , (v 1 ) 0 , (v 1 ) 0 ,u + 0 ,u − 0 , (ω 1 ) 0 , (ω 2 ) 0 , (˙ V 2+ ) 0 )(77)ρ 0 = (u + cos θ + u − sin θ)b(x) + (cos θI + sin θb × ) · ω ⊥ + (cos θI − sin θb × ) · w ⊥ + V 2+ − |B|(x) ∂ θ ρ (78) (v 1 ) 0 = |B|(x) v 2 (79) (v 2 ) 0 = −|B|(x) v 1(80)u + 0 = −|B|(x)u −(81)u − 0 = |B|(x)u + (82) (ω 1 ) 0 = 2|B|(x) ω 2 (83) (ω 2 ) 0 = −2|B|(x) ω 1 (84) (˙ V 2+ ) 0 = V 2+ × B(x) − |B|(x) ∂ θ V 2+ .(85)
The derivative D y f 0 (x, y) is therefore given by
D y f 0 (x, y)[δy] =(δ˙ ρ 0 , δ(v 1 ) 0 , δ(v 1 ) 0 , δu + 0 , δu − 0 , δ(ω 1 ) 0 , δ(ω 2 ) 0 , δ(˙ V 2+ ) 0 )(86)δ˙ ρ 0 = (δu + cos θ + δu − sin θ)b(x) + (cos θI + sin θb × ) · δω ⊥ + δ V 2+ − |B|(x) ∂ θ δ ρ (87) δ(v 1 ) 0 = |B|(x) δv 2 (88) δ(v 2 ) 0 = −|B|(x) δv 1(89)δu + 0 = −|B|(x)δu −(90)δu − 0 = |B|(x)δu + (91) δ(ω 1 ) 0 = 2|B|(x) δω 2 (92) δ(ω 2 ) 0 = −2|B|(x) δω 1 (93) δ(˙ V 2+ ) 0 = δ V 2+ × B(x) − |B|(x) ∂ θ δ V 2+ .(94)
In order to assess the invertibility of D y f 0 (x, y), first note that the space Y is given by
Y = ℓQ × R 2 × R 2 × R 2 × ℓ 2+ R 3 ,(95)
where ℓ 2+ R 3 is the set of loops in R 3 with zero'th and first Fourier harmonics equal to 0.
Now fix an arbitrary y s ∈ Y ("s" stands for "source") with components
y s = ( ρ s , (v 1 ) s , (v 2 ) s , u + s , u − s , (ω 1 ) s , (ω 2 ) s , ( V 2+ ) s ),(96)δv 1 = − 1 |B|(x) (v 2 ) s (97) δv 2 = 1 |B|(x) (v 1 ) s (98) δu + = 1 |B|(x) u − s (99) δu − = − 1 |B|(x) u + s (100) δω 1 = − 1 2|B|(x) (ω 2 ) s (101) δω 2 = 1 2|B|(x) (ω 1 ) s .(102)
In particular, δω ⊥ = δω 1 e 1 + δω 2 e 2 = 1 2|B|(x) b × (ω ⊥ ) s . By Eq. (94), the Fourier harmonics of δ V 2+ are determined by the sequence of equations
( V + k ) s =|B|(x)δ V + k × b − k|B|(x)δ V − k (103) ( V − k ) s =|B|(x)δ V − k × b + k|B|(x)δ V + k ,(104)
where k is any integer greater than or equal to 2. For each k, this linear system may be
solved for (δ V + k , δ V − k ), giving δ V + k = 1 k|B|(x) bb · ( V − k ) s + k 2 k 2 − 1 (b × ( V − k ) s ) × b + k k 2 − 1 ( V + k ) s × b (105) δ V − k = − 1 k|B|(x) bb · ( V + k ) s + k 2 k 2 − 1 (b × ( V + k ) s ) × b − k k 2 − 1 ( V − k ) s × b .(106)
Finally, upon decomposing Eq. (87) into first-and higher-order harmonics, the first-order harmonics of δ ρ may be expressed as
δ ρ + 1 = 1 |B|(x) ( ρ s ) − 1 + 1 |B| 2 (x) u + s b + 1 2|B| 2 (x) (ω ⊥ ) s (107) δ ρ − 1 = − 1 |B|(x) ( ρ s ) + 1 + 1 |B| 2 (x) u − s b + 1 2|B| 2 (x) b × (ω ⊥ ) s ,(108)
and the k'th-order harmonics (k ≥ 2) of δ ρ may be expressed as
δ ρ + k = − 1 |B|(x) ( ρ s ) + k + 1 k|B|(x) δ V + k (109) δ ρ − k = 1 |B|(x) ( ρ s ) − k − 1 k|B|(x) δ V − k ,(110)
with δ V + k and δ V − k given in Eqs. (105) and (106). It is now apparent that D y f 0 (x, y) is invertible for all (x, y) with explicit inverse given by Eqs.
D. Finding The Slow Manifold in Loop Space
Now that loop space dynamics associated with the Lorentz force have been identified with a fast-slow system, Proposition 1 implies that there is a unique slow manifold in loop space given by the formal series y * ǫ = y * 0 + ǫy * 1 + ǫ 2 y * 2 + . . . . Interestingly, because Theorem 1 shows that the slow variable x lives in a finite-dimensional space, this slow manifold is finitedimensional. (The dimension is 7.) Therefore the series y * ǫ may be interpreted as describing the shape of rigid loops in loop space. The term "rigid" is appropriate in this case because the loops on the slow manifold are determined by only 6 real parameters. (The loop shape is independent of S.) The first two terms in shape function series y * ǫ = y * 0 + ǫy * 1 + ǫ 2 y * 2 + . . . may be computed as follows.
1. The Leading-Order Shape Function y * 0 According to Eq. (38) in Proposition 1, the leading-order shape functions for the slow loops in loop phase space, i.e. y * 0 , are given by
ρ * 0 (θ) = sin θ w ⊥ |B(x)| − cos θ w ⊥ × b(x) |B(x)| (111) (v ⊥ ) * 0 =0 (112) (u + ) * 0 =0 (113) (u − ) * 0 =0 (114) (ω ⊥ ) * 0 =0 (115) ( V 2+ ) * 0 =0.(116)
The leading-order dynamics on the slow manifold, i.e. the ǫ → 0 limit ofẋ = g ǫ (x, y * ǫ (x)), are therefore governed by
∂ t w 1 =u b(x) · R w 2 + 1 2 w 1 u b · ∇ln|B| + 1 2 w 2 u b · ∇ × b (117) ∂ t w 2 = − u b(x) · R w 1 − 1 2 w 1 u b · ∇ × b + 1 2 w 2 u b · ∇ln|B| (118) ∂ t u = − 1 2 |w ⊥ | 2 |B| b · ∇|B| (119) ∂ t x =ub(x). (120) S =|B(x)|(121)
Note that these leading-order slow evolution equations have the exact conservation laws
∂ t |w ⊥ | 2 2|B| = 0,(122)
corresponding the conservation of action, and
∂ t 1 2 u 2 + 1 2 |w ⊥ | 2 = 0,(123)
corresponding to the conservation of energy. These conservation laws may be recovered as limiting forms of the exact conservation laws for action and energy for loop space dynamics.
It follows that Eqs. Again referring to Proposition 1, the first-order shape functions are determined by the system of equations
0 = −|B(x)| (u − ) * 1 + ub(x) · ∇b · w ⊥ + 2b · cos θṽ 0 × δB 0 dθ (124) 0 = |B(x)| (u + ) * 1 + u b(x) · ∇b · (w ⊥ × b) + 2b · sin θṽ 0 × δB 0 dθ (125) 0 = 2|B(x)|(ω 2 ) * 1 + (cos θe 1 + sin θe 2 ) ·ṽ 0 × δB 0 dθ (126) 0 = −2|B(x)|(ω 1 ) * 1 + (cos θe 2 − sin θe 1 ) ·ṽ 0 × δB 0 dθ (127) 0 = |B(x)|(v 2 ) * 1 + e 1 · ṽ 0 × δB 0 dθ − u 2 b · ∇b · e 1 (128) 0 = −|B(x)|(v 1 ) * 1 + e 2 · ṽ 0 × δB 0 dθ − u 2 b · ∇b · e 2 (129) 0 = ( V 2+ ) * 1 × B(x) − |B(x)|∂ θ ( V 2+ ) * 1 + π 2+ (ṽ 0 × δB 0 ) (130) u |B| cos θ 1 2 τ w ⊥ + 1 2 k w ⊥ × b − w ⊥ × κ + u |B| sin θ − 1 2 k w ⊥ + 1 2 τ w ⊥ × b − κ · w ⊥ b = [(u + ) * 1 cos θ + (u − ) * 1 sin θ]b(x) + (cos θI + sin θb × ) · (ω ⊥ ) * 1 +( V 2+ ) * 1 − |B|∂ θ ρ * 1 ,(131)
where 0 in a subscript denotes evaluation using the leading-order shape function y * 0 , and I have introduced the useful shorthand notation
τ = b · ∇ × b (132) κ = b · ∇b (133) k = b · ∇ln|B|(134)
The solution of these equations may be found with the help of the identities
F L =ṽ 0 × δB 0 =ub × ( ρ * 0 · ∇B) + ρ 0 ρ 0 : ∇B B − |B| ρ 0 ρ 0 · ∇|B| (135) F L dθ = − |w ⊥ | 2 2|B| ∇|B| (136) cos θ F L dθ = − u 2|B| b × ([w ⊥ × b] · ∇B) (137) sin θ F L dθ = u 2|B| b × (w ⊥ · ∇B) (138) π 2+ ( F L ) = |w ⊥ | 2 2|B| [cc − aa] : (∇B b − ∇|B|I) cos 2θ − |w ⊥ | 2 2|B| [ac + ca] : (∇B b − ∇|B|I) sin 2θ(139)
where a = w ⊥ /|w ⊥ | and c = w ⊥ × b/|w ⊥ |. Explicitly, the solution is given by
(u + ) * 1 = − uκ · w ⊥ × b |B| (140) (u − ) * 1 = uκ · w ⊥ |B| (141) (ω 1 ) * 1 = u 4|B| 2 w ⊥ · (∇B · b × − b × · ∇B) · e 1 (142) (ω 2 ) * 1 = u 4|B| 2 w ⊥ · (∇B · b × − b × · ∇B) · e 2(143)(v 1 ) * 1 = − (µ 0 ∇|B| + u 2 κ) × b |B| · e 1(144)(v 2 ) * 1 = − (µ 0 ∇|B| + u 2 κ) × b |B| · e 2(145)( V + 2 ) * 1 = 1 2 µ 0 [ac + ca] : ∇b b + µ 0 [ac + ca] · ∇ln|B| (146) ( V − 2 ) * 1 = 1 2 µ 0 [cc − aa] : ∇b b + µ 0 [cc − aa] · ∇ln|B| (147) ρ * 1 = − u|w ⊥ | |B| 2 1 4 a · (∇b + b × · ∇b · b × ) + 1 2 k a − 1 2 τ c + 2κ · a b cos θ + u|w ⊥ | |B| 2 1 4 a · (∇b · b × − b × · ∇b) − 1 2 τ a − 1 2 k c − 2κ · c b sin θ − 1 4 |w ⊥ | 2 |B| 2 1 2 [cc −
where ( V 2+ ) * 1 = ( V + 2 ) * 1 cos θ + ( V − 2 ) * 1 sin θ. Note in particular that Eqs. (144)-(145) lead to the following improved expression for the time derivative of x on the slow manifold:
x = u b(x) − ǫ (µ 0 ∇|B| + u 2 κ) × b |B| + O(ǫ 2 ).(149)
The O(ǫ) correction term reproduces the famous ∇B and curvature drifts from guiding center theory. 4 Thus, evidence is mounting that loop dynamics restricted to the slow manifold is closely related to guiding center dynamics.
E. Slow Loops Move as Guiding Centers
Apparently an explanation is required for the low-order coincidence of guiding center dynamics with loop space dynamics on the slow manifold. In order to show that motion on the slow manifold in loop space is in fact equivalent to guiding center dynamics to all orders in perturbation theory, it is useful to draw upon Kruskal's description 8 of guiding center dynamics based on near-identity coordinate transformations. In Kruskal's approach, first a set of coordinates (ζ, ξ 1 , . . . , ξ 5 ) on the 6-dimensional phase space for a single charged particle is found with the following property: in these coordinates, the Lorentz force equation
takes the formζ = 1 ǫ Ω −1 (ξ) + Ω 0 (ζ, ξ) (150) ξ =U 0 (ζ, ξ),(151)ǫ ) −1 •Φ (N +1) ǫ = id+O(ǫ N ).
In addition, approximate decoupling means that in the new coordinates (ζ N , ξ N ) the Lorentz force equation takes the forṁ
ζ N = 1 ǫ Ω −1 (ξ N ) + Ω (N ) ǫ (ξ N ) + ǫ N δΩ (N ) ǫ (ζ N , ξ N ) (152) ξ N =U (N ) ǫ (ξ N ) + ǫ N δU (N ) ǫ (ζ N , ξ N ),(153)
where δΩ
ζ = 1 ǫ Ω −1 (ξ) + Ω ǫ (ξ) (154) ξ = U ǫ (ξ),(155)
where Ω ǫ , U ǫ are each formal power series in ǫ. Equations (154)
∂ t ζ(θ, t) + 1 ǫ ω c ( ζ(t), ξ(t)) ∂ θ ζ(θ, t) = 1 ǫ Ω −1 ( ξ(θ, t)) + Ω 0 ( ζ(θ, t), ξ(θ, t)) (156) ∂ t ξ(θ, t) + 1 ǫ ω c ( ζ(t), ξ(t)) ∂ θ ξ(θ, t) =U 0 ( ζ(θ, t), ξ(θ, t)) (157) ω c ( ζ, ξ) =Ω −1 ξ(θ ′ ) dθ ′ (158) S(t) = 1 ǫ ω c ( ζ(t), ξ(t)),(159)
where ζ(θ) may be chosen to be of the form ζ(θ) = θ + ν(θ) with a single-valued ν. Equiva- N (θ, t)).
lently, the transformed loop ( ζ N (θ), ξ N (θ)) = Φ (N ) ǫ ( ζ(θ), ξ(θ)) satisfies ∂ t ζ N (θ, t) + 1 ǫ ω c ( Φ (N ) ǫ ) −1 ( ζ N (t), ξ N (t)) ∂ θ ζ N (θ, t) = 1 ǫ Ω −1 ( ξ N (θ, t)) + Ω (N ) ǫ ( ξ N (θ, t)) + ǫ N δΩ (N ) ǫ ( ζ(θ, t), ξ N (θ, t)) (160) ∂ t ξ N (θ, t) + 1 ǫ ω c ( Φ (N ) ǫ ) −1 ( ζ N (t), ξ N (t)) ∂ θ ξ N (θ, t) = U (N ) ǫ ( ξ N (θ, t)) + ǫ N δU (N ) ǫ ( ζ N (θ, t), ξ N (θ, t)) (161) S(t) = 1 ǫ ω c ( Φ (N ) ǫ ) −1 ( ζ N (t), ξ N (t)) (162) where the loop ( Φ (N ) ǫ ) −1 ( ζ N (t), ξ N (t))(θ) = (Φ (N ) ǫ ) −1 ( ζ N (θ, t), ξ
This result is a simple corollary of the fact that constructing loop space dynamics commutes with applying coordinate transformations. Indeed, suppose thatż = Y (z) is the infinitesimal generator of a smooth dynamical system on a manifold M ∋ z, and let φ : M → M : z → z be a diffeomorphism. We may apply the diffeomorphism to M, thereby obtaining the transformed infinitesimal generator Y = φ * Y , and then construct the corresponding loop space dynamics:
∂ t z(θ, t) + Ω( z(t)) ∂ θ z(θ, t) = Y ( z(θ, t)).(163)
Equivalently, we may first construct loop space dynamics associated with Y :
∂ t z(θ, t) + Ω( z(t)) ∂ θ z(θ, t) = Y ( z(θ, t)),(164)
and then inquire as to the dynamics of the transformed loop z(θ) = φ( z(θ)). Applying the chain rule gives (163) with
Ω( z) = Ω( φ −1 ( z)), where φ −1 ( z)(θ) = φ −1 ( z(θ)).
It is therefore a small step to replace Φ (N ) ǫ in Lemma 1 with the formal all-orders transformation Φ ǫ , and thereby obtain the following expression for the loop space infinitesimal generator that is valid to all orders in perturbation theory.
Lemma 2. The infinitesimal generator for loop space dynamics associated with the Lorentz
force is equivalent to the formal series
∂ t ζ(θ, t) + 1 ǫ ω c ( Φ ǫ ) −1 ( ζ(t), ξ(t)) ∂ θ ζ(θ, t) = 1 ǫ Ω −1 ( ξ(θ, t)) + Ω ǫ ( ξ(θ, t)) (165) ∂ t ξ(θ, t) + 1 ǫ ω c ( Φ ǫ ) −1 ( ζ(t), ξ(t)) ∂ θ ξ(θ, t) =U ǫ ( ξ(θ, t)) (166) S(t) = 1 ǫ ω c ( Φ (N ) ǫ ) −1 ( ζ(t), ξ(t)) ,(167)
where ζ(θ) = θ + ν(θ).
This perturbative characterization of loop space dynamics is useful because (a) the coefficients of the all-orders guiding center equations appear explicitly, and (b) the fast-slow split for loop space dynamics has become especially simple.
Proposition 2. In terms of the formulation given in Lemma 2, the fast and slow variables for loop space dynamics associated with the Lorentz force are given by
x =(ν, ξ, S) (168) y =( ν, ̺),(169)
where S = ǫ S and
ν = ν(θ ′ ) dθ ′ (170) ξ = ξ(θ ′ ) dθ ′ (171) ν(θ) = ν(θ) − ν(θ ′ ) dθ ′ (172) ǫ ̺(θ) = ξ(θ) − ξ(θ ′ ) dθ ′ .(173)
Proof. According to Eqs. (165)-(166), the time derivatives of ν, ξ, and S are given bẏ
ν = 1 ǫ Ω −1 ( ξ(θ ′ )) dθ ′ − ω c ( Φ ǫ ) −1 ( ζ(t), ξ(t)) + O(1) (174) ξ = U ǫ ( ξ(θ ′ )) dθ ′ (175) S =ω c ( Φ (N ) ǫ ) −1 ( ζ(t), ξ(t))(176)
The quantityξ = O(1) because U ǫ = O(1). The quantityν = O(1) because Φ 0 = id and ω c is given by Eq. (158). The quantityṠ is obviously O(1). Therefore x = (ν, ξ, S) is a reasonable candidate for the slow variable.
The leading-order contributions to the time derivatives of ν and ̺ are given by
∂ t ν(θ, t) = − 1 ǫ Ω −1 (ξ(t)) ∂ θ ν(θ, t) + O(1) (177) ∂ t ̺(θ, t) = − 1 ǫ Ω −1 (ξ(t))∂ θ ̺(θ, t) + O(1).(178)
Therefore, if y = ( ν, ̺), ǫẏ = f 0 (x, y) + O(ǫ), with f 0 (x, y) given by
f 0 (x, y) =(˙ ν 0 ,˙ ̺ 0 ) (179) ν 0 = − Ω −1 (ξ) ∂ θ ν (180) ̺ 0 = − Ω −1 (ξ) ∂ θ ̺.(181)
It follows that the derivative D y f 0 (x, y) is a non-vanishing multiple of the identity, and that (x, y) comprise a fast-slow split for loop space dynamics.
In fact, the fast-slow split has become so simple that the coefficients defining the slow manifold, as well as the infinitesimal generator for the slow dynamics, may be computed explicitly to all orders in ǫ.
Theorem 2. The slow manifold associated with the fast-slow split given in Proposition 2 is given by y * ǫ = 0. The infinitesimal generator on the slow manifold is given bẏ
ν =Ω ǫ (ξ) − δω ǫ (ν, ξ) (182) ξ =U ǫ (ξ) (183) S =ω c ( Φ ǫ ) −1 ( ζ * (t), ξ * (t)) (184) where δω ǫ (ν, ξ) = 1 ǫ Ω −1 (ξ) − 1 ǫ ω c ( Φ ǫ ) −1 ( ζ * (t), ξ * (t)) ,(185)
and
ζ * (θ) =θ + ν (186) ξ * (θ) =ξ.(187)
Note that δω ǫ = O(1) as ǫ → 0 because Φ 0 = id.
Proof. Because the slow manifold is unique, it suffices to check that y * ǫ = 0 is a solution of the invariance equation. To that end, note that wherever y = 0, ξ(θ) = ξ. The right-handside of the invariance equation ǫDy * ǫ (x)[g ǫ (x, y * ǫ (x))] = f ǫ (x, y * ǫ (x)) therefore vanishes when y * ǫ = 0 because both Ω −1 (ξ) + ǫΩ ǫ (ξ) and U ǫ (ξ) are independent of θ. Being linear in y * ǫ , the left-hand-side of the invariance equation also vanishes. The asymptotic series y * ǫ = 0 is therefore the unique slow manifold.
Theorem 2 establishes the equivalence between dynamics on the slow manifold in loop space and all-orders guiding center dynamics, Eqs. (154)-(155), upon making the simple identifications ξ = ξ, ζ = ν − S/ǫ, and then noting that the evolution of S decouples from the evolution of (ν, ξ) on the slow manifold. An immediate corollary of this observation is that ν represents the so-called adiabatic phase 21-23 associated with gyromotion.
IV. HAMILTONIAN STRUCTURE ON THE SLOW MANIFOLD
A superconducting ring conserves the magnetic flux threading the ring's center. Charged particles in strong magnetic fields approximately exhibit the same property, although in that context the phenomenon is usually referred to as adiabatic invariance instead of superconductivity. There is good reason for this change in nomenclature; regardless of the strength of the magnetic field, a charged particle is emphatically not a superconducting ring. Nevertheless, the proximity of the two concepts, flux conservation on the one hand and adiabatic invariance on the other, begs the following question. Since charged particles are not flux-conserving superconductors, what is the physical explanation for the behavioral similarity between the two sorts of objects?
The answer to the question is symmetry. Charged particle dynamics exhibit an approximate symmetry that, according to Noether's theorem, implies the presence of a conserved quantity that happens to be numerically equal to the magnetic flux at leading-order in perturbation theory. This explanation is of course very old and well-known. (See, for instance, Section E.5 of Ref. 8) However, the usual way of exhibiting this symmetry is rather technical and laborious, and therefore not as illuminating as one might hope from the physical point of view. In this final technical Section, I would like to elucidate the symmetry underlying a particle's approximate superconductivity, sometimes referred to suggestively as "gyrosymmetry," 24 in a manner that makes the symmetry itself appear almost obvious.
Naturally, I aim to do this using the loop space picture of guiding center dynamics that has been the subject of this article.
The starting point for this demonstrating is the action functional (24) for loop space dynamics associated with the Lorentz force. I aim to show that obvious symmetries of this functional ultimately give rise to the symmetry of particle dynamics associated with adiabatic invariance. To that end, there are two obvious symmetries worth discussing. (1) Because the Lebesgue measure on the circle dθ is translation invariant, the value of the action does not change when the loop ( x, v) is subject to the phase shift
x → x ψ 1 (188) v → v ψ 1 ,(189)
where ψ 1 ∈ S 1 is any angle. (2) Because the phase function S only appears in the action via its time derivative, the value of the action is also unchanged when S is translated according
to S → S + ψ 2 ,(190)
where ψ 2 ∈ S 1 is another arbitrary angle. This pair of obvious symmetries may be conveniently encoded as a single T 2 ≡ S 1 × S 1 -action on the phase space for loop dynamics
ℓP × S 1 , Ψ ψ 1 ,ψ 2 ( x, v, S) = ( x ψ 1 , v ψ 1 , S + ψ 2 ).(191)
I will now show that Ψ restricted to the subgroup ψ 2 = 0 is precisely the symmetry responsible for a charged particle's adiabatic invariance.
First it is convenient to leave behind the action functional (24) in favor of the 1-form Ξ on ℓP × S 1 given by
Ξ( x, v, S)[˙ x,˙ v,Ṡ] = 1 ǫ A( x(θ)) + v(θ) ·˙ x(θ) dθ +Ṡ 1 ǫ A( x(θ) + v(θ) · ∂ θ x(θ) dθ.(192)
This 1-form is clearly related to the action functional (24), for
A( x, v, S) =ˆt 2 t 1 Ξ( x, v, S)[˙ x,˙ v,Ṡ] dt −ˆt 2 t 1 H( x, v) dt,(193)
where H is the loop energy defined in Eq. (13). In fact, if X = (˙ x,˙ v,Ṡ) denotes the infinitesimal generator for loop space dynamics, c.f. Eqs. (20)- (22), the Euler-Lagrange equations associated with A may be written The invariance of A under the T 2 -action Ψ is equivalent to the pair of pullback relations
ι X dΞ = −dH,(194)Ψ * ψ 1 ,ψ 2 Ξ =Ξ (195) Ψ * ψ 1 ,ψ 2 H =H.(196)
Therefore, if we define the infinitesimal generators
∂ 1 ( x, v, S) = d dλ 0 Ψ λ,0 ( x, v, S) (197) ∂ 2 ( x, v, S) = d dλ 0 Ψ 0,λ ( x, v, S),(198)
Noether's theorem implies the functionals
I 1 =ι ∂ 1 Ξ (199) I 2 =ι ∂ 2 Ξ(200)
are each constant along trajectories of loop space dynamics. Curiously, these functionals are each equal to the (normalized) loop action (14),
I 1 = I 2 = 1 ǫ A( x(θ) + v(θ) · ∂ θ x(θ) dθ.(201)
Now consider the slow manifold Γ ⊂ ℓP × S 1 . Let F t denote the loop space dynamics flow. Being a formally invariant set, the flow on loop space maps Γ into itself, i.e. F t (Γ) = Γ.
Therefore the set Ψ ψ 1 ,ψ 2 (Γ) ≡ Γ ψ 1 ,ψ 2 satisfies F t (Γ ψ 1 ,ψ 2 ) = F t (Ψ ψ 1 ,ψ 2 (Γ)) = Ψ ψ 1 ,ψ 2 (F t (Γ)) = Γ ψ 1 ,ψ 2 ,(202)
where I have used the commutativity of the flow F t and the T 2 -action Ψ ψ 1 ,ψ 2 implied by Eqs. (195)-(196). In other words Γ ψ 1 ,ψ 2 is an invariant set for each (ψ 1 , ψ 2 ) ∈ T 2 . In the fast-slow coordinates (x, y) on ℓP × S 1 , and for sufficiently small (ψ 1 , ψ 2 ), Γ ψ 1 ,ψ 2 is therefore a formally invariant set given as the graph of some function Y ψ 1 ,ψ 2 (x). Moreover, the fact that Ψ ψ 1 ,ψ 2 does not depend on ǫ implies that Y ψ 1 ,ψ 2 must be a formal power series in ǫ. By the uniqueness of the slow manifold, this means that Γ ψ 1 ,ψ 2 = Ψ ψ 1 ,ψ 2 (Γ) = Γ is equal to the slow manifold, i.e. that Γ is
γ * (ι X dΞ) = −γ * dH ⇒ι X Γ dΞ Γ = −dH Γ ,(203)
where X Γ is the infinitesimal generator X restricted to the slow manifold Γ, and Ξ Γ , H Γ are the pullbacks of the 1-form Ξ and functional H along γ. Because Γ is T 2 -invariant, the pullback relations (195)-(196) imply analogous pullback relations on Γ:
Ψ * ψ 1 ,ψ 2 Ξ Γ = Ξ Γ (204) Ψ * ψ 1 ,ψ 2 H Γ = H Γ .(205)
Noether's theorem applied to Ψ ψ 1 ,ψ 2 restricted to the slow manifold therefore implies that
J 1 = ι ∂ 1 Ξ Γ (206) J 2 = ι ∂ 2 Ξ Γ .(207)
Are each constant along trajectories contained in the slow manifold. Because ι ∂ k Ξ Γ = γ * (ι ∂ k Ξ) = γ * I k , J 1 and J 2 are each equal to the normalized loop action restricted to the slow manifold. Because dynamics on the slow manifold is the same thing as guiding center dynamics, the obvious symmetry Ψ ψ 1 ,ψ 2 on the phase space for loop dynamics is now shown to be responsible for a nontrivial conservation law for guiding center dynamics. My argument will therefore be complete if I can show that J 1 = J 2 = J is equal to the magnetic flux at leading-order in ǫ.
To that end, I will demonstrate even more by explicitly recovering the Hamiltonian formulation of guiding center dynamics due to Littlejohn from the restricted 1-form Ξ Γ and the restricted Hamiltonian H Γ . First note that the equality of the two Noether invariants J 1 , J 2 has the remarkable consequence that the difference of the infinitesimal generators ∆ = ∂ 1 − ∂ 2 lies in the kernel of the closed 2-form dΞ Γ . Indeed,
ι ∆ dΞ Γ = ι ∂ 1 dΞ Γ − ι ∂ 2 dΞ Γ = L ∂ 1 Ξ Γ − L ∂ 2 Ξ Γ − dJ 1 + dJ 2 = 0.(208)
(Note that the identity L ∂ k Ξ Γ = 0 follows from differentiating Eq. (195).) Therefore dΞ Γ is not a symplectic form. This suggests that in order to recover Littlejohn's symplectic formulation of guiding center dynamics, it is necessary to first quotient ℓP × S 1 by the foliation tangent to the kernel of dΞ Γ .
Because it is not immediately clear whether the dimension of dΞ Γ 's characteristic foliation is greater than 1, it is helpful to first quotient by the subfoliation tangent to ∆. By a slight abuse of notation, I will denote the latter foliation by ∆. If the descendent of dΞ Γ on Γ/∆ is non-degenerate, this would imply that ∆ frames the kernel of dΞ Γ and that the quotient by ∆ produces a symplectic space. Otherwise, a further quotient may be necessary. In the case at hand it is reasonable to suspect that only the quotient by ∆ is necessary; because the dimension of the slow manifold Γ is 7 and the dimension of the foliation tangent to ∆ is 1, the quotient by ∆ will be 6-dimensional, which is the same dimension as Littlejohn's symplectic phase space.
In order to explicitly carry out the quotient by ∆, it is necessary to have an explicit expression for Ψ ψ 1 ,ψ 2 restricted to the slow manifold, and especially useful to have this expression in the natural coordinates x on Γ. To find this expression, observe that because Γ is T 2 -invariant, there must be a mapping ϕ ψ 1 ,ψ 2 : X → X such that
Ψ ψ 1 ,ψ 2 (x, y * ǫ (x)) = (ϕ ψ 1 ,ψ 2 (x), y * ǫ (ϕ ψ 1 ,ψ 2 (x)))(209)
for all x ∈ X. The mapping ϕ ψ 1 ,ψ 2 : X → X is precisely the T 2 -action restricted to Γ expressed in the coordinates x. Also observe that after applying Ψ ψ 1 ,ψ 2 to an arbitrary point (x, y), the slow variable x = (x, u, w 1 , w 2 , S) transforms according to
x → x (210) u → u (211) w 1 → w 1 cos ψ 1 + w 2 sin ψ 1 (212) w 2 → w 2 cos ψ 1 − w 1 sin ψ 1 (213) S → S + ǫψ 2 .(214)
(The transformation rules for w 1 , w 2 may be summarized in vector notation as w ⊥ → w ⊥ cos ψ 1 + w ⊥ × b sin ψ 1 .) In particular, the transformation of the slow variable x does not depend on the fast variable y. It follows that the T 2 -action on x-space is given by
ϕ ψ 1 ,ψ 2 (x, u, w 1 , w 2 , S) = (x, u, w 1 cos ψ 1 + w 2 sin ψ 1 , w 2 cos ψ 1 − w 1 sin ψ 1 , S + ǫψ 2 ),(215)
whence the quotient by ∆, π : Γ → Γ/∆, may be identified as
π(x, u, w 1 , w 2 , S) = (x, u, u 1 , u 2 ),(216)
where u ⊥ = u 1 e 1 (x) + u 2 e 2 (x) = cos(S/ǫ)w ⊥ + sin(S/ǫ)w ⊥ × b.
In particular, a useful section of π is given by s : Γ/∆ → Γ, s(x, u, u 1 , u 2 ) = (x, u, u 1 , u 2 , 0).
Because ι ∆ dΞ Γ = 0 and Ψ * ψ 1 ,ψ 2 Ξ Γ = Ξ Γ , the 2-form dΞ Γ on Γ descends to the 2-form dΞ Γ/∆ on Γ/∆, where the 1-form Ξ Γ/∆ = s * Ξ Γ = s * γ * Ξ = (γ • s) * Ξ. An explicit expression for Ξ Γ/∆ modulo an exact 1-form is
Ξ Γ/∆ (x, u, u 1 , u 2 )[ẋ,u,u 1 ,u 2 ] = 1 ǫ A(x) + ub(x) + v * ⊥ǫ +ˆ1 0 B(x + λǫ ρ * ǫ ) × ρ * ǫ dθ dλ ·ẋ + ǫ v * ǫ +ˆ1 0 B(x + λǫ ρ * ǫ ) × ρ * ǫ λdλ · D ρ * ǫ [ẋ,u,u 1 ,u 2 ] dθ(219)= 1 ǫ A(x) + ub(x) + ǫW ·ẋ + ǫ µ 0 u 2 du 1 − u 1 du 2 u 2 1 + u 2 2 + O(ǫ 2 ),(220)where µ 0 = |u ⊥ | 2 2|B(x)| and W = − (µ 0 ∇|B| + u 2 κ) × b |B| + 1 2 ( ρ * 0 · ∇B) × ρ * 0 dθ + 1 2 (∇ ρ * 0 ) · v * 0 dθ = − 3 2 µ 0 ∇|B| × b |B| − u 2 κ × b |B| − 1 2 µ 0 τ b − µ 0 R.(221)
Upon taking an exterior derivative, Eq. (220) reproduces the symplectic form for guiding center theory derived by Littlejohn modulo terms of O(ǫ) in W . This is not a contradiction.
Littlejohn's derivation made use of near-identity coordinate transformations that are not uniquely determined, i.e. the transformations depended on a number of arbitrary parameters. Different choices for those parameters would be necessary to recover the result (220) from the near-identity coordinate transformation approach.
The residual part of the symmetry Ψ that survives when passing to the quotient is the transformation u 1 →u 1 cos ψ + u 2 sin ψ (222)
u 2 →u 2 cos ψ − u 1 sin ψ.(223)
According to Eq.(220), the conserved quantity associated with this residual symmetry is given by
J 1 = ǫµ 0 + O(ǫ 2 ).(224)
Because µ 0 = |B|| ρ * 0 | 2 /2, J 1 is proportional to the magnetic flux passing through the loop x + ǫ ρ * 0 (θ), as claimed. The conclusion is that the adiabatic invariant for charged particle dynamics in a strong magnetic field is the Noether conserved quantity associated with the symmetry of loop space dynamics under phase shift z → z ψ .
V. DISCUSSION
The loop space picture of guiding center dynamics developed in this article is closely
x(t) = ∞ k=−∞ ǫ |k| X k (t) exp(ikC(t)/ǫ)(225)
for the spatial location of a charged particle, where X k and C were allowed to be formal power series in ǫ. Hazeltine and Waelbroeck introduced the ansatz
x(t) =X(t) + ǫρ(X(t), U (t), t, γ(t)) (226) v(t) =U (t) + u(X(t), U (t), t, γ(t)),(227)
for the spatial location and velocity of a charged particle, where U , ρ, u, γ were allowed to be formal power series in ǫ, and the profile functions ρ, u were assumed periodic in γ with zero average. Of course, being periodic, ρ, u may also be written
ρ = k =0 ρ k (X(t), U (t), t) exp(ikγ) (228) u = k =0 u k (X(t), U (t), t) exp(ikγ),(229)
which establishes a close link to Kruskal's ansatz (225). Apparently each of these representations of the solution to Newton's equation ǫẍ =ẋ × B(x) involve evaluating a time dependent, parameterized loop (characterized either by the coefficients X k or ρ k , u k ) at a rapidly rotating phase (either C or γ). Therefore there can be no doubt that loop space is playing a role in each of these approaches.
On the other hand, these approaches differ from the slow manifold approach described in this article for the following reasons.
(1) Neither approach recognizes the link between its collection of formal asymptotic series and the non-perturbative notion of loop space dynamics. In particular, neither approach establishes that guiding center dynamics is equivalent to loop space dynamics restricted to a slow manifold. The conceptual framework supported by the slow manifold picture is therefore missing from Kruskal's and Hazeltine and Waelbroeck's work. For instance, as discussed in Section IV, the genuine simplicity of the symmetry underlying adiabatic invariance only manifests itself in the context of loop space dynamics. Moreover, the prospect of using the numerical method from Ref. 15 to capture high-order guiding center effects without resorting to the laborious machinations of perturbation theory would not emerge in absence of the slow manifold picture.
(2) While in Kruskal's approach each of the X k 's is expanded in an asymptotic series, in the slow manifold approach only the fast variable restricted to the slow manifold y = y * ǫ is expanded in such a manner; the slow variable x is never subject to asymptotic expansion.
In fact Kruskal does not identify the fast slow split given in Theorem 1 at all. The main drawback of this "expand everything" approach is that it obscures the phase space geometry underlying the problem with needless additional algebraic manipulations. Indeed, Kruskal comments that additional technical work due to Gardner and Berkowitz is required to prove that his calculation produces a valid asymptotic expansion of a solution to Newton's equation. In contrast, from the perspective of fast-slow systems the error estimates required to prove such a result may be formulated in a general context using little more than Gronwall's inequality. 14 That said, Hairer and Lubich 25 have recently managed to apply Kruskal's ansatz to the problem of adiabatic invariance for a particular structure-preserving numerical integrator for charged particle dynamics; the ansatz is used in a proof that the integrator preserves a modified adiabatic invariant over extremely large time intervals.
(3) While Hazeltine and Waelbroeck aim to parameterize the fluctuating position ρ and fluctuating velocity u using the mean position X and mean velocity U , the slow manifold approach parameterizes the fast variable y using the slow variable x = (X, U ). In particular, the parameter x involves pieces of the first harmonic of the fluctuating velocity, and does not involve the perpendicular components of the mean velocity. A first guess at a resolution of this apparent discrepancy is that the implicit function theorem might be able reparameterize the graph y = y * ǫ (x) by the variables (x, u, v 1 , v 2 ), which are equivalent to Hazeltine and Waelbroeck's (X, U ). However, expressions (144)-(145) show that such an inversion is very poorly conditioned in general, and impossible in a uniform magnetic field. It therefore seems (X, U ) is a problematic choice for parameterizing the fluctuating position and velocity. That this is true can also be seen in the details of the guiding center calculation presented in Chapter 2 of Ref. 7. While the goal of the calculation was to determine the dependence of ρ and u on (X, U ), an unexpected constraint on the perpendicular components of U appears in Eq. (2.30). In addition, as a consequence of mischaracterizing the general solution of Eq. (2.28), the derivation fails to recognize that there are two undetermined parameters (instead of one) in the leading-order contribution to u. If the roles of the constrained components of U and the unconstrained components of u were merely exchanged, the method of Hazeltine and Waelbroeck would reproduce the steps in computing the slow manifold in loop space. In other words, while the goal of the calculation in Ref. 7 seems to be flawed, the calculation itself seems to be suggesting that identifying the fast slow split as in Theorem 1 is the way to fix the problem! It is also interesting to compare the approach used in Section IV to identify the noncanonical Hamiltonian structure of guiding center dynamics with Littlejohn's approach in Refs. 5 and 26. Because Littlejohn worked in particle space rather than loop space, his strategy for identifying the Hamiltonian structure was to exploit the coordinate covariance of the symplectic Hamilton's equation, i.e. that the equation ι X dθ = −dH has the same form in any coordinate system. In contrast, the strategy used in Section IV to find the Hamiltonian structure made use of the form-invariance of the (pre-)symplectic Hamilton's equation under restriction to invariant sets.
In a forthcoming publication, I will report on exploiting the slow manifold picture of guiding center dynamics for the sake of building a novel numerical scheme for efficiently simulating the slow drift of strongly magnetized charged particles. I have managed to show that applying the implicit-midpoint time integration scheme to loop space dynamics expressed in terms of the fast and slow variables (x, y) leads to a nonlinearly implicit energyconserving scheme that does not suffer from the preconditioning problem that usually plagues implicit integrators applied to stiff problems. Moreover, the scheme is provably accurate when taking timesteps much larger than the cyclotron period ǫ provided initial conditions are chosen to lie approximately on the slow manifold. This integrator is currently being optimized for the purpose of employing Gear et. al 's technique for numerically selecting initial conditions on the slow manifold with any desired accuracy. The ultimate goal of this undertaking is to develop the first charged particle simulation tool that is able to resolve high-order guiding center effects while stepping over the cyclotron period. Such high-order effects appear to play an important role in the dynamics of so-called runaway electrons generated in magnetic traps. 27
VI. ACKNOWLEDGEMENTS
Research presented in this article was supported by the Los Alamos National Laboratory LDRD program under project number 20180756PRD4.
Theorem 1 .
1When written in terms of the dependent variables x = (x, u, w 1 , w 2 , S) andy = ( ρ, v 1 , v 2 , u + , u − , ω 1 , ω 2 , V 2+ ),loop space dynamics associated with the Lorentz force is a fast-slow dynamical system.Proof. By Eqs. (63)-(70), the evolution equation for y is ǫẏ = f ǫ (x, y), where f ǫ depends
and consider solving the equation D y f 0 (x, y)[δy] = y s for δy. By Eqs. (88)-(93), (δv 1 , δv 2 ), (δu + , δu − ), and (δω 1 , δω 2 ) may be expressed in terms of y s by solving three separate 2 × 2 matrix equations, giving the result
Remark 3 .
3The factor of 2 appearing in (ω 1 ) 0 and (ω 2 ) 0 is caused by spinning the loops by the phase S.
(119) and (120) are identical to the leading-order equations describing guiding center dynamics. (See the discussion below Eq. (15) in Ref. 20) 2. The First-Order Shape Function y * 1
where ξ = (ξ 1 , . . . , x 5 ) and Ω −1 is nowhere vanishing. Here ζ is an angular variable, and the functions Ω 0 , U 0 are 2π-periodic in ζ. Next a sequence of near-identity coordinate transformations Φ (N ) ǫ : (ζ, ξ) → (ζ N , ξ N ) is found that decouples ζ from ξ with increasing accuracy. Here near-identity means Φ
=
O(1) as ǫ → 0. Assuming the magnetic field is C ∞ , the integer N may in principle be made as large as one would like. Therefore the difference between Eqs. (152) -(153) and those same equations with δΩ (N ) ǫ , δU (N ) ǫ = 0 may be be made arbitrarily small.The system of equations given by dropping δΩ : (ζ, ξ) → (ζ, ξ) that decouples ζ from ξ to all orders in ǫ. The formal power series for the transformed Lorentz force infinitesimal generator is given bẏ
-(155) are the all-orders guiding center equations. The first N terms in the all-orders guiding center equations agree with the guiding center equations of order N for each N. I will now demonstrate that the formal slow dynamics on the slow manifold in loop space agrees with the all-orders guiding center equations. As a way to motivate my argument, consider first the following characterization of loop space dynamics associated with the Lorentz force in terms of the special coordinate systems used in Kruskal's theory. Lemma 1. The coordinates (ζ, ξ) for the Lorentz force may be chosen so that loop space dynamics associated with the Lorentz force are equivalent to
where d denotes the exterior derivative on ℓP × S 1 . Equation (194) is an example of a presymplectic Hamilton's equation. Therefore Ξ, together with H, geometrically encode the information contained in the functional A.
T 2
2-invariant to all orders in ǫ. (Compare this argument with Kruskal's "Theorem of phase independence" in Section C.1 of Ref. 8.) Let γ : Γ → ℓP × S 1 be the inclusion map. Because Γ is a formally invariant set, the presymplectic Hamilton's equation (194) implies
related to the nonlinear WKB expansion of Kruskal 1 and the two-timescale technique described by Hazeltine and Waelbroeck in Ref. 7. Kruskal introduced the ansatz
aa] : ∇b b + [cc − aa] · ∇ln|B| cos 2θ+
1
4
|w ⊥ | 2
|B| 2
1
2
[ac + ca] : ∇b b + [ac + ca] · ∇ln|B| sin 2θ,
The gyration of a charged particle. M , PM-S-33 (NYO-7903Princeton UniversityProject Matterhorn ReportM. Kruskal, "The gyration of a charged particle," Project Matterhorn Report PM-S-33 (NYO-7903) (Princeton University, 1958).
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]
| [
"Alok Kumar ",
"Jagadev ",
"\nSOA University\nJagamohan Nagar\n",
"\nSOA University\nJagamohan NagarBhubaneswar -751030Jagamara, Jagamara, Bhubaneswar -751030\n",
"\nSOA University\nJagamohan Nagar\n",
"\nSOA University\nJagamohan Nagar, Jagamara, Bhubaneswar751030Jagamara, Bhubaneswar -751030\n",
"Alok Kumar ",
"Jagadev ",
"\nSOA University\nJagamohan Nagar\n",
"\nSOA University\nJagamohan NagarBhubaneswar -751030Jagamara, Jagamara, Bhubaneswar -751030\n",
"\nSOA University\nJagamohan Nagar\n",
"\nSOA University\nJagamohan Nagar, Jagamara, Bhubaneswar751030Jagamara, Bhubaneswar -751030\n"
]
| [
"SOA University\nJagamohan Nagar",
"SOA University\nJagamohan NagarBhubaneswar -751030Jagamara, Jagamara, Bhubaneswar -751030",
"SOA University\nJagamohan Nagar",
"SOA University\nJagamohan Nagar, Jagamara, Bhubaneswar751030Jagamara, Bhubaneswar -751030",
"SOA University\nJagamohan Nagar",
"SOA University\nJagamohan NagarBhubaneswar -751030Jagamara, Jagamara, Bhubaneswar -751030",
"SOA University\nJagamohan Nagar",
"SOA University\nJagamohan Nagar, Jagamara, Bhubaneswar751030Jagamara, Bhubaneswar -751030"
]
| [
"/ (IJCSE) International Journal on Computer Science and Engineering",
"/ (IJCSE) International Journal on Computer Science and Engineering"
]
| Wide implementation of IEEE 802.11 based networks could lead to deployment of localized wireless data communication environments with a limited number of mobile hosts, called ad hoc networks. Implementation of a proper routing methodology in ad hoc networks makes it efficient in terms of performance. A wide spectrum of routing protocols has been contributed by several researchers. Real time applications have been most popular among the applications, run by ad hoc networks. Such applications strictly adhere to the Quality of Service (QoS) requirements such as overall throughput, end-toend delay and power level. Support of QoS requirements becomes more challenging due to dynamic nature of MANETs, where mobility of nodes results in frequent change in topology. QoS aware routing protocols can serve to the QoS support, which concentrate on determining a path between source and destination with the QoS requirements of the flow being satisfied. We propose a protocol, called Power and Delay aware Temporally Ordered Routing Algorithm (PDTORA), based on Temporally Ordered Routing Algorithm (TORA) Protocol, where verification of power and delay requirements is carried out with a query packet at each node along the path between source and destination. Simulations justify better performance of the proposed new protocol in terms of network lifetime, end-to-end delay and packet delivery ratio as compared to TORA. | null | [
"https://arxiv.org/pdf/1012.0887v2.pdf"
]
| 18,451,390 | 1012.0887 | 7e9ad544bf2fa502da35c3d19c9125f99717c56b |
Power and Delay Aware On-Demand Routing For Ad Hoc Networks
2010
Alok Kumar
Jagadev
SOA University
Jagamohan Nagar
SOA University
Jagamohan NagarBhubaneswar -751030Jagamara, Jagamara, Bhubaneswar -751030
SOA University
Jagamohan Nagar
SOA University
Jagamohan Nagar, Jagamara, Bhubaneswar751030Jagamara, Bhubaneswar -751030
Power and Delay Aware On-Demand Routing For Ad Hoc Networks
/ (IJCSE) International Journal on Computer Science and Engineering
02042010Powerdelayon-demand routingad hoc networks
Wide implementation of IEEE 802.11 based networks could lead to deployment of localized wireless data communication environments with a limited number of mobile hosts, called ad hoc networks. Implementation of a proper routing methodology in ad hoc networks makes it efficient in terms of performance. A wide spectrum of routing protocols has been contributed by several researchers. Real time applications have been most popular among the applications, run by ad hoc networks. Such applications strictly adhere to the Quality of Service (QoS) requirements such as overall throughput, end-toend delay and power level. Support of QoS requirements becomes more challenging due to dynamic nature of MANETs, where mobility of nodes results in frequent change in topology. QoS aware routing protocols can serve to the QoS support, which concentrate on determining a path between source and destination with the QoS requirements of the flow being satisfied. We propose a protocol, called Power and Delay aware Temporally Ordered Routing Algorithm (PDTORA), based on Temporally Ordered Routing Algorithm (TORA) Protocol, where verification of power and delay requirements is carried out with a query packet at each node along the path between source and destination. Simulations justify better performance of the proposed new protocol in terms of network lifetime, end-to-end delay and packet delivery ratio as compared to TORA.
I. INTRODUCTION
Wide implementation of IEEE 802.11 based wireless networks could lead to deployment of localized wireless data communication environments called ad hoc networks. Such networks do not support wired communication and fixed infrastructure as well. The wireless nodes in MANETs are allowed to run applications, which share data of different types and characteristics. Applications running on MANETs may possess different characteristics like network size, frequency of topology change, communication requirements and data characteristics. Every node lies within the coverage area of the MANET and can communicate with any other node in the network within its own transmission range. However, nodes are free to move within the coverage area of the MANET. A node is allowed to communicate with another node not lying within its transmission range, via multi-hop routes, where each node along the route acts as a router of the message. At the same time, new nodes can join the network any time and existing nodes can leave the network any time too.
Design of communications and routing protocols becomes a challenging factor due to dynamic nature of MANETs. One of the major challenges arises around design of multi-hop routing communication protocols. Most of the existing routing protocols such as Dynamic Source Routing (DSR) protocol, Ad hoc On-demand Distance Vector (AODV) protocol, Temporally ordered Routing Algorithm (TORA) protocol and many other such protocols mostly rely on best effort service [1]. However, a best effort service may not be able to fulfill the purposes in routing for multimedia and real time applications, which strictly require the network to adequately provide the guarantees to QoS.
A variety of routing protocols has been proposed by different authors that effectively support multi-hop communications in MANETs. Such protocols can be globally categorized as: on-demand or reactive protocols like DSR, AODV and TORA; table-driven or proactive protocols such as Destination Sequenced Distance Vector protocol (DSDV). In on-demand routing protocols, a route is established between the required source and destination prior to the communication and removed after the communication is over. In table-driven routing protocols, each node implements a routing table, which permanently stores the routing information to all possible destinations, irrespective of whether a communication is initiated or not. In regular updates of routing information for routes, which might not be used for a longtime, and subsequently incurs a convincing overhead. In addition, this approach needs more memory space for the routing table, as more and more routing information are appended to the routing table. Another routing approach, called Zone Routing Protocol (ZRP), has been proposed, which incorporates the benefits of on-demand as well as table-driven approaches. It implements a proactive table-driven strategy for route establishment among the nodes of the same zone, and an on-demand reactive strategy is used for establishment of communication between nodes belonging to different zones. Such a protocol can be effectively implemented in larger ad hoc networks, where the applications exhibit a high degree of locality of communications, i.e. node with close proximity to each other communicate more frequently than the nodes lying farther.
The following sections of the paper are organized as follows. Section II covers a brief review of existing QoS aware routing protocols. In Section III, a power and delay aware TORA (PDTORA) protocol is described. Section IV comprises the simulations, Section V concludes the paper, and Section VI includes the probable future enhancements.
II. BACKGROUND
In QoS-aware routing protocols, the principal goal is to determine a path between a source and the desired destination with the specified QoS requirements being satisfied. The basic constraints around determining a QoS-aware path are minimal search, distance and trace conditions. Obviously, a QoS-aware routing protocol is so called since it performs path selection on the basis of a specified QoS. A brief overview of QoS aware routing protocols for MANETs is included in this section.
In [2], the authors have proposed Power Aware Multiple Access (PAMAS) protocol, where a node can switch off its radio link for a specific duration of time, if it perceives that it would not be able to send or receive packets due to multiple access interferences. Authors in [3] have introduced poweraware metrics resulting in power-efficient routes. Such metrics include maximizing the time of network partition and reducing the variance in power levels of nodes. These metrics can be directly implemented in a network with a centralized control, which can possibly use a routing algorithm, based on minimizing the power level (power per bit) to transmit a packet between the source and destination. One such routing algorithm proposed by authors in [4], conditional max-min battery capacity routing algorithm, chooses a route with minimal transmission power, where all nodes along the route, possess remaining battery capacity higher than a predefined threshold. In case at least one of the nodes along a route does not satisfy to the required minimum battery capacity, the route is rejected. QoS routing protocols such as Core Extraction Distributed Ad hoc Routing (CEDAR) protocol [5] are implemented for small to medium size ad hoc networks comprising tens to hundreds of nodes, where first of all, the core of the network is dynamically established and then the link states of stable high-bandwidth links are propagated to the nodes of the core. An on-demand route computation is performed using the local state by the nodes of the core. An on-demand route computation is performed using the local state by the nodes of the core. A Time Division Multiple Access (TDMA) based computation of available bandwidth for ad hoc networks is proposed by authors of [6]. It performs end-to-end bandwidth allocation following computation. This approach enables the source node to determine the availability of resources to support the desired QoS requirements.
Authors in [7] have proposed QoS-TORA protocol, based on link reversal best effort protocol TORA, which is designed for a TDMA network. In this approach, measurement of bandwidth of a link is made in terms of slot reservations during the data phase of a TDMA frame. It is implemented in Medium Access Control (MAC) as well as network layers. Simulations demonstrate its capability to establish a route with end-to-end QoS being maintained. It would be observed from the simulations that QoS-TORA provides a better throughput under highly mobile environments. Optimized Link State Routing (OLSR) protocol based solution proposed by authors in [8], performs delay and throughput aware QoS routing, and demonstrates better results in packet delivery ratio, packet loss ratio and delay, as compared to that in OLSR.
INORA, a QoS routing protocol [9], incorporates the features of INSIGNIA and TORA. In particular, it makes use of in-band signaling mechanism of INSIGNIA and QoS routing mechanism of TORA. QoS signaling in INORA is used to reserve and release resources, to set up, tear down and renegotiate flows in the network. This signaling mechanism operates independent of TORA routing protocol. In this approach, first of all, TORA determines a route between source and destination, and then, the signaling mechanism (INSIGNIA) performs reservation of resources along the route provided by TORA.
III. POWER AND DELAY AWARE TORA (PDTORA)
Implementation of QoS routing protocols in ad hoc networks serves to fulfill the purpose of reservation of sufficient resources along a route so as to meet the QoS requirements of a flow. On the other hand, the QoS routing protocol should be able to find the path that consumes minimum resources [10]. QoS metrics vary from application to application. Major QoS metrics for ad hoc networks are available bandwidth, cost, end-to-end delay, power, packet loss ratio and so on. The QoS metrics can be generally classified as, additive metrics, concave metrics and multiplicative metrics.
For a given link (s,d), let q(s,d) be the performance metric, with s as the source and d as the destination nodes. The path (s,s 1 ,s 2 ,….,s k , d ) connects s and d. A given constraint is said to be additive, if q (s,d) = q (s,s 1 ) + q(s 1 ,s 2 )+ …….. + q(s k ,d). Thus, end-to-end delay dl(s,d) along (s,d), is an additive constraint, since it comprises the delay incurred at each link along the path (s,d). Further, a constraint is said to be Example of a multiplicative metric [11] can be reliability on availability of a link based on certain criteria such as link breakage probability.
Delay
Communication delay of a packet across an ad hoc network is the latency consumed by a packet to reach the destination from the source. The components of end-to-end latency of a packet at the network layer are processing delay, packetization delay, transmission delay, queuing delay and the propagation delay. Subsequently, the end-to-end delay of a path represents the sum of delay incurred at each link along the path. Node delay involves the protocol processing time at node i for link ( i, j), and link delay is the latency consumed by the packet to travel from node i to node j, i.e. along link (i, j). For wireless ad hoc networks, propagation delays are negligibly small and almost equal for each hop along the path. The major factors involved in computation of node delay are the queuing delay and delay incurred at the MAC layer processing. Computation of MAC layer delay is elaborated in [12], and two dimension finite-state Markov models [13] can be used for estimation of queuing delay, which is determined from the queuing delay distribution Pr(D>t), where the average queuing delay is defined to be D, for which delay distribution is more than 90%. Conclusively end-to-end delay of a path can be obtained by adding up the node delays and link delays along the path.
Power
In the route discovery phase in the on-demand routing protocols like DSR, a shortest possible path is chosen and maintained until the path breaks. Hence, usage of such a path for communication for longer period of time may result in reduction of power at the nodes along the path. It is more likely, when a node belongs to multiple active routes. It results as a consequence of transmission and reception of each message causing the battery power being drained out. When a node runs out of battery power, it is unable to forward any message along the path of communication, and consequently falls out of the network. In such a case, the route breaks, and the protocol initiates another route discovery phase to find another alternative route. Such scenarios of dying nodes may adversely affect the operational life time of the ad hoc network. The principal goal of this protocol is to perform routing around nodes with higher battery power, which enhances life time of the network. The maximum power provided by the battery of a node, when fully charged, is considered to be the initial power of the node, which is taken to be the power metric.
Power and Delay Extension in TORA
As a source-initiated on-demand routing protocol, TORA relies on a link reversal algorithm and provides loop-free multipath routes to a specified destination [1]. In this approach, a node maintains the topology information involving its one-hop neighbors. During a reconfiguration process following a path break, TORA has the unique property to limit the control packets to a small region. The metrics such as delay, power and distance used in TORA, are depicted in Fig. 1. For a given node n, H(n) denotes its height from the destination node. Three major functions performed by TORA are: establishing, maintaining and erasing routes. Route establishment function is initiated, when a source node requires a path to a specific destination, to which it does not possess a directed link. During this process, a destinationoriented Directed Acyclic Graph (DAG) is established using a query / update mechanism. Prior to a communication, a source node sends a query packet to the destination, which incorporates the information regarding source address, destination address, minimum power level, maximum permissible delay (QRY (<source address>, <destination address>, <minimum power level>, <maximum delay>). The power extension in the query packet indicates the minimum power required to be available along the path during the communication. In addition, the delay extension specifies the maximum delay allowed between the source and destination. As depicted in Fig. 1, QoS power extension 0.2 indicates that a minimum of 20% initial power level be available along the path and a maximum allowable delay of 50 milli seconds (ms). The Verification for specified QoS power and QoS delay is made at each node as the query packet traverses the path from source to destination. A query packet is dropped if one of the constraints is not satisfied at any point of time.
As the query packet traverses the network, each node compares its available power level with the power level, mentioned in the query packet. If the available power level at a node is found to be less than the power level specified in the query packet, then the query packet is dropped. In case the QoS power holds perfect, then the delay to destination is estimated, and if the estimate exceeds the QoS delay as mentioned in the query packet, then the packet is dropped. If the delay constraint is satisfied, the node subtracts its Node Traverse Time (NTT) from the delay bound provided in the extension and the query packet is forwarded to next hop along the route. Fig. 2 demonstrates the sequence of operations during traversal of a query packet, which is forwarded by nodes 2,3,4,5,6 between node 1 (source) and node 7 (destination). Each node that terminates the query packet, replies with an update packet back to the source, indicating its distance from the destination and delay.
As shown in Fig. 1, the destination 7 originates an update packet. Each node along the path of this packet sets its distance to a higher value than the distance of sender of the update packet. In addition, each intermediate node adds its own NTT to the delay field of the packet. As a result, a set of directed links are created from the originator of the query packet to the destination node 7, resulting in the DAG, depicted in Fig. 1. After a path to the destination is established, it is presumed to exist as long as it is required, in spite of the changes in path lengths as a result of reconfigurations, taking place during the data transfer.
In case the route to the destination is found by an intermediate node to be invalid as shown in Fig. 3, it alters its distance value to a higher value than its neighbors and originates an update packet (node 6 in Fig. 3). On receiving the update packet from node 6, node 5 reverses its link with node 1 and forwards the update packet to it. It results in a change in DAG as compared to Fig. 1. In case none of the neighbors of the source node has a path to the destination, it needs to initiate a fresh query / update procedure. If the link between nodes 1 & 6 breaks, then node 5 reverses its path to node 6, which is in conflict with the earlier reversal, and hence a partition in the network can be inferred (Fig. 3). When a node detects a partition, it originates a clear message, which erases the information regarding existing path in the partition to the specified destination.
IV. SIMULATION AND PERFORMANCE EVALUATION
The proposed scheme is evaluated using ns-2 simulator [14]. It uses the random way point model for ad hoc networks. In this simulation, the ns-2 WaveLAN implementation for MAC 802.11 is used, with a channel access rate of 2 Mbps in an ad hoc network with 50 mobile nodes. Each mobile node has a mobility range of 670m x 670m. Radio transmission range of each node is set to 250m. The QoS constraints are set as 250ms for delay and 20% of initial power, with the initial power for each node being set to 20 joules, which means a combined network initial power is set to 1000 joules.
The performance metrics are chosen as follows:
Packet delivery ratio: It represents the ratio of number of packets received by the destination to the number of packets sent by the source.
Average end-to-end delay: It is defined as the end-to-end delay experienced by packets from source to destination, which includes route discovery latency, queuing delay at node, transmission delay at the MAC layer and the propagation delay across the wireless channel.
Packet Delivery Ratio
Packet delivery ratio for TORA and PDTORA protocols is depicted in Fig. 4, where speed of mobility taken into account is up to 100 meters/second with a pause time of 10 seconds. At low speeds of nodes, both the protocols demonstrate higher throughput. However, higher speeds may lead to frequent changes in links and probable link failures, ultimately reducing throughput. It can be observed from Fig. 4, that packet delivery ratio in PDTORA is 3% higher than that in TORA for high mobility up to 100 m/s. Packet delivery ratio with respect to number of nodes for different mobile speeds is depicted in Fig. 5. In Fig. 5 (a), for mobile speed of 10 m/s, PDTORA shows 15% improvement over TORA. Accordingly, in Fig. 5 (b), for mobile speed of 20 m/s, PDTORA possesses 23% improvement in packet delivery ratio over TORA. Please note that in the simulation, number of nodes is set up to 50.
End-to-end delay
A measure of end-to-end delay for the QoS requirement of 250 ms with different node mobility is depicted in Fig. 6. It can be noticed that the end-to-end delay increases with increasing speed of nodes. This phenomenon is a consequence of higher mobility causing frequent route changes and frequent link failures. PDTORA maintains the delay QoS within the specified limit (250 ms) and thus has 60% improvement over TORA. End-to-end delay with different mobile speeds for number of nodes from 10 to 50 is shown in Fig. 7. It increases with increasing number of nodes, since number of links increase too. In both the cases, PDTORA shows comprehensively better performance over TORA for higher number of mobile nodes.
Packet Loss
Effect of increasing node mobility on packet loss for both TORA and PDTORA is shown in Fig. 8. Packet loss ratio increases with increasing node speed in both the protocols, as a result of more link breakages. However, packet loss ratio in TORA remains much higher than that of PDTORA for the entire scenario. The difference is quite comprehensive at higher mobile speeds.
Node Lifetime
In the course of communication, nodes may happen to die out. Fig. 9 shows the number of nodes which die at some time instants using both TORA and PDTORA. It can be clearly noticed that nodes in TORA die earlier than PDTORA. It happens during forwarding of the query packet, when the power level of an intermediate node is found to be less than that mentioned in the QoS extension for power in the query packet. In TORA, the first node dies at t = 25 sec., whereas in PDTORA, the first node dies at t = 65 sec. Again, at time instant t = 100 sec., 41 nodes die in TORA, whereas only 6 nodes die in PDTORA. The current paper presents an extension of TORA protocol with power and delay aware modification (PDTORA). The nodes in the network which do not satisfy to the QoS requirements of maximum delay and minimum power levels, are eliminated from the route of communication, during query phase. Each intermediate node on receipt of the query packet determines whether to forward it or not, depending on the QoS requirements. At the destination, an update packet is generated. Form the simulations, it could be observed that improvement of QoS metrics in PDTORA over TORA is significant.
VI. FUTURE WORK
The same approach can be effectively used to improve the QoS metric for other on-demand QoS routing protocols. We are currently working over power and delay aware extensions over Dynamic Source Routing (DSR) protocol.
Figure 1 .
1Power and Delay Extension in TORA
Figure 2 .
2Algorithm of Power and Delay Extension in TORA
Figure 3 .
3Route Maintenance in TORA
Figure 4 .Figure 5 .Figure 6 .Figure 7 .
4567Effect of Mobility on Packet Delivery Ratio Effect of number of Node on Packet Delivery Ratio (a) mobility 10 m/s ; (b) mobility 20 m/s; Effect of Mobility on End-To-End Delay for Pause Time of 10s Effect of number of nodes on end-to-end delay (a) mobility 10 m/s; (b) mobility 20 m/s;
Figure 8 .Figure 9 .
89Effect of mobility on packet loss for pause time of 10 s Number of nodes dead vs. Time V. CONCLUSION
table-driven approach, latency involved in route acquisition is negligibly small. However, it includesAlok Kumar Jagadev et. al. / (IJCSE) International Journal on Computer Science and Engineering
Vol. 02, No. 04, 2010, 917-923
Alok Kumar Jagadev et. al. / (IJCSE) International Journal on Computer Science and EngineeringVol. 02, No. 04, 2010, 917-923
concave if q(s,d) = min { q(s,s 1 ), q(s 1 ,s 2 ), ….., q(s k ,d)}. Thus,
the bandwidth requirement bw(s,d), between s and d is
concave, since it comprises the minimum bandwidth between
the links along the path. Similarly, a constraint is
multiplicative if q(s,d) = q(s,s 1 ) x q(s 1 ,s 2 ) x …… q(s k ,d). For
example, the probability P(s,d) of a packet, sent from s to d is
multiplicative, as it is the product of probabilities of individual
links along the path. Hence, bandwidth and power are concave
metrics, whereas cost, delay and jitter are additive metrics.
A highly adaptive distributed routing algorithm for mobile wireless networks. V D Park, M S Corson, Proceedings of IEEE INFOCOM '97. IEEE INFOCOM '97V.D. Park and M.S. Corson. "A highly adaptive distributed routing algorithm for mobile wireless networks". Proceedings of IEEE INFOCOM '97 : PP 1405 -1413, April 1997.
PAMAS -power aware multi-access protocol with signaling for ad hoc networks. S Singh, C S Raghavendra, ACM Computer Communication Review (ACM CCR'98). S. Singh and C.S. Raghavendra. "PAMAS -power aware multi-access protocol with signaling for ad hoc networks". ACM Computer Communication Review (ACM CCR'98), July 1998.
Power-aware routing in mobile ad-hoc networks. S Singh, M Woo, C S Raghavendra, ACM/IEEE International Conference on Mobile Computing and Networking. S. Singh, M.Woo, and C.S. Raghavendra. "Power-aware routing in mobile ad-hoc networks". ACM/IEEE International Conference on Mobile Computing and Networking, pp. 181-190, October 1998.
Maximum battery life routing to support ubiquitous mobile computing in wireless ad hoc networks. C K Toh, IEEE Communication MagazineC.K. Toh, "Maximum battery life routing to support ubiquitous mobile computing in wireless ad hoc networks", IEEE Communication Magazine, pp. 2-11, June 2001.
CEDAR : a core extraction distributed ad hoc routing algorithm. R Sivakumar, P Sinha, V Bharghavan, IEEE Journal on selected Area in Communication. R. Sivakumar, P. Sinha and V. Bharghavan, CEDAR : a core extraction distributed ad hoc routing algorithm," IEEE Journal on selected Area in Communication, pp 1454-1466, August 1999.
QoS routing in ad hoc wireless networks. C Lin, J Liu, IEEE Journal on Selected Areas in Communication. C. Lin and J. Liu, "QoS routing in ad hoc wireless networks, "IEEE Journal on Selected Areas in Communication, pp 1426-1438, August 1999.
A bandwidth reservation mechanism for ondemand ad hoc path finding. R Gerasimov, Simon, IEEE/SCS 35 th Annual Simulation Symposium. San Diego, CAI Gerasimov and R. Simon. "A bandwidth reservation mechanism for on- demand ad hoc path finding". IEEE/SCS 35 th Annual Simulation Symposium, San Diego, CA, Pages 27-33, April 2002.
Delay and throughput aware proactive QoS routing in mobile adhoc networks. R Asokan, M Pushpavalli, A M Natarajan, Proceedings of the International Conference on Advanced Communication System (ICACs-2007. the International Conference on Advanced Communication System (ICACs-2007R. Asokan, M. Pushpavalli and A.M. Natarajan "Delay and throughput aware proactive QoS routing in mobile adhoc networks". Proceedings of the International Conference on Advanced Communication System (ICACs-2007) pp 1-6, January 2007.
INORA-a unified signaling and routing mechanism for QoS support in mobile ad hoc networks. D Dharmaraju, A Roy-Chowdhury, P Hovareshti, J S Baras, Proceedings. Pages 86-93. Pages 86-93Parallel Processing WorkshopsD. Dharmaraju, A. Roy-Chowdhury, P. Hovareshti, and J.S. Baras. "INORA-a unified signaling and routing mechanism for QoS support in mobile ad hoc networks". Parallel Processing Workshops, 2002 Proceedings. Pages 86-93, August 2002.
QoS routing for wireless ad hoc networks : problems, algorithms, and protocols. Prasant Mohapatra, L Jian, Chao Gui, IEEE Wireless Communications MagazinePrasant Mohapatra, Jian L, and Chao Gui, "QoS routing for wireless ad hoc networks : problems, algorithms, and protocols" IEEE Wireless Communications Magazine, pp 44-52, March 2003.
QoS routing for wireless ad hoc networks : problems, algorithms and protocols. Baoxian Zhang, Hussein T Mouftah, IEEE Communications MagazineBaoxian Zhang and Hussein T. mouftah, "QoS routing for wireless ad hoc networks : problems, algorithms and protocols" IEEE Communications Magazine, pp. 110-117 October 2005.
A novel delay oriented shortest path routing protocol for mobile ad hoc networks. S T Shen, J H Chen, proceedings of IEEE ICC. S.T. Shen and J.H. Chen, "A novel delay oriented shortest path routing protocol for mobile ad hoc networks, "proceedings of IEEE ICC 2001)
Adaptive QoS routing based on prediction of local performance in ad hoc networks. H Sun, H Hughes, Proceedings of IEEE WNCN. IEEE WNCNH. Sun and H. Hughes, "Adaptive QoS routing based on prediction of local performance in ad hoc networks". Proceedings of IEEE WNCN 2003.
| []
|
[
"Radiative generation of neutrino masses in a 3-3-1 type model",
"Radiative generation of neutrino masses in a 3-3-1 type model"
]
| [
"Arindam Das *[email protected]†[email protected]‡[email protected]§[email protected] \nDepartment of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan\n",
"Kazuki Enomoto \nDepartment of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan\n",
"Shinya Kanemura \nDepartment of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan\n",
"Kei Yagyu \nDepartment of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan\n"
]
| [
"Department of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan",
"Department of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan",
"Department of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan",
"Department of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan"
]
| []
| A new model for tiny neutrino masses is proposed in the gauge theory of SU (3) C ⊗ SU (3) L ⊗ U (1) X , where neutrino masses are generated via the quantum effect of new particles. In this model, the fermion content is taken to be minimal to realize the gauge anomaly cancellation, while the scalar sector is extended from the minimal 3-3-1 model to have an additional SU (3) L triplet field.the "Zee model" like diagrams are naturally induced, which contain sufficient lepton flavor violating interactions to reproduce current neutrino oscillation data. Furthermore, the remnant Z 2 symmetry appears after the electroweak symmetry breaking, which guarantees the stability of dark matter. It is confirmed that this model can satisfy current dark matter data. As an important prediction to test this model, productions and decays of doubly-charged scalar bosons at collider experiments are discussed in successful benchmark scenarios. | 10.1103/physrevd.101.095007 | [
"https://arxiv.org/pdf/2003.05857v2.pdf"
]
| 212,675,718 | 2003.05857 | 20da0c5fc4d273dc6244b98811408056f988c06f |
Radiative generation of neutrino masses in a 3-3-1 type model
12 Mar 2020
Arindam Das *[email protected]†[email protected]‡[email protected]§[email protected]
Department of Physics
Osaka University
560-0043ToyonakaOsakaJapan
Kazuki Enomoto
Department of Physics
Osaka University
560-0043ToyonakaOsakaJapan
Shinya Kanemura
Department of Physics
Osaka University
560-0043ToyonakaOsakaJapan
Kei Yagyu
Department of Physics
Osaka University
560-0043ToyonakaOsakaJapan
Radiative generation of neutrino masses in a 3-3-1 type model
12 Mar 2020
A new model for tiny neutrino masses is proposed in the gauge theory of SU (3) C ⊗ SU (3) L ⊗ U (1) X , where neutrino masses are generated via the quantum effect of new particles. In this model, the fermion content is taken to be minimal to realize the gauge anomaly cancellation, while the scalar sector is extended from the minimal 3-3-1 model to have an additional SU (3) L triplet field.the "Zee model" like diagrams are naturally induced, which contain sufficient lepton flavor violating interactions to reproduce current neutrino oscillation data. Furthermore, the remnant Z 2 symmetry appears after the electroweak symmetry breaking, which guarantees the stability of dark matter. It is confirmed that this model can satisfy current dark matter data. As an important prediction to test this model, productions and decays of doubly-charged scalar bosons at collider experiments are discussed in successful benchmark scenarios.
I. INTRODUCTION
The structure of the electroweak gauge symmetry SU(2) L ⊗ U(1) Y has been well established by various experiments. The spontaneous breaking SU(2) L ⊗ U(1) Y → U(1) em by the Higgs mechanism has been confirmed by the discovery of the Higgs boson at the LHC. However, the possibility of larger gauge groups including the SU(2) L ⊗ U(1) Y symmetry can also be considered at higher energies than the electroweak scale. In fact based on various physics motivations, a plethora of models with extended gauge symmetries has been proposed.
One of the simple but well-motivated extensions is the SU(3) C ⊗ SU(3) L ⊗ U(1) X (3-3-1) gauge symmetry [1,2]. This extension can naturally give the answer to the question in the Standard Model (SM) "Why are there three generations of chiral fermions?" by the gauge anomaly cancellation. Namely, the number of generation has to be the same as the fundamental color degrees of freedom or its multiples [3]. There are many new particles in models with the 3-3-1 gauge symmetry. Clearly, these models necessarily introduce new gauge bosons. Cancellation of the gauge anomaly requires additional chiral fermions. In order to break the gauge symmetry into the SM one, additional scalar fields have also to be introduced. A next question is whether these new particles can play a role to explain phenomena which cannot be explained in the SM, such as the neutrino oscillation, the existence of dark matter and the baryon asymmetry of the Universe.
Apart from the 3-3-1 models, the radiative seesaw mechanism has been known as the attractive scenario to naturally explain tiny neutrino masses due to loop suppression factors without introducing super heavy new particles. The original model was proposed by Zee [4], in which neutrino masses are generated at 1-loop level. The model proposed by Zee and Babu [5,6] generates neutrino masses at 2-loop level. After these models appeared, radiative seesaw models with dark matter particles have also been proposed, in which the stability of dark matter is guaranteed by a discrete symmetry which is simultaneously forbid tree level diagrams for neutrino masses. For instance, the model proposed by Krauss-Nasri-Trodden [7] and that by Ma [8] correspond to those along this line. Therefore, if we can construct models for generating tiny neutrino masses radiatively based on the 3-3-1 scenario, we can explain origins of tiny neutrino masses, dark matter and generation of chiral fermions simultaneously. This is the subject of the present paper.
Several models with radiative generation of neutrino masses in the 3-3-1 scenario have been proposed in Refs. [9][10][11][12]. These models, however, do not contain dark matter candidates. Recently, in Refs. [13,14] one-loop neutrino mass models including a dark matter candidate have been constructed by the extension of the minimal model based on the SU(3) C ⊗ SU(3) L ⊗ U(1) X ⊗ U(1) N gauge symmetry, so-called 3-3-1-1 models. In the minimal version of 3-3-1-1 models [15][16][17], three right-handed neutrinos are introduced in order to realize the anomaly cancellation, and tiny masses of active neutrinos are obtained by the seesaw mechanism at tree level. In addition, dark matter candidates are naturally obtained by the remnant discrete symmetry after the spontaneous breaking of the 3-3-1-1 gauge symmetry [15,17]. Radiative seesaw scenarios can also be considered in the framework of 3-3-1-1 models [13,14] by changing U(1) N charges for three right-handed neutrinos to avoid the tree level mass term while keeping the dark matter candidates. However, this model is a bit complicated, because many additional particles are further required for the anomaly cancellation and for making all sterile neutrinos massive [14].
In this paper, we construct a new model with the 3-3-1 gauge symmetry in order to explain tiny neutrino masses, dark matter and generation of chiral fermions simultaneously. We take the minimal content of fermions required for the gauge anomaly cancellation, while we introduce an additional scalar SU(3) L triplet field to break the lepton number by the scalar self-interactions. Our model then induces "Zee model" like diagrams after the spontaneous breaking of SU(3) L ⊗ U(1) X → SU(2) L ⊗ U(1) Y . It has been known that the neutrino oscillation data cannot be reproduced by the Zee model due to a too restricted structure of lepton flavor violating (LFV) interactions, see e.g., [18,19]. On the contrary, our model includes additional sources of LFV interactions, so that we can explain current neutrino data.
Another interesting feature of our model is that the appearance of an unbroken discrete Z 2 parity, which guarantees the stability of dark matter candidate, i.e., the lightest neutral particle with a Z 2 -odd charge. This discrete symmetry arises as the remnant symmetry of the 3-3-1 gauge symmetry and a global U(1) ′ symmetry 1 , where the latter is softly-broken and is introduced in order to avoid dangerous flavor changing neutral interactions between a SM fermion and an extra fermion. We find that the Z 2 -odd scalar dark matter can explain the thermal relic abundance satisfying the current direct search results. This paper is organized as follows. In Sec. II, we show the particle content in our model, and give the Higgs potential, kinetic terms for scalar fields and Yukawa interaction terms. In Sec. III, we discuss the generation mechanism for neutrino masses and their mixings. Constraints from LFV decays of charged leptons are considered in Sec. IV. We show numerical results for the correlations between branching ratios of the LFV decays in the parameter sets satisfying the current neutrino data. Sec. V is devoted for the discussion of the dark matter and the collider phenomenology. We conclude the article in Sec. VI. In appendices, we present the mass formulae of physical scalar bosons (App. A) and those for the decay branching ratios for the LFV decays (App. B).
II. MODEL
In this section, we define our model based on the 3-3-1 gauge symmetry. We first present the particle content and give expressions for component fields of SU(3) L (anti-)triplet fields.
We then discuss the Higgs potential, kinetic terms for scalar fields and Yukawa interaction terms in the following subsections in order.
A. Particle content
Fermion Scalar
Fields L i L e i R E i R Q a L Q 3 L d i R B R u i R U a R Φ 1 Φ 2 Φ 3 Φ ℓ SU (3) C 1 1 1 3 3 3 3 3 3 1 1 1 1 SU (3) L 3 1 13 3 1 1 1 1 3 3 3 3 U (1) X −2/3 −1 −1 1/3 0 −1/3 −1/3 2/3 2/3 1/3 −2/3 1/3 4/3 U (1) ′ 0 −q 2q 0 0 −q 2q q −2q q −q −2q 0(3) C ⊗ SU (3) L ⊗ U (1) X .
The U (1) ′ symmetry is a global, which is softly-broken by scalar interactions. The flavor indices i and a run over 1-3 and 1-2, respectively.
The particle content and the charge assignment under the 3-3-1 gauge symmetry are shown in Table I, in which the fermion content is taken to be minimal based on the requirement of the gauge anomaly cancellation. From the cancelation of the pure SU(3) L gauge anomaly, generation must be 3, number of the color, which can be regarded as the origin of the three-generation structure for leptons and quarks [3]. In addition to the gauge symmetry, we introduce a global U(1) ′ symmetry, which is softly-broken. This U(1) ′ symmetry is imposed to avoid the dangerous flavor changing neutral current, while it maintains necessary scalar interaction terms in the potential to generate one-loop induced neutrino masses.
The embedding scheme of leptons and quarks into 3-3-1 multiplets is the same as that given in Ref. [22], in which the electric charge Q is defined by
Q = T 3 + Y, with Y = 1 √ 3 T 8 + Q X ,(1)
where T 3 , T 8 and Q X are the third and the eighth components of the SU(3) L generator and the U(1) X charge, respectively. After the 3-3-1 symmetry is broken into the SU(2) L ⊗U(1) Y symmetry, Y in Eq. (1) is identified with the weak hypercharge.
The component fields of left-handed leptons and quarks are then determined as follows:
L i L = ν i e i E i L , Q a L = d a u a U a L , Q 3 L = t b B L ,(2)
where the flavor indices i and a run over 1-3 and 1-2, respectively. In Eq. (2), component fields denoted as lowercase (uppercase) letters can be identified as SM (extra) fermions.
We have three, two and one extra charged leptons, up-type quarks and down-type quark, respectively, whose electric charges are the same as the corresponding charged leptons and quarks in the SM. All these extra fermions have Dirac mass terms proportional to the Vacuum Expectation Value (VEV) which breaks the SU(3) L ⊗ U(1) X symmetry into the
SU(2) L ⊗ U(1) Y symmetry.
The scalar triplet fields are represented as
Φ 1 = φ + 1 φ 0 1 η 0 1 , Φ 2 = φ 0 2 φ − 2 η − 2 , Φ 3 = η + 3 η 0 3 φ 0 3 , Φ ℓ = η ++ ℓ η + ℓ φ + ℓ .(3)
In 3-3-1 gauge theories the Higgs sector contains at least three SU(3) L -triplet scalar fields [1,2,22], which give masses to the fermions except for the neutrinos after the electroweak symmetry breaking. In our model, we further introduce an additional triplet field Φ ℓ for the neutrino mass generation, which will be discussed in Sec. III.
B. Higgs potential
The most general Higgs potential is given by
V = i=1,4 m 2 i |Φ i | 2 + (m 2 13 Φ † 1 Φ 3 + ǫ αβγ µΦ α 1 Φ β 2 Φ γ 3 + h.c.) + i=1,4 λ i |Φ i | 4 + j>i i,j=1,4 λ ij |Φ i | 2 |Φ j | 2 + ρ ij |Φ † i Φ j | 2 + ξ 1 (Φ † 1 Φ 2 )(Φ † 3 Φ ℓ ) + ξ 2 (Φ † 2 Φ 3 )(Φ † ℓ Φ 1 ) + h.c.,(4)
where Φ 4 = Φ ℓ . The m 2 13 and µ terms softly break the U(1) ′ symmetry, by which the appearance of an additional Nambu Goldstone (NG) boson is avoided. The parameters m 2 13 , µ, ξ 1 and ξ 2 are complex in general, and these complex phases cannot be simultaneously taken to be zero by phase redefinitions of the scalar fields. For simplicity, we take these parameters to real.
The VEVs of the Higgs triplet fields can generally be taken as
Φ 1 = 1 √ 2 0 v 1 0 , Φ 2 = 1 √ 2 v 2 0 0 , Φ 3 = 1 √ 2 0 v ′ V .(5)
The VEV of the third component of Φ 1 can be taken to zero without any loss of generality by using the field rotation of Φ 1 and Φ 3 . The VEV v ′ causes phenomenologically dangerous mixing between the SM fermions and the extra ones. Therefore, we arrange µ = 0 and m 2 13 = 0 so as to have a remnant Z 2 symmetry, by which v ′ = 0 is guaranteed and such a dangerous mixing can be avoided. We call this Z 2 symmetry asZ 2 . The charge of theZ 2 symmetry can be defined as (−1) |Q ′ |/q with Q ′ being the U(1) ′ charge, by which Φ 1,2 and all the SM right-handed fermions are assigned to be odd, while the other fields are even. Thẽ Z 2 symmetry is spontaneously broken by the VEVs v 1 and v 2 , so that domain walls would appear in the early Universe [23,24]. We will briefly comment on this issue at the end of this subsection.
In the following, we assume that V ≫ v 1 , v 2 . Under the setup with v ′ = 0, SU(3) L ⊗U(1) X is spontaneously broken into SU(2) L ⊗ U(1) Y by the VEV V at higher energy scales than the electroweak scale. Then, the SU(2) L ⊗ U(1) Y symmetry is broken down to U(1) em by v 1
and v 2 at the electroweak scale. The Fermi constant G F is reproduced by
G F = ( √ 2v 2 ) −1 with v ≡ v 2 1 + v 2 2 .
For the later convenience, we introduce tan β = v 2 /v 1 as the analogue of two Higgs doublet models (THDMs).
After the spontaneous breakdown of the 3-3-1 gauge symmetry and theZ 2 symmetry, another remnant Z 2 symmetry, let us denote it as Z rem 2 , appears, whose charge can be defined as (−1) 2s+2 √ 3T 8 +|Q ′ |/q with s being the spin of the particle. From this definition, all the extra fermions and scalar bosons denoted by η in Eq. (3) are assigned to be odd under Z rem 2 . Because the Z rem 2 symmetry is unbroken, the lightest neutral Z rem 2 -odd particle can be a candidate of dark matter. We will discuss dark matter physics in Sec. V.
Because of the Z rem 2 symmetry, we can classify the physical scalar fields into the Z rem 2 -even and Z rem 2 -odd ones as follows. In the Z rem 2 -even sector, we have two-pairs of singly-charged scalar bosons H ± andĤ ± , one CP-odd Higgs boson A and three CP-even Higgs bosons H i (i = 1, 2, 3). The discovered Higgs boson with a mass of about 125 GeV can be identified with the H 1 state. On the other hand in the Z rem 2 -odd sector, we have one-pair of doublycharged scalar bosons η ±± ℓ , two-pairs of singly-charged scalar bosons η ± andη ± , and one complex neutral scalar boson η 0 . The other eight scalar states are the NG bosons which are absorbed into the longitudinal components of the massive gauge bosons (W µ , W ′ µ , Y µ , Z µ and Z ′ µ ), see Sec. II C. In Appendix A, we explicitly show the relation between the mass eigenstates and the weak eigenbasis of the scalar states and their mass formulae.
Let us discuss the effective theory of our 3-3-1 model in the large VEV limit V ≫ v with V µ ≡ M 2 . In this case, the masses of H ± , A and H 2 are determined by the M parameter 2 , while that of H 1 is determined by v. On the other hand, all the other physical Higgs bosons are decoupled from the theory, as their masses are determined by V . Therefore, the scalar sector effectively coincides with a THDM with a special flavor structure which cannot be realized in THDMs with a softly-broken Z 2 symmetry, see Sec. II D. Similar to the usual THDMs, we can define the decoupling limit by M ≫ v, where only the SM-like Higgs boson H 1 remains at the scale v. We can also define the so-called alignment limit, where the SMlike Higgs boson couplings with the SM gauge bosons and fermions become the same values as those of the SM Higgs boson at tree level. This alignment limit can be taken by choosing potential parameters such that the (1,2) element of the mass matrix of the CP-even Higgs bosons given in Eq. (A14) is zero. Therefore, our 3-3-1 model provides another important example that predicts the THDM as the low energy effective theory other than the minimal supersymmetric extension of the SM [25] and composite Higgs models [26][27][28].
As mentioned in the above, our model potentially has the domain wall problem. It has been known that the energy density of domain walls is only suppressed by the inverse of the radius of the Universe, which is much slower than the dilution of the energy density for ordinary matter and radiation. Therefore, the existence of domain walls could significantly change the history of the Universe. In Refs. [29][30][31][32], solutions for the domain wall problem have been discussed. According to Ref. [29], if a discrete symmetry which is spontaneously broken by Higgs VEVs (in our model, this corresponds to the electroweak symmetry breaking VEV v) is not restored at high temperature, the domain wall problem might not arise. Such situation can happen if finite temperature effects which are proportional to T 2 on a negative mass squared term are also negative [33]. In the SM, this does not happen, because there is only one scalar quartic coupling. Such quartic coupling gives a positive effect of finite temperature on the negative mass squared term, so that the broken symmetry at zero temperature is restored at high temperature as it is seen in the usual thermal history of the Universe. On the other hand, if we consider models with multi-scalar fields as in our model, this is not always the case, because some combinations of scalar quartic parameters can be taken to negative so as to realize the symmetry non-restoration scenario. Thus, we might be able to avoid the domain wall problem. Clearly, more dedicated discussions for this solution have to be done in order to ensure its justification, which is beyond the scope of this paper.
C. Kinetic terms for scalar fields
Kinetic terms for the scalar triplet fields are expressed as
L kin = i=1,4 |D µ Φ i | 2 ,(6)
where Φ 4 = Φ ℓ . The covariant derivative D µ for SU(3) L triplet fields is given by
D µ = ∂ µ − igA µ − ig X Q X X µ ,(7)
with g and g X being the SU(3) L and U(1) X gauge couplings, respectively. The SU(3) L gauge boson A µ is expressed by the 3 × 3 matrix form as: (4,5) and (6,7). We can identify W µ ≡ A µ 12 and W ′µ ≡ A µ 45 with the SM W boson and the additional charged gauge boson, respectively, while Y µ ≡ A µ 67 with a neutral complex gauge boson. Their masses are given by the VEVs of the Higgs triplet fields as follows:
A µ ≡ A Aµ T A = 1 2 (A µ 3 + A µ 8 √ 3 ) A µ 12 √ 2 A µ 45 √ 2 A * µ 12 √ 2 1 2 (−A µ 3 + A µ 8 √ 3 ) A µ 67 √ 2 A * µ 45 √ 2 A * µ 67 √ 2 − A µ 8 √ 3 , A = 1, . . . , 8,(8)where we introduced A µ ij ≡ (A µ i − iA µ j )/ √ 2 with (i, j) = (1, 2),m W = g 2 v, m W ′ = g 2 v 2 c 2 β + V 2 , m Y = g 2 v 2 s 2 β + V 2 ,(9)
where s θ = sin θ and c θ = cos θ. In addition to these complex states, there are three real neutral gauge bosons, where one of them can be identified with the massless photon γ µ . We can define the basis where γ µ is separated from the other two massive states (Z µ andZ ′µ ) as follows:
A µ 3 A µ 8 X µ = R V γ μ Z μ Z ′µ , R V = √ 3g X √ 3g 2 +4g 2 X 1 2 − 3g 2 √ 3g 2 +4g 2 X g X √ 3g 2 +4g 2 X − √ 3 2 − √ 3g 2 √ 3g 2 +4g 2 X √ 3g √ 3g 2 +4g 2 X 0 2g X √ 3g 2 +4g 2 X .(10)
TheZ andZ ′ states can be mixed. Their mass matrix is given by
M 2 V = V 2 4 g 2 (1 + c 2 β ǫ) g 3 3g 2 + 4g 2 X (1 − c 2 β ǫ) g 3 3g 2 + 4g 2 X (1 − c 2 β ǫ) 1 9 (3g 2 + 4g 2 X )[1 + (1 + 3s 2 β )ǫ] ,(11)with ǫ = v 2 /V 2 .
In the large V limit, the mass eigenvalues are expressed as
m 2 Z = g 2 v 2 4 3g 2 + 4g 2 X 3g 2 + g 2 X + O(ǫ) , m 2 Z ′ = V 2 9 3g 2 + g 2 X + O(ǫ) .(12)
From the expression of m 2 Z , we see that the weak mixing angle θ W can be identified with
c W ≡ cos θ W = 3g 2 + g 2 X 3g 2 + 4g 2 X .(13)
We then have the same expression of m Z as that of the mass of the SM Z boson. The U(1) em coupling e is consistently given by e = g sin θ W as that in the SM.
The electroweak rho parameter can be expressed at tree level by using the definition of θ W given in Eq. (13),
ρ = m 2 W m 2 Z c 2 W = 1 + O(ǫ).(14)
In order to satisfy |ρ − 1| ≤ 10 −3 , V has to be taken to be larger than around 8 TeV. We note that the calculation of one-loop corrections to the rho parameter is different from that in models with ρ = 1 at tree level, e.g., the SM. In models with ρ = 1 at tree level, the electroweak sector is described by four input parameters which can be chosen to be α em , m Z , G F and δρ with the last one being the deviation of the rho parameter from unity.
This means that the radiative correction to the rho parameter cannot be a prediction, because the additional parameter δρ provides an additional counterterm by which loop corrections to the rho parameter can be absorbed by imposing a renormalization condition.
The similar situation can also happen in models with higher isospin scalar multiplets with a non-vanishing VEV such as SU(2) L triplet scalar fields [34,35].
D. Yukawa interactions
Thanks to the global U(1) ′ symmetry, Yukawa interaction terms for the SM right-handed fermions and those for the extra right-handed fermions are separately given as follows:
L Y = ǫ αβγ f ij (L ci L ) α (L j L ) β (Φ ℓ ) γ + y ij eL i L Φ 1 e j R + y ai d2Q a L Φ * 2 d i R + y i d1Q 3 L Φ 1 d i R + y ai u1Q a L Φ * 1 u i R + y ai u2Q 3 L Φ 2 u i R + y ij EL i L Φ 3 E j R + y DQ 3 L Φ 3 B R + y ab UQ a L Φ * 3 U b R + h.c.,(15)
where ǫ αβγ is the complete antisymmetric tensor with α, β and γ being the indices for the SU(3) L triplet. Because of ǫ αβγ , the complex 3 × 3 matrix f ij has also to be antisymmetric.
From the structure of the VEVs and the Yukawa interactions given in Eqs.
M f = v √ 2 Y f , M F = V √ 2 y F ,(16)
where
Y e = y e c β , Y d = y 11 d2 s β y 12 d2 s β y 13 d2 s β y 21 d2 s β y 22 d2 s β y 23 d2 s β y 1 d1 c β y 2 d1 c β y 3 d1 c β , Y u = y 11 u1 c β y 12 u1 c β y 13 u1 c β y 21 u1 c β y 22 u1 c β y 23 u1 c β y 1 u2 s β y 2 u2 s β y 3 u2 s β .(17)
We note that M E , M U and M D are respectively the 3 × 3, the 2 × 2 and the 1 × 1 matrices.
They are diagonalized by bi-unitary transformations:
f L,R = V f L,R f ′ L,R , F L,R = V F L,R F ′ L,R .(18)
As we can see in Eq. (17), the mass matrix for the SM charged leptons has the same form as in the SM; i.e., only one of the Higgs fields gives their masses. On the other hand, the mass matrices for the up-type and the down-type SM quarks are given by two VEVs of the Higgs fields; i.e., 3 × 3 matrices are composed of the two independent Yukawa matrices.
This structure predicts characteristic flavor-dependent Higgs-boson couplings to quarks [36] in the THDM which is effectively deduced after the SU(3) L ⊗ U(1) X symmetry breaking as discussed in Sec. II B.
The f ij terms in Eq. (15) do not contribute to the masses of charged leptons, but they play an important role for the neutrino mass generation. From these interaction terms, we can assign two units of the lepton number to the Higgs triplet field Φ ℓ . This lepton number is explicitly broken by the ξ 1 and ξ 2 terms in the Higgs potential given in Eq. (4), and they turn out to be the source of the Majorana masses for the active neutrinos. We will discuss the neutrino mass generation in Sec. III.
III. NEUTRINO MASSES
Majorana neutrino masses are generated at one-loop level as shown in Fig. 1, in which the Z rem 2 -even and Z rem 2 -odd particles run in the loop in the left and right diagram, respectively. We here give relevant interaction terms among physical charged scalar bosons and fermions in their mass eigenbases for the calculation of one-loop induced neutrino masses; where The total contribution to the Majorana neutrino masses is then expressed as
L int = √ 2 V s β 1 + v 2 V 2 s 2 βν ′ L (W M diag E )E ′ R (c θη η + − s θηη + ) − √ 2 tan β vν ′ L M diag e e ′ R (c θ H H + − s θ HĤ + ) − 2[ν ′c L F e ′ L (s θ H H + + c θ HĤ + ) −ν ′c L F W E ′ L (s θη η + + c θηη + ) −Ē ′c L W T F e ′ L η ++ ℓ ] + h.c.,(19)ν L ν L ν L ν L e R e L E R E L φ + l φ + η + l η + φ 0 2 φ 0 1 φ 0 1 φ 0 3 φF ≡ V T L f V L , W = (V e L ) † V E L ,M ν = (M e ν + M E ν ) + (M e ν + M E ν ) T ,(20)
where M e ν and M E ν represent the contribution from the left and right diagram, respectively. They are calculated as
(M e ν ) ij = C e v F ij m 2 j , (M E ν ) ij = C E V (F W ) ik M 2 k G k (W † ) kj ,(21)where m i ≡ (M diag e ) ii and M i ≡ (M diag E ) ii , and G k = 1 2 ln m 2 η ± m 2 η ± + M 2 k + m 2 η ± M 2 k − m 2 η ± ln m 2 η ± M 2 k − M 2 k + m 2 η ± M 2 k − m 2 η ± ln m 2 η ± M 2 k .(22)
The flavor independent coefficients C e and C E are given by
C e = √ 2 8π 2 tan βc θ H s θ H ln m 2Ĥ ± m 2 H ± , C E = √ 2 8π 2 s β c θη s θη 1 + v 2 V 2 s 2 β .(23)
The mass matrix given in Eq. (20) is diagonalized by introducing the Pontecorvo-Maki-Nakagawa-Sakata matrix U PMNS as;
U T PMNS M ν U PMNS = diag(m 1 ν , m 2 ν , m 3 ν ),(24)
where m i ν (i = 1, 2, 3) are mass eigenvalues for neutrinos, and
U PMNS = 1 0 0 0 c 23 s 23 0 −s 23 c 23 c 13 0 s 13 e −iδ CP 0 1 0 −s 13 e iδ CP 0 c 13 c 12 s 12 0 −s 12 c 12 0 0 0 1 ,(25)
with s ij = sin θ ij , c ij = cos θ ij and δ CP being the CP phase. We consider both the cases for the orders of the neutrino masses; i.e., the normal hierarchy (|m 1 ν | < |m 2 ν | < |m 3 ν |) and the inverted hierarchy (|m 3 ν | < |m 1 ν | < |m 2 ν |). The flavor structure of M e ν is the same as that of the Zee model. It has been known that the Zee model cannot explain the current neutrino oscillation data because of the too restricted structure of flavor violating couplings which only arise from the antisymmetric 3 × 3 F matrix, see e.g., [18,19] 3 . Although the flavor structure of M E ν also takes the similar form to that of M e ν , another flavor violating source in the matrix W is inserted into the mass matrix. Consequently, M E ν has a different flavor mixing pattern. Hence, it can explain the current neutrino data. We note that such an additional flavor violating source vanishes if we take the masses of the extra leptons degenerate; i.e., M 1 = M 2 = M 3 , by which M 2 k and G k in Eq. (21) commute with W † and then the effect of the W matrix disappears by the unitarity. We also note that the contribution from M E ν is typically much larger than that from M e ν , because the latter is proportional to the mass squared of the SM charged leptons. Therefore, the neutrino masses and the mixings are determined essentially only by the contribution from M E ν . We thus switch off the contribution from M e ν hereafter for simplicity, which can be realized by taking C e = 0 or equivalently θ H = 0.
IV. LEPTON FLAVOR VIOLATION
In our model, new particles can contribute to LFV decays of the charged leptons. In this section, we discuss the constraints from ℓ → ℓ ′ γ and ℓ → ℓ ′ ℓ ′′ ℓ ′′′ types of LFV decays in the parameter sets which satisfy the current neutrino oscillation data. 3 If we take the general Yukawa interactions for leptons, then there is a corner of parameter space which can satisfy the current neutrino results [37,38].
ℓ ℓ ′ η −− ℓ E i γ ℓ ℓ ′ H − ν i γ ℓ ℓ ′ η 0 E i γ FIG. 2. Diagrams for ℓ → ℓ ′ γ processes. ℓ E k ℓ ′ ℓ ′′ ℓ ′′′ E l η −− ℓ η −− ℓ ℓ ν k ℓ ′ ℓ ′′ ℓ ′′′ ν l H −Ĥ − ℓ E k ℓ ′ ℓ ′′ ℓ ′′′ E l η 0 η 0 FIG. 3. Diagrams for ℓ → ℓ ′ ℓ ′′ ℓ ′′′ processes.
Diagrams of the ℓ → ℓ ′ γ processes are shown in Fig. 2. The branching ratios of these processes are calculated as
B(ℓ i → ℓ j γ) ≃ 48π 3 α em C ij G 2 F m 2 i |(a ϕ R ) ij | 2 + |(a ϕ L ) ij | 2 ,(26)
where (ℓ 1 , ℓ 2 , ℓ 3 ) = (e, µ, τ ) and (C 21 , C 31 , C 32 ) = (1, 0.1784, 0.1736). In the above expression, a ϕ R and a ϕ L represent ϕ-loop (ϕ = η 0 , η ±± ℓ , H ± andĤ ± ) contributions to the amplitude for the decays with the right-and left-handed initial charged lepton ℓ i , respectively. Their analytic expressions are presented in Appendix B. From the first diagram in Fig. 2, the structure of the W matrix is constrained. Since W is a unitary matrix, we cannot simply take small values for each component. We can instead take the small mass difference among E i in order to suppress the contribution from the first diagram. On the other hand, the magnitude of the contributions from the second and the third diagrams can be easily suppressed by taking small values for the F matrix elements. Typically, |F ij | 10 −3 is required from the constraint by the µ → eγ data. Fig. 3 shows one-loop box diagrams contributing to the ℓ → ℓ ′ ℓ ′′ ℓ ′′′ type of the LFV decay process. Among the six possible processes, the branching ratio of µ → 3e is most strongly constrained by the data. In our model, the branching ratio of µ → 3e is calculated as
B(µ → eeē) = 1 4G 2 F 1 16π 2 2 |a LRLR | 2 + |a LRRL | 2 + |a LLLL | 2 + |a RRRR | 2 + |a LLRR | 2 + |a RRLL | 2 ,(27)
where a ijkl (i, j, k, l = L or R) denote contributions from the diagrams with the i-handed muon and the j-handed electron in the µ-e current, and the k-handed positron and the l-handed electron in the e-e current. Detailed expressions for each contribution are given in Appendix B. We note that the µ → 3e data typically do not further constrain the parameter region allowed by the ℓ → ℓ ′ γ data.
Now, let us numerically show the prediction of the branching ratios of ℓ → ℓ ′ γ and µ → 3e decays in the parameter sets which satisfy the current neutrino data. As aforementioned in Sec. III, we take C e = 0 (or θ H = 0) in the numerical evaluation. We assume the CPconservation in the Yukawa interaction terms. The W matrix given in Eq.
We then take the following parameters as inputs;
F 12 , F 23 , F 13 , w 12 , w 23 , w 13 , C E , M 1 , M 2 , M 3 , m η ± , mη±, V, tan β, m η ±± ℓ , m η 0 , mĤ ± .(29)
The parameters in the first line are required for the neutrino mass calculation. For the calculation of the LFV decays, we also need to specify the parameters in the second line.
We refer to the neutrino oscillation data given in Ref. [42], and we apply the 3σ allowed ranges of two squared mass differences and three mixing angles to our analysis.
In Fig. 4, we show various correlations between the branching ratios of the LFV decays.
In these plots, we scan six parameters F ij and w ij , and fix other parameters as written in the caption of Fig. 4. The coefficient C E is determined so as to reproduce the mass squared difference of atmospheric neutrinos, i.e., ∆m 2 atm = |(m 3 ν ) 2 − (m 1 ν ) 2 |. All these points satisfy the current neutrino oscillation data assuming the normal hierarchy for neutrino masses,
where the black (red) points show the case with tan β = 30 (100). We see that the red points are given in the lower-left region of these planes as compared with the black points, because values of the branching ratios of ℓ → ℓ ′ γ are dominantly determined by the second term of Eq. (B1), which is proportional to c β . We note that the loop contributions of η ±± ℓ andĤ ± are unimportant as long as |F ij | become larger, whose magnitude is typically taken to be smaller than O(10 −3 ). We see that the µ → eγ data give the most sever constraint on the parameter space, because of its strongest upper bound on the branching ratio. For the other two modes τ → µγ and τ → eγ, our predictions are typically smaller than the current limit by two or more orders of magnitude, because the branching ratio has already been highly suppressed by the µ → eγ data. It is also seen that the branching ratio of µ → 3e is significantly lower than the current upper limit. We observe an anticorrelation between the branching ratios of µ → eγ and τ → eγ, see the most upper-left panel, which is predicted by our characteristic flavor structure of the Yukawa interactions. In addition, we find a very strong correlation between the branching ratios of µ → eγ and µ → 3e. This can be understood from the fact that the largest factor f L or f R given in Eq. (B4) mainly determines the size of these branching ratios.
Similar plots but for the inverted hierarchy case are shown in Fig. 5. We see that B(τ → µγ) tends to have similar values with the order of 10 −11 (10 −12 ) for tan β = 30 (100) as a function of the other branching ratios, which cannot be seen in the normal hierarchy case.
The other behavior is quite similar to the normal hierarchy case.
Let us give a comment on cases for the other sets of the fixed parameters in the above analysis. Among the fixed parameters, the mass differences between the extra leptons can significantly affect the results of the LFV branching ratios. For larger values of the mass difference, these branching ratios tend to become larger, because the suppression by the unitarity of the W matrix becomes weaker. Therefore, larger values of tan β or V are required to avoid the constraint from the µ → eγ data. Varying the other mass parameters such as m η ± does not change the above results significantly.
V. PHENOMENOLOGY
In this section, we discuss phenomenological consequences of our model.
A. Dark matter physics
As we discussed in Sec. II, our model has a dark matter candidate η 0 , which is the Z rem 2 -odd complex neutral scalar boson. This scalar boson has trilinear interaction terms with neutral Z rem 2 -and CP-even scalar bosons H i (i = 1, 2, 3), among which H 1 can be identified with the discovered Higgs boson (h) with the mass of 125 GeV. Therefore, the phenomenology of dark matter is similar to the Higgs portal scenario in which annihilation processes occur via the s-channel Higgs-boson mediations.
If the dark matter mass m η 0 is smaller than 2m W and if the additional Higgs bosons H 2 and H 3 are much heavier than 2m W , the dominant annihilation process is η 0 η 0 * → h ( * ) → ff with f = t, whose thermal averaged cross section is evaluated at the leading order as
σv ≃ f m f πv 1 − 4m 2 f m 2 η 0 3/2 λ 2 hηη (4m 2 η 0 − m 2 h ) 2 + m 2 h Γ 2 h ,(31)
where Γ h is the width of h (∼ 4 MeV) and λ hηη is the dimensionful η 0 η 0 * h coupling. From
Eq. (31), the annihilation cross section becomes significant when m η 0 is getting close to m h /2 due to the resonant effect of h, so that smaller values of λ hηη are required to keep the observed relic abundance of dark matter, Ω DM h 2 ≃ 0.12 [43].
On the other hand, constraints from dark matter direct detections have to be taken into account. In our scenario, the dark matter scattering with a nucleon N through the t-channel
Higgs mediation becomes to be most important. Using the effective vertex, which is given by
L N = g NN Nh,(32)
with g N ≃ 1.1 × 10 −3 [44], the scattering cross section is expressed as
σ N ≃ g 2 N λ 2 hηη 4π(m N + m η 0 ) 2 m 2 N m 4 h ,(33)
where m N is the mass of the nucleon. We here neglect the 3-momentum of the dark matter. The black curve shows the required value of λ hηη /v satisfying Ω DM h 2 = 0.12 as a function of the dark matter mass.
In Fig. 6, we show the combined results for the calculations of the relic abundance and the bound from the direct search experiment (XENON1T) [45]. The red curve represents the upper limit on the normalized hη 0 η 0 * coupling by the VEV λ hηη /v as a function of the dark matter mass m η 0 . The required value of λ hηη /v to satisfy Ω DM h 2 = 0.12 is shown as the black curve. As already mentioned, smaller values of λ hηη are required to keep the observed value of the abundance when the dark matter mass is around the resonance region ∼ m h /2. Our dark matter candidate can simultaneously satisfy both the relic abundance and the direct detection bounds at m η 0 ∼ m h /2 as it has been known in Higgs portal models, see e.g., [46][47][48]. Another scenario with a much larger mass ( a few TeV) may also be considered for the dark matter to satisfy both the dark matter data. However, we do not discuss details of this case because such a scenario strongly depends on the parameters of extra fields. Instead, we only have shown that there is at least a solution in our model to satisfy the dark matter data in addition to the neutrino data. Because of the special Yukawa interaction, they can decay into quarks with different flavors such as A/H 2 → tc and H ± → ts [36]. Dedicated simulation studies for these flavor violating decays of the extra Higgs bosons at the LHC have been performed in Ref. [49]. In addition, there are the other extra particles whose masses are proportional to V such as η ±± ℓ , η ± ,η ± , η 0Ĥ ± and H 3 . They can also be detected at the LHC if the associated coupling constants are small enough.
One of the most interesting signatures in our model arises from the doubly-charged scalar bosons η ±± ℓ . At the LHC, they can be created in pair via the Drell-Yan process pp → γ * /Z * → η ++ ℓ η −− ℓ and in associated with the singly-charged scalar bosons pp → W ± * → η ±± ℓη ∓ /η ±± ℓ η ∓ [50]. In Fig. 7, we show the cross sections for these production processes at the pp colliders with the collision energy of 13 TeV (left) and 27 TeV (right). We use Using the mass spectrum assumed in the analyses for the neutrino masses and LFV decays, the decay pattern of η ±±
ℓ is determined to η ±± ℓ → E ± k ℓ ± i → ℓ ± i ℓ ± j η 0 ,
where η 0 is the dark matter candidate. The intermediate state E ± k can be on-shell in this case. The flavor of the same-sign dilepton in the final state is determined by the F and W matrices, which are constrained by the neutrino oscillation data and the LFV data. Therefore, by measuring the flavor of the same-sign dilepton system, we can indirectly test the mechanism for the neutrino mass generation. The partial decay rates of E ± i and η ±± ℓ are calculated as
Γ(E ± i → ℓ ± j η 0 ) = M E 32π 1 − m 2 η 0 M 2 E 2 |(f R ) ij | 2 + |(f L ) ij | 2 ,(34)Γ(η ℓ ±± → ℓ ± i ℓ ± j η 0 ) = m η ℓ ±± 16π M E 32πΓ E 1 − m 2 η ±± ℓ M 2 E 2 1 − m 2 η 0 M 2 E 2 × |(2W T F f R ) ij | 2 + |(2W T F f L ) ij | 2 ,(35)
where f L and f R are given in Eq. (B4), and the mass of the SM charged leptons is neglected. All these benchmark points satisfy the neutrino oscillation data and the LFV data assuming the normal hierarchy case for the neutrino masses.
B ij denote B(η ℓ ±± → ℓ ± i ℓ ± j η 0 ),
Inputs
Outputs In these expressions, small differences of the masses and the widths of E i 4 are ignored; i.e.,
FM E ≡ M 1 (= M 2 = M 3 ) and Γ E ≡ Γ E 1 (= Γ E 2 = Γ E 3 )
. The total width Γ E is typically of order 0.1 GeV, so that the narrow width approximation is valid for the calculation of the decay rate of η ±± ℓ . In Table II and Table III, contains missing energies which are carried by the dark matter η 0 , the invariant mass distribution of the same-sign dilepton system does not have a peak at around the mass of η ±± ℓ . This property is different from that of doubly-charged Higgs bosons from an SU(2) L triplet or a singlet field, which can decay into the same-sign dilepton without missing energy. Therefore, the current bounds on the mass of such doublycharged Higgs bosons, around 800 GeV depending on the flavor of the final state leptons at the LHC [52], cannot be applied to that on η ±± ℓ . In order to extract the bound on the mass of η ±± ℓ from current experiments and its discovery potential at future experiments, dedicated simulation studies are needed, which are beyond the scope of this paper.
For the sake of completeness, let us discuss the phenomenology for the singly-charged scalar bosons η ± ,η ± andĤ ± . For simplicity, we neglect the effect of the small mixing angle θ η (θ H ) between η ± andη ± (H ± andĤ ± ). These bosons can be produced in pair via the Drell-Yan process at collider experiments. In addition,η ± can also be produced in association with η ±± ℓ as already discussed in the above text, see also Fig. 7 for its production cross section. Their decay processes can be η ± /η ± → ν i E ± k → ν i ℓ ± j η 0 5 andĤ ± → ℓ ± i ν j . Because of the Z rem 2 symmetry, the decays of η ± andη ± include the dark matter. The decay of η ± (η ± andĤ ± ) occurs via the Yukawa coupling y E (F ), so that the flavor dependence of the charged lepton in the final state could be different among the decaying particles. We note that the singly-charged scalar bosons in the inert doublet model can decay into the W boson and a lighter Z 2 -odd scalar particle. Therefore, the signatures from the decays of η ± andη ± can be different from those in the inert doublet model [53,54]. On the other hand, the decay property ofĤ ± is quite similar to that of singly-charged scalar bosons in the Zee 5 If η ±± ℓ are lighter thanη ± , theη ± → η ±± ℓ W ∓( * ) processes are also possible.
model [55][56][57].
VI. CONCLUSIONS
We have proposed a new model for the generation of tiny neutrino masses based on the 3-3-1 gauge symmetry within the minimal fermion content required for the gauge anomaly cancellation. In this model, the source for lepton number violation is obtained by extending the minimal Higgs sector of 3-3-1 models with an additional SU(3) L triplet scalar field.
Majorana masses for the active neutrinos are generated at one-loop level. We have found the parameter sets which satisfy the current neutrino data under the constraint from the LFV decays of the charged leptons such as µ → eγ and µ → 3e.
In our model, Z rem 2 appears as a remnant symmetry after the breaking of the electroweak symmetry and that of the global U(1) ′ symmetry, where the latter symmetry is introduced to avoid the dangerous flavor changing neutral current. The symmetry Z rem 2 guarantees the stability of the dark matter candidate which is the lightest neutral Z rem 2 -odd scalar particle η 0 . We have confirmed that the dark matter candidate can satisfy the relic abundance and the direct search results when the dark matter mass is taken to be at around the half of the discovered Higgs boson mass.
We then have discussed the collider phenomenology of our model. One of the most interesting signatures arise from productions and decays of the doubly-charged scalar bosons We first give the mass formulae for the Z rem 2 -even scalar bosons. The neutral components of the scalar triplet fields can be expressed as
φ 0 a = 1 √ 2 (φ R a + v a + iφ I a ), (a = 1, 2, 3),(A1)
with v 3 = V . There are 3 pairs of singly-charged, 3 CP-odd and 3 CP-even scalar states in the Z rem 2 -even sector. Their mass eigenstates are obtained by introducing the following orthogonal transformations:
φ ± 1 φ ± 2 φ ± ℓ = c β s β 0 −s β c β 0 0 0 1 G ± φ ± H ± = c β s β 0 −s β c β 0 0 0 1 1 0 0 0 c θ H −s θ H 0 s θ H c θ H G ± H ± H ± , (A2) φ I 1 φ I 2 φ I 3 = (x + 12 , x − 12 , x 3 ) G 0 G ′0 A , (A3) φ R 1 φ R 2 φ R 3 = R H H 1 H 2 H 3 ,(A4)
where G ± , G 0 and G ′0 are the NG bosons which are absorbed into the longitudinal component of W ± , Z and Z ′ , respectively. In Eq. (A4), R H is the 3 × 3 orthogonal matrix which can be expressed by three independent mixing angles. These mixing angles are determined from the mass matrix for the CP-even Higgs bosons given in Eq. (A14). In Eq. (A3),
x ± 12 ≡ (x 1 ± x 2 )/|x ± 12 | and x 3 are three component vectors defined as
x T 1 = − v V c β (1 + v 2 V 2 c 2 β ) −1/2 , 0, (1 + v 2 V 2 c 2 β ) −1/2 ,(A5)
x T 2 = (c β , −s β , 0) ,
x T 3 = s β (1 + v 2 V 2 s 2 β c 2 β ) −1/2 , c β (1 + v 2 V 2 s 2 β c 2 β ) −1/2 , (1 + V 2 v 2 s 2 β c 2 β ) −1/2 . (A7) ,(A6)M 2 H = 2v 2 λ 1 c 2 β + µV √ 2 tan β v 2 λ 12 s β c β − µV √ 2 vV λ 13 c β − µV √ 2 s β 2v 2 λ 2 s 2 β + µV √ 2 cot β vV λ 23 s β − µv √ 2 c β 2V 2 λ 3 + µv 2 √ 2V c β s β .(A13)
(A14)
In the above expressions, the lower-left elements are the same as the corresponding transposed elements.
Next, we present the mass formulae for the Z rem 2 -odd scalar states, in which there are one pair of doubly-charged, 3 pairs of singly-charged and one neutral complex scalar states.
The squared mass of the doubly-charged scalar bosons η ±± ℓ is given by
m 2 η ±± ℓ = m 2 4 + 1 2 V 2 λ 34 + v 2 (c 2 β λ 14 + s 2 β λ 24 + s 2 β ρ 24 ) .(A15)
The mass eigenstates of the singly-charged and neutral states are defined as follows:
η ± 2 η ± 3 η ± ℓ = 1 1 + v 2 s 2 β V 2 v V s β 1 0 −1 v V s β 0 0 0 1 G ′± η ± η ± ℓ = 1 1 + v 2 s 2 β V 2 v V s β 1 0 −1 v V s β 0 0 0 1 1 0 0 0 c θη −s θη 0 s θη c θη G ′± η ± η ± ,(A16) η 0 1 η 0 * 3 = 1 1 + v 2 c 2 β V 2 v V c β 1 −1 v V c β G 0 Y η 0 * ,(A17)
where G ′± and G 0 Y are the NG bosons which are absorbed into the longitudinal component of W ′ and Y , respectively. The squared masses of the physical scalar bosons and the mixing angle θ η are given by
m 2 η ± = c 2 θη (M 2 η ± ) 11 + s 2 θη (M 2 η ± ) 22 + s 2θη (M 2 η ± ) 12 ,(A18)m 2 η ± = s 2 θη (M 2 η ± ) 11 + c 2 θη (M 2 η ± ) 22 − s 2θη (M 2 η ± ) 12 ,(A19)m 2 η 0 = V 2 2 1 + v 2 V 2 c 2 β ρ 13 + √ 2 tan β µ V ,(A20)sin 2θ η = 2(M 2 η ± ) 12 m 2 η ± − m 2 η ± .(A21)
The mass matrix for the singly-charged state M 2 η ± is calculated as
M 2 η ± = V 2 2 ρ 23 + √ 2µ V tan β 1 + v 2 V 2 s 2 β vV ξ 2 c β 2 1 + v 2 V 2 s 2 β vV ξ 2 c β 2 1 + v 2 V 2 s 2 β m 2 4 + V 2 λ 34 +v 2 c 2 β (λ 14 +ρ 14 )+v 2 s 2 β λ 24 2 . (A22)
The amplitudes a ϕ L are obtained by taking f L ↔ f R in Eq. (B1) and m i ↔ m j in Eqs. (B2)-(B3).
For the µ → 3e process, we have
a LRLR = 3 k,l=1 (f † R ) ek M k (f L ) kµ F kl box (m η 0 )(f † R ) el M l (f L ) le ,(B6)a LRRL = 3 k,l=1 (f † R ) ek M k (f L ) kµ F kl box (m η 0 )(f † L ) el M l (f R ) le ,(B7)a LLLL = 3 k,l=1 (f † L ) ek (f L ) kµ G kl box (m η 0 )(f † L ) el (f L ) le + 16(F * W ) ek (W T F ) kµ G kl box (m η ±± )(F * W ) el (W T F ) le + 4 m 2Ĥ ± (F † ek F kµ )F † el F le ,(B8)a RRRR = 3 k,l=1 (f † R ) ek (f R ) kµ G kl box (m η 0 )(f † R ) el (f R ) le , (B9) a LLRR = 3 k,l=1 (f † L ) ek (f L ) kµ G kl box (m η 0 )(f † R ) el (f R ) le ,(B10)a RRLL = 3 k,l=1 (f † R ) ek (f R ) kµ G kl box (m η 0 )(f † L ) el (f L ) le ,(B11)
with F ij box (m) ≡
the mass matrices for the SM fermions (M f with f = u, d, e) and those for the extra fermions (M F with F = U, D, E) are given by:
FIG. 1 .
1One-loop diagrams for the neutrino mass generation. Charged scalar fields are written in the basis where the NG boson fields are separated, see Appendix A.
and θ H and θ η being the mixing angles of charged scalar fields, see Eq. (A2) and (A16). In the above expression, M diag e and M diag E are the diagonalized mass matrices for the SM charged leptons and those for the extra leptons, respectively. The dashed fields are related to the original one by Eq. (18). It is important to mention here that the appearance of the W matrix which is the 3 × 3 unitary matrix similar to the Cabibbo-Kobayashi-Maskawa matrix plays a crucial role to reproduce the current neutrino oscillation data as it will be clarified below.
(19) becomes the orthogonal matrix which can be parameterized by the three angles entered in the W matrix,
The current upper limits on the branching ratios of LFV decays of charged leptons are given at the 90% confidence level as B(µ → eγ) < 4.2 × 10 −13 (MEG[39]), B(τ → eγ) < 3.3 × 10 −8 (BaBar[40]), B(τ → µγ) < 4.4 × 10 −8 (BaBar[40]), B(µ → eee) < 1.0 × 10 −12 (SHINDRUM[41]).
FIG. 4 .
4Correlations between BRs of ℓ → ℓ ′ γ (upper figures) and those of ℓ → ℓ ′ γ and µ → 3e (lower figures) in the case of tan β = 30 (100) for black (red) points. The dashed vertical line shows the current upper limit on the branching ratio of µ → eγ. In these plots, we fix V = 10 TeV, (M 1 , M 2 , M 3 ) = (300, 301, 302) GeV, m η ± = 450 GeV, mη± = m η ±± ℓ = mĤ ± = 400 GeV and m η 0= 63 GeV. We scan the six parameters of F ij and w ij with the ranges of −10 −3 ≤ F ij ≤ 10 −3 and −π/2 ≤ w ij ≤ π/2. All points satisfy the neutrino oscillation data assuming the normal hierarchy case for the neutrino masses.
FIG. 6 .
6Combined results for the dark matter relic abundance and the constraint from the dark matter direct search experiment. The red curve shows the upper limit on the normalized coupling by the VEV λ hηη /v as a function of the dark matter mass given by the XENON1T experiment.
FIG. 7 .
7Production cross sections for the pp → γ * /Z * → η ++ ℓ η −− ℓ and pp → W ± * → η ±± ℓη ∓ processes as a function of m η ±± ℓ . The collision energy is taken to be 13 TeV (left) and 27 TeV (right). For the η ±± ℓη ∓ productions, we take mη± = m η ±± ℓ . FIG. 8. Production cross sections for the e + e − → γ * /Z * → η ++ ℓ η −− ℓ process at future lepton colliders. The left panel shows the dependence on the center of mass energy √ s with fixed values of the mass of η ±± ℓ to be 200, 400 and 600 GeV. The right panel shows the dependence on the mass of η ±± ℓ with fixed values of √ s to be 255 GeV, 500 GeV, 1 TeV and 3 TeV. B. Collider physics In our model, there are many new particles, which can potentially be produced at collider experiments. However, in the case with V ≫ v, our model effectively coincides with the THDM with a special Yukawa interaction, which gives new extra bosons H ± , A and H 2 .
NNPDF2. 3 -
3LO[51] for the parton distribution functions. We here neglect effects of the mixing angle θ η on the associated production cross section, because by the analyses for neutrino masses in Sec. IV we have typically θ η = O(10 −2 ) which is sufficiently small. It can be seen that the cross section for the pair production with m η ±±ℓ ≃ 400 GeV can be a few (a few tens of) fb at √ s =13 TeV (27 TeV). Slightly larger (smaller) values are obtained for the cross section of the associated production η ++ ℓη − (η −− ℓη + ). In Fig. 8, we also show the pair production cross section at future lepton colliders, e + e − → γ * /Z * → η ++ ℓ η −− ℓ , as a function of the center of mass energy √ s (left) and the mass m η ±± ℓ (right).
and Γ tot is the total width of η ℓ ±± . In this table, w ij , B ij and Γ tot are given in the units of rad, % and keV, respectively. The other input parameters are fixed to be V = 10 TeV, (M 1 , M 2 , M 3 ) = (300, 301, 302) GeV, m η ± = 450 GeV, mη± = m η ±± ℓ = mĤ ± = 400 GeV and m η 0 = 63 GeV.
we give several benchmark points which satisfy the neutrino data and the LFV data for the cases of the normal and the inverted hierarchies, respectively. The other input parameters are fixed to be V = 10 TeV, (M 1 , M 2 , M 3 ) = (300, 301, 302) GeV, m η ± = 450 GeV and mη± = m η ±± ℓ = mĤ ± = 400 GeV. The dark matter mass m η 0 is fixed to be 63 GeV in order to satisfy the relic abundance and the constraint from the direct search experiment, see Sec. V A. For each point, we show our predictions for the branching ratios of η ±± ℓ and its total width Γ tot . It is seen that the width of η ±± ℓ is typically given to be of the order of keV, because the couplings F ij are taken to be O(10 −3 -10 −4 ) in order to avoid the constraint from LFV decays of charged leptons. Depending on these benchmark points, η ±± ℓ can predominantly decay into the same-sign dilepton with various combinations of their flavors. Because the decay of η ±± ℓ
η
±± ℓ , because of the characteristic flavor dependence of the same-sign dilepton in the final state. Even if η ±± ℓ are too heavy to be detected at collider experiments, the Higgs sector of our model, which effectively coincides with the THDM at V ≫ v, predicts the special structure of the Yukawa interaction to the SM quarks. The extra Higgs bosons can then mainly decay into quarks with different flavors, so that the detection of such bosons would be important to test our model.In conclusion, our model can give an interesting testable example of the 3-3-1 scenario, where the number of generation of quarks and leptons, neutrino oscillation and dark matter can be explained simultaneously.
TABLE I .
IParticle content and charge assignment under the gauge symmetry SU
The appearance of such remnant unbroken Z 2 parity in 3-3-1 models has been pointed out in Refs.[20,21].
This parameter plays the similar role to a soft-breaking Z 2 parameter in THDMs.
The small mass difference is required in order to reproduce the neutrino mixing data and to avoid the constraints from LFV decays of charged leptons.
ACKNOWLEDGMENTSThe authors would like to thank Niko Koivunen and Katri Huitu for useful discussions.The squared masses of physical Higgs bosons and the mixing angle θ H are given byWe can identify m H 1 as the mass of the discovered Higgs boson, 125 GeV. The mass matricesAppendix B: Formulae for lepton flavor violating decays of charged leptonsFor the ℓ → ℓ ′ γ processes, we havewhere f L and f R are the coefficients of theĒ ′ R ℓ ′ L η 0 * andĒ ′ L ℓ ′ R η 0 * vertex, respectively, given asThe loop functions G 1 and G 2 are given by.(B13)
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| []
|
[
"Some Notes on Complex Symmetric Operators",
"Some Notes on Complex Symmetric Operators"
]
| [
"Marcos S Ferreira "
]
| []
| []
| In this paper we show that every conjugation C on the Hardy-Hilbert space H 2 is of type C = T * C 1 T , where T is an unitary operator and C 1 f (z) = f (z), with f ∈ H 2 . In the sequence, we extend this result for all separable Hilbert space H and we prove some properties of complex symmetry on H. Finally, we prove some relations of complex symmetry between the operators T and |T |, where T = U |T | is the polar decomposition of bounded operator T ∈ L (H) on the separable Hilbert space H. | 10.2478/ejaam-2021-0006 | [
"https://arxiv.org/pdf/1709.08616v3.pdf"
]
| 55,925,362 | 1709.08616 | 08df464d80f9f16b006140e894cb84d86fb22ee7 |
Some Notes on Complex Symmetric Operators
10 Feb 2019 February 12, 2019
Marcos S Ferreira
Some Notes on Complex Symmetric Operators
10 Feb 2019 February 12, 2019arXiv:1709.08616v3 [math.FA]
In this paper we show that every conjugation C on the Hardy-Hilbert space H 2 is of type C = T * C 1 T , where T is an unitary operator and C 1 f (z) = f (z), with f ∈ H 2 . In the sequence, we extend this result for all separable Hilbert space H and we prove some properties of complex symmetry on H. Finally, we prove some relations of complex symmetry between the operators T and |T |, where T = U |T | is the polar decomposition of bounded operator T ∈ L (H) on the separable Hilbert space H.
Introduction
Let L (H) be the space of bounded linear operators on a separable Hilbert space H. A conjugation C on H is an antilinear operator C : H → H such that C 2 = I and Cf, Cg = g, f , for all f, g ∈ H. An operator T ∈ L (H) is said to be complex symmetric if there exists a conjugation C on H such that CT = T * C (we will often say that T is C-symmetric). Complex symmetric operators generalize the concept of symmetric matrices of linear algebra. Indeed, it is well known ([5, Lemma 1]) that given a conjugation C, there exists an orthonormal basis {f n } ∞ n=0 for H such that Cf n = f n . Hence, if T is C-symmetric then
T f n , f m = Cf m , CT f n = f m , T * Cf n = T f m , f n ,(1)
that is, T has a symmetric matrix representation. The reciprocal of this fact is also true. That is, if there is an orthonormal basis such that T has a symmetric matrix representation, then T is complex symmetric. The complex symmetric operators class was initially addressed by Garcia and Putinar [5,6] and includes the normal operators, Hankel operators and Volterra integration operators. Now, let L 2 be the Hilbert space on the unit circle T and let L ∞ be the Banach space of all essentially bounded functions on T. It is known that {e n (e it ) := e int : n ∈ Z} is an orthonormal basis for L 2 . The Hardy-Hilbert space, denoted by H 2 , consists of all analytic functions f (z) = ∞ n=0 a n z n on the unit disk D such that ∞ n=0 |a n | 2 < ∞. It is clear that B := {e n (z) = z n : n = 0, 1, 2, . . .} is an orthonormal basis for H 2 .
For each φ ∈ L ∞ , the Toeplitz operator T φ : H 2 → H 2 is defined by
T φ f = P (φf ) ,
for each f ∈ H 2 , where P : L 2 → H 2 is the orthogonal projection. The concept of Toeplitz operators was initiated by Brown and Halmos [1] and generalizes the concept of Toeplitz matrices. In [7], Guo and Zhu raised the question of characterizing complex symmetric Toeplitz operators on H 2 in the unit disk. In order to obtain such characterization, Ko and Lee [8] introduced the family of conjugations C λ : H 2 → H 2 , given by
C λ f (z) = f (λz)
with λ ∈ T and proved the following result:
Theorem 1.1. Let φ(z) = ∞ n=−∞ φ(n)z n ∈ L ∞ . Then T φ is C λ -symmetric if, and only if, φ(−n) = λ n φ(n), for all n ∈ Z.
Canonical Conjugations
Our first objective in this paper is to study relations between an arbitrary conjugation C on H 2 and the conjugation C 1 f (z) = f (z). Once the conjugation C 1 is a kind of canonical conjugation on H 2 , we observe a close relationship between conjugations of H 2 and conjugation C 1 , namely: Theorem 2.1. If C is an conjugation on H 2 , then exists an unitary operator T : H 2 → H 2 such that T C = C 1 T.
Proof. Since C is an conjugation, there exists an orthonormal basis
B ′ = {f n } ∞ n=0 of H 2 such that Cf n = f n . Now, let B = {e n } ∞
n=0 the standard orthonormal basis of H 2 and the linear isomorphism T : H 2 → H 2 given by T ∞ n=0 a n f n = ∞ n=0 a n e n .
Note that T f n = e n , for all n ≥ 0, and therefore T is unitary. Now, for f (z) = ∞ n=0 a n e n ∈ H 2 , we get
C 1 f (z) = ∞ n=0 a n e n = ∞ n=0 a n T (f n ) = T ∞ n=0 a n Cf n = (T C) ∞ n=0 a n f n = (T C) ∞ n=0 a n T −1 (e n ) = T CT −1 f (z) , whence C 1 T = T C.
The previous theorem says that all complex symmetric Toeplitz operator is unitarily equivalent to a C 1 -symmetric operator. Indeed:
Remark 2.2. Let T φ : H 2 → H 2 an Toeplitz operator. Observe that, if T φ is C- symmetric, since the operator T of previous theorem is unitary, we have C 1 = T CT * ,
therefore the operator T 2 := T T φ T * is C 1 -symmetric (see [5, p. 1291]). This shows that T φ and T 2 are unitarily equivalent operators. Moreover, is obvious that, if T commutes with C 1 or C, then C = C 1 .
Corollary 2.3. Let A ∈ L (H 2 ) . Then A is C 1 -symmetric if, and only if, the matrix of A with respect the canonical basis of H 2 is symmetric. Proof. If A is C 1 -symmetric, then C 1 A = A * C 1 . Moreover, by previous theorem there exists an isomorphism T on H 2 such that T C 1 = C 1 T. Consider B = {e n } ∞ n=0 and B ′ = {f n } ∞ n=0 orthonormal basis of H 2 such that T f n = e n and C 1 f n = f n .
Thus, we must
C 1 e n = C 1 (T f n ) = T C 1 (f n ) = T f n = e n ,
that is C 1 e n = e n , ∀n ≥ 0. Therefore, by (1), follows that
[A] B = [A] t B .
Reciprocally, suppose that A is C-symmetric such that Ce n = e n . By previous theorem, T C = C 1 T and T e n = e n . Hence, T is the identity operator and so C = C 1 .
In fact, the reciprocal of the Theorem 2.1 is true:
Proposition 2.4. If T : H 2 → H 2 is an unitary operator, then C := T −1 C 1 T is an conjugation on H 2 .
Proof. It is easy to see that C is an antilinear operator. Now, since T is an unitary operator, considering B = {e n } ∞ n=0 the orthonormal basis of H 2 , we have We already know that every normal operator is complex symmetric and that the reciprocal in general is not true. However, for Toeplitz operators, Theorem 1.1 gives us:
Fact 2.8. If T φ is J -symmetric, then T φ is normal.
Now note that if T φ is normal not necessarily T φ is J -symmetric. In fact, if φ(z) = −z + z then T φ is normal, however is not J -symmetric.
Properties of Complex Symmetry
In the following, we present some properties of complex symmetry in Hilbert spaces. The first result gives us a way to get complex symmetric operators from another complex symmetric operator. First, we need some lemmas: Proof. We already know that U is unitary and C and J-symmetric and that UC = CJC is a conjugation, by Lemmas 3.1 and 3.2. Now since U * = U −1 = JC and T is Csymmetric, we have
UT (UC) = UT CU * = UCT * U * = UC(UT ) * .
Reciprocally, suppose that UC(UT ) * = UT (UC). Thus and so T is CJC-symmetric. Analogous, we prove that T is JCJ-symmetric. Proof. We already know that U = CJ is unitary and both C and J -symmetric. Now, note that
CT * U * = C(UT ) * = U * UC(UT ) * = U * UT UC = T UC = T CU * , whence CT * = T C.J T * C = (CJ ) * T ⇔ UT * C = CU * T.
First see that if T is C-symmetric, then UT * C = U(CT ) = (CU * )T . Reciprocally, we have
CT * = CU * (UT * C)C = CU * (CU * T )C = (UCCU * )T C = T C.
Complex Symmetry of Aluthge and Duggal Transforms
Recall that the polar decomposition of an operator T : H → H is uniquely expressed by T = U |T |, where |T | = √ T * T is a positive operator and U is a partial isometry such that Ker(U) = Ker |U| and U maps cl(Ran |T |) onto cl(Ran(T )). In this case, the Aluthge and Duggal Transforms are given, respectively, by T = |T | We already known that the Aluthge transform of a complex symmetric operator is also complex symmetric (see [4,Theorem 1]). In this section we study relations between complex symmetry of T and |T | with relation the conjugations C and J, as well as the operators T and T .
Ce n , Ce m = T * C 1 T e n , T * C 1 T e m = T T * C 1 T e n , C 1 T e m = T e m , T e n = e m , T * T e n = e m , e n .By other hand, once C 2 = (T −1 C 1 T ) (T −1 C 1 T ) = I, follow the desired.In short, the Theorem 2.1 and the Proposition 2.4 tell us that:Corollary 2.5. If T : H 2 → H 2 an linear isomorphism and C := T −1 C 1 T ,then T is unitary if, and only if, C is a conjugation on H 2 . Now, once every separable Hilbert space has an orthonormal basis, follows that the Corollary 2.5 is true for any separable Hilbert space H. In fact, if B = {f n } is an orthonormal basis on H, then J : conjugation on H. Thus, we have: Theorem 2.6. If T : H → H an linear isomorphism and C := T −1 J T , then T is unitary if, and only if, C is a conjugation on H. Proof. Analogous to Theorem 2.1 and Proposition 2.4. Remark 2.7. Note that in the Hardy-Hilbert space H 2 , we have J = C 1 .
Lemma 3.1. ([6, Lemma 1]) If C and J are conjugations on a Hilbert space H, then U = CJ is a unitary operator. Moreover, U is both C-symmetric and J-symmetric.Lemma 3.2. ([3, Lemma 2.2]) If U : H → H is a unitary and complex symmetric operator with conjugation C, then UC is a conjugation.
Proposition 3 . 3 .
33Let T : H → H an operator and C and J conjugations on H. Then T is C-symmetric if, and only if, UT is UC-symmetric, where U = CJ.
Lemma 3. 4 .
4If T : H → H is both C-symmetric and J-symmetric, then T is both CJC-symmetric and JCJ-symmetric. Proof. By Lemma 3.1, we have that U := CJ is unitary and C and J-symmetric. Hence, by Lemma 3.2, UC = CJC is a conjugation on H. Thus, since CT = T * C and JT = T * J we get (CJC) T = C (T J) C = T * (CJC) ,
Proposition 3. 5 .
5If T : H → H is both C and J-symmetric, then T U is C-symmetric, where U = CJ.Proof. In fact, once T is both C-symmetric and J-symmetric, we have by Lemma 3.4 that T is CJC-symmetric and so(T U) C = T (CJC) = CU * T * = C (T U) * .Proposition 3.6. Let A : H → H an invertible operator and C-symmetric. If T is an operator on H such that T A = AT , then T is C-symmetric if, and only if, T A is C-symmetric.Proposition 3.7. Let U : H → H an unitary operator J-symmetric. If T is an operator such that UT * = T U (that is, T and T * are unitarily equivalents), then:(i) JT * = T * J ⇔ T is UJ−symmetric. (ii) UJT = T JU * ⇔ T is J−symmetric.Proposition 3.8. An operator T : H → H is C-symmetric if, and only if, J T * C = (CJ ) * T .
Proposition 3. 9 .
9Let T : H → H an operator and C a conjugation on H. If T C = CT , then T is C-symmetric if, and only if, T is self-adjoint.
T = |T | U.
Proposition 4 . 1 .
41If T is complex symmetric, then |T | is also complex symmetric.Proof. If CT = T * C, we have by Remark of [4, Lemma 1] that T = CJ |T |, where J commutes with |T |. Thus, once that CJ is a unitary operator, follows that J |T | = C(CJ |T |) = |T | * (CJ) * C = |T | * J.
Corollary 4. 2 .
2If T is complex symmetric, then |T | is self-adjoint.
Proposition 4 . 3 .
43Let C and J conjugations on H such that T = CJ |T |. If |T | is C-symmetric, then T is also C-symmetric. Proof. First, let's show that |T | is J-symmetric. In fact, see that J(JC |T |) = C |T | = |T | * C = (|T | * CJ)J,and so JC |T | is J-symmetric. Thus, by Proposition 3.3, |T | is J-symmetric. Therefore, it is enough to see that:CT = C(CJ |T |) = |T | * J = (|T | * JC)C = (CJ |T |) * C = T * C.
Corollary 4. 4 .
4Let T = CJ |T |. If |T | is C-symmetric, then T = T .
Corollary 4. 5 .
5Let T = CJ |T |. Then |T | is C-symmetric if, and only if, T is Jsymmetric. Proposition 4.6. Let T = CJ |T |. If C |T | = |T | * C and CJ = JC, then T is Jsymmetric. Proof. In fact, we have that JT = J(CJ |T |) = C |T | = |T | * JJC = |T | * JCJ = (CJ |T |) * J = T * J.
Algebraic properties of Toeplitz operators. A Brown, P R Halmos, J.Reine Angew. Math. 213A. Brown, P.R. Halmos, Algebraic properties of Toeplitz operators, J.Reine Angew. Math. 213 (1963-1964) 89-102.
Banach algebra techniques in operator theory. R G Douglas, Graduate Texts in Mathematics. New YorkSpringer-Verlag179second ed.R. G. Douglas, Banach algebra techniques in operator theory, second ed., Graduate Texts in Mathematics, vol. 179, Springer-Verlag, New York, 1998.
M Fatehi, Complex symmetric weighted composition operators. ArXiv e-printsM. Fatehi, Complex symmetric weighted composition operators, ArXiv e-prints (2018).
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Complex symmetric operators and applications. S R Garcia, M Putinar, Trans. Amer. Math. Soc. 358S.R. Garcia, M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006) 1285-1315.
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| []
|
[
"muNet: Evolving Pretrained Deep Neural Networks into Scalable Auto-tuning Multitask Systems",
"muNet: Evolving Pretrained Deep Neural Networks into Scalable Auto-tuning Multitask Systems"
]
| [
"Andrea Gesmundo [email protected] ",
"Google Research ",
"Jeff Dean ",
"Google Research "
]
| []
| []
| Most uses of machine learning today involve training a model from scratch for a particular task, or sometimes starting with a model pretrained on a related task and then fine-tuning on a downstream task. Both approaches offer limited knowledge transfer between different tasks, time-consuming human-driven customization to individual tasks and high computational costs especially when starting from randomly initialized models. We propose a method that uses the layers of a pretrained deep neural network as building blocks to construct an ML system that can jointly solve an arbitrary number of tasks. The resulting system can leverage cross tasks knowledge transfer, while being immune from common drawbacks of multitask approaches such as catastrophic forgetting, gradients interference and negative transfer. We define an evolutionary approach designed to jointly select the prior knowledge relevant for each task, choose the subset of the model parameters to train and dynamically auto-tune its hyperparameters. Furthermore, a novel scale control method is employed to achieve quality/size trade-offs that outperform common fine-tuning techniques. Compared with standard fine-tuning on a benchmark of 10 diverse image classification tasks, the proposed model improves the average accuracy by 2.39% while using 47% less parameters per task. | 10.48550/arxiv.2205.10937 | [
"https://arxiv.org/pdf/2205.10937v2.pdf"
]
| 248,987,520 | 2205.10937 | 84a28c8c7cf7f6cca11fcf8b496660895a86b85d |
muNet: Evolving Pretrained Deep Neural Networks into Scalable Auto-tuning Multitask Systems
Andrea Gesmundo [email protected]
Google Research
Jeff Dean
Google Research
muNet: Evolving Pretrained Deep Neural Networks into Scalable Auto-tuning Multitask Systems
Most uses of machine learning today involve training a model from scratch for a particular task, or sometimes starting with a model pretrained on a related task and then fine-tuning on a downstream task. Both approaches offer limited knowledge transfer between different tasks, time-consuming human-driven customization to individual tasks and high computational costs especially when starting from randomly initialized models. We propose a method that uses the layers of a pretrained deep neural network as building blocks to construct an ML system that can jointly solve an arbitrary number of tasks. The resulting system can leverage cross tasks knowledge transfer, while being immune from common drawbacks of multitask approaches such as catastrophic forgetting, gradients interference and negative transfer. We define an evolutionary approach designed to jointly select the prior knowledge relevant for each task, choose the subset of the model parameters to train and dynamically auto-tune its hyperparameters. Furthermore, a novel scale control method is employed to achieve quality/size trade-offs that outperform common fine-tuning techniques. Compared with standard fine-tuning on a benchmark of 10 diverse image classification tasks, the proposed model improves the average accuracy by 2.39% while using 47% less parameters per task.
Introduction
ML techniques are increasingly successful in a growing number of applications, either by iteratively improving the state-of-the-art in impactful domains such as language [Brown et al., 2020] and vision [Dosovitskiy et al., 2021], or achieving new capabilities such as protein folding [Senior et al., 2020], chip design [Mirhoseini et al., 2020], superhuman performance in different competitions [Silver et al., 2016, Vinyals et al., 2019. Although successful, the standard ML methodology is based on practices that limit the quality and efficiency of the produced solutions. Some of these practices include:
Single task models The majority of ML practice, both in applications and research, aims to produce models that can solve a single task. Such models can be customized and tuned to the task at hand, and in some cases achieve state-of-the-art results with a self-contained and well-defined methodology.
Limited prior knowledge reuse A significant portion of all ML models are trained from a random initialized state. This inefficiency has been alleviated by the increased availability of reference pretrained models that can be used as a starting point for fine-tuning to produce a dedicated model for any target tasks with matching input modalities and task framing [Devlin et al., 2019, Raffel et al., 2020. However, the approach of training large base models, and then fine-tuning a separate copy of it on each downstream task, loses out on the potential benefits of incorporating knowledge of the downstream tasks into the core model and enabling this knowledge to be reused for related tasks.
Manual Tuning The standard process of training an ML model requires repeating the training multiple times to tune its hyperparameters and identify the configuration that yields the better results. Engineering efficiency Traditional large-scale software systems enable teams of hundreds or thousands of software engineers to work collectively on a single software artifact, through decomposition of the problem into many sub-problems, and through well-defined abstraction boundaries. We currently lack the ability to have thousands of ML engineers and researchers collectively contribute to a single model. By enabling automatic incorporation of new tasks and knowledge into a single running system, through evolutionary exploration, we see a direction where many people can all contribute to the improvement of a single overall ML model that is suited to a growing number of tasks, and that incorporates the learning and knowledge facilitated by many other people working on the same system. In addition to enabling thousands of engineers and researchers to contribute to a single system, it may even be possible to have tens of millions of people without knowledge of ML training to contribute to training a single ML model, by contributing new tasks and examples and building on the skills that have been taught to the model by others.
We propose a method designed to explore the identified opportunities for improvement of the standard ML methodology. This method can jointly solve multiple tasks to achieve increased efficiency and quality for each. The knowledge learned from each task is compartmentalized in components that can be reused by multiple tasks. As the system accumulates the ability to solve more tasks, it is able to find better solutions for subsequent tasks, and to do so with increasing efficiency, requiring fewer added parameters for new tasks. The knowledge compartmentalization allows to avoid common problems of multitask models such as catastrophic forgetting. The exploration of model architectures and identification of the subset of prior knowledge most relevant for each task is guided by an evolutionary algorithm designed to dynamically adjust the exploration/exploitation balance without need of manual tuning of meta-parameters. The same evolutionary logic is also employed to dynamically tune the hyperparameters of the components of the multitask model. The proposed auto-tuning approach identifies a schedule of values over time for each hyperparameter rather than a single value. We apply the proposed method to the domain of image classification, demonstrating empirically that it can achieve quality/size trade-offs that outperform common fine-tuning techniques. Furthermore, the proposed method can use of any pretrained model as a starting point of the evolution. Allowing to build on top of prior work and further increasing efficiency in terms of convergence time.
Method
Deep neural networks are commonly defined as a sequence of layers that maps the input data into a prediction over the output space. As a concrete example, we refer to the Visual Transformer (ViT) architecture [Dosovitskiy et al., 2021] that is used for the experimental phase. ViT is composed of a sequence of different types of layers:
1. Patch embedding: the first layer of the model maps the input image into a sequence of embedded tokens, each corresponding to a patch of the input image.
2. Class token: a classification token is prepended to the sequence. The final hidden state corresponding to this token is used as the aggregate sequence representation for classification tasks [Devlin et al., 2019]. 2). This model was generated by the experiment repetition achieving the max average test accuracy across the eight tasks. Each task is identified with a unique color. Top nodes represent the head layer of each task, and display the validation accuracy for that task. Each sequence of edges of the same color connecting a task input to its head, defines the layers sequence composing the model for each task.
Internal nodes are represented with the color of the task on which the parameters of the corresponding layer were trained last. The number of unique tasks each layer have been trained on through the sequence of its ancestors is displayed in the top right corner label, n .
3. Position embedding: the sequence representation is then augmented with an embedding that carries each patch positional information.
4. Transformer layers: the sequence representation generated by the input layers is iteratively transformed by a stack of transformer layers [Vaswani et al., 2017].
5.
Model head: a final fully connected layer mapping the representation produced by the top-most transformer layer for the class token into the logits.
Mutations
We define four types of mutation (see Figure 1). These mutations are designed to transform a preexisting model, parent, into a mutated version of it, child. The child model is defined by incremental changes to architecture, parameters and hyperparameters. A child model may become a parent in following generations. Each child model is trained on a single task, but can leverage the knowledge accumulated by its ancestors on different tasks. In the presented experiments (see Section 3), a pretrained ViT model is used as the initial pre-existing model, root model. During the first evolutionary iteration, the root model is mutated by applying a subset of the possible mutations. In subsequent iterations, ViT models that have already been subjected to mutations and training cycles can also be selected as a parent model. The presented method instantiation allows the following mutation types:
Layer cloning Any layer of a parent model is by default shared with the child model in a frozen state, so that the child models will not be able to apply gradient updates to the shared parameters, although they can flow gradients through the shared, frozen layers. The layer cloning mutation allows a child model to create a trainable copy of any of the parent layers. The head layer is always cloned since it always needs to be trainable. If a child model is trained on a task different from the parent's task, then a new head layer is created with output shape matching the number of classes of the new task and zero initialized following Dosovitskiy et al. [2021]. Notice that the parent model parameters and architecture are immutable and cannot be affected by child mutations and subsequent training. This is one of the features that provides strong guarantees against catastrophic forgetting.
Layer insertion
The model architecture can be mutated by inserting a new layer in between any two consecutive layers of the parent architecture. In the instantiation of the method presented in this paper, we allow the insertion of residual adapter layers , Houlsby et al., 2019. Residual adapters have been used with success as a parameter efficient method to adapt a pretrained model to a specific downstream task or domain. We define residual adapters as a sequence of two fully connected layers with variable inner dimension size. The Gelu non-linearity is applied on the inner representation [Hendrycks and Gimpel, 2016]. Layer normalization is applied to the input of the fully connected layers [Ba et al., 2016]. The second layer is zero initialized, to guarantee that its insertion does not alter the parent model representation at the start of the child training.
Layer removal This mutation removes a layer from the sequence that defines the parent model. In the instantiation presented in this paper, layer removals are constrained to be applicable only to the top transformer layer. This constraint avoids the knowledge and representation disruption that would result from removing internal layers, but still allows removing parameters and, combined with other types of mutations, to incrementally reach more parameter efficient configurations.
Hyperparameter change Every model is associated with a set of hyperparameters such as those of its optimizer, architecture and data preprocessing. Any of the parent hyperparameters can be changed into a value sampled from the set of possible values defined for each hyperparameter. In the instantiation of the method presented in this paper, the sampling of numerical hyperparameters is constrained to the values that are neighbouring the parent value in the sorted list of possible values. This constrains hyperparameters to be changed incrementally and biases the search toward the initial values of the root model, that in our application are the result of an extensive study aimed to identify a fine-tuning configuration for ViT models that is as generic as possible [Steiner et al., 2021].
Notice that, every hyperparameter of a child model is set to a single value. However, considering that a child model can be interpreted as a continuation of its ancestors training with different hyperparameters, then the method can be regarded as capable of learning a schedule for each hyperparameter. Furthermore, given that a different subset of layers is trainable for each ancestor, this approach can also be considered to be capable of learning a different optimizer schedule per layer.
Evolutionary algorithm
This section describes the novel evolutionary algorithm defined for the proposed method. We refer to the first parent model, used to initialize the evolutionary process, as the root model. The root model can be either pretrained or randomly initialized. During the evolutionary process, the algorithm searches for an improved model for a single task at a time, referred to as the active task. During the active phase of a task, a population of models for the active task is evolved, we refer to this as the active population. The active population is initialized with a set of seed parent models, that includes the root model and the best model generated for each prior task. Then, the active population is iteratively extended by:
1. sampling a parent model from the active population, 2. producing a child model by applying to the parent model a sampled set of mutations, 3. performing cycles of training and evaluation to train and score the child model.
Each model is assigned a score that can be a function of multiple factors such as the validation quality or model cost metrics. Early population pruning is performed by discarding the child models that did not achieve a better score than their parent. At the end of each active phase for a task, only the model achieving best score is kept as part of the multitask model. Tasks can become active multiple times, allowing for increased cross-tasks knowledge transfer. Details of the evolutionary algorithm are provided below and in Algorithm 1.
Parent sampling Parent models are sampled among the models in the active population. These candidate parent models are visited in order of score, starting with the highest scoring one. A candidate parent model, m, is selected as parent with probability:
p parent (m) = 1 2 #of f springs(m)(1)
Where #of f springs(m) denotes the number of child models that have been generated so far for the active task by selecting model m as parent. If the current candidate parent is rejected, then iteratively the model with the next best score is considered to be selected as parent with probability p parent (·). This approach can be interpreted as a back-off strategy following half-life exponential decay with t1 /2 = 1. This method prioritizes exploitation of high scoring models having few offsprings. But also, in combination with early pruning, it automatically transitions toward a more exploratory behavior in case the higher scoring models are unable to generate improved offsprings.
Mutations sampling
The mapping from a parent model into a child model is defined by a subset of the possible mutation actions. The set of possible mutation actions includes: a) one layer cloning action for each layer of the parent model, b) one residual adapter insertion for each pair of consecutive transformer layers, c) one top transformer layer removal action, d) one hyperparameter change for each hyperparameter. Each possible mutation is independently sampled for application with mutation probability, µ. For all the experiments reported in this paper, µ is set to 0.1.
Child training A newly sampled child model is trained on the active task for a given number of epochs. The model is evaluated on the validation set and scored after each epoch. After training, only the parameters of the version of the child model achieving best score are retained.
Scoring function Each trained model is scored. The scoring function can be defined to optimize a mixture of factors such as quality, inference latency, training compute or model size depending on the application requirements and can change over time. The experiments presented in this paper demonstrate the ability to control the quality/size trade-off by using the following scoring function:
score(m) = q(m) * s ( #accounted-params(m) #root-model-params )(2)
Where q(m) denotes the quality metric computed on the validation set. s ∈ ]0, 1] is the scale factor. #root-model-params is the total number of parameters of the root model. And #accounted-params(m) is the sum of parameters used by model m, dividing each parameter count by the number of models sharing its use:
#accounted-params(m) = p∈P (m) 1 #models(p) + 1(3)
Where P (m) denotes the set of all parameters of m, and #models(p) is the count of models for tasks different from the active task that are currently using this parameter. The scaling factor, s, allows to control the size of the generated multitask model, and achieve different quality/size trade-offs.
Note that, the defined evolutionary algorithm guarantees that once a model has been trained, its architecture and the parameters storing its knowledge and cannot be altered. Nonetheless, new models can access its knowledge or even extend it to improve it or specialize it. Therefore, this method provides immunity against common problems of multitask models: 1) catastrophic forgetting, since the knowledge of a trained model is always preserved, 2) negative transfer, since the method automates the selection of the knowledge most relevant for each new task, 3) gradients interference, since within each training cycle each parameter can receive gradients only from one source.
Experiments
This section describes the experiments conducted to analyze the properties of the proposed method and test their generality. The proposed method is referred to as "multitask network" or for brevity muNet. All the experiments reported are reproducible by using: 1) The ViT checkpoints and model definition library published by Steiner et al. [2021], 2) the published code of the proposed method, 3) datasets publicly available via the Tensorflow Datasets image classification catalog. All the experiments are executed on a TPUv3 machine with 8 cores [Jouppi et al., 2017].
The default ViT configuration used, is the one identified by Steiner et al. [2021] as the most generic and best performing for ViT fine-tuning: SGD optimizer with 0.9 momentum and 0.01 learning rate, using cosine decay schedule with 10% warm up, 512 batch size, no weight decay, and gradient clipping at global norm 1. During auto-tuning experiments, the evolutionary algorithm can change the following hyperparameters of the optimizer, image preprocessing and architecture:
• Bold values are the defaults. This search space is a parametrization of the ViT model configuration as defined by the published ViT model library, except for the "residual adapters inner dimension", which has been added for the layer insertion mutation introduced by our evolutionary algorithm.
Multitask Character Classification Benchmark
The first benchmark is designed to ease reproducibility and enable fast development iterations. It is composed of 8 publicly available character classification tasks (see Table 3). For each we identify non-overlapping training, validation and test splits (see Table 4).
The root model for this benchmark, is a ViT Ti/16 pretrained on the imagenet-21k dataset with weight decay 0.1, no stochastic depth, no dropout and "light1" augmentation as defined in Steiner et al. [2021]. As the intent of this benchmark is to allow for fast iterations, the root model architecture is capped to use only 3 of the 12 transformer layers provided by the referred configuration. The default image size for this benchmark is set to 32×32 pixels, since it is a multiple of the patch size (16) that is close to the resolution in which images are provided for most of the tasks in this benchmark (28).
Baseline models are initialized with the same parameters of the root model and are fine-tuned for a total of 80 epochs on each task. 8 model replicas are trained in parallel, one on each core. The version of the baseline model achieving the best validation quality among all the periodic evaluations performed by all replicas is evaluated on the test set. The muNet experiments are allocated an equivalent budget of training steps. Each task is given 2 active task iterations, and each such iteration is composed of 8 child model generation phases. During each generation phase, 8 child models are sampled and trained in parallel, one on each of the 8 TPUv3 cores. Each child model is trained for 5 epochs, in order to achieve a training budget equivalent to that allocated for the baseline model:
#baseline-epochs = #muN et-epochs * #generations * #task-iterations(4)
To smooth the distribution of compute over the set of tasks, the amount of the training performed between validations is capped to: min(1 epoch, 100 batches).
Results
The experiments demonstrate the ability of the proposed method to achieve better quality/size trade-offs compared to standard fine-tuning techniques. Metrics are summarized in Table 1 and 6. Figure 3 (left) displays a graphical summary of the trade-offs achieved by the different methods. The horizontal axis measures the average number of parameters per task. The vertical axis measures the mean accuracy achieved on the final test sets of the 8 tasks. Each experiment was repeated 5 times for each model configuration using different random seed values for each repetition. The vertical coordinate of the plotted curves represents the average quality and the shaded area represents the standard deviation computed across the experiment repetitions.
The rightmost point of the "Fine-tune top layers" curve represents quality/size achieved by fully fine-tuning a distinct copy of the root model for each each of the 8 tasks. The horizontal coordinate of full fine-tuning matches the size of the ViT Ti/16 3 layers model used as root model:~1.48M parameters. The next point following the "Fine-tune top layers" curve from right to left, represents the model configuration having all the layers before the first transformer layer frozen and shared across the 8 tasks. This configuration achieves a lower quality with fewer parameters per task compared to full fine-tuning. The same trend continues as more transformer layers are frozen and shared, until the last configuration matches the multi-head architecture, where all the layers are frozen and shared except for each individual task head.
The rightmost point of the "Residual adapters" curve represents the quality/size achieved by the architecture configuration that shares all the parameters of the root model, and only head and residual adapters with hidden dimension of 512 are trainable. Following the "Residual adapters" curve right to left, the hidden dimension of each residual adapter is halved at each step, resulting in a monotonic decrease of both quality and model size. Different trade-offs for the proposed method are are achieved by using different scale factors: s ∈ {0.02, 0.3, 0.7, 0.9, 0.95, 0.98, 1}. The rightmost point in the curve uses scale factor of 1, resulting in no size penalty: score(m) = q(m). We also compare against a version of the proposed method with ablated hyperparameters auto-tuning.
The proposed method achieves the best quality across the spectrum of model sizes. The best muNet model improves the best quality achieved by full fine-tuning by 13%, while using 21% fewer parameters per task. muNet with scale factor 0.3 outperforms all the baseline models of all types in both quality and size dimensions. The configuration with scale factor 0.3 outperforms the quality of the best full fine-tuning model, while using less parameter than the multi-head model. The smaller size is achieved because the evolutionary search converged to use only 2 of the 3 transformer layers for all tasks. Figure 2 displays the multitask architecture generated by the best experiment repetition with scale factor 0.3. The layers with most parameters are shared across all tasks: transformer layers (444,864 parameters) and patch embedding (147,648). While the smaller layers are branched for specialization: class token (192), position embedding (960), residual adapters (12,896). Comparing with muNet without auto-tuning allows us to assess the impact of the hyperparameter evolution.
Furthermore, residual adapters ablation experiments are then performed to analyze the effects of the layer insertion mutation (see Figure 4). For scale factors above 0.9, it is possible to achieve equivalent performance within noise even without residual adapters. However, for lower scale factors, residual adapters seem critical to achieve significantly better quality/size trade-offs. Nevertheless, muNet with neither residual adapters nor auto-tuning still outperforms the residual adapters baseline.
Visual Domain Decathlon Benchmark
The generality of the proposed method is tested with more challenging tasks, longer training, and bigger models. The Visual Domain Decathlon Benchmark consists of 10 image classification tasks that have been explicitly selected to represent different domains (see Table 3), thus providing a more challenging context for knowledge transfer. All datasets consist of images with 72×72 resolution. Thus, the default resolution is set to the next multiple of the patch size: 80×80.
While the experiments on the Multitask Character Classification Benchmark provided comparison in a fast training setup, this benchmark is configured for longer and more expensive training. Thus, baseline models are trained for a total of 480 epochs on each task. The proposed method is again provided with an equivalent training budget. Two iterations are performed over the task set. For every iteration on each active task, 8 batches of child models are generated. That, following equation 4, results in 30 epochs of training per child model. Epochs are again capped to 200 batches. As a root model, we use a ViT B/16 pretrained on the imagenet-21k dataset with weight decay 0.1, no stochastic depth, no dropout and "medium1" augmentation as defined in Steiner et al. [2021]. All ViT B/16 12 transformer layers are used, for a total of 85.6M parameters. This is~60 times bigger than the root model used for the experiments on the Multitask Character Classification Benchmark. The batch size is halved to 256, to fit the memory requirements. The number of experiment repetitions is decreased from 5 to 3 due to the higher experiment cost. Other configuration details are unchanged.
Results Figure 3 presents a graphical summary of the comparison between the different methods, while Table 2 reports a numerical summary. Again, we observe that the proposed method outperforms standard fine-tuning methods. Once more, auto-tuning provides a significant contribution to improvements in both quality and size dimensions. The evolution of the best "scale factor=0.3" shows that most of the layers specialization happens in the smaller layers and 1 to 2 transformer layers are dropped for each task (see Figure 9). Differently from the previous results, we observe that the best fine-tuning performance is obtained by fine-tuning only the layers above the first transformer layer.
Individual task accuracies (see Table 7) show that muNet models archives significant gains on the smaller tasks, this can be expected as the smaller tasks have fewer data to train on and can benefit the most from knowledge transfer. But also, significant gains can be observed on larger tasks like imagenet or cifar100. Baseline models validation curves (see Figure 5) show lower variance.
Analyzing the knowledge exchange dynamics (see Figures 6 and 7), we notice that specialized tasks with small datasets, such as vgg-flowers, aircraft and ucf101, reuse knowledge from multiple tasks, but no other task reuses the parameters fine-tuned by them. Inversely, generic tasks with large datasets, such as imagenet12, contribute to all other tasks, but their model results mostly trained on their data.
The distributions of the hyperparameters sampled by the auto-tuning algorithm (see Figure 8) show that the distributions shift in accordance with the reward system. For example, as the size penalty increases, distributions shift toward sampling smaller adapters and less transformer layers.
Related work
The success of transfer-learning applications hinge on adequate prior knowledge selection, that avoids common negative-transfer pitfalls [Rosenstein, 2005. Common solutions rely on data or model selection techniques [Zhang et al., 2020, Mensink et al., 2021. Models trained jointly on multiple tasks can be affected by negative gradients interference when parameters receive gradients from multiple sources [Chen et al., 2018, Yu et al., 2020, and by catastrophic forgetting of prior knowledge as new tasks are learned. Catastrophic forgetting is also critical for continual learning or life long learning applications [McCloskey andCohen, 1989, French, 1999]. These knowledge loss problems can be alleviated with weighted combination of tasks [Liu et al., 2019b, Sun et al., 2020 and gradient transformation methods [Chen et al., 2018, Sener and Koltun, 2018, Kendall et al., 2018. Stronger guarantees are provided by methods that compartmentalize task specific knowledge in dedicated parameter subsets , Houlsby et al., 2019, Rusu et al., 2016, Rosenfeld and Tsotsos, 2020.
The automation of hyperparameter tuning has been commonly addressed with Bayesian optimization [Srinivas et al., 2010, Bergstra et al., 2011, Snoek et al., 2012, evolutionary methods have also been explored [Jaderberg et al., 2017, Zhang et al., 2011. Hyperparameter tuning can be considered related to the neural architecture search (NAS), as architectures can be defined by the selection of architectural hyperparameters. Initially, NAS methods have been based on reinforcement learning Le, 2017, Tan et al., 2019]. Sample efficient evolutionary approaches have been also proposed [Real et al., 2019, Maziarz et al., 2018, Furthermore, more efficient parameter-sharing approaches have been proposed [Pham et al., 2018, Liu et al., 2019a, Kokiopoulou et al., 2019 that connect the NAS field with the one of routing networks , Maziarz et al., 2019.
Conclusion
We introduced a novel method that can evolve pretrained ML models into multitask systems capable of jointly solving many tasks. Empirical evidences show that the method can achieve improved quality and efficiency compared to common fine-tuning techniques. We also presented a novel evolutionary approach that is employed for both identifying the knowledge most suitable for any new task and also dynamically tuning the hyperparameters of the system components. Furthermore, the presented evolutionary method can automatically adjust the exploration/exploitation balance without requiring to tune any additional meta-parameter. The generated multitask systems are immune from catastrophic forgetting, negative transfer and gradients interference.
Overall, the approaches demonstrated in this work are encouraging signs that a much-more-automated, incrementally extensible machine learning system for handling thousands or millions of tasks is achievable. We show that starting with a high quality baseline model and combining this with a novel evolutionary search procedure can efficiently tackle new tasks and create a single multi-task model with high accuracy across all tasks in a completely automated manner. Future work can continue to build toward a system that can handle thousands and then millions of tasks, across multiple modalities.
Limitations To contain results variance, it is important to control the ratio between the size of the search space and the available exploratory budget. This can be challenging considered that any incremental addition to the search space causes an exponential increase in the number of possible model configurations. For example, adding one clonable layer doubles the number of possible configurations, and adding a new hyperparameter with 10 possible values increases it by a decimal order of magnitude. While, an increase in compute leads to a linear increase in exploration budget. Method details such as incremental hyperparameter mutation and exponential decay sampling, contribute to control the variance by implicitly imposing a prior distribution over the search space.
Societal impact Improvements in the efficiency, ease-of-use and automation of contemporary ML methodologies such as those proposed in this work, can broaden the accessibility of ML approaches to a much wider audience, as less ML-specific knowledge is needed to achieve a high-quality model for a new task. This approach can contribute to reducing the energy usage and carbon footprint of ML applications. Granting access to the broader community to novel techniques and systems allows to democratize the ground-breaking advancements in automation and achievements of new capabilities. S. Liu, Y. Liang, and A. Gitter. Loss-balanced task weighting to reduce negative transfer in multi-task learning.
In AAAI, 2019b.
A Assets details
This section reports details of the datasets, code and model checkpoints used for the presented empirical study. The code implementing the proposed method is published at https://github.com/googleresearch/google-research/tree/master/muNet and distributed under the Apache License 2.0. The ViT model definition and checkpoints published by Steiner et al. [2021] are available at https://github.com/google-research/vision_transformer and distributed under the Apache License 2.0. These resources allow to reproduce all the reported experiments. Tables 3, 4 and 5 report details of the datasets used. [Cohen et al., 2017] Creative Commons Attribution 4.0 License emnist/letters [Cohen et al., 2017] Creative Commons Attribution 4.0 License kmnist [Clanuwat et al., 2018] Attribution-ShareAlike 4.0 International mnist [LeCun et al., 1998] Creative Commons Attribution 4.0 License omniglot [Lake et al., 2015] The MIT License cmaterdb/bangla [Das et al., 2012b,a] Apache License 2.0 cmaterdb/devanagari [Das et al., 2012b,a] Apache License 2.0 cmaterdb/telugu [Das et al., 2012b,a] Apache License 2.0
Visual Domain Decathlon Benchmark aircraft Creative Commons Attribution 4.0 License cifar100 Creative Commons Attribution 4.0 License daimlerpedcls Creative Commons Attribution 4.0 License dtd Creative Commons Attribution 4.0 License gtsrb Creative Commons Attribution 4.0 License imagenet12 Creative Commons Attribution 4.0 License omniglot Creative Commons Attribution 4.0 License svhn Creative Commons Attribution 4.0 License ucf101 Creative Commons Attribution 4.0 License vgg-flowers Creative Commons Attribution 4.0 License
B Experimental details
This section reports additinoal details of the experiments discussed in the paper. The muNet experiments are repeated without the layer insertion action that is needed to insert residual adapters in the generated architectures. The residual adapter ablation is repeated also for the "muNet without auto-tuning" configuration. All other aspects of the experiment are unchanged, such as: scale factors considered: {0.02, 0.3, 0.7, 0.9, 0.95, 0.98, 1}, 5 repetitions for each scale factor, and equivalent train budget for each experiment. Curves plot top 1 test accuracy averaged across the 5 repetitions, and the shaded area represents standard deviation. Notice that for scale factors above 0.9 the corresponding curves with/without residual adapters are equivalent within noise, while for lower scale factors the absence of residual adapters leads to a significant worsening of the achieved quality/size trade-offs. For reference, the performance achieved by the residual adapters baseline is also plotted. Notice that muNet with neither residual adapters nor auto-tuning still outperforms the residual adapters baseline across the whole range. The cross markers, +, highlight the validation accuracy achieved by the best models. The circle markers, •, display the test accuracy achieved by the best models. Curves for the muNet repetitions are shorter since they include only the trainingvalidation cycles performed by the model (and its ancestors) that results being the best at the end of the final active task cycle. We observe that muNet models archive significant gains on the smaller tasks, this can be expected as the smaller tasks have fewer data to train on and can benefit the most from transfer of knowledge. But also we observe significant gains on larger tasks like imagenet12 or cifar100. Baseline models display lower variance. Figure 6: Graph representing the knowledge flow for the best models generated by muNet for the Visual Domain Decathlon Benchmark tasks. Each node represents a task. Each directional edge from a source task to a target task represents the usage by the target task best model of knowledge/parameters introduced by the source task. Knowledge transfer is quantified as the fraction of training cycles performed on the source tasks by the parameters included the best model for the target task. The thicker the edge and arrow, the more training has been performed on the source task. Edges starting and ending on the same node represent the fraction of the total training that has been performed on the target task itself. Specialized tasks with small dataset, such as vgg-flowers, aircraft and ucf101, reuse knowledge from multiple tasks, but no other task reuses parameters fine-tuned by them. And vice versa, generic tasks with large datasets, such as imagenet12, contribute to all other tasks, but their best model is mostly trained on their own dataset. Figure 7: Graph representing the knowledge flow for the best models generated by muNet for the Multitask Character Classification tasks. Each node represents a task. Each directional edge from a source task to a target task represents the usage by the target task best model of knowledge/parameters introduced by the source task. Knowledge transfer is quantified as the fraction of training cycles performed on the source tasks by the parameters included the best model for the target task. The thicker the edge and arrow, the more training has been performed on the source task. Edges starting and ending on the same node represent the fraction of the total training that has been performed on the target task itself. Compared to the results for Visual Domain Decathlon (see Figure 6), we notice a more distributed knowledge transfer, that can be expected within a set of tasks of similar domain. Also we notice the same pattern that tasks with more training data mostly rely on the knowledge learned from their own datasets. In this case the larger tasks are emnist/digits and emnist/letter, that train mostly on their own data with also a significant exchange on knowledge between the two tasks since they are very related and have images in similar format. Also interesting to notice that the 4 smaller tasks (omniglot and the 3 cmarterdb tasks) transfer knowledge from at least 7 of the 8 tasks. Notice that the configuration with scale factor 0.3 converges to distributions that allow to achieve a different quality/size trade-off in accordance with the size penalty integrated in the scoring function, such as: smaller adapters dimension and less transformer layers.
Figure 1 :
1Graphical representation of the four types of mutations defined by the proposed method. New models are generated by applying a subset of the possible mutations.
Figure 2 :
2Model graph representing the multitask network for the eight character classification tasks displayed on the bottom nodes, generated by muNet with scale factor=0.3 (see Section 2.
K
. Maziarz, A. Khorlin, Q. de Laroussilhe, and A. Gesmundo. Evolutionary-neural hybrid agents for architecture search. ArXiv, abs/1811.09828, 2018. K. Maziarz, E. Kokiopoulou, A. Gesmundo, L. Sbaiz, G. Bartók, and J. Berent. Gumbel-matrix routing for flexible multi-task learning. ArXiv, abs/1910.04915, 2019. M. McCloskey and N. J. Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. Psychology of Learning and Motivation, 24:109-165, 1989. T. Mensink, J. R. R. Uijlings, A. Kuznetsova, M. Gygli, and V. Ferrari. Factors of influence for transfer learning across diverse appearance domains and task types. IEEE transactions on pattern analysis and machine intelligence, PP, 2021.
Figure 4 :
4Residual adapters ablation experiments on the Multitask Character Classification Benchmark.
Figure 5 :
5Validation quality measured during training of the best model achieved by each of the 3 repetitions of muNet and full fine-tune baseline models on each of the Visual Domain Decathlon Benchmark tasks. The vertical axis measures the top 1 accuracy over the validation set. The horizontal axis measures the number of training-validation cycles.
Figure 8 :
8Distributions of the hyperparameters sampled for the models generated with the "muNet scale factor=0.3" and muNet (scale factor=1) configurations for the Multitask Character Classification Benchmark (top) and Visual Domain Decathlon Benchmark (bottom), aggregated across experiment repetitions.
Figure 3: Comparison between standard fine-tuning techniques and the proposed method, with and without hyperparameters auto-tuning enabled, on the Multitask Character Classification Benchmark (left: see Section 3.1) and the Visual Domain Decathlon Benchmark (right: see Section 3.2). image size ∈ [multiples of root model's patch size]• residual adapters inner dimension ∈[8, 16, 32, 64, 128] learning rate ∈ [0.0001, 0.0002, 0.0005, 0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5]
• learning rate schedule ∈ [constant, cosine, restarts]
• learning rate schedule warm up ratio ∈ [0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4]
• momentum ∈ [0.7, 0.8, 0.85, 0.9, 0.95, 0.98, 0.99]
• nesterov update ∈ [False, True]
• cropped area range min ∈ [0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0]
• cropped aspect ratio range min ∈ [0.25, 0.5, 0.75, 1.0]
• flip left/right ∈ [False, True]
• brightness delta ∈ [0.0, 0.01, 0.02, 0.05, 0.1, 0.2]
• contrast delta ∈ [0.0, 0.01, 0.02, 0.05, 0.1, 0.2]
• saturation delta ∈ [0.0, 0.01, 0.02, 0.05, 0.1, 0.2]
• hue delta ∈ [0.0, 0.01, 0.02, 0.05, 0.1, 0.2]
•
Table 1 :
1Models comparison on the Multitask Character Classification Benchmark. Quality is measured as the test accuracy averaged across the 8 tasks. The table reports the max, average and standard deviation of the quality achieved by 5 experiment repetitions. Model size is measured as the average number of parameters per tasks. Compute is measured as TPUv3 core-hours. Videos displays the evolution of the best experiment repetition. Tasks details are reported inTable 6.Test acc. (%)
Params/task Compute
Video
Model
Max
Avg.±Std.
(×10 6 )
(core-h) (youtu.be/... )
Multi-head
20.79 20.25±0.65
0.23
4.32
LIvOCmF1aRk
Full fine-tuning
82.20 81.46±0.40
1.53
4.91
_fikzbxS_ZY
Residual adapters dim=512 82.28 81.98±0.17
0.82
5.01
R3ETGxo9CWE
muNet w/o auto-tuning
88.96 88.57±0.37
1.27
4.81
CQdeP1mpr-8
muNet scale factor=0.3
84.94 82.01±1.88
0.20
4.63
Ld9gfmJT6Ig
muNet
92.98 91.41±1.06
1.20
5.02
-xOl3lJV4fw
Table 2 :
2Models comparison on the Visual Domain Decathlon Benchmark. Quality is measured as the test accuracy averaged across the 10 tasks. The table reports the max, average and standard deviation of the quality achieved by 3 experiment repetitions. Model size is measured as the average number of parameters per tasks. Compute is measured as TPUv3 core-hours. Videos displays the evolution of the best repetition. Tasks details are reported inTable 7.Test acc. %
Params/task Compute
Video
Model
max
avg.±std.
(×10 6 )
(core-h) (youtu.be/...)
Multi-head
49.77 49.73±0.03
8.81
176.34
7UgPZYgh53U
Unfreeze above 1st tr. layer 81.51 81.39±0.14
78.98
183.67
vNo-j150nA0
Full fine-tuning
81.03 80.81±0.15
85.91
185.34
BK7AW95ii4s
Residual adapters dim=512 80.64 80.54±0.11
18.28
180.86
oWiniz6F2Lw
muNet w/o auto-tuning
82.31 82.27±0.03
55.04
181.05
P0SBFOuyj0s
muNet scale factor=0.3
80.11 79.59±0.65
9.30
165.34
THyc5lUC_-w
muNet
83.58 83.20±0.29
45.00
185.45
2scExBaHweY
Table 3 :
3Datasets metadata: 1) name, 2) short description, 3) number of classes, 4) input image
resolution, 5) number of train samples, 6) number of validation samples, 7) number of test samples.
Name
Description
Cls. Res. Train Valid.
Test
Multitask Character Classification Benchmark
emnist/digits
classify digits
10
28
228k
12k
40k
emnist/letters
classify letters
37
28
84.4k
4.4k
14..8k
kmnist
classify Japanese chars.
10
28
57k
3k
10k
mnist
classify digits
10
28
57k
3k
10k
omniglot
cls. 50 alphabets chars. 1623 105
19.3k
2.7k
3.1k
cmaterdb/bangla
cls. Bangla numerals
10
32
4.75k
250
1k
cmaterdb/devanagari cls. Devangari numerals
10
32
2.38k
125
500
cmaterdb/telugu
cls. Telugu numerals
10
32
2.38k
125
500
Visual Domain Decathlon Benchmark
aircraft
classify aircrafts
100
72
3.33k 1.67k 1.67k
cifar100
classify image subject
100
72
50k
5k
5k
daimlerpedcls
classify pedestrians
2
72
23.5k
2.9k
2.9k
dtd
classify textures
47
72
1.9k
940
940
gtsrb
cls. German traffic signs
43
72
31.4k
3.9k
3.9k
imagenet12
cls. annotated concept
1000
72
1.23M 24.5k 24.5k
omniglot
cls. 50 alphabets chars. 1623
72
17.8k
3.2k
3.2k
svhn
classify house numbers
10
72
47.2k
13k
13k
ucf101
cls. dynamic images
101
72
7.6k
976
976
vgg-flowers
classify flowers
102
72
1k
510
510
Table 4 :
4Datasets splits configuration. Datasets names match with the Tensorflow Datasets Catalogs identification strings and link to the corresponding catalog page. Datasets splits are represented with the standard Tensorflow Datasets format. SplitsName
Train
Validation
Test
Multitask Character Classification Benchmark
emnist/digits
train[5%:]
train[:5%]
test
emnist/letters
train[5%:]
train[:5%]
test
kmnist
train[5%:]
train[:5%]
test
mnist
train[5%:]
train[:5%]
test
omniglot
train
small1
small2
cmaterdb/bangla
train[5%:]
train[:5%]
test
cmaterdb/devanagari train[5%:]
train[:5%]
test
cmaterdb/telugu
train[5%:]
train[:5%]
test
Visual Domain Decathlon Benchmark
aircraft
train
validation[:50%] validation[50%:]
cifar100
train
validation[:50%] validation[50%:]
daimlerpedcls
train
validation[:50%] validation[50%:]
dtd
train
validation[:50%] validation[50%:]
gtsrb
train
validation[:50%] validation[50%:]
imagenet12
train
validation[:50%] validation[50%:]
omniglot
train
validation[:50%] validation[50%:]
svhn
train
validation[:50%] validation[50%:]
ucf101
train
validation[:50%] validation[50%:]
vgg-flowers
train
validation[:50%] validation[50%:]
Table 5 :
5Datasets reference and license.
Name
Reference
License
Multitask Character Classification Benchmark
emnist/digits
Table 6 :
6Per task details of the models comparison on the Multitask Character Classification Benchmark reported inTable 1. For each task is reported the mean and standard deviation of the test and validation top 1 accuracy computed over the experiment repetitions.Model
emnist/
digits
emnist/
letters
kmnist
mnist
omniglot
cmaterdb/
bangla
cmaterdb/
devanagari
cmaterdb/
telugu
Test accuracy
Multi-head
37.1±2.1 12.1±2.2 20.7±1.0 38.0±1.4 0.3±0.3 18.8±1.5 18.6±3.3 16.5±3.0
Full fine-tuning
98.2±0.1 87.4±0.1 91.5±0.3 98.2±0.1 49.1±1.0 82.4±0.8 63.4±2.1 81.3±1.1
R.adapter dim=512 98.2±0.1 86.5±0.3 89.5±0.1 98.2±0.1 47.9±0.9 85.2±0.7 67.8±1.3 82.6±0.7
muNet w/o a.tuning 98.1±0.1 87.9±0.3 91.8±0.3 98.1±0.2 66.6±1.8 90.7±0.6 83.2±1.6 92.1±1.0
muNet scale=0.3
95.8±1.5 80.9±4.8 80.2±7.6 95.5±1.9 57.0±5.1 85.0±3.2 77.8±5.7 84.0±4.8
muNet
98.4±0.4 89.3±1.5 92.7±1.8 98.6±0.4 78.8±5.7 93.9±1.2 85.8±2.0 93.9±1.8
Validation accuracy
Multi-head
37.3±1.7 16.9±0.6 26.7±1.1 36.8±1.2 1.0±0.2 19.4±1.4 17.3±1.6 18.1±1.0
Full fine-tuning
98.2±0.1 88.3±0.2 96.9±0.2 98.2±0.1 48.2±0.8 83.0±1.1 62.9±1.2 80.6±0.6
R.adapter dim=512 98.3±0.0 87.8±0.1 96.1±0.1 98.2±0.1 47.3±0.7 89.0±0.2 73.1±0.8 82.6±1.2
muNet w/o a.tuning 98.1±0.1 89.4±0.3 96.8±0.2 97.9±0.2 63.2±1.8 94.3±1.2 86.2±1.6 93.9±2.2
muNet scale=0.3
96.0±1.3 83.4±4.7 90.6±3.8 95.4±2.1 56.6±5.4 86.9±3.7 81.4±7.5 86.4±6.1
muNet
98.4±0.3 90.8±1.7 97.4±0.7 98.4±0.4 76.7±5.7 96.5±1.2 92.6±2.2 95.8±2.4
Table 7 :
7Per task details of the models comparison on the Visual Domain Decathlon Benchmark reported inTable 2. For each task is reported the mean and standard deviation of the test and validation top 1 accuracy computed over the experiment repetitions.Validation accuracy Multi-head 36.6±0.1 41.6±0.3 56.7±0.2 73.9±0.1 90.5±0.1 22.6±0.2 40.2±0.2 18.3±0.1 48.1±0.1 77.3±0.4 Unfreeze above 1st 74.6±0.3 94.9±0.0 89.5±0.2 100±0.0 100±0.0 82.6±0.1 78.6±0.3 39.7±0.2 62.0±0.3 93.8±0.2 Full fine-tuning 74.0±0.2 94.2±0.0 88.4±0.1 100±0.0 99.9±0.0 82.6±0.1 77.2±0.4 40.7±0.3 60.1±0.4 93.2±0.1 R.adapters dim=512 70.2±0.2 94.3±0.1 90.4±0.0 100±0.0 100±0.0 82.3±0.1 75.2±0.1 39.7±0.1 62.9±0.2 94.1±0.0 muNet w/o a.tuning 72.9±0.5 94.5±0.2 90.7±0.2 100±0.0 100±0.0 83.5±0.1 79.0±0.4 43.0±1.0 65.8±0.3 97.0±0.2 muNet scale=0.3 69.8±0.8 92.8±1.1 89.4±0.9 98.0±0.8 99.8±0.1 77.4±1.8 71.9±1.4 39.3±3.1 63.7±1.0 95.2±0.6 muNet 75.8±2.2 95.6±0.4 91.4±0.7 100±0.0 100±0.0 84.9±1.8 78.9±1.4 47.0±1.4 65.6±0.4 96.7±0.4Model
imagenet12
svhn
cifar100
gtsrb
daimlerpedcls
omniglot
ucf101
aircraft
dtd
vgg-flowers
Test accuracy
Multi-head
34.6±0.1 40.5±0.2 55.9±0.1 73.3±0.2 91.5±0.1 22.1±0.3 39.6±0.7 17.3±0.5 48.4±0.6 74.1±0.7
Unfreeze above 1st 73.8±0.2 93.8±0.2 88.7±0.2 99.9±0.0 99.8±0.1 81.6±0.4 78.6±1.1 41.0±0.4 62.8±0.2 93.9±0.3
Full fine-tuning
73.2±0.1 93.5±0.1 87.5±0.4 99.9±0.1 99.8±0.1 81.5±0.1 78.6±0.1 41.2±1.0 60.2±0.5 92.8±0.6
R.adapters dim=512 69.5±0.1 93.7±0.1 88.7±0.2 99.9±0.0 99.9±0.0 81.2±0.2 76.0±0.2 40.8±0.2 61.4±1.1 94.4±0.2
muNet w/o a.tuning 71.6±0.4 93.8±0.2 89.7±0.2 99.9±0.1 99.9±0.0 81.9±0.6 80.3±1.9 43.2±0.4 66.1±0.6 96.3±0.4
muNet scale=0.3
68.1±0.7 92.3±1.4 88.4±1.2 98.2±0.8 99.7±0.2 76.9±1.6 72.7±2.1 40.2±2.9 64.7±0.6 94.6±1.5
muNet
74.3±2.3 94.6±0.4 90.2±0.9 99.9±0.0 99.9±0.1 84.0±1.7 79.7±1.9 47.2±0.8 65.9±1.1 96.3±0.8
Preprint.
Algorithm 1 Pseudocode for the generation of a multitask system for a given list of tasks 1: List of tasks: T ← [t 1 , t 2 , ..., t |T | ] 2: Set of all the models in the multitask system: M ← {root-model} 3: for #task-iterations do 4:for Active task: t ∈ T do
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| [
"https://github.com/googleresearch/google-research/tree/master/muNet",
"https://github.com/google-research/vision_transformer"
]
|
[
"SPIN REPRESENTATIONS OF WEYL GROUPS AND THE SPRINGER CORRESPONDENCE",
"SPIN REPRESENTATIONS OF WEYL GROUPS AND THE SPRINGER CORRESPONDENCE"
]
| [
"Dan Ciubotaru "
]
| []
| []
| We give a common framework for the classification of projective spin irreducible representations of a Weyl group, modeled after the Springer correspondence for ordinary representations. | 10.1515/crelle.2011.160 | [
"https://arxiv.org/pdf/1006.3823v2.pdf"
]
| 37,214,201 | 1006.3823 | 0f1f22a77f37281fd49525e1a9c18d46b95fde25 |
SPIN REPRESENTATIONS OF WEYL GROUPS AND THE SPRINGER CORRESPONDENCE
20 May 2011
Dan Ciubotaru
SPIN REPRESENTATIONS OF WEYL GROUPS AND THE SPRINGER CORRESPONDENCE
20 May 2011
We give a common framework for the classification of projective spin irreducible representations of a Weyl group, modeled after the Springer correspondence for ordinary representations.
Introduction
Let Φ = (V, R, V ∨ , R ∨ ) be a semisimple crystallographic R-root system (see §2) with Weyl group W and a choice of positive roots R + . Assume that V is endowed with a W -invariant inner product , , and define the dual inner product on V ∨ , denoted by , as well. The Weyl group W is a finite subgroup of O(V ) and therefore, one can consider the double cover W of W in Pin(V ), a double cover of O(V ). A classical problem is to classify the irreducible genuine W -representations (i.e., the representations that do not factor through W ). This is known case by case, and goes back to Schur ([21]), in the case of S n , and was completed about 30 years ago for the other root systems by Morris, Read, Stembridge and others (see [16,17,19,24] and the references therein). In this paper, we attempt to unify these classifications in a common framework, based on Springer theory ( [23,13]) for ordinary W -representations. Our point of view is motivated by the construction of the Dirac operator for graded affine Hecke algebras ( [1]).
The group W is generated by certain elements s α of order 4, α ∈ R + , with relations similar to the Coxeter presentation of W (see §2.4). Letα ∈ R ∨ denote the coroot corresponding to the root α ∈ R. The starting observation is the existence of a remarkable central element Ω W ∈ C[ W ] (see §2. 6):
Ω W = α>0,β>0 sα(β)<0
|α||β| s α s β ;
(1.0.1) this element, rather surprisingly, behaves like an analogue of the Casimir element for a Lie algebra. Every irreducible W -representation σ acts by a scalar σ(Ω W ) on Ω W . For example, W has one (when dim V is even) and two (when dim V is odd) distinguished irreducible representations, which we call spin modules ( §2.3). If S is one such spin module, then S(Ω W ) = 2ρ, 2ρ , whereρ = 1 2 α∈R +α.
Before stating the main result, we need to introduce more notation. Let g be the complex semisimple Lie algebra with root system Φ and Cartan subalgebra h = V ∨ ⊗ R C, and let G be the simply connected Lie group with Lie algebra g. Extend the inner product from V ∨ to h. Let us denote by T (G) the set of (1.0.2) For example, if G = SL(n), the nilpotent elements e that occur in T 0 (G) are those whose Jordan canonical form has all parts distinct. In general, every distinguished nilpotent element e (in the sense of Bala-Carter [6]) appears in T 0 (G).
For every class in T (G), we may (and will) choose a representative (e, h, f ) such that h ∈ h. For every nilpotent element e, let A(e) denote the A-group in G, and let A(e) 0 denote the set of representations of A(e) of Springer type. For every φ ∈ A(e) 0 , let σ (e,φ) be the associated Springer representation (see §3.2). Normalize the Springer correspondence so that σ 0,triv = sgn.
There is an equivalence relation ∼ on the space W gen of genuine irreducible W -representations: σ ∼ σ ⊗ sgn; here, sgn is the sign W -representation. where Ω W is as in (1.0.1). (2) Let (e, h, f ) ∈ T 0 (G) be given. For every Springer representation σ (e,φ) , φ ∈ A(e) 0 , and every spin W -module S, there exists σ ∈ Ψ −1 [(e, h, f )] such that σ appears with nonzero multiplicity in the tensor product σ (e,φ) ⊗ S. Conversely, for every σ ∈ Ψ −1 [(e, h, f )], there exists a spin W -module S and a Springer representation σ (e,φ) , such that σ is contained in σ (e,φ) ⊗ S. Since triv(Ω W ) = sgn(Ω W ), Theorem 1.0.1(1) says in particular that any two associate genuine W -types σ 1 ∼ σ 2 lie in the same fiber of Ψ. This is why we need to quotient by ∼ in (1.0.5). It is natural to ask if one could reformulate (2) in the theorem so that a bijection like (1.0.5) (with certain appropriate quotients in the right hand side) holds for non-distinguished e ∈ N 0 (g). This is almost always the case, but there are counterexamples, e.g., Remark 3.9.1.
We should make clear that while the main result may appear close to a Springer type classification for W , we do not provide here a geometric construction for genuine representations of W . As we explain in §3, this classification fits in the setting of elliptic representation theory of W (Reeder [20]) and W , and its connection with nilpotent orbits. A different relation, between elliptic conjugacy classes in W and a family of nilpotent orbits ("basic") is presented in a recent paper by Lusztig [14].
There are two directions in which one can generalize Theorem 1.0.1. Firstly, it is apparent that one can extend these results to the generalized Springer correspondence ( [13]), by using a Casimir element Ω W ,c for an appropriate parameter function c : R + → Z. We present the details in §3.10, the main result being Theorem 3.10.3, which is the exact analogue of Theorem 1.0.1.
Secondly, an analogous correspondence should hold, even in the absence of nilpotent orbits, for non-crystallographic root systems, and more generally, for complex reflection groups. There, one should be able to substitute the nilpotent orbits and Springer representations in the right hand side of the correspondence in Theorem 1.0.1, with the space of elliptic tempered modules (in the sense of [20,18]) for the corresponding graded Hecke algebra and their "lowest W -types". This problem will be considered elsewhere.
Acknowledgements. I am grateful to the referee for useful comments and particularly for pointing out a mistake in a previous version of section 3.3. This research was supported by nsf-dms 0968065 and NSA-AMS 081022.
Preliminaries
Root systems.
We fix an R-root system Φ = (V, R, V ∨ , R ∨ ). This means that V, V ∨ are finite dimensional R-vector spaces, with a perfect bilinear pairing
( , ) : V × V ∨ → R, R ⊂ V \ {0}, R ∨ ⊂ V ∨ \ {0} are finite subsets in bijection R ←→ R ∨ , α ←→α, such that (α,α) = 2. (2.1.1)
Moreover, the reflections
s α : V → V, s α (v) = v − (v,α)α, s α : V ∨ → V ∨ , s α (v ′ ) = v ′ − (α, v ′ )α, α ∈ R, (2.1.2) leave R and R ∨ invariant, respectively. Let W be the subgroup of GL(V ) (respec- tively GL(V ∨ )) generated by {s α : α ∈ R}.
We will assume that the root system Φ is reduced, meaning that α ∈ R implies 2α / ∈ R, and crystallographic, meaning that (α,α ′ ) ∈ Z for all α ∈ R,α ′ ∈ R ∨ . We also assume that R generates V . We will fix a choice of simple roots Π ⊂ R, and consequently, positive roots R + and positive coroots R ∨,+ . Often, we will write α > 0 or α < 0 in place of α ∈ R + or α ∈ (−R + ), respectively.
We fix a W -invariant inner product , on V . Denote also by , the dual inner product on V ∨ . If v is a vector in V or V ∨ , we denote |v| := v, v 1/2 .
2.2.
The Clifford algebra. A classical reference for the Clifford algebra is [4] (see also section II.6 in [2]). Denote by C(V ) the Clifford algebra defined by V and the inner product , . More precisely, C(V ) is the quotient of the tensor algebra of V by the ideal generated by
ω ⊗ ω ′ + ω ′ ⊗ ω + 2 ω, ω ′ , ω, ω ′ ∈ V.
Equivalently, C(V ) is the associative algebra with unit generated by V with relations:
ω 2 = − ω, ω , ωω ′ + ω ′ ω = −2 ω, ω ′ . (2.2.1)
Let O(V ) denote the group of orthogonal transformation of V with respect to , . This acts by algebra automorphisms on C(V ), and the action of
−1 ∈ O(V ) induces a grading C(V ) = C(V ) even + C(V ) odd . (2.2.2)
Let ǫ be the automorphism of C(V ) which is +1 on C(V ) even and −1 on C(V ) odd . Let t be the transpose automorphism of C(V ) characterized by
ω t = −ω, ω ∈ V, (ab) t = b t a t , a, b ∈ C(V ). (2.2.3)
The Pin group is
Pin(V ) = {a ∈ C(V ) : ǫ(a)V a −1 ⊂ V, a t = a −1 }. (2.2.4)
It sits in a short exact sequence
1 −→ Z/2Z −→ Pin(V ) p − −− → O(V ) −→ 1, (2.2.5)
where the projection p is given by p(a)(ω) = ǫ(a)ωa −1 .
2.3. The spin modules S. If dim V is even, the Clifford algebra C(V ) has a unique (up to equivalence) complex simple module (γ, S) of C(V ), of dimension 2 dim V /2 , endowed with a positive definite Hermitian form , S such that γ(a)s, s ′ S = s, γ(a t )s ′ S , for all a ∈ C(V ) and s, s ′ ∈ S.
(2.3.1)
When dim V is odd, there are two simple inequivalent unitary modules (γ + , S + ), 2] . In order to simplify the formulation of the results, we will often refer to any one of S, S + , S − , as a spin module. When there is no possibility of confusion, we may also denote by S any one of S + or S − , in order to state results in a uniform way. Via (2.2.4), a spin module S is an irreducible unitary Pin(V ) representation. It is well-known (e.g., section II.6 in [2]) that as Pin(V )-representations, we have: Therefore, W is a central extension of W :
(γ − , S − ) of dimension 2 [dim V /S ⊗ S ∼ = • V, when dim V is even, S ⊗ S ∼ = [dim V /2] i=0 2i V, when dim V is odd.1 −→ Z/2Z −→ W p − −− → W −→ 1. (2.4.2)
The group W has a Coxeter presentation similar to that of W . Recall that as a Coxeter group, W has a presentation:
W = s α , α ∈ Π| s 2 α = 1, (s α s β ) m(α,β) = 1, α = β ∈ Π , (2.4.3)
for certain positive integers m(α, β). Theorem 3.2 in [16] gives:
W = z, s α , α ∈ Π| z 2 = 1, s 2 α = z, ( s α s β ) m(α,β) = 1, α = β ∈ Π . (2.4.4)
We will also need the explicit embedding of W into Pin(V ).
i V , 0 ≤ i ≤ dim V , forms a set of irreducible, pairwise inequivalent W -representations.
2.6. The Casimir element of W . The notions in this subsection are motivated by the results of [1], where the element Ω W that we define here appeared naturally in the context of the Dirac operator for the graded affine Hecke algebra.
Let c : R + → R be a W -invariant function.
Definition 2.6.1.
Denote
Ω W ,c = α>0,β>0 sα(β)<0 c(α)c(β)|α||β| s α s β = α>0,β>0 α,β =0 α,β | cos(α, β)| c(α)c(β) s α s β . (2.6.1)
The equality holds because the contributions in the second sum of the pairs {α, β} and {s α (β), α} cancel out, whenever s α (β) > 0. If c ≡ 1, we write Ω W for Ω W ,1 .
If C w is the W -conjugacy class of w ∈ W , then there are two possibilities for p −1 (C w ) ⊂ W :
(1) p −1 (C w ) is a single W -conjugacy class, or (2) p −1 (C w ) splits into two conjugacy W -classes C w := {w ′ : w ′ ∈ C w } and z C w := {zw ′ : w ′ ∈ C w }.
One sees that if w = s α s β , then the second case holds ( [16]). This implies that we have
Ω W ,c ∈ C[ W ] W .
(2.6.2)
In particular, every σ ∈ W acts on Ω W ,c by a scalar, which we denote σ(Ω W ,c ).
We will refer to Ω W as the Casimir element of W . The justification for the name is given by Theorem 1.0.1(1). As a hint towards this result, let us recall (e.g., [16, p. 562
]) that tr S (s α s β ) = | cos(α, β)| dim S, α, β ∈ R + , (2.6.3)
for a spin module S. This means that we have
S(Ω W ) = α>0,β>0 α,β = 2ρ, 2ρ , (2.6.4) whereρ = 1 2 α>0α .
W -types
In this section, we prove our main results, Theorems 1.0.1 and 3.10.3. Before that, we recall certain elements from the theory of elliptic representations of a finite group. While these elements are not necessary for proving Theorems 1.0.1 and 3.10.3, they are useful for setting our result in the appropriate context.
For a finite group Γ, let R(Γ) denote the representation theory ring of Γ, and let Γ denote the set of irreducible representations of Γ.
3.1. Elliptic representations of a finite group. The reference for most of the results in this and the next subsection is [20]. Assume first that Γ is an arbitrary finite subgroup of GL(V ). An element γ ∈ Γ is called elliptic (or anisotropic) if V γ = 0. Let Γ ell denote the set of elliptic elements in Γ. This is closed under conjugation by Γ. Let L be the set of subgroups L ⊆ Γ, such that V L = 0. For every L ∈ L, let Ind Γ L : R(L) → R(Γ) be the induction map, and denote
R ind (Γ) = L∈L Ind Γ L (R(L)) ⊆ R(Γ), (3.1.1) R(Γ) = R(Γ)/R ind (Γ). (3.1.2)
One calls R(Γ) the space of elliptic representations of Γ. Define a bilinear pairing, called the elliptic pairing on Γ: [20] shows, in particular, that the radical of e Γ is precisely R ind (Γ), and thus e Γ induces a nondegenerate bilinear form on R(Γ). Moreover, if C ell (Γ) denotes the set of Γ-conjugacy classes in Γ ell , we have
e Γ (σ, σ ′ ) = i≥0 (−1) i dim Hom Γ ( i V ⊗ σ, σ ′ ), σ, σ ′ ∈ R(Γ). (3.1.3) Proposition 2.2.2 indim R(Γ) = |C ell (Γ)|. (3.1.4) Lemma 3.1.1. If σ ∈ R(Γ), then σ ⊗ dim V V − (−1) dim V σ is in R ind (Γ). Proof. We have i V ⊗ dim V V ∼ = dim V −i V , as Γ-representations, for all 0 ≤ i ≤ dim V. From this, it follows that e Γ (σ ⊗ dim V V, σ ′ ) = (−1) dim V e Γ (σ, σ ′ ), for all σ, σ ′ . Let R red (Γ) denote the quotient of R(Γ) by the subspace generated by σ ⊗ dim V V − (−1) dim V σ, for all Γ-types σ. Lemma 3.1.1 implies that the natural (surjective) map R red (Γ) → R(Γ) (3.1.5) is well-defined and preserves e Γ .
3.2.
Elliptic representations of W . We specialize to Γ = W here acting on the reflection representation V ; [20] analyzes the relation between R(W ) and Springer representations.
Let g be the complex Lie algebra determined by the root system Φ, and let G be the simply connected connected Lie group with Lie algebra g. For every x ∈ g, let Z G (x) denote the centralizer of x in G, and let Z G (x) 0 be the identity component. Define the A-group of x in G to be the quotient
A(x) = Z G (x)/Z G (x) 0 Z(G), (3.2.1) where Z(G) is the center of G.
Specialize to the case when x ∈ N (g), the set of nilpotent elements of g. By the Jacobson-Morozov theorem, there exists a Lie algebra homomorphism κ :
sl(2, C) → g such that κ 0 1 0 0 = e.
Let s 0 be a the semisimple part of the Lie algebra of a maximal torus in Z G (κ(sl(2, C)). As explained in §3.2 of [20], the group A(e) acts naturally on s 0 , i.e., A(e) ⊂ GL(s 0 ), and therefore we may define R(A(e)) with respect to this action.
Definition 3.2.1. An element e ∈ N (g) is called distinguished if Z G (e) contains no nontrivial torus. An element e ∈ N (g) is called quasi-distinguished if there exists a semisimple element t ∈ Z G (e) such that t exp(e) centralizes no nontrivial torus in G. In particular, every distinguished e is quasi-distinguished with t = 1.(3) If e is distinguished, then {H e (φ) : φ ∈ A(e) 0 } is an orthonormal basis of R e (W ) with respect to e W .
3.3. Elliptic representations of W . We specialize now to Γ = W acting also on the (nongenuine) reflection representation V . Let R gen ( W ) denote the subspace of R( W ) spanned by W gen , the irreducible genuine W -types. Every nongenuine W -type is a pullback of a W -type, so we may regard R(W ) naturally as a subspace of R( W ). Clearly, we have
e W ( σ, σ ′ ) = 0, if σ ∈ R gen ( W ), σ ′ ∈ R(W ), (3.3.1)
therefore we have an orthogonal decomposition R( W ) = R gen ( W ) ⊕ R(W ). As before, define R( W ) to be the quotient by the radical of e W , and let R gen ( W ) be the image of R gen ( W ) in R( W ). Consequently, there is an orthogonal decomposition
R( W ) = R gen ( W ) ⊕ R(W ). (3.3.2) From (3.1.4), we have dim R gen ( W ) = |C ell ( W )| − |C ell (W )|. Recall the projection p : W → W from (2.2.5). Since V is a nongenuine representation of W , an element w ∈ W is elliptic if and only if p( w) ∈ W is elliptic. Recall that if C w ⊂ W is a conjugacy class, then p −1 (C w )
is a single conjugacy class in W or it splits into two conjugacy classes in W . Let C 0 (W ) denote the set of conjugacy classes of W which split in W and set C 0
ell (W ) = C 0 (W ) ∩ C ell (W ). Then we have dim R gen ( W ) = |C 0 ell (W )| ≤ |C ell (W )| = dim R(W ). (3.3.3)
The dimension of the second space is as follows, see [5],[20, §3.1]: Since σ(z) = −1 for every genuine W -type σ, it is clear that if C w / ∈ C 0 (W ), then tr σ ( w) = 0, for all w ∈ p −1 (C w ) and all genuine W -types σ. (
A n−1 : 1,1) If dim V is even, then tr S ( w) = 0 if and only if det V (1 + p( w)) = 0. (2) If dim V is odd, then tr S ( w) = 0 if det V (1−p( w)) = 0 (i.e., if w is elliptic).
In particular, C 0 ell (W ) = C ell (W ) in this case. Proof. If dim V is even, and S is the spin module, we see by (2.3.2) that tr S ( w) 2 = tr • V (p( w)) = det V (1+p( w)), and this proves (1). If dim V is odd, and S + , S − are the two spin modules, (2.3.2) implies that tr
S + −S − ( w) 2 = 2⊕ dim V i=0 (−1) i tr i V (p( w)) = det V (1 − p( w)
). Since S + and S − are associate, tr S + −S − ( w) = 0 implies tr S ± ( w) = 0, and this proves (2).
Remark 3.3.2.
In type A n−1 , there is a single elliptic conjugacy class, consisting of the n-cycles in S n , and it is easy to check directly that it splits in S n . Theorem 4.1 and Lemma 6.4 in [19] for B n and D n respectively, and sections 6-9 in [17] for the exceptional groups, show that if dim V is even, the split elliptic conjugacy classes are precisely the ones on which S does not vanish. In terms of the classification of elliptic elements of W from Carter [5], when dim V is even, the set C 0 ell (W ) = C ell (W ) is explicitly as follows:
(1) in type B 2n , the elliptic conjugacy classes corresponding to partitions of 2n with only even parts; (2) in type D 2n , the elliptic conjugacy classes corresponding to partitions of 2n with only even parts, or partitions with only odd parts and multiplicity one;
(3) in G 2 : {A 2 , G 2 }; (4) in F 4 : {A 2 + A 2 , D 4 (a 1 ), B 4 , F 4 , F 4 (a 1 )}; (5) in E 6 : {3A 2 , E 6 , E 6 (a 1 ), E 6 (a 3 )}; (6) in E 8 : {A 8 , 2A 4 , 4A 2 , D 8 (a 1 ), D 8 (a 2 ), 2D 4 (a 1 ), E 6 (a 2 ) + A 2 , E 6 + A 2 , E 8 , E 8 (a 1 ), E 8 (a 2 ), E 8 (a 3 ), E 8 (a 4 ), E 8 (a 5 ), E 8 (a 6 ), E 8 (a 7 ), E 8 (a 8 )}.
Let S denote a spin W -module. One can consider the linear map:
ι S : R(W ) −→ R gen ( W ), ι S (σ) = σ ⊗ S. (3.3.5)
Since the genuine W -types are determined by their values on p −1 (C 0 (W )), the map ι S is surjective if and only if tr S does not vanish on any conjugacy class in p −1 (C 0 (W )). Using Lemma 3.3.1 and [19,17] again, we see that:
Lemma 3.3.3. The map ι S is surjective if and only if W is of type B n , D n , G 2 , F 4 , or E 8 .
Since R ind (W ) (resp. R gen,ind ( W )) can be identified with the vector subspace of virtual characters that vanish on W ell (resp. W ell ), we have
ι S (R ind (W )) ⊂ R gen,ind ( W ),(3.R(W ) ։ R gen ( W ) և R red gen ( W ). (3.4.1) Via ξ : R gen ( W ) → R gen ( W ), σ → σ ⊗ sgn + (−1) dim V σ, (3.4.2)
we may identify R red gen ( W ) with the image of ξ, i.e., with the subspace of R gen ( W ) spanned by { σ ⊗ sgn + (−1) dim V σ = 0 : σ ∈ W / ∼ } (here we think of W / ∼ as a system of representatives for the symmetry classes). This shows that dim R red gen ( W ) = | W / ∼ | when dim V is even, and | W / ∼ | − |{ σ ∈ W : σ ⊗ sgn ∼ = σ}| when dim V is odd. From [17,19,21], we see that the dimension of | W / ∼ | equals:
A n−1 : the number of partitions of n into distinct parts, By the Jacobson-Morozov theorem, we know that there is a one-to-one correspondence between G-orbits of nilpotent elements in g, and the set T (G) of G-conjugacy classes of Lie triples in g: This definition of T 0 (G) agrees with the one from the introduction by Proposition 2.4 in [3]. Every quasi-distinguished nilpotent element is in N 0 (g), but in types A, D, E 6 , not all e ∈ N 0 (g) are quasi-distinguished in the sense of Definition 3.2.1. For example, in sl(n), the only quasi-distinguished nilpotent orbit is the regular orbit, but N 0 (g) contains every orbit whose Jordan form has all blocks of distinct sizes.
3.6. Type A. We begin the proof of Theorem 1.0.1. This is a case-by-case verification, combinatorially for classical root systems, and a direct computation for exceptional.
The starting point is type A n−1 , W n = S n . Let P (n) be the set of all partitions of n, and let DP (n) be the set of distinct partitions. If λ = (λ 1 , λ 2 , . . . , λ m ) is a partition of n, written in decreasing order, we denote the length of λ by ℓ(λ) = m. We say that λ is even (resp. odd) if n − ℓ(λ) is even (resp. odd).
It is well-known that every partition λ parameterizes a unique S n -type σ λ , and this gives a one-to-one correspondence between P (n) and S n . The first part of Theorem 1.0.1 for S n is a classical result of Schur. Theorem 3.6.1 ( [21]). There exists a one-to-one correspondence S n / ∼ ←→ DP (n).
For every even λ ∈ DP (n), there exists a unique σ λ ∈ S n , and for every odd λ ∈ DP (n), there exist two associate σ + λ , σ − λ ∈ S n . The dimension of σ λ or σ ± λ is
2 [ n−m 2 ] n! λ 1 ! . . . λ m ! 1≤i<j≤m λ i − λ j λ i + λ j . (3.6.1)
Notice that DP (n) precisely parameterizes the set of quasi-distinguished orbits in type A n−1 (the only local systems of Springer type are the trivial ones here).
In order to simplify the formulas below, we write σ λ := σ + λ ⊕ σ − λ , if λ is an odd partition in DP (n).
The decomposition of the tensor product of an S n -type σ µ with the spin representation S = σ (n) is well-known (see [24, §9.3] for example). If λ = (n) (this case has been covered by §2.5 already), we have:
dim Hom Sn [ σ λ , σ µ ⊗ σ (n) ] = 1 ǫ λ ǫ (n) 2 ℓ(λ)−1 2 g λ,µ ,(3.
6.2)
where ǫ λ = 1 (resp. ǫ λ = √ 2) if λ is even (resp. odd), and g λ,µ are certain Kostka type numbers ([24, §9.3]). In particular, g λ,λ = 0, and this proves (2) in Theorem 1.0.1.
In order to verify claim (1) of Theorem 1.0.1, we need a formula for the character of σ λ on the conjugacy class represented by the cycle (123) in S n . This may be well-known, but I could not find a reference, so I include a combinatorial proof. Lemma 3.6.2. If λ = (λ 1 , . . . , λ m ) is a partition in DP (n), n ≥ 3, then we have
|C Sn (123) | tr σ λ ((123)) dim σ λ = m i=1 λ i (λ 2 i − 1) 6 − n 2 , (3.6.3)
where C n (123) denotes the conjugacy class of the cycle (123) in S n . Proof. The proof is by induction on n. One can immediately verify this for n = 3, and let us assume it holds for n − 1. By restriction to S n−1 , one has
tr σ λ = m i=1 tr σ λ i ,
where λ i = (λ 1 , . . . , λ i − 1, . . . , λ m ). One discards λ i if it is not in DP (n− 1). Using the induction hypothesis, and the ratios |C Sn (123) |/|C
Sn−1 (123) | = n n−3 , dim σ λ i / dim σ λ = λi n j =i (λi−λj −1)(λi+λj ) (λi−λj )(λi+λj −1)
, an elementary calculation leads to the following curious identity that we need to verify:
m i=1 λ 2 i (λ i − 1) j =i (λ i − λ j − 1)(λ i + λ j ) (λ i − λ j )(λ i + λ j − 1) = m i=1 λ 2 i (λ i − 1) − i =j λ i λ j .f (x) = (x 2 − x) m i=1 (x − λ i − 1)(x + λ i ) (x − λ i )(x + λ i − 1) ,(3.
6.5)
which has the expansion 1) . Notice that the coefficient of x −2 in the Laurent expansion of the right hand side of (3.6.6) is precisely (−2) times the left hand side of (3.6.4). Then one verifies (3.6.4) easily, by computing the coefficient of x −2 in the Laurent expansion of (3.6.5).
f (x) = (x 2 − x) + m i=1 A i − n i=1 A i λ i (λ i − 1) (x − λ i )(x + λ i − 1) , (3.6.6) where A i = −2λ i j =i (λi−λj −1)(λi+λj ) (λi−λj )(λi+λj −
Using the formula in Lemma 3.6.2, we immediately check that
σ λ (Ω Sn ) = n i=1 λi(λ 2 i −1) 3 .
Here, for simplicity, we assumed that the roots of type A n−1 are the standard ones. A middle element h of a Lie triple for the nilpotent orbit indexed by the partition λ is, in coordinates, (−(λ 1 − 1), . . . , (λ 1 − 1), . . . , −(λ m − 1), . . . , (λ m − 1)), and now claim (1) in Theorem 1.0.1 is established.
3.7.
Types B, C. The nilpotent orbits in sp(2n) and so(m) are parameterized (via an analogue of the Jordan canonical form) by partitions of 2n (resp. m), where the odd (resp. even) parts occur with even multiplicity. Such an orbit is in N 0 (sp(2n)) (resp. N 0 (so(m)) if and only if the associated partition has only even (resp. odd) parts, and all parts have multiplicity at most 2.
Let W n denote the Weyl group of type B n /C n . The group W n is a semidirect product W n = S n ⋊ (Z/2Z) n , and therefore, as it is well-known, its representations are obtained by Mackey theory. More precisely, let χ k = (triv ⊠ · · · ⊠ triv) n−k ⊠ (sgn ⊠ · · · ⊠ sgn) k be a character of (Z/2Z) n , and let S n−k × S k be the isotropy group of χ k in S n . For every partitions λ of n − k and µ of k, one constructs an irreducible W n representation σ (λ,µ) as
σ (λ,µ) = Ind Wn S n−k ×S k ×(Z/2Z) n (σ λ ⊠ σ µ ⊠ χ k ). (3.7.1)
This gives a bijection
W n ←→ BP (n), σ (λ,µ) ↔ (λ, µ), (3.7.2)
where BP (n) is the set of bipartitions of n. In particular, in this notation, if λ ∈ P (n), the representation σ (λ,∅) is obtained from the S n -type σ λ , by letting the simple reflections of W n not in S n act by the identity. Let W n denote the spin cover of W n . The genuine representations of W n were classified by [19], starting with the classification for S n -types.
]). There is a one-to-one correspondence
( W n ) gen / ∼ ←→ P (n).
For every λ ∈ P (n), there exist:
(1) one irreducible W n -type σ λ = σ (λ,∅) ⊗ S, if n is even;
(2) two associate W n -types σ ± λ = σ (λ,∅) ⊗ S ± , if n is odd. This realization is very convenient for computing the character of σ λ (and similarly σ ± λ ). Using the character of S and the characters of type A n−1 , we immediately see that:
tr σ λ ( s α s β ) dim σ λ = | cos(α, β)| trσ λ ((123)) dim σ λ , if α, β form an A 2 trσ λ ((12)) dim σ λ , if α, β form an B 2 /C 2 . (3.7.3)
The relevant formulas for tr σ λ in S n go back to Frobenius. In the form that we need, they are:
|C Sn (123) | tr σ λ ((123)) dim σ λ = p 2 (λ) − n 2 , |C Sn (12) | tr σ λ ((12)) dim σ λ = p 1 (λ), (3.7.4)
where p k (λ) is the sum of k powers of the content of λ.
The formula for σ λ (Ω W ) becomes very explicit. Assume for simplicity of notation that we take the standard Bourbaki coordinates for roots for type B and C. Set
ǫ = 1, if Φ = B n , ǫ = 1 2 , if Φ = C n . (3.7.5)
Then, we have
σ λ (Ω W ) = 4p 2 (λ + ǫ), (3.7.6)
where p 2 (λ + ǫ) denotes the sum of squares of the ǫ-content of λ, i.e., the content of the (i, j)-box is j − i + ǫ.
The set of all contents of λ + ǫ (with repetitions) represents the coordinates of one half of the middle element of a nilpotent orbit O λ,ǫ in so(2n + 1) (if ǫ = 1), respectively sp(2n) (if ǫ = 1/2). Moreover, the nilpotent orbit lies in fact in N 0 (g). (See [9, §4.4] for the combinatorics needed to verify this claim.)
The following algorithm (due to Slooten) attaches to each partition λ with content λ+ǫ a set of bipartitions. All the W n -types parameterized by these bipartitions are Springer representations for the nilpotent orbit O λ,ǫ . This can be seen by comparing Slooten's algorithm with Lusztig's algorithm with S-symbols for the Springer correspondence ([9, §4.4]). Moreover, the relation between this algorithm and the elliptic representation theory of W (via the elliptic representation theory of the graded affine Hecke algebra attached to Φ) is part of [8].
Algorithm ( [22]). Start with an empty bipartition (µ, µ ′ ), µ = ∅, µ ′ = ∅. Locate the largest content in absolute value in λ + ǫ. This could appear in the last box of the first row or the last box of the first column. Assume first that these two entries are distinct. If the largest content is in the first row, remove the row from λ and put its length in µ. If the largest content is in the first column, remove the column from λ and put its length in µ ′ . Continue with the remaining (smaller) partition.
If at this step there was an ambiguity, namely the largest entry in absolute value appeared twice, start two cases, and proceed in each one of them separately as above.
If at the last step, we are left with a single box, treat it as a row if its content is nonnegative, and as a column if its content is negative.
It is apparent that the number of W -types that the algorithm returns is always a power of 2. Moreover, it is known that the algorithm returns a unique W n -type if and only if the associated nilpotent orbit is distinguished.
Let σ (µ,µ ′ ) be one of the W n -types returned by the algorithm. It remains to check that the multiplicity of σ λ in σ (µ,µ ′ ) ⊗ S is nonzero (claim (2) of Theorem 1.0.1). We have:
Hom W [ σ λ , σ (µ,µ ′ ) ⊗S] = Hom W [σ (λ,∅) ⊗S, σ (µ,µ ′ ) ⊗S] = Hom W [ • V, σ (λ,∅) ⊗σ (µ,µ ′ ) ].
(3.7.7)
In bipartition notation, we have k V = σ ((n−k),(1 k )) . We claim that k V , where k = |µ|, occurs in σ (λ,∅) ⊗ σ (µ,µ ′ ) . To see this, notice that (3.7.1) implies:
σ (λ,∅) ⊗ σ ((n−k),(1 k )) = σ (λ,∅) ⊗ Ind Wn S n−k ×S k ×(Z/2Z) n (triv ⊠ sgn ⊠ (triv n−k ⊠ sgn k )) = Ind Wn S n−k ×S k ×(Z/2Z) n (σ λ | S n−k ×S k ⊗ (triv ⊠ sgn) ⊠ (triv n−k ⊠ sgn k )).
From the construction of µ (using the rows of λ) and µ ′ (using the columns of λ), and the induction/restriction rules in S n , we have:
σ µ ⊠ σ µ ′ t ֒→ σ λ | S n−k ×S k ,
where µ ′ t denotes the transpose partition of µ ′ . Since σ µ ′ t = σ µ ′ ⊗ sgn, as S krepresentations, the claim follows by applying (3.7.1) again.
3.8. Type D. Let W (D n ) denote the Weyl group of type D n . This is a subgroup of W n of index 2. An irreducible W n -representation σ (λ,µ) restricts to a representation of W (D n ) which is:
• irreducible if λ = µ; • a sum of two inequivalent irreducible W (D n )-types if λ = µ.
Moreover, σ (λ,µ) and σ (µ,λ) , λ = µ, restrict to the same W (D n )-representation. In our case, the only W (D n )-types that we consider are those associated via Springer's correspondence with nilpotent orbits in N 0 (D n ). In the Jordan form partition notation ( [6]), these are the orbits indexed by partitions of 2n where all parts are odd and appear with multiplicity at most 2. It turns out that for every Springer representation attached to one of these orbits, the bipartition (λ, µ) has the property that λ = µ.
The classification of W (D n ) gen is obtained from Theorem 3.7.1. We define the equivalence relation ∼ on P (n): λ ∼ λ t . This is the combinatorial equivalent of the relation ∼ on S n : σ λ ∼ σ λ ⊗ sgn = σ λ t . Theorem 3.8.1 ([19, §7,8]). There is a one-to-one correspondence
W (D n ) gen / ∼ ←→ P (n)/ ∼ .
More precisely, recall the irreducible representations σ λ of S n and σ λ of W n . We have:
(1) If n is odd, every irreducible genuine W n -representations σ λ restricts to an irreducible W (D n )-representation, and this gives a complete set of inequivalent irreducible genuine W (D n )-representations. Moreover, σ λ and σ λ t are associate as W (D n )-representations. (2) If n is even, and if λ = λ t , then σ λ restricts to a sum of two associate irreducible W (D n )-type; if λ = λ t , then σ λ restricts to an irreducible W (D n )representation.
The analysis of the map Ψ is now completely analogous to the cases B n /C n . The same formulas and algorithm hold with the convention that ǫ = 0.
3.9. Exceptional root systems. The character tables for exceptional W are in [17]. We reorganize them here so that the claims in Theorem 1.0.1 follow. The notation for W -types is borrowed from [17]. We put a * next to W -type to indicated that it has an associate W -type. We use Carter's notation for W -types and nilpotent orbits ( [6]). In each table, we give the correspondences (1), (2) in Theorem 1.0.1 between genuine W -types, nilpotent orbits, and Springer representations attached to nilpotent orbits. We also give the traces of the characters of the genuine W -types on elements of the form s α s β , where α, β form an A 2 , B 2 , or G 2 . Using these traces and the sizes of the corresponding conjugacy classes (which are listed in [5]), we computed σ(Ω W ) and verified assertion (1) in Theorem 1.0.1. For tensor product decompositions, we used the package chevie in the computer algebra system GAP.
Table 1. G 2 Nilpotent e ∈ N 0 σ e,φ ∈ W σ ∈ W gen tr σ (A 2 ) tr σ (G 2 ) G 2 (1, 0) 2 s 1 √ 3 G 2 (a 1 ) (2, 1) 2 sss −2 0 (1, 3) ′ 2 ss 1 − √ 3 Table 2. F 4 Nilpotent e ∈ N 0 σ e,φ ∈ W σ ∈ W gen tr σ (A l 2 ) tr σ (A s 2 ) tr σ (B 2 ) F 4 (1, 0) 4 s 2 2 2 √ 2 F 4 (a 1 ) (4, 1) 12 s 0 0 2 √ 2 (2, 4) ′′ 8 sss 4 −2 0 F 4 (a 2 ) (9, 2) 24 s 0 0 0 (2, 4) ′ 8 ssss −2 4 0 F 4 (a 3 ) (12, 4) 8 ss 0 0 −2 √ 2 (9, 6) ′ 12 ss −2 −2 0 (6, 6) ′′ 8 s −2 −2 0 (1, 12) ′ 4 ss 2 2 −2 √ 2 Remark 3.9.1. In E 6 , there is a nilpotent orbit D 4 (a 1 ) ⊂ N 0 (E 6 ), such that A(e) = S 3 .
There are three Springer representations attached to D 4 (a 1 ): σ e, (3) , σ e, (21) , and σ e,(111) . Since the rank is even, there is a single spin module S. The fiber Ψ −1 (e) contains three W -types σ 1 , σ 2 , σ 3 of dimensions 40, 20, 20, respectively. Moreover, σ 2 and σ 3 are associate. We compute that in the decomposition σ 1 ⊗ S occur all three σ e, (3) , σ e, (21) , and σ e,(111) , while in the decomposition of σ 2 ⊗ S (equivalently σ 3 ⊗ S) only σ e, (21) occurs among the three Springer representations.
3.10. The generalized Springer correspondence. The references for the construction of the generalized Springer correspondence, and for the results we need to use are [13,15]. Let G be a simply connected complex simple group, with a fixed Borel subgroup B, and maximal torus H ⊂ B. The Lie algebras will be denoted by the corresponding Gothic letter. Fix a nondegenerate G-invariant symmetric bilinear form , on g. Denote also by , the dual form on g * . Definition 3.10.1. A cuspidal triple for G is a triple (L, C, L), where L is a Levi subgroup of G, C is a nilpotent L−orbit on the Lie algebra l, and L is an irreducible G−equivariant local system on C which is cuspidal in the sense of [13,15].
Let us fix a cuspidal triple (L, C, L), such that H ⊂ L, and P = LU ⊃ B is a parabolic subgroup. Let T ⊂ L denote the identity component of the center of L, with Lie algebra t. Write an orthogonal decomposition with respect to , , h = t + a; here a is a Cartan subalgebra for the semisimple part of l. Let
pr C : h → t (3.10.1)
denote the corresponding projection onto t.
Following [15, §2], we attach to (L, C, L) an R-root system Φ = (V, R, V ∨ , R ∨ ) and a W -invariant function c : R + → Z. Let R ⊂ t * be the reduced part of the root system given by the nonzero weights of ad(t) on g; it can be identified with
(x), where x ∈ C. Let V be the R-span of R in t * . The Weyl group is W = N G (L)/L. (3.10.2)
This is a Coxeter group due to the particular form L must have to allow a cuspidal local system. Let R + is the subset of R for which the corresponding weight space lives in u. The simple roots Π = {α i : i ∈ I} correspond to the Levi subgroups L i containing L maximally: α i is the unique element in R + which is trivial on the center of l i . For every simple α i , c(α i ) ≥ 2 is defined to be the smallest integer such that
ad(x) c(αi)−1 : l i ∩ u → l i ∩ u is zero. (3.10.3)
This is a W -invariant function on Π and we extend it to R + .
The explicit classification of cuspidal triples (when G is simple), along with the corresponding values for the parameters c(α) can be found in the tables of [15, §2.13].
Define R ⊥ = {x ∈ t : α(x) = 0, for all α ∈ R}, and let t ′ be the orthogonal complement of R ⊥ in t. For every α ∈ R, defineα ∈ t to be the unique element of t ′ such that α(α) = 2. Let R ∨ denote the set of allα, and let V ∨ be the R-span of R ∨ in t ′ .
Consider the varietẏ
g = {(x, gP ) ∈ g × G/P : Ad(g −1 )x ∈ C + t + u}, (3.10.4)
on which G × C × acts via (g 1 , λ): x → λ −2 Ad(g 1 )x, x ∈ g, and gP → g 1 gP, g ∈ G.
Let pr 1 and pr 2 be the projections ofġ on the two factors. For every nilpotent element e ∈ g, let P e denote the preimage of {e} inġ under pr 1 . Via pr 2 , we may make the identification:
P e = {gP : Ad(g −1 )e ∈ C + u}.
LetL denote the pull-back of the local system L on C under the G × C ×equivariant projectionġ → C, (x, gP ) → Ad(g −1 )x.
Let π 1 (e) = Z G (e)/Z G (e) 0 be the fundamental group of G · e. The hypercohomology with compact support H • c (P e ,L) carries a π 1 (e) × W action ( [13], see also [15]). Let A(e) C denote the set of irreducible representations of π 1 (e) which appear in this way. Moreover, for φ ∈ A(e) C we have: is an irreducible W −representation. The correspondence ⊔ e∈G\N (g) A(e) C → W , (e, φ) → σ (e,φ) is the generalized Springer correspondence of [13], and it is a bijection. We normalize it so that σ (e,φ) = sgn, when e ∈ G · C (there is single φ that appears in that case). where Ω W ,c is as in (2.6.1) and pr C is as in (3.10.1); (2) Let (e, h, f ) ∈ T 0 (G, C) be given. For every Springer representation σ (e,φ) , φ ∈ A(e) C , and every spin W -module S, there exists σ ∈ Ψ −1 [(e, h, f )] such that σ appears with nonzero multiplicity in the tensor product σ (e,φ) ⊗ S. Conversely, for every σ ∈ Ψ −1 [(e, h, f )], there exists a spin W -module S and a Springer representation σ (e,φ) , such that σ is contained in σ (e,φ) ⊗ S. Proof. Again the proof amounts to verifying the assertions in every case. The interesting cases are when R is of type B n /C n , G 2 , or F 4 . For type G 2 , the cuspidal local system appears for the Levi subgroup L = 2A 2 in G = E 6 . Assuming that the simple roots of R of type G 2 are α, β with α long and β short, the parameter function is c(α) = 1, c(β) = 3. To give an explicit formula for Ω W ,c , let's normalize α, β such that such that α,α = 2 and β ,β = 6. Then we have 1 4 σ(Ω W ,c ) = 3 2 (c(α) 2 + 3c(β) 2 ) + 4 √ 3c(α)c(β) tr σ (G 2 ) dim σ + (c(α) 2 + 3c(β) 2 ) tr σ (A 2 ) dim σ ;
(3.10.9) recall that the notation for conjugacy classes is as in [5]. Then one can verify easily part (1) of Theorem 3.10.3.
For type F 4 , the cuspidal local system appears for the Levi subgroup L = (3A 1 ) ′ in G = E 7 . Assuming that the simple roots of R of type F 4 are α, β with α short and β long, the parameter function is c(α) = 2, c(β) = 1. We normalize α, β so that α,α = 2 and β ,β = 1. Then we have: 1 4 σ(Ω W ,c ) = 3(2c(α) 2 +c(β) 2 )+16c(α) 2 tr σ (A 2 ) dim σ +8c(β) 2 tr σ ( A 2 ) dim σ +18 √ 2c(α)c(β) tr σ (B 2 ) dim σ . Nilpotent e ∈ N 0,C σ e,φ ∈ W σ ∈ W gen E 6 (1, 0) 2 s E 6 (a 1 ) (1, 3) ′ 2 ss E 6 (a 3 ) (2, 1) 2 sss For types B n and C n , the combinatorics is similar to that in the proof of Theorem 1.0.1. More precisely, assume the notation from Theorem 3.7.1 and the discussion following it. Assume also that the roots of type B n and C n are in the standard Bourbaki coordinates. For a partition λ of n, viewed as a left justified Young tableau, define the content of the box (i, j) with parameters c 1 , c 2 to be the number m c (i, j) := c 1 (i−j)+c 2 . Let us denote by p 2 (λ, c 1 , c 2 ) the sum of squares of contents of boxes for λ and parameters c 1 , c 2 . The same computation as after Theorem 3.7.1 shows that σ λ (Ω W ,c ) = 4p 2 (λ, c(ǫ 1 − ǫ 2 ), c(ǫ n )), for type B n , σ λ (Ω W ,c ) = 4p 2 (λ, c(ǫ 1 − ǫ 2 ), 1 2 c(2ǫ n )), for type C n . Notice that σ λ (Ω W ,c ) for C n is identical with σ λ (Ω W ,c ) for B n if we set c(2ǫ n ) = 2c(ǫ n ). The geometric values of the parameters are as follows ( §2.13 in [15]):
(1) g = sp(2k + 2n), l = sp(2k) ⊕ C n , C = (2, 4, . . . , 2p) ⊕ 0, k = p(p + 1)/2, Φ = B n , c(ǫ 1 − ǫ 2 ) = 2, c(ǫ n ) = 2p + 1; (2) g = so(k + 2n), l = so(k) ⊕ C n , C = (1, 3, . . . , 2p + 1) ⊕ 0, k = p 2 , Φ = B n , c(ǫ 1 − ǫ 2 ) = 2, c(ǫ n ) = 2p + 2; (3) g = so(k + 4n), l = so(k) ⊕ sl(2) n ⊕ C n , C = (1, 5, 9, . . . , 4p + 1) ⊕ (2) n ⊕ 0, k = (p + 1)(2p + 1), c(ǫ 1 − ǫ 2 ) = 4, c(ǫ n ) = 4p + 3; (4) g = so(k + 4n), l = so(k) ⊕ sl(2) n ⊕ C n , C = (3, 7, 11, . . . , 4p + 3) ⊕ (2) n ⊕ 0, k = (p + 1)(2p + 3), c(ǫ 1 − ǫ 2 ) = 4, c(ǫ n ) = 4p + 5. For the partition λ with content m c (i, j), Slooten's algorithm is the same as in the case c ≡ 1. Also the analysis of the tensor product decomposition works in the same way as in the proof of Theorem 1.0.1 for B n and c ≡ 1. (The argument there only uses Slooten's algorithm and the Weyl group of type B n , and not the parameter function c.) The algorithm giving the nilpotent element e ∈ g from the partition λ with content m c (i, j), at geometric values of c, is again very similar to the one in [9, §4.4], and we skip the details.
1 G
1-conjugacy classes of Jacobson-Morozov triples (e, h, f ) in g. We set:T 0 (G) = {[(e, h, f )] ∈ T (G) : the centralizer of {e, h, f } in g is a toral subalgebra}.
.
There is a surjective map Ψ : W gen −→ T 0 (G), (1.0.3) with the following properties: (1) If Ψ( σ) = [(e, h, f )], then we have σ(Ω W ) = h, h , (1.0.4)
( 3 )
3If e is distinguished, then properties (1) and (2) above determine a bijection:Ψ −1 ([e, h, f ])/ ∼ ←→ {σ e,φ : φ ∈ A(e) 0 }.(1.0.5)
.
The spin cover W . The Weyl group W acts by orthogonal transformations on V , so one can embed W as a subgroup of O(V ). We define the group W in Pin(V ): W := p −1 (O(V )) ⊂ Pin(V ), where p is as in (2.2.5). (2.4.1)
τ
( s α ) = f α := α/|α|, α ∈ Π,(2.4.5) extends to a group homomorphism τ : W → Pin(V ). Moreover, we have τ ( s β ) = f β := β/|β|, for all β ∈ R + .
Corollary 3.2.3 in[20] shows that R(A(e)) = 0 if and only if e is quasi-distinguished in g.From Springer theory, recall B e , the variety of Borel subalgebras of g containing e. Let d e denote the complex dimension of B e . Since Z G (e) acts on B e , we get an action of A(e) on the cohomology H • (B e ). Let A(e) 0 be the set of irreducible A(e)-representations that appear in this action, and let R 0 (A(e)) be the subspace of R(A(e)) spanned by A(e) 0 . The Springer correspondence constructs an action of W on H • (B e ) which commutes with the action of A(e), and gives a map: G\{(e, φ) : e ∈ N (g), φ ∈ A(e) 0 } −→ W , (e, φ) → σ (e,φ) := Hom A(e) [φ, H 2de (B e )], (3.2.2) which is well-defined and bijective. For every φ ∈ A(e) 0 , set H e (φ) := Hom A(e) [φ, H • (B e )], (3.2.3) and regard it as an element of R(W ). Let R e (W ) be the span in R(W ) of H e (φ), φ ∈ A(e) 0 . The space {H e (φ) : e, φ} is basis of R(W ), and therefore, we have a decomposition R(W ) = e R e (W ), which induces a decomposition R(W ) = e R e (W ). Theorem 3.2.2 ([20]). The map H e : R 0 (A(e)) → R e (W ) induces a vector space isomorphism H e : R 0 (A(e)) → R e (W ). Moreover, we have: (1) The isomorphism H e is an isometry with respect to the elliptic pairings e W and e A(e) ; (2) The spaces R 0 (A(e)) and R e (W ) are nonzero if and only if e is quasidistinguished;
B
n : the number of partitions of n, D n : the number of partitions of n with even number of parts, G 2 : 3, F 4 : 9, E 6 : 5, E 7 : 12, E 8 : 30.
.
Let S be a spin module for W .
B
n : the number of partitions of n, D n : the number of equivalence classes of partitions of n under transposition, G 2 : 3, F 4 : 9, E 6 : 6, E 7 : 13, E 8 : 30.(3.4.3) In addition, in types B 2n+1 , D 2n+1 , E 7 , there are no self-associate W -types. In type A 2n−1 , the number of self-associate S 2n -types equals the number of partitions of odd length of 2n into distinct parts. Comparing(3.4.3) with (3.3.4), we see that if W is of type B n , G 2 , F 4 , or E 8 , then we have dim R(W ) = dim R red gen ( W ). (If the type is B 2n+1 ,then the maps in (3.4.1) are both isomorphisms.) Remark 3.4.1. One can ask if there is a natural linear map R(W ) → R red gen ( W ), having good properties with respect to the elliptic pairing. When dim V is odd, it is easy to check that such a map is σ → 1 √ 2 σ ⊗ (S + − S − ), and that this map is an injective metric with respect to the elliptic pairing on R(W ) and the standard pairing on R gen ( W ). When dim V is even, a similar construction exists, but instead of W , one needs to consider W even = { w ∈ W : sgn( w) = 1}. This fits naturally with the theory of the Dirac index for graded Hecke algebras and it is considered in [10]. 3.5. The classification of W -types. The rest of the section is dedicated to the proof of Theorem 1.0.1.
(e, h, f ) : [h, e] = 2e, [h, f ] = −2f, [e, f ] = h.(3.5.1) Definition 3.5.1. Let N 0 (g) denote the set of all nilpotent elements e whose centralizer in g is a solvable subalgebra. Let T 0 (G) denote the set of G-conjugacy classes of triples (e, h, f ) such that e ∈ N 0 (g).
identity arises when one proves Schur's dimension formula by induction ([12, Proposition 10.4]). We consider the function
σ
(e,φ) := Hom π1(e) [φ, H 2 dim Pe c (P e ,L)] (3.10.5)
Definition 3 .10. 2 .
32Denote T 0 (G, C) = {[(e, h, f )] ∈ T 0 : A(e) C = 0}.
Theorem 3 .10. 3 .
33There is a surjective map Ψ c : W gen −→ T 0 (G, C), (3.10.6) (T 0 (G, C) is as in Definition 3.10.2) with the following properties: (1) If Ψ( σ) = [(e, h, f )], then σ(Ω W ,c ) = pr C (h), pr C (h) . (3.10.7)
( 3 )
3If e is distinguished, then properties(1) and(2)above determine a bijection:Ψ −1 ([e, h, f ])/ ∼ ←→ {σ e,φ : φ ∈ A(e) C }.(3.10.8)
( 3 .
310.10) The correspondence from Theorem 3.10.3 for G 2 and F 4 is listed in Tables 6 and 7. The explicit values of pr C (h) in coordinates, for these cases, can be found in [7, Tables 1 and 2].
Corollary 3.3.5. The map ι S is an isomorphism if and only ifdim V is odd or W is of type A.Proof. This is immediate from Proposition 3.3.4 by comparing the dimension of the two spaces as in (3.3.3) and Lemma 3.3.1.3.4. Applying(3.1.5) to this setting, we get a surjective linear map R red gen ( W ) → R gen ( W ) which preserves the elliptic pairing. Thus we have constructed two maps:3.6)
and so ι S gives a linear map
ι S : R(W ) → R gen ( W ).
(3.3.7)
Proposition 3.3.4. The map ι S is surjective.
Proof. By Lemma 3.3.3, this is true when W is of type B n , D n , G 2 , F 4 , E 8 , since ι S
is surjective. The conclusion is implied if tr S is nonzero on every conjugacy class
in p −1 (C 0
ell (W )); this turns out to be the case for every irreducible W , by Lemma
3.3.1 and Remark 3.3.2.
Table 3 . E 6
3Nilpotent e ∈ N 0
σ e,φ ∈ W
σ ∈ W gen tr σ (A 2 )
E 6
(1, 0)
8 s
4
E 6 (a 1 )
(6, 1)
40 s
8
E 6 (a 3 )
(30, 3)
120 s
0
(15, 5)
72 s
0
D 5
(20, 2)
60 s *
6
D 5 (a 1 )
(64, 4)
80 s *
−2
A 4 + A 1
(60, 5)
64 s *
−4
D 4 (a 1 )
(80, 7), (90, 8), (20, 10)
40 ss
−4
(90, 8)
20 s *
−2
Table 4. E 7
Nilpotent e ∈ N 0
σ e,φ ∈ W
σ ∈ W gen tr σ (A 2 )
E 7
(1, 0)
8 s *
4
E 7 (a 1 )
(7, 1)
48 s *
12
E 7 (a 2 )
(27, 2)
168 s *
24
E 7 (a 3 )
(56, 3)
280 s *
20
(21, 6)
112 s *
8
E 7 (a 4 )
(189, 5)
720 s *
0
(15, 7)
120 s *
0
E 7 (a 5 )
(315, 7)
448 s *
−4
(280, 9)
560 s *
−20
(35, 13)
112 ss *
−16
E 6 (a 1 )
(120, 4), (105, 5)
512 s *
16
A 4 + A 1
(512, 11), (512, 12)
64 s *
−4
(512, 11), (512, 12)
64 ss *
−4
Table 5 .
5E 8 the root system of the reductive part of Z GNilpotent e ∈ N 0
σ e,φ ∈ W
σ ∈ W gen tr σ (A 2 )
E 8
(1, 0)
16 s
8
E 8 (a 1 )
(8, 1)
112 s
32
E 8 (a 2 )
(35, 2)
448 ss
80
E 8 (a 3 )
(112, 3)
1344 ss
168
(28, 8)
320 s
40
E 8 (a 4 )
(210, 4)
2016 s
144
(160, 7)
1680 s
120
E 8 (a 5 )
(700, 6)
5600 sss
160
(300, 8)
2800 s
80
E 8 (a 6 )
(1400, 8)
6480 s
0
(1575, 10)
9072 s
0
(350, 14)
2592 s
0
E 8 (a 7 )
(4480, 16)
896 s
−72
(5670, 18)
2016 sss
−72
(4536, 18)
2016 ss
−48
(1680, 22)
1344 s
−40
(1400, 20)
1120 s
−32
(70, 32)
224 s
−8
E 8 (b 4 )
(560, 5)
5600 ss
280
(50, 8)
800 s
40
E 8 (b 5 )
(1400, 7)
6720 s
128
(1008, 9)
7168 s
120
(56, 19)
448 s
8
E 8 (b 6 )
(2240, 10)
8400 s
−120
(840, 13)
5600 s
−80
(175, 12)
2800 ss
−40
D 5 + A 2
(4536, 13), (840, 13)
4800 s
−120
D 7 (a 1 )
(3240, 9), (1050, 10)
11200 s
−40
D 7 (a 2 )
(4200, 12), (3360, 13)
7168 ss
−160
E 6 (a 1 ) + A 1
(4096, 11), (4096, 12)
8192 s
−128
Table 6 .
62A 2 in E 6 , W = G 2 .
Table 7 .
7(3A 1 ) ′ in E 7 , W = F 4 .Nilpotent e ∈ N 0,C σ e,φ ∈ W σ ∈ W genE 7
(1, 0)
4 s
E 7 (a 1 )
(2, 4) ′′
8 sss
E 7 (a 2 )
(4, 1)
12 s
E 7 (a 3 )
(8, 3) ′
24 s
(1, 12) ′
4 ss
E 7 (a 4 )
(2, 4) ′
8 ssss
(4, 7) ′
12 ss
E 7 (a 5 )
(12, 4)
8 ss
(6, 6) ′′
8 s
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84112 E-mail address: ciubo@math. Salt Lake City, UTCiubotaru) Dept. of Mathematics, University of Utahutah.eduCiubotaru) Dept. of Mathematics, University of Utah, Salt Lake City, UT 84112 E-mail address: [email protected]
| []
|
[
"Fast Out-of-Sample Predictions for Bayesian Hierarchical Models of Latent Health States",
"Fast Out-of-Sample Predictions for Bayesian Hierarchical Models of Latent Health States"
]
| [
"Aaron J Fisher ",
"Yates Coley ",
"Scott L Zeger "
]
| []
| []
| Hierarchical Bayesian models can be especially useful in precision medicine settings, where clinicians are interested in estimating the patient-level latent variables associated with an individual's current health state and its trajectory. Such models are often fit using batch Markov Chain Monte Carlo (MCMC). However, the slow speed of batch MCMC computation makes it difficult to implement in clinical settings, where immediate latent variable estimates are often desired in response to new patient data. In this report, we discuss how importance sampling (IS) can instead be used to obtain fast, in-clinic estimates of patient-level latent variables. We apply IS to the hierarchical model proposed inColey et al. (2015)for predicting an individual's underlying prostate cancer state. We find that latent variable estimates via IS can typically be obtained in 1-10 seconds per person and have high agreement with estimates coming from longer-running batch MCMC methods. Alternative options for out-of-sample fitting and online updating are also discussed. | null | [
"https://arxiv.org/pdf/1510.08802v1.pdf"
]
| 88,521,642 | 1510.08802 | 97e597cd31f8d6984a532791e0d1f750fbeaa94c |
Fast Out-of-Sample Predictions for Bayesian Hierarchical Models of Latent Health States
October 30, 2015
Aaron J Fisher
Yates Coley
Scott L Zeger
Fast Out-of-Sample Predictions for Bayesian Hierarchical Models of Latent Health States
October 30, 2015
Hierarchical Bayesian models can be especially useful in precision medicine settings, where clinicians are interested in estimating the patient-level latent variables associated with an individual's current health state and its trajectory. Such models are often fit using batch Markov Chain Monte Carlo (MCMC). However, the slow speed of batch MCMC computation makes it difficult to implement in clinical settings, where immediate latent variable estimates are often desired in response to new patient data. In this report, we discuss how importance sampling (IS) can instead be used to obtain fast, in-clinic estimates of patient-level latent variables. We apply IS to the hierarchical model proposed inColey et al. (2015)for predicting an individual's underlying prostate cancer state. We find that latent variable estimates via IS can typically be obtained in 1-10 seconds per person and have high agreement with estimates coming from longer-running batch MCMC methods. Alternative options for out-of-sample fitting and online updating are also discussed.
Introduction
Hierarchical Bayesian models provide a natural statistical framework for precision medicine -allowing estimation of both patient-level latent variables associated with health status trajectory, as well as population-level causal effects of endogenous covariates and exogenous interventions. The use of a Bayesian framework encourages the inclusion of existing medical knowledge at each level. When such models are fit on a training dataset using Markov Chain Monte Carlo (MCMC), latent variable estimates are immediately available for any patient in the training dataset. For example, Coley et al. (2015) use a patient-level latent class to categorize patients in a training dataset as having either indolent or aggressive prostate cancer.
A computational challenge arises though when new patients enter the clinic or when existing patients accrue new measurements. Here, clinicians may wish to give patients fast, in-visit estimates of their latent variables and associated health states. However, batch MCMC estimation approaches that require refitting the entire model can take hours to complete. Additionally, if the model is refit on protected clinical data from multiple sites, then MCMC may require communication between fire-walled servers as the algorithm iterates, further increasing the computation cost.
Instead, sampling algorithms tailored for out-of-sample fitting can be used to get fast latent variable estimates in response to new patient data, while naturally avoiding the issue of server communication. Such algorithms can be based on conditional posteriors (Wu et al., 2015), Gibbs Sampling (GS), Importance Sampling (IS), or Rejection Sampling (RS) (Bishop et al., 2006). In this report, we specifically describe how IS can be used to obtain latent variable estimates for out-of-sample data. We also discuss conceptual parallels between these four types of approaches. We then apply IS to the prostate cancer model proposed by Coley et al. (2015) to get fast latent variable estimates for new, simulated patients. In this case, the IS procedure typically takes approximately 1-10 seconds per patient. This approach can be combined with periodic refitting of the entire model via MCMC, in order to update the posteriors for the population-level parameters (Lee and Chia, 2002).
This IS approach is related to online (or streaming) learning methods, which aim to continuously update population-level parameters with a constant computational cost over time. We avoid a fully online approach here though, due to additional challenges in online learning. Specifically, our use of IS can be viewed as a 1-step version of a sequential importance sampler (SIS), also known as particle filter. Employing a standard particle filter to update estimates of the population-level parameters would seem to be a natural extension. However, particle filters are known to suffer from the problem of degeneracy, which makes it difficult to estimate posteriors for "static" parameters that do not change as more data is acquired (Kantas et al. (2014), see section II of Andrieu et al. (2005) for an intuitive explanation). This concern applies in our case, as our population-level parameters are assumed to be static. Instead, we combine IS with periodic MCMC (Lee and Chia, 2002) to update the posteriors for all parameters and latent variables at all levels. Note that this is not a fully online method, as the computational cost of MCMC increases as more data is acquired. 1
Online model fitting has also been explored in the literature on topic modeling for corpuses of texts. Text corpuses are often too large to fit an entire model on at once, making online fitting a more feasible option. generally did not improve in accuracy as more data was incorporated.
Our specific context within precision medicine (Coley et al., 2015) is different than that of topic modeling in that, while the model is complex and contains several layers, the data can be fully stored in memory at once.
Thus, while the approach of combining IS with periodic MCMC is not fully online and not feasible for text analysis, it is still a feasible option for the limited sample sizes in our application. Relative to variational Bayes approaches, the formulas required to apply IS are simple to derive and can be easily ported to other applications within precision medicine.
The remainder of this document is organized as follows. In Section 1, we give an overview of our motivating data example of latent prostate cancer state estimation. In Section 2, we detail our approach for applying IS to hierarchical models. We use an abbreviated notation that can be readily generalized to other precision medicine settings. In Section 3, we conceptually compare our IS approach with RS, GS, and conditional posteriors (Wu et al., 2015). However, with the exception of RS, we do not explore the performance of these alternate methods here. In Section 4, we apply IS to simulated data, and compare the results to latent variable estimates obtained from batch MCMC.
Clinical Application & Motivation in Prostate Cancer
Our application is based on the clinical framework of Coley et al. (2015), who develop a latent class model to predict underlying prostate cancer state in men participating in an active surveillance program for low risk 1 See Kantas et al. (2014) for a recent literature review of particle methods in the context of static parameters.
disease. The latent cancer state is defined as being "indolent" or "aggressive", corresponding to the Gleason score (Gleason, 1977(Gleason, , 1992) that would be assigned if a patient's entire prostate were to be removed and analyzed.
Gleason scores < 6 are classified as indolent, and Gleason scores ≥ 7 are classified as aggressive. This latent state is known only for those patients who elect to undergo a prostatectomy while under active surveillance, resulting in a partially latent class problem. For those patients who do not elect to have their prostates removed, the model of Coley et al. (2015) is used to estimate posterior probabilities of latent class membership, or, in other words, the risk of having an aggressive cancer with the potential to metastasize. Latent variable predictions can then be used by clinicians and patients to make decisions about future treatment or biopsies. This prediction tool addresses a pressing need in prostate cancer care as the most common treatments for prostate cancer have a high risk of persistent side effects including erectile dysfunction and urinary incontinence, while prostate biopsies are painful and pose a risk of infection (Chou et al., 2011a,b).
The hierarchical model of Coley et al. (2015) includes sub-models for longitudinal prostate specific antigen (PSA) measurements and for longitudinal biopsy results. Both of these sub-models incorporate information about the patient's latent state. More specifically, log-transformed PSA measurements are modeled with a stratified random effects model where the distribution of patient-level random effects depends on latent state. Biopsy results are coded as binary outcomes denoting grade reclassification on a biopsy, that is, the biopsied tissue was assigned a Gleason score of 7 or higher. The log-odds of reclassification is modeled with a linear predictor whose value also depends on a patient's latent state, reflecting the imperfect sensitivity and specificity of the biopsy procedure. Each patient's latent class is assumed constant over the surveillance period. As a patient continues in active surveillance, additional PSA and biopsy measurements are accrued and the accuracy of latent class predictions improves. Sub-models are also included for informative observation processes associated with biopsies and prostatectomies.
In our context, the patient-level latent variables refer to the latent classes and the random effects used in the sub-model for PSA. The population-level parameters refer to the coefficients in each sub-model and the variance parameters for the patient-level variables. See Coley et al. (2015) for a full model description.
IS Algorithm for Fast Estimates from New Data
In this section we detail an IS algorithm that enables rapid estimates of patient-level variables, such as latent classes. This method is meant to be applied to out-of-sample data after MCMC has been applied to get a posterior sample based on current training data. We present the algorithm in a simple, abbreviated notation that is applicable in many clinical settings.
Let the joint posterior based on training data from n patients be denoted as
p(θ, b 1:n |y 1:n ) ∝ n i=1 [f (y i |b i , θ)g(b i |θ)]π(θ)(1)
where y i is the vector of clinical measurements (here, PSA and biopsy measurements) for patient i, y 1:n is the list of measurements for the first n patients, b i is a vector of latent variables (here, latent class and random effects) for patient i, b 1:n is a list of latent variables for the first n patients, θ contains the population-level parameters, π is the prior for θ, and f and g are multivariate probability distributions chosen based on the application and context. Estimation of b i is of primary interest in this report.
Let J n = {θ (j) , b(j)
1:n } J j=1 be a set of J draws from the posterior distribution in Eq 1 obtained via methods such as MCMC.
Core IS Algorithm
After posterior samples from the joint model (J n ) are obtained for current data, importance sampling to update these estimates given new data requires three steps: (1) generating proposal values for the latent variables to be estimated or updated, (2) calculating weights for proposed values, and (3) weighting proposed values to estimate an updated posterior. We first illustrate how this process can be used to quickly estimate latent variables for a new patient and then show how similar calculations can be done to incorporate newly measured data on existing patients in real-time.
For a new patient (indexed by i = n + 1), prediction of latent variables requires calculating expectations with respect to the posterior distribution based on all n + 1 patients (i.e. p(θ, b 1:(n+1) |y 1:(n+1) )). While we cannot immediately draw from this distribution, we can evaluate a function that is proportional to its density (based on Eq 1). We can also use the posterior distribution based on the first n patients as a proposal distribution (denoted by q) from which to generate candidate values of (θ, b 1:(n+1) ). Let q(θ, b 1:(n+1) ) := g(b n+1 |θ)p(θ, b 1:n |y 1:n )
Practically, this proposal step is achieved by conditioning on each θ (j) in J n and then drawing b
(j) n+1 from the distribution g(b (j) n+1 |θ (j) ). This results in the augmented set J n+1 := {θ (j) , b (j) 1:(n+1) } J j=1 . The importance weights w (j) are then proportional to w (j) ∝ p(θ (j) , b (j) 1:(n+1) |y 1:(n+1) ) q(θ (j) , b (j) 1:(n+1) ) ∝ n+1 i=1 [f (y i |b (j) i , θ (j) )g(b (j) i |θ (j) )]π(θ (j) ) g(b (j) n+1 |θ (j) ) n i=1 [f (y i |b (j) i , θ (j) )g(b i |θ (j) )]π(θ (j) ) = f (y n+1 |b (j) n+1 , θ (j) )(3)
The final weights w (j) are standardized to sum to 1. The new posterior for (θ, b 1:(n+1) ) can then be represented as the mixture distribution satisfying P (θ = θ (j) , b 1:(n+1) = b (j) 1:(n+1) ) = w (j) . Posterior means for b (n+1) can be calculated as
J j=1 w (j) b (j) (n+1) .
The approach is similar when we wish to incorporate new measurement data for a patient k who's previous data has already informed the posterior sample J n (i.e., k ≤ n). The set J n already contains proposals {b (j) k } J j=1 for patient k's latent variable values. Thus, we can use draws from J n as our proposal distribution q(θ (j) , b (j) 1:n ). Our goal then is to re-weight this set of proposals based on new data. Let y * 1:n refer to the data set after incorporating new data on patient k, such that y * i = y i if and only if k = i. The importance weights in Equation 3 then simplify to
w (j) ∝ p(θ (j) , b (j) 1:n |y * 1:n ) q(θ (j) , b (j) 1:n ) ∝ n i=1 [f (y * i |b (j) i , θ (j) )g(b (j) i |θ (j) )]π(θ (j) ) n i=1 [f (y i |b (j) i , θ (j) )g(b (j) i |θ (j) )]π(θ (j) ) = f (y * k |b (j) k , θ (j) ) f (y k |b (j) k , θ (j) )(4)
If the repeated measures for each patient are independent conditional on b i , as is the case in the proposed model from Coley et al. (2015), then the ratio in Eq 4 reduces to the likelihood of only the new data conditional on b (j) k and θ (j) .
Efficient Implementation
For implementation in clinical practice, proposals for new patients can be generated prior to actually observing new data, so that only weight calculation and re-weighting of the proposal distribution needs to be done in real-time.
By random chance, some new patients may have data such that very few of the pre-generated, proposed latent variables values receive high weights. This will reduce the effective sample size of the posterior 1/ J j=1 w (j) 2 , which in turn increases the Monte Carlo error of the posterior mean estimates. 2 However, we can use the effective sample size to flag patients who might have high error. When this effective sample size drops below a given threshold (e.g. 1000), we can repeat our procedure with a larger set of pre-generated proposals (J). If limited computing is available for MCMC, we can also approximate a larger set of proposals from Eq 2 by drawing multiple b n+1 values for each θ (j) , rather than drawing just one.
Comparison to Alternative Algorithms for Fast Estimates from New Data
In this section we outline some of the conceptual connections between our IS approach and out-of-sample fitting approaches based on Rejection Sampling (RS), Gibbs Sampling, and conditional posteriors (Wu et al., 2015).
Most directly related to our IS approach, RS can also be applied using the unstandardized weights in Eq 3.
While RS allows for fewer particles to be stored in memory, we found IS to be more computationally efficient in our scenario (see Section 4.1).
Out-of-sample estimation can also be done using Gibbs Sampling to update only the parameters associated with new patient data (i.e., b n+1 ). One simple implementation is to run separate MCMC chains, each initialized on a different element of J n . Another approximate implementation that combines these chains is to treat the set {θ (j) n } J j=1 as fixed and to create a proxy categorical parameter z according to the following hierarchical distribution:
p ∼ Dirichlet(α = 1 J ) z ∼ Categorical(p) b n+1 ∼ g(θ (z) ) y n+1 ∼ f (b n+1 , θ (z) )
where p is a J-length vector of probabilities, 1 J is a J-length vector of ones, and z is a scalar such that P (z = j) = p j for j = 1, 2, ...J. The above model can then be fit with traditional Gibbs Sampling, and the resulting posterior estimates for p are analogous to the weights in Eq 3.
Finally, our IS approach functions similarly to the out-of-sample estimation approach of Wu et al. (2015).
Their approach can be generalized to estimate the updated posterior probability that P (b n+1 = x|y 1:(n+1) ) using the estimatorP (b n+1 = x|y 1:(n+1) ) :
= 1 J J j=1 f (yn+1|bn+1=x,θ (j) )g(x|θ (j) ) f (yn+1|bn+1=x ,θ (j) )g(x |θ (j) )dx
. This approach is especially practical when the patient-specific variables b n+1 are discrete, and the integral in the denominator can be replaced with a summation. For cases with both continuous and discrete patient-specific variables, the approach can be combined with a proposal generation method based on Eq 2. We do not explore the performance of this approach, or of the above Gibbs Sampling approach, in this report.
Application
We applied the proposed IS approach to simulated data based on the Johns Hopkins Active Surveillance (JHAS) cohort. 1,298 men with very low or low risk prostate cancer diagnoses were enrolled in JHAS from January 1995 to June 2014. Results of all PSA tests and biopsies performed prior to enrollment and during active surveillance were collected. Patients were followed until grade reclassification, elective treatment, or loss to follow-up. Patients still active in the program were administratively censored at the time of data collection for this analysis (October 2014). The Gleason score determination based on pathologic analysis of the entire prostate specimen was also recored for patients who underwent prostatectomy. Details on the dataset are available in Coley et al. (2015).
Out simulated dataset consisted of 1,000 patients. The model proposed in Coley et al. (2015) was used as the data generating model, with parameter values set equal to their corresponding posterior mean estimates from fitting the model to the JHAS data. Covariates to the model (age and date of diagnosis) were each generated from a normal distribution with mean and variance equal to that observed in JHAS patients. See Coley et al. (2015) for more details on model specification and covariates.
Using this data as our initial sample (y 1:n ), we generate 500,000 draws (J n ) from the posterior for the population-level and patient-level variables (see Eq 1). Averaging over J n , we estimate the risk of having aggressive cancer for each patient who's latent class is unknown. The task of generating J n was run across 400 parallel jobs on a x86-based linux cluster, with as many as 200 jobs allowed to run simultaneously. The total elapsed computation time was 33 hours. Within each job, MCMC was implemented using the R2jags software package (Su and Yajima, 2015).
We then re-estimate each patient's risk using IS, taking as input only the population-level parameter posteriors from the MCMC step. When generating values for the patient-level variables b i , we further increase the diversity of the proposal set by we drawing 10 values from g(b i |θ (j) ) for each posterior draw θ (j) , for a total of 5 million proposals. We experimented with approaches of using only 50,000 proposals, using all 5 million proposals, or starting with 50,000 and increasing number of proposals until the effective sample size exceeds 1000. We refer to the first two approaches as "fixed" methods and the last approach as "dynamic". Within each approach, the final set of proposals were then weighted to obtain IS risk estimates. These simulation steps are meant to approximate the procedure of using IS to get risk estimates for a new patient, under the assumption that any individual patient has only a minor affect on the population-level parameter posteriors. In section 4.1, we assess coherence between IS risk estimates and MCMC risk estimates.
Results
We find a high degree of coherence between the estimated posterior probability of aggressive cancer from IS and from MCMC, as shown in Figure 1. With the dynamic proposal approach, the root mean square of the difference (rMSD) between these two sets of risk estimates was 1.3% (on the probability scale, from 0% to 100%).
The maximum absolute difference was 5.9%, with 99% of patients having a difference less than 4.5%. Of the approaches using a fixed number of proposals, the corresponding maximum and 99% quantile of differences were 16.6% and 5.6% for 50,000 proposals and 4.8% and 3.5% for 5 million proposal. We also considered a rejection sampling approach using the unstandardized weights in Eq 3 but found the results to have a greater deviation from MCMC estimates (rMSD = 1.6% for the dynamic approach). Figure 3 illustrates the roughly inverse relationship between the effective sample size used for IS and the difference between IS and MCMC risk estimates. 3 Here, all three approaches performed similarly, with the fixed 50,000-proposal approach being the fastest. Computation time per patient ranged from 1.5-13.5 minutes, 1.4 seconds -5.9 minutes, and 1.4-2.8 seconds for the 5-million-proposal, dynamic, and 50,000-proposal approaches, respectively. Interquartile ranges for the per-patient calculation times of three methods were 3.1-3.9 minutes, 2.3-4.6 seconds, and 2.3-2.5 seconds.
These findings suggest that the proposed IS algorithm can be an appropriate substitute for full MCMC runs in order to provide real-time updates in a clinical setting.
Discussion
The joint model of Coley et al. (2015) is among a growing number of statistical models for making individualized health predictions and recommendations. Development of such precision medicine methods must occur within a framework for clinical implementation. Specifically, concerns about convenience, security, and effective communication must be addressed alongside statistical considerations. In this technical report, we present a fast implementation of the of latent health state model proposed in Coley et al. (2015), using importance sampling to generate in-clinic predictions. This approach informs decision-making by enabling doctors and patients to access updated predictions in real-time in a clinical setting.
Supplemental Code
Code for simulating data, obtaining IS estimates, and comparing the results against MCMC estimates is available at: https://github.com/aaronjfisher/in-clinic-updates-PSA
Hoffman et al. (2010) propose a online variational Bayes approach for topic modeling. Canini et al. (2009) propose a particle filter approach, in a context where the static parameters can be integrated out. However, in the setting of Canini et al. (2009), even the best performing online methods were outperformed by batch (non-online) MCMC, and
Figure 1 :
1Agreement between IS and MCMC estimates for the posterior predictions of aggressive prostate cancer state in a new patient. Point color represents the number of candidate points used -50,000, 5 million, or dynamic. The dashed line indicates the axis of equality (i.e., perfect agreement).
Figure 2 :
2Difference between IS and MCMC risk estimates as a function of effective sample size for IS. Point color represents the number of candidate points used -50,000, 5 million, or dynamic. The dotted vertical line shows the threshold used for dynamic proposal generation at an effective sample size of 1,000. Both axes are shown with log-scale spacing.
The effective sample size is also known as the effective number of particles.
It is worth noting, however, that the differences did not tend to decrease at a rate proportional to the square root of the effective sample size.
On-line parameter estimation in general state-space models. C Andrieu, A Doucet, V B Tadic, Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC'05. 44th IEEE Conference on. IEEEAndrieu, C., Doucet, A., and Tadic, V. B. (2005). On-line parameter estimation in general state-space models. In Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC'05. 44th IEEE Conference on, pages 332-337. IEEE.
Pattern recognition and machine learning. C M Bishop, springer4New YorkBishop, C. M. et al. (2006). Pattern recognition and machine learning, volume 4. springer New York.
Online inference of topics with latent dirichlet allocation. K R Canini, L Shi, T L Griffiths, International conference on artificial intelligence and statistics. Canini, K. R., Shi, L., and Griffiths, T. L. (2009). Online inference of topics with latent dirichlet allocation. In International conference on artificial intelligence and statistics, pages 65-72.
Screening for prostate cancer: a review of the evidence for the U.S. Preventive Services Task Force. R Chou, J M Croswell, D Tracy, C Bougatsos, I Blazina, R Fu, K Gleitsmann, H C Koenig, C Lam, A Maltz, J B Rugge, Lin , K , Annals of Internal Medicine. 155Chou, R., Croswell, J. M., Tracy, D., Bougatsos, C., Blazina, I., Fu, R., Gleitsmann, K., Koenig, H. C., Lam, C., Maltz, A., Rugge, J. B., and Lin, K. (2011a). Screening for prostate cancer: a review of the evidence for the U.S. Preventive Services Task Force. Annals of Internal Medicine, 155:762-771.
Treatments for Localized Prostate Cancer: Systematic Review to Update the. R Chou, T Dana, C Bougatsos, R Fu, I Blazina, K Gleitsmann, J B Rugge, No. 91. ARHQ Publication No. 12-0516-EF-2U.S; Rockville, MDAgency for Healthcare Research and QualityPreventive Services Task Force. Evidence SynthesisChou, R., Dana, T., Bougatsos, C., Fu, R., Blazina, I., Gleitsmann, K., and Rugge, J. B. (2011b). Treatments for Localized Prostate Cancer: Systematic Review to Update the 2002 U.S. Preventive Services Task Force. Evi- dence Synthesis No. 91. ARHQ Publication No. 12-0516-EF-2. Rockville, MD: Agency for Healthcare Research and Quality.
Bayesian joint hierarchical model for prediction of latent health states with application to active surveillance of prostate cancer. R Y Coley, A J Fisher, M Mamawala, H B Carter, K J Pienta, L Zeger, S , Coley, R. Y., Fisher, A. J., Mamawala, M., Carter, H. B., Pienta, K. J., Zeger, and L, S. (2015). Bayesian joint hierarchical model for prediction of latent health states with application to active surveillance of prostate cancer. (http://arxiv.org/abs/1508.07511).
The Veteran's Administration Cooperative Urologic Research Group: Histologic grading and clinical staging of prostatic carcinoma. D Gleason, Urologic Pathology: The Prostate. Tannenbaum, M.PhiladelphiaLea and FebigerGleason, D. (1977). The Veteran's Administration Cooperative Urologic Research Group: Histologic grading and clinical staging of prostatic carcinoma. In Tannenbaum, M., editor, Urologic Pathology: The Prostate, pages 171-198. Lea and Febiger, Philadelphia.
Histologic grading of prostate cancer: a perspective. D F Gleason, Human pathology. 233Gleason, D. F. (1992). Histologic grading of prostate cancer: a perspective. Human pathology, 23(3):273-279.
Online learning for latent dirichlet allocation. M Hoffman, F R Bach, D M Blei, Advances in neural information processing systems. Hoffman, M., Bach, F. R., and Blei, D. M. (2010). Online learning for latent dirichlet allocation. In Advances in neural information processing systems, pages 856-864.
On particle methods for parameter estimation in state-space models. N Kantas, A Doucet, S S Singh, J M Maciejowski, N Chopin, arXiv:1412.8695arXiv preprintKantas, N., Doucet, A., Singh, S. S., Maciejowski, J. M., and Chopin, N. (2014). On particle methods for parameter estimation in state-space models. arXiv preprint arXiv:1412.8695.
A particle algorithm for sequential bayesian parameter estimation and model selection. D S Lee, N K Chia, IEEE Transactions on. 502Signal ProcessingLee, D. S. and Chia, N. K. (2002). A particle algorithm for sequential bayesian parameter estimation and model selection. Signal Processing, IEEE Transactions on, 50(2):326-336.
R2jags: A Package for Running jags from R. R package version 0. Y.-S Su, M Yajima, Su, Y.-S. and Yajima, M. (2015). R2jags: A Package for Running jags from R. R package version 0.05-01.
Partially latent class models for case-control studies of childhood pneumonia aetiology. Z Wu, M Deloria-Knoll, L L Hammitt, S L Zeger, Journal of the Royal Statistical Society: Series C (Applied Statistics). Wu, Z., Deloria-Knoll, M., Hammitt, L. L., and Zeger, S. L. (2015). Partially latent class models for case-control studies of childhood pneumonia aetiology. Journal of the Royal Statistical Society: Series C (Applied Statistics).
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|
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"Multiway Ensemble Kalman Filter",
"Multiway Ensemble Kalman Filter"
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| [
"Yu Wang \nUniversity of Michigan\nUniversity of Michigan\n\n",
"Alfred Hero [email protected] \nUniversity of Michigan\nUniversity of Michigan\n\n"
]
| [
"University of Michigan\nUniversity of Michigan\n",
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| In this work, we study the emergence of sparsity and multiway structures in second-order statistical characterizations of dynamical processes governed by partial differential equations (PDEs). We consider several state-of-the-art multiway covariance and inverse covariance (precision) matrix estimators and examine their pros and cons in terms of accuracy and interpretability in the context of physicsdriven forecasting when incorporated into the ensemble Kalman filter (EnKF). In particular, we show that multiway data generated from the Poisson and the convection-diffusion types of PDEs can be accurately tracked via EnKF when integrated with appropriate covariance and precision matrix estimators. | null | [
"https://arxiv.org/pdf/2112.04322v1.pdf"
]
| 244,954,563 | 2112.04322 | a63305c6465cba14637f9eca2de54ad8a20f1d79 |
Multiway Ensemble Kalman Filter
Yu Wang
University of Michigan
University of Michigan
Alfred Hero [email protected]
University of Michigan
University of Michigan
Multiway Ensemble Kalman Filter
In this work, we study the emergence of sparsity and multiway structures in second-order statistical characterizations of dynamical processes governed by partial differential equations (PDEs). We consider several state-of-the-art multiway covariance and inverse covariance (precision) matrix estimators and examine their pros and cons in terms of accuracy and interpretability in the context of physicsdriven forecasting when incorporated into the ensemble Kalman filter (EnKF). In particular, we show that multiway data generated from the Poisson and the convection-diffusion types of PDEs can be accurately tracked via EnKF when integrated with appropriate covariance and precision matrix estimators.
Introduction
There has recently been a resurgence of interest in integrating machine learning with physicsbased modeling. Much of the recent work has focused on black-box models such as deep neural networks [20,14,30,24,16,25]. However, seeking shallower models that capture mechanism in a physically interpretable manner has been a recurring theme in both machine learning and physics [28]. In this paper, we introduce a high-dimensional statistical approach that naturally integrates physics and machine learning through Kronecker-structured Gaussian graphical models. The learned representation can then be incorporated into a high dimensional predictive model using the ensemble Kalman filtering framework.
Multiway covariance/precision models. High-dimensional multiway/tensor-variate data arise naturally in physical sciences. For example, weather satellites measure spatio-temporal climate variables such as temperature, wind velocity, sea level, pressure, etc. Due to the non-homogeneous nature of these data, second-order information that encodes (conditional) dependency structure within the data is of interest. Assuming the data are drawn from a tensor normal distribution, a straightforward way to estimate this structure is to vectorize the tensor and estimate the underlying Gaussian graphical model associated with the vector. However, such an approach ignores the tensor structure and requires estimating a rather high dimensional precision matrix, often with insufficient sample size. In many scientific applications the sample size can be as small as one when only a single tensor-valued measurement is available.
To address sample complexity challenges in learning second-order representations for multiway (tensor) data, sparsity is often imposed on the covariance Σ or the inverse covariance Ω. Such approaches include the sparse Kronecker product (KP) or Kronecker sum (KS) decomposition of Σ or Ω. Statistical models and corresponding learning algorithms can be derived using generative models or matrix approximations. The former include: KGlasso/Tlasso [23,15] for estimating Ω = A ⊗ B, using a representation AXB = Z for data X when Z is white noise. Another generative model is SyGlasso/SG-PALM [27,26] that models the precision matrix as Ω = (A ⊕ B) 2 , which corresponds to assuming the data X obeys a Sylvester equation XA + BX = Z. Matrix approximation methods include: KPCA [22,8] that approximates the covariance matrix as Σ = r i=1 A i ⊗ B i . Another matrix approximation method is the TeraLasso [9] that models the precision matrix as Ω = A ⊕ B. TeraLasso is equivalent to approximation of the conditional dependency graph (encoded by the precision matrix) with a Cartesian product of smaller graphs 1 .
Multiway second-order characterization of dynamic processes. Physical systems often exhibits sparsity and low-rank structures in their covariance or inverse covariance matrix. This is due to the fact that many physical systems are governed by differential equations, which are characterized by sparse differential operators. For instance, Wang and Hero [26] showed that for multivariate data generated by the Poisson equation, the discretized data has an inverse covariance matrix equal to a squared Kronecker sum of smaller sparse matrices. In related work, Lindgren et al. [13] elucidated a link between certain classes of Gaussian Fields (GF) and Gaussian Markov Random Fields (GMRF) via stochastic partial differential equations, and showed that efficient learning algorithms can be developed using the fact that GMRFs have sparse precision matrices.
For multiway/tensor-variate Gaussian data, the aforementioned multiway (inverse) covariance estimators have been shown to be statistically consistent in high-dimensional regimes when sample sizes (N ) are much less than the dimensionality (d) of the covariates. An important question is whether these Kronecker structures can be integrated into the Kalman filter for tracking the states of a physical system that generates multiway data.
Numerical experiments: ensemble Kalman filtering
The Kalman filter is a well-known technique to track the states of a linear system over time, and many variants have been proposed to deal with non-linear systems, such as the extended and ensemble Kalman filters. The ensemble Kalman filter (EnKF) is particularly effective when the dynamical system is complicated and its gradient is infeasible to calculate, which is often the case in physical systems [5,1]. However, such systems are often high-dimensional and the EnKF operates in the regime where the number of ensemble members, N , is much less than the size of the state, d, suggesting that sparse inverse covariance models will be especially attractive. Hou et al. [11] introduced a sparsity-penalized EnKF, which uses an estimator of the forecasting covariance whose inverse is sparsity regularized. Here we propose incorporating the multiway covariance / inverse covariance models discussed in Section 1 into the EnKF of Hou et al. [11].
For motivation, we consider the Poisson equation, an elliptical PDE that governs many physical processes including electromagnetic induction, heat transfer, and convection [2]. On a rectangular region Ω = (0, d 1 ) × (0, d 2 ) in the 2D Cartesian plane, the Poisson equation with homogeneous Dirichlet boundary condition is expressed as
Du = (∂ 2 x + ∂ 2 y )u = f in Ω, u = 0 on ∂Ω(1)
where f : Ω → R is the given source function and u : Ω → R is the unknown. Using the finite difference method with a square mesh grid with unit spacing, the unknown and the source can be expressed as d 1 -by-d 2 matrices, U and F, respectively, that are related to each other via
U i+1,j + U i−1,j + U i,j+1 + U i,j−1 − 4U i,j = F i,j(2)
for any interior grid point (i, j). Defining n-by-n square matrix
A n = 2 −1 −1 2 . . . . . . . . . −1 −1 2 ,
the relation (2) can be expressed as the (vectorized) Sylvester equation with K = 2:
(A d1 ⊕ A d2 )u = f ,(3)
where u = vec(U), f = vec(F). Note that A is tridiagonal. In the case where f is white noise with variance σ 2 , the inverse covariance matrix of u has the form cov
−1 (u) = σ −2 (A d1 ⊕ A d2 ) T (A d1 ⊕ A d2
) and hence sparse. 1 Note that Tlasso, TeraLasso, Syglasso/SG-PALM are generalizable to precision matrices of the form
K k=1 A k , K k=1 A k , and ( K k=1 A k ) 2 , respectively, for K ≥ 2.
Here, we discuss two ways to extend the spatial Poisson equation described above to incorporate temporal dynamics, and illustrate how multiway (inverse) covariance models can be used to track spatio-temporal systems. The first extension, which we call the Poisson-AR(1) process, imposes an autoregressive temporal model of order 1 on the source function f in the Poisson equation (1). Specifically, we say a sequence of discretized spatial observations {U k ∈ R d1×d2 } k indexed by time step k = 1, · · · , T is from a Poisson-AR(1) process if
(A d1 ⊕ A d2 ) vec(U k ) = vec(Z k ),(4)vec(Z k ) = a vec(Z k−1 ) + vec(W k ), |a| < 1,(5)where Z 0 ∼ N (0, σ 2 z I) and {W k ∈ R d1×d2 } k is spatiotemporal white noise, i.e., W k i,j ∼ N (0, σ 2 w ), i.i.d.
The second time-varying extension of the Poisson PDE model (1) is the convection-diffusion process [2] ∂u
∂t = θ 2 i=1 ∂ 2 u ∂x 2 i − 2 i=1 ∂u ∂x i .(6)
Here, θ > 0 is the diffusivity; and ∈ R is the convection velocity of the quantity along each coordinate. Note that for simplicity of discussion here, we assume these coefficients do not change with space and time (see, Stocker [19], for example, for a detailed discussion). These equations are closely related to the Navier-Stokes equation commonly used in stochastic modeling for weather and climate prediction [2,19]. Coupled with Maxwell's equations, these equations can be used to model magneto-hydrodynamics [17], which characterize solar activities including flares.
A solution of Equation (6) can be approximated similarly as in the Poisson equation case, through a finite difference approach. Denote the discrete spatial samples of u(x, t) at time t k as a matrix U k ∈ R d1×d2 . We obtain a discretized update propagating u(x, t) in space and time, which locally satisfies
U k i,j − U k−1 i,j ∆t = θ U k i+1,j + U k i−1,j + U k i,j+1 + U k i,j−1 − 4U k i,j h 2 − U k i+1,j − U k i−1,j + U k i,j+1 − U k i,j−1 2h ,(7)
where ∆t = t k+1 − t k is the time step and h is the mesh step (spatial grid spacing). For ∆t = 1 and h = 1, U k can be shown to obey the Sylvester matrix update equation [21] A θ, U k + U k B T θ, = U k−1 , or equivalently,
(B θ, ⊕ A θ, ) vec(U k ) = vec(U k−1 ),(8)
where A θ, ∈ R d1×d1 and B θ, ∈ R d2×d2 are symmetric tridiagonal matrices whose entries depend on θ, , and h [7]. Note that, although {U k } k updates according to a sparse Kronecker sum, the inverse covariance of the marginal distribution that smooths over time is not sparse.
We turn Equation (8)
X k ) = H vec(U k ) + vec(V k ),(9)vec(U k ) = (B θ, ⊕ A θ, ) −1 vec(U k−1 ) + vec(W k ).(10)
We model the observed process X as an incomplete noisy version of the convection-diffusion state U t obeying the Sylvester matrix update equation above discretization the convection-diffusion becomes assume a linear Gaussian state-space model for the observed process X t governed by convection-diffusion dynamics:
A θ, U t + U t A θ, = U t−1 , X t = U t + V t ,
where V t ∼ N (0, σ 2 I) is an i.i.d. Gaussian white noise. Note that in general one might not fully observe the state, leading to a partially observed measurement process
vec(X t ) = H vec(U t ) + vec(V t ),
where H is a measurement matrix that can incorporate effects such as unobserved, masked, or superposed states. It is also possible to take into account the error in the dynamic process, i.e.,
vec(U t ) = (A θ, ⊕ A θ, ) −1 vec(U t−1 ) + vec(W t , )
where vec(W t ) is assumed to be white noise. Under the case of perfect observation (H is the identity matrix and W t = 0), Note that although the state variable evolves via a Sylvester equation, similar to the Poisson equation case, the state (inverse) covariance matrix at time step t k admits different structures. Specifically, the state precision matrix Ω k = cov −1 (vec(U k )) ∈ R d1d2×d1d2 evolves as Katzfuss et al. [12], for example). This matrix is not necessarily sparse for finite k but, assuming that the eigenvalues of the matrix B θ, ⊕ A θ, are in (−1, 1), the limiting precision matrix
Ω k = (B θ, ⊕ A θ, )Ω k−1 (B θ, ⊕ A θ, ) + σ −2 w I (seeΩ ∞ = lim k→∞ Ω k is Ω ∞ = (B θ, ⊕ A θ, )Ω ∞ (B θ, ⊕ A θ, ) + σ −2 I.
The Ω ∞ matrix is sparse because A θ, and B θ, are both tridiagonal.
To illustrate, we consider a 2D spatio-temporal process of dimension 64 × 64 where only half of the entries are observed, which leads to a measurement matrix H ∈ {0, 1} 2048×4069 . We generated the true states and the corresponding observations according to Convection-Diffusion and Poisson dynamics for T = 20 time steps. Several realizations of the true state variables are shown in Figure 1. At each time step, we generated an ensemble of size N = 25 and estimated the state covariance / inverse covariance using several sparse (multiway) inverse covariance estimation methods, including Glasso [6], KPCA [8], KGlasso [23], TeraLasso [9], SG-PALM [26]. Figure 2 shows evolution of the computed root mean squared errors (RMSEs) for the estimated states across all ensemble members. In Figure 3 we show the true and estimated (inverse) covariance matrices obtained for the last time step. The Poisson process is time-invariant, and at each time step the EnKF involves estimation of a sparse Kronecker sum squared inverse covariance matrix. Hence, the SG-PALM method operates under the correct model assumption in this situation. On the other hand, the inverse covariance structure under the convection-diffusion dynamics model is dense due to the smoothing effect of Kalman filtering and the nature of the temporal dynamics. But, its steady-state covariance has low-dimensional structures. The KPCA in this case was able to approximate this structure, as illustrated in Figure 3. Remark. The proposed multiway EnKF is able to track systems governed by elliptic (e.g., Poisson) parabolic, and hyperbolic (e.g., convection-diffusion) PDEs. There are indeed other important PDEs / dynamical systems that cannot be modelled by these types of equations. Furthermore, the Sylvester matrix equations arise when the finite-difference discretization is performed on a rectangular grid. The relations (2) and (7) might not hold for finite-difference on, for example, spherical coordinates.
Future work
Applications. Spatiotemporal PDEs are prominent techniques for modeling real-world physical systems. One such system arises in space physics, where solar flares and coronal mass ejections are associated with rapid changes in filed connectivity and are powered by partial dissipation of electrical currents in the solar atmosphere [18]. The nonlinear force-free filed model is often used to describe the solar coronal magnetic field [4,29] and can be derived from the convection-diffusion Hickmann et al. [10] introduced the Air Force Data Assimilative Photospheric Flux Transport model that uses localized ensemble Kalman filtering to adjust a set of photospheric simulations to agree with the available observations. In future work, we plan to incorporate our proposed multiway EnKF framework for tracking these solar physical systems.
Interpretability of factorization-based multiway model. While the Kronecker products expansion used in KPCA captures dense structures in the covariance matrix of data generated from more complex spatio-temporal physical processes, it lacks physical interpretability. In contrast to the case of Sylvester graphical model and Poisson processes, it is not obvious whether the sum of Kronecker products structure corresponds to any true physical models. Recent work in quantum informatics [3] has demonstrated a link between estimation of the density matrix for entangled quantum states and the structured tensor approximation via r i=1 A i ⊗ B i . Further characterizing these connections and extending them to study its connections with certain classes of discretized PDEs would be an interesting future direction.
Fourth
Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).
Figure 1 :
12D convection-diffusion (bottom) and Poisson (top) state variables at three different time stamps. Note that there is temporal correlation exists in the convection-diffusion states while the Poisson states are temporally independent.
Figure 2 :
2RMSEs of the estimated states via EnKF over 20 time steps using different (inverse) covariance estimators. RMSEs over all ensemble members are shown here with the mean highlighted using solid lines. Here, each state is of dimension 64 × 64 and is generated via either a convectiondiffusion (top row) or Poisson equation (bottom row). The best performers in terms of mean RMSE over all ensemble members are KPCA for convection-diffusion and SG-PALM for Poisson.
Figure 3 :
3Covariance/precision structures for Poisson and Convection-Diffusion dynamics and their estimates. Here, white/blank entries indicate zeros in the (inverse) covariance matrix. For Poisson dynamics the Sylvester graphical model matches the true structure of the precision matrix. For convection-diffusion dynamics the covariance instead of the precision matrix is structured and sparse. process described in this work. Additionally, global maps of the solar photospheric magnetic flux are fundamental drivers for simulations of the corona and solar wind. However, observations of the solar photosphere are only made intermittently over approximately half of the solar surface.
Acknowledgments and Disclosure of FundingThe authors thank Zeyu Sun at the University of Michigan for many helpful discussions of potential applications of the work on solar physics. The work is partially supported by NASA DRIVE Science Center grant 80NSSC20K0600 and by the National Nuclear Security Administration within the US Department of Energy under grant DE-NA0003921.
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| [
"https://github.com/ywa136/"
]
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[
"Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups",
"Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups"
]
| [
"Kanehisa Takasaki [email protected] \nDepartment of Fundamental Sciences\nKyoto University Yoshida\nSakyo-ku606-8501KyotoJapan\n"
]
| [
"Department of Fundamental Sciences\nKyoto University Yoshida\nSakyo-ku606-8501KyotoJapan"
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| Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the SU (N +1) N = 2 SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the A(2) 2N affine Toda system. Our construction, too, uses fractional powers of the superpotential W (x) that characterizes the curve. We also consider the uplane integral of topologically twisted theories on four-dimensional manifolds X with b + 2 (X) = 1 in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind. | 10.1142/s0217751x00002366 | [
"https://arxiv.org/pdf/hep-th/9901120v2.pdf"
]
| 13,927,818 | hep-th/9901120 | fc824b5bfd57ef05decf61b5edf485d0b840621d |
Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups
28 Jan 1999
Kanehisa Takasaki [email protected]
Department of Fundamental Sciences
Kyoto University Yoshida
Sakyo-ku606-8501KyotoJapan
Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups
28 Jan 1999
Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the SU (N +1) N = 2 SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the A(2) 2N affine Toda system. Our construction, too, uses fractional powers of the superpotential W (x) that characterizes the curve. We also consider the uplane integral of topologically twisted theories on four-dimensional manifolds X with b + 2 (X) = 1 in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.
Introduction
The geometry of Seiberg and Witten's low energy effective theory of the four-dimensional N = 2 SUSY Yang-Mills theories (with and without matters) [1] are based on complex algebraic curves now generally called the "Seiberg-Witten curves". Coordinates of the moduli space (Coulomb branch) U of these curves are given by the Casimirs u j (j = 1, · · · , N = rank(G)) of the scalar field φ of the N = 2 vector multiplet. Each curve carries a special meromorphic differential (the "Seiberg-Witten differential") dS SW . Given a suitable set of cycles A j , B j (j = 1, · · · , N), this differential dS SW induces a "special geometry" on the moduli space U. The "special coordinates" are given by the period integrals a j and a D j of dS SW along these cycles. The prepotential F of the low energy effective theory is determined by this special geometry.
Gorsky et al. [2] and Martinec and Warner [3] discovered that an integrable system, typically an affine Tida system, is hidden behind this geometric setup. This point of view soon turned out to be very useful not only for studying various four-dimensional N = 2 SUSY gauge theories [4,5,6,7,8,9,10,11,12,13] but also for elucidating a universal mathematical structure underlying those examples [14,15]. In fact, all the building blocks of the Seiberg-Witten theory fit to the language of integrable systems. The Seiberg-Witten curve is nothing but the spectral curve of the integrable system. The differential dS SW is related to its complex symplectic structure. The special coordinates a j are identified with the action variables; the angle variables live on an Abelian subvariety (called the "special" or "distinguished" Prym variety) of the Jacobi variety of the Seiberg-Witten curve. The Casimirs u j give an involutive set of Hamiltonians. The commuting flows generated by these Hamiltonians sweep out the aforementioned Prym variety. In a more geometric language, the phase space X has a Lagrangian fibration X → U by these N-dimensional complex "Lieouville tori". This link with integrable systems implies the existence of another kind of deformations of the Seiberg-Witten curves: Whitham deformations [2,3,4,9,10] The differential equations that characterize these deformations, are called "Whitham equations". These deformations are parametrized by an extra set of variables T n (n = 1, 2, · · ·). These variables are referred to as "slow variables" in the theory of Whitham equations; "fast variables" t n are the time variables of commuting Hamiltonian flows in the integrable system (or an integrable hierarchy).
Several interesting physical interpretations of these Whitham deformations have been proposed. Deformations by T 1 are identified to be the renormalization group flows [16,17,6,18,13]. In this sense, the other Whitham deformations may be thought of as generalized RG flows. Gorsky et al. [19] constructed an explicit solution of the Whitham equations for the SU(N + 1) Seiberg-Witten curve, and argued its relation to topologically twisted gauge theories. Edelstein et al. [20] interpreted the Whitham deformations of Gorsky et al. as soft breaking of N = 2 SUSY by spurion fields.
In this paper we generalize the construction of Gorsky et al. [19] to other classical gauge groups. We consider the N = 2 SUSY Yang-Mills theories in the vector representation of classical gauge groups. As Martinec and Warner pointed out [3], the Seiberg-Witten curves for these theories can be written in a unified form, see (2.1), which is essentially the spectral curves of affine Toda systems. This expression contains a function W (x) (called the "superpotential" in analogy with topological Landau-Ginzburg theories We also address another issue that is raised in our previous paper [21]. As discussed therein, an interesting interplay of the "slow variables" T n and the "fast variables" t n can be observed in the u-plane integral of the topologically twisted theories on fourdimensional manifolds X with b + 2 (X) = 1 [22,23,24]. (b + 2 (X) is the self-dual part of the 2nd Betti number.) Our consideration in the previous paper was limited to the case of SU(N + 1). We revisit this issue, now armed with the explicit construction of Whitham deformations for all classical gauge groups. This paper is organized as follows. Section 2 is a collection of basic mathematical notions concerning Seiberg-Witten curves. Particularly important are the Prym varieties and related differentials that are crucial in handling the cases other than SU(N + 1). Section 3 is a review of the construction of Gorsky et al. We present all technical details, which are used in the subsequent sections. In Section 4 we consider the case of the SO(2N) Seiberg-Witten curve in detail. Section 5 deals with the case of SO(2N + 1) and Sp(2N), along with some other cases that are not directly related to N = 2 SUSY Yang-Mills theories but can be treated similarly. In Section 6 we consider the u-plane integral of the topologically twisted theories. Section 7 is devoted to discussions. Appendix is added to show a precise form of the spectral curves of the affine Toda systems in the usual Lax formalism.
2 Curves, Differentials, and Prym Varieties
Various complex algebraic curves
The Seiberg-Witten curves of N = 2 SUSY Yang-Mills theories in the vector representation of classical gauge groups can be written in the following common form [3]:
z + µ 2 z = W (x). (2.1)
Here µ is some power of the renormalization group parameter Λ, and W (x) (the "superpotential") a polynomial or a Laurent polynomial of the following form:
SU(N + 1) : W (x) = x N +1 − N +1 j=2 u j x N +1−j . SO(2N + 1) : W (x) = x −1 x 2N − N j=1 u j x 2N −2j . Sp(2N) : W (x) = x 2 x 2N − N j=1 u j x 2N −2j + 2µ. SO(2N) : W (x) = x −2 x 2N − N j=1 u j x 2N −2j .
The polynomials
P (x) = x N +1 − N +1 j=2 u j x N +1−j (2.2)
for the SU(N + 1) gauge group and
Q(x 2 ) = x 2N − N j=1 u j x 2N −2j (2.3)
for the other gauge groups may be identified with the characteristic polynomial det(xI − φ) of the scalar field φ of the N = 2 SUSY vector multiplet. The coefficients u j are accordingly the expectation value of the Casimirs of φ. In the case of SU(N + 1) and SO(2N), W (x) coincides with the superpotential of the topological Landau-Ginzburg theories (or d < 1 strings) of singularities of the A and D type [25,26]. Because of this, W (x) is referred to as the "superpotential".
As Martinec and Warner pointed out [3], these curves coincide with the spectral curves of the affine Toda system of the type g (1)∨ dual to the the untwisted affine Lie algebra g (1) of the gauge group G. In particular, the affine Toda systems for the non-simply-laced gauge groups SO(2N + 1) and Sp(2N) are of the twisted affine type B
(1)∨ N = A (2) 2N −1 and C (1)∨ N = D (2)
N +1 . The affine Toda systems for the other classical affine Lie algebras, too, have spectral curves of the above form:
B (1) N , C (1) N -Toda : W (x) = x 2N − N j=1 u j x 2N −2j , A (2) 2N -Toda : W (x) = x x 2N − N j=1 u j x 2N −2j .
All these curves are hyperelliptic. The following is an equivalent expression in the usual expression y 2 = R(x) of hyperelliptic curves. Actually, it is in this form (or a quotient curve discussed later on) that the Seiberg-Witten curves for classical gauge groups were first derived [28,29,30,31].
1. The Seiberg-Witten curves:
SU(N + 1) : y 2 = P (x) 2 − 4µ 2 , z = (P (x) + y)/2. SO(2N + 1) : y 2 = Q(x 2 ) 2 − 4µ 2 x 2 , z = (Q(x 2 ) + y)/2x. Sp(2N) : y 2 = Q(x 2 )(x 2 Q(x 2 ) + 4µ), z = (2µ + x 2 Q(x 2 ) + xy)/2. SO(2N) : y 2 = Q(x 2 ) 2 − 4µ 2 x 4 , z = (Q(x 2 ) + y)/2.
2. Other affine Toda spectral curves:
B (1) N , C (1) N : y 2 = Q(x 2 ) 2 − 4µ 2 , z = (Q(x 2 ) + y)/2. A (2) 2N : y 2 = x 2 Q(x 2 ) 2 − 4µ 2 , z = (xQ(x 2 ) + y)/2.
The curves other than the SU(N + 1) Seiberg-Witten curve can be classified into two groups:
• A: The Sp(2N) Seiberg-Witten curve and the A (2) 2N -Toda curve.
• B: The other curves.
We shall show that the curves in the two groups exhibit different properties in many aspects. The first aspect that we now point out, is their genera:
• Case A: The curve has genus 2N.
• Case B: The curve has genus 2N − 1.
Involutions and Prym varieties
The above curves, which we denote by C, have several involutions. Common to all are the hyperelliptic involution
σ 1 : (x, y) → (x, −y), (x, z) → (x, µ 2 /z). (2.4)
All the curves other than the SU(N + 1) curves have the second involution
σ 2 : (x, y) → (−x, y), (x, z) → (−x, z). (2.5)
The quotient C 2 = C/σ 2 by the second involution is also a hyperelliptic curve. It has different properties in accordance with the above classification:
• Case A: C 2 has genus N, and the covering map C → C 2 is unramified.
• Case B: C 2 has genus N − 1, and the covering map C → C 2 is ramified.
The double covering C → C 2 determines the Prym variety Prym(C/C 2 ). This is an N-dimensional Abelian variety, which plays the role of the Jacobi variety Jac(C) for the SU(N + 1) Yang-Mills theory. (Fay's book [32] provides us with useful information on this kind of Prym varieties.) Following Fay's book, let us specify the structure of this Prym variety in more detail.
The algebro-geometric definition of this Prym variety is based on an automorphism An equivalent complex analytic expression is the complex torus
Prym(C/C 2 ) ≃ C N /(∆Z N + 2PZ N ), (2.6)
where ∆ is a diagonal matrix ∆ = diag(d 1 , · · · , d N ) with positive integers on the diagonal line, and P is a complex symmetric matrix (P jk ) with positive definite imaginary part.
This Prym variety is thus a polarized Abelian variety with the following polarization (d 1 , · · · , d N ):
• Case A: (d 1 , · · · , d N ) = (2, · · · , 2, 2).
• Case B: (d 1 , · · · , d N ) = (2, · · · , 2, 1).
In particular, Case A is substantially a principally polarized Abelian variety with period matrix P; the above expression is simply for dealing with the two cases in a unified way.
The matrix elements of P are period integrals of holomorphic differentials dω j (j = 1, · · · , N) that are "odd" under the action of σ 2 :
σ * 2 dω j = −dω j . (2.7)
These differentials are uniquely determined by the normalization condition 8) and the matrix elements P jk of P are given by
A j dω k = δ jk ,(2.P jk = d j 2 B j dω k . (2.9)
The 2N cycles A j , B j (j = 1, · · · , N) in these period integrals have to be chosen as follows: In particular, these cycles have the intersection numbers A j · A k = B j · B k = 0 and A j · B k = δ jk .
• Case A: The 4N cycles A j , −σ 2 (A j ), B j , −σ 2 (B j ) (j = 1, · · · , N) form
Seiberg-Witten differential
The Seiberg-Witten differential is given by
dS SW = x dz z = xW ′ (x)dx W (x) 2 − 4µ 2 . (2.
10)
The following list shows a more explicit form of this differential.
1. For the Seiberg-Witten curves:
SU(N + 1) : dS SW = xP ′ (x) dx y . SO(2N + 1) : dS SW = 2Q ′ (x 2 )x 2 − Q(x 2 ) dx y . Sp(2N) : dS SW = x 2Q ′ (x 2 )x 2 + 2Q(x 2 ) dx y . SO(2N) : dS SW = 2Q ′ (x 2 )x 2 − 2Q(x 2 ) dx y .
2. For the other Toda curves:
B (1) N , C (1) N : dS SW = 2Q ′ (x 2 )x 2 dx y . A (2) 2N : dS SW = x 2Q ′ (x 2 )x 2 + Q(x 2 ) dx y .
Here the prime means differentiating by x, i.e., P ′ (x) = dP (x)/dx, etc.
A fundamental property of this differential is that it generates holomorphic differentials on C as follows. Let (∂/∂u j ) · · ·| z=const. denote differentiating the quantity inside by u j while keeping z constant. For the Seiberg-Witten differential, this gives
∂ ∂u j dS SW | z=const. = ∂x ∂u j z=const. dz z = − ∂W (x)/∂u j ∂W (x)/∂x xW ′ (x)dx W (x) 2 − 4µ 2 . (2.11)
Here we have also used the relation
∂W (x) ∂u j + ∂W (x) ∂x ∂x ∂u j z=const. = 0,(2.12)
which follows from the equation of the curve C. One can verify, for each case presented above, that all these differentials for j = 1, · · · , N are holomorphic differentials. In the case of the SU(N +1) Seiberg-Witten curve, these N holomorphic differentials form a basis of the space of holomorphic differentials on C. In the other cases, these differentials are also linearly independent, but "odd" under the action of σ 2 , because the Seiberg-Witten differential itself is also "odd":
σ * 2 dS SW = −dS SW . (2.13)
The u j -derivatives (2.11) give a basis of "odd" holomorphic differentials (also called "Prym differentials").
Given a set of cycles A j and B j mentioned above, one can define the special coordinates a j and their duals a D j on the moduli space U as follows:
a j = A j dS SW , a D j = B j dS SW .
(2.14)
The N functions a j (j = 1, · · · , N), as well as the a D j 's, are functionally independent and give a local coordinate system on U. Differentiating the Seiberg-Witten differential now by a j 's give the normalized holomorphic differentials dω j :
∂ ∂a j dS SW z=const. = dω j . (2.15)
The prepotential F = F (a 1 , · · · , a N ) is defined by the differential equations
∂F ∂a j = a D j . (2.16)
The matrix elements P jk of P can be expressed as second derivatives of the prepotential:
P jk = ∂ 2 F ∂a j ∂a k . (2.17)
Quotient curve of genus N
The Prym variety Prym(C/C 2 ) is isogenous to the Jacobi variety Jac(C ′ ) of the quotient curve C ′ = C/σ ′ by the following involution σ ′ :
• Case A: σ ′ = σ 2 . • Case B: σ ′ = σ 1 σ 2 .
The quotient curve C ′ is also hyperelliptic and has genus N. The matrix P is actually the period matrix of the Jacobi variety Jac(C ′ ):
Jac(C ′ ) ≃ C N /(Z N + PZ N ). (2.18)
The differentials dS SW and dω j are "even" (i.e., invariant) under the action of σ ′ , so that they are the pull-back of differentials on C ′ . On the other hand, the homology classses
p ′ ([A j ]) = [A ′ j ], p ′ ([B j ]) = 2 d j [B ′ j ]. (2.19)
This is the origin of the factor d j /2 in the period integral representation of the matrix elements of P.
The equation of the quotient curve C ′ can be written out in terms of the following two invariants ξ and η of σ ′ :
• Case A: ξ = x 2 , η = y. • Case B: ξ = x 2 , η = xy.
The following equations of the quotient curves C ′ are thus derived.
1. For Seiberg-Witten curves:
SO(2N + 1) : η 2 = ξ Q(ξ 2 ) − 4µ 2 ξ . Sp(2N) : η 2 = Q(ξ) ξQ(ξ) + 4µ . SO(2N) : η 2 = ξ Q(ξ) 2 − 4µ 2 ξ 2 .
2. For the other Toda curves:
B (1) N , C (1) N : η 2 = ξ Q(ξ) 2 − 4µ 2 . A (2) 2N : η 2 = ξQ(ξ) 2 − 4µ 2 .
Note that the curves C and C ′ exhibit somewhat different characteristics. The curve C, viewed as a double covering of the x-sphere, has two points P ∞ and Q ∞ at infinity above x = ∞. These two points correspond to z = ∞ and z = 0, and mapped to each other by the hyperelliptic involution σ 1 . This is a typical characteristic of the spectral curves of affine Toda systems [33,34,35]. The curve C ′ , in contrast, has a single point at infinity above the ξ-shere. In particular, C ′ is branched over ξ = ∞. Hyperelliptic curves of this type arise in the KdV hierarchy [36,37]. As well known, the KdV hierarchy is a special case of the KP hierarchy with only the "odd" time variables t 2n+1 being left nontrivial [38].
The SU(2) curve is however exceptional: It has the involution σ 2 . By this accidental symmetry, one can construct the quotient curve C ′ = C/σ ′ with σ ′ = σ 1 σ 2 , which can be written
η 2 = ξ (ξ − u 2 ) 2 − 4µ 2 (2.20)
in terms of the invariants ξ = x 2 and η = xy. By shifting ξ → ξ − u 2 , this turns into the substantially the same curve
η 2 = (ξ + u 2 )(ξ − 2µ)(ξ + 2µ) (2.21)
as Seiberg and Witten first derived [1]. As Gorsky et al. noted [2], this curve appears in a classical study on modulations of elliptic solutions of the KdV equation.
= dω j , ∂ ∂T n dS z=const. = dΩ n . (3.1)
The precise setup is as follows:
1. The moduli u = (u 1 , · · · , u N ) are now understood to be functions u j ( a, T ) of a = (a 1 , · · · , a N ) and T = (T 1 , T 2 , · · ·). The new parameters T n are the "slow variables"
in the theory of Whitham equations. These deformed moduli u j = u j ( a, T ) are required to reduce to the Seiberg-Witten moduli u j = u j ( a) (i.e., the inverse of the period map u → a from the u-space to the a-space) at T = (1, 0, 0, · · ·).
2. dΩ n are meromorphic differentials of the second kind with poles at the two points P ∞ and Q ∞ only, and normalized to have zero A j periods:
A j dΩ n = 0 (j = 1, · · · , N). (3.2)
3. dω j are the normalized holomorphic differentials on the curve C.
4. The differential dS is a linear combination of these differentials of the form
dS = n≥1 T n dΩ n + N j=1 a j dω j ,(3.3)
and required to reduce to the Seiberg-Witten differential dS SW at the point T =
(1, 0, 0, · · ·).
The solution of Gorsky et al. [19] for the SU(N + 1) Seiberg-Witten curve is constructed by the following steps.
Consider the meromorphic differentials
dΩ n = R n (x) dz z , R n (x) = P (x) n/(N +1) + . (3.4)
Here (· · ·) + denotes the polynomial part of a Laurent series of x. The fractional power of P (x) is understood to be a Laurent series of the form x n + · · · at x = ∞.
Since R 1 (x) = x, dΩ 1 is nothing but the Seiberg-Witten differential. As in the case of the Seiberg-Witten differential (2.11), the u-derivatives of these meromorphic differentials turn out to be holomorphic differentials.
2. Consider the differential
dS = n≥1 T n dΩ n (3.5)
and its period integrals
a j = A j dS = n≥1 T n A j dΩ n . (3.6)
These period integrals are functions of the moduli u j and the deformation parameters T n . They determine a family of deformations of the Seiberg-Witten period map u → a with parametes T n .
3. The period map u → a from the u-space to the a-space is invertible if T is close to (1, 0, 0, · · ·), because the Seiberg-Witten period map at this point is invertible. The inverse map a → u = u 1 ( a, T ), · · · , u N ( a, T ) gives deformations of the Seiberg-Witten moduli u j = u j ( a), hence of the curve C, with parameters T n .
The differentials
dΩ n = dΩ n − N j=1 c (n) j dω j , c (n) j = A j dΩ n (3.7)
satisfy the required normalization condition.
5. dS is now a linear combination of dΩ n and dω j of the required form.
The outcome is the following theorem:
Theorem 1
The differential dS satisfies the Whitham equations (3.1) under the deformations of the curve C thus constructed.
We review the proof of this result in the rest of this section.
Differentiating dΩ n by moduli
The first, and most essential step is to derive the following property of dΩ n . Note that u j 's and T n 's are now understood to be independent variables.
Lemma 1 (∂/∂u j )dΩ n | z=const. are holomorphic differentials on C.
Proof. We first derive a set of conditions that the polynomial R n (x) should satisfy, and verify that they are indeed fulfilled. Differentiating the eqution of the curve C gives the
relation ∂P (x) ∂u j + ∂P (x) ∂x ∂x ∂u j z=const. = 0. (3.8)
Therefore, recalling that dz/z = P ′ (x)dx/y, one can rewrite the u j -derivative of dΩ n as follows:
∂ ∂u j dΩ n z=const. = ∂R n (x) ∂u j + R ′ n (x) ∂x ∂u j z=const. dz z = ∂R n (x) ∂u j + R ′ n (x) x N +1−j P ′ (x) P ′ (x)dx y = ∂R n (x) ∂u j P ′ (x) + R ′ n (x)x N +1−j dx y . (3.9)
has to satisfy the following condition:
deg ∂R n (x) ∂u j P ′ (x) + R ′ n (x)x N +1−j ≤ N − 1. (3.10)
Let us confirm that R n (x) = P (x) n/(N +1) + has this property. By the definition of (· · ·) + , R n (x) can be written
R n (x) = P (x) n/(N +1) + O(x −1 ).
Differentiating this relation by x and u j , respectively, gives
R ′ n (x) = n N + 1 P (x) (n−N −1)/(N +1) P ′ (x) + O(x −2 ), ∂R n (x) ∂u j = − n N + 1 P (x) (n−N −1)/(N +1) x N +1−j + O(x −1 ). (3.11)
From these relations, one can easily see that the above condition is certainly satisfied.
Q.E.D.
Deriving Whitham equations
Once dΩ n turns out to have the aforementioned property, deriving the Whitham equations (3.1) is rather straightforward. Let us present this calculation following the work of Itoyama and Morozov [9].
First, since that dω j (j = 1, · · · , N) give a basis of the space of holomorphic differentials, the holomorphic differentials in the above Lemma can be written
∂ ∂u k dΩ n z=const. = N j=1 σ (n) kj dω j . (3.12)
The coefficients are determined by integrating the both hand sides along A j :
σ (n) kj = A j ∂ ∂u k dΩ n z=const. = ∂ ∂u k A j dΩ n = ∂c (n) j ∂u k . (3.13)
Second, the derivatives of a j = a j ( u, T ) turn out to be written
∂a j ∂u k = n≥1 T n σ (n) kj , ∂a j ∂T n = c (n) j . (3.14)
This can be readily verified by directly differentiating
a j = A j dS = n≥1 T n c (n) j (3.15)
and using (3.13).
Now change the independent variables from ( u, T ) to ( a, T ). These two systems of coordinates are connected by the functions a j = a j ( u, T ) and u k = u k ( a, T ). In this setup, the following identities are satisfied:
N k=1 ∂u k ∂a i ∂a j ∂u k = δ ij , N k=1 ∂u k ∂T m ∂a j ∂u k = − ∂a j ∂T m . (3.16)
The first relation is obvious from the chain rule. The second one is rather confusing; this is obtained by differentiating the identity
a j = a j (u 1 ( a, T ), · · · , u N ( a, T ), T ) by T m .
Using the chain rule along with these relations (3.12), (3.14) and (3.16), one can verify the Whitham equations (3.1) as follows.
1. The first equation of (3.1):
∂ ∂a i dS z=const. = n T n ∂ ∂a i dΩ n z=const. = n,k T n ∂u k ∂a i ∂ ∂u k dΩ n z=const. = j,k ∂u k ∂a i · n T n σ (n) kj · dω j = j,k ∂u k ∂a i ∂a j ∂u k dω j = dω i .
2. The second equation of (3.1):
∂ ∂T m dS z=const. = dΩ m + n T n ∂ ∂T m dΩ n z=const. = dΩ m + n,k T n ∂u k ∂T m ∂ ∂u k dΩ n z=const. = dΩ m + j,k ∂u k ∂T m · n T n σ (n) kj · dω j = dΩ m + j,k ∂u k ∂T m ∂a j ∂u k dω j = dΩ m − j c (m) j dω j = dΩ m .
This completes the proof of the Theorem.
Prepotential and homogeneity
The prepotential F = F (a, T ) is defined by the equations
∂F ∂a j = B j dS, ∂F ∂T n = − P∞ f n (z)dS − Q∞ g n (z)dS.P (x) n/(N +1) = z + µ 2 z n/(N +1) . (3.19)
The singular part of Laurent expansion of the right hand side at z = ∞ or z = 0 determines the Laurent polynomials f n (z) and g n (z). Obviously the singular part is a Laurent polynomial with constant coefficients, and accordingly f n (z) and g n (z), too, turn out to have constant coefficients. The compatibility of the above defining equations for F is ensured by Riemann's bilinear relations. Second derivatives of the prepotential are also related to period integrals:
∂ 2 F ∂a j ∂a k = B j dω k , ∂ 2 F ∂a j ∂T n = B j dΩ n , ∂ 2 F ∂T m ∂T n = − P∞ f m (z)dΩ n − Q∞ g m (z)dΩ n . (3.20)
The construction of the Whitham deformations also ensures the homogeneity
n≥1 T n ∂F ∂T n + N j=1 a j ∂F ∂a j = 2F . (3.21)
To see this, first note that the period integrals a j = a j ( u, T ) have the obvious homogeneity
a j ( u, λ T ) = λa j ( u, T ). (3.22)
Acoordingly, u j = u j ( a, T ) are invariant under the rescaling of a j and T n , i.e., they are homogeneous functions of degree zero:
u j (λ u, λ T ) = u j ( u, T ). (3.23)
One can see, from these fact, that the period integrals on the right hand side of (3.17) are homogeneous functions of degree one, so that the prepotential becomes homogeneous of degree two (upon suitably normalizing the linear part).
4 Whitham Deformations of SO(2N ) curve
Setup and results
As warm-up, let us examine some fundamental properties of the Seiberg-Witten differential dS SW . The u j -derivatives of dS SW can be written
∂ ∂u j dS SW z=const. = x 2N −2j−2 dx x −4 Q(x 2 ) 2 − 4µ 2 =
x 2N −2j dx y (j = 1, · · · , N). = ξ N −j dξ 2η (j = 1, · · · , N). A new feature in the present setup is the parity; for the differential to be invariant under σ ′ , the polynomial R n (x) has to be odd,
R n (−x) = −R n (x). (4.4)
The superpotential W (x) is even, and the leading term is x 2N −2 . We are thus led to the following:
R n (x) = W (x) (2n−1)/(2N −2) + (n = 1, 2, · · ·). (4.5)
Note that these polynomials are indeed odd polynomials. Furthermore, R 1 (x) = x, so that in this case, too, dΩ 1 is equal to the Seiberg-Witten differential.
Lemma 2 (∂/∂u j )dΩ n | z=const. are holomorphic differentials on C, and invariant under the involution σ ′ .
Proof. We can proceed in mostly the same way as the case of the SU(N +1) Seiberg-Witten curve. The u j -derivatives of dΩ n can be written
∂ ∂u j dΩ n z=const. = x 2 ∂R n (x) ∂u j W ′ (x) − x 2 R ′ n (x) ∂W (x) ∂u j dx y . (4.6)
The proof now boilds down to verifying the following two statements:
1. The factor in front of dx/y is a polynomial.
2. The degree of this polynomial does not exceed 2N − 2.
Let us consider the first statement. The problem is that ∂W (x)/∂u j and W ′ (x) respectively have a pole of second and third order at x = 0. ∂R n (x)/∂u j and R ′ n (x), however, have a zero at x = 0, because R n (x) is an odd polynomial and has no constant term. Therefore, along with the factor x 2 , they cancel the pole of at most third order of the other two functions. The second statement can be verified by the same reasoning as the proof for the the SU(N + 1) Seiberg-Witten curve. Q.E.D.
Whitham deformations
Having obtained the meromorphic differentials dΩ n with the aforementioned properties, we can now construct a family of Whitham deformations.
The construction is parallel to the case of the SU(N + 1) Seiberg-Witten curve:
1. Consider the differential dS = n≥1 T n dΩ n .
(4.7)
and its period integrals
a j = A j dS = n≥1 T n A j dΩ n . (4.8)
The period integrals a j = a j ( u, T ) determine a period map u → a from the u-space to the a-space.
2. The period map u → a is invertible if T is close to (1, 0, 0, · · ·). The inverse map a → u = u 1 ( a, T ), · · · , u N ( a, T ) gives deformations of the curve C with parameters T .
The differentials
dΩ n = dΩ n − N j=1 c (n) j dω j , c (n) j = A j dΩ n ,(4.9)
satisfy the normalization condition A j dΩ n = 0 (j = 1, · · · , N). The following can be proven in exactly the same way as the proof for the SU(N + 1)
Seiberg-Witten curve:
Theorem 2 The differential dS satisfies the Whitham equations
∂ ∂a j dS z=const. = dω j , ∂ ∂T n dS z=const. = dΩ n ,(4.12)
under the deformations of the curve C thus constructed.
Relation to KdV hierarchy
By construction, the differentials dS, dΩ n , dΩ n and dω j are all invariant under the involution σ ′ . Accordingly, they actually descend to (or, equivalently, are the pull-back of) differentials on the quotient curve C ′ = C/σ ′ . In particular, the counterpart of dω j are also holomorphic, and, as already mentioned, form a normalized basis of holomorphic differentials on C ′ .
The meromorphic differentials dΩ n possess an even more interesting interpretation.
Thy correspond to meromorphic differentials on C ′ with a single pole at ξ = ∞ (which is the image of P ∞ and Q ∞ ). Recall that this is a branch point of the covering map of C ′ over the ξ-sphere. This is substantially the same setup that emerges in hyperelliptic solutions of the KdV hierarchy [36,37]. Those hyperelliptic solutions are constructed from a theta function and a series of meromorphic differentials dΩ KdV n (n = 1, 2, · · ·) with a single pole (of order 2n) at a fixed branch point (such as the point ξ = ∞) of the hyperelliptic curve.
In this respect, we should have numbered the deformation variables as T 1 , T 3 , T 5 , · · · rather than T 1 , T 2 , T 3 , · · ·, following the usual numbering of the time variables in the KdV hierarchy. Note that the odd indices correspond to the degrees 2n − 1 of powers of
W (x) 1/(2N −2) in the definition of R n (x).
We must, however, also add that our meromorphic differentials are not exactly the These remarks also apply to the other cases where the essential part of the theory is described by the quotient curve C ′ of the KdV type. The case of the non-simply-laced gauge groups SO(2N + 1) and Sp(2N) can be treated in almost the same way as the case of SO (2N). The curve C has the involution σ ′ , and the quotient curve C ′ = C/σ ′ has genus N. The Seiberg-Witten differential dS SW is invariant under this involution. Our first task is to construct meromorphic differentials dΩ n = R n (x) dz z , R n (x) = odd polynomial, (5.1) whose u j -derivatives are holomorphic differentials on C.
The polynomials R n (x) are again given by the polynomial part of fractional powers of the superpotential W (x):
SO(2N + 1) : R n (x) = W (x) (2n−1)/(2N −1) + . (5.2) Sp(2N) : R n (x) = W (x) (2n−1)/(2N +2) + . (5.3)
The u j -derivatives can be written as follows:
1. For SO(2N + 1),
∂ ∂u j dΩ n z=const. = x ∂R n (x) ∂u j W ′ (x) − xR ′ n (x) ∂W (x) ∂u j dx y . (5.4) ∂ ∂u j dΩ n z=const. = x −1 ∂R n (x) ∂u j W ′ (x) − x −1 R ′ n (x) ∂W (x) ∂u j dx y . (5.5)
The prefactor of dx/y turns out to be a polynomial of degree ≤ 2N − 2 and ≤ 2N − 1 for SO(2N + 1) and Sp(2N), respectively. Therefore the above differentials are holomorphic differentials on C.
The Whitham deformations of C are given by the inverse period map a → u of the period integrals
a j = A j dS = n≥1 T n A j dΩ n . (5.6) of the differential dS = n≥1 T n dΩ n . (5.7)
The normalized meromorphic differentials dΩ n are given by
dΩ n = dΩ n − n j=1 c (n) j dω j , c (n) j = A j dΩ n .
Some other cases
The same fractional power construction of Whitham deformations also applies to the Toda curves for other classical affine Lie algebras, namely, B
(1)
N , C(1)
N and A
2N . The meromorphic differentials dΩ n are obtained in the same form as the other cases. The polynomials R n (x) are given by the following:
B (1) N , C (1) N : W (x) = W (x) (2n−1)/(2N ) + , A (2) 2N : W (x) = W (x) (2n−1)/(2N +1) + . (5.11)
The case of the A
2N (also called BC N ) Toda curve, seems to be particularly interesting and mysterious. Although this does not appear in the list of ordinary N = 2 SUSY Yang-Mills theories, an M-theoretical interpretation [41] might be possible.
Actually, the same construction works with no problem for a more general rational superpotential, e.g.,
W (x) = P (x) + M k=1 v k x − c k . (5.12)
Note that the curve C is still hyperelliptic. This is a special case of M5-branes with semi-infinite D4 branes (arising from the poles of W (x)) on both sides of the stack of two NS 5-branes.
Yang-Mills theories with fundamental matters
z + µ 2 R(x) z = P (x), (5.13) where R(x) is the polynomial R(x) = N f i=1 (x + m i ). (5.14)
This curve can be formally converted into the form of (2.1) by changing coordinates from
x and z to x and
w = z/ R(x). (5.15)
The converted curve can be written
w + µ 2 w = W (x) = P (x) R(x) , (5.16)
thus an irrational superpotential arises. The role of z is now played by w. For instance, the Seiberg-Witten differential can be written
dS SW = x dw w . (5.17)
Since the superpotential itself is multi-valued, The fractional power construction requires a more careful handling. This multi-valuedness is however relatively harmless, simply affecting an overall multiplicative constant of the meromorphic differentials dΩ n . G is a transformation that generates a standard solution of the descent equations for observables [22,23]. The Donaldson-Witten function of these observables is the path
integral Z DW = exp I(S) + O(P ) . (6.2)
In the case with b + 2 (X) = 1, the Donaldson-Witten function becomes a sum of two pieces:
Z DW = Z SW + Z u . (6.
3)
The first piece Z SW is the contributions from the strong-coupling singularities of the moduli space U (which is called the "u-plane" by abuse of the terminology for the SU (2) and SO(3) gauge groups). The second piece Z u is called the u-plane integral, which is absent if b + 2 (X) > 1. This is the contributions from the whole moduli space U According to Moore and Witten [22] (for SU(2), SO(3) gauge groups) and Mariño and Moore [23] (for other gauge groups), the u-plane integral for the above Donaldson-Witten function Z DW can be written (6.4) where χ and σ are the Euler number and the signature of X; A and B are modular forms on U; U is a contribution from O(P ) only; T is a "contact term" which is induced by the intersection of the 2-cycle S with itself, accordingly multiplied by the self-intersection number S 2 ; Ψ is a lattice sum collecting the contributions of abelianized gauge fileds and other fermionic degrees of freedom.
Z u = U dadāA χ B σ exp(U + S 2 T )Ψ,
Blowup formula and tau function
Mariño and Moore [23] pointed out that the blowup formula of the u-plane integral contains a factor that can be interpreted as a special "tau function" of the affine Toda system (or, more precisely, an underlying integrable hierarchy). It should however be noted that this is the interpretation in the case of the SU(N + 1) gauge group.
The blowup formula connects the manifold X and its blowupX at a point Q. X is assumed to be a complex algebraic surface (e.g.,
e U → e U α β det ∂u k ∂a j 1/2 ∆ −1/8 e −t 2 T Θ γ,δ t 2π V | P (6.6)
in the u-plane integral for Z DW . Various terms on the right hand side of this rule have the following meaning: α and β are some numerical constants; ∆ is the discriminant of the family of the curves C over U; Θ γ,δ (Z | P) is the ordinary N-dimensional theta function with characteristic (γ, δ) and period matrix P; V is the gradient vector
V = ∂V ∂a j (6.7)
of the gauge invariant potential V from which the integrand G 2 V of I(S) and I(B)
are constructed (e.g., V = Tr φ 2 in the aforementioned usual setup of topological gauge theories). For the SU(N + 1) Seiberg-Witten curve, the matrix P is the period matrix of Jac(C); for the other classical gauge groups, the period matrix of Prym(C/C 2 ) (or Jac(C ′ )) appears. The characteristic (γ, δ) is determined by the physical setup; typically, γ = (0, · · · , 0) and δ = (1/2, · · · , 1/2).
It is the product of the last two terms in (6.6) that Mariño and Moore, in the case of the SU(N + 1) gauge group, identified to be the tau function of the affine Toda system:
τ γ,δ (t) = e −t 2 T Θ γ,δ t 2π V | P . (6.8)
Thus, the coupling constant t plays the role of a time variable in the A (1)
N Toda system. For the other classical gauge groups, however, the relation to the affine Toda systems is slightly more complicated, as we discuss later on.
Multi-time tau function as blowup factor
Our previous paper [21] proposes a "multi-time" analogue of the single-time tau function τ γ,δ (t) above. In the present setup including all classical gauge groups, the multi-time analogue can be written
τ γ,δ (t 1 , t 2 , · · ·) = exp 1 2 m,n≥1 q mn t m t n Θ γ,δ n≥1 t n V (n) | P . (6.9)
The coefficients q mn of the Gaussian factor and the components of the directional vectors
V (n) = V (n) j
are written in terms of period integrals of the meromorphic differentials dΩ n that we have considered:
q mn = − 1 2πi P∞ f n (z)dΩ m − 1 2πi Q∞ g n (z)dΩ m , V (n) j = − 1 2πi P∞ f n (z)dω j − 1 2πi Q∞ g n (z)dω j .V (n) = ∂V (n) ∂a j . (6.14)
This leads to the identification of the coefficients q mn as the "contact terms" C V (m) , V (n) of higher Casimir observables in the sense of Losev et al. [24].
Strong evidence supporting our proposal is that the above multi-time tau function has a good modular property under the symplectic transformations
B j → A jk B k + B jk A k , A j → C jk B k + D jk A k , A B C D ∈ Sp(2N, Z). (6.15)
of the cycles A j , B j . This is crucial for ensuring the correct modular property of the integrand of the u-plane integral. Under this symplectic transformation, indeed, the tau function τ γ,δ transforms as
τ γ,δ (t 1 , t 2 , · · ·) → ǫ det(CP + D) 1/2 τ γ ′ ,δ ′ (t 1 , t 2 , · · ·),(6.16)
where ǫ (an 8th root of unity), γ ′ and δ ′ are determined by the Sp (2N, Z) matrix. This fact can be confirmed in the same way as the proof in the case of the SU(N + 1) topological gauge theory [21]. It should be also mentioned that this modular property of the tau function has been known for years [42,43]. Now the point is that the modular property of τ γ,δ is independent of t 1 , t 2 , · · ·. In particular, the modular invariance of the u-plane integral in Mariño and Moore's setup at t 1 = t and t n = 0 (n > 1) is retained even if the higher coupling constants t n are turned on.
KdV again
What is new in the case of the orthogonal and symplectic groups is that the theta function in the tau function is not a theta function on Jac(C); this is a theta function on the Prym variety Prym(C/C 2 ) or, up to an isogeny, the Jacobi variety Jac(C ′ ) of the quotient curve C ′ . As remarked in the previous sections, the quotient curve C ′ and the meromorphic differentials dΩ n are of the KdV type. The tau function τ γ,δ for the orthogonal and symplectic gauge groups thus turns out to be a tau function of the KdV hierarchy (which is a special case of the algebro-geometric tau functions of the KP hierarchy [39,40]) rather than of the affine Toda system.
This conclusion might cause confusion, but here is no contradiction. It is well known that the affine Toda system can be mapped to linear flows on the Prym variety [35]. Apart from the case of A N .) Thus, our tau function τ γ,δ is something different from tau functions of the affine Toda systems, and our interpretation is that it is a special tau function of the KdV type.
Whitham deformations and prepotential
Let us now turn on the Whitham deformations with "slow variables" T n . The Seiberg-Witten prepotential F is also deformed and becomes a function F ( a, T ) of both a = (a 1 , · · · , a N ) and T = (T 1 , T 2 , · · ·). More precisely, F is defined by (3.17) for all cases considered in the preceding sections. As one can immediately see by comparing these equations with the definition of q mn , V (n) j and P jk in the form of period integrals, these fundamental quantities in our interpretation of the blowup formula can be written as second order derivatives of the prepotential:
V jn = 1 2πi ∂ 2 F ∂a j ∂T n , q mn = 1 2πi ∂ 2 F ∂T m ∂T n , P jk = ∂ 2 F ∂a j ∂a k . (6.17)
In particular, the potential V (n) of the observables I n (B) turn out to be first order derivatives of F :
V (n) = 1 2πi ∂F ∂T n . (6.18)
Of course the a j -derivatives give the dual special coordinates a D j = B j dS:
a D j = ∂F ∂a j . (6.19)
Thus the prepotential F = F ( a, T ) in the Whitham deformations, too, is a kind of "generating function".
Unlike the "fast variables" t n , the role of the "slow variables" T n is to deform the period map u → a that connects the u-space and the a-space. Presumably, this will be a kind of deformations of "background geometry" in the sense of string theory, but the precise meaning is still beyond our scope.
Discussions
We have seen that the construction of Whitham deformations by Gorsky et al. [19] can be extended to the Seiberg-Witten curves of all the classical gauge groups and some other complex algebraic curves. Although the construction is based on the somewhat special form (2.1) of the curves, the only requirement seems to be that W (x) be a rational function with a polynomial leading part. Actually, we have obtained partial evidence that an irrational superpotential might be allowed for at least in some special cases.
We have also extended our proposal in the previous paper [21]
Non-hyperelliptic curves
The first nontrivial step beyond rational superpotentials is irrational superpotentials of the form
W (x) = R 1 (x) + R 2 (x) R 3 (x),(7.1)
where R i (x)'s are polynomial or rational functions of x. This means that the curve C is no longer hyperelliptic, but can be written in a special quartic polynomial in z with rational coefficients.
Well known examples of curves of this form [3,44] The numbers on the right hand side are the exponents of E 6 .
For more general cases, however, an entirely new approach will be necessary. For instance, Witten's M5-brane construction [41] yields a non-hyperelliptic curve of the form
z k+1 + g 1 (x)z k + · · · + g k (x)z + 1 = 0 (7.3)
for the N = 2 SUSY gauge theory (coupled to bifundamental matters) with the product gauge group SU(N 1 ) × · · · × SU(N k ). The Seiberg-Witten differential is given by
dS SW = x dz z . (7.4)
A natural ansatz for the meromorphic differentials dΩ n of Whitham deformations is to seek for them in the form dΩ n = R n (x, z) dz z , R n (x, z) = polynomial. (7.5)
We do not know how to construct the polynomials R n (x, z). The problem becomes even harder for the elliptic models of M5-branes.
Relation to topological Landau-Ginzburg theories
The fractional power construction strongly suggests a direct link with topological Landan-Ginzburg theories of A-D-E singularities coupled to gravity or, equivalently, d < 1 topological strings [25,26]. The relation between the Seiberg-Witten theory and d < 1 topological strings has been studied from several aspects, such as the WDVV equations [46,47], flat coordinates and Gauss-Manin systems [48,49], etc. Of course the very notion of prepotentials itself is a bridge connecting the two worlds. In the d < 1 topological strings, the role of the Whitham equations is played by the dispersionless limit of integrable hierarchies [50,51,52]. The fractional powers of the superpotential are fundamental building blocks of the Lax representation therein. Nevertheless, the emergence of fractional powers in the Seiberg-Witten theory is quite surprising.
An interesting outcome of our Whitham deformations is that they have an exotic limit as µ → 0. In this limit, the Seiberg-Witten curve reduces to the rational curve z = W (x), (7.6) and the Seiberg-Witten differential turns into the rational differential
dS SW = x W ′ (x)dx W (x) . (7.7)
As we shall show below, the Whitham equations, too, have a well defined limit. Furthremore, these differential equations are similar, but not identical, to the following counterpart in d < 1 topological strings [50,51,52]:
∂ ∂T n m≥1 T m R m (x) W (x)=const.
= R n (x). (7.8) This difference stems from the difference of the two theories as Landau-Ginzburg models. Namely, whereas the Whitham deformations at µ = 0 is still related to a curve defined by (7.6), the Landau-Ginzburg description of d < 1 topological strings is based on a 0-dimensional manifold defined by the equation
W (x) = 0. (7.9)
Now, let us present the Whitham equations at µ = 0. For simplicity, we consider the case of the SU(N + 1) curve where W (x) = P (x); the other case can be treated similarly. Suppose, as usual, that the cycles A j are chosen to encircle the cuts between two neighboring roots e ± j of P (x) − 4µ 2 . As µ → 0, the j-th cuts shrink to a point at the j-th root e j of P (x) = N +1 j=1 (x − e j ). The period integrals a j = A j dS then reduce to residue integrals, which can be readily calculated:
a j = 2πi n≥1
T n R n (e j ) (n = 1, · · · , N).
(7.10)
This defines a map u → a from the u-space to the a-space with deformation parameters T n , and this map is invertible if T is close to (1, 0, 0, · · ·). (Note that the N + 1-th root e N +1 of P (x) is not independent; the roots of P (x) obeys the constraint N +1 j=1 e j = 0.) The inverse map determines, as in the case with µ = 0, a family of deformations of the rational curve z = P (x). Under these deformations, the following equations can be eventually derived:
∂ ∂T n m≥1 T m R m (x) P (x)=const. = R n (x) − N +1 k=1 R n (e k )P (x) (x − e k )P ′ (x) ∂ ∂a j m≥1 T m R m (x) P (x)=const. = 1 x − e j − 1 x − e N +1 P (x) 2πiP ′ (x) (7.11)
We omit the proof of these equations, but the following comment would be enough for understanding: These equations can be derived from the Whitham equations (3.1) if dΩ n and dω j are interpreted as follows:
dΩ n = R n (x) P ′ (x)dx P (x) − N j=1
2πiR n (e j )dω j ,
dω j = 1 x − e j − 1 x − e N +1 dx 2πi . (7.12)
In fact, they give a correct limit, as µ → 0, of the differentials on the µ = 0 curve.
A Spectral Curves of Affine Toda Systems
Here we present a list of the Toda spectral curves
N , C(1)
N and D
N are borrowed from the work of Adler and van Moerbeke [35]. The other cases associated with the twisted affine algebras A (2) N and D (2) N are derived by the following "folding" procedure:
D (1) 2N → A (2) 2N −1 , D (1) N +2 → D (2) N +1 , D (1) 2N +2 → A(2)
2N .
(A.
2)
The L-matrices in the following list have a "symmetric" form, i.e., L(z) T = L(z −1 ), as opposed to the L-matrices of Martinec and Warner [3]. Accordingly, the actual form of the spectral curves becomes L(x) = N +1 j=1 a j E j,j+1 + a 0 zE N 1 + N +1 j=1 b j E jj + N +1 j=1 a j E j+1,j +a 0 z −1 E 1N .
det L(z) − xI = (−1) N A(z + z −1 ) + (−1) N +1 P (x), A = a 0 a 1 · · · a N , µ = A.
B
(1) N : The L-matrix is (2N + 1) × (2N + 1).
L(x) = N j=1 a j (E j,j+1 − E 2N +1−j,2N +2−j ) + a 0 (zE 2N +1,2 − zE 2N,1 ) + N j=1 b j (E jj − E 2N +2−j,2N +2−j ) + N j=1 a j (E j+1,j − E 2N +2−j,2N +1−j ) + a 0 (z −1 E 2,2N +1 − z −1 E 1,2N ). det L(z) − xI = x 2(−1) N A(z + z −1 ) − Q(x 2 ) .
A = a 0 a 1 a 2 2 · · · a 2 N , µ = 2(−1) N A.
C
(1)
N : The L-matrix is 2N × 2N. L(x) = N −1 j=1 a j (E j,j+1 − E 2N −j,2N +1−j ) + a N E N,N +1 + a 0 zE 2N,1 + N j=1 b j (E jj − E 2N +1−j,2N +1−j ) + N −1
j=1 a j (E j+1,j − E 2N +1−j,2N −j ) + a N E N +1,N + a 0 z −1 E 1,2N . det L(z) − xI = (−1) N A(z + z −1 ) + Q(x 2 ).
A = a 0 a 2 1 · · · a 2 N −1 a N , µ = −(−1) N A.
D
(1) N : The L-matrix is 2N × 2N.
L(z) = N −1 j=1 a j (E j,j+1 − E 2N −j,2N +1−j ) + a N (E N,N +2 − E N −1,N +1 ) +a 0 (zE 2N,2 − zE 2N −1,1 ) + N j=1 b j (E jj − E 2N +1−j,2N +1−j ) + N −1 j=1 a j (E j+1,j − E 2N +1−j,2N −j ) + a N (E N +2,N − E N +1,N −1 ) +a 0 (z −1 E 2,2N − z −1 E 1,2N −1 ).
det L(z) − xI = −4(−1) N Ax 2 (z + z −1 ) + Q(x 2 ).
A = a 0 a 1 a 2 2 · · · a 2 N −2 a N −1 a N , µ = 4(−1) N A.
A
(2)
2N −1 : The L-matrix is 2N × 2N.
L(z) = N −1 j=1 a j (E j,j+1 − E 2N −j,2N +1−j ) + a N E N,N +1 +a 0 (zE 2N,2 − zE 2N −1,1 ) + N j=1 b j (E jj − E 2N +1−j,2N +1−j ) + N −1 j=1 a j (E j+1,j − E 2N +1−j,2N −j ) + a N E N +1,N +a 0 (z −1 E 2,2N − z −1 E 1,2N −1 ).
det L(z) − xI = 2(−1) N Ax(z + z −1 ) + Q(x 2 ).
A = a 0 a 1 a 2 2 · · · a 2 N −1 a N , µ = −2(−1) N A.
6. D
N +1 : The L-matrix is (2N + 2) × (2N + 2).
L(z) = N j=1 a j (E j+1,j+2 − E 2N +2−j,2N +3−j ) + a 0 (zE 2N +2,1 − E 12 ) + N j=1 b j (E j+1,j+1 − E 2N +3−j,2N +3−j ) + N j=1 a j (E j+2,j+1 − E 2N +3−j,2N +2−j ) + a 0 (z −1 E 1,2N +2 − E 21 ). det L(z) − xI = (−1) N A(z + z −1 − 2) + x 2 Q(x 2 ).
A = a 2 0 a 2 1 · · · a 2 N , µ = −(−1) N A.
A
2N : The L-matrix is (2N + 1) × (2N + 1).
L(z) =
N −1 j=1 a j (E j+1,j+2 − E 2N +1−j,2N +2−j ) + a N E N +1,N +2 +a 0 (zE 2N +1,1 − E 12 ) + N j=0 b j (E j+1,j+1 − E 2N +2−j,2N +2−j ) + N −1 j=1 a j (E j+2,j+1 − E 2N +2−j,2N +1−j ) + a N E N +2,N +1 +a 0 (z −1 E 1,2N +1 − E 21 ). det L(z) − xI = (−1) N A(z + z −1 ) + xQ(x 2 ). A = a 2 0 a 2 1 · · · a 2 N −1 a N , µ = −(−1) N A.
of A-D-E singularities). In the case of the SU(N + 1) Seiberg-Witten curve, W (x) is a polynomial. Gorsky et al. use fractional powers of W (x) very ingeniously. Actually, their method work for a general rational superpotential. We demonstrate it in the case of the Seiberg-Witten curves for other classical gauge groups and some other Toda spectral curves.
σ 2 :
2Jac(C) → Jac(C) induced by the involution σ 2 . Consider the Jacobi variety as the set of the linear equivalence classes of divisors D of degree zero. The Prym variety Prym(C/C 2 ), by definition, is the image of id − σ 2 (where id is the identity map), namely, consists of the linear equivalence classes of divisors of the form D − σ 2 (D).
•
Case B: The homology classes [A N ] and [B N ] are "odd" under the action of σ 2 , i.e., σ 2 ([A N ]) = −[A N ] and σ 2 ([B N ]) = −[B N ]. The 4N−2 cycles A j , −σ 2 (A j ), B j , −σ 2 (B j ) (j = 1, · · · , N − 1) and A N , B N altogether form a symplectic basis of cycles of C.
[
A j ] and [B j ] are mapped by the projection p ′ : C → C ′ to an integer multiple of the homology classes [A ′ j ] and [B ′ j ] of a symplectic basis on C ′ . More precisely,
3
Whitham Deformations of SU (N + 1) curve 3.1 Setup and results of Gorsky et al. The Whitham deformations for the SU(N + 1) Seiberg-Witten curve takes the form ∂ ∂a j dS z=const.
P
∞ and Q ∞ are the two points at infinity (z = ∞ and z = 0); P∞ and Q∞ are integrals along a small closed path encircling the indicated point once in the anti-clockwise direction. f n (z) and g n (z) are Laurent polynomials that represent the singular part of dΩ n at P ∞ and Q ∞ : dΩ n = df n (z) + holomorphic (P → P ∞ ), dΩ n = dg n (z) + holomorphic (P → Q ∞ ).(3.18)These Laurent polynomials have constant coefficients for all n.1 To see this, let us note that the fractional power of P (x) in R n (x) can be written
, they are holomorphic differentials on C. Furthermore, like dS SW itself, they are invariant under the involution σ ′ : (x, y) → (−x, −y), and can be identified with holomorphic differentials on the quotient curve C ′ = C/σ ′ with coordinates ξ = x
method of Gorsky et al., we now seek for a series of meromorphic differentials dΩ n of the formdΩ n = R n (x) dz z , R n (x) = polynomial, (4.3)with the same properties. As one can see by careful inspection of the proof in the SU(N + 1) Seiberg-Witten curve, the fractional power construction persists to be meaningful even if the polynomial P (x) is replaced by a rational (or Laurent series) superpotential W (x).
same as those in the standard formulation of the KdV hierarchy. = ∞. Our meromorphic differentials dΩ n are not of this form; they are a linear combination of dΩ KdV n . Accordingly, the "fast" and "slow" time variables are also a linear combination of the standard ones.
SO(2N + 1) and Sp(2N )
deformations of the curve C by the inverse period map a → u.
The
Seiberg-Witten curve does not take the previous form (2.1) if fundamental matters (hypermultiplets in the fundamental representation) exist. One can, however, rewrite the curve in this form if W (x) can be an irrational function. Let us examine this prescription in the case of the SU(N + 1) Yang-Mills theory with fundamental matters. If the theory contains N f (< 2N) fundamental matters, the Seiberg-Witten curve takes the form
-Witten function and u-plane integral Correlation functions of topologically twisted gauge theories on a four-dimensional manifold X can be collected into a generating function. This generating function is called the Donaldson-Witten function. For instance, consider a 2-cycle observable I(S), S ∈ H 2 (X, Z), and a 0-cycle observable O(P ), P ∈ H 0 (X, Z), of the form I(S) = const. S G 2 Tr φ 2 , O(P ) = k c k Tr φ k (P ). (6.1)
CP 2 ,
2CP 1 × CP 1 , del Pezzo surfaces, etc.). Let B denote the exceptional divisor (i.e., inverse image of Q) inX, and consider the following Donaldson-Witten function ofX: Z DW = exp tI(B) + I(S) + O(P ) . (6.5) Note that this path integral is over the fields onX; the pull-back of I(S) and O(P ) tõ X are denoted by the same notations. The new observable I(B) with support on B is inserted with the coupling constant t. The blowup formula then shows that the integrand of the u-plane integral forZ DW is obtained by replacing
in the previous paper (for the SU(N + 1) topological gauge theory) is to interpret this tau function as the counterpart of Mariño and Moore's blowup factor τ γ,δ (t) for the Donaldson-Witten functioñ Z DW = exp n≥1 t n I n (B) + I(S) + O(P ) . (6.12) with many 2-cycles observables I n (B) inserted. The observables I n (B) are of the form I n (B) = const. B G 2 V (n) , (6.13) and the directional vector V (n) is the gradient of the gauge invariant potential V (n) ,
N
, however, this does not imply that its solutions and tau functions can be written in terms of theta functions on the Prym variety. Actually, the flows are first linearized on the Jacobi variety Jac(C) of the affine Toda spectral curve itself, and then shown to be confined to a subspace that is parallel (but not identical) to the Prym variety embedded therein. All that one can expect is, accordingly, an expression in terms of theta functions on Jac(C). (Surprisingly, however, very few is known about such an explicit expression of solutions of the affine Toda systems other than the A
on the u-plane integral of the SU(N +1) topological gauge theory to all other classical gauge groups. A byproduct of the construction of Whitham deformations is to determine which flows of the underlying integrable hierarchy (the Toda hierarchy for the case of SU(N +1) and the KdV hierarchy for the other classical gauge groups) should be extracted; appropriate flows are those generated by the meromorphic differentials dΩ n that arise in the construction of Whitham deformations. This enables us to express the relevant quantities q mn etc. as derivatives of the prepotential F . Let us conclude this paper with discussions on possible extensions and implications of these results.
the text, along with the L-matrices L(z). The L-matrices are realized in a representation of minimal dimensions. The L-matrices for A
µ(z + z −1 ) = W (x). (A.3)upon removing an overall constant or a non-dynamical factor (the factor x the case ofB(1) N ). This curve, however, can be readily converted to the form of Marninec and Warner by recalling z → z/µ. Some comments on the notations in the list are in order. a j and b j (j = 1, · · · , N) are related to the canonical variables q and p (both in the Cartan subalgebra of the classical part of the affine algebra) of the affine Toda system as follows:a j = c j ge α j ·q , b j = p j = h j · p. (A.4)Here α j (j = 1, · · · , N) are the simple roots and α 0 is the affine root. (The null root is ignored.) c j are numerical constants related to the root system, and g the coupling constant. (The a j 's should not be confused with the special coordinates a j 's in the Seiberg-Witten theory.) E jk denotes the matrix with the only non-vanishing elements equal to 1 at (j, k):(E jk ) mn = δ jm δ kn .
are the Seiberg-Witten curves of SU(5) (A 4 ) in the 10-dimensional anti-symmetric representation, E 6 in the 27dimensional minimal representation, and G 2 in the 7-dimensional minimal representation.An obvious difficulty is that W (x) itself is multi-valued, so that the fractional powers of W (x) requires a more careful treatment. This difficulty, however, might be easily overcome, because the same fractional powers are used in the work of Eguchi and Yang[45] on the topological Landau-Ginzburg of the E 6 singularity. This work also predicts an interesting phenomena if the fractional power construction really works for the case of E 6 . Namely, as they observed in the topological Landau-Ginzburg theory, the admissible Whitham deformations will be limited to those associated with the fractional powersW (x) n/12 with
n ≡ 1, 4, 5, 7, 8, 11 mod 12.
(7.2)
for Scientific Research (No. 10640165) from the Ministry of Education, Science and Culture.
Our previous paper[21] contains a wrong comment on this fact.
AcknowledgementsI am grateful to Toshio Nakatsu for useful discussions, and to Yuji Shimizu for bibliographic information on Prym varieties. This work is partly supported by the Grant-in-Aid
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| []
|
[
"A blind method to detrend instrumental systematics in exoplanetary light-curves",
"A blind method to detrend instrumental systematics in exoplanetary light-curves"
]
| [
"G Morello [email protected] \nDepartment of Physics & Astronomy\nUniversity College London\nGower StreetWC1E6BTUK\n"
]
| [
"Department of Physics & Astronomy\nUniversity College London\nGower StreetWC1E6BTUK"
]
| []
| The study of the atmospheres of exoplanets requires a photometric precision, and repeatability, at the level of one part in ∼10 4 . This is beyond the original calibration plans of current observatories, hence the necessity to disentangle some of the instrumental systematics from the astrophysical signals in raw datasets. Most methods used in the literature are parametric, i.e. based on an approximate model of the instrument, and therefore they have many degrees of freedom, which are, most likely, the cause of several controversies in the literature. Non-parametric methods have been proposed to guarantee an higher degree of objectivity(Carter & Winn 2009;Thatte et al. 2010;Gibson et al. 2012;Waldmann 2012;Waldmann et al. 2013;Waldmann 2014). Recently, Morello et al. (2014, 2015have developed a non-parametric detrending method that gave coherent and repeatable results when applied to Spitzer/IRAC datasets that were debated in the literature. Said method is based on Independent Component Analysis (ICA) applied to individual pixel time-series, hereafter "pixel-ICA". The main purpose of this paper is to investigate the limits and advantages of pixel-ICA on a series of simulated datasets. We focus in particular on two mechanisms that cause systematics similar to the Spitzer/IRAC ones, then we generate several datasets to analyze, with different time scales, non-stationarity, sudden change points, etc. The performances of pixel-ICA detrending method are compared against the ones of a traditional polynomial centroid division (PCD) method. | 10.1088/0004-637x/808/1/56 | [
"https://arxiv.org/pdf/1503.05309v2.pdf"
]
| 117,276,619 | 1503.05309 | 374838505ce70901e4a1ed95464b02237f84599a |
A blind method to detrend instrumental systematics in exoplanetary light-curves
18 Mar 2015
G Morello [email protected]
Department of Physics & Astronomy
University College London
Gower StreetWC1E6BTUK
A blind method to detrend instrumental systematics in exoplanetary light-curves
18 Mar 2015Subject headings: methods: data analysis -techniques: photometric -planets and satellites: atmospheres -planets and satellites: individual(GJ436b)
The study of the atmospheres of exoplanets requires a photometric precision, and repeatability, at the level of one part in ∼10 4 . This is beyond the original calibration plans of current observatories, hence the necessity to disentangle some of the instrumental systematics from the astrophysical signals in raw datasets. Most methods used in the literature are parametric, i.e. based on an approximate model of the instrument, and therefore they have many degrees of freedom, which are, most likely, the cause of several controversies in the literature. Non-parametric methods have been proposed to guarantee an higher degree of objectivity(Carter & Winn 2009;Thatte et al. 2010;Gibson et al. 2012;Waldmann 2012;Waldmann et al. 2013;Waldmann 2014). Recently, Morello et al. (2014, 2015have developed a non-parametric detrending method that gave coherent and repeatable results when applied to Spitzer/IRAC datasets that were debated in the literature. Said method is based on Independent Component Analysis (ICA) applied to individual pixel time-series, hereafter "pixel-ICA". The main purpose of this paper is to investigate the limits and advantages of pixel-ICA on a series of simulated datasets. We focus in particular on two mechanisms that cause systematics similar to the Spitzer/IRAC ones, then we generate several datasets to analyze, with different time scales, non-stationarity, sudden change points, etc. The performances of pixel-ICA detrending method are compared against the ones of a traditional polynomial centroid division (PCD) method.
Introduction
The field of extrasolar planetary transits is one of the most productive and innovative subject in Astrophysics in the last decade. Transit observations can be used to measure the size of planets, their orbital parameters (Seager & Mallén-Ornelas 2003), stellar properties (Mandel & Agol 2002;Howarth 2011), to study the atmospheres of planets (Brown 2001;Charbonneau et al. 2002;Tinetti et al. 2007), to detect small planets (Miralda-Escudé 2002;Agol et al. 2005), and exomoons (Kipping 2009,b). In particular, the study of planetary atmospheres requires a high level of photometric precision, i.e. one part in ∼10 4 in stellar flux (Brown 2001), which is comparable to the effects of current instrumental systematics and stellar activity (Berta et al. 2011;Ballerini et al. 2012), hence the necessity of testable methods for data detrending. In some cases, different assumptions, e.g. using different instrumental information or functional forms to describe them, leaded to controversial results even from the same datasets; examples in the literature are Tinetti et al. (2007); Ehrenreich et al. (2007); Beaulieu et al. (2008); Désert et al. (2009Désert et al. ( , 2011 for the hot-Jupiter HD189733b, and Stevenson et al. (2010); Beaulieu et al. (2011); Knutson et al. (2011Knutson et al. ( , 2014 for the warm-Neptune GJ436b. Some of these controversies are based on Spitzer/IRAC datasets at 3.6 and 4.5 µm. The main systematic effect for these two channels is an almost regular undulation with pe-riod ∼3000 s, so called pixel-phase effect, as it is correlated with the relative position of the source centroid with respect to a pixel center (Fazio et al. 2004;Morales-Caldéron et al. 2006). Traditional parametric techniques correct for this effect dividing the measured flux by a polynomial function of the coordinates of the photometric centroid; some variants may include time-dependence (e.g. Stevenson et al. (2010); Beaulieu et al. (2011)), and/or spline interpolation to correct for high resolution spatial effects (Stevenson et al. 2012). The results obtained with these methods are strongly dependent on a few assumptions, e.g. the degree of the polynomial adopted, the photometric technique, the centroid determination, etc. Also, the very same method, applied to different observations of the same system, often leads to significantly different results. Morello et al. (2014Morello et al. ( , 2015 reanalyzed the 3.6 and 4.5 µm Spitzer/IRAC primary transits of HD189733b and GJ436b obtained during the cryogenic regime, so called "cold Spitzer" era, adopting a blind, non-parametric detrending technique, based on an Independent Component Analysis (ICA) of individual pixel time series, in this paper called "pixel-ICA". The results obtained with this method are repeatable over different epochs, and a photometric precision of one part in ∼10 4 in stellar flux is achieved. In the previous literature, the use of ICA to decorrelate the transit signals from astrophysical and instrumental noise, in spectrophotometric observations, has been proposed by Waldmann (2012); Waldmann et al. (2013); Waldmann (2014). The reason to prefer such non-parametric detrending methods is twofold: they require very little, if any, prior knowledge of the instrument systematics and astrophysical signals, therefore they also ensure a higher degree of objectivity compared to parametric methods. As an added value, they give stable results over several datasets, also in those cases where traditional parametric methods have been unsuccessful.
To understand better the advantages and limits of the pixel-ICA detrending algorithm, we tested it on simulated datasets, for which instrumental systematic effects are fully under control. In particular, in this paper, we:
1. elaborate some toy-models that can reproduce systematic effects similar to Spitzer/IRAC ones 1 ;
2. test the pixel-ICA method on simulated datasets;
3. explore the limits of its applicability;
4. compare its performances with the most common parametric method, based on division by a polynomial function of the centroid (PCD).
Instrument simulations
Instrument jitter only
We consider an ideal transit light-curve with parameters reported in Tab. 1, sampled at 8.4 s over 4 2 3 hr, totaling 2001 data points, symmetric with respect to the transit minimum. Fig. 1 shows the ideal light-curve. To each data point we associate a number of photons proportional to the expected flux, in particular 50,000 photons in the out-of-transit. We generate random gaussian coordinates for each photon, representing their positions on the plane of the CCD. Finally, we add a grid on this plane: each square of the grid represents a pixel, and the number of photons into a square at a time is the read of an individual pixel in absence of pixel systematics. To simulate the effect of instrumental jitter, we move the gridlines from one data point to another (it is equivalent to shift the coordinates of the photons). Fig. 2 shows the jitter effect at different levels, i.e. individual pixels, and small and large clusters. Pixel flux variations are related to the changing position of the PSF: the flux is higher when the centre of the PSF is closer to the centre of the pixel. The same is valid for the flux integrated over a small aperture, compared to the width of the PSF. These variations are not observed with a large aperture that includes all the photons at any time.
The effect of pixel systematics
Spitzer/IRAC datasets for channels 1 and 2 show temporal flux variations correlated to the Table 1: Transit parameter values adopted in all simulations: p = r p /R s is the ratio of planetary to stellar radii, a 0 = a/R s is the orbital semimajor axis in units of stellar radius, i is the orbital inclination, e is the eccentricity, P is the orbital period, γ 1 and γ 2 are quadratic limb darkening coefficients (Howarth 2011 centroid position, independently from the aperture selected (Fazio et al. 2004;Morales-Caldéron et al. 2006). Here we consider two effects that, combined to the instrument jitter, can produce this phenomenon:
1. inter-pixel quantum efficiency variations, simulated multiplying the photons in a pixel by a coefficient to get the read, being the coefficients not identical for all the pixels;
2. intra-pixel sensitivity variations, simulated by assigning individual coefficients dependent on the position of the photon into the pixel.
Description of simulations
We performed simulations for two values of (gaussian) PSF widths, σ P SF :
• σ P SF =1 p.u. (pixel side units);
• σ P SF =0.2 p.u.
The two sets of frames differ only in the scaling factor in photon coordinates, therefore no relative differences are attributable to random generation processes. We simulated several jitter time series as detailed in Tab. 2 and Fig. 3. For each case, we adopted:
1. a random generated quantum efficiency map with standard deviations of ∼10 −2 , to simulate the inter-pixel variations;
2. a non-uniform response function for each pixel, i.e. 1−0.1d, where d is the distance from the centre of the pixel, to simulate the intra-pixel variations.
Finally, we add white noise time series at an arbitrary level of 5 photon counts/pixel/data point for most cases. In a select case, we investigated the effect of different white noise levels. It is worth to note that the same quantum efficiency maps and noise time series (sometimes multiplied by a scaling factor) have been adopted for all the simulations with pixel arrays of the same size, to minimize possible aleatory effects when comparing the results.
Results
After having generated the simulated raw datasets, we applied both pixel-ICA and PCD detrending techniques to evaluate their reliability and robustness in those contexts. The main results are reported in this Section, we show in particular:
• the simulated raw light-curves and the corresponding detrended ones;
• the root mean square (rms) from the theoretical transit light-curve, before and after the detrending processes;
• the residual systematics in the detrended light-curves;
• the planetary, orbital, and stellar parameters estimated by fitting the light-curves;
• for a subsample of cases, the results of full parameter retrieval, including error bars.
3.1. Case I: inter-pixel effects, large PSF Fig. 4 shows the raw light-curves simulated with σ P SF =1 p.u., and inter-pixel quantum efficiency variations over 9×9 array of pixels, and the correspondent detrended light-curves obtained with pixel-ICA and PCD 2 methods. This array is large enough that the observed modulations are only due to the pixel effects. Tab. 3 reports the discrepancies between the detrended light-curves and the theoretical model. It is worth to note that in all cases, pixel-ICA method reduces the dicrepancies by a factor ∼3, while the parametric method reduces them by a factor ∼2 (for the selected binning). Fig. 5 shows how they scale for binning over n points, with 1 ≤ n ≤ 10. This behaviour suggests a high level of temporal structure in raw data, which is not present in ICAdetrended light-curves. Some systematics are still detected in residuals obtained with the parametric method. Fig. 6 shows the transit parameters retrieved from the detrended light-curves; in a few representative cases, we calculated the error bars as detailed in Morello et al. (2015), and App. A in this paper. Numerical results are reported in Tab. 7. (blue) raw light-curves simulated with σ P SF = 1p.u., and inter-pixel quantum efficiency variations over 9×9 array of pixels. Right panels: detrended transit light-curves obtained with (green 'x') polynomial centroid fitting method, and (red dots) pixel-ICA method. All the light-curves are binned over 10 points, except those in the bottom right, to make clearer visualization of the systematic effects. Table 3: Root mean square of residuals between the light-curves and the theoretical model for simulations with σ P SF = 1 p.u., and inter-pixel quantum efficiency variations over 9×9 array of pixels; in particular they are calculated for the raw light-curves, light-curves detrended with pixel-ICA, and PCD method, binned over 10 points.
Jitter -Root mean square of residuals for light-curves binned over 1 to 10 points, scaled to their non-binned values. The simulations were obtained with σ P SF =1 p.u., 9×9 array, and inter-pixel effects. The dashed black line indicates the expected trend for white residuals, blue dots are for normalized raw light-curves, red ' * ' are for pixel-ICA detrendend light-curves, and green 'x' for PCD detrended light-curves. best estimates of the planet-to-star radii ratio, p = r p /R s , for detrended light-curves with (red dots) pixel-ICA, and (green x) PCD method (σ P SF =1 p.u., inter-pixel effects over 9×9 array). Error bars are reported for a few representative cases of jitter signal, i.e. sin1, cos1 (chosen as examples of periodic functions with different phasing), saw1v3 (example with non-stationary amplitude), saw1vf2 (nonstationary frequency), and jump04c (sudden change). Middle panel: the same for the orbital semimajor axis in units of the stellar radius, a 0 = a/R s . Bottom panel: the same for the orbital inclination, i.
rms (raw − model) rms (ICA − model) rms (PCD − model) sin1 6.5×10 −4 2.0×10 −4 3.0×10 −4 cos1 6.9×10 −4 2.1×10 −4 3.2×10 −4 sin2 6.7×10 −4 2.0×10 −4 3.0×10 −4 cos2 6.5×10 −4 2.1×10 −4 3.1×10 −4 sin3 6.6×10 −4 2.0×10 −4 3.0×10 −4 cos3 6.6×10 −4 2.0×10 −4 3.1×10 −4 saw1 5.4×10 −4 2.1×10 −4 3.1×10 −4 saw1v1 6.0×10 −4 2.1×10 −4 3.0×10 −4 saw1v2 5.6×10 −4 2.1×10 −4 3.1×10 −4 saw1v3 5.9×10 −4 2.1×10 −4 3.1×10 −4 saw1vf1 5.5×10 −4 2.1×10 −4 3.0×10 −4 saw1vf2 5.3×10 −4 2.0×10 −4 3.0×10 −4 jump04c 6.9×10 −4 1.9×10 −4 3.0×10 −
For the same configuration, i.e. σ P SF =1 p.u., inter-pixel effects, we investigated the consequences of considering a smaller array (5×5), which does not include the whole PSF. Fig. 7 shows the raw light-curves, and the correspondent detrended ones, obtained with the two methods. Tab. 4 reports the discrepancies between those light-curves and the theoretical model. The discrepancies are higher than for the larger pixelarray by a factor 2 (for the raw light-curves), because of the additional effect. After pixel-ICA detrending, the discrepancies are reduced by a factor ∼5 (for the selected binning) in most cases, and ∼13 for the 'jump04c', while the performances of the parametric method are case dependent, and discrepancies are reduced by a factor between 2 and 7 in all cases, and also ∼13 for 'jump04c'. Fig. 8 shows how the residuals scale for binning over n points, with 1 ≤ n ≤ 10. The temporal structure due to jitter effect is dominant in raw data, but little traces of this behaviour (if any) are present after pixel-ICA detrending. Even for this aspect, the performances of the parametric method are case dependent. Fig. 9 shows the transit parameters retrieved from detrended light-curves; in a few representative cases, we calculated the error bars. Numerical results are reported in Tab. 8. In conclusion, the choice of a non-optimal pixel array introduces additional systematics, that worsen the parameter retrieval, but it is quite remarkable that the pixel-ICA technique gives consistent results in most cases, whereas the parametric technique appears to be less robust. (blue) raw light-curves simulated with σ P SF = 1p.u., and inter-pixel quantum efficiency variations over 5×5 array of pixels. Right panels: detrended transit light-curves obtained with (green 'x') polynomial centroid fitting method, and (red dots) pixel-ICA method. All the light-curves are binned over 10 points to make clearer visualization of the systematic effects. Table 4: Root mean square of residuals between the light-curves and the theoretical model for simulations with σ P SF = 1 p.u., and inter-pixel quantum efficiency variations over 5×5 array of pixels; in particular they are calculated for the raw light-curves, light-curves detrended with pixel-ICA, and PCD method, binned over 10 points.
Jitter best estimates of the planet-to-star radii ratio, p = r p /R s , for detrended light-curves with (red dots) pixel-ICA, and (green 'x') PCD method (σ P SF =1 p.u., inter-pixel effects over 5×5 array). Error bars are reported for representative cases of jitter signal, i.e. sin1, cos1, saw1v3, saw1vf2, and jump04c. Middle panel: the same for the orbital semimajor axis in units of the stellar radius, a 0 = a/R s . Bottom panel: the same for the orbital inclination, i. Fig. 10 shows the raw light-curves simulated with σ P SF =0.2 p.u., and inter-pixel quantum efficiency variations over 5×5 array of pixels, and the correspondent detrended light-curves, obtained with the two methods considered in this paper. The array is large enough that observed modulations are due only to the pixel effects. Tab. 5 reports the discrepancies between those lightcurves and the theoretical model. The pixel-ICA technique reduces the dicrepancies by a factor 3-10 (for the selected binning), depending on the original values, overperforming the parametric method by a factor 2-3, except for the case 'jump04c'. Fig. 11 shows how the residuals scale for binning over n points, with 1 ≤ n ≤ 10. Again, a significant temporal structure is present in the raw data, but not in the pixel-ICA detrended light-curves, while the performances of the parametric method are case dependent. Fig. 12 shows the transit parameters retrieved from detrended light-curves; in representative cases, we calculated the error bars. Numerical results are reported in Tab. 9. (blue) raw light-curves simulated with σ P SF =0.2 p.u., and inter-pixel quantum efficiency variations over 5×5 array of pixels. Right panels: detrended transit light-curves obtained with (green 'x') polynomial centroid fitting method, and (red dots) pixel-ICA method. All the light-curves are binned over 10 points to make clearer visualization of the systematic effects. Table 5: Root mean square of residuals between the light-curves and the theoretical model for simulations with σ P SF = 0.2 p.u., and inter-pixel quantum efficiency variations over 5×5 array of pixels; in particular they are calculated for the raw light-curves, light-curves detrended with pixel-ICA, and PCD method, binned over 10 points.
rms (raw − model) rms (ICA − model) rms (PCD − model) sin1 1.5×10 −3 3.1×10 −4 2.6×10 −4 cos1 1.4×10 −3 3.4×10 −4 8.1×10 −4 sin2 1.5×10 −3 3.2×10 −4 2.8×10 −4 cos2 1.5×10 −3 3.2×10 −4 6.2×10 −4 sin3 1.5×10 −3 3.1×10 −4 2.7×10 −4 cos3 1.4×10 −3 3.2×10 −4 4.9×10 −4 saw1 1.7×10 −3 3.1×10 −4 3.5×10 −4 saw1v1 1.7×10 −3 2.9×10 −4 2.5×10 −4 saw1v2 1.6×10 −3 3.6×10 −4 3.7×10 −4 saw1v3 1.7×10 −3 3.1×10 −4 4.0×10 −4 saw1vf1 1.7×10 −3 3.1×10 −4 2.9×10 −4 saw1vf2 1.6×10 −3 3.2×10 −4 2.6×10 −4 jump04c 3.3×10 −3 2.6×10 −4 2.6×10 −
Case II: inter-pixel effects, narrow PSF
Jitter the planet-to-star radii ratio, p = r p /R s , for detrended light-curves with (red dots) pixel-ICA, and (green 'x') PCD method (σ P SF =0.2 p.u., inter-pixel effects over 5×5 array). Error bars are reported for representative cases of jitter signal, i.e. sin1, cos1, saw1v3, saw1vf2, and jump04c. Middle panel: the same for the orbital semimajor axis in units of the stellar radius, a 0 = a/R s . Bottom panel: the same for the orbital inclination, i.
Case III: intra-pixel effects
For simulations with σ P SF =1 p.u., the effect of intra-pixel sensitivity variations is negligible, i.e. ∼10 −5 , unless we consider unphysical or very unlikely cases, where the quantum efficiency can assume both positive and negative values in a pixel, or it is zero for a significant fraction of the area of the pixel (in this case the systematics would be caused by loss of photons, similar to the case with small aperture and no pixel systematics). Intrapixel effects become significant when the PSF is narrower, therefore we analyzed only the relevant simulations with σ P SF =0.2 p.u. Fig. 13 shows the raw light-curves simulated with σ P SF =0.2 p.u., and intra-pixel quantum efficiency variations over 5×5 array of pixels, and the correspondent detrended light-curves. The array is large enough that the observed modulations are only due to the pixel effects. Tab. 6 reports the discrepancies between those light-curves and the theoretical model. The pixel-ICA technique reduces the dicrepancies by a factor 4-8 (for the selected binning) for the first 12 jitter series, and by a factor 83 for the case 'jump04c', overperforming the parametric method by a factor 2-4. Fig. 14 shows how the residuals scale for binning over n points, with 1 ≤ n ≤ 10. In this case, the temporal structure is preserved in all detrended light-curves, except for the case 'jump04c', which means that both methods have some troubles to decorrelate intra-pixel effects. Fig. 15 shows the transit parameters retrieved from detrended light-curves; in representative cases, we calculated the error bars. Detailed numerical results are reported in Tab. 10. While in some cases the parametric method may perform better than pixel-ICA, if adopting higher order polynomials, in some other cases higher order polynomials lead to worse results than lower order polynomials. The pixel-ICA method is less case dependent. It is also quite remarkable that, though the systematics are not well decorrelated, the parameter retrieval gives the correct results within the error bars. Fig. 13.-Left panels: (blue) raw light-curves simulated with σ P SF =0.2 p.u., and intra-pixel quantum efficiency variations over 5×5 array of pixels. Right panels: detrended transit light-curves obtained with (green 'x') polynomial centroid fitting method, and (red dots) pixel-ICA method. All the light-curves are binned over 10 points to make clearer visualization of the systematic effects. Table 6: Root mean square of residuals between the light-curves and the theoretical model for simulations with σ P SF = 0.2 p.u., and intra-pixel quantum efficiency variations over 5×5 array of pixels; in particular they are calculated for the raw light-curves, light-curves detrended with pixel-ICA, and PCD method, binned over 10 points.
Jitter -Root mean square of residuals for light-curves binned over 1 to 10 points, scaled to their nonbinned values. The simulations were obtained with σ P SF =0.2 p.u., 5×5 array, and intra-pixel effects. The dashed black line indicates the expected trend for white residuals, blue dots are for normalized raw light-curves, red ' * ' are for pixel-ICA detrendend light-curves, and green 'x' for PCD detrended light-curves. best estimates of the planet-to-star radii ratio, p = r p /R s , for detrended light-curves with (red dots) pixel-ICA, and (green 'x') PCD method (σ P SF =0.2 p.u., intra-pixel effects over 5×5 array). Error bars are reported for representative cases of jitter signal, i.e. sin1, cos1, saw1v3, saw1vf2, and jump04c. Middle panel: the same for the orbital semimajor axis in units of the stellar radius, a 0 = a/R s . Bottom panel: the same for the orbital inclination, i.
rms (raw − model) rms (ICA − model) rms (PCD − model) sin1 3.5×10 −3 5.1×10 −4 1.2×10 −3 cos1 3.5×10 −3 5.1×10 −4 1.8×10 −3 sin2 3.4×10 −3 4.9×10 −4 1.2×10 −3 cos2 3.5×10 −3 5.0×10 −4 1.5×10 −3 sin3 3.4×10 −3 5.0×10 −4 1.2×10 −3 cos3 3.4×10 −3 5.0×10 −4 1.4×10 −3 saw1 3.4×10 −3 7.4×10 −4 1.8×10 −3 saw1v1 3.8×10 −3 7.3×10 −4 1.7×10 −3 saw1v2 3.6×10 −3 6.8×10 −4 1.7×10 −3 saw1v3 3.7×10 −3 7.3×10 −4 1.7×10 −3 saw1vf1 3.2×10 −3 6.6×10 −4 1.7×10 −3 saw1vf2 3.0×10 −3 6.1×10 −4 1.5×10 −3 jump04c 6.8×10 −3 8.2×10 −5 1.6×10 −
Conclusions
We have tested the pixel-ICA algorithm, i.e. a non-parametric method proposed by Morello et al. (2014Morello et al. ( , 2015 to detrend Spitzer/IRAC primary transit observations, on simulated datasets. Systematics similar to the ones present in Spitzer/IRAC datasets are obtained by combining instrumental jitter with inter-or intra-pixel sensitivity variations. A variety of jitter time series is used to test the pixel-ICA method with:
1. periodic signals with different frequencies, phasing, and shape;
2. non-stationary signals with varying amplitudes or frequencies;
3. sudden change point.
The detrending performances of pixel-ICA method have been compared with the more common division by a polynomial function of the centroid, in this paper PCD method. Here we summarize the main results found:
1. Pixel-ICA algorithm can detrend non-stationary signals and sudden changes, as well as periodic signals with different frequencies and phasing, relative to the transit.
2. Inter-pixel effects are well-detrended with pixel-ICA method.
3. Even if the instrument PSF is not entirely within the array of pixels, pixel-ICA leads to quite robust results.
4. In most cases, pixel-ICA overperforms PCD method, especially if the instrument PSF is narrow, or it is not entirely within the photometric aperture.
5. Intra-pixel effects are only detectable if the PSF is relatively small.
6. Intra-pixel effects cannot be totally detrended by any of the two methods, but pixel-ICA, in most cases, overperforms PCD method, which is largely case-dependent. Also, pixel-ICA method provides consistent results within the error bars.
These facts support the reliability of the results obtained with pixel-ICA method in the literature, and explain the higher inter-epoch stability compared to previous ones, that were obtained with parametric detrending techniques.
The author would like to thank Prof. G. Tinetti and Dr. I. P. Waldmann for useful comments. G. Morello is funded by UCL Perren/Impact scholarship (CJ4M/CJ0T). This work was partially supported by Research Councils UK.
A. Brief outline of pixel-ICA algorithm
ICA is a statistical technique that transforms a set of signals into an equivalent set of maximally independent components. It is widely used in a lot of different contexts, e.g. Neuroscience, Econometrics, Photography, and Astrophysics, to separate the source signals present in a set of observations/recordings (Hyvärinen et al. 2001). The underlying assumption is that real signals are linear mixtures of independent source signals. The validity of this assumption for astrophysical observations has been discussed with more details in Morello et al. (2015) (App. A). The major strenghts of this approach are:
1. it requires the minimal amount of prior information; 2. even if the assumption is not valid, the method is able to detrend in part the source signals.
If additional information is available, some variants of ICA can be used to obtain better results (Igual et al. 2002;Stone et al. 2002;Barriga et al. 2011), but they are not considered in this paper.
Pixel-ICA method uses individual pixel time series from an array to decorrelate the transit signal in photometric observations of stars with a transiting planet. The main steps are:
1. ICA transformation of the time series to get the independent components; 2. identification, by eye, of the transit component;
3. fitting of the non-transit components plus a constant on the out-of-transit of the integral light-curve (sum over the pixel array); 4. subtraction of the non-transit components, with coefficients determined by the fitting, from the integral light-curve;
5. normalization of the detrended light-curve.
The normalized, detrended light-curve is model-fitted with Mandel & Agol (2002) formulas, which depend on several stellar and orbital parameters. We typically perform a Nelder-Mead optimisation algorithm (Lagarias et al. 1998) to obtain first estimates of the best parameters of the model, then we generate Monte Carlo chains of 20,000 elements to sample the posterior distributions (approximately gaussians) of the parameters. The updated best parameters are the mean values of the chains, and the zero-order error bars, σ par,0 , are their standard deviations.The zero-order error bars only accounts for the scatter in the detrended light-curve; they must be increased by a factor that includes the uncertainties due to the detrending process, in formulas:
σ par = σ par,0 σ 2 0 + σ 2 ICA σ 2 0 (A1)
where σ par is the final parameter error bar, σ 0 is the square root of the likelihood's variance (approximately equal to the standard deviation of residuals), and σ ICA is a term associated to the detrending process. Morello et al. (2014Morello et al. ( , 2015 suggest the following formula for σ ICA :
σ 2 ICA = f 2 j o 2 j ISR j + σ 2 ntc−f it (A2)
where ISR is the so-called Interference-to-Signal-Ratio matrix, o j are the coefficients of the non-transitcomponents, m is their number, σ ntc−f it is the standard deviation of residuals from the theoretical raw light-curve, out of the transit, f is the normalising factor for the detrended light-curve. The sum on the left takes into account the precision of the components extracted by the algorithm; σ ntc−f it indicates how well the linear combination of components approximates the out-of-transit. The MULTICOMBI code, i.e. the algorithm that we use for the ICA transformation, provides two Interference-to-Signal-Ratio matrices, ISR EF and ISR W A , associated to the sub-algorithms EFICA and WASOBI, respectively. Two approaches has been suggested to derive a single Interference-to-Signal-Ratio matrix:
ISR = ISR EF + ISR W A 2 (A3) ISR i,j = min ISR EF i,j , ISR W A i,j(A4)
Eq. A3 is a worst-case estimate, while Eq. A4 takes into account the outperforming separation capabilities of MULTICOMBI compared to EFICA and WASOBI. We adopt Eq. A4 throughout this paper, but results obtained with both options are reported in Tab. 7,8,9,and 10. In most cases the differences are negligible.
B. The effect of white noise on pixel-ICA detrending
Simulations analyzed in the main part of this paper contain the same amount of white noise, at a level of 5 photons/pixel/frame. In this Section, we discuss the effect of different white noise levels. In particular, we focus on the case with σ P SF =0.2, jitter 'sin1', and intra-pixel effects over a 5×5 array. The choice of a test case with intra-pixel effects is due to the greater challenge associated with detrending these effects. The same noisy time series are injected to each pixel in all cases, but with different scaling factors. Fig. 16 shows the raw and detrended light-curves obtained with white noise 10 and 100 times higher than in the first simulation. Fig. 17 shows how the residuals scale for binning over n points, with 1 ≤ n ≤ 10, in these two cases. As expected, binning properties depend on the amplitude of white noise relative to systematic signals; therefore, for particular configurations it may appear that systematics are removed in the detrending process. Fig. 18 shows the transit parameters retrieved from detrended light-curves; in two representative cases, we calculated the error bars. Note that the retrieved parameters deviates from the original values as linear functions of the white noise amplitude, which proves the detrending method is robust in presence of white noise. The last point, i.e. the one with the highest level of white noise, breaks this trend, because the gaussian posterior of the inclination is distorted by the limit of 90 • , and correlations affect the other parameter posterior distributions. In general, retrieved parameters are consistent with the original values within 1 σ. Fig. 16.-Left panels: (blue) raw light-curves simulated with σ P SF =0.2 p.u., intra-pixel quantum efficiency variations over 5×5 array of pixels, jitter 'sin1', and white noise at 50 and 500 photon counts/pixel/frame. Right panels: (red) detrended transit light-curves obtained with pixel-ICA method. All the light-curves are binned over 10 points, as in previous figures. Note the different vertical scales adopted. best estimates of the planet-to-star radii ratio, p = r p /R s , for detrended light-curves (σ P SF =0.2 p.u., intra-pixel effects over 5×5 array, jitter 'sin1'). Error bars are reported for representative cases. Middle panel: the same for the orbital semimajor axis in units of the stellar radius, a 0 = a/R s . Bottom panel: the same for the orbital inclination, i.
In Sec. 3.1, and 4 we state that inter-pixel effects are well detrended with pixel-ICA method, based on the binning properties of residuals (and consistent results). Given that binning properties can only prove that systematics are negligible compared to the actual white noise level, we performed a last test for a simulation with inter-pixel effects (σ P SF =1, 9×9 array, jitter sin1) and a reduced white noise level, by a factor of 10. Note that it is an extreme low value of 0.5 photon counts/pixel/data point, which is currently impossible to have in real datasets. Fig. 19 shows the raw and detrended light-curves for this simulation, and the binning properties of their residuals. Time structure is very high for the raw light-curve, but it is again well detrended by pixel-ICA. We also checked that all the retrieved parameters are consistent with the original values within 1 σ.
C. Tables
Fig. 1 .
1-Referent transit light-curve adopted in simulations.
Fig. 3 .
3-Top panel: jitter time series saw1 (black), saw1v1 (ecru, cross markers), saw1v2 (green, triangles), and saw1v3 (cyan, empty circles); markers are represented every 20 data points for reasons of visibility. Bottom panel: jitter time series saw1vf1 (orange), and saw1vf2 (grey). The other jitter time series are not reported, since their representations are obvious (seeTab.2).
Fig. 4.-Left panels: (blue) raw light-curves simulated with σ P SF = 1p.u., and inter-pixel quantum efficiency variations over 9×9 array of pixels. Right panels: detrended transit light-curves obtained with (green 'x') polynomial centroid fitting method, and (red dots) pixel-ICA method. All the light-curves are binned over 10 points, except those in the bottom right, to make clearer visualization of the systematic effects.
Fig. 5 .
5Fig. 5.-Root mean square of residuals for light-curves binned over 1 to 10 points, scaled to their non-binned values. The simulations were obtained with σ P SF =1 p.u., 9×9 array, and inter-pixel effects. The dashed black line indicates the expected trend for white residuals, blue dots are for normalized raw light-curves, red ' * ' are for pixel-ICA detrendend light-curves, and green 'x' for PCD detrended light-curves.
Fig. 6 .
6-Top panel:
Fig. 7.-Left panels: (blue) raw light-curves simulated with σ P SF = 1p.u., and inter-pixel quantum efficiency variations over 5×5 array of pixels. Right panels: detrended transit light-curves obtained with (green 'x') polynomial centroid fitting method, and (red dots) pixel-ICA method. All the light-curves are binned over 10 points to make clearer visualization of the systematic effects.
Fig. 8 .
8-Root mean square of residuals for light-curves binned over 1 to 10 points, scaled to their non-binned values. The simulations were obtained with σ P SF =1 p.u., 5×5 array, and inter-pixel effects. The dashed black line indicates the expected trend for white residuals, blue dots are for normalized raw light-curves, red ' * ' are for pixel-ICA detrendend light-curves, and green 'x' for PCD detrended light-curves.
Fig. 9 .
9-Top panel:
Fig. 10.-Left panels: (blue) raw light-curves simulated with σ P SF =0.2 p.u., and inter-pixel quantum efficiency variations over 5×5 array of pixels. Right panels: detrended transit light-curves obtained with (green 'x') polynomial centroid fitting method, and (red dots) pixel-ICA method. All the light-curves are binned over 10 points to make clearer visualization of the systematic effects.
Fig. 11 .
11-Root mean square of residuals for light-curves binned over 1 to 10 points, scaled to their nonbinned values. The simulations were obtained with σ P SF =0.2 p.u., 5×5 array, and inter-pixel effects. The dashed black line indicates the expected trend for white residuals, blue dots are for normalized raw light-curves, red ' * ' are for pixel-ICA detrendend light-curves, and green 'x' for PCD detrended light-curves.
Fig. 12 .
12-Top panel: best estimates of
Fig. 14 .
14Fig. 14.-Root mean square of residuals for light-curves binned over 1 to 10 points, scaled to their nonbinned values. The simulations were obtained with σ P SF =0.2 p.u., 5×5 array, and intra-pixel effects. The dashed black line indicates the expected trend for white residuals, blue dots are for normalized raw light-curves, red ' * ' are for pixel-ICA detrendend light-curves, and green 'x' for PCD detrended light-curves.
Fig. 15 .
15-Top panel:
Fig. 18.-Top panel: best estimates of the planet-to-star radii ratio, p = r p /R s , for detrended light-curves (σ P SF =0.2 p.u., intra-pixel effects over 5×5 array, jitter 'sin1'). Error bars are reported for representative cases. Middle panel: the same for the orbital semimajor axis in units of the stellar radius, a 0 = a/R s . Bottom panel: the same for the orbital inclination, i.
Table 2 :
2List of jitter time series adopted in simulations and their properties.Abbr.
Shape
Peak-to-peak amplitude (p.u.)
Period (s)
sin1
sinusoidal
0.6
4014.6
cos1
cosinusoidal
0.6
4014.6
sin2
sinusoidal
0.6
3011.0
cos2
cosinusoidal
0.6
3011.0
sin3
sinusoidal
0.6
2007.3
cos3
cosinusoidal
0.6
2007.3
saw1
∼sawtooth
0.6
2990.4
saw1v1
∼sawtooth
variable
2990.4
saw1v2
∼sawtooth
variable
2990.4
saw1v3
∼sawtooth
decreasing
2990.4
saw1vf1
∼sawtooth
0.6
variable
saw1vf2
∼sawtooth
0.6
decreasing
jump04c Heaviside step
0.4
mid-transit discontinuity
Raw 5x5, sin1, intra, PSF = 0.2, noise = 50 Raw 5x5, sin1, intra, PSF = 0.2, noise = 500−0.05
0
0.05
0.97
0.98
0.99
1
1.01
Φ
Normalized flux
−0.05
0
0.05
0.97
0.98
0.99
1
1.01
Detrended
Φ
−0.05
0
0.05
0.94
0.96
0.98
1
1.02
1.04
Φ
Normalized flux
−0.05
0
0.05
0.94
0.96
0.98
1
1.02
1.04
Detrended
Φ
Fig. 17.-Left panel: Root mean square of residuals for binning over 1 to 10 points, scaled to their nonbinned values, obtained for simulations with σ P SF =0.2 p.u., intra-pixel effects over a 5×5 array, jitter 'sin1', and white noise at 50 photon counts/pixel/frame. The dashed black line indicates the expected trend for white residuals, blue dots are for normalized raw light-curves, and red ' * ' are for pixel-ICA detrendend light-curves. Right panel: the same, but with white noise at 500 photon counts/pixel/frame.1
2
3
4
5
6
7 8 9 10
5x5 array , sin1, intra, PSF = 0.2, noise = 50
rms residuals
binning
pure white noise
raw − model
ICA − model
1
2
3
4
5
6
7 8 9 10
5x5 array, sin1, intra, PSF = 0.2, noise = 500
rms residuals
binning
pure white noise
raw − model
ICA − model
Raw 9x9, sin1, inter, PSF = 1, noise = 0.5Fig. 19.-Left panels: (blue) raw light-curves simulated with σ P SF =1 p.u., inter-pixel sensitivity variations over 9×9 array of pixels, jitter 'sin1', and white noise at 0.5 photon counts/pixel/frame; (red) correspondent detrended light-curve with pixel-ICA. Right panel: Root mean square of residuals for binning over 1 to 10 points, scaled to their non-binned values.−0.05
0
0.05
0.97
0.98
0.99
1
1.01
Φ
Normalized flux
−0.05
0
0.05
0.97
0.98
0.99
1
1.01
Detrended
Φ
1
2
3
4
5
6
7 8 9 10
9x9 array, sin1, inter, PSF = 1, noise = 0.5
rms residuals
binning
pure white noise
raw − model
ICA − model
Table 7 :
7Retrieved transit parameters for simulations with σ P SF =1, 9×9 array, inter-pixel effects (see Sec. 3.1). In representative cases, we report the partial error bars obtained by the residuals, the final error bars, and the worst case error bars (see App. A).Jitter
Parameters Best values
1-σ errors
1-σ errors
1-σ errors
(residual scatter only)
(ICA)
(ICA worst case)
p
0.15505
9×10 −5
1.4×10 −4
1.4×10 −4
sin1
a0
8.99
0.02
0.03
0.04
i
85.78
0.03
0.04
0.04
p
0.15501
1.0×10 −4
1.5×10 −4
1.5×10 −4
cos1
a0
9.05
0.03
0.04
0.04
i
85.85
0.03
0.04
0.05
p
0.15507
sin2
a0
8.98
i
85.78
p
0.15510
cos2
a0
8.94
i
85.74
p
0.15505
sin3
a0
8.99
i
85.79
p
0.15503
cos3
a0
9.02
i
85.82
p
0.15506
saw1
a0
8.98
i
85.78
p
0.15506
saw1v1
a0
8.99
i
85.79
p
0.15511
saw1v2
a0
9.02
i
85.81
p
0.15508
1.0×10 −4
1.5×10 −4
1.5×10 −4
saw1v3
a0
8.99
0.03
0.04
0.04
i
85.79
0.03
0.05
0.05
p
0.15509
saw1vf1
a0
8.98
i
85.77
p
0.15506
1.0×10 −4
1.5×10 −4
1.6×10 −4
saw1vf2
a0
8.99
0.03
0.04
0.05
i
85.79
0.03
0.05
0.05
p
0.15503
1.0×10 −4
1.4×10 −4
1.5×10 −4
jump04c
a0
9.00
0.03
0.04
0.04
i
85.80
0.03
0.04
0.05
Table 8 :
8Retrieved transit parameters for simulations with σ P SF =1, 5×5 array, inter-pixel effects (see Sec. 3.1). In representative cases, we report the partial error bars obtained by the residuals, the final error bars, and the worst case error bars (see App. A).Jitter
Parameters Best values
1-σ errors
1-σ errors
1-σ errors
(residual scatter only)
(ICA)
(ICA worst case)
p
0.15524
1.5×10 −4
2.2×10 −4
2.3×10 −4
sin1
a0
8.97
0.04
0.06
0.07
i
85.76
0.05
0.07
0.07
p
0.15505
1.6×10 −4
2.2×10 −4
2.3×10 −4
cos1
a0
9.18
0.05
0.07
0.07
i
85.99
0.05
0.07
0.07
p
0.15510
sin2
a0
9.08
i
85.88
p
0.15536
cos2
a0
8.89
i
85.66
p
0.15528
sin3
a0
8.99
i
85.79
p
0.15516
cos3
a0
9.07
i
85.86
p
0.15519
saw1
a0
8.97
i
85.77
p
0.15517
saw1v1
a0
9.02
i
85.81
p
0.15494
saw1v2
a0
8.92
i
85.74
p
0.15527
1.5×10 −4
2.4×10 −4
2.6×10 −4
saw1v3
a0
8.98
0.04
0.07
0.07
i
85.77
0.05
0.07
0.08
p
0.15524
saw1vf1
a0
9.00
i
85.78
p
0.15531
1.6×10 −4
2.2×10 −4
2.4×10 −4
saw1vf2
a0
8.99
0.04
0.06
0.07
i
85.79
0.05
0.07
0.07
p
0.15508
1.3×10 −4
2.6×10 −4
2.8×10 −4
jump04c
a0
9.04
0.04
0.07
0.08
i
85.83
0.04
0.08
0.09
Table 9 :
9Retrieved transit parameters for simulations with σ P SF =0.2, 5×5 array, inter-pixel effects (see Sec. 3.2). In representative cases, we report the partial error bars obtained by the residuals, the final error bars, and the worst case error bars (see App. A).Jitter
Parameters Best values
1-σ errors
1-σ errors
1-σ errors
(residual scatter only)
(ICA)
(ICA worst case)
p
0.15507
3×10 −5
6×10 −5
7×10 −5
sin1
a0
9.010
0.010
0.018
0.020
i
85.808
0.010
0.020
0.022
p
0.15505
4×10 −5
6×10 −5
7×10 −5
cos1
a0
9.018
0.010
0.016
0.019
i
85.815
0.011
0.017
0.020
p
0.15508
sin2
a0
9.009
i
85.806
p
0.15505
cos2
a0
9.004
i
85.801
p
0.15510
sin3
a0
9.001
i
85.798
p
0.15507
cos3
a0
9.015
i
85.812
p
0.15506
saw1
a0
8.993
i
85.791
p
0.15506
saw1v1
a0
9.000
i
85.798
p
0.15506
saw1v2
a0
8.997
i
85.795
p
0.15506
3×10 −5
5×10 −5
6×10 −5
saw1v3
a0
8.998
0.009
0.015
0.018
i
85.796
0.010
0.016
0.019
p
0.15508
saw1vf1
a0
8.999
i
85.797
p
0.15506
3×10 −5
5×10 −5
6×10 −5
saw1vf2
a0
9.001
0.009
0.015
0.018
i
85.798
0.010
0.016
0.019
p
0.1556
5×10 −4
5×10 −4
5×10 −4
jump04c
a0
9.12
0.16
0.16
0.16
i
85.94
0.17
0.17
0.17
Table 10 :
10Retrieved transit parameters for simulations with σ P SF =0.2, 5×5 array, intra-pixel effects (see Sec. 3.3). In representative cases, we report the partial error bars obtained by the residuals, the final error bars, and the worst case error bars (see App. A).Jitter
Parameters Best values
1-σ errors
1-σ errors
1-σ errors
(residual scatter only)
(ICA)
(ICA worst case)
p
0.1551
3×10 −4
4×10 −4
5×10 −4
sin1
a0
9.05
0.07
0.10
0.13
i
85.84
0.08
0.11
0.14
p
0.1550
2×10 −4
3×10 −4
3×10 −4
cos1
a0
9.12
0.07
0.08
0.09
i
85.92
0.08
0.08
0.09
p
0.1550
sin2
a0
9.13
i
85.93
p
0.1551
cos2
a0
9.10
i
85.89
p
0.1551
sin3
a0
9.01
i
85.81
p
0.1551
cos3
a0
9.02
i
85.82
p
0.1548
saw1
a0
9.10
i
85.93
p
0.1555
saw1v1
a0
8.92
i
85.71
p
0.1546
saw1v2
a0
9.27
i
86.11
p
0.1555
4×10 −4
4×10 −4
5×10 −4
saw1v3
a0
8.95
0.10
0.11
0.13
i
85.77
0.11
0.12
0.15
p
0.1555
saw1vf1
a0
8.78
i
85.54
p
0.1550
3×10 −4
3×10 −4
3×10 −4
saw1vf2
a0
8.99
0.09
0.10
0.10
i
85.79
0.10
0.11
0.11
p
0.15508
4×10 −5
3×10 −4
4×10 −4
jump04c
a0
9.001
0.011
0.10
0.12
i
85.796
0.012
0.10
0.13
we do not mean, by any chance, to emulate the Spitzer/IRAC system, but rather to study some mechanisms that, in particular configurations, may reproduce in part the effects observed in Spitzer/IRAC datasets.
More specifically, the decorrelating function adopted is a second-order polynomial of the centroid coordinates, but we checked, in several cases, that higher order polynomials lead to identical results.
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| []
|
[
"Local SGD Optimizes Overparameterized Neural Networks in Polynomial Time",
"Local SGD Optimizes Overparameterized Neural Networks in Polynomial Time"
]
| [
"Yuyang Deng \nThe Pennsylvania State University Wyze Labs Inc\nThe Pennsylvania State University\n\n",
"Mohammad Mahdi \nThe Pennsylvania State University Wyze Labs Inc\nThe Pennsylvania State University\n\n",
"Kamani Mehrdad Mahdavi \nThe Pennsylvania State University Wyze Labs Inc\nThe Pennsylvania State University\n\n"
]
| [
"The Pennsylvania State University Wyze Labs Inc\nThe Pennsylvania State University\n",
"The Pennsylvania State University Wyze Labs Inc\nThe Pennsylvania State University\n",
"The Pennsylvania State University Wyze Labs Inc\nThe Pennsylvania State University\n"
]
| []
| In this paper we prove that Local (S)GD (or FedAvg) can optimize deep neural networks with Rectified Linear Unit (ReLU) activation function in polynomial time. Despite the established convergence theory of Local SGD on optimizing general smooth functions in communication-efficient distributed optimization, its convergence on non-smooth ReLU networks still eludes full theoretical understanding. The key property used in many Local SGD analysis on smooth function is gradient Lipschitzness, so that the gradient on local models will not drift far away from that on averaged model. However, this decent property does not hold in networks with nonsmooth ReLU activation function. We show that, even though ReLU network does not admit gradient Lipschitzness property, the difference between gradients on local models and average model will not change too much, under the dynamics of Local SGD. We validate our theoretical results via extensive experiments. This work is the first to show the convergence of Local SGD on non-smooth functions, and will shed lights on the optimization theory of federated training of deep neural networks. | null | [
"https://arxiv.org/pdf/2107.10868v2.pdf"
]
| 236,318,091 | 2107.10868 | a15f47a6e06913d8d7108bad65d2c6b5f9def7d9 |
Local SGD Optimizes Overparameterized Neural Networks in Polynomial Time
Yuyang Deng
The Pennsylvania State University Wyze Labs Inc
The Pennsylvania State University
Mohammad Mahdi
The Pennsylvania State University Wyze Labs Inc
The Pennsylvania State University
Kamani Mehrdad Mahdavi
The Pennsylvania State University Wyze Labs Inc
The Pennsylvania State University
Local SGD Optimizes Overparameterized Neural Networks in Polynomial Time
In this paper we prove that Local (S)GD (or FedAvg) can optimize deep neural networks with Rectified Linear Unit (ReLU) activation function in polynomial time. Despite the established convergence theory of Local SGD on optimizing general smooth functions in communication-efficient distributed optimization, its convergence on non-smooth ReLU networks still eludes full theoretical understanding. The key property used in many Local SGD analysis on smooth function is gradient Lipschitzness, so that the gradient on local models will not drift far away from that on averaged model. However, this decent property does not hold in networks with nonsmooth ReLU activation function. We show that, even though ReLU network does not admit gradient Lipschitzness property, the difference between gradients on local models and average model will not change too much, under the dynamics of Local SGD. We validate our theoretical results via extensive experiments. This work is the first to show the convergence of Local SGD on non-smooth functions, and will shed lights on the optimization theory of federated training of deep neural networks.
Introduction
The proliferation of mobile devices and internet of things (IoT) have resulted in immense growth of data generated by users, and offer huge potential in further advancement in ML if harnessed properly. However, due to regulations and concerns about data privacy, col- lecting data from clients and training machine learning models on a central server is not plausible. To decouple the ability to do machine learning without directly accessing private data of users, the Local SGD (a.k.a. Federated Averaging (FedAvg)) algorithm proposed in [22] to train deep neural networks in a communication efficient manner, without leaking users' data. In Local SGD, the goal is to minimize a finite sum problem under the orchestration of a central server, where each component function is the empirical loss evaluated on each client's local data. Local clients perform SGD on their own local models and after every τ steps, server synchronizes the models by aggregating locally updated models and averaging them. This simple idea has been shown to be effective in reducing the number of communication rounds, while enjoying the same convergence rate as fully synchronous counterpart, and become the key optimization method in many federated learning scenarios. We refer readers to several recent surveys [15,16,17,12] and the references therein for a non-exhaustive list of the research. Although significant advances have been made on understanding the convergence theory of Local SGD [26,14,9,8,19,29,28], however, these works mostly focus on general smooth functions. It has been observed that Local SGD can also efficiently optimize specific family of non-smooth functions, e.g., deep ReLU networks [22,32,18,10]. Up until now, the theoretical understanding of Local SGD on optimizing this class of non-smooth functions remains elusive. Inspired by this, we focus on rigorously understanding the convergence of Local GD or Local SGD when utilized to optimize non-smooth objectives.
While numerous studies investigated the behavior of single machine SGD on optimizing deep neural networks [7,6,1,2,3,35,34], and established linear convergence when the neural network is wide enough, however, these results cannot be trivially generalized to Local SGD. In fact, in local methods, due to local updating and periodic synchronization, the desired analysis should be more involved to bound the difference between local models and (virtual) averaged model.
arXiv:2107.10868v2 [cs.LG] 22 Feb 2022
On general smooth functions, according to gradient Lipschitzness property, we know that local gradients are close to gradients on averaged model. However, due to non-smoothness of ReLU function, this idea is no longer applicable. This naturally raises the question of understanding why Local (S)GD can optimize deep ReLU neural networks, which we aim to answer in this paper.
Contributions. We show that, both Local GD and Local SGD can provably optimize deep ReLU networks with multiple layers in polynomial time, under heterogeneous data allocation setting, meaning that each client has training data sampled from a potentially different underlying distribution. In the deterministic setting, we prove that Local GD can optimize an Llayer ReLU network with Ω(n 16 L 12 ) neurons, with a linear convergence rate O(e −R ), where R is the total number of communication rounds. In the stochastic setting, we prove that Local SGD can optimize an Llayer ReLU network with Ω(n 18 L 12 ) neurons, with the rate of O(e −R/R0 ), where R 0 is some constant depending on the number of samples n and neurons m. To the best of our knowledge, this paper is the first to analyze the global convergence of the both Local GD and Local SGD methods on optimizing deep neural networks with ReLU activation, and the first to show that it can converge even on non-smooth functions. To support our theory, we conduct experiments on MNIST dataset and demonstrate that the results match with our theoretical findings.
From a technical perspective, a key challenge to establish the convergence of both methods appears to be the non-smoothness of objective. In fact, as mentioned before, in the analysis of Local SGD on general smooth functions, a crucial step is to leverage the gradient Lipschitzness property, such that we can bound the gap between gradients on local model and averaged model. However, deep ReLU networks do not admit such benign property which complicates bounding the drift between local models and virtual averaged model due to multiple local updates (i.e., infrequent synchronization). To overcome the difficulty resulting from non-smoothness, we discover a "semi gradient Lipschitzness" property that indicates despite the non-smooth nature of ReLU function, its gradient still enjoys some almost-Lipschitzness geometry and characterizes the second order Lipschitzness nature of the neural network loss. This allows us to develop techniques to bound the local model deviation under the dynamics of Local (S)GD.
Notations. We use boldface lower-case letters such as x and upper-case letters such as W to denote vectors and matrices, respectively. We use v to denote Euclidean norm of vector v, and use W and W F to denote spectral and Frobenius norm of matrix W, respectively. We use N (µ, δ) to denote the Gaussian distribution with mean µ and variance δ. We also use W to denote the tuple of all W 1 , ..., W L , i.e., W = (W 1 , ..., W L ). Finally, we use B(W, ω) to denote the Euclidean ball centered at W with radius ω. Convergence Theory of Neural Network. The empirical success of (deep) neural networks motivated the researchers to study the theoretical foundation behind them. Numerous studies take efforts to establish the convergence theory of overparameterized neural networks. While earlier works study the simple two layer network as the starting point [27,5,21,33,4], but these papers make strong assumption on input data or sophisticated initialization strategy. Li and Liang [20] study the two layer network with cross-entropy loss, and for the first time show that if the network is overparameterized enough, SGD can find the global minima in polynomial time. Furthermore, if the input data is well structured, the guarantee for generalization can also be achieved. Du et al [7] derive the global linear convergence of two-layer ReLU network with l 2 regression loss. They also extend their results to deep neural network in [6], but they assume the activation is smooth. Allen-Zhu et al [2] firstly prove the global linear convergence of deep RelU network, and derive a key semi-smoothness property of ReLU DNN, which advances the analysis tool for ReLU network. Zou and Gu [35] further improve Allen-Zhu's result. They reduce the width of the network to a small dependency on the number of training samples, by deriving a tighter gradient upper bound. Recently, some works further reduce this dependency to cubic, quadratic and even linear [25,24,23].
Local (S)GD on Neural Network. Recently, Huang et al [11] study the convergence of Local GD on 2-layer ReLU network, which is the most relevant work to ours. However, besides the analysis methods which are significantly different, [11] only considers deterministic algorithm (Local GD) on a simple two-layer network. In this paper, we establish convergence for both Local GD and Local SGD on an L-layer deep ReLU network.
Problem Setup
We consider a distributed setting with K machines.
Let S i = {(x i 1 , y i 1 ), .
.., (x i n , y i n )} denote the set of all n training data allocated at client i . We further let S = K i=1 S i to be the union of all clients' data. The goal is to solve the following finite sum minimization problem in a distributed manner:
min W L(W) = 1 K K i=1 L i (W),
where L i (W) = 1 n (x,y)∈Si (W; x, y) is the loss function evaluated on ith client data based on loss function (). The description of the network architecture and loss function type are presented next.
Deep ReLU network. We consider a L-layer neural network architecture with ReLU activation function:
f (W, V, x) = Vσ(W L σ(W L−1 · · · σ(W 1 x)))
where σ(x) = max(x, 0), W l ∈ R m×m is the weight matrix of lth layer (we set W 1 ∈ R m×d ), x ∈ R d is the input data. For ease of exposition, we assume the number of neurons is same for all layers. Also, following the prior studies [7, 2, 35], we fix the top layer V, and only train the parameters of the hidden layers W = (W 1 , W 2 , . . . , W L ).
We consider regression setting with squared losss (W, V; x, y) = 1 2 f (W, V, x) − y 2 where the gradient of L i (W) w.r.t. W l can be derived as:
∇ W l L i (W) = 1 n (xj ,yj )∈Si D j,l B j,l+1 (f j − y j )f j,l−1 ,
where f j,l = σ(W l σ(W l−1 · · · σ(W 1 x j ))),
f j = Vσ(W L σ(W L−1 · · · σ(W 1 x j ))), B j,l+1 = VD j,L W L · · · D j,l+1 W l+1 and D j,l ∈ R m×m is a diagonal matrix with entries D j,l (r, r) = 1[(W l f j,l−1 ) r ≥ 0] for r ∈ [m]
. For ease of exposition, we will express ∇ W L i (W) as the following tuple:
∇ W L i (W) = (∇ W1 L i (W), · · · , ∇ W L L i (W)).
Algorithm description. To mitigate the communication bottleneck in distributed optimization, a popular idea is to update models locally via GD or SGD, and then average them periodically [22,26]. The Local (S)GD algorithm proceeds for T iterations, and at tth iteration, the ith client locally performs the GD or SGD on its own model W (i) (t):
Local GD: W (i) (t + 1) = W (i) (t) − η∇ W Li W (i) (t) , Local SGD: W (i) (t + 1) = W (i) (t) − ηG (t) i , where G (t) i is the stochastic gradient such that E[G (t) i ] = ∇ W L i W (i) (t)
. After τ local updates (i.e., t divides τ ), the server aggregates local models W (i) (t + 1), i = 1, . . . , K and performs the next global model according to:
W(t + 1) = 1 K K i=1 W (i) (t + 1).
Then, the server sends the averaged model back to local clients, to update their local models and the procedure is repeated for T /τ stages. This idea can significantly reduce the communications rounds by a factor of τ , compared to fully synchronized GD/SGD. Even though it is a simple algorithm, and has been employed for distributed neural network training for a long time, we are not aware of any prior theoretical work that analyzes its convergence performance on deep ReLU neural networks. We note that the aggregated model at server cannot be treated as τ iterations of synchronous SGD, since each local update contains a bias with respect to the global model which necessities to bound the drift among local and global models. The bias issue becomes even more challenging when non-smooth ReLU is utilized compared to the existing studies that focus on smooth objectives.
Main Results
In this section, we present the convergence rates. We start with making the following standard separability assumption [2, 35] on the training data.
Algorithm 1: Local (S)GD Input: Synchronization gap τ , Number of iterations T . Initialization network parameter W (0) ∼ N (0, 2/mI) and v ∼ N (0, I/d) parallel for i = 1, ..., K do for t = 1, ..., T do
W (i) (t + 1) = W (i) (t) − η∇ W L i W (i) (t) # Local GD Sample a data (x,ỹ) uniformly from S i . Compute G (t) i = n∇ W W (i) (t);x,ỹ . W (i) (t + 1) = W (i) (t) − ηG (t) i # Local SGD
if t divides τ then all nodes send their local parameter W (i) (t + 1) to server.
W (t + 1) = 1 K K i=1 W (i) (t + 1)
; send W (t + 1) to all nodes to update their local models. each client initializes its local models: W (i) (t + 1) = W (t + 1). end end end Assumption 1. For any x ∈ S, x = 1, and for any x, x ∈ S, x − x ≥ φ.
The following theorem establishes the convergence rate of Local GD on deep ReLU network: where R = T τ is the total number of communication rounds, and W (T ) = 1
K K i=1 W (i) (T ).
The proof of Theorem 1 is provided in Appendix A. As expected, the fastest convergence rate is attained when the synchronization gap τ is one. Theorem 1 however precisely characterizes how large the number of neurons needs to picked to guarantee linear convergence rate. Here we require the width of network m to be O(Kn 16 L 12 ) to achieve linear rate in terms of communication rounds, which is linear in the number of clients and polynomial in n and L. The most relevant work to this paper is Huang et al [11], where they consider two-layer ReLU network, and achieve and O(e −R/K ) convergence rate with Ω(n 4 ) neurons.
Their convergence rate is strictly worse than us, while they require smaller number of neurons because they only consider simple two-layer architecture. An interesting observation from above rate is that the number of neurons per layer is polynomial in the number of layers which is also observed in our empirical studies. This implies that by adding to the depth of model, we also need to increase the number of neurons at each layer accordingly. We note that compared to analysis of single machine GD on deep ReLU networks [35,23], the width obtained here is worse, and we leave the improvement on either the dependency on n or K as a future work. Now we proceed to establish the convergence rate of Local SGD:
Theorem 2 (Local SGD). For Local SGD, under Assumption 1, if we choose m ≥ Kdn 18 L 12 (log m) 5 φ 3 , η = O dφ mτ n 3 log 2 m then with probability at least 1 − e −Ω((log m) 2 ) it holds that L(W (T )) ≤ (n log 2 m) · e −Ω(R/R0) L(W (0)), where R = T τ is the total number of commu- nication rounds, R 0 = n 5 log 2 m φ 2 , and W (T ) = 1 K K i=1 W (i) (T ).
Comparison to related bounds on Local SGD.
[9] established an O(1/ √ KT ) rate with O( √ KT )
communication rounds on general smooth nonconvex functions, while our result enjoys faster rate and better communication efficiency. We would also like to emphasize that our setting is more difficult, since 1) we study nonconvex and non-smooth functions; 2) we prove a global convergence, but their result only guarantees the convergence to a first order stationary point, and 3) our result is stated for last iterate, but theirs only guarantees that at least one of the history iterates vissits local minima.
Comparison to related work on ReLU networks. Since we are not aware of any related work of Local SGD on ReLU network, here we only discuss single machine algorithms. Compared to single machine SGD on optimizing ReLU network, the most analogous work to ours is [35], since both our and their analysis adapt the proof framework from [2]. They achieve linear convergence with Ω(n 17 ) neurons for achieving linear convergence, while we need Ω(n 18 ) neurons. We also noticed that recent works [24,23] have reduce the network width to a significantly small number, so we leave improving our results by adapting a finer analysis as future work.
In this section we will present an overview of our proof strategy for deterministic setting (Local GD). The stochastic setting shares the similar strategy. We let
W(t) = 1 K K i=1 W (i) (t) denote the virtual averaged iterates.
We use t c to denote the latest communication round, also the cth communication round.
Main Technique
Our proof involves three main ingredients, namely (i) semi gradient Lipschitzness, (ii) shrinkage of local loss, and (iii) local model deviation analysis as we discuss briefly below.
Semi Gradient Lipschitzness. 1 In the analysis of Local SGD on general smooth functions, one key step is to utilize the gradient Lipschitzness property, such that we can bound the gap between gradients on local model and averaged model by:
∇L(W) − ∇L(W) ≤ H W −W .
However, ReLU network does not admit such benign property. Alternatively, we discover a "semi-gradient Lipschitzness" property. For any parameterization W andW such that W, W ∈ B(W(0), ω):
1 K K i=1 ∇ W Li(W) − ∇WLi(W) 2 F ≤ O mL 4 d W −W 2 2 + O ω 2/3 L 5 m log m d L(W),
This inequality demonstrates that for any two models lying in the small local perturbed region of initialization model, ReLU network almost achieves gradient Lipschitzness, up to some small additive zeroth order offset. That is, if we can carefully move local models W (i) (t) such that they do not drift from the initialization and virtual average model W(t) too much, then the gradient at local iterate
∇ W (i) L i (W (i) (t))
is guaranteed to be close to the gradient at virtual averaged iterate ∇ W L i (W(t)).
Shrinkage of Local Loss.
Another key property of local loss is that the local loss is strictly decreasing, compared to the latest communication round. We show that with high probability, if we properly choose learning rate, the following inequality holds: for Local GD:
L i (W (i) (t)) ≤ L i (W (i) (t − 1)) ≤ · · · ≤ L i (W (i) (t c )),
1 Notice that this is not the semi-smoothness property derived by Allen Zhu et al [2], even though we also need that property in analysis. and for Local SGD:
L i (W (i) (t)) ≤ exp φ mn 2.5 log 2 m L i (W(t c )),
where t c ≤ t ≤ t c + τ − 1, and t c is the latest communication round of t. This nice property will enable us to reduce the loss at any iteration to its latest communication round.
Local Model Deviation Analysis. During the dynamic of Local (S)GD, the local models will drift from the virtual averaged model, so the other key technique in Local (S)GD analysis is to bound local model deviation W (i) (t) − W(t) F . However, in the highlynonsmooth ReLU network, this quantity is not a viable error to control. Hence, inspired by [11], we consider
the deviation W (i) (t)−W(t c ) F , where t c is the latest communication round of t,
and derive the deviation bound as:
1 K K i=1 W (i) (t) − W(tc) 2 F ≤ O η 2 τ 2 mn d L(W(tc)).
Here we bound the local model deviation by the loss at the last communication round, which is a key step that enables us to achieve linear rate.
Sketch of the Proof
In this section we are going to present the overview of our key proof techniques. The detailed proofs are deferred to appendix. Before that, we first mention two lemmas that facilitate our analysis.
Lemma 1 (Semi-smoothness [2]). Let ω ∈ Ω 1/(d 3/2 m 3/2 log 3/2 (m)) , O 1/(log 3/2 (m)) .
Then for any two weightsŴ andW satisfyinĝ W,W ∈ B(W (0) , ω), with probability at least 1 − exp(−Ω(mω 3/2 L)), there exist two constants C and C such that
L(W) ≤ L(Ŵ) + ∇L(Ŵ),W −Ŵ (1) + C L(Ŵ) · ω 1/3 m log(m) √ d · W −Ŵ (2) + C m d W −Ŵ 2 .(3)
Lemma 2 (Gradient bound [35]). Let ω = O φ 3/2 n −3 L −6 log −3/2 (m) , then for all W ∈ B(W(0), ω), with probability at least 1 − exp − Ω(mφ/(dn))), it holds that
∇L(W) 2 F ≤ O mL(W)/d , ∇L(W) 2 F ≥ Ω mφL(W)/(dn 2 ) .
The above two lemmas demonstrate that, if the network parameters lie in the ball centered at initial solution with radius ω, then the network admits local smoothness, and there is no critical point in this region.
The whole idea of the proof is that, we firstly assume each local iterates and virtual averaged iterates lie in the ω-ball centered at initial model, so that we can apply the benign properties (semi smoothness, bounded gradients and semi gradient Lipschitzness) of objective function. Then, with these nice properties we are able to establish the linear convergence of the objective as claimed. Lastly, we verify the correctness of bounded local iterates and virtual averaged iterates assumption.
The proof is conducted via induction. The inductive hypothesis is as follows: for any h ≤ t, we assume the following statements holds for ω = O φ 3/2 n −6 L −6 log −3/2 (m) :
(I) W(h) − W(0) ≤ ω, W (i) (h) − W(0) ≤ ω, ∀i ∈ [K]. (II) L(W(t c )) ≤ 1 − Ω ητ mφ dn 2 c L(W(0)).
The first statement indicates that the virtual iterates do not drift too much from the initialization, under Local GD's dynamic, if we properly choose learning rate and synchronization gap. The second statement gives the linear convergence rate of objective value. Now, we need to prove these two statements hold for t + 1.
Step 1: Boundedness of virtual average iterates. We first verify (I), the boundedness of virtual iterates during algorithm proceeding. The idea is to keep track of the dynamics of the average gradients on each local iterate. To do so, by the updating rule we have:
W(t + 1) − W(0) ≤ η t j=1 1 K K i=1 ∇L i (W (i) (j)) ≤ η t j=1 1 K K i=1 O m d L i (W(j)) ≤ ητ c j=1 O m d L(W(t c )),
where we apply the gradient upper bound (Lemma 2) and the decreasing nature of local loss. Now we plug in induction hypothesis II to bound L(W(t c )):
W(t + 1) − W(0) ≤ ητ c j=1 O m d 1 − Ω ητ mφ dn 2 c L(W(0)) ≤ ητ O m d c j=1 1 − Ω ητ mφ 2dn 2 c L(W(0)) = O 2 √ dn 2 √ mφ L(W(0)).
Since we choose m ≥ Kdn 16 L 12 log 3 m φ 5
, it can be concluded that W(t + 1) − W(0) ≤ ω.
Step 2: Boundedness of local iterates. The next step is to show that local iterates are also lying in the local perturbed region of initial model. This can be done by tracking the dynamic of the gradients on individual local model:
W (i) (t + 1) − W(0) ≤ η t j=1 ∇L i (W (i) (j)) ≤ η t j=1 O m d L i (W(j)) ≤ ητ c j=1 O m d KL(W(t c )) ≤ O 2 √ dn 2 √ mφ KL(W(0)) ≤ ω.
Step 3: Linear convergence of objective value.
We now switch to prove statement (II). Since we know that, W(t + 1) − W ≤ ω, we can apply Lemma 1 by letW = W(t c+1 ) andŴ = W(t c ) and gradient bound (Lemma 2). We have the following recursive relation over the loss at different communication stages:
L(W(tc+1)) ≤ 1 − Ω ητ mφ dn 2 L(W(tc)) + η 2 1 K K i=1 tc−1 t =t c−1 ∇Li(W(tc)) − ∇Li(W (i) (t )) 2 F + ηC · ω 1/3 L 2 m log(m) 2 √ d 1 K K i=1 tc−1 t =t c−1 ∇L(W(tc)) − ∇Li(W (i) (t )) 2 F ,
where t c is the latest communication round at iteration t. Now we can use semi gradient Lipschitzness property to reduce the difference between gradients to local model deviation. Further plugging in the local model deviation bound, and unrolling the recursion will complete the proof:
L(W(t c )) ≤ 1 − Ω ητ mφ dn 2 L(W(t c−1 )) ≤ e −Ω(R) L(W (0)).
as desired.
Experiment
In this section we present our experimental results to validate our theoretical findings. For this purpose, we run our experiments on MNIST dataset using a varying number of MLP layers with ReLU activation function on the hidden layer. We denote the number of neurons in the hidden layer with m. To run the experiments on a distributed setting, we create 50 clients. Then, we distribute the MNIST dataset on these clients in IID (homogeneous) or non-IID (heterogeneous) ways. For IID setting, each client has training data i.i.d. sampled from the whole dataset. For non-IID setting, we allocate only two classes of data to each client, and hence, different clients will have access to different distribution of data.
Effects of different model sizes m. We firstly train the model using Local SGD with the same synchronization gap and different number of hidden neurons, m. Figure 1 shows the results of this experiment on models with different hidden layer's size in homogeneous and heterogeneous settings. As it can be seen in both cases, the model with higher model size can achieve better final accuracy. This phenomenon has more impact in the heterogeneous data distribution compared to homogeneous setting.
Effects of different synchronization gap τ . Now, we fix the model size m = 50 and change the synchronization gap τ . We do the comparison between fully synchronous SGD and Local SGD with τ = 5, 10, 20, 50.
The results in Figure 2 shows that in both homogeneous and heterogeneous settings, the convergence rate becomes slower when synchronization gap increases. However, in the heterogeneous setting, increasing τ will decrease the convergence speed more significantly.
Effects of number of layers L. When we increase the number of layers L, based on condition of m in Theorem 2, we need to increase the number of neurons as well to achieve the same rate. For instance, Figure 3 shows L = 5), it is evident that the model performs poorly. By increasing the number of neurons per layer, we can see that m = 100 can make 5-layer model achieve the same performance of the single layer model, which has 10× more neurons and more than 3× bigger in terms of parameter size. This is consistent with Theorem 2, as increasing the number of layers requires significantly more neurons per layer compared to single layer counterpart to guarantees linear convergence rate.
Discussion and Future Works
In this paper, we proved that both Local GD and Local SGD that are originally proposed for communication efficient training of deep neural networks can achieve global minima of the training loss for overparameterized deep ReLU networks. We make the first theoretical trial on the analysis of Local (S)GD on training Deep ReLU networks with multiple layers, but we do not claim that our results, e.g., number of required neurons and dependency on the number of layers, are optimal in any sense.
A number of future works/improvements are still exciting to explore:
Tightening the condition on the number of neurons. In the bounds obtained for both Local GD and SGD, the required number of neurons to has a heavy dependency on the number training samples n, which is worse than the single machine case. We are aware of some recent works [25, 24, 23] that demonstrate sig- nificantly reduced number of required neurons, and we believe incorporating their results can also improve our theory to entail tighter bounds.
Extension of analysis to other federated optimization methods. To further reduce the harm caused by multiple local updates, a line of recent studies proposed alternative methods to reduce the local model deviation [13,30]. Establishing the convergence of these variants on deep non-smooth networks is another valuable research direction.
Extension to other neural network architectures. In this paper we only consider simple ReLU forward feed neural network, but as shown in the prior works [2, 36], single machine SGD can optimize more complicated neural network like CNN, ResNet or RNN as well. Hence, one natural future work is to extend our analysis on Local SGD to those neural network models.
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A Proof of Theorem 1 (Local GD)
In this section we present the proof of convergence rate of Local GD (Theorem 1). Similar to analysis of Local SGD for smooth objectives [26,14], we start from pertubed virtual iterates analysis. We let W(t) = 1 K K i=1 W (i) (t) denote the virtual averaged iterates. Before providing the proof, let us first introduce some useful lemmas.
A.1 Proof of Technical Lemma
The following lemma is from Allen-Zhu et al's seminal work [2], which characterizes the forward perturbation property of deep ReLU network:
Lemma 3 (Allen et al [2]). Consider a weight matricesW, W such thatW, W ∈ B(W(0), ω), with probability at least 1 − exp(−O(mω 2/3 )), the following facts hold:
f j,l−1 − f j,l−1 (0) ≤ O ωL 5/2 log m , f j,l−1 ≤ O(1), v V(D j,L W L · · · D j,l+1 W l+1 D j,l −D j,LWL · · ·D j,l+1Wl+1Dj,l ) ≤ O ω 1/3 L 2 m log m/d · v , f j,l−1 −f j,l−1 ≤ O L 3/2 W −W 2 , v V(D j,L W L · · · D j,l+1 W l+1 D j,l ≤ O m/d · v ,
where v is arbitrary vector, f j,l−1 (0) = σ(W l (0)σ(W l−1 (0) · · · σ(W 1 (0)x j ))).
The following lemma establishes a bound on the deviation between local models and (virtual) averaged global model in terms of global loss.
Lemma 4. For Local GD, let t c denote the latest communication stage of before iteration t. If the condition that L i (W (i) (t)) ≤ L i (W(t c )) for any t c ≤ t ≤ t c + τ − 1 holds, then the following statement holds true for t c ≤ t ≤ t c + τ − 1:
W (i) (t) − W(t c ) 2 F ≤ O (η 2 τ 2 + η 2 τ ) mn d L i (W(t c )).
where K is the number of devices, τ is the number of local updates between two consecutive rounds of synchronization, n is the size of each local data shard, and m is the number of neurons in hidden layer.
Proof. According to updating rule we have:
W (i) (t + 1) − W(t c ) 2 F = W (i) (t) − η∇L i (W (i) (t)) − W(t c ) 2 F = W (i) (t) − W(t c ) 2 F − 2η ∇L i (W (i) (t)), W (i) (t) − W(t c ) + η 2 ∇L i (W (i) (t)) 2 F = W (i) (t) − W(t c ) 2 F + 2η ∇L i (W (i) (t)), η t−1 t =tc ∇L i (W (i) (t)) + η 2 ∇L i (W (i) (t)) 2 F = W (i) (t) − W(t c ) 2 F + 2η 2 τ ∇L i (W (i) (t)), 1 τ t−1 t =tc ∇L i (W (i) (t)) + η 2 ∇L i (W (i) (t)) 2 F .
Applying the identity a, b = 1 2 a 2 + 1 2 b 2 − 1 2 a − b 2 on the cross term yields:
W (i) (t + 1) − W(t c ) 2 F = W (i) (t) − η∇L i (W (i) (t)) − W(t c ) 2 F ≤ W (i) (t) − W(t c ) 2 F + η 2 (t − t c ) ∇L i (W (i) (t)) 2 + 1 (t − t c ) t−1 t =tc ∇L i (W (i) (t)) 2 + η 2 ∇L i (W (i) (t)) 2 F ≤ W (i) (t) − W(t c ) 2 F + η 2 (t − t c ) ∇L i (W (i) (t)) 2 + 1 t − t c t−1 t =tc ∇L i (W (i) (t)) 2 + η 2 ∇L i (W (i) (t)) 2 F .
Plugging the gradient upper bound from Lemma 2 yields:
W (i) (t + 1) − W(t c ) 2 F = W (i) (t) − η∇L i (W (i) (t)) − W(t c ) 2 F ≤ W (i) (t) − W(t c ) 2 F + η 2 (t − t c ) O mn d L i (W (i) (t)) + 1 t − t c t−1 t =tc O mn d L i (W (i) (t )) + η 2 O mn d L i (W (i) (t)).
Since we assume L i (W (i) (t)) ≤ L i (W(t c )) for any t c ≤ t ≤ t c + τ − 1, so we have:
W (i) (t + 1) − W(t c ) 2 F = W (i) (t) − η∇L i (W (i) (t)) − W(t c ) 2 F ≤ W (i) (t) − W(t c ) 2 F + η 2 τ O mn d L i (W (i) (t c )) + O mn d L i (W (i) (t c )) + η 2 O mn d L i (W (i) (t c )).
Doing the telescoping sum from t + 1 to t c will conclude the proof:
W (i) (t + 1) − W(t c ) 2 F ≤ O (η 2 τ 2 + η 2 τ ) mn d L i (W (i) (t c )).
The next lemma is the key result in our proof, which characterizes the semi gradient Lipschitzness property of ReLU neural network.
Lemma 5 (Semi-gradient Lipschitzness). For Local GD, at any iteration t, if W,W ∈ B(W (0), ω), then with probability at least 1 − exp −Ω(mω 2/3 ) , the following statement holds true:
1 K K i=1 ∇ W L i (W) − ∇WL i (W) 2 F ≤ O mL 4 d W −W 2 2 + O ω 2/3 L 5 m log m d + ω 2 L 6 m log m d L(W),
where K is the number of devices, τ is the number of local updates between two consecutive rounds of synchronization, n is the size of each local data shard, and m is the number of neurons in hidden layer.
Proof. Observe that:
1 K K i=1 ∇ W L i (W) − ∇WL i (W) 2 F = 1 K K i=1 L l=1 ∇ W l L i (W) − ∇W l L i (W) 2 F . Let f (i) j = VD (i) j,L W (i) L · · · D j,1 W (i) 1 x j and L (i) j = f (i)
j − y j . Now we examine the difference of the gradients:
∇ W l L i (W) − ∇W l L i (W) = 1 n n j=1 (L (i) j VD j,L W L · · · D j,l+1 W l+1 D j,l ) (f (i) j,l−1 ) − (L (i) j VD j,LWL · · ·D j,l+1Wl+1Dj,l ) (f (i) j,l−1 ) = 1 n n j=1 ((L (i) j −L (i) j )VD j,L W L · · · D j,l+1 W l+1 D j,l ) (f (i) j,l−1 ) + 1 n n j=1 L (i) j V(D j,L W L · · · D j,l+1 W l+1 D j,l −D j,LWL · · ·D j,l+1Wl+1Dj,l ) (f (i) j,l−1 ) + 1 n n j=1 (L (i) j VD j,LWL · · ·D j,l+1Wl+1Dj,l ) (f (i) j,l−1 −f (i) j,l−1 ) .
According to Lemma 3 we know the following facts:
f (i) j,l−1 −f (i) j,l−1 ≤ O ωL 5/2 log m , f (i) j,l−1 ≤ O(1), L (i) j V(D j,L W L · · · D j,l+1 W l+1 D j,l −D j,LWL · · ·D j,l+1Wl+1Dj,l ) ≤ O ω 1/3 L 2 m log m/d · L (i) j , L (i) j −L (i) j F = Vσ(W L · · · σ(W 1 x j )) − Vσ(W L · · · σ(W 1 x j )) ≤ O L 3/2 W −W 2 , (L (i) j −L (i) j )VD j,L W L · · · D j,l+1 W l+1 D j,l ) ≤ O m/d · L 3/2 W −W 2 .
So we have the following bound for Frobenius norm:
∇ W l L i (W) − ∇W l L i (W) 2 F ≤ O m/d · L 3/2 W −W 2 + O ω 1/3 L 2 m log m/d · L (i) j + O ωL 5/2 m log m/d L (i) j 2 ≤ O m/d · L 3 W −W 2 2 + O ω 2/3 L 4 m log m/d + ω 2 L 5 m log m/d L (i) j 2 .
Hence we can conclude the proof:
1 K K i=1 ∇ W L i (W) − ∇WL i (W) 2 F = 1 K K i=1 L l=1 ∇ W l L i (W) − ∇W l L i (W) 2 F ≤ O mL 4 d W −W 2 2 + O ω 2/3 L 5 m log m d + ω 2 L 6 m log m d L(W).
Lemma 6. For Local GD, at any iteration t in between two communication rounds: t c ≤ t ≤ t c + τ − 1 and i ∈ [K], if W (i) (t) ∈ B(W (0), ω), then with probability at least 1 − exp −Ω(mω 2/3 ) , the following statement holds true:
L i (W (i) (t)) ≤ L i (W (i) (t − 1)) ≤ · · · ≤ L i (W (i) (t c )).
Proof. According to updating rule and the semi smoothness property:
L i (W (i) (t)) ≤ L i (W (i) (t − 1)) + ∇L i (W (i) (t − 1)), W (i) (t) − W (i) (t − 1) + C L i (W (i) (t)) · ω 1/3 L 2 m log(m) √ d · W (i) (t) − W (i) (t − 1) 2 + C L 2 m d W (i) (t) − W (i) (t − 1) 2 2 ≤ L(W (i) (t − 1)) − η ∇L(W (i) (t − 1)), ∇L i (W (i) (t − 1)) + ηC L i (W (i) (t − 1)) · ω 1/3 L 2 m log(m) √ d · ∇L i (W (i) (t − 1)) 2 + η 2 C L 2 m d E ∇L i (W (i) (t − 1)) 2 2 ≤ 1 − Ω ητ mφ dn 2 L i (W (i) (t − 1)),
where in we plug in the gradient upper bound from Lemma 2. According to our choice of η, we can conclude that
L i (W (i) (t)) ≤ L i (W (i) (t − 1)) ≤ · · · ≤ L i (W (i) (t c )).
A.2 Proof of Theorem 1
With the key lemmas in place, we now prove Theorem 1 by induction. Assume the following induction hypotheses hold for h ≤ t:
(I) W(h) − W(0) ≤ ω, W (i) (h) − W(0) ≤ ω, ∀i ∈ [K], (II) L(W(t c )) ≤ 1 − Ω ητ mφ dn 2 c L(W(0))
where ω = O φ 3/2 n −6 L −6 log −3/2 (m) , and t c is the latest communication round of h, which is also cth communication round. Then we shall show the above two statements hold for t + 1.
A.2.1 Proof of inductive hypothesis I
Step 1: Bounded virtual average iterates. First we prove the first hypothesis for t + 1:
W(t + 1) − W(0) ≤ ω.
By the updating rule we know that:
W(t + 1) − W(0) ≤ η t j=1 1 K K i=1 ∇L i (W (i) (j)) ≤ η t j=1 1 K K i=1 O( m d ) L i (W(j)) ≤ ητ c j=1 1 K K i=1 O( m d ) L i (W(t c )) ≤ ητ c j=1 O( m d ) L(W(t c )),
where we apply the gradient upper bound (Lemma 2) in and the decreasing nature of local loss (Lemma 6) in . Now we plug in induction hypothesis II to bound L(W(t c )):
W(t + 1) − W(0) ≤ ητ c j=1 O( m d ) 1 − Ω ητ mφ dn 2 c L(W(0)) ≤ ητ O( m d ) c j=1 1 − Ω ητ mφ 2dn 2 c L(W(0)) ≤ ητ O( m d )O 2dn 2 ητ mφ L(W(0)) = O 2 √ dn 2 √ mφ L(W(0)).
Since we choose m ≥ dn 16 L 12 log 3 m φ 5
, it follows that W(t + 1) − W(0) ≤ ω.
Step 2: Bounded local iterates. Then we prove the second hypothesis for t + 1:
W (i) (t + 1) − W(0) ≤ ω.
By the updating rule we know that:
W (i) (t + 1) − W(0) ≤ η t j=1 ∇L i (W (i) (j)) ≤ η t j=1 O m d L i (W(j)) ≤ ητ c j=1 O m d L i (W(t c )) ≤ ητ c j=1 O m d KL i (W(t c )),
where we apply the gradient upper bound (Lemma 2) and the decreasing nature of local loss (Lemma 6). Now we plug in induction hypothesis II to bound L(W(t c )):
W(t + 1) − W(0) ≤ ητ c j=1 O m d 1 − Ω ητ mφ dn 2 c L i (W(0)) ≤ ητ O K m d c j=1 1 − Ω ητ mφ 2dn 2 c L i (W(0)) ≤ ητ O K m d O 2dn 2 ητ mφ L i (W(0)) = O 2 √ dn 2 √ mφ KL i (W(0)).
Since we choose m ≥ Kdn 12 L 12 log 3 m φ 5
, it immediately follows that W (i) (t + 1) − W(0) ≤ ω as desired.
A.2.2 Proof of inductive hypothesis II
Step 1: One iteration analysis from Semi-smoothness. Now we proceed to prove that hypothesis II holds for t + 1. If t c ≤ t + 1 < t c+1 , then the statement apparently holds for t c . If t + 1 ≥ t c+1 , we have to examine the upper bound for L(W(t c+1 )). The first step is to characterize how global loss changes in one iteration. We use the technique from standard smooth non-convex optimization, but notice that here we only have semi-smooth objective. According to semi-smoothness (Lemma 1) and updating rule:
L(W(t c+1 )) ≤ L(W(t c )) + ∇L(W(t c )), W(t c+1 ) − W(t c ) + C L(W(t c )) · ω 1/3 L 2 m log(m) √ d · W(t c+1 ) − W(t c ) 2 + C L 2 m d W(t c ) − W(t c ) 2 2 ≤ L(W(t c )) − ∇L(W(t c )), ητ 1 τ K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) + ητ C L(W(t c−1 )) · ω 1/3 L 2 m log(m) √ d · 1 τ K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) + η 2 C L 2 m d 1 K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) 2 ≤ L(W(t c )) − ητ 2 ∇L(W(t c )) 2 F − ητ 2 1 τ K K i=1 tc+1−1 t =tc ∇L i (W (i) (t)) 2 F + ητ 2 ∇L(W(t c )) − 1 τ K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) 2 F + ητ C L(W(t c )) · ω 1/3 L 2 m log(m) 2 √ d × ∇L(W(t c )) F + 1 τ K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) − ∇L(W(t c )) F + η 2 C L 2 m d 1 K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) 2 F ≤ L(W(t c )) − ητ 2 ∇L(W(t c )) 2 F − η 2τ − η 2 C L 2 m d 1 K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) 2 F + η 2 1 K K i=1 tc+1−1 t =tc ∇L i (W(t c )) − ∇L i (W (i) (t )) 2 F + ηC L(W(t)) · ω 1/3 L 2 m log(m) 2 √ d ∇L(W(t c )) F + ητ C · ω 1/3 L 2 m log(m) 2 √ d ∇L(W(t c )) − 1 τ K K i=1 tc+1−1 t =tc ∇L i (W (i) (t )) F L(W(t c )) ,
where in we use the identity a, b = 1 2 a 2 + 1 2 b 2 − 1 2 a − b 2 . We plug in the semi gradient Lipschitzness from Lemma 5 and gradient bound from Lemma 2 in last inequality, and use the fact that η 2τ − η 2 C L 2 m d ≥ 0 to get:
L(W(t c+1 )) ≤ L(W(t c )) − Ω ητ mφ dn 2 L(W tc ) + ητ 2 O mL 4 d W (i) (t ) − W(t c ) 2 2 + O ω 2/3 L 5 m log m d + ω 2 L 6 m log m d L(W(t c )) + ητ C · ω 1/3 L 2 m log(m) 2d L(W(t c )) + ητ C · ω 1/3 L 2 m log(m) 2 √ d × O mL 4 d W (i) (t ) − W(t c ) 2 2 + O ω 2/3 L 5 m log m d + ω 2 L 6 m log m d L(W(t c ))L(W(t c )
,
Choosing ω = φ 3/2
Cωn 6 L 6 log(m) 3/2 where C ω is some large constant and plugging in local model deviation bound from Lemma 4, to get the main recursion relation as follows:
L(W(t c+1 )) ≤ 1 − Ω ητ mφ dn 2 L(W(t c )).(4)
Unrolling the recursion and plugging in η = O( dn 2 τ mφ ) will conclude the proof:
L(W(T )) ≤ exp(−R)L(W(0)) = ⇐⇒R = O log 1 .
B Proof of Theorem 2 (Local SGD)
In this section we will present the proof of convergence rate of Local SGD (Theorem 2). Before that, let us first introduce some useful lemmas.
B.1 Proof of Technical Lemma
The following lemma establishes the boundedness of the stochastic gradient. Lemma 7 (Bounded stochastic gradient). For Local SGD, the following statement holds true for stochastic gradient at any iteration t:
E St 1 K K i=1 G (t) i 2 ≤ O mL(W(t c )) dK , 1 K K i=1 G (t) i 2 ≤ O mnL(W(t c )) d , where S t = {(x i ,ỹ i )} K i=1
are the set of randomly sampled data to compute 1
K K i=1 G (t)
i , K is the number of devices, τ is the number of local updates between two consecutive rounds of synchronization, n is the size of each local data shard, d is the dimension of input data, and m is the number of neurons in hidden layer.
Proof. Observe the following facts:
E St 1 K K i=1 G (t) i 2 = 1 K 2 K i=1 E ∇ (W (i) (t);x i ,ỹ i ) 2 = 1 K 2 K i=1 1 n (xj ,yj )∈Si ∇ (W (i) (t); x j , y j ) 2 ≤ 1 K 2 K i=1 O(m/d)L i (W (i) (t)),
where we plug in the gradiet upper bound from Lemma 2. According to the shrinkage of local loss (Lemma 9), we can conclude that:
E St 1 K K i=1 G (t) i 2 ≤ O mL(W(t c )) dK .
Now we switch to prove the second statement by observing that:
1 K K i=1 G (t) i 2 = 1 K K i=1 ∇ (W (i) (t);x i ,ỹ i ) 2 ≤ 1 K K i=1 O(mn/d)L i (W (i) (t)) ≤ 1 K K i=1 O(mn/d)L i (W (i) (t c )),
which completes the proof.
The next lemma is similar to Lemma 4, but it characterizes the local model deviation under stochastic setting.
Hence, it will be inevitably looser than the deterministic version (Lemma 4). Lemma 8. For Local SGD, let t c denote the latest communication stage of before iteration t. If the condition that L i (W (i) (t)) ≤ L i (W(t c )) for any t c ≤ t ≤ t c + τ − 1 holds, then the following statement holds true for t c ≤ t ≤ t c + τ − 1:
W (i) (t) − W(t c ) 2 F ≤ O (η 2 τ 2 + η 2 τ ) mn d L i (W(t c )),
where K is the number of devices, τ is the number of local updates between two consecutive rounds of synchronization, n is the size of each local data shard, and m is the number of neurons in hidden layer.
Proof. According to updating rule:
E W (i) (t + 1) − W(t c ) 2 F = E W (i) (t) − ηG (t) i − W(t c ) 2 F = E W (i) (t) − W(t c ) 2 F − 2ηE ∇L i (W (i) (t)), W (i) (t) − W(t c ) + η 2 E G (t) i 2 F = E W (i) (t) − W(t c ) 2 F + 2η ∇L i (W (i) (t)), η t−1 t =tc ∇L i (W (i) (t)) + η 2 E G (t) i 2 F = E W (i) (t) − W(t c ) 2 F + 2η 2 (t − t c ) ∇L i (W (i) (t)), 1 (t − t c ) t−1 t =tc ∇L i (W (i) (t)) + η 2 E G (t) i 2 F .
Applying the identity a, b = 1 2 a 2 + 1 2 b 2 − 1 2 a − b 2 on the cross term we have:
E W (i) (t + 1) − W(t c ) 2 F ≤ E W (i) (t) − W(t c ) 2 F + η 2 (t − t c ) E ∇L i (W (i) (t)) 2 + 1 (t − t c ) t−1 t =tc ∇L i (W (i) (t)) 2 + η 2 1 n (xj ,yj )∈Si ∇ (W (i) (t); x j , y j ) 2 .
Plugging the gradient upper bound from Lemma 2 yields:
E W (i) (t + 1) − W(t c ) 2 F ≤ E W (i) (t) − W(t c ) 2 F + η 2 (t − t c ) O m d L i (W (i) (t)) + 1 t − t c t−1 t =tc O m d L i (W (i) (t )) + η 2 O m d L i (W (i) (t)).
Since we assume L i (W (i) (t)) ≤ L i (W(t c )) for any t c ≤ t ≤ t c + τ − 1, we have:
E W (i) (t + 1) − W(t c ) 2 F ≤ E W (i) (t) − W(t c ) 2 F + η 2 τ O m d L i (W (i) (t c )) + O m d L i (W (i) (t c )) + η 2 O m d L i (W(t c )).
Do the telescoping sum from t + 1 to t c will conclude the proof:
E W (i) (t + 1) − W(t c ) 2 F ≤ O (η 2 τ 2 + η 2 τ ) m d L i (W(t c )).
The next lemma will reveal the boundedness of objective in stochastic setting. The slight difference to the dynamic of objective in deterministic setting (Lemma 6) is that, we show the objective is strictly decreasing in Local GD, but here we only derive a small upper bound of it: L i (W (i) (t)) ≤ O(1) · L i (W(t c )), with high probability. Even though it is not a strictly decreasing loss, it is enough to enable us to prove linear convergence of objective.
Lemma 9. For Local SGD, at any iteration t in between two communication rounds: t c ≤ t ≤ t c + τ − 1 and i ∈ [K], if W (i) (t) ∈ B(W (0), ω), then with probability at least 1 − exp −Ω(mω 2/3 ) , the following statement holds true:
L i (W (i) (t)) ≤ O(1) · L i (W(t c )).
Proof. We examine the absolute value bound for L i (W (i) (t − 1)):
L i (W (i) (t)) ≤ L i (W (i) (t − 1)) + η ∇L i (W(t − 1)) G (i) (t) + ηC L i (W (i) (t − 1)) · ω 1/3 L 2 m log(m) √ d · G (i) (t) 2 + η 2 C L 2 m d G (i) (t) 2 2 ≤ L i (W (i) (t − 1)) + ηm √ n d L i (W (i) (t − 1)) + ηC ω 1/3 L 2 m mn log(m) d √ d L i (W (i) (t − 1)) + η 2 C L 2 m 2 n d 2 L i (W(t − 1)) ≤ 1 + O ηm √ n d L i (W (i) (t − 1))
≤ 1 + O φ mτ n 2.5 log 2 m τ L i (W(t c )) ≤ exp φ mn 2.5 log 2 m L i (W(t c )) ≤ O(1) · L i (W(t c )).
B.2 Proof of Theorem 2
With the above lemmas in hand, we can finally proceed to the proof of Theorem 2. We prove Theorem 2 by induction. Assume the following induction hypotheses hold for all h ≤ t, with probability at least 1 − e −Ω((log m) 2 ) :
(I) W(h) − W(0) ≤ ω, W (i) (h) − W(0) ≤ ω, ∀i ∈ [K],
(II) L(W(t c )) ≤ n log 2 m · e −R/R0 ,
where ω = O φ 3/2 n −6 L −6 log −3/2 (m) , t c is the latest communication round of h, also the cth communication round, and R 0 = n 5 log 2 m
Proceedings of the 25 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2022, Valencia, Spain. PMLR: Volume 151. Copyright 2022 by the author(s).
Theorem 1 (
1Local GD). For Local GD, under Assumption 1, if we choose m ≥ Kdn 16 L 12 (log m) probability at least 1 − e −Ω((log m) 2 ) it holds that L(W (T )) ≤ e −Ω(R) L(W (0)),
Figure 1 :
1the convergence rate of models with L = 5 and various m ∈ {10, 50, 100}, compared with the single layer model with m = 50. If we use the same number of neurons as the single layer (i.e. Comparing the effect of model size using Local SGD on homogeneous and heterogeneous data distribution. By changing the model size from m = 50 to m = 1000, the model converges faster. In heterogeneous setting the increase in the model size has more impact on the convergence rate than the homogeneous setting.
Figure 2 :Figure 3 :
23Comparing the effect of synchronization gap (i.e., number of local updates τ ) on the model convergence. In this experiment the model size is fixed on m = 50. τ = 20, m = 50, L = 1) Local (τ = 20, m = 10, L = 5) Local (τ = 20, m = 50, L = 5)Local (τ = 20, m = 100, L = 5) The effect of number of layer L. If we increase the number of layers to 5, compared to single layer, we need more neurons (m) to converge with the same rate as single layer.
2 Related Work
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from decentralized data. In Artificial Intelligence
and Statistics, pages 1273-1282, 2017.
AcknowledgementThis work was supported in part by NSF grant 1956276.. Then, we need to show that these two statements hold for t + 1.B.2.1 Proof of inductive hypothesis IStep 1:Bounded virtual average iterates. Now we prove the first hypothesis for t + 1: W(t + 1) − W(0) ≤ ω. By the updating rule we know that:Since we choose m ≥ n 18 L 12 d log 5 m φ 3 , we conclude that W(t + 1) − W(0) ≤ ω.Step 2: Bounded local iterates. Then we prove the second hypothesis for t + 1:By the updating rule we know that:where we apply the gradient upper bound (Lemma 2) and the decreasing nature of local loss (Lemma 9). Since we choose m ≥ Kdn 18 L 12 log 3 m φ 5, we know that W (i) (t + 1) − W(0) ≤ ω.B.2.2 Proof of inductive hypothesis IINow we proceed to prove that hypothesis II holds for t + 1. If t c ≤ t + 1 < t c+1 , then the statement apparently holds for t c . If t + 1 ≥ t c+1 , we have to examine the upper bound for L(W(t c+1 )). The first step is to characterizewhere in we use the identity a, b = 1 2 a 2 + 1 2 b 2 − 1 2 a − b 2 . We plug in the semi gradient Lipschitzness from Lemma 5 and gradient bound from Lemma 2 in last inequality to get:Choosing ω = φ 3/2Cωn 6 L 6 log(m) 3/2 where C ω is some large constant and plugging in local model deviation bound from Lemma 8, to get the main recursion relation as follows:Also by semi smoothness, we have:Taking log on the both sides of(5)and(6)yields:So we can apply martingale concentration inequality. With probability at least 1 − e −Ω(log 2 m)where in the last inequality we use the fact that 2aPlugging that η = dφ mτ n 3 log 2 m yields:at the last step. According to Allen-Zhu et al[2], log[L(W(0))] ≤ O(n log 2 m) with probability at least 1 − e − log 2 m , and using our choice R ≥ Ω n 5 log 2 m φ 2 log n log 2 m we have the following bound: log[L(W(T ))] ≤ O(n log 2 m) − Ω log n log 2 m ≤ log , so we conclude that L(W(T )) ≤ , or equavilently, L(W(T )) ≤ n log 2 m · e −R/R0 , where R 0 = n 5 log 2 m φ 2. | []
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[
"When Does the Inflaton Decay?",
"When Does the Inflaton Decay?"
]
| [
"C Armendariz-Picon \nDepartment of Physics\nSt. Lawrence University\n13617CantonNYUSA\n"
]
| [
"Department of Physics\nSt. Lawrence University\n13617CantonNYUSA"
]
| []
| In order for the inflaton to decay into radiation at the end of inflation, it needs to couple to light matter fields. In this article we determine whether such couplings cause the inflaton to decay during inflation rather than after it. We calculate decay amplitudes during inflation, and determine to what extent such processes have an impact on the mean and variance of the inflaton, as well as on the expected energy density of its decay products. Although the exponential growth of the decay amplitudes with the number of e-folds appears to indicate the rapid decay of the inflaton, cancellations among different amplitudes and probabilities result in corrections to the different expectation values that only grow substantially when the number of e-folds is much larger than the inverse squared inflaton mass in units of the Hubble scale. Otherwise, for typical parameter choices, it is safe to assume that the inflaton does not decay during inflation. * Electronic address: [email protected] 1 arXiv:1708.09755v2 [hep-th] | 10.1088/1475-7516/2018/05/018 | [
"https://arxiv.org/pdf/1708.09755v2.pdf"
]
| 119,059,173 | 1708.09755 | 17b61c98aad1cf2a82e8ec0c535945918ab6ba06 |
When Does the Inflaton Decay?
C Armendariz-Picon
Department of Physics
St. Lawrence University
13617CantonNYUSA
When Does the Inflaton Decay?
In order for the inflaton to decay into radiation at the end of inflation, it needs to couple to light matter fields. In this article we determine whether such couplings cause the inflaton to decay during inflation rather than after it. We calculate decay amplitudes during inflation, and determine to what extent such processes have an impact on the mean and variance of the inflaton, as well as on the expected energy density of its decay products. Although the exponential growth of the decay amplitudes with the number of e-folds appears to indicate the rapid decay of the inflaton, cancellations among different amplitudes and probabilities result in corrections to the different expectation values that only grow substantially when the number of e-folds is much larger than the inverse squared inflaton mass in units of the Hubble scale. Otherwise, for typical parameter choices, it is safe to assume that the inflaton does not decay during inflation. * Electronic address: [email protected] 1 arXiv:1708.09755v2 [hep-th]
I. INTRODUCTION
In order for inflation to be successful, its end has to be followed by reheating, a period during which the universe is populated by radiation [1][2][3]. Once this radiation thermalizes, the universe behaves as in the standard hot Big-Bang cosmology, and all its predictions, from Big-Bang nucleosynthesis to the decoupling of the Cosmic Microwave Background are naturally reproduced and recovered.
The simplest way to transfer the energy stored in the inflaton to that of radiation is for the inflaton to decay. This is typically accomplished by coupling the inflaton itself to lighter matter fields, although gravitational particle production also appears to be possible [4,5]. Yet once the inflaton is coupled to matter, there is no guarantee that its decay will happen after the end of inflation, rather than during inflation. Naively, we would expect the inflaton to decay whenever its decay rate in flat spacetime Γ is larger than the Hubble constant H. Since the inflaton decay rate is proportional to the square of a coupling constant, at "large" couplings the inflaton should decay during inflation, whereas at "small" couplings we would expect it to decay long after it. In fact, in a generally covariant theory the effective coupling constant that determines how rapidly the inflation decays depends on a positive power of the scale factor, which grows exponentially during inflation. As pointed out in [6,7], this suggests that quantum corrections may become large, and thus spoil the standard inflationary predictions.
In this manuscript we attempt to determine whether the inflaton decays during inflation.
The first question we need to face is what we mean by "decay." Even in flat spacetime, the definition of "unstable particle" is not straightforward. But in the end, a particle is unstable when the probability for a transition to a multi-particle final state (that of the decay products) is non-zero. Such transition probabilities are in fact what determines the decay rate of the inflaton during reheating [1][2][3]. Yet, as we shall see, matters are not as simple during inflation. Here we mostly consider three measures that we believe capture the concept of decay: i) The probabilities for the inflaton state to evolve into various multiparticle states, ii) the expectation value of the energy-momentum tensor of the inflaton and its decay products, and iii) the corrections that the couplings to matter introduce in the evolution of the inflaton. The first is the analogue of what is calculated in flat spacetime and in perturbative studies of reheating. The second quantifies the backreaction of the inflaton decay products on the universe's expansion, and the third directly addresses the impact on the evolution of the background (classical) inflaton field.
One of the main obstacles we encounter in the calculation of these measures is the appearance of divergences in the integrals over the modes the inflaton couples to. In flat spacetime, rigorous theorems guarantee that these divergences can be appropriately removed by renormalization of coupling constants and fields, and there are well-defined algorithms that detail how to do so [8]. But in curved spacetimes renormalization is much less developed. In this manuscript we adopt adiabatic subtraction as a regularization and renormalization scheme [9,10]. The main advantages of this scheme is that it can be directly implemented in a timedependent background, without the need to formulate the theory in a manifestly covariant way, that it accomplishes regularization and renormalization in one fell swoop, and that it is one of the main schemes used in the literature on the topic, particularly in the context of calculations of the expectation value of the energy-momentum tensor. As a result, the implementation of adiabatic subtraction in a cosmological setting is relatively simple and straightforward. The main disadvantages of the scheme are that manifest covariance is lost, and that the connection with the counterterms in the action is neither obvious nor manifest. Although adiabatic subtraction has been successfully applied in free field theories, it also remains unclear to what extent it is justified once interactions are included [11]. But, overall, these shortcomings just reflect the status of the renormalization program in curved spacetimes, rather than those of the adiabatic scheme itself.
There is a huge literature on the effect of quantum corrections on the evolution of the inflaton (see for instance [12] and references therein), but, to our knowledge, there are not many references that investigate the matter in the context of the inflaton's decay during inflation. Our work somewhat overlaps with reference [13], whose methods and focus on the inflaton fluctuations significantly deviate from our analysis. It is also related to articles that study axion-like couplings of the inflaton to gauge fields, such as [14], although the nature of these couplings, and the evolution of the matter fields are very different from what we consider here. Whether the inflaton decays or not during inflation may also have implications for the warm inflation scenario initially proposed in [15]. In the latter, the inflaton is assumed to decay during inflation, and the resulting radiation is argued to modify the dynamics of the inflaton and somehow prolong the duration of inflation. Although warm inflation is not our main focus, our results could be used to check its underlying assumptions.
II. ACTION
Our main goal is to study how the couplings of the inflaton needed to reheat the universe affect the background dynamics during inflation. We model the inflaton as an homogeneous scalar field φ, and its decay products as a single, massless scalar χ,
S = d 4 x √ −g − 1 2 ∂ µ φ∂ µ φ − 1 2 m 2 φ φ 2 − 1 2 ∂ µ χ∂ µ χ − λ 2 φχ 2 .(1)
We assume that the inflaton potential is quadratic because it is then simpler to identify quantum states that behave classically, although, actually, most of our results hold for an arbitrary potential. We expect the dominant couplings of the inflaton to be captured by renormalizable terms, namely, cubic and possibly quartic couplings if we model matter by a scalar. These are in in fact the couplings that have been mostly considered in the literature [1]. Note that we do not include any counterterms in the action, nor any of the additional possible operators compatible with the symmetries of the theory (general covariance and a χ → −χ symmetry). In the adiabatic subtraction scheme the divergences are removed directly from the corresponding mode integral, without the need to invoke any counterterms.
Leaving gravity aside, the only interaction in the action (1) consists of the cubic coupling proportional to λ. Our aim is to work in perturbation theory, so we shall assume that λ is "small." Since λ has dimensions of mass, it may not be clear what this means at first. In flat spacetime, at one loop, the renormalized effective potential for φ that follows from the action (1) is [16] V eff (φ) = 1 2
m 2 φ φ 2 + λ 2 φ 2 64π 2 log λφ φ 2 µ ,(2)
where φ µ is a renormalization scale. Because quantum corrections are small whenever the logarithmic term is subdominant, for the time being it appears that λ is small whenever λ m φ . Since gravitational couplings are suppressed by 1/M P , we expect gravitational corrections to be much smaller than those induced by renormalizable couplings, such as the one proportional to λ.
A. Hamiltonian
In order to obtain a manifestly unitary time evolution, we need to find the Hamiltonian of the theory first. Since we are interested in inflation, we shall consider a spatially flat FRW universe,
ds 2 = a 2 (t) −dt 2 + d x 2 ,(3)
where the time coordinate t is conformal time. In that case the canonical momenta are
π φ = a 2φ , π χ = a 2χ ,(4)
and, therefore, the Hamiltonian reads
H = d 3 x π 2 φ 2a 2 + a 2 2 ( ∇φ) 2 + a 4 m 2 φ 2 φ 2 + π 2 χ 2a 2 + a 2 2 ( ∇χ) 2 + a 4 λ 2 φχ 2 .(5)
We shall later be interested in quantities like the expectation value of the inflaton field in Fourier space, φ(t, k) . In an infinite universe the latter is proportional to δ( k) and therefore diverges for the zero mode k = 0. It is hence convenient to perform a transformation to a set of discrete canonical fields in a finite volume universe, in which such expectation values remain finite. We assume that our fields live in a finite universe of comoving volume V = L 3
and impose periodic boundary conditions on the latter. Then the fields can be expanded as
φ = 1 √ V k φ k (t)e i k· x , π φ = 1 √ V k π φ k (t)e −i k· x ,(6a)χ = 1 √ V k χ k (t)e i k· x , π χ = 1 √ V k π χ k (t)e −i k· x ,(6b)
where the sums run over k = 2π L n, n ∈ Z 3 , and the Fourier modes satisfy the Poisson bracket relations
{φ k , π φ k } = δ k k , {χ k , π χ k } = δ k k .(7)
Note that π φ k is the canonical momentum conjugate to φ k . The introduction of a finite volume universe is also useful to regularize the infrared divergences that we shall encounter below.
Translational invariance implies that only the k = 0 mode φ 0 can have a non-vanishing expectation value. We can isolate this mode by averaging the inflaton over the whole universe,
φ V ≡ 1 V V d 3 x φ( x) = φ 0 √ V .(8)
Because of the explicit volume factor, the expectation value of the inflaton is not that of its
zero mode φ 0 ≡ φ k=0 , but instead φ ≡ φ = φ V = φ 0 √ V ≡φ 0 √ V .(9)
Our goal is to study the evolution of the homogeneous inflaton, so it suffices to focus on the evolution of the zero mode φ 0 . The restriction of the Hamiltonian (5) to this mode results
in H = (π φ 0 ) 2 2a 2 + m 2 φ a 4 2 φ 2 0 + k π χ k π χ − k 2a 2 + a 2 k 2 2 χ k χ − k + a 4 λ 2 √ V φ 0 χ k χ − k .(10)
The zero mode φ 0 thus couples to χ through the cubic, momentum-conserving interaction
φ 0 χ k χ − k .
Setting the coupling λ to zero in equation (10) we recover the free Hamiltonian of the theory,
H 0 = (π φ 0 ) 2 2a 2 + m 2 φ a 4 2 φ 2 0 + k π χ k π χ − k 2a 2 + a 2 k 2 2 χ k χ − k .(11)
III. QUANTIZATION
The classical equations of motion of the model admit a solution along which the inflaton slowly rolls down its potential, while the matter fields remain equal to zero. Yet in the quantum theory, we cannot set χ = 0 because of the zero point fluctuations. Since the inflaton couples to χ 2 , such vacuum fluctuations end up modifying the evolution of the zero mode inflaton. Alternatively, we can think of the such couplings as inducing the decay of the inflaton into matter field quanta.
In order to determine the impact of quantum corrections on the background evolution, we obviously need to quantize the theory. We shall treat the inflaton interactions perturbatively by resorting to the interaction picture. In the latter, the fields follow the time evolution determined by the free Hamiltonian, which we study first.
A. Free Fields
In the free quantum theory, the Heisenberg operators φ 0 (t) and π φ 0 (t) satisfy the equaltime commutation relations [φ 0 (t), π φ 0 (t)] = i and the Heisenberg equations
iφ 0 = [φ 0 , H 0 ] = i π φ 0 a 2 ,(12a)iπ φ 0 = [π φ 0 , H 0 ] = −ia 4 m 2 φ φ 0 ,(12b)
where H 0 is the free (quadratic) Hamiltonian (11). In order to find a solution of these equations, we expand φ 0 and π 0 in creation and annihilation operators b † and b as usual,
φ 0 = u(t)b + u * (t)b † , π φ 0 = a 2 [u(t)b +u * (t)b † ],(13)
where [b, b † ] = 1. The time dependent coefficients u(t) are determined by the condition that the fields satisfy the Heisenberg equations and the canonical commutation relations. Thus, u needs to obey the equation of motion
u + 2Hu + m 2 φ a 2 u = 0,(14)
where H =ȧ/a, and satisfy the normalization condition
a 2 [uu * − u * u ] = i.(15)
That a 2 [uu * − u * u ] is constant follows from equation (14), which also happens to be the field equation satisfied by the background inflaton solution.
The quantization of the matter fields χ k proceeds along the same lines, with the minor difference that, in order for χ k to carry a well-defined momentum, the creation and annihilation operators must involve opposite momenta,
χ k = w k (t)c k + w * k (t) c † − k , π χ k = a 2 [ẇ k c − k +ẇ * k c † k ].(16)
In these expressions we have used isotropy, namely, that the equation of motion satisfied by w k ≡ w k only depends on the magnitude of k, as we shall see below. As before, in order for χ k and π χ k to satisfy the canonical commutation relations, the c k and c † k must satisfy the commutation relations [c k , c † k ] = δ k k and the modes w k must be properly normalized,
a 2 [w kẇ * k − w * kẇ k ] = i.
B. Coherent States
We would like the quantum state of the inflaton to reproduce the properties of a classical rolling scalar field. In the case of the harmonic oscillator, states with classical properties are known as "coherent states," and are defined to be eigenvectors of the annihilation operator. Since the free Hamiltonian of the inflaton (11) resembles that of an harmonic oscillator, we shall choose the state of the inflaton in analogy with such coherent states,
|β ≡ N exp(β b † )|0
, where β is a constant that characterizes the state, N is a normalization factor and b|0 = 0.
The expectation value of the inflaton field φ in such a coherent state is
φ = 1 √ V [βu(t) + β * u * (t)] ,(17)
which is the same as that of a rolling inflaton with field value 2Re[βu]/ √ V . In particular, φ satisfies the equation of motion of a classical scalar field in an expanding universe. The variance of the averaged inflaton in a coherent state is
∆φ 2 V ≡ (φ V −φ) 2 = u * u V ,(18)
which can be interpreted as stating that the variance of φ V is of order of the power spectrum on scales of the size of the universe. As long as ∆φ 2 V φ 2 the expectation of the inflaton thus behaves like a classical scalar field. In view of equation (17) this is satisfied for sufficiently large β 1, in analogy with the classical limit of the harmonic oscillator.
Here we are dealing in fact with a two-parameter class of coherent states: The parameter β determines the field expectation, and the overall magnitude of u determines its variance.
Note that, as defined, ∆φ 2 V is quite different from ∆φ 2 . The former is just the variance of the zero mode alone, whereas the latter is the sum of variances of all the modes,
∆φ 2 = (1/V ) k φ k φ − k .
We choose the state of the matter fields to be annihilated by the operators c k , c k |0 = 0.
In the free theory, at tree level, this implies that the expectation of χ is zero, χ = 0. In other words, in the classical theory the matter fields χ vanish. In the quantum theory χ remains zero because of the χ → −χ symmetry of the theory.
C. Shifted Inflaton
Because the expectation of φ in a coherent state is non-zero, these states are not particularly convenient for perturbative calculations, which largely rely on Feynman rules that assume vanishing φ . It is thus convenient to shift the canonical pair (φ 0 , π φ 0 ) by its expec-
tationφ 0 andπ φ 0 , φ 0 ≡φ 0 + ∆φ 0 , π φ 0 ≡π φ 0 + ∆π φ 0 .(19)
Since coherent states have Gaussian wave functions, it is easy to see, at least perturbatively, 1 that working with a field φ 0 in a coherent state with β|φ 0 |β =φ 0 is mathematically equivalent to working in a theory with a shifted field ∆φ 0 in a state |0 with 0|∆φ 0 |0 = 0 and correlation
0|∆φ 0 (t 1 )∆φ 0 (t 2 )|0 = u(t 1 )u * (t 2 ).(20)
The latter are in fact the relations we would obtain by expanding ∆φ 0 and ∆π φ 0 as in equation (13), with the same mode functions u.
Shifting the inflaton by its expectation somewhat changes the structure of the Hamiltonian. In terms of the shifted field, the Hamiltonian of the theory (10) becomes
H = (∆π φ 0 ) 2 2a 2 + m 2 φ a 4 2 ∆φ 2 0 + k π χ k π χ −k 2a 2 + a 2 k 2 2 χ k χ − k + a 4 λ 2 √ V (φ 0 + ∆φ 0 ) χ k χ − k ,(21)
where we have used thatφ 0 andπ φ 0 satisfy the free Hamiltonian equations (note that equations (19) define a canonical transformation.) In perturbation theory in λ we have the freedom to regard the terms proportional to λφ 0 χ k χ − k as part of the interaction, or as part of the free theory. The latter is a better approximation, so we shall choose the free Hamiltonian to be
H 0 = (∆π φ 0 ) 2 2a 2 + m 2 φ a 4 2 ∆φ 2 0 + k π χ k π χ −k 2a 2 + a 2 k 2 2 χ k χ − k + a 4 λ 2 √ Vφ 0 χ k χ − k .(22)
At this point it is important to notice that, once the transition to the shifted field ∆φ 0 has been accomplished, our results become applicable to any inflationary potential V (φ), provided that we simply identify
m 2 φ ≡ d 2 V dφ 2 ≡ V φφ .(23)
What is special about the quadratic potential is the absence of any inflaton self-couplings.
As long as the latter are weaker than the couplings to matter, our analysis should apply without modifications.
D. Mode Functions
By assumption, the background fieldφ satisfies the equation of motion of a scalar in an expanding universe. For simplicity we are going to look at solutions of the background equations in the limit in which the slow roll parameter ≡Ḣ/H 2 − 1 approaches zero and the universe expands as in de Sitter space, and also in the limit in which η ≡ V φφ /H 2 tends to zero and the inflaton field remains frozen (modulo a decaying solution),
a = − 1 Ht , (24a) φ 0 = const.(24b)
Recall that in de Sitter, conformal time t extends from t = −∞ to t = 0, and thus remains negative throughout history. Although it is essential to consider deviations from de Sitter when discussing inflationary perturbations, in our context there is not much loss of generality in the de Sitter limit. On the other hand, the simple structure of equations (24) will simplify many of our analytical calculations considerably.
The normalized solution of the mode equation (14) in the limit → 0 can be taken to be
u(t) = 1 a 1 √ 2k L (−k L t) 1/2−r 1 + i 2r (−k L t) 2r ,(25a)
where k L is an arbitrary constant with dimensions of inverse length, and we have defined
r = 9 4 − η, η ≡ V φφ H 2 .(25b)
In the limit η → 0, the mode function u(t) approaches a constant modulo a decaying term, just like the background solutionφ,
u(t) → H 2k 3 L 1 + i 3 (−k L t) 3−η/3 .(26)
Although it can often be neglected, in some cases cancellations force us to keep track of the decaying mode. In those instances, time integrals often diverge as η → 0, which is why we keep a non-zero η in the exponent of the decaying mode.
The value of k L has remained arbitrary so far. By its very nature, the zero mode always remains outside the horizon, so there is no way to determine its amplitude by tracing its evolution back to the short-wavelength limit during inflation. We shall instead assume that our finite volume universe is part of a larger inflationary patch, so that the mean square fluctuation of the scalar on scales of our finite universe is what one expects from inflation, namely, about H 2 . Since the mean square fluctuation of the scalar on scales of the volume of the universe is given by
∆φ 2 V = |u| 2 V ≈ 1 V H 2 2k 3 L 1 + (k L t) 6 9 .(27)
we shall simply set k L = 1/L and assume that the comoving size of the finite universe L is much larger than the comoving horizon, −k L t 1. In fact, u in equation (26) has the structure of the mode function of a light massive field in de Sitter space in the longwavelength limit, provided we identify the wave number of the mode with k L . In that sense, we can think of k L as the wave number of our finite universe, and of (−k L t) −1 ≡ e N L as the exponential of the number of e-folds since that mode left the horizon. For the same reason, we should not trust the form of u in the regime −k L t > ∼ 1. In order to keep our zero mode normalization explicit, we shall keep all the factors of k L in our equations. For a given arbitrary variance of the zero mode, the value of k L is then determined by equation (27).
With the inflaton shifted by its tree-level expectation, the equation of motion for the matter field mode functions becomes
w k + 2Hẇ k + k 2 + λφ 0 √ V a 2 w k = 0.(28)
The coupling to the inflaton has introduced an effective mass for the field χ,
m 2 χ ≡ λφ 0 √ V = λφ.(29)
In the de Sitter limit we can readily solve for the mode functions of matter w k , since the effective mass m χ = λφ remains essentially constant. In that case the mode functions are
w k (t) = 1 a √ −πt 2 exp iπν 2 H 1 (ν, −kt) ≡ −t/2 a v(−kt), ν ≡ 9 4 − λφ H 2 ,(30a)
where H 1 is the Hankel function of the first kind. We have included an apparently irrelevant phase in the mode function because for a sufficiently massive field, λφ > 9H 2 /4, the order of the Hankel function becomes imaginary, ν = iµ. In this case the mode functions are
w k (t) = 1 a √ −πt 2 exp − πµ 2 H 1 (iµ, −kt) ≡ −t/2 a v(−kt), µ = λφ H 2 − 9 4 . (30b)
In the limit of light matter fields, m 2 χ H 2 the mode functions (30a) approach those of a massless field,
w k = 1 a e −ikt √ 2k 1 − i kt .(31)
E. Interactions
In order to determine the evolution of the inflation in the presence of interactions, we need to solve the Heisenberg equations in the full interacting theory. Because this is not feasible, we resort instead to perturbation theory in the interaction picture. In this approach, operators O carry the free time evolution,
i dO I dt = [O I (t), H 0 (t)] + ∂O I ∂t ,(32)
and states evolve with the interaction Hamiltonian. At time t the state of the system is
|ψ(t) = U I (t, −T )|ψ(−T ) ,(33)
where −T is the time at which the interaction picture is introduced, and
U I (t, −T ) = T exp −i t −T H I (t 1 ) dt 1(34)
is the time evolution operator in the interaction picture. As usual T is the time-ordering operator, and H I is the interaction Hamiltonian in the interaction picture. In the case at hand, from equation (21) H
I = a 4 λ 2 √ V k ∆φ 0 χ k χ − k ,(35)
where the interaction picture fields ∆φ 0 and χ k are now free fields, as in subsection III A.
Perturbative calculations are carried out by expanding U I to the desired order in the coupling constants. Say, to second order in λ
U I (t, −T ) ≈ 1 − i t −T H I (t 1 ) dt 1 − t −T dt 1 t 1 −T dt 2 H I (t 1 )H I (t 2 ).(36)
Interactions also affect the vacuum state. We would actually like to calculate the expectation value of different operators in the vacuum state of the full interacting theory, rather than that of the free theory. In order to obtain the former from the latter, we shall use a well-known theorem by Gell-Mann and Low [17]: In the interaction picture we choose the initial state |ψ(−T ) to be the vacuum |0 of the free theory, allow the initial time −T to approach −∞, and switch on the interactions "infinitely slowly" by multiplying the coupling constant λ by e ε t , where ε is a positive parameter that is taken to zero at the end of the calculation. The inclusion of this factor not only recovers the interacting vacuum from that of the free theory, but also regularizes some of the oscillatory integrals in the limit T → ∞.
For simplicity we shall not write down this factor explicitly in our integrals, and its presence shall remain implicit.
IV. TRANSITION AMPLITUDES AND PROBABILITIES
One of the main focuses of particle physics is the S-matrix. The latter is the overlap between appropriately defined in and out particle states, but it can also be expressed as the matrix element of the time evolution operator in the interaction picture [18],
Φ out |Ψ in = φ|U I (+∞, −∞)|ψ ,(37)
where |φ and |ψ are free particle states whose quantum numbers match those of the out and in states. Whenever such such a matrix element is non-zero for single-particle state |ψ and a multi-particle state |φ , the particle described by |ψ is unstable.
In an inflating spacetime there is no static out region, because the time derivatives of the scale factor (24a) diverge in the asymptotic future t → 0 − . Therefore, it does not seem possible to define appropriate out states, and there is no full analogue of an S-matrix.
Nevertheless, the time evolution operator in the interaction picture between the asymptotic regions t → −∞ and t → 0 − is still well-defined, and one may compute its matrix elements between free states as well. If the transition amplitude between state of the inflaton and any other state is non-zero, that would be an indication that the inflaton is unstable.
To make this idea more precise, consider the vacuum expectation value of an arbitrary (hermitian) observable O in the presence of interactions,
O(t) = 0|U † I (t, −∞)O I (t)U I (t, −∞)|0 ,(38)
where U I is the time-evolution operator (34), and the interaction picture operator O I follows the free time evolution. There is no need to divide the expectation value by the in-in amplitude 0|U † I U I |0 because the latter equals one, as U I is unitary. For the same reason, in an expansion of the expectation value in terms of Feynman diagrams it suffices to consider connected diagrams alone.
It is convenient to expand the time-evolution operator U I into the identity plus a piece related to the interactions. Defining the operator T by the relation U I (t, −∞) = 1 + iT, and inserting appropriate resolutions of the identity, the expectation value (38) becomes
O(t) = 0|O I |0 − 2 Im ψ 0|O I |ψ T ψ + ψ,ψ T * ψ ψ |O I |ψ T ψ ,(39)
where we have introduced the transition amplitude between the vacuum and a free state |ψ
T ψ ≡ ψ|T |0 .(40)
Equation (39) shows that all we need to know to calculate the expectation value of any operator are its matrix elements in the free theory, ψ |O I |ψ , and the transition amplitudes to those states, T ψ . If the transition amplitude T ψ is non-zero, it is as if the inflaton state had made a transition from |0 to |ψ , as expected from a decay.
Actually, some of the summands in equation (39) cancel out and need not be considered.
Because U I is unitary, U † I U I = 1. Inserting in the last equation U I = 1 + iT yields the "optical theorem"
2 Im T 0 = ψ P ψ ≡ P tot ,(41)
where T 0 is the vacuum persistence amplitude, P ψ = |T ψ | 2 the transition probability to the state ψ, and P tot the total transition probability. Because of the optical theorem, the summand −2Im 0|O I |ψ T 0 in equation (39) is cancelled by the disconnected piece of ψ,ψ T * ψ ψ |O I |ψ T ψ . The disconnected piece of a matrix element is defined by the relation
ψ 2 |O I |ψ 1 disc ≡ 0|O I |0 ψ 2 |ψ 1 = 0|O I |0 δ ψ 2 ψ 1 .(42)
If we represent the matrix element ψ 2 |O I |ψ 1 diagrammatically, its disconnected piece is the contribution from disconnected diagrams, for which the external lines that represent the states ψ 1 and ψ 2 simply go through the diagram, and are hence disconnected from the operator insertion O I (see the example on figure 1.) This cancellation is essentially the same that allows disconnected diagrams to be disregarded in the in-in formalism. In field theories in Minkowski spacetime the instability of a particle is also signaled by the appearance of a non-zero imaginary component of its forward scattering amplitude. Here, the quantum state of the inflaton is not a one-particle state, but a coherent state with an infinite number of quanta. By shifting the inflaton field, the quantum this state becomes the vacuum |0 , whose stability we expect to be quantified by the vacuum persistence amplitude T 0 .
One of the main goals of this section is the calculation of transition amplitudes and probabilities, not only because the former capture the intuitive concept of "decay," but also because they automatically determine the expectation value of any observable. Two of the observables we shall be concerned about are the field deviation ∆φ 0 and its square ∆φ 2 0 . The former has a non-zero matrix element between the vacuum and a state with a single excitation of the inflaton zero mode, |L = b † |0 , so we shall be interested in the transition amplitude and probability to such an excited state,
T L ≡ L|T |0 , P L = |T L | 2 .
(43) Here, and in what follows, we shall refer to a single excitation of the inflaton's zero mode by "L." The matrix element 0|∆φ 2 0 |L vanishes. But since, ∆φ 2 0 has a non-zero matrix element with states that contain two long quanta, we shall also be interested in the transition amplitudes
T LL ≡ L, L|T |0 ≡ 1 √ 2 0|b b T |0 ,(44)
as well as in the amplitudes and probabilities
T L k− k ≡ L, k, − k|T |0 , P L k− k = |T L k− k | 2 ,(45)where |L, k, − k = b † c † k c † − k |0
is a state with a single inflaton zero mode and two matter quanta of opposite momenta k. The expectation value of ∆φ 2 0 in the state |L, k, − k does not depend on the value of k, which is why we shall also encounter the total decay probability into three quanta
P 3 = 1 2 k P L k− k .(46)
(We include a factor of 1/2 in the sum because |L, k, − k and |L, − k, k represent the same state.) At lowest order in λ, the sum of P L and P 3 is simply the total decay probability of the inflaton, P tot . For a cubic interaction of the form (35) the only two diagrams that contribute to T 0 at leading order (∝ λ 2 ) are those on figure 2. Cutting both diagrams vertically through the middle reveals the final states the inflaton can decay into, namely, a single zero mode quantum or a zero mode plus two matter quanta. These are of course the decays whose probability is captured by P L and P 3 . Finally, we shall also need to calculate the transition
amplitude into two matter fields | k, − k = c † k c † − k |0 , T k,− k ≡ k, − k|T |0 ,(47)
which is of order λ 2 and enters the leading order correction to the energy-momentum tensor of the inflaton decay products. In flat spacetime, all the transition amplitudes above would vanish because of energy conservation. In an expanding background energy is not conserved, so these transitions are allowed. Because spatial translations remain isometries, though, spatial momentum is conserved, which is why the matter fields quanta appear in pairs of opposite momenta. In some instance, relying on transition amplitudes reduces and better organizes the number of diagrams needed to be considered, and is thus computationally simpler than a direct calculation of the expectation value in the in-in formalism.
A. Decay at First order
At first order in λ, the only two possible final states are |L, k, − k and |L , as shown in figure 3. To render the analytical calculations somewhat more manageable, we are going to consider two opposite limits: The limit in which the effective mass of the decay products is much larger than the Hubble scale H, and the limit in which the effective mass is much smaller than H. By equation (29), the effective mass of matter particles is determined by the inflaton field, so for fixed λ, each limit can be regarded as a limit of large or small inflaton values or coupling constants. Note that our results only depend on the effective mass of the particles χ, and not on the origin of this mass. In the presence of a "bare" mass term m 2 0 χ 2 in the action of the theory (1), the effective mass becomes m 2 χ = m 2 0 + λφ, and all our results carry through by using the last expression instead of equation (29).
We begin by evaluating the transition amplitude to the state |L, k, − k to lowest order,
iT L k− k ≡ −i t −∞ dt 1 L, k, − k|H I (t 1 )|0 = −i λ √ V t −∞ dt 1 a 4 (t 1 )u * (t 1 )w * k 2 (t 1 ),(48)
where we have used that the interaction Hamiltonian is that of equation (35). Substituting the form of the mode functions and changing integration variables to z 1 = −kt 1 this becomes
T L k− k = − 1 2 2k 3 L V λ H ∞ −kt dz 1 z 1 1 − i 3 k L k z 1 3−η/3 v * 2 (z 1 ),(49)
which happens to depend on k only through the combination kt. We then obtain the decay probability into three quanta (46) by adding all the individual probabilities,
P 3 = 1 2 k |T L k− k | 2 ≈ V 4π 2 1 (−t) 3 ∞ 0 dz z 2 |T L k− k (z)| 2 ,(50)
where we have approximated the sum over k by an integral and, again, z = −kt. For large z 1 , the function v in equation (49) approaches the flat spacetime limit e iz 1 / √ z 1 , which implies that at large −kt the transition amplitude T L k− k (z) is of order e −2iz /z 2 . This renders the integral over z in (50) convergent.
Things are quite different for the transition amplitude T L in equation (43). At lowest order in λ, the latter is instead
T L = − t −∞ dt 1 L|H I (t 1 )|0 ≈ − λ √ V 2H 4 χ 2 I t −∞ dt 1 t 1 u * (t 1 ) t 3 1 ,(51)
where we have used that the expectation of χ 2 I (t, x) in de Sitter is space and time independent,
χ 2 I (t, x) = H 2 4π 2 ∞ 0 dz z 2 |v(z)| 2 .(52)
The proportionality of T L to χ 2 I at this order can be seen on diagram (b) in figure 3. Some care must be taken in the evaluation of the time integral in (51), because we shall later need its imaginary part, which is proportional to the decaying mode, whose integral diverges in the strict massless limit η = 0. Keeping track of the decaying mode we arrive at
t −∞ dt 1 t 1 u * (t 1 ) t 3 1 ≈ H k 3/2 L √ 2 1 3 1 (−k L t) 3 − i η 1 (−k L t) η/3 .(53)
To calculate χ 2 I in equation (52), we note that for large z, |v(z)| 2 approaches 1/z, and the integral over z diverges quadratically. To make sense of χ 2 I (and thus T L ) we need to regularize and renormalize. As we mentioned in the introduction, in this work we rely on adiabatic subtraction [9,10], which takes care of both steps at once. In this approach, we subtract from the transition amplitude the expression obtained by replacing the mode functions by adiabatic approximations. The adiabatic order of the approximations is simply set by the requirement that the subtracted expression be finite for any of the free parameters of the theory. In the case at hand, it thus suffices to subtract the second adiabatic order approximation v (2) ,
χ 2 I ren = H 2 4π 2 ∞ 0 z 2 |v(z)| 2 − |v (2) (z)| 2 .(54)
where the form of the adiabatic modes v (2) follows from the results in Appendix A,
|v (2) (−kt)| 2 = − 1 t 1 ω 0 − 1 2ω 3 0 3 4ω 2 0 ω 2 0 − 1 2ω 0 ω 0 −ä a .(55)
To emphasize that a given quantity has been renormalized in the adiabatic scheme, we append the superscript "ren" to it. Combining equations (51) and (53) we thus get a relation between T L and the renormalized value of χ 2 I ,
T ren L = − 1 6 k 3 L V 2 λ χ 2 I ren H 3 1 (−k L t) 3 1 − 3i η (−k L t) 3−η/3 .(56)
Note the divergence with 1/η, which is why we had to avoid the limit η → 0 in the decaying mode of u.
Limit of Heavy Fields
We proceed now to the evaluation of the different transition amplitudes and probabilities in the limit of heavy matter fields, m χ H. In this limit it is useful to approximate the Hankel function by the uniform expansion in Appendix B. In the heavy field limit, it suffices to keep just the first term in the expansion
v(z) ≈ e i µ ξ(z) (µ 2 + z 2 ) 1/4 + O(µ −2 ).(57)
The integral over z 1 in equation (49) is then highly oscillatory, but there are no points where the phase ξ(z) is stationary. We can evaluate the integral instead by repeated integration by parts, which also results in an asymptotic expansion in powers of the small parameter
µ −1 , z −∞ dz 1 f (z 1 ) e −2iµξ(z 1 ) = f (z 1 ) −2iµ dξ/dz 1 e −2iµξ(z 1 ) z −∞ − z −∞ dz 1 d dz 1 f (z 1 ) −2iµ dξ/dz 1 e −2iµξ(z 1 ) .(58)
At next to lowest order in µ −1 we find
T L k− k (z) ≈ i 4 2k 3 L V λ H e −2iµξ(z) µ 2 + z 2 1 − i 3 (−k L t) 3−η/3 + iz 2 (µ 2 + z 2 ) 3/2 + · · · ,(59)
which reaches its largest magnitude in the long-wavelength limit z = −kt → 0, and is suppressed by the large ratio µ 2 ≈ m 2 χ /H 2 . Inserting this amplitude into the decay probability (50) results in
P 3 ≈ 1 512π λ 2 m χ H 1 (−k L t) 3 1 + 9 η 2 (−k L t) 6−2η/3 ,(60)
where we only quote the leading terms.
Although the probability is suppressed by the small factor λ 2 /m χ H, this suppression factor is more than compensated by (−k L t) 3 ≡ e 3N L . The latter is simply the exponential of the number of e-folds since a mode of the size of the entire (finite) universe left the horizon.
Since our finite universe must encompass the visible universe, N L must be larger than about fifty. We thus conclude that for reasonable parameter choices, the decay probability should be exponentially large. At this point the reader may recall Weinberg's work on the future asymptotic behavior of quantum correlators during inflation [6,7]. He argued that as long as field interactions are not proportional to too many powers of the scale factor a, quantum corrections to in-in correlators cannot become large. Our transition amplitude is not an expectation value yet, but in any case, as also noted in [6], a non-derivative interaction of the form √ −g λφχ 2 does not satisfy Weinberg's conditions of convergence. A possible way to avoid the large enhancement of the decay probability could involve derivative couplings of the inflaton to matter, such as √ −g φ∂ µ χ∂ µ χ. Since the latter is proportional to a 2 , rather than a 4 , this is likely to tame the exponential growth of the decay probability. As already pointed out in [6] such derivative couplings are the only possible ones if χ is a Goldstone boson.
Let us turn our attention now to the renormalized transition amplitude (51), which we obtain by replacing χ 2 I by its renormalized counterpart. Using equation (55) we find that
|v (2) (z)| 2 = 1 m 2 H + z 2 1 + 9m 4 H + 22m 2 H z 2 + 8z 4 8(m 2 H + z 2 ) 3 ,(61)
where m H ≡ m χ /H. Then, from the asymptotic expansion (B1) with n = 4 and the second order adiabatic modes (61) we obtain by brute-force calculation that in the limit of large m χ ,
χ 2 (t, x) ren ≈ 29 60 H 2 m 2 χ H 2π 2 .(62)
Cancellations among the different terms in |v (2) (z)| 2 and |v(z)| 2 yield an integral of order 1/µ 2 , which is why we had to keep terms in the uniform expansion to order µ −4 .
To conclude, let us revisit the decay probability P 3 . Later on we shall need to calculate the expectation of an observable that depends on P 3 . The former turns out to contain an additional divergent integral that needs to be renormalized by subtraction of the zeroth order adiabatic modes. Therefore, because in the adiabatic scheme one subtracts from the divergent expectation value its adiabatic approximation, in that context we should subtract from P 3 the zeroth order adiabatic approximation too. Since the zeroth-order adiabatic modes in the heavy field limit have the same functional form as the exact modes, with µ simply replaced by m χ /H, it is easier instead to replace the integrand by its derivative with respect to µ times µ − m χ /H ∼ H/m χ . The derivative lowers the degree of divergence of the expression, which in the case at hand becomes P ren 3 = 9 4096π
λ 2 H 2 H 3 m 3 χ 1 (−k L t) 3 1 + 9 η 2 (−k L t) 6−2η/3 .(63)
Note the suppression by an additional factor of H 2 /m 2 χ as compared to (60), as could have been guessed from the the subtraction procedure we just described.
Limit of Light Fields
In the limit of light matter fields, mχ H, the mode functions are
v(z) ≈ e iz √ z 1 + i z .(64)
Therefore, in the long-wavelength limit −kt 1 the transition amplitude in equation (49) readily evaluates to
T L k− k = 1 6 2k 3 L V λ H 1 (−kt) 3 1 − 3i η (−k L t) 3−η/3 + · · · .(65)
In this case the amplitude is strongly enhanced in the long wavelength limit, by the characteristic factor 1/(−kt) 3 .
The different behavior of the transition amplitude in the long wavelength limit has drastic implications, namely, the total decay probability into a pair of matter quanta (50) blows up in the infrared. Infrared divergences are typical of massless theories, but in this case the divergence survives away from the massless limit m χ = 0. For a finite mass, at small z, v(z) ∼ z −ν and T L k− k ∼ (−kt) −2ν . Therefore, although the integral (50) converges in the ultraviolet z → ∞, where it has the same behavior as in the heavy field limit, it diverges in the infrared, whenever ν > 3/4. Since we are dealing with a finite universe here, the infrared divergence is just an artifact of our continuum approximation. In particular, in a finite universe of size L, the smallest (non-zero) value of k is k IR = 2π/L. Imposing an infrared cut-off at k IR , and focusing in the dominant contribution in the infrared, we obtain instead
P 3 ≈ 1 864π 2 λ 2 H 2 1 (−k 3 L t) 3 (−k IR t) 3 1 + 9 η 2 (−k L t) 6−2η/3 ,(66)
where we have returned to the limit m χ → 0. Incidentally, with k IR = 2π/L, equation (66) is basically what we would have gotten by including only the longest modes in the discrete sum of equation (46). We keep k L and k IR as separate quantities to convey how our results depend on the normalization of the zero mode and the value of the infrared cut-off, but note that both are expected to be of order 1/L. In that case, the probability grows as e 6N L , where N L is the number of e-folds of inflation since the universe left the horizon. Clearly, such an exponential enhancement is likely to overcome any eventual suppression of the probability by λ.
We turn our attention now to the amplitude for a transition between the vacuum and a single excitation of the inflaton zero mode, equation (51). It can be readily checked that for massless fields the difference between the exact mode functions and their second order adiabatic approximation vanishes, thus implying χ 2 I ren = 0. Away from the strict massless limit, the integral (54) remains finite, as can be checked by inspecting the integrand in the ultraviolet (z → ∞) and infrared (z → 0) limits. The integral is dominated by the exact modes z 2 |v| 2 ∼ z 2−2ν , whose contributions diverges as z → 0, while z 2 |v (2) | 2 approaches zero. Hence, in the limit ν → 3/2, provided that k IR m χ , the integral over z is of order
χ 2 (t, x) ren ≈ 3 2 H 2 m 2 χ H 2π 2 .(67)
Although this expression appears to blow up in the limit m χ → 0, this is just an artifact of the infinite volume limit k IR = 0, which is necessary for k IR m χ to hold for all masses m χ .
In a finite volume universe, as m χ → 0 the difference between the exact and adiabatic modes approaches zero at all values of k. In such a way, χ 2 ren remains continuous at m χ = 0.
It is in fact reassuring that we can derive some of the previous results using a somewhat different method. The calculation of the expectation value of χ 2 I (t, x) amounts to the calculation of the propagator of χ in the limit of coincident points. The latter diverges, with coefficients proportional to different curvature invariants. To eliminate these divergences, one subtracts from the coincident limit an appropriate "adiabatic" short-distance expansion of the propagator. Typically one is interested in calculating the renormalized action, or the renormalized energy-momentum tensor, and one needs to subtract an adiabatic expansion of the propagator to fourth order. The subtraction leaves a finite result that can be taken to be the renormalized value of the expectation value. Here, since we are just interested in χ 2 I by itself, it suffices to subtract an expansion to second adiabatic order. 2 In de Sitter spacetime, the renormalized value of χ 2 I calculated as described is (see equation (6.182) in [10])
χ 2 I (t, x) ren = H 2 16π 2 m 2 χ H 2 − 2 ψ 3 2 + ν + ψ 3 2 − ν − log m 2 χ H 2 − 1 + m 2 χ H 2 − 2 3 ,(68)
where ψ ≡ Γ /Γ is the digamma function and we have restored the (finite) term of fourth adiabatic order. In the limit of light and heavy fields this agrees with equations (62) and
(67). 2 In other words, the counterterms we would need to renormalize ∂H I /∂φ 0 ∝ χ 2 are not the same as those needed to renormalize T µν ⊃ √ −g χ 2 . The former would involve curvature invariants proportional to φ, whereas the latter would involve curvature invariants alone.
B. Decay at Second Order
At second order in λ, the vacuum can decay into a pair of quanta, as shown in figure 4.
The transition amplitude to a pair of matter quanta is the sum of diagrams (a) and (b) in figure 4,
T k− k = (a) T k− k + (b) T k− k .(69)
Looking at diagram (a), or directly from the corresponding expressions for the transition amplitude we find
(a) T k− k = −i 2k 3 L V λ H ∞ −kt dz 1 z 1 1 + i 3 − k L z k 3−η/3 |v(z 1 )| 2 T L k− k (z 1 ).(70)
The integral in (70) can be readily evaluated using our previous methods. Since there is no mode sum, no renormalization is required. In the limit of heavy fields, the subleading corrections in the limit −k L t 1 cancel, while they survive in the light field limit, in which we only quote the dominant terms when −kt 1,
(a) T k− k = −i k 3 L V λ 2 H 2 × 1 16 e −2iµξ(−kt) (µ 2 + k 2 t 2 ) 2 if H m χ , 1 72 1 (kt) 6 1 − 6i η (−k L t) 3−η/3 if m χ H.(71)
The contribution from diagram (b) can be evaluated along the same lines. Since there is a mode sum from the closed matter field loop, the latter needs to be renormalized, which is
why diagram (b) is proportional to χ 2 I ren , (b) T ren k− k = λ 2 χ 2 I ren H 4 1 (−k L t) 3 1 48 e −2iµξ µ 2 + k 2 t 2 1 − 3i η (−k L t) 3−η/3 if H m χ , −i 72 1 (−kt) 3 1 − 6i η (−k L t) 3−η/3 if m χ H.(72)
Note that in the limit of constant u, this amplitude obeys (b) T ren k− k = iT L k− k T ren L , where T L k− k and T ren L are the first-order transition amplitudes in equations (49) and (56). Such a relation could have been guessed from the structure of diagram (b).
The transition amplitude to two zero mode quanta T LL is the sum of the contributions from diagrams (c) and (d) in figure 4, It is relatively straightforward to evaluate the renormalized contribution to T LL from diagram (c) for any matter field mass,
T LL = (c) T LL + (d) T LL .(73)(c) T ren LL = i √ 2 k 3 L V 72 λ 2 χ 2 I 2 ren H 6 1 (k L t) 6 1 − 3i η (−k L t) 3−η/3 .(74)
Neglecting the decaying mode in u amounts to keeping only the leading term in the previous expression, the one proportional to (k L t) −6 . In this approximation, it is readily seen that
(c) T ren LL = i √ 2 (T ren L ) 2 , where T ren L
is the (renormalized) transition amplitude into an inflaton zero mode to first order in λ, which we have already calculated in the previous subsection.
This is in fact what diagram (c) appears to suggest. The second contribution, (d) T LL , cannot be that easily recovered from previous amplitudes, and needs to be evaluated explicitly,
(d) T LL = −i √ 2 λ √ V t dt 1 a 4 (t 1 )u * (t 1 ) k w 2 k (t 1 )T L k− k (t 1 ).(75)
If u were constant, the integrals in equation (75) would converge, but because of the decaying mode they do not, and it is necessary to renormalize by subtraction of the zeroth order adiabatic modes. As before, the subtraction does not have much of an effect on the dominant terms in the limit of light fields, which are dominated by the infrared cut-off,
(d) T ren LL ≈ (−k L t) −3 √ 2π 2 λ 2 H 2 × 3 256 H m χ 1 − 6i η (−k L t) 3−η/3 + 3πi 16 H 2 m 2 χ + · · · if H m χ , i 864 1 (−k IR t) 3 1 − 6i η (−k L t) 3−η/3 + · · · if m χ H.(76)
Note that we have kept the subleading term in the heavy field limit because (d) T ren LL will later appear in combination with P ren 3 , which is of order µ −3 .
V. EXPECTATION VALUES
We have framed our analysis so far in terms of decays of the inflaton into states with definite number of quanta. The concept of particle states plays a central role in S-matrix theory, but it faces its limits in curved spacetimes, due to the global nature of the particle concept [10]. In addition, as we have argued, in an inflationary spacetime there is no static out region, so it remains unclear what to make of the exponentially growing decay probabilities that we have encountered.
But in any case, given that inflation is formulated purely as a field theory, it is questionable whether particles should play any role in its description. In the end, all we are interested in is expectation values of different field operators, which is all we need to cast the predictions of the theory. This is the focus of the present section.
A. Zero mode
In the previous subsection we have seen that the probability for the inflaton zero mode to decay is sizable. In order to study the impact of these transitions on the zero mode itself, we shall calculate the expectation of ∆φ V = ∆φ 0 / √ V and ∆φ 2 V = ∆φ 2 0 /V . The first captures how these transitions affect the mean, "classical", value of the inflaton, whereas the latter tells us to what extent the field itself behaves classically. Of course, we could have derived both expectations directly from equation (38).
We begin by noting that in the free theory 0|∆φ 0 |ψ is nonzero only if |ψ describes a single excitation of the inflaton zero mode (this is represented diagrammatically in figure 5.) Therefore, using equation (39) we immediately infer that to leading order in the interaction
∆φ = −2 Im (u T L ) √ V ,(77)
where T L is the (renormalized) decay amplitude into a single zero mode excitation in equation (56). Inserting the latter into (77) we find, both in the limit of heavy and light fields that ∆φ ren ≈ − λ 2η
χ 2 I ren H 2 1 (−k L t) η/3 .(78)
Since T L grows as e 3N L , one may have naively expected ∆φ to grow similarly, since the non-decaying mode of u is constant. Instead, the leading (real) term in the −k L t 1 limit drops out, and we are left with a secular growth proportional to e N L η/3 stemming from the (imaginary) decaying mode of u. Thus, rather than growing with a power of the scale factor, as the transition amplitudes, in the limit N L η 1 the expectation value grows with the logarithm of a. We shall further analyze the implications of (78) in the conclusions. In the meantime, note that the next to leading correction to ∆φ is of order λ 3 .
On the other hand, the free operator ∆φ 2 has non-zero matrix elements between two states with a single inflaton quantum, or between the vacuum and a state with two inflaton quanta (recall that we do not need to consider vacuum diagrams.) In particular, at quadratic order in λ, from equation (39),
∆φ 2 V = |u| 2 V (1 + 2P L + 2P 3 ) − 2 √ 2 V Im u 2 T LL (79a)
where P L ≡ |T L | 2 is the (renormalized) decay probability into a single zero mode, P 3 the decay probability into a pair of matter quanta and a single inflaton, and T LL the transition amplitude into two inflaton quanta. To arrive at equation (79a) we have only included the connected piece of the different matrix elements, as discussed around equation (42). Looking back at our results for the transition amplitudes and probabilities, one may have expected the variance to grow exponentially with the number of e-folds, but it is easy to check that, in fact, the leading late time contributions to the variance cancel again. More precisely, using equations (43), (56) and (74) we find
2 |u| 2 V P ren L − 2 √ 2 V Im [u 2 (c) T ren LL ] ≈ λ 2 8η 2 χ 2 I 2 ren H 4 1 (−k L t) 2η/3 .(79b)
Therefore, the exponential growth of P ren L and (c) T ren LL has no effect on the variance of the zero mode. Incidentally, equation (79b) is the contribution to the expectation of ∆φ 2 V of diagram (a) in figure 6. As shown in reference [7], in the in-in formalism the expectation value of There is yet another cancellation between the two remaining contributions, which add up to 2|u| 2 V P ren
3 − 2 √ 2 V Im u 2 (d) T ren LL ≈ 9λ 2 128π 2 η 1 k 3 L V H m χ 1 (−k L t) η/3 if H m χ , λ 2 96π 2 η 2 1 k 3 IR V 1 (−k L t) 2η/3 if m χ H (79c)
As alluded to earlier, we need the subtracted probability P ren 3 because the expectation of ∆φ 2 V , which depends on (d) T LL , requires renormalization. By the way, the difference in equation (79c) is the contribution of diagram (b) on figure 6 in the in-in formalism.
B. Energy-Momentum Tensor
Our previous methods can be also employed to calculate the expectation of the energymomentum tensor of matter. Actually, since χ couples to the inflaton, it is not possible to separate the energy-momentum tensor of χ from that of the inflaton. Their combined energy-momentum tensor is
T µν = ∂ µ φ∂ ν φ + ∂ µ χ∂ ν χ − 1 2 g µν ∂ ρ φ∂ ρ φ + ∂ ρ χ∂ ρ χ + 2V (φ) + m 2 0 χ 2 + λφχ 2 ,(80)
where we have included a bare quadratic mass term for the field χ for later purposes, and φ ≡φ + ∆φ. Inserting this expansion into the inflaton potential we get
V (φ) = V (φ) + V φ (φ)∆φ + 1 2 V φφ (φ) ∆φ 2 + · · · ,(81)
which is exact for a quadratic potential. Our goal now is to calculate T µν to quadratic order in λ, by regarding the mass term λφχ 2 as part of the free theory.
a. At zeroth order (λ 0 ) in the interaction there is a contribution to T µν stemming from that of the scalar φ in de Sitter. This one would be present even in the absence of inflaton decays, so we shall ignore it here. There is also a non-perturbative contribution from a free scalar χ of mass m 2 χ = λφ in de Sitter. We already only calculated the (renormalized) expectation of χ 2 I using equation (54), but in this case we can directly borrow the desired result from the literature (see equation (6.183) in [10]),
T µν ren = g µν 64π 2 m 2 χ m 2 χ − 2H 2 ψ (3/2 + ν) + ψ (3/2 − ν) − 2 log m χ H + 4 3 m 2 χ H 2 − 29 15 H 4 ,(82)
where, again, ψ is the digamma function. Although this applies to a free scalar χ, it depends on λ because its mass (29) arises from its interactions with the inflaton. The expectation value is at most of order H 4 , which represents a negligible correction to the inflaton energy density as long as H is sub-Planckian. This is the only correction that does not depend on the normalization of the inflaton zero mode u, although it does depend on the value ofφ.
b. At first order (λ 1 ) there is a contribution from the terms linear in ∆φ in the energymomentum tensor. Because of slow-roll, the non-derivative ones ought to give the dominant contribution,
T µν ren ⊃ −V φ (φ) ∆φ ren g µν .(83)
The renormalized expectation of ∆φ ren at first order is quoted in equation (78).
c. At second order (λ 2 ) the number of terms proliferates significantly. There is a contribution from the expectation value of the cubic term λ∆φχ 2 . This expectation is determined by the transition amplitudes into a single inflaton zero mode T L , and into a zero mode plus two matter quanta, T L k− k . The first contribution simply reduces to
T µν ren L = − λ 2 g µν ∆φ ren χ 2 I ren ,(84)
where ∆φ ren is again that in equation (78)
T µν (t, x) L k− k ≈ λg µν V 3/2 k Im u w 2 k T L k− k .(85)
Because the integral over momenta in equation (85) logarithmically diverges in the ultraviolet, we need to subtract the zeroth order adiabatic approximation to render the integral finite. In the limit of heavy fields the ensuing integral over k can be evaluated exactly and happens to be purely imaginary. For light fields the correction diverges in the infrared, which is dominated by the contribution of the exact modes, which is the only one we keep.
We thus arrive at
T µν ren L k− k ≈ g µν × 9 256π 2 1 k 3 L V λ 2 m 2 χ H 4 if H m χ , 1 48π 2 η 1 k 3 IR V λ 2 H 2 (−k L t) η/3 if m χ H.(86)
There are additional contributions from the expectation of the quadratic terms in the energy-momentum tensor. We have already carried out some of the hard work to evaluate these contributions, because they can be readily calculated from the different transition amplitudes that we have found in section IV. But at this point the calculation becomes increasingly difficult, and we are likely to meet the limits of the adiabatic renormalization scheme. To gauge the contribution of these quadratic terms we shall limit ourselves to the simplest one, namely, that proportional to the "bare" mass m 2 0 . Since the terms with derivatives are accompanied by additional factors of a −2 , the former is expected to be the fastest growing. Using equation (39) and taking into account the possible decay channels up to second order we get
T µν ⊃ − m 2 0 2 χ 2 g µν = − m 2 0 2 g µν Re k T * L < L|χ 2 I |L k − k T L k− k − Im k 0|χ 2 I | k − k T k− k + 1 2 k L k − k|χ 2 I |L k − k |T L k− k | 2 . (87)
Each term in this sum has an interpretation in terms of diagrams in the in-in formalism.
Say, the first term corresponds to the diagram (a) in figure 7, the second to diagrams (a) and (b), and the third to diagram (b). The figure does not label the vertices, but recall that in the in-in formalism they are of two types. Hence, each diagram gives rise to several mode sums, and thus the multiple correspondence.
The mode sums that contain T * L T L k− k and (b) T k− k in equation (87) diverge in the ultraviolet, and it suffices to subtract the adiabatic modes of zeroth adiabatic order to render them finite. Following the adiabatic prescription, we also subtract from the remaining (finite) mode sums the appropriate zeroth order approximations. In addition, since the first term in equation (87) contains a divergent tadpole subdiagram, it appears reasonable to replace the latter by its renormalized counterpart, namely, the renormalized expectation of χ 2 at zeroth order. In that case, the expectation value in the heavy field limit becomes
χ 2 ren ≈ 87λ 2 10240π 4 η H m χ 4 1 (−k L t) η/3 if H m χ ,(88a)
which displays the characteristic suppression by powers of H/m χ , and a slow grow with the number of e-folds, unlike the transition probabilities it depends on. If we had not subtracted the zeroth order adiabatic approximation from the finite mode sums, the final results would have been proportional to (H/m χ ) 2 instead.
In the limit of light fields, on the other hand, the three mode sums in equation (87) diverge in the infrared when we approximate them by an integral. Concentrating on the infrared contribution, neglecting adiabatic subtraction, and keeping only the dominant term in the light field limit we find
χ 2 ren ≈ λ 2 128π 4 η 2 k L k IR 3 H 2 m 2 χ 1 (−k L t) 2η/3 if m χ H. (88b)
Again, there is a secular growth in the expectation value, and the latter is further enhanced by the large factor (H/m χ ) 2 . This can only have an impact on the energy-momentum tensor of matter for non-vanishing m 0 . But, of course, given the symmetries of the theory, there is no reason for m 0 to vanish. As we mentioned, one way to rule out a mass term is to assume that χ is a Goldstone boson. But in that case, its couplings to the inflaton would need to involve derivatives.
Finally, it is also instructive to check whether these results have any impact on the energymomentum tensor of the inflaton field itself. Because we are only concerned with the zero mode, and the latter slowly evolves during inflation, it suffices to consider the expectation of the non-derivative terms, namely, in full analogy with equation (87). We already have calculated ∆φ 2 V in equations (79). Inspection of those equations reveals a moderate impact on the variance of φ V , in the sense that there is no exponential growth in either mass limit.
T µν ⊃ − V φφ (φ) 2 ∆φ 2 V g µν ,(89)
VI. INFLATON DYNAMICS
Up to this point we have seen that the inflaton decay probability rapidly grows during inflation, but that such a growth does not directly impact the expectation values of the different field operators that we have studied, which in the limit η → 0 grow with the logarithm of the scale factor. But if we are interested in knowing whether it is a good approximation to assume that the inflaton evolves as in the absence of matter couplings, an approach that directly focuses on the evolution of the zero mode is somewhat more efficient.
A. Quantum Corrected Equation of Motion
The approach most widely used in the literature to study the impact of quantum corrections on the evolution of the inflation involves the quantum effective action Γ eff , and the effective evolution equation δΓ eff /δφ = 0. Because we are interested in the in-in expectation value of the field φ, in order to follow this venue one needs to work with the effective action in the in-in formalism, which makes the whole procedure fairly cumbersome. But this procedure is rather heavy-handed anyway. If one is interested in the evolution of the expectation value of the inflaton it suffices to consider the Heisenberg equations of motion
i d O dt = [O, H ] ,(90)
where we have assumed that the operator O does not depend explicitly on time. Focusing on the zero mode of the inflaton and its conjugate momentum, and using the Hamiltonian (10) we thus get
d φ 0 dt = π φ 0 a 2 ,(91a)d π φ 0 dt = −m 2 φ a 4 φ 0 − ∂H I ∂φ 0 ,(91b)
where we have split the Hamiltonian into a free piece H 0 (11) and an interaction H I ,
H = H 0 + H I .
Combining both equations in (91) and using (9) we get the "quantumcorrected" equation of motion
φ + 2H φ + m 2 φ a 2 φ + 1 a 2 √ V ∂H I ∂φ 0 = 0.(92)
Note that translational invariance implies that the expectation value of the non-zero modes of φ vanishes, which is why we can focus on the evolution of the zero mode. The term in the corrected equation of motion that does not contain time derivatives can be thought of as the derivative of the effective potential in an expanding universe. In particular, note that V −1/2 ∂/∂φ 0 ≡ ∂/∂φ V . If the inflaton potential is not quadratic, the quantum corrected equation of motion still has the form (92), with φ replaced by ∆φ , m 2 φ by V φφ (φ) and ∂/∂φ 0 by ∂/∂∆φ 0 .
Clearly, the expectation value of φ obeys the classical equation of motion, modulo corrections given by the expectation of ∂H I /∂φ 0 . For the interaction Hamiltonian (35), in particular, 1
a 2 √ V ∂H I ∂φ 0 = a 2 λ 2 χ 2 (t, x) .(93)
By construction, these quantum corrections are real, since the operator ∂H I /∂φ 0 is hermitian. In the light of the last equation, the quantum corrected equation of motion has a natural and simple interpretation: For arbitrary values of χ, the classical equation of motion of the homogeneous scalar φ has the formφ + 2Hφ + m 2 φ a 2 φ + λ 2 a 2 χ 2 = 0. When χ is in the vacuum state, we assign to χ the classical value χ = 0, and the previous equation reduces to that of the classical background fieldφ. But quantum-mechanically, we cannot set χ to zero, since it experiences vacuum fluctuations. The correction term in (92) is simply what we get when we replace χ 2 by its vacuum expectation value χ 2 . Note that the quantum-corrected equation of motion does not have the form that has been often quoted in the literature [1,2], φ + (2H + Γa) φ + m 2 φ a 2 φ = 0. The latter contains an additional damping term proportional to φ that does not appear in (92). In particular, as we shall see, in de Sitter space quantum corrections simply add an additional constant driving force to the equation of motion of the inflaton at first order.
The evaluation of ∂H I /∂φ 0 basically amounts to the calculation of χ 2 (t, x) , which is one of the main focuses of quantum field theory in curved spacetimes. In fact, there also is a nice parallel between the equation of motion (92) and the equations of semiclassical gravity in which such calculations are carried out. In the latter the gravitational field is sourced by the expectation value of the energy-momentum tensor, G µν = 8πG T µν . These semiclassical equations are the gravitational (non-linear) analogues of equation (92), with the classical metric playing the role of φ , and the energy momentum tensor playing the role of ∂H I /∂φ 0 .
In the cases we have analyzed, the expectation value of χ 2 grows with a power of time, (62) and (67), p = 0 at order λ 0 , and from equations (88) p = −η/3 or p = −2η/3 at order λ 2 . A particular solution of (92) in those instances is
χ 2 ∝ (−t) p . From equations∆φ = − λ 2 χ 2 m 2 φ + p(p − 3)H 2 ,(94)
which can be thought of as the correction to the inflaton background value due to quantum effects. Quantum corrections are negligible whenever ∆φ φ . This condition in some sense replaces the condition that the term proportional to λ in the effective potential (2) be subdominant. But comparison of both expressions shows that they are in fact very different in nature. Such a disagreement suggests that in some cases it may not be justified to apply quantum corrections derived in Minkowski spacetime to field theories in an expanding universe.
If we had tried to calculate ∆φ V to third order in λ within the in-in formalism, we would have had to evaluate a relatively complicated expression containing three time integrals of the expectation of a term cubic in the interaction. In the present approach, since ∂H I /∂φ 0 is already proportional to λ, it suffices to calculate χ 2 to second order, which considerably simplifies the analysis.
VII. SUMMARY AND CONCLUSIONS
In most inflationary scenarios the universe inflates until the inflaton reaches the vicinity of the bottom of its potential, where the violation of the slow-roll conditions triggers the end of inflation. It is after such an end that the inflaton is supposed to decay into matter and thus reheat the universe.
But in order for the previous picture to hold, the inflaton must survive the inflationary stage. The condition that is taken to signal the decay of inflaton after the end of inflation is ΓH −1 1, where Γ is the decay rate of the inflaton and H the Hubble constant. Since H −1 is proportional to cosmic time, this is equivalent to the demand that the total decay probability of the inflaton become large. Therefore, we would expect the inflaton to survive until the end of inflation as long as its total decay probability remains small.
In Section IV we have calculated various decay probabilities of the inflaton during inflation. We have seen that these probabilities grow rapidly, apparently implying that the inflaton should decay just after a relatively small number of e-folds. But closer inspection reveals that this growth is not translated into exponentially large corrections to the expectation value of the inflaton or the energy-momentum tensor of its decay products, because there are cancellations among the different terms that contribute to the expectation values.
Since we are dealing with a field theory anyway (and are not interested in S-matrix elements) evaluation of the expectation of different field operators appears to be a better strategy to discern whether the inflaton is effectively decaying during inflation.
As an illustration, we shall begin by looking at the impact of such decays on the expectation of the inflaton itself. From equation (78), and because m 2 χ = m 2 0 + λφ ≥ λφ, the leading correction is bounded by the model-independent limit
∆φ φ < ∼ 3 16π 2 1 η H 2 φ 2 1 (−k L t) 2η/3 1,(95)
which interestingly, does not depend on the coupling constant λ. Recall that η ≡ V φφ /H 2 is a slow-roll parameter, and that 1/(−k L t) equals e N L , where N L is the total number of e-folds of inflation. Because in any reasonable inflationary model the size of the field's quantum fluctuations ought to be much smaller than the field itself (H φ ), at least while observable scales are exiting the horizon, we expect the impact of the decay on the background field to be small, unless the number of e-folds is very large, N L η 1. On the other hand, these corrections could play an important role during self-reproduction in hilltop inflationary models [19], in which eternal inflation occurs at field values H/φ > ∼ η.
We obtain a similar constraint using the dynamical correction (94) at first order in λ. The demand that the order λ correction to the energy density (83) be smaller than that of the background also leads to a similar, albeit weaker, limit.
In order to bound the value of the coupling constant λ we need to consider corrections at order λ 2 . To avoid an excessive proliferation of parameters, let us assume that m 2 0 is negligible. Then, the only relevant contributions at second order are those to ∆φ 2 V . The quantum corrections that we have calculated are proportional to different positive powers of H/m χ , and are thus expected to be tighter for light matter fields. Demanding that the light field limit correction to the energy density associated with (79c) be smaller than that of the background we obtain λ φ 2 min (H/φ) 4 , 96π 2 η 2 (−k L t) 2η/3 .
There is an additional condition here because λ needs to be small enough for the field to be light, λφ = m 2 χ H 2 . Equation (96a) is thus not very constraining because it can be satisfied for small enough λ. Note that H/φ is typically very small, particularly in chaotic inflationary models. Therefore, for light matter fields quantum corrections are expected to be automatically small. In the heavy field limit the same analysis of equation (
Our results can be also interpreted in a different light. We have argued above that the zero mode φ k=0 can be thought of as a proxy for the longest mode that left the horizon during inflation. Therefore, they suggest that the inflaton couplings responsible for its decay cannot alter the power spectrum of the inflaton on the largest scales significantly, as long as N L η ≤ 1. This result agrees with the the conclusions of references [6,7], even though the interaction responsible for the inflaton decay is not one of the "safe" or "dangerous" interactions discussed therein.
Finally let us stress that since our conclusions mostly involve renormalized quantities, they depend on the validity of the adiabatic scheme for regularization and renormalization. This is why it would be to useful to repeat our analysis with a more rigorous renormalization scheme, although at this point there does not appear to be a clear consensus as to what the latter should be. Nevertheless, although adiabatic subtraction suppresses quantum corrections by additional factors of H/m χ in the limit of heavy fields, it does not have much of an impact in the opposite limit.
It is fair to say that reheating after inflation remains one of the least investigated aspects of inflation, in spite of the already significant amount of literature devoted to the topic.
Yet the impact on inflation from the couplings necessary for reheating essentially remained unexplored. In this work we have barely scratched the surface of the subject by analyzing one of the simplest decay-inducing interactions. In this simple case we have encountered potentially large corrections in the limit N L η 1, which could signal large corrections to the primordial spectrum of scalar perturbations on large scales. Because of this possibility alone, we believe that the topic deserves further scrutiny.
to W into the solution (A3). Therefore,
w (n) k = 1 a 1 √ 2W (n) exp −i t W (n) dt .(A7)
Once we have obtained the expansion of a given quantity to the desired order, we set of course α = 1.
FIG. 1 :
1The two diagrams that represent the matrix element L|φ 2 0 |L , where |L = b † |0 is the state with a single zero mode inflaton quantum. A dashed line stands for the inflaton zero mode, and a dot for the field insertion φ 2 0 . Diagram (a) is clearly connected. The disconnected part of the expectation, L|φ 2 0 |L disc , is the contribution of diagram (b).
FIG. 2 :
2The two diagrams that contribute to the vacuum persistence amplitude T 0 at leading order. A dashed line represents the inflaton zero mode and a solid line a matter field. By cutting these diagrams vertically through the middle we can identify the particles the vacuum can decay into, namely, a single inflaton zero mode, or a zero mode plus two matter quanta (see alsofigure 3.) Unless otherwise noted, all of our Feynman diagrams stand for transition amplitudes (and not in-in expectation values.)
FIG. 3 :
3The two decay channels at first order in λ. (a) Decay into two matter quanta plus an inflaton zero mode. (b) Decay into a single inflaton zero mode. To match the structure of the matrix elements of T , initial states appear on the right of the diagram, and final states on the left.
FIG. 4 :
4The two relevant decay channels at second order in λ. (a) and (b): Decay into two matter quanta. (c) and (d): Decay into a pair of inflaton zero mode quanta.
FIG. 5 :
5Diagrammatic representation of the first order correction to the expectation of ∆φ 0 in the in-in formalism. The dot denotes the insertion of ∆φ 0 . This tadpole diagram is essentially diagram (a) of figure 2 cut in half.
FIG. 6 :
6The two diagrams that contribute to ∆φ 2 0 at order λ 2 . A dashed line represents the inflaton zero mode and a solid line a matter field. The dot represent the insertion of ∆φ 2 0 . Note that in the in-in formalism each vertex can be of two types (not shown.) an observable can be expressed in terms of nested commutators that involve the interaction Hamiltonian and the observable itself. When one expands the nested commutators, some of the resulting expressions are proportional to [∆φ V (t 1 ), ∆φ V (t 2 )], from which the constant mode in u cancels. This appears to be the origin of the cancellations that we have observed.
and χ 2 I
2ren is the expectation of χ 2 at zeroth order, equation (56). Similarly, the contribution of the second transition to the expectation value of λ∆φχ 2 is
FIG. 7 :
7Contributions to the expectation of χ 2 to order λ 2 in the in-in formalism. The dot represents the insertion of χ 2 (t).
during inflation, where Γ = λ 2 /(32πm φ ) is the decay rate of the inflaton in flat spacetime, we obtain a criterion that is very different from those in equations (96), especially because the former does not involve the background fieldφ. On the other hand, if ΓH −1 > ∼ 1 does signal the decay of the inflaton after the end of inflation, equations (96) imply that there is a wide range of coupling constants for which the inflaton decays shortly after the end of inflation, but not during it. More precisely, the inflaton will typically decay long after inflation if the matter fields it couples to remain light during inflation, but it will decay shortly after the end of inflation if, on top of equation (96b) the coupling constant λ
A simple, though formal, proof can be obtained by changing variables in the generating functional for the n-point functions of φ 0 , Z[J(t)] ≡ Dφ 0 (t) exp(iS 2 [φ 0 ]) exp(i J(t)φ 0 (t)).
128π 2 9φ H (−k L t) η/3 η 4/3 ,(96b)meaning that the coupling constant λ has to be large enough for the matter field to be heavy, but not large enough for it to decay too rapidly. The upper limit in equation (96b) is again not very restrictive because we expectφ/H to be much larger than the other parameters. We could derive various similar constraints by applying the same methods to other expectation values, but these would not produce anything significantly different. We just note that since many of the quantum corrections are proportional to different powers of e ηN L , there appears to be an upper limit on the number of e-folds of inflation of the order N L ∼ 1/η.Alternatively, in the limit N L η ≤ 1 our results are not very sensitive to the normalization of the inflaton zero mode fluctuations, which is determined by k L . In the limit η → 0, the different corrections grow with the logarithm of the scale factor, as many other loop corrections to inflationary observables.
Appendix A: Adiabatic ModesIn the adiabatic subtraction scheme one needs to subtract from a divergent expectation value an adiabatic approximations obtained by replacing the exact mode functions by adiabatic approximations. The latter are solutions of the mode equation (28) expanded in powers of an appropriate slowness parameter. In order to present the form of these solutions, let us first introduce the scaled mode functionw k ≡ aw k , which obeys the equatioñThis is useful because the equation of motion forw k resembles that of an harmonic oscillator with a time-dependent frequency. The adiabatic approximation is essentially a limit of slow expansion. To study this limit we replace the scale factor a(t) by a(αt), and consider the limit of small α. Inserting this scale factor in the equation (A1) and changing variablesThe (positive frequency) normalized solution of this equation can be written down in WKB formwwhere W obeys the relationThis equation can be solved recursively by expanding in powers of α. To leading order,which we shall label as zeroth adiabatic order. The term of order α 0 vanishes, and that of order α iswhich we shall label as the second adiabatic order. Luckily, we shall not need higher orders here. We obtain the adiabatic mode to order n simply by inserting the order n approximation H 1 (iµ, z) = 2 π e −iπ/4 e πµ/2 e i µ ξ (µ 2 + z 2 ) 1/4where we have abbreviated ξ ≡ 1 + z 2 /µ 2 + log z µ + µ 2 + z 2 , p = 1and where the functions U s (p) are those of equation(7.10)in Chapter 10 of reference[21].We shall need to keep terms up to order µ −5 in the expansion at most, so we just gather here the first four functions
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| []
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[
"Effect of nonlinear dissipation on the basin boundaries of a driven two-well or catastrophic single-well Modified Rayleigh-Duffing Oscillator",
"Effect of nonlinear dissipation on the basin boundaries of a driven two-well or catastrophic single-well Modified Rayleigh-Duffing Oscillator"
]
| [
"C H Miwadinou ",
"A V Monwanou ",
"J B Chabi Orou "
]
| []
| []
| This paper considers effect of nonlinear dissipation on the basin boundaries of a diven two-well Modified Rayleigh-Duffing Oscillator where pure and unpure quadratic and cubic nonlinearities are considered. By analyzing the potential an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case the of our oscillator. It is found the effect of unpure quadratic parameter and amplitude of parametric excitation on the critical Melnikov amplitude µcr. Finally, we examine carefully the phase space of initial conditions in order to analyze the effect of the nonlinear damping, and particular how the basin boundaries become fractalized. | 10.1142/s0218127415500248 | [
"https://arxiv.org/pdf/1310.0958v3.pdf"
]
| 27,990,094 | 1310.0958 | f56fc2548ed1d344a9efba9bd7de3bf6deca7416 |
Effect of nonlinear dissipation on the basin boundaries of a driven two-well or catastrophic single-well Modified Rayleigh-Duffing Oscillator
C H Miwadinou
A V Monwanou
J B Chabi Orou
Effect of nonlinear dissipation on the basin boundaries of a driven two-well or catastrophic single-well Modified Rayleigh-Duffing Oscillator
Catastrophic single-well potentialtwo-well potentialmodified Rayleigh-Duffing oscillatormelnikov criterionbifurcationchaotic behaviorbasin boundaries
This paper considers effect of nonlinear dissipation on the basin boundaries of a diven two-well Modified Rayleigh-Duffing Oscillator where pure and unpure quadratic and cubic nonlinearities are considered. By analyzing the potential an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case the of our oscillator. It is found the effect of unpure quadratic parameter and amplitude of parametric excitation on the critical Melnikov amplitude µcr. Finally, we examine carefully the phase space of initial conditions in order to analyze the effect of the nonlinear damping, and particular how the basin boundaries become fractalized.
Introduction
The Rayleigh oscillator is one canonical example of self-excited systems. However, generalizations of such systems, such as the Rayleigh-Duffing oscillator, have not received much attention. The presence of a pure and unpure quadratic and cubic terms makes the Rayleigh-Duffing oscillator a more complex and interesting case to analyze. This oscillator is used to modelize the following phenomenon: a El Niño Southern Oscillation (EN SO) coupled tropical ocean-atmosphere weather phenomenon in which the state variables are temperature and depth of a region of the ocean called the thermocline (where the annual seasonal cycle is the parametric excitation and the model exhibits a Hopf bifurcation in the absence of parametric excitation), a M EM S device consisting of a 30µm diameter silicon disk which can be made to vibrate by heating it with a laser beam resulting in a Hopf bifurcation (where the parametric excitation is provided by making the laser beam intensity vary periodically in time) etc. [16] and [18].
The behavior of Rayleigh-Duffing oscillator with periodic forcing and/or parametric excitations has been investigated extensively by many researchers. For instance, in their work, the siewe siewe and their collaborators [20] have studied the nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well 6−Rayleigh oscillator combined to parametric excitations. Three years ago, Siewe Siewe and al. [2] investigated the effect of the nonlinear dissipation on the boundaries of a driven two-well Rayleigh-Duffing oscillator and in other paper [3], the same authors focussed their analyze on the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potential. However, in many situations, the nonlinear dynamics dominates the behavior of physical systems giving rise to multi-stable potentials or catastrophic monostable potentials. The authors in Refs. [2,3] and [20] showed with a rigorous theoretical consideration that the resonant parametric perturbation can remove chaos in low dimensional systems. They confirmed this prediction with numerical simulations. It is interesting to note that there is a situation analyzed in [21], where Melnikov analysis is applied to a nonlinear oscillator which can behave as a one-well oscillator, a two-well oscillator or three-well oscillator by simply modifying one of its parameters, which acts as a symmetry-breaking mechanism. Therefore, the chaotic behavior using the parametric perturbation in the modified Rayleigh-Duffing oscillator with a two-well potential still needs to be investigated further. Another good example is constituted by the generalized perturbed pendulum [22]. Our aim is to make a contribution in the study of the transition to chaos in the modified Rayleigh-Duffing oscillator by using the Melnikov theory, and then see how the fractal basin boundaries arise and are modified as the damping coefficient is varied. The last part of this work consists of a numerical investigation of the strange attractor at parameter values which are close to the analytically predicted bifurcation curves. In particular, the case of the two-well potential is considered.
The paper is organized as follows. In the next section, after describing the model, analyzing of the model, and some comparison with the simple Rayleigh-Duffing oscillator, the conditions for the existence of chaos are thoroughly analyzed. A convenient demonstration of the accuracy of the method is obtained from the fractal basin boundaries, and this is discussed in Section 4. We conclude in the last section. In the appendixe A we show the Melnikov integration procedures in details.
Desciption and analysis of the model
In this paper, we examine the dynamical transitions in parametric and periodically forced self-oscillating systems containing the cubic terms in the restoring force and the pure and hybrid quadratic and cubic in nonlinear damping function as follows:
x + µ(1 −ẋ 2 )ẋ + βẋ 2 + k 1ẋ x + k 2ẋ 2 x + (γ + α cos Ωt)x +λx 3 = F cos Ωt,(1)
where , µ, β, k 1 , k 2 , γ, λ, F and Ω are parametrs. Physically, µ, k 2 , β and k 1 represent respectively pure, unpure cubic and pure, unpure quadratic nonlinear damping coefficient terms, α and F are the amplitudes of the parametric and external periodic forcing, and √ γ and Ω are respectively natural and external forcing frequency. Moreover λ characterize the intensity of the nonlinearity and is the nonlinear damping parametr control. The nonlinear damping term corresponds to the Modified Rayleigh oscillator, while the nonlinear restoring force corresponds to the Duffing oscillator, hence its name Modified Rayleigh-Duffing oscillator.
In this section, we derive the fixed points and the phase portrait corresponding to the system Eq. (1) when it is unperturbed. If we let = α = F = 0, Eq. (1) is considered as an unperturbed system and can be rewritten aṡ
x = y,ẏ = −γx − λx 3 ,(2)
which corresponds to an integrable Hamiltonian system with the potential function given by
V (x) = 1 2 γx 2 + 1 4 λx 4 ,(3)
whose associated Hamiltonian function is
H(x, y) = 1 2 y 2 + 1 2 γx 2 + 1 4 λx 4 .(4)
From Eqs. (2) and (4), we can compute the fixed points and analyze their stabilities.
• If γ > 0, λ > 0 or γ < 0, λ < 0, the system have one fixed point (0, 0) which is a center.
• For γ > 0, λ < 0, there are three fixed points: two saddles connected by two heteroclinic orbits and one center. The potential defined by Eq. (3) has two-well (see Fig. 1(b)).
• For γ < 0, λ > 0, there are three fixed points: two saddles connected by two heteroclinic orbits and one center. The potential defined by Eq. (3) has catastrophic sigle-well (see Fig. 1(a) ).
Therefore, depending on the values of the external excitation, the system can escape over the potential barrier and dramatically suffers an unbounded motion. The two-well potential function of the unperturbed system (4).
γ = −1, λ = 1, α = 0, β = 0, µ = 0.2, = 0.1, k 1 = 0, k 2 = 0; (b) Phase portrait of the unforced Modified Rayleigh-Duffing oscillator with γ = −1, λ = 1, α = 0.5, β = 0.5, µ = 0.2, = 0.1, k 1 = 0.5, k 2 = 0.5
Taming chaotic behavior in the Modified Rayleigh-Duffing oscillator
In this section, we discuss the chaotic behavior of the system
x + µ(1 −ẋ 2 )ẋ + βẋ 2 + k 1ẋ x + k 2ẋ 2 x + (γ + α cos Ωt)x +λx 3 = F cos Ωt,(5)
where µ, k 1 , β, k 2 , γ, α, λ, Ω and F are assumed to be small parameters. Hence, our dynamical system may be written aṡ
x = y,ẏ = −γx − αx cos Ωt − λx 3 − µ(1 −ẋ 2 )ẋ − βẋ 2 − k 1ẋ x − k 2ẋ 2 x + F cos Ωt,(6)
where F = F When the pertubations are added, the homoclinic orbit might be broken transversely. And then, by the smale-Birkoff Theorem [19], horseshose type chaotic dynamics may appear. It is well known, that the predictions for the appearance of chaos are limited and only valid for orbits starting at points sufficiently close to the separatrix. On the other hand it constitutes a first order perturbation method. Although the chaos does not manifest itself in the form of permanent chaos, and some sorts of transient chaos may showup. Hower, it manifest itself in terms of fractal basin boundaries, as it was shown by [20]. We start our analysis form the unperturbed Hamiltonian Eq. (4). The potential V (x) Eq. (3) has the local peak ( Fig. 1 (b)) or local antipeak (see Fig. 1 (a)) at the saddle point x = 0. Esistence of this point with a horizontal tangent makes possible homoclinic or heteroclinic bifurcations to take place. At the saddle point x = 0, for an unperturbed system ( Fig. 1), the system velocity reaches zero in velocity y = 0 (for infinite time t = ±∞) so the total energy has only its potential part. In this paper, the homoclinic case is be study.
Transforming Eqs. (3,4), for a closen nodal energy (H = 0) and for γ < 0, λ > 0 we get the following expression for velocity:
y = dx dt = 2(− γ 2 x 2 − λ 4 x 4 ).(7)
Now one can perform integration over x:
t − t 0 = ± dx x −γ − λ 2 x 2 ,(8)
where t 0 represents an integration constant. Finally, we get so called homoclinic orbits ( Fig. 2):
x h = ± −2γ λ sech( √ −γ(t − t 0 )) y h = ± 2 λ γsech( √ −γ(t − t 0 )) tanh ( √ −γ(t − t 0 )),(9)
where ' + ' and ' − ' signs are related to left − and righ + sign orbits, respectively ( Fig. 2). Note, the central saddle point x 0 = 0 is reached in time t corresponding to +∞ and −∞ respectively. We apply the Melnikov method to our system in order to find the necessary criteria for the existence of homoclinic bifurcations and chaos. The Melnikov integral is defined as
M (t 0 ) = +∞ −∞ f (x h , y h ) ∧ g(x h , y h )dt,(10)
where the corresponding differential form f means the gradient of unperturbed hamiltonian while g is a perturbation from Eq. (6) after to put α = α and F = F . Eq. (10) can be rewritten as follows:
M ± (t 0 ) = −µ y 2 h dt + µ y 4 h dt − k 1 x h y 2 h dt − β y 3 h dt− k 2 x h y 3 h dt − α x h y h cos Ω(t + t 0 )dt+ F y h cos Ω(t + t 0 )dt,(11)
where t 0 is the cross-section time of the Poincare map and t 0 can be interpreted as the initial time of the forcing term. After substituting the equations of the homoclinic orbits x h and y h given in Eq. (9) into Eq. (11) and evaluating the corresponding integral, we obtain the Melnikov function given by
M ± (t 0 ) = −µI 0 + µI 1 − k 1 I 2 − βI 3 − k 2 I 4 − α sin Ωt 0 I 5 + F sin Ωt 0 I 6 ,(12)
where
I 0 = 2γ 2 λ +∞ −∞ sech 2 ( √ −γt) tanh 2 ( √ −γt)dt, I 1 = 4γ 4 λ 2 +∞ −∞ sech 4 ( √ −γt) tanh 4 ( √ −γt)dt, I 2 = ± 2γ 2 λ −2γ λ +∞ −∞ sech 3 ( √ −γt) tanh 2 ( √ −γt)dt, I 3 = ± 2γ 3 λ 2 λ +∞ −∞ sech 3 ( √ −γt) tanh 3 ( √ −γt)dt, I 4 = 4γ 3 λ 2 √ −γ +∞ −∞ sech 4 ( √ −γt) tanh 3 ( √ −γt)dt, I 5 = 2γ λ √ −γ +∞ −∞ sech 2 ( √ −γt) tanh ( √ −γt) sin Ωt 0 dt, I 6 = ±γ 2 λ +∞ −∞ sech( √ −γt) tanh ( √ −γt) sin Ωt 0 dt.(13)
After evaluation of these elementary integrals (see Appendix ), the Melnikov function is computed.
M ± (t 0 ) = 4µγ √ −γ 3λ − 16µγ 3 √ −γ 35λ 2 ∓ πk 1 γ 2 8 ( 2 λ ) 3 2 ∓ 23 70 πΩF 2 λ sech( πΩ 2 √ −γ ) sin Ωt 0 − παΩ 2 2λ √ −γ cosech( πΩ 2 √ −γ ) sin Ωt 0 .(14)
It is known, that the intersections of the homoclinic orbits are the necessary conditions for the existence of chaos. The Melnikov function theory measures the distance between the perturbed stable and unstable manifolds in the Poincaré section. If M ± (t0) has a simple zero, then a homoclinic bifurcation occurs, signifying the possibility of chaotic behavior. This means that only necessary conditions for the appearance of strange attractors are obtained from the Poincaré-Melnikov-Arnold analysis, and therefore one has always the chance of finding the sufficient conditions for the elimination of even transient chaos. Then the necessary condition for which the invariant manifolds intersect themselves is given by
µ cr = λ 2 4γ √ −γ( 1 3 − 4γ 2 35 ) × παΩ 2 2λ √ −γ cosech( πΩ 2 √ −γ ) ± 23 70 πΩF 2 λ sech( πΩ 2 √ −γ ) ± πk 1 γ 2 8 ( 2 λ ) 3 2(15)
Above this value µ ≥ µ cr the system transit through a global homoclinic bifurcation which is a necessary condition for ap-pearance of chaotic vibrations. This implies that if the perturbation is sufficiently small, the reduced Eq. (6) has transverse homoclinic orbits resulting in possible chaotic dynamics. We study the chaotic threshold as a function of only the frequency parameter Ω. A typical plot of µ against Ω is shown in Fig. 3 , in which the critical homoclinic bifurcation curves are plotted versus the frequency parameter Ω. The threshold of chaotic motion increases with the increasing of the external amplitude µ ( Fig. 3). The region below the homoclinic bifurcation curve corresponding to F = 0.3 (region (I) of Fig. 3) represents the periodic orbits. When µ crosses its first critical value, a homoclinic bifurcation takes place, so that a hyperbolic Cantor set appears in a neighborhood of the saddles (regions (II) and (III) of Fig. 3 ). The dynamics should therefore be chaotic only for large values of the damping. At the same time, when F = 0.5 (regions (I) and (II) of Fig. 3 ) it represents the periodic orbits, while the dynamics should therefore be chaotic in the region (III). Fig. 4 represents the effect of different amplitude parameters values on critical amplitude µ versus frequency showing chaotic regions. When α = 0 and k 1 = 0 the necessary condition for which the invariant manifolds intersect themselves corresponding exactly to the condition which is obtained by Siewe Siewe and al. for Rayleigh-Duffing oscillator [2] (see Fig. 4 (a)). Figs. 4 (b), (c) and (d) show respectively hybrid quadratic nonlinearity, amplitude of excitation parameter and these two parameters simultanious effect on critical amplitude µ cr for F = 0.5. In these case, we noticed that the parameters k 1 , α and F are several effect on melnikov critical amplitude which show when chaotic behavior appear in the modified Rayleigh-Duffing. Critical amplitude increases with the increasing of the parameter k 1 (see Figs. 4 (b)) but with α the critical amplitude are two extrema which show the effect of parametric excitation.
Bifurcation analysis, phase portraits and fractal basins
In this part, we study the behavior of the system given by Eq.1 as a function of the damping parameter for different values of the external perturbation. The bifurcation diagram and the maximal Lyapunov exponents have been represented for the variable x, and they can be seen in Fig.5. A positive Lyapunov exponent for a bounded attractor is usually a sign of chaos. We want to check the threshold of the external amplitude for the onset of possible chaos obtained in Section 3. For Ω = 1, the critical value of the external force has been obtained numerically for F cr = 0.5. Above this value, numerical simulations have been carried out for the selected parameter values F = 0.5 (see Figs.5 (a) and (c)) and F = 0.6 (see Figs. 5(b) and (d)). From these figures, one can see that the thresholds of damping amplitude for the onset of chaos increase when the external amplitude increases above F cr . After the chaotic motion in the small domain of µ ([0, 0.085] for F = 0.5 and [0, 0.13] for F = 0.6), the Lyapunov exponent changes from a negative value to a positive value when µ increases, signifying the appearance of homoclinic chaos motion. From Figs. 3 and 5, we can noticed that the Melnikov critrical value µ cr obtained in Section 3 is confirmed by numerical simulations. The phase portrait of chaotic and periodic orbits have been plotted in Fig. 7 with parameters of Fig.5. Clearly, we noticed that periodic appear when µ = 0.085 and perxist in means forms and is destroyed when this parameter is increasing which indicate the homoclinic chaos is appeared. Fig.6 illustrate the bifurcation diagram and the maximal Lyapunov exponents of our system when the modified parameters equals 0 (α = 0, β = 0, k 1 = 0, k 2 = 0) and its corresponding phase portrait have been plotted in Fig.8. These figures show that our results coincide exactly with the results when the modified parameters equals 0 which are obtained for Rayleigh-Duffing by Siewe Siewe and al.(see [2]). The effect of nonlinear damping parameters, parametric excitation and external forced amplitude are also seeked through these figures.
A basin of attraction is defined as the set of points taken as initial conditions, that are attracted to a fixed point or an invariant set. The basin of attraction in this case signals the points in phase space that are attracted to a safe oscillation within the potential well, and the set of points that escape outside the potential well to the infinity. In order to verify the analytical results obtained in the previous sections, we have numerically integrated the system by using a fourth order Runge-Kutta in order to investigate the homoclinic chaos in our model. We want to study what is the effect of using the nonlinear damping terms on the equation of the oscillator and how the basins of attraction are affected as the coefficient parameter µ is varied. To show the fractal structure, we consider the case of the bifurcation close to the resonance since it may undergo the limit cycles in the system. We see through from Figs.9, 10, 11, 12, 13, 14 , the basin boundaries become fractal, which means that the damping parameter value µ has contributed to the fractalization of the boundaries, with the corresponding uncertainty associated to this fact. For instance, as this control parameter increases above this critical value, the regular shape of basin of attraction is destroyed and the fractal behavior becomes more and more visible (see Fig.12 and 13). Such fractal boundaries indicate that whether the system is attracted to one or the other periodic attractor may be very sensitive to initial conditions. It is also found that even if µ is increased beyond the analytical critical value for the homoclinic bifurcation, it is still possible that the final steady motion could be periodic rather than chaotic. These results confirme our analytical result. Finally, we prove through from Figs. 13 and 14 the modified parameters of habituelly Rayleigh-Duffing oscillator are very effect on chaotic motions of this system.
Conclusion
In this paper, we have studied conditions of a global homoclinic bifurcation in a double well potential modified Rayleigh-Duffing system with pure cubic nonlinear damping coefficient term. Using the Melnikov method we have got the analytical formula for transition to chaos in a one degree of freedom, system subjected to self-excitation term with a non-symmetric stiffness with parametric excitation. In our case this effect is mutually introduced through the modified Rayleigh-Duffing damping and parametric excitation terms. The transition boundaries in the parameter space are obtained, which divide the space into different regions. In each region, the solutions are explored theoretically and numerically. The critical value of damping coefficient µ under which the system oscillates chaotically has been estimated, in the first step by means of the Melnikov method and later confirmed by calculating the corresponding Lyapunov exponent, bifurcation diagrams, and basin of attractions. Results were given for external periodic perturbation. By means of the basin of attraction, we have shown that for certain regions of parameter space, the deterministic system driven harmonically experiences behaviors that may be chaotic or non-chaotic. The Melnikov method, is sensitive to a global homoclinic bifurcation and gives a necessary condition when the damping coefficient µ = µ cr1 is larger than the critical homoclinic bifurcation values. Our analytical results are consistent with direct computations on homoclinic orbits. It is also investigated the effect of unpure quadratic nonlinear damping and parametric excitation amplitude on chaotic behavior through the melnikov criteria and attraction basin.
I 0 = 2γ 2 λ +∞ −∞ sech 2 ( √ −γt) tanh 2 ( √ −γt)dt = 2γ 2 λ +∞ −∞ tanh 2 ( √ −γt) cosh 2 ( √ −γt) dt.(16)
and simple algebraic manipulations:
τ = t √ −γ, tanh τ = ξ,(17)
we obtain
I 0 = 2γ 2 λ √ −γ 1 −1 ξ 2 dξ.(18)
Finally the result of one has a following expression:
I 0 = −4γ √ −γ 3λ .(19)
After same algebraic manipulations, I 1 , and I 4 become finally
I 1 = − 16γ 3 √ −γ 35λ 2(20)
and
I 4 = 0.(21)
On the the hand the integrals I 2 and I 3 can be evaluated by using following expressions:
sinh p τ cosh 2n+1 τ dτ = sinh p+1 τ 2n × [sech 2n τ +
and sinh 3 τ dτ = 1 3 cosh 3 τ − cosh τ
With these expressions, the finite values of I 2 and I 3 are:
I 3 = ± πγ 2 4λ 2 λ(26)
and
I 4 = 0.(27)
Now, we evaluate I 5 and I 6 .
I 5 = 2γ λ √ −γ +∞ −∞ sech 2 ( √ −γt) tanh ( √ −γt) cos Ω(t + t 0 )dt,(28)I 5 = 2γ √ −γ λ sin Ωt 0 +∞ −∞ tanh τ cosh 2 τ sin Ωτ √ −γ dt(29)
We put
I A 5 = +∞ −∞ tanh τ cosh 2 τ sin Ωτ √ −γ dt(30)
This integrale can be calculated by using the residue theorem
f (z)dz = 2πi N k=1 Res[f (z), z k ],(31)
where
Res[f (z), z k ] = 1 (m − 1)! lim z→z k d m−1 z m−1 [(z − z k ) m f (z)].(32)
In our case,
f (z) = 4 exp(z) − exp(−z) (exp(z) + exp(−z)) 3 exp iΩz √ −γ ,(33)
where on the real axis (Fig. 15)Rez = τ :
Imf (z) = tanh τ cosh 2 τ sin( Ωτ √ −γ ).(34)
The multiplicity of each pole of the complex function f (z) (Eq. (33):
z k = ( π 2 + πk)i f ork = 1, 2, ...
can be easily determined as m = 3. After summation of all poles (Fig. 15) we get:
I A 5 = 2πΩ 2 γ sin( Ωz0 √ −γ ) sinh( Ωπ 2 √ −γ )(36)
Finally, the result of the above analysis can be written:
I 5 = 4πΩ 2 λ sin(Ωt 0 ) sinh( Ωπ 2 √ −γ )(37)
Integral I 6 is calculated with the same algebraic manipulations and the can be written as follows: Figure 15: Deformed contour integration schema and imaginary poles.
I 6 = ∓ 23 70 πΩ 2 λ sech( πΩ 2 √ −γ ) sin Ωt 0 .(38)
Fig.represents the corresponding phase portraits between the unforced Rayleigh-Duffing oscillator and the unforced Modified Rayleigh-Duffing oscillator, respectively with single-well (left) and two-well (right) conditions.
Figure 1 :
1(a) The catastrophic single well potential of the unperturbed system (4); b
Figure 2 :
2(a) Phase portrait of the unforced Rayleigh-Duffing oscillator with
Figure 3 :
3Critical amplitude µ versus frequency for two different external amplitude parameters values.
Figure 4 :
4Effect of different amplitude parameters values on critical amplitude µ versus frequency showing chaotic regions.
Figure 5 :
5Bifurcation diagram and corresponding Maximal Lyapunov exponent of Modified Rayleigh-Duffing oscillator equation versus µ with parameters of Fig. 3; (a, c)F = 0.5 , (b, d)F = 0.6.
Figure 6 :
6Bifurcation diagram and corresponding Maximal Lyapunov exponent of Modified Rayleigh-Duffing oscillator equation versus µ when the modified parameters equals to 0 with parameters ofFig. 4(a); (a, c)F = 0.5, (b, d)F = 0.6.
Figure 7 :
7Phase portraits corresponding to Modified Rayleigh-Duffing oscillator with parameters of Fig.5; (a) chaotic orbit µ = 0.0001, (b) period-1 orbit µ = 0.1, (c) period-1 orbit µ = 0.5, (d) chaotic orbit µ = 0.8.
Figure 8 :
8Phase portraits corresponding to Modified Rayleigh-Duffing oscillator when the modified damping parameters equals to 0 with parameters ofFig.6; (a) period-1 orbit µ = 0.5, (b) chaotic orbit µ = 0.695.
Figure 9 :
9Basin of attraction corresponding to the system with µ = 0.0001, the others parameters are: γ = −1, λ = 1, α = 0.3, β = 0.05, Ω = 1, = −1, k 1 = 0.5, k 2 = 0.05 and F = 0.5.
Figure 10 :
10Basin of attraction corresponding to the system with µ = 0.045, and the parameters of 9.
Figure 11 :
11Basin of attraction corresponding to the system with µ = 0.6, and the parameters of 9.
Figure 12 :
12Basin of attraction corresponding to the system with µ = 0.8, and the parameters of 9.
Figure 13 :
13Basin of attraction corresponding to the system with µ = 0.85, and the parameters of 9.
Figure 14 :
14Basin of attraction corresponding to the system with µ = 0.6,and the parameters of 8.
− p − 1)(2n − p − 3)...(2n − p − 2k + 1) 2 k (n − 1)(n − 2)...(n − k) sech 2n−2k ]+ (2n − p − 1)(2n − p − 3)...(3 − p)(1 − p) − p − 2)(2n − p − 4)...(2n − p − 2k) (2n − 3)(2n − 5)...(2n − 2k − 1) sech 2n−2k−1 ]+ (2n − p − 2)(2n − p − 4)...(−p + 2)(−p) (2n − 1)!! sinh p τ dτ(23)sinh 2 τ cosh τ dτ = sinh τ − arctan(sinh τ ),
AcknowlegmentsThe authors thank IMSP-UAC and Benin gorvernment for financial support.Appendix AEvaluation of the integrals from I 0 , to I 6 is straightforward. After substitution x h (t), and y h (t) (Eq. (9)) we have:
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| []
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[
"Prepared for submission to JHEP Discussion of a possible corrected black hole entropy",
"Prepared for submission to JHEP Discussion of a possible corrected black hole entropy"
]
| [
"Miao He \nInstitute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China\n",
"Zi-Liang Wang \nInstitute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China\n",
"Chao Fang \nInstitute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China\n",
"Dao-Quan Sun \nInstitute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China\n",
"Jian-Bo Deng [email protected] \nInstitute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China\n"
]
| [
"Institute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China",
"Institute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China",
"Institute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China",
"Institute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China",
"Institute of Theoretical Physics\nLanZhou University\n730000LanzhouP. R. China"
]
| []
| Einstein's equation could be interpreted as the first law of thermodynamic near the spherically symmetric horizon. By using this method, we investigate the Eddingtoninspired Born-Infeld (EiBI) gravity. Without matter field, the EiBI gravity can also derive the first law. With an electromagnetic field, as the field equations have a more general spherically symmetric solution in EiBI gravity, we find that the entropy would have a correction. Through recalling the Einstein gravity with a more general static spherical symmetric, this correction of the entropy might be generalized to Einstein gravity. Furthermore, we point out that the Einstein gravity and EiBI gravity might be equivalent on the event horizon. At last, under EiBI gravity with the electromagnetic field, a specific corrected entropy of black hole is given. | 10.1155/2018/2315084 | [
"https://arxiv.org/pdf/1610.09762v3.pdf"
]
| 55,379,602 | 1610.09762 | 40b76111c72ddae1bab42042df85a6c2db6a6f26 |
Prepared for submission to JHEP Discussion of a possible corrected black hole entropy
4 Jan 2017
Miao He
Institute of Theoretical Physics
LanZhou University
730000LanzhouP. R. China
Zi-Liang Wang
Institute of Theoretical Physics
LanZhou University
730000LanzhouP. R. China
Chao Fang
Institute of Theoretical Physics
LanZhou University
730000LanzhouP. R. China
Dao-Quan Sun
Institute of Theoretical Physics
LanZhou University
730000LanzhouP. R. China
Jian-Bo Deng [email protected]
Institute of Theoretical Physics
LanZhou University
730000LanzhouP. R. China
Prepared for submission to JHEP Discussion of a possible corrected black hole entropy
4 Jan 20171 Corresponding author.Black HolesThermodynamicsOther Theories of Gravity
Einstein's equation could be interpreted as the first law of thermodynamic near the spherically symmetric horizon. By using this method, we investigate the Eddingtoninspired Born-Infeld (EiBI) gravity. Without matter field, the EiBI gravity can also derive the first law. With an electromagnetic field, as the field equations have a more general spherically symmetric solution in EiBI gravity, we find that the entropy would have a correction. Through recalling the Einstein gravity with a more general static spherical symmetric, this correction of the entropy might be generalized to Einstein gravity. Furthermore, we point out that the Einstein gravity and EiBI gravity might be equivalent on the event horizon. At last, under EiBI gravity with the electromagnetic field, a specific corrected entropy of black hole is given.
In this paper, we derive the first law of black hole thermodynamics from the Einstein's equation near the event horizon, which is analogous with the technique proposed by T. Padmanabhan [14,16]. we also use this technique in EiBI gravity and get the known first law, the results show a more general formula of entropy, which also holds for Ads Schwarzchild black hole and Ads R-N black hole. Motived by this, by supposing a more general static spherically symmetric, we also get the same result in Einstein gravity.
This paper is organized as follows: In Sec.II, there is a derivation of the thermodynamic identity from Einstein gravity. In Sec.III, we investigate the EiBI gravity by using the thermodynamic route to the field equation, and get the formula of entropy. In Sec.IV, the same result has been got by recalling the Einstein gravity for a more general static spherically symmetric. In Sec.V, a corrected entropy in EiBI gravity with electromagnetic field is given. Conclusions and discussions are given in Sec.VI.
Black hole thermodynamic identity from Einstein's equation
The Einstein's equation can be derived from the thermodynamics [13]. On the other side, it is possible to interpret the Enistein's equation near the spherical symmetric event horizon as the first law of thermodynamics. Considering a static spherically symmetric spacetime
ds 2 = −f (r)dt 2 + 1 f (r) dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) , (2.1)
and the event horizon r = r H satisfying f (r H ) = 0, then one can get its thermodynamic quantities
T = κ 2π = f (r H ) 4π , S = πr 2 H , V = 4π 3 r 3 H . (2.2)
If we consider a Anti-de-Sitter spacetime with a negative cosmological constant Λ, there would be a pressure P = −Λ/8π [24]. The mass of black hole was treated as enthalpy and the first law of black hole thermodynamics is
dH = dM = T dS + V dP . (2.3)
Through the Legendre transformation, one can get
dU = T dS − P dV . (2.4)
The Einstein's equation with a cosmological constant is
G µν + Λg µν = 8πT µν . (2.5)
If the metric has the form of eq.(2.1), in the case of T µν = 0, one can obtain the θθ component of the equation
− 1 + f (r) + rf (r) = −Λr 2 ,(2.6)
Setting r = r H , then multiplying above equation by dr H , we can rewrite eq.(2.6) as
f (r H ) 4π d(πr 2 H ) − d r H 2 = − Λ 8π d 4πr 3 H 3 . (2.7)
Noticing eq.(2.2), above equation can be regarded as the first law of black hole thermodynamics, since U = r H /2 for the Schwarzschild solution. For a charged Ads black hole, the metric also takes the form of eq.(2.1). The energymomentum tensor of electromagnetic field is
T µν = 1 4π (F µσ F σ ν − 1 4 g µν F σρ F σρ ) . (2.8)
Its none zero components are:
T tt = f E 2 0 /8π, T rr = −f −1 E 2 0 /8π, T θθ = r 2 E 2 0 /8π, T φφ = r 2 sin 2 θE 2 0 /8π,
where E 0 = Q/r 2 , and Q represents the charge of black hole. According to the Einstein's equations, one can also get
− 1 + f (r) + rf (r) = −Λr 2 − r 2 E 2 0 . (2.9)
By the same technique, and treating Q as a constant, we get
f (r H ) 4π d(πr 2 H ) − d r H 2 + Q 2 2r H = P d 4πr 3 H 3 , (2.10)
which can be also treat as the first law with the thermodynamic quantities in eq.(2.2), and one can verify that U = r H /2 + Q 2 /2r H for the Ads R-N black hole.
Here we keep Q as a constant, which means a chargeless particle falls into the Ads R-N black hole, eq.(2.10) is consist with the first law. For a charged particle falls into the Ads R-N black hole, the event horizon r H would arise due to changes of dM and dQ, then eq.(2.10) could be rewrite as [16]
f (r H ) 4π d(πr 2 H ) − d r H 2 + Q 2 2r H + Q r H dQ = P d 4πr 3 H 3 . (2.11)
Then it would adopt to the first law with the formulation
dU = T dS − P dV + ΦdQ . (2.12)
We should point out that the T µν contributes to the internal energy U . Thus the Einstein's equation can be interpreted as the first law of thermodynamic near the event horizon. The analogous technique was first proposed by T. Padmanabha, and some relevant comments about the meaning of thermodynamic quantities for this result were given [14][15][16]25]. Note that the structure of the equation itself allows us to read off the expression for entropy. This technique has been used for Gauss-Bonnet gravity and Lovelock gravity [16], in which their entropy expressiones are the same with [17,18], respectively. In the next part, we will show our discussion of this method in the Eddington-inspired Born-Infeld gravity [19].
The entropy in Eddington-inspired Born-Infeld gravity
The Eddington-inspired Born-Infeld theory of gravity is based on the Palatini formulation which treats the metric and connection as independent fields [19]. Its action can be written as
S = 1 8πκ d 4 x[ |g µν + κR µν (Γ)| − λ √ g] + S M (g, Γ, Ψ) , (3.1)
where g µν is the metric of spacetime and its determinant is g, R µν is the symmetric Ricci tensor related to Γ, the dimensionless parameter λ = 1 + κΛ, and the parameter κ has the inverse dimension of cosmological constant Λ.
By varying the action with respect to g µν and Γ, one obtains the equation of motion
q µν = g µν + κR µν , (3.2) |q|q µν = λ |g|g µν − 8πκ |g|T µν , (3.3)
where q µν is the auxiliary metric compatible to the connection with
Γ λ µν = 1 2 q λσ (q µσ,ν + q νσ,µ − q µν,σ ) . (3.4)
By combining eq.(3.2) and eq.(3.3), then expanding the field equations to 2nd order of κ [19]
R µν Λg µν + 8π(T µν − 1 2 T g µν ) + 8πκ[S µν − 1 4 Sg µν ] ,(3.5)
where S µν = T α µ T αν − 1 2 T T µν , one can find that the equation is the 1st order corrections to Einstein' equation. On the other hand, EiBI gravity can be interpreted as a correction of the matter term compared with Einstein gravity. Even the EiBI gravity is fully equivalent to the Einstein gravity in vacuum.
Let's consider the thermodynamics from the field equation in this gravity model. Generally, a static spherically symmetric for g µν could be 6) and the auxiliary metric q µν is assumed as [23]
ds 2 g = −ψ 2 (r)f (r)dt 2 + 1 f (r) dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) ,(3.ds 2 q = −G 2 (r)F (r)dt 2 + 1 F (r) dr 2 + H 2 (r)(dθ 2 + sin 2 θdφ 2 ) . (3.7)
The Ricci tensor was calculated as below
R tt = 2 GG H F 2 H + G 2 F F H H + 3 2 GG F F + GG F 2 + 1 2 G 2 F F , (3.8) R rr = −2 H H − F H F H − 3 2 G F GF − G G − F 2F , (3.9) R θθ = 1 − HH F − G G HH F − H 2 F − HH F , (3.10) R φφ = sin 2 θR θθ ,(3.11)
where the X = ∂X/∂r, which we will use in the rest of this paper. Without the matter fields, the eq.(3.3) reduces to
H 2 GF = λr 2 ψf , (3.12) GH 2 F = λr 2 ψf , (3.13) G = λψ ,(3.14)
thus one can obtian
G = λψ, F = λ −1 f, H 2 = λr 2 . (3.15)
Plugging these into eq.(3.2), then the θθ component is
1 − rf (r) − λf (r) = 1 κ (λ − 1)r 2 . (3.16) Near the event horizon r = r H (f (r H ) = 0), we have d r H 2 − f (r H ) 4π d(πr 2 H ) = −P d 4πr 3 H 3 . (3.17)
As the EiBI gravity is fully equivalent to the Einstein gravity in vacuum [19], the above equation could imply the first law. In fact, the black hole solution to EiBI gravity with no source is the same as Schwarzchild-de Sitter metric, which illustrates ψ = 1 and f = 1 − 2m/r + Λr 2 /3 [19]. The thermodynamic quantities of eq.(2.2) also hold in EiBI gravity here in eq.(3.17). Thus the first law can also be got from the EiBI gravity. Next, we would consider the EiBi gravity with matter field. With the electromagnetic field, One can get the energy-momentum tensor according to eq.(2.8) and eq.(3.6)
T tt = (ψ 2 f ) −1 E 2 0 /8π, T rr = −f E 2 0 /8π, T θθ = r −2 E 2 0 /8π, T φφ = r −2 sin −2 θE 2 0 /8π,
where E 0 = Q/r 2 and the Q represents the charge of black hole. Then the eq.(3.3) becomes
H 2 GF = (λ + κE 2 0 ) r 2 ψf ,(3.18)GH 2 F = (λ + κE 2 0 )r 2 ψf , (3.19) G = (λ − κE 2 0 )ψ ,(3.20)
and one can get
G = (λ − κE 2 0 )ψ, F = (λ − κE 2 0 ) −1 f, H 2 = (λ + κE 2 0 )r 2 . (3.21)
The θθ component of eq.(3.2) is written as
1 − λ + κE 2 0 + κrE 0 E 0 λ − κE 2 0 · rf − Y λ − κE 2 0 f = 1 κ (λ − 1)r 2 + E 2 0 r 2 , (3.22) where Y = 2κrE 0 E 0 + G G HH − H 2 . (3.23)
If we assume that the event horizon satisfies f (r H ) = 0 and ψ(r H ) = 0, then set r = r H in eq.(3.22) and multiply it by dr H , noting E 0 = −2E 0 /r, it gives
d r H 2 + Q 2 2r H − r H 2 f (r H )dr H = −P d 4πr 3 H 3 . (3.24)
This equation should be the first law since it can go back to eq.(3.17) when Q = 0. And one can also confirm this by noticing that dU = d(r H /2 + Q 2 /2r H ) and dV = d(4πr 3 H /3). Moreover, it gives the same result in Einstein gravity eq.(2.10). So we should identify that
T dS = r H 2 f (r H )dr H . (3.25)
However, as the metric takes the form of eq.(3.6), the surface gravity could be [26]:
κ = lim r→r H 1 2 ∂ r g tt √ g tt g rr = ψ(r H )f (r H ) 2 . (3.26)
One can obtain the temperature on the event horizon
T = ψ(r H )f (r H ) 4π . (3.27)
Then eq.(3.25) would imply that
dS = 2πr H ψ(r H ) dr H , (3.28) or S = 2πr H ψ(r H ) dr H . (3.29)
Obviously, when ψ = 1, one can get S = πr 2 H . Above all, it seems that the entropy eq.(3.29) is just for the EiBI gravity. However, we will show that it also holds in Einstein gravity once metric takes the form of eq.(3.6).
The entropy in Einstein gravity
In the Einstein gravity, we consider a more general static spherical symmetric configurations of the form
ds 2 = −ψ(r) 2 f (r)dt 2 + 1 f (r) dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) . (4.1)
One can calculate its nozero components of Ricci tensor
R tt = (ψ 2 f ) f 2 − (ψ 2 f ) f 4 − f f + (ψ 2 f ) ψ 2 f + (ψ 2 f ) f r , (4.2) R rr = − (ψ 2 f ) 2ψ 2 f + (ψ 2 f ) 4ψ 2 f − f f + (ψ 2 f ) ψ 2 f − f rf , (4.3) R θθ = 1 − rf 2 f f + (ψ 2 f ) ψ 2 f − f , (4.4) R φφ = sin 2 θR θθ . (4.5) R µν − Λg µν = 8π(T µν − 1 2 T g µν ) . (4.6)
For the Schwarzchild vacuum T µν = 0, according to eq.(4.6), eq.(4.2) and eq.(4.3) one can get ψ = C, a constant. We can always have ψ = 1 by choosing the coordinate time dt = Cdt without changing the killing vector, then the metric would back to the spherically symmetric eq.(2.1). For the charged black hole, we have got the Reissner-Nordstrom metric, which implies ψ = 1, thus the first law could be obtained. The reason of this result might be that the matter field leads to ψ = 1, according to the Einstein's equation.
To get a general case, we just consider the θθ component of eq.(4.6), which can be expressed as
1 − rf (r) 2 f (r) f (r) + (ψ 2 f (r)) ψ 2 f (r) − f (r) = Λr 2 + 8π(T θθ − 1 2 T r 2 ) . (4.7)
We assume the metric satisfies f (r H ) = 0 and ψ(r H ) = 0 on the event horizon. By setting r = r H and considering the matter field contributes to the internal energy U , then multiplying dr H one can get
dU − r H 2 f (r H )dr H = −P d 4πr 3 H 3 . (4.8)
There is nothing different from eq.(3.24) which was got from the EiBI gravity. It implies that the Einstein gravity and EiBI gravity might be equivalent on the event horizon from the view of black hole thermodynamics. The Hawking temperature on the event horizon becomes
T = κ 2π = ψ(r H )f (r H ) 4π . (4.9)
Now rewrite eq.(4.8) as
dU − T 2πr H ψ(r H ) dr H = −P dV .
(4.10)
One would find the entropy has to satisfy
dS = 2πr H ψ(r H ) dr H , (4.11) or S = 2πr H ψ(r H ) dr H ,(4.12)
once ψ = 1, it is obvious that S = πr 2 H = A/4, which is the well-known Bekenstein-Hawking black hole entropy. For the Schwarzschild black hole and R-N black hole, which all have ψ = 1, S = A/4. However, if there is a static spherically symmetric solution taking the form of eq.(4.1), we may conjecture that the entropy would be amended. In fact, the "dirty" black hole could have the spherical symmetric like eq.(4.1) [26]. Moreover, in ref [27,28], it has been shown that the entropy of the "dirty" black hole should be corrected with some terms of integrals over the event horizon.
A corrected entropy in Eddington-inspired Born-Infeld gravity
Now, let's return back to the EiBI gravity, as it has the solution taking the form of eq.(3.6). The black hole solution with electromagenatic field in EiBI gravity has been found, while f (r H ) = 0 and ψ(r H ) = 0 [19,23]. It is given as below
ψ(r) = √ λr 2 λr 4 + κQ 2 . (5.1)
With this result we can get the corrected entropy
S = 2πr H ψ(r H ) dr H = π 1 r 2 H r 4 H + κ λ Q 2 dr 2 H (5.2) = π r 4 H + κ λ Q 2 − π κ λ |Q| · ln κ λ |Q| r 2 H + 1 + κ λ Q 2 r 4 H . (5.3)
A logarithmic term occurs in this formula as a corrected entropy. When Q = 0, one gets S = πr 2 H , which is consistent with the vacuum case, and so is the same as Einstein gravity. When κ → 0, EiBI gravity would reduce to the Einstein gravity, and entropy becomes the Bekenstein-Hawking one. Moreover, the logarithmic term in the corrected entropy has been proposed from the loop quantum gravity [6], and conformal anomaly theory [29]. Even the corrected entropy-area relation for apparent horizon in FRW universe takes the similar form [12].
Conclusion and discussions
In Summary, in this paper, we started with the Einstein's equation and show that the first law of black hole thermodynamic could be obtained near the event horizon for a static spherically symmetric. The analogous technique was proposed by T.Padmanabhan [14][15][16]. Since it provided a convenient approach to study the black hole thermodynamics just from the field equation, we investigated the black hole thermodynamic in EiBI gravity. We found that there is nothing different from Einstein gravity in vacuum, but entropy could be different from the R-N black hole when considering the electromagnetic filed. The corrected entropy from EiBI gravity should be eq.(3.29), which can reduce to the Bekenstein-Hawking entropy when ψ = 1, and it is also the same as Einstein gravity when Q = 0 without the matter filed [19].
Next, we investigated a more general static spherically symmetric taking the form eq.(4.1) in Einstein gravity, due to the solution of EiBI gravity has the same form. It is surprised to found that the Einstein gravity implied the same result, which means that the entropy might also have a correction taking the form of eq.(4.12). Thus the correction of entropy from EiBI might also hold in the Einstein gravity for a static spherically symmetric like eq.(4.1).
Moreover, as the Einstein gravity and EiBI gravity hold the same result, we remarked that these two theories of gravity could be equivalent on the event horizon from the view of thermodynamics.
At last, as an example, a specific corrected entropy of the charged black hole in EiBI gravity was given. The logarithmic term in our corrected entropy is analogous to the theory of loop quantum gravity [4][5][6] and conformal anomaly theory [29].
AcknowledgmentsWe would like to thank the National Natural Science Foundation of China (Grant No.11571342) for supporting us on this work.
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"Kondo length in bosonic lattices"
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"Domenico Giuliano \nDipartimento di Fisica\nUniversità della Calabria Arcavacata di Rende I-87036\nCosenzaItaly (\n\n) I.N.F.N\nArcavacata di Rende I-87036\nGruppo collegato di CosenzaCosenzaItaly (\n",
"Pasquale Sodano \nInternational Institute of Physics\nUniversidade Federal do Rio Grande do Norte\n59078-400Natal-RNBrazil (\n\nDepartemento de Física Teorica e Experimental\nUniversidade Federal do Rio Grande do Norte\n59072-970Natal-RNBrazil (\n",
"Andrea Trombettoni \nCNR-IOM DEMOCRITOS Simulation Center\nVia Bonomea 265I-34136TriesteItaly (\n\n) SISSA and INFN\nSezione di Trieste\nVia Bonomea 265I-34136TriesteItaly\n"
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"Dipartimento di Fisica\nUniversità della Calabria Arcavacata di Rende I-87036\nCosenzaItaly (",
") I.N.F.N\nArcavacata di Rende I-87036\nGruppo collegato di CosenzaCosenzaItaly (",
"International Institute of Physics\nUniversidade Federal do Rio Grande do Norte\n59078-400Natal-RNBrazil (",
"Departemento de Física Teorica e Experimental\nUniversidade Federal do Rio Grande do Norte\n59072-970Natal-RNBrazil (",
"CNR-IOM DEMOCRITOS Simulation Center\nVia Bonomea 265I-34136TriesteItaly (",
") SISSA and INFN\nSezione di Trieste\nVia Bonomea 265I-34136TriesteItaly"
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| []
| Motivated by the fact that the low-energy properties of the Kondo model can be effectively simulated in spin chains, we study the realization of the effect with bond impurities in ultracold bosonic lattices at half-filling. After presenting a discussion of the effective theory and of the mapping of the bosonic chain onto a lattice spin Hamiltonian, we provide estimates for the Kondo length as a function of the parameters of the bosonic model. We point out that the Kondo length can be extracted from the integrated real space correlation functions, which are experimentally accessible quantities in experiments with cold atoms. 75.20.Hr , 72.15.Qm , 75.30.Kz . | 10.1103/physreva.96.033603 | [
"https://arxiv.org/pdf/1704.07485v2.pdf"
]
| 119,345,678 | 1704.07485 | e1cbf72e98d128ab3a531a8e9e4a32adb467788d |
Kondo length in bosonic lattices
5 Sep 2017
Domenico Giuliano
Dipartimento di Fisica
Università della Calabria Arcavacata di Rende I-87036
CosenzaItaly (
) I.N.F.N
Arcavacata di Rende I-87036
Gruppo collegato di CosenzaCosenzaItaly (
Pasquale Sodano
International Institute of Physics
Universidade Federal do Rio Grande do Norte
59078-400Natal-RNBrazil (
Departemento de Física Teorica e Experimental
Universidade Federal do Rio Grande do Norte
59072-970Natal-RNBrazil (
Andrea Trombettoni
CNR-IOM DEMOCRITOS Simulation Center
Via Bonomea 265I-34136TriesteItaly (
) SISSA and INFN
Sezione di Trieste
Via Bonomea 265I-34136TriesteItaly
Kondo length in bosonic lattices
5 Sep 2017(Dated: September 6, 2017)numbers: 6785-d7520Hr7215Qm7530Kz
Motivated by the fact that the low-energy properties of the Kondo model can be effectively simulated in spin chains, we study the realization of the effect with bond impurities in ultracold bosonic lattices at half-filling. After presenting a discussion of the effective theory and of the mapping of the bosonic chain onto a lattice spin Hamiltonian, we provide estimates for the Kondo length as a function of the parameters of the bosonic model. We point out that the Kondo length can be extracted from the integrated real space correlation functions, which are experimentally accessible quantities in experiments with cold atoms. 75.20.Hr , 72.15.Qm , 75.30.Kz .
I. INTRODUCTION
The Kondo effect has been initially studied in metals, like Cu, containing magnetic impurities, like Co atoms, where it arises from the interaction between magnetic impurities and conduction electrons, resulting in a net, low-temperature increase of the resistance 1-3 . It soon assumed a prominent role in the description of strongly correlated systems and in motivating and benchmarking the development of (experimental and theoretical) tools to study them 2,4 . Indeed, due to the large amount of analytical and numerical tools developed to attack it, the Kondo effect has become a paradigmatic example of a strongly interacting system and a testing ground for a number of different many-body techniques.
The interest in the Kondo effect significantly revitalized when it became possible to realize it in a controlled way in a solid state system, by using quantum dots in contacts with metallic leads, in which the electrons trapped within the dot can give rise to a net nonzero total spin interacting with the spin of conduction electrons from the leads, thus mimicking the behavior of a magnetic impurity in a metallic host [5][6][7][8] . An alternative realization of Kondo physics is recovered within the universal, low energy-long distance physics of a magnetic impurity coupled to a gapless antiferromagnetic chain 9,10 . In fact, though low-energy excitations of a spin chain are realized as collective spin modes, the remarkable phenomenon of "spin fractionalization" 11 implies that the actual stable elementary excitation of an antiferromagnetic spin-1/2 spin chain is a spin-1/2 "half spin wave" 12,13 (dubbed spinon). Spinons have a gapless spectrum and, therefore, for what concerns screening of the impurity spin, they act exactly as itinerant electrons in metals, as the charge quantum number is completely irrelevant for Kondo physics. A noticeable advantage of working with the spin chain realization of the Kondo effect is that a series of tools developed for spin systems, including entanglement witnesses and negativity, can be used to study the Kondo physics in these systems 14,15 .
Another important, long-lasting reason for interest in Kondo systems lies in that the multichannel "overscreened" version of the effect 16,17 provides a remarkable realization of non-Fermi liquid behavior 18 . Finally, the nontrivial properties of Kondo lattices provide a major arena in which to study many-body nonperturbative effects, related to heavy-fermion materials 19,20 . A recent example of both theoretical and experimental activity on multichannel Kondo systems is provided by the topological Kondo model [21][22][23][24] , based on the merging of several one-dimensional quantum wires with suitably induced and possibly controllable Majorana modes tunnel-coupled at their edges, and by recent proposals of realizing topological Kondo Hamiltonians in Y -junctions of XX and Ising chains [25][26][27] and of Tonks-Girardeau gases 28 . Finally, the effects of the competition between the Kondo screening and the screening from localized Majorana modes emerging at the interface between a topological superconductor and a normal metal has been recently discussed in [29] using the techniques developed in [30].
The onset of the Kondo effect is set by the Kondo temperature T K , which emerges from the perturbative renormalization group (RG) approach as a scale at which the system crosses over towards the strongly correlated nonpertubative regime 2,31 . The systematic implementation of RG techniques has clearly evidenced the scaling behavior characterizing the Kondo regime, which results in the collapse onto each other of the curves describing physical quantities in terms of the temperature T , once T is rescaled by T K 31,32 . The collapse evidences the one-parameter scaling, that is, there is only one dimensionful quantity, which is dynamically generated by the Kondo interaction and invariant under RG trajectories. Thus, within scaling regime, one may trade T for another dimensionful scaling parameter such as, for instance, the system size ℓ. In this case, as a consequence of one-parameter scaling, a scale invariant quantity with the dimension of a length emerges, the Kondo screening length ξ K , given by ξ K = v F /k B T K , where v F is the Fermi velocity of conduction electrons and k B is the Boltzmann constant 31 . Physically, ξ K defines the length scale over which the impurity magnetic moment is fully screened by the spin of conduction electrons, that is, the "size of the Kondo cloud" 33 . Differently from T K , which can be directly measured from the low-T behavior of the resistance in metals, the emergence of ξ K has been so far only theoretically predicted, as a consequence of the onset of the Kondo scaling 31 . Thus, it would be extremely important to directly probe ξ K , as an ultimate consistency check of scaling in the Kondo regime. As the emergence of the Kondo screening length is a mere consequence of the onset of Kondo scaling regime, ξ K can readily be defined for Kondo effect in spin chains, as well 9,15,34 . Unfortunately, despite the remarkable efforts paied in the last years to estimate ξ K in various systems by using combinations of perturbative, as well as nonperturbative numerical methods 10 , the Kondo length still appears quite an elusive quantity to directly detect, both in solid-state electronic systems as well as in spin chains 33 . This makes it desirable to investigate alternative systems in which to get an easier experimental access to ξ K .
A promising route in this direction may be provided by the versatility in the control and manipulation of ultracold atoms 35,36 . Indeed, in the last years several proposals of schemes in which features of the Kondo effect can be studied in these systems have been discussed. Refs. [37,38] suggest to realize the spin-boson model using two hyperfine levels of a bosonic gas 37 , or trapped ions arranged in Coulomb crystals 38 (notice that in general the Kondo problem may be thought of as a spin-1/2, system interacting with a fermionic bath 39 ). Ref. [40] proposes to use ultracold atoms in multi-band optical lattices controlled through spatially periodic Raman pulses to investigate a class of strongly correlated physical systems related to the Kondo problem. Other schemes involve the use of ultracold fermions near a Feshbach resonance 41 , or in superlattices 42 . More recently, the implementation of a Fermi sea of spinless fermions 43 or of two different hyperfine states of one atom species 44 interacting with an impurity atom of different species confined by an isotropic potential has been proposed 43 . The simulation of the SU (6) Coqblin-Schrieffer model for an ultracold fermionic gas of Yb atoms with metastable states has been discussed, while alkaline-earth fermions with two orbitals were also at the heart of the recent proposal of simulating Kondo physics through a suitable application of laser excitations 45 . Despite such an intense theoretical activity, including the investigation of optical Feshbach resonances to engineer Kondo-type spin-dependent interactions in Li-Rb mixtures 46 , and the remarkable progress in the manipulation of ultracold atomic systems, such as alkaline-earth gases, up to now an experimental detection of features of Kondo physics and in particular of the Kondo length in ultracold atomic systems is still lacking.
In view of the observation that optical lattices provide an highly controllable setup in which it is possible to vary the parameters of the Hamiltonian and to accordingly add impurities with controllable parameters 47,48 , in this paper we propose to study the Kondo length in ultracold atoms loaded on an optical lattice. Our scheme is based on the well-known mapping between the lattice Bose-Hubbard (BH) Hamiltonian and the XXZ spin-1/2 Hamiltonian 49 , as well as on the Jordan-Wigner (JW) representation for the spin 1/2 operators, which allows for a further mapping onto a Luttinger liquid model [50][51][52] . Kondo effect in Heisenberg spin-1/2 antiferromagnetic spin chains has been extensively studied [53][54][55] , though mostly for side-coupled impurities (i.e., at the edge of the chain). For instance, in Ref. [54], the Kondo impurity is coupled to a single site of a gapless XXZ spin chain, while in Ref. [9] a magnetic impurity is coupled at the end of a J 1 − J 2 spin-1/2 chain. At variance, in trapped ultracold atomic systems, it is usually difficult to create an impurity at the edge of the system. Accordingly, in this paper we propose to study the Kondo length at an extended (at least two links) impurity realized in the bulk of a cold atom system on a 1d optical lattice. In particular, we assume the lattice to be at half-odd filling, so to avoid the onset of a gapped phase that takes place at integer filling in the limit of a strong repulsive interaction between the particles. Since the real space correlation functions are quantities that one can measure in a real cold atom experiment, we address the issue of how to extract the Kondo length from the zeroes of the integrated real space density-density correlators. Finally, we provide estimates for ξ K and show that, for typical values of the system parameters, it takes values within the reach of experimental detectability (∼ tens of lattice sites).
Besides the possible technical advances, we argue that, at variance with what happens at a magnetic impurity in a conducting metallic host, where one measures T K and infers the existence of ξ K from the applicability of oneparameter scaling to the Kondo regime, in an ultracold atom setup one can extract from density-density correlation functions the Kondo screening length, that is in principle easier to measure, so that, to access ξ K , one has not to rely on verifying the one parameter scaling, which is what tipically makes ξ K quite hard to detect.
The paper is organized as follows:
• In section II we provide the effective description of a system of ultacold atoms on a 1d optical lattice as a spin-1/2 spin chain. In particular, we show how to model impurities in the lattice corresponding to bond impurities in the spin chain;
• In section III we derive the scaling equations for the Kondo running couplings and use them to estimate the corresponding Kondo length;
• In section IV we discuss how to numerically extract the Kondo length from the integrated real space densitydensity correlations and compare the results with the ones obtained in section III;
• In section V we summarize and discuss our results.
Mathematical details of the derivation and reviews of known results in the literature are provided in the various appendices.
II. EFFECTIVE MODEL HAMILTONIAN
Based on the spin-1/2 XXZ spin-chain Hamiltonian description of (homogeneous, as well as inhomogeneous) interacting bosonic ultracold atoms at half-filling in a deep optical lattice, in this section we propose to model impurities in the spin chain by locally modifying the strength of the link parameters of the optical lattice, eventually resorting to a model describing two XXZ "half-spin chains", interacting with each other via a local impurity. When the impurity is realized as a spin-1/2 local spin, such a system corresponds to a possible realization of the (two channel) Kondo effect in spin chains 9,54 . Therefore, our mapping leads to the conclusion that spin chain Kondo effect may possibly realized and detected within bosonic cold atoms loaded onto a one-dimensional optical lattice.
To resort to the spin-chain description of interacting ultracold atoms, we consider the large on-site interaction energy U -limit of a system of interacting ultracold bosons on a deep one-dimensional lattice. This is described by the extended BH Hamiltonian 56-58
H BH = − ℓ−1 j=−ℓ t j;j+1 (b † j b j+1 + b † j+1 b j ) + U 2 ℓ j=−ℓ n j (n j − 1) + V ℓ−1 j=−ℓ n j n j+1 − µ ℓ j=−ℓ n j .(1)
In Eq.(1), b j , b † j are respectively the annihilation and the creation operator of a single boson at site j (with j = −ℓ, · · · , ℓ) and, accordingly, they satisfy the commutator algebra [b j , b † j ′ ] = δ j,j ′ , all the other commutators being equal to 0. As usual, we set n j = b † j b j . Moreover, t j;j+1 is the hopping amplitude for bosons between nearest neighboring sites j and j + 1, U is the interaction energy between particles on the same site, V is the interaction energy between particles on nearest-neighboring sites. Typically, for alkali metal atomes one has V ≪ U while, for dipolar gases 59 on a lattice, V may be of the same order as U 60,61 . Throughout all the paper we take U > 0 and V ≥ 0. To outline the mapping onto a spin chain, we start by assuming that t j;j+1 is uniform across the chain and equal to t. Then, we discuss how to realize an impurity in the chain by means of a pertinent modulation of the t j;j+1 's in real space. In performing the calculations, we will be assuming open boundary conditions on the 2ℓ + 1-site chain and we will set the average number of particles per site by fixing the filling f = NT N where N T is the total number of particles on the lattice and N = 2ℓ + 1 is the number of sites.
In the large-U limit, one may set up a mapping between the BH Hamiltonian in Eq.(1) and a pertinent spin-model Hamiltonian H S , with H S either describing an integer, 61,62 , or an half-odd spin chain 63 , depending on the value of f . An integer-spin effective Hamiltonian is recovered, at large U , for f = n (with n = 1, 2, . . .), corresponding to µ = µ 0 (n) = n(U + 2V ) − U/2 and U ≫ t 60 , which allowed for recovering the phase diagram of the BH model in this limit by relating on the analysis of the phase diagram of spin-1 chains within the standard bosonization approach 62,64 . In particular, the occurrence of Mott and Haldane gapped insulating phases for ultracold atoms on a lattice has been predicted and discussed 61,65,66 .
Here, we rather focus onto the mapping of the BH Hamiltonian onto an effective spin-1/2 spin-chain Hamiltonian. This is recovered at U/t ≫ 1 and half-odd filling f = n + 1/2 (with n = 0, 1, 2, . . .), corresponding to setting the chemical potential so that µ = (U + 2V )(n + 1 2 ) − U 2 . In this regime, the effective low-energy spin-1/2 Hamiltonian for the system is given by 63 (2) has to be supplemented with the condition that j S z j = 0, implying that physically acceptable states are only the eigenstates of j n j belonging to the eigenvalue N T : this corresponds to singling out of the Hilbert space only the zero magnetization sector. As discussed in detail in Ref. [63], H spin−1/2 provides an excellent effective description of the low-energy dynamics of the BH model at half-odd filling. Although the mapping is done in the large-U limit, in Ref. [63] it is shown that it is in remarkable agreement with DMRG results also for U/J as low as ∼ 3 − 5 and for low values of N T such as N T ∼ 30.
H spin−1/2 = −J ℓ−1 j=−ℓ S + j S − j+1 + S + j+1 S − j + J∆ ℓ−1 j=−ℓ S z j S z j+1 ,(2)
Additional on-site energies ǫ i can be accounted for by adding a term ℓ j=−ℓ ǫ j n j to the right-hand side of Eq.(1). Accordingly, H spin−1/2 in Eq.(2) has to be modified by adding the term ℓ j=−ℓ ǫ j S z j . As soon as the potential energy scale is smaller than U , we expect the mapping to be still valid (we recall that with a trapping parabolic potential typically ǫ j = Ωj 2 with Ω ≡ mω 2 λ 2 /8, m being the atom mass, ω the confining frequency and λ/2 the lattice spacing 68 ). Yet, we stress that recent progresses in the realizations of potentials with hard walls 69,70 make the optical lattice realization of chains with open boundary conditions to lie within the reach of present technology.
Another point to be addressed is what happens slightly away from half-filling, that is, for f = n + 1/2 + ε, with ε ≪ 1. In this case, one again recovers the effective Hamiltonian in Eq. (2), but now with the constraint on physically acceptable states given by (1/N ) j S z j = ε. Since keeping within a finite magnetization sector is equivalent to having a nonzero applied magnetic field 71 , one has then to add to the right hand side of Eq.(2) a term of the form H j S z j , where H ∝ ε: again, we expect that the mapping is valid as soon as that the magnetic energy is smaller than the interaction energy scale U , and, of course, that the system spectrum remains gapless 72 .
To modify the Hamiltonian in Eq.(2) by adding bond impurities to the effective spin chain, we now create a link defect in the BH Hamiltonian in Eq.(1) by making use of the fact that optical lattices provide a highly controllable setup in which it is possible to vary the parameters of the Hamiltonian as well as to add impurities with tunable parameters 47,48 . This allows for creating a link defect in an optical lattice by either pertinently modulating the lattice, so that the energy barriers among its wells vary inhomogeneously across the chain, or by inserting one, or more, extra laser beams, centered on the minima of the lattice potential. In this latter case, one makes the atoms feel a total potential given by V ext = V opt + V laser , where the optical potential is given by V opt = V 0 sin 2 (kx), with k = 2π/λ and λ = λ 0 / sin (θ/2), λ 0 being the wavelength of the lasers and θ the angle between the laser beams forming the main lattice 47 (notice that the lattice spacing is d = λ/2). For counterpropagating laser beams having the same direction, θ = π and d = λ 0 /2, while d can be enhanced by making the beams intersect at an angle θ = π. V laser is the additional potential due to extra (blue-detuned) lasers: with one additional laser, centered at or close to an energy maximum of V opt , say at x ≡ x 0;1 among the minima x 0 = 0 and x 1 = d, the potential takes the form V laser ≈ V 1 e −(x−x0;1) 2 /σ 2 . When the width σ is much smaller than the lattice spacing, the hopping rate between the sites j = 0 and j = 1 is reduced and no on-site energy term appears, as shown in panels a) and b) of Fig.1. Notice that we use a notation such that the j-th minimum corresponds to the minimum x j = jd in the continuum space.
When x 0;1 is equidistant from the lattice minima x 0 and x 1 , corresponding to x 0;1 = λ/4 = d/2 and σ < d, then only the hopping t 0,1 is practically altered (see Fig.1a) ). When x 0;1 is displaced from d/2 one has an asymmetry and also a nearest neighboring link [e.g., t −1;0 in Fig.1 b)] may be altered (an additional on-site energy ǫ 0 is also present). With d ∼ 2 − 3µm, one should have σ 2µm, in order to basically alter only one link. Notice that barrier of few µm can be rather straightforwardly implemented 28,73 and recently a barrier of ∼ 2µm has been realized in a Fermi gas 74 .
As discussed in the following, this is the prototypical realization of a weak-link impurity in an otherwise homogeneous spin chain 75,76 .
In general, reducing the hopping rate between links close to each other may either lead to an effective weak link impurity, or to a spin-1/2 effective magnetic impurity, depending on whether the number of lattice sites between the reduced-hopping-amplitude links is even, or odd (see appendix A for a detailed discussion of this point). To "double" the construction displayed in panels a) and b) of Fig.1 to the one we sketch in panels c) and d) of Fig.1, we consider a potential of the form V laser ≈ V 1 e −(x−x0;1) 2 /σ 2 + V 2 e −(x−x−1;0) 2 /σ 2 with x −1;0 lying between sites j = −1 and j = 0: assuming again σ d, when V 1 = V 2 and x 0;1 = −x −1;0 = d/2 then only two links are altered, and in an equal way [the hoppings t −1;0 and t 0;1 in Fig.1 c)], otherwise one has two different hoppings [again t −1;0 and t 0;1 in Fig.1 d) ]. When σ is comparable with d, apart from the variation of the hopping rates, on-site energy terms enter the Hamiltonian in Eq.(1), giving rise to local magnetic fields in the spin Hamiltonian in Eq. (2). Though this latter kind of "site defects" might readily be accounted for within the spin-1/2 XXZ framework, for simplicity we will not consider them in the following, and will only retain link defects, due to inhomogeneities in the boson hopping amplitudes between nearest neighboring sites and in the interaction energy V . Correspondingly, the hopping amplitude t j;j+1 in Eq.(1) takes a dependence on the site j also far form the region in which the potential V laser is centered.
In the following, we consider inhomogeneous distributions of link parameters symmetric about the center of the chain (that is, about j = 0). Moreover, for the sake of simplicity, we discuss a situation in which two (symmetrically placed) inhomogeneities enclose a central region, whose link parameters may, or may not, be equal to the ones of the rest of the chain. We believe that, though experimentally challenging, this setup would correspond to the a situation in which the experimental detection of the Kondo length is cleaner. In fact, we note that all the experimental required ingredients are already available, as our setup requires two lasers with σ d (ideally, σ ≪ d) and centered with similar precision.
As we discuss in detail in Appendix A, an "extended central region" as such can either be mapped onto an effective weak link, between two otherwise homogeneous "half-chains", or onto an effective isolated spin-1/2 impurity, weakly connected to the two half-chains. In particular, in this latter case, the Kondo effect may arise, yielding remarkable nonperturbative effects and, eventually, "sewing together" the two half chains, even for a repulsive bulk interaction 53,54 . Denoting by G the region singled out by weakening one or more links, in order to build an effective description of G, we assume that the mapping onto a spin-1/2 XXZ-chain works equally well with the central region, and employ a systematic Shrieffer-Wolff (SW) summation, in order to trade the actual dynamics of G for an effective boundary Hamiltonian, that describes the effective degrees of freedom of the central region interacting with the half chains. One is then led to consider the Hamiltonian in Eq.(1), with link-dependent hopping rates t j;j+1 .
To illustrate how the mapping works, we focus onto the case of M = 2 altered links, corresponding to two bluedetuned lasers, and briefly comment on the more general case. To resort to the Kondo-like Hamiltonian for a spin-1/2 impurity embedded within a spin-1/2 XXZ-chain, we define the hopping rate to be equal to t throughout the whole chain but between j = −1 and j = 0, where we assume it to be equal to t L , and between j = 0 and j = 1, where we set it equal to t R , corresponding to panels c) and d) of Fig.1. On going through the SW transformation, one therefore gets the effective spin-1/2 Hamiltonian H s = H bulk + H K , with H bulk = H L + H R and
H L = −J −2 j=−ℓ S + j S − j+1 + S + j+1 S − j + J∆ −2 j=−ℓ S z j S z j+1 H R = −J ℓ−1 j=1 S + j S − j+1 + S + j+1 S − j + J∆ ℓ−1 j=1 S z j S z j+1 .(4)
The "Kondo-like" term is instead given by
H K = −J ′ L (S + −1 S − 0 + S − −1 S + 0 ) − J ′ R (S + 0 S − 1 + S − 0 S + 1 ) + J ′ zL S z −1 S z 0 + J ′ zR S z 0 S z 1 ,(5)
where
J ′ α = t α f and J ′ zα ≈ V − 3J ′ 2
α /4U (with α = L, R). Our choice for H K corresponds to the simplest case in which G contains an even number of links -or, which is the same, an odd number of sites, as schematically depicted in Fig.2b). We see that the isolated site works as an isolated spin-1/2 impurity S G , interacting with the two half chains via the boundary interaction Hamiltonian H (1)
B ≡ H K .
The other possibility, which we show in Fig.2a), corresponds to the case in which an odd number of links is altered and G contains an even number of sites. In particular, in Fig.2a) we have only one altered hopping coefficient. This latter case is basically equivalent to a simple weak link between the R-and the L-half chain, which is expected to realize the spin-chain version of Kane-Fisher physics of impurities in an interacting one-dimensional electronic system 77 . In Appendix A, we review the effective low-energy description for a region G containing an in principle arbitrary number of sites. In particular, we conclude that either the number of sites within G is odd, and therefore the resulting boundary Hamiltonian takes the form of H K in Eq.(5), or it is even, eventually leading to a weak link Hamiltonian 75,76 . Even though this latter case is certainly an interesting subject of investigation, we are mostly interested in the realization of effective magnetic impurities. Therefore, henceforth we will be using H s as the main reference Hamiltonian, to discuss the emergence of Kondo physics in our system.
ext V /V 0 ext V /V 0 ext V /V 0 ext V /V 0 x/d x/d x/d x/d a) b) c) d)
III. RENORMALIZATION GROUP FLOW OF THE IMPURITY HAMILTONIAN PARAMETERS
In this section, we employ the renormalization group (RG) approach to recover the low-energy long-wavelength physics of a Kondo impurity in an otherwise homogeneous chain. From the RG equations we derive the formula for the invariant length which we eventually identify with ξ K . In general, there are two standard ways of realizing the impurity in a spin chain, which we sketch in Fig.2. Specifically, we see that the impurity can be realized as an island containing either an even or odd number of spins. The former case is equivalent to a weak link in an otherwise homogeneous chain, originally discussed in Refs. [77,78] for electronic systems, and reviewed in detail in Ref. [79] in the specific context of spin chains. In this case, which we briefly review in Appendix C, when ∆ > 0 in Eqs.(4), the impurity corresponds to an irrelevant perturbation, which implies an RG flow of the system towards the fixed point corresponding to two disconnected chains, while for ∆ < 0 the weak link Hamiltonian becomes a relevant perturbation. Though this implies the emergence of an "healing length " for the weak link as an RG invariant length scale, with a corresponding flow towards a fixed point corresponding to the two chains joined into an effectively homogeneous single chain, there is no screening of a dynamical spinful impurity by the surrounding spin degrees of freedom and, accordingly, no screening cloud is detected in this case 79 .
At variance, a dynamical effective impurity screening takes place in the case of an effective spin-1/2 impurity 34 . In this latter case, at any ∆ such that −1 < ∆ < 1, the perturbative RG approach shows that the disconnected-chain weakly coupled fixed point is ultimately unstable. In fact, the RG trajectories flow towards a strongly coupled fixed point, which we identify with the spin chain two channel Kondo fixed point, corresponding to healing the chain but, at variance with what happens at a weak link for 0 < ∆ ≤ 1, this time with the chain healing taking place through an effective Kondo-screening of the magnetic impurity 53 .
A region containing an odd number of sites typically has a twofold degenerate groundstate and, therefore, is mapped onto an effective spin-1/2 impurity S G . The corresponding impurity Hamiltonian in Eq.(A2) takes the form of the Kondo spin-chain interaction Hamiltonian for a central impurity in an otherwise uniform spin chain 9 . To employ the bosonization formalism of appendix C to recover the RG flow of the impurity coupling strength, we resort to Eq.(B12), corresponding to the bosonized spin Kondo Hamiltonian H K given by
H K = α=L,R −J ′ α [S + 0 e − i √ 2 Φα(0) + S − 0 e i √ 2 Φα(0) ] + J ′ zα S z 0 1 √ 2π ∂Θ α (0) ∂x .(6)
The RG equations describing the flow of the impurity coupling strength can be derived by means of standard techniques for Kondo effect in spin chains 54 and, in particular, by considering the fusion rules between the various operators entering H K in Eq. (6). In doing so, in principle additional, weak link-like, operators describing direct tunneling between the two chains can be generated, such as, for instance, a term ∝ e
i √ 2 [ΦL(0)−ΦR(0)] , with scaling dimension h A = 1
g . However, one may safely neglect a term as such, since, for g < 1, it corresponds to an additional irrelevant boundary operator that has no effects on the RG flow of the running couplings appearing in H K . For g ≥ 1 it becomes marginal, or relevant, but still subleading, compared to the terms ∝ J ′ α , as we discuss in the following and, therefore, it can again be neglected for the purpose of working out the RG flow of the boundary couplings. This observation effectively enables us to neglect operators mixing the L and the R couplings with each other and, accordingly, to factorize the RG equations for the running couplings with respect to the index α .
More in detail, we define the dimensionless variables G α (ℓ) and G z,α (ℓ) as
G α (ℓ) = ℓ ℓ 0 1− 1 2g J ′ α J and G z,α (ℓ) = J ′ zα J ,(7)
(see Appendix B for a discussion on the estimate of the reference length ℓ 0 ) with α = L, R. The RG equations for the running couplings are given by
dG α (ℓ) d ln( ℓ ℓ0 ) = h g G α (ℓ) + G α (ℓ)G z,α (ℓ) dG zα (ℓ) d ln( ℓ ℓ0 ) = G 2 α (ℓ) ,(8)
with h g = 1 − 1/(2g). For the reasons discussed above, the RG equations in Eq.(8) for the L -and the R -coupling strengths are decoupled from each other. In fact, they are formally identical to the corresponding equations obtained for a single link impurity placed at the end of the chain ("Kondo side impurity") 9 . At variance with this latter case, as argued by Affleck and Eggert 53 , in our specific case of a "Kondo central impurity" the scenario for what concerns the possible Kondo-like fixed points is much richer, according to whether G L (ℓ 0 ) = G R (ℓ 0 ) ("asymmetric case"), or G L (ℓ 0 ) = G R (ℓ 0 ) ("symmetric case"), as we discuss below. To integrate Eqs.(8), we define the reduced variables X α (ℓ) ≡ G α (ℓ) and X z,α (ℓ) = G z,α (ℓ) + 1 − 1 2g for α = L, R (since the equations for the two values of α are formally equal to each other, from now on we will understand the index α). As a result, one gets
dX(ℓ) d ln( ℓ ℓ0 ) = X(ℓ)X z (ℓ); dX z (ℓ) d ln( ℓ ℓ0 ) = X 2 (ℓ).(9)
Equations (9) coincide with the RG equations obtained for the Kosterlitz-Thouless phase transition 80 . To solve them, we note that the quantity
κ = X 2 z (ℓ) − X 2 (ℓ) ,(10)
is invariant along the RG trajectories. In terms of the microscopic parameters of the BH Hamiltonan one gets
κ = κ(ℓ 0 ) = (V /J − 3J ′ 2 /(4U J) + 1 − 1/(2g)) 2 − (J ′ /J) 2 .
To avoid the onset of Mott-insulating phases, we have to assume that the interaction is such that g > 1/2. This implies h g > 0 and X z (ℓ 0 ) > 0: thus, we assume X(ℓ 0 ), X z (ℓ 0 ) > 0. This means that the RG trajectories always lie within the first quarter of the (X, X z )-parameter plane and, in particular, that the running couplings always grow along the trajectories.
Using the constant of motion in Eq.(10), Eqs.(9) can be easily integrated. As a result, one may estimate the RG invariant length scale ℓ * defined by the condition that, at the scale ℓ ∼ ℓ * , the perturbative calculation breaks down (which leads us to eventually identify ℓ * with ξ K ). As this is signaled by the onset of a divergence in the running parameter X(ℓ) 27 , one may find the explicit formulas for ℓ * , depending on the sign of κ, as detailed below:
• κ = 0. In this case, as the symmetry at ℓ = ℓ 0 between K and X z is preserved along the RG trajectories, it is enough to provide the explicit solution for X z (ℓ)(= X(ℓ)), which is given by
X z (ℓ) = X z (ℓ 0 ) 1 − X z (ℓ 0 ) ln( ℓ ℓ0 ) .(11)
From Eq. (11), one obtains
ℓ * ∼ ℓ 0 exp 1 X z (ℓ 0 ) ,(12)
which is the familiar result one recovers for the "standard" Kondo effect in metals 34 .
• κ < 0. In this case, the explicit solution of Eqs. (9) is given by
X z (ℓ) = √ −κ tan atan X z (ℓ 0 ) √ −κ + √ −κ ln ℓ ℓ 0 X(ℓ) = −κ + X 2 z (ℓ) ,(13)
which yields
ℓ * ∼ ℓ 0 exp π − 2 atan( Xz (ℓ0) √ |κ| ) 2 |κ| .(14)
• κ > 0. In this case one obtains
X z (ℓ) = − √ κ [X z (ℓ 0 ) − √ κ] ℓ ℓ0 2 √ κ + [X z (ℓ 0 ) + √ κ] [X z (ℓ 0 ) − √ κ] ℓ ℓ0 2 √ κ − [X z (ℓ 0 ) + √ κ] X(ℓ) = −κ + X 2 z (ℓ) .(15)
As a result, we obtain
ℓ * ∼ ℓ 0 X z (ℓ 0 ) + √ κ X z (ℓ 0 ) − √ κ 1 2 √ κ .(16)
To provide some realistic estimates of ℓ * , in Fig.3 we plot ℓ * /ℓ 0 as a function of the repulsive interaction potential V , keeping fixed all the other system parameters (see the caption for the numerical values of the various parameters). The two plots we show correspond to different values of J ′ . We see that, as expected, at any value of V /J, ℓ * decreases on increasing J ′ . We observe that with realistically small values of V /J, say between 0 and 0.5, one has a value of the Kondo length order of 20 sites (for J ′ /J = 0.2) and 5 sites (for J ′ /J = 0.6), that should detectable from experimental data.
Also, we note a remarkable decrease of ℓ * with V /J and, in particular, a finite ℓ * even at extremely small values of V , which correspond to negative values of J ′ z and, thus, to an apparently ferromagnetic Kondo coupling between the impurity and the chain. In fact, in order for the Kondo coupling to be antiferromagnetic, and, thus, to correspond to a relevant boundary perturbation, one has to either have both J ′ and J latter negative. In our case, the RG equations in Eqs. (9), show how the β-function for the running coupling X(= G) is proportional to X z G, rather than to G z G. Thus, what matters here is the fact that X z − G z = 1 − 1 2g > 0, which makes X z (ℓ 0 ) positive even though G z (ℓ 0 ) is negative. As a result, even when both J ′ and J ′ z are negative as it may happen, for instance, if one starts from a BH model with V ∼ 0, one may still recover a Kondo-like RG flow and find a finite ℓ * , as evidenced by the plots in Fig.3. Being an invariant quantity along the RG trajectories, here ℓ * plays the same role as ξ K in the ordinary Kondo effect, that is, once the RG trajectories for the running strengths are constructed by using the system size ℓ as driving variable, all the curves are expected to collapse onto each other, provided that, at each curve, ℓ is rescaled by the corresponding ℓ * 2,34,81,82 . In fact, in the specific type of system we are focusing onto, that is, an ensemble of cold atoms loaded on a pertinently engineered optical lattice, it may be difficult to vary ℓ by, in addition, keeping the filling constant (not to affect the parameters of the effective Luttinger liquid model Hamiltonian describing the system). Yet, one may resort to a fully complementary approach in which, as we highlight in the following, the length ℓ, as well as the filling f , are kept fixed and, taking advantage of the scaling properties of the Kondo RG flow, one probes the scaling properties by varying ℓ * . Indeed, from our Eqs. (12,14,16), one sees that in all cases of interest, the relation between ℓ * and the microscopic parameters characterizing the impurity Hamiltonian is known. As a result, one can in principle arbitrarily tune ℓ * at fixed ℓ by varying the tunable system parameter. As we show in the following, this provides an alterative way for probing scaling behavior, more suitable to an optical lattice hosting a cold atom condensate. In order to express the integrated RG flow equations for the running parameters as a function of ℓ and ℓ * , it is sufficient to integrate the differential equations in Eqs.(9) from ℓ * up to ℓ. As a result, one obtains the following equations:
• For κ = 0:
X(ℓ) = X z (ℓ) = X z (ℓ 0 ) − ln( ℓ ℓ * ) ;(17)
• For κ < 0:
X z (ℓ) = √ −κ tan π 2 − √ −κ ln ℓ * ℓ X(ℓ) = −κ + X 2 z (ℓ) ;(18)
• For κ > 0:
X z (ℓ) = √ κ ℓ * ℓ 2 √ κ + ℓ * ℓ 2 √ κ − 1 X(ℓ) = −κ + X 2 z (ℓ) .(19)
From Eqs. (17,18,19), one therefore concludes that, once expressed in terms of ℓ/ℓ * , the integrated RG flow for the running coupling strengths only depends on the parameter κ. Curves corresponding to the same values of κ just collapse onto each other, independently of the values of all the other parameters.
We pause here for an important comment. As discussed in 9 , in the spin chain realization of the Kondo model, one exactly retrieves the equation of the conventional Kondo effect at g = 1/2 only after adding a frustrating secondneighbor interaction, thus resorting to the so-called J 1 − J 2 model Hamiltonian. In principle, the same would happen for the XXX-spin chain with nearest-neighbor interaction only, except that, strictly speaking, the correspondence is exactly realized only in the limit of an infinitely long chains. In the case of finite chains, the presence of a marginally irrelevant Umklapp operator may induce finite-size violations from Kondo scaling which, as stated above, disappear in the thermodynamic limit. Yet, as this point is mostly of interest because it may affect the precision of numerical calculations, we do not address it here and refer to Ref. [9] for a detailed discussion of this specific topic.
Another important point to stress is that, strictly speaking, we have so far neglected the possible effects of the asymmetry (J In fact, the nature of the stable Kondo fixed point reached by the system in the large scale limit deeply depends on whether or not the bare couplings between the impurity and the chains are symmetric, or not. Nevertheless, as we argue in the following, one sees that, while the nature of the Kondo fixed point may be quite different in the two cases (two-channel versus one-channel spin-Kondo fixed point), one can still expect to be able to detect the onset of the Kondo regime and to probe the corresponding Kondo length by looking at the density-density correlations in real space, though the correlations themselves behave differently in the two cases. We discuss at length about this latter point in the next section. Here, we rather discuss about the nature of the Kondo fixed point in the two different situations, starting with the case of symmetric couplings between the impurity and the chains.
When J ′ L = J ′ R and J ′ zL = J ′ zR , since, to leading order in the running couplings, there is no mixing between the L -and the R coupling strengths, the L − R symmetry is not expected to be broken all the way down to the strongly coupled fixed point which, consequently, we identify with the two-channel spin-chain Kondo fixed point, in which the impurity is healed and the two chains have effectively joined into a single uniform chain. Due to the L − R symmetry, one can readily show that all the allowed boundary operators at the strongly coupled fixed point are irrelevant 53,54 , leading to the conclusion that the two-channel spin-Kondo fixed point is stable, in this case.
Concerning the effects of the asymmetry, on comparing the scale dimensions of the various impurity boundary operators, one expects them to be particularly relevant if the asymmetry is realized in the transverse Kondo coupling strengths, that is, if one has J ′ L ≫ J ′ R . We assume that this is the case which, moving to the dimensionless couplings, implies G L (ℓ 0 ) ≫ G R (ℓ 0 ). Due to the monotonicity of the integrated RG curves, we expect that this inequality keeps preserved along the integrated flow, that is, G L (ℓ) ≫ G R (ℓ) at any scale ℓ ≥ ℓ 0 . In analogy with the standard procedure used with multichannel Kondo effect with non-equivalent channels, one defines ℓ * as the scale at which the larger running coupling G L (ℓ) diverges, which is the signal of the onset of the nonperturbative regime. Due to the coupling asymmetry, we then expect G R (ℓ * ) ≪ 1, that is, at the scale ℓ ∼ ℓ * , the system may be regarded as a semi-infinite chain at the left-hand side, undergoing Kondo effect with an isolated magnetic impurity, weakly interacting with a second semi-infinite chain, at the right-hand side. To infer the effects of the residual coupling, one may assume that, at ℓ ∼ ℓ * , the impurity is "re-absorbed" in the left-hand chain 53,54 , so that this scenario will consist of the left-hand chain, with one additional site, connected with a link of strength ∼ G R (ℓ * ) to the endpoint of the right-hand chain. Within the bosonization approach, the weak link Hamiltonian is given by 77
V Asym B ∼ −G R (ℓ * )e i √ 2 [ΦL(0)−ΦR(0)] + h.c. .(20)
V Asym B has scaling dimension 1 g . Depending on whether g > 1, or g < 1, it can therefore be either relevant, or irrelevant (or marginal if g = 1). When relevant, it drives the system towards a fixed point in which the weak link is healed. When irrelevant, the fixed point corresponds to the two disconnected chains. In either case, the residual flow takes place after the onset of Kondo screening. We therefore conclude that Kondo screening takes place in the left-hand chain only and, accordingly, one expects to be able to probe ℓ * by just looking at the real space density-density correlations in that chain only. From the above discussion we therefore conclude that Kondo effect is actually realized at a chain with an effective spin-1/2 impurity whether or not the impurity couplings to the chains are symmetric, or not, though the fixed point the system is driven to along the RG trajectories can be different in the two cases.
IV. DENSITY-DENSITY CORRELATIONS AND MEASUREMENT OF THE KONDO LENGTH
In analogy to the screening length ξ K in the standard Kondo effect 83,84 , in the spin chain realization of the effect, the screening length ℓ * is identified with the typical size of a cluster of spins fully screening the moment of the isolated magnetic impurity, either lying at one side of the impurity itself (in the one-channel version of the effect-side impurity at the end of a single spin chain), or surrounding the impurity on both sides (two channel version of the effect-impurity embedded within an otherwise uniform chain).
So far, ℓ * showed itself as quite an elusive quantity to experimentally detect, both in electronic Kondo effect, as well as in spin Kondo effect 33 . In this section, we propose to probe ℓ * in the effective spin-1/2 XXZ chain describing the BH model, by measuring the integrated real-space density-density correlation functions. Real-space density-density correlations in atomic condensates on an optical lattice can be measured with a good level of accuracy (see e.g. Refs. [35,85].) Given the mapping between the BH-and the spin-1/2 XXZ spin Hamiltonian, real-space densitydensity correlation functions are related via Eq.(3) to the correlation functions of the z-component of the effective spin operators in the XXZ-Hamiltonian (local spin-spin susceptibility), which eventually enables us to analytically compute the correlation function within spin-1/2 XXZ spin chain Hamiltonian framework. The idea of inferring informations on the Kondo length by looking at the scaling properties of the real-space local spin susceptibility was put forward in Ref. [86]. In the specific context of lattice model Hamiltonians, the integrated real-space correlations have been proposed as a tool to extract ξ K in a quantum dot, regarded as a local Anderson model, interacting with itinerant lattice spinful fermions 81 . Specifically, letting S G denote the spin of the isolated spin-1/2 impurity and S j the spin operator in the site j, assuming that the impurity is located at one of the endpoints of the chain and that the whole model, including the term describing the interaction between S G and the spins of the chain, is spin-rotational invariant, one may introduce the integrated real-space correlation function Σ(x), defined as 81
Σ(x) = 1 + x y=1 S G · S y S G · S G .(21)
The basic idea is that the first zero of Σ(x) one encounters in moving from the location of the impurity, identifies the portion of the whole chains containing the spins that fully screen S G . Once one has found the solution of the equation Σ(x = x * ) = 0 , one therefore naturally identifies x * with ℓ * . It is important to stress that this idea equally applies whether one is considering the spin impurity at just one side of the chain (one-channel spin chain Kondo), or embedded within the chain (two-channel spin chain Kondo). Thus, while in the following we mostly consider the two-channel case, we readily infer that our discussion applies also to the one-channel case.
To adapt the approach of Ref. [81] to our specific case, first of all, since our impurity is located at the center of the chain, one has to modify the definition of Σ(x) so to sum over j running from −x to x. In addition, in our case both the bulk spin-spin interaction, as well as the effective Kondo interaction with the impurity, are not isotropic in the spin space. This requires modifying the definition of Σ(x), in analogy to what is done in Ref. [81] in the case in which an applied magnetic field breaks the spin rotational invariance. Thus, to probe ℓ * we use the integrated z-component of the spin correlation function, Σ z (x), defined as
Σ z (x) = 1 + x y=−x S z G S z y − S z G S z y S z G S z G − S z G 2 .(22)
In general, estimating ℓ * from Σ z (x) would require exactly computing the spin-spin correlation functions by means of a numerical technique, such as it is done in Ref. [81] -nevertheless one in general expects that the estimate of ℓ * obtained using perturbative RG differs by a factor order of 1 from the one obtained by nonperturbative, numerical means. For the purpose of showing the consistency between the estimate of ℓ * from the spin-spin correlation functions and the results from the perturbative analysis of Sec.III, one therefore expects it to be sufficient to resort to a perturbative (in J ′ z , J ′ ) calculation of Σ z (x), eventually improved by substituting the bare coupling strengths with the running ones, computed at an appropriate scale 34 . To leading order in the impurity couplings, we obtain
S z G S z y = −J ′ z,R ∞ 0 dτ G z,z (y, 1; τ |ℓ) , (y > 0) S z G S z y = −J ′ z,L ∞ 0 dτ G z,z (y, 1; τ |ℓ) , (y < 0) ,(23)
with the finite-τ correlation function G z,z (x, x ′ ; τ |ℓ) defined in Eq.(C2). To incorporate scale effects in the result of Eq.(23), we therefore replace the bare impurity coupling strengths with the running ones we derived in Sec.III, computed at an appropriate length scale, which we identify with the size x of the spin cluster effectively contributing to impurity screening. Therefore, referring to the dimensionless running coupling X z (λ) defined in Eqs. (8), we obtain Fig.3a). b): Same as before, but with V /J = 2.1875 (corresponding to ∆ = 0.2) and J ′ /J = 0.1. As expected, the lower value of J ′ yields a larger ℓ * ∼ 32.
Σ z (x) = 1 − 8J ′ z (x)ℓ πu x y=1 ∞ 0 dw G z,z y, 1; πuw ℓ ℓ = 1 − 8ϕ(∆) X z (x) + 1 2g − 1 ℓ x y=1 ∞ 0 dw G z,z y, 1; πuw ℓ ℓ ,(24)
with ϕ(∆) given by
ϕ(∆) = arcos ∆ 2 π 2 1 − ∆ 2 2 .(25)
Remarkably, ϕ(∆) → 1 as ∆ → 0. In Fig.4, we show Σ z (x) vs. x (only the positive part of the graph) for two paradigmatic situations: in Fig.4a) we consider the absence of nearest-neighbor "bare" density-density interaction (V = 0). In Fig.4b) we consider a rather large, presently not straightforward to be implemented in experiments, value of V (V /J ∼ 2.2) to show the results for the Kondo length with a positive value of the XXZ anisotropy parameter. We see that there is not an important dependence of the Kondo length upon V , since the main parameter affecting ℓ * is actually given by J ′ /J. From the analysis of Ref. [63], one sees that, even at V = 0, a nonzero attractive density-density interaction between nearest-neighboring sites of the chain is actually induced by higher order (in t/U ) virtual processes, which implies that, for V = 0, g keeps slightly higher than 1. At variance, for finite V , g can be either larger, or smaller than 1, as it is the case in the plot in Fig.4b). In both cases we see the effect of "Friedel-like" oscillations in the density-density correlation, which eventually conspire to set Σ z (x) to 0 at a scale x ∼ ℓ * (see the caption of the figures for more details on the numerical value of the various parameters).
In general, Eq.(24) has to be regarded within the context of the general scaling theory for Σ z (x) 34 . In our specific case, at variance with what happens in the "standard" Kondo problem of itinerant electrons in a metal magnetically interacting with an isolated impurity 34 , the boundary action in Eq.(B12) contains terms that are relevant as the length scale grows. In general, in this case a closed-form scaling formula for physical quantities cannot be inferred from the perturbative results, due to the proliferation of additional terms generated at higher orders in perturbation theory 87 . Nevertheless, here one can still recover a pertinently adapted scaling equation, as only dimensionless contributions to S B G effectively contribute Σ z (x) to any order in perturbation theory. The point is that, as we are considering a boundary operator in a bosonized theory in which the fields Φ L,R (x, τ ) obey Neumann boundary conditions at the boundary, the fields Θ L.R (0, τ ) appearing in the bosonized formula for S z 1,L , S z 1,R in Eqs.(B8) are pinned at a constant for any τ . As a result, the corresponding contribution to the boundary interaction reduces to the one in Eq.(B12), which is purely dimensionless and, therefore, marginal. As for what concerns the contribution ∝ J ′ L,R , it is traded for a marginal one once one uses as running couplings the rescaled variables X L and X R , rather than J ′ L , J ′ R . Now, from Eqs.(B8) we see that the bosonization formula for S z j contains a term that has dimension d 1 = 1 and a term with dimension d 2 = (2g) −1 . Taking into account the dynamics of the degrees of freedom of the chains comprised over a segment of length x, we therefore may make the scaling ansatz for Σ z in the form
Σ z [x, ℓ, X z , X] =ω 0 x ℓ , X z , X + ℓ 1−gω 1 x ℓ , X z , X ,(26)
with ω 0 , ω 1 scaling functions. Now, we note that, due to the existence of the RG invariant κ, which relates to each other the running parameters X z and X along the RG trajectories (Eq.(10) in the perturbative regime), we may trade ω 0,1
x ℓ , X z , X for two functions ω 0,1 of only x ℓ and X. As a final result, Eq.(26) becomes
Σ z [x, ℓ, X z , X] = ω 0 x ℓ , X z (x) + ℓ 1−g ω 1 x ℓ , X z (x) .(27)
Equation (27) provides the leading perturbative approximation at weak boundary coupling, as it can be easily checked from the explicit formula in Equation (C2). Eq. (27) illustrates how the function we explicitly use in our calculation can be regarded as just an approximation to the exact scaling function for Σ z (x). A more refined analytical treatment might in principle be done by considering higher-order contributions in perturbation theory in S B G . Alternatively, one might resort to a fully numerical approach, similar to the one used in Ref. [81]. Yet, due to the absence of an intermediate-coupling phase transition in the Kondo effect 2 , in our opinion resorting to a more sophisticated approach would improve the quantitative relation between the microscopic "bare" system parameters and the ones in the effective low-energy long-wavelength model Hamiltonian, without affecting the main qualitative conclusion about the Kondo screening length and its effects.
For this reason, here we prefer to rely on the perturbative RG approach extended to the correlation functions which, as we show before, already provides reliable and consistent results on the effects of the emergence of ℓ * on the physical quantities.
The obtained estimate of ξ K , although perturbative, provides, via the RG relation k B T K = v F /ξ K , an estimate of the Kondo temperature. When the measurements are done at finite temperature, of course thermal effects affect the estimate of ξ K : we anyway expect that if the temperature is much smaller than T K , then such effects are negligeable. Considering that T K has been estimated of order of tens of nK 28 , and that T K may be increased by increasing v F , which may be up to hundreds nK, and by increasing J ′ /J, we therefore expect that with temperatures smaller than the bandwidth one can safely extract ξ K . One should anyway find a compromise since by increasing J ′ /J the Kondo length decreases (and the Kondo effect itself disappears). A systematic study of thermal effects on the estimate of ξ K is certainly an important subject of future work.
V. CONCLUSIONS
In this paper we have studied the measurement of the Kondo screening length in systems of ultracold atoms in deep optical lattices. Our motivation relies primarily on the fact that the detection of the Kondo screening length from experimentally measurable quantities in general appears to be quite a challenging task. For this reason, we proposed to perform the measurement in cold atom setups, whose parameters can be, in principle, tuned in a controllable way to desired values.
Specifically, after reviewing the mapping between the BH model at half-filling with inhomogenous hopping amplitudes onto a spin chain Hamiltonian with Kondo-like magnetic impurities, we have proposed to extract the Kondo length from a suitable quantity obtained by integrating the real space density-density correlation functions. The corresponding estimates we recover for the Kondo length are eventually found to assume values definitely within the reach of present experiments (∼ tens of lattice sites for typical values of the system parameters). We showed that the Kondo length does not significantly depend on nearest-neighbor interaction V , and it mainly depends on the impurity link J ′ .
Concerning the Kondo length, a comment is in order for quantum-optics oriented readers: in a typical measurement of the Kondo effect at a magnetic impurity in a conducting metallic host, one has access to the Kondo temperature T K , by just looking at the scale at which the resistance (or the conductance, in experiments in quantum dots) bends upwards, on lowering T . The very existence of the screening length ξ K is just inferred from the emergence of T K and from the applicability of one-parameter scaling to the Kondo regime, which yields ξ K = v F /k B T K . However the latter relation stems from the validity of the RG approach. Thus, ultimately probing directly ξ K in solid-state samples would correspond to verifying the scaling in the Kondo limit, which is what makes it hard to actually perform the measurement. At variance, as we comment for solid-state oriented readers, in the ultracold gases systems we investigate here, one can certainly study dynamics (e.g., tilting the system) but a stationary flow of atoms cannot be (so far) established, so that the measure of T K may be an hard task to achieve. Rather surprisingly, as our results highlight, it is the Kondo length which can be more easily directly detected in ultracold gases and our corresponding estimates (order of tens of lattice sites) appear to be rather encouraging in this direction.
Several interesting issues deserve in our opinion further work: as first, it would be desirable to compare the perturbative results we obtain in this paper with numerical, nonperturbative findings in the Bose-Hubbard chain, to determine the corresponding correction to the value of ℓ * . It would be also important to understand the corrections to the inferred value of ξ K coming from finite temperature effects, that should be anyway negligeable for T (much) smaller that T K . Even more importantly, we mostly assumed that it is possible to alter the hopping parameters in a finite region without affecting the others. This led us to infer, for instance, the existence of the ensuing even-odd effect -however, having two lasers with σ ≪ d is a condition that may be straightforwardly implementable. In this case, one has to deal with generic space-dependent hopping amplitudes t j;j+1 . It would therefore be of interest to address, very likely within a fully numerical approach, the fate of the even-odd effect in the presence of a small modulation in space of the outer hopping terms. In particular, a theoretically interesting issue would be the competition between an extended nonlocal central region and the occurrence of magnetic and/or nonmagnetic impurities in the chain. Another point to be addressed is that an on-site nonuniform potential may in principle be present (event though its effect may be reduced by hard wall confining potentials) and an interesting task is to determine the interplay between the Kondo length and the length scale of such an additional potential.
In conclusion, we believe that our results show that the possible realization of the setup proposed in this paper could pave the way to the study of magnetic impurities and, in perspective, to the experimental implementation of ultracold realizations of Kondo lattices and detection of the Kondo length, providing, at the same time, a chance for studying several interesting many-body problems in a controllable way. In this appendix we review the description of a region G, singled out by weakening two links in a XXZ spin chain, in terms of an effective low-energy Hamiltonian H G . In particular, we show how, depending on whether the number of sites containined within G is odd, or even, either H G coincides with the Kondo Hamiltonian H K in Eq.(5), or it describes a weak link between two "half-chains" 75,76 .
In general, Kondo effect in spin-1/2 chains has been studied for an isolated magnetic impurity (the "Kondo spin"), which may either lie at the end of the chain (boundary impurity), or at its middle (embedded impurity) 53,54 . In the former case, the impurity can be realized by "weakening" one link of the chain, in the latter case, instead, it can be realized by weakening two links in the body of the chain. Following the discussion in Sec.III of the main text, here we mostly focus on the latter case. In general, in a spin chain, impurities may be realized as extended objects, as well, that is, as regions containing two, or more, sites. Whether the Kondo physics is realized, or not, does actually depend on whether the level spectrum of the isolated impurity takes, or not, a degenerate ground state. A doubly degenerate ground state is certainly realized in an extended region with an odd number of sites, without explicit breaking of "spin inversion" symmetry (that is, in the absence of local "magnetic fields"). For instance, let us consider a central region realized by three sites (j = −1, 0, 1), lying between the weak links. Let the central region Hamiltonian be given by
H Middle 3J = −J S + −1 S − 0 + S + 0 S − 1 + h.c. + J z S z −1 S z 0 + S z 0 S z 1 ,(A1)
and let the central region be connected to the left-hand chain (which, as in the main text, we denote by the label L ), and to the right-hand chain (denoted by the label R ) with the coupling Hamiltonian
H Coupling = − J ′ L S + 1,L S − −1 + J ′ R S + 1,R S − 1 + h.c. + J ′ z,L S z 1,L S z −1 + J ′ z,R S z 1,R S z 1 .(A2)
A simple algebraic calculation shows that the ground state of H Middle 3J is doubly degenerate and consists of the spin-1/2 doublet given by
| 1 2 2 = 1 √ 2 {sin( θ 2 )[↑↑↓ + | ↓↑↑ ] + √ 2 cos( θ 2 )| ↑↓↑ } ,(A3)
and
| − 1 2 2 = 1 √ 2 {sin( θ 2 )[↓↓↑ + | ↑↓↓ ] + √ 2 cos( θ 2 )| ↓↑↓ } ,(A4)
with
cos(θ) = J z √ 2J 2 + J z , cos(θ) = √ 2J √ 2J 2 + J z ,(A5)
whose energy is given by E
1 2 2 = −J z − J 2 z + 2J 2 .
Defining an effective spin-1/2 operator for the central region, S G , as
S + G ≡ | 1 2 2 2 − 1 2 | , S z G ≡ 1 2 b=±1 b|b 1 2 2 2 b 1 2 | ,(A6)V 3J B = −{[J ′ L sin(θ)S + 1,L + J ′ R S + 1,R ]S − G + [J ′ L S − 1,L + J ′ R S − 1,R ]S + G } + cos(θ)[J ′ z,L S z 1,L + J ′ z,R S z 1,R ]S z G . (A7)
Thus, we see that we got back to the spin-1/2 spin-chain Kondo Hamiltonian, with a renormalization of the boundary couplings, according to
J ′ L(R) −→ J ′ L(R) sin(θ) = √ 2J ′ L(R) J √ 2J 2 + J z , J ′ z,L(R) −→ J ′ z,L(R) cos(θ) = J ′ z,L(R) J z √ 2J 2 + J z .(A8)
A local magnetic field h may break the ground state degeneracy, thus leading, in principle, to the breakdown of the Kondo effect. However, in analogy to what happens in a Kondo dot in the presence of an external magnetic field 7,8,88 , Kondo physics should survive, at least as long as h ≪ E K , with E K (∼ k B T K ) being the typical energy scale associated to the onset of Kondo physics. At variance, when the central region is made by an even number of sites, the groundstate is not degenerate anymore. As a consequence, the central region should be regarded as a weak link between two chains. For instance, we may consider the case in which the central region is made by two sites. Using for the various parameters the same symbols we used above, performing a SW resummation, we obtain the effective weak link boundary Hamiltonian
V 2J B = −λ ⊥ S + L,1 S − R,1 + S + R,1 S − L,1 − λ z S z L,1 S z R,1 ,(A9)
with
λ ⊥ ∼ (J ′ ) 2 J + 2J z , λ z ∼ (J ′ z ) 2 2J . (A10)
Appendix B: bosonization approach to impurities in the XXZ spin chain
In this section we review the bosonization approach to the XXZ spin chain as it was originally developed in Refs. [53,54]. As a starting point, we consider a single, homogeneous spin-1/2 XXZ spin chain, with ℓ sites, obeying open boundary conditions at its endpoints, described by the model Hamiltonian H XXZ , given by
H XXZ = −J ℓ−1 j=1 S + j S − j+1 + S + j+1 S − j + J z ℓ−1 j=1 S z j S z j+1 .(B1)
The low-energy, long-wavelength dynamics of such a chain is described 53 in terms of a spinless, real bosonic field Φ(x, τ ) and of its dual field Θ(x, τ ). The imaginary time action for Φ is given by
S E [Φ] = g 4π β 0 dτ ℓ 0 dx 1 u ∂Φ ∂τ 2 + u ∂Φ ∂x 2 ,(B2)
where the constants g, u are given by
g = π 2(π − arccos( ∆ 2 )) , u = v f π 2 1 − ( ∆ 2 ) 2 arccos( ∆ 2 ) ,(B3)
with v f = 2dJ, d being the lattice step, and ∆ = J z /J. The fields Φ and Θ are related to each other by the relations
∂Φ(x,τ ) ∂x = 1 u ∂Θ(x,τ ) ∂x , and ∂Θ(x,τ ) ∂x = 1 u ∂Φ(x,τ ) ∂x
. A careful bosonization procedure shows that, in addition to the free Hamiltonian in Eq.(B2), an additional Sine-Gordon, Umklapp interaction arises, given by
H SG L = −G U ℓ 0 dx cos[2 √ 2Θ(x)] .(B4)
Since the scaling dimension of H SG L is h U = 4g, it will be always irrelevant within the window of values of g we are considering here, that is, 1/2 < g. In fact, H SG L becomes marginally irrelevant at the "Heisenberg point", g = 1/2, which deserves special attention 9 , though we do not consider it here. Within the continuous bosonic field framework, the open boundary conditions of the chain are accounted for by imposing Neumann-like boundary conditions on the field Φ(x, τ ) at both boundaries 76,[89][90][91] , that is
∂Φ(0, τ ) ∂x = ∂Φ(ℓ, τ ) ∂x = 0 . (B5)
Equation (B5) implies the following mode expansions for Φ(x, τ ) and Θ(x, τ )
Φ(x, τ ) = 2 g q − iπuτ ℓ P + i n =0 α(n) n cos πnx ℓ e − πn ℓ uτ Θ(x, τ ) = 2g θ + πx ℓ P + n =0 α(n) n sin πnx ℓ e − πn ℓ uτ ,(B6)
with the normal modes satisfying the algebra
[q, P ] = i , [α(n), α(n ′ )] = nδ n+n ′ ,0 .(B7)
The bosonization procedure allows for expressing the spin operators in terms of the Φ-and Θ-fields. The result is 92
S + j −→ c(−1) j e i √ 2 Φ(xj ,τ ) + be i √ 2 Φ(xj ,τ )+i √ 2Θ(xj,τ ) S z j −→ 1 √ 2π ∂Θ(x j , τ ) ∂x + a(−1) j sin[ √ 2Θ(x j , τ )] .(B8)
The numerical parameters a, b, c in Eq.(B8) depend only on the anisotropy parameter ∆ = J z /J 92-96 . While their actual values is not essential to the RG analysis in Sec.III, it becomes important when computing the real-space correlation functions of the chain within the bosonization approach, in which case one may refer to the extensive literature on the subject, as we do in Sec.IV.
To employ the bosonization approach to study an impurity created between the L and the R chain, we start by doubling the construction outlined above, so to separately bosonize the two chains with open boundary conditions (which is appropriate in the limit of a weak interaction strength for either H K in Eq.(5), or V 2J B in Eq.(A9)). Therefore, on introducing two pairs of conjugate bosonic fields Φ L , Θ L and Φ R , Θ R to describe the two chains, the corresponding Euclidean action is given by
S E [Φ L , Φ R ] = g 4π β 0 dτ ℓ 0 dx X=L,R 1 u ∂Φ X ∂τ 2 + u ∂Φ X ∂x 2 ,(B9)
supplemented with the boundary conditions
∂Φ L (x, 0) ∂x = ∂Φ L (ℓ, τ ) ∂x = 0 , ∂Φ R (x, 0) ∂x = ∂Φ R (ℓ, τ ) ∂x = 0 .(B10)
Taking into account the bosonization recipe for the spin-1/2 operators, Eqs.(B8), one obtains that, in the case in which G contains an even number of sites (and is, therefore, described by the prototypical impurity Hamiltonian V 2J B ), the effective weak link impurity between the two chains is described by the Euclidean action
S B G = −λ ⊥ β 0 dτ {e i √ 2 [ΦL(τ )−ΦR(τ )] + e − i √ 2 [ΦL(τ )−ΦR(τ )] } − λ z 2π 2 β 0 dτ ∂Θ L (τ ) ∂x ∂Θ R (τ ) ∂x ,(B11)
with Φ L,R (τ ) ≡ Φ L,R (0, τ ), and Θ L,R (τ ) ≡ Θ L,R (0, τ ). Similarly, in the case in which G contains an odd number of sites, in bosonic coordinates, the prototypical Kondo Hamiltonian H K yields to the Euclidean action given by Equation (B12) provides the starting point to perform the RG analysis for the Kondo impurity of Section III. To illustrate in detail the application of the RG approach to link impurities in spin chains, in the following part of this appendix we employ it to study the weak link boundary action in Eq.(B11). Following the standard RG recipe, to describe how the relative weight of the impurity interaction depends on the reference cutoff scale of the system, we have to recover the corresponding RG scaling equations for the running coupling strengths associated to λ z and to λ ⊥ . This is readily done by resorting to the Abelian bosonization approach to spin chains applied to the boundary action in Eq.(B11) 53 . From Eq.(B11) one readily recovers the scaling dimensions of the various terms from standard Luttinger liquid techniques, once one has assumed the mode expansions in Eqs.(B6) for the fields Φ L (x, τ ), Θ L (x, τ ), as well as Φ R (x, τ ), Θ R (x, τ ) 53,54 . Specifically, one finds that the term ∝ λ ⊥ has scaling dimension h ⊥ = 1 g , while the term ∝ λ z has scaling dimension h = 2. As we use the chain length ℓ as scaling parameter of the system, to keep in touch with the standard RG approach, we define the dimensionless running coupling strengths L ⊥ (ℓ) = ℓ ℓ0 1− 1 g λ ⊥ J
S B G = −
and L (ℓ) = ℓ ℓ0 −1 λz J , with ℓ 0 being a reference length scale (see below for the discussion on the estimate of ℓ 0 ). To leading order in the coupling strengths, we obtain the perturbative RG equations for the running parameters given by 77
dL ⊥ (ℓ) d ln ℓ ℓ0 = 1 − 1 g L ⊥ (ℓ) dL (ℓ) d ln ℓ ℓ0 = −L (ℓ) .(B13)
Equations (B13) encode the main result concerning the dynamics of a weak link in an otherwise uniform XXZ chain 53,76,79 . Leaving aside the trivial case g = 1, corresponding to effectively noninteracting JW fermions, which do not induce any universal (i.e., independent of the bare values of the system parameters) flow towards a conformal fixed point, we see that the behavior of the running strengths on increasing ℓ is drastically different, according to whether g < 1 (∆ > 0), or g > 1 (∆ < 0). In the former case, both h ⊥ and h are > 1, which implies that V 2J B is an irrelevant perturbation to the disconnected fixed point. The impurity interaction strengths flow to zero in the low-energy, long-wavelength limit, that is, under RG trajectories, the system flows back towards the fixed point corresponding to two disconnected chains. At variance, when g > 1, L ⊥ (ℓ) grows along the RG trajectories and the system flows towards a "strongly coupled" fixed point, which corresponds to the healed chain, in which the weak link has been healed within an effectively uniform chain obtained by merging the two side chains with each other 77 . The healing takes place at a scale ℓ ∼ ℓ Heal , with 75,76 ℓ Heal ∼ ℓ 0 1 L(ℓ 0 ) g g−1 .
(B14)
As we see from Eq.(B14), defining ℓ Heal requires introducing a nonuniversal, reference length scale ℓ 0 . ℓ 0 is (the plasmon velocity times) the reciprocal of the high-energy cutoff D 0 of our system. To estimate D 0 , we may simply require that we cutoff all the processes at energies at which the approximations we employed in Appendix B to get the effective boundary Hamiltonians break down. This means that D 0 must be of the order of the energy difference δE between the groundstate(s) and the first excited state of the central region Hamiltonian. From the discussion of Appendix B, we see that δE ∼ J, which, since we normalized all the running couplings to J, implies ℓ 0 ∼ d, d being the lattice step of the microscopic lattice Hamiltonian describing our spin system. To conclude, it is important to stress that, though an RG invariant length scale ℓ Heal emerges already at a weak link between two chains with ∆ < 0, there is no screening cloud associated to this specific problem. Indeed, in the case of a weak link impurity, the healing of the chain is merely a consequence of repeated scattering off the Friedel oscillations due to backscattering at the weak link [97][98][99] , which conspire to fully heal the impurity at a scale ℓ Heal . At variance, when there is an active spin-1/2 impurity, the density oscillations are no longer simply determined by the scattering by Friedel oscillations, but there is also the emergence of the Kondo screening cloud induced in the system 79 .
Appendix C: Bosonization results for the correlation functions between spin operators at finite imaginary time
In this appendix we provide the generalization of the equal-time spin-spin correlation functions on an open chain, derived in Ref. [92], to the case in which the spin operators are computed at different imaginary times τ, τ ′ . As discussed in the main text, such a generalization is a necessary step in order to compute the contributions to the spin correlations due to the impurity interaction in S B G . The starting point is provided by the finite-τ bosonic operators over a homogeneous, finite-size chain of length ℓ, which we provide in Eqs.(B6) of the main text. Inserting those
FIG. 1 :
1External potential Vext (in units of V0) as a function of x (in units of d) for different values of V1/V0 and V2/V0 (with σ = 0.2d). In panels a) and b), we consider V2 = 0 so that only two hopping parameters are altered: panel a) corresponds to x1/d = 0.5 and panel b) to x0;1/d = 0.25 (in both cases V1/V0 = 2). In panels c) and d), we have x0;1/d = 0.5 and x−1;0/d = −0.5 in both cases, but V1/V0 = V2/V0 = 2 for panel c) and V1/V0 = 2, V2/V0 = 3 for panel d).
FIG. 2 :
2Sketch of two different kinds of central regions in an otherwise uniform spin chain, respectively realizing an effective weak-link impurity (a)), and an effective spin-1/2 impurity (b)).
FIG
z positive, or the former one positive, the . 3: ℓ * /ℓ0 as a function of V /J for 0 ≤ V /J ≤ 0.5. The other parameters are chosen so that U/J = 4 and J ′ /J = 0.2 (panel a)), and J ′ /J = 0.6 (panel b)). As discussed in Appendix B, ℓ0 is of order of the lattice spacing d.
R ) in the bare couplings.
FIG
. 4: a): Plot of Σz(x) vs. x for U/J = 4, V = 0 (corresponding to ∆ = −0.1875) and J ′ /J = 0.2. From the plot one infers ℓ * ∼ 26, which is in good agreement with the value obtained from the plot in
We thank L. Dell'Anna, A. Nersesyan, A. Papa, and D. Rossini for valuable discussions. A.T. acknolweldges support from talian Ministry of Education and Research (MIUR) Progetto Premiale 2012 ABNANOTECH -ATOM-BASED NANOTECHNOLOGY. Appendix A: Effective weak link-and Kondo-Hamiltonians for a spin-1/2 XXZ spin chain
2
ΦR(τ ) ]S − G + h.c.
formulas for Φ(x, τ ) and Θ(x, τ ) in the bosonic formulas in Eqs.(B8) and computing the imaginary-time ordered correlation functions G +,− (x, x ′ ; τ |ℓ) = T τ S + x (τ )S − x ′ (0) and G z,z (x, x ′ ; τ |ℓ) = T τ S z x (τ )S z x ′ (0) , one obtains:as well asAs stated above, Eqs.(C1) and (C2) provide the finite-τ generalization of Eqs.(8a) and (8b) of Ref.[92], to which they reduce in the τ → 0 limit.
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| []
|
[
"Trifo-VIO: Robust and Efficient Stereo Visual Inertial Odometry using Points and Lines",
"Trifo-VIO: Robust and Efficient Stereo Visual Inertial Odometry using Points and Lines"
]
| [
"† Feng Zheng ",
"Grace Tsai ",
"Zhe Zhang ",
"Shaoshan Liu ",
"Chen-Chi Chu ",
"Hongbing Hu "
]
| []
| []
| In this paper, we present the Trifo Visual Inertial Odometry (Trifo-VIO), a tightly-coupled filtering-based stereo VIO system using both points and lines. Line features help improve system robustness in challenging scenarios when point features cannot be reliably detected or tracked, e.g. low-texture environment or lighting change. In addition, we propose a novel lightweight filtering-based loop closing technique to reduce accumulated drift without global bundle adjustment or pose graph optimization. We formulate loop closure as EKF updates to optimally relocate the current sliding window maintained by the filter to past keyframes. We also present the Trifo Ironsides dataset, a new visual-inertial dataset, featuring highquality synchronized stereo camera and IMU data from the Ironsides sensor [3] with various motion types and textures and millimeter-accuracy groundtruth. To validate the performance of the proposed system, we conduct extensive comparison with state-of-the-art approaches (OKVIS, VINS-MONO and S-MSCKF) using both the public EuRoC dataset and the Trifo Ironsides dataset. | 10.1109/iros.2018.8594354 | [
"https://arxiv.org/pdf/1803.02403v2.pdf"
]
| 3,743,447 | 1803.02403 | 9c6895fa458ca7b60951a41b3c39ef4f638190e9 |
Trifo-VIO: Robust and Efficient Stereo Visual Inertial Odometry using Points and Lines
† Feng Zheng
Grace Tsai
Zhe Zhang
Shaoshan Liu
Chen-Chi Chu
Hongbing Hu
Trifo-VIO: Robust and Efficient Stereo Visual Inertial Odometry using Points and Lines
In this paper, we present the Trifo Visual Inertial Odometry (Trifo-VIO), a tightly-coupled filtering-based stereo VIO system using both points and lines. Line features help improve system robustness in challenging scenarios when point features cannot be reliably detected or tracked, e.g. low-texture environment or lighting change. In addition, we propose a novel lightweight filtering-based loop closing technique to reduce accumulated drift without global bundle adjustment or pose graph optimization. We formulate loop closure as EKF updates to optimally relocate the current sliding window maintained by the filter to past keyframes. We also present the Trifo Ironsides dataset, a new visual-inertial dataset, featuring highquality synchronized stereo camera and IMU data from the Ironsides sensor [3] with various motion types and textures and millimeter-accuracy groundtruth. To validate the performance of the proposed system, we conduct extensive comparison with state-of-the-art approaches (OKVIS, VINS-MONO and S-MSCKF) using both the public EuRoC dataset and the Trifo Ironsides dataset.
I. INTRODUCTION
Motion tracking is the cornerstone for a wide range of applications, such as robotics, self-driving, AR/VR, etc. Due to complementary properties of cameras and inertial measurement units (IMUs) and the availability of these sensors in smartphones and off-the-shelf plug-and-play devices [3], [4], visual-inertial odometry (VIO) has become popular in recent years. Well-known examples that use VIO are Apple ARKit [1] and Google ARCore [2].
There are two common ways to categorize VIO approaches. Based on when visual and inertial measurements are fused, VIO approaches can be divided into looselycoupled and tightly-coupled approaches. Loosely-coupled approaches [22], [28], [37] estimate motions from images and inertial measurements, independently, and then fuse the two estimates to obtain the final estimate. Tightly-coupled approaches [11], [18], [19] fuse visual and inertial data directly at the measurement level to jointly estimate all IMU and camera states. While loosely coupling is flexible and tends to be more efficient, tightly-coupled approaches generally produce more accurate and robust motion estimates. Our proposed Trifo-VIO is a tightly-coupled approach.
Based on how visual and inertial measurements are fused, VIO approaches can be categorized into filteringbased and optimization-based approaches. Filtering based approaches [19], [33] typically employ the Extended Kalman † Feng Zheng, Zhe Zhang, Chen-Chi Chu, and Hongbing Hu are with Trifo, Inc., Santa Clara, CA 95054, USA. Email: {feng.zheng, zhe.zhang, jason.chu, hongbing.hu}@trifo.com * Grace Tsai and ‡ Shaoshan Liu emails are [email protected] and [email protected] respectively. Fig. 1: The Trifo Ironsides dataset capture setup. The Ironsides sensor [3] outputs synchronized stereo camera and IMU data, at 60Hz and 200Hz respectively. The 6-axis robot arm with a working radius of 850 mm provides both motion and millimeter-accuracy groundtruth. The dataset contains 9 sequences, featuring various motion types and textures, making it ideal for both evaluation and development.
Filter (EKF), where state propagation/prediction is made by integrating IMU measurements, and update/correction is driven by visual measurements. Contrarily, optimization based approaches [18], [27] use batch nonlinear optimization to directly minimize the errors between integrated motion from IMU measurements and camera motion estimated by the classic reprojection error minimization. Typically optimization-based approaches are more accurate but computationally more expensive due to repeated linearization. There are approaches that combines the advantages from both approaches. For example, PIRVS [41] performs EKF updates iteratively for efficient motion estimation while using optimization (bundle adjustment) at the backend to reduce long-term drifts. Our proposed Trifo-VIO is an efficient filtering-based VIO, and its accuracy as demonstrated by extensive evaluation is at the same level of or even better than state-of-the-art optimization based approaches.
Most VIO approaches mentioned above only rely on point features, e.g. FAST [29], Shi-Tomasi [31], as intermediate image measurements. The performance of these approaches suffer considerably in low-texture environments, or in scenarios when point features can not be reliably detected or tracked, e.g. lighting change. Many of such low-texture environments, however, contain planar elements that are rich in linear shapes [14], and the detection of edges is less sensitive to lighting changes in nature. Therefore, in the proposed Trifo-VIO, in addition to point features, we extract line segment features as useful image measurements to increase the motion constraints available for challenging scenarios, leading to better system robustness. Both stereo points and line features are processed over a sliding window at cost only linear in the number of features, by using the Multi-State Constraint Kalman Filter (MSCKF) [23].
In addition, visual or visual-inertial odometry systems typically operate at faster speed but are more prone to drift compared to SLAM (Simultaneous Localization And Mapping) systems because odometry systems do not maintain a persistent map of the environment. Therefore, in the proposed Trifo-VIO, we introduce a lightweight loop closing method to reduce long-term drift without any computationally expensive map optimization, e.g. bundle adjustment (BA).
We summarize our contributions as follow:
• To the best of our knowledge, the proposed Trifo-VIO approach is the first tightly-coupled filtering based stereo VIO that uses both point and line features. •
II. RELATED WORK
In this section, we review the state of art of odometry or SLAM approaches in terms of line or edge features and loop closure.
PL-SLAM [26] builds on top of ORB-SLAM [25] and extend its formulation to handle both point and line correspondences in monocular setup. In a similar vein, another joint point and line based work [14], termed PL-SLAM as well, aims at stereo camera setting, and additionally introduces a bag-of-words (BoW) place recognition method using both point and line descriptors for loop detection. In [34], Tarrio and Pedre propose an edge-based visual odometry for a monocular camera, with simple extension to using rotation prior obtained from gyroscope as regularization term within edge alignment error minimization. Most recently, Ling et al. present a tightly-coupled optimization-based VIO by edge alignment in the distance transform domain [20].
Within rich body of filtering-based VIO literatures, there are not many works using edge or line features. One of the earliest work along this line is [17], which uses only line observations to update the filter and also conducts observability analysis. In [38], the authors extend [17] with a new line parameterization which is shown to exhibit better linearity properties and support rolling-shutter cameras. The edge parametrization introduced in [39] allows non-straight contours. Similar to [17] and [38], we use straight line segments.
Direct methods, such as LSD-SLAM [10], DSO [9], rely on image intensities at high-gradient regions, which include but are not limited to image region of features and edges. Usenko et al. [35] extend the vision-only formulation of LSD-SLAM to tightly couple with IMU by minimizing a combined photometric and inertial energy functional. ROVIO [5], [6] is a direct filtering-based VIO method, using photometric error of image patches as innovation term in the EKF update. Our usage of line features, to some extent, lies in between direct and feature-based methods. Despite the advantage of feature-free operation, direct methods rely on brightness constancy assumption, usually suffering from environment lighting change and camera gain and exposure settings. In contrast, in particular to ROVIO, we use point reprojection error and point-to-line distance as the filter update innovation instead of photometric error.
Drift is an inhere issue in SLAM and odometry methods. Loop closure has proven to be effective to correct drift, and state-of-art approaches typically employ global pose graph optimization [25], [27], [14]. In particular, VINS-MONO [27] introduces a two-step loop closure method: (1) local tightly-coupled relocalization which aligns the sliding window with past poses, and (2) global pose graph optimization. Our lightweight loop closure resembles the first step employed by VINS-MONO, except that we realize it in a filtering framework and we exclude global optimization for efficiency. To our best knowledge, it is the first tightlycoupled filtering-based loop closure method. Furthermore, our proposed Trifo-VIO handles both stereo point and line features and loop closure in a consistent filtering framework.
III. ESTIMATOR DESCRIPTION
The backbone of our estimator is MSCKF whose key idea is to maintain and update a sliding window of camera poses using feature track observations without including features in the filter state [23]. Instead, 3D feature positions are estimated via least-squares multi-view triangulation and subsequently marginalized, which resembles structureless BA to some extent. The advantage of doing this is considerable reduction of computational cost, making MSCKF's complexity linear in the number of features, instead of cubic like EKF-SLAM [8].
We introduce two types of EKF updates: (1) joint point and line features based update to cope with challenging scenarios and to enhance robustness, and (2) loop closing update to reduce accumulated drift. Filter consistency is ensured by using the right nullspace of the observability Gramian to modify state transition matrix and observation matrix at each propagation and update step, following OC-EKF [15].
A. State Parameterization
We follow [23] and define the evolving IMU state as follows:
X B = [ B G q T b g T G v T B b a T G p T B B C q T B p T C ] T (1)
where B G q is the unit quaternion representing the rotation from the global frame {G} to the IMU body frame {B}, G p B and G v B are the IMU position and velocity in the global frame, and b g and b a denote gyroscope and accelerometer biases. Optionally, we include IMU extrinsics B C q and B p C in the state, which represent the rotation and the translation between the IMU body frame {B} and the camera frame {C}.
At time k, the full state of our estimator consists of the current IMU state estimate and N camera poseŝ
X k = [X T B kX T C1 ...X T C N ] T (2) whereX C = [ C Gq
T Gp T C ] T represents the camera pose estimate.
We use the error-state representation in order to minimally parameterize orientation in 3 degrees of freedom (DOF) and to avoid singularities [32]. Specifically, for the position, velocity, and biases, the standard additive error is employed, while for the orientations, the compositional update q = δq ⊗q is used, where δq is the 3DOF error quaternion as follows
δq = [ 1 2 δθ 1] T(3)
B. EKF Propagation
Whenever a new IMU measurement is received, it is used to propagate the EKF state and covariance estimates. We use the standard continuous-time IMU kinematics model as follows
B Gq = 1 2 Ω(ω) B Gq (4) b g = 0 3×1 (5) Gv = R( G Bq )â + G g (6) b a = 0 3×1 (7) Gṗ B = Gv (8) B Cq = 0 3×1 (9) Bṗ C = 0 3×1(10)
whereω andâ are angular velocity and linear acceleration from gyroscope and accelerometer respectively with biases removed, R denotes the corresponding rotation matrix of the quaternion, and Ω(ω) ∈ R 4×4 is the skew-symmetric matrix formed from the angular rate
Ω(ω) = −[ω × ]ω −ω T 0(11)
Our discrete-time implementation employs 4th order Runge-Kutta numerical method. We ignore earth rotation rate in the model as in most MEMS IMUs it cannot be sensed due to gyro bias instability and noise. For sake of simplicity, we also omit the description of state transition matrix and covariance propagation. Interested readers please refer to [23]. Green: tracked features. Line features help improve system robustness in challenging scenarios, e.g. low-texture environment (a) and motion blur (b). These two stereo frames are from the Trifo Ironsides dataset PI 3058.
C. Measurement Model for Point Features
In MSCKF, all the continuous measurements of the same 3D point, i.e. feature tracks, are used to update all involved camera poses that observe the point. The residual is the standard reprojection error:
r fi = z fi −ẑ fi(12)
where
z fi = [u i v i ] T isz fi = π( R( C Gq ) ( G p fi − Gp C ) )(13)
where π is the pinhole projection model (π :
R 3 → R 2 ) π(p) = 1 Z X Y(14)
We then linearize Eq. 12 about the estimates for the camera pose and for the feature position, and calculate the Jacobians with respect to the state and the feature position as H X f i and H fi respectively, following [23]. After that, we marginalize the feature position via nullspace projection to de-correlate it with the state. So far, we have described the measurement model for monocular camera. One tricky part is the estimation of 3D feature positions, which is typically computed by multi-view triangulation in least-squares fashion. There has to be enough baseline among the cameras observing the same feature in order to do the triangulation. Therefore, monocular MSCKF cannot estimate the 3D positions of features nor do EKF updates while being static or undergoes rotation dominant motion. This motivates us to adopt the more practical stereo camera setup to overcome this limitation, from which we can also easily get the true scale. For stereo feature measurements, we employ a simple yet effective representation, similar to [33], Note that the stereo camera is assumed to be calibrated beforehand, and the camera extrinsics relating the left and the right cameras is assumed to be constant.
z fi = π( C1p fi ) π( C2p fi ) = π( R( C1 Gq )( G p fi − Gp C1 ) ) π( R( C2 Gq )( G p fi − Gp C2 ) )(15)
D. Measurement Model for Line Features
We now present the measurement model of line features for updating the state estimates. We denote a line l i in image using point-normal form, l i = [z li n li ], where z i is any point on the line and n li ∈ R 2×1 is a unit vector denoting line's normal direction in image space. For a 3D line, L j , we over-parameterize it by using two 3D endpoints,
G L j = [ G p b G p e ],
where G p b and G p e are the beginning and ending endpoints on the 3D line in the global frame.
For the line feature residual, r li ∈ R 2×1 , we use the point to line distance, as follows
r li = (z li −ẑ l ib ) · n li (z li −ẑ lie ) · n li(16)
whereẑ l ib ∈ R 2×1 andẑ lie ∈ R 2×1 are the 2D projections of the beginning and ending endpoints on the 3D line, and · represents dot product. To conform to the standard form of EKF residual as in Eq. 12, we simplify Eq. 16 to
r li = n T li z li − n T liẑlib n T li z li − n T liẑlie(17)
Note n T l ib z l ib produces a scalar number, thus one 3D endpoint results in one dimensional residual. This is desirable as line features can only provide useful constraints in the normal direction. Therefore, every 3D line represented by two endpoints produces a two dimensional residual. The over-parameterization makes sure that, if the projected 3D line and the observed line do not perfectly align, at least in one dimension of the residual it will not be zero. This holds even when one projected endpoint is accidentally on the observed line.
Another benefit of this measurement model is that the line feature Jacobian becomes extremely easy to calculate and feature marginalization can be done in the same way as point features. Under the chain rule, we can derive the Jacobian for each line as follows where H l ib ∈ R 2×3 can be calculated in the same way as point feature Jacobian H fi and the "point" here is the beginning endpoint of the line. Likewise, H lie ∈ R 2×3 is the "point" Jacobian of the line ending endpoint. Note that n T li H l ib ∈ R 1×3 , hence H li ∈ R 2×3 . Similarly, the Jacobian of line feature with respect to the state can be derived as
H li = n T li H l ib n T li H lie(18)H X l i = n T li H X l ib n T li H X l ie(19)
where H X l ib and H X l ie are the Jacobians of the line's beginning and ending endpoints with respect to the state, and they share the same formula as point features.
To extend the measurement model to the stereo setting is straightforward. We follow the way we use for stereo point features, and represent the stereo line residual, r li ∈ R 4 , as follows [21], can significantly benefit from this operation.
E. EKF Update: Point and Line Features
We adopt a similar update strategy as [23]: whenever a point and/or line feature is no longer tracked, or the sliding window size exceeds the predefined maximum size, EKF update is triggered. Point and line features are subsequently marginalized since their positions are directly correlated with the state estimateX. This makes the algorithm complexity linear in the number of features. The marginalization is performed by using the left nullspace of feature Jacobian, which cancels out the feature term in the linearized residual. We then stack the transformed residuals and the state Jacobians of both points and lines to form the final residual and observation matrix.
F. EKF Update: Loop Closure
To reduce accumulated drift while being efficient for resource-constrained platforms which cannot afford global BA, we present a novel lightweight loop closure method, formulated as native EKF updates. As will be described in Section V, when a new camera state is added to the sliding window, we perform keyframe selection and trigger loop detection in a parallel thread if selected. If a loop is detected while the keyframe is still in the sliding window, loop closing updates will be triggered. Otherwise, the keyframe is added to the database along with its feature descriptors and 3D positions.
Since loop detection establishes feature matches between the current keyframe and the past, we use feature positions from the past keyframes for EKF updates instead of retriangulating them using current poses which suffer from drift. The update procedure is almost the same as the update with point features, except that we treat 3D positions of such loop closure features as prior knowledge, and thus do not perform feature marginalization. This makes sense given such "map" points have been marginalized in the past along with the keyframes which are inserted into the loop detection database.
As shown in Fig. 3, the accumulated drift is effectively corrected by the loop closing update. A benefit of doing loop closure as EKF updates is that the drift is corrected progressively across multiple consecutive camera frames as long as loop closure features are tracked, rather than immediately which often introduces sudden large jump in the subsequent pose estimate and it is not desirable for closed-loop control, e.g. of drones. While the introduced loop closing update is similar to [24] and [41] where map based update is employed, their maps are either pre-built or online estimated via the costly BA in a separate thread. To our best knowledge, the introduced tightly-coupled filteringbased loop closure using marginalized "map" points is novel.
IV. IMAGE PROCESSING
In this section, we describe our image processing pipeline for detection and tracking of point and line features. An example is shown in Fig. 2.
For each new image, we track existing point features via KLT optical flow (OF) [21] and for non-tracked image regions new features are detected via FAST feature detector [29]. We enforce uniform distribution of features in image by spatial binning and maintain a fixed number of high response features in each bin. To cope with fast motion, we obtain initial guess for optical flow using relative rotation computed from gyroscope measurements. We use KLT OF for stereo feature matching as well similar to [33] for efficiency. To reject outliers in both stereo and temporal matching, we use 2-point RANSAC and space-time circular matching [16].
For line features, we use Line Segment Detector (LSD) [36] to extract line segments. For each line segment detected, we extract binary descriptor using Line Band Descriptor (LBD) [40]. Both stereo and temporal matching of line features are based on LBD descriptor matching. To ensure best matches, we perform 4-way consistency check, i.e. left to right, right to left, previous to current, and current to previous. Furthermore, we prune putative matches by checking the length and orientation of lines.
To further enhance the robustness of feature tracking and matching, we introduce a fast brightness check between stereo and temporally consecutive images based on their mean brightnesses, and perform histogram matching to ensure consistent brightness and contrast across stereo and temporal images if necessary. An example is shown in Fig. 4. This considerably boosts stereo feature matching and temporal tracking performance under unfavorable conditions, e.g. auto-exposure mismatch between stereo cameras, dramatic lighting change. This is in contrast to using histogram equalization for each frame as done in [27] and [41], which incurs more computation and disregards brightness consistency between stereo and temporal images, and may result in over enhancement.
V. LOOP DETECTION
In this section, we describe our loop detection approach. For each new image, we do keyframe selection based on the number of features tracked and the pose distance to existing keyframes in the loop detection database. If a keyframe is selected, we extract ORB descriptors [30] for loop detection. Our loop detection is implemented based on DBoW2 [12] which is both fast and reliable, and it runs in a parallel thread to the main VIO thread. For candidate loops, similar to [27], we perform two-step outlier rejection: 2D-2D fundamental matrix test and 3D-2D PnP test both within the RANSAC framework. This outlier rejection strategy is effective as shown in Fig. 5. If the number of inlier feature matches is above the pre-defined threshold, we mark loop detected and trigger loop closing EKF updates as described in Sec. III-F. If the current keyframe does not contain loops, we add it to the database when it is marginalized from the active sliding window maintained by the filter along with its pose, 2D and 3D positions of features, and their descriptors. We set a maximum number of keyframes in the database considering memory requirement and detection speed to make sure that it returns result within one camera frame.
VI. EXPERIMENTS
We conduct two experiments to demonstrate the performance of the proposed Trifo-VIO approach. Both experiments compare Trifo-VIO to competitive state-of-the-art VIO approaches including OKVIS [18], VINS-MONO [27], and S-MSCKF [33]. OKVIS and VINS-MONO are optimization based tightly coupled VIO systems, while S-MSCKF is a tightly-coupled filtering-based stereo VIO system closely related to us. Both OKVIS and S-MSCKF support stereo camera hence we run them in stereo mode, while VINS-MONO is a monocular system. The first experiment is conducted with the public EuRoC MAV dataset [7], while the second is with our new Trifo Ironsides dataset. As all comparison approaches contain more or less non-determinism, e.g. due to RANSAC, we repeat all experiments five times and report median numbers.
A. EuRoC MAV Dataset
The EuRoC dataset contains eleven sequences in three categories (MH, V1, V2) collected on-board a Micro Aerial Vehicle (MAV). We select nine from them so that each category contains 3 datasets. For comparison approaches, we use their default parameters, as they all have been carefully tuned for the EuRoC dataset. In addition, we keep the global loop closure on for VINS-MONO, as we want to compare with it in its best form. Evaluation results are shown in Fig. 6a. It is evident that Trifo-VIO is among the best performing methods, leading results in MH 04, MH 05, V1 01, and V2 01.
For the V2 03 dataset, S-MSCKF produces poor results, mentioned in [33] as well, for the reason that "the continuous inconsistency in brightness between the stereo images causes failures in stereo feature matching". Hence, its result is not reported in Fig. 6a. In contrast, the histogram matching method employed by Trifo-VIO makes it robust to this challenging scenario, as demonstrated in Fig. 4. In addition, V2 03 has about 400 missing frames in the left camera data, resulting in OKVIS tracking failure. After we prune extra frames from the right camera data, OKVIS runs well. Note that we use the original V2 03 dataset for Trifo-VIO evaluation as our approach is robust enough to handle frame drop in either stereo or temporal frames. VINS-MONO is not affected as it is a monocular approach and uses only left camera data.
B. Trifo Ironsides Dataset
To further evaluate the performance of Trifo-VIO, we introduce a new public dataset. The dataset is recorded by the Trifo Ironsides [3] in a robot arm platform as shown in Fig. 1. We collect in total 9 sequences, featuring a wide range of motions and environmental conditions, from controlled slow motion around each axis under good visual conditions to fast random motion with motion blur and low texture. For details, please refer to Table I. We provide the entire dataset in two formats, ROS bag and zipped format, similar to the EuRoC dataset. The groundtruth provided by the robot arm is up to millimeter accuracy and precisely synced with the Ironsides sensor. Therefore, the Trifo Ironsides dataset is ideal for both VIO/SLAM development and evaluation.
The comparison result is shown in Fig. 6b. We tune parameters of all approaches to make them perform well as much as possible. Our Trifo-VIO is consistently among the top two best performing approaches. S-MSCKF results are close to us, except dataset PI 3058, where we show significantly better results due to the usage of additional line features. PI 3058 is the most challenging sequence in the dataset, containing fast motion and low texture in many parts of the sequence, making it hard for VIO approaches which rely on only point features. For OKVIS, it performs well for easy sequences (PI S1 X1, PI S1 Y1, and PI S1 Z1) whose dominant motions are slow translation. However, OKVIS produces poor results for PI 3058, PI S1 Y2, and PI S1 Z2, hence we omit reporting those numbers. We notice that OKVIS's feature matching suffers from repetitive textures in the scene. Note that we exclude VINS-MONO from this comparison as rotation dominant motion at the beginning of many datasets leads to poor initialization and tracking failure. To improve monocular SLAM initialization, delayed initialization till enough parallax and model selection between fundamental matrix and homography could help [13], [25].
VII. CONCLUSIONS
In this work, we have presented Trifo Visual Inertial Odometry (Trifo-VIO), a new tightly-coupled filtering-based stereo visual inertial odometry approach using both point and line features. Line features help improve system robustness in point-scarce scenarios, e.g. low texture and changing light. Both stereo point and line features are processed over a sliding window at cost only linear in the number of features. To reduce drift, which is inherent in any odometry approach, we have introduced a novel lightweight loop closure method formulated as native EKF updates to optimally relocate the current sliding window maintained by the filter to past We have also presented the Trifo Ironsides dataset, a new public visual-inertial dataset, featuring high-quality synchronized stereo camera and IMU data from the Trifo Ironsides sensor with various motion types and textures and millimeteraccuracy groundtruth. The extensive evaluation against competitive state-of-the-art approaches using this new dataset and the public EuRoC dataset clearly demonstrate the superior performance of the proposed Trifo-VIO approach.
Fig. 2 :
2Stereo point and line features. Magenta: new features.
the observation of the i-th feature in the image, whileẑ fi is the predicted measurement of the feature from projecting its estimated 3D position Gp fi = [ GX i GŶ i GẐ i ] T in global frame into the image based on the estimated camera pose and the projection model as followŝ
whereẑ fi ∈ R 4 ,
4C1p fi and C2p fi are the estimated 3D positions of the same feature point in left and right camera coordinates respectively, andX C1 = [ T are stereo camera poses at the same timestamp.
Fig. 3 :
3Results of loop closing EKF update. In this sequence, the Ironsides device is handheld, and we move it around the office and go back to the starting position. (a), (b) and (c) are visualizations of device poses at the beginning, around the end before the loop closure and after it, by rendering a fixedposition virtual cube in front of the first camera. From the rendered cube, it is evident that the pose drift is effectively corrected. In addition, (b) and (c) are not the same frame but a few frames apart, as the drift is corrected progressively rather than immediately. (d) shows the trajectory in the horizontal plane, where the red dot indicates the beginning position, and the magenta dot and the green dot are the positions before and after the loop closure, corresponding to (a), (b), and (c) respectively. Note that the red dot is largely covered by the green dot, indicating the drift is almost fully corrected.
Fig. 4 :
4Results of histogram matching (HM) to deal with dramatic brightness mismatch. (a) Stereo HM: top -stereo left image, middle -original stereo right image, bottomstereo right image after HM. The frames are from the EuRoC V2 03 difficult dataset. (b) Temporal HM: top -left image, middle -original left image at the next timestamp, bottomleft image at the next timestamp after HM. The frames are from the EuRoC V1 03 difficult dataset. It is obvious that middle images in both (a) and (b) exhibit strong brightness difference compared to the top ones, and the HM results at the bottom match the top ones well in terms of overall brightness and distribution. Intensity based feature tracking and matching, e.g. KLT optical flow
Fig. 5 :
5Two-step outlier rejection for loop-detection feature matches. The frames are from the EuRoC V2 02 medium dataset. (Best viewed in color.)
Fig. 6 :
6Absolute trajectory RMSE (Root Mean Square Error) results of our Trifo-VIO and competing approaches including OKVIS, S-MSCKF, and VINS-MONO on both the public EuRoC dataset and our new Trifo Ironsides dataset. Note that we exclude VINS-MONO from the second evaluation. (Best viewed in color.) keyframes. All of them (point features, line features, and loop closure) are handled in a consistent filtering-based framework.
We introduce a novel lightweight filtering-based loop closure method formulated as EFK updates, which optimally relocates the current sliding window maintained by the filter to the detected loops.• We conduct extensive evaluation of our Trifo-VIO with comparison to state-of-the-art open-source VIO approaches including OKVIS[18], VINS-MONO[27], and the recent S-MSCKF[33] using both the EuRoC dataset and our new Trifo Ironsides dataset. • We release the Trifo Ironsides dataset captured using the Ironsides[3], a high-quality device with synchronized stereo camera and IMU data, with millimeter-accuracy groundtruth from the robot arm.The dataset is available at https://github.com/
TrifoRobotics/IRONSIDES/wiki/Dataset.
TABLE I :
IThe Trifo Ironsides DatasetDataset
Motion Type
Speed
Texureness
Difficulty
PI S1 X1
Pure X trans.
Slow
Medium
Easy
PI S1 X2
Pure X rot.
Slow
Medium
Easy
PI S1 Y1
Pure Y trans.
Slow
Medium
Easy
PI S1 Y2
Pure Y rot.
Slow
Medium
Easy
PI S1 Z1
Pure Z trans.
Slow
Medium
Easy
PI S1 Z2
Pure Z rot.
Slow
Medium
Easy
PI S1 R1
Random
Medium
Medium
Medium
PI 3030
Random
Medium
Medium
Medium
PI 3058
Random
Fast
Low
Hard
ACKNOWLEDGMENT Thanks Yen-Cheng Liu for calibrating Ironsides devices and collecting the dataset. Thanks Chandrahas Jagadish Ramalad for preparing the Trifo Ironsides dataset for release.
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"Hidden symmetries induced by a canonical transformation and gauge structure of compactified Yang-Mills theories",
"Hidden symmetries induced by a canonical transformation and gauge structure of compactified Yang-Mills theories"
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"M A López-Osorio ",
"E Martínez-Pascual ",
"H Novales-Sánchez \nFacultad de Ciencias Físico Matemáticas\nUniversidad Michoacana de San Nicolás de Hidalgo\nAvenida Francisco J. Mújica S/N\n58060Morelia, MichoacánMéxico\n",
"J J Toscano ",
"\nFacultad de Ciencias Físico Matemáticas\nBenemérita Universidad Autónoma de Puebla\nApartado Postal 1152Puebla, PueblaMéxico. (\n"
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"Facultad de Ciencias Físico Matemáticas\nUniversidad Michoacana de San Nicolás de Hidalgo\nAvenida Francisco J. Mújica S/N\n58060Morelia, MichoacánMéxico",
"Facultad de Ciencias Físico Matemáticas\nBenemérita Universidad Autónoma de Puebla\nApartado Postal 1152Puebla, PueblaMéxico. ("
]
| []
| Compactified Yang-Mills theories with one universal extra dimension were found [Phys. Rev. D 82, 116012 (2010)] to exhibit two types of gauge invariances: the standard gauge transformations (SGTs) and the nonstandard gauge transformations (NSGTs). In the present work we show that these transformations are not exclusive to compactified scenarios. Introducing a notion of hidden symmetry, based on the fundamental concept of canonical transformation, we analyse three different gauge systems, each of which is mapped to a certain effective theory that is invariant under the socalled SGTs and NSGTs. The systems under discussion are: (i) four dimensional pure SU (3) Yang-Mills theory, (ii) four dimensional SU (3) Yang-Mills with spontaneous symmetry breaking, and (iii) pure Yang-Mills theory with one universal compact extra dimension. The canonical transformation, that induces the notion of hidden symmetry, maps objects with well defined transformation laws under a gauge group G to well defined objects under a non-trivial subgroup H ⊂ G. In the case where spontaneous symmetry breaking is present, the set of SGTs corresponds to the group into which the original gauge group is broken into, whereas the NSGTs are associated to the broken generators and can be used to define the unitary gauge. For the system (iii), the SGTs coincide with the gauge group SU (N, M 4 ), whereas the NSGTs do not form a group; in this system the 'fundamental' theory and the effective one are shown to be classically equivalent. | 10.1103/physrevd.88.036015 | [
"https://arxiv.org/pdf/1302.2981v2.pdf"
]
| 118,624,882 | 1302.2981 | ddde42cf210745a374c4b0671e16f27b7ff53267 |
Hidden symmetries induced by a canonical transformation and gauge structure of compactified Yang-Mills theories
31 Oct 2013
M A López-Osorio
E Martínez-Pascual
H Novales-Sánchez
Facultad de Ciencias Físico Matemáticas
Universidad Michoacana de San Nicolás de Hidalgo
Avenida Francisco J. Mújica S/N
58060Morelia, MichoacánMéxico
J J Toscano
Facultad de Ciencias Físico Matemáticas
Benemérita Universidad Autónoma de Puebla
Apartado Postal 1152Puebla, PueblaMéxico. (
Hidden symmetries induced by a canonical transformation and gauge structure of compactified Yang-Mills theories
31 Oct 2013numbers: 1110Kk1115-q1470Pw1480Rt
Compactified Yang-Mills theories with one universal extra dimension were found [Phys. Rev. D 82, 116012 (2010)] to exhibit two types of gauge invariances: the standard gauge transformations (SGTs) and the nonstandard gauge transformations (NSGTs). In the present work we show that these transformations are not exclusive to compactified scenarios. Introducing a notion of hidden symmetry, based on the fundamental concept of canonical transformation, we analyse three different gauge systems, each of which is mapped to a certain effective theory that is invariant under the socalled SGTs and NSGTs. The systems under discussion are: (i) four dimensional pure SU (3) Yang-Mills theory, (ii) four dimensional SU (3) Yang-Mills with spontaneous symmetry breaking, and (iii) pure Yang-Mills theory with one universal compact extra dimension. The canonical transformation, that induces the notion of hidden symmetry, maps objects with well defined transformation laws under a gauge group G to well defined objects under a non-trivial subgroup H ⊂ G. In the case where spontaneous symmetry breaking is present, the set of SGTs corresponds to the group into which the original gauge group is broken into, whereas the NSGTs are associated to the broken generators and can be used to define the unitary gauge. For the system (iii), the SGTs coincide with the gauge group SU (N, M 4 ), whereas the NSGTs do not form a group; in this system the 'fundamental' theory and the effective one are shown to be classically equivalent.
I. INTRODUCTION
Recently, the ATLAS [1] and CMS [2] experiments at the Large Hadron Collider reported the presence of a scalar boson with mass in the range 125 − 126 GeV that is compatible with the Standard Model Higgs boson. If couplings of this particle to pairs of W and Z weak gauge bosons are found to coincide with those predicted by the Standard Model in subsequent analysis of experimental data, the Higgs mechanism [3] will be firmly established as a genuine phenomenon of nature. Since the Higgs mechanism endows gauge bosons with mass through absorption of Goldstone bosons [4], resulting in spontaneous symmetry breaking (SSB), experimental data would confirm that the weak interaction is spontaneously broken. This in turn would validate the existence of a degenerate vacuum as the source of elementary particle masses and constitutes a good motivation to investigate new mechanisms of mass generation. In particular, it is interesting to study some kind of source, alternative to spontaneous symmetry breaking, that still allows the Higgs mechanism to operate. In this direction, it is already known that gauge theories formulated on spacetime manifolds with compact extra dimensions [5] enable endowing with mass the Kaluza-Klein gauge excitations in the absence of degenerate vacuum. Although in these theories there are pseudo-Goldstone bosons, thus allowing the Higgs mechanism to operate, they do not correspond to genuine Goldstone bosons in the sense of spontaneous breakdown of a global symmetry. The emergent pseudo-Goldstone bosons in Kaluza-Klein theories are directly generated by compactification of the spatial extra dimensions.
In this work we clarify the gauge structure of pure Yang-Mills theories formulated on spacetime manifolds with compact spatial extra dimensions through a novel notion of hidden symmetry. Some theoretical aspects of these theories have already been studied in Refs. [6][7][8][9]. Also, they have been the subject of important phenomenological interest in the context of dark matter [10], neutrino physics [11], Higgs physics [12], flavor physics [13], hadronic and linear colliders [14], and electroweak gauge couplings [15]. In Ref. [6], some results in the context of a pure Yang-Mills theory with one universal extra dimension (UED) were presented. For instance, the necessity of explaining the gauge structure of the compactified theory in order to quantize it was emphasized. In particular, it was noticed that as a consequence of compactification, the original gauge transformations split into two classes: the standard gauge transformations (SGTs) and the nonstandard gauge transformations (NSGTs). In the following pages these transformations are formulated, via a certain canonical transformation, within the framework of hidden symmetry.
The concept of hidden symmetry is usually associated to theories in which SSB occurs; however, we will show that this is not exclusive to theories of this kind. A symmetry, encoded in a gauge group G, can also be hidden if there is a canonical transformation that maps well defined objects under the group G to well defined objects under a subgroup H ⊂ G. As we will show below, this transformation is crucial to understanding a hidden symmetry in this context. It is at the level of the remaining symmetry that SGTs and NSGTs find a clear interpretation. The set of SGTs forms a group which coincides with the subgroup H, whereas the set of NSGTs does not form a group, as will be shown. Nevertheless, the phenomenon of SSB can be fit into this general scenario. When a scalar sector that leads to a degenerate vacuum which is invariant under the subgroup H is introduced, the NSGTs are associated to the broken generators of the group G.
The novel point of view of hidden symmetry given in this paper will be exemplified by studying in detail three gauge models. The first of them shows that our notion of hidden symmetry is not necessarily embedded either in a compactification scheme from a higher dimensional theory or in a theory that presents SSB. Using the four dimensional pure SU (3) Yang-Mills theory, we explicitly construct a canonical transformation which maps gauge fields of SU (3) into gauge fields, two doublets and a singlet with respect to the subgroup SU (2). In this case, the SU (3) symmetry is hidden into the SGTs and NSGTs, which correspond to transformations in SU (2) and transformations related with the five remainder generators of SU (3), respectively.
The second model under consideration is an SU (3) Yang-Mills theory with a renormalizable scalar sector that presents SSB; our canonical transformation will decompose the SU (3) Yang-Mills connection and the matter scalar into well defined objects with respect to SU (2). The new ingredients in the analysis of this model are twofold. On one hand, these will help us to compare a hidden symmetry arising from SSB with a hidden symmetry in terms of the suitable canonical transformation. On the other hand, they will provide a way to clarify the physical meaning of the NSGTs by showing that the unitary gauge corresponds to a particular transformation of this type.
The third system on which we focus our attention is an effective theory that results from compactifying the spatial extra dimension of a manifold M 5 on which a pure Yang-Mills theory is defined. The compactification scheme is achieved along the same lines of Ref. [6]. The basic fields of the higher dimensional theory are gauge fields under the gauge group SU (N, M 5 )-that is, the group SU (N ) with gauge parameters propagating in the bulk M 5 . Defined on the Minkowski spacetime M 4 , the effective theory existing alongside carries the four-dimensional Fourier modes of the five-dimensional gauge fields as basic fields; we show that Fourier expansions in this case determine a canonical transformation which maps well defined objects under SU (N, M 5 ) to well defined objects under SU (N, M 4 ). In our terminology, the gauge symmetry of the higher-dimensional theory is just hidden in the lower-dimensional theory -that is, SU (N, M 5 ) is codified in the SGTs and NSGTs, the former being represented by SU (N, M 4 ). It is worth noticing that in the effective theory there emerge a massless scalar bosons which can be removed by a specific NSGT, these correspond to pseudo-Goldstone bosons which remarkably do not arise from SSB mechanism; in this scenario, compactification does not involve broken generators; it entails a change in the support manifold of the group parameters. The specific NSGT to eliminate the pseudo-Goldstone bosons can hence be interpreted as a unitary gauge [6].
The suitable canonical transformations found in these systems, through which the original symmetry is hidden, permeates at the level of the Dirac algorithm for constrained systems. In each case, the primary Hamiltonian and each generation of constraints, of the theory manifestly invariant under the group G, are mapped onto the corresponding primary Hamiltonian and generation of constraints, present in the theory invariant under SGTs and NSGTs. Since every model we analyzed is a first-class constraint system, this implies that the canonical transformation for each case maps the gauge generator of the group G onto the gauge generators of the SGTs and NSGTs. This result is particularly interesting for the third system, as it implies that the gauge structure of the higher-dimensional theory is certainly rewritten in terms of SGTs and NSGTs; by counting degrees of freedom, we will show that the five-dimensional and the effective theory are equivalent at the classical level.
The rest of the paper has been organized as follows: In Sec. II, the pure SU (3) Yang-Mills theory is introduced, and canonical analysis of the theory both before and after considering a suitable canonical transformation are independently achieved. It is shown that both frameworks lead to the same theory with the same number of physical degrees of freedom and the same gauge transformations; i.e. , the canonical transformation simply recasts the system. In Sec. III a renormalizable scalar Higgs sector is added to the model presented in Sec. II, and the corresponding suitable canonical transformation is introduced. We show that the presence of spontaneous breakdown SU (3) → SU (2) allows us to use a specific NSGT as the unitary gauge. Section IV is devoted to the study of pure SU (N, M) Yang-Mills theory in an arbitrary number of dimensions; with the more tractable case of one UED, we explicitly present the suitable canonical transformation and compactification scheme that led us to the effective theory invariant under SGTs and NSGTs. We argue that both theories are equivalent as they have the same gauge transformations, simply written in different coordinates, and contain the same number of physical degrees of freedom. In Sec. V, a summary of our results is presented. Finally, in the Appendix, we collect the proof on the canonical nature of the Fourier transform.
II. THE TOY MODEL: PURE SU (3) YANG-MILLS THEORY
The purpose of this section is to illustrate the notion of hidden symmetry within the context described in the Introduction, for which we consider the situation where G = SU (3) and H = SU (2), and construct the desirable canonical transformation. This system neither presents SSB nor is involved in any compactification scheme. This model has attracted important phenomenological interest within the context of the so-called 331 models [16], and it is useful for us because the SU (2) group is completely embedded in the SU (3) one. This feature allows us to clearly illustrate all the peculiarities of the notion of hidden symmetry that we are introducing in this paper.
A. The SU (3) perspective of the model We consider the four-dimensional Yang-Mills theory based on the SU (3) group with the well-known Lagrangian
L SU(3) = − 1 4 F a µν F µν a , (II.1)
where the components of the Yang-Mills curvature are given in terms of the gauge fields A a µ by
F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν . (II.2)
In the special case of SU (3), the completely antisymmetric structure constants f abc have the following nonvanishing values:
f 123 = 1, f 147 = −f 156 = f 246 = f 257 = f 345 = −f 367 = 1 2 and f 458 = f 678 = √ 3 2 . The Lagrangian (II.1) is invariant under gauge transformations δA a µ (x) = D ab µ α b (x) , (II.3)
where α a are the gauge parameters of the group and D ab µ = δ ab ∂ µ − gf abc A c µ is the covariant derivative in the adjoint representation. The above gauge transformations imply that the components of the curvature transform in the adjoint representation of the group,
δF a µν = gf abc F b µν α c . (II.4)
As far as the Hamiltonian structure of the theory is concerned, the canonical momenta are defined by (II.5) where the dot over the fields denotes time derivative. This expression immediately leads to the following primary constraints:
π µ a ≡ ∂L SU(3) ∂Ȧ a µ = F µ0 a ,φ (1) a ≡ π 0 a ≈ 0 , (II.6)
where the symbol ≈ denotes weakly zero [17]. The time evolution along the motion of an arbitrary function on the phase space is dictated by the primary Hamiltonian
H (1) SU(3) = H SU(3) + µ a φ (1) a , (II.7)
where µ a are Lagrange multipliers, and H SU (3) is the canonical Hamiltonian. The latter is
H SU(3) = 1 2 π i a π i a + 1 4 F a ij F ij a − A a 0 φ (2) a . (II.8)
Any physically allowed initial configuration of fields and conjugate momenta must satisfy the primary constraints (II.6); hence the constraints must be constant in time. This consistency condition on the primary constraints leads to the following secondary constraints:
φ (2) a ≡ D ab i π i b ≈ 0 . (II.9)
Applying the consistency condition to secondary constraints yields no new constraints. In this case, all the constraints are of first-class type [17]; the Poisson brackets among the constraints are linear combinations of the constraints themselves. The nonvanishing Poisson brackets between first-class constraints are
{φ (2) a [u], φ (2) b [v]} SU(3) = gf abc φ (2) c [uv] , (II.10) where smeared form φ (2) a [u] := d 3 x u(x)φ (2)
a (x) of the constraints was used. The label SU (3) on the Poisson bracket indicates that it is calculated with respect to the canonical conjugate pairs (A a µ , π µ a ). As is well known [18] the number of true degrees of freedom, in a theory with first-class constraints only, corresponds to the total number of canonical variables minus twice the number of first-class constraints, all divided by two. Therefore, the number of true degrees of freedom is in this case 16 per spatial point x of Minkowski spacetime M 4 .
In this system all first-class constraints generate gauge transformations (II.3) through the gauge generator [19]
G = (D ab 0 α b )φ (1) a − α a φ (2) a (II.11)
via the Poisson bracket as follows
δA a µ = {A a µ , G} SU(3) . (II.12)
We now turn to formulate the same theory but from the perspective of SU (2).
B. The SU (2) perspective of the model
The fundamental representation of SU (3) has dimension 3. A particular choice of this representation is given by the well known Gell-mann matrices λ a , being the corresponding generators λ a /2. Since matrices λ 3 and λ 8 commute with each other, there are three independent SU (2) subgroups, whose generators are (λ 1 , λ 2 , λ), (λ 4 , λ 5 , λ) and (λ 6 , λ 7 , λ). For each case, λ is a different linear combination (with real coefficients) of λ 3 and λ 8 . In this work, we will consider the subgroup determined by the set of generators (λ 1 , λ 2 , λ 3 ) and the corresponding values of the structure constants will be denoted by fābc = ǫābc, whereā = 1, 2, 3. We will also use the notationâ = 4, 5, 6, 7 so that a = 1, . . . , 8 =ā,â, 8. From the SU (2) perspectiveā will label gauge fields, whereasâ and 8 will label tensorial representations of SU (2), see Eqs. (II.18).
In the configuration space, we consider the following point transformation:
Aā µ = Wā µ , (II.13a) A 4 µ = 1 √ 2 Y * µ1 + Y 1 µ , A 5 µ = −i √ 2 Y * µ1 − Y 1 µ , (II.13b) A 6 µ = 1 √ 2 Y * µ2 + Y 2 µ , A 7 µ = −i √ 2 Y * µ2 − Y 2 µ , (II.13c) A 8 µ = Z µ . (II.13d)
This mapping relates the coordinates of the SU (3) formulation to the coordinates we will use in the SU (2) perspective. The inverse is conveniently arranged as follows:
Wā µ = Aā µ , (II.14a) Y µ = Y 1 µ Y 2 µ = 1 √ 2 A 4 µ − iA 5 µ A 6 µ − iA 7 µ , (II.14b) Y † µ = (Y * µ 1 Y * µ 2 ) = 1 √ 2 (A 4 µ + iA 5 µ A 6 µ + iA 7 µ ) , (II.14c) Z µ = A 8 µ . (II.14d)
As will be confirmed below, see Eq.(II.18), fields Y µ and Y † µ transform as contravariant and covariant SU (2) objects, respectively, whereas Z µ becomes invariant under this group.
Using the above point transformation, the Yang-Mills curvature components (II.2) can be rearranged as follows:
Fā µν = Wā µν + ig Y † µ σā 2 Y ν − Y † ν σā 2 Y µ , (II.15a) Y µν = D µ Y ν − D ν Y µ + ig √ 3 2 (Y µ Z ν − Y ν Z µ ) , (II.15b) F 8 µν = Z µν + ig √ 3 2 Y † µ Y ν − Y † ν Y µ . (II.15c)
In these equations, Wā µν = ∂ µ Wā ν − ∂ ν Wā µ + gǫābc Wb µ Wc ν are the components of the su(2)-valued curvature, D µ = ∂ µ − ig σā 2 Wā µ is the covariant derivative in the fundamental representation of SU (2), and Z µν = ∂ µ Z ν − ∂ ν Z µ . The components Fâ µν are encoded into Y µν .
In terms of SU (2) objects, the Lagrangian (II.1) becomes 16) and the gauge transformations (II.3) are mapped onto
L SU(2) = − 1 4 Fā µν F µν a − 1 2 Y † µν Y µν − 1 4 F 8 µν F µν 8 ,(II.δWā µ = Dāb µ αb − ig β † σā 2 Y µ − Y † µ σā 2 β , (II.17a) δY µ = ig σā 2 αā Y µ + D µ − ig √ 3 2 Z µ β + ig √ 3 2 Y µ α Z , (II.17b) δZ µ = ∂ µ α Z − ig √ 3 2 β † Y µ − Y † µ β , (II.17c) where β † = 1 √ 2 α 4 + iα 5 1 √ 2 α 6 + iα 7 .
From the SU (2) perspective, the eight parameters of SU (3) split into three gauge parameters, αā, two doublets, β and β † , and a singlet, α Z , of SU (2). In Eq. (II.17a) the covariant derivative of SU (2), Dāb µ = δāb∂ µ − gǫābcWc µ , in its adjoint representation, emerges.
The standard gauge transformations (SGTs) are defined from the transformation laws Eq. (II.17) by setting the parameters β and α Z equal to zero,
δ s Wā µ ≡ Dāb µ αb , (II.18a) δ s Y µ ≡ ig σā 2 αā Y µ , (II.18b) δ s Z µ ≡ 0 . (II.18c)
From these expressions, it is shown Wā µ transform as gauge fields, Y µ as a doublet of SU (2), and Z µ as a scalar under SU (2). This means that the transformation defined by Eq. (II.13) constitutes an admissible point transformation, as well defined objects under SU (3) are mapped onto well defined objects under SU (2). Moreover, Eqs. (II.18b) and (II.18c) make manifest that Y µ and Z µ are matter fields. In the context of this description, there arise nonstandard gauge transformations (NSGTs), which are defined from Eq. (II.17) by setting αā = 0:
δ ns Wā µ = −ig β † σā 2 Y µ − Y † µ σā 2 β , (II.19a) δ ns Y µ = D µ − ig √ 3 2 Z µ β + ig √ 3 2 Y µ α Z , (II.19b) δ ns Z µ = ∂ µ α Z − ig √ 3 2 β † Y µ − Y † µ β , (II.19c)
These NSGT tell us that there is a gauge symmetry larger than SU (2), in our case SU (3). More precisely, the difference between SGTs and NSGTs in this model is that the former are associated with generators that constitute a group, whereas the latter have to do with generators that do not share this property. We will discuss further on this point at the end of this section. This differentiation is crucial to quantizing the theory, as it requires incorporating the gauge parameters as degrees of freedom from the beginning, so the use of only the SU (2) parameters or the complete set of the SU (3) parameters would lead to very different quantized theories. Of course, the theory can be quantized using the SU (2) basis but taking into account that Y µ and Z µ are also gauge fields, which means that the β and α Z parameters must be recognized as (ghost) degrees of freedom in the context of the BRST symmetry [20,21]. In order to justify that the components of Eq. (II.17) are actual gauge transformations, we wish to show the invariance of the Lagrangian Eq. (II.16) under these variations. Therefore, one may want to start exploring the behaviour of Eq. (II.15) under Eq. (II.17). After some algebra, one finds
δFā µν = gǫābcFb µν αc + ig Y † µν σā 2 β − β † σā 2 Y µν , (II.20a) δY µν = ig σā 2 Y µν αā − igFā µν σā 2 β + ig √ 3 2 Y µν α Z − F 8 µν β , (II.20b) δF 8 µν = ig √ 3 2 Y † µν β − β † Y µν . (II.20c)
It can be shown that the Lagrangian in the SU (2) description Eq. (II.16) is invariant under these transformations. Therefore it is also invariant under the transformations in Eq. (II.17) that decompose into the sum of SGTs and NSGTs.
We now place some technical comments. In the variation of Y µν Eq. (II.20b), the following extra term is explicitly found
B µν = − g 2 Y † µ σā 2 Y ν − Y † ν σā 2 Y µ σā 2 β + 3 4 Y † µ Y ν − Y † ν Y µ β + 3 4 β † Y µ − Y † µ β Y ν − 3 4 β † Y ν − Y † ν β Y µ + β † σā 2 Y µ − Y † µ σā 2 β σā 2 Y ν − β † σā 2 Y ν − Y † ν σā 2 β σā 2 Y µ .
It at first sight seems to be different from zero, but consistency between the SU (2) and the SU (
B r µν = − 1 4 T rs pq − T rs qp Y * µs Y q ν − Y * νs Y q µ β p − T rs pq Y p µ Y q ν − Y p ν Y q µ β s ,
where T rs pq ≡ (σā) r p (σā) s q + 3δ r p δ s q . Using the explicit values of the indices shows that T rs pq is symmetric in p and q, hence B r µν ≡ 0. A similar behavior is present when proving the invariance of the effective four-dimensional Yang-Mills Lagrangian obtained after compactification of the fifth spatial extra dimension described in Sec. IV. Finally, since Eqs. (II.20) are obtained from (II.17), at the curvature level the SGTs and NSGTs are induced; in particular the SGTs of Fā µν , Y µν , and F 8 µν are
δ s Fā µν = gǫābcFb µν αc , (II.21a) δ s Y µν = ig σā 2 Y µν αā , (II.21b) δ s F 8 µν = 0 , (II.21c)
which imply the previously enunciated fact: Fā µν , Y µν , and F 8 µν transform in the adjoint, fundamental, and trivial representation of SU (2), respectively.
It is interesting to note that one can fix the gauge for the Y µ fields in a covariant way under the SU (2) group. This is particularly useful in practical phenomenological applications [22]. To do this, let
fâ = δâb∂ µ − gfâbcAc µ A μ b (II.22)
be the corresponding gauge-fixing functions. In the SU (2) coordinates, these functions can be arranged in a doublet of this group as follows:
f Y = D µ Y µ , (II.23)
where D µ is the covariant derivative in the fundamental representation of SU (2).
We now proceed to study the Hamiltonian structure of the theory from the SU (2) point of view. So, to describe the system in phase space terms, the following conjugate momenta are defined:
π W μ a = ∂L SU(2) ∂Ẇā µ = F µ0 a , (II.24a) π Y µ r = ∂L SU(2) ∂Ẏ r µ = Y * µ0 r , (II.24b) π * Y µ r = ∂L SU(2) ∂Ẏ * µ r = Y µ0 r , (II.24c) π Z µ = ∂L SU(2) ∂Ż µ = F µ0 8 . (II.24d)
It is important to notice that π W μ a are not the canonical momenta associated with the pure SU (2) theory, whose Lagrangian is 25) and which leads to the conjugate momenta
L = − 1 4 Wā µν W µν a ,(II.p μ a = W µ0 a . (II.26)
These momenta differ from those derived from the Lagrangian in Eq. (II.16), which are explicitly given by
π W μ a = p μ a + ig Y †µ σā 2 Y 0 − Y †0 σā 2 Y µ . (II.27)
The relations between the canonical momenta in the SU (3) and the SU (2) descriptions are
π μ a = π W μ a , (II.28a) π µ 4 = 1 √ 2 π Y µ 1 + π * µ1 Y , π µ 5 = −i √ 2 π Y µ 1 − π * µ1 Y , (II.28b) π µ 6 = 1 √ 2 π Y µ 2 + π * µ2 Y , π µ 7 = −i √ 2 π Y µ 2 − π * µ2 Y , (II.28c) π µ 8 = π Z , (II.28d) whose inverses are π W μ a = π μ a , (II.29a) π Y µ = (π Y µ 1 π Y µ 2 ) = 1 √ 2 (π µ 4 + iπ µ 5 π µ 6 + iπ µ 7 ) , (II.29b) π † µ Y = π * µ1 Y π * µ2 Y = 1 √ 2 π µ 4 − iπ µ 5 π µ 6 − iπ µ 7 , (II.29c) π Z µ = π µ 8 . (II.29d)
From the conjugate momentum expressions Eq. (II.24) and using the notation Eq. (II.29), one can readily recognize the primary constraints as
φ (1) a = π W 0 a ≈ 0 , (II.30a) φ (1) Y = π Y 0 ≈ 0 , (II.30b) φ (1) † Y = π † 0 Y ≈ 0 , (II.30c) φ(1)Z = π Z 0 ≈ 0 . (II.30d) Notice that φ (1) Y and φ (1) † Y
are covariant and contravariant SU (2) doublets, respectively. Since π W μ a do not coincide with the conjugate momenta associated to the pure SU (2) theory Eq. (II.25), the primary constraints φ (1) a differ from the primary constraints p 0 a which emerge in the canonical analysis of Eq. (II.25). The same observation will apply for the secondary constraints.
The primary Hamiltonian, which governs the evolution of the system, takes the form
H (1) SU(2) = H SU(2) + µāφ (1) a + φ (1) Y µ Y + µ † Y φ (1) † Y + µ Z φ (1) Z . (II.31)
It corresponds to the sum of the canonical Hamiltonian
H SU(2) = 1 2 π W ī a π W ī a + π Y i π † i Y + 1 2 π Z i π Z i + 1 4 Fā ij F ij a + 2Y † ij Y ij + F 8 ij F ij 8 − Wā 0 φ (2) a − φ (2) † Y Y 0 − Y † 0 φ (2) Y − Z 0 φ (2) Z , (II.32)
and a linear combination of the primary constraints Eq. (II.30) where the Lagrange multipliers µ Y , µ † Y and µ Z are
µ Y = 1 √ 2 µ 4 − iµ 5 µ 6 − iµ 7 , (II.33a) µ † Y = 1 √ 2 (µ 4 + iµ 5 µ 6 + iµ 7 ) , (II.33b) µ Z = µ 8 . (II.33c)
By using the primary Hamiltonian Eq. (II.31), the consistency condition over the primary constraints Eq. (II.30) yields the following secondary constraints:
φ (2) a = Dāb i π W ī b − ig π Y i σā 2 Y i − Y † i σā 2 π † i Y ≈ 0 , (II.34a) φ (2) † Y = π Y i ↼ D i + ig √ 3 2 Z i − igY † i σā 2 π W ī a + √ 3 2 π Z i ≈ 0 , (II.34b) φ (2) Y = D i − ig √ 3 2 Z i π † i Y + ig σā 2 π W ī a + √ 3 2 π Z i Y i ≈ 0 , (II.34c) φ (2) Z = ∂ i π Z i − ig √ 3 2 π Y i Y i − Y † i π † i Y ≈ 0 , (II.34d)
where the action of
↼ D µ on a contravariant SU (2) doublet, say π Y µ , is another contravariant SU (2) doublet defined by π Y µ ↼ D µ ≡ ∂ µ π Y µ + igπ Y µ σā 2 Wā µ .
The consistency condition applied to each secondary constraint yields no new constraints. It turns out, that all primary and secondary constraints do form a set of first-class constraints; in fact, the relevant Poisson brackets between these first-class constraints are
{φ (2) a [u], φ (2) b [v]} SU(2) = gǫābc φ (2) c [uv] , (II.35a) {φ (2) a [u], φ (2)r Y [v]} SU(2) = ig (σā) r s 2 φ (2)s Y [uv] , (II.35b) {φ (2)r Y [u], φ (2)s Y [v]} SU(2) = g 2 T rs pq d 3 x (uv)(x) π * iq Y Y p i − π * ip Y Y q i (x) , (II.35c) {φ (2)r Y [u], φ (2) * Y s [v]} SU(2) = ig (σā) r s 2 φ (2) a [uv] + √ 3 2 δ r s φ (2) Z [uv] + g 2 (T rs pq − T rs qp ) d 3 x (uv)(x) π Y i s Y p i − Y * is π * ip Y (x) , (II.35d) {φ (2)r Y [u], φ(2)Z [v]} SU(2) = − ig √ 3 2 φ (2)r Y [uv] , (II.35e)
where {·, ·} SU(2) denotes the Poisson bracket that involves the SU (2) phase space coordinates. Due to the symmetries present in the lower indices of T rs pq , one finds that the terms proportional to g 2 on the right hand side of Eq. (II.35c) and Eq. (II.35d) do not contribute to the occurrence of tertiary constraints; instead these terms identically vanish, and the Poisson brackets among all the constraints give a linear combination of constraints themselves. A more elegant argument to show that such terms must identically vanish on the whole phase space is the following: Notice that Eq. (II.13) and Eq. (II.28) define a canonical transformation in the ordinary sense [23], and hence {·, ·} SU(3) = {·, ·} SU (2) . Moreover, it is easy to see that this canonical transformation maps the primary constraints Eq. (II.6) onto Eq. (II.30); hence the primary Hamiltonian in the SU (3) phase space coordinates Eq. (II.7) becomes the corresponding Hamiltonian in the SU (2) coordinates Eq. (II.31). As a consequence, the set of secondary constraints in both formalisms must match under the canonical transformation. Indeed, this can be proved by direct calculation. Since exclusively the primary Hamiltonian is employed to evolve the constraints in time through the Poisson bracket, one concludes that the Dirac algorithm in the SU (2) formulation must lack of tertiary constraints just as it does in the SU (3) formulation; this fact rules out the presence of the extra-terms proportional to g 2 in the gauge algebra Eq. (II.35). In conclusion the canonical transformation defined by Eq. (II.13) and Eq. (II.28) maps each stage of the Dirac algorithm in the SU (3) formulation into the corresponding stage in the SU (2) one. Notice that the number of physical degrees of freedom of the SU (2) effective theory matches with the corresponding number of the pure SU (3) Yang-Mills theory.
We end the Hamiltonian analysis from the SU (2) perspective by calculating the gauge generator G [19]. This generator is linear in all first-class constraints Eq. (II.30) and Eq. (II.34) with coefficients of the primary ones related to that of the secondary ones; the relation among the coefficients is obtained by imposing the condition that the total time derivative of G, (2) , must be a linear combination of the primary constraints only [24]. As a consequence one gets
∂G ∂t + {G, H SU(2) } SUG = Dāb 0 αb − ig β † σā 2 Y 0 − Y † 0 σā 2 β φ (1) a + φ (1) Y D 0 − ig √ 3 2 Z 0 β + ig σā 2 αā − √ 3 2 α Z Y 0 + β † ↼ D 0 + ig √ 3 2 Z 0 − igY † 0 σā 2 αā − √ 3 2 α Z φ (1) † Y + ∂ 0 α Z + ig βY † 0 − β † Y 0 φ (1) Z − αāφ (2) a − β † φ (2) Y − φ (2) † Y β − α Z φ (2) Z . (II.36)
This gauge generator is the sum of G s ≡ G| β=0,αZ =0 and G ns ≡ G| αā=0 which independently generate the SGTs and NSGTs, Eqs. (II.18) and (II. 19), respectively, via the Poisson brackets
δ s Wā µ = {Wā µ , G s } SU(2) , δ s Y µ = {Y µ , G s } SU(2) , δ s Z µ = {Z µ , G s } SU(2) , (II.37a) δ ns Wā µ = {Wā µ , G ns } SU(2) , δ ns Y µ = {Y µ , G ns } SU(2) , δ ns Z µ = {Z µ , G ns } SU(2) . (II.37b)
From these transformation laws and the constraint algebra Eq. (II.35), it is straightforward to see that on the constraint surface the Lie algebra among SGTs and NSGTs can be summarized as follows:
[ where [·, ·] denotes a Lie product. The first of these relations closes with structure constants and specifically follows from the Lie subalgebra Eq. (II.35a), therefore exponentiation of SGTs provides a Lie group which in fact corresponds to SU (2). Since the Lie product of NSGTs does not close, they do not exponentiate into a group. To conclude this subsection, we would like to emphasize the following: A hidden symmetry arises when an admissible canonical transformation is introduced. The canonical transformation is admissible in the sense that it maps well defined objects under some group G to well defined objects of a subgroup H of G. The gauge symmetry, which is manifest in G, is hidden in H. The gauge symmetries with respect to the group G that appear hidden from the H perspective are those associated with the generators of G that do not generate H. This is true independently of whether or not the G group is spontaneously broken down into H. In our toy model G = SU (3) and H = SU (2); after the canonical transformation, only the fields Wā µ = Aā µ explicitly continue being gauge fields under H. The rest of the fields, Y µ , Y † µ and Z µ , fulfill very different transformation laws under H; nevertheless, the latter fields can be mapped back with the canonical transformation to gauge fields with respect to G. This result is crucial for our study of passing from the SU (N, M 5 ) gauge group description to the SU (N, M 4 ) one via compactification, as in this case the phenomenon of spontaneous symmetry breaking is not present. Note that in this subtler case SU (N, M 4 ) is a subgroup of SU (N, M 5 ) not due to a difference in the number of generators, which is the same indeed, but because the gauge parameters of the group SU (N, M 5 ) are restricted to take values on the submanifold M 4 of M 5 . We will show that there exists an admissible canonical transformation in this case.
III. THE SU (3) YANG-MILLS THEORY WITH SPONTANEOUS SYMMETRY BREAKING
We now proceed to extend the study of the previous section to the case when the SU (3) group is spontaneously broken into the SU (2) in the usual sense. One of the two main purposes is to contrast the notion of hidden symmetry induced by a suitable canonical transformation with that coming from SSB. The other is to show how a specific NSGT can be used to define the unitary gauge. In this scenario, we will be able to make a precise analogy of this procedure with a similar one used in the context of extra dimensions.
A. The SU (3) perspective of the model To carry out the mentioned SSB, we add to the pure SU (3) theory given by the Lagrangian in Eq. (II.1) a renormalizable scalar sector L Φ , so that
L SU(3),Φ = L SU(3) + L Φ , (III.1) where L Φ = (D µ Φ) † (D µ Φ) − V (Φ † , Φ) . (III.2) In this expression D µ = ∂ µ − ig λ a 2 A a µ is the covariant derivative in the fundamental representation of SU (3), 1 and Φ is a complex contravariant Poincaré scalar triplet of SU (3). In addition, V (Φ † , Φ) is the renormalizable scalar potential given by V (Φ † , Φ) = µ 2 Φ † Φ + λ Φ † Φ 2 . (III.3)
It is straightforward to show that the Lagrangian in Eq. (III.1) is simultaneously invariant under Eq. (II.3) and the infinitesimal rotation of the triplet Φ in the isospin space,
δΦ = −iα a λ a 2 Φ . (III.4)
The gauge symmetries of the Lagrangian in Eq. (III.1) will be reflected in the occurrence of first-class constraints in the Hamiltonian setting. In order to formulate the theory in phase space terms, in addition to the canonical pairs (A a µ , π µ a ), cf. Eqs. (II.5), the conjugate pairs (Φ, π) and (Φ † , π † ) must be introduced, where
π = ∂L Φ ∂Φ = (D 0 Φ) † , (III.5a) π † = ∂L Φ ∂Φ † = D 0 Φ . (III.5b)
Note that π and π † correspond to covariant and contravariant SU (3) triplets, respectively. From the Eqs. (III.5) the velocitiesΦ † andΦ are expressible in terms of phase space variables; therefore they do not give rise to more primary constraints in addition to those defined in (II.6). To bring uniformity into the present section, primary constraints will be denoted by ϕ
(1) a ≡ φ(1)
a . The incorporation of the scalar sector into the pure SU (3) Yang-Mills Lagrangian does not have influence upon the primary constraints of the pure theory alone.
The canonical Hamiltonian associated with Eq. (III.1) will be the sum of Eq. (II.8) and the contribution from the Higgs sector L Φ , namely
H SU(3),Φ = H SU(3) + H Φ , (III.6) where H Φ = ππ † + igA a 0 π λ a 2 Φ − Φ † λ a 2 π † − D i Φ † (D i Φ) + V (Φ, Φ † ) . (III.7)
Notice that the term linear in A a 0 will modify the secondary constraints that are produced in the absence of the Higgs sector. Indeed, the primary Hamiltonian
H (1) SU(3),Φ = H SU(3),Φ + µ a ϕ (1) a (III.8)
allows us to obtain the consistency condition on the primary constraints Eq. (II.6) providing the following secondary constraints:
ϕ (2) a ≡ φ (2) a − ig π λ a 2 Φ − Φ † λ a 2 π † ≈ 0 , (III.9) where φ(2)
a corresponds to the secondary constraints Eq. (II.9) conveyed by the pure SU (3) Yang-Mills theory. The consistency requirement on ϕ (2) a does not bring more constraints, ending with the Dirac algorithm. The primary and secondary constraints of the theory, Eqs. (II.6) and (III.9), form a set of first-class constraints; the nonvanishing Poisson brackets between the constraints reveal the SU (3) symmetry of the theory
{ϕ (2) a [u], ϕ (2) b [v]} SU(3) = gf abc ϕ (2) c [uv] ,(III.10)
where {·, ·} SU (3) is the Poisson bracket in the SU (3) formulation which takes into account the conjugate pairs (A a µ , π µ a ), (Φ, π) and (Φ † , π † ). Since only secondary constraints are modified by the Higgs sector, one expects that once the SSB of SU (3) into SU (2) operates, the affected constraints will only be the secondary ones.
Before going into the SU (2) formulation of the theory, the gauge generator is presented. Linear in all first-class constraints, this corresponds to
G = (D ab 0 α b )ϕ (1) a − α a ϕ (2) a . (III.11)
Notice that the scalar contribution in the secondary constraints Eq. (III.9) is responsible for the appropriate transformation law that the scalar fields must follow, cf. Eq. (III.4); in fact, In this subsection we revisit the SSB [4] from what we have referred to as the SU (3) perspective. We consider the case µ 2 < 0, in which the vacuum is infinitely degenerate, so the theory presents SSB.
δA a µ = {A a µ , G} SU(3) (III.12a) δΦ = {Φ, G} SU(3) (III.
The extremum at Φ = 0 is not considered. We may presume that the expectation value of Φ in the vacuum does not vanish. The energy of the system is minimal on all the points of the spherical surface given by
Φ † min Φ min = − µ 2 2λ ≡ v 2 . (III.13)
All points on these surface are physically equivalent because they are connected through SU (3) transformations. To break down SU (3) into SU (2), one chooses a particular direction Φ min such that
λā 2 Φ min = 0 , (III.14a) λâ 2 Φ min = 0 , (III.14b) λ 8 2 Φ min = 0 . (III.14c)
The isotropy group, the one corresponding to unbroken symmetries, at Φ min is SU (2). It is convenient to choose a representative of the solutions to the Eq. (III.13) as Φ † min = (0 0 v). This choice means that five generators of SU (3), namely, λâ 2 and λ 8 2 , are broken. Within this formulation, two cases clearly arise depending on the nature of the gauge parameters α a (cf. Eq. (II.3)). These are (i) The Goldstone Theorem [4]. Assuming the parameters α a to be constant functions on Minkowski space, the invariant Lagrangian corresponds to
L SU(3),H = (∂ µ Φ) † (∂ µ Φ) − V (Φ † , Φ) .
When the theory is subjected to the translation Φ → ϕ ≡ Φ − Φ min , there arise five real massless scalars. These correspond to ϕ 1 , ϕ 2 and the imaginary part of ϕ 3 denoted as φ Z . In addition, a massive scalar H emerges, identified as the real part of ϕ 3 , that quantifies the normal excitations to the surface of the minimal energy. Hence, associated with each broken generator of SU (3) there is a massless scalar or Goldstone boson.
(ii) The Higgs Mechanism [3]. Assuming the parameters α a to be nonconstant functions on Minkowski space, the invariant Lagrangian corresponds to Eq. (II.1). In this case, besides the presence of five pseudo-Goldstone bosons, five massive gauge bosons (Aâ µ and A 8 µ ) arise. This is the celebrated Higgs mechanism. In this scenario, the pseudo-Goldstone bosons represent spurious degrees of freedom, as they can be removed from the theory in a special gauge, known as unitary gauge. In the following section, we will show that this mechanism has a natural description in the SU (2) coordinates, and that the unitary gauge can be understood as the action of fixing the parameters within what will be defined as NSGT on the scalar fields, Eq. (III.19b). (2) Eq. (II. 16), and the scalar sector L Φ is mapped onto L φ by decomposing the SU (3) triplet Φ into an SU (2) doublet and a scalar,
Φ 1 Φ 2 = φ 1 φ 2 , (III.15a) Φ 3 = φ 0 . (III.15b)
Therefore, the Lagrangian in Eq. (III.1) is recast in terms of well defined objects under the action of SU (2), 16) where the Higgs sector becomes
L SU(2),φ = L SU(2) + L φ ,(III.L φ = (D µ Φ) † (D µ Φ) Φ → φ Aµ→ Wµ,Yµ,Y † µ ,Zµ + V (Φ, Φ † ) Φ → φ . (III.17)
Gauge invariances of the theory in this formulation correspond to Eq. (III.1) together with
δφ = −i σā 2 αā + 1 2 √ 3 α Z φ − i √ 2 φ 0 β , (III.18a) δφ 0 = − i √ 2 β † φ + i √ 3 α Z φ 0 . (III.18b)
Notice that in the scalar sector of the theory, the SGTs and NSGTs also naturally arise. Indeed
δ s φ = −i σā 2 αāφ , δ s φ 0 = 0 ; (III.19a) δ ns φ = − i 2 1 √ 3 α Z φ + √ 2φ 0 β , δ ns φ 0 = − i √ 2 β † φ + i √ 3 α Z φ 0 . (III.19b)
We now proceed to the Hamiltonian formulation associated to the singular Lagrangian Eq. (III.16). Since the scalar sector does not contain spacetime derivatives of either gauge fields Wā µ , or SU (2) doublets Y µ , or the scalar Z µ , the canonical conjugate momentum associated with each of these fields coincides with those defined in Sec. II B. Hence, the conjugate momenta in the SU (2) formulation are given by Eqs. (II.24) and
π φ = ∂L φ ∂φ = φ † ↼ D 0 + ig 2 √ 3 Z 0 + ig √ 2 φ 0 * Y † 0 , (III.20a) π 0 = ∂L φ ∂φ 0 = ∂ 0 − ig √ 3 Z 0 φ 0 * + ig √ 2 φ † Y 0 , (III.20b) π † φ = ∂L φ ∂φ † = D 0 − ig 2 √ 3 Z 0 φ − ig √ 2 φ 0 Y 0 , (III.20c) π * 0 = ∂L φ ∂φ 0 * = ∂ 0 + ig √ 3 Z 0 φ 0 − ig √ 2 Y † 0 φ . (III.20d)
It is worth noticing that π φ and π † φ are covariant and contravariant SU (2) doublets, respectively, whereas, π 0 and its complex conjugate are SU (2) scalars. The relations among conjugate momenta (III.20) and the corresponding objects Eq. (III.5) are
π φ = (π 1 φ π 2 φ ) = (π 1 π 2 ) , (III.21a) π 0 = π 3 . (III.21b)
As expected, the scalar sector of the theory does not bring additional constraints into the SU (2) formalism either. Instead of going through the Dirac formalism using the Poisson bracket {·, ·} SU (2) , that in this case would include also the canonical pairs (φ, π φ ), (φ 0 , π 0 ), (φ † , π † φ ), and (φ 0 * , π * 0 ), we will make use of the arguments given after Eqs. (II.37) in the following way: First, notice that Eqs. (II.13), (II.28), (III.15) and (III.21) define a canonical transformation from SU (3)
(1) a ≡ φ (1) a , ϕ (1) Y ≡ φ (1) Y , ϕ (1) Z ≡ φ(1)
Z }; the transformation hence recasts the primary Hamiltonian Eq. (III.8) in terms of SU (2) variables as follows:
H (1) SU(2),φ = H SU(2) + H φ + µāϕ (1) a + ϕ (1) Y µ Y + µ † Y ϕ (1) † Y + µ Z ϕ (1) Z (III.22)
where H SU (2) is given by Eq. (II.16) and H φ is the Legendre transformation of L φ . As a consequence of these two observations, the set of secondary constraints that emerges in the SU (3) viewpoint must be faithfully mapped onto the set of secondary constraints given in terms of the SU (2) coordinates. These are
ϕ (2) a = φ (2) a − ig π φ σā 2 φ − φ † σā 2 π † φ ≈ 0 , (III.23a) ϕ (2) † Y = φ (2) † Y + ig √ 2 π * 0 φ † − φ 0 π φ ≈ 0 , (III.23b) ϕ (2) Y = φ (2) Y − ig √ 2 π 0 φ − φ 0 * π † φ ≈ 0 , (III.23c) ϕ (2) Z = φ (2) Z − ig √ 3 φ 0 * π * 0 − π 0 φ 0 + 1 2 (π φ φ − φ † π † φ ) ≈ 0 , (III.23d) where φ (2) a , φ(2)† Y , φ(2)
Y and φ
Z are given by Eqs. (II.34). Indeed, this can be proved by direct calculation. Finally, the set of equations that define the gauge algebra Eq. (III.10) can be expressed in terms of SU (2) variables using only the canonical transformation. The nonvanishing Poisson brackets are
{ϕ (2) a [u], ϕ (2) b [v]} SU(2) = gǫābc ϕ (2) c [uv] , (III.24a) {ϕ (2) a [u], ϕ (2)r Y [v]} SU(2) = ig (σā) r s 2 ϕ (2)s Y [uv] , (III.24b) {ϕ (2)r Y [u], ϕ(2)* Y s [v]} SU(2) = ig (σā) r s 2 ϕ (2) a [uv] + √ 3 2 δ r s ϕ(2)Z [uv] , (III.24c) {ϕ (2)r Y [u], ϕ(2)Z [v]} SU(2) = − ig √ 3 2 ϕ (2)r Y [uv] . (III.24d)
Since the canonical transformation connects the Dirac algorithm unfolded in the two different sets of coordinates at each step, we have that the gauge generator Eq. (III.11) must be translated into the corresponding one in the SU (2) variables, namely
G = Dāb 0 αb − ig β † σā 2 Y 0 − Y † 0 σā 2 β ϕ (1) a + ϕ (1) Y D 0 − ig √ 3 2 Z 0 β + ig σā 2 αā − √ 3 2 α Z Y 0 + β † ↼ D 0 + ig √ 3 2 Z 0 − igY † 0 σā 2 αā − √ 3 2 α Z ϕ (1) † Y + ∂ 0 α Z + ig βY † 0 − β † Y 0 ϕ (1) Z − αāϕ (2) a − β † ϕ (2) Y − ϕ (2) † Y β − α Z ϕ(2)
Z , (III. 25) from which the sectors that independently generate SGTs, G s ≡ G| β=0,αZ =0 , and NSGT, G ns ≡ G| αā=0 , are easily identified. Notice that it is due to the terms depending on the Higgs sector in each secondary constraint that Eqs. (III.19) are suitably recovered from the following brackets:
δ s φ = {φ, G s } SU(2) , δ s φ 0 = {φ 0 , G s } SU(2) , (III.26a) δ ns φ = {φ, G ns } SU(2) , δ ns φ 0 = {φ 0 , G ns } SU(2) . (III.26b)
The corresponding variations for Wā µ , Y µ and Z µ are given in Eqs. (II.37). Since the gauge algebra Eq. (III.24) is isomorphic to Eq. (II.35), it follows that on the constraint surface the algebra of SGTs and NSGTs also becomes Eq. (II.38). The finite version of SGTs corresponds to the action of SU (2), whereas the NSGTs are associated with broken generators.
In this subsection we have recast an SU (3) manifestly invariant theory as an SU (2) (2) symmetry is exact, whereas the SU (3) is hidden. We now turn to discuss the SSB of the SU (3) group into the SU (2) one, from the viewpoint of the latter.
D. SSB from the SU (2) perspective
We reconsider the case of infinite degeneracy of vacuum, µ 2 < 0. Configurations with minimal energy Eq. (III.13) lie on φ † min φ min + φ 0 * min φ 0 min = v 2 . As we have remarked, there is a natural separation of SU (3) parameters into those parameters of the isotropy group, αā, and those associated to the broken part of the group, αâ and α 8 . In fact, this split is what determines the SGTs and NSGTs previously defined. The functional form of the Lagrangian Eq. (III.16), where the SU (2) sector of SU (3) is manifest, suggests the study of the following cases:
(i) The Goldstone theorem. We assume the broken part of SU (3), generated by λâ 2 and λ 8 2 , to be global -that is, we allow αâ and α 8 to be spacetime independent. In other words, assume that the NSGTs are global, but not necessarily SGTs. In such a situation, the following Lagrangian is invariant under this class of transformations:
L g = − 1 4 Wā µν W µν a + (D µ φ) † (D µ φ) + (∂ µ φ 0 * )(∂ µ φ 0 ) + V | Φ→φ ,
where Wā µν are the components of the su(2)-valued curvature and D µ is the covariant derivative of SU (2) in the fundamental representation. There arise five massless scalars when the theory is developed around the particular minimum Φ min , which is decomposed into the doublet φ min = 0 and the scalar φ 0 min = v, by carrying out the shift φ 0 → H + iφ Z ≡ φ 0 − v. These scalars do correspond to φ, φ † and the singlet φ Z , which are identified with the so-called Goldstone bosons. The massive field H survives. Hence, there is a massless scalar associated with each independent NSGT.
(ii) The Higgs mechanism. Now assume the larger symmetry SU (3) -that is, that both the SGTs and NSGTs are local. In this scenario, the theory developed around the particular minimum is characterized by the Lagrangian given in Eq. (III.16), with φ 0 replaced by (v + H + iφ Z ). Five gauge fields, Y µ , Y † µ , and Z µ , acquire mass and simultaneously five pseudo-Goldstone bosons appear, namely φ, φ † and φ Z . Notice that all the mass terms are invariant under the SU (2) subgroup.
All pseudo-Goldstone bosons can be removed from the theory through the so-called unitary gauge; the degrees of freedom that they represent appear as the longitudinal polarization states of the gauge bosons associated with the broken generators. The implementation of the unitary gauge can be understood in terms of the NSGTs. Indeed, consider the NSGT (III.19b) with particular gauge parameters
β = − i √ 2 v φ , (III.27a) α Z = − √ 3 v φ Z , (III.27b)
which yields φ ′ = 0 and φ ′ Z = 0. Therefore, the unitary gauge corresponds to a particular NSGT which maps the pseudo Goldstone bosons onto zero. In addition, from the NSGT given by Eqs. (II.19), one finds
W ′ā µ = Wā µ , (III.28a) Y ′ µ = Y µ − i √ 2 v ∂ µ φ , (III.28b) Z ′ µ = Z µ − √ 3 v ∂ µ φ Z . (III.28c)
The incorporation of the pseudo Goldstone bosons as the longitudinal component of the massive gauge bosons Y ′ µ and Z ′ µ is evident from these expressions. We will come back to this latter on, when discussing this mechanism in the context of theories with compactified extra dimensions.
The unitary gauge can also be implemented via a finite NSGT. Consider the non-linear parametrization of the triplet Φ,
Φ(x) = U(x) 0 0 v + H , (III.29) with U(x) = exp i λâ 2 αâ + i λ 8 2 α 8 = exp − i 2v iλ 4 φ 1 − φ 1 * − λ 5 φ 1 + φ 1 * +iλ 6 φ 2 − φ 2 * − λ 7 φ 2 + φ 2 * + 3 2 λ 8 φ Z , (III.30)
where the parameter values given in (III.27) were used. The finite version of the NSGT (III.19b) are obtained by acting with U −1 (x) as follows:
Φ ′ (x) = U −1 (x)Φ = 0 0 v + H .
(III.31)
The components of Eq. (III.28) are recovered by entering the particular element U −1 (x) ∈ SU (3) into the finite gauge transformation of the connection,
A ′ µ = U (x)A µ U † (x) − i(∂ µ U )U † ,
and keeping the analysis at first order.
IV. YANG-MILLS THEORIES WITH COMPACTIFIED EXTRA DIMENSIONS
In this section, we introduce a pure higher-dimensional Yang-Mills theory with an underlying gauge group SU (N, M m ), whose parameters are allowed to propagate in the spacetime manifold M m = M 4 × N n . Gauge fields A a M , defined on M m , act as fundamental fields in the m-dimensional theory, where a and M are gauge and spacetime indices, respectively. We begin our discussion by noticing that the transition from the SU (N, M m ) gauge group description to SU (N, M 4 ) will simultaneously convey a certain transformation that maps well defined objects under the Poincaré group ISO(1, m − 1) onto well defined objects under the standard ISO (1,3). We now proceed to present a brief discussion on this issue.
A. The Poincaré group perspective
Let us consider the flat spacetime manifold M m = M 4 × N n , with mostly minus metric g MN and n spatial extra dimensions, with coordinates (X M ) = (x µ , xμ), where µ = 0, 1, 2, 3 andμ = 5, . . . , m. We introduce gauge fields A M (X) = A a M (X)T a , where T a are generators of the gauge group SU (N, M m ). In this m-dimensional spacetime, the Poincaré group ISO(1, m − 1) is defined through its 1 2 m(m + 1) generators. A number m of these generators (P M ) belong to the group of translations, and the 1 2 m(m − 1) remainder (J MN ) are associated with the Lorentz group SO(1, m − 1). These generators satisfy the following Poincaré algebra:
[P M , P N ] = 0 , (IV.1) [J MN , P R ] = i (g MR P N − g N R P M ) , (IV.2) [J MN , J RS ] = i (g MR J N S − g MS J N R − g N R J MS + g N S J MR ) . (IV.3)
It is not difficult to see that in this algebra there are two subalgebras merged. One of these algebras generates the Poincaré group ISO(1, 3):
[P µ , P ν ] = 0 , (IV.4) [J µν , P ρ ] = i (g µρ P ν − g νρ P µ ) , (IV.5) [J µν , J ρσ ] = i (g µρ J νσ − g µσ J νρ − g νρ J µσ + g νσ J µρ ) , (IV.6)
whereas the other one generates the inhomogeneous orthogonal group in n dimensions ISO(n):
[Pμ , Pν] = 0 , (IV.7) [Jμν , Pρ] = i (δνρPμ − δμρPν) , (IV.8) [Jμν , Jρσ] = i (δμσJνρ − δμρJνσ − δνσJμρ + δνρJμσ) . (IV.9)
An infinitesimal Poincaré transformation in M m is given by
δX M = ω MN X N + ǫ M , (IV.10)
where ω MN = −ω N M and ǫ M are the infinitesimal parameters of the group. This transformation induces the following variation:
δA M (X) = ω MN + g MN ω RS X S + ǫ R ∂ R A N (X) . (IV.11)
This relation can be naturally split into variations for A µ (X) and Aμ(X) components as follows:
δA µ (X) = [ω µν + g µν (ω ρσ x σ + ǫ ρ ) ∂ ρ ] A ν (X) + ωρσxσ + ǫρ ∂ρ + ω ρσ xσ∂ ρ − x ρ ∂σ A µ (X) + ω µν Aν (X) , (IV.12a)
δAμ(X) = ωμν + gμν ωρσxσ + ǫρ ∂ρ Aν (X)
+ (ω ρσ x σ + ǫ ρ ) ∂ ρ + ω ρσ xσ∂ ρ − x ρ ∂σ Aμ(X) + ωμ ν A ν (X) . (IV.12b)
It can be seen from these expressions that A µ and Aμ transform under the Lorentz group SO(1, 3) as a vector and as a scalar, respectively, whereas they transform as a scalar and as a vector under the orthogonal group SO(n). This means that before compactification, the m-dimensional Yang-Mills action S[A M ] (manifestly invariant under ISO(1, m − 1)) can be written in terms of well defined objects under ISO (1,3) and ISO(n). Thus we can recast this theory in terms of the action S[A µ , Aμ]. In the latter formulation the ISO(1, 3) and ISO(n) symmetries are manifest, but the ISO(1, m − 1) is hidden. In complete analogy with the ideas introduced in previous sections for unitary gauge groups, we can define two types of standard transformations, which correspond to the inhomogeneous subgroups ISO (1,3) and ISO(n). The former, which we will call standard Poincaré transformations (SPTs), are defined by setting ωμν = ω µν = ǫμ = 0 in Eqs. (IV.12):
δA µ (X) = [ω µν + g µν (ω ρσ x σ + ǫ ρ ) ∂ ρ ] A ν (X) , (IV.13a) δAμ(X) = (ω ρσ x σ + ǫ ρ ) ∂ ρ Aμ(X) .
(IV.13b)
The latter ones, which we will call standard orthogonal transformations (SOTs), arise when ω µν = ω µν = ǫ µ = 0 in Eqs. (IV.12):
δA µ (X) = ωρσxσ + ǫρ ∂ρA µ (X) , (IV.14a) δAμ(X) = ωμν + gμν ωρσxσ + ǫρ ∂ρ Aν (X) .
(IV.14b)
The action S[A µ , Aμ] is manifestly invariant under these standard spacetime transformations. However, this action is not manifestly invariant under transformations induced by the J µν generators. These are nonstandard Poincaré transformations (NSPTs), which are defined from (IV.12) by setting the parameters ω µν = 0 and the remaining ones equal to zero:
δA µ (X) =ω ρσ xσ∂ ρ − x ρ ∂σ A µ (X) + ω µν Aν (X) , (IV.15a) δAμ(X) =ω ρσ xσ∂ ρ − x ρ ∂σ Aμ(X) + ωμ ν A ν (X) . (IV.15b)
In the five-dimensional pure Yang-Mills theory with one compact spatial extra dimension, there arise massless bosons that are interpreted as pseudo-Goldstone bosons. These fields can be removed via a particular NSGT which is understood as a unitary gauge [6]. Although these pseudo-Goldstone bosons are present, in the switch from the gauge group SU (N, M 5 ) to SU (N, M 4 ) there is no SSB involved, because the number of generators in both groups is the same. So, in this class of theories the pseudo-Goldstone bosons needed to implement the Higgs mechanism have nothing to do with the unitary gauge group SU (N, M 5 ), but with the Poincaré group. The boson fields arise by compactification of the spatial extra coordinates which leads to an explicit breaking of the ISO(1, 4) group into ISO (1,3). This observation implies that the corresponding effective theory, which depends on the KK fields, is subject to satisfying only the SPTs. We expect a similar behavior when considering compactification of higher-dimensional pure SU (N, M m ) Yang-Mills theories into SU (N, M 4 ) effective theory. where D ab M = δ ab ∂ M − g m f abc A c M and the gauge parameters are allowed to propagate in the bulk. From Eq. (IV.17), the components of the curvature are transformed in the adjoint representation δF a MN = g m f abc F b MN α c (x, y) . The Hamiltonian description of the theory goes along the same line as Sect. II A. The conjugate momentum to A a M is denoted by π M a . The canonical analysis yields the following first-class constraints:
φ (1) a = π 0 a (x, y) ≈ 0 (IV.18a) φ (2) a = D ab I π I b (x, y) ≈ 0 (IV.18b)
where I labels all spatial components of M m . Therefore, the number of physical degrees of freedom is (
N 2 − 1)m − 2(N 2 − 1) = (N 2 − 1)(m − 2) per spatial point of M m .
The corresponding gauge algebra has the structure of Eq. (II.10) with the corresponding coupling constant g m : For the sake of simplicity, from now on we focus on the case n = 1; that is, the five-dimensional SU (N, M 5 ) Yang-Mills theory. The notion of hidden symmetry induced by a canonical transformation will be given in terms of Fourier transformations and the identification of G as SU (N, M 5 ) and H as SU (N, M 4 ). In five dimensions, the theory consists of 3(N 2 − 1) true degrees of freedom per spatial point of M 5 .
{φ (2) a [u], φ (2) b [v]} SU(N,M) = g m f abc φ (2) c [uv] ,(
The components A a M (x, y) of the connection find a natural split into A a µ (x, y) and A a 5 (x, y), and following Ref. [6], we assume the compact extra dimension homotopically equivalent to the circle S 1 of radius R. Fields A a µ (x, y) and A a 5 (x, y) are assumed to be periodic with respect to the fifth coordinate, so they can be expressed as Fourier series. In order to recover a pure four-dimensional Yang-Mills sector within the effective theory, we introduce a further symmetry in the compact extra dimension by replacing it with S 1 /Z 2 , hence y is identified with −y. We assume that A a µ (x, y) and A a 5 (x, y) are, respectively, even and odd under the reflection y → −y; these imply that curvature components F a µν (x, y) and F a µ5 (x, y) display even and odd parity in the extra dimension, respectively. Under these assumptions, the following Fourier expansions are allowed:
A a µ (x, y) = 1 √ R A (0)a µ (x) + 2 R ∞ m=1 A (m)a µ (x) cos 2π my R , (IV.21a) A a 5 (x, y) = 2 R ∞ m=1 A (m)a 5 (x) sin 2π my R , (IV.21b) F a µν (x, y) = 1 √ R F (0)a µν (x) + 2 R ∞ m=1 F (m)a µν (x) cos 2π my R , (IV.21c) F a µ5 (x, y) = 2 R ∞ m=1 F (m)a µ5 (x) sin 2π my R . (IV.21d)
In particular, it will be important to make the analogy between Eqs. (IV.21a) and (IV.21b) and the point transformations in Eq. (II.13).
Following the compactification scheme introduced in Ref. [6], one obtains the Fourier components of the curvature in terms of the gauge fields Fourier modes:
F (0)a µν = F (0)a µν + gf abc A (m)b µ A (m)c ν , (IV.22a) F (m)a µν = D (0)ab µ A (m)b ν − D (0)ab ν A (m)b µ + gf abc ∆ mrn A (r)b µ A (n)c ν , (IV.22b) F (m)a µ5 = D (0)ab µ A (m)b 5 + 2πm R A (m)a µ + gf abc ∆ ′ mnr A (r)b µ A (n)c 5 , (IV.22c) where D (0)ab µ = δ ab ∂ µ − gf abc A(0)
c µ , the coupling constant g = g 5 / √ R, and
F (0)a µν = ∂ µ A (0)a ν − ∂ ν A (0)a µ + gf abc A (0)b µ A (0)c ν .
(IV. 23) In addition
∆ mrn = 1 √ 2 (δ r,m+n + δ m,r+n + δ n,r+m ) , (IV.24a) ∆ ′ mrn = 1 √ 2 (δ m,r+n + δ r,m+n − δ n,r+m ) . (IV.24b)
Notice that there is a clear resemblance between Eqs. (II.15) and (IV.22). In the same fashion that the su(3)-valued curvature in our toy model was decomposed into well defined objects (Fā µν , Y µν , and F 8 µν ) under the SU (2) subgroup, we will show that the components of Eq. (IV.22) represent the decomposition of the pure SU (N, M 5 ) Yang-Mills curvature into well defined objects (F N, M 4 ). In our toy model, such decomposition was performed by means of the point transformation in Eq. (II.13); in the present case we will take advantage of Eqs. (IV.32). Moreover, in the present theory, the curvature decomposition is also a map from well defined objects under ISO(1, 4) onto well defined objects under ISO (1,3).
Integrating out the extra dimension after Fourier expanding Eq. (IV.16) yields the following effective Lagrangian, cf. (II.16):
L SU(N, M 4 ) = − 1 4 F (0)a µν F (0)aµν + F (m)a µν F (m)aµν + 2 F(δA (0)a µ = D (0)ab µ α (0)b + gf abc A (m)b µ α (m)c , (IV.26a) δA (m)a µ = gf abc A (m)b µ α (0)c + D (mn)ab µ α (n)b , (IV.26b) δA (m)a 5 = gf abc A (m)b 5 α (0)c + D (mn)ab 5 α (n)b , (IV.26c)
after the extra dimension is integrated out. The parameters α (0)a (x) and α (m)a (x) are the Fourier components in the expansion of α a (x, y) = α a (x, −y). In Eq. (IV.26) the following quantities have been defined:
D (mn)ab µ = δ mn D (0)ab µ − gf abc ∆ mrn A (r)c µ , (IV.27a) D (mn)ab 5 = − 2πm R δ mn δ ab − gf abc ∆ ′ mrn A (r)c 5 . (IV.27b)
In analogy with Eqs. (II.18) and (II.19), the SGTs and NSGTs are defined in this case. The SGTs correspond to Eq. (IV.26) after setting α (n)a = 0:
δ s A (0)a µ = D (0)ab µ α (0)b , (IV.28a) δ s A (m)a µ = gf abc A (m)b µ α (0)c , (IV.28b) δ s A (m)a 5 = gf abc A (m)b 5 α (0)c . (IV.28c)
In analogy with the gauge fields Wā µ under SU (2)
δF (0)a µν = gf abc F (0)b µν α (0)c + F (m)b µν α (m)c , (IV.30a) δF (m)a µν = gf abc F (m)b µν α (0)c + δ mn F (0)b µν + ∆ mrn F (r)b µν α (n)c , (IV.30b) δF (m)a µ5 = gf abc F (m)b µ5 α (0)c + ∆ ′ mrn F (r)b µ5 α (n)c .(∆ (m)a µν = − g 2 [f abc f bde (δ pq δ mn + ∆ rpq ∆ rmn ) + f adb f bce (δ nq δ mp + ∆ rnq ∆ rmp ) (IV.31a) +f abe f bcd (δ np δ mq + ∆ rnp ∆ rmq )] A (p)d µ A (q)e ν α (n)c , ∆ (m)a µ5 = − g 2 f abc f bde ∆ ′ rqp ∆ ′ rmn + f adb f bce ∆ ′ rqn ∆ ′ rmp +f abe f bcd (δ np δ mq + ∆ npr ∆ ′ mqr ) A (p)d µ A(δ s F (0)a µν = gf abc F (0)b µν α (0)c , (IV.32a) δ s F (m)a µν = gf abc F (m)b µν α (0)c , (IV.32b) δ s F (m)a µ5 = gf abc F (m)b µ5 α (0)c . (IV.32c)
The phase space description of this theory allows us to define the gauge generators associated to the so-called SGTs and NSGTs defined above. The canonical analysis of the effective Lagrangian Eq. (IV.25) goes along the same lines of reasoning as Sect. B2 of Ref. [6]. The conjugate momenta are given by
π (0)µ a = F (0)µ0 a , (IV.33a) π (n)µ a = F (n)µ0π µ a (x, y) = 1 √ R π (0)µ a (x) + 2 R ∞ m=1 π (m)µ a (x) cos 2π my R , (IV.34a) π 5 a (x, y) = 2 R ∞ m=1 π (mφ (1)(0) a = π (0)0 a ≈ 0 , (IV.35a) φ (1)(n) a = π (n)0 a ≈ 0 . (IV.35b)
The primary Hamiltonian takes the form (cf. (II.31))
H (1) SU(N, M 4 ) = H SU(N, M 4 ) + µ (0)a φ (1)(0) a + µ (n)a φ (1)(n) a , (IV.36)
where besides the linear combination of primary constraints, with Lagrange multipliers µ (0)a and µ (n)a as coefficients, the canonical Hamiltonian is (cf. (II.32))
H SU(N, M 4 ) = 1 2 π (0)i a π (0)i a + π (n)i a π (n)i a + π (n)5 a π (n)5 a + 1 4 F (0)ij a F (0)a ij + 2F (n)i5 a F (n)a i5 − A (0)a 0 φ (2)(0) a − A (n)a 0 φ (2)(n) a , (IV.37)
where φ
(2)(0) a and φ
(2)(n) a are functions of phase space that will be specified after presenting a couple of key results useful for the rest of the discussion. Yang-Mills theory and the effective theory based on SU (N, M 4 ), it immediately follows that both canonical Hamiltonians H SU(N,M) and H SU (N,M 4 ) are mapped into each other via such canonical transformation, as can be proved by direct calculation. However, in a singular theory, the time evolution is governed by the primary Hamiltonian and not by the canonical one. An important observation is the following: If in a general singular theory of fields there is a spacetime independent canonical transformation which connects two primary Hamiltonians corresponding to two different formulations of the same theory -that is, if such transformation maps one set of primary constraints into the other one, then both formulations must have the same number of generations of constraints (tertiary, quartic, etc.). This is an immediate consequence of the relation between the Poisson brackets in the two different formulations. Another consequence is that the set of secondary (tertiary, quartic, etc.) constraints in one of the formulations is necessarily mapped onto the corresponding set of constraints in the other formulation via the canonical transformation. The following result allows us to use these observations within the current analysis. Propositions IV.1 and IV.2 ensure that secondary constraints
φ (2)(0) a = D (0)ab i π (0)i b − gf abc A (n)c i π (n)i b + A (m)c 5 π (m)5 b ≈ 0 , (IV.38a) φ (2)(n) a = D (nm)ab i π (m)i b − D (nm)ab 5 π (m)5 b − gf abc A (n)c i π (0)i b ≈ 0 ,(IV.{φ (2)(0) a [u], φ (2)(0) b [v]} = gf abc φ (2)(0) c [uv] , (IV.39a) {φ (2)(0) a [u], φ (2)(n) b [v]} = gf abc φ (2)(n) c [uv] , (IV.39b) {φ (2)(m) a [u], φ (2)(n) b
[v]} = gf abc δ mn φ (2) [25]. This implies that there are no more gauge invariances of the Lagrangian Eq. (IV.25) than those altogether generated by Eq. (IV.40), which in turn correspond to Eq. (IV.26). Therefore, the effective Lagrangian Eq. (IV.25) must be invariant under these transformations, so that any extra term in the calculation of δL SU(N,M 4 ) must be either identically zero or a surface term. In this regard we argue that the extra terms Eqs. (IV.31a) and (IV.31b) must vanish since they do not include any derivative, hence they cannot be rewritten as a surface term.
G s = D (0)ab 0 α (0)b φ (1)(0) a + gf abc A (n)b 0 α (0)c φ (1)(n) a − α (0)a φ (2)(0) a , (IV.40a) G ns = gf abc A (m)b 0 α (m)c φ (1)(0) a + D (mn)ab α (n)b φ (1)(m) a − α (m)a φ(
We end this section with a heuristic counting of true degrees of freedom in the effective theory. Let us take for the moment "truncated Fourier expansions" up to some order K, so that, letting K → ∞ will precisely yield (A a µ (x, y), π µ a (x, y)) and (A a 5 (x, y), π 5 a (x, y)) in terms of (A (0)a µ (x), π (0)µ a (x)), (A (n)a µ (x), π (n)µ a (x)), (A (n)a 5 (x), π (n)5 a (x)) and trigonometric functions. In other words, K quantifies the contribution from the extra dimension in the "truncated Fourier expansions". The number of canonical pairs and first-class constraints in the truncated version are 2 × [4(N 2 − 1) + 4K(N 2 − 1) + K(N 2 − 1)] and 2(N 2 − 1) + 2K(N 2 − 1), respectively. Thus, the number of true degrees of freedom when K is large but finite is N 0 (K) = 2(N 2 − 1) + 3K(N 2 − 1) per spatial point of M 4 . Allowing K → ∞ causes this number of true degrees of freedom to diverge, precisely because one is also counting the continuum contribution of the extra dimension. In order to obtain the number of true degrees of freedom per spatial point of M 5 , one needs to take the ratio N 0 /K before considering K → ∞. After this process is done, we have that the number of true degrees of freedom per spatial point of M 5 is 3(N 2 − 1), which coincides with the corresponding number in the pure SU (N, M 5 ) Yang-Mills theory.
V. FINAL REMARKS
In order to clarify the gauge structure of pure five-dimensional Yang-Mills theories formulated on a spacetime manifold with a compact spatial extra dimension, a notion of hidden symmetry based on the fundamental concept of canonical transformation was introduced. Although the idea of hidden symmetry is well known in the context of theories with SSB, we have extended this notion to include more general scenarios. The canonical transformation under consideration maps well defined objects under a gauge group G to well defined objects under a non-trivial subgroup H ⊂ G. This transformation was constructed within two different categories depending whether the subgroup H is generated (a) by an appropriate subset of the generators of G, or (b) by the same set of generators of G, with its gauge parameters being the parameters of G restricted to a suitable submanifold. In both scenarios, all canonical pairs (q a , p a ) of the G-invariant theory are assumed to have well defined transformation laws under the group G. For instance, among the fields q a one may find gauge fields as well as matter fields; the canonical transformation that will hide the G symmetry, maps (q a , p a ) into (Q a , P a ) so that from the H perspective all Q's and P 's have well defined transformation laws under H. For instance, some Q's transform as gauge fields while the remainder arise in a tensorial representation of H.
In this paper we have analyzed two systems that fall into the category (a) described above; these correspond to pure SU (3) Yang-Mills theory, and SU (3) Yang-Mills theory coupled to a Higgs sector with SSB. In both cases G = SU (3) and H = SU (2). The former model allowed us to clarify the meaning of a suitable canonical transformation that lead us to the concept of hidden symmetry -such transformation maps gauge fields of SU (3) into gauge fields, two doublets and a singlet with respect to the SU (2) subgroup. The latter model was useful in order to formulate our notion of hidden symmetry within the context of a well known theory with SSB. The particular scenario of SSB gave an insight into the interpretation of NSGTs; a definite type of these transformations can be seen as the unitary gauge. In both cases, the original symmetry was hidden into the set of SGTs, which we showed corresponds to the SU (2) group, and the NSGTs, which do not form a group.
Pure Yang-Mills theory with one compactified UED falls into the category (b) described above. This theory is formulated to be invariant under the gauge group SU (N, M 5 ), and the corresponding Poincaré group ISO (1,4). Compactification maps the theory into an effective theory invariant under SU (N, M 4 ) and ISO (1,3). The suitable canonical transformation maps, in this case, gauge fields A a M of SU (N, M 5 ) onto gauge fields A We conclude after examination of the Lie algebra between SGTs and NSGTs that in the effective theory the SGTs can be identified with the SU (N, M 4 ) group, whereas the NSGTs do not exponentiate into any group. It is important to notice that since there are no broken generators in this scenario, the Higgs mechanism does not operate in the conventional sense; the pseudo-Goldstone bosons needed for this mechanism are provided by an explicit breaking of the Poincaré group ISO (1,4) into ISO (1,3). Extension of this analysis to theories with more than one compactified UED will be reported elsewhere.
In the Hamiltonian analysis of these models, we found that each canonical transformation translates all the relevant quantities -such as the set the of constraints and the primary Hamiltonian-from the G invariant theory to the theory invariant under SGTs and NSGTs. Since each model we analyzed is a first-class constraint system, each canonical transformation maps the gauge generator of the G-symmetry into gauge generators of the SGTs and NSGTs. These results are particularly interesting for the pure SU (N, M 5 ) Yang-Mills theory with one compactified UED and its effective theory; it implies that the gauge structure of the higher-dimensional theory has certainly been rewritten in terms of SGTs and NSGTs. Besides, by arguing that the five-dimensional and the effective theory have the same number of physical degrees of freedom, we conclude that the fundamental and the effective theory are equivalent at the classical level.
In this Appendix we will prove that Fourier expansion is a canonical transformation by showing that it maps conjugate canonical pairs to conjugate canonical pairs. In order to do that, we will explicitly calculate the nonvanishing Poisson brackets between the zero modes (A . We expect to find that this Poisson brackets yield the components of the canonical symplectic two form, proving in this way that the Fourier transformation is indeed a canonical transformation.
We will make use of the following Poisson brackets among the gauge fields and their canonical conjugate momenta, at a fixed time: Also, in order to properly deal with the distributional character of the Poisson brackets, we will use smooth smearing functions u and v defined on M 4 . We proceed to calculate the Poisson bracket between the zero modes with four-dimensional spacetime labels
{A a M (x, y), π N b (x ′ , y ′ )} SU(N,M) = δ a b δ N M δ(x − x ′ )δ(y − y ′ ) {A a M (x, y), A b N (x ′ ,y{A (0)a µ [u], π (0)ν b [v]} SU(N,M 4 ) = d 3 x d 3 x ′ u(x)v(x ′ ){A (0)a µ (x), π (0)ν b (x ′ )} SU(N,M 4 ) = d 3 x d 3 x ′ dy dy ′ u(x)v(x ′ ) 1 R {A a µ (x, y), π ν b (x ′ , y ′ )} SU(N,M) = d 3 x d 3 x ′ dy dy ′ u(x)v(x ′ ) 1 R δ a b δ ν µ δ(x − x ′ )δ(y − y ′ ) = δ a b δ ν µ [uv] . (A.1)
The corresponding calculation for the m modes with four-dimensional spacetime labels reads , π (m)5 a ) are canonical pairs, one obtains that (A a M , π M a ) are canonical pairs. This is achieved by using the Fourier transform and smear functions u and v defined on M 5 , therefore periodic in y. These functions will be asked to be even when calculating the Poisson brackets between A a µ and π ν b , so that they can be expanded as follows
{A (m)a µ [u], π (n)ν b [v]} SU(N,M 4 ) = d 3 x d 3 x ′ u(x)v(x ′ ){A (m)a µ (x), π (n)ν b (x ′ )} SU(N,M 4 ) = d 3 x d 3 x ′ dy dy ′ u(x)v(x ′ ) 2 R cos 2π my R cos 2π ny ′ R {A a µ (x, y), π ν b (x ′ ,yu(x, y) = 1 √ R u (0) (x) + 2 R ∞ m=1 u (m) (x) cos 2π my R ; (A.4)
and we will demand they be odd when calculating the Poisson brackets between A a 5 and π 5 b , and thus expanded as
u(x, y) = 2 R ∞ m=1 u (m) (x) sin 2π my R . (A.5)
In conclusion, from a set of conjugate pairs we obtain, via the Fourier transform, another set of conjugate pairs. This proof can easily be extended in the presence of more extra dimensions, provided each field has suitable periodic and parity properties on the extra dimensions.
SGT, SGT ] = SGT, [ SGT, NSGT ] = SGT + NSGT, [ NSGT, NSGT ] = SGT + NSGT, (II.38)
12b) faithfully reproduce Eqs. (II.3) and (III.4) -that is, the symmetries of the theory. B. SSB from the SU (3) perspective
C. The SU (2) perspective of the model In this subsection the description of the field theory Eq. (III.1) from the SU(2)perspective is achieved. The pure SU (3) Yang-Mills sector L SU(3) is mapped, by means of the point transformation Eq. (II.13), into L SU
to SU (2) coordinates; therefore {·, ·} SU(3) = {·, ·} SU(2) . Second, the canonical transformation maps the set of primary constraints {ϕ (1) a } onto the set of primary constraints {ϕ
B. Pure SU (N, M m ) Yang-Mills Theory The Lagrangian that describes pure SU (N, M m ) Yang-Mills theory is given by (cf. (II.1)) L SU(N, M) (x, y) this subsection (x, y) denotes the coordinates of M 4 × N n . The components F a MN are regarded as functions of gauge fields A a M (x, y) as in Eq. (II.2) except that in this case the coupling constant is denoted by g m , whose dimension is of [mass] (4−m)/2 . Gauge invariances of this theory are (cf. (II.3)) δA a M = D ab M α b (x, y) , (IV.17)
IV.19) where the Poisson bracket {·, ·} SU(N,M) is calculated in terms of canonical conjugate pairs (A a M , π M a ). In the same fashion, gauge transformations Eq. (IV.17) can be obtained via the corresponding gauge generator cf. Eq. (II.11) as follows: δA a M = {A a M , G} SU(N,M) . (IV.20) We now perform the transition from the SU (N, M m ) variables to the natural variables that arise in the effective theory after compactification. C. Compactified theory and the SU (N, M 4 ) description
under the subgroup SU (
m)a µ5 F
µ5(m)aµ5 . (IV.25) the effective Lagrangian Eq. (IV.25). In this framework, gauge transformations Eq. (IV.17) are mapped by Eqs. (IV.21a) and (IV.21b) onto
α
Eq. (II.18a), the Fourier component A to SU (N, M 4 ). Similarly, the matter field Y µ is comparable with the excited KK modes A (n)a µ , which transform in the adjoint representation of SU (N, M 4 ). In addition, A (n)a 5 transform as matter fields in the adjoint representation of SU (N, M 4 ). The NSGTs are obtained from Eq. (IV.26) by setting α (0)a ≡ 0, that is (cf. (II.19)) (n)b . (IV.29c) Gauge invariance of Eq. (IV.25) under Eq. (IV.26) is guaranteed, since the latter imply the following variations at the level of the Fourier components of the curvature:
IV.30c) It is not difficult to see that the effective Lagrangian L SU(N, M 4 ) is invariant under these transformations. Therefore, the components of Eq. (IV.26) are genuine gauge transformations of the effective theory. It is worth noticing that the scalar fields A (m)a 5 can be eliminated altogether via a particular NSGT. Consider a NSGT with infinitesimal gauge parameters given by α (m)a (x) = (R/2πm)A (m)a 5 (x), Ref. [6]. Then, from Eq. (IV.29c), we can see that A scalar fields are in fact pseudo Goldstone bosons. It is important to stress that the invariance of the effective theory Eq. (IV.25) under the transformations Eq. (IV.26) is by no means immediate. A direct calculation of the curvature variations Eq. (IV.30) from Eq. (IV.26) gives raise to the following extra terms quadratic in g:
q)e 5 α
5(n)c , (IV.31b) in Eqs. (IV.30b) and (IV.30c), respectively. These terms, that would destroy the invariance of the effective Lagrangian L SU(N,M 4 ) under Eq. (IV.26), are necessarily zero by consistency with the Fourier transformation in Eq. (IV.21). The variation of curvatures δF a MN = g 5 f abc F b MN α c is duly mapped onto Eqs. (IV.30) under the point transformation Eq. (IV.21). We will discuss further this point within the Hamiltonian formalism of the theory. The SGTs Eq. (IV.28) induce the corresponding transformations at the curvature level. From Eq. (IV.30), all Fourier components of F a MN do covariantly transform under the symmetry group of SGTs, SU (N, M 4 ):
worth noticing, from Eqs. (IV.28) and (IV.32), that canonical pairs are well defined objects with respect to SU (N, M 4 ). In addition, the Fourier expansions in Eqs. (IV.21c) and (IV.21d) together with π M a = F M0 a allow us to write
relate the conjugate momenta inherent in the pure SU (N, M 5 ) Yang-Mills theory and those present in the effective SU (N, M 4 ) theory. Moreover, they are analogous to Eq. (II.28). The temporal component of Eqs. (IV.33a) and (IV.33b) define the following primary constraints:
Proposition IV. 1
1The Fourier expansion of gauge fields and conjugate momenta, Eqs. (IV.21a), (IV.21b) and (IV.34), define a canonical transformation. The proof of this proposition is collected in the Appendix. This proposition ensures that {·, ·} SU(N,M) = {·, ·} SU(N,M 4 ) , where {·, ·} SU(N,M 4 ) indicates the Poisson bracket with respect to (A ). Because there exists a spacetime independent canonical transformation between the pure SU (N, M 5 )
Proposition IV. 2
2The set of primary constraints Eq. (IV.18a) of the five dimensional pure SU (N, M 5 ) Yang-Mills theory is faithfully mapped onto the set of primary constraints Eq. (IV.35) of the SU (N, M 4 ) Yang-Mills theory. The proof of this proposition is straightforward from Eq. (IV.34a) and the linear independence of trigonometric functions. Moreover, it can be extended to the case of m dimensional pure SU (N, M m ) Yang-Mills theory and its compactification down to four dimensions.
SU (N, M 4 ). As Lie groups SU (N, M 5 ) and SU (N, M 4 ) share the same number of generators, so the map from one to the other cannot involve SSB. However, SU (N, M 4 ) is a subgroup of SU (N, M 5 ) in the following sense: The parameters defining SU (N, M 4 ) are the parameters defining SU (N, M 5 ) restricted to the submanifold M 4 .
′ )} SU(N,M) = {π M a (x, y), π N b (x ′ , y ′ )} SU(N,M)
′ )} SU(N,M) = δ a b δ ν µ δ mn [uv] .
3) perspectives of the same theory indicates that it must vanish. Indeed, using the point transformation Eq. (II.14), the Eqs. (II.20a) to (II.20c) can be rearranged to Eq. (II.4) as required. Also, since B µν is linear in β, its occurrence in the variation of Y µν would spoil the invariance of the Lagrangian Eq. (II.16) under the NSGTs Eq. (II.19); however, one can see that the rth SU (2) component (r = 1, 2) of the doublet B µν is of the form
The complete transformations (II.17) are duly reproduced by the addition δ = δ s + δ ns . It is worth noticing that the gauge generator Eq. (II.36) is the image of the gauge generator Eq. (II.11) under the canonical transformation defined by Eqs. (II.13) and (II.28).
manifestly invariant theory, cf. Eqs.(III.1) and (III.16) via the admissible point transformation, Eqs. (II.13) and (III.15). In the context of theories with SSB, it is said that the SU
38b )
38bthat emerge in the canonical Hamiltonian Eq. (IV.37) can be also calculated from Eq. (IV.18b) via the canonical transformation mentioned in Prop. IV.1. Less trivial outcomes of the considerations above are the following: First, the effective theory must not present either tertiary or higher constraint generations. Second, the gauge algebra of the effective theory can be obtained via the canonical transformation from the gauge algebra Eq. (IV.19) of the pure five dimensional Yang-Mills theory. In fact,
The gauge generator that reproduces the gauge transformations in Eq. (IV.26) is the sum of the SGTs (G s ) plus the NSGTs (G ns ) generators, where(0)
c
[uv] + ∆ mnr φ (2)(r)
c
[uv] ,
(IV.39c)
which coincides with Eqs. (68)-(70) of Ref. [6].
From the transformation laws generated by G s and G ns , together with the constraint algebra Eq. (IV.39), one can infer the Lie algebra Eq. (II.38) on the constraint surface for the SGTs and NSGTs in this case. Due to the constraint algebra Eq. (IV.39a), the SGTs exponentiate into SU (N ), and since in G s the gauge parameters α (0)a are defined on M 4 , we have that exponentiation of SGTs provides SU (N, M 4 ); the algebra of NSGTs does not close, hence this transformations do not exponentiate into a group. The sum G s + G ns is the image under the canonical transformation mentioned in Prop. IV.1 of the gauge generator that reproduces gauge transformations Eq. (IV.17) in the five dimensional case.If a complete set of gauge transformations at the Hamiltonian level can be found, then a complete set of gauge transformations at the Lagrangian level can be recovered2)(m)
a
.
(IV.40b)
We trust that no confusion will arise with the symbol Dµ already used for the covariant derivative of SU (2) in its fundamental representation, as we think one can infer the nature of the covariant derivative depending on which object this is acting on.
AcknowledgmentsWe acknowledge financial support from CONACyT (México), and J. J. T. also acknowledges SNI (México).Appendix A: Fourier expansion as a canonical transformation
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| []
|
[
"Extensivity and nonextensivity of two-parameter entropies",
"Extensivity and nonextensivity of two-parameter entropies"
]
| [
"S Asgarani [email protected] \nDepartment of Physics\nIsfahan University of Technology (IUT) Isfahan\nIran\n",
"B Mirza \nDepartment of Physics\nIsfahan University of Technology (IUT) Isfahan\nIran\n"
]
| [
"Department of Physics\nIsfahan University of Technology (IUT) Isfahan\nIran",
"Department of Physics\nIsfahan University of Technology (IUT) Isfahan\nIran"
]
| []
| In this paper, we investigate two-parameter entropies and obtain some conditions for their extensivity. By using a generalized (k, r) − product, correlations for subsystems are related to the joint probabilities, so that the entropy remains extensive. * | 10.1016/j.physa.2006.10.100 | [
"https://arxiv.org/pdf/cond-mat/0608054v1.pdf"
]
| 11,828,365 | cond-mat/0608054 | 429bb479997956b7effe2acd02e7d43ede16c455 |
Extensivity and nonextensivity of two-parameter entropies
2 Aug 2006
S Asgarani [email protected]
Department of Physics
Isfahan University of Technology (IUT) Isfahan
Iran
B Mirza
Department of Physics
Isfahan University of Technology (IUT) Isfahan
Iran
Extensivity and nonextensivity of two-parameter entropies
2 Aug 20061
In this paper, we investigate two-parameter entropies and obtain some conditions for their extensivity. By using a generalized (k, r) − product, correlations for subsystems are related to the joint probabilities, so that the entropy remains extensive. *
Introduction
A quantity X(A) associated with a system A is said additive with regard to a specific composition of A and B if it satisfies
X(A + B) = X(A) + X(B)(1)
where + inside the argument of X precisely indicates that composition. suppose, instead of two subsystems A and B, we have N of them (A 1 , A 2 , ..., A N ). Then the quantity X is additive if we have
X( n i=1 A i ) = n i=1 X(A i )(2)
supposing that all subsystems are equal,
X(N) = NX(1)(3)
with the notation X(N) ≡ X( n i=1 A i ) and X(1) ≡ X(A 1 ). Another related concept is extensivity which corresponds to a weaker demand, namely that of,
lim N →∞ |X(N)| N < ∞(4)
Clearly, all quantities which are additive, are also extensive, whereas the opposite is not necessarily true. In other words, extensivity is defined as additivity when N → ∞. Of course, there are quantities that are neither additive nor extensive. They are called nonextensive . Boltzmann-Gibbs (BG) statistical mechanics is based on the entropy
S BG ≡ −k W i=1 p i ln p i(5)
with W i=1 p i = 1 (6) where p i is the probability associated with the i th microscopic state of the system and k is Boltzmann constant. From now on, and without loss of generality, we shall take k equal to unity.
Nonextesive statistical mechanics, first introduced by C. Tsallis in 1988 [1,2,3], is based on the so-called 'nonextensive' entropy S q defined as follows:
S q ≡ 1 − W i=1 p q i q − 1(7)
As we see this entropy depends on parameter q. Afterwards, some other entropies were suggested depending on one parameter [4,5,6,7] .
Recently, an entropy was introduced [8,9] that depends on two parameters, and in some special limits recovers other entropies that had been introduced previously. That is
S k,r ≡ − W i=1 p i ln k,r p i (8) with ln k,r (x) = x r x k − x −k 2k(9)
The concept of extensivity has been investigated mostly for systems with no correlation, namely independent systems. In that case, the probabilities belong to the composition system are defined as the product of the probabilities in each subsystem. If the composition law is not explicitly indicated, it is tacitly assumed that systems are statistically independent. In that case, for two systems A and B, it immediately follows that
S BG (A + B) = S BG (A) + S BG (B)(10)
hence, BG-entropy is additive and also extensive, but for q-entropy we have
S q (A + B) = S q (A) + S q (B) + (1 − q)S q (A)S q (B)(11)
hence, q-entropy is nonextensive for q = 1. In [10] Tsallis has illustrated the remarkable changes that occur when A and B are specially correlated. Indeed, he has shown that in such case
S q (A + B) = S q (A) + S q (B)(12)
for the appropriate value of q (hence extensive), whereas
S BG (A + B) = S BG (A) + S BG (B)(13)
hence BG-entropy isn't extensive in the case of correlated systems [14]. This paper is organized as follows. In sec. 2, the nonextensivity of S k,r is discussed, where the extensivity of BG-entropy is recovered in an special limit. In sec. 3, we investigate how to interpret entropy S k,r extensive and finally in sec. 4 extensivity of entropy with canonic ensemble is discussed when we have correlated subsystems.
2 nonextensivity of S k,r in the case of independent systems
As said, the entropy S k,r Eqs. (8) and (9) is more general than the other entropies introduced previously and in some special limits recovers them. We prove that this entropy is nonextensive in the case of independent subsystems. Supposing two independent subsystems A and B, for the probability in the composite system A + B we have
p A+B ij = p A i p B j ∀(i, j)(14)
with the definitions
S k,r (A) ≡ − W A i=1 p A i ln k,r p A i (15) S k,r (A + B) ≡ − W A i=1 W B j=1 p A+B ij ln k,r p A+B ij(16)
By adding and subtracting the phrase p r+k+1 i p r−k+1 j and using Eq. (9), we can find
S k,r (A + B) = W B j=1 p r−k+1 j S k,r (A) + W A i=1 p r+k+1 i S k,r (B)(17)
As we see
S k,r (A + B) = S k,r (A) + S k,r (B)(18)
hence S k,r isn't extensive in general. However, one may choose some special range of parameters where Eq. (17) is extensive. We study the extensivity of S k,r in some special limits.
q-entropy(Tsallis entropy)
The q-logarithm that is usually used is
ln q (x) ≡ x 1−q − 1 1 − q(19)
With this logarithm the q-entropy is defined as
S q (p) ≡ − W i=1 p q i ln q p i(20)
by choosing r = k and q = 1 + 2k in Eqs. (8) and (9), one has
S q (p) ≡ − W i=1 p i ln q p i (21) where ln q (x) ≡ x q−1 − 1 q − 1(22)
which gives an equivalent entropy to Eq. (20). For exp q (x) it is obtained
exp q (x) ≡ [ 1 + (q − 1)x ] 1 q−1 (23)
In the limit r = k and q = 1 + 2k from Eq. (17), one recovers
S q (A + B) = W B j=1 p j S q (A) + W A i=1 p q i S q (B) = S q (A) + S q (B) + (1 − q)S q (A)S q (B)
(24) that is the familiar expression for nonextensivity of q-entropy. In the limit q → 1 extensivity of BG-entropy is obtained.
k-entropy
The k-entropy introduced in [6,7] is
S k (p) ≡ − W i=1 p i ln k p i (25) where ln k (x) ≡ x k − x −k 2k (26)
It is clear that we can recover k-logarithm from Eq. (9) in the limit r → 0.
For exp k (x) we have exp k (x) ≡ ( √ 1 + k 2 x 2 + kx) 1 k(27)
In that limit Eq. (17) results in
S k (A + B) = W B j=1 p −k+1 j S k (A) + W A i=1 p k+1 i S k (B) = S k (A) + S k (B)(28)
that ensures the nonextnsivity of the k-entropy. It is clear that in the limit k → 0 the extensivity of BG-entropy is obtained again.
3 How to interpret the entropy S k,r extensive Suppose, we have N subsystems (A 1 , A 2 , ..., A N ). We define the probabilities in the composite system p A 1 +A 2 +...+A N i 1 i 2 ...i N that satisfy the condition
i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N = 1(29)
and marginal probabilities as follows
p As is ≡ i 1 i 2 ...i s−1 i s+1 i N p A 1 +A 2 +...+A N i 1 i 2 ...i N(30)
If
p A 1 +A 2 +...+A N i 1 i 2 ...i N also satisfies the condition p A 1 +A 2 +...+A N i 1 i 2 ...i N = exp k,r ( N s=1 ln k,r p As is )(31)
then for the entropy of the composite system with the definition
S k,r ( N s=1 A s ) ≡ − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N ln k,r p A 1 +A 2 +...+A N i 1 i 2 ...i N(32)
we have
S k,r ( N s=1 A s ) = − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N ln k,r [exp k,r ( N s=1 ln k,r p As is )] = − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N N s=1 ln k,r p As is = − N s=1 is p As is ln k,r p As is = N s=1 S k,r (A s )(33)
It is useful at this point to connect the present problem to some generalized algebra which have been discussed by many authors. We use the product introduced in [9]. It is defined as follows:
x ⊗ k,r y ≡ exp k,r ( ln k,r (x) + ln k,r (y))(34)
hence we can write (31) as
p A 1 +A 2 +...+A N i 1 i 2 ...i N = exp k,r ( N s=1 ln k,r p As is ) = p A 1 i 1 ⊗ k,r p A 2 i 2 ⊗ k,r . . . ⊗ k,r p A N i N(35)
So extensivity of the entropy is satisfied if we use logarithm, exponential and also the product based on (34). In the limit k → 0 and r → 0 (BG-limit), the usual product is recovered and (35) describes the probability of composite system in the case of independent subsystems and also extensivity of BGentropy in that case which is expected. Eq. (31) is a very special correlation for subsystems which leads to extensivity of entropy. however, it is possible to define a general correlation among subsystems so that the entropy remains extensive. Consider the following relation
p A 1 +A 2 +...+A N i 1 i 2 ...i N ≡ p i 1 A 1 ⊗ k,r p A 2 i 2 ⊗ k,r . . . ⊗ k,r p A N i N(36)
where p As is s are the probabilities of each subsystem, but p A 1 +A 2 +...+A N i 1 i 2 ...i N s are not necessarily represent the joint probabilities. Now the sum of subsystem entropies can be written as
N s=1 S k,r (A s ) = − N s=1 is p As is ln k,r p As is = − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N N s=1 ln k,r p As is = − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N ln k,r [exp k,r ( N s=1 ln k,r p As is )] = − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N ln k,r p A 1 +A 2 +...+A N i 1 i 2 ...i N(37)
So entropy is extensive if
S k,r ( N s=1 A s ) = − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N ln k,r p A 1 +A 2 +...+A N i 1 i 2 ...i N = − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N ln k,r p A 1 +A 2 +...+A N i 1 i 2 ...i N(38)
It is clear that
p A 1 +A 2 +...+A N i 1 i 2 ...i N and p A 1 +A 2 +...+A N i 1 i 2 ...i N
can be related to each other by the following relations
ln k,r p A 1 +A 2 +...+A N i 1 i 2 ...i N − ln k,r p A 1 +A 2 +...+A N i 1 i 2 ...i N = φ i 1 i 2 ...i N (39) i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N φ i 1 i 2 ...i N = 0 (40) where φ i 1 i 2 ...i N+ φ i 1 i 2 ...i N )(41)
In the Tsallis limit Eq. (41) can be written as (by using (22) and (23))
p A 1 +A 2 +...+A N i 1 i 2 ...i N = 1 − N + (q − 1)φ i 1 i 2 ...i N + N s=1 (p As is ) q−1 1 q−1 (42)
which is equivalent to the Tsallis proposal for the joint probabilities [10] if we choose φ (q)
i 1 i 2 ...i N = (q − 1)φ i 1 i 2 ...i N(43)
Consider two subsystems A and B where the probabilities of composite system and each subsystem are shown in the following table
A \ B 1 2 1 p A+B 11 p A+B 12 p A 1 2 p A+B 21 p A+B 22 1 − p A 1 p B 1 1 − p B 1 1
with the following relations
p A+B 11 + p A+B 12 = p A 1 (44) p A+B 21 + p A+B 22 = p A 2 = 1 − p A 1 (45) p A+B 11 + p A+B 21 = p B 1 (46) p A+B 12 + p A+B 22 = p B 2 = 1 − p B 1(47)
and also a constraint (40)
p A+B 11 φ 11 + p A+B 12 φ 12 + p A+B 21 φ 21 + p A+B 22 φ 22 = 0 (48)
Using Eq. (41), it is possible to write Eqs. (44) to (47) in terms of p A 1 , p A 2 , φ 11 , φ 12 , φ 21 and φ 22 . So φ ij s can be determined. For simplicity we use Tsallis limit and so (42) for the probabilities of the composite system. We also assume that both subsystems A and B are equal, namely
p A 1 = p B 1 = p. So we have p A+B 11 = [2p q−1 + (q − 1)φ 11 − 1] 1 q−1 (49) p A+B 12 = [p q−1 + (1 − p) q−1 + (q − 1)φ 12 − 1] 1 q−1 (50) p A+B 21 = [(1 − p) q−1 + p q−1 + (q − 1)φ 21 − 1] 1 q−1 (51) p A+B 22 = [2(1 − p) q−1 + (q − 1)φ 22 − 1][2p q−1 + (q − 1)φ 11 − 1] 1 q−1 + [p q−1 + (1 − p) q−1 + (q − 1)φ 12 − 1] 1 q−1 = p (53) [p q−1 + (1 − p) q−1 + (q − 1)φ 12 − 1] 1 q−1 + [2(1 − p) q−1 + (q − 1)φ 22 − 1] 1 q−1 = 1 − p (54) φ 11 [2p q−1 + (q − 1)φ 11 − 1] 1 q−1 + 2φ 12 [p q−1 + (1 − p) q−1 + (q − 1)φ 12 − 1] 1 q−1 +φ 22 [2(1 − p) q−1 + (q − 1)φ 22 − 1] 1 q−1 = 0 (55)
With a given value of q, above equations can be solved and one may obtain φ 11 (p), φ 12 (p) = φ 21 (p) and φ 22 (p). A few typical (q, φ 11 (1/2), φ 12 (1/2)) points are: where it should be noted that for p = 1/2 symmetries of equations ensures that φ 11 (p) = φ 22 (p). We will investigate more numerical estimates for twoparameter entropies in another paper.
4 Extensive entropy in the case of correlated subsystems, a constraint approach
In this section, a new approach is used where the condition (40) is entered to the entropy as a constraint and then the entropy is maximized . Parallel to what is done in [9], we introduce the entropy in the composite system as
S( N s=1 A s ) ≡ − i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N Λ(p A 1 +A 2 +...+A N i 1 i 2 ...i N )(56)
where Λ(x) is a generalization of the logarithm. We have the constraints
i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N = 1 (57) i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N E A 1 +A 2 +...+A N i 1 i 2 ...i N = U (58) i 1 i 2 ...i N p A 1 +A 2 +...+A N i 1 i 2 ...i N φ i 1 i 2 ...i N = 0(59)
For simplicity we use the notation {i} instead of {i 1 i 2 ...i N }. Then the entropic functional can be introduced as
F [p] = S(p) − β ′ {i} p {i} − 1 − β {i} p {i} E {i} − U − β ′′ {i} p {i} φ {i} (60)
where β, β ′ and β ′′ are Lagrange multipliers and it has been supposed that φ {i} isn't an explicit function of p {i} . If F [p] in Eq. (60) is stationary for variations of the probabilities p {j} ,
δ δp {j} F [p] = 0 (61) one finds d dp {j} [ p {j} Λ(p {j} )] = −β(E {j} − µ − µ ′ φ {j} )(62)
where µ = −β ′ /β and µ ′ = −β ′′ /β. Without loss of generality, we can express the probability distribution p j as
p {j} = αε − β λ (E {j} − µ − µ ′ φ {j} )(63)
where α and λ are two arbitrary, real and positive constants, and ε(x) an invertible function that can be a generalization of, and in some limit reduce to, the exponential function. If we require that ε(x) be the inverse of Λ(x), Eqs. (62) and (63) result in
d dp {j} [ p {j} Λ(p {j} )] = λε −1 ( p {j} α )(64)
that can be rewritten as [9] d dx
[ xΛ(x)] = λε −1 ( x α )(65)
So, for Λ(x) we have
Λ(x) = ln k,r (x) = x r x k − x −k 2k(66)
and the constants α and λ can be expressed in terms of k and r
α = ( 1 + r − k 1 + r + k ) 1/(2k) (67) λ = (1 + r − k) (r+k)/(2k) (1 + r + k) (r−k)/(2k)(68)
Eq. (66) indicates that by imposing the condition (59), the definition of logarithm dose not change and the only thing we must change is the definition of probability in the composite system. It is useful here to interpret each subsystem separately. By imposing the conditions
is p As is = 1 (69) is p As is E As is = U s(70)
For the subsystems, the entropic functional will be
F s [p] = S s (p) − β ′ s is p is − 1 − β s is p is E is − U s(71)
and by maximizing the entropic functional in the way similar to the case of composite system, we obtain
p is = α exp k,r − β s λ (E is − µ s )(72)
where µ s = −β ′ s /β s , exp k,r (x) is inverse function of ln k,r (x) and α and λ are defined in Eqs. (67) and (68). Using (63) and (72), Eq. (31) can be written as
ln k,r [α exp k,r (− β λ (E {j} − µ − µ ′ φ {j} ) )] = N s=1 ln k,r [α exp k,r (− β s λ (E is − µ s ) )]
(73) Where parameters φ {j} can be used to ensure extensivity of the twoparameter entropies. From Eq. (73), it is clear that extensivity of entropy dose not necessarily ensures extensivity of energy ( For a discussion in the case of q-entropy see [11] ) . In the Boltzmann-Gibbs limit Eq. (73) becomes
β(E {j} − µ − µ ′ φ {j} ) = N s=1 β s (E is − µ s )(74)
where only in a special case leads to the extensivity of energy.
Probabilities and effective number of states
Our motivation for studying such kind of correlations and extensivity of the two-parameter entropies of correlated subsystems was the following argument by Tsallis [10,12] which defines effective number of states. Suppose that the probability distribution in phase space is uniform within a volume W and also S q is given by
S q = ln q W(75)
With the help of q-product [13] defined as
x ⊗ q y ≡ exp q (ln q x + ln q y)
= (x 1−q + y 1−q − 1) 1/1−q(76)
it is possible to interpret S q extensive. Supposing that W A and W B be the number of states for subsystems A and B. Equation
W ef f A+B ≡ W A ⊗ q W B(77)
can be interpreted as a definition for effective number of states for the system A + B. Definition (76) ensures that
ln q W ef f A+B = ln q W A ⊗ q W B = ln q W A + ln q W B(78)
Eq. (78) shows extensivity of the entropy (75). If we suppose
W A 1 = W A 2 = ... = W A N = 1/p(79)
the probability in the composite system will be
(1/p A 1 +A 2 +...+A N i 1 i 2 ...i N ) = (1/p) ⊗ q (1/p) ⊗ q ... ⊗ q (1/p)(80)
and hence p A 1 +A 2 +...+A N
i 1 i 2 ...i N = p ⊗ 2−q p ⊗ 2−q ... ⊗ 2−q p(81)
where (81) is obtained from (80) by the properties of q-product. At this point it is appropriate to use the following q-product which is used in this paper
x ⊗ q ′ y ≡ exp q ′ (ln q ′ x + ln q ′ y)
= (x q ′ −1 + y q ′ −1 − 1) 1/q ′ −1
comparing Eq. (82) with Eq. (76) shows that q ′ = 2 − q. By our q-product Eq. (81) can be written as
p A 1 +A 2 +...+A N i 1 i 2 ...i N = p ⊗ q p ⊗ q ... ⊗ q p(83)
This is a hinting point to define the probability of composite system in terms of the probabilities of subsystems by a generalized (k, r)-product.
conclusion
In this paper, it is shown that two-parameter entropies S k,r are not in general extensive. A formulation is given where by (k, r)-products of subsystem probabilities one may obtain joint probabilities involving some functions φ i 1 i 2 ...i N . Demanding extensivity of the entropy imposes some constraints on φ i 1 i 2 ...i N s and so joint probabilities are identified. We believe this is the most general representation for obtaining extensive entropies in the case of correlated subsystems.
are arbitrary functions with (40) as a constraint. Eqs. (36) and (39) result in p A 1 +A 2 +...+A N i 1 i 2 ...i N
Eqs. (49) to (52) in (44) to (48), we obtain φ 12 = φ 21 and so
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| []
|
[
"Multi-Stage Feature Selection Based Intelligent Classifier for Classification of Incipient Stage Fire in Building",
"Multi-Stage Feature Selection Based Intelligent Classifier for Classification of Incipient Stage Fire in Building"
]
| [
"Allan Melvin Andrew \nCentre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia\n",
"Ammar Zakaria \nCentre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia\n",
"Shaharil Mad Saad [email protected]. \nCentre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia\n",
"Ali Yeon [email protected].*correspondence:[email protected] \nCentre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia\n",
"Md Shakaff \nCentre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia\n",
"Vittorio M N Passaro \nCentre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia\n"
]
| [
"Centre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia",
"Centre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia",
"Centre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia",
"Centre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia",
"Centre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia",
"Centre of Excellence for Advanced Sensor Technology (CEASTech)\nUniversiti Malaysia Perlis\n02600Jejawi, Arau, PerlisMalaysia"
]
| []
| In this study, an early fire detection algorithm has been proposed based on low cost array sensing system, utilising off-the shelf gas sensors, dust particles and ambient sensors such as temperature and humidity sensor. The odour or "smellprint" emanated from various fire sources and building construction materials at early stage are measured. For this purpose, odour profile data from five common fire sources and three common building construction materials were used to develop the classification model. Normalised feature extractions of the smell print data were performed before subjected to prediction classifier. These features represent the odour signals in the time domain. The obtained features undergo the proposed multi-stage feature selection technique and lastly, further reduced by Principal Component Analysis (PCA), a dimension reduction technique. The hybrid PCA-PNN based approach has been applied on different datasets from in-house developed system and the portable electronic nose unit. Experimental classification results show that the dimension reduction process performed by PCA has improved the classification accuracy and provided high reliability, regardless of ambient temperature and humidity variation, baseline sensor drift, the different gas concentration level and exposure towards different heating temperature range. | 10.3390/s16010031 | null | 13,366,994 | 1708.08750 | 194f47c5d3ae0698cae9f38ffbb991e07761b0b2 |
Multi-Stage Feature Selection Based Intelligent Classifier for Classification of Incipient Stage Fire in Building
Published: 19 January 2016
Allan Melvin Andrew
Centre of Excellence for Advanced Sensor Technology (CEASTech)
Universiti Malaysia Perlis
02600Jejawi, Arau, PerlisMalaysia
Ammar Zakaria
Centre of Excellence for Advanced Sensor Technology (CEASTech)
Universiti Malaysia Perlis
02600Jejawi, Arau, PerlisMalaysia
Shaharil Mad Saad [email protected].
Centre of Excellence for Advanced Sensor Technology (CEASTech)
Universiti Malaysia Perlis
02600Jejawi, Arau, PerlisMalaysia
Ali Yeon [email protected].*correspondence:[email protected]
Centre of Excellence for Advanced Sensor Technology (CEASTech)
Universiti Malaysia Perlis
02600Jejawi, Arau, PerlisMalaysia
Md Shakaff
Centre of Excellence for Advanced Sensor Technology (CEASTech)
Universiti Malaysia Perlis
02600Jejawi, Arau, PerlisMalaysia
Vittorio M N Passaro
Centre of Excellence for Advanced Sensor Technology (CEASTech)
Universiti Malaysia Perlis
02600Jejawi, Arau, PerlisMalaysia
Multi-Stage Feature Selection Based Intelligent Classifier for Classification of Incipient Stage Fire in Building
Published: 19 January 201610.3390/s16010031Received: 23 October 2015; Accepted: 18 December 2015;Article Academic Editor:electronic nosegas sensorsfire detectionfeature selectionfeature fusionnormalized dataPrincipal Component Analysis (PCA)Probabilistic Neural Network (PNN)
In this study, an early fire detection algorithm has been proposed based on low cost array sensing system, utilising off-the shelf gas sensors, dust particles and ambient sensors such as temperature and humidity sensor. The odour or "smellprint" emanated from various fire sources and building construction materials at early stage are measured. For this purpose, odour profile data from five common fire sources and three common building construction materials were used to develop the classification model. Normalised feature extractions of the smell print data were performed before subjected to prediction classifier. These features represent the odour signals in the time domain. The obtained features undergo the proposed multi-stage feature selection technique and lastly, further reduced by Principal Component Analysis (PCA), a dimension reduction technique. The hybrid PCA-PNN based approach has been applied on different datasets from in-house developed system and the portable electronic nose unit. Experimental classification results show that the dimension reduction process performed by PCA has improved the classification accuracy and provided high reliability, regardless of ambient temperature and humidity variation, baseline sensor drift, the different gas concentration level and exposure towards different heating temperature range.
Introduction
Fires can be categorized into two main groups: direct burning and indirect burning. Residential fires may happen indoors or outdoors [1]. Most fires start from an incipient stage and develop further to smouldering, flaming and fire stages [2]. In incipient and smouldering cases, fires have less flames and smoke, while in the flaming and fire stages, fires have more flames and radiate extreme heat.
According to the work published in the recent decade, fire research can be categorized mainly into four types; namely, fire detection, fire prediction, fire data analysis and reduction of false fire alarms [2]. Predicting or perceiving fire at the early stage is very challenging and crucial for both personal and commercial applications. Over the years, several methods have been proposed which utilise various sensing technologies to provide early fire detection [2]. The research conducted by Rose-Pehrsson is able to provide early fire detection using a Probabilistic Neural Network and achieves higher classification accuracy [3]. However, they were only able to demonstrate it as early as the smouldering stage. As for data analysis alone, various methodologies have been utilised. The most common methods used are related to clustering techniques and classification algorithms.
Several fire data analysis algorithms have been proposed. According to the research, most of these algorithms are based on time-fractal approaches to characterize the temporal distribution of detected fire sequences [4]. Some of the research has focused on utilizing unsupervised ways to detect fire from the signals [5]. In their paper, Chakraborty and Paul proposed a hybrid clustering algorithm using a modified k-means clustering algorithm. Although it required very little processing time and managed to detect the fire flames at fast speed, the proposed algorithm can be only be used in video image processing based on RGB and HSI colour models. Bahrepour et al., in their research, investigated the feasibility of spatial analysis of indoor and outdoor fires using data mining approaches for WSN-based fire detection purposes [6]. In their paper, they had investigated the most dominant feature in fire detection applications. Kohonen self-organizing map (kSOM) had been utilized as a feature reduction technique which can cluster similar data together. Experimentals result show that their method reduces the number of features representing the fire data features. They also performed analysis on residential fires and used artificial neural network, naive Bayes and decision tree classifiers to compute the best combination of sensor type in fire detectors. The outputs of various classifiers were fused using data fusion techniques to achieve higher fire detection accuracy. The reported results showed that 81% accuracy for residential fire detection and 92% accuracy for wildlife fire detection could be achieved.
Most of the proposed methods provide high classification rates in detecting fires, albeit they need to be in close vicinity to the source of the fire and only operate based on specific types of sensors [7][8][9][10][11][12][13]. Mimicking the human nose in early fire detection is still the biggest challenge for olfactory engineering. The present electronic nose systems have difficulties in detecting early fires, especially in large spaces, and cannot provide additional information regarding the burning stages and the scorching fire material. To overcome the mentioned weakness, bio-inspired approaches based on electronic nose technology is a promising method, which utilises artificial intelligence in detecting and predicting the possibility of fire occurrence. Although there are many proposed feature selection techniques and classifiers involved, the real question is whether it is possible to implement them in conventional fire detectors, yet to be determined, at a low cost. This paper focuses on investigating a multi-stage feature selection method using a bio-inspired artificial neural network and principal component analysis for data reduction, which can give the best detection accuracy, reduce misclassification and offer high reliability for indoor fire detection applications. This work is important to investigate the most suitable features and classification algorithm, which could be proved less computationally complex and having potential to be used in embedded applications.
The rest of this paper is organized as follows: Section 1 introduces the features of fires. Section 2 describes the proposed four-stage fire detection algorithm. Section 3 discusses the experimental results of the proposed method and compares the performance of the proposed method with those of other fire detection algorithms, and Section 4 presents the conclusions of our study.
Methods
In this section, the odour measurement technique, the feature extraction from sensor arrays using various data normalisation techniques, the artificial neural network-based feature selection, the feature reduction using PCA, and the classification stages are explained. Figure 1 shows the flowchart of the proposed multi-stage feature selection approach using PCA and PNN. The dashed line around PNN training on training dataset in Figure 1 indicates that the PNN training is conducted prior to the classification of fire sources. The training dataset is used by PNN in the fire sources classification process.
Datasets
In this study, two datasets have been used. The first dataset consists of odour signals which have been obtained from an in-house metal oxide gas sensor-based low cost (IAQ) system, consisting of oxygen (O2), volatile organic compound (VOC), carbon dioxide (CO2), ozone (O3), nitrogen dioxide (NO2), particulate matter up to 10 micrometres in size (PM10), temperature and humidity sensors. The prediction classifier for the early fire detection has been developed based on odours from various sample sources. The odour sources consist of five common fire sources and three common building construction materials. Information about the materials tested and their sample dimensions prepared according to the corresponding European Standard, is shown in Table 1. For each source, more than 100 odour measurement samples have been taken at seven different temperature points, starting from 50 °C up to 250 °C. About 200 ambient air measurement datapoints have been added to the dataset as a reference air sample. The ambient air samples are considered the 9th tested sample in this paper. The final IAQ system dataset is a matrix of 1000 rows and eight columns. The training set contains 600 samples (60% of the dataset), the validation set contains 100 samples (10% of the dataset), and the test set contains the remaining samples, which is 30% of the dataset. In order to estimate the true performance of the classifier, the test is based on the remaining samples which were not used during the training and validation process. The dataset has been referred as the IAQ dataset in this paper.
Datasets
In this study, two datasets have been used. The first dataset consists of odour signals which have been obtained from an in-house metal oxide gas sensor-based low cost (IAQ) system, consisting of oxygen (O 2 ), volatile organic compound (VOC), carbon dioxide (CO 2 ), ozone (O 3 ), nitrogen dioxide (NO 2 ), particulate matter up to 10 micrometres in size (PM 10 ), temperature and humidity sensors. The prediction classifier for the early fire detection has been developed based on odours from various sample sources. The odour sources consist of five common fire sources and three common building construction materials. Information about the materials tested and their sample dimensions prepared according to the corresponding European Standard, is shown in Table 1. For each source, more than 100 odour measurement samples have been taken at seven different temperature points, starting from 50˝C up to 250˝C. About 200 ambient air measurement datapoints have been added to the dataset as a reference air sample. The ambient air samples are considered the 9th tested sample in this paper. The final IAQ system dataset is a matrix of 1000 rows and eight columns. The training set contains 600 samples (60% of the dataset), the validation set contains 100 samples (10% of the dataset), and the test set contains the remaining samples, which is 30% of the dataset. In order to estimate the true performance of the classifier, the test is based on the remaining samples which were not used during the training and validation process. The dataset has been referred as the IAQ dataset in this paper. The second dataset obtained from a Portable Electronic Nose (PEN3) from Airsense Analytics GmbH (Schwerin, Germany) has been used as the control dataset. This set has 10 sensor inputs (10 columns). For each source, more than 100 samples of odour measurements have been taken at seven temperature points, starting from 50˝C up to 250˝C. Like IAQ, 200 ambient air measurement datapoints have been added to the dataset as a reference air sample. The final PEN3 dataset is a matrix of 1000 rows and 10 columns. The training set contains 600 samples (60% of the dataset), the validation set contains 100 samples (10% of the dataset), and the test set contains the remaining samples, which is 30% of the dataset, similar to the first dataset. A similar approach for performance analysis was followed for the above process as with IAQ. The dataset is referred to as PEN3 dataset in this paper.
Measurement of Odour Signals
In the IAQ dataset, the odour samples have been collected from the IAQ system placed at 2.1 m height in the testing room. The height of 2.1 m has been selected to deploy the in-house system in buildings based on few classification preliminary tests done at different heights in a standard sized room (33 m 3 in volume) in Malaysia. Heights of 0.7, 1.4 and 2.1 m have been tested in the preliminary tests. A height of 2.1 m was the most suitable and was been selected because the experimental results show that the gases generated at the incipient fire stage fill the top part of the room first since the density of the emitted gases are lesser than that of ambient air. For this experiment, the deployment of the sensor unit at this height gives the best chance in predicting an earlier fire event. Having the sensor units deployed at an inappropriate height in the building can cause it to miss useful data for fire data analysis and prediction, and thus, could trigger false fire alarms. That is also the main reason why conventional fire detectors are placed on the ceilings of buildings [14]. For realisation of a wireless sensing IAQ system, the data of the low cost system is sampled at the sampling rate of 10 sample/min [15]. The data has been recorded for 15 min each time. Each data measurement has been sent wirelessly to the server for processing and data storage using an available wireless sensor network. The data measurements have been recorded in websocket "sqlite" format and then converted to ".csv" format using a custom LabVIEW application. Afterwards, the odour signals have been translated into digital form by a custom MATLAB application.
In the PEN3 dataset, the data from PEN3 has been captured using a program supplied by AirSense Analytics GmbH. The PEN3 has been placed at 1.5 m distance from the smell source which has been heated in a vacuum oven. PEN3 has a sampling frequency of 1 sample/s. The data has been recorded for 15 min each. The data measurements have been recorded in ".nos" format and then converted to ".xls" format using a custom application. Then, the samples have been converted into digital format by a custom MATLAB application.
Normalised Feature Extraction
Baseline drift is a widespread phenomenon in signal analysis, which could also cause incorrect representation of data in subsequent feature extraction and feature selection processes of an odour signal, and baseline correction is the solution to the problem and the correct way of representing the signal when the analysis deals with sensor values from different conversion units. Baseline manipulation helps to pre-process the sensor output to free itself from the drift effect, the intensity dependence and, possibly, from non-linearity [7].
In this paper, for the feature extraction stage, five types of baseline correction algorithms have been executed on both datasets by converting the raw data value from Volts to unit ratio values. Unit ratio is a dimensionless unit. Each type of baseline correction has been considered as a feature. The ability to distinguish the fire event from the normalised data itself helps to reduce the computation complexity and classification time, thus it will be easier to implement it in the embedded system using C programming.
The first feature is Relative Logarithmic Sum Squared Voltage value (RLSSV). RLSSV is the division of logarithmic voltage by the logarithmic sum squared voltage value. The equation for calculating RLSSV is shown in Equation (1):
RLSSV " logv i logp ř v 2 q (1)
where v i is the voltage value at time i for each specific sensor. The second feature is the Relative Logarithmic Voltage value (RLV). RLV is the ratio between the logarithmic voltage and the instantaneous voltage value. It can be calculated using Equation (2):
RLV " logv i v (2)
where v i is the voltage value at time i for each specific sensor. The next feature is Relative Sum Squared Voltage value, referred to as RSSV. RSSV is obtained by dividing the instantaneous voltage value by the square root value of sum of squared voltages. Equation (3) shows the formula used in computing the RSSV:
RSSV " v i a ř v 2 (3)
where v i is the voltage value at time i for each specific sensor. The fourth feature is Relative Voltage value (RV). RV is calculated by finding the ratio of the voltage at time I and the average. It can be calculated using Equation (4):
RV " v i v o(4)
where v i is the voltage value at time i and v 0 is the baseline voltage value for each specific sensor. The final feature investigated is the Fractional Voltage Change value (FVC). FVC is directly proportional to the difference between the averaged baseline value and current value and indirectly proportional to the averaged baseline value, as shown in Equation (5):
FVC " v 0´vi v 0 (5)
where v i is the actual sensor value at time i and v 0 is the baseline value of each specific sensor.
A raw data example of the scorching smell generated by paper at 250˝C and its waveform after the RLSSV feature has been extracted are presented in Figure 2a,b, respectively.
Feature Selection
In this feature selection stage, the relative logarithmic sum squared voltage, the relative logarithmic voltage value, the relative sum squared voltage value, the relative voltage value, and the fractional voltage value, of the signal have been obtained. The selected features are chosen to investigate their performance on early fire data. The features have been tested for their reliability by examining the classification accuracy with a Probabilistic Neural Network (PNN). PNN and its function in this paper is explained further in Section 2.7. Out of the five features, the three best features with the highest classification accuracy are selected for dimensional reduction using PCA.
Feature Selection
In this feature selection stage, the relative logarithmic sum squared voltage, the relative logarithmic voltage value, the relative sum squared voltage value, the relative voltage value, and the fractional voltage value, of the signal have been obtained. The selected features are chosen to investigate their performance on early fire data. The features have been tested for their reliability by examining the classification accuracy with a Probabilistic Neural Network (PNN). PNN and its function in this paper is explained further in Section 2.7. Out of the five features, the three best features with the highest classification accuracy are selected for dimensional reduction using PCA.
Dimension Reduction Using PCA
PCA is a linear technique which transforms a dataset from its original m-dimensional form into a new and compressed n-dimensional form where n < m. Dimension reduction has been implemented to investigate its effects on classification. Since the number of observations is reduced after the dataset is dimensionally reduced, the training period of PNN classifier will be minimized [16]. Thus, PCA is helpful not only in reducing the input variables of a dataset, but it also indirectly increases the classification ability of a classifier.
PCA gives the same number of principal components as the number of input variables. For example, if the data matrix has a dimension of 100 rows and 10 columns, the data matrix could be reduced to a 100 rows and three column matrix of principal components, without removing any important information from the original dataset. The data is arranged according to the variances between the classes, starting from highest variances descending from first column up to n numbered columns. However, out of the n reduced principal components, not all the principal components are needed to represent the data. Thus, the principal components need to be tested to find the appropriate number of principal components required for feature fusion. As explained in previous studies the optimal number of principal components can be obtained using a few criteria, such as the Broken stick model, Velicer's partial correlation procedure, cross-validation, Bartlett's test for equality of eigenvalues, Kaiser's criterion, Cattell's scree test and cumulative percentage of variance [17], which basically explais how much variances we are about to retain in the data. Based on this, in this study, eight principal components have been selected to observe the effect on the classification accuracy of PNN. For each selected feature in IAQ dataset, eight principal components have been obtained from eight input variables while for PEN3 dataset, 10 principal components have been obtained from 10 input variables. The latent, proportion and cumulative percentage corresponding to the principal component value from the principal components for the relative voltage value feature in the IAQ dataset and PEN3 dataset are given in Tables 2 and 3 respectively.
Feature Fusion
In the feature fusion stage, the dimensionally reduced features have been fused to form the proposed IAQ-PCA hybrid feature for the IAQ dataset and the proposed PEN3-PCA hybrid feature for the PEN3 database. A similar approach was also reported by Luo who proposed an adaptive sensory fusion method for fire detection and isolation for intelligent building systems [18]. The proposed features have been tested and compared with the other normalised features mentioned in Section 2.3. The result of classification trials will be shown in Section 3. The feature fusion process for the IAQ-PCA hybrid features is shown in Figure 3. A similar process was also repeated for the PEN3-PCA hybrid features.
Feature Fusion
In the feature fusion stage, the dimensionally reduced features have been fused to form the proposed IAQ-PCA hybrid feature for the IAQ dataset and the proposed PEN3-PCA hybrid feature for the PEN3 database. A similar approach was also reported by Luo who proposed an adaptive sensory fusion method for fire detection and isolation for intelligent building systems [18]. The proposed features have been tested and compared with the other normalised features mentioned in Section 2.3. The result of classification trials will be shown in Section 3. The feature fusion process for the IAQ-PCA hybrid features is shown in Figure 3. A similar process was also repeated for the PEN3-PCA hybrid features.
Probabilistic Neural Network
Probabilistic Neural Network is highly regarded as a biologically inspired approach in classification as it functions similar to the human cognitive system. It requires less computational time and processing power compared to other classifiers. The human brain receives the input pattern from the nerves, compares it to the pattern in memory, and sums it together with other input patterns to find the probability that an the event will occur [3]. Thus, in this work, PNN has been selected and used as a core classifier.
PNN can be used for classifying different input patterns. It was proposed by Specht based on Bayesian classification and the probability density function using classical estimators. Compared to the conventional multi-layer perceptron (MLP) classifier which uses a sigmoidal activation function, PNN uses an exponential activation function in its algorithm. The computational time for PNN is also much less than for the MLP classifier [3]. For example, let us consider a simple two class problem:
Classifying two classes problem, class A and class B.
The estimator for the probability density function as given in Equation (6) has been used in PNN:
( ) = 1 (2 ) 1 exp − ( − ) ( − ) 2 (6)
where, XAi is the i th training pattern from class A, n is the dimension of the input vectors, mA is the number of training patterns in class A, T is the transpose of the value and σ is a smoothing parameter corresponding to the standard deviation of the Gaussian distribution. This is the standard probability density function estimator used commonly in PNN and other neural networks. There are also some works highlighting on the modification in the exponential power of Equation (6), for example, normal, log-normal, Rayleigh and Weibull probability density functions which intend to provide
Probabilistic Neural Network
Probabilistic Neural Network is highly regarded as a biologically inspired approach in classification as it functions similar to the human cognitive system. It requires less computational time and processing power compared to other classifiers. The human brain receives the input pattern from the nerves, compares it to the pattern in memory, and sums it together with other input patterns to find the probability that an the event will occur [3]. Thus, in this work, PNN has been selected and used as a core classifier.
PNN can be used for classifying different input patterns. It was proposed by Specht based on Bayesian classification and the probability density function using classical estimators. Compared to the conventional multi-layer perceptron (MLP) classifier which uses a sigmoidal activation function, PNN uses an exponential activation function in its algorithm. The computational time for PNN is also much less than for the MLP classifier [3]. For example, let us consider a simple two class problem:
Classifying two classes problem, class A and class B. The estimator for the probability density function as given in Equation (6) has been used in PNN:
f A pXq " 1 p2πq n{2 1 m A m A ÿ i"1 exp «´p X´X Ai q T pX´X Ai q 2σ 2 ff (6)
where, X Ai is the i th training pattern from class A, n is the dimension of the input vectors, m A is the number of training patterns in class A, T is the transpose of the value and σ is a smoothing parameter corresponding to the standard deviation of the Gaussian distribution. This is the standard probability density function estimator used commonly in PNN and other neural networks. There are also some works highlighting on the modification in the exponential power of Equation (6), for example, normal, log-normal, Rayleigh and Weibull probability density functions which intend to provide better estimations of unknown stochastic processes, which do not require either an a priori choice of a mathematical model or the elaboration of the data histogram, but only the computation of the variability range of each components of available data samples [19]. Similar to our biological brain, the probabilistic neural network has four operational units known as input units, pattern units, summation units and output units. When PNN is given an input, the pattern unit will calculate the distance between the input vector and the trained input vectors. A vector with the information regarding the distance between the input and the training input is produced and passed to the summation unit. The contributions for each class of input are summed by the summation unit and a net output is generated. The net output has the information of the maximum of the probabilities to indicate a 1 for the specific class or a 0 for the other class.
The steps involved in the PNN algorithm are described below:
Step 0: Initialize the weights
Step 1: For each training input to be classified, do Step 2 to 4
Step 2: Pattern units: Compute the net input to the pattern units:
Z inj " xpw j q " x T w j(7)
Compute output Equation (8) using Equation (7):
Z outj " exp " z inj´1 σ 2 (8)
Step 3: Summation unit: Sum the inputs from the pattern units to which they are connected. The summation unit for class B multiplies its total input by Equation (9):
V B "´P B C B m A P A C A m B(9)
Where: Step 4: Output (decision) unit:
P
The output unit sums the signals from f A and f B. The input vector is classified as Class A if the total input to the decision unit is positive. Based on the above example, the PNN network can classify two different classes when the input patterns of both classes are given to it. However, training the network with more sample inputs improves the ability of PNN. The degree of nonlinearity of the decision boundaries of PNN can be controlled by varying the spread factor, σ. Large values of σ make the decision boundary approach a hyperplane, while having a relatively small value approaching zero for σ gives a good approximation for highly nonlinear decision surfaces of PNN [3].
Consequently, in this paper, PNN is used to select the dominant features and to test the classification accuracy of the proposed and dominant features in distinguishing various materials involved in incipient fire cases.The PNN architecture is shown in Figure 4. The overall process flow of proposed multi-stage feature selection and fusion for both datasets is shown in Figure 5.
Results and Discussion
A Probabilistic Neural Network has been applied for classification of scorching smells generated from the different materials. In this application, both raw datasets have been subjected to the PNN classifier to select the most dominant features, prior to dimension reduction. The overall process flow of proposed multi-stage feature selection and fusion for both datasets is shown in Figure 5. The overall process flow of proposed multi-stage feature selection and fusion for both datasets is shown in Figure 5.
Results and Discussion
A Probabilistic Neural Network has been applied for classification of scorching smells generated from the different materials. In this application, both raw datasets have been subjected to the PNN classifier to select the most dominant features, prior to dimension reduction.
Results and Discussion
A Probabilistic Neural Network has been applied for classification of scorching smells generated from the different materials. In this application, both raw datasets have been subjected to the PNN classifier to select the most dominant features, prior to dimension reduction. The parameters used in PNN are shown in Table 4. As mentioned earlier in Section 2.7, the spread factor can be varied to control the degree of nonlinearity of the decision boundaries. It is the most important factor which influences the classification performance of the classifier. Therefore, the spread factor has been varied in these experiments to obtain the best classification performance [15]. The best value for spread factor for both datasets is recorded to be 0.08.
Classification performances have been computed for the nine classes for the IAQ dataset and PEN3 dataset as shown in Table 5. The classification accuracy of the each feature is clearly shown in the table. The classification result has been obtained by averaging the classification accuracy for 50 repetitions. For each dataset, the three best features with the highest classification accuracy have been selected for dimensional reduction with PCA. For the IAQ dataset, it is observed that RSSV, FVC and RV give the best accuracies, 98.90%, 98.84% and 98.81%, respectively. The PEN3 dataset, on the other hand, has RSSV, FVC and RLSSV with 99.75%, 99.51% and 99.29%, respectively, as its best features.
The three selected features have eight columns each (inputs from eight gas and electrochemical sensors). At this stage, the dimension of each feature has been reduced to remove the redundant data and to select only the optimal number of features with high variance between classes, which is sufficient to represent the fire signature. Reducing the dimensions of the original data indirectly increases the classification accuracy and reduces the processing time of the classifier. The selection of principal component values in PCA will determine how much the dimensions of the m-dimension dataset will be reduced. The performance of the classifier has been investigated by varying the principal component values and the results have been recorded in Table 6. As seen in Table 6, 6-8 principal components give the most successful classification results for the IAQ dataset, while 4-6 principal components give the most successful classification results for the PEN3 dataset. The range of classification accuracies range from a minimum of 98.13% to a maximum 99.02% for the IAQ dataset, and from a minimum of 97.49% to maximum of 100.00% for the PEN3 dataset. Out of this range, the best classification accuracies for the IAQ dataset have been observed to occur when the principal component value is seven, while, for the PEN3 dataset, the optimal principal component value has been observed to be five. Thus, the dimensions of the IAQ and PEN3 datasets have been reduced to seven and five principal components scores, respectively. The dimensionally reduced features have been fused to form the proposed IAQ-PCA hybrid feature for the IAQ dataset and the proposed PEN3-PCA hybrid feature for the PEN3 database. The fused feature for the IAQ dataset is a matrix of 1000 rows and 21 columns, while the fused feature for the PEN3 dataset is a matrix of 1000 rows and 15 columns.
The confusion matrixes of PNN of both the IAQ-PCA hybrid feature and the PEN3-PCA hybrid feature for classification trials and its respective mean classification accuracy for 50 repetitions have been tabulated in Tables 7 and 8. Both tables consist of the true positive, true negative, false positive and false negative counts, which are useful in computing performance evaluation of the PNN classifier. M1 denotes material 1, and NA denotes normal air. The performance evaluation of a classifier can be performed by examining a few statistical measures obtained by calculating the sensitivity, specificity and accuracy scores for the classifier [20]. The sensitivity is the division of the correctly selected decisions over the total decisions which are actually the deserved selections, as shown in Equation (10). The specificity (Equation (11)) indicates the division of correctly rejected decisions by the total decisions which actually deserve rejection. The accuracy is the score of correctly decided decisions over the total decisions made. The accuracy formula is shown in Equation (12):
Sensitivity " TP TP`FNˆ1 00%(10)
Specificity " TN TN`FPˆ1 00 %, and
Accuracy " TP`TN TP`TN`FP`FNˆ1 00% (12) where, the TP indicates the true positive decisions, FP is the false positive decisions, TN is the true negative decisions and FN is the false negative decisions. Based on Table 7, TP is 315, FP is 5, TN is 78 and FN is 2. Both hybrid features have been compared with the other best features selected as discussed earlier through Table 5 for both the IAQ and PEN3 datasets. Tables 9 and 10 show that the proposed IAQ-PCA and PEN3-PCA hybrid features have better performances compared to the standard normalised features. The IAQ-PCA hybrid feature recorded a highest accuracy value of 98.25%, while the PEN3-PCA hybrid feature recorded a highest accuracy of 100%. The proposed features have been compared with other common available classifiers. Feed-forward Neural Network (FFNN), Elman Neural Network (ENN) and k-Nearest Neighbour (kNN) classifiers have been selected for this purpose. The comparison results between the classifiers for the proposed PCA-based hybrid features are presented in Table 11. For FFNN and ENN, the number of hidden layers, the learning rate, the momentum factor, and the type of activation functions have been modified to obtain the best classification performance. The architectures of the classifiers have been modelled to have 21 input neurons, 45 hidden neurons and nine output neurons for the IAQ-PCA hybrid feature, and 15 input neurons, 32 hidden neurons and nine output neurons for the PEN3-PCA hybrid feature, respectively. The learning rate has been set at 0.001 and the momentum factor is 0.85 for both classifiers. In addition, the activation function, the testing tolerance and the maximum iteration have been tuned to log-sigmoid, 0.00001 and 1000, respectively. The backpropagation algorithm has been utilised for the weights training.
For the kNN classifier, the k value has been set to 3 for the IAQ-PCA feature. For the PEN3-PCA feature, the k value is set at 1. The k value in the kNN classifier is extremely training data dependent. Having cross-validation methods such as K-fold and leave-one-out are useful to find the k value which leads to the highest classification generalizability. In these paper, all the parameters involved in these classifiers have been selected based on trial and error to get the best classification accuracy. As seen on Table 11, the sensitivity, specificity and accuracy of each classifier have been tabulated for both features. From the table, it can be clearly seen that the dimensional reduction and fusion of the features to form hybrid features has deliberately increased the classification accuracy of the classifiers. The success rate of PCA-based hybrid features in the PNN classifier surpasses the performance of other common classifiers.
Conclusions
Feature selection and feature reduction have been demonstrated in detail. Both combined features from IAQ and PEN3 gives better classification accuracy. In this paper, a PCA-PNN-based feature selection technique has been proposed and investigated. The data has gone through various stages of processing such as normalised feature extraction, feature verification, binary data normalisation, PCA and data randomisation, before it is fed to the classifier. For investigation purposes, PNN has been selected as the classifier and the results have been further tested using other classifiers on the two datasets, The IAQ dataset from the in-house system and the PEN3 dataset from a commercial electronic nose system. As a result, the PEN3 dataset has better classification performance compared to the IAQ dataset for all the comparisons. This could be due to the sensitivity of the PEN3 electronic nose's gas sensors and the data capturing ability of the Winmuster software, which is used commercially. It is also observed from the analysis that the performance of the IAQ electronic nose is almost comparable to that of the PEN3 electronic nose. Thus, it is proven to be useful for early fire detection and prediction of various incipient stage scorching materials.
Figure 1 .
1A flowchart of the proposed multi-stage feature selection approach using PCA and PNN.
Figure 1 .
1A flowchart of the proposed multi-stage feature selection approach using PCA and PNN.
Figure 2 .
2(a) Example of raw data for a scorching smell generated by paper at 250 °C; (b) The RLSSV feature extracted from the scorching smell of paper at 250 °C in (a).
Figure 2 .
2(a) Example of raw data for a scorching smell generated by paper at 250˝C; (b) The RLSSV feature extracted from the scorching smell of paper at 250˝C in (a).
Sensors 2016 ,
201616, 31
Figure 3 .
3Feature Fusion Process for IAQ-PCA Hybrid Features.
Figure 3 .
3Feature Fusion Process for IAQ-PCA Hybrid Features.
A & P B are the priori probalility of occurrence of patterns in Class A and Class B, C A & C B are the cost associated with classifying vectors in Class A and B, and m A & m B are the number of training patterns in Class A and Class B.
Figure 4 .
4PNN Architecture.
Figure 5 .
5Multi-stage Feature Selection and Fusion Process Flow.
Figure 4 .
4PNN Architecture.
Figure 4 .
4PNN Architecture.
Figure 5 .
5Multi-stage Feature Selection and Fusion Process Flow.
Figure 5 .
5Multi-stage Feature Selection and Fusion Process Flow.
Table 8 .
8Confusion Matrix of PNN of proposed PEN3-PCA hybrid feature for 50 repetition.
Table 1 .
1The tested materials and its sample dimension prepared according to European Standard.Sample
Materials
Material Type
Dimension
Sample 1
Paper
Common Fire Source
16 pieces 5 cmˆ5 cm
90 gsm sheets stacked together
Sample 2
Plastic
Common Fire Source
4 cmˆ2 cmˆ40 cm (density
20 kg¨m´3) polyurethane
Sample 3
Styrofoam
Common Fire Source
4 cmˆ2 cmˆ40 cm styrofoam
Sample 4
Cotton
Common Fire Source
1 wick 18 cm long (approx. 0.17 g)
Sample 5
Cardboard
Common Fire Source
16 pieces 5 cmˆ5 cm stacked together
Sample 6
Wood
Building Construction Material
1 cmˆ1 cmˆ2 cm beech wood
Sample 7
Brick
Building Construction Material
1 piece brick
Sample 8
Gypsum board
Building Construction Material
1 cmˆ1 cmˆ2 cm gypsum board
Table 2 .
2Latent, proportion, and cumulative values of selected principal components for relative voltage value feature in the IAQ dataset.Principal Component
Latent
Proportion
Cumulative
1
0.1064
0.4813
0.4813
2
0.0474
0.2141
0.6954
3
0.0335
0.1517
0.8471
4
0.0144
0.0650
0.9121
5
0.0096
0.0435
0.9556
6
0.0073
0.0329
0.9886
7
0.0019
0.0085
0.9970
8
0.0007
0.0030
1.0000
Table 3. Latent, proportion, and cumulative values of selected principal components for relative
voltage value feature in the PEN3 dataset.
Principal Component
Latent
Proportion
Cumulative
1
7.8692
0.5338
0.5338
2
3.5164
0.2385
0.7723
3
1.8546
0.1258
0.8981
4
0.7612
0.0516
0.9497
5
0.4236
0.0287
0.9784
6
0.2476
0.0170
0.9954
7
0.0461
0.0030
0.9984
8
0.0176
0.0012
0.9996
9
0.0041
0.0003
0.9999
10
0.0015
0.0001
1.0000
Table 4 .
4PNN architectures.Parameters
Value for the IAQ Dataset
Value for PEN3 Dataset
Number of input neurons
8
10
Number of output neurons
9
9
Spread factor
0.08
0.08
Testing Tolerance
0.001
0.001
Number of training samples
600
600
Number of validation
samples
100
100
Number of testing samples
300
300
Total number of samples
1000
1000
Table 4 .
4PNN architectures.Parameters
Value for the IAQ Dataset
Value for PEN3 Dataset
Number of input neurons
8
10
Number of output neurons
9
9
Spread factor
0.08
0.08
Testing Tolerance
0.001
0.001
Number of training samples
600
600
Number of validation
samples
100
100
Number of testing samples
300
300
Total number of samples
1000
1000
Table 4 .
4PNN architectures.Parameters
Value for the IAQ Dataset
Value for PEN3 Dataset
Number of input neurons
8
10
Number of output neurons
9
9
Spread factor
0.08
0.08
Testing Tolerance
0.001
0.001
Number of training samples
600
600
Number of validation samples
100
100
Number of testing samples
300
300
Total number of samples
1000
1000
Table 5 .
5Average PNN classification accuracies of features for IAQ and PEN3 datasets.Features
IAQ
PEN3
Minimum
Classification
Accuracy (%)
Maximum
Classification
Accuracy (%)
Average
Classification
Accuracy (%)
Minimum
Classification
Accuracy (%)
Maximum
Classification
Accuracy (%)
Average
Classification
Accuracy (%)
RLSSV
97.11
99.41
98.75
97.15
99.54
99.29
RLV
97.64
98.65
98.31
97.43
99.02
98.84
RSSV
97.31
99.16
98.90
98.16
100.00
99.75
RV
97.36
99.43
98.81
98.19
99.45
99.12
FVC
97.42
99.14
98.84
98.41
99.55
99.51
Table 6 .
6Average PNN classification results in % for selecting principal component values in PCA for the IAQ and PEN3 datasets.Principal Component Value
IAQ
PEN3
RSSV
FVC
RV
RSSV
FVC
RLSSV
1
74.07
75.30
74.47
83.26
82.58
82.12
2
82.43
83.11
83.56
87.51
87.39
87.03
3
87.74
87.27
88.28
91.97
91.67
90.97
4
90.17
90.21
90.21
98.28
97.95
97.49
5
95.62
95.66
95.45
100.00
99.91
99.76
6
98.30
98.13
97.70
98.75
98.66
98.12
7
99.02
99.02
98.96
97.35
97.12
96.81
8
98.88
98.80
98.86
96.74
96.55
96.26
Table 7 .
7Confusion Matrix of PNN of proposed IAQ-PCA hybrid feature for 50 repetitions.
Table 9 .
9Average PNN classification results comparison between the best features for the IAQ dataset.Table 10. Average PNN classification results comparison between the best features for the PEN3 dataset.Feature
Sensitivity (%)
Specificity (%)
Accuracy (%)
IAQ-PCA Hybrid Feature
99.37
93.98
98.25
RSSV
99.05
91.67
97.50
FVC
98.74
91.57
97.25
RV
99.04
89.53
97.00
Feature
Sensitivity (%)
Specificity (%)
Accuracy (%)
PEN3-PCA Hybrid Feature
100.00
100.00
100.00
RSSV
99.85
96.17
99.75
FVC
99.63
96.85
99.51
RLSSV
99.51
96.09
99.29
Table 11 .
11Average classification results comparison between different classifiers for proposed PCA based hybrid features.Classifier
IAQ
PEN3
Sensitivity (%)
Specificity (%)
Accuracy (%)
Sensitivity (%)
Specificity (%)
Accuracy (%)
PNN
99.75
92.63
98.25
100.00
100.00
100.00
FFNN
98.71
91.53
97.16
99.88
95.47
99.75
ENN
98.53
91.64
97.65
99.78
94.57
99.74
kNN
99.41
91.42
97.89
99.89
95.91
99.85
Sensors 2016,16, 31
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Conflicts of Interest:The authors declare no conflict of interest.
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| []
|
[
"PARTIAL DIFFERENTIAL CHOW FORMS AND A TYPE OF PARTIAL DIFFERENTIAL CHOW VARIETIES",
"PARTIAL DIFFERENTIAL CHOW FORMS AND A TYPE OF PARTIAL DIFFERENTIAL CHOW VARIETIES"
]
| [
"Wei Li "
]
| []
| []
| We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then, based on the generic differential intersection theory, we define the partial differential Chow form for an irreducible partial differential variety V of Kolchin polynomial ω V (t) = (d + 1) t+m m − t+m−s m . And we establish for the partial differential Chow form most of the basic properties of the ordinary differential Chow form. Furthermore, we prove the existence of a type of partial differential Chow varieties. | 10.1080/00927872.2020.1737870 | [
"https://arxiv.org/pdf/1709.02358v2.pdf"
]
| 119,172,617 | 1709.02358 | 15556ae056011e569a65a0d21a1d64167543f8f6 |
PARTIAL DIFFERENTIAL CHOW FORMS AND A TYPE OF PARTIAL DIFFERENTIAL CHOW VARIETIES
Sep 2017
Wei Li
PARTIAL DIFFERENTIAL CHOW FORMS AND A TYPE OF PARTIAL DIFFERENTIAL CHOW VARIETIES
Sep 2017
We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then, based on the generic differential intersection theory, we define the partial differential Chow form for an irreducible partial differential variety V of Kolchin polynomial ω V (t) = (d + 1) t+m m − t+m−s m . And we establish for the partial differential Chow form most of the basic properties of the ordinary differential Chow form. Furthermore, we prove the existence of a type of partial differential Chow varieties.
Introduction
In their paper on Chow forms [3], Chow and van der Waerden described the motivation in these words:
It is principally important to represent geometric objects by coordinates. Once this has been done for a specific kind of objects G, then it makes sense to speak of an algebraic manifold or an algebraic system of objects G, and to apply the whole theory of algebraic manifolds. It is desirable to provide the set of objects G with the structure of an algebraic variety (eventually, after a certain compactification), thus to characterise G by algebraic equations in the coordinates.
Through developing the theory of Chow forms, they managed to represent projective algebraic varieties or algebraic cycles by Chow coordinates, thus generalised Plücker coordinates and Grassmann coordinates; and they also provided the set of algebraic cycles of fixed dimension and degree with the structure of Chow variety.
To be more specific, given an algebraic cycle V of dimension d in a projective space, its Chow form is the unique homogenous polynomial F , which states the condition when V and d+1 hyperplanes have a point in common. The coefficients of the Chow form are defined to be the Chow coordinates of V . Chow proved that the set of all algebraic cycles of fixed dimension and degree in the coordinate space is a projective variety, called the Chow variety. So Chow varieties are simply parameter spaces of algebraic cycles of fixed dimension and degree. As basic concepts of algebraic geometry, Chow forms, as well as Chow varieties, play an important role in both theoretic and algorithmic aspects of algebraic geometry and have fruitful applications in many fields, such as intersection theory, transcendental number theory and algebraic computational complexity theory [1,4,5,10,22,23,28]. Differential algebra, founded by Ritt and Kolchin, is a branch of mathematics aiming to study algebraic ordinary or partial differential equations in a similar way in which polynomial equations are studied in algebraic geometry [14,26]. The basic geometric objects of differential algebra are differential varieties. It is natural to ask how to represent differential varieties by coordinates and further provide specific sets of differential varieties with the structure of differential varieties. Also, in view of the importance of Chow forms and Chow varieties in algebraic geometry, it is desirable to develop the theory of differential Chow forms and differential Chow varieties in differential algebra and hope they play similar roles as their algebraic counterparts.
The work on differential Chow forms [9,20] could be regarded as the beginning of such a systematic development, where the theory of differential Chow forms is established for ordinary differential varieties in both affine and projective cases and the existence of differential Chow varieties is proved in very special cases. Then the existence of ordinary differential Chow varieties in general cases is finally proved with a model-theoretical proof [6]. However, the theory of partial differential Chow forms is not yet developed for partial differential varieties.
But unlike the ordinary differential case, we encounter an insuperable obstacle in the course of defining partial differential Chow forms: due to the more complicated structure of partial differential characteristic sets, it is impossible to define differential Chow forms for most of the irreducible partial differential varieties (see Example 4.2). Then comes a natural question, that is, to explore in which conditions on partial differential varieties that we can define partial differential Chow forms and provide a specific kind of partial differential varieties (after taking Kolchin closure) with a structure of partial differential varieties. This is what we will deal with in this paper. More specifically, we will give a sufficient condition for the existence of partial differential Chow forms, and for those partial differential varieties, we will define partial differential Chow forms and prove the basic properties of partial differential Chow forms similar to those of their ordinary differential counterparts. And finally, we will show a type of partial differential Chow varieties exist.
To give the definition of partial differential Chow form, we need the generic intersection theory in the partial differential case which is also interesting in itself. Intersection theory is a fundamental issue in both algebraic geometry and differential algebra. The intersection theorem is a basic result in algebraic geometry, which claims that every component of the intersection of two irreducible varieties of dimension r and s in the n-dimensional affine space has dimension greater than or equal to r + s − n. However, as pointed out by Ritt, the intersection theorem fails for differential algebraic varieties [26]. Recently, we proved a generic intersection theorem for ordinary differential varieties and generic ordinary differential hypersurfaces [9]. Freitag generalised our result to the partial differential case using more geometric and model theoretical languages [7]. In this paper, we prove the intersection theorem of differential algebraic varieties with quasi-generic differential hypersurfaces (to be defined in Definition 3.1) using pure differential algebraic arguments. In particular, when the quasi-generic partial differential hypersurface is a generic one, the proof gives more elementary and simplified proofs for generic intersection theorems either in the ordinary differential case [9,Theorem 3.6] or in the partial differential case [7,Theorem 3.7].
The rest of the paper is organised as follows. In section 2, the basic notions and preliminary results that will be used in this paper are presented. Then an intersection theory for quasi-generic partial differential polynomials will be given in section 3. In section 4, the definition of the partial differential Chow form and a sufficient condition for its existence are introduced. Basic properties of partial differential Chow form will be explored in section 5. In section 6, we show that a special type of partial differential Chow varieties exist.
Preliminaries
In this section, some basic notation and preliminary results in differential algebra will be given. For more details about differential algebra, please refer to [14].
Let F be a differential field of characteristic 0 endowed with a finite set of derivation operators ∆ = {δ 1 , . . . , δ m }, and let E be a fixed universal differential extension field of F . If m = 1, F , E are called ordinary differential fields; and if m > 1, they are called partial differential fields. Throughout the paper, unless otherwise indicated, all the differential fields (rings) we consider are partial differential fields (rings), and for simplicity, we shall use the prefix "∆-" as a synonym of "partial differential" or "partial differentially" when the derivation operators in problem are exactly {δ 1 , . . . , δ m }.
Let Θ be the free commutative semigroup (written multiplicatively) generated by δ 1 , . . . , δ m . Every element θ ∈ Θ is called a derivative operator and can be expressed uniquely in the form of a product m i=1 δ ai i with a i ∈ N. The order of θ is defined to be ord(θ) = m i=1 a i . The identity operator is of order 0. For ease of notation, we use Θ s to denote the set of all derivative operators of order equal to s and Θ ≤s denotes the set of all derivative operators of order not greater than s. For an element u ∈ U, denote u [s] = {θ(u) : θ ∈ Θ ≤s }.
A subset Σ of a ∆-extension field G of F is said to be ∆-dependent over F if the set (θα) θ∈Θ,α∈Σ is algebraically dependent over F , and is said to be ∆-independent over F , or a family of ∆-F -indeterminates in the contrary case. In the case Σ consists of one element α, we say that α is ∆-algebraic or ∆-transcendental over F respectively. The ∆-transcendence degree of G over F , denoted by ∆-tr.deg G/F , is the cardinality of any maximal subset of G which are ∆-independent over F . And the transcendence degree of G over F is denoted by tr.deg G/F .
Let F {Y} = F [Θ(Y)]
be the ∆-polynomial ring with ∆-indeterminates Y = {y 1 , . . . , y n } and coefficients in F . A ∆-monomial in Y is just a monomial in Θ(Y). A ∆-ideal in F {Y} is an ideal which is closed under the derivation operators. A prime (resp. radical) ∆-ideal is a ∆-ideal which is prime (resp. radical) as an ordinary algebraic ideal. Given S ⊂ F {Y}, we use (S) F {Y} and [S] F {Y} to denote the algebraic ideal and the ∆-ideal in F {Y} generated by S respectively.
In this paper, by a ∆-affine space A n , we mean the set E n . A ∆-variety over F is V(Σ) = {η ∈ E n : f (η) = 0, ∀f ∈ Σ} for some set Σ ⊂ F {Y}. The ∆varieties in A n defined over F are the closed sets in a topology called the Kolchin topology. Given a ∆-variety V defined over F , we denote I(V ) to be the set of all ∆-polynomials in F {Y} that vanish at every point of V . And we have a one-to-one correspondence between ∆-varieties (resp. irreducible ∆-varieties) and radical ∆ideals (resp. prime ∆-ideal), that is, for any ∆-variety V over F , V(I(V )) = V and for any radical ∆-ideal P in F {Y}, I(V(P)) = P. To distinguish from the notations in the differential case, for an algebraic ideal P ⊂ F [Y], we use V(P) to denote the algebraic variety in A n defined by P; and for an algebraic variety V ⊂ A n , we use I(V ) to denote the radical ideal in F [Y] corresponding to V . For a prime ∆-ideal P, a point η ∈ V(P) is called a generic point of P (or V(P)) if for any f ∈ F {Y}, f (η) = 0 implies f ∈ P. A ∆-ideal has a generic point if and only if it is prime.
A homomorphism ϕ from a differential ring (R, ∆) to a differential ring (S,
∆ ′ ) with ∆ ′ = {δ ′ 1 , . . . , δ ′ m } is a differential homomorphism if ϕ • δ i = δ ′ i • ϕ (∀i)
. Suppose ∆ ′ = ∆ and R 0 is a common ∆-subring of R and S, ϕ is said to be a ∆-R 0 -homomorphism if ϕ leaves every element of R 0 invariant. If, in addition R is a domain and S is a ∆-field, ϕ is called a ∆-specialization of R into S. For ∆specializations, we have the following lemma which generalizes the similar results both in the ordinary differential case ([9, Theorem 2.16]) and in the algebraic case ( [12, p.168-169] and [9, Lemma 2.13]). Lemma 2.1. Let P i ∈ F {U, Y} (i = 1, . . . , m) be ∆-polynomials in the independent ∆-indeterminates U = (u 1 , . . . , u r ) and Y. Let η be an n-tuple taken from some extension field of F free from F U 1 . If P i (U, η) (i = 1, . . . , m) are ∆-dependent over F U , then for any ∆-specialization U to U ∈ F r , P i (U, η) (i = 1, . . . , m) are ∆-dependent over F . A ranking on F {Y} is a total order on Θ(Y) = {θy j : j = 1, . . . , n; θ ∈ Θ} which is compatible with the derivation operators: 1) for any θy j ∈ Θ(Y) and δ i , δ i θy j > θy j and 2) θ 1 y i > θ 2 y j =⇒ δ k θ 1 y i > δ k θ 2 y j for θ 1 y i , θ 2 y j ∈ Θ(Y). By convention, 1 < θy j for all θy j ∈ Θ(Y). Two important kinds of rankings are often used: 1) Elimination ranking: y i > y j =⇒ θ 1 y i > θ 2 y j for any θ 1 , θ 2 ∈ Θ.
Proof: Assume k = max i ord(P i ). Since P i (U, η) (i = 1, . . . , m) are ∆-dependent over F U , there exists s ∈ N such that the P i (U, η) [s] are algebraically dependent over F (U [s+k] ). When U ∆-specializes to U ∈ F r , U [
2) Orderly ranking: k > l =⇒ for any θ 1 ∈ Θ k , θ 2 ∈ Θ l and i, j ∈ {1, . . . , n}, we have θ 1 y i > θ 2 y j .
Let f be a ∆-polynomial in F {Y} and R a ranking endowed on it. The greatest derivative θy j w.r.t. R which appears effectively in f is called the leader of f , denoted by ld(f ). Let d be the degree of f in ld(f ). The rank of f is ld(f ) d , denoted by rk(f ). The coefficient of rk(f ) in f is called the initial of f and denoted by I f . The partial derivative of f w.r.t. ld(f ) is called the separant of f , denoted by S f . For any two ∆-polynomials f , g in F {Y}\F , f is said to be of lower rank than g if either ld(f ) < ld(g) or ld(f ) = ld(g) and deg(f, ld(f )) < deg(g, ld(f )). By convention, any element of F is of lower rank than elements of F {Y}\F . We denote f g if and only if either f is of lower rank than g or they have the same rank. Clearly, is a totally ordering of F {Y}.
Let f and g be two ∆-polynomials and rk(f ) = θ(y j ) d . g is said to be reduced w.r.t. f if no proper derivatives of θ(y j ) appear in g and deg(g, θ(y j )) < d. Let A be a set of ∆-polynomials. A is said to be an autoreduced set if each ∆-polynomial of A is reduced w.r.t. any other element of A. Every autoreduced set is finite.
Let A be an autoreduced set. We denote H A to be the set of all the initials and separants of A and H ∞ A to be the minimal multiplicative set containing H A . The ∆-saturation ideal of A is defined to be
sat(A) = [A] : H ∞ A = {p ∈ F {Y} ∃h ∈ H ∞ A , s.t. hp ∈ [A]}.
The algebraic saturation ideal of A is denoted by asat(A) = (A) : H ∞ A . Let A =< A 1 , A 2 , . . . , A s > and B =< B 1 , B 2 , . . . , B l > be two autoreduced sets with the A i , B j arranged in nondecreasing ordering. A is said to be of lower rank than B, if either 1) there is some k (≤ min{s, l}) such that for each i < k, A i has the same rank as B i , and A k ≺ B k or 2) s > l and for each i ∈ {1, 2, . . . , l}, A i has the same rank as B i . It is easy to see that the above definition introduces really a partial ordering among all autoreduced sets. Any sequence of autoreduced sets steadily decreasing in ordering A 1 ≻ A 2 ≻ · · · A k ≻ · · · is necessarily finite.
Let A =< A 1 , A 2 , . . . , A t > be an autoreduced set with S i and I i as the separant and initial of A i , and F any ∆-polynomial. Then there exists an algorithm, called Ritt's algorithm of reduction, which reduces F w.r.t. A to a ∆-polynomial R that is reduced w.r.t. A, satisfying the relation
t i=1 S di i I ei i · F ≡ R, mod [A],
for d i , e i ∈ N (i = 1, 2, . . . , t). We call R the remainder of P w.r.t. A. We will need the following result in Section 3. If F 1 , . . . , F l ∈ F {Y}, then there exist ∆-polynomials E 1 , . . . , E l ∈ F {Y}, reduced with respect to A and of rank no higher than the highest of the ranks of F 1 , . . . , F l , and there exist natural numbers j A , t A (A ∈ A), such that
A∈A S jA A I tA A · F j ≡ E j , mod [A] (1 ≤ j ≤ l).
Let J be a ∆-ideal in F {Y}. An autoreduced set C ⊂ J is said to be a characteristic set of J , if J does not contain any nonzero element reduced w.r.t. C. All the characteristic sets of J have the same and minimal rank among all autoreduced sets contained in J . If J is prime, C reduces to zero only the elements of J and we have J = sat(C). An autoreduced set C is called coherent if whenever A, A ′ ∈ C with ld(A) = θ 1 (y j ) and ld(A ′ ) = θ 2 (y j ), the remainder of
S A ′ θ θ1 (A) − S A θ θ2 (A ′ ) w.r.t. C is zero, where θ = lcm(θ 1 , θ 2 ). (Here, if θ j = m i=1 δ aji i (j = 1, 2) and max(a 1i , a 2i ) = c i , then θ = m i=1 δ ci i and θ θj = m i=1 δ ci−aji i
.) The following result gives a criterion for an autoreduced set to be a characteristic set of a prime ∆-ideal.
2.2.
Kolchin polynomials of prime differential ideals. Let P be a prime ∆ideal in F {Y} with a generic point η ∈ A n . The ∆-dimension of P, denoted by ∆-dim(P), is defined as the ∆-transcendence degree of F η over F . Let A be a characteristic set of P w.r.t. some ranking. We use ld(A) to denote the set {ld(F ) : F ∈ A}. Call y j a leading variable of A if there exists some θ ∈ Θ such that θ(y j ) ∈ ld(A); otherwise, y j is called a parametric variable of A. The ∆-dimension of P is equal to the cardinality of the set of parametric variables of A.
For a prime ∆-ideal, its Kolchin polynomial contains more quantitative information than the ∆-dimension. To recall the concept of Kolchin polynomial, we need an important numerical polynomial associated to a subset E ⊆ N m .
Lemma 2.4. [15,16] For every set E = {(e i1 , . . . , e in ) : i = 1, . . . , l} ⊆ N m (m ≥ 1), let V E (t) denote the set of all elements v ∈ N m such that v is not greater or equal to any element in E relative the the product order on N m . Then there exists a univariate numerical polynomial ω E (t) such that ω E (t) = card(V E (t)) for all sufficiently large t. Moreover, ω E (t) satisfies the following statements:
1) deg(ω E ) ≤ m, and deg(ω E ) = m if and only if E = ∅. And if E = ∅, ω E (t) = t+m m ; 2) ω E (t) ≡ 0 if and only if (0, . . . , 0) ∈ E; 3) If min l i=1 e ik = 0 for each k, then deg(ω E (t)) < m − 1. Theorem 2.5. [15, Theorem 2] Let P be a prime ∆-ideal in F {y 1 , . . . , y n }.
There exists a numerical polynomial ω P (t) with the following properties:
1) For sufficiently large t ∈ N, ω P (t) equals the dimension of P ∩ F [Y [t] ]. 2) deg(ω P ) ≤ m = card(∆). 3) If we write ω P (t) = m i=0 a i t+i i
where a i ∈ Z, then a m equals the ∆dimension of P. 4) If A is a differential characteristic set of P with respect to an orderly ranking on F {y 1 , . . . , y n } and if E j denotes for any y j the set of points (l 1 , . . . , l m ) ∈ N m such that δ l1 1 · · · δ lm m y j is the leader of an element of A, then ω P (t) = n j=1 ω Ej (t). The numerical polynomial ω P (t) is defined to be the Kolchin polynomial of P. Prime ∆-ideals whose characteristic sets consist of a single polynomial are of particular interest to us. Lemma 2.6. Let P be a prime ∆-ideal in F {y 1 , . . . , y n }. Suppose A ∈ F {y 1 , . . . , y n } constitutes a characteristic set of P under some orderly ranking R. Then {A} is also a characteristic set of P under an arbitrary ranking. In this case, we call P the general component of A.
Proof: Suppose S A is the separant of A under R. Then P = [A] : S ∞ A . Let R ′ be an arbitrary ranking and θ(y k ) be the leader of A under R ′ . It suffices to show that there is no nonzero ∆-polynomial in P which is reduced with respect to A under the ranking R ′ . Suppose the contrary and let f ∈ P\{0} be a ∆-polynomial reduced with respect to A under R ′ . Then f is free from the proper derivatives of θ(y k ). Since f ∈ P, there exist l ∈ N and finitely many nonzero polynomials T τ for τ ∈ Θ such that S l
A f = τ T τ τ (A). For each τ = 1, τ (A) = S ′ A · τ θ(y k ) + L τ where S ′ A is the separant of A under R ′ .
Substitute τ θ(y k ) = −L τ /S ′ A for τ > 1 into both sides of the above identity and remove the denominators, then we get
S l A S l ′ A ′ f = T 1 A. Thus, A divides f which implies f = 0.
The contradiction shows that A is a also a characteristic set of P under any ranking.
Kolchin gave a criterion for a prime ∆-ideal to be the general component of some ∆-polynomial.
Lemma 2.7. [14, p. 160, Proposition 4] Let P be a prime ∆-ideal in F {y 1 , . . . , y n }.
Then a necessary and sufficient condition that P is the general component of some polynomial A of order s is that the Kolchin polynomial of P is of the form
ω P (t) = n t + m m − t + m − s m .
The following result on algebraic ideals will be used later.
Lemma 2.8. Let P be a prime ideal in the polynomial ring F [x 1 , . . . , x n ] of di- mension d > 0. Assume P ∩ F [x 1 ] = {0}. Then J = (P) F (x1)[x2,...,xn] is a prime ideal of dimension d − 1. Proof: Since P ∩ F [x 1 ] = {0}, J = F (x 1 )[x 2 , . . . , x n ]. If f 1 , f 2 ∈ F (x 1 )[x 2 , . . . , x n ] and f 1 f 2 ∈ J , then there exist M 1 , M 2 ∈ F [x 1 ] such that M i f i ∈ F [x 1 , . . . , x n ] and M 1 f 1 M 2 f 2 ∈ P. So either M 1 f 1 ∈ P or M 2 f 2 ∈ P, which implies that either f 1 ∈ J or f 2 ∈ J . Thus, J is a prime ideal. Since dim(P) = d and P ∩ F [x 1 ] = {0}, without loss of generality, we suppose {x 1 , x 2 , . . . , x d } is a parametric set of P. We claim that {x 2 , . . . , x d } is a parametric set of J , so dim(J ) = d − 1 follows. First, note that J ∩ F (x 1 )[x 2 , . . . , x d ] = {0}. For any other variable x k ∈ {x d+1 , . . . , x n }, P ∩ F [x 1 , x 2 , . . . , x d , x k ] = {0}, so J ∩ F (x 1 )[x 2 , . . . , x d , x k ] = {0}. Thus, {x 2 , . . . , x d } is a parametric set of J .
3. Quasi-generic intersection theory in partial differential algebra
In this section, we will prove the quasi-generic intersection theorem with an elementary proof using pure differential algebraic languages, which generalises generic intersection theorems in both ordinary and partial differential cases [9,7]. We should remark that the proof in the ordinary differential case could not be adapted here because of the complicated structure of differential characteristic sets in the partial differential case. However, the proof here we give could definitely simplify that of its ordinary differential analog.
Definition 3.1. A generic ∆-polynomial of order s and degree g is a ∆-polynomial which involves all ∆-monomials of order s and degree g with coefficients being ∆-F -indeterminates. To be more precise, a generic ∆-polynomial L of order s and degree g is of the following form
L = M∈Ms,g u M M,
where M s,g is the set of all ∆-monomials of order bounded by s and degree bounded by g and all the coefficients u M are ∆-F -indeterminates. The ∆-zero set of a generic ∆-polynomial is called a generic ∆-hypersurface. And a generic ∆-hyperplane is defined to be the ∆-zero set of a generic ∆-polynomial of the form u 0 + n j=1 u j y j . A quasi-generic ∆-polynomial of order s is a ∆-polynomial L of the form
L = M∈M L u M M,
where the coefficients u M are ∆-F -indeterminates and its support M L of ∆-monomials appearing in L satisfies the following conditions:
• 1 ∈ M L ; • for each j = 1, . . . , n, there exists some ∆-monomial M j (y j ) ∈ M L ∩ F {y j } with ord(M j (y j )) = s.
Now, we give the main quasi-generic intersection theorem in partial differential algebra, which generalises the generic intersection theorem in the ordinary case [9].
Theorem 3.2. Let V ⊂ A n be an irreducible ∆-variety over F . Let L be a quasigeneric ∆-polynomial of order s with the set of its coefficients u. Then
1) over F u , V ∩ V(L) = ∅ if and only if ∆-dim(V ) > 0. 2) if ∆-dim(V ) > 0, then the intersection of V and V(L) is an irreducible ∆-variety over F u and its Kolchin dimension polynomial is ω V ∩V(L) (t) = ω V (t) − t + m − s m . In particular, the ∆-dimension of V ∩ V(L) is equal to △-dim(V ) − 1. Proof: Let P = I(V ) ⊂ F {Y} be the prime ∆-ideal corresponding to V and η = (η 1 , . . . , η n ) be a generic point of P which is free from u (i.e., the u are ∆-F η - indeterminates). Let L = u 0 + n j=1 u j M j + Mα∈M L \{1,M1,...,Mn} u α M α
where each M j is a ∆-monomial in y j of order s, whose existence is guaranteed by the definition of quasi-generic ∆-polynomials. Let T = L− u 0 and set ζ 0 = −T| Y=η .
1) Let J 0 = [P, L] F1{Y,u0} , where F 1 = F u\{u 0 } . Then it is easy to show that (η, ζ 0 ) is a generic point of J 0 , so J 0 is a prime ∆-ideal. Let J = [P, L] F u {Y} . Clearly, J = [J 0 ] F u {Y} and J ∩ F 1 {Y, u 0 } = J 0 , so J = [1] if and only if J 0 ∩ F 1 {u 0 } = {0}, or equivalently, ζ 0 is ∆-transcendental over F 1 . We show that J = [1] (i.e., V ∩ V(L) = ∅) if and only if ∆-dim(V ) = 0.
If ∆-dim(V ) = 0, then for each j = 1, . . . , n, η j is ∆-algebraic over F . So
F 1 η is ∆-algebraic over F 1 . Since ζ 0 ∈ F 1 η , ζ 0 is ∆-algebraic over F 1 and J 0 ∩ F 1 {u 0 } = [0]
, which implies J = [1]. For the other direction, suppose J = [1], i.e., ζ 0 is ∆-algebraic over F 1 . For each j, by differentially specializing u j to 1 and all the other elements in u\{u 0 u j } to 0, by Lemma 2.1, M j (η j ), as well as η j , is
∆-algebraic over F . So ∆-dim(V ) = 0. Thus, J = [1] if and only if ∆-dim(V ) > 0. 2) Assume ∆-dim(V ) > 0. We will show that ω J (t) = ω V (t) − t+m−s m . For sufficiently large t, let I t = P ∩ F [Y [t] ], L [t−s] F1[Y [t] ,u [t−s] 0 ] . We claim that i) I t ∩ F 1 [u [t−s] 0 ] = {0}; ii) J ∩ F u [Y [t] ] = (I t ) F u [Y [t] ] .
If i) and ii) are valid, then by Lemma 2.8, we have
ω V ∩V(L) (t) = dim(J ∩F u [Y [t] ]) = dim (I t F1(u [t−s] 0 )[Y [t] ] ) = ω P (t)− t + m − s m .
So it remains to show the validity of claims i) and ii).
First note that (η [t] , ζ [t−s] 0 ) is a generic point of I t . Claim i) is equivalent to say that the ζ [t−s] 0
are algebraically independent over F 1 . This is indeed valid, for ζ 0 is ∆-transcendental over F 1 by 1).
For claim ii), it suffices to show that for each f ∈ J ∩F u [Y [t] ], f can be written as a linear combination of polynomials in
P ∩ F [Y [t] ] and L [t−s] with coefficients in F u [Y [t] ]. Let f ∈ J ∩ F u [Y [t] ]. Multiplying f by some nonzero polynomial in F 1 {u 0 } when necessary, we can assume f ∈ F 1 [Y [t] , u [t−s+k] 0 ] for some k ∈ N. So, f ∈ J 0 and f (η [t] , ζ [t−s+k] 0 ) = 0 follows. Let Z = ∪ k i=1 Θ t−s+i . Rewrite f as a polynomial in θ(u 0 ) θ∈Z with coefficients in F 1 [Y [t] , u [t−s] 0
], and suppose
f = α g α M α where g α ∈ F 1 [Y [t] , u [t−s] 0
] and the M α are finitely many distinct monomials in the variables θ
(u 0 ) θ∈Z . So f (η [t] , ζ [t−s+k] 0 ) = 0 implies that α g α (η [t] , ζ [t−s] 0 )M α (θ(ζ 0 )) θ∈Z = 0. If we can show that θ(ζ 0 ) θ∈Z are algebraically independent over F 1 (η [t] ), then obviously, g α (η [t] , ζ [t−s] 0 ) = 0 for each α and g α ∈ I t which implies that f ∈ (I t ) F u [Y [t] ] .
So it suffices to show that θ(ζ 0 ) θ∈Z are algebraically independent over F 1 (η [t] ). Suppose the contrary, then θ(ζ 0 ) θ∈Z are algebraically dependent over F 1 (η [t] ). Let A be a ∆-characteristic set of P with respect to some orderly ranking. Since ∆-dim(V ) > 0, there exists at least one j 0 such that y j0 is a parametric variable of A. By algebraically specializing u j0 to 1 and all the other derivatives in Θ ≤t−s+k (u\{u 0 }) to 0, and by the algebraic version of Lemma 2.1, θ(M j0 (η j0 )) θ∈Z are algebraically dependent over F (η [t] ). By multiplying some D(η [t] ) ∈ F [η [t] ] when necessary, we get a nonzero polynomial
G(Y) = l g l (Y [t] )T l (M j0 (y j0 )) vanishing at η, where the T l (M j0 (y j0 )) are distinct monomials in θ(M j0 (y j0 )) θ∈Z and for each l, g l (η [t] ) = 0. By Proposition 2.2, there exist h l ∈ F [Y [t] ]
, reduced with respect to A, and natural numbers
j A , k A (A ∈ A) such that A∈A I jA A S kA A · g l ≡ h l mod [A]
, for all l's.
Thus, H(Y) = l T l (M j0 (y j0 ))h l (Y [t]
) is a nonzero polynomial which is reduced with respect to A and satisfies H(η) = 0, a contradiction. Thus, θ(ζ 0 ) θ∈Z are algebraically independent over F 1 (η [t] ) and claim 2) is valid. Consequently, we have
proved that ω V ∩L (t) = ω V (t) − t+m−s m .
Remark 3.3. By the proof of Theorem 3.2, once we know a variable y i0 which is a parametric variable of a characteristic set of I(V ) under some orderly ranking, for those L whose support contains 1 and a ∆-monomial in y i0 of order s with coefficients ∆-F -indeterminates, we could still get
ω V ∩L (t) = ω V (t) − t+m−s m .
When L is a generic ∆-polynomial, as a corollary, we get the partial differential analog of [9, Theorem 1.1], which was proven by Freitag [7] with a model-theoretical proof.
Corollary 3.4. Let V be an irreducible ∆-variety over F with ω V (t) > t+m m .
Let L be a generic ∆-polynomial of order s and degree g with coefficient set u. Then the intersection of V and L = 0 is a nonempty irreducible ∆-variety over F u and its Kolchin polynomial is
ω V ∩L (t) = ω V (t) − t + m − s m .
The following result gives the information of the intersection of several quasigeneric ∆-polynomials.
Corollary 3.5. Let L i (i = 1, . . . , r; r ≤ n) be independent quasi-generic ∆- polynomials of order s i respectively. Suppose u i is the set of coefficients of L i . Then [L 1 , . . . , L n ] F u1,...,ur {Y} is a prime ∆-ideal with its Kolchin polynomial equal to ω(t) = n i=1 t + m m − t + m − s i m .
In particular, its ∆-dimension is 0, the differential type is m − 1 and the typical ∆-dimension is n i=1 s i .
Partial Differential Chow forms
In this section, we will introduce the definition of partial ∆-Chow forms and show for a specific kind of ∆-varieties, their ∆-Chow forms exist.
Let V ⊂ A n be an irreducible ∆-variety over F with ∆-dimension d. Let
L i = u i0 + u i1 y 1 + · · · + u in y n (i = 0, 1, . . . , d) be d+1 independent generic ∆-hyperplanes with coefficient vector u i = (u i0 , u i1 , . . . , u in ). Let J = [I(V ), L 0 , . . . , L d ] F {Y,u0,...,u d } . Lemma 4.1. J ∩ F {u 0 , . . . , u d } is a prime ∆-ideal of codimension 1.
Proof: Let η = (η 1 , . . . , η n ) be a generic point of V free from each u i and let
ζ i = − n k=1 u ik η k (i = 0, . . . , d). Denote ζ = (ζ 0 , u 01 , . . . , u 0n , . . . , ζ d , u d1 , . . . , u dn ) and u = ∪ d i=0 u i \{u i0 }. It is easy to show that (η, ζ) is a generic point of J , so J is a prime ∆-ideal. Thus, J ∩F {u 0 , . . . , u d } is a prime ∆-ideal with a generic point ζ. Since the ∆-dimension of P is d, by Lemma 2.1, any d of the ζ i are ∆-independent over F u . Note that F ζ ⊂ F u, η . So ∆-tr.degF ζ /F = (d + 1)n + d, i.e., the codimension of J ∩ F {u 0 , . . . , u d } is 1.
In the ordinary differential case, there always exists a unique irreducible δpolynomial such that J ∩ F {u 0 , . . . , u d } is the general component of this polynomial. This unique polynomial is defined to be the δ-Chow form of V . However, unlike the ordinary differential case, for a prime ∆-ideal of codimension 1, it may not be the general component of any single ∆-polynomial, as Example 4.2 shows. The above fact makes it impossible to define ∆-Chow forms for all the irreducible ∆-varieties. Below, we define ∆-Chow forms for irreducible ∆-varieties satisfying certain properties.
Definition 4.3. If J ∩ F {u 0 , . . . , u d } is the general component of some irreducible ∆-polynomial, that is, there exists an irreducible ∆-polynomial F (u 0 , . . . , u d ) such that J ∩ F {u 0 , . . . , u d } = sat(F ),
then we say the ∆-Chow form of V exists and we call F the ∆-Chow form of V or its corresponding prime ∆-ideal I(V ).
Following this definition, a natural question is to explore in which conditions on ∆-varieties such that their ∆-Chow forms exist. Now, we proceed to give a sufficient condition for the existence of ∆-Chow forms. Proof: For each j = 1, . . . , n, let E j denote the matrix whose row vectors are (a 1 , . . . , a m ) ∈ N m such that δ a1 1 · · · δ am m y j is the leader of an element of A. Here, if y j is not a leading variable, then set E j = ∅. Suppose the leading variables of A are y i1 , . . . , y i l . By Theorem 2.5,
ω P (t) = n j=1 ω Ej (t) = (n − l) t+m m + l j=1 ω Ei j = (d+1) t+m m − t+m−s m
. Since E ij = ∅, the degree of ω Ei j is less than m. Comparing the coefficient of t m of the both sides of the above equality, we get l = n − d.
For j = 1, . . . , n − d, let e ij = (e ij 1 , . . . , e ij n ) ∈ N m be a vector constructed from E ij with each e ij k the minimal element of the k-th column of E ij , and let H ij be the matrix whose row vectors are the corresponding row vectors of E ij minus e ij respectively. Denote s ij = n k=1 e ij k . Then clearly, ω Ei j (t) = ω ei j (t)+ ω Hi j (t− s ij ). By item 3) of Lemma 2.4, the degree of ω Hi j (t−s ij ) is strictly less than m−1. Thus,
ω P (t) = (d + 1) t+m m − t+m−s m = d t+m m + n−d j=1 ω ei j (t) + n−d j=1 ω Hi j (t − s ij ). So= s (m−1)! t m−1 + s(m+1)−s 2 2·(m−2)! t m−2 + o(t 3 ), we get s = n−d j=1 s ij , −s 2 /2 = − n−d j=1 s 2 ij /2 + (m − 2)! · n−d j=1 coeff ω Hi j , t m−2 .
If two of the s ij are nonzero, then obviously −s 2 /2 < − On the one hand, since ζ
[t] 0 ⊂ F (u [t] 01 , . . . , u [t] 0n , η [t] ), we have ω (ζ0,u01,...,u0n) (t) ≤ ω (u01,...,u0n,η) (t) = (n + 1) t + m m − t + m − s m .
On the other hand, by Lemma 4.
≥ n t + m m + t + m m − t + m − s m .
Thus, ω (ζ0,u01,.. We conjecture that for the existence of ∆-
Chow form of V , ω V (t) = (d+1) t+m m − t+m−s m
is also a necessary condition:
Conjecture 4.6. Let V ⊂ A n be an irreducible ∆-variety over F of differential dimension d. Then a necessary and sufficient condition such that the ∆-Chow form of V exists is that the Kolchin polynomial of V is
ω V (t) = (d + 1) t + m m − t + m − s m for some s ∈ N.
In the remaining sections of the paper, we focus on irreducible ∆-varieties of Kolchin polynomial ω(t) = (d + 1) t+m m − t+m−s m whose ∆-Chow forms exist guaranteed by Theorem 4.5. The following result is an easy fact, which could be used to compute ∆-Chow forms by pure algebraic computations. = t + 1. The ∆-Chow form of P is F (u 0 ) = δ 1 (u 00 ) 2 u 2 02 −2δ 1 (u 00 )u 00 δ 1 (u 02 )u 02 −δ 1 (u 00 )δ 1 (u 01 )u 01 u 02 +u 2 00 (δ 1 (u 02 )) 2 + δ 1 (u 00 )u 2 01 δ 1 (u 02 )+ u 00 (δ 1 (u 01 )) 2 u 02 − u 00 δ 1 (u 01 )u 01 δ 1 (u 02 )+ δ 1 (u 00 )δ 1 (u 01 )u 01 u 02 .
Properties of the partial differential Chow form
In this section, we will prove basic properties of ∆-Chow forms. In particular, we will show the ∆-Chow forms are ∆-homogenous and prove the ∆-Chow form has a Poisson-type product formula similar to its ordinary differential counterpart. 5.1. Partial differential Chow forms are differentially homogenous. In this section, we will show that the ∆-Chow form is also ∆-homogenous. Recall that F is a ∆-field with the set of derivations ∆ = {δ 1 , . . . , δ m } and set of derivative operators Θ. Given two derivatives θ 1 = m i=1 δ ai i and θ 2 = m i=1 δ bi i ∈ Θ, if a i ≤ b i for each i, then we denote θ 1 |θ 2 . In case θ 1 |θ 2 , we denote θ2 θ1 = m i=1 δ bi−ai i , and denote the product of binomial coefficients m i=1 bi ai by θ2 θ1 . It is easy to verify that θ(f g) = τ |θ θ τ · θ τ (f ) · τ (g) for all f, g ∈ F . Definition 5.1. A ∆-polynomial f ∈ F {y 0 , y 1 , . . . , y n } is said to be of ∆-homogenous of degree r if f (λy 0 , λy 1 , . . . , λy n ) = λ r f (y 0 , y 1 , . . . , y n ) holds for a ∆-indeterminate λ over F {y 0 , y 1 , . . . , y n }.
The following lemma is a partial differential analog of the Euler's criterion on homogenous polynomials, which was listed as an exercise in [14, p.71].
Lemma 5.2. A necessary and sufficient condition that f ∈ F {y 0 , y 1 , . . . , y n } be ∆-homogenous of degree r is that f satisfies the following system of equations:
(5.1) τ ∈Θ n j=0 τ θ θ τ (y j ) · ∂f ∂τ θ(y j ) = rf, θ = 1 0, θ ∈ Θ, θ = 1.
Proof: Denote Y = (y 0 , . . . , y n ) temporarily for convenience. Let λ be a ∆indeterminate over F {Y}. First we show the necessity. Since f is ∆-homogenous of degree r, then f (λY) = λ r f (Y). Differentiating both sides of this equality w.r.t. θ(λ), we get
n j=0 τ ∈Θ ∂τ θ(λy j ) θ(λ) ∂f ∂τ θ(y j ) (λY) = τ ∈Θ n j=0 τ θ θ τ (y j ) · ∂f ∂τ θ(y j ) (λY) = ∂f (λY) ∂θ(λ) = rf (Y)λ r−1 , θ = 1 0, θ ∈ Θ, θ = 1.
Setting λ = 1, we get (5.1).
For the sufficiency, we first choose an orderly ranking R. Obviously, ∂ ∂τ (λ) f (λY) = 0 for all τ ∈ Θ and ord(τ ) > ord(f ). Suppose θ ∈ Θ satisfies that for all τ ∈ Θ, if θ|τ and τ = θ, then ∂ ∂τ (λ) f (λY) = 0. Then
λ · ∂ ∂θ(λ) f (λY) = τ ∈Θ τ θ θ τ (λ) ∂ ∂τ θ(λ) f (λY) = τ ∈Θ τ θ θ τ (λ) n j=0 ξ∈Θ ξτ θ τ θ ξ(y j ) ∂ ∂ξτ θ(y j ) f (λY) = n j=0 τ ∈Θ ξ∈Θ,ξ|τ τ ξ θ θ τ ξ (λ) τ θ τ ξ θ ξ(y j ) ∂ ∂τ θ(y j ) f (λY) = n j=0 τ ∈Θ τ θ θ ξ∈Θ,ξ|τ τ ξ τ ξ (λ)ξ(y j ) ∂ ∂τ θ(y j ) f (λY) = n j=0 τ ∈Θ τ θ θ τ (λy j ) ∂ ∂τ θ(y j ) f (λY) = τ ∈Θ n j=0
τ θ θ τ (y j ) · ∂f ∂τ θ(y j ) Y=λY By (5.1), if θ = 1, then ∂ ∂θ(λ) f (λY) = 0, so f (λY) is free from θ(λ) for all 1 = θ ∈ Θ; and if θ = 1, we get λ · ∂ ∂λ f (λY) = rf (λY), so ∂λ −r f (λY)
∂λ = −rλ −r−1 f (λY) + λ −r ∂f (λY) ∂λ = 0 and f (λY) = λ r f (Y) follows. Thus, f (Y) is ∆-homogenous of degree r.
Now, we show the ∆-homogeneity of ∆-Chow forms. Proof: By the definition of ∆-Chow form, F (u 0 , . . . , u d ) has the symmetric property in the sense that interchanging u i and u j in F , the resulting polynomial and F differ at most by a sign. In particular, F is of the same degree in each u i . So it suffices to show the ∆-homogeneity of F for u 0 .
Let η = (η 1 , . . . , η n ) be a generic point of V and ζ i = − n j=1 u ij η j . From the definition of the ∆-Chow form, F (ζ 0 , u 01 , . . . , u 0n ; . . . ; ζ d , u d1 , . . . , u dn ) = 0. For each j = 1, . . . , n and θ ∈ Θ with ord(θ) ≤ s = ord(F ), take the partial derivatives of both sides of this equality with respect to θ(u 0j ), then we get
(5.2) ∂F ∂θ(u 0j ) − τ ∈Θ ∂F ∂τ θ(u 00 ) τ θ θ τ (η j ) = 0, (j = 1, . . . , n)
where ∂F ∂θ(u0j ) is obtained by substituting ζ i for u i0 (i = 0, . . . , d) in ∂F ∂θ(u0j ) . Fix a θ 1 ∈ Θ. For each θ ∈ Θ with θ 1 |θ, multiply (5.2) by θ (ζ 0 ) ∂F ∂θ(u 00 ) .
So (5.3) is reduced to
n j=0 θ1|θ θ θ1 θ θ1 (u 0j ) ∂F ∂θ(u0j ) = 0. Thus, the polynomial G θ1 = n j=0 θ1|θ θ θ1 θ θ1 (u 0j ) ∂F ∂θ(u0j ) vanishes at (ζ 0 , u 01 , . . . , u 0n ; . . . ; ζ d , u d1 , . . . , u dn ), which implies that G θ1 ∈ sat(F ). Since ord(G θ1 ) ≤ ord(F ) and deg(G θ1 ) = deg(F ), G θ1 = r ·F for some r ∈ F . For a fixed orderly ranking R on u 0 , we consider the lex monomial ordering induced by R. When θ 1 = 1, note that the leading monomial of F will definitely not appear in G θ1 , so G θ1 must be a zero polynomial. And when θ 1 = 1, G 1 and F can only differ by a nonnegative integer, so r ∈ N. Thus, by Lemma 5.2, F is differentially homogenous in u 0 of degree r.
Definition 5.4. The number r in Theorem 5.3 is defined to be the ∆-degree of the ∆-variety V or its corresponding prime ∆-ideal. 5.2. Factorization of partial differential Chow forms. In this section, we fix an orderly ranking R on u 0 , . . . , u n with u 00 greater than any other u ij . Suppose ld(F ) = θ(u 00 ) and θ is reserved for this derivative temporarily in this section. Let F u = F u 1 , . . . , u d , u 01 , . . . , u 0n and F 0 = F u τ (u 00 ) : τ ∈ Θ, θ ∤ τ .
Regard F as a univariate polynomial f θ(u 00 ) in θ(u 00 ) with coefficients in F 0 and suppose g = deg(F, θ(u 00 )). Then f θ(u 00 ) is irreducible over F 0 and in a suitable algebraic extension field of F 0 , f (θ(u 00 )) = 0 has g roots γ 1 , . . . , γ g . Thus
(5.4) f (θ(u 00 )) = A(u 0 , u 1 , . . . , u d ) g l=1 θ(u 00 ) − γ l where A(u 0 , u 1 , . . . , u d ) ∈ F {u 0 , . . . , u d } is free from θ(u 00 ).
For each l = 1, . . . , g, let
(5.5) F l = F 0 γ l
be an algebraic extension of F 0 defined by f θ(u 00 ) = 0. We will define derivations δ l,1 , . . . , δ l,m on F l so that F l , {δ l,1 , . . . , δ l,m } becomes a partial differential field. This can be done step by step in a very natural way. For the ease of notation,
for each τ = m k=1 δ d k k with (d 1 , . . . , d m ) ∈ N m , we denote τ l = m k=1 δ d k l,k .
In step 1, for each a ∈ F u , define τ l (a) = τ (a), in particular, δ l,k (a) = δ k (a) for each k = 1, . . . , m. In step 2, we need to define the derivatives of u 00 . For all τ ∈ Θ with θ ∤ τ or τ = θ, define τ l (u 00 ) as follows:
τ l (u 00 ) = τ (u 00 ) ∈ F l , θ ∤ τ γ l ∈ F l , τ = θ.
And for all τ ∈ Θ with θ|τ and τ = θ, we define τ l (u 00 ) inductively on the ordering of Θ(u 00 ) induced by R. Since F , regarded as a univariate polynomial f in θ(u 00 ), is a minimal polynomial of γ l , S f = ∂f ∂θ(u00) does not vanish at θ(u 00 ) = γ l . First, for the minimal τ = δ k θ for some k ∈ {1, . . . , m}, define
τ l (u 00 ) = δ l,k (γ l )= − T /S f θ(u00)=γ l ,
where δ k (f ) = S f · δ k θ(u 00 ) + T . This is reasonable, since all the derivatives of u 00 involved in S f and T have been defined in the former steps and we should have δ l,k f (γ l ) = S f θ(u00)=γ l δ l,k (γ l )+T θ(u00)=γ l = 0. Suppose all the derivatives of u 00 less than τ (u 00 ) = m k=1 δ d k k θ(u 00 ) have been defined, we can proceed in the similar way to define τ l (u 00 ) = m k=1 δ d k l,k (γ l ). Namely, use the differential polynomial τ (f ) = S f · τ (u 00 ) + T τ and define τ l (u 00 ) = −T τ /S f πθ(u00)=π(γ l ), πθ<τ . In this way, (F l , {δ l,1 , . . . , δ l,m }) is a partial differential field which can be considered as a finitely differential extension field of (F u , ∆).
Since F u is a finitely generated ∆-extension field of F contained in E. By the definition of universal differential extension fields, there exists a ∆-extension field F * ⊂ E of F u and a differential F u -isomorphism ϕ l from (F l , {δ l,1 , . . . , δ l,m }) to (F * , ∆). For a polynomial G ∈ F {Y} and a point η ∈ F n l , G(η) = 0 implies G(ϕ l (η)) = 0. For convenience, by saying η is in a ∆-variety V over F , we mean ϕ l (η) ∈ V . Summing up the above results, we have Lemma 5.5. (F l , {δ l,1 , . . . , δ l,m }) is a finitely differential extension field of (F u , ∆), which is differentially F u -isomorphic to a differential subfield of E.
Note that the above defining steps give a differential homomorphism φ l from (F {u 0 , . . . , u d }, ∆) to the differential field (F l , {δ l,1 , . . . , δ l,m }) for each l by mapping τ (u ij ) to τ l (u ij ). That is, for a ∆-polynomial p ∈ F {u 0 , . . . , u d }, φ l (p) is obtained from p by substituting τ θ(u 00 ) = τ l (γ l ). Then we have the following result.
Lemma 5.6. Let P ∈ F {u 0 , . . . , u d }. Then P ∈ sat(F ) if and only if φ l (P ) = 0.
Proof: If P ∈ sat(F ), then there exists m ∈ N such that S m F P ∈ [F ]. Since φ l is a differential homomorphism and φ l (F ) = 0, φ l (S m F P ) = 0. Note from the above that φ l (S F ) = 0, so φ l (P ) = 0 follows. For the other side, suppose φ l (P ) = 0. Let R be the differential remainder of P w.r.t. F under the ranking R. Since φ l (P ) = 0, φ l (R) = 0. Note that R is free from all the proper derivatives of θ(u 00 ) and deg(R, θ(u 00 )) < g. So R| θ(u00)=γ l = 0, which implies from the irreducibility of F that R is divisible by F . Thus, R = 0 and P ∈ sat(F ).
Remark 5.7. Similar to the ordinary differential case, in order to make F l a partial differential field, we need to introduce differential operator δ l,1 , . . . , δ l,m related to γ l and there does not exist a unique set of differential operators to make all F l (l = 1, . . . , g) differential fields.
Below, we now give the following Poisson-type product formula. . Fix an orderly ranking with u 00 > u ij and suppose ld(F ) = θ(u 00 ) and g = deg F, θ(u 00 ) . Then, there exist ξ l1 , . . . , ξ ln in a differential extension field (F l , {δ l,1 , . . . , δ l,m }) of (F u , ∆) such that
F (u 0 , u 1 , . . . , u d ) = A(u 0 , u 1 , . . . , u d ) g l=1 θ u 00 + n ρ=1 u 0ρ ξ lρ (5.6)
where A(u 0 , u 1 , . . . , u d ) is in F {u 0 , . . . , u d }. Note that equation (5.6) is formal and should be understood in the following precise meaning: θ(u 00 + n ρ=1 u 0ρ ξ lρ ) △ = θ(u 00 ) + θ l ( n ρ=1 u 0ρ ξ lρ . Proof: We will follow the notations above. By Lemma 5.6, φ l (S F ) = 0. Let ξ lj = φ l ( ∂F ∂θ(u0j ) ) φ l (S F ) for j = 1, . . . , n and ξ l = (ξ l1 , . . . , ξ ln ) ∈ F n l . We will prove γ l = −θ l n j=1 u 0j ξ lj .
Differentiating F (ζ 0 , u 01 , . . . , u 0n ; . . . ; ζ d , u d1 , . . . , u dn ) = 0 w.r.t. θ(u 0j ) on both sides, we have
(5.7) ∂F ∂θ(u 0j ) + ∂F ∂θ(u 00 ) · (−ξ j ) = 0,
where the ∂F ∂θ(u0j ) are obtained by substituting ζ i to u i0 (i = 0, . . . , d) in ∂F ∂θ(u0j) . Multiplying u 0j to the above equation and for j from 1 to n, adding them together, we have
n j=1 u 0j ∂F ∂θ(u 0j ) + ∂F ∂θ(u 00 ) · (− n j=1 u 0j ξ j ) = n j=1 u 0j ∂F ∂θ(u 0j ) + ∂F ∂θ(u 00 ) · ζ 0 = 0.
Thus, n j=0 u 0j ∂F ∂θ(u0j ) ∈ sat(F ). By Lemma 5.6, n j=1 u 0j φ l ∂F ∂θ(u 0j ) + φ l (u 00 )φ l ∂F ∂θ(u 00 ) = 0, so φ l (u 00 ) = − n j=1 u 0j ξ lj . Thus, θ l (φ l (u 00 )) = φ l (θ(u 00 )) = γ l = −θ l ( n j=1 u 0j ξ lj ). Substituting them into equation (5.4), (5.6) is proved.
Theorem 5.9. The points (ξ l1 , . . . , ξ ln ) (l = 1, . . . , g) in (5.6) are generic points of the ∆-variety V over F . If d > 0, they also satisfy the equations u σ0 + n ρ=1 u σρ y ρ = 0 (σ = 1, . . . , d).
Proof: Suppose P (y 1 , . . . , y n ) ∈ F {Y} is any ∆-polynomial vanishing on V . Then P (ξ 1 , . . . , ξ n ) = 0. From (5.7), ξ ρ = ∂f ∂θ(u0ρ) ∂f ∂θ(u00) , so we have
P ∂F ∂θ(u 01 ) ∂F ∂θ(u 00 ) , . . . , ∂F ∂θ(u 0n ) ∂F ∂θ(u 00 ) = 0,
where ∂F ∂θ(u0ρ) are obtained by substituting ζ i to u i0 (i = 0, 1, . . . , d) in ∂f ∂θ(u0ρ) . Thus, there exists an m ∈ N, such that
( ∂F ∂θ(u 00 ) ) m · P ∂F ∂θ(u 01 ) ∂F ∂θ(u 00 )
, . . . , ∂F ∂θ(u 0n ) ∂F ∂θ(u 00 ) ∈ sat(F ).
By Lemma 5.6, we have P (ξ l1 , . . . , ξ ln ) = 0, which means that (ξ l1 , . . . , ξ ln ) ∈ V . Conversely, for any Q ∈ F {Y} such that Q(ξ l1 , . . . , ξ ln ) = 0, by Lemma 5.6, there exists an l ∈ N such that Q = ( ∂F ∂θ(u00) ) l Q( ∂F ∂θ(u01) ∂F ∂θ(u00) , . . . , ∂F ∂θ(u0n) ∂f ∂θ(u00) ) ∈ sat(F ). So Q(ξ 1 , . . . , ξ n ) = 0. Thus, (ξ l1 , . . . , ξ ln ) is a generic point of V .
By equation (5.7), ∂F ∂θ(u0j ) + ∂F ∂θ(u00) · (−ξ j ) = 0, so we have n j=1 u σj ∂F ∂θ(u0j ) + ζ σ ∂F ∂θ(u00) = 0. Thus, n j=0 u σj ∂F ∂θ(u0j ) ∈ sat(F ). If σ = 0, then n j=0 u σj φ l ( ∂F ∂θ(u0j ) ) = 0. Consequently, u σ0 + n j=1 u σj ξ τ j = 0 (σ = 1, . . . , d). Remark 5.10. The leading differential degree could not be defined in the partial differential case, for the number g in Theorem 5.8 depends on the ranking we choose to get the Poisson-type product formula. Also, it may happen that under any orderly ranking, the leaders of the ∆-Chow forms of two irreducible ∆-varieties with the same Kolchin polynomial are alway different, so it is difficult to define partial differential cycles as we did in the ordinary differential case.
We conclude this section by showing that the vanishing of the ∆-Chow form gives a necessary and "sufficient" condition (in the sense of Kolchin closure) such that V and d + 1 number of ∆-hyperplanes have a nonempty intersection. 1) Let R be some elimination ranking satisfying u ij < u 00 < y 1 · · · < y n . Let ld(F ) = θ(u 00 ) and S F the separant of F . Then 2) For any given
(v 0 , . . . , v d ) ∈ (P n ) d+1 , if V ∩ i V(v i0 +v i1 y 1 +· · ·+v in y n ) = ∅, then F (v 0 , . . . , v d ) = 0. And if F (v 0 , . . . , v d ) = 0 and S F (v 0 , . . . , v d ) = 0
, then V and v i0 + v i1 y 1 + · · · + v in y n = 0 (i = 0, . . . , d) have at least one point in common.
Proof: The proof of item 1) is similar to the ordinary differential case. And item 2) is a direct consequence of item 1).
6. The existence of a type of partial differential Chow varieties
As mentioned in the introduction, to study a specific kind of geometric objects, it is important and useful to represent them by coordinates and further show that the set of objects is actually an algebraic system. For us, this specific kind of objects are irreducible ∆-varieties with Kolchin dimension polynomial (d + 1) t+m m − t+m−s m . As in the ordinary differential case, we could give these ∆-varieties coordinate representations via their ∆-Chow forms. Fix an index (n, d, s, r). Let G (n,d,s,r) be a functor from the category of ∆-fields to the category of sets which associates each ∆-field K with the set G (n,d,s,r) (K), consisting of all irreducible ∆-varieties V ⊂ A n over K with ω V (t) = (d + 1) t+m m − t+m−s m and ∆-degree r. If this functor is represented by some ∆-constructible set, meaning that there is a ∆-constructible set and a natural isomorphism between the functor G (n,d,s,r) and the functor given by this ∆-constructible set (regarded also as a functor from the category of ∆-fields to the category of sets), then we call this ∆-constructible set the ∆-Chow variety of index (n, d, s, r) of A n , and also say the ∆-Chow variety of index (n, d, s, r) exists.
In this section, we will show that ∆-Chow varieties of index (n, d, s, r) exist for all chosen n, d, s, r. Similar to the ordinary differential case, the main idea is to first definably embed G (n,d,s,r) into a finite disjoint union C of the chosen algebraic Chow varieties and then show the image of G (n,d,s,r) is a definable subset of C. So, the language from model theory of partial differentially closed fields (see [21,24,27]) will be used and we assume E is a ∆-closed field of characteristic 0 (i.e., E |= DCF 0,m ) throughout this section. 6.1. Definable properties and Prolongation admissible varieties. Here are some basic notions and results from model theory that we will be used in the proof of the main theorem. For more details and explantations, see [6].
We say that a family of sets {X a } a∈B is a definable family if there are formulae ψ(x; y) and φ(y) so that B is the set of realizations of φ (i.e., B = {ē ∈ E n : E |= φ(ē)}) and for each a ∈ B, X a is the set of realizations of ψ(x; a).
Given a property P of definable sets, we say that P is definable in families if for any family of definable sets {X a } a∈B given by the formulae ψ(x; y) and θ(y), there is a formula φ(y) so that the set {a ∈ B : X a has property P} is defined by φ.
Given an operation F which takes a set and returns another set, we say that F is definable in families if for any family of definable sets {X a } a∈B given by the formulae ψ(x; y) and θ(y), there is formula φ(z; y) so that for each a ∈ B, the set F (X a ) is defined by φ(z; a).
We will require the following facts about definability in algebraically closed fields. We also need to generalise results on prolongation admissible varieties [6] to the partial differential case. Notations τ l , ∇ l , B l should be specified beforehand. For an algebraic variety X = V(f 1 , . . . , f o ) ⊂ A n defined by polynomials f i ∈ F [y 1 , . . . , y n ], τ l (X) ⊆ A n( l+m m ) denotes the algebraic variety defined by (θ(f i )) θ∈Θ ≤l considered as algebraic polynomials in F [Θ ≤l (Y)] with Y = (y 1 , . . . , y n ). Thus, τ l A n = A n( l+m m ) with coordinates corresponding to variables (Y, Θ 1 (Y), . . . , Θ l (Y)). Given a point a ∈ A n , ∇ l (ā) denotes the point ā, Θ 1 (ā), . . . , Θ l (ā) ∈ τ l A n , and for a ∆-variety W ⊂ A n , B l (W ) is the Zariski closure of the set {∇ l (ā) :ā ∈ W}. In other words,
B l (W ) = V I(W ) ∩ F [Θ ≤l (Y)] ⊆ τ l A n .
Definition 6.4. Let V ⊂ τ s A n be an algebraic variety. We say V is prolongation admissible if B s V(I(V )) = V . Lemma 6.5. Let V ⊂ τ s A n be an irreducible prolongation admissible variety and A a characteristic set of V w.r.t. an ordering induced by some orderly ranking R on Θ(Y). For each k = 1, . . . , n, let E k = {θy k ∈ ld(A) : ∀ τ y k ∈ ld(A), τ |θ ⇒ τ (y k ) = θ(y k )}.
If E k = ∅, then for each τ y k ∈ Θ ≤s (y k ) which is a proper derivative of some element of E k , there exists A τ,k ∈ A such that ld(A τ,k ) = τ y k and A τ,k is linear in τ y k .
Proof: Let W = V I(V ) ⊂ A n and W = ∪ l i=1 W i be the irreducible decomposition of W . Since V is prolongation admissible, B s (W ) = V . So there exists some i 0 such that B s (W i0 ) = V . Suppose B is a ∆-characteristic set of W i0 w.r.t. R. Let C = Θ(B) ∩ F [Θ ≤s (Y)], C is a characteristic set of B s (W i0 ) = V .
Since C and A have the same rank, A should satisfy the desired property.
We now show that prolongation admissibility is a definable property. Lemma 6.6. Prolongation admissibility is definable in families.
Proof: Let (V b ) b∈B be a definable family of algebraic varieties in τ s A n with V b defined by f i b, (θ(y j ) θ∈Θ ≤s ,1≤j≤n ) = 0, i = 1, . . . , ℓ. By abuse of notation, let B s (V b ) be the Zariski closure of {∇ s (ā) : ∇ s (ā) ∈ V b } in τ s A n . Then deg(B s (V b )) has a uniform bound T in terms of the degree bound D of the f i , m, n, ℓ and s. Indeed, let z j,θ (j = 1, . . . , n; θ ∈ Θ ≤s ) be new ∆-variables and replace θ(x j ) by z j,θ in each f i to get a new differential polynomial g i . Consider the new differential system S := {g 1 , . . . , g ℓ , δ k (z j,θ ) − z j,δ k θ : k = 1, . . . , m; θ ∈ Θ ≤s−1 }. Regard S as a pure algebraic polynomial system in z jθ and δ k (z jθ ) temporarily, and let U be the Zariski closed set defined by S in τ (τ s A n ). Let Z = {c ∈ τ s A n : ∇(c) ∈ U }. [8,Corollary 4.5], the degree of the Zariski closure of Z, namely B s (V b ), is bounded by some number D 1 which depend on D, m, n, ℓ and s.
Clearly, Z = {∇ s (ā) : ∇ s (ā) ∈ V b }. By
By [11,Proposition 3], an irreducible algebraic variety V ∈ τ s A n can be defined by n s+m m + 1 + 1 polynomials of degree bounded by the degree of V . So B s (V b ) can be defined by at most n s+m m + 1 + 1 polynomials of degree bounded by
D 2 1 . Hence, (B s (V b )) b∈B is a definable family. Since V b is prolongation admissible if and only if V b = B s (V b )
, which implies that {b : V b is prolongation admissible} is a definable set. Thus, prolongation admissibility is definable in families.
V . A compo- nent W 1 of W is called a dominant component if B s (W 1 ) = V .
The following result shows how to get the desired unique irreducible ∆-varieties from irreducible prolongation admissible varieties. . We now show that V(P) ⊆ W and B s (V(P)) = V . Since V is an irreducible prolongation admissible variety, there exists a pointā ∈ A n such that ∇ s (ā) is a generic point of V . So as ∆-polynomials, B i vanishes atā while H B does not. Thus, P vanishes atā, and consequently, ∇ s (ā) ∈ B s (V(P)). So V ⊂ B s (V(P)). Since both V and B s (V(P)) are of the same dimension, and Y [s] induced by R. Suppose ld(F ) = θ(u 00 ) for some θ ∈ Θ s and S F = ∂F ∂θ(u00) . By Theorem 5.11, the polynomial G j = S F y j − ∂F ∂θ(u0j ) ∈ J and note that deg(G j ) = (d+1)r. We construct polynomials G j,θ ∈ J for θ ∈ Θ ≤s with rk(G j,θ ) = θ(y j ) and deg(G j,θ ) ≤ (ord(θ) + 1)(d + 1)r inductively on the order of θ. Set G j,1 = G j . Let G j,δi = rem(δ i (G j,1 ), G j,1 ) be the algebraic remainder of δ i (G j,1 ) with respect to G j,1 . Clearly, G j,δi ∈ J and is of the form G j,δi = S 2 F δ i (y j ) + T j,δi for some T j,δi ∈ F [u [s+1] ]. An easy calculation shows that deg(G j,δi ) ≤ 2(d + 1)r.
Suppose the desired G j,τ = S ord(τ)+1 F τ (y j ) + T j,τ (τ ∈ Θ ≤k ) have been constructed, we now define G j,τ (τ ∈ Θ k+1 ). For τ ∈ Θ k+1 , let G j,τ be the algebraic remainder of τ (G j ) with respect to < G j,τ : τ ∈ Θ ≤k ; k ≤ s >. Then G j,τ ∈ J and G j,τ = S k+2 F τ (y j ) + T j,τ ,where T j,τ ∈ F [u [k+1+s] ] satisfies deg(T j,τ ) ≤ (k + 2)(d + 1)r. In this way, polynomials G j,τ ∈ J (τ ∈ Θ ≤s ) are constructed.
Clearly, < F [s] , G j,τ : τ ∈ Θ ≤s > is an irreducible ascending chain under R s , Now, we are ready to prove that ∆-Chow varieties of index (n, d, s, r) exist for all n, d, s, r. As mentioned in the beginning of this section, we will use certain algebraic Chow varieties to parametrize ∆-varieties in G (n,d,s,r) . For the sake of later use, we briefly recall the concept of algebraic Chow varieties here. For an irreducible variety V ⊆ P n of dimension d, the algebraic Chow form of V is the polynomial G(u 0 , . . . , u d ) whose vanishing gives a necessary and sufficient condition for V and d + 1 hyperplanes having a nonempty intersection in P n . The Chow form of a d-cycle W in P n , W = l i=1 t i W i with t i ∈ N and dim(W i ) = d, is the product of Chow forms of W i with multiplicity t i . Its degree in each u i is called the degree of W and its coefficient vector is defined to be the Chow coordinate of W . The set of Chow coordinates of all d-cycles in P n of degree e is a projective variety in the Chow coordinate space [3], called the Chow variety, and denoted by Chow n (d, e). However, the affine Chow variety of all d-cycles in A n of degree e is not closed in the Chow coordinate space, but it is always a constructible set [ Let C 1 be the subset consisting of all points a ∈ C such that a is the Chow coordinate of an irreducible variety W which is prolongation admissible and additionally satisfies the following conditions:
so J s = F [s] , (G j,τ ) τ ∈Θ ≤s : S ∞ F is a prime ideal in F [Y [s] ,
(1) π s,0 (W ) is of dimension d + 1;
(2) the unique dominant component of the ∆-variety defined by equations of W is of ∆-degree g.
Theorem 6.11. The set C 1 is a ∆-constructible set and the map which associates an irreducible ∆-variety V ⊆ A n in G (n,d,s,r) with the Chow coordinate of the irreducible variety B s (V ) ⊆ τ l (A n ) identifies G (n,d,s,r) with C 1 . In particular, the : F c is irreducible is a definable set. Take an arbitrary c ∈ C 0 and the corresponding polynomial F c ∈ S e for an example. Let V c be the corresponding irreducible variety with Chow coordinate c. By item 5) of Fact 6.3, (V c ) c∈C0 is a definable family. And by Lemma 6.6 and Fact 6.3, C 2 = c ∈ C 0 : V c is prolongation admissible and dim(π s,0 (V c )) = d + 1 is a definable set. Then by Lemma 6.8, for each c ∈ C 2 , the ∆-variety corresponding to V c has a unique dominant component τ |θ θ τ θ τ (u ik )τ (y k ) regarded as a polynomial in variables Θ ≤s (y k ) and Θ ≤s (u ik ). Since B s (W c ) = V c , by Lemma 4.7, the Zariski closure of the image of U under the following projection map π : τ s A n × P (n+1)( s+m m ) d+1 −→ P (n+1)( s+m m ) d+1
is an irreducible variety of codimension 1, and the defining polynomial F of π(U ) is the ∆-Chow form of W c . By item 4) of Fact 6.3, the total degree of F is definable in families; this quantity is just the ∆-degree of W c . So the ∆-degree of W c is definable in families. Hence, C 1 is a definable set, and also a ∆-constructible set due to the fact that the theory DCF 0,m eliminates quantifiers [21,24,27]. By Lemma 6.8 and its proof, each irreducible variety V corresponding to a point of C 1 determines an irreducible ∆-variety W ∈ G (n,d,s,r) , where W is the unique dominant component of the ∆-variety corresponding to the prolongation admissible variety V . And on the other hand, each W ∈ G (n,d,s,r) determines the corresponding algebraic irreducible variety B s (W ), whose Chow coordinate is a point of C 1 guaranteed by Lemma 6.9. So we have established a natural one-to-one correspondence between G (n,d,s,r) and C 1 . Thus, G (n,d,s,r) is represented by the ∆-constructible set C 1 .
Conclusion
In this paper, a quasi-generic partial differential intersection theorem is first given. Namely, the intersection of an irreducible partial differential variety V with a quasi-generic differential hypersurface of order s is shown to be an irreducible differential variety with Kolchin polynomial ω V (t) − t+s+m m . Then partial differential Chow forms are defined for irreducible partial differential varieties of Kolchin polynomial (d + 1) t+m m − t+m−s m and basic properties similar to their algebraic and ordinary differential counterparts are presented. Finally, differential Chow coordinate representations are defined for such partial differential varieties, and the set of all irreducible partial differential varieties of fixed Kolchin polynomial and differential degree is shown to have a structure of differentially constructible set.
The above results have generalized the generic differential intersection theory and the theory of differential Chow forms and differential Chow varieties obtained for the ordinary differential case [9,6] to their partial differential analogs. However, the theory of partial differential Chow forms and partial differential Chow varieties far more complete and there are several problems left open for further research. As stated in Conjecture 4.6, we conjecture that Kolchin polynomial of the form (d + 1) t+m m − t+m−s m for some d, s ∈ N gives not only a sufficient condition, but also a necessary condition for the existence of partial differential Chow forms. Another problem left is how to represent general irreducible partial differential varieties by coordinates and further how to provide a set of partial differential varieties of fixed characteristics with a structure of differential constructible set.
Proposition 2. 2 .
2[14, p.80, Proposition 2] Let A be an autoreduced set of F {Y}.
Proposition 2.3. [14, p.167, Lemma 2] If A is a characteristic set of a prime ∆-ideal P ⊂ F {Y}, then P = sat(A), A is coherent, and asat(A) is a prime ideal not containing a nonzero element reduced w.r.t. A. Conversely, if A is a coherent autoreduced set of F {Y} such that asat(A) is a prime ideal not containing a nonzero element reduced w.r.t. A, then A is a characteristic set of a prime ∆-ideal in F {Y}.
Example 4 . 2 .
42Let m = 2 and V = V(δ 1 (y), δ 2 (y)) ⊂ A 1 . Let L 0 = u 00 + u 01 y and J = [I(V ), L 0 ] ⊂ F {y, u 00 , u 01 }. Then J ∩ F {u 00 , u 01 } = sat u 01 δ 1 (u 00 ) − u 00 δ 1 (u 01 ), u 01 δ 2 (u 00 ) − u 00 δ 2 (u 01 ) , which is of codimension 1 but not the general component of a single ∆-polynomial.
Lemma 4. 4 .
4Let P be a prime ∆-ideal in F {y 1 , . . . , y n } and A a characteristic set of P with respect to an orderly ranking R. Suppose the Kolchin polynomial of P is ω P (t) d, s ∈ N. Then there exist n − d distinct variables y i1 , . . . , y i n−d such that ld(A) = {y i1 , . . . , y i n−d−1 , θ(y i n−d )} for some θ ∈ Θ s .
Hi j (t − s ij ). Comparing the coefficients of t m−1 and t m−2 on the both sides and use the fact t+m m − t+m−s m
/2, which implies that the above system of equations is not valid. Thus, there exists only one i j such that s ij = s and all the other n − d − 1 of the s ij is equal to zero. Without loss of generality, suppose s i n−d = s. Hi j (t − s ij ). As a consequence, H ij = {(0, . . . , 0)}. Thus, each E ij has only one row vector, and ld(A) = {y i1 , . . . , y i n−d−1 , θ(y i n−d )} for some θ ∈ Θ s .The following result gives a sufficient condition for the existence of ∆-Chow forms.
Theorem 4. 5 ..
5Let V ⊂ A n be an irreducible ∆-variety over F with Kolchin polynomialω V (t) = (d + 1) t + m m − t + m − s m for some s ∈ N.Then the ∆-Chow form of V exists. And the order of the ∆-Chow form of V is s.Proof: Let P = I(V ) ⊂ F {Y}. Let P ⋆ = [P, L 1 , . . . , L d ] ⊂ F u 1 , . . . , u d {Y}. Then by Corollary 3.4, P ⋆ is a prime ∆-ideal of ∆-dimension 0 and ω P ⋆ = ω V (t) Let J 0 = [P ⋆ , L 0 ] F u1,...,u d {Y,u0} ∩ F u 1 , . . . , u d {u 0 }. Recall J = [P, L 0 , . . . , L d ] F {Y,u0,...,u d } ∩ F {u 0 , . . . , u d }. Then J and J 0 have such relations: J 0 = [J ] F u1,...,u d {u0} and J = J 0 ∩ F {u 0 , . . . , u d }. So J = sat(F ) F {u0,...,u d } for some ∆-polynomial F if and only if J 0 = sat(F ) F u1,...,u d {u0} . Thus, it suffices to consider for the case dim(V ) = 0, that is, to show the ∆-Chow form of V exists if ω V (t) = t+m m − t+m−s m for some s ∈ N. Now suppose dim(V ) = 0 and let η = (η 1 , . . . , η n ) be a generic point of V free from u 0 . Let ζ 0 = − n j=1 u 0j η j . Then (ζ 0 , u 01 , . . . , u 0n ) is a generic point of J = [I(V ), L 0 ] ∩ F {u 0 }.
the leading variables of a characteristic set of A with respect to an orderly ranking is {y i1 , . . . , y in−1 , θ(y in )} with θ ∈ Θ s . So {τ (η in ) : τ ∈ Θ ≤t , θ ∤ τ } is algebraically independent over F . By the contrapositive of the algebraic version of Lemma 2.1, S := {τ (ζ 0 ) : τ ∈ Θ ≤t , θ ∤ τ } is algebraically independent over F (u
.,u0n) (t) = (n + 1 .
1By Lemma 2.7, there exists an irreducible ∆-polynomial F of order s such that J = sat(F ), so the ∆-Chow form of V exists.
Lemma 4. 7 ..
7Let V ⊂ A n be an irreducible ∆-variety of Kolchin dimension polynomial ω V (t)Let F (u 0 , . . . , u d )be the ∆-Chow form of V . Then
d
] = F (u 0 , . . . , u d ) .Proof: Let Q = I(V ) ∩ F [Y [s] V ), L 0 , . . . , L d ] ∩ F {u 0 , .. . , u d } = sat(F ) and ord(F ) = s. So (F ) = sat(F ) as an algebraic autoreduced set with ld(θ(L i )) = θ(u i0 ), and let F 1 be the algebraic remainder of F with respect to A, thenF 1 ∈ [I(V ), L 0 , . . . , L d ]∩F [Y [s] , u [s] ] = I(V )∩F [Y [s] ] , where u = ∪ i u i \{u i0 }. So F ∈ Q and (4.1) follows.Below is an example of ∆-Chow forms.
Example 4. 8 .
8Let P = [δ 1 (y 1 ), y 2 − y 2 1 ] ⊂ F {y 1 , y 2 }. Clearly, ω P (
Theorem 5. 3 ..
3Let V ⊂ A n be an irreducible ∆-variety of Kolchin polynomial ω V (t) Let F (u 0 , . . . , u d ) be the ∆-Chow form of V .Then F (u 0 , . . . , u d )is ∆-homogenous of the same degree r in each u i .
Theorem 5 . 8 .
58Let F (u 0 , u 1 , . . . , u d ) be the ∆-Chow form of an irreducible ∆variety over F of Kolchin polynomial ω V (t)
Theorem 5. 11 .F
11Let V be an irreducible ∆-variety of Kolchin polynomial ω V (t(u 0 , . . . , u d ) the ∆-Chow form of V . The following assertions hold.
{F, S F y 1 − ∂F ∂θ(u 01 ) , . . . , S F y n − ∂F ∂θ(u 0n ) }isa characteristic set of [I(V ), L 0 , . . . , L d ] F {Y,u0,...,u d } w.r.t. R.
Definition 6 . 1 .
61Let V be an irreducible ∆-variety over F of Kolchin polynomial ω V (t) ∆-degree r. Let F (u 0 , . . . , u d ) be the ∆-Chow form of V .The coefficient vector of F , regarded as a point in a projective space determined by (n, d, s, r), is defined to be the ∆-Chow coordinate of V . Definition 6.2.
Fact 6. 3 .
3[6] Relative to the theory of algebraically closed fields (ACF), we have the the following statements.(1) The Zariski closure is definable in families.(2) The dimension and degree of the Zariski closure of a set are definable in families.(3) Irreducibility of the Zariski closure is a definable property. (4) If the Zariski closure is an irreducible hypersurface given by the vanishing of some nonzero polynomial, then the degree of that polynomial in any particular variable is definable in families. (5) The set of irreducible varieties in A n of dimension d and degree g is a definable family.
Definition 6 . 7 .
67Let V ⊂ τ l A n be an irreducible prolongation admissible variety and W = V(I(V )) be the ∆-variety defined by defining equations of
Lemma 6 . 8 .
68Let V ⊆ τ s (A n ) be an irreducible prolongation admissible variety of dimension (d + 1) s+m m − 1 and in the case s > 0, suppose π s,0 (V ) is of dimension d + 1. Then W = V(I(V )) has a unique dominant component W 1 and ω W1 (tTwo cases should be considered according to whether s = 0 or not. Case 1) s = 0. In this case, P = I(V ) is a prime ideal of F [y 1 , . . . , y n ] of dimension d. By [14, p.200, Proposition 10], {P} is a prime ∆-ideal of F {y 1 , . . . , y n } with Kolchin ∆-polynomial ω {P} (t) = d t+m m . Thus, W = V(P) itself is its dominant component and satisfies the desired property. Case 2) s > 0. Fix an orderly ranking R on F {Y} and denote R l to be the ordering on Θ ≤l (Y) induced by R. Since π s,0 (V ) is of dimension d + 1, a characteristic set of π s,0 (V ) w.r.t. R 0 is of the form B 1 , . . . , B n−d−1 where ld(B i ) = y σi for each i. Since V is irreducible and prolongation admissible, by Lemma 6.5, S = {θ(y σi ) : ord(θ) ≤ s, i = 1, . . . , n − d − 1} is a subset of the leaders of a characteristic set A of V w.r.t. R s . Since the dimension of V is (d + 1) s+m m − 1, ld(A) = S ∪ {τ (y σ n−d )} for some τ ∈ Θ s and σ n−d ∈ {1, . . . , n}\{σ 1 , . . . , σ n−d−1 }. So there exists B n−d ∈ A s.t. ld(B n−d ) = τ (y σ n−d ). Let B =< B 1 , . . . , B n−d >. Clearly, B is an irreducible coherent autoreduced set of F {y 1 , . . . , y n }, by [14, Lemma 2, p.167], B is a ∆-characteristic set of a prime ∆-ideal P ⊂ F {y 1 , . . . , y n } w.r.t. R. Clearly, P = sat(B) has Kolchin polynomial ω P (t)
B s (V(P)) = V. So, I(V ) = P ∩ F [Θ ≤s (Y)] ⊂ P, as a consequence, V(P) ⊂ V(I(V )) = W . Suppose W 0 is a dominant component of W . Given a generic point ξ ∈ W 0 , ∇ s (ξ) is a generic point of V . So, B vanishes at ξ and H B does not vanish at ξ.Thus, V(P) vanishes at ξ, i.e., W 0 ⊆ V(P). So W 0 = V(P). Thus, V(P) is the unique dominant component W and ω W1 (tProof of the main theorem. Before proving the main theorem, we need to bound the degree of B s (V ) to get the candidates of the algebraic Chow varieties which can be used to paramertrize ∆-varieties in G (n,d,s,r) .Lemma 6.9. Let V ⊂ A n be an irreducible ∆-variety in G (n,d,s,r) . Then B s (V ) is an irreducible variety in τ s (A n ) of dimension (d Proof: Clearly, B s (V ) is an irreducible variety in τ s (A n ) of dimension (d+1) s+m m − 1. For the degree bound, we first show deg(B s (V )) ≤ (s+1)(d+1)r n(s+1)( s+m m )+1 . Since V ∈ V (n,d,s,r) , the ∆-Chow form F (u 0 , . . . , u d ) of V exists, and we have ord(F ) = s and deg(F, u [s] 0 ) = r. Let J = [I(V ), L 0 , . . . , L d ] F {Y,u0,...,u d } .Let R be a ranking on F {u 0 , . . . , u d , Y} satisfying 1) θ(u ij ) < τ (y k ) for any θ and τ , and 2) R restricted to u 0 , . . . , u d is an orderly ranking. Let R s be the ordering on u
6 ,
6Proposition 3.4], also denoted by Chow n (d, e). All the Chow varieties we use here are affine ones. Let Chow n( s+m m ) (d+1) s+m m −1, e be the algebraic Chow variety in τ s (A n ) (s > 0) which is of dimension (d + 1) s+m m − 1 and degree e. Consider the disjoint union of algebraic constructible sets 1 , D 2 are the lower and upper bounds given in Lemma 6.9. So each point a ∈ C represents a [(d + 1) s+m m − 1]-cycle in τ s A n . To represent an irreducible ∆-variety V of the desired Kolchin polynomial and ∆-degree by a point in C, we only need to consider irreducible varieties with Chow coordinates in C.
In the case s = 0, the ∆-Chow form of each V ∈ G (n,d,0,r) is equal to the Chow form of B 0 (V ) ⊆ A n , so the set of ∆-Chow coordinates of ∆-varieties in G (n,d,s,r) is just the same as the set of Chow coordinates of all irreducible varieties in A n of dimension d and degree r. By item 5) of Fact 6.3, the latter set is a definable subset of Chow n (d, r), so G (n,d,0,r) is definable. Below, we suppose s > 0.First, we show C 1 is a ∆-constructible set. From the definition of Chow coordinates, we know each Chow n( are Chow forms of algebraic cycles in τ s A n of dimension (d + 1) s+m m − 1 and degree e. The algebraic cycle whose Chow coordinate is c is irreducible if and only if its Chow form F c is irreducible. Since irreducibility is a definable property, the set C 0 = c ∈ Chow n( s+m m ) ((d + 1) s+m m − 1, e)
W c and the Kolchin polynomial of W c is (Kolchin polynomial of W c is (d + 1) t+m m − t+m−s m , the ∆-Chow form of W c exists. Let U be the algebraic variety in τ s A n × P (n+1)( s+m m ) d+1 defined by the defining formulae of V c and θ(L i ) = 0 for θ ∈ Θ ≤s and i = 0, . . . , d with each θ(L i ) = θ(u i0 ) + n k=1
s+k] algebraically specializesto U
[s+k] . By [9, Lemma 2.13], P i (U, η)
[s] are algebraically dependent over F .
Thus, P i (U, η) (i = 1, . . . , m) are ∆-dependent over F .
2.1. Differential characteristic sets.
Thus, we have ω (ζ0,u01,...,u0n) (t) = tr.deg F (u).
Note that card(S) = t+m
m
− t+m−s
m
. [t]
01 , . . . , u
[t]
0n , ζ
[t]
0 )/F
= tr.deg F (u
[t]
01 , . . . , u
[t]
0n )/F + tr.deg F (u
[t]
01 , . . . , u
[t]
0n )(ζ
[t]
0 )/F (u
[t]
01 , . . . , u
[t]
0n )
u which is a component of V(F, (G j,τ ) τ ∈Θ ≤s ). By Beźout Theorem[11, Theorem 1],deg(J s ) ≤ [(d + 1)r] ( s+m m ) · Let J ′ s = J s ∩ F [Y [s] ]. We claim that J ′ s = I(V ) ∩ F [Y [s] ]. Indeed, on the one hand, J ′ s ⊂ J ∩ F [Y [s] ] = I(V ) ∩ F [Y [s] ]; on the other hand, for any polynomial H ∈ I(V ) ∩ F [Y[s] ], the algebraic remainder of H with respect to < G j,τ : τ ∈ Θ ≤s > is a polynomial H 1 ∈ J ∩ F {u 0 , . . . , u d } = sat(F ) with ord(H 1 ) ≤ 2s.Thus, H 1 ∈ asat(F[s] ) and H ∈ J s . So by[11, Lemma 2] or[19, Theorem 2.1], deg(I(V ) ∩ F [Y [s] ]) = deg(J ′ s ) ≤ deg(J s ). Now,we show deg(I(V )∩F [Y [s] ]) ≥ r/ s+m m . By Lemma 4.7, I∩F [Y [s] ], L , . . . , u [s] d ] = (F ). Similar to the procedures in [18, Theorem 6.25], the ∆-Chow form of I(V ) could be obtained from the algebraic Chow form of I(V ) ∩ F [Y [s] ] by algebraic specializations. So (d + 1)r ≤ (d + 1) s+m m deg(I(V ) ∩ F [Y [s] ]) and deg(B s (V )) = deg(I(V ) ∩ F [Y [s] ]) ≥ r/ s+m m . Remark 6.10. In the ordinary differential case, the construction of G j,k is much easier and each G j,k (k ≤ s) could be chosen from F [u [s] 0 , . . . , u [s] d , Y [s]]. However, we could not construct G j,θ in that way for there may exist τ ∈ Θ ≤s such that any derivative of τ (u 00 ) does not appear in F . Also, here G j,θ ∈ F [u[2s] 0 , . . . , u [2s] d , Y [s] ] for θ ∈ Θ ≤s .[2s]
0 , u
[2s]
1 , . . . , u
[2s]
d ],
n
j=1 θ∈Θ ≤s
deg(G jθ )
≤ [(d + 1)r] ( s+m
m ) ·
n
j=1
s
l=0
[(l + 1)(d + 1)r] ( s+m
m )
≤ [(s + 1)(d + 1)r] n(s+1)( l+m
m )+1 .
[s]
0 , . . . ,
L
[s]
d
∩ F [u
[s]
0
By saying η free from F U (resp. ∆-free from F U ), we mean that U is a set of ∆-F ηindeterminates.
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| []
|
[
"The FIMP-WIMP dark matter in the extended singlet scalar model",
"The FIMP-WIMP dark matter in the extended singlet scalar model"
]
| [
"Pritam Das \nDepartment of Physics\nTezpur University\nAssam-784028India\n\nDepartment of Physics\nIndian Institute of Technology Guwahati\n781039AssamIndia\n",
"Mrinal Kumar Das ",
"Najimuddin Khan \nDepartment of Physics\nTezpur University\nAssam-784028India\n\nSchool of Physical Sciences\nIndian Association for the Cultivation of Science 2A & 2B\nRaja S.C. Mullick Road700032KolkataIndia\n\nSchool of physics\nInstitute for Research in Fundamental Sciences (IPM)\nP.O.Box19395-5531TehranIran\n"
]
| [
"Department of Physics\nTezpur University\nAssam-784028India",
"Department of Physics\nIndian Institute of Technology Guwahati\n781039AssamIndia",
"Department of Physics\nTezpur University\nAssam-784028India",
"School of Physical Sciences\nIndian Association for the Cultivation of Science 2A & 2B\nRaja S.C. Mullick Road700032KolkataIndia",
"School of physics\nInstitute for Research in Fundamental Sciences (IPM)\nP.O.Box19395-5531TehranIran"
]
| []
| We explore the simplest viable dark matter model with a real singlet scalar, vector-like singlet and doublet fermions. The Yukawa couplings associated with the fermion sector play a crucial role in getting the current relic density through Freeze-in and Freeze-out mechanism. We discuss the constraints from the recent muon anomalous magnetic moment experimental data and relic density. We also perform the collider analysis for the FIMP dark matter in the context of 14 TeV LHC experiments with the MATHUSLA100/200 detector. Our analysis shows that one can get enough events > 3 for heavy charged fermion track at 14 TeV LHC with an integrated luminosity L = 10 6 fb −1 . * [email protected] † [email protected] ‡ [email protected] | 10.1016/j.nuclphysb.2022.115677 | [
"https://arxiv.org/pdf/2104.03271v3.pdf"
]
| 246,241,150 | 2104.03271 | c1a03f58cabdbffab6df531edfb7b269661f6202 |
The FIMP-WIMP dark matter in the extended singlet scalar model
24 Jan 2022
Pritam Das
Department of Physics
Tezpur University
Assam-784028India
Department of Physics
Indian Institute of Technology Guwahati
781039AssamIndia
Mrinal Kumar Das
Najimuddin Khan
Department of Physics
Tezpur University
Assam-784028India
School of Physical Sciences
Indian Association for the Cultivation of Science 2A & 2B
Raja S.C. Mullick Road700032KolkataIndia
School of physics
Institute for Research in Fundamental Sciences (IPM)
P.O.Box19395-5531TehranIran
The FIMP-WIMP dark matter in the extended singlet scalar model
24 Jan 2022
We explore the simplest viable dark matter model with a real singlet scalar, vector-like singlet and doublet fermions. The Yukawa couplings associated with the fermion sector play a crucial role in getting the current relic density through Freeze-in and Freeze-out mechanism. We discuss the constraints from the recent muon anomalous magnetic moment experimental data and relic density. We also perform the collider analysis for the FIMP dark matter in the context of 14 TeV LHC experiments with the MATHUSLA100/200 detector. Our analysis shows that one can get enough events > 3 for heavy charged fermion track at 14 TeV LHC with an integrated luminosity L = 10 6 fb −1 . * [email protected] † [email protected] ‡ [email protected]
I. INTRODUCTION
The standard model (SM) of particle physics is indeed one of the most successful theories of the last century; however, it still has a few unsolved empirical observations. Among those unsolved questions, strong CP problem, neutrino mass and mixing, matter-antimatter asymmetry, nature of dark matter and dark energy have been considered as shortfalls of the SM. The last few decades have seen an upheaval in astrophysics and cosmology. The Universe is filled with matter and dark matter (DM) and a large amount of dark energy [69]. From the SM model point of view, there is no sufficient candidate left to propose as a dark matter candidate. Therefore, one must go beyond the SM to explain the current dark matter density. The recent LHC Higgs signal strength data [70,71] also allows us to include additional fields of the new physics beyond the SM. Now one can get the exact relic density of the dark matter using various theories [69,72]. Many of the DM genesis theories are based upon the thermal Freeze-out mechanism. In this Freeze-out mechanism, the weakly interacting massive particles (WIMPs) are considered to be in thermal equilibrium in the early universe. When the primordial temperature drops below the mass of the DM, i.e., T < M DM , it dilutes away until the annihilation and/or co-annihilation to lighter particles becomes slower than the rate of expansion of the universe; hence, the co-moving DM number density becomes constant. In the literature, we have now seen that most of the single component WIMP dark matter model are tightly constrained by the recent direct detection limits [73]. The low mass region may give the relic density in the right ballpark Ωh 2 = 0.1198 ± 0.0026 [74]. However, the direct detection limits exclude most of the region [75][76][77][78]. Hence, multi-component dark matter models are more appealing at the current scenario [79].
Recently the authors of the Ref. [72], have introduced the idea of the Freeze-in mechanism. The dark matter interacts with the other particle feebly, called Feebly Interacting Massive Particles (FIMPs). Initially, we assume that the dark matter density remains small in the early universe. The idea is that the dark matter particle(s) get populated through the decay and/or annihilation of the other heavy particles in this model. After a certain temperature, the production of the dark matter through all the processes are ended (T < M M other P articles ), and the co-moving DM number density becomes constant. It is also confirmed that if the same couplings are involved in both the decay and annihilation scattering processes, then the latter case has negligible contribution to DM relic density over the former one [72,80,81]. There are two types of Freeze-in mechanisms that have been discussed in the literature. The infra-red (IR) Freeze-in [72,80,81] and ultraviolet (UV) Freeze-in [82]. The earlier one based on renormalizable theory and the latter based on theory consists of higher dimension operators in the Lagrangian. The relic density in the UV Freeze-in mechanism depends explicitly on the reheat temperature. It is also noted that the dark DM density depends on the partial decay width (DM production channels only) of the mother particles. The other decay channel may reduce the mother particles density at T < M DM rapidly. However, at T > M DM , the reverse processes can compensate the mother particles density from the other bath particles. But the forward DM production channels from the mother particles (p mothers → p others p DM , p mothers → p DM p DM ) are very slow due to the smaller coupling strengths, hence the reverse processes (p others p DM → p mothers , p DM p DM → p mothers ) cannot take place due to the reasons of coupling strengths as well as the initial density of the produce DM particles. Therefore, the other decay channels of the mother particles will not affect the relic density calculations in the Freeze-in mechanism as the mother particle was thermally equilibrium with the other particles in the early universe [72]. Thus, we have to solve only one Boltzmann equation for the evolution of the DM. In this case, one has to consider the sum of all DM production through the decay and annihilation channels of different mother particles. Suppose the mother particles are not in thermal equilibrium, then we need to solve the evaluation of the mother particles, and at the same time, we have to solve the evaluation for the DM particle [81]. Before or near the Freeze-in temperature, the mother particles' density should go very small so that the decay and annihilation contributions become zero. Hence, to explain the DM density through the IR Freezein mechanism, we need a tiny partial decay of the mother particles, i.e., and the coupling strength should be O(10 −9 ). One may ask the question about the naturalness of the theory, the answer itself lies in the Freeze-in mechanism. By default, we need such a small coupling to explain the DM density. In the UV Freeze-in mechanism, one could achieve these similar coupling strengths by adjusting the heavy cut-off scale in the Lagrangian. However, the dynamic reasons for such tiny coupling strengths are beyond the scope of this work.
The lepton flavor violating processes (µ → eγ) along with the muon and electron anomalous magnetic moment are also a striking indication of BSM. The discrepancy between the measured value and the SM predictions is there [83][84][85]: δa µ = a exp µ − a SM µ = (2.51 ± 0.59) × 10 −9 and δa e = a exp e − a SM e = −(8.8 ± 3.6) × 10 −13 . The recent results from Fermilab predicts that the muon anomalous magnetic moment deviates at 4.2σ from the SM prediction [83]. Earlier, it was measured with 3.7σ deviation from the SM prediction at Brookhaven National Laboratory. The muon magnetic moment is more sensitive hadronic and electroweak contributions, as well as BSM physics due to it's large mass, which influences such larger discrepancies between the δa µ . In the meanwhile, the electron anomalous magnetic moment results doesn't allow much deviation 1 [86]. The discrepancy of the muon magnetic moment can be solved by additional Higgs boson [87,88] and a light Z gauge boson associated with an extra U (1) Lµ−Lτ symmetry [89], or a light hidden photon [90], imposing discrete symmetries [91]. Apart from these, various models with collider searches have also been discussed in the literature [92][93][94][95][96][97][98], where both the anomalies and dark matter were discussed nicely.
The addition of the new fields to the SM is widespread in the literature. The lightest and stable particle (at least lifetime of DM should larger than the age of the universe) due to the imposed discrete Z n -charges (n ≥ 2, integer), behaves as a viable dark matter candidate [99]. Various study on minimal DM models considering scalar and fermion multiplets are available today [75,76,[100][101][102][103]. In particular, the addition of real singlet scalar, singlet as well as doublet fermion in a minimal model [104,105], have rich demand in DM study. The mixing of singlet and doublet fermions reduces the coupling to weak gauge bosons and other fermions, yielding the relic density at the right ballpark with allowed direct detection cross-section [106]. However, most models have discussed the Freeze-out mechanism with WIMP dark matter mass 50 GeV −300 TeV. The dark matter mass above 300 TeV violates the perturbativity bound [107], and the low dark matter mass region is either ruled out by the Higgs signal strength data or direct detection search limits.
In the previous work, [105], it was found that only the singlet scalar WIMP dark matter model with mass less than ∼ 550 GeV (except near the Higgs resonance region) are ruled out from recent direct detection limits [73]. We have also found a considerable increment in the WIMP dark matter parameter spaces in the presence of the new Yukawa couplings. One can reduce the Higgs portal coupling κ to avoid the direct detection limits. The additional t, u-channels through the new fermions can increase the dark matter annihilation cross-section to produce the exact relic density. Regarding the collider search studies, we have found that dilepton+ / E T signature can arise from the new fermionic sector and observed at the Large Hadron Collider (LHC). We have performed the collider analysis in the context of 14 TeV LHC experiments with a future integrated luminosity of 3000 fb −1 for the final state dilepton+ / E T in detail. The projected exclusion contour reaches up to 1050 − 1380 GeV for 3000 fb −1 for a light dark matter O(10) GeV from searches in the pp → E ± 1 E ∓ 1 , E ± 1 → l ± S → ll + / E T channel. In this study, the SM is extended with a real singlet scalar and singlet and doublet fermion [105] is revisited in the context of the low and high dark matter region.
There have been a rich literature of model studies [95,[108][109][110][111], where authors have incorporated the analysis of muon anomalous magnetic moments along with neutrino and dark matter mass models. These studies have also tested the validity of their model parameters and dark matter regions in collider searches. Motivated by these studies, in this minimal setup of our model work, we have identified the new parameter space relevant to low dark matter mass from 10 keV to 6 GeV via the Freeze-in mechanism in this present paper. The WIMP dark matter scenario is revisited keeping the collider constraints from the dilepton+ / E T searches [105]. In this minimal model, both low and high DM mass region are studied, including the allowed region from the latest muon anomalous magnetic moment data [83] from the Fermilab. We also carry out the collider study for the FIMP like scenario in the context of 14 TeV LHC experiments with the MATHUSLA100/200 detector. A charged track can be obtained in this model due to the decay of the heavy charged fermion E ± 1,2 into the SM fermions and dark matter. We find that one can get enough events > 3 at 14 TeV LHC with an integrated luminosity L = 10 6 fb −1 . Soon, if this type of minimal model turns out to be the FIMP and/or WIMP dark matter model realized in Nature, our present (including [105]) study could estimate a better parameter space. To the best of our knowledge, a detailed analysis of this model has not been done, which motivating us to study the model.
The rest of the work is organized as follows. We have given the complete model description in section II. The possible constraints from the lepton flavour violation decay and new results on muon magnetic moment are discussed in section III. The dark matter analysis through Freeze-out and Freeze-in mechanism is carried out in detail in section IV. The possible collider searches from the LHC charged tracker detector is analyzed in section VI. Finally, we have conclude our work in section VII.
II. MODEL
The model addressed here, contains (i) a real singlet scalar (S), (ii) a vector-like charged fermion singlet E − S and (iii) a vector-like fermion doublet,
F D = (X 0 1 E − D ) T .
It is to be noted that these additional fermions are vector-like, and hence, they do not introduce any new anomalies [105,112,113]. All the newly added particles are considered odd under discrete Z 2 symmetry (Ψ → −Ψ), such that these fields do not mix with the SM fields. Hence, the lightest and neutral particle is stable and can be considered as a viable dark matter candidate. The Lagrangian of the model read as [105],
L = L SM + L S + L F + L int ,(2.1)
where,
L S = 1 2 |∂ µ S| 2 − 1 2 kS 2 φ 2 − 1 4 m 2 S S 2 − λ S 4! S 4 , (2.2) L F = F D γ µ D µ F D + E S γ µ D µ E S − M N D F D F D − M N S E S E S , L int = −Y N F D φE S − Y f i ψ i,L F D S − Y f i l i,R E S S + h.c. (2.3)
Here, D µ stands for the corresponding covariant derivative of the doublet and singlet fermions. The left-handed lepton doublet is denoted by ψ i,L = (ν i , l i ) T L , whereas l i,R indicate the right-handed singlet charged leptons, with i = e, µ, τ three generation of leptons. L and R stand for left and right chirality of the fermions. The SM Higgs potential is given by, V SM (φ) = −m 2 φ 2 + λφ 4 , with, φ = (G + , H+v+iG √ 2 ) T is the SM Higgs doublet. G's stand for the Goldstone bosons and v = 246.221 GeV being the vacuum expectation value of the Higgs H fields. The mass matrix for these charged fermion fields is given by,
M = M N D M X M † X M N S , (2.4) where, M X = Y N v √ 2 .
The charged component of the fermion doublet (E ± D ) and the singlet charged fermion (E ± S ) mix at tree level. The mass eigenstates are obtained by diagonalizing the mass matrix with a rotation of the (E ± D E ± S ) basis,
E ± 1 E ± 2 = cos β sin β − sin β cos β E ± D E ± S , with tan 2β = 2M X M N S − M N D .M E ± 1 = M N D − 2(M X ) 2 M N S − M N D , M E ± 2 = M N S + 2(M X ) 2 M N S − M N D .
The masses of the neutral fermion scalar fields can be calculated as,
M X 0 1 = M N D , M 2 S = m 2 S + kv 2 2 and M 2 H = 2λv 2 . (2.6)
Hence, in this model, neutral fermion can not be the DM candidate as
M E ± 1 < M X 0 1 < M E ± 2 .
Only the scalar fields S for M S < M E ± 1 can behave as a viable DM candidate. We will provide a detailed discussion on the new region of the allowed parameter spaces and the effect of the additional Z 2 -odd fermion in the dark matter section IV.
The parameter space of this model is constrained by various bounds arising from theoretical considerations like absolute vacuum stability and unitarity of the scattering matrix, observation phenomenons like dark matter relic density. The direct search limits at LEP and electroweak precision measurements put severe restrictions on the model. The recent measurements of the Higgs invisible decay width and signal strength at the LHC put additional constraints. The dark matter (DM) requirement saturates the DM relic density all alone restricts the allowed parameter space considerably. These constraints are already discussed in our previous paper [105]. We discuss the lepton flavour violation (µ → eγ), incorporating the new anomalous magnetic moment result in the next section.
III. LEPTON FLAVOR VIOLATION (µ → eγ) AND ANOMALOUS MAGNETIC MO-
MENT
In this model, the LFV couplings (in our model, Y f i and Y f i with i = e, µ, τ ) and model parameters got severely constraint by the bounds from LFV processes. Hence, it is evident that LFV bounds can mimic other observable like DM parameter spaces (one can go through the WIMP dark matter analysis section, carried out in this work). The observed dark matter abundance is typically obtained for κ = Y all f i = O(0 − 1) through s-channel, t-channel annihilation and the combination of these two processes (co-annihilation, i.e., mass differences can also play a crucial role). The lepton flavour observable are expected to give stringent constraints on the parameter spaces.
Among the various LFV processes, the radiative muon decay Γ(µ → eγ) is one of the popular and restrictive one, which in the present model is mediated by charged particles E ± 1 , E ± 2 present in the internal lines of the one-loop diagram 1. The corresponding expression for the branching
ψ l ′ ,L E ± 1,2 γ S ψ l,L l ′ R E ± 1,2 γ S l R
FIG. 1: Muon and electron anomalous magnetic moment and LFV process µ → eγ decay diagrams mediated by charged particles E ± 1 and E ± 2 .
ratio is given by,
BR(µ → eγ) = 3α em 64πG 2 F A † f 1 A f 2 F (M 2 E ± 1 /M 2 S ) M 2 S + B † f 1 B f 2 F (M 2 E ± 2 /M 2 S ) M 2 S 2 , (3.1) where, F (x) = x 3 −6x 2 +3x+2++xln(x) 6 (x−1) 4
. The coupling strengths of the corresponding vertices are A f 1 = cos βY f 1 + sin βY f 1 A f 2 = cos βY f 2 + sin βY f 2 , B f 1 = sin βY f 1 + cos βY f 1 and B f 2 = sin βY f 2 + cos βY f 2 respectively. With the most recent experimental bounds for LFV [114], one can get allowed parameters from the lepton flavor violating decay constraints [114]
BR(µ → eγ) < 4.2 × 10 −13 at 90% CL., for Y f j and Y f j < O(10 −3 ) with j = e or µ.
Along with the LFV process, the muon anomalous magnetic moment provides new sensitivity for new physics contribution due to the higher level of precision in both theoretical and experimental measurements. The recent measurement of the muon anomalous magnetic moment, a µ = (g − 2) µ /2, at Fermilab shows a discrepancy w.r.t. the Standard Model (SM) prediction [83],
a F N AL µ = 116592040(54) × 10 −11 , (3.2) a SM µ = 116591810(43) × 10 −11 ,(3.3)
and, when it was combined with the previous Brookhaven result [84], a BN L µ = 116592089(63)×10 −11 leads to an observed excess of δa µ = 251(59) × 10 −11 at 4.2σ C.L. In the near future experiments at Fermilab, these discrepancies are expected to be clarified with more precise data. The electron anomalous magnetic moment sensitivity is [83][84][85]: δa e = a exp e − a SM e = −(8.8 ± 3.6) × 10 −13 . It is negative and one need a very sensitive parameters to solve the electron as well as muon anomalous magnetic moment simultaneously. In our model, with the vector-like fermions in hand, the new contribution to anomalous magnetic moment can be written as [115,116],
δa i = Re[A 2 f i ]m 2 l 8π 2 1 0 x(1 − x) 2 (x − x 2 )m 2 l + (x − 1)M 2 E ± 1 − xM 2 S dx + Re[B 2 f i ]m 2 l 8π 2 1 0 x(1 − x) 2 (x − x 2 )m 2 l + (x − 1)M 2 E ± 2 − xM 2 S dx,(3.4)
where i = e, µ, τ . As we can see directly from equation (3.4), the contribution to δa i depends on the masses (m E ± 1,2 , M S ), Yukawa couplings (Y f i and Y f i ) and the mixing angle (β). Depending on the new Yukawa couplings Y f i and Y f i , the loop diagram simultaneously contributes to the electron as well as muon anomalous magnetic moment. For simplicity, let us assume all the Y f i couplings are negligibly small. We find that, a very large Yukawa couplings (Y f i > 4π) are required to explain the present lepton anomalous magnetic moment data. For example, one can get δa µ = 2.51 × 10 −11 for Y f 2 = 29.32, M S = 1000 GeV, M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV. This choice of couplings and masses violate the LFV as well as perturbative limits. If we assume non-zero (20) to get δa µ = 2.51 × 10 −11 . Similarly, we also need a very large Y f 1 = Y f 1 to get electron anomalous magnetic moment. On the contrary, these large couplings are not healthy for this model as it will render a large negative potential (become unbounded from below) towards the singlet as well as the Higgs scalar fields due to radiative corrections 2 . To work out this model smoothly, we always kept Y f 2 and Y f 2 < O(10 −3 ). It is to be noted, the regions allow by LFV data are also allowed by these anomalous magnetic moment data. In the next section, we discuss the dark matter analysis and bound coming from the relic density.
Y f 2 = Y f 2 , then we also need Y f 2 ∼ O
IV. DARK MATTER
As pointed out in the previous section, the viable DM candidate in this model is the lightest Z 2 -odd singlet scalar S. Here, the dark matter can give exact relic density through the Freeze-out mechanism and/or Freeze-in mechanism, depending on the choice of parameter spaces. Suppose the dark matter is in thermal equilibrium in the early universe; in this case, the dark matter can annihilate to the SM particles when T > M DM , where T is the temperature of the universe. It Freezes out for T < M DM , and depending on the parameter spaces it could give the exact relic density. However, if it is not in thermal equilibrium in the early Universe, it could produce from some mother particles and give correct relic density through the Freeze-in mechanism. Dark matter resulting from a decay or annihilation of various mother particles, is in thermal equilibrium at early universe. This condition is given by,
Γ H(T ) ≥ 1, (4.1)
where, Γ is the relevant decay width and H(T ) is the Hubble parameter given by [72,81,117]
H(T ) = g * π 2 90 T 4 M 2 Pl 1/2 , (4.2)
where, M Pl = 2.4 × 10 18 GeV is the reduced Planck mass and T is the temperature (1 GeV = 1.16 × 10 13 Kelvin). If the production of mother particles occur mainly from the annihilation of other particles in the thermal bath, Γ will be replaced by [72,81,117] Γ = n eq < σv >,
(4.3)
where, n eq is their equilibrium number density and is given by [117] n eq =
g * mT 2π 3/2 e −m/T , for non-relativistic states T << M ζ 3 π 2 g * T 3 , for relativistic boson states T >> M 3 4 ζ 3 π 2 g * T 3 , for relativistic fermion states T >> M (4.4)
where, the Riemann zeta function has the value ζ 3 = 1.2 and g * is the effective degrees of freedom in this framework. Here, < σv > is the thermally averaged annihilation cross-section of the particles in the thermal bath and can be expressed as [117,118]
< σ xx v >= 2π 2 T ∞ 4m 2 ds √ s (s − 4m 2 ) K 1 ( √ s T )σ xx 4πm 2 T K 2 ( m T ) 2 , (4.5)
where, K 1,2 is themodified Bessel function of functions of order 1 and 2 respectively. The DM production through annihilation depends upon the Higgs portal couplings κ through s-and crosschannels (one can reverse the Figs. 2-(a), 2-(b) and 2-(c)) and the new Yukawa coupling Y f i through t-channel ( Figs. 2-(d)). Earlier we have checked [105] that for κ, Y f i ∼ O(0.001) with DM mass ∼ O(GeV), the dark matter is in thermal equilibrium at early Universe, i.e., neq<σxxv> H(T ) >> 1. These dark matter mass region give the exact relic density via the Freeze-out mechanism.
In this work, we find that the non-thermally produced singlet scalar can also serve as a viable dark matter candidate at O(1) GeV (depends on the parameters κ, Y f i and β) satisfying the dark matter relic density in the right ballpark. Those light dark matter can interact with other particles very feebly. For such very weakly interacting particles, called feebly interacting massive particles or FIMPs, one can invoke the non-thermal dynamics, so-called the Freeze-in mechanism. This mechanism needs weak interactions, which could be one reason to have a tiny fine-tuned coupling in this model. In this model, we find that the shallow dark matter mass region O(1) keV-MeV could also give the exact relic density in the right ballpark.
We will discuss both the dark matter regions coming from the Freeze-out and Freeze-in mechanism in the following two subsections keeping eye on all the constraints. We want to refer to our previous paper [105] for the detailed constraints arising from various theoretical and experimental point of view.
A. WIMP dark matter
The lightest Z 2 -odd singlet scalar S plays the role of viable WIMP dark matter candidate for the choice of region of the parameter spaces with mass O(100) GeV and annihilation coupling strength O(0.1) [105]. In our WIMP dark matter study, we use FeynRules [119] to get the input codes for micrOMEGAs [120] and compute the relic density of the scalar DM. We also verified the results using SARAH-4.14.3 [121] including SPheno-4.0.3 [122] mass spectrum in micrOMEGAs and get the same relic density of the scalar DM. The Higgs portal couplings κ controls the DM production and/or annihilation through s-and cross-channels (see figs. 2-(a), 2-(b) and 2-(c)). The Yukawa couplings associate with the singlet scalar (Y f i and Y f i ) and the charged fermions (Y N ) also have significant influences [105] in the WIMP DM parameter space. Depending on the Yukawa couplings (Y f i and Y f i ), DM can annihilate via t-and u-channels (see fig. 2-(d)) in our model. The relevant vertices for the dark matter annihilation and/or co-annihilation are given by:
g HSS = |κv|, g HHSS = |κ|, g SiE ± 1 = A f i = |(cos βY f i + sin βY f i )|, g SiE ± 2 = B f i = |(sin βY f i + cos βY f i )|, (4.6) g Sν i X 0 1 = C f i = |Y f i , with i = e, µ, τ |.
The interference between the s-channel, cross-channel and t, u-channels played a crucial role to achieve the correct DM density. The co-annihilation channels (e.g., see Fig. 3) also have an essential role in getting a viable region of allowed dark matter parameter space. The contributions from the S mediate co-annihilation t, u-channel diagrams is negligibly small. If the mass difference between the DM and other Z 2 -odd particles are within 2%−10%, it is expected that co-annihilation will dominate over DM self-annihilation [123]. We find a huge suppression to the relic density due to large co-annihilation for M E ± 1,2 ,X 0 1 − M S ≈ 50 GeV. Hence most of the higher mass region are ruled out due to the under abundance of a single component dark matter. It is to be noted that the Sommerfeld enhancement [124] do not play any role to enhance the current dark matter phenomenology due to M E ± 1,2 > M DM . We already have seen that dilepton+ / E T signature can arise from the new fermionic sector through pp → E ± 1 E ∓ 1 , E ± 1 → l ± S → ll + / E T channel. The charged fermion may observe at the Large Hadron Collider (LHC) with high luminosity. We have already done such analysis in our previous work [105] in the context of 14 TeV LHC experiments with a future integrated luminosity of 3000 fb −1 . By performing a detailed cut based collider analysis, we have seen that a large region of the parameter spaces can be probed/excluded by the LHC experiments. The projected exclusion contour have reached up to 1050 − 1380 GeV for 3000 fb −1 for a light dark matter O(10) GeV from searches in the pp
S S H f f (a) S S H V * V (b) S S H H (c) S S Y = E ± 1 , E ± 2 , X 0 1 f ′ f ′ (d)→ E ± 1 E ∓ 1 , E ± 1 → l ± S → ll + / E T channel.
Hence, in this analysis, we keep fixed the mass parameters M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV. For simplicity, we consider the effect from the interaction term − Y f i l i,R E S S (see eqn. 2.3) to be zero, with Y f i = 0. We will discuss the effect at the end of the WIMP dark matter section. We also assume Y f 1 = Y f 3 = Y f with Y f 2 = 0.001 3 to avoid flavor violating decays and fixed mixing angle as cos β = 0.995. This small mixing angle (β) diluted the contribution from the second charged fermion E ± 2 . We now scan a prominent region for the DM mass in this work, Higgs portal coupling κ and new Yukawa 3 In general, large Yukawa coupling are also allowed from the relic density. One can also get similar results for the Fig. 4 (left) remains almost identical; however, the right one gets modified due to having only one dominant contribution from the diagram with the second Yukawa coupling. , the coannihilation channels (Fig. 3 4 ) play an important role in the dark matter density calculation 5 . It is noted that the co-annihilation effects are completely absent here as ∆M ±,0 > 0.1M DM . We display the allowed parameters κ − M DM plane in Fig. 4(left). In the shallow mass region (below M H /2), t, u-channel annihilation processes play a key role in giving rise to correct relic bounds. Those regions are also allowed by Higgs decay width and direct detection bounds. SS → νν are the main dominant channels in this region. In the low DM mass region, annihilation via s-channel, cross-channel and other co-annihilation processes are negligibly small. The Higgs portal coupling is kept very small (κ ∼ 0) to avoid the Higgs signal strength constraints. On the other hand, all the processes, depending on the parameters in the high mass region, are important to provide the exact relic density. The above and below region correspond to Higgs portal coupling |κ| The blue points indicate the relic density within the 3σ range. These plot are generated by varying the dark matter mass, Higgs portal coupling κ and the new Yukawa coupling Y f . The empty region is in the second plot are mostly disfavoured by the relic density and direct detection constrainsts. The parameters allowed by relic density are also allowed by the recent LFV [114] BR(µ → eγ) < 4.2 × 10 −13 at 90% CL, electron as well as muon anomalous magnetic moment g − 2 data at Fermilab [83].
choice of Y f 2 = Y f 3 = Y f and Y f 1 = 0.001 or Y f 2 = Y f with Y f 1 = Y f 3 = 0.001. For different choice of couplings,S Y ν l , l V ν l , l V = W, Z Y = E ± 1,2 , X 0 1 (a) Y Y ′ V f f V = W, Z Y, Y ′ = E ± 1,2 , X 0 1 (b) Y S Y ′ V ν l , l V = W, Z Y, Y ′ = E ± 1,2 , X 0 1 (c)
GeV is ruled out by the present direct detection cross-section [73]. However, we do not have any DM signature in those experiments at the current date, and by adjusting Higgs portal coupling κ and the new Yukawa coupling Y f , one can work out this forbidden region. As we increase the viable allowed DM mass, the Higgs portal coupling κ keeps increasing if we neglect the Yukawa coupling Y f . Similarly, if we choose a small Higgs portal coupling κ, one has to increase the Yukawa coupling Y f with DM mass to get the relic density at the right ballpark. Variation of these couplings can exceedingly increase dark matter parameter spaces through the combination of s-,tand u-annihilation and co-annihilation channels.
The SS → V V, HH (with V = W ± , Z) channels keep on dominating in the mid − high (500-1000 GeV) DM mass regions. Various allowed benchmark points are shown in Table I. One can see, in the presence of DM annihilation via s-channel and t, u-channels, as most of the regions are giving the correct DM density, which are also allowed by other experimental constraints. For κ ∼ 0, the s-channel annihilation still dominates near Higgs resonance region ∼ M H 2 . The other mass regions give an overabundance for the same κ and Y f ∼ 0. For a small κ ∼ 0, the t+u-channels contributes to get the correct relic density at 3σ C.L., whereas s-channel and cross-channel processes dominate for large κ and small Y f . We have shown the viable region of parameters in the κ − Y f plane in Fig. 4(right). A blue circular eye-like pattern is obtained here. The same data points are also shown in the κ − M DM plane of Fig. 4(left). The blue regions are the allowed data points passes by all experimental and theoretical bounds [105] such as stability, perturbativity, unitarity, LHC diphoton signal strength, electroweak precision experiments, lepton flavour violation and lepton anomolus magnetic moments 6 . The empty region violates one of the constraints, such as the relic density of the dark matter, direct detection, and Higgs decay width for the DM mass < M H 2 . There is large excluded region for κ, Y f ∼ 0 compared to the presence of both κ and Y f . For example, the points Y f ≈ 0, κ ≈ 0 give exact relic density for the dark matters masses near Higgs resonance, i.e., 55 < M DM < 64 GeV. We get under abundance around M DM ≈ M h /2 ± 1. This can be seen in the left panel of the Fig. 4. The central region of the eye is also dominated mainly by the co-annihilation channels where ∆M ±,0 < 0.1M DM for high dark matter masses. The following circular empty region is produced under abundance due to a large co-annihilation cross-section for the high dark matter mass. The blue iris-eye region is now allowed for the other masses, where the co-annihilation effects are negligible. Both the side of this iris-eye region is also produced underabundance due to the effect of additional dark matter annihilation through the s, t and u channel processes for different dark matter masses. The small blue region (both the left and right side) gives the exact relic density for the low dark matter masses M DM < 100 GeV through the t and u channel processes. The bigger blue region has all possible effects on the relic density for various dark matter masses. The outer side of the blue region is mostly ruled out from the direct detection data. The low DM mass region was satisfying the current relic density within 3σ C.L., obtained via the t, u-channel processes. The s-channel annihilation processes in the low mass region were considerably absent; hence, very few allowed points for κ ∼ 0 were observed on the right side of Fig. 4. In those scenarios mainly E ± 1 E ± 1 → Y Y (with Y = W ± , Z, H) via t, u-channels and other annihilations and/or co-annihilations dominate and the relic density gets under-abundant for large Y f and become overabundant for Y f = 0. As we move towards the higher mass regions, both the annihilations and co-annihilations processes start dominating. Please note that, the coannihilations processes fully dominate in the region where we have ∆M ±,0 << 0.1M DM . Hence, large points satisfying the relic density within 3σ C.L. are observed for a wider variety of κ, Y f and M DM .
Channel M DM (GeV) κ M E ± 1 (GeV) Y f Ω DM h 2 Percentage σ(SS → νν) 98% BP-1 10 0.00 1500 0.29 0.115 σ(SS → ll) 2% σ(SS → νν) 98% BP-2 80 -0.01 1500 0.28 0.1142 σ(SS → ll) 2% σ(SS → νν) 46% σ(SS → W ± W ∓ ) 23% BP-3 270 0.058 1500 0.24 0.113 σ(SS → HH) 13% σ(SS → ZZ) 11% σ(SS → ll) 7% σ(SS → W ± W ∓ ) 48% BP-4 760 0.224 1500 0.09 0.1243 σ(SS → HH) 24% σ(SS → ZZ) 24% σ(SS → ll) 3% σ(SS → W ± W ∓ ) 49% BP-5 1000 0.31 1500 0.05 0.114 σ(SS → ZZ) 24% σ(SS → HH) 24% σ(SS → ll) 2% σ(E ± 1 S → W ± ν)
It is important to highlight that the small mixing angle (β) and large mass M E ± 2 = 3000 GeV diluted the contribution from the new second charged fermion E ± 2 . This eigenstate is mostly composed of the SU (2) singlet charged fermion E ± S due to this choice of small mixing angle. One can increase the percentage of E ± D component in the same eigenstate E ± 2 by decreasing the value of this mixing angle. At the same time the coupling strengths of the t, u-channels vertices g E ± 1 l ∓ S = cos βY f i and g E ± 2 l ∓ S = sin βY f i get modified. It influences the annihilation of the dark matter through t, u-channels; hence the relic density will change. For example, one can find the exact relic density allowed by all the other constraints for the dark matter mass M DM = 270 GeV (see BP-3 of table I) with Higgs portal coupling κ = 0.058 and Y f = 0.25 with fixed cos β = 0.995 and M E ± 2 = 3000 GeV. In this case, if we change cos β = 0.6, then we get an overabundance of the dark matter density Ω DM h 2 > 0.15. To achieve the exact relic density, we need to decrease the value of the mass of the second eigenstate and/or increase the new Yukawa coupling Y f . For example, by lowering the mass eigenstate of E ± 2 to M E ± 2 = 1510 GeV and keeping the same Y f = 0.24 or keeping M E ± 2 = 3000 GeV with Y f = 0.301, one can get Ω DM h 2 = 0.113. It is important to note that the left plot of the Fig. 4 will remain almost same for the choice of different mixing angle cos β and fixed M E ± 2 = 3000 GeV as the Higgs portal coupling |κ| is constrained from the dark matter direct detection. However, the right plot of Fig. 4 will be modified. One can have larger allowed region in the Y f − κ plane from the relic density and other constraints. The allowed values of new Yukawa coupling can reach up to Y f ≤ 0.391 for κ = 0 and cos β = 0.6 in the Y f − κ plane. We show the effect of the mixing angle in Fig. 5 (left) of the t, u-channels contribution in the relic density. One can also consider non-zero Y f i to get enhancement (for the same sign of Y f i ) in the dark matter annihilation as well as co-annihilation into the SM particles, and for cos β = 0.995 this enhancement is small. We show the effect in Fig. 5 (right) that, one needs smaller Y f to get relic density for non-zero Y f i . The blue and red lines satisfy the relic density Ω DM h 2 = 0.1198. The blue line correspond to Y f i = 0 whereas red line stand for the choice Y f i = 0.6. The Higgs portal coupling κ = 0, mixing angle cos β = 0.995 and masses M E ± 1 = 1500 GeV and M E ± 2 = 3000 are same in both cases.
It is also noteworthy that the non-zero Yukawa coupling Y f 2 (and Y f 1 = Y f 3 = Y f ) puts additional contribution to the processes Γ(µ → eγ). and throughout this analysis, we keep fixed Y f 2 = O(10 −3 ). The parameters shown in Fig. 4 are all allowed from the most stringent constraints of the flavour violating decay [114] BR(µ → eγ) < 4.2 × 10 −13 at 90% CL. We also get negligibly small contributions to the electron as well as muon anomalous magnetic moment for the allowed parameters. These contributions can have positive and/or negative impacts depending upon the choice of parameters. Possibly, with M S = 1000 GeV, M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV, one needs Y f 2 = 29.32, to reproduce the present experimental data δa µ = 2.51 × 10 −11 [83]. However, this choice of couplings and masses violate the LFV as well as perturbative limits Y f i > 4π. Hence, the parameters allowed by relic density are also allowed by the recent electron as well as muon anomalous magnetic moment g − 2 data at Fermilab [83]. However, the converse is not true in our model, as the region allowed by electron and/or muon anomalous magnetic moment g − 2 data violates the relic density, LFV as well as perturbative bounds.
B. FIMP dark matter
We again want to remind the readers about the FIMPs. The main idea is that the dark matter sector gets populated through decay or annihilation of heavy particles until the number density of the corresponding heavy particle species become Boltzmann-suppressed. We need to solve Boltzmann equations that dictate the final relic abundance for the dark matter candidate S. One can easily calculate the decay width or annihilation of the heavy mother particles from the other particles. The coupling strengths are O(1), and we get Γ/H >> 1 for the mother particles. Hence, all the mother particles (SM particles including heavy fermions) remain in the thermal equilibrium in the early Universe. Therefore we do not need to solve the Boltzmann equation for the evaluation of the mother particles, rather we have only solved the evaluation equation for the dark matter produces from the decay and annihilation of various mother particles.
This model can produce dark matter from the decay of the heavy fermions (X Heavy = E ± 1 , E ± 2 and X 0 1 ) and Higgs. It has already been noticed in the existing literature [72,80,81,125,126] that if the same couplings are involved in both decays as well as scattering processes, then the former has the dominant contribution to DM relic density over the latter one. The scattering processes have negligibly small contribution to DM relic density [72,80,81]. With reference to these past studies, we consider that the dark matter candidate is stable in our model and can produce only from the decay of the heavy vector fermions and Higgs.
The Boltzmann equation for the dark matter can be written as [72,81,117],
dn dt + 3Hn = − i S(X Heavy → SS, f SM S), (4.7)
where, X Heavy = E ± 1 , E ± 2 , X 0 1 , H and f SM is SM leptons. Here, the decay-based source term S can be written as,
S = Γ(X Heavy → f SM S, SS) K 1 ( m X Heavy T ) K 2 ( m X Heavy T ) n eq
Heavy,i (4.8)
where, K 1,2 is themodified Bessel function of the first and second kind. For x = m X Heavy T and Y = n T 3 , the Boltzmann equation from eqn. (4.7) now reads [117],
dY (x) dx = i g X Heavy 2π 2 Γ(X Heavy → f SM S, SS) H(x ≈ 1) x 3 K 1 (x),(4.9)
where, g X Heavy is the degrees of freedom of the heavy particle. We can integrate the dark matter production over the entire thermal history and look for the final yield Y (x 0 ) with the help of the appropriate integral [72,81,117]as,
Y (x 0 ) = 45M P l 6.64 π 4 g S √ g ρ i g X Heavy M 2 X Heavy,i Γ(X Heavy,i → f SM S, SS) xmax x min x 3 K 1 (x)dx, (4.10)
with, g S , √ g ρ are the effective numbers of degrees of freedom in the bath at the freeze-in temperature for the entropy, s, and energy density ρ. Finally the relic density 7 (M S ≡ M DM ) can be written as [72,81,117], 6: Dark matter production diagrams from the decay of the heavy particles contribute to the relic density.
Ωh 2 = h 2 3 H 2 0 M 2 P l M S 28 T 3 0 Y (x 0 ) ≈ 1.09 × 10 27 M S i g X Heavy,i Γ(X Heavy,i → f SM S, SS) M 2 X Heavy,i (4.11) f V = E ± 1 , E ± 2 , X 0 1 f SM S H S S H f * V S f SM f V FIG.
We will now follow up the eqn. (4.11) to calculate the relic density. The main production diagrams from the decay of the heavy particles are shown in Fig. 6. The last diagram is kinematically forbidden as we have considered M f V ∼ 1.5 TeV. It is to be noted that, the heavy fermion mass of M f V ∼ 1.5 TeV can lead to dilepton plus transverse missing energy signature 3σ at the LHC for √ s = 14 TeV with an integrated luminosity 3000 fb −1 [105]. The additional four-body decay
(H →f * V f * V , f * V → f SM S ⇒f SM f SM SS)
diagram are suppressed by the heavy fermion propagator. The partial decays of the heavy fermions and Higgs into the dark matter particle are given by,
Γ(f V → f SM S) = M f V 8π |g f V f SM S | 2 (4.12) Γ(H → SS) = 1 32π M H |g HSS | 2 1 − M 2 S M 2 H 1 2 , (4.13)
where, the coupling strengths are (see eqn. 4.7) g HSS = |κv|,
g SiE ± 1 = A f i = |(cos βY f i + sin βY f i )|, g SiE ± 2 = B f i = |(sin βY f i + cos βY f i )| and g Sν i X 0 1 = C f i = |Y f i |, with i = e, µ,
τ . Now we will discuss the numerical analysis for the FIMP dark matter region of the model parameter spaces. Let us first neglect the contribution from the decay and annihilation from the scalar sector, i.e., κ = 0, hence Γ(H → SS) = 0 and σ(SM particles → SS) = 0. It is to be noted that the mass of the dark matter can be set by the independent parameter m S (see eqn. (2.6)). We find that the t-channel annihilation processes The contour lines stand for the relic density in the new Yukawa coupling vs. dark matter mass plane. The red-lines indicate the relic density within the 3σ range. These region with Y f i << 1 are also allowed by the recent LFV [114] BR(µ → eγ) < 4.2 × 10 −13 at 90% CL, electron as well as muon anomalous magnetic moment g − 2 data at Fermilab [83]. The masses M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV and cos β = 0.995, hence M X 0 1 = 1514.9625 GeV.
σ(f V f V → SS) through f SM propagator and σ(f SM f SM → SS) through f V
(i = e, µ, τ ) with the mass of the dark matter are shown in Fig. 7. In the meanwhile, we have neglected the effect from the interaction term − Y f i l i,R E S S by considering Y f i = 0 in these plots.
The dominant contributions (> 95%) to the relic density mainly come from the decay of the fermions E ± 1 and X 0 1 . In both plots, the solid red line represents Ωh 2 = 0.1198, and the red dashed lines correspond to the 3σ variation in Ωh 2 . The lighter region corresponds to higher values of Ωh 2 , which over close the universe and these regions strictly forbidden. The left plot is kept for the dark matter mass region M S ≡ M DM = 6 MeV to 60 MeV, whereas the right plot corresponds to the dark matter mass region 60 MeV to 6 GeV. One can also get the exact relic density for the dark matter mass O(1) keV region, but it will create a problem for the structure formation. We calculate the free streaming length by following the formula given in equation (14) of the Ref. [127],
λ F S = 1 a F S da H 0 F (a) < p DM > a F S (< p DM > a F S ) 2 + (m DM a) 2 ,(4.14)
where, F (a) = Ω rad,0 + aΩ m,0 + a 4 Ω Λ,0 . We get < p DM >≈ M H 2 and a F S is calculated by equating the decay rate of the Higgs into two DM (Γ(H → SS)) to the Hubble expansion rate H ≈ T 2 M P lanck during radiation dominated era and H 0 is the present Hubble expansion rate. The latest values of the cosmological parameters can be found in Ref. [128]. We find the free streaming length larger than 10 Mpc; hence, we avoid showing these regions in this analysis. We also calculate the free streaming length [127,129] of the dark matter particles, and it is coming out to be less than O(100) kpc in the parameter space we have shown in both plots of Fig. 7. Therefore, the dark matter still behave like a cold one [129]. For the heavier dark matter mass region, we need smaller Yukawa coupling as it compensate by the dark matter mass in the decay width. For example, we get the relic density in the right ballpark for Y f i = 1.244 × 10 −9 with M DM = 1 MeV and Y f i = 1.759 × 10 −10 with M DM = 50 MeV and we also get the relic density in the right ballpark for Y f i = 3.935 × 10 −11 with M DM = 1 GeV and Y f i = 1.759 × 10 −11 with M DM = 5 GeV. Hence, we get the relic density for Y f i ∼ O(10 −10 ) through the Freeze-in scenario, and the contributions are tiny O(10 −32 ) to the recent LFV BR(µ → eγ) [114], electron as well as muon anomalous magnetic moment [83] g − 2 data.
We also change the mixing angle from cos β = 0.995 to cos β = 0.60, keep fixed value of the masses M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV. We find a tiny change in the relic density; however, the plots in Fig. 7 remain almost the same. Remarkably, the relic density gets hugely modified for the WIMP dark matter scenario in the Freeze-out mechanism, and they are shown in Fig. 5. The effect of the mixing angle is almost negligible in this Freeze-in mechanism where the dark matter gets produce from the decay of the heavy particles. For the fixed value of cos β = 0.995 and M E ± 1 = 1500 GeV, one can also varied the mass M E ± 2 , we find a very small effect in the relic density as the decay rate of E ± 2 is far smaller than the decay rate of E ± 1 . We now consider the contributions from the Higgs decay in the relic density. In this case, we still assume Y f i = 0, to diminish the contributions from decay of E ± 2 . The larger κ increases the Higgs decays' contributions, whereas large values of Y f i increases the contributions from the vector-like fermions E ± 1 and X 0 1 . We show such variations for two different dark matter masses M DM = 1 MeV and 1 GeV respectively in Fig. 8. The solid red line in both plots corresponds Ωh 2 = 0.1198, and the red dashed lines represent the 3σ variation in Ωh 2 . The lighter region will over close the Universe. Therefore, very small κ = O(10 −10 ) and Y f i = O(10 −10 ) are needed to get the exact relic density in the right ballpark. The additional fine-tune issues in these low dark matter mass regions are: (a) How could one get such a small dark matter mass? (b) How could one get such small couplings? One of the answers might be the Freeze-in mechanism for FIMP dark matter, and we need small coupling and mass to get the exact relic density. In the following collider section, we will discuss, whether it is possible to put additional constraints on DM mass from the indirect dark matter detection and/or from the collider experiments, if we have a tiny dark matter mass. Now, we check the effect for the interaction term Y f il i,R E S S with Y f i = 0. It will increase the contribution from the second charged fermion E ± 2 . If we assume the Y f i = 0 and κ = 0, then in the relic density, the dominant contribution come from the decay of this charged fermion E ± 2 . In Fig. 9, we show the contour regions for the relic density in Y f i vs. dark matter mass (both MeV and GeV region) plane. The red lines within the 3σ range. The contribution from the other charged fermion E ± 1 is less than one percent for this choice of cos β = 0.995.
In the presence of Y f i and κ, one can have additional contributions to the relic density from the decay of charged fermion E ± 1 and Higgs scalar. In these two contour plot in Fig. 10, we keep x-axis reserve for Y f i and y-axis for Y f i and κ respectively. We neglect the contribution from the Higgs decay in the left plot of Fig. 10, i.e., κ = 0 whereas the Y f l = 0 in right plot. We consider fixed dark matter mass at 1 MeV and we keep cos β = 0.995, M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV.
V. INDIRECT DETECTION LIMITS
The normal gamma-ray limits from the Fermi-LAT do not apply for the mass < O(100) MeV, although the data from several other satellites such as HEAO-1 [130], INTEGRAL [131], COMP-TEL [132], EGRET [133] are extremely sensitive to photons with energies well below a O(100) MeV. For a DM annihilation spectrum dN γ /dE, and a galactic DM density profile ρ(r), the galactic contribution to the differential photon flux per unit energy is given by [134],
dΦ γ,G dEdΩ = r ρ Γ A 4π 2 α−1 M DM J D,A dN γ dE , (5.1)
where, Γ A = ρ M DM < σv DM >, r ∼ 8.5 kpc is the Sun's distance from the Galactic center and ρ = 0.3 GeV cm −3 is the local DM density. α = 1(2) for DM decays (annihilations) and J A = l.o.s ds r ρ(s) ρ α is a dimensionless quantity that describes the density of decays or annihilations along the line-of-sight (l.o.s.), and ρ(s) = ρ(r, l, b) can be found in [135]. The extra-galactic photon spectrum arising from dark matter decays; thus, the contribution is zero here. The E 2 dΦ γ,T ot dEdΩ vs. E limits are shown in Fig. 1 of the Ref. [134]. We find the average dark matter annihilation cross-section < σv DM >≈ 10 −64 cm 3 /s in the exact relic density region. We have found that to explain above experimental data, we need a very large κ ∼ Y f l ∼ 25 which violates perturbativity. Hence, the regions considered in this analysis are allowed from these experimental limits.
VI. DIRECT DETECTION AT COLLIDER AS CHARGED TRACK
As we have mentioned that we need a tiny Higgs portal coupling κ ∼ 10 −10 to get the exact relic density; hence, the direct detection limit (e.g., XENON-IT limit [73]) is not applicable for the FIMP dark matter. The detailed collider bounds have been discussed in our previous paper [105] for the same model in the context of WIMP scenario. The authors of the Ref. [136] have discussed the method to prob the FIMP dark matter in the recent LHC collider with high integrated luminosity using the MATHUSLA surface detector. A charged track can be obtained in this model due to the vector fermions' decay into SM fermion and dark matter candidate at the collider. In this model, the length travelled by the charged fermion E ± 1,2 before its decay. Can we get enough events from this charged track? It mainly depends on the production crosssection σ LHC √ s of the mother particle (vector fermion or Higgs) and luminosity L at the detector. The number of events at the LHC is calculated in Ref. [136], and it is given by The author of Ref. [136], find the number of events N events ≥ 3 for √ s = 13 TeV with an integrated luminosity L = 3000 fb −1 using their model parameter spaces. They showed that the MATHUSLA100/200 detector could detect these mother particle up to 1 TeV. The dominant production of the vector fermions come through the Drell-Yan processes.
N events = σ LHC
In this model, we find the production cross-section of the vector like charged fermions with mass M E ± 1 = 1500 GeV is 6.53 × 10 −3 fb [105]. Hence we need large luminosity and/or energy to get a significant event at the present MATHUSLA100/200 surface detector. We find N events > 3 at 14 TeV LHC with an integrated luminosity L = 10 6 fb −1 . The 14 TeV HL-LHC will collect data around 3000 f b −1 , so this search will not be effective. We need wait for the 100 TeV with high luminosity collider.
VII. CONCLUSION
In this work, we study the possibility of dark matter in an extended singlet scalar model. The structure of the model shown here uses a minimum number of particle content. This model contains two additional vector-like charged fermion and a neutral fermion along with a real singlet scalar field. In the previous study [105], we have added one extra vector-like fermion doublet in the model to complete the neutrino framework. We also see that the additional heavy vector-like fermions from this doublet do not alter the dark matter phenomenology as the mixing was taken to be very small. However, the neutrino sector is skipped in this work and rather focus on the WIMP and FIMP dark matter analysis.
We revisited the dark matter analysis through a Freeze-out mechanism considering the collider bound as discussed in Ref. [105]. Here, we show a broad region of the WIMP dark matter parameter spaces which satisfy the relic density at the right ballpark. We choose different dark matter parameter spaces to get the new allowed region from the relic density through the Freeze-out mechanism. In the presence of vector-like fermions, one can get the correct relic density via co-annihilation, or one may have the interaction term such that the dark matter can annihilate into SM particles through additional cross-channel, t-and u-channels. The destructive and/or constructive interference among these channels helps to modify the effective average dark matter annihilation cross-section and provide the exact relic density in this model.
This study also focuses on the low dark matter parameter spaces where the dark matter density can explain through the Freeze-in mechanism. As we know, one cannot get the same viable dark matter parameter space in the low dark matter region as it will either violate the relic density, direct detection constraints or the perturbative limit. Interestingly, in this study, we showed that a single model could explain the low and high dark matter mass region allowed from all the other phenomenological constraints. If, in the future, we able to get any signature at a very low or high dark matter mass region (keV-TeV mass region), our present (including the previous) study could help in estimating a better parameter space.
A few comments on the recent muon anomalous magnetic moment experimental results are made in this model. We see that both the charged particles and the dark matter candidate played a crucial role in providing additional contributions. It was found that one can explain the experimental data throughout the parameter spaces. However, the lepton violating decay channels, perturbative and unitary limits put stringent constraints on the parameter spaces. The dark matter mass region allowed by relic density through Freeze-in and Freeze-out mechanisms is also allowed by the recent muon g − 2 data at Fermilab. However, the converse is not valid in our model, as the region allowed by muon g − 2 data violates the relic density bound. We also perform the collider analysis to search the new particles in the FIMP like scenario in the context of the same 14 TeV LHC experiments with the MATHUSLA100/200 detector. A charged track can be obtained due to the decay of the heavy charged fermion E ± 1,2 into SM fermion and dark matter. One can get events larger than N = 3, when the LHC operated with integrated luminosity L = 10 6 fb −1 .
eqn. 2.4 gives the following eigenvalues for the charged leptons (M N S − M N D M X ) as,
FIG. 2 :
2The DM annihilation diagrams give the relic density. V stands for gauge bosons W, Z; f represents the SM leptons and f are SM leptons and quarks.
FIG. 3 :
3The co-annihilation and annihilation diagrams of the DM and the other Z 2 -odd fermion fields. f are SM leptons and quarks. coupling Y f . The dark matter mass M DM have been scanned from ∼ 5 GeV to 1500 GeV with a step size of 5 GeV while Higgs portal coupling changes from −0.6 to 0.6 with step size 0.001 and new Yukawa coupling from −0.5 to 0.5 with step 0.001. For ∆M ±,0 < 0.1M DM [123] (∆M ± = M E ± 1 − M DM and ∆M 0 = M N − M DM )
left) are strictly ruled out from the direct detection data. In the absence of new fermion interaction (Y f = 0), DM-mass region in between 70 GeV to 450 4 The DM does not participate in diagram 3(b), but this diagram plays an essential role in relic density calculation through co-annihilation processes when M Y − M S ≈ (2 − 10)% of M S . 5 There is a missing diagram corresponding to Sf V → f SM H, mediated by the t-channel exchange of a f V , however, due to propagator-vertex suppression, the annihilation channel have very tiny contribution, therefore we have not included that diagram FIG. 4:
45% BP-6 1285 0.372 1500 0.035 0.112 σ(X 1 S → W ± l) 32% σ(SS → νν) 9% σ(SS → W ± W ∓ ) 2%TABLE I: The benchmark points allowed by all the theoretical and experimental constraints.σ(SS → νν) is mainly dominated by the t + u-channel annihilation processes whereas σ(SS → Y Y ), Y = W, Z, H, t dominated by the s + cross-channel annihilation processes. We consider Y f 1 = Y f 3 = Y f to avoid flavor violating decay processes.
FIG. 5 :
5The blue and red lines indicate the relic density Ω DM h 2 = 0.1198. These plots are generated by varying the dark matter mass and the new Yukawa coupling Y f for different cos β and Y f i . The values Y f i = 0 (left panel) and cos β = 0.995 (right panel) remain fixed. The masses M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV and κ = 0 are same for both cases. These parameters allowed by relic density are also allowed by the recent LFV, electron as well as muon anomalous magnetic moment g − 2 data.
propagator are also suppresses as compared to the decay Γ(f V → f SM S) contributions. The variation of the new vector-like Yukawa coupling Y
FIG. 8 :
8The contour lines stand for the relic density in the new Yukawa coupling vs. dark matter mass plane. The red-lines indicate the relic density within the 3σ range. These region are also allowed by the recent LFV, electron as well as muon anomalous magnetic moment g − 2 data. The masses M E ± 1 = 1500 GeV and M E ± 2 = 3000 GeV and cos β = 0.995, hence M X 0 1 = 1514.9625 GeV.
FIG. 9 :FIG. 10 :
910The contour lines stand for the relic density in the new Yukawa coupling vs. dark matter mass plane. The red-lines indicate the relic density within the 3σ range. The contour lines stand for the relic density in the new Yukawa coupling (Y f i ) vs. the old Yukawa coupling (Y f i ) and κ respectively. The red-lines indicate the relic density within the 3σ range.
In this work, we take both the discrepancies as input value to test whether our model can fit those anomalies or not.
The detailed analysis with radiative corrections is beyond scope of this work
We are getting δα i < 10 −15 (with i = e, µ) hence, lepton anomalous moment bounds are satisfied here. On the contrary, to get exact value of δα i , we need very large Yukawa coupling, which is restricted by the perturbativity bounds.
We also crosscheck by using the output of FeynRules[119] into micrOMEGAs[120] and get the same results.
VIII. ACKNOWLEDGEMENTSThe research work of P.D. and M.K.D. is supported by the Department of Science and Technology, Government of India under the project grant EMR/2017/001436. NK would like to thank to Prof. Dilip Kumar Ghosh for his support at IACS. This project has also received funding/support from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska -Curie grant agreement No 860881-HIDDeN".
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| []
|
[
"A Freely Available Wide Coverage Morphological Analyzer for English",
"A Freely Available Wide Coverage Morphological Analyzer for English"
]
| [
"Daniel Karp \nDepartment of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA\n",
"Yves Schabes \nDepartment of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA\n",
"Martin Zaidel \nDepartment of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA\n",
"Dania Egedi \nDepartment of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA\n"
]
| [
"Department of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA",
"Department of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA",
"Department of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA",
"Department of Computer and Information Science\nUniversity of Pennsylvania Philadelphia PA\n19104-6389USA"
]
| [
"Appears in the Proceedings of the 14th International Conference on Computational Linguistics (COLING '92)"
]
| This paper presents a morphological lexicon for English that handle more than 317000 in ected forms derived from over 90000 stems. The lexicon is available in two formats. The rst can be used by an implementation of a two-level processor for morphological analysis(Karttunen and Wittenburg, 1983;Antworth, 1990). The second, derived from the rst one for e ciency reasons, consists of a disk-based database using a UNIX hash table facility(Seltzer and Yigit, 1991). We also built an X Window tool to facilitate the maintenance and browsing of the lexicon. The package is ready to be integrated into an natural language application such as a parser through hooks written in Lisp and C.To our knowledge, this package is the only available free English morphologicalanalyzer with very wide coverage. | 10.3115/992383.992409 | [
"https://arxiv.org/pdf/cmp-lg/9410024v1.pdf"
]
| 5,480,885 | cmp-lg/9410024 | e8732223b218ec0487827f709c7b54a9e704b4a6 |
A Freely Available Wide Coverage Morphological Analyzer for English
August 1992
Daniel Karp
Department of Computer and Information Science
University of Pennsylvania Philadelphia PA
19104-6389USA
Yves Schabes
Department of Computer and Information Science
University of Pennsylvania Philadelphia PA
19104-6389USA
Martin Zaidel
Department of Computer and Information Science
University of Pennsylvania Philadelphia PA
19104-6389USA
Dania Egedi
Department of Computer and Information Science
University of Pennsylvania Philadelphia PA
19104-6389USA
A Freely Available Wide Coverage Morphological Analyzer for English
Appears in the Proceedings of the 14th International Conference on Computational Linguistics (COLING '92)
Nantes, FranceAugust 1992cmp-lg/9410024 24 Oct 94
This paper presents a morphological lexicon for English that handle more than 317000 in ected forms derived from over 90000 stems. The lexicon is available in two formats. The rst can be used by an implementation of a two-level processor for morphological analysis(Karttunen and Wittenburg, 1983;Antworth, 1990). The second, derived from the rst one for e ciency reasons, consists of a disk-based database using a UNIX hash table facility(Seltzer and Yigit, 1991). We also built an X Window tool to facilitate the maintenance and browsing of the lexicon. The package is ready to be integrated into an natural language application such as a parser through hooks written in Lisp and C.To our knowledge, this package is the only available free English morphologicalanalyzer with very wide coverage.
Introduction
Morphological analysis has experienced great success since the introduction of two-level morphology (Koskenniemi, 1983;Karttunen, 1983). Two-level morphology and its implementation are now well understood both linguistically and computationally (Karttunen, 1983;Karttunen and Wittenburg, 1983;Koskenniemi, 1985; Barton et al., 1987;Koskenniemi and Church, 1988). This computational model has proved to be well suited for many languages. Although there are some proprietary wide coverage morphological analyzers for English, to our knowledge those that are freely available provide only very small coverage.
Working from the 1979 edition of the Collins Dictionary of the English Language available through ACL-DCI (Liberman, 1989), we constructed lexicons for PC-KIMMO (Antworth, 1990), a public domain implementation of a two-level processor. Using the morphological rules for English in ections provided by Karttunen and Wittenburg (1983) and our lexicons, PC-KIMMO outputs all possible analyses of each input word, giving its root form and its in ectional This work was partially supported by DARPA Grant N0014-90-31863, ARO Grant DAAL03-89-C-0031, and NSF Grant IRI90-16592. We thank Aravind Joshi for his support for this work. We also thank Evan Antworth, Mark Foster, Lauri Karttunen, Mark Liberman, and Annie Zaenen for their help and suggestions. y Visiting from Stanford University.
attributes. To improve performance, we used PC-KIMMO as a generator on our lexicons to build a diskbased hashed database with a UNIX database facility (Seltzer and Yigit, 1991). Both formats, PC-KIMMO and database, are now available for distribution. We also provide an X Window tool for the database to facilitate maintenance and access. Each format contains the morphological information for over 317000 English words. The morphological database for English runs under UNIX; PC-KIMMO runs under UNIX and on a PC. This package can be easily embedded into a natural language parser; hooks for accessing the morphological database from a parser are provided for both Lucid Common Lisp and C. This morphological database is currently being used in a graphical workbench (XTAG) for the development of tree-adjoining grammars and their parsers (Paroubek et al., 1992).
Lexicons for PC-KIMMO
We used the set of morphological rules for English described by Karttunen and Wittenburg (1983). The rules handle the following phenomena (among others 1 ): epenthesis, y to i correspondences, s-deletion, elision, i to y correspondences, gemination, and hyphenation. In addition to the set of rules, PC-KIMMO requires lexicons. We derived PC-KIMMO-style lexicons from the 1979 edition of the Collins Dictionary of the English Language. The 90000-odd roots 2 in the lexicon yield over 317000 in ected forms.
The lexicons use the following parts of speech: verbs (V), pronoun (Pron), preposition (Prep), noun (N), determiner (D), conjunction (Conj), adverb (Adv), and adjective (A). Figure 1 shows the distribution of these parts of speech in the two formats: The rst column is the distribution of the root forms in the PC-KIMMO lexicon les, and the second column is the distribution for the in ected forms derived from the lexicons and stored in the database. For each word, the lexicon lists its lexical form, a continuation class, and a parse. The continuation class speci es which in ections the lexical form can undergo. At most, a noun root engenders four in ections (singular, plural, singular genitive, plural genitive); an adjective root, three (base, com-parative, superlative); and a verb root, ve (in nitive, third-person singular present, simple past, past participle, progressive). The exact number generated by any given root depends on its continuation class.
Adjectives
The continuation classes for adjective specify that the word can undergo the rules of comparative and superlative. For example, the lexicon entry for the adjectivè funky' is:
funky A Root2 "A(funky)"
The entry consists of a word funky, followed by the continuation class A Root2, and a parse "A(funky)". The continuation class speci es that the word can undergo the normal rules of comparative and superlative, and the parse states that the word is an adjective with root`funky'. The following is a sample run of PC-KIMMO's recognizer: recognizer funky funky A(funky) recognizer funkier funky+er A(funky) COMP recognizer funkiest funky+est A(funky) SUPER
The output line contains the root form and any afxes, separated by`+'s. Thus, a`+' in the output indicates a morphological rule was used; its absence means no rule was used, and the parse was returned as found in the lexicon. PC-KIMMO will automatically add attributes such as COMP and SUPER to the parse, depending on the morphological rule matched by the surface form. But for irregularly in ected forms, special continuation classes indicate that the complete parse (viz., part of speech, root, and attributes) should be takeǹ as is' from the lexicon entry. For example:
better A Root1 "A(good) COMP" best A Root1 "A(good) SUPER" good A Root1 "A(good)"
The class A Root1 tells PC-KIMMO not to apply the morphological rules to`better',`best', and`good'. Thus,`gooder' is not recognized as`good+er'. The attributes (such as COMP) can later be translated into feature structures with the help of templates as in PATR (Shieber, 1986). The list of attributes is found in Appendix A.
Nouns
In ections of nouns, such as the formation of plural and genitive, are handled by morphological rules (unless the formation is idiosyncratic). In the lexicon for nouns, the continuation class N Root1 indicates that the formation of genitive applies regularly and that no other in ection applies. The continuation class N Root2 indicates that the formation of the plural and of the genitive apply regularly. mice N Root1 "N(mouse) PL" mouse N Root1 "N(mouse) SG" ambassador N Root2 "N(ambassador)" Thus, the above lexicon entries are recognized as below:
Verbs
Given the in nitive form of a verb, the formation of the third person singular (+s), its past tense (+ed), its past participle (+ed), and its progressive form (+ing) is handled by morphological rules unless lexical idiosyncrasies apply. In order to encode all possible idiosyncrasies over the three verb endings, eight continuation classes are de ned (see Figure 2). Each continuation class speci es the in ectional rules which can apply to the given lexical item. Continuation class Applicable rules V Root1 none V Root2 +ed V Root3 +s V Root4 +s, +ed V Root5 +ing V Root6 +ing, +ed V Root7 +ing, +s V Root8 +ing, +s, +ed Examples of lexical entries for verbs follow:
admire V Root8 "V(admire)" dyeing V Root1 "V(dye) PROG" dye V Root4 "V(dye)" zigzagging V Root1 "V(zigzag) PROG" zigzagged V Root1 "V(zigzag) PAST WK" zigzagged V Root1 "V(zigzag) PPART WK" zigzag V Root3 "V(zigzag)" tangoes V Root1 "V(tango) 3SG PRES" tango V Root6 "V(tango)" taught V Root1 "V(teach) PAST STR" taught V Root1 "V(teach) PPART STR" teach V Root7 "V(teach)"
Examples of runs follow: The attributes WK (for \weak") and STR (for \strong") mark whether the verb forms its past tense regularly or irregularly, respectively. The distinction enables unambiguous reference to homographs|words spelled identically but with di erent semantic and syntactic properties. For example, the verb`lie' with the meaning`to make an untrue statement' and the verb lie' with the meaning`to be prostrate' have di erent syntactic and morphological behavior: the rst one is regular, while the second one is irregular:
He has lain on the floor. He has lied about everything.
Usually, it su ces to index the syntactic properties of each verb by its root form alone. However, homographs require addition information. In English, the attributes WK and STR are su cient to distinguish homographs with di erent morphological behavior.
Lexicons as a Database
PC-KIMMO builds in memory a data structure from the complete lexicon. Consequently, our large lexicons occupy more than 19 Mbytes of process memory. Further, the large size of the structure implies long search times as PC-KIMMO swaps pages in and out. Thus, to solve both the time and space problems simultaneously, we compiled all in ectional forms into a disk-based database using a UNIX hash table facility (Seltzer and Yigit, 1991).
To compile the database, we used PC-KIMMO as a generator, inputting each root form and all the endings that it could take, as indicated by the continuation class. The resulting in ected form became the key, and the associated morphological information was then inserted into the database.
For example, the PC-KIMMO lexicon le contains the entry: saw N Root2 "N(saw)"
The class N Root2 indicates that the noun`saw' forms its plural, singular genitive, and plural genitive regularly. Thus, we send to the generator three lexical forms and the three su xes for each in ection, extracting three in ected surface forms:
Lexical saw+s saw+'s saw+s+'s Surface saws saw's saws'
The root form of a noun is identical with the singular in ection, so we have a total of four in ected forms. Since we know which su x we added to the root, we also know the attributes for that in ection. The in ected form becomes the key, while the part of speech, root, and attributes are stored as the content in the database. Hence, the lexicon entry for the nouǹ saw' produces four key{content pairs in the database: (saw, saw N SG), (saws, saw N PL), (saw's, saw N SG GEN), (saws', saw N PL GEN).
Likewise, the verb lexicon contains the entries:
saw V Root8 "V(saw)" saw V Root1 "V(see) PAST STR"
The continuation class V Root8 indicates four in ections besides the in nitive: third-person singular (+s), past (+ed), weak past participle (+ed), and present participle (+ing). Hence, the generator produces: Lexical saw+s saw+ed saw+ing Surface saws sawed sawing The class V Root1 allows no in ections, but builds the in ection{feature pair directly: (saw, see V PAST STR).
Hence, morphological analysis is reduced to sending the surface forms to the database as keys and retrieving the returned strings. Figure 3 lists the database keys and content strings produced by the three lexicon lines given above. Note that distinct entries are separated by`#'. Since multiple lexical forms can map to the same surface form, the actual number of keys (ca. 292000) is less than the number of lexical forms (ca. 317000). Also, with the database residing on the disk, access times average 6 to 10 milliseconds, which greatly improves upon PC-KIMMO.
Implementation Considerations
The large number of keys implies a very large disk le. To reduce the size of the le, we take advantage of the morphological similarity in English between an in ected form and its lexical root form. Indeed, the root is often contained intact within the in ected form.
Key
Contents saw saw N SG#saw V INF#see V PAST STR saws saw N PL#saw V 3SG PRES saw's saw N SG GEN sawing saw V PROG sawed saw V PAST WK#saw V PPART WK saws' saw N PL GEN
Figure 3: Database pairs
Hence, instead of storing the root, we store the number of shared characters along with any di ering characters, and reassemble the root from the in ected form on each database query. Further, despite the large set of attributes, relatively few combinations (ca. 80) are meaningful, and can be encoded in a single byte. Since a large proportion of roots are wholly contained within the surface form, and since 92% of the keys have one lexical entry, the average content string is only three bytes long. Consequently, the total disk le is under 9Mbytes. We anticipate further compaction in the near future.
Accompanying Utilities
Besides the PC-KIMMO lexicons, we currently maintain the database le and an ASCII-character \ at" version for on-line database browsing. One program converts the lexicons into the database format, while others dump the database into the at le or reconstruct the database from the at le. We have also built a X Windows tool to perform maintenance on the database le (see Figure 4). This tool automatically maintains the consistency between the at le and the database le. We have built hooks in C and Lisp (Lucid 4.0) to access either the database or PC-KIMMO from within a running process. The PC-KIMMO lexicons, the database les, the LISP and C access functions, programs for converting between formats, and the X Window maintenance tool are available without charge for research purposes. Please send e-mail to [email protected].
Conclusion
We have presented freely available morphological tables and a morphological analyzer to handle English in ections. The tables handle approximately 317000 in ected forms corresponding to 90000 stems.
These tables can be used by an implementation of a two-level processor for morphological analysis such as PC-KIMMO.
However, these large tables degrade the performance of PC-KIMMO's current implementation, requiring about 18 Mbytes of RAM while slowing the access time.
To overcome these shortcomings, we created a morphological analyzer consisting of a disk-based database using a UNIX hash table facility. With this database, access times average 6 to 10 milliseconds while moving all of the data to the disk. We also provide an X Window tool for facilitating the maintenance and access to the database.
The package is ready to be integrated into an application such as a parser. Hooks written in Lisp and C for accessing these tables are provided.
To our knowledge, this package is the only available free English morphologicalanalyzer with very wide coverage.
Figure 2 :
2Continuation classes for verbs
Figure 4 :
4Morphological
We refer the reader toKarttunen and Wittenburg (1983) orAntworth (1990) for more details on the morphological rules.2 Proper nouns were not included in the tables.
PC-KIMMO: a two-level processor for morphological analysis. Evan L Antworth, Summer Institute of LinguisticsEvan L. Antworth. 1990. PC-KIMMO: a two-level pro- cessor for morphological analysis. Summer Institute of Linguistics.
G Edward Barton, Robert C Berwick, Eric Sven Ristad, Computational Complexity and Natural Language. MIT PressG. Edward Barton, Robert C. Berwick, and Eric Sven Ristad. 1987. Computational Complexity and Natu- ral Language. MIT Press.
A twolevel morphological analysis of English. Lauri Karttunen, Kent Wittenburg, Texas Linguistic Forum. 22Lauri Karttunen and Kent Wittenburg. 1983. A two- level morphological analysis of English. Texas Lin- guistic Forum, 22:217{228.
KIMMO: A two-level morphological analyzer. Lauri Karttunen, Texas Linguistic Forum. 22Lauri Karttunen. 1983. KIMMO: A two-level morpho- logical analyzer. Texas Linguistic Forum, 22:165{ 186.
Two-level morphology: a general computational model for word-form recognition and production. Kimmo Koskenniemi, Helsinki, FinlandUniversity of HelsinkiTechnical reportKimmo Koskenniemi. 1983. Two-level morphology: a general computational model for word-form recogni- tion and production. Technical report, University of Helsinki, Helsinki, Finland.
An application of the twolevel model to Finnish. Kimmo Koskenniemi, Computational Morphosyntax: Report on Research. Fred KarlssonUniversity of HelsinkiKimmo Koskenniemi. 1985. An application of the two- level model to Finnish. In Fred Karlsson, editor, Computational Morphosyntax: Report on Research 1981-1984. University of Helsinki.
Complexity, two-level morphology and Finnish. Kimmo Koskenniemi, Kenneth W Church, Proceedings of the 12 th International Conference on Computational Linguistics (COLING'88). the 12 th International Conference on Computational Linguistics (COLING'88)Kimmo Koskenniemi and Kenneth W. Church. 1988. Complexity, two-level morphology and Finnish. In Proceedings of the 12 th International Conference on Computational Linguistics (COLING'88).
Text on tap: the ACL data collection initiative. Mark Liberman, Proceedings of DARPA Workshop on Speech and Natural Language Processing. DARPA Workshop on Speech and Natural Language ProcessingMorgan KaufmanMark Liberman. 1989. Text on tap: the ACL data col- lection initiative. In Proceedings of DARPA Work- shop on Speech and Natural Language Processing, pages 173{188. Morgan Kaufman.
XTAG { a graphical workbench for developing tree-adjoining grammars. Patrick Paroubek, Yves Schabes, Aravind K Joshi, Third Conference on Applied Natural Language Processing. Trento, ItalyPatrick Paroubek, Yves Schabes, and Aravind K. Joshi. 1992. XTAG { a graphical workbench for developing tree-adjoining grammars. In Third Conference on Applied Natural Language Processing, Trento, Italy.
A new hashing package for UNIX. Margot Seltzer, Ozan Yigit, USENIX. Margot Seltzer and Ozan Yigit. Winter 1991. A new hashing package for UNIX. In USENIX.
An Introduction to Unication-Based Approaches to Grammar. Center for the Study of Language and Information. M Stuart, Shieber, Stanford, CAStuart M. Shieber, 1986. An Introduction to Uni - cation-Based Approaches to Grammar. Center for the Study of Language and Information, Stanford, CA.
| []
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"Universal Trimers from Three-Body Interactions in One-Dimensional Lattices",
"Universal Trimers from Three-Body Interactions in One-Dimensional Lattices"
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"Arthur Christianen \nInstitute for Molecules and Materials\nRadboud University\nNijmegenThe Netherlands\n",
"John Sous \nDepartment of Physics and Astronomy\nUniversity of British Columbia\nV6T 1Z1VancouverBritish ColumbiaCanada\n\nITAMP\nHarvard-Smithsonian Center for Astrophysics\n02138CambridgeMassachusettsUSA\n\nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n"
]
| [
"Institute for Molecules and Materials\nRadboud University\nNijmegenThe Netherlands",
"Department of Physics and Astronomy\nUniversity of British Columbia\nV6T 1Z1VancouverBritish ColumbiaCanada",
"ITAMP\nHarvard-Smithsonian Center for Astrophysics\n02138CambridgeMassachusettsUSA",
"Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA"
]
| []
| We investigate the formation of trimers in an infinite one-dimensional lattice model with singleparticle hopping t and hard-core two-body U and three-body V interactions of relevance to Rydberg atoms and polar molecules. For sufficiently attractive U ≤ −2t and positive V > 0 a large trimer is stabilized, which persists as V → ∞, while both attractive U ≤ 0 and V ≤ 0 bind a small trimer. Surprisingly, the excited state above this small trimer is also bound and has a large extent; its behavior as V → −∞ resembles that of the large ground-state trimer. | null | [
"https://arxiv.org/pdf/1806.10647v2.pdf"
]
| 155,100,279 | 1806.10647 | 5a83f4a99a5eeabb3cb3a4282f763292a06afd88 |
Universal Trimers from Three-Body Interactions in One-Dimensional Lattices
Arthur Christianen
Institute for Molecules and Materials
Radboud University
NijmegenThe Netherlands
John Sous
Department of Physics and Astronomy
University of British Columbia
V6T 1Z1VancouverBritish ColumbiaCanada
ITAMP
Harvard-Smithsonian Center for Astrophysics
02138CambridgeMassachusettsUSA
Department of Physics
Harvard University
02138CambridgeMassachusettsUSA
Universal Trimers from Three-Body Interactions in One-Dimensional Lattices
We investigate the formation of trimers in an infinite one-dimensional lattice model with singleparticle hopping t and hard-core two-body U and three-body V interactions of relevance to Rydberg atoms and polar molecules. For sufficiently attractive U ≤ −2t and positive V > 0 a large trimer is stabilized, which persists as V → ∞, while both attractive U ≤ 0 and V ≤ 0 bind a small trimer. Surprisingly, the excited state above this small trimer is also bound and has a large extent; its behavior as V → −∞ resembles that of the large ground-state trimer.
Introduction.-Few-body physics forms the basis of our understanding of the microscopic building units of the universe [1]. It contributes to a plethora of fundamental phenomena, including Efimov's universality [2], quantum impurities in cold gases [3,4], quasiparticles [5,6] and quasiparticle pairing [7][8][9] in nanoscale systems, the fractional quantum Hall effect [10], nuclear systems [11] and neutrons [12].
A principal problem in this field is one of particles in a central potential, and the ensuing binding of bound states. One intriguing effect prevalent in continuum systems is the formation of shallow bound states, which extend beyond the range of the potential. Such a feeble bound state can be the lowest-energy state of the system, such as one formed in a delta-function potential in lower dimensions, or an excited state, such as Feshbach molecules [13] and halo states [14]. Lattice systems with local two-body interactions do not host shallow excited bound states [15,16]. It is therefore important to determine whether conditions exist under which shallow excited bound states can form in lattice systems in presence of higher-body interactions, e.g. three-body interactions.
In this work, we demonstrate that lattice systems with purely local nearest-neighbor two-and three-body interactions host bound states that extend well beyond the range of the binding forces, giving way to an emergent universality in one dimension [17][18][19] distinct from Efimov's universality. Namely, we demonstrate that a combination of two-site U and three-site V interactions stabilize universal large three-body bound states, which are either the ground state (for V > 0) or the first excited state (for V < 0) of the system. Tuning the strengths of interactions allows control over the size of the bound states, providing access to the crossover between universal and non-universal few-body physics in experiments.
Model.-We consider a minimal one-dimensional model of structureless fermions (e.g. spinless electrons) or hard-core bosons with nearest-neighbor (NN) hopping, and two-and three-body interactionŝ
H =−t i (ĉ † iĉ i+1 +ĉ † i+1ĉ i ) + U in ini+1 +V in ini+1ni+2 ,(1)
where t is the hopping amplitude, U is the NN two-body interaction and V is the NN three-body interaction, i is the site index,ĉ † (ĉ) is the particle creation (annihilation) operator, andn is the particle number operator. This model in the NN approximation serves to provide insight into the physics of the dominant three-body interactions in a wide range of experiments.
Dimers.-A nonzero value of |U | > 2t is required to bind a dimer state, so as to compensates for the kinetic energy lost in binding [15,20].
Trimers.-We study three-particle states in the infinite chain by solving the equation of motion for the Green's functionĜ(ω) = (ω + iη −Ĥ) −1 . We derive an exact hierarchy of equations of motion for three-particle propagators G(m 1 , m 2 ; n 1 , n 2 ; K, ω) = K, m 1 , m 2 |Ĝ(ω)|K, n 1 , n 2 defined for states |K, n 1 , [20]. A stable attractively (repulsively) bound trimer (also known as trion) corresponds to the appearance of a discrete pole in the Green's function below (above) the continuum of scattering states.
n 2 = 1 √ N i e iKRiĉ † i−n1ĉ † iĉ † i+n2 |0
Trimer stability diagram.-To identify stable trimers we search for discrete peaks outside of the three-particle continuum. This consists of scattering states of three free particles, 1 + 1 + 1, and those of a dimer and a free particle, 2 + 1. In the current work, we discuss trimers formed below the continuum (U/t < 0), i.e. attractively bound trimers.
In Figure 1, we plot the stability diagram for bound states with total quasimomentum K = k 1 + k 2 + k 3 = 0. The solid blue line identifies the stability behavior of attractive trimers. To characterize different regimes of physical behavior, we compute the average size of the trimer M , where M = n 1 + n 2 is the distance between the two outer particles in a given configuration of the
trimer.
First consider the upper-left quadrant of the diagram (V ≤ 0, U ≥ −2t). For U = 0, a bound trimer (blue region of Figure 1) appears for V −5.5t with particles tightly bound in the trimer state M ≤ 3 as expected of the short-range three-body attraction. Increasingly attractive U values lead to more tightly bound trimers and naturally lowers the V needed for binding. Now consider the lower-right quadrant (U ≤ −2t, V ≥ 0). Surprisingly, for sufficiently attractive U ≤ −2t, trimers are alway stable regardless of the magnitude of the repulsive V . This behavior persists for extremely large V (not shown). The large V effectively pushes the particles in the trimer apart as it becomes energetically costly to occupy three consecutive sites, but fails to completely break down the trimer. These exotic large trimers with M > 3 are bound by non-perturbative higher-order interactions.
We now discuss the lower-left quadrant of Figure 1, (U ≤ −2t, V ≤ 0). As expected, these strongly attractive U and V bind a small trimer. Interestingly, however, a second bound state appears below the continuum, see also the inset of single lattice spacing, demonstrating the possibility of an emergent long-wavelength continuum description and universal low-energy physics, see discussion below.
U =−2.5t V =0 V =t V =3t V =9t (b)U =−2.5t V =−9t V =−5t V =−3t V =−t
We note in passing that, for K = 0, only small, and no large, repulsively bound trimers appear above the continuum (not shown).
Trimer structure.-To shed light on the mechanism behind the formation of trimers and their structure, we analyze the probability density of the trimer eigenstates |0, α T at quasimomentum K = 0.
P (M ) = n1+n2=M | 0, n 1 , n 2 |0, α T | 2(2)
We study P (M ) as a function of V for a fixed U = −2.5t in Figure 2 for (a) the GS trimers and (b) the ES trimers.
The size of the GS trimer evolves with V from small ( M ≈ 3) to large ( M > 3) (Figure 2(a)), see also the dotted line in Figure 1. This crossover behavior is characterized by a shift in the maximum of P (M ) to larger values. In comparison, the ES trimer is much more extended, however its spread also grows with V ( Figure 2(b)). That these bound states extend over several lattice sites is an indication of universality in the sense that the binding energy depends only on the two-body a 2 and three-body a 3 scattering lengths [17].
To corroborate this picture, we study the binding energy E B of the trimer bound states. In Figure 3, we plot E B along with M and its spread for the GS (blue) and ES (salmon) trimers as a function of V for an exemplary U = −2.5t at K = 0. As expected, for V < 0, E B (solid line) of the GS trimer grows with |V |, saturating at the smallest possible size of M = 2 with essentially no spread. For repulsive V > 0, the binding energy decreases, asymptotically approaching E B ≈ 0.0225t (horizontal solid line), and both M and its spread increase, saturating at M ≈ 10.14. Intriguingly, we find the same asymptotic behavior for the ES trimer as V → −∞ (we have verified this numerically).
We can understand this behavior as follows. In the limit V → −∞, the ground-state trimer |Ψ GS asymptotically approaches the state with the smallest possible size and no spread, i.e. |K, 1, 1 . The ES trimer must be orthogonal to the GS trimer, and in this limit we find Ψ ES |Ψ GS → Ψ ES |K, 1, 1 = 0. On the other hand, in the limit V → ∞, the NN configuration |K, 1, 1 in the trimer wavefunction is energetically forbidden. This reflects in the relation Ψ GS |K, 1, 1 = 0 . The problem of finding the Hamiltonian spectrum requires diagonalizing the Hamiltonian operator whose structure then takes the same exact form in these two asymptotic limits, explaining the resemblance between the asymptotic forms of the ES and GS trimers.
The asymptotic saturation of E B and M of the GS and ES trimers is yet another indication of universal behavior, which we now discuss. We find that the large GS and ES trimers asymptotically behave as [21]. In the long-wavelength limit, the trimer's binding energy depends only on a 3 , E B = 1/ma 2 3 . Identifying m −1 = 2ta 2 , we find, for U = −2.5t, a 3 ≈ 2 0.0225 a exp − 1 2 γ(−2.5t) t V . This exponential dependence of a 3 on 1/V (or equivalently, the inverse logarithmic dependence of V on a 3 , i.e. V ∝ −1/ ln(a 3 Λ), where Λ ∼ a −1 is the momentum cutoff) is a signature of three-body universality in one dimension [17][18][19].
E B → E 0 (U ) exp γ(U )t/V , where E 0 (−2.5t) ≈ 0.0225t and γ(−2.5t) ≈ 0.5π
Binding mechanism of large trimers.-We now turn to the binding mechanism of the large GS trimers, which are stable despite the strong three-body repulsion. In Figure 4 we analyze E B along with the corresponding M of the GS trimer for V = 1000 t at K = 0 as a function of U . As expected E B increases with increasingly attractive U , but only up to U ∼ −3.9t. At this U , E B develops a maximum followed by a rapid decrease. This striking behavior accompanies an opposite trend in M which has a minimum roughly coinciding with the maximum in E B .
V =1000t U =−2.2t U =−2.4t U =−3t U =−4t U =−10t
To explain this maximum in the binding energy as a function of U , we consider the probability density
P (∆) = | 0, ∆, M − ∆|0, α T | 2 P (M )(3)
to find the central particle at a distance ∆ from the outer left particle in the trimer for a given M component of the wavefunction. In Figure 5, we plot P (∆) for the M = 8 component of the GS trimer wavefunction at a fixed V = 1000t for different values of U . Simple perturbative arguments suggest that binding should be facilitated by the formation of configurations with NN particles ∆ = 1, 7 as a result of the attractive NN twobody interaction. Surprisingly, for small |U | the central particle is only slightly more likely to be NN to either outer one and has a large probability to be anywhere in between. This is an example of the rare occurrence where a perturbatively small term has a large effect on the behavior of the system. A larger attractive U naturally favors NN configurations with ∆ = 1, 7.
With this insight in hand we can qualitatively explain the behaviour in Figure 4. Larger attractive U forces the central particle closer to either of the two outer particles. For moderate U > −3.9t in Figure 4 this favors a smaller trimer, as intuitively expected. Larger U < −3.9t, however, forces the trimer into configurations with two NN particles and the third further apart (e.g. U = −10t result in Figure 4) accompanied by an increase in M . This trimer configuration is a weakly bound state of a strongly bound dimer and a particle. Remarkably, M ± σ (shaded region of Figure 4), where σ is the standard deviation of P (M ), shows larger spread for more attractive U corroborating this picture of a dimer and a loosely bound third particle.
These results point to a non-perturbative binding mechanism: The large timers are bound by higher-order interactions that mediate long-range binding yet avoid the forbidden M = 2 configuration. Furthermore, this pattern of decrease in E B for large trimers composed of NN pairs and a loosely bound particle indicates that configurations with the central particle 'free' in between the outer two play a crucial role in binding. There, the central particle mediates a three-body force through pairwise interactions with the outer two. This is most efficient in configurations with the central particle close to both the outer two, a situation favorable in smaller trimers formed for modest U . Larger U forces the central particle closer to one of the outer two, ultimately weakening the binding to the other one, which leads to a larger trimer with a 2 + 1-like structure.
Concluding remarks.-We studied the interplay of two-and three-body interactions in a minimal onedimensional lattice model. We constructed three-body bound-state stability diagram identifying regions in parameter space of attractively bound trimers. Trimers form even in the limit of infinite three-body repulsion. An ES bound trimer appears for attractive V and persists as V → −∞, where it develops asymptotic behavior similar to that of the GS trimer as V → ∞.
These large trimers are bound by non-perturbative long-range forces mediated by short-range interactions, which favor large configurations with the central particle free in between the outer two. They extend over several lattice spacings pointing to an emergent long-wavelength universality, and are thus of great interest to efforts targeting the creation of large coherent quantum objects with non-trivial internal structure.
Our analysis applies to few-body bound states realized, for example, with polar molecules in optical lattices [22] or Rydberg atoms in tweezers [23], and to systems with three-site blockade (V → ∞ limit), such as Coulomb blockaded Rydberg gases [24] and quantum dots [25]. Other potential experimental systems with few-body interactions include trapped ultracold gases [26][27][28], ultracold atoms in optical lattices [29][30][31][32][33], Rydberg excitations in cold gases [34][35][36][37][38][39], Rydberg slow light polaritons [40][41][42][43], ion traps [44], optics coupled-cavity arrays [45], and circuit QED systems [46], where many of the ideas we discuss and others [47] can be investigated. We note that our method allows the simulation of spectroscopy in the frequency domain (inset of Figure 1) and can be extended to analyze the time-resolved response in one and higher dimensions.
Note that our results imply universality for fermionic trimers in one dimension.
An interesting question arises whether statistics play a role in universality in one dimension when the equivalence between hard-core bosons and spinless fermions [48] breaks down, e.g. for soft-core interactions. Another emergent line of inquiry is whether the universal correspondence between GS and ES complexes persists for larger number of particles.
FIG
. 1. (color online) Trimer stability diagram at K = 0 as a function of V and U in units of t. The grey area indicates the continuum. The solid blue line identifies the boundary of the stability region of attractively bound states, while dashed red lines identify regions where an excited-state (ES) trimer co-exists with the ground-state (GS) trimer. The crossover from small ( M ≤ 3) to large ( M > 3) GS trimers is indicated by the dotted blue line. In the inset, we show the spectral function A(ω) = − 1 π Im G(1, 1; 1, 1; 0, ω) for the parameter values indicated by the cross: V = −2.5t, U = −2.5t, demonstrating the appearance of the GS (blue) and ES (red) trimer peaks below the edge of the continuum (dashed line).
Figure 1 .
1These feebly bound excitedstate (ES) trimers are extended ( M > 3) similar to the ground-state (GS) trimers at large repulsive V . The large trimer states extend beyond the scale of a
FIG. 2 .
2(color online) Analysis of the size of trimers: The probability P (M ) = n 1 +n 2 =M | 0, n1, n2|0, αT | 2 for the two outer particles in a trimer to be M = n1 + n2 sites apart at U = −2.5t and various values of V for the (a) groundstate (GS) trimers (lower-right quadrant ofFigure 1) and (b) excited-state (ES) trimers (lower-left quadrant ofFigure 1). The two trimers exhibit qualitatively similar behavior with increasing V (compare lines of the same colors in (a) and (b)). Further analysis of the M = 8 component of the GS trimer is presented inFigure 5.
FIG. 3 .
3(color online) Binding of ground-state (GS) (blue) and excited-state (ES) (salmon) trimers for U = −2.5t at K = 0 as a function of V . We plot the binding energy EB (solid and dashed lines) and the average trimer size M (dotted lines) with M ± σ (boundary of the shaded regions), where σ is the standard deviation of P (M ). EB approaches the horizontal black line in the asymptotic limit V → −∞(∞) for the ES (GS) trimer.
online) Binding mechanism of the large GS trimer for V = 1000t at K = 0 as a function of U . We plot the binding energy EB (solid green line), the average trimer size M (dashed green line), and M ± σ (boundary of shaded regions), where σ is the standard deviation of P (M ). The shaded region shows the spread of P (M ) about the average M .
FIG. 5 .
5(color online) Analysis of the internal structure of the large GS trimer through the probability P (∆) =| 0,∆,M −∆|0,α T | 2 P (M )to find the central particle ∆ sites apart from the outermost left particle for the M = 8 component of the trimer wavefunction for V = 1000t at K = 0 and for different values of U .
Author to whom correspondence should be addressed. Current address: Department of Physics. New York, New York 10027, USAColumbia UniversityEmail: [email protected]* Author to whom correspondence should be addressed. Current address: Department of Physics, Columbia University, New York, New York 10027, USA. Email: [email protected]
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[
"The fractal dimension of music: Melodic contours and time series of pitch",
"The fractal dimension of music: Melodic contours and time series of pitch"
]
| [
"Maria H Niklasson \nCathedral School\nBetel Music Institute\nCampus Bromma, Åkeshovsvägen 29SE-75375, SE-16839Uppsala, BrommaSweden, Sweden\n",
"Gunnar A Niklasson \nDepartment of Engineering Sciences\nThe Ångström Laboratory\nSolid State Physics\nUppsala University\nP.O. Box 534SE-75121UppsalaSweden\n"
]
| [
"Cathedral School\nBetel Music Institute\nCampus Bromma, Åkeshovsvägen 29SE-75375, SE-16839Uppsala, BrommaSweden, Sweden",
"Department of Engineering Sciences\nThe Ångström Laboratory\nSolid State Physics\nUppsala University\nP.O. Box 534SE-75121UppsalaSweden"
]
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| We analyze the fractal dimension of melodic contours and pitch time series of classical music and folk music tunes. The fractal dimensions obtained from box counting and detrended fluctuation analysis show significant differences. They are ascribed to the low accuracy of box counting for dimensions close to two as well as to a possible bias because the pitches in the time series are connected by lines to obtain the melodic contour used in the box counting analysis. We observe a tendency that folk music exhibits lower fractal dimensions than classical music, but further studies are needed in order to assess cutoff effects in the comparatively short folk music tunes. We conclude that detrended fluctuation analysis is the preferable method for fractal analysis of music, and this verifies previous studies of analysis of short time series. | null | [
"https://arxiv.org/pdf/2004.02612v1.pdf"
]
| 214,802,075 | 2004.02612 | deb36a0e6165b48bcc6a0c81582927790cb46259 |
The fractal dimension of music: Melodic contours and time series of pitch
Maria H Niklasson
Cathedral School
Betel Music Institute
Campus Bromma, Åkeshovsvägen 29SE-75375, SE-16839Uppsala, BrommaSweden, Sweden
Gunnar A Niklasson
Department of Engineering Sciences
The Ångström Laboratory
Solid State Physics
Uppsala University
P.O. Box 534SE-75121UppsalaSweden
The fractal dimension of music: Melodic contours and time series of pitch
* Present address: Royal College of Music, SE-11591, Stockholm. Sweden
We analyze the fractal dimension of melodic contours and pitch time series of classical music and folk music tunes. The fractal dimensions obtained from box counting and detrended fluctuation analysis show significant differences. They are ascribed to the low accuracy of box counting for dimensions close to two as well as to a possible bias because the pitches in the time series are connected by lines to obtain the melodic contour used in the box counting analysis. We observe a tendency that folk music exhibits lower fractal dimensions than classical music, but further studies are needed in order to assess cutoff effects in the comparatively short folk music tunes. We conclude that detrended fluctuation analysis is the preferable method for fractal analysis of music, and this verifies previous studies of analysis of short time series.
Introduction
Since the pioneering work of Mandelbrot [1] fractal geometry has been widely used in the characterization of structures in the natural world as well as in a variety of scientific fields. It was first noticed by Voss and Clarke [2,3] that the power spectral density of audio recordings show major similarities to 1/f noise. This can be understood in terms of fractal, or more precisely self-affine, structural properties of music, which may, at least to a first approximation, be modeled by fractional Brownian motion [4,5]. The discovery of fractal structure in music was popularized by Gardner [6]. As pointed out already in the early work [3,4] such analyses are important for the development of algorithmic composition [7], i.e. computer generated music, which is often used as a tool for developing novel ideas in musical composition.
The statistical properties of music can be analyzed using different methodologies. Measurements of the spectral density of audio power and frequency fluctuations in audio recordings is a versatile and fast method [2,3,8]. However, the results of such a procedure will depend both on the musical composition and how it is performed. In the present paper we take another approach and analyze the actual musical scores, in order to focus on the intention of the composer. It seems that Hsü and Hsü [9] made the first attempt to analyze fractal properties of musical scores. They defined a fractal dimension from the distribution of melodic intervals. However, this concept is suspect and has subsequently been strongly criticized [10]. A conceptually much better method is to digitize a musical score to create a time series of the pitch, a so called "melodic contour" or "music walk" [5].
In addition, different methods to determine the fractal dimension of music have been employed and the obtained fractal dimensions sometimes display significant differences. Box counting [11] is a well-known fractal analysis method which has been used in early work [12]. Fourier transformation has been used to obtain a power spectral density from the melodic contour [5] and from measures of rhythm [13]. Methods based on the variance of the audio amplitude [8] or the root-mean square (RMS) fluctuation [5] of the melodic contour have also been used to determine the fractal dimension, D. In the case of fractional Brownian motion the square root of the variance (i.e. the RMS fluctuation) scales as a power of the length of the considered time interval [4]. The scaling exponent, called the Hurst exponent, is given by the relation H=2-D [4]. The RMS analysis can be improved by removing trends from the data and the resulting Detrended Fluctuation Analysis (DFA) [14,15], was recently used to analyze music [16]. It should also be noted that a variety of more complex methods have been used to study scaling in music, ranging from multifractal analysis [17][18][19], concepts from chaos theory [20] and a variety of other metrics [21] to determinations of special note patterns or fractal generators responsible for the scaling [22,23].
Recently we showed [16] that the distribution of melodic intervals, i.e. the increments in pitch between successive time steps in the melodic contour is strongly non-Gaussian and can to a fair approximation be fitted by Levy-stable distributions [24]. This suggests that a basic theme in the structure of music can be described as a Levy motion [25,26]. This finding is of potential importance for the analysis of the fractal structure of music since it has been claimed that methods based on the variance are not reliable in cases of non-Gaussian long-tailed distribution of increments [26]. Although many methods have been used to study fractal structures in music a coherent picture has not yet emerged. There still exist considerable uncertainties regarding the applicability and accuracy of the methods used to extract even the most basic parameter, i.e. the fractal dimension D. In the present paper we compare different methods to calculate the fractal dimension from the melodic contour of a number of classical pieces and folk tunes. We discuss the possible reasons for differences in the obtained numerical values and find a tendency that classical works exhibit a higher fractal dimension than folk music.
Fractal analysis methods
The music scores were digitized manually to obtain a time series of pitches, a so called melodic contour. Each note in the script was given an integer number characterizing the pitch and a number of time steps characterizing the duration of the note. The pitch is defined according to the twelve tone scale of Western music, in which each octave, which ranges from frequency f to 2f, is divided into twelve intervals between tones, each tone being associated with a pitch value. The pitch is related to the logarithm of the frequency of the tone and is represented by an integer. The frequencies in an octave are related to the pitches i by [5]
0 = 2 /12 (1)
where f0 is the frequency of the base note in an octave. Since musical pieces span over several octaves, we number the pitches starting from the lowest note in the score. The shortest note in the score was taken as the fundamental time step. Hence a note may be assigned to one or more time steps, depending on its duration. Pauses were assigned an appropriate number of time steps but no pitch, hence they were treated as interruptions of the music. Figure 1 shows two examples of melodic contours, specifically for a classical work, Bach's Concerto for two violins in D minor, movement 2, and for a folk music tune, "Leksand gift tune". It should be noted that folk tunes generally are much shorter than classical ones, as seen from the different scales on the horizontal axes.
Box counting [11] is a method specially developed for self-similar fractals, but it can also be applied to self-affine surfaces and profiles [27], like melodic contours. An image of the structure is covered by boxes of size L and the number of boxes, N, covering the contour is determined as a function of box size. For a fractal structure, [11,27] ( ) ~ − .
In order to avoid crossover effects between "local" and "global" dimensions, it is essential that the units in the vertical direction (pitch) and in the horizontal direction (time) are chosen so that the image to be analyzed exhibits an overall square shape [27,28]. We have analyzed images of melodic contours, like those in figure 1, by the box counting routine in the image analysis program ImageJ [29]. Time series of pitch, like those of Figure 1, were considered analogous to random walks (a "music walk"), in order to apply RMS analysis. In this method the scaling of the RMS fluctuations was obtained by dividing the time series into non-overlapping subseries ("bins") of length L and equal number of data points, n, and using the relation [27] (
< 1 ∑ [ℎ( ) − ℎ ̅ ] 2 >) 1/2~ ,(3)
where h(t) is the pitch, ℎ ̅ its average, the sum is over all the data in a bin and the brackets denote an average over all bins of size L. It is very important to decouple the analysis of a fluctuating signal from any underlying trend in the data. Detrended Fluctuation Analysis (DFA) [14,15] is a well-established method for advanced analysis of Hurst exponents. It should be realized that the pitch series h(t) corresponds to the cumulative time series of the DFA method [14,15]. In the DFA method the trend of the cumulative time series is determined by least-squares fitting of a polynomial to the data in each bin. In the present study we use linear and quadratic polynomials as trend curves. The RMS fluctuations of the detrended time series is calculated by replacing the average ℎ ̅ in eq. (3) by the polynomial trend curve. Calculations are performed for each bin and averaged over all bins of the same size, in analogy to eq. (3). In the scaling range the DFA estimator, which is proportional to the RMS fluctuation, is ~L H [14,15,30]. We have determined the Hurst exponent by the DFA method, using a Matlab function due to Weron [30,31]. The algorithm was modified to encompass the RMS method (DFA0), linear trend subtraction (DFA1) as well as quadratic trend subtraction (DFA2).
The accuracy of the fractal analysis is affected by crossovers between the fractal scaling range (D=2-H) in eq. (2) and (3) and a non-fractal range (D=2) at large L. This is because the behavior for long-time intervals is affected by the fact that we have a finite set of data with a finite span of pitch values. The DFA estimator cannot increase above the maximum given by the range of the data and this can lead to a crossover which is different for each data set. An advantage with detrending is that it may shift the crossover and extend the fractal scaling range towards larger L, however sometimes at the expense of introducing another crossover at small L [15].
The above computations were compared to alternative algorithms from the work of Russ [32]. His Kolmogorov method is very similar to box counting and gives slightly larger (0.05-0.1) values of D than found with our method. In addition, the RMS fluctuation routine of Russ was found to be in good agreement with our DFA0 calculations. Another method to determine H is the so-called rescaled range analysis, which was carried out by the R/S routine in the software SELFIS [33]. However, we found that this method exhibits ill-defined crossovers, which makes results very uncertain. Figure 2 shows box counting results for the two musical contours depicted in figure 1. The data follow closely a straight line according to eq. (2) with crossover effects at small box sizes, where the one-dimensional lines making up the melodic contour will eventually dominate the results. It is seen that the Bach piece exhibits a much higher fractal dimension (D=1.58) than the Leksand tune (D=1.36). Indeed, there is a tendency in all our data ( Table 1) that classical music has a higher fractal dimension than folk music tunes. However, a problem is that crossover effects are more apparent in the smaller data sets of the folk tunes ( Figure 2b). The Table also lists characteristic exponents, of Levy-stable distributions, that were fitted to the distribution of melodic intervals, from our previous work [16]. It is observed that high fractal dimensions are correlated with low Lévy exponents. Figure 3 shows the RMS fluctuations as a function of time interval for two classical musical scores. These calculations using the DFA0 algorithm show a very good scaling behavior over the whole range of bin sizes, with fractal dimensions of 1.80 and 1.83 . For classical music pieces applying detrending did not improve the scaling; instead the scaling range was slightly restricted. Hence it seems that there is no underlying linear or quadratic trend in these datasets. The situation is different for folk music, though, and in at least two cases linear detrending improved the accuracy of determining the scaling exponent. Figure 4 shows calculations by the DFA0 and DFA1 algorithms for the Leksand piece. There is a scaling range for short time intervals that crosses over to a rather constant behaviour for large time intervals. However, the scatter of the data points in significantly larger for the DFA0 method. Therefore the DFA1 data was used to compute D for the folk music tunes, giving acceptable scaling ranges extending almost decade in bin size. Fig. 4 thus illustrates one advantage of detrending. The slope of the scaling range gives Hurst exponents H=0.18 and 0.50 for the Bach and Leksand pieces, respectively, equivalent to fractal dimensions of 1.82 and 1.50, respectively. Hence fractal dimensions obtained by DFA are consistently higher than box counting dimensions. These differences merit a thorough discussion. First it must be remembered that box counting computes D of the melodic contour, in which the pitch values are joined by straight lines, while DFA gives the dimension of a time series of pitch values. Secondly, it is known that box counting underestimates D close to the dimension of the embedding space (in our case 2) [34]. Thirdly, we are using relatively short time series in the present work. Hence the accuracy of various methods in this case is of prime importance.
Results and discussion
Detrended Fluctuation Analysis (DFA) has been found to perform better than a variety of other methods in comparisons using synthetic data [30,35]. In particular, it is accurate for short fractional Brownian motion time series of up to 2048 data points [35]. Our present state of knowledge indicates that DFA should be the preferred method to analyze musical contours, although we stress that it needs to be validated also for fractional Lévy motions. The lower dimensions obtained by box counting may be due partly to the low accuracy of box counting for small values of H and partly to a bias introduced by the image analysis of the contour lines connecting the points in the "music walk".
We now compare our fractal dimensions with previous studies, focusing on those that have used similar methods. Analysis of audio signals have yielded fractal dimensions in the range 1.6 -1.7 from box counting [12] and 1.7 -1.9 from variance methods [8]. A variation of D between different music genres was found in the more extensive of these studies [8]. The RMS fluctuation method (DFA0) has been used mostly for classical music. Su and Wu [5] obtained fractal dimensions in the range 1.62-1.78 by analyzing the melodic contours of a number of classical pieces. However, their results showed strong crossover effects in most cases and probably more accurate values could have been obtained with the DFA1 method. Despite the methodical differences and uncertainties, literature values are generally in acceptable agreement with our results for classical music. It seems that classical music exhibits fractal dimensions in the range 1.6 -1.9 and that folk music tunes, at least in some cases, would appear to have lower fractal dimensions. However, folk tunes are shorter, which results in small data sets, and the fractal dimensions may be significantly affected by crossover effects and cutoffs to the fractal range. In general, our results should be validated by analysis of a larger corpus of music. In that case, automated analysis, for example by using the MIDI Toolbox of the University of Jyväskylä [36], which computes melodic contours for monophonic MIDI files, will be necessary.
Conclusion
We have shown that melodic contours and time series of pitches exhibit a self-affine fractal scaling range. Fractal dimensions were computed by box counting and detrended fluctuation analysis. Our results confirm the trends obtained in a previous comparison of fractal analysis methods [35]. In particular, we infer that the DFA method is the best one, with the analyzed data showing wide scaling ranges with minor crossover effects. We observe that folk music tunes exhibit lower fractal dimensions than classical music, but the importance of crossover effects needs to be more accurately assessed in these cases. Recent results indicate that music can be modelled as a Levy motion [16]. Hence the accuracy of variance-based methods like DFA need to be re-examined for time series whose increments can be approximated by Levystable distributions. However, the present study substantiates the use of the DFA method in [16].
Fig. 1 .
1Melodic contour showing pitch as a function of time step for (a) Bach's Concerto for two violins in D minor, movement 2, and (b) Leksand gift tune. The length of the vertical axis is 35 pitch units in (a) and 30 pitch units in (b).
Fig. 2 .
2Number of boxes as a function of box size for (a) Bach's Concerto for two violins in D minor, movement 2, and (b) Leksand gift tune. The dashed lines denote fits to the data points and the slope gives the fractal dimension.
Fig. 3 .
3Logarithm (base 10) of DFA estimator as a function of logarithm of length of time interval (bin size) for (a) Bach's Concerto for two violins in D minor, movement 2, and (b) Vivaldi's Concerto in A Minor 1st Movement. The lines are fits to the scaling ranges in the plots.
Fig. 4 .
4Logarithm (base 10) of DFA estimator as a function of logarithm of length of time interval (bin size) for the Leksand gift tune, computed by the DAF0 and DFA1 methods. Full lines are fits to the scaling ranges as well as to the horizontal trend for sizes larger than the crossover.
Table 1 .
1Number of data points, N, as well as fractal dimension, D, obtained by box counting and by DFA methods for the music pieces considered in this paper. Values from the DFA0 method were used for classical music and DFA1 method for folk music. In addition, the characteristic exponent, of Levy-stable distributions that were fitted to the distribution of melodic intervals[16], are given for comparison. Uncertainties in D are estimated to be of the order of ± 0.03, while error bars of are approximately ±0.10 for the classical pieces and ±0.20 for the folk tunes.N
D (box) D(DFA)
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| []
|
[
"DDO68-V1: an extremely metal-poor LBV in a void galaxy",
"DDO68-V1: an extremely metal-poor LBV in a void galaxy",
"DDO68-V1: an extremely metal-poor LBV in a void galaxy",
"DDO68-V1: an extremely metal-poor LBV in a void galaxy"
]
| [
"Yulia Perepelitsyna \nSpecial Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia\n",
"Simon Pustilnik \nSpecial Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia\n",
"Yulia Perepelitsyna \nSpecial Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia\n",
"Simon Pustilnik \nSpecial Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia\n"
]
| [
"Special Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia",
"Special Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia",
"Special Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia",
"Special Astrophysical Observatory Russian Academy of Sciences\nNizhnij Arkhyz, Karachai-Circessia369167Russia"
]
| []
| The lowest metallicity massive stars in the Local Universe with Z ∼(Z⊙/50-Z⊙/30) are the crucial objects to test the validity of assumptions in the modern models of very lowmetallicity massive star evolution. These models, in turn, have major implications for our understanding of galaxy and massive star formation in the early epochs. DDO68-V1 in a void galaxy DDO68 is a unique extremely metal-poor massive star. Discovered by us in 2008 in the HII region Knot3 with Z = Z⊙/35 [12+log(O/H)∼7.14], DDO68-V1 was identified as an LBV star. We present here the LBV lightcurve in V band, combining own new data and the last archive and/or literature data on the light of Knot3 over the 30 years. We find that during the years 2008-2011 the LBV have experienced a very rare event of 'giant eruption' with V-band amplitude of 4.5 mag (V ∼ 24.5 m − 20 m ). | 10.1017/s1743921318006439 | [
"https://arxiv.org/pdf/1812.03686v1.pdf"
]
| 119,273,997 | 1812.03686 | 9d7d4fb78341720bba582275b08560bc76ccdb6c |
DDO68-V1: an extremely metal-poor LBV in a void galaxy
10 Dec 2018
Yulia Perepelitsyna
Special Astrophysical Observatory Russian Academy of Sciences
Nizhnij Arkhyz, Karachai-Circessia369167Russia
Simon Pustilnik
Special Astrophysical Observatory Russian Academy of Sciences
Nizhnij Arkhyz, Karachai-Circessia369167Russia
DDO68-V1: an extremely metal-poor LBV in a void galaxy
10 Dec 2018Dwarf Galaxies: From the Deep Universe to the Present Proceedings IAU Symposium No. 344, 2018 K. McQuinn, S. Stierwalt, eds.stars: individual (DDO68-V1)stars: supergiantsstars: abundancesstars: mass lossstars: variables: LBVgalaxies: individual (DDO68UGC5340)
The lowest metallicity massive stars in the Local Universe with Z ∼(Z⊙/50-Z⊙/30) are the crucial objects to test the validity of assumptions in the modern models of very lowmetallicity massive star evolution. These models, in turn, have major implications for our understanding of galaxy and massive star formation in the early epochs. DDO68-V1 in a void galaxy DDO68 is a unique extremely metal-poor massive star. Discovered by us in 2008 in the HII region Knot3 with Z = Z⊙/35 [12+log(O/H)∼7.14], DDO68-V1 was identified as an LBV star. We present here the LBV lightcurve in V band, combining own new data and the last archive and/or literature data on the light of Knot3 over the 30 years. We find that during the years 2008-2011 the LBV have experienced a very rare event of 'giant eruption' with V-band amplitude of 4.5 mag (V ∼ 24.5 m − 20 m ).
Introduction
Luminous Blue Variable (LBV) stars represent a short (about or less than 0.1 Myr) transient phase of massive star evolution from the main sequence hydrogen burning O stars to the core-helium burning Wolf-Raye (WR) stars.
Evolution of massive stars with the lowest known metallicities is crucial for understanding the early galaxy formation and evolution at high redshifts due to their great energy release/feedback (e.g., Barkana & Loeb (2001)).
The most metal-poor massive stars are currently identified in several extremely metalpoor (Z ∼ Z ⊙ /45-Z ⊙ /35) dwarf galaxies. Most of these extreme galaxies are found in nearby voids. Stellar evolution models (including those with the fast rotation) have substantially advanced during the last decade. However, the direct comparison of the model predictions with the properties of real extremely metal-poor massive stars is still absent. Such studies should await for the next generation extremely large telescopes.
Overview
DDO68, at the distance D=12.75 Mpc, is one of the most metal-poor galaxies (Z ∼ Z ⊙ /35) residing in the nearby Lynx-Cancer void. DDO68 is a merger of low-mass gasrich components (Ekta, Chengalur, Pustilnik (2008), Makarov et al. (2017)). Its very low-Z gas was identified with BTA spectra in 2005. Most of SF regions are found at the periphery, mainly in the 'Northern ring' and the 'Southern tail' (Pustilnik, Kniazev, & Pramskij (2005), Izotov & Thuan (2007)).
In 2008 we discovered in its SF Knot 3 ( Fig. 1) , see also Izotov & Thuan (2009)). Hubble Space Telescope (HST) images of DDO68 were obtained in May 2010 with ACS for Proposal GO 11578 (PI A.Aloisi) and presented in papers by Sacchi et al. (2016) and by Makarov et al. (2017). The lightcurve of Knot 3 (Fig. 2) in DDO68 in V and B bands since 1988 is based on the new and archive data and the data from Pustilnik et al. (2017). All magnitudes are for the aperture with r=2.5 ′′ . The dotted lines at V = 20.20 and B = 20.25 correspond to the minimal observed light of the entire Knot 3. These minimal levels were slightly reduced due to a more advanced background determination with respect of that adopted in paper by . These magnitudes are consistent, in particular, with Knot 3 light on the night 2005.01.12, when the LBV was too faint and did not show up in the spectrum of Knot 3.
With except of one direct photometry (the HST image), all other magnitudes are derived as the 'residual light' via subtraction of the constant luminosity of the underlying HII region (V = 20.20) from the lightcurve on the Fig. 3. We observe a very rare case of LBV 'giant eruption' (Smith & Owocki (2006)) during the years 2008-2011, with the DDO68-V1: an extremely metal-poor LBV in a void galaxy (Sterken, 2003). In the right panel of Figure 3, the photometric variability is observed up to 2.5 m over the periods of 0.5-2 years. (c) If the photometric behavior of the most metal-poor LBV is similar to that of more typical LBVs, the DDO68-V1 light variations during the last 28 years suggest that it underwent a 'giant eruption' during the years 2008 -2011.
Implications and conclusions
(d) We call to the community for the campaign of DDO68-V1 multiwavelength monitoring that can give the new insights in the lowest metallicity LBV properties and prove the substantial increase of its bolometric luminosity.
(e) Having in mind other known examples of extragalactic SN impostors, one can occasionally catch this unique object in the SN impostor phase. Moreover, in the case of the great luck, we can catch even the unique case of a nearby SNII explosion related to the extremely low-Z massive star.
The full-format paper presenting all details of observational data and their analysis as well as a wider discussion of all available data is prepared for publication in MNRAS.
Figure 1 .
1The part of the HST image of DDO68 in W 606 (V ) band centered on the region Knot 3 with the used aperture superimposed (Daper = 5 ′′ ). DDO68-V1 is in the center of the aperture.
3 Figure 2 .
32The lightcurve of Knot 3 in DDO68 in V and B bands since 1988 based on new and archive data and the data fromPustilnik et al. 2017. All magnitudes are for the aperture with r = 2.5 ′′ .
Figure 3 .
3Left panel: Light curve of the LBV in V -band (filled lozenges). With except of one direct photometry (the HST W606 image in May 2010), all other magnitudes are derived as the 'residual light' via the subtraction of the constant luminosity of the underlying Hii region (V = 20.20) from the previous lightcurve. Lozenges with arrows indicate 3σ upper limits. Right panel: Close-up of the LBV light curve in V -band for period of 2015 -2018. There is an indication of the phenomenon of S Doradus type variations (Sterken, 2003). total amplitude of the LBV optical variability δV∼4.5 m , reaching M V = -10.5. Series of 'giant eruptions' in LBVs which form several expanding shells, can precede their SN explosions at rather short time scale. Observations of light variations of DDO68-V1 after the 'giant eruption', since Year 2015 reveal the behaviour resembling the phenomenon of S Doradus
a) We extend the recently published lightcurve for the period of 2005 -2015 for DDO68-V1, adding our fresh (years 2016-2018) Zeiss-1000 and BTA telescopes photometry of the HII region Knot 3 (containing the LBV = V1) and the photometry from the archive images at ten epochs with ten different telescopes over the period of 1988 -2013.(b) The data allow us at the first time to determine the reliable amplitude of this LBV lightcurve. All available data suggest that the LBV V -band light varied during
AcknowledgementsThe work was supported by the grant of Russian Science Fund No. 14-12-00965. The authors thank O. Spiridonova, V. Goransky and A. Moskvitin for their help with DDO68 observations at the SAO 1m telescope. The authors are grateful to L. van Zee, D. Hunter, B. Elmegreen, U. Hopp, L. Makarova, R. Swaters, B. Mendez, V. Taylor, R. Jansen, R.A. Windhorst, S.C. Odewan, J.E. Hibbard for providing archival CCD images of DDO68 obtained for their observational programs. We are pleased to thank P. Kaigorodov and D. Kolomeitsev for their kind help in extracting the data from archive tapes.
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| []
|
[
"Provably robust blind source separation of linear-quadratic near-separable mixtures *",
"Provably robust blind source separation of linear-quadratic near-separable mixtures *"
]
| [
"Christophe Kervazo ",
"Nicolas Gillis ",
"Nicolas Dobigeon "
]
| []
| []
| In this work, we consider the problem of blind source separation (BSS) by departing from the usual linear model and focusing on the linear-quadratic (LQ) model. We propose two provably robust and computationally tractable algorithms to tackle this problem under separability assumptions which require the sources to appear as samples in the data set. The first algorithm generalizes the successive nonnegative projection algorithm (SNPA), designed for linear BSS, and is referred to as SNPALQ. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions of the mixing are mitigated, thus improving the separation quality. SNPALQ is shown to be able to recover the ground truth factors that generated the data, even in the presence of noise. The second algorithm is a brute-force (BF) algorithm, which is used as a post-processing step for SNPALQ. It enables to discard the spurious (mixed) samples extracted by SNPALQ, thus broadening its applicability. The BF is in turn shown to be robust to noise under easier-to-check and milder conditions than SNPALQ. We show that SNPALQ with and without the BF postprocessing is relevant in realistic numerical experiments. | 10.1137/20m1382878 | [
"https://arxiv.org/pdf/2011.11966v1.pdf"
]
| 227,151,057 | 2011.11966 | eafb262577a0c8631f8a3b287aff4120c05cceab |
Provably robust blind source separation of linear-quadratic near-separable mixtures *
Christophe Kervazo
Nicolas Gillis
Nicolas Dobigeon
Provably robust blind source separation of linear-quadratic near-separable mixtures *
non-linear blind source separationnonnegative matrix factorizationnon-linear hyperspectral un- mixinglinear-quadratic modelsseparabilitypure-pixel assumption AMS subject classifications 15A2365F5568Q2565D18
In this work, we consider the problem of blind source separation (BSS) by departing from the usual linear model and focusing on the linear-quadratic (LQ) model. We propose two provably robust and computationally tractable algorithms to tackle this problem under separability assumptions which require the sources to appear as samples in the data set. The first algorithm generalizes the successive nonnegative projection algorithm (SNPA), designed for linear BSS, and is referred to as SNPALQ. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions of the mixing are mitigated, thus improving the separation quality. SNPALQ is shown to be able to recover the ground truth factors that generated the data, even in the presence of noise. The second algorithm is a brute-force (BF) algorithm, which is used as a post-processing step for SNPALQ. It enables to discard the spurious (mixed) samples extracted by SNPALQ, thus broadening its applicability. The BF is in turn shown to be robust to noise under easier-to-check and milder conditions than SNPALQ. We show that SNPALQ with and without the BF postprocessing is relevant in realistic numerical experiments.
1. Introduction. Blind source separation (BSS) [8,7,25] is a powerful paradigm with a wide range of applications such as remote sensing [37], biomedical and pharmaceutical imaging [1,36], and astronomy [39]. BSS aims at decomposing a given data set into a set of unknown elementary signals to be recovered, generally referred to as the sources. Because it is simple and easily interpretable, many works [8] have focused on the linear mixing model (LMM) which assumes that the ith data set samplex i ∈ R m for i ∈ [[n]] can be written as
x i = r k=1 h ki w k + n i ,
where w k is the kth source for k ∈ [[r]] = {1, 2, . . . , r}, and h ki its the associated mixing coefficient in the ith (mixed) observation. The vector n i accounts for any additive noise and/or slight mismodelings in the ith pixel. Using a standard matrix formulation, the LMM can thus be rewritten asX = WH + N, whereX = [x 1 ,x 2 , ...,x m ] ∈ R m×n is the data set, W = [w 1 , w 2 , ..., w r ] ∈ R m×r are the sources, H ∈ R r×n is the mixing matrix containing the coefficients h ki 's, and N = [n 1 , n 2 , ..., n m ] ∈ R m×n is the noise. We denote by X = WH the noiseless version ofX. The goal of BSS is to recover W and H from the sole knowledge ofX. This is in general an ill-posed problem [8]. Hence, in most works, additional constraints are imposed on the unknown matrices W and H to make the problem better posed: for instance, orthogonality in principal component analysis (PCA - [23]), independence in independent component analysis (ICA - [8]), and sparsity in sparse component analysis (SCA - [41,7,25]). We will here focus on nonnegativity constraints, akin to nonnegative matrix factorization (NMF) [27]. Although NMF is NP-hard in general [40], and its solution non-unique [16], Arora et al. [4,5] have introduced the subclass of near-separable non-negative matrices for which NMF can be solved in a polynomial time with weak indeterminacies. This subclass corresponds to data sets in which each source appears purely in at least one data sample. Building on near-separable NMF, several provably robust algorithms have been proposed [4,14,35,19]. Among them, one can cite the successive projection algorithm (SPA) [3], which is a fast greedy algorithm provably robust to noise [20], or an enhanced version, the successive nonnegative projection algorithm (SNPA) [18], which is more efficient when W is ill-conditioned and is applicable when W is rank-deficient.
1.1. LQ mixing model. In various applications, the LMM may however suffer from some limitations and can only be considered as a first-order approximation of non-linear mixing models [6,13,11]. In such situations, linear-quadratic (LQ) [9] models can for instance better account for the physical mixing processes by including termwise products of the sources [12,22]. This model can be written as (1.1)x i = r k=1 h ki w k + r p=1 r l=p β ipl (w p w l ) + n i .
In (1.1), the linear contribution associated to LMM is complemented by a set of second-order interactions w p w l between the sources, where denotes the Hadamard product and β ipl is the amount of the interaction w p w l within the ith observation. It is worth mentioning the closely-related so-called bilinear mixing model [13,9], which is a particular instance of the LQ mixing model, from which the squared terms w p w p for p ∈ [[r]] in (1.1) are removed; see Application 1.1 below for a discussion in the context of blind hyperspectral unmixing where the LQ and bilinear models are widely used. The LQ mixing model (1.1) can also be rewritten in a matrix form (1.2)X = Π 2 (W)H + N where Π 2 (W) ∈ R m×r is the extended source matrix containing the sources and their secondorder products as its columns, withr = r(r + 3)/2, and H ∈ Rr ×n is the matrix gathering all the mixing coefficients associated with the linear (h ki 's) and nonlinear (β ipl 's) contributions.
Written in such a matrix form, the similarity between the LQ and linear models is easily visible: the LQ mixings can be written in a linear form by considering the quadratic terms w p w l as new sources, additional to the usual ones w k . Following this line of thought, the w p w l terms are often called virtual sources. In the sequel of this paper, this terminology will be adopted and the non-virtual sources w i will be referred to as primary.
Application 1.1 (Hyperspectral imaging). To illustrate the BSS of LQ-mixtures (LQ-BSS), we consider throughout this paper the example of hyperspectral (HS) imaging. Despite having a finer spectral resolution than conventional natural images, HS images generally suffer from a limited spatial resolution. Therefore, several materials are generally present in each pixel, and thus the acquired spectra correspond to mixtures of the different pure material spectra, called endmembers. This mandates the use of BSS methods -more specifically of NMF -to perform spectral unmixing. To be more precise, using the terminology of HS unmixing [13], w k in (1.1) corresponds to the spectral signature of the kth endmember and h ki to the abundance of the kth endmember in the ith pixel. The spectral signature of a source is the fraction of light reflected by that source depending on the wavelength, and hence 0 ≤ w k ≤ 1 for k ∈ [[r]]. Concerning the model choice, the linear BSS model is often a too rough approximation in HS: in particular, when the light arriving on the sensor interacts with several materials, nonlinear mixing effects may occur [6,13,11]. Specifically, this is often the case when the scene is not flat, for instance in the presence of large geometric structures, such as in urban [30] or forest [12] scenes. In such a context, it has been shown [12,22] that LQ models enable to better account for multiple scatterings. While it is further possible to include higher-order terms, most of the works neglect the interactions of order larger than two since they are expected to be of significantly lower magnitudes [2,29] as 0 ≤ W ≤ 1.
Identifiability issue in LQ-BSS.
Despite source identifiability issues in the general context of non-linear BSS problems [8,10,24], it was recently showed [9] that the non-linearity inherent to bilinear mixtures leads to an essentially unique solution in the noiseless case. More precisely, it was shown that for a data matrix X following the bilinear model in the absence of noise (and under some appropriate assumptions, see below), anyŴ andĤ such that X = Π 2 (Ŵ)Ĥ satisfyŴ = W andĤ = H up to a scaling and permutation of the columns ofŴ and the rows ofĤ. However, this identifiability result suffers from some limitations:
• It relies on two strong assumptions: 1. rowrank(X) = r(r+1) 2 , requiring thatĤ has full row rank and hence that every extended source is present in the data set. In other words, all possible interactions of two primary sources must be present in some observation. This is unlikely to happen in practice. 2. the products of the sources up to order four must be linearly independent. It requires the family to be linearly independent. As its size is r(r+1) 24 (r − 1)(r − 2) + 12 , such a linear independence assumption might not be satisfied in real-world scenarios, since the number of observations m must be of order Θ(r 4 ). • It does not apply to mixings with squared terms [9, section 7], that is, LQ mixings instead of bilinear ones. • No guarantee is given in the presence of noise. Moreover, finding an exact factorization Π 2 (Ŵ)Ĥ of X is a difficult problem. The algorithm used in [9] is a heuristic and does not find an exact solution (see [9,Fig. 4]), leading to errors on the recovered sources.
Application 1.2 (Hyperspectral imaging (cont'd)). In HS imaging, the assumption that H has full row rank is unlikely to be satisfied as many endmembers do not interact, because they are located far apart in the image.
For the second assumption, even with r = 10 endmembers, which is a relatively small number, at least m ≥ 385 spectral bands would be required to ensure the linear independence of the family (1.3). This is not satisfied for typical HS sensors dedicated to Earth observation. As an example, the Airborne Visible / Infrared Imaging Spectrometer (AVIRIS) operated by the Jet Propulsion Laboratory (JPL, NASA), acquires HS images composed of m = 224 spectral bands, among them several dozens are inexploitable due to low signal-to-noise ratios.
1.3. Near-separable LQ mixings. To overcome the above identifiability issues, we propose in this work to tackle BSS problems of the form (1.1) under a near-separable NMF-like paradigm. In particular, the rationale is to convert the linear independence condition on the family (1.3) into a non-negative independence condition, which is significantly less restrictive. Consider for instance the family of points located on a circle within the unit simplex in three dimensions, that is, distinct points within the set {x ∈ R 3 + | x 1 = 1, x 2 = q} for some q < 1. Although the rank of this family is 3, no point is within the convex cone of other points, and hence this family is non-negatively independent.
More specifically, denoting ∆ = {x ∈ Rr|x ≥ 0, r i=1 x i ≤ 1} and Π 2 (W) \{j} the submatrix of Π 2 (W) excluding w j , we assume the following constraints: min x∈∆ w j − Π 2 (W) \{j} x 2 > 0 (order-2 α-robust simplicial).
h ki ≥ 0 for all i ∈ [[
The two first constraints ensure the mixing coefficients for each pixel to be nonnegative and to sum to at most one, and can be equivalently written as h i ∈ ∆ for all i ∈ [[r]]. The last one ensures that no source lies within the convex hull formed by the other ones, their second order product and the origin. It is thus an extension of the α-robust simplicial 1 definition of [5] which requires that
α 1 (W) = min j∈[[r]] min x∈∆ w j − W \{j} x 2 > 0.
In addition, extending the subclass of r near-separable mixings of [17] to the LQ model, we will assume the mixing to be r-LQ near-separable, as defined below.
Definition 1.3. The matrixX is said to be r-LQ near-separable if it can be written as:
X = Π 2 (W) I r 0 r(r−1) 2 ×r H P H +N,
where W ∈ R m×r is order-2 α-robust simplicial, I r is the r-by-r identity matrix, 0 p×q is the p-by-q matrix of zeros, P is a permutation matrix, and H ∈ Rr ×m−r is a matrix satisfying the sum to at most one and nonnegativity conditions. It is important to note that contrary to the sources W, the virtual sources (w p w l ) p,l∈ [[r]],l≤p are not required to appear in some samples.
Application 1.4 (Hyperspectral imaging (cont'd)). It has been shown [13] that bilinear and LQ models enable to better account for multiple scatterings. Examples of such models include the Fan model [15], the generalized bilinear model [21], the polynomial post-nonlinear model [2]; see [13] and the references therein for more details. In this work, we will focus on the so-called Nascimento model [33,38], which is a bilinear-based model that naturally extends the classical linear model and the sum-to-at-most one constraint on the abundances.
The near-separable assumption in HS is referred to as the pure-pixel assumption, as it requires each endmember to appear at least once purely within a pixel. This hypothesis is common and realistic [20,28], provided that the spatial resolution is not too low.
Contributions.
In this paper, we introduce two algorithms which, given a r-LQ near separable mixture (Definition 1.3), approximately recovers the factors W and H. As such, our results are (i) theoretical: we show the identifiability of this problem even in the presence of noise, and (ii) practical: in contrast to [9], the two algorithms run in polynomial time. More specifically, the contributions -graphically summarized in Figure 1 -are the following:
• We introduce the successive nonnegative projection algorithm for linear-quadratic mixtures (SNPALQ), which generalizes SNPA [18] to linear-quadratic (LQ) mixings by explicitly modeling the presence of quadratic products within its greedy search process. • The conditions under which SNPALQ is provably robust to noise are detailed in Section 3.1.2. In particular, such conditions encompass the linear case (see Section 3.1.1), which is important as the LQ model we consider generalizes the linear one. • To further mitigate the robustness conditions of SNPALQ and broaden its applicability, we introduce a second algorithm dubbed brute force (BF), that we use as a post-processing step to enhance SNPALQ results (which we denote SNPALQ+BF). In Section 3.2, we prove that BF lead to robustness guarantees under weaker conditions than SNPALQ. • In Section 4, the effectiveness of the proposed algorithms is attested through extensive numerical experiments, in which among others SNPALQ is shown to obtain better results than SNPA on LQ mixings, and the SNPALQ+BF to obtain a very high rate of perfect recovery of the ground truth factors.
Remark 1.5. Near-separable algorithms have often been used to initialize NMF algorithms that do not rely on the separability assumption [18]. In particular, the initializations of many LQ-BSS algorithms are often (and paradoxically) performed with the output of near-separable Figure 1. Graphical summary of the contributions, explaining which algorithm to use in which setting. We call a provably robust algorithm an algorithm which is proved to recover the sources even in the presence of noise. algorithms assuming linear mixtures; see for example [2,30]. Therefore, beyond their intrinsic interest, the two algorithms proposed in the next section are fast and theoretically wellgrounded initialization strategies for LQ-BSS algorithms in the absence of the separability assumption.
1.5. Notation. In the following, we denote [[r]] = {1, 2, .., r}, |K| the number of elements in the set K whose ith element is denoted K(i). The ith column of a matrix A ∈ R m×r is denoted a i . The submatrix formed by the columns indexed by K is denoted A K , and the submatrix formed by all the columns of A except the ones indexed by K as A \K . The set ∆ r , for which the superscript is omitted when clear from the context, is ∆ r = {x ∈ R r |x ≥ 0, r i=1 x i ≤ 1}. In addition, we denote by Π q (W) the matrix containing all the columns of W and their products up to order q ∈ N * . We will use Π 2 (W) which denotes the matrix containing the products up to order 2, that is, Π 2 (W) = [w 1 , w 2 , . . . , w r , w 1 w 1 , w 2 w 1 , w 3 w 1 , w 2 w 2 , w 3 w 2 , . . . , w r w r ]
= (w i ) i∈[[r]] , (w i w j ) i,j∈[[r]] i≤j ,
and Π 4 (W) which contains the products up to order 4. Additional notations, specific to the theoretical and proof sections, will be introduced later for the sake of readability.
2. Two algorithms for LQ-BSS: SNPALQ and BF. To perform near-separable BSS of LQ mixtures, a first (naive) approach is to use an LMM-based near-separable NMF algorithm to identify ther extended sources. Since the quadratic terms (
w i w j ) i,j∈[[r]] j≤i
can be considered as virtual sources (see Eq. (1.2)), they could be retrieved along with the columns of W, provided that they appear purely in the data set. One could for instance resort to SNPA [18], an LMMbased algorithm which has shown to yield very good separation performances compared to state-of-the-art LMM-based algorithms such as VCA [32] and SPA [3], and admits robustness guarantees. SNPA is a greedy algorithm: it iteratively constructs the near-separable NMF solution K by sequentially adding a new source to the current set of sources already identified. More precisely, after initializing the index set K = ∅ and a residual matrix R =X, each iteration of SNPA consists of the following two steps:
• selection: the index of the column of R maximizing a score function f is added to K.
• projection: the residual is updated by projecting the columns ofX onto the convex hull formed by the columns ofX K and the origin. During the selection step, the function f aims at selecting the most relevant column of R to be identified as a source. This function, which can for example be the 2 -norm, needs to fulfill the following assumption:
Assumption 2.1. The function f : R m → R + is µ-strongly convex, its gradient is L-Lipschitz and its global minimizer is the all zero vector 0 m , that is, f (0 m ) = 0.
The projection step is a convex optimization problem and can be solved for example using a fast gradient method [34]. We refer the reader to [18, Appendix A] for more details.
Nevertheless, the bottleneck of the above naive approach consisting in using SNPA for LQ mixtures is that the presence of all the virtual sources as pure data samples is too strong. Indeed all virtual sources are not likely to be observed purely in the data set. As such, the recovery of the extended sources by SNPA is not guaranteed, calling for algorithms specifically designed for LQ mixtures.
To overcome this limitation, we propose two new algorithms 2 enabling to tackle LQ mixtures. The first algorithm, referred to as SNPALQ, is a variant of SNPA specifically designed to handle LQ mixings; see Section 2.1. The second one is a brute-force (BF) algorithm, extending the work of [5] to LQ mixtures and exhibiting robustness guarantees under milder conditions than SNPALQ; see Section 2.2. As BF is however computationally more expensive than SNPALQ, we propose to use it as a post-processing of the output provided by SNPALQ.
Combining both algorithms in a single method, which we refer to as SNPALQ+BF, allows us to benefit from the best of each of these algorithms.
2.1. SNPALQ. The rationale behind SNPALQ is that we are interested by recovering the primary sources only, w i for i ∈ [[r]]. The virtual sources w i w j (i, j ∈ [[r]]) can be considered as nuisance. We propose to take them into account in the separation process only to improve the extraction of the primary sources. At each iteration of SNPALQ, we perform the following two steps (see Algorithm 2.1):
• Selection step (unchanged compared to SNPA): the column of the residual matrix R maximizing a function f fulfilling Assumption 2.1 is selected. • Projection step (different from SNPA): SNPALQ performs the projection onto the convex hull formed by the origin, the sources extracted so far and their second-order products. Therefore, if two sources w i and w j (i = j) are extracted during the iterative process of SNPALQ, the contribution of the virtual sources w i w j , w i w i and w j w j are removed. Beyond the advantage that these virtual sources will not be extracted in the subsequent steps, their non-linear contribution is reduced, giving more weight to the linear part. Recall that SNPA projects each column ofX onto the convex hull formed by the origin and all the sources extracted so far to compute the residual R, and does not take into account the virtual sources. Thus, the primary sources defining W are more likely to be extracted by SNPALQ in the early steps of the iterative process; see Figure 2 for an illustration. SNPALQ Figure 2. Example of a bilinear mixing for which SNPALQ is successful at recovering W but SNPA is not (the principle is the same for LQ, except that there are more virtual sources). There are three primary sources, represented with the red X markers, and three virtual sources, namely wi wj for i = j and 1 ≤ i, j ≤ 3, represented with the blue + markers. The columns ofX are made of the primary sources and the mixed points represented with the blue circles. The red dashed line is the convex hull of the origin and the sources extracted after two iterations of SNPA. The plain blue line is the convex hull of the origin and the sources extracted by SNPALQ after two iterations, as well as the corresponding virtual source. Only the last primary source lies outside of the blue convex hull. Therefore, SNPALQ extracts it in its third iteration and then stops, returning the primary sources only. On the other hand, at the third iteration, SNPA fails to extract the last primary source because some of the (mixed) columns ofX lie further away from the red dashed convex hull. Moreover, it will need in total 8 iterations to terminate because the convex hull of the columns ofX has 8 vertices (we assume the virtual sources do not appear purely in the data set).
will be proved in Section 3.1.2 to extract the primary sources in the first r steps, under specific conditions. SNPALQ alternates the two above steps until one of the following two criteria is met:
• A maximum of r max columns have been extracted. If an upper bound is not available, one can take r max = n so that SNPALQ relies on the second stopping criterion only.
Our theoretical results will rely on this criterion assuming r is know. • R F ≤ t X F : the algorithm stops when the relative reconstruction error is sufficiently small. The choice of a good value for the tolerance parameter t is important: if t is too large, the SNPALQ could stop before the extraction of all the sources. If t is too low, the SNPALQ could extract too many source candidates in the presence of noise, making the whole algorithm computationally expensive. Theoretical results concerning the choice of t are left for future work.
Algorithm 2.1 Successive Nonnegative Projection Algorithm for LQ mixtures (SNPALQ) 1: Input:X ∈ R m×r : a r-LQ r-near-separable matrix following Definition 1.3 and Constraints (1.4), f : a strongly convex function satisfying Assumption 2.1, r max : number of sources, t ≥ 0: stopping criterion on the norm of the residual.
h j = argmin h∈∆ |K|(|K|+3) 2 f (x j − Π 2 (X K )h) 8: r j =x j − Π 2 (X K )h j 9:
end for 10: k = k + 1 11: end while 12: Output: A set K of indices such thatX K W up to a permutation.
2.2. Brute force algorithm. The conditions ensuring SNPALQ to recover the sources might not be satisfied in practice (see Sections 3.1.3 and 4.2.3). Therefore, we propose here a second algorithm, BF, inspired by the algorithm of Arora et al. [5] for linear mixtures. As we will see in Section 3.2, it requires milder assumptions for the source recovery. Noise-free mixtures -For the sake of simplicity, the rationale underlying BF is first exposed in the absence of noise. Let us assume w.l.o.g. that there are no duplicated columns in the data set X. Due to the separable assumption, X can be written as:
(2.1) X = W,X P ∈ R m×n ,
where P is a permutation andX contains the LQ mixings of W. Let us consider a column
s k = min h∈∆ n(n+3) 2 −1 x k − Π 2 (X) \{k} h 2 .
If x k is not a column of W, we have s k = 0 under the r-LQ separable mixing model (Definition 1.3 with N = 0). Moreover, under the assumption that W is order-2 α-robust simplicial, that is, α 2 (W) > 0, x k is a source, that is, a column of W, if and only if s k > 0. For sake of consistency with SNPALQ, this condition can be generalized to any function f fulfilling Assumption 2.1. Adopting this generalization, x k is as primary source if and only if
(2.2) min h∈∆ n(n+3) 2 −1 f x k − Π 2 (X) \{k} h > 0.
Noisy mixtures -We here extend the above principles to make the BF algorithm able to recover an approximation of W from noisy mixturesX = X + N for a bounded noise fulfilling max i∈[[t]] n i 2 ≤ for some ≥ 0; see Algorithm 2.2. To do so, we need to modify (2.2) in two ways.
• In the noise-free case, we assumed that no duplicated columns are present within X, and it is easy to discard such duplicates. In the noisy setting, when evaluating the residual (2.2), not only the columnx k should be removed from Π 2 (X) but also all columns close tox k (see Figure 3 for an illustration). • Moreover, as the noise might shift mixed data points outside the convex hull formed by Π 2 (W) and the origin, s k might be nonzero for a mixed columnx (that is,x k =x j for some j ∈ [[n − r]]); see Figure 3 for an illustration.
Therefore, the condition (2.2) in the noiseless case should be modified to
(2.3) min h∈∆ n(n+3) 2 −1 f x k − Π 2 (X) \{i∈[[n]] | f (x i −x k )>d} h * > L 2 2 (3 + ) 2 ,
with L the Lipschitz constant of f and d a threshold parameter discussed in Appendix B; see (B.22) for an explicit value. The right-hand side stems from the fact that the noise is corrupting both the data columns (with a maximum energy of ) and their quadratic products (with a maximum energy of 2 + 2 if the columns of X have a unit norm); see Definition B.12.
Following [5], the columns ofX satisfying the condition (2.3) are called the LQ-robust loners. Section 3.2 will show that these columns exactly correspond to good approximations of the sources. To approximately recover the sources, the BF algorithm then amounts to check which columns ofX are LQ-robust loners. However, due to the noise, different LQ-robust loners may be candidates for estimating the same source. Therefore, at the end of BF, the LQ-robust loners need to be clustered to obtain a single estimate of each source. Fortunately, such a clustering -described in Algorithm 2.2 -is easy and does not lead to any indeterminacy as the LQ-robust loners are located close to the sources, which are comparatively further from each others.
Algorithm 2.2 Brute force (BF)
1: Input: A r-LQ r-near-separable matrixX ∈ R m×r following Definition 1.3 and constraints (Eq.
h k = argmin h∈∆ f x k − Π 2 (X) \{i∈[[t]] | f (x i −x k )>d} h 5: if f x k − Π 2 (X N ) \{i∈[[t]] | f (x i −x k )>d} h k > L 2 2 (3 + ) 2 then 6: K = K ∪ {k} 7:
end if 8: end for 9: Clustering onX K : assign two columnsx i andx j ofX K to the same cluster if and only if x j −x k 2 ≤ 2 2 µ (d + L(2K(X) + )). Update K by keeping only one column for each cluster.
10: Output: A set K of indices such thatX K W up to a permutation.
BF algorithm as a post-processing -Even if the BF algorithm can be used per se to perform separation from LQ near-separable mixtures, it can also serve as a post-processing to refine the results provided by SNPALQ. This strategy is particularly appealing when SNPALQ robustness conditions are not met, in which case SNPALQ may extract mixed data columns or virtual sources in addition to the sought-after primary sources. Given an SNPALQ solution X K , assume r columns correspond to the primary sources W, and the |K| − r remaining ones to (spurious) columns in which the primary sources are mixed along with their quadratic products. Up to a permutation, the SNPALQ solution can be written as
(2.4)X K W,X ∈ R m×|K| ,
where P is a permutation, andX ∈ R m×(|K|−r) are data points. This matches the form of (2.1). Therefore, instead of using the BF algorithm directly on the data setX, it can be applied on the SNPALQ solutionX K , which has in practice a significantly smaller number of columns, that is, |K| n. Using BF as a post-processing step significantly reduces the computational cost; see Section 2.3. Furthermore, it is worth noting that SNPALQ already identifies as sources columns ofX lying far from each other. Thus, in our experiments, the clustering step in BF, whenever used as a post-processing, was never necessary since each cluster contained exactly one point.
Remark 2.2. While we advocate BF as a post-processing enhancing SNPALQ results, the reciprocal point of view can be also adopted: SNPALQ can be seen as a screening (or pruning) method, enabling to select only a few number of potential candidates and lightening the computational burden of BF.
2.3. Computational cost. The computational costs of the two proposed algorithms are as follows:
• SNPALQ: The complexity of the kth iteration is dominated by computing the projection step, which requires the projection of a m-by-n matrix onto a convex hull with k(k + 3)/2 + 1 vertices, requiring O mnk 2 operations with a first-order method [18, Appendix A]. • BF : Solving (2.3) for the n data points with a first-order method (as for SNPALQ) requires O mn 2 operations. This is computationally rather heavy. For example, for HS images, n is the number of pixels and typically of the order of millions. • SNPALQ+BF : Assuming SNPALQ extracts |K| indices, it requires O mn|K| 2 operations for SNPALQ, and O m|K| 2 operations for the post-processing with BF. Hence BF used as a post-processing has a smaller computational cost than SNPALQ which further justifies its use.
Remark 2.3 (Handling simpler models). As the LQ mixing model encompasses in particular the linear and bilinear ones, both SNPALQ and BF can be employed to separate these (simpler) mixtures. However, in practice, SNPALQ+BF should be specifically tailored in agreement with the target mixing model. For instance, bilinear mixtures can be handled by SNPALQ+BF by removing the projections on the squared sources in the projection steps, reducing the computational burden while improving the separation performance, avoiding the projections on the non-existing quadratic terms.
3. Theoretical results. This section reports the theoretical results associated with the recovery of the sources by SNPALQ and BF, even in the presence of noise. More specifically, in Section 3.1.1, we first derive robustness guarantees for SNPALQ when applied to linear mixings. These guarantees are then extended to LQ mixings in Section 3.1.2. The required conditions for these recovery results are discussed in Section 3.1.3. In Section 3.2, we derive and discuss the recovery guarantees for BF. For the sake of simplicity, the results derived in this section are stated for the particular choice f (·) = · 2 . Our results are stated in a more general setting for any function f (·) satisfying Assumption 2.1 in Appendix B, where the proofs are given.
3.1. Robustness of SNPALQ. As the LQ model is a generalization of the linear one (see Section 1), we first prove robustness of SNPALQ with respect to (w.r.t.) noise for linear mixings in Section 3.1.1. However, as expected, we will see that the derived bounds on the admissible noise levels and the corresponding error on the source estimates are slightly worse than those associated with SNPA because of the additional projections on the (non-existing) virtual sources. In Section 3.1.2, robustness of SNPALQ is proved in the case of LQ mixings.
3.1.1. Linear mixtures. Before stating the main result of this section in Theorem 3.1, let us introduce additional notations. For a matrix A ∈ R m×r A , we define 3
K(A) = A 1,2 = max i∈[[r A ]] a i 2 ,
which is the maximum of the 2 norm of the columns of a matrix A. We denote P f A (x) the projection of x onto the convex hull formed by the columns of A and the origin w.r.t. the semimetric induced by the function f (see Assumption 2.1):
P f A (x) = Ay * with y * = argmin y∈∆ f (x − Ay).
The residual of the projection is denoted R f A , that is,
R f A (x) = x − P f A (x)
. When used on matrices, both the projection and residual operators are applied column-wise
(for instance, for all i ∈ [[t]], R f A (X) i = R f A (x i ))
. Furthermore, we define the following quantities associated with the minimal norm of the residuals
ν f,Π 2 (A) (A) = min j∈[[r A ]] R f Π 2 (A) \{j} (a j ) 2 , γ f,Π 2 (A) (A) = min i,j∈[[r A ]] i =j R f Π 2 (A) \{i,j} (a j ) − R f Π 2 (A) \{i,j} (a i ) 2 , β Lin Π 2 (A) (A) = min ν f,Π 2 (A) (A), √ 2 2 γ f,Π 2 (A) (A) .
As such, β Lin Π 2 (A) (A) is the minimum between the smallest residual of the column of A and the smallest difference between the residuals of the columns of A after the projection onto Π 2 (A).
The following theorem states the robustness of SNPALQ in the case of linear mixtures. As mentioned earlier, it is here stated in a simplified formulation by assuming that f (·) = · 2 . Its generalized counterpart for any f (·) satisfying Assumption 2.1, as well as the corresponding detailed proof, are reported in Appendix B (see Theorem B.6).
Theorem 3.1 (Robustness of SNPALQ when applied on linear mixings -Simplified version). LetX
= WH + N ∈ R m×n be a near-separable [17] linear mixing with α Π 2 (W) (W) > 0 and β Lin Π 2 (W) (W) > 0. Let n i 2 ≤ for all i ∈ [[t]] with < O β Lin Π 2 (W) (W) 4 K(W) 2 . Then SNPALQ (Algorithm 2.1) with f = · 2 identifies in r steps all the columns of W up to error O K(W) 2 β Lin Π 2 (W) (W) 2 .
As in [18], Theorem 3.1 can be proved by induction: we show that SNPALQ extracts a new column of W at each iteration.
3.1.2. LQ mixings. We now extend the above result to the case of LQ mixings. Similarly to the linear case, we define
β LQ Π 2 (W) (A) = min ν f,Π 2 (W) (A) 2 µ L 1 − 1 G , γ f,Π 2 (W) (A) ,
for some constant G > 1 upper-bounded by a quantity depending on the mixtures; see (3.1) below. The robustness of SNPALQ when analyzing LQ mixings is stated below for f (·) = · 2 . In Appendix B, Theorem B.11 generalizes this statement to any f (·) satisfying Assumption 2.1.
Theorem 3.2 (Robustness of SNPALQ when applied on LQ mixings -Simplified version). Let
X = Π 2 (W)H + N ∈ R m×n
be an LQ mixing satisfying Definition 1.3 with α Π 2 (W) (W) > 0 and β LQ Π 2 (W) (W) > 0. Let
n i 2 ≤ with < O β LQ Π 2 (W) (W) 4 K(Π 2 (W)) 2
. Furthermore, let us assume that at each iteration of SNPALQ the following condition is fulfilled:
(3.1) K R · 2 Π2(B) (A) ≥ 2GK R · 2 Π2(B) (bi)i∈[ [s]] , (a i a j ) i≤j i∈[[k]] j∈[[k]] , (b i b j ) i≤j i∈[[s]] j∈[[s]] , (a i b j ) i∈[[k]] j∈[[s]]
where B contains the columns of W already extracted by SNPALQ andB the corresponding columns with noise, A contains the remaining columns of W still-to-be extracted, and G > 1 is a constant. Then, SNPALQ (Algorithm 2.1) with f = · 2 identifies in r steps the columns
of W up to an error O K(Π 2 (W)) 2 β LQ Π 2 (W) (W) 2 .
Similarly to the robustness result for linear mixtures, the above theorem is shown by induction. The main difference is that, in the LQ case, the virtual sources (and the mixed data columns for which their contribution is nonzero) might have a large residual and hence be extracted, whereas we would like to extract only the primary sources. Therefore, we must introduce the additional condition (3.1). Roughly speaking, it requires the energy of the residual of a non-already extracted source to be higher than twice the maximum of (i) the largest energy of the virtual sources, which prevents SNPALQ to extract a virtual source, and (ii) the largest energy of the already-extracted sources, which precludes extracting two columns ofX corresponding to the same source.
3.1.3. Interpretation of SNPALQ recovery conditions. In addition to the mixing constraints described in Section 1.3, among which near-separability, we here give more insights concerning some of the conditions for SNPALQ robustness when applied on LQ mixtures.
Condition on α Π 2 (W) (W) -The condition α Π 2 (W) (W) > 0 is of uttermost importance.
It ensures that no column of W lies within the convex hull of the other columns of W, the origin, and the second order products of the columns of W. On the contrary, α Π 2 (W) (W) = 0 would mean that at least one columns of W would be indistinguishable from the mixed data columns. Compared to SNPA, this condition is more restrictive for linear mixings. For example, let us consider the noiseless mixtures X = WH with
W = 1 1 1 1 0 0 0 1 0 for which α Π 2 (W) (W) = 0.
During its two first iterations, SNPALQ extracts the two first columns of W. But as w 1 w 2 = w 3 , all data columns in X can be written as a nonnegative combination of [w 1 , w 2 , w 1 w 2 ], and hence SNPALQ stops after the second iteration (the residual being zero) without extracting w 3 . On the contrary, SNPA is able to extract the thre columns of W since rank(W) = 3.
On the other hand, even if the virtual sources appear purely in the mixture, trying to solve the LQ problem using the naive approach explained at the beginning of Section 2, namely applying SNPA on a LQ-mixing with the hope to extract both sources and virtual sources and then rejecting the virtual ones, would require α Π 2 (W) (Π 2 (W)) > 0, which is a stronger condition than the one of SNPALQ. Indeed, this would require all the virtual sources not to lie within the convex hull of the other columns of Π 2 (W) and the origin, which should not be required as we do not need to estimate them. Condition on β LQ Π 2 (W) (W) -The condition β LQ Π 2 (W) (W) > 0 is stronger than the corresponding condition of SNPA which requires β Lin W) (W) > 0. As discussed for SNPA in [18], this condition is most often satisfied as long as α Π 2 (W) (W) > 0. Condition on the noise level -When applied to linear mixings, the admissible noise levels are lower with SNPALQ than SNPA, which requires < O
β Lin W (W) 2 K(W) 2
. This is expected, and will be confirmed in the numerical experiments of Section 4, since SNPALQ then performs useless additional projections on non-existing virtual sources. On the other hand, when applied to LQ mixings, the admissible noise levels are larger with SNPALQ than SNPA, since the recovery conditions of SNPA involve β Π 2 (W) (Π 2 (W)). Moreover, SNPALQ does not need the virtual sources to be present in the data set, while SNPA would require each virtual source to appear as a column ofX. Condition on K R · 2 Π 2 (B) (A) -At each iteration of SNPALQ, the following condition is required: with B the columns of W already extracted by SNPALQ (B their noisy approximation) and A the other columns of W. This means that at each iteration, a new column of W must have a larger residual than the already extracted sources and the virtual sources. This condition is the most difficult one to fulfil. In particular the difficulties might arise for a large number of sources, as more terms are present in the right-hand side (see Section 4.2.3), or when W has large entries. However,
K R · 2 Π 2 (B) (A) ≥ 2GK R · 2 Π 2 (B) (bi)i∈[ [
• The condition is sufficient but not necessary (see Section 4.2.3), making that SNPALQ can work even if it is not fulfiled. • Some terms in the right-hand side are or might be negligible, as
K R · 2 Π 2 (B) (b i ) i∈[[s]] ≤ K R · 2 B (b i ) i∈[[s]] and K R · 2 Π 2 (B) (b i b j ) i≤j i∈[[s]] j∈[[s]] ≤ K R · 2 (b i b j ) i≤j (b i b j ) i≤j i∈[[s]] j∈[[s]] ,
and the norm of both right-hand side terms is of the order of the noise level .
• The two remaining terms are driven by the correlation of the columns of W. If such a correlation is limited, the condition is expected to be more likely fulfilled. • Even if SNPALQ extracts spurious columns ofX, the post-processing with BF will discard them as it does not need this condition to be satisfied.
3.2. Robustness of BF on LQ mixings. We now study the robustness of the BF step. First, Theorem 3.3 below states that BF identifies the columns of W, provided some bounds on the admissible noise levels. The maximum corresponding source estimation error is also given. Then the recovery conditions are discussed. The proof of the above theorem closely follows the proof of [5] in the linear case. In particular it extends two definitions of [5] to LQ mixtures: i) the LQ-robust loners (see condition (2.3)) and ii) the canonical columns which are, roughly speaking, the columns corresponding to the sources (up to the noise) in the data set. It then amounts to show that the LQ-robust loners are approximately the canonical columns, which is done by Lemma B. 16 (showing that all the robust loners are close to a canonical column) and B.20 (showing that all canonical columns are robust loners). Extracting the robust loners thus enables to approximately recover the sources, as shown by Theorem B.21.
3.2.2. Discussion on the BF recovery conditions. Let us discuss the conditions to ensure the source recovery by BF. Condition on α Π 4 (W) (W) -Assuming α Π 4 (W) (W) > 0 is the counterpart of Deville's result in [9], which required the family (1.3), containing the products up to order four of the sources, to be linearly independent. Here, this condition is turned into a nonnegative independence, which is significantly less restrictive in general. In fact, this condition is most likely a necessary condition for LQ unmixing since α Π 4 (W) (W) = 0 implies that some columns of W can be written as mixtures of other observations. Condition on -The condition 4 d + 2 (2 + ) < α W (W) with d = O α Π 4 (W) (W) 2 is a limit on the admissible noise level. Roughly speaking, the better some sources can be approximated by a non-negative combination of the other terms of family (1.3), the smaller the noise power can be. Comparison with SNPALQ -The conditions of recovery of BF are very mild. For example, in the noiseless case, BF only requires α Π 4 (W) (W) > 0, while SNPALQ relies on much stronger conditions. This will be confirmed in the numerical experiments in Section 4. However, BF is computationally much more demanding (see Section 2.3), which motivates its use as a post-processing for SNPALQ.
Numerical results.
We here study the behaviors of SNPALQ and BF as a postprocessing on simulated yet realistic data sets in the specific applicative context of HS unmixing. The observed mixtures are supposed to follow the Nascimento model [33]. The function f (·) used by the algorithms is here chosen as f (·) = · 2 (in-depth study of other choice for f (·) is left for future work). The code is available from https://bit.ly/SNPALQv1. Section 4.2 dwells on noiseless mixtures. More precisely, we show in Section 4.2.1 that SNPALQ yields very good practical results in this setting, which are enhanced by the BF postprocessing in Section 4.2.2. We further show in Section 4.2.3 that the condition (B.15) is only sufficient: it does not need to be fulfilled for SNPALQ to provide reliable results. Lastly, Section 4.2.4 confirms that, beyond the usual Nascimento model involving bilinear mixtures, both SNPALQ and the BF generalize well to LQ models. In Section 4.3, the robustness of SNPALQ in the presence of noise is studied for different non-linearity levels.
SNPA [18] and SPA [20], two well-known algorithms for near-separable NMF, are used to benchmark the results of the proposed algorithm.
4.1. Experimental setting and metrics. Experiments are conducted on realistic LQ nearseparable nonnegative data sets X following Definition 1.3. The parameters of the model are chosen as follows.
• The primary sources (referred to as endmember spectra in the HS literature) defining the columns of W are defined as spectral signatures extracted from the USGS database 4 . They correspond to reflectance spectra associated with materials from diverse origins (such as minerals, soils, and plants) and naturally follow 0 ≤ W ≤ 1. • The matrix H is generated in the following way:
-The columns of a first matrixH of the same dimension as H are generated randomly using a Dirichlet distribution D(α, . . . , α) with α = 0.5, which is standard in HS imaging [31]. -The r first rows (corresponding to the linear contribution) are multiplied by 1 − ν, while the remaining rows (corresponding to the virtual endmembers) are multiplied by ν to enable various non-linearity levels:
(4.1) H = H [[r]] × (1 − ν) H [r+1,r] × ν .
Note that, acccordingly to the Nasciemento model [33] we consider here, we mostly focus on bilinear mixtures in this experimental section. In this case, the lines ofH [r+1,r] corresponding to squared sources (s j s j ) j∈ [[r]] are enforced to be all-zero lines. -The previous transformation does not preserve the sums of the entries in each column ofH which were equal to one (Dirichlet distribution). Since the columns are assumed to sum to (at most) one, the last step divides each column of H by its 1 norm. • The elements of the matrix N are independently and identically drawn from a centered Gaussian distribution with a variance corresponding to a given signal-to-noise ratio (SNR). • The matrixX is finally created ensuring all the entries to be non-negative:X = [Π 2 (W)H + N] + , where [.] + is the elementwise projection on the non-negative orthant. The quality of an algorithm is assessed using the minimum spectral angle distance (SAD) between the true and the estimated endmembers:
θ min = min i∈[[r]] SAD(w i , x K(i) ), where SAD(u, v) = u T v u 2 v 2
and where the set of indices K is permuted to maximize θ min . We consider a perfect separation is achieved if θ min > 0.999.
Numerical results on noiseless mixtures.
Study of SNPALQ.
We first explore the behavior of SNPALQ as a function of the number of endmembers r in a noiseless setting. We consider n = 1000 mixed pixels with m = 20 and the non-linearity parameter is chosen as ν = 0.5. We conducted 100 Monte-Carlo experiments, each time generating a new dataset. Fig. 4 reports the percentage of full recovery by the different algorithms. In this experiment, SNPALQ obtains much better results than SNPA or SPA, and achieves in more than 90% of the experiments a perfect separation. While an initial improvement of the results when r increases might look surprising, it is probably not to be linked directly with the r value itself, but rather with the generated H . Indeed, when r is small, the data columns are more spread within the convex hull formed by the endmembers and these are therefore more difficult to extract; see Section 4.2.3.
On the other hand, the results of SNPA and SPA are rather bad on this non-linear data set, and deteriorate quickly when the number of endmembers increases. While both algorithms obtain close results, it is interesting to note that SPA becomes worse than SNPA when r becomes closer to m, which is expected as SNPA has an interest over SPA mainly when the matrices W are either not full-rank or ill conditioned [18].
Study of BF as a post-processing step.
In this section, we analyze the relevance of the introduction of BF as a post-processing conducted after SNPALQ. Figure 4 displays in orange the separation quality when applying the BF to SNPALQ. This result show that BF enables to achieve perfect results for all experiments by improving SNPALQ results, especially for low r values. A natural question is however the cost of such a post-processing; see Section 2.2. Table 1 thus displays the number of columns extracted by SNPALQ (2nd line) as a function of the actual number r of sources. These columns are the input of the BF, and therefore they determine its computational time. Interestingly enough, on average, SNPALQ does not need to extract more than r + 1 components to extract all the columns of W. As such, the post-processing step is applied on a small number of columns ofX and is cheap. 3. Discussion about condition (3.1). The introduction of condition (3.1) is one of the major difference compared to the linear case, for which it does not appear explicitly 5 . As such, we here aim at discussing its validity on real noiseless HS data. It is important to notice that this context might be favorable, since W naturally fulfills 0 ≤ W ≤ 1. To do so, we propose the following complementary experiment: for each of the 100 Monte-Carlo experiments, we draw a new W matrix from the USGS database and split the columns of W into two disjoints matrices:
A = [(w i ) i∈J ,J ⊆[[r]] ] and B = [(w i ) i∈[[r]
]\J ]. We then check whether these matrices A and B fulfill condition (3.1). By repeating the process with all the possible A and B, we can thus obtain a percentage of subsets A and B for which condition (3.1) is fulfilled in the USGS database. In this experiment, we consider n = 1 000 samples with m = 50 observations. Figure 2, the condition is observed to be slightly less restrictive in general for SNPALQ than for SNPA.
Most importantly, the results become quite bad for relatively small r values: for r = 10, the condition is fulfilled for only slightly more than 5% of the tested subsets A and B. Thus it might be surprising that SNPALQ algorithm achieves perfect results in almost all experiments. Such a discrepancy appears because Condition 3.1 is only a sufficient condition. The reason is twofold:
• Condition 3.1 considers all the possible ways to split the matrix W into two submatrices A and B. This allows to prove the recovery of SNPALQ regardless of the order in which the columns of W are extracted. However, in practice, SNPALQ only needs this condition to be satisfied for the order in which it extracts the indices, and hence it is in general much milder. • The virtual endmembers typically do not not appear purely, which makes the condition too conservative (recall that this condition is not necessary in the linear case; see Theorem 3.1). In other words, Condition 3.1 considers the worst case scenario for any possible mixing matrix H while, in practice, the non-linearity can be mild. In summary, while Condition 3.1 might seem restrictive, SNPALQ can yield excellent results in settings in which it is not fulfilled. In particular, it could be of interest to include the non-linearity level ν in a study of necessary conditions for SNPALQ, which is left for future work.
Differences between LQ and bilinear mixtures.
To conclude this section, we now study the slight differences of behavior of SNPALQ+BF when analyzing LQ or bilinear mixtures. The experiment settings are similar to the one associated with Figure 4. We consider 100 Monte-Carlo runs of n = 1000 pixels with m = 20 and the non-linearity parameter is chosen as ν = 0.5. The difference is that the data sets are now LQ, instead of bilinear: squared sources are included in the mixtures. Figure 6 displays the results obtained by two variants of SNPALQ + BF:
• The orange curve ( markers) displays the results of the algorithm when no squared sources are included in the projection steps of both SNPALQ and the BF; • The yellow curve (+ markers) displays the results of the algorithm when squared sources are included in the projection steps of both SNPALQ and the BF. As can be seen with the orange curve, SNPALQ+BF (LQ version) almost perfectly handles LQ mixtures, similarly to what was shown above for bilinear ones. The slightly deteriorated results (which are still much better than the ones obtained by the linear algorithms) shown with the yellow curve (+ markers) were expected: by not incorporating the presence of squared sources during the unxming process, the algorithm introduces errors. As such, the user of SNPALQ+BF should use as much as possible prior knowledge to determine beforehand whether the data set results from bilinear or LQ mixings.
Robustness study: noisy mixtures.
The impact of the noise and non-linearity levels is now studied. We generated bilinear data setsX with 7 different SNR levels and 12 values for the non-linearity parameter ν. For each pair of SNR and ν values, 24 Monte-Carlo experiments are conducted on nonlinear mixtures characterized by m = 50 spectral bands, r = 10 endmembers and n = 1 000 pixels. Figure 7a. However, SNPALQ+BF performs worse than SNPA in the presence of a stronger noise (SNR ∈ [25dB, 30dB]), which is expected as it projects the residual onto non-existing virtual endmembers, leading to a loss of information (the norm of the residual decreases faster). SNPALQ+BF shows its benefit over SNPA when the non-linearity level increases and the noise level is not too large (upper-right corner of the figures). More precisely, when ν ≥ 0.3 and SNR ≥ 40 dB, SNPALQ+BF always obtains a perfect recovery, which represents a significant improvement over SNPA, up to 20%. In the lower-right corner of the figure, when the SNR decreases, the results of both algorithms deteriorate as the problem is highly difficult.
Conclusion.
In this paper, we have considered the problem of linear-quadratic blind source separation, under the near-separable assumption which requires the primary sources to appear purely in the data set. We first introduced SNPALQ, an extension of SNPA [18], which takes into account the presence of quadratic terms in the projection step. SNPALQ is guaranteed to recover the sources for linear-quadratic under appropriate conditions. We then introduced a second algorithm, namely brute-force (BF), and extension of the algorithm of Arora et al [5], which provably recovers the sources under milder conditions than SNPALQ. It is recommended to use BF as a post-processing of SNPALQ (denoted by SNPALQ+BF) due to its high computational cost. Finally, we illustrated the performance of SNPALQ and SNPALQ+BF in various settings, and showed that they obtained good separation results on realistic hyperspectral data sets, and for various experimental settings, including linear, bilinear and linear quadratic mixtures. Improving SNPALQ+BF results for low SNR while still alleviating recovery conditions of both algorithms is left for future work.
Appendix A. A few useful results of [20,18].
Lemma A.1 (Lemma 3.3 in [18]). For any B ∈ R m×s , x ∈ R m , and f satisfying Assump-
tion 2.1, we have R f B (x) 2 ≤ L µ x 2 .
Lemma A.2 (Lemma 3.4 in [18]). Let B ∈ R m×s and B =B + N with N 1,2 ≤ˇ , and f satisfy Assumption 2.1. Then,
max j R f B (b j ) 2 ≤ L µˇ .
Lemma A.3 (Lemma 3.7 in [18]). Let A ∈ R m×k , B, andB ∈ R m×s satisfy B −B 1,2 ≤ , and let f satisfy Assumption 2.1. Then,
ν R f B (A) ≥ α [A,B] ([A, B]) − min(s, 2)ˇ .
Lemma A.4 (Lemma 3.13 in [18]). Let B ∈ R m×s , A ∈ R m×k , n ∈ R m , and z ∈ ∆ k , and let f satisfy Assumption 2.1. Then,
f R f B (Az + n) ≤ f R f B (Az + n) and f R f B (Az + n) ≥ f R f B (A)z + n .
Lemma A.5 (Lemma 3 in [20]). Let the function f satisfy Assumption 2.1. Then, for any x 2 ≤ K and n 2 ≤ ≤ K,
f (x) − KL ≤ f (x + n) ≤ f (x) + 3 2 KL.
Lemma A.6 (Lemma 2 in [20]). Let Z = [P, Q], where P ∈ R m×k and Q ∈ R m×s , and let f satisfy Assumption 2.1. If ν(P) > 2 L µ K(Q), then, for any 0 ≤ δ ≤ 1 2 , [18]). Let x ∈ R m , B andB ∈ R m×s satisfy the inequality B −B 1,2 ≤ˇ ≤ B 1,2 , and let f satisfy Assumption 2.1. Then,
f * = max x∈∆ f (Zx) such that x i ≤ 1 − δ for 1 ≤ i ≤ k satisfies f * ≤ max i f (p i ) − 1 2 µ(1 − δ)ω(P) 2 . Lemma A.7 (Lemma B.1 inR f B (x) − R f B (x) 2 2 ≤ 12 L µ¯ B −B 1,2 .
Lemma A.8 (Lemma B.2 in [18]). Let x, y ∈ R m , B, andB ∈ R m×s be such that B −B 1,2 ≤ˇ ≤ B 1,2 , and let f satisfy Assumption 2.1. Then,
R f B (x) − R f B (y) 2 2 ≥ R f B (x) − R f B (y) 2 2 ≥ 4 3KL µˇ .
Appendix B. Proofs of our main results: SNPALQ and BF are provably robust in the presence of noise.
In this section, we study the robustness of SNPALQ (Section B.1 and B.2) and of BF (Section B.3). But before, let us introduce a few additional notations. For two matrices A ∈ R m×r A and B ∈ R m×r B , we define
(B.1) ν R f Π 2 (B) (A) ≥ α Π 2 ([A,B]) ([A, B]).
Proof. The result follows from the definitions of these quantities:
α Π 2 ([A,B]) ([A, B]) = min j∈[[k+s]] y∈∆ [A, B] j − Π 2 ([A, B]) \{j} y 2 ≤ min j∈[[k]] y∈∆ a j − Π 2 ([A, B]) \{j} y 2 ≤ min j∈[[k]] y∈∆ a j − Π 2 (B)y 2 ≤ ν R f Π 2 (B) (A) .
Lemma B.2 (Extension of [18]-Lemma 3.6). Let Z andZ ∈ R m×r satisfy Z −Z 1,2 ≤ˇ . Then,
α Π 2 (Z) (Z) ≥ α Π 2 (Z) (Z) −ˇ (1 + max(1, 2K(Z) +ˇ )).
Proof. We have
α Π 2 (Z) (Z) = min j∈[[r]] y∈∆ z j − Π 2 (Z) \{j} y 2 = min j∈[[r]] y∈∆ z j − n j − Π 2 (Z − N) \{j} y 2 = min j∈[[r]] y∈∆ z j − n j − Π 2 (Z) \{j} − [N, 0] \{j} − 2[0, (z k n l ) l≤k ] \{j} + [0, (n k n l ) l≤k ] \{j} y 2 ≥ min j∈[[r]] y∈∆ z j − Π 2 (Z) \{j} y 2 − n j 2 − [N, 2(z k n l ) l≤k − (n k n l ) l≤k ] \{j} y 2 ≥ α Π 2 (Z) (Z) −ˇ (1 + max(1, 2K(Z) +ˇ )).
Corollary B.3 (Extension of [18]-Corollary 3.7)
. Let A ∈ R m×k , B andB ∈ R m×s satisfy B −B 1,2 < C , and let f satisfy Assumption 2.1. Then,
(B.2) ν R f Π 2 (B) (A) > α Π 2 ([A,B]) ([A, B]) − C (1 + max(1, 2K([A, B]) + C )).
Proof. The cases s = 0 and s = 1, with s the number of columns of B, are direct extensions of Lemma A.3 as Π 2 (B) =B and
α Π 2 ([A,B]) ([A, B]) ≤ α [A,B] ([A, B]).
For the case s > 1, Lemma B.1 and B.2 imply that: 2K([A, B]) + C )).
ν R f Π 2 (B) (A) ≥ α Π 2 ([A,B]) ([A,B]) ≥ α Π 2 ([A,B] ([A, B]) − C (1 + max(1,
Lemma B.4 (Extension of [18]-Lemma B-3). Let
A ∈ R m×k , B ∈ R m×s ,B ∈ R m×s , f satisfy Assumption 2.1, and letˇ be such that B −B 1,2 ≤ˇ ≤ B 1,2 andˇ ≤ −1+ √ 1 + K. Then,
ω(R f Π 2 (B) (A)) ≥ β Lin Π 2 ([A,B]) ([A, B]) − 2 6 L µ B 1,2ˇ (2 +ˇ ).
Proof. For all i, we have
β Π 2 ([A,B]) ([A, B]) − R f Π 2 (B) (a i ) 2 ≤ R f Π 2 (B) (a i ) 2 − R f Π 2 (B) (a i ) 2 ≤ R f Π 2 (B) (a i ) − R f Π 2 (B) (a i ) 2 ≤ 12 L µˇ (ˇ + 2) Π 2 (B) 1,2 ,
where the last line is obtained using Lemma A.7 by noting that
Π 2 (B) − Π 2 (B) 2 ≤ˇ (ˇ + 2) ≤ Π 2 (B) 1,2 .
Furthermore, for all i, j,
1 √ 2 R f Π 2 (B) (a i ) − R f Π 2 (B) (a j ) 2 ≥ 1 √ 2 R f Π 2 (B) (a i ) − R f Π 2 (B) (a j ) 2 − 4 √ 2 3 Π 2 (B) 1,2 L µˇ (ˇ + 2) ≥ β Π 2 ([A,B]) ([A, B]) − 2 6 B 1,2 L µˇ (ˇ + 2),
where the last line is obtained by Lemma A.8, since Π 2 (B) 1,2 = B 1,2 (as b i 2 ≤ 1 for i ∈ [[s]]).
Theorem B.5 (Robustness of SNPAB when applied on linear mixings -induction step). Let the following hold:
• f satisfies Assumption 2.1 with strong convexity parameter µ and a gradient Lipschitz constant L. •X follows a linear mixing model. Precisely,X is near-separable [17] with
X = WH + N, W = [A, B] and A ∈ R m×k , B ∈ R m×s , H = [I r , H ] ∈ R r×n + where ∀j ∈ [[n]], h j ∈ ∆.
Let further assume that the noise is bounded with
n i 2 ≤ for all i ∈ [[t]]. • W = [A, B] is such that α Π 2 (W) (W) > 0 and β Lin Π 2 (W) (W) > 0. • The error onB ∈ R m×s satisfies B −B 1,2 ≤ˇ = C for some C > 0.
• is sufficiently small and satisfies
C < min α Π2(W) (W)µ 2(L + µ) , 2L + µ 2µ + K(W) + 2L + µ 2µ + K(W) 2 + α Π2(W) (W), β Lin Π2(W) (W) 2 µ 3/2 C 144K(W)L 3/2 , −1 + 1 + β Lin Π2(W) (W) 2 96K(W) µ L , 1 + K(W) − 1, CK(W) .
Then the index i corresponding to a columnx i ofX maximizing the function f (R f Π 2 (B) (.)) satisfies
(B.3) x i = Wh i = [A, B]h i where h il ≥ 1 − δ for some l ∈ [[k]],
where δ = 72 K(W)L 3/2 β Lin Π 2 (W) (W) 2 µ 3/2 . This implies
x i − w l 2 = x i − a l 2 ≤ + 2K(W)δ = 1 + 144 K(W) 2 β Lin Π 2 (W) (W) 2 L 3/2 µ 3/2 . (B.4)
Proof. The result (B.3) is proved by contradiction. Let us assume that the column ofX
maximizing f (R f Π 2 (B) (.)) satisfiesx i = Wh i + n i with h il < 1 − δ for 1 ≤ l ≤ k. We have f R f Π 2 (B) (x i ) ≤ Lemma A.4 f R f Π 2 (B) (W) h i + n i ≤ Lemma A.5 f R f Π 2 (B) (W) h i + 3 2 K R f Π 2 (B) (W) L
Furthermore, due to Lemma A.1,
R f Π 2 (B) (W) h i 2 ≤ max i R f Π 2 (B) (w i ) 2 ≤ Lemma A.1 L µ K(W).
Therefore,
f R f Π 2 (B) (x i ) ≤ f R f Π 2 (B) (W) h i + 3 2 K(W) L 3/2 µ 1/2 ≤ max x∈∆ r x l ≤1−δ 1≤l≤k f R f Π 2 (B) (W) x + 3 2 K(W) L 3/2 µ 1/2 . (B.5) Now, to bound f R f Π 2 (B) (W) x using f R f Π 2 (B) (A) , we use Lemma A.6. To do that, we must check that ν R f Π 2 (B) (A) > 2 L µ K R f Π 2 (B) (B)
, enabling to use the lemma with
P = R f Π 2 (B) (A) and Q = R f Π 2 (B) (B): ν R f Π 2 (B) (A) ≥ Cor. B.3 α Π 2 (W) (W) −ˇ (1 + max(1, 2K(W) +ˇ )) ≥ ≤ α Π 2 (W) (W)µ 2(L+µ) ≤ 2L+µ 2µ +K(W)+ 2L+µ 2µ +K(W) 2 +α Π 2 (W) (W) 2 L µˇ ≥ Lemma A.2 2 L µ K R f B (B) ≥ 2 L µ K R f Π 2 (B) (B) .
Thus, as δ < 1/2 when < β Π 2 (W) (W) 2 µ 3/2 144KL 3/2 , Lemma A.6 applies and we obtain
max x∈∆ r x l ≤1−δ 1≤l≤k f R f Π 2 (B) (W) x ≤ max j f R f Π 2 (B) (a j ) − 1 2 µδ(1 − δ)ω R f Π 2 (B) (A) 2 .
Therefore,
f R f Π2(B) (x i ) ≤ max j f R f Π2(B) (a j ) − 1 2 µδ(1 − δ)ω R f Π2(B) (A) 2 + 3 2 K(W) L 3/2 µ 1/2 < max j f R f Π2(B) (ā j ) − n j − 1 8 µδ(1 − δ)β Π2(W) (W) 2 + 3 2 K(W) L 3/2 µ 1/2 ,
where the last line is obtained by Lemma A.4 and the fact that (see Lemma B.4
)
ω R f Π 2 (B) (A) ≥ β Π 2 (W) (W) − 2 6K(W)Lˇ (ˇ + 2) µ ,
and thus ω R f Π 2 (B) (W) > β Π 2 (W) (W)/2 whenˇ < −1 + 1 + β Π 2 (W) (W) 2 96 B 1,2 µ L . Lastly, using again Lemma A.5 and the fact that, if < K(W),
(B.6) R f Π 2 (B) (ā j ) 2 ≤ L µ ā j 2 ≤ L µ (K(W) + ) ≤ 2 L µ K(W),
we obtain
f R f Π 2 (B) (x i ) < max j f R f Π 2 (B) (ā j ) − 1 8 µδ(1 − δ)β Π 2 (W) (W) 2 + 9 2 K(W) L 3/2 µ 1/2 . (B.7) Since 1 8 µδ(1 − δ)β Π2(W) (W) 2 ≥ 1 16 µβ Π2(W) (W) 2 δ = 1 16 µβ Π2(W) (W) 2 72 K(W)L 3/2 β Π2(W) (W) 2 µ 3/2 = 9 2 K(W) L 3/2 µ 1/2 , (B.8) we obtain f R f Π 2 (B) (x i ) < f R f Π 2 (B) (ā j ) , which is a contradiction sincex i should maxi- mize f R f Π 2 (B) (.)
among the X columns and the a i are among these columns. The proof of (B.4) follows the exact same lines as in [18]: we use (B.3), implying that
x i = (1 − δ )w l + k =l γ k w k for some l and 1 − δ ≥ 1 − δ, so that k =l γ k ≤ δ ≤ δ. Therefore x i − w l 2 = −δ w l + k =l γ k w k 2 ≤ 2δ max j w j 2 ≤ 2δ K(W) ≤ 2K(W)δ,
which leads to, when considering the noisy version of X,
x i − w l 2 ≤ (x i − x i ) + (x i − w l ) 2 ≤ + 2K(W)δ for some 1 ≤ l ≤ k.
Theorem B.6 (Robustness of SNPALQ when applied on linear mixings). Let X = WH + N ∈ R m×n be a near-sepable [17] linear mixing with α Π 2 (W) (W) > 0 and β Lin Π 2 (W) (W) > 0. Let furthermore f satisfy Assumption 2.1 and the noise be bounded:
n i 2 ≤ for all i ∈ [[t]] with C < min Cβ Lin Π 2 (W) (W) 2 µ 3/2 144K(W)L 3/2 , α Π 2 (W) (W)µ 2(L + µ) , 2L + µ 2µ + K(W) + 2L + µ 2µ + K(W) 2 + α Π 2 (W) (W), −1 + 1 + β Lin Π 2 (W) (W) 2 96K(W) µ L , 1 + K(W) − 1, K(W) ,
where C = 1 + 144
K 2 β Lin Π 2 (W) (W) 2 L 3/2 µ 3/2
and L and µ defined in Assumption 2.1. Then, SNPALQ (Algorithm 2.1) identifies in r steps all the columns of W up to error C . Precisely, denoting by K the index set extracted by SNPALQ after r steps, there exists a permutation π of [[r]] such that:
max 1≤j≤r x K(j) − w π(j) 2 ≤ˇ = C .
Proof. The result follows by applying Theorem B.5 inductively using
C = 1 + 144 K(W) 2 β Lin Π2(W) (W) 2 L 3/2 µ 3/2 .
The matrix B of Theorem B.5 corresponds to the columns extracted so far by SNPA, while A corresponds to the columns of W remaining to be extracted. Note that the initialisation of the induction is done with B being the empty matrix.
(B.9) K(P) ≥ 2G L µ K(Q) with G > L µ ≥ 1,
then, for any δ ∈ 0, 1 2 ,
(B.10) f * = max x∈∆ f (Zx) such that x i ≤ 1 − δ for 1 ≤ i ≤ k satisfies (B.11) f * ≤ max i f (p i ) − 1 2 µ(1 − δ)δΩ(P) 2
with Ω(P) = min γ(P), K(P)
2 µ L 1 − 1 G .
Proof. First, let us provide a lower bound for f * . Remember that due to the strong convexity of f with parameter µ, its gradient Lipschitz continuity and the fact that f (0 m ) = 0, we have that for all y ∈ R m (B.12)
µ 2 y 2 2 ≤ f (y) ≤ L 2 y 2 .
Consequently, since (1 − δ)p i is an admissible solution, we have that
(B.13) f * ≥ f ((1 − δ)p i ) ≥ 1 2 µ(1 − δ) 2 p i 2 2 ≥ µ 8 p i 2 2 ,
where the last inequality is due to the assumption δ ≤ 1/2. Let us now discuss upperbounds of f . By strong convexity of f , the optimal solution x * of (B.10) is attained at a vertex of the feasible domain {x ∈ R r
+ | r i=1 x i ≤ 1 and x i ≤ 1 − δ for 1 ≤ i ≤ r}. Here are the different cases a) x * = 0 r ; b) x * = e i for k + 1 ≤ i ≤ r; c) x * = (1 − δ)e j for 1 ≤ j ≤ k; d) x * = δe i + (1 − δ)e j for 1 ≤ i, j ≤ k; e) x * = δe i + (1 − δ)e j for k + 1 ≤ i ≤ r and 1 ≤ j ≤ k
Let us analyze them separately. a) This first case is clearly impossible, as f (0 m ) = 0 and f (y) > 0 for all y = 0; see Eq. (B.12). b) Zx * = q i for some i. Using Eq. (B.12), we obtain (B.14)
f
* = f (q i ) ≤ L 2 K(Q) ≤ Hyp. (B.9) µ 8G 2 K(P) 2 < Eq. (B.13) f * ,
which is a contraction. c) Zx * = (1 − δ)p i for some i. Let us then distinguish two subcases:
• If p j 2 2 ≤ µ 4L K(P) 2 , then:
f * = f ((1 − δ)p j ) < f (0m)=0 f strongly convex (1 − δ)f (p j ) ≤ Eq. (B.12) (1 − δ) L 2 p j 2 2 ≤ (1 − δ) µ 8 K(P) 2 < µ 8 K(P) 2 ≤ f * ,
which is a contradiction. • If p j 2 2 > µ 4L K(P) 2 , then, by strong convexity of f ,
f * ≤ (1 − δ)f (p j ) − 1 2 µδ(1 − δ) p j 2 2 = f (p j ) − δf (p j ) − 1 2 µδ(1 − δ) p j 2 2 . Since f (p j ) ≥ Eq. (B.12) µ 2 p j 2 2 ≥ 1 2 µ(1 − δ) p j 2 2 , f * < f (p j ) − µδ(1 − δ) p j 2 2 ≤ f (p j ) − 1 2 µδ(1 − δ) µ 2L K(P) 2 ≤ f (p j ) − 1 2 µδ(1 − δ) K(P) 2 µ L 1 − 1 G 2 ,
which satisfies the bound of the theorem. d) Zx * = δp i + (1 − δ)p j for some i = j. Then, by strong convexity of f ,
f * ≤ δf (p i ) + (1 − δ)f (p j ) − 1 2 µδ(1 − δ) p i − p j 2 2 ≤ K(P) − 1 2 µδ(1 − δ)γ(P) 2 e) Yx * = δq i + (1 − δ)p j for some i, j.
First (similarly to case c), let us distinguish two subcases:
• Let us assume p j 2 2 ≤ µ 4L K(P) 2 . Then,
f * = f (δq i + (1 − δ)p j ) < Strong convexity δf (q i ) + (1 − δ)f (p j ) ≤ Eq. (B.12) δ L 2 q i 2 2 + (1 − δ) L 2 p j 2 2 < Hyp. (B.9) δ µ 8 K(P) 2 + (1 − δ) µ 8 K(P) 2 = µ 8 K(P) 2 ≤ B.13 f * , a contradiction. • If p j 2 2 > µ 4L K(P) 2 , then, by strong convexity of f , f * ≤ δf (q i ) + (1 − δ)f (p j ) − 1 2 µδ(1 − δ) q i − p j 2 2
Using the triangle inequality, we obtain
q i − p j 2 ≥ p j 2 − q i 2 , since q i 2 ≤ K(Q) < Hyp. (B.9) 1 g µ 4L K(P) ≤ p j 2 . Thus, q i − p j 2 ≥ 1 2 µ L K(P) − 1 2G µ L K(P) = 1 2 µ L K(P) 1 − 1 G . Furthermore, f (q i ) ≤ L 2 q i 2 2 ≤ L 2 K(Q) 2 ≤ µ 8G 2 K(P) 2 < L 2G 2 p j 2 2 ≤ L µG 2 f (p j ).
Putting the above expression altogether, we have
f * < f (p j ) + δ L µG 2 f (p j ) − δf (p j ) − 1 2 µδ(1 − δ) 1 2 µ L K(P) 1 − 1 G 2 ≤ f (p j ) + L µG 2 − 1 δf (p j ) − 1 2 µδ(1 − δ) 1 2 µ L K(P) 1 − 1 G 2 ≤ f (p j ) − 1 2 µδ(1 − δ) 1 2 µ L K(P) 1 − 1 G 2
where the last line, which satisfies the bound of the theorem, requires L < µG 2 .
Lemma B.8. Let A ∈ R m×k andB ∈ R m×s be such that B −B = N and N 2 < , and let f satisfy Assumption 2.1. Then,
Ω(R f B (A)) 2 ≥ Ω(R f B (A)) 2 − 4 (K(A) + K(B)).
Proof. Let us look at the two terms of Ω(R f B (A)):
• Denoting z a j = argmin x∈∆ a j −Bx 2 , we have, for j ∈ [[k]], R f B (a j ) 2 = a j −Bz a j 2 = a j − (B + N)z a j 2 ≥ a j − Bz a j 2 − Nz a j 2 Thus, for j ∈ [[k]], R f B (a j ) 2 2 ≥ a j − Bz a j 2 − Nz a j 2 2 ≥ a j − Bz a j 2 2 − 2 a j − Bz a j 2 Nz a k 2 + Nz a j 2 2 ≥ R f B (a j ) 2 2 − 2(K(A) + K(B)) ,
where the last line is obtained since z a j ∈ ∆. This yields
K(R f B (A)) 2 ≥ K(R f B (A)) 2 − (K(A) + K(B)) ,
and, as µ
L 1 − 1 G 2 ≤ 1, K(R f B (A)) 2 4 µ L 1 − 1 G 2 ≥ K(R f B (A)) 2 4 µ L 1 − 1 G 2 − (K(A) + K(B)) 2 .
• Denoting z a i = argmin x∈∆ a i −Bx 2 and z a j = argmin x∈∆ a j −Bx 2 ,
we have, for i = j, i, j ∈ [[k]], R f B (a i ) − R f B (a j ) 2 ≥ a i − Bz a i − (a j − Bz a j ) 2 − N(z a i − z a j ) 2 .
This yields
R f B (a i ) − R f B (a j ) 2 2 ≥ γ(R f B (A)) 2 − 4 (K(A) + K(B)),
which gives the result.
Lemma B.9. Let A ∈ R m×k such that K(A) ≤ 1 andB ∈ R m×s be such that B −B = N and N 2 < < 1, and let f satisfy Assumption 2.1. Then,
Ω(R f Π 2 (B) (A)) 2 ≥ Ω(R f Π 2 (B) (A)) 2 − 4(K(A) + K(Π 2 (B))) max( , 2 K(B) + 2 ).
Proof. The result follows directly from the previous Lemma B.8, by noting that
Π 2 (B) − Π 2 (B) 2 ≤ max( , 2 K(B) + 2 ).
Theorem B.10 (Robustness of SNPAB when applied on linear-quadratic mixings -induction step). Let the following hold:
• X follows a LQ mixing model. Precisely,X satisfies Definition 1.3 with
X = Π 2 (W)H + N, W = [A, B] and A ∈ R m×k , B ∈ R m×s , H ∈ R r(r+1) 2 ×n + with ∀j ∈ [[n]], h j ∈ ∆.
Let us further assume the noise to be bounded as N 1,2 ≤ , and denote by X = Π 2 (W)H the noiseless version ofX. Note that, with these notations,
Π 2 (W) = (ai) i∈[[k]] , (b i ) i∈[[s]] , (a i a j ) i≤j i∈[[k]] j∈[[k]] , (b i b j ) i≤j i∈[[s]] j∈[[s]] , (a i b j ) i∈[[k]] j∈[[s]] . •B ∈ R m×s satisfies B −B 1,2 ≤ C for some C > 0. • W = [A, B] is such that α Π 2 (B) (A) > 0, γ(R f Π 2 (B) (A)) > 0.
We further assume w.l.o.g. that K(W) ≤ 1.
• For some G > 1, the matrix W and the considered A and B satisfy: • f satisfies Assumption 2.1 with strong convexity parameter µ and gradient Lipschitz constant L such that L < µG 2 . • is sufficiently small so that (B.16)
(B.15) K R f Π 2 (B) (A) ≥ 2GK R f Π 2 (B) (bi)i∈C < min −40L 3/2 − 16µ 3/2 CK(W) + (40L 3/2 + 16µ 3/2 CK(W)) 2 + 32µ 3 C 2 Ω(R f Π 2 (B) (A)) 2 K(Π 2 (W)) 16µ 3/2 C , µ 3/2 CΩ(R f Π 2 (B) (A)) 2 K(Π2(W))(40L 3/2 + 8Cµ 3/2 ) , min(M, Ω(R f Π 2 (B) (A))) 2 8K(Π2(W)) , 1 + min(M, Ω(R f Π 2 (B) (A))) 2 8K(Π2(W)) − K(W) with M a constant
and
Ω(R f Π 2 (B) (A)) = min γ(R f Π 2 (B) (A)), K(R f Π 2 (B) (A)) 2 µ L 1 − 1 G .
Then the index i corresponding to a columnx i ofX maximizing the function f (R f Π 2 (B) (.)) satisfies
x i = Π 2 (W)h i with h il ≥ 1 − δ for some l ∈ [[k]], (B.17) and (B.18) δ = 20 K(Π 2 (W)
)
Ω(R f Π2(B) (A)) 2 − 8K(Π 2 (W))C max(1, 2K(W) + C ) L 3/2 µ 3/2 ≤ 1 2 .
This implies
(B.19) x i − w l 2 = x i − a l 2 ≤ 1 + 40K(Π 2 (W)) 2 Ω(R f Π 2 (B) (A)) 2 − M 2 L 3/2 µ 3/2 .
Proof. The robustness is proved by contradiction. Let us assume that the column ofX
maximizing f (R f Π 2 (B) (.)) satisfiesx i = Π 2 (W)h i + n i with h il < 1 − δ for 1 ≤ l ≤ k. We have f R f Π 2 (B) (x i ) ≤ Lemma A.4 f R f Π 2 (B) (Π 2 (W)) h i + n i ≤ Lemma A.5 f R f Π 2 (B) (Π 2 (W)) h i + 3 2 K R f Π 2 (B) (Π 2 (W)) h i L.
Using Lemma A.1,
R f Π 2 (B) (Π 2 (W)) h i 2 ≤ max i R f Π 2 (B) (Π 2 (W) i ) 2 ≤ Lemma A.1 L µ K(Π 2 (W)),
we obtain
f R f Π2(B) (x i ) ≤ f R f Π2(B) (Π 2 (W)) h i + 3 2 K(Π 2 (W)) L 3/2 µ 1/2 ≤ max x∈∆ r x l ≤1−δ 1≤l≤k f R f Π2(B) (Π 2 (W)) x + 3 2 K(Π 2 (W)) L 3/2 µ 1/2 ≤ Lem (B.7) Eq. (B.15) max j f R f Π2(B) (a j ) − 1 2 µδ(1 − δ)Ω R f Π2(B) (A) 2 + 3 2 K(Π 2 (W)) L 3/2 µ 1/2 ≤ Lemma A.4 max j f R f Π2(B) (ā j ) − n j − 1 2 µδ(1 − δ)Ω R f Π2(B) (A) 2 + 3 2 K(Π 2 (W)) L 3/2 µ 1/2 ≤ Lemma A.5 max j f R f Π2(B) (ā j ) − 1 2 µδ(1 − δ)Ω R f Π2(B) (A) 2 + 9 2 K(Π 2 (W)) L 3/2 µ 1/2 ≤ Lem (B.9) max j f R f Π2(B) (ā j ) − 1 2 µδ(1 − δ) Ω R f Π2(B) (A) 2 −4(K(A) + K(Π 2 (B)))C max(1, 2K(B) + C )] + 9 2 K(Π 2 (W)) L 3/2 µ 1/2 . ≤ Lem (B.9) max j f R f Π2(B) (ā j ) − 1 2 µδ(1 − δ) Ω R f Π2(B) (A) 2 − 8K(Π 2 (W))C × max(1, 2K(W) + C )] + 9 2 K(Π 2 (W)) L 3/2 µ 1/2 . (B.20)
The fifth inequality follows from Lemma A.1 since
R f Π 2 (B) (ā j ) 2 ≤ L µ ā j 2 ≤ L µ (K(Π 2 (W)) + ) ≤ 2 L µ K(Π 2 (W)),
if ≤ K(Π 2 (W)), which implies that
f * < max j f R f Π2(B) (ā j ) − 1 2 µδ(1 − δ) Ω R f Π2(B) (A) 2 − 8K(Π 2 (W))C max(1, 2K(W) + C ) + 10 2 K(Π 2 (W)) L 3/2 µ 1/2 .
Then, replacing δ by its expression (B.18), we obtain
1 2 µδ(1 − δ) Ω(R f Π 2 (B) (A)) 2 − 8K(Π 2 (W))C max(1, 2K(W) + C ) ≥ 1 4 µδ Ω(R f Π 2 (B) (A)) 2 − 8K(Π 2 (W))C max(1, 2K(W) + C ) = 1 4 µ 20 K(Π 2 (W)) Ω(R f Π 2 (B) (A)) 2 − 8K(Π 2 (W))C max(1, 2K(W) + C ) L 3/2 µ 3/2 × Ω(R f Π 2 (B) (A)) 2 − 8K(Π 2 (W))C max(1, 2K(W) + C ) = 10 2 K(Π 2 (W)) L 3/2 µ 3/2 .
Therefore we finally obtain a contradiction since we should have
f R f Π 2 (B) (x i ) < max j f R f Π 2 (B) (ā j ) ,
which is impossible sincex i should maximize f R f Π 2 (B) (.) among the columns ofX and thē a j are among these columns.
Note that in the previous reasoning, we have assumed δ to be in 0, 1 2 , which is satisfied if:
C < min −40L 3/2 − 16µ 3/2 CK(W) + 40L 3/2 + 16µ 3/2 CK(W) 2 + 32µ 3 C 2 Ω(R f Π 2 (B) (A)) 2 K(Π2(W)) 16µ 3/2 C , Cµ 3/2 Ω(R f Π2(B) (A)) 2 K(Π 2 (W))(40L 3/2 + 8Cµ 3/2 )
,
Ω(R f Π2(B) (A)) 2 8K(Π 2 (W)) , −K(W) + K(W) 2 + Ω(R f Π2(B) (A)) 2 8K(Π 2 (W))
The proof of (B.19) follows from result (B.17). We have
x i = (1 − δ )w l + k =l γ k w k + i,j
g ij w i w j for some l and 1 − δ ≥ 1 − δ so that k =l γ k + i,j g ij ≤ δ ≤ δ. Hence
x i − w l 2 = −δ w l + k =l γ k w k + i,j g ij w i w j 2 ≤ 2δ max j Π 2 (W) j 2 = 2δ K(Π 2 (W)) ≤ 2δK(Π 2 (W)),
which gives, when considering the noisy version of X,
x i − w l 2 ≤ ( x i − x i ) + (x i − w l ) 2 ≤ + 2K(Π 2 (W))δ for some 1 ≤ l ≤ k.
To conclude the proof, we use the fact that
+ 2K(Π 2 (W))δ ≤ 1 + 40K(Π 2 (W)) 2 Ω(R f Π 2 (B) (A)) 2 − M 2 L 3/2 µ 3/2 =Ĉ ,
where M 2 is a constant 6 chosen such that M 2 = 8K(Π 2 (W))C max(1, 2K(W) + C ) , which requires C < M 2 8K(Π 2 (W)) and
C < −K(W) + K(W) 2 + M 2 8K(Π 2 (W))
.
TheoremC < min −40L 3/2 − 16µ 3/2Ĉ K(W) + 40L 3/2 + 16µ 3/2Ĉ K(W) 2 + 32µ 3Ĉ2 β LQ Π 2 (W) (W) 2 K(Π 2 (W)) 16µ 3/2Ĉ , µ 3/2Ĉ β LQ Π 2 (W) (W) 2 K(Π2(W))(40L 3/2 + 8Ĉµ 3/2 ) , min(M, β LQ Π 2 (W) (W)) 2 8K(Π2(W)) , K(W) 2 + min(M, β LQ Π 2 (W) (W)) 2 8K(Π2(W)) − K(W) whereĈ = 1 + 40K(Π 2 (W)) 2 β LQ Π2(W) (W) 2 − M 2 L 3/2 µ 3/2 ,
with M a constant 7 (the smaller M , the more restrictive the condition on the noise, but the better the estimation). Furthermore, let us assume that at each iteration of SNPALQ the following condition is fulfilled:
K R f Π2(B) (A) ≥ 2GK R f Π2(B) (bi)i∈[ [s]] , (a i a j ) i≤j i∈[[k]] j∈[[k]] , (b i b j ) i≤j i∈[[s]] j∈[[s]] , (a i b j ) i∈[[k]] j∈[[s]] , 6
The reader might wonder why such a constant M does not appear in SNPALQ robustness proof for linear mixing. Actually, it was implicitly chosen as M 2 = Ω(R f Π 2 (B) (A)) 2 /2. 7 Despite a slight loss of generality, the reader can think of M 2 = Ω(R f Π 2 (B) (A)) 2 /2 to create a link with the linear case.
where B contains the columns of W already extracted by SNPALQ andB the corresponding columns with noise, A contains the remaining columns of W still-to-be extracted, and L < µG 2 is a constant. Then, SNPALQ identifies in r steps the columns of W up to an error C . Precisely, denoting K the index set extracted by SNPALQ after r steps, there exists a permutation π of [[r]] such that:
max 1≤j≤r x K(j) − w π(j) 2 ≤Ĉ .
Proof. The result follows by induction.
• In the initialization step, B is the empty matrix.
• The induction step is given by Theorem B.10: the B matrix corresponds to the columns of W extracted so far by SNPALQ, while the columns of A the ones still-to-be extracted. LettingĈ
= 1 + 40K(Π 2 (W)) 2 β LQ Π 2 (W) (W) 2 − M 2 L 3/2 µ 3/2
in Theorem B.10, we obtain that if the already extracted columns are at a distance at mostĈ of some columns of W (more exactly, B − B 2 ≤Ĉ ), then the next extracted column will be at distance at mostĈ from a new column of W (that is, a column of A), provided that is small enough. where V = L 2 µα Π 4 (W) (W) 2 K(X) 2 Y 3 (2K(X)+ ) 2 /(2Y ) + K(X) + max(1, 2K(X)+ )( +K(X)) + 3 2 L(4K(X) + ), with Y = 1 + max(1, K(X)). We callx j a robust loner if
min h * ∈∆ f (x j − Π 2 (X [[t]
]\L )h * ) > L 2 2 (1 + max(1, 2K(X) + )) 2 .
Definition B.13 (Canonical columns). LetX be r-LQ near-separable; see Definition 1.3. We call canonical columns (associated to i ∈ [[r]]), the columnsX k(i) , k(i) ∈ [[t]], ofX such that all the columns of H k(i) have a single nonzero entry located in their ith row. Note that by definition of near-separability, there exists at least a canonical column for all i ∈ [[r]]. Moreover, all the canonical columnsx k(i) associated to i ∈ [[r]] satisfy f (x k(i) −w i ) < L Lemma B.14. LetX be r-LQ near-separable (Definition 1.3). Considering all the canonical columns, written asX K (that is, the canonical columns associated to all i ∈ [[r]]), every column x j ofX is such that
min h * ∈∆ f x j − Π 2 X K h * ≤ L 2 2 (1 + max(1, 2K(X) + )) 2 .
Proof. For all h ∈ ∆, we have: Proof. The result is proved by contraposition. We want to show that If ∀i ∈ [[r]], f (x j − w i ) > d + (2K(X) + ), thenx j is not a robust loner.
f x j − Π 2 X K h ≤ L 2 x j − Π 2 X K h 2 2 ≤ L 2 x j − x j 2 + x j − Π 2 (X K )h 2 + Π 2 (X K ) − Π 2 X K h 2 2 . Moreover x j − x j 2 + x j − Π 2 (X K )h 2 + Π 2 (X K ) − Π 2 X K h 2 ≤ + x j − Π 2 (X K )h 2 + max( , 2K(X) + 2 ). Thus, min h * ∈∆ f x j − Π 2 X K h * ≤ L 2 2 (1 + max(1, 2K(X) + )) 2 .
Ifx j is such that ∀i ∈ [[r]], f (x j − w i ) > d + L(2K(X) + ), then ∀i ∈ [[r]], f (x j −x k(i) ) > d, withx k(i) the canonical columns associated to i; see lemma B.15. As such,denotingX K all the canonical columns, to be a robust lonerx j must satisfy min h * ∈∆ f x j − Π 2 (X K )h * > L 2 2 (1 + max(1, 2K(X) + )) 2 .
This is however not the case according to Lemma B.14. Thus, by definition,x j is not a robust loner.
Lemma B.17. LetX = Π 2 (W)H + N be r-LQ near-separable (Definition 1.3), i ∈ [[r]] and x k(i) a canonical column associated to i. If, for some k ∈ [[t]], f (x k −w i ) ≤ d− 3 2 L(2K(X)+ ), then f (x k −x k(i) ) ≤ d.
Proof. We have Proof. We want to prove that x j a ij > 1 − 2d − 3 L(4K(X) + ) LK(X) 2 [1 + max(1, K(X))] 2 ⊆ x j f (x j − w i ) ≤ d − 3 2 L(2K(X) + )
f (x k −x k(i) ) = f (x k − w i − n k(i) ) ≤ f (x k − w i ) + 3 2 LK(x k − w i ) = d −
Let us consider a columnx j ∈ x j a ij > 1 − 2d−3 L(4K(X)+ )
LK(X) 2 [1+max(1,K(X))] 2 . We have (looking at the noiseless version x j ofx j ) that where the second inequality is obtained using 1 1−a ij r k =i a kj + r l=1 r q=l b jlq = 1. Therefore,
f (x j − w j ) ≤ L 2 x j − w i 2 2 = L 2 w i − a ij w i −f (x j − w j ) ≤ L 2
(1 − a ij ) 2 K(X) 2 [1 + max(1, K(X))] 2 .
To conclude the proof, let us consider the noisyx j , we have
f (x j − w i ) = f (x j + n j − w i ) ≤ f (x j − w i ) + 3 2 L K(x j − w i ) ≤ L 2
(1 − a ij ) 2 K(X) 2 [1 + max(1, K(X))] 2 + 3L K(X) Proof. We have Proof. Letx k(i) be a canonical column associated to i ∈ [[r]]: we have that f (x k(i) − w i ) ≤ L 2 2 . To check whetherx k(i) is a robust-loner, we must leave out of consideration the columns x k such that f (x k −x k(i) ) ≤ d. This particularly excludes all the columns satisfying
f (x k(i) − Π 2 (X J )h) ≥ f (x k(i) − Π 2 (X J )h) − L max(1, 2K(X) + )K(x k(i) − Π 2 (X J )h) ≥ f (w i − Π 2 (X J )h) − L K(w i − Π 2 (X J )f (x k − w i ) ≤ d − 3 2 L(2K(X) + ),
see Lemma B.17. In particular, only the columnsx j , j ∈ J with x j = r k=1 a kj w k + r l=1 r q=l+1 b jlq w l w q and a ij ≤ 1− 2d−3 L(4K(X)+ )
LK(X) 2 [1+max(1,K(X))] 2 are taken into account (Lemma B.18). Since the 2 distance of w i to the convex hull of Π 4 (W) is at least α Π 4 (W) (W), the distance between w i and the convex hull of the retained X J columns and their quadratic product is at least 2d−3 L(4K(X)+ ) LK(X) 2 [1+max(1,K(X))] 2 α Π 4 (W) (W). As for all h ∈ ∆
f (w i − Π 2 (X J )h) ≥ µ 2 w i − Π 2 (X J ) 2 2 , we obtain f (w i − Π 2 (X J )h) ≥ µ 2L
2d − 3 L(4K(X) + ) K(X) 2 [1 + max(1, K(X))] 2 α Π 4 (W) (W) 2 .
Thus, f (x k(i) − Π 2 (X J )h) ≥ L 2 2 (3 + ) 2 (see Lemma B.19) and hencex k(i) is a robust loner. Then, BF with f satisfying Assumption 2.1 identifies the columns of W up to a 2 error of 2 µ (d + L(2K(X) + )).
Proof. By Lemma B.20, all canonical columns are robust loners. Moreover, Lemma B.16 shows that every robust-lonerx j satisfies f (x j − w i ) ≤ d + L(2K(X) + ) for some i ∈ [[r]]. As such, identifying the robust loners enables to approximately identify the columns of W. Since several robust-loners can correspond to the same source, we need to apply a clustering step to regroup them. This is done easily, as two robust lonersx j andx k correspond to the same source if and only if they satisfy x j −x k 2 ≤ 2 2 µ (d + L(2K(X) + )). In fact, • If two robust lonersx j andx k correspond to the same source (in the sense that f (x j − w i ) ≤ d + L(2K(X) + ) and f (x k − w i ) ≤ d + L(2K(X) + )), they must
(w i w j ) i,j∈[[r]] j<i , (w i w j w k ) i,j,k∈[[r]] k<j<i , (w i w j w k w l ) i,j,k,l∈[[r]] l<k<j<i ,
Figure 3 .
3Illustration of condition (2.3) with f (·) = · 2 . The point under scrutinyx k is represented in violet ('X' marker). The dots are the columns of X\{k}, and the yellow cross ('+' marker) correspond to the quadratic products of the columns of X. The plain line ball of radius d and centerx k contains the columns ofX which are discarded in (2.3). The dotted polygon is the convex hull of the origin and the columns of Π2(X) \{k} that are not contained in the ball of radius d aroundx k . The dashed circle of radius 2 (3 + ) 2 indicates the distance at which the point must be located from the dotted convex hull to be considered an LQ-robust loner. On the figure (a), the dashed circle does not intersect the convex hull, and hence the cross is an LQ-robust loner. On figure (b), the dashed circle overlaps the convex hull, making that its center point is not a robust loner. of X, x k for k ∈ [[n]]. We can check whether it is contained in the convex hull of the other columns of X, their LQ mixtures and the origin by solving
3.2.1. Main result. The following theorem characterizes the robustness of BF. It is stated in a simplified form by considering f (·) = · 2 . Its generalized counterpart handling any f (·) satisfying Assumption 2.1 is reported in Appendix B (see Theorem B.21).
Theorem 3. 3 (
3Robustness of BF when applied on LQ mixings -Simplified version). LetX = Π 2 (W) + N, satisfying Definition 1.3 with n i 1 ≤ for i ∈ [[n]]. Let further assume that satisfies4 d + 2 (2K(X) + ) < α W (W), with d = O α Π 4 (W) (W) 2 (seeEquation (B.22) in Appendix B for the full expression). Then, BF (Algorithm 2.2) applied onX with f = · 2 identifies the columns of W up to a 2 error of d + 2 (2K(X) + ).
Figure 4 .
4Percentage of experiments in which a perfect separation is achieved, among 100 Monte-Carlo of noiseless bilinear synthetic data sets. The parameters are: m = 20 observations, n = 1000 pixels and ν = 0.5.
Figure 5 .
5Comparison between theoretical conditions ensuring endmember recovery by SNPALQ (resp. SNPA) and actual results. The dashed line correspond to the percentage of submatrices A and B for which condition 3.1 is fulfilled and the plain lines correspond to the actual proportion of perfect recovery by SNPALQ (resp. SNPA).
Figure 5 (
5dashed lines) depicts, as a function of the number of endmembers r, the proportion of the different realizations of W for which condition 3.1 is fulfilled. This proportion of sub-matrices A and B fulfilling condition (3.1) decreases with r, which was expected as the number of elements in the right-hand side increases. Then, as exemplified in
Figure 6 .Figure 7 .
67Percentage of experiments in which a perfect separation is achieved. There are 100 Monte-Carlo experiments, with m = 20 observations and n = 1000 pixels. The non-linearity parameter is ν = 0.5, and the mixtures are linear quadratic: they include squared sources. In addition to the results of SNPALQ bilinear (in which the projection step does not include the source auto-products) and SNPALQ (LQ), the results of SNPA and SPA are included. As a function of SNR and non-linearity level ν, percentage of perfect separation using a) SNPALQ+BF, b) SNPA.
Figure 7
7depicts the recovery performances of SNPALQ and SNPA. For low non-linearity levels, the mixtures approximately follow the LMM: in agreement with their robustness guarantees, the results of SNPALQ+BF are then perfect when the SNR is high (SNR ≥ 40dB); see upper-left corner of
αG
B (A) = min j∈[[r A ]] x∈∆ a j − B \{j} x 2 .For instance, in the special case A = B, α A (A) is the minimum distance between a column of A and the convex hull formed by the other columns of A and the origin. Let us also denote ν(A) = min i∈[[r A ]] a i 2 , γ(A) = min i,j∈[[r A ]]i =j a i − a j 2 , ω(A) = min ν(A), , γ(A) , where µ, L and G are some constants that will be specified later.B.1. Proof of SNPALQ robustness for linear mixtures. The proof is conducted by induction. We first derive a few useful lemmas, which are then used to prove the induction step in Theorem B.5. The main result is then stated in Theorem B.6.Lemma B.1 (Extension of[18]-Lemma 3.5). Let A ∈ R m×k , B ∈ R m×s and f satisfy Assumption 2.1. Then,
B. 2 .
2Proof of SNPALQ robustness for LQ mixtures. Similarly to the above derivations, after stating a few useful lemmas, the induction step of the proof of SNPALQ robustness for LQ mixtures is given in Theorem B.10 and the main result is stated in Theorem B.11. Lemma B.7 ([18]-Lemma 15 extended). Let Z = [P, Q], where P ∈ R m×k and Q ∈ R m×r−k , and let f satisfy Assumption 2.1. If
B. 3 .
3Proof for the Brute Force algorithm (BF).Definition B.12 (LQ-robust loner). LetX be an LQ mixing satisfying Definition 1.3 and j ∈ [[t]]. Let us denote L the set of indices k ∈ [[t]] such that f (x k −x j ) ≤ d = V, (B.22)
Lemma B. 15 .
15LetX = Π 2 (W)H + N be r-LQ near-separable (Definition 1.3). Let us denotex k(i) any robust loner associated toi ∈ [[r]]. If f (x j − w i ) > d + L(2K(X) + ) for some j ∈ [[t]], then f (x j −x k(i) ) > d.Proof. We havef (x j −x k(i) ) = f (x j − w i − n k(i) ) ≥ f (x j − w i ) − K(x j − w i )L > d + L(2K(X) + ) − (2K(X) + )L = d.Lemma B.16. LetX = Π 2 (W)H + N be r-LQ near-separable (Definition 1.3). If a column x j is a robust loner, then there is an index i ∈ [[r]] such that f (x j − w i ) ≤ d + L(2K(X) + ).
.
LetX = Π 2 (W)H + N be r-LQ near-separable (Definition 1.3). All the columnsx j , j ∈ [[t]], with x j = r k=1 a kj w k + r l=1 r q=l+1 b jlq w l w q and a ji > 1 − 2d−3 L(4K(X)+ ) LK(X) 2 [1+max(1,K(X))] 2 for some i ∈ [[r]] satisfy f (x j − w i ) ≤ d − 32 L(2K(X) + ).
≤
K(X) + max(K(X), K(X) 2 ),
Lemma B.19. LetX be r-LQ near-separable (Definition 1.3), J ⊆ [[t]] andx k(i) a canonical column associated to i. Iffor some i ∈ [[r]], f (w i − Π 2 (X J )h) ≥ µ 2L 2d − 3 L(4K(X) + ) K(X) 2 [1 + max(1, K(X))] 2 α Π 4 (W) (W) 2 , then f (x k(i) − Π 2 (X J )h) ≥ L 2 2 (3 + ) 2 .
h ) −
)L max(1, 2K(X) + ) ( + K(X)[1 + max(1, K(X))]) ≥ f (w i − Π 2 (X J )h) − L K(X)[1 + max(1, K(X))]− L max(1, 2K(X) + ) ( + K(X)[1 + max(1, All canonical columns are robust-loners.
Theorem B.21 (Robustness of BF when applied on LQ mixings). Let X = Π 2 (W) + N satisfying Definition 1.3 with n i 1 ≤ for i ∈ [[n]]. Let also satisfy 4 2 µ (d + L(2K(X) + )) < α W (W).
n]] and k ∈ [[r]] (nonnegativity condition), k=1 h ki ≤ 1 for all i ∈ [[n]] (sum to at most one condition), (1.4)r
α 2 (W) = min
j∈[[r]]
1.4), r: number of sources, and f a strongly convex function satisfying Assumption 2.1, = max i∈[[t]] n i 2 , d given by Equation (B.22).2: Initialization: K = {}
3: for k ∈ [[t]] do
4:
s]] , (ai aj) i≤ji∈[[k]]
j∈[[k]]
, (bi bj) i≤j
i∈[[s]]
j∈[[s]]
, (ai bj) i∈[[k]]
j∈[[s]]
Table 1
11st line: actual r value. 2nd and 3rd lines: average number |K|, over 100 Monte-Carlo experiments, of endmembers extracted by SNPALQ and SNPALQ+BF.r
2
3
4
6 8 10 15 20
|K|
SNPALQ
2.08 3.16 4.05 6 8 10 15 20
SNPALQ+BF
2
3
4
6 8 10 15 20
4.2.
[ [s]] , (ai aj) i≤ji∈[[k]]
j∈[[k]]
, (bi bj) i≤j
i∈[[s]]
j∈[[s]]
, (ai bj) i∈[[k]]
j∈[[s]]
.
B.11 (Robustness of SNPALQ when applied on LQ mixings). Let X = Π 2 (W)H + N ∈ R m×n be an LQ mixing satisfying Definition 1.3 with α Π 2 (W) (W) > 0 and β LQ Π 2 (W) (W). Let f satisfy Assumption 2.1 and n i 2 ≤ for all i ∈ [[t]], with (B.21)
The denomination "α-robust simplicial" is slightly abusive here, as the coefficients of x sum to at most one, in contrast to[5] in which they sum to exactly one.
The algorithms will be made available online at https://sites.google.com/site/nicolasgillis/code
Note that in the signal processing literature, such a norm is sometimes denoted as A ∞,2 , see for instance[26]. We prefer to keep the original notation of[18].
https://www.usgs.gov/
More exactly, in the linear case this condition is replaced by one on the admissible noise levels; see[18].
2 .
≤ 2 2 µ (d + L(2K(X) + )).• If two robust loners satisfy x j −x k 2 ≤ 2 2 µ (d + L(2K(X) + )), they must correspond to the same source w i . This follows by contradiction: suppose thatx j corresponds to a source w i (f (x j − w i ) ≤ d + L(2K(X) + )) andx j to another source w l ,). Then we obtain that:, andfrom which it can be deduced thatwhich is a contradiction. Therefore, once the robust-loners are found and the clustering described above performed, each source can be identified by picking a point from each cluster. The 2 -norm error is then at most 2 µ (d + L(2K(X) + )). References.
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| []
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[
"Entropy of Black Holes in D = 5, N = 2 Supergravity and AdS Central Charges",
"Entropy of Black Holes in D = 5, N = 2 Supergravity and AdS Central Charges"
]
| [
"S Cacciatori [email protected]†email:[email protected]‡email:[email protected]§email:[email protected] \nDipartimento di Fisica dell'Università di Milano and INFN\nSezione di Milano\nVia Celoria 1620133MilanoItaly\n\nIntroduction\n\n",
"D Klemm \nDipartimento di Fisica dell'Università di Milano and INFN\nSezione di Milano\nVia Celoria 1620133MilanoItaly\n\nIntroduction\n\n",
"† ",
"W A Sabra \nCenter for Advanced Mathematical Sciences (CAMS) and Physics Department\nAmerican University of Beirut\nLebanon\n",
"‡ ",
"D Zanon \nDipartimento di Fisica dell'Università di Milano and INFN\nSezione di Milano\nVia Celoria 1620133MilanoItaly\n\nIntroduction\n\n"
]
| [
"Dipartimento di Fisica dell'Università di Milano and INFN\nSezione di Milano\nVia Celoria 1620133MilanoItaly",
"Introduction\n",
"Dipartimento di Fisica dell'Università di Milano and INFN\nSezione di Milano\nVia Celoria 1620133MilanoItaly",
"Introduction\n",
"Center for Advanced Mathematical Sciences (CAMS) and Physics Department\nAmerican University of Beirut\nLebanon",
"Dipartimento di Fisica dell'Università di Milano and INFN\nSezione di Milano\nVia Celoria 1620133MilanoItaly",
"Introduction\n"
]
| []
| We consider general black holes in D = 5, N = 2 supergravity coupled to vector multiplets, and discuss the issue of microstate counting from various viewpoints. The statistical entropy is computed for the near-extremal case using the central charge of the AdS 2 factor appearing in the near-horizon geometry. Furthermore, we explicitly construct the duality transformation connecting electrically charged black holes to magnetically charged black strings, under which the AdS 2 × S 3 near horizon geometry becomes AdS 3 × S 2 . For AdS 3 the counting of microstates correctly reproduces the Bekenstein-Hawking entropy, thus resolving the discrepancy previously found for AdS 2 . * | 10.1016/s0550-3213(00)00461-2 | [
"https://arxiv.org/pdf/hep-th/0004077v2.pdf"
]
| 14,151,028 | hep-th/0004077 | 279d4a09c231db54449f81a29973aa46c35d5672 |
Entropy of Black Holes in D = 5, N = 2 Supergravity and AdS Central Charges
May 2000
S Cacciatori [email protected]†email:[email protected]‡email:[email protected]§email:[email protected]
Dipartimento di Fisica dell'Università di Milano and INFN
Sezione di Milano
Via Celoria 1620133MilanoItaly
Introduction
D Klemm
Dipartimento di Fisica dell'Università di Milano and INFN
Sezione di Milano
Via Celoria 1620133MilanoItaly
Introduction
†
W A Sabra
Center for Advanced Mathematical Sciences (CAMS) and Physics Department
American University of Beirut
Lebanon
‡
D Zanon
Dipartimento di Fisica dell'Università di Milano and INFN
Sezione di Milano
Via Celoria 1620133MilanoItaly
Introduction
Entropy of Black Holes in D = 5, N = 2 Supergravity and AdS Central Charges
May 2000arXiv:hep-th/0004077v2 2 IFUM 657/FT CAMS/00-04 hep-th/0004077
We consider general black holes in D = 5, N = 2 supergravity coupled to vector multiplets, and discuss the issue of microstate counting from various viewpoints. The statistical entropy is computed for the near-extremal case using the central charge of the AdS 2 factor appearing in the near-horizon geometry. Furthermore, we explicitly construct the duality transformation connecting electrically charged black holes to magnetically charged black strings, under which the AdS 2 × S 3 near horizon geometry becomes AdS 3 × S 2 . For AdS 3 the counting of microstates correctly reproduces the Bekenstein-Hawking entropy, thus resolving the discrepancy previously found for AdS 2 . *
Introduction
The study of black hole solutions in N = 2 five-dimensional supergravity coupled to vector and hypermultiplets plays an important role in the understanding of the non-perturbative structure of string and M-theory [1,2]. In this setting the interplay between classical and quantum results is exemplified at its best.
In this paper we consider general charged black holes of the D = 5, N = 2 theories, not necessarily those obtained from compactification of eleven-dimensional supergravity on a Calabi-Yau threefold. The analysis is simplified by the rich geometric structure of the N = 2 theories. Black hole solutions are given in terms of a rescaled cubic homogeneous prepotential which defines very special geometry [3]. In the extremal BPS case, half of the vacuum supersymmetries are preserved, while at the horizon supersymmetry is fully restored [4].
In five dimensions the supergravity action contains a Chern-Simon term which allows the existence of black holes with nonvanishing angular momenta (but still nonrotating horizon) [5,6,7,8].
These issues have been the object of recent, extensive studies. In the present paper we focus on the asymptotic symmetries of the near-horizon geometry of the general near-extremal solution: the aim is the computation of the entropy from a counting of microstates to be compared to the macroscopic, thermodynamical entropy.
We will see that the calculation of the microscopic entropy of small excitations above extremality is equivalent to a microstate counting for certain black holes in two-dimensional anti-de Sitter space. The latter, however, is problematic: AdS 2 has two timelike boundaries, but when applying Cardy's formula for the density of states only one boundary is taken into account. This procedure leads to a statistical entropy result which is off by a √ 2 factor with respect to the Bekenstein-Hawking entropy [9]. Up to now, no satisfactory explanation of this mismatch is known. Our results support the point of view that only the ground state has an effective description in terms of a quantum-mechanical system [10], whereas the excitations above extremality are described by a two-dimensional conformal field theory [11].
In our work we address these issues in a constructive approach. The main result that we present is an explicit duality transformation, which realizes an invariance of the N = 2 supergravity action. This duality turns the AdS 2 × S 3 near horizon geometry of the extremal black hole solution into AdS 3 × S 2 . The key point underlying the duality is the fact that the three-sphere can be written as a Hopf fibration over the base S 2 . For AdS 3 , the counting of microstates is performed using Cardy's formula and it is shown that it reproduces correctly the Bekenstein-Hawking entropy, thus resolving the discrepancy previously found for AdS 2 .
In the case where the D = 5, N = 2 supergravity action is obtained by Calabi-Yau (CY) compactification of M-theory, the considered duality transformation, which maps electrically charged black holes onto magnetically charged black strings, corresponds to the duality between M2 branes wrapping CY two-cycles and M5 branes wrapping CY four-cycles. According to [12], M-theory compactified on AdS 3 × S 2 × M, where M denotes some Calabi-Yau threefold, is dual to a (0, 4) superconformal field theory living on an M5 brane wrapping some holomorphic CY four-cycle. This fact has been used in [13] to compute the entropy of five-dimensional BPS black holes 1 . We stress that our method for microstate counting applies to any near-extremal black hole in N = 2, D = 5 supergravity, independent of whether it is obtained by CY compactification or not.
Our paper is organized as follows: in section 2, the basic notions of N = 2, D = 5 supergravity and very special geometry relevant to our analysis are summarized. In section 3 we review the black hole solutions and consider the ST U model as a simple example, which nonetheless retains all the interesting features of the general solutions. In section 4 we present the isometry superalgebra which arises in the near horizon limit, while in section 5 we show that the motion of a particle which moves near the horizon of the extremal rotating black hole is described by conformal quantum mechanics. This indicates that the ground state may have a description in terms of conformal quantum mechanics [15,16], even when rotation is included. In section 6 we compute the statistical entropy of small excitations near extremality, using the AdS 2 central charge [9], and find a √ 2 factor of discrepancy as compared to the thermodynamical Bekenstein-Hawking result. In section 7 we construct the duality transformation for the supergravity action, and in the following section we finally perform the state counting, using the fact that the nearhorizon geometry of the dual solution includes an AdS 3 factor. In this way, we obtain a microscopic entropy which agrees precisely with the corresponding thermodynamical result. We conclude with some final remarks.
2 D = 5, N = 2 Supergravity and Very Special Geometry
The theory of N = 2 supergravity theory coupled to an arbitrary number n of Maxwell supermultiplets was first considered in [17]. In the analysis of [17], it was established that the scalar fields of the vector multiplets parametrize a Riemannian space. The homogeneous symmetric spaces take the form
M = Str 0 (J) Aut(J) ,
where Str 0 (J) is the reduced structure group of a formally real unital Jordan Algebra of degree three, Aut(J) is its automorphism group. The scalar manifold can be regarded as a hypersurface, with vanishing second fundamental form of an (n + 1)-dimensional Riemannian space G whose coordinates X are in correspondence with the vector multiplets including that of the graviphoton. The equation of the hypersurface is V = 1 where V, the prepotential, is a homogeneous cubic polynomial in the coordinates of G,
V(X) = 1 6 C IJK X I X J X K . (2.1)
Non-simple Jordan algebras of degree three are of the form R ⊕ Σ n , where Σ n is the Jordan algebra associated with a quadratic form. The corresponding symmetric scalar manifolds are
M = SO(1, 1) × SO(n − 1, 1) SO(n − 1) . (2.2)
In this case, V(X) is factorizable into a linear times a quadratic form in (n−1) scalars, which for the positivity of the kinetic terms in the Lagrangian, must have a Minkowski metric. For simple Jordan algebras, one obtains four sporadic locally symmetric spaces related to the four simple unital formally real Jordan algebras over the four division algebras R, C, H, O. For more details we refer the reader to [17].
For M-theory compactification on a Calabi-Yau threefold with Hodge numbers h (1,1) and h (2,1) , the five dimensional theory contains the gravity multiplet, h (1,1) − 1 vector multiplets and h (2,1) + 1 hypermultiplets. The (h (1,1) − 1)-dimensional space of scalar components of the abelian vector supermultiplets coupled to supergravity can be regarded as a hypersurface of a h (1,1) -dimensional manifold whose coordinates X I (φ) are in correspondence with the vector bosons (including the graviphoton). The defining equation of the hypersurface is as in (2.1) V(X) = 1 6 C IJK X I X J X K = X I X I = 1, I, J, K = 1, . . . , h (1,1) .
(2.3)
Here C IJK are the topological intersection numbers of the Calabi-Yau, X I are the so called "dual" special coordinates. The bosonic part of the ungauged supersymmetric N = 2 Lagrangian which describes the coupling of vector multiplets to supergravity is given by
e −1 L = 1 2 R − 1 4 G IJ F µν I F µνJ − 1 2 G ij ∂ µ φ i ∂ µ φ j + e −1 48 ǫ µνρσλ C IJK F I µν F J ρσ A K λ . (2.4)
The corresponding vector and scalar metric are completely encoded in the function V(X),
G IJ = − 1 2 ∂ I ∂ J ln V(X)| V=1 G ij = G IJ ∂ i X I ∂ j X J | V=1 (2.5)
where ∂ i and ∂ I refer, respectively, to partial derivatives with respect to the scalar fields φ i and X I = X I (φ i ).
Further useful relations are
∂ i X I = − 2 3 G IJ ∂ i X J , X I = 2 3 G IJ X J . (2.6)
It is worth pointing out that for Calabi-Yau compactifications, V represents the intersection form, X I and X I = 1 6 C IJK X J X K correspond, respectively, to the size of the twoand four-cycles of the Calabi-Yau threefold.
Black Holes in the ST U = 1 Model
In the last few years considerable progress has been made in the study of BPS black hole states of the low-energy effective actions of compactified string and M-theory. This was mainly motivated by the important role that these states play in the understanding of the non-perturbative structure of string theory. The magnetic and electric BPS solutions of five-dimensional N = 2 supergravity models coupled to vector and hypermultiplets can be regarded as solitons interpolating between two vacua: Minkowski flat space at infinity and AdS 3 × S 2 and AdS 2 × S 3 near the horizon. At a generic point in space-time, the BPS solution breaks half of supersymmetry. However, near the horizon supersymmetry is enhanced and fully restored.
In M-theory compactified on a Calabi-Yau threefold, electrically charged point-like and magnetically charged string-like BPS states correspond to the two and five-branes of M-theory wrapped around the two-and four-cycles of the Calabi-Yau space respectively. Though the details of the low-energy Lagrangian depend very much on the geometric and topological data of the compactified Calabi-Yau space, the analysis of the BPS solutions is considerably simplified by the rich geometric structure based on "very special geometry" underlying the N = 2 five-dimensional theories with vector supermultiplets [18].
The metric for the BPS black hole solutions can be brought to the form [8]
ds 2 = −e −4V (dt + w m dx m ) 2 + e 2V d x 2 , F I mn = ∂ m (X I Q n ) − ∂ n (X I Q m ), F I tm = −∂ m (e −2V X I ), e 2V X I = 1 3 H I ,(3.1)
where H I are harmonic functions, H I = h I + Q I r 2 , h I are constants and Q I denote the electric charges. Furthermore one has Q n = e −2V w n , and the field strength of w m is self-dual. If one defines the rescaled coordinates Y I = e V X I , then the underlying very special geometry implies that
e 3V = 1 6 C IJK Y I Y J Y K .
As a general magnetic string solution of D = 5, N = 2 supergravity, one obtains [4]
ds 2 = e −W (−dt 2 + dz 2 ) + e 2W d x 2 , F I mn = −ǫ mnp ∂ p H I , e W X I = H I , e 3W = 1 6 C IJK H I H J H K , (3.2)
with the harmonic functions
H I = h I + P I r , (3.3)
where h I are constants and P I are magnetic charges.
In the electric case, the near-horizon geometry is given by AdS 2 × S 3 and the black hole entropy, related to the horizon volume S 3 , is given in terms of the extremized electric central charge. For the magnetically charged D = 5 BPS black string, with near-horizon geometry AdS 3 × S 2 , one similarly finds that the extremized value of the BPS tension is related to the volume of the S 2 .
As an example we consider the ST U = 1 model [1,18]. This can be obtained by compactification of heterotic string theory on K 3 × S 1 [19]. The tree-level prepotential of this model is given by
V = ST U = 1, (3.4)
and corresponds to the scalar manifold (2.2) for n = 2. Taking S = X 0 , T = X 1 and U = X 2 , one gets for the matrix G IJ
G IJ = 2 S 2 0 0 0 T 2 0 0 0 U 2 . (3.5)
Considering S as the dependent field, i. e. S = 1/(T U), we find
G ij = 1 T 2 1 2T U 1 2T U 1 U 2 , G ij = 4 3 T 2 − T U 2 − T U 2 U 2 . (3.6)
The field equations following from the action (2.4) admit the non-extremal static black hole solution [20]
ds 2 = −e −4V f dt 2 + e 2V (f −1 dr 2 + r 2 dΩ 2 3 ), F I rt = −H −2 I ∂ rHI , (3.7) X I = H −1 I e 2V ,
where dΩ 2 3 = dθ 2 + sin 2 θdφ 2 + cos 2 θdψ 2 (3.8) denotes the metric on the three sphere S 3 . The H I are harmonic functions given by
H I = 1 + Q I r 2 ,(3.9)
and V reads
e 2V = (H 0 H 1 H 2 ) 1/3 . (3.10)
Furthermore we have
f = 1 − µ r 2 (3.11)
with the nonextremality parameter µ, and
H I = 1 +Q I r 2 , (3.12)
where theQ I denote the physical electric charges. They are related to the Q I appearing in (3.9) by the equations
Q I = µ 2 sinh β I tanh β I 2 , Q I = µ 2 sinh β I . (3.13)
The extremal (BPS) limit is reached when β I → ∞, µ → 0, with µ sinh β I kept fixed. For the ADM mass M ADM , the Bekenstein-Hawking entropy S BH , and the Hawking temperature T H one obtains
M ADM = π 4G 5 ( I Q I + 3 2 µ), (3.14) S BH = A hor 4G 5 = π 2 2G 5 I (µ + Q I ) 1/2 , (3.15) T H = µ π I (µ + Q I ) 1/2 . (3.16)
In the extremal case, also a rotating generalization of (3.7) can be obtained from the general form (3.1). Its metric is given by
ds 2 = −e −4V (dt + w φ (r, θ)dφ + w ψ (r, θ)dψ) 2 + e 2V (dr 2 + r 2 dΩ 2 ), (3.17) where w φ (r, θ) = − α sin 2 θ r 2 , w ψ (r, θ) = α cos 2 θ r 2 . (3.18)
The gauge fields are
A I t = e −2V X I , A I φ = e −2V X I w φ , A I ψ = e −2V X I w ψ . (3.19)
The moduli X I and the functions V and H I are as in (3.7), (3.10) and (3.9) respectively, and the ADM mass is given by (3.14) for µ = 0. The Bekenstein-Hawking entropy and the angular momenta read [8]
S BH = A hor 4G = π 2 2G 5 Q 0 Q 1 Q 2 − α 2 , J φ = −J ψ = απ 4G 5 .
Near-Horizon Limit and Isometry Superalgebra
In the following two sections, we shall be particularly interested in the near-horizon limit of (3.17). For r → 0 we can write
e 2V = Z hor 3r 2 ,(4.1)
where Z = Q I X I is the central charge, and Z hor = 3(Q 0 Q 1 Q 2 ) 1/3 is its value at the horizon. Introducing the horospherical coordinates (τ, ρ),
τ = t 2 3 Z hor , ρ = 2r 2 3 Z hor , (4.2)
one gets for the near-horizon metric
ds 2 = −ρ 2 dτ 2 + Z hor 12ρ 2 dρ 2 + 6α Z hor ρdτ (sin 2 θdφ − cos 2 θdψ) + Z hor 3 dΩ 2 − 27α 2 Z 3 hor (sin 2 θdφ − cos 2 θdψ) 2 . (4.3)
We observe that, in contrast to the case of vanishing rotation parameter, the spacetime does not split into a product AdS 2 × S 3 . Although the AdS 2 part is the same as without rotation, there are nondiagonal elements, and the three-sphere is distorted. The isometry superalgebra of the near-horizon supergravity configuration was determined in [21], where the fact that the residual isometry supergroup can be determined (modulo bosonic factors) from a knowledge of the Killing spinors [22,23,24] has been used. In this way, one obtains that the near-horizon geometry is invariant under the superalgebra su(1, 1|2) ⊕ u(1) in the rotating case, and under su(1, 1|2) ⊕ su(2) for α = 0 [21]. Thus, for α = 0, the bosonic subalgebra is su(1, 1) ⊕ su(2) L ⊕ u(1) R . In fact, the near-horizon spacetime is a homogeneous manifold of the form [SO(2, 1) [21]. The conformal algebra su(1, 1) ∼ = so(2, 1) is generated by the Killing vectors [21]
× SU(2) L × U(1) R ]/[U(1) × U(1)]h = ∂ τ , d = ρ∂ ρ − τ ∂ τ , (4.4) k = − Z hor 12ρ 2 (1 − 27α 2 Z 3 hor ∂ τ ) − τ 2 ∂ τ + 2τ ρ∂ ρ − 3α 2Z hor ρ (∂ φ − ∂ ψ ), satisfying [d, h] = h, [d, k] = −k, [h, k] = 2d. (4.5)
Thus, although the manifold is not a product AdS 2 × S 3 , we find the so(2, 1) symmetry inherent to AdS 2 . In the following section, we will see that this symmetry, which is the conformal symmetry in 0 + 1 dimensions, occurs also in the action of a particle charged under the vectors moving in the near-horizon regime.
Particle Motion near the Horizon
We now consider a particle of mass m, carrying the charges q I under the abelian vectors, which moves in the background (4.3). Like in [15], we introduce the new coordinate q,
ρ = Z hor 3q 2 . (5.1)
We use a Hamiltonian formalism, and define
H = 1 2 g µν (Π µ − q I A I µ )(Π ν − q I A I ν ),(5.2)
where the Π µ denote generalized momenta. For our configuration, this leads to
H = − 9q 4 2Z 2 hor Π 2 τ 1 − 27α 2 Z 3 hor + 27q 2 α Z 3 hor Π τ Π φ − Π ψ + Z 2 hor 9α q I X I hor − 1 2 (q I X I hor ) 2 + 3q 2 2Z hor Π 2 q + 3L 2 2Z hor ,(5.3)
where
L 2 = Π 2 θ + Π 2 φ sin 2 θ + Π 2 ψ cos 2 θ (5.4)
denotes the conserved angular momentum. As the coordinates τ , φ and ψ are cyclic, the associated conjugate momenta are constants of motion. If H solves the mass-shell constraint 2H = −m 2 , −Π τ is to be identified with the particle Hamiltonian H. Setting Π q = p and defining u = pq, one obtains
H = p 2 2F (u) + mg 2q 2 F (u) , (5.5) with mg = L 2 + Z hor 3 (m 2 − (q I X I hor ) 2 ),(5.6)
and the function F (u) given by
F (u) = 1 2 C 2 + (1 − 27α 2 Z 3 hor )( 3 Z hor (u 2 + L 2 ) + m 2 − (q I X I hor ) 2 ) + C . (5.7)
In (5.7), the constant C is defined by
C = 9α Z 2 hor (Π φ − Π ψ ) + q I X I hor . (5.8)
One observes that in the limit
Z hor → ∞, m − q I X I hor → 0,(5.9)
with Z hor (m − q I X I hor ) kept fixed, we have F (u) → m, and (5.5) reduces to the DFF model [25]
H = p 2 2m + g 2q 2 .
(5.10)
Note that also the general Hamiltonian (5.5) describes a model of conformal mechanics. To see this, we write it in the form
H = p 2 2f (u) , (5.11) with f (u) = u 2 F (u) u 2 + mg . (5.12)
The generators of the conformal group are then given by [26]
H = p 2 2f , D = 1 2 u, K = 1 2 q 2 f,(5.
Statistical Entropy from AdS 2 Central Charge
In order to determine the central charge of the boundary CFT, we proceed along the lines of [27,28,29], and reduce the bosonic part of the D = 5, N = 2 supergravity action to two dimensions. In this section we shall only consider nonrotating black holes carrying electric charge. This means that we can consistently truncate the Chern-Simons term in (2.4), so that the bosonic part of the action in five dimensions reads
I = 1 16πG 5 d 5 x √ −g R − 1 2 G IJ F I µν F Jµν − G ij ∂ µ φ i ∂ µ φ j ,(6.1)
where G 5 = l 3 denotes Newton's constant. The matrices G IJ and G ij for the ST U model were given in section 3. The reduction ansatz for the metric is
ds 2 = ds 2 (2) + l 2 Φ 2 dΩ 2 ,(6.2)
where Φ denotes the dilaton and dΩ 2 is given by (3.8).
One now assumes that the gauge fields, scalars and dilaton do not depend on the coordinates on the internal S 3 . In this way, one arrives at the two-dimensional effective action
I = Ω 16π d 2 x √ −g Φ 3 R + 6Φ(∇Φ) 2 + 6Φ l 2 − Φ 3 G ij ∂ α φ i ∂ α φ j − 1 2 Φ 3 G IJ F I αβ F Jαβ ,(6.3)
where Ω = 2π 2 denotes the volume of the unit S 3 , and early greek indices α, β, . . . refer to two-dimensional spacetime. We now wish to integrate out the field strength F I αβ (which in two dimensions must be a multiple of the volume form ǫ αβ ) from the action. This can be done using the Lagrange multiplier method of [30]. Let us briefly sketch how this works: Instead of looking at the gauge field action
I g = − Ω 16π d 2 x √ −g 1 2 Φ 3 G IJ F I αβ F Jαβ ,(6.4)
one looks at the formally extended actioñ
I g = Ω 16π d 2 x − 1 2 √ −gΦ 3 G IJ F I αβ F Jαβ + λ I (F I αβ − ∂ α A I β + ∂ β A I α )ǫ αβ ,(6.5)
where the definition of F I as a field strength associated with A I is implemented by means of the Lagrange multiplier λ I . Note that the three variables F I , A I and λ I are considered as independent in this setting. Variation with respect to A I yields ∂ β (λ I ǫ αβ ) = 0, (6.6) so that Λ I := λ I / √ −g are constants. Due to (6.6), the term λ I (−∂ α A I β + ∂ β A I α )ǫ αβ in the action (6.5) is a boundary term and can be dropped. We can then integrate out the field strength F I , using its equation of motion −Φ 3 G IJ F Jαβ + Λ I ǫ αβ = 0.
(6.7)
This yieldsĨ
g = − Ω 16π d 2 x √ −g G IJ Λ I Λ J Φ 3 ,(6.8)
so that we are left with the total two-dimensional action
I = Ω 16π d 2 x √ −g Φ 3 R + 6Φ(∇Φ) 2 + 6Φ l 2 − Φ 3 G ij ∂ α φ i ∂ α φ j − G IJ Λ I Λ J Φ 3 . (6.9)
The dilaton kinetic term in (6.9) can be eliminated by a conformal rescalinḡ
g αβ = Φ 2 g αβ . (6.10) DefiningΦ = Φ 3 , we obtain I = Ω 16π d 2 x √ −ḡ ΦR + 6 l 2Φ1/3 −ΦG ij ∂ α φ i ∂ α φ j − G IJ Λ I Λ J Φ 5/3 . (6.11)
Let us now consider the nonextremal black hole solution (3.7) of the action (6.1), and expand it near extremality. To this end, we introduce an expansion parameter ǫ (ǫ → 0), and set t =t ǫ , r = √ ǫr, µ = µ 0 ǫ,
Φ =Φ 0 + ǫϕ, φ i = φ i 0 + ǫφ i , (6.12) whereΦ 0 = (Q 0Q1Q2 ) 1/2 l 3 , φ i 0 =Q −1 i (Q 0Q1Q2 ) 1/3 . (6.13)
Introducing the new coordinate
x = (Q 0Q1Q2 ) 1/6 2l 2 (r 2 − µ 0 2 ),(6.14)
we arrive at
ds 2 = −(λ 2 x 2 − a 2 )dt 2 + (λ 2 x 2 − a 2 ) −1 dx 2 (6.15)
for the rescaled two-dimensional metric, with λ and a given by we obtain for the action at lowest order in the expansion parameter ǫ,
λ = 2l (Q 0Q1Q2 ) 1/3 ,(6.I = 1 2 d 2 x √ −ḡη[R + 2λ 2 ],(6.19)
so the leading order is governed by the Jackiw-Teitelboim (JT) model [31]. (6.15), together with the linear dilaton η = η 0 λx, (6.20)
η 0 = Ωǫ 16πl 2 (Q 0Q1Q2 ) 2/3 IQ −1 I ,
represents a black hole solution of this model [9], with mass and thermodynamical entropy given by
M (2) = 1 2 η 0 a 2 λ, S (2) = 2πη hor = 2πη 0 a. (6.21)
This black hole spacetime has constant curvature, i. e. it is locally AdS 2 . Now it is known that the asymptotic symmetries of two-dimensional anti-de Sitter space form a Virasoro algebra [9], similar to the case of AdS 3 , where one has two copies of Virasoro algebras as asymptotic symmetries [32]. When realized canonically in the Hamiltonian formulation of JT gravity, this algebra was shown to exhibit a central charge [9,33] c = 24η 0 . (6.22)
Using this central charge in Cardy's formula, the authors of [9] were able to give a microscopic derivation of the entropy of the two-dimensional black holes (6.15) in the JT model. Our aim is now to perform a similar calculation for the near-extremal five dimensional black hole under consideration, making use of the fact that the dimensionally reduced supergravity action coincides with the JT model at leading order in the nonextremality parameter, and that the relevant two-dimensional metric is given by ( Comparing this with the two-dimensional results (6.21), one finds ∆S BH = S (2) and ∆M ADM = ǫM (2) . The factor ǫ appearing in the relation between the two masses stems from the fact that M ADM was computed with respect to the Killing vector ∂ t , whereas M (2) is related to ∂t = ǫ∂ t . This means that up to these normalizations the five-and two-dimensional energies and entropies match. Expanding also the Hawking temperature (3.16) for small values of the nonextremality parameter µ, one finds for the temperature dependence of ∆M ADM
∆M ADM = π 3 T 2 H 32l 3Q 0Q1Q2 IQ −1 I , (6.27)
so the energy of the excitations above extremality is that of an ideal gas of massless particles in 1 + 1 dimensions. This suggests that the microstates should be described by a two-dimensional field theory rather than a quantum mechanical system. Let us now proceed with the computation of the statistical entropy, using the central charge (6.22). The Virasoro generator L 0 for the black hole (6.15) is given by [9]
L 0 = M (2) λ = Ωǫµ 2 0 128πl 4 (Q 0Q1Q2 ) 1/3 IQ −1 I . (6.28)
Inserting this together with the central charge (6.22) into Cardy's formula, we get for the statistical entropy
S stat = 2π cL 0 6 = Ωµ 8 √ 2l 3 (Q 0Q1Q2 ) 1/2 IQ −1 I , (6.29)
which agrees, up to a factor √ 2, with the thermodynamical entropy ∆S BH of the small excitations above extremality. The same mismatch by a factor √ 2 has been found in [9], where the authors proposed an explanation of this for the case when the model (6.19) comes from dimensional reduction of three-dimensional AdS gravity. Although in our case AdS 2 arises as near-horizon geometry of a higher-dimensional black hole with no intermediate AdS 3 geometry involved, we shall see in the next section that by means of a duality transformation the near-horizon geometry AdS 2 × S 3 of the extremal black hole becomes AdS 3 × S 2 . We will then be able to use Strominger's counting of microstates [34] in order to reproduce correctly the Bekenstein-Hawking entropy of the black hole.
Duality Invariance of the Supergravity Action
In this section we will show that in presence of a Killing vector field ∂ z , the supergravity action (6.1) is invariant under a certain generalization of T-duality 3 . The key observation is then that the three sphere S 3 appearing in the black hole geometry can be written as a Hopf fibration, i. e. as an S 1 bundle over CP 1 ≈ S 2 . Performing then a duality transformation along the Hopf fibre untwists the S 3 , and transforms the electrically charged black hole into a magnetically charged black string, which has AdS 3 × S 2 as near-horizon limit in the extremal case.
To begin with, we reduce the action (6.1) to four dimensions, using the usual Kaluza-Klein reduction ansatz for the five-dimensional metric,
ds 2 = e k/ √ 3 ds 2 4 + e −2k/ √ 3 (dz + A α dx α ) 2 ,(7.1)
where k denotes the dilaton, and early greek indices α, β, . . . refer to four-dimensional spacetime. Assuming that the fields appearing in (6.1) are independent of z, one arrives at the four-dimensional action
I 4 = L 16πG 5 d 4 x √ −g 4 R 4 − 1 2 (∇k) 2 − 1 4 e − √ 3k F 2 − 1 2 e −k/ √ 3 F 2 − G ij ∂ α φ i ∂ α φ j ,(7.2)
where L denotes the length of the circle parametrized by z, F is the field strength associated to the Kaluza-Klein vector potential A, and
F 2 = F αβ F αβ , F 2 = G IJ F I αβ F Jαβ . (7.3)
We now dualize both F and F I , using again the Lagrange multiplier method of [30]. Dropping boundary terms, we arrive at the dualized action
I 4 = L 16πG 5 d 4 x √ −g 4 [R 4 − 1 2 (∇k) 2 − 1 4 e √ 3k ( ⋆ F ) 2 − 1 2 e k/ √ 3 1 4 G IJ ⋆ F Iαβ ⋆ F αβ J − G ij ∂ α φ i ∂ α φ j ],(7.4)
where we defined
⋆ F αβ = 1 2 e − √ 3k ǫ αβγδ F γδ , (7.5) ⋆ F Iαβ = e −k/ √ 3 G IJ ǫ αβγδ F Jγδ . (7.6)
Comparing (7.4) with (7.2), we observe that the gravitational and gauge field parts of the four-dimensional action, as well as the dilaton kinetic energy, are invariant under the Z 4 transformation
k → −k, F αβ → ⋆ F αβ , F I αβ → ⋆ F Iαβ , G IJ → 1 4 G IJ . (7.7)
The Z 4 is actually a subgroup of the usual symplectic Sp(2m+ 2, R) duality group [35,36] of D = 4, N = 2 supergravity (coupled to m vector multiplets) generated by
S = 0 1 −1 0 . (7.8)
Note that the transformation G IJ → G IJ /4 means that
X I → 3X I = 1 2 C IJK X J X K , X I → 1 3 X I ,(7.9)
so essentially the special coordinates go over into their duals. The fact that this dualization implies G IJ → G IJ /4 can be shown using the expression
G IJ = 9 2 X I X J − 1 2 C IJK X K ,(7.10)
as well as the "adjoint identity"
C IJK C J ′ (LM C P Q)K ′ δ JJ ′ δ KK ′ = 4 3 δ I(L C M P Q) (7.11)
of the associated Jordan algebra [17]. It can also be seen that this duality transformation is consistent with the relations (2.6). Furthermore, making use of the equation
G ij ∂ α φ i ∂ α φ j = G IJ ∂ α X I ∂ α X J ,(7.12)
one checks that (7.9) does not change the kinetic term of the scalar fields, so (7.7), (7.9) represent in fact a duality invariance of the four-dimensional action (7.2). In the special case of the ST U = 1 model, (7.9) implies that the moduli φ i go over into their inverse,
φ i → 1 φ i . (7.13)
We now wish to apply the duality (7.7), (7.9) to the black hole solution (3.7). To this end, we consider the S 3 as an S 1 bundle over S 2 , and write for its metric
dΩ 2 = 1 4 dϑ 2 + sin 2 ϑdϕ 2 + (dζ + cos ϑdϕ) 2 ,(7.14)
where ζ (0 ≤ ζ ≤ 4π) parametrizes the S 1 fibre. Introducing the coordinate z = λζ, where λ denotes an arbitrary length scale, one can write the 5d metric in the KK form (7.1), where
ds 2 4 = re V 2λ −e −4V f dt 2 + e 2V f −1 dr 2 + e 2V r 2 4 (dϑ 2 + sin 2 ϑdϕ 2 ) , e −k/ √ 3 = re V 2λ ,(7.
15)
A = λ cos ϑdϕ.
(Note that F = dA is essentially the Kähler form on S 2 ). We now dualize in 4d according to (7.7), and then relift the solution to five dimensions. This yields the configuration
ds 2 = e −2V µ 4λ 2 dt 2 + 2dzdt + 4λ 2 r 2 dz 2 + r 2 4λ 2 e 4V f −1 dr 2 + r 2 4 dΩ 2 2 , F I ϑϕ =Q I 4λ
sin ϑ, (7.16)
X I = H I e −2V .
One effect of the duality transformation is thus the untwisting of the Hopf fibration 4 . Although the metric in (7.16) contains nondiagonal elements proportional to dzdt, there is no rotation present. To see this, one observes that the nondiagonal elements come from the vector potential A in four dimensions, which gives rise to the field strength F . The equations of motion for F following from the action (7.2) read
∇ α (e −k √ 3 F αβ ) = 0, (7.17)
so there exists an associated conserved charge
J = S 2 ∞ d 2 S αβ e −k √ 3 F αβ . (7.18)
For the solution (7.16) under consideration, however, one easily verifies that J (which, up to a normalization factor, represents the angular momentum) vanishes. One can further simplify (7.16) by an SL(2, R) transformation
t ′ z ′ = 0 − 2λ √ µ √ µ 2λ 2λ √ µ t z . (7.19)
Introducing also the new radial coordinate ρ = r 2 /(4λ), we then get for the metric
ds 2 = e −2V (−f dt ′2 + dz ′2 ) + e 4V (f −1 dρ 2 + ρ 2 dΩ 2 2
). (7.20) (7.20), together with the gauge and scalar fields given in (7.16), represents a nonextremal generalization of the supersymmetric magnetic black string found in [4]. The duality (7.7) thus maps electrically charged black holes onto magnetically charged black strings. Now a short comment on the SL(2, R) transformation (7.19) is in order. The orbits of the Killing vector
∂ z ′ = 2λ √ µ ∂ t (7.21)
are non-compact since the time coordinate is non-compact. This means that globally the spacetimes in (7.16) and (7.20) are not equivalent. To make the transformation (7.19) a symmetry, we have to compactify the orbits of ∂ z ′ . We shall see below however, that the temperature and entropy of one black string can be deduced from the other, which indicates that the two solutions (7.16) and (7.20) are in the same universality class [40]. The Bekenstein-Hawking entropy of the black string (7.20) results to coincide precisely with that of the dual black hole given by (3.15), if we assign to z ′ the period ∆z2λ/ √ µ,
where ∆z = 4πλ denotes the period of z. The Hawking temperature can be computed by requiring the absence of conical singularities in the Euclidean metric, yielding
T H = 2λ √ µ π I (µ + Q I ) 1/2 ,(7.22)
i. e. 2λ/ √ µ times the black hole temperature (3.16). The factor 2λ/ √ µ stems from the rescaling of the time coordinate contained in (7.19). Thus, up to this normalization, the temperature and entropy of the black string (7.20) coincide with that of the dual black hole (3.7), i. e. they are duality invariant.
Microstate Counting from AdS 3 Gravity
We now want to use the near-horizon geometry of the dual solution (7.20) to count the microstates giving rise to the Bekenstein-Hawking entropy. In [4] it was shown that in the extremal case, the geometry becomes AdS 3 × S 2 near the event horizon. The idea is now to use the central charge of AdS 3 gravity [32] in Cardy's formula, in order to compute the statistical entropy, like it was done by Strominger [34] for the BTZ black hole 5 . As only the AdS 3 part is relevant, we would like to reduce the supergravity action from five to three dimensions. To this end, we first Hodge-dualize the magnetic two-form field strength in (7.16). This yields for the action (6.1)
I = 1 16πG 5 d 5 x √ −g R − 1 2 G IJ H Iµνρ H µνρ J − G ij ∂ µ φ i ∂ µ φ j ,(8.1)
where
H Iµνρ = − 1 2 √ 3 G IJ ǫ µνρλσ F Jλσ . (8.2)
Note that for the solution under consideration, the H I do not depend on the coordinates of the internal S 2 . Furthermore, in 3d the three-forms H I are proportional to the volume form and can be integrated out. For the metric, we use the reduction ansatz
ds 2 = ds 2 3 + l 2 Φ 2 dΩ 2 2 ,(8.3)
where G 5 = l 3 as before, and dΩ 2
We recognize (8.11) as the BTZ black hole [42], with event horizon atr =r + . One easily verifies that the period of the coordinate z ′′ is 2π. Λ ef f = −1/l 2 ef f is the effective cosmological constant. The effective 3d Newton constant can be read off from the action (8.8), yielding 1
16πG ef f = 1 4l Φ 3/2 hor ,(8.13)
where the subscript indicates that the dilaton Φ is to be evaluated at the horizon. In this way, we get 1
G ef f = 4π l 5/2 e 3V hor ρ 3/2 hor . (8.
14)
The Bekenstein-Hawking entropy of the BTZ black hole (8.11) is given by We can now apply Strominger's counting of microstates [34] to reproduce the Bekenstein-Hawking entropy. To this end, one first observes that the central charge appearing in the asymptotic symmetry algebra of AdS 3 [32] in our case reads c = 3l ef f 2G ef f .
S (3) = A hor 4G ef f = π 2 2l 3 I (µ + Q I ) 1/2 ,(8.
Final Remarks
The conclusions we have drawn are valid for general black holes of D = 5, N = 2 supergravities. In particular they apply also to the case of theories obtained from compactifications on Calabi-Yau spaces. In different contexts there has been a discussion of dualities [43,44,45] which connect various black hole solutions. We have exhibited an explicit duality transformation which is an invariance of the action: it turns the AdS 2 ×S 3 near horizon geometry into AdS 3 × S 2 . Our calculation shows that the correct statistical entropy is given by the counting of microstates from AdS 3 , where both L 0 andL 0 are different from zero. Using instead the central charge of the AdS 2 Virasoro algebra, with only right-movers, gives a factor √ 2 mismatch between statistical and thermodynamical entropy. Within the AdS 2 approach we were able (up to the mentioned factor √ 2) to capture only the entropy of the small excitations above extremality, not that of the ground state itself. The reason for this was the fact that in two dimensions the Einstein-Hilbert term is a topological invariant, and does not contribute to the central charge computed in [9]. In the extremal limit ǫ → 0 the AdS 2 central charge (6.22) vanishes, whereas the central charge (8.18) for AdS 3 is given by
c = 3π 16l 3 λ 3Q 0Q1Q2 . (9.1)
This is in agreement with Strominger's observation [46] that the AdS 2 Virasoro algebra is related to the right-moving AdS 3 Virasoro algebra by a topological twist which shifts the central charge to zero. It might be that the degeneracy of the ground state itself is effectively captured by a model of conformal quantum mechanics [16]. However, our results support the point of view that the excitations above extremality are described by a two-dimensional conformal field theory [47,11].
, H] = H, [D, K] = −K, [H, K] = −2D. (5.14)
15) which, as it should be, equals the entropy (3.15) of the five-dimensional black hole we started with. The BTZ black hole mass M (3) can be computed using the formulãr 2 + = 8G ef f M (3)
S
mass and angular momentum. For (8.11) one has J = 0, so L 0 =L 0 = 1 2 l ef f M (3) . Plugging this, together with the central charge (8.18), into Cardy'stat = πλ 3/2 l 1/2 G ef f = π 2 2l 3 I (µ + Q I ) 1/2 , (8.22) which coincides precisely with the thermodynamical entropy (3.15) of the 5d black hole (3.7).
The work in[13] includes as a special case also the results obtained in[14].
Cf.[27] for similar computations in the case of heterotic 4D string black holes.
By considering (6.1) we assumed that the Chern-Simons term does not contribute. One can easily generalize the discussion below to nonvanishing CS term. This results in a θ term in four dimensions, which does not spoil the considered duality invariance.
The fact that Hopf bundles can be untwisted by T-dualities was observed in[37,38]. The idea of untwisting and twisting fibres to relate strings and black holes, and thus to gain new insights into black hole microscopics, was also explored in[39].
Cf. also[41], where similar computations for black strings in six dimensions with BT Z × S 3 nearhorizon geometry were performed.
2 denotes the standard metric on the unit S 2 . This gives the reduced actionwhere early greek indices α, β, . . . refer to three-dimensional spacetime. Using the procedure described in section 6, the three-forms H I can be integrated out. In this way, one finally obtainswhere we introduced the magnetic chargesof the black string (7.16). We find it convenient to conformally rescale the metric,yieldingfor the action. The conformally rescaled 3d metric readsThe idea is now to expand this metric near the horizon and near extremality. This can be done by setting(8.10)and taking the limit ǫ → 0. This leads to the metricwhere we introducedr
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| []
|
[
"Seesaw Models with Minimal Flavor Violation",
"Seesaw Models with Minimal Flavor Violation"
]
| [
"Xiao-Gang He \nDepartment of Physics\nINPAC, SKLPPC\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nPhysics Division\nNational Center for Theoretical Sciences\nDepartment of Physics\nNational Tsing Hua University\n300HsinchuTaiwan\n\nCTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan\n",
"Chao-Jung Lee \nCTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan\n",
"Jusak Tandean \nCTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan\n",
"Ya-Juan Zheng \nCTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan\n"
]
| [
"Department of Physics\nINPAC, SKLPPC\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Physics Division\nNational Center for Theoretical Sciences\nDepartment of Physics\nNational Tsing Hua University\n300HsinchuTaiwan",
"CTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan",
"CTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan",
"CTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan",
"CTS, CASTS\nDepartment of Physics\nNational Taiwan University\n106TaipeiTaiwan"
]
| []
| We explore realizations of minimal flavor violation (MFV) for leptons in the simplest seesaw models where the neutrino mass generation mechanism is driven by new fermion singlets (type I) or triplets (type III) and by a scalar triplet (type II). We also discuss similarities and differences of the MFV implementation among the three scenarios. To study the phenomenological implications, we consider a number of effective dimension-six operators that are purely leptonic or couple leptons to the standardmodel gauge and Higgs bosons and evaluate constraints on the scale of MFV associated with these operators from the latest experimental information. Specifically, we employ the most recent measurements of neutrino mixing parameters as well as the currently available data on flavor-violating radiative and three-body decays of charged leptons, µ → e conversion in nuclei, the anomalous magnetic moments of charged leptons, and their electric dipole moments. The most stringent lower-limit on the MFV scale comes from the present experimental bound on µ → eγ and can reach 500 TeV or higher, depending on the details of the seesaw scheme. With our numerical results, we illustrate some striking differences among the seesaw types. Particularly, we show that in types I and III there are features which can bring about potentially remarkable effects which do not occur in type II. In addition, we comment on how a few of the new effective operators can induce flavor-changing dilepton decays of the Higgs boson. | 10.1103/physrevd.91.076008 | [
"https://arxiv.org/pdf/1411.6612v4.pdf"
]
| 118,478,576 | 1411.6612 | 33fb33a2e5ddef4dc7c8c3a846d229a6ad955373 |
Seesaw Models with Minimal Flavor Violation
1 Dec 2014
Xiao-Gang He
Department of Physics
INPAC, SKLPPC
Shanghai Jiao Tong University
200240ShanghaiChina
Physics Division
National Center for Theoretical Sciences
Department of Physics
National Tsing Hua University
300HsinchuTaiwan
CTS, CASTS
Department of Physics
National Taiwan University
106TaipeiTaiwan
Chao-Jung Lee
CTS, CASTS
Department of Physics
National Taiwan University
106TaipeiTaiwan
Jusak Tandean
CTS, CASTS
Department of Physics
National Taiwan University
106TaipeiTaiwan
Ya-Juan Zheng
CTS, CASTS
Department of Physics
National Taiwan University
106TaipeiTaiwan
Seesaw Models with Minimal Flavor Violation
1 Dec 2014
We explore realizations of minimal flavor violation (MFV) for leptons in the simplest seesaw models where the neutrino mass generation mechanism is driven by new fermion singlets (type I) or triplets (type III) and by a scalar triplet (type II). We also discuss similarities and differences of the MFV implementation among the three scenarios. To study the phenomenological implications, we consider a number of effective dimension-six operators that are purely leptonic or couple leptons to the standardmodel gauge and Higgs bosons and evaluate constraints on the scale of MFV associated with these operators from the latest experimental information. Specifically, we employ the most recent measurements of neutrino mixing parameters as well as the currently available data on flavor-violating radiative and three-body decays of charged leptons, µ → e conversion in nuclei, the anomalous magnetic moments of charged leptons, and their electric dipole moments. The most stringent lower-limit on the MFV scale comes from the present experimental bound on µ → eγ and can reach 500 TeV or higher, depending on the details of the seesaw scheme. With our numerical results, we illustrate some striking differences among the seesaw types. Particularly, we show that in types I and III there are features which can bring about potentially remarkable effects which do not occur in type II. In addition, we comment on how a few of the new effective operators can induce flavor-changing dilepton decays of the Higgs boson.
I. INTRODUCTION
The standard model (SM) of particle physics has been immensely successful in describing a vast amount of experimental data at energies up to O(100) GeV [1]. One of the major implications is that in the quark sector flavor-dependent new interactions beyond the SM can readily be ruled out if they give rise to substantial flavor-changing neutral currents (FCNC). This motivates the formulation of the principle of so-called minimal flavor violation (MFV), which postulates that the sources of all FCNC and CP violation reside in SM renormalizable Yukawa couplings defined at tree level [2,3]. The MFV framework offers a predictive and systematic way to explore new physics which does not conserve quark flavor and CP symmetries.
The implementation of the MFV principle for quarks is straightforward, but for leptons there are ambiguities, as the SM by itself does not predict lepton-flavor violation. Since there is now compelling empirical evidence for neutrino masses and mixing [1], it is of interest to formulate MFV in the lepton sector by incorporating ingredients beyond the SM that can account for this observation [4]. Yet, we lack knowledge regarding not only the origin of neutrino mass, but also the precise nature of massive neutrinos. They could be Dirac fermions, like the electron and quarks, as far as their spin properties are concerned. However, neutrinos being electrically neutral, there is also the possibility that they are Majorana particles. The mass-generation mechanisms and Yukawa couplings for the neutrinos in the two cases differ significantly. Since the MFV hypothesis is closely associated with Yukawa couplings, one expects that the resulting phenomenologies in the two scenarios are also different.
In the Majorana neutrino case, there have been studies in the literature on some aspects of MFV realizations in various seesaw scenarios [4][5][6][7], especially in the well-known simplest models of types I, II, and III [8][9][10][11]. In this work, we take another look at these three seesaw schemes to investigate the contributions of new interactions organized according to the MFV principle. We adopt a model-independent approach, where such contributions consist of an infinite number of terms which are built up from leptonic Yukawa couplings and their products. It turns out that the infinite series can be resummed into only seventeen terms [12]. This formulation allows one to have a much more compact understanding of the terms that pertain to a given process. We find that for the specific processes to be considered only a few of them are relevant. We apply this to extract lower limits on the scale of MFV in the three seesaw scenarios using the latest experimental data, including the existing bounds on flavor-violating leptonic processes and the most recent measurements of neutrino mixing parameters. Also, we will examine the similarities and differences among the three seesaw types in relation to their MFV phenomenologies. Especially, we demonstrate that in types I and III there are features which can bring about potentially remarkable effects which do not happen in type II.
The plan of this paper is as follows. In the next section, we describe the MFV framework for leptons in the case that neutrinos are Dirac fermions. In Section III, we discuss the implementation of the MFV principle in scenarios involving Majorana neutrinos with masses generated via the seesaw mechanism of types I, II, and III. This is applied in Section IV, where we explore some of the phenomenology with effective dipole operators involving leptons and the photon. We evaluate constraints on the MFV scale associated with these operators from currently available data on flavor-violating radiative decays of charged leptons, µ → e conversion in nuclei, flavor-violating three-body decays of charged leptons, and their anomalous magnetic moments and electric dipole moments. With our numerical results, we illustrate some striking differences among the three seesaw types. In Section V, we look at several other leptonic operators satisfying the MFV principle. A few of them can cause flavor-violating decay of the Higgs boson. We make our conclusions in Section VI.
II. LEPTONIC MFV WITH DIRAC NEUTRINOS
Let us begin by describing how to arrange interactions under the MFV framework for leptons assuming that neutrinos are of Dirac nature. Following the MFV hypothesis that renormalizable Yukawa couplings defined at tree level are the only sources of FCNC and CP violation, we need to start with such couplings for the neutrinos and charged leptons. We slightly extend the SM by including three right-handed neutrinos, ν k,R , which transform as (1, 1, 0) under the SM gauge group G SM = SU(3) C ×SU(2) L ×U(1) Y . The Lagrangian responsible for the lepton masses can then be written as
L m = −(Y ν ) klLk,L ν l,RH − (Y e ) klLk,L E l,R H + H.c. ,(1)
where summation over k, l = 1, 2, 3 is implicit, Y ν,e are Yukawa coupling matrices, L k,L represents left-handed lepton doublets, ν k,R and E k,R denote right-handed neutrinos and charged leptons, H is the Higgs boson doublet, andH = iτ 2 H * involving the second Pauli matrix τ 2 . Under the SM gauge group, L k,L , E k,R , and H transform as (1, 2, −1/2), (1, 1, −1), and (1, 2, 1/2), respectively.
The MFV hypothesis [3,4] implies that L m is formally invariant under the global group
U(3) L ×U(3) ν ×U(3) E = G ℓ ×U(1) L ×U(1) ν ×U(1) E with G ℓ = SU(3) L ×SU(3) ν ×SU(3) E .
This entails that L k,L , ν k,R , and E k,R belong to the fundamental representations of the SU(3) L,ν,E , respectively,
L L → V L L L , ν R → V ν ν R , E R → V E E R , V L,ν,E ∈ SU(3) L,ν,E ,(2)
and under G ℓ the Yukawa couplings transform in the spurion sense according to
Y ν → V L Y ν V † ν ∼ (3,3, 1) , Y e → V L Y e V † E ∼ (3, 1,3) .(3)
Taking advantage of the requirement that the final effective Lagrangian be invariant under G ℓ , without loss of generality one can always work in the basis where Y e is diagonal,
Y e = √ 2 v diag m e , m µ , m τ(4)
with v ≃ 246 GeV being the Higgs' vacuum expectation value (VEV), and ν k,L , ν k,R , E k,L , and E k,R refer to the mass eigenstates. Consequently, one can express L k,L and Y ν in terms of the Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix U PMNS as
L k,L = (U PMNS ) kl ν l,L E k,L , Y ν = √ 2 v U PMNSmν ,m ν = diag m 1 , m 2 , m 3 ,(5)
where m 1,2,3 are the light neutrino eigenmasses and in the standard parametrization [1] U
with δ being the Dirac CP -violation phase, c kl = cos θ kl , and s kl = sin θ kl . Based on the transformation properties of the fields and Yukawa spurions, one then uses an arbitrary number of Yukawa coupling matrices to put together G ℓ -invariant objects which can induce the desired FCNC and CP -violating interactions. Thus, for operators involving two lepton fields, the pertinent building blocks arē
L L γ α ∆ ℓ L L ,ν R γ α ∆ ν8 ν R ,Ē R γ α ∆ e8 E R ,ν R 1, σ αβ ∆ ν L L ,Ē R 1, σ αβ ∆ e L L . (7)
For these to be G ℓ invariant, the ∆'s should transform according to
∆ ℓ ∼ (1 ⊕ 8, 1, 1) , ∆ ν8 ∼ (1, 1 ⊕ 8, 1) , ∆ e8 ∼ (1, 1, 1 ⊕ 8) , ∆ ν ∼ (3, 3, 1) , ∆ e ∼ (3, 1, 3) .(8)SinceL L γ α ∆ ℓ L L ,ν R γ α ∆ ν8 ν R , andĒ R γ α ∆ e8 E R must be Hermitian, ∆ ℓ,ν8
,e8 must be Hermitian as well. To be acceptable terms in the Lagrangian, the above objects should be combined with appropriate numbers of other SM fields into singlets under the SM gauge group, with all the Lorentz indices contracted.
The MFV principle dictates that these ∆'s are built up from the Yukawa coupling matrices Y ν,e and Y † ν,e . Let us first discuss a nontrivial ∆ which transforms as (1 ⊕ 8, 1, 1) under G ℓ and consists of terms in powers of
A = Y ν Y † ν = 2 v 2 U PMNSm 2 ν U † PMNS , B = Y e Y † e = 2 v 2 diag m 2 e , m 2 µ , m 2 τ ,(9)
both of which also transform as (1 ⊕ 8, 1, 1). Formally, ∆ is a sum of infinitely many terms, ∆ = ξ jkl··· A j B k A l · · · with coefficients ξ jkl··· expected to be at most of O(1). Under the MFV framework, these coefficients must be real because otherwise they would introduce new sources of CP -violation beyond those in the Yukawa couplings. With the Cayley-Hamilton identity X 3 = X 2 TrX + 1 2 X TrX 2 − (TrX) 2 + 1 1DetX for an invertible 3×3 matrix X, one can resum the infinite series into a limited number of terms [12]
∆ = ξ 1 1 1 + ξ 2 A + ξ 3 B + ξ 4 A 2 + ξ 5 B 2 + ξ 6 AB + ξ 7 BA + ξ 8 ABA + ξ 9 BA 2 + ξ 10 BAB + ξ 11 AB 2 + ξ 12 ABA 2 + ξ 13 A 2 B 2 + ξ 14 B 2 A 2 + ξ 15 B 2 AB + ξ 16 AB 2 A 2 + ξ 17 B 2 A 2 B ,(10)
where 1 1 stands for the 3×3 unit matrix. Though one starts with all ξ jkl··· being real, the resummation process generally renders the coefficients ξ r in Eq. (10) complex due to imaginary parts created among the traces of the matrix products A j B k A l · · · with j+k+l+· · · ≥ 6 after the application of the Cayley-Hamilton identity. The imaginary contributions turn out to be reducible to factors proportional to a Jarlskog invariant quantity, Im Tr A 2 BAB 2 = (i/2) Det[A, B], which is much smaller than unity [12].
With Eq. (10), one can devise the objects in Eq. (8). Thus, the first of the Hermitian combinations can be ∆ ℓ = ∆ + ∆ † . To obtain nontrivial ∆ ν,e , one can take ∆ ν = Y † ν ∆ and ∆ e = Y † e ∆. The construction of ∆ ν8,e8 can be carried out in a similar way, except A and B are replaced bỹ
A = Y † ν Y ν andB = Y † e Y e .
Sinceà andB are diagonal, so are any powers of them. Therefore, ∆ ν8,e8 do not produce any FCNC and CP -violation effects.
We end this section by mentioning that the above discussion can be easily applied to the quark sector with the renormalizable Yukawa Lagrangian given by
L m = −(Y u ) klQk,L U l,RH − (Y d ) klQk,L D l,R H + H.c. ,(11)
where Y u,d are Yukawa coupling matrices, Q k,L represents left-handed quark doublets, and U k,R (D k,R ) denote right-handed up-type (down-type) quarks. These fields transform as (3, 2, 1/6), (3, 1, 2/3), and (3, 1, −1/3), respectively, under the SM gauge group G SM . In the basis where Y d is diagonalized,
Y d = √ 2 v diag m d , m s , m b , Y u = √ 2 v V † CKMmu ,m u = diag m u , m c , m t ,(12)
where V CKM is the Cabibbo-Kobayashi-Maskawa matrix which has the same standard parametrization as in Eq. (6).
For MFV interactions, employing Y u,d along with A = Y u Y † u and B = Y d Y † d
as building blocks, one can construct objects such as ∆ q , ∆ u , and ∆ d , which are the quark counterparts of ∆ ℓ , ∆ ν and ∆ e , respectively [13].
III. SEESAW MODELS WITH MFV
If neutrinos are Majorana particles, the Yukawa couplings that take part in generating their masses differ from those in the Dirac neutrino case and depend on the model details. In this section we discuss how to realize the MFV hypothesis in the well-motivated seesaw models. The seesaw mechanism endows neutrinos with Majorana mass and provides a natural explanation for why they are much lighter than their charged partners. If just one kind of new particles are added to the minimal SM, there are three different scenarios [8][9][10][11]: the famous seesaw models of type I, type II, and type III. A crucial step in the implementation of MFV in a given model is to identify the quantities A and B in terms of the relevant Yukawa couplings. This will be the emphasis of the section.
A. MFV in type-I seesaw model
In the type-I seesaw model the SM is slightly expanded with the inclusion of three gaugesinglet right-handed neutrinos, ν k,R , which are allowed to possess Majorana masses [8]. The renormalizable Lagrangian for the lepton masses is
L I m = −(Y ν ) klLk,L ν l,RH − (Y e ) klLk,L E l,R H − 1 2 (M ν ) kl ν c k,R ν l,R + H.c. ,(13)
where M ν = diag(M 1 , M 2 , M 3 ) contains the right-handed neutrinos' Majorana masses and the superscript c refers to charge conjugation. 1 Accordingly, the masses of the neutral fermions make up the 6×6 matrix which transforms as (1, 3, 1) under the SM gauge group G SM . Accordingly, the Lagrangian describing the Yukawa couplings of leptons is
L II m = −(Y e ) klLk,L E l,R H − 1 2 (Y T ) klLk,LT L c l,L + H.c. ,(19)
withT = iτ 2 T * . It respects lepton-number conservation if T is assigned a lepton number of −2.
After the VEV of the neutral component of T becomes nonzero, T 0 = v T / √ 2, one obtains in L II m the neutrino mass matrix
m ν = 1 √ 2 v T Y T = U PMNSmν U T PMNS(20)
in the basis where the charged lepton's Yukawa coupling matrix, Y e , has been diagonalized. If the nonzero elements of Y T are of O(1), the tiny size of neutrino masses then comes from the suppression of v T due to certain choices of the parameters in the scalar potential V. One can express it as [15]
V = −m 2 H H † H + M 2 T Tr(T † T ) + 1 2 λ 1 (H † H) 2 + 1 2 λ 2 [Tr(T † T )] 2 + λ 3 H † H Tr(T † T ) + λ 4 Det(T † T ) + λ 5 H † T † T H − 1 √ 2 µ TH † T † H + µ * T H † TH ,(21)
where m 2 H > 0 and M 2 T > 0. The µ T part explicitly breaks lepton-number symmetry and causes T 0 to develop a nonzero VEV. From the minimization of V, one gets
λ 1 v 2 = 2m 2 H + 2|µ T |v T − λ 3 v 2 T , |µ T |v 2 = 2M 2 T v T + λ 3 v 2 v T + λ 2 v 3 T .(22)
For |µ T |v T ≪ m 2 H and v T ≪ v, the first equality simplifies to λ 1 v 2 ≃ 2m 2 H like in the SM, and with the additional conditions v ≪ M T /|λ 3 | 1/2 and v T ≪ |µ T | the second relation in Eq.
(22) translates into v T ≃ |µ T |v 2 2M 2 T ,(23)
which is small if |µ T |v ≪ M 2 T . This turns into the seesaw form v T ∼ v 2 /M T if |µ T | ∼ M T ≫ v, but the prerequisites just mentioned do not preclude a scenario with a relatively lighter triplet, in which M T can be as low as the TeV level [7], provided that |µ T | ≪ v.
Since the triplet couples to SM gauge bosons, a nonzero v T will make the ρ 0 parameter deviate from unity [15],
ρ 0 = m 2 W m 2 Z cos 2 θ W = v 2 + 2v 2 T v 2 + 4v 2 T ≃ 1 − 2v 2 T v 2 .(24)
Its empirical value ρ 0 = 1.00040 ± 0.00024 [1] then implies, at the 2σ level, that v T < 1.6 GeV. This is much weaker than the requirement for the v T range that can produce neutrino masses of
O(0.1 eV) if the elements of Y T are of O(1).
To implement the MFV hypothesis in this seesaw scheme, one observes that the Lagrangian in Eq. (19) possesses formal invariance under the global group
U(3) L ×U(3) E = G ′ ℓ ×U(1) L ×U(1) E , with G ′ ℓ = SU(3) L ×SU(3) E , if L L and E R belong to the fundamental representation of SU(3) L,E , respectively, L L → V L L L , E R → V E E R , V L,E ∈ SU(3) L,E ,(25)
and the Yukawa couplings are spurions transforming according to
Y e → V L Y e V † E , Y T → V L Y T V T L .(26)
Here the building block ∆ still has the expression in Eq. (10), with B = Y e Y † e being the same as in Eq. (9), but unlike before
A = Y T Y † T = 2 v 2 T U PMNSm 2 ν U † PMNS ,(27)
where from Eq. (20)
Y T = √ 2 v T U PMNSmν U T PMNS .(28)
It is interesting to notice that A in Eq. (27) is the same as its Dirac-neutrino counterpart in Eq. (9), up to an overall factor. Due to this difference, whereas the elements of the latter are tiny, those in Eq. (27) can be of O(1) if v T is similar in order of magnitude to the neutrino masses. This will in fact be realized in our numerical analysis, as we will again choose the largest eigenvalues of A to be unity, which amounts to imposing v T = √ 2 max m 1 , m 2 , m 3 . Compared to Eq. (17) in the type-I case, A in Eq. (27) is in general much simpler. In particular, it no longer depends on the Majorana phases in U PMNS which have canceled out due tom ν being diagonal and does not involve the O matrix which can supply potentially major extra effects including CP -violating ones [13].
C. MFV in type-III seesaw model
In the type-III seesaw model the new particles consist only of three fermionic SU(2) L triplets [11]
Σ k = Σ 0 k / √ 2 Σ + k Σ − k −Σ 0 k / √ 2 , k = 1, 2, 3 ,(29)
which transform as (1, 3, 0) under the SM gauge group G SM . The Lagrangian responsible for the lepton masses is then
L III m = −(Y e ) klLk,L E l,R H − √ 2 (Y Σ ) klLk,L Σ lH − 1 2 (M Σ ) kl Tr Σ c k Σ l + H.c.(30)
where Σ c k is the charge conjugate of Σ k . For convenience, we define the right-handed fields E k,R = Σ − k and N k,R = Σ 0 k and left-handed fields E k,L = Σ + k c and N k,L = Σ 0 k c . In terms of matrices containing them and SM leptons, one can express the mass terms in L III m after electroweak symmetry breaking as
L III m ⊃ − Ē LĒL M ℓ √ 2 M D 0 M Σ E R E R − 1 2 ν ′ LNL 0 M D M T D M Σ (ν ′ L ) c N R + H.c. ,(31)
where M ℓ = vY e / √ 2 and M D = vY Σ / √ 2 are 3×3 matrices and ν ′ L = U PMNS ν L . For M Σ ≫ M D in their nonzero elements, a seesaw mechanism like that in type I becomes operational to generate the light neutrinos' mass matrix
m ν = − v 2 2 Y Σ M −1 Σ Y T Σ .(32)
Hence it is tempting simply to write Y Σ in a similar way to Y ν in type I,
Y Σ = i √ 2 v U PMNSm 1/2 ν OM 1/2 Σ ,(33)
and use Y e in Eq. (4) like before
One, however, needs to justify this approximation because the light charged leptons, E k , mix with the heavy ones, E k , as can be deduced from Eq. (31). They are related to the mass eigenstates E ′ k and E ′ k by
E C E C = (U EE ) C (U EE ) C (U EE ) C (U EE ) C E ′ C E ′ C , C = L, R .(34)
This
alters U PMNS in Eq. (33) to (U EE ) † L U PMNS as well as Y e to (U EE ) † L Y e (U EE ) R . At leading order [16], (U EE ) L = 1 1 − M D M −2 Σ M † D and (U EE ) R = 1 1 for M D ≪ M Σ .
Thus, the deviations of (U EE ) L,R from the unit matrix are negligible, and the approximation of Y Σ in Eq.(33) is justified. Accordingly, in type III with MFV we work with
A = Y Σ Y † Σ = 2 v 2 U PMNSm 1/2 ν OM Σ O †m1/2 ν U † PMNS(35)
and, as in the previous scenarios, B in Eq. (9). These are no different from those in type I, with
M Σ = diag(M 1 , M 2 , M 3 )
. Also, we fix the biggest eigenvalue of A to one.
IV. LEPTONIC DIPOLE OPERATORS IN SEESAW MODELS WITH MFV
Having set up the basics of the MFV realizations in the minimal seesaw models of types I, II, and III, we now study some of the phenomenological implications and point out possible differences among them. It is clear from the last section that as far as MFV phenomenology is concerned type I and type III will be virtually alike because, with the new fermion masses being far above the TeV level, the building blocks for the quantity ∆ are the same in both cases. However, within the MFV context we expect that marked differences can materialize between these two models and type II.
To explore the phenomenological consequences of MFV, one can adopt an effective theory approach [3,4], assuming that the heavy degrees of freedom in the full theory have been integrated out. This is especially suitable for the seesaw scenarios under consideration, where the masses of the new particles are much greater than the energies of the processes which we examine in this paper. A number of higher-dimensional effective operators involving leptons have been listed in the leptonic MFV literature [4,5]. Here we focus on dimension-six operators which generate dipole interactions between SM leptons and gauge bosons. We deal with several other leptonic operators in the next section.
The dipole operators of interest are [4] O (e1)
RL = g ′Ē R Y † e ∆ ℓ1 σ κω H † L L B κω , O (e2) RL = gĒ R Y † e ∆ ℓ2 σ κω H † τ a L L W κω a ,(36)
where W and B stand for the usual SU(2) L ×U(1) Y gauge fields with coupling constants g and g ′ , respectively, τ a are Pauli matrices, summation over a = 1, 2, 3 is implicit, and ∆ ℓ1,ℓ2 have the same form as ∆ in Eq. (10), but with generally different ξ r . One can write the effective Lagrangian for O
(e1,e2) RL as L eff = 1 Λ 2 O (e1) RL + O (e2) RL + H.c. ,(37)
where Λ is the scale of MFV and their own coefficients in this Lagrangian have been absorbed by the ξ r in their respective ∆'s.
The terms in Eq. (37) with the photon have the general form
L E k E l γ = √ απ Λ 2Ē k σ κω m E k (∆ ℓ ) kl + m E l (∆ ℓ ) * lk − m E k (∆ ℓ ) kl − m E l (∆ ℓ ) * lk γ 5 E l F κω ,(38)
where α ≃ 1/137 is the fine-structure constant, F κω denotes the electromagnetic field-strength tensor, (E 1 , E 2 , E 3 ) = (e, µ, τ ), and hereafter ∆ ℓ = ∆ ℓ1 − ∆ ℓ2 . These interactions contribute to the flavor-changing decays E l → E k γ, E k E − j E + j and nuclear µ → e conversion, as well as to the anomalous magnetic moments (g − 2) and electric dipole moments (EDMs) of charged leptons. We ignore the contributions from L eff to µ → e conversion and E l → E k E − j E + j that are mediated by the Z boson due to the suppression by its mass. The flavor-changing transitions and lepton EDMs have been searched for over the years, but with null results so far, leading to increasingly severe bounds on their rates. For the electron and muon g − 2, the predictions and measurements have reached high precision which continues to improve, implying that the inferred room for new physics is small and progressively decreasing. Thus the experimental information on these processes can offer very stringent restrictions on the scale Λ in Eq. (37). We address this in the rest of the section.
A. Flavor-changing transitions and anomalous magnetic moments
We treat first observables that are not sensitive to CP -violating effects. In this case, we can retain no more than three of the 17 terms of ∆ in Eq. (10), as the others are suppressed by comparison. Since we pick the largest eigenvalue of the A matrix to be one in order to enhance the impact of new physics, the elements of A are much greater than those of the B matrix, whose biggest eigenvalue is 2m 2 τ /v 2 ≃ 1.0 × 10 −4 . As a consequence, the matrix elements of terms in Eq. (10) with at least one power of B are in general significantly smaller due to this suppression factor than terms without any B. Thus we can make the approximation ∆ = ξ 1 1 1 + ξ 2 A + ξ 4 A 2 in dealing with such observables.
Among the flavor-changing decays E l → E k γ, one can expect the most severe constraint from µ → eγ which has the strictest measured limit [1]. With its amplitude derived from Eq. (38), one determines its branching ratio to be
B(µ → eγ) = ατ µ m 5 µ Λ 4 (∆ ℓ ) 21 2 ,(39)
where τ µ is the muon lifetime, the electron mass has been neglected, and
(∆ ℓ ) jk = ξ ℓ 2 A jk + ξ ℓ 4 (A 2 ) jk .(40)
From Eq. (39), one can easily obtain the corresponding formulas for τ → eγ and τ → µγ by appropriately changing the flavor indices. These tau decays can place complementary restraints on the dipole couplings. Searches for µ → e conversion in nuclei can offer constraints on new physics competitive to those from µ → eγ measurements [17]. Assuming that the flavor-changing transition is again due to the dipole operators alone, one can write the conversion rate in nucleus N as [18]
B(µN → eN ) = απ m 5 µ (∆ ℓ ) 21 D N 2 Λ 4 ω N capt ,(41)
where D N is the dimensionless overlap integral for N and ω N capt the rate of µ capture in N . Based on the existing experimental limits on µ → e transition in various nuclei [1] and the corresponding D N and ω N capt values [18], one expects that the data on µTi → eTi and µAu → eAu may supply important restrictions. To calculate their rates, we will employ D Ti = 0.087, D Au = 0.189, ω Ti capt = 2.59 × 10 6 /s, and ω Au capt = 13.07 × 10 6 /s [18]. Another kind of flavor-changing process which receives contributions from the interactions in Eq. (38) is the three-body decay E l → E k E − j E + j . If there are no other contributions, one can express its rate as
Γ E l →E k E jĒj = α 2 m 5 ℓ l (∆ ℓ ) lk 2 4π Λ 4 I E l →E k E jĒj ,(42)
where the m E k terms in Eq. (38) have been neglected and the factor I E l →E k E jĒj , from the phasespace integration, can be calculated using formulas available in the literature [5]. The factors relevant to the processes we will examine are I µ→eeē = 9.885, I τ →µµμ = 3.264, I τ →µeē = 16.97, I τ →eeē = 17.41, and I τ →eµμ = 3.01.
For numerical analysis, we need in addition the values of the various neutrino parameters, especially their masses and the elements of the mixing matrix U PMNS . For these, we adopt the numbers quoted in Table I from a recent fit to global neutrino data [19]. Most of them depend on whether neutrino masses fall into a normal hierarchy (NH) or an inverted one (IH). Since empirical information on the absolute scale of m 1,2,3 is still far from precise [1], for definiteness we set m 1 = 0 (m 3 = 0) in the NH (IH) case. As for the Majorana phases α 1,2 , there are still no data available on their values.
To proceed, we also need to specify the A matrix, which is model dependent, as seen in the preceding section. For the type-I or -III seesaw scenario, A in Eq. (17) [19], in terms of best-fit values and allowed 1σ ranges of the mass-mixing parameters. The neutrino mass hierarchy may be normal m 1 < m 2 < m 3 or inverted m 3 < m 1 < m 2 .
different realizations, depending on M ν and O. We consider first the least complicated possibility that the right-handed neutrinos ν k,R are degenerate, with M ν = M1 1, and O is a real orthogonal matrix, in which case
A = 2M v 2 U PMNSmν U † PMNS(43)+ e iδ c 23 c 13 s 13 ξ ℓ 2Â3 + ξ ℓ 4Â 2 3 ,(45)
Later on we will also provide examples for a case in which O is complex. With (∆ ℓ ) kl specified, we can determine lower limits on the MFV scale Λ from the experimental information on the observables described above. The particular data we use are listed in the first two columns of Table II. Since in our model-independent approach ξ 2,4 are free constants, for simplicity we assume that only one of them is nonzero at a time. For ξ 4 = 0, we scan the ranges of the neutrino mass and mixing parameters quoted in Table I Table II, which are the strictest ones to date in their respective groups of processes with the same flavor changes. Subsequently, we apply the acquired values of (∆ ℓ ) 21,32,31 to obtain limits from the other experimental bounds in this table. We collect theΛ numbers in the third column which correspond to the NH (IH) of neutrino masses. For comparison, employing the central values in Table I would give us results which are smaller by up to 30%. It is worth mentioning that in all this computation the right-handed neutrino mass is M ≃ (6.0-6.3) × 10 14 GeV.
For the type-II scheme, A is given only in Eq. (27), which has eigenvalues 1,2,3 = 2m
and its (∆ ℓ ) 31,32 counterparts, which are analogous to those in Eqs. (45) and (46), respectively. Utilising these matrix elements, we take steps similar to those elaborated in the previous paragraph, while adjusting v T such that max  1 , 2 , 3 = 1, in order to extract from data the lower limits onΛ for ξ ℓ 4 = 0. We collect the results in the fourth column of Table II, which correspond to v T ∼ 0.07 eV. With ξ ℓ 2 = 0 instead, we obtain comparable numbers for Λ min /|ξ ℓ 4 | 1/2 , specifically 285 (320) TeV from the B(µ → eγ) data. Now, since 2 k M 2 v 2 T = 2 k v 4 , one realizes that the numbers in the fourth column of Table II are also the limits on Λ/|ξ ℓ 4 | 1/2 in the type-I case of the last paragraph with ξ ℓ 2 = 0. It is clear from Table II that to date the most stringent constraint on the dipole operators in Eq. (37) comes from the measured bound on µ → eγ among processes that change lepton flavor. It is instructive to entertain the consequence of this for the calculated branching ratios of the other transitions if other operators are absent or have only minor impact. Thus, employing thê Λ min numbers belonging to B(µ → eγ) in Table II and the corresponding neutrino parameter values, we compute the results listed in Table III. The ratios of any two of them and the relative size of any one of them with respect to B(µ → eγ) are, therefore, predictions of the particular scenario considered. They can be checked experimentally if two or more of these processes are detected in the future, as the presence of other operators with nonnegligible effects would likely lead to a different set of predictions. Since the numbers in Table III are at least two orders of magnitude below their current experimental bounds, it is of interest to make comparison with the projected sensitivities of future experiments on lepton flavor violation.
There are planned searches for µ → eγ with projected sensitivity reaching a few times 10 −14 within the next five years [21]. If they come up empty, the predictions in Table III will decrease somewhat, probably by up to an order of magnitude. Nevertheless, the prediction for µ → 3e in Table III will likely still be testable with new experiments looking for it which will start running in a couple of years and may be able to probe a branching ratio as low as 10 −16 after several years [22,23]. Complementary checks may be available from upcoming searches for flavor-violating tau decays which can improve their current empirical limits by two orders of magnitude [22]. Potentially severe restrictions will be supplied by future measurements on µ → e conversion in nuclei which will begin in a few years and are expected to achieve sensitivity at the level of 10 −17 or better eventually [22,23]. As another significant observation from Table II, it indicates that no remarkable differences in the bounds on Λ appear among the three types of seesaw models if in type I or III the righthanded neutrinos are degenerate and the O matrix is real, with A in Eq. (43). If O is complex and/or the right-handed neutrinos are nondegenerate, A is less simple which may give rise to more pronounced deviations from the type-II results. To illustrate this, we next explore the possibility that O is complex.
B(µ − → e − e − e + ) 3.3 (3.3) × 10 −15 3.3 (3.3) × 10 −15 B(τ → µγ) 7.9 (14) × 10 −13 1.7 (1.4) × 10 −12 B(τ − → µ − µ − µ + ) 1.5 (2.7) × 10 −15 3.3 (2.6) × 10 −15 B(τ − → µ − e − e +
With ν k,R still degenerate, M ν = M1 1, but O complex, A is more complicated,
A = 2 v 2 M U PMNSm 1/2 ν OO †m1/2 ν U † PMNS .(48)
One can always write OO † = e 2iR with a real antisymmetric matrix
R = 0 r 1 r 2 −r 1 0 r 3 −r 2 −r 3 0 .(49)
Since OO † is not diagonal, A generally has dependence on the Majorana phases in U PMNS if they are not zero. To concentrate first on demonstrating how O can bring about significant new effects, we switch off the Majorana phases, α 1,2 = 0. Subsequently, for illustration, we pick r 1 = r 2 = r 3 = ρ and again scan the parameter ranges in Table I in order to get the highest Λ/|ξ ℓ 2,4 | 1/2 from the experimental bound on B(µ → eγ), with the condition that the largest eigenvalue of A in Eq. (48) is unity. We exhibit the resulting dependence on ρ in Figure 1. It reveals that, in this example, the contribution of O can boost Λ min by up to 80% with respect to its value when O is real, which implies that the predicted branching ratio is enhanced by an order of magnitude. Turning our attention now to the impact of the Majorana phases, we make the same choice of r 1,2,3 = ρ as in the preceding paragraph, select α 1 = 0 and ρ = −1, and letΛ min vary as a function of α 2 , in similar steps to those in the last example. We display the variation in Figure 2, which shows that although the Majorana phases in this instance can increase the lower limits on Λ only moderately, they can induce sizable reduction of it. Hence the associated decay rate is affected in roughly the same way. All these examples demonstrate that the O matrix and Majorana phases in types I and III can produce striking effects which do not occur in type II.
Before finishing this subsection, we examine limitations from the anomalous magnetic moments, a ℓ , of charged leptons. The Lagrangian for a ℓ is L a ℓ = √ απ a ℓl σ κω ℓF κω / 2m ℓ , which gets contributions from the flavor-diagonal couplings in Eq.(38). Accordingly
a E k = 4 m 2 E k Λ 2 ξ ℓ 1 δ kk + ξ ℓ 2 A kk + ξ ℓ 4 (A 2 ) kk ,(50)
where we have ignored the tiny Im ξ ℓ 1,2,4 . Since a e is much suppressed by the electron mass, and since the measurement of a τ is not yet precise [1], the strongest restrictions from anomalous magnetic moments can be expected from a µ . Currently its experimental and SM values differ by a exp µ − a SM µ = (249 ± 87) × 10 −11 [24], which suggests that we can require the new contributions to satisfy a µ < 3.4 × 10 −9 . Assuming again that the right-handed neutrinos are degenerate and the O matrix is real, if only one of ξ ℓ 1,2,4 is nonzero at a time, from this a µ bound we infer Λ/|ξ ℓ 1,2,4 | 1/2 > 3.6, 2.7, 2.5 TeV. Upon comparing these numbers with those in Table II, we conclude that the muon g − 2 cannot at present compete with the flavor-violating leptonic transitions in restraining Λ. Variation of the lower limit onΛ = Λ/|ξ ℓ 2 | 1/2 , subject to B(µ → eγ) data, versus α 2 for α 1 = 0, degenerate ν k,R , and complex-O parameter ρ = r 1 = r 2 = r 3 = −1 (solid curves), as explained in the text. The dashed curves depict the corresponding variation of the lower limit on Λ/|ξ ℓ 4 | 1/2 for ξ ℓ 2 = 0.
B. Lepton EDMs
The interactions in Eq. (38) also contribute to a charged lepton's electric dipole moment, denoted by d ℓ , which is a sensitive probe for the presence of new sources of CP violation, as the SM prediction is very small [25]. The Lagrangian for d ℓ is L d ℓ = −(id ℓ /2)lσ κω γ 5 ℓF κω , and so
d E k = √ 2 e v Λ 2 Im Y † e ∆ ℓ kk .(51)
Unlike the observables treated in the previous subsection, the pertinent terms in ∆ ℓ are those with higher orders in A and B. Thus for the electron [13]
d e = √ 2 e v Λ 2 ξ ℓ 12 Im Y † e ABA 2 11 + ξ ℓ 16 Im Y † e AB 2 A 2 11 ,(52)
where we have again neglected Im ξ ℓ r . Hereafter we drop ξ ℓ 16 term which is suppressed by a factor of B relative to the ξ ℓ 12 term. The latest analysis on d e under the MFV framework has been performed in Ref. [13] for the Dirac neutrino case as well as the type-I (and hence also type-III) seesaw model. If neutrinos are Dirac particles, d e has the form
d D e = 32e m e Λ 2 v 8 m 2 µ − m 2 τ m 2 1 − m 2 2 m 2 2 − m 2 3 m 2 3 − m 2 1 ξ ℓ 12 J ℓ ,(53)
where J ℓ = Im U e2 U µ3 U * e3 U * µ2 is a Jarlskog invariant for U PMNS . This turns out to be independent of the m 1,2,3 individually because the neutrino squared-mass differences defined in Table I imply that m 2 1 − m 2 2 m 2 2 − m 2 3 m 2 3 − m 2 1 = δm 2 ∆m 2 2 − 1 4 δm 2 3 . On the other hand, in types I and III with degenerate ν k,R and a real O matrix, in which case A is given by Eq. (43),
d I,III e = 32e m e M 3 Λ 2 v 8 m 2 µ − m 2 τ m 1 − m 2 m 2 − m 3 m 3 − m 1 ξ ℓ 12 J ℓ ,(54)
Since m k ≪ M, one can see that d D e is considerably suppressed relative to d M e . In contrast, for type II one derives
d II e = 32e m e Λ 2 v 2 v 6 T m 2 µ − m 2 τ m 2 1 − m 2 2 m 2 2 − m 2 3 m 2 3 − m 2 1 ξ ℓ 12 J ℓ = v v T 6 d D e ,(55)
which is far above d D e due to v T ≪ v. From these formulas, one can readily find those for d µ,τ by cyclically changing the mass subscripts.
Numerically, d D e = 1.3 × 10 −99 e cm GeV 2 /Λ 2 [13], which is too minuscule to yield any useful restraint onΛ from the newest data |d e | exp < 8.7 × 10 −29 e cm from the ACME experiment [26]. In the Majorana neutrino case, the type-I (or -III) prediction in Eq. (54) has been evaluated in Ref. [13] to yield the limitΛ > 0.36 (0.12) TeV corresponding to M = 6.16 (6.22) × 10 14 GeV for the NH (IH) of neutrino masses.
In type II, from Eq. (55) we arrive at (57)
With v T ≃ 0.069 eV from the requirment that the largest eigenvalue of A in Eq. (27) be unity, it follows thatΛ
> 0.17 TeV ,(58)
which is roughly comparable to its counterparts in type I (or III) quoted above.
In the preceding discussion, d e is caused by the CP -violating Dirac phase δ in U PMNS , and the Majorana phases α 1,2 therein do not take part. However, if O is complex, the phases in it may give rise to an extra contribution to d e and the Majorana phases can modify it further. As investigated in detail in Ref. [13], these new CP -violating contributions to d e can be more important than those of δ. Such effects do not occur in type II, as d II e does not have dependence on O or α 1,2 .
V. ADDITIONAL LEPTONIC OPERATORS
Besides the dipole operators, there are other dimension-six operators that can arise in the three simplest seesaw scenarios under the MFV framework [5]. Focusing on operators that are purely leptonic or couple leptons to SM gauge and Higgs bosons, we have
L ′ eff = 1 Λ 2 3 m=1 O (m) 4L + 1 Λ 2 O (e3) RL + H.c. + 1 Λ 2 2 n=1 O (n) LL ,(59)
where
O (1) 4L =L L γ µ ∆ (1) 4L L LLL γ µ ∆ (1) 4L ′ L L , O(2)4L =L L γ µ ∆ (2) 4L τ a L LLL γ µ ∆ (2) 4L ′ τ a L L , O (3) 4L =L L γ µ ∆ (3) 4L L LĒR γ µ E R , O (e3) RL = (D µ H) †Ē R Y † e ∆ RL D µ L L , O (1) LL =L L γ µ ∆ (1) LL L L H † iD µ H , O(2)LL =L L γ µ τ a ∆ (2) LL L L H † τ a iD µ H ,(60)
with the ∆'s having the approximate expression ∆ = ξ 1 1 1 + ξ 2 A + ξ 4 A 2 like before, but with generally different coefficients ξ 1,2,4 of their own, and D µ being the usual covariant derivative involving SM gauge bosons.
The first three of these operators contribute directly to
E − k → E − l E − l E + l and E − k → E − l E − j E + j
with l = j, whereas the last three contribute mainly via diagrams mediated by the Z boson. Here we assume that the dipole operators O (1,2) RL treated in the last subsection are absent. To see what constraints can be derived from the experimental information on these decays, for simplicity we select ∆ (1,2) 4L ′ = 1 1. We then find their branching ratios to be [5], respectively,
B E k → E l E lĒl = τ E k m 5 E k 1536 π 3 Λ 4 (A + ) lk 2 + 2 (A − ) lk 2 , B E k → E l E jĒj = τ E k m 5 E k 1536 π 3 Λ 4 (A + ) lk 2 + (A − ) lk 2 ,(61)
where the final masses have been neglected relative to the initial one, τ E k is the lifetime of E k , and the matrices A ± are combinations of the ∆'s,
A + = ∆ (1) LL + ∆ (2) LL sin 2 θ W + ∆ (3) 4L , A − = ∆ (1) LL + ∆ (2) LL sin 2 θ W − 1 2 + ∆ (1) 4L + ∆ (2) 4L ,(62)
with sin 2 θ W = 0.23 and the contributions from O (e3)
RL having been neglected due to suppression by m 2 E /m 2 Z . To illustrate the lower limits on Λ obtainable from the data on these decays, already quoted in Table II, for definiteness we further assume that either only O
(1,2,3) 4L with ∆ (1) 4L = ∆ (2) 4L = ∆ (3) 4L or O (1,2) LL with ∆ (1) LL = ∆ (2)
LL are contributing at a time, with ξ 4 = 0 in the ∆'s, and that in type I (or III) the A matrix is given by Eq. (43). Using the maximal A kl determined earlier, we infer the lower bounds onΛ presented in Table IV.
Obviously, for these operators the measured limit on B(µ − → e − e − e + ) provides the strictest constraint among the flavor-violating processes. To see the implication of this for the predictions on the tau three-body modes, we calculate their branching ratios with theΛ min numbers belonging to µ − → e − e − e + in Table IV and the neutrino parameter values employed to extract them. We display the results in Table V, which are larger than their counterparts in Table III by roughly two orders of magnitude. This considerable variation in predictions will help make it easier to identify the underlying physics if one or more of the flavor-violating transitions we study are observed in the future.
The first two of the operators in Eq. (59) also contribute to
E − k → E − l E − l E + j with l = j. Its branching ratio is B E k → E l E lĒj = τ E k m 5 E k 768 π 3 Λ 4 D E k →E l E lĒj 2 ,(63)
where For simplicity, we choose ∆ (1)
D E k →E l E lĒj = ∆ (1) 4L lk ∆ (1) 4L ′ lj + ∆ (1) 4L lj ∆ (1) 4L ′ lk + ∆ (2) 4L lk ∆ (2) 4L ′ lj + ∆ (2) 4L lj ∆ (2) 4L ′ lk .(O (1,2) LL B(τ − → µ − µ − µ + ) 14L lk ∆ (1) 4L ′ lj = ∆ (2) 4L lk ∆ (2) 4L ′ lj = ξ 2 A lk A lj .
Subsequently, we scan the parameter ranges in Table I to maximize A lk A lj , while setting the largest eigenvalue of A to unity. With the results, we extract from the experimental bounds B(τ − → e − e − µ + ) < 1.5 ×10 −8 and B(τ − → µ − µ − e + ) < 1.7×10 −8 [1], respectively, the limitsΛ min > 2.9 (2.7) and 6.0 (5.8) TeV in type I or III for the normal (inverted) hierarchy of neutrino masses. The corresponding results for type II areΛ min = 2.7 (2.7) and 5.5 (5.8) TeV, respectively. If the above choice for these operators is to satisfy the measured limit on B(µ − → e − e − e + ) as well, we arrive atΛ min in the (27-146) TeV range instead and the branching ratios of τ − → e − e − µ + , µ − µ − e + below 10 −10 , like those in Table V.
We note that the renormalizable couplings of the scalar triplet to leptons as described by Eq. (19) induce at tree level T -mediated diagrams that correspond to extra operators such as (Y T ) km (Y T ) * lnLk,L γ µ L l,LLm,L γ µ L n,L /M 2 T [4,7], which we do not analyze explicitly in this work. They also contribute to the three-body charged-lepton decays, and so for (Y T ) kl = O(1) the lower bounds on M T are comparable in order of magnitude to those onΛ min in Table IV, although M T in general is not the same as Λ. Thus our requirement in type II that the biggest eigenvalue of A = Y T Y † T be unity translates into the limitation M T > O(100 TeV) according to the table. With such a mass, the triplet scalars would be undetectable at the LHC. If we relax the condition on A, the minimum of M T can be lowered, but at the same timeΛ min also becomes weakened. Specifically, for M T at the TeV level, which may be within LHC reach, Y T needs to be two orders of magnitude smaller Finally, we address the potential impact of O on the decay of the recently discovered Higgs boson. Their presence can bring about modifications to the standard decay modes of the Higgs and/or cause it to undergo exotic decays [27]. As data from the LHC will continue to accumulate with increasing precision, they may uncover clues of new physics in the couplings of the Higgs.
The latest LHC measurements have begun to reveal the Higgs couplings to leptons. The ATLAS and CMS Collaborations have observed h → τ + τ − and measured its signal strength to be σ/σ SM = 1.42 +0.44 −0.38 and 0.91 ± 0.27, respectively [28,29]. In contrast, the only experimental information available on h → µ − µ + to date are the bounds B(h → µ − µ + ) < 1.5 × 10 −3 and 1.6 × 10 −3 from ATLAS and CMS, respectively [30,31]. On the other hand, CMS [32] has intriguingly reported the detection of a slight excess of h → µ ± τ ∓ events with a significance of 2.5σ. If the finding is interpreted as a statistical fluctuation, it translates into the limit B(h → µτ ) = B(h → µ − τ + ) + B(h → µ + τ − ) < 1.57% at 95% CL [32]. In view of these data, we demand nonstandard contributions to respect
0.7 < Γ h→ττ Γ SM h→ττ < 1.8 , Γ h→µμ Γ SM h→µμ < 6.7 , Γ h→µτ Γ SM h < 1.5% ,(65)M h→E kĒl =ū E k y L kl P L + y R kl P R v E l ,(66)
where u and v are the leptons' spinors and in the SM at tree level y L,R kl = δ kl m E k /v. Its decay rate is then
Γ h→E kĒl = m h 16π y L kl 2 + y R kl 2 ,(67)
where in the kinematic factor the lepton masses have been neglected compared to m h . Including the contributions of O LL , we have
y L kl = δ kl m E k v − m E k m 2 h 2Λ 2 v (∆ RL ) kl − m E k + m E l v 2Λ 2 ∆ (1) LL + ∆ (2) LL kl , y R kl = δ kl m E l v − m E l m 2 h 2Λ 2 v (∆ RL ) * lk .(68)
Assuming that either only O with ∆ (1) LL = ∆ (2) LL are contributing at a time and maximizing the relevant elements of A, which is in Eq. (43) for type I and Eq. (27) for type II, we get the numbers collected in Table VI for ξ 1 = ξ 4 = 0, which fullfil the conditions in Eq. (65). These results are much weaker than those in Table IV do not directly affect the dilepton Higgs decays, but contribute to the three-body lepton decays and hence can cancel the contributions of O (1,2) LL to the latter if both of them are present simultaneously, which implies the need for some degree of fine-tuning. Now, in calculating the numbers in Table VI, we treated the flavor-diagonal modes independently of the µ ± τ ∓ channels. If the requirements from h → µ − µ + , τ − τ + data, which led to the higher Λ min values in the table, are to be satisfied also by the contributions to h → µ ± τ ∓ , we find that Γ h→µτ /Γ SM h cannot be more than about 0.15%.
VI. CONCLUSIONS
The application of the MFV hypothesis to the lepton sector provides a framework for systematically analyzing the predictions of different models in which lepton-flavor nonconservation and CP violation arise from the leptonic Yukawa couplings. We have explored this in the simplest seesaw scenarios where neutrino mass generation is mediated by new fermion singlets (type I) or triplets (type III) and by a scalar triplet (type II). Taking a model-independent effective-theory approach, we consider the phenomenological implications by analyzing the contributions of new interactions that are organized according to the MFV hypothesis and consist of only a limited number of terms which have been resummed from an infinite number of them.
More specifically, we evaluate constraints on the MFV scale Λ associated with leptonic dipole operators from the latest experimental information on flavor-violating E k → E l γ decays, nuclear µ → e conversion, flavor-violating three-body decays of charged leptons, muon g − 2, and the electron's EDM. We find that the existing data, especially the bound on B(µ → eγ), can restrict the lower limit on Λ to over 500 TeV or more, depending on the details of the seesaw scheme. In types I and III, this corresponds to the new fermions responsible for the seesaw mechanism being superheavy, with masses roughly of order 10 15 GeV. On the other hand, it is interesting to point out that in type II, although the VEV of the scalar triplet needs to be v T ∼ 0.07 eV in our approach, its mass does not have to be 10 15 GeV and can be as low as a few hundred TeV. If M T is to be within LHC reach, in the TeV range, v T has to be of O(10 eV) instead. Another major difference between type I (or III) and type II is that in the former the Yukawa couplings of the new fermions contain features which can have substantially enhancing effects, including CP -violating ones, and which do not exist in type II.
Beyond the dipole operators, we look at additional dimension-six operators that involve only leptons or couple them to the SM gauge and Higgs bosons. Since these operators contribute to the flavor-changing three-body decays as well, the associated MFV scale must satisfy their experimental bounds. Based on the resulting strongest constraints, we estimate predictions on some of these processes which are markedly distinguishable from the corresponding predictions from the dipole operators.
It is interesting that some of the extra operators can also contribute to the flavor-conserving and flavor-violating leptonic decays of the Higgs boson and are therefore subject to constraints from future Higgs measurements at the LHC which will continue to improve in precision. Upcoming searches for other processes that violate lepton flavor will offer complementary tests on the different seesaw scenarios we have studied. The examples we have presented serve to illustrate the great importance of making such experimental efforts.
13 e −iδ −s 12 c 23 − c 12 s 23 s 13 e iδ c 12 c 23 − s 12 s 23 s 13 e iδ s 23 c 13 s 12 s 23 − c 12 c 23 s 13 e iδ −c 12 s 23 − s 12 c 23 s 13 e iδ c 23 c 13 ,
with eigenvalues 1 ,2, 3
13= 2Mm 1,2,3 /v 2 . Explicitly, including this in Eq. (40) yields (∆ ℓ ) 21 = −c 12 c 13 s 12 c 23 + e iδ c 12 s 23 s 13 ξ ℓ 2Â1 + ξ ℓ 4Â 2 1 + s 12 c 13 c 12 c 23 − e iδ s 12 s 23 s 13 ξ ℓ 2Â2 + ξ ℓ 4Â 2 2 + e iδ s 23 c 13 s 13 ξ ℓ 2Â3 + ξ ℓ ) 31 = c 12 c 13 s 12 s 23 − e iδ c 12 c 23 s 13 ξ ℓ 2Â1 + ξ ℓ 4Â 2 1 − s 12 c 13 c 12 s 23 + e iδ s 12 c 23 s 13 ξ ℓ 2Â2 + ξ ℓ 4Â 2 2
(∆ ℓ ) 32 = − s 12 c 23 + e −iδ c 12 s 23 s 13 s 12 s 23 − e iδ c 12 c 23 s 13 ξ ℓ 2Â1 + ξ
3212 c 23 − e −iδ s 12 s 23 s 13 c 12 s 23 + e iδ s 12 c 23 s 13 ξ ℓ 2Â2 + ξ
inferred from experimental upper-bounds on the branching ratios of flavor-violating leptonic transitions, as explained in the text. In this and the remaining tables, the unbracketed (bracketed) results correspond to the normal (inverted) hierarchy of light neutrino masses with m 1(3) = 0. (∆ ℓ ) 21,32,31 separately, while ensuring that the values of M make max  1 , 2 , 3 = 1. This allows us extract the maximal lower-limits onΛ = Λ/ ξ ℓ 2 1/2 from the measured bounds on B(µ → eγ), B(τ → µγ), and B(τ → eγ) listed in
(
∆ ℓ ) 21 = −c 12 c 13 s 12 c 23 + e iδ c 12 s 23 s 13 ξ ℓ 2Â1 + ξ ℓ 4Â 2 1 + s 12 c 13 c 12 c 23 − e iδ s 12 s 23 s 13 ξ ℓ 2Â2 + ξ
B
(τ − → e − e − e + ) 2.5 (5.7) × 10 −16 8.7 (4.7) × 10 −16 B(τ − → e − µ − µ + ) 4.3 (9.9) × 10 −17 15 (8.0) × 10 −17
FIG. 1 :
1Variation of the lower limit onΛ = Λ/|ξ ℓ 2 | 1/2 , subject to B(µ → eγ) data, versus complex-O parameter ρ = r 1 = r 2 = r 3 in the absence of the Majorana phases, α 1,2 = 0, for ξ ℓ 4 = 0 and degenerate ν k,R (solid curves), as explained in the text. The dashed curves depict the corresponding variation of the lower limit on Λ/|ξ ℓ 4 | 1/2 for ξ ℓ 2 = 0.
FIG. 2: Variation of the lower limit onΛ = Λ/|ξ ℓ 2 | 1/2 , subject to B(µ → eγ) data, versus α 2 for α 1 = 0, degenerate ν k,R , and complex-O parameter ρ = r 1 = r 2 = r 3 = −1 (solid curves), as explained in the text. The dashed curves depict the corresponding variation of the lower limit on Λ/|ξ ℓ 4 | 1/2 for ξ ℓ 2 = 0.
NH (IH) case. Then |d e | exp < 8.7 × 10 −29 e
B
(τ − → µ − e − e + ) 7.7 (14) × 10 −13 8.8 (16) × 10 −13 1.7 (1.3) × 10 −12 1.9 (1.5) × 10 −12 B(τ − → e − e − e + ) 4.3 (9.9) × 10 −14 4.3 (9.9) × 10 −14 15 (8.1) × 10 −14 15 (8.1) × 10 −14 B(τ − → e − µ − µ + ) 2.4 (5.5) × 10 −14 2.7 (6.3) × 10 −14 8.4 (4.5) × 10 −14 9.5 (5.1) × 10 −14
h
→ µ ∓ τ ± 0.08 (0.09) 0.24 (0.25) 0.09 (0.09) 0.25 (0.25)
[or (35)] can have manyParameter
NH
IH
sin 2 θ 12
0.308 ± 0.017
0.308 ± 0.017
sin 2 θ 23
0.437 +0.033
−0.023
0.455 +0.139
−0.031
sin 2 θ 13
0.0234 +0.0020
−0.0019
0.0240 +0.0019
−0.0022
δ/π
1.39 +0.38
−0.27
1.31 +0.29
−0.33
δm 2 = m 2
2 − m 2
1
7.54 +0.26
−0.22 × 10 −5 eV 2
7.54 +0.26
−0.22 × 10 −5 eV 2
∆m 2 = m 2
3 − m 2
1 + m 2
2 /2
(2.43 ± 0.06) × 10 −3 eV 2 (2.38 ± 0.06) × 10 −3 eV 2
TABLE I :
IResults of a recent analysis of global three-neutrino oscillation data
in order to maximizeObservable
Experimental
upper boundΛ
min /TeV
Types I & III
Type II
B(µ → eγ)
5.7 × 10 −13 [1]
338 (307)
294 (312)
B(µTi → eTi)
6.1 × 10 −13 [20]
85 (77)
73 (78)
B(µAu → eAu)
7.0 × 10 −13 [1]
80 (73)
70 (74)
B(µ − → e − e − e + ) 1.0 × 10 −12 [1]
81 (74)
70 (75)
B(τ → µγ)
4.4 × 10 −8 [1]
22 (24)
23 (23)
B(τ − → µ − µ − µ + ) 2.1 × 10 −8 [1]
5.6 (5.9)
5.9 (5.9)
B(τ − → µ − e − e + )
1.8 × 10 −8 [1]
8.7 (9.3)
9.2 (9.3)
B(τ → eγ)
3.3 × 10 −8 [1]
15 (13)
13 (13)
B(τ − → e − e − e + )
2.7 × 10 −8 [1]
4.9 (4.2)
4.3 (4.2)
B(τ − → e − µ − µ + )
2.7 × 10 −8 [1]
3.2 (2.7)
2.8 (2.7)
TABLE II :
IILower limits onΛ = Λ/ ξ ℓ
2
1/2 associated with dipole operators O
(e1,e2)
RL
TABLE III :
IIIPredictions calculated from the contributions of the dipole operators alone, with theΛ min numbers from the experimental bound on B(µ → eγ) inTable II and the neutrino parameter values used to determine them.
TABLE IV :
IVLL inferred from data on flavor-violating decays of charged leptons, as explained in the text. Only OLower limits onΛ = Λ/|ξ 2 | 1/2 associated with operators O
(1,2,3)
4L
and O
(1,2)
(1,2,3)
4L
or O
(1,2)
LL are
assumed to be present at a time.
Observable
Prediction
Types I & III
Type II
O
(1,2,3)
4L
O
(1,2)
LL
O
(1,2,3)
4L
TABLE V :
VPredictions calculated from the contributions of either O LL alone, with theΛ min numbers from the experimental bound on B(µ − → e − e − e + ) inTable IVand the neutrino parameter values used to determine them.(1,2,3)
4L
or O
(1,2)
where Γ SM h→ττ = 257 keV, Γ SM h→µμ = 894 eV, and Γ SM h = 4.08 MeV [33] are the SM widths for a Higgs mass m h = 125.1 GeV, which reflects the average of the newest measurements[28,34].One can write the amplitude for h → E − k E + l as
, but do not necessarily conflict with them. The reason is that O(1,2,3)
4L
TABLE VI :
VILower limits onΛ = Λ/|ξ 2 | 1/2 associated with operators O LL inferred from measurements on dilepton Higgs decays, as explained in the text. Only O LL are assumed to be contributing at a time.(e3)
RL and O
(1,2)
(e3)
RL or O
(1,2)
The presence of M ν breaks the global U(3) ν completely if M 1,2,3 are unequal and partially into O(3) ν if M 1,2,3are equal[4].
AcknowledgmentsThis research was supported in part by the MOE Academic Excellence Program (Grant No. 102R891505) and NSC of ROC and by NNSF (Grant No. 11175115) and Shanghai Science and Technology Commission (Grant No. 11DZ2260700) of PRC.
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"The Connection of Polaritons and Vacuum Rabi Splitting",
"The Connection of Polaritons and Vacuum Rabi Splitting"
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"David Snoke \nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n3941 O'Hara St15260PittsburghPA\n"
]
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"Department of Physics and Astronomy\nUniversity of Pittsburgh\n3941 O'Hara St15260PittsburghPA"
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| Polaritons, in particular microcavity exciton-polaritons, have attracted much attention in recent years, as the phenomena of Bose-Einstein condensation and superfluidity have been observed for these quasiparticles. While the basic physics of these systems is well understood, there has been confusion over the connection of these systems to other phenomena, namely the Jaynes-Cummings Hamiltonian, Rabi flopping, and the vacuum Rabi splitting of atoms in a cavity. This paper reviews the basic theory of polaritons and shows these connections explicitly. arXiv:1509.01468v1 [cond-mat.quant-gas] 4 Sep 2015 | null | [
"https://arxiv.org/pdf/1509.01468v1.pdf"
]
| 118,498,277 | 1509.01468 | 85b1dcf5a1bdcb4cc83443c453876f9f06fe9b21 |
The Connection of Polaritons and Vacuum Rabi Splitting
David Snoke
Department of Physics and Astronomy
University of Pittsburgh
3941 O'Hara St15260PittsburghPA
The Connection of Polaritons and Vacuum Rabi Splitting
Polaritons, in particular microcavity exciton-polaritons, have attracted much attention in recent years, as the phenomena of Bose-Einstein condensation and superfluidity have been observed for these quasiparticles. While the basic physics of these systems is well understood, there has been confusion over the connection of these systems to other phenomena, namely the Jaynes-Cummings Hamiltonian, Rabi flopping, and the vacuum Rabi splitting of atoms in a cavity. This paper reviews the basic theory of polaritons and shows these connections explicitly. arXiv:1509.01468v1 [cond-mat.quant-gas] 4 Sep 2015
Introduction
Numerous recent works [1,2,3,4] have reviewed the fascinating effects seen in experiments on Bose-Einstein condensation and superfluidity of microcavity exciton-polaritons. The field has progressed from simply observing evidence of Bose-Einstein condensation [5,6] to various effects such as Josephson junctions [7,8], phase locking of two condensates [9], superfluid motion past a barrier [10], and quantized circulation in a ring [11]. The lifetime of the polaritons in the cavities has steadily progressed, from about a picosecond in early experiments up to about 200 ps in present systems [12].
Amidst all this activity, some basic confusions remain about the nomenclature for the system and the connection to other systems with light-matter coupling. For example, does the "Rabi splitting" of exciton-polariton systems have anything to do with Rabi oscillations or vacuum Rabi splitting? We will see here that indeed these are variations of the same phenomenon. Similarly, is the term "strong coupling" used in the polariton field in the same way as in other fields of optics? Again, we will see that this is the case.
The basic concept of a polariton is simple. One starts with any oscillating dipole which can couple to the electromagnetic field. This can be, for example, an optical phonon in a solid, an exciton in a solid, or a two-level quantum oscillator consisting of two states in an atom. The coupling of the dipole oscillation to the electromagnetic field allows both radiation from the dipole or absorption of radiation by the dipole.
Phonon-polaritons
The polariton effect is most easily introduced by examining the case of phonon-polaritons, when an optical phonon in a solid couples to the electromagnetic field. In this section we review the standard theory for phonon-polaritons.
We start with the interaction energy of a dipole in an external electric field, H dipole = −q x· E. For a polarization field P = (N/V )q x, the total interaction energy is then
H int = − d 3 r P · E.(1)
The polarization, which is proportional to the local displacement, and the electric field can be written in terms of the phonon and photon operators. Dropping the vector notation for simplicity, the electric field is [13]
E(r) = −i k hω k 2 ∞ V a k e ik·r − a † k e −ik·r ,(2)
where ∞ is the dielectric constant of the medium not counting the contribution of the optical phonons, and the polarization field is [14]
P (r) = N V qx(r) = qN V h 2m r N ω 0 k c k e ik·r + c † k e −ik·r ,(3)
where we have used creation and destruction operators c † k and c k for the phonons, and we assume an optical phonon with constant frequency ω 0 . This may be written more simply as
P (r) = h ∞ Ω 2 2V ω 0 k c k e ik·r + c † k e −ik·r ,(4)
where Ω = q 2 N/ ∞ m r V . Substituting these formulas for E and P into H int , we obtain
H int = ih Ω 2 ω k ω 0 1 V k,k d 3 r a k e ik·r − a † k e −ik·r c k e ik ·r + c † k e −ik ·r .(5)
The integral of the exponential factors over r gives us a δ k,k which eliminates one momentum sum, so that we have the total Hamiltonian
H = k hω k a † k a k +hω 0 c † k c k + i 2h Ω ω k ω 0 a k c −k + a k c † k − a † k c k − a † k c † −k .(6)
This can be simplified by defining new operators which are linear superpositions of the creation and destruction operators which appear in the Hamiltonian. We define the new bosonic destruction operator ξ k = α k a k + β k c k + γ k a † −k + δ k c † −k and its Hermitian conjugate for the creation operator, with the coefficients α k . . . δ k chosen such that
H = kh ω ξ † k ξ k , which implies [ξ k , H] =hωξ k = α k [a k , H] + β k [c k , H] + γ k [a † −k , H] + δ k [c † −k , H] = α kh ω k a k − i 2h Ω ω k ω 0 c k − i 2h Ω ω k ω 0 c † −k +β kh ω 0 c k + i 2h Ω ω k ω 0 a k − i 2h Ω ω k ω 0 a † −k +γ kh −ω k a † −k − i 2h Ω ω k ω 0 c k − i 2h Ω ω k ω 0 c † −k +δ kh −ω 0 c † −k − i 2h Ω ω k ω 0 a k − i 2h Ω ω k ω 0 a † −k .(7)
The condition (7) is equivalent to the matrix equation
ω k i 2 Ω ω k /ω 0 0 − i 2 Ω ω k /ω 0 − i 2 Ω ω k /ω 0 ω 0 − i 2 Ω ω k /ω 0 0 0 − i 2 Ω ω k /ω 0 −ω k i 2 Ω ω k /ω 0 − i 2 Ω ω k /ω 0 0 − i 2 Ω ω k /ω 0 −ω 0 α k β k γ k δ k = ω α k β k γ k δ k .(8)
Following the standard diagonalization procedure of setting the determinant to zero, we have
ω 4 − ω 2 (ω 2 0 + ω 2 k ) + ω 2 k ω 2 0 − Ω 2 ω 2 k = 0.(9)
This is equivalent to the standard phonon-polariton equation [15]
ω 2 = c 2 k 2 (∞) ω 2 T − ω 2 ω 2 L − ω 2 ,(10)
with the bare photon frequency ω k = ck/ √ ∞ and ω 2 T = ω 2 L − Ω 2 , using
q 2 N mV = (∞)(ω 2 L − ω 2 T ).(11)
and our definition of Ω above. There are two positive-frequency solutions.
When ω k = ω 0 , it is easy to show that in the limit ω 0 Ω, the energies of the eigenstates arehω =hω 0 ±hΩ, and the corresponding eigenstates are ξ k = (a k ∓ ic k )/ √ 2. Away from the crossover region, when either ω k ω 0 or ω k ω 0 , the eigenmodes correspond to nearly pure photon a k and phonon c k operators.
Exciton-polaritons: ensemble of two-level oscillators (Frenkel limit)
We adopt the same picture as the phonon picture, but now imagine an ensemble of two-level electron oscillators at discrete locations i. The polarization of each oscillator is [16]
P i = qN V x i = qN V c| x|v b † iv b ic + c| x|v * b † ic b iv ,(12)
where we use b † in and b in for the fermionic creation and destruction operators of the electrons, and n = c, v represent the conduction and valence bands, respectively; since in the Frenkel picture, each oscillator has no spatial overlap with the others, we could equally well call these e and g for the excited and ground states of each oscillator, as in the case of an ensemble of atoms.
We define the exciton operator in the Frenkel limit as
C † k = 1 √ N i e ik·ri b † ic b iv .(13)
The inverse Fourier transform is
b † ic b iv = 1 √ N k e −ik·ri C † k ,(14)
so we have
P i = −i qN V 1 mω 0 √ N k p cv C k e ik·ri − p * cv C † k e −ik·ri ,(15)
where we write p cv = c| p|v = imω 0 c| x|v [17]. We then have
H int = − d 3 r P · E = hω k 2 V qN V 1 mω 0 √ N k,k d 3 r a k e ik·r − a † k e −ik·r p * cv C † k e −ik ·r − p cv C k e ik ·r =h 2 q 2 N mV 2 mhω 0 ω k ω 0 k −p cv a k C −k + p * cv a k C † k + p cv a † k C k − p * cv a † k C † −k .(16)
As for the phonon-polaritons, we write
ξ k = α k a k + β k C k + γ k a † −k + δ k C † −k . Then H = kh ωξ † k ξ k = k (hω k a † k a k +hω 0 C † k C k ) + H int(17)
and
[ξ k , H] =hωξ k = α k [a k , H] + β k p cv [C k , H] + γ k [a † −k , H] + δ k p * cv [C † −k , H] =hα k ω k a k + Ω 2 ω k ω 0 C k − Ω * 2 ω k ω 0 C † k +hβ k ω 0 C k + Ω * 2 ω k ω 0 a k − Ω * 2 ω k ω 0 a † −k +hγ k −ω k a † −k + Ω 2 ω k ω 0 C k − Ω * 2 ω k ω 0 C † −k +hδ k −ω 0 C † −k + Ω 2 ω k ω 0 a k − Ω 2 ω k ω 0 a † −k ,(18)
with
Ω = q 2 N mV F(19)
and F = p cv 2/mhω 0 ; |F | 2 is the oscillator strength. (We have used the approximation that the C k operators are purely bosonic.) This is equivalent to the matrix equation
ω k 1 2 Ω * ω k /ω 0 0 1 2 Ω ω k /ω 0 1 2 Ω ω k /ω 0 ω 0 1 2 Ω ω k /ω 0 0 0 − 1 2 Ω * ω k /ω 0 −ω k − 1 2 Ω ω k /ω 0 − 1 2 Ω * ω k /ω 0 0 − 1 2 Ω * ω k /ω 0 −ω 0 α k β k γ k δ k = ω α k β k γ k δ k . (20) The determinant is ω 4 − ω 2 (ω 2 0 + ω 2 k ) + ω 2 k ω 2 0 − |Ω| 2 ω 2 k = 0.(21)
This is equivalent to the standard polariton equation
ω 2 = c 2 k 2 ∞ ω 2 T − ω 2 ω 2 0 − ω 2(22)
with ω 2 T = ω 2 0 − Ω 2 and ω k = ck/ √ ∞ . Cavity Polaritons. In a cavity we use ω k = (c/n) k 2 + k 2 ⊥ ) (c/n)(1 + k 2 /2k ⊥ ), with k ⊥ = π/L. For the resonant case we set ω k = (c/n)π/L = ω 0 . Then the determinant equation is
ω 4 − 2ω 2 ω 2 0 + ω 2 0 (ω 2 0 − |Ω| 2 ) = 0,(23)
which has the solutions
ω 2 = ω 2 0 ± ω 0 Ω,(24)
or
ω = ω 0 1 ± Ω/ω 0 ω 0 ± Ω 2 ,(25)
where the final approximation is valid when Ω ω 0 . These correspond to the lower polaritons and upper polaritons so familiar in microcavity polariton optics. There will also be "dark," uncoupled states at ω 0 which lie in the middle of the upper-lower polariton splitting. This uncoupled ω 0 was ω L in the case of the phonon-polaritons.
Exciton-polaritons: Wannier picture
The Wannier limit of excitons consists of the case in which the electron and hole (empty electron state) are no longer confined to the same oscillator, but instead, due to coupling between the oscillators, the electron and hole can migrate to different oscillators. In this case there will be a Coulomb attraction between the free electron and the hole, which effectively acts as a particle with positive charge. This leads to bound states of the free electron and hole that are exactly the same as the Rydberg bound states of a hydrogen atom, but with the energy scaled by the dielectric constant of the medium.
For this calculation we use the interaction Hamiltonian in k-space instead of in real space:
H int = − q m k,k h 2 V ω k c| p|v a k b † c,k +k b vk + a † k b † c,k −k b v,k + + c| p|v * a k b † v,k +k b ck + a † k b † v,k −k b c,k .(26)
The exciton creation operator in the Wannier case is [18]
C † k = k φ(k/2 − k )b † c,k−k b v,−k .(27)
The Fourier transform of the 1s wave function is
φ(k) = 1 √ V 8 √ πa 3 (1 + a 2 k 2 ) 2 ,(28)
where a is the exciton Bohr radius. It is not easy to invert this to write the Hamiltonian in terms of the exciton operators. Therefore instead of an exact diagonalization, we write a matrix on the states |ex = C † k |0 and |phot = a † k |0 . The off-diagonal term is
ex|H|phot = − 0| k φ(k/2 − k )b † v,−k b c,k−k q m k ,k h 2 V ω k (29) × c| p|v a k b † c,k +k b vk + a † k b † c,k −k b v,k(30)+ c| p|v * a k b † v,k +k b ck + a † k b † v,k −k b c,k a † k |0 = − 0| k φ(k/2 − k )b † v,−k b c,k−k (31) × q m k h 2 V ω k c| p|v b † c,k +k b vk + c| p|v * b † v,k +k b ck |0 = − k φ(k/2 + k ) q m h 2 V ω k c| p|v .(32)
The sum over k is
V (2π) 3 2πk 2 dk d(cos θ) 1 √ V 8 √ πa 3 (1 + a 2 | k/2 + k | 2 ) 2 = √ V (2π) 3 2πk 2 dk d(cos θ) 8 √ πa 3 (1 + a 2 ( 1 4 k 2 + k 2 + kk cos θ)) 2 = √ V (2π) 3 ∞ 0 2πk 2 dk 32 √ πa 3 2 + 1 8 a 4 (k 2 − 4k 2 ) 2 + a 2 (k 2 + 4k 2 ) = √ V √ πa 3 .(33)
Therefore ex|H|phot =h 2
1 √ πa 3 q 2 m ω 0 ω k c| p|v 2 mhω 0 =h 2 Ω ω 0 ω k ,(34)
where Ω = q 2 mπa 3 F
and F is defined as above for the oscillator strength.
We thus have the matrix ex|H|ex ex|H|phot phot|H|ex phot|H|phot =h ω 0
1 2 Ω ω 0 /ω k 1 2 Ω * ω 0 /ω k ω k(36)
which has the determinant equation
ω 0 ω k − ω(ω k + ω 0 ) + ω 2 − 1 4 |Ω| 2 ω 0 /ω k = 0.(37)
This has the same behavior as the previous determinant equation (21) for the exact diagonalization, except near ω k = 0 where it breaks down. See Figures 1 and 2. Note that the Rabi frequency (35) depends on the exciton Bohr radius a through the volume πa 3 . The Frenkel limit can be viewed as the Wannier picture of excitons in the case when the exciton Bohr radius becomes equal to the unit cell size of the underlying crystal. This, in the Frenkel limit, the volume πa 3 becomes the unit cell size a 3 L = V /N , in which case the Rabi frequency (35) of the Wannier limit becomes exactly the same as (19) in the Frenkel limit.
Comparison to atomic vacuum Rabi splitting
As mentioned above, the Frenkel exciton limit is no different from the case of an ensemble of isolated atoms. We can therefore compare the case of an ensemble of atoms in vacuum to the above results with just a small change in notation.
The standard Hamiltonian for vacuum Rabi splitting with a two-level atom is [19]
H = ih ω 0 b † ic b iv +hω k a † k a k + ih g(a k b † ic b iv + a † k b † iv b ic ),(38)
where
g = ω 0 d 2 2h V = ω 0 q 2 x 2 2h V = 1 2 q 2 mV 2 p 2 mhω 0 ,(39)
and we have again used p = imω x . The Hamiltonian (38) is equivalent to (16) with the rescaling of the photon operator
a k → ie ik·r a k ,(40)
assuming p cv is real, and dropping the two terms a k b ic b † iv and a † k b iv b † ic , which is comparable to the approximation made in Section 4, which is valid when ω 0 Ω. Agerwal [19] found that the splitting is
Ω = 2g √ N ,(41)
which is exactly the same as found for our definition (19) for the Rabi splitting of a two-level Frenkel exciton system. It really is the same system, namely an ensemble of independent twolevel oscillators coupled only by the electromagnetic field, just solved in a different way. It may at first seem as though the result (35) is not the same as the coupling (41), since the latter is proportional to √ N , where N is the number of atoms, while (35) is not proportional to the number of excitons. It is important to keep in mind, however, that the number of oscillators in an excitonic system is not given by the number of excitons, but by the number of atoms in the underlying medium. In the Frenkel case, the number of oscillators is exactly the number of atoms, while in the Wannier case, the effective number of oscillators is reduced by the ratio (a L /a) 3 , since each Wannier exciton is spread over many lattice sites.
Final Remarks
We have seen that the description of the interaction of electromagnetic field with all three types of electronic oscillation, namely Frenkel excitons, Wannier excitons, and exceptions of atomic states, can be described by the same formalism. The coupling term in each case can be viewed as the natural unit of frequency for the susceptibility of an ensemble of classical oscillators [20] g ∼ q 2 mV osc ,
where V osc is the volume per oscillator. In each of the cases, "strong coupling" can be defined as the limit when the Rabi frequency Ω is much greater than γ = 1/τ , where τ is the decay time.
We have also seen that the standard approximation using a 2 × 2 matrix to represent the mixing of the photon states and electronic excitation is only valid in the limit when ω Ω. When Ω is comparable to ω, sometimes called the "ultrastrong" coupling limit [21], the full Hamiltonian must be used.
The use of the term "vacuum Rabi frequency" for the state splitting Ω is connected to the standard Rabi frequency. The standard Rabi frequency corresponds to the rate at which an electronic oscillator will flip states in the presence of a classical driving field, and is given by [22]
ω R = q|p cv | mhω 0 E 0 ,(43)
where E 0 is the classical field amplitude. To get the vacuum Rabi frequency, we use the amplitude of the field which corresponds to the average electric field amplitude in a vacuum. From (2) we have
|E k | 2 =h ω 0 2 V 2a † k a k + 1 ,(44)
which for a vacuum gives
|E k | = hω 0 2 V .(45)
Then (43) becomes
ω R = q|p cv | mhω 0 hω 0 2 V = 1 2 q 2 m 2|p cv | 2 mhω 0 = 1 2 Ω,(46)
which is the same as the coupling term we have used above. One of the important physical implications of these calculation is that the coupling energy for exciton-polaritons does not depend on the number of polaritons, to lowest order. The number of excitons is not analogous to the number of atoms in the atomic case; rather, the number of excitons corresponds to the fraction of atoms in excited states. At high density, the number of excitons can affect the splitting between the upper and lower polariton branches, giving an effectively reduced coupling Ω, due to phase space filling, which reduces the oscillator strength, which comes about when we can no longer use the approximation that the exciton operators are bosonic, as we did above. This has been seen [23] to lead to a collapse of a microcavity system to weak coupling, leading to standard lasing.
Figure 1 :
1Plot of the solutions of Eq. (37) for ω 0 = 1, Ω = 0.1.
Figure 2 :
2Plot of the solutions of Eq. (21) for ω 0 = 1, Ω = 0.1.
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. S Gambino, M Mazzeo, A Genco, O Di Stefano, S Savasta, S Patane, D Ballarini, F Mangione, G Lerario, D Sanvitto, G Gigli, ACS Photonics. 11042S. Gambino, M. Mazzeo, A. Genco, O. Di Stefano, S. Savasta, S. Patane, D. Ballarini, F. Mangione, G. Lerario, D. Sanvitto, and G. Gigli, ACS Photonics 1, 1042 (2014).
D W Snoke, Solid State Physics: Essential Concepts. Pearson/Addison-WesleySection 9.3D.W. Snoke, Solid State Physics: Essential Concepts, (Pearson/Addison-Wesley, 2009), Section 9.3.
. R Balili, B Nelsen, D W Snoke, L Pfeiffer, K West, Physical Review B. 7975319R. Balili, B. Nelsen, D.W. Snoke, L. Pfeiffer, and K. West, Physical Review B 79, 075319 (2009).
| []
|
[
"New Sequence Sets with Zero-Correlation Zone",
"New Sequence Sets with Zero-Correlation Zone"
]
| [
"Xiangyong Zeng [email protected]. \nFaculty of Mathematics and Computer Science\nDepartment of Electrical and System Engineering\nLiu is with\nHubei University\n11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q\n",
"Lei Hu \nFaculty of Mathematics and Computer Science\nDepartment of Electrical and System Engineering\nLiu is with\nHubei University\n11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q\n",
"MemberQingchong Liu \nFaculty of Mathematics and Computer Science\nDepartment of Electrical and System Engineering\nLiu is with\nHubei University\n11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q\n",
"X Zeng \nOakland University\n48309RochesterMIUSA\n"
]
| [
"Faculty of Mathematics and Computer Science\nDepartment of Electrical and System Engineering\nLiu is with\nHubei University\n11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q",
"Faculty of Mathematics and Computer Science\nDepartment of Electrical and System Engineering\nLiu is with\nHubei University\n11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q",
"Faculty of Mathematics and Computer Science\nDepartment of Electrical and System Engineering\nLiu is with\nHubei University\n11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q",
"Oakland University\n48309RochesterMIUSA"
]
| []
| A method for constructing sets of sequences with zero-correlation zone (ZCZ sequences) and sequence sets with low cross correlation is proposed. The method is to use families of short sequences and complete orthogonal sequence sets to derive families of long sequences with desired correlation properties. It is a unification of works of Matsufuji and Torii et al., and there are more choices of parameters of sets for our method. In particular, ZCZ sequence sets generated by the method can achieve a related ZCZ bound. Furthermore, the proposed method can be utilized to derive new ZCZ sets with both longer ZCZ and larger set size from known ZCZ sets. These sequence sets are applicable in broadband satellite IP networks.Index Termszero-correlation zone (ZCZ), low correlation, perfect sequence, orthogonal sequence, broadband satellite IP networks. | null | [
"https://arxiv.org/pdf/cs/0508115v1.pdf"
]
| 6,428,998 | cs/0508115 | 20dc6d81376c15280a7c0b8676a889461dceb1f1 |
New Sequence Sets with Zero-Correlation Zone
25 Aug 2005 February 1, 2008
Xiangyong Zeng [email protected].
Faculty of Mathematics and Computer Science
Department of Electrical and System Engineering
Liu is with
Hubei University
11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q
Lei Hu
Faculty of Mathematics and Computer Science
Department of Electrical and System Engineering
Liu is with
Hubei University
11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q
MemberQingchong Liu
Faculty of Mathematics and Computer Science
Department of Electrical and System Engineering
Liu is with
Hubei University
11 Xueyuan Road430062, 100049Wuhan, BeijingP. R. China, P. R. China. Q
X Zeng
Oakland University
48309RochesterMIUSA
New Sequence Sets with Zero-Correlation Zone
25 Aug 2005 February 1, 2008arXiv:cs/0508115v1 [cs.IT] 1 DRAFT 2
A method for constructing sets of sequences with zero-correlation zone (ZCZ sequences) and sequence sets with low cross correlation is proposed. The method is to use families of short sequences and complete orthogonal sequence sets to derive families of long sequences with desired correlation properties. It is a unification of works of Matsufuji and Torii et al., and there are more choices of parameters of sets for our method. In particular, ZCZ sequence sets generated by the method can achieve a related ZCZ bound. Furthermore, the proposed method can be utilized to derive new ZCZ sets with both longer ZCZ and larger set size from known ZCZ sets. These sequence sets are applicable in broadband satellite IP networks.Index Termszero-correlation zone (ZCZ), low correlation, perfect sequence, orthogonal sequence, broadband satellite IP networks.
I. INTRODUCTION
Recent practice in broadband satellite IP networks has demanded sequences of low correlation in a small detection aperture [1]- [3]. In a broadband satellite IP network, there is always a control channel broadcasting the system time along with other system information [1], [2]. A user terminal listens to the control channel, adjusts its timing accordingly so that its packet arrives at the central receiver within a few symbols [1]. Because a broadband satellite IP network has to support a very large number of terminals, which can be over 10 million, there can be many simultaneous transmissions from different terminals [1], [2]. Sequences need to be designed as signature sequences to differentiate terminals, which are often called unique words [4]. Such sequences need to have low values of both autocorrelation and crosscorrelation within the detection aperture. These requirements translate to the design of sequences of zero-correlation zone (ZCZ), or sequences of low correlation in an aperture.
Polyphase ZCZ sequence sets were constructed by Suehiro for the first time [5]. The concept of zero-correlation zone (ZCZ) was introduced in [6]. Several classes of ZCZ sequences were derived based on complementary pairs [5], [7], [8]. The construction was extended in [9], [10] by employing complementary sets. Binary sequence sets make hardware implementation much easier than polyphase sequences. However, binary ZCZ sequence sets can not achieve the upper bound of set size [11]. Sets of ternary ZCZ sequences with entries in {±1, 0} can achieve the upper bound. Their hardware implementation is similar to binary sequences. Several classes of ternary ZCZ sequences have been constructed [12]- [16]. Important progress was also achieved in the design of sequences using short sequences to construct a set of long sequences with the desired correlation properties [17]- [20]. By extending the concept of ZCZ to the two-dimensional case, families of ZCZ arrays where the one-dimensional ZCZ becomes a rectangular ZCZ can also be synthesized [21]- [23]. More references can be found in [24]. In addition, some polyphase sequences with low correlation zone have been constructed in [25]- [27].
In this correspondence, we present a general method for constructing sets of ZCZ sequences and sequence sets with low cross correlation. The original idea of this method was proposed by Gong in 1995 [17], where she obtained new low-correlation sequence sets by interleaving two ideal sequences with two-level periodic autocorrelation property [18]. Different from Gong's approach, we use two sets of sequences to replace Gong's ideal sequences and their shift equivalent sequences. One set of sequences is complete orthogonal, while the other set can consist of either shift equivalences of a fixed perfect sequence or sequences inequivalent and with some good correlation properties.
Our method also unifies the constructions presented in [19], [20]. The ZCZ sequence sets constructed in [19], [20] can be constructed by our method. Furthermore, there are more choices of parameters of sequence sets for our method. In particular, ZCZ sequence sets generated by our method can achieve the bound of set size, while those constructed in [20] can not. Furthermore, our method can be utilized to derive new ZCZ sets with both longer ZCZ and larger set size from known ZCZ sets.
The remainder of this correspondence is organized as follows. Section II gives some definitions and introduce a basic method for constructing sequences. Sets of sequences with desired correlation properties are constructed by using the proposed method in Section III and IV. Section V concludes the study.
II. PRELIMINARIES AND A BASIC CONSTRUCTION METHOD
Throughout the correspondence all sequences, except shift sequences, have entries in the complex field.
A. Some Definitions
Let S = {s 0 , s 1 , · · · , s M −1 } be a set of M sequences of length N, where s h = (s h,0 , s h,1 , · · · , s h,N −1 ) for 0 ≤ h < M.(1)
The periodic cross-correlation function R s h ,s k (τ ) of s h and s k ,
0 ≤ h, k ≤ M − 1, is defined as R s h ,s k (τ ) = N −1 i=0 s h,i · s * k,i+τ , τ = 0, 1, · · ·(2)
where the symbol * denotes a complex conjugate and the subscript addition is performed modulo
N. If h = k, then R s h ,s k (τ ) is called the periodic autocorrelation function of s h , denoted by R s h (τ ).
The energy of s h is defined as
E s h = N −1 i=0 |s h,i | 2 .(3)s h is called a perfect sequence if R s h (τ ) = E s h , if τ ≡ 0 mod N 0, otherwise(4)
The sequence s h is called a p-phase sequence if for any
0 ≤ i ≤ N − 1, s h,i = exp( 2l i πj p ) for some 0 ≤ l i < p, where j 2 = −1.
In the case of p = 2, the sequence is called a binary sequence.
The sequence set S is said to be orthogonal if the set has the following characteristic:
R s h ,s k (τ ) = E s h , if h = k, τ = 0 0, if h = k, τ = 0(5)
for any 0 ≤ h, k < M. In the case of N = M, the set S is said to be complete orthogonal.
S is said to be an (N, M; Zcz)-ZCZ set and has a zero-correlation zone of size Zcz if
R s h ,s k (τ ) = E s h , if h = k, τ = 0 0, if h = k, τ = 0 0, if 1 ≤ |τ | ≤ Zcz(6)
for any 0 ≤ h, k < M. Thus, for any s h , s k in S, R s h (τ ) = 0 for 1 ≤ |τ | ≤ Zcz and R s h ,s k (τ ) = 0 for 0 ≤ |τ | ≤ Zcz and h = k.
Such a set is also said to be Z-orthogonal [10].
A mathematical bound
Zcz ≤ N M − 1(7)
holds for an arbitrary (N, M; Zcz)-ZCZ set [11]. It is derived from the Welch lower bound [28], and is later introduced by Matsufuji et al. with a more simple proof [19].
Set δ = max|R s h ,s k (τ )|,(8)
where 0 ≤| τ |< N, 0 ≤ h, k < M and the maximization excludes the case of τ = 0 and h = k.
We call δ the maximal correlation of S and S an [N, M; δ] sequence set.
For the sequence s h = (s h,0 , s h,1 , · · · , s h,N −1 ), the left shift operator L on s h is defined as
L(s h ) = (s h,1 , · · · , s h,N −1 , s h,0 ).(9)
February 1, 2008 DRAFT For any i > 0, iteratively define
L i (s h ) = L(L i−1 (s h ))(10)
where L 0 (s h ) = s h . Sequences s h and s k are called (cyclically) shift equivalent if there exists an integer k such that s h = L k (s k ).
For a sequence set A = {a 0 , a 1 , · · · , a n−1 }, define
L i (A) = {L i (a 0 ), L i (a 1 ), · · · , L i (a n−1 )}.(11)
Two sequence sets A and B are called shift equivalent if there exists an integer i such that
A = L i (B)
. When all sequences in the set A are shift equivalent to a fixed sequence a, namely,
A = {L e 0 (a), L e 1 (a), · · · , L e n−1 (a)},(12)
the integer sequence e = (e 0 , e 1 , · · · , e n−1 ) is called a shift sequence.
The following notations are used in the rest of this correspondence:
• m|n: the integer n is a multiple of m;
• gcd(p, q): the greatest common divisor of integers p and q.
• ⌊z⌋: the largest integer not exceeding z.
B. A Basic Construction of Sequence Set
A basic method to construct a sequence set is given in the following procedure.
Procedure 1:
(1) Let A ={a 0 , a 1 , · · · , a n−1 } be an ordered set of sequences, where a i = (a i,0 , a i,1 , · · ·,
a i,m−1 ). Choose a complete orthogonal sequence set B = {b 0 , b 1 , · · · , b n−1 }, where b i = (b i,0 , b i,1 , · · ·, b i,n−1 ).
(2) Let U be the m × n matrix whose j-th column sequence is a j . Listing all entries of U row by row (from left to right and from top to bottom), we obtain a sequence of length mn, u = (u 0 , u 1 , · · · , u mn−1 ),
which is called the sequence associated with the ordered set A, and U is called its matrix form.
s h,i = u i b h,imodn(14)
for 0 ≤ i < mn. The original idea of this construction was proposed by Gong in 1995 [17], and later she employed it to construct more sequence sets [18]. In her construction, she assumes that Eq. (12) and Eq. (16) below
B = {b, L 1 (b), · · · , L n−1 (b)}(16)
hold for two ideal two-level autocorrelation sequences a and b and a shift sequence e = (e 0 , e 1 , · · · , e n−1 ).
A similar construction was given by Torii et al. in [20], where the divisibility condition n|m or m|n was required. However, none of the sequence sets generated by that construction achieves the bound (7). Recently, a new method for generating sets of polyphase ZCZ sequences achieving the bound (7) was given in [19].
Procedure 1 unifies the constructions in [19], [20]. In Sections III and IV, we concentrate on constructing sequence sets with good correlation property.
The following formula on correlation of sequences constructed from Procedure 1 is a basis on which we prove the results in the correspondence.
Let 0 ≤ τ < mn and write τ = rn + s with 0 ≤ r < m and 0 ≤ s < n.
Proposition 1: For 0 ≤ h, k < n,
R s h ,s k (τ ) = n−1 j=0 d i R a i ,a s+i−ϕ(s+i)n (r + ϕ(s + i))(17)
where d i = b h,i b * k,s+i−ϕ(s+i)n and ϕ(s + i) is 0 if s + i < n and is 1 otherwise. The proof of this proposition is presented in Appendix I.
III. A CLASS OF ZCZ SEQUENCE SETS
In this section, we first use the procedure in Section II-B to construct new ZCZ sequences.
Next, we give a brief comment on some known constructions of perfect sequences and complete orthogonal sequence sets. Finally, we study how to construct sets of long ZCZ sequences with both longer ZCZ and larger set size, based on known sets of short ZCZ sequences.
A. Construction of ZCZ Sequences Based on Perfect Sequences
In this subsection all sequences of the set A are assumed to be shift equivalent, i.e., Eq. (12) holds. In this case, Eq. (17) is further simplified to
R s h ,s k (τ ) = n−1 i=0 d i R a i ,a s+i−ϕ(s+i)n (r + ϕ(s + i)) = n−1 i=0 d i R a (e i+s − e i + r)(18)
where we define e i = 1 + e i−n for subscript i ≥ n.
By Eq. (18), if Eq. (16) holds, and if both a and b have an ideal two-level autocorrelation function, then the correlation value R s h ,s k can be completely determined by the shift sequence e = (e 0 , e 1 , · · · , e n−1 ). Based on this nice observation, Gong [18] presented two methods to construct the shift sequences with desirable combinational property, which guarantees the constructed signal sets have low correlation property.
Gong's idea is extended to construct families of ZCZ sequences in this subsection. If a is a perfect sequence and B is complete orthogonal, then we can choose the shift sequences such that the sequences sets as described in Theorems 1 and 2 have desirable ZCZ property.
Set r 0 = min{m + e 0 − e n−1 , e i+1 − e i − 1 | 0 ≤ i ≤ n − 2}.(19)
Theorem 1: Let m ≥ n and 0 ≤ e 0 < e 1 < · · · < e n−1 < m.
If a = (a 0 , a 1 , · · · , a m−1 ) is a perfect sequence, then S is an (mn, n; r 0 n + n − 2)-ZCZ set.
Furthermore,
(1) When n|(m + 1) and e i = i(m+1) n for 0 ≤ i < n, S is an (mn, n; m − 1)-ZCZ set.
(2) When n|m and
(i 1 − i 2 )m n ≤ e i 1 − e i 2 ≤ (i 1 − i 2 )m n + 1(20)
for any 0 ≤ i 2 < i 1 ≤ n − 1, S is an (mn, n; m − 2)-ZCZ set. Each of the shift sequences e (0) , · · · , e (n−2) , and e (n−1) , defined by
e (i) = (0, m n , · · · , (n − i − 1)m n , (n − i)m n + 1, (n − i + 1)m n + 1, · · · , (n − 1)m n + 1),(21)
satisfies Eq. (20).
The proof of this theorem is presented in Appendix II.
When n|(m + 1), we obtain ZCZ sequence sets which achieve the bound (7) by Theorem 1(1). The polyphase ZCZ sets constructed in [6] also attain this bound, however, the number of phases is equal to the length of the sequences in the set. Applying Theorem 1(1), we can obtain polyphase ZCZ sets with fewer number of phases than the sequence length. Precisely, if a and those sequences in B are of pand q-phase respectively, then the sequences in S are of pq gcd(p,q)phase. This number of phases may be independent on the sequence length. A disadvantage is that the condition n|m + 1 restricts us in the construction of binary, three-phase, or quadriphase ZCZ sequence sets according to Theorem 1(1).
When n|m, the ZCZ sets constructed by Theorem 1(2) do not achieve the bound (7). The absolute values of cross-correlation functions R s 0 ,s 1 (τ ) and R s 1 ,s 0 (τ ) are given by |R s 0 ,s 1 (τ )| = |R s 1 ,s 0 (τ )| = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).
Example 3:
Consider quadriphase sequence sets. Let 0, 1, 2 and 3 represent +1, +j, −1 and −j, respectively. In the above terminology, the (64, 4; 14)-ZCZ set C 1 in P. 561 of [20] is constructed from the perfect quadriphase sequence a = (0000012302020321), a complete orthogonal se- and Theorem 2: Let m ≥ n, a = (a 0 , a 1 , · · · , a m−1 ) be a perfect sequence, and e = (e 0 , e 1 , · · · , e n−1 ) be a shift sequence defined by e i = m − 1 − i. Then S is an (mn, n; n mod m)-ZCZ set.
quence set B = {0000,|R s 0 ,s 1 (τ )| = |R s 1 ,s 0 (τ )| = |R s 0 ,s 3 (τ )| = |R s 3 ,s 0 (τ )| = |R s 1 ,s 2 (τ )| = |R s 2 ,s 1 (τ )| = |R s 2 ,s 3 (τ )| = |R s 3 ,s 2 (τ )| = (0,|R s 0 ,s 2 (τ )| = |R s 2 ,s 0 (τ )| = |R s 1 ,s 3 (τ )| = |R s 3 ,s 1 (τ )| = (0,
Furthermore, when m = n + 1, S is an (mn, n; m − 1)-ZCZ set.
The proof of Theorem 2 is presented in Appendix III.
For m and n, by choosing appropriate shift sequences, we can construct more sequence sets with parameters different from those constructed in [20].
Example 4: Suppose m = 16, n = 12, e = (15,14,13,12,11,10,9,8,7,6,5,4), and a is the perfect sequence shown in Example 3. The complete orthogonal sequence set B is supposed to consist of row sequences of the following matrix:
+ + − + + + − − − + − − + − + + + − − − + − − + + + + + − − − + − − + − + + + − − − + − − + − + + + − − − + − − + − + + + − − − + − − + − + + + + − − + − − + − + + + − + − + − − + − + + + − − + + − − + − + + + − − − + − − + − + + + − − − + + − + − + + + − − − + − + + + + + + + + + + + +
where the symbols + and − represent +1 and −1, respectively. By Procedure 1, we obtain a quadriphase (192, 12; 12)-ZCZ set. Since the length of these sequences is very long and the set size is large, we only list the first two sequences in S as follows: Other pairs of sequences in S have the same cross-correlation property concerning the ZCZ.
B. Perfect Sequences and Orthogonal Sequence Sets
The construction presented in III-A depends on the existence of perfect sequences and complete orthogonal sequence sets. In this subsection, we give a comment on their existence.
For an arbitrary integer n ≥ 3, there is always a perfect sequence of length n [29], and for some special n, there are many methods to construct them [30], [31]. For ternary (but not three-phase) sequences with entries in {0, ±1}, a lot of perfect sequences have been constructed [32], [33], [34], [35], [36]. More generally, in [37] an analytical method is proposed to find unimodular perfect sequence of any length.
For binary sequence set, the case is different. The following formula (22) The proof of this proposition is presented in Appendix IV.
Orthogonal sequence sets can be derived from unitary matrices and Hadamard matrices [20].
Hadamard matrices are a type of square {−1, 1}-matrices and exist for many orders n. Let q = 0 or q be an odd prime, and t and l be arbitrary positive integers such that n = 2 l (q t +1) ≡ 0 mod 4.
Then there is always a Hadamard matrix of order n, and it can be constructed by Paley's method.
A unitary matrix can be obtained by multiplying each entry of a Hadamard matrix of order n by a same number 1 √ n . For an arbitrary positive integer n, a ternary complete orthogonal set of n sequences can be constructed from Hadamard matrices [38].
Thus, although it can generate low-phase ZCZ sets, the method presented in the previous subsection is more suitable for constructing high-phase and ternary ZCZ sets.
C. Constructing Sets of ZCZ Sequence With Longer ZCZ and Larger Set Size
Due to the sparsity of known low-phase perfect sequences, the low-phase ZCZ sets constructed in Section III-A do not possess a longer ZCZ and a larger family size simultaneously. In this subsection, we utilize the basic construction in Procedure 1 to present two methods, by combining which we can obtain ZCZ sets with both longer ZCZ and larger family size from known ZCZ sets.
Let d be a fixed positive integer and C be a set of l sequences c k (0 ≤ k < l), where c k = (c k,0 , c k,1 , · · · , c k,m−1 ). For each given c k , we define an ordered sequence set A k = {a k 0 , a k 1 , · · · , a k n−1 } where a k i = (a k i,0 , · · · , a k i,m−1 ) and a k i,j = c k,(j(n+d)+i+d⌊ i+1 n ⌋)modm .
By Procedure 1, for any 0 ≤ k < l, we use A k and a fixed complete orthogonal set B of n sequences to produce a sequence set S k . We combine the sets S 0 , S 1 , · · · , S l−1 to get a larger set 0≤k<l S k , which has ln sequences.
Theorem 3: Let C be an (m, l; Zcz)-ZCZ set, d be an integer with 0 ≤ d < Zcz, and m be relatively prime to n + d. Set
r 0 = ⌊ Zcz−d n+d ⌋; s 0 = min{n − 1, Z CZ − d − (n + d)r 0 }.(24)
Then 0≤k<l S k is an (mn, ln; r 0 n + s 0 )-ZCZ set. Furthermore, (1) Assume d = 0 and gcd(m, n) = 1. Then 0≤k<l S k is an (mn, ln; Zcz)-ZCZ set.
(2) Assume l = 1 and Zcz = m − 1, i.e., C consists of a single perfect sequence. If n > Zcz, then S 0 is an (mn, n; m − 1 − d)-ZCZ set.
The proof of this theorem is presented in Appendix V.
The hypothesis on the relation between m and n in Theorem 3 is very weak, so it is convenient to apply the theorem to construct large ZCZ sets from the small ones. For the case that m and n are relatively prime, we can take d = 0, thus getting an (mn, n; m − 1)-ZCZ set which achieves the bound (7) can be constructed by applying Theorem 3(2).
Example 5:
Suppose that m = 8, n = 12, d = 1, a perfect quadiphase sequence a = (01022122) whose entries represent the same elements as those in Examples 3 and 4, and B is the complete orthogonal set in Example 4. By Theorem 3(2), we can obtain a quadiphase (96, 12; 6)-ZCZ set S. We actually only show the sequences s 0 and s 1 . 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 16, 24, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 88, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 32, 24, 0, 0, 0, 0, 0, 0).
Other pairs of sequences in S have the same cross-correlation property concerning the ZCZ. Theorem 3 provides a method to construct a ZCZ set with larger family size from that with smaller family size. The following theorem is directly derived from the basic construction in Procedure 1. It is a method to construct a ZCZ set with longer ZCZ from a set with shorter ZCZ.
Theorem 4: If
A is an (m, n; Zcz)-ZCZ set, then S constructed by Procedure 1 is an (mn, n; nZcz)-ZCZ set.
The proof of Theorem 4 is presented in Appendix VI.
A variant of Theorem 4 is proved in [20], where the divisibility condition n|m or m|n is assumed and the variant is used to iteratively construct new ZCZ set from the old ones. Our variant of Theorem 4 removes this divisibility assumption, and hence it is applicable to construct more ZCZ sets.
The application of Theorems 3 and 4 depends not on the existence of perfect sequences but on that of complete orthogonal sequence sets. This relaxes restriction on the application of the two theorems since complete orthogonal sequence sets are easily constructed. Applying them to known ZCZ sets, we will obtain new sets with both longer ZCZ and larger set size. Moreover, for the construction of some even-phase, say p-phase, ZCZ sets, we choose some specific n such that n = 2 l (q t + 1) ≡ 0 mod 4 as mentioned in III-B; since there always exists a Hadamard matrix of order n, the new set constructed for this kind of parameter n has the same number of phases as the old one, that is, it is still of 2p gcd(p,2) = p-phase.
D. The comparison of parameters on known constructions
In this correspondence, some new ZCZ sequences obtained can reach the bound (7) IV. SEQUENCES WITH LOW CROSS CORRELATION Some sequence sets with low cross correlation are found, e.g., those from Gold-pair construction and interleaved construction [39], [17], [18], Kasami sequences [40], and bent function sequences [41]. Using the basic method in Section II-B, we can also construct sequences with low out-of-phase autocorrelation and cross-correlation.
Let A = {L e 0 (a), L e 1 (a), · · · , L e n−1 (a)} be an ordered set of shift equivalent sequences, and m, n, B and S be as in Procedure 1. Set N r,s = |{0 ≤ j < n| e j+s − e j + r ≡ 0 mod m}| and N 0 = max{N r,s | 0 ≤ r < m, 0 ≤ s < n}.
for any 1 ≤ s < n, then S is an [mn, n; 2E a ] sequence set.
(2) If m ≥ 2n and e = (e 0 , e 1 , · · · , e n−1 ), where 0 ≤ e i < n for all i, satisfies
|{(e j+s − e j )mod n | 0 ≤ j < n − s}| = n − s(26)
for any 1 ≤ s < n, then S is an [mn, n; 2E a ] sequence set.
The proof of this theorem is presented in Appendix VII.
Gong provides two methods in [18] to construct shift sequences e = (e 0 , e 1 , · · · , e n−1 ) satisfying equality (25). In her methods, 0 ≤ e i < n for 0 ≤ i < n and the length of the perfect sequence is a prime or is of the form p l −1 for some prime p. If we take m = n in Theorem 5(1), then new sequence set can be constructed by Gong's methods. However, this reduces applicability of Procedure 1 since perfect sequences with such a length may exist in small quantities. Theorem 5(2) is especially suitable for constructing ternary sequence sets in the sense that there are a lot of ternary perfect sequences [36] and a ternary complete orthogonal set of n sequences exists for an arbitrary positive integer n [38].
Let a = (a 0 , a 1 , · · · , a m−1 ) be a ternary perfect sequence, and B be a ternary complete orthogonal set of n sequences, where n ∈ [ m 4 , m 2 ] is a prime or is of the form p s − 1 where p is a prime. It is easy to find such n and B for a given m ≥ 4. We can use Gong's methods in [18] to find a shift sequence e = (e 0 , e 1 , · · · , e n−1 ) satisfying Eq. (26), and an [mn, n; 2E a ] sequence set S is obtained by Theorem 5 (2). Different from Gong's [n 2 , n; 2n + 3] set [18], our construction provides sequence sets with different choices of parameters of set size and sequence length. 9,18,0,9,18,9,9,0,9,9,9,9,9,9,9,0,0,9,9,0,9,0,9,0,0,0,0,9,0,0,0,0,9,9,0,0,9,0,9,0,9,0,0,0,0,9,0,0,0,0,9,18,9,0,0,0,0,9,0,0,0,0,9,0,9,0,9,0,0,9,9,0,0,0,0,9,0,0,0,0,9,0,9,0,9,9,0,0,9,9,9,9,9,9,9,0,9,9,18,9,0,18,9) and |R s 0 ,s 1 (τ )| = |R s 1 ,s 0 (τ )| = (0, 9,18,0,9,0,9,9,0,9,9,9,9,9,9,9,0,0,9,9,0,9,0,9,0,0,0,0,9,0,0,0,0,9,9,0,0,9,0,9,0,9,0,0,0,0,9,0,0,0,0,9,18,9,0,0,0,0,9,0,0,0,0,9,0,9,0,9,0,0,9,9,0,0,0,0,9,0,0,0,0,9,0,9,0,9,9,0,0,9,9,9,9,9,9,9,0,9,9,0,9,0,18,9) V. CONCLUSION In this correspondence we consider how to construct sequence sets with zero-correlation zone or with low correlation. We present a general method to generate new sequence sets with these correlation properties. We prove two theorems (Theorems 3 and 4), and by applying them to known ZCZ sequence sets, we can obtain new ZCZ sets with both longer ZCZ and larger family size. More sets of sequences seem to be likely to be generated if appropriate parameters are carefully chosen in Procedure 1.
|R s 0 (τ )| = |R s 1 (τ )| = (72,
APPENDIX I PROOF OF PROPOSITION 1
Proof: Let u be the sequence associated with the ordered set A (see Procedure 1 (2)), and entry in the sequence L τ (u) is a (s+i)modn,(r+ϕ(s+i))modm and it is exactly the first element of T i .
So,
T i = L (r+ϕ(s+i))modm (a s+i−ϕ(s+i)n ).(27)
Since the i-th column of the matrix form of s k is b k,i a i , the i-th column of the matrix form
of L τ (s k ) is L (r+ϕ(s+i))modm (b k,s+i−ϕ(s+i)n a s+i−ϕ(s+i)n ).(28)
Thus, one has
R s h ,s k (τ ) = n−1 i=0 m−1 l=0 (a i,l b h,i )(a * s+i−ϕ(s+i)n,(l+r+ϕ(s+i))modm b * k,s+i−ϕ(s+i)n ) = n−1 i=0 m−1 l=0 (a i,l a * s+i−ϕ(s+i)n,(l+r+ϕ(s+i))modm )(b h,i b * k,s+i−ϕ(s+i)n ) = n−1 i=0 d i R a i ,a s+i−ϕ(s+i)n (r + ϕ(s + i))(29)
where
d i = b h,i b * k,s+i−ϕ(s+i)n .
APPENDIX II
PROOF OF THEOREM 1
Proof: Set t i = e i+s − e i + r. Then by Eq. (18),
R s h ,s k (τ ) = n−1 i=0 d i R a (t i ).
where τ = rn + s.
(i) Assume 0 ≤ r ≤ r 0 and s = 0. Then
R s h ,s k (rn) = R a (r) n−1 i=0 d i = R a (r)R b h ,b k (0).
Thus R s h ,s k (0) = E a · n for h = k and r = 0, and R s h ,s k (rn) = 0 otherwise.
(ii) Assume 0 ≤ r ≤ r 0 and s > 0. If i + s < n,
If i + s ≥ n, −m < e i+s − e i + r = 1 + e i+s−n − e i + r
≤ 1 + e i−1 − e i + r ≤ 1 + e i−1 − e i + r 0 ≤ 0.
So, either 0 < t i ≤ m or −m < t i ≤ 0 holds, and if the equality holds then s = n − 1 and r = r 0 . Thus, for any 0 < τ ≤ r 0 n + n − 2, either 0 < t i < m or −m < t i < 0 holds for all i ∈ {0, 1, · · · , n − 1}, and consequently, R s h ,s k (τ ) = 0.
Combining the above two cases, we have R s h ,s k (τ ) = 0 for 0 < τ ≤ r 0 n + n − 2 and for τ = 0 and h = k. By
R s h ,s k (−τ ) = R s k ,s h (τ ) * ,(30)
we conclude that S is an (mn, n; r 0 n + n − 2)-ZCZ set.
(1) Assume n | (m + 1). Equalities
e i+1 − e i = m + 1 n for 0 ≤ i ≤ n − 2
imply r 0 = m+1 n − 1, and S is an (mn, n; m − 1)-ZCZ set. (2) Assume n | m and holds for some i 0 . Thus r 0 = m n − 1 and S is an (mn, n; m − 2)-ZCZ set. If e = e (i) , it is easy to check that r 0 = m n − 1 and S is an (mn, n; m − 2)-ZCZ set.
(i 1 − i 2 )m n ≤ e i 1 − e i 2 ≤ (i 1 − i 2 )m n + 1 for 0 ≤ i 2 < i 1 ≤ n − 1.
APPENDIX III
PROOF OF THEOREM 2
Proof: It is easy to check that S is an (mn, n; 0)-ZCZ set if m = n, so we only need to consider the case of n < m.
By Eq. (18), for r = 0 and τ = s,
R s h ,s k (τ ) = n−1 i=0 d i R a (e i+s − e i )
.
By e i = m − 1 − i, we have e i+s − e i ≡ −s if i + s < n; n + 1 − s if i + s ≥ n.(31)
Thus, for 0 < s < n, R a (e i+s − e i ) = 0 for all 0 ≤ i ≤ n − 1;
for r = 1 and s = 0,
R s h ,s k (τ ) = n−1 i=0 d i R a (1) = 0. For h = k, R s h ,s k (0) = n−1 i=0 d i R a (0) = R a (0)R b h ,b k (0) = 0.
By Eq. (30), S is an (mn, n, n mod m)-ZCZ set.
APPENDIX IV
PROOF OF PROPOSITION 2
Proof: Let a = (a 0 , a 1 , · · · , a m−1 ) be a binary perfect sequence, m ≥ 4. Take a positive integer n such that n = 2 t ≤ m for some integer t ≥ 1, and a complete orthogonal set B derived from a Hadamard matrix of order 2 t .
(i) If n = m, we use a and B to construct an (m 2 , m; m − 2)-ZCZ set [20]. Then by the bound (22), m − 2 ≤ m 2 , i.e., m ≤ 4. (ii) If n < m, we assume m 2 s+1 ≤ n < m 2 s for some s ≥ 0. (ii.1) If n = m 2 s+1 , then by Theorem 1, an (mn, n; m − 2)-ZCZ set is constructed. Similarly as in (i), one has m ≤ 4.
(ii.2) If m 2 s+1 < n < m 2 s , then a (2 s mn, 2 s n; 2 s n)-ZCZ set is constructed form a and B by Theorem 2. In such a case, m 2 < 2 s n < m, and it contradicts the assumed bound (22). Thus m ≤ 4.
APPENDIX V PROOF OF THEOREM 3
Proof: We only need to prove that
R s k i ,s k ′ i ′ (τ ) = 0 for 1 ≤ τ ≤ r 0 n + s 0 and R s k i ,s k ′ i ′ (0) = 0 for k = k ′ or i = i ′ .
It is easy to check that if s k i and s k ′ i ′ are the same sequences, then k = k ′ and i = i ′ . So, 0≤k<l S k has ln different sequences. By (9) and (14), one has (b i,t 2 b * i ′ ,(s+t 2 )modn )(a k t 2 ,t 1 a k ′ * (s+t 2 )modn,(t 1 +r+ϕ(s+t 2 ))modm ) = n−1
t 2 =0 m−1 t 1 =0 (b i,t 2 b * i ′ ,(s+t 2 )modn )(c k,p c * k ′ ,q ) = n−1 t 2 =0 d t 2 R c k ,c k ′ (τ + d(r + ⌊ s+t 2 +1 n ⌋ − ⌊ t 2 +1 n ⌋))(32)
where τ = rn + s, d t 2 = b i,t 2 b * i ′ ,s+t 2 , and the subscripts p = (t 1 (n + d) + i 2 + d⌊ t 2 + 1 n ⌋)modm and q = ((t 1 + r + ϕ(s + t 2 ))(n + d) + (s + t 2 )mod n + d⌊ ((s + t 2 )modn) + 1 n ⌋)mod m.
The last equality in Eq. (32) holds since q − p = (t 1 + r + ϕ(s + t 2 ))(n + d) + s + t 2 − ϕ(s + t 2 )n + d⌊ s+t 2 −ϕ(s+t 2 )n+1 n ⌋ − (t 1 (n + d) + t 2 + d⌊ t 2 +1 n ⌋) = (t 1 + r)(n + d) + s + t 2 + d⌊ s+t 2 +1 n ⌋ − (t 1 (n + d) + t 2 + d⌊ t 2 +1 n ⌋) = τ + dr + d⌊ s+t 2 +1 n ⌋ − d⌊ t 2 +1 n ⌋. and gcd(m, n + d) = 1 implies that, for any fixed 0 ≤ t 2 < n, |{(t 1 (n + d) + t 2 + d⌊ t 2 + 1 n ⌋)modm | 0 ≤ t 1 < m}| = m.
By (18),
R s k i ,s k ′ i ′ (0) = R c k ,c k ′ (0)R b i ,b i ′ (0) = 0 if k = k ′ or i = i ′ . When 1 ≤ τ ≤ r 0 n + s 0 , 1 ≤ τ + d(r + ⌊ s+t 2 +1 n ⌋ − ⌊ t 2 +1 n ⌋) ≤ r 0 n + s 0 + dr 0 + d ≤ Zcz.
Thus, R s k i ,s k ′ i ′ (τ ) = 0 for 1 ≤ τ ≤ r 0 n + s 0 .
APPENDIX VI
PROOF OF THEOREM 4
Proof: Since B is a complete orthogonal sequence set, any two sequences in S are different.
Let τ = rn + s and s h , s k ∈ S.
(i) For 0 < r < Zcz and 0 ≤ s < n, since A is an (m, n; Zcz)-ZCZ set, R a i ,a s+i−ϕ(s+i)n (r + ϕ(s + i)) = 0.
So by Proposition 1, R s h ,s k (τ ) = 0;
(ii) For r = Zcz and s = 0, ϕ(s + i) = 0, and by Proposition 1, one also has R s h ,s k (τ ) = 0;
(iii) For r = 0 and any 0 < s < n, R a i ,a s+i−ϕ(s+i)n (ϕ(s + i)) = 0.
Thus, By Eq. (30), we conclude S is an (mn, n; nZcz)-ZCZ set.
R s h ,s k (τ ) = n−1 i=0 d i R a i ,
APPENDIX VII
PROOF OF THEOREM 5
Proof: Obviously, S has n different sequences.
|R s h ,s k (τ )| = | n−1 i=0 d i R a (t i )| ≤ n−1 i=0 |R a (t i )| ≤ N 0 E a .
where t i = e i+s − e i + r. for 0 ≤ i 1 = i 2 < n − s. Thus, there is at most one integer i 0 (0 ≤ i 0 < n − s) such that t i 0 ≡ 0 mod m. Similarly, there is at most one integer i ′ 0 (n − s ≤ i ′ 0 < n) such that t i ′ 0 ≡ 0 mod m. Then, one has N 0 ≤ 2. is at most one integer i ′ 0 (n − s ≤ i ′ 0 < n) such that t i ′ 0 ≡ 0 mod m. Then, one has N 0 ≤ 2.
( 4 )
4The sequence set S = S(A, B) is defined as S = {s 0 , s 1 , · · · , s n−1 }. Example 1: Let A = {a 0 , a 1 }, where a 0 = (1, −1, −1), a 1 = (1, 1, −1), and B consist of two row sequences of the Hadamard matrix 1, S = {s 0 , s 1 } where s 0 = (1, 1, −1, 1, −1, −1), s 1 = (1, −1, −1, −1, −1, 1).
Example 2 :
2Let m = 9, n = 2, and e = (0, 5). three-phase perfect sequence where 0, 1 and 2 represent 1, exp( 2πj 3 ) and exp( 4πj 3 ), respectively, and B is the same as in Example 1. A six-phase (18, 2; 8)-ZCZ set S shown as S = {(040004022040004022), (010301052343034325)} is obtained by applying Theorem 1(1), where an entry i represents a six-phase complex number exp( 2iπj 6 ). The absolute values of autocorrelation functions R s 0 (τ ) and R s 1 (τ ) are given by |R s 0 (τ )| = |R s 1 (τ )
0123, 0202, 0321} and the shift sequence e (0) = (0, 4, 8, 12). By applying Theorem 1(2), there exist other 3 quadriphase (64, 4; 14)-ZCZ sets which are constructed from and R s 3 (τ ) are given by |R s 0 (τ )| = |R s 1 (τ )| = |R s 2 (τ )| = |R s 3 absolute values of cross-correlation functions are given by
The absolute values of autocorrelation function of s 0 are given by|R s 0 (τ )16, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). The other sequences in S have the same autocorrelation property concerning the ZCZ. The absolute values of cross-correlation function of s 0 and s 1 are given by |R s 0 ,s 1
However, in these constructions the number of phases of the sequences increases with the sequence length. For a fixed (and ideally, small) number of phases, we have few knowledge on the existence of perfect sequences of arbitrary length. For example, up to now only in length 2, 4, 8 and 16 have quadriphase perfect sequences been found.
Proposition 2 :
2is expected to hold for binary (N, M; Zcz)-ZCZ sets where M ≥ 2 [11]: Let M ≥ 2. If Eq. (22) holds for any binary (N, M; Zcz)-ZCZ set, then binary perfect sequences exist only in length 4.
by Theorem 3(2), provided a perfect sequence of length m and a complete orthogonal set of n sequences exist. For the case that m and n are not relatively prime, some ZCZ sequences are still constructed by our method, which extends Theorem 3 of [19]. For example, if C consists of a single perfect sequence of length m, and assume n > m − 1 and gcd(m, n + 1) = 1, then S 0 is an (mn, n; m − 2)-ZCZ set. It is an improvement on work of Torii et al. [20] since the divisibility condition n|m or m|n is not necessarily required. Some new quadriphase ZCZ sequence sets with (L, M; Zcz) = (96, 12; 6), (320, 20; 14), (384, 24; 14)
For
example, the absolute values of autocorrelation function of s 0 are given by |R s 0 (τ )The other sequences in S have the same autocorrelation property concerning the ZCZ. The absolute values of cross-correlation function of s 0 and s 1 are given by |R s 0 ,s 1
Example 6 :
6Let C be the quadriphase (64, 4; 14)-ZCZ set shown in Example 3(1). Take d = 1 and n = 16. Let H be the matrix in Eq. (15) and B consist of row sequences of H ⊗ H ⊗ H ⊗ H which is a Hadamard matrix of order 16, where ⊗ denotes the Kronecker product. Applying Theorem 3, we obtain a quadriphase (1024, 64; 13)-ZCZ set.
Example 7 :
7Suppose that A is the quadriphase (1024, 64; 13)-ZCZ set constructed in Example 6. By Theorem 4 we get a quadriphase (65536, 64; 832)-ZCZ set since there exists a complete orthogonal set of 64 sequences derived from a Hadamard matrix of order 64.
and 3 .
3But there are very strict limitations on the parameters: the length of perfect sequences and the length of the shift sequences and the order of the Hadamard matrices. Those quadriphase (N, M; Zcz)-ZCZ sequences obtained in this correspondence can not reach the bound (7), however, for given N and M, their zero correlation zone Zcz are the maximal among all known results about quadriphase (N, M; Zcz)-ZCZ sequences. Tables I and II summarize some of the best known quadriphase (N, M; Zcz)-ZCZ sequences, and the symbol "△" mean that the quadriphase (N, M; Zcz △ )-ZCZ sequences can be constructed by our method but not by others. Utilizing Theorems 3 and 4 to these sequences, some new sequence sets are obtained.
Theorem 5 :
5Assume a is a perfect sequence and the absolute value of each entry in the sequences of set B does not exceed 1. Then S is an [mn, n; N 0 E a ] sequence set. Furthermore, (1) If m ≥ n and the shift sequence e = (e 0 , e 1 , · · · , e n−1 ), where 0 ≤ e i < m for all i, satisfies |{(e j+s − e j )mod m | 0 ≤ j < n − s}| = n − s
Example 8 :
8Let and B be a complete orthogonal set derived from H ⊗ H ⊗ H. Then S constructed by Procedure 1 is a quadriphase [64, 8; 16] sequence set. We list the first two sequences in S The absolute values of autocorrelation function of s 0 and the absolute values of cross-correlation function of s 0 and s 1 are given by | R s 0 (τ ) |= (64, Similarly, let m = 16, n = 8, a be the quadriphase perfect sequence of length 16 listed in Example 3, and e and B be the same as above, then by Theorem 5(2) we can obtain a quadriphase [128, 8; 32] sequence set.
Example 9 :
9Let m = 13, n = 8, e = (0, 5, 6, 5, 7, 7, 3, 6), a = (−1, −1, −1, −1, 0, 1, −1, 1, 0, 0, −1, 0, 1) be a ternary perfect sequence, and B be the complete orthogonal set in Example 8. By Theorem 5 we get a ternary [104, 8; 18] sequence set S. The first two sequences in S are shown as s 0 = (− + − + + + − − − − + − 000 + − + 0 + 00 + 0 − 000 − − − 000− 000 + − + −0 − + + 00 − 0 + 0 − −0 + + + − + − − − − 0 − − − −− 0 − 0 − − − − − + − − − − − 00 − −0 − 0 − + + −0 + 0 + 0 − − − +) and s 1 = (− − − − + − − + − + + + 000 − − − 0 − 00 + 0 − 000 − + − 000− 000 + + + +0 + + − 00 − 0 + 0 − +0 − + − − − − + − + 0 + − + −+ 0 + 0 + − + − + + + − + − + 00 − +0 + 0 + + − −0 + 0 + 0 − + − −) where + and − represent +1 and −1, respectively. The absolute values of auto-and crosscorrelation of s 0 and s 1 are given by
1 ≤ e i − e i−1 − 1 ≤ m n for 1 ≤ i ≤ n − 1.If m + e 0 − e n−1 = m n and e i − e i−1 − 1 = m n for 1 ≤ i ≤ n − 1, then e n−1 − e 0 = n−1 j=1 (e j − e j−1 ) = (n − 1)m n + (n − 1)and it conflicts to m + e 0 − e
a s+i−ϕ(s+i)n (ϕ(s + i)) = 0. If s = 0 and h = k, R s h ,s k (R a i (0) = E a 0 · R b h ,b k (0) = 0.
e i+s − e i )mod m | 0 ≤ i < n − s}| = n − s hods for all 1 ≤ s < n, then (e i 1 +s − e i 1 )mod m = (e i 2 +s − e i 2 )mod m and (e i 1 +s − e i 1 + r)mod m = (e i 2 +s − e i 2 + r)mod m
e i+s − e i )mod n | 0 ≤ i < n − s}| = n − s for all 1 ≤ s < n, thene i 1 +s − e i 1 = e i 2 +s − e i 2 and −m ≤ −2n < t i 1 − t i 2 < 2n ≤ m for 0 ≤ i 1 = i 2 < n − s. Thus(e i 1 +s − e i 1 + r)mod m = (e i 2 +s − e i 2 + r)mod m and there is at most one integer i 0 (0 ≤ i 0 < n − s) such that t i 0 ≡ 0 mod m. Similarly, there
TABLE I
ISOME KNOWN QUADRIPHASE (N,M,ZCZ)-ZCZ SEQUENCES SETS FOR M ≤ N M ≤ 16N
M
4
4
8
8
8
16
16
16
16
16
M
2
4
2
4
8
2
4
8
12
16
Zcz
2
2
6
6
6
14
14
14
12 △
14
TABLE II
SOME KNOWN QUADRIPHASE (N,M,ZCZ)-ZCZ SEQUENCES SETS FOR 8 ≤ N
M ≤ 16 < M ≤ 36
N
M
8
16
8
16
8
16
8
16
8
16
M
20
20
24
24
28
28
32
32
36
36
Zcz
6 △
14 △
6
14 △
6 △
14 △
6
14
6 △
14 △
February 1, 2008DRAFT
e(1) , e(2) and e(3) , respectively. They are not shift equivalent to C 1 and the four sets are listed as follows:(1) e (0) = (0, 4, 8, 12) s 0 = (0000012302020321000012302020321000002301020221030000301220201032) s 1 = (0123020203210000012313132103333301232020032122220123313121031111) s 2 = (0202032100000123020210322222301202022103000023010202321022221230) s 3 = (0321000001230202032111112301313103212222012320200321333323011313) (2) e (1) = (0, 4, 8, 13) s 0 = (0003012202010320000012302020321000012302020321000002301020221030) s 1 = (0122020103200003012313132103333301202021032222230121313321011113) s 2 = (0201032000030122020210322222301202032100000123020200321222201232) s 3 = (0320000301220201032111112301313103222223012020210323333123031311) (3) e (2) = (0, 4, 9, 13) s 0 = (0023010202210300003012202010320000012302020321000012302020321000) s 1 = (0102022103000023011313032133332301202021032222230131310321111123) s 2 = (0221030000230102023210222212300202032100000123020210322222301202) s 3 = (0300002301020221031111012331312103222223012020210333330123131321)(4) e (3) = (0, 5, 9, 13) s 0 = (0123020203210000023010202210300003012202010320000012302020321000) s 1 = (0202032100000123031311032333312300202321022221230131310321111123) s 2 = (0321000001230202003212222012320201032000030122020210322222301202) s 3 = (0000012302020321011113012131332102222123002023210333330123131321)For each sequence set, the absolute values of autocorrelation functions R s 0 (τ ), R s 1 (τ ), R s 2 (τ )February 1, 2008 DRAFT
T = (T 0 , T 1 , · · · , T n−1 ) the matrix form of L τ (u). By Procedure 1 (2), for 0 ≤ i < n, the i-thFebruary 1, 2008 DRAFT
≤ r < e i+s − e i + r ≤ e n−1 − e 0 + r 0 ≤ m;February 1, 2008 DRAFT
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| []
|
[
"FINDING A CYCLE BASE OF A PERMUTATION GROUP IN POLYNOMIAL TIME",
"FINDING A CYCLE BASE OF A PERMUTATION GROUP IN POLYNOMIAL TIME"
]
| [
"Mikhail Muzychuk ",
"Ilia Ponomarenko "
]
| []
| []
| A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved that a cycle base of a permutation group of degree n can be constructed in polynomial time in n. | 10.1016/j.jalgebra.2018.03.028 | [
"https://arxiv.org/pdf/1702.05292v1.pdf"
]
| 119,651,077 | 1702.05292 | c97eef1651124b576e0fdcad33b351cf35bf3f11 |
FINDING A CYCLE BASE OF A PERMUTATION GROUP IN POLYNOMIAL TIME
17 Feb 2017
Mikhail Muzychuk
Ilia Ponomarenko
FINDING A CYCLE BASE OF A PERMUTATION GROUP IN POLYNOMIAL TIME
17 Feb 2017
A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved that a cycle base of a permutation group of degree n can be constructed in polynomial time in n.
Introduction
It is well known that the graph isomorphism problem is polynomial-time equivalent to finding the automorphism group of a graph. However, it is not clear whether the automorphism group given as the input can help to test isomorphism. A byproduct of our main result says that it does help if the input graphs (or any other combinatorial object) is circulant. To be more precise, we need the concept of cyclic base explained below.
Any permutation group K ≤ Sym(n) acts by conjugation on the set cyc(K) = {G ≤ K : G is regular and cyclic}.
A cycle base of K is a set B ⊆ cyc(K) intersecting each K-orbit in this action in exactly one element; in other words, B is a maximal set of pairwise non-conjugate regular cyclic subgroups of K. In slightly different form, this notion was first used in [13] for efficient recognizing and isomorphism testing of circulant tournaments. Then with the help of the classification of finite simple groups, it was proved in [12] that |B| ≤ ϕ(n)
for every cycle base B of the group K, where ϕ is the Euler function. Finally, a cycle base technique was applied for polynomial-time recognizing and testing isomorphism of arbitrary circulant graphs [3,11]. In particular, an efficient algorithm was proposed in [3] to construct a cycle base of the automorphism group of a graph.
The idea to use the cyclic base for Cayley graph isomorphism testing goes back to Babai's lemma [1,Lemma 3.1], which establishes a one-to-one correspondence between the Cayley representations 1 of a graph X over group G and regular subgroups of the group K = Aut(X) that are isomorphic to G. Moreover, two Cayley representations of X are equivalent if and only if the corresponding subgroups are conjugate in K. In this terminology, the above mentioned result [3] shows that if the group G is cyclic, then given the group K, one can efficiently find a full system of pairwise nonequivalent Cayley representations of the graph X. 1 By a Cayley representation of a graph X over a group G, we mean a Cayley graph over G isomorphic to X; two such representations are called equivalent if some isomorphism of the corresponding Cayley graphs belong to Aut(G).
It should be mentioned that not every permutation group K is 2-closed, i.e., is the automorphism group of a graph; for instance, Sym(n) is a unique 2-transitive group of degree n, which is 2-closed. In particular, this may occurs if K = Aut(X), where X is a combinatorial object defined by a set of relations of arity r > 2. Therefore, known algorithms cannot be used to find a full system of pairwise nonequivalent Cayley representations of such X over a cyclic group. The main result of the present paper says that one can find such a system efficiently if the group K is given. It is assumed that the input permutation group K is given by a set of generators and the cardinality of this set is polynomial in n, see [14]. The output B is a set of full cycles contained in K; in particular, B is empty if and only if K contains no regular cyclic subgroup.
For solvable permutation groups, a polynomial-time algorithm for finding cyclic base was constructed in [3,Theorem 6.3]. Therefore, to prove Theorem 1.1, it suffices to be able, given a group K ≤ Sym(n) to find efficiently a solvable group K 0 ≤ K which controls regular cyclic subgroups of K, i.e., for each H ∈ cyc(K) there exists k ∈ K such that H k ≤ K 0 (indeed, in this case, one can find the set B as a subset of cyclic base of K 0 , for details, see [3,Subsection 6.2]). The existence of a solvable group K 0 which controls regular cyclic subgroups was proven in [12], but no algorithm for finding K 0 was provided there. In the present paper, we fill this gap in the following theorem, the proof of which occupies the rest of the paper. Theorem 1.2. Given a group K ≤ Sym(n), one can construct in time poly(n) a solvable subgroup of K which controls its regular cyclic subgroups.
The Main Algorithm for proving Theorem 1.2 is described in Section 5. The basic idea of the algorithm is, as in the case of 2-closed groups, to construct a solvable subgroup by "removing" non-abelian composition factors of K step by step. This reduces the problem to the case, where all non-abelian composition factors are isomorphic to a simple group T and contained in the socle of K. In contrast to the case of 2-closed groups, where T is an alternating group, here T might be also a projective special linear group. A relevant theory for the "removing" part is developed in Sections 3 and 4. It is based on the classification of primitive groups containing a regular cyclic subgroup obtained by G. A. Jones in [6]. This classification is also used in Section 2 providing algorithmic tools to find a regular cyclic subgroup of a primitive groups.
Let X be a combinatorial object, i.e., an object in a concrete category in the sense of [1]. It is said to be a circulant object if the group Aut(X) contains a regular cyclic subgroup. A circulant representation of X is defined in the same way as for graphs. Then in view of the one-to-one correspondence between the circulant representations of X and regular cyclic subgroups of Aut(X), the following statement is an immediate consequence of Theorem 1.1. Corollary 1.3. Given a combinatorial object X of size n and the group Aut(X), one can test in time poly(n) whether X is a circulant object and (if so) find a full system of pairwise nonequivalent circulant representations of X within the same time.
Using Corollary 1.3, it is a routine task to construct a canonical form of a circulant object X (see e.g. the proof of [3, Theorem 1.2]).
Corollary 1.4. The problem of finding a canonical form of a circulant object X is polynomial-time reduced to constructing the group Aut(X).
All undefined notations and standard facts concerning permutation groups can be found in the monographs [2] and [16]. Throughout the paper, we freely use known polynomial-time algorithms for permutation groups [14, Section 3.1].
Notation.
Hereinafter, Ω denotes a set of cardinality n and Sym(Ω) = Sym(n) is the symmetric group on Ω.
The orbit set of a group K ≤ Sym(Ω) is denoted by Orb(K) = Orb(K, Ω). The restriction of the group K to a K-invariant set ∆ ⊆ Ω is denoted by K ∆ . The pointwise and setwise stabilizers of the set ∆ in the group K are denoted by K ∆ and K {∆} , respectively; we also set K
∆ = (K {∆} ) ∆ .
For an imprimitivity system D of a group K ≤ Sym(Ω), we denote by K D and K D the permutation group induced by the action of K on the blocks of D and the subgroup of K leaving each block of D fixed.
The holomorph Hol(G) of a regular group G ≤ Sym(Ω) is identified with the subgroup of Sym(Ω) induced by the right multiplications and automorphisms of G.
The normalizer of G ≤ Sym(Ω) in Sym(Ω) is denoted by N Ω (G).
2. Finding a cyclic base of a primitive group 2.1. Classification. The primitive groups containing a regular cyclic subgroup were completely described in paper [6, Theorem 3] modulo the classification of finite simple groups. Below, we cite the corresponding result.
Theorem 2.1. Let K ≤ Sym(n) be a primitive group containing a regular cyclic subgroup. Then one of the following statements holds:
(1) C p ≤ K ≤ AGL 1 (p), where n = p is prime, (2) K = Sym(n) for some n ≥ 2 or K = Alt(n) for some odd n ≥ 3, (3) PGL d (q) ≤ K ≤ PΓL d (q) and n = (q d − 1)/(q − 1) for some d ≥ 2, (4) K = PSL 2 (11), M 11 , or M 23 , and n = 11, 11, or 23, respectively.
The following auxiliary statement follows from Theorem 2.1 and will be used in Section 4. The authors are thankful to Prof. E. Vdovin for his help with handling the centralizers of graph automorphisms of the projective special linear groups. Lemma 2.2. Let K ≤ Sym(n) be a primitive group and G ∈ cyc(K). Then
C Aut(K) (G) ≤ Inn(K).
Proof. The group N := N Sym(n) (K) is embedded into the group Aut(K). Moreover, N = Aut(K) in all cases mentioned in Theorem 2.1 unless K = Sym(6) or case (3) occurs. In the former case, the statement of lemma is easily checked with the help of GAP [4]. In the remaining case,
PGL d (q) ≤ K ≤ PΓL d (q)
and hence the group Aut(K) is embedded into Aut(PSL d (q)). Then any external automorphism of K can be written as the product of diagonal, field, and graph automorphisms. The automorphisms of the first two types are realized in Sym(n), and one can apply the above argument. Let now σ be a graph automorphism of S. According to [10,
C S (σ) ∈ {S d (q), O d (q), O ± d (q)},
where the parity of d determines which group on the right-hand side occurs as the centralizer C S (σ). The order of each of these group is not divisible by the Zsigmondy prime for (q, d). Therefore, none of this groups contains an element of order n = |G|. This shows that no graph automorphism of K centralizes G.
2.2.
Recognizing. Our first goal is to recognize the groups K appearing in Theorem 2.1, and then in each case, to construct a regular cyclic subgroup of K. This is done in more or less standard way in the following statement, where we use the fact that the socle of a subgroup of Sym(n) can be found in polynomial time in n [14, Section 3.1]. Lemma 2.3. Given a primitive group K ≤ Sym(n), one can test in time poly(n) whether cyc(K) = ∅, and (if so) find H ∈ cyc(K) within the same time.
Proof. Let S = Soc(K). If the number n is prime and S ∼ = C n , then case (1) of Theorem 2.1 occurs and we output H = S. Next, if |S| = (n!)/2, then S = Alt(n) and case (2) occurs. Here we output cyc(K) = ∅ if K = S and n is even, and the group H generated by a full cycle of Sym(n), otherwise. In the remaining two cases, the group S is determined by its order up to isomorphism [8, Theorem 5.1]. If now case (4) occurs, then the group H can be found by the inspection of the elements of K. Finally, if S ∼ = PSL d (q) for some d and q, for which n = (q d − 1)/(q − 1), then cyc(K) = ∅.
To complete the proof, we assume that S ∼ = PSL d (q) for suitable d and q. Then using the main algorithm from [7] 2 , one can find a d-dimensional vector space V over GF(q) and an explicit isomorphism
f : S → PSL(V )
given by the images of generators of S. This isomorphism is induced by a bijection from {1, . . . , n} onto the lines of V . Let us extend f to an isomorphism
f ′ : S ′ → PGL(V ),
where S ′ is a unique subgroup of K, which is isomorphic to PGL d (q) and contains S. Using the natural basis of V , one can construct a Singer subgroup H ≤ GL(V ) given by an explicit generator matrix. Then (f ′ ) −1 (H) is a regular cyclic subgroup of K, as required.
Let K ≤ Sym(Ω) be a primitive group and H a regular cyclic subgroup of K. Denote by p the largest prime divisor of the number n. The group H has a unique subgroup P = P (H) of order p and the set P = Orb(P, Ω) is an imprimitive system of H. Note that H normalizes the direct sum of the permutation groups N Θ (P Θ ), Θ ∈ P. It is easy to see that the group
(1) N (H) = H Θ∈P N Θ (P Θ ),
2 Note that this algorithm is polynomial in n, because q ≤ n.
is permutation isomorphic to the wreath product N Θ (P Θ ) ≀ H P in the imprimitive action. In particular, it is solvable. Note also that the setwise stabilizer of Θ in N (H) contains P .
Theorem 2.4. Let K ≤ Sym(n) be a primitive group containing a regular cyclic subgroup H. Then given C ∈ cyc(K), there exists s ∈ Soc(K) such that
C s ≤ N (H).
Proof. In what follows, we set P = P (H), N = N (H), and S = Soc(H). Assume first that n = p is a prime. Then P = H and N = N Sym(n) (P ). Moreover, cyc(K) = Syl p (S) and by the Sylow theorem, every C ∈ cyc(K) is S-conjugate to P ≤ N . This proves the required statement in the considered case. Assume now that n is composite. Then by Theorem 2.1, the group S is either projective special or alternating. Let us consider these two cases separately.
Let S = PSL d (q) for appropriate d and q. If (d, q) = (2, 8) and K = PΓL 2 (8), then n = 9, p = 3, and the required statement follows by a direct calculation in GAP [4]. Otherwise, from [6, Corollary 2] it follows that cyc(K) = cyc(PGL d (q)) and every two groups in cyc(K) are conjugate in PGL d (q). On the other hand, the group HS ≤ K contains a regular cyclic group H and hence contains the group PGL d (q) again by [6, Corollary 2]. Thus, every C ∈ cyc(K) is conjugate to H ≤ N in HS and hence in S, as required. where Q = P (C). Thus, the groups C and H can be treated as subgroups of the wreath product Sym(p) ≀ Sym(m) in imprimitive action. This product contains the elements
c ′ = (1, . . . , 1; c m ) and h ′ = (1, . . . , 1; h m ),
where c m and h m are generators of the groups C P and G P , respectively. Each of the permutations c ′ , h ′ is the disjoint union of p cycles of length m. Therefore, (c ′ ) s ′ = h ′ for some s ′ ∈ S. Thus, we may assume that
C ≤ Q ≀ H P and H ≤ P ≀ H P .
If p = 2, then Q = P and we are done, because in this case N = P ≀ H P . Let p > 2. Then Q and P are Sylow subgroups of Alt(p) and hence conjugate in Alt(p). Therefore, there exists s ∈ S such that
C s ≤ (Q ≀ H P ) s = P ≀ H P = N, as required.
3. Imprimitivity systems of feasible groups 3.1. Feasible groups. Let K ≤ Sym(Ω) be a transitive group and D a minimal imprimitivity system of K. We assume that K is feasible with respect to D, by which we mean that D is normal, i.e.,
Orb(K D , Ω) = D,
and K ∆ is a non-solvable group containing a regular cyclic subgroup for some (and hence for all) ∆ ∈ D. In what follows, S = Soc(K D ).
Lemma 3.1. In the above notation, Orb(S, Ω) = D. Moreover, for each ∆ ∈ D, the following statements hold:
(1) S ∆ = Soc(K ∆ ), (2) S ∆ is a 2-transitive non-abelian simple group.
Proof. The characteristic subgroup S of the group K D K is normal in K. Therefore, Orb(S, Ω) is a nontrivial imprimitive system of K that is a refinement of D. Since the imprimitive system is minimal and normal, this implies that
Orb(S, Ω) = Orb(K D , Ω) = D. Next, let ∆ ∈ D. Then Soc(K ∆ ) is a simple group (Theorem 2.1). Since S ∆ is a nontrivial normal subgroup of K ∆ , we have Soc(K ∆ ) ≤ S ∆ .
Since S ∆ is a direct product of simple groups and Soc(K ∆ ) is normal in S ∆ , we obtain that S ∆ = Soc(K ∆ )C S ∆ (Soc(K ∆ )). It follows from Theorem 2.1 that Soc(K ∆ ) is a non-abelian 2-transitive simple group with trivial C Sym(∆) (Soc(K ∆ ). Therefore Soc(K ∆ ) = S ∆ implying both statements (1) and (2).
From Theorem 2.1, it follows that if a permutation group K contains a regular cyclic subgroup, then K is feasible with respect to every minimal imprimitivity system D such that the group K ∆ is not solvable for some (and hence for all) ∆ ∈ D.
3.2. An imprimitivity system induced by stabilizers. Throughout this subsection, K is a feasible group with respect to a minimal imprimitivity system D.
In the notation of Subsection 3.1, we define a binary relation ∼ on the set Ω by setting α ∼ β ⇔ S α and S β are S-conjugate.
Clearly, this is an equivalence relation. Since S is normal in K, the equivalence classes of ∼ form an imprimitivity system E of K.
From Lemma 3.1, it follows that D is a refinement of E. In a natural way, this induces an equivalence relation on D, which is denoted again by ∼; thus for all ∆ and Γ in D, we have ∆ ∼ Γ if and only if S α and S β are conjugate in S for all α, β belonging to ∆ ∪ Γ. In the following statement, we identify a bijection with its graph treated as a binary relation. Proof. To implication ⇐ in formula (2) immediately follows from the equality S δ = S f (δ) that holds for all δ ∈ ∆ whenever f ∈ Orb(S, ∆ × Γ). Conversely, assume ∆ ∼ Γ and take δ ∈ ∆. Then there exists γ ∈ Γ such that S δ = S γ : indeed, since the groups S δ and S γ ′ are S-conjugate for all γ ′ ∈ Γ, we have
S δ = (S γ ′ ) s = S γ ,
for some s ∈ S, where γ = (γ ′ ) s . Next, the mapping f : δ s → γ s , s ∈ S, is well-defined, because if δ s = δ t for some t ∈ S, then S γ s = S δ s = S δ t = S γ t and hence γ s = γ t by the 2-transitivity of S Γ (statement (2) of Lemma 3.1). Similarly, the 2-transitivity of S ∆ implies that the mapping f is a bijection. This proves equality (2), because from [15, Corollary 13, p. 86] it follows that the group S being 2-transitive on ∆ and Γ has at most two orbits in the action on ∆ × Γ.
For a class Λ ∈ E, denote by 1 DΛ the identity subgroup on the set D Λ of all classes Γ ∈ D contained in Λ. Given a class ∆ ∈ D Λ , we define a bijection
(3) f Λ : Λ → ∆ × D Λ , γ → (f Γ,∆ (γ), Γ),
where Γ ∈ D is a unique block containing γ and f Γ,∆ is the inverse to the bijection f defined in Lemma 3.2.
Theorem 3.3. Let K be a feasible group with respect to a minimal imprimitivity system D. Then for each Λ ∈ E and each ∆ ∈ D Λ , the bijection
(3) induces a permutation isomorphism from (K D ) Λ onto the direct product (K D ) ∆ × 1 DΛ .
Proof. Note that the set ∆ is of cardinality at least five by statement (2) of Lemma 3.1. In the notation of Lemma 3.2, this immediately implies that |f | < |f c |. Therefore, the bijection f is K D -invariant by the normality of the group S in K D . It follows that for each Γ ∈ D Λ , all k ∈ K D and γ ∈ Γ, we have
(γ k ) fΛ = (f (γ k ), Γ) = (f (γ) k , Γ) = (f (γ), Γ) (k,1) = (γ fΛ ) (k,1) ,
where f = f Γ,∆ . This proves that f is a permutation isomorphism from (K D ) Λ onto (K D ) ∆ × 1 DΛ .
3.
3. An imprimitivity system associated with S. The group S is the direct product of, say d, simple groups S i , i = 1, . . . , d. By Lemma 3.1, every set ∆ ∈ D is S i -invariant. Therefore the group (S i ) ∆ is either trivial or isomorphic to S i . Set
E ′ = {Ω 1 , . . . , Ω d }, where Ω i = (Si) ∆ =1 ∆.
It is easy to see that D is a refinement of E ′ and the restriction of S i to Ω i is isomorphic to S i . Furthermore, by statement (2) of Lemma 3.1 the sets Ω i are pairwise disjoint. Consequently, E ′ is a partition of Ω. This shows that
(4) S = d i=1 (S i ) Ωi .
Note that by the transitivity of K and statement (1) of Lemma 3.1, the groups Soc(S ∆ ) with ∆ ∈ D are pairwise isomorphic. Since S i ∼ = (S i ) ∆ , this implies that the groups S i are pairwise isomorphic too.
Lemma 3.4. Let K be a feasible group with respect to the imrimitivity system D.
Then the set E ′ is an imprimitivity system of K. Moreover, E is a refinement of E ′ .
Proof. The simple groups S 1 , . . . , S d are uniquely determined up to permutation of indices. Therefore, the group K acts on the set of all of them by conjugation and hence permutes the sets Ω i . This proves the first statement of the lemma. Next, in view of formula (4) for each i and δ ∈ Ω i , we have
(5) S δ = (S i ) δ j =i S j .
Since the S i are isomorphic non-abelian simple groups, this implies that the groups S δ and S γ are not S-conjugate unless γ ∈ Ω i . Therefore, the class of E containing δ is a subset of Ω i . Thus, E is a refinement of E ′ .
Imprimitive groups containing regular cyclic subgroups
4.1. The imprimitivity systems E and E ′ are equal. In this subsection, we apply the theory developed in Section 3 to feasible groups containing a regular cyclic subgroup.
Theorem 4.1. Let K ≤ Sym(Ω) be a feasible group with respect to a minimal imprimitivity system D. Suppose that K contains a regular cyclic subgroup. Then the imprimitivity systems E and E ′ defined in Subsections 3.2 and 3.3 coincide.
Let C be a regular cyclic subgroup of the group K. Then in the notation of Section 3, the group (S i ) Ωi C Ωi satisfies the hypothesis of Theorem 4.2 below. Therefore in view of Remark 3.5, the statement of Theorem 4.1 is an immediately consequence of Theorem 4.2, the proof of which occupies the rest of this subsection. of the group K is minimal. Then for any α, β ∈ Ω, the groups S α and S β are conjugate in S.
Proof. To prove the first statement, without loss of generality we may assume that K = SC, where C is a regular cyclic subgroup of K. It is easy to see that in this case (6) K D = SC D . and the cyclic group C D is regular. Note that by statement (2) of Lemma 3.1, for any set ∆ ∈ D, the group (K D ) ∆ ≥ S ∆ is primitive and contains a regular subgroup C ∆ .
Recall that the group S being simple acts on a set ∆ ∈ D faithfully. Therefore, the intersection of S with the pointwise stabilizer K ∆ of the set ∆ in the group K is trivial. Since K = SC, this shows that K ∆ K D is a cyclic group. Consequently,
H = (K ∆ ) c : c ∈ C
is a normal solvable subgroup of K D . Therefore, H intersects S trivially and hence is cyclic. It follows that each group (K ∆ ) c ≤ H is also cyclic. Since all these groups are of the same order, they must be equal. Thus, the group K ∆ and hence the group (K D ) ∆ fixes each point of the set Ω. This proves the following statement. Each c ∈ C induces the automorphism σ c : k → k c of the group K D that centralizes the group C D . Therefore π −1 σ c π is an automorphism of the primitive group (K D ) ∆ that centralizes the regular cyclic subgroup (C D ) ∆ (Lemma 4.3). By Lemma 2.2, this implies that π −1 σ c π is an inner automorphism of the group (K D ) ∆ corresponding to a certain element k ′ ∈ (K D ) ∆ . Thus, σ c equals the inner automorphism of K D corresponding to the element k = π −1 (k ′ ).
Lemma 4.4. For each c ∈ C, there exists k ∈ K D such that x c = x k for all x ∈ K D .
To complete the proof of the first statement of Theorem 4.2, let α, β ∈ Ω. By the transitivity of C, there exists c ∈ C such that β = α c . In view of Lemma 4.4, one can find k ∈ K D , for which
(7) S β = S α c = (S α ) c = (S α ) k = S α k .
Note that by the definition of the group K D , the points α and α k belongs to the same orbit of the group S. Therefore, α k = α s for some s ∈ S. Thus,
S α k = S α s = (S α ) s ,
which together with (7) shows that S β = (S α ) s , as required.
4.2.
The embedding into the wreath product. Under the hypothesis of Theorem 4.1, we fix a K-block ∆ ∈ D and arbitrary elements k Λ ∈ K taking the class Λ ∆ ∈ E containing ∆ to the class Λ ∈ E (here we assume that k Λ∆ = 1). Let us define a bijection
(8) f * : Ω → ∆ × D, γ → (γ * , Γ)
with γ * = γ kΛ fΛ ∆ , where Λ is the class of E that contains γ (and hence Γ) and f Λ∆ is the bijection defined in formula (3) for Λ = Λ ∆ . Thus if K * = K f * , then
(9) K * ≤ Sym(∆) ≀ Sym(D),
where the wreath product on the right-hand side is considered in the imprimitive action.
Theorem 4.5. Under the identification of K and K * via the bijection f * , the following statements hold:
(1) K D is the direct sum ot the permutation groups (K D ) Λ , Λ ∈ E,(2)(K D ) Λ = (K D ) ∆ × 1 DΛ for all Λ ∈ E.
Proof. Statement (1) follows from Theorem 4.1 and formula (4), whereas statement (2) is a straightforward consequence of Theorem 3.3.
Let H be a regular cyclic subgroup of the group K ∆ . The wreath product on the right-hand side of inclusion (9) contains the subgroup Note that this group contains the group N (H) defined in (1). The following statement is crucial for our arguments.
Theorem 4.6. Let K ≤ Sym(Ω) be a transitive group and ∆ a minimal K-block. Suppose that K ∆ is a non-solvable group containing a regular cyclic subgroup H. Then (1) K is a feasible group with respect to the imprimitivity system D = ∆ K , (2) the group K * ∩ W * controls the regular cyclic subgroups of K * .
Proof. The first statement follows from the definition. To prove the second one, we identify K and K * via the bijection f * . In what follows, we assume that the imprimitivity system E consists of d ≥ 1 blocks, say Λ 1 , . . . , Λ d . The number e of the blocks of D contained in Λ i does not depend on i; these blocks are denoted by ∆ i1 , . . . , ∆ ie . In this notations, |D| = de and ∆ ij = ∆.
It suffices to verify that for every regular cyclic subgroup C ≤ K there exists an element k ∈ K such that C k ≤ W * . To this end, we make use of Theorem 4.5 to write a generator c of the group C D ≤ K D in the form
(12) c = (c Λ1 , . . . , c Λ d ) = (c 1 , . . . , c 1 e , · · · , c d , . . . , c d e )
where c i ∈ (K D ) ∆ for i = 1, . . . , d. Note that K ∆ is a primitive group containing a cyclic regular subgroup H. Therefore by Theorem 2.4, one can find an element s i ∈ S ∆ such that
(13) (c i ) si ≤ N (H), i = 1, . . . , d,
where N (H) is the group defined by formula (1). By statement (1) of Theorem 4.5, the permutation s ∈ Sym(Ω) such that s ∆ij = s i for all i, j, belongs to the group S ≤ K. In particular, c s ∈ K. Together with formula (13), this shows that
(14) (C s ) D ≤ N (H) × · · · × N (H) de and (C s ) D ≤ K D .
At this point, we make use of an obvious permutation isomorphism from the group N * (H) onto the wreath product N Θ (P Θ ) ≀ N P (H P ) with a fixed Θ ∈ P to identify the set ∆ with Θ × P. Then by the associativity of the wreath product, we have
(15) W * = N Θ (P Θ ) ≀ (N P (H P ) ≀ K D ).
Note that by the first inclusion in (14), the set Θ is a block of the group C s ; denote by P * the corresponding imprimitivity system.
To complete the proof of Theorem 4.6, we use Lemma 6.1 and Corollary 6.2 proved in the Appendix (Section 6). Namely, in view of the second inclusion of (14), the groups A ′ = N P (H P ), B ′ = K D , C ′ = (C s ) P * and the sets ∆ ′ = P and Γ ′ = D satisfy the hypothesis of Lemma 6.1. Next, by the definition of N (H), the group C 0 = (C s ) P = H P is regular and cyclic. Therefore,
N P (C 0 ) = N P (H P ) = A ′ .
Thus the condition of Corollary 6.2 follows from the first inclusion in (14). Thus, by this corollary we obtain (16) (C s ) P * ≤ N P (H P ) ≀ K D .
Now we again apply Corollary 6.2 but this time to the groups
A ′ = N Θ (P Θ ), B ′ = N Θ (H P ) ≀ K D , C ′ = C s
and the sets ∆ ′ = Θ and Γ ′ = P * . Note that the hypothesis of Lemma 6.1 follows from the definition of P * and formula (16), respectively. The first inclusion (14) and the definition of N (H) imply that (C s ) Θ is a regular cyclic subgroup of the group N (H) Θ = N Θ (P Θ ) ∼ = AGL(1, p) Since p = |Θ| is a prime, this subgroup is unique and hence the condition of Corollary 6.2 is also satisfied. Thus, by this corollary and formula (15) we conclude that
G s ≤ A ′ ≀ B ′ = N Θ (P Θ ) ≀ (N P (H P ) ≀ K D ) = W * as required.
The main algorithm and poof of Theorem 1.2
The Main Algorithm below finds a solvable subgroup M of a given permutation group K that controls its regular cyclic subgroups. In particular, cyc(M ) = ∅, whenever cyc(K) = ∅. Except for the algorithm constructed in the proof of Lemma 2.3, we use the standard algorithms for computing with permutation groups [14,Section 3.1] and the algorithm in [9,Corollary 6.4] finding the intersection of two groups in Sym(n) in time poly(n), whenever one of them is solvable.
Main Algorithm.
Input: a transitive permutation group K ≤ Sym(n). Output: a solvable group M ≤ K that controls the regular cyclic subgroups K.
Step 1. If n = 1, then output M = K. Find a minimal K-block ∆ and the imprimitive system D containing ∆. If D is not normal, then output M = {id Ω }.
Step 2. Recursively apply the algorithm to the group K D ≤ Sym(D); replace K by the full preimage in Sym(∆) of the resulting group.
Step 3. If the group K is solvable or intransitive, then output M = K or {id Ω }, respectively.
Step 4. Apply Lemma 2.3 to check whether K ∆ contains a regular cyclic subgroup H. If there is no such H, then output M = {id Ω }.
Step 5. Find the bijection f * : Ω → ∆ × D and the group W * = W * (∆, H), defined by formulas (8) and (10), respectively.
Step 6. Output the full f * -preimage M of the group K * ∩ W * , where K * = K f * .
Let us prove the correctness of the algorithm. The output at Step 1 is obviously correct. After Step 2, we may assume by induction on n that K D is a solvable group such that if G is a regular cyclic subgroup of the input group, then
(G D ) k D ∈ cyc(K D )
for some element k of the input group. Therefore, G k ∈ cyc(K). This means that the group K controls the regular cyclic subgroups of the input group. Thus, the output at Step 3 is correct. The correctness of Step 4 follows from the fact that G ∆ ∈ cyc(K ∆ ) for all G ∈ cyc(K) and ∆ ∈ D.
Lemma 5.1. At Step 5, the imprimitivity system D of the group K is still minimal and K is a feasible group with respect to D.
Proof. Indeed, the group cyc(K ∆ ) = ∅ by Step 4. Moreover, K ∆ is not solvable, for otherwise K is isomorphic to a subgroup of a solvable group K ∆ ≀ K D and hence is solvable in contrast to Step 3. This implies that the group
(K 0 ) ∆ ≥ K ∆
is also non-solvable, where K 0 denotes the input group. Therefore, K 0 is a feasible group with respect to D. By Lemma 3.1, this implies that
D = Orb(S 0 , Ω),
where S 0 = Soc((K 0 ) D ). Since obviously S 0 ≤ K, the group K ∆ ≥ (S 0 ) ∆ is primitive by statement (2) of that lemma. Thus D is a minimal imprimitive system of K.
By Lemma 5.1 and statement (2) of Theorem 4.6, the group K * ∩W * constructed at Step 6 controls the regular cyclic subgroups of K * . Moreover, it is solvable as a subgroup of a solvable group W * = N * (H) ≀ K D : indeed, N * (H) is solvable by its construction, whereas K D is solvable by the induction. Thus, the ouput group M ≤ K at Step 6 is solvable and controls the regular cyclic subgroups K. The correctness of the Main Algorithm is completely proved.
To estimate the running time f (n) of the Main Agorithm, we note that Steps 1 and 3 run in polynomial time in n. The same is true for Step 4 by Lemma 2.3.
Finding the bijection f * at Step 5 is reduced to finding the orbits of S = Soc(K D ) on the set Ω × Ω (Lemma 3.2) and hence can efficiently be implemented in time poly(n). Finally, within the same time one can find the intersection of K * and solvable group W * (see the remark before the Main Algorithm). Since Step 2 can easily be implemented in time f (n/m) + n c , where m = |∆| is a divisor of n and c is a constant, we get f (n) ≤ f (n/m) + n c . Taking into account that m ≥ 2, we conclude that f (n) = n O(1) , as required.
Appendix. Lemma on the wreath product
In this section, we establish a sufficient condition for a permutation group to be a subgroup of a wreath product in the imprimitive action. Below given sets ∆ and Γ, we denote by D(∆, Γ) the partition of ∆ × Γ into the subsets
(17) ∆ γ = ∆ × {γ}, γ ∈ Γ,
which are identified with ∆ with the help of the bijection (δ, γ) → δ. A permutation g ∈ Sym(∆ × Γ) belongs to the wreath product Sym(∆) ≀ Sym(Γ) if and only if it preserves the partition D(∆, Γ). In this case, g permutes the blocks (17) via
(∆ γ ) g = ∆ γ g .
For each γ ∈ Γ, the permutation g induces a permutation g(γ) ∈ Sym(∆) such that (δ, γ) g = (δ g(γ) , γ g ). From the definition, it immediately follows that
(18) (f −1 )(γ f ) = f (γ) −1 and (f g)(γ) = f (γ)g(γ f ) for all f, g ∈ Sym(∆) ≀ Sym(Γ).
In what follows, we set C(γ) = {c(γ) : c ∈ C} for all C ⊆ Sym(∆) ≀ Sym(Γ) and γ ∈ Γ.
The well-known Kaloujnine-Krasner embedding Theorem [2, Theorem 2.6A] implies that an imprimitive permutation group is embedded into a wreath product of its block restriction and the quotient. The statement below gives necessary and sufficient conditions for a permutation group to be contained in a given wreath product of two permutation groups. In the sequel to avoid a confusion, we denote by Fun(X, Y ) (not by Y X ) the set of all functions from X to Y . (1) if C ≤ A ≀ B, then for each γ ∈ Γ, there exist t ∈ Fun(Γ, A) and C 0 ≤ A such that (tCt −1 ) ∆γ = C 0 × 1 {γ} ; (2) if C 0 ≤ A is such that C ∆γ = C 0 × 1 {γ} for all γ, then tCt −1 ≤ C 0 ≀ B for some t ∈ Fun(Γ, N ∆ (C 0 )).
Proof. Throughout the proof, we fix an arbitrary γ 0 ∈ Γ. By the transitivity of the action of C on Γ, for each γ ∈ Γ one can find c γ ∈ C such that γ cγ 0 = γ. We assume that c γ0 = 1 ∆×Γ . Set t to be a permutation in Fun(Γ, Sym(∆)) such that (19) t(γ) := c γ (γ 0 ) for all γ ∈ Γ.
To prove statement (1), assume that C ≤ A ≀ B. Then c(γ) ∈ A for all c ∈ C and γ ∈ Γ. Therefore the permutation t defined in (19) belongs to Fun(Γ, A). To show that t is the required element, we first note that C γ (γ) is a subgroup of the group A and C ∆γ = C γ (γ) × 1 {γ} , where C γ = C {∆γ } . Now, from the obvious equality c −1 γ C 0 c γ = C γ , where C 0 = C γ0 . Therefore, by the second formula in (18), we obtain C 0 (γ 0 )c γ (γ 0 ) = (C 0 c γ )(γ 0 ) = (c γ C γ )(γ 0 ) = c γ (γ 0 )C γ (γ cγ 0 ) and hence C 0 (γ 0 ) cγ (γ0) = C γ (γ).
Thus, ((tCt −1 ) γ )(γ) = t(γ)C γ (γ)t −1 (γ) = c γ (γ 0 )C γ (γ)c γ (γ 0 ) −1 = C 0 (γ 0 ) = C 0 , as required.
To prove statement (2), assume that C D ≤ B. Since D is an imprimitive system of C, this implies that C ≤ Sym(∆) ≀ Sym(Γ) and the permutations c(γ) ∈ Sym(∆) are well-defined for every c ∈ C. For each d ∈ C and γ ∈ Γ, formulas (18) imply that
(20) (tdt −1 )(γ) = t(γ)d(γ t )t −1 (γ td ) = c γ (γ 0 )d(γ)t −1 (γ d ) = c γ (γ 0 )d(γ)t((γ d ) −1 ) = c γ (γ 0 )d(γ)(c γ d (γ 0 )) −1 = (c γ d(c γ d ) −1 )(γ 0 ),
where t is the permutation defined in (19). However, γ This together with (20) implies that (tdt −1 )(γ) ∈ C 0 . Since (tdt −1 ) ∈ C D ≤ B, we conclude that tdt −1 ∈ C 0 ≀ B. It remains to show that t ∈ Fun(Γ, N ∆ (C 0 )), or, equivalently, that c γ (γ 0 ) ∈ N ∆ (C 0 ) for each γ ∈ Γ. To this end, take arbitrary γ. Then by the assumption of statement (2), given f ∈ C 0 , f × 1 {γ} ∈ C ∆γ and hence there exists d ∈ C {∆γ } such that d(γ) = f . Note that γ d = γ, and also (tdt −1 )(γ) ∈ C 0 . Thus, by formula (20), we conclude that
c γ (γ 0 ) f c γ (γ 0 ) −1 = c γ (γ 0 ) d(γ) c γ d (γ 0 ) −1 ∈ C 0 ,
whence c γ (γ 0 ) ∈ N ∆ (C 0 ), as desired.
Theorem 1.1. A cycle base of any permutation group of degree n can be constructed time poly(n).
Finally
, let S = Alt(Ω), h a generator of H, and c ∈ Sym(Ω) a full cycle. Then the cyclic representations of the permutations h m and c m , where m = n/p, consist of m > 1 cycles of length p. Therefore, these permutations are conjugate in S (see, e.g. [5, Lemma 1.2.10]). Thus, without loss of generality, we may assume that Orb(Q, Ω) = Orb(P, Ω) = P,
Lemma 3. 2 .
2Let ∆, Γ ∈ D. Then ∆ ∼ Γ if and only if there exists a bijection f : ∆ → Γ such that (2) Orb(S, ∆ × Γ) = {f, f c }, where f c is the complement of f in ∆ × Γ.
Remark 3. 5 .
5From Lemma 3.4 and formula 3.4, it immediately follows that E = E ′ if and only if (S i ) δ = (S i ) γ for all i = 1, . . . , d and all δ, γ ∈ Ω i .
Theorem 4 . 2 .
42Let K ≤ Sym(Ω) be a group containing a regular cyclic subgroup and S K a non-abelian simple group. Assume that the imprimitivity system D = Orb(S, Ω)
Lemma 4 . 3 .
43For each ∆ ∈ D, the restriction epimorphism π : K D → (K D ) ∆ is an isomorphism.
( 10 )
10W * = W * (H, ∆) = N * (H) ≀ K Dwhere the group N * (H) is defined as follows. Let the group P = P (H) and the imprimitivity system P be as in Subsection 2.2. Then(11) N * (H) = N Ω (H) Θ∈P N Θ (P Θ ).
Lemma 6 . 1 .
61Let A ≤ Sym(∆) and B ≤ Sym(Γ), and let C be a transitive subgroup of Sym(∆ × Γ). Suppose that D = D(∆, Γ) is an imprimitivity system of C and C D ≤ B. Then
γ d(c γ d ) −1 )(γ 0 ) ∈ C {∆γ 0 } (γ 0 ) = C 0 .
Corollary 6 . 2 .
62Under the hypothesis of statement (2) of Lemma 6.1, assumeN ∆ (C 0 ) ≤ A. Then C ≤ A ≀ B.
Table 4 .
45.1],
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| []
|
[
"SET-VALUED SORTING INDEX AND JOINT EQUIDISTRIBUTIONS",
"SET-VALUED SORTING INDEX AND JOINT EQUIDISTRIBUTIONS"
]
| [
"Sen-Peng Eu ",
"ANDYuan-Hsun Lo ",
"Tsai-Lien Wong "
]
| []
| []
| Recently Petersen defined a new Mahonian index sor over the symmetric group Sn and proved that (inv, rmin) and (sor, cyc) have the same joint distribution. Foata and Han proved that the pairs of set-valued statistics (Cyc, Rmil), (Cyc, Lmap), (Rmil, Lmap) have the same joint distribution over Sn.In this paper we introduce the set-valued statistics Inv, Lmil, Sor and Lmic1 and generalize simultaneously results of Petersen and Foata-Han and find many equidistributed triples of set-valued statistics and quadruples of statistics.2010 Mathematics Subject Classification. 05A05, 05A19. | null | [
"https://arxiv.org/pdf/1403.2165v1.pdf"
]
| 119,630,820 | 1403.2165 | 701507d28e7fc19f5ba1b464906f670283fff970 |
SET-VALUED SORTING INDEX AND JOINT EQUIDISTRIBUTIONS
Sen-Peng Eu
ANDYuan-Hsun Lo
Tsai-Lien Wong
SET-VALUED SORTING INDEX AND JOINT EQUIDISTRIBUTIONS
Recently Petersen defined a new Mahonian index sor over the symmetric group Sn and proved that (inv, rmin) and (sor, cyc) have the same joint distribution. Foata and Han proved that the pairs of set-valued statistics (Cyc, Rmil), (Cyc, Lmap), (Rmil, Lmap) have the same joint distribution over Sn.In this paper we introduce the set-valued statistics Inv, Lmil, Sor and Lmic1 and generalize simultaneously results of Petersen and Foata-Han and find many equidistributed triples of set-valued statistics and quadruples of statistics.2010 Mathematics Subject Classification. 05A05, 05A19.
Introduction
Let S n be the symmetric group of [n] := {1, 2, . . . , n}. For σ = σ 1 σ 2 . . . σ n ∈ S n , define the inversion statistic inv by inv(σ) := #{(i, j) : i < j and σ i < σ j }, and the cycle statistic cyc(σ) by cyc(σ) := the number of cycles in the cycle decomposition of σ.
We say a permutation statistic over S n is Mahonian if it is equidistributed with inv, and is Stirling if with cyc. For example, it is well known that the right to left minimum statistic rmin, defined by rmin(σ) := #{σ i : σ i < σ j for all j > i}, is Stirling. Recently Petersen found a new Mahonian statistic sor, called the sorting index (see Section 2 for definition), and proved the following: Theorem 1.1 (Petersen [8]). The pairs of statistics (inv, rmin) and (sor, cyc) have the same joint distribution over S n . Also, we have σ∈Sn q inv(σ) x rmin(σ) = σ∈Sn q sor(σ) x cyc(σ)
= x n r=2 (x + [r] q − 1),
where [r] q := 1 + q + q 2 + · · · + q r−1 .
A combinatorial proof is found by Chen et al. [2] via a bijection φ : S n → S n such that (inv, rmin)σ = (sor, cyc)φ(σ), where (inv, rmin)σ means (inv(σ), rmin(σ)). The bijection φ turns out to be of the form
φ = B(σ) −1 • A(σ),
a composition of B-code and A-code introduced by Foata and Han [6]. By defining the set-values statistics right-to-left minimum letters Rmil, left-to-right maximum places Lmap and the cycle set Cyc respectively by (1) For σ ∈ S n , we have
(Rmil, Lmap)σ = (Cyc, Lmap)φ(σ).
(2) The set-valued statistics (Cyc, Rmil), (Cyc, Lmap), (Rmil, Lmap) are symmetric and joint equidistributed over S n .
The motivation of this work is to generalize above two theorems, to see if there is a set-valued version of Petersen's result or other pairs of set-valued statistics having the same distributionà la Foata and Han. It turns out that we can have them both.
Throughout the paper a statistic is set-valued if and only if the first letter is in capital. By introducing new set-valued statistics Inv, Lmil, Sor and Lmic 1 (see Section 2 for definitions) and corresponding ordinary statistics lmin, lmic 1 , our first main theorem extends simultaneously both Petersen and Foata-Han's results. Theorem 1.3. We have:
(1) For σ ∈ S n , the following holds:
(Inv, Rmil, Lmap, Lmil)σ = (Sor, Cyc, Lmap, Lmic 1 )φ(σ).
(2) The quadruple statistics (inv, rmin, lmax, lmin) and (sor, cyc, lmax, lmic 1 ) have the same joint distribution over S n , and σ∈Sn q inv(σ) x rmin(σ) y lmin(σ) = σ∈Sn q sor(σ) x cyc(σ) y lmic 1 (σ) = xy n r=2 (x + [r] q + yq r−1 − 1 − q r−1 ). Theorem 1.3 generalizes Theorem 1.1 and Theorem 1.2 (1). We will see that Theorem 1.2 (2) will be generalized in a later theorem.
By abuse of terminology, a set-valued statistic is called Mahonian (or Stirling) if the corresponding ordinary statistic is so. Our second result is to find triples of set-valued Stirling statistics having the same joint distribution as (Rmil, Lmap, Lmil) and (Cyc, Lmic 1 , Lmap). By switching between certain set-valued statistics by applying invese, reverse, or complement operations on permutations, we obtain 8 more (and 4 partial) triples of set-valued statistics having the same joint distribution. See Theorem 4.2 for the complete list. From these we obtain two sets of symmetric and equidistributed pairs of set-valued statistics, the first of which includes those three pairs in Theorem 1.2 (2).
The third part of the work is to derive quadruples of statistics which are joint equidistributed with (inv, rmin, lmax, lmin) and (sor, cyc, lmic 1 , lmax). Note that the first statistic is Mahonian and the others are Stirling. Again by switching among statistics, in Theorem 5.2 we derive 10 more (and 11 partial) quadruples of statistics having the same joint distribution.
The rest of the paper is organized as follows. Definitions and preliminary results will be put in Section 2. In Section 3 we prove Theorem 1.3. Section 4 is devoted to triples of set-valued statistics, and Section 5 to quadruples of ordinary statistics.
Preliminary results
2.1.
A-code and B-code. We first introduce the A-and B-code of Foata and Han [6], which are the key tools of this paper. Given σ ∈ S n , define its Lehmer code [7] by Leh(σ) := ( 1 , 2 , . . . , n ),
where i = |{j : 1 ≤ j ≤ i, σ j ≤ σ i }|. Let L n := {( 1 , 2 , . . . , n ) : 1 ≤ i ≤ i for 1 ≤ i ≤ n}.
It is clear that Leh : S n → L n is a bijection. The A-code of a permutation σ is defined by
A(σ) := Leh(σ −1 ).
For example, let σ = 2413765. Then σ −1 = 3142765 and A(σ) = (1, 1, 3, 2, 5, 5, 5).
The B-code of σ is defined in the following way. For each i = 1, . . . , n, let k i ≥ 1 be the smallest integer such that
σ −k i (i) ≤ i. Define B(σ) := (b 1 , b 2 , . . . , b n ) with b i = σ −k i (i).
Equivalently, B(σ) can be determined from the cycle decomposition of σ. Assume that i appears in a cycle c. If i is the smallest element of c, then set b i = i; otherwise, choose b i to be the first element j in c with respect to the reverse direction such that j < i. For example, let σ = 2431756 = (124)(3)(576). Then B(σ) = (1, 1, 3, 2, 5, 5, 5). By definition it is easy to see that B-code is a bijection from S n to L n .
2.2.
The set-valued statistic Lmic 1 . Define the left-to-right minimum letters statistic Lmil, the set-valued left-to-right minimum places statistic Lmip and the left to right minimun statistic lmin respectively by
Lmil(σ) := {σ i : σ i < σ j for all j < i}, Lmip(σ) := {i : σ i < σ j for all j < i},
and lmin(σ) := #Lmil(σ) (or #Lmip(σ)).
It is easy to see that Lmil(σ) = Lmip(σ −1 ). For an integer sequence ( 1 , . . . , n ) ∈ L n , define O(( 1 , . . . , n )) := {i : i = 1}, the set of indices with values 1. It turns out that Lmip and Lmil correspond to Lehmer code and A-code respectively. The proof of the following lemma is directly by definition.
Lemma 2.1. We have Lmip(σ) = O(Leh(σ))
and
Lmil(σ) = O(A(σ)).(1)
Hence it is natural to consider the statistic corresponding to B-code. Define the set-valued statistic Lmic 1 by
Lmic 1 (σ) := O(B(σ)).(2)
For example, Lmic 1 (579328164) = O((1, 1, 3, 3, 1, 6, 2, 6, 3)) = {1, 2, 5}.
We have the following combinatorial interpretation of Lmic 1 , which explains the somewhat awkward notation, standing for the left-to-right minimum of the shifted cycle containing 1. Also, Lmil and Lmic 1 are related via φ.
Lemma 2.3. We have Lmil(σ) = Lmic 1 (φ(σ)).
Proof. From (1), (2) and the definition of φ, we have
Lmic 1 (φ(σ)) = O(B(φ(σ))) = O(A(σ)) = Lmil(σ).
2.3. The set-valued statistics Inv and Sor. The goal of this subsection is to define and investigate the set-valued statistics Sor and Inv. First we need the concept of the induced set.
Definition 2.4. Given ( 1 , 2 , . . . , n ) ∈ L n , define its induced set ( 1 , 2 , . . . , n ) according to the following algorithm:
(1) Set S, U with the initial values S = {1, 2, . . . , n} and U = ∅.
(2) For i from n down to 1 do the followings: (a) let i be the i -th smallest element among S, Now we review the sorting index sor of Petersen [8]. Given σ, decompose it uniquely into the product of transpositions σ = (i 1 j 1 )(i 2 j 2 ) . . . (i k j k ) with j 1 < j 2 < · · · < j k and i r < j r for 1 ≤ r ≤ k, and then define In [2] it is proved that inv(σ) = sor(φ(σ)) and
(b) add ordered pairs ( i , j) into U for those j ∈ S with j > i . If there is no such j then skip this step. (c) delete i from S. (3) Define ( 1 , 2 , . . . , n ) := U.sor(σ) = n i=1 (i − b i ),(3)
As for the Inv, since inv(σ) := #{(i, j) : i < j and σ i < σ j }, it is natural to define the set-valued statistic inversion set Inv by Inv(σ) := {(i, j) : i < j and σ i < σ j }.
Similar to the relation between Sor and B-code, we have the following:
Proposition 2.5. For σ ∈ S n , we have Inv(σ) = A(σ) .(5)
Proof. Let σ = σ 1 · · · σ n be the permutation with A(σ) = (a 1 , . . . , a n ). Since A(σ) = Leh(σ −1 ), hence σ −1 n = a n , σ −1 n−1 = (a n−1 )-th smallest element in [n] \ {σ −1 n }. In general, for 1 < i < n we have σ −1 n−i = (a n−i )-th smallest element in [n] \ {σ −1 n , . . . , σ −1 n−i+1 }. Thus σ can be rebuilt from A(σ) as follows. At the initial stage there are n vacancies from left to right. For i from n down to 1, we recursively put letter i into the a i -th vacancy from the left. The resulting permutation is exactly σ.
For example, if A(σ) = (1, 1, 3, 2, 5, 5, 5), then σ can be recovered in the following way:
→ 7 → 76 → 765 → 4 765 → 43765 → 2 43765 → 2143765
Now observe that in each step above, the position we choose for the letter i is exactly i in Definition 2.4. In other words, we can synchronize the rebuilding of σ and the construction of the induced set A(σ) . Moreover, note that the ordered pair ( i , j) is added to U if and only if it is an inversion of σ, for the letter i must be larger than σ j . Thus it must have Inv(σ) = A(σ) .
And finally there is a set version of inv(σ) = sor(φ(σ)): Lemma 2.6. We have Inv(σ) = Sor(φ(σ)).
Proof. From (4), (5) and the definition of φ, we have
Sor(φ(σ)) = B(φ(σ)) = A(σ) = Inv(σ).
Proof of Theorem 1.3
Proof of Theorem 1.3. (1) is obtained by combining Lemma 2.3, 2.6 and Theorem 1.2. The first statement of (2) is directly from (1) and in the following we look at the generating function. For n ≥ 1 let
F n (q, x, y) := σ∈Sn q sor(σ) x cyc(σ) y lmic 1 (σ) .
It is clear that F 1 (q, x, y) = xy. We claim that for n ≥ 2 one has F n (q, x, y) = xy n r=2 (x + [r] q + yq r−1 − 1 − q r−1 ).
Let t i j denote the transposition (ij) and let η 1 = 1, η 2 = 1 + t 12 and η j = 1 + i<j t i j for j ≥ 3. Petersen [8] showed that
η 1 η 2 · · · η n = σ∈Sn σ.(6)
Now define the linear map Θ : S n → Z[q, x, y] by Θ(σ) := q sor(σ) x cyc(σ) y lmic 1 (σ) . Hence by (6) it suffices to show that
Θ(η 1 η 2 · · · η n ) = xy n r=2 (x + [r] q + yq r−1 − 1 − q r−1 ).
It is easy to see that Θ(η 1 ) = xy and Θ(η 1 η 2 ) = xy(x + yq). Let n ≥ 3. We proceed by induction. Suppose
Θ(η 1 η 2 · · · η n−1 ) = xy n−1 r=2 (x + [r] q + yq r−1 − 1 − q r−1 ).
Take σ = σ 1 σ 2 · · · σ n−1 ∈ S n−1 . It can be embedded in S n as σ = σ 1 σ 2 · · · σ n−1 n. Let σ := σt i n for some 1 ≤ i ≤ n and it is clear that sor(σ ) = sor(σ) + (n − i).
Assume that the cycle decomposition of σ is c 1 · · · c m for some m and c t contains the letter i. Hence σ = c 1 · · · c m (n) if i = n and c 1 · · · c t · · · c m if i = n, where c t = (. . . , i, n, σ(i), . . .). Thus
cyc(σ ) = cyc(σ) + 1 if i = n,
cyc(σ) otherwise;
and lmic 1 (σ ) = lmic 1 (σ) + 1 if i = 1,
lmic 1 (σ) otherwise.
So we have Θ(σ · η n ) = Θ(σt n n + σt n−1 n + · · · + σt 1 n ) = Θ(σ)(x + q + q 2 + · · · + q n−2 + yq n−1 ), and therefore
Θ(η 1 η 2 · · · η n ) = Θ σ∈Sn, σ(n)=n σ · η n = σ∈Sn, σ(n)=n Θ(σ · η n )
= (x + q + q 2 + · · · + q n−2 + yq n−1 ) σ∈S n−1
Θ(σ)
= (x + [n] q + yq n−1 − 1 − q n−1 )Θ(η 1 η 2 · · · η n−1 ).
The proof is then completed by induction.
Set-valued joint equidistribution
In this section we seek for more set-valued statistics having the same joint distribution as (Rmil, Lmil, Lmap) and (Cyc, Lmic 1 , Lmap). In the introduction we have defined Rmil (rightto-left minimum letters) and Lmap (left-to-right maximum places) while in Section 2. The idea is quite simple: we look at the relations between these statistics by performing operations of "inverse", "complement", or "reverse" on permutations.
For σ = σ 1 σ 2 . . . σ n ∈ S n , let σ −1 denote its inverse, if Stat 1 (σ) = Stat 2 (χ(σ)) for all σ ∈ S n . Also we define Stat * := {n + 1 − i : i ∈ Stat} for a set-valued statistic Stat, if applicable. It turns out that these eight set-valued statistics are related via the mappings i, r and c. The proof of the following proposition is straightforward and is omitted. (1) A solid edge between Stat 1 and Stat 2 means Stat 1 (σ) = Stat 2 (χ(σ)) and Stat 2 (σ) = Stat 1 (χ(σ)) for σ ∈ S n , with χ = i, r or c as labeled. (2) A dotted edge between Stat 1 and Stat 2 means Stat 1 (σ) = Stat * 2 (χ(σ)) and Stat 2 (σ) = Stat * 1 (χ(σ)) for σ ∈ S n , with χ = i, r or c as labeled.
For examples, we have Rmap (
1) (Rmil, Lmil, Lmap) (2) (Cyc, Lmic 1 , Lmap) (3) (Lmap, Lmip, Rmil) (4) (Rmip * , Rmap * , Lmal * ) (5) (Lmal * , Rmal * , Rmip * ) (6) (Lmil, Rmil, Rmap * ) (7) (Rmal * , Lmal * , Lmip) (8) (Rmap * , Rmip * , Lmil) (9) (Lmip, Lmap, Rmal * ) (10) (Lmap, -, Cyc) (11)(Lmic 1 , Cyc, -) (12) (Cyc, -, Rmil) (13) (Rmil, -, Cyc),
Proof. By Proposition 4.1, we have
(1) i − → (3), (1) i•r•c − −− → (4), (1) r•c − − → (5), (1) r − → (6), (1) c − → (7), (1) i•r − − → (8), (1) i•c − − → (9).
Therefore, (3) to (9) are joint equidistributed with (1). Moreover, by Theorem 1.3, we have
(1) φ − → (2),(3)φ − → (10),(6)φ − → (11).
Finally, (2) i − → (12) and (10) i − → (13) follow from the fact that Cyc(σ) = Cyc(σ −1 ), and the proof is completed.
From the theorem we can read off the following many pairs of set-valued statistics which are symmetric and joint equidistributed. Note that (1) includes those pairs of Foata and Han.
joint equidistributed quadruples
In this section we look at the ordinary number-valued statistics. The goal is to find quadruples of statistics joint equidistributed with (inv, rmin, lmax, lmin) and (sor, cyc, lmax, lmic 1 ). Most of the materials in this section are well known. Our contribution is to relate them with the (sor, cyc, lmax, lmic 1 ) and derive the generating functions with respect to the first three statistics.
The statistics rmin, rmax, lmin, lmax are defined in the obvious way. We may take more familiar Mahonian statistics into consideration. Let Des(σ) := {i : σ i > σ i+1 } be the descent set of σ. It is well known that the statistics major maj, inverse major imaj, reverse major rmaj, charge chg, and cocharge cochg, defined by
maj(σ) := i∈Des i, imaj(σ) := maj(σ −1 ), rmaj(σ) := maj(σ r•c ), chg(σ) := i∈Des(σ −1 ) (n − i), cochg(σ) := i / ∈Des(σ −1 ) (n − i),
are all Mahonian [9]. Again, given two statistics stat 1 , stat 2 and one bijection χ : S n → S n , the notation
stat 1 χ − → stat 2
means stat 1 (σ) = stat 2 (χ(σ)) for all σ ∈ S n . The equidistribution of maj and inv can be proved combinatorially from the celebrated fundamental bijection ψ : S n → S n of Foata [4], that is, one has
maj ψ − → inv.
Moreover, a close look of ψ will also show that (rmax, rmin) ψ − → (rmax, rmin).
We now perform the "inverse-reverse-complement" trick on these statistics. We are ready to state the main result of this section. Theorem 5.2. The following quadruple statistics are joint equidistributed over S n :
(1) (inv, rmin, lmin, lmax) (2) (sor, cyc, lmic 1 , lmax) (3) (inv, lmax, lmin, rmin) (4) (inv, rmin, rmax, lmax) (5) (inv, lmax, rmax, rmin) (6) ( n 2 − inv, lmin, rmin, rmax) (7) ( n 2 − inv, rmax, lmax, lmin) (8) ( n 2 − inv, rmax, rmin, lmin) (9) ( n 2 − inv, lmin, lmax, rmax) (10) (sor, lmax, lmic 1 , cyc) (11) ( n 2 − sor, lmic 1 , cyc, -) (12) ( n 2 − sor, -, lmax, lmic 1 ) (13) ( n 2 − sor, -, cyc, lmic 1 ) (14) ( n 2 − sor, lmic 1 , lmax, -) (15) (-, cyc, -, rmin) (16) (-, rmin, -, cyc) (17) (maj, rmin, rmax, -) (18) (imaj, lmax, rmax, -) (19) (rmaj, lmax, lmin, -) (20) (chg, rmin, lmin, -) (21) (cochg, lmin, rmin, -), and the generating function with respect to the first three statistics in each quadruple is F n (q, x, y) = xy n r=2 (x + [r] q + yq r−1 − 1 − q r−1 ).
Proof. Observe that inv(σ −1 ) = inv(σ) and inv(σ r ) = inv(σ c ) = n 2 − inv(σ) for σ ∈ S n . The proof is done via the following mappings.
Rmil(σ) := {σ i : σ i < σ j for all j > i}, Lmap(σ) := {i : σ i > σ j for all i > j}, and Cyc(σ) := {the smallest number in each cycle of the cycle decomposition}, Foata and Han derived the following set-valued joint equidistribution results.
Theorem 1 . 2 (
12Foata, Han[6]). The followings hold.
Lemma 2 . 2 .
22For σ ∈ S n , write the the cycle containing 1 in the way that 1 is at the end of the cycle and denote the resulting cycle c. ThenLmic 1 (σ) = Lmil( c) by regarding c as a word. Proof. By the definition of B-code, b i = 1 if and only if i ∈ c and all letters on the left of i in c are larger than i. In other words, i is a left-to-right minimum letter in c. Then we have O(B(σ)) = Lmil( c).
For the running example, σ = 579328164 = (1527)(394)(68) and the shifted cycle is c = (5271), hence Lmic 1 (579328164) = Lmil(5271) = {1, 2, 5}.
r − i r ).For example, since σ = 2431765 = (12)(24)(56)(57) we have sor(σ) = (2 − 1) + (4 − 2) + (6 − 5) + (7 − 5) = 6. In other words, sor(σ) measures the total distance of the letters needed to move during the bubble-sorting process. In this example,
which clarifies the relation between sorting index and the B-code (b 1 , b 2 , . . . , b n ) of σ. Observe that in the step (2)(b) of 2.4 we add exactly (i − i ) ordered pairs into U for each i, hence from (3) it makes sense to define the set-valued statistic sorting set Sor by Sor(σ) := B(σ) .
2
2Lmil (left-to-right minimum letters) and Lmip (left-to-right minimum places). Similarly we can defined set-valued statistics Rmip, Lmal, Rmap and Rmal by Rmip(σ) := {i : σ i < σ j for all j > i}, Lmal(σ) := {σ i : σ i > σ j for all j < i}, Rmap(σ) := {i : i > σ j for all j > i}, Rmal(σ) := {i : σ i > σ j for all j > i}.
σ r := (σ n , σ n−1 , . . . , σ 1 ) its reverse and σ c := (n + 1 − σ 1 , n + 1 − σ 2 , . . . , n + 1 − σ n ) its complement. For example, if σ = 364152, then σ −1 = 461352, σ r = 251463 and σ c = 413625. It is clear that the mappings i, r, c : S n → S n , defined by i(σ) := σ −1 , r(σ) := σ r and c(σ) := σ c , are bijections. Given two set-valued statistics Stat 1 , Stat 2 and one bijection χ : S n → S n ,
Figure 1 .
1Relations between eight statistics
Proposition 4 . 1 .
41The relations between the eight set-valued statistics Lmil, Lmip, Rmil, Rmip, Lmal, Lmap, Rmal and Rmap are illustrated by the graph (with edges solid or dotted and labeled by i, r or c) inFigure 1.
We come to the main result of this section.
Theorem 4 . 2 .
42The following triples of set-valued statistics have the same joint distribution over S n . A dash means the statistic is omitted.
Corollary 4 . 3 .
43In each of the following items, the pairs of set-valued statistics are symmetric and joint equidistributed over S n .(1) (Rmil, Lmap), (Rmip d , Lmal d ), (Rmal d , Lmip), (Rmap d , Lmil), (Cyc, Rmil) and (Cyc, Lmap) (2) (Rmil, Lmil), (Lmap, Lmip), (Rmip d , Rmap d ), (Rmal d , Lmal d ), and (Cyc, Lmic 1 ).
Proof. (1),(2) and (4) are obvious. For (3), if i ∈ Des(σ r•c ), then (n+1)−σ r i < (n+1)−σ r i+1 and thus σ n−i > σ n−i+1 , which implies that n − i ∈ Des(σ). Hence we have rmaj(σ) = maj(σ r•c ) =
In this short paper we generalize simultaneously Petersen and Foata-Han's results to more than two statistics and find many triples or quadruples of statistics having the same joint distribution over S n . In all quadruples, the first statistic is Mahonian while the others are Stirling, and we then read off many symmetric equidistributed pairs of Stirling statistics. However, it is well known that there are pairs of Mahonian statistics with a symmetric joint distribution as well[9], for examples, (inv, maj) and (maj, imaj) are two of them. Hence it would be interesting to generalize our results further to include more Mahonian statistics.On the other hand, the generating function obtained in Theorem 1.3 only involves three of the four statistics, hence a natural question is to find a four-variable generating function including lmax as well. We leave them to the interested reader.
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New permutation coding and equidistribution of set-valued statistics. D Foata, G.-H Han, Theoret. Comput. Sci. 410D. Foata, G.-H. Han, New permutation coding and equidistribution of set-valued statistics, Theoret. Comput. Sci., 410 (2009), pp. 3743-3750.
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The sorting index. T Petersen, Kyle, Advances in Applied Mathematics. 47Petersen, T. Kyle, The sorting index, Advances in Applied Mathematics 47.3 (2011): 615-630.
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An interesting new Mahonian permutation statistic. M C Wilson, Electron. J. Combin. 17147M.C. Wilson, An interesting new Mahonian permutation statistic, Electron. J. Combin., 17 (2010), p. R147.
| []
|
[
"Optical cold damping of neutral nanoparticles near the ground state in an optical lattice",
"Optical cold damping of neutral nanoparticles near the ground state in an optical lattice"
]
| [
"Mitsuyoshi Kamba \nDepartment of Physics\nTokyo Institute of Technology\nOokayama 2-12-1, Meguro-ku152-8550Tokyo\n",
"Ryoga Shimizu \nDepartment of Physics\nTokyo Institute of Technology\nOokayama 2-12-1, Meguro-ku152-8550Tokyo\n",
"Kiyotaka Aikawa \nDepartment of Physics\nTokyo Institute of Technology\nOokayama 2-12-1, Meguro-ku152-8550Tokyo\n"
]
| [
"Department of Physics\nTokyo Institute of Technology\nOokayama 2-12-1, Meguro-ku152-8550Tokyo",
"Department of Physics\nTokyo Institute of Technology\nOokayama 2-12-1, Meguro-ku152-8550Tokyo",
"Department of Physics\nTokyo Institute of Technology\nOokayama 2-12-1, Meguro-ku152-8550Tokyo"
]
| []
| We propose and demonstrate purely optical feedback cooling of neutral nanoparticles in an optical lattice to an occupation number of about 1. The cooling force is derived from the optical gradients of displaced optical lattices produced with two sidebands on the trapping laser. To achieve highly accurate position observations required for cooling near the ground state, we reduce the laser intensity noise to a relative power noise of 6 × 10 −8 /Hz in a frequency band of 30 kHz to 600 kHz. We establish a reproducible method for neutralizing nanoparticles at high vacuum via a combination of discharging and irradiating an ultraviolet light. Our results form an important basis for the investigation of quantum mechanical properties of ultracold nanoparticles and are also useful for precision measurements with neutral nanoparticles. | null | [
"https://arxiv.org/pdf/2205.00902v1.pdf"
]
| 248,496,360 | 2205.00902 | 4a32811b6a001fa9e5e6e48b0cc67801374a80af |
Optical cold damping of neutral nanoparticles near the ground state in an optical lattice
May 2022
Mitsuyoshi Kamba
Department of Physics
Tokyo Institute of Technology
Ookayama 2-12-1, Meguro-ku152-8550Tokyo
Ryoga Shimizu
Department of Physics
Tokyo Institute of Technology
Ookayama 2-12-1, Meguro-ku152-8550Tokyo
Kiyotaka Aikawa
Department of Physics
Tokyo Institute of Technology
Ookayama 2-12-1, Meguro-ku152-8550Tokyo
Optical cold damping of neutral nanoparticles near the ground state in an optical lattice
2May 2022(Dated: May 3, 2022)
We propose and demonstrate purely optical feedback cooling of neutral nanoparticles in an optical lattice to an occupation number of about 1. The cooling force is derived from the optical gradients of displaced optical lattices produced with two sidebands on the trapping laser. To achieve highly accurate position observations required for cooling near the ground state, we reduce the laser intensity noise to a relative power noise of 6 × 10 −8 /Hz in a frequency band of 30 kHz to 600 kHz. We establish a reproducible method for neutralizing nanoparticles at high vacuum via a combination of discharging and irradiating an ultraviolet light. Our results form an important basis for the investigation of quantum mechanical properties of ultracold nanoparticles and are also useful for precision measurements with neutral nanoparticles.
I. INTRODUCTION
Optically levitated nanoparticles have attracted great interests in recent years for their diverse applications, ranging from precision measurements [1] and test of fundamental physics [2][3][4][5][6] to various sensory devices such as accelerometers [7][8][9] and magnetometers [10]. Extensive studies have realized cooling of their center-of-mass motion to near the ground state [11][12][13][14][15], opening an intriguing possibility of exploring quantum mechanical properties of the motion of mesoscopic and macroscopic objects [3,16,17].
Up to now, three cooling methods have been demonstrated. The first method is parametric feedback cooling (PFC), where the motion of nanoparticles is optically observed and decelerated by modulating the intensity of the trapping laser [18,19]. The second method is cavity cooling, where a high-finesse optical resonator placed near the trapped nanoparticles removes their kinetic energy [11,[20][21][22]. The third method is electric feedback cooling, or cold damping, where charged nanoparticles are optically observed and their motion is attenuated by an external electric field synchronized to their motion [23][24][25]. While the lowest temperature obtained with PFC is limited to several 100 µK [26], the other two methods are able to reach temperatures of the order of 10 µK near the ground state [11][12][13][14][15].
The recent progresses in cooling nanoparticles draw attention to the possibility of exploring the quantum mechanical properties of their motion. One of the promising approaches to reveal such properties is the quantum state tomography with time-of-flight measurements [17], where nanoparticles are released from the trapping optical potential and recaptured into the same potential. From the amplitude of the oscillation after they are recaptured, their initial momenta are deduced. A previous experimental study reported such measurements with nanoparticles moderately cooled via PFC and found that the stray electric fields near surfaces can exert a strong acceleration on nanoparticles [27]. Stray electric fields near surfaces have been a serious issue also in the field of trapped ions [28]. Such stray fields can be a great obstacle for time-of-flight experiments with nanoparticles cooled via cold damping because this approach is applicable only to charged nanoparticles. While cavity cooling can in principle cool neutral nanoparticles, the presence of a high-finesse res-onator may influence on their motion after the release from the trapping potential.
Here, we propose and demonstrate a new efficient cooling scheme for neutral nanoparticles, which we call optical cold damping. By applying purely optical forces on neutral nanoparticles in an optical lattice, we realize feedback cooling to an occupation number of about 1. We present important advances in optical techniques; first, we enhance the efficiency of collecting the light scattered by nanoparticles with careful alignments of optical layouts. Second, we reduce the noise floor of observing nanoparticles' motion via the active intensity stabilization of the trapping laser to near the shot noise level in a frequency band of 30 kHz to 600 kHz. Laser intensity stabilization near the shot noise limit at such a high frequency range has been relatively unexplored, in spite of numerous previous studies in diverse fields, including gravitational wave detection [29] and cold atom experiments [30]. Third, we develop an optical setup that allows us to apply controllable optical forces on nanoparticles for cooling their motion. The force originates from the optical gradients of displaced optical lattices that are produced by the weak sidebands on the trapping laser. The present work is also an important advance in terms of the wavelength of the laser; instead of the commonly used wavelength of 1064 nm, we use 1550 nm, at which the absorption of light in nanoparticles is much lower and motional heating via photon recoils is also less than at 1064 nm. Despite the difficulty that an optimized objective lens is not readily available at this wavelength, we demonstrate highly accurate observation of the position of nanoparticle, thereby enabling to reach the occupation number of about 1 in a room-temperature environment.
II. THEORETICAL DESCRIPTION OF COLD DAMPING
For the comprehensive understanding of our results, we hereby briefly introduce the theoretical description of the motion of nanoparticles in the presence of feedback cooling and various heating mechanisms [19,23,24,31]. We ignore heating via the laser phase noise (LPN), which is reduced to a negligibly small value in our setup [13]. The one-dimensional equation of motion in the presence of fluctuating forces and damping mechanisms is given by
q + Γ totq + Γ cqn + Ω 2 0 q = F BG + F r m(1)
with Γ tot = Γ BG + Γ c + Γ r . Here q and q n denote the position of nanoparticles and the noise in the feedback signal, respectively, while Γ BG , Γ c , and Γ r denote the damping rate due to collisions with background gases, the damping rate due to feedback cooling, and the damping rate due to photon recoils, respectively. In addition, Ω 0 ,m, F BG , and F r denote the oscillation frequency along an optical lattice, the mass of trapped nanoparticles, the stochastic force from background gases, and the stochastic force from photon scattering, respectively.
The effective motional temperature T eff for the particle following Eq.(1) is given as
T eff =T 0 Γ BG Γ tot + mΩ 2 0 S n Γ 2 c 2k B Γ tot +h ω 0 P sc 5mc 2 k B Γ tot(2)
where T 0 , S n , k B ,h, ω 0 , P sc , c are the temperature of background gases, the power spectral density (PSD) of q n , the Boltzmann constant, the reduced Planck constant, the frequency of the trapping light, the optical power scattered by nanoparticles, and the speed of light, respectively. At sufficiently low pressures, we can ignore the first term in Eq.(2) and write the occupation number in the presence of feedback cooling n eq as
n eq + 1 2 = 1 2Γ ch Ω 0 2γ tot + mΩ 2 0 S n Γ 2 c (3)
γ tot =h ω 0 P sc 5mc 2 (4) which provides a minimum value of n eq + 1/2 = √ 2γ tot mS n /h at an optimum feedback gain of Γ c = 2γ tot /(mΩ 2 0 S n ). For our typical experimental parameters, Γ c is approximately 2π × 5 kHz.
Thus, we find that enhancing the signal-to-noise ratio (SNR) for observing the motion of nanoparticles is crucially important to reach the ground state of the trapping potential. In the context of the theory for a quantum feedback control, it has been known that the efficiency of collecting photons scattered by nanoparticles is directly connected to n eq [32,33].
III. EXPERIMENTAL SETUP
A. Optical and vacuum systems
Our apparatus mainly consists of an optical system and a vacuum chamber. The schematic of our experimental setup is shown in Fig. 1. A single-frequency fiber laser at a wavelength of 2πc/ω 0 = 1550 nm (NKT Photonics, Koheras C15) with a power of 140 mW is incident on a vacuum chamber and is focused by an objective lens with a numerical aperture of 0.85. A one-dimensional optical lattice is formed along the z direction by retro-reflecting the laser with a partially reflective mirror placed inside a vacuum chamber. The distance between the mirror and the trap position d = 14 mm is made as short as possible for minimizing the impact of the LPN on the motion of nanoparticles [13].
We load silica nanoparticles with a radius of R = 166(3) nm and a mass of 4.4(2) × 10 −17 kg in an optical lattice at around 500 Pa via the pulsed laser deposition of powdery samples placed underneath the trap region [13]. We observe nanoparticles with two independent photodetectors; one of which is placed in a feedback loop for cooling and is called in-loop (IL), while the other is used only for estimating T eff in the presence of feedback and is called out-of-loop (OL) [24,25]. We turn on feedback cooling at around 10 Pa and evacuate the chamber to high vacuum. For the charge neutralization of trapped nanoparticles, a deuterium lamp and an electrode for inducing a corona discharge are installed in a vacuum chamber.
In the present study, we focus on the realization of optical cold damping in the z direction, while we cool the motions in the x and y directions via PFC to avoid the escape of trapped nanoparticles at low pressures. We enhance the SNR both by optimizing the optical layouts around the IL and OL photodetectors and by reducing the relative intensity noise (RIN) of the fiber laser. In the following subsections, we explain these crucial aspects of our optical setup. As compared to our previous work [13], where we demonstrated n eq ∼ 3, these modifications improved the SNR by about a factor of 6, implying cooling to the ground state (n eq < 1) is feasible.
B. Enhancement of the signal from nanoparticles
Feedback cooling of nanoparticles to near the ground state requires a high efficiency in collecting scattered photons. In our setup, the scattered light from nanoparticles is collected by the objective lens for focusing the laser and is extracted through an isolator. Approximately 85 % of the extracted light is incident on the IL photodetector and the remaining light is incident on the OL photodetector.
The main difference of the present setup from recent studies by other groups with a single-beam optical trap [11,12,14,15] is that the scattered light (of the order of 100 µW) is nearly overlapped with the strong retro-reflected beam (about 40 mW). While we cannot separate these lights, we recognize that their spatial profiles slightly differ with each other, which is presumably due to the complicated spatial profile of the light scattered by nanoparticles [34]. As a result, the amplitude of the signal in the absence of feedback cooling obtained with photodetectors depends on the focal length of the lens placed in front of the photodetectors and the distance between the lens and the photodetectors. We carefully optimized the optical layouts to maximize the amplitude of the signal in the absence of feedback cooling.
C. Reduction of the RIN of the laser
The RIN of a commercially available low noise fiber laser is much larger than the PSD of nanoparticles near the ground state. In addition, in our setup, the frequency stabilization introduced for reducing the LPN adds intensity noises at around a few 100 kHz, close to Ω 0 /2π. Therefore, most of previous studies, including ours, have used a balanced detection scheme for subtracting the RIN from the photodetector signals [19,23]. However, this scheme is ultimately limited by the fact that the subtracting light adds the shot noise on the observation signal and degrades the SNR by 3 dB, which is a crucial degradation for cooling to the ground state.
In order to reduce the impact of the RIN on the signal without relying on a balanced detection scheme, we implement the direct reduction of the RIN with electro-optic amplitude modulators (EO-AMs). The advantage of EO-AMs over acoustooptic modulators (AOMs) lies in their large feedback bandwidth of several MHz. In our setup, Ω 0 is about 2π × 200 kHz, comparable to the typical bandwidth of AOMs. Therefore, EO-AMs are the only choice in our optical setup. We use two free-space EO-AMs; the first EO-AM plays a dominant role in decreasing the RIN in a frequency range from DC to 5 MHz, while the second EO-AM is added to further reduce the RIN in a frequency range from 10 kHz to 3 MHz. The feedback signals for the stabilization are generated by homebuild PID electronics. In this manner, we obtain the RIN of 6 × 10 −8 /Hz, about 40 % above the shot noise level, in a frequency band of 30 kHz to 600 kHz (Fig. 2).
With the active intensity stabilization, we achieve a higher value of SNR than with a balanced detection scheme. However, as shown in Fig. 2, the feedback introduces intensity noises at high frequencies of more than 1 MHz. Such noises are an obstacle to the measurement of T eff with the OL photodetector, in particular when the PSD is suppressed to near the noise floor via feedback cooling. To avoid this issue, we employ a balanced detection scheme for the OL photodetector. Thanks to the flatness of the noise floor, determined purely by the shot noise, we are able to clearly measure the small signal of nanoparticles near the ground state.
IV. APPLICATION OF AN OPTICAL COOLING FORCE
For the realization of efficient cold damping, it is crucial to apply an external force proportional to the velocity of nanoparticles. As shown in Section II, the magnitude of the cooling force has to be larger than a value determined by γ tot , m, Ω 0 and S n . In our previous work with charged nanoparticles, an electric field with a moderate amplitude of a few V, providing a force of the order of 1 × 10 −15 N, was sufficient to reach n eq ∼ 3 [13]. We require a new alternative approach that can provide a force of such a magnitude without relying on the charge of nanoparticles. We propose and demonstrate a mechanism based on the gradient of an optical potential. Figure 3(a) shows how our optical potentials exert controllable forces on neutral nanoparticles. We fully take advantage of the standing wave structure that allows us to manipulate trapped nanoparticles via the phase modulation of the laser. We generates weak sidebands (about 1.4 % in amplitude) at frequencies of ±ω 1 on the trapping laser. While nanoparticles are trapped at the intensity maximum of the optical lattice with ω 0 , the optical lattices with ω 0 ± ω 1 produce gradients at the position of nanoparticles. Thus, the two sidebands exert forces on trapped nanoparticles in the opposite orientations with each other. The magnitudes of these forces depend on the magnitudes of sidebands, implying that an oscillatory cooling force is yielded when the ratio of the magnitudes of the two sidebands are time-varying. We modulate the ratio of In this way, we realize the generation of controllable optical forces on trapped nanoparticles with a bandwidth of a few MHz. The frequency of the RF has to carefully chosen. For a given amplitude of the sideband, the largest gradient force is obtained at ω 1 = πc/4d, at which each sideband produces an optical lattice displaced by a quarter of a lattice cite at the trapped position. This frequency corresponds to 2π × 2.7 GHz in our setup. In the present study, we employ ω 1 = 2π × 450 MHz because our free-space EO modulators have a bandwidth of a few 100 MHz. With the present setup, we expect that the magnitude of the optical force is on the order of 1 × 10 −14 N, which is larger than we need for realizing cold damping.
To confirm that our idea is properly implemented in our setup, we first measure the amplitude of the cooling force by observing the time variation of the amplitude of the oscillation signal from nanoparticles when the feedback signal is turned on. The feedback signal is obtained by feeding the IL photodetector signal through band-pass filters and an amplifier with a gain of G (Fig. 1). Figure 3(b) shows an example of such measurements. After we let nanoparticles heated by photon recoils for 100 ms, we abruptly turn on the feedback signal and observe how fast the signal amplitude decays. The time constant of the decay directly reveals Γ c in Eq.(1). Figure 3(c) shows the measured values of Γ c for various values of G. We find that Γ c is proportional to the gain over a wide gain range, suggesting that our modulation scheme is valid in this range and an appropriate feedback signal is generated without any measurable nonlinearity of the DBM. The most important aspect of this measurement is that the maximum value of Γ c is turned out to be more than 2π × 10 kHz, which is larger than the value we need for cooling nanoparticles to near the ground state [13,15]. Thus, we find that our scheme is a promising method for cold damping of neutral nanoparticles to near the ground state.
V. OPTICAL COLD DAMPING
We investigate the limit of our new cooling approach by measuring n eq for various experimental conditions. In these measurements, we first measure T eff by comparing the PSDs with and without feedback cooling [ Fig. 4(c)] [19,35], after which n eq is calculated from T eff . Figure 4(a) shows the measured n eq with respect to the pressure, while Figure 4(b) shows the measured n eq with respect to Γ c . At pressures lower than 2 × 10 −5 Pa, we find that the observed n eq agrees well with the theoretically expected value of Eq.(3) with our experimental parameters. The calculation with Eq.(3) is also in good agreement with the measurements of Fig. 4(b) without any fitting parameter. The lowest T eff and the lowest n eq are T eff ∼ 13±2 µK and n eq ∼ 0.85±0.20, respectively, where the error indicates systematic errors from thermal fluctuations.
For comparison, we test conventional cold damping for charged nanoparticles in the present setup, which is readily realized by applying the feedback signal to the electrode along the z direction (the right lens in Fig. 1). We find that the ob- tained n eq is comparable to that with optical cold damping. Furthermore, we find that optical cold damping is even more stable than conventional cold damping. Upon decreasing the pressure, we often observe that the charge number varies at pressures between 1 and 1 × 10 −4 Pa and, in some cases, the charge is inverted, resulting in the loss of nanoparticles under conventional cold damping. Even if they are not lost, with conventional cold damping, the feedback gain has to be carefully adjusted because the ratio of the charge to the mass depends on nanoparticles and can vary during experiments. Such issues are of never concern with optical cold damping. The amplitude of the optical gradient force on nanoparticles is proportional to their polarizability, which is proportional to m [19], indicating that the acceleration of nanoparticles caused by a given amplitude of the feedback signal is independent from m. Therefore, fixed experimental parameters can be used for various nanoparticles. It is obvious that, although our scheme is developed for cooling neutral nanoparticles, it works similarly well for charged nanoparticles.
VI. CHARGE NEUTRALIZATION
Charge neutralization has been an imperative issue in diverse fields, including gravitational wave detection [36], inertial sensors [37], and precision measurements with levitated microparticles [1,4,9]. In many of these studies, irradiat- ing UV lights has been a powerful means for neutralization. In a recent work with nanoparticles, a corona discharge with a voltage of several kV was employed for the charge management [38]. In our setup, we find that the charge number of nanoparticles often varies at pressures between 1 and 1 × 10 −4 Pa. In particular, a frequent charge variation is observed when nanoparticles are brought to low pressures for the first time after they are loaded into an optical lattice. In many cases, nanoparticles are negatively charged after the evacuation. We infer that this behavior is relevant to outgassing from the surface of nanoparticles. Therefore, charge neutralization has to be carried out at pressures of < 1 × 10 −4 Pa. We tested both methods, discharging and irradiating a UV light, in our setup and found that neither of which is suitable for our application. On the one hand, a UV light from a deuterium lamp is found to be strong enough for neutralizing positively charged nanoparticles even at high vacuum, while it turns out that it cannot neutralize negatively charged nanoparticles because it provides dominantly negative charges. On the other hand, although a corona discharge readily occurs at a voltage of 500 V at low to medium vacuum and can neutralize both positively and negatively charged nanoparticles, a discharge hardly occurs with our high voltage capability at high vacuum.
We find that a combined method is the easiest and the most reproducible way to neutralize nanoparticles and to keep them neutralized for many hours. We first apply a high voltage at around 350 Pa, where nanoparticles are positively charged with a typical charge of about 50e with e being the elementary charge. The charge number is measured by observing the response of nanoparticles to a sinusoidal electric field oscillating at a frequency close to Ω 0 [ Fig. 5(a)]. After we confirm that nanoparticles are positively charged, we evacuate the chamber to below 1 × 10 −4 Pa and apply a UV light. Typically nanoparticles are neutralized in several minutes [ Fig. 5(b)]. Nanoparticles neutralized in this manner stay neutralized for tens of hours, although rarely they get charged after several hours. In such a case, we repeat the procedure stated above.
VII. CONCLUSION
We have developed a purely optical feedback cooling scheme for neutral nanoparticles in an optical lattice and demonstrated cooling their motion to n eq ∼ 1. For this purpose, we improved the SNR of observing the motion of nanoparticles both by enhancing the efficiency of collecting the light scattered by nanoparticles and by lowering the RIN of the laser to near the shot noise in a frequency band of 30 kHz to 600 kHz. The strong optical force required for efficient cooling is derived from the optical gradients of the displaced optical lattices generated by two weak sidebands on the trapping laser. By modulating the relative amplitudes of the two sidebands, we exert oscillatory cooling forces on trapped nanoparticles. In addition to realizing the new cooling approach, we established a reproducible procedure to neutralize nanoparticles suitable for experiments at high vacuum.
Our scheme is superior to PFC, the most commonly used optical cooling approach, in terms of the following aspects. First, our scheme modulates the laser at frequencies of a few 100 MHz, much higher than typical oscillation frequencies of nanoparticles. This is in contrast to PFC, which introduces an intensity noise at twice the oscillation frequency with an amplitude higher than that of the shot noise by orders of magnitude. Second, due to the strong damping provided by our scheme, the lowest n eq achieved with our scheme is about two orders of magnitude lower than with PFC [26].
In comparison with cavity cooling, which can also be a powerful approach for neutral nanoparticles, our scheme pro-vides comparable n eq , while the high accuracy of observing the motion of nanoparticles, about 10 fm/ √ Hz, which is naturally provided by the optics for feedback cooling, is strongly beneficial for the application in sensitive accelerometers. Furthermore, the simplicity of our scheme that the trapping laser also carries the mechanism of cooling may facilitate the momentum spectroscopy via time-of-flight. With our scheme, time-of-flight measurements are readily implemented just by turning off the trapping laser, while a high-finesse resonator required for cavity cooling may intervene the rapid turning off of the optical potential. Our results also form the important basis for applications in precision measurements and accelerometers, where ultracold neutral particles are required [4,9].
In the present work, we demonstrate optical cold damping near the ground state only in one direction, while the motions in other directions are cooled via PFC and are still high. For future experiments, it is desirable to cool the motions in all directions to near the ground state, which can be realized by producing controllable optical gradients perpendicular to the light propagation directions either via a two-dimensional AOM, or via three dimensional optical lattices.
The present work represents the unique possibility of manipulating nanoparticles in an optical lattice via the phase modulation of the trapping laser, which can be potentially useful for accelerating microscopic particles such as atoms and molecules. We envision that the phase modulation technique will open further exciting possibilities such as the generation of anharmonic potentials [39,40] and the exploration of the physics with a time-dependent optical lattice [41][42][43].
FIG. 1 .
1(a) Schematic of the experimental setup. PD and SG denote a photodetector and a signal generator, respectively. (b) Single nanoparticles are trapped in an optical lattice formed by retroreflecting the laser.
FIG. 2 .
2RIN as a function of the frequency. The RIN of a free run laser is compared with the RIN with the feedback to the first EO-AM and the RIN with the feedback to both the first and the second EO-AMs. The shot noise level determined by the optical powers on the photodetectors for the stabilization and for the observation is indicated by a dashed line.
FIG. 3 .
3(a) Schematic showing the mechanism of exerting optical forces on neutral nanoparticles. Nanoparticles are trapped in an optical lattice with a frequency of ω 0 , while the light carries weak sidebands at ω 0 ± ω 1 as well. At the position of nanoparticles, the two sidebands exert optical gradient forces with opposite orientations on nanoparticles. By modulating the relative amplitudes of the sidebands, we can apply an oscillatory force on neutral nanoparticles. (b) Time variation of the amplitude of the oscillation signal along the optical lattice at a pressure of 2 × 10 −6 Pa. When the feedback force is turned off at t = −100 ms, the signal increased because of photon recoil heating. When the feedback force is turned on at t = 0 ms, the oscillation signal rapidly decays as shown in the inset. An exponential fit on the measured decay is shown by a solid line. (c) The damping rate via the cooling force as a function of the feedback gain. The solid line is a linear fit. the two sidebands with the combination of an EO-AM and an electro-optic phase modulator (EO-PM). The sidebands produced by an EO-AM are in phase with each other, while the sidebands produced by an EO-PM are out of phase with each other. Therefore, the ratio of the magnitudes of the sidebands after an EO-AM and an EO-PM can be controlled by the amplitude of the radio frequency (RF) signal with a frequency of ω 1 applied to the EO-PM. A double balanced mixer (DBM) allows us to control the amplitude and the phase of the RF signal.
occupation number with respect to the pressure for Γ c = 2π × 3.7 kHz. The error bars indicate systematic thermal fluctuations in measuring T eff . The calculated value at low pressures obtained with Eq.(3) is shown by the solid line. (b) The occupation number with respect to Γ c at a pressure of 5×10 −6 Pa. The error bars indicate systematic thermal fluctuations in measuring T eff . The solid line is not a fit and shows the calculated values with Eq.(3) for our experimental parameters. (c) PSDs of the OL signal with and without feedback cooling obtained at 6 × 10 −6 Pa and 6 Pa, respectively.
FIG. 5 .
5(a) PSD of singly charged nanoparticles with a sinusoidal modulation via an external electric field at a pressure of 4 × 10 −5 Pa. The peak at 230 kHz arises due to the modulation. (b) Typical time variation of the charge of a trapped nanoparticle under the illumination of a UV lamp at around 4 × 10 −5 Pa. In the inset, an expanded plot near the moment of neutralization is shown.
ACKNOWLEDGMENTSWe thank M. Kozuma and H. Kanamori for fruitful discussions. We are grateful to S. Nakano and N. Kagatani for their experimental assistance. This work is supported by the Murata Science Foundation, the Mitsubishi Foundation, the Chal-
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| []
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[
"Identifying Metastases in Sentinel Lymph Nodes with Deep Convolutional Neural Networks",
"Identifying Metastases in Sentinel Lymph Nodes with Deep Convolutional Neural Networks"
]
| [
"Richard Chen \nProscia Inc\nProscia Inc\nProscia Inc\n\n",
"Yating Jing \nProscia Inc\nProscia Inc\nProscia Inc\n\n",
"Hunter Jackson \nProscia Inc\nProscia Inc\nProscia Inc\n\n"
]
| [
"Proscia Inc\nProscia Inc\nProscia Inc\n",
"Proscia Inc\nProscia Inc\nProscia Inc\n",
"Proscia Inc\nProscia Inc\nProscia Inc\n"
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| []
| Metastatic presence in lymph nodes is one of the most important prognostic variables of breast cancer. The current diagnostic procedure for manually reviewing sentinel lymph nodes, however, is very time-consuming and subjective. Pathologists have to manually scan an entire digital wholeslide image (WSI) for regions of metastasis that are sometimes only detectable under high resolution or entirely hidden from the human visual cortex. From October 2015 to April 2016, the International Symposium on Biomedical Imaging (ISBI) held the Camelyon Grand Challenge 2016 to crowd-source ideas and algorithms for automatic detection of lymph node metastasis. Using a generalizable stain normalization technique and the Proscia Pathology Cloud computing platform, we trained a deep convolutional neural network on millions of tissue and tumor image tiles to perform slide-based evaluation on our testing set of whole-slide images images, with a sensitivity of 0.96, specificity of 0.89, and AUC score of 0.90. Our results indicate that our platform can automatically scan any WSI for metastatic regions without institutional calibration to respective stain profiles. | null | [
"https://arxiv.org/pdf/1608.01658v1.pdf"
]
| 12,903,060 | 1608.01658 | b9a04c9ddad98c9ca6899390aeba95fc9d9959fa |
Identifying Metastases in Sentinel Lymph Nodes with Deep Convolutional Neural Networks
Richard Chen
Proscia Inc
Proscia Inc
Proscia Inc
Yating Jing
Proscia Inc
Proscia Inc
Proscia Inc
Hunter Jackson
Proscia Inc
Proscia Inc
Proscia Inc
Identifying Metastases in Sentinel Lymph Nodes with Deep Convolutional Neural Networks
Metastatic presence in lymph nodes is one of the most important prognostic variables of breast cancer. The current diagnostic procedure for manually reviewing sentinel lymph nodes, however, is very time-consuming and subjective. Pathologists have to manually scan an entire digital wholeslide image (WSI) for regions of metastasis that are sometimes only detectable under high resolution or entirely hidden from the human visual cortex. From October 2015 to April 2016, the International Symposium on Biomedical Imaging (ISBI) held the Camelyon Grand Challenge 2016 to crowd-source ideas and algorithms for automatic detection of lymph node metastasis. Using a generalizable stain normalization technique and the Proscia Pathology Cloud computing platform, we trained a deep convolutional neural network on millions of tissue and tumor image tiles to perform slide-based evaluation on our testing set of whole-slide images images, with a sensitivity of 0.96, specificity of 0.89, and AUC score of 0.90. Our results indicate that our platform can automatically scan any WSI for metastatic regions without institutional calibration to respective stain profiles.
INTRODUCTION
Digital Pathology
Pathology is a 150-year-old medical specialty [1] that has seen a paradigm shift over the past few years with the advent of digital pathology. The digitization of tissue slides introduces a plethora of opportunities to leverage computer-assisted technologies to aid pathologists in diagnosing cancer [2]. While proliferation of digital pathology is at an all time high, the industry has not crossed the rubicon into clinical diagnostics due to lack of standardization of image formats, system noise, and lack of clinical and technical studies on digital pathology systems.
The inherent problem in pathology is subjectivity. The discipline is plagued with human variability from tissue acquisition, improper staining techniques, and subjectivity in diagnosing under a microscope. A pathologist looks for patterns in a tissue sample and uses his/her medical training to interpret those patterns and make a diagnosis [3]. As evidenced through many applications in myriad industries, computer-assisted pattern recognition software can match or even supersede a human's ability to recognize patterns [4].
Though digital pathology is on the precipice of wide-spread adoption, the difficulties in hardware scanning variability and fear of "black-box" computational tools has lead to a longer adoption curve than seen in other medical specialties that have gone totally digital, e.g. radiology [5].
Region of Interest (ROI) detection algorithms, like the one proposed here have the potential to act as an intermediary clinical decision support tool, rather than immediately moving to a fully computerized diagnostic procedure [6]. ROI detection tools cut down time for analyzing whole slide tissue sections by reducing the signal-to-noise ratio and directing the pathologist to those regions that are flagged for containing certain properties indicative of the region of interest, as learned by the model.
While computerized primary diagnosis is still in its nascent stages, these such tools provide an expedient route to enabling precision medicine in pathology by automating the ROI detection and alleviating diagnostic subjectivity.
Camelyon Competition
From October 2015 to April 2016, the International Symposium on Biomedical Imaging (ISBI) held the Camelyon Grand Challenge 2016 to crowd-source ideas and algorithms for automatic detection of lymph node metastasis [7] [8]. The following two metrics in the challenge were used to evaluate the performance of the algorithms: 1) Slide-based Evaluation: algorithm performance on discriminating between normal slides and metastasis slides 2) Lesion-based Evaluation: algorithm performance on lesion detection and localization
The dataset in the challenge contains a total of 400 whole-slide images (WSIs) of sentinel lymph node from two independent datasets collected in Radboud University Medical Center (UMC) (Nijmegen, the Netherlands) and the UMC Utrecht (Utrecht, the Netherlands). Whole-slide images are large image files organized in a multi-resolution pyramid structure, in which each image in the pyramid is a downsampled version of the highest resolution image. The first training dataset consists of 170 WSIs of lymph node (100 Normal and 70 Tumor) and the second training dataset consists of 100 WSIs (60 Normal and 40 Tumor). The first testing dataset consists of 80 WSIs and the second testing dataset consists of 50 WSIs. The ground truth data of the training set came from a pathologist who manually drew contours of regions of lymph node metastasis.
Because the labeled data for the test set was never made public in the Camelyon Challenge, we performed a 60-20-20 train-validation-test split on the 270 WSIs provided as training data. In addition, we only tested the performance of our algorithms using slide-based evaluation. Our results are based on a small subset of the entire image set; however, these initial results are motivation for further work on the entire dataset and completion of the Camelyon challenge.
METHOD
Our framework for detecting metastases in sentinel lymph nodes can be modularized into four components: image preprocessing and tiling of WSIs, color deconvolution and stain normalization of WSI tiles, tile-based classification using convolutional neural networks (CNNs), and postprocessing of tumor probability heatmaps of the WSIs. The libraries and scientific packages used were OpenSlide, NumPy, Pillow, OpenCV, Caffe, and pyCaffe. For each WSI, we perform a connected components analysis to draw contours around only tissue regions of the slide, cutting down total pixels processed by over 70%. Specifically, we transformed the WSI's RGB color vector to the HSV colorspace and performed Otsu's Binarization on only the saturation channel to separate brightlystained tissue regions from the gray space of the background region. We also used a median blur to filter out fatty tissue and background artifacts. To reduce computational run-time, we performed all of our segmentation on a downsampled version of the original image and scaled the pixel coordinates of the contour regions to the dimension with the highest resolution. Specifically, we performed segmentation on the thumbnail dimension of all the WSI's. Using OpenSlide [9], we were able to compute level-based dimensions which are used to create a linear scaling function to scale contours created at any resolution to be represented in a higher dimensional resolution space without losing any contour information. Knowing the downsample factor from the thumbnail dimension to the level 1 dimension of the WSI, we perform a scale transformation and rapidly calculate the contours of the tissue region at the highest spatial resolution of the image. Finally, within each contour region, we partitioned the tissue and tumor regions into 256 x 256 tiles at 40x (level 1) using a fully parallel computing architecture in the Pathology Cloud Computing Platform that expedites the process of creating the training set.
Image Preprocessing & Tiling
(a) (b) (c) (d) (e) (f)
Stain Normalization
Since the Camelyon challenge dataset was generated from two separate institutions, we wanted to eliminate any stain variability that could negatively affect training and testing. By performing stain normalization, we can perform a non-linear color feature mapping that maps all of the respective training images by to a target stain. Without elimination of the stain variability, the heterogeneity of the stains will introduce bias to our model, inducing a bimodal stain vector color distribution. We performed stain normalization on each 256 x 256 tile generated during WSI tiling.
Tile-Based Classification using Convolutional Neural Networks
After creating a training set of 256 x 256 tiles, we trained a convolutional neural network to discriminate between tissue and tumor tiles. Namely, we chose AlexNet [10]. The architecture of AlexNet consists of five convolutional and three fully-connected learned layers, 60 million parameters, and a two-way softmax to assign probabilities to the tissue and tumor class labels. In addition to stain normalization, we preprocessed each training image by subtracting the mean activity over the training set from each pixel. We also extracted random 224 x 224 patches and their horizontal reflections from each 256 x 256 tile in our training set and trained our model on these patches to prevent overfitting.
Post-Processing of Tumor Probability Heatmap
Given a WSI in the Camelyon training data, we generated its corresponding tumor probability heatmap, with each pixel assigned a value p for p ∈ [0, 1], indicating the probability of metastasis. To analyze these heatmaps efficiently, we generated tumor probability heatmaps at level 4 instead of level 0 due to computational complexity. To accomplish this, for every tile within the tissue and contour regions at level 0, we used our AlexNet model to assign a probability to each tile, and color-mapped that tile's probability to its corresponding downsampled tile at level 4. In the pyramidal structure of each WSI, the image at level 4 is 8 downsamples greater than the image at level 0. Specifically, for a 256 x 256 tile at level 4, it would take 64 256 x 256 tiles at level 0 to represent the same region in the image. As a result, probabilities assigned to 256 x 256 tiles at level 1 can be mapped and colored to 32 x 32 pixel regions at level 4. After generating the tumor probability heatmap, we performed post-processing for slidebased evaluation. For slide-based evaluation, a 60-20-20 trainvalidation-test split was done on 270 tumor probability heatmaps generated from the WSIs provided as training data, with each partition having an approximately equal proportion of tumor to tissue tiles. For each heatmap, we computed geometrical and morphological features about its tumor probability distribution. Such features include: max, mean, variance, skewness, and kurtosis of (area, perimeter, compactness, rectangularity, solidity of tumor regions), average prediction across tumor region, total number of tumor regions, and the total number pixels with probability greater than 0.90. After computing these features, we trained a Random Forest classifier, and used it to evaluate our test set of WSIs.
EXPERIMENTAL RESULTS
We first evaluated tile-based classification accuracy with and without stain normalization. Under this evaluation metric, where we were able to accurately classify all given tiles in the test set as belonging to a tissue region or a metastatic region with 92.7% without stain normalization, and 96.6% accuracy with stain normalization. Because we achieved a higher tile-based classification accuracy under stain normalization, we performed slidebased evaluation with this extra pre-processing step.
For slide-based evaluation, the Random Forest classifier yielded an average image-based classification sensitivity of .96, average specificity of .89, and a computed area under the Receiver Operating Characteristic Curve (AUC) score of 0.90.
CONCLUSION
The results of the study show that stain normalization is a crucial part of building a generalizable deep learning model for identifying metastatic regions in sentinel lymph nodes, as it eliminates the variability induced by different stains from the two locations. This process significantly improved classification performance of our model and indicates that this model could be extended to any breast cancer digitized images without institutional recalibration. Our algorithms and model will be subject to additional training and validation on the full image set provided by ISBI and the Camelyon Challenge 2016. We are grateful to the organizers and all those involved in the competition, and we look forward to improving our results and computing a tumor localization score for the fully trained model.
Fig. 1 :
1Image preprocessing steps for tissue segmentation. (a) Original Image (b) HSV colorspace (c) Saturation channel (d) Median blur (e) Otsu's Binarization (e) Original Image with contour overlay
Fig. 2 :
2AlexNet Layer Architecture
Fig. 3 :
3An example of stain normalization on a heavily overstained source image (a) Source Image (b) Target Image (c) Output Image
Fig. 4 :
4A comparison between the (a) ground truth data of Tumor 087 and (b) tumor probability heatmap of Tumor 087
Rudolf virchow (1821-1902). Paolo Scarani, Virchows Archiv. 4422Paolo Scarani. Rudolf virchow (1821-1902). Virchows Archiv, 442(2):95-98, 2003.
Digital pathology: current status and future perspectives. Shaimaa Al-Janabi, Andr Huisman, Paul J Van Diest, Histopathology. 611Shaimaa Al-Janabi, Andr Huisman, and Paul J Van Diest. Digital pathology: current status and future perspectives. Histopathology, 61(1):1-9, 2012.
Diagnostic concordance among pathologists interpreting breast biopsy specimens. Joann G Elmore, Gary M Longton, Patricia A Carney, Berta M Geller, Tracy Onega, Anna N A Tosteson, Heidi D Nelson, Margaret S Pepe, Kimberly H Allison, Stuart J Schnitt, Jama. 313111122Joann G. Elmore, Gary M. Longton, Patricia A. Carney, Berta M. Geller, Tracy Onega, Anna N. A. Tosteson, Heidi D. Nelson, Margaret S. Pepe, Kimberly H. Allison, Stuart J. Schnitt, and et al. Diagnostic concordance among pathologists interpreting breast biopsy specimens. Jama, 313(11):1122, 2015.
Digital pathology imaging offers more benefits than glass slides and microscopes. Carol P Joseph P Houghton, Michael J Wilson, Dolaghan, BMJ. 354Joseph P Houghton, Carol P Wilson, and Michael J Dolaghan. Digital pathology imaging offers more bene- fits than glass slides and microscopes. BMJ, 354, 2016.
An industry perspective: An update on the adoption of whole slide imaging. Michael, Montalto, Journal of Pathology Informatics. 7118Michael. Montalto. An industry perspective: An update on the adoption of whole slide imaging. Journal of Pathology Informatics, 7(1):18, 2016.
Histopathological image analysis: A review. N Metin, Senior Gurcan, Laura E Member, Ali Boucheron, Anant Can, Madabhushi, Senior Member, Nasir M. Rajpoot, Bulent Yener, and Senior Member. Metin N. Gurcan, Senior Member, Laura E. Boucheron, Ali Can, Anant Madabhushi, Senior Member, Nasir M. Rajpoot, Bulent Yener, and Senior Member. Histopatho- logical image analysis: A review. IEEE Reviews in Biomedical Engineering, pages 147-171, 2009.
Isbi challenge on cancer metastasis detection in lymph node. Isbi challenge on cancer metastasis detection in lymph node.
Deep learning for identifying metastatic breast cancer. Dayong Wang, Aditya Khosla, Rishab Gargeya, Humayun Irshad, Andrew H Beck, ArxivDayong Wang, Aditya Khosla, Rishab Gargeya, Hu- mayun Irshad, and Andrew H. Beck. Deep learning for identifying metastatic breast cancer. Arxiv, 2016.
OpenSlide: A vendor-neutral software foundation for digital pathology. Adam Goode, Benjamin Gilbert, Jan Harkes, Drazen Jukic, Mahadev Satyanarayanan, Journal of pathology informatics. 4Adam Goode, Benjamin Gilbert, Jan Harkes, Drazen Jukic, and Mahadev Satyanarayanan. OpenSlide: A vendor-neutral software foundation for digital pathology. Journal of pathology informatics, 4, 2013.
Imagenet classification with deep convolutional neural networks. Alex Krizhevsky, Ilya Sutskever, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. Imagenet classification with deep convolutional neural networks. pages 1097-1105, 2012.
| []
|
[
"K-ST: A Formal Executable Semantics of PLC Structured Text Language",
"K-ST: A Formal Executable Semantics of PLC Structured Text Language"
]
| [
"Kun Wang ",
"Jingyi Wang ",
"Christopher M Poskitt ",
"Xiangxiang Chen ",
"Jun Sun ",
"Peng Cheng "
]
| []
| []
| Programmable Logic Controllers (PLCs) are responsible for automating process control in many industrial systems (e.g. in manufacturing and public infrastructure), and thus it is critical to ensure that they operate correctly and safely. The majority of PLCs are programmed in languages such as Structured Text (ST). However, a lack of formal semantics makes it difficult to ascertain the correctness of their translators and compilers, which vary from vendor-to-vendor. In this work, we develop K-ST, a formal executable semantics for ST in the K framework. Defined with respect to the IEC 61131-3 standard and PLC vendor manuals, K-ST is a high-level reference semantics that can be used to evaluate the correctness and consistency of different ST implementations. We validate K-ST by executing 509 ST programs extracted from Github and comparing the results against existing commercial compilers (i.e., CODESYS, CX-Programmer, and GX Works2). We then apply K-ST to validate the implementation of the open source OpenPLC platform, comparing the executions of several test programs to uncover five bugs and nine functional defects in the compiler. | null | [
"https://arxiv.org/pdf/2202.04076v1.pdf"
]
| 246,680,075 | 2202.04076 | 545b10b076741c603d7bf373c2575620371811e2 |
K-ST: A Formal Executable Semantics of PLC Structured Text Language
Kun Wang
Jingyi Wang
Christopher M Poskitt
Xiangxiang Chen
Jun Sun
Peng Cheng
K-ST: A Formal Executable Semantics of PLC Structured Text Language
1Index Terms-Formal executable semanticsPLC programmingStructured textK-frameworkOpenPLC
Programmable Logic Controllers (PLCs) are responsible for automating process control in many industrial systems (e.g. in manufacturing and public infrastructure), and thus it is critical to ensure that they operate correctly and safely. The majority of PLCs are programmed in languages such as Structured Text (ST). However, a lack of formal semantics makes it difficult to ascertain the correctness of their translators and compilers, which vary from vendor-to-vendor. In this work, we develop K-ST, a formal executable semantics for ST in the K framework. Defined with respect to the IEC 61131-3 standard and PLC vendor manuals, K-ST is a high-level reference semantics that can be used to evaluate the correctness and consistency of different ST implementations. We validate K-ST by executing 509 ST programs extracted from Github and comparing the results against existing commercial compilers (i.e., CODESYS, CX-Programmer, and GX Works2). We then apply K-ST to validate the implementation of the open source OpenPLC platform, comparing the executions of several test programs to uncover five bugs and nine functional defects in the compiler.
INTRODUCTION
P ROGRAMMABLE Logic Controllers (PLCs) are responsible for automating process control in several modern industrial systems, e.g. in manufacturing and public infrastructure. It is critical to ensure that PLCs are operating correctly, as any functional or security-related defects may lead to serious incidents in the system. This has most famously been demonstrated by the Stuxnet worm [1], while many other less-known safety and security incidents [2], [3], [4] and potential hazards [5], [6] related to PLCs have resulted in significant consequences, with an estimated $350,000 in damage on average [7].
The majority of PLCs are programmed using languages defined in the IEC 61131-3 open international standard [8]. Programs can be written in graphical languages such as Function Block Diagrams (FBD), but the standard also defines Structural Text (ST), a fully textual language based on the idea of organizing code into 'function blocks' and designed with a syntax similar to Pascal. ST is a particularly important IEC 61131-3 language given its utility for data processing [9], and the fact that snippets of ST are actually required in FBD and other graphical languages. It is therefore important that translators and compilers for ST are correctly implemented and exhibits only expected behavior This has motivated a surge of research on analyzing and verifying PLC programs [7], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], although few work focuses on ST implementations/compilers. Specifically, Zhang et al. [22] propose VetPLC, a temporal context-aware, program-based approach to produce timed event sequences that can be used for automatic safety vetting. McLaughlin et al. [19] propose TSV which translates assembly-level code into an intermediate language (ILIL) to verify safety-critical code executed on PLCs. Mader and Wupper [24] translate Instruction List (IL) into timed automata [28]. Bauer et al. [23] similarly use timed automata as the formalism for Sequential Function Chart (SFC). In [25], the proposed method transforms IL to Petri-nets [29], and manually builds two additional Petri-nets for modeling the PLC and its environment. Xiong et al. [21] propose an algorithm based on variable state analysis for automatically extracting the behavior model (BM) from an ST program. These works attempt to transform PLC programs into an intermediate language or another programming language (i.e., C) which is suitable for verifying or detecting potential issues using existing associated verifiers or checkers. The issue of these approaches is that there lacks analysis and proof of equivalence in the conversion process. In addition, the analyses they perform are often limited (since the existing tools are not designed for PLCs) and do not offer the feedback to the level of source code. Canet et al. [20] propose formal semantics for a significant fragment of the IL language, and a direct coding of this semantics into a model checking tool. Huuck [26] develops a formal operational semantics and abstract semantics for IL, which allows approximating program simulation for a set of inputs in one simulation run. However, IL is a low level assembly-like language that has been deprecated from the IEC61131-3 standard.
To the best of our knowledge, a practical and complete semantics for the ST language does not exist, which makes it difficult to ascertain the correctness of ST translators and compilers (e.g. by comparing executions). There are a number of reasons why such a reference semantics is yet to emerge. First, there is insufficient documentation defining or describing the complete features of the ST language [9]. For instance, the official documentation introduces language features by only a few examples, based on which it is difficult for readers to fully understand the behavior of the language. Second, the ST compilers provided by different vendors (e.g. Allen-Bradley, Siemens) can implement the language differently, and their closed source solutions make it difficult to fully assess how they behave systematically (other than through manual observation). A preliminary attempt at defining a high-level semantics for ST was made by Huang et al. [30]. However, it falls short of a reference semantics as it misses several important features of the language, e.g. certain data types, and key sentences.
In this work, we develop K-ST, a formal executable reference semantics for ST in the K framework [31]. Our highlevel semantics is both executable and machine readable, and can be used by the K framework to generate interpreters, compilers, state-space explorers, model checkers, and deductive program verifiers. Our principal goals for the design of K-ST are as follows:
1) Validated reference semantics. K-ST is designed to cover all the main features of ST, and is validated against hundreds of different real-world ST programs extracted from Github. 2) General and extendable. The semantics is highlevel (rather than tied to a particular compiler), with the goal of supporting different ST implementations as well as extensions for vendor-specific functions.
3) Analyses of ST compilers. Most importantly, K-ST
can be used to check the correctness and consistency of different ST implementations, and thus ensure that a compiler is not introducing an unintended behavior or compile-time threat [32], [33] into a critical industrial system.
Given the absence of complete feature descriptions for the ST language in official documentation, we not only refer to the definitions and code samples in the official documents, but also extensively consult the guidance manuals provided by multiple vendors to better define the semantics of the ST language. For example, there is no specific documentation on how integer overflow is handled in the official documents. Through investigating multiple instruction manuals, we found that existing ST compilers generally use truncation to handle integer overflow without any warning. The rewriting rule of the K framework provides a good mechanism for expanding the unique features of ST. For example, we can rewrite REPEAT to WHILE to achieve the execution effect of REPEAT.
We validate K-ST by extracting 567 real-world ST code samples from Github and comparing their executions in our semantics against their executions under various commercial compilers (i.e., CODESYS, CX-Programmer, and GX Works2). We find that K-ST is sufficiently complete to support 509 of these programs (with 26,137 lines), missing only vendor-specific functions and hardware-related functions that we did not formalize; and it executes those programs correctly (i.e., producing the same outputs as the corresponding existing compiler). Furthermore, to evaluate the utility of K-ST for testing ST compilers, we compared the executions of the 567 programs (and several mutants) under K-ST and OpenPLC [34], a popular open source PLC program compiler. Through this semantics-based testing, we are able to uncover five bugs and nine functional defects in the OpenPLC compiler, all of them are previously unknown. Fig. 1 summarises the high-level workflow of this process.
In summary, we make three main contributions.
•
We propose an executable formal reference semantics for ST;
•
We collect a set of 567 complete ST program samples, and validate the correctness of our executable semantics by running those programs in the semantics and in existing compilers (CODESYS, CX-Programmer, and GX Works2), comparing the results.
•
We test OpenPLC, an open source PLC program compiler, using our proposed semantics, and find five bugs and nine functional defects.
The remaining part of this paper is organized as follows. Section 2 introduces the background of ST and the Kframework. The proposed executable operational semantics of ST formalized in K is introduced in Section 3. Section 4 shows some practical applications of formal semantics that we proposed. The evaluation results of the proposed semantics are introduced in Section 5. Section 6 concludes this work.
BACKGROUND
In this section, we briefly introduce the background of the Structured Text and the K-framework.
Structured Text
The Programmable Logic Controller, invented in 1969 by Dick Morley, is specially designed for applications in industrial environments, e.g. assembly lines, robotic devices, or public infrastructure. These kinds of applications all require high reliability and ease of programming.
Early PLCs were represented as a series of logic expressions in some kind of Boolean format. With the development of programming terminals and the complexity of existing control procedures, Ladder Diagrams (LD) were developed to program PLCs. As of 1993, the IEC 61131-3 standard developed by the International Electrotechnical Commission (IEC) defined five programming languages, including two textual programming languages-ST and IL-as well as three graphical languages-LD, Function Block Diagrams (FBD), and SFC. A simple example in Fig. 2 [35] shows a ST code example which can be used for linear scaling of an analog sensor signal.
ST is a high-level PLC programming language which is similar to Pascal [36] (widely used from 1980 to 2000), C/C++ and Java. While it contains common constructs from modern programming languages such as FUNCTION, IF/ELSIF/ELSE and CASE branches, WHILE and FOR loops, it has its own characteristics, such as the lack of recursion, capitalized keywords, REPEAT statement, and FUNCTION BLOCK structure. For instance, FUNCTION BLOCK as an important part of ST, has its own state. Its main purpose is to modularize and structure a straightforwardly defined portion of the program. It is analogical to the class-object manifestation in the object oriented programming. Function blocks exist in two forms: as a type or as an instance, but only the instance can be called. For each function block, the local variables retain their values between each "call". TABLE 1 shows the common elements of ST.
ST as the only textual programming language supported by new IEC standard, has a number of advantages compared to other PLC languages. First, ST programs can be copied relatively easily. Second, compared with the other four languages, it is more convenient for mathematical calculations, formulas and algorithms, and for managing large amounts of data [9]. Third, compared with 20 years ago, PLC solutions are more in demand today and ST can better adapt to this change. Finally, LD, SFC and FBD also require parts of the program to be written in ST anyway [37], [38].
Unfortunately, the absence of documents defining or describing the complete features of the ST language and the implementation method customized by the vendor can lead to inconsistent implementations of ST. In addition, understanding the semantics of the ST language, and ensuring that it is formally defined is difficult for end users accustomed to graphical programming. A formal executable semantics of ST not only provides a standard, but also helps PLC engineers verify the completeness and correctness of these implementations.
The K Framework
K is a formal logic framework based on rewriting logic [39]. It was developed with the overarching goal of pursuing the ideal language framework, where all programming languages have formal semantic definitions and all language tools are automatically derived in a correct-by-construction manner at no additional cost. The K backends, such as the Isabelle theory generator, the model checker, and the deductive verifier, can be utilized to prove properties based on the semantics and generated verification tools [40]. Several executable semantics in K have been developed for mainstream programming languages, including C [41], Java [42], JavaScript [43], Rust [44], Solidity [45], and IMP [46].
A language semantics definition in the K consists of three parts: the language syntax, the configuration, and a set of semantics constructed based on the syntax and the configuration. Given the semantics definition for a programming language and some source programs, K executes these programs like a translator. For illustration, in the following we take a strict subset of the ST language, i.e., ST demo shown in Fig. 2 as an example to illustrate how to define language semantics in K.
Configuration. The whole configuration cell T of ST demo contains two cells, namely k and state. The cell k is used to store the source program $P GM for execution, and the cell state is used to record the mapping from a variable identifier to its value. The configuration simulates the memory status and environmental changes during runs of the program.
$P GM : P gm k .M ap state T With the configuration defined, we present the syntax of ST demo in Fig. 3, which includes some numerical operations, logic operations and commonly used statements. Based on the configuration and the syntax of ST demo , we introduce some basic rules in the semantics. The role of the semantics is to tell K how to execute the source code, where K executes the code and updates the configuration sentenceby-sentence after parsing the source program.
Here, we show the semantics of Allocate, Lookup and Assignment in Fig. 4 as they are the most commonly used constructs in programming languages. Table 2 describes some common semantic notations. Take Allocate as an example: when K runs to lines 9-13 in Fig. 2, the content in k cell is VAR a : REAL; V Bs END VAR · · · k , where V Bs stands for b : REAL; Error : BOOL := FALSE;. Then, K will rewrite VAR a : REAL; V Bs END VAR · · · k to VAR V Bs END VAR · · · k , which means that a : REAL; has been executed according to rule Variable Allocate. Meanwhile, it adds the mapping between the variable name and the corresponding value (a → 0.0) in the current state cell Rho. Besides, "requires notBool (X in keys (Rho))" guarantees that the variable will not be re-declared. Similarly, variables b and Error will be allocated separately. After that, the content in the k cell is VAR .V arBodys END VAR · · · k , where .V arBodys represents an empty variable declaration list, that is, no additional variable needs to be allocated. The rule Variable Finish Allocate will be called to convert "VAR .V arBodys END VAR" in k to ".", which means that there is no more code to execute in the VAR block and K will continue to execute the subsequent code.
FORMAL SEMANTICS OF STRUCTURED TEXT IN THE K-FRAMEWORK
In this section, we introduce K-ST, the executable operational semantics of ST formalized in K. Note that in practice the PLC programming environment is provided by specific PLC manufacturers including Codesys and Siemens's TIA portal (TIA, Structured Control Language (SCL)). As a consequence, the implementations of different manufacturers may be different and may include their own unique functions or structures. Our approach is therefore to focus on the common features, allowing other unique functions of the environment The beginning of a semantic rule.
a requires b Execute a when b is true. a b k
k stands for k cell in configuration. a b means a will be rewritten by b. · · · a · · · · · · represents the content in the a context.
. . stands for empty.
Any value.
a : b Variable a is type b. a → b, a ← b Mapping from a to b. a b
The execution of a, followed by execution of b.
a ⇒ b Similar to a b , it can only be used outside .
to be implemented by extending the operational semantics. Specifically, the syntax of ST is constructed based on the official IEC 61131-3 standard [38]. The configuration is specifically designed for ST. Based on the syntax and the configuration, we then formalize the semantic rules for the language features with rewriting logic. Next, we present each component of the semantic one by one.
The Syntax of ST
The Configuration of ST
The execution of an ST programs need to update the following kinds of state: data segment, code segment and stack. Among them, the data segment is used to store global variables, the code segment is used to store program execution code, and the stack is used to store local variables of the program. Note that running environment switching caused by function calls is also achieved by the operation of stack. The overall runtime configuration of ST in K is presented in Fig. 5. We highlight our careful design choices as follows.
Overview. There are 11 main cells in the configuration T , i.e., k, control, allenv, genv, gvenv, store, type, constant input, output and nextLoc. The value of each cell is initialized according to its specified type. For instance, for cells with a mapping relationship, their values are initialized to M ap type, and for cells that store a collection, they are initialized to List type. A '.' followed by any type means an empty set of this type. For instance, .M ap in the cell genv represents that genv is initialized with an empty map.
Enumeration type. By default, when an enumeration type is defined in ST, PLC compilers will automatically associate a number (indexed from 0 and added by 1 each time) to each variable in the enumeration. For repeated declarations, we use count cell to record the value of the current enumeration.
Global variables. There are two types of global variables. One is the POUs and customized types that users define. These variables can be accessed anywhere in the program. We store these variables in genv cell as the basis for program operation. The other is the variables defined in VAR GLOBAL. These variables cannot be directly accessed in the program unless they are declared with VAR EXTERNAL. We store these variables in gvenv cell and provide them on demand.
Program execution. The source code parsed by the syntax SourceU nit, called $P GM , are stored in the cell k for execution. Then the $P GM will be executed unit by unit. If the program terminates normally, there will be a '.' in the k cell, denoting that no more unit needs to be executed. In the preprocessing phase (the first pass of K), the k cell only contains the token execute. Afterwards, K will start executing from the MAIN program.
Stack operations. The cell control contains seven subcells-f stack, env, temp, count, gvid, print and breakwhich record the operating environment of the currently running code segment. Specifically, the function stack f stack is a list used to store the environment before executing other POUs, including variables in the current environment and subsequent program. After that, the cell env is to store the mapping relationship between variables and indexes in the current environment during program execution. Besides, cells temp and count are used in ENUM and STRUCT, where temp is for temporary mapping and count is used as a counting pointer. The cell gvid is to record all identifiers of global variables to assist the generation of global variables. The cell print is to record variables which need to be output. Finally, break stores the program after the loop in order to support the implementation of the EXIT statement in FOR, WHILE and REPEAT loops.
Execution environment. The allenv cell is to cache the execution environment before function calls (for strict type checking of parameter passing in function calls 1 ). The cell genv is to record the result of the pre-processing (including POUs and custom types) and will be copied to env when env is refreshed. The last cell related to the environment is called gvenv and is used to index global variables.
Memory operation. The store cell is used to simulate memory to record the mapping relationships of indexes and variables values. After that, the cells input and output are used to realize external inputs and external output respectively. And the last cell nextLoc ensures that the index of variable can always be incremented without duplication. The design consideration behind is that for complex languages, it is more effective to explicitly manage arbitrarily large memory than use garbage collection [48].
Semantics of the Core Features
We implement the executable semantics covering most core features of ST and leave the vendor-specific functionalities 1. This is optional but recommended for ST compilers.
T ype ::= IN T |DIN T |SIN T |LIN T |U IN T |U DIN T |U SIN T |U LIN T |BY T E|W ORD|DW ORD|REAL |LREAL|ST RIN G|ST RIN G [Expression] |W ST RIN G|W ST RIN G [Expression] |T IM E|DAT E |T IM E OF DAY |DAT E AN D T IM E|Id|ARRAY [Expression] OF T ype Variable types V arT ype ::= V AR GLOBAL | V AR | V AR IN P U T | V AR OU T P U T | V AR IN OU T | V AR T
Variable declaration
Operation :
:= + | − | * | / | * * | M OD | < | > | = | <= | >= | <> | AN D | & | AN D T HEN | XOR | OR | OR ELSE | .. Expression ::= Int | F loat | String | Bool | Bit | AllT ime | Id | Expression Operation Expression Expression (Expressions) | Expression.Expression | Expression [Expressions] | (Expression)
Loop statements
Return ::= RET U RN ;
Return statement
Exit ::= EXIT ; Exit statement
Statement ::= Expression; | Assignment | If | Case | W hile | F or | Repeat | Return | Exit
Statements ::= Statement * Statements as potential extensions. For example, some compilers would use additional keywords to distinguish the declaration part and the execution part of the program. In the following, we provide an overview of four core semantic features of ST, including 1) data types, 2) main control statements, 3) declarations and calls of POUs and 4) memory operations. Before diving into the details, we present the notations as follows. The rule symbol represents the beginning of a semantic rule. The symbol ⇒ means "rewritten by", for instance A ⇒ B denotes that A can be replaced by B.
Extended Data Types
The K framework supports diverse data types including identifiers (Id), integers (Int), bools (Bool), floats (F loat) and strings (String), etc, which cover most of the requirements. However, there are still some unsupported data types needing additional implementation in K-ST, which we call extended data types. These extended data types can be categorized into two kinds: 1) elementary types (TIME, BYTE, WORD, DWORD, TIME OF DAY, DATE and DATE AND TIME) and 2) compound types (ENUM and STRUCT). We implement these extended data types by the composition of built-in types and methods in K as follows. We take TIME OF DAY as an example to introduce the realization of extending elementary types. There are two types of TIME OF DAY in ST, e.g., TIME OF DAY#23 : 45 : 56.30 and TOD#23 : 45 : 56.30. Fig. 6 shows our implementation of TIME OF DAY type together with its relevant operations. Line 1 and 2 defines the syntax of TIME OF DAY and how to parse it (Get TIME OF DAY) respectively. Line 3 is used to convert Get TIME OF DAY to TIME OF DAY, which is achieved by two steps-Gtd2T d and Standardizationwhere Gtd2T d realizes the conversion of the format and Standardization realizes content conversion, e.g., replacing 60 minutes with 1 hour. Lines 4-11 define some arithmetic and relational operations of TIME OF DAY.
For compound types, we take STRUCT as an example and show its semantics in Fig. 7 including STRUCT declaration and instantiation. Declarations are shown in rule Struct Declaration, where we allocate memory for
Main Control Statements
Control statements are important in ST for achieving complex program logic (as in most other programming languages). We show the rules for CASE, REPEAT and EXIT in Fig. 8 (as the semantics of IF, WHILE and FOR are typical). A CASE statement can be rewritten as a combination of an IF and CASE through rule Case. The rule REPEAT is implemented as follows. We first store the subsequent statements outside the loop (recorded as K) in cell break to deal with the EXIT statement that may appear, and then rewrite it into the form of WHILE for further execution. During the execution of the loop body, once EXIT is executed, all the statements in the current cell k are discarded and rewritten to K (storing the subsequent statements), as shown in rule Exit.
The Declaration and Call of POUs
In ST programs, statements are inside Program Organization Units (POUs), i.e., FUNCTION, FUNCTION BLOCK or PROGRAM. A FUNCTION is a stateless POU type, comparing to a FUNCTION BLOCK which stores its own state after execution. The design of FUNCTION BLOCK is to similar to a class object in the object-oriented programming (OOP) for better modularization. FUNCTION BLOCKs exist in two forms: as a type or as an instance, and only the instance can be called. For FUNCTION BLOCK instance, the local variables retain their values between each "call". PROGRAMs are defined by the IEC 61131-3 standard as a "logical assembly of all the programming language elements and constructs necessary for the intended signal processing required for the control of a machine or process by a PLC-system" [38]. Due to space limit, we show the declaration, call and return operation of FUNCTION BLOCKs in Fig. 9 as an example for illustration (FUNCTION and PROGRAM are shown in Fig. 10 and explained only when necessary).
Declaration. The declaration of FUNCTION BLOCK is similar to the STRUCT. As shown in rule Function Block Declaration, we first assign an index in memory for FUNCTION BLOCK X, set the type to the built-in F unctionBlock, and convert the entire declaration statement to the built-in type f unblambde(X, void, V ds, S) for storage, where void means no return value, V ds and S are variable declarations and operations in X respectively. The purpose of setting const to true is to prevent it from being modified. Note that FUNCTION and PROGRAM set type and store to F unction, f unblambde(X, T, V ds, S) and P rogram, plambde(X, void, V ds, S, .M ap).
Instantiation. The instantiation of FUNCTION BLOCK is achieved through variable declarations, as shown in rule Function Block Instantiation. However, the value is set to runf unblambde(X, void, V ds, S, .M ap) to distinguish it from f unblambde and .M ap is designed to store the FUNCTION BLOCK environment for next call and external query. This is because a FUNCTION BLOCK can only be called after instantiation, i.e., runf unblambda can be executed but f unblambda can not. Since FUNCTION and PROGRAM have no such restrictions, f unlambde and plambda can be directly called and executed.
Call. There are two cases when a FUNCTION BLOCK is called. The first case is that the FUNCTION BLOCK is called for the first time, as shown in rule Function Block Call First. Since there is no initial environment (the last value of runf unblambda is .M ap), we will first store the current execution environment inf o in f stack, including subsequent statements K, the Allenv of the current environment, and the parameters C in cell control. Then we reset parameters C through renew. After that, K executes the variable declaration V ds (including index application, initialization and assignment) and statements S in the function block. Besides, U pdate is used to update the .M ap in runf unblambde to record the current environment. Finally, RETURN can return to the calling program and configure the corresponding environment. In other cases (not called for the first time), as shown in rule Function Block Call Others, there is already a mapping relationship between related variables and values in cell store and the mapping relationship between identifiers and indexes is also stored in the runf unblambde. Therefore, no new memory allocation will be made during the execution process and the existing environment will be used. Note that the value of the variable in the FUNCTION BLOCK will not be initialized, which means that the execution result for the same input may be different.
Regardless of whether RETURN appears in the FUNCTION BLOCKk, we add a RETURN by default for each FUNCTION BLOCK as a sign that the FUNCTION BLOCK has finished running and returned to the calling POUs. Since FUNCTION BLOCKs and PROGRAMs do not have a return value, we set null as the return value. Note that a FUNCTION has a return value, and the returned value is the value corresponding to the function identifier, so we need to use ClearEnv to clean up the memory environment corresponding to the function identifier after calling procedure renew and add the declaration of the function identifier variable in V ds.
Memory Operations
Here, we present the rules for memory operations on elementary types in ST, such as built-in types and extended elementary types. What elementary types have in common is that they take only one memory slot. For complex types, such as enums, structs, arrays, etc, which are compositions of elementary types, the memory operation can be regarded as a set of memory operations on elementary types. For instance, the assignment to struct can be equivalent to assign value for each variable of this struct. Similar to ST demo , main memory operations of ST are still composed of allocation, lookup, assignment and additional clearenv. Where allocation implements the allocation of memory for variables in the store, lookup is used to find variable values in store cell, assignment implements the assignment of variables, and clearenv implements the recovery of memory in the store. However, because the complete ST semantics has a more complex type design, they will involve more cells in configuration, and are more complicated, as shown in Fig. 11.
Note that HOLE is just a variable, but it has special meaning in the context of sentences with the "heat" or "cool" attribute. In short, "heat" is to lookup the corresponding content of the HOLE in the formula, and "cool" is to put the recheck results back into the formula. For example, in expression a + b where a is represented HOLE, "heat" is to take a out of the formula and find its corresponding value. If it is 3, and "cool" puts 3 back into the original formula, at the formula becomes 3 + b.
Let us start with the assignment operation (we omit lookup as it is straightforward). The assignment of ST divides the assignment of ST demo into two steps, where context and rule Find Index are used to determine the index L of the assigned variable X in store, rule Assignment implements the update of the store at index L. The purpose of this division is to make the assignment operation better applicable to complex types, because in some cases the index of the assigned variable can not be directly obtained and multiple queries are required. For instance, when assigning a value to A [3,5,7], where A is a multi-dimensional array, we need to look up one by one dimension to finally determine the index. In addition, we refer to the state of X in type and constant during the assignment process. On the one hand, we use Limit to ensure that the assigned value meets the type requirements, and on the other hand, we ensure that the variable can be assigned through the value in constant, that is, this variable is not a constant. Although the memory cleaning operation is not necessary for ST in K, a simple clearenv operation can effectively reduce repetitive code and improve code readability. For rule clearenv, what needs attention is the operation on cell env, it replaces the index L of variable X with undef which means null in the map supported by K. ST has relatively complex and strict type definition, therefore the rule Allocation of ST involves more cells and operations, such as type and constant for storing variable types and whether they are constants, where U ndef ined is used to generate the default of the specified type. In addition, according to the content in TABLE 1, not only VAR will be used in the variable declaration process, but also other keywords, such as VAR INPUT, VAR IN OUT, etc. In order to reduce the complexity of the code, we also implement these declarations through VAR declaration. For instance, Fig. 12 shows the implementation of VAR GLOBAL and CONSTANT. We realize regional change (from env cell to gvenv cell) through letogv, and SetConstant realizes the modification of the value in const cell. Remark that, K-ST covers 259 core features with 876 rules in total and 2315 lines of K code.
TESTING AND ANALYSING ST COMPILERS
In addition to providing formal references for defined languages, our formal semantics also has several applications that use language-independent tools provided by K, such as state space exploration, model checking, symbol execution and deductive program validation. We omit demonstration of these applications in this paper since they have been well illustrated in related works [43], [44]. In this work, we introduce the testing of ST implementation/compiler based on executable semantics K-ST.
As discussed earlier, because ST compilers are typically provided by vendors, the execution behavior of compilers may be different, and may even be inconsistent with respect to the high-level semantics [49]. One of the main applications of the proposed semantics is to define the correct and secure 'reference' execution behavior of ST, which can help programmer detect bugs in existing ST compilers.
To explore this application (and given the closed nature of commercial compilers), we choose OpenPLC 2 as our test object, which is open source and supports ST programming. Our overall workflow of our testing approach is depicted in Fig. 13. It includes three parts: program variation, program execution and result comparison. First, seed programs are mutated is to improve the diversity of test samples. Program execution uses the mutated program as input to run OpenPLC and executable semantics respectively, and the result comparison is to compare the consistency of the two running results. It should be noted that we use a policy similar to [50], that is, the program does not need input, TABLE 4 shows the measure of consistency, where Q and Q represent the values of each variable after the program executes, I and I represent the commands corresponding to the exception termination, and represent consistency and inconsistency respectively. Specifically, unless K-ST and OpenPLC successfully execute the program and the result is exactly the same or terminated abnormally due to the same statement, the performance of both is considered to be inconsistent. In order to better mutate seed programs to improve the diversity of test samples, we propose specific mutation operations in TABLE 5 to generate mutated test samples. These mutation operations can enrich the test samples while minimizing program errors. Our method for generating mutant ST programs is shown in Algorithm 1. Given an ST program S i , the algorithm makes a copy, assigns randomly initial values to all variables at the time of declaration, and applies some applicable mutation operators to randomly selected lines in the program. The test is done by comparing results of these samples on K-ST and OpenPLC. It should be noted that the correct program and the error program in the test sample are meaningful for checking the consistency of execution behavior. Because K-ST and OpenPLC report errors to a program at the same time also belong to the category of consistency verification. In addition, considering the lag of OpenPLC updates, we also tested it on the latest
Arithmetic Operator Replacement a + b a − b Arithmetic Operator Insertion a + b a + b − c Arithmetic Operator Deletion a + b − c a + b
Relational Operator Replacement
EVALUATION
In order to evaluate the semantics of ST which defined in K, we deployed K-ST on the K version 5.1.11 (Intel(R) Core(TM) i7-9750H CPU @ 2.60GHz). In the following, we design multiple experiments to systematically answer the following research questions (RQs).
• RQ1: How much of the ST language is the K-ST covering? Completeness of the semantics is an important indicator to measure executable formal semantics. The lack of key semantics will seriously affect the usefulness of formal semantics.
• RQ2: Is K-ST correct? Semantic correctness is the basis for ensuring the usability of executable formal semantics, so we need to analyze the correctness of formal semantics implemented.
• RQ3: Can K-ST be used to discover bugs in the compiler? This is important since an important application of executable formal semantics is to identify compiler bugs.
Test Setting
For the purpose of evaluating the coverage and the correctness of K-ST, the test data set that we used comes from GitHub. We searched 4853 programs in GitHub through keywords in the ST language. Then we screened these samples. We first automatically screen out samples containing other programming languages (2516) and XML forms (1542). After that, we manually splice the remaining programs and remove samples that lack the components required for operation (such as POUs). After screening, 567 complete programs written in pure ST formed our test set. In other words, these 567 samples contain all the components required for operation and do not use other languages, such as C, and Python. With the aim of testing the correctness of the execution behavior of OpenPLC comprehensively, we use two sample sets, including test samples collected from Github (Github set) and test samples obtained through mutation (Mutated set). The Github set is the sample set with 567 test samples mentioned before. The Mutation set is generated by Algorithm 1. We selected 30 high-quality samples from Github set as initial mutant seeds. These 30 samples contain all the key features of ST. Then three rounds of iterative mutation are carried out through Algorithm 1. Each round of iteration produces 10 mutation samples per seed. Except for the initial seed used in the first round, the seeds of each round of mutation are the result of the previous round of mutation. We get a set containing 33,330 mutation samples.
Experiment Result and Analyses
Semantic completeness (RQ1)
We executed K-ST on 567 test samples collected from Github. Among these 567 test samples, K-ST supports the execution of 509 of them. For these 509 tests which K-ST can support, Fig. 14 As indicated in Fig. 14, compared with the FUNCTION, the FUNCTION BLOCK is more favored by programmers (the PROGRAM is necessary for ST program operation). For Declaration types, the most used is the VAR (with a ratio of 470/509), followed by VAR INPUT (386/509), VAR OUTPUT (360/509) and VAR IN OUT (313/509). Among all the Data types, BOOL is the most used, followed by unsigned integer and ARRAY varied slightly, constituting with 322/509 and 311/509, respectively. For the Data types, BOOL is the most common type. In addition, we must remark that we do not count the type of array members. At last, IF is the most common statement in all the tests considered. This is also in line with the main working scenarios of PLC.
Remark that, we do not consider the vendor-based functions. Because these functions vary not only from vendor to vendor, but even from product to product. In particular, Mitsubishi PLC provides completely different data types, including Bit, Word[Signed/Unsigned], Double Word[Signed/Unsigned], Bit STRING [16-bit/32bit], FLOAT, STRING [32] and Time. Siemens PLC supports keyword BEGIN to represent the end of variable declaration and the beginning of operation instructions. In addition, there are also obvious differences between different products of the same vendor. For example, the S7-1500 and the S7-1200 of Siemens support different type conversion methods 5 . Where the S7-1500 only provides explicit conversions of types, and the S7-1200 provides both explicit and implicit conversions.duiy
Semantics Correctness (RQ2)
On the other hand, in order to evaluate the correctness of K-ST, we compared the running results of K-ST with those of vendor compilers CODESYS, CX-Programmer and GX 5 Works2 respectively. We consider the proposed semantics is correct if the execution behaviors of K-ST are consistent with the ones of the CODESYS, CX-Programmer and GX Works2 compilers. We list the coverage of the K-ST semantics in TABLE 6 from the perspective of each feature specified by the official ST documentation, where FC, C and N mean "Fully Covered and Consistent with Compilers", "Covered and Consistent with Compilers" and "Not Covered", respectively.
From TABLE 6, we can see clear that for POUs, we fully cover the declaration and call. In variable declarations, AT is related to input and output. We remark, however, that the storage mode of variables in K is very different from that in real PLCs, so we just support simple computer-side input and output. In addition, RETAIN and PERSISTENT are related to the actual situation in the PLC, so they are not implemented. For instance, AT is used to bind the actual point of the PLC, RETAIN and PERSISTENT support the preserve of variable value after a power failure or power loss. The Array is the only one which be covered but not fully covered in all data types. Limited by the realization of arrays, it is temporarily impossible to achieve the array for enum and struct, and to assign values to multi-dimensional arrays as a whole. In statements, ⇒ has been used in the K and can be replaced by :=. For built-in functions, we show a list which we supported, including 30 numerical functions, 9 logical functions, 9 string functions and 160 translate function.
In the process of comparing with CODESYS, CX-Programmer and GX Works2, the following points need to be explained. Firstly, due to the closed nature of these compilers, they cannot be simply called, so we have to manually fill the code in the specified way into the compiler to compile and run, and compare the results, which is laborious and tedious work. This also hinders us from testing these commercial compilers in an extensively large scale. After that, different vendors have obvious differences in the implementation of compilers, so the source code needs to be adapted to a certain extent. For example, only 10 basic data types Bit, Word[Signed/Unsigned], Double Word[Signed/Unsigned], Bit STRING[16-bit/32-bit], FLOAT, STRING [32] and Time are provided in the GX Works2 compiler, so we need to adapt the variable types of the source program.
The Result of Testing for OpenPLC (RQ3)
We execute OpenPLC and K-ST with Github set and Mutation set as input. The execution results of the two data sets are shown in TABLE 7. Where K p O f is the number of programs that K-ST can be executed normally and OpenPLC cannot compile and run; K f O p is the number of programs that K-ST cannot run normally, and OpenPLC can execute normally.
For the Github set, K-ST supports 509 of them, and OpenPLC supports 490. Through analysis, we found that the reason for this phenomenon is that OpenPLC has some functional deficiencies. For example, OpenPLC does not support the initialization of variables using formulas at the time of declaration; numerical calculations of BYTE, WORD, DWORD types are not supported, etc.
For the Mutation set, there is a big difference between the execution results of K-ST and OpenPLC. First of all, we filter 2271 timeout programs that timed out both in OpenPLC and K-ST with 10 seconds as the time limit. After that, we manually analyzed these samples with inconsistent results to determine the causes. For the large K p O f value, functional deficiencies remain the main reason. We do find an interesting bug of OpenPLC. The bug is a "VAR" parsing exception in OpenPLC. If the first operation instruction starts with "VAR", such as "VAR0 := 1;", OpenPLC terminates abnormally. The interesting phenomenon is when an error statement appears in the unexecuted part of the program, such as after the "RETURN;", K-ST can provide the execution of the program, while OpenPLC cannot. The main reason for this phenomenon is that K adopts an operationbased detection mechanism. Because the error code will not be executed, it will not lead to the termination of executable semantics. The case study is shown in APPENDIX A.
CON ST AN T FC LREAL FC Assignment statement V AR GLOBAL FC BOOL FC := FC V AR FC BY T E FC ⇒ N V AR IN P U T FC W ORD FC Branch statement V AR OU T P U T FC DW ORD FC IF FC V AR IN OU T FC ST RIN G FC CASE FC V AR EXT ERN AL FC W ST RIN G FC Loop statement V AR T EM P FC T IM E FC W HILE FC AT C DAT E FC F OR FC RET AIN N T IM E OF DAY FC REP EAT FC P ERSIST EN T N DAT E AN D T IM E FC Break statement Typed constant Enum RET U RN FC Type # Data FC Enum declaration FC EXIT FC Built − in function
After that, by analyzing those programs that have different results on K-ST and OpenPLC, we find that the reasons for the different results are mainly due to the differences in underlying implementations between K and OpenPLC. For example, for integer mode operation −7 MOD 3, the running result of K-ST is −1, and the running result of OpenPLC is 2. From a mathematical point of view, both results are correct, but they will have a completely different impact on the following operations. Then we run the program again in CODESYS, and the results of CODESYS are the same as K-ST. For those samples that K-ST cannot run normally, but OpenPLC can execute normally, our analysis found some bugs in OpenPLC, which cause some error programs can be executed in OpenPLC. For example, OpenPLC can check explicit division 0 operations but allow the execution of implicit division 0 operations. TABLE 8 details all functional deficiencies and bugs we found in OpenPLC. We show some relevant case studies in APPENDIX B. Considering that Beremiz can be regarded as an updated version of OpenPLC, we have retested the inconsistencies we found in Beremiz. We found that in the latest Beremiz, it fixes some problems, including negative MOD operation results and "VAR" parsing exceptions. But other bugs and shortcomings still exist. So in response to these problems in OpenPLC, we have submitted them to OpenPLC and Beremiz developers and are waiting for their confirmation 6 .
RELATED WORK
In this section, we discuss some other PLC program analysis techniques, summarize their characteristics, and distinguish them from our work.
Keliris et al. [17] propose a framework (ICSREF) which can automate the reverse engineering process for PLC binaries. They instantiate ICSREF modules for reversing binaries compiled with CODESYS and getting the complete Control Flow Graph (CFG). And they provide an end-to-end case study of dynamic payload generation and attack deployment. Tychalas et al. [7] analyze the binary files generated by all control system programming languages in CODESYS to understand the differences and even the vulnerabilities introduced during the program compilation process. And based on this analysis, they provide a fuzzing framework (ICSFuzz) to perform security evaluation of the PLC binaries. Our work differs from them because we focus on the 6. https://bitbucket.org/automforge/matiec git/issues?status=new &status=open source code and do not rely on any specific compilation environment.
Kuzmin et al. [51] propose to use linear-time temporal logic (LTL) to guide program behavior and check whether ST program meets the corresponding temporal logic through Cadence SMV. Darvas et al. [13] propose rulebased reductions and Cone of Influence (COI) reduction variant for state explosion problems that may be encountered in the formal analysis of ST code, and use NuSMV model checker to verify temporal logic. After that, they [52] provide a state machine and data-flow-based formal specification method for PLC modules. In addition, they [35] analyze the feasibility of converting between the 5 PLC programming languages provided by Siemens, and point out that the extended SCL (a vendor-defined ST) can be used as the target language for conversion. Adiego et al. [53] propose an intermediate model-based method which can transform PLC programs written to different modeling languages of verification tools to facilitate checking temporal logic. Hailesellasie et al. [54] provide the UBIS which converts potential intrusion ST program and trusted version of the program into attributed graphs through UPPAAL and compares their nodes and edges to detect stealthy code injections. Bohlender et al. [55] apply formal verification and falsification of temporal logic specifications to analyze chemical plant automation systems. Rawlings et al. [56] use symbolic model checking tools st2smv and SynthSMV to verify and falsify a ST program controlling batch reactor systems. Xiong et al. [21] use the behavior model (BM) to specify the behavior of ST programs, and provide an method based on automatic theoretical to verify LTL attributes on BM. Our work differs from them because these works attempt to transformed PLC programs into intermediate languages or other programming language which are suitable for verifying or detecting potential issues in associated versifiers or checkers lack analysis and proof of equivalence in the conversion process, and the analysis they can perform is very limited. In addition, these methods do not offer the feedback to the level of source code.
Huang et al. [30] maybe the closest work to ours. They first defined the executable semantics of ST language in the K and use it to check some security properties. Our work differs from them because we covered more complete ST language and we can use it to discover the error of ST compilers.
CONCLUSION
In this paper, we introduce an executable operational semantics of ST formalized in the K-framework. We present the semantics of the core features of ST, namely data types, memory operations, main control statements and function calls. Experiment results show that the proposed ST semantics has already covered the main core language features and correctly implements 26,137 lines of public ST code on Github. Furthermore, the applications of the proposed semantics in the testing and analysing of PLC compiler are discussed. By comparing and analyzing the execution results of OpenPLC and executable semantics, we found five bugs and some functional deficiencies in OpenPLC. In the future, we hope to further improve the definition of ST semantics to adapt to the programming environment provided by different vendors. Because the vendor may customize keywords (Bit STRING of GX Works2), add additional structures (LABEL of Siemens), and even extend ST widely (ExST of CODESYS) when implementing.
Fig. 1 :
1High-level workflow of our approach when the code is being run on a PLC.
Fig. 2 :
2An ST programming example.
Fig. 3 :
3The syntax of ST demo .
Fig. 4 :
4The partial semantics of ST demo .
Ids | IdV al * EnumDeclarationExp ::= Id : (EnumBlock) ; | Id : (EnumBlock) := Id; StructDeclarationExp ::= Id : ST RU CT V arDeclarationExp * EN D ST RU CT Enum and Struct declaration F unction ::= F U N CT ION Id : T ype V arDeclaration * Statements EN D F U N CT ION Function declaration F unctionBlock ::= F U N CT ION BLOCK Id V arDeclaration * Statements EN D F U N CT ION Function block declaration P rogram ::= P ROGRAM Id V arDeclaration * Statements EN D P ROGRAM Program declaration
::= Ids : T ype; | Ids : T ype := Expression; V arDeclaration ::= V arT ype V arDeclaration EN D V AR
Fig. 5 :Fig. 6 :
56The runtime configuration of ST in K Implementation of TIME OF DAY in K.
Fig. 7 :
7The partial semantics of STRUCT in K.each defined data structure. The instantiation of STRUCT consists of four main steps: 1) rule Struct Instantiation: CreatStruct allocates memory for I1, 2) StructInits generates each variable in turn according to V ds in STRUCT, 3) Set assigns values to the corresponding variables according to Idvs, and finally, 4) U pdate stores the mapping relationship of variables related to I1 into the memory of I1 to facilitate subsequent use.
Fig. 8 :
8The partial semantics of main control statements.
Fig. 9 :
9The partial semantics of FUNCTION BLOCK.
Fig. 10 :
10The partial semantics of FUNCTION and PROGRAM.
Fig. 11 :
11The partial semantics of memory operations.
Fig. 12 :
12The partial semantics of variables declaration.
Fig. 13 :
13Overview of the test process.
category of result consistency comparison includes the values of all variables in the program.
Beremiz 3 which uses the same underlying implementation (MATIEC 4 ) as OpenPLC. The specific results of the test are shown in Section 5.
lists the number of tests for some important features which showed in TABLE 1 used in the evaluation. Specially speaking, there are six kinds of features, namely FUNCTION, FUNCTION BLOCK, PROGRAM, Declaration types, Date types and Statements. For Declaration types, we list the number of tests for CONSTANT, VAR GLOBAL, VAR, VAR INPUT, VAR OUTPUT, VAR IN OUT, VAR TEMP and VAR EXTERNAL. For Data types, we list the number of tests for elementary types -signed integer (INT, DINT, SINT, LINT), unsigned integer (UINT, UDINT, USINT, ULINT), float (REAL, LREAL), boolean (BOOL), byte (BYTE, WORD, DWORD), string (STRING, WSTRING), time (TIME, DATE, TIME OF DAY, DATE AND TIME), compound type -enum (ENUM) and struct (STRUCT) and array type -ARRAY. For Statements, we list the number of tests for main control statements -IF, CASE, FOR, WHILE, REPEAT, EXIT and RETURN.
Fig. 14 :
14Number of tests for each features in the ST.
N umerical f unction ( 30 )
30ADD, SU B, M U L, SQR, IN C, DEC, M AX, M IN , M U X, ABS, SQRT , T RU N C, F RAC, F LOOR, LN , LOG, EXP , SIN COS, T AN , COS, T AN ASIN , ACOS, AT AN , N EG, EXP T , DIV , M OD, LIM IT Logical f unction (9) GT , LT , GE, LE, EQ, N E, AN D, OR, SEL String f unction (9) CON CAT , IN SERT , DELET E, REP LACE, F IN D, LEN , LEF T , RIGHT , M ID T ranslate f unction (160) FC: Fully Covered and Consistent with Compilers (256/262) C: Covered and Consistent with Compilers (3/262) N: Not Covered (3/262)
TABLE 1 :
1The common elements of ST languageType
Element
Type
Element
Type
Element
Program Organization Unit
FUNCTION BLOCK
Built-in Data Type
INT
Built-in Data Type
ARRAY
FUNCTION
DINT
...
PROGRAM
SINT
Declaration Type
VAR GLOBAL
Main Statement
IF
LINT
VAR
CASE
UINT
VAR INPUT
WHILE
UDINT
VAR OUTPUT
FOR
USINT
VAR IN OUT
REPEAT
ULINT
VAR EXTERNAL
EXIT
REAL
VAR TEMP
RETURN
LREAL
AT
...
BOOL
RETAIN
User Data Type
ENUM
STRING
PERSISTENT
STRUCT
WSTRING
CONSTANT
Built-in Data Type
TIME
TIME OF DAY
...
DATE
DATE AND TIME
...
TABLE 2 :
2Summary of semantic notations.Notation
Description
rule
TABLE 3
3presents the syntax of ST defined in K-ST, which
covers most of the core syntax. We remark that TABLE 3
only contains the main part of K-ST while omitting oth-
ers, e.g., some built-in functions (LEN, DELETE and so on)
for space reason. The syntax is described by a dialect of
Extended Backus-Naur Form (EBNF) [47] according to the
grammar of ST, where * means zero or more repetitions.
In the ST, the top grammatical structures mainly includes
user-defined type -TYPE statements and three Program
Organization Units (POUs) -FUNCTION, FUNCTION BLOCK
and PROGRAM. The other parts are included in the top gram-
matical structures.
TABLE
Expressions ::= Expression * Assignment ::= Expression := Expression; Assignment statement ElseIf Block ::= ELSE Statements | ELSE IF Expression T HEN Statements ElseIf Block * If ::= IF Expression T HEN Statements ElseIf Block * EN D IF ; CaseBlock ::= Expression : Statements | Expression .. Expression : Statements Case ::= CASE Expression OF CaseBlock * EN D CASE; | CASE Expression OF CaseBlock * ELSE Statements EN D CASE; Branch statements W hile ::= W HILE Expression DO Statements EN D W HILE; F or ::= F OR Expression T O Expression DO Statements EN D F OR; | F OR Expression T O Expression BY Expression DO Statements EN D F OR; Repeat ::= REP EAT Statements U N T IL Expression EN D REP EAT ;Expressions
TABLE
TABLE 5 :
5Mutation operationsMutation Operation
Example
Variable Random Assignment
a : IN T ; a : IN T := 3527;
Scalar Variable Replacement
a := b; a := c | 30;
. https://support.industry.siemens.com/dl/dlmedia/272/109742272/att 918238/v6/93516999691/zh-CHS/index.html#ae443583b99950f7cca0d7237fe81ad4
TABLE 6 :
6Coverage of the proposed ST semanticsFeature
Coverage Feature
Coverage Feature
Coverage
POUs(core)
Data types(core)
Enum instantiation
FC
P OU s declaration
SIN T
FC
Struct
F U N CT ION
FC
IN T
FC
Struct declaration
FC
F U N CT ION BLOCK
FC
DIN T
FC
Struct instantiation
FC
P ROGRAM
FC
LIN T
FC
F unction block
P OU s calls
U SIN T
FC
F unction block instantiation
FC
F U N CT ION
FC
U IN T
FC
Array
F U N CT ION BLOCK
FC
U DIN T
FC
One − dimensional array
C
P ROGRAM
FC
U LIN T
FC
M ulti − dimensional array
C
Variable Declaration(core)
REAL
FC
Statements(core)
TABLE 7 :
7The results of K-ST and OpenPLC.Data Set
Github Set
Mutation Set
Number of samples
567
31059 (2271)
Number of program
K-ST
509
15850
run completely
OpenPLC
490
11581
Inconsistent
KpO f
30
5664
K f Op
11
1395
Different Result
0
735
TABLE 8 :
8The bugs and functional deficiencies of OpenPLC VAR" parsing exception The first operation instruction starts with "VAR", and OpenPLC terminates abnormally.Division by zeroOpenPLC can check explicit division 0 but allow the execution of implicit division 0.Overflow access OpenPLC can check explicit overflow access but allow the execution of implicit overflow access.MOD by zeroOpenPLC provides MOD 0 operation, and the result is 0.MOD ExceptionThe divisor of MOD operation can be empty.OpenPLC does not support normal numerical calculation * * . calculations of BYTE, WORD, DWORD types are not supported.Array functions defects Parentheses are not allowed in array assignments.instantiation defects Multiple instantiation of function blocks in one statement is not allowed. defects OpenPLC does not support normal assignment of ENUM type.Some non-keyword strings cannot be used as variable names, such as "ramp", "LocalVar0 ", etc.OpenPLC can not support formula and other variables previously declared as initial value.Structural defects Without operation or variable declarations, OpenPLC cannot compile ST program. Without statements in FOR, WHILE, IF, CASE, REPEAT, OpenPLC cannot compile ST.Type
Problem
Description
Bug
"Functional
deficiencies
Numerical
calculation defects
Numerical FUNCTION BLOCK
ENUM Variable
declaration defects
. https://www.openplcproject.com/
. https://beremiz.org/ 4. https://github.com/thiagoralves/OpenPLC Editor/tree/master/matiec
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| [
"https://github.com/thiagoralves/OpenPLC"
]
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[
"Colloquium: Structural, electronic and transport properties of silicon nanowires",
"Colloquium: Structural, electronic and transport properties of silicon nanowires"
]
| [
"Riccardo Rurali \nDepartament d'Enginyeria Electrònica\nInstitut de Ciència de Materials de Barcelona (CSIC)\nUniversitat Autònoma de Barcelona\nCampus de Bellaterra08193, 08193Bellaterra, Bellaterra, BarcelonaSpain, Spain\n"
]
| [
"Departament d'Enginyeria Electrònica\nInstitut de Ciència de Materials de Barcelona (CSIC)\nUniversitat Autònoma de Barcelona\nCampus de Bellaterra08193, 08193Bellaterra, Bellaterra, BarcelonaSpain, Spain"
]
| []
| In this paper we review the theory of silicon nanowires. We focus on nanowires with diameters below 10 nm, where quantum effects become important and the properties diverge significantly from those of bulk silicon. These wires can be efficiently treated within electronic structure simulation methods and will be among the most important functional blocks of future nanoelectronic devices. Firstly, we review the structural properties of silicon nanowires, emphasizing the close connection between the growth orientation, the cross-section and the bounding facets.Secondly, we discuss the electronic structure of pristine and doped nanowires, which hold the ultimate key for their applicability in novel electronic devices. Finally, we review transport properties where some of the most important limitations in the performances of nanowire-based devices can lay. Many of the unique properties of these systems are at the same time defying challenges and opportunities for great technological advances. | 10.1103/revmodphys.82.427 | [
"https://arxiv.org/pdf/0910.2553v1.pdf"
]
| 4,946,328 | 0910.2553 | 113db0680d48e654265abef5a56fa69000d73b42 |
Colloquium: Structural, electronic and transport properties of silicon nanowires
14 Oct 2009
Riccardo Rurali
Departament d'Enginyeria Electrònica
Institut de Ciència de Materials de Barcelona (CSIC)
Universitat Autònoma de Barcelona
Campus de Bellaterra08193, 08193Bellaterra, Bellaterra, BarcelonaSpain, Spain
Colloquium: Structural, electronic and transport properties of silicon nanowires
14 Oct 2009(Dated: October 14, 2009)
In this paper we review the theory of silicon nanowires. We focus on nanowires with diameters below 10 nm, where quantum effects become important and the properties diverge significantly from those of bulk silicon. These wires can be efficiently treated within electronic structure simulation methods and will be among the most important functional blocks of future nanoelectronic devices. Firstly, we review the structural properties of silicon nanowires, emphasizing the close connection between the growth orientation, the cross-section and the bounding facets.Secondly, we discuss the electronic structure of pristine and doped nanowires, which hold the ultimate key for their applicability in novel electronic devices. Finally, we review transport properties where some of the most important limitations in the performances of nanowire-based devices can lay. Many of the unique properties of these systems are at the same time defying challenges and opportunities for great technological advances.
Figures 50
I. INTRODUCTION
One-dimensional nanostructured systems have attracted a great attention in the last two decades, with this interest extraordinary boosted by the facile synthesis of carbon nanotubes (CNTs) reported in the beginning of the 90s (Iijima, 1991). The reason is twofold: on the one hand they have proved to be an excellent test-bed to study the most intriguing physical effects, whereas on the other hand they are believed to be among the most important building blocks of the next generation of electronic devices.
CNTs are hollow cylinders obtained by rolling up one or more graphene sheets, a oneatom-thick allotrope of carbon (Charlier et al., 2007). The symmetry and the electronic structure of graphene (Neto et al., 2009) are such that the properties of the CNT depends critically on the exact way it is wrapped up, and it can be either metallic or semiconducting.
This confer the CNTs with a richer physics, but it is clearly far from ideal from the viewpoint of applications, especially when -as it is the case-a simple route to selectively grow one type of CNT or the other is lacking.
Nanowires are an extremely attractive alternative to CNTs, because it is much easier to control their electrical properties and, as long as the surface is properly passivated -something that occurs naturally during or right after growth-, they are invariably semiconducting 1 .
Silicon nanowires (SiNWs), in particular, look like a very appealing choice, since they provide the ideal interface with the existing Si devices, while taking advantage from a tractable material technology. SiNWs are commonly grown by the vapor-liquid-solid technique (Wagner and Ellis, 1964;Westwater et al., 1997), where a Au nanoparticle is used to catalyze SiH 4 decomposition. Briefly, the Au particle is deposited onto a Si surface and react with the Si atoms of the substrate, forming Au-Si alloy droplets. These droplets adsorb Si from the vapor phase, resulting in a supersaturated state where the Si atoms precipitate and the SiNW starts nucleating 2 .
As David K. Ferry illustrates in an enlightening paper (Ferry, 2008), nanowires could provide the paradigm shift needed to continue improving the density and the performances of electronic circuits. For almost four decades the increase in computing power has been described by the well-known Moore's law (Moore, 1965), which has been standing on three pillars: (a) the increase of the size of the microchips; (b) the reduction of the transistor size, and (c) the circuit cleverness, that is the reduction of the number of devices required to perform a certain function. While the first of these driving forces played a significant role only in the pioneering years of solid state electronics, the reduction of device size has a pivotal role, since the physical limit of material scaling is rapidly approaching. Nanowires can lead to an obvious benefit concerning the miniaturization, thanks to bottom-up growth that allows overcoming the limit of conventional lithography-based top-down design. Subtler are the perspective advantages concerning circuit cleverness, which can be significantly improved by taking advantage of the coexisting nature of interconnection and active device of nanowires.
In particular, a replacement of metallic vias with vertical transistors is envisaged. These new circuits could be easily reconfigurated to perform different operations, achieving a much higher level of integration (Ferry, 2008). Additionally, compared to classical planar device technology, nanowires can better accommodate all-around gates (see Fig. 1), which improve field-effect efficiency and device performances (Colinge, 2004;Ng et al., 2004) and mobilities of ∼ 1000 cm 2 V −1 s −1 , substantially larger than those obtained in conventional Si devices, have been obtained 3 .
Several promising applications have already been demonstrated, ranging from electron devices Cui and Lieber, 2001;Cui et al., 2003;Goldberger et al., 2006;Hu et al., 2008;Lu et al., 2008b;Wang et al., 2006b;Yu et al., 2000;Zheng et al., 2004), logic gates (Huang et al., 2001), non-volatile memories (Duan et al., 2002), photovoltaics (Kempa et al., 2008;Tian et al., 2009Tian et al., , 2007, photonics (Gudiksen et al., 2002; 2 See Wang et al. (2008) for a comprehensive review of the growth techniques.
3 It is difficult to make rigorous comparisons, because the mobility has a strong inverse dependence on the dopant density which is seldom known with accuracy in nanowires. However, the peak value of 1350 cm 2 V −1 s −1 obtained for the hole mobility by Cui et al. (2003) must be compared with the typical values for bulk Si of ∼ 400 cm 2 V −1 s −1 and ∼ 100 cm 2 V −1 s −1 for an acceptor concentration of 10 16 cm −3 and 10 18 cm −3 , respectively. Pauzauskie and Yang, 2006), to biological sensors (Cui et al., 2001b;Hahm and Lieber, 2004;Zhong et al., 2003). On top of that, giant piezoresistance effect (He and Yang, 2006) and enhanced thermoelectric performances (Boukai et al., 2008;Hochbaum et al., 2008) have recently been reported. The interested reader is encouraged to check some of the many experimental reviews (Kumar, 2007;Li et al., 2006;Lu and Lieber, 2006;Patolsky and Lieber, 2005;Thelander et al., 2006;Wu et al., 2008a;Xia et al., 2003).
In this paper we will review the theory of SiNWs. Clearly, we will make several references to experiments, whenever they support or challenge the theoretical predictions. Sometimes the comparisons are difficult to make, because SiNWs that are routinely grown range from 50 to 200 nm, while those that can be efficiently studied within electronic structure methods are 2-3 nm thick, at most. Luckily, this gap is slowly narrowing and thin SiNWs with diameters below 10 nm have been successfully grown by several groups (Coleman et al., 2001a,b;Cui et al., 2001aCui et al., , 2003De Padova et al., 2008;Holmes et al., 2000;Ma et al., 2003;Morales and Lieber, 1998;Wu et al., 2004;Zhong et al., 2005). The theoretical results that we discuss outline the most urgent problems that will have to be dealt with within the next generation of nanowires, those with characteristic sizes approaching the quantum limit.
Although many of the features that we will discuss are common to other types of semiconducting nanowires, for the sake of clarity we will restrict to SiNWs throughout the paper. It should be at least pointed out, however, that in recent years tremendous progresses are being made with compound semiconductors nanowires -mainly III-V nanowires-especially for what concerns photonics application (Björk et al., 2002;Dick et al., 2004;Thelander et al., 2003).
A final remark concerns the computational methodologies. Although the main goal of the paper is giving a complete overview of the most important results that have been obtained within atomistic simulations, we will not enter in technical details, unless where it is necessary. Most of the results have been obtained within density functional theory (DFT), whose theoretical grounds are clearly out of the scope of this work. The interested reader can look at both the original papers (Hohenberg and Kohn, 1964;Kohn and Sham, 1965), excellent reviews (Jones and Gunnarsson, 1989;Payne et al., 1992) and comprehensive books (Martin, 2004). Less frequently, we will refer to the tight-binding formalism (Colombo, 2005;Goringe et al., 1997;Slater and Koster, 1954) or to empirical interatomic potentials (Justo et al., 1998;Stillinger and Weber, 1985;Tersoff, 1989).
II. STRUCTURAL PROPERTIES
A. Growth orientations and monocrystallinity
The extraordinary impact that the discovery of carbon nanotubes (Iijima, 1991;Oberlin et al., 1976;Radushkevich and Lukyanovich, 1952) had on condensed matter and nanoscience at first biased the research on Si quasi one-dimensional systems to the pursuit of tubular structures. Hollow structures resembling carbon nanotubes (Li et al., 2002), structures based on hollow elements (Menon and Richter, 1999) or on fullerene-like system (Marsen and Sattler, 1999) have been proposed. Although these -or other structures inspired by cluster assemble (Sen et al., 2002)-are stable within a total energy framework, they have not been observed experimentally to date.
In the meanwhile Si nanotubes have been indeed successfully synthesized (Sha et al., 2002), while their use for nanoelectronics still remains troublesome (Perepichka and Rosei, 2006), and things with Si nanowires turned out to be simpler than speculated. Convincing experimental evidence soon indicated that SiNWs are rod-like structures constructed around a bulk Si single-crystalline core (Holmes et al., 2000;Morales and Lieber, 1998;Teo et al., 2003;Wu et al., 2004;Zhang et al., 2000).
An important consequence of their single-crystal nature is that SiNWs grow along very well defined crystalline directions (see Fig. 2). Wu et al. (2004) preference that leads to 110 over 111 SiNWs at small diameters is also supported by first-principles calculations (Akiyama et al., 2006). More recently, a continuum model that allows studying how growth begins and evolves toward steady-state wire growth has been presented (Schwarz and Tersoff, 2009). The advantage of this approach is that complex situations such as catalyst coarsening and interrupted growth can be easily handled.
Ideally, nonetheless, one would like to be able to control the wire orientation at growth
time. An important achievement in this sense was the demonstration that the growth orientation can also be controlled externally by adjusting the growth pressure (Holmes et al., 2000;Lugstein et al., 2008). Alternatively, the use of different techniques can bias somehow the growth along certain crystal axis. For instance, the less common oxide-assisted growth method, generally yielding a broader diameter distribution (Wang et al., 1998), might favor different orientations for ultra-thin SiNWs (Teo et al., 2003). Significantly, the thinnest SiNW reported to date (Ma et al., 2003) was synthesized with this technique and was a 112 wire (see Fig. 3).
B. Surface reconstructions in pristine nanowires
The next major issue one has to face when studying the structure of a SiNW is the shape of its cross-section which, as we shall see briefly, is intimately related with the growth orientation. Although one can pictorially imagine nanowires as cylindrical structures, clearly, when going down to the atomic-scale detail, this is not the structural arrangement that they assume -or even can assume. The analogous problem in solids and small particles (Wang et al., 1984;Zhdanov and Kasemo, 1998) is elegantly solved by means of the Wulff construction or Wulff rule (Marks, 1994), which relates the equilibrium shape with the surface free energy of the facets involved. Solving the energy minimization problem min sγ s , where s is the number of surface unit cells and γ s the corresponding energy, leads to the optimum shape. Zhao and Yakobson (2003) have reexamined the use of Wulff construction within the determination of the equilibrium cross-sections of SiNWs. They showed that the conventional formulation of the Wulff criterion lacks of two important aspects: (i) in solids and smooth spherical particles the energy of the edges between facets is neglected compared to the surface contribution; (ii) the bulk is assumed already at its minimum and thus invariant.
Hence, they propose the following generalization for the Wulff energy:
F = E e + s sγ s + E b(1)
where they include the energy of matching adjacent facets E e , i.e. the energy of the edges, and the energy of the bulk E b , releasing the constraint on the innermost part of the wire which can now change.
They investigated different faceting arrangements for SiNWs grown along the 110 axis comparing them on the basis of Eq. 1. They found that the ground-state structure for SiNWs up to 5 nm is a pentagonal cross-section constructed joining five prisms cut out of a [110] Si plane [see Fig. 5(d)]. This structure has seldom been detected experimentally [a remarkable observation by Takeguchi et al. (2001) is shown in Fig. 5(e)]), probably because it is not constructed around a bulk-core, which seems to be the favored situation at growth time.
However, if one restricts to wires with a strictly bulk core the model of Zhao and Yakobson (2003) correctly predicts hexagonal over square cross-sections for 110 SiNWs, in agreement with the experiments (Ma et al., 2003;Wu et al., 2004) (see Fig. 7).
The most important result of the work of Zhao and Yakobson (2003) is emphasizing the role of the edges and how the interplay between edges and surfaces play a key role in determining the reconstruction of Si one-dimensional structures. Before its formalization, this effect had been already pointed out by Ismail-Beigi and Arias (1998) Unfortunately, a word of care should be spent concerning the above discussion. Down at the ultimate nanoscale limit it is delicate to give general rules and for extremely thin SiNWs counterexamples can be found to the general trends discussed previously. For instance, Cao et al. (2006) showed that the faceting arrangement proposed by Ismail-Beigi and Arias (1998) for 100 wires and later followed by other authors (Lee and Rudd, 2007b;Rurali, 2005;Rurali and Lorente, 2005a;Vo et al., 2006) is favored only beyond a 1.7 nm diameter, whereas tiny SiNWs prefer sharp edges, i.e. the removal of the edges does not pay back.
In the spirit of the work of Zhao et al. (2004), Justo et al. (2007) carried out an interesting and systematic study of SiNWs grown along the 100 , 110 , and 112 crystal axis, carrying out extensive calculations based on an interatomic potential (Justo et al., 1998) Silicon has a very rich phase diagram (Kaczmarski et al., 2005) and many solid phases other than the diamond structure are known. Among them is the so-called clathrate phase that becomes stable at negative pressures. The stability of such a phase for quasi one-dimensional nanostructure has been investigated by Ponomareva and coworkers (Ponomareva et al., 2006(Ponomareva et al., , 2005. They studied cage-like SiNWs carved out of a
Si clathrate structure and compared them with both tetrahedral diamond-like and polycrystalline SiNWs. Their results indicate that also in these nanostructures the tetrahedral structure is favored. Nevertheless, the difference in energy is rather small and it is suggested that clathrate based SiNWs might have better conductive properties.
C. Passivated nanowires
The study of the structure of pristine SiNWs has been a fertile ground where to start the theoretical research of these fascinating systems. However, quite soon it became clear that the wires grown experimentally have always passivated facets. Silicon form highly directional covalent bonds according to the know sp 3 tetrahedral pattern. Silicon atoms at the surface have dangling bonds (DB), unsaturated bonds that make the atom highly reactive and that induce strong reconstruction of the surfaces. Generally speaking, surface passivation consists of the termination of DBs on the surface with elements that assure their chemical stability.
Hence the surface is chemically passive.
Surface passivation in SiNWs mainly originates from two causes: (i) the growth of a thin layer of SiO 2 by thermal oxidation of silicon; (ii) presence of hydrogen in the growth environment during the synthesis or HF attack of the oxidized wires after growth, a process yielding removal of the SiO 2 layer and H passivation. Hydrogen passivation is rather simple to model. If a sufficient amount of hydrogen is supplied the H atoms readily terminate each Si DB by forming a stable Si-H system. Passivation by oxidation is more complex.
Thermal oxide is amorphous and then difficult to model at the nanoscale, because of the large amount of atoms required to describe the disordered phase. In the study of SiNWs, for most practical effects, hydrogen termination is a reasonable approximation to oxide passivation and this is the strategy adopted in most of the theoretical studies reviewed
here. This approach is also justified by the fact that it is easy to remove the oxide layer after the growth and to induce H passivation by simply etching it with HF. This procedure is often followed (Guichard et al., 2006;He and Yang, 2006;Ma et al., 2003;Ross et al., 2005;Wang et al., 2008;Wu et al., 2004) in order to work with cleaner structures where the passivation rely on an individual termination of the DBs, rather than a less controllable and more defective oxide coverage (Baumer et al., 2004). Furthermore, it has also proven to leave the morphology of the nanowire essentially intact, except for the removal of the oxide layer (Zhang et al., 2000), allowing inspection of the underlying atomic scale structure (see Fig. 3). Yet, more attention is likely to be devoted in the near future to the specific nature of SiO 2 passivation, beyond the simple models considered so far (Avramov et al., 2007).
The passivation has a crucial effect on the electronic structure of the wires and it is essential to provide the wires with predictable band gap widths and an invariably semiconducting character. We will discuss these topics in detail in Section III.
Surface passivation has also an important effect on the structural arrangements of SiNWs.
Besides preventing complex reconstructions, it also influences the structure of the sub-surface and innermost part of the wires. H-passivated SiNWs grown along different orientations have been found to maintain remarkably the bulk symmetry (see Fig. 4), with negligible deviations of the Si-Si bond lengths; the deviation increases close to the surface, depending on the level of surface rearrangement (Vo et al., 2006). The limit case in this sense are pristine wires, where the absence of passivation results in major surface rearrangements and large deviations of the Si-Si bond length also in the wire core (Kagimura et al., 2005).
An interesting path to the determination of the structure of H-passivated 110 SiNWs has been proposed by Chan et al. (2006). Their optimization procedure is based on a genetic algorithm. With this method, in principle suitable for any other growth orientation, they identified a pool of magic structures 4 for 110 wires. Although some of them have not been observed experimentally, their hexagonal structure provided a good agreement with the STM image of the wire facet of Ma et al. (2003).
A more systematic approach was followed by Zhang et al. (2005) in a study analogous to the one performed by Justo et al. (2007) for pristine nanowires. They carried out a comprehensive study of the possible low-index facets in H-passivated SiNWs grown along the 100 , 110 , 111 , and 112 axis. While many choices are possible for 100 , 110 , and 111 wires, they showed that there is only one low index configuration -with two {111}
and two {110} facets-for 112 wires. It is suggested that this would ease the controlled growth with a predetermined cross-section and could have important consequences on the engineering of devices based on SiNWs. In Section III.B, however, we will see that it has been recently suggested that the exact cross-section shape is less important than other parameters -such as the effective diameter and the surface-to-volume ratio-when it comes to determining the electronic properties of SiNWs.
Another important aspect to consider in H-passivated SiNWs is the surface structure of the hydrogenated facets. This issue has been tackled by Vo et al. (2006), where a systematic study of the effects of varying the diameter and the growth direction has on the structure of the hydrogenated surfaces of 100 110 and 111 SiNWs. In particular, they studied the relative stability of symmetric SiH 2 dihydrides, canted SiH 2 dihydrides and a (2×1) surface reconstruction (where first reconstruction is allowed and then passivation occur), see Fig. 6. They deliberately chose simple, round cross-sections, as their scope was focusing on the atomic scale structure of the facet. Their wires were constructed selecting all the atoms falling inside a virtual cylinder placed in bulk silicon, in such a way that the facets approximated a circular cross-section. This procedure agrees with the smoothness prescription described above, which -more importantly-seems also to be confirmed by the experiments (Ma et al., 2003). They found that, in agreement with bulk Si (100) surfaces (Northrup, 1991), the canted dihydride surface is more stable than the symmetric dihydride structure, because canting allows a larger H-H separation. Additionally, faceting confers an increased stability to the canted dihydride surface, because at the facets' edges the SiH 2 groups are free to rotate. Relief of the surface strain through bending as an additional mechanism has been explored by Zdetsis et al. (2007).
A perhaps more flagrant effect of the surface induced strain is the fact that the axial lattice parameter of thin SiNWs is in general different from bulk Si. Ng et al. (2007) reported contraction along the wire axis for 100 111 and 112 SiNWs, and elongation for 110 growth orientation.
Other types of surface passivation -including OH (Aradi et al., 2007;Ng et al., 2007;Nolan et al., 2007), NH 2 (Nolan et al., 2007), F (Ng et al., 2007), or Br, Cl, andI (Leu et al., 2006)-have been considered. While changing the passivation has a limited effect the structural properties of the nanowire, it can affect in a more significant way the electronic band structure. We will come back on this topic in Section III.B.3.
As a conclusive remark one should notice that, despite the intensive research carried out to find the equilibrium shapes for the different growth orientations -proposing structures that range from fullerene-like (Marsen and Sattler, 1999) to star-shaped (Sorokin et al., 2008)in most cases the experimentally observed cross-sections of passivated SiNWs are deceptively simple (see Fig. 3 and 7), whereas unpassivated SiNWs have never been reported.
Furthermore, as we shall see in Section III, although the cross-section shape has captured great attention and has been the object of many studies, in realistic, passivated wires the growth orientation and the average diameter turned out to have a more significant impact on the electronic properties of SiNWs.
D. Mechanical properties of nanowires
If one carves out of bulk Si a rod-shaped system like a nanowire, there is no apparent reason to expect an enhanced stiffness, while the larger surface-to-volume ratio is rather suspected to be detrimental. A simple way of understanding these effects is that there is a layer of material at the surface and edges whose mechanical properties differ from those of the bulk including different elastic moduli and eigenstrains.
These intuitive ideas have been rigorously tested by Lee and Rudd (2007b), by means of an exhaustive study of 100 SiNWs with increasing diameters. They calculated the Young's modulus, finding that it softens from the bulk value as the surface-to-volume ratio increases, going through a steep decrease around 2-2.5 nm diameter (see Fig. 8). They showed that the origin of this behavior is the compressive surface stress. To get a better insight into these atomic scale mechanisms the Young's modulus can be decomposed into a core (Si core atoms) and a surface contribution (Si surface atoms, H-H and Si-H systems).
This decomposition allows highlighting the insensitivity to the facet ratio, as the contributions to the Young's modulus that are strongly facet dependent are very small. These first-principles results are in good agreement with empirical atomistic potentials and continuum techniques (Lee and Rudd, 2007a), unless for the smallest wires where these simplified approaches fail (see Fig. 8).
The Young's modulus, as many other properties of ultra-thin SiNWs reviewed in this paper, is strongly anisotropic. Ma et al. (2008) These results are in good agreement with the work of Leu et al. (2008) where the Poisson ratio is also considered.
We note that to calculate the Young's modulus a definition of the cross-sectional area must be assumed and it is not univocal. We will run into a similar problem concerning the definition of the wire diameter in Section III.B.2 when dealing with quantum confinement. Lee and Rudd (2007b) The error in the Young's modulus is 2δr 0 /r 0 and goes to zero in the limit of large wires (r 0 → ∞). The Poisson ratio, on the other hand, is much more sensitive, because the error is −δr 0 /r 0 + (δr − δr 0 )/(r − r 0 ), r being the radius at a strain ǫ; while the first term vanishes for large radius, the other is always present and can be significant as (r − r 0 ) is typically small.
Bending has received comparatively less attention, although a few experimental measurements have been reported (Hoffmann et al., 2006;Hsin et al., 2008;Tabib-Azar et al., 2005;Zheng et al., 2009). This is probably due to the difficulty of studying a bended nanowire within atomistic simulations that normally relies on periodic boundary conditions. The fab-rication of ingenious mechanical structures with enhanced elastic properties suggests that this could be a promising research direction (San Paulo et al., 2007).
Beyond elastic deformation, materials undergo non reversible, plastic deformation which directly precede fracture. This regime has been studied for 100 pristine
SiNWs (Justo et al., 2007) and for 111 and cage-like SiNWs (Menon et al., 2004), using two different interatomic potentials (Justo et al., 1998;Stillinger and Weber, 1985).
At small strains ǫ the stress increases linearly, as expected in the elastic regime, while at larger deformation the plastic behavior appears until the fracture occurs at ǫ ∼ 0.10, with a good agreement between the two different models. Experimentally, however, the fracture is much delayed with respect to theoretical predictions and the wire breaks at et al., 2005). It should be mentioned that both these theoretical studies considered pristine nanowires, while the wires in the experiments are coated with a thin layer of oxide, thus it is difficult to make a rigorous comparison. At the same time it is not surprising that a different surface treatment can produce a noticeable difference in the mechanic response, because it is just at the surface that the nanoscale signature emerges.
ǫ ∼ 0.25 (Kizuka
Correlating structural deformations and changes in the electronic properties is an active field of research (Rurali et al., 2008a) and the use of strain to enhance carrier mobility has been investigated (Hong et al., 2008;Huang et al., 2008;Leu et al., 2008). Furthermore, a giant piezoresistance effect -the application of a strain to a crystal that results in a change in the electrical resistance-has been reported recently (He and Yang, 2006). The underlying atomic scale mechanism is still poorly understood, however, and the attempts made so far proved to be somehow elusive (Cao et al., 2007;Rowe, 2008).
III. ELECTRONIC PROPERTIES
A. Pristine Nanowires
The reason for the attention devoted to geometrical features such as the growth orientation, the faceting arrangement, and the surface structure, is that they are crucial when it comes to the electronic properties of the nanowire. Clearly, the thinner is the nanowire, the more it is sensitive to the structure details, as in the limit of very large diameter -no matter which is its crystal axis or cross-section shape-its properties converge to those of bulk Si.
As we discussed in the previous section, pristine nanowires turned out to have a limited relevance, at least to date, because experimentally grown SiNWs are always passivated.
However, the study of bare, unpassivated wires is still interesting for two reasons: (a) it leads to the important conclusion that passivation is essential to obtain nanowires with predictable and easy to control electrical properties; (b) it sheds a light on some atomic scale mechanisms of high fundamental interest.
An interesting example in this sense is the electronic structure of 100 SiNWs with {100}
facets. While other facets, like the {111} facets, have an electronic structure similar to the corresponding infinite surface (Pandey, 1981;Rurali et al., 2006), {100} facets can be very different. In the Si(100) surface each surface atom has two DBs. The surface is known to reduce its energy by forming dimers, thus halving the number of DBs (Chadi, 1979).
The reconstruction of {100} facets follows the same pattern, but wires dominated by such facets have been reported to be metallic. Rurali and Lorente (2005a) showed that thin 100
SiNWs sustain two different reconstructions of the {100} facet that turn the wire metallic or semimetallic, in agreement with what previously suggested by Ismail-Beigi and Arias (1998).
The metallic behavior can be ascribed to a modified coordination of the {100} facet atoms, leading to a distortion of the surface dimers, with respect to the Si(100) surface . The metallicity rapidly vanishes as the diameter is increased and the facets recover the coordination and the semiconducting electronic structure of the Si(100) surface.
The 100 wires with sharp corners studied by Cao et al. (2006) can be metallic too.
Interestingly, the edge metallic states decay slower with the diameter compared to the facet metallic states. Consequently, wires thicker than those considered by Rurali and Lorente (2005a), where edges were absent, can still be metallic. In both cases the metallic states are related directly or indirectly with the edges -in one case purely edge states, in the other dimer rows with an altered coordination near the edges; as the wire size increases the relative number of atoms at the edges decreases rapidly and the facet recovers the semiconducting character of the infinite surface.
Besides the fundamental interest of these findings -in one case a metallicity driven by the finite size of the facet, in the other sustained by the edges-it is clear that such wires are not desirable for electronics application. On the one hand one wants to work with semiconducting systems; on the other hand, although some application can be envisaged for metallic SiNWs, e.g. interconnects, the metallicity should be much more robust, so that it is not destroyed by small variations of the diameter and does not depend critically on the atomic scale structure of the wire.
A comprehensive study of the surface reconstruction and electronic structure of pristine 110 wires has been carried out by Singh et al. (2005). The cross-section chosen for these wires is such that they have {100} Contrarily to what we shall discuss in Section III.B.2 concerning quantum confinement, here the thinner is the wire, the smaller is the effective band gap. This looks like a general feature of pristine, unpassivated nanowires, where band gaps are smaller than in bulk (Rurali, 2005;Rurali et al., 2006). In thin wires the surface-to-volume ratio is larger and surface states, which often lie in the gap, dominate the electronic structure and result in an effective narrowing of the energy gap.
B. Passivated nanowires
Band structure and band gap anisotropy
We have already mentioned a few times that surface passivation is required to obtain ultra-thin nanowires that are semiconducting and have a predictable and controllable band gap. Notwithstanding, the electronic structure of the nanowires still depends on the growth orientation, on the cross-section shape and on the diameter. The band gap is strongly anisotropic (Leu et al., 2006(Leu et al., , 2008Ng et al., 2007;Niquet et al., 2006;Rurali et al., 2007a;Singh et al., 2006;Vo et al., 2006;Yan et al., 2007;Zhao et al., 2004) and, for wires of com-parable diameters, it follows the ordering
E 100 g > E 111 g ∼ E 112 g > E 110 g ,(2)
with the orientation effect still sizeable up to 3 nm diameter (Ng et al., 2007). The band gap of 112 wires is of the same order of 111 wires, though it has been reported to be slightly larger (Leu et al., 2006;Rurali et al., 2007a) or slightly smaller (Ng et al., 2007).
This anisotropy has been qualitatively tracked back to the different geometrical structure of the wires in the 100 , 111 , and 110 directions (Bruno et al., 2005(Bruno et al., , 2007b. While the 100 and 111 wires appear as a collection of small clusters connected along the axis, the 110 wires resemble a linear chain (see Fig. 11 where the case of Ge nanowires is shown).
Therefore, one expects that quantum confinement effects are larger in the 100 and 111
wires, because of their quasi zero-dimensionality, with respect to the 110 wires.
Bulk Si has an indirect band gap, with the valence band maximum at the Γ point and the conduction minimum at about 85% along the Γ to X direction, and a phonon is required to conserve the momentum in any electronic transition. Remarkably, however, SiNWs grown along most of the crystallographic orientations have a direct band gap, meaning that the maximum of the valence band and the minimum of the conduction band occur at the same point in k-space. This property has allowed to envisage the use of SiNWs as optically active materials for photonics applications (Canham, 1990;Guichard et al., 2006).
In 100 SiNWs, the confinement plane contain four of the six equivalent conduction band valleys. These minima at ±y and ±z are then projected onto Γ due to band folding, thus resulting in a direct band gap. When the axis is along a lower symmetry direction the confinement plane cannot contain four conduction band valleys and it will contain at most two. This is the case of 110 SiNWs. The minima at ±z are projected onto Γ again. Now, both the large and the small masses appear in the confinement plane, with the larger longitudinal mass being the relevant effective mass for describing the confinement effect in the cross-section plane. On the other hand, the four remaining minima will be projected to a point between Γ and the zone boundary Z, with the effective mass on the confinement plane being a value between the longitudinal and transverse masses. Therefore, the conductionband edge at Γ is expected to have a smaller upward shift induced by confinement and the band gap becomes direct (Yan et al., 2007).
Although the projection along the 111 axis would lead to an indirect band gap, the thinnest 111 SiNWs have a direct band gap (Rurali et al., 2007a;Vo et al., 2006;Zhao et al., 2004). One should bear in mind that, besides the band folding arguments given above, the effective masses play an important role. In the quantum confinement regime (see below, Section III.B.2) the conduction band states are shifted upward, the smaller the diameter, the larger the shift. Nonetheless, the magnitude of this energy shift is different for each k-point of the band structure and depends on the effective mass. In bulk Si the effective mass at Γ is heavier than at X or L. Hence, upon confinement, one expects the conduction band energy at X and L to increase more than at Γ. This simple considerations based on effective mass theory (EMT) describe well the transition from direct to indirect band gap experienced by 111 SiNWs that occur around 2 nm: as the diameter increases the quantum confinement effect vanishes (see Section III.B.2), Γ and X and L points are not shifted and the gap remains indirect. Among the studied orientations 112 SiNWs are the only ones that have an indirect band gap also for the thinnest diameters (Aradi et al., 2007;Huang et al., 2008;Lu et al., 2008a;Ng et al., 2007;Rurali et al., 2007a;Scheel et al., 2005).
Although the band gap is highly anisotropic and, as we shall see in the next section, strongly dependent on the wire diameter, it is very interesting to observe that it is rather insensitive to the shape of the cross-section. Ng et al. (2007) have studied the effect of the variation of the cross-section in thin 110 SiNW, generating 13 different cross-sections obtained by modifications of a reference 1 nm wire. They found that the band gap is practically constant and changes are within 0.09 eV. Later, it was demonstrated that wires of even utterly different cross-sections can have the same band gap, provided that their surface-to-volume ratio is the same (Yao et al., 2008). The effect of the surface-to-volume ratio on the band gap can be described by the universal expression
E gap = E bulk gap + aS(3)
where E bulk gap is the gap of bulk Si, a is an adjustable parameter and S is the surface-to-volume (in nm −1 ).
One of the most intriguing physical effect that arise in confined systems like SiNWs is the so-called quantum confinement. Such a regime is conveniently described through the particlein-a-box model system in most quantum mechanics text books (Bransden and Joachain, 2000). The simplified situation considered is an infinite potential well where the motion of the particles is restricted to be in the direction of the confinement. As the motion of the particles is restricted, their kinetic energy increases and it is readily shown that the eigenstate energies are given by the following relation:
E n =h 2 n 2 π 2 2m * d 2(4)
where m * is the effective mass and d the width of the potential well. According to Eq. 4, not only the energy levels, but also the spacing between them increases as the confinement becomes more pronounced, i.e. the smaller is d. Quantum confinement has a critical impact on semiconductors because it affects directly their most important electronic property: the energy band gap.
Semiconducting nanowires provide a very good approximation of the model situation described above. Clearly, the potential well is not infinitely deep and realistic wire crosssections like those described in Sections II.B and II.C are difficult to describe analytically, thus there is a need for a detailed electronic structure modeling.
The first experimental proofs of quantum confinement in nanostructured Si were reported in the pioneering works of Canham (1990) and Lehmann and Gösele (1991), where a simple electrochemical etching process was used to create crystalline Si nanostructures with visible luminescence at room temperature. As TEM images revealed later (Cullis and Canham, 1991), the etched structures consisted of rather disordered bundles of nanowires, though it is interesting to note that ordered structures like those speculated in the first place (Canham, 1990) have been recently proposed for the fabrication of ordered arrays of quantum wires (Rurali et al., 2007b) and to achieve enhanced thermoelectric effect . Read et al. (1992) and Buda et al. (1992) performed DFT calculations of the band gap upshifts in perfect H-terminated SiNWs as a function of wire thickness, modeling porous
Si (Canham, 1990) with rectangular columns oriented along the 100 axis. Both these works showed that the fundamental gap is direct at the Γ point. This makes by itself the probability of radiative recombination higher than in bulk Si, since no phonon is required in the electronhole recombination process. Unfortunately, as it is well-known, standard local and semi-local implementations of DFT fail to account quantitatively for the band gap of semiconductors and one must resort to self-energy corrections to the Kohn-Sham gap to obtain a good agreement with the experimental values. Yet, the trends are expected to be qualitatively correct (Williamson et al., 2002) and Read et al. (1992) reported band gap upshift of up to 2 eV for wires of ∼ 12Å diameter. They also shown that a generalization of Eq. 4 gives a good description of the quantum confinement for wires wider than 23Å, whereas thinner wires show significant deviations from this idealized EMT picture. In such a range Buda et al. (1992) showed that with the more realistic DFT potential the band gap scales as the inverse of the diameter d, rather than 1/d 2 as predicted by particle-in-a-box arguments where infinitely hard walls are assumed. Subsequent studies that the interested reader could look at include the works by Ohno et al. (1992), Sanders and Chang (1992), Hybertsen and Needels (1993), Xia and Chang (1993), Yeh et al. (1994), Saitta et al. (1996), Xia and Cheah (1997), and Ossicini et al. (1997).
A first step toward a quantitative evaluation of SiNW band gaps in the quantum confinement regime was given by Delley and Steigmeier (1995), including a constant self-energy correction independent on the size. Namely, they increased all their calculated band gaps by 0.6 eV, the self-energy correction for bulk Si. They also shown that EMT can predict with great accuracy the band gap of relatively thin nanowires, provided that the potential well is not assumed to be infinite. SiNWs. They showed that the self-energy is indeed anisotropic and is larger for thinner wires.
The dependence of the band gap on the wire diameter can be described as
E gap = E bulk gap + C × (1/d) α ,(5)
where E bulk gap is the calculated band gap of bulk silicon, d is the effective diameter of the wires, while C and α are fitting parameters. This formula is derived within a simple particle-in-a-box effective mass approximation, where α = 2 when barrier height is infinite. The GW results can be fitted to this formula, yielding values of α ranging from 0.9 to 1.1, much lower than those expected within EMT and depending on the growth orientation (see Fig. 10), so that the band gap and the dielectric response are anisotropic (Bruneval et al., 2005;Bruno et al., 2007a;Zhao et al., 2004).
Although GW is in principle the best suited methodology to calculate the band gap in semiconductor systems, it suffers from the serious inconvenience of a considerable computational load. In the works of Zhao et al. (2004), Bruno et al. (2007a) and Yan et al. (2007), for instance, only relatively small SiNWs can be calculated directly and the band gaps of larger, more realistic wires are obtained by numerically fitting the available data to Eq. 5.
Furthermore, an alternative to many-body GW calculations is mandatory when it comes to calculate doping levels, a task that requires large computational cells. In the remainder of this section we discuss two possible approaches.
A successful way to improve the DFT band gaps consists in using hybrid-functionals for the exchange-correlation energy, where a certain amount of exact Hartree-Fock exchange is mixed to conventional LDA/GGA functionals. The amount of Hartree-Fock exchange (typically 12-15%) is chosen to reproduce some parameters of the bulk system (the band gap, among them), rather than being based on solid theoretical grounds. Hence, strictly speaking, one cannot claim to solve the electronic structure from first-principles. In such a theoretical framework the band gap of SiNWs with diameters up to 3 nm can be calculated directly (Aradi et al., 2007;Ng et al., 2007;Rurali et al., 2007a). These results are important because they allow direct comparison with the only experimental measurements available to date (Ma et al., 2003). A direct comparison of the experimental data with GW calculations is not possible for two reasons: the diameters of the wires grown experimentally are larger than those that could be simulated and most of the available measurements are for for 112 SiNWs, whose larger primitive cell precludes GW calculations even for the thinner wires.
Alternatively, the band structure of nanowires can be calculated with a semiempirical tight-binding method, where the self-energy is obtained within a simpler semiclassical treatment of the image charge effects (Niquet et al., 2006). This is a very powerful method because, due its reduced computational load, it allows calculating SiNWs with diameters up to 10 nm with good accuracy. As we will see in some more detail below (Section III.C.3), a great advantage of this method is that it allows dealing with different dielectric surroundings, which is very important in systems with abrupt dielectric interfaces like nanowires .
So, how should the band gap of SiNWs be calculated? The accurate calculation of band gap is one of the most challenging problems in semiconductor theoretical physics, so it is not surprising that it is not easy to answer this question. GW calculations provide in principle the most accurate estimations. However, they are restricted to very thin SiNWs.
Semiempirical tight-binding, on the other hand, is a very attractive choice for larger wires, especially for those diameters where quantum confinement become small and gap broadening is dominated by dielectric mismatch effects (Pereira et al., 2009). Hybrid-functional DFT calculations are an interesting compromise for those wires that are too large for GW (too many atoms in the primitive cell) and too small for tight-binding (relaxation effects cannot be neglected and the use of a parametrization obtained for bulk Si could be questioned).
It is difficult to asses on the accuracy of each of these methods, since the experimental measurement of the energy gap of SiNWs is extremely challenging, and only the data of Ma et al. (2003) are available to date. More experimental results are needed to clarify this important point.
Surface chemistry
We have seen above -and will see again below-that many properties of SiNWs are determined by their large surface-to-volume ratio. Hence, it is natural that most of the exciting physics takes place at the wires' surface (Kobayashi, 2004;Zhong and Stocks, 2006). In Section III.A we have seen, for instance, that wires bounded by facets derived by semiconducting surfaces can exhibit surface metallicity. Passivated nanowires are more predictable, in this sense, and it is just because they are always semiconducting that they are expected to play an important role in the next generation of electronic devices. Yet, the surface has a relevant role that merits some considerations.
An important case is that of chemical sensors where the adsorption of a molecule yields measurable variations of the electrical conductance Cui et al., 2001b). Upon adsorption, the molecular orbitals can hybridize with the wire states, resulting in a sizeable modifications of its electronic structure. Whether the ef-fectiveness of this process depends on the facet where adsorption takes place is the topic addressed by Leão et al. (2007). They studied the sensitivity of different facets of a 110
SiNW, showing the existence of a specific relation between the way surface atoms are bonded to core atoms and the relative contribution of these surface atoms to band edge states. These observations are important concerning the optimal design of those chemical sensors where the adsorption of a molecule directly modify the wire transmission. A broader class of sensors, however, seems to work on simpler basis. The dipole induced by molecule adsorption can act as a gate voltage, opening or closing the conductive channel in a field-effect transistor set-up.
In the quantum confinement regime the band gap width depend critically on the diameter. The possibility of controlling the band gap width is tremendously attractive for optoelectronics applications: not only SiNWs can have a direct band gap, which per se increases the optical efficiency, but its width can in principle be tuned. It is not difficult to imagine, however, that controlling the wire diameter with tolerances within 1-3 nm is a more than challenging task. A simpler route to band gap tuning is controlling the chemical composition and the coverage density of the wire surface. Halogens such as Cl, Br, and I can be used as surface passivation agents instead of H and, while not altering the semiconducting character of the wires, they result in a significant shrinking of band gap (Leu et al., 2006). The strongest reduction of the band gap is provided by I, followed by Br and Cl, in the opposite order of the bonding strength of these species and SiNWs. Interestingly, the surface coverage is a further degree of freedom and one can span all the band gap values between a H-and halogen-passivated wire by varying the H:halogen ratio. Also, increasing the halogen surface concentration the band edge states, concentrated in the wire core in presence of H-passivation, progressively spread to the surface.
Analogous results have been reported for OH and NH 2 (Aradi et al., 2007;Nolan et al., 2007). It should be noted that the passivating species do not contribute significantly to the states close to the band edges, so that the reduction of the gap is not caused by the introduction of additional bands. It rather comes from the hybridization of the valence band states with the frontier orbitals of the different passivating functional groups that cause a significant band gap reduction relative to H-passivated wires.
These results indicate that the band gap width in SiNWs can be tailored not only by controlling the wire diameter, but also by an appropriate choice of the surface termination.
C. Doped and defective nanowires
Semiconductors are privileged materials for electronics applications because their resistivities can be can be varied by design with great control 5 . Equally important, they can be designed to conduct one of two types of carriers: electrons and holes. These two features are the core of device design, which relies on the interaction of adjacent semiconductors with different densities of electrons and holes. The most efficient way to control the carrier density is doping the semiconductor, that is incorporating substitutional impurities (dopants) in the lattice (Muller and Kamins, 1986).
In the simplest model, a substitutional defect with minor relaxation effects forms four bonds with the neighbor atoms in the Si crystal. For a group-V element, such as P or As, the fifth valence electron is not covalently bonded to near neighbors and it is only weakly bonded by the excess positive charge of the impurity nucleus. Hence, a small amount of energy is required to break this weak interaction and this electron is free to wander about the crystal and contribute to conduction. These impurities are called donors because they donate an electron; analogous arguments apply to group-III elements which are acceptors.
There are at least two reasons that make the physics of impurities in nanowires different with respect to bulk systems: (i) the lattice sites are no longer equivalent in the direction of confinement and (ii) in the quantum confined regime all the eigenstates, including those associated to defects, are shifted in energy with important consequences on the dopant activation. Below, we discuss in some detail these and other topics relevant to dopant efficiency. We conclude the section with a generalization of the formation energy for defects in nanowires.
Surface segregation, surface traps and dopant aggregation
In bulk Si all the lattice sites are equivalent. In a nanowire this is true only for the axial direction, because the lateral confinement breaks the translational symmetry. In other words, given one arbitrary Si atom in the nanowire, as one moves along the wire axis he finds an infinite number of equivalent atoms, whereas as one moves along the radial direction he finds a finite number of nonequivalent atoms. Therefore, the substitutional defects at these non-equivalent sites will have in general different formation energies and doping levels.
This problem was first tackled by Fernández-Serra et al. (2006b), who studied B and P substitutional in 100 and 110 SiNWs. They revealed a tendency to surface segregation of these impurities, which means that substitution close to the surface is energetically cheaper than substitution in the innermost part of the wire. The effect is especially pronounced in presence of DB defects, so that the dopant impurities are effectively trapped by these surface defects. Most importantly, the dopant-DB complex is electrically inactive, reducing the carrier concentration at room temperature.
The segregation energy of P is significantly larger than that of B. This means that for the same impurity concentration, a much larger fraction of P atoms will be captured and neutralized by surface traps, resulting in a lower conductance. This is in agreement with the experimental results (Cui et al., 2000;Yu et al., 2000) where, for similar doping levels, B-doped SiNWs present a lower resistance than P-doped ones. Similar studies have been carried out for 110 SiNWs of different diameters (Leão et al., 2008;Peelaers et al., 2006). They showed that B and P prefer to sit at edge or near edge sites (the most external lattice sites with all-Si nearest neighbors), depending on the surface facet and on the atomic surface structure (Leão et al., 2008).
In perfectly passivated wires the surface segregation originates from a simple relaxation effect. At these edge and near edge positions it is easier to release the strain induced by the substitution, whereas in the center of the wire the allowed relaxation is smaller due to the constraint of the surrounding Si lattice. In surface defective wires, on the other hand, the presence of DBs greatly enhances the tendency to surface segregation, with the impurity atoms moving at surface sites. Here the driving force to surface segregation, yielding a much more sizeable effect, is an electronic effect: the formation of a stable dopant-DB complex.
The surface trapping of dopants has a dramatic impact for two reasons: (a) the impurities trapped at the surface are deep-state defects and are electrically inactive, thus not contributing to the carrier concentration at room temperature; (b) due to their large surface-to-volume ratio and considering a typical dopant concentration of 5×10 18 cm −3 and an estimate of of 10 12 cm −2 interface DB defects, one finds that for wires of less than 4 nm diameter there are always enough DBs to trap all the dopants (Peelaers et al., 2006). Luckily, the difference in formation energies among surface and core substitutional sites has been shown to decrease rapidly as the diameters grow larger (Leão et al., 2008). Hence, there is a twofold benefit in enlarging the wire diameter: on the one hand, the surface-to-volume ratio decreases, and so does the density of DBs with respect to the dopants concentration; on the other hand, the trapping efficiency of these reduced density of surface defects is lower, as the formation energies of core substitutionals and dopant-DB complexes become of the same order. Leão et al. (2008) estimate that the populations of core and surface dopants will be similar for wires with diameters of 3 nm or more.
The importance of surface impurities has been further highlighted by Durgun et al. (2007a). They considered various impurities such as Al, Ga, C, Si, Ge, N, P, As, Te, and Pt, focusing on adatoms configurations, rather than on substitutionals. They found that the energetically most favorable adsorption site of the six considered depends on the group of the Periodic Table that the impurity belongs to. All the configurations studied, however, give rise to deep state in the gap and are not viable choices as active dopants.
Another source of dopant deactivation is the formation of electrically inactive dopant complexes. Two nearest neighbor dopants can form a bound state, so that the weakly bonded electrons that contributed to the conductance are now participating in the dopantdopant bond. Yet, this energy gain has to compete with the energy cost that results from the strain accumulated around the dopant-pair defect. This strain is more easily released in nanowires than in bulk, due their large surface-to-volume ratio. Moon et al. (2008) reported a high stability of P pairs, which increases as the wire diameter is reduced. Interestingly, this is not the case of B. When two B atoms occupy nearest neighbor sites the lattice undergoes a significant relaxation, the B impurities move far apart and assume a planar threefold coordinated configuration. This is possible because, unlike P, B can present either sp 2 or sp 3 hybridization. Again, p-type doping seems to be more robust than n-type doping, at least as far as one considers B and P. A similar mechanism leads to mutual passivation when both a B and a P impurity are present at the same time. Besides the obvious compensation of having an n-and a p-type dopant, Peelaers et al. (2006) showed that these impurities favor aggregation at the wire surface. Also, the use of B/P co-doping has been proposed to reduce in a controllable way the band gap (Iori et al., 2008).
In Section III.B.2 we have seen how one of the most important quantities of a semiconductor, its band gap, depends critically on the wire diameter in the regime of quantum confinement. This is a very important parameter, because it determines the amount of carriers -intrinsic carriers-that can be thermally excited from the valence band to the conduction band. Intrinsic carriers, however, are not very important at typical device operation temperatures and the conduction is dominated by extrinsic carriers, those carriers that are thermally excited from a dopant level. Hence, dopants to be such must be very shallow, meaning that the impurity electronic states have to be only a few meV from the band edge. Now, if the band gap broadens as an effect of quantum confinement, what happens to dopant levels? In a purely effective mass picture they will be shifted, like any other state, becoming deeper, i.e. the upshift of a donor level will be less than of the conduction band edge. Clearly, this fact has dramatic consequences on the dopant efficiency. Namely, a dopant impurity which is known to be very shallow in bulk Si, becomes deeper as the diameter shrinks down, and it will not eventually be usable to dope ultra-thin SiNWs. At which diameter does this happen?
From EMT, one can deduce the effective Bohr-radius of the ground state a B ≈ (ǫ/m * )a 0 , where a 0 is the Bohr-radius of the isolated hydrogen atom. This results in about 2.2 nm (thus a 4.4 nm diameter) for P. A crude estimate of the extension of the wave function is taking twice this diameter, thus ∼ 9 nm. Yet, EMT neglects relaxation effect which can be important in very thin SiNWs and the dopant levels should be calculated directly.
The trend of the ionization energies vs diameter can be qualitatively obtained from DFT calculations (Durgun et al., 2007a;Leão et al., 2008). As we discussed in Section III.B.2, however, the local and semi-local approximations commonly used for the exchange-correlation energy severely underestimate the band gap and likewise the gap states and the related ionization energies. While the best suited approach for correct band gap calculations was the GW methodology (see Section III.B.2), it does not seem a viable solution for defective systems. Due to the need of simulating isolated impurities, computational supercells have to be large enough to allow neglecting the interaction of a defect with its periodic images. This implies a large number of atoms which is normally beyond the current computational capabilities of GW based codes.
Hybrid functionals -where a certain amount of exact Hartree-Fock exchange is mixed to conventional LDA/GGA functionals-provide accurate estimations of defect states in bulk Si (Deák et al., 2005) and have also been used to calculate P donors in 110 and 111
SiNWs (Rurali et al., 2009). As expected, these calculations yielded ionization energies that are deeper than the values obtained by DFT (Leão et al., 2008), though the difference decreases for larger wires. Remarkably, P behaves as an EMT dopant down to diameters of 1.5 nm, its wave function highly localized, whereas it breaks down for wires of 1.0 nm diameter. For such small wires the wave function is qualitatively different: it significantly interacts with the surface and cannot be described as quasi-one dimensional confined EMT state.
Dielectric confinement
The estimation of the ionization energies of dopants in nanowires has also been tackled efficiently at the tight-binding level (Diarra et al., 2008(Diarra et al., , 2007. We have already seen in Section III.B.2 that this approach can be complementary to the calculations at GW and hybrid-functional level (Niquet et al., 2006). Its quantitative reliability could be questioned for ultra-thin wires (diameters smaller than 2 nm), because this model neglects relaxation effects, important for such wires, and relies on a parametrization obtained for bulk Si. On the other hand, it is the best alternative to deal with larger wires where confinement still produces sizeable effects. A remarkable feature of this approach is the flexibility with which the screening properties of the surrounding dielectric medium can be manipulated, allowing to study in detail the so-called dielectric confinement.
The Coulomb potential of an impurity gives rise to a bound state in the energy gap.
In bulk Si this potential is strongly screened (ǫ r = 11.3), the Bohr radius is large, and the ionization energies amount to a few meV, so that the impurities are ionized at room temperature. The screening of a nucleus charge +e leaves a total charge +e/ǫ at the impurity site, whereas the remaining charge +e(1 − 1/ǫ) is repelled at infinity. In a one dimensional system the screening properties are different. The charge +e(1 − 1/ǫ) must be repelled at the surface of the nanowire, leading to an extra term in the potential.
The physics of screening in one dimensional systems is straightforwardly incorporated in the tight-binding Hamiltonian of Diarra et al. (2008Diarra et al. ( , 2007:
H = H 0 + U imp + (6)
where H 0 is the Hamiltonian of the undoped wire, U imp = ±V (r, r 0 ) is the potential at r of an impurity at r 0 , and is the self-energy potential, which accounts for the interaction between the carrier and the surface polarization charges induced by its own presence.
On the basis of this tight-binding model Diarra et al. (2007) showed that the ionization energies of typical donors are significantly deeper than in bulk, even for large wires (d > 10 nm) where the effects of quantum confinement are weak. This effect is due to the interaction between the electron and the surface polarization charge +e(1 − 1/ǫ). These results (i) indicate that dielectric confinement can be stronger than quantum confinement and that donor levels deepen more than how much the band gap broaden; (ii) the dielectric mismatch can be used to vary the ionization energies. In particular, a metallic or high-permittivity surrounding gate, present in realistic applications, is expected to reduce significantly the ionization energies. These predictions have been recently supported by the experimental results of Björk et al. (2009).
Metallic impurities
Although much of the attention has been devoted to dopants so far, metal impurities are becoming increasingly important for the reliability (Bailly et al., 2008;Hannon et al., 2006;den Hertog et al., 2008;Oh et al., 2008) and functionalization of SiNWs. Transition metals have attracted some interest because of their possible use in designing nanoscale dilute magnetic semiconductors (Durgun et al., 2008;Giorgi et al., 2008;Xu et al., 2008).
Room-temperature ferromagnetism in SiNWs has indeed been reported , although the annealing conditions to stabilize the magnetization are very critical. The ferromagnetic coupling of Mn impurities is confirmed by electronic structure calculations.
At variance with typical dopants these impurities do not segregate to the surface, at least in absence of surface DBs (Xu et al., 2008), and they favor aggregation. This tendency has been reported to be important to stabilize the magnetism, because only when the Mn-Mn distance is below a certain cutoff ferromagnetic coupling is favored over antiferromagnetic coupling (Giorgi et al., 2008).
It is interesting to observe that specific TM impurities can drive the nanowire to a halfmetallic ground state. A half-metal is a system where one spin is metallic, whereas the other is insulating (de Groot et al., 1983). Such systems have the greatest interest for spintronics applications, because naturally all the conduction electrons belong to the same spin and the spin polarization P is maximum 6 . In the framework of a comprehensive analysis of surface adsorption of TM atoms in SiNWs, Durgun et al. (2007b) discovered that wires decorated with Co and Cr can be ferromagnetic half-metals. Upon adsorption of a Co or Cr atom at the surface, the spin degeneracy is lifted and, while the bands of majority spin continue being semiconducting, two minority spin bands made of hybridized TM-3d and Si-3p states cross the Fermi level driving the wire to half-metallicity. There is a sizeable charge transfer from the TM atom to the wire and the high values of the binding energies indicate strong bonds, which is important to prevent uncontrolled clustering that would be detrimental to the magnetic ordering. The half-metallicity is obtained for huge coverages (one impurity for primitive cell) typical of dilute magnetic semiconductors. As the coverage is reduced, the gap of the minority spins starts closing and the system is no longer half metallic (P < 1).
However, the total spin polarization remains very high and close to its maximum permitted value.
Formation energy
The energetic cost of creating a defect is given by the formation energy. The formation energy is the main computational quantity describing the stability and energetics of a defect in a host material and it is essential to determine impurity equilibrium concentrations (Zhang and Northrup, 1991), solubilities (Van de Walle et al., 1993) or diffusivities (Fahey et al., 1989). In bulk systems the formation energy can be calculated according to the well established formalism of Zhang and Northrup (1991):
∆E f = E D tot − i n i µ i + q(ε v + µ e ),(7)
where E D tot is the total energy of the defective system, n i is the number of atoms belonging to species i with chemical potential µ i , q is the net charge of the system, ε v is the energy of the top of the valence band of the clean host and µ e is the chemical potential for electrons; the sum runs over all the species present in the system. However, the extension of Eq. 7 to one dimensional systems presents some subtleties. In particular, the chemical potential of the host species is ill-defined.
To see why, let us focus on the formation of a Si vacancy and let us play the movie of the wire growth: Si atoms start precipitating from the supersaturated Au-Si droplet and contribute to the nucleation of the wire; once in a while one Si atom does not fill the proper lattice site and a vacancy is formed. When calculating the formation energy one has to estimate the contribution to the total energy of this atom, the one that left the vacant site and was incorporated in the wire somewhere else. In bulk this is easy, because all the lattice sites are equivalent and then each atom contributes equally to the total energy. In a nanowire, on the other hand, this quantity is not well defined, because it depends where the misplaced atom is thought to end up, as the lattice sites are non-equivalent. Rurali and Cartoixà (2009) have proposed a way to circumvent this problem that also deals with the passivating agents, if present. They showed that rewriting the equations for the formation of N defects at a time, being N the number of Si atoms in the primitive cell, leads to the definition of an effective chemical potential of the wire primitive cell. In this way, whenever the formation of a defect involves the addition/removal of an atom of the host species, e.g. vacancies, self-interstitials, substitutionals, it is transferred to/from the correct reservoir -the wire itself-and only easily computable quantities are involved.
A further problem arises when dealing with charged defects. In a periodic boundary condition formalism point charges result in an infinite electrostatic energy. This inconvenience can be obviated by using a compensating jellium background. The errors in the total energy are often corrected a posteriori by means of the Madelung energy (Makov and Payne, 1995), though other methods have been proposed. Again, this correction has to be properly generalized to be used in nanowires. In particular, in solids the Madelung correction is scaled by the value of the (isotropic) macroscopic dielectric constant of the host material. In nanowires, on the other hand, a dielectric tensorǭ will be needed for the correct description of the interaction between the different instances of the charged defect (Rurali and Cartoixà, 2009).
Notice that the value of the dielectric tensor will depend on the ratio between the axial lattice parameter, the lateral vacuum buffer and the chosen values of the axial (Hamel et al., 2008) and transverse components of theǭ tensor and therefore cannot be looked up in tables.
IV. TRANSPORT PROPERTIES
The study of electron and heat transport is one of the most rapidly growing research field in nanowires. The reason is twofold: on the one hand transport measurements often are the most direct and simplest way to test the theoretical predictions (Björk et al., 2009;Chen et al., 2008;Cui et al., 2000;Li et al., 2003;Sellier et al., 2006;Yu et al., 2000); on the other hand, the behavior can be much different from bulk Si and can be exploited for enhanced performances in applications, whereas other times it can be detrimental.
A. Electron transport
Surface roughness disorder
An important cause of the degradation of the electrical conductance in SiNW-based devices is the scattering occurring at the surface in presence of surface defects or surface roughness (Luisier et al., 2007;Wang et al., 2005). This is not unexpected, since we have seen that many properties are ruled by the large surface-to-volume ratio of SiNWs. Considering non-smooth surfaces has indeed a great importance, as SiNWs exhibiting either tapering (Kodambaka et al., 2006;Wang et al., 2006c;Wu et al., 2008b) or fancier saw-tooth faceting (Ross et al., 2005) are often reported.
The effects of surface roughness on electron transport have first been addressed by Svizhenko et al. (2007). They modeled the surface disorder adding with probability 1/2 one monolayer at each facet of a given unit cell. Since the position of these surface bumps is uncorrelated, they obtained a white-noise roughness. The surface roughness originates strong irregularities in the density of states along the wire axis, which in turn causes reflection of carriers and a strong reduction of the conductance. When these effects sum up in very long wires the disorder quickly drives the transport into the Anderson localization regime (Anderson, 1958).
A complementary approach to the description of the surface roughness disorder has been followed by Persson et al. (2008) and Lherbier et al. (2008). They modeled the roughness as random fluctuations δr of the wire radius around its average value r 0 , through a Lorentzian autocorrelation function, obtaining a correlated disorder. They showed that the backscattering strongly depends on the nanowire orientation, the anisotropy coming from the differences in the underlying band structure. In particular, electrons are less sensitive to surface roughness in 110 SiNWs, whereas holes are better transmitted in 111 SiNWs . Also, as the disorder correlation length -roughly the length scale of the diameter fluctuations-increases, the lowest-lying states of the conduction band get trapped into the largest sections of the wire 7 . The modified extent of the electron wave function affect many key quantities for transport, such as the mean free path and the localization length. Interestingly, the room temperature mobility of electrons and holes seems rather insensitive to short length scale fluctuations, as well to very long length scale fluctuations, a case in which the surface experienced by the carriers is locally smooth.
Single-impurity scattering
Besides surface disorder, the other main critical source of scattering is the presence of impurities. Surface scattering has a stronger impact on the transport in SiNWs than in bulk Si because of the much larger surface-to-volume ratio. The case of impurity scattering seems different, since it should solely depend on the impurity density and should affect in a similar way bulk Si and SiNWs. This is not the case, though. So, where is the catch?
With the reduction of the wire size below 10 nm, the impurity cross-sections become of the same order of the wire characteristic dimension and can result in total backscattering. In the semiclassical picture used to study transport in bulk materials impurities are point-like centers that scatter randomly the incoming carriers. This chaotic process slows down the carrier flow and results in a reduction of the conductance. The quantum picture in a thin one-dimensional medium is slightly different: impurities have to go through to a scattering potential that often extends throughout most of the wire cross-section and, following with the semiclassical analogy, the trajectories of the carriers are not simply deviated, but they can be entirely backscattered.
Impurity is a fairly generic denomination when referred to semiconductors. In fact, it refers to both undesired defects, by-products of an imperfect growth, and to dopants, which are purposely introduced to provide the material with tailor-made electric features. Clearly, this case is the most challenging: dopants increase the carrier density at device operation 7 A similar mechanism has been reported recently for selectively strained nanowires (Wu et al., 2009). temperature, but at the same time might induce a significant scattering which leads to a drop in the conductance. Fernández-Serra et al. (2006a) studied the resistance associated with a substitutional P impurity in the wire core, at the wire surface, and with a DB+impurity, a complex whose importance was discussed in a previous work of theirs (Fernández-Serra et al., 2006b). Resonant backscattering -a strong reduction of the conductance in correspondence to impurity related bound states-is the main signature of substitutional impurities, though P in the core or at the surface yield different results. On the other hand, DB+impurity complexes are transparent to the incoming carriers and the transport is ballistic.
Therefore, donor impurities such as P either segregate to the surface where they are likely to form an electrically inactive complex with a DB or they stay in the wire core where they produce a strong backscattering, particularly at certain resonant energies. In both cases the current is reduced.
Multiple-impurity scattering
The calculations of Fernández-Serra and co-workers opened the field of dopant scattering in SiNWs, but they have two limitations: (i) they study the scattering properties of an individual impurity, while in realistic SiNWs the wire resistance results from multiple scattering events; (ii) impurities can be ionized, the typical situation for dopants, and the proper charge state must be taken into account in the conductance calculation.
The first of these two issues has been tackled by comparing the conductance evaluated directly in long wires, with a certain distribution of impurities, with the predictions that can be made on the basis of single-dopant calculations (Markussen et al., 2007). This is a challenging task, because to make such a comparison on equal footing the long wire too has to be treated within a first-principle formalism, which involves an extraordinary computational load. This difficulty can be circumvented thanks to an ingenious method that allows constructing the Hamiltonian of the long wire assembling building blocks obtained from the single-dopant calculations . In this way the electronic structure problem has not to be solved directly in the long wire.
The surprising result is that the properties of long, realistic wires -such as mean free path, resistance vs length-can be entirely predicted from single-impurity conductances. So, the resistance of a wire with an arbitrary distribution of impurities is obtained by classically adding the resistances of each individual scatterer according to Ohm's law:
R(L, E) = R c (E) + R s (E) L/l (8)
where R s (E) is the average resistance that can be evaluated from the single-dopant calculations, R c (E) is the contact resistance, L is the length of the wire and l the average separation between dopants. Interestingly, a similar approach has been proposed also for phonon transport (Markussen et al., 2009;Savić et al., 2008).
This method allows easy comparisons of the conductance of wires with different distributions of defects. The case of P substitutionals, for instance, has been addressed by Markussen et al. (2008b), where a uniform radial distribution was compared to a mainly surface distribution, in accordance with the previously reported indications of P surface segregation (Fernández-Serra et al., 2006b).
Charged impurity scattering
Addressing charged impurities poses well-known problems related to the use of periodic boundary conditions. Large supercells, out of the current capability of first-principles methods, are needed to allow the correct screening of the electrostatic potential of the impurity.
The conductance depends more critically than other quantities on this incomplete screening.
Such systems have been dealt with within an approximate method that combine firstprinciple methods with finite element calculations of the electrostatic potential . The idea is very simple. If a charged dopant is approximated with a point charge, its electrostatic potential can be obtained in a very cheap way with a finite element calculation.
Far from the impurity this is a reasonable approximation -a P + impurity gives rise to essentially the same Coulomb potential of a As + impurity-and the agreement with a selfconsistent electronic structure calculation is indeed very good. Close to the impurity, on the other hand, quantum electronic structure accounts properly for the different chemical nature of different impurities, a task not accomplished by a finite element model.
The part of the potential that converges slowly with the cell size is the long-range Coulomb potential. So, here comes the simple idea: the local potential around the impurity is calculated at the first-principle level in a large, but tractable computational cell. Then the long-range tails are calculated within a finite element model (which it is known to yield the same result than a first-principle calculation). In this way one can in principle engineer arbitrary boundary conditions, as the long range part of the potential can be obtained at a negligible computational cost.
This approximated approach allows explicitly addressing the calculation of the conductance associated to impurities with different charge states and comparing majority carriers (electrons in a n-type wire, holes in p-type wire) vs minority carriers (electrons in a p-type wire, holes in n-type wire). The results are indeed utterly different and it is shown that in sufficiently thin wires minority carriers transmission is entirely suppressed . What happens is that in the case of minority carriers the dopant constitutes an effective barrier in the potential landscape. When the energy of the electron is less than that of the barrier height, it must tunnel through the potential and the transmission is therefore exponentially suppressed.
B. Heat transport
Recently, there has been a lot of excitement around the thermal conductive properties of SiNWs. Surprisingly, this excitement stems from the poor thermal conductance of SiNWs.
The reason is that the use of SiNWs as materials with enhanced thermoelectric properties has been demonstrated independently by two groups (Boukai et al., 2008;Hochbaum et al., 2008).
While in some devices one wants to get the heat away as efficiently as possible and a high thermal conductance is desirable, in thermoelectrics one wants a thermal conductance as small as possible (Cahill et al., 2003). It has been suggested that at the nanoscale the thermoelectric efficiency could be increased with respect to bulk materials, since the electrical mobilities are expected to be higher, while the surface scattering of the phonons should decrease the lattice thermal conductance (Vo et al., 2008). Recent experimental results have reported thermal conductivities of ∼ 1.6 W m −1 K −1 , two orders of magnitude lower than the value for bulk Si (150 W m −1 K −1 at room temperature).
Heat is transmitted by phonons, the vibrations of the crystal lattice. Calculating the phonon modes of a SiNW with a first-principles method is a demanding task and can be done only for the thinnest wires (Peelaers et al., 2009). Fortunately, this is not too serious an inconvenience and the phonon band structure can be calculated with a great level of accuracy within simple empirical interatomic potentials. It has been shown that phonon dispersions calculated with DFT and with the bond-order Tersoff potential (Tersoff, 1989) yield thermal conductances in excellent agreement (Markussen et al., 2008a). It is indeed on the Stillinger-Weber potential that the first atomistic calculations of the thermal conductance of SiNWs were based on (Volz and Chen, 1999).
The decrease of the thermal conductance with the reduction of the diameter comes from the interplay of two factors: (a) phonon confinement, i.e. the change in the phonon spectra (Adu et al., 2005) and (b) the increase of the inelastic phonon scattering at the surface. The dependence of the phonon dispersion on the wire size has been studied for 100 wires by Wang and Wang (2007), who showed that the thermal conductance decreases as the wire diameter is reduced. Ponomareva et al. (2007) obtained similar results for 111 wires, although the thermal conductance steeply increases again for diameters below 2 nm (Ponomareva et al., 2007), a direct signature of phonon confinement. Namely, as the diameter is reduced the lowest frequency excited mode is severely affected by the confinement; this long wavelength mode dominates the low frequency spectrum and, carrying a larger amount of energy, determines the enhanced thermal conductance at small diameters 8 .
It should be bear in mind, however, that the effects of phonon confinement are normally studied in nanowires with an ideal structure, whereas in the recent reports of the enhanced thermoelectric figures of SiNWs surface corrugation seems to play an important role. Recently, Donadio and Galli (2009) showed that the computed thermal conductance strongly depends on the surface structure, whereas it can be insensitive to variations of the diameter in the size range investigated (d < 4 nm). Phonon confinement do not necessarily lead to low values of the thermal conductance, which in some cases can even increase as a function of size, due to the presence of long wavelength phonons with very long mean free paths, like shown by Ponomareva et al. (2007).
It is useful to consider these studies together, because of their complementary methodological approaches. Wang and Wang (2007) calculate the phonon dispersion with the usual procedure consisting in displacing each atom along ±x, ±y, and ±z to obtain the dynam-8 A similar feature is also present in the 100 wires of Wang and Wang (2007), where for temperatures lower than 50 K the thermal conductance of a 1.54 nm wire is larger than the thermal conductance of a 4.61 nm wire. However, it is difficult to be more quantitative at this respect, due to the scale of the plots used in Wang and Wang (2007).
ical matrix by finite differences. Then they calculate the thermal conductance with non equilibrium Green's functions. In this way they rely on the harmonic approximation and thereby neglect any phonon-phonon scattering, an approach valid in the low to mid temperature limit. Similar calculations for thicker wires, d > 35 nm (Mingo, 2003;Mingo et al., 2003), where phonon confinement effects are unimportant, yielded an excellent agreement with experimental results . Ponomareva et al. (2007) and Donadio and Galli (2009), on the other hand, calculate the thermal conductance within a molecular dynamics simulation from Fourier's law J z = −σ∂T /∂z, where J z is the heat flow along the wire axis z and ∂T /∂z is the thermal gradient (Schelling et al., 2002). Within this scheme, one does not calculate explicitly the full phonon dispersion and anharmonic effects are automatically included in the simulation. The anharmonic forces are increasingly important at higher temperatures, since the atomic displacements get bigger. This means that phonon-phonon scattering becomes more and more important at higher temperatures and dominates over the effects of including more conducting channels. The drawback is that classical molecular dynamics is reliable only above the Debye temperature of the material (645 K for Si) where quantum effects in the ionic dynamics can be neglected and below it the results must be interpreted with some care.
It should not come as a surprise that the thermal conductance is also anisotropic, like many other important quantities that we have discussed throughout the paper. At low energy the phonon dispersion features four acoustic branches, one dilatational, one torsional, and two flexural modes (Thonhauser and Mahan, 2004). The torsional mode is related to rotational invariance around the wire axis, and it is similar for all the orientations. At higher energies, however, one can notice that the bands in 110 SiNWs have a larger slope, i.e. larger velocities, than the other orientations which feature mostly flat bands (see Fig. 12).
Hence, at a given energy there are more bands in the 110 wire than in the 100 or 111 and consequently one expects a larger thermal conductance. Heat conductance is indeed strongly anisotropic. Up to ≈ 20 K, where the phonon dispersion is dominated by the acoustic modes, the thermal conductance is independent on the growth orientation, but then 110 SiNWs stand out, with an up to twofold increase in their conductance (Markussen et al., 2008a).
V. CONCLUSIONS
In this paper we have reviewed the major advances in the theoretical study of the structural, electronic and transport properties of silicon nanowires. While the geometry and the electronic structure of nanowires are relatively well-understood, many open questions remain on the possibility of effectively dope ultra-thin nanowires and on many of the atomic scale mechanisms ruling electrical current and heat transport.
Silicon nanowires are rod-like system constructed around a single-crystalline core. The most important consequence of their monocrystallinity is that they grow along well-defined crystallographic orientations, and at sufficiently small diameters a strong anisotropy of most of their properties emerge: the band gap, the Young's modulus, the electrical conductance or the specific heat, just to name some, are different for wires grown along different orientations. The cross-sections are intimately related with the growth orientations -given a growth orientation only certain sets of bounding facets are allowed-, although their impact on the electronic properties of the wires seems limited, while other magnitudes such as the diameter or the surface-to-volume ratio have a greater influence. The band gap can be direct, opening the way to the use of Si in photonics, and can be tuned by varying the wire diameter, choosing the growth orientation or controlling the surface passivation. Extrinsic carrier conduction seems to be highly problematic and whether is feasible or not for ultrathin nanowires is not clear yet. The reason is that dopant efficiency is bedeviled by multiple factors: surface segregation and clustering with consequent neutralization, deepening of the doping level due to dielectric and quantum confinement.
Many promising applications have already been demonstrated. Although the nanowires used in these applications are smaller than any device that can be fabricated with lithography based techniques, they are still larger than those studied theoretically, where quantum effects leave their clear signature. Whereas it is clear that nanowires will play an important role in the next generation of electronic devices, it is difficult to say if the use of such ultra-thin wires will be practical. Many of the properties of these extremely thin nanowires pose severe technological challenges, but at the same time represent extraordinary opportunities. Anisotropic band gaps that critically depend on the wire diameter are apparently incompatible with any standardized technological process, to give an example. However, once the growth orientation and wire thickness can be controlled with great precision this would open up the possibility of band gap engineering, which would be extremely attractive for optoelectronics applications.
Joints efforts in theory and experiments hold the key to nanowires' future.
took the area bounded by the outermost atoms, i.e. the passivating H atoms, Ma et al. (2008) made a similar choice, but excluding the H atoms. Aware of this degree of arbitrariness in the possible choices, Leu et al. (2008) studied the variation of the calculated mechanical properties as a function of the uncertainty δr 0 in the estimation of the radius r 0 .
and {110} facets, at variance with the SiNWs of Rurali et al. (2006) which have {111} and {110}. This variation results in significant structural differences, because of the comparatively larger readjustment of {100} facets, which involve the formation of surface dimers (Chadi, 1979) and therefore a noticeable reconstruction; in {110} and {111} facets, on the other hand, no new bonds are formed and the overall reorganization of surface atoms is moderate. These 110 SiNWs turned out to be indirect band gap semiconductors, with the states of the top of the valence band and the states of the bottom of the conduction band originating at different facets. Yet, it should be noted that a metallic reconstruction for 110 SiNWs have been reported by Fernández-Serra et al. (2006b). Once again, small variations of the atomic scale structure or of the cross-section can result in major changes in the electronic structure.
The self-energy correction to the Local Density Approximation (LDA) or the Generalized Gradient Approximation (GGA) band-gap, however, is expected to depend on the wire diameter and on the growth orientation.Zhao et al. (2004),Bruno et al. (2007a), andYan et al. (2007) carried out calculations within the many-body perturbation method based on the GW approximation(Aryasetiawan and Gunnarsson, 1998) for 100 , 110 , and 111
FiguresFIG. 1 (
1Color online) (a) Cartoon and (b) experimental realization of a ZnO nanowire-based fieldeffect transistor with an all-around (or surrounding) gate. The channel length is 200 nm. From Ng et al. (2004). FIG. 7 (Color online) (a) TEM images of 3.8 nm SiNWs grown along the 110 direction, (c) high resolution TEM cross-sectional image, and equilibrium shapes for the (b) NW and the (d) NW cross-sections predicted by Wulff construction. The scale bars are 5 nm. From Wu et al. (2004). FIG. 9 (Color online) Band structures of 100 110 111 and 112 SiNWs with a diameter of ∼ 3.0 nm (cross-sections in the insets). The arrows indicate the fundamental band gap which is direct for 100 and 110 SiNWs and indirect for 111 and 112 SiNWs. As discussed in the text the band gap of 111 SiNWs becomes direct when the diameter is reduced below 2 nm. Adapted from Ng et al. (2007). FIG. 10 (Color online) Quasi-particle GW gaps for 100 (circles), 111 (squares), and 110 (diamonds) SiNWs as a function of wire size compared with experimental results (triangles) from scanning tunneling spectroscopy (Ma et al., 2003). The gray region represents the LDA electronic gaps from 110 (bottom) to 100 (top) wires. From Bruno et al. (2007a). FIG. 12 Phonon band structures calculated within the Tersoff bond-order potential of 2 nm diameter SiNWs grown along the (a) 100 (b) 110 and (c) 111 axis. The phonon wave vectors, q, are all in the respective wire directions and are shown in units of the reciprocal unit cell lengths, with a 100 = 5.4, a 110 = 3.8, and a 111 = 9.4Å. From Markussen et al. (2008a).
carried out an interesting and extensive study of the growth orientations, showing a connection between the diameter and the favored crystal axis in the Au-catalyzed synthesis of SiNWs: the smallest-diameter nanowires grow primarily along the 110 direction, whereas larger nanowires favor the 111 direction; intermediate diameters, 10 to 20 nm, on the other hand, are dominated by 112 wires. Thermodynamic models have been proposed to account for this diameterdependent growth direction (Schmidt et al., 2005; Wang et al., 2006a) with consistent results in good agreement with the experiments, fixing the cross-over from 110 to 111 growth at 20-25 nm ( 112 orientation was not considered in those studies). The stacking sequence
a few years earlier. In their work they considered a pristine SiNW grown along the 100 axis. This orientation favors a square cross-section with {100} facets, an energetically cheaper solution than a square cross-section with {111} facets (Rurali and Lorente, 2005b). The abrupt match between the {100} facets results in an energetically expensive edge, a large value of E e in Eq. 1, which can be reduced by forming smaller, transition {110} facets that allow a smoother match between the dominant {100} facets and partially release the stress accumulated at the edge(examples can be seen in Figs. 4(c), 6(a) and 8.).
. In order to elucidate the role of the different facets for the stability, for each growth orientation they examined cross-sections bounded by different facet compositions. For the 100 SiNWs, for instance, they considered both all-{100} facets, all-{110} facets and three intermediate combinations. Proceeding in this way they were able to formulate a universal scaling law in terms of the wire perimeter, according to which the nanowire energy per atom always lies within two limiting energy lines, which are directly related to the character of the prevailing facets. Interestingly, in the limit of thick wires, the edge energy become negligibleas suggested by Zhao et al. (2004) and the energy scales linearly with the inverse of the wire
perimeter.
they found the highest values for 110 SiNWs, while 100 SiNWs give the lowest values.extended the study of the stiffness vs. diam-
eter to wires grown along the 110 , 111 , and 112 orientations. While their results are
in good agreement with those of Lee and Rudd (2007b) for 100 wires, they showed that
wires of similar diameter, but with different orientations, differ considerably. In particular
Below we will discuss explicitly a few cases where nanowires derived from semiconducting solids can be metallic, whereas nanowires made of metal atoms are beyond the scope of this paper and are not being discussed.
Magic structures is used in this context to refer to distinct types of wire configurations with low formation energies that emerge as the number of atoms per length is increased(Chan et al., 2008).
. Quantum confinement
Conventionally, materials with resistivities less than about 10 −2 Ω cm are considered conductors and materials with resistivities greater than about 10 5 Ω cm are considered insulators(Muller and Kamins, 1986).
. Quantum confinement
The spin polarization is normally defined as P = (N ↑ − N ↓ )/(N ↑ + N ↓ ), where N ↑,↓ are the densities of states at the Fermi level.
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Color online) Transmission electron microscopy (TEM) image of a single crystalline SiNW grown along (a) the 111 and (b) the 112 axis. The high resolution TEM micrograph of the crystalline core shows clearly the Si(111) and Si(224) planes, respectively, together with the Fourier transform of the image. Z Zhong, D Wang, Y Cui, M W Bockrath, C M Lieber, 1377. FIG. 2Science. 3025649In case of the 111 SiNW a magnified view of the sidewalls of the wire show the presence of Au-Si particles about 7 nm in size. From Lugstein et al.Zhong, Z., D. Wang, Y. Cui, M. W. Bockrath, and C. M. Lieber, 2003, Science 302(5649), 1377. FIG. 2 (Color online) Transmission electron microscopy (TEM) image of a single crystalline SiNW grown along (a) the 111 and (b) the 112 axis. The high resolution TEM micrograph of the crystalline core shows clearly the Si(111) and Si(224) planes, respectively, together with the Fourier transform of the image. In case of the 111 SiNW a magnified view of the sidewalls of the wire show the presence of Au-Si particles about 7 nm in size. From Lugstein et al. (2008).
Red and large blue circles represent the H atoms and Si atoms, respectively. Small blue circles correspond to Si atoms on the underlying layers. The 112 wire in panel (a), with a diameter of 1.3 nm, is the thinnest SiNW reported to date. Ma et al.Color online) (a) Constant-current scanning tunneling microscope (STM) image of the the 110 direction; (d) Schematic view of the dihydride phase on Si (001). Reprinted with permission from AAASFIG. 3 (Color online) (a) Constant-current scanning tunneling microscope (STM) image of the the 110 direction; (d) Schematic view of the dihydride phase on Si (001). Red and large blue circles represent the H atoms and Si atoms, respectively. Small blue circles correspond to Si atoms on the underlying layers. The 112 wire in panel (a), with a diameter of 1.3 nm, is the thinnest SiNW reported to date. Adapted from Ma et al. (2003). Reprinted with permission from AAAS.
Color online) Optimized structures of possible cross-sections of H-passivated SiNWs grown along (a,b) the 110 (c,d) the 100 and (e) 112 orientation. FIG. 4 (Color online) Optimized structures of possible cross-sections of H-passivated SiNWs grown along (a,b) the 110 (c,d) the 100 and (e) 112 orientation. Adapted from Singh et al. (2006).
Hexagonal 110 wire with four {111} and two {100} (a) unreconstructed and (b) reconstructed facets. (c) Energy of different types of wires (see Zhao and Yakobson (2003)) as a function of their diameter d. The most stable structure for d < 6 nm is. ) , Takeguchi , Takeguchi et al.Panel (d) shows a high resolution TEM image of a pentagonal nanowire grown byFIG. 5 Hexagonal 110 wire with four {111} and two {100} (a) unreconstructed and (b) recon- structed facets. (c) Energy of different types of wires (see Zhao and Yakobson (2003)) as a function of their diameter d. The most stable structure for d < 6 nm is [solid pentagons in panel (c)] is shown in panel (d). Panel (d) shows a high resolution TEM image of a pentagonal nanowire grown by Takeguchi et al. (2001). Adapted from Zhao and Yakobson (2003) and Takeguchi et al. (2001).
Color online) (a) Cross-section view of 3 nm SiNWs grown along three different directions 100 , 110 , and 111 . (b) Side view of three different surface structures; in the last configuration the surface first reconstructs and then is passivated. FIG. 6 (Color online) (a) Cross-section view of 3 nm SiNWs grown along three different directions 100 , 110 , and 111 . (b) Side view of three different surface structures; in the last configuration the surface first reconstructs and then is passivated. From Vo et al. (2006).
For comparison values of continuum formula are also plotted. The solid curve E = E DF T bulk − C/w, where w is the width of the wire and C = 66.11 GPa/nm, is the best fit to a pure surface area to volume size dependence. (Insets) Cross sections of some of the SiNWs studied, where each Si atom is colored corresponding to its transverse relaxation inÅ. Color online) Young's modulus calculated within DFT as a function of wire size. The widths of wires are (a) 1.49, (b) 2.05, (c) 2.80, and (d) 3.92 nm. Adapted from Lee and RuddFIG. 8 (Color online) Young's modulus calculated within DFT as a function of wire size. For comparison values of continuum formula are also plotted. The solid curve E = E DF T bulk − C/w, where w is the width of the wire and C = 66.11 GPa/nm, is the best fit to a pure surface area to volume size dependence. (Insets) Cross sections of some of the SiNWs studied, where each Si atom is colored corresponding to its transverse relaxation inÅ. The widths of wires are (a) 1.49, (b) 2.05, (c) 2.80, and (d) 3.92 nm. Adapted from Lee and Rudd (2007b).
Large spheres represent Ge atoms; small spheres are hydrogen atoms used to saturate the dangling bonds. Color online) Geometrical structures of 0.4 nm Ge nanowires along the 110 (top). 111 (middle), and 100 (bottom) directions shown from the side (left) and from the top (right)FIG. 11 (Color online) Geometrical structures of 0.4 nm Ge nanowires along the 110 (top), 111 (middle), and 100 (bottom) directions shown from the side (left) and from the top (right). Large spheres represent Ge atoms; small spheres are hydrogen atoms used to saturate the dangling bonds. Adapted from Bruno et al. (2005).
| []
|
[
"Behavior of Analog Quantum Algorithms",
"Behavior of Analog Quantum Algorithms"
]
| [
"Lucas T Brady \nJoint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA\n\nJoint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA\n",
"Lucas Kocia \nSandia National Laboratories\n94550LivermoreCaliforniaUSA\n",
"Przemyslaw Bienias \nJoint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA\n\nJoint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA\n",
"Aniruddha Bapat \nJoint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA\n\nJoint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA\n",
"Yaroslav Kharkov \nJoint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA\n\nJoint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA\n",
"Alexey V Gorshkov \nJoint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA\n\nJoint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA\n"
]
| [
"Joint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Sandia National Laboratories\n94550LivermoreCaliforniaUSA",
"Joint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Center for Quantum Information and Computer Science\nNIST/University of Maryland\n20742College ParkMarylandUSA",
"Joint Quantum Institute\nNIST/University of Maryland\n20742College ParkMarylandUSA"
]
| []
| Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the quantum approximate optimization algorithm (QAOA). These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters. In this work, we explore connections between these analog algorithms, as well as limits in which they become approximations of the optimal procedure. Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state that effect the coherent error cancellation of diabatic transitions. Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern. Additionally, the ratios of the QAOA bangs are determined by the smooth, non-oscillatory part of the optimal procedure. We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure. With these arguments, we conclude that different analog algorithms can emulate the optimal protocol under different limits and approximations. Finally, we present a new algorithm for better approximating the optimal protocol using the analytic and numeric insights from the rest of the paper. In practice, numerically, we find that this algorithm outperforms standard QAOA and naive quantum annealing procedures. | 10.2172/1856736 | [
"https://arxiv.org/pdf/2107.01218v1.pdf"
]
| 235,732,198 | 2107.01218 | 721a7b73c1a3f8aaa8fce1b0ecffcd065a0b0461 |
Behavior of Analog Quantum Algorithms
2 Jul 2021
Lucas T Brady
Joint Center for Quantum Information and Computer Science
NIST/University of Maryland
20742College ParkMarylandUSA
Joint Quantum Institute
NIST/University of Maryland
20742College ParkMarylandUSA
Lucas Kocia
Sandia National Laboratories
94550LivermoreCaliforniaUSA
Przemyslaw Bienias
Joint Center for Quantum Information and Computer Science
NIST/University of Maryland
20742College ParkMarylandUSA
Joint Quantum Institute
NIST/University of Maryland
20742College ParkMarylandUSA
Aniruddha Bapat
Joint Center for Quantum Information and Computer Science
NIST/University of Maryland
20742College ParkMarylandUSA
Joint Quantum Institute
NIST/University of Maryland
20742College ParkMarylandUSA
Yaroslav Kharkov
Joint Center for Quantum Information and Computer Science
NIST/University of Maryland
20742College ParkMarylandUSA
Joint Quantum Institute
NIST/University of Maryland
20742College ParkMarylandUSA
Alexey V Gorshkov
Joint Center for Quantum Information and Computer Science
NIST/University of Maryland
20742College ParkMarylandUSA
Joint Quantum Institute
NIST/University of Maryland
20742College ParkMarylandUSA
Behavior of Analog Quantum Algorithms
2 Jul 2021(Dated: July 6, 2021)
Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the quantum approximate optimization algorithm (QAOA). These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters. In this work, we explore connections between these analog algorithms, as well as limits in which they become approximations of the optimal procedure. Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state that effect the coherent error cancellation of diabatic transitions. Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern. Additionally, the ratios of the QAOA bangs are determined by the smooth, non-oscillatory part of the optimal procedure. We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure. With these arguments, we conclude that different analog algorithms can emulate the optimal protocol under different limits and approximations. Finally, we present a new algorithm for better approximating the optimal protocol using the analytic and numeric insights from the rest of the paper. In practice, numerically, we find that this algorithm outperforms standard QAOA and naive quantum annealing procedures.
I. INTRODUCTION
Analog quantum algorithms come in a variety of forms, from Adiabatic Quantum Computing [1] and Quantum Annealing [2] to variational algorithms such as the quantum approximate optimization algorithm (QAOA) [3]. Analog quantum algorithms are particularly relevant in the Noisy Intermediate Scale Quantum device [4] era, where they are capable of running effectively on smallscale devices.
All these analog quantum algorithms use the same basic ingredients but combined in different ways that obfuscate the connections between these algorithms. Adiabatic Quantum Computing slowly changes the system from an initial Hamiltonian, whose ground state you start in, to a final Hamiltonian, whose ground state you want to know. By going slowly, Adiabatic Quantum Computing can rely on the adiabatic theorem [6] which ensures this state transfer so long as the ramp is smooth and monotonic and the runtime scales as an inverse polynomial of the spectral gap. Quantum Annealing is a broader algorithm that allows for non-adiabatic effects, and this has recently led to the field of diabatic quantum annealing [7] that explicitly uses excitations above the ground state to solve problems faster, the difficulty being how to control and utilize these excitations. * Electronic address: [email protected] QAOA is based upon a different, variational framework where the Hamiltonian evolution obeys bang-bang structure at all times. Here, the quantum optimization problem is solved by optimizing the lengths of Hamiltonian pulses in a hybrid, quantum-classical loop. Not much is known about how QAOA relates to other analog quantum algorithms and how its performance scales with the number of variational parameters. While originally motivated by a Trotterization of Adiabatic Quantum Computing and Quantum Annealing, QAOA performs quite differently in practice. The numerical results of [8,9] show that QAOA variational parameters fall along certain smooth curves as the depth of the circuit increases. These curves superficially resemble a Trotterization of an annealing path, but the size of the Trotter steps is insensitive to the circuit depth, invalidating standard Trotter error arguments. There is also numerical evidence [8] that these curves, when interpreted as annealing paths, exhibit properties of diabatic speedups.
More recently, techniques from optimal control theory [10] have been applied to analog quantum algorithms [11][12][13][14][15], specifically in the context of the variational approach of QAOA. These optimal control techniques were applied to the more generalized problem of analog quantum algorithms in Ref. [15]. This optimal protocol takes on a bang-anneal-bang form with guaranteed bangs at the beginning and end that become vanishingly smaller as the allowed time for the protocol increases. In the middle, the protocol often takes on an annealing-like form with a smooth control function. We refer to this opti-mized protocol as an optimal curve/protocol throughout the paper. This paper focuses on analytically proving the connections among all the analog algorithms mentioned previously. First, we show that QAOA emulates the optimal curve, acting as a large-time-step Trotterization of this curve. Second, we show that, in the limit of long time, the optimal curve resembles an optimal adiabatic path similar to the annealing schedule of Roland and Cerf [26], which was optimized to ensure adiabaticity with respect to the instantaneous spectral gap. Therefore, the asymptotic curves discovered for QAOA in Refs. [8,9] are derived from the optimal adiabatic path of the system. In the short time and low circuit depth limit, the optimal curve and QAOA are still connected and begin to resemble excited state computation seen in diabatic quantum annealing [7].
These results rely on the fact that the optimal schedules have large annealing-like regions, which [15] showed are common in optimal curves. In practical terms, the relationship between QAOA and the optimal curves means that QAOA can safely be scaled up by bootstrapping, or using the results of lower circuit depth optimization to produce a good guess for the variational parameters in a higher circuit depth setting. This bootstrapping method was suggested by Refs. [8,9], and our results in this paper seek to understand why this method is valid. This relationship also means that QAOA parameters can be used to form an initial ansatz for the optimal schedule, which in general has better performance than QAOA. Furthermore, our results contradict the common design philosophy that annealing paths should be monotonic (see reverse annealing as a notable exception [17,18]). Monotonicty is a holdover from the infinite-time adiabatic limit [1], and a monotonic schedule improves the energy of the state over doing nothing [19]. However, our results show that adding an oscillation to the annealing schedule, with a frequency dependent on the spectral gap, improve performance by coherently cancelling the error due to leakage to excited states. The amplitudes of these oscillations vanish in the infinite-time limit.
Finally, we present a new practical algorithm for approximating the bang-anneal-bang optimal control protocol. This algorithm uses the analytic and numeric results from the rest of the paper to create an ansatz for the form of the optimal protocol. This ansatz has a small number of variational parameters, and the number of parameters can be taken to be independent of system size or scaled up with system size depending on the available resources. We demonstrate that this new ansatz outperforms both QAOA and a naive monotonic annealing schedule.
We begin in Section II by reviewing the relevant algorithms and providing background information on them. Section III provides the original numerical motivation for this work, presenting the QAOA asymptotic curves of [8,9] and the numerical connection between these curves and the optimal schedules of [15]. In order to prove this connection, our analytics are broken up into two parts.
The first analytic part in Section IV relates to the optimal curves themselves, showing how the oscillatory behavior arises. This section explores the properties of the initial and final bangs, which serve to spread the population out into more than just the ground state and then bring it back together, with the intermediate annealing region providing a nearly-adiabatic procedure. The oscillations result from properties similar to counter-diabatic driving terms from shortcuts to adiabaticity [20][21][22]. The second analytic part in Section V presents work involving product-formula expansions. We show that, if the underlying annealing curve consists of a smooth slow curve and a fast small oscillation, a product formula of that annealing curve gets a reduction in its error bounds when the product formula step sizes match the period of the oscillations. This reduction, combined with potential coherent error effects and additional optimization, can help explain the step size of QAOA and how it relates to the optimal curves. We present our new bang-anneal-bang ansatz algorithm in Section VI. Finally, in Section VII, we summarize and review the implications and caveats of our work, providing possible directions for future study and development.
II. THE ALGORITHMS
All of the analog quantum algorithms considered here fit within a linear control framework described by the HamiltonianĤ (t) = u(t)B + (1 − u(t))Ĉ.
(1)
The HamiltonianB is often described as the "mixer" and encodes quantum mixing (e.g. a uniform transverse field on qubits).Ĉ is known as the "problem" Hamiltonian and encodes the optimization task (e.g. a diagonal Hamiltonian with the target cost function along the diagonal). In all examples, the initial state of the system is taken to be the ground state ofB, and the target state of the system is taken to be the ground state ofĈ. The control function u(t) : [0, t f ] → [0, 1] specifies the time evolution protocol of the algorithm over its total runtime t f . The analog algorithms studied here each come with a different design ansatz for this control function.
A. Adiabatic Quantum Computing
Adiabatic Quantum Computing was originally proposed to solve combinatorial optimization problems [1]. The function u(t) is taken to be a monotonic function, starting at u(0) = 1 and ending at u(t f ) = 0. If the change in u(t) is slow enough, the quantum adiabatic theorem [6] guarantees adiabaticity, which means that the system will stay in the same relative eigenstate throughout the evolution. Notably, this is usually employed to ensure that a system starting in the ground state ofB at t = 0 will evolve into the ground state ofĈ at t = t f . This necessitates that the Hamiltonian,Ĥ(t), maintains a non-zero spectral gap throughout, with some exceptions (e.g. ground state degeneracy for all t or just at t = t f ) [1]. A commonly cited condition for adiabaticity is that [1]
t f ≫ dĤ(t) d (t/t f ) min t ∆(t) 2 ,(2)
where ∆(t) is the spectral gap of the Hamiltonian at time t. This is a simplified condition that often works in practice, and its formal version, while more complicated [6], depends roughly on the same parameters, potentially with worse exponents. Therefore, a large part of the analysis of adiabatic quantum algorithms involves spectral theory to determine the size of ∆(t). During most of the anneal, the spectral gap is usually independent of n, but during avoided level crossings, which often correspond to phase transitions, the gap can close polynomially or exponentially with n. In hard optimization problems, this spectral gap is exponentially small in the vicinity of avoided level crossings.
Often the monotonic annealing schedule, u(t), is taken to be a linear ramp (or some other hardware-determined shape), but the ramp can be optimized to slow down when the gap is small and speed up when the gap is large. This optimization, originally proposed by Roland and Cerf [26], is necessary to recover the Grover quadratic speed-up for unstructured search, and there is good evidence that optimization of the schedule in general can lead to a similar quadratic speed-up over unoptimized schedules [35]. The optimized Roland and Cerf schedule is specific to the unstructured search problem, but it can be generalized by methods such as the quantum adiabatic brachistochrone [27]. One problem with these optimized schedules is that they require full knowledge of the spectral gap to construct. The unstructured search problem has the same spectral structure in all problem instances, but knowledge of the spectral gap is generally hard to find a priori. Another problem of such optimized schedules is that, if the minimal spectral gap is exponentially small, they might require realistically unachievable exponential precision [36].
B. Quantum Annealing
Quantum Annealing was originally proposed [2] before Quantum Adiabatic Computing and was justified not by the adiabatic theorem, but instead by comparison to classical thermal annealing. In practice, the setup of Annealing is roughly the same as Adiabatic, with u(0) = 1, u(t f ) = 0, and a smooth, usually monotonic ramp in between.
The relative definitions of Annealing and Adiabatic are slightly ambiguous and vary throughout the field. In this paper, we will use one of the more common definitions of Quantum Annealing as a generalization of Adiabatic Quantum Computing, with Adiabatic being a subclass of Annealing. Whereas Adiabatic Quantum Computing requires adiabaticity, meaning the state of the system must always track the ground state, Quantum Annealing allows for either non-adiabatic effects or adhering to adiabaticity. These non-adiabatic effects might be due to thermal noise or simply going too fast (in the present paper, we will consider only unitary dynamics, so there will be no thermal noise). These non-idealities could mean that the final state is an excited state that is deemed good enough for practical purposes.
It is also possible to utilize the sped up behavior and engineer the dynamics to depopulate the ground state and then repopulate it [37,38], utilizing the power of higher excited states for intermediate computational steps. This is the basis of diabatic quantum annealing [7]. While diabatic algorithms show promise, it is currently unclear how to efficiently engineer the desired effects. This paper could be interpreted as addressing this question, and we point interested readers to Section VI where we describe a practical algorithm for engineering a useful diabatic evolution.
C. Quantum Approximate Optimization Algorithm (QAOA)
While QAOA is sometimes described in the digital quantum circuit framework, it is ultimately an analog quantum algorithm. The control function is no longer smooth but instead takes on a pulsed, bang-bang form where u(t) can only equal 0 or 1, meaning we are only applying eitherB orĈ but not linear combinations of them. The original [3] formulation of QAOA is best described using unitaries where the system starts in an initial state |x(0) (the ground state ofB) and ends at a final time, t f , in the state
|x(t f ) = p i=1 e −iβiB e −iγiĈ |x(0) .(3)
The (positive) times γ and β, also known as angles, describe how long to apply each bang, with the label γ referring to evolution times underĈ and β referring to evolution times underB by convention. The total runtime for this algorithm is
t f = p i=1 (γ i + β i ).
The number of layers in QAOA, p, is usually fixed, while the angles γ i and β i are allowed to vary freely.
As a hybrid variational algorithm, QAOA uses a classical optimizer to optimize the angle parameters and a quantum computer to sample and estimate the final energy E = x(t f )|Ĉ |x(t f ) . The goal is to prepare a state that is close to the target state by finding the γ i and β i that minimize E .
As it was originally proposed, QAOA was conceived as a generalized discretization of Quantum Annealing. Indeed, a Suzuki-Trotter expansion of an Annealing sched-ule would result in a bang-bang pattern similar to QAOA. However, numerical results [8,9] have shown that the optimal angles do not approach a Trotterization. This is evident from the observation that the ideal bang lengths remain roughly constant as p is increased, whereas under Trotterization, they become vanishingly small as p → ∞.
A key goal of the current paper is to explain this phenomenon and describe this large-p behavior.
D. Optimal Schedules
In a previous study [15], the analog quantum algorithm problem was analyzed through the lens of optimal control theory. We asked what properties an optimal u(t) must have in order to produce the lowest possible E within a given amount of time t f .
The resulting schedule takes on a form with a finitelength u = 0 bang at the beginning and a finite-length u = 1 bang at the end. Our analytics suggested multiple possibilities in the middle, but in all numerics tested (mostly focusing on the Ising model with some additional data for the Heisenberg model), the middle region was dominated by a smooth non-monotonic annealing region. The form of this annealing schedule was not studied extensively, and the exact shape of this region, as well as a heuristic picture of the evolution, is one of the main contributions of the current paper.
The initial and final bangs in such a bang-anneal-bang procedure are guaranteed to decrease as t f increases and vanish in the limit t f → ∞ [15]. In fact, this corresponds to recovering the adiabatic limit. Our results in the current paper can be used to interpret the initial and final bangs as exciting the system into a diabatic annealing regime.
III. NUMERICALLY COMPARING OPTIMAL CURVES AND QAOA
Our main results are inspired by two separate pieces of numerical evidence. The first is the asymptotic largep structure of QAOA, as has already been presented in Refs. [8,9]. The second is the asymptotic large-t f behavior of the optimal curves. This behavior of the optimal curves was explored partially in the Appendices of the previous paper [15], but here we formalize those results and connect them to the behavior of QAOA.
A. QAOA Curves
One of the primary sources of excitement with QAOA is the ability to predict the γ i and β i from similar problem instances. It has been observed [39] that QAOA angles give rise to similar performance across similar problem instances. More relevant for our purposes, when considering a fixed problem instance, the optimized QAOA angles form a certain pattern, and the QAOA protocol approaches an asymptotic continuous limit with increasing number of layers p.
Specifically, suppose that the optimal QAOA angles for a given p are given by γ i and β i , then we can construct continuous functions γ p (s) and β p (s) for s ∈ [0, 1] such that
γ p i − 1 p − 1 = γ i ,(4)β p i − 1 p − 1 = β i .(5)
As p increases, it has been noted numerically [8,9] that these functions γ p (s) and β p (s) converge to asymptotic functions γ(s) and β(s) that become independent of p in the limit p → ∞. These asymptotic curves should not be confused with a simple Suzuki-Trotter expansion of some underlying annealing curve. In order to guarantee an accurate approximation of a Hamiltonian time evolution by a Suzuki-Trotter product formula, the time steps are required to be of vanishing order. However, the asymptotic curves prescribe angles of constant order, so if they were interpreted as a simple Suzuki-Trotter expansion, the expected error would be non-vanishing as the number of QAOA rounds goes to inifinity.
It is possible to construct an annealing curve from the asymptotic QAOA curve, as done in Zhou et al. [8]. Specifically, they define u(s) = β(s) β(s)+γ(s) as an annealing curve, which is well-motivated in part because it is commonly seen that β(s) is dominant at the beginning and γ(s) is dominant at the end (the reason for this is connected to the asymptotic shape of the optimal curves that QAOA is emulating as we discuss in Sec. IV). The resulting annealing curve captured a well-known effect from diabatic quantum annealing, so-called diabatic cascades [37], providing an empirical link between QAOA and diabatic quantum annealing.
An example of these asymptotic QAOA curves is given in Fig. 1. These numerics indicate that there is some asymptotic curve for each problem instance that QAOA angles are converging to. Notably, this means that QAOA can bootstrap itself up, using lower p parameters to create good guesses for what the higher p parameters are.
B. Bang-Anneal-Bang Oscillations
The new numerics that inspired this current study involve the bang-anneal-bang behavior of the optimal curves when compared to QAOA. The runtime of a QAOA protocol can be defined in terms of its variational parameters as Plotted are the γi, βi, and ui = β i β i +γ i . The x-axis is the normalized QAOA layer i−1 p−1 . The lighter curves are for lower p (starting at p = 10), and the darker curves are for higher p (ending at p = 30). These curves do vary slightly, but especially at higher p, they settle into some smooth asymptotic curve. This data was gathered forB being a transverse field andĈ being a randomized Ising model with all-to-all couplings drawn at random uniformly from the range [−1, 1] on n = 8 qubits (exact couplings given in Appendix C).
t f = p i=1 (γ i + β i ).(6)
It is natural then to ask what the optimal curve is for that length of time. The numeric answer is exemplified in Fig. 2.
In Fig. 2, we plot the QAOA bangs for a particular instance of a randomized Ising model alongside the optimal curve, with a bang-anneal-bang structure, that takes the same length of time as QAOA. For ease of optimization, the QAOA instance here uses the same time length, (γ i + β i ), for each layer (with that length also being treated as a variational parameter), but all the qualitative properties apply in the normal QAOA setting as well. Also plotted is the QAOA curve defined by u i = βi βi+γi with these points plotted on the x-axis at the midpoint of the corresponding QAOA layer.
There are two key qualitative points to be made here. First, the optimal curve oscillates about some base curve. The period of these oscillations matches up with the length of the QAOA layers, with there being p = 14 QAOA layers and 14 oscillations of the optimal curve. Second, the underlying curve that is being oscillated about matches up with the QAOA curve. These are very general properties and were seen in every numerical instance studied.
This behavior suggests a connection between QAOA and bang-anneal-bang form of the optimal procedures. Furthermore, when the optimal procedure is given a long time, it approaches an adiabatic procedure, with the initial and final bangs becoming vanishingly small and the Fig. 1, and the time for the optimal protocol is fixed based on the time taken by the QAOA protocol. Notice first that the optimal protocol oscillates in such a way that it fits 14 oscillations into this time frame. Also, in green, we plot the QAOA variables defined as ui = β i β i +γ i which track the underlying annealing portion of the optimal curve (for details see main text). These properties have been seen numerically in every Ising model we have studied.
amplitude of the oscillations approaching zero. The rest of the paper will be devoted to explaining this connection by focusing on the two parts of this problem.
In Section IV, we explain where these oscillations come from, employing an asymptotic near-adiabatic perturbative analysis. In the long-t f limit, the period of oscillations turns out to be inversely proportional to the instantaneous spectral gap; although, the smaller p used in current QAOA implementations result in a t f such that this limit is not reached and the periods do not correspond to the spectral gap. It could be possible to extract spectral gap information from a long enough QAOA procedure, but that is likely to be outside the regime of near-term quantum computers. Numerically, the examples we can access also are not in this asymptotic regime, but the same analytic mechanism can explain the origin of these oscillations even if the timescales are not long enough for the spectral gap to govern the oscillation period.
Then in Section V, we explain the connection between these oscillations and QAOA by interpreting QAOA as a large-time-step product formula (a.k.a. Trotterization) of the underlying optimal curve. Due to the large timescales involved, QAOA cannot ordinarily be interpreted as a product formula without incurring untenable errors. We show that a product formula aligning with an underlying oscillation incurs less error overall; though, our method does still have scaling with p that could potentially be mitigated by coherent cancellation of Trotter errors.
Because these properties of the optimal curve are qualitatively universal, we can utilize them to produce an ansatz for the optimal protocol. We do this in Section VI and show that this ansatz, which includes only a small number of variational parameters, can outperform both naive annealing and QAOA.
IV. DERIVING THE OSCILLATIONS
First we consider how to characterize the optimal curve, specifically the oscillatory pattern in the annealing portion of its bang-anneal-bang form. The interior annealing region mostly has a smooth annealing form which is quite apparent for transverse field Ising models but appears to varying degrees in other models [15].
In the large-runtime limit, these optimal curves become a monotonic annealing schedule, and the oscillations have vanishing amplitude. This is consistent with the adiabatic theorem [6], which guarantees that a monotonically decreasing control function will transform an initial ground state to a final ground state. Interestingly, there are conjectures that, in the space of the Lie algebra generated byB andĈ, the shortest path that transforms between the terminal ground states is precisely the adiabatic path which transfers all eigenstates in the initial Hamiltonian to the equivalent eigenstates in the final Hamiltonian [25]. Their conjecture was proven in the adiabatic and near-adiabatic limit, but is harder to prove far away from this limit. Furthermore, the result considers paths that prepare the exact final ground state, which is a valid assumption in our setting only in the limit of long runtimes. Therefore, we expect the optimized annealing schedules for long times to approach an optimized adiabatic schedule, similar to what was derived by Roland and Cerf [26] for the unstructured search problem or in the quantum adiabatic brachistochrone [27].
It is possible to emulate an adiabatic protocol in a shorter period of time, using shortcuts to adiabaticity and counter-diabatic protocols [22], and most notably, it is even possible to emulate the effects of a CD addition to the Hamiltonian using only the original Hamiltonians with a fast oscillation of the control function [28]. These fast oscillations rely on user-defined periods and parameters and so do not describe the properties seen in the numerics for the optimal schedule. This method also relies on full knowledge of the counter-diabatic driving term which we lack and which is difficult to find for large system sizes.
Before we proceed, we comment on whether we should expect an adiabatic evolution, or at least one that keeps the instantaneous eigen-distribution constant, potentially only at certain points (at least constant between the beginning and end of the annealing region). Numerically, we do see this in the optimal curves. For long times, as stated previously, the anneal is just an optimized adiabatic schedule with very small oscillation amplitudes. Whereas for shorter times, the oscillations Probabilities
t/t f |C 0 (t)| 2 |C 1 (t)| 2 u(t)
FIG. 3: Here we zoom in on the optimal curve from Fig. 2 during the annealing region. In addition, we plot the probabilities of being in the instantaneous ground state, |C0(t)| 2 , and first excited state, |C1(t)| 2 . This plot shows that this annealing region is transferring the states adiabatically with the populations roughly maintained from the beginning to end of the anneal. There is variation in these amplitudes but they roughly return to themselves after a full oscillation of the annealing curve.
are quite pronounced, and an examination of the eigendistribution, such as in Fig. 3 reveals that the instantaneous eigen-distribution does indeed remain relatively constant, matching up at the beginning and end of the anneal as well as at points roughly in line with the periods of the oscillations. Therefore, it seems natural that the annealing region of the optimal curves is emulating a shortcut to adiabaticity approach. The Magnus expansion method of Ref. [28] uses the frequency of their oscillations as a fit parameter with their method relying on a perturbative approach as this period becomes small. Since our goal is to derive the frequency of the oscillations rather than impose a frequency, this Magnus expansion method is not useful in our circumstance. Here we consider the following different approach to derive such an oscillatory counterdiabatic procedure.
A. Near Adiabatic Approximation
To demonstrate this approach, we will first restrict to the setting where only the ground state and first excited state are relevant. This holds in the near-adiabatic limit and is supported by our numerics, shown in Fig. 3.
The methods we use here are similar to those used in adiabatic boundary cancellation methods [29,30] and are generally connected to adiabatic analysis [31,32] and the analysis of shortcuts to adiabaticity [22]. We specifically follow results from [33].
Consider a case where we have some control function,
u(t) = u 0 (t) + c(t) ,(7)
so that the Hamiltonian iŝ
H(t) =Ĥ 0 (t) +Ĥ c (t) (8) = u 0 (t)B + (1 − u 0 (t))Ĉ + c(t)(B −Ĉ) .
Here, u 0 (t) is a function determined by the adiabatic nature of the problem. Any sufficiently slow procedure is adiabatic, but the annealing schedule can be optimized, such as the analytic fine-tuning of Roland and Cerf [26], to improve performance and ensure the onset of adiabaticity at smaller t f . The function c(t) represents our control freedom, and we can choose it so that the adiabatic passage described by u 0 (t) is followed as precisely as possible.
We will furthermore label the instantaneous eigenstates ofĤ 0 (t) by |j 0 (t) with eigenvalues λ j (t) so that
H 0 (t) |j 0 (t) = λ j (t) |j 0 (t) ,(9)
and we ignore any degeneracies (to account for degeneracies, we could work in a subspace defined by the symmetries of our Hamiltonian and our initial ground state). These eigenstates are defined up to a phase choice which will be set below. Throughout this section (unless otherwise noted), we use the 0 subscript to indicate that these quantities are relative to the eigenframe determined by u 0 (t) rather than the full eigenframe determined by u(t). Now, we can express our current state in terms of these eigenstates by
|ψ(t) = j C j (t) |j 0 (t) .(10)
We make the assumption that |C 0 (t)| and |C 1 (t)| are much larger than all other probability amplitudes. This assumption implies thatu 0 is small so that the system is evolving approximately adiabatically. Also it implies that c(t) is small so that the small deviations from the base curve also do not break the approximation of a twolevel system.
Applying the Schrödinger equation produces
j i d C j (t) d t |j 0 (t) + C j (t) d d t |j 0 (t) (11) = jĤ (t)C j (t) |j 0 (t) .
Using the orthonormality of the eigenstates |j 0 (t) we can reduce this to a system of coupled differential equations. For instance, the coefficient of the eigenstate |k 0 (t) in the left hand side of the above equation is
i d C k (t) d t + j =k C j (t) k 0 (t)| d d t |j 0 (t) ,(12)
where we have set the phases of the eigenstates by requiring that k 0 (t)| d d t |k 0 (t) = 0. This choice of phase is fairly common in adiabatic analysis and shortcuts to adiabaticity where it is often referred to as part of the adiabatic frame. To see why this phase can be chosen, consider 0 =
d d t k 0 (t)|k 0 (t) = ( d d t k 0 (t)|) |k 0 (t) + k 0 (t)| d d t |k 0 (t)
. Therefore, Re( k 0 (t)| d d t |k 0 (t) ) = 0 is always automatically satisfied, while the phase of the state can always be chosen such that Im( k 0 (t)| d d t |k 0 (t) ) = 0. If the Hamiltonian were stoquastic (Hamiltonians where the off-diagonal elements are all real and nonpositive), this choice of phase would mean that the instantaneous ground state maintain the same phase throughout the evolution, which we take to be real and positive. Similarly, the stoquastic first excited state can be represented using only real amplitudes throughout.
The instantaneous eigenvalues of the Hamiltonian are defined by Eq. 9, and we set λ 0 (t) = 0 at all times without loss of generality. The time derivative of Eq. 9 yields
dĤ 0 (t) d t |j 0 (t) +Ĥ 0 (t) d d t |j 0 (t) (13) = d λ j (t) d t |j 0 (t) + λ j (t) d d t |j 0 (t) .
The inner product of this equation with another eigenstate |k 0 (t) such that k = j:
k 0 (t)| dĤ 0 (t) d t |j 0 (t) ) + k 0 (t)|Ĥ 0 (t) d d t |j 0 (t) (14) = d λ j (t) d t k 0 (t)|j 0 (t) + λ j (t) k 0 (t)| d d t |j 0 (t) .
We can act on the bra states with the Hamiltonian and eliminate one element through orthogonality to get
k 0 (t)| dĤ0(t) d t |j 0 (t) ) (λ j (t) − λ k (t)) = k 0 (t)| d d t |j 0 (t) .(15)
In Eq. (12), this time derivative of eigenstates is multiplied by the amplitudes C j (t). By our assumptions, only C 0 (t) and C 1 (t) will be relevant, and we can discard cases where j = 0, 1. Now consider the right-hand side of Eq. (11). If we were in the true adiabatic reference frame of the fullĤ(t) instead of justĤ 0 (t), the Hamiltonian would just scale each eigenstate by its eigenvalue, but instead we get
k 0 (t)|Ĥ(t) j C j (t) |j 0 (t) (16) = C k (t)λ k (t) + j C j (t) k 0 (t)|Ĥ c (t) |j 0 (t) .
We define
γ(t) ≡ 0 0 (t)| (B −Ĉ) |1 0 (t) ), ∆(t) ≡ λ 1 (t), κ i (t) ≡ i 0 (t)| (B −Ĉ) |i 0 (t) ,
where λ 0 (t) = 0. Thus, ∆(t) has a meaning of the instantaneous spectral gap for the Hamiltonian H 0 (t). In the stoquastic setting, all of these quantities are real, and we will treat them as such going forward.
Putting everything together, the Schrödinger equation for the ground state and first excited state amplitudes give
i d C 0 (t) d t + C 1 (t) γ(t)u 0 (t) ∆(t) (17) = c(t) (C 0 (t)κ 0 (t) + C 1 (t)γ(t)) , i d C 1 (t) d t − C 0 (t) γ(t)u 0 (t) ∆(t) (18) = ∆(t)C 1 (t) + c(t) (C 0 (t)γ(t) + C 1 (t)κ 1 (t)) .
With these equations, we can separate out the amplitudes and phases so that
C i (t) = A i (t)e iϕi(t) .(19)
Separating the real and imaginary parts of the differential equations, the resulting differential equations are (suppressing all functional dependencies for brevity)
ϕ ≡φ 0 −φ 1 = ∆ + c(κ 0 − κ 1 )(20)+ A 2 0 − A 2 1 A 0 A 1 cγ cos(ϕ) − γu 0 ∆ sin(ϕ) , A 0 = − cγ sin(ϕ) + γu 0 ∆ cos(ϕ) A 1 ,(21)A 1 = cγ sin(ϕ) + γu 0 ∆ cos(ϕ) A 0 .(22)
As we already mentioned, the assumption that the Hamiltonian is stoquastic ensures that the γ and κ i functions are real. In the absence of stoquasticity, these functions could be complex-valued which would have just made the algebra above to separate our amplitudes and phases slightly more complicated without fundamentally changing the results.
The equations (21,22) for the amplitudes A 0,1 can be integrated to give
A 0 (t) = a cos t 0 dt ′ cγ sin(ϕ) + γu 0 ∆ cos(ϕ) + ϑ ,(23)A 1 (t) = a sin t 0 dt ′ cγ sin(ϕ) + γu 0 ∆ cos(ϕ) + ϑ ,(24)
where a and ϑ are constants that can be set such that a cos ϑ is the initial population of the ground state and a sin ϑ is the initial population of the first excited state. The signs here do not matter because any sign can be absorbed into the ϕ phase.
Maintaining the same populations of |A 0 (t)| and |A 1 (t)| throughout the evolution translates to the trig argument in Eqs. (23)
& (24) at time t f , Θ 0 [u(t)] = t f 0 dt cγ sin(ϕ) + γu 0 ∆ cos(ϕ) ,(25)
being close to a multiple of π. However in practice, a non-zero multiple of π would correspond to swapping the populations back and forth during the anneal which is inconsistent with the assumptions we made about being near-adiabatic with low leakage. Therefore, we want Θ 0 to be as close to zero as possible, meaning that the problem has simplified to finding the c(t) that ensures Θ 0 ≈ 0.
In the numerical examples shown, the oscillations fit neatly into the time allowed, giving an integer number of oscillations. This is largely because we look at cases where the time for the optimal procedure is the same as the time that QAOA takes. In other cases when the time does not match the time from a QAOA protocol, the oscillations are not regular. The point here is that we only expect c(t) to have a nice, simple oscillatory structure when t f neatly divides into periods of the oscillations.
In Appendix A 1, we work in the perturbative limit oḟ u 0 ≪ 1 to derive c(t) which ensures that Θ is zero:
c(t) =u 0 (t) 2 ∆(u 0 (t)) 2 d ln ∆(u0(t)) 2 γ(u0(t)) d u 0 (t) cos(∆(u 0 (t))t)+O(u 3 0 ). (26)
The cosine here follows the oscillations of the phase, ϕ(t), and mean that we go faster when the phase difference is large and slower when the phase difference is small. Essentially, these oscillations are designed to take advantage of the natural phase oscillations to speed up the procedure. The dependence of the amplitude on the gap reflects the fact that adiabaticity is easier (and therefore these oscillations are not necessary) when the gap is large. So long as t f is a multiple of the period of oscillations τ = 2π ∆(u0(t)) + O(u 0 (t)), then these oscillations will ensure that the amplitudes follow the eigenbasis associated with u 0 (t) up to corrections of O(u 3 0 ). As a note here, it is well known that oscillations, such as these, can eliminate the asymptotic nature of adiabatic theorem [32,34]. If oscillations are present, then in the infinite-time limit, the system will no longer be in the ground state, no matter how slowly it evolved. However, this is not a problem for us because the amplitude of the oscillations is decreasing withu 2 0 (and hence with 1/t 2 f ) which is small enough for the deleterious effects to not manifest [32].
As shown in the Appendix, this perturbative expression relies on the cancellation that occurs when c(t) is out of phase with the oscillations of the phase difference ϕ(t). Specifically, c(t) and cos(ϕ(t)) need to be in phase (up to integer multiples of π) to ensure cancellation. This can be seen by looking at Eq. (25) where having c(t) ∝ cos(ϕ(t)) maximizes the effect of the c(t) term, in the perturbative limit, allowing us to cancel out more of the contributions from theu 0 term. Later in Eq. (27), we can see that having oscillations in the control field that change like cos(ϕ(t)) (differentiating to a sin(ϕ(t))) will counteract the cos(ϕ(t)) in that integrand leading to a smaller contribution to the total integral. Up to first order inu 0 , the phase difference scales like ϕ(t) = ∆(u 0 (t))t, so we wind up with oscillations with period inversely proportional to the spectral gap.
Unfortunately, the numerics shown in Figs. (1)-(3) for the optimal curves do not fall into the perturbative regime described above. In these numerics,u 0 is large enough (t f is small enough) thatφ is no longer dominated by the spectral gap and begins oscillating at a higher frequency. As stated in the numerics section, determining the exact optimal curves for larger t f becomes unfeasible due to the difficulty of determining the gradient when so many solutions are good up to numerical precision.
While it is not possible to solve for this frequency perturbatively any more, based on the analysis in Appendix A 1, as well as numerical simulations of Eqs. (20)(21)(22) outside of the perturbative regime ofu 0 , the best way to follow adiabaticity along u 0 (t) still relies on the phase ϕ(t). In this nonperturbative regime, it becomes easier to deal with the full adiabatic reference frame that follows the eigenstates of the full u(t).
In Appendix A 2, we derive the near-adiabatic differential equations again, this time following the full control function u(t) instead of u 0 (t). In this setting, we need to impose boundary conditions that u(0) = u a at the start of the relevant region and that u(t f ) = u b at the end of the region. The resulting equations are similar, and the key quantity is still given by an argument that is functionally similar to Θ 0 :
Θ[u(t)] ≡ t f 0 dt γ(u(t))u ∆(u(t)) cos(ϕ(t)).(27)
Unfortunately, this form does not lend itself to a perturbative approach anymore because the small quantitẏ u now contains information about both the base curve and the oscillations. Fortunately, this form makes it even more clear how to ensure that this quantity should be close to zero. Namely, by roughly having γ(u(t))u ∆(u(t)) ∝ sin(ϕ(t)) we ensure that the integrand in Eq. (27) changes sign with a frequency twice that of ϕ(t). This allows the integral to cancel itself out over the oscillations, resulting in a Θ[u(t)] which is small. This intuition coincides perfectly with the numerics in the problem.
To see this in practice, in Fig. 4, we plot u(t), cos(ϕ(t)), and γ(u(t))u ∆(u(t)) as determined numerically for an optimal curve resulting from a model using a randomized Ising model. This plot was constructed so that the time allotted for the annealing curve corresponds to the amount of time that p = 10 QAOA needed.
In Fig. 4, the frequency of the phase oscillations matches the frequency of the optimal curve. Note that this is still for relatively low t f where the annealing curve Controls
t/t f u(t) cos(ϕ(t)) γ(u(t))u(t) ∆(u(t)) cos(ϕ(t)) γ(u(t))u(t) ∆(u(t))
FIG. 4: This plot shows the optimal curve, u(t), as well as two of the quantities that go into Eq. (27). This plot uses the same problem instance from Fig. 1, and the initial and final bits of time have been cut off to focus on just the annealing region of interest. Notably, from this curve, we can see that the phase difference between the ground and first excited state matches up exactly with the oscillatory pattern of u(t) (with a π phase shift) and is out of phase (π/2 phase shift) witḣ u(t) as we expect from the analytic arguments surrounding Eq. (27).
has relatively large amplitude oscillations, meaning the resulting oscillations are not exactly sinusoidal in shape and the period does not exactly mesh up with the asymptotic expectation of the spectral gap.
V. PRODUCT FORMULA ERROR
Based on the numerics presented in Section III, we see that QAOA is emulating the behavior of the optimal curve that takes the same amount of time. This section will seek to elucidate how QAOA can emulate the optimal curve, discussed in the previous section, despite the step sizes being large enough to throw off the usual error analysis of product formulas, also known as Trotterization.
To further see how QAOA is emulating the optimal curve, compare Figs. 3 & 5. These show the probabilities of being in the ground state and first excited state of the instantaneous evolutions for the optimal curve and QAOA, respectively. For QAOA, the "instantaneous eigenbasis" is determined by u i = βi βi+γi , the proportion of the QAOA layer dedicated toB. This eigenbasis is not physically related to QAOA, which still consists of large bangs, but it catches the effective Hamiltonian being emulated by the pairs of bangs. Both procedures roughly track the ground state with some leakage, mostly into the first excited state. QAOA is a rougher procedure with more leakage, compared to the optimal protocol. Probabilities QAOA Layer
1 − |C 0 (t)| 2 |C 1 (t)| 2 |C 2 (t)| 2 |C 3 (t)| 2 |C 4 (t)| 2 FIG. 5:
The instantaneous ground state and first excited state probabilities are plotted versus the QAOA layer. The instantaneous eigenbasis is defined based off ui = β i β i +γ i at the end of the ith layer. The probabilities are measured after the full QAOA layer (both theĈ and theB bangs). QAOA roughly follows an adiabatic-like procedure with the ground state population mostly being preserved. The problem instance displayed here is the same as in Fig. 1. Let's suppose that there is some optimal curve, u(t), defined such that the evolution governed by Eq. (1) brings the state as close to the target state as possible in time t f . Based off the previous section, we will here assume that this optimal control function can be described approximately by
u(t) = u 0 (t/t f ) + c(t, t f ),(28)
where c(t) is some oscillatory function. For concreteness, we take
c(t, t f ) = −c 0 (t f ) sin 2π τ t + φ ,(29)
where c 0 is some amplitude, τ some period, and φ some phase. We have included the negative sign and specified down to sine since this will correspond to φ = 0 later on. In essence, we have oscillations whose pattern matches our preexisting pattern of switchings in QAOA protocols and also the pattern of oscillations in the optimal protocols.
For the purposes of this section, we focus on a small region of the annealing curve whereu 0 is small and approximately constant. Then we ask how accurate a product series approximation is to the true evolution.
The actual evolution will be governed by the unitary time evolution operator,
U (t f , 0) = exp T −i t f 0 dtĤ(t) ,(30)
where exp T denotes the time ordered exponential. We approximate this unitary by breaking it up into a product formula that has a QAOA-like format,
U P F (t f , 0) = p−1 k=0Û 1 (k∆t + ∆t, k∆t) ,(31)
where ∆t is the length of the QAOA layer. TheÛ 1 here are operators corresponding to a single Trotter slice of the evolution. In this section, we assume that every QAOA layer uses the same ∆t, and as we will see, this corresponds to the frequency of oscillations in the optimal curve being constant with time (again assuming thatu 0 is small enough). Therefore, we simply set ∆t = t f p . It is appropriate to interpret this section as looking at a small region of the optimal curve in the adiabatic limit where the oscillations occur on much shorter timescales than the gross changes in the curve. When we specify down to our specific control problem, we get
U 1 (t 0 + ∆t, t 0 ) = exp −iB t0+∆t t0 dt u(t) (32) × exp −iĈ t0+∆t t0 dt (1 − u(t)) .
Our core result is that taking ∆t = τ , the size of the Trotter slice equal to the period of the annealing oscillation, while keeping the ratio of the bang lengths proportional to u 0 (t), leads to a smaller upper bound on the Trotterization error than if we picked a different ∆t. In this way it becomes advantageous for QAOA to match its layer length to the period of the optimal curve oscillations and its ratio ofB bang lengths andĈ bang lengths to the value of the base annealing function u 0 (t).
We examine this enhancement in two different settings described in the appendices. First, in Appendix B 1, we show this enhancement in the context of the standard operator error for product formulas. The main arguments in the appendix center around the standard error formula for the Trotter approximation known as the product formula. We zoom in on a small region of the annealing curve and consider the control function given in Eq. (28) vs. the case without the added oscillations. We find that these added oscillations can be accounted for in the product formula, and if the oscillation period and phase match up, it can lead to a lower error bound. If the oscillation period does not match, the error bound numerically matches up with the error from the case without oscillations at all.
We find (see the Appendix B 1 for details) that
||Û (t f , 0) −Û P F (t f , 0)|| (33) ≤ B ,Ĉ ∆t 2 p 2 1 − c 0 π ,
where we assumed ∆t = τ and φ = 0. In the case of c 0 = 0, this is equal to the standard error bound for product formulas. This improvement decreases when ∆t = τ , so the enhancement is specifically dependent on matching the size of the Trotter steps to the period of the annealing oscillations. This washing out can be seen in Fig. 6, but notice that there is also a small enhancement if ∆t = mτ or m∆t = τ for any m ∈ Z + , m > 1. Unfortunately, this bound on the error from unitaries scales linearly with p, the number of QAOA layers. Therefore, rather than getting tighter with more QAOA layers, as we expect, the bound gets looser. This scaling is because this worst-case bound assumes that adjacent layers of the product formula have errors that accumulate coherently.
Our second approach is restricted to an adiabatic anneal where the goal is to maintain the populations of eigenstates, specifically the ground state in our setting. The overall Trotter error bound in this setting was recently tightened by Yi and Crosson [40]. The same oscillatory enhancement found in the case of operator errors can be shown to occur in this setting as well, but the method requires a perturbative limit which does not hold for the QAOA angles. Specifically, the method requires ∆t ∈ O(n −1 ) which is not consistent with what we see numerically from QAOA with step sizes remaining roughly constant and large as the system size n increases, β i , γ i ∈ O(1). We rederive and extend the previous results and modify them for our setting in Appendix B 2. This extension consists of considering the setup where the underlying annealing schedule takes on the form of Eq. (28). This oscillatory annealing schedule is accounted for in the context of the adiabatic product formula analysis. The key result of the method of Ref. [40] is a reduction in the scaling of the Trotter error from O(∆t 2 p) down to O( 1 p ) + O( ∆t p ) when trying to simulate an adiabatic evolution. We show that this error scaling does not vanish when an oscillating schedule is considered (for small enough oscillations that the adiabatic theorem still holds) and show that there is an enhancement to the error scaling when the product formula step size matches the oscillation period.
It is not possible to fully apply this second approach to our setting because of the perturbative ∆t issues. Our product formula enhancement works partially in this setting and inherits the improved p scaling that the adiabatic Trotter method [40] naturally has over the operator error scaling, Eq. (33).
These two approaches are limited to unoptimized product formula approximation of the underlying optimal curve. Of course, QAOA has more freedom than this and can modify the parameters to do a smarter approximation than just a product formula. It is allowed to modify the angles away from what a product formula would do in order to achieve more enhancement. Specifically, it could be possible to coherently match up the leakage between multiple QAOA layers. All the upper bounds described above assume a worst case scenario that assumes the errors from adjacent QAOA layers add coherently via the triangle inequality, but it may be possible to design the . The oscillation period in this case was taken to be τ = 0.2, and the annealing function was taken to be a simple linear ramp with superposed oscillatory function. The blue dots with lines represent the case with oscillations, and the red squares represent the error bound when no oscillations are present. Eq. (33) is specifically for the case when ∆t = τ with this plot being more general. As can be seen, the proportional amount of enhancement is greatest when ∆t = τ , but there is a weaker enhancement when there is an integer multiple difference between these two quantities.
protocol so that the errors subtract coherently to some degree. Such an approach has been proposed recently [42] where the Trotterization error itself is used to engineer counter-diabatic driving terms.
We note that the results in this section should all be taken as analytic evidence supporting the numeric evidence from Sec. III. These bounds do exhibit an enhancement when we match up the Trotter step size and the oscillation period, but the bounds are not tight enough to describe the exact setting we see in the numerics. We leave it up to future work to tighten these bounds further to the setting of QAOA.
VI. BANG-ANNEAL-BANG ANSATZ ALGORITHM
One of the leading problems with the optimal curves is how to construct them efficiently. These optimal curves always seem to have the same qualitative structure, but working out the exact shape and length of various features is key to implementing these schedules effectively. Formally, these schedules can be found via a gradient descent procedure, using the analytically constructed gradient Φ(t) = δ E(t f ) δu(t) . This requires information from experimentally inaccessible intermediate times, and nu-merically estimating this gradient can prove cumbersome for an entire continuous function.
To address these issues, we here present a variational algorithm that produces a good approximation of the bang-anneal-bang optimal path. This algorithm will not produce the exact optimal procedure but will approximate it, and in our numerical trials it produces better results, given fixed time, than either QAOA or a simple, linear annealing schedule. The number of variational parameters can be adjusted depending on the available resources.
This algorithm is based off the insight that the asymptotic curve derived from QAOA angles coincides with the base curve in the annealing region of the optimal curves. Specifically, if QAOA is parameterized in terms of p layers with β i , the angles for mixerB bangs, and γ i , the angles for problemĈ bangs, then the asymptotic QAOA curve can be found in the large p limit by
v i − 1 p − 1 = β i β i + γ i ,(34)
where v(s) has the meaning of the control function. This behavior was noted numerically in Refs. [8,9]. The current work provides justification for the existence of these asymptotic curves and links them to the optimal protocols. Specifically, [8] interpreted this v(s) as an annealing curve which resulted in a good annealing procedure that actually captured well-known effects from diabatic quantum annealing. Our current algorithm is an improvement on this that captures even more of the structure and power of the optimal procedure. In optimal protocols, this v(s) has roughly the same functional form as the base curve u 0 (s) that determines the shape of the annealing region, up to a superposed oscillatory pattern. Furthermore, the period of that oscillatory pattern coincides with the duration of the QAOA layers. Therefore, it should be possible to use an existing QAOA procedure to get a good guess as to what the optimal procedure should look like. The initial and final bangs are vanishingly small for longer procedures and so are not well captured by QAOA. These bangs can be inserted in later as variational parameters. Therefore, we propose the following hybrid variational algorithm for approximating the optimal curves. 1. Find QAOA angles for large enough p to be able to identify the shape of v i−1 p−1 . In practice, we have found that at p ∼ 5 it is already possible to start identifying the pattern, with p ∼ 10 − 20 clearly identifying the pattern. in the middle, and a u = 1 bang at the end. Furthermore, superpose an oscillatory curve c(t) = A(t) sin(ω(t)t + φ) in the annealing region so that this region is described by v(t) + c(t).
• The lengths of the initial,γ, and final,β, bangs are variational parameters.
• There are multiple ways to parameterize the interior anneal: -The length of the anneal can be fixed to be the same as the time the QAOA procedure took minus the bang lengths, T QAOA −γ −β; the frequency of oscillations can be chosen to be ω(t) = 2πp/T QAOA ; and the amplitude of the oscillations A(t) = A is taken to be a variational parameter. -The length of the anneal can be fixed to be T QAOA −γ −β; the frequency of oscillations ω(t) = ω and amplitude of oscillations A(t) = A are taken to be static variational parameters. -The length of the anneal can be fixed to be T QAOA −γ −β; the frequency of oscillations ω(t) is chosen to be a variable function so that the period of a given oscillation matches the length of the corresponding QAOA layer. The amplitudes of oscillation can either be fixed to be the same or treated as seperate variational parameters in each oscillation. -The length of the anneal, T A , can be treated as a variational parameter [43]. The frequency can be taken as fixed ω(t) = 2πp/T QAOA or allowed to vary as a free fitted parameter as in previous versions. The amplitude of oscillation is a single variational parameter or can be binned into different regions with the amplitude in each region being treated as a variational parameter. -Adjust this ansatz as suits the system at hand and how many variational parameters the specific setting is capable of handling.
• Based on analytics, the optimal phase φ should be 0, but for optimization purposes it might be beneficial to treat this phase as a variational parameter as well.
• In the end, this procedure will result in an ansatz with at least three (β,γ, and A), but possibly more, variational parameters.
4. Using the constructed anstaz, run a variational algorithm to determine the optimal values of the selected variational parameters, attempting to optimize with respect to the final energy of the state with respect toĈ. This procedure will always produce a better protocol than just interpreting u(s) = v(s), and the number of variational parameters can be small. The most intensive part of this from a variational standpoint is the initial QAOA procedure to discover the shape of v(s). Given the asymptotic nature of this curve, it is possible to find v(s) for a given p (corresponding to a QAOA procedure that takes time t f ) and then scale it up into a banganneal-bang ansatz for a larger t f .
Since the base annealing curve is related asymptotically to an optimized adiabatic schedule, it could be possible to use insight from the adiabatic limit to bypass the QAOA step entirely and create an ansatz for v(s) a priori. For instance, in the unstructured search problem, it could be possible to use Roland & Cerf's [26] optimized adiabatic annealing schedule as a guess for the u 0 (t) base curve. If knowledge of the spectral gap is available, similar curves could be constructed for other problem instances. T ← p−1 i=0 |βi| + |γi| ⊲ Total QAOA time 5:β,γ, ω, φ ← Initial Guess 6: while Optimizing do return v bab (t) 12: function ConstructAnsatz(v0(s),β,γ, T, ω, φ) 13: for t ∈ [0,β] do ⊲ Initial Bang
Algorithm 1 Bang-Anneal-Bang Ansatz Optimization
for t k ∈ [β, T −γ] do ⊲ Interior Anneal 16: v bab (t) ← v0( t−γ T −γ−β ) + A cos (ω t + φ) 17: for t k ∈ [T −γ, T ] do ⊲ Final Bang 18: v bab (t k ) ← 1 19:
return v bab (t)
Below we present some of the results for this algorithm, simulated on a classical computer, solving directly the Schrödinger equation. Our algorithm could be implemented on a quantum computer, replacing this simulation of the Schrödinger equation with actual quantum evolution. Three different levels of the above algorithm are used. The first listed as "Basic Interpolation" is the form used by [8] where the QAOA derived asymptotic curve v(s) is just interpreted as an annealing curve. The second, "Sine Interpolation," superposes on top of this a sine curve whose period is equal to the average duration of a QAOA layer. Finally, "BAB Ansatz" is our full Bang-Anneal-Bang Ansatz, here treatingγ (the initial bang),β (the final bang), ω (the frequency of the oscillations), A (the amplitude of the oscillations), and φ (the phase of the oscillations) to all be variational parameters. The BAB Ansatz approximately follows the optimal procedure found using gradient descent, "GD." The energies resulting from each procedure can be found in Tab. I. The problem instance in this case is the same as in Fig. 1 Fig. 7. The more information is used in constructing the ansatz, the closer we get to the optimal energy possible for this time. Notably, our full BAB Ansatz is required to perform better than the underlying QAOA protocols used to kickstart this procedure. Based on our numerics, this ordering is representative of the relative qualities of the algorithms with BAB outperforming QAOA but not quite reaching the quality of the Optimal procedure.
This version of the algorithm is outlined in pseudocode in Alg. 1. For all procedures, the time allotted is t f , the same as the time that the QAOA procedure took. Fig. 7 shows an example. The kickstarting QAOA procedure used p = 6, and also shown are a basic linear ramp, and the exact optimal procedure found via a gradient descent ("GD") procedure. The resulting energies of all the relevant procedures are summarized in the accompanying Table. I. While this represents results for a single problem instance, these results are typical of what we see in other instances. Notably, the relative ordering of the energies achieved by each procedure in Table I is typical of all instances we examined.
Note that we have attempted to run this algorithm without the initial QAOA procedure to find an estimate for v(s) and instead using just a linear ramp for v(s). The resulting procedure performs poorly compared to any of our procedures employing the QAOA derived asymptotic curve but is still favorable when compared to a simple linear ramp. As stated before, if anything is known a priori about the shape of this base annealing curve, that information can be used instead of the QAOA procedure.
VII. CONCLUSION
The optimal protocol is by construction the most efficient way to operate a quantum annealer or analog quantum computer. This protocol demonstrates structure including the initial and final bangs explored in Ref. [15] that vanish in the long-t f limit.
This work explored the structure of the annealing region in more detail. Due to previous results regarding the optimality of the adiabatic path [25], we expect and indeed see that in the long-t f limit, the annealing region approaches an optimized adiabatic schedule, similar to what was derived by Roland and Cerf [26]. Furthermore, that optimal curve's annealing section has an oscillatory pattern superposed on top of it. In the adiabatic limit, the amplitude of these oscillations should vanish to recover a monotonic annealing ramp. However, in the near-adiabatic limit, these oscillations are helpful in managing the leakage between the ground state and first excited state. We derive the near-adiabatic form of these oscillations and describe their dependence on the phase difference between the ground state and first excited state amplitudes outside of this perturbative limit.
This analysis of the near-adiabatic limit of the annealing curve should be of interest in itself since it can be used to potentially enhance adiabatic protocols with little additional a priori information. This enhancement can be implemented by our algorithm in Sec. VI using the original annealing curve instead of a QAOA-derived curve.
Furthermore, we explore the connections between QAOA and this oscillatory structure of the optimal curves. Numerically, we see that optimal QAOA schedule incorporates the structure of the underlying optimal curve. The length of the QAOA layers matches up with the oscillation period of the annealing curve, and the ratio of the bang lengths within the QAOA layer matches up with the average value of the annealing curve within that period.
This behavior provides an explanation for the QAOA asymptotic curve behavior at large p seen in Refs. [8,9]. The behavior of the optimal curve can be understood asymptotically where it approaches an optimized adiabatic procedure with a fixed curve form. If QAOA is emulating this optimal curve, then QAOA should also be approaching a fixed asymptotic form.
We sought to provide analytic evidence for this matching up between the QAOA curve and the optimal proce-dure. Our results do show that there is an decrease in the error of a product formula if the product formula step size matches the oscillations in an annealing curve being Trotterized. Furthermore, this enhancement requires that the ratio of the bangs follows the annealing curve, just as we see in the numerics. Unfortunately, this error bound scales unfavorably with p, the number of QAOA layers, failing to match up with the scaling in practice. Based on other methods, we provide further arguments for how this additional scaling behavior could occur, but it remains an open question how to tighten this analysis to match the exact scaling seen in QAOA in practice.
As a result of this analytic and numeric work, we not only achieve an explanation for the asymptotic large-p behavior of QAOA, but we also better understand the optimal procedure. One of the main difficulties with the optimal curve is that it is not feasible to construct this protocol on real hardware. The protocol requires too much information about the intermediate quantum state and requires treating an entire smooth curve as a variational parameter. To address these issues, in Section VI, we constructed a new algorithm that uses the results of this paper to create an ansatz with very few variational parameters that outperforms naive quantum annealing and QAOA.
This algorithm uses a QAOA procedure to find the form of the annealing region of the optimal procedure and then uses this to create an ansatz. This ansatz then treats the lengths of the initial bang, final bang, and some basic properties of the oscillatory pattern as variational parameters in an ansatz. In practice, this algorithm outperforms QAOA and quantum annealing but falls slightly short of the full optimal protocol. istration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. parameter, the algorithm will likely find a local minimum. Our analytics indicate that the underlying annealing curve will be the same shape regardless of the number of QAOA layers, p, or the running time t f . Therefore, increasing the guess for TA, keeping everything else the same, should still be valid and will likely put your optimization into another local minimum, corresponding to a longer and more accurate procedure. Assuming the original QAOA had enough parameters to accurately estimate the annealing curve, this can be a good way to increase the accuracy of the procedure without introducing additional variational parameters.
Appendix A: Near-Adiabatic
In this section of the appendix, we explore additional features of the near-adiabatic limit. In section A 1, we carry out a perturbative analysis of the near-adiabatic equations to find the form of the oscillations, c(t), in the limit where the base ramp is changing slowlyu 0 ≪ 1. The end result of this section is the derivation of Eq. (26) from the main text.
The derivation of the near-adiabtic limit in the main paper relies on following the base annealing curve without the oscillations. It is possible to derive the nearadiabatic frame following the exact adiabatic frame, following the control function, oscillations and all. We do this derivation in Section A 2. This formulation is more useful outside the perturbative limit. This connects to Eq. (27) from the main text.
Section A 3 derives the optimal control equations in the near-adiabatic limit. These differential equations, if solvable, would give not only the oscillatory portion c(t) but the entire function u(t) = u 0 (t) + c(t), enforcing u(0) and u(t f ). These equations might be of interest to ex-perts or numericists but no longer lend themselves to a perturbative analysis, making them less useful within the current context.
Perturbative Limit
Recall the Schrödinger equation, Eq. (17) and (18), for the ground state and first excited state amplitudes,
i d C 0 (t) d t + C 1 (t) γ(t)u 0 (t) ∆(t) (A1) = c(t) (C 0 (t)κ 0 (t) + C 1 (t)γ(t)) , i d C 1 (t) d t − C 0 (t) γ(t)u 0 (t) ∆(t) (A2) = ∆(t)C 1 (t) + c(t) (C 0 (t)γ(t) + C 1 (t)κ 1 (t)) .
Next, we will Taylor expand ∆(u 0 (t)) and γ(u 0 (t)) around u 0 (0) = u
(0) 0 so that ∆(u 0 (t)) ≈ ∆(u (0) 0 ) + d ∆(u 0 (t)) d u 0 (t) u0(t)→u (0) 0u 0 t, (A3) γ(u 0 (t)) ≈ γ(u (0) 0 ) + d γ(u 0 (t)) d u 0 (t) u0(t)→u (0) 0u 0 t.(A4)
For convenience, we shorten the notation here so that
∆(u 0 (t)) ≈ ∆ 0 + ∆ 1u0 t, (A5) γ(u 0 (t)) ≈ γ 0 + γ 1u0 t.(A6)
Officially, we would need to do the same expansion with the κ variables, but it turns out that the κ j are not relevant to lowest non-zero order in perturbation theory.
The strategy here will be to do time dependent perturbation theory, specifically keeping track of orders ofu 0 . Furthermore, we will assume that we are zoomed in on one section of the u 0 (t) curve whereu 0 can be treated as approximately constant.
We will furthermore make an ansatz that our additional control function c(t) = c 0 sin(ωt + θ).
(A7)
In the new notation, we have two-level Hamiltonian that we will analyse within time-dependent perturbation theory with
H 0 = 0 0 0 ∆ 0 ,(A8)
and perturbation
V = c(t)κ 0 (γ 0 + γ 1u0 t) c(t) − iu 0 ∆0+∆1u0t (γ 0 + γ 1u0 t) c(t) + iu 0 ∆0+∆1u0t c(t)κ 1 + ∆ 1u0 t . (A9)
If we assumed that c 0 ∈ O(u 0 ) and did a perturbative expansion to first order inu 0 , we would find that c 0 = 0 leads to no leakage between the ground state and first excited state. Therefore, in order to get a nontrivial solution, we assume that c 0 ∈ O(u 2 0 ). Furthermore, looking at V , we can see that it is V ∈ O(u 0 ). Therefore, if we look at time dependent perturbation theory to second order in V , we will extract all the second order dependence onu 0 . Up to the second order in the perturbation V,
|ψ I (t) = 1 − i t 0 dτV(t)− t 0 dτV(τ ) τ 0 dτ 2 V(τ 2 ) |ψ(0) , with V(t) = W (t) † V (t)W (t) where W (t) = 1 0 0 e −i∆0t .(A10)
In order to transform the wave-function from the interaction picture ψ I → ψ back to original representation, we use relation |ψ(t) = W (t) |ψ I (t) . We can perform these integrals keeping terms up to second order inu 0 . We specifically want there to be no leakage between the ground state and first excited state after one oscillation period. The natural frequency in this system is ∆ 0 , so we expect
ω = ∆ 0 + δω (A11) and t f = 2π ∆0 − δt f where δt f , δω ∈ O(u 0 ).
Because ω only appears inside the trig function in Eq. (A7) which is multiplied by c 0 ∈ O(u 2 0 ), δω ∈ O(u 0 ) does not matter to the level of perturbation theory considered in this section. To zeroth order inu 0 , t f = 2π ∆0 . It is possible that ω could be farther away from ∆ 0 , but numerical analysis of these equations as well as other evidence presented in the main body suggests that ω ≈ ∆ 0 leads to the smallest correction that still works. Given the assumptions of the near-adiabatic approximations, a smaller correction term is preferable.
Let's define a unitary operator U (t) via |ψ I (t) = U (t) |ψ(0) , so that |ψ(t f ) = W (t f )U (t f ) |ψ(0) . We want no leakage between the ground state and the first excited state by the end of this evolution which means that we want U 12 (t f ) = 0 up to the relevant order.
After a lengthy calculation, one can show that
U 12 (t f ) = γ 0 ∆ 0 δt fu0 − 2π 2 γ 0 ∆ 1 ∆ 4 0u 2 0 − πγ 0 cos θ ∆ 0 c 0 (A12) + i 4πγ 0 ∆ 1 ∆ 4 0u 2 0 − i 2πγ 1 ∆ 3 0u 2 0 − i πγ 0 sin θ ∆ 0 c 0 + O(u 3 0 ).
Due to our assumptions about stoquasticity, all the quantities represented here are real which means that we can separate Eq. (A12) out into real and imaginary components and require those two sets to sum to zero independently. So we need the terms on the first line of Eq. (A12) to sum to zero independently and the terms on the second line to sum to zero independently. Looking at the imaginary parts of Eq. (A12), we can see that the leakage is zero if
c 0 =u 2 0 csc θ ∆ 2 0 2∆ 1 ∆ 0 − γ 1 γ 0 . (A13)
The smallest amplitude of oscillations corresponds to θ = π 2 , that results in
c 0 =u 2 0 ∆ 2 0 2∆ 1 ∆ 0 − γ 1 γ 0 + O(u 3 0 ).(A14)
By substituting θ → π/2 in Eq. (A12) and requiring the real part of the resulting expression to vanish we arrive at
δt f = 2π 2 ∆ 1 ∆ 3 0u 0 + O(u 2 0 ).(A15)
So with this oscillation in the control function, we have proven that after a time t f = 2π ∆0 − δt f , the amplitudes will return to themselves, resulting in perfect adiabatic transfer up to second order inu 2 0 in this near-adiabatic limit.
We can now plug into our ansatz, Eq. (A7), all the results of this section to get
c(t) =u 0 (t) 2 ∆(u 0 (t)) 2 d ln ∆(u0(t)) 2 γ(u0(t)) d u 0 (t)
cos(∆(u 0 (t))t)+O(u 3 0 ), (A16) which uses the results of Eq. (A14) as well as the definitions of ∆ 1 and γ 1 from Eqs. (A3-A6). The log derivative is used to compress notation. This form of oscillations will cancel out the deleterious effects ofu 0 = 0 up to second order.
The ansatz for ω in Eq. A11 is ultimately based off the zeroth order approximation of ϕ(t) which determines the natural frequency in this problem. The correction to the total evolution time δt f could, therefore, be derived by considering the phase difference between the first excited state and ground state. We start with theφ equation, Eq. (20), and look for the solution to this equation to first order inu 0 (which means that c(t) will be negligibly small). Firstly, since the A i terms only appear attached to small parameters, we can approximate them by their zeroth order constants, and the same goes for ∆ and γ. The sin ϕ and cos ϕ terms are troublesome, but they are already multiplied by small parameters, so we can approximate them by the zeroth order solution to this equation ϕ(t) ≈ ∆ (0) t (using as a boundary condition that ϕ(0) = 0). With these approximations and iterations, the first order solution to the equation is
ϕ(t) ≈ ∆ 0 t (A17) + 1 2 ∆ 1 t 2 + γ 0 (cos(∆ 0 t) − 1) ∆ 2 0 tan(2ϑ) u 0 .
We expect the populations to return after one full cycle of the system, both in terms of the phases of the eigenstate populations and the frequency of the ansatz ω. We can then ask what value of t corresponds to a full period of oscillation such that ϕ(t f ) = 2π + O(u 2 0 ). To zeroth order inu 0 it is obvious that t f = 2π/∆ 0 . Using this, it is easy to show that
t f = 2π ∆ 0 − 2π 2 ∆ 1 ∆ 3 0u 0 + O(u 2 0 ). (A18)
corresponds to one period of oscillation for the phase. Note that this time does indeed correspond to Eq. (A15) derived through by requiring no leakage from the unitary matrix.
Adiabatic Frame
In the main text, we derived the near-adiabatic limit for the case of the instantaneous eigenframe evolving alongside the base curve u 0 (t) so that the full control function was given by u(t) = u 0 (t) + c(t). In this setting c(t) was our actual free function with u 0 (t) fixed and c(0) = c(t f ) = 0. It is possible to treat this entire problem a control problem and seek out the u(t) that maintains populations the best between t = 0 and t = t f subject to the constraint that u(0) = u a and u(t f ) = u b .
The full version of this problem would just result in optimal curves in general, but here we are interested in just the smooth annealing region. One feature of this smooth annealing region is that the bangs have already excited up some of the state into the first excited state, making the near-adiabatic approximation even more relevant. For the purposes of this section, we will zoom in on one small region of the annealing curve and still implicitly assume that u a and u b are not that far apart.
We consider the probability amplitudes, C i (t) of being in the instantaneous eigenstates of a system, |j(u(t)) , with instantaneous eigenenergies, λ j (u(t)). So our state can be written as
|ψ(t) = j C j (t) |j(u(t)) ,(A19)
Applying the Schrödinger equation to this state yields a set of coupled differential equations
i d C k (t) d t + j C j (t) k(u(t))| d d t |j(u(t)) (A20) = λ k (u(t))C k (t).
Now, a few assumptions will be made, the first being that the ground and first excited states are nondegenerate. This could in general be satisfied by going into a symmetric subspace and looking at the relevant probability amplitudes within that symmetric subspace (for instance with the transverse field Ising model, we will consider the subspace defined by the usual Ising rotational symmetry). The second and more relevant assumption is that |C 0 | ≫ |C 1 | ≫ |C 2 | ≫ . . . which is just a statement that we are in the near-adiabatic limit of evolution. For our purposes, we will assume that the amplitudes for the second excited state and above are small enough throughout the evolution to be negligible. Furthermore, we will later consider |C 1 | to be a small quantity relative to |C 0 | for approximation purposes. The last requirement is that we set λ 0 (t) = 0 which can be done without loss of generality.
Applying these assumptions and following the calculations of [33], we derive the equations:
d C 0 (t) d t = − γ(u(t))u(t) ∆(u(t)) C 1 (t), (A21) d C 1 (t) d t = γ(u(t))u(t) ∆(u(t)) C 0 (t) − i∆(u(t))C 1 (t), (A22)
where the new functions represent
γ(u) ≡ ϕ 0 (u)| (B −Ĉ) |ϕ 1 (u) , (A23) ∆(u) ≡ λ 1 (u) − λ 0 (u).(A24)
This leaves us with two coupled, complex differential equations. To make things more explicit, we now split the C variables into real amplitudes and phases such that
C 0 (t) = e iϕ0(t) A 0 (t), C 1 (t) = e iϕ1(t) A 1 (t).
These can be inserted into the differential equations. After some algebra, including separating out real and imaginary components, the differential equations reduce to the real equationsφ
= ∆ − A 2 0 − A 2 1 A 0 A 1 γu ∆ sin(ϕ),(A25)A 0 = − γu ∆ cos(ϕ)A 1 ,(A26)A 1 = γu ∆ cos(ϕ)A 0 ,(A27)
where ϕ(t) ≡ ϕ 0 (t) − ϕ 1 (t)
The A equations can be integrated to give
A 0 (t) = a cos t 0 dt ′ γu ∆ cos(ϕ) + ϑ , (A28) A 1 (t) = a sin t 0 dt ′ γu ∆ cos(ϕ) + ϑ . (A29)
These equations are similar to what was seen in the main text, and once again we are left with the conclusion that at the end of the evolution, we want
Θ[u(t)] ≡ t f 0 dt γu ∆ cos(ϕ) (A30)
to be close to a multiple of π. Though, we again have the caveat that having Θ[u(t)] equal to any multiple of π other than zero would violate the assumptions of nearadiabaticity.
Optimal Control
Our setup is to take a procedure that goes from time 0 to time t f moving from u(0) = u 1 at the beginning to u(t f ) = u 2 at the end. We want to ensure that the instantaneous eigenstate populations are maintained during that evolution, at least from the beginning to the end (but not necessarily in the middle), so we want to minimize
J = |C * 0 (t f )C 0 (t f ) − C * 0 (0)C 0 (0)| (A31) + |C * 1 (t f )C 1 (t f ) − C * 1 (0)C 1 (0)|.
The actual form of whether we are looking at the L 1 or L 2 norm of the difference between the probabilities is largely irrelevant, and another choice could be made with little consequence. We have also written out the probabilities explicitly as |C| 2 = C * C which will be helpful shortly. Now, we will treat this as an optimal control problem, seeking to find the conditions on u(t) such that J is minimized. In order to enforce Eqs. A21 & A22, we introduce Lagrange Multipliers D 0 (t) and D 1 (t) so that
J = |C * 0 (t f )C 0 (t f ) − C * 0 (0)C 0 (0)| (A32) + |C * 1 (t f )C 1 (t f ) − C * 1 (0)C 1 (0)| + t f 0 dt D 0 (t) Ċ 0 (t) + γ(u(t))u(t) ∆(u(t)) C 1 (t) + D 1 (t) Ċ 1 (t) − γ(u(t))u(t) ∆(u(t)) C 0 (t) + i∆(u(t))C 1 (t) + c.c.
where the final c.c. indicates that we need to complex conjugates of the third and forth lines, just to treat the variables and their complex conjugates equally (remember that u(t) is purely real). Now, we just perform a Calculus of Variations analysis of this using C 0 (t), C 1 (t), C * 0 (t), C * 1 (t), and u(t) as the variational parameters. In doing this procedure, it is important to remember that the C variables are fixed at t = 0 but not at t f and that u(t) is fixed at both end points.
Note that under a full optimal control theory analysis, such as [15], there would be restrictions on u(t) such as u(t) ∈ [0, 1]. In this setting, we will ignore this restriction for ease of analysis, and this ignoring is justified by the fact that we are interested specifically at how this system behaves in an annealing region. We are using this analysis explicitly to look at the annealing rather than bangbang portions of the control function, and any of our results here should be taken explicitly within that context. Also note that any restrictions on the C variables is already taken care of by the fact that Eqs. A21 & A22 are being enforced by the Lagrange multipliers. These equations came from the Schrödinger equation, so the Cs will obey all necessary properties of probability amplitudes.
The resulting end point equations yield the boundary conditions for the D variables
D 0 (t f ) = −sgn(|C 0 (t f )| 2 − |C 0 (0)| 2 )C * 0 (t s ), D 1 (t f ) = −sgn(|C 1 (t f )| 2 − |C 1 (0)| 2 )C * 1 (t s ).
Any changes to using the L 1 or L 2 norm originally would have shown up here and would have just resulted in slightly different boundary conditions. The variational procedure for the D Lagrange multipliers just results in Eqs. A21 & A22 again as expected, and the variational procedure for the C variables results inḊ
0 (t) = − γ(u(t))u(t) ∆(u(t)) D 1 (t),(A33)D 1 (t) = γ(u(t))u(t) ∆(u(t)) D 0 (t) − i∆(u(t))D 1 (t).(A34)
These are essentially following their own Schrödinger evolution. Also note that based on the boundary conditions for the Ds, we have D 1 (t) roughly the same size as C 1 (t) and D 0 (t) roughly the same size as C 0 (t). Hence we can use the same hierarchy of C 0 ≫ C 1 with these new variables. The last equation, resulting from the variations of u(t) is the one that is actually important here. Assuming the gap is nonzero, the resulting condition can be written as (suppressing functional dependencies for space reasons)
γ D 0Ċ1 +Ḋ 0 C 1 − C 0Ḋ1 −Ċ 0 D 1 = iC 1 D 1 ∆∆ ′ ,(A35)
where ∆ ′ = ∂ ∆(u(t) ∂ u(t) . The natural next step is to use Eqs. A21, A22, A33, & A34 to eliminate the time derivatives of the C and D variables:
(C 1 (t)D 0 (t) + D 0 (t)C 1 (t)) γ(u(t)) (A36) = − C 1 (t)D 1 (t)∆ ′ (u(t)),
This gives us the full set of optimal control equations that are necessary for an optimal procedure. Unfortunately, this formalism does not lend itself to the perturbative analysis discussed in the previous sections. These results are presented for completeness.
Appendix B: Trotterization Error
In this appendix we examine product formula errors and how they interact with the oscillations observed in the optimal curve.
In Sections B 1 and B 2 we derive directly how these oscillations influence the product formula. Section B 1 focuses on the standard product formula error formulated in terms of operators while Section B 2 follows the arguments of Ref. [40] and examines the product formula error for an adiabatic evolution. In section B 1 we derive Eq. (33) from the main text.
The last two sections provide additional background for Section B 1 with Section B 3 rederiving the basic known formulas necessary to bound the Product Formula errors. Section B 4 looks at the robustness of our oscillatory enhancement to perturbations of the product formula parameters.
Standard Operator Error Scaling
In this section, our goal will be to find the bound on the matrix norm error between unitaries given in Eqs. 30 & 31 under the assumption that the evolution is governed by the oscillatory function given in Eq. 28 This sequence of arguments will initially follow the appendix of [23]. For another good reference on this, try [24].
Finding the error between these two unitaries is fairly straightforward and is laid out well in Ref. [23]. We rederive this result in Appendix B 3. To cite the result
||Û (t f , 0) −Û P F (t f , 0)|| (B1) ≤ p−1 k=0 (k+1)∆t k∆t ds s k∆t dr Ĥ 0 (r),Ĥ 1 (s) .
The matrix norm used in this proof was the standard operator norm. Also, notably, this result does not rely on perturbative methods like the Baker-Campbell-Hausdorff equation or the Magnus expansion. Now, it is fairly straightforward to specify down to the form we are using in which case
||Û (t f , 0) −Û P F (t f , 0)|| (B2) ≤ B ,Ĉ p−1 k=0
(k+1)∆t k∆t ds s k∆t dr u(r)(1 − u(s)).
Next, the form in Eq. (B2) is a little unruly to work with. It actually is much easier to go over to Fourier space where
u(t) = ∞ −∞ dξũ(ξ)e 2πitξ .(B3)
Of course if we take the Fourier transform of Eq. (28), we would get
u(ξ) =ũ 0 (ξ) + c 0 2i e iφ δ ξ + 1 τ − e −iφ δ ξ − 1 τ .
(B4) Putting this Fourier transformed version in allows us to easily do the integrals over s and r, resulting in
||Û (t f , 0) −Û P F (t f , 0)|| (B5) ≤ B ,Ĉ p−1 k=0 ∞ −∞ dξ ∞ −∞ dηũ(ξ) (δ(η) −ũ(η)) × e 2iπ∆tη 1 − e 2iπ∆tξ η − 1 − e 2iπ∆tη ξ e 2iπ∆tk(η+ξ) 4π 2 ηξ(η + ξ) .
Notice that all the k dependence is in the last line, so we can carry out the k sum fully to get
||Û (t f , 0) −Û P F (t f , 0)|| (B6) ≤ B ,Ĉ ∞ −∞ dξ ∞ −∞ dηũ(ξ) (δ(η) −ũ(η)) × η −1 + e 2iπ∆tξ + ξ 1 − e 2iπ∆tη 1 − e 2iπ∆tp(η+ξ) 4π 2 ηξ(η + ξ) (e 2iπ∆tξ − e −2iπ∆tη )
.
From this point, putting in Eq. (B4) and simplifying down is quite possible; although, the fully general expression is a bit messy and not horribly informative. One possible simplification that is quite informative is the case where τ → ∆t in which case manipulation can simplify all of this nicely down to
||Û (t f , 0) −Û P F (t f , 0)|| (B7) ≤ B ,Ĉ ∞ −∞ dξ ∞ −∞ dηũ 0 (ξ) (δ(η) −ũ 0 (η)) × η −1 + e 2iπ∆tξ + ξ 1 − e 2iπ∆tη 1 − e 2iπ∆tp(η+ξ) 4π 2 ηξ(η + ξ) (e 2iπ∆tξ − e −2iπ∆tη ) − B ,Ĉ c 0 ∆t 2 p cos(φ) 2π .
If we undo the Fourier transform, this further reduces to
||Û (t f , 0) −Û P F (t f , 0)|| (B8) ≤ B ,Ĉ p−1 k=0 (k+1)∆t k∆t ds s k∆t dr u 0 (r)(1 − u 0 (s)) − B ,Ĉ c 0 ∆t 2 p cos(φ) 2π .
In other words, the sine function we added onto the control function essentially becomes decoupled from the rest of the error in the case that its period is the same as the Trotter slice size. Furthermore, the first line of the error bound will always be positive (remember that u 0 ∈ [0, 1]), but the second line can be negative, effectively reducing the error in the Trotterization. As promised, choosing φ = 0 results in the maximum improvement. The improvement in the error is proportional to ∆t 2 p which is coincidentally the same rough scaling as the term above, so this term will actually be competitive and could contribute greatly to the error bound. To see this more precisely, note that the first term in the bound can be upper bounded quite easily by 1 2 ∆t 2 p so that
||Û (t f , 0) −Û P F (t f , 0)|| (B9) ≤ B ,Ĉ ∆t 2 p 2 1 − c 0 π cos(φ) .
Note that c 0 ≤ 0.5 at the very worst to ensure that u(t) ∈ [0, 1], so it is not possible for this bound to be below zero.
In Appendix B 4 we explore the robustness of this effect to perturbations.
Adiabatic Trotter Error
In this subsection our goal will be to bound the Trotter error by looking at the error on the ground state fidelity directly. It should be noted that our analysis indicates that the underlying annealing curve adiabatically transfers not just the ground state but also higher excited states, with this reducing down to just ground state adiabaticity in the limit of t f → ∞. The results in this section focus on just the ground state, but similar results can be derived for any excited states, and those results can be simultaneously applicable.
The methods in this section closely follow the results of Yi and Crosson [40] who themselves draw inspiration from [6] and [41]. Specifically, this result can be thought of as a modification of their Proposition 1 (proven in their Appendix F) to the setting where the underlying annealing curve has an oscillatory structure. In practice, this modification is exactly the same as the modification to the usual Trotter operator error formula, meaning we can recover the oscillatory enhancement and still have the improved scaling analysis of Yi and Crosson.
As a reminder, the control function is
u(t) = u 0 (t/t f ) + c(t, t f ),(B10)
where u 0 (s) is a smooth monotonically decreasing function, and
c(t, t f ) = −c 0 (t f ) sin 2π τ t + φ .(B11)
We discretize our adiabatic evolution over time t f into M steps with "short" timestep ∆t = t f /M :
U 1 (t + ∆t, t) = exp −i t+∆t t dt ′ (u(t ′ )B + (1 − u(t ′ ))Ĉ) . (B12)
We want to evaluate the integral
t+∆t t dt ′ u(t ′ ) = t+∆t t dt ′ u 0 ( t t f ) + c(t, t f ) .
(B13) For our purposes, we will assume that u 0 (t/t f ) is approximately constant over this interval, which is valid for large t f . In other words, we assumed here that u ′ 0 ( t t f ) is approximately constant over this small interval. We will also now introduce s ≡ t t f as a normalized time. In this case
t+∆t t dt ′ u(t ′ ) (B14) = ∆tu 0 (s) + c 0 τ 2π (cos( 2π τ (t + ∆t) + φ) − cos( 2π τ t + φ)).
We can use trig identities to reduce this further to
t+∆t t dt ′ u(t ′ ) = ∆tu 0 (s) (B15) − c 0 τ π sin( π∆t τ ) sin( 2π τ (t + ∆t 2 ) + φ).
We will now define σ ≡ τ /t f and ∆s ≡ ∆t/t f so that t+∆t t dt ′ u(t ′ ) = ∆tU (s).
where for convenience, we have defined U (s) ≡ u 0 (s)−c 0 σ ∆sπ sin π∆s σ sin 2π σ (s + ∆s 2 ) + φ .
(B17) In the limit of ∆s → 0, this just reduces to U (s) → u(s).
To recover the original results of [40], take u 0 = 1 − s for a linear ramp and c 0 = 0 for no oscillations. This is part of the discretization error that we will later assume is smaller than the adiabatic error.
It is also important to note that c 0 ≪ u ′ 0 (s) (specifically elsewhere we found that c 0 ∈ O(u 2 0 )). This means that the sinusoids can be counted as a correction to the u 0 (s) terms rather than their own term. This will be useful when bounding quantities since we can then treat these two as a whole rather than seperate quantities to bound Each such discrete unitary is then Trotterized to first order:Û ′ 1 (t + ∆t, t) = exp −i∆tU (s)B (B18)
× exp −i∆t(1 − U (s))Ĉ , we define the effective Hamiltonian for this Trotterized evolution bỹ H(t) = i log Û ′ 1 (t + ∆t, t) /∆t. (B19)
In the limit of the discretization step size ∆t → 0, there is a continuous Hamiltonian defined by this. This effective Hamiltonian has the nice property thatH(0) =B and H(t f ) =Ĉ, so evolution underH(t) for slow t f can be described as an adiabatic process. The optimal curves approach an adiabatic procedure with vanishing initial and final bangs in the large t f limit, so this is appropriate in our setting for large t f (corresponding to large p for QAOA).
The core idea of this method then is to bound the error on the evolution, not using operator errors but using the adiabatic theorem directly. This will result in tighter scaling in terms of the number of Trotter or QAOA slices p but will introduce scaling with the p-independent spectral gap ofH(t). This technique will use the adiabatic theorem of Jansen, Ruskai, and Seiler [6].
We split the effective Hamiltonian into two parts such that e −i∆tH(s) = e −i∆tU(s)B e i∆tU(s)Ĉ e −i∆tĈ (B20)
≡ e −i∆tĜ(U(s)) e −i∆tĈ .
To reiterate our ultimate goal, we would want to show that there is an enhancement when ∆t = τ . As we will see, this goal is not consistent with the assumptions of this method, which we will discuss later in this section.
In order to utilize the Adiabatic condition bounds of [6], it is necessary to compute the matrix norms of derivatives ofH(s) with respect to s. Using Magnus expansion techniques, [40] Here the F functions are defined to be
F 0 (x) = ∞ j=0 x j = 1 1 − x ,(B24)F 1 (x) = ∞ j=1 x j−1 j = − ln(1 − x)/x,(B25)F 2 (x) = ∞ j=2 x j j 2 = − x 0 dx ′ ln(1 − x ′ ).(B26)
We present both norm derivatives here, but since our results only effect the constant prefactors and not overall scaling, it will be enough to keep track of || d d sH || because our results do not alter the dominant term in Ref. [40] which depends on || d d sH ||. The next step in this process is to bound the norms of the derivatives ofĜ. This is possible by looking at The d U d s is the only new portion of our results compared to Ref. [40], and to recover their results exactly, we would need to set d U d s → 1. In our setting, this derivative is just d U d s = u ′ 0 (s) − 2c 0 ∆s cos 2π σ (s + ∆s 2 ) + φ sin π∆s σ .
(B29)
Again in the limit of small ∆s, this reduces to just
d U d s → d u d s = u ′ 0 (s) − 2πc 0 σ cos 2π σ s + φ .(B30)
A nice feature of Eq. B27 is that it has an exact solution in the form of the Magnus Expansion. The terms in the Magnus expansion can then be bounded as in Ref. [40], and we follow a similar bounding but now keeping track of d U d s . For instance, we can work out that the first two derivatives ofĜ can have their norms bounded by
||Ĝ|| ≤ U (s)D − + 1 2∆t F 2 (2∆tD − U (s)) ,(B31)
||Ĝ ′ || ≤ U ′ (s)D − + ∆tD 1 U (s)U ′ (s)F 1 (2∆tD − U (s)) .
Here we define D − ≡ ||B −Ĉ||, D 0 ≡ ||Ĉ||, and D 1 ≡ || B ,Ĉ ||. Now, we can finally get back to the bound on ||H ′ || from Eq. B22.
||H ′ || ≤ ||Ĝ ′ || F 1 (2∆t||Ĉ|| + 2∆t||Ĝ||) (B33) ≤ [U ′ (s)D − + ∆tD 1 U (s)U ′ (s)F 1 (2∆tD − U (s))] × F 1 (2∆tD 0 + 2∆tU (s)D − + F 2 (2∆tD − U (s))) .
These functions can be bounded if their arguments are x < 1/2
F 1 (x) ≤ 1 + x,(B34)F 2 (x) ≤ x 2 2 (1 + x).(B35)
This allows us to bound We make the same assumption as Yi and Crosson that D 0 , D − ∈ O(n), D 1 ∈ O(n 2 ). The other essential assumption here is that ∆t ∈ O(n −1 ) in order to make the arguments of the F functions small. Unfortunately, this assumption is extremely hard to justify in our QAOA setting since we observe that ∆t ≈ τ . As discussed elsewhere in this paper, τ is inversely proportional to the spectral gap of the problem, and the spectral gap often scales as an inverse polynomial or exponential in the number of qubits during phase transitions. At the moment, we will still assume that ∆t ∈ O(n −1 ), but this is the point in the argument where this method breaks down in our setting. For now, we continue the argument under the asusmption that ∆t ∈ O(n −1 ) in order to complete the analysis. With these assumptions, the second term is proportional to O(1) and so we drop it. On the other hand, these assumptions mean that the first term is O(n).
With these assumptions, 2∆tD − U (s) ≤ 1/2 and so
||H ′ || ≤ U ′ (s)D − + 3 2 ∆tD 1 U (s)U ′ (s) .(B37)
Finally, ǫ tro ≤ǫ ′ adb + ǫ ′ first is just the base error due to u 0 alone without the superposed oscillations. This section will not consider this portion of the error because we are only concerned with the enhancement due to the oscillations. The second portion is the cross terms between u 0 (t) and the oscillations, which we write as E CT . In the main portion of the paper, we found that E CT = 0. The last portion, due to just the behavior of the oscillatory portions, is E Os which is what is responsible for the enhancement we see in the actual results.
Luckily, for both the single oscillation and the aggregate case, simple algebra shows that the cross term error, E CT , cancel out at the first order in ǫ as well. Therefore, those terms are consistent with being in an extremum. For the single oscillation case, the error due to the ǫ shift is (setting φ = 0)
E (1) CT = B ,Ĉ ∞ −∞ dλ ic k ǫ 2 1 − e 2iπλτ k ũ 0 (λ) 2λτ k + O(ǫ 3 ).
(B62) Here c k refers to the amplitude of the oscillations during the kth time step, and τ k is the period of the oscillations during the kth time step.
When we look at the error in the cross terms for the entire procedure over multiple oscillations, each with amplitude c 0 and period τ , it becomes
E CT = B ,Ĉ ∞ −∞ dλ ic 0 ǫ 2ũ 0 (t) 2λτ (1 − e 2πiλτ )(B63)
× (e 2iπλτ + (2p − 1)e 2iπλ(p+1)τ − (2p + 1)e 2iπλpτ + 1)
+ O(ǫ 3 ).
When we consider the case of a single oscillation, the correction from the oscillation, E Os , has no term that is linear in ǫ, meaning that ∆t k = τ k is an exact minimum. To be exact this term comes out to be (again setting φ = 0)
E (1) Os ≤ B ,Ĉ − c k τ 2 k 2π − πc k ǫ 3 3τ k + O ǫ 4 . (B64)
This result is mildly problematic because it means that this enhancement is not a minimum here but a higher order critical point. This problem gets fixed when we consider more oscillations and is a result just of the fact that we are considering a single oscillation here.
To understand how to fix this, we go instead to the aggregate case in the main text where all the oscillations are considered together (under the approximation that the frequency of the oscillations is constant). There, the expansion results in
E Os = B ,Ĉ − c 0 pτ 2 2π (B65) + 1 6 πc 0 p 2p 2 − 3p + 1 ǫ 2 + O(ǫ 3 ).
Note that this is all before coherence or any additional optimization arguments have been made.
FIG. 1 :
1This plot shows the QAOA variational parameters for a single problem instance at several different values of p.
FIG. 2 :
2This plot numerically demonstrates some of the key points of this paper, showing a p = 14 QAOA protocol and the optimal protocol that takes the same length of time. These numerics correspond to the same problem instance shown in
ure 5 shows QAOA probabilities only after a full layer of bangs; the intermediate probabilities deviate even more from an adiabatic transfer.
FIG. 6 :
6This plot shows the upper bound on the Trotter error in the unitaries from Eq. (33) for a fixed t f , changing p (and hence ∆t = t f p )
3 .
3Create an ansatz for the bang-anneal-bang curve that has a u = 0 bang at the beginning, the annealing curve defined by v i−1 p−1
E
bab ← Evolution under v bab ⊲ Energy of Protocol 9:β,γ, ω, φ ← Update Based on Optimization
FIG. 7 :
7Various versions of our algorithm, with "BAB Ansatz" being the most advanced in terms of number of parameters.
U(s)B (B −Ĉ)e −i∆t U(s)Ĉ . (B28)
||H ′ || ≤ [U ′ (s)D − + ∆tD 1 U (s)U ′ (s) (1 + 2∆tD − U (s))] 2∆tD − U (s)) . (B36)
TABLE I :
IEnergies related to the procedures shown inbut
is run for a time corresponding to p = 6 QAOA.
Model Energy
Linear -5.987
Basic -6.257
Sine -6.522
QAOA -6.578
BAB -6.636
GD
-6.705
Ground -7.214
bounds the norms of these derivatives by||
d
d sH
|| ≤||
d
d sĜ
|| F 1 (2∆t||Ĉ|| + 2∆t||Ĝ||),
(B22)
||
d 2
d s 2H || ≤||
d 2
d s 2Ĝ || F 1 (2∆t||Ĉ|| + 2∆t||Ĝ||)
(B23)
+ 2∆t||
d
d sĜ
|| 2 F 0 (2∆t||Ĉ|| + 2∆t||Ĝ||).
t k t k−1 B(s)ds exp − i t k t k−1 A(s)ds + i exp − i t k t k−1 B(s)ds A(t k ) exp − i t k t k−1 A(s)ds .
AcknowledgmentsWe would like to thank Chris Baldwin and Minh Tran for helpful discussions.The research of L.T.B. was partially supported by a National Institute of Standards and Technology (NIST) National Research Council (NRC) Research Postdoctoral Associateship Award in the Information Technology Lab (ITL). The research was supported by the U.S. Department of Energy Award No. DE-SC0019449 for work analyzing the structure of QAOA curves, AFOSR for work analyzing and exploring the perturbative near-adiabatic limit, and ARO MURI for work associated with the asymptotic behavior and continuous limit of QAOA.This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under the Quantum Computing Application Teams program. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Admin-where G(t f ,H) = 1 t f H′ (0) λ (0) 2 + H′ (1) λ (1) 2 (B45)which comes from Ref.[6]and encapsulates the adiabatic condition. In the above Eq. (B39) we usedǫ ′ adb is the error from a finite time implementation of an adiabatic process, ǫ adb , plus discretization error, ǫ disc . ǫ ′ tot is the error from a finite time implementation of an adiabatic process plus discretization error plus Trotter error. ǫ tot is the error in doing the discrete and Trotterized evolution adiabatically.ǫ disc is the error in discretizing and Trotterizing the adiabatic discretized and Trotterized process.The dominant term in Eq. (B44) is the second term, and for our purposes, we are interested in how the U (s) 2 U ′ (s) 2 portion effects this as opposed to just taking U (s) → 1 − s. We specifically want to look at the maximum value of this expression as a function of s. For the original result where U (s) = 1 − s, the maximum value of U (s) 2 U ′ (s) 2 is just one at s = 0.In this setting, the enhancement from meshing up with the period of oscillations is ironically not an enhancement so much as a lack of detriment from the oscillations. It is easy to see that if ∆s = σ, then U (s) = u 0 (s) and therefore U (s) 2 U ′ (s) 2 = u 0 (s) 2 u ′ 0 (s) 2 , so matching up with the oscillations just makes us to recover the error that would have existed without the oscillations. On the other hand, not matching up with the oscillations can lead to severe detriments to the error term.If we choose φ = − π∆s σ such that the argument of the oscillations is zero at s = 0 and then choose the linear ramp u 0 (s) = 1 − s, we can look at the value of this function at s = 0. For the linear ramp without the oscillation, s = 0 is the maximum value of this function, so it will be demonstrative:For any ∆s < σ, the result in a quantity > 1 and lead to a worsening of the bound. Even for ∆s > σ, there are regions of the curve other than s = 0 that are detrimentally effected by the oscillations. For a function other than the linear ramp, the maximum of this quantity could occur somewhere else in s. The only way to ensure that the oscillations will not be deleterious to the Trotterization is to have ∆s = σ.Unfortunately, as stated already, having ∆s = σ violates the assumption that ∆t ∈ O(n −1 ). It is possible that QAOA angles will start scaling with n differently as we scale up these algorithms, but there is currently no numeric or experimental evidence of this type of scaling.Trotterization Errorwhere we used the identityIntegrating the last equation and usingProduct Formula PerturbationsOne important question is whether ∆t k = τ k is a true minimum or just an enhancement. In other words, is it beneficial to wiggle slightly away from this. Furthermore, if it is a true minimum, how much can we wiggle away before messing up the fact that we are in a minimum well.The results in Section V look only at the case where the product formula step size ∆t k matches up with the periods of the oscillations τ k . What happens if we consider small perturbations such thatfor a small ǫ.For the purpose of this section, we break the error, ||Û (0, ∆t k ) −Û P F (0, ∆t k )||, up into three portions. The
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If the length of the anneal, TA is treated as a variational. If the length of the anneal, TA is treated as a variational
| []
|
[
"Industrial machine tool component surface defect dataset",
"Industrial machine tool component surface defect dataset"
]
| [
"Tobias Schlagenhauf [email protected]. \nKarlsruhe Institute of Technology\nGermany\n",
"Magnus Landwehr \nKarlsruhe Institute of Technology\nGermany\n"
]
| [
"Karlsruhe Institute of Technology\nGermany",
"Karlsruhe Institute of Technology\nGermany"
]
| [
"Data in Brief"
]
| a b s t r a c tUsing machine learning (ML) techniques in general and deep learning techniques in specific needs a certain amount of data often not available in large quantities in technical domains. The manual inspection of machine tool components and the manual end-of-line check of products are labor-intensive tasks in industrial applications that companies often want to automate. To automate classification processes and develop reliable and robust machine learning-based classification and wear prognostics models, one needs real-world datasets to train and test the models. The presented dataset consists of images of defects on ball screw drive spindles showing the progression of the defects on the spindle surface. The dataset is analysed via an initial object detection model available under: https://github.com/2Obe?tab= repositories . The reuse potential of the dataset lays in the development of failure detection and failure forecasting models for the purpose of condition monitoring and predictive maintenance. The dataset is available under https://doi.org/ 10.5445/IR/10 0 0129520 . | 10.1016/j.dib.2021.107643 | null | 232,335,366 | 2103.13003 | a2a9818adb515fe00337ff94cdcd129d88603aa6 |
Industrial machine tool component surface defect dataset
2021
Tobias Schlagenhauf [email protected].
Karlsruhe Institute of Technology
Germany
Magnus Landwehr
Karlsruhe Institute of Technology
Germany
Industrial machine tool component surface defect dataset
Data in Brief
391076432021Article history: Received 20 May 2021 Revised 19 November 2021 Accepted 23 November 2021Contents lists available at ScienceDirect Data in Brief Data Article a r t i c l e i n f o * Corresponding author.Condition monitoring Deep learning Machine learning Object detection Semantic segmentation Instance segmentation Classification Dataset
a b s t r a c tUsing machine learning (ML) techniques in general and deep learning techniques in specific needs a certain amount of data often not available in large quantities in technical domains. The manual inspection of machine tool components and the manual end-of-line check of products are labor-intensive tasks in industrial applications that companies often want to automate. To automate classification processes and develop reliable and robust machine learning-based classification and wear prognostics models, one needs real-world datasets to train and test the models. The presented dataset consists of images of defects on ball screw drive spindles showing the progression of the defects on the spindle surface. The dataset is analysed via an initial object detection model available under: https://github.com/2Obe?tab= repositories . The reuse potential of the dataset lays in the development of failure detection and failure forecasting models for the purpose of condition monitoring and predictive maintenance. The dataset is available under https://doi.org/ 10.5445/IR/10 0 0129520 .
Specifications
Table Subject
Manufacturing Engineering Specific subject area
The subject area is condition monitoring and lays in the intersection between the fields of Machine Learning (Computer Science) and Manufacturing Engineering/Mechanical Engineering. The subject area is of special importance for engineers who want to build intelligent and autonomous condition monitoring systems for the supervision of machine tool components. Type of data Image How data were acquired The data were acquired by a classical camera system mounted close to the system of interest during the operation of the system. Since the camera is mounted during operation different failure states are recorded which cannot be found in the literature so far. See section Experimental Design, Materials and Methods Data format Raw Analyzed Parameters for data collection For data collection a machine tool element like a ball screw drive like it is found in machine tools during industrial operation is considered.
Description of data collection
The data is collected by mounting the camera system onto the nut of the ball screw drive such that the camera looks vertically on the surface of the spindle.
Value of the Data
• For industrial companies it is very important to keep the availability of machines as high as possible which makes it necessary to supervise the condition of machine tool components. The automation of this process saves cost and is necessary to build autonomous machines. Though, building autonomous systems requires large amounts of data showing the effects of interest. This is important because intelligent systems based on machine learning techniques need sufficient data to learn from. In the context of the automatic detection of surface defects, Cum grano salis the machine learning mode learns how images with defect and images without defect are looking. If there is not sufficient data then the model can't learn the specific characteristics. Since having data of defective components implies that a company has defective components (which is costly and should be prevented), this data is often rare in technical domains which in turn contradicts the need of large dataset for performant models. • Especially companies developing (intelligent) condition monitoring systems for machine tool components benefit from the data. Since the availability of machines is of high importance for most industries, the dataset addresses a large circle of users. • The dataset can be used by every company which wants to develop intelligent systems for failure detection and condition monitoring. The dataset can be used for transfer learning to enrich datasets from other technical domains supervising the condition of metallic surfaces. Examples could be the renewable energy sector e.g. to find defects on turbines or the railway sector e.g. to find defects on rails. • The novel dataset shows image data of worn ball screw drives in a timely context. • The dataset shows the progression of failures and delivers failures at different steps in time which is of large value for practitioners who want to detect failures as soon as possible.
• The dataset contains worn and not worn surfaces for classification. The images are annotated and the failures are provided with a segmentation mask indicating the size and location of the failures. (e) and without (f) pollution. Hence, the whole spectrum of conditions is covered. Figs. 2 and 3 show a larger subset of images with and without pitting. It is obvious that the correct classification of images needs a substantial amount of domain knowledge.
Data Description
Dataset for defect classification
Dataset for defect detection/segmentation
Besides the classification of images, the authors introduce a dataset for instance segmentation which addresses the research problem of image-based size extraction and stands out from the already available datasets for metal surface defect detection like NEU-DET [3] , GC10-DET [4] , or SD-saliency-900 saliency [5] with a more suitable representation of real-world problems due to containing a high-class imbalance and pixel-wise annotation masks. Furthermore, this dataset is ideally suited for application areas, namely models that are trained with little data and therefore need to have a high model efficiency.
Condition monitoring enabled by image-based size extraction to detect the current state of a machine tool element, according to [6] , can, for example, lead to the reduction of equipment failure cost, improved plant reliability, and optimized maintenance intervals towards a conditionbased maintenance strategy and is therefore obviously worthwhile considering. The automatic detection and evaluation of a failure is a critical step towards autonomous production machines.
The introduced dataset is not only valuable for condition-based surface damage detection models on BSDs but also through a size progress detection on image sequences for analysis of wear development over time. This provides the community with a useful dataset for the development and test of wear detection algorithms for all machine tool elements prone to wear which can be recorded by a camera. Three important features are worth noting in particular. The dataset contains tiny damages and hence is suited to develop models especially for the detection of small, respectively early defects. In addition to that, the dataset also includes pollution origin from soil which makes detection more difficult together with foreign materials originating from e.g. the production process. As a third feature, the dataset contains the development of the same failures over a period of time. This feature can be used to develop models for the forecasting of failure progressions. To the best of our knowledge, such dataset does not exist in the literature right now. In Fig. 4 , one exemplary course of an annotated size progress of the dataset is displayed.
As shown, the graph first remains for approx. 2/3 of the documented time interval at zero due to the fact that there is no surface damage. As soon as a pitting occurs, it will only continuously increase its size, in this figure represented by the pixel amount of the pitting in relation to the total pixels in an image. The drawn circles render the size of a single pitting shown in the image cutouts on the left to give an idea about the increasing pitting size. You can also see in the images increasing soiling of the surface and, therefore, there is an increasing difficulty to correctly annotate the pitting. This explains why the shown graph also contains decreasing parts, which is obviously not possible in the real application and opens the possibility to develop models able to cope with this situation. While classification requires that its data(-points) are assigned to discrete values, such as categories [7] , and detection can be used for localization of objects within images [8] , it is recommendable to combine both to detect and classify single objects in images to get as close as possible to the perfect description of an image. Since these dataset annotations can be used for classification as well as detection problems, it is attainable to detect the size of an object and further, with the given wear developments, forecast the pitting size of the future. Generally, computer vision classification and detection tasks can be divided into four types ( Fig. 5 ).
Instance segmentation (d) as for classification and detection is a pixel-wise object detection method useful for computer vision research tasks like extraction of shape and the exact size of surface damage. Known as one of the most fundamental and challenging tasks in the computer vision research area [9] , this dataset can also be used for semantic segmentation (a) as a pixelwise classification with no possibility to distinguish two or more adjacent objects from the same class, an image classification (b) for pitting recognition, and object detection (c) for single object detection.
While most of the related research datasets for damage detection on the metal surface are not annotated for pixel-wise object detection, the introduced dataset cannot only be used for instance segmentation but moreover for the analysis of developments of surface damage over time. The (a) NEU-DET [10] , shown in Fig. 6 , for instance, with its 1800 200 × 200 × 1 pixel images and six annotation classes (rolled-in scale, patches, crazing, pitted surface, inclusion, scratches) or the (c) GC10-DET [4] with its 3570 2048 × 10 0 0 × 1 big images and 10 annotation classes (cresent gap, welding line, water spots, silk spot, inclusion, oil spot, crease, punching, waist folding, rolled pit) can only be used for object detection problems.
Compared with the instance segmentation (d) SD-saliency-900 dataset [5] with its 900 200 × 200 × 1 samples, the introduced dataset contains more irrelevant surface information which is an important challenge to address since many real-world problems contain a high-class imbalance [11] .
The dataset contains 1104 channel-3 images with 394 image annotations for the surface damage type "pitting". The annotations made with the annotation tool labelme [12] are available in JSON format and hence convertible to VOC and COCO format. All images come from two BSD types.
The dataset is divided into two folders, data with all images as JPEG, labeled with all annotations, and saved_model with a baseline model. The authors also provide a python script to divide the data and labels into three different split types -"train_test_split", which splits images into the same train and test data-split the authors used for the baseline model, "wear_dev_split", which creates all 27 wear developments, and "type_split", which splits the data into the occurring BSD types. One of the two mentioned BSD types is represented with 69 images and 55 different image sizes. All images with this BSD type come either in a clean or soiled condition.
The other BSD type is shown on 325 images with two image sizes. Since all images of this type have been taken with continuous time, the degree of soiling is evolving.
Also, the dataset contains the above-mentioned 27 pitting development sequences. Fig. 7 shows the evolving pitting development with and without the shown annotations from one of the 27 pitting developments. For convenience, only every third image starting at the beginning of the pitting formation is displayed.
Experimental Design, Materials and Methods
Sensor system
The sensor system used for the creation of the image dataset is depicted in Fig. 8 . The system is mounted onto the nut of the BSD using a mounting adapter numbered with #3. The camera (#1) looks through a hole in the so-called diffusor (#4) onto the spindle. Since turning the spindle leads to a linear motion of the nut and the spindle is turning underneath, the camera gets to see all raceways of the spindle. Using this setup, the whole spindle can be photographed. #2 is a manufactured housing enclosing the spindle which is used to ensure uniform lighting conditions during the experiment. Additionally, the housing protects the camera from pollution. An important part of the system which is responsible for lightning of the images is the so-called diffusor which also implements the light sources. The light sources are two standards LED stripes mounted onto the surface where #4 is located. The diffusor itself is 3Dprinted and consists of a semitransparent plastic leading to diffuse light. Since the LEDs are not pointed onto the spindle but directly onto the housing, the light does not get directly onto the spindle but is reflected by the housing and then further made more diffuse bypassing the diffusor. During tests, this setup was found to be yielding the best results for our purpose. The used camera system is a standard Raspberry Pi V2 microcontroller camera which is a good tradeoff between resolution, costs, and necessary mounting space. The camera is set up to take images with a resolution of 2592 × 1944 pixels per image.
Test setup
The dataset is generated on a test bench located at the Institute of Production Science at the Karlsruhe Institute of Technology. The test bench is depicted together with the mounted camera systems in Fig. 9 .
The test bench is constructed such that a maximum of five spindles can be worn in parallel. The spindles are positioned like the five on a dice, with the middle spindle being the leading spindle connected to the motor. The other four spindles are operated by a chain drive connected to the central spindle, thus it is ensured that all spindles are operated in the same way. The spindles used are standard 32mm diameter spindles with no special treatment or prestress. Each spindle is preloaded with 70% of the C a given by the manufacturer, where 100% of the C a is the axial load at which the manufacturer ensures a safe operation of 10 6 revolutions. In this case, the C a is chosen with 12kN. With this setup, the camera automatically triggers a complete surface recording every four hours. Between each image, the spindle is turned by an additional 22.5 °, and an area of 150 × 150 pixels is cropped automatically from the large image.
Data analysis baseline
Regarding the introduced dataset, the authors also present a baseline model. The here used model architecture is a Mask R-CNN (regional Convolutional Network) [13] with an on the COCO dataset [14] pretrained Inception ResNet v2 [15] . The Mask R-CNN architecture is composed of two stages, a faster R-CNN with a deep convolutional network composed of Inception v4 and ResNet building blocks united in an Inception ResNet v2 architecture and an FCN (fully convolutional network). Here the authors used the standard implementation as used in [13] . For further implementation details please consider this source.
With the chosen architecture, the authors achieved a mIoU (mean intersection over union) baseline score of 0.316. It is noticeable that the model has difficulties predicting small pitting in general ( Figs. 10 and 11 ).
Examining the horizontal and vertical development of pitting and relating it to a binarized model prediction, a zero-one principle -where zero corresponds to "not detected", we can see that pitting detection becomes more reliable as development increases. In Fig. 10 , the circumstance just described can be readily understood. The relative horizontal spread of the pitting (width) is described on the x-axis and the relative vertical spread (height) is described on the y-axis. The binarization of the detection is represented by the coloring of the points. Fig. 11 visualizes the just mentioned circumstance on selected examples.
The pitting shown in image cutout (a) was due to its large horizontal and vertical spread detected. While the not detected pitting in cutouts (b), (d), and the detected pitting in (c) are relatively small. For convenience, the trained model will be provided. The code for the baseline detection model is available under: https://github.com/2Obe/BSData .
Ethics Statement
The authors read and follow the ethical duties of authors.
Declaration of Competing Interest
This work was supported by the German Research Foundation (DFG) under Grant FL 197/77-1.
CRediT Author Statement
Fig. 1 .
1Subset of the image data taken during the destruction test.
Fig. 2 .
2Subset of images without pitting.
Fig. 3 .
3Subset of images with pitting.
Fig. 4 .
4Annotated pitting size over time of a specific pitting development.
Fig. 5 .
5Different Image classification and Object detection types supported by the dataset.
Fig. 6 .
6Different datasets for metal surface damage.
Fig. 7 .
7Pitting process.
Fig. 8 .
8Sensor system used for image generation.
Fig. 9 .
9Test bench with mounted camera systems for image generation.
Fig. 10 .
10Relationship between pitting detection and its relative size.
Fig. 11 .
11Prediction examples from the author's model.
Tobias Schlagenhauf :
SchlagenhaufConceptualization, Methodology, Data curation, Software, Writingoriginal draft; Magnus Landwehr: Data curation, Software, Writing -original draft.
The authors further declare that they have no competing interests. The authors declare that they have no known competing financial interests or personal relationships which have or could be perceived to have influenced the work reported in this article.
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| A B S T R A C TReductions of the self-consistent mean field theory model of amphiphilic molecules in solvent leads to a singular family of functionalized Cahn-Hilliard (FCH) energies. We modify the energy, removing singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the "morphological complexity" observed in experiments. These benchmarks investigate the delicate balance between the rate of arrival of amphiphilic materials onto an interface and a least energy mechanism to accommodate the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces grow either by a regularized motion against curvature, pearling bifurcations, or curve-splitting directly into networks of interfaces. We evaluate a number of schemes that use second order backward differentiation formula (BDF2) type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a preconditioned steepest descent (PSD) nonlinear solver or upon linearly implicit time discretization based on the standard implicit-explicit (IMEX) and the scalar auxiliary variable (SAV) approaches. We add an exponential time differencing (ETD) scheme for comparison purposes. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper "preconditioning" to achieve robust performance that can enhance efficiency by several orders of magnitude. The nonlinear PSD scheme achieves the smallest global discretization error at fixed local truncation error, however the IMEX and SAV schemes are the most computationally efficient as measured by the number of Fast Fourier Transform (FFT) calls required to achieve a desired global error. Indeed the performance of the SAV scheme directly mirrors that of IMEX, modulo a factor of 1.4 in FFT calls for the auxiliary variable system. | null | [
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| 219,531,860 | 2006.04784 | 48ffc81e68d029ac4ae0f8adaf7ca6eda6e21c39 |
Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow
Andrew Christlieb
Department of Mathematics
Michigan State University
48824East LansingMIUSA
Department of Computational Mathematics, Science and Engineering
Michigan State University
48824East LansingMIUSA
Keith Promislow
Department of Mathematics
Michigan State University
48824East LansingMIUSA
Zengqiang Tan
Department of Computational Mathematics, Science and Engineering
Michigan State University
48824East LansingMIUSA
School of Mathematics and Statistics
Huazhong University of Science and Technology
Wuhan 430074the P.R. of China
Sulin Wang
Department of Mathematics
Michigan State University
48824East LansingMIUSA
Brian Wetton
Department of Mathematics
University of British Columbia
V6T 1Z2VancouverBCCanada
Steven Wise
Department of Mathematics
University of Tennessee
37996KnoxvilleTNUSA
Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow
Phase Field ModelBenchmark ComputationsAdaptive Time SteppingFunctionalized Cahn-Hilliard
A B S T R A C TReductions of the self-consistent mean field theory model of amphiphilic molecules in solvent leads to a singular family of functionalized Cahn-Hilliard (FCH) energies. We modify the energy, removing singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the "morphological complexity" observed in experiments. These benchmarks investigate the delicate balance between the rate of arrival of amphiphilic materials onto an interface and a least energy mechanism to accommodate the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces grow either by a regularized motion against curvature, pearling bifurcations, or curve-splitting directly into networks of interfaces. We evaluate a number of schemes that use second order backward differentiation formula (BDF2) type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a preconditioned steepest descent (PSD) nonlinear solver or upon linearly implicit time discretization based on the standard implicit-explicit (IMEX) and the scalar auxiliary variable (SAV) approaches. We add an exponential time differencing (ETD) scheme for comparison purposes. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper "preconditioning" to achieve robust performance that can enhance efficiency by several orders of magnitude. The nonlinear PSD scheme achieves the smallest global discretization error at fixed local truncation error, however the IMEX and SAV schemes are the most computationally efficient as measured by the number of Fast Fourier Transform (FFT) calls required to achieve a desired global error. Indeed the performance of the SAV scheme directly mirrors that of IMEX, modulo a factor of 1.4 in FFT calls for the auxiliary variable system.
Introduction
In this paper we present a series of physically motivated computational benchmark problems addressing the evolution of the functionalized Cahn-Hilliard (FCH) gradient flow possessing a family of equilibria with rich morphological structure separated by slightly different energies. The faithful resolution of final end states requires significant computational accuracy. There has been considerable recent attention to the development of energy stable computational schemes for gradient descent flows [16,18,25,35,36,43,44,45]. Gradient flows are defined by the dissipation of a free energy, and it is essential that numerical schemes preserve that property. Energy stable schemes have the desirable property that the energy, or a modified energy, decreases at every time-step irrespective of time-step size.
We argue that energy decay should be a consequence of accuracy, indeed in some situations energy decay without accuracy can mask poor performance by leading to plausible but incorrect computational outcomes. Accuracy should be balanced against computational cost, which motivates us to compare computational efficiency between schemes as measured by the minimal computational cost required to achieve a desired global discretization error.
Meaningful assessment of computational efficiency can be achieved from gradient flows with strong nonlinear interactions that generate selection mechanisms that choose between distinct outcomes with small energy differences.
For motivation, we emulate the "morphological complexity" experiments conducted by [29]. By strongly dispersing (stirring) amphiphilic diblock polymers in solvent, and then allowing the mixture to relax, they observe the formation of a wide variety of structures whose evolution and end-state depend sensitively upon the polymer chain and mixture properties. See Figure 1 and [1,2]. Reductions of the self-consistent mean field theory models of amphiphilic molecules in solvent leads to a singular family of FCH energies, [7]. We modify these energies, mollifying the singularities to produce a family of computationally tractable, but highly nonlinear, FCH gradient flows similar to those studied earlier, [14,15,21]. We present a series of benchmark problems that recover the onset of morphological complexity. These benchmarks reveal a delicate balance between the rate of arrival of amphiphilic materials onto an interface and the gradient flow's selection of a least energy mechanism to redistribute the amphiphilic mass after its absorption. This rate-based selection mechanism yields a trichotomy of responses in which two-dimensional interfaces either grow by a regularized motion against curvature, under-go pearling bifurcations, or directly curve-split into networks of interfaces. We present four numerical schemes, each combining at least second order temporal discretization and pseudo-spectral spatial discretization. The FCH energy is computationally stiff due to the strength of its nonlinear terms. Each of the second order methods considered here balances implicit and explicit terms. Their efficiency is sensitive to the choice of the implicit terms, with improvements of several orders of magnitude possible when the methods are well balanced. These methods include an implicit-explicit (BDF2-IMEX) method, a second order exponential time differencing Runge-Kutta method (ETDRK2), and a scalar auxiliary variable approach (BDF2-SAV ). The latter scheme features provably unconditional modified energy stability properties. All of these schemes are linear in their implicit stage. We compare these with a fully implicit, second order, backward differentiation scheme based upon a preconditioned steepest descent with approximate line search (BDF2-PSD) for the nonlinear solve. We will drop the 'BDF2' and 'RK2' components of the acronyms in the sequel for the sake of brevity.
The FCH gradient flows possess many distinct, emergent timescales that preclude a fixed time-stepping approach.
For each scheme a specified target local truncation error is used to generate an adaptive time-stepping procedure. The first set of benchmarks, the sub-critical, critical, and super-critical, use a relatively well conditioned form of the FCH energy, and vary the initial data to trigger the bifurcations of morphological complexity that engender many possible outcomes in the super-critical benchmark. A proper resolution of the time evolution requires considerable accuracy.
The second set of benchmarks enhance the stiffness of the FCH energy by increasing the second derivative of the well at the background state, mimicking the singular nature of the FCH energy as reduced form the self-consistent mean field theory. This adds a small foot to the left minima of the well, see Figure 2, hence these benchmarks are called Foot 1 and Foot 2. The stiffness increases the ratio of the absorption time-scale to the redistribution time-scale thereby inducing morphological complexity.
Each of the second order schemes we consider requires an appropriate choice of implicit terms or preconditioner.
This choice is typically based upon the linearization of a residual about a spatially constant equilibrium solution.
The linearly implicit IMEX and SAV accommodate the increase in stiffness for the Foot 1 and Foot 2 benchmarks without significant adjustment. The nonlinear solve in the PSD scheme requires optimization of internal parameters, in particular an error tolerance associated to the iterative nonlinear solver, to converge. Moreover the efficiency of the PSD scheme decrease in comparison to the two linear implicit schemes with increasing numerical stiffness. Other preconditioning schemes, for example based upon non-constant coefficient linear terms, could improve the efficiency of the PSD scheme, however this is not considered here. The ETD approach is not efficient at handling the nonlinear stiffness in the super-critical benchmark and is not pursued for the Foot 1 and Foot 2 benchmarks.
We conduct grid refinement studies to verify that each benchmark has an adequate spatial resolution and develop highly accurate solutions for each benchmark by an extensive computation with a very small local truncation error.
Once spatially resolved, all four schemes yield concordant results for sufficiently small specified local truncation error.
We adjust the local truncation error restriction and use short runs to tune performance parameters in each scheme for each benchmark, and record the accuracy and cost of each optimized scheme. At given local truncation error we find that the PSD approach is generically the most accurate with IMEX and SAV generally the least accurate, as measured by global error at the final time. However, at fixed local truncation error the IMEX and SAV schemes require less computational effort than the PSD and ETD, with the IMEX and SAV schemes performing almost identically, modulo a fixed factor in extra computational effort required by SAV due to the extra system for the auxiliary variable. For these benchmarks a global L 2 relative discretization error of 2.5 × 10 −3 is found to be a harbinger of accuracy, and within this constraint we view the local truncation error as an internal parameter to be adapted for each scheme to optimize global performance. For the sub-critical, critical, and super-critical benchmarks, all schemes except ETD achieved this global accuracy with comparable efficiency although at quite different values of the local truncation error. The ETD scheme is not competitive, likely due to the strength of the nonlinearity in the FCH system. As presented in Figure 15, achieving this accuracy for the super-critical benchmark requires 1.5 × 10 5 , 2 × 10 5 , and 2.1 × 10 5 FFT calls for IMEX , PSD and SAV respectively, while ETD requires 4.6 × 10 6 FFT calls. As the global error target is further tightened, the PSD scheme requires increased computational effort, first increasing rapidly and then saturating. Conversely the computational effort of the IMEX and SAV schemes increases linearly with global discretization error. For the more strongly nonlinear Foot 1 and Foot 2 benchmarks the efficiency of the linear-implicit schemes continues its linear relationship to global discretization error. As depicted in Figure 16, for the stronger nonlinearity the efficiency of PSD deteriorated in comparison to the linear-implicit methods.
The SAV scheme is specifically designed to be energy stable with respect to an associated modified energy.
This property either assumes fixed time-stepping, which is impractical for the FCH gradient flows when accuracy is paramount, or an adaptive time stepping based upon modifications by factors of two and nesting. This latter strategy is implemented for the super-critical benchmark within the BDF2-SAV scheme. This is found to provide no benefit for accuracy while increasing computational cost by a factor of two to three. We also implement the second order Crank-Nicolson approach in combination with the SAV strategy but find that it is not computationally efficient. In all cases all convergent schemes preserve the energy decay property of the gradient flow.
Remark 1. The work [45] directly compares the PSD and SAV methods described herein, but in the context of uniform, fixed time step setting. Based upon their experience with FCH-type simulations, the authors state that "ultimately adaptive time stepping algorithms should be compared." The present study seeks to fill this gap, using time step adaptivity to make quantitative comparison of accuracy against efficiency for a variety of numerical schemes.
Moreover, the family of regularized FCH models presented here allow for interpolation between the smooth versions of the FCH considered in earlier analytical and numerical studies and the singular versions arising as reductions from self-consistent mean field analysis whose inherent numerical stiffness makes them more challenging than the models considered in [45].
This paper is organized as follows. In section 2, we briefly sketch the derivation of a singular FCH model from a random phase approximation of self-consistent mean field theory, outline the regularization of the singular model and its use to calibrate the family of regularized FCH models studied herein. We also present the initial data and motivate the benchmark problems. This derivation illuminates the incorporation of the well-stiffness in the Foot 1 and Foot 2 benchmarks that is the initial motivation for this computational study. In section 3, we present the second order adaptive numerical schemes that we use to resolve the benchmark problems and highlight the sensitivity of efficiency to choice of implicit terms. In section 4, we present an overview of the simulations of each of the five benchmark problems for a fixed local truncation error, showing the conditions under which the schemes agree and disagree. In section 5, we contrast the performance of the schemes, particularly with respect to accuracy in the far-field of the domain, energy decay, evaluation of the precise critical value for onset of defects, and comparison of time-stepping performance and computational efficiency. We summarize the performance in section 6.
Mean field approximation of amphiphilic diblock suspensions
The self consistent mean-field (SCMF) approach derives density functional models that approximate the bulk interactions of collections of polymers represented by molecular units, [20]. When applied to amphiphilic diblock polymers suspended in a solvent the reduction yields a free energy for the three density components, φ i , for i = A, B, S , which represent the hydrophilic head, A, and the hydrophobic tail, B, of the diblock polymer, and the solvent, S , respectively. Considering a suspension of n s solvent molecules and n P polymer diblocks, each comprised of N A and N B monomers of molecule A and B, respectively, [10,41] used the self-consistent mean field reduction to derive the free energy to a continuum phase-field model. More specifically, they introduced the mean densities
φ A = n P N A |Ω| , φ B = n P N B |Ω| , φ S = n S |Ω| ,(1)
and derived a bilinear approximation to the SCMF free energy expressed in terms of the variance from the mean
φ i0 = φ i − φ i , F (2) UD (φ 0 ) = i j Ω a i j φ i φ j (D −1 φ i0 )(D −1 φ j0 ) + b i j φ i φ j + χ i j φ i0 φ j0 + δ i j c i j φ i |∇φ i0 | 2 dx.(2)
Here a = (a i j ), b = (b i j ), c = (c i j ), with i, j ∈ {A, B, S }, denote material parameters and δ i j is the usual Kronecker delta function. Their derivation is similar to [11], with both approaches incorporating long-range interaction terms through the operator D := (−∆) 1 2 , the square-root of the negative Laplacian operator, subject to periodic boundary conditions.
The long-range terms describe entropic effects of chain folding and volume exclusion derived from the interactions of the polymer chains with effective mean fields. A similar energy was proposed as a model of a microemulsions of oil, water, and surfactant by [40], who argued directly, and somewhat phenomenologically, from a Landau theory for a scalar density. This scalar model was extended to a more nonlinear one by [23] and [24], who proposed a density dependence on the coefficients. Uneyama and Doi also proposed a nonlinear extension, [42], for their vector model in which the average density φ k was replaced with the local density φ k . This extrapolation yielded a family of models that include the Ohta-Kawasaki free energies. In [7] the nonlinear extrapolation approach was modified, first through a shift in dependent variables to the spatially averaged density ψ k := D −1 φ k0 , and then by an extrapolation step in which the average density φ k is replaced with the slowly varying average density,
φ k → φ k (1 + ψ k ).(3)
The three-component model is then reduced to a scalar field similar to [24] by requiring a point-wise incompressibility, ψ A +ψ B +ψ S = 0, and replacing the global constraint on the A-and B-polymer fractions with the point-wise constraint,
φ A /N A = φ B /N B . Choosing the parameterization ψ A = ψ B = (b r − b l )u + (b r + b l ) 2m f , ψ S = 1 − (b r − b l )u + (b r + b l ) 2 ,
in terms of the free variable u, for choices of b r > b l made below that normalize the range of u. The resulting model depends upon N P := N A + N B , the polymer fractions α A = N A /N P and α B = 1 − α A , the polymer-solvent molecular mole fraction m f := n P N P /n S , and the dimensionless parameter ε = l L N 1/2 P 1 which rescales the Kuhn length l of the diblock polymer into a mean-square end-to-end length a single ideal diblock polymer chain expressed as a ratio of the domain length L. The amphiphilicity of the diblock molecules is expressed in terms of a weighted Flory-Huggins parameter
χ w := α A χ AS + α B χ BS − α A α B χ AB > 0,(4)
where for k, m ∈ {A, B, S } the Flory-Huggins parameters χ km > 0 record the strength of the repulsive interaction between a k-monomer and an m-monomer. The value of χ w depends upon the composition of the polymer diblock chain, but not on its length. With these reductions and notation, the Uneyama-Doi bilinear energy (2) reduces to the singular functionalized Cahn-Hilliard (S-FCH) form
F S−FCH (u) = 1 2 Ω ε 2 ∆u − W S (u) 2 + P(u)dx,(5)
where the singular potential W S is defined via its derivative,
W S (u) = m f 24 ln (b r − b l )u + (b r + b l ) + 2m f − 6N P ln (b r − b l )u + (b r + b l ) − 2 + χ w (b r − b l )u + C 0 .(6)
The condition χ w > 0 guarantees that W S has three zeros on its domain. The parameters b r and b l are chosen to map the left and right zeros to −1 and +1 respectively, and the potential W S is defined as the primitive of W S that has a double zero at u = −1. The first derivatives of the well W S are singular at the endpoints where the corresponding to pure solvent and pure polymer phases. The perturbative potential P takes the form
P(u) := 9(b r − b l ) α A α B u 2 u(b r − b l ) + 2m f − W S (u) 2 .(7)
The constant C 0 does not impact the value of the energy and is chosen to minimize the perturbative potential P.
Regularized FCH and experimental motivation for the benchmark problems
We draw motivation for the benchmark simulations from the complexity observed in the experiments conducted in [29]. In that study the authors prepared well-stirred dispersions of amphiphilic diblock of Polyethylene oxide (PEO) -Polybutadiene (PB) in water, and allowed the mixture to relax and come to quasi-equilibrium. The weight fraction of polymer was fixed at 1%, and they considered a long and a short polymer chain, characterized by a fixed molecular length of the hydrophobic PB, with N PB (= N B ) taken as 45 and 170. They varied the aspect ratio α A = N A /N B , characterized by the weight fraction, w PEO , of the amphiphilic PEO component. They recovered a bifurcation diagram, presented in Figure 1 (left), which shows that for the short chains the well-mixed dispersions largely formed codimension one spherical bilayer interfaces, codimension two solid tubes, or codimension three solid spherical micelles, with some overlap depending upon the aspect ratio. However for α A ∈ (0.3, 0.5) the suspensions of long chains form structures that are loaded with defects, such as the network structures and endcaps depicted in Figure 1 (right -top and bottom).
The self-assembly of spatially extended morphologies from a relatively dilute suspension can be viewed as an arrival and a redistribution process. The dispersed amphiphilic molecules are generically too dilute to self assemble, but may diffuse until they arrive at localized structure, insert themselves and lowering their contribution to the system energy by isolating their hydrophobic tail from contact with the solvent. Within the FCH model, the rate of arrival determines the final outcome of this growth phase. The selection mechanism for the end state is delicate, with many possible outcomes separated by slightly different final energies. This landscape affords an excellent diagnostic to benchmark the performance of computational tools.
To stabilize the benchmark problems we make several changes to the initial configuration and the model. In particular we replace the well-stirred initial dispersion, typically modeled with random initial data, with a fixed bilayer interface configuration with an asymmetric shape and a spatially constant background density of amphiphilic diblock that emulates the reservoir of dispersed molecules. The asymmetry in the shape seeds the motion against curvature.
In a benchmark problem this is best not left to random fluctuations as would be the case for a perfectly circular initial shape. For computational reproducibility we smooth the well, replacing the singular well W S with
W q (u) := (u − b − ) 2 2 + q ε 1 − sech u − b − ε (u − b + ) 2 2 + γ 3 u − 3b + − b − 2 ,(8)
where the parameter q regulates the second derivative W q (b − ), as depicted in Figure 2 (right). This allows a range of approximation of the singularity of the left well of W S . We fix b ± = ±1 and take the asymmetry parameter γ = 0.3 to match the shape of W S . The perturbative potential P is also singular, and is regularized via replacement with the standard FCH functionalization terms to facilitate comparison to prior analytical results. This yields the non-singular FCH free energy model
E FCH (u) := Ω 1 2 ε 2 ∆u − W q (u) 2 − ε 2 2 η 1 |∇u| 2 + η 2 W q (u) dx.(9)
The values of the functionalization parameters η 1 and η 2 are determined from a least-square fit of P for the long-chain data. All parameter values for each benchmark are recorded in Table 1. For the critical case, the value of η 2 is tuned to enhance the strength of the pearling transient.
The FCH equation is given by the
H −1 gradient flow of E FCH u t = ∆ δE FCH δu ,(10)
which takes the explicit form
u t = ∆ ε 2 ∆ − W q (u) (ε 2 ∆u − W q (u)) − −ε 2 η 1 ∆u + η 2 W q (u) .(11)
The regularized form of the FCH possesses several advantages. It encompasses both the smooth q = 0 and the stiff q > 0 models, naturally allowing for a quantification of the impact of nonlinear stiffness on the computational schemes. While the stiff version mimics the SCMF reduction, the smooth FCH model has been much better studied [5,6,14,15] and has advantages in applications which require a simple model that stabilize higher codimensional morphologies with a minimum of numerical stiffness. These applications include the hybrid phase field models for fluid-structure interactions [28].
FCH model calibration and benchmark motivation
To calibrate the parameters in the regularized well it is convenient to exploit a rescaling of the FCH-SCMF energy that leaves the associated gradient flow invariant:
ε → ε √ ν , W S → W S ν , P → P ν 2 , t → ν 2 t.
The rescaling of ε is equivalent to a change in domain size L → √ νL. We take each monomer to have equal weight, equal to the molecular weight of the solvent. Correspondingly the weight fraction of PEO, w PEO , equals the molar fraction, α A , and the polymer weight fraction within the solvent reduces to the molar fraction of polymer,
m f = n P N P n s = 1 100 .
For the short-chain polymer benchmark we take N P = 45 and C 0 = 0.8 and for the long-polymer benchmark we take Figure 2 (left) and compared to the regularized well W q used in the benchmark simulations. Intuitively, both a high density of dispersed diblock polymers or a high energy associated to an isolated diblock molecule correspond to a high rate of absorption of the dispersed polymers onto the bilayer interface. The arrival rate is a key quantity controlling defect formation. When the arrival rate is slow, the bilayer interface can grow in size to accommodate the new mass. The growth process is adiabatic and has been studied rigorously, [5], deriving a motion against curvature, regularized by a higher order Willmore term that includes surface diffusion. If the rate of arrival increases beyond a critical threshold, then defects, such as pearling, endcaps, and loop formation are observed. At moderate rates, a pearling bifurcation can be triggered, the onset of which is well understood within the context of the FCH gradient flow, [12]. The pearling can be transient, subsiding as the dilute suspension of amphiphilic material is consumed. The pearling can also be lead to the formation of end-cap type defects, essentially micelles that remain connected to the underlying structure from which they emerged. The endcaps form most readily at points of high curvature of the bilayer interface. The stem of the endcap can grow, forming a long trailing bilayer-type stem and may ultimately reconnect with the initial structure, forming a loop. At yet higher arrival rates the bilayer interface itself may undergo curve splitting -directly forming closed loops and network structures. The rich array of possible outcomes, and the wide variety of end-states of the gradient flow, provide an excellent diagnostic of the accuracy of the proposed schemes.
The benchmark problems introduce two methods to control the rate of arrival of surfactant at the interface. The first is through background level of amphiphilic molecules, controlled by the parameter d in (13), increasing d corresponds to adding more amphiphilic material to the dispersion. The second is through the convexity of the left well in W q , controlled by the parameter q and the value of ε. Increasing the value of q increases W q (−1), leading to an increase in the energy of dispersed amphiphilic molecules, which also increases their rate of arrival. The energy of dispersed chains increases with chain length due to the exposure of a longer hydrophobic tail to solvent, [9]. This is evident within the singular model through the scaling of W S with N P via m f .
In the first three benchmarks we take q = 0, corresponding to shorter chains, and induce bifurcation by raising the background density. At the low background level in the sub-critical benchmark the initial bilayer interface absorbs amphiphilic material and increases its length, however the rate of absorption is sufficiently slow that there is no generation of defects. In the super-critical benchmark the background level is raised and the elevated rate of arrival induces formation of several defects that coalesce and merge over time. In the critical benchmark the aspect ratio parameter η 2 is tuned to extend the duration of the pearling transient within the bilayer interface. Accurate simulations of this benchmark approach the formation an endcap defect before relaxing back to a smooth bilayer profile as the reservoir of dispersed diblock molecules is depleted. In the Foot 1 and Foot 2 benchmarks, we return to the low dispersion level of initial data and systems parameters of the sub-critical case, but increase the value of value of q within the well. This corresponds to lengthening the polymer chains, increasing the rate of absorption without adjusting the total amount of material absorbed. In both Foot 1 and Foot 2 this induces defect formation.
The initial data
Space is discretized through the standard Fourier pseudo-spectral method assuming periodic boundary conditions on square domains. For the benchmark computations it is useful to have smooth periodic initial conditions on uniform grids. To begin, we fix Ω = [0, L] 2 , with L = 4π, and set the number of grid points along the x 1 and x 2 axes to be N o = 256, corresponding to a mesh spacing h o = L/N o . Given a simple non-intersecting parametric curve
Γ = x 1 (t), x 2 (t) , t 0 ≤ t ≤ t 1 ,
we construct a region Γ R of uniform width R about Γ, with outer and inner boundaries Γ ± defined by
Γ ± = x 1 (t) ± x 2 (t) s(t) R, x 2 (t) ∓ x 1 (t) s(t) R t 0 ≤ t ≤ t 1 ,(12)
where s = s(t) is the arc-length of Γ. We construct the piece-wise constant function φ Γ to be 1 inside Γ R and −1 outside, and smooth it by convolution with the filter F :
L 2 (Ω) → C ∞ per (Ω), defined via F[φ Γ ](x) = k 1 ,k 2 ∈I Nφ o,Γ (k 1 , k 2 ) exp −λ 0 (k 2 1 + k 2 2 ) exp 2πi L (x 1 k 1 + x 2 k 2 ) , whereφ o,Γ is the discrete Fourier transform (DFT) of φ Γ interpolated to the N o × Nu 0 N,i, j = φ 256 (ih, jh) + ε d α 2 m (0) , 0 ≤ i, j ≤ N,(13)
where d ∈ R is a parameter that varies in the benchmarks and α m (0) = W q (b − ) q=0 . Clearly, u 0 N will be a periodic grid function. The curve Γ is defined through polar variables as Γ = ρ(θ) cos(θ)
+ L 2 , ρ(θ) sin(θ) + L 2 θ ∈ [0, 2π) , where ρ(θ) = 3 − ε 2 cos 6(θ − π 11 ) − ε 2 cos θ − 3π 11 .
The initial data u 0 N corresponding to N = 256 with this choice of Γ is shown in Figure 3 (right) for d = 0. The curve Γ is chosen to break any symmetry with the periodic domain and to seed the curvature growth of the bilayer interface. The mass, m 0 , of the initial data, defined via the relation
m 0 := 1 2 Ω (u 0 N + 1) dx,
is reported in Table 1.
The numerical schemes
As we indicated in the previous section, we use the Fourier pseudo-spectral method to discretize space and simplify the spatial differential operators. The details are standard and skipped for the sake of brevity. In what follows, for simplicity we will write the numerical schemes semi-discretely, using the spatially continuous differential operators, though in practical computations these are replaced by their standard pseudo-spectral approximations.
We use the second order backward differentiation formula (BDF2) to produce the IMEX, PSD, and SAV schemes, and use the solution from the third order Adams-Moulton (AM3) scheme as a predictor to control the local error to resolve the benchmark problems described in Section 2.2.
Variable step size BDF2 and AM3 schemes
Consider the initial value problem,
u (t) = F(u(t)), u(t 0 ) = u 0 , for t 0 ≤ t ≤ T . Let us denote the temporal step size via k n := t n − t n−1 , where t 0 ≤ t n−1 < t n ≤ T .
Suppose the second order variable step size BDF2 scheme has the form
au n+1 + bu n + cu n−1 = F(u n+1 ),(14)
where, upon Taylor expanding and comparing the coefficients, we may identify
a = 1 k n+1 + 1 k n+1 + k n b = − 1 k n+1 − 1 k n c = 1 k n − 1 k n+1 + k n .(15)
Introducing the time-step ratio γ := k n+1 k n , the general variable step size BDF2 scheme can be written as
1 + 2γ 1 + γ u n+1 − (1 + γ) 2 1 + γ u n + γ 2 1 + γ u n−1 = k n+1 F(u n+1 ),(16)
which recovers the classical uniform version 3u n+1 − 4u n + u n−1 = 2kF(u n+1 ) when γ = 1.
Suppose the third order variable step size AM3 scheme has the form
u n+1 = u n + ω 1 F(u n+1 ) + ω 2 F(u n ) + ω 3 F(u n−1 ) .
To identify the coefficients {ω i } 3 i=1 , we make the approximation
u(t n+1 ) − u(t n ) =
where the quadratic polynomial P(t) is the interpolant of F(u(t)) at t n−1 , t n and t n+1 . Therefore the variable step size AM3 is
u n+1 = u n + k n+1 6 3 + 2γ 1 + γ F(u n+1 ) + (3 + γ)F(u n ) − γ 2 1 + γ F(u n−1 ) ,(17)
which recovers the uniform version u n+1 = u n + k 5 12 F(u n+1 ) + 2 3 F(u n ) − 1 12 F(u n−1 ) when γ = 1. Further details about these two methods can be found in [26].
Adaptive schemes
The FCH gradient flow (10), which may be written as u t = F(u), where F(u) = ∆ δE FCH δu , undergoes bifurcations that trigger hidden timescales. As these events occur at unpredictable times, an adaptive approach to time-stepping is required to balance accuracy and efficiency. To initialize the algorithm, we set a target local truncation error tolerance, σ tol , and the minimal and maximal time-step values k min and k max .
Given initial data u 0 , initial time t 0 , and some final time T , we fix the temporal step size k 1 := k min and compute the first time-step approximation u 1 at time t 1 for the FCH equation (10) via an appropriate version of (locally) second order method. The adaptive algorithm, based upon [37,39], then proceeds as follows.
Step 0: Given time index n ∈ N + , and approximations u n−1 , u n at times t n−1 and t n , respectively, with time step sizes k n = t n − t n−1 and initialk n+1 := k n .
Step 1: Compute a second order accurate primary approximationũ n+1 using one of the BDF2 schemes (from the next three sections) using step sizes k n andk n+1 .
Step 2: Compute the time step ratio γ =˜k n+1 k n and a third order accurate approximation, u p , via the AM3 scheme:
u p := u n +k n+1 6 3 + 2γ 1 + γ F(ũ n+1 ) + (3 + γ)F(u n ) − γ 2 1 + γ F(u n−1 ) .(18)
Step 3: Calculate the relative error approximation
e n+1 := ũ n+1 − u p L 2 u p L 2 .
Step 4: If e n+1 ≤ σ tol ork n+1 = k min , then
Accept the primary approximation, u n+1 =ũ n+1 .
Recalculate k n+1 = max k min , min{A d p (e n+1 ,k n+1 ), k max } , and update the current time, t n+1 = t n + k n+1 .
Update the time step index: n ← n + 1.
Goto
Step 0.
Else
Recalculate the time step sizek n+1 = max k min , min{A d p (e n+1 ,k n+1 ), k max } .
Goto
Step 1.
Endif
Here
A d p (e, k) := ρ s σ tol e 1/3 k,
and we take the safety coefficient ρ s = 0.9, and k min = 10 −9 for all simulations. For the IMEX and SAV schemes k max is taken to be ∞, while for the PSD scheme, the optimal value of k max depends upon q, as shown in the Table 3. As discussed in [26], to ensure zero-stability for the variable step size BDF2 in (16), A d p (e, k) needs to be bounded from above by 1 + √ 2 k. Numerical exploration with this bound on A d p showed it afforded no significant impact on the benchmark problems.
Remark 2. We have chosen the time step adaptivity to directly enforce that the approximate solutions are accurate to a desired local error tolerance, σ tol . We employ an algorithm similar to that in [37], though there are several others that have a similar design and purpose, including for example, [22,26,27,38]. The method of [34] is different in that the energy is monitored in time as a surrogate error indicator. When the preliminary steps indicate a rapid change in energy, the algorithm reduces the time step size with the goal of capturing the corresponding dynamics of the density field, the motivation is that abrupt changes in the energy correspond to topological changes in the density field. A preliminary comparison of the two disparate approaches gives us reason to favor the direct method. First, our objective is accurate density field calculations, and the direct method controls the density field explicitly, rather than implicitly through the energy. The energy functional is scalar valued, and many classes of deformation do not locally change the value of the energy. This makes the performance of the energy monitoring time-stepping method very sensitive to choices in the time stepping control parameters. Second, the computation of the energy is an added expense that makes the use of an energy-based error indicator less attractive.
The BDF2-PSD scheme
The BDF2-PSD scheme uses a fully implicit variable time-step BDF2 for the numerical approximation of the system (11) which takes the form
au n+1 + bu n + cu n−1 = ∆ δE FCH δu n+1 ,(19)
where the coefficients a, b, c are given in (15). The solution u n+1 in (19) can be solved in terms of a zero residual,
R(u n+1 ; u n , u n−1 ) := Π 0 δE FCH δu n+1 −∆ −1 (au n+1 ) − ∆ −1 (bu n + cu n−1 ) = 0,(20)
where Π 0 denotes the linear zero-mass orthogonal projection operator. Given u n−1 and u n , to solve u n+1 from (20), this method is accompanied by a preconditioned steepest descent (BDF2-PSD) solver, with an approximate line search (ALS) to invert the highly nonlinear system of equations. This solver is referred to the PSD with ALS, see [3,19].
We refer to this method as PSD for brevity.
The preconditioned steepest descent method solves nonlinear system (20) iteratively through a series of linear systems. The strictly positive, self-adjoint operator L PSD is the linearization of (20) about the spatially constant state u ≡ b − after dropping the small η 1 and η 2 terms,
L PSD := ε 4 ∆ 2 − 2α m ε 2 ∆ + α 2 m − a∆ −1 ,
which is well-defined on mass-less functions, and preconditions the iterative scheme. Here α m = W q (b − ) depends strongly on q. The solution u n+1 is thus defined as the limit of the sequence {u n+1 s } ∞ s=0 , constructed through the ALS recurrence relation u n+1 0 := u n + k n+1 k n (u n − u n−1 ),
u n+1 s+1 = u n+1 s + λd n+1 s , s = 0, 1, 2, . . .(21)
where the search direction d n+1 s at u n+1 s is defined as
d n+1 s := −L −1 PSD R(u n+1 s , u n , u n−1 ).
For a prescribed iterative stopping tolerance i tol , the ALS procedure is terminated once Table 2. The values used in the simulations are determined by linear interpolation. The iterative stopping tolerance, i tol , impacts the accuracy and computational cost of the PSD scheme. Numerical optimization finds that an optimal choice of i tol is sensitive to both the well stiffness, q, and the local truncation error, σ tol . We determine this relation through the ratio
i tol = ν(q)σ tol ,
and determine an optimal value of ν(q). This requires balance, as overly small values of i tol lead to excessive iterations that do not improve the scheme's accuracy. On the other hand i tol must be small enough to ensure that numerical error from the iterative solver does not pollute the adaptive time-stepping and does not impede the convergence of the iterative solver at subsequent time-steps. Instructively, the iterative convergence rate is found to depend upon the upper limit, k max , imposed on the adaptive time-stepping algorithm. This leads to a coupled numerical optimization study, presented in Table 3 which shows the sensitively of iterations numbers upon k max for the three values of q, and the optimal value of ν. The iteration counts increase considerably with q, while ν decreases exponentially with q.
If the upper bound k max is removed then the iteration count may increase considerably, with associated increase in computational effort. The tuning of k max and ν with q is the most unpredictable element of the optimization process for any of the schemes. The FCH equation (11) can be rewritten as
u t = ∆ L IMEX u + N IMEX (u) + ε 2 η 1 ∆u − η 2 W q (u) ,(23)
where we introduce the linear positive operator
L IMEX := ε 4 ∆ 2 − 2α m ε 2 ∆ + α 2 m ,(24)
obtained by linearizing δE FCH δu in (11) about u = b − and dropping the small, negative η 1 and η 2 terms. The term N IMEX is genuinely nonlinear with zero linearization about u = b − ,
N IMEX (u) := ε 2 α m − W q (u) ∆u + ε 2 ∆ α m u − W q (u) + W q (u)W q (u) − α 2 m u.
The resulting second order semi-implicit IMEX scheme is chosen to stabilize the spatially constant background state u ≡ b − . To this end we take the dominant linear terms implicit and the remainder, including smaller linear terms involving η 1 and η 2 , explicit,
au n+1 + bu n + cu n−1 = ∆ L IMEX u n+1 + N IMEX (u * ,n+1 ) + ε 2 η 1 ∆u * ,n+1 − η 2 W q (u * ,n+1 ) ,(25)
where u * ,n+1 can be chosen as any explicit (locally) second order approximation of u(t n+1 ) to make the scheme consistent, for instance,
u * ,n+1 = u n + k n+1 k n (u n − u n−1 ).(26)
Now we can isolate and solve u n+1 in (25) from
a − ∆L IMEX u n+1 = −bu n − cu n−1 + ∆ N IMEX (u * ,n+1 ) + ε 2 η 1 ∆u * ,n+1 − η 2 W q (u * ,n+1 ) .(27)
The BDF2-SAV scheme
Computational schemes based upon the SAV formulation have been applied to the FCH gradient flow, see [45].
The version presented here is a slight variation. We rewrite the FCH energy functional E FCH (u) in (9) in the form:
E FCH (u) = Ω ε 4 2 (∆u) 2 − η 1 2 + ζ ε 2 |∇u| 2 + G(u) dx,(28)
where ζ > 0 is a parameter and
G(u) := −ε 2 ∆u W q (u) + ζu + 1 2 (W q (u)) 2 − η 2 W q (u).(29)
The choice of principle linear operator for the SAV scheme is a bit less intuitive than for the IMEX or PSD schemes.
We introduce
L SAV = ε 4 ∆ 2 + ε 2 (η 1 + 2ζ) ∆ = L 0 + L 1 ,(30)
where the sub-operators are parameter dependent
L 0 (β 1 , β 2 ) = ε 4 ∆ 2 − β 1 α m ε 2 ∆ + β 2 α 2 m ,(31)L 1 (β 1 , β 2 ) = ε 2 (η 1 + 2ζ) ∆ + β 1 α m ε 2 ∆ − β 2 α 2 m ,(32)
where α m = α m (q) and the constants β 1 , β 2 ≥ 0 are the stabilization parameters. The operator L 0 defines the principle linear implicit terms in the SAV scheme. The default choice for these parameters is β 1 = 2 and β 2 = 1.
Introducing the auxiliary energy
E 1 (u) = Ω G(u)dx,
the FCH energy (28) takes the form
E FCH (u) = 1 2 (u, L SAV u) L 2 (Ω) + E 1 (u).(33)
For fixed time-steps the SAV scheme is known to be energy stable for a modified energy, if the functional E 1 (u) can be shown to be uniformly bounded from below over H 2 per (Ω), [35]. This is achieved by choice of ζ = ζ(q). Specifically
E 1 (u) ≥ Ω W q (u) + ζ |∇u| 2 dx + Ω 1 2 (W q (u)) 2 − η 2 W q (u) dx,
and choosing ζ larger than the negative of the minimum value of the W q , we estimate
E 1 (u) ≥ |Ω| min u 1 2 (W q (u)) 2 − η 2 W q (u) > −D 0 ,
where D 0 > 0 only depends upon the domain Ω, the value of η 2 and W q .
For the energy splitting approach, we introduce the scalar auxiliary variable
r = r(t) := E 1 (u) + D 0 ,
then the FCH equation can be rewritten as
∂u ∂t = ∆µ, µ := L SAV u + r √ E 1 (u) + D 0 V[u],(34)dr dt = 1 2 √ E 1 (u) + D 0 Ω V[u] ∂u ∂t dx,(35)
where V[u] = δE 1 /δu = G (u). For choosing u * ,n+1 as in (26), the SAV scheme takes the form au n+1 + bu n + cu n−1 = ∆µ n+1 , µ n+1 = L 0 u n+1 +L 1 u * ,n+1 + r n+1
E 1 (u * ,n+1 ) + D 0 V[u * ,n+1 ],(36)
ar n+1 + br n + cr n−1 = Ω V[u * ,n+1 ] 2 E 1 (u * ,n+1 ) + D 0 au n+1 + bu n + cu n−1 dx.
We remark that the r n+1 variable in (37) also contributes to the implicit equation for u n+1 . The full resolution of u n+1 from (36)-(37) is presented in [37,45], but is driven by the inversion of the operator L 0 − a∆ −1 . With a fixed time-step k, the SAV scheme is unconditionally energy stable for the auxiliary energy E aux u n , u n−1 , r n , r n−1 := 1 2 (u n , L SAV u n ) L 2 (Ω) − u n − u n−1 , L 1 (u n − u n−1 )
L 2 (Ω) + 1 2 2u n − u n−1 , L SAV (2u n − u n−1 ) L 2 (Ω) + r n 2 + 2r n − r n−1 2 .
Theorem 1. When implemented with a fixed time step size k > 0, the SAV scheme (36)-(37) is unconditionally modified-energy stable in the sense that the discrete modified-energy law holds, E aux u n+1 , u n , r n+1 , r n ≤ E aux u n , u n−1 , r n , r n−1 , n ≥ 1,
The proof of Theorem 1 is given in Appendix A. Details on energy stability properties of SAV schemes can be found in [37,45].
The stabilization parameters make L 0 a strictly positive operator and play an essential role in the convergence, accuracy, and efficiency of the SAV scheme. The operator L 0 agrees with L IMEX for the choice β 1 = 2 and β 2 = 1 that we take here.
The ETDRK2 scheme
For the temporal discretization, the FCH equation (11) can be viewed as an infinite dimensional ODE written in the following operator splitting form
du dt = Lu + N(u),(39)
where L is a negative-definite linear differential operator and N(u) is remaining nonlinearity. For time-step τ = t n+1 − t n , we may multiply both sides of (39) by the linear semigroup e −Lt , to obtain the "exponentiated" form of (39) e −Lt u t = e −Lt N(u).
Integrating (40) over [t n , t n+1 ] yields the time-differenced system,
u(t n+1 ) = e L(t n+1 −t n ) u(t n ) + t n+1 t n e L(t n+1 −s) N(u(s))ds,(41)
= e Lτ u(t n ) + τ 0 e L(τ−s) N(u(t n + s))ds,
The exponential time differencing (ETD) approach uses this formulation, arriving at an iterative scheme by approximating the integrals with finite differences, more details can be found in [13,16,17,31]. Precisely, the explicit first order ETD Runge-Kutta (ETDRK1) scheme uses the approximation N u(t n + s) ≈ N(u n ) for s ∈ [0, τ]. This yields u n+1 = e Lτ u n +
(1 − s τ )N(u n ) + s τ N(ũ n+1 ) ds.(44)
Evaluating the integrals exactly, we find
ũ n+1 = e Lτ u n + L −1 e Lτ − I N(u n ), u n+1 =ũ n+1 + L −1 L −1 (e Lτ − I) − τI N(ũ n+1 ) − N(u n ) τ .(45)
For the FCH equation (11), we mirror the IMEX and SAV approach, choosing
L = ∆L I MEX − κI = ∆(ε 4 ∆ 2 − 2α m ε 2 ∆ + α 2 m ) − κI, N(u) = ∆N I MEX (u) + κI,(46)
where κ is some positive constant and I is the identity operator. Numerical tests show that FTT calls of ETDRK2 is not sensitive to the choice of κ > 0. However, accuracy improves slightly for choices of κ ∈ [10 −4 , 10]. We take κ = 1 in the simulations. We refer to ETDRK2 as ETD for brevity.
Benchmark simulations
We present an overview of the Benchmark simulations for local truncation error σ tol = 10 −5 , for which the PSD scheme is accurate while the IMEX, SAV, and ETD schemes are borderline accurate. Generically we find that a global L 2 (Ω) relative discretization error of 2.5 × 10 −3 is sufficient to ensure that each scheme is quantitatively accurate, with the correct numbers, types, and placements of defects.
Sub-critical benchmark
The sub-critical benchmark has a low level of dispersed diblock polymer material, controlled by the parameter d in (13), while the relatively mild concavity of W q at u = b − , controlled by α m (0) = W q (b − ) q=0 , leads to a gentle absorption rate. The bilayer interface profile does not pearl and remains a simple closed curve from initial data to its final equilibrium shape, as shown in Figure 5 at times T = 10 and T = 250. As shown in [5], gentle absorption drives motion against curvature, regularized by surface diffusion, which relaxes to a curvature driven flow as the background material is depleted. All schemes are in quantitative agreement, as can be verified by the contour plot comparison in Figure 8 (left) and the data of Table 4. All schemes agree to within L 2 relative error 7 × 10 −3 as reported in Table 4. The red number on colorbar indicates max u. Table 4. The red number on colorbar indicates max u.
Critical benchmark
Super-critical benchmark
The sub-critical and super-critical benchmarks differ only in the level of the background material, controlled by the parameter d in (13). The elevated value of this parameter in the super-critical benchmark increases the rate of arrival of mass to the interface, exceeding the interface's capacity to absorb the arriving mass via a curve lengthening flow or by pearl generation. The interface undergoes defect generation. For the super-critical benchmark with σ tol = 10 −5 the output from the four schemes do not agree at leading order, as can be seen in Figure 7. For the PSD and ETD schemes the bilayer interface absorbs material from the background and pearls locally at points of high curvature, and then ejects 8 endcap defects, five of which intersect back with the underlying interface, forming closed loops.
Two of the loops subsequently merge to form an extended loop which grows into a cisternal structure characterized by two long parallel interfaces. The IMEX and SAV simulations differ from the PSD and ETD , but agree with each-other. They also produce 8 endcap defects initially, however only four of them subsequently form closed loops.
Two of these loops merge, forming a cisternal structure, however there are two small endcaps in the IMEX and SAV simulations, in contrast to the one small endcap in the PSD and ETD simulation. At longer times the cisternal region grows, consuming structures and at time T = 250 it leaves one loop, one long endcap, and one short endcap in all The top row presents the PSD simulation and the bottom row represents the SAV simulation. The IMEX and SAV simulations are very similar, and the PSD and ETD simulations are very similar, but the two groups of simulations disagree, being separated by an L 2 relative error of 9 × 10 −2 , as reported in Table 4. The red number on colorbar indicates max u. simulations -however in the SAV and IMEX simulations the distance between cisternal region and small loop is significantly longer than in PSD and ETD simulations. Figure 8 shows the levels sets corresponding to u = −0.12 for the sub-critical and super-critical benchmarks with σ tol = 10 −5 and N = 256, showing their agreement in the sub-critical benchmark and their disparity in the super-critical benchmark. In the super-critical benchmark the higher rate of absorption driven by the higher initial background level of u produces dynamic choices associated to endcap formation that require greater accuracy than the linearly implicit schemes can achieve at σ tol = 10 −5 . If σ tol is reduced to 10 −6 , then PSD and ETD simulations do not change quantitatively, while the SAV and IMEX simulations move into quantitative agreement with the PSD and ETD schemes. The value of u in the far field, away from the interfacial structure, is asymptotically constant at equilibrium and has been shown to be a key bifurcation parameter for the onset of pearling, [32,33]. Faithful resolution of this value is essential to an accurate simulation. Figure
Foot 1 benchmark
The Foot 1 and sub-critical benchmarks, are identical in initial data and parameters with the exception of the value of the concavity of the well W q , controlled by the parameter q. For Foot 1 we take q = 0.2 which increases the value of α m (q) = W q (b − ), as depicted in Figure 2. This adjustment raises the energy associated to small, spatially uniform values of u, thereby increasing the rate of absorption of material from the bulk. Although the total amount of material in the background is the same in both benchmarks, the increased absorption rate in the Foot 1 benchmark leads to defect formation. We consider only the PSD , IMEX , and SAV schemes, and each capture these events with quantitative accuracy, as shown in Table 4. In Figure 10 (left) the pearling and defect formation are visible in the lower-right of the bilayer interface already at time T = 1.5. At time T = 50 the simulations produce six closed loops place roughly symmetrically around the bilayer interface. This structure is quasi-stable, but eventually evolves onto a double-sheeted bubble similar to that depicted in the right-most panel of the top row of Figure 18. Table 4. The red number on colorbar indicates max u.
Foot 2 benchmark
The Foot 2 and sub-critical benchmarks have an identical setup with the exception of the value of q, which is taken to be q = 0.5 in Foot 2. This introduces a very strong, nonlinear stiffness, and the large value of α m = W q (b − ) q=0.5 significantly increases the energy penalty associated to dispersed amphiphilic material. As a consequence its rate of absorption onto the bilayer interface increases, inducing a curve-splitting bifurcation in which the bilayer interface splits directly in two, as shown in Figure 11 (left) at T = 1. All three schemes agree qualitatively on the 512 × 512 mesh, producing four loops and two double loops. Grid refinement in Table 5 shows that the N = 256 grid is insufficient to produce accurate results. Further grid refinement to N = 1024 yields quantitative agreement with the N = 512 simulations. The large value of W q (b − ) for q = 0.5 yields a profile that is much less smooth. The spatial convergence to the far-field value occurs at the exponential rate W q (b − ) ε, which is significantly greater for q = 0.5, necessitating the higher spatial resolution. Table 4. The red number on colorbar indicates max u.
The time-trace of the background levels, u − b − evaluated at the domain center (solid) and domain corner (dashed), are presented for the Foot 2 benchmark in Figure 13 (left). It has several notable differences from the sub-critical benchmark presented in Figure 9 (left). The most salient distinction is that the large value of α m (0.5) greatly increases the temporal rate of absorption of amphiphilic material from the background. For the Foot 2 benchmark the background state begins to achieve its equilibrium value at T = 1 and is fully equilibrated around T = 7 ∼ 8. This is roughly 10-15 times faster than the relaxation for the q = 0 sub-critical benchmark, depicted in Figure 9.
Computational accuracy and efficiency
The four schemes presented are second order accurate, as verified by the convergence study presented in Table 6. Nevertheless, the performance of the schemes is not equally accurate nor efficient, particularly as the nonlinear stiffness parameter q is increased. Generally the ETD scheme requires substantially smaller time steps to achieve competitive local truncation errors. This is consistent with analysis in [8] which showed that Runge-Kutta based schemes, even fully implicit ones, can lead to larger truncation errors. It is clear that the ETD scheme achieves second order accuracy, however it incurs a larger constant from amplification of error in the stages due to the presence of large space gradients in the bilayer morphologies. We discuss the relation of accuracy to energy decay, global discretization error, and computational efficiency.
Energy decay
A major feature of gradient schemes is the decay of the overall system energy. Much attention has been given to the construction of gradient stable schemes for which energy decay is unconditional with respect to the temporal stepsize. However in gradient flows that generate a rich variety of structures issues of accuracy move to the forefront and energy decay ideally becomes a consequence of accuracy. For the super-critical benchmark, the various competing outcomes are significantly different but have only marginally different energies and considerable accuracy is required for a scheme to differentiate between the available options. As shown in Figure 12 (left), with σ tol = 10 −5 for each of the 5 benchmarks the energy decay behavior is very similar and decays uniformly. There are however important differences. As the middle inset shows, for the super-critical benchmark the energy trace for the IMEX and SAV simulations are almost indistinguishable from each-other, but diverge from the more accurate PSD simulation with roughly a 1% relative error. The differences in energy decay, and solution u, are largely erased for IMEX and SAV when σ tol is reduced to 10 −6 . The ETD has an energy trace that is more faithful to the PSD, but has a notable excursion for T ∈ [750, 850] that is eliminated for the reduced value σ tol = 10 −6 . The second inset shows detail of the Foot 1 benchmark. In this case the PSD , IMEX , and SAV schemes have reasonable quantitative agreement. And error is further reduced by taking σ tol to be 10 −6 in IMEX and SAV. These features emphasize that system energy can be a poor proxy for accuracy, and that energy decay is generally a minor benchmark for a gradient flow. The time-stepping profiles for the IMEX and SAV schemes are remarkably similar, and differ in important ways from that of the PSD scheme. As shown in Figure 13 (right), the PSD generically takes the largest time step-sizes, and typically hits the maximum step-size ceiling k max shortly after the resolution of the initial transient. This ceiling is required to insure the convergence of the nonlinear iterative scheme and to optimize its performance as measured by FFT per time unit. This value is smaller for the stiffer Foot 2 benchmark than for the super-critical benchmark as reported in Table 3. Indeed the time-step profile for PSD is largely equivalent for the super-critical and the Foot 2 benchmarks, until it hits the lower value of k max for the Foot 2 benchmark. This is in contrast to the IMEX and SAV profiles which are different for the two benchmark problems, but largely agree with each other. Each of the schemes has swings in step size of roughly one order of magnitude during the various defect generation and merging events that occur after the initial transient. The step sizes for the IMEX and SAV schemes are generically smaller than those for PSD, by as much as two orders of magnitude for the stiffer Foot 2 benchmark. However this is offset by the growing number of iterations required for solving the stiffer nonlinear system in this problem. The ETD scheme has the smallest time steps, typically over an order of magnitude smaller than any of the BDF2 schemes.
An excellent proxy for accuracy is to determine the lowest (critical) value of the background level, as measured by the initial data parameter d in (13), at which a defect is generated within the flow. The onset of a defect is easily detected through the maximum value of u, as the maximum value of the bilayer profile in these simulations occurs at u = 0.3566, while defects and higher codimensional structures such as micelles reside much more deeply in the right well of W q , with maximum values close to u = 0.74. Tracking the temporal evolution of max u yields a strong dichotomy. We fixed the parameters as in the critical benchmark problem but slightly adjusted the value of d to modify the amount of amphiphilic material in the bulk. The critical d value, reported in Table 7 depends upon the local truncation error, but converges to a common value of d = 0.7526 with decreasing σ tol . Indeed the PSD scheme is very close to identifying the correct critical value with σ tol = 10 −5 while IMEX and SAV require a value of σ tol of 10 −7 or 10 −8 to achieve similar accuracy. The time evolution of max(u(·, t)) under PSD for the critical benchmark parameters and for seven different value of d is presented in Figure 14.
Global discretization error verses computational cost
The definitive measure of accuracy is to compute the global discretization error of a simulation as measured against a known highly accurate answer. To produce these highly-accurate solutions we conduct a spatial grid refinement study for each benchmark problem and each computational scheme. For all but the stiffest Foot 2 benchmark increasing the grid from N = 256 to N = 512 produces consistent results, with solution differences reported in Table 5.
We present results only to the accuracy determined within this grid refinement study. Specifically the highly accurate simulations are calculated with the PSD scheme with σ tol = 10 −9 for q = 0, and with σ tol = 3 × 10 −8 for q = 0.2.
For q = 0.5, the IMEX scheme with σ tol = 3 × 10 −9 is used. The output of these simulations are taken as the highly accurate simulation against which others are compared. For all three schemes, sufficient refinement of σ tol lead to a global error that is within the anticipated accuracy of the scheme. Indeed our computations find that a global L 2 (Ω) relative discretization error of 2.5 × 10 −3 is generically sufficient to ensure that scheme is quantitatively accurate, with the correct numbers, types, and placements of defects.
We measure the computational efficiency of the three schemes in two ways. First as global relative truncation error, G rte , verses σ tol , and then more meaningfully as global discretization error verses FTT calls. This latter is euphemistically referred to as the dollars-per-digit metric. The first result, presented in Figure 15 (left), shows the decay in global L 2 relative error with decreasing σ tol . The blue curves, corresponding to the left (blue) vertical axis,
show that all four schemes improve in global accuracy with decreasing σ tol . For the super-critical benchmark the linear-implicit IMEX and SAV schemes are inaccurate for σ tol > 4 × 10 −6 and then have global discretization errors that decay linearly on a log-log plot, corresponding to a global discretization error roughly proportional the the 2/3 power of the local truncation error. The ETD scheme is more accurate than IMEX and SAV for σ tol = 10 −5 but becomes somewhat less accurate than IMEX and SAV when decreasing σ tol . Conversely, the PSD is accurate for all σ tol < 1 × 10 −3 , but its global accuracy at first improves sub-linearly with σ tol on the log-log scale before setting into the 2/3 power law relation between global discretization and local truncation errors. For the linear-implicit schemes the workload as measured by total FFT calls is remarkably linear as function of local truncation error on the loglog curve. Their workload grows approximately as a −1/2 power of the local truncation error over three orders of magnitude, with the IMEX more efficient than the SAV by a fixed factor of 1.4 over this range. The ETD scheme has a significantly higher workload, often by more than an order of magnitude, across all ranges of σ tol . The PSD workload starts out significantly higher than the linear-implicit schemes, but grows more slowly, becoming comparable at very small values of σ tol .
A more intuitive comparison of the performance arises from plotting the FTT calls verses the global discretization error, with σ tol acting as a parameterization of the curve. This is the dollars-per-digit plot, shown in Figure 15 (right).
In this plot, the lowest curve attains the desired global discretization error with the least computation cost. Setting G rte = 2.5 × 10 −3 as an acceptable upper limit, we find that all schemes except ETD achieve this global tolerance at comparable computational costs that correspond to disparate local truncation errors. The IMEX scheme is generally the most efficient, hitting the global accuracy mark with 1.5 × 10 5 FFT calls at σ tol = 3 × 10 −6 , while PSD does so with 2 × 10 5 FFT calls at a much lower σ tol = 10 −3 , and SAV with 2 × 10 5 FFT calls at σ tol = 3 × 10 −6 . However the efficiency of the PSD decays with global relative error above this acceptable upper limit, recovering only at very small global error. The overall result is a large interval in which the linear-implicit schemes slightly outperform PSD.
The ETD scheme is not competitive, requiring considerably more computational effort to achieve the same accuracy.
For the stiffer Foot 1 benchmark simulations with q = 0.2 the linear-implicit schemes perform at a similar level to the q = 0 benchmarks, while the nonlinearly implicit PSD experiences slower convergence in its nonlinear solver.
As shown in Figure 16 (left), the global error for each scheme is an approximately linear function of local truncation error on the log-log scale, corresponding to a power law exponent in the range 0.5 ∼ 0.6 that is slightly reduced from the 2/3 exponent observed for the super-critical benchmark. The computational efficiency plot, Figure 16 (right), the data for both Foot 1 and Foot 2 benchmarks are compared. The linear-implicit schemes substantially outperform the nonlinear-implicit PSD , with the IMEX scheme preserving its proportional efficiency over SAV over two orders of magnitude of global discretization error. For the linear-implicit schemes the computational cost is very similar for Foot 1 and 2, with the Foot 2 simulations slightly more accurate due to the increase in spatial resolution to N = 512.
Conversely, the nonlinear-implicit PSD requires significantly more effort with increasing q as the iteration count in the nonlinear solver increases significantly. The minimal cost for SAV to achieve the acceptable global discretization error is roughly 1.4 that of IMEX. It is worth noting that PSD is comparably more efficient at lower global error;
indeed it requires only 5 and 20 times the computational effort of IMEX to achieve an error of 7 × 10 −4 for the Foot 1 and Foot 2 benchmarks, respectively.
In Figure 17 the temporal trace of the global error is plotted for local truncation errors of σ tol = 10 −5 , 10 −6 , and 10 −7 . In all cases the PSD is the most accurate, generically by an order of magnitude at the same local truncation error. However the accuracy for PSD increases only modestly with decreasing σ tol while SAV and IMEX schemes have more significant improvements. For the sub-critical benchmark the global error accumulates slowly in each of the schemes as the shape of the interface evolves and inaccuracies in its location accumulate. For the super-critical benchmark the error has peaks at each of the major defect merging events that occur at t = 50, 150, 185, 210. These peaks reflect the impact of slight timing errors in the defect merging events and in the spatial structure of the merging transient. Each scheme shows about a half-order of magnitude loss of accuracy during the merging that is recovered afterwards. This holds except for the SAV and IMEX schemes with σ tol = 10 −5 which are both insufficiently accurate to capture the correct sequencing of the defect evolution.
Conclusion
We have demonstrated that the morphological complexity that develops within the gradient flows of the FCH energy requires accuracy for faithful representation. The benchmark problems place a complex labyrinth of saddle points between the initial data and the end state solution. The saddle points' energies differ by algebraically small orders of ε 1. Unlike problems in one space dimension which manifest exponentially long residence times, [8],
resolving the algebraically small differences in the energy landscape makes these benchmarks ideal: simple to code, quick to simulate, and effective at exposing the trade-offs between accuracy and efficiency in a stiff, highly non-convex problem.
These benchmarks model the chemical and material science problems for which computational accuracy is crucial. Small errors in the resolution of the structure of different configurations generate divergent alternate temporal evolutions and errors that grow to become leading order. The impact of this is magnified as the nonlinear stiffness in the model is increased. The nonlinear solve requires in the more strongly implicit PSD approach tends to raise the overall accuracy of the scheme, and for less-stiff forms of the model this compensates for the increased computational effort required for the iterative solver. The result is that the linear-implicit and nonlinear-implicit models are comparable. However for the more nonlinearly stiff versions of the model, the linear-implicit schemes require no tuning and experience only modest decline in efficiency, while the nonlinear-implicit PSD requires tuning of the error tolerance and maximum time-step parameters to optimize its performance. Despite this tuning the efficiency of the PSD scheme falls behind the linear-implicit schemes by a factor that is comparable to the increase in stiffness, as measured by the left-well concavity α m (q) = W q (b − ).
Within the linear-implicit schemes the performance of the IMEX and SAV schemes are almost indistinguishable.
Their global accuracy as a function of local truncation error are almost identical. The only discrepancy lies in the computational effort which is routinely a factor of 1.4 larger for the SAV scheme. This is a result of the extra steps required to resolve the larger SAV system of equations. Beyond the guarantee of the decay of the associated modified energy, it is difficult to identify a feature in the SAV scheme in which it improves upon the simpler IMEX approach.
Far and away the most important step in balancing the linear-implicit schemes is selecting a proper linear term for the implicit step. Given the theoretical understanding of the importance of the background state (the value of u away from non-trivial structures), it is reasonable and efficient to use the linearization about the spatially constant state u ≡ b − . We generalize this to the family of operators presented in (31), and find that the choice of β 1 + β 2 ≈ 3 provides optimal performance, with the choice β 1 = 2 and β 2 = 1 corresponding to the linearization about the spatially constant background state. These constant coefficient linear operators are trivially inverted in the spatially periodic setting considered herein. It certainly may not be the case that such a convenient and efficient linear-implicit operator is available in all systems. The ETD scheme does not have competitive accuracy for the FCH system. The ETD formulation has been proven effective at handling linear stiffness. It places the higher-order differential operators into a semi-group where they are more stable to discretization error. This advantage does not translate to the nonlinear stiffness that arises within the FCH model. Even in the more mild q = 0 model the large spatial gradients presented by the bilayers of the super-critical benchmark problem lead to amplification of error in the Runge-Kutta approximation. Figures 3I and 3N from [29], reprinted with Permission from the AAAS. Figures 5A, 5B, and 9C. Reprinted (adapted) with permission from [30]. Copyright (2004) American Chemical Society.
As a final demonstration of the complexity possible within the FCH gradient flow, we present a series of compu-tations that show a putative equilibrium state resulting from the gradient flow of the initial data from the super-critical benchmark, see Figure 18. The only variation is in the value of the parameter η 2 , which represents the aspect ratio of the amphiphilic molecule. The decreasing values of η 2 correspond to the increasing values of w PEO in the horizontal axis of the experimental bifurcation diagram presented in Figure 1. Perhaps the fundamental contribution of this numerical study lies in the suggestion that the shapes of the final structures produced in these casting problems are not uniquely determined by the properties and densities of the molecules they are composed of, but also depend upon the history of the morphology. Once defects are induced by transient flow, they become an integral part of the energy landscape and can entrap the gradient flow at a rich variety of local minima. These gradient flow transients form an intriguing phylogenesis, whose resolution requires significant accuracy.
µ n+1 = L 0 u n+1 +L 1ū n+1 +r n+1 X n+1 ,
3r n+1 − 4r n + r n−1 = Ω 1 2 X n+1 (3u n+1 − 4u n + u n−1 )dx,
whereū n+1 = 2u n − u n−1 and X n+1 =
V[ū n+1 ] √ E 1 [ū n+1 ]+D 0
. Taking the L 2 inner product of (47) with µ n+1 , and (48) with 3u n+1 −4u n +u n−1 , we have −2k ∇µ n+1 2 = (3u n+1 −4u n +u n−1 , µ n+1 ) = (L 0 u n+1 , 3u n+1 −4u n +u n−1 ) + (L 1ū n+1 , 3u n+1 −4u n +u n−1 ) + (r n+1 X n+1 , 3u n+1 −4u n +u n−1 )
=: I 1 + I 2 + I 3 .(50)
From the identity 2a(3a − 4b + c) = |a| 2 + |2a − b| 2 − |b| 2 + |2b − c| 2 + |a − 2b + c| 2 ,
we deduce that I 1 = 1 2 L 0 u n+1 , u n+1 + 1 2 L 0 (2u n+1 − u n ), 2u n+1 − u n − 1 2 (L 0 u n , u n ) − 1 2 L 0 (2u n − u n−1 ), 2u n − u n−1 + 1 2 L 0 (u n+1 − 2u n + u n−1 ), u n+1 − 2u n + u n−1 .
From the identity 2(2b−c)(3a−4b+c) = |a| 2 +|2a−b| 2 −2|a−b| 2 − |b| 2 +|2b−c| 2 −2|b−c| 2 − 3|a − 2b + c| 2 ,
we rewrite I 2 as I 2 = 1 2 L 1 u n+1 , u n+1 + 1 2 L 1 (2u n+1 − u n ), 2u n+1 − u n − L 1 (u n+1 − u n ), u n+1 − u n − 1 2 (L 1 u n , u n ) − 1 2 L 1 (2u n − u n−1 ), 2u n − u n−1 + L 1 (u n − u n−1 ), u n − u n−1 + 3 2 −L 1 (u n+1 − 2u n + u n−1 ), u n+1 − 2u n + u n−1 .
Multiplying (49) by 2r n+1 and using identity (51), we get I 3 = 2r n+1 3r n+1 − 4r n + r n−1 = |r n+1 | 2 + |2r n+1 − r n | 2 − |r n | 2 + |2r n − r n−1 | 2 + r n+1 − 2r n + r n−1 2 .
Finally, since L SAV := L 0 + L 1 we may combine (50), (52), (54) and (55) to deduce 0 ≥ − 2k ∇µ n+1 2 = 1 2 L SAV u n+1 , u n+1 + 1 2 L SAV (2u n+1 − u n ), 2u n+1 − u n − L 1 (u n+1 −u n ), u n+1 −u n + |r n+1 | 2 + |2r n+1 − r n | 2 − 1 2 L SAV u n , u n − 1 2 L SAV (2u n − u n−1 ), 2u n − u n−1 + L 1 (u n −u n−1 ), u n −u n−1 − |r n | 2 − |2r n − r n−1 | 2 + 1 2 L 0 (u n+1 − 2u n + u n−1 ), u n+1 − 2u n + u n−1 + 3 2 −L 1 (u n+1 − 2u n + u n−1 ), u n+1 − 2u n + u n−1 + |r n+1 −2r n + r n−1 | 2 .
= E aux u n+1 , u n , r n+1 , r n − E aux u n , u n−1 , r n , r n−1 + |r n+1 −2r n + r n−1 | 2 + 1 2 L 0 (u n+1 − 2u n + u n−1 ), u n+1 − 2u n + u n−1 + 3 2 −L 1 (u n+1 − 2u n + u n−1 ), u n+1 − 2u n + u n−1 .
Dropping the last three non-negative terms in (56), yields (38).
Fig. 1 :
1(left) Experimentally observed bifurcation diagram for the morphology of blends of Polyethylene oxide (PEO) -Polybutadiene (PB) amphiphilic diblock in water. The horizontal axis, w PEO , is the weight fraction of PEO as a percent of the total diblock weight, and the vertical axis denotes the molecular weights of the PB component of the diblock, fixed at N PB = 45 or 170 (vertical axis). Morphological Complexity is observed for N PB = 170 but not for the shorter N PB = 45 chains. (right) Experimental images from the morphological complexity regime showing (top) network structures and (bottom) a mixture of end caps and Y-junction morphology corresponding to regions marked N and C Y in the bifurcation diagram. FromFigures 1 and 2ACof[29], Reprinted with permission from AAAS.
N
P = 170 and C 0 = 3.0, and rescale the well W S by a factor of ν = 4.4. For the short-chain and long chain polymers the respective choices b l = −0.0097 and b l = −0.01 + 10 −7 sets the left well of W S at u = −1. The scaled W S is presented in
Fig. 2 :
2(left) Graph of scaled singular well W S as recovered by reduction of SCMF for N P = 45 (red) and N P = 170 (blue-dotted). (right) Graph of the regularized well, W q for q = 0, 0.2, 0.5.
o mesh with spacing h o = L/N o and λ 0 = 7.0269 × 10 −3 . With the choice R = 0.14725 the total mass of F[φ Γ ] per unit length of Γ approximates the mass of an exact bilayer dressing of Γ. For a fixed curve Γ we define φ 256 (x) := F[φ Γ ](x), which is clearly smooth and Ω-periodic. Now, let N be an arbitrary positive integer (typically a power of 2 in the Fourier pseudo-spectral setting), with h = L/N. For each of the benchmark cases we define the initial data to be
Fig. 3 :
3(left) A 1D cross-section of the grid function u 0 256 , along with finer mesh realizations u 0 512 and u 0 1024 . (right) The initial data u 0 512 constructed from (13) with width R = 0.14725 and d = 0. The red number on the colorbar indicates max
2 < i tol . The parameter λ in (22) is the search-step-size. Numerical investigations show that the optimal value of λ is somewhat sensitive to the value of α m = α m (q) and temporal step size k. This dependence is determined by minimizing the average number of PSD iterations for a fixed k over the first 50 temporal steps of the simulation. Optimal values of λ for different values of q and k are reported in
q λ k ≤ 10 −6 5 · 10 −6 10 −5 5 · 10 −5 10 −4 5 · 10 −4 10 −3 5 · 10
Figure 4 showsFig. 4 :
44FFT counts for simulations of IMEX and SAV using the dominant implicit term based on L 0 . Overall the schemes preform well if β 1 + β 2 = 3, with performance deteriorating dramatically for smaller values and slowly for larger values of this sum. Indeed values of β 1 + β 2 < 3 can lead to FFT counts that are several orders of magnitude higher per time-unit at a fixed local truncation error. The left panel provides total FFT counts for the IMEX scheme with β 2 = 3 − β 1 , showing that the performance is optimal so long as neither β 1 nor β 2 are too small. The right panel shows performance of the SAV scheme for each of the five benchmark problems. The choice β 1 = 2 is taken as the default for both IMEX and SAV. Total FFT calls verses stabilization parameters β 1 with β 2 = 3 − β 1 for IMEX and SAV for each of the 5 benchmark simulations. (left) Total FFT calls verses β 1 for IMEX. (right) Total FFT calls verses β 1 for SAV. The choice β 1 = 2 used in both IMEX and SAV simulations herein is indicated with black arrow.
τ 0 e
0L(τ−s) ds N(u n ) = e Lτ u n + L −1 e Lτ − I N(u n ). (43) The explicit second order ETD Runge-Kutta (ETDRK2) uses a linear approximation for N u(t n + s) ≈ (1− s τ )N(u n )+ s τ N(u n+1 ) for s ∈ [0, τ]. This yields the scheme
Fig. 5 :
5Simulation of the sub-critical benchmark with q = 0, σ tol = 10 −5 and N = 256 at times T = 10(left) and T = 250(right).
For the critical case the value of η 2 and d are tuned to create a strongly pearled interface and a long pearled transient, lasting roughly from T = 4 to T = 21. The bilayer interface pearls transiently, forming 21 pearls, whose discrete count generates a thresholding effect that slows the absorption of the dispersed amphiphilic polymer as the interface must generate new pearls to lengthen. During the 21-pearl transient period the pearled bilayer interface undergoes a "bicycle chain" meander in which adjacent pearls move in opposite directions, either in towards the center or out towards the boundary of the domain, as can be seen inFigure 6(left). At time T = 21 the pearls have reduced in size, and two extra pearls form at the points of highest curvature. The formation of the additional pearls facilitates an absorption of mass. As the background level of amphiphilic material is depleted the rate of absorption slows and the the interface returns to an unpearled state, similar to that depicted inFigure 5(right) that is able to move freely under a curvature driven motion. No endcap defects are formed in the critical benchmark, and each of the computational schemes are in quantitative agreement.
Fig. 6 :
6Simulation of the critical benchmark with q = 0, σ tol = 10 −5 and N = 256 at times T = 15(left) and T = 21(right). All schemes agree to within L 2 relative error 2 × 10 −3 as reported in
Fig. 7 :
7Simulations of the super-critical benchmark with q = 0, σ tol = 10 −5 and N = 256 at time T = 50(left) and T = 250(right).
Fig. 8 :
8Contour curves from each of the simulations of each of the schemes with σ tol = 10 −5 and N = 256. The level set u = −0.12 for (left) the sub-critical simulation at T = 10 and (right) the super-critical benchmark at T = 50.
Fig. 9 :
9Value of u − b − at center point (solid) and corner point (dashed) of computational domain for the sub-critical (left) and super-critical (right) benchmarks with σ tol = 10 −5 . Horizontal axis is log of time.
9 traces the evolution of the value of u − b − at the domain center (solid lines) and domain corner (dashed lines) for each of the simulation strategies. For the sub-critical simulation no defects are formed and the far-field values of u relax to a tight range of equilibrium values over the time frame T = 75 ∼ 100. The super-critical simulations have various defect merging events and each is associated with a small excursion in the background levels. In the inset of Figure 9 (right) these excursions can be seen at T = 210, 330, and 460 for the PSD scheme. For the ETD scheme the excursions are similar but can be delayed by up to T = 20. Conversely for the IMEX and SAV schemes the background levels are in close agreement, recording excursions T = 150, 350, and 500, but differ in both timing and in number of events from the more accurate PSD and ETD simulations.
Fig. 10 :
10Simulation of the Foot 1 benchmark with q = 0.2, σ tol = 10 −5 and N = 256 at times T = 1.5(left) and T = 50(right). All three schemes agree to within L 2 relative error 3 × 10 −3 as reported in
Fig. 11 :
11Simulation of the Foot 2 benchmark with q = 0.5, σ tol = 10 −5 at times T = 1 (left) and T = 50 (right) for N = 512. All three schemes agree to within L 2 error 1 × 10 −2 as reported in
Fig. 12 :
12(left) System energy verses time on a semilog-x scale for each of the five benchmark problems for each scheme with σ tol = 10 −5 . The boxed insets for the super-critical (middle) and foot 1 (right) benchmarks show more detail and include results for IMEX and SAV with σ tol = 10 −6 .
Fig. 13 :
13(left) Value of u − b − at center point (solid) and corner point (dashed) of the computational domain for the q = 0.5 Foot 2 benchmark for σ tol = 10 −5 and N = 512. (right) Evolution of the adaptive temporal step-size on a log-log scale for each of the four schemes for the q = 0 super-critical benchmark (solid) and the q = 0.5 Foot 2 benchmark (dashed). Horizontal axis is log of time.
Fig. 14 :
14Running value of max(u) from the PSD scheme for the critical benchmark problem with d = 0.7523, ..., 0.7728 in increments of 0.0001 when σ tol = 10 −7 . When accurately resolved the defect onset occurs at the critical value d = 0.7526.
Fig. 15 :
15(left: blue y-axis and lines) Global L 2 relative error verses σ tol at T = 250 for the q = 0 super-critical benchmark as measured by comparison to the most accurate solution. (left: red y-axis and lines) Computational cost verses σ tol as measured by total number of FFT calls. (right) "Dollars-per-digit" or computational cost verses global L 2 relative error, plotted parametrically in σ tol .
Fig. 16 :
16(left: blue y-axis and lines) Global L 2 relative error verses σ tol at T = 50 for the q = 0.2 Foot 1 benchmark as measured by comparison to the most accurate solution. (left: red y-axis and lines) Computational cost verses σ tol as measured by total number of FFT calls. (right) "Dollars-per-digit" or computational cost verses global L 2 relative error for Foot 1 and Foot 2 benchmarks, plotted parametrically in σ tol .
Fig. 17 :
17Time evolution of the global L 2 relative error between output of the three schemes and the highly accurate solution for σ tol = 10 −5 , 10 −6 , and 10 −7 for (top) the sub-critical benchmark and (bottom) the super-critical benchmark.
Fig. 18 :
18(top-row) Approximate equilibrium states computed from super-critical benchmark initial data and parameters except for values of η 2 taken as 2.55ε, 2.6ε, 2.65ε, 2.8ε, 2.85ε (left-to-right). Final times are T = 5K, 50K, 100K, 50K, 3K respectively. (bottom-row) Experimental comparisons showing (left-to-right) bubbles, bubbles with endcaps, bubbles and branched endcaps, long-branched filaments with endcaps, and double-sheeted bubbles (bubble inside of bubble).
Acknowledgement A. Christlieb acknowledges support from NSF grant DMS-1912183. K. Promislow acknowledges support from NSF grant DMS-1813203. Z. Tan recognizes support from the China Scholarship Council under 201906160032. B. Wetton recognizes support from a Canadian NSERC grant. S.M. Wise recognizes support from NSF grant DMS-2012634. Appendix A. Proof of the energy decay in SAV -Theorem 1 From the relations (34)-(35) the SAV scheme with fixed time-step k > 0 takes the form 3u n+1 −4u n +u n−1 2k = ∆µ n+1 ,
Table 1 :
1Parameters for Benchmark Cases.Case\Param
q
η 1
η 2
d
ε
γ
α m (q)
N
Mass
Sub-critical
0
1.45ε
3ε
0.2 0.1 0.3
1.7
256 6.11
Critical
0
1.45ε 1.5ε 0.75 0.1 0.3
1.7
256 7.61
Super-critical
0
1.45ε
3ε
0.5 0.1 0.3
1.7
256 6.93
Foot 1
0.2 1.45ε
3ε
0.2 0.1 0.3
5.1
256 6.11
Foot 2
0.5 1.45ε
3ε
0.2 0.1 0.3
10.2
512 6.11
Table 2 :
2Dependence of optimal value of search-step-size λ on temporal step size k.
Table 3 :
3Dependence of PSD iteration count on q, ν and k max .Iteration count/1000
ν(q)
Value of k max
optimal k max
0.009 0.01
0.02 0.03 0.04 0.05 0.06
q = 0
1.E-03
36.3
34.8 36.5 36.7 38.2 41.4
0.05
q = 0.2
2.E-05
43.3
42.8 43.7
0.02
q = 0.5
1.E-06 162.0 161.7 162.0
0.01
3.4. The BDF2-IMEX scheme
Table 4 :
4L 2 relative error between (PSD, IMEX, SAV, ETD) for each benchmark simulations at final time.Benchmark
IMEX/PSD SAV/PSD SAV/IMEX ETD/PSD
T
Sub-Critical
7.276E-03 7.315E-03 4.103E-05
250
Critical
2.204E-03 2.212E-03 1.796E-05
250
Super-Critical 8.817E-02 8.819E-02 4.156E-05
1.514E-02 250
Foot 1
2.358E-03 2.359E-03 1.702E-06
50
Foot 2
3.318E-04 3.322E-04 6.195E-07
50
Table 5 :
5L 2 grid refinement (absolute) error with the PSD scheme.
N
256 / 512 512 / 1024
Sub-Critical
6.218E-04
Critical
3.827E-04
Super-Critical 2.589e-04
Foot 1
8.502E-02
Foot 2
1.008
5.762E-04
Table 6 :
6L 2 temporal convergence errors and rates. The error is determined by comparison to PSD with a fixed temporal step size
k = 10 −6 and i tol = 10 −11 .
Schemes
IMEX
PSD
SAV
ETD
fixed k
L 2 Error Rate L 2 Error Rate L 2 Error Rate L 2 Error Rate
8 × 10 −2
2.20E-01
7.03E-05
2.20E-01
4 × 10 −2
5.38E-02 2.03 1.77E-05 1.99 5.38E-02 2.03
2 × 10 −2
1.37E-02 1.98 4.43E-06 2.00 1.37E-02 1.98
1 × 10 −2
3.54E-03 1.95 1.11E-06 2.00 3.54E-03 1.95
5 × 10 −3
9.31E-04 1.93 2.79E-07 1.99 9.31E-04 1.93 3.39E-01
2.5 × 10 −3
2.46E-04 1.92 8.36E-08 1.74 2.46E-04 1.92 1.37E-01 1.30
1.25 × 10 −3
6.47E-05 1.93 6.89E-08 0.28 6.46E-05 1.93 4.92E-02 1.48
6.25 × 10 −4
1.69E-05 1.94 6.62E-08 0.06 1.69E-05 1.94 1.61E-02 1.61
3.125 × 10 −4 4.37E-06 1.95 5.81E-08 0.19 4.37E-06 1.95 4.94E-03 1.71
1.563 × 10 −4 1.12E-06 1.96
1.12E-06 1.96 1.45E-03 1.77
3.907 × 10 −5
1.14E-04 1.83
9.769 × 10 −6
8.33E-06 1.89
2.442 × 10 −6
5.67E-07 1.94
Table 7 :
7The dependence of the critical value of d in (13) upon σ tol for each scheme.
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| []
|
[
"Robust D-stability of uncertain MIMO systems: LMI criteria *",
"Robust D-stability of uncertain MIMO systems: LMI criteria *"
]
| [
"Wang Long \nDepartment of Mechanics and Engineering Sciences\nCenter for Systems and Control\nInstitute of Automation\nPeking University\n100871BeijingP.R.China\n",
"Wang Zhizhen \nDepartment of Mechanics and Engineering Sciences\nCenter for Systems and Control\nInstitute of Automation\nPeking University\n100871BeijingP.R.China\n",
"Wu Baoyu \nDepartment of Mechanics and Engineering Sciences\nCenter for Systems and Control\nInstitute of Automation\nPeking University\n100871BeijingP.R.China\n",
"Yu Wensheng \nChinese Academy of Sciences\n100080BeijingP.R.China\n"
]
| [
"Department of Mechanics and Engineering Sciences\nCenter for Systems and Control\nInstitute of Automation\nPeking University\n100871BeijingP.R.China",
"Department of Mechanics and Engineering Sciences\nCenter for Systems and Control\nInstitute of Automation\nPeking University\n100871BeijingP.R.China",
"Department of Mechanics and Engineering Sciences\nCenter for Systems and Control\nInstitute of Automation\nPeking University\n100871BeijingP.R.China",
"Chinese Academy of Sciences\n100080BeijingP.R.China"
]
| []
| The focal point of this paper is to provide some simple and efficient criteria to judge the D-stability of two families of polynomials, i.e., an interval multilinear polynomial matrix family and a polytopic polynomial family. Taking advantage of the uncertain parameter information, we analyze these two classes of uncertain models and give some LMI conditions for the robust stability of the two families. Two examples illustrate the effectiveness of our results. | null | [
"https://arxiv.org/pdf/math/0211011v1.pdf"
]
| 2,264,843 | math/0211011 | ae81fa74c835bfb66c7811464582802b387179c5 |
Robust D-stability of uncertain MIMO systems: LMI criteria *
1 Nov 2002
Wang Long
Department of Mechanics and Engineering Sciences
Center for Systems and Control
Institute of Automation
Peking University
100871BeijingP.R.China
Wang Zhizhen
Department of Mechanics and Engineering Sciences
Center for Systems and Control
Institute of Automation
Peking University
100871BeijingP.R.China
Wu Baoyu
Department of Mechanics and Engineering Sciences
Center for Systems and Control
Institute of Automation
Peking University
100871BeijingP.R.China
Yu Wensheng
Chinese Academy of Sciences
100080BeijingP.R.China
Robust D-stability of uncertain MIMO systems: LMI criteria *
1 Nov 2002Interval multilinear polynomial matrixrobust D-stabilityPolytopic polynomial matrixLinear Matrix InequalitiesParametric uncertainty
The focal point of this paper is to provide some simple and efficient criteria to judge the D-stability of two families of polynomials, i.e., an interval multilinear polynomial matrix family and a polytopic polynomial family. Taking advantage of the uncertain parameter information, we analyze these two classes of uncertain models and give some LMI conditions for the robust stability of the two families. Two examples illustrate the effectiveness of our results.
Introduction
The study of robust stability problems under parameter uncertainties has been pioneered by the Russian scientist Kharitonov(1978). A rich array of useful results have been developed over the last twenty years [2]− [5], [7]− [14] . Generally speaking, by dealing directly and effectively with the real parameter uncertainties in control systems, we can identify apriori the critical subset of the uncertain parameter set over which stability will be violated. The seminal theorem of Kharitonov [2] points out: any real parameter interval polynomial family is Hurwitz if and only if a special subset (called Kharitonov set) is Hurwitz. To general uncertain systems, edge theorem gives a positive answer [3] , which is an one-dimensional test.
Consider the unity feedback system with an interval plant and a fixed controller in forward path, its characteristic polynomial is a multilinear function of certain interval variables [9,10,13] . That is to say, when all but one variables are fixed, all coefficients of the polynomial are affine linear in the remaining variable. The collection of all such models is called multilinear uncertainties structure. For an MIMO system, if all relationships between any input and any output belong to corresponding polytopes (the simplest form is a line), all admissible models form a polytopic polynomial matrix family. In the past one or two decades, there is continually growing interest in the robust analysis of matrix [6,9,10,11,13] . Unlike polynomial case, the vertex result does not hold for interval matrix. In fact, there does not even exist a result similar to Edge Theorem for general matrix. Many problems still remain open until now. Work reported to date shows that a reduced dimensional test holds [13] for polytopic polynomial matrices. Limited by the complexity of such problems, several methods have been proposed, such as eigenvalues estimation, Lyapunov approach, algebraic approach and spectrum theory, etc. [6]− [9] . Usually, the algebraic approach based on Kharitonov theorem is hardly effective and convenient when used for the robust analysis of matrix directly. Lyapunov approach is an appealing method developed in the context of Lyapunov theory, which presents in the form of Linear Matrix Inequalities(LMI). However, until now there is few useful result on robust test for matrix families under parameter uncertainties.
The purpose of this paper is to address the D-stability of interval multilinear polynomial matrix family and polytopic polynomial family. Even though they are nonlinear problems, we still can establish some efficient robust stability tests, which are usually negative for a general nonlinear case. Recent work addressed the D-stability problem for polytope of matrices using Lyapunov approach [9] and spectrum theory [6] . In this paper, we adopt both of two methods to analyze two classes of uncertain MIMO models and give several LMI criteria on robust stability. Our technique not only captures the uncertain parameter information, but also is easy to use. In the end two examples demonstrate the effectiveness of our results.
Definitions and Notations
In this paper, we use the following standard notations and definitions. Definition 1 Given an open convex region D in the complex plane, a scalar matrix is termed D-stable, if all its eigenvalues lie in D; and a polynomial matrix is termed D-stable, if all roots of its determinant lie in D; a matrix family is termed D-stable, if all its members are D-stable.
Definition 2 Let M be an arbitrary set, we define convM as the convex hull of M , i.e., the smallest convex set which contains M .
Definition 3 For a matrix A, its right null-space N A is defined as the space whose every element N A satisfies AN A = 0. For simplicity, we denote N A a basis for the right null-space of A.
Definition 4 A polynomial matrix is a matrix with all of its entries being polynomials; an interval multilinear polynomial matrix is a polynomial matrix with all of its entries being multilinear dependent on some interval coefficients; a family of such matrices is called interval multilinear polynomial matrix family, such as the model in (5); a polytopic polynomial matrix is a matrix with all of its entries being polytopic polynomials; a family of such matrices is called polytopic polynomial matrix family, such as the model in (6).
D = s ∈ C : 1 s * B 1 s < 0 ,
where B is an 2 × 2 matrix and B * = B. D is called an LMI region([8] [16]).
Definition 6
For every i ∈ {1, . . . , N }, A i (s) is an n × n polynomial matrix of the form
A i (s) = A i 0 + A i 1 s + . . . + A i l s l ,(1)
and the n × nl scalar matrix A i
△ = (A i 0 , . . . , A i l ) is the coefficient matrix of A i (s).
Definition 7 For every i ∈ {1, . . . , n}, j ∈ {1, . . . , n}, P ij (s) is a polytope of polynomials , i.e.,
P ij (s) = m k=1 λ k p (k) ij (s) : λ k ≥ 0, m k=1 λ k = 1 ,(2)
where p
(1) ij (s), . . . , p (m)
ij (s) are fixed kth-order polynomials. Apparently, the vertex set of every P ij (s) is
K ij (s) = {p (k) ij (s), k = 1, . . . , m}(3)
and the exposed edge set of every P ij (s) is included in the following set
E ij (s) = λp (k) ij (s) + (1 − λ)p (t) ij (s), k, t = 1, . . . , m, k = t, λ ∈ [0, 1] .(4)Definition 8 Denote Q = (q 1 , . . . , q m ) T : q i ∈ [q L i , q U i ] . a 1 (q 1 , . . . , q m ), . . . , a N (q 1 , . . . , q m ) are multilinear functions of q 1 , . . . , q m . Definition 9 R = I . . . I O . . . O O . . . O I . . . I
is a 2nl × (l + 1)n-dimensional scalar matrix and I, O are n × n unity, zero matrices respectively. As usual, ⊗ stands for the Kronecker product.
Preliminary results
In what follows, two kinds of uncertain models are considered:
MA(s) = N i=1 a i (q 1 , . . . , q m )A i (s), (q 1 , . . . , q m ) T ∈ Q (5) PA(s) = {(p ij (s)) n×n : p ij (s) ∈ P ij (s), i, j = 1, . . . , n}(6)
The vertex sets of MA(s) and PA(s), respectively, are
K MA (s) = N i=1 a i (q 1 , . . . , q m )A i (s), q j ∈ {q L j , q U j }, j = 1, . . . , m. K PA (s) = {(p ij (s)) n×n : p ij (s) ∈ K ij (s), i, j = 1, . . . , n}.(7)
Let P n n be the collection of all permutations of 1, 2, . . . , n, and define E PA (s) = (p ij (s)) n×n : p kl k (s) ∈ E kl k (s), (l 1 , . . . , l n ) ∈ P n n , k = 1, . . . , n p ki k (s) ∈ K ki k (s), i k = 1, . . . , l k − 1, l k + 1, . . . , n
It is easy to see that, E PA (s) is a subset of PA(s) produced by taking only one entry from its exposed edge set in every row/column and all other entries from their vertex sets.
The lemma below is due to Henrion, et al.
Lemma 1[9] A polynomial matrix A i (s) is stable if there exists a matrix P i solving the LMI feasibility problem N * A i R * (B ⊗ P i )RN A i < 0, P i = P * i > 0. Where N A i is the right null-space of A i ,
and * denotes the conjugate transpose operator. Lemma 2 [3] (Edge Theorem) Suppose D ⊂ C is a simply-connected region, for any polynomial polytope Ω without degree dropping, the root set of Ω is contained in D if and only if the root set of all exposed edges of Ω is contained in D.
Another lemma is on the D-stability of the family PA(s),
Lemma 3[13] PA(s) is D-stable if and only if E PA (s) is D-stable. Proof: For all A 0 (s) ∈ PA(s), let A 0 (s) = (p ij (s)) n×n , where p ij (s) ∈ P(s). For simplicity, we write p ij for every p ij (s) ∈ P ij (s).
In what follows, we will construct several sets in terms of
A 0 (s). Let A k = {A k (i 1 , . . . , i n ; s), i 1 , . . . , i k ∈ {1, . . . , n}} (k = 0, . . . , n), where A k (i 1 , . . . , i n ; s) = q 11 . . . q 1k . . . . . . . . . q i 1 1 . . . q i 1 k . . . . . . . . . (p 0 vt ) n×(n−k) q i k 1 . . . q i k k . . . . . . . . . q n1 . . . q nk q lt ∈ E itt (s), l = i t K itt (s), l = i t p 0 lv are entries of A t = 1, . . . , k l = 1, . . . , n v = k + 1, . . . , n
It is easy to see that A 0 (s) = A 0 and A k ⊂ A k+1 . In the sequel, we will prove our statement in two steps:
1) Firstly, we will show that A n is D-stable if E PA (s) is D-stable. By definition, we have A n = A n (i 1 , . . . , i n ; s) = q 11 . . . q 1n . . . . . . . . . q n1 . . . q nn q lt ∈ E itt (s), l = i t K itt (s), l = i t t = 1, .
. . , n l = 1, . . . , n :
i 1 , . . . , i n ∈ {1, . . . , n} .
For all A n (i 1 , . . . , i n ; s) ∈ A n , if (i 1 , . . . , i n ) ∈ P n n , then A n (i 1 , . . . , i n ; s) ∈ E PA (s). Otherwise, there must exist some pair i s , i t satisfying i s = i t . Without loss of generality, suppose i 1 = i 2 = 1, namely
A n (i 1 , . . . , i n ; s) = q 11 q 12 . . . q 1n . . . . . . . . . . . . q n1 q n2 . . . q nn q 11 ∈ E 11 (s) q 12 ∈ E 12 (s)
By using Laplace formula on the first row of A n (i 1 , . . . , i n ; s), we have det(A n (i 1 , . . . , i n ; s)) = q 11 M 11 + q 12
M 12 + n i=3 q 1i M 1i
where M 1i is the algebraic complement of q 1i . By Lemma 2, A n (i 1 , . . . , i n ; s) is D-stable ⇔ q 11 M 11 + q 0 12 M 12 + n i=3 q 1i M 1i and q 0 11 M 11 + q 12 M 12 + n i=3 q 1i M 1i are D-stable. where q 0 11 ∈ K 11 , q 0 12 ∈ K 12 . For these two uncertain matrices, if they do not belong to E P A (s), then there must exist at least two equal indexes. Repeat the same process to them, in the end, we have
E PA (s) is D-stable ⇒ A n is D-stable.
2) Secondly, for all A n (i 1 , . . . , i n ; s) ∈ A n , by using Laplace formula on the n-th column of A n (i 1 , . . . , i n ; s) and by Lemma 2, we have
A n is D-stable ⇒ A n−1 is D-stable.
Continuing this process, we have
A k is D-stable ⇒ A k−1 is D-stable. Since A 0 (s) = A 0 , 1) and 2) imply that A 0 (s) is D-stable. That is to say, if E PA (s) is D-stable, then, for all A 0 (s) ∈ PA(s), A 0 (s) is D-stable. By definition, this means that PA(s) is D-stable.
This completes the proof of Sufficiency.
Necessity is obvious because E PA (s) is a subset of PA(s). ⋄
In this paper, we assume that both PA(s) and MA(s) have fixed degrees.
Main results
Although overbounding is a bit conservative, it still offers a powerful tool to solve the stability problem for control systems with multilinear uncertainties. Theorem 1 MA(s) ⊂ conv{K MA (s)}. Proof: For any A 0 (s) ∈ MA(s), by induction, we will show A 0 (s) ∈ conv{K MA (s)}. Denote q = (q 1 , . . . , q t , q t+1 , . . . , q m ) T ∈ Q, where q 1 , . . . , q t are interval parameters and q t+1 , . . . , q m are fixed.
If t = 1, i.e., q 1 ∈ [q L 1 , q U 1 ] and q 2 , . . . , q m are are fixed. Then, in this case,
MA(s) = N i=1 a i (q 1 , . . . , q m )A i (s), q 1 ∈ [q L 1 , q U 1 ] K MA (s) = N i=1 a i (q 1 , . . . , q m )A i (s), q 1 ∈ {q L 1 , q U 1 } .
Since a 1 (q 1 , . . . , q m ), . . . , a N (q 1 , . . . , q m ) are linear in q 1 , N i=1 a i (q 1 , . . . , q m )A i (s) is also linear in q 1 . Clearly, A 0 (s) ∈ conv{K MA (s)}.
Assume that the claim holds for t = k. When t = k + 1, we have, in this case, Since a 1 (q 1 , . . . , q m ), . . . , a m (q 1 , . . . , q m ) are linear in q k+1 , we have
MA(s) = N i=1 a i (q 1 , . . . , q m )A i (s), q i ∈ [q L i , q U i ], ia i (q 0 1 , . . . , q 0 k , q 0 k+1 , . . . , q 0 m ) = λ 0 a i (q 0 1 , . . . , q 0 k , q L k+1 , q 0 k+2 , . . . , q 0 m ) +(1 − λ 0 )a i (q 0 1 , . . . , q 0 k , q U k+1 , q 0 k+2 , . . . , q 0 m )
for some λ 0 ∈ [0, 1]. Therefore,
A 0 (s) = N i=1 λ 0 a i (q 0 1 , . . . , q 0 k , q L k+1 , q 0 k+2 , . . . , q 0 m ) + (1 − λ 0 )a i (q 0 1 , . . . , q 0 k , q U k+1 , q 0 k+2 , . . . , q 0 m ) A i (s) = λ 0 N i=1 a i (q 0 1 , . . . , q 0 k , q L k+1 , q 0 k+2 , . . . , q 0 m )A i (s) +(1 − λ 0 ) N i=1 a i (q 0 1 , . . . , q 0 k , q U k+1 , q 0 k+2 , . . . , q 0 m )A i (s) Since both N i=1 a i (q 0 1 , . . . , q 0 k , q L k+1 , q 0 k+2 , . . . , q 0 m )A i (s) and N i=1 a i (q 0 1 , . . . , q 0 k , q U k+1 , q 0 k+2 , . . . , q 0 m )A i (s) belong to the set conv N i=1 a i (q 1 , . . . , q m )A i (s), q i ∈ {q L i , q U i }, i = 1, . . . , k q k+1 , . . . , q m are fixed. , we conclude A 0 (s) ∈ conv N i=1 a i (q 1 , . . . , q m )A i (s), q i ∈ {q L i , q U i }, i = 1, . . . , k + 1 q k+2 , . . . , q m are fixed.
.
That is to say, the claim holds also for t = k +1. Therefore, our conclusion is verified inductively. ⋄ With Lemma 1 and Theorem 1, we get an LMI condition for robust stability of interval multilinear polynomial matrix family MA(s):
Theorem 2 MA(s) is robust D-stable if there exist some matrices P i = P * i > 0, Q solving the LMI feasibility problem R B i * B ⊗ P i Q Q * 0 R B i < 0, i = 1, . . . , 2 m .(9)
Proof: For every A(s) ∈ MA(s), by virtue of theorem 1, there exist λ 1 , . . . ,
λ 2 m ∈ [0, 1] such that 2 m i=1 λ i = 1 and A(s) = 2 m i=1 λ i B i (s). Moreover, for all i ∈ {1, . . . , 2 m }, R B i * B ⊗ P i Q Q * 0 R B i < 0 ⇔ R * (B ⊗ P i ) R + B * i Q * R + R * QB i < 0 ⇒ 2 m i=1 λ i (R * (B ⊗ P i )R + B * i Q * R + R * QB i ) < 0 ⇔ R * B ⊗ ( 2 m i=1 λ i P i ) R + ( 2 m i=1 λ i B i ) * Q * R + R * Q( 2 m i=1 λ i B i ) < 0
Multiplying N A from the right and N * A from the left, the inequality becomes
N * A R 2 m i=1 λ i B i * B ⊗ ( 2 m i=1 λ i P i ) Q Q * 0 R 2 m i=1 λ i B i N A < 0 Because of N A = N 2 m i=1 λ i B i , we have N * A R * B ⊗ ( 2 m i=1 λ i P i ) RN A < 0 From P * i = P i > 0, we have that 2 m i=1 λ i P i = ( 2 m i=1 λ i P i ) * > 0
. Now by the Lemma 1, the conclusion is obvious.
⋄ Remark 1 The standpoint of Theorem 2 is to transform stability problem into a positive real-like condition, and the latter can be solved using the LMI toolbox.
In regard to Lemma 1 and Lemma 3, we claim that the stability of uncertain family PA(s) can be inferred from whether an LMI condition holds or not for K PA (s). This is shown by the following two theorems. Theorem 3 PA(s) is D-stable ⇔ conv(K PA (s)) is D-stable. Proof: Sufficiency: We will show E PA (s) ⊂ conv(K PA (s)).
For all A 1 (s) ∈ E PA (s), by the definition of E PA (s), there exists (l 1 , . . . , l n ) ∈ P n n such that A 1 (s) = (p ij (s)) n×n : p kl k (s) ∈ E kl k (s), k = 1, . . . , n, p ki k (s) ∈ K ki k (s) i k = 1, . . . , l k − 1, l k + 1, . . . , n Using addition of matrices, we have
A 1 (s) = m t=1 λ 1t A (t) 1l 1 where A (t)
1l 1 is the matrix that all its entries coincide with A 1 (s) except one, which lies in the first row and the l 1 -th column and equals to p For every A (t) 1l 1 (t = 1, . . . , m), applying the same process to p 2l 2 , we can find m uncertain matrix families, and for every matrix which belongs to one of those families, all its entries in the first and second rows belong to the corresponding vertex sets. Continuing this procedure, we will get A 1 (s) ∈ conv(K PA (s)). Therefore, E PA (s) ⊂ conv(K PA (s)). Then,
conv(K PA (s)) is D-stable ⇒ E PA (s) is D-stable ( by Lemma 3) ⇔ PA(s) is D-stable.
Necessity: The relationship between PA(s) and conv(K PA (s)) can be easily established, thereby Necessity is proved. For all A 1 (s) ∈ conv(K PA (s)), there exist n 2 numbers λ ij ∈ [0, 1] n i,j=1 λ ij = 1 and n 2 matrices F ij (s) ∈ K PA(s) such that
A 1 (s) = n i,j=1 λ ij F ij (s). Denote F ij (s) = f ij hl (s) n×n
with f ij hl (s) ∈ K hl (s) for all h, l ∈ {1, . . . n}. By addition of matrices,
A 1 (s) = n i,j=1 λ ij f ij hl (s) n×n
For every h ∈ {1, . . . , n}, l ∈ {1, . . . , n}, n i,j=1 λ ij f ij hl (s) still belongs to P hl (s) whenever λ ij ∈ [0, 1], n i,j=1 λ ij = 1 and f ij hl (s) ∈ K hl (s). Thus conv(K PA (s)) ⊂ PA(s). This completes the proof. ⋄
For all A(s) ∈ PA(s), we have A(s) = (p ij (s)) n×n , where p ij (s) ∈ P ij (s) with degree l.
Rewriting it as
A(s) = A 0 + A 1 s + . . . + A l s l . Denote A △ = (A 0 , . . . , A l )
, then A is an n × nl scalar matrix. By Theorem 3, we have Theorem 4 PA(s) is robust D-stable if there exist some matrices P A = P * A > 0, Q solving the LMI feasibility problem
R A * B ⊗ P A Q Q * 0 R A < 0, A ∈ K PA (s).(10)
Proof: By Theorem 3, this problem is equivalent to the D-stability of conv(K PA (s)). For the latter, for all A(s) ∈ conv(K PA (s)), there exist n 2 numbers λ ij ∈ [0, 1], n i,j=1 λ ij = 1 and n 2 matrices F ij (s) ∈ K PA(s) such that
A(s) = n i,j=1 λ ij F ij (s).
Now by a similar argument as in the proof of Theorem 2, we get the desired result. ⋄
Illustrative Examples
In this section, we give examples to illustrate the utility of our main results. Example 1 is considered in the context of robust stability of interval multilinear polynomials with respect to left half plane. Example 1 (n=1, N=2, l=3, m=3) Let A 1 (s), A 2 (s) be two given polynomials A 1 (s) = s 3 + 2.64s 2 + 1.82s + 0.37 (11) A 2 (s) = s 3 + 5.57s 2 + 9.04s + 3.85 (12) And the uncertain model is A(s) = {a 1 (q 1 , q 2 , q 3 )A 1 (s) + a 2 (q 1 , a 2 , q 3 )A 2 (s)}, where q 1 ∈ [1, 2], q 2 ∈ [3, 3.8], q 3 ∈ [0.5, 0.8] and a 1 (q 1 , q 2 , q 3 ) = 0.6q 1 + 0.1q 2 − q 3 + 0.1q 1 q 2 , a 2 (q 1 , q 2 , q 3 ) = −0.6q 1 − 0.1q 2 + q 3 + 1 − 0.01q 2 q 3 . For Hurwitz stable, the 2 × 2 Hermite matrix B corresponds to 0 1 1 0 . Applying Theorem 2 to this problem, it suffices to solve 16 linear matrix inequalities.
Using the LMI Toolbox in M atlab, then it is easy to check that the corresponding LMI problem is feasible. Thus, we conclude that the whole polynomials family is robust Hurwitz stable. Example 2 (n=3, N=2, l=3, m=3) Consider the third order uncertain model below A(s) = {a 1 (q 1 , q 2 , q 3 )A 1 (s) + a 2 (q 1 , a 2 , q 3 )A 2 (s)} A 1 (s) = , q 3 ∈ [1.5, 1.8] and a 1 (q 1 , q 2 , q 3 ) = q 1 − q 2 + q 3 + 0.1q 1 q 2 , a 2 (q 1 , q 2 , q 3 ) = −q 1 + q 2 − q 3 + 1 − 0.01q 2 q 3 . In this Example, the quadratic stability region is the unity circle, thus the associated matrix is −1 0 0 1 . Solving the corresponding LMI in Theorem 2, it is easy to show that the uncertain model is robust Schur stable. Remark 2 Our results can also be verified by the plots of root loci of the whole polynomials family in following figures. From the plots of root loci, we can see that our LMI criteria are not very conservative, and can provide correct, effective information on robust stability of uncertain systems.
Conclusion
In this paper, we have dealt with the performance robustness of interval multilinear polynomial matrix families and polytopic polynomial matrix families. Some computationally tractable and nonconservative sufficient conditions for these two classes of system models have been obtained.
Definition 5
5Let D ⊂ C be an open convex set of the form
= 1 ,
1. . . , k + 1 q k+2 , . . . , q m are fixed.K MA (s) = N i=1 a i (q 1 , . . . , q m )A i (s), q i ∈ {q L i , q U i }, i = 1, .. . , k + 1 q k+2 , . . . , q m are fixed.
For all A 0
0(s) ∈ MA(s), there exists an m-dimensional vector q 0
For
all B i (s) ∈ K MA (s), rearrange it as B i (s) = B i 0 +B i 1 s+. . .+B i l s l . Take B i = (B i 0 , . . . , B i l ) the coefficient matrix of B i (s). It is easy to see that there exist 2 m distinct B i (s). Hence, conv{K A (s)} = conv{B 1 (s), . . . , B 2 m (s)}.
For p 1l 1
1(s) ∈ E 1l 1 (s), we know that there exist m real numbers λ 11 , . . . , λ 1m ∈ [
for every t ∈ {1, . . . , m}. Thus, for everyone of {A
}, all of its entries in the first row belong to vertex sets.
12 . . . q 1n . . . . . . . . . . . . q n1 q n2 . . . q nn 1n . . . . . . . . . . . . q n1 q n2 . . . q nnThe corresponding matrices are
q 0
11 q
and
q 11 q 0
12 . . . q
,
− 16s 2 + 15s + 7.6 25s 3 − 2.2s 2 − 3.8s + 0.035 20s 3 − 13s 2 − 18s − 0.96 −9.8s 2 + 14s − 4.5 6s 2 + 17s + 7.1 −3.7s 3 + 13s 2 + 0.078s − 6.7 19s 3 + 0.35s 2 − 7.5s + 3.4 −1.3s 2 + 1.1s + 1.5 −20s 3 − 4.5s 2 + 9.4s − 0.27 −11s 3 − 5.5s 2 − 20s + 3.8 10s 3 − 1.9s 2 − 10s − 5.1 with q 1 ∈ [1, 1.2], q 2 ∈ [2.1, 2.4]1
10
15s 3 + 2.5s 2 + 12s − 1
10s 2 − 7.4s + 2.8
12s 2 − 1.1s − 4.7
13s 3 + 12s 2 + 0.22s − 1.1
23s 3 + 4.7s 2 − 1.3s + 3
2.8s 2 + 0.99s − 0.5
2.7s 3 − 7.3s 2 − 7.9s − 3.7
11s 3
A 2 (s) = 1
10
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| []
|
[
"Six-dimensional Supergravity and Projective Superfields",
"Six-dimensional Supergravity and Projective Superfields"
]
| [
"William D Linch Iii ",
"Gabriele Tartaglino-Mazzucchelli [email protected]. ",
"\nDepartamento de Ciencias Físicas\nFacultad de Ciencias Exactas\nSantiago de Chile and Center for String and Particle Theory\nDepartment of Physics\nTheoretical Physics\nDepartment of Physics and Astronomy\nUniversidad Andres Bello\nUniversity of Maryland\n20472College ParkMD\n",
"\nSchool of Physics M013\nUppsala University\nBox 516SE-751 20UppsalaSweden\n",
"\nThe University of Western\nAustralia\n",
"\nStirling Highway\nCrawley W.A. 6009Australia\n"
]
| [
"Departamento de Ciencias Físicas\nFacultad de Ciencias Exactas\nSantiago de Chile and Center for String and Particle Theory\nDepartment of Physics\nTheoretical Physics\nDepartment of Physics and Astronomy\nUniversidad Andres Bello\nUniversity of Maryland\n20472College ParkMD",
"School of Physics M013\nUppsala University\nBox 516SE-751 20UppsalaSweden",
"The University of Western\nAustralia",
"Stirling Highway\nCrawley W.A. 6009Australia"
]
| []
| We propose a superspace formulation of N = (1, 0) conformal supergravity in six dimensions. The corresponding superspace constraints are invariant under super-Weyl transformations generated by a real scalar parameter. The known variant Weyl super-multiplet is recovered by coupling the geometry to a super-3-form tensor multiplet. Isotwistor variables are introduced and used to define projective superfields. We formulate a locally supersymmetric and super-Weyl invariant action principle in projective superspace. Some families of dynamical supergravity-matter systems are presented. | 10.1007/jhep08(2012)075 | [
"https://arxiv.org/pdf/1204.4195v3.pdf"
]
| 118,830,807 | 1204.4195 | 0738751546531f7e669a6027f6ca08bc8f45b8f3 |
Six-dimensional Supergravity and Projective Superfields
12 Jun 2012 June, 2012
William D Linch Iii
Gabriele Tartaglino-Mazzucchelli [email protected].
Departamento de Ciencias Físicas
Facultad de Ciencias Exactas
Santiago de Chile and Center for String and Particle Theory
Department of Physics
Theoretical Physics
Department of Physics and Astronomy
Universidad Andres Bello
University of Maryland
20472College ParkMD
School of Physics M013
Uppsala University
Box 516SE-751 20UppsalaSweden
The University of Western
Australia
Stirling Highway
Crawley W.A. 6009Australia
Six-dimensional Supergravity and Projective Superfields
12 Jun 2012 June, 20121 Corresponding address as of June, 2012.
We propose a superspace formulation of N = (1, 0) conformal supergravity in six dimensions. The corresponding superspace constraints are invariant under super-Weyl transformations generated by a real scalar parameter. The known variant Weyl super-multiplet is recovered by coupling the geometry to a super-3-form tensor multiplet. Isotwistor variables are introduced and used to define projective superfields. We formulate a locally supersymmetric and super-Weyl invariant action principle in projective superspace. Some families of dynamical supergravity-matter systems are presented.
Introduction
Recently, new off-shell superspace techniques have been developed to study general supergravity-matter systems with eight real supercharges in various space-time dimensions. These are based on the use of projective superspace, introduced in the 1980s by Karlhede, Lindström, and Roček to study rigid 4D, N = 2 supersymmetry [1,2]. 1 Analogously to harmonic superspace [6,7], projective superspace is based on the extended superspace R 4|8 × CP 1 where the projective coordinates are related to the SU(2) Rsymmetry group of the extended supersymmetry algebra, an idea first introduced in the seminal paper [8].
The first attempt to extend projective supermultiplets to curved supersymmetry was undertaken in 2007 in a study of matter couplings in 5D, N = 1 anti de-Sitter superspace [9]. This was subsequently adapted to supergravity in various dimensions in a series of papers, chronologically: 5D in [10,11]; 4D in [12,13]; 2D in [14]; and 3D in [15]. The formalism is based on two central ingredients: (i) a covariant geometric description in superspace of the supergravity multiplet; (ii) the existence of covariant projective multiplets, which are generalizations of the superconformal projective multiplets introduced by Kuzenko in [16,17], and a locally supersymmetric and super-Weyl invariant action principle that is consistently defined on the curved geometry of ingredient (i).
In many respects, the curved projective superspace formalism has proven to resemble the covariant Wess-Zumino superspace approach to 4D, N = 1 supergravity [18]. However, while a prepotential description of 4D, N = 2 conformal supergravity was given in harmonic superspace in [19], its relationship to standard geometrical methods of curved superspace has not yet been elaborated in detail. A synthesis of curved harmonic and projective superspace could provide a coherent superspace description of 4D, N = 2 supergravity, along the lines of the Gates-Siegel approach to the 4D, N = 1 case [20]. Besides the calculational advantages this affords (e.g. background field quantization), such an understanding has applications in covariant string theory. These descriptions are (necessarily) closely related to the projective [21,22] and harmonic superspace formalisms [23,24]. A particularly relevant example is that of the pure spinor formalism [25] compactified on a K3 surface, where the physical state conditions on the unintegrated vertex operators are automatically formulated in terms of analyticity conditions in 6D, N = (1, 0) projective superspace [26,27]. Addition of the "missing" harmonics as non-minimal variables allows for a (simpler) description of the physical state conditions and the integrated vertex operators in harmonic superspace [27]. This paper is devoted to the continuation of the aforementioned program and to the demonstration that a projective superspace formalism can be efficiently implemented also in the case of six space-time dimensions. As a step toward the definition of six-dimensional curved projective multiplets, one first needs to indentify a proper geometric description in superspace of off-shell, N = (1, 0) supergravity. In a standard fashion, a starting point to describe off-shell supergravity systems is the coupling of the Weyl multiplet of conformal supergravity to matter compensators. This is possible both in components, through the superconformal tensor calculus techniques (see [28] for standard references), and in superspace. In components, the Weyl multiplet of 6D, N = (1, 0) conformal supergravity was constructed in reference [29]. To our knowledge, however, a geometric description of the Weyl multiplet in six dimensions, analogous in spirit to the 4D, N = 2 geometry of Howe [30], has hitherto not been fully developed. 2 In this paper, we begin to fill this gap by presenting a superspace geometry suitable to the description of N = (1, 0) conformal supergravity in six dimensions. Specifically, our geometry naturally describes the 40+40 components of the Weyl supermultiplet of [29] in superspace, in the form having the "matter" components of the multiplet described by an anti-self-dual 3-form W − abc , a positive-chirality spinor χ αi , and a real scalar D. We will refer to this Weyl multiplet as the type-i multiplet. In reference [29], it was shown that there is a second 40+40 Weyl multiplet possessing as matter fields a scalar σ, a 2-form tensor B ab , and a negative chirality spinor ψ αi ; we will refer to this as the type-ii Weyl multiplet. Such a formulation is engineered by coupling the type-i multiplet to an on-shell tensor multiplet [32,33] and solving for the type-i matter fields in terms of the fields of the tensor by using the equations of motion of the latter. The same mechanism can be used to describe the type-ii Weyl multiplet in superspace as we will show by coupling the type-i superspace geometry to a tensor multiplet described in terms of a closed super 3-form (first introduced in the flat case in [34]).
Having constructed a superspace geometry suitable to the description of six-dimensional Weyl multiplets, the consistent definition of six-dimensional covariant projective superfields in this supergravity background proceeds exactly as in the lower-dimensional cases. In this paper, we will focus on such technical problems as the construction of the 6D, N = (1, 0) multiplets, the projective action principle, and the analytic projection operator. We leave the applications of our results, some of which we set out in the Conclusion (section 4), for future investigation. Our hope is that the techniques we are starting to develop here will be of use not only in extending classic results (e. g. [35,36]) but also newer ones which have arisen in the resurgence of interest in 6D, N = (1, 0) supersymmetry and supergravity; see, for example, [37,38,39,40,41]. This work is organized into two parts with the supergeometrical part concentrated in section 2 and the projective superspace part presented in section 3. We begin the first part with the construction of the curved superspace geometry and give the dimension ≤ 3 2 commutator algebra and torsion constraints in section 2.1. In section 2.2, we give the super-Weyl transformations compatible with this geometry and use them to elucidate the relation to the type-i multiplet of the superconformal tensor calculus. In section 2.3, we solve the Bianchi identities of a closed super-3-form in the type-i background and re-interpret the resulting supergravity-matter system in terms of the type-ii Weyl multiplet. The second part begins with the construction of covariant projective superfields in six dimensions and the analytic projection operator in section 3.1. In section 3.2, we define the projective action principle, prove its consistency, and give families of examples of dynamical projective supergravity-matter systems. We conclude in section 4 with some reflection on our results and a description of future work and open problems. Our conventions are defined in appendix A and the requisite properties of the analytic projection operator are demonstrated in appendix B.
6D, N = (1, 0) Supergravity in Superspace
In this section, we present a new curved superspace geometry suitable to the description of N = (1, 0) conformal supergravity in six dimensions. In the spirit of Howe and Tucker [42], we will see that the geometry is invariant under super-Weyl transformations generated by an unconstrained real scalar superfield. For this reason, the geometry will describe the 40+40 components of the type-i Weyl multiplet and, once coupled to a tensor multiplet super 3-form, the type-ii Weyl multiplet. We refer the reader to the following list of references for previous work on flat superspace and multiplets in six dimensions: [33,32,43,44,45,46,47]. For the use of curved superspace to describe supergravity multiplets in six dimensions, see [48,49,50,51,52,53,54].
Our goal is to develop a formalism of differential geometry in a curved six-dimensional, N = (1, 0) superspace M 6|8 that is locally parametrized by real bosonic (x m ) and real fermionic (θ µ i ) coordinates
z M = (x m , θ µ i ) , m = 0, · · · , 3; 5, 6 , µ = 1, 2 , i = 1, 2 . (2.1)
A natural condition on such a geometry is that it reduce to six-dimensional, N = (1, 0) Minkowski superspace in the flat limit . Let us, to this end, recall that the spinor covariant derivatives D i α associated with 6D, N = (1, 0) Minkowski superspace satisfy the anticommutation relations
{D i α , D j β } = −2i ε ij (γ c ) αβ ∂ c . (2.2)
An explicit realization of D i α is given by the expression
D i α = ∂ ∂θ α i + i (γ b ) αβ θ βi ∂ b . (2.3)
Given a superfield F of Grassmann parity ǫ(F ), the conjugation rule of its covariant derivative is
(D i α F ) = −(−) ǫ(F ) D αiF ,(2.4)
withF := (F ) * the complex conjugate of F . Details of our notation and conventions are given in appendix A.
The Algebra of Covariant Derivatives
For our curved geometry, we choose the structure group to be SO(5, 1) × SU(2). The covariant derivative (D A ) = (D a , D i α ) expands as
D A = E A + Ω A + Φ A , (2.5) where E A = E A M (z)∂ M , Ω A = 1 2 Ω A bc (z)M bc , Φ A = Φ A ij (z)J ij ,(2.6)
denote the frame form, the spin connection, and the SU(2) connection, respectively. Here, ∂ M = ∂/∂z M , M ab = −M ba is the Lorentz generator and J ij = J ji is the SU(2) Rsymmetry generator. These are defined by their action on the spinor covariant derivatives as
[M ab , D k γ ] = − 1 2 (γ ab ) γ δ D k δ , [J ij , D k γ ] = ε k(i D j) γ . (2.7)
It follows that
[M ab , D c ] = 2η c[a D b] . (2.8)
The supergravity gauge group is generated by local transformations of the form
δ K D A = [K, D A ] where K = K C (z)D C + 1 2 K cd (z)M cd + 1 2 K kl (z)J kl ,(2.9)
with all the gauge parameters obeying natural reality conditions but otherwise arbitrary. Given a tensor superfield T (z), its transformation rule is
δ K T = KT . (2.10)
The covariant derivatives satisfy the (anti)commutation relations
[D A , D B } = T AB C D C + 1 2 R AB cd M cd + F AB kl J kl ,(2.11)
with T AB C the torsion, R AB cd the Lorentz curvature, and F AB kl the SU(2) R-symmetry field-strentgh. These tensor fields are related to each other by the Bianchi identities:
[ABC) [D A , [D B , D C }} = 0 . (2.12)
To describe conformal supergravity, we impose conventional constraints on the torsion. In the six-dimensional case we are considering, they can be chosen to be
T i α j β c = −2iε ij (γ c ) αβ , (dimension-0) (2.13a) T i α j β γ k = 0 , T i α b c = 0 , (dimension-1 2 ) (2.13b) T a b c = 0 , T a β(j β k) = 0 . (dimension-1) (2.13c)
These constraints are similar to the four-dimensional, N = 2 superspace geometry of [30], formally identical to the five-dimensional conformal supergravity constraints of [11], and closely related to the six dimensional off-shell geometry of reference [48].
With the constraints so introduced, the solution of the Bianchi identities can be shown to imply that the torsion and curvature tensors in (2.11) are expressed in terms of a small number of mass-dimension-1 real tensor superfields C abc and C ij c , and their covariant derivatives. The torsion component C ij c = C ji c is an iso-triplet and C abc = W abc + N abc is a 3-form, which we split into anti-self-dual (W ) and self-dual (N) parts.
In terms of these basic torsions, the graded commutation relations of the covariant derivatives are given by
{D αi , D βj } = 2iε ij (γ a ) αβ D a + 2iC a ij (γ abc ) αβ M bc + 4iε ij W abc (γ a ) αβ M bc (2.14a) +4iε ij N abc (γ a ) αβ M bc − 6iε ij C kl a (γ a ) αβ J kl − 8i 3 N abc (γ abc ) αβ J ij , [D γk , D a ] = −C b kl (γ ab ) γ δ D l δ + W abc (γ bc ) γ δ D δk + N abc (γ bc ) γ δ D δk + i 2 (γ a ) γδ T bc δ k − i(γ b ) γδ T ca δ k M bc (2.14b) + (γ a ) γδ C δ ij k − 6δ i k C a γ j + 5(γ a ) γδ δ i k C δj − 1 3 W δj J ij ,
where T ab γ k , C α i , and W α i are defined below (c.f. eq. 2.17, 2.15a, and 2.15c, resp.). The dimension-1 superfields C a ij , W αβ := 1 6 W abc (γ abc ) αβ , and N αβ := 1 6 N abc (γ abc ) αβ satisfy additional constraints which follow from the Bianchi identities. To display the content of these constraints more clearly, we first define their Lorentz-and isospin-irreducible components
D γk C a ij =: C a γk ij + (γ a ) γδ C δ ijk + ε k(i C a γj) + ε k(i (γ a ) γδ C δ j) , (2.15a) D γk N αβ =: N γk αβ +Ň γk αβ , (2.15b) D γk W αβ =: W γk αβ + δ (α γ W β) k . (2.15c)
Multiple isospin indices are fully symmetrized as are multiple Lorentz indices of the same height (except for the caseŇ , which has a part proportional to a γ-matrix; c.f. eq. 2.16), Lorentz indices at different heights have had their traces removed, and fields with both vector and spinor indices are γ-traceless. These properties are reflected in their explicit forms as solutions to the Bianchi identities:
C a γk ij = 0 , N γk αβ = 0 , C δ ijk = − 1 6 (γ b ) δβ D β(k C b ij) ,Ň γk αβ = − 3 4 (γ a ) γ(α C a β)k , C a βj = 2 3 Π c γ a β D i γ C a ij , W γk αβ = D γk W αβ − 2 5 δ (α γ D δk W β)δ , C γk = − 1 9 D δl C δγ lk , W αi = 2 5 D i β W βα .
(2.16)
Here, Π bβ aα = δ b a δ β α + 1 6 (γ aγ b ) α β is the projector onto the γ-traceless subspace (i.e. (γ a ) γα Π bβ aα = 0 = Π bβ aα (γ b ) βγ ). We note that, of the two components of DN, one vanishes and the other is related to an irreducible part of DC.
Finally, the dimension- 3 2 torsion is given in terms of the remaining fields as
T ab γk = i 2 (γ ab ) β δ W k βγ δ + 7i 4 (γ [a ) γδ C b] δ k + i(γ ab ) δ γ C δk − 1 6 W δk . (2.17)
With this, the dimension-1 and dimension-3 2 commutators are completely specified. It has been verified that the Bianchi identities are satisfied up to and including dimension 3 2 . Futher details of the geometry are not required for the purposes of this paper and will be expounded upon elsewhere.
Super-Weyl Transformations and the Type-i Weyl Multiplet
A short calculation shows that the constraints (2.13a)-(2.13c) are invariant under arbitrary super-Weyl transformations defined by
δD αi = 1 2 σD αi − 2(D βi σ)M α β + 4(D α j σ)J ij , (2.18a) δD a = σD a − i 2 (D k σ)γ a D k − (D b σ) M ab − i 8 (D iγ a D j σ) J ij , (2.18b)
where the parameter σ(z) is a real, unconstrained superfield. The components of the dimension-1 torsion are required to transform as
δC a ij = σC a ij + i 8 (D (iγa D j) σ) , (2.19a) δW abc = σW abc , (2.19b) δN abc = σN abc − i 32 (D kγ abc D k σ) . (2.19c)
The transformations of C a ij and N abc contain in-homogeneous terms which can be used to gauge away many of their components. The anti-self-dual 3-form W abc transforms homogeneously and represents a superspace generalization of the Weyl tensor. It can be shown that, by properly choosing a Wess-Zumino gauge for our superspace geometry, the surviving physical components embedded in the geometry contain the SU(2) fieldstrength, the gravitino curl, an anti-self-dual auxiliary 3-form of mass dimension-1, an auxiliary spinor of positive chirality of mass dimension- 3 2 , the Weyl tensor, and a real auxiliary scalar field of mass-dimension-2. The resulting multiplet describes the (40+40)component Weyl supermultiplet [29] e a m , ψ m αi , Φ a ij , W − abc , χ αi , D , (2.20) to which we will refer as the type-i multiplet. Here, e a m is the sechsbein, ψ m αi the gravitino, Φ a ij the SU(2) connection, and W − abc , χ αi , D are the "matter" fields. Here, the component gauge fields and the gravitino are related to the θ = 0 components of the supersechsbein and superconnections whereas the matter fields of the Weyl multiplet arise in our geometry as components of the Weyl superfield: W − abc = W abc | θ=0 , χ αi = W αi | θ=0 and D = D αi W αi | θ=0 . As originally defined [29], this Weyl multiplet contains an additional dilatation gauge field b m (x) but this degree of freedom is pure gauge and one can choose to work in the gauge in which it vanishes. Such a gauge arises naturally in the superspace geometry we have introduced here. This situation is similar to the 5D conformal supergravity in superspace described in [11] and to Grimm's formulation of 4D supergravity [55], as explained in detail in [13]. In these superspace treatments, as with our geometry, the b m field does not arise.
The Tensor Multiplet and the Type-ii Weyl Multiplet
There is a second formulation of the Weyl supermulitplet in which the anti-self-dual 3-form W , auxiliary positive chirality spinor χ, and auxiliary scalar D, are replaced by a tensor multiplet consisting of a propagating scalar σ, a gauge 2-form tensor B, and a negative chirality tensorino χ [29]. This alternate formulation, to which we will refer as the type-ii Weyl multiplet, plays an important role in six-dimensional supergravity since it is the one that, within the superconformal tensor calculus approach, can be consistently used to construct actions for general matter-coupled supergravity systems. (See, for example, [38] for a recent discussion of six-dimensional Poincaré supergravity obtained by coupling the type-ii Weyl multiplet to a linear multiplet.) In this subsection, following the same logic used in the component case, we work out the superspace version of the type-ii formulation by coupling the type-i formulation to a tensor multiplet [32,33]. In flat space, the tensor multiplet has been constructed as a closed 3-form in superspace in [34]. It is natural to formulate the consistent curved tensor multiplet constraints extending such a construction to our curved superspace geometry. To this end, we must work out the mass dimension ≤ 3 part of the 3-form Bianchi identities in the supergravity background.
The super-3-form H can be written in local coordinates as
H = 1 3! dZ M 3 dZ M 2 dZ M 1 H M 1 M 2 M 3 = 1 3! E A 3 E A 2 E A 1 H A 1 A 2 A 3 . (2.21)
This form is closed, dH = 0, iff its components satisfy the Bianchi identities
1 3! D [B H A 1 A 2 A 3 ) − 1 2! · 2! T [BA 1 | C H C|A 2 A 3 ) = 0 . (2.22)
The dimension-2 condition is consistent with the constraint
H αiβjγk = 0 ,(2.23)
provided that
H αiβjc = 2iε ij (γ c ) αβ Φ ,(2.24)
where Φ is an arbitrary real scalar superfield. Next, the dimension-5 2 identity is solved by
H αibc = −(γ bc ) α β D βi Φ . (2.25)
Finally, the dimension-3 identity gives the expression for the 3-form
H abc = H + abc + H − abc H + abc = i 8 D kγ abc D k − 16N abc Φ , (2.26a) H − abc = −16W abc Φ , (2.26b)
divided here into its self-dual and anti-self-dual parts. Additionally, it implies that Φ satisfies
D (iγ a D j) Φ + 16iC a ij Φ = 0 . (2.27)
This constraint is super-Weyl invariant iff Φ has scaling-dimension equal to 2, that is, [34]. Indeed, using (2.25)-(2.26b), one derives the Bianchi identity
δΦ = 2σΦ. It is the curved-space analogue of the flat space constraint D (i α D j) β Φ = 0 which describes the tensor multiplet consisting of a scalar 3 σ(x) ∼ Φ(z)| θ=0 , a tensorino ψ αi (x) ∼ D αi Φ(z)| θ=0 , and a self-dual 3-form field-strength h + abc (x) ∼ D kγ abc D k Φ(z)| θ=01 3! D [a H bcd] − 1 2! · 2! T [ab γk H cd]γk = 0 , (2.28)
which implies that, up to spinorial torsion terms, the 3-form superfield H abc ∼ D [a B bc] is locally exact. Finally, we note that the constraint (2.27) puts the tensor multiplet on-shell. This is most easily checked by taking the flat-space limit, D
(i α D j)
β Φ = 0, and showing that it implies, for example, ∂ a ∂ a Φ = 0. In the supergravity case, the equations are covariantized by the supergravity fields which provide additional interaction terms.
In fact, it can be shown that the constraint (2.27) is equivalent to the condition
D α(i V β j) − 1 4 δ β α D γ(i V γ j) = 0 , (2.29)
on a spinor potential superfield V αi , provided we identify In flat superspace, this multiplet was first introduced by Sokatchev in [31]. It is straightforward to verify that the new constraint is invariant under super-Weyl transformations iff V is a super-Weyl tensor of scaling dimesion-3 2 : δ σ V αi = 3 2 σV αi . Furthermore, it is invariant under a gauge transformation so that, in the components of this new multiplet, the superfield 3-form field-strength H abc = H + abc +H − abc is replaced with the exterior derivative of its gauge 2-form potential B ab ∼ D iγab V i [31]. It is non-trivial that Φ, as defined in (2.30), is a super-Weyl tensor of scaling dimension-2.
Φ = D αi V αi .
As first pointed out in reference [29], provided that the scalar component σ(x) ∼ Φ| θ=0 is everywhere non-vanishing, the equations of motion can be solved for the components {W − , χ, D} in terms of the components {σ, B, ψ}. The result is a description in terms of the components of the type-ii Weyl multiplet [29]
e a m , ψ m αi , Φ a ij , σ, ψ αi , B ab . (2.31)
This formulation can be interpreted as arising by taking the (40 + 40)-component type-i multiplet, coupling to the 11 + 8 fields {σ, B, ψ}, and then imposing the 11 + 8 degrees of freedom of the equations of motion as constraints. In this interpretation, the tensor supermultiplet does not add any degrees of freedom to the type-i multiplet overall. In our superspace language, assuming the superfield Φ(z) = 0 is everywhere non-vanishing, this is equivalent to solving for the dimension-1 torsion superfields of the type-i geometry in terms of the tensor multiplet 3-form superfields.
This suggests a second mechanism to remove the newly added tensor-multiplet degrees of freedom: Whenever the scalar field in the superfield Φ is nowhere-vanishing on the body of the supermanifold, it is evidently possible to use the super-Weyl parameter to gauge Φ → 1. Equation (2.27) then reduces to C ij a = 0 and equations (2.23)-(2.26a) become
H αiβjγk = 0 , H αiβjc = 2iε ij (γ c ) αβ , H αibc = 0 , H + abc = −16N abc , H − abc = −16W abc . (2.32)
This super-Weyl gauge corresponds to further strengthening the second conventional constraint in equation (2.13c) by imposing T a β(j γ k) → 0. (Equivalent observations were made already in reference [54].) The residual Weyl transformations are constrained by (2.19a) to satisfy D α(i D βj) σ = 0. There is still enough of the Weyl parameter left to remove all of the remaining components of the self-dual field N αβ . This leaves only the anti-self-dual part W αβ which, in this formulation, carries all of the off-shell degrees of freedom of the type-i multiplet.
It is interesting to note that in five dimensions there is a mechanism similar to the one just described to formulate a variant Weyl multiplet. In fact, by coupling the fivedimensional Weyl multiplet to an abelian vector multiplet constrained to satisfy the curved Chern-Simons equation of motion, one can solve it for the auxiliary fields of the standard Weyl multiplet and end up with the so-called dilaton-Weyl multiplet [56,57]. See reference [11] for a description of this mechanism in superspace.
We conclude this section by comparing the six-dimensional variant to the lowerdimensional cases. In D = 4 and 5, vector multiplets with eight supercharges are of primary importance for conformal supergravity since they possess a scalar field as their lowest component. For this reason, off-shell vector multiplets are the most natural conformal compensators in 4D, N = 2 and 5D, N = 1 supergravity and, once coupled to the Weyl multiplet, give rise to the so-called minimal multiplets [30,58,59]. In six dimensions, on the other hand, the lowest component of an off-shell vector multiplet is a positive-chirality Weyl spinor. In superspace, the 6D off-shell vector multiplet is described by a dimension-3 2 superfield-strength F α i constrained by [60]
D (i α F βj) − 1 4 δ β α D (i γ F γj) = 0 and D i α F α i = 0 (2.33)
which, compared with the tensor multiplet constraint (2.29), is missing the scalingdimension-2 scalar superfield (2.30). 4 Due to the differences just mentioned, there is no direct analogue of the minimal multiplet in six dimensions. In some respects, the 6D tensor multiplet closely mimics features of the lower-dimensional vector multiplets. It has a scalar that naturally plays the role of a dilaton but the crucial difference is that the 6D tensor multiplet is on-shell. 4 Note that the tensor multiplet defined by the constraint (2.29), which is the first of the two vector multiplet constraints in (2.33), has on-shell physical fields while the vector multiplet is off-shell. The V αi -multiplet includes the following physical fields:
A 2-form gauge potential B ab ∼ (γ ab ) α β D αi V βi | θ=0 ; a scalar dilaton σ ∼ D αi V αi | θ=0 ; and a fermion χ αi ∼ D αi D j β V β j | θ=0 .
The vector multiplet, on the other hand, consists of the following physical fields:
A gaugino λ αi ∼ F αi | θ=0 ; a gauge field strength F ab ∼ (γ ab ) α β D αi F βi | θ=0 ; and an auxiliary iso-triplet Y ij ∼ D (i α F αj)
. One can check that the components of the vector multiplet that are also components of the V αi superfield are pure gauge in the latter case. On the other hand, the physical fields of V αi that are responsible for putting the multiplet on-shell are precisely those killed by the second constraint in (2.33).
Six-dimensional Curved Projective Superspace
Covariant projective supermultiplets have been used recently to efficiently describe matter couplings in extended supergravity. This was first done in five dimensions [10,11], then applied to the four-dimensional case [12,13], and recently extended to two [14] and then three [15] dimensions. In this section, we continue this program by showing that the existence of covariant projective supermultiplets is consistent with the geometry of section 2. (Projective superfields in flat 6D, N = (1, 0) Minkowski superspace were first introduced in [46] and further studied in [47].) We then conclude with a presentation of a locally supersymmetric and super-Weyl invariant action principle.
6D, N = (1, 0) Covariant Projective Superfields
In defining curved projective multiplets, we follow the same procedure that has been successfully developed in the 2 ≤ D ≤ 5 supergravity cases [10,11,12,13,14,15]. We start by introducing an
isotwistor variable v i = (v 1 , v 2 ) ∈ C 2 \ {0}, defined to be inert under the action of the supergravity structure group: [M ab , v i ] = [J kl , v i ] = 0.
Using this isotwistor, we define the covariant derivatives
D (1) α := v i D i α . (3.1) Note that the D (1)
α derivative is homogeneous of degree one in v i . Our curved superspace is then extended to M 6|8 × CP 1 , with the isotwistor variable interpreted as providing homogeneous coordinates [v 1 : v 2 ] of the complex projective line. Tensor superfields on this extension are called isotwistor superfields. A weight-n isotwistor superfield U (n) (z, v) is defined to be holomorphic on an open domain of C 2 \{0} with respect to the homogeneous coordinates v i for CP 1 and is characterized by the conditions:
(i) It is a homogeneous function of v of degree n, that is,
U (n) (z, cv) = c n U (n) (z, v) , c ∈ C * ,(3.2)
(ii) The supergravity gauge transformations act on U (n) as follows:
δ K U (n) = K C D C + 1 2 K cd M cd + K kl J kl U (n) , (3.3a) J kl U (n) (v) = − 1 (v, u) v (k v l) u i ∂ ∂v i − nv (k u l) U (n) (v) , (3.3b) where (v, u) := v i u i , δ i j = 1 (v, u) v i u j − v j u i . (3.4)
The auxiliary variable u i is constrained by (v, u) = 0 but is otherwise completely arbitrary. By definition, U (n) is a function only of v and not u; the same should be true for its variation. Indeed, due to (3.2), the superfield (J kl U (n) ) can be seen to be independent of u i even though the transformations in (3.3b) explicitly depend on it.
With the definitions (i) and (ii) assumed, the set of isotwistor superfields is closed under products and the action of the D (1) α derivative. More precisely, given weight-m and weight-n isotwistor superfields U (m) and U (n) , the superfield (U (m) U (n) ) is a weight-(m + n) isotwistor superfield and the superfield (D (1) α U (n) ) is a weight-(n + 1) isotwistor superfield. Note that, as implicitly indicated in (3.3a), general isotwistor superfields are not restricted to be Lorentz scalar. Ultimately, the use of the extra isotwistor variable should be interpreted as an efficient way to deal with superfields that are (in general, infinite-dimensional) representations of the SU(2) group; see [12,13] for more details.
The most important property of isotwistor superfields is that the anti-commutator of D (1) α covariant derivatives is zero when acting on a Lorentz scalar, isotwistor superfield U (n) . In fact, from (2.14a), one obtains the anti-commutation relation
{D (1) α , D(1)β } = −8iC (2) γ(α M β) γ − 16iN αβ J (2) ,(3.5)
where we have defined
C (2) αβ := v i v j C ij αβ and J (2) := v i v j J ij . (3.6)
The SU(2) generators appear in the previous algebra only in the combination defined by J (2) which can easily be shown to vanish when acting on general isotwistor superfields J (2) U (n) = 0. If one imposes that U (n) be a Lorentz scalar, then
{D (1) α , D(1)β }U (n) = 0 . (3.7)
We define a weight-n, covariant projective superfield Q (n) (z, v) to be an isotwistor superfield (i.e. satisfying (i) and (ii)) constrained by the analyticity condition
D (1) α Q (n) = 0 . (3.8)
The consistency of the previous constraint is guaranteed by equation (3.7) which takes the form of an integrability condition for the analyticity constraint (3.8).
In conformal supergravity, the important issue is whether the projective multiplets can be made to vary consistently under the super-Weyl transformations. Suppose we are given a weight-n, projective superfield Q (n) that transforms homogeneously: δ σ Q (n) ∝ σQ (n) . Its transformation law is, then, uniquely fixed to be
δ σ Q (n) = 2nσ Q (n) ,(3.9)
by imposing super-Weyl invariance of the constraint (3.8).
Given a projective multiplet Q (n) , its complex conjugate is not covariantly analytic. However, one can introduce a generalized analyticity-preserving conjugation Q (n) →Q (n) , defined as
Q (n) (v) ˘≡Q (n) v →v ,v = i σ 2 v , (3.10) withQ (n) (v) the complex conjugate of Q (n) (v)
. It follows thatQ (n) = (−1) n Q (n) so that real supermultiplets can be consistently defined when n is even. The superfield Q (n) is called the smile-conjugate of Q (n) . Note that, geometrically, this conjugation is a composition of complex conjugation and the antipodal map on CP 1 . A fundamental property is that
D (1) α Q (n) ˘= (−1) ǫ(Q (n) ) D (1) αQ (n) ,(3.11)
implying that the analytic constraint (3.8) is invariant under smile conjugation.
A simple class of 6D projective superfields is defined as G (m) (z, v) = v i 1 · · · v im G i 1 ···im (z). These are constructed in terms of the completely symmetric isotensor superfields G i 1 ···im (z) = G (i 1 ···im) (z) and describe regular holomorphic tensor fields on the complex projective space CP 1 parametrized by the homogeneous coordinates v i . Provided that the SU(2) transformation rule of G i 1 ···im is the standard one
J kl G i 1 ···im = δ (i 1 (k G l) i 2 ···im) ,(3.12)
the superfield G (m) satisfies all the conditions to be isotwistor. Moreover, the analyticity condition D
α G (m) = 0 is equivalent to the following constraint on G i 1 ···im :
D (j α G i 1 ···im) = 0 . (3.13)
This constraint is consistent with the super-Weyl transformation rule δ σ G i 1 ···im = 2mσG i 1 ···im . When m = 2p, one can further constrain G (2p) to be smile-real which is equivalent to the condition (G i 1 ···i 2p ) = G i 1 ···i 2p . This kind of multiplet is known in 4D, N = 2 supersymmetry literature as an O(2p)-multiplet. It is a generalization of the well-known linear multiplet G ij = G ij that has p = 1; for an incomplete list of references see [61,62,63,2,64,65]. Note that when m = 1, G (1) = v i q i , the (necessarily complex) superfield q i satisfies D (i α q j) = 0 and describes a six-dimensional extension of the Fayet-Sohnius hypermultiplet [66]. It is necessarily on-shell as a consequence of the impossibility of adding a central charge to the 6D, N = (1, 0) algebra.
Instead of the homogeneous coordinates [v 1 : v 2 ], it is often useful to work with an inhomogeneous local complex variable ζ that is invariant under arbitrary projective rescalings v i → c v i , with c ∈ C * . In such an approach, one should replace
Q (n) (z, v) with a new superfield Q [n] (z, ζ) ∝ Q (n) (z, v), where Q [n] (z, ζ)
is holomorphic with respect to ζ. Its explicit definition depends on the supermultiplet under consideration. The space CP 1 can naturally be covered by two open charts in which ζ can be defined, and the simplest choice is: (i) the north chart characterized by v 1 = 0; (ii) the south chart with v 2 = 0. In the projective superspace literature, the classification of multiplets normally proceeds by restricting to the north chart and depends on the pole structure of Laurent expansion in ζ. Analogously to the curved cases in dimensions 2 ≤ D ≤ 5 [10,11,12,13,14,15], six-dimensional projective superfields generically possess an infinite number of standard superfields. As an example, consider off-shell charged hypermultiplets. In projective superspace these have a natural description in terms of the so-called arctic superfield: A weight-n polar multiplet is described in terms of arctic superfields Υ (n) (z, v), and their antarctic smile-conjugates Υ (n) (z, v). By definition, Υ (n) is a projective superfield that is well-defined in the whole north chart of CP 1 (converselyΥ (n) (z, v) is well-defined in the whole south chart). In the north chart, Υ (n) = (v 1 ) n Υ [n] andΥ (n) = (v 2 ) nΥ[n] = (v 1 ) n ζ nΥ[n] are represented as
Υ [n] (z, ζ) = ∞ k=0 ζ k Υ k (z) ,Υ [n] (z, ζ) = ∞ k=0 (−1) k ζ kῩ k (z) , ζ := v 2 v 1 ,(3.14)
in terms of an infinite sequence of ordinary superfields {Υ k (z)} ∞ k=0 and their complex conjugates. The spinor covariant derivative D (1) α can be represented as
D (1) α = (v 1 )D [1] α (ζ) , D [1] α (ζ) = D 2 α − ζD 1 α . (3.15)
From this representation, and the representation of the arctic multiplet in the north chart (3.14), it follows that the analyticity condition (3.8) nontrivially relates the superfield coefficients Υ k (z) in the series.
Another important example not of the polar type and mentioned later is the smilereal tropical multiplet. A weight-0, real, tropical superfield
V (0) (z, v) = V [0] (z, ζ) = ∞ k=−∞ ζ k V k (z)
is required to be well-defined only on C * , that is, CP 1 with both north and south poles removed. The reality condition V (0) =V (0) implies that V k = (−1) k V −k . A special case of this is given by the product of a weight-0 arctic field and its smile-conjugate V (0) ∼Υ (0) Υ (0) . A more detailed classification of 6D covariant projective superfields will be considered elsewhere. (See [46,47] for a discussion in the flat case. In particular, it is shown in [47] how the flat six-dimensional vector multiplet is described in terms of a prepotential tropical superfield.)
For applications, it is crucial that the analyticity constraint defining projective superfields can be solved in terms of unconstrained isotwistor superfields and an analytic projection operator. We introduce the fourth-order operator
∆ (4) := D (4) − 5i 6 C (2)γδ D (2) γδ − 5iC (3)γ D (1) γ − i 4 (D (2) γδ C (2)γδ ) + 3C (2)γδ C (2) γδ , (3.16) where D (4) := − 1 96 ε αβγδ D (1) α D (1) β D (1) γ D (1) δ , D(2)αβ := D (1) [α D (1) β] = −D (2) βα ,(3.17)
and ε αβγδ C
(2) γδ = ε αβγδ (γ a ) γδ C (2) a = 2(γ a ) αβ C (2) a = 2C (2)αβ , (3.18a) ε αβγδ (D (1) β C (2) γδ ) = −12C (3)α = −12C αijk v i v j v k , C (3)α := − 1 12 ε αβγδ (D (1) β C (2) γδ ) . (3.18b)
The superfield C α ijk is the dimension-3 2 torsion component defined in (2.15a). The crucial property of the analytic projection operator is that, given an arbitrary weight-(n − 4) isotwistor, Lorentz scalar superfield U (n−4) , the superfield Q (n) defined by
Q (n) (z, v) := ∆ (4) U (n−4) (z, v) ,(3.19)
is a weight-n projective superfield: It is worth noting that the analytic projection operator can be also used to build a weight-4 projective multiplet P (4) (z, v) from a scalar, v-independent superfield P (z). In fact, for any P (z), the superfield P (4) (z, v) := ∆ (4) P (z) is a weight-4 projective superfield. Moreover, if one wants both P and P (4) to transform homogeneously under super-Weyl transformations then they have to satisfy: δ σ P = 6σP and δ σ P (4) = 8σP (4) . A derivation of these statements is given in Appendix B. We conclude this subsection by giving the analytic projection operator in an equivalent form: 5
D (1) α Q (n) = 0 .∆ (4) = ε αβγδ − 1 96 D (1) α D (1) β D (1) γ D (1) δ − 5i 12 D (1) α C (2) βγ D (1) δ − i 8 (D (2) αβ C (2) γδ ) + 3 2 C (2) αβ C (2) γδ . (3.22)
This expression will be useful in the next subsection.
The Action Principle
In this subsection, we give a projective superfield action principle invariant under the supergravity gauge group and super-Weyl transformations and such that, in the flat limit, it reduces to the six-dimensional action of [46,47]. The latter is an extension of the one originally introduced in four dimensions in [1] and reformulated in a projective-invariant form in [67]. The result is a simple generalization of the action principle in the curved projective superspaces in 2 ≤ D ≤ 5 [10,11,12,13,14,15].
Let L (2) be a real projective multiplet of weight-2. We assume that L (2) possesses the super-Weyl transformation δ σ L (2) = 4σL (2) , (3.23) which complies with the rule (3.9). We also introduce a real isotwistor superfield Θ (−4) such that
δ σ Θ (−4) = −2σΘ (−4) , ∆ (4) Θ (−4) = 1 . (3.24)
Associated with L (2) and Θ (−4) is the following functional
S = 1 2π C (v, dv) d 6 x d 8 θ E Θ (−4) L (2) , E −1 = Ber (E A M ) . (3.25) 5
It is instructive to compare the six-dimensional analytic projection operator with the five-dimensional one of [10,11]. There, the projector was presented in the gauge C iĵ a = 0 (â = 0, · · · , 4 is the 5D vector index in the notation of [10]) with only the 5D scalar torsion S ij appearing in the projector. With an appropriate dimensional truncation, one can see that the coefficients in the 6D and 5D projectors match.
This functional is invariant under arbitrary re-scalings
v i (t) → c(t) v i (t), ∀c(t) ∈ C * ,
where t denotes the evolution parameter along the integration contour. By using that under super-Weyl transformations, E transforms as (3.26) and the transformation properties (3.23)-(3.24), we find that the functional S is super-Weyl invariant. The action (3.25) is also invariant under arbitrary local supergravity gauge transformations (2.9), (2.10) and (3.3a). The invariance under general coordinate and Lorentz transformations is trivial given that both Θ (−4) and L (2) are Lorentz scalars.
δ σ E = −2σE
The invariance under the SU(2) transformations can be proved similarly to the 2 ≤ D ≤ 5 cases: First, we note that
U (−2) := Θ (−4) L (2) (3.27)
is a isotwistor multiplet of weight −2. Then, one verifies that
K ij J ij U (−2) = −∂ (−2) K (2) U (−2) , ∂ (−2) := 1 (v, u) u i ∂ ∂v i . (3.28)
Next, since K (2) U (−2) has weight zero, it is easy to see that
(v, dv) K ij J ij U (−2) = −dt d dt K (2) U (−2) , (3.29)
where, again, t denotes the evolution parameter along the integration contour in (3.25).
Since the integration contour is closed, the SU(2)-part of the transformations of U (−2) (3.3a) does not contribute to the variation of the action (3.25).
The isotwistor superfield Θ (−4) is used to ensure the invariance of the action under super-Weyl and SU (2) transformations. An important point is that, in general, the supersymmetric action (3.25) is independent of the explicit form of Θ (−4) , which is just an auxiliary constructive tool. To prove this, we need one observation about the analytic projection operator ∆ (4) (3.16) or (3.22). Specifically, let us show that ∆ (4) is symmetric under integration-by-parts. In the geometry of section 2, the rule for integration-by-parts is
d 6 x d 8 θ E D A V A = 0 , (3.30)
with V A = (V a , V α i ) an arbitrary superfield. Introducing arbitrary isotwistor superfields Ψ (−n) and Φ (n−6) , and by using the form of the analytic projection operator given in (3.22), we find the symmetry relation
1 2π C (v, dv) d 6 x d 8 θ E Ψ (−n) ∆ (4) Φ (n−6) − Φ (n−6) ∆ (4) Ψ (−n) = 0 . (3.31)
Now, let U (−2) be a real isotwistor prepotential for the Lagrangian L (2) in (3.25):
L (2) = ∆ (4) U (−2) . (3.32)
By using (3.31) and ∆ (4) Θ (−4) = 1, we can re-express the action (3.25) in the form
S = 1 2π C (v, dv) d 6 x d 8 θ E U (−2) . (3.33)
If the Lagrangian L (2) , and hence U (−2) , is independent of Θ (−4) then (3.33) makes manifest the independence of (3.25) on Θ (−4) .
We point out that there is a freedom in the choice of Θ (−4) . For instance, given a real weight-m isotwistor superfield Γ (m) , Θ (−4) may be defined as
Θ (−4) := Γ (m) ∆ (4) Γ (m) , δ σ Γ (m) = 2(m + 3)σΓ (m) . (3.34)
Additionally, one can consider a real Lorentz scalar and SU(2) invariant superfield P = P (z) such that
Θ (−4) := P ∆ (4) P ; δ σ P = 6σP . (3.35)
Note that the use of P is inequivalent to that of a general, weight-0 isotwistor superfield Γ (0) which may have non-trivial dependence on the projective parameter ζ and is, as such, not invariant under SU(2) transformations.
Let us take the flat limit of the action (3.25). This, up to total flat vector derivatives, can be written as
S = 1 2π C (v, dv) d 6 x d 8 θΘ (−4) L (2) = 1 2π C (v, dv) d 6 x D (−4) D (4)Θ(−4) L (2) θ=0 = 1 2π C (v, dv) d 6 x D (−4) L (2) θ=0 , (3.36)
with L (2) ,Θ (−4) , and D (4) the flat-superspace limit of the Lagrangian L (2) , the density Θ (−4) , and the analytic projector ∆ (4) (3.16), respectively. Here, we have also introduced the operator
D (−4) := − 1 96 ε αβγδ D (−1) α D (−1) β D (−1) γ D (−1) δ , D (−1) α := u i (v, u) D i α . (3.37)
The flat action is invariant under arbitrary projective transformations of the form:
(u i , v i ) → (u i , v i ) R , R = a 0 b c ∈ GL(2, C) . (3.38)
As it is explicitly independent of u, the same invariance holds for the curved-superspace action (3.25). This invariance is a powerful tool in superspace theories with eight supercharges. For example, in 5D, N = 1 [10] and 4D, N = 2 [68] supergravity it has been used to reduce the projective action principle to components. Clearly, the same techniques can be used in the six-dimensional case to reduce the action (3.25).
One can rewrite the contour integral in the north chart of CP 1 , v 1 = 0, in terms of the inhomogeneous complex variable ζ
v i = v 1 ζ i , ζ i = (1, ζ) , ζ i = ε ij ζ j = (−ζ, 1) , ζ = v 2 v 1 ∈ C . (3.39)
The Lagrangian L (2) (z, v) in the north chart can be rewritten as
L (2) (z, v) := i(v 1 ) 2 ζL(z, ζ) . (3.40)
Since the action and the Lagrangian are independent of u i , we can make the conventional choice
u i = (1, 0) , u i = ε ij u j = (0, −1) . (3.41)
The action (3.36) is, then, rewritten as
S = C dζ 2πi d 6 x ζ (D 1 ) 4 L θ=0 , (D 1 ) 4 := − 1 96 ε αβγδ D 1 α D 1 β D 1 γ D 1 δ . (3.42)
This expression is the rigid supersymmetric action in the 6D, N = (1, 0) projective superspace of [46,47]. Thus, our curved projective action principle is, as expected, a generalization of the known flat one.
Some Matter Systems
We conclude this section with examples of supergravity-matter systems. We start by considering two classes of projective superfield conformal compensators: an O(2) multiplet, given by the real, linear superfield G (2) := G ij v i v j and a weight-1, arctic multiplet Υ (1) and its smile conjugateΥ (1) that describes the off-shell, charged hypermultiplet.
Note that to use the linear multiplet as a proper compensator, G ij should be nowherevanishing which is equivalent to G := G ij G ij = 0. This composite scalar and SU (2) invariant superfield, which has scale dimension 4, δG = 4σG, can be used to choose the super-Weyl gauge G = 1. In this gauge, D i α G = D i α 1 = 0 which, together with the analyticity constraint D (i α G jk) = 0, implies that G ij = w ij is covariantly constant D i α w jk = 0 wherefore also the SU(2) group is broken to the U(1) subgroup that leaves w ij invariant. By imposing the consistency of the supergravity algebra with the covariant constancy of w ij , {D i α , D j β }w kl = 0, one can easily see that, in this gauge, the dimension-1 torsions satisfy 6
N abc = 0 , C ij a = C a w ij . (3.43)
The Lagrangian for the O(2) multiplet compensator is given by
L (2) SG−linear = −G (2) ln G (2) iΥ (1) Υ (1) . (3.44)
It encodes the dynamics of a massless improved linear multiplet coupled to conformal supergravity. It has the same form as the 4D, N = 2 counterpart given in [70] as a locally-supersymmetric extension of the projective-superspace formulation [1] for the 4D, N = 2 improved tensor multiplet [71,72]. The action (3.44) is independent of the (ant-)arctic superfields Υ (1) ,Υ (1) which turn out to be pure-gauge superfields. The Lagrangian (3.44) can be shown to be dual to the Lagrangian for an arctic compensator coupled to conformal supergravity:
L (2) SG−hyper = −i Υ (1)Υ(1) . (3.45)
The duality map is the same as in reference [70].
By using the compensators, we can couple supergravity to general matter. We consider a few examples which are familiar from the lower-dimensional cases; we refer the reader to [12,70] for the geometric interpretation of the models that follow. (3.46)
6 Similar gauges in superspace were used before in 4D in [68,69] and in 3D in [15].
Here, K(Φ I ,ΦJ ) is a real function of n complex variables Φ I , with I = 1, . . . , n, that satisfies the homogeneity condition
Φ I ∂ ∂Φ I K(Φ,Φ) = K(Φ,Φ) . (3.47)
This Lagrangian represents a conformal non-linear sigma-model as in [17].
Given a system of m weight-0 arctic multiplets Ξ i , i = 1, · · · , m, and the conformal compensator Υ (1) , one can write the Lagrangian
L (2) NLSM−hyper = Υ (1)Υ(1) exp K(Ξ i ,Ξj) .
(3.48)
The real function K(Ξ i ,Ξj) can be interpreted as a Kähler potential since the Lagrangian is invariant under the transformation
Υ (1) → e −Λ(Ξ) Υ (1) , K(Ξ,Ξ) → K(Ξ,Ξ) + Λ(Ξ) +Λ(Ξ) ,(3.49)
with Λ a holomorphic function. In the dual picture, where the compensator is given by a linear superfield, the previous Lagrangian is equivalent to
L (2) NLSM−linear = G (2) K(Ξ i ,Ξj) . (3.50)
Next, we consider a system of n linear O(2) multiplets G where L is a real homogeneous function of degree-1:
G (2) I ∂ ∂G (2) I L = L . (3.52)
More generally, it is possible to couple linear O(2) multiplets and hypermultiplets in an arbitrary way provided that the Lagrangian L (2) (G (2) ,
Υ (1) ,Υ (1) , Ξ,Ξ) is weight-2 in the sense that L (2) (c 2 G (2) , cΥ (1) , cΥ (1) , Ξ,Ξ) = c 2 L (2) (G (2) , Υ (1) ,Υ (1) , Ξ,Ξ) with c ∈ C * .
We conclude by considering some composite, weight-2, scaling-dimension-4, real projective superfields built from tensors and vector field-strengths. We begin by taking two tensor multiplets in the representations Φ and V α i , introduced in section 2.3, and coupling them through the composite O(2) superfield
G (2) := i(D (1) α Φ)V α(1) + i 4 ΦD (1) α V α(1) , V α(1) := v i V αi . (3.53)
That this combination is analytic follows from a short calculation:
D (1) β G (2) = i(D (1) β D (1) α Φ)V α(1) − i(D (1) α Φ)D (1) β V α(1) + i 4 (D (1) β Φ)D (1) α V α(1) + i 4 ΦD (1) β D (1) α V α(1) = 4C (2) βα ΦV α(1) + i 4 ΦD (1) β D (1) α V α(1) = 0. (3.54)
Here, we are using the constraints (2.27) and (2.29) in the second equality. The third equality uses D
(1) β D (1) α V α(1) = 16iC (2)
βα V α(1) , which follows from the tensor constraint (2.29). Additionally, it is non-trivial but easy to check that this composite field is a super-Weyl tensor of scaling dimension 4: δG (2) = 4σG (2) . Of course, all the previous arguments also hold in the case that the two tensor multiplets are not independent one of one another but satisfy Φ = D αi V αi as in (2.30).
Comparison of the constraints to those of the vector multiplet (2.33) shows that the same arguments work if we formally replace the tensor potential V α i → F α i with the vector field-strength. Thus, the coupling of a vector and a tensor multiplet naturally gives rise to the weight-2 projective composite superfield [29]
F (2) := i(D (1) α Φ)F α(1) + i 4 ΦD (1) α F α(1) . (3.55)
If one, furthermore, considers a vector multiplet prepotential, which can be shown to be described by a weight-0, real, tropical superfield V := V (0) , then it is possible to construct the Lagrangian
L (2) = V F (2) . (3.56)
This should be compared with the five-dimensional vector multiplet Lagrangian coupled to supergravity [10,11].
Finally, we point out that we can further extend the construction of the previous bilinear: Consider a real weight-0 isotwistor superfield Φ (0) (z, v) and a real weight-1 isotwistor superfield V α(1) constrained by
(γ a ) αβ D (1) α D (1) β + 16iC (2) a Φ (0) (z, v) = 0 , δ σ Φ (0) = 2σΦ (0) , (3.57a) D (1) α V β(1) − 1 4 δ β α D (1) α V α(1) = 0 , δ σ V α(1) = 3 2 σV α(1) . (3.57b)
Then, analogously to the previous cases, the composite superfield
L (2) := i(D (1) α Φ (0) )V α(1) + i 4 Φ (0) D (1) α V α(1) , (3.58)
is a real, weight-2 projective superfield such that δ σ L (2) = 4σL (2) . Note that, in this case, L (2) is not an O(2) multiplet.
The Lagrangian (3.58) appears to be the projective superspace analogue of the harmonic superspace Lagrangian introduced by Sokatchev to describe an off-shell tensor multiplet [31]. The latter was constructed by first taking a tensor multiplet of the Φ-type and an independent tensor multiplet of the V αi -type and lifting them to harmonic superspace by allowing them arbitrary dependence on the harmonics. The construction of the projective action (3.58) is analogous: We started with two copies of the tensor multiplet (in different representations) and took them off-shell by allowing them to have arbitrary dependence on the isotwistor variable v i .
Conclusion
In this paper, we have initiated the study of six-dimensional, N = (1, 0) supergravity in projective superspace. Beginning with the conventional constraints (2.13a)-(2.13c), we provided the solution (2.14a)-(2.17) of the Bianchi identities up to and including dimension- 3 2 . Super-Weyl transformations (2.18a, 2.18b) preserving this geometry were computed and used to recover the components of the type-i Weyl multiplet of 6D, N = (1, 0) conformal supergravity. Coupling this multiplet to a closed super-3-form, we recovered the type-ii Weyl multiplet. With the supergeometry understood, projective isotwistor variables were introduced and used to define projective superfields. The defining constraint of such fields was solved by constructing the analytic projection operator (3.16), which was subsequently used to define a projective superspace action principle (3.25). This was checked to be invariant under super-Weyl, local super-Poincaré, and local SU(2) transformations and reduced to its flat limit which agrees with the flat actions of [46,47]. We concluded with the presentation of families of examples of such action principles for supergravity-matter systems.
Clearly, much remains to be done to complete our understanding of N = (1, 0) supergravity in six-dimensional projective superspace. Perhaps the most pressing open problem is the construction of the projective superspace analogue of Sokatchev's harmonic supergravity [31]. This construction exploits a remarkable combination of harmonic superspace prepotentials, both representations of the tensor multiplet, and the dynamical equations of the "matter" components of the type-i multiplet to avoid multiplet doubling. The Lagrangian (3.58) is similar to the doubled-tensor compensator Lagrangian central to that construction. It would be of interest to confirm their equivalence and work out the detailed relation between these constructions.
Additional directions of study include compactification to five dimensions and comparison with the work of [10,11] and the recovery of our geometry from the gauging of the six-dimensional superconformal group along the lines of references [73,74] which develope a direct link between superspace and superconformal tensor calculus. 7 More straightforward work in need of completion includes: the presentation of the complete solution of the Bianchi identities for the supergeometry of section 2; the investigation of six-dimensional supersymmetric backgrounds and projective superspace matter couplings as in the research on 4D and 5D anti-de-Sitter supergeometries [9,75,76,77,78]; a more systematic classification of covariant projective superfields in six dimensions; and the component reduction of the 6D projective action principle, for example along the lines of [10,68], which, within our formalism, is a first step towards the analysis of various supergravity-matter systems in components (see e. g. [79]).
Finally, we mention that new results on the construction of higher-derivative supergravity actions in six-dimensions have been obtained in [40]. It would be interesting to understand how classes of higher-derivative actions are constructed in six-dimensional projective superspace.
project No. DE120101498.
A Six-dimensional Notation and Conventions
Our six-dimensional superspace conventions are obtained by lifting the five-dimensional conventions established in references [80,9,10]. The procedure is to first define γ a := −Γ a C −1 andγ a = −CΓ a for a = 0, . . . , 3; 5. Then we take γ 6 = C −1 andγ 6 = −C. 8 The relative sign has been chosen so that the six 8 × 8 Dirac matrices satisfy The overall sign is chosen so that, in terms of explicit indices, the formulae are
(γ a ) αβ = (Γ a ) αβ , (γ a ) αβ = −(Γ a ) αβ for a = 0, 1, 2, 3; 5 (γ 6 ) αβ = ε αβ , (γ 6 ) αβ = −ε αβ . (A.3)
In terms of Pauli-type matrices, the Dirac matrices take the form Γ a = 0 (γ a ) αβ (γ a ) βα 0 (A.4) with α = 1, . . . , 4. We can give an explicit representation of γ a ,γ a in terms of the 4D Pauli matrices. In particular, denoting the 4D, SL(2, C) spinor indices by α = 1, 2 anḋ α = 1, 2, we use the representation
γ a = 0 −(σ a ) αβ (σ a )α β 0 (A.5)
for a = 0, . . . , 3 and
γ 5 = iε αβ 0 0 iεαβ , γ 6 = ε αβ 0 0 −εαβ (A.6) 8
Keeping this procedure in mind, it is easier to verify certain statements using formulae from five dimensions. For example, since γ T a = (C −1 ) T Γ T a = −C −1 CΓ a C −1 = −γ a for a = 0, 1, . . . , 3; 5 we need inspect only γ 6 to conclude that these matrices are anti-symmetric. and similarly for matrices with upper indices. They obey the Pauli-type algebra
(γ a ) αβ (γ b ) βγ + (γ b ) αβ (γ a ) βγ = −2η ab δ γ α , (γ a ) αβ (γ b ) βγ + (γ b ) αβ (γ a ) βγ = −2η ab δ α γ . (A.7)
Due to our sign choices, the five-dimensional subalgebra agrees with that of references [80,9,10].
Note that the six-dimensional Pauli-type matrices are antisymmetric
(γ a ) αβ = −(γ a ) βα , (A.8)
implying an isomorphism between the space of six-dimensional vectors and antisymmetric 4 × 4 spin matrices
V αβ := (γ a ) αβ V a = −V βα ⇔ V a = 1 4 (γ a ) αβ V αβ . (A.9)
The second relation is a consequence of the analysis below and equation (A.15) in particular. Similarly, six-dimensional 2-forms are in one-to-one correspondence with traceless 4 × 4 matrices and (anti-)self-dual 3-forms are in correspondence with symmetric rank-2 spin matrices with their indices (down) up as we now work out in detail.
To begin, it is useful to define the normalized anti-symmetrized products of Pauli-type matrices On the other hand, a more commonly used convention regarding the 2-form matrix is as the spinor representation (2.7) of the Lorentz generator M ab which is related by
(Σ ab ) α β = − 1 2 (γ ab ) α β . (A.12)
In terms of these matrices, we define
F α β := 1 2 (Σ ab ) α β F ab ⇒ F ab = −(Σ ab ) β α F α β . (A.13)
The second relation is a consequence of (A.17) which follows from the analysis below.
Using the second type of matrix, we can constructF α β := (γ ab ) α β F ab but (γ ab ) α β = −(γ ab ) β α so that this second matrix is not essentially new.
Finally, the third-rank antisymmetric tensors can be separated into (anti-)self-dual parts which are then in one-to-one correspondence with symmetric 4 × 4 matrices. To see how this works in detail, we must first establish some Fierz identities. There is a completeness relation
1 2 (γ a ) αβ (γ a ) γδ = ε αβγδ .
(A.14)
Contraction with ε γ ′ δ ′ γδ implies the completeness relation
1 2 (γ a ) αβ (γ a ) γδ = δ γ α δ δ β − δ γ β δ δ α (A.15)
and that
1 2 ε αβγδ (γ a ) γδ = (γ a ) αβ ⇒ (γ a ) αβ = 1 2 ε αβγδ (γ a ) γδ . (A.16)
Contraction of (A.15) with itself gives
1 4 (γ ab ) α β (γ ab ) γ δ = − 1 2 δ α β δ δ γ + 2δ δ β δ α γ . (A.17)
Another contraction with (A.15) gives (γ abc ) αβ (γ abc ) γδ = 24(δ α γ δ β δ + δ α δ δ β γ ) , (A. 18) while contraction with (A.14) shows that (γ abc ) αβ (γ abc ) γδ = 0 and (γ abc ) αβ (γ abc ) γδ = 0 .
(A.19)
Thus we see thatγ abc and γ abc correspond to (anti-)self-dual 3-forms. To show that (γ abc ) γ abc is (anti-)self-dual, one can use the identities 20) to conclude that, for example, γ 012 = −ǫ 012 345 γ 345 whereasγ 012 = ǫ 012 345γ 345 . Therefore, to conform to the accepted conventions on (anti-)self-duality, we normalize ǫ 012356 = 1.
γ 0γ1 γ 2γ3 γ 5γ6 = −1 andγ 0 γ 1γ2 γ 3γ5 γ 6 = +1 ,(A.
From the trace relation on the 3-forms 21) it follows that the (anti-)self-dual parts of a 3-form N satisfy
tr(γ abc γ def ) = 4! δ d [a δ e b δ f c] + 1 3! ǫ abc def , (A.N αβ := 1 3! N abc (γ abc ) αβ ⇒ N (+) abc = 1 8 N αβ (γ abc ) αβ , N αβ := 1 3! N abc (γ abc ) αβ ⇒ N (−) abc = 1 8 N αβ (γ abc ) αβ . (A.22)
In six dimensions, Hodge duality on 3-forms is an involution of order 2:
1 3! ǫ abcrst ǫ def rst = −3!δ d [a δ e b δ f c] . (A.23)
Following [80], the components of the SU * (4) spinor and its complex conjugate
(Ψ α ) = ψ ᾱ φα and (Ψ * α ) = ψα φ α (A.24)
are defined in terms of 4D SL(2, C) spinors. Introducing the unitary matrix
B αβ = 0 ε αβ −εαβ 0 ⇒ B Tβ α = 0 εβα −ε βα 0 , (A.25)
it can be checked explicitly that, using the representation defined by (A.5) and (A.6),
B αᾱ B ββ (γ * a )ᾱβ = (γ a ) αβ . (A.26)
We may, therefore, define the covariant conjugate 9
(Ψ α ) := (B αᾱ Ψ * α ) = φ α −ψα . (A.27)
The complex conjugate B * Finally, we define the doublet
(Ψ i α ) such that Ψ 1 α = Ψ α and Ψ 2 α =Ψ α . (A.30)
This combination satisfies the SU(2)-Majorana-Weyl reality condition
(Ψ i α ) = B αᾱ ((Ψ i ) * )ᾱ = Ψ αi , (A.31)
where we used the normalization ε 21 = 1. It follows from this that
Ψ i = Ψ i . (A.32)
Finally, the following conjugation relation holds
(D i α Φ) = −(−1) |Φ| D αiΦ . (A.33)
B On the Analytic Projector
In this appendix we derive the results stated at the end of section 3 about the analytic projection operator ∆ (4) equations (3.16) and (3.22).
First of all we, want to prove that, given a general scalar weight-n isotwistor superfield U (n) , the superfield Q (n+4) := ∆ (4) U (n) satisfies D (1) α Q (n+4) = 0. The derivation goes along the same lines of the 5D case of [9,10,11]. 10 The starting point is to observe that by construction
β }D (1) γ D (1) δ D (1) ρ + 3D (1) γ {D (1) α , D (1) β }D (1) δ D (1) ρ +2D (1) γ D (1) δ {D (1) α , D(1)β }D (1) ρ + D (1) γ D (1) δ D (1) ρ {D (1) α , D(1)β } . (B.2)(1)
Then, one applies the previous equation on the superfield U (n) and compute the anticommutators. Since [J (2) , D
α ] = J (2) U (n) = 0, the SU(2) part of the anti-commutator algebra (3.5) does not contribute at all in the computation. On the other hand, from the Lorentz part of (3.5) we need to systematically move to the right the Lorentz generator by 10 The analysis in this section generalize and in principle can be used to derive the analytic projection operator for the conformal supergravity geometry of [11] in the case where the 5D C iĵ a torsion component is nonzero. Such case was not presented in [11].
using [M α β , D γk ] = 1 4 δ β α D γk − δ β γ D αk to then hit U (n) and use M α β U (n) = 0. Moreover, one needs to elaborate on relations involving multiple D (1) derivatives of C (2) αβ . It is easy to prove that
{D (1) α , D(1)
β }C
(2)
γδ = 0 ⇔ D (1) α D (1) β C (2) γδ = −D (1) β D (1) α C (2) γδ = D (2) αβ C (2) γδ . (B.3)
On the other hand, due to (2.16), we know that
D (1) α C (2) βγ = D(1)
[α C (2) βγ] ,
D (1) α C (2) βγ = 2ε αβγδ C (3)δ . (B.4)
It is then clear that the following equation holds
D (1) α D (1) β C (2) γδ = D(1)
[α D (1) β C (2) γδ] = 1 12 ε αβγδ (D (2) ρτ C (2)ρτ ) .
(B.5)
One can also derive this further result
D (1) α D (2) βγ C (2) δρ = −12iD (1) α C(2)
[βγ C (2) δρ] .
(B.6)
At this point, after some algebra, one can obtain
D (1) α D (4) U (n) = ε βγδρ 5i 12 C (2) βγ D (1) α D (2) δρ − 5i 36 (D (1) β C (2) γδ )D (1) α D (1) ρ + i 48 (D (2) βγ C (2) δρ )D (1) α − 3 2 C (2) βγ C (2) δρ D (1) α U (n) (B.7)
It is then easy to get the following results
D (1) α ε βγδρ C(2)
βγ D (2) δρ U (n) = ε βγδρ C
D (1) α ε βγδρ (D (1) β D (1) γ C (2) δρ )U (n) = ε βγδρ (D (2) βγ C (2) δρ )D (1) α − 12i D (1) α C (2) βγ C (2) δρ U (n) , (B.8c) D (1) α ε βγδρ (C (2) βγ C(2)
δρ )U (n) = ε βγδρ (C As a next step we want to compute the super-Weyl transformations of the superfield Q (n+4) = ∆ (4) U (n) supposing that U (n) transforms homogeneously δ σ U (n) = wσU (n) . To do that, we need some straightforward intermediate results. In particular, we have Finally, after some algebra, one can obtain the following equation δ σ ∆ (4) U (n) = (w + 2)σ∆ (4) U (n)
δ σ D (1) α = 1 2 σD (1) α − 2(D(1)+(w − 2n − 6) − 1 24 ε αβγδ σ (1) α D (1) β D (1) γ D (1) δ U (n) − 1 16 ε αβγδ σ (2) αβ D (2) γδ U (n) − 1 24 ε αβγδ σ (3) αβγ D (1) δ U (n) − 1 96 ε αβγδ σ (4) αβγδ U (n) − 5i 3 σ (1) α C (2)αβ D (1) β U (n) − 5i 6 σ(2)
αβ C (2)αβ U (n) + 5i σ (1) α C (3)α U (n) . (B.14)
It is clear that by choosing w = 2(n + 3) , (B.15) the weight-(n + 4) projective superfield Q (n+4) = ∆ (4) U (n) transforms homogenously, δ σ Q (n+4) = 2(n + 4)σQ (n+4) , in agreement with equation (3.9).
To conclude this appendix, we point out that, in the case n = 0, all the previous results are exactly the same if instead of a weight-0 isotwistor superfield U (0) one considers a vindependent superfield P (z) such that δ σ P = 6σP . Then, the superfield P (4) := ∆ (4) P is a weight-4 projective superfield. To convince oneself of this, one has only to notice that for both U (0) and P the conformal weight is 6 and that J (2) U (0) = J (0) U (0) = J (2) P = J (0) P = 0 holds.
3
To follow the nomenclature normally used in the literature for the component fields of the tensor multiplet, we call the lowest component field σ(x). Context should serve to distinguish this component from the super-Weyl parameter superfield σ(z).
both Q (n) and U (n−4) can be required to transform homogeneously with respect to super-Weyl transformations in which case the transformations are fixed to be δ σ U (n−4) = (2n − 2)σU (n−4) , δ σ Q (n) = 2n σQ (n) .(3.21)
conf = i K(Υ (1)I ,Υ (1)J ) .
{Γ a , Γ b } = −2η ab 1 , (A.1)with a = 0, . . . ,
γ ab := γ [aγb] := 1 2 (γ aγb − γ bγa ) , γ abc := γ [aγb γ c] := 1 3! (γ aγb γ c ± perm.) , γ ab :=γ [a γ b] := 1 2 (γ a γ b −γ b γ a ) ,γ abc :=γ [a γ bγc] := 1 3! (γ a γ bγc ± perm.) . (A.10) For example, this normalization implies γ ab γ c = γ abc + 2η c[a γ b] ,γ c γ ab =γ abc − 2η c[aγb] . (A.11)
BB
* = −1 = B * B . (A.28) This implies that performing the conjugation twice, Ψ = (BΨ * ) = B(B * Ψ) = −Ψ . (A.29)
D
(1) α D (4) = − 1 480 ε βγδρ 4{D (1) α , D
the equations (B.7)-(B.8d) one can then observe that the combination of operators in the analytic projection operator (3.16) is such that D (1) α ∆ (4) U (n) = 0 . (B.9)
term in (B.10a) does not actually enter into this computation since J(2) commutes with the D
It is worth pointing out that a prepotential formulation of the Weyl multiplet was given by Sokatchev in 6D harmonic superspace[31].
An early attempt in six dimensions was made in reference[48].
The signs have been chosen so that the covariant conjugate reduces in five dimensions to the (transpose of the) Dirac conjugate in the conventions of[80].
Acknowledgements We are grateful to Sergei M. Kuzenko
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[
"Recent experimental results from the relativistic heavy-ion collisions at LHC and RHIC",
"Recent experimental results from the relativistic heavy-ion collisions at LHC and RHIC"
]
| [
"Ilya Selyuzhenkov \nResearch Division and ExtreMe Matter Institute EMMI\nGSI Helmholtzzentrum für Schwerionenforschung Planckstraße\n\n\n64291DarmstadtGermany\n"
]
| [
"Research Division and ExtreMe Matter Institute EMMI\nGSI Helmholtzzentrum für Schwerionenforschung Planckstraße\n",
"64291DarmstadtGermany"
]
| []
| A new era has started in the field of relativistic heavy-ion physics with lead beams delivered by the Large Hadron Collider (LHC) in November 2010. In this proceedings I highlight the main results from experimental measurements with Pb-Pb collisions at the incident energy of 2.76 TeV/nucleon recorded by the LHC experiments. Recent experimental developments from the Relativistic Heavy Ion Collider (RHIC) at the GeV incident energy scale are also discussed. All together LHC and RHIC measurements provide new insights on the properties and features of the new hot and dense form of matter created in the course of the relativistic heavy-ion collision. | null | [
"https://arxiv.org/pdf/1109.1654v1.pdf"
]
| 117,685,007 | 1109.1654 | 7665725e7a644e7f783ddbd13aee0c079a661b0a |
Recent experimental results from the relativistic heavy-ion collisions at LHC and RHIC
Ilya Selyuzhenkov
Research Division and ExtreMe Matter Institute EMMI
GSI Helmholtzzentrum für Schwerionenforschung Planckstraße
64291DarmstadtGermany
Recent experimental results from the relativistic heavy-ion collisions at LHC and RHIC
A new era has started in the field of relativistic heavy-ion physics with lead beams delivered by the Large Hadron Collider (LHC) in November 2010. In this proceedings I highlight the main results from experimental measurements with Pb-Pb collisions at the incident energy of 2.76 TeV/nucleon recorded by the LHC experiments. Recent experimental developments from the Relativistic Heavy Ion Collider (RHIC) at the GeV incident energy scale are also discussed. All together LHC and RHIC measurements provide new insights on the properties and features of the new hot and dense form of matter created in the course of the relativistic heavy-ion collision.
Introduction
The main goal of the more than a ten years of operation of the Relativistic Heavy Ion Collider (RHIC), and of the heavy-ion program recently launch at the Large Hadron Collider (LHC) is to study the properties of the hot and dense matter, the so called quark-gluon plasma (QGP), which is believed to have existed a few microseconds after the big bang. By colliding heavy nuclei at relativistic energies we heat up the normal cold matter and transfer it from the hadronic phase to fireball of deconfined quarks and gluons, which allows us to probe the QGP properties in the laboratory.
Theoretically the time evolution of the system created in the course of heavy-ion collisions is described by a sequence of several stages. It starts from the initial pre-equilibrium state when hard parton scattering occurs and gluonic fields are formed. The next stage is the formation and then expansion of a thermalized state of matter, the quark-gluon plasma, which is conventionally described by hydrodynamics. Consequently, the quarks and gluons are coupled (hadronize) into hadrons, which ends with the phase of chemical and then kinetic freeze out (all interactions are ceased at this moment). The only data which directly accessible for experimentalist is the information on this last hadronic stage. Evidently, it is not possible to constrain theoretical models which pretend to describe the evolution of the heavy-ion collision and identify the one which most precisely reflects the nature of the collision with a single measurement, and so it is necessary to study many experimental observables to pin down the properties of the QGP.
In this proceedings I highlight the new results from the ALICE, CMS, and ATLAS Collaborations from the first heavy-ion run at LHC in November 2010, as well as recent experimental developments by the STAR and PHENIX Collaborations at RHIC. I start with a discussion of global properties of the collisions at LHC energies, such as charged hadrons multiplicity, particle yields, and a measurement of the collision freeze out volume from the Bose-Einstein correlations of identical pions. I then briefly review the results for anisotropic flow which reflect collectivity in particles production and allows us to experimentally constrain the possible initial conditions of the collision and the QGP viscosity. Switching to hard probes, I present the main results on the suppression of particle production at high transverse momenta in heavy-ion collision. I continue the discussion with a few highlights from the beam energy scan program at RHIC aimed to probe the properties of the phase boundary and search for the critical point in the QCD phase diagram. I complete my proceedings with a few remarks on probes of local parity violation in the strong interaction which shows the potential to go beyond the scope of the QGP physics with the heavy-ion programs at RHIC and LHC. Figure 1(a) shows a compilation of the results for charged particle multiplicity density measured for heavy-ion collisions at the LHC and lower energies at RHIC, SPS, and AGS, as well as for proton-proton collisions. The charged particle multiplicity density in central Pb-Pb collisions at 2.76 TeV/nucleon is measured to be dN ch /dη ≈ 1600, what is larger by a factor of 2.15 than that at top RHIC energy. Compared to pp collisions at the same energy the charge density is increased by a factor of 1.9. The measured multiplicity and spectra correspond to an increase by 2.5-3 times in energy density from RHIC to LHC, which for central Pb-Pb collisions at LHC is measured to be dE t /dη ∼ 2 TeV per unit of rapidity [2]. As it can be seen from Fig. 1(b) the shape of the charged particle production per participant pair versus centrality is almost identical for RHIC and LHC energies, what may indicate that saturation effects do not significantly change despite shifting toward smaller Feynman x f at LHC. Figure 2(a) shows the transverse momentum spectra of identified particle measured by the ALICE Collaboration for the 0-5% most central Pb-Pb collisions at 2.76 TeV/nucleon. The spectra slopes change dramatically compared to RHIC data (open symbols in Fig. 2(a)), especially for protons. This reflects a significantly stronger radial flow. The radial flow velocity reaches about 60% of the speed of light with a simultaneous reduction of the kinetic freeze-out temperature down to 80 MeV ( Fig. 2(b)). system lifetime are deduced from the Hanbury-Brown-Twiss (HBT) momentum-space twoparticle correlations of identical pions. The HBT homogeneity region, which is connected to the HBT out-long-and side-radii via (2π) 3 2 R out R side R long , increases by a factor 2 ( Fig. 3(a)) compared to the top RHIC energy of 0.2 TeV/nucleon pair. The system lifetime also increases by more than 30% ( Fig. 3(b)). These trends are consistent with hydrodynamical model calculations for LHC energies using parameters tuned to reproduce the RHIC data.
Particle yields
Bose-Einstein correlations
Anisotropic transverse flow
Azimuthal anisotropic flow is a key observable indicating collectivity among particles produced in relativistic heavy-ion collisions. Figure 4 shows the integrated (a) and p t differential (b) elliptic flow, v 2 , measured in Pb-Pb collisions at 2.76 TeV/nucleon. The immediate conclusion to be drawn from the comparison of v 2 results measured at LHC by the ALICE, ATLAS, and CMS Collaborations to that at lower RHIC energies is that the integrated v 2 increases by 30%, see Fig. 4(a). Figure 4(b) shows the differential flow results as a function of charged particle's transverse momentum, p t . The results from RHIC and the LHC are similar in both magnitude and the shape of p t dependence. This behavior is a consequence of a stronger radial flow at the LHC as was already discussed in Sec. 2 above. The strong particle collectivity reflected by large v 2 at LHC shows that the system Qualitatively the validity of the hydrodynamic description for LHC and effects of stronger radial flow can be tested with the anisotropic flow measurement of identified particles and its dependence on the mass of different species. Figure 5 shows the p t differential elliptic flow of charged pions, kaons, and protons measured by the ALICE Collaboration in Pb-Pb collisions at 2.76 TeV/nucleon. The observed larger than at RHIC mass splitting of v 2 agrees well with a picture of increased radial flow and follows viscous hydrodynamic predictions (solid lines in Fig. 5(a)) except for anti-protons in the most central collisions. Anti-protons also fall out of the universal scaling with number of quarks seen at RHIC energies.
|>1} η ∆ {2,| 2 , v ± π |>1} η ∆ {2,| 2 , v ± K |>1} η ∆ {2,| 2 , v p hydro LHC (CGC initial conditions) /s=0.2) η (|>1} η ∆ {2,| 2 , v ± π |>1} η ∆ {2,| 2 , v ± K |>1} η ∆ {2,| 2 , v p (a) (b)
One of the main highlights of recent anisotropic flow results from both RHIC and LHC experiments is establishing the connection between the measured event anisotropy in the momentum space (anisotropic flow) and the fluctuations of the energy density in the initial state of the heavy-ion collisions. Fig. 6(a)), vs. non-zero v 3 measured with respect to the participant plane -the plane determined by the event-by-event fluctuating shape of the initial energy density (blue squares in Fig. 6(a)). Triangular and higher order harmonic flow also explains the double hump structure seen originally at RHIC in the two-particle azimuthal correlations and often referred to as the Mach Cone effect. Figure 6(b) shows that for the most central Pb-Pb collisions at 2.76 TeV/nucleon the whole shape of the two particle azimuthal correlations is driven by the interplay between various anisotropic flow components, mainly v 2 and v 3 .
0.1 ALICE > 1} η ∆ {2, 2 v > 1} η ∆ {2, 3 v > 1} η ∆ {2, 4 v {4} 3 v RP Ψ 3/ v 2 2 Ψ 3/ v × 100
Particle production at large transverse momenta
Production of particles with very large transverse momentum, p t , in heavy-ion collisions happens very early in the collision history and therefore these particles have to propagate through the hot and dense medium created in the collision. Consequently, the modification of high p t particle production compared to the production without medium (e.g. in protonproton collisions) carries information about the medium properties such as the energy loss mechanism and its dependence on the path length. Quantitatively the modification of particle production is described by the nuclear modification factor, R AA , which presents the ratio of the particle yields in heavy-ion collision to that of the proton-proton collisions scaled by the corresponding number of binary collisions. Figure 7(a) shows the charged particle nuclear modification factor, R AA , for Pb-Pb collisions at 2.76 TeV/nucleon measured by the ALICE and CMS Collaborations compared to results for charged and identified neutral pions from RHIC and SPS experiments. The deviation of R AA from unity reflects the effect of medium modification, and reveals strong suppression (R AA << 1) of particle production in heavy-ion collisions compared to that in proton-proton interactions. As a function of transverse momentum, R AA shows a minimum around 5-7 GeV/c, and then rises significantly towards higher transverse momentum but even at p t ∼ 100 GeV/c the particle production is largely suppressed (R AA ∼ 0.5). Compared to different model calculations (color lines in Fig. 7(a)) these new results provide strong constrains on models with different parton energy loss. Figure 7(b) shows that even heavy quark (J/ψ) production is strongly suppressed at RHIC and LHC, though at LHC the suppression is reduced in accord with the expectations from the statistical model [9]. Important probes of the nuclear parton densities created in heavy-ion collision are the colorless objects such as prompt photon and Z boson, since they are produced directly from hard parton interactions and propagate through the medium of quarks and gluons without modification. Figure 8(a) shows the R AA of direct photons as a function of the photon transverse energy for the most central Pb-Pb collisions measured by the CMS Collaboration. Photon R AA measured at LHC is consistent with unity (no modification), which is similar to the recent results for prompt photons by the PHENIX Collaboration (orange points in Fig. 8(b)). The R AA of another colorless probe, the Z boson, is also measured by the CMS Collaboration and found to be consistent with no medium modification: R (Z) AA = 1.2 ± 0.29(stat.)±0.16(syst.) [10].
RHIC beam energy scan and the search for the critical point
Another frontier of the heavy-ion physics program, which complements the study of the QGP properties at RHIC and LHC, is to determine the nature of the phase transition between confined (hadrons) and deconfined (quark-gluon plasma) matter with a search for the critical point on the QCD phase diagram. These two objectives are the main goals of the Beam Energy Scan program at the RHIC facility. The features of the phase transition and proximity of the critical point can be studied by looking at irregular changes in the degrees of freedom of the system created in heavy-ion collisions. Experimentally this should be reflected in non-monotonic behavior of the sensitive physics observables. Examples of sensitive observables are particle collectivity such as anisotropic flow or HBT correlations, or fluctuations in the system (e.g. fluctuations of the conserved quantities such as baryon number or strangeness). Figure 9(a) shows the relative variation in the identified particle and anti-particle elliptic flow measured by the STAR Collaboration for different collision energies ranging from 11.5 GeV/nucleon up to the top RHIC energy of 200 GeV/nucleon. With decreasing collision energy the baryon and anti-baryon elliptic flow difference increases dramatically compared to that for mesons. Another change in the elliptic flow pattern at lower energies is the breaking of the constituent quark scaling for the elliptic flow which seems to hold at top RHIC energy. Figure 9(b) shows number of constituent quark scaled elliptic flow of identified particles for Au-Au collisions at 11.5 GeV/nucleon. Despite the large statistical errors, results for the φ-meson (which carries the information about the strange quark production) deviates significantly from the overall scaling of other particles, which probably indicate the breaking of the quark collectivity at lower energies and change in the degrees of freedom in the system.
Probes of local parity violation in strong interactions
The strong magnetic field created in the interaction zone of non-central relativistic heavy-ion collisions may interact with the topologically non-trivial gluonic field configurations of QCD such as instantons and sphalerons. It is predicted [13] that experimentally this may lead to charge separation of hadrons produced in the collision along the magnetic field, which itself is aligned perpendicular to the reaction plane of the collision. Since instanton and sphaleron configurations breaks the parity symmetry of QCD, the measurement of charge separation provides a unique experimental test of how well the parity symmetry is preserved by the strong interaction [14].
Experimentally the effects of charge separation can be quantified by the charge dependent azimuthal correlations with respect to the reaction plane. Figure 11 shows the experimental results for the cos(φ α + φ β − 2Ψ RP ) correlator, which is the parity even observable but directly sensitive to the event-by-event charge fluctuations, and thus to the possible local parity violation in strong interactions. The STAR and PHENIX Experiments at RHIC, and now the ALICE Collaboration at LHC [15] observe a significant charge separation at higher collision energies ( Fig. 11(a)), which seems to disappears between 11.5 and 7.7 GeV energies (lower panel in Fig. 11(a)). The experimental situation is significantly complicated by the presence of the parity conserving background correlations which may contribute to the measured charge dependent azimuthal correlations at RHIC and LHC. Recent progress in understanding the possible parity even backgrounds, such as identifying the large flow fluctuations in the first harmonic flow helps to better understand background contributions, but there is still a long way to go before we will be able to conclude whether the observed charge separation is indeed connected to the effects of local parity violation, or it is just a complicated interplay of yet unidentified background sources.
Summary
The results by the ALICE, ATLAS, and CMS Collaborations from the first heavy-ion run at the Large Hadron Collider in November 2010 opened a new era of experimental studies of the quark-gluon plasma in the laboratory. Together with the new high statistics data collected during the past few years by the STAR and PHENIX Experiments at the Relativistic Heavy Ion Collider this provides an extremely rich experimental data set which allows us to study the properties of the quark-gluon plasma in the great detail, and let us to learn more about the features of the universe a few microseconds just after the Big Bang. I am looking forward to further experimental developments and more exciting results from the LHC and RHIC scientific communities!
Figure 1 :
1(a) Charged particle multiplicity per participant pair measured for Pb-Pb collisions by the ALICE and ATLAS Collaborations at LHC, and compared to results for proton-proton and heavy-ion collisions at lower energies. (b) Centrality dependence of the multiplicity per participant pair measured by the ALICE and ATLAS Collaborations at the LHC, and the STAR Collaboration at RHIC. Figures taken from[1].
Figure 2 :
2(a) Identified charged particle spectra measured by the ALICE Collaboration for heavy-ion collisions at the LHC in comparison with results for top RHIC energy. (b) Freeze-out temperature, T f o , and radial velocity, β t , extracted from the blast wave fits to the identified charged particle spectra measured at RHIC and LHC. Figures taken from[3].
Figure 3
3shows the decoupling time and the size of the freeze-out (homogeneity) region of the fireball created in Pb-Pb collisions at LHC and at lower energies. The volume and the
Figure 3 :
3(a) The homogeneity volume (triple product of the pion HBT radii which is proportional to the system volume via (2π)3 2 coefficient). (b) The decoupling time of the system created in the heavy-ion collision. Results are for Pb-Pb collisions at 2.76 TeV/nucleon measured by the ALICE Collaboration, and for central Au-Au and Pb-Pb collisions for lower AGS and RHIC energies. Figures taken from [4].
Figure 4 :
4(a) Integrated elliptic flow v 2 as a function of the collisions energy. (b) v 2 as a function of transverse momentum for the 40-50% centrality range measured in heavy-ion collisions at RHIC and LHC. Figure (a) taken from [2], figure (b) from [1].created in heavy-ion collision at TeV energy scale behaves as a strongly interacting, close to be a perfect fluid -similar to the properties of the QGP observed at RHIC. All this speaks towards applicability of the hydrodynamic model description of the heavy-ion collisions at LHC energies.
Figure 5 :
5(a) Elliptic flow of pions, kaons and anti-protons vs. transverse momentum for the 10-20% centrality range. The lines are hydrodynamical model calculations. (b) Elliptic flow versus transverse kinetic energy are both scaled with the number of constituent quarks for the 40-50% centrality range. Figures taken from [5].
Figure 6
6
Figure 6 :
6Alver, Gombeaud, Luzum & Ollitrault, Phys. Rev. C82 034813 (2010) (a) Integrated elliptic (v 2 ), triangular (v 3 ), and quadrangular (v 4 ) flow measured for Pb-Pb collisions at 2.76 TeV/nucleon by the ALICE Collaboration. (b) Two-particle correlation function measured for 1% most central collisions by the ATLAS Collaboration. The measured 2-particle correlations are reproduced well by the combination of the Fourier coefficients from the anisotropic flow measurement (solid lines). Figure (a) taken from [5], figure (b) from [1]. anisotropic flow (v 3 , v 4 ) together with the largest v 2 component for Pb-Pb collisions at 2.76 TeV/nucleon measured by the ALICE Collaboration. The geometrical origin of the v 3 component is established by comparison of the vanishing triangular flow, v 3 , when measured with respect to the collision reaction plane (green symbols in
Figure 7 :
7(a) Nuclear modification factor, R AA , as a function of transverse momentum for neutral pions and charged hadrons for the most central heavy-ion collisions. (b) R AA of J/ψ as a function of the number of participants measured at RHIC and LHC. Figure (a) taken from [2], figure (b) from[8].
Figure 8 :
8(a) Nuclear modification factor, R AA , of isolated photons as a function of transverse momentum for 0-10% central events measured by the CMS Collaboration. (b) Results by the PHENIX Collaboration for R AA of several mesons and direct photons for the 0-10% central Au-Au collisions. Figure (a) taken from [2], figure (b) from[7].
Figure 9 :
9(a) Fractional difference between elliptic flow of particles and anti-particles for 0-80% Au-Au collisions plotted vs. collision energy. (b) Elliptic flow of identified particles for Au-Au collisions at 11.5 GeV/nucleon versus transverse kinetic energy both scaled by the number of constituent quarks. Figures taken from[11].
Figure 10 (Figure 10 :
1010a) shows results for the higher order moments (the skewness, S, and kurtosis, κ) of the net proton number. Small deviation of the conserved quantity, the baryon number, from the Hadron Gas Resonance (HGR) model below 39 GeV/nucleon may suggest a hint of proximity to the critical point.Figure 10(b) shows the charged particle R (a) Higher order moments of the net proton number measured for Au-Au collisions for different incident energies. (b) The nuclear modification factor of neutral pions as a function of transverse momentum for Cu-Cu collisions at three different energies. Figure (a) taken from [11] and figure (b) taken from [12].an energy of 22.4 GeV/nucleon and it illustrates the significant change in pattern of the particle suppression at lower energy compare to the top RHIC energy. Overall, there are some hints from the Beam Energy Scan program at RHIC on the possible critical point between energies of 7 and 20 GeV/nucleon. Hopefully the upcoming measurements from RHIC energy scan will allow us to make a conclusive statement about existence of the QCD critical point.
Figure 11 :
11Charged dependent azimuthal correlations with respect to the reaction plane of heavy-ion collision vs. centrality: (a) comparison between results for Pb-Pb collisions at 2.76 TeV/nucleon vs. the measurement for Au-Au at 200 GeV/nucleon, (b) results for Au-Au collisions for 5 different collision energies in the range of 7.7 -200 GeV/nucleon. Note the inverted centrality scale in figures (a) and (b). Figure (a) taken from [15] and figure (b) from[11].
AcknowledgementsThis work was supported by the Helmholtz Alliance Program of the Helmholtz Association, contract HA216/EMMI "Extremes of Density and Temperature: Cosmic Matter in the Laboratory".
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| []
|
[
"Secure communication with single-photon two-qubit states",
"Secure communication with single-photon two-qubit states"
]
| [
"Almut Beige ",
"Berthold-Georg Englert ",
"Christian Kurtsiefer ",
"Harald Weinfurter ",
"\n†Max-Planck-Institut für Quantenoptik\n§Sektion Physik\nTechnische Universität Wien\nHans-Kopfermann-Str. 1, Germany ‡Atominstitut, Stadionallee 285748, 1020Garching, WienAustria\n",
"\nUniversität München\nSchellingstrasse 480799MünchenGermany\n"
]
| [
"†Max-Planck-Institut für Quantenoptik\n§Sektion Physik\nTechnische Universität Wien\nHans-Kopfermann-Str. 1, Germany ‡Atominstitut, Stadionallee 285748, 1020Garching, WienAustria",
"Universität München\nSchellingstrasse 480799MünchenGermany"
]
| []
| We propose a cryptographic scheme that is deterministic: Alice sends single photons to Bob, and each and every photon detected supplies one key bitno photon is wasted. This is in marked contrast to other schemes in which a random process decides whether the next photon sent will contribute to the key or not. The determinism is achieved by preparing the photons in two-qubit states, rather than the one-qubit states used in conventional schemes. In particular, we consider the realistic situation in which one qubit is the photon polarization, the other a spatial alternative. Further, we show how one can exploit the deterministic nature for direct secure communication, that is: without the need for establishing a shared key first. | 10.1088/0305-4470/35/28/103 | [
"https://arxiv.org/pdf/quant-ph/0101066v4.pdf"
]
| 19,014,297 | quant-ph/0101066 | 26194e9bd95b81914ac279626bbd3b0dbf1e9ebe |
Secure communication with single-photon two-qubit states
0101066v4 23 Jul 2002
Almut Beige
Berthold-Georg Englert
Christian Kurtsiefer
Harald Weinfurter
†Max-Planck-Institut für Quantenoptik
§Sektion Physik
Technische Universität Wien
Hans-Kopfermann-Str. 1, Germany ‡Atominstitut, Stadionallee 285748, 1020Garching, WienAustria
Universität München
Schellingstrasse 480799MünchenGermany
Secure communication with single-photon two-qubit states
0101066v4 23 Jul 2002Submitted to: J. Phys. A: Math. Gen.arXiv:quant-ph/PACS numbers: 0367Dd, 4279Sz
We propose a cryptographic scheme that is deterministic: Alice sends single photons to Bob, and each and every photon detected supplies one key bitno photon is wasted. This is in marked contrast to other schemes in which a random process decides whether the next photon sent will contribute to the key or not. The determinism is achieved by preparing the photons in two-qubit states, rather than the one-qubit states used in conventional schemes. In particular, we consider the realistic situation in which one qubit is the photon polarization, the other a spatial alternative. Further, we show how one can exploit the deterministic nature for direct secure communication, that is: without the need for establishing a shared key first.
Introduction
Cryptographic schemes based on the exchange of single photons, each carrying one bit of information, have been widely discussed in the literature [1]. In some of the schemes, Alice and Bob share entangled photon pairs [2]. In others, Bob performs measurements on photons that Alice sends him [3]. They always need to communicate via a classical channel as well. Experiments have shown that secure key distribution is possible indeed, even over a distance of several kilometers [4,5,6].
The standard procedures, such as the so-called BB84 protocol of [3], are not deterministic in the sense that Bob may or may not get a key bit for the next photon that Alice will send; on average one key bit is obtained for every two photons transmitted ‡. By contrast, the scheme we propose here, is deterministic: Bob gets a key bit for each and every photon sent by Alice.
This determinism is the main advantage of our new scheme. It offers, in particular, the option of secure communication without first establishing a shared key.
To achieve the determinism, Alice sends Bob photons prepared in certain twoqubit states, rather than photons carrying one-qubit states. She uses, for example, the spatial binary alternative of the photon with the basis states |R and |L and the two polarization states |v and |h . Here, |R and |L describe a photon traveling in the "right" or the "left" fiber, respectively, and |v and |h refer to photons with vertical and horizontal polarization.
With the aid of unitary two-qubit gates [8], Alice can turn either one of the simple product states |Rv = |R ⊗ |v , |Rh , |Lv , and |Lh into any desired superposition thereof, so that she can send each photon in the single-photon two-qubit state of her choosing. Likewise, Bob's measurements of certain sets of four mutually orthogonal two-qubit states are achieved by appropriate unitary gates. They transform the states of the measurement basis in question into the four basic product states, which are then easily discriminated.
Since each photon carries two qubits now, the new scheme is not more efficient in terms of qubits than the standard ones: Each qubit pair sent gives one key bit.
Quantum key distribution
We present the scheme for key distribution first. It has, of course, a number of features in common with the BB84 protocol, but generates a key bit for every transmitted photon. We discuss its security against eavesdropping and observe that, despite the deterministic nature, it cannot be used for direct communication, that is: for sending a message without establishing a shared key first. We then introduce a second scheme, with more involved state preparation and analysis, that does enable Alice and Bob to communicate directly and confidentially.
Ð ÔÖ Ô Ö Ø Ö ÓÒ Ó ¦ × Ò Ð Ô ÓØÓÒ× ÖÖÝ Ò ØÛÓ ÕÙ Ø× Ð ×× Ð ÓÑÑÙÒ Ø ÓÒ Ó Ø Ø × × Ø Ø ¼ × × ½ ¾ ¿ ¼ ½ ¼ ¾ ¼ ¿ ¼ Ñ ÖÖÓÖ Ñ ×ÔÐ ØØ Ö Figure 1
. Schematic view of the setup. To transmit the bit "+" or "−" Alice prepares a photon in one of the states |i± and sends it to Bob. For the purpose of cryptography, two pairs of states are sufficient [e.g., those of (5)], whereas communication requires four pairs [those of (9)]. A beam splitter reroutes the photon randomly, and then Bob measures in which state |B j or |B ′ j it is. In some cases, communication via the classical channel is required to decode the bit.
The experimental setup of the cryptographic scheme we propose is sketched in Fig. 1. To transmit one bit, "+" or "−", Alice sends Bob a photon prepared in one of the four states |i± (i = 1, 2). For a "+" bit she chooses randomly between |1+ and |2+ ; for a "−" bit between |1− and |2− . When the photon arrives at Bob's end, he randomly chooses between two different two-qubit bases for his analysis of the photon state. Experimentally, this can be achieved by sending the incoming photon through a beam splitter and rerouting it to different measurement devices as shown in Fig. 1. Bob measures either the basis states |B 1 , . . . , |B 4 or |B ′ 1 , . . . , |B ′ 4 , and he can always infer the bit Alice sent. Depending on the outcome of his measurement, he might be able to deduce the incoming bit immediately. In some cases, classical communication is required, and Alice has to tell Bob whether the photon state she prepared was of type "1" or "2".
For illustration, let us consider the simplest version in which the states sent by Alice and the states detected by Bob are all product states § such as (|1+ , |1− ; |2+ , |2− ) = (|Rs , |La ; |Sv , |Ah ) ,
(|B 1 , |B 2 , |B 3 , |B 4 ) = (|Rv , |Rh , |Lv , |Lh ) , (|B ′ 1 , |B ′ 2 , |B ′ 3 , |B ′ 4 ) = (|Ss , |As , |Sa , |Aa )(1)
where
|S |A = 1 √ 2 (|R ± |L ) , |s |a = 1 √ 2 (|v ± |h )
(2) § Therefore, this particular example could also be realized by exploiting the polarization qubits of paired photons.
B 1 or B ′ 1 B 2 or B ′ 2 B 3 or B ′ 3 B 4 or B ′ 4 type 1 + + − − type 2 + − + −
are symmetric (S and s) and antisymmetric (A and a) superpositions of the basic alternatives. Note that each of Bob's states is orthogonal to either the "+" state or the "−" state of each pair; this is the essential property for the deterministic transmission. Suppose, for instance, that Bob detects state |B 3 = |Lv ; it is orthogonal to |1+ = |Rs and |2− = |Ah and therefore it signifies "−" if a photon of type "1" was sent and "+" if it was of type "2". These matters are summarized in Table 1.
Let us now exhibit the basic general features of our deterministic scheme, as they are illustrated by this particular example. How can Bob always know which bit Alice sent? He can distinguish the "+" states from the "−" states unambiguously if, for all state pairs |i+ /|i− , each possible measurement result can only be caused by |i+ or |i− , but not by both. This must be the case for every basis measured by Bob. Then he can infer the bit transmitted as soon as Alice identifies the type of pair used (that is: she tells him the value of the pair label i).
For the security of the scheme it is important that the state pairs |1± and |2± sent by Alice are neither identical nor orthogonal. It is equally important that Bob has more than one basis at his disposal, because this is what renders possible the detection of an eavesdropper. In the example (1), the two bases are in fact even complementary since the transition probabilities | B i |B ′ j | 2 = 1 4 do not depend on the quantum numbers i, j. This maximal incompatibility is not really needed, but the bases should not be very similar to each other in order to ensure that an eavesdropper will surely cause a substantial number of false detections at Bob's end, as we discuss below.
To analyze this in more detail, we consider a scheme that is somewhat more general than the one based on the products states (1). Here Bob's bases are related to each other by
(|B ′ 1 , |B ′ 2 , |B ′ 3 , |B ′ 4 ) = (|B 1 , |B 2 , |B 3 , |B 4 )K(3)
where the 4 × 4 matrix K is given by
K = 1 1 + k 2 1 k k k 2 k k 2 −1 −k k −1 k 2 −k k 2 −k −k 1 (4)
with a real parameter k. For brevity and simplicity, we are satisfied with discussing the most elementary version of the scheme, where Alice makes use of two state pairs only that are given by
|1+ = (|B 1 + k|B 2 )/ √ 1 + k 2 , |1− = (k|B 3 − |B 4 )/ √ 1 + k 2 ,|2+ = (|B 1 + k|B 3 )/ √ 1 + k 2 , |2− = (k|B 2 − |B 4 )/ √ 1 + k 2 ,(5)
More generally, she could always use four pairs, and even six pairs in some versions [9]. Relations (5) remain valid if the |B j 's are replaced by the |B ′ j 's. Note that the inverse of the transformation (3) is also furnished by K since this matrix is both Hermitian and unitary. For k = 1, in particular, we return to the situation of (1) where the two bases |B j and |B ′ j are complementary. Table 1 continues to apply, irrespective of the value of k.
Let us now imagine that Evan, the eavesdropper, is listening in. He intercepts each photon sent by Alice, performs a measurement on it, and then forwards a replacement photon to Bob. Evan will not be able to infer with certainty which two-qubit state is carried by the intercepted photon, and so he has to make an educated guess based on his measurement result. Then he prepares the replacement photon accordingly, namely in the two-qubit state that has the best chance of avoiding wrong detector clicks at Bob's end. If, for instance, Alice has sent a |1+ photon, then the detectors for |B 3 and |B 4 as well as |B ′ 3 and |B ′ 4 would yield wrong clicks and reveal the interference of the eavesdropper. Thus, Evan has to solve a two-fold problem: Which basis should he measure, and which states should be forwarded to Bob, such that the probability for a wrong click is minimal?
These questions can be answered systematically [9], also for more general interceptresend strategies, much like the corresponding studies [10] for the BB84 protocol. (The generalizations do not offer a real advantage to Evan, however.) In an optimal strategy then, the probability that Bob will detect a wrong click is
p (2) min = 1 2 − 1 2 √ 1 + k 4 1 + k 2 .(6)
All other intercept-resend strategies that Evan might employ result in larger error rates. The largest value obtains for k = ±1, namely p (2) min = (2 − √ 2)/4 = 14.6%; for k = 0 and k → ±∞ the minimal error rate vanishes. Both limiting cases are easily understood: for k = ±1 Bob's measurement bases are complementary and therefore maximally incompatible, and for k = 0 or k → ±∞ they are essentially identical. We note in passing that, if four pairs of states are used rather than just the two pairs of (5), the minimal error rate increases to
p (4) min = 1 2 min{1, k 2 } 1 + k 2 ,(7)
which can be as large as 25%, and an eavesdropper's presence can then be noticed more easily.
For the purposes of this letter, we continue to focus on the two-pair scheme and assume that Alice and Bob have wisely chosen a k value near k = 1, say. Suppose they want to establish a key of 1000 bits, and Alice sends 1100 photons in two-qubit states, randomly chosen from the four states of (5). Bob detects all photons, then selects a random subset of 100 and tells Alice in which states he found them. If some of Bob's measurement results are inconsistent with the states Alice sent, such as detecting |B ′ 3 for a |1+ photon, then Alice doesn't trust the transmission and they start all over. If, however, Bob's results are all right, then Alice concludes that the likelihood that Evan has listened in is less than (1 − p (2) min ) 100 = 1.3 × 10 −7 , which she and Bob have earlier decided to be sufficiently small for the security level they'd like to have. Alice then reveals the type of each photon, "1" or "2", and Bob infers the bits sent with the aid of Table 1. Thereafter they share a secure 1000-bit key string. A confidential message of this length can then be exchanged.
Secure communication without first establishing a shared key
Given the deterministic nature of the scheme, one might wonder if Alice couldn't send a message directly to Bob without first establishing a shared cryptographic key. That would require that Evan cannot infer the transmitted bits before Alice and Bob become aware of his presence. Now, Table 1 tells us that Evan could acquire correct knowledge of every second bit sent by just performing the same measurements as Bob because knowledge of the photon type is not needed in the 1st and 4th columns. In fact, Evan can improve his educated guesses by choosing his measurement more cleverly [9], since the "+" states sent by Alice are distributed differently over the two-qubit Hilbert space than the "−" states. For the example of (5), he can systematically exploit the difference between the two-dimensional subspaces spanned by the "+" states and the "−" states to achieve odds as large as 1 2 + 1 2 / √ 1 + k 2 for guessing the bits right, which exceeds 85% for k = 1. Clearly, secure direct communication is not possible under these circumstances.
But there is a modified scheme that does enable Alice and Bob to communicate directly and confidentially. Again we focus on the simplest version, in which Bob's measurement bases are related to each other by
B 1 | B 2 | B 3 | B 4 | = i √ 3 0 1 1 1 −1 0 −1 1 −1 1 0 −1 −1 −1 1 0 B ′ 1 | B ′ 2 | B ′ 3 | B ′ 4 | .(8)
Just like K of (4), the 4 × 4 transformation matrix appearing here is Hermitian and unitary, so that it also furnishes the inverse transformation. The states sent by Alice now are identical with Bob's basis states, grouped into four pairs of orthogonal states in accordance with
|i+ = |B i , |i− = |B ′ i for i = 1, 2, 3, 4 .(9)
The basic features discussed above in the paragraphs between (2) and (3) are here present as well. Table 2. For the states of (8) and (9): Key bits as inferred by Bob upon learning which type of photon was sent by Alice. photon sent state detected by Bob by Alice Table 3. Direct confidential communication. Alice chooses a random key sequence of 1, 2, 3, 4 (1st row) and matches it with the bit sequence of the message (2nd row) interspersed with randomly located control bits (boxed) to determined the sequence of states to be sent (3rd row). Bob obtains a sequence of detected states (4th row). The control bits are used to test for the presence of an eavesdropper. After Alice reveals the random sequence of the 1st row, Bob can then reconstruct the message of the 2nd row.
B 1 B 2 B 3 B 4 B ′ 1 B ′ 2 B ′ 3 B ′ 4 type 1 + − − − − + + + type 2 − + − − + − + + type 3 − − + − + + − + type 4 − − − + + + + −
Alice's key 1 3
4 4 1 2 1 3 3 · · · message + + − − − + − + − · · · states sent 1+ 3+ 4− 4− 1− 2+ 1− 3+ 3− · · · Bob finds B 1 B ′ 1 B ′ 4 B 2 B 2 B ′ 4 B 4 B 3 B ′ 3 · · ·
How Bob infers the bits sent is summarized in Table 2. Consider, for example, that he found a certain photon in state |B 3 . He'll infer that "+" was sent if Alice tells him that it was a type-3 photon because |B 3 is orthogonal to |3− , and that "−" was sent if it was of type 1, 2, or 4 because |B 3 is orthogonal to |1+ , |2+ , and |4+ . Now, the minimal error rate resulting from eavesdropping of the intercept-resend kind is 1 6 = 16.7% for (9) with (8), which is less than the 25% of the four-pair key-sharing scheme to which (7) refers, but more than the 14.6% of the two-pair version . Thus, Evan's interference can be detected just as easily in the present scheme of (9) and (8) as in the previous one of (5) and (3) or of (1). Therefore, the scheme defined by (8) and (9) could be used for secure key distribution.
But this scheme is also well suited for direct communication, since the four "+" states span the whole two-qubit Hilbert space uniformly, and the four "−" states do so as well. Thus, Evan cannot distinguish "+" photons from "−" photons here without knowing the photon type. In particular, although the columns of Table 2 have 3:1 ratios of the signs, both kinds carry equal weight. If, for example, |B 3 is detected then |3+ is as likely as |1− , |2− , and |4− together.
Direct confidential communication is achieved as follows; see Table 3.
Step one:
These error rates refer to the situation in which Evan wishes to find out the value of each bit transmitted. Instead, he could settle for just a reasonable likelihood for guessing the bit value right and bargain for a reduced error rate in return. A detailed discussion of compromises of this kind will be presented elsewhere.
Alice generates, at her end, a random sequence of 1, 2, 3, 4 that will serve as the cryptographic key. Only Alice knows this key.
Step two: She matches this sequence with the string of +/− message bits, interspersed with a fair number of control bits at random positions, and so determines the two-qubit states to be sent to Bob. Only Alice knows which bits are control bits and which are message bits.
Step three: Alice sends the photons in these states, and Bob detects them in one of the states of his measurement bases.
Step four: Alice tells Bob which photons carried control bits, and he tells her in which state he found them.
Step five: Alice verifies that Bob's findings are consistent with what she sent. If no inconsistencies -that is: errors -are noticed, Alice concludes that the transmission was secure and continues with step six; otherwise she repeats the procedure beginning with step one.
Step six: Alice reveals the key sequence of step one, and Bob reconstructs the message with the aid of Table 2.
This scheme for direct communication is secure because Alice does not reveal her key sequence until she has convinced herself that Evan has not been listening in. Without this classical information, Evan cannot infer a single bit of the message. The only bits he might decode before his presence is detected are the control bits which, however, are not part of the confidential message.
Final remarks
Experimental implementations of our schemes for key distribution and direct communication can be realized with the aid of the universal two-qubit gates that where introduced recently [8]. Concerning practical aspects, we remark that we gain a factor of two compared to other cryptography schemes even for imperfect transmission and detection. Redundant encoding can overcome the losses in the communication scheme without revealing information to the eavesdropper. We'd also like to note that, rather than using two binary alternatives of single photons, one could, of course, equally well exploit the states of any other four-dimensional Hilbert space.
In summary, we propose a new cryptographic scheme. Under ideal conditions, the scheme is deterministic: Alice and Bob get a key bit for each photon sent, whereas other schemes [2,3] need at least two photons and are not deterministic. A significant percentage of Bob's measurement results will be wrong if an eavesdropper intercepts the transmission, so that his presence can surely be noticed. In addition, we show how the encoding and deterministic decoding of qubits in a four-dimensional Hilbert space allows direct communication, even without first establishing a shared key.
Table 1 .
1For the states of (1): Key bits as inferred by Bob upon learning which type of photon was sent by Alice. Note that Bob does not need this classical information if he detects the states of the 1st and 4th columns.photon sent
state detected by Bob
by Alice
‡ Only few protocols try to go beyond this 50% efficiency[7].
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| []
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[
"A Learning Approach to Natural Language Understanding",
"A Learning Approach to Natural Language Understanding"
]
| [
"Roberto Pieraccini \nAT&T Bell Laboratories\n600 Mountain Avenue Murray Hill07974NJ\n",
"Esther Levin \nAT&T Bell Laboratories\n600 Mountain Avenue Murray Hill07974NJ\n"
]
| [
"AT&T Bell Laboratories\n600 Mountain Avenue Murray Hill07974NJ",
"AT&T Bell Laboratories\n600 Mountain Avenue Murray Hill07974NJ"
]
| []
| In this paper we propose a learning paradigm for the problem of understanding spoken language. The basis of the work is in a formalization of the understanding problem as a communication problem. This results in the definition of a stochastic model of the production of speech or text starting from the meaning of a sentence. The resulting understanding algorithm consists in a Viterbi maximization procedure, analogous to that commonly used for recognizing speech. The algorithm was implemented for building a module, called conceptual decoder for the decoding of the conceptual content of sentences in an airline information domain. The decoding module is the basis on which a complete prototypical understanding system was implemented and whose performance are discussed in the paper. The problems, the possible solutions and the future directions of the learning approach to language understanding are also discussed in this paper. | 10.1007/978-3-642-57745-1_25 | [
"https://arxiv.org/pdf/cmp-lg/9406003v1.pdf"
]
| 15,549,741 | cmp-lg/9406003 | 8e2847f7665d786b6f9f231428efd46eac1496fa |
A Learning Approach to Natural Language Understanding
Jun 1994
Roberto Pieraccini
AT&T Bell Laboratories
600 Mountain Avenue Murray Hill07974NJ
Esther Levin
AT&T Bell Laboratories
600 Mountain Avenue Murray Hill07974NJ
A Learning Approach to Natural Language Understanding
Jun 1994
In this paper we propose a learning paradigm for the problem of understanding spoken language. The basis of the work is in a formalization of the understanding problem as a communication problem. This results in the definition of a stochastic model of the production of speech or text starting from the meaning of a sentence. The resulting understanding algorithm consists in a Viterbi maximization procedure, analogous to that commonly used for recognizing speech. The algorithm was implemented for building a module, called conceptual decoder for the decoding of the conceptual content of sentences in an airline information domain. The decoding module is the basis on which a complete prototypical understanding system was implemented and whose performance are discussed in the paper. The problems, the possible solutions and the future directions of the learning approach to language understanding are also discussed in this paper.
Introduction
A natural language understanding system is a machine that produces an action as the result of an input sentence (speech or text). There are examples [14] of systems that are able of modeling and learning the relationship between the input sentence and the action in a direct way. However, when the task is rather complicated, i.e. the set of possible actions is extremely large, we believe that it is necessary to rely on a intermediate symbolic representation. Fig. 1 depicts a natural language understanding as composed of two components. The first, called semantic translator analyzes the input sentence in natural language (N-L) and generates a representation of its meaning in a formal semantic language (S-L). The action transducer converts the meaning representation into statements of a given computer language (C-L) for executing the required action.
Although there are several and well established ways [5] of performing the semantic translation with relatively good performance, we are interested in investigating the possibility of building a machine that can learn how to do it from the observation of examples. Traditional non-learning methods are based on grammars (i.e. set of rules) both at the syntactic and semantic level. Those grammars are generally designed by hand. Often the grammar designers rely on corpora of examples for devising the rules of a given application. But the variety of expressions that are present in a language, even though it is restricted to a very specific semantic domain, makes the task of refining a given set of rules an endless job. Any additional set of examples may lead to the introduction of new rules, and while the rate of growth of the number of rules decreases with the number of examples, larger and larger amounts of data must be analyzed for increasing the coverage of a system. Moreover, if a new different application has to be designed, very little of the work previously done can be generally exploited. The situation is even more critical for spoken rather than written language. Written language generally follows standard grammatical rules more strictly than spoken language, that is often ungrammatical and idiomatic. Besides, in spoken language, there are phenomena like false starts and broken sentences that do not appear in written language. The following is a real example from a corpus of dialogues [21] within the airline information domain, (DARPA ATIS project [13], see section 3). It is clear from this example that rules for analyzing spontaneously spoken sentences can hardly be foreseen by a grammarian. We believe that a system that learns from examples will ease the work of a designer of text or speech understanding system giving the possibility of analyzing big corpora of sentences. The question remains on how to collect those corpora, which kind of annotation is needed, and what is the amount of manual work that has to be carried on.
The basis of the work exposed in this paper is a semantic translator, called CHRONUS. 2 CHRONUS is based on a stochastic representation of conceptual entities resulting from the formalization of the speech/text understanding problem as a communication problem. The paper is structured as follows. In section 2 we formalize the language understanding problem and we propose an algorithm based on the maximum a posteriori decoding. In section 3 we explain how the described algorithm for conceptual decoding can be part of a complete understanding system and we give a short description of all the modules that were implemented for an information retrieval application. In section 4 we discuss experimental performance of the system as well as issues related to the training of the conceptual decoder. Finally in section 5 we conclude the paper with a discussion on the open problems and the future developments of the proposed learning paradigm.
Formalization of the Language Understanding Problem
In this section we propose a formalization of the language understanding problem in terms of the noisy channel paradigm. This paradigm has been introduced for formalizing the general speech recognition problem [2] and constitutes a basis for most of the current working speech recognizers. Recently, a version of the paradigm was introduced for formalizing the problem of automatic translation between two languages [8]. The problem of translating between two languages has the same flavor of the problem of understanding a language [19]. In the former, both the input and the output are natural languages, while in the latter the output language is a formal semantic language apt to represent meaning.
The first assumption we make is that the meaning of a sentence can be expressed by a sequence of basic units M = µ 1 , µ 2 , . . . , µ N M and that there is a sequential correspondence between each µ j and a subsequence of the acoustic observation A = a 1 , a 2 . . . a N A , so that we could actually segment the acoustic signal into consecutive portions, each one of them corresponding to a phrase that express a particular µ i . The second assumption consists in thinking of the acoustic representation of an utterance as a version of the original sequence of meaning units corrupted by a noisy channel whose characteristics are generally unknown. Thus, the problem of understanding a sentence can be expressed in this terms: given that we observed a sequence of acoustic measurements A we want to find which semantic message M most likely produced it, namely the one for which the a posteriori probability P (M | A) is maximum. Hence the problem of understanding a sentence is reduced to that of maximum a posteriori probability decoding (MAP).
For the actual implementation of this idea we need to represent the meaning of a sentence as a sequence of basic units. A simple choice consists in defining a unit of meaning as a keyword/value pair m j = (k j , v j ), where k j , is a conceptual category (i.e. a concept like for instance origin of a flight, destination, meal) and v j is the value with which k j is instantiated in the actual sentence (e.g. Boston, San Francisco, breakfast). Given a certain application domain we can define a concept dictionary Γ and for each concept γ j ∈ Γ we can define a set of values Υ j = {ϕ1 j , ϕ2 j , . . . , ϕN j v }. Examples of meaning representation for phrases in the airline information domain are given in Table 1.
For information retrieval applications, the number of concepts Γ is relatively small (about 50, in the ATIS application), while the dictionary of concept values Υ can be relatively large (consider for instance all the possible flight numbers in the airline reservation domain). Moreover, the limited amount of training data available at the time we developed the system (a few thousand sentences) suggested to keep the number of model parameters relatively small. Therefore, in the decoding process, we considered only concepts k j ∈ Γ; the concept values v j ∈ Υ are derived through pattern matching functions at a later stage of processing.
In the remaining of the section we will use the following notations: W = w 1 , . . . , w N W is the sequence of words actually uttered in the sentence represented by the acoustic observation A and C = c 1 , . . . , c N W , c i ∈ Γ is the corresponding sequence of concepts labels. A consecutive set of words w i , . . . , w i+k labeled by the same concept label, constitutes a phrase expressing a concept, thus defining a segmentation of the sentence into concepts, referred to, in the following, as conceptual segmentation.
Hence, according to the maximum a posteriori decoding criterion, given the sequence of acoustic observations A, we want to find the sequence of conceptual labelsC and the sequence of wordsW that maximize the a posteriori probability P (W, C | A), namely
P (W,C | A) = max W,C P (W, C | A).(1)
Using the Bayes inversion formula, the conditional probability can be rewritten as:
P (W, C | A) = P (A | W, C)P (W | C)P (C) P (A) ,(2)
and since P (A) is constant:
arg max W,C P (W, C | A) = arg max W,C P (A | W, C)P (W | C)P (C)(3)
In this formula P (C) represents the a-priori probability of the sequence of concepts, P (W | C) is the probability of the sentence (intended as a sequence of words), given that the sentence conveys the sequence of concepts C, and P (A | W, C) is the acoustic model. The three components of the conditional probability in 2 can be thought of also as a semantic (probability of meanings), syntactic (probability of words given a meaning), and acoustic component respectively. We can then reasonably assume that the acoustic representation of a word is independent from the conceptual relation it belongs to, hence:
P (A | W, C) = P (A | W),(4)
and this is the criterion that is usually maximized in stochastic based speech recognizers, for instance those using hidden Markov modeling [2] [29] for the acoustic/phonetic decoding. Both the syntactic and the semantic probabilities can be rewritten as:
P (W | C) = P (w 1 ) N W i=2 P (w i | w i−1 , . . . , w 1 , C) (5) P (C) = P (c 1 ) N W i=2 P (c i | c i−1 , . . . , c 1 ).(6)
We assume that the probability of a word w i given all the previous words in the sentence and the sequence of concepts C, depends only upon the n most recent words and the concept c i that the word w i contributes to express. Under this assumption, equation 5 can be rewritten in terms of the n-gram concept conditional word probabilities:
P (w i | w i−1 , . . . , w 1 , C) = P (w i | w i−1 , . . . , w i−n+1 , c i )(7)
Analogously, the sequence of concept labels C can be regarded as an m − th order Markov process under the assumption:
P (c i | c i−1 , . . . , c 1 ) = P (c i | c i−1 , . . . , c i−m )(8)
For n = m = 1 we can represent the probabilities in equations 7 and 8 as a first order hidden Markov model, whose states represent the conceptual labels and whose observations are the words. Given the representation of the conceptual structure as a traditional HMM, the decoding of the conceptual content of a sentence can be carried out with the Viterbi algorithm. If the input is a text sentence, Viterbi decoding is used for finding the sequence of statesC such that:
P (W,C) = max C P(W, C).(9)
If the input is speech, we want to findW andC that maximize equation 3. In principle, the solution to this problem can be found by substituting each state of the conceptual HMM with a finite state network representing the corresponding bigram structure, and by substituting each word with its corresponding acoustic HMM. The integrated network, obtained in this way, can be used for decoding the input speech with Viterbi algorithm.
Implementation of a Speech Understanding System
For putting into practice the idea of MAP conceptual decoding, or at least to show its effectiveness, one needs a corpus of training examples (sentences and their correspondent meaning) and a test set with a criterion for validating the results. A relatively large corpus expressively designed for speech understanding (but not for learning) is being developed within the DARPA ATIS project [13]. ATIS stands for Air Travel Information System and the task is built around a subset of the OAG (Official Airline Guide) database, including 10 American cities. A corpus of spontaneous sentences is being collected and annotated by different sites [21]. The corpus is collected through a Wizard of Oz paradigm. Each subject is given a scenario and a travel planning problem to solve. The subjects are requested to solve the problem by interacting with a machine (that is actually a human wizard). The partial and the final responses of the machine are presented to the subjects via a display or a speech synthesizer. The sentences uttered by the subjects are recorded, transcribed and annotated carefully. Although the ATIS corpus may not be the best corpus for testing a semantic learning paradigm, it is readily available and it includes some kind of meaning annotation that can be indirectly used for our purpose. In this section we will give details about the design and test of a complete speech understanding system for the ATIS task. The system can work both from speech and text input. The lexical parser preprocesses sentence transcription in order to assign words to word categories (like numbers and acronyms). The conceptual decoding provides a segmentation of the sentence into phrases associated with conceptual units according to the theory explained in chapter 2. The template generator assigns to each concept the proper value, while the dialog manager takes care of keeping the history of the dialog and including in the current template the missing information. Finally the SQL 3 translator generates the query. The appropriate information, stored in a relational database, is then retrieved and displayed.
The Conceptual Decoding
Although MAP decoding of concepts is purely based on semantics, and the final structure of the conceptual model is in principle completely data driven, a certain amount of structure can be imposed during the design of the concept dictionary Γ. Imposing a certain structure to the model certainly alleviates the problems of the locality of the model and that of concept embedding, as explained later. For designing the concept dictionary, we had in mind the typical structure of a query that is represented as in Fig. 3.
In a typical sentence there is a phrase that generally represent the question , then a subject and finally a restriction on the query. For example, in the sentence:
SHOW ME THE FLIGHTS TO SAN FRANCISCO IN THE MORNING
3 SQL, Structured Query Language, is a high level language for accessing data in a relational database. SHOW ME THE MORNING FLIGHTS TO SAN FRANCISCO THE MORNING in this sentence plays exactly the same semantic role as the phrase IN THE MORNING of the previous sentence, but it has a different syntactic connotation. Therefore when a restriction is placed in front of the subject of the question we call it an attribute. Many different concepts can play both the role of restriction and attribute, hence we represent them by separate entities. For instance there is a fare concept and an a fare concept that are semantically indistinguishable but with the syntactic role of restriction and attribute respectively. Giving some syntactic connotations to the concepts helps to have better and sharper stochastic models. Without distinguishing between attribute and restriction concepts, the transition probabilities tend to be more uniform and the concept dependent bigram language models tend to be larger since the expressions used for an attribute concept are often different than those used for a restriction concept (IN THE MORNING vs. THE MORNING).
Another problem is that of concept embedding. The basic assumption we made in section 2, namely that the meaning can be expressed as a sequence of meaning units is not generally true 4 Take for instance the following sentence:
WHAT TYPE OF ECONOMY FARE COULD I GET FROM SAN FRANCISCO TO DALLAS
The concept of question is represented by two different phrases, namely WHAT TYPE OF and COULD I GET. Other concepts are embedded in between. Since we are using a flat representation for the meaning (a sequence rather than a tree) one of the possible solution for coping with this problem is to define additional symbols to account for separate phrases in which a single concept can be broken into. For example, in the previous sentence, we introduce a q attr (question attribute) concept, and the corresponding conceptual segmentation will be as follows: TO DALLAS
There are other special concepts like dummy, that accounts for phrases that do not carry information that is relevant for the application (e.g. courtesy forms, etc.), and and that represent conjunctives (e.g. and, or, also, etc.). The concept dictionary used in this implementation counts a total of 48 concepts.
The Lexical Parser
When dealing with a speech understanding system there are different issues related to lexicon that may not be encountered in text understanding.
• Although in both systems the vocabulary is generally limited in size, the limitations in a speech system are more severe than in a text system.
• A text system can generally deal with unknown (out of vocabulary) words. Once an unknown word is spotted, the system can take some action, like for instance asking the user to rephrase the sentence using a different word. In a speech recognition system an unknown word cannot generally be detected reliably by the system; it is generally confused and substituted for a known word.
• Numbers and acronyms are generally written in an unambiguous form but they can be uttered in different ways and allow to ambiguous interpretations. For instance, the acronym DC10 can be uttered as D C TEN or D C ONE ZERO, or D C ONE OW and can be interpreted as DC10, or D C10, or DC 10, etc.
The lexical parser analyzes the input transcription and generates a lattice of word categories called superwords. A superword can be one of the following: c. a grammar, represented by a finite state automaton (FSA); for instance, the grammar for natural numbers (e.g. THIRTY SEVEN), the grammar for airport acronyms (e.g. D F W ), the grammar for airport names (e.g. SAN FRANCISCO INTERNATIONAL AIR-PORT), etc. Each sequence recognized by an FSA is characterized by the corresponding FSA identifier and a normalized form of the compound word (e.g. THIRTY SEVEN is represented as ((number)37), D F W as ((airport)DFW)). As far as the stochastic model is concerned, two words with the same FSA identifier are represented by the same super-word. When there is ambiguity in deciding which superword to assign to a sequence of words, the lexical parser generates all the possible interpretations and arranges them in a lattice 5 .
Besides, articles (i.e. THE and A ) are deleted by the lexical parser, hence they play no role in the conceptual decoding process. Concept conditional language models are thus estimated among content words only. The reason for doing this is that in this particular application there is no relevant information carried by an article in front of a noun or an adjective. Besides, articles (like other short function words) can be easily misrecognized or deleted by the speech recognizer. Hence, due to the locality of the language model (i.e. bigrams of words), many bigrams will carry the probability of a noun or an adjective preceded by an article, missing the more important correlation to the preceding content word.
Using super-words reduces the number of parameters to be estimated and increases the robustness of the system. The probabilities associated to a given super-word are shared between all the words that are represented by the same super-word. This has the effect of allowing the system to generalize the statistics gathered in the training phase to all the words belonging to the same super-word. This is shown in an experiment where we implemented the conceptual decoding in two different situations. In the first implementation there was no super lexicon, the vocabulary consisted of 501 words, including three word classes (i.e. numbers, city names, and acronyms), and the articles were accounted for in the bigrams. In this case, the input to the system was a textual transcription of each utterance, where the numbers and the acronyms were unambiguously represented (e.g. What's the fare for US-AIR 4393). In the second experiment a super-lexicon including 753 words and 18 grammars was used. In this case the input to the system was a speech transcription of each utterance. It has to be noticed that although the lexicon in the second experiment was larger than the lexicon in the first experiment, the word coverage over the test set was the same in both cases. The test set consisted of 148 context independent sentences (included in the official February 1991 test set [16]). The 148 sentences were hand segmented into concepts. The set of sentences corresponded to an overall number of 713 concepts. The performance were assessed based on the hand segmentation. The percentage of correct concepts (when the conceptual segmentation agreed with the one provided by hand) and of correct sentences (sentence for which all the included concepts are correctly decoded and segmented) is reported in Table 2.
Interfacing with a Speech Recognizer
The most natural way of interfacing the conceptual decoder with a speech recognizer is by implementing the maximization of equation 3. This requires to implement a decoder that explores a network obtained by explicitly instantiating acoustic HMMs [11] representing words (or superwords) of the vocabulary for any concept. For a task like ATIS the dimension of the resulting network can be rather large. In theory, if there are 50 conceptual states and about 1,000 words, each one of them represented by a HMM with an average number of 15 states, the overall network is bound by a total number of 750,000 acoustic HMM states, with a number of connections of the order of 50,000,000 (each conceptual conditional bigram model is represented by 1000 × 1000 connections). Of course not all the bigrams are observed or even possible in each state. If only those words and bigrams that were observed during the training are represented in a conceptual state, a more reasonable model can be obtained. In an experimental version of CHRONUS we estimated an integrated model with a total of 2400 HMM word models (corresponding to about 36,000 HMM acoustic states) and nearly 46,000 connections. This size of the model can be easily managed by a beam search recognizer [12]. The problem in using such a model is that while it constitute a reasonably good model for decoding the semantic message of a sentence, the limited amount of training data used for its estimation makes it a quite coarse model for constraining the speech recognition process. When bigrams of words that were not observed in the training data are actually uttered, the recognizer is forced to substitute them for known bigrams. Hence the recognition errors are propagated along the sentence, resulting in relatively poor recognition performance.
Smoothing techniques can be applied for estimating the probability of unobserved bigrams, like for instance methods relying on the Good-Turing estimation of probabilities [6]. This will increase the complexity of the model by allowing all the possible bigrams in each state. However a factorization of the maximization of equation 3 [18] can still lead to reasonably good results at an acceptable complexity. Hence several solutions could be implemented, like best first coupling (the best first recognized sentence is given to the conceptual decoding), N-best coupling [15] and word lattice coupling [9].
The Template Generator
The goal of the template generator module is that of analyzing the conceptual segmentation and generating the final representation of the meaning (i.e. the template) by supplying the correct value to each concepts detected during the decoding stage. For every relevant concept a look-up table was built for performing the mapping between phrase templates and conceptual values. The look-up table associates keywords or short phrases to concept values. Values are then assigned to the decoded concepts according to the result of a pattern matching procedure with the keyword stored in the appropriate tables. The values that are eventually associated to the decoded concepts belong to different categories. They can be actual database items (e.g. Boston, American airlines, breakfast), database attributes, (e.g. flight, stop, meal), logic values (e.g. null, not null) or operators, (e.g. minimum, maximum). The design of pattern matching tables is still manual. More details on the template generator can be found in [26].
The Dialog Manager
The dialog manager implements the function of keeping the dialog history and allowing the resolution of anaphoric and elliptical sentences. The simple strategy implemented in our system consists in keeping a current context template with all the information that has been used for specifying the actual query. When a new sentence is presented to the system, the dialog manager tries to merge the current template with the current context template, in order to get the missing information. The merging of the two templates follows application specific rules. For instance, when a concept is mentioned in a new sentence, with a different value than the corresponding concept in the context template, all concepts in the context template at a lower hierarchical level are deleted. This assumes a predefined hierarchy of the concepts. The origin and the destination of a flight have the highest level in the hierarchy. Then, when either the origin or the destination are different from those specified in the context template, the context template is deleted and a new context is started. This strategy, although very simple, has proven to be effective in the majority of sentences of this task.
The SQL translator
The last part of the interpretation process, namely the access to the required information, is implemented through a translator that dynamically generates the SQL query in order to retrieve the data. The template produced by the template generator is processed according to the value of its subject concept. If a subject concept is not found in the template a default subject is used. The subject of the query is used for selecting the right table of the database. If there is more than one subject or the subject is not directly related to a particular table, a link function is invoked in order to perform the correct joins. Once the table (or a joint table) is selected, the rest of the template tokens are interpreted accordingly.
Putting it into Practice
Assessing the performance of a language understanding system is still an open problem mainly because the concept of correct answer is generally ambiguous and must be based on defined conventions that are not task independent. The DARPA community agreed upon scoring answers by comparison with given reference answers that are produced for each valid sentence of the corpus. The answers can be made up of data extracted from the flight information database, numbers, or logical values (yes/no). In case of ambiguity of the question, multiple reference answers are given. Of course the problem of the definition of a correct answer still remains. For instance, for a question like
SHOW THE LATE EVENING FLIGHTS BETWEEN BOSTON AND DALLAS
the correctness of the answer depends upon the conventional definition of late evening. Then, once a time interval has been defined for late evening, it is still not clear what is the information to be listed. It could be the airline and flight number of each flight, but it could also include the departure time, the arrival time, the fare, and so on. A special committee within the DARPA [21], that should rule the majority of cases. Besides, it was also agreed on using two reference answers, namely a minimal and a maximal reference answer. An answer is thus considered correct if it contains all the information included in the minimal reference answer and no more than the information included in the maximal reference answer.
Experimental Performance
The described system was tested on a set of 687 sentences called February 92 test set [22]. For increasing the robustness we provided the system with a simple rejection criterion. The rejection heuristic is based on the measure of success in the operation of template generation (how many decoded concepts are successfully matched to a value), although more sophisticated heuristics can be developed. The results [24] on the complete test set, from text input, account for 68% of correct answers, 18% wrong answers and 14% rejects. When the system was coupled with a speech recognizer through the best first hypothesis, the performance dropped to 52% of correct answers, 26% wrong answers and 22% rejects. However, the contribution of the different modules to the overall error rate is far more interesting than it absolute value. All the 121 sentences that produced a wrong answer (in the text understanding experiment) were carefully analyzed and the errors were classified according to Table 3. From this analysis it results that most of the errors are due to the parts of the system that are not trained. The template generator errors reflect a lack of entries in the look-up tables. The dialog manager errors are due to the fact the the simple strategy for merging the context and the current template should be refined with more sophisticated rules. The SQL translator should be an error-free module. Its only function is that of translating between two different representation in a deterministic fashion. However, in this test, the SQL module faced two kinds of problems. The first is that the interpretation rules used for generating the answers were not exactly the same ones used for the official test, and this accounts for roughly half of the errors. The other half of the error is due to the limited power of the template representation, and this will be discussed in section 5.
Training the Conceptual Model
Smoothing of Bigram Models
The conceptual model, as explained in section 2, is defined by two sets of probabilities, namely the concept conditional bigrams P (w i | w i−1 , c i ) and the concept transition probabilities P (c i | c i−1 ). In the first experiments these probabilities were estimated using a set of 532 sentences whose conceptual segmentation was provided by hand. The accuracy of the system in the experiments carried out using the model estimated with such a small training set, although surprisingly high [17], shows a definite lack of training data. Smoothing the estimated model probability provides an increase of the performance. The knowledge of the task can be introduced through a supervised smoothing of the concept conditional bigrams. The supervised smoothing is based on the observation that, given a concept, there are several words that carry the same meaning. For instance, for the concept origin, the words
DEPART(S) LEAVE(S) ARRIVE(S)
can be considered as synonyms, and can be interchanged in sentences such as:
THE FLIGHT THAT DEPART(S) FROM DALLAS THE FLIGHT THAT LEAVE(S) FROM DALLAS THE FLIGHT THAT ARRIVE(S) FROM DALLAS.
A number of groups of synonyms were manually compiled for each concept. The occurrence frequencies inside a group were equally shared among the constituting words, giving the same bigram probability for synonymous words.
Using a Larger Training Corpus
If one wants to use a larger corpus than the initial handlabeled few hundred sentences and wants to avoid an intensive hand segmentation labor, one has to capitalize on all the possible information associated to the sentences in the corpus. Unfortunately, when the corpus is not expressively designed for learning, like the ATIS corpus, the information needed may not be readily available. In the remaining of this section we analyze solutions that, although particularly devised for ATIS, could be generalized to other corpora and constitute a guideline for the design of new corpora. A training token consists of a sentence and its associated meaning. The meaning of sentences in the ATIS corpus is not available in a declarative form. Instead, each sentence is associated with the action resulting from the interpretation of the meaning, namely the correct answer. One way of using this information for avoiding the handlabeling and segmentation of all the sentences in the corpus consists in creating a training loop in which the provided correct answer serves the purpose of a feedback signal. In the training loop all the available sentences are analyzed by the understanding system obtained with an initial estimate of the conceptual model parameters. The answers are then compared to the reference answers and the sentences are divided into two classes. The correct sentences, for which we assume that the conceptual segmentation obtained with the current model is correct, and the problem sentences. Then the segmentation of the correct sentences is used for reestimating the model parameters, and the procedure is repeated again. The procedure can be repeated until it converges to a stable number of correct answers. Eventually, the remaining problem sentences are corrected by hand and included in the set of correct sentences for a final iteration of the training algorithm. This procedure proved effective for reducing the amount of handlabeling. In the experiment described in [25] we showed that the performance increase obtained with the described training loop, without any kind of supervision (the remaining problem sentences were excluded from the training corpus) is equivalent to that obtained with the supervised smoothing. This means that the training loop, although is not able to learn radically new expressions or new concepts, is able to reinforce the acquired knowledge and to infer the meaning of semantically equivalent words. In a set of 4500 sentences the training loop automatically classified almost 80% of the sentences, leaving the remaining 20% to the manual segmentation.
The Sequential Correspondence Assumption
In section 2 we based our formalization of the speech understanding problem on the assumption that there is a sequential correspondence between the representation of a sentence (words or acoustic measurements) and the corresponding representation of meaning. This assumption is not generally true for any translation (semantic or not) task. An interesting example (reported in [27]) of a task where there is no sequential correspondence between a message and its semantic representation, is that of roman numbers (e.g. I, XXIV, XCIX) and their correspondent decimal representation (e.g. 1,24,99). Fortunately, in a natural language understanding task, we may have the freedom of choosing the semantic representation, like we did in the implementation of CHRONUS explained above. But in general, if we are dealing with a large corpus of sentences that have not been expressively designed for the purpose of learning a semantic translator, and we would like to take advantage of some kind of semantic annotation already available, we may have to face the problem of the not sequentiality of the representation. For instance, in the ATIS corpus, each sentences is associated with the intermediate representations used by the annotators for obtaining the reference correct answers. In fact the annotators rephrase each valid sentence in an artificial language that is a very restricted form of English. This pseudo-English rephrasing (called win or wizard input) constitute the input of a parser, called NLparse [10], that unambiguously generates the SQL query. For instance, for a sentence like:
I'D LIKE TO FIND THE CHEAPEST FLIGHT FROM WASHINGTON D C TO ATLANTA
The win rephrasing is:
List cheapest one direction flights from Washington and to Atlanta and the corresponding associated SQL statement is: SELECT airport service.airport code FROM airport service WHERE airport service.city code IN (SELECT city.city code FROM city WHERE city.city name = 'WASHING-TON' )) AND flight.to airport IN (SELECT airport service.airport code FROM airport service WHERE airport service.city code IN (SELECT city.city code FROM city WHERE city.city name = 'ATLANTA' )))))))) AND (flight.from airport IN (SE-LECT airport service.airport code FROM airport service WHERE airport service.city code IN (SELECT city.city code FROM city WHERE city.city name = 'WASHINGTON' )) AND flight.to airport IN (SELECT airport service.airport code FROM airport service WHERE airport service.city code IN (SELECT city.city code FROM city WHERE city.city name = 'AT-LANTA' )))));.
Both the SQL query and the win sentence can be considered semantic representations of the original sentence. In fact the SQL query is the final target of the understanding system and can be unequivocally obtained from the win sentence through an existing parser. Obviously the sequential correspondence assumption is strongly violated for the SQL representation. However a sequential correspondence can be easily found between the pseudo-English win sentence and the original message, at least for the shown examples. Since all the valid sentences in the ATIS corpus have a win annotation, the pseudo-English language can be thought of as an alternate candidate for the meaning representation in our learning framework. Using win for representing the meaning may lead to two different solutions. In the first we can think of developing a system that learns how to translate natural language sentences into pseudo-English sentences and then use the existing parser for generating the SQL query. In the second solution each win sentence in the corpus can be translated in the corresponding conceptual representation used for CHRONUS. This translation is unambiguous (win is an unambiguous artificial language by definition). A parser can be easily designed for performing the translation or, simply use CHRONUS itself for performing the translation 6 . Unfortunately also for the win representation, the sequential correspondence assumption is violated for a good percentage of the sentences in the corpus. A typical example is constituted by the following sentence:
COULD YOU PLEASE GIVE ME INFORMATION CONCERNING AMERICAN AIRLINES A FLIGHT FROM WASHINGTON D C TO PHILADELPHIA THE EARLIEST ONE IN THE MORNING AS POSSIBLE
whose corresponding win annotation is:
List earliest morning flights from Washington and to Philadelphia and American.
The problem of reordering the words of win representation for aligning it with the original sentence is a complex problem that cannot be solved optimally. Suboptimal solutions with satisfactory perfomance can be developed based on effective heuristics. We will not discuss the details of how the reordering can be put into practice. Rather we want to emphasize the fact that an iterative algorithm based on a model similar to that explained in section 2 led to almost 91% correct alignments between English sentences and corresponding win representations on a corpus of 2863 sentences. With additional refinements this technique can be used, integrated in the training loop, for automatically processing the training corpus of the conceptual model.
Discussion and Conclusions
In this paper we propose a new paradigm for language understanding based on a stochastic representation of semantic entities called concepts. An interesting way of looking at the language understanding paradigm is in term of a language translation system. The first block in Fig. 1 translates a sentence in natural language (N-L) into a sentence expressed in a particular semantic language (S-L). The natural language characteristics are generally unknown, while the semantic language designed to cover the semantic of the application is completely known and described by a formal grammar. The second step consists in the translation of the sentence in S-L into computer language code C-L for performing the requested action. This second module can be generally (but not necessarily) designed to cover all the possible sentences in S-L, since both S-L and C-L are known. However, the boundary between the first and the second module is quite arbitrary. In [28], for instance, an automatic system is designed for translating ATIS English sentences directly into the SQL query, and in [14] there is an example of a system that goes from an English sentence to the requested action without any intermediate representation of the meaning. However, the closer we move the definition of S-L to N-L, the more complicate becomes the design of the action transducer, reaching in the limit the complexity of a complete understanding system. Conversely, when we move the definition of S-L closer to C-L, we may find that learning the parameters of the semantic translator becomes quite a difficult problem when the application entails a rather complex semantics. The subject of this paper deal with the investigation of the possibility of automatizing the design of the first block (i.e. the semantic translator) starting from a set of examples. The semantic language chosen for the experiments reported in this paper is very simple and consists of sequences of keyword/value pairs (or tokens). There is no syntactic structure in the semantic language we use. Two sentences for which the difference in the semantic representation is only in the order of the tokens are considered equivalent. In this way we cover a good percentage of sentences in the domain, but still there are sentences that would require a structured semantic language. For instance the two following sentences are indistinguishable when represented by our semantic language, and obviously they have a different meaning.
IS THE EARLIEST FLIGHT GOING TO BOSTON ON A SEVEN FOUR SEVEN
IS THE EARLIEST FLIGHT ON A SEVEN FOUR SEVEN GOING TO BOSTON
The representation of this kind of sentences requires a more sophisticated semantic language that allows the use of bracketing for delimiting the scope of modifiers.
Although the system we propose uses a very simple intermediate semantic representation, we showed that it can successfully handle most of the sentences in a database query application like the ATIS task. When this simple representation is used and when the problem of semantic translation is formalized as a communication problem, a MAP criterion can be established for decoding the units of meaning from text or speech. The resulting decoder can then be integrated with other modules for building a speech/text understanding system.
An understanding system based on a learning paradigm, like the one proposed in this paper, can evolve according to different dimensions of the problem. One dimension goes with the increase in complexity of the semantic language S-L. Rather than using a sequential representation on could think of a tree representation of the meaning. However, this poses additional problems both in the training and decoding stage, and requires the use of algorithms designed for context-free grammars, like for instance the inside-outside algorithm [3][23] that have a higher complexity that those explained in this paper. Another dimension of the problem goes toward a complete automatization of the system, also for those modules that, at the moment, require a manual compilation of some of the knowledge sources. One of these modules is the template generator. Both [20] and [28] report examples of systems where the decision about the actual values of the conceptual entities (or an equivalent information) is drawn on the basis of knowledge acquired automatically from the examples in the training corpus. The kind of annotation required for the training corpus is also another dimension along with the research on learning to understand language should move. A strategy for learning the understanding function of a natural language system becomes really effective and competitive to the current non-learning methods when the amount of labor required for annotating the sentences in a training corpus is comparable or inferior to the amount of work required for writing a grammar in a traditional system. This requires the development of a learning system the does not require any other information than the representation of the meaning associated to each sentence (e.g. it does not require an initial segmentation into conceptual units, like in CHRONUS, for bootstrapping the conceptual models). Moreover, the representation of the meaning should be made using a pseudo-natural language, for making easier and less time consuming the work of the annotators. An example of this kind of annotation was introduced in section 4.2.3 with the pseudo-English win rephrasing. This suggests a possible evolution of the learning strategy for understanding systems toward a system starting with the limited amount of knowledge required for understanding a small subset of the whole language (e.g. the win language). Then the system can evolve to understanding larger subsets of the language using the language already acquired for rephrasing new and more complex examples. But, of course, the science of learning to understand is still in its infancy, and many more basic problems must be solved before it becomes an established solution to the design of a language interface.
Figure 1 :
1Understanding as a translation process
FROM uh sss FROM THE PHILADELPHIA AIRPORT um AT ooh THE AIR-LINE IS UNITED AIRLINES AND IT IS FLIGHT NUMBER ONE NINETY FOUR ONCE THAT ONE LANDS I NEED GROUND TRANSPORTATION TO uh BROAD STREET IN PHILELD PHILADELPHIA WHAT CAN YOU AR-RANGE FOR THAT 1
Figure 2 :
2Block diagram of the understanding system A block diagram of the speech understanding system is shown inFig. 2.
Figure 3 :
3Typical structure of a query the question phrase is SHOW ME, the subject is THE FLIGHTS and the restriction is TO SAN FRANCISCO IN THE MORNING. The same sentence could be rephrased as:
a. a word, like ABOUT, MONTH, RETURN, etc. b. a word with optional morphological inflections, like AIRFARE(S), DAY(S), ADVANCE(D), etc. (e.g. there is no distinction between AIRFARE and AIRFARES; both words are represented by the super-word AIRFARE(S)).
(
SELECT DISTINCT flight.flight id FROM flight WHERE (flight.flight id IN (SELECT flight fare.flight id FROM flight fare WHERE flight fare.fare id IN (SELECT fare.fare id FROM fare WHERE fare.one direction cost = (SELECT MIN ( fare . one direction cost ) FROM fare WHERE fare.fare id IN (SELECT flight fare.fare id FROM flight fare WHERE flight fare.flight id IN (SELECT flight.flight id FROM flight WHERE (flight.from airport IN (
SHOW ME THE FLIGHTS TO BOSTON (question,display) (subject,flight) (destin,BBOS) HOW MUCH IS THE PRICE OF THE FLIGHT FROM ATLANTA (question,display) (subject,fare) (destin,MATL) Table 1: Example of keyword/pair representations of simple phrases within the ATIS domain.IS BREAKFAST SERVED ON THE
FLIGHT?
(question,yes-no) (subject,breakfast)
Table 2 :
2Effect of the super-lexicon in conceptual decoding
Table 3 :
3Analysis of the errors for the NL ATIS February 92 test community agreed upon a certain number of rules, called principles of interpretation
Semantics can be represented, in general, by a tree[1] or by a network[4]. The assumption made here is that the semantics represented by the sentences in the application we are considering is simple enough and can be represented with a flat structure like a sequence of symbols
It is quite straightforward to modify the Viterbi algorithm for performing the decoding from a lattice rather than from a sequence of observations. A discussions on this subject can be found in[7]
win is a subset of natural English. Only a little adaptation was needed for developing a win translator based on the existing CHRONUS
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| []
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[
"REPRESENTATIONS OF NON-NEGATIVE POLYNOMIALS VIA CRITICAL IDEALS",
"REPRESENTATIONS OF NON-NEGATIVE POLYNOMIALS VIA CRITICAL IDEALS"
]
| [
"Tuan Dang ",
"Hiep "
]
| []
| []
| This paper studies the representations of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its critical ideal. Under the assumptions that the semi-algebraic set K is regular and f satisfies the boundary Hessian conditions (BHC) at each zero of f in K, we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if f ≥ 0 on K. In particular, we focus on the polynomial ring R[x]. | null | [
"https://arxiv.org/pdf/1112.4072v1.pdf"
]
| 13,344,051 | 1112.4072 | e7979d3f938b5f6572a840b3744c20c9fbeed178 |
REPRESENTATIONS OF NON-NEGATIVE POLYNOMIALS VIA CRITICAL IDEALS
17 Dec 2011
Tuan Dang
Hiep
REPRESENTATIONS OF NON-NEGATIVE POLYNOMIALS VIA CRITICAL IDEALS
17 Dec 2011
This paper studies the representations of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its critical ideal. Under the assumptions that the semi-algebraic set K is regular and f satisfies the boundary Hessian conditions (BHC) at each zero of f in K, we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if f ≥ 0 on K. In particular, we focus on the polynomial ring R[x].
introduction
We know that a polynomial in one variable f (x) ∈ R[x] satisfies f (x) ≥ 0, for all x ∈ R, then f (x) = m i=1 g 2 i (x), where g i (x) ∈ R[x], i.e., f is a sum of squares in R[x] (SOS for short). However, in multi-variate cases, this is not true. A counterexample was given by Motzkin in 1967. If f (x, y) = 1 + x 4 y 2 + x 2 y 4 − 3x 2 y 2 , then f (x, y) ≥ 0, for all x, y ∈ R. But f is not a SOS in R [x, y]. To remedy that, we will consider the polynomials that are positive on K, where K is a semi-algebraic set in R n . For example, Schmüdgen's theorem [Schm] states that for a compact semi-algebraic set, every strictly positive polynomial belongs to the corresponding finitely generated preordering. Afterward, Putinar [Pu] simplified this representation under an additional assumption by using the quadratic module instead of the preordering. However, these results of Schmüdgen and Putinar have two restrictions. Firstly, the polynomials are positive, not merely non-negative. Secondly, K must be a compact semi-algebraic set. Hence we seek to identify the representations of the non-negative polynomials on the non-compact semi-algebraic sets.
In [NDS], the authors presented a representation of the non-negative polynomials on the whole space modulo their gradient ideals. Afterward, in [DNP], the authors proved a similar representation on the arbitrary semialgebraic sets. These results were achieved under the condition of the corresponding ideals must be radical. However, it is not simple to check this condition. In order to overcome such limitation, in [M], Marshall considered another condition -the boundary Hessian condition (BHC). He proved that the result in [NDS] still held true if the radical condition is replaced by the BHC condition. In [Hi], the author presented an extension of theorem 2.1 in [M] in the same way that the result in [DNP] was the extension of the corresponding result in [NDS].
However, in [Hi] and [DNP] the authors considered a larger polynomial ring R[x, λ], i.e., they added Lagrange multipliers to the representations. This paper will help us overcome this. We will present the representations of the non-negative polynomials via their critical ideals. In particular, we focus on the polynomial ring R[x].
Preliminaries
In this section, we present some notions and results from algebraic geometry and real algebra needed for our discussions. The readers may consult [BCR], [CLO], and [PD] for more details.
Throughout this paper, denote by R[x] the ring of polynomials in x = (x 1 , . . . , x m ) with real coefficients. Given an ideal I ⊆ R[x], define its complex variety to be the set
V (I) = {x ∈ C m | p(x) = 0, ∀p ∈ I},
and its real variety to be
V R (I) = V (I) ∩ R m . A nonempty variety V = V (I) ⊆ C m is irreducible if there do not exist two proper subvarieties V 1 , V 2 ⊂ V such that V = V 1 ∪ V 2 .
The readers should note that in this paper, "irreducible" means that the set of complex zeros cannot be written as a proper union of subvarieties defined by real polynomials.
Given any ideal I of R[x], its radical ideal √ I is defined to be the following ideal:
√ I = {q ∈ R[x] | q l ∈ I for some l ∈ N}.
Clearly, I ⊆ √ I; I is a radical ideal if √ I = I. As usual, for a variety V ⊆ C m , I(V ) denotes the ideal in C[x] of polynomials vanishing on V . We will write I R (V ) for the ideal
I(V ) ∩ R[x].
We need versions of the Nullstellensätz for varieties defined by polynomials in R[x]. The following two theorems are normally stated for ideals in C[x]; however, keeping in mind that V (I) lies in C m , they hold as stated for ideals in R[x].
Theorem 2.1 ( [CLO]). If I is an ideal in R[x] such that V (I) = ∅, then 1 ∈ I.
Theorem 2.2 ( [CLO]). If I is an ideal in R[x], then I R (V (I)) = √ I.
Let g 1 , . . . , g s ∈ R[x]. We define the preordering generated by g 1 , . . . , g s as follows: We also define the semi-algebraic set generated by g 1 , . . . , g s as follows:
P =K = {x ∈ R n | g i (x) ≥ 0, i = 1, . . . , s}.
Definition 2.1 (see [NW], Definition 12.1). For each x ∈ R n , let J x be the set of indices j for which g j vanishes at x. The semi-algebraic set K is called regular, if for each x ∈ K, the vectors ∇g j (x), j ∈ J x , are linearly independent.
Throughout this paper, we always assume that the semi-algebraic set K is regular.
The critical variety
Definition 3.1. The critical variety of f on K is defined as follows:
C(f, K) := {x ∈ R n | there exist real numbers λ i such that ∇f (x) − s i=1 λ i ∇g i (x) = 0, λ i g i (x) = 0, i = 1, . . . , s}.
Remark 3.1.
(i) In the global case, i.e., when the semi-algebraic set K is the whole space R n , we have
C(f, K) = {x ∈ R n | ∇f (x) = 0},
which is the real gradient variety of f (see [NDS]). (ii) Consider the projection π : R n × R s → R n , (x, λ) → x, where variables λ = (λ 1 , . . . , λ s ) are Lagrange multipliers. Then C(f, K) = π(V KKT ), here
V KKT := {(x, λ) ∈ R n × R s | ∇f (x) − s i=1 λ i ∇g i (x) = 0, λ i g i (x) = 0, i = 1, . . . , s},
is the real KKT variety of f on K (see [DNP]).
In this section, we will study the properties of the critical variety C(f, K).
Proposition 3.1. The following statements hold true
(i) C(f, K) = C(f + a, K), for all a ∈ R.
(ii) If f attains its infimum at x * ∈ K, then x * ∈ C(f, K).
Proof.
(i) We see clearly that ∇f = ∇(f +a), for all a ∈ R. Then, by definition of the critical variety, we have C(f, K) = C(f + a, K), for all a ∈ R. (ii) By Karush-Kuhn-Tucker theorem (see e.g. [NW]), if f attains its infimum at x * ∈ K, then there exist λ * 0 , λ * 1 , . . . , λ * s at least one of which is different from zero, such that
λ * 0 ∇f (x * ) − s i=1 λ * i ∇g i (x * ) = 0, λ * i g i (x * ) = 0, i = 1, . . . , s. Since K is regular, then we can choose λ * 0 = 1. Thus x * ∈ C(f, K).
We will use the following notations in the remainder of the paper.
g J (x) := j∈J g j (x) , J = ∅, 1 , J = ∅. If J = {j 1 . . . , j k }, we will denote by h J ∈ R[x] the following polynomial h J (x) := det(A J (x)A T J (x)), where A J (x) := ∂f ∂x 1 ∂f ∂x 2 · · · ∂f ∂xn ∂g j 1 ∂x 1 ∂g j 1 ∂x 2 · · · ∂g j 1 ∂xn . . . . . . · · · . . . ∂g j k ∂x 1 ∂g j k ∂x 2 · · · ∂g j k ∂xn is a (k + 1) × n-matrix. Observe that h J (x) = 0 if and only if the vectors ∇f, ∇g j , j ∈ J are linearly dependent.
Proposition 3.2. The critical variety C(f, K) is an algebraic set. More precisely we have
C(f, K) = {x ∈ R n | g J (x)h J c (x) = 0, ∀J ⊆ {1, . . . , s}},
where we use the notation J c := {1, . . . , s}\J.
Proof. The proof is similar as that of Proposition 3.1 in [HP2] and therefore is omitted here.
Boundary Hessian Conditions, gradient ideals and KKT ideals
We say f satisfies the BHC (boundary Hessian conditions) at the point x * in K if there are some k ∈ {1, . . . , n}, and v 1 , ..., v k ∈ N with 1 ≤ v 1 < ... < v k ≤ s such that g v 1 , . . . , g v k are parts of a system of local parameters at x * , and the standard sufficient conditions for a local minimum of
f | L at x * hold, where L is the subset of R n defined by g v 1 (x) ≥ 0, . . . , g v k (x) ≥ 0. This means that if t 1 , . . . , t n are local parameters at x * chosen so that t i = g v i for i ≤ k, then in the completion R[[t 1 , . . . , t n ]] of R[x] at x * , f decomposes as f = f 0 + f 1 + f 2 + · · · (where f i is homogeneous of degree i in the variables t 1 , . . . , t n with coefficients in R), f 1 = a 1 t 1 + · · · + a k t k with a i > 0, i = 1, . . . , k, and the (n−k)-dimensional quadratic form f 2 (0, . . . , 0, t k+1 , . . . , t n ) is positive definite.
Theorem 4.1 (Marshall [M]). If f satisfies the BHC at each zero of f in K, then f ∈ P + f 2 .
Example 4.1. Let f, g 1 ∈ R[x, y, z] be given by
f (x, y, z) = x; g 1 (x, y, z) = x − y 2 − z 2 . Then K = {(x, y, z) ∈ R 3 | z − y 2 − z 2 ≥ 0}.
Clearly, f ≥ 0 on K, and the unique zero of f in K occurs at (0, 0, 0). Furthermore, f satisfies the BHC at (0, 0, 0). Indeed, let t 1 = g 1 = x − y 2 − z 2 , t 2 = y and t 3 = z. These form a system of local parameters at (0, 0, 0).
Then f = x = (x − y 2 − z 2 ) + y 2 + z 2 , so f = f 1 + f 2 , where f 1 (t 1 , t 2 , t 3 ) = t 1 ,
and f 2 (t 1 , t 2 , t 3 ) = t 2 2 + t 2 3 . Also, the coefficient of t 1 in f 1 is positive (it is 1), and t 2 , t 3 do not appear in f 1 . The quadratic form f 2 (0, t 2 , t 3 ) = t 2 2 + t 2 3 is positive definite (when viewed as a quadratic form in the two variables t 2 , t 3 ). So, according to the definition, f satisfies the BHC at (0, 0, 0). Here f has a representation as follows:
f = σ 0 + σ 1 g 1 + hf 2 ,
where σ 0 = y 2 + z 2 , σ 1 = 1, h = 0. Now we define the gradient ideal of f as follows:
I grad = ∂f ∂x 1 , . . . , ∂f ∂x n .
Under the assumption that I grad is radical, we have the following result.
Theorem 4.2 (Nie-Demmel-Sturmfels [NDS]). Suppose that
(i) f ≥ 0 on R n , (ii) I grad is radical.
Then f is a sum of squares modulo I grad .
If we replace the radical condition of I grad by an another condition that f satisfies the BHC at each zero of f , then we will have the following result. Similar to generalization of the gradient ideal, we define the KKT ideal of f as follows:
I KKT = F 1 , . . . , F n , λ 1 g 1 , . . . , λ s g s ,
where F i = ∂f ∂x i − s j=1 λ j ∂g j ∂x i , ∀i = 1, . . . , n.
Two following results are generalizations of theorem 4.2 and theorem 4.3 in the same way.
Theorem 4.4 (Demmel-Nie-Powers [DNP]). Suppose that
(i) f ≥ 0 on K, (ii) I KKT is radical. Then f ∈ P + I KKT .
Theorem 4.5 (Hiep [Hi]). Suppose that
(i) f ≥ 0 on K, (ii) f satisfies the BHC at each zero of f in K.
Then f ∈ P + I KKT .
Remark 4.1. The radical condition and the BHC condition are different. This means that there exist polynomials which satisfy the radical condition, but do not satisfy the BHC condition and conversely. The following example will demonstrate this difference. 1. Let n = 1 and s = 0 (so that K = R). Then the polynomial in one variable f (x) = 6x 2 + 8x 3 + 3x 4 satisfies the BHC condition, but it does not satisfy the radical condition. Indeed, ∂f ∂x = 12x(x + 1) 2 , f (x) ≥ 0 on R, f has a zero at x = 0, and ∂ 2 f ∂x 2 (0) = 12 > 0. However, the gradient ideal I = 12x(x + 1) 2 which also is the KKT ideal, is not radical, because g(x) = x(x + 1) ∈ √ I, but g ∈ I. 2. Let n = 2 and s = 0 (so that K = R 2 ). Then the polynomial in two variables f (x, y) = x 2 does not satisfy the BHC condition, but it satisfies the radical condition. Indeed, the Hessian matrix of f is not positive definite at any zero of f in K. However, the gradient ideal I = 2x which also is the KKT ideal, is radical.
Remark 4.2. If we leave both the radical condition and the BHC condition, then we will have the corresponding representations of strictly positive polynomials.
Theorem 4.6 (Nie-Demmel-Sturmfels [NDS]). If f > 0 on R n , then f is a sum of squares modulo I grad .
Theorem 4.7 (Demmel-Nie-Powers [DNP]). If f > 0 on K, then f ∈ P + I KKT .
Remark 4.3. In the proof of theorem 4.4, theorem 4.5 and theorem 4.7, we must work in a larger polynomial ring R[x, λ], i.e., we must add the Lagrange multipliers to our representations.
Sums of squares modulo critical ideals
In this section, we present our main results. These are similar to theorem 4.4 and theorem 4.5, but without modulo I KKT . It is replaced by modulo another ideal -the critical ideal of f on K. In its proof, we work particularly in the polynomial ring R[x].
Let us start with some notations. The ideal
I(f, K) := g J h J c , ∀J ⊆ {1, . . . , s} generated by g J h J c is called the critical ideal of f on K. By Proposition 3.2, we have C(f, K) = V R (I(f, K)).
Theorem 5.1. Suppose that
(i) f ≥ 0 on K,
(ii) f satisfies the BHC at each zero of f in K.
Then f ∈ P + I(f, K).
To prove the theorem 5.1, we need the following lemma.
Lemma 5.1. Let W be an irreducible component of V (I(f, K)). If W ∩R n = ∅, then f is constant on W .
Proof. This follows from the proof of lemma 3.6 in [HP3].
Proof of theorem 5.1. We decompose V (I(f, K)) into its irreducible components and let W 0 be the union of all the components whose intersection with K is empty. We note that this includes all components W with W ∩ R n = ∅. Thus, by lemma 5.1, f is constant on each of the remaining components. We group together all components for which f takes the same value. Then we have pairwise-disjoint subsets W 1 , . . . , W r of W such that for each i, f takes a constant value a i on W i , with the a i being distinct. Further, since each W i contains a real point and f is non-negative on C(f, K) ∩ K, the value of f on each W i is real and non-negative. We assume a 1 > · · · > a r ≥ 0. We fix a primary decomposition of I(f, K), for each i ∈ {0, 1..., r}, let J i be the intersection of those primary components corresponding to the irreducible components occurring in W i . Thus, V (J i ) = W i , ∀i = 0, 1, . . . , r.
Since W i ∩ W j = ∅, we have J i + J j = R[x] by theorem 2.1. Therefore the Chinese remainder theorem (see, e.g., [E]) implies that there is an isomorphism
ϕ : R[x]/I(f, K) −→ R[x]/J 0 × R[x]/J 1 × · · · × R[x]/J r . Lemma 5.2. There is q 0 ∈ P such that f ≡ q 0 mod J 0 .
Proof. According to the argument presented above, V (J 0 ) ∩ K = ∅, hence there exists u 0 ∈ P such that −1 ≡ u 0 mod J 0 . This result is a special case of theorem 8.6 in [Lam].
We
write f = f 1 − f 2 for SOS polynomials f 1 = (f + 1 2 ) 2 and f 2 = (f 2 + 1 4 ). Hence f ≡ f 1 + u 0 f 2 mod J 0 . Let q 0 = f 1 + u 0 f 2 ∈ P . Then f ≡ q 0 mod J 0 .
Lemma 5.3. f is a sum of squares modulo J i , for all i = 1, . . . , r − 1.
Proof. According to the argument presented above, on each W i , 1 ≤ i ≤ r − 1, f = a i > 0, and hence the polynomial u = f /a i − 1 vanishes on W i . Then by theorem 2.2 there exists some integer k ≥ 1 such that u k ∈ J i . From the binomial identity, it follows that
1 + u = k−1 j=0 1/2 j u j 2 + qu k .
The reader can see clearly in lemma 7.24 in [Lau]. Thus f = a i (u + 1) is a sum of squares modulo J i . Now we continue the proof of theorem 5.1. If a r > 0, then by the proof of lemma 5.3, we imply that f is a sum of squares modulo J r .
Lemma 5.4. If a r = 0, then there is q r ∈ P such that f ≡ q r mod J r .
Proof. By the assumption that f satisfies the BHC at each zero of f on K and by theorem 4.1, there exist g ∈ P and h ∈ R[x] such that f = g + hf 2 , i.e., f (1−hf ) = g. Since f vanishes on W r , f m ∈ J r for some positive integer
m. Let t = hf, v = m−1 i=0 t i . Then t, v ∈ R[x]
, t m ∈ J r , and (1−t)v ≡ 1 mod J r .
By the binomial theorem, there exist c i ∈ Q, i = 0, 1, . . . , m − 1, such that
v ≡ m−1 i=0 c i t i 2 mod J r .
This yields q r ∈ P satisfying
f ≡ f (1 − hf )v = gv ≡ q r mod J r .
To finish the proof of theorem 5.1, we claim the following lemma.
Lemma 5.5. Given q 0 , q 1 , . . . , q r ∈ R[x], there exists q ∈ R[x] such that q − q i ∈ J i , ∀i = 0, 1, . . . , r. Moreover, if each q i ∈ P , then q ∈ P .
Proof. The proof is by induction on r ≥ 1. Assume r = 1. As J 0 + J 1 = R[x], 1 = u 0 + u 1 for some u 0 ∈ J 0 , u 1 ∈ J 1 . Set q := u 2 0 q 1 + u 2 1 q 0 ; thus q ∈ P . Moreover, q − q 0 = u 2 0 q 1 + q 0 (u 2 1 − 1) = u 2 0 q 1 − u 0 (u 1 + 1)q 0 ∈ J 0 . Analogously, q − q 1 ∈ J 1 . Let t be the constructed polynomial, satisfying t − q 0 ∈ J 0 and t − q 1 ∈ J 1 . Consider now the ideals J 0 ∩ J 1 , J 2 , . . . , J r . As (J 0 ∩ J 1 ) + J i = R[x](i ≥ 2), we can apply the induction assumption and deduce the existence of q ∈ R[x] for which q − t ∈ J 0 ∩ J 1 , q − q i ∈ J i (i ≥ 2). Moreover, q ∈ P if t, q 2 , ..., q r ∈ P , which concludes the proof.
Using lemma 5.2, lemma 5.3, lemma 5.4 and lemma 5.5, we imply that there is q ∈ P such that f ≡ q mod I(f, K), i.e., f ∈ P + I(f, K).
Remark 5.1. If we replace the BHC condition by the radical condition of I(f, K), then we will have the following result.
Theorem 5.2. Suppose that
(i) f ≥ 0 on K, (ii) I(f, K) is radical. Then f ∈ P + I(f, K).
Proof. From the proof of theorem 5.1, by our definition of irreducibility, each W i is conjugate symmetric (i.e., a point X ∈ C n belong to W i if and only if its complex conjugateX ∈ W i ). By lemma 1 in [NDS], there exist polynomials p 0 , p 1 , . . . , p r ∈ R[x] such that p i (W j ) = δ ij , where δ ij is the Kronecker delta function.
We consider the polynomial
q := q 0 p 2 0 + r i=1 a i p 2 i ,
where q 0 is as in lemma 5.2. By construction, q ∈ P . Moreover, f − q vanishes on C(f, K), since f (x) = q 0 (x) = q(x) for X ∈ W 0 (by lemma 5.2) and f (x) = a i = q(x) for X ∈ W i , ∀i = 1, . . . , r.
By the assumption that I(f, K) is radical and using Hilbert's Nullstellensätz (see in [CLO]), we deduce that f − q ∈ I(f, K). This implies that f ∈ P + I(f, K).
Remark 5.2. If we leave both the radical condition of I(f, K) and the BHC condition, then we will have the corresponding representations of strictly positive polynomials.
Theorem 5.3. If f > 0 on K, then f ∈ P + I(f, K).
Proof. This follows similar argument in the proof of theorem 5.1. However, we can assume a 1 > · · · > a r > 0. Thus, by lemma 5.3, f is a sum of squares modulo J i , for all i = 1, . . . , r. Also by lemma 5.2 and lemma 5.5, we imply that there is q ∈ P such that f ≡ q mod I(f, K), i.e., f ∈ P + I(f, K).
Applications in optimization
In this section, we present a result that is similar to theorem 4.1 in [DNP] and theorem 6.1 in [Hi].
We consider the following optimization problem: Find
(1)
f * := inf x∈K f (x).
In the case where K is compact, the SOS methods are based on representations of positive polynomials on compact semi-algebraic sets, which were presented in the theorems of Schmüdgen [Schm] and Putinar [Pu]. However, these theorems do not hold in the case where K is not compact. A more traditional approach in numerical optimization methods uses the first order optimality conditions. Using theorem 5.1 and theorem 5.3, we combine these two methods to give a procedure for approximating f * in the case where the semi-algebraic set is not necessarily compact. In order to implement membership in P + I(f, K) as a SDP, we need a bound on the degrees of the sums of squares involved. Thus, for d ∈ N, we define the truncated preordering as follows: P d = e∈{0,1} s σ e g e 1 1 . . . g es s | deg(σ e g e 1 1 . . . g es s ) ≤ 2d , and the truncated critical ideal as follows:
I d (f, K) = J⊆{1,...,s} φ J g J h J c | deg(φ J g J h J c ) ≤ 2d .
Then we define a sequence {f * d } of SOS relaxations of the optimization problem (1) as follows:
(
2) f * d = max Γ∈R Γ,(3)
s.t.f (x) − Γ ∈ P d + I d (f, K).
Obviously each Γ feasible in (3) is a lower bound of f * . So f * d ≤ f * . When we increase d, the feasible region defined by (3) is increasing, and hence the sequence of lower bounds {f * d } is also monotonically increasing. Thus we have f * 1 ≤ f * 2 ≤ f * 3 ≤ · · · ≤ f * . It can be shown that the sequence of lower bounds {f * d } obtained from (2) and (3) converges to f * in (1), provided that f * is attained at one point x * ∈ K. We summarize in the following theorem:
Theorem 6.1. Assume f has a minimum f * := f (x * ) at one point x * ∈ K.
Then lim d→∞ f * d = f * . Furthermore, if f satisfies the BHC at each zero of f − f * in K, then there exists some d ∈ N such that f * d = f * , i.e., the SOS relaxations (2) and (3) converge in a finite number of steps.
Proof. The proof is similar to that of theorem 6.1 in [Hi] (see also theorem 4.1 in [DNP]). However, we only consider the polynomial ring R[x].
e = (e 1 , . . . , e s ) ∈ {0, 1} s and σ e are sums of squares of polynomials in R[x].
Definition 3 . 2 .
32For each subset J of {1, . . . , s}, we consider the polynomial
Theorem 4. 3 (
3Marshall[M]). Suppose that(i) f ≥ 0 on R n , (ii) f satisfies the BHC at each zero of f .Then f is a sum of squares modulo I grad .
Acknowledgment. The author would like to thank Prof. Murray Marshall, Prof. Ha Huy Vui and Assoc. Prof. Pham Tien Son for many interesting and helpful discussions on the topic of this work.
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| []
|
[
"Spectrum and electromagnetic transitions of bottomonium",
"Spectrum and electromagnetic transitions of bottomonium"
]
| [
"Wei-Jun Deng ",
"Hui Liu ",
"Long-Cheng Gui ",
"Xian-Hui Zhong ",
"\nDepartment of Physics\nHunan Normal University\n410081ChangshaChina\n",
"\nSynergetic Innovation Center for Quantum Effects and Applications (SICQEA)\n410081ChangshaChina\n",
"\nKey Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education\n410081ChangshaChina\n"
]
| [
"Department of Physics\nHunan Normal University\n410081ChangshaChina",
"Synergetic Innovation Center for Quantum Effects and Applications (SICQEA)\n410081ChangshaChina",
"Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education\n410081ChangshaChina"
]
| []
| Stimulated by the exciting progress in the observation of new bottomonium states, we study the bottomonium spectrum. To calculate the mass spectrum, we adopt a nonrelativistic screened potential model. The radial Schrödinger equation is solved with the three-point difference central method, where the spin-dependent potentials are dealt with nonperturbatively. With this treatment, the corrections of the spin-dependent potentials to the wave functions can be included successfully. Furthermore, we calculate the electromagnetic transitions of the nS (n ≤ 4), nP (n ≤ 3), and nD (n ≤ 2) bottomonium states with a nonrelativistic electromagnetic transition operator widely applied to meson photoproduction reactions. Our predicted masses, hyperfine and fine splittings, electromagnetic transition widths and branching ratios of the bottomonium states are in good agreement with the available experimental data. In particular, the EM transitions of Υ(3S ) → χ b1,2 (1P)γ, which were not well understood in previous studies, can be reasonably explained by considering the corrections of the spin-dependent interactions to the wave functions. We also discuss the observations of the missing bottomonium states by using radiative transitions. Some important radiative decay chains involving the missing bottomonium states are suggested to be observed. We hope our study can provide some useful references to observe and measure the properties of bottomonium mesons in forthcoming experiments. | 10.1103/physrevd.95.074002 | [
"https://arxiv.org/pdf/1607.04696v3.pdf"
]
| 118,870,902 | 1607.04696 | f9eff022986c1a9957570bf282fd099fbabe6e2d |
Spectrum and electromagnetic transitions of bottomonium
1 Apr 2017
Wei-Jun Deng
Hui Liu
Long-Cheng Gui
Xian-Hui Zhong
Department of Physics
Hunan Normal University
410081ChangshaChina
Synergetic Innovation Center for Quantum Effects and Applications (SICQEA)
410081ChangshaChina
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education
410081ChangshaChina
Spectrum and electromagnetic transitions of bottomonium
1 Apr 2017arXiv:1607.04696v3 [hep-ph]numbers: 1440Pq1320Gd1239Jh
Stimulated by the exciting progress in the observation of new bottomonium states, we study the bottomonium spectrum. To calculate the mass spectrum, we adopt a nonrelativistic screened potential model. The radial Schrödinger equation is solved with the three-point difference central method, where the spin-dependent potentials are dealt with nonperturbatively. With this treatment, the corrections of the spin-dependent potentials to the wave functions can be included successfully. Furthermore, we calculate the electromagnetic transitions of the nS (n ≤ 4), nP (n ≤ 3), and nD (n ≤ 2) bottomonium states with a nonrelativistic electromagnetic transition operator widely applied to meson photoproduction reactions. Our predicted masses, hyperfine and fine splittings, electromagnetic transition widths and branching ratios of the bottomonium states are in good agreement with the available experimental data. In particular, the EM transitions of Υ(3S ) → χ b1,2 (1P)γ, which were not well understood in previous studies, can be reasonably explained by considering the corrections of the spin-dependent interactions to the wave functions. We also discuss the observations of the missing bottomonium states by using radiative transitions. Some important radiative decay chains involving the missing bottomonium states are suggested to be observed. We hope our study can provide some useful references to observe and measure the properties of bottomonium mesons in forthcoming experiments.
I. INTRODUCTION
Heavy quarkonium is considered to be an excellent laboratory to study quantum chromodynamics (QCD) at low energies [1][2][3]. Due to a large mass of the heavy bottom quark, the bottomonium system is essentially nonrelativistic, which makes it relatively easy for us to study the perturbative and nonperturbative QCD via the bottomonium spectroscopy with a nonrelativistic approximation. In the past few years, great progress has been achieved in the study of the bottomonium spectroscopy [4][5][6][7]. A fairly abundant bottomonium spectroscopy has been established in experiments [8](see Tab. I). Furthermore, many new experiments are being and/or to be carried out at LHC and Belle. In near future, more missing bottomonium states will be discovered and more decay channels will be observed in experiments. Thus, it is necessary to carry out a comprehensive study of the bottomonium states according to the recent progress. On the one hand we can obtain more knowledge of bottomonium states from experimental observations. On the other hand, the predicted properties can provide some useful references for our search for the missing bottomonium states in experiments.
In the past years, stimulated by the exciting progress in experiments, many theoretical studies of bottomonium spectrum have been carried out with different methods, such as the widely used potential models [9][10][11][12][13][14][15][16][17], lattice QCD [18][19][20][21], effective Lagrangian approach [22], nonrelativistic effective field theories of QCD [23][24][25], various coupled-channel quark models [26][27][28], and light front quark model [29][30][31][32]. Although some comparable predictions from different mod- * E-mail: [email protected] † E-mail: [email protected] els have been achieved, many properties of the bottomonium states are still not well understood. For example, the recent calculations with the relativized quark model [12] obtain a successful description of the masses for the low-lying excitations, however, the predicted mass for the higher excitation Υ(6S ) is about 100 MeV higher than the data if Υ(11020) is identified as Υ(6S ); while the recent nonrelativistic constituent quark model [13] gives a good description of the mass of Υ(6S ), however, the predicted masses for the ground states Υ(1S ) and η b (1S ) are about 50 MeV larger than the experimental values. Furthermore, there are puzzles in the electromagnetic (EM) transitions of bottomonium states. For example, about the M1 transitions of Υ(2S , 3S ) → η b (1S )γ, the predictions from the relativistic quark model [16] and nonrelativistic effective field theories of QCD [25] are about an order of magnitude smaller than the recent predictions from the relativized quark model [12] and nonrelativistic constituent quark model [13]; while about the EM transitions of Υ(3S ) → χ b1,2 (1P)γ, the predicted partial widths in the literature [11][12][13] are inconsistent with the data. Thus, to deepen our knowledge about the bottomonium spectrum, more theoretical studies are needed.
In this work, first we use the nonrelativistic screened potential model [11,[33][34][35] to calculate the masses and wave functions. In this model, the often used linear potential br is replaced with the screened potential b(1 − e −µr )/µ. The reason is that the linear potential, which is expected to be dominant at large distances, is screened or softened by the vacuum polarization effect of the dynamical light quark pairs [36,37]. Such a screening effect might be important for us to reasonably describe the higher radial and orbital excitations. Considering the corrections of the spin-dependent interactions to the space wave functions cannot be included with the perturbative treatment, we treat the spin-dependent interactions as nonperturbations in our calculations. With the nonperturbative treatment, we can reasonably include the effect of spin-dependent interactions on the wave functions, which is important for us to gain reliable predictions of the decays.
Moreover, using the obtained wave functions, we study the EM transitions between bottomonium states. Difference of our method from the often used potential models is that the EM transition operator between initial and final hadron states is used a special nonrelativistic form h e ≃ j [e j r j · ǫ − e j 2m j σ j · (ǫ ×k)]e −ik·r j [38], which has been well developed and widely applied to meson photoproduction reactions [39][40][41][42][43][44][45][46][47][48][49][50]. In this operator, the effect of binding potential between quarks is considered. Furthermore, the possible higher EM multipole contributions to a EM transition process can be included naturally.
The paper is organized as follows. In Sec. II, we calculate the masses and wave functions within a screened potential model. In Sec. III, the EM transitions between the bottomonium states are calculated, and our analysis and discussion are given. Finally, a summary is given in Sec. IV. [8], and the theoretical predictions with the previous screened potential model (SNR model) [11], relativized quark model (GI model) [12], and nonrelativistic constituent quark model (NR model) [13] are also listed in the same table.
n 2S +1 L J name J PC PDG [8] SNR [11] GI [12] NR [13] Ours
1 3 S 1 Υ(1S ) 1 −1 1 F 3 h b3 (1F) 3 +− 10355 10322 10339 1 3 F 4 χ b4 (1F) 4 ++ 10358 10340 1 3 F 3 χ b3 (1F) 3 ++ 10355 10321 10340 1 3 F 2 χ b2 (1F) 2 ++
10350 10315 10338
II. MASS SPECTRUM
As a minimal model of the bottomonium system we use a nonrelativistic screened potential model [11,[33][34][35]. The effective potential of spin-independent term V(r) is regarded as the sum of Lorentz vector V V (r) and Lorentz scalar V s (r) contributions [4], i.e., [8]. The theoretical predictions with the previous screened potential model [11], relativized quark model [12], relativistic two-body calculation [14], and nonrelativistic constituent quark model [13,15] are also listed in the same For the Lorentz vector potential V V (r), we adopt the standard color Coulomb form:
V(r) = V V (r) + V s (r).(1)V V (r) = − 4 3 α s r .(2)
To take into account the screening effects, which might originate from the vacuum polarization of the dynamical light quark pairs [36,37], we replace the widely used linear scalar potential br with a special form
V s (r) = b(1 − e −µr ) µ ,(3)
as suggested in Refs. [11,[33][34][35]. Here µ is the screening factor which makes the long-range scalar potential of V s (r) behave like br when r ≪ 1/µ, and become a constant b/µ when r ≫ 1/µ. The main effect of the screened potential on the spectrum is that the masses of the higher excited states are lowered. Such a screening effect might be important for us to reasonably describe the higher radial and orbital excitations. We include three spin-dependent potentials as follows. For the spin-spin contact hyperfine potential, we take [51]
H S S = 32πα s 9m 2 bδ σ (r)S b · S¯b,(4)
where S b and S¯b are spin matrices acting on the spins of the quark and antiquark. We takeδ σ (r) = (σ/ √ π) 3 e −σ 2 r 2 as in Ref. [51]. The five parameters in the above equations (α s , b, µ, m b , σ) are determined by fitting the spectrum.
For the spin-orbit term and the tensor term, we take the common forms [4]:
H S L = 1 2m 2 b r 3 dV V dr − dV s dr L · S,(5)
and
H T = 1 12m 2 b 1 r dV V dr − d 2 V V dr 2 S T ,(6)
where L is the relative orbital angular momentum of b andb quarks, S = S b + S¯b is the total quark spin, and the spin tensor S T is defined by [4] S T = 6 S · rS · r r 2 − 2S 2 .
In the | 2S +1 L J basis, the matrix element for the spin-spin operator S b · S¯b is
S b · S¯b = 1 2 S (S + 1) − 3 4 .(8)
For the spin-orbit operator L · S, its matrix element is
L · S = 1 2 [J(J + 1) − L(L + 1) − S (S + 1)].(9)
The element of the tensor operator S T can be written in the form [52]
S T = 4 S 2 L 2 − 3 2 L · S − 3(L · S) 2 (2L + 3)(2L − 1)
.
To obtain masses and wave functions of the bottomonium states, we need to solve the radial Schrödinger equation
d 2 u(r) dr 2 + 2µ R E − V bb (r) − L(L + 1) 2µ R r 2 u(r) = 0,(11)
with V bb (r) = V(r) + H S S + H S L + H T ,
where µ R = m b m¯b/(m b +m¯b) is the reduced mass of the system, and E is the binding energy of the system. Then, the mass of a bb state is obtained by
M bb = 2m b + E.(13)
In the literature, the spin-dependent interactions were usually dealt with perturbatively. Although the meson mass obtains perturbative corrections from these spin-dependent potentials, the wave functions obtain no corrections from these spin-dependent potentials. To reasonably include the corrections from these spin-dependent potentials to both the mass and wave function of a meson state, we deal with the spindependent interactions nonperturbatively.
In this work, we solve the radial Schrödinger equation by using the three-point difference central method [53] from central (r = 0) towards outside (r → ∞) point by point. In this method, we need to know the role of u(r → 0). When r → 0 we easily obtain u(r → 0) ∝ r L+1 if we neglect the contributions of the spin-orbit and tensor terms. However, including the spin-orbit and tensor potential contributions, we have a term ∝ 1/r 3 in the potential. In the limit r → 0, the potential V bb (r) ∝ 1/r 3 . In this case, we do not know the role of u(r → 0), thus, we cannot solve the radial Schrödinger equation with the three-point differential central method. To overcome this problem, we assume that in a small range r ∈ (0, r c ), the V bb (r) ∝ 1/r 3 c , which is a finite constant. Then, the role of u(r → 0) is still ∝ r L+1 . The price of our method is that a cutoff distance r c should be introduced in the calculation, which is determined by fitting the spectrum. The details of the method for solving Eq. (11) are outlined in the Appendix.
For the model parameters, we take α s = 0.368(3), b = 0.206(2) GeV 2 , µ = 0.056(11) GeV, m b = 4.757(2) GeV, and σ = 3.10(25) GeV. This parameter set is slightly different from that suggested in Ref. [11]. In our calculation, the cutoff distance r c = 0.060 (12) fm is adopted. The uncertainties for these determined parameters mean that if one changes one of the parameter within its uncertainty, the mass change of one state is less than 5 MeV. It should be mentioned that the masses of the 3 P 0 states are sensitive to the cutoff distance r c . Thus, in the present work we use the mass of χ b0 (1P) to determine the cutoff distance r c . With the determined cutoff distance r c = 0.06 fm, the calculated masses of the other 3 P 0 states are in good agreement with the measurements and the other model predictions.
With the determined parameter set, by solving the radial Schrödinger equation we obtain the masses of the bottomonium states, which have been listed in Tab I. From the table, we see that our results are compatible with the previous screened potential model predictions [11], which indicates that our numerical method is reliable. The recent relativized quark model can successfully describe the low-lying bottomonium states, however, their predicted mass for the higher excitations Υ(6S ) is about 100 MeV larger than the experimental measurements [12]. Although the recent nonrelativistic constituent quark model systematically improve the descriptions of the higher mass spectrum, the predicted masses for the ground states Υ(1S ) and η b (1S ) are about 40 ∼ 50 MeV higher than the data [13]. Interestedly, it is found that the screened potential model obtains a fairly good description of the masses not only for the low-lying states, but also for the higher excitation Υ(6S ).
Furthermore, in Tab. II, we give our predictions of the hyperfine splittings for some S -wave states, and fine splittings for some P-wave states. It is found that our predicted splittings are in good agreement with the world average data [8].
Comparing the model predictions [11][12][13][14][15] with each other, we find obvious model dependencies of the predicted mass splittings. Thus, to better understand these nonperturbative strong interactions in the bottomonium system, more modelindependent studies are needed.
In order to clearly see the properties of the wave functions, we plot the radial probability density of the states as a function of the interquark distance r in Fig. 1. It is found that the spindependent potentials have notable corrections to the S -and triplet P-wave states; however, the corrections to the triplet Dwave states are tiny. The strong attractive spin-spin potential H S S shifts the wave functions of the 1 S 0 states towards the center, while the strong attractive tensor potential H T shifts the wave functions of the 3 P 0,1 states towards the center.
III. ELECTROMAGNETIC TRANSITIONS
Using these obtained wave functions of the bottomonium states, we further study their EM transitions. The quarkphoton EM coupling at the tree level is adopted as
H e = − j e jψ j γ j µ A µ (k, r)ψ j ,(14)
where ψ j stands for the jth quark field in a hadron. The photon has three momentum k, and the constituent quark ψ j carries a charge e j .
To match the nonrelativistic wave functions of the bottomonium states, we should adopt the nonrelativistic form of Eq. (14) in the calculations. For the EM transition of a hadron, in the initial-hadron-rest system the nonrelativistic expansion of H e in Eq. (14) becomes [38,41]
h e ≃ j e j r j · ǫ − e j 2m j σ j · (ǫ ×k) φ,(15)
where m j , σ j , and r j stand for the constituent mass, Pauli spin vector, and coordinate for the jth quark, respectively. The vector ǫ is the polarization vector of the photon. For emitting a photon, we have φ = e −ik·r j , while for absorbing a photon, we have φ = e +ik·r j . It is found that the first and second terms in Eq.(15) are responsible for the electric and magnetic transitions, respectively. The main feature of this EM transition operator is that the effects of binding potential between quarks are considered. Furthermore, the possible higher EM multipole contributions are included naturally. This nonrelativistic form has been widely applied to meson photoproduction reactions [39][40][41][42][43][44][45][46][47][48][49][50]. It should be mentioned that, at the order of 1/m j , we have neglected the contributions from the term e j r j · ǫp j ·k/m j as suggested in Refs. [39,40] for a strong suppression of p j ·k/m j . Then, one obtains the standard helicity amplitude A of the radiative decay process by the relation
A = −i ω γ 2 f |h e |i .(16)
Finally, we can calculate the EM decay width by
Γ = |k| 2 π 2 2J i + 1 M f M i J f z ,J iz |A J f z ,J iz | 2 ,(17)
where J i is the total angular momentum of an initial meson and J f z and J iz are the components of the total angular momenta along the z axis of initial and final mesons, respectively. In our calculation, for the well-established bottomonium states, the experimental masses are adopted [8]; while for the missing bottomonium states, their masses are adopted from our theoretical predictions.
A. Υ(1S ) → η b (1S )γ
The Υ(1S ) → η b (1S )γ decay process is a typical M1 transition at tree level, which is strongly suppressed by the constituent bottom quark mass m b . Our predicted partial width III: Partial widths of the M1 radiative transitions for some low-lying S -and P-wave bottomonium states. For comparison, the measured values from the PDG [8], and the theoretical predictions with the relativistic quark model [16], nonrelativistic effective field theories of QCD (EFT model) [25], relativized quark model (GI model) [12], and nonrelativistic constituent quark model (NR model) [13] are also listed in the same table.
Initial meson Final meson
E γ (MeV) Γ M1 (eV) Γ M1 (eV) state state
Ref. [16] GI [12] ours Ref. [16] GI [12] EFT [25] NR [13] Ours
PDG [8] Υ(1 3 S 1 ) η b (1 1 S 0 ) 60 62 62 5.8 10 15.2 9.34 10 Υ(2 3 S 1 ) η b (2 1 S 0 ) 33 24 24 1.4 0.59 0.67 0.58 0.59 η b (1 1 S 0 ) 604 606 606 6.4 81 6 +26 −6 56.5 66 12.5 ± 4.9 η b (2 1 S 0 ) Υ(1 3 S 1 ) 516 524 524 12 68 ∼ 80 45.0 64 Υ(3 3 S 1 ) η b (3 1 S 0 ) 27 18 18 0.8 0.25 0.66 3.9 η b (2 1 S 0 ) 359 350 350 1.5 0.19 11.0 11 < 14 η b (1 1 S 0 ) 911 913 913 11 60 57.0 71 10 ± 2 η b (3 1 S 0 ) Υ(2 3 S 1 ) 301 309 309 2.8 9.1 9.20 8.7 Υ(1 3 S 1 ) 831 840 840 24 74 51.0 60 χ b2 (1 3 P 2 ) h b (1 1 P 1 ) 13 13 9.6 × 10 −2 0.12 8.9 × 10 −2 9.5 × 10 −2 h b (1 1 P 1 ) χ b1 (1 3 P 1 ) 6 6 1.0 × 10 −2 9.0 × 10 −3 1.15 × 10 −2 9.4 × 10 −3 χ b0 (1 3 P 0 ) 40 40 0.89 0.96 0.86 0.90 χ b2 (2 3 P 2 ) h b (1 1 P 1 ) 363 363 0.24 1.78 4.5 χ b1 (2 3 P 1 ) 350 350 2.2 0.17 0.18 χ b0 (2 3 P 0 ) 329 329 9.7 2.39 16 h b (2 1 P 1 ) χ b2 (1 3 P 2 ) 342 342 2.2 6.91 × 10 −3 1.1 χ b1 (1 3 P 1 ) 360 360 1.1 1.28 2.5 χ b0 (1 3 P 0 ) 393 393 0.32 36.4 10 is Γ[Υ(1S ) → η b (1S )γ] ≃ 10 eV.(18)
Combining this partial width with the measured total width of Υ(1S ) [8], we obtain
B[Υ(1S ) → η b (1S )γ] ≃ 2.0 × 10 −4 .(19)
Our predictions are in good agreement with the recent results of the relativized quark model [12] and nonrelativistic constituent quark model [13] (see Tab. III). However, our predicted Γ[Υ(1S ) → η b (1S )γ] is larger than the value 5.8 eV from the relativistic quark model [16], while smaller than the recent prediction 15.2 eV from the pNRQCD approach [25]. It should be mentioned that this decay rate is extremely sensitive to the masses of Υ(1S ) and η b (1S ). If all of the models adopt the experimental masses, the predictions of Γ[Υ(1S ) → η b (1S )γ] from different models might be consistent with each other.
B. Radiative transitions of 2S states
1. Υ(2S )
The allowed EM transitions of
Υ(2S ) are Υ(2S ) → χ bJ (1P)γ and Υ(2S ) → η b (1S , 2S ) γ. The Υ(2S ) → χ bJ (1P)γ processes are governed by the E1 transitions, while Υ(2S ) → η b (1S , 2S ) γ are typical M1 transitions.
From Tab. IV, it is found that our predicted partial decay widths for the Υ(2S ) → χ bJ (1P)γ processes are in good agreement with the world average data from the PDG [8], and also are consistent with predictions from various potential models [11][12][13][14][15][16].
For the M1 transitions Υ(2S ) → η b (1S , 2S )γ, our predicted partial decay widths have been listed in Tab. III. From the table, it is seen that our predicted Γ[Υ(2S ) → η b (2S )γ] is in agreement with the other model predictions. It should be pointed out that although our predicted Γ[Υ(2S ) → η b (1S )γ] is compatible with the recent potential model predictions [12,13], it is about 5 times larger than the average value 1.25(49) × 10 −2 keV from the PDG [8] and the recent lattice NRQCD result 1.72(55) × 10 −2 keV [18]. More studies of the M1 transition Υ(2S ) → η b (1S )γ are needed in both theory and experiments.
2. η b (2S )
The η b (2S ) resonance can decay into h b (1P)γ and Υ(1S )γ channels by the E1 and M1 transitions, respectively. Our predicted partial decay width of
Γ[η b (2S ) → h b (1P)γ] ≃ 3.41
keV is in good agreement with the predictions of the relativistic quark model [16], potential model [12], and nonrelativistic constituent quark model [13] (see Tab. IV). However, it is about a factor 1.8 smaller than the previous SNR model prediction [11]. This difference might come from the corrections of the spin-dependent potentials to the wave function of
η b (2S ).
Furthermore, from Tab. III it is seen that our predicted partial decay width for the M1 transition η b (2S ) → Υ(1S )γ is compatible with the recent predictions of potential models [12,13], and with the pNRQCD approach [25]. However, our prediction is notably larger than Γ[η b (2S ) → Υ(1S )γ] ≃ 12 eV predicted with a relativistic quark model [16] (see Tab. III).
C. Radiative transitions of 1P states
The typical radiative transitions of χ bJ (1P) are χ bJ (1P) → Υ(1S )γ. From the Tab. V, it is found that the partial widths Γ[χ bJ (1P) → Υ(1S )γ] predicted by us are in agreement with the predictions in [11-13, 15, 16]. Combining our predicted partial widths with the measured branching ratios
B[χ b0 (1P) → Υ(1S )γ] ≃ (1.76 ± 0.48)%, B[χ b1 (1P) → Υ(1S )γ] ≃ (33.9 ± 2.2)%, and B[χ b2 (1P) → Υ(1S )γ] ≃ (19.1 ± 1.2)% [8]
, we easily estimate the total widths for χ b0 (1P), χ b1 (1P) and χ b2 (1P), which are
Γ total χ b0 (1P) ≃ 1.56 +0.59 −0.33 MeV,(20)Γ total χ b1 (1P) ≃ 94 ± 7 keV,(21)Γ total χ b2 (1P) ≃ 166 +12 −9 keV,(22)
respectively. It is interesting to find that the estimated width for χ b0 (1P) is consistent with the recent measurement 1.3±0.9
MeV from the Belle Collaboration [54]. It should be mentioned that these widths predicted by us strongly depend on the measured branching ratios. It is found that B[χ b0 (1P) → Υ(1S )γ] still bears a large uncertainty. Thus, to determine finally the width of χ b0 (1P), more accurate measurements are needed. For the singlet state h b (1P), its main radiative transi-
tion is h b (1P) → η b (1S )γ. We predict that Γ[h b (1P) → η b (1S )γ] ≃ 35.8 keV,
which is consistent with the predictions in Refs. [11,12] (see Tab. V). A relatively large partial width was also predicted in Ref [16]. Combining the measured branching ratio
B[h b (1P) → η b γ] ≃ 49 +8 −7 % [8]
with our predicted partial width, we estimate that the total width of
h b (1P) might be Γ total h b (1P) ≃ 73 +12 −10 keV,(23)
which could be tested in future experiments. Finally, we give our estimations of the typical M1 transitions h b (1P) → χ b0,1 (1P)γ, which are listed in Tab. III. The rates of these M1 transitions are very weak. Our results are consistent with those obtained in the framework of the relativized quark model [12] and nonrelativistic constituent quark model [13].
D. Radiative transitions of 1D states
In the 1D bottomonium states, only the 2 −− state Υ 2 (1D) with a mass of M Υ 2 (1D) = 10164 MeV is confirmed in exper-iments [55]. The other 1D states are still missing. The discovery of the Υ 2 (1D) state provides a strong constrain on the masses of the other 1D states. In our calculations, we predict the mass splittings
M Υ 3 (1D) − M Υ 2 (1D) ≃ 4, M Υ 2 (1D) − M Υ 1 (1D) ≃ 7, and M Υ 2 (1D) − M η b2 (1D) ≃ 0 MeV.
Combining these predicted multiplet mass splittings with the measured mass of M Υ 2 (1D) = 10164 MeV, one can predict the masses for the
Υ 1 (1D), Υ 3 (1D) and η b2 (1D) states, which are M Υ 1 (1D) ≃ 10157 MeV, M Υ 3 (1D) ≃ 10168 MeV, and M η b2 (1D) ≃ 10164 MeV, respectively. 1. Υ 2 (1D) For the established 2 −− state Υ 2 (1D) [i.e., Υ 2 (10164)], the EM transitions are dominated by Υ 2 (1D) → χ b1,2 (1P)γ.
We calculate their partial decay widths, which are listed in Tab. V. Combining with the predicted partial widths of Γ[Υ 2 (1D) → ggg] ≃ 0.62 keV and Γ[Υ 2 (1D) → ππΥ(1S )] ≃ 0.29 keV from Ref. [13], we estimate the total width of Υ 2 (1D), Γ tot ≃ 30 keV. With this estimated width, we further predict the branching ratios
B[Υ 2 (1D) → χ b1 (1P)γ] ≃ 73%,(24)B[Υ 2 (1D) → χ b2 (1P)γ] ≃ 24%.(25)
Our results are in agreement with the predictions obtained with the previous SNR model [11], relativistic quark model [16], and nonrelativistic constituent quark model [13]. The large branching ratios indicate the Υ 2 (1D) → χ b1,2 (1P)γ transitions may be observed in forthcoming experiments.
The missing 1D states
According to the predicted mass M Υ 1 (1D) = 10157 MeV of Υ 1 (1D), we calculate the partial decay widths of Γ[Υ 1 (1D) → χ b0,1,2 (1P)γ], which are listed in Tab. V. In Ref. [13], the total width of Υ 1 (1D) is predicted to be Γ tot ≃ 44 keV. Using it as an input, we predict
B[Υ 1 (1D) → χ b0 (1P)γ] ≃ 45%,(26)B[Υ 1 (1D) → χ b1 (1P)γ] ≃ 30%,(27)B[Υ 1 (1D) → χ b2 (1P)γ] ≃ 2%.(28)
These branching ratios are consistent with those from the recent works [12,13]. The fairly large branching ratios indicate that the missing Υ 1 (1D) state is most likely to be observed through the radiative transitions Υ 1 (1D) → χ b0,1 (1P)γ. While taking the mass of Υ 3 (1D) with M Υ 3 (1D) = 10168 MeV, we calculate the partial decay widths of Γ[Υ 3 (1 3 D 3 ) → χ bJ (1P)γ]. Our results are listed in Tab. V. It is found that the EM decays of Υ 3 (1D) are governed by the χ b2 (1P)γ channel, and the decay rates into the χ b0,1 (1P)γ channels are negligibly small. Our prediction of Γ[Υ 3 (1D) → χ b2 (1P)γ] ≃ 32.1 keV is consistent with the predictions from the potential models [11,12] and relativistic quark model [16] (see Tab. V). According to the predictions in Refs. [12,13], the partial widths of Γ[Υ 2 (1D) → ggg] and Γ[Υ 2 (1D) → ππΥ(1S )] are too small to compare with Γ[Υ 3 (1D) → χ b2 (1P)γ], thus, the branching fraction of B[Υ 3 (1D) → χ b2 (1P)γ] ∼ 100%. To establish Υ 3 (1D), the decay channel χ b2 (1P)γ is worth observing in future experiments.
For the singlet 1D state η b2 (1D), our predicted partial width Γ[η b2 (1D) → h b (1P)γ] ≃ 30.3 keV is close to the predictions from the other potential models [11,12,16] (see Tab. V). Combining with the predictions Γ[η b2 (1D) → gg] ≃ 1.8 keV and Γ[η b2 (1D) → ππη b (1S )] ≃ 0.35 keV in Ref. [12], we obtain the total width of Υ 1 (1D), Γ tot ≃ 32.5 keV, with which we further estimate that
B[η b2 (1D) → h b (1P)γ] ≃ 93%.(29)
The large radiative transition rate indicates that the missing η b2 (1D) state is most likely to be observed in the h b (1P)γ channel.
E. Radiative transitions of 2P states
The 2P bottomonium states have been established in experiments. The branching ratios of B[χ b0,1,2 (2P) → Υ(1S , 2S )γ] and B[h b (2P) → η b (1S , 2S )γ] have been measured. These measured branching ratios give us a good chance to study the radiative transitions of the 2P bottomonium states, and test our model.
1. χ b0 (2P)
The allowed EM decay modes of χ b0 (2P) are Υ(1S , 2S )γ, Υ 1 (1D)γ and h b (1P)γ. We calculate their partial widths and list them in Tab. IV. From the table, one can see that our predictions are compatible with the other model predictions. Taking the predicted total width Γ tot ≃ 2.5 MeV of χ b0 (2P) from Ref. [12] as an input, we further predict that
B[χ b0 (2P) → Υ(1S )γ] ≃ 2.2 × 10 −3 ,(30)B[χ b0 (2P) → Υ(2S )γ] ≃ 5.8 × 10 −3 .(31)
Our prediction is compatible with the recent results obtained from potential models [12,13], and the previous results obtained from SNR 1 model [11]. However, the predicted branching ratio B[χ b0 (2P) → Υ(2S )γ] is about an order of magnitude smaller than the data from the PDG [8]. To test our predictions, more accurate measurements are needed in experiments.
We also study the typical M1 transition χ b0 (2P) → h b (1P)γ. Our predicted partial decay width Γ[χ b0 (2P) → h b (1P)γ] ≃ 1.6 × 10 −2 keV is close to the recent predictions with the GI potential model [12] (see Tab. III).
χ b1 (2P)
The χ b1 (2P) state can decay into Υ(1S , 2S )γ, Υ(1 3 D 2,3 )γ and h b (1P)γ via radiative transitions. Our predicted partial widths for these transitions are listed in Tab. IV. From the table it is found that the decay rates of χ b1 (2P) into the Dwave states Υ 1,2 (1D) are much weaker than those into the S -wave states. Our predicted partial widths of Γ[χ b1 (2P) → Υ(1S , 2S )γ] are consistent with observations from the CLEO Collaboration [56]. Combining our predicted partial widths with the total width Γ tot ≃ 133 keV predicted in Ref. [13], we obtain that
B[χ b1 (2P) → Υ(1S )γ] ≃ 8.1%,(32)B[χ b1 (2P) → Υ(2S )γ] ≃ 11.5%,(33)
which are close to the measured values B[χ b1 (2P) → Υ(1S )γ] ≃ 9.2 ± 0.8% and B[χ b1 (2P) → Υ(2S )γ] ≃ 19.9 ± 1.9% [8]. The branching fraction ratio
Γ[χ b1 (2P) → Υ(2S )γ] Γ[χ b1 (2P) → Υ(1S )γ] ≃ 1.4,(34)
is slightly smaller than the world average value 2.2 ± 0.4 from the PDG [8]. From Tab. IV, we can find that this ratio has a strong model dependency. To test the predictions from various models, more accurate measurements are needed in experiments. Furthermore, the typical M1 transition χ b2 (2P) → h b (1P)γ is also studied. The predicted partial decay width
Γ[χ b2 (2P) → h b (1P)γ] ≃ 1.8 × 10 −4 keV,(35)
is about an order of magnitude smaller than the recent prediction 2.2 × 10 −3 keV in Ref. [12]. However, the recent prediction 1.7 × 10 −4 keV with a nonrelativistic constituent quark model [13] is in good agreement with our prediction. The Lattice QCD study may be able to clarify this puzzle.
χ b2 (2P)
The χ b2 (2P) state can decay into Υ(1S , 2S )γ, Υ 1,2,3 (1D)γ and h b (1P)γ channels. In these decays, the χ b2 (2P) → Υ(1S , 2S )γ processes play dominant roles. From Tab. IV, it is seen that our predicted partial widths of Γ[χ b2 (2P) → Υ(1S , 2S )γ] are compatible with the observations from the CLEO Collaboration [56] and other model predictions [11][12][13]16]. Combining our predicted partial widths with the estimated total width of χ b2 (2P) according to the CLEO observations [56], i.e., Γ tot ≃ 143 keV, we have
B[χ b2 (2P) → Υ(1S )γ] ≃ 9.5%,(36)B[χ b2 (2P) → Υ(2S )γ] ≃ 11%,(37)
which are close to the average data from the PDG [8]. The estimated partial width ratio
Γ[χ b2 (2P) → Υ(2S )γ] Γ[χ b2 (2P) → Υ(1S )γ] ≃ 1.2,(38)
is also close to the lower limit of the world average data 1.51 +0.59 −0.47 from the PDG [8]. This ratio has strong model dependencies. Thus, more accurate measurements are needed to test various model predictions.
The decay rates of χ b2 (2P) → Υ 1,2,3 (1D)γ are much weaker than those of χ b2 (2P) → Υ(1S , 2S )γ. Our predicted results are close to the predictions in Refs. [11,13,16] (see Tab. IV). Combining the estimated total width of χ b2 (2P) with our predicted partial widths, we have
B[χ b2 (2P) → Υ 1 (1D)γ] ≃ 1.8 × 10 −4 , (39) B[χ b2 (2P) → Υ 2 (1D)γ] ≃ 2.9 × 10 −3 ,(40)B[χ b2 (2P) → Υ 3 (1D)γ] ≃ 1.7 × 10 −2 .(41)
To look for the missing Υ 3 (1D) state, the three-photon decay chain χ b2 (2P) → Υ 3 (1D)γ → χ b2 (1P)γγ → Υ(1S )γγγ is worth observing. The combined branching ratio can reach up to O(10 −3 ).
h b (2P)
The h b (2P) state can decay into η b (1S , 2S )γ, η b2 (1D)γ, and χ b0,1,2 (1P)γ via EM transitions, in which the η b (1S , 2S )γ decay modes are dominant. We calculate the partial decay widths of Γ[h b (2P) → η b (1S , 2S )γ], which are listed in Tab. IV. Our results are compatible with the other model predictions [11][12][13]16]. Our predicted partial width ratio,
Γ[h b (2P) → η b (2S )γ] Γ[h b (2P) → η b (1S )γ] ≃ 1.0,(42)
is close to the lower limit of the measurement 1.0 ± 4.3 from the Belle Collaboration [57]. Furthermore, combining the measured branching ratio
B[h b (2P) → η b (1S )γ] ≃ 22.3 ± 3.8 +3.1 −3.
3 % with our predicted partial width, we estimate the total width of h b (1P), which is
Γ total h b (2P) ≃ 72 +34 −17 keV.(43)
It could be tested in future experiments. We also study the transition of h b (2P) → η b2 (1D)γ. The predicted partial width Γ[h b (2P) → η b2 (1D)γ] ≃ 2.24 keV is compatible with the predictions from the relativized quark model [12] and the relativistic quark model [16]. Using this predicted total width in Eq. (43) as an input, we further predict
B[h b (2P) → η b2 (1D)γ] ≃ 3%.(44)
Combining this ratio with our predicted ratio of
B[η b2 (1D) → h b (1P)γ] ≃ 93% and the measured ratios of B[h b (1P) → η b γ] ≃ 49%
, we obtain the combined branching ratio for the three-photon cascade
h b (2P) → η b2 (1D)γ → h b (1P)γγ → η b γγγ: B[h b (2P) → η b2 (1D)γ → h b (1P)γγ → η b γγγ] ≃ 1.4%. (45)
Thus, to establish the missing η b2 (1D) this three-photon cascade is worth observing. Finally, we give our predictions for the typical M1 transitions h b (2P) → χ b0,1,2 (1P)γ. Our results are listed in Tab. III. It is seen that concerning these M1 transitions, there are obvious differences in various model predictions. From the table, it is found that for the EM transitions Υ(3S ) → χ bJ (1P)γ, the predicted partial widths are in good agreement with the world average data from the PDG [8]. Note that the transition widths for Υ(3S ) → χ b1,2 (1P)γ calculated from the previous screened potential model [11] are too large as compared with experimental data. These problems have been overcome in our calculations by considering the corrections of the spin-dependent interactions to the wave functions. It indicates that the corrections of the spindependent interactions to the wave functions are important to understand these EM transitions, which was also found in Ref. [58].
While for the EM transitions Υ(3S ) → χ bJ (2P)γ, from Tab. IV it is found that our predicted partial widths of Γ[Υ(3S ) → χ bJ (2P)γ] are in good agreement with the experimental data and the predictions in Refs. [11][12][13][14][15][16]. Combining our predicted partial widths with the measured width of Υ(3S ), we estimate that
B[Υ(3S ) → χ b0 (2P)γ] ≃ 5.5%, (46) B[Υ(3S ) → χ b1 (2P)γ] ≃ 12.8%, (47) B[Υ(3S ) → χ b2 (2P)γ] ≃ 15.6%,(48)
which are also in good agreement with the data from the PDG [8].
For the typical M1 transitions Υ(3S ) → η b (1S , 2S )γ, our predicted partial widths are listed in Tab. III. Our results are the same order of magnitude as the predictions from the recent nonrelativistic constituent quark model [13]. However, our prediction of the Γ[Υ(3S ) → η b (1S )γ] ≃ 71 eV is notably larger than the world average data 10 ± 2 eV [8]. To clarify this puzzle, more studies are needed. Tabs. III and IV. From Tab. IV, it is found that with the corrections of the spin-dependent potentials to the wave functions, our predicted partial widths for the E1 transitions η b (3S ) → h b (1P, 2P)γ are about a factor 2 smaller than the previous screened potential model predictions [11]. Furthermore, it should be mentioned that our predicted partial width ratio
Γ[η b (3S ) → h b (2P)γ] Γ[η b (3S ) → h b (1P)γ] ≃ 6.1,(49)
is notably different from the other model predictions [11][12][13]16]. From Tab. III, it is found that our predicted partial widths for the M1 transitions η b (3S ) → Υ(1S , 2S )γ are compatible with the recent predictions in Refs. [12,13], however, our predictions are about a factor 3 larger than the predictions with the relativistic quark model [16]. These radiative transitions should be further studied in theory.
G. Radiative transitions of 2D states
Until now, no 2D bottomonium states have been observed in experiments. In our calculations, their masses are adopted from our potential model predictions.
Υ 3 (2D)
The radiative transitions of Υ 3 (2D) are dominated by the χ b2 (2P)γ channel, and the partial width decaying into the χ b2 (1P)γ channel is also sizeable. Taking the mass of M Υ 3 (2D) = 10436 MeV predicted by us, we calculate the partial widths of Γ[Υ 3 (2D) → χ b2 (1P, 2P)γ]. The results compared with the other model predictions are listed in Tab. V, where we can see that our predictions are compatible with the other model predictions. In Ref. [12], the total width of Υ 3 (2D) is predicted to be Γ tot ≃ 25 keV. With this predicted width, we further estimate the branching ratios:
B[Υ 3 (2D) → χ b2 (1P)γ] ≃ 21%, (50) B[Υ 3 (2D) → χ b2 (2P)γ] ≃ 68%.(51)
To establish the Υ 3 (2D) state, the χ b2 (1P, 2P)γ channels are worth observing.
Υ 2 (2D)
The radiative transitions of Υ 2 (2D) are dominated by the χ b1 (2P)γ channel, and the partial widths decaying into the χ b2 (2P)γ, χ b1 (1P)γ and χ b2 (1P)γ channels are also sizeable. With the predicted mass M Υ 2 (2D) = 10432 MeV, we predict the partial widths for these radiative transitions. Our results compared with the other model predictions are listed in Tab. V. From the table, it is seen that the partial widths predicted by us are comparable with the other model predictions in magnitude [11][12][13]16]. However, it should be mentioned that the predicted ratios from different models are very different. In Ref. [12], the total width of Υ 2 (2D) is predicted to be Γ tot ≃ 23 keV. With this predicted total width, we further estimate that
B[Υ 2 (2D) → χ b1 (2P)γ] ≃ 50%,(52)B[Υ 2 (2D) → χ b2 (2P)γ] ≃ 16%,(53)B[Υ 2 (2D) → χ b1 (1P)γ] ≃ 17%, (54) B[Υ 2 (2D) → χ b2 (1P)γ] ≃ 5%.(55)
Observation of the χ b1,2 (2P)γ and χ b1 (1P)γ channels may be crucial to establish the missing Υ 2 (2D) state.
3. Υ 1 (2D)
The radiative transitions of Υ 1 (2D) are dominated by the χ b0,1 (2P)γ channels, and the partial widths decaying into the χ b0,1,2 (1P)γ and χ b2 (2P)γ channels are also sizeable. Taking the mass of M Υ 1 (2D) = 10425 MeV, we calculate the partial decay widths. Our predicted partial widths for the transitions Υ 1 (2D) → χ b0,1,2 (1P, 2P)γ compared with the other model predictions are listed in Tab. V. From the table, it is found that most of our predictions are compatible with the other potential predictions in magnitude. In Ref. [12], the total width of Υ 1 (2D) is predicted to be Γ tot ≃ 38 keV, with this input, we estimate the branching ratios for the dominant radiative transitions of Υ 1 (2D), which are
B[Υ 1 (2D) → χ b0 (2P)γ] ≃ 25%, (56) B[Υ 1 (2D) → χ b1 (2P)γ] ≃ 18%, (57) B[Υ 1 (2D) → χ b0 (1P)γ] ≃ 15%, (58) B[Υ 1 (2D) → χ b1 (1P)γ] ≃ 7%.(59)
There may be hope for observing the missing Υ 1 (2D) state in the χ b0,1 (2P)γ and χ b0,1 (1P)γ channels.
η b2 (2D)
The main EM decay channels of η b2 (2D) are h b (2P)γ and h b (1P)γ. With the mass M η b2 (2D) = 10432 MeV predicted by us, the partial widths of the transitions η b2 (2D) → h b (1P, 2P)γ are calculated. The results compared with the other model predictions are listed in Tab. V. It is found that the predicted partial widths roughly agree with the potential model predictions [11][12][13]. Using the predicted total width of η b2 (2D) (Γ tot ≃ 25 keV) from [12], we predict that
B[η b2 (2D) → h b (1P)γ] ≃ 23%,(60)B[η b2 (2D) → h b (2P)γ] ≃ 62%.(61)
To determine the missing η b2 (2D) state in experiments, its transitions into the h b (1P, 2P)γ channels are worth observing.
H. Radiative transitions of 3P states
In the past several years, obvious progress has been achieved in the observations of the 3P states. In 2011, the AT-LAS Collaboration first discovered the χ b (3P) through its radiative transitions to Υ(1S , 2S ) with Υ(1S , 2S ) → µ + µ − at the LHC [59]. Only a few months after that, the χ b (3P) state was confirmed by the D0 Collaboration [60]. Recently, the LHCb Collaboration also carried out a precise measurement of the χ b (3P) state, identifying χ b (3P) as the χ b1 (3P) state [61,62]. The measured mass of χ b1 (3P) is M χ b1 (3P) ≃ 10516 MeV. In our calculations, the mass splittings are predicted to be
M χ b2 (3P) − M χ b1 (3P) ≃ 13 MeV, M χ b1 (3P) − M χ b0 (3P) ≃ 25 MeV, and M h b (3P) − M χ b1 (3P) ≃ 4 MeV.
Combining these predicted mass splittings with the measured mass of χ b1 (3P), we estimate the masses of χ b2 (3P), χ b0 (3P) and h b (3P), which are M χ b2 (3P) ≃ 10529 MeV, M χ b0 (3P) ≃ 10491 MeV, and M h b (3P) ≃ 10520 MeV, respectively.
1. χ b1 (3P)
The Υ(1S , 2S , 3S )γ are the main EM decay channels of χ b1 (3P). From Tab. IV, it is seen that our predicted partial widths for these channels are close to the recent predictions with the nonrelativistic constituent quark model [13], and the predictions with the previous SNR potential models [11]. Furthermore, taking the total width of χ b1 (3P), Γ tot ≃ 117 keV, predicted in Ref. [12] as an input, we estimate that
B[χ b1 (3P) → Υ(1S )γ] ≃ 5.4%, (62) B[χ b1 (3P) → Υ(2S )γ] ≃ 4.8%, (63) B[χ b1 (3P) → Υ(3S )γ] ≃ 8.8%.(64)
These large branching ratios may explain why χ b (3P) is discovered through its radiative transitions into Υ (1S , 2S ). Taking the masses of 2D waves calculated by us, we predict the partial widths for the transitions χ b1 (3P) → Υ 1,2 (2D)γ. Our results are listed in Tab. IV. From the table, it is found that our results are close to the potential model predictions [11,12]. Similarly, with the predicted total width χ b1 (3P) from [12], we estimate that
B[χ b1 (3P) → Υ 1 (2D)γ] ≃ 9.0 × 10 −3 , (65) B[χ b1 (3P) → Υ 2 (2D)γ] ≃ 8.0 × 10 −3 .(66)
The sizeable branching ratios of B[χ b1 (3P) → Υ 1,2 (2D)γ] indicate that one may discover the missing D-wave states Υ 1 (2D) and Υ 2 (2D) through the radiative transition chains χ b1 (3P) → Υ 1,2 (2D)γ → χ b1 (1P, 2P)γγ → Υ(1S , 2S )γγγ. We further estimate the branching ratios for these decay chains. The results are listed in Tab. VII. It is found that the important chains involving
Υ 1 (2D) are χ b1 (3P) → Υ 1 (2D)γ → χ b1 (2P, 1P)γγ → Υ(1S , 2S )γγγ [B ∼ O(10 −4 )]. While the important chains involving Υ 2 (2D) are χ b1 (3P) → Υ 2 (2D)γ → χ b1 (2P)γγ → Υ(2S )γγγ [B ≃ 4.6 × 10 −4 ], χ b1 (3P) → Υ 2 (2D)γ → χ b1 (1P)γγ → Υ(1S )γγγ [B ≃ 4.6 × 10 −4 ], and χ b1 (3P) → Υ 2 (2D)γ → χ b1 (2P)γγ → Υ(1S )γγγ [B ≃ 3.2 × 10 −4 ]. 2. χ b2 (3P)
With M χ b2 (3P) = 10529 MeV for the χ b2 (3P) state, we calculate its radiative decay properties. Our results are listed in Tab. IV. For comparison, the other model predictions are also listed in the same table. It is found that the radiative decay ratios of χ b2 (3P) into the 1D-wave states are negligibly small, while the partial widths for the transitions χ b2 (3P) → Υ(1S , 2S , 3S )γ and χ b2 (3P) → Υ 3 (2D)γ are sizeable. Most of our results are consistent with the other predictions. Taking the total width of χ b2 (3P), Γ tot ≃ 247 keV, predicted in Ref. [12] as an input, we estimate that
B[χ b2 (3P) → Υ(1S )γ] ≃ 3.3%, (67) B[χ b2 (3P) → Υ(2S )γ] ≃ 2.7%, (68) B[χ b2 (3P) → Υ(3S )γ] ≃ 4.4%.(69)
These fairly large branching ratios indicate the missing χ b2 (3P) state is most likely to be established via the radiative decays χ b2 (3P) → Υ(1S , 2S , 3S )γ. Furthermore, we find that the branching ratio
B[χ b2 (3P) → Υ 3 (2D)γ] ≃ 1.9%(70)
is sizeable. Thus, χ b2 (3P) might be a good source when looking for the missing Υ 3 (2D). According to our analysis, the important radiative decay chains involving
Υ 3 (2D) are χ b2 (3P) → Υ 3 (2D)γ → χ b2 (2P)γγ → Υ(1S , 2S )γγγ,
and their combined branching ratios can reach up to B ≃ 1.3 × 10 −3 .
χ b0 (3P)
With the predicted mass M χ b0 (3P) = 10491 MeV for the χ b0 (3P) state, we calculate its radiative decay properties. Our results are listed in Tab. IV. It is found that the partial radiative decay widths of χ b0 (3P) into the S -wave states Υ(1S , 2S , 3S ) are comparable to those of χ b1,2 (3P). In Ref. [12], the total width of χ b0 (3P) is predicted to be Γ tot ≃ 2.5 MeV, with which we estimate that
B[χ b0 (3P) → Υ(1S )γ] ≃ 7.5 × 10 −4 , (71) B[χ b0 (3P) → Υ(2S )γ] ≃ 1.0 × 10 −4 , (72) B[χ b0 (3P) → Υ(3S )γ] ≃ 3.2 × 10 −4 .(73)
These branching ratios are about an order of magnitude smaller than those of B[χ b1,2 (3P) → Υ(1S , 2S , 3S )γ], which may indicate that χ b0 (3P) is relatively difficult to observe in the Υ(1S , 2S , 3S )γ channels.
h b (3P)
For the singlet h b (3P) state, with the predicted mass M h b (3P) = 10520 MeV, we calculate the radiative decay properties. Our results are listed in Tab. IV. The EM decays of h b (3P) are dominated by the η b (3S )γ channel, while the partial widths into the η b (1S , 2S )γ and η b2 (2D)γ channels are sizeable as well. Our predicted partial decay widths into the S -wave states are the same order of those from various potential models [11][12][13] (see Tab. IV). Taking the predicted width of h b (3P), Γ tot ≃ 83 keV, from Ref. [12] as an input, we obtain
B[h b (3P) → η b (1S )γ] ≃ 12.9%,(74)B[h b (3P) → η b (2S )γ] ≃ 9.2%,(75)B[h b (3P) → η b (3S )γ] ≃ 17.0%.(76)
IV. SUMMARY
In the nonrelativistic screened potential quark model framework, we study the bottomonium spectrum. The radial Schrödinger equation is solved with the three-point difference central method, where the spin-dependent potentials are dealt with non-perturbatively. In our calculations, the corrections of the spin-dependent interactions to the wave functions are successfully included as well. It is found that the corrections of spin-dependent interactions to the wave functions of the Swave and 3 P 0,1 states are notably big. The bottomonium spectrum predicted within our approach is in a global agreement with the experimental data.
Moreover, using the obtained wave functions we study the EM transitions of nS (n ≤ 4), nP (n ≤ 3), and nD (n ≤ 2) bottomonium states with a nonrelativistic EM transition operator widely applied to meson photoproduction reactions, in which the effects of binding potential between quarks are considered, and the possible higher EM multipole contributions are included. It is found that (i) except for some M1 transitions, our predictions for the EM transitions are in good agreement with the experimental data. (ii) The corrections of the spin-dependent interactions are important to understand some EM transitions. For example, the EM transitions of Υ(3S ) → χ b1,2 (1P)γ, which were not well understood in previous studies, can be reasonably explained in the present work by considering the corrections of the spin-dependent interactions to the wave functions. (iii) Strong model dependencies exist in various model predictions of some transition widths, especially for the partial width ratios. To test the various model predictions more observations are expected to be carried out in forthcoming experiments.
Additionally, we discuss the observations of the missing bottomonium states by using radiative transitions. (i) We suggest our experimental colleagues observe the three-photon decay chains
χ b2 (2P) → Υ 3 (1D)γ → χ b2 (1P)γγ → Υ(1S )γγγ [B ∼ O(10 −3 )] and h b (2P) → η b2 (1D)γ → h b (1P)γγ → η b γγγ (B ≃ 1.4%),
where the missing Υ 3 (1D) and η b2 (1D) states are most likely to be observed. (ii) We also suggest our experimental colleagues observe the following three-photon decay chains: The LHC and Belle experiments have demonstrated the ability to observe and measure the properties of bottomonium mesons. In the near future, more missing bottomonium states are to be discovered and more decay channels will be measured in experiments. We expect that our theoretical predictions in this paper will be helpful for experimental exploration of the bottomonium mesons.
χ b1 (3P) → Υ 1 (2D)γ → χ b1 (2P, 1P)γγ → Υ(1S , 2S )γγγ [B ∼ O(10 −4 )], χ b1 (3P) → Υ 2 (2D)γ → χ b1 (2P)γγ → Υ(2S )γγγ [B ≃ 4.6 × 10 −4 ], χ b1 (3P) → Υ 2 (2D)γ → χ b1 (1P)γγ → Υ(1S )γγγ [B ≃ 4.6 × 10 −4 ], and χ b1 (3P) → Υ 2 (2D)γ → χ b1 (2P)γγ → Υ(1S )γγγ [B ≃ 3.2 × 10 −4 ],
Finally, to determine the binding energy E, we adopt the following method. As we know if E 0 is a trial value near the eigenvalue of the binding energy E, the asymptotic form of the numerical solution of the radial wave function u(r, E 0 ) at large r is given by the linear combination of the regular solution g(E 0 )e −k 0 r and irregular solution f (E 0 )e +k 0 r with k 2 0 = 2µE 0 . Thus, we can take the radial wave function u(r, E 0 ) at large r as [53] u(r,
E 0 ) = f (E 0 )e +k 0 r ,(80)
Similarly, for another trial value E 1 , we have
u(r, E 1 ) = f (E 1 )e +k 1 r ,(81)
with k 2 1 = 2µE 1 . If f (E) is an analytic function, we can expand f (E 1 ) as
f (E 1 ) = f (E 0 ) + f ′ (E 0 )(E 1 − E 0 ) + · · ·.(82)
If |E 1 − E 0 | is small enough, we can only keep the first two terms. Then, we have
f ′ (E 0 ) = f (E 1 ) − f (E 0 ) E 1 − E 0 = u(r, E 1 )e −k 1 r − u(r, E 0 )e −k 0 r E 1 − E 0 .
(83) Note that, if E 1 is just the eigenvalue of the binding energy E, f (E 1 ) should be zero. Thus, from Eq.(82) we have
E = E 0 − f (E 0 )/ f ′ (E 0 )(84)
In the numerical calculations, the recurrence method is used to calculate the eigenvalue E. Letting E 1 → E 0 , u(r, E 1 ) → u(r, E 0 ) and E → E 1 , then we calculate new u(r, E 1 ) and new E with Eqs. (78) and (79). The recurrence is stopped when |E − E 0 | ≤ ǫ, where ǫ stands for the accuracy that we need. [8], the predictions from the relativistic quark model [16], relativized quark model (GI model) [12], nonrelativistic constituent quark model (NR model) [13], and the previous screened potential model (SNR model) [11] are listed in the table as well. SNR 0 and SNR 1 stand for the results calculated by the zeroth-order wave functions and the first-order relativistically corrected wave functions with the screened potential model [11], respectively. V: Partial widths of the radiative transitions for the 1P-, 1D-and 2D-wave bottomonium states. For comparison, the predictions from the relativistic quark model [16], relativized quark model (GI model) [12], nonrelativistic constituent quark model (NR model) [13], and previous screened potential model (SNR model) [11] are listed in the table as well. SNR 0 and SNR 1 stand for the results calculated by the zeroth-order wave functions and the first-order relativistically corrected wave functions with the screened potential model [11], respectively. VI: Partial widths of the radiative transitions for the higher 4S states. For comparison, the predictions from the relativized quark model (GI model) [12], nonrelativistic constituent quark model (NR model) [13], and the previous screened potential model (SNR model) [11] are listed in the table as well. SNR 0 and SNR 1 stand for the results calculated by the zeroth-order wave functions and the first-order relativistically corrected wave functions with the screened potential model [11], respectively. Decay chain B 1 B 2 B 3 B 3 3 P 1 → 2 3 D 1 → 2 3 P 0 → Υ(2S ) 9.0 × 10 −3 25% 5.8 × 10 −3 1.3 × 10 −5
FIG. 1 :
1(Color online) Predicted radial probability density |u(r)| 2 for S -, P-and D-wave bottomonium states.
) is well established in experiments. Its mass and width are M Υ(3S ) = 10355.2 ± 0.5 MeV and Γ = 20.32 ± 1.85 keV, respectively. The EM transitions Υ(3S ) → χ bJ (1P, 2P)γ and Υ(3S ) → η b (1S , 2S )γ have been observed in experiments. We calculate the partial widths and compare them with the data in Tab. IV.
0 state, η b (3S ), is still missing. The predicted mass splitting between 3 3 S 1 and 3 1 S 0 is about 17 MeV. Combining it with the measured mass of 3 3 S 1 , we predict that the mass of η b (3S ) might be M η b (3S ) ≃ 10338 MeV. Using this predicted mass, we study the E1 transitions η b (3S ) → h b (1P, 2P)γ and M1 transitions η b (3S ) → Υ(1S , 2S )γ. Our results have been listed in
To look for the missing h b (3P) state, the transitions h b (3P) → η b (1S , 2S )γ are worth observing.I. Radiative transitions of 4S states Υ(4S ) is established in experiments. Its mass and width are M Υ(4S ) ≃ 10579 MeV and Γ ≃ 20.5 MeV, respectively. However, the η b (4S ) is still missing. We predict their radiative properties. The results compared with the other predictions are listed in Tab. IV. From the table, it is found that obvious model dependencies exist in these predictions. Our calculations give relatively large decay rates for the Υ(4S ) → χ bJ (3P)γ transitions. Thus, the missing χ bJ (3P) states might be produced by the radiative decay chains of Υ(4S ) → χ bJ (3P)γ → Υ(1S , 2S , 3S )γγ. Combining the predicted branching ratios of χ bJ (3P) → Υ(1S , 2S , 3S )γ and Υ(4S ) → χ bJ (3P)γ, we further estimate the combined branching ratios, which have been listed in Tab. VI. From the table, one can see that the most prominent two-photon decay chains are Υ(4S ) → χ b1 (3P)γ → Υ(1S , 2S , 3S )γγ [B ∼ O(10 −5 )], followed by Υ(4S ) → χ b2 (3P)γ → Υ(1S , 2S , 3S )γγ [B ∼ O(10 −6 )]. There are few chances for χ b0 (3P) to be observed in the radiative decay chains of Υ(4S ).
where the missing Υ 1 (2D) and Υ 2 (2D) states might have chances to be observed. (iii) The missing χ bJ (3P) states might be produced via the radiative transitions of Υ(4S ). The most prominent decay chains are Υ(4S ) → χ b1 (3P)γ → Υ(1S , 2S , 3S )γγ [B ∼ O(10 −5 )], followed by Υ(4S ) → χ b2 (3P)γ → Υ(1S , 2S , 3S )γγ [B ∼ O(10 −6 )].
VII: Three-photon decay chains of 3 3 P 2 . The combined branching fractions of the chain are defined byB = B 1 × B 2 × B 3 with B 1 = B[3 3 P 1 → 2 3 D J γ], B 2 = B[2 3 D J → m 3 P J γ], and B 3 = B[m 3 P J → Υ(1S , 2S )γ].
TABLE I :
IPredicted masses (MeV) of bottomonium states. For com-
parison, the measured masses (MeV) from the PDG
P 2 χ b2 (2P) 2 ++ 10269 10269 10261 10246 10264 2 3 P 1 χ b1 (2P) 1 ++ 10255 10251 10246 10236 10249 2 3 P 0 χ b0 (2P) 0 ++ 10233 10226 10226 10221 10220 2 1 P 1 h b (2P) 1 +− 10260 10256 10250 10240 10254 3 3 P 2 χ b2 (3P) 2 ++11011 11097
10988
1 3 P 2 χ b2 (1P) 2 ++ 9912
9918
9897 9886 9921
1 3 P 1 χ b1 (1P) 1 ++ 9893
9897
9876 9874 9903
1 3 P 0 χ b0 (1P) 0 ++ 9859
9865
9847 9855 9864
1 1 P 1 h b (1P) 1 +− 9899
9903
9882 9879 9909
2 3 10540 10550 10521 10528
3 3 P 1 χ b1 (3P) 1 ++ 10516 10524 10538 10513 10515
3 3 P 0 χ b0 (3P) 0 ++
10502 10522 10500 10490
3 1 P 1 h b (3P) 1 +−
10529 10541 10516 10519
1 3 D 3 Υ 3 (1D) 3 −−
10156 10155 10127 10157
1 3 D 2 Υ 2 (1D) 2 −− 10164 10151 10147 10122 10153
1 3 D 1 Υ 1 (1D) 1 −−
10145 10138 10117 10146
1 1 D 2 η b2 (1D) 2 −+
10152 10148 10123 10153
2 3 D 3 Υ 3 (2D) 3 −−
10442 10455 10422 10436
2 3 D 2 Υ 2 (2D) 2 −−
10438 10449 10418 10432
2 3 D 1 Υ 1 (2D) 1 −−
10432 10441 10414 10425
2 1 D 2 η b2 (2D) 2 −+
10439 10450 10419 10432
TABLE II :
IIHyperfine and fine splittings in units of MeV for bottomonium in our calculation. The experimental data are taken from the PDG
TABLE
TABLE IV :
IVPartial widths of the radiative transitions for the nS -and nP-wave (n = 2, 3) bottomonium states. For comparison, the measured values from the PDG
TABLE
Initial meson Final mesonE γ (MeV) Γ E1 (keV) Γ EM (keV) state state Ref.[16] SNR 0,1 [11] GI [12] ours Ref.[16] SNR 0 [11] SNR 1 [11] GI [12] NR [13] Oursχ b2 (1 3 P 2 )
Υ(1 3 S 1 )
442
442
442 442 40.2
38.2
32.6
32.8
39.15
31.8
χ b1 (1 3 P 1 )
422
423
424 424 36.6
33.6
30.0
29.5
35.66
31.9
χ b0 (1 3 P 0 )
391
391
391 391 29.9
26.6
24.3
23.8
28.07
27.5
h b (1 1 P 1 )
η b (1 1 S 0 )
480
501
488 488 52.6
55.8
36.3
35.7
43.66
35.8
Υ(1 3 D 3 )
χ b2 (1 3 P 2 )
244
240
257 253 24.6
26.4
24.5
24.3
24.74
32.1
χ b1 (1 3 P 1 )
271
0
1.1 × 10 −2
χ b0 (1 3 P 0 )
304
0
9.2 × 10 −5
Υ(1 3 D 2 )
χ b2 (1 3 P 2 )
241
236
249 249 6.35
6.29
5.87
5.6
6.23
7.23
χ b1 (1 3 P 1 )
262
255
267 267 23.3
23.8
19.8
19.2
21.95
21.8
χ b0 (1 3 P 0 )
300
0
0.83 × 10 −2
Υ(1 3 D 1 )
χ b2 (1 3 P 2 )
235
230
240 242 0.69
0.65
0.61
0.56
0.65
1.02
χ b1 (1 3 P 1 )
256
249
259 261 12.7
12.3
10.3
9.7
12.29
13.3
χ b0 (1 3 P 0 )
280
282
292 294 23.4
23.6
16.7
16.5
20.98
19.8
η b2 (1 1 D 2 )
h b (1 1 P 1 )
254
246
263 262 28.4
42.3
36.5
24.9
17.23
30.3
Υ(2 3 D 3 )
χ b2 (1 3 P 2 )
517
529 511
4.01
3.73
2.6
3.80
5.22
χ b1 (1 3 P 1 )
535
0
0.16
χ b0 (1 3 P 0 )
567
0
0.08
χ b2 (2 3 P 2 )
172
184 166
18.0
15.9
16.4
10.70
17.0
χ b1 (2 3 P 1 )
185
0
0.34 × 10 −2
χ b0 (2 3 P 0 )
207
0
0.66 × 10 −3
Υ(2 3 D 2 )
χ b2 (1 3 P 2 )
513
523 507
0.98
0.68
0.4
0.80
1.11
χ b1 (1 3 P 1 )
531
541 525
3.26
4.46
2.6
3.43
4.00
χ b0 (1 3 P 0 )
555
0
0.89 × 10 −2
χ b2 (2 3 P 2 )
168
178 162
4.17
3.82
3.8
2.55
3.75
χ b1 (2 3 P 1 )
181
192 175
15.7
12.1
12.7
9.10
11.4
χ b0 (2 3 P 0 )
197
0
1.7 × 10 −3
Υ(2 3 D 1 )
χ b2 (1 3 P 2 )
507
516 500
0.11
0.05
0.9
0.061
0.44
χ b1 (1 3 P 1 )
525
534 518
1.76
1.87
2.9
1.58
2.17
χ b0 (1 3 P 0 )
557
566 551
2.79
6.20
1.6
3.52
5.56
χ b2 (2 3 P 2 )
162
171 155
0.42
0.39
0.4
0.24
0.47
χ b1 (2 3 P 1 )
175
184 167
7.87
6.35
6.5
4.84
6.74
χ b0 (2 3 P 0 )
198
206 190
15.1
9.49
10.6
8.35
9.58
η b2 (2 1 D 2 )
h b (1 1 P 1 )
522
536 519
6.19
7.30
3.0
4.15
5.66
h b (2 1 P 1 )
181
188 171
31.3
25.4
16.5
11.66
15.6
TABLE
state state SNR[11] GI[12] Ours SNR 0 /SNR 1[11] NR[13] GI[12] Ours Υ(4S ) χ b2 (1P)Initial Final
E γ (MeV)
Γ E1 (keV)
Γ EM (keV)
646
646
0.14/0.56
0.012
0.66
χ b1 (1P)
664
664
0.10/0.20
0.047
0.017
χ b0 (1P)
695
695
0.04/0.001
0.059
0.14
χ b2 (2P)
306
305
0.14/0.56
0.11
0.34
χ b1 (2P)
319
319
0.09/0.001
0.18
0.024
χ b0 (2P)
341
340
0.04/0.21
0.17
0.44
χ b2 (3P)
40
51
50
0.55/0.52
1.45
0.82
4.4
χ b1 (3P)
55
63
64
0.91/0.74
1.17
0.84
4.9
χ b0 (3P)
77
79
89
0.82/0.54
0.61
0.48
3.4
η b (4S ) h c (1P)
669
663
0.90/5.64
1.98
h c (2P)
334
319
0.95/2.16
1.56
h c (3P)
67
48
65
1.24/5.68
1.24
17.4
TABLE
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| []
|
[
"A Foreground Inference Network for Video Surveillance Using Multi-View Receptive Field",
"A Foreground Inference Network for Video Surveillance Using Multi-View Receptive Field"
]
| [
"Thangarajah Akilan [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of Windsor\nWindsorCanada\n"
]
| [
"Department of Electrical and Computer Engineering\nUniversity of Windsor\nWindsorCanada"
]
| []
| Foreground (FG) pixel labeling plays a vital role in video surveillance. Recent engineering solutions have attempted to exploit the efficacy of deep learning (DL) models initially targeted for image classification to deal with FG pixel labeling. One major drawback of such strategy is the lacking delineation of visual objects when training samples are limited. To grapple with this issue, we introduce a multi-view receptive field fully convolutional neural network (MV-FCN) that harness recent seminal ideas, such as, fully convolutional structure, inception modules, and residual networking. Therefrom, we implement a system in an encoder-decoder fashion that subsumes a core and two complementary feature flow paths. The model exploits inception modules at early and late stages with three different sizes of receptive fields to capture invariance at various scales. The features learned in the encoding phase are fused with appropriate feature maps in the decoding phase through residual connections for achieving enhanced spatial representation. These multi-view receptive fields and residual feature connections are expected to yield highly generalized features for an accurate pixel-wise FG region identification. It is, then, trained with database specific exemplary segmentations to predict desired FG objects.The comparative experimental results on eleven benchmark datasets validate that the proposed model achieves very competitive performance with the prior-and state-of-the-art algorithms. We also report that how well a transfer learning approach can be useful to enhance the performance of our proposed MV-FCN. | null | [
"https://arxiv.org/pdf/1801.06593v1.pdf"
]
| 25,504,593 | 1801.06593 | 1e3519b0820ff5453633a96d8f07cef9b0e2d43a |
A Foreground Inference Network for Video Surveillance Using Multi-View Receptive Field
Thangarajah Akilan [email protected]
Department of Electrical and Computer Engineering
University of Windsor
WindsorCanada
A Foreground Inference Network for Video Surveillance Using Multi-View Receptive Field
Deep learningForeground-background clusteringVideo-surveillance
Foreground (FG) pixel labeling plays a vital role in video surveillance. Recent engineering solutions have attempted to exploit the efficacy of deep learning (DL) models initially targeted for image classification to deal with FG pixel labeling. One major drawback of such strategy is the lacking delineation of visual objects when training samples are limited. To grapple with this issue, we introduce a multi-view receptive field fully convolutional neural network (MV-FCN) that harness recent seminal ideas, such as, fully convolutional structure, inception modules, and residual networking. Therefrom, we implement a system in an encoder-decoder fashion that subsumes a core and two complementary feature flow paths. The model exploits inception modules at early and late stages with three different sizes of receptive fields to capture invariance at various scales. The features learned in the encoding phase are fused with appropriate feature maps in the decoding phase through residual connections for achieving enhanced spatial representation. These multi-view receptive fields and residual feature connections are expected to yield highly generalized features for an accurate pixel-wise FG region identification. It is, then, trained with database specific exemplary segmentations to predict desired FG objects.The comparative experimental results on eleven benchmark datasets validate that the proposed model achieves very competitive performance with the prior-and state-of-the-art algorithms. We also report that how well a transfer learning approach can be useful to enhance the performance of our proposed MV-FCN.
I. INTRODUCTION
Foreground region labeling is a crucial task in video surveillance used to detect moving objects in challenging conditions. It requires robust algorithms to tackle with varying environmental factors, like illumination changes and dynamic backgrounds [1]. It is also an integral part of various machinevision problems, such as object segmentation [2], [3], [4], image quality assessment [5], object discovery [6], visual tracking [7], and human-robot/machine interaction [8]. The primary objective of FG labeling is to place a tight mask on the most probable regions, in which moving objects mostly humans and vehicles can be identified. Such mask is, in many ways, very informative than a simple detection with bounding box as it allows close localization of objects, which is essential for retrieval, recognition, autonomous driving, and object preserved data compression for cloud-based systems [9]. Besides, obtaining pixel-level foreground labels is also an important step towards general machine understanding of scenes. An example application setup is drawn in Fig. 1 to conceive the importance of this work.
Much attention has been paid to automate this process; and thus, there has been myriad of algorithms proposed that mainly include statistical approaches. For instance, Gaussian mixture models (GMM) [10], clustering algorithms, like conditional random field (CRF) [11] and graph-cut [12]. However, some researchers focus on neural network (NN) models, like Self-Organizing Maps (SOM) [13] for this task. Here, a reasonable approach is to formulate it as a structured output problem that can be solved by training a system in an image-to-image fashion. This approach has been adopted in recent years' deep convolutional neural networks (DCNN/ deep convnets) for FG region labeling and gained wider acceptance.
One of the main challenges in DCNN-based methods is that dealing with objects of very different scales and the dithering effect at bordering pixels of FG objects. To deal with these challenges, we propose a new model inspired by Google introduced inception module [14] that performs convolution of multiple filters with different scales on the same input by simulating human cognitive processes in perceiving multi-scale information and Microsoft introduced ResNet [15] that acts as lost feature recovery mechanism. To enhance the knowledge of proposed network, we exploit intra-domain transfer learning that boosts the correct FG region prediction. Using this methodology is also inspired by human-like reasoning, in which the network learns new task precisely and more quickly by applying already learned knowledge, i.e., the weights and biases [16].
On a historical perspective, the theories of neural networks (NN) for the visual-based problems, arguably come from a pioneer computer vision system, the Mark I Perceptron machine by Rosenblatt in late 1950s [17]. Presumably, concurrent with that Hubel and Wiesel's [18] discovery of neural connectivity pattern of cat's visual cortex, inspired Fukushima to introduce an NN referred to as Neocognitron [19], which is invariance to image translations. Later, the Neocognitron was devised with backpropagation mechanism that structures modern-day deep convnets. That is a multi-layered NN containing layers of several convolution, rectification, sub-sampling, and normalization operations. In which, the low-level convolutional layers operate as Gabor filters and color blob detectors [20] that extract the information, such as, edges and/or textures, Subsequently, the deep CNNs have been effectively exploited for semantic segmentation/labeling [22], instance partitioning [9], and medical image sectioning [23], [24]. Thereupon we are interested in implementing a DCNN for the problem of FG object/region identification. Thus, the key insight of this paper is to propose a deep convnet that enhances feature learning for a better FG-region localization based on novel strategies introduced in the recent literature. We formulate this problem as a binary classification task with a DCNN, where the network is trained end-to-end with exemplary FG-BG segmentations to predict the most probable FG region for a given input frame. The proposed model is a multi-view receptive field fully convolutional neural network (MV-FCN) having two architectural phases: an encoder and a decoder that mainly combines inception modules and residual connections. Besides, the network is fully convolutional without any max pooling and fully connected layers.
In the following, we review the literature in Section II that covers sufficient CNN architectural information to understand the proposed model described in Section III. Sections IV presents details of the experimental setup and results along with discussion on compared FG detection algorithms. Finally, Section V concludes the paper with future directions.
II. LITERATURE REVIEW
DCNNs have shown state-of-the-art performance over traditional methods, like GMM and graph-cut for the problem of FG detection/segmentation/localization. Here, the Fully convolutional networks FCN [22] is a pioneer model that reinterprets the standard visual classification convnets as fully convolutional computation. It has been well exploited in many present-day applications, like sliding window-based detection [26], [27], semantic segmentation [28], [29] and image retrieval/restoration [30], and spatial model for 3D face pose estimation [31]. It is trained end-to-end and pixels-to-pixels [22] using the whole image as input at a time. This model enhances pixel predictions at the last layer trough feature-level augmentation with a skip architecture that fuses the feature hierarchy to combine deep, coarse, semantic information and shallow, fine, appearance information from selected midlayers. In contrast, our model does the coarse-level feature fusion in the inference path, like ResNet [24].
In 2015, researchers from Microsoft introduced a CNN architecture with residual connections termed as ResNet that won the 1st place in the ILSVRC image classification competition with 3.57% top-5 error. This network was built upon the philosophy of increasing depth of the network instead of widening, through residual connections to provide a better data representation. The ResNet architecture negates the vanishing gradient issue raises in deep networks by carrying important information in the previous layer to the next layer. Although such connection seems like an addition to the traditional CNN approach, it alleviates the training of the network and reduces number of parameters [15]. An illustration for the ResNet connection is given in Fig. 2 (b), where X is input feature, H(X) is any desired mapping, and F (X) is a residual mapping. In [15], the residual feature fusion operation H(X) = F (X) + X is performed by a shortcut connection and element-wise addition. Contrastingly, our model stacks the futures depthwise as H(X) = F (X) X, like shown in Fig. 2 (c), where denotes feature-map concatenation. This favors to have less number of filters in convolutional layers at the same time to carry forward earlier layer's features intact.
The inception module was a micro-architecture first introduced in [25] by Szegedy et al., following the success of ResNet [14], [32]. The module acts as computation of multiple filters with different scales on the same input as in Fig. 2 (d). Also, it performs average pooling at the same time. Finally, all the outcomes are aggregated along the channel dimension that to take advantage of multi-level feature representation, resulting in a higher discriminatory encoding. We simplify the inception connection with three different filters using stride of 1 (S1) convolution on the input image as shown in Fig. 2 (e). We also extract features through down-sampling operation on the three feature maps generated by the kernels and stack the features channel-wise at matching mid-layers through residual connections as depicted in Fig. 3.
Researchers in [24] extend the FCN [22] to function like an encoder-decoder CNN named U-net for an application of biomedical cell segmentation. In that, the activation maps after convolution at the encoding stage are concatenated with the activation maps at the decoding stage. Such structuring allows the network to exploit the original context information to supplement the features after up-sampling at the higher layers. In other words, it is a remedy for the lost spatial resolution due to max-pooling and striding at consecutive layers. The advantage of this model is that it works elegantly with less training samples and yields precise cell partitioning for 2D samples of Magnetic resonance imaging (MRI). Milletari et al. [33] extend this model to be trained end-to-end on MRI volumes to predict segmentation for the whole volume at once. They name the network V-net as it does volumetric medical image segmentation. The major difference of V-net from Unet is the volumetric convolutions as the input is a slice-wise volume (3D patches). Our model, on the other hand, has the following variations from the U-net: the traditionally deployed max-pooling operation in the contraction path achieves invariance feature but has a toll on localization accuracy [34]. Thus, we perform subsampling process by using 2D convolution with a stride rate of 2, a kernel size of 3 × 3, and zero padding. Hence, our model enhances feature learning through early and late stages micro inception modules. Figure 3 abstracts away details of the proposed multi-view receptive field fully convolutional network (MV-FCN) through a schematic. The MV-FCN integrates two complementary feature flows (CFF) and a pivotal feature flow (PFF). The PFF is essentially an encoder-decoder CNN while CFF1 and CFF2 complement its learning ability. The PFF only uses convolution kernels size of 3×3, while CFF1 and CFF2 utilize filters size of 5 × 5 and 9 × 9 respectively in their first conv layers. However, after their first sub-sampling convolutional operations they use filter size of 3 × 3 in their subsequent layers, so their output activation maps match a middle layer in the PFF for a successful feature-level augmentation. Thus, the features learned in the complementary layers are merged with appropriate intermediate feature maps in the PFF through residual connections Using such heterogeneous convolutional kernels captures information available from different scales and provides both local and global context [35] and the fusion of feature maps from encoding layers that hold high-frequency detail resulting to sharper foreground boundaries.
III. PROPOSED MV-FCN ARCHITECTURE
In the encoding phase of PFF, four convolutional layers are networked sequentially after the very first conv layer that generates 16 channels with spatial dimension same as the input. Each of the four conv layers performs spatial subsampling by using a kernel size 3 × 3 with stride of 2 such that the encoding process outputs activation map with a dimension of 15 × 20 × 96. The decoding phase, on the other hand, employs four transpose convolutional layers interspace with residual feature concatenations and regular conv layers. Consequently, the decoding stage ends up with an inception module (layer 28 in Table I) that merges the first stage activations from PFF, CFF1, and CFF2 with the final stage decoding activations resulting to a feature map of 240 × 320 × 112.
The extracted features from various conv layers in the encoding path are also combined with spatially matching upsampled feature maps in the decoding path systematically. As stated earlier in section II, this strategy is an elegant solution for the lost of spatial resolution due to series of subsampling and convolutional operations carried out over the encoding process [24]. Hence, all the convolutions are immediately followed by ReLU activation functions, except the transpose convolution (it is generally referred as deconvolution) and the final layer. Top classification layers consist of a batchnormalization, conv with 128 channels followed by drop out of 0.3, and finally a single channel output conv with Sigmoid activation function. Table I summarizes the network detail, where conv2D and conv2DT denote 2D convolution and its transpose, respectively. The integers in the parentheses in layer type refer the kernel size and stride rate in the order while the N one in output shape refers the mini-batch size. In total, the proposed model takes 494,337 trainable parameters.
In summary, the MV-FCN does not employ max pooling or hidden fully connected (FC) layers, but subsumes convolu- tional (conv), transpose convolutional (CTrans), and symmetric expanding paths with inception and residual connections to capture contextual information for an accurate FG inferencing. The network is capable of taking any spatial dimensions of input images and resize them into 240 × 320 by using nearestneighbor scaling algorithm to match with the input layer dimension. The convolutional layers use stride rate of 1 in all directions, except the sub-sampling layers that perform convolution with a stride rate of k − 1, where K is the kernel size. At this juncture, it is vital to discuss about the intricacies of the core components and the functions utilized in the proposed network.
A. Convolutional Layer
The convolutional layer is the core unit of modern deep learning architectures that is determined by its kernel weights that are updated during training via backpropagation. All the filter weights are fixed like a system memory; some literature refer it as anchor vectors since they serve as reference visual patterns in the testing phase. Output feature map C w.r.t. a kernel ω, its associated bias b, and an input image/patch x the convolutional operation is performed as
C(m, n) = b + K−1 k=0 K−1 l=0 ω(k, l) * x(m + k, n + l),(1)
where * , K, {m, n}, and {k, l} represent the convolutional operation, size of the kernel, first coordinate or origin of the image/patch, and element index of the kernel respectively. Hence, feature map dimension of the conv layer is given by (I s − K s + 2 × P )/S + 1, where I s , K s , P , and S denotes size of input image/path, filter size, number of zero-padded pixels, and stride rate respectively.
B. Transpose Convolution
Transpose convolutional layers perform up-sampling, i.e., the transpose/gradient of 2D convolution such that its output spatial dimension becomes twice as the input and dense activation map, as illustrated in Fig. 4 without losing the connectivity pattern. Although it looks like an image resizing process, it has trainable parameters for the up-sampling stage, and these parametric quantities are updated during training. It is achieved by inserting zeros between consecutive neurons in the input receptive field, then sliding the convolutional kernel with unit strides [36]. To elaborate it, if a desired convolution is governed by kernel size of K, stride rate of S, zero padding size of P , and its output has size of i then the associated transpose convolution can be computed with such a kernel K = K, stride S = 1, padding P = K − P − 1,ĩ , and α, whereĩ is the size of the diluted input obtained by imputing S − 1 zeros between each input neuron, and α = ((i + 2P − K) mod S) represents the number of zeros inserted to the top and right edges of the input that results an output feature size of:
O = S(i − 1) + α + K − 2P.
(2) Fig. 4: Illustration of convolution and its transpose operations: A 2D conv with K = 3, S = 2, and P = 1, and its corresponding transpose conv with K = K, S = 1, and P = K − P − 1 [37].
The modern deep neural networks implemented for imageto-image learning use multiple layers of transpose conv for generating images or feature maps from a series of lower resolution descriptions [37]. This idea has a long history in signal processing domain, originally developed for the efficient computation of the undecimated wavelet transform (UWT) known as 'algorithme a trous' [38]. More detail and computational information on transpose convolutional operation can be referred from [37] and [36].
C. Activation Functions
The adaptation of activation functions in neural networks (NN) can be referenced to the work of McCulloch and Pitts in late 1943 [39], where the activation function rectify the input to either 1 or −1 if its value is positive or negative, respectively.
ReLUs are nonlinear activations generally used after convolutional operations. It can be formally defined as in (3) when taken a case where there are K number of anchor vectors, denoted by w k ∈ R N , k = 1, 2, . . . , K. For a given input x, the correlations with a k and k = 1, 2, . . . , K, defines a nonlinear rectification to an output y = (y 1 , . . . , y K ) T , where
y k (x, a k ) = max(0, a T k x) ≡ ReLU(a T k x),(3)
i.e., it clips negative values to zero while keeping positive ones intact. The benefit of ReLU is sparsity, overcoming vanishing gradient issue, and efficient computation than other activations.
Sigmoid function, on the other hand, has output in the range [0, 1] for an input x and it is defined by
f (x) = 1 1 + exp(−x)
.
Therefore, it is very appropriate for binary classification tasks, like in this work and linear regression problems. A thorough exposition of the purpose of activation functions in NN with graphical examples can be found in [40].
D. Batch normalization
The batch normalization (layer 29 in Table I) before the final classification layer has multifaceted benefits: (i) reducing internal covariate shift -During training, there is a change in the distribution of activation maps as network parameters are being tuned. Such condition challenges the learning, but the BN alleviate pressure by maintaining the mean and standard deviation of the activation close to 0 and 1, respectively. (ii) Effect of regularization -Since the batch of examples given in the training are normalized, it increases the generalization of the model. It is also claimed that BN allows to reduce the strength of dropout. (iii) Counterbalancing vanishing or exploding gradients -When the BN is located prior to nonlinearity, it avoids an undesirable situation, where the training saturates areas of non-linearities, solving the issues of vanishing exploding gradients.
Mathematically it can be defined as follows. Let the output of a layer X ∈ R N,D , where N is the number of samples available in the mini-batch and D is the number of hidden neurons, then normalized matrixX is given as in (5) [41].
X = X − µ B σ 2 B + ,(5)
where µ B , σ 2 B , and refer to the mean and variance of the mini-batch, and a small value of 0.001 to prevent division by zero, respectively. Then the layer maintains its representational strength by testing the identity transform as:
y = γX + β,(6)
where, β and γ are trainable parameters that are initialized with β = 0 and γ = 1, in this work. Note that, when β = µ B and γ = σ 2 B + Eqn. (6) returns the previous layer's activation map.
E. Training strategy
Exclusive sets: We target the widely used benchmark database the change detection 2014 [42]. Table II briefs the properties of the datasets. Since the database has less number of annotated ground truths, where both the FG and BG are presented in the same frame, we employ data augmentation by applying random transformations with rotation within 10 degrees, translation vertically and horizontally with a fraction of 0.1 from the total height and width, and zooming in range of 0.1 inside image samples. These data augmentations are done on training images and the corresponding ground truths during training. Naturally, this allows the network to learn invariance to such transformations, without a need to see these variant samples in the annotated benchmark datasets. To form exclusive sets of training and test data, the available samples are divided in sequence order, whereby training set takes 70% while test set takes 30% of the total number of samples that have ground truths with FG and BG information in a particular dataset. This way of data splitting is more appropriate rather than a random selection since the images in the datasets are frames from video sequences. Because, a random choice of samples may pick a f rame t for training set while picking a temporally closest frame, like f rame t+1 or f rame t−1 for test set. There can be many such instances in random selection resulting in mere exclusiveness of training and test sets. Figure 5 demonstrates the data split used in this work, in which n is the total number of samples that has ground truths in the sequence and k = n × 0.7 that is the dividing point (frame no.) for the ordered split.
Optimizer: The MV-FCN is trained by using Adamoptimizer that minimizes binary cross-entropy loss defined by (7), where optimizer takes a base learning rate of 0.0002 with a learning rate scheduler that reduces the learning rate by factor of 0.8 over the training.
E = −1 n N n=1
[p n logp n + (1 − p n ) log(1 −p n )], (7) where it takes two inputs; first one is the output from the final layer of the network (layer 32 in Table I) with dimension of N × C × H × W , which maps the FG pixel probabilitieŝ p n = σ(x n ) ∈ [0, 1] using Sigmoid non-linearity function σ(.) defined earlier in Eqn. 4. And the second one is target p n ∈ [0, 1] with the same dimension as the first one, where N, C, H, and W represent the batch size, the number of channels, hight, and width respectively of the inputs. In this case, p n is the ground truth segmentation images whose pixel values are normalized. The network is trained on each video sequence separately. [43] and [44].
Training Environment: Python with Keras libraries (Tensorflow backend) is used as a software platform for the implementation of the model. The network is then mainly trained on a GeForce GTX 1060-6 GB GPU with Intel(R) Core(TM) i7-4770 CPU @ 3.40 GHz and 32 GB memory (RAM). In average the training takes about 2 hours on the GPU for each dataset when batch size is 8 and maximum of 30 epochs. The testing is purely carried out on CPU and it takes about 0.445 second per sample in average.
F. Binary Foreground Mask
It is also crucial to create a binary mask that localizes the interested FG region in a given frame. We apply a threshold to the score-map generated by the trained MV-FCN at frame-level to form a FG mask like shown in Fig. 6, where the threshold τ is a dataset-specific global parameter set empirically in the range [0.05, 0.75]. Then to clean noisy artifacts, we post process the thresholded binary image through neighborhood pixel connectivity that removes regions with less than 50 pixels. In another approach, We employ the Otsu's clusteringbased model to choose appropriate threshold automatically, since the score-map is a representation of bi-modal image. Otsu's algorithm iteratively finds a threshold τ that lies in between two peaks of the intensity histogram such that the intra-class variances of FG and BG classes are minimum. There, the weighted sum of within-class variances is defined as:
σ 2 ρ (τ ) = ρ 0 (τ )σ 2 0 (τ ) + ρ 1 (τ )σ 2 1 (τ ),(8)
where the weights ρ 0 and ρ 1 are the probabilities of BG and FG classes clustered by a threshold τ , and the variances of these two classes are σ 2 0 and σ 2 1 respectively. An explicit derivation of the method can be found in [45]. Note that the binarization process is not part of the MV-FCN training procedure, but exclusive for testing stage as the numerical analysis is made on the binary masks.
IV. EXPERIMENTAL SETUP, RESULTS, AND DISCUSSION
To provide a better understanding of the performance, we select eleven various sequences from the change detection database [42] that consists of diversified change and motion, including benchmarks of baseline, dynamic background, camera jitter, shadow, videos shot with PTZ camera, and low framerate. A succinct description of the datasets is given in Table II. Hence, the general nature of the datasets as follows. The baseline benchmark represents a mixture of mild challenges, like subtle background motion, isolated shadows, swaying tree branches, and natural illumination changes. The dynamic background category includes scenes with strong (parasitic) BG motion: boats and canoes on shimmering water, or a man walking on a shore of a shimmering water body. The camera jitter datasets contain outdoor videos captured by vibrating cameras due to high wind and unstable mount. The jitter magnitude varies from one video to another. The shadow category comprises indoor video exhibiting strong as well as faint shadows. Here, some shadows are cast by moving objects. Lastly, in PTZ camera recordings, adjustments in camera strongly changes the backgrounds of a recorded video. Such conditions break the assumption of traditional BG modeling algorithms that the recording devices are relatively static, or move slowly, and thus it challenges the most algorithms.
A. Qualitative Analysis
To limit the number of pages of this report, three sample results 1 from a selected video sequence per category from Table II are shown in Fig. 7 -Fig. 12. Hence, The impact of intraclass transfer learning is visualized using one sample from the Office dataset in Fig. 13. Such transfer learning technique allows the network to produce stronger discrimination of FG regions from BG as the distribution of probability falls around two distinct peaks, generally with the intensity values of 0 (dark as BG) and 255 (bright as FG). Visual results of proposed MV-FCN appear close to the ground-truth references; however, it has to be quantitatively analyzed for further validation. The following subsection IV-B provides the numerical analysis in terms of Figure-of-Merit (FoM).
B. Quantitative Analysis: FoM
The goal of FG inferencing is to label the pixels of a given video frame as being part of an expected FG object or not (i.e., BG). In such problem domain, the standard performance measure used is Figure-of-merit or FoM in short. The FoM measures the similarity between the predicted FG region and the ground-truth for a concerned FG object present in the image, and is defined as a weighted harmonic mean measure of recall and precision, i.e., a region of intersection divided by the union of predicted and actual FG regions. It is also referred as intersection-over-union (IoU) as in (9).
F oM = 2 × (P recision × Recall) P recision + Recall 2 ,(9)
where recall is the detection rate defined by T P/(T P + F N ) and precision is the percentage of correct prediction compared to the total number of detections as positives, given by T P/(T P + F P ), where T P, F N , and F P refer true positive, false negative, and false positive respectively. For a given output X from the proposed MV-FCN, i.e., the probabilities over a set of pixels V = {1, 2, · · · , N } in the input image, and Y ∈ {0, 1} the ground-truth assignment for the set V , 1 Rest of the results will be available in the project web page. where 0 and 1 refer the BG and FG object pixels respectively, then (9) can be formalized as (10).
F oM = 2 × I(X) U (X) ,(10)
where I(X) and U (X) can be approximated as follows:
I(X) = v∈V X v * Y v + ≡ T P,(11)U (X) = v∈V (X v +Y v )+ ≡ (T P +F P )+(T P +F N ),(12)
where is a very small value set to 1e − 08.
The Table IV quantitatively compares the performance of our MV-FCN with some of the results recorded in the literature from prior-art and state-of-the-art techniques. These methods include the probabilistic-based approaches as well as neural network (NN)-based learning algorithms in recent years. Figure 14, on the other hand, summarizes the results, where the best performance of the proposed model is compared with the best results from other methods listed in Table IV. Note that, not all the models have reported results for all the datasets we tested, at the same time, there are not many literature that use NN for the task we are intended. A brief is given for the compared methods to serve as an introduction to them. The technical aspect of the methods are not analyzed, here since this work mainly revolves around the implementation of a DCNN that has the potentiality to localize the FG regions with comparable performance to the literature. a binary scene modeling based on density analysis. Whereby, the SDAE encodes the intrinsic structural information of a scene. The encoded features of image patches are then hashed in Hamming space, and then based on the hash method a binary scene is modeled through density analysis, which captures the spatiotemporal distribution information (measured by Hamming distance evaluation). Similarly, Zhao et al. [13] also take advantage of NN with a stacked multilayer Self-Organizing Map (SOM) to model the BG. In which, the model is initially trained using some BG samples, and then, based on this pre-trained model, FG detection is conducted for a new test sample, at the same time the BG model is updated using the test image online as a procedure for BG maintenance. Gemignani and Rozza [51] extend the basic SOM model of Zhao et al. with a self-balancing multi-layered SOM that tracks a long time pixel dynamics for better FG detection. Authors in [52] approach the background modeling as evidence collection of each pixel in a scene with a weight-sample-based method. They also use a minimumweight and reward-and-penalty weighting strategy to account rapidly changing scenarios in such a way that most inefficient sample is replaced instead of the oldest sample or a random sample. Then a pixel is classified as a BG if the sum of the weights of the active samples is larger than a manually set specific threshold; otherwise, it is classified as a FG. Although their method's computational speed is quite similar to ours (2 FPS), they record a poor FG detection accuracy. On the other hand, Varadarajan et al. [47] propose an algorithm, where the spatial relationship between pixels is taken into account by using a region-based GMM but traditional GMMs use pixel- based Gaussian distributions [3].
Charles et al. [48] coined a system as SuBSENSE, short for Self-Balanced SENsitivity SEgmenter that adapts and integrates Local Binary Similarity Pattern (LBSP) features as additional clues to pixel intensities in a nonparametric BG model that is then automatically tuned using pixel-level feedback loops. In [55], the authors exploit Local Binary Pattern (LBP) with local Singular Value Decomposition (SVD) operator to extract invariant feature representation that is similar to the LBSP in [48]. Then, they use SAmple CONsensus(SACON) approach for building the BG model based on statistics of the pixel processes (about 300 frames). Then they employ the Hamming distance, like applied in [49] with a threshold to classify each pixel as either FG or BG.These models require static and clean background samples to build up the dictionary; thus, they lack application for real-world problems. Similarly, Allebosch et al. [56] also employ local features, such as, Local Ternary Pattern (LTP) based edge descriptors and RGB color cues to classify individual pixels. They form two backgrounds based on the aforesaid edge descriptors and color cues and create two FG masks. Then, using a pixel-wise logical AN D operation they refine the detected FG region. Meanwhile, Babaee et al. [54] employ a conventional CNN, train the network with randomly selected video frames and their ground-truth segmentations patch-wise, like in [49], and carrie out spatial-median filtering as the postprocessing of the network outputs. Although over the past two decades many algorithms have been proposed, none of [50], 0.9620 [48] 0.9605 [13], 0.9606 [51] Canoe 0.8492 0.8400 0.9404 0.9315 0.7923 [48], 0.6131 [52] 0.7258 [49], 0.6337 [13] Boats 0.8493 0.8403 0.8727 0.8600 0.8324 [47], 0.7532 [53] 0.6932 [48], 0.6401 [52] 0.8121 [54], 0.6017 [13], 0.6560 [ them can be the ultimate model for FG inferencing. Therefore, Bianco et al. [53] explore a way of harnessing multiple stateof-the-art foreground detection algorithms for improving the classification accuracy. They obtain a solution tree by Genetic Programming (GP); however, this approach also cannot be acclaimed as an universal solution for the same problem.
In general, most of the state-of-the-art methodologies use patch-wise processing and multi-modality based algorithms for BG establishment and a feedback-based approach as postprocessing to refine the primarily detected FG regions. Such setup ensues complex computations and higher processing time due to the time-consuming iterative pursuit of low-rank matrix or sparse matrix. On the contrary, the proposed model processes the whole input image as a single entity during inferencing. Then it refines the output by a none iterative postprocessing, resulting approximative 0.445s (mean average processing time) per frame, i.e., 2.25 FPS (see Fig. 15) on Intel(R) Core(TM) i7-4770 CPU @ 3.40 GHz with 32.0 GB memory for FG inferencing once the network is trained.
V. CONCLUSION
This work put an NN forwards for foreground inferencing that is inspired by recent innovations in deep learning, such as, ResNet, Inception modules, and Fully convolutional network with skip connections. The proposed model utilizes a heterogeneous set of convolutions to capture invariance features at different scales.
In traditional approaches, much time has been spent on complex mathematical modeling to optimize background generation and postprocessing to get a few more valid foreground pixels. Besides that, feature engineering and manual parameter tuning of traditional methods become unneeded since the network parameters can be learned from exemplar FG segmentations during training. For these reasons, we advocate that the proposed multi-view receptive field FCN is a novel addition to neural-based FG detection algorithms.
The qualitative and quantitative performance evaluations of the proposed MV-FCN on some challenging video sequences collected from benchmark datasets demonstrate that the model performs better than or very competitively to the prior-and state-of-the-art methods. However, the limitation of the network comes with a high number of trainable parameters. We leave this for our future direction, where we plan to optimize the network to achieve better results with less number parameters.
In application point of view, the MV-FCN can be applicable to many other applications, like MRI slice partitioning and path segmentation for autonomous vehicles. Finally, it must be considered that a perfect FG inferencing is still an intriguing task and a good FG detection system should use the knowledge derived from its ultimate purpose.
Fig. 1 :
1A Probable Application Environment of the Proposed MV-FCN. while the top-level layers provide the abstractive meaning of input data. The DCNNs have become the front-runner technology for various computer vision-based applications followed by Krizhevsky et al.'s [21] successful campaign with big achievements on the ImageNet 2012 Large-Scale Visual Recognition Challenge (ILSVRC-2012).
Fig. 2 :
2ResNet and Inception Modules: (a) Standard CNN Connection, (b) ResNet Connection[24], (c) Proposed Connection, (d) Original Inception Module[25] and (e) Our Inception Module.
Fig. 3 :
3Schematic Diagram of the MV-FCN: Convk, Si, CTransk, Concat, and BN stand for convolution using kernel size of k and stride of i, transpose convolution with filter size of k, activation maps concatenation, and batch normalization operations, respectively.
Fig. 5 :
5Ordered Exclusive Split of Training and Test Sets: G.Tground truth, I-RGB raw input image, S.q#-sequence ID.
Fig. 6 :
6Creating FG Mask: Applying an appropriate threshold to the score-map generated from the last classification layer of MV-FCN for a frame taken from the Office dataset.
2
2×T P/(T P +F N )×T P/(T P +F P ) T P/(T P +F N )+T P/(T P +F P ) = 2×T P 2 T P (T P +F P +T P +T N ) = 2×T P (T P +F P )+(T P +F N )Fig. 7: Sample results for the Office dataset. Col. 1-5: Sample input frames, MV-FCN generated score-maps, binary FG masks with empirical and Otsu's thresholds. Col. 6: training and validation FoM and loss respectively in the top and bottom.
Fig. 8 :
8Sample Results for the Overpass Dataset. Col. 1-5: Sample input frames, MV-FCN generated score-maps, binary FG masks with empirical and Otsu's thresholds. Col. 6: Training and validation FoM and loss respectively in the top and bottom.
1 )
1Discussion on compared methods: In the literature [49], Zhang et al. develop a neural network (NN) that has a stacked denoising autoencoder (SDAE) learning module andFig. 9: Sample Results for the Traffic Dataset. Col. 1-5: Sample input frames, MV-FCN generated score-maps, binary FG masks with empirical and Otsu's thresholds. Col. 6: Training and validation FoM and loss respectively in the top and bottom.
Fig. 10 :
10Sample Results for the PeopleInShade Dataset. Col. 1-5: Sample input frames, MV-FCN generated score-maps, binary FG masks with empirical and Otsu's thresholds. Col. 6: Training and validation FoM and loss respectively in the top and bottom.
Fig. 11 :
11Sample Results for the TwoPositionPTZCam Dataset. Col. 1-5: Sample input frames, MV-FCN generated score-maps, binary FG masks with empirical and Otsu's thresholds. Col. 6: Training and validation FoM and loss respectively in the top and bottom.
Fig. 12 :
12Sample Results for the Turnpike 0 5fps Dataset. Col. 1-5: Sample input frames, MV-FCN generated score-maps, binary FG masks with empirical and Otsu's thresholds. Col. 6: Training and validation FoM and loss respectively in the top and bottom.
Fig. 13 :
13MV-FCN score map when: (b) trained from scratch, (c) fine-tuned with intra-class transfer learning.
Fig. 14 :
14FoM vs dataset.
Fig. 15 :
15Infer. Speed of the Proposed MV-FCN.
TABLE I :
ILayer detail of the MV-FCN.
TABLE II :
IIDataset Summary.
Transfer learning: To improve the network's learningModel Fine-tuned for
Model Transferred from
Highway
Turnpike 0 5fps
Office
CopyMachine
Canoe
Boats
Boats
Canoe
Overpass
Pedestrians
Traffic
Highway
Boulevard
TwoPositionPTZCam
CopyMachine
Office
PeopleInShade
Pedestrians
TwoPositionPTZCam
Turnpike 0 5fps
Turnpike 0 5fps
TwoPositionPTZCam
TABLE III :
IIITransfer Learning Dataset Pairs.experience we incorporate intraclass domain transfer.
Table III lists the fine-tuning dataset pairs. For instance, the
pre-trained network with TwoPositionPTZCam is fine-tuned
for Turnpike 0 5fps. Here, both the domain have moving
vehicles as FG objects. The theoretical and philosophical
expositions of transfer learning can be found in
TABLE IV :
IVPerformance Comparison in terms of FoM: S-training from scratch, P-pre-trained model fine-tuning, Global and Otsu stand for the two used thresholding methods. Values in red are the best FoM while the ones in blue are the second best.
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| []
|
[
"CP violation in B s mixing from heavy Higgs exchange",
"CP violation in B s mixing from heavy Higgs exchange"
]
| [
"Bogdan A Dobrescu \nTheoretical Physics Department\nFermi National Accelerator Laboratory\nBataviaIllinoisUSA\n",
"Patrick J Fox \nTheoretical Physics Department\nFermi National Accelerator Laboratory\nBataviaIllinoisUSA\n",
"Adam Martin \nTheoretical Physics Department\nFermi National Accelerator Laboratory\nBataviaIllinoisUSA\n"
]
| [
"Theoretical Physics Department\nFermi National Accelerator Laboratory\nBataviaIllinoisUSA",
"Theoretical Physics Department\nFermi National Accelerator Laboratory\nBataviaIllinoisUSA",
"Theoretical Physics Department\nFermi National Accelerator Laboratory\nBataviaIllinoisUSA"
]
| []
| The anomalous dimuon charge asymmetry reported by the D0 Collaboration may be due to the tree-level exchange of some spin-0 particles that mediate CP violation in Bs −Bs meson mixing. We show that for a range of couplings and masses, the heavy neutral states in a two Higgs doublet model can generate a large charge asymmetry. This range is natural in "uplifted supersymmetry", and may enhance the B − → τ ν and Bs → µ + µ − decay rates. However, we point out that on general grounds the reported central value of the charge asymmetry requires new physics not only in Bs−Bs mixing but also in ∆B = 1 transitions or in B d −B d mixing.Introduction.-The Standard Model (SM) predicts that the violation of CP symmetry in B −B meson mixing is very small [1], and various measurements have so far confirmed this prediction in the B d system. Experimental sensitivity to the properties of B s mesons has improved within the last few years, with well-understood data sets from pp collisions at the Tevatron analyzed by the D0 and CDF Collaborations. The large ratio of the s and d quark masses, and also the large V ts /V td ratio make the B s system more sensitive to new physics than the B d system. We explore here the possibility that tree-level exchange of new particles induces a sizable CP violation in B s −B s mixing.Recently[2], the D0 Collaboration has reported evidence for CP violation in final states involving two muons of the same charge, arising from semileptonic decays of b hadrons. The like-sign dimuon charge asymmetry, measured by D0 with 6.1 fb −1 of data, is defined by | 10.1103/physrevlett.105.041801 | [
"https://arxiv.org/pdf/1005.4238v2.pdf"
]
| 42,763,302 | 1005.4238 | 8d7128961259b88f372e08519ec4a8ac31d822d4 |
CP violation in B s mixing from heavy Higgs exchange
18 Jun 2010
Bogdan A Dobrescu
Theoretical Physics Department
Fermi National Accelerator Laboratory
BataviaIllinoisUSA
Patrick J Fox
Theoretical Physics Department
Fermi National Accelerator Laboratory
BataviaIllinoisUSA
Adam Martin
Theoretical Physics Department
Fermi National Accelerator Laboratory
BataviaIllinoisUSA
CP violation in B s mixing from heavy Higgs exchange
18 Jun 2010(Dated: May 24, 2010; revised June 14, 2010)
The anomalous dimuon charge asymmetry reported by the D0 Collaboration may be due to the tree-level exchange of some spin-0 particles that mediate CP violation in Bs −Bs meson mixing. We show that for a range of couplings and masses, the heavy neutral states in a two Higgs doublet model can generate a large charge asymmetry. This range is natural in "uplifted supersymmetry", and may enhance the B − → τ ν and Bs → µ + µ − decay rates. However, we point out that on general grounds the reported central value of the charge asymmetry requires new physics not only in Bs−Bs mixing but also in ∆B = 1 transitions or in B d −B d mixing.Introduction.-The Standard Model (SM) predicts that the violation of CP symmetry in B −B meson mixing is very small [1], and various measurements have so far confirmed this prediction in the B d system. Experimental sensitivity to the properties of B s mesons has improved within the last few years, with well-understood data sets from pp collisions at the Tevatron analyzed by the D0 and CDF Collaborations. The large ratio of the s and d quark masses, and also the large V ts /V td ratio make the B s system more sensitive to new physics than the B d system. We explore here the possibility that tree-level exchange of new particles induces a sizable CP violation in B s −B s mixing.Recently[2], the D0 Collaboration has reported evidence for CP violation in final states involving two muons of the same charge, arising from semileptonic decays of b hadrons. The like-sign dimuon charge asymmetry, measured by D0 with 6.1 fb −1 of data, is defined by
A b sl ≡ N ++ b − N −− b N ++ b + N −− b ,(1)
where N ++ b is the number of events with two b hadrons decaying into µ + X. The D0 result, A b sl = −[9.57 ± 2.51(stat.)± 1.46(syst.)]× 10 −3 is 3.2σ away from the SM prediction of −0.2 × 10 −3 . The CDF [3] measurement of A b sl , using 1.6 fb −1 of data, has a positive central value, A b sl = (8.0 ± 9.0 ± 6.8) × 10 −3 , but is compatible with the D0 measurement at the 1.5σ level because its uncertainties are 4 times larger than those of D0. Combining in quadrature (including the systematic errors) the D0 and CDF results for A b sl we find a 3σ deviation from the SM:
A b sl ≃ −(8.5 ± 2.8) × 10 −3 .(2)
Another test of CP violation in B s −B s mixing is provided by the measurement of the "wrong-charge" asymmetry in semileptonic B s decays,
a s sl ≡ Γ(B s → µ + X) − Γ(B s → µ − X) Γ(B s → µ + X) + Γ(B s → µ − X) .(3)
The D0 measurement in this channel [4], a s sl = −(1.7 ± 9.1 +1.4 −1.5 )×10 −3 , is consistent with the SM. Assuming that the CP asymmetry in B d −B d mixing is negligible, the like-sign dimuon charge asymmetry is entirely due to B s − B s mixing and is related to a s sl : A b sl = (0.494 ± 0.043) a s sl , where the coefficient depends on the fraction ofb antiquarks which hadronize into a B s meson [2]. This allows the extraction of a s sl from Eq. (2), which then can be combined with the D0 measurement of a s sl , resulting in
(a s sl ) combined ≈ −(12.7 ± 5.0) × 10 −3 .(4)
Even though the inclusion of the CDF dimuon asymmetry and the D0 semileptonic wrong-charge asymmetry reduces the deviation in a s sl derived from the D0 dimuon asymmetry, the above result is still about 2.5σ away from the SM value [5] of (a s sl ) SM ≈ 0.02 × 10 −3 . The D0 [6] and CDF [7] Collaborations have also reconstructed B s →J/ψ φ decays and measured angular distributions as a function of decay time, and reported some deviation consistent with CP violation in B s −B s oscillations (see [8] for a fit to earlier B s data). The sign [9] and size of this deviation are compatible with Eq. (4), further strengthening the case for physics beyond the SM.
Generic new physics.-The matrix element of some new physics Hamiltonian, H NP , contributing to B s −B s mixing may be parameterized as [10], [5], [8]
B s |H NP |B s = C Bs e −iφs − 1 2M Bs (M SM 12 ) * ,(5)
where C Bs > 0 and −π ≤ φ s ≤ π. The magnitude of the off-diagonal element of the B s −B s mass matrix due to SM box diagrams is: |M SM 12 | ≃ (9.0 ± 1.4) ps −1 , where we used the same inputs as in [5] except for the updated values of the B s decay constant f Bs = (231±15) MeV and bag parameter B = 0.86 ± 0.04 computed on the lattice with 2+1 flavors [11]. The phase of M SM 12 is negligible. The measured mass difference of the B s mass eigenstates depends linearly on C Bs , ∆M s = 2|M SM 12 | C Bs . The combination [12] of the CDF and D0 measurements is ∆M s = (17.78 ± 0.12) ps −1 , so that we find
C Bs = 0.98 ± 0.15 .(6)
The semileptonic wrong-charge asymmetry is given by
a s sl = 2|Γ 12 | ∆M s sin φ s ,(7)
where Γ 12 is the off-diagonal element of the B s −B s decaywidth matrix. New physics contributing to ∆B = 1 processes may affect Γ 12 , but the effects are typically negligible compared to the SM b → ccs transition due to tree-level W exchange, which is suppressed only by V cb . The SM prediction for |Γ 12 | is given by |Γ SM 12 | = (1/2)(0.090 ± 0.024) ps −1 , where we again used the results of [5] with updated values for f Bs and B (this is consistent with the result of [13]). Using the a s sl value from Eq. (4), we find that Eq. (7) gives
sin φ s = −2.5 ± 1.3 .(8)
This is a somewhat troubling result: the central value is more than 1σ away from the physical region | sin φ s | ≤ 1. This tension arises because the absolute value of B s −B s mixing is constrained by the measured ∆M s , not allowing enough room for an asymmetry as large as the central value of a s sl shown in Eq. (4). This suggests that the central value of a s sl will be reduced by a factor of more than two when the error bars will become small enough.
Alternatively, the assumptions about new physics employed here may need to be relaxed. For example, the wrong-charge asymmetry in semileptonic B d decays, a d sl , may be non-negligible. Its value given by measurements at B factories is (−4.7 ± 4.6) × 10 −3 [12], so including it would change the relation between A b sl and a s sl as discussed in [2]. This possibility is intriguing, but one should keep in mind that new physics contributions to (bd)(bd) operators are often suppressed by additional powers of m d /m s and V td /V ts compared to those to (bs)(bs) [14].
Another possibility is that there are sizable new contributions to Γ 12 . This is problematic because the SM tree-level contribution is CKM-favored, while new particles that induce ∆B = 1 effects are constrained by various limits on flavor-changing neutral currents (e.g., b → sγ or K −K mixing) and by collider searches. Nevertheless, examples of relatively large shifts in Γ 12 can be found [13,15]. Consider for example two operators,
(b R γ µ c R )(ū R γ µ s R ) and (b R γ µ u R )(c R γ µ s R )
, which may be induced by W ′ exchanges. The main effect of these ∆B = 1 operators is to enhance the rate for B d → DK decays. Given that these dominant decay modes of B d involve a form factor which is not known precisely, these operators may account for a significant fraction of the measured decay width. If the scale of the new operators is 0.9 TeV then Γ 12 is enhanced by 30%. In what follows we will focus on ∆B = 2 transitions [see Eq. (5)], ignoring new contributions to |Γ 12 |.
New physics models for B s −B s mixing.-Although more experimental studies are required before concluding that physics beyond the SM contributes to B s −B s mixing, it is useful to analyze what kind of new physics could induce CP-violating effects as large as sin φ s ≈ −1. Given that the SM B s −B s mixing is a 1-loop effect, it is often assumed that new physics contributes also at one loop, for example via gluino-squark box diagrams in the MSSM [16]. However, the large effect indicated by the data is more likely to be due to tree-level exchange of new particles which inducebsbs operators. These particles must be bosons (with spin 0, 1, or 2 being the more likely possibilities) carrying baryon number 0 or ±2/3. In the first case they must be electrically neutral and color singlets or octets. The bosons of baryon number ±2/3 are diquarks of electric charge ∓2/3 and transform under SU (3) c as3 or 6 (3 or6 for charge +2/3).
The new bosons may be related to electroweak symmetry breaking, as in the case of the heavy Higgs states in two Higgs doublet models. We concentrate in what follows on a spin-0 boson H 0 d = (H 0 + iA 0 )/ √ 2, which is electrically neutral and a color singlet (and part of a weak doublet). The Yukawa couplings of H 0 d to b and s quarks in the mass eigenstate basis are given by
− H 0 d y bsbR s L + y sbsR b L + H.c.(9)
Let us assume for simplicity that the vacuum expectation value (VEV) of H 0 is negligible at tree level (the coupling to quarks induces a small VEV at one loop), so that H 0 and A 0 have the same mass M A . Examples of theories with these features are the MSSM in the uplifted region [17], as discussed later, and composite Higgs models [18]. Tree-level H 0 d exchange gives rise to a single term in the Lagrangian which contributes to B s −B s mixing:
y bs y * sb M 2 A (b R s L )(b L s R ) ,(10)
where the quark fields are taken in the mass eigenstate basis. If the VEV of H 0 is taken into account, then additional operators contribute [19], most importantly (b R s L ) 2 ; we will ignore these contributions in what follows. The matrix element of operator (10) is
B s |H NP |B s = − y bs y * sb η M 2 A M 4 Bs f 2 Bs B 4 2(m b + m s ) 2 .(11)
The bag parameter for operator (10) has been estimated using the quenched aproximation on the lattice [20], B 4 ≈ 1.16. The parameter η ≈ 4 takes into account the running of operator (10) between the M A and M Bs scales [21]. For the sum of quark masses we use m b + m s ≈ 4.3 GeV. Comparing Eqs. (5) and (11) we find M A |y bs y sb |η = (147 ± 15) TeV
C 2 Bs + 1 − 2C Bs cos φ s 1/4 , arg(y bs y * sb ) = tan −1 C Bs sin φ s 1 − C Bs cos φ s .(12)
The off-diagonal coupling y bs is expected to be suppressed by V ts compared to the diagonal y b Yukawa coupling of H 0 d tob R b L , while y sb is suppressed by an additional factor of m s /m b , so that we take |y bs | 10 −2 and |y sb | 2 × 10 −4 . When y bs and y sb saturate these upper bounds, the experimental constraint Eq. (6) on C Bs gives M A ≈ (0.65 ± 0.07) TeV and arg(y bs y * sb ) = −1.3 ± 0.3 for φ s = −π/6. Figure 1 shows the range of M A / |y bs y sb | as a function of φ s .
The H 0 d exchange that induces CP violation in B s −B s mixing contributes to the B s → µ + µ − branching fraction, provided the coupling of H 0 d to muons is not negligible. The coupling y µ H 0 dμ R µ L leads to
B B s → µ + µ − = (|y bs | 2 + |y sb | 2 ) |y µ | 2 M 4 A η 2 b τ Bs M 5 Bs f 2 Bs 64π(m b + m s ) 2 ≈ 1.3×10 −8 |y bs | 10 −2 2 |y µ | 10 −2 2 1 TeV M A 4 .(13)
QCD corrections are taken into account by η b ≈ 1.5 [22]. The experimental limit B (B s → µ + µ − ) < 4.3 × 10 −8 [23] imposes |y µ | < 0.018 for M A = 1 TeV. Given this constraint, the impact on B → Kµ + µ − observables is relatively small [24].
Uplifted supersymmetry.-Let us describe a renormalizable gauge-invariant theory that includes the interactions of Eq. (9) without violating current limits on flavor processes. The MSSM parameter space contains a region where the down-type fermion masses are induced at one loop by the VEV of the up-type Higgs doublet H u . In this so-called uplifted Higgs region [17,25,26] the ratio of H u and H d VEVs is very large, v u /v d ≡ tan β 100, but all Yukawa couplings remain perturbative. The physical states of this uplifted two Higgs doublet model include a SM-like Higgs boson, h 0 , which is entirely part of H u in the tan β → ∞ limit, the two neutral states H 0 and A 0 of mass M A , and a charged Higgs boson H ± of mass
M H + = (M 2 A + M 2 W ) 1/2 ≈ M A .
The heavy states H 0 , A 0 and H ± are almost entirely part of H d .
The Yukawa terms in the superpotential give rise to H d couplings to down-type fermions in the Lagrangian:
− H d (d cŷ d Q + e cŷ ℓ L) + H.c. ,(14)
where the quark and leptons shown here are gauge eigenstates, and their generation index is implicit. Theŷ d andŷ ℓ couplings are 3 × 3 matrices in flavor space. Various 1loop diagrams involving superpartners generate couplings of H † u to down-type fermions,
− H † u (d cŷ′ d Q + e cŷ′ ℓ L) + H.c. ,(15)
inducing masses for down-type quarks and charged leptons. The dominant contributions, from gluino and wino loops, to the effective quark Yukawa matrix are
(ŷ ′ d ) ij ≈ − α s 4π e −iθµ (ŷ d ) ij f ij .(16)
The complex coefficients f ij have magnitude of order one:
f ij ≈ 8|µ|e iθg 3Md i F Mg MQ j , Md i MQ j − 3αe iθW 2s 2 W α s F MW MQ j , |µ| MQ j (17) where 0 < F (x, y) < 1 is a function given in Eq. (3.2)
of [17]. The phases of the gluino and wino masses are explicitly displayed here, so that Mg, MW > 0.
We assume that the communication of supersymmetry breaking to squarks is flavor blind. In the absence of renormalization group (RG) effects of the Yukawa couplings, the squark mass matrices at the weak scale are proportional to the 3 × 3 unit matrix, so that theŷ ′ d matrix is given byŷ d times a complex number which depends on superpartner masses. However, the large t, b and τ Yukawa couplings have substantial RG effects, driving MQ 3 < MQ 1 = MQ 2 and Md 3 < Md 1 = Md 2 , which breaks the alignment betweenŷ ′ d andŷ d in the 3j and j3 elements. After diagonalization of the down-type quark masses (i.e., ofŷ ′ d ), the neutral component of H d acquires off-diagonal couplings as in Eq. (9). Assuming that the unitary matrix which transforms between the gauge and mass eigenstate bases of right-handed down-type quarks is approximately the unit matrix we find
y bs = y 0 (a 33 − a 31 )(V d L ) 33 (V d L ) * 23 , y sb = y 0 m s m b a 13 (V d L ) 23 (V d L ) * 33 , y b = y 0 1 + a 31 + (a 33 − a 31 ) (V d L ) 33 2 ,(18)
where
a ij ≡ f 11 /f ij − 1, and y 0 ≡ −e iθµ 4πm b /(α s v h f 11 ), with v h ≈ 174
are small enough to satisfy the limits from ε K and a d sl for M A > 100 GeV.
In the uplifted Higgs region the τ Yukawa coupling to H d (at the weak scale) must be large, |y τ | ≈ 1.3, in order for the observed m τ to be generated by wino and bino diagrams [17]. The b Yukawa coupling to H d may be smaller, |y b | ≈ 0.5 − 1, due to the large contribution to m b from a 1-loop gluino diagram. However, if there is a partial cancellation between the two terms in Eq. (17), then a larger Yukawa coupling |y b | > 1 is needed.
The small m µ leaves more room for its possible origin, and consequently a wider range of values for y µ . If m µ is generated entirely by the Yukawa coupling to H d , then |y µ | ≈ |y τ |m µ /m τ ≈ 0.08, which is compatible with the current limit on B(B s → µ + µ − ) provided M A 1.7 TeV. Such a large mass would imply φ s ≈ 0.1, which is too small to accommodate a significant charge asymmetry. On the other hand, m µ may be due to loop-induced couplings of the muon to H † u which exist even for y µ → 0. For example, in models of gauge mediate supersymmetry breaking [27], which fit well the requirements of uplifted supersymmetry, there is a vectorlike chiral superfield d m with the quantum numbers of weak-singlet down-type squarks. The scalar components of this messenger superfield,d m andd c m may couple to the SM fermions [28]: κd mμ c L t L and κ ′dc mt c R µ R which at 1-loop give [29]
m µ ≃ m t 3κκ ′ 32π 2 ∆M 2 dm M 2 dm .(20)
A typical splitting between the messenger scalar squaredmasses is ∆M 2
dm ≈ 0.2M 2 dm , where M dm ∼ O(100)
TeV is the messenger fermion mass. The muon mass may be generated entirely through this mechanism if κκ ′ ≈ 0.3. A similar mechanism is used in [30]. Thus, the y µ coupling, which determines the heavy Higgs contribution to B(B s → µ + µ − ), is sensitive to physics at the 100 TeV scale, and can be significantly smaller than 0.08.
The dominant contributions to (g − 2) µ , due to winoslepton diagrams, tend in the uplifted region to enhance the discrepancy between the SM and experiment [25]. We point out, though, that the wino-slepton diagrams become small if the slepton doublet of the second generation is sufficiently heavier than MW , while the bino-slepton diagrams can explain the discrepancy if y µ 10 −2 .
Flavor-changing charged currents due to H ± exchange are important independent of RG effects. The couplings
m b V ub y b y b v d + y ′ b v u H −b R u L + m τ y τ y τ v d + y ′ τ v u H −τ R ν L +H.c. ,(21)
(y ′ b and y ′ τ are the 33 eigenvalues ofŷ ′ d andŷ ′ e ) may significantly affect the rate for the B ± → τ ± ν decay:
B(B − → τ ν) B(B − → τ ν) SM = 1 − y * b y τ v 2 h m b m τ M 2 B + M 2 H + 2 .(22)
Unlike the usual MSSM where B(B − → τ ν) is smaller than in the SM, the uplifted region allows an enhancement compared to the SM [25,31], depending on the phase of y * b y τ . This is interesting because the measurement of this branching fraction is larger than the SM prediction by a factor of 2, a ∼2σ discrepancy [32]. For y * b y τ = −1 and M H + = 1 TeV, B(B − → τ ν) increases by 24% compared to the SM prediction.
Conclusions.-We have shown that the evidence for CP violation reported by the D0 Collaboration may be explained in part by the exchange of the neutral states of a two Higgs doublet model contributing to B s −B s mixing. In particular, in the uplifted Higgs region of the MSSM [17], a large CP-violating effect in B s −B s mixing implies that the B s → µ + µ − decay could be discovered in the near future, and that, unlike in the usual MSSM, the rate for B − → τ ν may be enhanced compared to the SM prediction. Independent of the new physics interpretation, however, the reported central value of the charge asymmetry requires new physics beyond B s −B s mixing, for example in ∆B = 1 transitions or in B d −B d mixing.
FIG. 1 :
1Range for MA compatible with a CP asymmetry in Bs −Bs mixing described by the φs angle. The vertical size of the shaded band accounts for the 1σ experimental uncertainty in ∆Ms and for the theoretical uncertainties in fB s and |M SM 12 |. The off-diagonal Yukawa couplings are expected to satisfy |y bs | 10 −2 and |y sb | 2 × 10 −4 . The running between MA and MB s is parametrized by η ≈ 4.
GeV. The unitary matrix V d L transforms the d Li quarks from gauge to mass eigenstates.Fory b = O(1) and V d L ≃ (V CKM ) † , we obtain |y bs | ≈ 10 −2 , |y sb | = O(y bs m s /m b ), confirming the bounds used after Eq.(12). The combination of couplings that control K −K and B d −B d mixing,|y sd y ds | = |y bs y sb | m d |V 2 td a 13 | m b |a 33 − a 31 | O(10 −13 ) ,|y bd y db | = |y bs y sb | m d |V td | 2 m s |V ts | 2 2 × 10 −9 ,
Acknowledgments: We thank C. Bauer, L. Dixon, E. Eichten, E. Gamiz, E. Lunghi, and A. Petrov for helpful discussions. We are grateful to A. Buras and A. Kronfeld for comments on the manuscript. Fermilab is operated by Fermi Research Alliance, LLC, under Contract DEAC02-07CH11359 with the US Department of Energy.
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| []
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[
"ON THE HAMILTONIAN NUMBER OF A PLANAR GRAPH",
"ON THE HAMILTONIAN NUMBER OF A PLANAR GRAPH"
]
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"Thomas M Lewis "
]
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| []
| The Hamiltonian number of a connected graph is the minimum of the lengths of the closed, spanning walks in the graph. In 1968, Grinberg published a necessary condition for the existence of a Hamiltonian cycle in a planar graph, formulated in terms of the lengths of its face cycles. We show how Grinberg's theorem can be adapted to provide a lower bound on the Hamiltonian number of a planar graph.2010 Mathematics Subject Classification. 05C10. | null | [
"https://arxiv.org/pdf/1508.06892v1.pdf"
]
| 119,306,854 | 1508.06892 | 413783db40db70456bbbd66894cac779a15ec28d |
ON THE HAMILTONIAN NUMBER OF A PLANAR GRAPH
Thomas M Lewis
ON THE HAMILTONIAN NUMBER OF A PLANAR GRAPH
The Hamiltonian number of a connected graph is the minimum of the lengths of the closed, spanning walks in the graph. In 1968, Grinberg published a necessary condition for the existence of a Hamiltonian cycle in a planar graph, formulated in terms of the lengths of its face cycles. We show how Grinberg's theorem can be adapted to provide a lower bound on the Hamiltonian number of a planar graph.2010 Mathematics Subject Classification. 05C10.
Introduction
A Hamiltonian cycle in a graph is a closed, spanning walk that visits each vertex exactly once; a graph is called Hamiltonian provided that it contains a Hamiltonian cycle. While not every graph is Hamiltonian, every connected graph contains a closed, spanning walk. A closed, spanning walk of minimum length is called a Hamiltonian walk, and the Hamiltonian number of a connected graph G, denoted by h(G), is the length of a Hamiltonian walk in G. Thus the Hamiltonian number of a graph can be understood as a measure of how far the graph deviates from being Hamiltonian.
In 1968, Grinberg [11] published a necessary condition for the existence of a Hamiltonian cycle in a planar graph, formulated in terms of the lengths of its face cycles. The sole purpose of this paper is to show how Grinberg's theorem can be adapted to provide a lower bound on the Hamiltonian number of a planar graph. Before we state this theorem, it is helpful to place our work in context.
In general, determining the Hamiltonian number of a graph is difficult, but for a connected graph G of order n, the elementary bounds n ≤ h(G) ≤ 2(n − 1) are easily obtained. A Hamiltonian walk on G must visit each vertex, which gives the lower bound. On the other hand, a pre-order, closed, spanning walk on a spanning tree of G has length 2(n − 1), yielding the upper bound. Over the years, much of the research on the Hamiltonian number has advanced along two fronts: developing tighter bounds for the Hamiltonian number in terms of natural graph parameters, or evaluating the Hamiltonian numbers of some special graphs or families of graphs.
Goodman and Hedetniemi [9] initiated the study of the Hamiltonian number of a graph. They proved, among other things, properties of Hamiltonian walks, upper and lower-bounds for the Hamiltonian number of a graph, and a formula for the Hamiltonian number of a complete n-partite graph. Their most accessible result is this: let G be a k-connected graph on n vertices with diameter d; then
h(G) ≤ 2(n − 1) − k/2 (2d − 2),
which improves the elementary upper bound. In another result, they related the Hamiltonian numbers of G and G − U , the graph obtained by deleting the unicliqual points of G.
Soon after the publication of the seminal paper of Goodman and Hedetniemi, Bermond [3] published a theorem on the Hamiltonian number problem inspired by Ore's theorem. Ore's theorem gives a sufficient condition for a graph to be Hamiltonian in terms of the sums of the degrees of non-adjacent vertices; see, for example, Theorem 6.6 of [7]. Bermond showed the following: let G be a graph of order n and let c ≤ n; if deg(v) + deg(w) ≥ c for every pair of non-adjacent vertices v and w in V (G), then h(G) ≤ 2n − c.
Chartrand, Thomas, Zhang, and Saenpholphat [8] introduced an alternative approach to the Hamiltonian number. Let G be a connected graph of order n. Given vertices u and v, let d(u, v) denote the length of a shortest path from u to v. A cyclic ordering of the vertices of G is a permutation s :
v 1 , v 2 , . . . , v n , v n+1 of V (G), where v n+1 = v 1 . Given a cyclic ordering s, let d(s) = n i=1 d(v i , v i+1 ). The set H(G) = {d(s) : s is a cyclic ordering of V (G)}
is called the Hamiltonian spectrum of G. Chartrand and his colleagues showed that h(G) = min H(G). This paper contains two other notable results: first, that a connected graph G of order n satisfies h(G) ≤ 2(n − 1) with equality if and only if G is a tree; second, that for each integer n ≥ 3, every integer in the interval [n, 2(n − 1)] is the Hamiltonian number of some graph of order n. Král, Tong, and Zhu [12] and Liu [13] conducted additional research on the Hamiltonian spectra of graphs. Various authors have studied the Hamiltonian number of special graphs and families of graphs. Punnim and Thaithae [19,17] studied the Hamiltonian numbers of cubic graphs. A graph of order n with Hamiltonian number n + 1 is called almost Hamiltonian. Punnim, Saenpholphat, and Thaithae [16] characterized the almost Hamiltonian cubic graphs and the almost Hamiltonian generalized Petersen graphs. Asano, Nishizeki, and Watanabe [2,14] established a simple upper bound for the Hamiltonian number of a maximal planar graph of order n ≥ 3 and created an algorithm for finding closed, spanning walks in a graph with length close to its Hamiltonian number. G. Chang, T. Chang, and Tong [5] studied the Hamiltonian numbers of Möbius double loop networks. The Hamiltonian number problem has a variety of cognates: Vacek [20,21] analyzed open Hamiltonian walks; Araya and Wiener [1,22] investigated hypohamiltonian graphs; Goodman, Hedetniemi, and Slater [10] studied the the Hamiltonian completion problem; T. Chang and Tong [6] considered the Hamiltonian numbers of strongly connected digraphs; and, Okamoto, Zhang, and Saenpholphat [18,15] studied the upper traceable numbers of graphs.
The Grinberg number of a planar graph
Let G be a planar graph and let the faces (including the unbounded component) be labeled F 1 , F 2 , . . . , F N . Given a face F , let |F | denote its length, that is, the number of its edges (or vertices). Let G (G) be the set of all sums of the form
N i=1 ε i (|F i | − 2)
where (ε 1 , ε 2 , . . . , ε N ) ∈ {1, +1} N , excluding the two cases where all of the components have the same sign. Let g(G) = min G (G). will call G (G) the Grinberg set of G and g(G) the Grinberg number of G. To develop a feel for this, consider the graph presented in Figure 1. This graph has five interior faces, each a hexagon, and the exterior face has 18 edges. The Grinberg set for this graph is {4, 12, 20, 28} and the Grinberg number is 4.
Grinberg's theorem can be stated as follows: if a planar graph G is Hamiltonian, then g(G) = 0; see [11]. Our main result can be seen as a natural extension of Grinberg's theorem.
Theorem 2.1. Let G be a planar graph. A closed, spanning walk on G must contain g(G)/2 repeats of vertices. Proof. Our proof is an adaptation of the customary proof of Grinberg's theorem; see, for example, Theorem 18.2 of [4]. Let the vertices of the planar graph G be labeled {v 1 , v 2 , ..., v n } and let σ be a closed, spanning walk on G, given as a list of these vertices. First we will produce a new planar multigraph G , called the reduction of G, relative to σ. The reduced graph is created through applications of the following procedures:
Edge removal: Remove any edge from G that was not traversed by σ. Edge duplication: If an edge of G is traversed more than once by σ, then create an additional edge in G for each additional traversal of this edge by this walk.
Thus, while the reduced graph G is still planar, it may no longer be simple. The faces of G are labeled with a + or a − sign as follows: the unbounded component is marked +; thereafter, if a face of G is adjacent to (shares an edge with) a + region, then it is marked −, and if a face of G is adjacent to a − region, then it is marked +. An example of a planar graph and its reduction relative to a closed, spanning walk is presented in Figure 2.
There is a simple relationship between the number of faces of G and the number of repeats of vertices in the closed, spanning walk σ. we will show that
(1) Φ = 2 + n i=1 m i .
The degree of the vertex v i is 2m i +2 and therefore the number of edges of G is half of the sum of the degrees taken over all vertices, that is
n i=1 (m i + 1). Thus n − n i=1 (m i + 1) + Φ = 2,
from which formula (1) follows.
Our argument now moves into a second phase, culminating in a simple formula relating the lengths of the positively and negatively signed faces of G . Let the faces of G be divided according to their signs and let {A + : 1 ≤ n + } be the labels for the positively signed faces and let {A − : 1 ≤ n − } be the labels for the negatively signed faces. Since each edge of G is adjacent to a positively and a negatively signed face, it follows that
n + i=1 |A + | = n − i=1 |A − |.
Let ∆ = n − − n + , the difference between the number of negatively and positively signed faces of G . Then
(2) n + i=1 (|A + | − 2) − n − i=1 (|A − | − 2) = 2|∆|.
We will modify this formula to incorporate the faces of G. Let the faces of G be labeled {F i : 1 ≤ i ≤ N }. Each face of G is contained by a unique face of G . For each i, 1 ≤ i ≤ N , let ε i be sign of the face of G that contains F i . We will show that
(3) N i=1 ε i (|F i | − 2) = 2|∆|.
To verify this claim, we will follow Grinberg's strategy: we will add to G , one at a time, those edges of G that were not traversed by σ. Such an edge must split a face of G into two sub-faces, each with the same sign as the parent face. For the sake of argument, let us say that a face labeled A + i is divided by an edge of G into two sub-faces, labeled A + i 1 and A + i 2 . Since the two sub-faces share exactly one edge, we have
|A + i | − 2 = (|A + i 1 | − 2) + (|A + i 2 | − 2)
. Hence we can substitute (|A + i 1 | − 2) + (|A + i 2 | − 2) for |A + i | − 2 in equation (2) and retain equality. We continue this process until all of these edges have been added. We have almost arrived at equation (3). The only difference corresponds to those faces of G that were created because an edge was traversed more than once by σ, the closed, spanning walk. Such a face has only two edges and thus contributes 0 in the sum. In this way, we have transformed equation (2)
into equation (3).
Our proof is nearly complete. By the definition of the Grinberg number of a graph and equation (3),
g(G) ≤ N i=1 ε i (|F i | − 2) = 2|∆|
Recall that n − and n + count the number of negatively and positively signed faces in G , respectively, and each of these must be at least 1. Since they sum to Φ, it must be that |∆| ≤ Φ − 2. Bearing in mind equation (1), we obtain
1 2 g(G) ≤ n i=1 m i ,
as was to be shown.
Some examples
In this section we present several examples of Theorem 2.1.
(1) The graph in Figure 1 has Grinberg number 4; thus, any closed, spanning walk on this graph must have two repeats of vertices. It is easy to find a closed, spanning walk with two repeats and thus this is optimal. (2) Consider the simple grid graph presented in Figure 3. The exterior face has 8 edges; each of the four interior faces has 4 edges. The Grinberg number of this graph is 2, and any closed, spanning walk on this graph must contain at least one repeated vertex. It is easy to find an example of a closed, spanning walk with one repeated vertex. (4) The graph presented in Figure 5 has an outside face with 26 edges and three interior faces, each with 14 edges. The Grinberg number is 12; thus, any closed, spanning walk on this graph must have at least 6 repeats. It is easy to find a closed, spanning walk with 6 repeats. For example, begin at a and walk around the outside of the graph, ending at a.
Additional observations
Given a closed, spanning walk on the planar graph G, let ρ count the number of repeats of vertices, let ν = n − − 1, and let π = n + − 1. In other words, ν is one less than the number of negatively signed regions in G and π is one less than the number of its positively signed regions in G .
Since the reduced graph must have at least one of each type of each signed region, ν and π are nonnegative. By equations (1) and (3), there a b c Figure 6. The graph on the left has Grinberg number 6. A Hamiltonian walk on this graph (with 3 repeats) is shown on the right.
exists an element f ∈ G (G) for which
f = ν + π 1 2 f = max{ν, π} − min{ν, π}
In particular, this shows that ρ = 1 2 f + 2 min{ν, π}.
This can be a helpful observation when searching for Hamiltonian walks. For example, consider a planar graph G with 8 interior faces, each an octagon, and an exterior face with 20 edges. The Grinberg set for this graph is {6, 18, 30, 42, 54}; hence, the number of repeats of vertices in a Hamiltonian walk on G must be at least 3 and odd. The graph pictured in Figure 6 has a Hamiltonian walk with the minimal number of repeats, while the graph pictured in Figure 7 has a Hamiltonian walk that requires 5 repeats. Figure 7. A graph, composed of octagons, that has a Hamiltonian walk with 5 repeats.
Figure 1 .
1A graph with Grinberg set {4, 12, 20, 28} an Grinberg number 4.
Figure 2 .
2For each i, 1 ≤ i ≤ n, let m i count the number of times vertex v i is repeated in the walk σ. To be clear about this, if the vertex v i appears only once in the walk σ, then m i = 0. Let Φ count the number of faces of the reduced graph G . By way of the Euler characteristic formula, A planar graph (left) and its reduction (right) based on the closed, spanning walk a, b, c, d, e, f, g, h, a, i, j, r, j, k, l, m, n, p, q, p, n, o, a. Notice that the reduced graph has 4 repeated vertices and 6 faces.
Figure 3 .
3A simple grid graph with Grinberg number 2.(3) The Grinberg number of a tree can be computed by duplicating each edge and computing the Grinberg number of the resulting planar graph. The altered version of the tree inFigure 4has an outside face with 20 edges and 10 interior faces with 2 edges each. The Grinberg number of this tree is 18 and any closed, spanning walk on this tree must contain at least 9 repeats of vertices.
Figure 4 .
4A tree and its alteration.
Figure 5 .
5Now augment the walk by adding a, b, c, d, c, b, a, e, f, g, f, e, a. This walk has 6 repeats of vertices: a (twice), b, c, e, and f . A graph with Grinberg number 12. A closed, spanning walk on this graph must have at least 6 repeats of vertices. (5) Figure 6 exhibits a graph with 8 interior faces, each an octagon, and an exterior face with 20 edges. The Grinberg number of this graph is 6. Any closed, spanning walk on this graph must have at least 3 repeats of vertices. A closed, spanning walk with exactly 3 repeats is pictured in the figure.
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. Furman University, Greenville, Usa E Sc, mail address: [email protected] University, Greenville, SC, USA E-mail address: [email protected]
| []
|
[
"Diophantine equations and the monodromy groups",
"Diophantine equations and the monodromy groups"
]
| [
"Dijana Kreso [email protected] \nGraz University of Technology\nSteyrergasse 30/II8010GrazAustria\n",
"Robert F Tichy [email protected] \nGraz University of Technology\nSteyrergasse 30/II8010GrazAustria\n"
]
| [
"Graz University of Technology\nSteyrergasse 30/II8010GrazAustria",
"Graz University of Technology\nSteyrergasse 30/II8010GrazAustria"
]
| []
| We study Diophantine equations of type f (x) = g(y), where both f and g have at least two distinct critical points and equal critical values at at most two distinct critical points. Some classical families of polynomials (f n ) n are such that f n satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K, at least two distinct critical points and all distinct critical values, and char(K) ∤ deg f , then the monodromy group of f is a doubly transitive permutation group. This is the same as saying that (f (x) − f (y))/(x − y) is irreducible over K. In particular, f cannot be represented as a composition of lower degree polynomials. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if f (x) = g(h(x)) with g, h ∈ K[x] and deg g > 1, then either deg h ≤ 2, or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points. | null | [
"https://arxiv.org/pdf/1601.07316v1.pdf"
]
| 119,164,412 | 1601.07316 | 0dd1aa81ed8d3b687167db1f74eb02b5a71a96e8 |
Diophantine equations and the monodromy groups
27 Jan 2016
Dijana Kreso [email protected]
Graz University of Technology
Steyrergasse 30/II8010GrazAustria
Robert F Tichy [email protected]
Graz University of Technology
Steyrergasse 30/II8010GrazAustria
Diophantine equations and the monodromy groups
27 Jan 2016Diophantine equationsmonodromy grouppermutation groupspolynomial decomposition
We study Diophantine equations of type f (x) = g(y), where both f and g have at least two distinct critical points and equal critical values at at most two distinct critical points. Some classical families of polynomials (f n ) n are such that f n satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K, at least two distinct critical points and all distinct critical values, and char(K) ∤ deg f , then the monodromy group of f is a doubly transitive permutation group. This is the same as saying that (f (x) − f (y))/(x − y) is irreducible over K. In particular, f cannot be represented as a composition of lower degree polynomials. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if f (x) = g(h(x)) with g, h ∈ K[x] and deg g > 1, then either deg h ≤ 2, or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points.
Introduction
Diophantine equations of type f (x) = g(y) have been of long-standing interest to number theorists. A defining equation of an elliptic curve is a prominent example of such equations. By Siegel's classical theorem [38], it follows that an affine algebraic curve defined over a number field has only finitely many S-integral points, unless it has genus zero and no more than two points at infinity. Ever since Siegel's theorem, one of the driving questions was to classify polynomials f, g for which the equation f (x) = g(y) has infinitely many solutions in S-integers x, y. The classification was completed by Bilu and Tichy [5] in 2000, building on the work of Fried and Schinzel. It turns out that for the curve f (x) − g(y) = 0 to have genus zero and no more than two points at infinity, f and g must be representable as a composition of lower degree polynomials in a certain prescribed way.
The possible ways of writing a polynomial as a composition of lower degree polynomials were studied by several authors, starting with Ritt [34] in the 1920s. Results on this topic have many applications to various areas of mathematics. See [35,47] for an overview of the theory and applications.
The theorem of Bilu and Tichy was used to prove the finiteness of integral solutions to various equations of type f (x) = g(y) with f, g ∈ Q[x], see our recent survey paper [25] and the references therein. In this paper, we prove two theorems which simultaneously generalize many of these results.
For a number field K, a finite set S of places of K that contains all Archimedean places and the ring O S of S-integers of K, we say that the equation f (x) = g(y) has infinitely many solutions x, y with a bounded O S -denominator if there exists a nonzero δ ∈ O S such that there are infinitely many solutions x, y ∈ K with δx, δy ∈ O S .
For a polynomial f , the roots of the derivative f ′ are called critical points, and the values of f at critical points are called critical values. If for critical points β i 's of f , one has f (β i ) = f (β j ) when β i = β j , then f is said to have all distinct critical values. Corollary 1.2. Let K be a number field, S a finite set of places of K that contains all Archimedean places and O S the ring of S-integers of K. Let a 1 , a 2 , a 3 , b 1 , b 2 ∈ K with a 1 a 2 b 1 b 2 = 0. Let further n 1 , n 2 , m 1 , m 2 ∈ N be such that n 1 > n 2 , m 1 > m 2 , gcd(n 1 , n 2 ) = 1, gcd(m 1 , m 2 ) = 1 and n 1 , m 1 ≥ 3. Then the equation Corollary 1.2 follows from Theorem 1.1. Namely, if f (x) = a 1 x n 1 + a 2 x n 2 + a 3 , then clearly f ′ (x) = x n 2 −1 (a 1 n 1 x n 1 −n 2 + a 2 n 2 ), so f ′ has at least two distinct critical points. Also, xf ′ (x) = n 1 (f (x) − a 3 ) + a 2 (n 1 − n 2 )x n 2 . If f (α) = f (β) for distinct critical points α and β of f , then α n 2 = β n 2 . Then from f ′ (α) = f ′ (β) = 0 it follows that α n 1 = β n 1 . Since gcd(n 1 , n 2 ) = 1, we have α = β. It can be shown that if (1.4) holds, then either µ(0) = 0, or deg f = deg g ≤ 3. (Details can be found in [22], where equations of type (1.3) with one or both trinomials replaced by polynomials with an arbitrary but fixed number of nonconstant terms, are studied.) Corollary 1.2 generalizes the main result of Péter, Pintér and Schinzel [33,Thm. 1], who proved it in the case when K = Q and O S = Z. They generalized the results of Mignotte and Pethő [29,Thm. 1], of Bugeuad and Luca [8,Thm 6.2], and of Luca [28,Prop. 3].
Polynomials with a fxed number of nonconstant terms, but with the degrees of the terms and the coefficients that may vary, are called lacunary. Such polynomials have been studied from various viewpoints. In [22,23], equations of type f (x) = g(y), where f and g are arbitrary lacunary polynomials, are studied. One can study such questions via methods presented in this paper. In such investigations, of importance are also results about the behavior of lacunary polynomials with respect to functional composition. The latter topic has been of interest for a long time, and some remarkable results have been achieved in the last decade. For an account of the theory, we direct the reader to [20,45,46]. Theorem 1.1 implies the finiteness of integral solutions to the equation x n + x n−1 + · · · + x + 1 = y m + y m−1 + · · · + y + 1, with m > n ≥ 3. Indeed, let f (x) = x n + x n−1 + · · · + x + 1. Then f (x) + (x − 1)f ′ (x) = (n + 1)x n . So, f has all distinct critical values unless there exist two distinct critical points α and β of f with α n = β n . If so, then α n−1 + · · · + α + 1 = β n−1 +· · ·+β+1, and hence (1−α n )/(1−α) = (1−β n )/(1−β). Thus, α = β. The finiteness of integral solutions to this equation was shown by Davenport, Lewis and Schinzel [10]. Further corollaries of Theorem 1.1 are given in Section 6. In the sequel we explain our methods.
For a field K and f ∈ K[x] with f ′ (x) = 0, the Galois group of f (x) − t over K(t), where t is transcendental over K, seen as a permutation group of the roots of this polynomial, is called the monodromy group of f , and is denoted by Mon(f ). Polynomials with all simple critical points and all distinct critical values are called Morse. Serre [37] showed that for an arbitrary field K and Morse f ∈ K[x] such that char(K) ∤ deg f , the monodromy group of f is symmetric. The same was previously shown in [21] and [7] for the cases K = C, and K a finite field, respectively. Turnwald [43] showed that in Serre's result the condition on f can be relaxed from requiring that it has all simple critical points to requiring that it has one simple critical point (and all distinct critical values). In Section 5, we prove Proposition 1.5 and recover these related results . Proposition 1.5 is equivalent to saying that if f ∈ K[x] with char(K) ∤ deg f has at least two distinct critical points and all distinct critical values, then (f (x) − f (y))/(x − y) is irreducible over K. In particular, such f cannot be represented as a composition of lower degree polynomials. In relation to that, we mention some recent results of Pakovich [31]. For complex rational functions f = f 1 /f 2 , g = g 1 /g 2 , he analysed the irreducibility of the curve f 1 (x)g 2 (y) − f 2 (x)g 1 (y) = 0 (obtained by equating to zero the numerator of f (x) = g(y)), and showed several results in the case when f and g have "few" common critical values. He further showed that if a complex rational function f = f 1 /f 2 cannot be represented as a composition of lower degree rational functions, and has at least one simple critical point
Proposition 1.5. Let K be a field. If f ∈ K[x], char(K) ∤ deg f ,x 0 such that f (x 1 ) = f (x 0 ) for any other critical point x 1 of f , then the curve (f 1 (x)f 2 (y) − f 2 (x)f 1 (y))/(x − y) = 0 is irreducible.
In Section 3, we discuss some relations to our results.
The above results are proved in Section 3, and then used in Section 5 together with the finiteness criterion of Bilu and Tichy [5] [8,28,33]. Proposition 1.6. Let K be a field with char(K) = 0 and f ∈ K[x] with at least two distinct critical points and equal critical values at at most two distinct critical points. If f (x) = g(h(x)) with g, h ∈ K[x] and t = deg g > 1, then either k = deg h ≤ 2, or the derivative of f satisfies either
(1.7) f ′ (x) = a ′ (x − x 0 ) k 0 t−1 (x − x 1 ) k 1 t−1 (kx − k 0 x 1 − k 1 x 0 ), with a ′ ∈ K, k 0 , k 1 ≥ 1 such that k 0 + k 1 = k ≥ 3 and distinct x 0 , x 1 ∈ K, or (1.8) f ′ (x) = a ′ (x − x 0 ) 2t 0 +1 (x − x 1 ) t 0 (x − y 0 ) 2t 1 +1 (x − y 1 ) t 1 , with a ′ ∈ K, deg h = 3, t 0 , t 1 ≥ 1 such that t 0 + t 1 = t − 1, and distinct x 0 , x 1 , y 0 , y 1 ∈ K that satisfy 3x 0 − y 0 = 2y 1 , 3y 0 − x 1 = 2x 1 .
Some well-known families of polynomials (f n ) n satisfy that for all n, f n has at least two distinct critical points and equal critical values at at most two distinct critical points. Stoll [39] observed that this is the case for the families of polynomials (f n ) n with real coefficients that satisfy a differential equation
σ(x)f ′′ n (x) + τ f ′ n (x) − λ n f n (x) = 0, n ≥ 0 for some σ, τ ∈ R[x] with deg σ ≤ 2, deg τ ≤ 1, λ n ∈ R \ {0} and nonvanishing σ ′ − 2τ .
Classical orthogonal polynomials such as Hermite, Laguerre, Jacobi, Gegenbauer and Bessel polynomials satisfy such a differential equation. In [2], it is shown that x(x + 1) · · · (x + n − 1) for n ≥ 3 has at least two distinct critical points and equal critical values at at most two distinct critical points. There are many results in the literature on Diophantine equations of type f (x) = g(y) with f (x) = x(x + 1) · · · (x + n − 1), see e.g. [2,3,16]. For instance, by the celebrated theorem of Erdős and Selfridge, the equation x(x + 1) · · · (x + n − 1) = y n for m, n ≥ 2 has no solutions in positive integers x, y. Further families of polynomials with this property can be found in Section 6.
By Proposition 1.6 it follows that if K is a field with char(K) = 0, f ∈ K[x] has at least three simple critical points, equal critical values at at most two distinct critical points, and f (x) = g(h(x)) with g, h ∈ K[x] and deg g > 1, then deg h ≤ 2. It is easy to see (see Lemma 3.8 and the text below) that this holds if f has only simple critical points and equal critical values at at most two distinct critical points. This fact was used in [2,11,14,39] in the study of Diophantine equations of type f (x) = g(y) via Bilu and Tichy's theorem, to find the possible decompositions of f and g. The proofs in those papers are completed by a lengthy analyzis of subcases implied by the criterion, and rely on particular properties of f and g. Results of these papers are, to the most part, generalized by the following theorem. Theorem 1.9. Let K be a number field, S a finite set of places of K that contains all Archimedean places, O S the ring of S-integers of K and f, g ∈ K[x] with deg f ≥ 3, deg g ≥ 3 and deg f < deg g.
If f and g both have at least two distinct critical points and equal critical values at at most two distinct critical points, and do not satisfy (1.7) nor (1.8), then the equation f (x) = g(y) has finitely many solutions with a bounded O S -denominator unless either (deg f, deg g) ∈ {(3, 4), (3,5), (4,5), (4,6)}, or f is indecomposable and g(x) = f (ν(x)) for some quadratic
ν ∈ K[x].
In particular, if f and g have at least three simple critical points and equal critical values at at most two distinct critical points, then the equation f (x) = g(y) has finitely many solutions with a bounded O S -denominator, unless (deg f, deg g) = (4, 5), or f is indecomposable and g(x) = f (ν(x)) for some quadratic ν ∈ K[x]. Theorem 1.9 is proved in Section 5. In relation to Theorem 1.9, we further list all pairs of polynomials (f, g) with (deg f, deg g) ∈ { (3,4), (3,5), (4,5), (4,6)}, with at least two distinct critical points and equal critical values at at most two distinct critical points, for which the equation f (x) = g(y) has infinitely many solutions with a bounded O S -denominator. See Theorem 5.5. The case g(x) = f (ν(x)) with indecomposable f and deg ν = 2 in Theorem 1.9, can be examined by comparison of coefficients of the involved polynomials. It is usually simple to check if this holds. A different way to handle this special case can be found in Section 5. This approach relies on Ritt's [34] and Engstrom's [15] results about the essential uniqueness of prime factorization of polynomials over fields of characteristic zero with respect to composition. In Section 5, we further address the case deg f = deg g of Theorem 1.9 . In Section 6, we discuss applications of this theorem. Theorem 1.1 and Theorem 1.9 are ineffective since they rely on the main result of [5], which is ineffective.
Finiteness Criterion
In this section we present the finiteness criterion of Bilu and Tichy [5].
Let K be a field, a, b ∈ K \ {0}, m, n ∈ N, r ∈ N ∪ {0}, p ∈ K[x] be a nonzero polynomial (which may be constant) and D n (x, a) be the n-th Dickson polynomial with parameter a given by
(2.1) D n (x, a) = ⌊n/2⌋ j=0 n n − j n − j j (−a) j x n−2j .
Standard pairs of polynomials over K are listed in the following table.
kind standard pair (or switched) parameter restrictions first (x m , ax r p(x) m ) r < m, gcd(r, m) = 1, r + deg p > 0 second (
x 2 , (ax 2 + b)p(x) 2 ) - third (D m (x, a n ), D n (x, a m )) gcd(m, n) = 1 fourth (a −m 2 D m (x, a), −b −n 2 D n (x, b)) gcd(m, n) = 2 fifth ((ax 2 − 1) 3 , 3x 4 − 4x 3 ) -
We further call the pair D m x, a m/d , −D n x cos(π/d), a n/d (or switched), with d = gcd(m, n) ≥ 3 and cos(2π/d) ∈ K, a specific pair over K. If b, cos(2α) ∈ K, then D n (x cos α, b) ∈ K[x]. (This follows from b n D n (x, a) = D n (bx, b 2 a) which holds for any a, b, see [5,Sec. 3].) Thus, a specific pair over K is indeed a pair of polynomials with coefficients in K.
Theorem 2.2. Let K be a number field, S a finite set of places of K that contains all Archimedean places, O S the ring of S-integers of K, and f, g ∈ K[x] nonconstant. Then the following assertions are equivalent.
-The equation f (x) = g(y) has infinitely many solutions with a bounded O S -denominator;
-We have
(2.3) f (x) = φ (f 1 (λ(x))) & g(x) = φ (g 1 (µ(x))) , where φ ∈ K[x], λ, µ ∈ K[x]
are linear polynomials, and (f 1 , g 1 ) is a standard or specific pair over K such that the equation f 1 (x) = g 1 (y) has infinitely many solutions with a bounded O S -denominator.
We remark that in [4], in relation to Theorem 2.2, the authors asked and answered the following question: Given f, g ∈ Q[x] and δ ∈ Z, is it true that all but finitely many rational solutions to f (x) = g(y) with denominator δ also satisfy the equation f 1 (λ(x)) = g 1 (µ(y))? Unfortunately, this is not true in general, and some counterexamples are not hard to find. In [4,Thm.4], the authors found all counterexamples to this statement.
Dickson polynomials
For various properties of Dickson polynomials see [5,Sec. 3]. We now list some, which will be of importance in the sequel in relation to Theorem 2.2. Here, K is any field of characteristic zero. For n ≥ 2, n-th primitive root of unity ζ n ∈ K, α k = ζ k n + ζ −k n and β k = ζ k n − ζ −k n , we have:
D n (x, a) − D n (y, a) = (x − y) (n−1)/2 k=1 (x 2 − α k xy + y 2 + β 2 k a) when n is odd, D n (x, a) − D n (y, a) = (x − y)(x + y) (n−2)/2 k=1 (x 2 − α k xy + y 2 + β 2 k a)
when n is even.
(2.4)
Dickson polynomials further satisfy the following differential equation
(2.5) (x 2 − 4α)D ′′ n (x, a) + xD ′ n (x, a) − n 2 D n (x, a) = 0, n ≥ 0. By letting f (x) := D n (x, a) 2 − (x 2 − 4a)/n 2 D ′ n (x, a) 2 , from (2.5) it follows that f ′ (x) = 0, so f is constant. This implies that D n (x, a) has at most two distinct critical values. In fact, it is well known that if D ′ n (x 0 , a) = 0, then D n (x 0 , a) ∈ {±2a n/2 }, see [5, Sec. 3].
It follows that Dickson polynomial D n (x, a) with a = 0 has only simple critical points. We have the following corollary.
Polynomial decomposition via Galois theory
Throughout this section K is an arbitrary field with char(K) = 0.
A polynomial f ∈ K[x] with deg f > 1 is called indecomposable (over K) if it cannot be written as the composition f (x) = g(h(x)) with g, h ∈ K[x], deg g > 1 and deg h > 1.
Otherwise, f is said to be decomposable. Any representation of f as a functional composition of polynomials of degree > 1 is said to be a decomposition of f .
If µ ∈ K[x] is of degree 1, then there exists µ −1 ∈ K[x] such that µ • µ −1 (x) = µ −1 • µ(x) = x.
(By comparison of degrees one sees that no such polynomial exists when deg µ > 1). Then µ −1 is said to be the inverse of µ with respect to functional composition. This explains the assumption deg g > 1, deg h > 1 in the definition of indecomposable polynomials.
Note that for decomposable f ∈ K[x] we may write without loss of generality
f (x) = g(h(x)) with g, h ∈ K[x], deg g ≥ 2, deg h ≥ 2, h(x) monic and h(0) = 0. (3.1) Namely, if f = g • h with g, h ∈ K[x] \ K, then there exists linear µ ∈ K[x] such that µ • h is monic and µ(h(0)) = 0. Clearly f = g • µ −1 • (µ • h). Proposition 3.2. Let K be a field with char(K) = 0. Then f is indecomposable over K if and only if it is indecomposable over K.
Proposition 3.2 is due to Fried and McRae [18]. To see that it holds,
let f ∈ K[x] and g, h ∈ K with deg ≥ 2, deg h ≥ 2, h monic and h(0) = 0 be such that f = g • h, as in (3.1). Comparison of coefficients yields g, h ∈ K[x].
We now recall the definition of the monodromy group given in the introduction. By Gauss's lemma it follows that f (X) − t from Definition 3.3 is irreducible over K(t), so Mon(f ) is a transitive permutation group. Since char(K) = 0, f (X) − t is also separable. Let x be a root of f (X) − t in its splitting field L over K(t). Then t = f (x) and Mon(f ) = Gal(L/K(f (x))) is viewed as a permutation group on the conjugates of x over K(f (x)).
Lüroth's theorem (see [35, p. 13]) states that for fields K, L satisfying
K ⊂ L ⊆ K(x) we have L = K(f (x)) for some f ∈ K(x)
. This theorem provides a dictionary between decompositions of f ∈ K[x] and fields between K(f (x)) and K(x). These fields correspond to groups between the two associated Galois groups -Gal(L/K(f (x))) = Mon(f ) and Gal(L/K(x)) (stabilizer of x in Mon(f )). Find more about the Galois theoretic setup for addressing decomposition questions in [26] and [47].
In [47], Ritt's [34] Galois theoretic approach to decomposition questions is presented in a modernized and simplified language, and various new results are proved. In [26], the authors adopted this modernized language and examined the different ways of writing a cover of curves over a field K as a composition of covers of curves over K of degree at least 2 which cannot be written as the composition of two lower-degree covers. By the generalization to the framework of covers of curves, which provides a valuable perspective even when one is only interested in questions about polynomials, several improvements on previous work were made possible.
The monodromy group
We now list some well-known properties of the monodromy group that will be used in the sequel, sometimes without particular reference. Here, K is any field of characteristic zero.
A transitive permutation group G acting on a set X is called primitive if it preserves no nontrivial partition of X (trivial partitions are those consisting either of one set of size #X or of #X singletons). A permutation group G acting on a set X with #X ≥ 2 is called doubly transitive when, for any two ordered pairs of distinct elements (x 1 , y 1 ) and (x 2 , y 2 ) in X 2 , there is g ∈ G such that y 1 = gx 1 and y 2 = gx 2 . See [9] for a reminder about transitive group actions. The following two lemmas are due to Ritt [34] and Fried [17].
Lemma 3.4. If K is a field with char(K) = 0 and f ∈ K[x], then f is indecomposable if and only if Mon(f ) is primitive.
A transitive permutation group is primitive if and only if point stabilizers are maximal subgroups, see [9]. By Lüroth's theorem, f ∈ K[x] is indecomposable if and only if there are no proper intermediate fields of the extension K(x)/K(f (x)). By the Galois correspondence, this is the same as saying that there are no proper subgroups between Mon(f ) and its point stabilizers. This proves Lemma 3.4.
Lemma 3.5. If K is a field with char(K) = 0 and f ∈ K[x], then (f (x) − f (y))/(x − y) ∈ K[x, y] is irreducible over K if and only if Mon(f ) is doubly transitive. Let φ(x, y) = (f (x) − f (y))/(x − y) ∈ K[x, y].
In short, Lemma 3.5 follows from the fact that a group is doubly transitive on X if and only if point stabilizer of any x 0 ∈ X acts transitively on X \ {x 0 }, see [9]. Thus, Mon(f ) is doubly transitive if and only if φ(x, x 0 ) is irreducible over K(x 0 ). Since x 0 and x are algebraically independent over K, this is equivalent to irreducibility of φ(x, y) over K(y), which is by Gauss Lemma equivalent to irreducibility of φ(x, y) over K. For a detailed proof, see [43]. Lemma 3.6 has been long known in the case K = C, but derived in the language of Riemann surfaces. Turnwald [43] gave an elementary proof. The proofs of all the above mentioned results can be found in [43] and [35].
Every doubly transitive group is primitive. This translates to saying that if φ(x, y) = (f (x)−f (y))/(x−y) ∈ K[x, y] is irreducible over K, then f is indecomposable, which clearly holds. On the other hand, if Mon(f ) is primitive it is doubly transitive as soon as it is of composite degree n. This follows by a theorem of Schur (see [44, p. 34]), which states that a primitive permutation group of composite degree n which contains an n-cycle, is doubly transitive. Burnside showed (see [32, p. 127]) that if a transitive permutation group of prime degree is not doubly transitive, it may be identified with a group of affine transformations of Z/pZ. The latter two results of Schur and Burnside are classical results about permutation groups and were among the main ingredients of Fried's paper [17] in proving the following theorem.
Theorem 3.7. Let K be a field with char(K) = 0 and f ∈ K[x] with deg f ≥ 3. The following assertions are equivalent.
i) (f (x) − f (y))/(x − y) is irreducible over K, ii) f (x) is indecomposable and if n is an odd prime then f (x) = e 1 D n (c 1 x + c 0 , α) + e 0 with e i , c i , α ∈ K with α, a, b, c ∈ K, with a = 0 if n = 3, where D n (x, a) is the n-th Dickson polynomial with parameter a.
Here are the main ingredients of the proof of Theorem 3.7, as presented by Turnwald [43].
Note that if f is decomposable, then φ(x, y) = (f (x) − f (y))/(x − y) is clearly reducible over K.
Since also (2.4) holds, the first statement clearly implies the second. If f is of composite
degree and f (x) = e 1 D n (c 1 x + c 0 , α) + e 0 with e i , c i , α ∈ K, i.e. f is linearly related to Dickson polynomial, then f is decomposable by D mn (x, a) = D m (D n (x, a), a n ) for m, n ∈ N.
To prove the converse, assume that f is indecomposable. Then Mon(f ), where f is seen as with coefficients in K, is primitive, by Proposition 3.2 and Proposition 3.4. Assume that Mon(f ), where f is seen as with coefficients in K, is not doubly transitive. By Lemma 3.5, this is the same as saying that
(f (x) − f (y))/(x − y) is reducible over K. Then Mon(f ) is of prime degree p by Schur's result. By Burnside's result, Mon(f ) may be identified with a group of affine transformations ax + b of Z/pZ. If a = 1, b = 0, this permutation is identity, if a = 1, b = 0 it is a p-cycle,
and if a = 1, then it is of cycle type 1, r, . . . , r, where r is the least positive integer such that a r = 1. By Lemma 3.6 it follows that for any y 0 ∈ K, f (x) − y 0 is either a p-th power or has one simple root and (n − 1)/r roots of multiplicity r. The only polynomials that satisfy the latter property are those linearly related to a Dickson polynomial. The proof is technical and can be found in [43].
Polynomials with distinct critical values
In this section as well, K is an arbitrary field of characteristic zero. Lemma 3.8. Let K be a field with char(K) = 0 and f, g, h ∈ K[x] such that f (x) = g(h(x)) and deg g > 1. Then for every γ 0 ∈ K a root of g ′ and γ = g(γ 0 ) we have that every root of
h(x) − γ 0 is a root of both f (x) − γ and f ′ (x). Proof. If h(x 0 ) = γ 0 , then f (x 0 ) = g(h(x 0 )) = g(γ 0 ) = γ and f ′ (x 0 ) = g ′ (h(x 0 ))h ′ (x 0 ) = g ′ (γ 0 )h ′ (x 0 ) = 0.
Recal that a polynomial is called Morse (initially by Serre [37, p. 39]) if it has all simple critical points and all distinct critical values.
Note that if f ∈ K[x] is Morse, then f is indecomposable by Lemma 3.8. If f ∈ K[x]
has all simple critical points and equal critical values at at most two distinct critical points, by Lemma 3.8
it follows that if f (x) = g(h(x)) with deg g > 1, then deg h ≤ 2.
By following the approach of Turnwald [43] and by using Fried's techniques for proving Theorem 3.7, described in the previous section, we show the following result. Proof. We first show that f is indecomposable. Assume to the contrary and write f (x) = g(h(x)) with deg g ≥ 2, deg h ≥ 2, h monic and h(0) = 0 (as in (3.1)). Let γ 0 ∈ K be a root of g ′ and γ = g(γ 0 ). Then every root of h(x) − γ 0 is a root of both f (x) − γ and f ′ (x) by Lemma 3.8. If there exist two distinct roots of h(x) − γ 0 , say x 0 and x 1 , then
f ′ (x 0 ) = f ′ (x 1 ) = 0 and f (x 0 ) = f (x 1 ) = γ, which cannot be by assumption. Thus h(x) − γ 0 does not have two distinct roots, i.e. h(x) = (x − x 0 ) k + γ 0 , where k = deg h ≥ 2.
Also, if there exist two distinct roots of g ′ , say γ 0 and γ 1 , then analogously h(
x) = (x − x 1 ) k + γ 1 for some x 1 ∈ K. Then (x − x 0 ) k + γ 0 = (x − x 1 ) k + γ 1 . By taking derivative, we get k(x − x 0 ) k−1 = k(x − x 1 ) k−1 , wherefrom x 0 = x 1 , since k − 1 ≥ 1. Then also γ 0 = γ 1 . Thus g ′ (x) = a(x − γ 0 ) t−1 , where t = deg g ≥ 2, a ∈ K. Then f ′ (x) = g ′ (h(x))h ′ (x) = ak(h(x) − γ 0 ) t−1 (x − x 0 ) k−1 = = ak(x − x 0 ) k(t−1) (x − x 0 ) k−1 = ak(x − x 0 ) n−1 .
However, this contradicts the assumption that f ′ has at least two distinct roots. Thus, Mon(f ) is primitive.
Assume that Mon(f ) is not doubly transitive and deg f > 3. By Fried's proof of Theorem 3.7 (given below the theorem), it follows that for any y 0 ∈ K, f (x) − y 0 is either a p-th power, or has one simple root and (p − 1)/r roots of multiplicity r. The former cannot be since f has at least two distinct critical points. Assume the latter. If x 0 is a critical point of f , then the multiplicities of the roots of f (x) − f (x 0 ) are 1, 1, . . . , 1, k, where k ≥ 2 is the multiplicity of x 0 , since f has all distinct critical values. By assumption, k = p − 1, where p = deg f . If x 1 = x 0 is another root of f ′ , then in the same way the multiplicity of x 1 is p − 2. So, 2(p − 2) ≤ p − 1, and p ≤ 3, a contradiction. If p = 3, then k = 2, and Mon(f ) contains an element of cycle type 1, 2 by Lemma 3.6. Since Mon(f ) is a primitive permutation group and contains a transposition, it is symmetric by Jordan's theorem [44,Thm. 13.3]. In particular, Mon(f ) is doubly transitive.
Remark 3.10. To show that Mon(f ) in Theorem 3.9 is doubly transitive, after it is shown that it is primitive, it suffices to show that f is not linearly related to Dickson polynomial, by Theorem 3.7. By Corollary 2.6, if f is of type f (x) = e 1 D n (c 1 x + c 0 , α) + e 0 with n > 3, e i , c i , α ∈ K and α = 0, then f has two distinct critical points with equal critical values, which contradicts the assumption on f . If α = 0 and n ≥ 3, then f (x) = e 1 (c 1 x + c 0 ) n + e 0 has no two distinct critical points, a contradiction with the assumption on f . If f has two distinct critical points, but has at two equal critical values, then f can be decomposable.
Indeed, f (x) = (x 2 − 1) 2 , f ′ (x) = 4x(x 2 − 1), f (1) = f (−1) = 0.
If K is a field with char(K) = 0 and f ∈ K[x] has a critical point of multiplicity at most 2 and all distinct critical values, then Mon(f ) is either alternating or symmetric. Namely, one easily sees that for such f , Mon(f ) is primitive (since for such f either deg f ∈ {2, 3} or Proposition 3.9 applies). If x 0 is a root of f ′ of multiplicity at most 2, it follows that all the roots of f (x) − f (x 0 ), but x 0 , are of multiplicity 1 (since f has all distinct critical values), and x 0 is of multiplicity ≤ 3. So, Mon(f ) contains either a 2-cycle or a 3-cycle by Lemma 3.6. Since Mon(f ) is primitive and contains a 2-cycle or 3-cycle, it is either alternating or symmetric by [44,Thm. 13.3]. If it contains a 2-cycle it is symmetric. In this way Turnwald [43] showed that if f ∈ K[x] has one simple critical point and all distinct critical values, then Mon(f ) is symmetric. This in particular implies that a trinomial f (x) = a 1 x n 1 + a 2 x n 2 + a 3 , with gcd(n 1 , n 2 ) = 1 and a i 's in a field K with char(K) ∤ deg f , has symmetric monodromy group (via proof given below the Corollary 1.2). Also, the monodromy group of f (x) = x n + x n−1 + · · · + x + 1 is symmetric, since it is Morse (by the proof given in the introduction).
Clearly, if f ∈ K[x] is indecomposable and has a critical point x 0 of multiplicity at most 2 such that f (x 1 ) = f (x 0 ) for any other critical point x 1 of f , then Mon(f ) is either alternating or symmetric by the same argument as above. If a group is symmetric or alternating, then it is doubly transitive, as soon as it is of degree at least 4, see [9]. In particular, if f ∈ K[x] with deg f ≥ 4 is indecomposable and has a critical point x 0 of multiplicity either 1 or 2 such that f (x 1 ) = f (x 0 ) for any other critical point We mention some sufficient conditions for f to be indecomposable. Clearly, f is inde- [12,13], it is shown that if f (x) = a n x n + a n−1 x n−1 + · · · + a 1 x + a 0 ∈ Z[x] and gcd(n, a n−1 ) = 1, or f is an odd polynomial and gcd(n, a n−2 ) = 1, then f is indecomposable.
x 1 of f , then (f (x) − f (y))/(x − y) is irreducible. If deg f = 3,composable if deg f is prime. If f ∈ K[x] is such that the derivative f ′ is irreducible over K, then f is indecomposable over K by f ′ (x) = g ′ (h(x))h ′ (x). In
Pakovich [31] further showed that if f, g ∈ K[x] have at most one common critical value, then f (x) − g(y) ∈ K[x, y] is irreducible. In relation to Theorem 2.2, this shows that for such f and g, there does not exist φ ∈ K[x] with deg φ > 1 such that Equation (2.3) holds. So, in order to show the finiteness of solutions with a bounded denominator of the equation f (x) = g(y) for such f and g, one needs to check if f and g are linearly related to some standard or specific pair.
Positive characteristic
Throughout the paper, K is a field of characteristic zero. We restricted to this case for simplicity and since our main results, namely Theorem 1.1 and Theorem 1.9, hold over number fields. However, several results hold, under certain assumptions, over fields of positive characteristic. We now show that Proposition 1.5 holds when K is an arbitrary field and f ∈ K[x] is such that char(K) ∤ deg f .
Recall that for an arbitrary field K, and f ∈ K[x] with f ′ (x) = 0, the monodromy group of f is defined as the Galois group of f (x) − t over K(t), where t is transcendental over K, and is seen as a permutation group of the roots of this polynomial. Lemma 3.4 and Lemma 3.5 hold whenever f ′ (x) = 0, see [43].
One easily sees that by the same proof as in Proposition 1.5, if K is a field and f ∈ K[x] with char(K) ∤ deg f and at least two distinct critical points and all distinct critical values, then Mon(f ) is primitive.
Over an arbitrary field K, for a ∈ K and Dickson polynomial D n (x, a) the following holds: D n (λx, λ 2 ) = λ n D n (x, 1) for λ 2 = a and (D n (x, 1) 2 − 4) · n 2 = (x 2 − 4)D ′ n (x, 1) 2 . See e.g. [6]. Thus, D n (x, a) has at most two distinct distinct critical values. If n ≥ 4, D n (x, a) has at least two equal critical values.
Fried proved Theorem 3.10 assuming that char(K) ∤ deg f and that char(K) does not divide the multiplicites of zeros of f (x)−c ∈ K[x] for any c ∈ K. By the results of Müller [30], it follows that Theorem 3.7 holds if one assumes only char(K) ∤ deg f , see also [35, p. 57]. Then by the same proof as in Remark 3.10, it follows that Proposition 1.5 holds also when K is arbitrary and char(K) ∤ deg f .
Polynomials with at most two equal critical values
Proof of Proposition 1.6. Assume f (x) = g(h(x)) with deg g ≥ 2, deg h > 2, and without loss of generality that h is monic and h(0) = 0 (as in (3.1)).
Let γ 0 ∈ K be a root of g ′ and γ = g(γ 0 ). Then every root of h(x) − γ 0 is a root of both f (x) − γ and f ′ (x) by Lemma 3.8. If there exist three distinct roots of h(
x) − γ 0 , say x 0 , x 1 , x 2 , then f ′ (x 0 ) = f ′ (x 1 ) = f ′ (x 2 ) = 0 and f (x 0 ) = f (x 1 ) = f (x 2 ) = γ, which cannot be by assumption. Thus h(x) − γ 0 does not have three distinct roots, i.e. h(x) = (x−x 0 ) k 0 (x−x 1 ) k 1 + γ 0 for some distinct x 0 , x 1 ∈ K, and k 0 + k 1 = k = deg h ≥ 3, k 0 , k 1 ≥ 0.
If there do not exist two distinct roots of g ′ , then
g ′ (x) = a(x−γ 0 ) t−1 , where t = deg g ≥ 2, a ∈ K, and f ′ (x) = g ′ (h(x))h ′ (x) = a(h(x) − γ 0 ) t−1 h ′ (x) = = a(x − x 0 ) k 0 (t−1) (x − x 1 ) k 1 (t−1) (x − x 0 ) k 0 −1 (x − x 1 ) k 1 −1 (kx − k 0 x 1 − k 1 x 0 ) = = a(x − x 0 ) k 0 t−1 (x − x 1 ) k 1 t−1 (kx − k 0 x 1 − k 1 x 0 ),
so (1.7) holds. If so, then k 0 , k 1 ≥ 1, since otherwise f ′ has no two distinct roots.
Assume henceforth that there exist two distinct roots of g ′ , say γ 0 and γ 1 . Since h(x) = (x − x 0 ) k 0 (x − x 1 ) k 1 + γ 0 for some distinct x 0 , x 1 ∈ K, and k 0 + k 1 = k = deg h, k 0 , k 1 ≥ 0, then analogously h(x) = (x − y 0 ) l 0 (x − y 1 ) l 1 + γ 1 for some y 0 , y 1 ∈ K, and l 0 + l 1 = k = deg h, l 0 , l 1 ≥ 0. Assume without loss of generality that k 0 ≥ k 1 and l 0 ≥ l 1 . If k 1 = 0 and l 1 = 0, i.e. if both h(x) − γ 0 and h(x) − γ 1 do not have two distinct roots, then h(
x) − γ 0 = (x − x 0 ) k and h(x) − γ 1 = (x − y 0 ) k . Then h ′ (x) = k(x − x 0 ) k−1 = k(x − y 0 ) k−1 , and x 0 = y 0 since k − 1 > 1.
Then also γ 0 = γ 1 , a contradiction. If h(x) − γ 0 does not have two distinct roots, but h(x) − γ 1 does, so if k 1 = 0 and l 1 > 0, then
h ′ (x) = k(x − x 0 ) k−1 = (x − y 0 ) l 0 −1 (x − y 1 ) l 1 −1 (kx − l 0 y 1 − l 1 y 0 ).
It follows that kx 0 = l 0 y 1 + l 1 y 0 , l 1 = 1, and x 0 = y 0 , l 0 = k − 1. Then x 0 = y 1 , and y 0 = y 1 , a contradiction. We conclude that k 0 , k 1 , l 0 , l 1 ≥ 1, and
(4.1) h(x) = (x − x 0 ) k 0 (x − x 1 ) k 1 + γ 0 = (x − y 0 ) l 0 (x − y 1 ) l 1 + γ 1 .
By taking derivative h ′ (x) we get
(4.2) (x − x 0 ) k 0 −1 (x − x 1 ) k 1 −1 (kx − k 0 x 1 − k 1 x 0 ) = (x − y 0 ) l 0 −1 (x − y 1 ) l 1 −1 (kx − l 0 y 1 − l 1 y 0 ).
If (4.2) holds with k 1 = 1, then l 1 = 1, k 0 = l 0 = k − 1 > 1 and
(x − x 0 ) k 0 −1 (kx − k 0 x 1 − x 0 ) = (x − y 0 ) k 0 −1 (kx − k 0 y 1 − y 0 )
.
If x 0 = y 0 , then kx − x 0 − k 0 x 1 = kx − x 0 − k 0 y 1
, so x 1 = y 1 and γ 0 = γ 1 , a contradiction. Thus, k 0 = 2, kx 0 − y 0 − 2y 1 = 0 and ky 0 − x 0 − 2x 1 = 0, so k = 3, 3x 0 = y 0 + 2y 1 and 3y 0 = x 0 +2x 1 . Then from (4.1) it follows that γ 1 = γ 0 +y 2 0 y 1 −x 2 0 x 1 and 2(γ 1 −γ 0 ) = (x 0 −y 0 ) 3 . Moreover, there are exactly two distinct roots of g ′ , i.e. g ′ (
x) = a(x − γ 0 ) t 0 (x − γ 1 ) t 1 , where t 0 + t 1 = t − 1 = deg g − 1, a ∈ K \ {0} and t 0 , t 1 ≥ 1. Therefore, f ′ (x) = a(h(x) − γ 0 ) t 0 (h(x) − γ 1 ) t 1 h ′ (x) = 3a(x − x 0 ) 2t 0 +1 (x − x 1 ) t 0 (x − y 0 ) 2t 1 +1 (x − y 1 ) t 1 ,
and because of 3x 0 = y 0 + 2y 1 and 3y 0 = x 0 + 2x 1 , it follows that x 0 , x 1 , y 0 , y 1 are all distinct. Namely, otherwise f has no two distinct critical points. Thus, (1.8) holds, Assume henceforth that in (4.2) we have k 0 , k 1 , l 0 , l 1 > 1. If x 0 = y 0 and x 1 = y 1 , or x 0 = y 1 and x 1 = y 0 , then γ 0 = γ 1 , a contradiction. If not, then k 1 = l 1 = 2, k 0 = l 0 = k − 2. If k 0 , l 0 > 2, then kx 1 − k 0 x 1 − k 1 x 0 = 0, ky 1 − k 0 y 1 − k 1 y 0 = 0, x 0 = y 0 , and x 1 = y 1 , γ 0 = γ 1 , a contradiction. If k 1 = l 1 = k 0 = l 0 = 2, there is also a possibility that 4x 0 = 2y 1 + 2y 0 , 4y 1 = 2x 1 + 2x 0 , x 1 = y 0 . Then x 0 = x 1 = y 0 = y 1 , and γ 0 = γ 1 , a contradiction.
In the sequel, we discuss some aspects of Proposition 1.6. If in Proposition 1.6, Equation (1.7) holds, then
f (x) = c 1 (x − x 0 ) k 0 (x − x 1 ) k 1 t + c 0 , c 0 , c 1 ∈ K, c 1 = 0, for some distinct x 0 , x 1 ∈ K, k 0 , k 1 ≥ 1, k 0 + k 1 = k ≥ 3 and t ≥ 2. If f (x) = g(h(x))
with h monic and h(0) = 0 (which we can assume without loss of generality by (3.1)), then (1.8) holds, and f (x) = g(h(x)) with h monic and h(0) = 0, then
g(x) = c 1 (x − γ 0 ) t + c 0 , h(x) = (x − x 0 ) k 0 (x − x 1 ) k 1 + γ 0 and (−1) k−1 x k 0 0 x k 1 1 = γ 0 . Ifh(x) = (x − x 0 ) 2 (x − x 1 ) + γ 0 = (x − y 0 ) 2 (x − y 1 ) + γ 1 , g ′ (x) = c 1 (x − γ 0 ) t 0 (x − γ 1 ) t 1 ,
for c 1 = 0, t 0 , t 1 ≥ 1 such that t 0 + t 1 = deg g − 1, distinct x 0 , x 1 , y 0 , y 1 ∈ K with 3x 0 = y 0 + 2y 1 and 3y 0 = x 0 + 2x 1 , and distinct γ 0 , γ 1 ∈ K with γ 0 = x 2 0 x 1 , γ 1 = y 2 0 y 1 . Then also 2(γ 1 − γ 0 ) = (x 0 − y 0 ) 3 .
It is possible that f has at least two distinct critical points, equal critical values at at most two distinct critical points, is not of forbidden types in Proposition 1.6, and can be represented as f = g • h with deg g > 1 and deg h = 2. Indeed,
f (x) = (1 + x) 5 − x 5 = 5x 2 + 5x + 1 • (x 2 + x),
and f has three simple critical points since f ′ (x) = 5(2x + 1)(2x 2 + 2x + 1), and the critical values are not all equal. Moreover, one can show that
(1 + x) n − x n =P n,n−1 (x) • (x 2 + x),P n,n−1 (x) := n ′ j=1 ((2 − ω j − ω j )x + 1) , n = 2n ′ + 1.
for all odd n ≥ 3, and (1 + x) n − x n has all simple critical points and equal critical values at at most two distinct critical points. This is shown in [11] and recalled in Section 6.
Proofs of the main theorems
Proof of Theorem 1.1. If the equation f (x) = g(y) has infinitely many solutions with a bounded O S -denominator, then by Theorem 2.2 we have λ(x))), g(x) = φ(g 1 (µ(x))), (5.1) where (f 1 , g 1 ) is a standard or specific pair over K, φ, λ, µ ∈ K[x] and deg λ = deg µ = 1.
f (x) = φ(f 1 (
By assumption and Proposition 3.9 it follows that Mon(f ) and Mon(g) are primitive permutation groups. Thus f and g are indecomposable.
Assume that h := deg φ > 1. Then deg f 1 = 1 and deg g 1 = 1, since f and g are indecomposable. From (5.1) it follows that f (x) = g(µ(x)) for some µ ∈ K[x].
If deg φ = 1, then from (5.1) it follows that
(5.2) f (x) = e 1 f 1 (c 1 x + c 0 ) + e 0 , g(x) = e 1 g 1 (d 1 x + d 0 ) + e 0 ,
where c 1 , c 0 , d 1 , d 0 , e 1 , e 0 ∈ K, and c 1 d 1 e 1 = 0. Let deg f = deg f 1 =: k and deg g = deg g 1 =: l. By assumption k, l ≥ 3.
Note that (f 1 , g 1 ) cannot be a standard pair of the second kind, since k, l > 2. If (f 1 , g 1 ) is a standard pair of the fifth kind, then either f 1 (x) = (ax 2 − 1) 3 or g 1 (x) = (ax 2 − 1) 3 . By (5.2) it follows that either f or g are decomposable, a contradiction.
If (f 1 , g 1 ) is a standard pair of the first kind, then either f 1 (x) = x k or g 1 (x) = x l . Since f ′ and g ′ have at least two distinct roots, we have a contradiction.
If (f 1 , g 1 ) is a standard pair of the third or of the fourth kind, then
f (x) = e 2 D k (c 1 x + c 0 , α) + e 0 , g(x) = e ′ 2 D l (d 1 x + d 0 , β) + e 0 , (5.3)
where gcd(k, l) ≤ 2 and e 2 , e ′ 2 , α, β ∈ K \{0}. However, this cannot be. Namely, since k, l ≥ 3 and gcd(k, l) ≤ 2, it follows that either k ≥ 4 or l ≥ 4. Assume k ≥ 4. By Proposition 3.9, it follows that also when we consider f as with coefficients in K, the monodromy group of f over K is doubly transitive. Then (f (x) − f (y))/(x − y) is irreducible over K by Lemma 3.5. This is in contradiction with (2.4) . We conclude analogously if l ≥ 4.
If (f 1 , g 1 ) is a specific pair, then
f (x) = e 2 D k (γ 1 x + γ 0 , α) + e 0 , g(x) = −e 2 D l (δ 1 x + δ 0 , β) + e 0 ,(5.4)
for some γ 1 , δ 1 , γ 0 , δ 0 ∈ K, e 2 , α, β ∈ K \ {0}, where gcd(k, l) ≥ 3. This, by the same argument as above, cannot be unless (k, l) = (3,3). In this case, gcd(k, l) = 3, so f 1 (x) = D 3 (x, a) = x 3 − 3xa, g 1 (x) = −D 3 (1/2x, a) = −1/8x 3 + 3/2xa. Then g 1 (−2x) = f 1 (x) and from (5.2) it follows that g(µ(x)) = f (x) for some µ ∈ K[x].
Theorem 5.5. Let K be a number field, S a finite set of places of K that contains all Archimedean places, O S the ring of S-integers of K and f, g ∈ K[x] with deg f ≥ 3, deg g ≥ 3 and deg f < deg g.
Assume that f and g both have at least two distinct critical points and equal critical values at at most two distinct critical points, and do not satisfy (1.7) nor (1.8). Then the equation f (x) = g(y) has finitely many solutions with a bounded O S -denominator, unless either (deg f, deg g) ∈ {(3, 4), (3,5), (4,5), (4,6)}, or f is indecomposable and g(x) = f (ν(x)) for some quadratic ν ∈ K[x].
If (deg f, deg g) ∈ {(3, 4), (3,5), (4,5), (4, 6)}, then the equation has infinitely many solutions with a bounded O S -denominator when
f (x) = e 1 f 1 (c 1 x + c 0 ) + e 0 , g(x) = e 1 g 1 (d 1 x + d 0 ) + e 0 ,
for some c 1 , c 0 , d 1 , d 0 , e 1 , e 0 ∈ K, and c 1 d 1 e 1 = 0, and (f 1 , g 1 ) ∈ (D 3 (x, a 4 ), D 4 (x, a 3 )), (D 3 (x, a 5 ), D 5 (x, a 3 )), (D 4 (x, a 5 ), D 5 (x, a 4 )) ,
where D 3 (x, a) = x 3 − 3xa, D 4 (x, a) = x 4 − 4x 2 a + 2a 2 and D 5 (x, a) = x 5 − 5ax 3 + 5a 2 x are
Dickson polynomials, or f 1 (x) = 3x 4 − 4x 3 and g 1 (x) = (ax 2 − 1) 3 for some nonzero a ∈ K.
Proof. If the equation f (x) = g(y) has infinitely many solutions with a bounded O Sdenominator, then f (x) = φ(f 1 (λ(x))), g(x) = φ(g 1 (µ(x))), (5.6) where (f 1 , g 1 ) is a standard or specific pair over K, φ, λ, µ ∈ K[x] and deg λ = deg µ = 1.
Assume deg φ > 1. Since f and g are such that neither (1.7) nor (1.8) holds, by Proposition 1.6 if follows that deg f 1 ≤ 2 and deg g 1 ≤ 2.
Since deg f < deg g, it follows that deg f 1 = 1 and deg g 1 = 2. Since deg g 1 = 2, by Proposition 1.6 it further follows that φ is indecomposable. Then from (5.6) it follows that f (ν 1 (x)) = φ(x) for some ν 1 ∈ K[x] with deg ν 1 = 1, and then g(x) = f (ν(x)) for some ν ∈ K[x] with deg ν = 2.
Assume further that deg φ = 1. If so, then from (5.6) it follows that
(5.7) f (x) = e 1 f 1 (c 1 x + c 0 ) + e 0 , g(x) = e 1 g 1 (d 1 x + d 0 ) + e 0 ,
where c 1 , c 0 , d 1 , d 0 , e 1 , e 0 ∈ K, and c 1 d 1 e 1 = 0. Let deg f = deg f 1 =: k and deg g = deg g 1 =: l. By assumption l > k ≥ 3. Note that since f and g both have at least two distinct critical points and equal critical values at at most two distinct critical points, by (5.7) the same holds for f 1 and g 1 .
Note that (f 1 , g 1 ) cannot be a standard pair of the second kind, since k, l > 2. If (f 1 , g 1 ) is a standard pair of the fifth kind, then g 1 (x) = (ax 2 −1) 3 and f 1 (x) = 3x 4 −4x 3 . In this case, all the assumptions are satisfied and the equation f (x) = g(y) has infinitely many solutions with a bounded O S -denominator.
If (f 1 , g 1 ) is a standard pair of the first kind, then either f 1 (x) = x k or g 1 (x) = x l , so either f 1 or g 1 do not have two distinct critical points, a contradiction.
If (f 1 , g 1 ) is a standard pair of the third or fourth kind, then
f (x) = e 2 D k (c 1 x + c 0 , α) + e 0 , g(x) = e ′ 2 D l (d 1 x + d 0 , β) + e 0 , (5.8)
where gcd(k, l) ≤ 2 and e 2 , e ′ 2 , α, β ∈ K \ {0}. If either k ≥ 6 or l ≥ 6, this cannot be, since D k (x, a) with k ≥ 6 has equal critical values at three distinct critical points, by Corollary 2.6. If k, l < 6, since gcd(k, l) ≤ 2 and k < l it follows that (k, l) ∈ {(3, 4), (3,5), (4,5)}.
Recall that D k (x, α) has all simple critical points, so in particular has at least two distinct critical points, when k, l ≥ 3. Moreover, one easily sees that D k (x, α) has equal critical values at at most two distinct critical points for k ≤ 5. So, under the assumptions of the theorem we have infinitely many solutions with a bounded O S -denominator to the equation f (x) = g(y) also when f (x) = e 1 f 1 (c 1 x + c 0 ) + e 0 , g(x) = e 1 g 1 (d 1 x + d 0 ) + e 0 and (f 1 , g 1 ) ∈ (D 3 (x, a 4 ), D 4 (x, a 3 )), (D 3 (x, a 5 ), D 5 (x, a 3 )), (D 4 (x, a 5 ), D 5 (x, a 4 )) ,
where D 3 (x, a) = x 3 − 3xa, D 4 (x, a) = x 4 − 4x 2 a + 2a 2 , D 5 (x, a) = x 5 − 5ax 3 + 5a 2 x.
If (f 1 , g 1 ) is a specific pair, then f 1 (x) = e 2 D k (γ 1 x, α), g 1 (x) = e 2 D l (γ 2 x, β) for some γ 1 , γ 2 ∈ K with gcd(k, l) ≥ 3. If either k ≥ 6 or l ≥ 6, this cannot be by the same argument as above (since D k (x, a) with k ≥ 6 has at least three critical points with equal critical values by Corollary 2.6). The case (k, l) = (3, 3) is impossible, since k < l.
The following result is a corollary of Proposition 1.6 and Theorem 5.5.
(x) = f (ν(x)) for some quadratic ν ∈ K[x].
If The case deg f = deg g in Theorem 1.9 is somewhat harder to handle. Namely, in the proof of Theorem 5.5 we used that deg f < deg g when we concluded that if f (x) = φ(f 1 (λ(x))) and g(x) = φ(g 1 (µ(x))) with deg φ > 1, then deg f 1 = 1 and deg g 1 = 2 by Proposition 1.6. If we had allowed deg f = deg g, then we would have also had the possibility deg f 1 = 1 and deg g 1 = 1, which is easy to handle, and the possibility deg f 1 = 2 and deg g 1 = 2. In the latter case, we couldn't express easily the relation between f and g.
In the sequel we discuss how one can show that for f, g ∈ K[x], where f is indecomposable, there does not exist quadratic ν ∈ K[x] such that g(x) = f (ν(x)). One may first find if there exists a ∈ K such that g(x) = g 1 (x 2 + ax) for some g 1 ∈ K[x] (as in (3.1)). If deg g = m and g(x) = a m x m + a m−1 x m−1 + · · · , then a m ma = a m−1 , which determines a. Then g 1 , if it exists, is uniquely determined by g and a.
If g(x) = g 1 (x 2 + ax) for some decomposable g 1 ∈ K[x], then it is not possible that g(x) = f (ν(x)) for some indecomposable f and quadratic ν. Namely, by Ritt's [34] and Engstrom's [15] results (see also [47,Cor. 2.12]), it follows that any representation of g, which has coefficients in a field of characteristic zero, as a composition of indecomposable polynomials, consists of the same number of factors. If g(x) = g 1 (x 2 + ax) for some indecomposable
g 1 ∈ K[x]
, and g(x) = f (ν(x)) for some indecomposable f and quadratic ν, then by Ritt's and Engstrom's results (see [47,Cor. 2.9]), we have that f = g 1 • µ 1 and ν = µ −1 1
• (x 2 + ax) for some linear µ 1 ∈ K[x], since g = g 1 (x 2 + ax) = f (ν(x)), all factors are indecomposable and deg g 1 = deg f . Now one can also compare the roots of f and g 1 to reach contradiction. We will later illustrate this approach on a concrete example (see Theorem 6.6).
Corollaries of the main theorems
We now present several corollaries of Theorem 1.1 and Theorem 1.9. Most of these corollaries are results of published papers [2,11,14,27,33,39]. In most cases, our proofs of Theorem 1.1 and Theorem 1.9 are shorter than the proofs in those papers. Also, those proofs depend on particular properties of the involved polynomials, such as their coefficients.
We first list some corollaries of Theorem 1.1.
As we have seen in the introduction, Theorem 1.1 implies immediately Corollary 1.2, which generalizes the main of result of Péter, Pintér and Schinzel [33,Thm. 1]. They proved it using other tools: Hajós lemma on the multiplicites of roots of lacunary polynomials (see [35, p. 187]), a result of Fried and Schinzel [19] about indecomposability of polynomials in (1.3), and by comparison of coefficients. Theorem 1.1 implies the finiteness of integral solutions to the equation x n + x n−1 + · · · + x + 1 = y m + y m−1 + · · · + y + 1, with m > n ≥ 3. This result was shown by Davenport, Lewis and Schinzel [10], by a finiteness criterion developed by them, which is weaker than the later one of Bilu and Tichy [5].
For positive integers k ≤ n − 1 put
(6.1) P n,k (x) := k j=0 n j x j = n 0 + n 1 x + n 2 x 2 + · · · + n k x k .
The polynomial P n,k is said to be a truncated binomial expansion (polynomial ) at the k-th stage.
Corollary 6.2. Let n, k, m, l ∈ N be such that 3 ≤ k ≤ n − 1, 3 ≤ l ≤ m − 1 and k = l. If P n−1,k−1 and P m−1,l−1 are such that they have no two distinct roots whose quotient is a k-th, respectively l-th, root of unity, then the equation P n,k (x) = P m,l (y) has only finitely many integral solutions x, y.
Proof. Since,
(6.3) P ′ n,k (x) = nP n−1,k−1 (x) & P n,k (x) − (x + 1) P ′ n,k (x) n = n − 1 k x k ,
it follows that P n,k has all distinct critical points and all distinct critical values, unless it has two critical points whose quotient is a k-th root of unity. Thus, the statement follows by Theorem 1.1.
In [11], Dubickas and Kreso studied the equation P n,k (x) = P m,l (y) from Corollary 6.2. They showed that this equation has only finitely many integral solutions when 2 ≤ k ≤ n − 1, 2 ≤ l ≤ m − 1, and k = l, by assuming irreducibility of P n−1,k−1 and P m−1,l−1 . Irreducibility of truncated binomial expansions has been studied by several authors, and the results suggest that P n,k is irreducible for all k < n − 1. The existence of two distinct roots of P n−1,k−1 whose quotient is a k-th root of unity is an open problem when k < n − 1. Computations show that for n ≤ 100 and k < n − 1 no such two roots exist. The problem is solved in the case k = n − 1 in [11]. We will discuss this case later, when we will list some corollaries of Theorem 1.9.
Corollary 6.4. For m > n ≥ 3, the equation
(6.5) x n n! + · · · + x 2 2! + x + 1 = y m m! + · · · + y 2 2! + y + 1,
has only finitely many integral solutions.
Kulkarni and Sury [27] proved Corollary 6.4. If f is the polynomial on the left hand side of (6.5), then f (x) = f ′ (x) + x n /n!, and f thus has only simple critical points. To see that f has all distinct critical values it suffices to show that no two roots of f ′ are such that their quotient is an n-th root of unity. It is shown in [27] that this holds by using the fact that the Galois groups of f and f ′ are either symmetric or alternating, which is a result of Schur.
Note that Theorem 1.1 applies to equations of type f (x) = g(y), where f and g are any of the above mentioned polynomials. In particular, the equation
x n + x n−1 + · · · + x + 1 = y m m! + · · · + y 2 2! + y + 1, with m = n and m, n ≥ 3, has only finitely many integral solutions. We now discuss applications of Theorem 1.9. To get complete statements of some of the results in the literature, we still need to examine the exceptional cases in Theorem 1.9: If deg f < deg g we need to examine the cases when (deg f, deg g) ∈ {(3, 4), (3,5), (4,5), (4, 6)}, and g(x) = f (ν(x)) with quadratic ν and indecomposable f . All these cases are easy to handle (the former via Theorem 5.5 by direct analysis and comparison of polynomials, and the latter in the way described at the end of Section 5).
The following results can be found in [11].
Theorem 6.6. For m > n ≥ 3, the equation
(1 + x) n − x n = (1 + y) m − y m ,
has only finitely many integral solutions x, y.
Lemma 6.7. For positive integer n ≥ 3, the polynomial (1+x) n −x n has at least two distinct critical points and equal critical values at at most two distinct critical points.
Proof. Note that (1 + x) n − x n = P n,n−1 (x) by (6.1). Take two roots α and β of P ′ n,n−1 (x) = n((x + 1) n−1 − x n−1 ) such that P n,n−1 (α) = P n,n−1 (β). The former implies (α + 1) n−1 = α n−1 and (β + 1) n−1 = β n−1 , and so the latter yields α n−1 = β n−1 . Note that the roots of (x+ 1) n−1 −x n−1 lie on the vertical line ℜ(z) = −1/2. Then from α n−1 = β n−1 it follows that α and β are complex conjugates, since they are distinct but have equal absolute values.
Proof of Theorem 6.6. By Theorem 1.9 it follows that the equation has finitely many integral solutions, unless either n, m ≤ 6, or (
1 + x) m − x m = ((1 + x) n − x n ) • ν(x) for some quadratic ν ∈ K[x]
. We now show that the latter case cannot occur. One easily verifies that
(1 + x) n − x n =P n,n−1 (x) • (x 2 + x),P n,n−1 (x) := n ′ j=1 ((2 − ω j − ω j )x + 1) , n = 2n ′ + 1.
By Lemma 6.7 and Proposition 1.6,P n,n−1 is indecomposable for all odd n > 2, and if n > 2 is even, then (1 + x) n − x n is indecomposable. [47,Cor. 2.9] (see the end of Section 5). This cannot be since all the roots ofP n,n−1 (x) are real and the roots of (1 + x) m − x m are, except for at most one (which is −1/2 when m is even), all complex. Using Theorem 5. Lemma 6.8. Let (y n ) n be a sequence of polynomials with real coefficients that satisfy a differential equation
If (1 + x) m − x m = ((1 + x) n − x n ) • ν(x) for some quadratic ν, then (1 + x) m − x m = P n,n−1 (x) • µ 1 (x) with µ 1 ∈ Q[x] by(6.9) σ(x)y ′′ n (x) + τ y ′ n (x) − λ n y n (x) = 0, n ≥ 0, with σ, τ ∈ R[x], deg σ ≤ 2, deg τ ≤ 1, λ n ∈ R \ {0} and nonvanishing σ ′ − 2τ .
Then for all n ≥ 3, y n has equal critical values at at most two distinct critical points.
Proof. By letting λ n f (x) := λ n y n (x) 2 − σy ′ n (x) 2 we get λ n f ′ (x) = −(σ ′ (x) − 2τ (x))y ′ n (x) 2 , and from deg(σ ′ − 2τ ) ≤ 1 and y ′ n (x) 2 ≥ 0 for all x, it follows that there exists x 0 ∈ R such that f ′ (x) ≥ 0 for all x ≥ x 0 and f ′ (x) ≤ 0 for all x ≤ x 0 , or vice versa f ′ (x) ≤ 0 for all x ≥ x 0 and f ′ (x) ≥ x 0 for all x ≤ x 0 . This together with λ n f (x) := λ n y n (x) 2 − σy ′ n (x) 2 , shows that y n has equal critical values at at most two distinct critical points. Lemma 6.8 is due to Stoll [39]. He used it to find the possible decompositions of some classical orthogonal polynomials, namely Hermite, Laguerre, Jacobi, Gegenbauer and Bessel polynomials. They satisfy a differential equation of type (6.9) with nonvanishing σ ′ − 2τ . These polynomials also have all simple real zeros, and thus also all simple critical points, by Rolle's theorem. Stoll studied Diophantine equations with these polynomials in [40,41,42]. Lemma 6.10. Let K be a field with char(K) = 0, a 1 , a 2 , a 3 ∈ K with a 1 a 2 = 0, and n 1 , n 2 ∈ N with gcd(n 1 , n 2 ) ≤ 2. Then a 1 x n 1 + a 2 x n 2 + a 3 has at least two distinct critical points and equal critical values at at most two distinct critical points
Proof. Let f (x) = a 1 x n 1 + a 2 x n 2 + a 3 . Then the statement follows by xf ′ (x) = n 1 (f (x) − a 3 ) + a 2 (n 1 − n 2 )x n 2 and gcd(n 1 , n 2 ) ≤ 2.
By Lemma 6.10, we may apply Theorem 5.5 to the equation in Corollary 1.2, with the assumptions weakened to gcd(n 1 , n 2 ) ≤ 2 and gcd(m 1 , m 2 ) ≤ 2. Schinzel [36] characterized when this equation, with no assumptions on the greatest common denominators of n i 's and m i 's, but with K = Q and O K = Z, has infinitely many solutions with a bounded denominator.
Beukers, Shorey and Tijdeman [2] proved the following theorem.
Theorem 6.11. For m > n ≥ 3 and d 1 , d 2 ∈ Q, the equation
x(x + d 1 ) · · · (x + (m − 1)d 1 ) = y(y + d 2 ) · · · (y + (n − 1)d 2 )
has only finitely many integral solutions x, y.
Theorem 6.11 follows, to the most part, by Theorem 5.5 and the following lemma proved in [2] as a step in finding the possible decompositons of the polynomial x(x + d 1 ) · · · (x + (m − 1)d 1 ) with m ∈ N and d ∈ Q. Lemma 6.12. For nonzero d ∈ Q, and m ≥ 3, the polynomial x(x + d) · · · (x + (m − 1)d) has at least two distinct critical points and equal critical values at at most two distinct critical points.
Proof. We show that x(x + 1) · · · (x + (m − 1)) for m ≥ 3 has equal critical values at at most two distinct critical points . Then the same follows for x(x + d) · · · (x + (m − 1)d).
Let α 1 , α 2 , . . . , α m−1 be the critical points of f (x) = x(x + 1) · · · (x + (m − 1)). By Rolle's's theorem they are simple and real and can be ordered so that −(m − 1) < α m−1 < −(m − 2) < α m−2 < . . . < −1 < α 1 < 0. Note that |f (x)| asumes its unique maximal value on the interval [−i, −(i − 1)] at α i . Hence, |f (α i−1 )| ≥ |f (α i + 1)|. Therefore Theorem 6.13. Let G 0 (x) = 0, G 1 (x) = 1, and for nonzero integer B let G n+1 (x) = xG n (x) + BG n−1 (x) for n ∈ N. For m > n ≥ 3, the equation G m (x) = G n (y) has only finitely many integral solutions x, y. Theorem 6.13 is due to Dujella and Tichy [14]. It is easy to check that G n (x) = µ 1 (U n−1 (µ 2 (x))), where µ 1 , µ 2 ∈ K[x] are linear polynomials and U n is the n-th Chebyshew polynomial of the second kind, given by a differential equation (1 − x 2 )U ′′ n (x) − 3xU ′ n (x) ′ + n(n + 2)U n (x) = 0. One easily finds that U n has simple real roots (since U n (cos x) = sin(n + 1)x/ sin x), and thus simple critical points as well by Rolle's theorem. In a similar way as in Lemma 6.12, Dujella and Tichy showed U n has equal critical values at at most two distinct critical points. Since (1 − x 2 )U ′′ n (x) − 3xU ′ n (x) ′ + n(n + 2)U n (x) = 0 and (1 − x 2 ) ′ + 2 · 3x does not vanish, this immediately follows by Lemma 6.8. Thus, Theorem 6.13 follows to the most part by Theorem 5.5. (As usual, it remains to analyse the cases (m, n) ∈ {(4, 3), (5, 3), (5, 4), (6, 4)} and the case G m (x) = G n (ν(x)), where ν is quadratic. See [14] for details.)
|f (α i−1 )| |f (α i )| ≥ |f (α i + 1)| |f (α i )| = |α i + 1||α i +
It seems likely that the well-known Bernoulli and Euler polynomials satisfy the conditions of Theorem 5.5. As is well known, the k-th power sum of the first n − 1 positive integers S k (n) = 1 k + 2 k + · · · + (n − 1) k and the alternating k-th power sum of the first n − 1 positive integers T k (n) = −1 k + 2 k + · · · + (−1) n−1 (n − 1) k can be expressed in terms of Bernoulli polynomial B k (x) and Euler polynomials E k (x), as S k (n) = 1 k + 1 (B k+1 (n) − B k+1 ) , T k (n) = 1 2 E k (0) + (−1) n−1 E k (n) .
In various papers, of which we mention [1,3,24], equations of type µ 1 (B k (µ 2 (x))) = λ 1 (B n (λ 2 (x))), and µ 1 (E k (µ 2 (x))) = λ 1 (E n (λ 2 (x))), where µ i , λ i ∈ Q[x] are linear and k, n ≥ 3, have been studied, corresponding to equations with the above introduced power sums. We do not have a proof at hand, but if Bernoulli and Euler polynomials are such that they have equal critical values at at most two distinct critical points, then Theorem 5.5 would yield a unifying proof of the results in these papers. It is well known that Bernoulli polynomials have simple roots and that B ′ n (x) = nB n−1 (x), so that they have all simple critical points as well. Also, E ′ n (x) = nE n−1 (x) and the only Euler polynomial with a multiple root is of degree 5 and has one simple root and two double roots. If Bernoulli and Euler polynomials are such that at least they have equal critical values at at most two distinct critical points, then Theorem 5.5 would also apply to equations of type µ 1 (B k (µ 2 (x))) = λ 1 (E n (λ 2 (x))) with linear µ i , λ i ∈ Q[x].
Theorem 1 . 1 .
11Let K be a number field, S a finite set of places of K that contains all Archimedean places, O S the ring of S-integers of K, and f, g ∈ K[x] with deg f ≥ 3, deg g ≥ 3. If f and g both have at least two distinct critical points and all distinct critical values, then the equation f (x) = g(y) has infinitely many solutions with a bounded O Sdenominator if and only if f (x) = g(µ(x)) for some linear µ ∈ K[x].
1 x n 1 + a 2 x n 2 + a 3 = b 1 y m 1 + b 2 y m 2 has infinitely many solutions with a bounded O S -denominator if and only if for some linear µ ∈ K[x] we have(1.4) a 1 x n 1 + a 2 x n 2 + a 3 = (b 1 x m 1 + b 2 x m 2 ) • µ(x).
Corollary 2 . 6 .
26Let K be a field with char(K) = 0. If n ≥ 4 and a = 0, there exist two distinct critical points of D n (x, a) with equal critical values. If n ≥ 6 and a = 0, there exist three distinct critical points of D n (x, a) with equal critical values.
Definition 3. 3 .
3Given f ∈ K[X] with char(K) = 0 and deg f > 1, the monodromy group Mon(f ) of f is the Galois group of f (X) − t over the field K(t), where t is transcendental over K, viewed as a group of permutations of the roots of f (X) − t.
Lemma 3 . 6 .
36If K is a field with char(K) = 0 and e 1 , e 2 , . . . , e k are the multiplicities of the roots of f (x) − x 0 , where f ∈ K[x] with char(K) = 0 and x 0 ∈ K, then Mon(f ) contains an element having cycle lengths e 1 , e 2 , . . . , e k . Furthermore, if n = deg f , then Mon(f ) contains an n-cycle.
Proposition 3. 9 .
9Let K be a field with char(K) = 0 and f ∈ K[x] with at least two distinct critical points and all distinct critical values. Then Mon(f ) is doubly transitive. In particular, Mon(f ) is primitive, i.e. f is indecomposable.
Remark 3. 11 .
11Let K be a field with char(K) = 0 and f ∈ K[x]. If f has no two distinct critical points, then f ′ (x) = a(x − x 0 ) n−1 , and thus f (x) = a/n(x − x 0 ) n + const. Such polynomial is indecomposable if and only if n is prime.
Corollary 5 . 9 .
59Let K be a number field, S a finite set of places of K that contains all Archimedean places, O S the ring of S-integers of K and f, g∈ K[x] with deg f ≥ 3, deg g ≥ 3 and deg f < deg g.If f and g have at least three simple critical points and equal critical values at at most two distinct critical points, then the equation f (x) = g(y) has finitely many solutions with a bounded O S -denominator, unless (deg f, deg g) = (4, 5), or f is indecomposable and g
(deg f, deg g) = (4, 5), then the equation f (x) = g(y) has infinitely many solutions with a bounded O S -denominator also when f (x) = e 1 D 4 (c 1 x + c 0 , a 5 ) + e 0 , g(x) = e 1 D 5 (d 1 x + d 0 , a 4 ) + e 0 for some a, c 1 , c 0 , d 1 , d 0 , e 1 , e 0 ∈ K, and ac 1 d 1 e 1 = 0. From Theorem 5.5 and Corollary 5.9, Theorem 1.9 follows immediately.
2| . . . |α i + m| |α i ||α i + 1| . . . |α i + (m − 1)| = |α i + m| |α i | for i = 1, 2, . . . , m − 1. Note that f (−d 1 (m − 1)/2 − x) = f (−d 1 (m − 1)/2 + x)and hence |f (α i )| = |f (α m−i )| for all i = 1, 2, . . . , m − 1. For i ≤ m/2 we have |f (α i−1 )| > |f (α i )|, and by symmetry |f (α i−1 )| < |f (α i )| for i ≥ m/2 + 1.
and f has at least two distinct critical points and all distinct critical values, then Mon(f ) is a doubly transitive permutation group.
to prove Theorem 1.1. In [23], it is shown that two Morse polynomials with rational coefficients, of distinct degrees which are both ≥ 3, cannot have infinitely many equal values at integer points. This result, generalized by Theorem 1.1, does not imply Corollary 1.2, nor the aforementioned results in
the same holds, unless f has no two distinct critical points. One can compare these observations with Pakovich's results [31, Prop. 3.4 & Cor 3.1.] for rational functions. Pakovich's techniques are analytic, and thus completely different from ours.
AcknowledgementsThe authors are grateful for the support of the Austrian Science Fund (FWF) via projects W1230-N13, FWF-P24302 and F5510.
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| []
|
[
"Three-dimensional carbon nanotube networks from beta zeolite templates: Thermal stability and mechanical properties",
"Three-dimensional carbon nanotube networks from beta zeolite templates: Thermal stability and mechanical properties"
]
| [
"Eliezer F Oliveira1 ",
"Leonardo D Machado4 ",
"Ray H Baughman5 ",
"Douglas S Galvao1 [email protected] ",
"\n2Center for Computational Engineering & Sciences (CCES)\n1Group of Organic Solids and New Materials (GSONM)\nGleb Wataghin Institute of Physics\nUniversity of Campinas (UNICAMP)\nCampinasSPBrazil\n",
"\n3Department of Material Science and NanoEngineering\nUniversity of Campinas (UNICAMP)\nCampinasSPBrazil\n",
"\n4Department of Theoretical and Experimental Physics\nRice University\n77005HoustonTexasUnited States\n",
"\n5Alan G. MacDiarmid NanoTech Institute\nFederal University of Rio Grande do Norte (UFRN)\nNatalRNBrazil\n",
"\nThe University of Texas at Dallas\n75080-3021DallasTexasUnited States\n"
]
| [
"2Center for Computational Engineering & Sciences (CCES)\n1Group of Organic Solids and New Materials (GSONM)\nGleb Wataghin Institute of Physics\nUniversity of Campinas (UNICAMP)\nCampinasSPBrazil",
"3Department of Material Science and NanoEngineering\nUniversity of Campinas (UNICAMP)\nCampinasSPBrazil",
"4Department of Theoretical and Experimental Physics\nRice University\n77005HoustonTexasUnited States",
"5Alan G. MacDiarmid NanoTech Institute\nFederal University of Rio Grande do Norte (UFRN)\nNatalRNBrazil",
"The University of Texas at Dallas\n75080-3021DallasTexasUnited States"
]
| []
| We here investigated the thermal and mechanical behaviors of three-dimensional beta zeolitetemplated carbon nanotube networks (BZCN). These networks are topologically generated by inserting carbon nanotubes (CNTs) into zeolite channels and connecting them using X-type junctions. We considered two cases, one with the tubes filling all zeolite channels (HD-BZCN) and the other with just partial filling (LD-BZCN). Fully atomistic reactive molecular dynamics (MD) simulations show that the networks exhibit high thermal stability (up to 1000 K). When compressed, the structures can withstand very large strains without fracturing (>50% for HD-BZCN and >70% for LD-BZCN). LD-BZCN can be stretched over 100% without fracturing. | 10.1016/j.commatsci.2020.109781 | [
"https://arxiv.org/pdf/1912.10281v1.pdf"
]
| 209,444,974 | 1912.10281 | d3db4b5b896fc4915b8008c337b81fbd7ff31584 |
Three-dimensional carbon nanotube networks from beta zeolite templates: Thermal stability and mechanical properties
Eliezer F Oliveira1
Leonardo D Machado4
Ray H Baughman5
Douglas S Galvao1 [email protected]
2Center for Computational Engineering & Sciences (CCES)
1Group of Organic Solids and New Materials (GSONM)
Gleb Wataghin Institute of Physics
University of Campinas (UNICAMP)
CampinasSPBrazil
3Department of Material Science and NanoEngineering
University of Campinas (UNICAMP)
CampinasSPBrazil
4Department of Theoretical and Experimental Physics
Rice University
77005HoustonTexasUnited States
5Alan G. MacDiarmid NanoTech Institute
Federal University of Rio Grande do Norte (UFRN)
NatalRNBrazil
The University of Texas at Dallas
75080-3021DallasTexasUnited States
Three-dimensional carbon nanotube networks from beta zeolite templates: Thermal stability and mechanical properties
*Corresponding authors:zeolite templatescarbon nanotube networksmolecular dynamics simulations
We here investigated the thermal and mechanical behaviors of three-dimensional beta zeolitetemplated carbon nanotube networks (BZCN). These networks are topologically generated by inserting carbon nanotubes (CNTs) into zeolite channels and connecting them using X-type junctions. We considered two cases, one with the tubes filling all zeolite channels (HD-BZCN) and the other with just partial filling (LD-BZCN). Fully atomistic reactive molecular dynamics (MD) simulations show that the networks exhibit high thermal stability (up to 1000 K). When compressed, the structures can withstand very large strains without fracturing (>50% for HD-BZCN and >70% for LD-BZCN). LD-BZCN can be stretched over 100% without fracturing.
INTRODUCTION
The search for new carbon-based nanostructures remains a very active research area [1][2][3][4][5][6][7][8][9][10]. These novel structures can have different dimensionalities (0D, 1D, 2D, and 3D) and exhibit a wide range of electrical, thermal, and mechanical properties [11,12]. Of particular interest are 3D structures with well-ordered porous frameworks [13][14][15]. Previous studies predict that these carbon-based 3D networks display interesting electronic and mechanical properties, in addition to their large porosities and surface areas [13][14][15][16]. These properties could be exploited for a variety of applications in multiple technologies, including gas storage, catalysis, molecular sieving, and others [14,15]. However, their synthesis has proved to be very challenging, especially for frameworks with covalently bonded building units [16].
One technique commonly employed to synthesize designed carbon-based structures is the use of sacrificial templates. Templating methods have been used since the 80s [17][18][19] to obtain carbon materials having different structural complexities [14]. Porous templates successfully used to obtain 3D carbon nanostructures include silica opals [20], metal-organic frameworks [21,22], Ni foams [23], and mesoporous silica [24]. However, these 3D structures are in general disordered and/or with many structural defects [14][15]. In general, two major factors determine the quality of the resulting structures: (1) the method of synthesis and (2) the shape, length, and diameter of the template pores, which preclude or facilitate the diffusion of carbon atoms [14].
Recent studies have pointed out that it might be feasible to produce 3D carbon nanostructures with long-range order and few defects by using zeolites as sacrificial templates [13-15, 24, 25]. In general, zeolites are aluminosilicates containing molecular size pores and channels.
These porous structures can enable the synthesis of 3D nanostructures within their interior [13,14,26]. In the most common synthesis methodology of zeolite-templated carbons (ZTC), carbon atoms are introduced into the zeolites via chemical vapor deposition (CVD) of carbon-containing precursors, and then the template is sacrificed/removed [14]. Currently, there are more than 200 types of zeolites [14,27], although some are unsuitable for the production of covalently bonded 3D nanostructures, due to the absence of interconnected pores [14]. Also, it may be difficult to introduce precursors in zeolites having small pores, because of the difficulty of gas diffusion [15].
It has been suggested that only zeolites with pore sizes greater than 5 Å could be amenable for the experimental synthesis of ZTCs [15]. Consequently, for the synthesis of new 3D carbon-based nanostructures using zeolites, an important task is to properly screen zeolite databases to identify promising candidates.
Up to now, experimentally synthesized ZTCs consist of curved graphene nanosheets forming 3D structures [13,14,24,28]. However, the synthesis of 3D carbon networks (CNs) of interconnected carbon nanotubes (CNTs) inside zeolite channels remains elusive [14] and only in the realm of theory [13,14]. Such structures could be used for gas storage applications, as gas molecules strongly interact with curved surfaces [14][15][16]29]. Theoretical studies have shown that CNT-based CNs present electronic and mechanical properties that are dominated by the network ordering [16]. Consequently, the properties of CNs could be tuned by engineering the network geometry [16]. In particular, charge transport calculations have shown that charges follow specific paths through the 3D structure, suggesting applications in nanoelectronics circuits [16].
The beta zeolite [27,30], as an example shown in Figure 1, is an interesting candidate for CN synthesis, since this zeolite has equally spaced interconnected channels having the same diameter (5.6 Å). This uniform channel size, which is not a general feature of zeolites, might allow the fabrication of a regular structure comprising fused nanotubes having the same diameter (see Figure 2).
We here investigate families of CNs that can be formed inside the channels of beta zeolite.
Firstly, we carry out fully atomistic simulations to calculate the energy of different diameter nanotubes within the zeolite channels. Next, we create different 3D frameworks using selected CNTs. Finally, we examine the thermal stability and mechanical properties of the proposed structures.
MATERIALS AND METHODS
For this study, a beta zeolite (BEA framework type) composed solely of silicon and oxygen atoms was used (see Figure 1). The BEA space group is P422, with unit cell parameters a=b=12.63 Å, c=26.18 Å, and α=β=૪=90° [30]. This zeolite has two orthogonal and equivalent channels (along the y and z directions, see Figure 1) that partially intersect. Figure 1(c) illustrates the relationship between the channels in BEA. In order to obtain single wall CNT candidates for forming the nets inside BEA, we determined which carbon nanotubes could fill the zeolite channels while providing a low energy structure. After selecting these CNT candidates, we then created energetically favorable junctions between the CNTs. The resulting building unit composed of interconnected nanotubes was then replicated to generate a periodic 3D structure. Finally, we removed the zeolite template so that the thermal stability and mechanical properties of the resulting CNs could be investigated. From here on we will use the acronym BZCN to refer to CNs templated from BEA.
To determine which CNT are most energetically suitable for filling the channels of BEA, we carried out Molecular Mechanics (MM) simulations using the well-known Universal Force Field (UFF) [31], as implemented in the Forcite software [32]. For these calculations, geometry optimization was performed with an energy convergence tolerance of 0.001 kcal/mol and a force convergence tolerance of 0.5 kcal/mol/Å. The size and shape of the unit cell was also optimized.
This procedure was repeated for single wall nanotubes having different diameters. The used BEA unit cell contains 3840 atoms. The total number of atoms in the tested carbon nanotubes depended on their diameter, but the tube length was kept fixed (10 nm). We used a large periodic cell (12.3 nm) along the CNT axis direction, in order to prevent spurious inter-nanotube interactions.
After examining the CNT candidates, the next step was to generate junctions among the nanotubes. Two nanotubes were inserted into adjacent BEA channels, along the y and z directions, respectively (see Figure 1). The position of these nanotubes was fixed, and they were removed from the zeolite. In the region where the nanotubes were close to each other, atoms from both tubes were removed and placed into the space between them. Molecular dynamics (MD) simulations were then performed to create the junction, using the ReaxFF force field [33], as implemented in the computational package LAMMPS [34]. In these simulations, the temperature was controlled using a chain of three Nosé-Hoover thermostats, and a time step of 0.2 fs was used. During these simulations, the positions of the nanotubes was kept fixed and various cycles of minimization/heating/cooling (from 300 K to 1500 K) were performed to anneal defects in the junction region, until only energetically favorable pentagons, hexagons, and heptagons of carbon remained. It should be stressed that transverse channels in BEA do not intersect, and the nanotubes are connected by a "X" type junction, comprised of heptagons and pentagons [16], as displayed in By inserting the covalently connected nanotubes back inside BEA, we determined if the created junction would fit inside the void that initially existed between CNTs. This process was used to create unit cells that could be replicated to generate a periodic 3D structure of arbitrary size.
After these steps, we carried out MD simulations to analyze thermal stability and mechanical properties of BZCN structures. After removing the zeolite, we minimized the energy of each BZCN using the conjugate gradient technique, and then equilibrated the system in an NPT ensemble with T = 300 K and P = 0 GPa and periodic boundary condition (PBC). Next, we applied either tensile or compressive uniaxial strain to deform the structure and evaluated its mechanical response to beyond the fracture limit. We used strain rates of 10-6 fs-1 and -10-6 fs-1 for tension and compression, respectively. The deformation tests were performed at 300 K, and the pressure was set to 0 GPa along directions perpendicular to the external strain. The virial stress tensor component and the engineering strain at each compressive/tensile direction was used for building the stress-strain curve. The local stress distribution was evaluated through the von Mises stress analysis, using the second invariant of the deviatoric stress tensor [35]. This analysis allows us to determine where stress is spatially concentrated during compression/tensile tests, providing insights into the fracture dynamics. All deformation tests were performed using the ReaxFF force field [33], with a time step of 0.25 fs. The overall methodology applied here has been successfully used to study other carbon nanostructures [8,9,36].
RESULTS AND DISCUSSIONS
Obtaining BZCNs:
Initially, we inserted CNTs having different chiral indices (n,m) into BEA to determine the CNT with the lowest energy per carbon atom. The used unit cells contain seven channels, with a single nanotube embedded into the central channel (see Figure 2(a)). Figure 2(b) provides energy values versus nanotube diameter for (n, m) nanotubes. To obtain these energy values, first we determined the energy of the zeolite + nanotube structure, then we subtracted the energy of the isolated zeolite, and finally we divided the result by the number of carbon atoms. The results presented in Figure 2 BZCNs were created by adding multiple (6,0) carbon nanotubes to the BEA channels and then connecting them with "X" type junctions (Figure 2(c)) [37]. We considered two cases, with total and partial filling of BEA channels (hereafter name high density and low-density nets, respectively). These structures are shown in Figure 3 Energy per carbon atom for a CNT inserted into BEA versus nanotube diameter. Etotal is the total energy of a zeolite + nanotube system, Ezeolite is the energy of an isolated zeolite, and nCarbon is the number of carbon atoms in a given nanotube. (c) An "X" type junction [37] between two (6,0) CNTs. Carbons in hexagons, heptagons and pentagons are indicated by green, red, and yellow spheres, respectively.
In order to verify whether both 3D networks remain stable inside BEA, we performed 30000 MD equilibration steps at 300K followed by 30000 steps at 600 K. After verifying that they remained stable, we removed the zeolite and minimized the energy of the isolated carbon networks, to confirm the stability of the obtained net structures. Both BZCN are less dense than other usual 3D carbon allotropes such as graphite (~2.20 g/cm3) and diamond (~3.50 g/cm3) [11]. The
Mechanical properties of BZCNs:
We also investigated the mechanical behavior of BZCN under compressive and tensile strains. As the structures are porous and have large unit cells, it is important to determine whether the supercells used in the mechanical analyses are large enough to avoid spurious size-effects. For
HD-BZCN, we tested structures with supercells of 1x2x2 (1284 atoms), 2x2x3 (3852 atoms), 2x3x3 (5778 atoms), 2x4x4 (10272 atoms), and 3x4x4 (15408 atoms). For LD-BZCN, we tested structures with supercells of 2x1x1 (1078 atoms), 2x2x1 (3576 atoms), 2x2x2 (7152 atoms), 3x2x2 (10728 atoms), and 3x3x2 (16092 atoms). For each structure, we first verified their stability at 300, 500, and 1000 K. All BZCN considered in these tests remained stable, even at high temperatures.
Then, we applied to each structure a uniaxial tensile force along the y-direction (see Figure 3 for axis orientation). The results are presented in Figure 5. We obtained converged values from 10272 and 10728 atoms for HD-BZCN and LD-BZCN, respectively. Results are presented and discussed only for these structures. Figure 6(a)), and its distribution was no longer homogeneous, until fracture and amorphization occurred (Figures 7(a) and (b)).
Brittle behavior is observed in tensile strain simulations for HD-BZCN (Figure 6(b)). Also, the stress-strain curves are similar for the y and z directions, but rather different for the x direction. This can be explained by considering how the CNTs are arranged along the three Cartesian directions. Figures 3 and 4 show that nanotubes are parallel to the y and z directions, whereas all nanotubes are perpendicular to the x direction. The von Mises stress distribution during uniaxial tensile deformation is presented in Figures 7(c) and (d). For all directions, the stress is spatially well distributed for low strain values (< 8%). For intermediate strain values (~13%), the stress is more equally dispersed along the x-direction, although with higher values at the junctions, as expected. For the y direction, stress accumulates in the nanotubes that are parallel to the tensile direction, and failure occurs when these CNTs break (Figure 7(d)). For the x-direction, failure occurs not at the junctions, but at the nanotubes that are perpendicular to the tensile stress direction (see Figure 7(c)). Results for the z direction are not displayed in this figure, because both the stress arrangement and the fracture process are quite similar in the y and z orientations. Figures 8(a) and (b)). It should be stressed that although large deformations without structural failures are commonly observed in non-crystalline foam-like materials, this behavior is rare for defectless single crystalline structures [38], which makes LD-BZCN a good candidate for applications that require large deformations without fracture.
The tensile stress-strain curves for LD-BZCN show that it is a brittle (except for stretch in the x-direction) and highly anisotropic (Figure 6(d)). The tensile stress-strain curve is quite similar when the strain is applied along the y or z direction, but quite different when it is applied along the x-direction. This behavior was also observed for HD-BZCN and can be explained by the same structural features: LD-BZCN has some nanotubes that are parallel to the y and z directions, while all CNTs are perpendicular to the x direction ( Figure 4). For the y and z directions, the estimated values of the ultimate tensile strength (14.3 and 14.4 GPa, respectively) and strain (~15%) are similar. The results for the x-direction are more interesting. The stiffness is low, but very large strains can be applied. The strain value at the threshold of structural failure was 134%, and stress was ~9 GPa (see the inset of Figure 6(d)).
In Figures 8(c) and (d) we present representative MD snapshots detailing the von Mises stress values during the tensile process of LD-BZCN. For low strains, the structural stress is relatively well distributed for every direction (x,y, and z). As strain increases for the y and z directions, the stress is accumulated in the nanotubes that are parallel to the tensile direction, eventually leading to the structural failure of these CNTs (Figure 8(d)). For the x-direction, the deformation processes are more complex. As the strain increases, the nanotubes that were initially perpendicular to the tensile direction start to buckle, becoming increasingly aligned with the xdirection, as can be seen in Figure 8(c). For this orientation, the stress is initially concentrated in the junctions. Then, as the realignment process of nanotubes continues, the stress is concentrated in the CNTs, which eventually fail. It is informative to compare HD-BZCN and LD-BZCN with other carbon allotropes. In Table 1 we present the ultimate strength, ultimate strain, and Young's modulus of HD-BZCN, LD-BZCN, diamond, graphene, and carbon nanotube. All results presented in this table were obtained using ReaxFF [36], following the same methodology. When compared to other carbon allotropes, HD-BZCN and LD-BZCN exhibit lower values of ultimate strength and Young's modulus. These results are expected, considering that these BZCN phases have porous structures. The calculated ultimate strengths for these BZCN phases are higher than for many other materials, such as silicon, steel, and titanium alloys [39]. The Young's modulus of HD-BZCN is in the range found for some metals, while LD-BZCN is much less stiff. Finally, the calculated ultimate strains are similar to the other carbon phases in Table 1, with the exception of the LD-BZCN x-direction, where we have the unusual result of low value for Young's modulus, but a very high value for the ultimate strain. As discussed above, this results from the CN topology.
SUMMARY AND CONCLUSIONS
In this work, we investigated beta zeolite-templated nanotube-based 3D carbon networks (BZCNs). These networks are topologically generated by inserting carbon nanotubes (CNTs) into the zeolite channels and connecting them using X-type junctions, which are composed of only pentagons and heptagons. Our molecular mechanics calculations show there are many candidate tubes to generate the networks, with similar energy values. The results presented here are for the case of the (6,0) nanotube. We considered two cases, one with the tubes filling all zeolite channels (HD-BZCN) and one with partial filling (LD-BZCN).
We carried out fully atomistic reactive molecular dynamics (MD) simulations to investigate thermal stability and mechanical behavior under compressive and tensile loadings. Our results show that the networks exhibit high thermal stability (up to 1000 K). For compressive loadings, the structures were able to withstand very large strains (>50% for HD-BZCN and >70%
for LD-BZCN) without fracture. With respect to tensile loadings, both structures exhibited brittle and anisotropic behaviors. The anisotropy is more significant for the lower density structure, with
Young's modulus varying by a factor of ~30. For the low stiffness direction, the ultimate tensile strain value was very high, 134.1%.
Considering that there have been important advances in the zeolite-template synthesis [14,15], the production of large-size BZCN structures could be a reality in the coming years.
Recently 3D printed versions of porous carbon-based models were reported [40] and surprisingly some mechanical behaviors proved to be scale independent. This could be also the case for BZCN.
Works along these lines are in progress.
Figure 1 :
1Structure of beta zeolite with BEA framework from (a) xz and (b) yz plane views. The channels along the y and z directions are orthogonal and partially cross each other. All channels are equivalent, with a diameter of 5.6 Å. (c) Representation of the channel disposition of beta zeolite BEA. Cylinders in purple and yellow represents the channels along the y and z directions, respectively.
Figure 2 .
2Figure 2.
. The minimum energy nanotube was the (6,0), which was used to design our carbon networks. Although we have not considered BZCN structures composed of nanotubes with mixed chiralities, it is important to remark that these structures are likely relevant, given that energies inside BEA are quite similar for a range of chiral indices.
. In Figures 3(a) and 3(b), we present unit cells that can be replicated to generate the BZCN. Hereafter, we use HD-BZCN/LD-BZCN to refer to high-and low-density frameworks, respectively. The unit cell of HD-BZCN has 321 carbon atoms (a=13.4 Å, b=13.4 Å, c=29.3 Å, and α=β=૪=90º) and LD-BZCN has 894 carbon atoms (a=39.5 Å, b=39.5 Å, c=28.5 Å, and α=β=૪=90º).
Figure 2 :
2(a) Representative MD snapshot of a (6,0) carbon nanotube inside a BEA channel. (b)
calculated density values for HD-BZCN and LD-BZCN are 1.28 g/cm3 and 0.44 g/cm3, respectively.
Figure 4 provides
4MD snapshots showing the porosity HD-BZCN and LD-BZCN. For HD-BZCN (Figure 4(a)), almost circular-like channels were observed along the xy plane, results for LD-BZCN are presented in Figure 4(b). The shapes of the channels are very similar for the xy and xz planes; these rectangular channels have a crosssectional area of ~351.0 Å2 (~9 Å X 39.0 Å). For the yz plane, there are differently sized channels, namely: (i) small square channels with a cross-sectional area of ~132.2 Å2 (~11.6 Å x 11.4 Å); (ii) rectangular channels with a cross-sectional area of ~182 Å2 (~11.5 Å x 15.8 Å); and (iii) large square channels with a cross-sectional area of 251.2 Å2 (15.9 Å x 15.8 Å). Considering these results, these structures could potentially be used for gas storage, molecules and fast ion and molecular diffusion.
Figure 3 :
3Different views of BZCN unit cells for (6,0) CNTs, for high (a) and low (b) density (a) unit cells. These cells are composed of 321 (a) and 894 (b) carbon atoms. The multiple snapshots correspond to the same structure viewed from different orientations. (c) and (d) display the corresponding BZCN with and without the zeolite template.
Figure 4 :
4Different views of the channels in: (a) HD-BZCN and (b) LD-BZCN for (6,0) CNTs.
Figure 5 :
5Mechanical properties for a tensile strain along the y-direction for HD-BZCN and LD-BZCN (6,0) CNT structures having different sizes. (a) Stress-strain curves and (b) the convergence of the ultimate strength and strain for HD-BZCN as a function of the number of atoms. Corresponding results are provided in (c) and (d) for LD-BZCN.
Figure 6
6presents results for compressive and tensile strains (the entire deformation process can be seen in videos 1-8 in Supplementary Materials). Figures 6(a) and 6(b) provide results for HD-BZCN, and show its anisotropic behavior. During compression (Figure 6(a)) the stress remained almost constant (~10 GPa) from ~10% to ~40% strain. Interestingly, the stress level per atom remained well distributed throughout this strain range, as evidenced in the snapshots in Figures 7(a) and (b). This stress distribution can be attributed to the fact that deformation remained largely elastic for such large strain values. For larger strain values, stress rapidly increased (
Figure 6 :
6(a) Compression and (b) tensile stress-strain curves for HD-BZCN. Molecular dynamics simulations with the strain applied along the x, y, and z directions are indicated by using different color curves. (c) Compression and (d) tensile stress-strain curves for LD-BZCN. The inset in (d)shows the complete stress-strain curve for the x direction, where the tensile stress on the y-axis is in GPa and the percent strain is on the x-axis.
Figure 7 :
7Representative MD snapshots for different compressive (a and b) and tensile (c and d) strains in the x and y directions of HD-BZCN. The color indicates the per atom von Mises stress value. As deformation along the y and z directions yield similar stress distributions, results for the z orientation were omitted.
Figures 6
6(c) and 6(d) provide corresponding results for LD-BZCN. The stress-strain curves for compression (Figure 6(c)) are similar in all directions for strains between ~5% and ~60%. For each direction, the stress to realize ~60% strain was low (no higher than ~3.0 GPa), and there was a relatively uniform distribution of local stress values (see Figures 8(a) and (b)). Amorphization only began when strain reached ~75% (see
Figure 8 :
8Representative MD snapshots for different compressive (a and b) and tensile (c and d) strains in the x and y directions of LD-BZCN. The color indicates the per atom von Mises stress value. As deformation along the y and z directions yield similar stress distributions, results for the z orientation were omitted.
Table 1 :
1Predicted values for the ultimate strength, strain, and Young's modulus at room temperature for high density and low density BZCN. For comparison, the corresponding values for graphene, diamond, and nanotubes, which were obtained by using the same methodology[36], are also presented.Structure
Ultimate Strength
(GPa)
Ultimate Strain
(%)
Young's
Modulus (GPa)
HD-BZCN (x-direction)
46.2
16.8
157.0
HD-BZCN (y-direction)
34.1
15.3
143.6
HD-BZCN (z-direction)
28.3
15.4
109.8
LD-BZCN (x-direction)
9.1
134.1
1.5
LD-BZCN (y-direction)
14.3
15.2
44.9
LD-BZCN (z-direction)
14.4
15.8
41.5
Printed Schwarzites, Adv. Mater. 30 (2018) 1704820.
ACKNOWLEDGMENTSWe would like to thank the Brazilian agencies CNPq and FAPESP(Grants 2013/
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| []
|
[
"MOSAIC: a Multi-Object Spectrograph for the E-ELT",
"MOSAIC: a Multi-Object Spectrograph for the E-ELT"
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| [
"Andreas Kelz [email protected] \nLeibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D-14482PotsdamGermany\n",
"Francois Hammer [email protected] \nGEPI\nObservatoire de Paris\nCNRS\nUniv. Paris Diderot\nPlace Jules Janssen92190MeudonFrance\n",
"Pascal Jagourel \nGEPI\nObservatoire de Paris\nCNRS\nUniv. Paris Diderot\nPlace Jules Janssen92190MeudonFrance\n",
"\nMOSAIC consortium\n\n"
]
| [
"Leibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D-14482PotsdamGermany",
"GEPI\nObservatoire de Paris\nCNRS\nUniv. Paris Diderot\nPlace Jules Janssen92190MeudonFrance",
"GEPI\nObservatoire de Paris\nCNRS\nUniv. Paris Diderot\nPlace Jules Janssen92190MeudonFrance",
"MOSAIC consortium\n"
]
| []
| The instrumentation plan for the European-Extremely Large Telescope foresees a Multi-Object Spectrograph (E-ELT MOS). The MOSAIC project is proposed by a European-Brazilian consortium, to provide a unique MOS facility for astrophysics, studies of the inter-galactic medium and for cosmology. The science cases range from spectroscopy of the most distant galaxies, mass assembly and evolution of galaxies, via resolved stellar populations and galactic archaeology, to planet formation studies. A further strong driver are spectroscopic follow-up observations of targets that will be discovered with the James Webb Space Telescope. | null | [
"https://arxiv.org/pdf/1512.00777v1.pdf"
]
| 119,298,795 | 1512.00777 | 8c90e9d66d10b1f2c2df8694650874a1e294594f |
MOSAIC: a Multi-Object Spectrograph for the E-ELT
2 Dec 2015
Andreas Kelz [email protected]
Leibniz-Institut für Astrophysik Potsdam (AIP)
An der Sternwarte 16D-14482PotsdamGermany
Francois Hammer [email protected]
GEPI
Observatoire de Paris
CNRS
Univ. Paris Diderot
Place Jules Janssen92190MeudonFrance
Pascal Jagourel
GEPI
Observatoire de Paris
CNRS
Univ. Paris Diderot
Place Jules Janssen92190MeudonFrance
MOSAIC consortium
MOSAIC: a Multi-Object Spectrograph for the E-ELT
2 Dec 2015
The instrumentation plan for the European-Extremely Large Telescope foresees a Multi-Object Spectrograph (E-ELT MOS). The MOSAIC project is proposed by a European-Brazilian consortium, to provide a unique MOS facility for astrophysics, studies of the inter-galactic medium and for cosmology. The science cases range from spectroscopy of the most distant galaxies, mass assembly and evolution of galaxies, via resolved stellar populations and galactic archaeology, to planet formation studies. A further strong driver are spectroscopic follow-up observations of targets that will be discovered with the James Webb Space Telescope.
Motivation
The workhorse instruments of the current 8-10m class observatories are multi-object spectrographs (MOS), providing comprehensive follow-up of ground-based and spaceborne imaging data. With the advent of even deeper imaging surveys from, e.g., HST, VISTA, JWST and Euclid, many science cases require complementary spectroscopy with high sensitivity and good spatial resolution to identify the objects and to measure their astrophysical parameters. The light-gathering power of the 39m E-ELT and its spatial resolution, combined with a MOS, will enable the large samples necessary to tackle some of the key scientific drivers of the E-ELT project, ranging from studies of stellar populations out to the highest-redshift galaxies. Consequently, a MOS-facility is foreseen within the E-ELT instrumentation plan (Ramsay et al. 2014).
The MOSAIC consortium (lead by GEPI) proposes a MOS-instrument that is based on two previous concepts, EAGLE and OPTIMOS-EVE (Hammer et al. 2014). It foresees various observing modes that either provide a high multiplex, while using the E-ELT ground-layer adaptive optics system (GLAO), or a high-definition mode for fewer targets, but with better spatial resolution, enabled by multi-object adaptive optics (MOAO). The spectroscopy covers both the optical and the near-infrared.
Science cases for an ELT-MOS
For the EAGLE and OPTIMOS-EVE Phase A studies, top-level scientific questions were identified. These were re-examined and a unified set of requirements was pre-sented (Evans et al. 2014) for a versatile MOS-instrument that would exploit both the excellent spatial resolution in the near-infrared envisaged for EAGLE, combined with aspects of the spectral coverage and large multiplex of EVE. Further consolidation of the science cases led to the ELT-MOS white paper (Evans et al. 2015). Table 1 lists the eight top-level science cases which are briefly described below. • Most distant galaxies: MOSAIC shall enable detailed studies of the very first galaxies. The analysis of this light will provide vital information for the study of the early epoch when the Universe was re-ionized and its gas changed from a universally neutral to ionized state.
• Evolution of large-scale structures: MOSAIC shall map the 3d-structures of the gas between galaxies, which acts as a reservoir of matter from which protogalaxies can form, or which can feed gas into existing galaxies, aiding star formation. Clusters of galaxies lie in the highest density regions in which MOSAIC will probe the dynamical mechanisms governing galaxy formation and evolution at different epochs.
• Galaxy evolution with cosmic time: MOSAIC shall dissect galaxies over the full lifetime of the observable Universe, gathering information on their physical and chemical properties. This will allow an understanding of the origins of presentday massive galaxies, such as the Milky Way or Andromeda. MOSAIC will also provide new insights into low-mass dwarfs and low-surface brightness galaxies, which are out of reach of current facilities, but may play a major role in shaping galaxy evolution.
• AGN-galaxy co-evolution: The correlated growth of super-massive black holes (thought to be at the centre of most present-day galaxies) and their host galaxies by some self-regulated feedback processes, is a key question for galaxy evolution models. This feedback is thought to be related to massive outflows, driven by active galactic nuclei (AGN) and supernova explosions. MOSAIC shall provide substantial samples of galaxies in which the physical and geometrical parameters of such outflows can be measured.
• Galaxy archaeology and stellar populations in the Milky Way and beyond: As stars encapsulate information about the chemical composition of the gas they formed from, they retain some of the history of their host galaxy. As the elements are produced on different time scales for different stars, they also provide a timing of events. MOSAIC will observe stellar populations in the Milky Way, in nearby galaxies, and bright stars out to tens of Mega parsecs and measure their elemental abundances and radial velocities. This yields direct estimates of the chemistry for a large volume of the local Universe. Some of the old and metal-poor stars will provide a connection to the first galaxies.
• Exploring the centre of the Milky Way: One of the most spectacular results of the past decade was the observed orbits of stars around Sgr A⋆, the massive black hole at the centre of the Galaxy. Surrounding this central region are puzzling structures of gas, dust, and associated star formation, but these remain out of reach of current facilities. MOSAIC will provide insights into the physical conditions of this region.
• Planet formation in different environments: The number of known exo-solar planets grows rapidly and demands studies regarding the importance of the environment -specifically stellar density and metallicity -on their formation. MOSAIC shall undertake comprehensive radial-velocity studies of stars in considerably more diverse environments than currently possible, e.g. in open and globular clusters, spanning a wide spread of densities and metallicities, at a range of distances from the centre of the Galaxy. Figure 1. Design view of the current MOSAIC concept at the Nasmyth platform of the E-ELT (the light from the telescope enters from the left). The focal plate features multi-object adaptive optics modules and is mapped by hundreds of tiles to account for the non-telecentricity of the current ELT optical design. The spectrographs are mounted to the stable platform and linked by optical fibre bundles.
Technical Concept for MOSAIC
The design of a MOS-instrument that preserves and combines the capabilities of the previous EAGLE and EVE concepts and simultaneously complies with the conditions set by the E-ELT, such as its non-telecentricity, is highly complex. In the current design concept (Fig. 1), MOSAIC features a focal plate with 200 tiles that carry deployable elements (fibre bundles and mirrors) to pick-up the AO-corrected light. The Adaptive Optics (MOAO) correction modules (for HDM) and the natural guide star sensing devices are at the focal plate, while the laser guide star modules are further upstream. Altogether, eight spectrographs (5 for the visible, 3 for the near-infrared) are located at the gravity-stable Nasmyth platform and fed by optical fibres. This combination enables the three main observing modes, listed in Table 2.
Fiber-studies for the Science Optical Signal Transport System
MOSAIC will use miniaturized (micro-lens coupled) fibre-bundles of various sizes and multiplex. During prototype testing, the physical properties, the relative throughput and the focal ratio degradation (FRD) of such deployable bundles was studied (Guinouard et al. 2012), (Kelz et al. 2014). While the use of fibre-bundles has many advantages, such as target allocation flexibility and remote spectrograph location, their performance and stability is crucial, e.g. for accurate sky background subtraction ).
Table 1 .
1The top-level science cases for MOSAICSC1 Spectroscopy of the most distant galaxies
SC2 Evolution of large-scale structures
SC3 Mass assembly of galaxies through cosmic times
SC4 AGN-galaxy co-evolution and AGN-feedback
SC5 Resolved stellar populations beyond the local group
SC6 Galaxy archaeology
SC7 Galactic centre
SC8 Planet formation in different environments
Table 2 .
2MOSAIC observing modesHigh-Multiplex Mode
(HMM)
Multiplex
200
Spatial aperture
0.9 arcsec
Wavelength coverage
400-1800 nm
Spectral Resolution
5000 and 15,000
High-Definition Mode
(HDM)
Multiplex
10 IFUs
IFU field of view
2 × 2 arcsec
Spaxel size
75 milli arcsec
Wavelength coverage
800-1800 nm
Spectral Resolution
5000
Inter Galactic Medium (IGM)
Multiplex
10 IFUs
IFU field of view
2 × 2 arcsec
Spaxel size
0.3 arcsec
Wavelength coverage
400-1000 nm
Spectral Resolution
5000
Acknowledgments. AIP gratefully acknowledges support through the funding of the BMBF Verbundforschung grant no. 05A14BA1.
C J Evans, M Puech, B Barbuy, Proc. SPIE. SPIE914796Evans, C. J., Puech, M., Barbuy, B. et al. 2014, Proc. SPIE 9147, 96
. C J Evans, M Puech, J Afonso, arXiv:1501.04726Evans, C. J., Puech,M., Afonso, J. et al. 2015, arXiv:1501.04726
I Guinouard, L S De Oliviera, A C De Oliveira, Proc. SPIE. SPIE84503Guinouard, I., de Oliviera, L. S., de Oliveira, A. C. et al. 2012, Proc. SPIE 8450, 3
F Hammer, B Barbuy, J.-G Cuby, Proc. SPIE. SPIE914727Hammer, F., Barbuy, B., Cuby, J.-G. et al. 2014, Proc. SPIE 9147, 27
A Kelz, T Jahn, J Neumann, Proc. SPIE. SPIE9151190Kelz, A., Jahn, T., Neumann, J. et al. 2014, Proc. SPIE 9151, 190
M Puech, M Rodrigues, Y Yang, Proc. SPIE. SPIE91476Puech, M., Rodrigues, M., Yang, Y. et al. 2014, Proc. SPIE 9147, 6
S K Ramsay, M M Casali, J C González, N Hubin, Proc. SPIE. SPIE91471Ramsay, S. K., Casali, M. M., González, J. C., Hubin, N. 2014, Proc. SPIE 9147, 1
| []
|
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"The masses of young stars: CN as a probe of dynamical masses. ⋆",
"The masses of young stars: CN as a probe of dynamical masses. ⋆"
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| [
"S Guilloteau \nLAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance\n\nLAB\nUMR 5804\nCNRS\nF-33270FloiracFrance\n",
"M Simon \nDepartment of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNYUSA\n",
"V Piétu \nIRAM\n300 rue de la piscineF-38406Saint Martin d'HèresFrance\n",
"E Di Folco \nLAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance\n\nLAB\nUMR 5804\nCNRS\nF-33270FloiracFrance\n",
"A Dutrey \nLAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance\n\nLAB\nUMR 5804\nCNRS\nF-33270FloiracFrance\n",
"L Prato [email protected] \nLowell Observatory\n1400 West Mars Hill Road86001FlagstaffAZUSA\n",
"E Chapillon \nLAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance\n\nLAB\nUMR 5804\nCNRS\nF-33270FloiracFrance\n\nIRAM\n300 rue de la piscineF-38406Saint Martin d'HèresFrance\n\nAcademia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-14110617TaipeiTaiwan\n"
]
| [
"LAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance",
"LAB\nUMR 5804\nCNRS\nF-33270FloiracFrance",
"Department of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNYUSA",
"IRAM\n300 rue de la piscineF-38406Saint Martin d'HèresFrance",
"LAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance",
"LAB\nUMR 5804\nCNRS\nF-33270FloiracFrance",
"LAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance",
"LAB\nUMR 5804\nCNRS\nF-33270FloiracFrance",
"Lowell Observatory\n1400 West Mars Hill Road86001FlagstaffAZUSA",
"LAB\nUMR 5804\nUniv. Bordeaux\nF-33270FloiracFrance",
"LAB\nUMR 5804\nCNRS\nF-33270FloiracFrance",
"IRAM\n300 rue de la piscineF-38406Saint Martin d'HèresFrance",
"Academia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-14110617TaipeiTaiwan"
]
| []
| Aims. We attempt to determine the masses of single or multiple young T Tauri and HAeBe stars from the rotation of their Keplerian disks. Methods. We used the IRAM PdBI interferometer to perform arcsecond resolution images of the CN N=2-1 transition with good spectral resolution. Integrated spectra from the 30-m radiotelescope show that CN is relatively unaffected by contamination from the molecular clouds. Our sample includes 12 sources, among which isolated stars like DM Tau and MWC 480 are used to demonstrate the method and its accuracy. We derive the dynamical mass by fitting a disk model to the emission, a process giving M/D the mass to distance ratio. We also compare the CN results with higher resolution CO data, that are however affected by contamination. Results. All disks are found in nearly perfect Keplerian rotation. We determine accurate masses for 11 stars, in the mass range 0.5 to 1.9M ⊙ . The remaining one, DG Tau B, is a deeply embedded object for which CN emission partially arises from the outflow. With previous determination, this leads to 14 (single) stars with dynamical masses. Comparison with evolutionary tracks, in a distance independent modified HR diagram, show good overall agreement (with one exception, CW Tau), and indicate that measurement of effective temperatures are the limiting factor. The lack of low mass stars in the sample does not allow to distinguish between alternate tracks. | 10.1051/0004-6361/201423765 | [
"https://arxiv.org/pdf/1406.3805v1.pdf"
]
| 55,046,680 | 1406.3805 | e48b12f03bf21a1d8d21f71f652bdc3218db5f3e |
The masses of young stars: CN as a probe of dynamical masses. ⋆
15 Jun 2014 June 17, 2014
S Guilloteau
LAB
UMR 5804
Univ. Bordeaux
F-33270FloiracFrance
LAB
UMR 5804
CNRS
F-33270FloiracFrance
M Simon
Department of Physics and Astronomy
Stony Brook University
11794-3800Stony BrookNYUSA
V Piétu
IRAM
300 rue de la piscineF-38406Saint Martin d'HèresFrance
E Di Folco
LAB
UMR 5804
Univ. Bordeaux
F-33270FloiracFrance
LAB
UMR 5804
CNRS
F-33270FloiracFrance
A Dutrey
LAB
UMR 5804
Univ. Bordeaux
F-33270FloiracFrance
LAB
UMR 5804
CNRS
F-33270FloiracFrance
L Prato [email protected]
Lowell Observatory
1400 West Mars Hill Road86001FlagstaffAZUSA
E Chapillon
LAB
UMR 5804
Univ. Bordeaux
F-33270FloiracFrance
LAB
UMR 5804
CNRS
F-33270FloiracFrance
IRAM
300 rue de la piscineF-38406Saint Martin d'HèresFrance
Academia Sinica Institute of Astronomy and Astrophysics
P.O. Box 23-14110617TaipeiTaiwan
The masses of young stars: CN as a probe of dynamical masses. ⋆
15 Jun 2014 June 17, 2014Received / AcceptedAstronomy & Astrophysics manuscript no. printer c ESO 2014Stars: circumstellar matter -planetary systems: protoplanetary disks -individual: -Radio-lines: stars
Aims. We attempt to determine the masses of single or multiple young T Tauri and HAeBe stars from the rotation of their Keplerian disks. Methods. We used the IRAM PdBI interferometer to perform arcsecond resolution images of the CN N=2-1 transition with good spectral resolution. Integrated spectra from the 30-m radiotelescope show that CN is relatively unaffected by contamination from the molecular clouds. Our sample includes 12 sources, among which isolated stars like DM Tau and MWC 480 are used to demonstrate the method and its accuracy. We derive the dynamical mass by fitting a disk model to the emission, a process giving M/D the mass to distance ratio. We also compare the CN results with higher resolution CO data, that are however affected by contamination. Results. All disks are found in nearly perfect Keplerian rotation. We determine accurate masses for 11 stars, in the mass range 0.5 to 1.9M ⊙ . The remaining one, DG Tau B, is a deeply embedded object for which CN emission partially arises from the outflow. With previous determination, this leads to 14 (single) stars with dynamical masses. Comparison with evolutionary tracks, in a distance independent modified HR diagram, show good overall agreement (with one exception, CW Tau), and indicate that measurement of effective temperatures are the limiting factor. The lack of low mass stars in the sample does not allow to distinguish between alternate tracks.
Introduction
To understand the diversity among the many known planetary systems it is important to study the evolution of their protoplanetary disks and to establish a reliable clock for the very early (< 10 Myr) phases. Ages of the young stars that host the disks can provide the clocks. Unfortunately, the age of individual stars is not directly observable, and must rely on the comparison between the observed stellar properties and theoretical models of early stellar evolution, a model dependent derivation. From an observational point of view, stars can be characterized by their mass M * , radius R * , luminosity L * (or the equivalent combinations including effective temperature T eff and surface gravity g), and detailed spectrum. Ages of young stars are usually derived through their location in, for example, an HR (L * vs T eff ) diagram.
The existing stellar evolution models (Baraffe et al. 1998;D'Antona & Mazzitelli 1994Palla & Stahler 1999;Siess et al. 2000), the Y2 models from Demarque et al. (2004Demarque et al. ( , 2008, the Dartmouth models (Dotter et al. 2008), and the A&A proofs: manuscript no. printer and pulsation modes from astero-seismologic measurements can also be age indicators, but are restricted to (very) limited ranges of age and mass for young stars. The only prime parameter that can be unambiguously compared with observations remains the stellar mass.
The Keplerian rotation of disks (Guilloteau & Dutrey 1998) is the only method that can be used to measure M * , or more precisely M * /D, the Mass to Distance ratio for single stars. As L * scales as D 2 , stars can be accurately placed in a modified HR diagram: L/M 2 vs T eff , thereby canceling the impact of the distance uncertainty (that can be large for the star formation regions). Our pioneering work using this simple method indeed suggested that some of the available evolutionary models did not agree with these direct mass determinations (Simon et al. 2000). It was based on only 8 stars, but little progress has been made since. For isolated sources, CQ Tau was measured by Chapillon et al. (2008), and MWC 758 by Isella et al. (2010) using CO (but both stars suffer from large distance uncertainty, see Chapillon et al. 2008), while the mass of HH 30 (unfortunately a binary) was derived by Pety et al. (2006) from 13 CO. For embedded sources, CO disk detections were reported by Schaefer et al. (2009) for LkHa 358, GO Tau, Haro 6-13 and Haro 6-33, and by Guilloteau et al. (2011) for FT Tau, but no accurate masses could be derived because of contamination of the disk emission by emission or absorption from their surrounding environments (clouds, envelopes and/or outflows) or from molecular clouds along the line of sight. The effectiveness of dynamical mass measurements is attested by the work of Rosenfeld et al. (2012), who compared the dynamical mass derived from CO observations of the circumbinary disk of VX 4046 Sgr to the mass obtained from the analysis of the radial velocity curves of this spectroscopic binary. Piétu et al. (2007) and Dutrey et al. (2008) improved the results on DM Tau, LkCa 15, MWC 480, and GM Aur using CO isotopologues, and these 4 stars remain the only single young low mass stars with accurate masses.
Beating the contamination problem is a pre-requisite to determine accurate dynamical masses. In Guilloteau et al. (2013), we showed through a survey of 40 stars that CN N=2-1 transition is a good tracer for this purpose. It appears in general free of contamination from clouds, and is strong enough in many disks to be a sensitive tracer of the disk kinematics. We use here this property to study a sample of 12 stars in CN N=2-1 using high angular and spectral resolution spectro-imaging with the IRAM Plateau de Bure interferometer, and derive accurate masses for 11 of them.
Observations and Analysis
Source Sample
Our sample is derived from the study of Guilloteau et al. (2013). It includes all "bona-fide" disks with strong enough CN emission to be imaged in a short (4 hours per source) time with the IRAM interferometer. Sources exhibiting potential contamination from outflows or envelopes were deliberately excluded at this stage, with the exception of the enigmatic embedded object DG Tau B.
Our sample contains 12 stars: 9 T Tauri stars, one HAe (MWC 480) and two embedded objects, IRAS04302+2247 (the Butterfly star) and DG Tau B. All stars are single, except HV Tau, which is a triple system. Data for the well known, isolated (from any surrounding cloud), objects like DM Tau, LkCa 15 and MWC 480 are taken from Chapillon et al. (2012). These sources have been observed in many other molecular lines, and thus serve as a probe of the reliability of CN as a dynamical mass tracer.
Newly observed sources include objects for which previous attempts to derive dynamical masses were affected by low S/N and contamination from molecular clouds: CY Tau, DL Tau by (Simon et al. 2000), and GO Tau by Schaefer et al. (2009). The remaining sources had no previous dynamical mass measurements: DN Tau and IQ Tau (which had been observed in 12 CO by Schaefer et al. (2009) but not detected, presumably because of contamination by a molecular cloud), HV Tau, IRAS04302+2247 and DG Tau B.
Observations
All observations were carried out with the IRAM interferometer. DM Tau, LkCa 15 and MWC 480 have been reported by Chapillon et al. (2012). The characteristic angular resolution is 1.6 × 1.0 ′′ for DM Tau, and 1.3 × 0.8 ′′ for MWC 480 and LkCa 15 (1.0 × 0.65 ′′ with uniform weighting) which have baselines as long as 330 m (250 kλ).
CI Tau, CY Tau and GO Tau were observed on the night of Nov 6 to 7, 2010 in C configuration in a track sharing mode. The single-sideband, dual polarization receivers were tuned to cover the CN N=2-1 transition around 226.784 GHz. This transition has 19 hyperfine components, with relative intensities spanning 2 orders of magnitude. The high resolution backend covered the 6 strongest hyperfine components (which account for 81.25 % of the total line intensity, see Table 1), with a channel separation of 39 kHz (0.052 km/s at this frequency), and an effective spectral resolution about 1.6 times coarser given the apodization applied in the correlator. The effective integration time is about 2 hours per source, leading to an rms noise about 20 mJy/beam, or 0.3 K after resampling at 0.206 km/s spectral resolution. In addition, the wideband correlator provided a coverage of 4 GHz in each polarization.
With a longest baseline about 180 m (130 kλ), the C configuration provides an effective angular resolution of 1.3 × 1.0 ′′ at PA near 30 • (with a slight dependency on exact UV coverage). Skatrud et al. (1983); we use here the fitted values from the CDMS Database (Müller et al. 2001).
The same instrumental setting and antenna configuration were also used for the 6 other sources, which were observed on 3 contiguous nights from Nov 19, 2012 to Nov 22, 2012, two sources at a time. The effective integration time is about 3 hours per source, the rms noise 35 mJy/beam, or 0.6 K at the full spectral resolution. Phase noise were 20 to 50 • on DL Tau and HV Tau, 15 to 40 • on DN Tau and IRAS042302+2247, and 20 to 60 • on IQ Tau and DG Tau B.
All data was calibrated using the CLIC program in the GILDAS package. Bandpass calibration was made on strong quasars (3C84 or 3C454.3). Standard phase and amplitude calibrations were made using nearby quasars 0400+258 and 0507+179. Both quasars were weakly polarized, and the measured polarization information was taken into account in the amplitude calibration process. The flux calibration is based on a model flux for MWC 349 of 1.96 Jy at this frequency. The repeatability was better than 3% for the 3 consecutive nights where the same quasars were used.
In the continuum, the expected thermal noise is around 0.4 -0.6 mJy/beam for the newly observed sources. However, the original images are dynamic range limited. As the sources are compact and strong, phase-only self calibration was used to improve the on-source phase noise. This brought the final noise to within a factor 1.5-2 of the expected value. The spectral line data is essentially noise limited, so that the application of the self-calibration solution does not significantly affect the results. Thus, for CN, the reported values only use the original (non selfcalibrated) data set, with the exception of CY Tau, which will be discussed in more details.
Analysis Method
The continuum data were fit by a simple 2-D elliptical Gaussian model. Position angle of the dust disk major axis and inclination are reported in Table 2. Dust disk inclinations are derived from minor to major axis ratio. For highly inclined objects, they thus underestimate the true inclination, because of the flared disk geometry.
For the spectral line data, we use the DiskFit tool (Piétu et al. 2007) to fit a parametric disk model to the calibrated visibilities. The disk model assumes power laws for all major quantities: CN column density, CN rotation temperature, CN scale height distribution, as well as for the rotation velocity. Power laws are expressed in the from
F(r) = F 100 (r/100 AU) − f(1)
A single model has a priori 16 free parameters: -five geometric parameters: position x 0 , y 0 , rotation axis position angle PA, inclination i and systemic velocity V LSR -two parameters (value at 100 AU and exponent) for each power law: temperature T 100 , q, surface density Σ 100 , p, rotation velocity V 100 , v and scale height H 100 , h -the inner and outer radius, R int and R out -and finally, the local linewidth dV, which includes the thermal and turbulent broadening. For Keplerian rotation, we expect an exponent v = 0.50. We report instead the departure from Keplerian rotation, δv defined as v = 0.50 + 0.01δv. We neglect any radial dependency of the local linewidth. For the orientation, we follow the convention presented in Piétu et al. (2007) by giving the PA of the rotation axis oriented by the disk rotation, which is thus defined between 0 and 360 • .
Among these 16 parameters, some have negligible influence, such as the inner radius, R int , which can in general be set to an arbitrary low value < 20 AU. Others may be too strongly coupled together to be separately derived, such as those controlling the column density and temperature profiles. On the contrary, the position (x 0 , y 0 ), which is mostly determined by the phases, is essentially completely decoupled from the other parameters which are determined by the amplitude of the visibilities. In practice, the only strong coupling which matters for our objective (the stellar mass measurement) is between V 100 and sin(i). The scale height parameters, H 100 and h, will be discussed more Notes. Geometric parameters derived from CN and 1.3 mm continuum data. Orientation and inclinations are those of the rotation axis, following the convention of Piétu et al. (2007). thoroughly in Sec.3.2. For a given velocity resolution, the mass precision depends, to first order, on the product of the signal to noise ratio of the line brightness and the ratio of disk size to angular resolution, provided that each of these is substantially greater than 1. The velocity resolution must be sufficient to sample the local line width dV, which is typically 0.2 -0.4 km.s −1 ; undersampling this width would degrade the method precision, but using higher spectral resolutions provide no further improvement.
Minimization is performed using a modified Levenberg-Marquardt method, with multiple restarts to avoid being trapped in local minima. Error bars were computed from the covariance matrix. They thus should be interpreted with some caution in case of non-Gaussian distributions. However, for V 100 , the problem is well behaved and the covariance matrix provides a good estimate of its error, provided the inclination i is moderate (25 − 30 • < i < 80 • , so that the error on i remains to first order symmetric). For the two sources which have extreme inclinations, CY Tau and HV Tau, we used a different procedure to derive the uncertainty on the inclination: we instead computed the χ 2 curve as a function of i, by minimizing over all other parameters. Piétu et al. (2007) present a thorough discussion of the merit of fitting continuum subtracted data or fitting the continuum together with the spectral line. As the observed transitions are essentially optically thin, we use here continuum subtracted data.
All sources were analyzed in the same way, except DG Tau B (see Sec.3.3.4).
CO Data
We also re-analyzed CO J=2-1 data from (Simon et al. 2000), completed with the high angular resolution data obtained during the continuum survey of Guilloteau et al. (2011), for 3 sources: CI Tau, CY Tau and DL Tau. All sources suffer from contamination. Based on inspection of the images, we avoided the velocity range [4.20, 7.20] km s −1 for DL Tau (see also Fig.B.22 in Guilloteau et al. 2013, for the contaminated range in 13 CO), the range [5.50, 7.25] km s −1 for CY Tau, and [3.90, 6.50] km s −1 for CI Tau. This masking procedure makes the derivation of the systemic velocity more uncertain: formal errors are not reliable for this parameter because of the bias introduced by the channel selection. The possible bias on the systemic velocity also affects the fitted rotation velocity, but the resulting bias is not included in the formal error. Disk size and inclination may also be biased if the masked velocity range is large and encompasses the systemic velocity. Comparison with the CN results is given in Table 3. Unlike the CN data, the CO results are dominated by high resolution data, but both agree within the noise. The agreement also applies for the disk size: although CO and CN may have different radial distributions, they have identical outer radii, ∼ 460 AU for DL Tau and ∼ 280 AU for CY Tau (with typical formal errors about 15 AU). Figure 1 presents the self-calibrated continuum images.
Results and method limitations
For each source, we have two or three data cubes of 128×128 pixels and 460 channels each. Signal is spread over 50 to 100 channels, but the signal to noise per channel is in general rather low (see for example Fig.A.4). It is hopeless to present the full data cubes. For each source, we present instead two "optimal" quantities. The first quantity is the set of spectra for each (group of) hyperfine component, integrated over the disk area defined below. The second one are images of optimally filtered spectra, as computed for N 2 H + by Dutrey et al. (2007). Each channel is multiplied by the intensity of the integrated spectrum predicted by the best fit model. All channels are then summed together: the resulting quantity has the dimension of an intensity squared summed over velocity, and no simple physical interpretation, but gives the signal-to-noise for detecting line emission which matches the model profile. This signal to noise image shows where the emission is located. The process is applied separately for each (group of) hyperfine component, and then globally. The global S/N map serves as a mask to compute the integrated spectra, using a 2 σ threshold.
An example of these integrated spectra and signal to noise maps is given in Fig.2. All other sources are shown in Appendix A in Figs.A.1-A.11, except for DG Tau B which is discussed in Sect.3.3.4. Unless noted, the spectra have been smoothed to 0.206 km s −1 resolution for better clarity. The agreement between the observed line profiles and the best fit results appears sometimes limited, but this is a result of the difficulty to deconvolve low signal to noise data combined with synthesized beams with substantial sidelobes. Under such circumstances, the deconvolution cannot recover the total flux.
All sources show clear evidence for rotation, but illustrating the velocity gradient is not straightforward because of the multiple (and for the strongest, blended) hyperfine components (see Fig.B.1). We show in Fig.3 the first moment map derived from the isolated hfs component near 290 km s −1 in GO Tau. The fitting procedure, which takes all hyperfine components into account, retrieves the related information (orientation, velocity field and inclination) with much better precision.
Relevant parameters from the fit results are presented in Table 4. To illustrate the fit quality, we present in Appendix (Fig.B.1) channel maps for the strongest group of hyperfine components in GO Tau: observations, best model and residuals. On average, there is no systematic dependency of the residuals with velocities, and channels with strong emission have similar residuals than channels without. Although a few channels have some systematic residuals, this can be ascribed by the limitations of our disk model, such as the assumption of power law for the CN surface density, and should not affect the derived velocity field. For all other sources, we obtained significantly less signal to noise than for GO Tau, and noise is the limiting factor in the velocity field derivation.
A comparison between geometric parameters derived from CN and dust emission is given in Table 2. The stellar mass is derived from the Keplerian rotation of the disks. Because the maps provide only an angular scale the derived masses are proportional to the distance to the star. The masses listed in Table 4 are given at the star-forming region's (SFR) average distance, 140 pc (Kenyon et al. 1994).
CN as a dynamical mass tracer
A first important result from this study is an unambiguous confirmation that CN is essentially unaffected by contamination from the molecular clouds. The full kinematic pattern of the disk is visible, leading to accurate determination of the systemic velocity. However, the disk interpretation does not apply for the two embedded (presumably younger) objects DG Tau B and IRAS04302+2247.
A second essential result from Table 4 is that all sources appear in Keplerian rotation. The weighted mean deviation from the Keplerian exponent v = 0.50 + 0.01δv is δv = 0.5 ± 0.6. Thus, we can safely interpret the rotation pattern as being driven by a central mass.
The third result is the good agreement between geometric parameters (position angle and inclination) derived from other tracers. This is shown for CI Tau, DL Tau and CY Tau in Table 3. This agreement is important, because the geometric parameters are affected by different systematic effects due to calibration uncertainties. For continuum data, phase errors (for the high resolution data of Guilloteau et al. 2011) or amplitude errors (for our self-calibrated data) can bias the result beyond the thermal noise. For spectral line data, the orientation is defined by the velocity gradient, and thus depends on the phase bandpass calibration.
Finally, in previously studied isolated sources, the systemic velocity derived from CN is (within the noise induced uncertainties) in agreement with values derived from other molecular transitions (see Piétu et al. 2007, for DM Tau, LkCa 15 and MWC 480).
We thus conclude that, within the uncertainties of the measurements reported here, CN is a good tracer of the dynamical mass. This dynamical mass may however overestimate the stellar mass: given the angular scale of the study, it includes any contribution from any compact (< 20 − 30 AU radius) disk that may surround the star.
The vertical distribution of CN
In our analysis, CN is assumed to be homogeneously distributed in the vertical direction, i.e. to be distributed vertically following a gaussian whose characteristic size is the disk scale height. The scale height was initially arbitrarily fixed to H 100 = 16.5 AU and h = −1.25 (mildly flared disk). However, we expect CN to be located at the top of the molecular layer, between the highly irradiated surface layer where molecules are photo-dissociated and the colder disk plane where molecules condense on grains. Although this expected distribution has so far not been confirmed by imaging studies, and even faces some difficulty when considering the expected temperature in this molecular layer (Chapillon et al. 2012), it is important to evaluate if this can affect the derived disk inclination and hence, the stellar mass measurement.
We did that in 3 steps. First, we explored different disk thickness. This did not significantly affect the derived dynamical mass. Second, we treated the scale height as a free parameter. We found in general larger values for H 100 , and flatter disks h ≈ −1.0, which is consistent with the molecules being closer 3.11 ± 0.10 47.0 ± 1.3 567 ± 39 1.09 ± 0.07 0.0 ± 2.4 CI Tau 2.67 ± 0.03 51.0 ± 0.9 520 ± 13 0.80 ± 0.02 -2.7 ± 2.0 CY Tau 2.36 ± 0.12 24.0 ± 2.0 295 ± 11 0.63 ± 0.05 1.7 ± 1.7 GO Tau 2.07 ± 0.01 54.5 ± 0.5 587 ± 55 0.48 ± 0.01 4.0 ± 2.0 HV Tau C 3.76 ± 0.10 89.1 ± 3.0 256 ± 51 1.59 ± 0.08 -0.0 ± 2.9 DL Tau 2.83 ± 0.04 44.1 ± 2.6 463 ± 6 0.91 ± 0.02 1.9 ± 1.1 IQ Tau 2.64 ± 0.02 56.3 ± 3.9 225 ± 21 0.79 ± 0.02 -0.3 ± 4.9 DN Tau 2.91 ± 0.25 29.2 ± 3.0 241 ± 7 0.95 ± 0.16 -0.6 ± 1.8 04302+2247 4.18 ± 0.09 58.9 ± 2.1 750 ± 56 1.97 ± 0.08 -0.4 ± 2.3 Notes. δv is the departure from Keplerian rotation: v(r) ∝ r −(0.50+0.01 δv) .
to the disk surface. However, the number of free parameters becomes large, and the fits sometimes converge towards unrealistic solutions (e.g. very large H 100 and h > −1.0). Third, we used the method described by Guilloteau et al. (2012) for the analysis of CS in DM Tau. We assumed CN molecules to be absent at any point where the H 2 column disk towards the disk surface Σ o (r, z) is larger than a given value, Σ d . Σ o (r, z) density is computed using the prescribed scale height (H 100 = 16.5 AU and h = −1.25), and with the surface density profile corresponding to the best fit viscous disk model to the 226 GHz continuum data:
Σ g (r) = Σ 0 r R 0 −γ exp −(r/R c ) 2−γ .(2)
Despite the much more limited angular resolution of the new data, we reached a similar precision on γ and R c as Guilloteau et al. (2011), thanks to the higher sensitivity and lower phase noise due to self-calibration. Furthermore, the values found for γ and R c were within the errors equal to those measured by Guilloteau et al. (2011). We varied the depletion scale height Σ d between 10 21 and 10 24 cm −2 . The most extreme values provided significantly worse fit to the data, with a best fit Σ d around 10 23 found for most sources. The derived disk inclinations and dynamical masses only change by small amounts as a function of Σ d : less than ∼ σ/3 over the acceptable range of values for Σ d , and even at most about 1σ for more extreme values. We thus conclude that the uncertainties in the spatial distribution of CN due to our limited knowledge of the disk structure and chemistry do not affect the reliability of the CN N=2-1 transition as a tracer of the dynamical mass.
Special sources
We derived accurate dynamical masses for most sources in our sample. However, a few sources are peculiar in this respect, be-cause of unfavorable inclination and orientation, evolutionary status or multiplicity. We discuss here these sources which we omit in further analysis.
HV Tau
HV Tau is a triple system: AB is a close visual binary with angular separation around 70 mas (Simon et al. 1996), and HV Tau C is a much fainter T Tauri star located 4 ′′ NE of AB. A nearly edge-on compact dust disk surrounds HV Tau C (Monin & Bouvier 2000), but AB shows no IR excess and no sign of accretion (White & Ghez 2001). The system was modeled in detail by Duchêne et al. (2010), who showed that mmemission only comes from the HV Tau C circumstellar disk, which also exhibit 12 CO emission compatible with Keplerian rotation, although only the line wings are clearly visible.
CN emission was detected using the IRAM 30-m by Guilloteau et al. (2013), who attributed it to HV Tau C based on the previous non-detection of any circumstellar material around AB by Duchêne et al. (2010). Our images confirm this association. The derived dynamical mass, ∼ 1.6M ⊙ , is much larger than suggested by apparent spectral type (K6, White & Hillenbrand 2004). Given the system complexity, a more complete study of the HV Tau multiple system will be presented in a separate paper.
With IQ Tau, HV Tau C is another good example of disk detection through CN while CO emission is heavily contaminated by the molecular cloud. Note that both stars are close to each other, and affected by the same molecular cloud. The higher inclination of HV Tau C allows the line wings to remain visible in CO.
CY Tau
CY Tau and DN Tau are the two sources which combine a small (280 AU) disk with a low inclination (25-30 • ). The DN Tau disk is favorably oriented with respect to the synthesized beam, allowing the inclination to be precisely measured. On the contrary, for CY Tau, the synthesized beam major axis is within 30 • of the disk rotation axis, so that the limited angular resolution in this direction makes the inclination, and consequently the deprojected rotation velocity, rather uncertain. Without self calibration for CN, the χ 2 curve as a function of inclination for CY Tau has two minima, at ≃ 21 • and ≃ 30 • , separated by about 3σ. After application of the self-calibration solution from the continuum data, this CN χ 2 curve now exhibits a single minimum, indicating an inclination of ≃ 24 ±2 • . However, low inclinations are not strictly excluded, and the only safe conclusion is i < 28 • at the 3σ level, which implies M * > 0.45M ⊙ from CN only. From the CN χ 2 curve, we derive M * = 0.63 +0.11 −0.06 M ⊙ . However, using all other available measurements, the weighted mean of the inclinations is 27 ± 2 • , and the weighted v sin i = 0.95 ± 0.04 km s −1 , which suggest a somewhat lower value for the most likely mass, ∼ 0.51M ⊙ .
In our nominal solution from CN, we find an orientation which agree with previous higher resolution measurements made from continuum and CO (Guilloteau et al. 2011), see Table 3. The orientation derived from the (self-calibrated) continuum is different (although marginally, i.e. at the 2σ level, the errors being large). Such a difference can be due to inaccuracies in the amplitude calibration as a function of time (and hence hour angle). Similar calibration inaccuracies cannot significantly affect other disks, since their axis ratio is much larger than for CY Tau (or better constrained because of a more favorable orientation in the case of DN Tau).
The Butterfly star, IRAS 04302+2247
In this embedded source, CN is clearly not peaking towards the continuum (see Fig.4), and does not originate from the inner disk traced by dust and CO isotopologues. Although the morphology could be the trace of asymmetric emission from the envelope surrounding the central disk, we also fitted CN emission by a flared, rotating disk. The derived centroid, orientation, and inclination differ strongly from those of the continuum disk, reflecting the apparent asymmetry. We find an inner radius around 150 AU, similar to the large grains disk size found by Gräfe et al. (2013), thus indicating that CN is not present in the dense disk, but only in the outer envelope. Despite this different morphology, the rotation pattern still appears nicely Keplerian, and the derived total mass, 1.9M ⊙ , is consistent with the sum of the star (1.7M ⊙ , Dutrey et al. 2014, in prep.) and disk mass.
The continuum emission clearly reveals a weak, resolved, secondary source 4 ′′ south of the main disk (at PA 171 • ), with a total flux density of 12 mJy. The main disk has a flux of 138 mJy. IRAS 04302+2247 may actually be the progenitor of a binary (or higher multiplicity) system.
DG Tau B
DG Tau B is another embedded object, with a powerful, onesided, molecular outflow (Mitchell et al. 1997). The rather low signal to noise limits our ability to interpret the CN distribution. However, the CN emission is not centered on the strong continuum emission. On the contrary, we detect absorption towards the continuum source, and emission predominantly east of the continuum peak, see Fig.5. The absorption profile is very uncertain, but could be due to a single narrow component at V LSR ≃ 6.2 km s −1 , the hyperfine structure leading to the impression of two velocity features. The cloud velocity towards DG Tau B is 6.44 km s −1 from C 17 O measurements (Guilloteau et al. 2013). The peak continuum flux is 220 mJy/beam, or about 3.7 K, and the absorption dip has a depth of about 60 mJy/beam. Assuming the absorbing material fills the synthesized beam, this implies an upper limit on the excitation temperature of 7 K (taking into account the deviations from the Rayleigh Jeans approximation), obtained for optically thick absorbing medium. This is consistent with the expected temperatures in the surrounding dense cloud.
On the contrary, the emission has a much larger linewidth, of order 3-4 km s −1 . The integrated line flux is ∼ 0.6Jy km s −1 , about 50 % of the flux detected by the 30-m, which suggests that some low level emission has not been recovered properly 1 .
Discussion: Implications for stellar evolution models
Comparison of the measured masses with theoretical evolutionary tracks is best made on a modified HRD in which the distanceindependent quantity L/M 2 * is plotted vs T eff (Simon et al. 2000). Table 5 lists the stellar parameters of the stars we consider for 1 This does not imply missing flux, but can be entirely caused by the limits of deconvolution due to low signal to noise ratio a comparison with models of PMS stellar evolution. For completeness, Table 5 also lists the masses for several of stars that were reported previously. Figure 6 shows the locations of these stars within the SFR. Of the stars in Table 4, Table 5 does not include MWC 480, HV Tau C, and 04302+2247. Piétu et al. (2007) have already carefully considered MWC 480 on the modified HRD. We will discuss the multiple system HV Tau and its masses in a forthcoming paper (Guilloteau et al 2014, in prep.), and 04302+2247 is a Class I YSO with inadequately known stellar parameters (Gräfe et al. 2013). All spectral types (col. 3), effective temperatures, T eff (col. 4), and luminosities (col. 5) are from Andrews et al. (2013) except for Haro 6-33 which is from White & Hillenbrand (2004). We assumed T eff uncertainties corresponding to ±1 spectral type and used the spectral type-T eff look-up table in Pecaut & Mamajek (2013). To calculate uncertainties in L/M 2 * we propagated the uncertainties in L and M according to the usual procedure for the uncertainty of a ratio when one of the variables is squared.
For stars in Table 5 with mass uncertainties less than or equal to ±5%, Fig.7a plots their L/M 2 * vs T eff on a modified HRD using the evolutionary tracks calculated by Dotter et al. (2008, Dartmouth). Figure 7b is a similar plot for stars with mass precisions worse than ±5%. Figure 7b does not include Haro 6-33 because the uncertainty of its luminosity is not available. Figs.C.1-C.3 in Appendix C show similar plots for the evolutionary tracks of Baraffe et al. (1998, BCAH), Siess et al. (2000, SDF), and Tognelli et al. (2011, Pisa). The numbering of the stars follows that of Table 5.
Three features are apparent in all the modified HRD plots:
1. No stars with mass below ∼ 0.5M ⊙ appear. Mass measurement by the circumstellar disk technique of lower mass stars in Taurus requires higher sensitivity, presumably because their disks are less massive (e.g. Schaefer et al. 2009;Andrews et al. 2013) but also much smaller (Piétu et al. 2014). 2. T eff uncertainties severely compromise the comparisons with the tracks because most uncertainties span tracks separated by more than 0.1M ⊙ . 3. Uncertainties in L/M 2 * affect comparisons with evolutionary tracks very little because, at the ages displayed, the tracks are nearly parallel to the L/M 2 * axis. However, uncertainties in L/M 2 * can affect the age estimate. Table 6 summarizes, qualitatively, a comparison of the best masses with the 4 theoretical tracks. Columns 1 and 2 identify the stars and column 3 repeats, for convenience, their measured dynamical masses at 140 pc distance. Columns 4-7 give the mass range corresponding to the ±1σ uncertainties of the (T e f f , L/M 2 * ) values. Table 6 indicates that the measured masses of DM Tau, CI Tau, and LkCa 15 (0.53, 0.80, and 1.01M ⊙ , respectively) are in agreement with all 4 sets of tracks if they are actually at the fiducial distance 140 pc. However, IQ Tau with a measured mass indistinguishable from that of CI Tau, and GM Aur with a measured mass indistinguishable from that of LkCa 15, are inconsistent with their nominal mass tracks 0.80M ⊙ and 1.00M ⊙ , respectively. Two reasons are likely. Their published spectral types may not provide an accurate indication of their T eff . Also, their distances may be greater or less than 140 pc which would not affect their L/M 2 * values but would change their mass derived from the angular measure of their disk rotation. For example, Fig.6 shows that the position of GM Aur in the L1517/19 region is close to NTTS 045251+3016 at distance 158.7 ± 3.9 pc (Simon et al. 2013). If this distance applies to GM Aur, its mass For the stars with the most precise mass measurements, Table 7 summarizes the ages indicated by their locations on the modified HRDs. All 4 theoretical calculations indicate the stars are younger than 10 Myr, and that a representative age for most of the stars is between 2 and 4 Myr, consistent with prior estimates (e.g. Kenyon et al. 2008). Improved values of stellar parameters will yield more accurate age estimates.
A&A proofs: manuscript no. printer In conclusion, the reliability of these comparisons of the measured masses with the theoretical tracks is compromised by the uncertainties on the effective temperatures and distances. Also, a definitive assessment of the theoretical evolutionary tracks must await mass measurements of stars with masses below 0.5M ⊙ . We look ahead to GAIA to provide precise distances to stars in the Taurus star forming region and have started measurements that we hope will improve the determinations of T eff and will extend mass measurements to masses smaller than those presented in this paper.
Conclusion
We have analyzed ∼ 1.2 ′′ arcsecond resolution observations of the CN N=2-1 line in a sample of T Tauri stars, and performed a comparison with ∼ 0.6 ′′ resolution data obtained in 12 CO for several sources. The results show that the CN transition is a good, sensitive, contamination free, dynamical mass tracer for stars in the M1 -A4 spectral type range that are surrounded by disks larger than about 250 AU. The striking ability of CN to overcome the contamination problem is best shown by the detection of the disk in IQ Tau that escaped detection in CO at similar angular resolution.
The largest uncertainty in the mass derivation comes from the determination of the inclination, although the impact is small for sources with i > 45 • . This uncertainty can be reduced by higher angular observations, at the expense of more observing time. Another pending problem is that the more embedded, presumably younger, objects like DG Tau-B or IRAS04302+2247, have more complex structure and CN does not appear to be a very good disk tracer in these sources.
We used the derived dynamical masses in conjunction with similar measurements performed using other tracers to compare with the evolutionary tracks. Agreement of the measured masses with the evolutionary tracks is quite good. The discrepancies are most likely attributable to inaccurate effective temperatures and, to a lesser extent, distances different from the 140 pc average value. We hope to remove some of these discrepancies and to extend the mass measurements below 0.5M ⊙ (spectral type earlier than M2) where the differences among the theoretical calculations are the greatest. Baraffe et al. (1998) tracks. Stars with dynamical mass precisions measured to better than 5% appear in the left panel and stars with lower precions appear in the right panel.
offprint requests to: S.Guilloteau, e-mail: [email protected] ⋆ Based on observations carried out with the IRAM Plateau de Bure interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).
Fig. 3 .
3Velocity gradient for GO Tau. Contour spacing is 0.2 km s −1 .
Fig. 1 .
1Continuum images for all newly observed sources. Contour levels are logarithmically spaced, by a factor 2: -2 and 2, 4, 8, 16, 32, 64, 128 and 256 σ. Rms noise (in mJy/beam) and peak signal to noise are: CI Tau 0.6 and 180, CY Tau 0.5 and 190, GO Tau 0.26 and 180, HV Tau 0.12 and 230, DL Tau 0.23 and 550, IQ Tau 0.23 and 270, DG Tau B 0.7 (dynamic range limited) and 450, DN Tau 0.25 and 360, and IRAS04302+2247 0.4 and 180
Fig. 2 .
2Results for GO Tau. Top: Integrated line flux for CN N=2-1 hyperfine components (histogram) with the best fit profile superimposed (red curve). Bottom: Signal to noise maps for each group of component, and for all observed components (rightmost panel); contour spacing is 2σ.
Fig. 4 .
4Superposition of the CN N=2-1 emission (in red contours) over the continuum emission (in grey scale) in IRAS04302+2247. Contour levels are 2,4,6,8 σ for CN, and 2, 4, 8, 16 and 32 σ for the 1.4 mm continuum emission.
Fig. 5 .
5Top: (continuum subtracted) integrated emission from CN N=2-1 main group of hyperfine components in DG Tau B. Note the negative contours towards the continuum source at (0,0). Middle: CN N=2-1 spectrum toward the continuum source. Bottom: integrated flux density over the emission region.
Fig. 6 .
6Location of the observed stars in the Taurus Auriga region. The contours are low lying levels of CO integrated intensity from the Dame et al. (2001) survey. The red dots are stars in common between Ducourant et al. (2005) and Kenyon et al. (2008) master list of PMS stars in Taurus. The red crosses show the stars with precise distances (VLBA or VB+SB2) The blue crosses show the stars with masses measured by disk rotation.
Fig. 7 .
7Stars on the modified, distance-independent, HR diagram L/M 2 vs T eff from Dotter et al. (2008) evolutionary tracks. Left: stars with dynamical masses accurate to < 5%, right: other stars.
Fig
. C.1. Stars on the modified, distance-independent, HR diagram L/M 2 vs T eff for the
FigFig
. C.2. As for Fig.C.1, but for the Siess et al. . C.3. As for Fig.C.1, but for the Tognelli et al. (2011) tracks.
Table 1 .
1Frequencies of observed CN N=2-1 transitionsFrequency
Hyperfine
Relative
(MHz)
Transition
Intensity
226659.5584 J=3/2-1/2, F=5/2-3/2
0.1667
226663.6928 J=3/2-1/2, F=1/2-1/2
0.0494
226679.3114 J=3/2-1/2, F=3/2-1/2
0.0617
226874.1908 J=5/2-3/2, F=5/2-3/2
0.1680
226874.7813 J=5/2-3/2, F=7/2-5/2
0.2667
226875.8960 J=5/2-3/2, F=3/2-1/2
0.1000
Notes. CN N=2-1 line frequencies were measured in laboratory by
Table 2 .
2Geometric parametersSource
Orientation ( • )
Inclination ( • )
name
CN
Cont.
CN
Cont.
CI Tau
281.5 ± 0.5
282.7 ± 1.1 51.0 ± 1.9
45.7 ± 1.1
CY Tau
62.5 ± 1.8
35.6 ± 6.2 24.4 ± 2.4
34.4 ± 9.0
GO Tau
111.1 ± 0.3
114.8 ± 2.3 54.5 ± 0.5
48.6 ± 2.6
HV Tau C
197.8 ± 0.9
199.5 ± 0.8 89.1 ± 3.0
75.8 ± 1.1
DL Tau
322.6 ± 0.6
320.5 ± 0.3 43.6 ± 2.5
42.3 ± 0.3
IQ Tau
309.9 ± 1.0
313.8 ± 1.1 57.9 ± 6.7
58.2 ± 1.6
DG Tau B
25.7 ± 0.3
58.0 ± 0.4
DN Tau
174.2 ± 2.2
185.8 ± 5.7 29.7 ± 2.7
26.8 ± 3.9
04302+2247
297.6 ± 2.1
254.2 ± 1.4 58.9 ± 2.1
78.1 ± 3.0
Table 3 .
3Comparison between CO and CN results. Notes. (a) This work with 1.3 ′′ resolution. (b) From Guilloteau et al. (2011) with ≃ 0.5 ′′ resolution. (*) Guilloteau et al. (2011) incorrectly reported the orientation modulo 180 • for this source.Tracer
Orientation
Inclination
V LSR
V 100 sin(i)
( • )
( • )
( km s −1 )
( km s −1 )
CY Tau
CN
62.5 ± 1.8
24.0 ± 2.4 7.26 ± 0.01
0.99 ± 0.06
Cont. (a)
36 ± 6
34 ± 9
-
CO
63 ± 1
29 ± 5
7.27 ± 0.02
0.95 ± 0.05
Cont. (b)
63 ± 5
34 ± 3
-
DL Tau
CN
322.6 ± 0.6
43.6 ± 2.5 6.10 ± 0.01
1.94 ± 0.02
Cont. (a)
320.5 ± 0.3
42.3 ± 0.3
-
CO (*)
321.0 ± 2.4
39.6 ± 1.3 6.00 ± 0.10
2.04 ± 0.10
Cont. (b)
321 ± 3
38 ± 2
-
CI Tau
CN
281.5 ± 0.5
50.1 ± 1.9 5.73 ± 0.02
2.61 ± 0.04
Cont. (a)
282.7 ± 1.1
45.7 ± 1.1
-
CO
285.2 ± 0.8
53.3 ± 1.9 5.77 ± 0.03
2.46 ± 0.05
Cont. (b)
286.0 ± 2.1
53.8 ± 1.7
-
Table 4 .
4Disk and Star parameters derived from CNSource
V 100
i
R out
M *
δv
name
(km s −1 )
( • )
AU
(M ⊙ )
DM Tau
2.31 ± 0.17 -30.9 ± 2.9 641 ± 19 0.60 ± 0.09 0.1 ± 1.5
MWC 480
4.03 ± 0.41 36.3 ± 2.5 539 ± 39 1.83 ± 0.37 -2.2 ± 2.0
LkCa 15
Table 5 .
5Dynamical masses for single starsNotes. References for Mass: M0, this work; M1,Piétu et al. (2014); M2,Piétu et al. (2007); M3,Dutrey et al. (2008); M4,Schaefer et al. (2009). References for Spectral Type and Luminosity: L0,Andrews et al. (2013); L1,White & Hillenbrand (2004) would become (158.7/140) times greater, i.e. 1.13 ± 0.02M ⊙ , decreasing the discrepancy with respect to the 1.1M ⊙ theoretical track but still more than ∼ 1σ too high with respect to all the 1.1M ⊙ tracks.Fig.6also shows that the position of CI Tau in the L1529 region is close to HP Tau G2 for whichTorres et al. (2009) measured the precise distance 161.2 ± 0.9 pc. At this distance the measured mass of CI Tau would be (161.2/140) times greater or 0.92 ± 0.02M ⊙ . If this were its actual mass, the quality of consistency with the Dartmouth and Pisa evolutionary would be degraded. It seems reasonable to conclude that a definitive comparison of the measured masses with the available theoreti-cal evolutionary tracks should await a more accurate determination of effective temperatures of the target stars, mass measurements of stars with masses below 0.5M ⊙ , and accurate distances to all.Name
SpType T eff
L *
M *
Comments
Number
(K)
L ⊙
M ⊙
& References
1 CW Tau
K3
4840 +200
−220
2.42 ± 1.40 0.69 ± 0.14 M1, L0, L1495
2 CY Tau
M1
3615 ± 65 0.40 ± 0.09 0.63 ± 0.05 M0, L0, L1495
3 IQ Tau
M0.5
3765 ± 85 0.81 ± 0.26 0.79 ± 0.02 M0, L0
4 Haro 6-13 M0
3850 +200
−170
0.69 ± 0.22 1.00 ± 0.15 M4, L0. L1529
5 DL Tau
K7
4050 +150
−200
0.74 ± 0.38 0.91 ± 0.02 M0, L0
6 DM Tau
M1
3680 +170
−130
0.23 ± 0.02 0.53 ± 0.02 M2, L0, L1551
7 CI Tau
K7
4050 +150
−200
0.93 ± 0.35 0.80 ± 0.02 M0, L0, L1529
8 DN Tau
M0
3850 +200
−170
0.79 ± 0.16 0.95 ± 0.16 M0, L0, L1529
9 LkCa 15
K5
4450 +170
−250
0.81 ± 0.21 1.01 ± 0.03 M2, L0
10 Haro 6-33 M0.5
3765 +115
−85
0.76
0.5 ± 0.1
M4, L1
11 GO Tau
M0
3850 +200
−170
0.29 ± 0.09 0.48 ± 0.01 M0, L0
12 DS Tau
K5
4450 +170
−250
0.76 ± 0.34 0.68 ± 0.12 M1, L0
13 GM Aur
K3
4850 +200
−220
1.23 ± 0.32 1.00 ± 0.02 M3, L0, L1517/9
Table 6 .Table 7 .
67Comparison of best Masses with tracks IQ Tau 0.79 ± 0.02 F 0.65-0.70 F 0.45-0.70 F 0.48-0.72 P 0.50-0.60 5. DL Tau 0.91 ± 0.02 E 0.72-0.92 E 0.60-0.92 F 0.62-0.88 F 0.68-0.88 6. DM Tau 0.53 ± 0.02 E 0.50-0.70 E 0.35-0.59 E 0.43-0.62 E 0.44-0.62 7. CI Tau 0.80 ± 0.02 E 0.70-0.88 E 0.58-0.90 E 0.60-0.82 E 0.59-0.80 9. LkCa 15 1.01 ± 0.03 E 0.92-1.30 E 0.95-1.30 E 0.90-1.25 E 0.92-1.30 11. GO Tau 0.48 ± 0.01 P 0.70-0.92 E 0.45-0.75 F 0.52-0.82 E 0.48-0.70 13. GM Aur 1.00 ± 0.02 P 1.35-? Notes. Qualitative agreement: E= Excellent G= Good F= Fair P= Poor Approximate ages (Myr)# Star
Dynamical
Agreement & Evolutionary Track Mass range
Mass
BCAH
SDF
Pisa
Dartmouth
3. P 1.51-?
P 1.30-?
P 1.20-?
# Star
Evolutionary Tracks
BCAH SDF Pisa Dartmouth
3. IQ Tau
2
3
3
2
5. DL Tau
3
4
4
3
6. DM Tau 5
8
5
3.5
7. CI Tau
1.5
2.5
2.5
1.5
9. LkCa 15 3
4
4
3
11. GO Tau 2
4
3
3
13. GM Aur 2
2
2
1.5
Acknowledgements. This work was supported by "Programme National de Physique Stellaire" (PNPS) and "Programme National de Physique Chimie du Milieu Interstellaire" (PCMI) from INSU/CNRS. The work of MS was supported in part by NSF grant AST 09-07745. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.Appendix C: Evolutionary TracksTo compare the masses we have measured to different evolutionary models, we show inFig.C.1-C.3 the position of the stars with accurate masses on modified HR diagram for theBaraffe et al. (1998, BCAH), theSiess et al. (2000)and theTognelli et al. (2011, PISA)tracks; see alsoFig.7for the Dartmouth(Dotter et al. 2008)tracks. Taken together, the four different models differ the most with respect to each other at masses less than 0.5M ⊙ and have different evolutionary time scales for all masses at ages less than 5 Myr.
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| []
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[
"Self-organising Dissipative Polymer Structures",
"Self-organising Dissipative Polymer Structures"
]
| [
"Andrei N Yakunin [email protected] \nLaboratory of Polymer Materials\nKarpov Institute of Physical Chemistry\nFederal State Unitary Enterprise\nMoscowRussia\n"
]
| [
"Laboratory of Polymer Materials\nKarpov Institute of Physical Chemistry\nFederal State Unitary Enterprise\nMoscowRussia"
]
| []
| In frameworks of the scaling theory of phase transitions and critical phenomena the quantitative dependence of macroscopic properties on nanostructural parameters in a polymeric material is revealed. The draw ratios at neck and at break are referred to the macroscopic properties. The structure is characterized by an average thickness of amorphous layers in isotropic melt-crystallized linear high density polyethylene which is chosen as an example. The square of the neck draw ratio is equal to the product of the square of the draw ratio at break and the chain ends collision probability. This probability in its turn is proportional to the average thickness of amorphous layers in the isotropic material. The neck draw ratio is a parameter of order. Polymers with flexible chains are solutions in solid state as well as in melt, the interacting ends of a marked chain serving as a solvent. At critical polymerization degree all the phases are identical. This research is an important contribution to the molecular theory of polymer liquids. It has been found that the melt viscosity vs the molecular weight of linear flexible-chain polymer follows the power law with the 3.4-exponent within the reptation model near the critical point. This is different from the value 3 expected for the melt of ring macromolecules. The rotation vibration precession motion of chain ends about the polymer melt flow direction were taken into account to find better agreement with the experiment. | null | [
"https://export.arxiv.org/pdf/1210.7745v1.pdf"
]
| 117,411,190 | 1210.7745 | da7097901af6a671378d02e4192894ad77ec35ea |
Self-organising Dissipative Polymer Structures
Andrei N Yakunin [email protected]
Laboratory of Polymer Materials
Karpov Institute of Physical Chemistry
Federal State Unitary Enterprise
MoscowRussia
Self-organising Dissipative Polymer Structures
critical phenomenapolymer meltsentanglementsphase transitionscritical exponents
In frameworks of the scaling theory of phase transitions and critical phenomena the quantitative dependence of macroscopic properties on nanostructural parameters in a polymeric material is revealed. The draw ratios at neck and at break are referred to the macroscopic properties. The structure is characterized by an average thickness of amorphous layers in isotropic melt-crystallized linear high density polyethylene which is chosen as an example. The square of the neck draw ratio is equal to the product of the square of the draw ratio at break and the chain ends collision probability. This probability in its turn is proportional to the average thickness of amorphous layers in the isotropic material. The neck draw ratio is a parameter of order. Polymers with flexible chains are solutions in solid state as well as in melt, the interacting ends of a marked chain serving as a solvent. At critical polymerization degree all the phases are identical. This research is an important contribution to the molecular theory of polymer liquids. It has been found that the melt viscosity vs the molecular weight of linear flexible-chain polymer follows the power law with the 3.4-exponent within the reptation model near the critical point. This is different from the value 3 expected for the melt of ring macromolecules. The rotation vibration precession motion of chain ends about the polymer melt flow direction were taken into account to find better agreement with the experiment.
INTRODUCTION
The quantitative relationship of structural and macroscopic characteristics (respectively, an average thickness of amorphous layers and draw ratios at neck and at break) is revealed in linear melt-crystallized (MC) polyethylene (PE). This acquires a great significance for modern chemical nanoengineering when it is necessary to understand clearly how to obtain functional nanostructures in order to prepare the material with sought properties by self-organization and selfassembly. The model is associated with such phenomena as drawing polymeric materials and dissipating the energy during the transition from isotropic to oriented state. The results obtained are in a good agreement with experimental data.
It has been revealed [1] that the neck draw ratio, i.e. the ratio of initial cross-section area to final, λ n , in linear MC PE decreases with increasing the mean-weight molecular mass, M w , as lnλ n = A -βlnM w where A is a positive constant and β ≈ 0.3 is the critical exponent of the fluctuation theory of the second order phase transition (PT). It has been also shown [1] that the transformation to the oriented state is the first order PT forming 3 stages: swelling the polymer under applied stress, dissolving crystallites and crystallizing extended macromolecules. From these data one can suppose that no drawing will occur (λ n = 1) if the polymerization degree (the number of chain monomers per one macromolecule, PD) is more than a some critical meaning, N cr . Otherwise, the free energy of oriented state is less than the free energy of isotropic state [1] if N < N cr .
The polymer melt viscosity dependence, η(N), is commonly described in terms of the reptation model [2][3][4] suggested by P.-G. de Gennes in 1971. The reptation theory explains why the power law for the melt viscosity relation to the molecular weight of polymer can be observed. However, this is not fully supported by the experimental data where the exponent is equal to 3. 3-3.4 or even is out of the range compared to the theoretical value of 3. In order to achieve a better agreement with the experiment one may assume that the length of the tube created by entanglement chains along which a macromolecule can crawl making the reptation motions, can be subjected to fluctuations [3]. Wool [5] obtained a similar result using a different concept. Other attempts have been aimed at modifying the underlying equations [6,7], they still continue [8,9]. Similar techniques [8] are believed to be applicable to study more complex systems such as phaseseparate polymer solutions, blends, block and graft copolymer mesophases and other fluids. However, in such approaches the number of monomers between neighbour entanglements along the chain, N e , is introduced phenomenologically or ignored completely. Moreover, the recent Monte Carlo results [10] have shown that reptation motions prevail in melts where entropic trapping is absent in contrast to swollen gels [11]. At last, there are models in which an exponential increase of the reptation time with increasing PD can be observed if the polymer chains are long enough N > N e 3 [12,13]. Thus, the questions dealing with polymer melt viscosity and entanglements have been extensively discussed [8][9][10][12][13][14]. This indicates that the problems still exist although the entanglement concentration has been already estimated [5,15].
One of aims of the article is to construct a solution in such a way that would enable to achieve a good agreement with the well-known experimental data within the classical reptation model [2] without any additional simplifying assumptions.
The structure of the article is following. We will define the critical PD or the critical point. Then we will recall the main principles of the reptation theory [2][3][4] and try to understand what ways can be used to modify the theory. We will demonstrate that the model [16] based on fluctuations of the This work is financially supported by RFBR, grants 11-03-00669а. Dedicated to the memory of Dr. V. A. Buchin from Moscow State University. tube length may be not correct and suggest that a mechanical field be introduced to correct it. In that case, a chain can probably "swell" in a medium of other chains and the vector connecting the chain ends can rotate making a vibration precession motion about the direction of the polymer melt flow. Finally, we will obtain a good agreement with the experimental data using the modified reptation theory. Then we will compare correlation radii of concentration fluctuation above and below the melting temperature near the critical point, N cr . They will be found approximately equal, this would further confirm our theory. Physical clarity of underlying concepts makes this approach very attractive in understanding and explaining the nature of energy dissipation in polymer systems.
II. RESULTS AND DISCUSSION
A. Calculation of Critical Polymerization Degree
It has been found [15] that the ordering parameter can be defined for the chain with excluded volume as w 2 = (B/C) N −2β and w cr 2 = (B/C) 2 at N = N cr where B ≈ 0.2068 is the constant for the Fisher probability density, C ≈ 2842.45, C/B ≈ 13744.9, the number of components of ordering field n = 0 [17] for the Wilson ε -expansion [18]. If we determine
λ n 2 = w 2 /w cr 2 = (C/B) N −2β (1)
and use the value of ν = 1−6 −1/2 , β = 0.2998 [1,15] then lnN cr ≈15.89. At N = N cr λ n = 1 and the oriented state cannot be observed. These conclusions agree with the results of [1].
Receiving (1) it has been assumed [15] that the form of swelling coil is not spherical since there exists an fluctuation attraction between the chain ends owing to screening volume interactions. The mechanism of the interaction in polymer solutions [4] as well as in melts [19] was described.
Further we will use the Vidom-Kadanoff relation [19] γ − 1 + (1 − νd) = − 2β.
(
ν is the critical exponent of correlation radius, d is the space dimension. Let us define the mean magnetic correlation for a magnetic system connecting with polymer one [19] <M(r 0 )M(r)
> = (r 0 /r) 1+ζ ( 3 )
where ζ is the critical exponent of ordering field and the following relationship is true [19] 2β = ν (d − 2 + ζ). Taking into account that at critical point the correlation radius [19] r ~ aN ν where a is the diameter of monomer we see from (3) that
<M(r 0 )M(r)> = (r 0 /a) 1+ζ N −2β .( 4 )
We connect the mean magnetic correlation with λ n 2 (1).
Multiplying ν -1 ln(r 0 /r) by the Vidom-Kadanoff relation (2) we can find the exact thermodynamic equation <M(r 0 )M(r)> = g E (r) P C (r) ( 5 ) where g E (r) = (r 0 /r) d−1/ν is proportional to the Edwards correlation function, P C (r) = (r/r 0 ) (γ−1)/ν is the des Cloiseuaxs probability of collision of chain ends [19].
Multiplying ν -1 ln(n 0 r 0 /100aN ν ) by the Vidom-Kadanoff relation (2) we can obtain the formulae P C (r) = (λ n /λ br ) 2 = l a /l a cr ( 6 ) where n 0 r 0 /100a = (C/B) 1/(1+ζ) (compare with (1)), n 0 /100 is a constant, l a is the average thickness of amorphous layer in isotropic material, these layers containing the ends of a marked chain, and
λ br 2 = (n 0 /100) d−1/ν N es 2 N 1−νd( 7 )
λ br is the draw ratio at break. If n 0 = 18.5 then r 0 /a ≈ 65631 and (18.5/100) d/2−1/2ν ≈ 0.33, N es = (r 0 /a) d/2−1/2ν ≈ 1430.4 is another constant, the dependence l a ≈ 0.5 l K (100/n 0 ) (γ−1)/ν N γ−1 agrees with experimental data [1], l K = 2nm is the Kuhn segment in linear PE, l a cr ≈ 0.5 l K (r 0 /a) (γ−1)/ν ≈ 26.9nm is the value of l a at N = N cr . These results (7) are also in accordance with experimental data [20].
We see that at critical PD the correlation radius of concentration fluctuation r 0 /a ≈ 6.6·10 4⎯ is a macroscopic value. It should have approximately the same meaning above the melting temperature.
B. Coefficient of rotation diffusion in polymer melts
A mechanism of motions associated with the chain ends has been considered by M. Doi [16]. He has suggested that on a short time scale the chain end moves around quickly within the distance aN 1/2 . But on the terminal relaxation time scale the mean-square displacement of the centre of gravity for the coil should be equal approximately to the mean-square displacement for a separate Rouse segment [4]. The important conclusion follows this reasoning: no translational motions give additional contributions in viscosity. It should be taken into account rotation Brownian motions in order to use the reptation theory [2] without any changes.
Let us consider the mechanical field in which the frequency, ω, of rotation vibration motion of chain about the polymer melt flow direction is much less than the reciprocal value of τ: ϕ = ωτ << 1 where τ ∝ N 3 is the terminal relaxation time [2]. Then the arising phase difference, ϕ, can be assumed [21] to be connected with a resulting torque which tends to return the chain ends in their initial position. The scaling expression for the coefficient of rotation diffusion D rot ∝ τ −1 can be written as follows D rot ~ Tη −1 g in order to satisfy the required condition for the true relaxation time, τ tr ,
τ tr ∼ τN 1/2 .(8)
Here, T is the temperature expressed in energy units, η is the viscosity, g ~ a -2 r -1 is a pair correlation function [15] and we state that the torque has always to be applied with a certain (lever) length r ~ aN 1/2 where r is the mean end-to-end distance of chain in melts [4,19]. In more realistic case r ~ aN 1-ν [15], since, as we will see below, screening the volume interactions vanishes if the mechanical field is applied to a polymer melt. Consequently, the exponent sought ≈ 3.41. Thus, the terminal time of relaxation (8) increases resulting in the hydrodynamic effect of velocity, V, of a ball-shaped body, rotating effectively in a high-molecular-weight liquid with the viscosity η, on the friction force f. A characteristic size of this body is r.
Let us write all the factors due to the rotation motions:
r ∝ τ tr /τ = N 1-ν ,( 9 )V ∝ N -3ν , ( 1 0 ) f 1 aN/T ∝ N 4(1/2-ν) ( 1 1 )
where ν = 0.5 for ideal chains and > 0.5 for excluded volume chains, f 1 is the friction force per one monomer [2][3][4]19]. In order to observe the reptation motions we see that the following inequality should be held: f 1 aN/T ≤ 1, i.e. the work against the friction force is less than the thermal energy, approximately. It is true for flexible enough chains (f 1 aN ~ T). Their conformations may be changed easy, and such chains will make these motions. Other assumptions lead to polymer glass with frozen conformations. Multiplying the both hands of this inequality by aN, we find
τ R V ≤ aN ( 1 2 )
where τ R ~ ζ 1 a 2 N 2 /T is the relaxation time of the first Rouse mode, ζ 1 = T/D 1 is the friction coefficient of one monomer, τ 1 = a 2 /D 1 and D 1 are the time of relaxation and the diffusion coefficient typical for low-molecular-weight liquids, respectively. Then another form of the inequality (12) can be written as follows:
ξ 2 ≥ R g 2 ( 1 3 )
where ξ 2 = Ta/f 1 and the gyration radius R g ~ N 2ν-1/2 for the excluded volume chains [15]. For the ideal chains [2][3][4]19]
r ~ R g ~ R 0 ~ aN 1/2 .
It is the most important and interesting result of the present work. In reptational dynamics a characteristic spatial scale appears. Although it is connected with the friction force per one monomer, at least, it is of order of mean size of polymer coil.
We have seen from (13) that there exists a spatial ξ ~ R g . This gives an opportunity to assume that if the reptation motions due to the action of a mechanical field in polymer melts take place indeed then i) screening the volume interactions [22] vanishes, ii) the fluctuation attraction of chain ends and other monomers [4,19] remains, iii) the correlation radius can increase.
Note that although all the motions associated with the chain ends are fast at small scales of order of several atom radii, an inhibition of such motions of ends at large scales (of order of end-to-end distance) reduces the overall chain mobility. It is clear that the fluctuation attraction results in an inhibition of large-scale motion of chain ends as an average effect.
C. Ring macromolecules
It is impossible to compare the recent Monte Carlo research on linear and ring chains [23,24] with our results for 2 reasons, although the investigations of ring macromolecules are continued expansively as seen from the literature [25][26][27][28]. The first one is reported by the authors. They are not capable of considering the case N >> N e . The second one is the absence of hard criteria for decoupling the rotation and translation motions in their model for linear chains.
The self-diffusion coefficient for a melt of linear molecules can be estimated as follows: D = R 0 2 /τ tr ~ N (2ν-1/2)2 /N 4-ν ~ N −5(1−ν)) ~ N -2.05 where we have assumed R 0 ~ R g and have used the expression for the gyration radius R g ~ N 2ν-1/2 obtained by the author [15] and used above. The reptation law D ~ N -2 [19] can be expected for unknotted rings since there is no fluctuation attraction of the chain ends.
If N < N e then D ~ (aN) 2 /τ 1 N 2 ~ D 1 (R 0 ~ r ~ aN). Thus, for N ~ N e one can observe any exponent ranged from -2.05 to 0. In this respect the estimation N e ≈ 283 for linear chains was obtained [15].
D. Correlation radii near the melting temperature
In order to compare the correlation radii of concentration fluctuation below (in solid state) and above (in melt) the melting point let us recall the definition of critical point [1,29]. Below the melting temperature at N = N cr the neck draw ratio and the crystallinity tend to 1 and 0, respectively, lnN cr ≈15.89 [1,15,29]. In [29] the ratio of the correlation radius of concentration fluctuation to the monomer diameter is equal to r 0 /a ≈ 6.6·10 4 . From (13) we can see that ξ/a ≥ R g /a ≈ 5.2·10 4 at N = N cr , i.e. r 0 /a ≈ ξ/a. This confirms our considerations. At the critical PD the correlation radii of concentration fluctuation above and below the melting temperature are equal to each other approximately.
E. Conclusion remarks
The rotation vibration motion cannot be observed at macroscopic scales due to decreasing the field with increasing the distance. An antisymmetric stress seems can appear as a macromolecule is capable of making rotation vibration precession motions. At large scales it transforms to an average tensor which will be symmetric probably. We have also found that a non-equilibrium parameter of polymer melts such as the viscosity is indirectly connected with the equilibrium pair correlation function due to the finite length of macromolecules [15]. We can indeed neglect the fluctuation attraction of the chain ends, supposing it is little, for long enough chains, i.e. if N → ∞ and the physical state of the melt is not changed as in [12,13], but, in this case, at N > N cr the solid state and the melt are identical to each other.
Based on the molecular recognition of the ends of a marked chain due to their fluctuation attraction [4,15,19] the present theory enables not only to estimate critical exponents and to find the critical PD [15] but also to obtain the pair correlation function of concentration fluctuation (5) in order to connect the nanostructural characteristics with macroscopic polymer properties (6). The ends of a marked chain serve a solvent since in the solid state, on the one hand, the exponent γ testifies to this fact, on the other hand, the viscous flow of melt weakens screening the monomer-monomer interactions and leads to enhanced effective attraction between the chain ends.
Both lamellar and fibrillar structures are dissipative since the crystal of infinite size has a minimum of free energy [30]. Thus, a methodology of research of non-equilibrium polymer systems has been elaborated.
ACKNOWLEDGMENT
Thanks are expressed to Professor A. R. Khokhlov and to Professor I. Ya. Erukhimovich from Moscow State University for useful discussions, to Professor S. N. Chvalun and to Professor A. N. Kraiko from Moscow Institute of Physics and Technology for helpful remarks, to Professor V. N. Pokrovskii from University of Malta for numerous comments and for the reprint [9], and also to Dr. A. V. Mironov and A. Titkov for technical assistance in preparing the manuscript.
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| []
|
[
"Effects of electron-phonon interaction on thermal and electrical transport through molecular nano-conductors",
"Effects of electron-phonon interaction on thermal and electrical transport through molecular nano-conductors"
]
| [
"Jing-Tao Lü \nSchool of Physics\nHuazhong University of Science and Technology\n430074WuhanPeople's Republic of China\n",
"Hangbo Zhou \nDepartment of Physics and Center for Computational Science and Engineering\nNational University of Singapore\n117551SingaporeRepublic of Singapore\n\nNUS Graduate School for Integrative Sciences and Engineering\nNational University of Singapore\n117456SingaporeRepublic of Singapore\n",
"Jin-Wu Jiang \nShanghai Institute of Applied Mathematics and Mechanics\nShanghai Key Laboratory of Mechanics in Energy Engineering\nShanghai University\n200072ShanghaiPeople's Republic of China\n",
"Jian-Sheng Wang \nDepartment of Physics and Center for Computational Science and Engineering\nNational University of Singapore\n117551SingaporeRepublic of Singapore\n"
]
| [
"School of Physics\nHuazhong University of Science and Technology\n430074WuhanPeople's Republic of China",
"Department of Physics and Center for Computational Science and Engineering\nNational University of Singapore\n117551SingaporeRepublic of Singapore",
"NUS Graduate School for Integrative Sciences and Engineering\nNational University of Singapore\n117456SingaporeRepublic of Singapore",
"Shanghai Institute of Applied Mathematics and Mechanics\nShanghai Key Laboratory of Mechanics in Energy Engineering\nShanghai University\n200072ShanghaiPeople's Republic of China",
"Department of Physics and Center for Computational Science and Engineering\nNational University of Singapore\n117551SingaporeRepublic of Singapore"
]
| []
| The topic of this review is the effects of electron-phonon interaction (EPI) on the transport properties of molecular nano-conductors. A nano-conductor connects to two electron leads and two phonon leads, possibly at different temperatures or chemical potentials. The EPI appears only in the nano-conductor. We focus on its effects on charge and energy transport. We introduce three approaches. For weak EPI, we use the nonequilibrium Green's function method to treat it perturbatively. We derive the expressions for the charge and heat currents. For weak system-lead couplings, we use the quantum master equation approach. In both cases, we use a simple single level model to study the effects of EPI on the system's thermoelectric transport properties. It is also interesting to look at the effect of currents on the dynamics of the phonon system. For this, we derive a semi-classical generalized Langevin equation to describe the nano-conductor's atomic dynamics, taking the nonequilibrium electron system, as well as the rest of the atomic degrees of freedom as effective baths. We show simple applications of this approach to the problem of energy transfer between electrons and phonons. | 10.1063/1.4917017 | [
"https://arxiv.org/pdf/1501.06343v2.pdf"
]
| 119,191,323 | 1501.06343 | 5b8c94d602554ab2ecb005d4b709a6314e88b3ed |
Effects of electron-phonon interaction on thermal and electrical transport through molecular nano-conductors
Jing-Tao Lü
School of Physics
Huazhong University of Science and Technology
430074WuhanPeople's Republic of China
Hangbo Zhou
Department of Physics and Center for Computational Science and Engineering
National University of Singapore
117551SingaporeRepublic of Singapore
NUS Graduate School for Integrative Sciences and Engineering
National University of Singapore
117456SingaporeRepublic of Singapore
Jin-Wu Jiang
Shanghai Institute of Applied Mathematics and Mechanics
Shanghai Key Laboratory of Mechanics in Energy Engineering
Shanghai University
200072ShanghaiPeople's Republic of China
Jian-Sheng Wang
Department of Physics and Center for Computational Science and Engineering
National University of Singapore
117551SingaporeRepublic of Singapore
Effects of electron-phonon interaction on thermal and electrical transport through molecular nano-conductors
(Dated: January 26, 2015)numbers: 8535Gv8585+j8565+h0560Gg7363-b
The topic of this review is the effects of electron-phonon interaction (EPI) on the transport properties of molecular nano-conductors. A nano-conductor connects to two electron leads and two phonon leads, possibly at different temperatures or chemical potentials. The EPI appears only in the nano-conductor. We focus on its effects on charge and energy transport. We introduce three approaches. For weak EPI, we use the nonequilibrium Green's function method to treat it perturbatively. We derive the expressions for the charge and heat currents. For weak system-lead couplings, we use the quantum master equation approach. In both cases, we use a simple single level model to study the effects of EPI on the system's thermoelectric transport properties. It is also interesting to look at the effect of currents on the dynamics of the phonon system. For this, we derive a semi-classical generalized Langevin equation to describe the nano-conductor's atomic dynamics, taking the nonequilibrium electron system, as well as the rest of the atomic degrees of freedom as effective baths. We show simple applications of this approach to the problem of energy transfer between electrons and phonons.
I. INTRODUCTION
Electron-phonon interaction (EPI) is one of the most important many-body interactions in condensed-matter and molecular systems 1 , responsible for a variety of phenomena, from electrical, thermal conduction, superconductivity to Raman scattering, polaron formation, just to list a few [2][3][4][5][6][7][8] . Its effects on the electrical, thermal, and optical properties of bulk semiconductors and metals have been intensively studied along with the development of many-body theories and experimental techniques. Recent advances in experimental fabrication of meso-and nano-scopic structures have generated tremendous efforts in understanding the effects of EPI on transport properties of reduced-dimensional systems [9][10][11] .
Of special interest are current-induced forces and Joule heating in low-dimensional systems, especially in molecular nano-conductors . On the one hand, the electrical transport signature of EPI is an invaluable spectroscopic tool to study the structural information of molecular nano-conductors 22,24 . On the other hand, these processes are crucial in maintaining the stability of these conductors 25 , relevant to the continuous scaling down of modern electronic devices. Different theoretical approaches have been developed to study these problems, in many cases separately. Recently, it was realized that non-conservative nature of current-induced forces provides an alternative, deterministic way of energy transfer between electrons and phonons, or more generally atomic motions 15 . It is fundamentally different from the stochastic Joule heating. These advances have motivated the development of methods treating current-induced forces and Joule heating on the same footing [17][18][19]34 .
Equally significantly, there has been an increasing interest in the thermoelectric properties of low dimensional systems [50][51][52][53][54] . A starting point of the theoretical treatment is to ignore the effect of EPI, and study the transport of electrons and phonons separately. But how important the effect of EPI is is a pertinent question, on which much of recent work is devoted to [55][56][57][58][59] . Here, we will look at this problem using the various approaches we have developed.
EPI is a genuine many-body interaction, the exact treatment of which is challenging, if possible at all. One natural approach is to perform perturbation calculation over a certain small parameter. In the most common multi-probe transport setup (see Fig. 1 and Sec. II), this small parameter can be chosen according to the strength of EPI. This strength can be roughly characterized by the ratio between two time scales: the first one corresponds to the phonon period, and the second one corresponds to the electron dwell time 60 in the nano-conductor. If the time electrons spend in the nano-conductor is much shorter than the phonon period, the system is in the weak EPI regime. The small parameter is the EPI matrix. In the other limit, the coupling of the nano-conductor to electrodes is the small parameter, over which one can perform the perturbation expansion. arXiv:1501.06343v2 [cond-mat.mes-hall] 24 Mar 2015
In this review, we summarize our own effort in developing and/or utilizing different theoretical approaches to study the aforementioned problems in different parameter regimes. We discuss some relevant results when possible, but we make no effort on reviewing all of them considering the huge amount of literature. The paper is organized as follows: In Sec. II, we give a brief introduction of the EPI problem starting from the Born-Oppenheimer approximation. We then introduce the system setup and Hamiltonian we use in this paper. In Sec. III, we briefly summarize our use of the nonequilibrium Green's function (NEGF) method to study electron, phonon transport and their interaction perturbatively. We consider several applications of the method. The first one is the effects of EPI on the thermoelectric transport coefficients in a single level model. The second one is the heat transport between electrons and phonons due to EPI. The use of simple models enables us to approach the problems semi-analytically. The last example is a numerical study of the Joule heating and phonondrag effect in carbon nanotubes. In Sec. IV, we consider the case of strong EPI using the quantum master equation (QME) approach. After reviewing the earlier work, the same thermoelectric transport model is re-visited focusing on how the strength of EPI affects the results. In Sec. V, we focus on the current-induced dynamics. Based on the Feynman-Vernon influence functional approach, we derive the semi-classical Langevin equation, taking into account the equilibrium phonon and nonequilibrium electron baths. The final section is our conclusion and remarks.
II. BORN-OPPENHEIMER APPROXIMATION AND ELECTRON-PHONON INTERACTIONS
To discuss the meaning and formulation of the electron and phonon systems and their mutual interaction, we need to start from the Born-Oppenheimer approximation 2,3 . Consider an electron-ion system with a total Hamiltonian H = P i + H e , where P i is the kinetic energy operator for the ions, and H e = P e + U is electron Hamiltonian with kinetic energy of the electrons, P e , and potential energy U = U ee + U ei + U ii , which includes the Coulomb interactions among the electrons and ions. Since the ions are much heavier than the electrons, one can treat the ion kinetic energy term as a small perturbation with the expansion parameter 2
m e m p 1/4 ,(1)
where m e is the mass of an electron and m p mass of an ion (assuming all have the same mass). If the ions are considered infinitely heavy, the ions will not move and the electron wavefunctions satisfy
H e φ α (x; R) = E e α (R)φ α (x; R),(2)
where x represents the set of all coordinates of the electrons, R the positions of all the ions, and α is the electronic state quantum number. The eigen-functions and the eigenvalues depend on R parametrically. We assume an orthonormal set {φ α } that satisfies Eq. (2) has been obtained. To take into account the effects of the ions, we consider a trial full wavefunction in a factored form
Ψ(x, R) = φ α (x; R)χ β;α (R) = |αβ ,(3)
and consider the variational solution 4 of the full Hamiltonian, min χ Ψ|H|Ψ , subject to the normalization χ|χ = 1. This variational approach is equivalent to omitting the off-diagonal elements (which is the Born-Oppenheimer approximation, see Ref. 2, App. VIII), giving an equation for the ions
P i + E e α (R) + φ α |P i |φ α − 2 m p φ α |∇ R |φ α · ∇ R χ = Eχ,(4)
where · · · means the x-dependence is integrated out but still R-dependent; ∇ R is a multi-dimensional gradient operator with respect to R. Since the left-hand side depends on the electronic quantum number α, the full eigen-energy E and functions also depend on α parametrically, e.g., we may write E β;α . If we assume that the electrons are in its instantaneous ground state, the ions move in a potential surface generated by the electrons. There are no explicit electronphonon interaction (EPI) terms. To account for the EPI, we need to go back to the basis, Eq. (3), and consider the matrix elements αβ|H|α β .
The off-diagonal terms are interpreted as the EPI 5,6 , which are small. If the off-diagonals are omitted, the electrons stay in a given quantum state α. The off-diagonal terms describe the scattering of the electrons to different state α . If ion displacements are small, the most important contribution is from the linear term in the displacement
− 2 m p φ α χ β;α |∇ R |φ α · ∇ R |χ β ;α , (α, β) = (α , β ).
(6) These off-diagonal matrix elements can be used, e.g., in a Fermi-Golden rule calculation of scattering processes. However, the identity (in the sense of effective Hamiltonians) of the electrons and phonons and their mutual interaction are not at all clear. Although EPI plays major role in many physical processes 7 , such as electronic transport and superconductivity, its conceptual foundation is still not very solid. Within the Born-Oppenheimer scheme, it is not clear at all how to transform the original Hamiltonian H into a form of an electron system and independent phonon system and their interaction unambiguously. The problem is related to the fact that in deriving the phonon Hamiltonian (the potential surfaces), the effect of electrons is already used. Thus, putting the electrons back amounts to double counting, see Refs. 4 and 8 for some of the modern treatments.
Instead of pursuing a self-consistent theory of EPI from the Born-Oppenheimer approximation, here in this review, and also in many of the practical applications 61-65 , we adopt a phenomenological point of view, and use the model Hamiltonians as given below in Eqs. (8) and (10). Focusing only the term linear in the displacements away from the equilibrium positions of the ions, we can think of the single electron Hamiltonian H e below having a Rdependence. Taylor expanding it,
R = R 0 + u/ √ m p , we obtain M k ij = 1 √ m p i| ∂H e (R) ∂R k |j ,(7)
where |j is the single particle state when ionic system is in equilibrium position R 0 . The extra factor of square root of ion mass m p is because of our convention of displacement variable u. This form of interaction is intuitively understandable and originally proposed by Bloch 66 . In Chap. 4 of Ref. 6, a derivative from Eq. (6) to (7) is given, but the reasoning does not seem to be rigorous. Thus, our starting point of a derivation is a tightbinding Hamiltonian for the electrons, harmonic couplings for the phonons and a standard EPI term. They are taken as given and exact. The charge redistributions and self-consistency for the electrons are not part of the discussion. Symbolically, the total many-body Hamiltonian is given as
H tot = H 0 e + H p + H ep ,(8)
where the electron part is H 0 e = c † Hc, the phonon part H p = 1 2 (p T p + u T Ku) + V n (u C ). The variable u is mass normalized, u j = √ m j (R j − R 0 j ). Because of this, the conjugate momentum is p =u. V n is the nonlinear force contribution. c is a column vector of the electron annihilation operators, which we can separate into three regions, the left, center, and right, c = (c L , d, c R ) T , T stands for matrix transpose. Similarly u = (u L , u C , u R ) T . Accordingly, the matrices H and K are partitioned into nine regions (submatrices), e.g.,
H = H L H LC 0 H CL H C H CR 0 H RC H R ,(9)
such that H 0
e = H L e + H R e + H C e + V e , with V e = V L e + V R e , V L e = c L † H LC d + H.c.
. Note that we assume no interaction between the left and right leads (See Ref. 67 for transport when there is a lead-lead coupling). We do a similar partitioning for K using the notation of Ref. 68. The EPI takes the form
H ep (d, u C ) = ijk M k ij u C k d † i d j = k u C k d † M k d. (10)
We assume that the EPI appears only in the central region. A schematic representation of the system setup is shown in Fig. 1.
The separation of the electron and phonon leads makes the theoretical development easier. In reality, they could either be physically separated, or built into one. For example, one electrode could serve both as an electron and a phonon lead, but we assume that we have independent control over their temperatures T α e and T α p , α = L, R.
FIG. 1. Model system considered in this review. The center device, including both electrons, phonons, and their interactions, is coupled with two electron and two phonon leads. Each electron lead is characterized by its chemical potential µα and temperature T α e , and each phonon lead by temperature T α p .
III. WEAK EPI REGIME: NONEQUILIBRIUM GREEN'S FUNCTION METHOD
A. Theory
We first consider the case where EPI is weak, so that we can perform a perturbation expansion over the interaction matrix M . In order to do so, we use the NEGF method. Detailed introduction is given in our previous work 32, [68][69][70] . This section can be considered as an application of the general approach developed in Refs. [68][69] to the EPI problem. We use similar notations therein, and only give a brief outline of the approach here.
We denote the electron device Green's function without and with EPI by G 0 and G, the corresponding phonon Green's functions by D 0 and D, and the lead Green's functions without coupling to the center as g α and d α , respectively. The couplings of the device with the leads and that between the electrons and phonons are described by self-energies, with Σ and Π representing that of electron and phonon, respectively. For example, we define the time-ordered electron Green's function including EPI on the Keldysh contour [ Fig. 14
(b)] G ij (τ, τ ) = − i T C c i (τ )c † j (τ ) .(11)
Here, τ /τ is time on the contour, and i/j is index of the electronic states. The contour time order operator T C puts the operators later in the contour to the left. The average · is with respect to the density matrix of the full Hamiltonian. The contour ordered Green's function can be divided into different groups according to the spatial position of i/j, similar to the Hamiltonian. The most interesting one is G C , where i and j are both at the center device region. At the same time, it can be written as a 2 × 2 matrix in time space
G(τ i , τ j ) = G t (t i , t j ) G < (t i , t j ) G > (t i , t j ) Gt(t i , t j ) ,(12)
with G t , Gt, G > , G < the time-ordered, anti-timeordered, greater and lesser Green's functions. The retarded and advanced Green's functions are obtained from them, i.e., G r = G t − G < , and G a = G < − Gt. For the definition and relations among these Green's functions, we refer to the book by Haug and Jauho 71 , and our previous publications 32,68,69 .
To calculate the Green's functions, we use a process of two-step adiabatic switch on. We start from the decoupled system and leads. Each of the electron and phonon leads is at its own equilibrium state, characterized by the temperature T α and/or chemical potential µ α . The corresponding equilibrium Green's functions can thus be defined according to the equilibrium canonical distribution. The initial state of the system is arbitrary and not important in most cases (e.g., for steady state).
At the first step, we switch on the interaction of the center Hamiltonian with the electron and phonon leads. We wait until the electron and phonon subsystem reaches their own nonequilibrium steady state, since the temperature and/or chemical potential of each lead can be different. The two subsystems are quadratic and exactly solvable, and we get the non-interacting center Green's functions G 0 and D 0 from the Dyson equation (we omit the superscript C)
G 0 (1, 2) = g C (1, 2) + g C (1, 3)Σ b (3, 4)G 0 (4, 2), (13) D 0 (1, 2) = d C (1, 2) + d C (1, 3)Π b (3, 4)D 0 (4, 2). (14)
Here, we have used a single number to represent the matrix indices and contour time arguments, i.e., G 0 (1, 2) = G 0j 1j2 (τ 1 , τ 2 ). Summation or integration over repeated indices is assumed. g C (d C ) is the center electron (phonon) Green's function without coupling to the L and R leads. The self-energy Σ b = Σ L + Σ R includes contributions from L and R, with Σ α (1, 2) = H Cα g α (1, 2)H αC ; similarly for Π b .
At the second step, we adiabatically switch on the EPI in the center. We perform a perturbation expansion over the interaction Hamiltonian H ep , using Feynman diagramatics. The interacting Green's functions are expressed using similar Dyson equations as Eqs. (13)(14), (4,2). (16) Here, Σ ep and Π ep are electron and phonon self-energies due to EPI. Using Eq. (12), at steady state, we can get the following useful relations in energy/frequency domain
G(1, 2) = G 0 (1, 2) + G 0 (1, 3)Σ ep (3, 4)G(4, 2), (15) D(1, 2) = D 0 (1, 2) + D 0 (1, 3)Π ep (3, 4)DG r (ε) = (ε + i0 + )I − H C − Σ r tot (ε) −1 ,(17)D r (ω) = (ω + i0 + ) 2 I − K C − Π r tot (ω) −1 ,(18)Σ r tot (ε) = Σ r b (ε) + Σ r ep (ε),(19)Π r tot (ω) = Π r b (ω) + Π r ep (ω).(20)
We use ε for the energy of electron and ω for the angular frequency of phonon, respectively, and I is the identity matrix. To get an expression for the current, we also need the greater and lesser version of the Green's functions 71
G >,< (ε) = G r (ε)Σ >,< tot (ε)G a (ε),(21)D >,< (ω) = D r (ω)Π >,< tot (ω)D a (ω).(22)
The electrical current (I e ) is expressed as the change rate of the electron number in one of the leads (N α ) times the charge of electron (−e). For example,
I e = −e dN L (t) dt = − 2e ImTr V LC e c L † (t)d(t) = 2eReTr V LC e G <,CL (t = 0) .(23)
It can be expressed by the Green's function of the center region and the lead self-energies [71][72][73] ,
I e = e +∞ −∞ dε 2π Tr G > Σ < L − G < Σ > L .(24)
Similarly for the heat current carried by electrons (I h ) and phonons (I p )
I h = 1 +∞ −∞ dε 2π (ε − µ L )Tr G > Σ < L − G < Σ > L ,(25)I p = − +∞ −∞ dω 4π ωTr D > Π < L − D < Π > L .(26)
We have defined the positive current direction as electrons going from the lead to the center. We dropped the argument of the Green's functions for simplicity. We ignore the spin degrees of freedom, since it is not relevant here. Currents out of the right lead are obtained by replacing index L by R. One can symmetrize the expressions based on energy and charge conservation. The set of coupled equations Eqs. (17)(18)(19)(20)(21)(22) is difficult to solve, due to the many-body EPI. Since the EPI is weak, we consider only the lowest order Feynman diagrams shown in Fig. 2. The expressions for the selfenergies are as follows. The electron Fock self-energies from phonons are
Σ F,<,> mn (ε) = i M k mi G <,> 0,ij (ε − )D <,> 0,kl (ω)M l jn dω 2π ,(27)Σ F,r mn (ε) = i M k mi G r 0,ij (ε − )D < 0,kl (ω) + G > 0,ij (ε − )D r 0,kl (ω) M l jn dω 2π .(28)
The Hartree self-energy does not depend on energy
Σ r mn = −iM i mn D r ij (ω = 0) M j kl G < lk (ε) dε 2π .(29)
The phonon self-energies from electrons are
Π <,> mn (ω) = −i dε 2π Tr M m G <,> 0 (ε)M n G >,< 0 (ε − ) ,(30)Π r,a mn (ω) = −i dε 2π Tr M m G r,a 0 (ε)M n G < 0 (ε − ) + M m G < 0 (ε)M n G a,r 0 (ε − ) ,(31)
with ε − = ε− ω. Summation over repeated indices is assumed here. Different charge and energy conserving approximations have been developed in the literature. We will use two of them. In Subsec. III B, we perform an expansion of the current up to the second order in M , following the idea of Ref. 29. In the numerical model calculation in Subsec. III C, we use the self-consistent Born approximation (SCBA), which means we replace G 0 , D 0 by G, D respectively in the above equations.
B. Thermoelectric transport through a single electronic level
We consider a single electronic level H C e = ε 0 d † d, coupled to the left and right electrodes, characterized by the constant level-width broadening Γ α with energy cutoff ε D (see Eq. (40) for the general definition). It interacts with an isolated phonon mode with frequency ω 0 , and H ep = m 0 d † du. In the linear regime, we introduce an infinitesimal change of the chemical potential or temperature at lead L, e.g., µ L = µ + δµ, T L σ = T + δT σ , with σ = e or p, µ and T are the corresponding equilibrium values. We look at the response of the charge and heat current due to this small perturbation. The result, up to the 2nd order in M , is summarized as follows,
Ie e I h = L 0 L 1 L 1 L 2 δµ δTe T .(32)
The linear conductance and the Seebeck coefficient are
G e = e 2 L 0 ,(33)S ≡ − δV δT = − L 1 eL 0 T .(34)
The coefficients L n are
L n = 3 i=1 L (i) n ,(35)
with
L (1) n = 1 dε 2π (ε − µ) n (AᾱΓ α ) f ,(36)L (2) n = 1 dε 2π (ε − µ) n (∆A ᾱ Γ α ) f ,(37)L (3) n = 1 dε 2π (ε − µ) n ∆A ᾱ Γ α + 2G r 0 ImΣ r ep G a 0 Γ α f .(38)
We have defined
f = − ∂f ∂ε ,(39)Γ α = i (Σ r α − Σ a α ) ,(40)A α = G r 0 Γ e α G a 0 ,(41)∆A ᾱ = G r 0 ReΣ r ep Aᾱ + AᾱReΣ a ep G a 0 ,(42)∆A ᾱ = iG r 0 ImΣ r ep Aᾱ + iAᾱImΣ a ep G a 0 ,(43)
andᾱ means the lead different from α. L (1) n is the single electron Landauer result. L (2) n is the quasi-elastic term.
L (3) is the inelastic term. f is the Fermi-Dirac distribu- tion function f α (ε) = exp ε − µ α k B T α + 1 −1 .(44)
Since we are looking at the linear response regime, f L = f R , we dropped the subscript in Eq. (39). We will also use the Bose-Einstein distribution later
n B (ω, T ) = exp ω k B T − 1 −1 .(45)
When there is no ambiguity, we will also drop the argument T .
In the following, we set the position of the electronic level to ε 0 = 0, and look at the dependence of the conductance on the chemical potential µ. We firstly write down the expressions for the self-energies, and make some observations based on their functional forms.
The Hartree self-energy is real and does not depend on energy 32
Σ r H = − α m 2 0 Γ α 2πω 2 0 ε D −ε D f α (ε) ε 2 + Γ 2 /4 dε.(46)
At T = 0, we get
Σ r H = − α m 2 0 Γ α πω 2 0 Γ tan −1 2ε Γ µα −ε D ,(47)
with Γ = Γ L + Γ R . For large enough ε D , the lower limit term turns to −m 2 0 /(2ω 2 0 ), which is the polaron energy shift. Note here that the 1/2 is due to the fact we use u C k in our definition of H ep (Eq. 10). This is different from the common definition that uses the creation and annihilation operatore a † + a (Eq. 74). We have subtracted the polaron shift term in the following calculation, since it is a constant. After this subtraction, Σ r H is odd in µ with a negative slope near µ = 0. We focus on the µ > 0 regime. It saturates to −m 2 0 /(2ω 2 0 ) for large µ, e.g., µ Γ. At non-zero temperature, the slope and the saturation value change, but the shape of the curve is similar to the T = 0 case. This means that, the Hartree term shifts the electronic level, and reduces the conductance. On the other hand, when µ Γ, the conductance tends to zero, whether we include the EPI or not. Thus, the correction to the conductance due to the Hartree term ∆G eH ≤ 0. It starts from zero at µ = 0, goes back to zero at µ Γ, and it reaches a maximum magnitude at some point in the middle. The described behaviour is schematically shown in Fig. 3 (a). This effectively reduces the broadening of the single level spectral function, also the conductance peak. Since the Seebeck coefficient is related to the logarithmic derivative of the conductance, we expect it to increase the magnitude of the Seebeck coefficient near resonance, and to reduce it off resonance [ Fig. 4].
The imaginary part of the retarded Fock self-energy is 32
ImΣ r F (ε) = − m 2 0 4ω 0 α,s=± sA α (ε − s ω 0 ) × 1 + n B (sω 0 ) − f α (ε − s ω 0 ) . (48)
It is negative and even in ε. Its role on the differential conductance at the phonon threshold (eV = ω 0 ) has been discussed extensively [74][75][76][77] . The main conclusions are: it reduces the differential conductance at eV = ω 0 for resonant case (µ ∼ 0), where the bare transmission without EPI (T 0 ∼ 1), while it does the opposite for far off resonance case (µ Γ), where T 0 → 0. The transition point between the two opposite behaviors is T 0 = 1/2 if the electronic density of state (DOS) is flat. But, in general, it depends on the system parameters. At nonzero temperature, the sharp threshold broadens out, and the linear conductance is affected: its correction to the conductance ∆G eF I is negative for µ ∼ 0, and positive for µ Γ [ Fig. 3 (c)]. Physically, ImΣ r F gives rise to phonon scattering processes. Its effect can be understood as follows: At T = 0, for small bias (eV < ω 0 ), phonon emission is not possible due to Pauli blocking, while phonon adsorption is not possible due to zero phonon population. So, ImΣ r F does not affect the linear conductance. At high enough temperature, both phonon emission and adsorption are possible even at small bias, due to the broadening of the Fermi distribution, and finite population of phonon modes. The phonon scattering process decreases the conductance on resonance, but increases it far off resonance. As a result, the Seebeck coefficient becomes smaller.
The real part ReΣ r F (ε) is obtained by the Hilbert transform of the imaginary part. At zero temperature, it diverges logarithmically at ε − µ = ± ω 0 77 . Its effects on the conductance and Seebeck coefficient are difficult to analyze. We rely on the numerical result [ Fig. 3 Figure 3 shows the correction to the linear conductance of different self-energy terms as a function of µ. These numerical results confirm our qualitative analysis. By comparing the total conductance at low [ Fig. 3 (e)] and high temperature [ Fig. 3 (f)], we see that, (1) the Hartree term dominates at low temperature, and the G e -µ peak becomes narrower. (2) the Fock term becomes important at high temperature, and the G e -µ peak broadens out. Their effects on the Seebeck coefficient (S) are shown in Fig. 4. At low temperature, when the EPI is included, the magnitude of S gets larger for µ ∼ 0, and smaller for |µ| Γ. At high temperature, the effect of the Fock term results in drop of S. In any case, the correction to S is small for weak EPI. But for the case of strong EPI, the correction could be large (see Subsec. IV C).
(b)].
C. Heat transport between electrons and phonons
Let us look back at the setup in Fig. 1. We want to study the heat transport between electrons and phonons at finite temperature bias, but zero voltage bias. The simplest setup is that the system couples to one electron and one phonon lead, each at its own temperature, see
Q = i dε 2π dω 2π ω G > nm (ε)M k mi D < kl (ω)G < ij (ε − )M l jn ,(49)
where, again, summation over repeated indices is assumed. For the ease of analysis, we perform an expansion of the above expression to 2nd order in M , and it becomes
0.04 G e G (c) (d) 0.8 1.0 4 2 0 2 4 Μ Ω 0 4 2 0 2 4 Μ Ω 0 0.8 1.0 0.2 0.4 0.6 0.8 G e G 0 0.2 0.4 0.6 0.8 G e G 0 (e) (f) 4 2 0 2 4 0.0 0.2 Μ Ω 0 4 2 0 2 4 0.0 0.2 Μ Ω 0 (e) (f)Q (2) = +∞ 0 dω 2π ωTr Λ(ω, T e )A(ω) × n B (ω, T e ) − n B (ω, T p ) ,(50)ε ε ε ε t t t T H ep ε ε ε ε K K K μ L , T L T R FIG. 5.
The model system we consider to study the energy transport between electrons and phonons.
where
Λ(ω, T e ) = dε 2π Tr M A(ε)M A(ε − ) × f (ε, T e ) − f (ε − , T e ) ,(51)
and
A(ω) = i D r 0 (ε) − D a 0 (ε) .(52)
Now the question we ask is whether there is a diode behaviour for the heat transport between electrons and phonons 52 , e.g., Q(∆T ) = Q(−∆T ), with ∆T = T e − T p . This is relevant because in some special situation, e.g., at metal-insulator interface, or insulating molecular junctions, EPI becomes the bottleneck of heat transfer [78][79][80][81][82] . We can define the rectification ratio as
R = Q(∆T ) + Q(−∆T ) Q(∆T ) − Q(−∆T ) .(53)
If we assume a constant electron DOS, Λ(ω) does not depend on T , we get Q(−∆T ) = −Q(∆T ), and R = 0. The physical reason is that in this case, it is possible to map the electron-hole pair excitation into harmonic oscillators 79,83,84 . Then, it is equivalent to heat transport within a two-terminal harmonic system. We do not expect any rectification effect. To make R = 0, the electronic DOS has to be energy-dependent within the broadening of the Fermi-Dirac distribution given by k B ∆T . This effectively introduces anharmonicity into the system, consistent with previous studies [79][80][81] . We can go one step further, by making a Taylor expansion of the spectral function A(ε) about the Fermi energy, we find that the sign of R is determined by the sign of ∂ 2 A ∂ε 2 (The 1st order term is zero). To check this argument, we calculated the heat current across one-dimensional (1D) metal-insulator junction. The metal side is represented by a 1D tight-binding chain, with hopping element t = −0.1 eV, and onsite energy ε 0 = 0. The insulator side is represented by a 1D harmonic chain with the spring constant K = 0.1 eV/(Å 2 u). The insulator and metal couple through their last two degrees of freedom. Their interaction matrices are
M k = (−1) k m 0 0 1 1 0 , k = 1, 2.(54)
Here, m 0 = 0.05 eV/(Å √ u), k represents the phononic degrees of freedom. This means that the system couples Phonon modes can be excited by the mobile electrons due to the EPI effect; i.e., a high bias over the system leads to self-heating (Joule heating). In nanoscale electric devices, the electric current density can be much larger than that in the macroscopic system. The high current density will generate strong Joule heating, which may eventually break the device. In this sense, Joule heating becomes a bottleneck for further increase of the electric current density. Hence, lots of theoretical and experimental efforts have been devoted to understanding the Joule heating phenomenon in the nanoscale electric devices. In experiment, Joule heat can be measured via the thermal-mechanical expansion technique, which records the Joule heat induced temperature rise 85 . The Joule heat can increase the temperature of the molecular junction from room temperature to 463 K, which has been examined through the inelastic electron tunneling spectroscopy 86 . Grosse et al. investigated the nanoscale Joule heating in phase change memory devices 85 . The Joule heating leads to the temperature rise in the phase change memory device, which results in an obvious volume expansion. In another experiment, Joule heating is found to be responsible for the correlated breakdown of nanotube forests 39,41 .
(a) (b) ( ) (b) p e μ (eV) ΔT (K) e p (c) (d) μ ( ) ΔT (K) μ (eV) Energy (eV)
For a system without localized phonon modes, all phonon modes have important contribution to the Joule heat. The Joule heat contributed by these propagating phonon modes have important effects on the electric devices. For example, in graphene transistors, the output electric current will saturate with increasing source-drain voltage 87,88 . This saturated current density can be reduced by 16.5% due to the Joule heating 88 .
The localized phonon modes exist around some defects or nonuniform configurations, such as the free edge, the isotropic doping, interface, etc. This particular type of phonon modes has no direct contribution to the thermal conduction, but localized phonon modes play a particularly important role in the Joule heat phenomenon. They are characteristic for their exponentially decaying vibration displacement; i.e., only a small portion of atoms are involved in the localized vibration. For instance, there are some localized edge phonon modes at graphene nanoribbon's free edge. In these modes, edge atoms vibrate with large amplitude, but the vibrational displacement decays exponentially from the edge into the interior region.
The localized-phonon-mode-induced Joule heat was observed in graphene nanoribbons in experiment, and explained theoretically. Jia, et al. utilized Joule heating to trigger the edge reconstruction at the free edges in the graphene nanoribbons 89 . Engelund, et al. attributed this phenomenon to the Joule heating of the edge phonon modes 90 . There are two conditions for the important Joule heating of the edge phonon modes. First, these localized edge phonon modes can spatially confine the energy at the edges. Second, the electrons interact strongly with the localized edge phonon modes. The mean steadystate occupation of the edge phonon mode can be calculated from the ratio of the current-induced phonon emission rate and damping rate. The effective temperature for the free edge can be extracted by assuming this occupation to be Bose distributed. The effective temperature was found to be as high as 2500 K for bias around 0.55 V. This high effective temperature was proposed to be the origin for the edge reconstruction.
Although Joule heating might be used for selectively bond-breaking 91,92 , its most common outcome is a disaster of device breakdown. The effective temperature is a suitable quantity to describe the Joule heating. In 1998, Todorov studied Joule heating problem in a molecular junction 12 . In his work, the Einstein model is applied to represent the phonon modes in the system, and the electron-electron interaction is ignored as the system size is much smaller than the electron mean free path. Part of the EPI-induced Joule heat will be delivered out of the system by the phonon heat conduction, while the remaining Joule heat gives a high effective temperature. For low ambient temperature, the effective temperature scales with voltage V as T 4 eff ≈ γ 4 V 2 , with γ as an EPIdependent constant. It was shown that the effective temperature can be above 200 K for a very low ambient temperature around 4 K 93 . But at very high bias, the scaling law could differ from this 31 .
There are several experimental approaches to investigate the effective temperature of the electric device induced by Joule heating. The effective temperature can be extracted by measuring some quantities that are temperature-dependent. For example, the breaking force of the single molecular junction is related to the temperature. This force-temperature relationship can be used to estimate the effective temperature 30 . The Raman spectroscopy also depends on the temperature. Hence, it can be used to deduce the effective temperature of Ramanactive phonon modes 35,37,38 .
It has also been shown that EPI has an important effect on the thermal conductance in single-walled carbon nanotubes (SWCNTs) 94 . For them, we apply the Born approximation to consider the EPI effect using the NEGF approach, as the SCBA is computationally more expensive. The phonon thermal current can be calculated by considering the three EPI contributions shown in the Feynman diagrams in Fig. 2. The phonon thermal current flowing from the left lead into the center is given by Eq. (26). The expression for the right lead is analogous. The Joule heat is generated in the system and flows into the leads, so the total Joule heat is the sum of heat currents into both leads, Q = −(I L p + I R p ). The thermal current from Eq. (26) also includes that induced by the temperature gradient, which satisfies I L p = −I R p . Hence, Q gives solely the Joule heat.For metallic SWCNT (10, 10), both electrons and phonons are important heat carriers. The EPI only slightly reduces the electron thermal conductance, but it has a strong effect on the phonon thermal conductance. More specifically, Fig. 7 (a) shows an 'electron-drag' effect on the phonon thermal conductance at 150 K. The phonon thermal conductance becomes negative for high chemical potential value µ > 2.0 eV, which indicates that electrons can help to drag phonons from cold temperature region to the hot temperature region. The 'electron-drag' phenomenon happens at low temperature and high chemical potential, and it does not happen at a higher temperature 300 K as shown in Fig. 7 (b). For semiconductor SWCNT (10, 0), the electronic thermal conductance contributes less than 10% of the total thermal conductance at low bias (e.g. µ = 0.3 eV), while phonons make most significant contribution to the total thermal conductance. Similar 'electron-drag' phe-nomenon also exists in the semiconductor SWCNT (10, 0) at low temperature as shown in Fig. 8 (a).
IV. STRONG EPI REGIME: QUANTUM MASTER EQUATION APPROACH
A. Quantum Master equation formulism
In this section, we introduce the QME approach to consider the case of strong EPI. Before doing that, we should mention that the NEGF method has also been used to treat the strong EPI [95][96][97][98][99][100] . Since the idea behind it is very similar to that of the master equation approach, we choose not to introduce it here.
To simplify the formula, we ignore the coupling of molecular phonon modes to the phonon leads. The model Hamiltonian simplifies to
H tot = H S + H L e + H R e + V e(55)
where H S = H C p + H C e + H ep denotes the system Hamiltonian, and V e = V L e + V R e is the system-lead coupling. In the QME formalism we assume the system-lead coupling V e is weak so we can do perturbation on it. We work in the interaction picture with H 0 = H − V e as non-interacting part and V e as the interaction. For simplicity, in this section we use V to represent V e since we don't have V p . The equation of motion for the full density matrix follows the von Neumann equation
i ∂ρ I (t) ∂t = [V I (t), ρ I (t)].(56)
Here, the subscript I denotes operator in the interaction picture. The time argument in the parentheses means non-interacting evolution O(t) = e iH0t/ Oe −iH0t/ . The above equation can be written in an integral form as
ρ I (t) = −i t t0 dt [V I (t ), ρ I (t )] + ρ I (t 0 ).(57)
One can recursively apply the above equation to get a series expansion of the full density matrix in power of V I . We truncate the series to the second order and differentiate it with respect to time t at both sides of the equation to get the following integro-differential equation
∂ρ I (t) ∂t ≈ −i [V I (t), ρ I (t 0 )]− 1 2 t t0 dt [V I (t), [V I (t ), ρ I (t 0 )]].(58)
We prepare the initial state as a product state of the system and each lead, ρ(t 0 ) =ρ(t 0 ) ⊗ ρ L e ⊗ ρ R e . For the system-lead coupling V we assume it can be written as a product of system operator S and lead operator B as V = α S α ⊗ B α . In such cases we can trace over the lead degrees of freedom to get
∂ρ I (t) ∂t = −1 2 α,β t t0 dt [S α I (t), S β I (t )ρ I (t 0 )]C αβ (t−t )+H.c. (59) where C αβ (t − t ) = Tr[ρ L e ⊗ ρ R e B α I (t)B β I (t )]
is the correlation function of the leads. Here we have used the condition that the expectation value of a single lead operator B α is zero. We can now transform back to the Schrödinger picture and extend the initial time t 0 to −∞ to get the QME of Redfield type 101-103
∂ρ ∂t = − i [H S ,ρ] − 1 2 α,β(60)× t −∞ dt [S α , S β (t − t)ρ]C αβ (t − t ) + H.c. .
Here we have replacedρ(t 0 ) byρ, which is essential and correct only when one intends to get the 0thorder reduced density matrixρ 0 by solving the above QME 69,104,105 . In the application to the EPI problem, by exact diagonalizing the system Hamiltonian, this Redfield QME can take into account the coherence between electrons and phonons, in contrast to the usual rate equation approach 106,107 . We write the above equation in the eigenbasis of the system Hamiltonian H S to obtain 102,108
dρ nm dt = − i ∆ nmρnm + ij R ij nmρij ,(61)
where the relaxation tensor reads 109
R ij nm = 1 2 α,β S α ni S β jm W αβ ni −δ jm l S α nl S β li W αβ li + H.c.(62)
The transition coefficients are given by
W αβ kj = t −∞ dt e i∆ kj (t −t)/ C αβ (t − t ),(63)
where ∆ kj = E k − E j are the energy spacings of the system Hamiltonian.
Since we are only interested in the steady state, we impose the condition dρ/dt = 0 at t = 0 and solve the above equation order by order with respect to V . One can find that all the off-diagonal elements of the 0thorder reduced density matrix vanish in steady state and the diagonal elements can be evaluated via the matrix equation 109 i R ii nnρ (0) ii = 0, together with the constraint of Tr[ρ (0) ] = 1.
For the calculation of currents, we go through a similar derivation as the QME. The electronic current operator J e and heat current operator J h can be written in the form J e(h) = α S α ⊗ B α e(h) . The expectation value of currents can be calculated according to I e(h) = Tr[ρ I (t)J I (t)]. Since we are interested in the lowest order of current, we truncate Eq. (57) to the lowest order and plug in to get
I e(h) = −i t t0 dt Tr [V I (t ), ρ I (t 0 )]J I (t) .(64)
By taking the trace over the leads and transforming back to the Schrödinger picture one obtains the current at t = 0 as
I e(h) = 1 2 α,β 0 −∞ dt Tr[ρS α S β (t )]C αβ e(h) (−t ) + H.c.,(65)
where C αβ e(h) (t) = B α (t)B β e(h) (0) is the correlation function between the lead operators occurring in the systemlead coupling Hamiltonian and the current operator definition. For the same reason as the derivation of master equation, hereρ(t 0 ) needs to be replaced byρ to get correct steady state results. Since we are calculating lowest order of current, we can use the 0th order reduced density matrixρ (0) . Written in the eigenbasis of system Hamiltonian, the above equation becomes I e(h) = Tr ρ (0) I r e(h) with the reduced current operator defined as
(I r e(h) ) ij = 1 2 α,β,k S α ik S β kj W αβ e(h) (∆ kj ) + c.c. ,(66)
where the transition coefficients are
W αβ e(h) (∆ kj ) = 0 −∞ dτ e i∆ kj τ / C αβ e(h) (−τ ).(67)
.
Up to now the QME formalism is general and not restricted to any specific form of system or leads Hamiltonians. For the application to transport problems with EPI as concerned in this review, the system-coupling Hamiltonian is considered as a tunneling Hamiltonian 72 . In such case the system operator S, leads operator B and B can be specified
as S = d, d † , B = k∈L,R V k c † k , k∈L,R V k c k , B e = e k∈L V k c † k , −e k∈L V k c k and B h = k∈L (ε k − µ L )V k c † k , − k∈L (ε k − µ L )V k c k .
The infinite nature of the leads can be specified by defining a continuous spectra function for the leads
Γ α (ε) = −2ImΣ r α (ε) = 2π k∈α |V k | 2 δ(ε − ε k ), α = L, R.(68)
Throughout this section we use a wide band spectra function for the electronic leads with Lorentzian cut-off as
Γ α (ε) = η α 1 + (ε/ε D ) 2 , α = L, R.(69)
The non-vanishing correlation functions can be evaluated via
C 12 α (t) = ∞ −∞ dε 2π Γ α (ε)f α (ε)e iεt/ ,(70)C 21 α (t) = ∞ −∞ dε 2π Γ α (ε) 1 − f α (ε) e −iεt/ ,(71)C 12 h (t) = ∞ −∞ dε 2π (ε − µ L )Γ L (ε)f L (ε)e iεt/ ,(72)C 21 h (t) = ∞ −∞ dε 2π (ε − µ L )Γ L (ε) 1 − f L (ε) e −iεt/ ,(73)
and C 12 (t) = C 12 L (t) + C 12 R (t), C 12 e (t) = −eC 12 L (t), C 21 e (t) = eC 21 L (t). We note that here the upper index 1 or 2 refers to the two components of B and B e/h given above. For the system Hamiltonian, we focus only on the single electronic level coupled to a single phonon mode representing the center of mass of the molecule. In such case, the system Hamiltonian will reduce to
H S = ε 0 d † d + ω 0 a † a + λd † d(a † + a)(74)
with ω 0 the angular frequency of the phonon mode and λ denotes the EPI strength. This type of Hamiltonian has been well-studied in the context of molecular junction 32,33,55,97,99,110-119 . The QME formalism treats the nonlinearity of EPI exactly. Therefore in this section we will focus on strong EPI regime with emphasis on the EPI strength dependence of the electron/heat current. In the following we will discuss the effect of EPI on the electronic transport properties, including the phonon sidebands, negative differential resistance, thermoelectric properties and local heating effects.
B. Phonon sidebands and negative differential resistance
One of the earliest findings of the vibrational effects on the electronic transport through a molecular quantum dot is the appearance of the phonon sidebands in the I − V characteristics. When electrons transport through a single electronic level, the differential conductance (dI/dV ) will manifest a peak at resonant level when plotted against the voltage bias (V ). However, when the electronic level is interacting with a vibrational mode, replica side peaks will appear at the side of the resonant peaks. These side peaks are called phonon sidebands. A simple reason for the appearance of phonon sidebands is due to the fact that the electrons can emit or absorb phonons when they pass through the molecule. Therefore, the distance between each adjacent peaks is always equal to a single phonon energy. The phonon sidebands attract wide attentions in molecular junction systems. Experimentally, phonon sidebands were found in 1980s 120 and then were utilized to identify vibrational modes in molecular junctions [121][122][123][124] and quantum wires 24,[27][28][29]125 . Theoretically, at the beginning the sidebands were investigated by using scattering theory, which gives the transmission probability T (ε, ε ) for an electron to passing through an EPI system. The electron is coming from vacuum at energy ε and leaving at energy ε 126 . The scattering theory predicts side peaks in the transmission probability, which qualitatively justifies the phonon sidebands in molecular junction systems. However, prediction of phonon sidebands in the lead-molecule-lead junctions is a much more difficult task. A simple generalization to take the Fermi-Dirac statistics nature of the electron leads into account is to weight the exact transmission probability with the Fermi-Dirac distribution of each lead, i.e., by multiplying the transmission probability T (ε, ε ) by a factor f L (ε)[1 − f R (ε )] as a new transmission probability for electron going from the left lead to the right lead through the nano-conductor. This approximation is called single particle approximation (SPA). Plenty of earlier work is in this framework and it predicts Lorentzian type of phonon sidebands with the same width. But obviously this method assumes each electron transports independently through the junction, where the many-body effects are ignored. As a result, it overestimates the currents and it is not able to predict the quantized conductance e 2 /h either.
Based on the NEGF technique, more rigorous methods merged in dealing with the nonlinearity in EPI, such as the Green's function equation of motion method (EOM) 95 , SCBA [96][97][98]127 and nearest neighbor crossing approximation (NNCA) 100 . All of the above are Green's function based formalisms with different kinds of approximations. In general these approaches predict that the phonon side peaks are much sharper than the SPA approach. This sharpness is closely related to the Pauli exclusion, which the SPA approach failed to take into account. Other than Green's function based methods, another approach is to use rate equation of electron occupation probability in the molecule, via calculating the transition coefficients of the electron to tunnel from the molecule to each lead and vice versa. This method assumes the transport is an electron tunneling process and the electron will lose its phase information when it resides in the molecule. Therefore it will be valid when the molecule-lead coupling is weak and the coherence of the electron and phonon in the molecule can be neglected. For all these formalisms, we would like to point out that one should take care of the phonon distribution. Treating the phonon at equilibrium distribution at a fixed temperature could be valid when the EPI strength is much weaker than the coupling strength between the phonon and its environment. However, when the environmental influence is weak, one should consider phonons in nonequilibrium states. This nonequilibrium treatment of phonon distribution can have pronounced effect on I − V characteristics due to the fact that the current induced vibrational excitation can be significant 127,128 . Besides the peak distances and peak width discussed above, other aspects characterizing the sidebands include the weights of the zero-phonon band and the number of peaks. The investigation of these properties mainly focuses on the effects of EPI strength, Fermi energy of the molecule, chemical potential and temperature of the leads. In general, the higher order peaks will be suppressed by the Frank-Condon factor 77,128,129 . In the framework of NEGF, Chen et al. found that the weight from zero-phonon band will decrease monotonically with increase of EPI strength and temperature while the weights of higher order sidebands will increase and then decrease 97 . The chemical potentials of the leads will influence the presence of the sidebands at both sides of the zero phonon peaks [ Fig. 9, panel (a) and (b)]. If one keeps the chemical potential of one lead fixed and increases the chemical potential of the other lead, the phonon sidebands will appear only at one side of the 0th order peak [Fig 9, panel (c)]. However, if one fixes the Fermi-level of the molecule and changes the chemical potentials of both leads, phonon sidebands will appear at both sides 97,98 . We note that, in this section, Fermi-level of the molecule is defined as (µ L + µ R )/2.
In the framework of QME formalism described earlier, which is exact in the weak system-lead coupling limit, we find phonon sidebands for the single electronic level interacting with a single phonon mode. In this case the zero-phonon peak will occur at the renormalized resonant level due to polaron shift (ε 0 − λ 2 / ω 0 ) and the sidebands will appear at every distance of ω 0 . The peaks will appear at each side of zero-phonon peak under symmetric change of the lead chemical potentials while only appear at one side if we fix chemical potential of one lead. The EPI strength will not only shift the peaks, but also modulate the weights of the peaks. We find that upon increasing either EPI strength or temperature, the weight of the zero-phonon peak will decrease, which is consistent with the previous work 97 . However, interesting phenomena happen when the renormalized level of the quantum dot gets close to the Fermi-level of the dot, [ε 0 − λ 2 /ω 0 ≈ (µ L + µ R )/2], i.e., additional peaks appear at each side. Those peaks have distance ω 0 with the zero-order peak at the opposite side. However, the two major peaks really merge together, and those additional peaks disappear again. In the case of asymmetric change of lead chemical potential (right most panel of Fig. 9 (c)), we found peaks appear at both sides when the zero-phonon peaks merge together.
Another interesting perspective of the I − V characteristics is the phenomenon of the negative differential resistance (NDR), where the current decreases with the increase of voltage bias. The NEGF formalism predicts that NDR is impossible in ballistic electronic transport, but it will emerge in the presence of EPI. NDR has been both theoretically investigated 76,128,130 and experimentally measured 131 . An important reason of NDR is due to the redistribution of the molecular states. As discussed in the previous section, the zero-phonon band carries major portion of electronic current. The probability for the molecule to be in that state will be related to the chemical potential of the leads. If one increases the bias via increasing the chemical potential of one lead, one actually lifts the Fermi-level of the molecule as well. If one brings the Fermi-level of molecule far away from the eigenenergy of zero phonon state, the probability of the molecule to be in that state will decrease and thus the current will decrease. Based on this analysis one can draw several immediate conclusions: 1. If one increases the bias simultaneously for both leads in pace and keeps the Fermi-level of the molecule fixed, there will be no NDR. 2. If one treats the phonon fixed in the equilibrium distribution, NDR will not appear 128 . 3. If the chemical-potentialvarying lead couples stronger to the molecule than the chemical-potential-fixed lead, the redistribution will be more sensitive, thus the NDR will be enhanced. Figure 10 shows the NDR predicted in the QME formalism. We find that the NDR appears in the symmetric system-lead coupling. Moreover, NDR will be enhanced if the chemical-potential-varying lead couples stronger to the molecule than the other lead, but it will disappear in the other way around. We also find that NDR is most pronounced in the moderate EPI regime, while it is less significant in both weak and strong EPI regimes 128 .
C. Vibrational effect on thermoelectricity
In this part, we study the effects of EPI on the thermoelectric properties of the nano-conductor. We first look at the thermoelectric current, which is the electronic current induced by a temperature difference between the leads. Thermoelectric current exhibits quite different features comparing with voltage-bias current. It will increase monotonically and smoothly with the increasing of the temperature bias. Therefore, there will be no phonon sidebands. This is expected, because under temperature bias, the tunneling channel of the electron is always restricted to the molecular state that is close to the chemical potential of the leads. The increasing of temperature will only excite more conducting electrons, but not be able to extend extra tunneling channels. Therefore, there will be no sudden change of thermoelectric current. Due to the same reason, there will be no NDR effect as the increasing of the temperature bias will always make more electrons to be involved in tunneling. The restriction on tunneling channels also makes the thermoelectric current much smaller than the voltage-bias current.
The dependence of electronic conductance on the chemical potentials of the leads has been discussed in the Subsec. III B. The sign of the Seebeck coefficient indicates that the currents will change direction with different chemical potential. Another interesting perspective is to find the dependence of currents on EPI strength. Figure 11 shows the plot of voltage-bias current [panel (a)] and thermoelectric current [panel (b)] with the EPI strength λ. For voltage-bias current, we find that the current decays with EPI strength in general. More precisely, the maximum current one can achieve via adjusting the ε 0 is decreasing monotonically with EPI strength. This is due to the Frank-Condon blockage of the current. However, for each ε 0 we can find the enhancement of current due to EPI, and that is mainly because of the polaron shift which can bring the electron resonance level into the conduction band from outside. For the thermoelectric current, the profile is quite different. We see that the current can change sign with the increase of EPI strength when ε 0 is higher than the Fermi level, which indicates that the EPI can switch the charge carriers of the quantum dot between electrons and holes 59 . For each ε 0 there exist two optimized values of EPI strength such that the thermoelectric current maximizes. This optimized λ shifts left with decrease of ε 0 until disappear one by one at the λ = 0 end.
One important quantity to describe the efficiency of the thermoelectric material is the figure of merits Z e T . It is related to the electronic conductance G e , Seebeck coefficient S, thermal conductance G h via the formula Z e T = G e S 2 T /G h . Here we use the notation Z e T to denote the figure of merits of the system by ignoring the thermal conductance due to phonons. So G h here only takes account the thermal conductance due to electrons. The effect of phonons on the figure of merit Z e T is rather complicated, which is closely related to the electron Fermi energy 55,118,119,132 , the phonon energy 113 , the temperature and chemical potentials of the leads 118,119,132,133 . Figure 12 shows the dependence of the electronic conductance, thermal conductance, Seebeck coefficient and figure of merit on the electron energy ε 0 and EPI strength λ. The major effect of EPI is on the thermal conductance of the molecular junction 113 .
The EPI can open extra channels, from which high energy electrons can tunnel from hotter lead to colder lead, while low energy electrons tunnel in the opposite direction. Therefore, the thermal conductance is enhanced while the electronic conductance is not affected too much. Due to the increase of the thermal conductance, Z e T will be suppressed drastically 113,119 . Though the figure of merits Z e T will be reduced quickly under the influence of phonon scattering in weak EPI regime, it will gradually saturate at strong EPI regime. We also would like to point out that when the electron energy is close to the Fermi energy of the quantum dot, the Seebeck coefficient and hence the Z e T will become very small. However, the phonon scattering can renormalized the electron energy via polaron shift and thus the Seebeck coefficient can be enhanced. As a result, EPI can enhance Z e T in this particular parameter regime.
D. Local heating
Local heating is an important phenomenon in molecular junction, not only due to its own importance affecting the stability of the system, but also due to its close relation to the phonon sidebands 127 , NDR and thermoelectric effect 56,133 . The distribution of the phonon states in current-carrying system can be far away from equilibrium 127 , or in some cases, may even lead to phonon instability 17,43,134,135 . Phonons can be excited significantly by the voltage bias 128 , and in turn affect the I − V characteristics. At each peak of the phonon sidebands, one can actually find a vibrational excitation event 128,136 . Previous study also found that the local heating can enhance the thermoelectric efficiency 56,133 . However, in most cases local heating is not preferred because it will affect the stability of the system 94 and introduce noise to the measurements [137][138][139][140] . Therefore, lot of effort has been put into cooling the system using electronic current, such as using superconducting single electron transistor 141 , or double quantum dots 140 .
Here we study the effects of the electronic current on the phonon mode of the nano-conductor. To investigate the local heating, effective temperatures are usually defined in various ways 9,32,51,[142][143][144] . In this part, since we only have one single phonon mode, instead of defining the effective temperature, we use average phonon number to characterize the local heating effect. The way to specify the electronic current induced heating is to compare the nonequilibrium phonon number n neq with equilibrium phonon number n eq . When the molecule is weakly coupled to the leads, the molecular states statistics will follow canonical distributionρ = e −βH S /Tr(e −βH S ) and thus the equilibrium phonon number can be calculated exactly as 59
n eq = 1 e β ω0 − 1 + λ 2 /( ω 0 ) 2 e β(ε0−λ 2 /( ω0)) + 1 .(75)
The first term is the Bose-Einstein distribution function, the second term is a correction due to the polaron energy shift. The nonequilibrium phonon number can be calculated from the nonequilibrium reduced density matrix obtained from the QME. Figure 13 shows the difference of phonon numbers under voltage bias (top) and temperature bias (down). For voltage bias, ∆n is always positive which indicates that the system is always heated up. The yellow regime where ∆n ≈ 0 is the regime where the electronic current vanishes. When there is electronic current passing through, the heating effect is generally more pronounced in stronger EPI regime. However, for the temperature bias, we find both heating (∆n > 0) and cooling regimes (∆n < 0). Therefore, the local heating effect is not only related to the magnitude of electronic current, but also related to the energy each electron carries when it tunnels into the molecule. For the thermoelectric current, low energy electron can tunnel from the cooler lead to the molecule, absorb a phonon and tunnel to the hotter lead, resulting in cooling of the molecule. Such process is impossible in the voltage-bias case in the present setup as the electron is always flowing from the higher chemical potential side to the lower chemical potential side.
V. CURRENT-INDUCED SEMI-CLASSICAL LANGEVIN DYNAMICS
In the previous two sections, we mainly look at the effect of phonons on the electric and thermoelectric transport properties of electrons, which is also the focus of most published work. But, to study the current-induced forces, and their effect on atomic dynamics, we need to turn around. In this section, starting from the total Hamiltonian H tot , we derive a semi-classical Langevin equation to describe the atomic dynamics of the system, coupled to both phonon and (nonequilibrium) electron leads. The Langevin equation applies to the weak EPI regime, since similar to Sec. III our expansion parameter is the interaction matrix M . However, the derivation is not limited to the form of H tot . Following the same procedure we discuss below, we can also do an adiabatic expansion of the electron influence functional over the velocity of the ions. In that case, the Langevin equation applies to slow ions, but the EPI could be of arbitrary magnitude [17][18][19]145 .
The advantage of the Langevin equation approach is that, we can easily include the anharmonic phonon interaction, as in other molecular dynamics method. The anharmonic interaction is crucial in dealing with currentinduced dynamics. This is because the phonon modes that interact with electrons are normally high-frequency ones, while the low frequency modes conduct heat from the system to surrounding electrodes more effectively. The energy transfer from high to low frequency modes is possible only when anharmonic interaction is included. Although possible, it is not a trivial task to incorporate anharmonic interactions in the NEGF or QME approach.
A. Initial states and reduced density matrices
We assume at a remote past t 0 and earlier time, the central system is decoupled with the leads and there is also no EPI, so that the electrons and phonons are also decoupled. The density matrix is assumed to be product of known equilibrium states. For example, the left lead is specified by
ρ L e ∝ e −β L (H L e −µ L N L ) ,(76)ρ L p ∝ e −β L H L p .(77)
Exactly what to take for the center is not important as for steady state with t 0 → −∞, the results do not depend on it (except maybe in very subtle cases).
The density matrix (of the whole system) is governed by the von Neumann equation, and formally we can write
ρ(t) = U (t, t 0 )ρ(t 0 )U (t 0 , t),(78)
where
U (t, t 0 ) = T e −(i/ ) t t 0 Htot(t )dt(79)
assuming t > t 0 . For the other case of t < t 0 , the timeorder operator T should be replaced by the anti-time order operator. We are interested only in the center, so the leads degrees of freedom will be traced out. For notational simplicity, we assume only one left lead. The result for two or more leads is trivially generalized. We defineρ
(t) = Tr L ρ(t),(80)ρ p (t) = Tr eρ (t),(81)
where the first reduced density matrix only eliminates the lead, while the last one eliminates the electrons as well, leaving only an effective density matrix for the phonons. The procedure to eliminate the lead for both the electrons and phonons follows the standard method of Feynman and Vernon 146 , except that we need to be careful for the electrons which are fermions. Since there is no coupling between electrons and phonons in the leads, the phonon and electron degrees of freedom can be done separately (the initial density matrix is a product of the two).
B. Influence functional for phonons
The matrix elements of the density operator ρ(t) is taken in the basis of the coordinates u. Following the standard treatment 103,147 , the density matrix is then given by
u |ρ(t)|u ∝ D[u]e (i/ ) C 2 L(u,u)dτ ρ 0 (u 0 , u 0 ). (82)
The path C 2 consists of two segments (see Fig. 14(a)) running from time t with coordinate variable u back to t 0 at variable u 0 , and then second segment running from To eliminate the lead, the phonon Lagrangian is split into L = L C + L L − V where V is the coupling potential energy term between the lead and center, given by
V = (u C ) T V CL u L . The integration volume elements are also split D[u] = D[u C ]D[u L ]. The initial distribution is assumed product type ρ L 0 ⊗ ρ C 0 .
Taking the trace of Eq. (82) means that we identify u L and (u ) L as the same variable u L and integrate it out. As far as variable u L is concerned, the path is of Keldysh type, i.e., running from t 0 to t from above and then back from t to t 0 , as shown in Fig. 14(b). The reduced density matrix is theñ
ρ(t) = D[u C ]e (i/ ) C 2 L C dτ I[u C (τ )]ρ C 0 ((u ) C 0 , u C 0 ),(83)
with the influence functional
I[u C (τ )] = D[u L ]ρ L 0 ((u ) L 0 , u L 0 )e (i/ ) C (L L −V )dτ .(84)
The above expression can be further simplified. Firstly, the initial distribution of the lead is in thermal equilibrium, ρ L 0 ∝ e −β L H L . Secondly, we can rewrite the path integral formula back into the operator form; thus, we obtain
I[u C (τ )] = Tr e −β L H L Z L T C e −(i/ ) C (H L +V )dτ = T C e −(i/ ) C (H L +V )dτ eq.L ,(85)
where Z L is the canonical partition function, T C is contour order operator, the subscript eq.L stands for equilibrium average with respect to the left lead. One more transformation can be made to simplify it further. The time-dependence (or rather, contour time τ dependence) is understood to be in the Heisenberg picture governed by the Hamiltonian H L + V (τ ) which is different in the forward and backward direction. We can work in the interaction picture, thus eliminate the explicit H L from the formula. The resulting equation is
I[u C (t)] = T C e −(i/ ) C V I (τ )dτ eq.L .(86)
This is the same equation [Eq. (5)] given in Ref. 148.
The influence functional can be calculated explicitly since the interaction V is a quadratic form, and the contour operator naturally leads to contour ordered Green's functions and Wick's theorem is valid. We define the contour ordered phonon Green's function of the lead as
d L (τ, τ ) ≡ − i T C u L (τ )u L (τ ) T eq.L .(87)
Expanding the exponential (or a cumulant expansion, expanding and taking logarithm), we get
I[u C (t)] = T C 1 − i C V I (τ )dτ − 1 2 2 C dτ C dτ V I (τ )V I (τ ) + · · · = T C 1 − 1 2 2 C C u C (τ ) T V CL T C u L (τ )u L (τ ) T V LC u C (τ )dτ dτ + · · · = T C 1 − i 1 2 C dτ C dτ u C (τ ) T Π L (τ, τ )u C (τ ) + · · · = e − i 1 2 C dτ C dτ u C (τ ) T Π L (τ,τ )u C (τ ) .(88)
The first order term (and all odd order in V terms) vanishes, because u L = 0. We have defined the ubiquitous contour ordered self-energy due to the lead as
Π L (τ, τ ) = V CL d L (τ, τ )V LC .(89)
It can be shown that the last line in the above derivation is an exact result. It is instructive to clarify and compare with other notations used in the literature. A commonly used notation is (e.g., Ref. 103)
ln I = − 1 t t0 dt 1 t1 t0 dt 2 u + (t 1 ) − u − (t 1 ) T × L(t 1 , t 2 )u + (t 2 ) − L * (t 1 , t 2 )u − (t 2 ) .(90)
Using the rules C dτ → σ t t0 σdt, and Π(τ, τ ) → Π σ,σ (t, t ), u C (τ ) → u σ (t) where σ = + or − for the upper and lower branch, the relations among the various self-energies, and symmetry relation Π > ij (t, t ) = Π < ji (t , t), we can rewrite Eq. (88) in the form of Eq. (90). By comparison, we find
L(t, t ) = iΠ > L (t, t ), L * (t, t ) = iΠ < L (t, t ).(91)
α(t, t ) ≡ L(t, t ) is the notation used by Schmid 149 .
C. Influence functional for electrons
The derivation of the influence functional for the electrons is similar except that we have to deal with grassmann integrals [150][151][152][153] . We follow the approach of Weinberg 154 . To do the trace over the leads we need a specific representations for the operator ρ. For the electrons, we use the coherent state characterized by grassmann numbers such that
c i |c = c i |c .(92)
The hat denotes operator, without hat, it is a grassmann number. The state |c has explicit form given as
|c ≡ e iĉ † i ci |0 ,(93)c| ≡ 0| iĉ i e i ciĉ † i .(94)
The orthogonality is in the form
c |c = i (c i − c i ).(95)
Similar states for the creation operatorĉ † j can be constructed with eigenvalue (a grassmann number)c, given the following results (similar to inner product of eigen states of u and its conjugate momentum p and completeness of the eigenstates).
c|c = · · · e − jc j cj ,(96)c|c = e + jc j cj ,(97)1 = |c ˜ j (−dc j ) c|,(98)1 = |c ˜ j (dc j ) c|.(99)
The · · · is an extra + or − sign factor which we'll not keep track. The tilde over the product sign means that order is in exactly the opposite canonical order [e.g., that of Eq. (95)]. With the above very sketching outline, the fermion evolution operator
U (t, t 0 ) = T e −(i/ ) t t 0 H(τ )dτ(100)
can be represented as a path integral of the form
D[c, c]e iSe/ ,(101)
with the action
S e = − dτ c T Hc − i c T ∂c ∂τ .(102)
The lead influence functional can be then obtained with the same procedure as that for the phonons. The result involves an integral kernel which is exactly the contour ordered self-energy of the lead Σ L (τ, τ ).
D. Reduced density matrix for phonon in center
Putting things together, the reduced density matrix, when the leads are eliminated, has the form
(u ) C , (c ) C , (c ) C |ρ(t)|u C ,c C , c C ∝ D[u C ,c C , c C ]e i S/ ρ 0 ((u 0 ) C , (c 0 ) C , (c 0 ) C ; u C 0 ,c 0 C , c C 0 ) where S = S C p + S C e + S C ep + S I p + S I e ,(103)S C p = C2 dτ 1 2u 2 − 1 2 u T K C u − V n (u) ,(104)S C e = C2 dτ i c T ∂c ∂τ −c T H C c ,(105)S C ep = − C2 dτ k u kc T M k c,(106)S I p = − 1 2 C2 dτ C2 dτ u(τ ) T Π L (τ, τ )u(τ ),(107)S I e = − C2 dτ C2 dτ c(τ ) T Σ L (τ, τ )c(τ ).(108)
The ordinary number (column vector) u and grassmann numberc, c involve only the degrees of freedom of the center. For notational simplicity, we have dropped the superscript C. Note that the electron terms do not have the characteristic factor 1/2 asc and c are independent variables. The dependence onc and c is a bi-linear form, thus the path integral over them can be done analytically. This gives the reduced density matrix of the phonon only, as
u |ρ p (t)|u ∝ D[u]e (i/ )(S C p +S I p ) I p ρ 0,p .(109)
The influence functional to the phonons due to electrons is given by
I p ∝ det δ(τ, τ ) Ii ∂ ∂τ − H C − u k (τ )M k − Σ L (τ, τ ) .
(110) Interpreting the τ in the above as Keldysh variable defined on C has a problem. As agreed, the contour is supposed to be on C 2 with the t 0 end connected with the initial distribution of the electrons ρ 0 at the center. However, if we assume that in the limit t 0 → −∞ the results should not depend on the distribution of the center, we can ignore this initial distribution and it is completely fixed by the lead. But we cannot give a mathematically sound justification here.
Similar to that in Ref. 154 for the field theory of quantum electrodynamics, we want to put the influence functional in an exponential form. This can be done using the formula, Det(A) = e TrlnA , and the expansion of the function ln(1 + x) = x − x 2 /2 + · · · for small x, given
I p = det(G −1 0 +y) = det(G −1 0 ) exp ∞ n=1 (−1) n+1 n Tr[(G 0 y) n ] ,(111)
where
G −1 0 = δ(τ, τ ) Ii ∂ ∂τ − H C − Σ L (τ, τ ), (112) y = −δ(τ, τ ) k u k (τ )M k .(113)
G −1 0 and y are matrices indexed by lattice sites j as well as contour time τ . And, if the proper metric for a discretization of the time is chosen so that det(G −1 0 ) can be meaningful, we can identify G 0 as the electron contour ordered Green's function when there is no EPI defined in Sec. III. Since det(G −1 0 ) is independent of u, the effective action for the phonon is only determined by the exponential factor, which is a polynomial (functional) in u. With some caveat regarding the initial distribution, Eqs. (109) and (111) offer a formally exact solution to the problem.
The first two terms, written out explicitly in terms of the contour ordered Green's functions are
Tr [G 0 y] = − k Tr G 0 (τ, τ )M k u k (τ ),(114)− 1 2
Tr (G 0 y) 2 = − 1 2 j,l,m,n dτ dτ dτ dτ G 0jl (τ, τ ) y lm (τ , τ )G 0mn (τ , τ )y mj (τ , τ )
= − i 2 dτ dτ u(τ ) T Π ep (τ, τ )u(τ ),(115)
with Π kk ep (τ, τ ) = −i Tr G 0 (τ , τ )M k G 0 (τ, τ )M k . (116)
E. Semi-classical approximation
If we ignore the linear term in u which produces a constant force, the effect of which is to shift the equilibrium positions, and also neglect higher order contributions, we end up with a quadratic form for the effective action
S eff = C2 dτ 1 2u 2 − 1 2 u T K C u − V n (u) − 1 2 C2 dτ C2
dτ u(τ ) T Π tot (τ, τ )u(τ ). (117) with Π tot = Π L + Π R + Π ep as in Eq. (20). We have also included the right phonon lead. A generalized Langevin equation can be derived from the above action 18,149
u = −K C u + F n − t Π r tot (t − t )u(t )dt + ξ,(118)
where F n = −∂V n /∂u, Π r tot is the retarded total selfenergy, and the noises satisfy
ξ(t) = 0,(119)ξ(t)ξ T (t ) = i 1 2 Π > tot (t − t ) + Π < tot (t − t ) = i Π tot (t − t ).(120)
We note that the effect of the electron leads to the phonons has exactly the same form as that of the phonon leads. The self-energy consists of a sum of contributions of the two sources.
F. Applications
Before discussing the applications, we note that similar generalized Langevin equation as Eq. (118) can be derived by doing an adiabatic expansion over the momenta of the ions to the 2nd order [17][18][19]155 .
These equations have been used in different perspective [17][18][19]34,134,145,[155][156][157][158][159][160][161][162][163][164][165][166][167][168][169][170][171] .
Its most important feature is the inclusion of the quantum nature of the electron and phonon leads. For example, the zero point motions of atoms are correctly taken into account, and proved critical in determining the thermal and structural properties of materials made from light elements [158][159][160][161][162][163]170,171 .
Including the correct Bose distribution of the phonons opens a way to study the quantum ballistic phonon transport by doing classical molecular dynamics. In Refs. 156 and 157 the transition from ballistic to diffusive phonon thermal transport is studied using this approach. In Refs. 17 and 34, including the nonequilibrium electrons, current-induced dynamics have been studied. Several interesting effects have been predicted or confirmed, and their effects on the stability of the system are studied. For example, it has been shown that (1) the current-induced forces are not conservative 15 , (2) the atoms feel an effective magnetic force, originating from the Berry phase of the electrons 17 . Moreover, the power of the Langevin approach is to be able to include the anharmonic phonon-phonon interactions classically, and treat the EPI quantum-mechanically. This enables one to study the energy transport between different phonon modes, between electrons and phonons at the same time. The exploration of its power and potential is still under way.
As an example, we consider the heat generation in a 4 × 2 graphene armchair ribbon, see Ref. 157 for the definition of structure parameters. The electronic structure and EPI matrix are obtained from a combined SIESTA 172 + TranSIESTA 173 + Inelastica 64 calculation, while the Brenner potential is used for the inter-atomic interaction. To reduce the simulation time, we have ignored the energy-dependence of the electronic structure. As a result, the electronic friction becomes time-local. The Langevin equation becomes (after an integration by part) (121) where F C is the force from the second-generation Brenner potential, dΓ(t)/dt = Π r (t) is the phonon retarded self-energy due to two leads, V is applied bias voltage. The eV ξ − u term gives a nonconservative force as ξ − is antisymmetric. ξ = ξ L + ξ R is the noise due to left and right phonon leads, while f is the noise due to electron bath. The expression for the electronic friction and noise correlation is the same as Eqs. (56)(57)(58)(59)(60)(61)(62)(63) in Ref. 18. Further implementation details can be found in Ref. 157. The phonon heat current is calculated using (122) where α = L, R. In steady state, the energy flow balances, and the heat generation is calculated according to Q = −I L p − I R p . The rate of heat generation is plotted in Fig. 15. The result is for the same configuration as shown in Ref. 157 of Fig. 4. Each data point takes about 4 days on a typical Opteron CPU. The error bars are quite small. The heat generation at zero bias should be zero. However, we get a small value. This has to do with the cut-off used in the noise for the electrons. We have used an abrupt cut-off for the spectrum at ω = 1.29 eV. The calculation demonstrates the feasibility of computing the Joule heating current, intrinsically a quantum effect at nanoscale, by classical molecular dynamics.
u = F C − t Γ(t−t )u(t )dt − ηu−eV ξ − u+ξ+f,I α p = u T − t Γ α (t − t )u(t )dt + ξ α ,
VI. CONCLUSIONS
In summary, using a tight-binding-like Hamiltonian for the EPI, Eq. (8), we have introduced three different approaches to study the effect of EPI in different parameter regimes. We focused on the electronic, phononic, and thermoelectric transport properties of nano-conductors, in a general multi-probe setup. For each approach, we started with the theoretical derivation of the main equations. This was followed by applications in models or simple systems, mainly for illustration purpose.
Applications of these approaches to more interesting problems are straightforward. Examples of such prob-lems are: (1) application of the NEGF and QME approach to nonequilibrium thermoelectric transport to study the thermoelectric efficiency at finite power output, (2) application of the QME approach to look at system where both EPI and electron-electron interaction are important, (3) combining the generalized Langevin equation with first-principles or tight-binding electronic structure package to study current-induced dynamics in realistic nano-conductors, especially to explore how the electron-dissipated heat is transferred in and out of the nano-conductor. With these available tools, more interesting and important systems can be investigated.
FIG. 2 .
2Feynman diagrams due to electron-phonon interaction. The first two are Hartree and Fock diagram for electrons, and the last one is the polarization bubble for phonons. The expressions of these diagrams can be found in Eqs.(27)(28)(29)(30)(31).
Fig. 5 .
5The expression for the energy current from electrons to phonons can be obtained from Eq. (26) and the expressions of the self-energies Eqs.(27)(28)(29)(30)(31) 32
FIG. 3 .
3Chemical potential dependence of the conductance correction due to the Σ r H (a), ReΣ r F (b), ImΣ r F (c), and the sum of them (d). The blue, red, purple, and black lines correspond to kBT = 0.05, 0.15, 0.25, 0.35 ω0, respectively. (e)-(f) The electrical conductance as a function of chemical potential (µ) at kBT = 0.05 ω0 (e) and kBT = 0.35 ω0 (f). Solid lines include EPI, while the dashed lines do not. Other parameters used: ΓL = ΓR = ω0 = 1, m0 = 1 ω0/(Å √ u).
FIG. 4 .
4Change of the Seebeck coefficient due to EPI at different temperatures. The parameters and meaning of colors are the same asFig. 3.
FIG. 6 .
6(a) The energy current with µ = 0 (red, solid) and µ = 0.1 eV (blue, dotted). We have set Te = T + ∆T /2, Tp = T − ∆T /2, with T = 300 K. (b) Fermi level (µ) dependence of the energy current at fixed temperature difference ∆T = 300 K. (c) Rectification ratio R as a function of µ. (d) The electronic spectral function A(ε) as a function of energy ε. to only one electron and one phonon lead [Fig. 5]. Figure 6 summarizes our result. InFig. 6 (a), we show the energy current as a function of temperature difference between the metal and insulator (∆T ), at two Fermi levels (µ = 0, 0.1 eV). The energy current is asymmetric with respect to the sign change of ∆T . But the rectification ratio R has opposite sign. This is further highlighted in the plot of the energy current and the rectification ratio R as a function of µ for fixed temperature bias ∆T = ±300 K [Fig. 6 (b)and (c), respectively]. The sign of R is well correlated with the sign of ∂ 2 A ∂ε 2 [Fig. 6 (d)].D. Effect of EPI on thermal transport in single-walled carbon nanotubes
FIG. 7 .
7(Color online) The phonon thermal conductance versus chemical potential for metallic SWCNT (10, 10) at (a) 150 K and (b) 300 K. Solid line is for ballistic phonon thermal conductance without EPI effect. Reprinted from J. Appl. Phys., 110, 124319 (2011).
FIG. 8 .
8(Color online) The phonon thermal conductance versus chemical potential for semiconductor SWCNT (10, 0) at (a) 150 K, (b) 300 K, and (c) 1000 K. Solid line is for ballistic phonon thermal conductance without EPI effect. Reprinted from J. Appl. Phys., 110, 124319 (2011).
FIG. 9 .
9The phonon sidebands with different EPI strength with (a) low temperature (T = 0.02 ω0/kB) and symmetrically biased voltage (V L/R = ±V bias ), (b) high temperature (T = 0.1 ω0/kB) and symmetrically biased voltage, (c) low temperature and asymmetric biased voltage (VL = V bias and VR = 0). Other parameters include ε0 = 1.0 ω0, εD = 10 ω0 for all plots.
FIG. 10 .
10Prediction of NDR using QME formalism with the coupling strength (a) ηL = 0.1ηR, (b) ηL = ηR and (c) ηR = 0.1ηL. For all plots VR is fixed at 0 and VL = V bias . The temperature is fixed at T = 0.02 ω0/kB for all leads. Other parameters are the same asFig. 9
FIG. 11 .
11The dependence of voltage-bias current (a) and thermoelectric current (b) on the EPI strength under different ε0. For panel (a), µL = ω0, µR = − ω0 and T L = T R = 0.02 ω0. For panel (b), T L = 0.08 ω0, T R = 0.02 ω0 and µL = µR = 0.
FIG. 12 .
12The contour plot of electronic conductance Ge, thermal conductance G h , Seebeck coefficient S and figure of merit ZeT with respect to ε0 and λ. The parameters are: µ = 0, T = 5 ω0/kB. For this plot, the phonon mode is coupled to its environments at temperature T with coupling strength ηE = 0.1η.Figureadapted with permission from Phys. Rev. B, 91, 045410 (2015). Copyrighted by the American Physical Society.
FIG. 13 .
13The correction of phonon number due to electronic current under voltage bias (a) and temperature bias (b). The parameters are: µL = −µR = 2 ω0, T L = T + ∆T , T R = T − ∆T , ∆T = 3 ω0/kB and T = 5 ω0/kB. The phonon is weakly coupled to its own environment at temperature T . Figure adapted with permission from Phys. Rev. B, 91, 045410 (2015). Copyrighted by the American Physical Society.
FIG. 14 .
14Two types of paths in the Feynman-path integrals. (a) two-segment path C2. (b) Keldysh type closed contour C. The ticks represent the discretized integration variables; the open circles are not integrated. t 0 with variable u 0 forward to time t with variable u . Since the arbitrary constant in the proportionality can be fixed by normalization (Trρ = 1), we need not specify precisely the measure associated with D[u]. The path integral integrates all the intermediate variables except the two open ends at time t.
FIG. 15 .
15The heat generated per unit time versus bias voltage of the electrons for a 4 × 2 graphene armchair configuration.
We do not distinguish vibrations in isolated molecules and phonons in periodic lattices. We do not distinguish vibrations in isolated molecules and phonons in periodic lattices.
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| []
|
[
"Proof of the thermodynamical stability of the E ′ center in SiO 2",
"Proof of the thermodynamical stability of the E ′ center in SiO 2"
]
| [
"Carlo Maria Carbonaro \nDipartimento di Fisica\nINFM\nUniversità di Cagliari\nCittadella UniversitariaI-09042MonserratoCAItaly\n",
"Vincenzo Fiorentini \nDipartimento di Fisica\nINFM\nUniversità di Cagliari\nCittadella UniversitariaI-09042MonserratoCAItaly\n",
"Fabio Bernardini \nDipartimento di Fisica\nINFM\nUniversità di Cagliari\nCittadella UniversitariaI-09042MonserratoCAItaly\n"
]
| [
"Dipartimento di Fisica\nINFM\nUniversità di Cagliari\nCittadella UniversitariaI-09042MonserratoCAItaly",
"Dipartimento di Fisica\nINFM\nUniversità di Cagliari\nCittadella UniversitariaI-09042MonserratoCAItaly",
"Dipartimento di Fisica\nINFM\nUniversità di Cagliari\nCittadella UniversitariaI-09042MonserratoCAItaly"
]
| []
| The E ′ center is a paradigmatic radiation-induced defect in SiO2 whose peculiar EPR and hyperfine activity has been known since over 40 years. This center has been traditionally identified with a distorted, positively-charged oxygen vacancy V + O . However, no direct proof of the stability of this defect has ever been provided, so that its identification is still strongly incomplete. Here we prove directly that distorted V + O is metastable and that it satisfies the key requirements for its identification as E ′ , such as thermal and optical response, and activation-deactivation mechanisms. | 10.1103/physrevlett.86.3064 | [
"https://arxiv.org/pdf/cond-mat/0009465v1.pdf"
]
| 890,763 | cond-mat/0009465 | 0064a082d1276af2ed7eb9398747b8c406ff7e00 |
Proof of the thermodynamical stability of the E ′ center in SiO 2
29 Sep 2000
Carlo Maria Carbonaro
Dipartimento di Fisica
INFM
Università di Cagliari
Cittadella UniversitariaI-09042MonserratoCAItaly
Vincenzo Fiorentini
Dipartimento di Fisica
INFM
Università di Cagliari
Cittadella UniversitariaI-09042MonserratoCAItaly
Fabio Bernardini
Dipartimento di Fisica
INFM
Università di Cagliari
Cittadella UniversitariaI-09042MonserratoCAItaly
Proof of the thermodynamical stability of the E ′ center in SiO 2
29 Sep 2000(November 21, 2018)
The E ′ center is a paradigmatic radiation-induced defect in SiO2 whose peculiar EPR and hyperfine activity has been known since over 40 years. This center has been traditionally identified with a distorted, positively-charged oxygen vacancy V + O . However, no direct proof of the stability of this defect has ever been provided, so that its identification is still strongly incomplete. Here we prove directly that distorted V + O is metastable and that it satisfies the key requirements for its identification as E ′ , such as thermal and optical response, and activation-deactivation mechanisms.
Understanding defects in solids is a key factor in improving device performance and materials quality. Defect identification combines experimental observation and theoretical predictions, and a major ingredient in this process is the theoretical justification of the thermodynamical stability of the defect. If this information is missing, the identification is arguably incomplete or uncertain. Surprisingly, this is the case for the E ′ center [1,2], a radiation-induced defect first observed experimentally as early as 45 years ago [1] in SiO 2 , a material of prime current importance in microelectronics and fiber optics [3]. E ′ is traditionally identified with a positively-charged distorted oxygen vacancy V + O [4], with important support from calculations of hyperfine couplings [5] and optical activity [6]; its thermodynamical stability, however, was never theoretically proven, and the mechanisms involved in its activation and deactivation are still debated. Here, using the ab initio theory of defect formation in solids [7], we show that the conditions for E ′ stability are naturally realized in stoichiometric or neutron-irradiated SiO 2 , and conclusively put on firm ground the identification of the E ′ center.
Our model of the stability of E ′ is based on two native defects: the oxygen vacancy V O and the oxygen interstitial O i . The motivation is that E ′ is observed chiefly (though not only) in neutron-irradiated material [1][2][3], where V O and O i are both abundant. Indeed, we find that it is their concurrent presence that produces the conditions for the existence of E ′ , in neutron-irradiated as well as non-neutron-irradiated material. In the former, vacancies V O and interstitials O i are essentially produced in pairs by knock-on, kick-out events; in the latter, they form in thermal equilibrium and, as it turns out, in similar concentrations. As will become apparent below, our argument on E ′ applies to both cases without modifications, except for the fact that neutron-produced defects appear in concentrations determined by the irradiation dose, whereas the concentration of thermally formed defects depends on formation energies, which can be directly predicted.
At a growth temperature T g and with N s available sites, the equilibrium concentration of a defect is D= N s exp(−F form /k B T g ). The formation free energy F form = E form -T S form depends [7] on the chemical potentials of atoms added or removed, on the defect charge state, i.e. the charge released to or captured from the thermodynamic reservor constituted by the surrounding crystal, and on the electronic chemical potential µ e of the latter. Once the formation energies of the all relevant defects (vacancy and interstitial in our case) are known, the defect concentrations and the chemical potential µ e are determined self-consistently, subject to charge neutrality, as detailed in [7]. A specific defect configuration or charge state is predicted to exist if its formation energy is lower than that of all other defect states for some value of µ e . Also, the defect is metastable if a non-zero energetic barrier prevents its deactivation or disappearance into other lower-energy configurations of the same defect, or recombination with other defects. The formation energy for our defects in charge state Q reads
E form (Q) = E def tot (Q) − E undef tot + Qµ e + M(Q) + P (1)
where E def tot and E undef tot are the total energies of the defected and undefected system, respectively, µ e is the electron chemical potential (equaling the Fermi level E F in our T=0 calculations), M(Q) is the defect-dependent multipole correction for charge state Q of Ref. [8], P = µ O for V O and P = -µ O for O i , and µ O is the oxygen chemical potential. The latter is fixed to stoichiometric conditions, i.e. at the center of its variation range µ mol /2 + ∆H/2 < µ O < µ mol /2 determined by the total energy µ mol of the O 2 molecule, and the calculated formation enthalpy ∆H of SiO 2 . Ionization energies, i.e. the energy needed to promote (say) an electron from the valence band to an empty acceptor level, are defined via total energy differences of different charge states. Formation entropies are beyond the scope of the method used here; plausible estimates are used when needed.
Energies and forces are accurately calculated from first-principles within density-functional theory in the local approximation, using the ultrasoft pseudopotential plane-wave method as implemented in VASP [9]. An iso-lated defect is simulated in periodic boundary conditions via the repeated supercell approach. We use crystalline α-quartz SiO 2 supercells of tetragonal symmetry, comprising 71 to 73 atoms and of linear dimensions 18.49, 16.02, and 20.44 atomic units (theoretical lattice parameters [10], matching experiment to about 0.5%). Atomic geometries of the defects are optimized for each Q (obtained by removing or adding electrons as appropriate) until all residual force components in the system are below 0.02 eV/Å. No symmetry restriction is imposed on geometry optimization. Improving slightly on the setting of Ref. [5], a (222) mesh is used [9] for k-space summation (4 special points in the supercell Brillouin zone). Our use of a crystalline SiO 2 -based model of the E ′ defect, which is observed both in amorphous and crystalline phases, is justified by the closely similar behavior of several E ′ variants in crystalline and amorphous SiO 2 in experiment [2,11] as well as theory [4,5]. In addition, the simulated structure of amorphous silica [12] deviates moderately from that of crystalline a-quartz SiO 2 . doping conditions, but it is in fact neutral for most of the Fermi level range. As to the interstitial, we find that an oxygen atom initially placed near the center of the hexagonal channels of quartz, relaxes sideways towards the helical chains and, after overcoming a small (≃ 0.2 eV) barrier, it stabilizes into a split-interstitial (s-O i ) configuration with the nearest bridging oxygen in the helical tetrahedra chain, with ≃ 1 eV of energy gain with respect to the starting site. Details will be discussed elsewhere: here we note the main consequences of this result: a) the stabilization in the s-O i configuration (which is found to be quite close to that suggested in Ref. [13]) prevents O i from easily recombining with vacancies, because the detachment from the split-interstitial configuration costs about 1.2 eV; b) as seen in Fig.1, s-O i is a (negative-U ) deep double acceptor with second ionization energy at 3.3 eV above the valence band top Ev. The Fermi level, calculated as in [7], is pinned at E F = E v + 3.3 eV (vertical solid line in Fig.1). Thus a consequence of vacancy-interstitial pair formation, is that moderately p-type conditions are achieved; this is indeed often observed in irradiated samples [2,11]. In the absence of s-O i , the Fermi level would be at midgap, E v + 4.4 eV. From the formation energies one can estimate the chemical concentrations of s-O i and V O : for a typical T g of 1500 K, and assuming a reasonable formation entropy S form = 5 k B , the concentration of both defects is ∼10 14 cm −3 . This figure matches well E ′ concentrations measured [11] in non-neutron irradiated samples after UV, γ, or X illumination fall, and therefore supports the hypothesis that the vacancy is the parent defect of E ′ . Neutron irradiation of course produces dose-dependent [2], typically much higher concentrations (> 10 19 cm −3 ). The key point is that, since the concentrations of vacancies and interstitial are essentially the same in both cases, and because only the interstitial has electrical activity (via its double acceptor level), the Fermi level is pinned at the same value in both cases, so that the theory of E ′ stability and activation discussed below applies identically to both cases.
It appears from Fig.1 that the +1 charge state of the vacancy, V + O , is not among the thermodynamically stable ground states of the defect. Therefore, if this state of V O is to be identified with E ′ , it must at least be proven metastable; if it is metastable, a mechanism for its creation starting from the ground state (the neutral vacancy) must be identified. As to the first point, since the +1 vacancy has a formation energy that increases linearly with the Fermi level while the neutral one remains constant, V + O or a distorted variant thereof, may only be metastable in a limited range of that variable: we show below that the Fermi level pinning due to split-O i produces naturally the conditions for the metastable existence of E ′ . As to the second point, experiments indicate that E ′ is activated by ionizing radiation [2,3,11] such as γ, X, or UV photons shone onto vacancy-containing samples, or concurrently with neutron irradiation (causing knock-on vacancy creation). Indeed, since the vacancy ground state is the neutral undistorted configuration, the distorted +1 state (alias E ′ ) can only be accessed by ionization of V 0 O : our picture provides consistently such activation mechanism.
We proceed to study the behavior of the vacancy when subjected to the undistorted-to-puckered transition as proposed in earlier studies [4][5][6]. In accord with the results discussed above, we fix the Fermi level at E v + 3.3 eV. The creation of a vacancy starting from the perfect lattice results in moderate local distortions in both the neutral and +1 charge states. The puckered configuration is obtained by moving one of the two vacancyadjacent Si 1 and Si 2 atoms (specifically the "long-bond [4,5] Si 2 ) away from the vacant site, and pushing it across the basal plane of the incomplete tetrahedron centered on Si2 itself. When Si 2 pokes through this triangular constriction, it gets strongly and suddenly bound to a backbonding oxygen, O b . Upon completion of the distortion, Si 2 regains 4-fold coordination, and backbonding O b becomes 3-fold coordinated (see also [4][5][6]), while it was originally 2-fold coordinated as all tetrahedron-bridging oxygens in SiO 2 . Si 1 remains instead 3-fold coordinated: in the +1 charge state, its dangling bond is half-filled, and causes the observed [1,2] and predicted [5] EPR signature which identifies E ′ [2]. It may appear at first sight that the puckering distortion should be symmetric in Si 1 and Si 2 . This is not the case, however, because of the intrinsic asymmetry of the quartz structure. As already noted earlier on [4], only Si 2 finds the backbonding oxygen O b in the correct position. This applies largely also to amorphous silica, whose structure is moderately different at the local level from that of quartz [12]. The backbonding oxygen is therefore the main stabilizing agent of the E ′ defect.
The total energy of the system in charge state Q is calculated as a function of the separation between Si 1 and Si 2 . Only the modulus d Si−Si of the Si 1 -Si 2 connecting vector is constrained, and all other degrees of freedom are fully relaxed: the minimum energy path is thus mapped out for the undistorted-puckered transformation in the constrained-d Si−Si configurational subspace. In Fig. we display the full level diagram for the neutral, +1, and +2 charge states of the vacancy as a function of the puckering distortion, quantified by d Si−Si . All energy curves depend on the Fermi level through Eq.1; they can be directly compared on the same energy scale because E F is fixed at the natural value of E v + 3.3 eV determined above.
The outstanding feature of Fig. is that for the natural Fermi level of stoichiometric or neutron-irradiated silica, the candidate E ′ , i.e. distorted V + O , is indeed the stable defect state for the distorted geometry. We stress that the Fermi level position is essential here: if E F were at midgap, the +1 curve would be 1.1 eV above its position in Fig.. Then E ′ would be unstable towards magneticallyinactive V 0 O . Globally, E ′ is metastable with a confining barrier of 0.8 eV. The barrier to enter the metastable state is 1.1 eV, and the undistorted state is lower than the distorted by 0.3 eV [this difference to Ref. [5], where the distorted state was lower by the same amount, is possibly due to our improved k-sampling]. Clearly, in the absence of excitation, V + O will remain trapped in the metastable E ′ state and will show EPR activity. When thermally activated to overcome the barrier, the puckered center will transform into undistorted, and by electron capture it will become neutral. Therewith, E ′ disappears permanently, and so does its magnetic activity, because Si1 and Si2 combine their dangling bonds to bind into a dimer [4][5][6]10]. (The same deactivation route is not readily available for the level ordering of Ref. [5], which implies that i) E ′ remains activated at equilibrium, and that ii) concurrent barrier jump and electron capture are required to quench it.) The calculated puckered-to-undistorted barrier is now compared with an estimate extracted from the measured relative drop in E ′ population upon isocronal thermal annealing [11] in irradiated silica. In the simplest model, the distorted-state population N 0 diminishes by a factor of N/N 0 = exp (-R(T) τ ) upon annealing over a time τ at temperature T, with escape rate R = ν exp (-∆F/k B T), with ∆F the free energy barrier for escape from the puckered state, and ν an average vibrational frequency in that state. Using the data [11] for E ′ γ and assuming ν ∼ 50 THz, we obtain an experimental ∆F∼1.1 eV, in reasonable agreement with our calculated ∆F∼ ∆E=0.8 eV; account for the transition entropy, which we neglect, should further improve agreement since the transition state (through the tetrahedron basal plane) is severely constrained geometrically, and has a higher average vibrational frequency.
Let us now come to E ′ activation. In the present picture, E ′ is created transforming undistorted V 0 O into distorted V + O via two routes. The first proceeds on path A in Fig. with two successive one-photon ionizations of V 0 O into V +2 O , followed by non-radiative decay into E ′ . This path is efficient since the +1 undistorted state is kept populated by sustained illumination (a much less efficient double-photon excitation of V 0 O into V +2 O may also occur). The excitation energies for path A are both near 4-4.5 eV if the ionized electron is transfered to the Fermi level, i.e. to the E F -pinning impurity; if it is promoted to the conduction band, the energies are instead about 7 eV. Both processes are possible with X or γ radiation, whose energy vastly exceeds that needed in the transition. In UV irradiation, the center is often activated by pumping at 5 eV, and clearly only Fermi-level capture matches this figure. (There are, however, qualitative indications that the 5 eV excitation may activate E ′ via alternate routes involving other pre-existing defects.)
The second excitation route, path B, involves an optical excitation of V 0 O into undistorted V + O , and a thermal excitation of the latter into the puckered state. The energy difference (0.3 eV) between the two V + O states implies that only a fraction of 10 −5 of the vacancies gets promoted into the distorted state in equilibrium at room temperature, on sustained illumination. Therefore, though admissible, this path is preempted by path A. With the level ordering of Ref. [5], also path B competes with path A. (Our ordering, however, matches better the thermal behavior, as discussed above.)
Other calculated observables of our E ′ model are consistent with previous studies [4][5][6]14]. For instance, the optical absorption of the neutral undistorted state at 6.9 eV correlates well with the 7.6 eV absorption band usually attributed to the neutral vacancy [6]. For E ′ (metastable puckered V + O ) we find an absorption at 4.7 eV (defect-to-conduction promotion) followed by slow non-radiative decay back into E ′ . Since it is not followed by any emission or E ′ deactivation, this absorption correlates with the broad 5.8-eV band typical of E ′ [11,14], which exhibits the same behavior in experiment.
In summary, we conclusively put on firm ground the identification of the singly positive O vacancy in SiO 2 with the E ′ center proving its thermodynamical stability via first principles calculations. Our picture provides naturally activation and deactivation mechanisms, and other optical signatures, in agreement with known experimental observations. In addition, our picture naturally explains the moderate p conditions produced by irradiation in SiO 2 .
We acknowledge support from Istituto Nazionale per la Fisica della Materia under "Iniziativa Trasversale Calcolo Parallelo".
FIG. 1 .
1Minimum formation energies (eV) of the oxygen vacancy (dash-dotted line) and split-interstitial oxygen (dashed) as a function of the Fermi level, and calculated Fermi levels position (vertical solid). Fermi level zero is the valence band top.
Fig. 1
1shows the ground state formation energies of the interstitial O i and the vacancy V O . The vacancy acts as a double acceptor (double donor) in extreme n (p)
FIG. 2 .
2Total energy of neutral, positive (E ′ ), and doubly positive oxygen vacancies as a function of the puckering distortion, at the calculated Fermi level value.
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| []
|
[
"ON POSITIVITY OF EHRHART POLYNOMIALS",
"ON POSITIVITY OF EHRHART POLYNOMIALS"
]
| [
"F U Liu "
]
| []
| []
| Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.2010 Mathematics Subject Classification. Primary 52B20; Secondary 05A15, 90C57. | 10.1007/978-3-030-05141-9_6 | [
"https://arxiv.org/pdf/1711.09962v2.pdf"
]
| 119,166,161 | 1711.09962 | 0129207971c250d3c7cbad0af4b10e1ac6a64668 |
ON POSITIVITY OF EHRHART POLYNOMIALS
Aug 2018
F U Liu
ON POSITIVITY OF EHRHART POLYNOMIALS
Aug 2018
Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.2010 Mathematics Subject Classification. Primary 52B20; Secondary 05A15, 90C57.
Introduction
A polyhedron in the D-dimensional Euclidean space R D is the solution set of a finite set of linear inequalities:
P = x ∈ R D : D j=1 a i,j x j ≤ b i for i ∈ I ,
where a i,j ∈ R, b i ∈ R and I is a finite set of indices. A polytope is a bounded polyhedron. Equivalently, a polytope in R D can also be defined as the convex hull of finitely many points in R D . We assume readers are familiar with basic definitions such as faces and dimensions of polytopes as presented in [108]. In this paper, the letter d usually denotes the dimension of a polytope and D denotes the dimension of the ambient space. For majority of the examples presented here, we either have d = D or d = D − 1.
A lattice point or an integral point is a point in Z D . Counting lattice points inside polytopes is a fundamental and useful step in many mathematical analyses. A lot of combinatorial structures can be counted as lattice points of polytopes. For example, matchings on graphs [63], t-designs [74], (semi-)magic squares [10,Chapter 6], and linear extensions of posets [100] are all of this form. Counting lattice points not only appears in combinatorial problems, it also appears, for instance, in the context of representation theory [52,89], algebraic geometry [37], statistics [34,36], and number theory [7,75].
One approach to study the question of computing the number of lattice points in a polytope P is to consider a more general counting problem: For any nonnegative integer t, let tP := {tx : x ∈ P } be the t th dilation of P , and then consider the function i(P, t) := |tP ∩ Z D |, which counts the number of lattice points in tP. We say two polytopes P and Q are unimodularly equivalent 1 if there exists an affine transformation from the affine hull aff(P ) of P to the affine hull aff(Q) of Q that induces a bijection from lattice points in aff(P ) to lattice points in aff(Q). Such an affine transformation is called a unimodular transformation. It is easy to see that if two polytopes P and Q are unimodularly equivalent, then i(P, t) = i(Q, t). An integral polytope (or a lattice polytope) is a polytope whose vertices are lattice points. In the 1960's Eugène Ehrhart [35] discovered that the function i(P, t) has nice properties when P is an integral polytope. Theorem 1.1 (Ehrhart). For any integral d-polytope P, the function i(P, t) is always a polynomial (with real coefficients) of degree d in t.
Thus, we call i(P, t) the Ehrhart polynomial of an integral polytope P , and call the coefficients of i(P, t) the Ehrhart coefficients of P. Note that Ehrhart's theorem can be extended to rational polytopes with the concept of a quasi-polynomial; however, we will focus on integral polytopes in this article.
There is much work on the Ehrhart coefficients of integral polytopes. In the 1990's, many people studied the problem of counting lattice points inside integral (or more generally rational) polytopes [17,22,50,81,85] by using the theory of toric varieties. Although explicit formulas for coefficients of Ehrhart polynomials can be deduced from these results, they are often quite complicated. Only three coefficients of i(P, t) have simple known forms for arbitrary integral polytopes P : the leading coefficient is equal to the normalized volume of P , the second coefficient is one half of the sum of the normalized volumes of facets, and the constant term is always 1.
Although these three coefficients can be described in terms of volumes (recall 1 is the normalized volume of a point), and thus are positive, it is not true that all the coefficients of i(P, t) are positive. (The first counterexample comes up in dimension 3 known as the Reeve tetrahedron; see §4.1.) We say a polytope has Ehrhart positivity or is Ehrhart positive if it has positive Ehrhart coefficients. It is natural to ask the following question: This turns out to be a challenging question. Even though multiple families of polytopes have been shown to be Ehrhart positive in the literature, the techniques involved are (almost) all different. In Section 2, we will survey families of polytopes with the Ehrhart positivity property, discussing different reasons why they have this property. In particular, as a consequence of the techniques discussed in §2.4, one can show that any combinatorial type of rational polytopes can be realized as an integral polytope that is Ehrhart positive (see Theorem 2.4.8). This result indicates that Ehrhart positivity is not a combinatorial property. Therefore, it is desirable to find more geometric methods to prove Ehrhart positivity. In Section 3, we introduce such a tool called McMullen's formula, which we use to give a refinement of Ehrhart positivity, called α-positivity. We then use this tool to attack the Ehrhart positivity conjecture of "generalized permutohedra", a family of polytopes introduced by Postnikov [82] and report partial progress on this conjecture. In Section 4, we include negative results on Question 1.2, presenting examples with negative Ehrhart coefficients. In particular, we will discuss progress on a question asked and studied by Hibi, Higashitani, Tsuchiya and Yoshida [48] on all possible sign patterns of Ehrhart coefficients (see §4. 2). Note that this question can be considered to be a refinement of Question 1.2. Finally, in Section 5, we include various conjectures on Ehrhart positivity, and pose related questions.
We finish our introduction with the following remark on the coefficients of the h * -polynomial, which is closely related to the Ehrhart polynomials.
Remark on h * -vector. One method of proving Ehrhart's theorem (Theorem 1.1) is by considering the Ehrhart series of a d-dimensional integral polytope P :
Ehr P (z) := t≥0 i(P, t)z t .
It turns out that Theorem 1.1 is equivalent to the existence of a polynomial h * P (z) of degree at most d such that h * P (1) = 0 and
Ehr P (z) = h * P (z) (1 − z) d+1 .
See [102, Chapter 4] for a statement for the above equivalence result and a proof for Ehrhart's theorem.
We call h * P (z) the h * -polynomial of P , and the vector (h * 0 , h * 1 , . . . , h * d ), where h * i is the coefficient of z i in h * P (z), the h * -vector of P. One can recover the Ehrhart polynomial of a d-dimensional integral polytope P easily from its h * -vector:
(1.1) i(P, t) = d j=0 h * j t + d − j d .
It is a well-known result due to Stanley that the entries in h * -vectors are all nonnegative integers [93] in contrast to the fact that Ehrhart coefficients could be negative. As a consequence, positivity is not such an interesting question for h * -polynomials. Nevertheless, active research have been conducted in other directions. The most natural question probably is: for each d, can we give a complete characterization for all possible h * -vector of d-dimensional integral polytopes? For d = 2, the answer was first provided in 1976 by Scott [90] known as Scott's condition. However, for d ≥ 3, the question is wide open. A lot of work has been done in the literature on searching for inequalities and equalities satisfied by h * -vectors. Most of them were discovered by Hibi [43,45,46,47] and Stanley [93,95] in the 1990s using commutative algebra and combinatorial methods. In 2009, Stapledon [103] contributes more inequalities using the idea of degree and codegree of a polytope. Known equalities on h * -vectors include
(1.2) d i=0 h * i = d!Vol(P ), h * 0 = 1, h * 1 = i(P ) − (d + 1), h * d = |relint(P ) ∩ Z D |.
Please see [8,103] for lists of known inequalities. Recently, instead of focusing on inequalities satisfied by all polytopes, much work has been done on finding inequalities for polytopes under certain constraints. For example, Treutlein [105] shows that the necessary statement of Scott's condition holds for any integral polytope whose h * -polynomial is of degree at most 2, i.e., h * i = 0 for all i > 2. Most recently, Balletti and Higashitani [5] improve the result further to any integral polytope whose h * -polynomial satisfies h * 3 = 0.
In the literature, there are multiple interesting families of polytopes shown to be Ehrhart positive using very different techniques. In this section, we put together a collection of such families, separating them into four categories based on the reasons why they are Ehrhart positive. However, we make no attempt to give a comprehensive account of all families with this property. We also note that as the leading coefficient of i(P, t) is the volume of P, one can often extract a formula for volume from descriptions for Ehrhart polynomials we give below. However, we will focus only on results on Ehrhart polynomials here, and omit related formulas for volumes.
In this article, we use bold letters to denote both vectors and points in R D . For example, e i denotes both the ith vector in the standard basis and the point (0, . . . , 0, 1, 0, . . . , 0) where 1 is in the ith position.
For convenience, we use N to denote the set of nonnegative integers, and P the set of positive integers.
2.1. Products of positive linear polynomials. In this part, we present families of polytopes whose Ehrhart polynomials can be described explicitly, which can be shown to have positive coefficients using the following naive lemma. We start with the two simplest families of polytopes: unit cubes and standard simplices, whose Ehrhart polynomials fit into situation (a) of Lemma 2.1.1, and thus Ehrhart positivity follows. As these are the first examples of Ehrhart polynomials in this article, and the computations are straightforward, we include all the details. For most of the remaining examples we discuss in this paper, we only state the results without providing detailed proofs.
2.1.1. Unit cubes. The d-dimensional unit cube, denoted by d , is the convex hull of all points in R d with coordinates in {0, 1}, i.e.,
d := conv{x = (x 1 , x 2 , . . . , x d ) ∈ R d : x i = 0 or 1 for i = 1, 2, . . . , d}.
It is easy to verify that the unit cube is the solution set to the following linear system of inequalities:
d = {x = (x 1 , x 2 , . . . , x d ) ∈ R d : 0 ≤ x i ≤ 1 for i = 1, 2, . . . , d},
Then for any t ∈ N,
t d = {x = (x 1 , x 2 , . . . , x d ) ∈ R d : 0 ≤ x i ≤ t for i = 1, 2, . . . , d}.
Thus,
t d ∩ Z d = {x = (x 1 , x 2 , . . . , x d ) ∈ Z d : 0 ≤ x i ≤ t for i = 1, 2, . . . , d}. For each i, the number of integers x i such that 0 ≤ x i ≤ t is t + 1. Thus, i( d , t) = (t + 1) d .
2.1.2. Standard simplices. The d-dimensional standard simplex, denoted by ∆ d , is the convex hull of all the elements in the standard basis e 1 , e 2 , . . . , e d+1 of R d+1 : ∆ d := conv{e 1 , e 2 , . . . , e d+1 }.
One checks that ∆ d can also be defined by the following linear system: d+1 j=1
x j = 1, and x i ≥ 0 for i = 1, 2, . . . , d + 1.
Hence, for any t ∈ N,
t∆ d = x = (x1, x2, . . . , x d+1 ) ∈ R d+1 : d+1 j=1 xj = t, and xi ≥ 0 for i = 1, 2, . . . , d + 1 , and t∆ d ∩Z d+1 = x = (x1, x2, . . . , x d+1 ) ∈ Z d+1 : d+1 j=1 xj = t, and xi ≥ 0 for i = 1, 2, . . . , d + 1 .
Hence, i(∆ d , t) counts the number of nonnegative integer solutions to
x 1 + x 2 + · · · + x d+1 = t.
This is a classic combinatorial problem which is the same as counting the number of weak compositions of t into d + 1 parts (see [102, Page 18]), and the answer is given by
i(∆ d , t) = t + d d .
As we mentioned above, Ehrhart positivity of unit cubes and standard simplices follows from situation (a) of Lemma 2.1.1. Next, we present two families of examples with Ehrhart polynomials in the form of situation (b) of Lemma 2.1.1.
2.1.3. Pitman-Stanley polytopes. Let a = (a 1 , . . . , a d ) ∈ N d . The following polytope is introduced and studied by Pitman and Stanley [79]:
PS d (a) := x ∈ R d : x i ≥ 0 and i j=1 x j ≤ i j=1 a i , for i = 1,I d := i = (i1, i2, . . . , i d ) ∈ N d : d j=1 ij = d, and k j=1 ij ≥ k for k = 1, 2, . . . d − 1 .
Then the Ehrhart polynomial of PS d (a) is given by
(2.1) i(PS d (a), t) = i∈I d a 1 t + 1 i 1 d k=2 a k t i k .
For each i, both a1t+1 i1 and a k t i k are products of linear polynomials in t with positive coefficients, so it follows from Lemma 2.1.1/(b) that any Pitman-Stanley polytope PS d (a) is Ehrhart positive.
Pitman-Stanley polytopes are contained in two different more general families of polytopes: flow polytopes and generalized permutohedra. For each of these two bigger families of polytopes, formulas for Ehrhart polynomials of some subfamily have been derived, generalizing Formula (2.1). We present results on flow polytopes in the next part below, while the results on generalized permutohedra are postponed to §3.1.2 as part of a general discussion on the Ehrhart positivity conjecture of generalized permutohedra in Section 3.
2.1.4. Subfamilies of flow polytopes. Let G be a (loopless) directed acyclic connected graph on [n + 1] = {1, 2, . . . , n + 1} such that each edge {i, j} with i < j is always directed from i to j. Hence, we denote the edge by (i, j) to indicate the orientation. For any a = (a 1 , a 2 , . . . , a n ) ∈ N n , we associate to it another vector
(2.2)ā := a 1 , . . . , a n , − n i=1 a i .
Anā-flow on G is a vector f = (f (e)) e∈E(G) ∈ (R ≥0 ) E(G) such that for i = 1, 2, . . . , n, we have
e=(g,i)∈E(G) f (e) + a i = e=(i,j)∈E(G) f (e),
that is, the netflow at vertex i is a i . Note these conditions imply that the netflow at vertex n + 1 is − n i=1 a i . The flow polytope F G (ā) associated to G and the integer netflowā as the set of allā-flows f on G. For each edge e = (i, j) of G, we associate to it the positive type A n root α(e) = α(i, j) = e i − e j . For any b ∈ Z n+1 , the Kostant partition function KP G evaluated at b is
KP G (b) := # f = (f (e)) e∈E(G) ∈ N E(G) : e∈E(G) f (e)α(e) = b .
It is straightforward to verify that for a ∈ N n ,
KP G (ā) = |F G (ā) ∩ Z E(G) |,
i.e., KP G (ā) counts the number of lattice points in the flow polytope F G (ā). In the literature, various groups of people [55,84,4,70] obtained formulas for Kostant partition functions, or equivalently, the number of lattice points in flow polytopes.
As a consequence, we can easily obtain formulas for the Ehrhart polynomial of F G (ā).
Theorem 2.1.4 (Lidskii, Postnikov-Stanley, Baldoni-Vergne, Mészáros-Morales).
Suppose G is a connected graph on the vertex set [n + 1], with m edges directed i → j if i < j, and with at least one outgoing edge at vertex i for i = 1, . . . , n. Let out k (and in k , respectively) denote the outdegree (and the indegree, respectively) of vertex k in G minus 1.
Then for any a = (a 1 , . . . , a n ) ∈ N n , the Ehrhart polynomial of F G (ā) is given by
i(F G (ā), t) = j n k=1 a k t + out k j k · KP G (j 1 − out 1 , j 2 − out 2 , . . . , j n − out n , 0), (2.3) = j n k=1 a k t − in k j k · KP G (j 1 − out 1 , j 2 − out 2 , . . . , j n − out n , 0), (2.4)
where each summation is over all weak compositions j = (j 1 , . . . , j n ) of m − n that are ≥ (out 1 , . . . , out n ) in dominance order.
We remark that Lidskii [55] gives a formula for computing Kostant partition functions associated to the complete graph K n+1 , which yields Formula (2.3) above with G = K n+1 . Postnikov and Stanley [84, unpublished] were the first to discover Formula (2.3) for arbitrary graphs G using the Elliott-MacMahon algorithm. Baldoni and Vergne [4] give a proof for both formulas in Theorem 2.
a k t − in k j k = a k t − in k +j k − 1 j k
is a product of linear polynomials in t with positive coefficients as long as in k = 0 or −1. Also, note that KP G (j 1 − out 1 , j 2 − out 2 , . . . , j n − out n , 0) is nonnegative and i(F G (ā), t) = 0. The following result immediately follows from Lemma 2.1.1/(b). Proof. Let a > 0. If −a < 0 is a real root of p(t), then t + a is a factor of p(t). If −a + bi is a complex root of p(t) for some b ∈ R, then −a − bi must be a root of p(t) as well, which implies that
(t + a − bi)(t + a + bi) = (t 2 + 2at + a 2 + b 2 )
is a factor of p(t). Hence, p(t) is a product of factors with positive coefficients. Thus, our conclusion follows.
We say that a polynomial (with real coefficients) is negative-real-part-rooted or NRPR if all of its roots have negative real parts. The above lemma implies that if i(P, t) is NRPR, then P is Ehrhart positive. Ehrhart polynomials of unit cubes and standard simplices are all trivially NRPR, as they factor into linear polynomials with positive real coefficients. Hence, we would like to rule them out, and are only interested in examples of Ehrhart polynomials that are nontrivially NRPR.
It turns out that the following theorem which establishes a connection between roots of the h * -polynomial and roots of the Ehrhart polynomial of a polytope is very useful. ). Let P be a d-dimensional integral polytope, let k be the degree of the polynomial h * P (z) (so that 0 ≤ k ≤ d), and suppose that every root of h * P (z) lies on the circle {z ∈ C : |z| = 1} in the complex plane. Then there exists a polynomial f (t) of degree k such that
i(P, t) = f (t) · d−k i=1 (t + i),
and every root of f (t) has real part −(1 + (d − k))/2.
We say a polytope P is h * -unit-circle-rooted if the h * -polynomial h * P (z) of P has all of its roots on the unit circle of the complex plane. Below we introduce three families of polytopes, and show that each polytope P in these families is h * -unitcircle-rooted. Therefore, Ehrhart positivity for these families follows from Theorem 2.2.2 and Lemma 2.2.1.
2.2.1.
Cross-polytopes. The d-dimensional cross-polytope, denoted by ♦ d , is a polytope in R d defined by
♦ d := conv{±e 1 , ±e 2 , . . . , ±e d },
or equivalently by the following linear system:
±x 1 ± x 2 ± · · · ± x d ≤ 1.
Hence, i(♦ d , t) counts the number of integer solutions to
|x 1 | + |x 2 | + · · · |x d | ≤ t.
Counting the number of integer solutions with exactly k nonzero x i 's for k = 0, 1, 2, . . . , d, we obtain that
i(♦ d , t) = d k=0 2 k d k t k .
Unfortunately, it is not clear whether the above expression expands positively in powers of t. We compute its Ehrhart series instead. First, notice that i(♦ d , t) counts the number of integer solutions to
|x 1 | + |x 2 | + · · · |x d | + y = t. Hence, i(♦ d , t) = f (a 1 )f (a 2 ) · · · f (a d )f (b),
where the summation is over all weak compositions (a 1 , . . . , a d , b) of t into d + 1 parts, g(b) = 1 for all b ≥ 0 and f (a) = 1 if a = 0 and f (a) = 2 if a > 0. Therefore,
t≥0 i(♦ d , t)z t = d i=1 ai≥0 f (a i )z ai · b≥0 z b = 1 + z 1 − z d · 1 1 − z = (1 + z) d (1 − z) d+1 .
Thus, (1 + z) d is the h * -polynomial of the cross-polytope ♦ d . Hence, ♦ d is h * -unitcircle-rooted, and thus are Ehrhart positive. (1,q) . Let q = (q 1 , q 2 , . . . , q d ) ∈ P d be a sequence of positive integers. For each such a vector q, we define a simplex
Certain families of ∆
∆ (1,q) := conv e 1 , e 2 , . . . , e d , − d i=1 q i e i .
In [29], Conrads studied simplices of this form and showed that ∆ (1,q) is reflexive if and only if
q i divides 1 + d j=1 q j , for i = 1, 2, . . . , d.
Recently, Braun, Davis and Solus studied ∆ (1,q) in their investigation of a Conjecture by Hibi and Ohsugi, and they provided a number-theoretic characterization of the h * -polynomial of ∆ (1,q)
(2.5) h * ∆ (1,q) , z = q1+q2+···+q d b=0 z ω(b) , where ω(b) := b − d i=1 q i b 1 + q 1 + · · · + q d .
Formula (2.5) allows us to compute the h * -polynomial for ∆ (1,q) with special choices of q easily. We give two examples below such that ∆ (1,q) satisfies the hypothesis of Theorem 2.2.2 with k = d.
Example 2.2.4 (Standard reflexive simplices). If we choose q = 1 = (1, 1, . . . , 1) ∈ P d , then ∆ (1,q) is the d-dimensional standard reflexive simplex. Note that in this case, we have that q 1 + q 2 + · · · + q d = d. Furthermore, for each b ∈ {0, 1, 2, . . . , d}, one can verify that w(b) = b. Hence,
h * ∆ (1,1) , z = d b=0 z b = 1 + z + z 2 + · · · + z d .
Example 2.2.5 (Payne's construction). In [78], Payne constructed reflexive simplices that do not have unimodal h * -vectors. His construction is equivalent to the simplices ∆ (1,q) with the following choices of q : Given r ≥ 0, s ≥ 3 and k ≥ r + 2, let d = r + sk and 2.2.3. One family of order polytopes. Given a finite poset (partially ordered set) P, the order polytope, denoted by O(P), is the collection of functions x ∈ R P satisfying • 0 ≤ x a ≤ 1, for all a ∈ P, and • x a ≤ x b , if a ≤ b in P. The order polytope O(P) was first defined and studied by Stanley [100]. Here we consider a family of order polytopes constructed from a certain family of posets.
h * ∆ (1,q) , z = (1 + z k + z 2k + · · · + z (s−1)k )(1 + z + z 2 + · · · + z k+r
Let P k be the ordinal sum of k copies of 2 element antichains, equivalently, P k is the poset on the 2k-element set {a i,j : i = 1, 2 and j = 1, 2, . . . , k} satisfying
a i,j ≤ a i ′ ,j ′ if and only if j < j ′ or (i, j) = (i ′ , j ′ ). For any t ∈ N, the t th dilation tO(P k ) of O(P k ) is the collection of x = (x i,j : i = 1, 2 and j = 1, 2, . . . , k) ∈ R 2k satisfying 0 ≤ x i,j ≤ t, and x i,j ≤ x i ′ ,j ′ if j < j ′ .
Hence, i(O(P k ), t) counts the number of integer solutions x satisfying the above two conditions. Note that each solution gives a weak composition (y 1 , z 1 , y 2 , . . . , y k , z k , y k+1 ) of t into 2k + 1 parts, where
y j = min(x 1,j , x 2,j ) − max(x 1,j−1 , x 2,j−1 ), for j = 1, 2, . . . , k + 1, z j = max(x 1,j , x 2,j ) − min(x 1,j , x 2,j ) = |x 1,j − x 2,j |, for j = 1, 2, . . . , k,
and by convention let max(x 1,0 , x 2,0 ) = 0 and min(x 1,k+1 , x 2,k+1 ) = t. Thus,
i(O(P k ), t) = g(y 1 )f (z 1 )g(y 2 )f (z 2 ) · · · f (z k )g(y k+1 ),
where the summation is over all weak compositions of t into 2k + 1 parts, g(y) = 1 for all y ≥ 0, and f (z) = 1 if z = 0 and f (z) = 2 if z > 0. Therefore, similar to the calculation for cross-polytopes, we obtain Therefore, the conclusions we draw above for the order polytope O(P k ) all hold for the chain polytope C(P k ).
t≥0 i(O(P k ), t)z t = (1 + z) k (1 − z) 2k+1 . Thus, the h * -polynomial of O(P k ) is (1 + z) k .
It turns out that the polytopes studied in §2.2.1 and §2.2.2 are "reflexive" polytopes, and the order polytopes studied in §2.2.3 are "Gorenstein" polytopes. These are not coincidences as we will discuss below.
Connection to reflexivity and Gorensteinness. An integral polytope P in R D is reflexive (up to lattice translation) if the origin is in the interior of P and its dual
P ∨ := {y ∈ (R D ) * : y, x ≤ 1 ∀x ∈ P } is also an integral polytope, where (R D ) * is the dual space of R D .
It follows from the Macdonald Reciprocity Theorem [64] that if an integral polytope P is reflexive, then the roots of i(P, t) are symmetrically distributed with respect to the line {z ∈ C : Re(z) = −1/2} in the complex plane. Bey, Henk and Wills show that the converse is true if we include polytopes that are unimodularly equivalent to reflexive polytopes [ Reflexive polytopes are special cases of a more general family of polytopes: Gorenstein polytopes. Recall that the codegree of P is defined to be codeg(P ) := dim(P ) + 1 − deg (h * P (z)) . It is (again) a consequence of the Macdonald Reciprocity Theorem [64] that codeg(P ) is the smallest positive integer s such that sP contains a lattice point in its interior (see, for example, [44]). A Gorenstein polytope is an integral polytope P of codegree s such that sP is a reflexive polytope. The work [99] of Stanley gives a nice characterization for the h * -polynomials of Gorenstein polytopes: a d-dimensional integral polytope P is a Gorenstein polytope if and only if its h * -polynomial is symmetric, Furthermore, if the above condition holds, the integer s is the codegree of P.
that is, if h * P (z) = k i=0 h * i z i with h * k = 0, then h * i = h * k−i for i = 0, 1, 2, . . . , k.
Proof. The conclusion of the lemma follows from the observation that a number t 0 is a root of i(P, t) if and only if t 0 /s is a root of i(sP, t).
Corollary 2.2.9. Suppose P is a d-dimensional polytope that is h * -unit-circle- rooted.
Then P is a Gorenstein polytope (up to unimodular transformation). Moreover, if the degree of the h * -polynomial h * P (z) is d, then P is reflexive. Proof. Let k be the degree of h * P (z). By Theorem 2.2.2, among all the roots of i(P, t), k of them have real parts −(1 + (d − k))/2, and the other (d − k) roots are −1, −2, . . . , −(d − k). As i(P, t) is a polynomial with real coefficients, the sum of roots of i(P, t) is the sum of the real parts of roots of i(P, t), which is
−(1 + (d − k))/2 · k + d−k i=1 (−i) = − 1 2 d(d − k + 1).
Then the conclusions follow from Lemma 2.2.8.
Therefore, when an integral polytope P is h * -unit-circle-rooted, we not only get Ehrhart positivity for P but can also conclude that P is a Gorenstein polytope of codegree d − k + 1, where k is the degree of the h * -polynomial of P.
Coefficients with combinatorial meanings.
2.3.1. Zonotopes. In this part, we introduce a special family of polytopes, zonotopes, whose Ehrhart coefficients can be described combinatorially. As a consequence, Ehrhart coefficients of a zonotope are not only positive but also positive integers.
The Minkowski sum of two polytopes (or sets) P and Q is
P + Q := {x + y : x ∈ P, y ∈ Q.}. Let v 1 , v 2 , . . . , v n ∈ Z D be a set of integer vectors. The zonotope Z(v 1 , v 2 , . . . , v n ) associated with this set of vectors is the Minkowski sum of intervals [0, v i ], where [0, v i ] is the line segment from the origin to v i . Hence, Z(v 1 , · · · , v n ) := n i=1 [0, v i ] = n i=1 c i v i : 0 ≤ c i ≤ 1 for i = 1, 2, . . . , n .
In [96, Theorem 2.2], Stanley gives a combinatorial description for the Ehrhart coefficients of zonotopes.
Theorem 2.3.1 (Stanley). The coefficient of t k in i(Z(v 1 , · · · , v n ), t) is equal to X h(X),
where X ranges over all linearly independent k-subsets of {v 1 , . . . , v n }, and h(X) is the greatest common divisor of all k × k minors of the matrix whose column vectors are elements of X.
The main ingredient for the proof of the above theorem is that Z(v 1 , . . . , v n ) can be written as a disjoint union of half open parallelepiped C X ranging over all linearly
independent subsets X = {v j1 , . . . , v j k } of {v 1 , . . . , v n }, where C X is generated by ǫ 1 v j1 , . . . , ǫ i v j k for certain choices of ǫ 1 , . . . , ǫ k ∈ {−1, 1}. (See [96, Lemma 2.1].)
The theorem then follows from the fact that the number of lattice points in the half open parallelepiped C X is the volume of C X , which can be calculated by h(X).
The simplest examples of zonotopes are unit cubes considered in §2.1.1. We may recover the Ehrhart polynomial of a unit cube using Theorem 2.3.1. However, a more interesting example is the regular permutohedron.
Example 2.3.2.
The regular permutohedron, denoted by Π d , is the convex hull of all permutations in S d+1 ; that is,
Π d := conv{(σ(1), σ(2), · · · , σ(d + 1)) ∈ R d+1 : σ ∈ S d+1 }. It is straightforward to check that Π d is a translation of the zonotope 1≤i<j≤d+1 [0, e j − e i ].
For
any subset X of Φ d := {e j −e i : 1 ≤ i < j ≤ d+1}, we let G X be the graph on vertex set [d + 1] and {i, j} (with i < j) is an edge if and only if e j − e i ∈ X.
Then it follows from matroid theory that X is linearly independent if and only if G X is a forest on [d + 1]. (Recall that a forest is a collection of trees, or equivalently, is acyclic.) Furthermore, if X is linearly independent, then G X is a forest of d+1−|X| trees, and h(X) (described in Theorem 2.3.1) is 1.
Therefore, we conclude that the coefficient of t k in i(Π d , t) counts the number of forests on [d + 1] that contain exactly d + 1 − k trees. Therefore, we can compute, for example, i(Π 3 , t) = 16t 3 + 15t 2 + 6t + 1.
2.3.2.
Positivity of a generalized Ehrhart polynomial. The polynomial we discuss in this part is not exactly an Ehrhart polynomial. However, it is closely related to the result on zonotopes we have presented in the last part, and thus is included here. Galashin, Hopkins, McConville and Postnikov, in their study of root system chip firing [38], considered the following lattice points counting problem: Given an integral polytope P and a set of integer vectors v 1 , v 2 , . . . , v n , describe the number of lattice point in Then there exists a polynomial L(t) = L(t 1 , . . . , t n ) in n variables with nonnegative integer coefficients such that |(P + tZ) ∩ Z D | = L(t).
P + v 1 + v 2 + · · · + v n = P + Z(v 1 , . . . , v n ).
In particular, if we take t = (t, t, . . . , t), then
|(P + tZ) ∩ Z D | = |(P + tZ) ∩ Z D | = L(t, t, . . . , t)
is a polynomial in t of degree dim(Z) with positive integer coefficients.
Note that the second part of the above theorem was not explicitly stated in [38, Theorem 16.1]; but it was a consequence of the techniques used in its proof.
One sees that if we choose P to be the origin, then the above theorem recovers the Ehrhart positivity of zonotopes. However, in contrast with Stanley's results, no explicit formulas were given in [38] for the positive/nonnegative integer coefficients asserted in Theorem 2.3.3. Recently, Hopkins and Postnikov [49] analyzed techniques used in [38] further, and provided the desired explicit formula, completing the generalization of Theorem 2.3.1.
Theorem 2.3.4 (Hopkins-Postnikov). The homogeneous degree k part of the polynomial L(t) assumed by Theorem 2.3.3 is given by
X | quot X (P ) ∩ quot X (Z D )| · h(X) · vi∈X t i ,
where X ranges over all linearly independent k-subsets of {v 1 , . . . , v n }, quot X :
R D → R D /span R (X)
is the canonical quotient map, and h(X) is the greatest common divisor of all k × k minors of the matrix whose column vectors are elements of X.
2.4.
Higher integrality conditions. In this part, we will introduce families of polytopes whose Ehrhart coefficients are always volumes of certain projections of the original polytopes and are hence positive.
2.4.1. Cyclic polytopes. We start with a well-known family of polytopes: cyclic polytopes: The moment curve in R d is defined by
ν d : R → R d , x → ν d (u) = (u, u 2 , . . . , u d ).
Let U = {u 1 , . . . , u n } < be a linear ordered set. Then the cyclic polytope C d (U ) = C d (u 1 , . . . , u n ) is the convex hull of n > d distinct points ν d (t i ), 1 ≤ i ≤ n, on the moment curve:
C d (U ) := conv{ν d (u 1 ), ν d (t 2 ), . . . , ν d (u n )}.
Cyclic polytopes form an interesting family of polytopes. For instance, its facets are determined by the Gale evenness condition [108, Theorem 0.7], and the number
of i-dimensional faces of C d (U ) (where |U | = n)
is the upper bound for the number of i-dimensional faces of all d-dimensional polytopes with n vertices [66].
The following theorem on the Ehrhart polynomial of integral cyclic polytopes was initially conjectured in [8] by Beck, De Loera, Develin, Pfeifle and Stanley, and then proved in [56] by the author.
Theorem 2.4.1 (L.). For any d-dimensional integral cyclic polytope P = C d (U ) ⊂ R d , we have that (2.6) i(P, t) = Vol d (P )t d + i(π(P ), t) = d k=0 Vol k π (d−k) (P ) t k , where π (d−k) : R d → R k is the map that ignores the last d − k coordinates of a point, and Vol k (Q) is the volume of Q in the k-dimensional space R k .
The first step of the proof is to reduce the problem to simplices by using triangulations. For the simplex case, we consider the set obtained by removing the lower envelope of C d (U ) (with |U | = d + 1), and we decompose this set into d! signed (convex) half-open sets S σ , each of which corresponds to a permutation σ in the symmetric group S d . One important feature of this decomposition is that the number of lattice points in each piece S σ can be expressed in a simple formula involving the permutation σ, which makes it possible to compute the summation of all d! terms.
2.4.2. k-integral polytopes. Since the work in [56], the author generalized the family of integral cyclic polytopes to a bigger family of integral polytopes, "latticeface polytopes", and showed that their Ehrhart polynomials are also in the simple form of (2.6) [57, 58]. Later in [59], the author improved her results by introducing a notion of "higher integrality", which we will detail below.
Recall that a lattice point is also called an integral point. A point can be considered as a 0-dimensional affine space. We first extend this concept of integrality to higher dimensional affine spaces:
An ℓ-dimensional affine space W in R d is integral if π (d−ℓ) (W ∩ Z d ) = Z ℓ .
Note that this definition with ℓ = 0 is consistent with the original definition of an integral point. Note that in the above example, even though L 1 is not integral, after the projection, we still get a 1-dimensional lattice, which has the same dimension as L 1 .
L 1 , we have π (2−1) (L 1 ∩ Z 2 ) ∼ = Z/4Z. For the vertical red line, say L 2 , we have π (2−1) (L 2 ∩ Z 2 ) ∼ = Z 0 .
In this case, we say L 1 is in general position. On the contrary, L 2 is not in general position.
Definition 2.4.3. Suppose 0 ≤ k ≤ d. A d-dimensional polytope is k-integral P if
for any face F of P of dimension less than or equal to k, the affine hull aff(F ) of F is integral.
In particular, when k = d, we call P a fully integral polytope.
Remark 2.4.4. With the above definition, lattice-face polytopes, introduced in [57, 58], can be defined as polytopes that can be triangulated into fully integral simplices, which is a property any (integral) cyclic polytope has. Therefore, any cyclic polytope or lattice-polytope is fully integral.
The main result in [59] is a complete description for the Ehrhart coefficients of a k-integral polytope in terms of volumes of projections and Ehrhart polynomials of slices.
Definition 2.4.5. For any y ∈ π (d−k) (P ), we define the slice of P over y, denoted by π d−k (y, P ), to be the intersection of P with the inverse image of y under π (d−k) .
Recall that [t k ]f (t) denotes the coefficient of t k of a polynomial f (t).
Theorem 2.4.6 (L.). If P is a k-integral polytope, then
[t ℓ ]i(P, t) = Vol(π d−ℓ (P )) if 0 ≤ ℓ ≤ k, [t ℓ−k ] y∈π (d−k) (P )∩Z k i(π d−k (y, P ), t) if k + 1 ≤ ℓ ≤ d.
Therefore, if P is fully integral, the Ehrhart polynomial of P is in the form of (2.6), and thus P is Ehrhart positive.
Because both cyclic polytopes and lattice-face polytopes are fully integral polytopes, the above theorem generalizes results in [56,57,58].
The following is an example showing how to use Theorem 2.4.6 to obtain the Ehrhart polynomial of a 1-integral polytope. For the higher Ehrhart coefficients of P , we need to compute the Ehrhart polynomials of slices of P over lattice points in π (2) (P ) = [0, 4]. In the picture, the three shaded triangles are the slices of P over the lattice points 1, 2 and 3. The slices of P over lattice points 0 and 4 are the single points (0, 0, 0) and (4, 0, 0), respectively. We calculate the Ehrhart polynomials of all five slices, by summing which up we obtain 8t 2 + 10t + 5. Then the second part of Theorem 2.4.6 says that Therefore, i(P, t) = 8t 3 + 10t 2 + 4t + 1.
Recall that the face poset of a polytope P is the set of all faces of P ordered by inclusion. We say two polytopes have the same combinatorial type if they have the same face poset. As a byproduct of the study of Ehrhart polynomials of fullintegral polytopes, we can also show that Ehrhart positivity is independent from combinatorial types of polytopes [58].
Theorem 2.4.8 (L.). For any rational polytope P, there exists a polytope P ′ with the same face lattice such that P ′ satisfies the higher integrality condition and thus is Ehrhart positive.
Sketch of proof. First, by choosing appropriate bases for our underlying lattice Z d , we may assume that the affine hull of any face of P is in general position.
Next, for any s = (s 1 , . . . , s d ) ∈ Z d and x ∈ R d , we define
s ⋆ x = (s 1 x 1 , s 2 x 2 , . . . , s d x d ).
So s is an operator on R d that dilates points with different scalars at different coordinates. We observe that for any ℓ-dimensional affine space W ⊂ R d that is in general position, there exist (positive) integer scalars c 1 , . . . , c ℓ such that for any Since P has finitely many faces, we can apply the above operations inductively on dimensions of faces to obtain a full integral polytope P ′ that actually defined as s ⋆ P for some s ∈ Z d =0 .
s ∈ Z d =0 , if
Remark 2.4.9. There are a lot of properties of polytopes people study other than Ehrhart positivity, such as "normality", "integer decomposition property" (or IDP), "existence of a (regular) unimodular triangulation". For the majority of them, even if you start with a polytope P that does not have a certain property, dilating P with a large enough scalar often yields a polytope with the desired property (see, for example, [19,30,39]). Clearly, simple dilations won't change the answer to the Ehrhart positivity question for any polytope. After all, i(kP, t) = i(P, kt). Hence, the Ehrhart coefficients of a dilation of P have exactly the same sign pattern as Ehrhart coefficients of P. However, our proof of Theorem 2.4.8 says that dilating in different directions with different scalars can change a non-Ehrhart-positive polytope to a Ehrhart-positive one.
McMullen's formula and positivity of generalized permutohedra
The main purpose of this section is to study the Ehrhart positivity conjecture for generalized permutohedra. After reviewing previously known results supporting this conjecture, we introduce McMullen's formula, which is a formula for computing the number of lattice points inside polytopes. This provides us a way of attacking the question of Ehrhart positivity by reducing the problem to "α-positivity". We then discuss the author's joint work [26,24] with Castillo on the Ehrhart positivity conjecture of generalized permutohedra using this approach.
3.1. Motivation and evidence. In this part, we discuss the motivation for considering the Ehrhart positivity conjecture of generalized permutohedra and prior work by Postnikov which provides evidence for this conjecture. We start by formally defining generalized permutohedra, the main family of polytopes we study in this section. Whenever we talk about generalized permutohedra, we have D = d + 1.
3.1.1. Definition and first positivity conjecture. Given a strictly increasing sequence α = (α 1 , α 2 , · · · , α d+1 ) ∈ R d+1 , we define the usual permutohedron associated with α as Perm(α) := conv (α π(1) , α π(2) , · · · , α π(d+1) ) : π ∈ S d+1
In particular, if α = (1, 2, . . . , d + 1 Indeed, due to Postnikov's work, a big family of generalized permutohedra is already known to be Ehrhart positive, which provides a strong evidence to the above conjecture. We describe his work below. y I ∆ I as the Minkowski sum of the simplices ∆ I dilated by the factor y I . Postnikov shows that P Y d (y) is always a generalized permutohedron [82, Proposition 6.3]; however not every generalized permutohedron can be expressed as P Y d (y) for some y [82, Remark 6.4]. Therefore, we call P Y d (y) a type-Y generalized permutohedron. Postnikov then reformulates the construction of P Y d (y) using bipartite graphs: Let G be a subgraph of the bipartite graph K c,d+1 without isolated vertices. Label the vertices of G on the left by l 1 , l 2 , . . . , l c and vertices on the right by r 1 , r 2 , . . . , r d+1 . For each 1 ≤ j ≤ c, we let In [82, Section 11], Postnikov defines the "trimmed generalized permutohedron" as the "Minkowski difference" of P G (y 1 , . . . , y c ) and the simplex ∆ [d+1] . By providing a formula for the number of lattice points in a trimmed generalized permutohedron, he obtains a formula for the number of lattice points in P G (y 1 , . . . , y c ) [82,Theorem 11.3], which leads to an expression for the Ehrhart polynomial of P G (y 1 , . . . , y c ) as a summation over G-draconian sequences. , we have |I G j1 ∪ · · · ∪ I G j k | ≥ g j1 + · · · + g j k + 1. Theorem 3.1.4 (Postnikov). Suppose G is a subgraph of K c,d+1 without isolated vertices such that I G 1 = [d + 1]. Let y 1 , . . . , y c ∈ N. Then the Ehrhart polynomial of P G (y 1 , . . . , y c ) is given by
I G j = {i ∈ [d + 1] : {l j , r i } is an edge of G}.i(P G (y 1 , y 2 , . . . , y c ), t) = g y 1 t + 1 g 1 c k=2 y k t g k ,
where the summation is over all G-draconian sequences g = (g 1 , . . . , g c ).
Similar to the results discussed in §2.1.3 and §2.1.4, it follows from Lemma 2.1.1/(b) that the Ehrhart polynomial described in the above theorem has positive coefficients. Thus, by Remark 3.1.2, we immediately have the following: Note that as we pointed out above, type-Y generalized permutohedra do not contain all generalized permutohedra. Thus, Conjecture 3.1.1 does not follow from the above result.
Example 3.1.6 (Pitman-Stanley polytopes again). Let G be a subgraph of K d+1,d+1 where for each j ∈ [d+1], the left vertex l j is adjacent to right vertices r j , r j+1 , . . . , r d+1 . Then for any y = (y 1 , . . . , y d+1 ) ∈ N d+1 , P G (y) = P G (y 1 , . . . , y d+1 ) = It is easy to see the map π : R d+1 → R d that ignores the last coordinate of a point induces a unimodular transformation from P G (y) to the Pitman-Stanley polytope PS d ( y) considered in §2.1.3, where y = (y 1 , y 2 , . . . , y d ).
One can also check that the G-draconian sequences for the graph G given in this example are those g = (g 1 , . . . , g d+1 ) ∈ N d+1 satisfying d j=1 g j = d, g d+1 = 0, and k j=1 g j ≥ k for k = 1, 2, . . . , d − 1.
Hence, it can be verified that Theorem 2.1.2 is a special case of Theorem 3.1.4.
The family of type-Y generalized permutohedra not only includes the Pitman-Stanley polytope as we have seen in the example above, but also includes associahedra, cyclohedra, and more (see [82,Section 8]). However, it follows from work by Ardila, Benedetti and Doker that type-Y generalized permutohedra do not contain all matroid base polytopes [1, Proposition 2.3 and Example 2.6]. Therefore, Corollary 3.1.5 does not settle either Conjecture 3.1.1 or the Ehrhart positivity conjecture on matroid base polytopes by De Loera et al [31].
3.2.
McMullen's formula, α-positivity, and a reduction theorem. The goal of this part is to introduce McMullen's formula and discuss why it is a good tool to show Ehrhart positivity of a family of polytopes constructed from a fixed projective fan when an α-construction satisfies certain valuation properties. (We will discuss in §3.3 that generalized permutohedra form a family of polytopes constructed from the Braid fan. Hence, the techniques introduced here are relevant to our question.)
Throughout the rest of this section, we let V be a subspace of R D and V * be the dual space of V. For any polytope P , we use the notation lin(P ) to denote the linear space obtained by shifting the affine span aff(P ) to the origin.
Cones.
We need the concepts of cones, particularly feasible cones and normal cones, before we start our discussion.
A (polyhedral) cone is the set of all nonnegative linear combinations of a finite set of vectors. A cone is pointed if it does not contain a line. (ii) Given any face F of P , the normal cone of P at F with respect to V is ncone V (F, P ) := {u ∈ V * : u, p 1 ≥ u, p 2 , ∀p 1 ∈ F, ∀p 2 ∈ P } .
Therefore, ncone V (F, P ) is the collection of linear functionals u in V * such that u attains maximum value at F over all points in P.
The normal fan Σ V (P ) of P with respect to V is the collection of all normal cones of P .
Normal cones and pointed feasible cones are related by polarity. [24]). Suppose P is a polytope satisfying lin(P ) ⊆ V and F is a face of P. Then (ncone V (F, P )) • is a pointed cone, and is invariant under the choice of V . So we may omit the subscript V and just write (ncone(F, P )) • . Furthermore, ncone(F, P ) • = fcone p (F, P ).
3.2.2.
McMullen's formula and a refinement of positivity. In 1975 Danilov asked, in the context of toric varieties, whether it is possible to construct a function α such that for any integral polytope P , we have
(3.1) |P ∩ Z D | = F : a face of P α(F, P ) nvol(F ),
where α(F, P ) depends only on the normal cone of P at F, and nvol(F ) is the volume of F normalized to the lattice aff(F ) ∩ Z D .
McMullen [67] was the first to confirm the existence of (3.1) in a non-constructive way. Hence, we refer to the above formula as McMullen's formula. Pommersheim and Thomas [80] provide a canonical construction based on choices of flags. Berline and Vergne [11] give a construction in a computable way. Most recently, Ring-Schürmann [87] give another construction for McMullen's formula based on a choice of fundamental cells.
Before discussing a specific construction, even the existence of McMullen's formula has interesting consequences. In fact, it was one of the ingredients used in proving the results on higher integrality conditions discussed in §2.4.2. More importantly, it provides another proof for Ehrhart's theorem (Theorem 1.1) as well as a refinement of Ehrhart positivity. Note that pointed feasible cones do not change when we dilate a polytope. Thus, applying McMullen's formula to tP and rearranging coefficients, we obtain a formula for the function i(P, t) :
i(P, t) = dim P k=0 F :k-dimensional face of P α(P, F )nvol(F ) · t k .
Hence, i(P, t) is a polynomial in t of degree dim P, and the coefficient of t k in i(P, t) is given by Note that [t 0 ]i(P, t) is the constant term of the Ehrhart polynomial of P, which is known to be 1 for any integral polytope P. Furthermore, the normalized volume of any vertex is 1. Hence, the above equation becomes v:vertex of P α(P, v) = 1.
See Figure 2 for α-values of the vertices of the triangle P = conv((0, 0), (2, 0), (2, 1)) arising from different constructions. Berline-Vergne Since nvol(F ) is a positive number, it follows from (3.2) that α-values refine Ehrhart coefficients. We say a polytope P is α-positive for k-faces if α(F, P ) is positive for all k-dimensional faces F of P, and say P is α-positive if all α's associated to P are positive. The following result immediately follows from expression (3.2). Lemma 3.2.5. Suppose α is a solution to McMullen's formula. Let P be an integral polytope. For a fixed k, if P is α-positive for k-faces, then the coefficient of t k in the Ehrhart polynomial i(P, t) of P is positive.
Pommersheim-Thomas
Hence, if P is α-positive, then P is Ehrhart positive.
BV-construction and the Reduction Theorem.
At the first glance, αpositivity, being a refinement of Ehrhart-positivity, is a more difficult question to consider. However, for α-constructions that satisfy certain properties, studying α-positivity instead does not necessarily make the problem harder. Berline and Vergne [11] give such an α-construction, of which we give a quick review below. We refer to Berline-Vergne's construction of Ψ and α as BV-construction and BV-α-valuation, respectively. If α is the BV-α-valuation, we use the terminology BV-α-positivity instead of α-positivity.
Recall that the indicator function of a set
Properties (P1) and (P2) are the "certain valuation properties" we mentioned at the beginning of §3.2. The following Reduction Theorem lays out a consequence of these two properties.
Theorem 3.2.7 (Castillo-L., Reduction Theorem [24]). Suppose Ψ is a function from the set of indicator functions of rational cones C in V to R such that properties (P1) and (P2) hold, and suppose α is defined as in (3.3).
Let P and Q be two polytopes such that lin(P ) and lin(Q) are both subspaces of V. Assume the normal fan Σ V (P ) of P with respect to V is a refinement of the normal fan Σ V (Q) of Q with respect to V .
Then for any fixed k, if P is α-positive for k-faces, then Q is α-positive for k-faces.
One important implication of the Reduction Theorem is that we can reduce the problem of α-positivity of a family of polytopes constructed from a fan to the problem of α-positivity of a single polytope in the family. Definition 3.2.8. Let Σ be a projective fan in V * , i.e., a fan that is a normal fan of some polytope. Let Poly(Σ) be the set of polytopes Q whose normal fan Σ V (Q) with respect to V coarsens Σ.
Corollary 3.2.9. Assume the hypothesis on Ψ and α in Theorem 3.2.7. Let Σ be a projective fan in V * , and let P be a polytope such that Σ V (P ) = Σ. Then α-positivity (for k-faces) of P implies α-positivity (for k-faces) of Q for any Q ∈ Poly(Σ).
Assume further that α is a solution to McMullen's formula. Then for any integral polytope Q ∈ Poly(Σ), α-positivity for k-faces of P implies the coefficient of t k in i(Q, t) is positive. Hence, α-positivity of P implies Ehrhart-positivity of Q.
Proof. The first part follows directly from the Reduction Theorem, and the second assertion follows from the first part and Lemma 3.2.5.
Therefore, even though proving α-positivity is more difficult than proving Ehrhartpositivity for an individual polytope, it could be easier if we consider a family of polytopes Poly(Σ) associated to a fixed projective fan Σ, as we only need to prove α-positivity for one polytope in the family. Finally, because the BV-construction satisfies properties (P1), (P2) and (P4), all the results discussed above apply to the BV-construction or the BV-α-valuation. These ideas are illustrated by Example 3.3.3 below.
3.3. Positivity of generalized permutohedra. In this part, we apply the Reduction Theorem to reduce our first conjecture -Conjecture 3.1.1 -to a conjecture on α-positivity of regular permutohedra. Then we report partial progress made on both conjectures by using McMullen's formula with BV-α-valuation [26, 24].
3.3.1. Second positivity conjecture. Postnikov, Reiner, and Williams give several equivalent definitions for generalized permutohedra, one of which uses concepts of normal fans [83,Proposition 3.2]. Recall that the Braid fan, denoted by Br d , is the complete fan in R d+1 given by the hyperplanes x i − x j = 0 for all i = j. Proposition 3.3.1 (Postnikov-Reiner-Williams). A polytope P in V = R d+1 is a generalized permutohedron if and only if its normal fan Σ V (P ) with respect to V is refined by the Braid fan Br d .
Using the notation we give in Definition 3.2.8, the above result precisely says that the family of generalized permutohedra in R d+1 is Poly(Br d ). Furthermore, it follows from [82, Proposition 2.6] that any usual permutohedron in R d+1 has the Braid fan Br d as its normal fan. In particular, the normal fan of the regular permutohedron Π d is Br d . In [24], Castillo and the author use these results together with the Reduction theorem and its consequence (i.e., Corollary 3.2.9) to reduce Conjecture 3.1.1 to the following conjecture: The following example demonstrates how Corollary 3.2.9 works and why Conjecture 3.1.1 can be reduced to Conjecture 3.3.2. Figure 3. One notices that P is the regular permutohedron Π 2 whose normal fan is Br 2 , and each Q i is a generalized permutohedron whose normal fan coarsens Br 2 .
All the BV-α-values of the six vertices of P are 1/6. Since Q 1 has the same normal fan as P , all of its six vertices also have the same BV-α-values. Now the normal fan of Q 2 coarsens that of P. In particular, if we let v be the vertex on the bottom-left of Q 2 , then the normal cone ncone(v, Q 2 ) of Q 2 at v is the union of the normal cones of P at two of its vertices. It is a consequence of the "valuation properties" (P1) and (P2) that the BV-α-values α(v, Q 2 ) is the sum of the BV-α-values of these two vertices of P. Therefore, as shown in the figure, α(v, Q 2 ) = 1/6 + 1/6 = 1/3. One sees that similar phenomenon happens for the polytope Q 3 .
The above discussion shows that even if we did not know the BV-α-values of vertices of P, because each BV-α-value arising from Q i is a summation of a subset of BV-α-values of vertices of P, BV-α-positivity of vertices of the regular permutohedron P = Π 2 would imply BV-α-positivity of vertices of the generalized permutohedron Q i , and thus would imply the constant Ehrhart coefficient of Q i is positive. Conjecture 3.3.2 was the main conjecture studied in [24], and partial progress was made on proving it, which gave us corresponding partial results on Conjecture 3.1.1.
3.3.2.
Partial results. The first approach of attacking Conjecture 3.3.2 is to directly compute BV-α-valuations. In order to do that, we need to compute the BV-construction Ψ. One obvious benefit of considering Conjecture 3.3.2 instead of Conjecture 3.1.1 is that in each dimension there is only one regular permutohedron, and thus there are a limited number of BV-α-values or Ψ-values to be computed, especially for small d. Therefore, by explicit computation, we obtain the following result. Next, instead of focusing on all the coefficients of Ehrhart polynomials, we study certain special coefficients. Note that the first, second and last Ehrhart coefficients are always positive, so we only consider other Ehrhart coefficients. Correspondingly, we need to know how to compute the BV-construction Ψ(C) of cones C of dimension 2, 3, . . . , d − 1. The computation of the function Ψ is carried out recursively. Hence, it is quicker to compute Ψ for lower dimensional cones. As a result, the value of α(F, P ) is easier to compute if F is a higher dimensional face.
In general, the computation of Ψ(C) is quite complicated. However, when C is a unimodular cone computations are greatly simplified. In dimensions 2 and 3, with the help of Maple code provided by Berline and Vergne, simple closed expression for Ψ of unimodular cones can be obtained [24, Lemmas 3.9 and 3.10]. Applying these formulas to Π d , we obtain the second partial result towards Conjectures 3.3.2 and 3.1.1: Hence, the third and fourth Ehrhart coefficients of any integral generalized permutohedron (including matroid base polytopes) are positive.
Finally, the last partial result presented in [24] is the following:
E of Π d , we have α(E, Π d ) is positive, where α is the BV-α-valuation.
Hence, the linear Ehrhart coefficient of any integral generalized permutohedron (including matroid base polytopes) of dimension at most 500 is positive.
As we have discussed above, in order to compute the BV-α-values for an edge of a d-dimensional polytope, we have to compute the Ψ-value of a (d − 1)-dimensional cone, which is extremely difficult for large d if we use Berline-Vergne's algorithm directly. Therefore, we use a completely different strategy. Recall Property (P3) of the BV-construction, which says that Ψ is symmetric about coordinates. Note that the regular permutohedron Π d is a polytope with much symmetry. So a lot of BV-α-values of Π d coincide. In particular, we can separate edges of Π d into to d 2 groups, where edges in each group share the same BV-α-values.
Idea of Proof for Lemma 3.3.6. If we know the α-values for a give polytope P, Equation (3.2) gives us a way to compute the Ehrhart coefficients. However, we can also use (3.2) in the other direction: Suppose we know the linear coefficient of i(P, t), Equation (3.2) gives us an equation for α-values arising from edges of P :
E:edge of P α(E, P )nvol(E) = [t 1 ]i(P, t).
The α-values for the regular permutohedron also appear in other generalized permutohedra as all of them are in the family Poly(Br d ). Therefore, if we can find d 2 "independent" generalized permutohedra for which we know their linear Ehrhart coefficients, then we can set up a d 2 × d α-values arising from edges of Π d . Solving the system, we obtain all these α-values. See [24,Example 3.15] for an example of how we can solve a linear system to find α's.
Recall that Postnikov gives explicit formulas for the Ehrhart polynomials of type-Y generalized permutohedra (see Theorem 3.1.4). Among all the non-trivial Ehrhart coefficients, the linear terms can be easily described. Using these, we were able to set up, for each d, a desired linear system which is actually triangular.
Solving the system for d ≤ 500, we confirmed positivity of all d 2 α's arising from edges of Π d .
Equivalence Statements. In addition to the partial results discussed above, two equivalent statements to Conjecture 3.3.2 were discovered. The first states that
Negative Results
In this section, we will discuss examples and constructions of polytopes with negative Ehrhart coefficients. We start in §4.1 with the well-known Reeve tetrahedra, a family of 3-dimensional polytopes with negative linear coefficients. Constructions given in §4.2 were motivated by a refinement of Question 1.2, considering all possible sign patterns of Ehrhart coefficients. Examples studied in §4.3 and §4.4 provide negative answers to Question 1.2 for different families of polytopes (such as smooth polytopes and order polytopes), which will be summarized in §4.5. Finally in §4.6, we give negative examples addressing the question of whether Minkowski summation preserves Ehrhart positivity.
As mentioned before, due to the fact that the first, second, and last Ehrhart coefficients are always positive, given a d-dimensional polytope P, we need to ask the positivity question only for the coefficients of t d−2 , t d−3 , . . . , t 1 in i(P, t). We call these coefficients the middle Ehrhart coefficients of P.
i(T m , t) = m 6 t 3 + t 2 + 12 − m 6 t + 1.
One sees that the linear coefficient is 0 when m = 12 and is negative when m ≥ 13.
4.2.
Possible sign patterns. Motivated by the example of Reeve tetrahedra, Hibi, Higashitani, Tsuchiya and Yoshida study possible sign patterns of middle Ehrhart coefficients, and ask the following question:
Question 4.2.1 (Question 3.1 of [48]). Given a positive integer d ≥ 3 and integers 1 ≤ i 1 < · · · < i q ≤ d − 2, does there exist a d-dimensional integral polytope P such that the coefficients of t i1 , . . . , t iq of i(P, t) are negative, and the remaining coefficients are positive?
The following is the main result in [48] providing a partial answer to Question 4.2.1. The proof of both parts of the theorem is by construction. We will briefly discuss the construction for Theorem 4.2.2/(a), and refer interested readers to the original paper [48] for the other construction.
Sketch of proof for Theorem 4.2.2/ (a). Let L n := [0, n], which is a 1-dimensional polytope and its Ehrhart polynomial is i(L n , t) = nt + 1. Define the polytope P (d) m be the direct product of (d−3) copies of L d−3 and one copy of the Reeve tetrahedron
T m . Then P (d) m is a d-dimensional polytope with Ehrhart polynomial i P (d) m , t = i(L d−3 , t) d−3 ·i(T m , t) = ((d − 3)t + 1) d−3 · m 6 t 3 + t 2 + 12 − m 6 t + 1 .
The coefficients of the Ehrhart polynomial of P Figure 4 for an example.) Using inclusion-exclusion and the fact that the BV-α-values of cubes and standard simplices can be obtained easily due to property (P3), we obtain explicit formulas for all BV-α-values arising from P d (a, b), which we use to search for negative BV-α-values. The first negative values appear at d = 7, suggesting that we might have a negative Ehrhart coefficient in Q 7 (a, b). By direct computation, we are able to show that for some choices of (a, b), e.g., (5,2), the polytope Q d (a, b) has a negative linear Ehrhart coefficient for any d ≥ 7. Therefore, we have the following result [27, Proposition 1.3]: Q d (a, b) is not only a smooth polytope, but also a "type-B generalized permutohedron". The generalized permutohedra considered in Section 3 are of type A as the corresponding normal fan, Br d , is constructed from the type A root system. As a consequence, a polytope P is a (type-A) generalized permutohedron if and only if each edge direction of P is of the form of e i − e j for some i = j. Similarly, we can define a polytope P in R d is a type-B generalized permutohedron if each edge direction of P is in the form of e i ± e j for some i = j or of the form ±e i for some i. It is then straightforward to verify that Q d (a, b) is a type-B generalized permutohedron.
Using the idea of iterated chiseling cubes, we then improve the dimension range of our counterexamples from d ≥ 7 to d ≥ 3. (See [27, Section 2] for details.) Note that the above theorem is a stronger version than part (a) of Theorem 4.2.2. Even though the original purpose of the paper [27] was to answer Bruns' question, in the process of searching for a counterexample, we obtained a separate result answering a different question. For positive integers a > b, we let P d (a, b) be the polytope obtained by chiseling one vertex off a d at distance b. It is clear that P d (a, b) share the same BV-α-values with Q d (a, b). Hence, it has negative BV-αvalues at d ≥ 7. However, it turns out any d-dimensional integral polytope P that has the same normal fan as P d (a, b) is Ehrhart positive [27,Lemma 3.9]. Therefore, BV-α-positivity is strictly stronger than Ehrhart-positivity.
Finally, we studied a weaker version of Brun's question by requiring the smooth polytopes to be reflexive. More precisely, we asked whether all smooth reflexive polytopes have positive Ehrhart coefficients. Unfortunately, the answer to this question is still negative.
In fixed dimension d, there are only finitely many reflexive polytopes up to unimodular transformations. Because of their correspondence to toric Fano manifolds, smooth reflexive polytopes were completely classified up to dimension 9 [76,62]. We used polymake [2] to verify that up to dimension 8 all of them are Ehrhart positive, but in dimension 9 the following counterexample came up [27]:
Example 4.3.5. Let P be the polytope in R 9 defined by 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1 −1 0 0 0 0 4 0 0 0 0 −1 −1 −1 −1 −4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 ≤ 1 1 1 1 1 1 1 1 1 i(O(Q k ), t) = t+1 i=1 i k .
One can compute that
[t 1 ]i(O(Q 20 ), t) = −168011/330 < 0.
Hence, the linear Ehrhart coefficient of O(Q k ) is negative when k = 20.
Based on Stanley's example, the author and Tsuchiya studied the Ehrhart positivity question on polytopes O(Q k ) for any k, and gave a complete description of which Ehrhart coefficients of O(Q k ) are negative [61]. The following theorem is an immediate consequence to this description. Stanley's example and its extension are very interesting as O(Q k ) belongs to a lot of different families of polytopes. First of all, it is an order polytope, and thus is a (0, 1)-polytope.
Recall that a Gorenstein polytope of codegree s is an integral polytope such that sP is reflexive. It follows from a result by Hibi [42] that an order polytope is Gorenstein if and only if the underlying poset is pure, i.e., all maximal chains have the same length. Clearly, Q k is pure. Thus, O(Q k ) is a Gorenstein polytope.
Finally, Mészáros, Morales and Striker proved a result observed by Postnikov establishing a connection between flow polytopes of planar graphs and order polytopes [72,Theorem 3.8]. Using this connection, Morales (private communication) observes that the order polytope O(Q k ) is unimodularly equivalent to the flow polytope F G (1, 0, . . . , 0, −1), where G is the black graph on k + 1 vertices in Figure 5. (Note the red part of the figure is Q k .) (1,19,19,20).
It is easy to see that both P and Q are Ehrhart positive. However, one can check that P + Q is a 4-dimensional polytope with Ehrhart polynomial i(P + Q, t) = 10/3t 4 + 7/6t 3 − 1/3t 2 + 17/6t + 1, which has a negative quadratic coefficient. Since P is unimodularly equivalent to the standard 5-simplex, it is Ehrhart positive. Moreover, the Ehrhart polynomial of Q is i(Q, t) = 1/8t 5 + 5/12t 4 + 17/24t 3 + 19/12t 2 + 13/6t + 1, which also has positive coefficients. However, P + Q is a 5-dimensional polytope with Ehrhart polynomial i(P + Q, t) = 3007/40t 5 + 359/24t 4 − 255/24t 3 + 193/24t 2 + 89/20t + 1, which has a negative coefficient.
Notice that the polytopes given in Example 4.6.1 satisfy dim(P ) + dim(Q) = dim(P + Q), and those in Example 4.6.2 satisfy dim(P ) = dim(Q) = dim(P + Q). These are the two extreme situations in terms of dimensions. Therefore, even with some restrictions on the dimensions of P , Q and P + Q, the answer to the question above is false. Note that if r = 1, we obtain a polytope that is unimodularly equivalent to the standard d-simplex. The family of base-r d-simplices are introduced by Solus in his study of simplices for numeral systems [91], in which he shows that the h *polynomial of B (r,d) is real-rooted and thus is unimodal. Based on computational evidence, Solus makes the following conjecture [91, Section 5]:
Further discussion
Conjecture 5.1.1 (Solus). The base-r d-simplex is Ehrhart positive.
We remark that this family of ∆ (1,q) is very different from the ones constructed by Payne discussed in Example 2.2.5. If r > 1, the base-r simplex B (r,d) always contains the origin as an interior point, and it follows from (1.2) that the degree of h * -polynomial of B (r,d) is d. Since B (r,d) is not reflexive, by Corollary 2.2.9 the roots of its h * -polynomial are not all on the unit circle in the complex plane. Therefore, the techniques used to prove Ehrhart positivity for Payne's construction would not work here. 5.1.2. Birkhoff polytopes. The Birkhoff polytope B n is the convex polytope of n × n doubly-stochastic matrices; that is, the set of real nonnegative matrices with all row and column sums equal to one. Equivalently, B n can also be defined as the convex hull of all n × n permutation matrices. (See [106, Chapters 5 and 6] for a detailed introduction to B n .) There has been a lot of research on computing the volumes and Ehrhart polynomials of Birkhoff polytopes [9,21,32,77] By checking the available data [9], the first nine i(B n , t) have the property that all the roots have negative real parts. More importantly, Figure 6 in [8] suggests that the roots of i(B n , t) form a certain pattern. Hence, it could be promising to use Lemma 2.2.1 to attack this conjecture.
We also remark that B n is a Gorenstein polytope (up to lattice translation) of codegree n. However, with aforementioned data, one can see that B n is not h *unit-circle-rooted. Hence, we cannot apply Theorem 2.2.2 to show that all roots of i(B n , t) have negative real parts. (m k,k + m k,k+1 + · · · + m k,n ) − (m 1,k + m 2,k + · · · + m k−1,k ).
For each a = (a 1 , . . . , a n ) ∈ N n , Mészáros, Morales, and Rhoades [71] define the Tesler polytope, denoted by Tes n (a), to be the set of all n × n upper triangular matrices M with nonnegative entries and of hook sum a, i.e., the k th hook sum of M is a k . The lattice points in Tes n (a) are exactly Tesler matrices of hook sum a. When a = (1, 1, . . . , 1), these are important objects in Haglund's work on diagonal harmonics [40]. Therefore, we call Tes n (1, 1, . . . , 1) the Tesler matrix polytope as Tesler matrices of hook sum (1, 1, . . . , 1) were the original Tesler matrices defined by Haglund.
Another interesting example of a Tesler polytope is Tes n (1, 0, . . . , 0), which turns out to be the Chan-Robbins-Yuen polytope or CRY polytope, a face of the Birkhoff polytope. The CRY polytope, denote by CRY n , is the convex hull of all the n × n permutation matrices M = (m i,j ) such that m i,j = 0 if i ≥ j + 2, i.e., all entries below the sub-diagonal are zeros. It was initially introduced by Chan, Robbins and Yuen in [28], in which they made an intriguing conjecture on a formula for the volume of CRY n as a product of Catalan numbers. It was since proved by Zeilberger [107], Baldoni-Vergne [4], and Mészáros [68,69].
Using the Maple code provided by Baldoni, Beck, Cochet and Vergne [3], Morales computed the Ehrhart polynomials of both CRY polytopes and Tesler matrix polytopes for small n, and made the following conjecture [73]. We call the coefficients c ν λ,µ in the above expression the Littlewood-Richardson coefficients or LR coefficients. There are many different ways of computing c ν λ,µ . For example, it counts the number of semistandard Young tableaux T of shape ν/λ with content µ such that the reading word of T satisfies the "Yamanouchi word condition" [65]. One immediate consequence of these descriptions is that the LR coefficients are nonnegative integers. In this article, we use the hive model [20,53,54] to describe the LR coefficients.
A hive of size n is a triangular array of numbers a i,j with 0 ≤ i, j, i + j ≤ n arranged on a triangular grid consisting of n 2 small equilateral triangles. See the left side of Figure 6 for how a hive of size 4 should look like. For any adjacent triangles {a, b, c} and {b, c, d} in the hive, they form a rhombus {a, b, c, d}. The hive condition for this rhombus is (HC) b + c ≥ a + d.
Suppose |ν| = |λ| + |µ| with l(ν), l(λ), l(µ) ≤ n. A Littlewood-Richardson-hive or LR-hive of type (ν, λ, µ) is a hive {a i,j ∈ N : 0 ≤ i, j, i + j ≤ n} with nonnegative integer entries satisfying the hive condition (HC) for all of its rhombi, with border entries determined by partitions ν, λ, µ in the following way: a 0,0 = 0 and for each j = 1, 2, . . . , n, a j,0 − a j−1,0 = ν j , a 0,j − a 0,j−1 = λ j , a j,n−j − a j−1,n−j+1 = µ k .
With this definition, the LR-coefficient c ν λ,µ counts the number of LR-hives of type (ν, λ, µ). (Note that this is independent from n as long as l(ν), l(λ), l(µ) ≤ n.)
For example, if ν = (4, 3, 1), λ = (3, 2) and µ = (2, 1), then the border of a corresponding LR-hive of size 4 is shown on the right side of Figure 6. In fact, the hive condition will force a 2,1 = 8 and a 1,2 = 7. So it will be reduced to a hive of size 3. Finally, it follows from the hive condition that 6 ≤ a 1,1 ≤ 7. Thus, we have two LR-hives of this type, and we conclude that c (4,3,1) (3,2),(2,1) = 2. From the above description, it is not hard to see that c ν λ,µ counts the number of lattice points inside a polytope P ν λ,µ determined by the border condition and the hive condition. Furthermore, for any positive integer t, the LR-coefficient c tν tλ,tµ counts the number of lattice points inside the t th dilation of P ν λ,µ : c tν tλ,tµ = |tP ν λ,µ ∩ Z D |. King, Tollu and Toumazet studied c tν tλ,tµ , which they call the stretched Littlewood-Richardson coefficients, and made the following conjecture [ One notices that if P ν λ,µ is an integral polytope, then the polynomiality part of the above conjecture follows from Ehrhart's theorem. However, in general, P ν λ,µ is a rational polytope, which only implies that c tν tλ,tµ is a quasi-polynomial with some period. Nevertheless, the assertion of polynomiality in the above conjecture was established first by Derksen and Weyman [33] using semi-invariants of quivers, and then by Rassart [86] using Steinberg's formula [104] and hyperplane arrangements. Hence, the polynomial asserted in Conjecture 5.1.4 can be considered to be an Ehrhart polynomial, and positivity assertion in the conjecture (which is still open) is exactly an Ehrhart-positivity question.
Other questions. Many questions related to Ehrhart positivity remain open.
We include a few below. 5.2.1. Modified Bruns question. As we have discussed in §4.3, the answer to Bruns' question of whether all smooth polytopes are Ehrhart positive is negative, where counterexamples are constructed for each dimension d ≥ 3. Furthermore, we verify, with the help of polymake [2], that all smooth reflexive polytopes of dimension up to 8 are Ehrhart positive, and that there exists a non-Ehrhart-positive smooth reflexive polytopes of dimension 9. However, we did not investigate smooth reflexive polytopes of higher dimensions. Therefore, one can ask: Bruns' question can be rephrased using the language of fans: For any smooth projective fan Σ, is it true that any polytope with normal fan Σ is Ehrhart positive. Since the answer is false, a weaker version of this question can be asked: Question 5.2.2. Is it true that for any smooth projective fan Σ, there exists one integral polytope P with normal fan Σ that is Ehrhart positive? 5.2.2. h * -vector for 3-dimensional polytopes. Here instead of studying Ehrhart positivity question for families of polytopes in which dimensions vary, we focus on polytopes with a fixed dimension, and ask the following question: Since integral polytopes of dimension at most 2 are always Ehrhart positive, dimension 3 is a natural starting point.
We have mentioned in the introduction that various inequalities for h * -vectors have been found. So we may use Formula (1.1) which gives a connection between the h * -vector and Ehrhart coefficients together with known inequalities to study Question 5.2.3. Note that in dimension 3, only the linear Ehrhart coefficient could be negative. Applying (1.1), we obtain that P is Ehrhart positive (equivalently the linear Ehrhart coefficient of P is positive) if and only if the h * -vector (h * 0 , h * 1 , h * 2 , h * 3 ) of P satisfies (5.1) 11h * 0 + 2h * 1 − h * 2 + 2h * 3 > 0. In [6], Balletti and Kasprzyk give classifications for 3-dimensional polytopes with 1 or 2 interior lattice points, using which they extract all possible h * -vectors. Assume the number of interior lattice points is fixed to be 1 or 2. Applying (1.2), we obtain h * 0 = 1 and h * 3 = 1 or 2. Hence, only h * 1 and h * 2 change. Balletti and Kasprzyk then plot all occurring pairs of (h * 1 , h * 2 ) in [6, Figure 5]. The black part of Figure 7 is their figure, which we modify to include a red line representing the inequality (5.1), where points below the red line arise from polytopes with the Ehrhart positivity property. Note that in each part of the figure, the big triangular area is bounded by three known inequalities for h * -vectors. It is clear from the figure that these inequalities are far from optimal. Comparing the red line with the plotted data, one sees that a very high percentage of data points correspond to Ehrhart positive polytopes. However, if we only look at the triangular region (without the data points), then the area below the red line has a much lower percentage of the region. Therefore, improving the inequality bounds for h * -vectors will be helpful in understanding the Ehrhart positivity problem, in particular, in giving a more accurate answer to Question 5.2.3. . The plot of (h * 1 , h * 2 ) of 3-dimensional polytopes with 1 or 2 interior lattice points.
Question 1 . 2 .
12Which families of integral polytopes have Ehrhart positivity?
Lemma 2.1.1. Suppose a polynomial f (t) is either (a) a product of linear polynomials with positive coefficients, or (b) a sum of products of linear polynomials with positive coefficients. Then f (t) has positive coefficients.
1.4 using residue computation. Most recently, Mészáros and Morales [70] recover Baldoni-Vergne's result by extending ideas of Postnikov and Stanley on the Elliott-MacMahon algorithm and polytopal subdivisions of flow polytopes. Formula (2.4) is useful in obtaining positivity results since
.
Let p(t) be a polynomial in t with real coefficients. If the real part Re(r) is negative for every root r of p(t), then all the coefficients of p(t) are positive.
q
= (q 1 , q 2 , . . . , q d ) = (1, 1, . . .
Remark 2.2.6. Stanley also defined a "chain order" polytope C(P) for each poset P [100, Definition 2.1], and showed that C(P) is unimodularly equivalent to O(P) [100, Theorem 3.2/(b)], from which it follows that i(C(P), t) = i(O(P), t).
12, Proposition 1.8]. Recently, Hegedüs, Higashitani and Kasprzyk, in their study of roots of Ehrhart polynomials of reflexive polytopes, give the following result [41, Lemma 1.2]. Lemma 2.2.7 (Hegedüs-Higashitani-Kasprzyk). A d-dimensional integral polytope P is reflexive (up to unimodular transformation) if and only if the summation of the d roots of i(P, t) equals to −d/2.
Extending
Stanley's idea of decomposing zonotopes into half open parallelepiped, they show [38, Proof of Theorem 16.1] that P + Z(v 1 , . . . , v n ) can be written as disjoint union of sets in the form of F + C X where F is an open face of P and C X is a half open parallelepiped determined by a linearly independent set X of {v 1 , . . . , v n }.Theorem 2.3.3 (Galashin-Hopkins-McConville-Postnikov).Suppose P is an integral polytope in R D and v 1 , . . . , v n ∈ Z D is a set of integer vectors. Let Z = Z(v 1 , . . . , v n ). For any t = (t 1 , . . . , t n ) ∈ N n , we define tZ = Z(t 1 v 1 , . . . , t n v n ).
Example 2.4.2 (lines in R 2 ). See the left side of Figure 1 for examples of 1dimensional affine space in R 2 . The black lines are integral while the red lines are not integral. For the slanted red line, say
Figure 1 .
1Examples of higher integrality conditions.
} ⊂ R 3 ,which is illustrated on the right side ofFigure 1. One checks that P is 1-integral.Clearly π (2) (P ) = [0, 4] and π (3) (P ) = 0. By the first part of Theorem 2.4.6, [t 1 ]i(P, t) = Vol 1 ([0, 4]) = 4, and [t 0 ]i(P, t) = Vol 0 (0) = 1.
[t 3
3]i(P, t) = 8 and [t 2 ]i(P, t) = 10.
c m s m divides s m+1 for each m ∈ {1, 2, . . . , ℓ}, then s ⋆ W := {s ⋆ w : w ∈ W } is integral. For example, for the slanted red line L 1 appeared in Example 2.4.2, one checks that whenever s = (s 1 , s 2 ) satisfies 4s 1 divides s 2 , the affine space s ⋆ L 1 is integral. Hence, we can choose c 1 = 4.
Conjecture 3.1.1 (Castillo-L.). All integral generalized permutohedra are Ehrhart positive.
3. 1 . 2 .
12Ehrhart positivity of type-Y generalized permutohedra. In [82], Postnikov considers Minkowski sums of dilated simplices: For any nonempty subset I ⊆ [d + 1], define the simplex ∆ I := conv{e i : i ∈ I}. Let y = (y I : ∅ = I ⊆ [d + 1]) ∈ (R ≥0 ) 2 d+1 −1 be a vector indexed by nonempty subsets of [d + 1] with nonnegative entries. We define the polytope P Y d (y) := ∅ =I⊆[d+1]
For any (y 1 , y 2 , . . . , y c ) ∈ (R ≥0 ) c , we define the polytope P G (y 1 , . . . , y c )
[82]). A sequence of nonnegative integers g = (g 1 , g 2 , . . . , g c ) is a G-draconian sequence if c j=1 g j = d and for any subset {j 1 , . . . , j k } ⊆ [c]
.
Any integral type-Y generalized permutohedron is Ehrhart positive.
∆ [j,d+1] , where [j, d + 1] = {j, j + 1, . . . , d + 1}. It follows from Proposition 6.3 of [82] that the inequality description of this polytope isPG(y) = x ∈ R d+1 : xi ≥
.
Suppose P is a polytope satisfying lin(P ) ⊆ V. (i) The feasible cone of P at F is: fcone(F, P ) := {u ∈ V : x + δu ∈ P for sufficiently small δ} , where x is any relative interior point of F. (It can be checked that the definition is independent of the choice of x.)The pointed feasible cone of P at F is fcone p (F, P ) = fcone(F, P )/lin(F ).
.
Let K ⊆ V * be a cone, and let W be the subspace of V * spanned by K. (So W * is a quotient space of V .) The polar cone of K is the coneK • = {y ∈ W * : x, y ≤ 0, ∀x ∈ K}.
F
:k-dimensional face of P α(P, F )nvol(F ).
Example 3 .2. 4 .
34Setting k = 0 in (3.2), we obtain [t 0 ]i(P, t) = v:vertex of P α(P, v)nvol(v).
Figure 2 .
2Different α-constructions.
A ⊆ V is the function [A] : V → R defined as [A](x) = 1 if x ∈ A, and [A](x) = 0 if x ∈ A. The algebra of rational cones, denoted by C(V ), is the vector space over Q spanned by the indicator functions [C] of all rational cones C ⊂ V. We consider C(V ) a subspace of the vector space of all functions on V. Hence, in general, the indicators [C] of rational cones do not form a basis of C(V ) since there are many relations among them.Theorem 3.2.6 (Berline-Vergne). There exists a function Ψ from the set of indicator functions [C] of rational cones C in V to R with the following properties: (P1) Ψ induces a valuation on the algebra of rational cones in V , i.e., Ψ induces a linear transformation from C(V ) to R. (P2) If a cone C contains a line, then Ψ([C]) = 0. (P3) Ψ is invariant under orthogonal unimodular transformation, thus, is symmetric about coordinates, that is, invariant under rearranging coordinates with signs. (P4) Setting
F, P ) := Ψ([fcone p (F, P )]), gives a solution to McMullen's formula.
Conjecture 3.3.2 (Castillo-L.). Every regular permutohedron Π d is BV-α-positive.
Figure 3 .
3Examples for Corollary 3.2.9.
.
Let P, Q 1 , Q 2 and Q 3 be the 2-dimensional polytopes together with their normal fans shown in
Theorem 3.3.4(Castillo-L.). For d ≤ 6, the regular permutohedron Π d is BVα-positive. Therefore, all the integral generalized permutohedra (including matroid base polytopes) of dimension at most 6 are Ehrhart positive.
Castillo-L.). For any d, and any face F of Π d of codimension 2 or 3, we have α(F, Π d ) is positive, where α is the BV-α-valuation.
Castillo-L.). For any d ≤ 500, and any edge
Conjecture 3.3.2 holds if and only if the mixed lattice point valuation on hypersimplices is positive [24, Corollary 5.6]. The second equivalent statement is in terms of Todd classes. The BV-construction gives one way to write the Todd class of the permutohedral variety in terms of the toric invariant cycles. We can show that if there is any way of writing such class as a positive combination of such cycles, then the BV-α-valuation is one of them. (See [25, Proposition 7.2] or [23].)
4. 1 .
1Reeve tetrahedra. For d ≤ 2, there are no middle Ehrhart coefficients. Hence, possible examples with negative Ehrhart coefficients can appear only in dimension 3 or higher. The first example comes in dimension 3 : The Reeve tetrahedron T m is the polytope with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0) and (1, 1, m), where m is a positive integer. Its Ehrhart polynomial is
Theorem 4.2.2 (Hibi-Higashitani-Tsuchiya-Yoshida). Let d ≥ 3. The following statements are true.(a) There exists an integral polytope P of dimension d such that all of its middle Ehrhart coefficients are negative. (b) For each 1 ≤ k ≤ d − 2, there exists an integral polytope P of dimension d such that [t k ]i(P, t) is negative and all the remaining Ehrhart coefficients are positive.
be explicitly described, from which one can show that all middle Ehrhart coefficients are negative for sufficiently large m. In addition to Theorem 4.2.2, Hibi et al also show that answer to Question 4.2.1 is affirmative for d ≤ 6 [48, Proposition 3.2]. Note that for d ≤ 6, there are at most 3 middle Ehrhart coefficients. Later, Tsuchiya (private communication) improved their result showing that any sign pattern with at most 3 negatives is possible for the middle Ehrhart coefficients. Unfortunately, it is not currently clear how to extend the techniques used to prove this result to attack the question of whether any sign pattern with 4 negatives can occur. So Question 4.2.1 is still wide open. 4.3. Smooth polytopes. A d-dimensional integral polytope P is called smooth (or Delzant ) if each vertex is contained in precisely d edges, and the primitive edge directions form a lattice basis of Z d . In [18, Question 7.1], Bruns asked whether all smooth integral polytopes are Ehrhart positive. In[27], Castillo, Nill, Paffenholz, and the author show the answer is false by presenting counterexamples in dimensions 3 and higher. The main ideas we used was chiseling cubes and searching for negative BV-α-values.
Figure 4 .
4From 3 2 to Q 2 (3, 1). The first set of examples we construct is as follows: For positive integers a > 2b, we let Q d (a, b) be the polytope obtained by chiseling all vertices of a d at distance b. (See
Proposition 4.3.1 (Castillo-L.-Nill-Paffenholz). Let N d be the normal fan of Q d (a, b).For d ≤ 6, any d-dimensional smooth integral polytope with normal fan N d is Ehrhart positive. For any d ≥ 7, there exists a d-dimensional smooth integral polytope with normal fan N d whose linear Ehrhart coefficient is negative.
Theorem 4.3.3 (Castillo-L.-Nill-Paffenholz). For any d ≥ 3, there exists a ddimensional smooth integral polytope P such that all of its middle Ehrhart coefficients are negative.
Corollary 4.3.4 (Castillo-L.-Nill-Paffenholz). For d ≥ 7, there exists a smooth projective fan Σ, such that its associated BV-α-values contains negative values, but any smooth integral polytope in Poly(Σ) is Ehrhart positive.
Theorem 4.4.2 (L.-Tsuchiya). The order polytope O(Q k ) (defined in Example 4.4.1) is Ehrhart positive if and only if k ≤ 19.
Figure 5 .
5Q k and its corresponding planar graph G. 4.5. Non-Ehrhart-positive families. For each of the families listed below, it is not true that all the integral polytopes in the family are Ehrhart positive. (i) Smooth polytopes. (ii) Type-B generalized permutohedra. (iii) (0, 1)-polytopes. (iv) Order polytopes. (v) Chain polytopes. (vi) Flow polytopes. (vii) Gorenstein polytopes. (viii) Reflexive polytopes. (ix) Smooth reflexive polytopes. Furthermore, non-Ehrhart-positive examples were constructed for family (i) for each dimension d ≥ 3, for family (ii) for each dimension d ≥ 7, and for families (iii), (iv), (v), (vi), (vii) and (viii) for each dimension d ≥ 21. Proof. The conclusion for (i) and (ii) follows from Theorem 4.3.3, Proposition 4.3.1 and Remark 4.3.2. Notice that the order polytope O(Q k ) considered in §4.4 has dimension k + 1. Then the conclusion for (iii), (iv), (vi), (vii) follows directly from discussion in §4.4. Next, (v) follows from (iv) and Remark 2.2.6, and (viii) follows from (vii), the connection between Gorenstein polytopes and reflexive polytopes and the fact that Ehrhart positivity is invariant under dilating operations. Finally, (ix) follows from Example 4.3.5. 4.6. Minkowski sums. Recall that in §3.1.2, we learned that the type-Y generalized permutohedra which are defined to be Minkowski sums of dilated standard simplices are Ehrhart positive. Noticing that standard simplices are Ehrhart positive (see §2.1.2), we asked the following question in the first version of this survey: Is it true that if two integral polytopes P and Q are Ehrhart positive, then their Minkowski sum P + Q is Ehrhart positive? Tsuchiya (private communication) constructed a few examples, which gave a negative answer to the above question, shortly after it was posted. Below are two of his examples.
5. 1 .
1Ehrhart positivity conjectures. We list several families of polytopes that are conjectured to be Ehrhart positive. 5.1.1. Base-r simplices. Recall the definition of ∆ (1,q) given in §2.2.2. For any positive integer r ∈ P, we let q r := (r − 1, (r − 1)r, (r − 1)r 2 , . . . , (r − 1)r d−1 ) ∈ P d , and then define the base-r d-simplex to be B (r,d) := ∆ (1,q r ) .
5. 1 . 3 .
13Tesler polytopes. For any n × n upper triangular matrix M = (m i,j ), the k th hook sum of M is the sum of all the elements on the k th row minus the sum of all the elements on the k th column excluding the diagonal entry:
Figure 6 .
6A hive of size 4.
Conjecture 5 .1. 3 (
53Morales). For each n, the CRY polytope CRY n = Tes n (1, 0, . . . , 0) and the Tesler matrix polytope Tes n (1, 1, . . . , 1) are both Ehrhart positive.Connection to flow polytopes. Mészáros et al show in [71, Lemma 1.2] that for any a ∈ N n , the Tesler polytope Tes n (a) is unimodularly equivalent to the flow polytope F Kn+1 (ā), where K n+1 is the complete graph on [n + 1] andā is defined as in (2.2). Therefore, Tesler polytopes are flow polytopes associated to complete graphs. Note that the complete graph does not satisfy the hypothesis of Corollary 2.1.5. So Conjecture 5.1.3 does not follow. 5.1.4. Stretched Littlewood-Richardson coefficients. The Schur functions s λ form a basis for the ring of symmetric functions. (See [101, Chaprter 7] for background on symmetric functions.) Therefore, the product of two Schur functions s λ and s µ can be uniquely expressed as s λ · s µ = ν:|ν|=|λ|+|µ| c ν λ,µ s ν .
King-Tollu-Toumazet). For all partitions λ, µ, ν such that c ν λ,µ > 0, there exists a polynomial f (t) = f ν λ,µ (t) in t such that f (0) = 1 and f (t) = c tν tλ,tµ for all positive integers t. Furthermore, all the coefficients of f (t) are positive.
Does there exist a smooth reflexive polytope of dimension d with negative Ehrhart coefficients, for any d ≥ 10?
For each d, how likely is an integral polytope Ehrhart positive?
Figure 7
7Figure 7. The plot of (h * 1 , h * 2 ) of 3-dimensional polytopes with 1 or 2 interior lattice points.
of lattice points in PS d (a), from which a formula for the Ehrhart polynomial of PS d (a) immediately follows. Recall x2, . . . , d
,
hence we call it a Pitman-Stanley polytope.
Pitman and Stanley gave an explicit formula [79, Formula (33)] for computing
the number y
= x+y−1
y
.
Theorem 2.1.2 (Pitman-Stanley). Let
Example 2.1.3. Let G PS d be the graph on [d + 1] with edge set{(i, i + 1), (i, d + 1) : i = 1, 2, . . . , d}.Baldoni and Vergne[4, Example 16] show that F G PS d (ā) is unimodularly equivalent to the Pitman-Stanley polytope PS d (a).
Roots with negative real parts. In this part, we show examples with Ehrhart positivity using the following lemma. We use Re(z) to denote the real part of a complex number z.Corollary 2.1.5. Assume the hypotheses of Theorem 2.1.4. Assume further that
for each vertex i ∈ [n] = {1, 2, . . . , n}, the indegree of i is either 0 or 1. Then the
flow polytope F G (ā) is Ehrhart positive.
We remark that the graph G PS
d
defined in Example 2.1.3 satisfies the hypothesis
of the above corollary. Hence, Ehrhart positivity of the Pitman-Stanley polytope
is a special case of Corollary 2.1.5.
2.2.
) .
)Applying Theorem 2.2.3, one can obtain
) .
)Both Example 2.2.4 and Example 2.2.5 are h * -unit-circle-rooted. Hence, they are Ehrhart positive. We remark that the Ehrhart positivity of ∆ (1,q) considered in Example 2.2.5 was first proved by the author and Solus [60, Theorem 3.2].
By Lemma 2.2.1 and Theorem 2.2.2, the order polytope O(P k ) is Ehrhart positive.
), we obtain the regular permutohedron Π d considered in Example 2.3.2. In[82], Postnikov defined generalized permutohedra to be polytopes that can be obtained from usual permutohedra by moving vertices while preserving all edge directions. (Note that in this definition, edges are allowed to degenerate, and hence vertices can collapse.)In[31], De Loera, Haws, and Koeppe study the Ehrhart polynomials of matroid base polytopes, and conjecture those all have positive coefficients. However, it turns out that every matroid base polytope is a generalized permutohedron[1, Section 2]. In[26,24], Castillo and the author generalize the conjecture of De Loera et al. to all integral generalized permutohedra:
that left vertices of G are indexed by nonempty subsets I of [d + 1], and the left vertex l I is adjacent to the right vertex r i if and only if i ∈ I.Remark 3.1.2. It is clear that P G (y 1 , y 2 , . . . , y c ) is the type-Y generalized permuto-
hedron P Y
d (y) where y I = j:Ij =I y j . Conversely, the type-Y generalized permuto-
hedron P Y
d (y) is the polytope P G (y) where G is the subgraph of K 2 d+1 −1,d+1 such
. The following conjecture was made by Stanley in a talk [97]: Conjecture 5.1.2 (Stanley). Birkhoff polytopes are Ehrhart positive.
Unimodular equivalence is sometimes called integral equivalence, e.g., in[79].
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[
"Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann",
"Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann"
]
| [
"Martin Hecht \nInstitute for Computational Physics\nUniversity of Stuttgart\nPfaffenwaldring 2770569StuttgartGermany\n",
"Jens Harting \nInstitute for Computational Physics\nUniversity of Stuttgart\nPfaffenwaldring 2770569StuttgartGermany\n\nDepartment of Applied Physics\nDen Dolech 25600Eindhoven, EindhovenTUThe Netherlands\n"
]
| [
"Institute for Computational Physics\nUniversity of Stuttgart\nPfaffenwaldring 2770569StuttgartGermany",
"Institute for Computational Physics\nUniversity of Stuttgart\nPfaffenwaldring 2770569StuttgartGermany",
"Department of Applied Physics\nDen Dolech 25600Eindhoven, EindhovenTUThe Netherlands"
]
| []
| On-site boundary conditions are often desired for lattice Boltzmann simulations of fluid flow in complex geometries such as porous media or microfluidic devices. The possibility to specify the exact position of the boundary, independent of other simulation parameters, simplifies the analysis of the system. For practical applications it should allow to freely specify the direction of the flux, and it should be straight forward to implement in three dimensions. Furthermore, especially for parallelized solvers it is of great advantage if the boundary condition can be applied locally, involving only information available on the current lattice site. We meet this need by describing in detail how to transfer the approach suggested by Zou and He [1] to a D3Q19 lattice. The boundary condition acts locally, is independent of the details of the relaxation process during collision and contains no artificial slip. In particular, the case of an on-site no-slip boundary condition is naturally included. We test the boundary condition in several setups and confirm that it is capable to accurately model the velocity field up to second order and does not contain any numerical slip. | 10.1088/1742-5468/2010/01/p01018 | [
"https://arxiv.org/pdf/0811.4593v7.pdf"
]
| 15,541,565 | 0811.4593 | f9d49a5fc2c37316dd4aa5e319fc5b657d8e0c82 |
Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann
26 Feb 2010 (Dated: February 26, 2010)
Martin Hecht
Institute for Computational Physics
University of Stuttgart
Pfaffenwaldring 2770569StuttgartGermany
Jens Harting
Institute for Computational Physics
University of Stuttgart
Pfaffenwaldring 2770569StuttgartGermany
Department of Applied Physics
Den Dolech 25600Eindhoven, EindhovenTUThe Netherlands
Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann
26 Feb 2010 (Dated: February 26, 2010)arXiv:0811.4593v5 [physics.flu-dyn]PACS numbers: 0270-c -Computational techniques; simulations 4711-j -Computational methods in fluid dynamics
On-site boundary conditions are often desired for lattice Boltzmann simulations of fluid flow in complex geometries such as porous media or microfluidic devices. The possibility to specify the exact position of the boundary, independent of other simulation parameters, simplifies the analysis of the system. For practical applications it should allow to freely specify the direction of the flux, and it should be straight forward to implement in three dimensions. Furthermore, especially for parallelized solvers it is of great advantage if the boundary condition can be applied locally, involving only information available on the current lattice site. We meet this need by describing in detail how to transfer the approach suggested by Zou and He [1] to a D3Q19 lattice. The boundary condition acts locally, is independent of the details of the relaxation process during collision and contains no artificial slip. In particular, the case of an on-site no-slip boundary condition is naturally included. We test the boundary condition in several setups and confirm that it is capable to accurately model the velocity field up to second order and does not contain any numerical slip.
I. INTRODUCTION
The lattice Boltzmann method (LBM) is a widely used method for the simulation of fluid flow [2]. It solves the Boltzmann equation on a discrete lattice and it has been proven that the Navier Stokes equations can be recovered [3,4]. The method has been successfully applied to the simulation of flow in porous media [5,6], colloidal suspensions [7][8][9][10], liquid-gas phase transitions and multicomponent flows [11][12][13], spinodal decomposition [14,15], and many more applications.
In spite of the wide range of applications of the LBM there is sitll little consensus on how to implement boundary conditions in the LBM. For some applications, especially for complex geometries in technical applications, rather simple approaches like on-site bounce back [16] rules are preferred [17], and on the other hand quite complex methods to implement exact boundary conditions have been proposed [18,19]. A promising approach of velocity boundary conditions by Zou and He [1] for 2D simulations has been generalized to 3D with the restriction to the inflow being perpendicular to the boundary plane by Kutay et al. [5]. However, to our knowledge, a generalization to 3D with variable inflow direction has not yet been presented. Apart from this restriction, in the terms used in Ref. [5] some of the prefactors have to be revised (the correct ones can be found in Ref. [20]), but for the application studied by Kutay et al. , the terms used in their work might be appropriate. However, especially if the influx direction is not aligned with the computational lattice, our slightly more general expressions have to be used. We follow the ideas of Zou and He [1] and derive flux boundary conditions with variable influx direction for a D3Q19-lattice [21] meaning that in three dimensions the velocity space contains 19 discrete vectors. Zou and He [1] have demonstrated a derivation for the D2Q9 model and shortly sketched the application to pressure boundaries in a D3Q15i model, where i stands for an incompressible model of equilibrium distribution functions [22]. Already Zou and He point out that besides the basic idea of applying a bounce back rule to the non-equilibrium part, a further modification is necessary to achieve the correct transversal momentum. A suitable choice for this correction depends on the lattice type. Zou and He give an expression for the D3Q15i lattice for the case of pressure boundaries. In the subsequent publications on D3Q19 lattices [5,20], which is one of the most commonly used lattice types nowadays, it is assumed that the flux direction is restricted to the direction normal to the boundary plane and that this symmetry is also reflected in the distribution functions on the boundary nodes. In our generalization we drop this restriction and consistently derive the transversal momentum corrections. We investigate the accuracy of this boundary condition and highlight the special case of on-site no-slip boundary conditions included in this approach by simply setting the velocity equal to zero. Examples for possible applications are microfluidic devices [23], i.e., microscopic channel structures which are specially designed to modify a given flow profile by the roughness or wettability of the walls or the geometry of the channels. One example for those structures are micromixers [24]. In general, if simulating such devices, it is not always possible to align all walls with the Cartesian planes. In these cases one needs a boundary condition which is capable to specify the velocity in an arbitrary direction, depending on the orientation of the channel to be simulated. One might also think of applications in porous media [5], where flow through discretized samples of stones are simulated. On the boundaries of the microscopic pores, a no-slip condition has to be applied. This is a special case of velocity boundary conditions with the velocity being zero. Since the channels cannot be aligned with the computational lattice, the question raises, how large the error introduced by the discretization is. The on-site velocity boundary conditions brought forward in the present paper can be used as a replacement of the bounce-back rule for the no-slip condition if the velocity is set to zero. In contrast to the usual bounce-back rule the position of the wall is independent of the BGK relaxation time. This fact is of great advantage when analyzing the permeability of a discretized sample. Although the assumption of the fluid velocity being zero on the boundaries does not hold for several cases in microfluidics, the no-slip condition is highly important for many cases. Therefore, a considerable effort of research has been spent to develop no-slip boundary conditions [2,19,25,26]. Some approaches turned out to contain an artificial slip length depending on various details of the simulation method, whereas other attempts involve non-local calculations like the evaluation of a velocity gradient to extrapolate the flow field beyond the boundary. In contrast to that, the boundary condition we propose is of local type and allows to specify the velocity on the node exactly with vanishing slip length. Further, the boundary condition is of great benefit for hybrid simulations, i.e., simulations in which two simulation methods are coupled to simulate fluid flow [27][28][29]. The main goal of such hybrid simulations is to save computing time. A computationally cheap method is applied to simulate the flow on a more coarse-grained level, whereas in some regions, where for example interactions on the atomistic level are relevant, a different simulation method comprising more details is applied. The two simulation methods are coupled for example by exchange of mass, momentum and energy between the two domains via their respective boundary conditions. One can think of different setups: an LB simulation can be embedded into a finite element based Navier Stokes solver and resolve one region in more detail. Another example case is that in an LB simulation one region is resolved even on the molecular level by means of a Molecular Dynamics simulation. Practically, in current hybrid simulations the coupling is implemented within an overlapping region [30] of the two simulations, but it would be a great advance if one could manage simply to couple two boundary conditions without any overlap being needed. The possibility to generally determine the velocity on a boundary node in an LB simulation is one step towards this goal. The remainder of this paper is structured as follows: in the following section we describe the simulation method in general and introduce our notation of the lattice vectors. Then, we shortly review different boundary conditions in the literature in Sec. III. After that, we derive and discuss the velocity boundary condition for the D3Q19 lattice in Sec. IV. We separately discuss the special case of the no-slip condition in Sec. V. Numerical results are presented and discussed in Sec. VI and finally, we draw a conclusion in the closing section of the current paper.
II. SIMULATION METHOD
The lattice Boltzmann method is a numerical method to solve the Boltzmann equation Eq. (1) on a discrete lattice. The Boltzmann equation describes the dynamics of a gas from a microscopic point of view: in a gas, particles, each with velocities v i , collide with a certain probability and exchange momentum among each other. For ideal collisions total momentum and energy are conserved in the collisions. The Boltzmann equation expresses how the probability f (x, v, t) of finding a particle with velocity v at a position x and at time t evolves with time:
v · ∇ x f + F · ∇ p f + ∂f ∂t =Ω(f ) ,(1)
where F denotes an external body force, ∇ x,p the gradient in position and momentum space, andΩ(f ) denotes the collision-operator. Bhatnagar, Gross, and Krook [31] proposed the so-called BGK dynamics, where the collision operatorΩ is chosen as a relaxation with a characteristic time τ to the equilibrium distribution f (eq) (v, ρ)
.
Ω(f ) = − 1 τ f − f (eq) .(2)
The equilibrium distribution function for athermal models depends on the local density ρ(x, t) and the velocity field v(x, t). The lattice Boltzmann method [32] discretizes the probability density f in space and time.
The discrete Boltzmann equation, which is solved by the LBM can be rigorously derived from the Boltzmann equation [33]. The discretization, and especially the analytic expression for the equilibrium distribution f (eq) depends on the lattice type. We use a D3Q19-lattice which is a very popular lattice type for 3D LB-simulations. On each lattice site 19 values f i (x, t) are stored, each of them assigned to a lattice vector c i . We use the notation that The geometry is shown in Fig. 1 [49]. The local density at a lattice point can be obtained by summing up all f i ,
ρ(x, t) = 19 i=1 f i (x, t) ,(4)
and the streaming velocity is given by
v(x, t) = 1 ρ(x, t) 19 i=1 f i (x, t)c i .(5)
We express all quantities in lattice units, i.e., time is measured in units of update intervals and length is measured in units of the lattice constant. For practical applications a suitable mapping to physical units based on a dimensional analysis has to be applied. In the lattice Boltzmann method two steps are performed in an alternating way:
1. The "streaming step": propagate each of the distribution functions f i to the next lattice site in the direction of its assigned lattice vector c i .
2. The "collision step": on each lattice site relax the probability functions f i towards the equilibrium value f (eq) i
(v, ρ). In BGK dynamics this is according to Eq. (2).
The equilibrium value f (eq) i is obtained by discretizing the Boltzmann distribution. Several expressions of different order have been proposed, where we use the popular form involving terms in the velocity up to the second order [1,[34][35][36][37]
f (eq) i (ρ, v) = w i ρ 1 + c i · v c 2 s + (c i · v) 2 2c 4 s − v 2 2c 2 s (6)
with the lattice speed of sound c s = 1 √ 3 for the D3Q19 lattice and the lattice weights
w i = 2 36 , i = 1 . . . 6 1 36 , i = 7 . . . 18 12 36 , i = 19(7)
The pressure p = c 2 s ρ turns out to be proportional to the density and the dynamic shear viscosity is given by [2,38]
η = c 2 s ρ τ − 1 2 .(8)
III. BOUNDARY CONDITIONS
On the boundary nodes, the distribution function assigned to vectors c i pointing out of the lattice move out of the computational domain in the propagation step, and the ones assigned to the opposing vectors are undetermined because there are no nodes which the distributions could come from. Therefore, on the boundary nodes, special rules have to be applied. These boundary conditions can be chosen in various manners. Periodic boundaries are realized by propagating the f i leaving the computational domain on the one boundary to the boundary nodes located on the opposite side of the domain. Closed boundaries are commonly implemented by a so-called mid-grid bounce-back rule [2], which means that the distributions f i pointing out of the domain are copied to f j , for which c j = −c i , i.e., locally, on each lattice site, the undetermined values are filled with the ones which would stream out of the domain without collision on the boundary node. They enter one time step later into the simulation domain again [39]. However, for many questions in fluid dynamics it is required to determine the pressure or the velocity field at the boundary. The first is known as Dirichlet boundary condition, and the latter as Neumann boundary condition. In the Neumann case the flux on the boundary of the domain is fixed, whereas in the Dirichlet case the pressure is given as a boundary condition. Zou and He [1] have proposed how to implement Dirichlet and Neumann boundary conditions on a D2Q9 lattice and shortly sketched how to apply it for a D3Q15i simulation. Kutay et al. [5] have transferred this proposal to a D3Q19 lattice. However, their approach is derived under the assumption that the in-and outflow velocity is always perpendicular to the boundary plane, and oriented along one of the main lattice directions (c i , i = 1 . . . 6).
We generalize this to inflow with arbitrary direction in Sec. IV.
Often more elaborated boundary conditions are applied. Chen and co-workers [37] and Ginzbourg and d'Humières [19] suggested extrapolation of the f i on the first and second layer of the lattice to the nodes outside the domain. These extrapolated values can be thought of as the lattice populations propagating into the domain and arriving on the boundary nodes in the next streaming step. Inamuro and co-workers have introduced a counter-slip to compensate a numerical slip which occurs when applying on-site bounce-back [25]. Skordos came up with an approach where additional differential equations are solved on the boundary nodes to calculate the unknown populations [18]. Ansumali and Karlin have developed a LB no-slip boundary condition from kinetic theory [40], and, more recently, d'Orazio et al. [41] and Tang et al. [35] came up with thermal boundary conditions which also involve an extrapolation scheme and bounce-back with counter-slip respectively. Ladd and Verberg have developed a boundary condition with a resolution of the position of the wall on a sub-grid level, which is especially required if suspended particles are modeled [9,42,43]. Schiller and Dünweg [44] use a reduced set of distribution functions on the boundary nodes. For their reduced D3Q19 model they derive equilibrium distributions and propose a multi relaxation time dynamics and a special collision operator on the boundary.
Latt et al. have compared and discussed several of these approaches in Ref. [45]. They also include the boundary condition by Zou and He [1] in their discussion. As indicated by Latt et al. a generalization of the boundary conditions proposed by Zou and He is still not provided. However, a general local boundary rule which can be applied in a simple way on each node separately, would be desirable. We derive such a boundary condition in the following section. Our generalization of the velocity boundary condition proposed in Ref. [1] only involves the distribution functions defined on the local boundary node and allows by very simple and computationally cheap steps to set the velocity on the node to a distinct vector. The desired value is obtained exactly and we cannot detect any artifacts like a numerical slip length or bends in the velocity profile.
IV. GENERAL ON-SITE VELOCITY BOUNDARY CONDITION
As mentioned in the previous section, we extend the boundary condition by Zou and He [1] to a D3Q19 lattice. We derive the boundary condition for the bottom plane (z = 0) in detail and give the results for the other planes in the appendix. They can be derived following the same steps. The boundary conditions are derived by using the set of equations consisting of Eq. (4) and the components of Eq. (5):
ρv x = f 1 + f 7 + f 8 + f 9 + f 10 −(f 2 + f 11 + f 12 + f 13 + f 14 ) , (9) ρv y = f 3 + f 7 + f 11 + f 15 + f 16 −(f 4 + f 8 + f 12 + f 17 + f 18 ) , (10) ρv z = f 5 + f 9 + f 13 + f 15 + f 17 −(f 6 + f 10 + f 14 + f 16 + f 18 ) .(11)
Due to the continuity relation ∂ρ ∂t + ∇ · (ρv) = 0, we are free to specify only three of the four variables (ρ and the three components of v) on the boundary. If we fix the tangential velocity v x , v y on the bottom-layer of the lattice, and the density to a given value ρ 0 , the z-component of the inflow velocity v z can be calculated from Eq. (11) and Eq. (4),
v z = 1 − 1 ρ 0 [f 1 + f 2 + f 3 + f 4 + f 7 + f 8 + f 11 + f 12 + f 19 (12) + 2(f 6 + f 10 + f 14 + f 16 + f 18 )] ,
where the f i pointing out of the system appear with a prefactor of 2, and all in-plane components appear with weight 1. The components pointing into the system, f 5 , f 9 , f 13 , f 15 , and f 17 , which are undetermined after the streaming step, do not appear at all. With Eq. (12) Neumann (or pressure) boundary conditions can be applied by specifying ρ 0 on the boundary and using Eq. (12) to calculate v z . If Eq. (12) is written in the form
ρ = 1 1 − v z [f 1 + f 2 + f 3 + f 4 + f 7 + f 8 + f 11 + f 12 + f 19 (13) + 2(f 6 + f 10 + f 14 + f 16 + f 18 )] ,
all three components of the velocity can be specified and Eq. (13) is used to calculate the density ρ. This is the Dirichlet case, or flux-boundary condition. Again, the undetermined populations f 5 , f 9 , f 13 , f 15 , and f 17 do not enter the calculation. We have used two out of four equations (Eqns. (9)- (11) and Eq. (4)), but we still have to compute the five f i pointing into the computing domain. Following Zou and He [1] we assume that on the boundary the bounce-back condition is still valid for the non-equilibrium part f * i of the single particle distribution f i :
f * i = f i − f (eq) i .(14)
The bounce-back condition in +z-direction (in normal direction to the boundary) would read as
f * 5 = f 5 − f (eq) 5 = f 6 − f (eq) 6 = f * 6 ,(15)
which leads by taking f (eq) 5
and f (eq) 6
from Eq. (14) to
f 5 = f 6 − w 6 ρ 1 − v z c 2 s + v 2 z 2c 4 s + w 5 ρ 1 + v z c 2 s + v 2 z 2c 4 s = f 6 + 2w 5 c 2 s ρv z = f 6 + 1 3 ρv z .(16)
Here we make use of the fact that the distribution functions in Eq. (6) are approximated by taking only terms up to 2 nd order in v into account. However, this approximation could be applied directly to Eq. (16) as well. For the derivation of the boundary condition it is needed, otherwise the higher order terms would introduce anisotropic effects in the boundary rule.
Generally, in the collision step (in Eq. (2)) higher order terms may be taken into account for the bulk, but for the boundary conditions a qualitatively different approach, like a higher order extrapolation scheme, has to be considered when aiming for higher order accuracy.
For the D3Q19 lattice, however, we need two more equations. To keep the symmetry of the problem, we assume bounce-back of the non-equilibrium part for all populations f i . This results in four equations,
f 9 = f 14 + 2w 9 c 2 s ρ(v z + v x ) ,(17)f 13 = f 10 + 2w 13 c 2 s ρ(v z − v x ) ,(18)f 15 = f 18 + 2w 15 c 2 s ρ(v z + v y ) ,(19)f 17 = f 16 + 2w 17 c 2 s ρ(v z − v y ) ,(20)
so that the system of equations is overdetermined.
Therefore, following the Ansatz by Zou and He [1] for the pressure boundary condition on a D3Q15i lattice, we introduce two new variables N z x and N z y , the transversal momentum corrections on the z-boundary for distributions propagating in x and y-direction, respectively. These terms turn out to vanish in equilibrium, but they are non-zero, if velocity gradients are present, e.g., when shear flow is imposed by the particular choice of the boundary conditions. It turns out that these expressions appear again in the stress tensor. They reflect the fact that by imposing a transversal velocity component on the boundary, also stress is imposed to the system. The transversal momentum corrections involve the populations propagating in the boundary plane in the update rule of the boundary condition. We add the terms to the right hand side and assume that the same expression with opposite sign is needed for two of the vectors in the same plane. Our Ansatz thus reads as follows:
f 9 = f 14 + ρ 6 (v z + v x ) − N z x ,(21)f 13 = f 10 + ρ 6 (v z − v x ) + N z x ,(22)f 15 = f 18 + ρ 6 (v z + v y ) − N z y ,(23)f 17 = f 16 + ρ 6 (v z − v y ) + N z y .(24)
The system of equations (21)-(24), together with Eq. (9) and Eq. (10) is now a closed system. By Eq. (9) and Eq. (10) we specify the tangential components of the velocity v x and v y , which do not need to be equal to zero in our approach. Inserting Eqns. (21)-(24) into Eq. (9) and Eq. (10), gives an exact solution for N z x , and N z y , respectively:
N z x = 1 2 [f 1 + f 7 + f 8 − (f 2 + f 11 + f 12 )] − 1 3 ρv x ,(25)N z y = 1 2 [f 3 + f 7 + f 11 − (f 4 + f 8 + f 12 )] − 1 3 ρv y(26)
These transversal momentum corrections can be inserted into Eqns. (21)-(24) again and together with Eq. (16) we find explicit expressions for all unknown populations. Note that in Eq. (25) and (26) it is required to sum over all in-plane contributions to the velocity in x-and ydirection and the weights are consistent with the lattice weights of the f i appearing in the above expressions. As expected, N z x and N z y vanish (up to 2 nd order which is our precision within this derivation), if we set all f i to their equilibrium value. The results for the other planes are given in the appendix. A general form for all boundary planes can be written down by introducing the normal vector on the boundary n, the tangential vectors t i = c i − (c i · n)n, and the notation f −i denoting the direction to which a population is bounced back c −i = −c i . From the populations f i assigned to a direction c i pointing into the wall, the new populations f −i in opposite direction can be calculated as in Eq. (6) no approximations are made. Therefore, we have derived a way to implement explicit local on-site boundary conditions which model the fluid field up to second order in the velocity. A similar scheme as ours has been proposed by Halliday et al. [46] for a D2Q9 lattice. These authors construct the unknown distributions locally on each lattice site starting from a Chapman-Enskoog analysis. During their derivation they have to choose a set of variables they consider as free variables. This is similar to the approach of introducing the transversal momentum corrections in order to be able to solve the system of equations. Halliday et al. find results for the unknown populations which involve the components of the strain rate tensor calculated from the known populations. From this point of view it might be possible to apply a scheme similar to the one proposed by Halliday et al. to a D3Q19 lattice. However, in three dimensions the systems of equations, in the generality of Ref. [46], might become difficult to handle. Special care has to be taken when connecting the inand outflux boundary conditions at the corners and edges of the simulation domain with other types of boundary conditions that are applied on other boundary planes. If no-slip boundary conditions are assumed on the x-and y-boundary, one has to take care that the influx velocity tends to zero at the edges. We discuss the special case of no-slip boundaries as a subset of velocity boundaries in the following section.
f −i = f i − ρ 6 c i ·v− ρ 3 t i ·v+ 1 2 19 j=1 f j (t i · c j ) (1 − |c j · n|) .(27)
V. ON-SITE NO-SLIP BOUNDARY CONDITION
The on-site velocity boundary condition proposed in this paper includes an important special case: setting the velocity v = 0 results in a no-slip-boundary for non-moving boundaries. Therefore, this boundary condition can also be used as a replacement of the mid-grid bounce back rule. However, even more generally, moving boundaries, e.g., moving shear plates, can be implemented by imposing the wall velocity v on the boundary nodes. The position of the wall is exactly on the lattice nodes. This is in contrast to most no-slip boundaries proposed in the literature, where the wall position is assumed at half the distance between two nodes. However, in many of those approaches the exact position of the wall depends on the BGK relaxation time. This is not the case for our approach. One of the pillars of the LBM is local mass conservation [45], which should be fulfilled not only in the bulk, but also on closed boundaries. However, some extrapolation schemes may be less accurate in this point [44], whereas for our on-site approach mass conservation is strictly fulfilled at the closed walls. The transversal momentum corrections similar to those given in Eq. (25) and Eq. (26) are given in the appendix for each coordinate plane, and both velocity components in each of those planes. They are corrections to the onsite bounce back rule. With these corrections taken into account, the velocity is exactly zero on the node. For edge nodes with both boundaries being implemented as described here, we suggest to first apply the bounce back rule for all f i pointing out of the computational domain and then to calculate the transversal momentum correction. On an edge node only one tangential vector along the edge can be used for ensuring no-slip. Consider for example the edge between the xy-plane and the yz-plane, where the y-axis forms the edge. Contributions in the boundary planes known after bounce back are f 1 , f 7 , and f 8 in the xy-plane and f 5 , f 15 , and f 17 in the yz-plane. However, to ensure no-slip one can define
N xz y = 1 4 [f 3 − f 4 ] .(28)
The correction to the distributions f i with i = 7, 8, 15, and 17 then is N xz y c i · t i which has to be added to the distributions. The prefactor in Eq. (28) takes into account that the remaining slip velocity after bounce back is distributed among four populations obtained from the bounce back rule. Similar expressions can be written down for each edge. A general expression for the modified bounce back rule is
f −i = f i − 1 4 19 j=1 f j (t i · c j ) 1 − c j · n (1) 1 − c j · n (2) ,(29)
where n (1) and n (2) denote the two normal vectors on the two boundary planes meeting at the edge under consideration. On the edges and corners, apart from the incoming populations, there are so-called "buried links" [26], i.e., lattice vectors c i for which the opposing lattice vector c −i points out of the domain, as well. The two lattice vectors c (1,2) = ± n (1) − n (2) make up the buried link on an edge node. In the following, lattice vectors with subscript, c i , denote distinct vectors as defined in Eq. (3), whereas vectors with superscript, c (i) , denote vectors which belong to the buried links, and which depend on the normal vectors on the individual boundary planes. The distribution functions assigned to the buried links have to be assigned separately. We choose them such that they contribute to the same density according to their lattice weight:
f (1,2) = 1 22 18 i=1 f i 1 − c i · n (1) × n (2)(30)· 1 − c i · n (1) − n (2) √ 2 .
Similarly, the distribution on the resting node is chosen as f 19 = w19 w7 f (1,2) = 12 f (1,2) . The weights are always determined by the number of f i which contribute to the sum and their respective lattice weights w i according to Eq. (7). In Eq. (30) six f i with lattice weight 1 18 and ten f i with weight 1 36 contribute, which makes up an overall contribution of 22 36 . To reduce this to the desired lattice weight, we have to divide by 22, and to obtain a value for the resting node, we multiply by 12, because of the twelve times larger lattice weight of the resting node. At the edges surrounding the in-and outlet planes, on the other hand, one needs either pressure or velocity boundary conditions, depending on the boundary type used for in-or outlet. For velocity boundaries one has to take care that the velocity profile decays to zero, so the noslip boundary condition just described can be used on all edges. For pressure boundaries one prescribes a density ρ = ρ 0 which we can be used to calculate the distribution assigned to the buried link:
f (1,2) = ρ 0 − 22f (1,2) 14 ,(31)
where f 19 is then calculated as f 19 = 12f (1,2) . According to Maier et al. [26] no-slip boundaries cannot be enforced on convex boundary nodes. However, slip along the edge can be reduced by correcting all distributions f i traveling into the interior of the system by
N i = 1 4 c i · n (1) × n (2) 19 j=1 f j c j · n (1) × n (2) ,(32)
which follows the same idea as Eq. (29) and Eqns. (21)-(24): momentum in a direction parallel to the surface, which would remain on a node after applying the boundary rule, is removed by modifying those populations that will afterwards propagate back into the bulk of the system. In principle, one could split Eq. (29) into two steps: first, apply bounce back for all populations leaving the system, and then correct the populations traveling away from the edge by the term given in Eq. (32). For convex edges these are the populations traveling into the bulk, and for concave edges they propagate in the boundary planes. This opens a possibility to implement all rules in a single procedure, for which the normal vector n is stored on each lattice site by an integer number. The vector is obtained from the matrix M defined in Eq. (3). For values between 1 and 6 Eq. (27) is applied, for values between 7 and 18 either Eq. (32) applies or depending on the values stored on the neighboring nodes, the bounce-back rules corrected according to Eq. (29) may be applied instead. This information, which expresses if the edge is concave or convex, can be obtained once, when the lattice is generated and may be stored in the sign of the lattice vector index. The normal vector points into the bulk and indicates the direction of the symmetry plane on the edge nodes. Finally, on the corner nodes one can define three normal vectors on the boundary planes meeting there, n (1) , n (2) , and n (3) . Similar to the buried links, there is a complete plane in which six vectors are located, that only couple to the simulation in the collision step. The buried vectors c (1...6) are the ones for which c (i) · n (1) + n (2) + n (3) = 0 . Since the normal vector on this plane is not contained in the set of vectors for the D3Q19 lattice additional indices are needed to mark the corner nodes. After bouncing back the known f i , the distributions assigned to buried vectors are set to (29) is not necessary on the corner nodes [50]. In complex geometries there are points in which edges (convex or concave ones) meet planes which are oriented perpendicular to the direction of the edge. There, we propose to use bounce back for those populations which would leave the computational domain and to assign an appropriate value to the resting node, similar as described for the corner nodes. There are no buried links, because those links are located inside the boundary plane which the edge connects to, so the resting node must be set to f 19 = 12 36 18 i=1 f i . In total there are 6 planes and 4 possible orientations of the edges, each of them either convex or concave, making up 48 more cases. However, since there are no buried links and a momentum correction is not necessary either, the rules can be implemented easily in only a few lines of code. In the following section we show the results of tests of the boundary condition in simple geometries like Poiseuille flow between two plates, where the exact solution is known. As an example for more complex geometries we simulate the flow through a rectangular channel, where also edges and corners are involved.
f (1...6) = 1 18 18 i=1 f i 1 − c i · n (1) + n (2) + n (3) √ 3(
VI. NUMERICAL RESULTS
We test our boundary condition by simulating a Poiseuille flow through a tilted channel. The size of the computational domain is 64 × 8 × 128 LB nodes, where the channel has a width of 20 nodes and is tilted by an angle α = arctan( 40 127 ) ≈ 17.48 • . This angle is chosen such that both ends of the channel intersect the xy-plane at the top and the bottom of the computational domain and that there are two lattice sites of wall at the left and at the right of the channel at the bottom and the top plane respectively. The flow through our test channel is simulated in three dimensions. However, for convenience, the y-direction is periodic.The walls (only in this test) are implemented as simple bounce-back nodes. Here we apply the boundary conditions derived in Sec. IV as in-and outflux conditions and compare them to other implementations.
We choose this simple test because the analytical solution for the flow field is known and so we can estimate the numerical error. Usually, one would avoid to have walls not aligned with the computational lattice because of the staircase like discretization of the walls, which brings an additional discretization error into the simulation. This discretization error can be avoided by simply aligning the channel with the computational lattice. However, if more complex structures, like, e.g., Y-channels for applications in microfluidics are simulated, it may happen that always at least one channel is not aligned with one of the Cartesian directions. A technical workaround, if appropriate boundary conditions are missing, is to simulate a very long channel so that in the first section of the channel, the fluid can relax to a steady flow profile, and only afterwards enters the actual simulation domain. However, this causes the computational effort to increase substantially. Knowing the width of the channel, the center of the inand outlet, and a given velocity v 0 on the center line, one can calculate a Poiseuille flow field inside the channel,
v P (x) = v 0 1 − x − x 0 − γ(z − z 0 ) ∆x 2 ,(34)
where x 0 and z 0 denote the center of the simulation space, ∆x is the half width of the channel measured along the xdirection, and γ denotes the increment due to the tilting angle, which is related to the components of the velocity by γ = tan α = vz vx = 40 127 . We simulate flow through such a channel and apply different in-and outlet boundary conditions. We use a relaxation time τ = 1 in all simulations presented here. However, we have checked that the results do not depend on this particular choice. After 5000 time steps a steady flow field is reached. However, to be sure that the simulations have converged, we simulate 20000 time steps until we evaluate data. To visualize the difference between simulation and theoretical prediction we subtract the velocity on each lattice node and draw the resulting vector field in Fig. 2. The value of the velocity is scaled by a factor of 1500 for drawing the arrows. The colors are assigned the absolute value of the velocity after scaling the difference field. In Fig. 2 a) we apply the boundary condition used by Kutay et al. [5] to a case, where the restriction of the in-and outflow velocity parallel to the z-direction introduces an error in the region close to the boundary. Note that in Ref. [5], apart from assuming the velocity perpendicular to the boundary, the authors underestimate the transversal components, which may be of no importance in this case. We use the correct coefficients as presented very recently in Ref. [20], but keep the restriction to the inflow perpendicular to the boundary, which has a much larger influence on the flow field. Not only the first and second layer of nodes close to the boundary are affected, but the boundary condition introduces vortices which have approximately the size of the diameter of the chan- In-and outflow velocity constraint to the direction perpendicular to the boundary plane (a), In-and outflow velocity tilted and of parabolic shape as in the analytic solution, but with N z x and N z y set equal to zero (b), the boundary conditions derived in Sec. IV with the correct choice for N z x and N z y . difference field (c). The velocity difference vectors are scaled for drawing by a factor of 1500 and the absolute value of the velocity difference is color coded from blue (small) to red (large).
nel. Therefore, the first step to generalize the boundary condition from Ref. [5] to a case where the in-and outflow velocity has an arbitrary orientation, is to use Eqns. (17)- (20), as used in the simulation for which the result is shown in Fig. 2 b). It is obvious that this boundary condition still introduces vortices close to the in-and outflow. The strength of the vortices is smaller compared to the case shown in Fig. 2 a). However, the size of the vortices is comparable to the width of the channel here as well. The value of the tangential velocity on the boundary nodes differs from the value one inserts into the equations. By introducing the transversal momentum corrections N z x and N z y in Eqns. (21)-(24), the vortices disappear and the velocity takes exactly the value one specifies with Eqns. (9)-(11) as one can see in Fig. 2 c). The remaining differencefield can be mostly ascribed to the discretization on the lattice. Each single step of the wall discretized to individual steps can be found in the flow profile. However, at the in-and outflux boundary no additional artifacts can be seen, which demonstrates the strength of our boundary condition. The velocity on the boundary nodes takes exactly the value which we specify, and therefore, no vortices are generated. The transversal momentum corrections N z x and N z y could also be understood in terms of a counter-slip similar to the approach of Inamuro et al. [25], but the Ansatz how to obtain the unknown populations f i is different: we assume a bounce back rule for the non-equilibrium part of the distributions and end up with a linear correction to the reflected populations, whereas the authors of Ref. [25] construct the unknown distributions based on kinetic theory where the correction appears not on the level of the distribution functions but as a counter-slip on the level of the wall velocity. The values for the density and the velocity inserted into the equilibrium distributions in Inamuro's method are different from the ones used for bulk nodes. In our approach, however, the boundary nodes are treated similar as the bulk nodes: the velocity on the boundary node can be calculated by inserting Eq. (25) and Eq. (26) into Eqns. (21)- (24) and the obtained distributions f i together with the one from Eq. (16) and the density from Eq. (13) into Eq. (5). It turns out that the velocity calculated from Eq. (5) is exactly the one which is imposed at the boundary node by Eqns. (21)- (24). In all simulations we kept the tilting angle of the channel constant, because the error of our boundary conditon is angle independent. We can quantify the quality of the boundary conditions by computing the ratio of the absolute value of the difference field and the calculated velocity field on each node. The obtained values are averaged over the first twenty layers of LB nodes from the boundary.
ξ = V v(x) − v P (x) |v P (x)| dV ,(35)
Where the volume V contains those layers of lattice nodes, which are at most a distance of the channel width apart from the boundary of the computational domain. This captures approximately the vortices and provides a measure for the quality of the boundary condition. The results for the different cases shown in Fig. 2 are listed in the following tabular:
Boundary condition relative error ξ on-site velocity (Fig. 2 c) 0.0996 N z x and N z y set to zero (Fig. 2 b) 0.126 v x = v y = 0 (Fig. 2 a) 0.175
Good agreement with the expected Poiseuille flow profile (Eq. (34)) is reflected in small relative errors. Large numbers indicate deviations in the area, where the fluid fields are compared. We ascribe the remaining deviations to the discretization error of the wall and the accompanying uncertainty in the exact wall position in the present case of the staircase like discretization. We check this by increasing the resolution of the simulation by a factor of two. As we expect, the numerical error due to the staircase-like discretization decreases roughly by a factor of two to ξ = 0.051. This shows that the staircase-like discretization introduces a first order error. Therefore, we need further investigations to see the second order accuracy of the in-and outflux boundary condition. As another test for the flux boundary condtion we simulate a straight channel aligned parallel to the computational lattice, again in a 64 × 8 × 128-domain with the same boundary conditions as for the inclined channel. The remaining relative error decreases to 0.00235, which is typical for Poiseuille flow simulations at this resolution in combination with a mid-grid bounce back rule on the boundary. We can further measure the quality of our boundary condition in a shear simulation. On a 32 3 lattice we apply periodic boundaries in x-and y-direction and impose a shear velocity of v x = ±0.02 with opposite sign on the top and bottom plane. We obtain a linear flow profile within floating point precision. There are no notable jumps between the first and second layer of LB nodes, which confirms that the strain rate tensor Π is set up correctly on the boundary nodes.
In a next step we simulate Poiseuille flow again, but this time we use a 32 3 lattice with periodic boundaries in y-and z-direction. We apply a body force [14,47] by adding a force term ∆v = τ F ρ (36) to the velocity in Eq. (6) in the whole simulation volume. The Poiseuille profile we expect is of the form
v = F 2η 1 − x − x 0 ∆x 2 ,(37)
where the viscosity is given by Eq. (8). The velocity profile found in the simulation together with the expected Poiseuille profile is plotted in Fig. 3. The parabola contains no fit parameters. The velocity is exactly zero on the boundary nodes, whereas with a simple bounce-back a numerical slip can be observed, which results in a velocity of 4 × 10 −5 for the same simulation setup without using the transversal momentum corrections N z x and N z y . We have carried out this test with τ = 1, but the data presented in Fig. 3 is obtained with τ = 2 to ensure that our boundary conditions are not restricted to the special case of τ = 1. Apart from the influence of τ on the viscosity (Eq. (8)), our simulation results are not affected by the relaxation time. In particular, we do not see any τ -dependent (numerical) slip. Also in this second-order test we find that the numerical error is of the size of the floating point precision on the computer. This underlines that our boundary condition reproduces the velocity field up to second order. Finally, we test our implementation by simulating flow through a square channel. The analytical solution for the velocity of a pressure driven flow in a b × b square channel is [48] v(x, y) = − ∇p 2η
b 2 4 − y 2 − C(x) , with(38)C(x) = 8b 2 π 3 ∞ n=0 (−1) n cosh (2n+1)πx b cos (2n+1)πy b (2n + 1) 3 cosh (2n+1)π 2 where x ∈ [−b/2, b/2] and y ∈ [−b/2, b/2]
are the coordinates in the cross-section, with the origin in the center of the channel. The pressure gradient ∇p is imposed by pressure boundaries and the dynamic viscosity is known from Eq. (8). The infinite sum in Eq. (38) can be truncated when a given accuracy is reached. We sum up 50 terms and compare this approximation to our numerical results on a 32 3 domain, i.e., b = 15 plus one layer of boundary nodes. In Fig. 4 a) we compare the analytical solution from Eq. (38) with our simulation results. The velocity in z-direction is averaged over the y-and z-direction and the averaged value is plotted against the position in x-direction. A very good agreement of the numerical result with the analytical solution can be seen. For comparison, results for the node based bounce back rule are shown. For this boundary condition it is known that it is only first oderer accurate, which can be seen in the kink in the velocity profile close to the boundary nodes. It can however be made second order accurate by choosing the position of the wall somewhere (depending on the BGK relaxation time) in between two nodes, which is known as the mid-grid bounce back [2]. As one can see in Fig. 4 a), if the wall-position is chosen correctly for the bounce-back rule, a satisfiying accuracy can be achieved, too (top-down triangles). Note that the position of the wall is shifted by half a lattice unit due to the different approach at the wall. In Fig. 4 b) the relative error depending on the size of the simulation is studied. There are three different errors involved. The truncation errors of the sum in Eq. (38) can not be seen in this figure. If the sum is truncated after just a few terms, the error increases on two of the edges.
In the corners of the simulation domain the errors due to the discretization on the lattice remains. This is what determines the accuracy for relatively small simulations. However, if the lattice is refined, this error decreases. It decreases with the square of the lattice constant which is typical for second order accurate schemes. Another error which dominates for large systems is the floating point b) The relative error for system sizes between 8 and 64 nodes: in the corners next to the boundary the largest relative deviations occur. Note that on the boundary nodes, where the truncation error of Eq. (38) is largest, the deviation between simulation and approximated analytical solution is negligible. Therefore, the deviation can be taken as a measure for the quality of the numerical result. c) The averaged error for different lattice resolutions versus the number of lattice nodes in each dimension confirming the second order accuracy of the no-slip boundary condition.
precision which is reflected in noisy data in the center of the simulation domain. This error is independent on the lattice constant. In Fig. 4 b) the relative error is shown for different system sizes. In the corners the error decreases with the system size, whereas the noise in the center is independent by the system size. In Fig. 4 c) we plot the mean error averaged over the whole system against the number of lattice nodes used for computation in each dimension. The slope of approximately 2 confirms the second order accuracy of the boundary condition. For the simulations presented in Fig. 4 pressure boundaries according to Eq. (12) are used and on the walls and edges we apply no-slip conditions as described in Sec. V. For this plot we use only the range in which the lattice size dependent error dominates. For 64 lattice nodes in each dimension, the floating point precision in one of our post processing steps dominates the overall error. Therefore, we only use the smaller systems for this investigation. The inset in Fig. 4 shows the analytical velocity profile in a cross section perpendicular to the extension of the square channel.
VII. CONCLUSION
We have derived an explicit local on-site flux boundary condition for LB simulations on a D3Q19 lattice. Velocity terms up to second order enter the derivation and this accuracy is also confirmed in the numerical tests. The in-and outflux velocity underlies no restrictions to any peculiar direction. We have demonstrated the numerical accuracy by comparing simulation results for a flow through a tilted channel with the theoretical expectation of a Poiseuille flow. Remaining errors can be assigned to the discretization on the lattice and to rounding errors due to the floating point representation. We have tested the boundary condition in simulation of Poiseuille flow between two planar walls and in shear flow. In those tests the simulation data fits exactly the analytical solutions without any slip-parameter and independent on the BGK relaxation time. For this test we have used no-slip boundary conditions which are a special case included in the general velocity boundary conditions. Finally, we have tested the boundary condition by simulating the flow through a square channel. The scaling of the numerical error with the lattice resolution again confirms the second order accuracy.
Figure 1 :
1the vectors c i are the i th column vector of the matrix The geometry of the D3Q19 lattice with lattice vectors ci as defined in Eq. (3).
Due to the particular choice of Eqns. (21)-(26) or Eq.(27) respectively, it is possible to specify the velocity to an exact value on the lattice site. The rules presented here are independent on the relaxation rate in the collision step, since all calculations involve only the known values of the f i and equillibrium functions. Relaxation is calculated separately after all unknown f i are calculated and the macroscopic velocity and density are preserved during collision. There are no restrictions on the orientation of the inflow direction. Furthermore, all calculations are local on each lattice site. Apart from using only terms of first and second order in v for the equilibrium distributions f (eq) i
Figure 2 :
2Velocity difference fields for different approaches:
Figure 3 :
3Poiseuille flow between two parallel walls driven by a body force. The simulation data are averaged in each lattice plane parallel to the walls and agree up to floating point precision with the calculated Poiseuille profile.
Figure 4
4: a) Velocity profile in a square channel averaged along the yz-planes for the noslip boundary condition (squares), for the node based bounce back rule (triangles), mid-link bounce back (top down triangles), and the analytical solution. The noslip boundary condition collapses with the analytical solution, whereas the on-site bounce back boundary condition shows a kink in the profile close to the boundary nodes. The numerical results are obtained on a 32 3 lattice and for the analytical solution the sum in Eq. (38) is truncated after 50 terms. A 2D profile of the z-velocity in a xy-cross section is displayed as an inset.
if pressure boundaries are chosen. f 19 is set to 12 f (1...6) or 12f (1...6) , respectively. A correction similar to Eq.33)
if velocity boundaries are applied, or tof (1...6) = ρ0
18 −
f (1...6)
AcknowledgmentsThe German Research Foundation (DFG) is acknowledged for financial support (SFB 716 and EAMatWerk). We thank A. Narváez, B. Dünweg, and U. D. Schiller for fruitful discussions.AppendixHere we give the expressions for the other boundaries not treated explicitly in the text. We start with the topplane where we implement outflux for our simulations. We obtainwith v z defined in positive z-direction. Here, the undetermined populations after the streaming step arewith N z x and N z y defined as previously, in Eq. (25) and Eq.(26). For the left, right, front and back boundaries, which we do not use in this work one finds the following expressions. For the left (x = 0) boundary,andandAt the front (y = 0) boundary, one findsandAt the back the density is given byor the velocity readsand the distributions are
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. K Wieghardt, Theoretische Strömungslehre, Universitätsverlag GöttingenGöttingenK. Wieghardt, Theoretische Strömungslehre (Göttingen: Universitätsverlag Göttingen, 1957).
], use a different notation in which the vectors c8−c12 are permutated, c13 and c14 as well as c17 and c18 are exchanged. Note that some authors, e.g. in Ref. [5. Sometimes c19 is denoted as c0Note that some authors, e.g. in Ref. [5], use a different notation in which the vectors c8−c12 are permutated, c13 and c14 as well as c17 and c18 are exchanged. Sometimes c19 is denoted as c0.
Values of 19 onwards as vector index can be used to distinguish the different corner nodes. Values of 19 onwards as vector index can be used to dis- tinguish the different corner nodes
This figure "Fig3_new.jpg" is available in "jpg. This figure "Fig3_new.jpg" is available in "jpg" format from: http://arxiv.org/ps/0811.4593v5
| []
|
[
"Qubit decoherence due to detector switching",
"Qubit decoherence due to detector switching"
]
| [
"I Serban \nIQC and Dept. of Physics and Astronomy\nUniversity of Waterloo\n200 University Ave WN2L 3G1WaterlooONCanada\n\nInstituut-Lorentz\nUniversiteit Leiden\nP.O. Box 95062300 RALeidenThe Netherlands\n",
"F K Wilhelm \nIQC and Dept. of Physics and Astronomy\nUniversity of Waterloo\n200 University Ave WN2L 3G1WaterlooONCanada\n"
]
| [
"IQC and Dept. of Physics and Astronomy\nUniversity of Waterloo\n200 University Ave WN2L 3G1WaterlooONCanada",
"Instituut-Lorentz\nUniversiteit Leiden\nP.O. Box 95062300 RALeidenThe Netherlands",
"IQC and Dept. of Physics and Astronomy\nUniversity of Waterloo\n200 University Ave WN2L 3G1WaterlooONCanada"
]
| []
| We provide insight into the qubit measurement process involving a switching type of detector. We study the switching-induced decoherence during escape events. We present a simple method to obtain analytical results for the qubit dephasing and bit-flip errors, which can be easily adapted to various systems. Within this frame we investigate potential of switching detectors for a fast but only weakly invasive type of detection. We show that the mechanism that leads to strong dephasing, and thus fast measurement, inverts potential bit flip errors due to an intrinsic approximate time reversal symmetry. | 10.1140/epjqt/s40507-015-0020-6 | [
"https://arxiv.org/pdf/0905.3045v1.pdf"
]
| 59,452,415 | 0905.3045 | 33a2d7b7e680b8e465a90f3c99ce92dfa7646f09 |
Qubit decoherence due to detector switching
(Dated: May 19, 2009)
I Serban
IQC and Dept. of Physics and Astronomy
University of Waterloo
200 University Ave WN2L 3G1WaterlooONCanada
Instituut-Lorentz
Universiteit Leiden
P.O. Box 95062300 RALeidenThe Netherlands
F K Wilhelm
IQC and Dept. of Physics and Astronomy
University of Waterloo
200 University Ave WN2L 3G1WaterlooONCanada
Qubit decoherence due to detector switching
(Dated: May 19, 2009)numbers: 0540-a8525Cp0365Yz8585+j
We provide insight into the qubit measurement process involving a switching type of detector. We study the switching-induced decoherence during escape events. We present a simple method to obtain analytical results for the qubit dephasing and bit-flip errors, which can be easily adapted to various systems. Within this frame we investigate potential of switching detectors for a fast but only weakly invasive type of detection. We show that the mechanism that leads to strong dephasing, and thus fast measurement, inverts potential bit flip errors due to an intrinsic approximate time reversal symmetry.
We provide insight into the qubit measurement process involving a switching type of detector. We study the switching-induced decoherence during escape events. We present a simple method to obtain analytical results for the qubit dephasing and bit-flip errors, which can be easily adapted to various systems. Within this frame we investigate potential of switching detectors for a fast but only weakly invasive type of detection. We show that the mechanism that leads to strong dephasing, and thus fast measurement, inverts potential bit flip errors due to an intrinsic approximate time reversal symmetry. Noise-activated switching out of a metastable state is a common phenomenon in a wide range of physical systems, including Josephson junctions, nanomechanical devices, chemical reactions [1,2]. Starting with Kramers seminal work [3], such processes have been studied close to equilibrium [4], as well as in driven systems [5]. The activated escape paths have been studied theoretically and observed experimentally [6,7].
Recently, noise-activated switching has gained attention due to its role in quantum measurement, in particular for qubit detection. Examples of switching detectors include the superconducting quantum interference device (SQUID) [8,9,10], where switching occurs between the superconducting and dissipative state. The Josephson bifurcation amplifier [11,12,13] has been recently employed in the delicate task of detecting a qubit state in a minimally invasive fashion [14]. In this case, the detector can switch between different, weakly dissipative, dynamical states. Using an appropriate choice of a reference frame, switching between such dynamical states can also be described as escape from a static metastable potential well [15,16].
Switching is a highly nonlinear phenomenon, driven by large, non-equlibrium environmental fluctuations, so this type of detection is far from the weak measurement scenario. Some understanding for the switching type of detectors has been provided by numerical studies [17,18] and in a simplified two-state detector version in Ref. [19]. However, a full description of the qubit decoherence during, and induced by the switching event is still missing.
In this paper we propose a simple and analytical method to investigate qubit decoherence due to a switching type of detector. We model the detector as a classical, overdamped particle trapped in a metastable potential. The escape of the particle is driven by large, rare fluctuations in the environment. We investigate the qubit dephasing and bit flip errors induced by the switching, during the escape event. This allows novel insights into the measurement process and reveals the specific conditions during the switching event that lead to a combination of strong coherence loss and low error rate. These are classical particle performs Brownian motion according tȯ
x = K(x) + f (t),(1)
where K is the deterministic force experienced by the particle due to the metastable potential and f is white Gaussian noise with a probability density functional given by [20] P
[f (t)] = exp − t f 0 f (t) 2 /(2D)dt ,(2)
where we assume, [21] the noise intensity D to be small compared to the barrier height ∆U , see Fig. 1
P [x(t)] = exp − S[x(t)] D , S[x(t)] = 1 2 t f 0 dt(ẋ − K(x)) 2 .
(3) For the study of qubit decoherence during a switching event one will need expectation values of the type
O(t 0 ) = exp (λφ[x(t), s(t, t 0 )]) sw , (4) φ[x[(t), s(t, t 0 )] = t f 0 x(t)s(t, t 0 )dt,(5)
were s(t, t 0 ) is a time dependent modulation of x(t). We are interested in the qubit decoherence during switching. Thus, the averaging sw is performed only over switching trajectories of the detector, which satisfy the boundary conditions x(0) = x m and x(t f ) = x f , with x m inside and x f outside the metastable well. By choosing s(t, t 0 ) = 0 at t f > t > t 0 , the average becomes post-conditioned by a switching event taking place at the final time t f . Since the exact trajectory between the initial and final point remain unknown, we average over all possible paths
O(t 0 ) = (x f ,t f ) (xm,0) Dx(t) exp λφ[x(t), s(t, t 0 )] − S[x(t)] D × × P (x m , 0|x f , t f ) −1 ,(6)
where the total switching probability is
P (x m , 0|x f , t f ) = (x f ,t f ) (xm,0) Dx(t) exp (−S[x(t)]/D) .(7)
The switching trajectories form a narrow tube in the phase space centered around the optimal trajectory [22,23] which minimizes S, and for the present case satisfies
x opt = K(x opt )K (x opt ), x opt (0) = x m , x opt (t f ) = x f . (8) Thus S[x(t)] = S[x opt (t)] + S 2 [x(t) − x opt (t)] and we ap- proximate S 2 [x(t)] ≈ 1 2 t f 0 dt(ẋ(t) 2 − Λ(t) 2 x(t) 2 ),(9)
where
Λ(t) 2 = −(K (x) 2 + K(x)K (x))| x=xopt(t)
. Divergences due to the emergence of a slow mode on the barrier top [21,24] are avoided by the appropriate choice of the initial (kinetic) energy 0 <ẋ(0) 2 /2 ∆U which satisfies the boundary conditions (8). Thus, the switching event takes place, with non-vanishing probability, within a finite time t f . One can show that
O(t 0 ) = exp (λφ[x opt (t) + x 0 (t)/2, s(t, t 0 )]) ,(10)
where x 0 is the solution of
x 0 + Λ 2 x 0 + Dλs(t, t 0 ) = 0, x 0 (0) = x 0 (t f ) = 0. (11)
The two linearly independent solutions of the homogeneous part of Eq. (11) are
x 1 (t) =ẋ opt (t), x 2 (t) =ẋ opt (t) t 0 dt ẋ opt (t ) −2 ,(12)
and the full x 0 (t) = x 1 (t)c 1 (t) + x 2 (t)c 2 (t) can be determined by variation of constants. We consider the case of a metastable potential described by U (x)/Ω = x/2 − x 3 /6, which can represent a Josephson junction DC-biased at half the critical current. The characteristic frequency of the detector is given by Ω = K (x m ).
Pure dephasing: We assume a qubit Hamiltonian of the formĤ
= ωσ z + ησ z x(t),(13)
where x(t) is the coordinate of the classical particle. The only effect of the environment in this case is the irreversible decay of the phase coherence C(t 0 ) = O(t 0 ), see Eq. (4), for the specific value of λ = η/(i ) and coherence remains at an almost constant value. We observe that the optimal noise becomes stronger for shorter values of t f . However, the strongest drop in qubit coherence was observed for the longer t f . In this case the optimal trajectory spends more time close to the barrier top, where the motion is diffusive, driven by low amplitude noise.
s(t, t 0 ) = 1, t < t 0 0, t > t 0 .(14)
Bit flip errors: We consider a qubit-environment coupling which allows for energy exchange, and can induce bit flip errorsĤ
= ωσ z + ηx(t)σ x .(15)
The probability of noise induced errors during a switching event, at t 0 < t f is given by
P ↑→↓ (t 0 ) = | ↓ |Û I (t 0 )| ↑ | 2 P (x m , 0|x f , t f ) −1 ,(16)
where in the limit of short time and weak couplinĝ
U I (t 0 ) = T exp t0 0 dt H I (t) i ≈ 1 + t0 0 dt H I (t) i , H I (t) = ηÛ † 0 (t)σ xÛ0 (t)x(t),(17)
andÛ 0 describes the free qubit evolution. We obtain
P ↑→↓ (t 0 ) = lim λ→0 ∂ 2 ∂λ 2 η 2 2 exp (λφ[x(t), s(t, t 0 )R(t)]) + exp (λφ[x(t), s(t, t 0 )I(t)]) sw , R(t) + iI(t) = ↓ |Û † 0 (t)σ xÛ0 (t)| ↑ .(18)
In Fig. 3 (a) we observe, similar to the pure dephasing case, a sharp feature in P ↑→↓ (t 0 ) at the point in time where the most probable trajectory (c) reaches the steepest point on the potential barrier, and the most probable noise (b) reaches it maximum strength. Despite the optimal noise being strongest for short switching time t f , the peak in P ↑→↓ (t 0 ) is higher for longer t f . Another notable feature is the quasi-reversibility of the bit flip error which occurs at this point. This feature cannot be explained by the single, deterministic trajectory x opt alone. It causes only the steady increase of P ↑→↓ (t 0 ).
Prehistory density distribution: The results presented above can be understood from the distribution of switching trajectories. We calculate the probability P h (x, t) for the classical particle to occupy the position x at time t during a switching event, in the form of a prehistory density distribution [21]
P h (x, t) = P (x m , 0|x, t)P (x, t|x f , t f ) P (x m , 0|x f , t f ) .(19)
Within the approximation (9), the probability for a transition between any pair of points (x 1 , t 1 ) and (x 2 , t 2 ), with t 1,2 < t f reads 2 1 implies that the time integral is taken between t 1 and t 2 and δx 1,2 = x 1,2 − x opt (t 1,2 ). One can show that
P (x 1 , t 1 |x 2 , t 2 ) = (δx1,t2) (δx1,t1) Dx(t) exp − S[x opt (t)] 2 1 + S 2 [x(t)] 2 1 D ,(20)where S[x(t)]P (x 1 , t 1 |x 2 , t 2 ) = exp − S[x opt (t)] 2 1 + S 2 [x b (t)] 2 1 D · F (t 1 |t 2 ),(21)
whereẍ b + Λ 2 (t)x b (t) = 0 and x b (t 1,2 ) = δx 1,2 and
F (t 1 |t 2 ) = (0,t2) (0,t1) Dx(t) exp − S 2 [x(t)] D (22) = 2πDẋ opt (t 1 )ẋ opt (t 2 ) t2 t1ẋ opt (t) −2 dt −1/2 .
We obtain a Gaussian distribution, centered around
x opt (t) P h (x, t) = 1 πw(t) exp − (x − x opt (t)) 2 w(t) ,(23)
where Fig. 4 shows a narrow tube of trajectories close to the bottom of the well. This is followed by a strong widening of the distribution in the process of climbing up the potential barrier. This event is driven by a sharp noise pulse. On the barrier top we see again a fairly localized density distribution, driven by low-amplitude noise. The tube narrows even more on the outer side of the barrier. These results are in agreement with the findings of Ref. [25], for a different system. We found, see Figs. 2 and 3 that the qubit suffers the strongest decoherence at the point in time when the optimal trajectory reaches the steepest point on the barrier wall. This is true for both bit flip errors and dephasing. The magnitude of both effects depends strongly on the total time necessary for the switching event, such that longer t f leads to enhanced coherence loss, and higher bit flip rate. The observed effect can be explained by the strong widening of the prehistory distribution P h (x, t) at the same point in time, see Fig. 4.
w(t) = F (0|t f ) 2 /(F (0|t) 2 F (t|t f ) 2 ).
The peak in P ↑→↓ originates in the strong widening of the trajectory tube. Thus, large excursions around x opt , see Fig. 1, are very probable during this time. However, since each of these switching trajectories must return to the narrow tube of trajectories on the other side of the widening, i.e. in the region close to the barrier top, they all show an approximate time reversal symmetry. Thus any induced bit flip errors will be quasi reversed when the particle reaches the barrier top.
In conclusion, a switching detector, such as the one modeled here, presents several qualities which make it desirable as a nonlinear qubit detector. The strong decoherence suffered by the qubit as the classical particle climbs up the potential barrier affects strongly the coherence, leading to a fast measurement. The equally strong bit flip errors acquired during the process are reversed by the quasi time-reversal symmetry of most trajectories.
We observe that a fast switching event causes less decoherence, despite stronger noise being required for surmounting the barrier. Fig. 4 reveals that longer switching time t f allows more freedom in the choice of the particular time the particle climbs up the potential barrier and such incoherent behavior causes decoherence.
We note that the widening of the trajectory tube at the inflexion point of the potential appears throughout literature as a general feature of the tube of escape trajectories out of metastable potentials. Having identified it as the major cause for the features observed in the qubit decoherence, we expect these features to be common to various potentials. Therefore we expect that our results have applicability to existing experimental setups, e.g. the JBA and the DC-SQUID.
We are grateful to M.I. Dykman for pointing out the prehistory probability distribution approach and for many useful suggestions. We acknowledge useful discussions with J. Gambetta, W.A. Coish and T.C. Wei. This work has been supported by NSERC Discovery Grants and Quantum Works.
PACS numbers: 05.40.-a, 85.25.Cp, 03.65.Yz, 85.85.+j
FIG. 1 :
1Prehistory probability density P (x, t) for metastable potential U (x), where U (x) = −K(x), and optimal trajectory xopt(t). Here xm,M,i are the positions of the minimum, maximum and inflexion points of the potential.
FIG. 2 :
2Qubit coherence during a switching event (t0 < t f ) (a), optimal noise trajectory fopt(t0) (b) and most probable switching trajectory xopt(t0) (c) for various values of t f .
Fig. 2 FIG. 3 :
23shows that the escape process is driven by strong noise (b), with a maximum intensity at the time where the optimal trajectory (c) reaches the inflexion point of the barrier. In the vicinity of the same point, a sharp drop in qubit coherence (a) is observed. After the inflexion point the motion is slowed down, and becomes diffusive close to the barrier top. During this stage, the qubit Probability for an induced bit flip error as a function of t0, for various values of t f (a), the corresponding most probable noise (b) and the most probable driven trajectory (c). Here ω = 10Ω.
FIG. 4 :
4Width of the prehistory probability distribution P h (x, t) (a) and optimal trajectory (b).
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| []
|
[
"Optimizing Higher-order Lagrangian Perturbation Theory for Cold Dark Matter Models",
"Optimizing Higher-order Lagrangian Perturbation Theory for Cold Dark Matter Models"
]
| [
"Arno G Wei ",
"Stefan Gottl ",
"Thomas Buchert ",
"\nAstrophysikalisches Institut Potsdam\nMax-Planck-Institut f ur Astrophysik\nKarl-Schwarzschild-Str. 1, An der Sternwarte 16D{85748, D{14482Garching, PotsdamF.R.G\n",
"\nTheoretische Physik\nLudwig-Maximilians-Universit at, Theresienstr. 37\n"
]
| [
"Astrophysikalisches Institut Potsdam\nMax-Planck-Institut f ur Astrophysik\nKarl-Schwarzschild-Str. 1, An der Sternwarte 16D{85748, D{14482Garching, PotsdamF.R.G",
"Theoretische Physik\nLudwig-Maximilians-Universit at, Theresienstr. 37"
]
| []
| We report on the performance of Lagrangian perturbation | null | [
"https://arxiv.org/pdf/astro-ph/9512106v1.pdf"
]
| 116,905,263 | astro-ph/9512106 | f014d424402cd4c912a5e06412e9c89feba2f516 |
Optimizing Higher-order Lagrangian Perturbation Theory for Cold Dark Matter Models
Arno G Wei
Stefan Gottl
Thomas Buchert
Astrophysikalisches Institut Potsdam
Max-Planck-Institut f ur Astrophysik
Karl-Schwarzschild-Str. 1, An der Sternwarte 16D{85748, D{14482Garching, PotsdamF.R.G
Theoretische Physik
Ludwig-Maximilians-Universit at, Theresienstr. 37
Optimizing Higher-order Lagrangian Perturbation Theory for Cold Dark Matter Models
astro-ph/9512106 15 Dec 95
We report on the performance of Lagrangian perturbation
theory up to the second order for the standard cold dark matter (SCDM) and broken scale invariance (BSI) scenarios. We normalize both models to the COBE data, the BSI model serves as an example of models which t the small-scale power of galaxy surveys. We optimize Lagrangian perturbation solutions by removing small-scale power from the initial data and compare the results with those of numerical simulations. We nd an excellent performance of the optimized Lagrangian schemes down to scales around the correlation length or smaller, depending on the statistics used for the comparison. The optimization scheme can be expressed in a way which is independent of the type of uctuation spectrum and of the size of the simulations.
Lagrangian Perturbation Theory and Optimization
For an in-depth discussion of the Lagrangian perturbation approach relevant to this work see (Buchert 1994), and for the optimization approach , Melott et al. 1995, Wei et al. 1995 and references therein.
Denoting comoving Eulerian coordinates byq and Lagrangian coordinates byX, the eld of trajectoriesq =F (X; t) up to the second order on an Einsteinde Sitter background reads: F =X + q 1 (a) r X (1) (X) + q 2 (a) r X (2) (X);
(1) where the time-dependent coe cients can be expressed in terms of the expansion function a(t) = ( t t i ) 2=3 : q 1 = 3 2 (a 1);
(2) 555 q 2 = 3 2 2 ( 3 14 a 2 + 3 5 a 1 2 + 4 35 a 3 2 ):
The perturbation potentials have to be constructed by solving iteratively the two boundary value problems:
X (1) = I( ;i;k )t i ; (4) X (2) = 2II( (1) ;i;k );(5)
where I and II denote the rst and second principal scalar invariants of the tensor gradient ( (1) ;i;k ). Under the restriction of parallelism of the initial peculiarvelocity eld and the peculiar-acceleration eld, we can set (1) = t i ( being the initial peculiar-velocity potential), and the rst-order ow-eld (1) reduces to Zel'dovich's approximation, as discussed in (Buchert 1994) and references therein.
The optimization scheme used here for the Lagrangian perturbation theory was discussed in Coles et al. (1993) and further investigated for higher orders in Melott et al. (1995). The idea is to smooth away some of the small-scale uctuations in the initial data. In a previous work, Melott et al. (1995), we found that convolution of the initial density eld with a Gaussian window function W (k; k gs ) = exp( k 2 =k 2 gs ) of appropriate width k gs considerably improves on the performance of the Lagrangian perturbation schemes. Speci cally, here we run a search-loop over di erent smoothing lengths applied to the initial data of the rst and second order Lagrangian approximations. Then we compare the resulting density elds with those obtained with a PM-code (Kates et al. 1991) by determining the cross-correlation coe cient S = h 1 2 i= 1 2 of the two elds (here, the mean h: : :i is evaluated over all cells of the discrete density elds). As optimal we de ne that smoothing length k gs which maximizes S.
In a further analysis, we compute a scale-dependent cross-correlation S(R g ) by smoothing the two nal density elds 1 , 2 with various lter widths R g .
Numerical simulations with COBE-normalized SCDM and BSI initial data
In order to follow the nonlinear evolution of the formation of structure we have performed N-body simulations using a standard PM code (Kates et al. 1991) with 128 3 particles on a 256 3 grid (Kates et al. 1995). The universe is assumed to be spatially at ( = 1). It is dominated by cold dark matter. We consider here simulations which were performed in boxes of 500 h 1 Mpc, 200 h 1 Mpc and 75 h 1 Mpc. The simulations were started with the power spectrum P (k) of density perturbations calculated at z = 25,
P (k) = 2 k = 4 9 ( kR h 2 ) 4 2 (k)T 2 (k);(6)
where R h = 2H 1 denotes the horizon scale. The primordial perturbation spectrum is either the Harrison-Zel'dovich spectrum ( = const) of the SCDM 556 model or the spectrum with broken scale invariance calculated from a double in ationary model (Gottl ober et al. 1991). In the BSI model the primordial spectrum is of Harrison-Zel'dovich type both in the limit of very large and very small scales. We have used the CDM transfer function T (k) of Bond and Efstathiou (1984). For all simulations we have normalized our spectra using the 10 -variance of the CMB uctuations T = (30 7:5) K of the rst year COBE data (Smoot et al. 1992). Thus, the power spectrum of the BSI model shows less power on small scales than the SCDM model. The scales of nonlinearity de ned in eq.(7) below are SC DM nl = 27h 1 Mpc and B SI nl = 7h 1 Mpc at the time z = 0. In Fig. 1 we show the linear BSI and SCDM spectra and indicate the box sizes and the resolution of our simulations.
Results and Discussion
In an analysis of both models at di erent times (z = 0; 1; 2) we nd that the optimal k gs can consistently be estimated from the scale of nonlinearity k nl of the considered spectrum and time evolution, k nl being de ned by a 2 (t) (2 ) 3 Z k nl 0 d 3 k P (k) = 1:
As can be seen from Fig. 2, k gs ' 1:45k nl for the rst-order and k gs ' 1:2k nl for the second-order Lagrangian perturbation solutions. It can be seen from the lines in Fig. 2, which are regression ts to the scatter plot of k gs =k nl versus n(k nl ), that the optimal smoothing length does practically not depend on the slope n(k nl ) of the power spectrum at the nonlinearity scale, and is thus independent 557 Figure 2. The optimal smoothing length k gs in units of k nl versus the slope n(k nl ) of the power spectrum at k nl for the BSI and SCDM models.
of the exact shape of the power spectrum. While the smoothing length for rst-order shows some random scatter with n(k nl ), the optimal smoothing of the second order is much more robust against variations in the initial power spectrum.
The scale-dependent cross-correlation function S(R g ) measures primarily whether mass is moved to the right place. Fig. 3 shows its value for the comparison of the density elds at time z = 0. The performance of the optimized Lagrangian perturbation schemes is very good down to scales of 1 : : :2h 1 Mpc for both the SCDM and BSI models. This quality can clearly not be met by Figure 3. Scale-dependent cross-correlation function S(R g ) of rst-order (dashdotted lines) and second-order (dashed lines) optimized Lagrangian perturbation theory as well as of \chopped linear theory" (dash-dot-dot-dotted lines) correlated with the results of the numerical simulations for the SCDM (left) and BSI (right) models, evolved to z = 0. Thin lines are for the 500h 1 Mpc boxes, medium thick lines for 200h 1 Mpc and thick lines for the 75h 1 Mpc boxes, respectively.
the Eulerian linear theory. (For this comparison we use an improvement on the Eulerian linear theory as proposed in Coles et al. (1993), called \chopped linear theory"; it complies to both clin 1 and h clin i = 0, even at late times. In terms of the linearly evolved lin we set 1 + clin = (1 + lin ) if lin > 1 and 0 otherwise, where is a normalization constant keeping the total mass the same.)
The two-point correlation function (r) of the numerical simulations is reproduced well by the optimized Lagrangian perturbation schemes down to about the \correlation length" r 0 , (r 0 ) = 1 (Fig. 4). Below r 0 , which itself is underestimated by about 10%, the amplitude of (r) of the optimized Lagrangian perturbation schemes drops well below that of the numerical simulations. Still, the use of our optimization scheme leads to a considerable improvement in (r) on small scales, especially for models with a large amplitude of small-scale uctuations, like the SCDM model. The optimized Lagrangian perturbation schemes reproduce the density eld very good down to the scales of groups and clusters of galaxies, even deep into the nonlinear regime (i.e. today). Their inherent advantage is their high speed of execution, their complexity is comparable to one time step of a conventional particle-mesh N-body code. Thus, they allow the quick production of a large ensemble of simulations, e.g. for the evaluation of di erent cosmological models or for doing statistics over an ensemble of di erent realizations of a single model. Furthermore, the analyticity of the mapping (1) allows not only the local determination of the resulting shear stresses, e.g. for the introduction of local biasing schemes, it can also be used to enhance the particle density by interpolation of the mapping (1), so that one can use statistics with a high selection e ect, like 559 the simulation of galaxy catalogues with geometry and luminosity selection, even for very large-scale realizations (on Gpc-scales). The analysis of the singularity structure of the mapping (1) in order to identify building blocks of large-scale structure o ers a way to analytically relate initial data to present-day structure .
Figure 1 .
1The linear power spectra of our SCDM (dashed line) and BSI (solid line) models at time z = 0.The rst and second year COBE data were analyzed by many authors using di erent statistical techniques. The new normalization of G orski et al.(1994) is about 25 % higher than the normalization which we used in the simulations. Consequently, in this normalization the scales of nonlinearity would increase
Figure 4 .
4Two-point correlation function for the SDCM (left) and BSI (right) models at z = 0. Thick solid lines are for the numerical simulations, the dotted line is the prediction of the Eulerian linear theory, the thick (thin) dash-dotted and dashed lines are for rst-and second-order optimized (unoptimized) Lagrangian perturbation solutions, respectively.
Acknowledgments. We wish to thank Adrian L. Melott (University of Kansas) for valuable discussions as well as for the permission to use his programs for the cross{correlation statistics, which were also used in andMelott et al. (1995). AGW wishes to thank the AIP in Potsdam for the opportunity to work on this project during a stay at the AIP. TB acknowledges support of the \Sonderforschungsbereich 375 f ur Astro{Teilchenphysik der Deutschen Forschungsgemeinschaft". SG wishes to thank the MPA in Garching for its hospitality. AGW and TB like to thank the University of Valencia for hospitality during a working visit, which was nancially supported by \acciones integradas", project AI95-14.
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| []
|
[
"CLASSIFICATION OF DIFFERENTIAL SYMMETRY BREAKING OPERATORS FOR DIFFERENTIAL FORMS",
"CLASSIFICATION OF DIFFERENTIAL SYMMETRY BREAKING OPERATORS FOR DIFFERENTIAL FORMS"
]
| [
"Toshiyuki Kobayashi ",
"Toshihisa Kubo ",
"Michael Pevzner "
]
| []
| []
| We give a complete classification of conformally covariant differential operators between the spaces of differential i-forms on the sphere S n and j-forms on the totally geodesic hypersphere S n−1 by analyzing the restriction of principal series representations of the Lie group O(n + 1, 1). Further, we provide explicit formulae for these matrix-valued operators in the flat coordinates and find factorization identities for them. This note was published in C. R. Acad. Sci. Paris, Ser. I, (2016), http://dx. | 10.1016/j.crma.2016.04.012. | [
"https://arxiv.org/pdf/1605.05722v1.pdf"
]
| 119,663,577 | 1605.05722 | 08e92888d199b3e0b5b1208de5d6dba8c2cc9c14 |
CLASSIFICATION OF DIFFERENTIAL SYMMETRY BREAKING OPERATORS FOR DIFFERENTIAL FORMS
18 May 2016
Toshiyuki Kobayashi
Toshihisa Kubo
Michael Pevzner
CLASSIFICATION OF DIFFERENTIAL SYMMETRY BREAKING OPERATORS FOR DIFFERENTIAL FORMS
18 May 201610.1016/j.crma.2016.04.012arXiv:1605.05722v1 [math.DG]and phrases: Symmetry breaking operatorsbranching lawsF-methodconformal geometryVerma moduleLorentz group
We give a complete classification of conformally covariant differential operators between the spaces of differential i-forms on the sphere S n and j-forms on the totally geodesic hypersphere S n−1 by analyzing the restriction of principal series representations of the Lie group O(n + 1, 1). Further, we provide explicit formulae for these matrix-valued operators in the flat coordinates and find factorization identities for them. This note was published in C. R. Acad. Sci. Paris, Ser. I, (2016), http://dx.
Introduction
Suppose a Lie group G acts conformally on a Riemannian manifold (X, g). This means that there exists a positive-valued function Ω ∈ C ∞ (G×X) (conformal factor ) such that L * h g h·x = Ω(h, x) 2 g x for all h ∈ G and x ∈ X, where L h : X → X, x → h·x denotes the action of G on X. Since Ω satisfies a cocycle condition, we can form a family of representations ̟ (i) u for u ∈ C and 0 ≤ i ≤ dim X on the space E i (X) of differential i-forms on X by (1.1) ̟ (i) u (h)α := Ω(h −1 , ·) u L * h −1 α (h ∈ G). The representation ̟ (i) u of the conformal group G on E i (X) will be simply denoted by E i (X) u .
If Y is a submanifold of X, then we can also define a family of representations ̟
(j) v on E j (Y ) (v ∈ C, 0 ≤ j ≤ dim Y ) of the subgroup G ′ := {h ∈ G : h · Y = Y } ,
which acts conformally on the Riemannian submanifold (Y, g | Y ). We study differential operators D : E i (X) −→ E j (Y ) that intertwine the two representations ̟
(i) u | G ′ and ̟ (j) v of G ′ . Here ̟ (i)
u | G ′ stands for the restriction of the G-representation ̟ (i) u to the subgroup G ′ . We say that such D is a differential symmetry breaking operator, and denote by Diff
G ′ (E i (X) u , E j (Y ) v )
the space of all differential symmetry breaking operators. We address the following problems:
Problem A. Determine the dimension of the space Diff G ′ (E i (X) u , E j (Y ) v ).
In particular, find a necessary and sufficient condition on a quadruple (i, j, u, v) such that there exist nontrivial differential symmetry breaking operators.
Problem B. Construct explicitly a basis of Diff
G ′ (E i (X) u , E j (Y ) v ).
In the case where X = Y , G = G ′ , and i = j = 0, a classical prototype of such operators is a second order differential operator called the Yamabe operator
∆ + n − 2 4(n − 1) κ ∈ Diff G (E 0 (X) n 2 −1 , E 0 (X) n 2 +1 ),
∆ is the Laplace-Beltrami operator, where n is the dimension of X, and κ is the scalar curvature of X. Conformally covariant differential operators of higher order are also known: the Paneitz operator (fourth order) [11], which appears in four dimensional supergravity [2], or more generally, the so-called GJMS operators [3] are such examples. Analogous conformally covariant operators on forms (i = j case) were studied by Branson [1]. On the other hand, the insight of representation theory of conformal groups is useful in studying Maxwell's equations, see [10], for instance. Let us consider the more general case where Y = X and G ′ = G. An obvious example of symmetry breaking operators is the restriction operator Rest Y which belongs to Diff
G ′ (E i (X) u , E i (Y ) u ) for all u ∈ C. Another elementary example is Rest Y • ι N Y (X) ∈ Diff G ′ (E i (X) u , E i−1 (Y ) v ) if v = u + 1 where ι N Y (X)
denotes the interior multiplication by the normal vector field to Y when Y is of codimension one in X.
In the model space where (X, Y ) = (S n , S n−1 ), the pair (G, G ′ ) of conformal groups amounts to (O(n + 1, 1), O(n, 1)) modulo center, and Problems A and B have been recently solved for i = j = 0 by Juhl [4], see also [5,7] and [9] for different approaches by the F-method and the residue calculus, respectively.
Problems A and B for general i and j for the model space can be reduced to analogous problems for (nonspherical) principal series representations by the isomorphism (2.2) below. In this note we shall give complete solutions to Problems A and B in those terms (see Theorems 3.1 and 4.1).
Notation: N = {0, 1, 2, · · · }, N + = {1, 2, · · · }.
2. Principal series representations of G = O(n + 1, 1)
We set up notations. Let P = MAN be a Langlands decomposition of a minimal parabolic subgroup of G = O(n + 1, 1). For 0 ≤ i ≤ n, δ ∈ Z/2Z, and λ ∈ C, we extend the outer tensor product
representation i (C n )⊗(−1) δ ⊗C λ of MA ≃ (O(n)× O(1)) × R to P by letting N act trivially, and form a G-equivariant vector bundle V i λ,δ := G × P i (C n ) ⊗ (−1) δ ⊗ C λ over the real flag variety X = G/P ≃ S n .
Then we define an unnormalized principal series representations
(2.1) I(i, λ) δ := Ind G P i (C n ) ⊗ (−1) δ ⊗ C λ of G on the Fréchet space C ∞ (X, V i λ,δ )
of smooth sections. In our parametrization, I(i, n − 2i) δ and I(i, i) δ have the same infinitesimal character with the trivial one-dimensional representation of G. Then, for all u ∈ C, we have a natural G-isomorphism
(2.2) ̟ (i) u ≃ I(i, u + i) i mod 2 .
Similarly, for 0 ≤ j ≤ n−1, ε ∈ Z/2Z and ν ∈ C, we define an unnormalized principal series representation J(j, ν) ε :
= Ind G ′ P ′ j (C n−1 ) ⊗ (−1) ε ⊗ C ν of the subgroup G ′ = O(n, 1) on C ∞ (Y, W j ν,ε ), where W j ν,ε := G ′ × P ′ j (C n−1 ) ⊗ (−1) ε ⊗ C ν is a G ′ -equivariant vector bundle over Y = G ′ /P ′ ≃ S n−1 .
Existence condition for differential symmetry breaking operators
A continuous G ′ -intertwining operator T : I(i, λ) δ −→ J(j, ν) ε is said to be a symmetry breaking operator (SBO). We say that T is a differential
operator if T satisfies Supp(T f ) ⊂ Suppf for all f ∈ C ∞ (X, V i λ,δ ), and Diff G ′ (I(i, λ) δ , J(j, ν) ε )
denotes the space of differential SBOs. We give a complete solution to Problem A for (X, Y ) = (S n , S n−1 ) in terms of principal series representations:
Theorem 3.1. Let n ≥ 3. Suppose 0 ≤ i ≤ n, 0 ≤ j ≤ n − 1, λ, ν ∈ C,
and δ, ε ∈ Z/2Z. Then the following three conditions on 6-tuple (i, j, λ, ν, δ, ε) are equivalent:
(i) Diff O(n,1) (I(i, λ) δ , J(j, ν) ε ) = {0}. (ii) dim Diff O(n,1) (I(i, λ) δ , J(j, ν) ε ) = 1.
(iii) The 6-tuple belongs to one of the following six cases:
Case 1. j = i, 0 ≤ i ≤ n − 1, ν − λ ∈ N, ε − δ ≡ ν − λ mod 2. Case 2. j = i − 1, 1 ≤ i ≤ n, ν − λ ∈ N, ε − δ ≡ ν − λ mod 2. Case 3. j = i + 1, 1 ≤ i ≤ n − 2, (λ, ν) = (i, i + 1), ε ≡ δ + 1 mod 2. Case 3 ′ . (i, j) = (0, 1), −λ ∈ N, ν = 1, ε ≡ δ + λ + 1 mod 2. Case 4. j = i−2, 2 ≤ i ≤ n−1, (λ, ν) = (n−i, n−i+1), ε ≡ δ+1 mod 2. Case 4 ′ . (i, j) = (n, n − 2), −λ ∈ N, ν = 1, ε ≡ δ + λ + 1 mod 2.
We set Ξ := {(i, j, λ, ν): the 6-tuple (i, j, λ, ν, δ, ε) satisfies one of the equivalent conditions of Theorem 3.1 for some δ, ε ∈ Z/2Z}.
Construction of differential symmetry breaking operators
In this section, we describe an explicit generator of the space of differential SBOs if one of the equivalent conditions in Theorem 3.1 is satisfied. For this we use the flat picture of the principal series representations I(i, λ) δ of G which realizes the representation space
C ∞ (X, V i λ,δ ) as a subspace of C ∞ (R n , i (C n )) by trivializing the bundle V i λ,δ −→ X on the open Bruhat cell R n ֒→ X, (x 1 , · · · , x n ) → exp n j=1 x j N − j P.
Here {N − 1 , · · · , N − n } is an orthonormal basis of the nilradical n − (R) of the opposite parabolic subalgebra with respect to an M-invariant inner product. Without loss of generality, we may and do assume that the open Bruhat cell R n−1 ֒→ Y ≃ G ′ /P ′ is given by putting x n = 0. Then the flat picture of the principal series representation J(j, ν) ε of G ′ is defined by realizing C ∞ (Y, W j ν,ε ) as a subspace of C ∞ (R n−1 , j (C n−1 )). For the construction of explicit generators of matrix-valued SBOs, we begin with a scalar-valued differential operator. For α ∈ C and ℓ ∈ N, we define a polynomial of two variables (s, t) by
I ℓ C α ℓ (s, t) := s ℓ 2 C α ℓ t √ s ,
where C α ℓ (z) is the renormalized Gegenbauer polynomial given by
C α ℓ (z) := 1 Γ α + ℓ+1 2 [ ℓ 2 ] k=0 (−1) k Γ(ℓ − k + α) k!(ℓ − 2k)! (2z) ℓ−2k .
Then C α ℓ (z) is a nonzero polynomial for all α ∈ C and ℓ ∈ N, and a (normalized) Juhl's conformally covariant operator C λ,ν : C ∞ (R n ) −→ C ∞ (R n−1 ) is defined by
C λ,ν := Rest xn=0 • I ℓ C λ− n−1 2 ℓ −∆ R n−1 , ∂ ∂x n ,
for λ, ν ∈ C with ℓ := ν − λ ∈ N. For instance,
C λ,ν = Rest xn=0 • id if ν = λ, 2 ∂ ∂xn if ν = λ + 1, ∆ R n−1 + (2λ − n + 3) ∂ 2 ∂x 2 n if ν = λ + 2.
For (i, j, λ, ν) ∈ Ξ, we introduce a new family of matrix-valued differential operators
C i,j λ,ν : C ∞ (R n , i (C n )) −→ C ∞ (R n−1 , j (C n−1 )), by using the identifications E i (R n ) ≃ C ∞ (R n )⊗ i (C n ) and E j (R n−1 ) ≃ C ∞ (R n−1 )⊗ j (C n−1 )
, as follows. Let d * R n be the codifferential, which is the formal adjoint of the differential d R n , and ι ∂ ∂xn the inner multiplication by the vector field ∂ ∂xn . Both operators map E i (R n ) to E i−1 (R n ). For α ∈ C and ℓ ∈ N, let γ(α, ℓ) := 1 (ℓ is odd); = α + ℓ 2 (ℓ is even). Then we set
C i,i λ,ν := C λ+1,ν−1 d R n d * R n − γ(λ − n 2 , ν − λ) C λ,ν−1 d R n ι ∂ ∂xn + 1 2 (ν − i) C λ,ν for 0 ≤ i ≤ n − 1. C i,i−1 λ,ν := − C λ+1,ν−1 d R n d * R n ι ∂ ∂xn − γ(λ − n − 1 2 , ν − λ) C λ+1,ν d * R n + 1 2 (λ + i − n) C λ,ν ι ∂ ∂xn for 1 ≤ i ≤ n.
We note that there exist isolated parameters (λ, ν) for which C i,i λ,ν = 0 or C i,i−1 λ,ν = 0. For instance, C 0,0 λ,ν = 1 2 ν C λ,ν , and thus C 0,0 λ,ν = 0 if ν = 0. To be precise, we have the following:
C i,i
λ,ν = 0 if and only if λ = ν = i or ν = i = 0; C i,i−1 λ,ν = 0 if and only if λ = ν = n − i or ν = n − i = 0. We renormalize these operators by
C i,i λ,ν := Rest xn=0 if λ = ν, C λ,ν if i = 0, C i,i λ,ν otherwise, and C i,i−1 λ,ν := Rest xn=0 • ι ∂ ∂xn if λ = ν, C λ,ν • ι ∂ ∂xn if i = n, C i,i−1 λ,ν
otherwise.
Then C i,i λ,ν (0 ≤ i ≤ n − 1) and C i,i−1 λ,ν
(1 ≤ i ≤ n) are nonzero differential operators of order ν − λ for any λ, ν ∈ C with ν − λ ∈ N.
The differential operators C i,i+1 λ,ν and C i,i−2 λ,ν are defined only for special parameters (λ, ν) as follows.
C i,i+1 λ,i+1 := Rest xn=0 • d R n for 1 ≤ i ≤ n − 2, λ = i, d R n−1 • C λ,0 for i = 0, λ ∈ −N, C i,i−2 λ,n−i+1 := Rest xn=0 • ι ∂ ∂xn d * R n for 2 ≤ i ≤ n, λ = n − i, −d * R n−1 • C n,n−1 λ,0 for i = n, λ ∈ −N.
Then we give a complete solution to Problem B for the model space (X, Y ) = (S n , S n−1 ) in terms of the flat picture of principal series representations as follows:
Theorem 4.1. Suppose a 6-tuple (i, j, λ, ν, δ, ε) satisfies one of the equivalent conditions in Theorem 3.1. Then the operators C i,j λ,ν : C ∞ (R n ) ⊗ i (C n ) −→ C ∞ (R n−1 ) ⊗ j (C n−1 ) extend to differential SBOs I(i, λ) δ −→ J(j, ν) ε , to be denoted by the same letters. Conversely, any differential SBO from I(i, λ) δ to J(j, ν) ε is proportional to the following differential operators: in Case 4 ′ .
C i,i λ,ν in Case 1, C i,i−1 λ,ν in Case 2, C i,i+1 i,i+1 in Case 3, C 0,1 λ,1 in Case 3 ′ , C i,i−2 n−i,
Matrix-valued factorization identities
Suppose that T X : I(i, λ ′ ) δ → I(i, λ) δ or T Y : J(j, ν) ε → J(j, ν ′ ) ε are G-or G ′intertwining operators, respectively. Then the composition T Y •D X→Y or D X→Y •T X of a symmetry breaking operator D X→Y : I(i, λ) δ → J(j, ν) ε gives another symmetry breaking operator:
I(i, λ) δ D X→Y / / ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ J(j, ν) ε T Y I(i, λ ′ ) δ T X O O 6 6 ♠ ♠ ♠ ♠ ♠ ♠ ♠ J(j, ν ′ ) ε
The multiplicity-free property (see Theorem 3.1 (ii)) assures the existence of matrix-valued factorization identities for differential SBOs, namely, D X→Y • T X must be a scalar multiple of C i,j λ ′ ,ν , and T Y • D X→Y must be a scalar multiple of C i,j λ,ν ′ . We shall determine these constants explicitly when T X or T Y are Branson's conformally covariant operators [1] defined below. Let 0 ≤ i ≤ n. For ℓ ∈ N + , we set
T (i) 2ℓ := (( n 2 −i−ℓ)d R n d * R n +( n 2 −i+ℓ)d * R n d R n )∆ ℓ−1 R n = (−2ℓ d R n d * R n −( n 2 −i+ℓ)∆ R n )∆ ℓ−1 R n .
Then the differential operator T
(i) 2ℓ : E i (R n ) −→ E i (R n )
induces a nonzero O(n + 1, 1)-intertwining operator, to be denoted by the same letter T (i) 2ℓ , from I i, n 2 − ℓ δ to I i, n 2 + ℓ δ , for δ ∈ Z/2Z. Similarly, we define a G ′ -intertwining operator T ′ (j) 2ℓ :
if a = 0 ±2 if a = 0 , q = i + ℓ − n−1 2 if i = 0, a = 0 −2 if i = 0, a = 0 − ℓ + n−1 2 if i = 0 , r = i − ℓ − n+1 2 if i = n, a = 0 2 if i = n, a = 0 − ℓ + n+1 2 if i = n , K ℓ,a := ℓ k=1 a 2 + k .
Then the factorization identities for differential SBOs C i,j λ,ν for j ∈ {i − 1, i} and Branson's conformally covariant operators T (i) 2ℓ or T ′ (j) 2ℓ are given as follows. Theorem 5.1. Suppose 0 ≤ i ≤ n − 1, a ∈ N and ℓ ∈ N + . Then In the case where i = 0, C i,i λ,ν is a scalar-valued operator, and the corresponding factorization identities in Theorem 5.1 were studied in [4,8,9].
The main results are proved by using the F-method [5,6,9]. Details will appear elsewhere.
.
Suppose 1 ≤ i ≤ n, a ∈ N and ℓ ∈ N + . Then
Acknowledgements: The first named author was partially supported by Institut des Hauteś Etudes Scientifiques, France and Grant-in-Aid for Scientific Research (A) (25247006), Japan Society for the Promotion of Science. All three authors were partially supported by CNRS Grant PICS n o 7270.J j, n−1 2 − ℓ ε −→ J j, n−1 2 + ℓ ε for 0 ≤ j ≤ n − 1 and ε ∈ Z/2Z as the lift of the differential operator T ′ (j) 2ℓ : E j (R n−1 ) −→ E j (R n−1 ) which is given byConsider the following diagrams for j = i and j = i − 1:where parameters δ and ε ∈ Z/2Z are chosen according toTheorem 3.1 (iii). In what follows, we put
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Families of conformally covariant differential operators, Q-curvature and holography. A , http:/link.springer.com/book/10.1007/978-3-7643-9900-9/page/1Progr. Math., 275. Birkhäuser. A. Juhl, Families of conformally covariant differential operators, Q-curvature and holography. Progr. Math., 275. Birkhäuser, Basel, 2009.
F-method for symmetry breaking operators. T Kobayashi, 10.1016/j.difgeo.2013.10.003Diff. Geometry and its Appl. 33T. Kobayashi, F-method for symmetry breaking operators, Diff. Geometry and its Appl. 33, (2014), pp. 272-289.
Differential symmetry breaking operators. I. General theory and F-method. T Kobayashi, M Pevzner, 10.1007/s00029-015-0207-9Selecta. Math. (N.S.). 22T. Kobayashi, M. Pevzner, Differential symmetry breaking operators. I. General theory and F-method, Selecta. Math. (N.S.), 22, (2016), pp. 801-845.
Differential symmetry breaking operators. II. Rankin-Cohen operators for symmetric pairs. T Kobayashi, M Pevzner, 10.1007/s00029-015-0208-8Selecta. Math. (N.S.). 22T. Kobayashi, M. Pevzner, Differential symmetry breaking operators. II. Rankin-Cohen oper- ators for symmetric pairs, Selecta. Math. (N.S.), 22, (2016), pp. 847-911.
Symmetry Breaking for Representations of Rank One Orthogonal Groups. T Kobayashi, B Speh, 10.1090/memo/1126Memoirs of American Mathematical Society. 238T. Kobayashi, B. Speh, Symmetry Breaking for Representations of Rank One Orthogonal Groups, Memoirs of American Mathematical Society, vol. 238, 2015. 118 pp. ISBN: 978-1- 4704-1922-6.
Branching laws for Verma modules and applications in parabolic geometry. T Kobayashi, B Ørsted, P Somberg, V Souček, 10.1016/j.aim.2015.08.020I. Adv. Math. 285T. Kobayashi, B. Ørsted, P. Somberg, and V. Souček, Branching laws for Verma modules and applications in parabolic geometry. I. Adv. Math., 285, (2015), pp. 1796-1852.
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Kavli IPMU (WPI) and Graduate School of Mathematical Sciences. : T Addresses, Kobayashi, Komaba. The University of TokyoJapan; [email protected]: T. Kobayashi. Kavli IPMU (WPI) and Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan; [email protected].
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba. T Kubo, Meguro, TokyoJapan; [email protected]. Kubo. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan; [email protected].
M Pevzner, FR 3399 CNRS, F-51687. Reims, FranceLaboratoire de Mathématiques de Reims, Université de [email protected]. Pevzner. Laboratoire de Mathématiques de Reims, Université de Reims-Champagne-Ardenne, FR 3399 CNRS, F-51687, Reims, France; [email protected].
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[
"Doping control of magnetic anisotropy for stable antiskyrmion formation in schreibersite (Fe,Ni)3P with S4 symmetry",
"Doping control of magnetic anisotropy for stable antiskyrmion formation in schreibersite (Fe,Ni)3P with S4 symmetry"
]
| [
"Kosuke Karube [email protected] ",
"Licong Peng ",
"Jan Masell ",
"Mamoun Hemmida ",
"Hans-Albrecht Krug Von Nidda ",
"István Kézsmárki ",
"Xiuzhen Yu ",
"Yoshinori Tokura ",
"Yasujiro Taguchi ",
"K Karube ",
"L C Peng ",
"X Z Yu ",
"Y Taguchi ",
"J Masell ",
"M Hemmida ",
"H.-A Krug Von Nidda ",
"I Kézsmárki ",
"Y Tokura ",
"\nRIKEN Center for Emergent Matter Science (CEMS)\nRIKEN Center for Emergent Matter Science (CEMS)\nInstitute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT)\nExperimental Physics V\nRIKEN Center for Emergent Matter Science (CEMS)\nDepartment of Applied Physics\nUniversity of Augsburg\n351-0198, 351-0198, 76049, 86135, 351-0198Wako, Wako, Karlsruhe, Augsburg, WakoJapan., Japan., Germany., Germany., Japan\n",
"\nUniversity of Tokyo\nBunkyo-ku 113-8656Japan\n",
"\nTokyo College\nUniversity of Tokyo\nBunkyo-ku 113-8656Japan\n"
]
| [
"RIKEN Center for Emergent Matter Science (CEMS)\nRIKEN Center for Emergent Matter Science (CEMS)\nInstitute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT)\nExperimental Physics V\nRIKEN Center for Emergent Matter Science (CEMS)\nDepartment of Applied Physics\nUniversity of Augsburg\n351-0198, 351-0198, 76049, 86135, 351-0198Wako, Wako, Karlsruhe, Augsburg, WakoJapan., Japan., Germany., Germany., Japan",
"University of Tokyo\nBunkyo-ku 113-8656Japan",
"Tokyo College\nUniversity of Tokyo\nBunkyo-ku 113-8656Japan"
]
| []
| Magnetic skyrmions, vortex-like topological spin textures, have attracted much interest in a wide range of research fields from fundamental physics to spintronics applications. Recently, growing attention has also been paid to antiskyrmions emerging in opposite topological charge in non-centrosymmetric magnets with D2d or S4 symmetry. In these magnets, complex interplay among anisotropic Dzyaloshinskii-Moriya interaction, uniaxial magnetic anisotropy, and magnetic dipolar interactions generates a variety of magnetic structures. However, the relation between the stability of antiskyrmions and these magnetic interactions remains to be elucidated.In this work, we control the uniaxial magnetic anisotropy of schreibersite (Fe,Ni)3P with S4 symmetry by doping and investigate its impact on the stability of antiskyrmions. Our magnetometry study, supported by ferromagnetic resonance spectroscopy, shows that the variation of the Ni content and slight doping with 4d transition metals considerably change the magnetic anisotropy. In particular, doping with Pd induces easy-axis anisotropy, giving rise to | 10.1002/adma.202108770 | [
"https://arxiv.org/pdf/2202.08483v1.pdf"
]
| 245,986,869 | 2202.08483 | a3ff715c5474b56ae2d2e17687064eb573efa30d |
Doping control of magnetic anisotropy for stable antiskyrmion formation in schreibersite (Fe,Ni)3P with S4 symmetry
Kosuke Karube [email protected]
Licong Peng
Jan Masell
Mamoun Hemmida
Hans-Albrecht Krug Von Nidda
István Kézsmárki
Xiuzhen Yu
Yoshinori Tokura
Yasujiro Taguchi
K Karube
L C Peng
X Z Yu
Y Taguchi
J Masell
M Hemmida
H.-A Krug Von Nidda
I Kézsmárki
Y Tokura
RIKEN Center for Emergent Matter Science (CEMS)
RIKEN Center for Emergent Matter Science (CEMS)
Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT)
Experimental Physics V
RIKEN Center for Emergent Matter Science (CEMS)
Department of Applied Physics
University of Augsburg
351-0198, 351-0198, 76049, 86135, 351-0198Wako, Wako, Karlsruhe, Augsburg, WakoJapan., Japan., Germany., Germany., Japan
University of Tokyo
Bunkyo-ku 113-8656Japan
Tokyo College
University of Tokyo
Bunkyo-ku 113-8656Japan
Doping control of magnetic anisotropy for stable antiskyrmion formation in schreibersite (Fe,Ni)3P with S4 symmetry
1antiskyrmionS4 symmetryschreibersitemagnetic anisotropyLorentz transmission electron microscopyspintronics
Magnetic skyrmions, vortex-like topological spin textures, have attracted much interest in a wide range of research fields from fundamental physics to spintronics applications. Recently, growing attention has also been paid to antiskyrmions emerging in opposite topological charge in non-centrosymmetric magnets with D2d or S4 symmetry. In these magnets, complex interplay among anisotropic Dzyaloshinskii-Moriya interaction, uniaxial magnetic anisotropy, and magnetic dipolar interactions generates a variety of magnetic structures. However, the relation between the stability of antiskyrmions and these magnetic interactions remains to be elucidated.In this work, we control the uniaxial magnetic anisotropy of schreibersite (Fe,Ni)3P with S4 symmetry by doping and investigate its impact on the stability of antiskyrmions. Our magnetometry study, supported by ferromagnetic resonance spectroscopy, shows that the variation of the Ni content and slight doping with 4d transition metals considerably change the magnetic anisotropy. In particular, doping with Pd induces easy-axis anisotropy, giving rise to
formation of antiskyrmions, while a temperature-induced spin reorientation is observed in a Rhdoped compound. In combination with Lorentz transmission electron microscopy and micromagnetic simulations, we quantitatively analyze the stability of antiskyrmion as functions of uniaxial anisotropy and demagnetization energy, and demonstrate that subtle balance between them is necessary to stabilize the antiskyrmions.
Introduction
Vortex-like spin swirling objects, termed magnetic skyrmions and characterized by an integer topological winding number Nsk, have been extensively studied in the last decade, both in the fields of fundamental science and applications to spintronics devices. [1,2] One of the wellknown mechanisms of skyrmion formation is the competition between ferromagnetic exchange interaction and the Dzyaloshinskii-Moriya interaction (DMI) arising from the lack of inversion symmetry. Skyrmions with helical (Bloch type) and cycloidal (Néel type) spin configurations have been observed in non-centrosymmetric magnets with chiral (T or O class) and polar (Cnv class) structures, respectively. [3][4][5][6][7][8][9] Recently, a new topological spin texture, the antiskyrmion, with opposite sign of Nsk for the same polarity has attracted much attention. The antiskyrmion consists of both Bloch and Néel walls with the opposite helicities along two orthogonal axes, and its formation is attributed to the anisotropic DMI present in non-centrosymmetric tetragonal crystals belonging to the D2d or S4 symmetry group both containing four-fold rotoinversion (4 ̅ ). [10,11] In real materials, antiskyrmions were first found in Heusler compounds with D2d symmetry, Mn1.4PtSn and Mn1.4Pt0.9Pd0.1Sn. [12,13] More recently, the formation of antiskyrmions was also found in Fe1.9Ni0.9Pd0.2P [Pd-doped (Fe,Ni)3P] with S4 symmetry [14] and Fe/Gd-based multilayers [15] . Lorentz transmission electron microscopy (LTEM) observation for thin plates of these D2d and S4 magnets shows that the antiskyrmions are square shaped, and transform into bullet-shaped non-topological bubbles and elliptically deformed Bloch skyrmions, depending on magnetic fields, temperature, and lamella thickness. [13,14,16] According to numerical simulations, magnetic dipolar interaction (demagnetization energy) plays a dominant role in the formation of square-shaped antiskyrmions; [13,16,17] Néel walls (Bloch lines) at the corners of an antiskyrmion tend to shrink so as to reduce the magnetic volume charge. [18] Magnetic force microscopy studies have shown that the size of antiskyrmions increases significantly as the crystal thickness is increased, [14,19] and anisotropic fractal-like domains with 4 ̅ symmetry are induced near the surface of bulk crystals [14,20] by the interplay between DMI, uniaxial magnetic anisotropy and dipolar interactions to reduce the magnetic charge on the surface. [21] Despite these findings in the two families of antiskyrmion-hosting magnets, the relation between the stability of antiskyrmions and magnetic interactions (DMI, magnetic anisotropy, and dipolar interaction) is still unclear due to the lack of experimental studies where the relative strength of magnetic interactions is tuned systematically. In this work, we succeeded in controlling the magnetic anisotropy of schreibersite (Fe,Ni)3P with S4 symmetry by chemical doping to observe the formation of stable antiskyrmions. First, we searched for the appropriate solid solution of Fe and Ni to reduce the easy-plane anisotropy. Next, using 4d transition metals with strong spin-orbit coupling, we further modified the magnetic anisotropy to easy-axis type, leading to the formation of stable antiskyrmions and skyrmions. We finally identified the stable region of antiskyrmions and skyrmions on the plane of uniaxial magnetic anisotropy and demagnetization energy, thereby demonstrating the subtle balance between them to give rise to the antiskyrmion spin texture.
Results and Discussions
2-1. Structual properties
We synthesized bulk single crystals of schreibersite (Fe1-xNix)3P (0 ≤ x ≤ 0.66) and those doped with small amounts of 4d transition metals (Ru, Rh and Pd) listed in Table 1 by a selfflux method (see Note S1 and Table S1 for details of sample preparation). Using powder X-ray diffraction, we confirmed that all the compounds crystalize in a non-centrosymmetric tetragonal structure with the space group of 4 ̅ (No. 82, 4 2 ) as shown in Figure 1a (see Note S2, Figure S1, and Figure S2 for details). The tetragonal lattice constants (a, c) obtained by Rietveld analysis are plotted in Figure 1b. The parameter a decreases linearly with increasing Ni concentration, while c is almost unchanged below x = 0.47, in good agreement with a previous study. [22] When a small amount of Pd is substituted for Ni, both a and c increase linearly following Vegard's law. Similar increase in the lattice constants is also observed with Ru and Rh doping. with the non-centrosymmetric tetragonal space group of 4 ̅ . The structure viewed from the caxis is shown at the bottom. Three crystallographically inequivalent M sites are denoted as M1 (red), M2 (green) and M3 (blue). b) Lattice constants a and c at room temperature obtained from powder X-ray diffraction and Rietveld analysis are plotted as a function of x of (Fe1-xNix)3P and y of (Fe0.63Ni0.37-yPdy)3P. The data for (Fe1-xNix)3P from literature are also plotted with cross symbols. [22] The data points of (Fe0.62Ni0.29Pd0.09)3P plotted with open symbols deviate from the linear fit due to the slightly lower Fe concentration.
2-2. Magnetometry
To
where Ms is the saturation magnetization, and ( ) represents the magnetic field along the iaxis. Therefore, Ku is equal to the area enclosed by the magnetization curves along the two directions. In the present case, the magnetization process along the easy axis is dominated by the displacement of domain walls, but the contribution of this process is excluded in Equation x. The monotonous decrease in Tc and Ms with increasing x is in accord with the previous report on polycrystalline samples. [22] While there have been several studies on the magnetic properties of (Fe1-xMx)3P (M: Cr, Mn, Co, Ni), [22,[24][25][26] magnetocrystalline anisotropy has been reported only for (Fe1-xCox)3P. [27] As shown in Figure 2m, the large negative value of Ku for Fe3P (−848 kJ/m 3 at 2 K) determined in the present study coincides with the previous report. [27] The absolute value of Ku is rapidly suppressed by Ni substitution, reaching a minimum at x = 0.37 (−39 kJ/m 3 at 2 K). Nevertheless, the easy-plane anisotropy (Ku < 0) persists throughout the whole Ni concentration range in the present study of (Fe1-xNix)3P. Ni concentration x dependence of (k) ferromagnetic transition temperature Tc, (l) saturation magnetization Ms, and (m) uniaxial anisotropy constant Ku at 300 K and 2 K obtained from the magnetization measurements. The value of Ms was determined as the magnetization at μ0H = 3 T for Fe3P, 2 T for x = 0.18, and 1 T for x ≥ 0.37. For the conversion of Ms and Ku to unit volume, the lattice constants obtained by powder X-ray diffraction at room temperature were used. Tc and Ms of polycrystalline (Fe1-xNix)3P in the literature are indicated by cross symbols. [22] Ms and Ku of Fe3P at 300 K and 5 K in the literature are shown with open symbols. [27] Having thus established that the strong in-plane magnetic anisotropy (negative Ku) in (Fe1-xNix)3P can be appreciably reduced around x ~ 0.4 while keeping high Tc above room temperature, next we aim to further control Ku and to reverse its sign via doping with 4d
transition metals. such as MnBi, [28] Fe3Sn2, [29,30] and Mn1.4PtSn. [31] The spin reorientation temperature in the Rhdoped compound is determined to be TSR ~ 100 K where the kink-like anomaly is observed. As Figure S4). The power law with an exponent close to 3 agrees with the theoretical models for uniaxial anisotropy. [32][33][34] For the Pd 7% compound that shows Ms ~ 474 kA/m and Ku ~ 28 kJ/m 3 at room temperature, the quality factor is estimated to be Q ~ 0.2,
where Q is defined as = u /( 0 s 2 /2), the ratio of the uniaxial anisotropy constant to the maximum of demagnetization energy. These Ku and Q values are small as compared with those of the industrially-important commercial permanent magnets, such as (Sr,Ba)Fe12O19, Nd2Fe14B, SmCo5, [35] and the recently reported Heusler compound Mn1.4PtSn (Ku ~ 171 kJ/m 3 , Q ~ 1.7 at room temperature). [36] The values of Ku for the Rh-doped compound are in-between those for the Ru-and Pd-doped ones, and change from very small positive values at room temperature to negative values at low temperatures, leading to the temperature-induced spin reorientation. The significant change in the magnetic anisotropy from easy-plane to easy-axis with the Pd doping indicates the importance of enhanced spin-orbit coupling in the 4d element.
The systematic variation in the magnetic anisotropy with change in the dopant Ru, Rh, Pd is probably dominated by band filling of 4d orbitals, or position of the Fermi level.
While the systematic change of Ku with composition and temperature was identified by the magnetometry study using Equation (1), we performed ferromagnetic resonance (FMR) spectroscopy experiments on the Pd 7% compound to determine Ku by another method, and thus further validate our magnetometry-based approach for the quantification of anisotropy. .07)3P (equivalent to the notation Fe1.9Ni0.9Pd0.2P as adopted in our earlier work [14] ), and (e) (Fe0.62Ni0.29Pd0.09)3P. f-j) Magnetic field dependence of magnetization at 2 K for the same compositions as shown in the panels a-e. The magnetization data for Pd 9% are reproduced with permission from our earlier work. [14] (Copyright 2021, Springer Nature). The schematic figure of a rectangular bulk single crystal used for the measurements is shown at the top of panel f. k, l) Temperature dependence of (k) Ms and (l) Ku obtained from the magnetization measurements for the various compositions. The value of Ms was determined as the magnetization at 1 T for all the compounds. The values of Ku in (Fe0.63Ni0.30Pd0.07)3P obtained from the FMR measurement are also plotted with yellow star symbols.
2-3. Ferromagnetic resonance spectroscopy
We carried out FMR spectroscopy measurements at 9.4 GHz on a cylindrical disk prepared from a Pd 7% crystal and shown in the inset of Figure 4b. axes. The maxima of the microwave absorption curves correspond to the FMR field, Hres, whose angular dependence is displayed in Figure 4b for a full rotation of the field in the plane of the Methods section and in Ref. [37−39] , are plotted in Figure 3l. These values are in excellent agreement with those obtained from the analysis of the magnetization, supporting the fully quantitative determination of Ku in the present work. [37−39] , as described in the text.
2-4. Magnetic textures observed by Lorentz transmission electron microscopy
In order to observe magnetic structures in real space, LTEM measurements were carried out on (001) thin plates. We present the magnetic induction fields as deduced by transport-ofintensity equation (TIE) analyses for the compounds with Pd 7% (thickness t ~ 130 nm), Pd 4%
(t ~ 140 nm), and Rh 8% (t ~ 180 nm) in Figure 5. The stripe pattern with a few hundred nm periods observed for all the compounds at room temperature and zero magnetic field ( Figure 5a,d,g) corresponds to anisotropic helices with opposite helicities propagating along the [110] and [1 ̅ 10] axes.
In the Pd 7% sample, square antiskyrmions appear uniformly over the plate under a magnetic field perpendicular to the specimen (Figure 5b). Note that high-density antiskyrmions are observed only after the sample plate is tilted under a magnetic field to initially create nontopological bubbles, and tilted back to the original perpendicular position as detailed in the caption for Figure 5. [14] Similar antiskyrmion lattices are observed at low temperatures down to 100 K (Figure 5c).
In the Pd 4% sample, square antiskyrmions and elliptically deformed skyrmions coexist under a magnetic field (Figure 5e). This coexistence state is also observed for t ~ 170 nm, 190 nm, and 220 nm, but a homogeneous antiskyrmion lattice is not formed. At the thickness of t ~ 80 nm, only elliptic skyrmions are observed. The (co)existence of skyrmions in spite of the anisotropic DMI indicates that the dipolar interaction is dominant over the DMI in this system.
In the Rh 8% sample, while dense elliptic skyrmions with mixed helicities are stabilized under a magnetic field (Figure 5h), no antiskyrmions are observed at any field for any sample thickness from 70 nm to 240 nm. As the temperature is lowered in zero field, the stripe pattern changes to large (~ several micrometers) domains with in-plane magnetizations (Figure 5j), which is consistent with the behavior of spin reorientation in the bulk magnetization measurement (Figure 3b). The in-plane domains are arranged in such a way that the magnetic flux is closed as indicated with white arrows. The transition from the anisotropic helices to the in-plane closure domains starts at higher temperatures for smaller thickness, e.g., 200 K for t ~ 70 nm, 143 K for t ~ 110 nm and 150 nm, and 130 K for t ~ 180 nm. Furthermore, this transformation is accompanied by a coexistence region and a large thermal hysteresis; the stripe patterns remain partially at low temperatures, but once the change into the in-plane domains is complete, they hardly recover in a subsequent heating process ( Figure S5).
In Figure 5f and 5i, enlarged views of an elliptic skyrmion are presented. The elliptic deformation is due to the cooperative interplay between the dipolar interaction and the anisotropic DMI inherent to the crystal structure with D2d or S4 symmetry, [13,14,16] and hence distinct from elliptic skyrmions induced by artificial anisotropy. [40,41] From the observed ellipticity, the magnitude of DMI is roughly estimated to be ~ 26% of the demagnetization energy. [13] In addition, the variation of the DMI with composition is negligible since the ellipticity of the skyrmion is comparable for the Pd 4% and Rh 8% compounds. Although the change in the DMI for these compounds investigated in this work is small, it might vary significantly in a wider compositional range, similarly to the Co/Pt bilayers where the interfacial DMI depends on the band filling of 5d orbitals. [42] To determine the anisotropic DMI more quantitatively and understand its compositional dependence comprehensively, further studies (e.g. Brillouin light scattering [43] , spin-wave spectroscopy [44] ) are necessary. Therefore, while the anisotropic DMI in this system is indeed important, we focus hereafter on the effects of larger and composition/temperature-dependent demagnetization energy and magnetic anisotropy on the formation of the topological spin textures.
2-5. Stability of antiskyrmions governed by demagnetization energy and uniaxial anisotropy
To understand the effect of the thickness-dependent dipolar interactions quantitatively, we estimate demagnetization energy Ed for the thin plates using the following theoretical equation for a helical stripe (Bloch wall type) with a period of λ and a film thickness of t, [19,45] of the anisotropic helices at room temperature and zero field for the composition with Pd 7%, Pd 4%, and Rh 8%. The thickness dependence is well described by the Kittel's law ( ∝ 1/2 ), [46] which is attributed to the competition between demagnetization energy and domain wall energy. The calculated Ed by using Equation (2) at room temperature for each compound is plotted against t in Figure 6b, which increases monotonically with decreasing thickness. demonstrated in multilayers. [15] The phase diagram in Figure 6c also describes the temperature-induced change from the anisotropic helices to the in-plane domains when the sign of Ku changes to negative in the case of the Rh 8% doping (although initially formed anisotropic helices at room temperature with
d = 0 2 2 ( 2 )(2)
Ku > 0 can survive to the region of Ku < 0 as a metastable state). The phase boundary between the helices and the in-plane domains shows a positive slope, indicating that the dipolar interaction also contributes to the spin reorientation. The easy-plane anisotropy may also give rise to more interesting in-plane topological magnetism such as bimeron. [47,48] All the spin textures are plotted together, which include anisotropic helix and in-plane domains at zero field, and antiskyrmions and skyrmions under fields, for the Pd 7%, Pd 4%, and Rh 8% compounds with various thicknesses shown in panel b. Low-temperature data for Pd 7% (t = 130 nm) and Rh 8% (t ≤ 180 nm) are also included. The data points for Pd 7% are reproduced in part from our earlier work. [14] (Copyright 2021, Springer Nature).
2-6. Micromagnetic simulations
The experimental observation of the antiskyrmions stabilized by uniaxial anisotropy is in qualitative agreement with our micromagnetic simulations, presented in Figure 7. In Figure 7a, we show simulation results of magnetic textures at various Ku and external magnetic fields with a constant dipolar energy density of Ed = 66 kJ/m 3 (note that the horizontal axis is different from that in Figure 6c). We start from an initially triangular antiskyrmion lattice and relax it at
Conclusion
We have systematically studied the magnetic properties of schreibersite (Fe,Ni)3P with S4 symmetry by performing magnetometry, FMR spectroscopy, LTEM, and micromagnetic simulations, demonstrating that the magnetic anisotropy can be finely controlled by the composition to stabilize antiskyrmions. The strong easy-plane anisotropy of Fe3P is rapidly suppressed by partial Ni substitution, and when additionally doped with a small amount of Pd, the magnetic anisotropy switches to an easy-axis type, leading to the formation of antiskyrmions.
The Rh-doped compound exhibits a temperature-induced spin reorientation transition, which is directly observed by LTEM. The phase diagram regarding the magnetic textures clearly shows stable regions of antiskyrmions and elliptic skyrmions on the plane of the uniaxial anisotropy and the demagnetization energy. These findings unveil that easy-axis type uniaxial magnetic anisotropy and dipolar interaction with appropriate balance are both necessary to stabilize antiskyrmions, and will help to design new antiskyrmion systems towards applications in spintronics.
Methods
Sample preparation: Bulk single crystals of M3P (M: Fe, Ni, Ru, Rh, and Pd) were synthesized by a self-flux method from the initial molar ratio of M : P ~ 3.5 : 1 (Note S1 and Table S1).
Chemical compositions of the obtained crystals were determined by the energy dispersive Xray analysis (Table S1). Lattice constants of M3P with the non-centrosymmetric tetragonal structure were determined by powder X-ray diffraction and Rietveld analysis (Note S2, Figure S1, and Figure S2). To obtain the uniaxial anisotropy parameter Ku at these temperatures, the angular dependence of the resonance field was fitted using the Smit-Suhl formula, [49] similarly to the procedure followed in Ref. [38] . using over-and under-focus LTEM images. [50] Micromagnetic simulations: For the theoretical description of the magnetization we consider the same continuum model as in Ref. [14] , i.e., including the magnetic stiffness Aex, S4-symmetric Dzyaloshinskii-Moriya interaction D (DMI), uniaxial anisotropy Ku, Zeeman field H, and dipolar interactions due to the saturation magnetization Ms. For the numerical implementation, we use our modified version of MuMax3 [51,52] after which the textures appear to be in quasi-equilibrium (t = 10 ns). This method yields an approximation for the energetically most favorable texture which otherwise is a complex highdimensional optimization problem with many local minima. The resulting thickness-averaged magnetization is shown in Figure 7 together with the skyrmion number which is the twodimensional winding number [53] up to a sign. The dipolar energy density Ed = 66 kJ/m 3 is determined by deterministically finding the optimized helical wavelength at every value of the anisotropy which turns out to be qualitatively constant in the regime of interest.
[ Details of the loaded compositions are given in Table S1. Table S1 for details of temperatures and cooling rates). After this slow cooling, single crystals of M3P with metallic luster ranging in size from ~ 0.5 mm to ~ 2 mm were found in the ingot. It was found that larger single crystals could be obtained by Bridgman growth. Since the flux (mixture of M-rich cubic phase and M3P) remained in the ingot and could not be completely removed by centrifugation, the ingot was finally crushed and cut to separate the single crystals of M3P.
The final chemical composition of the obtained M3P crystals was determined by an energy dispersive X-ray (EDX) spectrometer (Bruker, XFlash6) equipped with a scanning electron microscope (JEOL, JSM-6701F). The standardless ( ) method was used to quantify the EDX spectra. The results of the EDX analysis are summarized in Table S1.
Note S2. Structural characterizations
The non-centrosymmetric tetragonal crystal structure of M3P with the space group of 4 ̅ (No. 82, 4 2 ) was confirmed by powder X-ray diffraction (Rigaku, RINT-TTR III) with Cu Kα radiation as shown in Figures S1 and S2. The lattice constants a and c were determined by Rietveld analysis using the RIETAN-FP program, [S1] and are summarized in Table 1 in the main text. In the M3P structure, there are three different crystallographic M sites (M1, M2, M3).
It is difficult to determine the exact site occupancy in the three sites by X-ray diffraction because of the similar scattering powers of Fe and Ni, and the small concentration of 4d metals. In the Rietveld calculations shown in Figures S1 and S2, we assume that M is evenly distributed over the three sites, and good fits were obtained. Previous synchrotron radiation diffraction study on (Fe,Ni)3P reported that Fe and Ni preferentially occupy the M1 and M3 sites, respectively, while the M2 site is randomly occupied by both Fe and Ni. [S2] However, the refinements after setting this preferential site occupancy for Fe and Ni did not show further improvement.
The (110) and (001) planes of the single crystals were determined by X-ray backreflection Laue photography (Rigaku, RASCO-BL II). In order to confirm the quality of the single crystals, X-ray diffraction on the (001) surface of a single crystal of (Fe0.63Ni0.30Pd0.07)3P was carried out. As shown in Figure S3, the full width at half maximum (FWHM) of the rocking ( ) scan of the (004) peak was ~ 0.08°. The single peak and the narrow FWHM indicate that this single crystal is of high quality. Figure S1. Powder X-ray diffraction patterns and Rietveld analysis of (Fe1-xNix)3P at room temperature. Figure S2. Powder X-ray diffraction patterns and Rietveld analysis of compounds doped with 4d transition metals at room temperature.
Figure 1 .
1Structural properties of (Fe1-xNix)3P. a) Crystal structure of M3P (M: transition metal)
( 1 )
1. The sign of Ku indicates the direction of the anisotropy; Ku > 0 for the easy-axis anisotropy while Ku < 0 for the easy-plane type.The temperature dependence M(T) at 0.01 T and the magnetic field variation M(H) at 2 K in (Fe1-xNix)3P are presented in Figure 2. Fe3P exhibits a ferromagnetic transition around 680 K and shows strong easy-plane anisotropy, as indicated by smaller M values and larger saturation field in the [001] axis than those in the [110] axis (Figure 2a,f). Partial substitution of Fe with Ni lowers the magnetic transition temperature Tc (Figure 2b-e) and decreases the saturation value of magnetization Ms (Figure 2g-j). Furthermore, the difference between the magnetization curves along the [110] and [001] axes becomes smaller as the Ni substitution proceeds, and is the smallest at the Ni concentration of 37%. M(T) of (Fe0.34Ni0.66)3P decreases below 40 K (Figure 2e) probably due to the spin glass behavior of the diluted Fe moments. The magnetic parameters, Tc, Ms, and Ku, obtained from the magnetization measurements using Equation (1) are plotted in Figure 2k-m as a function of Ni concentration
Figure 2 .
2Magnetic properties of (Fe1-xNix)3P. a-e) Temperature (T) dependence of magnetization M under the magnetic field μ0H = 0.01 T for (a) Fe3P, (b) (Fe0.82Ni0.18)3P, (c) (Fe0.63Ni0.37)3P, (d) (Fe0.53Ni0.47)3P, and (e) (Fe0.34Ni0.66)3P. The data of field cooling (FC) and field warming after a zero-field cooling (ZFC) are denoted with solid and dotted lines, respectively. f-j) Magnetic field dependence of magnetization at 2 K for the same compositions as shown in the panels a-e. Magnetizations for the [110] and [001] axes are shown by red and blue lines, respectively. The schematic figure of a rectangular bulk single crystal used for the measurements is shown at the top of panel f. k-m)
Figure 3
3shows the effect of 4d transition-metal doping on the magnetic properties. In the Ru-doped compound, (Fe0.59Ni0.32Ru0.09)3P (Figure 3a,f), weak easy-plane anisotropy is observed in the whole temperature range below Tc ~ 340 K, as in the case of (Fe,Ni)3P. On the other hand, the Rh-doped sample, (Fe0.60Ni0.32Rh0.08)3P (Figure 3b,g), shows a complex temperature dependence. Below Tc ~ 400 K, the magnetization along the [110] axis increases gradually upon cooling, whereas that along the [001] axis is almost independent of temperature. The magnetization along the [110] axis saturates at 100 K, below which the magnetization along the [001] axis slightly decreases. This behavior is a typical indicator of spin reorientation from the c-axis to the ab-plane, which is often seen in uniaxial ferromagnets
for another dopant Pd, 4% doped (Fe0.63Ni0.33Pd0.04)3P (Figure 3c,h) does not show spin reorientation and exhibits weak easy-axis anisotropy in the whole temperature range below Tc ~ 400 K. In Pd 7% doped (Fe0.63Ni0.30Pd0.07)3P (Figure 3d,i) and 9% doped (Fe0.62Ni0.29Pd0.09)3P (Figure 3e,j), the easy-axis anisotropy is more clearly seen than in the Pd 4% doped sample. The values of Ms and Ku in the investigated compounds are plotted against temperature in Figure 3k and 3l. While the Ms values are similar for the different compositions, the sign and the absolute value of Ku considerably change with the 4d metal dopant species. In particular, easy-axis anisotropy (Ku > 0) is induced by small amount of Pd doping, and the value of Ku increases with decreasing temperature and increasing the Pd concentration. The temperature dependence of Ku and Ms in the Pd 7% and 9% doped compounds are found to obey the relation u ( ) ∝ [ s ( )] 2.7 (
Figure 3 .
3Magnetic properties of 4d-metal doped compounds. a-e) Temperature dependence of magnetization under the magnetic field of μ0H = 0.01 T for (a) (Fe0.59Ni0.32Ru0.09)3P, (b) (Fe0.60Ni0.32Rh0.08)3P, (c) (Fe0.63Ni0.33Pd0.04)3P, (d) (Fe0.63Ni0.30Pd0
Figure 4a displays representative field-swept FMR spectra recorded at room temperature for various orientations of the magnetic field applied in the plane of the disk, which contains both the [110] and [001]
Figure 4 .
4Ferromagnetic resonance (FMR) spectroscopy on (Fe0.63Ni0.30Pd0.07)3P at 300 K. a) FMR spectra, representing the absorbed power Pabs as a function of the magnetic field strength, μ0H, at a constant microwave frequency 9.4 GHz, displayed at selected θ angles of the magnetic field. The vertical bars, marking the resonance field positions, are guides to the eyes for tracing the angular periodicity of the spectra. b) Dependence of the resonance field at room temperature on the orientation of the magnetic field, Hres(θ), upon its rotation in the plane spanned by the [001] and [110] axes. The angle θ is measured from the [001] axis as indicated in the inset, which depicts the single crystal under investigation prepared as a thin cylindrical disk. The solid black line shows a fit by uniaxial magnetocrystalline anisotropy
Figure 5 .
5Magnetic structures observed by LTEM. a-j) Color mapping of in-plane magnetic induction fields deduced by transport-of-intensity equation (TIE) analyses of over-and underfocus LTEM images. [50] The color-coded wheel and the schematic figure of the experimental configuration (crystal axes of the thin plate and the external field) at the bottom right side are common for all the panels. The images of antiskyrmions and skyrmions (panels b,c,e,h) were obtained after tilting the plate under the external field and then back to the original position, as described in the following, where the tilt angles around the [1 ̅ 10] and [110] axes are denoted as α and β, respectively. a-c) Field images for (Fe0.63Ni0.30Pd0.07)3P with thickness t ~ 130 nm at (a) 295 K and 0 mT, (b) 295 K and 350 mT [process: 0 mT (α ~ 0°) → 350 mT (α ~ 12°) → 350 mT (α ~ 0°)], and (c) 100 K and 650 mT [process: ZFC → 100 K and 0 mT (α ~ 0°) → 550 mT (α ~ 12°) → 550 mT (α ~ 0°) → 650 mT (α ~ 0°)]. d-f) Field images for (Fe0.63Ni0.33Pd0.04)3P with t ~ 140 nm at (d) 295 K and 0 mT, and (e, f) 295 K and 385 mT [process: 0 mT (α ~ 0°, β ~ 0°) → 385 mT (α ~ 2.4°, β ~ 1°) → 385 mT (α ~ 0°, β ~ 0°)]. g-j) Field images for (Fe0.60Ni0.32Rh0.08)3P with t ~ 180 nm at (g) 295 K and 0 mT, (h, i) 295 K and 450 mT [process: 0 mT (α ~ 0°) → 450 mT (α ~ 5°) → 450 mT (α ~ 0°)], and (j) 94 K and 0 mT. Panels f and i display enlarged views of the elliptic skyrmions shown in panels e and h, respectively. The ellipticity (ratio of the length of the major axis to that of the minor axis) of the skyrmion in panels f and i are 1.70 and 1.73, respectively. The magnetic induction fields in the in-plane domains (panel j) are also indicated with white arrows.
where the function ( ) is defined as ( ) = (1 − − )/ . For small x (i.e. ≪ ), ( ) is simplified as ( ) ≈ 1 − /2, while ( ) ≈ 1/ for large x (i.e. ≫ ).
Figure 6ashows λ(t)
Finally
, we map out the magnetic objects observed by LTEM onto the plane of Ku and Ed in Figure 6c. Here, anisotropic helices and in-plane domains at zero field, and antiskyrmions/skyrmions under magnetic fields are plotted all together. The Ku−Ed phase diagram indicates that antiskyrmions are stabilized when Ku is positive, sufficiently large and comparable to Ed. On the other hand, when Ku is very small as compared to Ed, as in the cases of Rh doping, or Pd doping with small thickness, elliptic skyrmions become more dominant. This result proves that the strong dipolar interaction destabilizes the antiskyrmions with magnetic volume charge, and favors the Bloch skyrmions without it. Therefore, uniaxial anisotropy and dipolar interaction are both important ingredients for stabilizing antiskyrmions in S4 magnets with anisotropic DMI (and probably D2d magnets as well). The significant role of uniaxial anisotropy and dipolar interaction for stabilizing antiskyrmions has been also
Figure 6 .
6Demagnetization energy and magnetic phase diagram. a, b) Thickness (t) dependence of (a) magnetic periodicity of the anisotropic helix and (b) demagnetization energy Ed at room temperature for (Fe0.63Ni0.30Pd0.07)3P, (Fe0.63Ni0.33Pd0.04)3P, and (Fe0.60Ni0.32Rh0.08)3P. c) Magnetic phase diagram observed by LTEM on the plane of uniaxial anisotropy constant Ku and Ed.
finite temperature. As a result, we obtain a field-polarized ferromagnetic state at large magnetic fields μ0H ≥ 500 mT, in-plane states at low anisotropy Ku ≤ 30 kJ/m 3 , and antiskyrmions at large anisotropies Ku ≥ 70 kJ/m 3 ≈ Ed of the order of the dipolar energy density Ed and average external magnetic fields μ0H ≤ 375 mT. This antiskyrmion phase pocket is surrounded by a belt of dipolar stabilized skyrmions, including a crossover regime where skyrmions and antiskyrmions coexist. A particularly instructive example is displayed in the panel for μ0H = 300 mT and Ku = 80 kJ/m 3 (enlarged view is shown in Figure 7c), which contains the characteristically distorted anti-/skyrmions: two square-shaped antiskyrmions in the lower row and in the upper row two elliptical skyrmions with opposite handedness and consequently, perpendicular elongation axes. Moreover, we checked that for thicker films the onset of the antiskyrmion phase shifts to lower anisotropies, in agreement with both our experimental observations and the argument that DMI-stabilized antiskyrmions benefit from the larger volume fraction.
Figure 7 .
7Thickness-averaged magnetization as obtained from three-dimensional micromagnetic simulations for a 100 nm thick film. a) Simulated magnetic textures on the plane of uniaxial anisotropy energy Ku and external magnetic field μ0H. The color encodes the orientation of the magnetization with white (black) indicating the +z (-z) component. The color of the frame of every panel encodes the total skyrmion number from +4 (blue, four skyrmions) to -4 (red, four antiskyrmions). The dipolar energy density in the simulations is Ed = 66 kJ/m 3 . b-c) Enlarged views of the magnetic textures at μ0H = 300 mT and (b) Ku = 140 kJ/m 3 , (c) 80 kJ/m 3 , (d) 40 kJ/m 3 , where the skyrmion number for each magnetic object is also indicated (+1 for skyrmion, -1 for antiskyrmion).
Magnetization measurement :
measurementFor magnetization measurements, single crystals were cut into a 1-mm-scale rectangle with flat (110) and (001) surfaces and approximately the same length along the [110] and [001] axes (Figure S3). Magnetization measurements were carried out using a superconducting quantum interference device magnetometer (MPMS3, Quantum Design) equipped with an oven option.
FMR measurement :
measurementAngular dependent ferromagnetic resonance (FMR) experiments were performed for the temperature range 250 K ≤ T ≤ 380 K, using a Bruker ELEXSYS E500 CW X-band (9.4 GHz) spectrometer. The sample of (Fe0.63Ni0.30Pd0.07)3P was in a continuous nitrogen gas-flow cryostat with temperature stability of about 1 K. The measurements were performed on a cylindrical thin disk (diameter/thickness = 5), whose plane contains both the tetragonal [001] axis and the [110] axis perpendicular to it. The orientation of the sample was controlled by a programmable goniometer in 5° steps during a full rotation of the magnetic field in the plane of the cylindrical disk at selected temperatures: 250 K, 300 K, 350 K and 380 K.
LTEM measurement :
measurementFor Lorentz transmission electron microscopy (LTEM) measurements, (001) thin plates with various thickness were thinned from bulk single crystals by a focused ion beam (FIB) system (NB5000, Hitachi); t ~ 70, 110, 150, 180, and 240 nm for (Fe0.60Ni0.32Rh0.08)3P; t ~ 80, 140, 170, 190, and 220 nm for (Fe0.63Ni0.33Pd0.04)3P; t ~ 50, 70, 100, 130, 160, and 210 nm for (Fe0.63Ni0.30Pd0.07)3P. LTEM measurements were performed with a transmission electron microscope (JEM-2100F, JEOL) equipped with a double-tilt liquidnitrogen holder (Gatan 636) and a double-tilt heating holder (Protochips: Fusion select). External magnetic fields applied to the (001) plates were obtained by tuning the objective lens current of JEM-2100F, which are parallel to the incident electron beam. The distribution of inplane magnetic induction fields was obtained by a transport of intensity equation (TIE) analysis
in which we flipped the sign of the derivatives in the DMI-field in one spatial direction. Thus, the DMI favors right-handed helices in the horizontal direction and left-handed helices in the vertical direction. We choose the micromagnetic parameters as Aex = 8.1 pJ/m, D = 0.2 mJ/m 2 , and Ms = 600 kA/m. For obtaining the results in Figure 7, we first relax a triangular lattice of antiskyrmions at zero temperature at some convenient values of the magnetic field and anisotropy. The simulated system with periodic boundary conditions in the x-y-plane measures 800 nm x 700 nm x 100 nm, discretized on 128 x 112 x 32 lattice sites, and contains 4 antiskyrmions. In a second step, we use this texture as the starting point for simulations at finite temperature T (T = 600 K) for a timespan t
The excess amount of M acts as a flux for the growth of M3P and avoids formation of M2P. The BN crucible was sealed in an evacuated quartz tube and heated slowly in a standard box furnace to 1100 ~ 1200℃ over 3 ~ 4 days, held for 12 ~ 24 hours, and then cooled to room temperature over 12 ~ 24 hours. At this preliminary reaction stage, the ingot contains polycrystalline M3P and other phases (M-rich cubic phase and M2P). The ingot was then crushed into small pieces, sealed together with the BN crucible again in an evacuated quartz tube, heated above the melting point and then slowly cooled in a standard box furnace or a Bridgman furnace (see
Figure S3 .
S3Rocking curve of the (004) X-ray diffraction peak (at 2θ = 86.78 deg) at room temperature on the surface of the single crystal of (Fe0.63Ni0.30Pd0.07)3P. The inset shows the photograph of the single crystal on a mm-scale grid sheet.
Figure S4 .
S4The relation between normalized uniaxial anisotropy constant Ku(T)/Ku(2 K) and the saturation magnetization Ms(T)/Ms(2 K) for (Fe0.63Ni0.30Pd0.07)3P and (Fe0.62Ni0.29Pd0.09)3P at various temperatures is plotted on a logarithmic scale. Power-law fits are indicated by lines.
Figure S5 .
S5Temperature variations of LTEM images for (Fe0.60Ni0.32Rh0.08)3P with a thickness of t ~ 110 nm at zero field. The cooling process from room temperature (RT) down to 123 K and the subsequent warming process back to RT are presented. The domain walls of the inplane ferromagnetic state at 123 K are indicated with the light blue arrows.
Table 1 .
1Lattice constants (a, c) at room temperature, magnetic transition temperature (Tc),
saturation magnetization (Ms) and uniaxial anisotropy constant (Ku) at 300 K and 2 K for
(Fe,Ni)3P and doped with 4d transition metals.
Composition
a
[Å]
c
[Å]
Tc
[K]
Ms (300 K)
[kA/m]
Ms (2 K)
[kA/m]
Ku (300 K)
[kJ/m 3 ]
Ku (2 K)
[kJ/m 3 ]
Fe3P
9.1018(8)
4.4587(4)
681
1080
1160
−647
−848
(Fe0.82Ni0.18)3P
9.0713(4)
4.4627(2)
557
801
925
−98.4
−205
(Fe0.63Ni0.37)3P
9.0425(5)
4.4607(3)
412
518
743
−6.73
−38.8
(Fe0.53Ni0.47)3P
9.0266(5)
4.4561(3)
334
306
632
−4.06
−44.6
(Fe0.34Ni0.66)3P
9.0006(5)
4.4429(3)
152
-
381
-
−62.4
(Fe0.59Ni0.32Ru0.09)3P 9.0934(6)
4.4907(3)
337
363
693
−5.65
−53.2
(Fe0.60Ni0.32Rh0.08)3P 9.0931(7)
4.4855(4)
394
503
731
1.89
−20.1
(Fe0.63Ni0.33Pd0.04)3P 9.0797(6)
4.4745(3)
399
511
750
22.2
51.3
(Fe0.63Ni0.30Pd0.07)3P 9.1122(7)
4.4865(4)
398
474
707
27.9
83.8
(Fe0.62Ni0.29Pd0.09)3P 9.1199(6)
4.4895(3)
392
451
704
33.5
114
characterize bulk magnetic properties, magnetization (M) was measured under magnetic fields applied parallel to the [110] and [001] axes, being perpendicular and parallel to the 4 ̅ axis, respectively. For the magnetization measurements, single crystals were cut into arectangular shape (Figure S3) so that the demagnetization factors in the [110] and [001]
directions were the same. By adopting this sample shape with the same demagnetization factors,
the uniaxial magnetic anisotropy constant (Ku) is directly obtained, without explicitly taking the
demagnetization effect into account, from the difference between the Helmholtz magnetic free
energy along the [110] and [001] axes, [23]
u = ∫ [ [110] ( ) − [001] ( )]
s
0
disk. The minimum of Hres is observed for magnetic field along the [001] axis, while the maximum is reached for field parallel to the [110] axis, clearly demonstrating the easy-axis nature of the anisotropy with the [001] direction being the easy axis. Due to the cylindrical shape of the sample, the demagnetization factor is unchanged upon the field rotation, hence the angular dependence of Hres is solely governed by Ku. Similar Hres(θ) curves were recorded at 250 K, 350 K and 380 K. The Ku values obtained by fitting these curves, as described in the
53] B. Berg, M. Lüscher, Nucl. Phys. B 1981, 190(2), 412-424. Doping control of magnetic anisotropy for stable antiskyrmion formation in schreibersite (Fe,Ni)3P with S4 symmetry Kosuke Karube*, Licong Peng, Jan Masell, Mamoun Hemmida, Hans-Albrecht Krug von Nidda, István Kézsmárki, Xiuzhen Yu, Yoshinori Tokura, Yasujiro Taguchi Note S1. Sample preparations Bulk single crystals were prepared by the following self-flux method. Pure metal M (Fe, Ni, Ru, Rh, Pd) powders (except for wire cuts of Pd) and red phosphorous P were loaded into a boron nitride (BN) crucible in a molar ratio of M : P ~ 3.5 : 1, with a total mass of ~ 3 g.Supporting Information
Table S1 .
S1Summary of single crystal growth methods and EDX results of M3P Slow transfer from the hot zone (990°C) to the cold zone (930°C) in a Bridgman furnace at a rate of 0.5 mm/h for 10 days Slow transfer from the hot zone (990°C) to the cold zone (930°C) in a Bridgman furnace at a rate of 0.5 mm/h for 10 days Slow transfer from the hot zone (950°C) to the cold zone (900°C) in a Bridgman furnace at a rate of 0.5 mm/h for 6 days Slow transfer from the hot zone (970°C) to the cold zone (910°C) in a Bridgman furnace at a rate of 0.5 mm/h for 9 days Slow transfer from the hot zone (970°C) to the cold zone (910°C) in a Bridgman furnace at a rate of 0.5 mm/h for 9 daysComposition
Loading mol ratio
Single crystal growth process
after the second melting
EDX results
Fe3P
Fe : P =
3.6 : 1
Slow cool from 1160°C to 1050°C
at a rate of 0.4°C/h in a box furnace
Fe3.026(3)P0.974(3)
(Fe0.82Ni0.18)3P
Fe : Ni : P =
2.8 : 0.7 : 1
Slow transfer from the hot zone (1030°C) to
the cold zone (950°C) in a Bridgman furnace
at a rate of 0.5 mm/h for 11 days
Fe2.489(8)Ni0.534(5)P0.977(6)
(Fe0.63Ni0.37)3P
Fe : Ni : P =
2.05 : 1.45 : 1
Slow transfer from the hot zone (1025°C) to
the cold zone (950°C) in a Bridgman furnace
at a rate of 0.4 mm/h for 13 days
Fe1.911(4)Ni1.103(7)P0.985(3)
(Fe0.53Ni0.47)3P
Fe : Ni : P =
1.75 : 1.75 : 1
Slow cool from 1020°C to 960°C
at a rate of 0.2°C/h in a box furnace
Fe1.609(7)Ni1.409(5)P0.982(2)
(Fe0.34Ni0.66)3P
Fe : Ni : P =
1.15 : 2.35 : 1
Slow cool from 990°C to 950°C
at a rate of 0.2°C/h in a box furnace
Fe1.025(4)Ni1.987(8)P0.988(5)
(Fe0.59Ni0.32Ru0.09)3P
Fe : Ni : Ru : P =
2.0 : 1.2 : 0.3 : 1
Fe1.776(7)Ni0.967(8)Ru0.267(2)P0.989(2)
(Fe0.60Ni0.32Rh0.08)3P
Fe : Ni : Rh : P =
2.0 : 1.2 : 0.3 : 1
Fe1.795(3)Ni0.975(4)Rh0.239(1)P0.991(4)
(Fe0.63Ni0.33Pd0.04)3P
Fe : Ni : Pd : P =
2.0 : 1.2 : 0.3 : 1
Fe1.911(4)Ni0.997(4)Pd0.121(1)P0.971(3)
(Fe0.63Ni0.30Pd0.07)3P
Fe : Ni : Pd : P =
1.8 : 1.1 : 0.6 : 1
Fe1.897(7)Ni0.892(8)Pd0.217(3)P0.994(5)
(Fe0.62Ni0.29Pd0.09)3P
Fe : Ni : Pd : P =
1.75 : 1.05 : 0.7 : 1
Fe1.877(6)Ni0.898(4)Pd0.259(2)P0.966(3)
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| []
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[
"Uniform estimates for a Modica-Mortola type approximation of branched transportation",
"Uniform estimates for a Modica-Mortola type approximation of branched transportation"
]
| [
"Antonin Monteil "
]
| []
| []
| Models for branched networks are often expressed as the minimization of an energy M α over vector measures concentrated on 1-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation M α ε , introduced by Edouard Oudet and Filippo Santambrogio in 2010, which is defined over H 1 vector measures. These energies induce some pseudo-distances between L 2 functions obtained through the minimization problem min{M α ε (u) :We prove some uniform estimates on these pseudo-distances which allow us to establish a Γ-convergence result for these energies with a divergence constraint. | 10.1051/cocv/2015049 | [
"https://arxiv.org/pdf/1503.03735v3.pdf"
]
| 54,837,065 | 1503.03735 | 7df1072ad0b4253fb59034b0bff85967d73fb78d |
Uniform estimates for a Modica-Mortola type approximation of branched transportation
23 Mar 2015 March 24, 2015
Antonin Monteil
Uniform estimates for a Modica-Mortola type approximation of branched transportation
23 Mar 2015 March 24, 2015
Models for branched networks are often expressed as the minimization of an energy M α over vector measures concentrated on 1-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation M α ε , introduced by Edouard Oudet and Filippo Santambrogio in 2010, which is defined over H 1 vector measures. These energies induce some pseudo-distances between L 2 functions obtained through the minimization problem min{M α ε (u) :We prove some uniform estimates on these pseudo-distances which allow us to establish a Γ-convergence result for these energies with a divergence constraint.
Introduction
Branched transportation is a classical problem in optimization: it is a variant of the Monge-Kantorovich optimal transportation theory in which the transport cost for a mass m per unit of length is not linear anymore but sub-additive. More precisely, the cost to transport a mass m on a length l is considered to be proportional to m α l for some α ∈]0, 1[. As a result, it is more efficient to transport two masses m 1 and m 2 together instead of transporting them separately. For this reason, an optimal pattern for this problem has a "graph structure" with branching points. Contrary to what happens in the Monge-Kantorovich model, in the setting of branched transportation, an optimal structure cannot be described only using a transport plan, giving the correspondence between origins and destinations, but we need a model which encodes all the trajectories of mass particles.
Branched transportation theory is motivated by many structures that can be found in the nature: vessels, trees, river basins. . . Similarly, as a consequence of the economy of scale, big roads are proportionally cheaper that big ones and it follows that the road and train network also present this structure. Surprisingly the theory has also had theoretical applications: recently, it has been used by Bethuel in [4] so as to study the density of smooth maps in Sobolev spaces between manifolds.
Branched transportation theory was first introduced in the discrete framework by E. N. Gilbert in [14] as a generalization of the Steiner problem. In this case an admissible structure is a weighted graph composed of oriented edges of length l i on which some mass m i is flowing. The cost associated to it is then i l i m α i and it has to be minimized over all graphs which transport some given atomic measure to another one. More recently, the branched transportation problem was generalized to the continuous framework by Q. Xia in [23] by means of a relaxation of the discrete energy (see also [24]). Then, many other models and generalizations have been introduced (see [15] for a Lagrangian formulation, see also [1], [2], [3] for different generalizations and regularity properties.). In this paper, we will concentrate on the model with a divergence constraint, due to Q. Xia. However, this is not restrictive since all these models have been proved to be equivalent (see [3] and [19]).
In this model, a transport path is represented as a vector measure u on some open set Ω ⊂ R d such that ∇ · u = µ + − µ − for two probability measures µ + and µ − . Then the energy of u is defined as M α (u) =´M θ α dH 1 if u is a vector measure concentrated on a rectifiable 1-dimensional set M on which u has multiplicity θ w.r.t. the Hausdorff measure (see [3] for more details). In this framework, u must be considered as the momentum (the mass θ times the velocity) of a particle at some point. Then (∇ · u)(x) represents the difference between incoming and outcoming mass at each point x.
In this paper, we are interested in some approximation of branched transportation proposed by E. Oudet and F. Santambrogio few years ago in [18] and which has interesting numerical applications. This model was inspired by the well known scalar phase transition model proposed by L. Modica and S. Mortola in [16]. Given u ∈ H 1 (Ω, R d ) for some bounded open subset Ω ⊂ R d , E. Oudet and F. Santambrogio introduced the following energy:
M α ε (u) = ε −γ 1ˆΩ |u| β + ε γ 2ˆΩ |∇u| 2 ,
where β ∈ (0, 1) and γ 1 , γ 2 > 0 are some exponents depending on α (see (2.3)). If u does not belong to the set H 1 (Ω), the value of M α ε is taken as +∞. We recall the heuristic which shows why M α ε is an approximation of M α (see [18]): assume that µ − (resp. µ + ) is a point source at S 1 (resp. S 2 ) with mass m. Then, it is clear that the optimal path for M α between these two measures is the oriented edge S = (S 1 , S 2 ) with a mass m flowing on it. We would like to approximate this structure, seen as a vector measure u concentrated on S, by some H 1 vector fields v which are more or less optimal for M α ε . What we expect is that v looks like a convolution of u with a kernel ρ depending on ε: v = u * ρ R , where
ρ R (x) = R −d ρ(R −1 x) (1.1)
for some fixed smooth and compactly supported radial kernel ρ ∈ C ∞ c (R d ). Then the support of v is like a strip of width R around S so that |v| is of the order of m/R d−1 and |∇v| is of the order of m/R d . This gives an estimation of M α ε (v) like
M α ε (v) ≃ ε −γ 1 R d−1 (m/R d−1 ) β + ε γ 2 R d−1 (m/R d ) 2 . (1.2)
With our choice for the exponents γ 1 , γ 2 and β, the optimal choice for R is
R = ε γ m 1−γ d−1 , (1.3) where γ = 2 2d − β(d − 1) = γ 2 d + 1 .
This finally leads to M α ε (v) ≃ m α as expected. It was proved in [18] that, at least in two dimensions, the energy sequence (M α ε ) ε>0 Γ-converges to the branched transportation functional c 0 M α for some constant c 0 and for some suitable topology (see Theorem 2.1 page 5). This result has been interestingly applied to produce a numerical method. However, rather than a Γ-convergence result on M α ε we would need to deal with the functionals M α ε , obtained by adding a divergence constraint: it should be shown that M α ε (u) := M α ε (u) + I ∇·u=fε Γ-converges to M α (u) := M α ε (u) + I ∇·u=µ + −µ − , where f ε ∈ L 2 is some suitable approximation of µ + − µ − and I A (u) is the indicator function in the sense of convex analysis that is 0 whenever the condition is satisfied and +∞ otherwise. Even if this property was not proved in [18], the effectiveness of the numerical simulations made the authors think that it actually holds true. Note that an alternative using a penalization term was proposed in [20] to overcome this difficulty.
In section 2 we recall Xia's formulation of branched transportation and its approximation M α ε introduced by E. Oudet and F. Santambrogio. The longest part of this paper, section 3, is devoted to a local estimate which gives a bound on the minimum value d α ε (f + , f − ) := min{M α ε (u) : ∇ · u = f } depending on f L 1 , f L 2 and diam(Ω) (see Proposition 3.2 page 6). In section 4, we deduce a comparison between d α ε and the Wasserstein distance with an "error term" involving the L 2 norm of f + −f − . As an application of this inequality, in the last section, we will prove the Γ-convergence result which was lacking in [18], of functionals M α ε to M α (with a divergence constraint on ∇ · u):
this answers the Open question 1 in [20,18] and validates their numerical method.
2 Mathematical setting.
The branched transportation energy
In all what follows, we will use the model proposed by Q. Xia (see [23] and [24]): Let d ≥ 1 be an integer and Ω be some open and bounded subset of R d . Let us denote by M div (Ω) the set of finite vector measures on Ω such that its divergence is also a finite measure:
M div (Ω) := u measure on Ω valued in R d : u M div (Ω) < +∞ , where u M div (Ω) := |u|(Ω) + |∇ · u|(Ω) with |u|(Ω) := sup ˆΩ ψ · du : ψ ∈ C(Ω, R d ), ψ ∞ ≤ 1
and, similarly,
|∇ · u|(Ω) := sup ˆΩ ∇ϕ · du : ϕ ∈ C 1 (Ω, R), ϕ ∞ ≤ 1 .
In all what follows, ∇ · u has to be thought in the weak sense, i.e.´ϕ∇ · u =´∇ϕ · du for all ϕ ∈ C 1 (Ω). Since we do not ask ϕ to vanish at the boundary, ∇ · u may contain possible parts on ∂Ω which are equal to u · n, where n is the external unit normal vector to ∂Ω. In other words, ∇ · u is the weak divergence of u1 Ω in R d , where 1 Ω is the indicator function of Ω, equal to 1 on Ω and 0 elsewhere. M div (Ω) is endowed with the topology of weak convergence on u and on its divergence: i.e. u n M div (Ω) −→ u if u n ⇀ u and ∇ · u n ⇀ ∇ · u weakly as measures.
Given 0 < α < 1, the energy of branched transportation can be represented as follows for measures u ∈ M div (Ω):
M α (u) = ´M θ α dH 1 if u can be written as u = U (M, θ, ξ), +∞ otherwise,
where U (M, θ, ξ) is the rectifiable vector measure u = θξ · H 1 |M with density θξ with respect to the H 1 −Hausdorff measure on the rectifiable set M . The real multiplicity is a measurable function θ : M → R + and the orientation ξ :
M → S d−1 ⊂ R d is such that ξ(x) is tangential to M for H 1 -a.e. x ∈ M .
Given two probability measures µ + and µ − on Ω, the problem of branched transportation consists in minimizing M α under the constraint ∇ · u = µ + − µ − :
inf M α (u) : u ∈ M div (Ω) and ∇ · u = µ + − µ − . (2.1)
Note that, if µ ± (∂Ω) = 0, the divergence constraint implies a Neumann condition on u: u · n = 0 on ∂Ω.
Functionals M α ε
For the minimum value in (2.5) to be finite whatever µ + and µ − in the set of probability measures, we will require α to be sufficiently close to 1. More precisely, we make the following assumption:
1 − 1 d < α < 1. (2.2)
Q. Xia has shown in [23] that, under this assumption, there exists at least one vector measure u ∈ M div (Ω) such that M α (u) < +∞.
We are interested in the following approximation of M α which was introduced in [18]: for all u ∈ M div (Ω) and for all open subset ω ⊂ Ω,
M α ε (u, ω) := ε −γ 1ˆω |u(x)| β dx + ε γ 2ˆω |∇u(x)| 2 dx if u ∈ H 1 (ω) +∞ otherwise, (2.3)
where β, γ 1 and γ 2 are three exponents depending on α and d through:
β = 2 − 2d + 2αd 3 − d + α(d − 1)
and
γ 1 = (d − 1)(1 − α) and γ 2 = 3 − d + α(d − 1).
Note that inequality 1 − 1/d < α < 1 implies that 0 < β < 1. When ω = Ω, we simply write M α ε (u, Ω) =: M α ε (u). We point out the 2-dimensional case where M α ε rewrites as
M α ε (u) = ε α−1ˆΩ |u(x)| β dx + ε α+1ˆΩ |∇u(x)| 2 dx, (2.4) where β = 4α−2 α+1 . Given two densities f + , f − ∈ L 2 + (Ω) := {f ∈ L 2 (Ω) : f ≥ 0} such that´f + =´f − , we are interested in minimizing M α ε (u) under the constraint ∇ · u = f + − f − : inf M α ε (u) : u ∈ H 1 (Ω) and ∇ · u = f + − f − . (2.5)
The classical theory of calculus of variation shows that this infimum is actually a minimum. A natural question that arises is then to understand the limit behavior for minimizers of these problems when ε goes to 0. A classical tool to study this kind of problems is the theory of Γ-convergence which was introduced by De Giorgi in [12]. For the definition and main properties of Γ-convergence, we refer to [11] and [8]. In particular, if M α ε Γ-converges to some energy functional M α 0 and if (u ε ) is a sequence of minimizers for M α ε admitting a subsequence converging to u, then, u is a minimizer for M α 0 . By construction of M α ε , we expect that, up to a subsequence, M α ε Γ-converges to c 0 M α . In the two dimensional case, we have the following Γ-convergence theorem proved in [18]: Theorem 2.1. Assume that d = 2 and α ∈ (1/2, 1). Then, there exists a constant c > 0 such that (M α ε ) ε>0 Γ-converges to cM α in M div (Ω) when ε goes to 0.
Nevertheless, this does not imply the Γ-convergence of M α ε (u)
+ 1 ∇·u=f + −f − to M α ε (u) + 1 ∇·u=f + −f − .
Indeed, the Γ-convergence is stable under the addition of continuous functionals but not l.s.c. functionals. Consequently, we cannot deduce, from this theorem, the behavior of minimizers for (2.5). For instance, it is not clear that there exists a recovery sequence (u ε ), i.e. u ε converges to u in M div (Ω) and M α ε (u ε ) converges to M α (u) as ε → 0, with prescribed divergence ∇ · u ε = f + − f − . To this aim, we require some estimations on these energies and this is the purpose of this paper.
Distance of branched transportation
We remind our hypothesis 1 − 1/d < α < 1.
In [23], Q. Xia has remarked that, as in optimal transportation theory, M α induces a distance d α on the space P(Ω) of probability measures on Ω:
d α (µ + , µ − ) = inf M α (u) : u ∈ M div (Ω) such that ∇ · u = µ + − µ − ,
for all µ + , µ − ∈ P(Ω). Thanks to our assumption α > 1 − 1/d, d α is finite for all µ ± ∈ P(Ω) and it induces a distance on the set P(Ω) which metrizes the topology of weak convergence of measures. Actually, d α has a stronger property which is a comparison with the Wasserstein distance: Proposition 2.2. Let µ + and µ − be two probability measures on Ω. We denote by W p the Wasserstein distance associated to the cost (x, y) → |x − y| p for p ≥ 1. Then, one has
W 1/α (µ + , µ − ) ≤ d α (µ + , µ − ) ≤ C W 1 (µ + , µ − ) 1−d(1−α) ,
for a constant C > 0 only depending on d, α and the diameter of Ω.
We refer to [17] for a proof of this property (see also [3], and [9] for an alternate proof) and [22], [21] for the definition and main properties of the Wasserstein distance. In the same way, we define d α ε as follows:
d α ε (f + , f − ) = inf M α ε (u) : u ∈ H 1 (R d ) such that ∇ · u = f + − f − ,(2.6)
where
f + , f − ∈ L 2 + (Ω) satisfy´Ω f + =´Ω f − .
Although d α is a distance, it is not the case for d α ε which does not satisfy the triangular inequality. Actually, because of the second term involving |∇u| 2 , M α ε is not subadditive. However, for u 1 , . . . , u n in M div (Ω), the inequality |∇u 1 + · · · + ∇u n | 2 ≤ n{|∇u 1 | 2 + · · · + |∇u n | 2 } implies
M α ε n i=1 u i ≤ n n i=1 M α ε (u i ).
In particular, d α ε is a pseudo-distance in the sense that the three properties in the following proposition are satisfied:
Proposition 2.3. Let f + , f − and f 1 ,.
. . , f n be L 2 densities, i.e. L 2 non negative functions whose integral is equal to 1. Then one has
1. d α ε (f + , f − ) = 0 implies f + = f − , 2. d α ε (f + , f − ) = d α ε (f − , f + ), 3. d α ε (f 0 , f n ) ≤ n d α ε (f 0 , f 1 ) + d α ε (f 1 , f 2 ) + · · · + d α ε (f n−1 , f n ) .
Local estimate
We remind our assumption (2.2) which insures that d α (µ + , µ − ) is always finite. Our goal is to prove that d α ε enjoys a property similar to the following one.
Proposition 3.1. Let Q 0 = (0, L) d ⊂ R d be a cube of side length L > 0.
There exists some constant C > 0 only depending on d and α such that for all non negative Borel finite measure µ of total mass θ > 0,
d α (µ, θδ 0 ) ≤ C θ α L,
where δ 0 is the Dirac measure at the point c Q 0 , the center of Q 0 .
Since d α ε (f + , f − ) is only defined on L 2 functions f ± we first have to replace θδ 0 by some kernel which concentrates at the origin when ε goes to 0. Let ρ ∈ C 1 c (B, R + ) be a radial non negative function such that´R d ρ = 1, where B ⊂ R d is the unit ball centered at the origin, and define ρ θ,ε := ρ R as in (1.1), where
R =: R θ,ε = ε γ θ 1−γ d−1 .
Let Q be a cube in R d centered at some point c Q ∈ R d and f ∈ L 2 + (Q) such that Q f =: θ Q . Then, we will denote by ρ Q the kernel θρ θ,ε refocused at c Q :
ρ Q (x) = θ Q ρ θ Q ,ε (x − c Q ).
We are going to prove Proposition 3.2 (Local estimate). Let us set Q 0 = (0, L) d for some L > 0. There exists C > 0 only depending on α, ρ and d such that for all f ∈ L 2
+ (Q 0 ) with´Q 0 f = θ, we have • If supp ρ Q 0 ⊂ Q 0 then, there exists u ∈ H 1 0 (Q 0 ) such that ∇ · u = f − ρ Q 0 and d α ε (f, ρ Q 0 ) ≤ M α ε (u) ≤ C θ α L + ε γ 2 f 2 L 2 and u L 1 ≤ C L θ. • Otherwise, there exists u ∈ H 1 0 ( Q 0 ) such that d α ε (f, ρ Q 0 ) ≤ M α ε (u) ≤ Cε γ 2 f 2 L 2 and u L 1 ≤ C L θ, where Q 0 = 2 supp ρ Q 0 := B(c Q 0 , 2R θ,ε ).
Remark 3.3. The Dirichlet term, ε γ 2 f 2 L 2 , in the estimates above is easily understandable. Indeed, if ε is very large so that one can get rid of the first term in the energy M α ε , then, one can use a classical Dirichlet type estimation. On the contrary, for ε very small, the Γ-limit result on M α ε tells us that these energies are close to M α so that it is natural to hope a similar estimate to that of M α : that is to say an estimation from above by θ α L (see [3]).
The main difficulty to prove Proposition 3.2 is the non subadditivity of the pseudodistances d α ε . Indeed, our proof is based on a dyadic construction used by Q. Xia in [23] to prove Proposition 3.1 (see also [3]). This gives a singular vector measure u which is concentrated on a graph. Since M α ε contains a term involving the L 2 norm of ∇u, we have to regularize u by taking a convolution with the kernel ρ θ,ε on each branch of the graph (θ being the mass traveling on it) so that two different branches are no longer disjoints.
It is useful to see that we have a first candidate for the minimization problem (2.6). This candidate is of the form u = ∇φ, where φ is the solution of the Dirichlet problem
∆φ = f + − f − in Q, φ = 0 on ∂Q.
Theorem 3.4 below gives a better result in the sense that the candidate u vanishes at the boundary:
Theorem 3.4. Let Q 0 = (0, L) d be a cube of side length L > 0. There exists C > 0 only depending on d such that for all f ∈ L 2 0 (Q 0 ), there exists u ∈ H 1 0 (Q 0 , R 2 ) solving ∇ · u = f and satisfying u L 1 (Q 0 ) ≤ CL f L 1 (Q 0 ) together with u H 1 0 (Q 0 ) := ˆQ 0 |∇u| 2 1/2 ≤ C f L 2 (Q 0 ) , where L 2 0 (Q 0 ) = f ∈ L 2 (Q 0 ) :´Q 0 f (x) dx = 0 .
For a proof of this theorem, see, for instance, Theorem 2 in [7]: the only difference with Theorem 3.4 is that we add the estimation u L 1 (Q 0 ) ≤ CL f L 1 (Q 0 ) which can be easily obtained following the proof of J. Bourgain and H. Brezis. The corresponding property formulated on a Lipschitz bounded connected domain Ω is also true (see Theorem 2' in [7]) except that the constant C could depend on Ω in this case.
Of course, this candidate is usually not optimal for (2.6) and this does not allow for a good estimate because of the first term in the definition of M α ε . For this reason, we have to use the dyadic construction of Q. Xia up to a certain level ("diffusion level") from which we simply use Theorem 3.4.
3.1 Dyadic decomposition of Q 0 and "diffusion level" associated to f Let us call "dyadic descent" of Q 0 = (0, L) d the set Q = j≥0 Q j , where Q j is the j th "dyadic generation":
Q j = (x 1 , . . . , x d ) + 2 −j Q 0 : x i ∈ {k2 −j L : 0 ≤ k ≤ 2 j − 1} for i = 1, . . . , d .
Note that Card(Q j ) = 2 jd . For each Q ∈ Q, let us define • D(Q): the descent of Q, the family of all dyadic cubes contained in Q.
• A(Q): the ancestry of Q, the family of all dyadic cubes containing Q.
• C(Q): the family of children of Q composed of the 2 d biggest dyadic cubes strictly included in Q.
• F (Q): the father of Q, the smallest dyadic cube strictly containing Q.
We now remind the dyadic construction described in [23] which irrigates f from a point source. We first introduce some notations: fix a function f ∈ L 2 + (Q 0 ) with integral θ and let Q ∈ Q be a dyadic cube centered at c Q ∈ R d . Then we introduce θ Q the mass associated to the cube Q as
θ Q =ˆQ f .
If θ Q = 0, we also define the kernel associated to Q through
ρ Q (x) = ρ R (x), (3.1) where ρ R is defined in (1.1) for R = R Q := ε γ θ 1−γ d−1 Q .
Define also the weighted recentered kernel by
ρ Q (x) = θ Q ρ Q (x − c Q ) (3.2)
if θ Q = 0 and ρ Q (x) = 0 otherwise. Lastly, we introduce the point source associated to the cube Q as
δ Q = θ Q × Dirac measure at point c Q .
We are now able to construct a vector measure X such that M α (X) < +∞. First define the measures X Q as below:
X Q = Q ′ ∈C(Q) θ Q ′ n Q ′ H 1 |[c Q ,c Q ′ ] , (3.3) where n Q ′ = c Q ′ − c Q c Q ′ − c Q . Then, we have ∇ · X Q = Q ′ ∈C(Q) δ Q ′ − δ Q and the energy estimate M α (X Q ) ≤ 2 d−2 θ α Q diam(Q), where diam(Q) stands for the diameter of Q. Finally, the measure X = Q∈Q X Q solves ∇ · X = f − δ Q 0 and satisfies M α (X) ≤ Cθ α diam(Q 0 ).
Indeed, it is enough to apply the following lemma with λ = α:
Lemma 3.5. Let Q ∈ Q and λ ∈]1 − 1/d, 1]. There exists a constant C = C(λ, d) such that Q ′ ∈D(Q) θ λ Q ′ diam(Q ′ ) ≤ Cθ λ Q diam(Q).
Proof. Let j 0 ≥ 0 be such that Q ∈ Q j 0 . The definition of D(Q), the Jensen inequality and the fact that
d − 1 − λd < 0 give Q ′ ∈D(Q) θ λ Q ′ diam(Q ′ ) = j≥0 2 −j diam(Q) Q ′ ∈D(Q)∩Q j 0 +j θ λ Q ′ ≤ diam(Q) j≥0 2 −j 2 jd 2 −jd Q ′ ∈D(Q)∩Q j 0 +j θ Q ′ λ ≤ θ λ Q diam(Q) j≥0 2 j(d−1−λd) ≤ Cθ λ Q diam(Q).
Then, the idea is to replace each term in (3.3) by its convolution with the kernel ρ Q ′ . Unfortunately, this will make appear extra divergence around each node. We have to modify X so as to make this extra divergence vanish using, for instance, Theorem 3.4. Furthermore, we cannot follow the construction for all generations j ≥ 1 otherwise, the "enlarged edges" (convolution of a dyadic edge and the kernel ρ θ,ε ) may overlap. This is the reason why we introduce the following definition:
Definition 3.6 ("Diffusion level"). For a cube Q 0 and f ∈ L 2 + (Q 0 ) we define the set D(Q 0 , f ) or D(f ) ⊂ Q as the maximal element for the inclusion in the set Λ = {D ⊂ Q : ∀Q ∈ D, A(Q) ∪ C(F (Q)) ⊂ D and supp ρ Q ⊂ Q} . If Λ = ∅, that is supp ρ Q 0 Q 0 , we take the convention D(f ) = ∅.
For all x ∈ Q 0 , define also the "generation index" of x associated to f as
j(f, x) = max {j : ∃Q ∈ D(f ) ∩ Q j , x ∈ Q} ∈ N ∪ {±∞},
where the convention max(∅) = −∞ has been used.
In this way, each cube in D(f ) contains the support of its associated kernel. Moreover, if Q is element of D(f ), then all its ancestry and its brothers (i.e. elements of the set C(F (Q))) are elements of D(f ). D(f ) can be constructed by induction as follows: first take j = 0 and D(f ) = ∅. If supp ρ Q 0 ⊂ Q 0 then add Q 0 to the set D(f ) and j is replaced
by j + 1. For all cubes Q in Λ ∩ Q j−1 : if all cubes Q ′ ∈ C(Q) ⊂ Q j are such that there associated kernel is supported on Q ′ then D(f ) is replaced by D(f ) ∪ C(Q).
If D(f ) has been changed at this stage j is replaced by j + 1 and the preceding step is reiterated. This process is repeated for j ≥ 1 until it fails.
Let D min (f ) be the set of all cubes in D(f ) which are minimal for the inclusion. If
D min (f ) = ∅, we also define D(f ) = Q∈D min (f ) Q .
Note that this is actually a disjoint union: two distinct cubes in D min (f ) are disjoint.
Indeed, for Q, Q ′ ∈ D min (f ) ⊂ Q, either Q∩Q ′ = ∅ or Q and Q ′ are comparable: Q ⊂ Q ′ or Q ′ ⊂ Q.
In the last case, since Q and Q ′ are minimal, we deduce that Q = Q ′ . Moreover, it is not difficult to see that, if D min (f ) = ∅, then D(f ) = {x ∈ Q 0 : j(f, x) is finite} and also that f (x) = 0 whenever j(f, x) = +∞, where f is the precise representation of f (i.e. the limit of the mean values of f on small balls). Indeed, assume that Q ∈ D(f ) is a cube of side length L Q . Then, by definition, supp ρ Q ⊂ Q and for some constant C depending on ρ and for ν = 1−γ d−1 , one has ε γ θ ν Q ≤ CL Q and so
Q f := L −d Q θ Q ≤ ε −γ/ν L 1/ν−d Q . Since 1/ν − d = (d−1)(αd−d+1) d+1 is positive, we deduce that L Q cannot be arbitrary small if there exists x ∈ Q such that f (x) > 0. Moreover, if f (x) ≥ η a.
e. for some η > 0, then there exists some constant C η > 0 depending on η, ε, d and α such that
∀Q ∈ D(f ), L Q ≥ C η . (3.4) In particular, one can deduce that D min (f ) = ∅ if and only if D(f ) = ∅ or f (x) = 0 a.e. Indeed, if D(f ) = ∅, then it is clear that D min (f ) = ∅. Conversely, assume that D min (f ) = ∅ (i.e. Q 0 ∈ D(f )) and that there exists x ∈ Q 0 such that f (x) > 0.
Since
Q∈D(f ) ∂Q is negligible for the Lebesgue measure, one can assume that x ∈ Q∈D(f ) Q. Then 0 ≤ j(f, x) < +∞ and so there exists a minimal cube Q ∈ D(f ) containing x. Then Q ∈ D min (f ). Indeed, if Q ′ ∈ D(f ) and Q ′ Q, then A(Q) ⊂ D(f ) and there exists Q ′′ ∈ A(Q) such that Q ′′ Q and x ∈ Q ′′ which is a contradiction.
We are now able to define two approximations of f very useful for our problem. The first is a dyadic approximation of f by an atomic measure,
D ε f = Q∈D min (f ) δ Q if D min (f ) = ∅, δ Q 0 otherwise,
where we recall the definition of δ Q := θ Q δ c Q . We also define an approximation in
H 1 (Q 0 ), d ε f = Q∈D min (f ) ρ Q if D min (f ) = ∅, ρ Q 0 otherwise,
where ρ Q is defined in (3.2). The following result shows in which sense d ε f is an approximation of f and justifies the term "diffusion level". Indeed, this proposition indicates that we get a good estimation by using a local diffusion from d ε f to f , i.e. minimizinǵ
Q |∇u| 2 over the constraint ∇ · u = d ε f − f for all Q ∈ D min (f ) (see Theorem 3.4). Proposition 3.7.
There exists a constant C > 0 depending on d and ρ such that for all
f ∈ L 2 + (Q 0 ), d α ε (d ε f, f ) + d α (D ε f, f ) ≤ C ε γ 2 f 2 L 2 (Q 0 ) . More precisely, if supp ρ Q 0 ⊂ Q 0 , there exists u ∈ H 1 0 (Q 0 ) such that ∇ · u = f − d ε f as well as M α ε (u) ≤ C ε γ 2 f 2 L 2 and u L 1 ≤ C diam(Q 0 ) f L 1 .
If supp ρ Q 0 Q 0 the same estimations hold but the condition u ∈ H 1 0 (Q 0 ) has to be replaced by u ∈ H 1 0 ( Q 0 ), where Q 0 is a cube containing Q 0 and supp ρ Q 0 . Proof. First assume that supp ρ Q 0 ⊂ Q 0 i.e. Q 0 ∈ D(f ). If D min (f ) = ∅, then f (x) = 0 a.e. and the proposition is trivial. Hence, one can assume that D min (f ) = ∅. Then f is supported on D(f ) and D min (f ) =: {Q i } i∈I is a finite or countable partition of D(f ).
Denote for simplicity
D i := diam(Q i ), f i := f 1 Q i (restriction of f to Q i ), θ i := θ Q i and ρ i := ρ Q i = θ i ρ R i for i ∈ I, where R i := R Q i = ε γ θ 1−γ d−1 i .
Since Q i is minimal in D(f ), we deduce that, for some constants C, C ′ > 0,
C ′ R i ≤ D i ≤ CR i . (3.5)
Indeed, the first inequality follows from the fact that supp ρ i ⊂ Q i and diam(supp ρ i ) = cR i for some constant c depending on ρ. For the second inequality observe that, since Q i is minimal, there exists Q ∈ C(Q i ) such that supp ρ Q Q and hence R Q ≥ c ′ diam(Q) = c ′ /2D i for some constant c ′ > 0 depending on ρ. Since θ Q ≤ θ Q i = θ i , one has R Q ≤ R i and the second inequality follows. Now, Theorem 3.4 allows us to find
u i ∈ H 1 0 (Q i ) such that ∇ · u i = g i , u i H 1 ≤ C g i L 2 and u i L 1 ≤ C D i g i L 1 , where g i := f i − ρ i . Since u i vanishes at ∂Q i , one can extend u i by 0 out of Q i to get a function in H 1 (R d ):
for the sake of simplicity, this function is still denoted by u i . Consequently, u = i u i belongs to H 1 0 (Q 0 ) and ∇ · u = f − d ε f . It remains to estimate M α ε (u) and u L 1 . First of all,
u L 1 ≤ i u i L 1 ≤ C diam(Q 0 ) i g i L 1 and the inequality g i L 1 ≤ 2θ i leads to u L 1 ≤ 2C diam(Q 0 ) f L 1 as required.
Let us compute the L 2 -norm of ρ i :
ρ i 2 L 2 = θ 2 i ρ R i 2 L 2 = θ 2 i R −d i ρ 2 L 2 = Cθ 2 i R −d i .
By a Cauchy-Schwarz inequality,
θ 2 i = ˆQ i f i 2 ≤ |Q i | f i 2 L 2 = D d i f i 2 L 2 (3.6)
which, together with (3.5), gives
ρ i 2 L 2 ≤ C R d i f i 2 L 2 R −d i = C f i 2 L 2 . Since u i H 1 ≤ C f i − ρ i L 2 , we get u i H 1 ≤ C f i L 2 . Now, because the energy M α ε is local and since each u i is supported on Q i , one has M α ε (u) = n i=1 M α ε (u i ) = n i=1 ε −γ 1ˆQ i |u i | β + ε γ 2ˆQ i |∇u i | 2 By construction of u i , we haveˆQ i |∇u i | 2 ≤ f i 2 L 2 .
It remains to estimate the first term. First of all, we use the Hölder and Poincaré inequalities as follows:
Q i |u i | β ≤ |Q i | 1−β/2 ˆQ i |u i | 2 β/2 ≤ D d−dβ/2 i D 2 iˆQ i |∇u i | 2 β/2 ≤ D ν i f i β L 2 (Q i ) , where ν = β + d − dβ 2 .
In view of (3.6) and (3.5), we have
D i ≤ CR i ≤ Cε γ θ 1−γ d−1 i ≤ Cε γ (D d 2 i f i L 2 ) 1−γ d−1
and, introducing δ :
= 1 − d(1−γ) 2(d−1) , D δ i ≤ Cε γ f i 1−γ d−1 L 2 . (3.7)
Finally, since −γ 1 + γν δ = γ 2 and β + ν(1−γ) δ(d−1) = 2, we get
ε −γ 1ˆQ i |u i | β ≤ Cε −γ 1 + γν δ f i β+ ν(1−γ) δ(d−1) L 2 = Cε γ 2 f i 2 L 2 .
The proof of the second inequality is quite similar but easier:
d α (D ε f, f ) ≤ n i=1 d α (δ Q i , f i ) ≤ n i=1 θ α i D i .
Once again, applying (3.6) and then (3.7), we get
d α (D ε f, f ) ≤ C ε γ 2 f 2 L 2 .
In the case where supp
ρ Q 0 Q 0 , i.e. R Q 0 := ε γ θ 1−γ d−1 Q
≥ CL (L being the side length of Q 0 and C a constant depending on ρ), the proof is the same. Indeed, we just apply Theorem 3.4 to g = f − ρ Q 0 and the same computations as above lead to the same result.
Proof of Proposition 3.2
Let Q 0 = (0, L) 2 , L > 0 and f ∈ L 2 + (Q 0 ) with´Q 0 f = θ.
In the case where supp ρ Q 0 Q 0 , Proposition 3.2 is a particular case of Proposition 3.7. Consequently, one can assume that supp ρ Q 0 ⊂ Q 0 i.e. Q 0 ∈ D(f ). In the case where D(f ) = {Q 0 }, one has d ε f = ρ Q 0 and Proposition 3.2 is a consequence of Proposition 3.7 as well. For this reason, one can assume that C(Q 0 ) ⊂ D(f ). Moreover, up to replacing f by f +η for some small constant η > 0 and passing to the limit when η → 0, one can assume that D(f ) is finite. Indeed, in view of (3.4),
D(f + η) is finite since for all Q ∈ D(f + η), diam(Q) ≥ C η > 0.
Our aim is to prove that there exists C > 0 only depending on α, d and ρ such that
d α ε (f, ρ Q 0 ) ≤ C θ α L + ε γ 2 f 2 L 2 .
The idea of the proof is to approximate the vector field X = X Q of the previous section (see (3.3)) by a vector field in H 1 using the kernel ρ. In this part, we will use the notations of the previous section: in particular, the definition of D(f ) in definition 3.6, the measures X Q in (3.3) and X = Q∈D(f ) X Q .
Since C(Q 0 ) ⊂ D(f ), we can construct the regularized vector field Y by the formula
Y = Q∈D(f ) Q =Q 0 Z Q ,
where, for all Q ∈ D(f ) such that Q = Q 0 (see Figure 1),
Z Q := θ Q n Q ρ Q * H 1 |[c F (Q) ,c Q ] , (3.8) n Q being the normalized vector n Q = c Q −c F (Q) c Q −c F (Q)
and ρ Q being defined in (3.1). and Z Q satisfies the two estimations
Q 1 Q 2 Q 3 Q 4 Q Z Q 1 Z Q 2 Z Q 3 Z Q 4 ρ Q 1 ρ Q 2 ρ Q 3 ρ Q 4Z Q L ∞ ≤ Cθ Q R 1−d Q and ∇Z Q L ∞ ≤ Cθ Q R −d Q . (3.11)
Then, the same computations as in (1.2) and the fact that R Q ≤ diam(Q) give (3.9). Let us estimate the L 1 norm of Y which has to be controlled by θ as stated in Proposition 3.2:
Y L 1 ≤ j≥1 Q∈D(f )∩Q j Z Q L 1 ≤ j≥1 Q∈D(f )∩Q j θ Q L 2 −j = L θ. Note that ∇ · Y = ρ Q 0 − h − d ε f,
where h stands for the extra divergence. h can be written as
h = Q∈D f ath. ρ Q − Q ′ ∈C(Q) ρ Q ′ ,Q ,
where ρ Q ′ ,Q represents the kernel ρ Q ′ translated at c Q , center of Q, and, for the sake of simplicity, the set of all cubes Q such that C(Q) ⊂ D(f ) has been denoted by D f ath. :
D f ath. := {Q ∈ D(f ) : C(Q) ⊂ D(f )}. Since ∇ · Y = ρ Q 0 − f + (f − d ε f ) − h = ρ Q 0 − f ,
we have to slightly modify the vector field Y . This will be done replacing Y by
V = Y + V 1 + V 2 , where V 1 , V 2 ∈ H 1 (Ω, R 2 ) are constructed so that ∇ · V 1 = h and ∇ · V 2 = d ε f − f .
The construction of V 1 and the estimation of M α ε (V 1 ), V 1 L 1 will be the object of the first step. In the second step we prove that M α ε (Y ) ≤ Cθ α L. Then, Proposition 3.7 allows us to construct V 2 ∈ H 1 such that ∇ · V 2 = d ε f − f with an estimation on M α ε (V 2 ) and V 2 L 1 .
First step: Correction at the nodes, construction of V 1 . For all Q ∈ D f ath. , let B Q be the support of ρ Q . Since supp ρ has been supposed to be the unit ball centered at the origin and ρ Q (
x) = θ Q ρ R Q (x − c Q ), we have B Q = B(c Q , R Q ) ⊂ Q.
Let us define the extra divergence corresponding to this node,
h Q = ρ Q − Q ′ ∈C(Q) ρ Q ′ ,Q .
For each Q ∈ D f ath. , thanks to Theorem 3.4, we can find
V Q ∈ H 1 0 (B Q ) such that ∇ · V Q = h Q and V Q H 1 ≤ C h Q L 2 .
But in this case, because h Q is radial up to a translation, we essentialy use the proposition in dimension 1 which is quite easy and gives better estimates. Let us give more details on this point:
(B, R d ) such that ∇ · V = F and ∇V ∞ ≤ C F ∞ and ∇V L 1 ≤ C F L 1 .
Proof. First of all, by a scaling argument, one can assume that R = 1. The vector field
V : B → R d defined by V (x) = v(|x|)x for some Lipschitz function v : R + → R satisfies ∇ · V (x) = r 1−d [r d v(r)] ′ in the distributional sense. Thus, if v is chosen as v(r) = r −dˆr 0 f (s)s d−1 ds,
then, V solves the following problem:
∇ · V (x) = F (x) on B, V (x) = 0 on ∂B.
Moreover, for a.e. x ∈ B, we have ∇V (x) = v ′ (|x|)x⊗ x |x| +v(|x|) Id, where Id is the matrix identity. In particular, we get
∇V ∞ ≤ C( rv ′ (r) ∞ + v ∞ ) ≤ C f ∞ = C F ∞ and ∇V L 1 ≤ C( r d v ′ (r) L 1 + r d−1 v(r) L 1 ) ≤ C F L 1 . Applying Lemma 3.8 to F = h Q and R = R Q gives V Q ∈ H 1 0 (B Q ) such that ∇ · V Q = h Q and ∇V Q L ∞ ≤ Cθ Q R −d Q and ∇V Q L 1 ≤ Cθ Q . (3.12)
Observe by the way that, in this particular case, the second inequality in (3.12) is a consequence of the first one. Moreover, since V Q is supported on
B Q = B(c Q , R Q ), we deduce that V Q L ∞ ≤ R Q ∇V Q L ∞ ≤ Cθ Q R 1−d Q
so that V Q satisfies the same estimations as (3.11). In particular, we get M α ε (V Q ) ≤ Cθ α Q diam(Q). Now define
V 1 = Q∈D f ath. V Q . Since V Q L 1 ≤ CR Q ∇V Q L 1 ≤ C diam(Q)θ Q , Lemma 3.5 implies V 1 L 1 ≤ C diam(Q 0 )θ Q 0 ≤ C ′ L f L 1
as required. Then, using the definition of M α ε in (2.3) and the subadditivity of x → |x| β , one gets
M α ε (V 1 ) ≤ ε −γ 1 Q∈D f ath.ˆ| V Q | β + 2 ε γ 2ˆ Q,Q ′ ∈D f ath. : Q ′ ⊂Q ∇V Q ′ : ∇V Q ,(3.13)
where A : B stands for the euclidian product of two matrices
A = (A ij ) 1≤i,j≤d , B = (B ij ) 1≤i,j≤d of size d × d: A : B := ij A ij B ij . For the estimation of |∇V 1 | 2 , we have used the identity |∇V 1 | 2 = ∇V 1 : ∇V 1 = Q,Q ′ ∈D f ath ∇V Q : ∇V Q ′ . Since V Q is supported on Q, ∇V Q : ∇V Q ′ vanishes except when Q ∩ Q ′ = ∅, i.e. Q ⊂ Q ′ or Q ′ ⊂ Q,
thus justifying the factor 2 and the inclusion Q ′ ⊂ Q in (3.13).
We need to estimate the two terms in (3.13). Since M α ε (V Q ) ≤ Cθ α Q diam(Q), thanks to Lemma 3.5, this term is less or equal than Cθ α L as required. Using the inequality f g L 1 ≤ f ∞ g L 1 , one can estimate the second term of (3.13) by
2 ε γ 2 Q,Q ′ ∈D f ath. : Q ′ ⊂Q ∇V Q L ∞ ∇V Q ′ L 1 .
Note that it would be more natural to use a Cauchy-Schwarz inequality (L 2 -L 2 ) at this step but, using it, we were not able to deduce the estimation by θ α L. Once again, since
R Q ′ ≤ diam(Q ′ ), we have ∇V Q ′ L 1 ≤ Cθ Q ′ ≤ diam(Q ′ )R −1 Q ′ θ Q ′ = C diam(Q ′ )ε −γ θ 1− 1−γ d−1 Q ′ .
(3.14)
Since 1 − 1 d < 1 − 1−γ d−1 < 1, Lemma 3.5 gives Q ′ ∈D f ath. : Q ′ ⊂Q ∇V Q ′ L 1 ≤ Cε −γ diam(Q)θ 1− 1−γ d−1 Q .
Now, elementary computations on exponents α, γ 2 , γ and Lemma 3.5 give successively
γ 2 = (d + 1)γ, α = 2 − (d + 1) 1−γ d−1 and
Cε γ 2
Q∈D f ath. diam(Q)θ Q R −d Q ε −γ θ 1− 1−γ d−1 Q = C Q∈D f ath. diam(Q)θ α Q ≤ Cθ α L.
Finally, we have obtained the desired inequality: M α ε (V 1 ) ≤ C θ α L.
Second step: estimation of the energy of Y on the nodes set. In order to get estimations on Y , it is convenient to divide Q 0 into 2 domains: the nodes set N and its complementary N c , where N :=
Q∈D(f ) B(c Q , cR Q )
and c > 0 is a constant which will be chosen later. By analogy with V 1 , one can write Y |N as a sum of vector fields Y Q , where
Y Q = 1 B(c Q ,cR Q ) Z Q − Q ′ ∈C(Q) Z Q ′ if Q ∈ D f ath. (see (3.8)), 1 B(c Q ,cR Q ) Z Q otherwise.
Now, from (3.11), we deduce that estimations (3.12) satisfied by V Q are also true for Y Q and consequently, we obtain M α ε (Y, N ) ≤ C θ α L as well (see (2.3) for the definition of M α ε (Y, N )).
Third step: estimation of the energy of Y out of the nodes set.
Reminding that Y = Q∈D(f ) Q =Q 0 Z Q ,
and because M α ε is not subadditive (due to the term |∇Y | 2 ) the first thing to do is to understand to which extent the supports of Z Q can intersect. To this aim, let us note that if the constant c > 0 in (3.2) is chosen equal to √ d or more, due to (3.10), then each Z Q restricted to N c is supported on Q (see figure 1): supp Z Q ∩ N c ⊂ Q. In particular, this implies that
supp Z Q ∩ supp Z Q ′ ∩ N c = ∅ =⇒ Q ∩ Q ′ = ∅ =⇒ Q ⊂ Q ′ or Q ′ ⊂ Q.
For this reason, M α ε (Y, N c ) can be estimated exactly in the same way as we did for the estimation of M α ε (V 1 ) in (3.13). Moreover, the Young inequality, f * µ L p ≤ f L p |µ|(R d ) for all f ∈ L p (R d ), µ ∈ M(R d ), and the definition of Z Q in (3.8) easily give
∇Z Q ′ L 1 ≤ Cθ Q ′ R −1 Q ′ diam(Q ′ ).
Since this estimation (which is the same as (3.14)) and (3.9) are the only ones we have used in the first step for the estimation of M α ε (V 1 ), we get M α ε (Y, N c ) ≤ Cθ α L as well.
End of the proof of Proposition 3.2 Finally, the vector field
V = Y + V 1 + V 2 , where V 2 is given by Proposition 3.7, satisfies ∇ · V = ρ Q 0 − f , M α ε (V ) ≤ 3{M α ε (Y ) + M α ε (V 1 ) + M α ε (V 2 )} ≤ C{θ α L + ε γ 2 f 2 L 2 } and V L 1 ≤ Y L 1 + V 1 L 1 + V 2 L 1 ≤ CL f L 1 .
Estimation between d α ε and the Wasserstein distance
Our aim is to prove an estimation on the pseudo-distances d α ε similar to Proposition 2.2. Because of the Dirichlet term in the definition of M α ε , d α ε cannot be estimated only by the Wasserstein distance W 1 but one has to add a term involving f + − f − L 2 . Using Proposition 3.2, we are going to prove the following theorem: Theorem 4.1. Let Q = (0, L) d be a a cube of side length L > 0 in R d and ε ∈ (0, 1). There exists C > 0 only depending on α, d and L such that for all f + , f − ∈ L 2
+ (Q) with Q f + =´Q f − = 1, there exists u ∈ H 1 (R d ) compactly supported on the set Q ε := {x ∈ R d : dist(x, Q) ≤ Cε γ } satisfying ∇ · u = f := f + − f − as well as d α ε (f + , f − ) ≤ M α ε (u) ≤ C H W 1−d(1−α) 1 (f + , f − ) + ε γ 2 f 2 L 2 and u L 1 ≤ C, (4.1)
where H : R + −→ R + is the scalar function defined by H(x) = x + x λ for some λ ∈ (0, 1) depending on α, and W 1 stands for the Wasserstein distance associated to the Monge cost (x, y) → |x − y|.
Remark 4.2. One can replace the condition´f ± = 1 by´f ± = θ ≥ 0. Then, the constant C will also depend on θ: C = C(θ, α, d, L). However, we can easily check that C is locally bounded with respect to θ, i.e. it is uniform for bounded values of θ.
(f + , f − ) ≃ d α (f + , f − ) ≤ CW 1 (f + , f − ) 1−d(1−α)
. On the contrary, when ε is very large, because of Theorem 3.4, one can expect that d α
ε (f + , f − ) ≃ ε γ 2 f 2 L 2 .
However, for technical reasons, due to the lack of subadditivity of the second term (Dirichlet energy) in the definition of M α ε , we were not able to reach the case H(x) = x.
Proof. Up to replacing
(f + , f − ) by (f + − f + ∧ f − , f − − f + ∧ f − ), one can assume that f + ∧ f − = 0, where for all x ∈ Q, (f − ∧ f + )(x) = inf(f − (x), f + (x))
. Indeed, it is sufficient to note that, if µ ± are two measures with the same mass and ν is a positive measure on Q then we have
W 1 (µ + + ν, µ − + ν) = W 1 (µ + , µ − ).
Our method to prove this proposition is an adaptation of that of J.-M. Morel and F. Santambrogio in [17] (see also Proposition 6.16. page 64 in [3]).
Let f + , f − ∈ L 2 + (Q) be two densities on the cube Q = (0, L) d such that´Q f ± = 1. Chose an optimal transport plan Π between f + and f − for the Monge-Kantorovich problem associated to the cost c(x, y) = |x − y|. Hence Π satisfies the constraint P ± # Π = f ± (x) dx where P + (resp. P − ) is the projection on the first variable x (resp. the second variable y) and dx is the Lebesgue measure. And we havê
Q |x − y| dΠ(x, y) = W 1 (f + , f − ) =: W. (4.2)
So as to use the local estimation of the previous part, let us classify the set of ordered pairs (x, y) with respect to the distance |x − y|. More precisely, for j ≥ 0, set
X j = {(x, y) ∈ Q 2 : d j ≤ |x − y| < d j+1 },
where d j = (2 j − 1) w and w ∈ (0, 1) will be chosen later. In particular, d 0 = 0 and X j is empty if d j > diam(Q), i.e. j > J := ln 2 diam(Q) w + 1 . For this reason, one can restrict to integers j ≤ J ≤ 1 + | ln w|: we will assume that d j ≤ diam(Q). Moreover, (4.2) immediately gives the estimation j d j θ j ≤ W , where θ j = Π(X j ). Next, for each integer j ∈ [1, J], consider a uniform partition of Q into cubes Q j k , k = 1, . . . , K j , with side length d j+1 . It is easy to estimate K j by
K j ≤ Cd −d j+1 .(4.4)
For j ≥ 0, set
Π j = Π |X j ; θ j = Π(X j ); f ± j = P ± # Π j and f j = f + j − f − j , Clearly, one has Π = j Π j and f ± = j f ± j .
In the same way, for j ≥ 0 and 1 ≤ k ≤ K j , set
Π j,k = Π |X j ∩ Q j k ×Q ; θ j,k = Π j,k (Q 2 ) and f ± j,k = P ± # Π j,k so that Π j = k Π j,k ; θ j = k θ j,k and f ± j = k f ± j,k .
Π j,k represents the part of the transport plan Π corresponding to points in Q j k which are sent at a distance comparable to d j+1 . In particular, f + j,k is supported on Q j k and f − j,k is supported on the cube Q j k with the same center but twice the side length of Q j k . As we did in (3.2), let us define ρ j k the kernel associated to Q j k by
ρ j k (x) = (R j k ) −d ρ(R j k (x − c j k )), where ρ ∈ C 1 c (R d , R + ), R j k = ε γ [θ j k ] 1−γ d−1 and c j k is the center of Q j k .
For the sake of simplicity, let us assume that supp ρ is the unit ball centered at the origin. Let B j k := B(c j k , r j k ) be the smallest ball containing Q j k and supp ρ j
k = B(c j k , R j k ): i.e. r j k = max{R j k , diam(Q j k )}. Thanks to Proposition 3.2, it is possible to find a vector field u j k ∈ H 1 0 (B j k ) satisfying ∇ · u j k = f j k := f + j,k − f − j,k , u j k L 1 ≤ Cθ j,k and E j k := E α ε (u j k ) ≤ C {θ α j,k d j+1 + ε γ 2 f j k 2 L 2 }. (4.5)
Moreover, if R j k ≥ d j+1 /2, the first term in the right-hand side of (4.5) can be omitted since one has
θ α j,k d j+1 ≤ Cε γ 2 f j k 2 L 2 . (4.6)
Indeed, in this case, writing θ := θ j,k and R := R j k , one has θ α d j+1 ≤ 2θ α R and, since
2 − α = (1 − γ) d+1 d−1 , θ α R = [θ α−2 R 1+d ][θ 2 R −d ] = ε γ 2 R −d θ 2 .
Then, (4.6) follows from the fact that, by the Cauchy-Schwarz inequality, we have
R −d θ 2 ≤ R −d |B j k |ˆB j k (f j k ) 2 ≤ CˆB j k (f j k ) 2 .
Now, let us define the vector field u = j,k u j,k , which satisfies
∇ · u = j,k ∇ · u j k = j,k f j k = f := f + − f − . First note that u L 1 ≤ C u j,k L 1 ≤ 2C θ j,k = 2C.
In order estimate the energy of u, a similar development of | ∇u j,k | 2 as in (3.13) and the Cauchy-Schwarz inequality give
E α ε (u) ≤ J J j=1 E α ε K j k=1 u j k ≤ C J j k E j k + (k,l)∈I j E j k E j l ,(4.7)
where I j stands for the set of pairs (k, l) satisfying k = l, θ j,k ≥ θ j,l and B j k ∩ B j l = ∅. We have to estimate the two terms in the right-hand side of (4.7).
Estimation of the first term in (4.7) We recall that E j k ≤ θ α j,k d j+1 + ε γ 2 f j k 2 L 2 . For the second term, note that
j,k f j k 2 L 2 ≤ f 2 L 2 . (4.8)
Indeed, since f + ∧ f − = 0, for all j, k, one has f + j,k ∧ f − j,k = 0 as well. In particular, f j k 2
L 2 = f + j,k 2 L 2 + f − j,k 2 L 2 , f 2 L 2 = f + 2 L 2 + f − 2 L 2
and (4.8) follows from the super-additivity of the power function x → |x| p for p ≥ 1: |x + y| p ≥ |x| p + |y| p for x, y ∈ R whenever xy ≥ 0.
For the first term, applying successively the inequality 0 < α < 1, the Hölder inequality, (4.3) and the fact that K j d j+1 = Cd 1−d j+1 (see (4.4)), one gets
j,k θ α j,k d j+1 ≤ j d j+1 K j [θ j /K j ] α = j [d j+1 θ j ] α [d j+1 K j ] 1−α ≤ j θ j d j+1 α j d j+1 K j 1−α ≤ C(w + W ) α j [w(2 j+1 − 1)] 1−d 1−α ≤ C ′ (w α + W α )w (1−d)(1−α) since θ 0 d 1 ≤ d 1 = w (we cannot estimate this term by W because d 0 = 0) and, because of (4.3), j≥1 θ j d j+1 ≤ 2 j≥1 θ j d j ≤ 2W . Finally, we get j,k E j k ≤ C w 1−d(1−α) + W α w −(d−1)(1−α) + ε γ 2 f 2 L 2 . (4.9)
Estimation of the second term in (4.7) Before following these computations, we need to understand what the condition "B j k ∩B j l = ∅ " is meaning. Assume that (k, l) ∈ I j . From Q j k ∩ Q j l = ∅, we see that supp ρ j k or supp ρ j l is not included in Q j k (resp. Q j l ). Since, by definition of I j , θ j,k ≥ θ j,l , this implies that R j k ≥ d j+1 /2. Therefore, as we noticed after formula (4.5),
E j k ≤ ε γ 2 f j k 2 L 2
and (4.6) also implies that
θ α j,l d j+1 ≤ θ α j,k d j+1 ≤ Cε γ 2 f j k 2 L 2 .
Now, (4.5), the subadditivity of the square root function, the preceding inequality, (4.8) and Cauchy-Schwarz inequality give in turn
(k,l)∈I j E j k E j l ≤ C (k,l)∈I j ε γ 2 f j k 2 2 ε γ 2 f j l 2 2 + θ α j,l d j+1 ≤ Cε γ 2 (k,l)∈I j f j k 2 L 2 + f j k L 2 f j l L 2 ≤ Cε γ 2 K j f j 2 L 2 + k,l f j k 2 L 2 k,l f j k 2 L 2 ≤ 2Cε γ 2 K j f j 2 L 2 . From K j ≤ d −d j+1 ≤ 2 −dj w −d and f j 2 L 2 ≤ f 2 L 2 , we obtain in the end that j (k,l)∈I j E j k E j l ≤ Cw −d ε γ 2 f 2 L 2 . (4.10)
End of the proof Let F = ε γ 2 f 2 L 2 . If W + F ≥ 1, (4.1) trivially follows from Proposition 3.2 since H(x) ≥ x. Otherwise, if W + F ≤ 1, (4.7), (4.9), (4.10) and the fact that J ≤ C(1 + ln w) give
E α ε (u) ≤ C(1 + | ln w|) w ν + W α w ν−α + w −d F , where ν := 1 − d(1 − α) ∈ (0, 1) and so α − ν = −(d − 1)(1 − α) < 0.
Let us fix some δ ∈ (0, 1) small enough so that 0 < ν ± δ < 1 and ν − α ± δ < 0. For some constant c depending on δ, one has 1 + | ln w| ≤ c(w δ + w −δ ) and so
E α ε (u) ≤ C w ν±δ + W α w ν−α±δ + w −d±δ F ,
where the sum is taken over the values of ±1 (+1 or −1) in the right-hand side. Then, we make the choice w = W + F λ ∈ [0, 1] for some λ > 0.
Since 0 < ν ± δ < 1, we get w ν±δ ≤ W ν±δ + F λ(ν±δ) and, because −d ± δ < 0,
ν − α ± δ < 0, we have w ν−α±δ ≤ W ν−α±δ and w −d±δ ≤ F λ(−d±δ) which gives E α ε (u) ≤ C W ν±δ + F λ(ν±δ) + W ν±δ + F 1+λ(−d±δ) .
We fix λ > 0 small enough that 1 + λ(−d ± δ) > 0: in this way, all the exponents in the preceding formula are positive. Finally, (4.1) follows from the fact, since W ,
F ≤ 1, we have W , F ≤ W 1−d(1−α) + F .
Remark 4.4. Since min{w ν + w −d F : w ∈ (0, 1)} = cF 1 d+ν and 1 d+ν < 1, one cannot obtain an estimation of the form E α ε (u) ≤ C(W + F ) as expected. However, one could improve a bit (4.1) by a better estimation of the number of indices l such that (k, l) ∈ I j .
A Γ-convergence result
Let Ω ⊂ R 2 be a bounded open set and µ = µ + − µ − for two probability measures µ + and µ − compactly supported on Ω. We recall the definition of the set M div (Ω) = {u : Ω → R 2 : u and ∇ · u are finite measures on Ω} which is endowed with the topology of weak star convergence on vector measures and their divergence. As weak star topology is never metrizable, the set M div (Ω) is not metrizable. However, for the natural norm u M div (Ω) = |∇ · u|(Ω) + |u|(Ω) given by the total variation of u and its divergence, we know that all bounded sets are metrizable: for all M > 0, there exists a metric d M for the weak star convergence of u and ∇ · u on the set M M (Ω) = {u ∈ M div (Ω) : |u|(Ω) + |∇ · u|(Ω) ≤ M }.
In [18] the Γ-convergence of the functional sequence M α ε to M α was proved. Our aim is to prove that this property remains true when adding a divergence constraint. Since, for u ∈ H 1 (Ω), one has ∇ · u ∈ L 2 , one cannot prescribe ∇ · u = µ if µ is not in L 2 . For this reason, we first have to define a regularization of µ. Let (f ε ) ε>0 ⊂ L 2 be a sequence of L 2 functions weakly converging to µ as measures and satisfying the estimation
ε γ 2 f ε 2 L 2 −→ ε→0 0. (5.1)
This estimation is going to be useful for the proof of Theorem 5.1. For example, we can define f ε as
f ε := ρ ε * µ,
where ρ ε (x) = ε −2γ ρ(ε −γ x) for some compactly supported ρ ∈ C 1 (R d , R + ) such that Ω ρ = 1 and γ is still defined as γ = γ 2 d+1 = α+1 3 . Now, let us define the functionals M α ε (resp M α ) adding a divergence constraint on u ∈ M div (Ω):
M α (u) = M α (u) if ∇ · u = µ +∞ otherwise M α ε (u) = M α ε (u) if ∇ · u = f ε +∞ otherwise
We are going to prove the following Γ-convergence result: We first remind how to build a recovery sequence in the case of a mass θ flowing on a single segment S, i.e. u = θH 1 |S . To this aim, we need to find a structure close to u which is almost optimal for M α ε . We proceed by a slicing argument: Let u be any vector measure in M div (Ω). Take some ν ∈ S 1 := {x ∈ R 2 : |x| = 1} which has to be thought as the tangent vector to S in the case where u = θH 1 |S . Let us consider v = [(u · ν) + ] |ν ⊥ (restriction on ν ⊥ of the positive part of u · ν) the flux of u across the hyperplane ν ⊥ = {x ∈ R 2 : x · ν = 0} and assume that´v = θ. Then M α ε (u) can be controlled from below by integrals on subintervals of Rν of the following Cahn-Hilliard type energy (see [10] for physical motivations):
F β ε (v) = ε −γ 1ˆR |v| β + ε γ 2ˆR |∇v| 2 .
This kind of models for droplets equilibrium was studied by G. Bouchitté, C. Dubs and P. Seppecher in [5] for instance (see also [6]). F β ε (v) can be renormalized through the
formula: v(x) = θR −d θ,ε w(R −1 θ,ε x), where R θ,ε = ε γ θ 1−γ d−1 . Then, the constraint´v = θ turns into´w = 1 and F β ε (v) = θ α F β (w), where F β (w) = ˆR |w| β +ˆR |∇w| 2 .
Then, the existence of an optimal profile w is given by
Lemma 5.2.
There exists a compactly supported profile z ∈ C 1 c (R, R + ) ∩ W 2,1 (R, R + ) solution of the minimization problem min ˆR |u| β +ˆR |u ′ | 2 : u ∈ H 1 (R, R + ) andˆR u = 1 .
(5.2)
Remark 5.3. Although we restrict to the two dimensional case, some works by G. Bouchitté, C. Dubs and P. Seppecher (see [13]) suggest that Lemma 5.2 and Theorem 5.1 could be generalized in every dimension as well. However, the aim of this paper is to use the tools of section 4 so as to establish the Γ − lim sup property for functionals M α ε (with divergence constraint) and, from the point of vue of the complexity of the proof, this is independent of the dimension. Since the Γ − lim inf property was only established in the 2D case in [18], we prefer to stay in this framework. Actually, the difficulty to prove a Γ-convergence result of M α ε (resp. M α ε ) to M α (resp. M α ) in every dimension would concern the Γ − lim inf part and this is not the purpose of this paper. Now let z θ,ε be defined by z θ,ε (x) = θR −d θ,ε z(R −1 θ,ε x) and let us introduce the kernel ρ θ,ε associated to z θ,ε given by Lemma 5.4. There exists a radial kernel ρ θ,ε ∈ H 1 (R 2 , R + ) ∩ C 0 c (R 2 , R + ) such that z θ,ε is the projection of ρ θ,ε on the axis (x 1 = 0):
π 2 ♯ ρ θ,ε (x) dx = z θ,ε (x 2 ) dx 2 ,
where π 2 stands for the projection on the second variable, dx (resp. dx 2 ) is the Lebesgue measure on R 2 (resp. R) and π ♯ µ stands for the pushforward of some measure µ by π : R 2 → R.
Proof. First renormalize the problem writing ρ θ,ε (x) = R −2 θ,ε ρ(R −1 θ,ε x) so that it is enough to find ρ satisfying π 1 ♯ ρ(x)dx = z(x 2 ) dx 2 It is easy to see that a radial solution is given by the formula
ρ(x) =ˆ∞ r −z ′ (s) π √ s 2 − r 2 ds.
for all x ∈ R 2 such that |x| = r.
As a consequence, in the case where u = θH 1 |S , a recovery sequence, i.e. a sequence (u ε ) such that u ε → u in M div (Ω) and M α ε (u ε ) → M α (u) as ε → 0, is obtained as
u ε = ρ θ,ε * u.
In the case of a finite energy configuration, i.e. u ∈ M div (Ω) such that M α (u) < ∞, thanks to classical properties in the theory of Γ-convergence, it is enough to find a recovery sequence for u belonging to a class of measures which are dense in energy.
Thanks to the work of Q. Xia in [23] (see also [18]), we know that the class of vector measures concentrated on finite graphs is dense in energy so that one can restrict to this case. This was used in [18] to prove the Γ-convergence of M α ε toward M α . In the setting of functionals with divergence constraint, we need the following lemma:
Lemma 5.5. Let u ∈ M div (Ω) be such that M α (u) < ∞. For all λ > γ, there exists a sequence (u ε ) ⊂ H 1 0 (Ω) converging to u in M div (Ω) such that M α ε (u ε ) −→ ε→0
M α (u) and ε λ ∇ · u ε L 2 is bounded.
Before proving this statement, we are going to investigate the case where u is concentrated on a finite graph. First of all, let us give some details on what "a vector measure concentrated on a finite graph G" is. Let G = (V (G), E(G), θ) be a weighted directed graph: V (G) ⊂ Ω is a finite set of vertices, E(G) is the finite set of oriented edges e = (e, τ e ), where e = [a e , b e ] ⊂ Ω and τ e is a unit vector representing the direction of e, and θ : E(G) → (0, +∞) is the weight function. Then the "vector measure associated to G" is given by
u G = e=(e,τe)∈E(g)
θ(e)τ e dH 1 |e .
These measures u G belong to M div (Ω), i.e. ∇ · u G is a measure, and they are called "transport paths" (see Definition 2.1 in [23]). When u is a transport path, we have the following lemma:
Lemma 5.6. Let u = u G ∈ M div (Ω) for some weighted directed graph G. Then, there exists a sequence (u ε ) ε>0 converging to u in M div (Ω) and a constant C depending on u such that, for ε small enough, u ε ∈ H 1 0 (Ω) and 1.´Ω |u ε | ≤ |u|(Ω) + C ε γ , 2.´Ω |∇ · u ε | ≤ |∇ · u|(Ω),
3. ε γ ∇ · u ε L 2 ≤ C, 4. |M α ε (u ε ) − c 0 M α (u)| ≤ Cε γ .
Proof. By definition, such a vector measure u can be written as a finite sum of measures u i = θ i τ i H 1 |S i concentrated on a segment S i ⊂ Ω directed by τ i with multiplicity θ i for i = 1, . . . , I. We first define a regularized vector fied v ε by v ε := i v i , where v i = ρ θ i ,ε * u i . Then, for ε small enough, v ε is compactly supported on Ω and satisfies ´Ω |v ε | ≤ |u|(Ω),
|M α ε (v ε ) − c 0 M α (u)| ≤ Cε γ .
The first statement is a consequence of the fact that´ρ θ i ,ε = 1 and the inequality f * µ L 1 ≤ f L 1 |µ|(Ω) for f ∈ C c (Ω) and for a finite measure µ on Ω. For the second statement, by definition of the kernel ρ θ,ε we know that, out of the nodes set N = i supp(∇ · v i ), M α ε (v ε , N c ) = c 0 M α (v, N c ). As a result, we just have to estimate these energies on N which is a finite union of balls: the supports of ρ θ i ,ε recentered at each end-point of the segment S i . Since the radius of these balls is of the order of ε γ , this immediately gives the fact that M α (u, N ) ≤ Cε γ for some constant C > 0 depending on u. For the sake of simplicity, in the rest of this proof, C > 0 will denote some constant depending on u which is large enough so that all the inequalities below are true. We are going to prove that
M α (u, N ) + M α ε (v ε , N ) ≤ Cε γ . It remains to estimate M α ε (v ε , N ). Since M α ε (v ε , N ) ≤ I i M α ε (v i , N ), it is enough to estimate M α ε (v i , N ). But v i L ∞ (N ) = ρ θ i ,ε * u i L ∞ (N ) ≤ Cε −2γ ρ L ∞ |u i |(N i ),
where N i := N + supp ρ θ i ,ε := {x + y : x ∈ N, y ∈ supp ρ θ i ,ε }. Note that supp ρ θ i ,ε is a ball centered at the origin with radius smaller than Cε γ so that N i is a finite union of balls with radii smaller than Cε γ as well. In particular, using the fact that u i = θ i τ i H 1 |S i , we get |u i |(N i ) ≤ Cε γ and so v i L ∞ (N ) ≤ Cε −γ . Similarly, one has ∇v i L ∞ (N ) = ∇ρ θ i ,ε * u i L ∞ (N ) ≤ Cε −3γ ∇ρ L ∞ |u i |(N i ) ≤ Cε −2γ . Now, the definition (2.4) gives M α ε (v ε , N ) = ε α+1ˆN |∇v ε | 2 + ε α−1ˆN |v ε | β ≤ |N |{ε α+1 ∇v ε 2 L ∞ + ε α−1 v ε β L ∞ }.
From the inequality |N | ≤ Cε 2γ and the equalities α = 3γ − 1, βγ = (4α−2)γ α+1 = 4γ − 2, we deduce M α ε (v ε , N ) ≤ Cε 2γ {ε 3γ−4γ + ε 3γ−2−βγ } ≤ 2Cε γ as required. For v ε to satisfy all the properties of Lemma 5.6, it remains to prove the second and third statements. The second statement says that the total variation of ∇ · u ε is lower than that of ∇ · u. Since, for instance, the divergence of v ε does not vanish at the nodes even if ∇ · u = 0, v ε has to be replaced by u ε := v ε − w ε where w ε ∈ H 1 0 (N ) is constructed as follows:
The node set is a finite union N = n j=1 B j , where each node B j is a ball centered at the end-point a i of some segment S i = [a i , b i ]. Let assume that ε is small enough so that these balls are non-overlapping. Then, on each node B j , g j := ∇ · v ε is a finite superposition of kernels like ρ θ,ε recentered at c j , the center of B j . In particular g j L 2 ≤ Cε −γ and´B j g j =´B j ∇ · v ε = (∇ · u)(B j ) =: θ j . If θ j = 0, then Theorem 3.4 allows to find w j ∈ H 1 0 (B j ) satisfying ∇ · w j = g j and w j H 1 (B j ) ≤ C ε −γ . If θ j = 0, let say θ j > 0, we rewrite g j as g j = g + − g − = λg + + (1 − λ)g + − g − where g + (resp. g − ) stands for the positive part (resp. negative part) of g and λ ∈ (0, 1] is chosen such that (1 − λ)´B g + =´B g − , i.e. θ j = λ´B j g + . Applying Theorem 3.4, we get w j ∈ H 1 0 (B j ) satisfying ∇ · w j = (1 − λ)g + − g − and w j H 1 (N ) ≤ C ε −γ . Let us define w ε = j w j and u ε := v ε − w ε . Since´B j |∇ · u ε | =´B j |g j − ∇ · w j | = λ´B j g + = θ j for all j, we havê Ω |∇ · u ε | =ˆN |∇ · u ε | ≤ j θ j = |∇ · u|(Ω).
Moreover, to estimate ∇·u ε L 2 , note that |∇·w ε | L 2 ≤ |w ε | H 1 ≤ Cε −γ and, because ∇·v ε is only composed of a finite sum of translated kernels of the form ρ θ i ,ε , ∇ · v ε L 2 ≤ Cε −γ as well. In particular ε γ ∇ · u ε L 2 is bounded. Then, from a Sobolev inequality, we deduce that w ε L 2 = j w j L 2 (B j ) ≤ C j ε γ ∇w j L 2 ≤ C ′ .
Consequently, by the Cauchy-Schwarz inequality, we get Ω |u ε | ≤ˆΩ |v ε | +ˆN |w ε | ≤ |u|(Ω) + |N | 1/2 w ε L 2 ≤ |u|(Ω) + Cε γ .
Similarly, by a Hölder inequality, we havê N |w ε | β ≤ ε γ(2−β) w ε β L 2 ≤ Cε γ(2−β) .
Once again, since α = 3γ − 1 and βγ = 4γ − 2, we deduce M α ε (w ε ) = ε α+1ˆN |∇w ε | 2 + ε α−1ˆN |w ε | β ≤ C{ε α+1−2γ + ε α−1+γ(2−β) } = 2Cε γ .
Since M α (u, N ) ≤ Cε γ , we get M α ε (u ε , N ) ≤ 2[M α ε (v ε , N ) + M α ε (w ε , N )] ≤ Cε γ which finally gives |M α ε (u ε ) − c 0 M α (u)| = |M α ε (u ε , N ) − c 0 M α (u, N )| ≤ Cε γ .
Figure 1 :
1A square Q and its 4 dyadic children Q i with the associated vector field Z Q By definition of the kernel ρ Q , one hasM α ε (Z Q ) ≤ Cθ α Q diam(Q). (3.9)This a consequence of the choice of R Q as a minimizer in (1.2). Indeed, for the sake of simplicity, let us assume that supp ρ is the unit ball centered at the origin. Then Z Q is concentrated on a strip of width R Q around the segment S = [c F (Q) , c Q ], i.e.supp Z Q ⊂ {x ∈ R d : dist(x, S) ≤ R Q } (3.10)
Lemma 3 . 8 .
38Let d ≥ 1 and B = B(0, R) ⊂ R d be a ball centered at the origin. There exists a constant C > 0 only depending on d such that the following holds: Let F ∈ L ∞ (B) be a radial function: i.e. for a.e. x ∈ B, F (x) = f (|x|) for some f ∈ L ∞ (0, R). Assume that´B F = 0. Then, there exists a radial function V ∈ W 1,∞ 0
Remark 4. 3 .
3It is tempting to think that estimation (4.1) also holds when H(x) = x which is the natural choice. Indeed, if ε is taken very small, since M α ε Γ-converge to M α and because of Proposition 2.2, one can expect that d α ε
Theorem 5. 1 .
1There exists a constant c 0 such that the functional sequence (M α ε ) ε>0 Γ-converges to c 0 M α as ε → 0. Moreover c 0 is given by the minimum value for the minimization problem (5.2).
Proof of Lemma 5.5. First fix a vector field u ∈ M div (Ω) and construct a sequence (u n ) n≥1 converging to u such that u n = u Gn is a vector measure associated to some weighted directed graph G n ⊂ Ω and M α ε (u n ) converges to M α (u). Since (u n ) weakly converges in M div (Ω), the total variations of both measures u n and ∇ · u n are bounded by some constant M > 0. In the following, we use a metric d on the space M M +1 (Ω). Extracting a subsequence if necessary, one can suppose that the two following estimations holdFor each n ≥ 1, let u ε,n be a sequence converging to u n as ε → 0 and satisfying all properties in Lemma 5.6 for some constant C = C n . Then, one can construct by induction a decreasing sequence (ε n ) n≥1 → 0 such that for all n ≥ 1 and ε ≤ ε n , u ε,n ∈ H 1 0 (Ω) and 1. u ε,n ∈ M M +1 (Ω),satisfies all properties of Lemma 5.5.Proof of Theorem 5.1. It is already shown in[18]We just have to prove that the Γ − lim sup property still holds when we add the divergence constraint. In other words, it remains to prove that for all u ∈ M div (Ω) such that ∇·u = µ, there exists a sequence (v ε ) ε>0 ⊂ M div (Ω) weakly converging to u as measures, satisfyingFirst of all, take a sequence (u ε ) ε>0 ⊂ H 1 0 (Ω) converging to u given by Lemma 5.5= f ε − ∇ · u ε the residual divergence. In particular,´Ω g ε = 0 and, from our hypothesis on (f ε ) (see (5.1)) and from the fact that γ 2 = 3γ, we obtain that ε γ 2 g ε 2 L 2 goes to 0 when ε → 0. Moreover, since f ε and ∇ · u ε weakly converge to µ as ε goes to 0, we know that g ε weakly converges to 0. In order to satisfy the divergence constraint, we may correct u ε with a vector field w ε , given by Theorem 4.1 (together with Remark 4.2), such thatand w ε M div (Ω) is bounded, where H(x) = C(x + x δ ) for some C > 0 and δ ∈ (0, 1). We deduce that (w ε ) is relatively compact in M div (Ω). From (5.3) and the Γ − lim inf property, this implies that w ε converges to 0 in M div (Ω). Now, by construction, vM α (u). Indeed, this last limit is a consequence of0 and assume that M α ε (v ε ) is bounded. Then,Proof. Let ν > 0 be some constant. For all real matrices A and B of size d × d, by the Young inequality, we haveWriting u ε = v ε + u ε − v ε , we use the preceding inequality for A = ∇v ε , B = ∇(u ε − v ε ) and the subadditivity of x → |x| β to getSince M α ε (v ε ) < C for some constant C < +∞, we deduce thatLet us take ν = M α ε (u ε − v ε ). Hence, taking the lim sup when ε → 0, we getis bounded as well. Then we can apply all the preceding computations exchanging u ε and v ε to get lim sup Aknowledgment. I would like to thank my advisor, Filippo Santambrogio, for discussions and advices which have proved to be very useful. This work was partially supported by the PGMO research project MACRO "Modèles d'Approximation Continue de Réseaux Optimaux".
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| []
|
[
"ANALYTIC MAPS OF PARABOLIC AND ELLIPTIC TYPE WITH TRIVIAL CENTRALISERS",
"ANALYTIC MAPS OF PARABOLIC AND ELLIPTIC TYPE WITH TRIVIAL CENTRALISERS"
]
| [
"Artur Avila ",
"Davoud Cheraghi ",
"Alexander Eliad "
]
| []
| []
| We prove that for a dense set of irrational numbers α, the analytic centraliser of the map e 2πiα z + z 2 near 0 is trivial. We also prove that some analytic circle diffeomorphisms in the Arnold family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers. | 10.4171/aihpc/22 | [
"https://arxiv.org/pdf/2003.13336v2.pdf"
]
| 214,713,993 | 2003.13336 | f52b4b1ba6e7d8e1f319b423d145df043d4e1df8 |
ANALYTIC MAPS OF PARABOLIC AND ELLIPTIC TYPE WITH TRIVIAL CENTRALISERS
21 Apr 2020
Artur Avila
Davoud Cheraghi
Alexander Eliad
ANALYTIC MAPS OF PARABOLIC AND ELLIPTIC TYPE WITH TRIVIAL CENTRALISERS
21 Apr 2020
We prove that for a dense set of irrational numbers α, the analytic centraliser of the map e 2πiα z + z 2 near 0 is trivial. We also prove that some analytic circle diffeomorphisms in the Arnold family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers.
Introduction
For α ∈ R, let H ω α denote the set of germs of holomorphic diffeomorphisms of (C, 0) of the form h(z) = e 2πiα z + O(z 2 ), defined near 0. We also consider the class C ω α of orientation preserving analytic diffeomorphisms of the circle R/Z with rotation number α. Let H ω = ∪ α∈R H ω α and C ω = ∪ α∈R C ω α . The analytic centraliser of an element h ∈ H ω α , denoted by Cent(h), is the set of elements of H ω which commute with h near 0. From dynamical point of view, any element of Cent(h) is a conformal symmetry of the dynamics of h, that is, the conformal change of coordinates g which conjugate h to itself, g −1 • h • g = h. Evidently, Cent(h) forms a group, where the action is the composition of the elements. For every k ∈ Z, a suitable restriction of the k-fold composition h •k is defined near 0 and belongs to Cent(h). If the only elements of Cent(h) are of the form h •k for some k ∈ Z, it is said that h has a trivial centraliser. In the same fashion, for h ∈ C ω , the collection Cent(h) of elements of C ω which commute with h enjoys the same features.
Theorem 1.1. There is a dense set of α ∈ R \ Q such that Cent(e 2πiα z + z 2 ) is trivial.
The above theorem is proved using a successive perturbation argument and the following statement for parabolic maps which we prove in this paper.
Theorem 1.2. For every p/q ∈ Q, Cent(e 2πip/q z + z 2 ) is trivial.
The main idea we employ to prove the above theorems also allows us to deal with analytic circle diffeomorphisms in the Arnold family,
S a,b (x) = x + a + b sin(2πx),
for a ∈ R and b ∈ (0, 1/(2π)). Theorem 1.3. For every b ∈ (0, 1/(2π)) there is a ∈ R such that Cent(S a,b ) is trivial and the rotation number of S a,b belongs to R \ Q.
Indeed, we prove that for each fixed b ∈ (0, 1/(2π)), the set of rotation numbers of the maps S a,b which have an irrational rotation number and a trivial centraliser is dense in R. The above theorem is obtained from a successive perturbation argument and the analogue of Theorem 1.2 for maps S a,b with a parabolic cycle.
Date: 22nd April 2020. 2010 Mathematics Subject Classification. Primary 37F50; Secondary 37E10, 37F10.
The main tool used to deal with parabolic maps is Ecalle cylinders and horn maps, first studied and applied by Douady-Hubbard [DH84] and Lavaures [Lav89].
To our knowledge, Theorems 1.1 and 1.3 provide the first examples in H ω and C ω with irrational rotation numbers and trivial analytic centralisers. Below we briefly explain how these results fit in the frame of the dynamics of such analytic diffeomorphisms.
When an element h ∈ H ω α , for α ∈ R \ Q, is locally conformally conjugate to its linear part near 0, Cent(h) is a large set. That is, if φ −1 • h • φ(w) = e 2πiα w near 0, for some φ ∈ H ω , then for any µ ∈ C \ {0}, h commutes with the map z → φ(µφ −1 (z)). Indeed, here, Cent(h) is isomorphic to C \ {0}. The problem of understanding Cent(h) precedes the problem of local conjugation of h to its linear part. That is because, the space of solutions for the conjugation problem are the right-cosets of Cent(h). In this spirit, the size of Cent(h) may be thought of a measure of linearisability of h near 0. The same argument applies to analytic circle diffeomorphisms.
For h ∈ H ω , Cent(h) projects onto a subgroup of R/Z through g → log g ′ (0)/(2πi). Similarly, for h ∈ C ω , one maps g ∈ Cent(h) to its rotation number. Let G(h) ⊂ R/Z denote the image of this projection.
By remarkable results of Siegel and Herman [Sie42,Her79] there is a full-measure set C ⊂ R \ Q such that for every α ∈ C , any h ∈ H ω α ∪ C ω α is analytically linearisable. But, for generic choice of α, there are h ∈ H ω α and h ∈ C ω α which are not linearisable [Cre38,Arn61]. We note that if f and g commute, and one of them is linearisable at 0, then the other one must also be linearisable through the same map. This implies that if h ∈ H ω α ∪ C ω α is not linearisable, then G(h) ⊆ (R \ C )/Z. However, by a profound result of Moser [Mos90], G(h) may not be any subgroup of that set. That is because there is an arithmetic restriction on the rotations of commuting non-linearisable maps. The optimal size of G(h), for nonlinearisable h in H ω α and C ω α , remains open. This complication is due to the rich structure of the local dynamics of such maps near 0, see [PM95,Che17] and the references therein. However, a complete solution for smooth circle diffeomorphisms is presented in [FK09].
In [Her79,Yoc95,Yoc02], Herman and Yoccoz carry out a ground breaking study of the centraliser and conjugation problem for circle diffeomorphisms and germs of holomorphic diffeomorphisms of (C, 0). In particular, Herman proves the existence of C ∞ circle diffeomorphisms with irrational rotation number having uncountably many C ∞ symmetries, and Yoccoz proves the existence of C ∞ circle diffeomorphisms with irrational rotation numbers and trivial centralisers. Perez-Marco in [PM95] elaborated a construction of Yoccoz to build elements h ∈ H ω and h ∈ C ω , with irrational rotation number, such that G(h) is uncountable. His construction provides remarkable examples where G(h) contains infinitely many elements of finite order. In this paper we close the problem of the existence of maps in H ω and C ω with irrational rotation number and trivial centraliser. In light of the above discussions, our result shows that quadratic polynomials and the Arnold family provide the least linearisable elements in H ω and C ω , respectively. This is consistent with the spirit of Yoccoz's argument in [Yoc95], that is, if some e 2πiα z + z 2 is linearisable, then any element of H ω α is linearisable. It is worth noting that the commutation problem for rational functions of the Riemann sphere was already studied by Fatou and Julia in 1920's [Jul22,Fat23] using iteration methods. A complete classification of such pairs was successfully obtain by Ritt [Rit23], using topological and analytic methods, and was reproved by Eremenko [Ere89] using modern iteration techniques. If iterates of g and h are not identical, modulo conjugation, they are either power maps, Chebyshev polynomials, or Lattès maps. The global commutation problem for entire functions of the complex plane still remains open, although substantial progress has been made so far, see for instance [GI59,Bak62,Lan99,Ng01,BRS16]. The global commutation problem on higher dimensional complex spaces has been widely studied using iteration methods in recent years, see [DS02,DS04,Kau18] and the references therein. For an extensive discussion on the centraliser and conjugation problems in low-dimensions one may refer to [Kop70] and the more recent survey article [OR10].
parabolic case
Fix an arbitrary rational number p/q ∈ Q with (p, q) = 1. Also fix an arbitrary g in Cent(Q p/q ). The map F = Q •q p/q has a parabolic fixed point at 0 with multiplier +1, and there are q attracting directions. It follows that the parabolic fixed point of F at 0 has multiplicity q + 1. That is, the Taylor series expansion of F near 0 is of the form
(1) F (z) = Q •q p/q (z) = z + 2 q k=q+1 a k z k ,
with a q+1 = 0.
Lemma 2.1. We have g ′ (0) q = 1.
Proof. Let g(z) = ∞ k=1 b k z k denote the Taylor series expansion of g about 0. First we show that b 1 = 0. Assume, for a contradiction, that n ≥ 2 is the smallest positive integer with b n = 0. Note that F • g = g • F near 0. By identifying the coefficient of z n+q in the Taylor series expansion of F • g and g • F we conclude that b n+q + nb n a q+1 = b n+q . Since a q+1 = 0, that gives us b n = 0, which contradicts the choice of n.
Now we identify the coefficients of z q+1 in the power series expansions of F • g and g • F , and obtain b q+1 + b q+1 1 a q+1 = b q+1 + b 1 a q+1 . This implies that (b q+1 1 − b 1 )a q+1 = 0. Since a q+1 = 0 and b 1 = 0, we must have b q 1 = 1. By Lemma 2.1, there is an integer j with 0 ≤ j ≤ q − 1 such that (Q •j p/q • g) ′ (0) = 1. Consider the holomorphic map
(2) G(z) = Q •j p/q • g, which is defined near 0 and commutes with F .
Lemma 2.2. The multiplicity of G at 0 is q + 1. That is,
G(z) = z + ∞ i=q+1 b i z i , with b q+1 = 0. Proof. Assume that G(z) = z + b n+1 z n+1 + b n+2 z n+2 + . . . is a convergent Taylor series with b n+1 = 0. Observe that F • G(z) = z + b n+1 z n+1 + b n+2 z n+2 + . . . + a q+1 z + b n+1 z n+1 + b n+2 z n+2 + . . . q+1 . . . + a q+j z + b n+1 z n+1 + b n+2 z n+2 + . . . q+j . . . = z + b n+1 z n+1 + b n+2 z n+2 + . . . + a q+1 z q+1 + b n+1 (q + 1)z q+n+1 + . . . . . . + a q+j z q+j + b n+1 (q + j)z q+n+j + . . . . . .. The coefficient of z q+n+1 in the above expansion is b q+n+1 + a q+1 b n+1 (q + 1) + a q+n+1 .
Similarly, the coefficient of z n+q+1 in the expansion of G • F is
a q+n+1 + b n+1 a q+1 (n + 1) + b q+n+1 .
Since F • G = G • F near 0, the above values must be identical. Using a q+1 = 0 and b n+1 = 0, we conclude that q = n.
We shall use the theory of Leau-Fatou flower, Fatou coordinates, and horn maps to exploit the local dynamics of F near 0. One may refer to [Mil06] and [Dou94] for the basic definitions and constructions we present below, although conventions may be different.
For s > 0, define the open sets
Ω s att = {ζ ∈ C | Re ζ > s − | Im ζ|}, Ω s rep = {ζ ∈ C | Re ζ < −s + | Im ζ|}. Also, consider the map I : C \ {0} → C \ {0}, I(z) = −1 qa q+1 z q .
For s > 0 there are holomorphic and injective branches of I −1 defined on Ω s att and Ω s rep . Consider two complex numbers v att and v rep such that
qa q+1 v q att = −1, v rep = e −πi/q v att . Evidently, I(v att ) = +1 and I(v rep ) = −1.
For s > 0, there is an injective and holomorphic branch of I −1 defined on Ω s att such that I −1 (Ω s att ) contains εv att , for sufficiently small ε > 0. Similarly, there is an injective branch of I −1 defined on Ω s rep such that I −1 (Ω s rep ) contains εv rep , for sufficiently small ε > 0. From now on, we shall fix these choices of inverse branches for I −1 on Ω s att and Ω s rep . This is independent of s > 0.
Let
W att = {z ∈ C \ {0} | | arg(z/v att )| ≤ π/q} , W rep = {z ∈ C \ {0} | | arg(z/v rep )| ≤ π/q} , W ′ att = {z ∈ C \ {0} | | arg(z/v att )| ≤ π/q − π/(4q)} , W ′ rep = {z ∈ C \ {0} | | arg(z/v rep )| ≤ π/q − π/(4q)} , where arg denotes a branch of argument with values in [−π, +π].
Let U be a Jordan neighbourhood of 0 such that G is defined on U and both G and F are injective on U . Since F ′ (0) = 1 and G ′ (0) = 1, there is δ > 0 such that B(0, δ) ⊂ U and
(3) F (W ′ att ∩ B(0, δ)) ⊂ W att , F (W ′ rep ∩ B(0, δ)) ⊂ W rep , G(W ′ att ∩ B(0, δ)) ⊂ W att , G(W ′ rep ∩ B(0, δ)) ⊂ W rep . We may choose r > 0 such that (4) I −1 (Ω r att ) ⊂ W ′ att ∩ B(0, δ), I −1 (Ω r rep ) ⊂ W ′ rep ∩ B(0, δ). Now we may lift F : W ′ att ∩B(0, δ) → W att and F : W ′ rep ∩B(0, δ) → W rep via the change of coordinate I(z) = ζ to define injective holomorphic maps F att : Ω r att → C, andF rep : Ω r rep → C. Straightforward calculations show thatF is of the form F att (ζ) = ζ + 1 + O(1/|ζ| 1/q ),F rep (ζ) = ζ + 1 + O(1/|ζ| 1/q ),
as |ζ| → +∞. There is s > 0 such that,
|F att (ζ) − (ζ + 1)| ≤ 1/4, ∀ζ ∈ Ω s att , |F rep (ζ) − (ζ + 1)| ≤ 1/4, ∀ζ ∈ Ω s rep .
There are injective holomorphic maps
Φ att : Ω s att → C, Φ rep : Ω s rep → C, such that Φ att •F att = Φ att + 1, on Ω s att , Φ rep •F rep = Φ rep + 1, onF −1 rep (Ω s rep ). It is known that (5) |Φ att (ζ)/ζ − 1| → 0, as Re ζ → +∞, (6) |Φ rep (ζ)/ζ − 1| → 0, as Re ζ → −∞.
Let us define
P s att = I −1 (Ω s att ), P s rep = I −1 (Ω s rep )
. Then, the injective holomorphic maps
φ att = Φ att • I : P s att → C, φ rep = Φ rep • I : P s rep → C, satisfy (7) φ att • F = φ att + 1, on P s att , φ rep • F = φ rep + 1, on F −1 (P s rep )
. The map φ att is an attracting Fatou coordinate for F , and φ rep is a repelling Fatou coordinate for F . Let
µ = b q+1 /a q+1 . Lemma 2.3. There is t ≥ 0 such that (i) G(z) = φ −1 att • T µ • φ att (z), for all z ∈ P t att , (ii) G(z) = φ −1 rep • T µ • φ rep (z), for all z ∈ P t rep .
Proof. By Equations (3) and (4), we may lift G : W ′ att ∩ B(0, δ) → W att via the change of coordinate I(z) = ζ to define an injective holomorphic mapG att : Ω r att → C. We note thatG att is of the form
G att (ζ) = ζ + b q+1 a q+1 + O 1 |ζ| 1/q , as |ζ| → +∞. In particular, if |ζ| is large enough, |G att (ζ) − (ζ + µ)| ≤ 1. This implies that there is t > s such that G att (Ω t att ) ⊂ Ω s att . Let V = Φ att (Ω t att ). Note that sinceF att (Ω t att ) ⊂ Ω t att , V + 1 ⊂ V . By Equation (5), if Re ζ is large enough, |Φ att (ζ) − ζ| ≤ |ζ|/3. This implies that V /Z = C/Z.
Consider the injective holomorphic map
G att = Φ att •G att • Φ −1 att : V → C.
Since F commutes with G near 0,F att commutes withG att on the common domain of definition Ω t att . Therefore, for w ∈ V , we havê
G att • T 1 (w) = Φ att •G att • Φ −1 att • T 1 (w) = Φ att •G att •F att • Φ −1 att (w) = Φ att •F att •G att • Φ −1 att (w) = T 1 • Φ att •G att • Φ −1 att (w) = T 1 •Ĝ att (w)
. Since V /Z = C/Z, the above relation implies thatĜ att induces a well-defined injective holomorphic map from C/Z to C/Z. Thus,Ĝ att is a translation on V /Z, and hence,Ĝ att is a translation on V , say T τ . However, since Φ ′ att (ζ) → +1, as Re ζ → +∞, andG att (ζ) is asymptotically a translation by µ near +∞, we must have τ = µ. That is,Ĝ att = T µ .
For z ∈ P t att , we have
φ −1 att • T µ • φ att = I −1 • Φ −1 att • T µ • Φ att • I = I −1 • Φ −1 att •Ĝ att • Φ att • I = I −1 •G att • I = G.
Part (ii): As in the previous part, we may lift G : W ′ rep ∩ B(0, δ) → W rep to obtain an injective holomorphic mapG rep : Ω r rep → C of the formG rep = ζ + µ + o(1), as |ζ| → +∞. Then, one may repeat the argument in part (i) withF rep and Φ rep .
Let B denote the set of z ∈ C such that F •n (z) → 0, as n → +∞. Evidently, P s att is contained in B. Let B 1 denote the connected component of B which contains P s att . (That is, B 1 is the immediate basin of attraction of 0 in the direction of v att .) For every z ∈ B 1 , there is k ∈ N with F •k (z) ∈ P s
att . By the maximum principle, B 1 is a simply connected subset of C. We may employ the functional relation in Equation 7, to extend φ att : P s att → C to a holomorphic map φ att : B 1 → C, such that φ att • F = φ att + 1 over all of B 1 . Consider the trip Π = {w ∈ C | −t − |µ| − 1 < Re w < −t} ⊂ Ω t rep . By the estimate in (6), if w ∈ Π with Im w sufficiently large, Φ −1 rep (w) ∈ Ω s att , and hence φ rep (w) ∈ B 1 . On the other hand, for some w ∈ Π, φ rep (w) does not belong to B 1 . Otherwise, a neighbourhood of 0 lies in B 1 , which is not possible since 0 belongs to the Julia set of F .
Let Π ′ denote the connected component of the set {w ∈ Π | φ −1 rep (w) ∈ B 1 } which contains the top end of Π. We may consider the map Proof. Let c 1 denoted the unique critical point of F within B 1 . The map φ att has a simple critical point at c 1 . It follows from Equation (7) that any z ∈ B 1 which is mapped to c 1 under some iterate of F is a critical point of φ att . The closure of the set of such points is equal to the boundary of B 1 .
h = φ att • φ −1 rep : Π ′ → C.
On the other hand, by Equation (7), those critical points are mapped to φ att (c 1 ), φ att (c 1 ) − 1, φ att (c 1 ) − 2, . . . . Since φ −1 rep is conformal on Π ′ ⊂ Ω t rep , we conclude that the only critical values of h are at φ att (c 1 ), φ att (c 1 ) − 1, φ att (c 1 ) − 2, . . . . All those points project to the same value in C/Z.
Lemma 2.5. The map H commutes with ξ → e 2πiµ ξ near 0.
Proof. By Lemma 2.3 Proof of Theorem 1.2. By Lemma 2.3, G = φ −1 att • T µ • φ att on P t att , and by Lemma 2.6, µ is an integer. Thus, on P t att ,
, G = φ −1 att • T µ • φ att on P t att , and G = φ −1 rep • T µ • φ rep on P t rep . Thus, φ −1 att • T µ • φ att = φ −1 rep • T µ • φ rep , at any point in P t att ∩ P t rep where both sides of the equation are defined. Equivalently, T µ • φ att • φ −1 rep = φ att • φ −1 rep • T µ ,G = φ −1 att • T •µ 1 • φ att = (φ −1 att • T 1 • φ att ) • (φ −1 att • T 1 • φ att ) • · · · • (φ −1 att • T 1 • φ att ) = F •µ . As P t
att is a non-empty open set, we must have G = F •µ on a neighbourhood of 0. Looking back at definitions (1) and (2), we conclude that (Q •q p/q ) •µ = Q •j p/q • g, on a neighbourhood of 0, for some 0 ≤ j ≤ q − 1. Thus, g = Q •(qµ−j) p/q near 0.
Elliptic case
Let g(z) = ∞ k=1 g k z k ∈ Cent(Q α ). It is easy to see that |g 1 | = 1. Let us say that g is r-good, if |g k | ≤ r 1−k for all k ≥ 1. Note that if g is r-good, then it is defined and holomorphic on the disk |z| < r.
Lemma 3.1. For every p/q ∈ Q and every r > 0, Q •k p/q is r-good for only finitely many values of k ∈ Z.
Proof. As Q p/q has a parabolic fixed point at 0, the family of iterates {Q •k p/q } k≥0 and {Q •−k p/q } k≥0 have no uniformly convergent subsequence on any neighbourhood of 0.
We let K(p/q, r) = k ∈ Z ; Q •k p/q is r-good . By the above lemma, K(p/q, r) is a finite set.
Lemma 3.2. For every p/q ∈ Q and every r > 0, there exists δ(p/q, r) > 0 such that for every p ′ /q ′ ∈ Q with |p ′ /q ′ − p/q| ≤ δ(p/q, r) we have K(p ′ /q ′ , r) ⊆ K(p/q, r).
Proof. By the compactness of the set of r-good holomorphic maps, there is N (r) such that any r-good map has less than N (r) critical points is the disk |z| < r/2.
As L tends to +∞, the set of the critical points of Q •L p/q increases, and accumulates on 0. Let L ∈ N be such that Q •L p/q has at least N (r) critical points in the open disk |z| < r/2. If p ′ /q ′ is close enough to p/q, then Q •L p ′ /q ′ has at least N (R) critical points in the open disk |z| < r/2. For l ≥ L, Q •l p ′ /q ′ has at least all those critical points, so it is not r-good.
Let M ∈ N be such that Q •−M p/q , and hence Q •−m p/q for any m ≥ M , does not extend to the open disk |z| < r. Then, the same is true for p ′ /q ′ close to p/q.
Finally, if k / ∈ K(p/q, r) and −M ≤ k ≤ L, Q •k p ′ /q ′ may not be r-good if p ′ /q ′ is too close to p/q, because otherwise one could take limits to conclude that Q •k p/q is r-good.
Lemma 3.3. For every p/q ∈ Q, every r > 0, and every ǫ > 0, there exists κ(p/q, r, ǫ) > 0 which satisfies the following. For every α ∈ R\Q with |α−p/q| ≤ κ(p/q, r, ǫ), and every g(z) = e 2πiβ z +O(z 2 ) which commutes with Q α and is r-good, there exists k ∈ K(p/q, r) such that |β − kp/q| < ǫ mod Z.
Proof. If the result does not hold, we may take a sequence α n → p/q and r-good maps g n (z) = e 2πiβn z + O(z 2 ) which commute with Q αn . By the compactness of the set of r-good maps, we may choose a convergent subsequence of the g n converging to a limit g which is r-good and commutes with Q p/q . Then, g will not be of the form Q •k p/q for some k ∈ K(p/q, r). This contradicts Theorem 1.2 and Lemma 3.1.
Lemma 3.4. For every α ∈ R \ Q, if a holomorphic germ of the form g(z) = e 2πikα z + O(z 2 ), for some k ∈ Z, commutes with Q α , then g = Q •k α near 0.
Proof. By considering Q •−k α • g instead, we may assume that k = 0. Then, by an inductive argument, one may show that the coefficients of the Taylor series expansion of g, except the first term, must be 0. That is, g(z) = z.
proof of Theorem 1.1. Start with any rational number p 1 /q 1 . We inductively define a strictly increasing sequence of rational numbers p n /q n , for n ≥ 1, so that for all 1 ≤ l ≤ j < n we have
(8)
|p n /q n − p j /q j | < δ(p j /q j , 1/j), (9) |p n /q n − p j /q j | < κ(p j /q j , 1/l, 1/j),
(10) |p n /q n − p j /q j | < 1/q 2 j .
Let α = lim n→∞ p n /q n . Since the sequence p n /q n is strictly increasing, it follows from Equation (10) that q n → ∞, as n → ∞, and α ∈ R \ Q.
Taking limit as n → ∞ in Equation (9), we note that |α − p j /q j | ≤ κ(p j /q j , 1/l, 1/j), for every 1 ≤ l ≤ j.
Assume that g(z) = e 2πiβ z + O(z 2 ) is a germ of a holomorphic map which commutes with Q α . There is l ≥ 1 such that g is 1/l-good.
By Equation (8) and Lemma 3.2, we obtain K(p j /q j , 1/l) ⊆ K(p l /q l , 1/l), for 1 ≤ l ≤ j. By Lemma 3.3, for every j ≥ l, there exists k ∈ Z with k ∈ K(p j /q j , 1/l) ⊆ K(p l /q l , 1/l) such that |β − kp j /q j | < 1/j mod Z. Taking limits of the latter inequality, as j → ∞, we obtain β = kα, for some k in the same range. Combining with Lemma 3.4, we conclude that g = Q •k α near 0.
Circle maps
We shall employ techniques from complex dynamics to study the analytic symmetries of the maps S a,b . So we consider the complexified family of maps S a,b (z) = z + a + b sin(2πz), for z ∈ C, but real values of a and b. Using the projection z → e 2πiz from C to C * = C \ {0}, S a,b induces the holomorphic map f a,b (w) = e 2πia we πb(w−1/w) from C * to C * . Evidently, f a,b preserves the unit circle T = {w ∈⊂ C ; |z| = 1}, and for a ∈ R and b ∈ (0, 1/(2π)), f a,b is a diffeomorphism of T. Below we always assume that a ∈ R and b ∈ (0, 1/(2π)).
Theorem 4.1. Assume that f a,b has a parabolic cycle on T, for some a and b. Then, Cent(f a,b ) is trivial.
Let us fix an arbitrary f a,b which has a parabolic cycle on T, say {w i } n i=1 , of period n ≥ 1. By relabelling if necessary, we may assume that f a,b (w i ) = w i+1 , with the subscripts calculated modulo n. Consider the map F a,b = f •n a,b : C * → C * . Each w i is a parabolic fixed point of F a,b with multiplier +1. For 1 ≤ i ≤ n, let U i ⊂ C * denote the immediate basin of attraction of w i for the iterates of F a,b . That is, U i is the union of the connected components of the basin of attraction of w i which contain w i on their boundary. The following lemma is a special case of a more general result by Geyer [Gey01,thm 4.4]. Since F a,b is τ -symmetric, it follows that τ (U i ) = U i , for 1 ≤ i ≤ n. Moreover, since F a,b (w i ) = w i , every connected component of each U i is invariant under F a,b . By a classical result of Fatou, see [Mil06], every connected component of each U i contains at least one critical point of F a,b . On the other hand, the critical points of F a,b are the pre-images of the critical points of f a,b . Since f a,b (U i ) = U i+1 , it follows that there is j with 1 ≤ j ≤ n, such that U j contains the critical points c 1 and c 2 . Moreover, c 1 and c 2 are the only critical points of F a,b inside U j . Then, the critical values of f a,b belong to U j+1 , which is distinct from U j .
By the maximum principle, every connected component of each U i is a simply connected region. Since the critical values of f a,b belong to U j+1 , any other U i does not contain any critical values of f a,b . These imply that for 1 ≤ l ≤ n − 1 there is a conformal branch of f •−l a,b from U j to U j−l . Therefore, each U i contains exactly two critical points of F a,b .
Every connected component of each U i is invariant under F a,b and τ , and contains at least one critical point of F a,b . Therefore, the number of the critical points in U i is two times the number of the connected components of U i . Since U i contains exactly two critical points of F a,b , U i consists of a single connected component containing both critical points.
Since each U i has a single connected component, each w i has a single attraction vector and a single repulsion vectors. As T is invariant, the attraction and repulsion vectors are the tangent vectors to T at w i . Fix an arbitrary i and consider an arc of T cut off by w i and w i+1 which does not contain any other w l . This arc is invariant under F a,b , and the orbit of any point on this arc must converge to either w i or w i+1 . Otherwise, there will be another fixed point of F a,b on this arc which is distinct from w i and w i+1 , and is either attracting or parabolic. This is a contradiction since such a cycle requires its own critical points distinct from the grand orbit of c 1 and c 2 .
By relabelling the points w i , and U i accordingly, we may assume that U 1 contains the critical points c 1 and c 2 of f a,b .
Since there is only one attracting direction for F a,b at w 1 , it follows that the multiplicity of the parabolic fixed point at w 1 is equal to +2. As in the previous section, there are attracting and repelling Fatou coordinates φ att : P att → C, φ rep : P rep → C, satisfying the functional equations
φ att • F a,b = φ att + 1, φ rep • F a,b = φ rep + 1,
with φ att (P att ) = Ω s att and φ rep (P rep ) = Ω s rep for some s > 0, F •j a,b converges to w 1 uniformly on compact subsets of P att as j → +∞, and F •j a,b converges to w 1 uniformly on compact subsets of P rep as j → −∞. The attracting coordinate may be extended to a holomorphic map φ att : U 1 → C using the above functional equation.
The map
h = φ att • φ −1 rep
has a maximal domain of definition, which is φ −1 rep (U 1 ) + Z. This induces a holomorphic map H defined on a neighbourhood of 0, with H(0) = 0. Proof. Any pre-image of c 1 and c 2 under F a,b within U 1 is a critical point of φ att . The set of the accumulation points of those pre-images is equal to the boundary of U 1 (which is contained in the Julia set of F a,b ). By the functional equation for φ att , φ att maps those critical points into the set φ att (c 1 ) + Z or φ att (c 2 ) + Z. On the other hand, φ rep is conformal on Ω s rep . This implies that the only critical value of h are contained in (φ att (c 1 ) + Z) ∪ (φ att (c 2 ) + Z).
Since F a,b is τ -symmetric, both φ att and φ rep are τ -symmetric. That is, φ att • τ = φ att and φ rep • τ = φ rep . This is due the uniqueness of a Fatou-coordinate up to translation by a constant. Combining with the above paragraph, we conclude that φ att (c 1 ) = φ att (c 2 ), and hence the critical values of H have the same argument.
Proof of Theorem 4.1. The proof already starts at the beginning of this section. Fix an arbitrary f a,b with a parabolic cycle {w i } n i=1 of period n. Let us also fix an arbitrary g ∈ Cent(f a,b ). The commutation implies that g(w 1 ) is a periodic point of period n for f a,b . By Lemma 4.2, f a,b has a unique periodic cycle, which is {w i } n i=1 . Therefore, there is an integer k ≥ 1 such that f •k a,b •g(w 1 ) = w 1 . Let us define the analytic map G = f •k a,b • g : T → T. As F a,b commutes with G, F a,b (w 1 ) = w 1 , F ′ a,b (w 1 ) = 1 we may repeat Lemma2.1 to conclude that G ′ (w 1 ) = 1. On the other hand, since the multiplicity of F a,b at w 1 is equal to +2, we may repeat Lemma 2.2 to conclude that the multiplicity of G at w 1 is also equal to +2. That is, G is of the form
G(w) = G(w 1 ) + (w − w 1 ) + b 2 (w − w 1 ) 2 + . . . ,
near 0, with b 2 = 0. As in the previous section, we must have G = φ −1 att • T µ • φ att on P att and
G = φ −1 rep • T µ • φ rep on P rep , where µ = 2b 2 /F ′′ a,b (0)
. Repeating Lemma 2.5, we conclude that H must commute with the rotation ξ → e 2πiµ ξ near 0. Now, as in the proof of Lemma 2.6, we use Lemma 4.3 instead of Lemma 2.5, to say that if c is a critical point of H, then we must have arg H(c) = arg(e 2πiµ H(c)). This implies that Re µ ∈ Z. On the other hand, if Im µ = 0, since the domain of definition of H is invariant under ξ → e 2πiµ , we conclude that H is defined over all of C. But this is a contraction since H has infinitely many critical points in a bounded region of the plane. Therefore, µ ∈ Z, and hence G = F •µ a,b . This completes the proof of Theorem 4.1
Fix an arbitrary b ∈ (0, 1/(2π)). By a general theorem of Poincaré, f a,b has a period point on T if and only if its rotation number ρ(f a,b ) ∈ Q. Moreover, by classical results, the map a → ρ(f a,b ) is an increasing function of a ∈ (0, 1). It is locally strictly increasing at irrational values, that is, if ρ(f a,b ) ∈ R \ Q for some a, then for a ′ ∈ (0, 1) with a ′ > a, ρ(f a ′ ,b ) > ρ(f a,b ). However, at rational values, the map is constant on a closed interval. 2 Given r > 1, we say that an analytic homeomorphism g : T → T is r-good, if g is holomorphic on the annulus 1/r < |z| < r and maps that annulus to the annulus 1/2 < |z| < 2. Evidently, every analytic homeomorphism of T is r-good for some r > 1. Moreover, by Schwarz-Pick lemma, for every r > 1, the class of r-good analytic homeomorphisms of T forms a compact class of maps.
Let us consider the sets P = {(a, b) ∈ (0, 1) × (0, 1/(2π)) ; f a,b has a parabolic cycle on T}.
and for each b ∈ (0, 1/(2π)),
P b = {a ∈ (0, 1) ; (a, b) ∈ P }.
Lemma 4.4. For every (a, b) ∈ P , f •k a,b is r-good for only finitely many values of k. Proof. The proof of Lemma 3.1 may be repeated here to show this statement.
For (a, b) ∈ P , we define K ′ (a, b, r) = k ∈ Z ; f •k a,b is r-good . Lemma 4.5. For every (a, b) ∈ P and every r > 0, there exists δ ′ (a, b, r) > 0 such that for every a ′ ∈ P b with |a ′ − a| ≤ δ ′ (a, b, r) we have K ′ (a ′ , b, r) ⊆ K ′ (a, b, r).
Proof. This is the same as the proof of Lemma 3.2.
Lemma 4.6. For every (a, b) ∈ P , every r > 0, and every ǫ > 0, there exists κ ′ (a, b, r, ǫ) > 0 which satisfies the following. For every a ′ ∈ P b with |a ′ − a| ≤ κ ′ (a, b, r, ǫ) and ρ(f a ′ ,b ) ∈ R \ Q, and every r-good map g which commutes with f a ′ ,b , there exists k ∈ K ′ (a, b, r) such that |ρ(g) − kρ(f a,b )| < ǫ mod Z.
Proof. The proof is identical to the one for Lemma 3.3. Here one uses the continuity of the map x → ρ(f x,b ), for x ∈ R.
Lemma 4.7. Assume that ρ(f a,b ) ∈ R \ Q. If g : T → T is an analytic map which commutes with f a,b and ρ(g) = kρ(f ) for some k ∈ Z, then g = f •k on T.
Proof. By considering f •−k a,b • g instead, we may assume that ρ(g) = 0. By Poincaré's theorem, g has a fixed point, and then by the commutation of f a,b and g, any iterate of that fixed point by f a,b must be a fixed point of g. Since the orbit of any point in T by f a,b is dense on T, g has a dense set of fixed points. Thus, g is the identity map on T.
Proof of Theorem 1.3. The proof is similar to the one for Theorem 1.1, using Theorem 4.1 instead of Theorem 1.2.
Fix an arbitrary b ∈ (0, 1/(2π)), and start with an arbitrary a ∈ P b . We inductively define an strictly increasing sequence of parameters a n ∈ P b , for n ≥ 1, so that for all 1 ≤ l ≤ j < n we have (11) |a n − a j | < δ ′ (a j , b, 1/j), (12) |a n − a j | < κ ′ (a j , b, 1/l, 1/j), 2 The set of a and b where ρ(f a,b ) is a rational number has non-empty interior, and is known as Arnold tongues.
One may refer to [Arn61], [Fag99] for basic features of those loci, and the global dynamics of the complexified standard family.
(13) |ρ(f an,b ) − ρ(f aj,b )| < 1/q 2 j ,
where p j /q j = ρ(f aj ,b ) ∈ Q and (p j , q j ) = 1. Let a = lim n→∞ a n . Since the sequence a n is strictly increasing, the sequence p n /q n must be increasing with at most two consecutive terms identical. It follows from Equation (13) that q n → ∞, as n → ∞, and ρ(f a,b ) ∈ R \ Q.
Taking limit as n → ∞ in Equation (12), we note that |a − a j | ≤ κ ′ (a j , b, 1/l, 1/j), for every 1 ≤ l ≤ j.
Assume that g is a orientation preserving analytic homeomorphism of T which commutes with f a,b . There is l ≥ 1 such that g is 1/l-good.
By Equation (8) and Lemma 3.2, we obtain K ′ (a j , b, 1/l) ⊆ K ′ (a l , b, 1/l), for 1 ≤ l ≤ j. By Lemma 3.3, for every j ≥ l, there exists k ∈ Z with k ∈ K ′ (a j , b, 1/l) ⊆ K ′ (a l , b, 1/l) such that |ρ(g) − kp j /q j | < 1/j mod Z. Taking limits of the latter inequality, as j → ∞, we obtain ρ(g) = kρ(f a,b ), for some k in the same range. Combining with Lemma 3.4, we conclude that g = f •k a,b on T.
This is a horn map of F . By the functional equations for φ att and φ rep , we must have h(ζ+1) = h(ζ)+1, whenever both side of the equation are defined. Thus, h induces a holomorphic mapH : Dom H → C,on a punctured neighbourhood of 0 so that H • e 2πiζ = e 2πih(ζ) . By the estimates in (5) and (6), Im h(ζ) → +∞, as Im ζ → +∞. This implies that H has a removable singularity at 0. That is Dom H contains a neighbourhood of 0. 1 Lemma 2.4. The map H has infinitely many critical points, all mapped to the same value.1 The map H is only defined modulo pre-composition and post-composition by linear maps of the form w → λw. This is due to the freedom in the choice of φatt and φrep up to post-compositions with translations. However, we are not concerned with those choices here.
whenever both sides of the equation are defined. We note that T −1 µ (Π ′ ) ∩ Π ′ is a non-empty open set, where both sides of the above equation are defined. This implies that the horn map h commutes with T µ . Hence, H commutes with the map ξ → e 2πiµ ξ.Lemma 2.6. We have µ ∈ Z.Proof. First note that Dom H is invariant under multiplication by e 2πiµ . That is, on the set e 2πiµ · Dom H we may define H as ξ → e 2πiµ H(e −2πiµ ξ). This matches H on (e 2πiµ · Dom H) ∩ Dom H.Let c denote a critical point of H. Differentiating H(e 2πiµ ξ) = e 2πiµ H(ξ) at c, we note that e 2πiµ c is a critical point of H. However, H(e 2πiµ c) = e 2πiµ H(c) is a critical value of H. By Lemma 2.4, we must have H(c) = e 2πiµ H(c), which using H(c) = 0, we conclude that µ ∈ Z.
Lemma 4 . 2 .
42For every 1 ≤ i ≤ n, U i consists of a single connected component, which is invariant under τ , and contains precisely two distinct critical points of F a,b . Moreover, ∪ n i=1 U i = T. Proof. The critical points of f a,b are the solutions of the equation f ′ a,b (w) = e 2πia e πb(w−1/w) (1+πb(w + 1/w)) = 0. Evidently, if w is a solution of this equation, then w, 1/w and 1/w are all solutions of the equation. Thus, w = w, and hence, the distinct solution of the equation are of the form c 1 and c 2 = τ (c 1 ), for some c 1 ∈ (−1, 0).
Lemma 4 . 3 .
43The horn map H has infinitely many critical points, which are mapped to critical values v 1 and v 2 satisfying arg v 1 = arg v 2 .
Acknowledgement: A. Avila and D. Cheraghi gratefully acknowledge funding from EPSRC (UK); grant EP/M01746X/1 -rigidity and small divisors in holomorphic dynamics, while carrying out this research.
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| We use 89 Y NMR in YBa2Cu3O6+y in order to evaluate with high sensitivity the distribution of hole content p in the CuO2 planes. For y = 1 and y = 0.6, this hole doping distribution is found narrow with a full width at half maximum smaller than ∆p = 0.025. This rules out any large static phase separation between underdoped and optimally doped regions in contrast with the one observed by STM in Bi2212 and by NQR in LaSrCuO. This establishes that static electronic phase separation is not a generic feature of the cuprates. | 10.1103/physrevlett.89.157002 | [
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Absence of static phase separation in the high T
24 Sep 2002 (November 3, 2018)
C Cuprate Yba
LLB
CE-Saclay
CEA-CNRS
91191Gif sur YvetteFrance
LEMHE
UMR 8647
Université Paris-Sud
91405OrsayFrance
Cu
J Bobroff
Laboratoire de Physique des Solides
UMR 8502
Université Paris-Sud
91405OrsayFrance
H Alloul
Laboratoire de Physique des Solides
UMR 8502
Université Paris-Sud
91405OrsayFrance
S Ouazi
Laboratoire de Physique des Solides
UMR 8502
Université Paris-Sud
91405OrsayFrance
P Mendels
Laboratoire de Physique des Solides
UMR 8502
Université Paris-Sud
91405OrsayFrance
A Mahajan
Laboratoire de Physique des Solides
UMR 8502
Université Paris-Sud
91405OrsayFrance
N Blanchard
Laboratoire de Physique des Solides
UMR 8502
Université Paris-Sud
91405OrsayFrance
G Collin
LLB
CE-Saclay
CEA-CNRS
91191Gif sur YvetteFrance
V Guillen
LEMHE
UMR 8647
Université Paris-Sud
91405OrsayFrance
J.-F Marucco
LEMHE
UMR 8647
Université Paris-Sud
91405OrsayFrance
Absence of static phase separation in the high T
24 Sep 2002 (November 3, 2018)
We use 89 Y NMR in YBa2Cu3O6+y in order to evaluate with high sensitivity the distribution of hole content p in the CuO2 planes. For y = 1 and y = 0.6, this hole doping distribution is found narrow with a full width at half maximum smaller than ∆p = 0.025. This rules out any large static phase separation between underdoped and optimally doped regions in contrast with the one observed by STM in Bi2212 and by NQR in LaSrCuO. This establishes that static electronic phase separation is not a generic feature of the cuprates.
In the cuprates, the hole doping of the CuO 2 planes induces the unusual features of the phase diagram: high-T c superconductivity, strange metal behavior, or pseudogap. A large body of theoretical work argues that this hole doping could be intrinsically strongly inhomogeneous in the planes, forming segregated hole-rich and hole-poor regions on a nanoscale [1]. This phase separation has been proposed to be essential to explain the unusual properties of the cuprates, appearing for example as stripes as argued from neutron scattering experiments [2]. Such proposals have been recently highlighted by STM studies which reveal strong inhomogeneities of the superconducting properties at the surface of Bi2212 [3] [4] [5] [6] [7]. The Berkeley STM group have imaged a spatial distribution of the superconducting gap which they associated with a distribution of concentration of holes ranging from p = 0.1 to p = 0.2 holes/planar unit cell [5].
The archetype of the cuprate families is YBa 2 Cu 3 O 6+y (YBaCuO). If these electronic inhomogeneities do not exist in this YBaCuO family, then they result of specific disorder in other compounds. However, STM measurements are not as extensive in YBaCuO due to surface cleaving problems and oxygen loss in vacuum. The existing STM studies display an inhomogeneous or relatively homogeneous surface depending on the surface preparation procedure [8] [9]. These limitations do not occur if one uses other local probes such as nuclear magnetic (NMR) or quadrupolar resonance (NQR). The huge body of NMR/NQR studies done so far concentrated on the p−variation of the average values of specific quantities (i.e. the static NMR shift K or the relaxation time T 1 ). However, as will be emphasized hereafter, the NMR/NQR spectroscopy also allows to determine the distribution of these quantities in the bulk samples. NMR is then sensitive to the local distribution of electronic properties like STM, but not to the specificities of the surface. As an appealing example, Singer et al [10] evidenced recently a distribution of T 1 over the Cu NQR spectrum in bulk LaSrCuO, which can be attributed to a distribution of p as large as the one observed on the Bi2212 surface.
We present an NMR static study of the YBaCuO family using 89 Y NMR spectra. This allows us to estimate the shape of the hole distribution in the bulk of the compound. We study two doping compositions (underdoped O 6.6 and slightly overdoped O 7 ). At O 7 the oxygen chains reservoirs are full, thus well ordered. At O 6.6 , the various properties are weakly oxygen content dependent. So both compositions should produce a minimal disorder of doping in the planes. We show indeed that our results reveal a very narrow hole distribution. These results will be compared with other experiments in the various cuprate families.
We synthesized a large sample batch of single crystal grains of YBaCuO. It was oxidized at T = 350 • C, and then slowly cooled under oxygen atmosphere from 350 • C to 270 • C at a rate of 0.4K/hr. After this procedure, the sample has a nominal maximum oxygen content y = 1 with a reduced T c = 89.1 K with respect to optimal doping (T c = 92.5K). A part of this batch was reduced to y = 0.60 ± 0.02 (T c = 55.8K) under primary vacuum (P = 0.13 mbar) at T = 385 • C . The oxygen content was measured by thermogravimetry, i.e. measurement of weight loss during the deoxidisation process until equilibrium is reached in the above conditions. For both samples, the crystallites were then aligned in stycast epoxy in the field H ext (≃ 7.5 Tesla) of the NMR spectrometer [11]. This field corresponds to a reference NMR resonance 89 ν = 15634.67 kHz for a liquid YCl 3 solution. Spectra were obtained for H ext perpendicular to the c cristallographic axis using a standard π/2 − π pulse sequence and Fourier transform of half of the spin echo [12]. The repetition time of the pulse sequence was long enough at each temperature to recover the saturation signal for any value of p. For a given T, the maximum of T 1 (p) varies from 17 to 110 sec when decreasing T from 300K down to 120K [13]. 89 Y NMR probes sensitively any doping distribution in the CuO 2 planes because the Y nucleus is coupled to Cu of its adjacent CuO 2 planes through the oxygen orbitals [13]. This results in a shift of the NMR line related to the Cu spin susceptibility χ given by
89 K(T ) = 8A hf χ(T ) µ B + 89 δ (1)
Here, 89 K(T ) is strongly dependent on hole content because of the strong hole doping dependence of χ(T ). In contrast, the T −independent chemical shift 89 δ and hyperfine coupling A hf = −1.95 kOe/µ B between the 89 Y and one Cu do not change much with hole doping (µ B is the Bohr magneton) [13]. This relation is illustrated in Fig.1 which displays the shift of the peak value of the 89 Y NMR line. It has been taken for a series of samples prepared from a single batch for different y. The shift 89 K(T ) displays the well known behaviour for χ(T ). It is nearly constant above optimal doping. It displays a marked decrease at low T which is the signature of the pseudogap for the underdoped cases. The oxygen content y can be converted into the hole content per plane using the parabolic dependence of T c [14] given by Note that near optimal doping, the large p dependence of 89 K makes it a sensitive probe to any doping variation: for example, at T = 120 K, a change in p as small as 0.01 corresponds to a change in 89 K of about 20 ppm, easily detectable. In the underdoped regime, the sensitivity to the doping is smaller but still sizeable.
p = 0.16 ± (1 − T c (y)/T max
The NMR spectrum of a given sample consists of a histogram of the NMR shifts throughout the sample. Note that, as STM, the NMR local probe is sensitive to variations nearly on the atomic scale, as each Y nucleus is coupled only to its 8 nearest neighbour Cu sites. Then, from the correspondence shift-doping of fig.1, any local distribution of doping will lead to a similar distribution in the shift, hence to a broadening of the spectrum.
Let us now present the experimental 89 Y NMR spectra for y = 0.6 and 1 from which we will extract quantitatively the actual doping distribution. The spectra are plotted for T = 120 K and 300 K for the optimally and underdoped samples in fig.2. As can be observed immediately there is no significant overlap between the two spectra at T = 120 K . This already proves that the O 7 (p = 0.18) sample contains no appreciable amount of underdoped nanoscale regions (equivalent to p = 0.1) and vice versa. In fig. 2 we also display the hole doping scales deduced from the relation between p and 89 K obtained from fig.1. From these hole doping scales it can be seen roughly that the full width at half maximum FWHM of the doping distribution is necessarily smaller than ∆p ≃ 0.04 at optimal doping and 0.02 for the underdoped case. This clearly proves without any further analysis that the observed doping distribution in YBaCuO is much smaller than in Bi2212 and LaSrCu3O where ∆p ≃ 0.1.
For a more quantitative analysis, we need to consider besides the distribution of χ(T ) any distribution of the chemical shift δ and hyperfine coupling A hf which enter Eq.1. Such distributions will come from any local structural disorder. These distributions are fully responsible for the FWHM ∆K = 32 ± 5 ppm of the undoped compound for T > T N [15]. Indeed, for oxygen content y < 0.15, no hole distribution broadening is expected [16]. This results from the fact that xoxygens introduced in the Cu reservoir layer mainly convert the two adjacent 3d 10 Cu(1) into 3d 9 and do not yield any hole doping of the CuO 2 planes, so that p is strictly 0. It is confirmed by the fact that the spectrum has the narrowest width amoung all dopings and is not found significantly dependent on oxygen content up to y = 0.15. For higher y, if ∆δ and ∆A hf are the FWHM of the respective distributions assumed to be uncorrelated and gaussian for simplicity, this will lead to an additional broadening with FWHM ∆K(y, T ):
∆K 2 (y, T ) = ∆δ 2 (y) + ∆K 2 hf (y, T )(2)
where ∆K hf = 8µ −1 B ∆A hf χ(T ). We note that the y = 0 compound is the more ordered as it is tetragonal with no twin boundaries and empty chains, hence ∆δ and ∆A hf should increase at higher y. As these broadenings add to the doping induced broadening, taking their y = 0 estimates at higher dopings will lead to an overestimate of the doping distribution. As ∆K hf is T-dependent, we can evaluate ∆A hf and ∆δ separately. Indeed, at y = 0.6, χ(T ) doubles between 120 K and 300 K so that ∆K hf doubles as well, whereas the FWHM increases only by 11% as seen in fig.2. As this T-dependence might as well be due to the doping distribution, this leads to an upper bound ∆A hf /A hf 0.13. Such an upper bound would be explained in a naive hybridization computation by a random displacement of Cu by 0.04Å from its ideal position, a typical value from Rietveld measurements. We will consider the two extreme cases:
(i) ∆A hf /A hf = 0 corresponding to ∆δ = 32 ppm (ii) ∆A hf /A hf = 0.13 corresponding to ∆δ = 29 ppm. In order to extract the actual doping distribution, we start from a distribution of oxygen doping P (y). We convert it for different temperatures into a shift distribution through the phenomenological 89 K vs y variation deduced from Fig.1. We convolute this K distribution with the δ and A hf gaussian distributions with FWHM either (i) or (ii). This process is iterated until an optimal P (y) is found to fit the experimental spectra for all temperatures with no additional free parameter.
In the underdoped y = 0.6 compound, this fitting procedure is limited by the fact that the NMR shift is nearly insensitive to the doping distribution for y < 0.6 as seen in fig.1. However, the thermogravimetry procedure during deoxydation constrains the measure of the average oxygen level y = 0.60 ± 0.02. We thus assume P (y) to be symetric around y. The best fit is then obtained for a gaussian distribution P (y) = exp −(y − 0.6) 2 /σ 2 with σ = 0.05 for (i) and σ = 0.1 for (ii). This oxygen dis-tribution P (y) and the corresponding hole distribution P(p) are plotted in fig.3 together with the usual T c diagram. The corresponding simulations fit perfectly the experimental ones at all temperatures (examples of fits are given as dotted lines in fig.2). For YBaCuO 7 , the chains are completely full. Further oxidation is prohibited. Therefore the distribution P (y) is expected to be non symmetric [17]. To take this limitation into account, we model P (y) as a convolution of a gaussian by exp(− |y − 1| /λ ± ) where λ ± are allowed to differ for y > 1 and y < 1. The best fit is found for σ = 0.01, λ − = 0.06, and λ + = 0.025 (i) or 0.004 (ii). P (y) and the corresponding fit of the spectra are respectively plotted in fig.3 and fig.2. The high asymmetric shape towards y < 1 found for P (y) confirms that no large overdoping of the planes is produced locally, as expected. In order to illustrate the high accuracy of our method, we choose P (y) slightly broader than our best fit, plotted as the dotted line in the upper panel of fig.3. The corresponding simulation plotted as a dashed line in the upper panel of fig.2 clearly fails to fit the experimental spectrum.
In summary, our results demonstrate that the maximum possible distribution of doping is quite sharp with typical width ∆p 0.025 for optimal doping and ∆p 0.01 for the underdoped sample. The distribution is much smaller than in the LaSrCuO and Bi2212 cuprates. In LaSrCuO, at optimal doping, Singer et al. find ∆p ≃ 0.09 as figured by the arrow in fig.3 [10]. Another NMR study [18] using 17 O and 63 Cu NMR allowed to evidence a short length scale spatial modulation in underdoped LaSrCuO compatible with Ref. [10]. In Bi2212, the STM experiments reveal regions on the surface with the usual superconducting gap while others display a pseudogap [3] [4] [5] [6] [7]. This has been interpreted to be due either to some disorder effect [4] or to superconducting and insulatinglike underdoped regions [5] [6] [7]. The corresponding doping distribution of width ∆p = 0.085 extracted in Ref. [5] is plotted in fig.3. Similar widths are found for an optimally and an underdoped compound in Ref. [7]. Therefore, YBaCuO appears much more homogeneous than both LaSrCuO and Bi2212. In other cuprates, no key experiment was performed to probe locally the doping distribution, to our knowledge. However, in Hg1201 and Tl2201, typical 17 O NMR widths are similar to those in YBaCuO and much smaller than in the Bi and La compounds [19]. Following the analysis done above and the proportionality between 17 O and 89 Y shifts, we then expect the Hg and Tl family not to exhibit any strong doping distribution either.
In the LaSrCuO family, the large distribution seen by NQR occurs in the bulk of the material and might be associated with tilt of the oxygen octaedra, buckling of the planes, or stripes. In the Bi material, the STM results might be only a surface specificity. In YBaCuO, the doping distribution P (y) measured here is not only much narrower but might even have only a macroscopic origin. As we used powders of cristallites with sizes smaller than 30 µm, any oxygen gradient within each cristallite, or a correlation between y and the cristallite size could lead to the observed P (y). At O 7 , such oxygen gradients are explained by the fact that oxygen diffusion between chains becomes limited at low temperature. This naturally leads to the asymmetry of P (y) towards y < 1 seen in fig.3. This effect has been observed systematically in the various studies published by our group using 89 Y, or 17 O NMR spectra in many different YBCO 7 powders. At O 6.6 , the observed broad P (y) is very narrow when plotted versus p ( fig.3). Indeed a change in oxygen content does not strongly modify the actual hole doping of the planes near y = 0.6 hence leading to the plateau of T c . This is to be associated with the fact that extra oxygens at such composition occupy empty chains, and do not modify the hole content, in analogy with the situation at y = 0. Therefore, the actual distributions would be much narrower on a submicron size region similar to the one sampled by STM. For both dopings, the key factor to the good homogeneity encountered is then probably the presence of the chains. The existing chain disorder does not lead to a sizeable distribution of hole content in the planes. Furthermore, those chains are probably sufficiently far from the planes not to stabilize a charge segregation. This specificity makes YBa 2 Cu 3 O 6.6 or 7 some of the best prototypes of clean homogeneous cuprates.
In conclusion, the nanoscale static segregations observed so far cannot be considered as an intrinsic phenomenon common to all cuprates. Contrarily to the observations in Bi2212 and LaSrCuO, the maximum hole doping distribution ∆p found in YBCO is small enough to conclude that our YBCO samples do not consist in interleaved regions with qualitatively different physical properties (metal versus insulator, with or without pseudogap, etc). The only remaining possibility for charge segregation or stripe-like scenarii to apply inYBaCuO is then a dynamical process where the phase separation would have a lifetime smaller than our timescale of observation. For the present experiment, it corresponds to the inverse spectral width, typically 1 msec. Such dynamics might have been detected in inelastic neutron scattering experiments, but the existing results are still under debate [20] [21]. fig.1 : 89 Y NMR shift 89 K for H ext ⊥c for different oxygen contents y versus temperature. In the inset, T c for each sample is plotted versus hole doping with the same symbol as the one used for K. fig.2 : 89 Y NMR Fourier Transform spectra are plotted at different temperatures and dopings in YBa 2 Cu 3 O 6+y for H ext ⊥c (black solid lines) with arbitrary normalization. In each panel, a doping scale gives the relation between the shift 89 K and hole doping p. Dotted lines are simulated spectra (see text). At O 7 :T = 120 K, an additional dashed line is plotted, which corresponds to the simulation made using the dotted distribution of the upper panel of fig.3. fig.3 : The full black dots represent T c versus oxygen content (upper panel) and hole doping (lower panel) for YBa 2 Cu 3 O 6+y . Distributions of oxygen content P (y) and the corresponding distribution of hole content P(p) were used to fit the spectra for y = 1 and y = 0.6 samples (see text). The dark and light gray enveloppes are obtained with a minimal distribution of chemical shift and hyperfine coupling within assumptions (i) and (ii) respectively. The dashed distribution and the arrow shown in the lower panel represent the distributions measured in Bi2212 [5] and LaSrCuO [10].
*present adress : Dept of Physics, IIT Bombay 400076 India.
corresponding p and experimental T c are plotted in the inset of fig.1.
Fig. Legend :
Fig. Legend : fig.1 : 89 Y NMR shift 89 K for H ext ⊥c for different oxygen contents y versus temperature. In the inset, T c for each sample is plotted versus hole doping with the same symbol as the one used for K. fig.2 : 89 Y NMR Fourier Transform spectra are plotted at different temperatures and dopings in YBa 2 Cu 3 O 6+y for H ext ⊥c (black solid lines) with arbitrary normalization. In each panel, a doping scale gives the relation between the shift 89 K and hole doping p. Dotted lines are simulated spectra (see text). At O 7 :T = 120 K, an additional dashed line is plotted, which corresponds to the simulation made using the dotted distribution of the upper panel of fig.3.
figure 1 (Bobroff et al. 2001)
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| []
|
[
"Defending Against Adversarial Attacks in Transmission-and Distribution-level PMU Data",
"Defending Against Adversarial Attacks in Transmission-and Distribution-level PMU Data"
]
| [
"Jun Jiang ",
"Xuan Liu ",
"Scott Wallace ",
"Member, IEEEEduardo Cotilla-Sanchez ",
"Member, IEEERobert Bass ",
"Member, IEEEXinghui Zhao "
]
| []
| []
| Phasor measurement units (PMUs) provide highfidelity data that improve situation awareness of electric power grid operations. PMU datastreams inform wide-area state estimation, monitor area control error, and facilitate event detection in real time. As PMU data become more available and increasingly reliable, these devices are found in new roles within control systems, such as remedial action schemes and early warning detection systems. As with other cyber physical systems, maintaining data integrity and security pose a significant challenge for power system operators. In this paper, we present a comprehensive analysis of multiple machine learning techniques to detect malicious data injection within PMU data streams. The two datasets used in this study come from two PMU networks: an inter-university, research-grade distribution network spanning three institutions in the U.S. Pacific Northwest, and a utility transmission network from the Bonneville Power Administration. We implement the detection algorithms with TensorFlow, an opensource software library for machine learning, and the results demonstrate potential for distributing the training workload and achieving higher performance, while maintaining effectiveness in the detection of spoofed data. | null | [
"https://arxiv.org/pdf/2008.09153v1.pdf"
]
| 221,246,032 | 2008.09153 | bcd6bc681c2de332948dced6888a25b27bb21b70 |
Defending Against Adversarial Attacks in Transmission-and Distribution-level PMU Data
Jun Jiang
Xuan Liu
Scott Wallace
Member, IEEEEduardo Cotilla-Sanchez
Member, IEEERobert Bass
Member, IEEEXinghui Zhao
Defending Against Adversarial Attacks in Transmission-and Distribution-level PMU Data
1Index Terms-Smart gridcyber securitymachine learningdata analyticsphasor measurement unitPMUsupport vector machineSVMartificial neural networkANNTensorFlow
Phasor measurement units (PMUs) provide highfidelity data that improve situation awareness of electric power grid operations. PMU datastreams inform wide-area state estimation, monitor area control error, and facilitate event detection in real time. As PMU data become more available and increasingly reliable, these devices are found in new roles within control systems, such as remedial action schemes and early warning detection systems. As with other cyber physical systems, maintaining data integrity and security pose a significant challenge for power system operators. In this paper, we present a comprehensive analysis of multiple machine learning techniques to detect malicious data injection within PMU data streams. The two datasets used in this study come from two PMU networks: an inter-university, research-grade distribution network spanning three institutions in the U.S. Pacific Northwest, and a utility transmission network from the Bonneville Power Administration. We implement the detection algorithms with TensorFlow, an opensource software library for machine learning, and the results demonstrate potential for distributing the training workload and achieving higher performance, while maintaining effectiveness in the detection of spoofed data.
I. INTRODUCTION
Over the past decade, smart grid technology has become an emerging and fast-growing field within both research and industry. The fundamental concept of the smart grid is to enable and enhance wide-area monitoring, control and protection by leveraging advances in modern sensing, communication and information technologies. A core device of the smart grid is the phasor measurement unit (PMU), invented by Phadke and Thorp in the 1980s [1]- [3]. These devices provide near realtime measurements of Steinmetz's current and voltage phasors, which represent the real-time status of an electric grid. Measurements from widely-distributed geographical locations are synchronized with a precise clock using the global positioning system (GPS), providing each PMU data point with a precise time stamp aligned to a common time reference [4]. This time stamp allows PMU data from disparate locations to be J. Jiang synchronized, thereby providing a precise and comprehensive view of the entire grid.
Because of the enhanced monitoring capability enabled by PMUs, these devices have been widely adopted by electric utilities, balancing authorities, and transmission operators. From 2009 to 2014, PMU deployment in the U.S. increased from 200 PMUs [5] to approximately 1700 [6]. However, along with the value these devices bring to the power grid, they also introduce new challenges. The volume of data PMUs generate presents challenges for the traditional workflow of grid operations. Specifically, PMU sampling and recording rates range from 10-60 samples per second [7]. This is much higher than conventional monitoring technologies such as supervisory control and data acquisition (SCADA) [8], which only sample once every two to four seconds. As a result, the volume of data to be stored, retrieved, processed, and analyzed is significantly larger than that of conventional systems. For instance, the Bonneville Power Administration's PMU network generates about 1.5 terabytes of data per month. This number is increasing as new PMUs are added to the system.
Besides the big data challenges, data security and integrity are also critical concerns. Data integrity can be compromised due to various causes, such as data drops, clock drifts, or injection of deceptive data signals. If PMU data are used to inform power systems operations, these types of deterioration can affect control operations, ultimately leading to major problems such as cascading failures.
To address these challenges, research developed within the fields of big data analytics and machine learning can be applied to efficiently process and analyze PMU data, as well as detect any data disruption as it happens. In this paper, we present our work on evaluating two widely-used machine learning methods, Support Vector Machines (SVM) and Artificial Neural Networks (ANN), on detecting malicious data injections in PMU data streams. We use two datasets containing PMU data collected from real power systems. The first dataset, BPA Data, comes from PMUs within Bonneville Power Administration's 500 kV transmission network.The second dataset, OSU Data, was collected from the interuniversity PMU network we built with three universities in the region: Washington State University in Vancouver, Portland State University, and Oregon State University and contains data from the distribution level. Using these two datasets, we conducted a comprehensive evaluation of the effectiveness of SVM and ANN in detecting spoofed signals within PMU data streams.
The contributions of this paper are multifold. First, we have identified a set of features derived from historical PMU measurements, that can be used to reliably train both SVMs and ANNs to effectively identify spoofs without extensive parameter turning. Additionally, because our method identifies spoofs at the phasor data concentrator (using data from mutiple PMUs) it is immune to attacks that involve physical penetration of substations, individual pmus, or individual private network links. Second,to the best of our knowledge, this is the first comprehensive evaluation of multiple machine learning methods for spoof detection within PMU data streams. Third, we showed that the performance of training an ANN can be improved significantly by leveraging the techniques of distributed computing. This provides potential for future work in real-time detection of spoofed signals. And lastly, to the best of our knowledge, this paper is the first to apply machine methods to PMU data at both the transmission level and the distribution level. Our results show that SVM and ANN are effective for detecting spoofed data within datasets. We expect these results to demonstrate the value of using machine learning and data analytics techniques to solve engineering problems withing the electric power industry. The organization of the rest of the paper is as follows. In Section II, we present related work in the cyber security aspect of the power grid, as well as applications of machine learning methods in this field. Section III introduces the two datasets used in this study and discusses their characteristics. In Section IV, we describe the methodology of our work, specifically for data processing, feature selection, and evaluation metrics. Section V presents the evaluation results of both SVM and ANN, in terms of their performance in both detection, and training timespan. Finally, Section VI concludes the paper and presents potential future directions for this work.
II. RELATED WORK
Cyber-security has long been a major concern for critical infrastructure, defined as assets essential for the functioning of a complex society [9]. Examples of critical infrastructure include electric power systems, natural gas and oil pipelines, and water supply systems. These facilities are monitored and controlled using Supervisory Control and Data Acquisition (SCADA) systems. SCADA systems collect measurement data from widely-distributed remote terminal units and issue control actions to the device-level layers within the infrastructure.
Cyber-security incidents involving energy delivery systems and associated theoretical attack structures are well documented [10]- [12]. These attacks, sometimes intelligently designed and executed, are difficult to detect, especially when disguised via spoofing techniques. For instance, in 2010, Stuxnet [13]- [15], a computer worm designed to infiltrate industrial equipment, altered the setpoint speed of centrifuges drives in an Iranian nuclear facility near Natanz, thereby causing the centrifuges to malfunction. During attacks, Stuxnet used spoofed data to mask malicious activities so that operators were not aware of the setpoint changes. Another example of using false data to facilitate a cyber attack was the multifaceted 2015 disruption of generation facilities in Ukraine [16]. Operators were prevented from sending commands to remote SCADA devices because the attackers had corrupted those devices with malicious firmware. This prevented operators from bringing generation resources back online after the initial attack. A third example is a series of cyber-attacks reported by McAfee in 2011 called 'Night Dragon.' These attacks targeted global energy and oil firms, and exfiltrated critical data such as operational blueprints [17]. These attacks had been ongoing for more than two years before they were identified because the attackers used a set of tools to compromise the target computers and mask their identity.
Recently, with more information and networking technologies being integrated within smart grids, cyber security concerns in power systems now receive an increasing amount of attention. Because PMUs are critical data sources [18], [19], these devices and their communication channels could become targets of cyber-physical attacks. Most PMUs in service are limited to providing data that informs intermediate algorithms, such as state estimation or voltage-stability assessment. The threat of cyber attack becomes unsettling when the target PMUs provide data for direct control of a protection system. For example, the Bonneville Power Administration pioneered the activation of Remedial Action Schemes (RAS) based on PMU data [20]- [22]. A cyber attack on these PMUs would render the RAS inoperable.
As noted previously, PMU data have GPS time stamps. Jian et al. demonstrate this feature is a potential attack vulnerability [23]. Zhang et al. examine the consequences of an attack on the time stamps of data collected within a smart grid widearea network [24]. Shepard et al. surveyed and evaluated the vulnerabilities of PMUs to GPS spoofing attacks [25]. Ng and Gao developed strategies to increase robustness of PMU time estimation [26]. Introduction of PMUs at the distribution level (aka micro-PMUs) allowed recent studies to identify attacks by coupling voltage and current data with network traffic data analysis [27], [28].
Within in the power systems community, PMU data spoofing outside of GPS signal manipulation has received attention for at least the last ten years. For example, detection of PMU data manipulation attacks by monitoring the impedances of transmission lines [29]. It has been shown in [30] that phasor measurement analysis and state estimation methods can be combined to detect spoofed data. Real-time detection has also been addressed [31].
Compared with SCADA systems, PMUs provide sample data at a much higher rate, normally at 30, 60, or 120 samples per second, as opposed to traditional SCADA refresh rate of seconds or even minutes. As a result, the amount of data generated by PMUs is significantly larger than that of a SCADA-based system. Therefore, big data challenges, in particular for online applications [32] are inevitably introduced to these systems. To address these challenges, big data and machine learning techniques may be leveraged and applied to PMU data storage and processing. A variety of machine learning techniques have been applied to analyze PMU data for the purpose of recognizing patterns or signatures of events. Both classification [33] and clustering [34] are effective methods in analyzing PMU data streams for event detection. And, one-class learning has the potential to identify anomalies in PMU data [35]. Machine learning techniques have proven effective at detecting security attacks in cyberphysical systems, including electric power systems [36]- [38]. However, none of these approaches have been evaluated on PMU data from a functioning power system.
The majority of prior work examining spoof detection on PMU signals are similar to ours in that they are designed to run at a central datacenter with access to multiple sensor/PMU signals (e.g., [38]- [41]). The benefit of this approach is that intruders cannot evade detection by injecting their spoof upstream of the detector. This means that any physical attack against the substation, its PMUs or the network connecting the substation to the data center are defended. Further, although our approach cannot run at the edge of the network (e.g., at an individual PMU), it is does not need full observations from all PMUs. Instead, it is designed to run on a small subset of PMU signals which improves scalability as the number of PMUs increase and allows the algorithm to run before all PMU signals have been aggregated.
The work presented here is distinguished in the following ways. First, spoofs are identified by observing changes in PMU signal correlation between pairs of PMUs [40]. Magiera and Katulski used a similar approach for spoof detection, though focused on Global Navigation Satellite Systems signals instead of PMU data [42]. Second, to the best of our knowledge, no previous work has evaluated machine learning methods using real PMU data streams at both the transmission and distribution levels. Third, our method explicitly considers, and avoids, the two limitations from previous work on detecting PMU spoofs identified above. Finally, we demonstrate the potential for achieving more efficient training performance by leveraging a distributed computing framework. This is the critical initial step to apply these methods to real-time PMU data streams for online spoof detection, which has not been addressed in previous literature.
III. DATASETS
Within an electrical grid, PMUs may be deployed at both the transmission and distribution levels to enhance situational awareness. These devices, as well as their communications network, may be targets of a cyber attack. In order to develop an effective approach for detecting such attacks, it is essential to evaluate at both the transmission level and the distribution level. Therefore, we have collected and analyzed PMU data from both levels for this study; dataset BPA Data came from a transmission network, while dataset OSU Data came from a distribution network.
A. Transmission Level Dataset
BPA Data was provided by Bonneville Power Administration, one of the first transmission operators to implement a comprehensive adoption of synchrophasors within a wide-area monitoring system. This dataset contains data collected by ten PMUs within BPA's 500 kV PMU network. Note that there are more PMUs deployed in BPA's transmission network. We chose to use ten PMUs for the following reasons. First, based on historical cyber-security incidents, these attacks usually have one specific target, either a device or a network channel. Therefore, to mimic these attacks and evaluate the spoof detection approaches, using data from a small set of PMUs is sufficient. Second, being able to detect cyber attacks using only local information from nearby PMUs is critical. This enables efficient detection. The same approach can be deployed to cover the whole system using a divide-and-conquer approach. The ten PMUs selected for this study are electrically-close to each other, based on their electrical distance, which has been shown to be a useful representation of power system connectivity [43]. Third, selecting 10 PMUs keeps the dataset at the same scale as our distribution level dataset, for the purpose of comparison.
B. Distribution Level Dataset
OSU Data comes from seven PMUs from our researchgrade, inter-university PMU network. This dataset provides PMU data samples from the distribution level. Our research PMU network consists of seven PMUs, one each at the Washington State University-Vancouver (WSU-V) and Portland State University (PSU) campuses, with the remaining five placed at multiple locations across the Oregon State University (OSU) campus. The PMUs provide monitoring at the utilization level (120/208 V), with the exception of two PMUs at OSU, which monitor a 4 kV and a 20 kV distribution substation. All PMUs monitor three phase services.
In our distribution level PMU network, all the PMUs report data at 60 samples per second to a Phasor Data Concentrator (PDC) located at the OSU campus. The data management scheme on the PDC emulates real PDC setups. A local archive stores 60 days of data on the PDC. The files are archived to another server for permanent storage.
IV. METHODOLOGY
One lesson learned from major cyber-security incidents, including Stuxnet and Night Dragon, is that attackers often mask their malicious activities via spoofing. In other words, they inject spoofed signals into the system to disguise cyber attacks. These spoofed signals are designed in a way that the system cannot easily identify the data as falsified. However, given that the falsified data is not an accurate measurement of the system state, there are limits in power systems to how well a spoof can be disguised. We developed three strategies for generating spoofed signals. In this section, we present these strategies, our methods for feature extraction, two machine learning techniques, and our evaluation metrics.
A. Spoofing Strategies
To describe our spoofing strategies, we use the following notation. A signal measured at time step k is denoted s(k). If a spoof begins at time step t, then s(k) = s true (k) for k < t and s(k) = s spoof (k) for k >= t. That is, prior to t the measured signal is exactly the true signal, otherwise the signal is provided by some spoofing method.
Our spoofing strategies aim to prevent easy detection and are designed in consideration of the following constraints:
(1) spoofs should provide historically-reasonable values for the target signals; (2) the onset of a spoof should not create a discontinuity-that is, a signal, s, sampled immediately before the spoof, s(t − 1) = s true (t − 1), and at the start of the spoof, s(t) = s spoof (t), should show a change in value that is consistent with historical variation of that signal;
(3) spoofed data should only require knowledge of the local signals, and not, for example, knowledge of signals at other PMUs. Below, we describe the three strategies employed for this study for generating s spoof . These strategies all meet the criteria above and have been explored in prior work (e.g., [41], [44]): Repeated-Last-Value, Mirroring and Time-Dilation.
The Repeated-Last-Value (RLV) spoof represents the strategy consistent with the constraints outlined above. This spoof requires knowledge of the true signal values immediately prior to the spoof onset, and repeats these values for the duration of the spoof. That is, assuming a spoof begins at time t, the signal
s(t + i) = s spoof (t + i) = s true (t − 1) for all i > 0.
The main short-coming of the RLV spoof is that it provides a constant value signal. Although the signal can clearly be guaranteed to remain within historic range, and will produce no discontinuity at the spoof onset, the distribution of values during the spoofed period will clearly be abnormal.
The Mirroring spoof improves upon the shortcoming of the RLV. To stage this attack, the adversary records the target signal for a period of time u prior to the spoof, and then when the spoof begins at time t, sets
s(t + i) = s spoof (t + i) = s true (t − i) for i = 0 . . . u.
However, mirroring imposes an immediate sign change on the signal's first derivative (s(t) − s(t − 1) = −s(t + 1) + s(t)). Thus, mirroring produces an inflection point in the signal data when the spoof begins, violating our second constraint.
The Time-Dilation spoof plays recorded data back at a slower than normal rate. A time dilation of 2x records data for n time steps (cycles), but plays these back over 2n cycles. Like mirroring, time-dilation upholds the three constraints while providing a historically-reasonable distribution of signal values. Unlike mirroring, however, time-dilation also improves continuity of the signals' first derivative. Time-dilation preserves the sign of the signal's derivative, but impacts the derivative's magnitude proportionally to the amount of dilation. In addition, unlike RLV and Mirroring, which provide an immediate window for malicious action when the spoof begins, time-dilation requires that the attacker continues to monitor, record, and resample live signal data even as the spoof begins. As a result, the attacker's window for malicious action occurs sometime after the spoof has begun, instead of at the onset. Figure 1 illustrates the spoof; the true signal is plotted as a dashed line, the spoof as a solid line. The spoof begins at cycle 250, illustrated in the plot by the fact that the solid line begins to diverge from the dashed line at this point. From cycle 250 forward, s(t) = s spoof (t) = s true (250 + (t − 250)/2). Thus, to perform the spoof, the true signal must be recorded for some portion of time after the spoof begins. That is, in this illustration, the true signal has been recorded from cycles 250 through 1125 and that waveform is played back between cycles 250 and 2000.
It should be noted that for this spoof, and all other spoofs described above, our focus is on the spoof onset. That is, we are interested in detecting the spoof as early as possible. Our spoofs are generated on the last 30 seconds of a one minute sample and we make no attempt to force the spoof and the true signal to re-align once the spoof is complete. This experimental setup strongly favors the spoofer by assuming that any discontinuity created at the end of the spoof between the last cycle of the spoofed data and the first cycle of real signal post-spoof (e.g., in Figure 1 between the spoofed signal at cycle 2000 and whatever comes next) cannot be detected. We designed our spoof strategies, e.g., mirror, time dilation etc., using real PMU data to demonstrate the effectivenss of our detection algorithm. A spoof that is created using real high-resolution PMU data is harder to detect. If the attacker does not have the real PMU data, it is easier for the algorithm to detect the spoof.
B. Feature Extraction
Feature extraction is a critical step in applying machine learning techniques to solve a problem. When performing analysis of complex data, one of the major problems stems from the number of variables/features involved. Analysis with a large number of variables generally requires a large amount of memory and computation power. Also, it may cause a classification algorithm to overfit to training samples and generalize poorly to new samples. Feature extraction aims for constructing combinations of the features while still describing the data with sufficient accuracy. In our work, one of the main objectives is to develop a generic set of features from PMU data that can be used by multiple machine learning algorithms for detecting spoofed signals.
PMUs measure phasors of line voltages and line currents for all voltages (A, B, C) and currents (A, B, C, N). From these are derived a number of other parameters, including magnitude and phase angle for the positive, negative and zero sequence voltages and currents; frequency; and, rate of change of frequency (ROCOF ); among others [7]. In [40], we describe how intra-PMU parameters (i.e., correlation between different signals from the same PMU) are usually weakly correlated, yet the inter-PMU parameters (i.e., the same signal from different PMUs) are often highly correlated, especially when the PMUs are electrically close to each other, as would be expect from an electrical network featuring a range of electrical distances between nodes [45]. These observations indicate that the correlations between PMU signals have great potential to serve as features to construct the models in machine learning techniques.
To quantify the degree of correlation between PMU parameters, we calculate the Pearson Correlation Coefficient (PCC) between two data streams, X(x 1 , x 2 , ..., x n ) and Y (y 1 , y 2 , ..., y n ). Specifically, given PMUs numbered 1, 2 . . . , p we develop p 2 vectors of correlation values between a specific signal for every pair of PMUs i < j. This is repeated for eight signals: positive sequence voltage magnitude |V + |, negative sequence voltage magnitude |V − |, zero sequence voltage magnitude |V 0 |, positive sequence phase angle φ + , negative sequence phase angle φ − , zero sequence phase angle φ 0 , frequency f , and ROCOF .
After carefully examining the correlation data for these eight signals in a previous study, we made the following observations [40]. First, correlation vectors r(|V + |), r(φ + ) and r(f ) are good candidates for detecting spoofing attacks, as these consistently exhibit moderate to high correlation values over wide ranges of time. The r(φ + ) correlation values are exceptionally high, near 1.0 under normal circumstances. Second, ROCOF correlation between PMUs is very poor, likely due to the fact that it is the second derivative of the positive sequence phase angle, and hence more susceptible to noise. Third, correlations on other signals, including r(|V − |), r(|V 0 |), r(φ − ) and r(φ 0 ), do not exhibit consistent moderate correlation.
Based on our observations, we chose five correlation features to use in the machine learning techniques. These include the three strongly correlated features, r(|V + |), r(φ + ) and r(f ), and two moderately correlated features, r(φ − ) and r(φ 0 ). These features are used in different learning algorithms to evaluate their effectiveness. Note that these correlation values fluctuate with time since the correlation is performed using data windows of a fixed length. In this work, we chose a fixed window of 5-seconds (300 cycles).
C. Parallel Data Preprocessing
The correlation features derived from the raw PMU data stream must be computed prior to training and then on a continuing basis during deployment. As a result, the cost of extracting features from the raw data has a real impact on computational resources that need to be continually dedicated to a tool such as this.
Computing the signal correlations is a parallizeable task, though the number of correlations that need to be computed grows quadratically in relation to the number of nearby PMUs. Thus, the constant factors associated with the preprocessing are relevant for determining real-world scalability.
We compared the cost of a data preprocessing pipeline built using two common Python frameworks for parallel/distributed processing: Celery [46] and ipyparallel [47]. Our results suggest that while inter-process queueing with Celery can add a significant performance penalty to parallelizing the feature extraction, the same is not true for ipyparallel. With ipyparallel, we were able to scale feature extraction linearly with the number of correlation values and number of CPUs. Computing the correlations on a sliding 300 cycle window over 1 minute of data required 1.85 seconds of CPU time and scaled linearly up to 24 cores at which time computation became I/O limited on our system. This speed is well within realtime requirements: 10 PMUs each with 5 signals yields 225 correlations and requires 8 CPU cores to complete in real-time.
Note that these preprecessing experiments are based on continuously calculating correlations using a 300-cycle sliding window with a step of 1 cycle. In reality, the sliding window can be adjusted by using larger steps, representing a trade off between the granularity of the correlations and the computational performance. Also, our previous study shows that for the spoof detection purposes, it is not necessary to calculate correlations for all PMU pairs, because a small number of PMUs that are electrically close to the spoofed PMU will be enough for the algorithm to trigger the detection. Therefore, the scalability issues can be addressed for a larger PMU network.
D. Machine Learning Techniques
Using the correlation features we derived from the raw PMU data streams, we carried out a comprehensive evaluation of two widely-used machine learning techniques, Support Vector Machine (SVM) [48], and Artificial Neural Networks (ANN) [49].
An SVM is supervised learning model with associated learning algorithms that analyzes data for classification and regression analysis. Given a set of training examples, each marked as belonging to one or the other of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a nonprobabilistic binary classifier. Specifically, an SVM model is a representation of the examples as points in space, mapped so that the examples of the separate categories are divided by a clear margin that is as wide as possible. New examples are then mapped into that same space and predicted to belong to a category based on which side of the margin they fall. Here in our case, we use the two-class SVM to learn a relationship that differentiates spoofed PMU signals from the normal signals. We leverage the Python library sci-kit learn for an SVM implementation based on libsvm [50], [51].
Similar to the SVM, an ANN is another widely-used technique for supervised learning. It is a machine learning model inspired by the biological nervous system. This technique has been widely applied in the fields of computer vision, speech recognition, anomaly detection, etc. However, to the best of our knowledge, it has not been evaluated using power systems data in the context of spoof detection.
An ANN is based on a collection of connected units called artificial neurons. Each connection (synapse) between neurons can transmit a signal to another neuron. The receiving (postsynaptic) neuron can process the signal and then send it to the downstream neurons connected to it. Neurons have a state, generally represented by real numbers, typically between 0 and 1. Neurons and synapses also have a weight that varies as learning proceeds, which can increase or decrease the strength of the signal that it sends downstream. Further, they have an activation function that determines how the aggregate input signals trigger an output signal.
Typically, neurons are organized in layers. Layers are made up of a number of interconnected neurons which contain an activation function. Data features are presented to the network via the input layer, which communicates to one or more hidden layers where the actual processing is done via a system of weighted connections. The hidden layers then link to an output layer where the results of the learning are made available to the users. In our work, we built an ANN which has two hidden layers, each with 100 neurons.
E. Evaluation Metrics
To evaluate the effectiveness of the two machine learning techniques on detecting spoofed signals in PMU data, we carried out a number of experiments and took measurements on multiple metrics. Below, we describe the performance metrics we use in this study:
Accuracy (1-Precision). FDR ranges from 0% to 100%; an ideal classifier has 0% FDR.
Latency: measures how long it takes to identify that a spoof is occurring. Recall that all the input features are signal correlation values calculated on a 5-second (300 cycle) window. Assuming a spoof begins at time t, the correlation value at time t contains one spoofed value and 299 nonspoofed values. The correlation value at time t + 299 is built from a window of entirely spoofed data. Our latency calculation measures how many cycles must occur before the classifier begins to consistently identify the spoof (marked by a contiguous set of thirty positive classifications). Thus, a latency of 0 indicates that as soon as the correlation window contains one spoofed data element the classifier will subsequently detect ≥ 30 consecutive cycles of spoofs. PMU data can be unreliable, and this lack of reliability can affect the aforementioned metrics. For instance, a PMU may go offline, resulting in so-called "data drop," which appears as a stream of zero-value data. Other times, a PMU may produce the same measurements over a continuous period, often termed "data drift." When processing PMU data for analysis, data drop and drift periods need to be removed. Correlation can help. As demonstrated by Meier et al., both data drop and data drift are easily identified through rapid decorrelation between the faulting PMU and near-by PMUs [52].
F. Common Limitations
As noted in Section II, much of the previous work suffers from practical limitations in that: (1) it models spoof detection using dc power flow state estimators; and (2) assumes that the only telemetry signals involved in spoofing/detection are voltage magnitude and phase angle. Below we describe each of these limitations in more detail and describe how our methodology avoids these issues.
First, as noted by Liu and Lie, most research on state estimation attacks use dc power flow models for H, whereas utility state estimators use ac models [39]. In essence, this means that much of the earlier work on spoof detection makes assumptions that are invalid in practice. Linearized state estimators neglect reactive power flow, which has a significant impact on bus voltage magnitudes. Moreover, a state estimator using an ac H will produce large residues when subjected to attack vectors derived from a dc H. Our methods do not depend on a explicit system model H. Instead, an implicit model is captured by the learning algorithms in the course of analyzing historical time-series data. In our approach, correlation of physical phenomena at multiple measurement points would need to be known in order to construct an undetectable attack vector. This is a significant hurdle for attackers. That is, even if the attacker used an accurate and complete ac H to derive an attack vector, the correlation of state measurements would still collapse as soon as the false data are injected into the measurements.
Second, both dc-and ac-based state estimators only use positive sequence voltage magnitudes and phase angles; no consideration is given to other data, such as negative and zero sequence voltages, frequency, or ROCOF , which are also calculated by PMUs. In contrast, to defeat a classifier built using the approaches we describe, an attacker would need to compromise and correlate all of the PMU data streams, not just |V + | and φ + . This means that our detection approach, which works outside of the state-estimation process, should be less susceptible to spoofing than a system that works as part of the state-estimation.
V. PERFORMANCE EVALUATION
Experiments have been carried out using both the BPA and OSU datasets to evaluate the effectiveness of SVM and ANN in terms of detecting spoofed signals. In addition, we also investigated the potential of distributing the training task to increase the computational performance. In this section, we present the process of preparing training and testing datasets, as well as the experimental results. . The normalization transforms from the training features are saved so they can later be used to transform testing data prior to being classified. Note that for the parameter selection in SVM, we have performed a grid search as follows. We first split the 11 training minutes into two sets (8 and 3 minutes respectively) and performed a grid search over the C, γ parameter space by training on the former set and testing on the later. We observed high performance (F1 > .95) across a wide range of parameter settings for both datsets. Thus, in subsequent sections, our results are obtained using the same set of parameters, C = 1.0, γ = 0.2.
A. Training and Testing Data
B. Spoof Detection Performance
For each of the spoof procedures, we train a two-class SVM and an ANN using 11 minutes of data, and then test its detection performance using the other 3 minutes of data. For each experiment, we measure the performance metrics described in Section IV-E. The experimental results for the three spoof procedures, i.e., RLV, Mirroring, and Time Dilation, are shown in Table I, II, and III, respectively. For all cases, we achieved high overall accuracy (ranging from 97.01% to 99.97%) and specificity (ranging from 99.67% to 100.00%), indicating that both techniques are effective in correctly identifying normal signals across the transmission level and distribution level. As for precision and FDR, in general both methods perform better on the BPA data. This is likely because the transmission level dataset has less noise compared to the distribution level dataset.
For all cases, the sensitivity is relatively lower than other metrics, ranging from 85.00% to 99.76%. This metric measures the percentage of spoofed signals being correctly identified by the learning algorithms. This is attributed to the following two reasons. First, since there is only one PMU being spoofed, there are more normal examples than the spoofed ones (3.5-4 times in both datasets). Therefore the algorithms learn better in terms of identifying normal data. Second, the calculation of these metrics is based on individual examples/time points. Each cycle is an example. If a cycle is within the later 30 seconds of the spoofed minute, it is labeled as spoofed. However, the features representing this example are the correlations from a 300-cycle time window before this time point, which means that some of the spoofed examples have correlation features composed of mainly nonspoofed data. This fact may also affect the sensitivity.
The overall performance of both techniques are high on both datasets, indicating that the features we use are generic across transmission level and distribution level.
C. ANN Training
A key feature of neural networks is an iterative learning process in which examples are presented to the network one at a time, and the weights associated with the input values are adjusted each time. Typically, this process is repeated for multiple iterations. In our experiments, we trained the network over 100 such iterations. However, there is a potential that the neural network achieves good performance with even less training time. To this end, we have measured the performance metrics after each time the network is trained, and the results are shown in Figure 2. For the larger BPA dataset, our neural network achieves near optimal performance after being trained 600 times (equivalent to approximately 6 minutes of CPU time on 1-core), while for the smaller OSU dataset, this number is decreased to approximately 150 (equivalent to less than 1 minute of CPU time on 1-core). This indicates that for a system with smaller numbers of PMUs, the neural network may achieve optimal performance with a much smaller number of epochs. Regardless, because the classifier is expected to only require retraining when the system configuration changes significantly, training can be viewed as an offline process.
D. Spoof Detection Latency
For each of our experiments, we have also measured the spoof detection latency, i.e., number of cycles after the spoof begins but before the classifier correctly identifies a string of 30 consecutive cycles as spoofed. This metric represents the promptness of our spoof detection method, which is critical in terms of applying this methodology in a real operational environment.
Specifically, we calculated the minimum and maximum latency for detecting spoofs in one minute data, for each of the spoof procedures and datasets. We have also calculated the "min of the max", which is the minimum of the max latency among all sites in each minute (the minimum latency of a spoof being detected by all sites). The latency results for SVM and NN are shown in Table IV, and V, respectively.
The experimental results show that the spoof detection methods works better on BPA data than on the OSU data in terms of latency. This is expected as the distribution-level OSU data are noisier than the transmission-level BPA data. Among the spoof procedures, Time Dilation is the most difficult to detect, indicated by longer latencies in all cases. The Time Dilation spoof uses real data to generate spoof signals, which are very similar to the real signals that a PMU would observe.
VI. CONCLUSION
With information and communication technologies being integrated in modern power systems, big data and cyber security challenges become more pronounced. Historical cyber attack incidents indicate that spoofing is a common approach for disguising malicious activities. Spoofed data injected into a normal data stream make it difficult to identify the underlying attack. To address the challenges, we applied machine learning techniques to PMU data for the purpose of detecting spoofs. Specifically, we developed a generic set of correlation data based on features from PMU data. We then trained a SVM and an ANN. To perform a comprehensive evaluation, we used two datasets, one from BPA's transmission level 500 kV network, and the other from our inter-university, distribution level PMU network. Experimental results showed that both techniques perform well after being trained using the same set of features. In addition, the neural network approach demonstrates a good potential in achieving highly efficient training performance, via minimizing training iterations, or leveraging distributed resources.
Work continues in multiple directions. First, we will explore classification accuracy and sensitivity (true positive rate) across a variety of spoofing circumstances. Additionally, we will explore specificity (true negative rate) across a large, contiguous sample. Second, we will investigate the possibilities of adjusting the performance of the spoof detection algorithm based on anticipated threat levels in response to cyber-defense intelligence. We expect to be able to modify detectability at the cost of increased computational complexity or increased false positive rate in response to anticipated events. Third, we will use principal component analysis to improve learning by reducing redundant information from the PMU data.
Fig. 1. 2x dilation spoof
: is calculated as the number of correct predictions, i.e., true positives (normal examples identified as such) plus true negatives (spoofed examples identified as such), divided by the total number of examples. Accuracy ranges from 0% to 100% with an ideal classifier measuring 100%. Sensitivity: measures the ability to correctly detect spoofed signals, and is calculated as the number of true positives (spoofed examples identified as such) divided by the number of total positives (the total number of spoofed examples which is the sum of true positives and false negatives). Sensitivity ranges from 0% to 100% with an ideal classifier measuring 100%. Precision: measures how many of the positively classified were relevant and is calculated as the number of true positives (spoofed examples identified as such) divided by the number of detected spoofs (false positives plus true positives). Precision ranges from 0% to 100% with an ideal classifier measuring 100%. Specificity: measures the ability to correctly identify normal signals. It is calculated as the number of true negatives (normal examples identified as such) divided by the number of total negatives (the total number of normal examples which is the sum of true negatives and false positives). Specificity ranges from 0% to 100% with an ideal classifier measuring 100%. F1: measures performance as a single value when classes are not equally prevalent. It is the harmonic mean of Sensitivity and Precision. Scores range from 0.0 to 1.0; higher values are better. False Discovery Rate (FDR): measures the propensity to spuriously identify a spoof. This value is calculated as the number of false positives (normal examples identified as spoofs) divided by the number of detected spoofs (false positives plus true positives). False Discovery Rate is equivalent to
For both the BPA and OSU datasets, we prepare the training and testing examples as follows. First, we choose 14 independent minutes of normal data from the dataset. We then apply one of the spoof procedures (Repeated-Last-Value, Mirroring, and Time-Dilation) to the last 30 seconds of one selected PMU signal on each of 14 different minutes of data. Finally, we calculate the pair-wise correlation features, as described in Section IV-B. For each spoof procedure, this approach generates roughly 2 ·10 6 examples from the 14 minutes of data and the 45 PMU pairs (i.e., 10 PMUs) in the BPA dataset, and roughly 1 · 10 6 examples from the 14 minutes of data and the 21 PMU pairs (i.e., 7 PMUs) in the OSU dataset. Examples are "Spoofed" in the last half of each minute if i is the spoofed PMU, and are "Normal" otherwise. Given the 14 minutes of data, we use 11 minutes (roughly 1.6 · 10 6 examples for BPA data and 8 · 10 5 examples for OSU data) for training, and 3 minutes (roughly 4.5 · 10 5 examples for BPA data and roughly 2.1·10 5 examples for OSU data) for testing. During training, all correlations features are standardized (normalized to 0 mean and standard deviation of 1)
Fig. 2 .
2ANN Performance vs. Training Times
, X. Liu, S. Wallace, and X. Zhao are with the School of Engineering and Computer Science, Washington State University, Vancouver, WA, 98686 USA e-mail: {jun.jiang2,xuan.liu2,wallaces,[email protected]} E. Cotilla-Sanchez is with the School of Electrical Engineering & Computer Science, Oregon State University, Corvallis, OR, 97331 USA email: [email protected] R. Bass is with the Maseeh College of Engineering and Computer Science, Portland State University, Portland, OR, 97201 USA email: [email protected]
TABLE I SPOOF
IDETECTION PERFORMANCE (RLV)Metrics
SVM BPA
NN BPA
SVM OSU
NN OSU
Accuracy
99.90%
99.97%
99.05%
99.26%
Sensitivity
99.13%
99.76%
94.99%
96.16%
Precision
99.96%
100.00%
99.75%
99.74%
Specificity
99.99%
100.00%
99.95%
99.94%
F1
0.995
0.999
0.973
0.979
FDR
0.04%
0.00%
0.25%
0.26%
TABLE II
SPOOF DETECTION PERFORMANCE (MIRRORING)
Metrics
SVM BPA
NN BPA
SVM OSU
NN OSU
Accuracy
99.81%
99.93%
98.53%
98.92%
Sensitivity
98.36%
99.36%
92.24%
94.35%
Precision
99.94%
99.99%
99.65%
99.69%
Specificity
99.99%
100.00%
99.93%
99.94%
F1
0.991
0.997
0.958
0.969
FDR
0.06%
0.01%
0.35%
0.31%
TABLE III SPOOF
IIIDETECTION PERFORMANCE (TIME DILATION)Metrics
SVM BPA
NN BPA
SVM OSU
NN OSU
Accuracy
99.39%
99.73%
97.01%
97.18%
Sensitivity
96.77%
98.71%
85.00%
85.54%
Precision
97.57%
98.84%
98.30%
98.78%
Specificity
99.71%
99.86%
99.67%
99.77%
F1
0.972
0.988
0.912
0.917
FDR
2.43%
1.16%
1.70%
1.22%
TABLE IV SPOOF
IVDETECTION LATENCY (SVM)Metrics
Min
Max
Min of Max
OSU mirror
34
135
108
OSU RLV
35
294
64
OSU dilation
27
443
147
BPA mirror
0
80
44
BPA RLV
0
118
6
BPA dilation
0
174
67
TABLE V SPOOF
VDETECTION LATENCY (NN)Metrics
Min
Max
Min of Max
OSU mirror
10
155
115
OSU RLV
12
297
99
OSU dilation
31
518
157
BPA mirror
7
24
19
BPA RLV
2
43
24
BPA dilation
11
64
35
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| []
|
[
"Spatial birth-and-death processes in random environment",
"Spatial birth-and-death processes in random environment",
"Spatial birth-and-death processes in random environment",
"Spatial birth-and-death processes in random environment"
]
| [
"Roberto Fernández ",
"Pablo A Ferrari ",
"Gustavo R Guerberoff ",
"Roberto Fernández ",
"Pablo A Ferrari ",
"Gustavo R Guerberoff "
]
| []
| []
| We consider birth-and-death processes of objects (animals) defined in Z d having unit death rates and random birth rates. For animals with uniformly bounded diameter we establish conditions on the rate distribution under which the following holds for almost all realizations of the birth rates: (i) the process is ergodic with at worst power-law time mixing; (ii) the unique invariant measure has exponential decay of (spatial) correlations; (iii) there exists a perfect-simulation algorithm for the invariant measure. The results are obtained by first dominating the process by a backwards oriented percolation model, and then using a multiscale analysis due to Klein to establish conditions for the absence of percolation. | null | [
"https://arxiv.org/pdf/math/0410191v1.pdf"
]
| 2,819,569 | math/0410191 | 39f65ce6713ee3531974c60055c14effc7a14ad8 |
Spatial birth-and-death processes in random environment
7 Oct 2004 August 28, 2018
Roberto Fernández
Pablo A Ferrari
Gustavo R Guerberoff
Spatial birth-and-death processes in random environment
7 Oct 2004 August 28, 2018birth-and-death processesrandom environmentbackwards ori- ented percolationmultiscale analysisrandom point processesrandom loss net- works
We consider birth-and-death processes of objects (animals) defined in Z d having unit death rates and random birth rates. For animals with uniformly bounded diameter we establish conditions on the rate distribution under which the following holds for almost all realizations of the birth rates: (i) the process is ergodic with at worst power-law time mixing; (ii) the unique invariant measure has exponential decay of (spatial) correlations; (iii) there exists a perfect-simulation algorithm for the invariant measure. The results are obtained by first dominating the process by a backwards oriented percolation model, and then using a multiscale analysis due to Klein to establish conditions for the absence of percolation.
Introduction
A number of reasons explain the standing interest in the study of birth-and-death processes. First, they are important probabilistic constructions in their own right, with proven potential to generate and test new mathematical approaches. Second, they offer the correct framework to formalize and study several important statistical applications. Among the more recent ones, we mention point processes (Baddeley and van Lieshout, 1995;Strauss, 1995;Baddeley, Kendall and van Lieshout, 1996) and loss networks (Kelly, 1991, Ferrari and. Third, they have become useful tools for other mathematical endeavours. Examples of this are Kendall's (1998) perfect simulation scheme and its offsprings (see Møller, 2001 for a review) and the study of hard-core statistical mechanical systems presented in Garcia (1998, 2001).
While the non-random version of these processes has a well developed theory (see, for instance, Møller and Waagepetersen, 2004), little is found in the literature for processes with random rates. Yet, in many situations a process with spatially homogeneous randomness is a more realistic model than a spatially homogeneous deterministic one. In general, the behavior of processes with "strong" disorder is expected to be qualitatively different from that of the non-disordered counterpart. In this paper, however, we tackle the complementary issue. We determine sufficient conditions for the disorder to be "weak" in the sense that it leads to processes exhibiting ergodic properties not far from those of the non-random versions.
We set our processes on a lattice that for convenience is taken to be Z d . The objects being born are called animals. They are supported on finite subsets of Z d , which, for technical reasons, are assumed to be uniformly bounded in size. Each animal γ has an associated birth rate w J (γ) parametrized by a random variable J defined on a certain probability space. Each realization of J is a random environment. In addition, there is an incompatibility relation between animals which is also assumed to be finite-range. The birth-and-death process is defined as follows: for each fixed environment, each animal γ attempts to appear at rate w J (γ) but to succeed, it must past a test, in general of stochastic nature, involving incompatible animals present at the moment of the attempt. Once born, animals disappear at a unit rate. Our main result, Theorem 2.15, establishes two conditions on the disorder [hypotheses (2.16) and (2.18) below] under which the process is ergodic and space-time mixing for almost all random environments. As in other models evolving in the presence of frozen-disorder (Klein, 1995;Gielis and Maes, 1996), while space mixing remains exponential, time relaxation is only proven to be faster than any power, with a bound of the form exp[−m ln q (1 + t)], for some m, q > 0.
Two conditions are imposed on the disorder. On the one hand, in (2.16) the average local rate of attempted births is asked to have some finite logarithmic moment. This is a relatively mild condition which is independent of the incompatibility relation. Its role [see Lemma 5.92 and formulas (5.81)-(5.82)] is to ensure that there is a not-too-small probability that locally no animal is found to be alive throughout a sufficiently thin time slice, even in the absence of incompatibility restrictions. The second disorder condition (2.18) requires a weighted birth-rate of incompatible animals to have a sufficiently small mean. Weights are defined by a certain size function S(γ) which in principle can be chosen in any way leading to the validity of the condition. This is a situation very much in the spirit of cluster-expansion formalisms; see, for instance, Dobrushin (1996). Usually -but not optimally-the size function is chosen as the number of sites in the support of the animal. This second disorder condition guarantees that large fluctuations in rate values are sufficiently sparse to have a negligible effect at sufficiently large scales.
In a subsequent corollary (Corollary 2.28), we determine a convenient sufficient criterium for both disorder hypotheses. We show that, barring very exceptional settings, there exists an ε (which we do not try to optimize) such that if
E sup x θ∋x |H(θ)| w J (θ) ≤ ε (1.1)
the process is almost-surely ergodic and mixing. Here |H(γ)| is the cardinality of the halo, which is the region around γ that determines incompatibility with other animals [see (2.24)]. Our analysis is based on a construction of birth-and-death processes described in Fernández, Garcia (1998, 2001). In this approach, the process is constructed by resorting first to a free process obtained by turning off incompatibilities and allowing every attempted birth to succeed. This free process corresponds to a marked Poisson process, where the marks are the lifetime of the animal and a random variable to be used in the compatibility test. It is useful to visualize each realization of this free process as the collection of cylinders determined by the animals alive during a certain lifespan. These cylinders inherit the incompatibility relation of the animlas forming their sections. Of particular importance for the construction of the interacting process are those incompatible cylinders that are alive when a given cylinder is born. They are called ancestors of the latter. The interacting process can be constructed if each cylinder has only a finite number of generations of ancestors. In this case, one can descend down genealogical trees deciding, by means of succesive incompatibility tests, which cylinders of the free process remain for the interacting one. Formally, the relation "being ancestor of" defines a (backwards) oriented percolation model in the space of cylinder configurations. The construction scheme succeeds if this model does not exhibit infinite percolation.
A sufficient condition for this lack of percolation is, in turns, obtained by resorting to a dominant multitype branching process whose subcriticality allows the construction to work. The subcriticality condition is, precisely,
sup γ 1 S(γ) θ incomp. with γ S(θ) w(θ) < 1 . (1.2)
The left-hand side of this inequality is the mean number of branches of the dominating branching process, and acts as a driving parameter: Time and space rates of convergence and mixing rates can be explicitly obtained in terms of it.
This approach can not be directly applied for random birthrates because, except in trivial cases, condition (1.2) is violated with probability one. An additional argument is needed showing that these violations are so sparse that the process is basically driven by the behavior in the overwhelmingly present "good" (regular) regions. As often in the study of disordered systems, this additional argument takes the form of a multiscale analysis. In this paper we adapt a time-tested multiscale argument whose origin can be traced to von Dreifus' (1987) dissertation. The argument was later adapted by , and Campanino, Klein and Perez (1991) to the study of (d + 1)-dimensional systems with d-dimensional disorder, which is our setting here. In the present work we follow the more general version due to Klein (1994) which has also been the basis of Gielis and Maes' (1996) study of spin-flip dynamics. Our proofs are similar to those of these references, and the informed reader will recognize many points in common, such as conditions (2.17), (5.43) and (5.44) below. But there is a number of small but frequent adaptations needed to accomodate "blob" rather than edge percolation. For the sake of clarity and completeness we have prefered to do a self-contained exposition of the proof, rather than a catalogue of differences with respect to preceding references. The paper size is comparable in both cases. We have, nevertheless, kept most of the notation, and of the architecture of the proof, adopted by Klein (1994) and Gielis and Maes (1996) for the benefit of readers familiar with these papers.
The initial ingredient of the multiscale scheme is a sequence of boxes of increasing size. The linear sizes of the boxes are the scales. Sites are then classified as regular or singular at a given scale according to whether there is an appropriate decay of the integrated connectivity function to the boundary of a box around the site. For the procedure to succeed there must exist a choice of scales that noticeably improves the probability of a site to be regular at succesive scales. This requires the singular regions to be sparsely distributed within a sea of regular sites; the distribution becoming more and more diluted as the observation scale grows. In the present setting of space-time (backwards) percolation, boxes are space-time parallelepipeds with a cubic spatial section. The space scales grow as a power law, but the time scale needs to grow faster. This is a consequence of the d-dimensional nature of the disorder. While regular and singular regions alternate in spatial directions, they are frozen in the time direction. Their effects are limited only by the deaths and lack of births of the concerned animals, which are processes of a more correlated nature than the (spatial) disorder distribution. Hence, the connectivity function decays much more slowly in time than in space. The time scale must satisfy two complementary requirements. First, it must grow fast enough to ensure a faster-than-power decay in the percolation probability from bottom to top inside a box. But, second, the growth must be suficiently slow to guarantee that time-like connections be mostly established inside a box, rather than by creeping through a sequence of spacially consecutive boxes. Mathematically, the first requirement is used in the proof of Theorem 5.29, to arrive to formula (5.38), while the second one appears explicitly in the proofs of Sublemma 5.63 and Lemma 5.92. The compromise is achieved by chosen a function T (L) so the bottom-top and left-right percolation probabilities inside a box of size L × T (L) decrease at a comparable rate, namely exponentially with L. This corresponds to a stretched-exponential dependence [(see (5.114)-(5.116)].
In statistical mechanical terms, our results are, somehow, the low-temperature counterpart of those of Gielis and Maes (1996). While these authors study single-spin-flipping evolutions leading to high-temperature invariant measures, here we focus on gases of "deffects" characteristic of low-temperature invariant measures. The comparison is not fully valid, however, because we are able to study only families of uniformly bounded deffects.
Definitions, examples and results
Basic definitions
Animals We consider the lattice Z d , d ≥ 1, (or, in general, the vertices of a graph with uniformly bounded coordination number) endowed with some norm. For concreteness we adopt x = max{|x i | : i = 1, 2, . . . , d}, x ∈ Z d . An animal model in Z d is defined by a countable family G of objects, the animals, for which there exists a map
G −→ P(Z d ) γ → V (γ) (2.1)
such that (i) V (γ) is a finite set -whose elements are called the vertices or sites of γ-and (ii) there is a finite number of animals asociated to each fixed set of vertices: V −1 (V (γ)) is finite for every γ ∈ G. Animal configurations are elements ξ of N G . Geometry is introduced through the map V in the obvious manner: The diameter of an animal γ is the diameter of its set V (γ), the distance between two animals is the distance between the corresponding sets of vertices, etc. To abbreviate we shall often denote x ∈ γ, for x ∈ Z d , rather than x ∈ V (γ). For a region Λ ⊂ Z d (of finite or infinite cardinality), we denote G Λ the set formed by all animals γ with V (γ) ⊂ Λ (animals in Λ). The corresponding configuration will get a subscript Λ: ξ Λ ∈ N G Λ . The omission of a subscript Λ indicates Λ = Z d . Animals are also characterized by a size function. In general terms, a size function is any funcion S : G → [1, ∞) that can be used in the convergence condition (2.18) below. In the examples of next section, this function is just the number of sites in V (γ).
In this paper we suppose that there exists a number ℓ 1 such that
diam (γ) ≤ ℓ 1 for all γ ∈ G . (2.2)
Animal interactions We introduce an interaction function M( · | · ) : G × N G → [0, 1], such that M(γ|ξ) is the probability that an attempted birth of γ ∈ G actually occurs when the current configuration of animals is ξ. If this function takes only the values 0 or 1 we refer the interaction as deterministic. This happens, for instance, for conditions such as volume-or perimeter-exclusion. The function M determines the (binary) interaction matrix
Int(γ, θ) := 1 sup ξ M(γ|ξ) − M(γ|ξ + δ θ ) = 0 (2.3)
where 1{A} is the indicator function of the set A, δ θ is the configuration in N G having only θ present, and ξ + δ θ is the configuration obtained by adding the animal θ to the configuration ξ. We say that γ is incompatible with θ, and denote γ ∼ θ, iff Int(γ, θ) = 1; otherwise we say that γ is compatible with θ. A family of animals Γ ⊂ G is a compatible family [herd in Dobrushin's (1996) nomenclature] if its elements are pairwise compatible. We assume that there exists a finite ℓ 2 ∈ R such that
dist(γ, θ) > ℓ 2 =⇒ Int(γ, θ) = 0 . (2.4)
The random environment Each animal γ has an associated birth rate w J (γ) which depends on a variable J, the random environment, belonging to a certain probability space (J , P). We assume that the disorder measure P is such that
Γ compatible =⇒ {w J (γ) : γ ∈ Γ} independent. (2.5)
Birth-and-death processes in random environment For each fixed environment J and each Λ ⊂ Z d we consider the interacting birth-and-death processes formally defined by the generator:
A J Λ F (η) = γ∈G Λ M(γ|η) w J (γ) [F (η + δ γ ) − F (η)] + η(γ) [F (η − δ γ ) − F (η)] , (2.6)
where F is a real continuous function on N G Λ .
In words, (2.6) says that when the current configuration of animals is η, each animal γ appears at rate M(γ|η) w J (γ) and disappears at rate 1 (if it is present). This is an interacting birth-and-death process of animals with quenched disorder. The factor w J (γ) acts as the rate of an internal Poissonian clock marking possible birth instants. The actual birth takes place only if a further test, determined by M(γ| · ), is passed. From the definition (2.3) of the interaction matrix we see that if Int(γ, θ) takes the value 1 (resp. 0) the presence of θ may have (resp. does not have) an influence on the birth rate of γ.
Unit death-rates are no loss of generality. In general, our w J (γ) stands for the ratio "birth rate / death rate" of γ. In fact, a similar treatment is possible for interacting-birth interacting-death processes. These are processes with a generator as in (2.6) but with a factor M death (γ|η) w J death (γ) multiplying the last summand. Our theory can be easily adapted if the death rates w J death (γ) are bounded below by a strictly positive number, uniformly in γ and η.
In general, there may be forbidden cases for which the matrix M takes value 0. An extreme example of this is provided by deterministic interactions (M = 0 or 1), such as those defining fixed-routing loss networks and some statistical mechanical models (see examples in Section 2.2). In these cases, the configurations resulting from forbidden cases -that is those of the form ξ + δ θ with M(θ|ξ) = 0-are not acceptable as initial configurations for the interacting process (thus, they will not be generated by it). The remaining configurations will be referred to as acceptable configurations.
Further notation The support of a function F on N G is the set
Supp (F ) = x ∈ Z d : ∃ γ ∋ x , η ∈ N G such that F (η) = F (η + δ γ ) .
(2.7)
In this paper we reserve the symbol P for the probability measure on the random environment, while the combination Q J (perhaps with further embellishments) will denote probability with respect to processes defined by a fixed environment J. We shall use superindices for the time coordinate of space-time animal configurations: η t Λ ∈ N G Λ × R. For finite sets A, the symbol |A| will denote the cardinality of A. For animals, the notation |γ| will mean |V (γ)|. We shall use a capital letter to denote a space-time point. As a default, we shall use the corresponding lowercase letter to denote its space component, and a subscript to identify its time component, ex. X = (x, t X ) ∈ Z d × R.
Examples
Some of the processes fitting our framework are the following.
Point processes
In these models the animals are subsets of Z d , hence V (γ) = γ. A large variety of point processes has been introduced in the literature. Our framework applies to their spacediscretized version. The area-interaction point processes of Baddeley and van Lieshout (1995) involve identical animals defined by the translations of a fixed compact set G ⊂ Z d . An animal of the form γ = x + G is called a grain of germ x. Animal configurations are labelled by the corresponding germs through the identification G ⊃ {x + G : x ∈ A} ←→ A ⊂ Z d . With this identification, the interaction function takes the form
M(x|A) = F (x + G) ∩ (A ⊕ G) (2.8) for some F : N → [0, 1], where A ⊕ G = y∈A {y + G}.
The process is attractive if F is increasing and repulsive otherwise. The former case corresponds to the penetrable sphere model introduced by Widom and Rowlison (1970).
In addition, the model is specified by internal-clock rates w(x). In the non-random version they are usually independent of x. For the disordered version, the independence hypotheses (2.5) leads to site disorder:
J = {J x : x ∈ Z d } (2.9)
for independent random variables {J x }. The internal clock rates should be of the form w J (x) = w({J y : y ∈ x + G}) . (2.10)
As further examples of point processes within our framework we mention: (i) the Strauss process (Strauss, 1995), which does not involve grains and the interaction function depends on the number of pairs of points closer than a fixed threshold r, and (ii) the perimeter-interaction process where the interaction matrix depends on the overlapping of perimeters of grains. The latter is a particular instance of the generalization propossed by Baddeley, Kendall and van Lieshout (1996). Our results apply to the corresponding lattice versions with site disorder.
Fixed-routing loss networks
These models are defined on a graph G whose links define possible calls or connections. An example is Z d with the usual nearest-neighbor links, or with a more general family of links (x, y) ∈ Z d × Z d with vertices x and y not necessarily at distance one. An animal γ is a connected finite subgraph possibly subjected to some further restriction, depending on the model (eg. not having a loop, or forming a closed circuit). The interaction matrix is often deterministic, preventing calls that would cause a link to be used beyond a predetermined maximal capacity. More generally, the interaction matrix embodies some penalization scheme for the multiple use of links.
The non-random rates w(γ) usually decrease with the number of links in γ. A general type of disorder satisfying the independence condition (2.5) is site-link disorder:
J = {J x : x ∈ V (G)} {J x,y : (x, y) ∈ L(G)} (2.11)
with all the random variables J x , J x,y being independent. By V (G) and L(G) we denote, respectively, the set of vertices and of links of the graph G. Condition (2.5) is satisfied by rates of the form
w J (γ) = w({J x : x ∈ V (γ)} , {J x,y : (x, y) ∈ L(γ)}) . (2.12)
We emphasize that, due to our hypothesis (2.2), our disordered loss network must involve connections not exceeding a radius ℓ 1 .
Models motivated by statistical mechanics
These are models of random geometrical objects coming from the study of spin systems in statistical mechanics. The associated birth-and-death processes have the statistical mechanical measure as invariant measure, and can be seen both as a convenient tool to study such measure and as a feasible simulation algorithm for it. We mention two of these models. Due to requirement (2.2), the disordered version considered here correspond to a "chopped" variant of the corresponding models.
Random cluster model This model plays an essential role in the study of the Potts model. We refer the reader to Grimmett (1995) for its detailed study as well as for references to the original articles. The animals, called clusters, are all finite subgraphs of a given graph G (whose vertices are usually the sites of Z d , but whose links may include non-nearest-neighbor pairs). The interaction matrix forbides the appearance of two clusters having a common vertix. Furthermore, the process has internal-clock rates of the form
w(γ) = (x,y)∈L(γ) J x,y 1 − J x,y x∈V (γ) 1 J x (2.13)
for appropiate parameters J x,y and J x [often denoted p(x, y) and q(x)]. A disordered version, satisfying the independence condition (2.5), is obtained by turning these parameters into independent random variables [site-link disorder (2.11)].
Ising contour model This is the model resulting from mapping into Peierls contours the typical spin configurations of one of the low-temperature pure phases of the Ising model. Contours are hypersurfaces formed by (d − 1)-dimensional unit cubes centered at points of Z d and perpendicular to the edges of the dual latice Z d + ( 1 2 , · · · , 1 2 ). These cubes are called plaquettes. Two plaquettes are adjacent if they share a (d −2)-dimensional face. A contour is a finite family of plaquettes such that (i) the family can not be partitioned into two subfamilies with no adjacent plaquettes, and (ii) every (d − 2)-dimensional face is covered by an even number of plaquettes of the family. Geometrically, a contour corresponds to a connected closed (hyper)surface. The set of vertices V (γ) is the set of centers of the plaquettes forming γ.
The contour-model equilibrium measure is the invariant measure of a birth-and-death process with a deterministic interaction matrix which prevents the appearance of two contours with adjacent plaquettes. The clock rates take the form
w(γ) = e −β x∈V (γ) J(x) , (2.14)
where the parameter β is interpreted as inverse temperature and each J(x) is a coupling constant associated to the plaquette centered at x. In the disordered version, these constants are independent random variables [site disorder (2.10)].
Main result Theorem 2.15 Consider an animal model as defined above, in particular satisfying (2.2), (2.4) and (2.5). Assume the disorder satisfies
(i) ℵ := E ln a 1 + sup x γ∋x w J (γ) < ∞ (2.16)
for some
a > 2d 2 1 + 1 + 1 d + 1 2d . (2.17)
Then there exists an ε = ε(d, a, ℵ) > 0, with a monotonically decreasing dependence
ℵ → ε(d, a, ℵ), such that if there exists a function S : G → [1, ∞) with (ii) E sup γ∈G 1 S(γ) θ:θ ∼γ S(θ) w J (θ) ≤ ε , (2.18)
then there exist constants m > 0, q > 0 such that for P-almost all configurations J the following is true. There exist constants C J x ≥ 0, x ∈ Z d , such that:
1. Existence and uniqueness. For each Λ ⊂ Z d there exists a unique time-invariant process Q J Λ on N G Λ × R with generator (2.6). The process has a unique invariant measure µ J Λ . 2. Time convergence. For each Λ ⊂ Z d and each acceptable configuration ξ Λ ∈ N G Λ ,
there exists a unique process Q J ( · | ξ Λ ) on N G Λ × R + defined by the generator (2.6) and the initial configuration ξ Λ . Furthermore, the process converges at a superpolynomial rate to the measure µ J Λ on N G Λ . Explicitly, the following inequality holds for any function f on N G Λ :
sup ξ Λ µ J Λ f − Q J ( f (η t Λ ) | ξ Λ ) ≤ f ∞ C J (Supp (f )) exp −m ln q (1 + t) (2.19) with C J (Supp (f )) = x∈Supp (f ) C J x . The supremum is taken over all acceptable configurations in N G Λ . 3. Space convergence. As Λ → Z d , µ J
Λ converges weakly to µ J := µ J Z d at an exponential rate. More precisely, for any function f depending only on animals contained in a finite set Λ:
µ J f − µ J Λ f ≤ f ∞ x∈Supp (f ) C J x exp −m dist(x, Λ c ) . (2.20)
4. Exponential mixing. For any functions f and g depending on animals contained in an arbitrary set Λ ⊂ Z d :
µ J Λ (f g) − µ J Λ f µ J Λ g ≤ f ∞ g ∞ x∈Supp (f ), y∈Supp (g) C J x C J y exp −m x − y /2 . (2.21)
5. Perfect simulation. The invariant measure µ J (or any of the µ J Λ ) can be perfectly simulated using the ancestors algorithm introduced by Fernández, Ferrari and Garcia (2002).
These properties are similar to those of the deterministic process obtained by a fixed translation-invariant asignment of J. The major difference is the possible slowing-down of the time convergence due to the subexponential time dependence in the right-hand side of (2.19).
We remark that while the size function S involved in hypothesis (2.18) is completely arbitrary, it is clear that it conveys some idea of the "mass" or "might" (Dobrushin's, 1996, terminology) of animals, in the sense that the larger S(γ) the larger the set of animals incompatible with γ. In fact, the natural choice for the examples presented in Section 2.2 is the size of the relevant set of vertices of each animal (area of the grain, length of the call, perimeter of the contour, etc.). But, of course, "natural" is not a synonym of "optimal".
In most cases of interest, condition (2.18) is morally more limitant than (2.16). To formalize this fact, let us denote
Υ J = sup x γ∋x w J (γ) (2.22) and Ψ J = sup γ∈G 1 S(γ) θ:θ ∼γ S(θ) w J (θ) ,(2.23)
and let us introduce the halo of an animal, namely the region of Z d around it beyond which no incompatible is possible:
H(γ) := W ⊂ Z d : V (θ) ⊂ W =⇒ θ ∼ γ c . (2.24)
This halo is a finite set because of assumption (2.4).
In the examples of Section 2.2 H(γ) = V (γ). In addition we denote
u 1 = inf γ∈G:γ =∅ |H(γ)| S(γ) , u 2 = sup γ∈G |H(γ)| S(γ) (2.25) and define Ξ J = sup x θ∋x |H(θ)| w J (θ) . (2.26)
In terms of these quantities we have the bounds
Υ J ≤ Ξ J and Ψ J ≤ u 2 u 1 Ξ J (2.27)
which lead to the following corollary of our main result.
Corollary 2.28 Consider an animal model as defined in Section 2.1 such that
u 1 > 0 and u 2 < ∞ . (2.29)
Then, there exists ε > 0 such that if E(Ξ J ) ≤ ε there exist constants m, q > 0 such that properties 1 to 5 of Theorem 2.15 hold for P-almost all disorder configurations J.
The conditions (2.29) are trivially satisfied by all the examples in Section 2.2, for which, in fact, u 1 = u 2 = 1. In our bounded-diameter setting, they are also satisfied if the function size is chosen translation invariant.
Proof. We first observe that for any a ≥ 1 there exists C a such that ln a (1 + x) ≤ C a x for all x ≥ 0. Therefore,
E ln a (1 + Υ J ) ≤ C a E (Υ J ) . (2.30)
Pick an a as in (2.17) and take
ε = min 1 C a , u 1 u 2 ε(d, a, 1) , (2.31)
where ε(d, a, ℵ) is the function given in Theorem 2.15. From (2.30) and the leftmost inequality in (2.27) we obtain
ℵ = E ln a (1 + Υ J ) ≤ C a ε ≤ 1 . (2.32)
From the rightmost inequality in (2.27) and the monotonicity of the function ℵ → ε(d, a, ℵ) we obtain
E[Ψ J ] ≤ u 2 u 1 ε ≤ ε(d, a, 1) ≤ ε(d, a, ℵ
The basic construction and the key lemma
The proof of Theorem (2.15) relies on two ingredients: a graphical construction of birthand-death processes propossed by Garcia (1998, 2001), and a multiscale argument following Klein (1994). In this section we review the graphical construction and state the key result obtained from Klein's argument.
Graphical construction
The construction is performed for each fixed environment J.
The free process We associate to each animal γ an independent marked Poisson process N γ with rate w J (γ). We call T k (γ), γ ∈ G, the ordered time-events of N γ with the convention that T 0 (γ) < 0 < T 1 (γ). For each occurrence time T i (γ) of the process N γ we choose independent marks S i (γ) exponentially distributed with mean 1 and
Z i (γ) uniformly distributed in [0, 1].
The free animal process is the process in which at each Poisson time-event T i (γ) (a copy of) the animal γ appears and lasts S i (γ) time units. It is convenient to identify each marked point (γ,
T k (γ), S k (γ), Z k (γ)) with (γ × [T k (γ), T k (γ) + S k (γ)] , Z k (γ)), the cylinder with basis γ, birth time T k (γ) lifetime S k (γ) and mark Z k (γ)
. This identification turns the free process into a measure on the space of cylinders. Denoting C = (γ, t, s, z), we use the notation
Basis (C) = γ, Birth (C) = t, Life (C) = [t, t + s], Mark (C) = z
The marks will be used later on to define the (interacting) birth-and-death process with generator (2.6). Let us denote Q J the probability measure on Z d × R corresponding to the free process (it always exist, being a countable product of marked Poisson processes).
Backwards oriented percolation We first extend our definition of incompatibility to cylinders in the natural way: Two cylinders C and C ′ are incompatible if they have incompatible bases and they are simultaneously alive at some instant of time. That is,
C ∼ C ′ if and only if Basis (C) ∼ Basis (C ′ ) and Life (C) ∩ Life (C ′ ) = ∅; otherwise C ∼ C ′ .
Let us now fix a family of cylinders C (for instance, obtained as a realization of the free process of the previous paragraph). We define the ancestors of a cylinder C ∈ C as the set
A C 1 = C ′ ∈ C : C ′ ∼ C and Birth (C ′ ) ≤ Birth (C) . (3.1)
Recursively for n ≥ 1, the nth generation of ancestors of C (in C) is
A C n = C ′′ : C ′′ ∈ A C ′ 1 for some C ′ ∈ A C n−1 . (3.2)
[In fact, A C n = A C n (C), but we shall omit this dependence except if it is crucially needed.] Likewise, for an arbitrary space-time point (x, t) ∈ Z d × R we define its set of ancestors as the set of cylinders that contain it
A (x,t) 1 = C ∈ C : Basis (C) ∋ x, Life (C) ∋ t ,(3.3)
and, recursively,
A (x,t) n = C ′′ : C ′′ ∈ A C ′ 1 for some C ′ ∈ A (x,t) n−1 . (3.4) Definition 3.5 We say that there exists backwards oriented percolation in C if there exists a space-time point (x, t) such that A (x,t) n
= ∅ for all n, that is, there exists a point with infinitely many generations of ancestors.
The clan of a space-time point (x, t), resp. of a cylinder C, is the union of the corresponding ancestors:
A (x,t) = n≥1 A (x,t) n , A C = n≥1 A C n . (3.6)
Other quantities that will be used later are the time-length, space-diameter and spacesize of the clan of a point (x, t):
TL (A (x,t) ) = t − inf s : Life (C) ∋ s, for some C ∈ A (x,t) (3.7) SD (A (x,t) ) = diam C∈A (x,t) Basis (C) (3.8) SS (A (x,t) ) = C∈A (x,t) Basis (C) . (3.9)
The interacting birth-and-death process If
Q J ({no backwards percolation}) = 1 ,(3.10)
the stationary process with generator (2.6) can be constructed by "cleaning" the free process defined above. For completeness we present a summary of this construction here. The reader is referred to Fernández, Ferrari and Garcia (2001) for details. The idea is to start from first ancestors ("Eves") and classify cylinders into kept or erased according to the test determined by the interaction function. Cylinders that are born in presence of a kept ancestor and that fail the test are erased, all the others are kept.
Explicitly, let C be a cylinder configuration in the set
C = {C without backwards percolation} . (3.11)
Since all clans in C are finite, each cylinder has a well defined, finite number of ancestors. Therefore the configuration can be decompossed in the form C = ∪ n≥0 C n , where C n :=
{C ∈ C : A C n = ∅, A C n+1 = ∅}.
The sets K and D of kept and erased cylinders are defined inductively as follows. Starting with K 0 = C 0 and D 0 = ∅ (cylinders without ancestors are kept, as they do not need to pass any test), we define, recursively,
K n = C ∈ C n \ ∪ n−1 i=0 (D i ∪ K i ) : Mark (C) ≤ M (C| ∪ n−1 i=0 K i ) (3.12) D n = C n \ [K n ∪ ∪ n−1 i=0 (D i ∪ K i )] where M (C|K ′ ) = M(Basis (C)|{Basis (C ′ ) : C ′ ∈ K ′ , Birth (C) ∈ Life (C ′ )}) .
We denote the set of kept cylinders as K(C) = ∪ n K n and the set of erased cylinders as D(C) = ∪ n D n . Clearly
(C1) C is the disjoint union of K(C) and D(C), and (C2) The event {C ∈ K(C)} is measurable with respect to the sigma field generated by A C , in fact K(A C ) = K(C) ∩ A C . In words, it is sufficient to know the (finite) clan of C to know whether C is kept or erased.
The stationary animal process Q J is defined by the sections of the kept cylinders:
η J,t (γ, C) = C∈K(C) 1 Basis (C) = γ, Life (C) ∋ t . (3.13)
If Q J (C) = 1, the process Q J is Markovian and has generator (2.6); that is,
d dt E J f (η t (C)) = E J A J Λ f (η t (C)) (3.14)
This fact is proven in Fernández, Ferrari and Garcia (2001) for the homogeneous case. This proof extends to the inhomogeneous case in an obvious manner. Let us denote µ J the distribution of any t-section η J,t , which, by construction, is independent of t. We shall determine the properties of µ J by studying the law of η J,0 , the stationary (interacting) birth-and-death process at time zero. As observed in (C2) above, the presence or absence of contours intersecting a region V ⊂ Z d at time t depends only on the clans of the cylinders alive at time t whose bases intersect V , that is, on
A V,t := C ′ ∈ A C : Basis (C) ∩ V = ∅, Life (C) ∋ t . (3.15)
In particular the function η J,t (γ, · ) defined by (3.13) is in fact a (deterministic) function only of A γ,t [= A V (γ),t ]. More precisely, if C and C ′ are two cylinder configurations such that A γ,t (C) = A γ,t (C ′ ), then η J,t (γ, C) = η J,t (γ, C ′ ). We code this fact as the identity (slightly abusive from the notational point of view):
η J,t (γ, C) = η J,t (γ, A γ,t ) . (3.16)
The key lemma
Let us call a sequence of cylinders C 1 , C 2 , . . . , C n an open path if C 2 ∼ C 1 , . . . , C n ∼ C n−1 and Birth (C i+1 ) < Birth (C i ) for all i. Given a cylinder configuration C and two spacetime points X = (x, t) and Y = (y, s), with s ≤ t, we say that X and Y are connected (in the configuration C) if there exists an open path C 1 , C 2 , . . . , C n such that x ∈ Basis (C 1 ), t ∈ Life (C 1 ), y ∈ Basis (C n ), s ∈ Life (C n ). [Equivalently, Y is in (the interior of a cylinder belonging to) the clan of ancestors of X.] The existence of such a connection defines an event denoted X → Y . For a given realization of the environment, the connectivity function is defined by
G J (X, Y ) = Q J {X → Y } . (3.17)
Many of the properties stated in the main theorem are a direct consequence of the following result there exist m > 0 and q 0 (a, d) > 1 such that for all q ∈ (1, q 0 ) there is a value ε(d, a, m, q, ℵ) > 0 so that the validity of (2.18) implies that for every
x ∈ Z d G J (x, t), (y, s) ≤ C J x exp −m max x − y , ln q (1 + |t − s|) (3.20)
for all y ∈ Z d and t, s ∈ R, where the constants C J x = C J x (ℓ 1 + ℓ 2 ) are finite for P-almost every environment J.
The fact that the inequality (3.19) is used implies that the function ℵ → ε( · · · , ℵ) can be chosen to be decreasing, as stated in Theorem 2.15.
We will prove this lemma by performing a multiscale analysis similar to the one used by Klein (1994) in his work on extintion of contact process in a random environment. That will be done in Section 5. We first discuss how Theorem 2.15 follows from the bound (3.20) for the connectivity function.
4 How the theorem follows from the key lemma 4
.1 Relation with percolation properties
The following theorem relates properties of the measure µ J with properties of the percolation model.
sup ξ Λ µ J Λ f − E J f (η t Λ ) ξ Λ ≤ 2 f ∞ x∈Supp (f ) Q J TL (A (x,0) ) > bt + e −(1−b)t E J SS (A (x,0) ) (4.2)
for any b ∈ (0, 1).
Space convergence. As
Λ → Z d , µ J Λ converges weakly to µ J . More precisely, if Supp (f ) ⊂ Λ, then µ J f − µ J Λ f ≤ 2 f ∞ x∈Supp (f ) Q J SD (A (x,0) ) ≥ dist ({x}, Λ c ) . (4.3)
4.
Mixing. For f and g with finite support, Proof. Items 1, 2 and 3 follow from displays (4.6), (4.7) and (4.9) of Theorem 4.1 of Fernández, Ferrari and Garcia (2001). The analogous of item 4 is stated in that theorem with the extra assumption that there exists a time h such that there is no space-time percolation in (0, h). We provide here a proof without this hypothesis.
µ J Λ (f g) − µ J Λ f µ J Λ g ≤ 4 f ∞ g ∞ × x∈Supp (f ), y∈Supp (g) Q J SD (A (x,0) ) ≥ x − y /2 + Q J SD (A (y,0) ) ≥ x − y /2 .
We consider functions f and g such that Supp (f ) = {x} and Supp (g) = {y}, the general case follows by telescoping. Let's fix a partition {Γ, Γ ′ } of G Λ . Below we choose Γ formed by animals "closer to x". Let A and B be Q J -independent realizations of {C ∈ C : Basis (C) ∈ Γ} and A ′ and B ′ independent realizations of {C ∈ C : Basis (C) ∈ Γ ′ }.
Then A ∪ A ′ , A ∪ B ′ , B ∪ A ′ , B ∪ B ′ have the same law as C Λ and A ∪ B ′ is independent of B ∪ A ′ . Let A (x,t) (C)
be the random variable defined in (3.6) and
X(A, A ′ ) = f η J,0 ( · , A ∪ A ′ ) = f η J,0 ( · , A (x,0) (A ∪ A ′ )) . Analogously Y (A, A ′ ) = g η J,0 ( · , A ∪ A ′ ) = g η J,0 ( · , A (x,0) (A ∪ A ′ )) .
With these definitions, we obtain
µ J Λ (f g) − µ J Λ f µ J Λ g = E X(A, A ′ ) Y (A, A ′ ) − X(A, B ′ ) Y (B, A ′ ) ,(4.5)
where E corresponds to a four-fold product of the measure Q J . This expression leads us to the bound
µ J Λ (f g) − µ J Λ f µ J Λ g ≤ 2 f ∞ g ∞ P(A c ) (4.6) with A = X(A, A ′ ) = X(A, B ′ ) and Y (A, A ′ ) = Y (B, A ′ ) . (4.7)
We now choose Γ as the set of animals intersecting {z ∈ Z d : z − x ≤ z − y } and Γ ′ as its complement. Then the event A is verified whenever the bases of the cylinders in both A (x,0) (A ∪ A ′ ) and A (x,0) (A ∪ B ′ ) are contained in Γ and those of A (y,0) (A ∪ A ′ ) and A (y,0) (B ∪ A ′ ) are contained in Γ ′ . The complement of the intersection of these four events yields
P(A c ) ≤ 2 Q J SD (A (x,0) ) ≥ x − y /2 + 2 Q J SD (A (y,0) ) ≥ x − y /2 .
For comparison purposes, let us present an alternative mixing bound.
Proposition 4.8 Assume that for a given J there is no backwards oriented percolation with Q J -probability one and consider functions f and g with finite support. Then,
µ J Λ (f g) − µ J Λ f µ J Λ g ≤ 2 f ∞ g ∞ × x∈Supp (f ), y∈Supp (g) Q J SD (A (x,0) ) + SD (A (y,0) ) ≥ x − y ,(4.
9)
where Q J is the free process obtained doubling the birth rates of Q J .
The bound (4.9) corresponds to standard high-temperature results in statistical mechanics (see, for instance, the main result in Bricmont and Kupiainen, 1996). Its proof relies on the very popular technique of "duplication of variables". In contrast, (4.4) is proven by "tetra-plication of variables". In our general setting, however, we are able to exploit better our first bound (4.4).
Proof. As above, it is enough to assume Supp (f ) = {x} and Supp (g) = {y}. Let C and C ′ be Q J -independent realizations of C Λ . We denote
X(C) = f η J,0 ( · , C)
and Y (C) = g η J,0 ( · , C) .
The duplication-of-variables identity is:
µ J Λ (f g) − µ J Λ f µ J Λ g = 1 2 E X(C) − X(C ′ ) Y (C) − Y (C ′ ) ,(4.10)
where E corresponds to the measure Q J × Q J . Let us now consider the event 11) and the transformation T : (C, C ′ ) → (T C, T C ′ ), that interchanges the (x, 0)-ancestors in C and C ′ :
B = Basis A (x,0) (C ∪ C ′ ) Basis A (y,0) (C ∪ C ′ ) = ∅ ,(4.T C = C \ A (x,0) (C) ∪ A (x,0) (C ′ ) T C ′ = C ′ \ A (x,0) (C ′ ) ∪ A (x,0) (C) . (4.12)
Conditioned to B being true, the distribution of (C, C ′ ) coincides with that of (T C, T C ′ ) (the processes inside and outside each realization of Basis A (x,0) (C ∪ C ′ ) are independent). Furthermore, the event B is T -invariant. Hence,
E F (C, C ′ ) 1{B} = E F (T C, T C ′ ) 1{B} (4.13)
for each local function F on G Λ × G Λ . But, in the presence of B, the function F involved in (4.10) is odd under this transformation. We conclude that
E X(C) − X(C ′ ) Y (C) − Y (C ′ ) 1{B} = 0 , (4.14)
which, by (4.10), implies
µ J Λ (f g) − µ J Λ f µ J Λ g ≤ 2 f ∞ g ∞ P(B c ) . (4.15)
The proof follows from the observation that
P(B c ) ≤ Q J SD (A (x,0) ) + SD (A (y,0) ) ≥ x − y .
Bounds for the size of the clan
The key inequality (3.20) leads to the following bounds for the probabilities of the timelength and space diameter and size of the clan of a space-time point. Inequality (4.17) is then a straightforward consequence of the fact that
Q J TL (A (x,0) ) > T ≤ y G J (x, 0) → (y, −T ) ,(4.21)
while (4.18) follows from
Q J SD (A (x,0) ) > L ≤ y: x−y ≥L ∞ 0 G J (x, 0) → (y, −t) dt . (4.22)
To obtain (4.19) we use (4.18) to bound the right-hand side of the inequality
Q J SS (A (x,0) ) > L ≤ Q J SD (A (x,0) ) > L 1/d (4.23)
and sum over L.
Proof of Theorem 2.15
The bound (3.20) provided by the key Lemma 3.18 implies, by Borel-Cantelli, the absence of backwards oriented percolation. Thus, we can apply Theorem 4.1 for almost-all realizations of the disorder.
Existence and uniqueness.
Under the hypothesis of no percolation in any region Λ, the process (3.13) defines a stationary process η t in Λ by considering those cylinders in C with basis contained in Λ. Uniqueness follows from the next item.
2. Time convergence. We use (4.17) and (4.19) to bound the right hand side of (4.2) by
2 f ∞ x∈Supp (f ) C J x exp −m ln q (1 + bt) + C J x e −(1−b)t ≤ 2 f ∞ x∈Supp (f ) C J x exp{−m ln q (1 + bt)},(4.24)
for some constant C J x < ∞.
Proof of Lemma 3.18
The lemma is proven by adapting Klein's (1994) multiscale analysis. The analysis involves several steps that will be studied separately.
General scheme and notation
The scheme is based on a sequence of linear sizes, called scales, defining boxes of increasing size. Sites are classified on regular or singular according to the decay of connectivity functions on a surrounding box. The definitions must be tunned up so that the probability for a site to be regular increase sufficiently fast with the scale. The presentation of this scheme is organized as follows:
(i) In Section 5.2 we recopilate a number of inequalities needed to prove next inductive step. In particular, the Hammersley-Lieb-Simon inequality is the key to pass from local to global decay of the connectivity function. The uniform bound on the size of the animals is needed to ensure its validity.
(ii) In Section 5.3 we define regularity and determine "good enough" probabilities for sites to be regular at each scale. Regularity at all scales with such probabilities guarantee, by Borel Cantelli, almost sure regularity.
(iii) Section 5.4 shows the crucial inductive step: A "good enough" probability of being regular at a given scale implies a "good enough" probability of being regular at the next scale. This is the heart of the argument.
(iv) The last step of the proof is the determination of conditions so the origin has a "good enough" probability of being regular at some initial scale. This is done in Section 5.5.
We introduce some notation. We fix ℓ 0 = ℓ 1 + ℓ 2 , the maximal size of the animals plus the radius of incompatibility -to avoid trivialities we assume ℓ 0 > d + 1. For L > 0 and
x ∈ Z d , we denote Λ[x; L] = y ∈ Z d : x − y ≤ L (5.1)
and define the ℓ 0 -boundary of Λ[x; L] as the set
∂ ℓ 0 Λ[x; L] = Λ[x; L + δ] \ Λ[x; L],(5.2)
where δ > 1 is chosen so the following is true: For any sequence of animals γ 1 , γ 2 , . . . , γ n with γ i ∼ γ i+1 , i = 1, 2, . . . , n − 1 connecting a point inside Λ[x; L] with a point outside Λ[x; L+δ], at least two animals in the sequence are contained in ∂ ℓ 0 Λ[x; L]. This condition is necessary to satisfy the Hammersley-Lieb-Simon (see next). A possible choice is δ = δ(ℓ 0 ) = 3(ℓ 0 −2) 2(d−1) . For X = (x, t) ∈ Z d × R, L > 0 and T > 0 we set
B L,T (X) = Λ[x; L] × [t − T, t] . (5.3)
The vertical, horizontal and complete boundaries of the box B L,T (X) are defined respectively as:
∂ V B L,T (X) = ∂ ℓ 0 Λ[x; L] × [t − T, t], (5.4) ∂ H B L,T (X) = Λ[x; L + δ] × {t − T }, (5.5) ∂B L,T (X) = ∂ V B L,T (X) ∪ ∂ H B L,T (X). (5.6)
Note that, as we are considering backwards oriented percolation, the face Λ[x; L + δ] × {t} is excluded from the boundary. For any integrable function H :
Z d × R → R we denote Z∈∂B L,T (X) H(Z) = z∈Λ[x;L+δ] H(z, t − T ) + z∈∂ ℓ 0 Λ[x;L] t t−T H(z, s)ds. (5.7)
We introduce the notion of connection within a box. Given a cylinder configuration C, its restriction to a box B = Λ × I, with Λ ⊂ Z d and I a real interval, is the family C B of cylinders obtained by "restricting" to I those cylinders of C with bases inside Λ. That is, each C = (γ, t, s, z) ∈ C with γ ∈ G Λ defines a cylinder C B = (γ, t I , s I , z) ∈ C B with t I = max(t, inf I) and s I = min(t + s, sup I) − t I . Given two space-time points, X = (x, t) and Y = (y, s), X, Y ∈ B with s ≤ t, the event X → B Y is formed by all cylinder configurations C such that the configuration C B exhibits an open path connecting X with Y . If this event is true, we say that X and Y are connected in B. The event defines the connectivity function in the region B:
G J B (X, Y ) = Q J {X → B Y } . (5.8)
As usual, we shall omit the subscript if B = Z d × R. We write
G J B L+δ,T (X) (X, ∂) = Z∈∂B L,T (X) G J B L+δ,T (X) (X, Z) . (5.9)
A toolbox: Inequalities
The first two inequalities we need have been basically proven in the literature. The minor adaptations needed for our setting do not justify a detailed exposition of their proofs. We content ourselves with providing appropriate references and indications. We consider the natural partial order in the space of cylinder configurations: C ≤ C ′ if C ′ contains all the cylinders in C. Events are said increasing, resp. decreasing, if their characteristic functions are nondecreasing, resp. nonincreasing with respect to this partial order.
Q J (A ∩ B) ≥ Q J (A) Q J (B) . (5.11)
In the context of continuous-time percolation, an inequality of this sort was first proven by Bezuidenhout and Grimmett (1991). They did so by showing that the process is the weak limit of discrete-time processes that satisfy the corresponding inequality. Their approach provides at the same time a proof of the van den Berg-Kesten inequality, but has the inconvenience of imposing an additional topological condition on events, namely that their boundary -in some suitable metric topology-have measure zero. While such requirement is satisfied for the events of interest to us, a different approach, based on the martingale convergence of expectations of the relevant events, yields the result as stated above. A proof along this line is provided in the book by Meester and Roy (1996) (see their Theorem 2.2) for the percolation of spheres of random radii on a continuous space (Poisson Boolean model). The proof is easily adaptable to our setting of percolating cylinders.
The second inequality refers to increasing events happening in a disjoint manner. For brevity, we state it only for the type of events needed in the sequel. Let us consider a box B = Λ × I, with Λ ⊂ Z d and I a closed finite interval in R, and space-time regions B 1 , . . . , B n ⊂ B. We denote {B 1 → B B 2 } • · · · • {B n−1 → B B n } the event of having n − 1 open paths, respectively connecting in B some point in B i to some point in B i+1 , 1 ≤ i ≤ n − 1, such that no two paths share the same cylinder.
Q J {B 1 → B B 2 } • · · · • {B n−1 → B B n } ≤ Q J B 1 → B B 2 · · · Q J B n−1 → B B n . (5.13)
This inequality is a consequence of the more general inequality proven by van den Berg (1996) for increasing events of marked Poisson processes. The only subtlety is that van den Berg's result requires the events to depend on Poisson clocks ringing within a bounded region of R d+1 . The connectivity events in (5.13) do not seem to fit in this framework because they may be determined also by cylinders born in an arbitrarily remote past. To obtain (5.13) we must, therefore, apply van den Berg's result to the marked process in B obtained by adding, independently, an invariant initial distribution of cylinders. This corresponds to a further, independent, spatial Poisson marked process on Λ × {inf I} with rates {w J (γ) : γ ∈ G Λ }. Less direct proofs are also possible either by adapting the Bezuidenhout and Grimmett (1991) approach mentioned above, or the proof of Theorem 2.3 of Meester and Roy (1996).
Our last inequality is obtained in Klein (1994) as a corollary of van den Berg's inequality. In our oriented setting, we can present a totally different argument.
Proposition 5.14 (Hammersley-Lieb-Simon inequality) Let L, T ∈ R + , δ as defined below (5.2)
and W ⊂ Z d × R. Then, for every X = (x, t X ) ∈ W and Y = (y, t Y ) ∈ W \ B L+δ,T (X), with t Y ≤ t X , we have G J W (X, Y ) ≤ Z ∈ ∂B L,T (X) ∩ W G J B L+δ,T (X) (X, Z) G J W (Z, Y Z ) (5.15) with Y Z = (y, min(t Y , t Z )).
Proof. The enclosing character of B L+δ,T (X) and the choice of δ, implies that every connection from X to Y includes a connection within B L+δ,T (X) to a point Z ∈ ∂B L,T (X) ∩ W . It is here where the assumption of bounded animal sizes is critically needed. This point Z is subsequently joined by an open path to a final cylinder whose section contains y. If t Y ≤ t Z , this open path determines the event Z → W Y (recall that only connections that are backwards in time are considered). If on the contrary t Z ≤ t Y , then this final cylinder must have been born before t Z and stood alive at least up to t Y . In particular, it contains the point (y, t Z ). Both situations are summed up in the inequality
G J W (X, Y ) ≤ Z ∈ ∂B L,T (X) ∩ W Q J X → B L+δ,T (X) Z ; Z → W Y Z . (5.16) (
The sum in the right-hand side involves, in fact, a time integral. Its justification requires, therefore, a limit of conectivities involving time-discretizations of ∂ V B L,T (X). These connectivities are continuous functions, so the limit converges to the integral. We omit the details.)
The desired inequality (5.15) follows from (5.16) and the fact that the events X → B L+δ,T (X) Z and Z → W Y Z are independent. Indeed, they are determined, respectively, by which cylinders are alive for t < t Z and which are alive for t > t Z . Such events are independent due of the exponential character of cylinder lives. Perhaps the simpler way to see this is by resorting to an alternative construction of the cylinder process, where for each animal γ the birth and death events are generated independently with respective exponential rates w J (γ) and 1. Cylinders are born at a birth event and live up to the next death event, neglecting intermediate events of the wrong type. With this construction, connectivity events before and after a given time t Z are determined by different, hence independent, birth and death events.
Corollary 5.17 Let L, T ∈ R + , δ as defined below (5.2) and W ⊂ Z d × R. Then, for every X = (x, t X ) and Y = (y, t Y ) in W with t Y ≤ t X , we have that,
G J W (X, Y ) ≤ Z 1 ∈ ∂B L,T (X) ∩ W G J B L+δ,T (X) (X, Z 1 ) Z 2 ∈ ∂B L,T (Z 1 ) ∩ W G J B L+δ,T (Z 1 ) (Z 1 , Z 2 ) · · · · · · Z N ∈ ∂B L,T (Z N−1 ) ∩ W G J B L+δ,T (Z N−1 ) (Z N −1 , Z N ) G J W (Z N , Y Z N ) (5.18) for every N ≤ integer part of max x − y L + δ , |t X − t Y | T . (5.19)
Proof. This is just an iteration of (5.15). The number of times such iteration can be performed is at least equal to the right-hand side in (5.19).
Regularity and "good enough" probabilities
In this section we introduce the main notions defining the multiscale approach. From now on we consider boxes as in (5.3) where the temporal height T is an increasing function of L. In Section 5.4 the function T (L) will be eventually chosen as a stretched exponential, but the following results do not depend on such a particular choice. To simplify the notation we characterize the boxes by its spacial lenght L and denote
B L (X) = B L+δ,T (L) (X).
(5.20)
Following Klein (1994) we separate Z d in regular and singular regions for a fixed realization of the environment J.
Definition 5.21 Let m > 0 and L > 1. A site x ∈ Z d is said to be (m, L)−regular if G J B L (x,0) (x, 0) , ∂ ≤ e −m(L+δ) . (5.22) Otherwise x is called (m, L)−singular. A set Λ ⊂ Z d is said (m, L)−regular if every x ∈ Λ is (m, L)−regular; otherwise it is (m, L)−singular.
Regularity will be used in conjunction with the Hammersley-Lieb-Simon inequality through the following crucial result.
Lemma 5.23 Let Λ be a (m, L)-regular region, W ⊃ Λ × R. Then, for every X = (x, t X ) and Y = (y, t Y ) in W , G J W (X, Y ) ≤ exp −m (L + δ) N Λ (X, Y ) (5.24) with N Λ (X, Y ) = integer part of min dist(x, Λ c ) L + δ , max x − y L + δ , |t X − t Y | T . (5.25)
Proof. This follows from Corollary 5.17. The value of N Λ (X, Y ) satisfies the constraint (5.19) and guarantees that all the intermediate sites z i are in Λ and, hence, they are regular. We then bound the right-hand side of (5.18) starting from the right: G J W (Z N , Y Z N ) is bounded by one, and each of the preceding sums by exp{−m (L + δ)}.
We also formalize the notion of scale Definition 5.26 A scaled sequence is a triple (L 0 , α, T ) where L 0 , α > 1 and T : (0, ∞) → (0, ∞) is a function that grows faster than any power. Each such triple defines an increasing sequence of sizes L k+1 = L α k for k = 0, 1, 2, . . .. The length L k is the k-th scale of the sequence.
Finally, we associate "good probabilities" to scales.
Definition 5.27 A scaled sequence (L 0 , α, T ) has m ∞ -good-enough probabilities if there exists p > αd such that P x is (m ∞ , L k ) − regular ≥ 1 − 1 L p k (5.28)
for all k = 0, 1, 2, . . ., for all x ∈ Z d .
We end this section with the proof of the "easy part" of the multiscale argument.
Theorem 5.29 If a scaled sequence (L 0 , α, T ) has m ∞ -good-enough probabilities, then, for any m ∈ (0, m ∞ ) there exist constants
{C J x (m) : x ∈ Z d } with P{C J x (m) < ∞ : x ∈ Z d } = 1, such that G J (X, Y ) ≤ C J x (m) exp −m max x − y , T −1 (t X − t Y ) (5.30) for all X, Y ∈ Z d × R with t X ≥ t Y ∈ R.
Proof. We follow Klein (1994). Take b > 1 (to be determined later) and consider
S k := Λ[x; b(L k+1 + δ)] is not a (m ∞ , L k ) − regular region . (5.31)
This event is verified if at least one of the sites in Λ[x; b(L k+1 + δ)] is (m ∞ , L k )-singular. As, by hypothesis, the probability for a given site to be singular is at most L −p k , we obtain
P{S k } ≤ y∈Λ[x;b(L k+1 +δ)] P y is (m ∞ , L k ) − singular ≤ [2b(L k+1 + δ)] d L p k ≤ (4bL α k ) d L p k = (4b) d L α k (p−αd) 0 .
As p > αd, this bound shows that the probabilities P{S k } are summable in k. Therefore, by Borel-Cantelli, with probability 1 there exists k 1 = k 1 (x, ℓ 0 , b, J) < ∞, such that Λ[x; b(L k+1 + δ)] is (m ∞ , L k )-regular region for all k ≥ k 1 . Now fix X, and classify the sites Y into regions
R = Y : max x − y , T −1 (t X − t Y ) < b(L k 1 + δ) (5.32) and R k = Y : b(L k + δ) ≤ max x − y , T −1 (t X − t Y ) < b(L k+1 + δ) (5.33) for k ≥ k 1 .
Let us first consider Y ∈ R k for some k ≥ k 1 . In this case we have x − y < b(L k+1 + δ) and so y ∈ Λ[x; b(L k+1 + δ)] which is a (m ∞ , L k )-regular region. It follows, from Lemma 5.
23 (for W = Z d × R and Λ = Λ[x; b(L k+1 + δ)]), that G J (X, Y ) ≤ e −m∞(L k +δ) N (5.34) with N = integer part of min b(L k+1 + δ) L k + δ , max x − y L k + δ , t X − t Y T (L k ) . (5.35)
A first bound of (5.34) comes from the observation that, as
Y ∈ R k implies x − y ≤ b(L k+1 + δ), N ≥ integer part of x − y L k + δ .
Therefore,
G J (X, Y ) ≤ exp −m ∞ (L k + δ) x − y L k + δ − 1 (5.36)
(we bounded the integer part of a number by the number minus one). This bound can be improved in cases where the temporal part dominates in the sense that
T −1 (t X − t Y ) > x − y . In this case we use that b(L k + δ) ≤ T −1 (t X − t Y ) for Y ∈ R k ,
so that, as T grows faster than any power,
t X − t Y T (L k ) ≥ T (b(L k + δ)) T (L k ) ≥ b(L k+1 + δ) L k + δ ≥ x − y L k + δ
if k 1 , and hence k, is chosen large enough. Hence
N ≥ integer part of b(L k+1 + δ) L k + δ ≥ integer part of T −1 (t X − t Y ) L k + δ (5.37)
where the last inequality comes from the definition of R k . From this and (5.34) we obtain that
G J (X, Y ) ≤ exp −m ∞ (L k + δ) T −1 (t X − t Y ) L k + δ − 1 (5.38) whenever T −1 (t X − t Y ) > x − y .
Inequalities (5.36) and (5.38) can be combined in the expresion
G J (X, Y ) ≤ exp −m ∞ max x − y , T −1 (t X − t Y ) − (L k + δ) . (5.39)
The additive correction (L k + δ) can be turned into a factor (
1 − b −1 ) because (L k + δ) ≤ (1/b) max{ x − y , T −1 (t X − t Y )}, by definition of R k . In addition we choose b = m∞ m∞−m so that m = m ∞ (1 − 1 b )
. In this way (5.39) yield
G J (X, Y ) ≤ exp −m max x − y , T −1 (t X − t Y ) (5.40) uniformly in k ≥ k 1 .
Finally we consider Y ∈ R. They satisfy max{ x − y ,
T −1 (t X − t Y )} < b (L k 1 + δ), so we can write G J (X, Y ) ≤ G J (X, Y ) exp mb (L k 1 + δ) exp −m max x − y , T −1 (t X − t Y ) ≤ C J x (ℓ 0 , m) exp −m max x − y , T −1 (t X − t Y ) (5.41)
where C J x (ℓ 0 , m) = e mb(L k 1 +δ) . The desired bound (5.30) follows from (5.40) and (5.41).
The change of scale
The change-of-scale theorem
This section contains the heart of the multiscale argument. Its main result is the following theorem, related to Theorem 3.2 of Klein (1994), which establishes sufficient conditions for a scaled sequence to have good-enough probabilities.
, m ∞ ) < ∞ such that if for some L 0 > L P x is (m 0 , L 0 ) − regular ≥ 1 − 1 L p 0 , (5.45)
for all x ∈ Z d , then the scaled sequence (L 0 , α, e L ν ) has m ∞ -good-enough probabilities.
Note that condition (2.17) guarantees the existence of ν satisfying (5.43), which in turns implies the existence of p as in (5.44).
The combination of this theorem and Theorem 5.29, implies that the key Lemma 3.18 -and hence all the properties stated in Theorem 4.1-follow once (5.45) is satisfied. Note that the logarithmic time-dependence in the key lemma is a consequence of the choice Assuming A is true, take ∆ > 0, b such that 0 < b < αν and κ > max{1, ν + θ 0 }. Define the event B ∆ = J :
T (L) = exp(L ν ) ,(5.P x is (m, l) − regular ≥ 1 − 1 l p (5.47) for x ∈ Z d ,γ∈G Λ K J ∆ (γ) ≥ e −l b , (5.52) where Λ = R j=1 Λ[x j ; l κ ] Λ[x; L + δ] (5.53)
-the union being taken on the points involved in the definition of the event Aand
K J ∆ (γ) = e −(1+∆)w J (γ) + (1 − e −w J (γ) )(1 − e −∆ ) e −∆w J (γ) . (5.54)
Then, there exist a 0 = a 0 (d, δ, α, ν, m 0 , R) ≥ 0 and l 0 = l 0 (d, δ, α, ν, κ, m 0 , θ, θ 0 , R) < ∞ such that if l > l 0 the following holds: If A and B ∆ are true for some ∆ ∈ (0, 1] then the
site x ∈ Z d is (M, L)-regular with M ≥ m − a o l θ 0 ≥ 1 L θ . (5.55)
This lemma is a direct consequence of the following Sublemas 5.56 and 5.63, where we analyze separately the connectivity of the site X = (x, 0) to sites in the vertical and horizontal boundary. The regularity stated below (5.54) follows immediately by summing the corresponding bounds over both boundaries. Sublemma 5.56 involves estimations purely in the spatial direction, hence only the event A is relevant and the choice of the time scale T (L) actually plays no role (though, for concreteness, we stick to the subexponential dependence). Event B ∆ is required to control the time-like percolation studied in Sublemma 5.63. It provides a lower bound for the probability of not having long towers of ancestors based around "deffective" sites [see (5.82)-(5.83) below]. The stretched-exponential choice (5.46) becomes essential for this sublemma which is in fact the hardest estimation of the paper.
Sublemma 5.56
There exist l 1 = l 1 (d, δ, α, ν, m 0 , θ, R) < ∞ and a 1 = a 1 (d, δ, α, ν, m 0 , R) > 0 such that for all l > l 1 , all Y ∈ ∂ V B L (X) and all J ∈ A
G J B L (X) (X, Y ) ≤ e −M 1 (L+δ) (5.57) with M 1 = m − a 1 l θ 0 ≥ 1 L θ . (5.58)
Proof. We work with a fixed environment J ∈ A. First we group the deffect-sets Λ[x j ; 2(l + δ) + 1] into larger cubes which absorb "pockets" totally surrounded by original deffects. This leaves us with a much simpler geometrical situation, where the regular region is the complement of a finite family of cubes. More precisely, elementary geometrical considerations show that there exists a possibly smaller collection of sites y 1 , y 2 , . . . , y R ′ ∈ Λ[x; L + δ], with R ′ ≤ R, n 1 , n 2 , . . . , n R ′ ∈ {1, 2, . . . , R} and n 1 + n 2 + . . . + n R ′ ≤ R, such that the sets Λ[y i ; n i (2(l + δ) + 1)], i = 1, 2, . . . , R ′ , are at a distance strictly larger than one, R j=1 Λ[x j ; 2(l + δ)
+ 1] ⊂ R ′ i=1 Λ[y i ; n i (2(l + δ) + 1)], and Λ ′ := Λ[x; L + δ] \ R ′ i=1 Λ[y i ; n i (2(l + δ) + 1)] is a connected (m, l)-regular region.
These problematic regions define cylinders of radius n i (2(l + δ) + 1) and temporal height e L ν centered at the points y i : ((y i , 0)) , i = 1, 2, . . . , R ′ [recall the notation in (5.3)]. We shall control the connectivity function in the spacial direction, on the (m, l)-regular region
B i = B n i (2(l+δ)+1),e L νB ′ = B L (X) \ R ′ i=1 B i (5.59)
[we use the notation resulting from (5.20) and (5.46)].
If X ∈ B ′ , we denote ∂B 0 = {X}; otherwise X ∈ B i ′ for some i ′ and we put ∂B 0 = ∂B i ′ . Similarly, if Y ∈ B ′ , we denote ∂B R+1 = {Y }; otherwise Y ∈ B i ′′
for some i ′′ and we put ∂B R+1 = ∂B i ′′ . Every connection from X to Y can be decomposed into disjoint connections among some of the cylinders B i . Therefore,
X → B L (X) Y ⊆ R ′ r=1 {i 1 ,i 2 ,...,ir}⊂{1,2,...,R ′ } ∂B 0 → B ′ ∂B i 1 • ∂B i 1 → B ′ ∂B i 2 • · · · • ∂B ir → B ′ ∂B R ′ +1 . (5.60)
As B ′ ⊂ Λ ′ × R with Λ ′ a (m, l)-regular region, we can apply (5.24) to obtain
Q J {∂B j 1 → B ′ ∂B j 2 } ≤ (R[2(l + δ) + 1]) d e L ν 2 exp − m(l + δ) D j 1 ,j 2 l + δ − 1 . (5.61)
We have denoted D j 1 ,j 2 = min x 1 − x 2 : (x 1 , t) ∈ ∂B j 1 , (x 2 , s) ∈ ∂B j 2 for some t, s and used the bounds
max x j 1 − x j 2 l + δ , |t − s| e L ν ≥ x j 1 − x j 2 l + δ ≥ D j 1 ,j 2 l + δ .
¿From (5.60), (5.61) and the van den Berg-Kesten inequality (5.13) we have
G J B L (X) (X, Y ) ≤ R ′ r=1 {i 1 ,i 2 ,...,ir}⊂{1,2,...,R ′ } (R[2(l + δ) + 1]) d e L ν 2(R ′ +1) × exp −m(l + δ) D 0,i 1 + D i 1 ,i 2 + . . . + D ir,R ′ +1 l + δ − (r + 1) .
This inequality, together with the bounds D 0,
i 1 + D i 1 ,i 2 + . . . + D ir,R ′ +1 ≥ L − 2[2(l + δ) + 1][n i 1 + n i 2 + . . . + n ir ] ≥ L − 2[2(l + δ) + 1]R and R ′ r=1 {i 1 ,i 2 ,...,ir}⊂{1,2,...,R ′ } 1 ≤ (R + 1)!, yield G J B L (X) (X, Y ) ≤ (R + 1)! (R[2(l + δ) + 1]) d e L ν 2(R+1) × exp −m(l + δ) L − 2[2(l + δ) + 1]R l + δ − (R + 1) . (5.62) Therefore G J B L (X) (X, Y ) ≤ e −M ′ 1 (L+δ) ; with M ′ 1 = m − c 1 m 0 l α−1 − c 2 l α(1−ν)
for some fixed constants c 1 , c 2 > 0 depending on d, δ, α, ν and R. This yields the proposed bound (5.57) because, as m ≥ 1 l θ ,
M ′ 1 ≥ M 1 = m − a 1 l θ 0 ≥ 1 L θ
for some a 1 = a 1 (d, δ, α, ν, m 0 , R) > 0 and for l large enough (the meaning of "large enough" depends on d, δ, α, ν, m 0 , θ and R).
Sublemma 5.63
Suppose the events A and B ∆ in (5.51) and (5.52) are true. Pick τ such that ν < τ < min{κ − θ 0 , αν}. Then there exists l 2 = l 2 (d, δ, α, ν, m 0 , θ, κ, b, τ, R) < ∞, such that for l > l 2 we have
G J B L (X) (X, Y ) ≤ exp{−M 2 e l τ 4 } for all Y ∈ ∂ H B L (X), with M 2 = m − e − l τ 3 ≥ 1 L θ . (5.64)
Note that it is possible to choose τ satisfying the hypothesis because κ > ν + θ 0 and ν < αν.
Proof. Let us fix an environment J ∈ A ∩ B ∆ . The proof relies on the introduction of additional space and time scales: δ(l) = l κ and τ (l) = exp(l τ ) .
(5.65)
The time scale τ (l) is intermediate between T (l) and T (L) = T (l α ) because 1 < τ < αν. The space scale δ is used to construct the region Λ defined in (5.53). For the time being it is only required to grow faster than l (κ > 1), but the hypotheses of Lemma 5.92 below will force κ to be strictly smaller than α [see condition ( We note that events in different S j are independent (see the paragraph preceding Corollary 5.17). We introduce the notation
B Λ = Λ × [−T (L), 0] for any Λ ⊂ Z d . Let Λ = R j=1 Λ[x j ; 2(l + δ) + 1] Λ[x; L + δ] (5.68)
be the set of "deffective" or "irregular" sites. The region Λ defined in (5.53) corresponds to a "buffer zone" around, and including, these irregular sites. We decompose the event {X → B L (X) Y } according to whether there exists at least one slice such that no deffective point within it is connected to a preceding slice. That is, we write
X → B L (X) Y = X → B L (X) Y ∩ A X → B L (X) Y ∩ A c (5.69) with A = N j=1 A j = N j=1 B Λ ∩ S j −→ S j S j+1 c . (5.70)
The probability of the first event on the right of (5.69) is damped by regularity. Indeed, the occurrence of {X → B L (X) Y } ∩ A implies the existence of a connection in B Λ[x;L+δ]\ Λ of vertical height at least equal to τ (l). Therefore,
{X → B L (X) Y } ∩ A ⊂ (y 1 , s 1 ) −→ B Λ[x;L+δ]\ Λ (y 2 , s 2 ) ,(5.Q J {X → B L (X) Y } ∩ A ≤ exp −m(l + δ) τ (l) T (l) − 1 + 2d ln(2L + 2δ) + 2 ln T (L) . (5.72)
Therefore,
Q J {X → B L (X) Y } ∩ A ≤ exp −m(l + δ) e l τ −l ν − 1 + 2d ln(2l α + 2δ) + 2l αν ≤ exp −(m − e − l τ 3 ) e l τ 2 ,(5.73)
for l large enough (recall that ν < τ ).
To prove that under our hypotheses A c is sufficiently improbable, we bound
A j ⊃ F j ∩ D −(j−1/2) τ (l) , (5.74)
where F j is the event that there is no connection inside the slice S j from points in the deffective set to regular points outside the buffer zone Λ: We are assumming that l is sufficiently large so that τ (l)/2 > 1, and hence τ (l)/2 > ∆ for all ∆ ∈ (0, 1]. Since each F c j involves connections within regular regions, we can apply Lemma 5.23 to obtain where c 1 , c 1 and c 3 are positive constants -depending on d, δ, κ, τ, θ and R-and l is large enough. We have used that m ≥ l −θ , θ < θ 0 and τ < κ − θ.
F j = ∂B Λ ∩ S j −→ S j \B Λ B Λ[x;L+δ]\ Λ ∩ S j c ,(5.Q J {F c j } ≤ e −h(l) ,(5.
On the other hand, We observe that
D s ⊃ γ∈G Λ {[E 1 (γ) ∪ E 2 (γ)] ∩ E 3 (γ)} ,(5.Q J [E 1 (γ) ∪ E 2 (γ)] ∩ E 3 (γ) = Q J [E 1 (γ)] + Q J [E 2 (γ)] Q J [E 3 (γ)] = e −w J (γ) + (1 − e −w J (γ) )(1 − e −∆ ) e −∆w J (γ) = K J ∆ (γ) . (5.81)
This result follows from well-known properties on Poisson processes and from the fact that the events in which an animal γ is activated at time −t and survives until time 0 define an inhomogeneous Poisson process of rate w J (γ)e −t . Therefore, due to the Q J -independence of events pertaining to different animals,
Q J {D s } ≥ γ∈G Λ K J ∆ (γ) ≥ e −l b . (5.82)
By the FKG-inequality we get, from (5.74), (5.77) and (5.79)-(5.81),
Q J {A j } ≥ Q J {F j } Q J {D −(j−1/2) τ (l) } ≥ 1 − e −c 3 l κ−θ 0 e −l b ≥ e −2l b (5.83)
for l large enough. Thus, the independence of the events A j , j = 1, 2, . . . , N leads to the bound
Q J (A c ) = N j=1 1 − Q J (A j ) ≤ (1 − e −2l b ) N ≤ exp −N e −2l b . (5.84)
The condition τ < αν implies that, for l large enough, N ≥ exp(l αν /2), and as, besides, b < αν, we obtain from (5.84) that
Q J (A c ) ≤ exp −e l αν /4 ,(5.G J B L (X) (X, Y ) ≤ Q J {X → B L (X) Y } ∩ A + Q J (A c ) ≤ exp −(m − e − l τ 3 ) e l τ 2 + exp −e l αν 4 ≤ exp −(m − e − l τ 3 )e l τ 4 ,(5.86)
with m − e −l τ /3 ≥ 1/L θ , for l larger than a certain threshold that depends on d, δ, α, ν, m 0 , θ, κ ,b, τ and R.
Probabilistic estimates: Good events have high probability
In this section we show that, assuming (5.47), the events A and B ∆ introduced in (5.51) and (5.52) have good enough probability for l sufficiently large. Hypothesis (3.19) is used to bound the probability of the second event.
Lemma 5.87 Let m > 0, l > δ, α > 1 and p > αd. Put L = l α . Pick a positive integer R > αp/(p − αd) and define the event A as in (5.51). If
P x is (m, l) − regular ≥ 1 − 1 l p (5.88)
for all x ∈ Z d , then
P{A} ≥ 1 − 1 2L p ,(5.89)
assuming l large exceed a certain minimum value that depends on d, α, p and R.
Proof. Let us say that two sites x 1 , x 2 ∈ Z d are l-nonoverlapping if the distance between the boxes Λ[x 1 ; l + δ] and Λ[x 2 ; l + δ] is strictly larger than one. In this case, the events {x i is (m, l) − regular}, i = 1, 2, are independent. We have
A c ⊂ (l) x 1 ,...,x R+1 ∈Λ[x;L+δ] R+1 i=1
x i is (m, l) − singular , (5.90) where the index (l) reminds us that the union is over collections of R + 1 sites which are l-nonoverlapping. Hence for l larger than a certain threshold that depends on α, p, κ, b, d, η, ℵ.
Proof. We start with the following observations:
(i) For any collection of sites {x i : i = 1, 2, . . . , R} with Θ G := sup x |{γ ∋ x}| < ∞. Therefore, for ∆ = e −l η , 0 < η < b − κd, and assuming l large enough in every step we get P γ∈G Λ[y;l κ +δ]
G ∪ R i=1 Λ[x i ,l κ ] ⊂ R i=1 G Λ[x i ;l κ +δ] .ln 1 K J ∆ (γ) > l b R ≤ P ∆ sup x γ∋x w J (γ) > l b−κd 4 d R + Θ G ln( ∆ 2 ) ≤ P sup x γ∋x w J (γ) > l b−κd 4 d+1 R ∆ ≤ P ln 1 + sup x γ∋x w J (γ) > l η 2 ≤ 2 a l aη E ln a 1 + sup x γ∋x w J (γ) ,(5.98)
where in the last step we have used Chebyshev inequality. Finally, (5.95), (5.98) and the fact that aη > α(p + d) imply that
P{B c ∆ } ≤ 1 L p 2 (a+2d) ℵ l aη−α(p+d) ≤ 1 2L p ,(5.99)
for l large enough.
Discussion on optimal choices
We pause to discuss the possible choices for the parameters involved in Lemmas 5.49, 5.87 and 5.92. The parameter a in Lemma 5.92 involves explicitly the birth rates of the animals. We observe that the smaller the value of a the weaker the condition on the birth rates. Thus, the goal is to choose the smallest possible value of this parameter compatible with the hypotheses. Let us first list the requirements on the different parameters: Reversing the preceding analysis, we see that the best strategy is to take α and a acording to (5.112) and (5.113) and to choose ν and p satisfying (5.43) and (5.44). This yields the smallest possible choices of a that allow choices of η, b and κ respecting the constraints (5.100)-(5.105). This explains the definition adopted for the parameters in the statement of the change-of-scale Theorem 5.42. This is also a good opportunity to comment on choices for time scales. They are determined by the demands impossed throughout the proofs of Sublemma 5.63 and Lemma 5.92. There are two major constraints: First, the scales δ(l) and τ (l) must be truly intermediate in the sense that l ≺ δ(l) ≺ L = l α and T (l) ≺ τ (l) ≺ T (L) = T (l α ) , (5.114) where f (l) ≺ g(l) means that f (l)/g(l) tends to zero as l tends to infinity. Second, the exponent h(l) defined in (5.77)/(5.78) must grow with l. This implies that τ (l) ≺ e δ(l) .
(5.115)
The combination of the two preceding displays yields the relations T (l) ≺ τ (l) ≺ e δ(l) ≺ e l α (5.116) which shows that at best the time scale can grow as a stretched exponential.
Conclusion of the proof of Theorem 5.42
We prove (5.28) by induction in k. The case k = 0 follows from hypothesis (5.45) because (m 0 , L)-regularity gives (m ∞ , L)-regularity for m ∞ < m 0 . Suppose now (5.28) is true for some k > 0. Lemmas 5.87 and 5.92 -with the replacements (m, l) → (m k , L k ) and L → L k+1 -and the preceding discussion on optimal choices imply that if
P x is (m k , L k ) − regular ≥ 1 − 1 L p k ,(5.
A last probabilistic estimate: The choice of initial scale
To finish the proof of the key Lemma 3.18 we show that for every real L (though we are interested in L large) there exists L 0 ≥ L such that (5.45) holds. It is only here that hypothesis (2.18) is needed. Let us suppose that a function S : G → [1, ∞) has been selected and consider the resulting quantity Ψ J defined in (2.23).
Lemma 5.125 Given L, p > 0, for each ρ ∈ [0, 1) there exist m(ρ), ε(ρ) > 0 and L(ρ) ≥ L such that P Ψ J > ρ < ε(ρ) =⇒ P x is (m(ρ), L(ρ)) − regular ≥ 1 − 1 L(ρ) p . (5.126)
definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The basic construction and the key lemma14 3.1 Graphical construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 How the theorem follows from the key lemma 19 4.1 Relation with percolation properties . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Bounds for the size of the clan . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Proof of Theorem 2.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Proof of Lemma 3.18 24 5.1 General scheme and notation . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 A toolbox: Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 Regularity and "good enough" probabilities . . . . . . . . . . . . . . . . . 28 5.4 The change of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.4.1 The change-of-scale theorem . . . . . . . . . . . . . . . . . . . . . . 32 5.4.2 Geometrical estimates: Good events imply good behavior . . . . . . 33 5.4.3 Probabilistic estimates: Good events have high probability . . . . . 40 5.4.4 Discussion on optimal choices . . . . . . . . . . . . . . . . . . . . . 42 5.4.5 Conclusion of the proof of Theorem 5.42 . . . . . . . . . . . . . . . 44 5.5 A last probabilistic estimate: The choice of initial scale . . . . . . . . . . . 45
Lemma 3. 18
18Under the hypotheses of Theorem 2.15 [namely, (2.2), (2.4), (2.5) and (2.17)], but replacing (2.16) by an inequality E ln a 1
Theorem 4. 1
1Assume that for a given J there is no backwards oriented percolation with Q J -probability one. Then, 1. Uniqueness. The measure µ J is the unique invariant measure for the process η J,t .2. Time convergence.For any function f with finite support,
Proposition 4. 16
16Under the hypotheses of Theorem 2.15 there exist m, m > 0 and constants C J x , C Jx , and C J x , x ∈ Z d , which are finite for P-almost every environment, such thatQ J TL (A (x,0) ) > T ≤ C J x exp −m ln q (1 + T ) (4.17) Q J SD (A (x,0) ) > L ≤ C J x exp{− mL} (4.18) E J SS (A (x,0) ) ≤ C J x . (4.19)Proof. As max (a, b) ≥ (a + b)/2, inequality (3.20) leads toG J (x, t), (y, s) ≤ C J x exp − m 2x − y + ln q (1 + |t − s|) .(4.20)
3 .
3Space convergence. Inequality (2.20) follows immediately from (4.3) and (4.18). 4. Exponential mixing. The bound (2.21) is just the combination of (4.4) and (4.18). 5. Perfect simulation. The construction proposed in Fernández, Ferrari and Garcia (2001) works, for almost all disorder realizations, due to the absence of backwards oriented percolation.
Proposition 5 . 10 (
510Harris-Fortuin-Kasteleyn-Ginibre inequality) If A and B are both increasing or both decreasing events,
Proposition 5 .
512 (particular case of the van den Berg-Kesten inequality) For B, B 1 , . . . , B n as above,
m > 0 and l sufficiently large, then we also have P x is (M, L) − regular ≥ 1 − 1 L p (5.48) where L = l α and M is an appropiate function of m. The inductive step has two ingredients. First, in Subsection 5.4.2 we show that the inductive regularity holds in the presence of two events A and B ∆ . This part of the argument is based on geometric considerations and the inequalities of Section 5.2. Subsequently, in Subsection 5.4.3 these events are proven to hold with high-enough probability. The success of the approach relies on the careful definition of the events A and B ∆ . 5.4.2 Geometrical estimates: Good events imply good behavior Lemma 5.49 Consider some fixed x ∈ Z d . Let ν, α, l be such that 0 < ν < 1, α > 1 and l > δ. Set θ 0 := min{α − 1, α(1 − ν)} (5.50) and take m 0 , θ, m with m 0 > 0, 0 < θ < θ 0 and l −θ < m < m 0 . Put L = l α , pick a positive integer R and define the event A = J : there exists x 1 , x 2 , . . . , x R ∈ Λ[x; L + δ] such that Λ[x; L + δ] \ R j=1 Λ[x j ; 2(l + δ) + 1] is a (m, l) − regular region . (5.51)
5.102)], making δ a scale intermediate between l and L = l α . The intermediate time scale defines a partition of the box B L (X) into N = integer part of T (L) τ (l) (5.66) "slices" of time-heigth τ (l): S j = B L+δ, τ (l) (x, −(j − 1) τ (l)) , j = 1, 2, . . . , N . (5.67)
l) = m{ δ(l) − (3l + 4δ + 1)} − 2 ln τ (l) − d ln δ(l) − d ln(2l + 3δ + 1) − 2 ln R . (5.78) Hence h(l) ≥ m c 1 l κ − c 2 l τ ≥ c 3 l κ−θ ≥ c 3 l κ−θ 0 , (5.79)
(γ) = {γ is not present at time s − ∆} , E 2 (γ) = {γ is present at time s − ∆ but it does not survive until time s} , E 3 (γ) = { there is no birth of γ in the interval [s − ∆, s]} .
Lemma 5.92 Let l > δ, α > 1 and p > 0. Put L = l α . For κ and b such that 0 < κ < b/d take a collection of sites {x i : i = 1, 2, . . . , R} and introduce the event B ∆ as in(5.52) for ∆ = e −l η , with 0 < η < b − κd. If ℵ < ∞ for some a > α(p + d)/η, then P{B ∆ } ≥ 1 − 1 2L p (5.93)
Λ[y;l κ +δ] ∆w J (γ) − ln(1 − e −∆ ) ≤ z∈Λ[y;l κ +δ] γ∋z ∆w J (γ) − ln(1 − e −∆ )
, and hence L k , sufficiently large. On the other hand, Lemma 5.49 -with (m, l) → (m k , L k ) and (M,L) → (m k+1 , L k+1 )-implies that if L −θ k < m k < m 0 for 0 < θ < θ 0 = α(1 − ν), then A ∩ B ∆ ⊆ x is (m k+1 , L k+1 ) − regular ,, and hence L k , sufficiently large. ¿From (5.118) and (5.119) we get that P x is (m k , L k ) − regular ≥ 1 natural k, for l suffciently large. To conclude we must check that m ∞ ≤ m k , for k = 1, 2, . . .
Thus, both hypothesis (2.16) and (2.18) of Theorem 2.15 are satisfied.) .
(2.33)
71 )
71the union being taken over all (y 1 , s 1 ), (y 2 , s 2 ) ∈ B Λ[x;L+δ]\ Λ with | s 1 − s 2 |≥ τ (l). By regularity (Lemma 5.23) we get, after simple computations
75 )
75and D s represents the lack of connection within a strip of height ∆ involving deffective sites:D s = Λ × {s} −→ Λ×[s−∆,s] Λ × {s − ∆}c
.
(5.76)
Inequalities (5.102) tell us that for choice of κ and b to be possible we must have that αν > d and hence that [see(5.50)] This identity, in combination with the extreme and leftmost inequalities in (5.102), respectively implies thatOn the other hand, if we combine the rightmost inequality in (5.102) with (5.108) and (5.103) we obtainη < ν[α − d + αd] − αd < α − d (5.110)[the last inequality is due to (5.101)]. Finally, we use this last bound, in combination with (5.104) and (5.105) to obtain a lower bound for the parameter a:Small values of a compatible with this restriction are obtained whenα > 1 ;
(5.100)
0 < ν < 1 ;
(5.101)
max{1, ν + θ 0 } < κ <
b
d
<
αν
d
;
(5.102)
0 < η < b − κd ;
(5.103)
p > αd ;
(5.104)
a >
α(p + d)
η
(5.105)
θ 0 = α(1 − ν) .
(5.106)
ν >
αd
α − d + αd
,
(5.107)
and
κ ≥ ν + α(1 − ν) .
(5.108)
Note that (5.107) yields, in view of (5.101), that
α > d .
(5.109)
a >
αd(α + 1)
α − d
with α > d .
(5.111)
α = d +
√
d 2 + d
(5.112)
and satisfy the inequality
a > 2d 2 1 + 1 +
1
d
+
1
2d
.
(5.113)
AcknowledgementsThe essential part of this work was done while G.R.G. was a visiting scholar at the Instituto de Matemática e Estatística, Universidade de São Paulo. He would like to acknowledge his warm gratitude to this institution and to FAPESP, the funding agency. He also thanks the Laboratoire de Mathématiques Rapahël Salem, UMR 6085, CNRS Université de Rouen for inviting him during the completion of his work. R.F. thanks the aforementioned Instituto de Matemática e Estatística de la Universidade de São Paulo and the Newton Institute for the Mathematical Sciences at Cambridge University, for hospitality during his work in this paper. The authors wish to thank J. van den Berg, G. GrimmettProof. By Theorem 5.1 in Fernández,Ferrari and Garcia (2001), the condition Ψ J ≤ ρ implies that there exist c(ρ), m(ρ) > 0 such that Chebyshev inequality shows that (2.18) is a sufficient condition for the validity of the left-hand side of (5.126), as long as ε ≤ ρ ε(ρ), for any given choice of ρ. This observation, the preceding Lemma 5.125 and Theorems 5.42 and 5.29 constitute the proof of the main Lemma 3.18. We observe that, in our proof, q = 1 ν and q 0 = a[α−d+αd] αd[α+a+1] with α = d + √ d 2 + d.
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| []
|
[
"Enhancement of thermoelectric properties by energy filtering: Theoretical potential and experimental reality in nanostructured ZnSb",
"Enhancement of thermoelectric properties by energy filtering: Theoretical potential and experimental reality in nanostructured ZnSb"
]
| [
"Kristian Berland \nCentre for Materials Science and Nanotechnology (SMN)\nUniversity of Oslo\nBlindernP.O.B. 1126NO-0318OSLONorway\n",
"Xin Song \nDepartment of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway\n",
"Patricia A Carvalho \nSINTEF Materials and Chemistry\nForskningsveien 1NO-0314OSLONorway\n",
"Clas Persson \nCentre for Materials Science and Nanotechnology (SMN)\nUniversity of Oslo\nBlindernP.O.B. 1126NO-0318OSLONorway\n\nDepartment of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway\n",
"Terje G Finstad \nCentre for Materials Science and Nanotechnology (SMN)\nUniversity of Oslo\nBlindernP.O.B. 1126NO-0318OSLONorway\n\nDepartment of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway\n",
"Ole Martin Løvvik \nDepartment of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway\n\nSINTEF Materials and Chemistry\nForskningsveien 1NO-0314OSLONorway\n"
]
| [
"Centre for Materials Science and Nanotechnology (SMN)\nUniversity of Oslo\nBlindernP.O.B. 1126NO-0318OSLONorway",
"Department of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway",
"SINTEF Materials and Chemistry\nForskningsveien 1NO-0314OSLONorway",
"Centre for Materials Science and Nanotechnology (SMN)\nUniversity of Oslo\nBlindernP.O.B. 1126NO-0318OSLONorway",
"Department of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway",
"Centre for Materials Science and Nanotechnology (SMN)\nUniversity of Oslo\nBlindernP.O.B. 1126NO-0318OSLONorway",
"Department of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway",
"Department of Physics\nUniversity of Oslo\nBlindernP.O.B. 1048NO-0316OSLONorway",
"SINTEF Materials and Chemistry\nForskningsveien 1NO-0314OSLONorway"
]
| []
| Energy filtering has been suggested by many authors as a means to improve thermoelectric properties. The idea is to filter away low-energy charge carriers in order to increase Seebeck coefficient without compromising electronic conductivity. This concept was investigated in the present paper for a specific material (ZnSb) by a combination of first-principles atomic-scale calculations, Boltzmann transport theory, and experimental studies of the same system. The potential of filtering in this material was first quantified, and it was as an example found that the power factor could be enhanced by an order of magnitude when the filter barrier height was 0.5 eV. Measured values of the Hall carrier concentration in bulk ZnSb were then used to calibrate the transport calculations, and nanostructured ZnSb with average grain size around 70 nm was processed to achieve filtering as suggested previously in the literature. Various scattering mechanisms were employed in the transport calculations and compared with the measured transport properties in nanostructured ZnSb as a function of temperature. Reasonable correspondence between theory and experiment could be achieved when a combination of constant lifetime scattering and energy filtering with a 0.25 eV barrier was employed. However, the difference between bulk and nanostructured samples was not sufficient to justify the introduction of an energy filtering mechanism. The reasons for this and possibilities to achieve filtering were discussed in the paper. | 10.1063/1.4944716 | [
"https://arxiv.org/pdf/1603.06629v1.pdf"
]
| 55,284,876 | 1603.06629 | 1249961ac64b4d86308ef4d534a2e5873b3691d8 |
Enhancement of thermoelectric properties by energy filtering: Theoretical potential and experimental reality in nanostructured ZnSb
21 Mar 2016
Kristian Berland
Centre for Materials Science and Nanotechnology (SMN)
University of Oslo
BlindernP.O.B. 1126NO-0318OSLONorway
Xin Song
Department of Physics
University of Oslo
BlindernP.O.B. 1048NO-0316OSLONorway
Patricia A Carvalho
SINTEF Materials and Chemistry
Forskningsveien 1NO-0314OSLONorway
Clas Persson
Centre for Materials Science and Nanotechnology (SMN)
University of Oslo
BlindernP.O.B. 1126NO-0318OSLONorway
Department of Physics
University of Oslo
BlindernP.O.B. 1048NO-0316OSLONorway
Terje G Finstad
Centre for Materials Science and Nanotechnology (SMN)
University of Oslo
BlindernP.O.B. 1126NO-0318OSLONorway
Department of Physics
University of Oslo
BlindernP.O.B. 1048NO-0316OSLONorway
Ole Martin Løvvik
Department of Physics
University of Oslo
BlindernP.O.B. 1048NO-0316OSLONorway
SINTEF Materials and Chemistry
Forskningsveien 1NO-0314OSLONorway
Enhancement of thermoelectric properties by energy filtering: Theoretical potential and experimental reality in nanostructured ZnSb
21 Mar 2016
Energy filtering has been suggested by many authors as a means to improve thermoelectric properties. The idea is to filter away low-energy charge carriers in order to increase Seebeck coefficient without compromising electronic conductivity. This concept was investigated in the present paper for a specific material (ZnSb) by a combination of first-principles atomic-scale calculations, Boltzmann transport theory, and experimental studies of the same system. The potential of filtering in this material was first quantified, and it was as an example found that the power factor could be enhanced by an order of magnitude when the filter barrier height was 0.5 eV. Measured values of the Hall carrier concentration in bulk ZnSb were then used to calibrate the transport calculations, and nanostructured ZnSb with average grain size around 70 nm was processed to achieve filtering as suggested previously in the literature. Various scattering mechanisms were employed in the transport calculations and compared with the measured transport properties in nanostructured ZnSb as a function of temperature. Reasonable correspondence between theory and experiment could be achieved when a combination of constant lifetime scattering and energy filtering with a 0.25 eV barrier was employed. However, the difference between bulk and nanostructured samples was not sufficient to justify the introduction of an energy filtering mechanism. The reasons for this and possibilities to achieve filtering were discussed in the paper.
Energy filtering has been suggested by many authors as a means to improve thermoelectric properties. The idea is to filter away low-energy charge carriers in order to increase Seebeck coefficient without compromising electronic conductivity. This concept was investigated in the present paper for a specific material (ZnSb) by a combination of first-principles atomic-scale calculations, Boltzmann transport theory, and experimental studies of the same system. The potential of filtering in this material was first quantified, and it was as an example found that the power factor could be enhanced by an order of magnitude when the filter barrier height was 0.5 eV. Measured values of the Hall carrier concentration in bulk ZnSb were then used to calibrate the transport calculations, and nanostructured ZnSb with average grain size around 70 nm was processed to achieve filtering as suggested previously in the literature. Various scattering mechanisms were employed in the transport calculations and compared with the measured transport properties in nanostructured ZnSb as a function of temperature. Reasonable correspondence between theory and experiment could be achieved when a combination of constant lifetime scattering and energy filtering with a 0.25 eV barrier was employed. However, the difference between bulk and nanostructured samples was not sufficient to justify the introduction of an energy filtering mechanism. The reasons for this and possibilities to achieve filtering were discussed in the paper.
I. INTRODUCTION
Thermoelectric materials allow for the conversion of temperature gradients to electricity and vice versa. They are today mainly used within sectors such as automotive, aerospace, defense, industrial and self-powered sensors. For direct power generation the low efficiency is the major technical factor limiting the growth of the market. [1][2][3][4] Good thermoelectric materials are distinguished by low thermal conductivity κ, high electronic conductivity σ and high Seebeck coefficient (S) at a given temperature T . This can be quantified by the dimensionless figure of merit ZT ZT = σS 2 T κ e + κ l .
Due to the Wiedemann-Franz law linking σ closely together with the electron part of the thermal conductivity κ e , 5 much emphasis is put on lowering the lattice thermal conductivity κ l . The power factor P F = σS 2 should furthermore be maximized by choosing the optimal charge carrier concentration. We have in this paper demonstrated that this last requirement entails electronic conditions favoring transport of high-energy over low-energy carriers. Nanostructured materials offer new mechanisms to selectively scatter phonons and low-energetic electrons without strongly affecting the transport of energetic electrons. 3,6,7 Efficient bulk thermoelectric materials are a good starting point for further nano-enhancements; yet even poor ones may serve -nanostructured silicon have for example shown promising thermoelectric properties. 8 A particularly interesting concept is that of energy-filtering. By introducing potential barriers or strongly energy-dependent scattering mechanisms lowenergetic carriers can be blocked, greatly enhancing the Seebeck coefficient. 7,[9][10][11][12][13][14][15][16][17][18] ZnSb has been known as a thermoelectric material for a long time. 19 When Caillat reported a figure of merit of 1.4 for Zn 4 Sb 3 in 1997 that composition got the most attention due to the remarkably low thermal conductivity. 20 ZnSb was then mostly regarded as an annoying phase impurity. However, two phase transitions, one from the α to β phase at 250 K, and one from β to γ at 767 K, 21 make Zn 4 Sb 3 difficult to use in applications. ZnSb has received renewed interest 22,23 for a number of reasons. There is an increased awareness of environmental concerns, where Zn and Sb score well for abundancy and low toxicity. There is also a lack of other good alternative materials for operation in the temperature-range 400 − 650 K, where ZnSb performs well. Further, the thermoelectric properties of bulk ZnSb are suitable for improvement by nanostructuring. 24,25 Several reports on densely packed pellets of ZnSb have appeared recently, utilizing techniques like ball-milling, 26-30 spark plasma sintering, 31 and cryogenic milling. 32 Optimization of doping levels and alloying elements have significantly enhanced the thermoelectric properties of ZnSb, 22,27,33,34 utilizing the potential of the impurity band. 33,[35][36][37] This has led to an improvement of the figure-of-merit from 0.3 in the 1960's 38 to consistent reports of zT > 0. 9. 22,23,27 A number of theoretical studies of ZnSb have been reported in recent years. Ab initio band structure calculations have been reported by several groups. 30,33,[39][40][41][42][43][44][45] These have e.g. allowed comparisons with experimental effective masses, 33 the stability of the material, 46 vacancy formation energies revealing the nature of the bonding 44 and charge transfer to bonds or neighbor atoms. 30,47 Also, a few phonon dispersion results and studies addressing thermal properties of ZnSb from first principles have recently appeared. 44,[48][49][50] In this paper, we have quantified the theoretical potential of energy filtering in ZnSb, demonstrating that it is indeed possible, from a theoretical point of view, to greatly enhance the power factor of ZnSb. In an attempt to verify this experimentally, we prepared nanostructured ZnSb samples by a combination of cryomilling and rapid hot pressing, which has been shown previously to generate densely packed pellets with very small grain size and significantly reduced thermal conductivity. 32 Our hypothesis was that such processing could introduce energy filtering from grain boundaries or nanoinclusions associated with grain boundaries. 13,15,17 The transport properties of these samples were then compared to the theoretical predictions with and without energy filtering.
This paper is organized as follows: First, a brief description of the sample preparation and experimental methods are provided. Then follows the theoretical approach to calculating thermoelectric properties and solving the Boltzmann transport equation with different scattering models including energy filtering. This is followed by an analysis of the potential of energy filtering of ZnSb. A comparison between theory and experiment for a bulk reference sample is then presented, validating the approach qualitatively and indicating quantitative shortcomings. The final part is a comparison between theory and experiment for nanostructured samples.
II. EXPERIMENTAL METHODS
Starting from stoichiometric Zn and Sb sealed in evacuated quartz tube without any intentional dopants, the initial materials were synthesized by melting and solidification. The mix was melted at 970 K followed by quenching in cold water. Two thermo-mechanical processing routes were then followed: (i) a "nanostructured" sample was produced by ball milling at 77 K and hot-pressing at 740 K for 30 min and cooling to RT within 2 h; (ii) a "bulk" sample, used as reference, was produced by ball milling at room temperature and hot-pressing at 740 K for 30 min and cooling to RT within more than 20 h. Further details on the fabrication method, reduction of thermal conductivity in nanostructured samples, etc. are described in Ref 32. A number of different methods were used to characterize the samples: The microstructure was investigated by transmission electron microscopy and energy dispersive spectroscopy (EDS) using an FEI Titan G2 60-300 instrument operated at 300 kV. For better statistics, the average grain size was estimated from the full-width half maximum (FWHM) of X-ray diffraction peaks using the TOPAS software, which includes information about the instrument contributions in the peak shape analysis. 51 The Seebeck coefficient was measured with the uniaxial four-point method in vacuum. 52 Finally, the electrical conductivity and carrier concentration were measured in vacuum with the Van der Pauw and Hall methods using a custom-built instrument. 53
III. THEORY
The Boltzmann transport equation in the relaxationtime approximation was used to calculate thermoelectric properties. As input for these calculations, we used the electronic band structure from density functional theory calculations together with a specified energy filtering and constant relaxation time τ . These results were also compared with results obtained with a simple energy (ǫ)dependent scattering of the form
τ (ǫ) = τ s (ǫ/k B T ) s ,(2)
where the scattering parameter s determines the energy dependency and thus the specific scattering mechanism. k B is the Boltzmann constant. Important examples include acoustic-phonon scattering (s = −0.5), polar optical phonon scattering (s = 0.5), and ionized impurity scattering (s = 1.5). 54 The net effect of less energy-dependent scattering mechanisms, such as scattering from neutral defects, can be represented by a constant lifetime contribution (s = 0). The various possibilities represented by equation (2) can account reasonably well for typical scattering mechanisms existing in bulk materials, at least for scattering around nondegenerate band minima. 55 Energy filtering was implemented in these calculations by simply removing contributions to the thermoelectric transport properties that arise from charge carriers close to the valence band edge. According to theoretical considerations, energy filtering can arise from extended barriers such as heterostructures, nanocomposites, nanoinclusions, or grain boundaries. [9][10][11][12][13][15][16][17] A. Electronic structure calculations
The structure and electronic properties of ZnSb were calculated utilizing the plane wave code VASP, working at the density functional theory (DFT) level and using the projector augmented wave approximation for atomic core regions. [56][57][58][59] The generalized gradient PBE 60 exchange-correlation functional was used and spin-orbit coupling was ignored.
To obtain the atomic and crystal structure, we relaxed the structure with DFT with an energy cutoff of 500 eV, which is 80% larger than the standard recommended maximum pseudopotential cutoff. Such high cutoffs are needed to accurately determine the structure. The kpoint sampling was set to 10 × 8 × 8 and due to the low PBE band gap, the Gaussian smearing was set to 0.03 eV. The structure was relaxed until forces became smaller than 0.02 eV/Å. The calculated lattice parameters of the orthorhombic unit cell, 6.28Å, 7.82Å, and 8.22Å, agree well with previous calculations. 61,62 For comparison, the experimental values at room temperature are 6.218Å, 7.741Å, and 8.115Å. 63 To obtain the electronic structure, we first generated the electronic charge density n(r), using an energy cutoff of 276 eV, corresponding to the recommended maximum pseudopotential cutoff and a dense k-mesh of 20 × 16 × 16 integrated using the tetrahedron method with Blöchl corrections. The total energy was converged to 10 −6 eV. Following this step, we generated the band structure with a non-selfconsistent DFT calculations with a k-mesh of 50 × 50 × 50, as such very dense meshes are required for accurate transport properties. Figure 1 shows the electronic band structure of ZnSb (left), density of states ρ(ǫ) (middle), and diagonal elements of the tensorial transport spectral functions Σ(ǫ) 64 for a constant relaxation time (right). Σ(ǫ) is a 3 × 3 tensor, and its diagonal elements are defined in the following manner:
Z Γ X S R Γ Y /0 wave vectorΣ αα (ǫ) = 1 V N k,i (ν i α (k)) 2 τ i (k) δ ǫ − ǫ i (k) ,(3)
where V is the volume, N is the number of majority charge carriers, τ i (k) is the relaxation time for band number i, ǫ i (k) is the energy of band i at reciprocal vector k, and ν i α (k) is the group velocity in the α-direction (α = x, y, z).
The most relevant region for low-field transport is that close to the band edges (for energies less than e.g. 0.5 eV away from the Fermi level). We first note that, close to the band edges, the level of anisotropy for Σ(ǫ) is somewhat higher (the relative difference between the diagonal components is larger) in the valence-band region than in the conduction-band region. We have in the remainder of the paper neglected the anisotropy by assuming that the samples are multicrystalline and isotropic on average. This was imposed by using the mean of the diagonal elements of the transport spectral function: Σ(ǫ) = Tr (Σ(ǫ)) /3. The spectral functions are on the other hand larger in magnitude above the conduction band minimum (CBM) than below the valence band maximum (VBM). This can be rationalized from the shape of the band structure (left) having a single dominant peak near the VBM and multiple ones of relatively similar energy near the CBM.
The presence of an impurity band originating from Zn defects can explain many of the features of ZnSb at low temperatures, and a model involving single parabolic bands including an explicit impurity band was rather successful in reproducing transport properties of intentionally undoped ZnSb. 35 In the present study, we have chosen to include contributions from such impurities as an effective scattering model combined with adapting the charge carrier concentration by changing the Fermi level. The alternative, introducing an explicit impurity band to the calculated band structure as in Ref. 35, would imply ambiguities related to the position and size of the impurity band. One could include the impurity band indirectly by adding Zn vacancies (the most stable intrinsic impurity in ZnSb) as in Ref. 44, but this would make it difficult to fine-tune the doping level, particularly without involving prohibitively large supercells. Also, our choice gave the ability to directly compare contributions from impurity scattering with other mechanisms.
B. Boltzmann transport equation
Key thermoelectric quantities can be expressed in terms of integrals of the transport-spectral function Σ(ǫ) as follows
σ = e 2 ∞ −∞ dǫ − ∂f FD (ǫ) ∂ǫ Σ(ǫ) , T σS = e ∞ −∞ dǫ − ∂f FD (ǫ) ∂ǫ Σ(ǫ)(ǫ − µ F ) , T κ 0 = ∞ −∞ dǫ − ∂f FD (ǫ) ∂ǫ Σ(ǫ)(ǫ − µ F ) 2 . (4)
Here the derivative of the Fermi-Dirac distribution function − ∂fFD(ǫ) ∂ǫ is the Fermi window, a symmetric function peaked when the energy ǫ is equal to the Fermi level, µ F .
Our calculated PBE band gap of ZnSb was 0.06 eV, which is consistent with previous studies at the same level of theory. 30,40,44 This level of theory is known to severely underestimate the gap compared to experimental values. The typical experimental value of the band gap for single crystal ZnSb is 0.5 − 0.6 eV 19,45,65,66 . However, there are also experimental reports of a ZnSb band gap around 0.3 eV. 67 We chose to enlarge the calculated band gap by 0.5 eV in order to be consistent with recent ab initio studies employing the more reliable Heyd-Scuseria-Ernzerhof (HSE) hybrid functional, where the band gap was predicted to be 0.56 eV. 30,47 The adjustment was implemented by a simple scissor operator widening the gap in Σ(ǫ) and ρ i (ǫ) by 0.5 eV, keeping their shapes otherwise fixed.
Energy filtering corresponding to a nonplanar potential 13 was implemented by removing the contributions from the top of the valence band region (ǫ = 0) in a width ∆, as expressed in terms of Heaviside step functions h as follows:
Σ(ǫ) → Σ(ǫ) (h(−ǫ − ∆) + h(ǫ)) .(5)
This kind of energy filtering is crude, but rather common in the literature. 13,16 Figure 2 shows calculated thermoelectric properties of ZnSb as a function of the Fermi level µ F . The left side presents results at T = 300 K, the right at T = 500 K. Panels a) show the Hall carrier concentration, b) the Seebeck coefficient, c) the conductivity, and d ) the power factor. The full black curves show the constant relaxation time results for bulk ZnSb including the band gap correction specified above. The stark contrast with the dashed one, based on the bare PBE gap, underlines the importance of this correction. With the low PBE gap, minority carrier contributions become significant for low and moderate doping, severely reducing the peak Seebeck value. Further, the asymmetry of Σ(ǫ), as seen in figure 1, reflects a favoring of electron transport over hole carrier transport, resulting in a negative Seebeck coefficient at Fermi levels close to the band edges. The asymmetry is also reflected in the shape of the conductivity and power factor, indicating that ZnSb could be a better n-type thermoelectric than a p-type, 30,43,44 provided that stable n-type ZnSb with suitable doping concentration could be prepared. So far no successful n-type has been reported while the difficulty has been rationalized by the easy formation of Zn vacancy type defects acting as acceptors. In this paper, emphasis has thus been on the regular p-type variant.
The effect of various degrees of energy filtering is shown with the thin green and dotted red curves in figure 2. Energy filtering drastically increases the peak Seebeck coefficient and power factor, but also shifts the peak positions to a lower Fermi level corresponding to higher p-doping concentrations. The particularly high peak with an energy filtering parameter of ∆ = 0.5 eV can be linked to the shape of the band structure and to the density of states and transport spectral function in figure 1. At energies around 0.5 eV additional bands start contributing causing a kink-like feature in these two functions.
Comparing the left and right subfigures, we find that for a given Fermi level, the Seebeck coefficient is lower at 500 K than at 300 K, but as far as the power factor is concerned, this is more than compensated by the increased conductivity, resulting in a higher value at 500 K.
In the comparison with experimental data (in Sec. IV), we will use the measured Hall carrier density at different temperatures n Hall (T ) to determine the Fermi level µ F (T ). We have then assumed that the holes and electrons scatter equally (but possibly depending on the energy of the band). This is a minor approximation, since the transport properties are dominated by the majority carriers for the Hall carrier concentrations and temperatures considered here (when assuming the band gap is 0.56 eV). In the case of constant scattering time, the Fermi level could thus be obtained for each temperature by solving the following equation:
n Hall (T )r H = ∞ −∞ dǫ f FD (ǫ − µ F )ρ(ǫ) sign(ǫ) + N val .(6)
Here N val is the number of valence electrons in the system and r H is the Hall factor. For simple energydependent scattering (equation (2)), we used the Hall factor 55 r H (s) = Γ(2s + 5/2)Γ(5/2)/ (Γ(s + 5/2)) 2 and related the Hall mobility to the drift mobility. Here, Γ is the gamma-function. For reference, r H (0) = 1, r H (−0.5) ≈ 1.18, and r H (1) = 1.4. This expression ignores non-parabolicity. This is in line with the use of simple scattering models also derived for parabolic bands. Care must be taken in determining the Fermi level when energy filtering is included in the model, since filtered electrons do not contribute to the Hall carrier concentration. Thus, if a filter is used on ρ(ǫ) in equation (6), the reference number of valence electrons N val should be adjusted accordingly. Further, the Hall correction factor and simple energy-dependent relaxation time approximations become inappropriate as they are developed for parabolic bands. We have therefore only combined energy-filtering models with the constant relaxation-time approximation.
The thermoelectric transport properties were calculated using the BoltzTraP 68 software package to generate the density of states ρ i (ǫ) and the transport spectral functions Σ i (ǫ) for each band i at constant scattering time. Next, equations (4,6) were solved in a postprocessing step using scipy 69 routines in python.
C. Potential of energy filtering for ZnSb
Energy filtering greatly enhances the peak Seebeck coefficient of ZnSb, as shown in figure 2. At the same time it severely reduces the electrical conductivity at a given Fermi level, since a significant number of charge carriers do not contribute to the transport anymore. However, the Fermi level may be manipulated if the doping level can be controlled. In that case, as the Fermi level approaches the filtered region, conductivity can be considerably increased, resulting in a strongly enhanced power factor. This is particularly so when filtering allows additional bands to contribute, as discussed above for the case of ∆ = 0.5 eV.
Energy-dependent scattering can also enhance the Seebeck effect. In fact, filtering can be viewed as an extremely energy-dependent form of scattering, as e.g. discussed by Bahk and coworkers. 13 Whereas filtering may be appropriate as a crude model of the scattering or trap- ping caused by extended energy barriers such as grain boundaries, 7,12,70 energy-dependent expressions are better suited to account for scattering by charged impurities such as acceptors or even charged nanoinclusions. 13 In figure 3, we compare the Seebeck coefficient as a function of temperature for different Hall carrier concentrations and different scattering/filtering accounts. In the upper panel, we compare the Seebeck coefficient for constant scattering time with energy-dependent scattering following equation (2) with s = 1/2 and 3/2. In the lower panel, we repeat the comparison for two different energy filtering parameters (equation (5)) ∆ = 0.25 eV and 0.5 eV. The figures illustrate how both energydependent scattering and filtering generally enhance the Seebeck coefficient. The picture is somewhat more complex with energy filtering: the Seebeck coefficient is not always enhanced and the largest filtering parameter affects the results far more than the smallest. These effects arise because the Fermi level is shifted to keep the Hall carrier concentration fixed and multiple bands start contributing to the conduction for the largest filtering
parameter.
The results of the energy filtering shown here are consistent with the data in figure 2. For instance, it is evident that decreasing the carrier concentrations (move to the right in figure 2 a)) leads to increasing the Seebeck coefficient (move to the right in figure 2 b)).
The slight dip in the Seebeck coefficient at T = 700 K (red curve) for the lowest Hall carrier concentrations arises from minority carrier contributions. When energy filtering is included this dip is absent; one effect of energy filtering is to increase the effective band gap by the same amount as the filtering parameter. Figure 4 shows the conductivity and power factor as a function of temperature for the same filtering parameters and Hall carrier concentrations as in figure 3. In the upper panel, we find as expected that conductivity increases with the Hall carrier concentration. That filtering seems to enhance conductivity reflects that we have compared conductivities for different Hall carrier concentrations, only accounting for mobile holes and electrons. Depending on the physical mechanism causing filtering-like effect-instead of merely being passive, elec-tron states could for instance also be removed from the active region-the effective doping concentration could dwarf the Hall carrier concentration. Compare, for instance, the Hall carrier concentration curves with and without energy filtering in figure 2(a). With such high hole densities, the true potential profile in a sample with filtering barriers present could be strongly interconnected with the hole concentration. 10 The lower panel of figure 4 shows the corresponding power factors. The crossing curves demonstrate that the optimal Hall carrier concentration for a given filtering parameter depends strongly on the target temperature. Further, the optimal doping concentration for the Seebeck coefficient differs widely from the optimal one for the power factor (figure 3); for instance, at 700 K and a filtering parameter of ∆ = 0.5 eV, the highest Hall carrier concentration considered (2 × 10 20 cm −3 ) results in both the lowest Seebeck coefficient and the highest power factor. Conversely the curve with the highest Seebeck coefficient corresponds to the lowest power factor. Figure 5 presents the optimal power factor and accompanying carrier concentration as function of the filtering width. In the lower panel, the optimal power factor is shown as a function of the filtering parameter ∆. A filtering parameter of 0.5 eV e.g. results in a tenfold increase in the power factor at 300 K. The relative enhancement is somewhat lower at higher temperature, but the power factor is nonetheless significantly higher than for lower temperatures. In the upper panel, the solid curves show the optimal Hall carrier concentration for the given filtering parameter and the dashed ones show the corresponding hole concentration (under the assumption that the filtering mechanism simply blocks propagation of filtered electrons). As the filtering parameter increases, the optimal hole concentration can easily become more than ten times larger than the Hall carrier concentration. Thus extremely high hole concentrations are required to optimize the power factor. This is the reason we have not evaluated filter widths beyond 0.5 eV, even if the power factor continues to increase as the filter width is increased further. At a certain point it is not realistic to obtain the carrier concentration required to optimize the power factor. We have somewhat arbitrarily selected 0.5 eV as the limit, since this would require an order of magnitude higher carrier concentration than the Hall concentration. However, for small filtering parameters, the optimal carrier concentration might be slightly lower than without filtering. In this case, the enhancement of the Seebeck coefficient outweighs the reduction in the conductivity.
IV. THERMOELECTRIC PROPERTIES OF BULK AND NANOSTRUCTURED ZNSB
A. Comparison with bulk reference sample
In comparing theory and experiment, we first considered a nominally undoped bulk-like sample with a sig- FIG. 5. The optimal charge carrier concentration (a) corresponding to the optimized power factor (b) for different filtering parameters ∆. In (a), the solid and dotted curves represent the optimal Hall concentration and hole concentration for 300 (blue, diamonds), 500 (green, squares), and 700 K (red, circles). It is shown in (b) how the optimal power factor increases with filtering parameter at the same temperatures as in the upper panel.
nificant intrinsic carrier concentration. The grain size of this sample was measured to be 0.2 µm by using the FWHM from the X-ray diffractogram. Note that the grain size distribution is also very important for thermoelectric properties; however, this was not available with our methods. In the calculations, we used the measured Hall carrier concentration to determine the Fermi level at each temperature, while the value of the constant relaxation time τ 0 was subsequently obtained by fitting the temperature-dependent calculated electrical conductivity to the experimentally measured one. In figure 6, the upper panel compares the calculated Seebeck coefficient (full curves) with the measured one (dashed curve), while the lower panel compares the experimental conductivity with the calculated one, using the fitted relaxation time. The constant relaxation time was used as a parameter to fit the calculated to experimental conductivity curves in the temperature range between 300 and 500 K, and was then found to be τ 0 = 1.35 × 10 −14 s. The dotted curve shows the measured Hall carrier concentration (right axis).
The reasonable agreement between theory and exper- iment for scattering parameters s = 0 and 0.5 indicates that our relatively simple model based on full bands generated with DFT and with a constant-time scattering reproduces the experimental temperature-dependent conductivity and Seebeck coefficient quite well. The small discrepancies could arise partly from the crude scattering account and partly from inaccurate band curvatures obtained with the PBE functional, which could affect the effective mass and nonparabolicity. Finally, the Hall carrier concentration varies strongly as a function of temperature, and any error in this measurement would influence the theoretical predictions. The Seebeck coefficient as a function of temperature in figure 6 reaches a maximum value at around 450 K before decreasing. This is qualitatively different from the monotonously increasing one for fixed carrier concentration, shown above in figure 3. The difference can most easily be rationalized by the rapid increase in Hall carrier concentration that was used in calculating the Seebeck coefficient in figure 6. The strong dependence of the Seebeck coefficient on the carrier concentration can e.g. be seen be comparing pan- els a) and b) in figure 2. It is worth noting that a turning point of S like the one seen in figure 6 is often used to estimate the band gap, using the Goldsmid formula. 71 In our case the turning point can be explained solely by the strongly increasing majority carrier concentration as a function of temperature, illustrating one of the potential pitfalls when using the Goldsmid formula for band gap assessment. 72
B. Including filtering for reference carrier concentration
The carrier concentration we obtained from the nominally undoped bulk sample could be regarded as a typical one. But how would the performance be affected if we included energy filtering assuming that the hole concentration is kept fixed? Figure 7 shows that in this case the power factor is reduced as the filtering parameter ∆ increases. This comparison differs inherently from that of figure 4, where the power factor was calculated for different Hall carrier concentrations. This may be useful for comparing with experiment, but does not explore the effect of energy filtering for a given hole concentration. The effective Hall carrier concentration may be significantly reduced by energy filtering, which is illustrated by the green and black curves in the upper panels of figure 2. To achieve a high power factor, the hole concentration must be high enough to maintain a relatively high number of mobile carriers.
C. Comparison for nanostructured ZnSb
Having established the potential of energy filtering in ZnSb in Sec. III C, we now explore whether nanostructuring of ZnSb can be seen to induce energy filtering. The filtering mechanism could for instance be potential barriers at the grain boundaries, thus relying heavily on the grain size. To this end, we investigated experimentally the transport properties of two different ZnSb samples with average grain size of 70 nm (nanostructured) and 0.2 µm (bulk), respectively. The processing of powders and pellet samples was briefly described in Sec. II and in more detail in a previous paper. 32 Figure 8 shows a TEM image from the nanostructured ZnSb pellet, depicting a number of small grains as well as clustering of oxygen containing precipitates close to the grain boundaries. Such clusters could give rise to barriers hindering transport of low-energy charge carriers, making it a possible source of the filtering effect. The mean grain size indicated by the XRD FWHM was 70 nm, 32 consistent with the TEM image in figure 8.
Transport properties of these nanostructured samples were then measured, and figure 9 shows a comparison between theory and those experiments. Three different scattering mechanisms are compared: constant relaxation time; an energy dependent scattering (equation (2)) with s = 0.5, corresponding to polar optical phonon scattering; and a combination of constant relaxation time with energy filtering (equation (5)) with ∆ = 0.25 eV. Like above, the Hall carrier concentration was used as input to determine the Fermi level at each temperature and scattering mechanism, followed by adjusting the relaxation time τ to fit the temperature dependent conductivity σ to experiment in the temperature range between 300 and 500 K. The Seebeck coefficient is independent of the specific relaxation time.
We first note that we can achieve a reasonable agreement between theory and experiment for all the scattering mechanisms in figure 9. The constant τ and s = 0.5 mechanisms yield too fast increase of σ when T > 500 K. Also, constant τ yields a too low Seebeck coefficient for all temperatures when compared with experiment. The best fit is thus achieved with the combination of con-stant τ with energy filtering, using a filtering parameter of 0.25 eV. The constant τ was found to be slightly lower in the nanostructured sample (10 fs) than that found for the bulk sample (13.5 fs).
The model using constant τ with energy filtering exhibits a good match with the experimental curves of the Seebeck coefficient and electronic conductivity. However, the fit is not so good for the power factor. This is because of small deviations contributing in the same directions of both S and σ and being magnified for the product. Because of cancellation of errors both the constant τ and the s = 0.5 mechanisms appear to give a better fit to the power factor.
This reflects that the difference in quality between the different models is not huge. Also, the deviation in the Seebeck coefficient from experiment of the nanostructured sample using the constant relaxation time model is similar to that of the bulk sample shown in figure 6. This simply reflects that the two samples display quite similar carrier densities, since the Hall concentration is decisive for the Seebeck coefficient in this material. This was demonstrated by performing similar experiments with other bulk and nanostructured samples (not shown here); the quantitative success of the constant-time scattering model in bulk samples was highly dependent on the charge carrier concentration, and the Seebeck coefficient was quite similar in bulk and nanostructured samples at similar carrier concentration. Also, the power factor was not enhanced by nanostructuring.
Thus, no new scattering mechanism can be seen to appear when going from bulk to nanostructured samples. In other words, there is no need to involve energy filtering or more energy-dependent scattering resulting from grain refinement as part of the mechanisms explaining the transport properties of the nanostructured samples in this study.
It would be interesting to repeat the measurements with even smaller grains, preferably comparable in size to the energy relaxation length. This might be feasible, since the average particle size of the as-milled powder from the cryomill is ∼ 10 nm. 32 The energy relaxation length is not known for ZnSb. It is significantly larger than 10 nm in lightly doped bulk silicon (0.89 µm at 270 K with a charge carrier concentration of ∼ 10 15 cm −3 ), 73 but may be in the same order of magnitude in nanostructured, heavily doped systems. 74 To achieve such small grains would require a faster annealing technique than the rapid hot press used in the present study, and a close eye should be kept on grain growth by limiting the temperature used in the experiments.
It may also be interesting to perform similar experiments with lower amount of precipitates clustered around the grain boundaries. Even if nanoinclusions may yield more predictable filtering barriers than grain boundaries, 75 a system featuring only grain boundaries might give a more pure signal of filtering which is easier to interpret.
The current study relied on undoped ZnSb to simplify the analysis and focus on the effect of nanostructuring on the scattering properties. If one succeeds creating a sample displaying clear signs of filtering, the next important step would be to combine this with intentional doping. This is required to move towards the peak power factor as seen in figure 2(d). It remains to see if any dopant has sufficient solubility in ZnSb to reach this regime.
V. CONCLUSION
We investigated the theoretical potential of energy filtering in the promising thermoelectric material ZnSb. It was shown to be considerable, with up to an order of magnitude increases in the power factor compared to bulk samples. This required a filtering parameter of 0.5 eV and high Hall carrier concentration. Our theoretical analysis also indicated that energy filtering would yield very high Seebeck coefficients at low Hall carrier concentrations.
The theoretical predictions were then tested against experiments on nanostructured ZnSb. The assumption was that nanostructuring could lead to energy filtering, enhancing thermoelectric properties by selectively hindering the conduction of low-energy charge carriers. Nanostructured ZnSb samples were processed by cryogenic milling of ZnSb into very fine powder and pressing pellets with a rapid hot press. They were nominally undoped, but still featured charge carrier concentrations in the order of 10 18 − 10 19 cm −3 .
The samples displayed a relatively large variation of the Hall concentration as function of temperature, which resulted in the Seebeck coefficient displaying a quite flat behavior. Thus, to obtain meaningful comparison between experiments and theoretical modeling, we adjusted the Fermi level of the calculations to reproduce experimental carrier concentrations for each temperature. Furthermore, the observed electrical conductivity at moderate temperatures (300 − 500 K) was used to calibrate the scattering parameters (constant scattering time τ and filtering parameter ∆). With those parameters fixed, the measured Seebeck coefficient and the power factor served as benchmarks of the various scattering models, in the hope that distinct features of the different models could rule out or support any of them.
Reasonable correspondence with the experimental data was obtained when using any of the following scattering models: (i) constant scattering time, (ii) constant scattering time combined with a filtering with height 0.25 eV, and (iii) polar optical phonon scattering (s = 0.5). The constant time combined with filtering (ii) exhibited a slightly better correspondence with experiment, but not enough to support the introduction of an extra adjustable parameter (the filtering height) in addition to a hypothetical physical mechanism.
Our conclusion is that an average grain size of around 70 nm is not small enough to obtain filtering with substantial effects on the scattering properties and power factor of ZnSb. Whether it is possible to obtain filtering in ZnSb, and whether a smaller grain size would render the effects of filtering observable are still open questions.
FIG. 1 .
1(eV −1 per u.cell) Σ αα (ǫ) (arb. units) Electronic band structure of ZnSb obtained using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional (left), corresponding density of states (DOS) (middle), diagonal elements of the tensorial transport spectral function (right). See text for explanation.
FIG. 2 .
2Calculated (Hall) carrier concentration (a), Seebeck coefficient (b), conductivity (c), and power factor (d) of ZnSb at T = 300 K (left panels) and T = 500 K (right panels) as a function of the Fermi level µF. A constant scattering time with τ0 = 1 · 10 −14 s was used. The dashed blue curves are based on the PBE band gap (0.06 eV), while the thick black curves rely on the experimental band gap (0.56 eV). The thin green and dotted red curves are results with a valence band energy filter of 0.25 eV and 0.5 eV.
calculated using energydependent scattering (a) and energy filtering (b). The red and blue curves are for fixed Hall concentrations of 10 19 and 10 20 cm −3 , respectively. The solid curves are for a constant scattering time, τ0 = 1.0 × 10 −14 s. Energy-dependent scattering according to equation (2) is shown with an exponent of s = 3/2 (dotted curves), s = 1/2 (dashed curves), and s = −1/2 (dash-dotted curves). In (b), energy filters (equation (5)) of respectively 0.25 eV (dashed curves) and 0.5 eV (dotted curves) are introduced.
FIG. 4 .
4The effect of Hall concentration and energy filtering on the conductivity σ (a) and power factor PF (b) in ZnSb. The constant relaxation time is τ0 = 10 −14 s. Following the conventions in figure 3(b), the red and blue curves are for fixed Hall concentrations of 10 19 and 10 20 cm −3 , respectively. The full curves are for a constant scattering time, while the dashed (dotted) curves have an energy filter of 0.25 (0.5) eV. The figures demonstrate how the power factor can be greatly enhanced with energy filtering.
FIG. 6 .
6Seebeck coefficient S (a), conductivity σ (b), and experimental carrier concentration (dotted blue curve in (b), right axis) of a bulk ZnSb sample. Crosses connected by black, dashed lines correspond to experimental data, while the filled symbols connected by solid lines correspond to calculated results based on the measured Hall carrier concentration using energy dependent scattering mechanisms according to equation (2) with s = 3/2 (purple, circles), 1/2 (green, squares), 0 (red, diamonds), and −1/2 (cyan, pentagons).
FIG. 7 .
7Power factor as function of temperature for constant scattering time (black, solid curve) and filtering parameters ∆ = 0.1 eV (blue, dashed curve) and ∆ = 0.15 eV (purple, dotted curve).The hole concentration was fixed to that of the bulk sample.
FIG. 8 .
8Transmission electron microscopy (TEM) image of the nanostructured sample. The upper part depicts several grains of typical size, as well as a number of oxide precipitates. (Proven by electron diffraction on several different precipitates, not shown here.) The lower part has zoomed in on precipitates located along a grain boundary.
FIG. 9 .
9Thermoelectric properties of nanostructured ZnSb: Seebeck coefficient S (a), electrical conductivity σ (b), experimental carrier concentration (dotted blue curve in (b), right axis), and power factor PF (c). Black crosses connected by dashed lines correspond to experimental data, while the filled symbols connected by solid lines correspond to calculated results based on the measured Hall carrier concentration using different scattering mechanisms: constant relaxation time (red, diamonds), constant relaxation time with a filter of ∆ = 0.25 eV added (green, stars), and an energy dependent scattering according to equation (2) with s = 1/2 (blue, squares).
ACKNOWLEDGEMENTSWe are grateful for enlightening discussions with Espen Flage-Larsen and for access to experimental facilities at California Institute of Technology via G. Jeff Snyder. We acknowledge the Research Council of Norway for financial support through the projects NanoThermo and Thelma. The computations were carried out using a grant from the Notur consortium.
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| []
|
[
"HI Epoch of Reionization Arrays",
"HI Epoch of Reionization Arrays"
]
| [
"S Komonjinda \nCenter for Astrophysics\n60 Garden St02138CambridgeMAUSA\n",
"Y Kovalev \nCenter for Astrophysics\n60 Garden St02138CambridgeMAUSA\n",
"eds.D Ruffolo \nCenter for Astrophysics\n60 Garden St02138CambridgeMAUSA\n",
"L J Greenhill [email protected] \nCenter for Astrophysics\n60 Garden St02138CambridgeMAUSA\n",
"G Bernardi [email protected] \nCenter for Astrophysics\n60 Garden St02138CambridgeMAUSA\n"
]
| [
"Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Center for Astrophysics\n60 Garden St02138CambridgeMAUSA"
]
| [
"The 11 th Asian-Pacific Regional IAU Meeting 2011 NARIT Conference Series"
]
| There are few data available with which to constrain the thermal history of the intergalactic medium (IGM) following global recombination. Thus far, most constraints flow from analyses of the Cosmic Microwave Background and optical spectroscopy along a few lines of sight. However, direct study of the IGM in emission or absorption against the CMB via the 1S hyperfine transition of Hydrogen would enable broad characterization thermal history and source populations. New generations of radio arrays are in development to measure this line signature. Bright foreground emission and the complexity of instrument calibration models are significant hurdles. How to optimize these is uncertain, resulting in a diversity in approaches. We discuss recent limits on line brightness, array efforts including the new Large Aperture Experiment to Detect the Dark Ages (LEDA), and the next generation Hydrogen Reionization Array (HERA) concept. | null | [
"https://arxiv.org/pdf/1201.1700v1.pdf"
]
| 117,757,021 | 1201.1700 | cf59c3b950c03f025fe1e622454246d366989430 |
HI Epoch of Reionization Arrays
9 Jan 2012
S Komonjinda
Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Y Kovalev
Center for Astrophysics
60 Garden St02138CambridgeMAUSA
eds.D Ruffolo
Center for Astrophysics
60 Garden St02138CambridgeMAUSA
L J Greenhill [email protected]
Center for Astrophysics
60 Garden St02138CambridgeMAUSA
G Bernardi [email protected]
Center for Astrophysics
60 Garden St02138CambridgeMAUSA
HI Epoch of Reionization Arrays
The 11 th Asian-Pacific Regional IAU Meeting 2011 NARIT Conference Series
120129 Jan 2012
There are few data available with which to constrain the thermal history of the intergalactic medium (IGM) following global recombination. Thus far, most constraints flow from analyses of the Cosmic Microwave Background and optical spectroscopy along a few lines of sight. However, direct study of the IGM in emission or absorption against the CMB via the 1S hyperfine transition of Hydrogen would enable broad characterization thermal history and source populations. New generations of radio arrays are in development to measure this line signature. Bright foreground emission and the complexity of instrument calibration models are significant hurdles. How to optimize these is uncertain, resulting in a diversity in approaches. We discuss recent limits on line brightness, array efforts including the new Large Aperture Experiment to Detect the Dark Ages (LEDA), and the next generation Hydrogen Reionization Array (HERA) concept.
Introduction
Characterization of the intergalactic medium (IGM) and early generations of luminous compact objects before global reionization (z > 6) is a frontier in observational cosmology. For the first ∼ 100 Myr following recombination and decoupling of radiation and matter, evolution was dominated by gravity acting on diffuse distributions of dark and baryonic matter. Models are robust owing to linearity of interaction. For the remainder of the first ∼ 1 Gyr, nonlinear processes become increasingly relevant, e.g., with the widespread star formation, and understanding is limited.
There are no data with which to directly constrain evolution of the IGM prior to formation of luminous objects (a.k.a. the "dark age," despite diffuse radiation). Even during reionization of small pockets initially and large expanses later, there are few data available to constrain the thermal history of the IGM, source populations, and how these formed. An integrated global estimate of the redshift of reionization assuming a sudden transition (z = 11.0 ± 1.4) stems from measured large scale polarization of the cosmic microwave background (CMB, [1]). Recent observations of the kinetic Sunyaev-Zeldovich effect on the CMB anisotropies have given a model dependent constrain on the duration of reionization ∆z < 7.9 with reionization beginning at z beg < 13.1 at 95% CL ( [2]).
Limits on the neutral Hydrogen (HI) column have been inferred from optical and near-infrared (NIR) spectra of luminous objects, where a trough blueward of Lyα appears for z > 6.3 (the "Gunn-Peterson Trough"). Neutral fraction is difficult to infer over a wide range of z owing to the high absorption cross section and ready saturation. However, inference may be drawn from modeling of the Lyα damping wing, including the effective radius of the proximate ionized zone. As of this writing, only two quasars are known for z > 6.4 ([4]), the most distant at z ∼ 7.09 ( [3]). Troughs have also been observed toward gamma ray bursts, and there are 3 known for z > 6.4, up to z ∼ 8.3 ([5], [6] and [7]). Modeling of spectroscopic data for quasars and GRBs enables inference for a handful of lines of sight. Broad sky coverage may be limited by nature -there are relatively few supermassive black holes that evolved in the first ∼ 1 Gyr -and instrumental reasons -wide-field NIR surveys and high sensitivity spectroscopic follow up is difficult (and subject to atmospheric windows.) Figure 1. Average sky brightness temperature of the λ21 cm line (i.e., spin temperature × optical depth) up to the end of reionization [9]. Variation with redshift is driven by varying coupling between radiation (the CMB) and gas kinetic temperatures. The λ21 cm transition of Hydrogen at high redshift has not yet been detected, but as it would trace the diffuse IGM horizon to horizon without the need for illuminating compact objects, study of this transition has the potential to provide a new avenue by which to infer the evolution of the IGM and source populations. This tracer complements CMB studies and optical/NIR spectroscopy and spectral-line origin in principle allows discrimination by redshift, possibly into the Dark Age. The physics of the λ21 cm transition has been described in reviews (e.g., [8]; [10], and references therein), but we summarize here salient points, referring to Figures 1 and 2.
Following recombination, the IGM cooled (adiabatically) faster than the CMB. When there was effective collisional coupling between λ21 cm transition and gas temperature, the line will be seen in absorption against the CMB (both in the sense of a sky average and fluctuations). This occurred for z ≫ 40, when gas density was high, but before the first stars. Coupling faded with expansion of the Universe, reducing contrast between the HI line and CMB. With early star formation, leakage of Lyman photons into the IGM recoupled the spin and kinetic temperatures via the Wouthysen-Field effect, resulting in a second absorption signature (e.g., [9]). The absorption depth and redshift of the minimum depend on the extent and evolution of heating sources (e.g., star formation, X-rays from black holes, shocks), which may involve "exotic" physics (e.g., dark matter annihilation). Eventually, IGM heating drives the HI signature into emission before the HI component is largely destroyed; in this regime, the sky-averaged and fluctuations signal will be more extended in fractional redshift and weaker in magnitude.
Observations of the λ21 cm line
Sensitivity to the anticipated unpolarized mK signal will be limited by systematics. These will arise principally from errors in instrument calibration and foreground models subtracted from the data. The two will be derived from similar data and likely to be coupled. Unpolarized foregrounds will be smoothspectrum mix of extragalactic point sources and Galactic diffuse emission with angular scales overlapping those anticipated for the λ21 cm signal ( [12]). Foreground brightness well exceeds the λ21 cm signal. The former rises to O(10 3 ) K at 100 MHz even away from the Galactic plane (proportional to approximately ν −2.5 [13]). This is O(10 4−5 ) times the λ21 cm signal in skyaveraged power at ∼ 70 and 140 MHz ( Figure 1). Fluctuations in foreground emission for "quiet" patches of sky are O(1-10) K) ( [11]) and may be only O(10 2−4 ) times the λ21 cm fluctuations of O(1-10) mK) at ∼ 140 MHz ( Figure 2). The brightness of individual foreground sources extends down to the λ21 cm signal. This can be subtracted where it is above limits imposed by source confusion and contamination by sidelobes. However, this exceeds thermal noise limits for even modest integrations with arrays matched to the large angular scales of fluctuations. Below this nonthermal limit, other approaches are required (e.g., subtraction of an extant high-resolution sky or a statistical model). The limit reported by [11] is a few mJy at 150 MHz and ∼ 2 ′ resolution (∼10 K).
A more detailed study of foreground structure is still necessary for EoR observations. At present, prediction of the unpolarized diffuse component requires interpolation between widely spaced frequencies. Diffuse emission dominates the polarized sky (cf. pulsars and compact AGN). There are no wideangle low-frequency polarized sky surveys or consistent picture from the few patches that have been studied, and angular and frequency structure in Stokes Q and U can differ considerably from that of Stokes I ( [12]). The point source population is somewhat better understood. Catalogs of low-frequency point sources with arcminute resolution are available from the Cambridge surveys at 151-178 MHz (e.g., [16]) and from the VLA 74 MHz survey ( [17]). In the Southern Hemisphere, Culgoora Surveys at 80 and 160 MHz ( [18]) provide measurements at arcmin resolution. A GMRT survey that will provide finer resolution at 150 MHz 1 is underway.
For single dipole instruments, a careful instrumental design coupled with a polynomial fit has been demonstrated to be effective in foreground subtraction ( [19]). However, uncertainty in antenna gain patterns (e.g., in response to the environment) and unmodeled responses of electronics have thus far limited sensitivity to the global 21cm signal other than for sudden reionization ( [19]).
For dipole arrays, a higher degree of calibration is possible, benefitting efforts to detect λ21 cm fluctuations, or total power via array elements capable of measuring calibrated total power. Bright compact sources are used to calibrate direction and frequency dependent instrument response, and they may be subtracted directly from visibility data ( [41], [50], [11], [22], [23]). Foregrounds are mapped in parallel and, given a smooth and slow variation with frequency, galactic diffuse emission and sub-confusion level point sources can in principle be filtered ( [24], [25], [26], [27], [28]) provided that calibration errors do not result in pixel to pixel spectral fluctuations (e.g., [29]). Equivalently, calibration errors can couple a fraction of the polarized emission from Galactic sources into the unpolarized visibilities (as noted above), contaminating the λ21 cm signal ( [11], [30]). For idealized models of sky and instruments, it has been possible to filter out the polarization leakage by identifying the effects of different Faraday screens along various lines of sight and subsequently filtering the data ( [31]). Active areas of investigation are in correction of ionospheric distortions ( [32]), wide field polarization ( [33]), correction for antenna gain patterns ( [34], [35]) and direction dependent deconvolution ( [36], [37]).
Ongoing λ21 cm Experiments
Arrays thus far reflect a diversity of architectures, which is a consequence of incomplete knowledge of the low-frequency sky, complex coupling of dipole elements to the sky (and man-made transmissions), and uncertainty regarding optimization of foreground subtraction. The RF hierarchy (i.e., layout of antennas) dictates the challenges to be met in signal processing and in calibration and imaging. Equivalently, the available computing budget where signal processing is implemented with general purpose platforms (e.g., BlueGene, GPUs) constrains what RF hierarchies are practical. We briefly review ongoing experiments (Table 1), emphasizing differences in architecture and signal processing. Recent limits on the λ21 cm signal are presented in Figure 3. for the breadth of reionization (∆z) for a "sudden reionization" model, estimated from single-dipole total-power measurements by EDGES ( [19]). Shading denotes 68% and 95% C.L. (right)-Upper limits on 3D power spectra (z=8-9) from Westerbork data (⋆) ( [11]), and the GMRT for two different foreground subtraction techniques (adapted from [38]). Linear fits over 2.0 (✷) and 0.5 MHz (•) provide a foreground subtraction model. The 0.5 MHz limit may preclude a cold IGM at z∼9 (dot-dashed line; [26]).
Giant Metrewave Radio Telescope (GMRT)-EOR experiment
Fourteen GMRT dishes are located in a 1 km 2 area, providing the low surface brightness sensitivity required to attempt detection of the λ21 cm line. A 16 MHz wide 150 MHz receiver band corresponds to z∼8.5. The narrow ∼3.3 • field of view (FOV; half-power) of the dishes has enabled the experiment to begin without wide-field polarization calibration and correction for ionospheric distortions. The target field is centered on a pulsar and in a novel calibration scheme, gated visibilities (pulsar ON -OFF) are differenced, thus eliminating the other (unswitched) foregrounds and simplifying phase and polarization calibration ( [39]). Foreground subtraction has thus far depended on fitting a frequency linear baseline on 0.5 and 2 MHz scales in visibility space, leveraging the anticipated smooth spectrum of the foregrounds ( [38]).
Low Frequency Array (LOFAR)-EoR experiment
Twenty-three high frequency (120-240 MHz) LOFAR stations are arranged in the ∼3 km core for use by the EOR experiment. Each station comprises two clusters of 24 tiles of 16 dual polarization dipoles (768 total). This is the deepest hierarchy of RF-element clustering for any EOR instrument. Two levels of beam forming within the LOFAR stations enables good signal rejection but at the cost of complex sidelobe structure that may be difficult to map in assessment of coupling to the sky brightness distribution (cf. the GMRT). Calibration and visibility plane foreground subtraction will be executed via generalized selfcalibration techniques that incorporate direction dependent effects ( [40], [41]). This is supported by stations far from the core, which are used to catalog point sources and to accumulate enough lines of sight to enable 3D tomography of the ionosphere ( [42]). The EOR experiment is in its first year of data collection.
Murchison Wide-field array (MWA)
The MWA will comprise 128 tiles of 16 dual polarization dipoles, each with a beam former establishing a single ∼20 • half-power FOV at 150 MHz. EOR observing will utilize a strongly centrally condensed pseudo-random configuration of ∼ 1 km 2 ( [43]). Dense u,v-plane sampling and co-planarity will enable imaging via co-adding of warped snapshots ( [33]). As such, wide-field correction can be achieved with geometric precision, and ionospheric distortion (approximated as a 2D rubber sheet) can be implemented in image space. The large number of baselines will enable detection of many calibrators per snapshot, a running estimation of the many, variable tile gain patterns, and correction during gridding of visibility data. Source subtraction down to nearly the confusion limit will be accomplished via peeling in visibility space and image-based Forward Modeling will enable deconvolution ( [36], [37]). Among EOR projects, so great reliance on image-based processing is unusual, but if successful, it may be relevant to much larger arrays, for which the volume of of visibilities, ∝O(N 2 dipole ), may preclude traditional techniques.
Precison Array to Probe the Epoch of Reionization (PAPER)
PAPER comprises 64 broadband dipoles (100-200 MHz) with plans to build out to 128 or 256 ( [44]). A shaped ground screen for each reduces what would otherwise be horizon-to-horizon dipole response to ∼60 • . LOFAR, MWA, and PAPER represent a progression to flatter hierarchies vis-a-vis RF and signal processing architecture. PAPER operates in a drift scan mode and benefits from the stability of the dipole gain patterns (cf. MWA and LOFAR). The array can be reconfigured, enabling experimentation with pseudo random distributions for building a sky model and redundant distributions that can benefit calibration and concentrate sensitivity on particular angular scales ( [45]) as may be desired for first detection of λ21 cm brightness fluctuations.
3.5. Large-Aperture Experiment to Detect the Dark Ages (LEDA) LEDA 2 is distinct among arrays in that it will attempt detection of the λ21 cm signal at z ∼ 20 MHz band) at the end of the Dark Ages and thus to constrain initial conditions for reionization (e.g., [9]). The target is the angleaveraged signal, and a large-N array is used enable joint estimation of instrument calibration and a sky model (in contrast to what is possible with a single dipole). LEDA will attempt to leverage the O(10×) magnitude of the λ21 cm absorption trough at z∼20 compared to emission at lower redshift. A 512-input full-Stokes, FPGA/GPU-enabled correlator (e.g., [46]) will be integrated into the 100×110-m Long Wavelength Array (station 1) for 30-88 MHz observing. Outrigger dipoles will be instrumented for total-power measurement using calibrated noise sources. Their gain patterns will be calibrated using cross-correlation data for baselines to the core stations. The outriggers will be well separated from the core to reduce mutual coupling effects that contribute high-order terms in gain patterns, to provide baselines that resolve most diffuse Galactic emission, and to improve sensitivity to the point source population.
Hydrogen Epoch of Reionization Array (HERA)
The Hydrogen Epoch of Reionization Array (HERA) is a planned 2 nd generation (mid-decade) experiment that targets the detailed evolution of the λ21 cm power spectrum (e.g., [47]), and to a lesser extent imaging ionized "bubbles" around selected luminous early quasars 3 . In concept, HERA will have similar sensitivity to phase 1 of the Square Kilometer Array, though HERA will be an experiment optimized for study of the λ21 cm line at z > 6. It will be timed to follow current generation US projects (PAPER, LEDA, MWA) and exploit lessons learned.
The HERA conceptual design comprises a centrally condensed, filled aperture with a sparsely sampled "halo" of outlying antennas. In seeking to characterize λ21 cm power spectra at a detailed level, emphasis is on constraint of systematics during instrument design and in foreground characterization and removal. The primary specifications affecting sensitivity are FOV, pass band, total collecting area, and area per element ( Table 1). The geographic distribution of elements enters vis-à-vis the quality of foreground mapping and source subtraction. FOV and pass band dictate directly the range of k-space to which the array is sensitive. For large k, i.e., fine angular scales, artifacts from subtraction of point-sources may become increasingly serious ( [49]). Errors in reconstruction and subtraction of diffuse foreground emission, particularly in the presence of ionospheric fluctuations, will also reduce sensitivity over a range of k, although quantitative modeling of the effect has been limited. The LOFAR architecture presents one model by which to accumulate collecting area (a sparse array with beam forming of many-dipole stations). In contrast, HERA is anticipated to use a "flatter" array configuration. This can be more computationally challenging, starting with cross-correlation, which is an O(N 2 ) problem. For a collecting area of 10 m 2 per antenna per polarization (more than for PAPER and LWA, and half that for MWA), 10 4 elements are required, for which correlation of 100 MHz demands 40 PFlop s −1 . A scalable solution for correlation drawing on GPU computing has been proposed by [46], with demonstration planned by LEDA. Calibration carries a similar computational order (e.g., [50], [51]), and scalable demonstration relevant to HERA may be achieved by LEDA and or MWA.
Summary-We have reviewed efforts to detect the high redshift λ21 cm signal with radio arrays. Diversity of approach reflected in the design of first
Figure 2 .
2Power in fluctuations as a function of redshift for four spatial frequencies on the sky (adapted from[10]).
Figure 3 .
3Experimental constraints on reionization thus far. (left)-Lower limits
Figure 4 .
4Indicative 3-D power spectrum measurement error vs a model, for a filled-aperture 10 5 m 2 array, assuming a neutral IGM, 1000 h integration, 8ṀHz bandwidth, T sky = (470, 1250) K, and 5000 antennas in a filled core and 3km diameter envelope. Symbols indicate independent bins. The vertical dotted line corresponds to bandwidth in k units. The effects of calibration error and systematics are ignored. Adapted from ([48]).
Table 1 .
1High Redshift λ21 cm ExperimentsExpt. (1)
Loc. (2)
Elements
Area (3) Bsln (4)
Band (5)
Array
Lone
(m 2 )
(km)
(MHz) z
PAPER
ZA
128
-
1000
0.2
110-180 6-12
MWA
AU
128×16
-
3000
1
80-200 6-15
LEDA
US
256
4
3000/30
0.3
30-88 17-42
DAWN
US
256
-
3000
0.1
30-88 17-42
GMRT
IN
14
-
8000
1
139-156 8-9
LOFAR
NL
24×768
-
18000
3
120-200 6-10
HERA
c. 2016 O(10 4 ) TBD
O(10 5 )
O(3)
<100-200 6-12+
(1) Shaded denotes efforts to measure the angle-averaged HI spectrum. LEDA and
DAWN share the LWA1 aperture. DAWN applies beam forming on cold patches of the
sky. LEDA instead cross-correlates the aperture ( §3.5). (2) IN: Pune, India; NL: Exloo,
Netherlands; WA: Boolardy, Western Australia; ZA: Karoo, South Africa; US: VLA
site, New Mexico. (3) Approximate collecting area, mid-band, per polarization. (4)
Maximum array baseline.
(5) Redshifts omit a 10% guard band.
http://tgss.ncra.tifr.res.in
http://www.ledatelescope.org
http://reionization.org/RFI2 HERA.pdf. This described a roadmap of instruments. Here we refer to the next, mid-decade experiment, dubbed HERA-2, simply as HERA.
Greenhill & Bernardi generation devices marks an "epoch of experimentation" from which strategies and optimizations for next generation arrays, such as HERA, will be based.
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| []
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[
"A Service of zbw The impact of government health and education expenditure on income inequality in EU The impact of government health and education expenditure on income inequality in European Union",
"A Service of zbw The impact of government health and education expenditure on income inequality in EU The impact of government health and education expenditure on income inequality in European Union"
]
| [
"Ionuț Jianu [email protected] \nLeibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics Jianu\nBucharest University of Economic Studies\nIonuțRomania\n"
]
| [
"Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics Jianu\nBucharest University of Economic Studies\nIonuțRomania"
]
| []
| Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Abstract. This research aims to provide an overview of the existing inequalities and their drivers in the member states of the European Union as well as their developements in the 2002-2008 and 2009-2015 sub-periods. It also analyses the impact of health and education government spending on income inequality in the European Union over the 2002-2015 period. In this context, I applied the EstimatedGeneralized Least Squares method using panel data for the 28-member states of the European Union. | null | [
"https://arxiv.org/pdf/2007.11409v1.pdf"
]
| 159,307,245 | 2007.11409 | d8a8ad8702b7ebae9d988274cbac344109e6ecce |
A Service of zbw The impact of government health and education expenditure on income inequality in EU The impact of government health and education expenditure on income inequality in European Union
Ionuț Jianu [email protected]
Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics Jianu
Bucharest University of Economic Studies
IonuțRomania
A Service of zbw The impact of government health and education expenditure on income inequality in EU The impact of government health and education expenditure on income inequality in European Union
econstor Make Your Publications Visible. Conference Paper -Published Version Suggested Citation: Jianu, Ionuț (2018) : The impact of government health and education expenditure on income inequality in EU, In: International Finance and Banking Conference FI BA 2018, XVIth Edition, Theoretical and Applied Economics. Special Issue, General Association of Economists From Romania (GAER), Bucharest, pp. 121-134, This Version is available at: http://hdl.handle.net/10419/194296 Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. Theoretical and Applied Economics. Special Issue 122 122inequalityincomeeducationhealthunemployment JEL Classification: D63I14I24
Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Abstract. This research aims to provide an overview of the existing inequalities and their drivers in the member states of the European Union as well as their developements in the 2002-2008 and 2009-2015 sub-periods. It also analyses the impact of health and education government spending on income inequality in the European Union over the 2002-2015 period. In this context, I applied the EstimatedGeneralized Least Squares method using panel data for the 28-member states of the European Union.
Introduction
One objective of the European Union agenda consists in the economic, social and territorial cohesion and recalls the higher social purpose of the European model of integration than that of the American or the Asian model. However, national social realities vary widely between the EU member states in terms of education, health, income, and employment.
The social situation does not respresents only a source for expansion of the mandate of politicians and received a special attention from the European citizens in the recent years. According to the European Commission (Reflection paper on the social dimension of Europe), 8 of 10 European citizens see unemployment, social inequalities and migration as the main challenges of the European Union. More than 50% of Europeans also believe that the next generation will be exposed to more difficult situations. In this context, the situation of social imbalances (income inequalities) across the European Union are one of the most widely debated concepts in recent years. Income inequality is natural given that people are naturaly different, have different capacities, visions, concerns, and different behaviors. For example, people's adaptability to the labour market favors high wage earnings, while rigidities in the workforce can lead to wage cuts. Rigid labour is limited in terms of the manifestation of wage discontent and, in the case of a wage cut, the employee will respect the management decision, while flexible workforce can present other employment opportunities to the management in the wage negotiation process, which can lead even to an increase of the wage of the employee by the management.
The motivation for choosing this theme consists mainly in the intensification of the economic debates on this topic, especially those related to pros and cons of income inequalities. Some researchers consider income inequality to be beneficial (given the wage disparities resulting from the different performances of the population on the labour market), while other economists support their reduction. Also, social concern regarding income gap and the methods used in order to reduce it represents an additional motivation for choosing this theme.
The objective of the paper is the examination of the situation of income inequalities in the European Union and the assesment of the impact of government health and education expenditures on income inequalities, the robustness of the assessment being strengthened by the integration of other control variables into the analysis.
Literature review
Even if many studies were made in this area, the results have sometimes been contradictory. Some economists see income inequality as an undemanding preoccupation of economists, while others consider the social sphere to be an illustration of the economic policies implemented.
According to the European Commission (2010), the socio-economic inequalities recorded in the post-2000 years were higher than those recorded in 1980, despite the economic growth achieved by the European Union. Social and human capital development policies did not have the desired impact on income inequalities due to the high labour polarization, this being a result of economic modernisation and labour market deregulation.
Eurofund (2017) analysed the evolution of pre-crisis and post-crisis incomes inequalities and found that the European Union has made significant progress in terms of convergence by 2008, but in the post-crisis period some efforts have been canceled by the impact of the economic and financial crisis that has spread stronger economic shocks in the peripheral states of the European Union than in developed member states using effective adjustment mechanisms. The impact of the crisis on the convergence process has also led to a significant decline in incomes across the European Union. The Eurofund (2017) also found that unemployment is the main channel by which the economic downturn has increased income inequality in the European Union, affecting different categories of population.
On the other hand, a series of studies (Benabou, 2000(Benabou, , 2002Bleaney, Gemmell and Kneller, 2001) highlighted the pro-growth and pro-income inequality character of some public expenditure categories, such as: government expenditure on health and education, and government expenditure on infrastructure. However, other categories of expenditure may offer inadequate incentives, which implies assuming some compromises in the budget execution. Dabla-Norris et al. (2015) identified the improvement of educational qualifications, removing financial barriers to third-country education and providing support for apprenticeship programs as factors that improve the quality of the education system and have a significant impact on income inequality. Also, in the OECD (2012) vision, educational policies that increase graduation rates in upper secondary education and in tertiary education play a fundamental role in reducing income inequality. The Organization believes that structural reforms in the labour market and those that enhance the quality of the education system are key factors in moderating income inequalities.
According to O'Donnell et al. (2013), health can influence the distribution of income through several channels, including through the one related to labour market. In this context, the labour productivity of people suffering from certain diseases is lower, which leads, generally, to lower wages. Discrimination is also another factor that can deepen income inequality between people with disabilities and healthy people. On the other hand, some researchers have also shown that the increase in income inequality is associated with high mortality (Wagstaff and Doorslaer, 2000), homicide and violence (Lynch et al., 2001). In this context, this reverse causality relationship brings significant challenges to a country in the event of an increase in income inequality, which can only be addressed by implementing effective structural reforms on the labour market (including education structural reforms). Ward et al. (2009) demonstrated the existence of a positive relationship between the unemployment rate and the level of income inequality in the European Union. At the same time, Boltanski and Chiapiello (2005) found that labour mobility, and its adaptability, are included into the category of labour market specificities that influence income inequality. In their view, rigidity of the workforce has unfavorable consequences on the income distribution, since workers who are not willing to change their residence in order to find a better job are susceptible to commply with their modest activity and to accept lower wages.
As regards the measurement of income inequality through the Gini coefficient, Solt (2016) found that most of the statistical data sources providing this indicator is dealing with data comparability and coverage issues. In many cases, the national Gini coefficients provided by international databases are computed through different methodologies or the number of observations is low, numerous data missing from the samples. The author mentioned the only source that publishes the Gini coefficient (The Luxembourg Income Study), computed on the basis of a uniform set of definitions and assumptions, respectively on the basis of harmonised microdata that ensures maximization of their comparability. In this context, the Standardized World Income Inequality Database uses the Luxembourg Income Study standardized data series. On the other hand, the missing data was computed through the 5-year weighted moving average algorithm, the uncertainty of the results being reduced by the Monte Carlo simulation and application of the algorithm for each simulation.
Methodology
This section presents the research methodology for quantifying the impact of government health and education spending on income inequalities in the European Union. For this purpose, I used yearly data for the 2002-2015 period for each member states of the European Union (392 initial observations) as follows: Statistical data on income inequality, defined in this paper through Gini coefficient, are limited and the integration of this variable into an econometric model may be difficult. Eurostat does not cover the entire analysis period for this coefficient (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015) and it has been necessary to consult another credible statistical platform for this indicator. I chose to use the data provided by the Standardized World Inequality Database given that the authors of the study created a new harmonised database for the Gini coefficient from several statistical sources (international studies and scientific research). Thereby, I obtained a complete set of the indicator, excepting the data for 2015 (Italy and Luxembourg -for which I used OECD data). Also, the method used maximizes the comparability of data between countries. The analysis period was limited to 2015 due to missing data for general government expenditure by function for 2016 at Eurostat level.
In the first phase I analysed the evolution of the five indicators (Table 1) In the step of identifying the influence factors and equation estimation method, I tested the stationarity of the variables through the "Summary" technique, which provides an overview of the main stationary tests: (i) Common root -Levin, Lin & Chu, (ii) Individual root -Im, Pesaran and Shin, (iii) Common root Breitung, (iv) Individual root -ADF-Fisher and (v) Individual PP-Fisher root. The variables used proved to be stationary at level and at first difference, which argued the use of the autoregressive term in the equation. Afterwards, I processed the data in Eviews 9.0 software to estimate the impact of the education and health expenditures of government on the Gini coefficient. For this purpose, I used EGLS -Estimated Generalized Least Squares method in panel window. In order to increase the feasibility of the method used, I applied the Period SUR option (for the ex-ante correction of heteroscedasticity and of the general correlations between the cross sections) on the following estimated equation:
Gini α β Gini β health β education β wages β un ε(1)
where: capture income inequality, Gini represents the autoregressive term, health and education surprise government health and education speding lagged by one year (expressed as a share of GDP), wages and un represents the contribution of wages to GDP, respectively the unemployment rate and ε is the error term. Also, α represents the coefficient of the constant and β captures the impact coefficients of the exogenous variables on Gini. For independent variables, I selected the lags according to the specificities of the economic theory.
Following the estimation of the equation, it was necessary to identify the method of estimating the effects. The use of autoregressive term rejects the random effects method. To confirm this hypothesis, I used Hausman test which indicated the rejection of this method. To identify the extent to which the Fixed Effect Method is appropriate for this model, I used the Redundant fixed effects test. However, dealing with issues related to multicollinearity or autocorrelation of residuals or heteroskedasticity, depending on the structure of the equation, the number of cross sections and the number of observations, may suggest using a standard panel model without fixed or random effects. Although the Fixed Effect method proved to be appropriate (Redundant fixed effects test), I rejected this technique because Period SUR option (used to correct heteroscedasticity and general cross-sectional correlations) could not be applied. I have also not been able to apply an alternative option (Cross-section SUR) given that the number of observations per cross-section (14) is less than the number of cross-sections (28) -Eviews software does not allow such an estimation.
A number of 364 observations resulted from the adjustments made for the application of this method. Further, in order to test the main assumptions for the validation of this model, I performed the following tests: (i) F test (verifying the statistical validity of the model); (ii) Jarque-Berra (examination of the normal distribution of the residuals); (iii) Breusch-Godfrey (testing the autocorrelation of residuals); (iv) Breusch-Pagan and Pesaran CD (testing cross-section dependence);
(v) Breusch-Pagan-Godfrey (testing of heteroskedasticity); (vi) Klein's criterion (testing for multicollinearity).
Given that the panel window do not support heteroskedasticity and autocorrelation tests, I used the theoretical framework to obtain the results of these tests. Thus, for testing the autocorrelation of residuals (iii), I estimated the following equation:
res1 γ δ Gini δ health δ education δ wages δ un res1 1 res1 2 ε (2)
where: res1 represents the residuals from the initial estimated model, and res1(-1) and res1(-2) represent the series of residuals lagged by one and two years.
In order to evaluate the probability of the autocorrelation test, I used the Microsoft Office Excel 2016 function -CHISQ.DIST.RT, a function that took into account: a) the product of the R-squared corresponding to the equation (2) and the number of observations, respectively b) the degrees of freedom (number of lags for res1).
As regards, the heteroskedasticity test (v), I estimated the following equation:
res1^2 λ μ Gini μ health μ education μ wages μ un ε (3) where res1 ^ 2 represents the square of the residuals resulting from the initial estimated model.
Finally, I used the CHISQ.DIST.RT function to evaluate the probability of the heteroskedasticity test. The function involved the following factors: a) the product of the R-squared corresponding to equation (3) and the number of observations, respectively b) the degrees of freedom (the number of independent variables, excluding the constant).
Results and interpretations
In this section I analysed the dynamics of the Gini coefficient in the European Union during 2002-2015, as well as its explanatory factors. Finally, I estimated the impact of government education and health spending and other control variables (the influence of the autoregressive term, the unemployment rate and the contribution of wages to GDP formation) on income inequality. sub-period, while 19 experienced increases. The most significant increases in income inequality were recorded in Croatia (2.86 deviation points), Slovenia (2.11 deviation points) ) and Spain (1.90 deviation points). On the other hand, all the 9 countries that have experienced cuts in the indicator in the second sub-period recorded insignificant decreases, below 1 deviation points. As can be seen, in most of the EU member states, the crisis had a negative impact on the income gap. However, the inclusive feature of the European integration model is also reflected in the positions occupied by the member states of the European Union in a ranking made by the Development Finance International in partnership with the international confederation of charities -Oxfam (the inequality reduction commitment index). As can be seen, the ranking (Table 2) is in line with the situation of income inequalities (Gini) in the member states of the European Union, the countries recording high levels of Gini index being also among the countries that make insufficient efforts to reduce income inequalities. The first countries in the world that make full efforts to reduce income inequality through government spending on health, education and social protection, progressive taxation and tax incidence, respectively labour market policies are Sweden, Belgium and Finland. On the other hand, Romania, Bulgaria and Lithuania occupy the last three places in the European Union from the point of view of the comittment to reduce income inequality.
The economic crisis has also played a decisive role in the dynamics of income inequality. According to Eurostat, in 2009, the EU-28 GDP fell by 4.3%, evolution accompanied by a large budget deficit of 6.6% of GDP, which required the adoption of fiscal consolidation policies by the member states. Although member states adopted austerity policies, many of these has not reduced the share of government spending on education and health in GDP. The dynamic of government health expenditure highlighted the same preference of member states for rising expenditure in the second sub-period, with only 5 countries of European Union opting for a decrease in this budgetary function: Hungary (-0.37% of GDP), Portugal (-0.27% of GDP), Greece (-0.26% of GDP), Bulgaria and Malta (both -0.13% of GDP). This preference of member states for increasing government health expenditure also facilitated an increase of the aggregate indicator at European Union level by 0.7% of GDP in the 2009-2015 sub-period, compared to the value recorded in the previous sub-period. The most significant increases in the second sub-period analysed were recorded in the Netherlands (+2.04% of GDP), Finland (+1.31% of GDP) and the UK (+1.31 of GDP). In 2015, Denmark, France and the Netherlands had the highest government health spending, while Cyprus, Latvia and Romania made the lowest government health spending from the European Union.
Some countries have laid the foundations of fiscal consolidation on reducing government social expenditures and thus, the income gap has further increased. However, unemployment had a strong influence on income discrepancies. Even if the unemployment rate at EU level in the analysed period has reached a maximum level in 2013 (10.93%), its level is still high (9.42% in 2015) and poses an important challenge for the European Union as a whole. In the post-crisis period (2009)(2010)(2011)(2012)(2013)(2014)(2015), the EU average unemployment rate was higher than in the previous period by 1.82 percentage points (from 8.08% to 9.90%).
According to Eurostat, in 2015 the highest unemployment in the European Union was recorded in Greece (24.9%), Spain (22.1%) and Croatia (16.1%), while the lowest level of this indicator was found in Germany (4.6%), Czech Republic (5.1%) and the United Kingdom (5.3%). Only in seven EU member states unemployment rate declined in the 2009-2015 subperiod compared to the previous one. The largest decline was found in Poland (-6.14 percentage points), followed by Germany (-3.61 percentage points) and Slovakia (-1.77 percentage points). On the other hand, the highest increases in unemployment were recorded in Spain (+12.21 percentage points), Greece (+11.11 percentage points) and Ireland (+8.23 percentage points). At first glance, it may be said that the rise in unemployment has led to the jobs loss of individuals with higher incomes too, and income inequality should not have changed. However, people who lost their jobs and previously earned better than other individuals, have been able to use their savings to generate a substitute income for their wage achieved in the past.
Unemployment also affected vulnerable groups, especially the people who have attained the International Standard Classification of Education taxonomic classes -ISCED 0-2 (less than primary, primary and lower secondary education) and ISCED 3-4 (upper secondary and postsecondary non-tertiary education). In this context, the EU-28 unemployment rate of the 15-74 age group who attained ISCED 0-2 increased in 2009 from the level recorded in 2008 by 3.2 percentage points, reaching 14.4%. On the other hand, the unemployment rate of the 15-74 age group with ISCED level 3-4 in the EU-28, increased by 1.8 percentage points, from 6.5% in 2008 to 8.3 % in 2009. Regarding the population with tertiary education (ISCED 5-8), the economic recession led to an increase in the unemployment rate by 1.1 percentage points in 2009 (4.9%), compared to the rate recorded in the previous year, which also argues the higher resilience to economic shocks of these categories of people.
As regards the share of wages in GDP, the indicator was constant during the analysed period and fluctuated around 37% of GDP. However, there were large discrepancies between the indicator recorded at the level of each member state. As a result, in 2015, the highest share of wages in GDP was recorded in Denmark (47.9% of GDP), Slovenia (41.7% of GDP) and Luxembourg (41.5% of GDP) while Ireland (24.8% of GDP), Greece (25.0% of GDP) and Romania (27.3% of GDP) had the weakest position of the indicator. In the second sub-period analysed, the largest increases of the contribution of wages to GDP formation compared to the previous sub-period were found in Bulgaria (+5.11% of GDP), Cyprus (+2.74% of GDP) and Finland (+2.56% of GDP). Contrariwise, Romania (-3.37% of GDP), Ireland (-2.43% of GDP) and Portugal (-1.87% of GDP), recorded the largest decline in the average of the indicator processed for the sub-period 2009-2015, compared to the previous sub-period.
As the increasing evolution of inequalities poses new challenges to the social dimension of the European Union, in the second phase of the research, I assessed the impact of the drivers of income inequalities on the Gini coefficient. Initially, I checked the stationarity of the variables included in the model (using the Summary method mentioned in the methodology -the lag being automatically chosen by Eviews software using the Schwarz information criterion), these being stationary at level and first difference, which required the inclusion of the Gini coefficient lagged by 1 year in the regression. Therefore, following the examination of the stationarity tests, I obtained the following results: Gini coefficient -stationarity identified at I(1); government expenditure on health as a share of GDP -stationarity identified at I(1); government expenditure on education as a share of GDP -stationarity identified at I(0); unemployment rate -the use of the Summary method did not provide a clear picture of the results (the number of tests that identified stationarity at I(0) was equal to the number of tests that identified stationarity at I(1)); Consequently, I applied the Hadri test for both level and first difference of unemployment rate. I rejected the stationarity assumption for I(0) as the probability of 0% was below the 5% threshold. The probability of 56% associated with Hadri Z-stat for I(1) argued the acceptance of the null hypothesis of stationarity. Finally, I have identified stationarity at I(0); contribution of wages to GDP formation -stationarity identified at I(1).
Next, I estimated the model, starting from the structure previously presented and I analysed its results. According to Figure 2, all variables used are statistically significant, their probability being less than the significance threshold of 5%. However, the risk of the estimator of the constant to be null is greater than 5% and less than 10% (8.21%), but this does not raise any questions with respect to the appropriate representation of the model given that the coefficient of determination (R-squared) is high, which demonstrates that the dynamic of the selected independent variables explains 99.72% of the dependent variable variance. Also, the low standard errors have increased the confidence in estimators. In order to check the validity of the model, I used the F-test and its associated probability created the premises for confiming the statistical validity of the model. The analysis of the impact coefficients of the variables included in the model is based on the "caeteris-paribus" hypothesis. According to the results of the model attached in Figure 3, the increase of the Gini coefficient recorded in the previous year by 1 deviation point leads to an increase of the actual Gini coefficient by 0.989 deviation points, this being also caused by the higher yield of higher incomes (people who earn higher income than other categories of people may use these additional resources to generate other types of income, a situation that lead to an increase in income gap).
Returning to the main focus of the analysis, I identified a negative impact of the increase in government education and health expenditure on the Gini coefficient in the European Union. Thus, I found that an increase in government health expenditure in the previous year by 1% of GDP leads to a decrease by 0.019 deviation points in the Gini coefficient, a lower impact than the one manifested through government education expenditure channel (an increase of 1% of GDP in government education expenditure lagged by one year led to a decline of 0.024 deviation points in Gini coefficient). Education offers to population the opportunity to alling with people who earn high income, within certain limits, through knowledge. Supporting education through higher funding, (one condition of it being the effectiveness of the structural reforms) can stimulate young people's insertion on labour market or their involvement in sustainable projects. However, the effects of the investments in education are not observed on the short term. This budget function generates new sources of income for the population by offering opportunities on the medium or long term, reason for which I included the 1-year lagged series on the list of the independent variables. This is also the case of government health expenditure that support education over time. In general, there is a significant link between education and health, since an educated person tends to take care of his health condition and a healthy person has the ability to continue his studies, not being constrained by factors related to weak health condition. Also, a healthy person does not face health barriers in the accession on the labour market and its performances are not conditioned or limited by the current condition of health. However, this is not a rule, the career success also depending on the severity of the health problem and other specific psychological factors.
According to Figure 2, an increase of the contribution of wages to the GDP formation by 1% of GDP leads to an increase of the Gini coefficient by 0.007 deviation points. This effect can be caused by the fact that in many countries from the European Union wage hikes occurred unequally, favouring the population earning high wages.
The estimate shows a positive impact of 0.011 deviation points on the Gini coefficient at a 1 percentage point increase in the unemployment rate, based on the higher impact of unemployment on vulnerable groups.
The coefficient of the constant term shows that, when all components of the equation remain constant, the Gini coefficient increases by 0.268 deviation points. However, given that the level of R-squared is high and the risk of the estimator to be null is greater than 5%, I ignored its coefficient as it can be inaccurate. As can be seen in Figure 3, the residuals are normally distributed, given that the probability of Jarque-Bera test is higher than 5% (52.46%).
In order to test the autocorrelation of the residuals, I did not use the Durbin-Watson test as it became invalid when I introduced the autoregressive term, as an exogenous variable, in the panel model. In this context, I performed the Breusch-Godfrey test (Annex 2), starting from equation (2) which includes 2 degrees of freedom (the number of lags for residuals) and a number of 308 observations (following the adjustments performed) and I obtained a R-squared value of 0.012849 and a n*R-square value of 3.957586. Using the CHISQ.DIST.RT function and previously processed data, I computed the probability of the Breusch-Godfrey autocorrelation test (13.82%), which argued the acceptance of the hypothesis according to which there is not autocorrelation between the residuals.
For enhancing the examination of the model's accuracy, I performed the cross-section dependence tests: Breusch-Pagan and Pesaran CD. Both the Breusch-Pagan (100.00%) and the Pesaran CD test (63.76%) are superior to the significance threshold of 5%, which confirms the absence of cross-section dependence.
The verification of heteroskedasticity (Annex 3) involved the estimation of the equation (3) and the computation of the probability of Breusch-Pagan-Godfrey test based on the n*Rsquared value (364 * 0.012786 = 4.6541049) and the degrees of freedom taken into consideration (5 -number of exogenous variables). Therefore, I accepted the hypothesis of homoskedasticity since the Breusch-Pagan-Godfrey probability of 45.95% is higher than 5%. Concerning multicoliniarity, I accepted the hypothesis related to its absence from the model, since the Pearson statistical correlations between the exogenous variables are lower than the coefficient of determination of the equation (1) -the Klein criterion.
Finally, I validated the model and its coefficients, given that there is no reason to have doubts on the maximum verisimilitude of the estimators.
Conclusions
This paper targeted the estimation of the impact of government health and education expenditure on income inequality. According to the results, a 1 percentage point increase in government health expenditure (express as a share of GDP) leads to a reduction in the Gini coefficient by 0.019 deviation points in the next year, while the same dynamic of government education expenditure causes a decrease of Gini coefficient by 0.024 deviation points in the next year. Therefore, this analysis confirms the inverse relationship between these two functions of budget expenditures and income inequality. The analysis of the EU member states' commitments to reduce inequalities has also confirmed this hypothesis. The model has proved to be statistically valid, all the tests performed in order to confirm the maximum verisimilitude of the estimators providing results that were in the normal parameters.
Supporting social cohesion through government spending on education and health is essential, but it is necessary to make it more effective by implementing structural reforms that bring both social and economic benefits outweighting the budgetary costs resulting from the implementation of these measures. Spending public money inefficiently can have negative consequences on the living standards of future generations, given that at some point the population will have to comply with their obligations resulted from high public debt, as a consequence of large budget deficits.
At both European Union and Romanian level, it is necessary to identify an optimal threshold of income inequality, a lower level than it -representing the natural income inequality and a higher level than the threshold -being the inequality induced by the national institutions and governments. In this context, for European Union member states it would be beneficial to assess the impact of budgetary measures on income distribution in the context of the annual budget proposals and to prevent the accumulation of disparities between incomes (for instance, Finland have such an approach, even if this activity is not mandated by law). Otherwise, there is a risk that the actual level of income inequality will continue on a increasing trend.
in the European Union on two sub-periods: 2002-2008 and 2009-2015. The second sub-period aims to surprise the evolution of indicators during the economic crisis and the first sub-period captures the evolution of the indicators in the pre-crisis period.
Figure 1 .
1Evolution of income inequality in EU-28Source: Own calculations using Standardized World Inequality DatabaseFigure 1 shows that there are large differences between member states of the European Union in terms of income inequality. According to the Standardized World Inequality Database, in the post-crisis period, the European Union experienced a trend of increasing income inequalities (at the end of 2015, the European Union average of the Gini coefficient was 30.0). Also, I would like to mention that for all indicators analysed, I have processed their final value at European Union level, taking into account the year of accession of each member state to European Union. In this context, the average Gini coefficient for the 2009-2015 sub-period increased by 0.67 deviation points compared to the one recorded in the previous sub-period (from 29.16 to 29.83). In 2015, the highest Gini coefficient was recorded in Latvia (36.8), Portugal (34.8) and Spain (34.3), while the lowest levels of the indicator have been observed in Denmark (25.4), Czech Republic (25.5), Slovakia (25.5) and Finland (25.5). In the 2009-2015 sub-period, only 9 states recorded decreases in the indicator compared to the 2002-2008
Annex 1 highlights the developments at European level in the2002-2008 and 2009-2015 subperiods regarding government expenditure on education and health, unemployment rate and wages. As regards government education expenditures in European Union, their share in GDP of 4.84% in 2015 was the lowest value in the analysed period. However, the average of the indicator for the 2009-2015 sub-period was higher than the one recorded in the previous subperiod by 0,08% of GDP. In 2015, Romania has recorded the lowest government spending on education (% of GDP), followed by Ireland, Bulgaria and Italy. On the other hand, the member states of the European Union spending the most on education are: Denmark, Sweden, Belgium and Finland. In the 2009-2015 sub-period, only 8 countries reduced the share of government education spending in GDP compared to the values recorded in the 2002-2008 sub-period, the most significant being: Hungary (-0.71% of GDP), Romania (-0.54% of GDP) and Poland (-0.53% of GDP). The most significant increases in education spending were found in Denmark (+0.63% of GDP), Luxembourg (+0.54% of GDP) and Belgium (+0.53% of GDP).
Figure 2 .
2Results of the modelSource: Own calculations using Eviews 9.0
Figure 3 .
3Distribution
Annex 1 .
1The dynamic of government health and education expenditure, wage share in GDP and unemployment rate in the EU-28
Table 1 .
1Statistical data series usedIndicators
Table 2 .
2The commitment to reducing inequality rank (152 countries)BE BG CZ DK DE EE IE EL ES FR HR IT CY LV LT LU HU MT NL AT PL PT RO SI SK FI SE UKCountry
Commitment to
reducing inequality
Health, education and
social protection
expenditure
Progressive structure
and incidence of tax
Labour market policies
to address inequality
Sweden
1
9
8
8
Belgium
2
4
3
24
Denmark
3
8
9
12
Germany
5
2
17
6
Finland
6
3
23
10
Austria
7
6
40
1
France
8
5
19
21
The Netherlands
9
19
13
9
Luxembourg
10
12
21
11
Ireland
13
1
53
19
Italy
16
17
14
29
Source:Own calculations using Eviews 9.0Series: Standardized Residuals
Sample 2003 2015
Observations 364
Mean
0.028024
Median
-0.050875
Maximum
2.702713
Minimum
-3.234350
Std. Dev.
0.989017
Skewness
0.038892
Kurtosis
3.281112
Jarque-Bera 1.290293
Probability
0.524586
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Autocorrelation of the residuals test -Breusch-Godfrey Source: Own calculations using Eviews 9. Annex 2Annex 2. Autocorrelation of the residuals test -Breusch-Godfrey Source: Own calculations using Eviews 9.0
Annex 3. Heteroskedasticity test -Breusch-Pagan-Godfrey Source: Own calculations using Eviews 9. Annex 3. Heteroskedasticity test -Breusch-Pagan-Godfrey Source: Own calculations using Eviews 9.0
| []
|
[
"Microcavity-enhanced Kerr nonlinearity in a vertical-external-cavity surface-emitting laser",
"Microcavity-enhanced Kerr nonlinearity in a vertical-external-cavity surface-emitting laser"
]
| [
"C Kriso \nFaculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n",
"S Kress \nFaculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n",
"T Munshi \nFaculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n",
"M Großmann \nInstitut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany\n",
"R Bek \nInstitut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany\n",
"M Jetter \nInstitut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany\n",
"P Michler \nInstitut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany\n",
"W Stolz \nFaculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n",
"M Koch \nFaculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n",
"A Rahimi-Iman \nFaculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n"
]
| [
"Faculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"Faculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"Faculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"Institut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany",
"Institut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany",
"Institut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany",
"Institut für Halbleiteroptik und Funktionelle Grenzflächen\nUniversität Stuttgart\nD-70569StuttgartGermany",
"Faculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"Faculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"Faculty of Physics\nMaterials Sciences Center\nPhilipps-Universität Marburg\nD-35032MarburgGermany"
]
| []
| Self-mode-locking has become an emerging path to the generation of ultrashort pulses with vertical-external-cavity surface-emitting lasers. In our work, a strong Kerr nonlinearity that is so far assumed to give rise to mode-locked operation is evidenced and a strong nonlinearity enhancement by the microcavity is revealed. We present wavelength-dependent measurements of the nonlinear absorption and nonlinear-refractive-index change in a gain chip using the Z-scan technique. We report negative nonlinear refraction up to 1.5⋅10 -11 cm 2 /W in magnitude in the (InGa)As/Ga(AsP) material system close to the laser design wavelength, which can lead to Kerr lensing. We show that by changing the angle of incidence of the probe beam with respect to the gain chip, the Kerr nonlinearity can be wavelength-tuned, shifting with the microcavity resonance.Such findings may ultimately lead to novel concepts with regard to tailored self-mode-locking behavior achievable by peculiar Kerr-lens chip designs for cost-effective, robust and compact fspulsed semiconductor lasers.The Kerr effect is at the basis of many important device concepts like all-optical switching 1 , optical limiting 2 and soliton mode-locking of lasers and microresonators 3,4 . The capability to accurately measure and model the nonlinear refractive index changes associated with the Kerr effect is crucial for improved device operation, where a specifically tailored nonlinear refractive index is required, e.g. for the intensity-dependent Kerr lensing. Several measurement schemes have been developed in the past for the characterization of the nonlinear refractive index 5-7 , with the Z-scan technique 8 being undoubtedly the most prominent one due to its simplicity and high sensitivity. This method continues to be of high experimental value with the rise of novel material classes like graphene and other two-dimensional semiconductors which often exhibit a very strong refractive nonlinearity 9-11 .Kerr-lens mode-locked Ti:Sapphire lasers have dominated the field of ultra-short high-power modelocked lasers since their initial discovery nearly three decades ago 12,13 . In these lasers, the intensitydependent refractive index of the gain crystal leads to self-focusing of the laser beam for high intensities. When part of the continuous-wave (cw) beam profile is suppressed by inserting a slit into C. Kriso et al.(2018)2 | 10.1364/oe.27.011914 | [
"https://arxiv.org/pdf/1808.03077v1.pdf"
]
| 119,070,410 | 1808.03077 | 658edd3d9bd83cb9d2d61ec721f3ba0f0d987a2d |
Microcavity-enhanced Kerr nonlinearity in a vertical-external-cavity surface-emitting laser
2018
C Kriso
Faculty of Physics
Materials Sciences Center
Philipps-Universität Marburg
D-35032MarburgGermany
S Kress
Faculty of Physics
Materials Sciences Center
Philipps-Universität Marburg
D-35032MarburgGermany
T Munshi
Faculty of Physics
Materials Sciences Center
Philipps-Universität Marburg
D-35032MarburgGermany
M Großmann
Institut für Halbleiteroptik und Funktionelle Grenzflächen
Universität Stuttgart
D-70569StuttgartGermany
R Bek
Institut für Halbleiteroptik und Funktionelle Grenzflächen
Universität Stuttgart
D-70569StuttgartGermany
M Jetter
Institut für Halbleiteroptik und Funktionelle Grenzflächen
Universität Stuttgart
D-70569StuttgartGermany
P Michler
Institut für Halbleiteroptik und Funktionelle Grenzflächen
Universität Stuttgart
D-70569StuttgartGermany
W Stolz
Faculty of Physics
Materials Sciences Center
Philipps-Universität Marburg
D-35032MarburgGermany
M Koch
Faculty of Physics
Materials Sciences Center
Philipps-Universität Marburg
D-35032MarburgGermany
A Rahimi-Iman
Faculty of Physics
Materials Sciences Center
Philipps-Universität Marburg
D-35032MarburgGermany
Microcavity-enhanced Kerr nonlinearity in a vertical-external-cavity surface-emitting laser
20181
Self-mode-locking has become an emerging path to the generation of ultrashort pulses with vertical-external-cavity surface-emitting lasers. In our work, a strong Kerr nonlinearity that is so far assumed to give rise to mode-locked operation is evidenced and a strong nonlinearity enhancement by the microcavity is revealed. We present wavelength-dependent measurements of the nonlinear absorption and nonlinear-refractive-index change in a gain chip using the Z-scan technique. We report negative nonlinear refraction up to 1.5⋅10 -11 cm 2 /W in magnitude in the (InGa)As/Ga(AsP) material system close to the laser design wavelength, which can lead to Kerr lensing. We show that by changing the angle of incidence of the probe beam with respect to the gain chip, the Kerr nonlinearity can be wavelength-tuned, shifting with the microcavity resonance.Such findings may ultimately lead to novel concepts with regard to tailored self-mode-locking behavior achievable by peculiar Kerr-lens chip designs for cost-effective, robust and compact fspulsed semiconductor lasers.The Kerr effect is at the basis of many important device concepts like all-optical switching 1 , optical limiting 2 and soliton mode-locking of lasers and microresonators 3,4 . The capability to accurately measure and model the nonlinear refractive index changes associated with the Kerr effect is crucial for improved device operation, where a specifically tailored nonlinear refractive index is required, e.g. for the intensity-dependent Kerr lensing. Several measurement schemes have been developed in the past for the characterization of the nonlinear refractive index 5-7 , with the Z-scan technique 8 being undoubtedly the most prominent one due to its simplicity and high sensitivity. This method continues to be of high experimental value with the rise of novel material classes like graphene and other two-dimensional semiconductors which often exhibit a very strong refractive nonlinearity 9-11 .Kerr-lens mode-locked Ti:Sapphire lasers have dominated the field of ultra-short high-power modelocked lasers since their initial discovery nearly three decades ago 12,13 . In these lasers, the intensitydependent refractive index of the gain crystal leads to self-focusing of the laser beam for high intensities. When part of the continuous-wave (cw) beam profile is suppressed by inserting a slit into C. Kriso et al.(2018)2
the cavity or reducing the pump spot on the crystal, an artifical ultrafast saturable absorber can be formed and can lead to very short pulse emission down to a few fs in duration 14,15 .
In recent years, a similar behavior has been observed in saturable-absorber-free mode-locked vertical-external-cavity surface-emitting lasers (VECSELs) [16][17][18][19][20] with very good mode-locking properties being reported for quantum-well-based (QW) 21 as well as for quantum-dot-based VECSELs 22 . These "self-mode-locked" VECSELs could lead to cheap and compact, high-power pulsed sources with sub-GHz to multi-GHz repetition rates for frequency metrology, spectroscopy and nonlinear imaging, rendering expenses for and limitations of saturable-absorber mirrors obsolete. Thus, one can expect self-mode-locking to open up new application scenarios for these techniques. However, the primary question which arose in this context yet remains unanswered, namely which effects are governing and promoting stable mode-locked operation with ultrafast sub-ps pulse emission.
Here, we present wavelength-dependent measurements of the nonlinear refractive index around the resonance of a multi-quantum-well VECSEL gain structure designed for lasing at around 960 nm. We perform Z-scan measurements in the range of 930 nm to 975 nm in order to reveal the interplay of nonlinear refraction and absorption in this spectral range. Changing the angle of incidence allows us to discuss the role of the longitudinal confinement factor, i.e. the cavity resonance, in the observed enhancement of the nonlinearity. We report a negative nonlinear refraction up to 1.5⋅10 -11 cm 2 /W in magnitude close to the laser design wavelength. This leads to a defocusing lens which is sufficient to perturb the cavity beam and give rise to Kerr-lens mode-locking. Figure 1a displays a possible cavity configuration to exploit this measured Kerr-lens for mode-locking where the defocusing lens in the gain chip leads to beam narrowing at the end mirror, thus, favoring mode-locking when a slit is inserted there. A similar cavity has been used in Ref. 21 to obtain self-mode-locking and has been investigated numerically in Ref. 23 to support in principle Kerr-lens mode-locking for a defocusing nonlinear lens in the gain chip.
Investigations on the self-mode-locking effect in VECSELs
The claim of the observed "self-mode-locking" behaviour being governed by Kerr-lens mode-locking 17 has been supported by investigations of the nonlinear refractive index in VECSEL gain structures 24-26 , reporting refractive index changes sufficiently strong to possibly perturb the cavity beam profile in a way to cause mode-locking. However, these measurements were exclusively performed at arbitrarily selected (experimentally available) single wavelengths. Such experiments were not taking into account the microcavity resonance as well as the strong dispersion of the nonlinear refractive index C. 3 around the band edge, which is characteristic for contributions to the nonlinear refractive index, both, from the bound-electronic Kerr effect (BEKE) existing below the bandgap 27 and from freecarrier-related nonlinearities (FCN) existing above the band edge 28 .
While the nonlinear optical properties of semiconductors below the band gap have been extensively studied and modeled using the Z-scan method 27,29,30 , nonlinear optical processes above and around the band gap have mostly been investigated by pump-probe techniques 28,31 or linear absorption measurements 32 . These methods cannot access the nonlinear refractive index directly but use nonlinear Kramers-Kronig relations to calculate it from nonlinear absorption spectra. Very few reports on direct measurements of the nonlinear refractive index properties of QW structures exist so far which solely can account fully for nonlinear pulse propagation of a single pulse through the medium 33,34 .
Although giving a rough estimate of the strength of the nonlinearity in multiple-quantum-well structures, these measurement results cannot arbitrarily be transferred to other sample designs as the nonlinear refractive index constitutes an effective one, composed of different material nonlinearites and their respective interaction containing both BEKE and FCN. Thus, the direct determination of the nonlinear refractive index by Z-scan measurements becomes inevitable whenever its precise value for a particular semiconductor heterostructure is of interest. We define the effective nonlinear refractive index 2 as
( ) = + ,(1a)
with ( ) being the total refractive index, 0 the intensity independent refractive index and the optical peak power density. Complementary to the definition of the intensity dependent refractive index 2 the nonlinear absorption can be defined as
( ) = + ,(1b)
with 0 being the linear absorption coefficient and ( ) the total absorption coefficient. It is important to point out that compared to 2 only comprises ultrafast
VECSEL chip design and Z-scan measurements
The investigated VECSEL chip sketched in For Z-scan data acquisition and modeling, we refer to the Methods section. The experimental setup is schematically shown in the Supplementary Information together with exemplary measurements. The angle of incidence of the probe beam on the VECSEL chip was varied in the course of the experiment from 10 to 20 and 30°, in order to reveal the influence of the microcavity on the measured nonlinearity. At each angle of incidence, Z-scan measurements were performed for different probe center wavelengths from 930 nm to 975 nm and varying probe intensites.
Experimental results and discussion
In order to determine very accurately the value of nonlinear absorption and refraction in the VECSEL chip, we performed Z-scan measurements for different probe-peak intensities at center wavelengths from 930 to 975 nm. If a refractive (or an absorbing) third-order nonlinearity is present, the nonlinear phase shift ΔΦ (or the normalized nonlinear absorption q 0 ) will vary linearly with peak probe power density. In such case, the strength of the nonlinearity is proportional to the slope of the nonlinear phase shift (or the normalized nonlinear absorption) with respect to the peak probe power density, Figure 2 for the incidence angles of 10, 20 and 30°.
For both nonlinear absorption and nonlinear refraction, a strong enhancement of the nonlinearity can be observed at wavelengths between 950 to 960 nm. Strikingly, this corresponds very well to the field enhancement at the QWs due to the microcavity resonance, which is represented by the surface photoluminescence peak at that on-chip angle and is also plotted into Figure 2a and b for comparison. When increasing the angle of incidence from 10 to 20 and further to 30°, the peak of the nonlinearity spectrum follows the corresponding surface PL spectrum peak, which arises from the longitudinal confinement factor in the microcavity, i.e. the Fabry-Pérot cavity's filter function. This unambiguously demonstrates that the microcavity resonance is strongly contributing to the enhancement of the respective nonlinearity. This enhancement takes place slightly below the linear absorption band edge, which can be estimated from the reflectivity spectra plotted in Figure 1d) as well as the different angle-dependent LCF peaks (cf. Figure S.1a). The corresponding surface PL at given angle of incidence, which is subdue to the microcavity resonance, is plotted in both graphs for comparison (shaded plots) as well as the corresponding reflectivity spectra (line plots), both normalized to 1 and with respect to the right axes. An inset represents the probe geometry with respect to the VECSEL chip. The errors were determined by performing a linear least square fit to the normalized nonlinear absorption 0 and nonlinear phase shift (as displayed in Figure S.3) and taking the 95% confidence interval of the fit as error.
The absolute magnitude of the effective nonlinear refractive index rises up to 1.5⋅10 -11 cm 2 /W, which is more than an order of magnitude larger than reported in Ref. 26 . This can be attributed to the fact that the authors did not use a probe wavelength close to the maximum of the microcavity resonance.
Moreover, the chip exhibited nearly half the number of quantum wells compared to our chip design.
The strong tunability of the nonlinearity with the microcavity resonance indicates that the (InGa)As/Ga(AsP) material system, which showed best self-mode-locking behavior so far 19 , can be possibly operated in the self-mode-locking regime over a large wavelength range exceeding 10 nm thus supporting both broad-band operation and tunability. This is consistent with the results of Ref. 33 for a multiple-quantum-well structure which showed a comparable bandwidth of strong nonlinear refraction (up to 8.5⋅10 -10 cm 2 /W in magnitude) where no microcavity effect was present.
In order to evaluate our measurement results with respect to possible self-mode-locking in VECSELs due to the Kerr effect, we plot the ratio of real and imaginary part of the third-order nonlinearity (3) ,
( ( ) ) ( ( ) ) = | | = | |,(4)
in Figure 3 (left axis). This describes how strong nonlinear refraction dominates over nonlinear absorption 8 . We only include wavelengths above 945 nm to cover mostly the laser-relevant wavelength range.
Here, a nearly constant ratio of real and imaginary part of (3) shows that both 2 and are equally affected by the cavity-resonance enhancement. In contrast to this, in pure quantum-well structures without any cavity, this ratio varies considerably as a function of the wavelength around the bandgap 33 . However, optical loss by two-photon absorption is quite significant in our measurements when rising up to more than 15% at the highest probe intensity in Supplementary Figure S.2c.
Nevertheless, this effect might be reduced in a pumped system, in which population inversion is maintained by the typical above-band high-power continuous-wave pumping scheme. Also, longer pulses -typical pulse lengths of self-mode-locked VECSELs are around 1 ps 21,22 -might reduce the occurrence of two-photon absorption 35 .
The typical focal length of a Kerr lens, which would occur in a self-mode-locked VECSEL cavity as reported in Ref. 21 , with the here measured value of nonlinear refraction is plotted in the same diagram (right axis of Figure 3). For sub-ps pulses with 1-kW peak power, the focal length of the induced Kerr nonlinearity should be in the order of magnitude of the laser cavity length, which ranges usually between 10 to 30 cm. Our calculated values are between a few cm up to 30 cm and more. This information supports the attribution of the observed self-mode-locking being Kerr-lens mode-locking and is key to designing a cavity where the angle of incidence of the laser mode on the gain chip is, apart from geometric constraints, a free design parameter in a V-or Z-cavity.
On the origin of nonlinear lensing and the implications for mode-locking
The respective contributions of FCN and BEKE to the total, measured nonlinearity will define the pulse-shaping mechanism in a Kerr-lens mode-locking scenario due to the different response times.
FCN has a slower response time, which is mostly due to the ps relaxation time of excited carriers, compared to BEKE, which can be regarded as instantenous with respect to fs pulses. Both FCN and BEKE exhibit characteristic dispersions when regarded separately as depicted in Figure 4a and c, with both effects showing strongest changes close to the band gap.
In the theory described by Ref. 27 , nonlinear refraction is inherently related to nonlinear absorption by the nonlinear Kramers-Kronig relations. Although in most resonant excitation-and-probe scenarios, carrier-related nonlinearities are believed to dominate the third-order nonlinear response 36 , bound-electronic contributions even continue to play a role closely above the band gap when no significant excitation or deexcitation of carriers occur 37 . Strong deexcitation of free carriers could be described with an additional use of the Bloch equations. This has been done in Ref. 38 and showed that a probe pulse will indeed experience a significant phase shift both from FCN and BEKE with the same order of magnitude when the semiconductor material is excited with a high-intensity pump pulse. These findings have also been supported by comparison to experiments 39 .
This indicates that also in mode-locked VECSELs a non-negligible amount of BEKE might be present.
The occurence of two-photon absorption in our measurements, also above the linear absorption edge, further supports this argument as it indicates that probe-power densities are sufficiently high to significantly perturb pulse propagation by ultrafast (3) effects, both from its imaginary part resulting in two-photon absorption as well as its real part resulting in nonlinear refraction, which as 10 absorber term in the common mode-locking model 42 (c.f. Figure 4b on pulse shaping). Additionally, self-phase modulation would have to be incorporated into the model, for which the value of the effective n 2 would be obtained from Z-scan measurements. As a consequence of this, the required group-delay dispersion (GDD) for shortest pulses and stable Kerr-lens mode-locking might differ from the slightly positive GDD values required for optimal SESAM-VECSEL mode-locking, which could become a key advantage of self-mode-locked devices towards ultra-short pulse generation after more detailed investigations triggered by the findings reported here.
In contrast, FCN might act as a slow saturable absorber with timescales similar to SESAMs and correspondingly similar mode-locking characteristics. It is important to point out that this would not be due to real saturable absorption occurring in the gain chip. Instead, the refractive-index change induced by changes in the excited-carrier densities would lead to nonlinear lensing and only lead to some kind of artificial saturable-absorber action (c.f. Figure 4b) when employed in an appropriate cavity, where nonlinear lensing reduces cavity losses, as indicated in Figure 1a.
If both effects were present simultaneously with comparable strength, a combined action of slow and fast saturable absorber would have to be incorporated into a mode-locking model. Also, a situation as in soliton mode-locking might occur in this case 3 .
A precise knowledge of the different timescales involved in the effective third-order refractive nonlinearity n 2 will ultimately enable us to explain and model the mechanisms behind Kerr-lens induced self-mode-locking, with important implications for future chip designs.
Conclusions
We have presented unique, systematic and direct measurements of the effective nonlinear refractive index change and nonlinear absorption in a vertical-external-cavity surface-emitting-laser chip using the Z-scan technique. In support of recent self-mode-locking achievements, strong enhancement of the Kerr nonlinearity is observed in the spectral region of the quantum-well band gap in clear correlation to a microcavity effect, which can shape Kerr lensing in the active region significantly.
Here, wavelength-tuning of the nonlinearity with the on-chip angle in accordance to the shift of the microcavity resonance is demonstrated. Negative nonlinear refraction up to 1.5⋅10 -11 cm 2 /W close to the laser design wavelength is obtained. While two-photon absorption poses a loss channel for lasing operation, the refractive index changes are sufficiently strong to perturb the cavity beam profile in favour of self-mode-locking. Moreover, by comparing our results to existing theories about the nonlinear-refractive-index changes in semiconductors, we believe that the ultrafast electronic Kerr effect contributes significantly to the nonlinearity while the share of free-carrier nonlinearities compared to the amount of the ultrafast Kerr effect yet remains to be investigated for a pumped system. We expect our results to lead to a better understanding of self-mode-locking in VECSELs and to pave the way to reliable, ultra-short, higher-power pulsed semiconductor-laser sources based on this method.
Methods
Fabrication of the gain chip:
The semiconductor structure used was fabricated by metal-organic vapor-phase epitaxy (MOVPE) in an AIXTRON FT 3x2" closed coupled shower-head reactor using the standard sources trimethylgallium, trimethyl-indium, trimethyl-aluminum, arsine and phosphine. Deposition occured at pressures of 100 mbar on 6° misoriented GaAs substrates. Following a 30 pair AlAs/GaAs DBR the GaAs spacer layer as well as the active region are grown and the whole structure is completed using a GaAs capping layer.
The active region comprises 20 (InGa)As quantum wells grouped into five packages which are separated using (AlGa)As spacer layers. The compressive strain induced by the QWs is mostly compensated by the surrounding tensile-strained Ga(AsP) barriers and the QW packages are placed according to a resonant periodic gain design at the antinodes of the standing wave electric field.
E-field simulations and longitudional confinement factor:
To calculate the electric-field distribution in the structure for a particular wavelength, we use the matrix formalism of Ref. 43
Z-scan setup:
For Z-scan measurements, the chip structure was mounted on a piece of copper without any active cooling. A wavelength tunable, mode-locked Ti:Sapphire laser emitting pulses of approximately 150 fs pulse length and a spectral width of about 8 nm was used to probe the chip structure at different center wavelengths around the design wavelength of the gain chip.
For each center wavelength, the pulse length was determined with a frequency-resolved optical gating (FROG) device and the laser spectrum was monitored with an optical spectrometer.
A lens with 50 mm focal length was used to focus the probe beam down to a spot size of 12 µm which was confirmed by a beam profile measurement around the focus of the probe beam. After reflection from the sample and a folding mirror, which were both mounted on a translation stage, a beam splitter and two detectors were used to record the position-dependent transmittance of the probe laser when the sample was moved through the focus of the lens. One detector was covered by a partially closed aperture, which was aligned for maximal transmission. The second detector collected the whole probe beam using a focusing lens. For further setup details and a schematic drawing of the experiment, we refer to the Supplementary Information.
Data modeling:
In order to extract the nonlinear absorption coefficient , we fit the open scan data with the model from Ref. 8, Additionally, the focal length, which would occur in a typically self-mode-locked VECSEL configuration, is plotted here in the right column. The error bars stem from the errors in nonlinear absorption and refraction as depicted in Figure 2. agreement with the linear fit shows that measured nonlinear absorption and nonlinear phase shift are indeed mainly caused by a third-order nonlinearity. It is important to note, that nonlinearrefractive-index changes caused by two-photon-absorption-generated free carriers would appear as an effective fifth-order nonlinearity 8 causing a deviation from the linear fit. Due to the good agreement to the linear fit, this mechanism can be neglected for our measurements.
( ) = ∑ [− ( )] ( + ) ∞ = ,(2)
Figure Captions
S3: Simultaneous presence of saturable absorption and two-photon absorption in open scan data
Saturable absorption, which becomes important above the linear absorption edge at wavelengths below 950 nm is phenomenologically included to the model applied on the open-scan data by adding the simple model for saturable absorption in transmission,
( ) = − + ( ) ,(S1)
to Eq. (2). Here, 0 is again the linear absorption at the respective center wavelength of the probe laser spectrum, and the saturation intensity which is determined by the fit. Eq. (S1) is normalized to 1 at the edges of the scan, added to Eq.
S4: Determination of fit uncertainty in open scan and Z-scan
To estimate the uncertainty in the determination of the fit parameter 0 or ΔΦ, we vary them in Eq.
(2) and (3)
( 3 ) 4 Figure 1 |
341effects, as carrier-related absorption would lead to saturation and thus more complex relation on incident intensity as detailed in the Supplementary Information. The goal of this work is the determination of this effective 2 of a VECSEL chip as well as of the nonlinear absorption acting as possible loss mechanism. C. Kriso et al. (2018) VECSEL-chip characteristics: (a) Illustration of a Kerr-lens mode-locked VECSEL with the dashed lines indicating the cw beam profile while the shaded area traces the beam profile when altered by the nonlinear defocusing lens in the VECSEL chip. Such an intensity-dependent lens can lead to beam narrowing at the end mirror of the cavity and favor modelocking when a slit inserted there adds losses to the cw beam by truncating it laterally. (b) Chip structure with standingwave electric field at the design wavelength of 960 nm. (c) Band-gap configuration of the different materials used in the gain chip compared to the stopband of the chip and the investigated wavelength range (930 -975 nm). (d) Measured reflectivity spectrum, surface and edge photoluminescence (PL), and calculated longitudional confinement factor (LCF) of the investigated VECSEL chip.
Figure 1b consists of 20 (InGa)As/Ga(AsP) quantum wells embedded within GaAs spacer layers to provide partial overlap of the antinodes of the standing wave electric field in the microcavity with the quantum wells. Subsequent 30 GaAs/AlAs distributed Bragg reflector (DBR) pairs provide the high reflectivity needed for efficient laser operation together with external mirrors. The chip is designed for lasing operation around 960 nm. Figure 1c shows the band gap alignment of the various materials composing the gain chip. It can be seen that the probe wavelengths, which range from 930 nm to 975 nm, are scanning across the quantum-well band gap, thus, our measurements are particularly sensitive to the nonlinear refractive index changes which are characteristic for near-resonance probing. The wavelength-dependent longitudinal confinement factor (LCF) characterizes the strength of a microcavity's standing-wave electric field at the location of the quantum wells (see Methods) and is depicted together with the reflectivity spectrum, the surface as well as the edge photoluminescence (PL) in Figure 1d. C. Kriso et al. (2018) 5
values for the nonlinear absorption and refraction 2 can be extracted. These are plotted in
Figure 2 6 Figure 2 |
262for comparison. We consider the contribution of the GaAs spacer layer negligible with respect to the contributions of the (InGa)As/Ga(AsP) gain region, as the theoretical nonlinear refractive index of GaAs 27 is in the order of -6⋅10 -13 cm 2 /W in the investigated wavelength range and thus significantly smaller than the effective n 2 values measured here. C. Kriso et al. (2018) Wavelength and angle-dependent nonlinear absorption and nonlinear refraction: (a) Nonlinear absorption and (b) nonlinear refraction 2 as a function of the wavelength (left axes), measured for the incidence angles of 10, 20 and 30°. The lines between the measurement points may serve as guides to the eyes. Arrows atop the diagram indicate the wavelength of the QW PL (cf.
Figure 3 |
3Ratio of real and imaginary part of the measured third order nonlinearity ( ) and typical focal length of a Kerr lens: The ratio of nonlinear refraction with respect to nonlinear absorption is plotted as a function of the wavelength in the left column of the diagram for the three different angles of incidence. Additionally, the focal length, which would occur in a typically self-mode-locked VECSEL configuration, is plotted here in the right column. The error bars stem from the errors in nonlinear absorption and refraction as depicted inFigure 2.
mentioned already above, are linked by nonlinear Kramers-Krönig relations. It is worth noting that also the authors of Ref.17 attribute their self-mode-locking results with a VECSEL to the ultrafast Kerr effect of bound electrons, without having performed any experimental characterization of the nonlinearity.However, our measurements did not investigate the effect of cw-pumping, which takes place in actual laser operation. The resulting generation of free carriers, which are probed by the laser pulse and lead to strong stimulated deexcitation of carriers might change the magnitude of the nonlinearity as well as the relative contributions from FCN and BEKE. Preliminary investigations concerning optical pumping and probe-induced nonlinear-refractive-index changes suggest little influence in the absolute value of the nonlinear refraction26 . The effect of the microcavity resonance on the nonlinearity enhancement revealed by this work will persist in a pumped system.To ultimately distinguish the possible two contributions from each other, that are the ultrafast bound-electronic Kerr effect (i.e. BEKE) and the non-instantaneous free-carrier nonlinearities (i.e. FCN), time-resolved Z-scan measurements would be required40 . In addition, the variation of the probe-pulse duration while maintaining equivalent peak powers could provide insight into the nature of the carrier dynamics including the effect of non-equilibrium many-body effects, which become important at sub-ps time-scales41 . This will be the subject of our future work.If BEKE is the dominating contribution to the nonlinear lensing, self-mode-locking of VECSELs has to be modeled by substituting the slow, highly nonlinear saturable absorber term with a fast saturable C.
Figure 4 |
4Illustration of possible saturable absorber mechanisms shaping the pulse: A dominating bound-electronic contribution (BEKE) in the nonlinear response, with its dispersion given in (a) as adapted from Ref.27 , would lead to a fast saturable absorber mechanism (red solid line in the schematic time trace of action shown in (b)) similar to traditional Kerrlens mode-locking. In contrast, a dominating free-carrier-related nonlinear response (FCN), with its dispersion around the band gap given in (c) for different carrier densities as adapted from Ref.28 , would lead to slow saturable absorber action (green dashed line in (b)) where pulse formation relies on its combination with fast gain saturation (black solid line in (b)).
. The overlap of the electric field with the position of the quantum wells is then described by the longitudional confinement factor, and − are the forward and backward traveling wave amplitudes of the vector potential of the electric field at the q th quantum well in the i th material layer, respectively.is the respective wave vector and the position of the q th QW. 0 + and 0 − are the vector potential amplitudes before the first material layer. The matrix formalism is solved for TE-polarization in accordance with the polarization of the probe laser.
-probe power density as a function of z and 0 , and 0 being the peak probe power density, the effective length of the sample and the Rayleigh length of the focused probe beam, respectively. The effective length = 1− − 0 takes into account the different strength of linear absorption 0 at different wavelengths. The total length is here the accumulated thickness of the different material layers including the penetration depth into the DBR and taking into account the double-pass through the chip with the respective angle of incidence. This leads to lengths L from 4.4 to 5.1 µm for angles of incidence from 10 to 30°. Saturable absorption, which becomes important at wavelengths below 950 nm, is taken into account with an additional term as detailed the nonlinear phase shift induced by the nonlinear refractive index 2 . The sign of the nonlinear phase shift can be easily determined by the position of peak and valley in the Zscan trace around the focus, which in our measurements always leads to a negative, thus defocusing, nonlinearity. emitting semiconductor lasers. Appl. Phys. B Lasers Opt. 75, M. et al. Temporal , spectral , and polarization dependence of the nonlinear optical response of carbon disulfide. Optica 1, 436-445 (2014).
Figure 1 |
1VECSEL-chip characteristics: (a) Illustration of a Kerr-lens mode-locked VECSEL with the dashed lines indicating the cw beam profile while the shaded area traces the beam profile when altered by the nonlinear defocusing lens in the VECSEL chip. Such an intensity-dependent lens can lead to beam narrowing at the end mirror of the cavity and favor mode-locking when a slit inserted there adds losses to the cw beam by truncating it laterally. (b) Chip structure with standing-wave electric field at the design wavelength of 960 nm. (c) Bandgap configuration of the different materials used in the gain chip compared to the stopband of the chip and the investigated wavelength range (930 -975 nm). (d) Measured reflectivity spectrum, surface and edge photoluminescence (PL), and calculated longitudional confinement factor (LCF) of the investigated VECSEL chip.
Figure 2 |
2Wavelength and angle-dependent nonlinear absorption and nonlinear refraction: (a) Nonlinear absorption and (b) nonlinear refraction 2 as a function of the wavelength (left axes), measured for the incidence angles of 10, 20 and 30°. The lines between the measurement points may serve as guides to the eyes. Arrows atop the diagram indicate the wavelength of the QW PL (cf. Figure 1d) as well as the different angle-dependent LCF peaks (cf.Figure S.1a). The corresponding surface PL at given angle of incidence, which is subdue to the microcavity resonance, is plotted in both graphs for comparison (shaded plots) as well as the corresponding reflectivity spectra (line plots), both normalized to 1 and with respect to the right axes. An inset represents the probe geometry with respect to the VECSEL chip. The errors were determined by performing a linear least square fit to the normalized nonlinear absorption 0 and nonlinear phase shift (as displayed inFigure S.3) and taking the 95% confidence interval of the fit as error.
Figure 3 |
3Ratio of real and imaginary part of the measured third order nonlinearity ( ) and typical focal length of a Kerr lens: The ratio of nonlinear refraction with respect to nonlinear absorption is plotted as a function of the wavelength in the left column of the diagram for the three different angles of incidence.
Figure 4 | 8 .
48Illustration of possible saturable absorber mechanisms shaping the pulse: A dominating boundelectronic contribution (BEKE) in the nonlinear response, with its dispersion given in (a) as adapted from Ref.27 , would lead to a fast saturable absorber mechanism (red solid line in the schematic time trace of action shown in (b)) similar to traditional Kerr-lens mode-locking. In contrast, a dominating free-carrier-related nonlinear response (FCN), with its dispersion around the band gap given in (c) for different carrier densities as adapted from Ref.28 , would lead to slow saturable absorber action (green dashed line in (b)) where pulse formation relies on its combination with fast gain saturation (black solid line in (b)).Supplementary InformationMicrocavity-enhanced Kerr nonlinearity in a vertical-external-cavity surface-emitting laserC. Kriso 1 , S. Kress 1 , T. Munshi 1 , M. Großmann 2 , R. Bek 2 , M. Jetter 2 , P. Michler 2 , W. Stolz 1 , M. Physics and Materials Sciences Center, Philipps-Universität Marburg, D-35032 Marburg, Germany 2 Institut für Halbleiteroptik und Funktionelle Grenzflächen, Universität Stuttgart, D-70569 Stuttgart, Germany S1: Angle-dependent VECSEL-chip properties Figure S.1| Angle-dependent chip characterization: (a) Measured reflectivity spectra for different angles of incidence (solid lines, left axis) as well as calculated longitudinal confinement factor (LCF) for the same angles (dashed lines, right axis). (b) Zoom on the longitudinal confinement factor (dashed lines) around the relevant wavelength range in comparison to the measured surface photoluminescence (PL) at the same angles (shaded areas). The angle-dependent characteristics of the Bragg reflector and the resonant-periodic-gain structure are displayed in Figure S.1. Figure S.1a shows the full stopband of the DBR for different angles of incidence together with the corresponding calculated longitudinal confinement factors. For an increasing angle of incidence, both the stopband and the cavity resonance experience a significant blue shift. C. Kriso et al. (2018) 19The surface PL spectra in theFigure S.1b (the same as inFigure 2a and b)are recorded by placing the collection optics (collimating and focusing lens aligned in a lens tube) at the respective angle of incidence with respect to the gain chip and by fiber coupling the signal to an optical spectrum analyzer. These spectra correspond well with the respective calculated longitudinal confinement factors, thus, validating the use of the surface PL for determing the spectral position of the microcavity resonance.S2: Experimental setup and exemplary dataFigure S.2 | Experimental Z-scan setup and data: Different neutral-density (ND) filters are used to systematically vary the incident probe power on the VECSEL chip. The sample is translated through the focus of the 150-fs pulsed probe beam, which is chopped and analyzed by two detectors using a lock-in scheme. (b) Exemplary Z-scan measurements at 955 nm for different peak probe intensities irradiated at an angle of 20° are shown, with the corresponding open scan recorded on detector 1 (c). The solid lines are the respective fits to the measurement data according to the models from Ref. The experimental setup is schematically shown in Figure S.2a. The sample moves through the focus of the mode-locked laser beam which can be attenuated by introducing neutral-density filters. After reflection from the VECSEL chip, detector 1 collects the total beam cross section and thus accounts only for nonlinear absorption. The transmittance as a function of the sample position will be referred to as open scan in the following. Detector 2, partially covered by an aperture, is sensitive to, both, nonlinear lensing and nonlinear absorption. The resulting trace will be referred to as closed scan. Following Ref. 8, we obtain the final Z-scan curve by dividing the closed scan by the open scan, performing a background subtraction with a low-intensity scan and normalizing to a transmittance of 1 for transmission at positions far away from the focus.
Figure S .
S2b shows exemplary Z-scan measurements at a center wavelength of 955 nm for different peak probe intensities at 20° angle of incidence together with the respective fits. They show good agreement between model and experiment including the symmetric decrease of peak and valley transmission for decreasing peak probe-power density. The decrease in transmission close to the focus in the open scan(Figure S.2c)is attributed to two-photon absorption.
Figure S. 3 |
3Coefficients for the extraction of nonlinear absorption and refraction: (a) Normalized nonlinear absorption 0 and (b) nonlinear phase shift as a function of the incident peak probe intensity for different probe center wavelengths irradiated at 20°. The errors for the respective extracted magnitudes represent the fit tolerance and were determined by varying the fit parameter ( 0 or ) until the fit model did not match the measurements anymore (similar to Ref. 44 ).
Figure S. 3
3shows, both, the normalized nonlinear absorption 0 (Figure S.3a) and the nonlinear phase shift ΔΦ (Figure S.3b) as a function of the incident peak power density. The very good
( 2 )
2and subtracted by 1 to align the total open-scan fit function to a transmission of 1 for z positions far away from the focus. An exemplary fit of an open scan at 940 nm for 20° angle of incidence is displayed in Figure S.4a. The resulting normalized nonlinear absorption (the positive one attributed to two-photon absorption) exhibits a good linear trend as shown in Figure S.4b, thus validating our method.
Figure S. 4 |
4Example of extraction of normalized nonlinear absorption in the presence of saturable absorption: (a) Open scan at 940 nm and 30° angle of incidence for different probe powers. The fit (solid line) is obtained as described in section 3 using =0.2 GW/cm 2 and 0 =1.2e-5 m -1 for all probe intensities. (b) Resulting nonlinear absorption as a function of the peak probe density with linear fit (solid line).
until there is a significant deviation from the measurement data. Upper and lower bounds are taken as equally spaced from the fit. An example of the limiting cases is displayed in Figures S.5a and b for open scan and Z-scan of Figure S.2. The bounds of 0 and ΔΦ are then taken into account in a weighted linear least square fit where the weights are the inverse of the respective variances meaning they are proportional to the inverse of the square of the error bounds. The resulting fits are plotted in Figure S.3. The subsequent error of the value of nonlinear refraction 2 or as displayed in Figure 2 is then the 95 % confidence interval of this fit.
Figure S. 5 |
5Measurement data with fit model as well as manually-determined symmetrical upper and lower bounds of the fit: This is a zoom in of Figure S.2b and c. (a) Open scan and (b) Z-scan with fit (solid line) and upper and lower limit of the respective fits (dashed lines) for the estimation of the data-evaluation uncertainties.
Authors' contributionsCorresponding [email protected]' statement/Competing interestsThe authors declare no conflict of interestAdditional informationSupplementary Information accompanies this paper
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[
"GAUGE EQUIVALENCE AND THE INVERSE SPECTRAL PROBLEM FOR THE MAGNETIC SCHRÖDINGER OPERATOR ON THE TORUS",
"GAUGE EQUIVALENCE AND THE INVERSE SPECTRAL PROBLEM FOR THE MAGNETIC SCHRÖDINGER OPERATOR ON THE TORUS"
]
| [
"G Eskin ",
"J Ralston ",
"\nDEPARTMENT OF MATHEMATICS\nUCLA, LOS ANGELES\n90095-1555CAUSA\n",
"\nIn memory of Mark Iosifovich Vishik\n\n"
]
| [
"DEPARTMENT OF MATHEMATICS\nUCLA, LOS ANGELES\n90095-1555CAUSA",
"In memory of Mark Iosifovich Vishik\n"
]
| []
| We study the inverse spectral problem for the Schrödinger operator H on the two-dimensional torus with even magnetic field B(x) and even electric potential V (x). V.Guillemin[11]proved that the spectrum of H determines B(x) and V (x). A simple proof of Guillemin's results was given by the authors in [3]. In the present paper we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of H determine the magnetic field and the electric potential. We also show that generically the spectrum and the magnetic field determine the "extended" gauge equivalence class of the magnetic potential. The proof is a modification of the proof in [3] with some corrections and clarifications. | 10.1134/s1061920813040043 | [
"https://arxiv.org/pdf/1312.4008v1.pdf"
]
| 119,165,407 | 1312.4008 | 237d5bb73ec0dfcff70be30fae4e85f171ca08d4 |
GAUGE EQUIVALENCE AND THE INVERSE SPECTRAL PROBLEM FOR THE MAGNETIC SCHRÖDINGER OPERATOR ON THE TORUS
14 Dec 2013
G Eskin
J Ralston
DEPARTMENT OF MATHEMATICS
UCLA, LOS ANGELES
90095-1555CAUSA
In memory of Mark Iosifovich Vishik
GAUGE EQUIVALENCE AND THE INVERSE SPECTRAL PROBLEM FOR THE MAGNETIC SCHRÖDINGER OPERATOR ON THE TORUS
14 Dec 2013
We study the inverse spectral problem for the Schrödinger operator H on the two-dimensional torus with even magnetic field B(x) and even electric potential V (x). V.Guillemin[11]proved that the spectrum of H determines B(x) and V (x). A simple proof of Guillemin's results was given by the authors in [3]. In the present paper we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of H determine the magnetic field and the electric potential. We also show that generically the spectrum and the magnetic field determine the "extended" gauge equivalence class of the magnetic potential. The proof is a modification of the proof in [3] with some corrections and clarifications.
Introduction
Let L = {m 1 e 1 + m 2 e 2 : m = (m 1 , m 2 ) ∈ Z 2 } be a lattice in R 2 . Here {e 1 , e 2 } is a basis in R 2 . We assume that the lattice L has the following property:
(1.1) For d, d ′ ∈ L, if |d| = |d ′ |, then d ′ = ±d.
Let L * = {δ ∈ R 2 : δ · d ∈ Z for all d ∈ L} be the dual lattice. We consider a Schrödinger operator of the form
(1.2) H = − i ∂ ∂x 1 − A 1 (x) 2 + − i ∂ ∂x 1 − A 2 (x) 2 + V (x), x ∈ R 2 ,
where A(x) = (A 1 (x), A 2 (x)) is the magnetic potential and V (x) is the electric potential. Let B(x) be the magnetic field,
(1.3) B(x) = curl A(x) = ∂A 2 ∂x 1 − ∂A 1 ∂x 2 . 1
We assume that B(x) and V (x) are periodic, i.e.
B(x + d) = B(x), V (x + d) = V (x), ∀d ∈ L,
i.e. B(x) and V (x) are smooth functions on T 2 = R 2 /L. We also assume that B(x) and V (x) are even, i.e. B(−x) = B(x) and V (−x) = V (x). Denote by G(T 2 ) the gauge group of complex-valued functions g(x) ∈ C ∞ (T 2 ) such that |g(x)| = 1. Any g(x) ∈ G(T 2 ) has the following form (1.4) g(x) = exp(2πiδ · x + iϕ(x)),
where δ ∈ L * and ϕ(x) is periodic, ϕ(x + d) = ϕ(x), ∀d ∈ L. The operator of multiplication by g(x) transforms the equation Hu = λu to the equation
H ′ u ′ (x) = λu ′ (x), where H ′ has the form (1.2) with A(x) replaced by A ′ (x), (1.5) A ′ (x) = A(x) − ig −1 (x)∇g(x) = A(x) + 2πδ + ∇ϕ(x), δ ∈ L * ,
and u ′ (x) = g −1 (x)u(x). The magnetic potentials A ′ (x) and A(x) related by (1.5) are called gauge equivalent. Since H and H ′ are unitarily equivalent, they have the same spectrum.
Let
(1.6) B(x) = β∈L * b β e 2πiβ·x
be the Fourier series expansion of B(x). We assume that the coefficient
(1.7) b 0 = |D| −1 D B(x)dx,
is not zero. Here D is a fundamental domain for the lattice L given by
(1.8) D = {t 1 e 1 + t 2 e 2 , |t j | ≤ 1 2 , j = 1, 2}
and |D| is the area of D. Note that if x ∈ D, then −x ∈ D. Given B(x), we let
(1.9) A(x) = A 0 (x) + a 0 + β∈(L * \0) a β e 2πiβ·x ,
where a 0 = (a 01 , a 02 ) is a constant,
(1.10) A 0 (x) = b 0 2 (−x 2 , x 1 )
and
(1.11) a β = b β (2πi) −1 (β 2 1 + β 2 2 ) −1 (−β 2 , β 1 ).
Note that curl A = B(x). Let A ′ (x) be any magnetic potential that is gauge equivalent to A(x).
Since curl ∇ϕ = 0, B ′ (x) = B(x),
where B ′ (x) = curl A ′ (x). Therefore A ′ (x) can be also represented in the form (1.9) with a ′ 0 not necessarily equal to a 0 . For the gauge equivalence of A ′ (x) and A(x), in addition to the equality of the magnetic fields, one needs (cf. (1.5)) (1.12) a ′ 0 − a 0 = 2πδ, for some δ ∈ L * . The main question in this paper is: For the class of magnetic potentials considered here, to what extent do the spectra of magnetic Schrödinger operators and the gauge equivalence classes of magnetic potentials determine the magnetic and electric fields?
In §2 we will describe the domain on which H is a self-adjoint operator with compact resolvent and hence has a discrete spectrum. Here we will describe the gauge equivalence classes of the magnetic potential assuming that the magnetic field B(x) is fixed.
Let γ j , j = 1, 2, be the basis of the homology group of the torus, given by γ j = {te j , 0 ≤ t ≤ 1}. Let
α j = γ j a 0 · dx = a 0 · e j ,
where a 0 is the constant vector in (1.9). For any d = m 1 e 1 + m 2 e 2 ∈ L we have a 0 · d = m 1 α 1 + m 2 α 2 , i.e. knowing {α 1 , α 2 } determines a 0 · d for any d ∈ L.
Let A ′ (x) be a magnetic potential of the form (1.9) with a 0 replaced by a ′ 0 . Define
α ′ j = γ j a ′ 0 · dx = a ′ 0 · e j .
Let {e * 1 , e * 2 } be the basis in L * dual to {e 1 , e 2 }, i.e. e j · e * k = δ jk . The potentials A(x) and A ′ (x) are gauge equivalent if and only if curl A = curl A ′ and (cf. (1.12)) (1.13) α ′ j − α j = (a ′ 0 − a 0 ) · e j = 2πδ · e j , j = 1, 2, for some δ ∈ L * . Short equivalent forms of (1.13) are e iα j = e iα ′ j , j = 1, 2, and (1.14) e ia 0 ·d = e ia ′ 0 ·d for any d = n 1 e 1 + n 2 e 2 ∈ L.
Changing x to −x we get the operator H ′ which is just H with a 0 changed to a ′ 0 = −a 0 . Note that H and H ′ have the same spectrum but their magnetic potentials are not gauge equivalent when a 0 = 0. Since we are looking for consequences of isospectrality, we introduce a weaker notion of gauge equivalence, namely (1.15) cos a 0 · d = cos a ′ 0 · d, ∀d ∈ L The condition (1.15) is equivalent to cos α j = cos α ′ j , j = 1, 2. Since
cos α j − cos α ′ j = 2 sin( α j − α ′ j 2 ) sin( α j + α ′ j 2 ),
cos α j = cos α ′ j implies that either α j − α ′ j or α j − α ′ j is an integer multiple of 2π. Thus there are two choices for each j: e ia 0 ·e j = e ia ′ 0 ·e j or e ia 0 ·e j = e −ia 0 ·e j . We will say that a ′ 0 and a 0 belong to the same "extended gauge equivalence class" if (1.15) holds. Thus for every extended gauge equivalence class of magnetic potentials, there are four choices of a 0 , including a ′ 0 = a 0 and a ′ 0 = −a 0 , giving distinct gauge equivalence classes when a 0 = 0.
Our first result gives conditions for the spectrum of H and gauge equivalence class of A to determine the fields:
Theorem 1.1. Let B(x)
, V (x) be periodic and even smooth functions, and assume L satisfies the condition (1.1). Suppose
(1.16) D B(x)dx = 2π.
Consider the spectrum of the Schrödinger operator H with A(x) having the form (1.9). Suppose that
(1.17) |B(x) − b 0 | < |b 0 |,where b 0 = 2π/|D|. Then the spectrum of H determines uniquely B(x)
and V (x) assuming that cos a 0 · d, d ∈ L, is given and It follows from Theorem 1.2 that if B(x), V (x) are fixed and the extended gauge equivalence classes of A(x) and A ′ (x) are different, i.e. if cos a 0 ·d = cos a ′ 0 ·d for some d ∈ L, then the corresponding operators H and H ′ have different spectra. This confirms the Aharonov-Bohm effect stating that different gauge equivalence classes have a different quantum mechanical effects, for example, the corresponding Schrödinger operators have different spectra(cf. [5]).
(1.18) cos a 0 · d = 0 for all d ∈ L.
The case when B(x) and V (x) are even and a 0 = 0 (cf. (1.9)) was proven in an important paper of Guillemin [11]. In [3] we reproved the result of [6] by a different and simpler method.
The method of the present paper is a modification of the method of [3] with some clarifications and corrections.
We mention briefly some related results on the inverse spectral problems in two and higher dimensions: The magnetic Schrödinger operator on T 2 with D B(x)dx = 0 and A(x) periodic was studied in [1]. In [10] Gordon et al. generalized [11] to the case of n-dimensional tori. Guillemin and Kazhdan [13] studied the inverse spectral problem for negatively curved manifolds. Guillemin [12] studied the inverse spectral problem on S 2 . Zeldich [18] solved the inverse spectral problem for analytic bi-axisymmetric plane domains. In [6] the inverse spectral problems on the torus for the Schrödinger operator −∆ + q(x) were studied. See also [4], [8], [16].
Gordon [7] and Gordon-Schuth [9] gave many interesting examples of isospectral manifolds which were not isometric.
The singularities of the wave trace
We introduce the "magnetic translation operators" (cf. [17])
(2.1) T j u(x) = e −iA 0 (e j )·x u(x + e j ), j = 1, 2,
where A 0 (x) is from (1.10). These operators are required to commute with each other and with H. This implies that
(2.2) A 0 (e 1 ) · e 2 = −A 0 (e 2 ) · e 1 = πl,
where l is an integer. Using (1.7), (1.10) we get that (2.2) is equivalent to
(2.3) D B(x)dx 1 dx 2 = 2πl. 5
Later we shall assume that l = 1. Having that T 1 , T 2 and H commute we denote by D 0 the subspace of the Sobolev space H 2 (R 2 ) consisting of u(x) ∈ H 2 (R 2 ) such that T j u = u, j = 1, 2. Then the operator H is self-adjoint in L 2 (D) on the restriction of D 0 to the fundamental domain D. We shall denote this operator by H D . Let λ 1 ≤ λ 2 ≤ λ 3 ≤ ... be the spectrum of H D . and let E D (x, y, t) be the fundamental solution for the wave equation on R 2 /L. Then the wave trace formula gives the equality as distributions in t
(2.4) ∞ j=1 cos t λ j = D E D (x, x, t)dx.
The distribution E D (x, y, t) is defined as follows: Let E(x, y, t) be the fundamental solution for the wave equation on R 2 :
∂ 2 E(x, y, t) ∂t 2 + HE(x, y, t) = 0, x ∈ R 2 , y ∈ R 2 , (2.5) E(x, y, 0) = 0, ∂E(x, y, 0) ∂t = δ(x − y), x ∈ R 2 , y ∈ R 2 . Then (2.6) E D (x, y, t) = (m,n)∈Z 2 T m 1 T n 2 E(x, y, t).
Note that
(2.7) T m 1 T n 2 E(x, y, t) = e −iA 0 (d)·x E(x + d, y, t), where d = me 1 + ne 2 .
We used in (2.7) that A 0 (e j ) · e j = 0. Since E(x, y, t) is singular only when |x−y| 2 = t 2 and since the condition (1.1) holds, the singularities of the trace (2.4) at t = |d|, d = m 1 e 1 + m 2 e 2 come only from two terms
(2.8) D (T m 1 T n 2 E(x, x, t) + T −m 1 T −n 2 E(x, x, t))dx.
To compute the singularities in (2.8) we will use as in [3] and [4] the Hadamard-Hörmander parametrix (cf. [14], [15]). We have
(2.9) E(x, y, t) = ∂ ∂t (E + (x, y, t) − E + (x, y, −t),
where E + (x, y, t) is the forward fundamental solution:
(2.10) ∂ 2 ∂t 2 + H E + (x, y, t) = δ(t)δ(x − y), E + (x, y, t) = 0 for t < 0. 6 It follows from [14] that Using that A 0 (d) · d = 0 and that A 0 (d) · x = −A 0 (x) · d we can rewrite (2.14) in the form
(2.11) E + (x, y, t) = m 0 (x, y) 1 2π (t 2 − |x − y| 2 ) − 1 2 + + m 1 (x, y)2 −2 π −1 (t 2 − (x − y) 2 ) 1 2 + + O((t 2 − (x − y) 2 ) 3 2 ), where (2.12) m 0 (x, y) = exp i 1 0 (x − y) · A(y + s(x − y))ds , (2.13) m 1 (x, y) = −m 0 (x, y) 1 0 V (y + s(x − y))ds + b(x, y) , where (2.14 ′ ) b(x, y) = − i ∂ ∂x − A(x) 2 m 0 m −1 0 (x, y). Let (2.14) I(d) = D exp i − A 0 (d) · x + 1 0 (A(x + sd) · d)ds dx(2.16) I(d) = D exp i 2A 0 (x) · d + a 0 · d + 1 0 (A 1 (x + sd) · d)ds dx, where (2.17) A 1 (x) = β∈L * \0 a β e 2πiβ·x ,
and a β are defined in (1.11).
Let {e * 1 , e * 2 } be the basis in L * dual to the basis {e 1 , e 2 }, i.e. (2.18) e * j · e k = δ jk , 1 ≤ j, k ≤ 2.
We shall construct e * j , j = 1, 2, explicitly. Let e ⊥ j = (−e j2 , e j1 ), Denote by ∆ the determinant e 11 e 12 e 21 e 22 . We assume that ∆ > 0. Note that ∆ is the area of the fundamental domain D : ∆ = |D|. Now we define
(2.19) e * 1 = − 1 ∆ e ⊥ 2 , e * 2 = 1 ∆ e ⊥ 1 .
We have d = me 1 + ne 2 = k(m 0 e 1 + n 0 e 2 ), where k ≥ 1, is an integer and m o , n 0 have no common factors. Then
A 0 (d) = b 0 2 d ⊥ = b 0 k 2 (m 0 e ⊥ 1 + n 0 e ⊥ 2 ) = b 0 k∆ 2 (m 0 e * 2 − n 0 e * 1 ) = b 0 k∆ 2 δ, where δ = −n 0 e * 1 + m 0 e * 2 . Using (1.7), (2.3) we get (2.20) A 0 (d) = πklδ. If β · d = 0, then 1 0 e 2πisβ·d ds = 0. When β · d = 0, we have β = pδ, p ∈ Z \ O, δ = −n 0 e * 1 + m 0 e * 2 .
We shall compute the inner product d · a pδ , where a β is given by (1.11). Note that d = k(m 0 e 1 + n 0 e 2 ), δ ⊥ = −n 0 (e * 1 ) ⊥ + m 0 (e * 2 ) ⊥ = 1 ∆ (n 0 e 2 + m 0 e 1 ) (cf. (2.19). Therefore
d · a pδ = kb pδ p|m 0 e 1 + n 0 e 2 | 2 ∆2πip 2 |δ ⊥ | 2 = kb pδ ∆ 2πip ,
since |δ ⊥ | 2 = 1 ∆ 2 |m 0 e 1 + n 0 e 2 | 2 . It follows from (1.7) and (2.3) that ∆ = |D| = 2πl b 0 . Hence
(2.21) d · a pδ = klb pδ ipb 0 .
Therefore I(d) has now the form
(2.22) I(d) = e ia 0 ·d D exp 2πikl((x · δ) + A 1 δ (δ · x)]dx, where (2.23) A 1 δ (δ · x) = p =0 b pδ 2πipb 0 e 2πip(δ·x) .
Potentials of the form (2.23) are called " directional potentials" in [6].
Note that d ds A 1 δ (s) = 1 b 0 B δ (s), where (2.24) B δ (s) = p =0 b pδ e ips 8
is a directional potential for B(x). ¿From here on we assume that l = 1 and set d 0 = 1 k d = m 0 e 1 + n 0 e 2 . Choose δ ′ ∈ L * so that (δ, δ ′ ) is a basis in L * and let (γ, γ ′ ) be the basis in L dual to (δ, δ ′ ) ∈ L * . We let D ′ be the fundamental domain for L with respect to the basis {γ, γ ′ } in the form {sγ + s ′ γ ′ , − 1 2 ≤ s, s ′ ≤ 1 2 }, and continue to let D be the fundamental domain for L from (1.8).
Setting x = sγ + s ′ γ ′ , we x · δ = s, x · δ ′ = s ′ .
Since the image of D is D ′ , using this change of variables we get (2.25) D exp 2πik((x·δ)+A 1 δ (x·δ))dx =
1/2 −1/2 1/2 −1/2 [exp 2πik(s+A 1 δ (s))]c 0 dsds ′ , where c 0 = ∂(x 1 ,x 2 ) ∂(s,s ′ )
is the Jacobian. Note that c 0 is the area of a fundamental domain for L. Therefore, integrating in s ′ we get
I(d) = c 0 1/2 −1/2 exp 2πik s + a 0 · d 0 2π + A 1 δ (s) ds.
When a 0 = 0 and A 1 δ (s) is an odd function of s then I(−d) = I(d) and the computations are simplified. When a 0 = 0 the spectral invariant is I(d) + I(−d) and we get, changing k to −k: δ · d = 0, the directional potential B δ is given by
B δ (x) = 1/2 −1/2 (B(x + sd) − b 0 )ds.
Hence, by the assumption |B(
x) − b 0 | < |b 0 | (2.27) max |B δ (x)| ≤ 1/2 −1/2 max |B(x) − b 0 |ds < |b 0 |.
Letting
(2.28) y = s + A 1 δ (s), 9 we have
dy ds = 1 + d ds A 1 δ (x) = 1 + B δ (s) b 0 > 0.
Therefore the inverse function s = s(y), y ∈ R 1 , is defined. Since y = y(s) is odd, the inverse function s = s(y) is also odd. Since y(s + 1) = y(s) + 1 we have s(y + 1) = s(y) + 1. Differentiating in y we get s ′ (y +1) = s ′ (y), i.e. s ′ (y) is periodic of period 1. The function s ′ (y) is even since s(y) is odd. Let e(y) = s(y) − y. Since s(y + 1) = s(y) + 1 we get that e(y + 1) = e(y), i.e. e(y) is periodic and odd. Note that s ′ (y) = 1 + e ′ (y).
After the change of variables s = s(y) in (2.26), we have
I(d) + I(−d) = 2c 0 1/2+A 1 δ (1/2) −1/2+A 1 δ (−1/2) cos(2πky + ka 0 · d 0 )s ′ (y)dy.
Since cos(2πky + ka 0 · d 0 )s ′ (y) is periodic and (cos ka 0 · d 0 )(cos 2πky)s ′ (y)dy
A 1 δ (1/2) = A 1 δ (−1/2) we get
As an even smooth function, s ′ (y) is a sum of its Fourier cosine series on (−1/2, 1/2). Suppose cos(ka 0 · d 0 ) is known and nonzero for all k ≥ 1. Then we know the Fourier cosine coefficients for k ≥ 1. This uniquely determines s ′ (y) up to a constant. Therefore s ′ (y) = C +s 1 (y), where 1/2 −1/2 s 1 (y)dy = 0. Since s(y) = y + e(y), where e(y) is periodic and odd we get s ′ (y) = 1 + e ′ (y). Knowing s ′ (y) we can find s(y) and subsequently A 1 δ (s) from the knowledge of the spectrum of H and cos(ka 0 · d 0 ), k ≥ 1. Repeating the same arguments for any d ∈ L we can recover A 1 (x). Now we can recover V (x) assuming that D V (x)dx = 0. One can check from (2.14 ′ ) that b(x, y) does not depend on a 0 : the only term in m 0 (x, y) (cf. (2.12)) that contains a 0 has the form e ia 0 ·(x−y) . Therefore y) will not contain a 0 . Therefore b(x, y) is known once we know A 1 (x). Since
−i ∂ ∂x m 0 (x, y) = (a 0 + c 1 (x, y))m 0 (x, y) where c 1 (x, y) is independent of a 0 . Hence (−i ∂ ∂x − A)m 0 (x, y) = (a 0 + c 1 (x, y) − (a 0 + b 0 2 x ⊥ + A 1 (x))m 0 (x,1 0 V (x + sd)ds = V δ (x), we have (2.31) J(d) = J 1 (d) + J 2 (d), where (2.32) J 1 (d) = D V δ (x · δ) exp[2πik(x · δ + a 0 · d 0 2π + A 1 δ (x · δ))]dx,
and J 2 (d) is the term containing b(x, y), i.e. J 2 (d) is known. Therefore J 1 (d) + J 1 (−d) is determined by the spectrum assuming that cos ka 0 · d is known.
Making the change of variables x = sγ + s ′ γ ′ as in (2.25) and integrating in s ′ we get
(2.33) J 1 (d) + J 1 (−d) = 2c 0 1/2 −1/2 V δ (s) cos 2πk(s + a 0 · d 0 2π + A 1 δ (s))ds.
Making the change of variables y = s + A 1 δ (s) as in (2.29) we get
(2.34) J 1 (d) + J 1 (−d) = 2c 0 1/2 −1/2 V δ (s(y))s ′ (y) cos(2πky + ka 0 · d 0 )dy,
where s = s(y) is the inverse to y = s + A 1 δ (s). Note that s ′ (y) is even periodic function of period 1, and V δ (s(y)) is also even periodic, since V (x) is even periodic and s(y) is an odd function satisfying s(y + 1) = s(y) + 1.
Since V δ (s(y))s ′ (y) is an even function, 1 −1 V δ (s(y))s ′ (y) sin 2πkydy = 0. Thus, as in (2.30) we have: (2.35)
J 1 (d) + J 1 (−d) = 2c 0 cos(ka 0 · d 0 ) 1/2 −1/2 V δ (s(y))s ′ (y) cos 2πkydy, k ≥ 1.
Knowing J 1 (d) + J 1 (−d) and cos(ka 0 · d 0 ) we know the Fourier cosine coefficients of the even function V δ (s(y))s ′ (y) for k ≥ 1. Therefore we can determine V δ (s(y))s ′ (y) up to a constant where s(y) = y + e(y), and e(y) is periodic with period 1. Thus we can recover V δ (s) for each δ ∈ L * and therefore we can recover V (x). This concludes the proof of Theorem 1.1. Now we shall prove Theorem 1.2. We shall assume that the magnetic field is generic in the following sense: There are two directions δ 1 and δ 2 which form a basis for L * such that the directional fields B δ 1 (s) and B δ 2 (s) are not identically zero. In this case the functions s 1 (x) and s 2 (x) (cf. (2.28)) corresponding to δ 1 and δ 2 respectively, are not identically zero. We make the additional generic assumption that a 1j = 0, j = 1, 2, where a kj are the Fourier cosine coefficients of s ′ j (y), k ≥ 0. Then from the main relation (2.30) we can recover cos a 0 · d j , j = 1, 2, where {d 1 , d 2 } is the dual basis to {δ 1 , δ 2 }. Given d ∈ L, there are are integers m and n such that d = md 1 + nd 2 . Hence cos a 0 · d = Re{e ia 0 ·(md 1 +nd 2 ) } = Re{(cos(a 0 ·d 1 )±i 1 − (cos(a 0 · d 1 ) 2 ) m (cos(a 0 ·d 2 )±i 1 − (cos(a 0 · d 2 ) 2 ) n }, and, since the ±'s disappear when one takes the real part, cos(a 0 · d) is determined by cos(a 0 · d 1 ) and cos(a 0 · d 2 ). Thus, cos a 0 · d = cos a ′ 0 · d for all d ∈ L as in (cf. (1.15)). This proves Theorem 1.2.
Theorem 1. 2 .
2Assume that the conditions (1.1), (1.16), (1.17) hold, and that the spectrum of H and the magnetic field B(x) are given. If B(x) satisfies a generic condition (stated in the proof ), then cos a 0 · d is determined for all d ∈ L, i.e. the exended gauge equivalence class of A(x) is determined by the spectrum and the magnetic field.
[
(V (x+sd)+b(x+sd, x)]ds exp −iA 0 (x+sd)·d)ds dx It follows from (2.8)-(2.13) that I(d) + I(−d) and J(d) + J(−d) are determined by the spectrum of H.
e
2πi(x+sd)·δ ′ ds = 0 if d · δ ′ = 0 and it is equal e 2πix·δ when
2πky + ka 0 · d 0 ))s ′ (y)dy.Since (sin 2πky)s ′ (y) is an odd function on (ka 0 · d 0 )(sin 2πky)s ′ (y)
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L Hörmander, The Analysis of Linear Partial Differential Equations, III. ViennaSpringer-VerlagL. Hörmander, The Analysis of Linear Partial Differential Equations, III, Springer-Verlag, Vienna, 1985
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Dynamics of electrons in solids in external fields. J Zak, Phys. Rev. 168J. Zak, Dynamics of electrons in solids in external fields, Phys. Rev., 168 (1968), 686-695
Spectral determination of analytic, bi-axisymmetirc plane domains. S Zeldich, Geometric Funct. Anal. 10S. Zeldich, Spectral determination of analytic, bi-axisymmetirc plane domains, Geometric Funct. Anal. 10 (2000), 628-677
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"Upper Covers of 2-Chains and of 2-Antichains in Sets of Indecomposable Subsets",
"Upper Covers of 2-Chains and of 2-Antichains in Sets of Indecomposable Subsets"
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"Bernd S W Schröder \nSchool of Mathematics and Natural Sciences\nThe University of Southern Mississippi\n118 College Avenue#5043, 39406HattiesburgMS\n"
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"School of Mathematics and Natural Sciences\nThe University of Southern Mississippi\n118 College Avenue#5043, 39406HattiesburgMS"
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| We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C ⊂ T such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains. | null | [
"https://arxiv.org/pdf/1803.02319v2.pdf"
]
| 119,589,056 | 1803.02319 | b0dbe529bd0cfecd7b48b36289674c25429e89da |
Upper Covers of 2-Chains and of 2-Antichains in Sets of Indecomposable Subsets
29 Nov 2018 December 3, 2018
Bernd S W Schröder
School of Mathematics and Natural Sciences
The University of Southern Mississippi
118 College Avenue#5043, 39406HattiesburgMS
Upper Covers of 2-Chains and of 2-Antichains in Sets of Indecomposable Subsets
29 Nov 2018 December 3, 2018arXiv:1803.02319v2 [math.CO]AMS subject classification (2010): 06A07 Key words: ordered set, indecomposable
We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C ⊂ T such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains.
Introduction
In [3], Schmerl and Trotter proved the following (in a more general context). Theorem 1.1 (See Theorem 2.2 in [3].) Let T be a finite indecomposable ordered set and let P ⊂ T be an indecomposable ordered subset of T with 4 ≤ |P | ≤ |T | + 2. Then there is an indecomposable ordered set U such that P ⊂ U ⊆ T and |U| = |P | + 2.
2-chains and 2-antichains satisfy the definition of indecomposability and, for them, Theorem 1.1 fails. Hence the requirement that |P | ≥ 4 is needed. This note characterizes all the ways in which Theorem 1.1 fails for 2-chains and for 2-antichains.
b ′ r ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ❏ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ✡ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ N Y X Z
Upper Covers of Nonadjacent 2-Chains
We first consider certain 2-chains C 2 = {a < b} such that a is not a lower cover of b. Note that, for the family X below, the ordered sets P such that there are a, b with (P, a, b) ∈ X include the ordered sets that are obtained as finite convex indecomposable subsets with at least 4 elements of the infinite ordered sets in [1] as well as the ordered sets obtained from the 3-irreducible ordered sets G n , J n and H n (see [4], p. 65) by deleting the elements c and d.
Definition 2. 1 We define the family X of triples (P, a, b) of a finite ordered set P , a minimal element a ∈ P and a maximal element b ∈ P by saying that (P, a, b) ∈ X iff one of the following hold.
1. P is an N and a and b are placed as in Figure 1.
2. There is a ( P , a, b) ∈ X such that P is obtained from P by attaching a as a new minimal element below P \ { a} and b = b.
3. There is a ( P , a, b) ∈ X such that P is obtained from P by attaching b as a new maximal element above P \ { b} and a = a.
We could immediately show that, if (P, a, b) ∈ X , then the only indecomposable subset of P that contains a and b is P itself. However, a direct argument is more technical than needed. Hence we delay this discussion until after the proof of Proposition 3.2. We start by proving that certain 2-chains {a, b}, in which the elements are not covers of each other, will be contained in sets P ⊆ T such that (P, a, b) ∈ X . The hypothesis is a bit technical, but the overall situation for 2-chains will be resolved in the proof of Proposition 3.2.
Lemma 2.2 Let T be a finite indecomposable ordered set with |T | > 2 and let C 2 = {a < b} ⊂ T be a chain with 2 elements such that T \{t ∈ T : a < t < b} is series-decomposable. Then C 2 is contained in a subset H of T such that (H, a, b) ∈ X and H is not isomorphic to N.
Proof. Let R := {t ∈ T : a < t < b} and suppose, for a contradiction, that T is a finite indecomposable ordered set with |T | > 2 such that the result does not hold and such that R is as small as possible.
Consider the ordered set S := T \ R. By assumption, S is seriesdecomposable. Hence, there are nonempty subsets L, U ⊂ S such that S = L ⊕ U. If b ∈ L, then T = (L ∪ R) ⊕ U, which is not possible. If a ∈ U, then T = L ⊕ (R ∪ U), which is not possible. Thus a ∈ L and b ∈ U. Because S contains no elements strictly between a and b, a is maximal in L and b is minimal in U. Moreover, because T is indecomposable, we conclude that R = ∅.
Let R a := {r ∈ R : r ≥ L} and let R b := {r ∈ R : r ≤ U}. Note that both sets are nonempty, because otherwise T would be series-decomposable into L ⊕ (R ∪ U) or into (L ∪ R) ⊕ U, respectively. Pick r a ∈ R a and r b ∈ R b such that, if R a ∩ R b = ∅, then r a = r b =: r. Then there is an a ′′ ∈ L such that r a ≥ a ′′ . Let a ′ ∈ L be a maximal element of L such that a ′ ≥ a ′′ . Then a ′ is not comparable to r a , because a ′ > r a implies a ′ > r a > a (contradicting maximality of a in L) and a ′ < r a implies r a > a ′ ≥ a ′′ (contradicting the choice of a ′′ ). Similarly, there is a b ′ ∈ U that is not comparable to r b and minimal in U.
If R a ∩ R b = ∅, then {a, a ′ , r, b, b ′ } is isomorphic to the ordered set X in Figure 1 and (X, a, b) ∈ X , a contradiction to the choice of T . For the remainder, we can assume that R a ∩ R b = ∅. That is, every element of R is below U or above L. Therefore, because T = L ∪ R ∪ U, every element of T is below U or above L. Now let a ∈ L be maximal in L and let b ∈ U be minimal in U, chosen so that R := {t ∈ T : a < t < b} is as small as possible. Note that R intersects neither L nor U, which means that R ⊆ R. Because R ′ := {t ∈ T : a ′ < t < b ′ } ⊆ R \ {r a , r b }, we have that | R| ≤ |R ′ | < |R|, which means R R and hence we have a = a or b = b.
Because every element of T is below U or above L, every element of T is comparable to a or to b. Because R ⊂ R, we have a < R\{ a} and b > R\{ b}. Let L := {t ∈ T \ R : t < b} and let U := {t ∈ T \ R : t > a}. Then a is maximal in L, b is minimal in U , and L ∩ U = ∅. Moreover, because every element of T is comparable to a or b, we have L ∪ U = T \ R.
Consider the case that a, b can be chosen so that T \ R is co-connected. If L < U, then we would have T \ R = L ⊕ U , which cannot be. Hence L < U , which means that there is a maximal ℓ ∈ L that is not comparable to a minimal u ∈ U . If ℓ ∈ L, then u > L, but u ∈ U also gives that a < u, so that, because u ∈ R, u is not smaller than b, implying that u < U, which contradicts the fact that u must be above L or below U. Similarly we exclude u ∈ U. Thus, ℓ, u ∈ R, and, in particular, ℓ = a and u = b. However, then N := { ℓ < b > a < u} is such that ( N , a, b) ∈ X , and subsequently ( N ∪ {a, b}, a, b) ∈ X , contradicting the choice of T .
Therefore, a, b can only be chosen so that T \ R is series-decomposable. Hence, by choice of T and because | R| < |R|, there is an ordered set P ⊆ T such that ( P , a, b) ∈ X . With P := P ∪ {a, b}, we have (P, a, b) ∈ X , independent of whether |P | = | P | + 1 or |P | = | P | + 2, a final contradiction to the choice of T .
Upper Covers of Adjacent 2-Chains
Now we prove that, if a chain in which both elements cover each other is contained in an indecomposable ordered set, then the ordered set must contain one of the ordered sets in Figure 2 or its dual.
Lemma 3.1 Let T be a finite indecomposable ordered set with |T | > 2 and let C 2 = {a < b} ⊂ T be a chain with 2 elements such that a is a lower cover of b. Then C 2 is contained in a subset of T that is isomorphic to one of the ordered sets in Figure 2 or to its dual.
Proof. First note that, if every strict upper bound of a is an upper bound of b, and if every strict lower bound of b is a lower bound of a, then C 2 is an order-autonomous subset of T , which is not possible. Thus, without loss of generality, there is a p ∈ T such that p > a and p ≥ b. (The dual case is not addressed in Figure 2 an upper cover of a, b is not comparable to any element x with a < x ≤ p.
❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ r r r r r r r r r a a a a a a a b b b b b b b x x x x x x x N N B B B B ′ B ′′ ℓ ℓ w w w w w v < 2 v < 2 w v 1 v 1 v 1 v 1 v < 2 v < 2 v < 2 ℓ ℓ ℓ ℓ ℓ u u u u u ❍ ❍ ❍ ❍ ❍ ❍ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✁ ✁ ✁ ✁ ✁ ✁ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ✦ ✦ ✦ ✦
In particular, a has an upper cover x = b. Let
C := {x ∈ T :
x is an upper cover of a} and note that |C| ≥ 2. Consider the case that there are x, y ∈ C that do not have the same strict lower bounds. Because we can, if needed, replace one of x, y with b and rename elements, we can assume that y = b. If b has a lower bound ℓ < b that is not a lower bound of x, then {ℓ < b > a < x} is an ordered set N, see Figure 2. If x has a lower bound ℓ < x that is not a lower bound of b, then {ℓ < x > a < b} is an ordered set isomorphic to N, but, compared to Figure 2, the positions of b and x are interchanged. Thus, from here on, we can assume that any two x, y ∈ C have the same strict lower bounds.
The argument above shows two characteristics of this proof. First, we will continue to add hypotheses on the set T , typically indicated by "from here on, we can assume." These hypotheses typically are regarding the comparability of sets. In Figure 2, elements of sets that will be introduced in the future will be denoted with the corresponding lowercase letter and, possibly, a superscript. Second, although b is an element of C 2 and x will be used to denote a generic upper cover of a that is not equal to b, the roles of x and b in the sets depicted in Figure 2 will be interchangeable, similar to the two versions of N above. This interchangeability will be indicated as needed.
Because any two x, y ∈ C have the same strict lower bounds (and because any element of C is above a), any ℓ ∈ T that is not comparable to either of a or b is not comparable to any element of C.
Suppose, for a contradiction, that, for every upper bound w ∈ A :=↑ a \ {a}, every lower bound ℓ < w is comparable to a or b. Let p ∈ T \ A be such that p is comparable to an element of A. By definition of A, there is a w ∈ A such that p < w. By assumption, p is comparable to a or b. By definition of A, we have p ≤ a or p < b. In either case, because any two elements of C have the same strict lower bounds, we infer that p < C, and hence p < A. We conclude that A is nontrivially order-autonomous in T , a contradiction. Thus there is an upper bound w ∈ A =↑ a \ {a} that has a lower bound ℓ < w that is not comparable to either of a or b. Let
W := {w ∈ T : w > a ∧ (∃ℓ < w)ℓ ∼ a ∧ ℓ ∼ b} = ∅.
Because any two x, y ∈ C have the same strict lower bounds, we have that W ∩ C = ∅ and that any ℓ as in the definition of W is not comparable to any element of C. Also note that any element that is above an element of W must be an element of W .
If there is a w ∈ W such that w > C, let ℓ < w be not comparable to either of a and b and choose x ∈ C such that w is comparable to one of b and x, but not the other. Because ℓ is not comparable to any element of C, we have that {a, b} is contained in an ordered set isomorphic to N , see Figure 2 for the case w > b (as indicated earlier, the case w > x is obtained by switching b and x). Thus, from here on, we can assume that every w ∈ W is comparable to all elements of C, that is, W > C.
No two x, y ∈ C can have the same strict upper bounds, because, otherwise, {x, y} would be order-autonomous in T , which cannot be. Hence, for any two x, y ∈ C, there is a u ∈ T such that u is a strict upper bound of one, but not the other. Let
U := {u ∈ T : (∃x ∈ C)[u > b ∧ u > x] ∨ [u > x ∧ u > b]} = ∅.
Note that, because W > C, we have that U ∩ W = ∅. If there is a u ∈ U that is not comparable to an element w ∈ W , let x ∈ C be such that u is above one of b and x, but not the other. Then {a, b} is contained in an ordered set isomorphic to B, see Figure 2 for the case u > b (again, the case u > b is obtained by switching b and x). Thus, from here on, we can assume that U < W . Next, we claim that
U = {u ∈ T : (∃x, y ∈ C)u > x ∧ u > y}.
The containment "⊆" follows from the definition. For the containment "⊇," Let t ∈ T and x, y ∈ C be such that t > x and t > y. In case t > b, we conclude that t ∈ U because of the presence of y, and, in case t > b, we conclude that t ∈ U because of the presence of x. This proves the equality.
Let
V 1 := {v 1 ∈ T \ W : v 1 > C, v 1 > U}.
Note that no element of U can be greater than or equal to any element of V 1 . Suppose, for a contradiction, that V 1 = ∅. Let H := C ∪ U and let p ∈ T \ H be comparable to an h ∈ H. If p < h, then, by definition of H and because U ∩ W = ∅, c is below an upper cover of a and hence p < C and then p < H. If p > h, then p is above an element of C. Because p ∈ U, we obtain p > C, and then, because V 1 = ∅ and W > U, we have p > U, which means that p > H. We conclude that H is nontrivially order-autonomous in T , a contradiction. Thus V 1 = ∅. Note that, because W > U, no element of V 1 is above any element of W .
If V 1 < W , then there are a v 1 ∈ V 1 and a w ∈ W that are not comparable. Because v 1 > U, there is a u ∈ U that is not comparable to v 1 . For this u ∈ U, there is an x ∈ C such that u is above one of b and x, but not the other. This means that {a, b} is contained in an ordered set isomorphic to B, see Figure 2 for the case that u > b (the case u > b is obtained by switching b and x). Thus, from here on, we can assume that V 1 < W .
Let
V 2 := {v 2 ∈ T \ W : v 2 > U}.
Note that no element of V 2 is above any element of W and that no element of V 1 is above any element of V 2 . Moreover, because V 2 ∩ U = ∅, we have V 2 > C. Suppose, for a contradiction, that V 1 < V 2 , and consider the set H := C ∪ U ∪ V 1 . Let p ∈ T \ H. If there is an h ∈ H such that p > h, then p is above an element of C, so, because p ∈ U, we have p > C, because p ∈ V 1 and W > U, we have p > U, and hence p ∈ V 2 ∪ W > H. If there is an h ∈ H such that p < h, then (because h ∈ W ) p is below an upper cover of a.
Hence p < C, which implies p < H and H is nontrivially order-autonomous, a contradiction. Thus,
V 1 < V 2 . Let V < 2 := {v 2 ∈ V 2 : v 2 < W } V < 2 := V 2 \ V < 2
Note that no element of V < 2 is below any element of V < 2 . Now consider the case that there is a v < 2 ∈ V < 2 that is not an upper bound of V 1 . Then there is a w ∈ W that is not above v < 2 , there is a v 1 ∈ V 1 that is not below v < 2 (and hence not comparable to it) and there is a u ∈ U that is not comparable to v 1 . Consequently, using the comparabilities between the various sets that are already established, we conclude that {a, b} is contained in an ordered set isomorphic to B, see Figure 2 for the case that u > b (the case u > b is obtained by switching b and x). Thus, from here on, we can assume that
V 1 < V < 2 . Because V 1 < V 2 , this means that V 1 < V < 2 . Suppose, for a contradiction, that V < 2 < V < 2 , and consider the set H := C ∪ U ∪ V 1 ∪ V < 2 . Let p ∈ T \ H.
If there is an h ∈ H such that p > h, then p is above an element of C, so, because p ∈ U, we have p > C, because p ∈ V 1 and W > U, we have p > U, so, because p ∈ V < 2 , we have p ∈ V < 2 ∪ W > H. If there is an h ∈ H such that p < h, then (because h ∈ W ) p is below an upper cover of a. Hence p < C, which implies p < H and H is nontrivially order-autonomous, a contradiction. Thus, V < 2 < V < 2 , which means that there are v < 2 ∈ V < 2 and v < 2 ∈ V < 2 that are not comparable. In particular, and this is all that will be used in the following, this means that neither set is empty.
Let v < 2 ∈ V < 2 be such that there is a v 1 ∈ V 1 that is not below v < 2 (and hence not comparable to it). Then there is a u ∈ U that is not comparable to v 1 .
First, consider the case that there is a v < 2 ∈ V < 2 that is not comparable to v < 2 . Then there is a w ∈ W that is not above v < 2 . Consequently, using the comparabilities between the various sets that are already established, we conclude that {a, b} is contained in an ordered set B ′ , see Figure 2 for the case that u > b (the case u > b is obtained by switching b and x).
Finally, if this is not the case, then there is a v < 2 ∈ V < 2 that is greater than v < 2 . Then there is a w ∈ W that is not above v < 2 . Consequently, using the comparabilities between the various sets that are already established, we conclude that {a, b} is contained in an ordered set B ′ , see Figure 2 for the case that u > b (the case u > b is obtained by switching b and x).
We can now summarize the situation for 2-chains.
Proposition 3.2 Let T be a finite indecomposable ordered set with |T | > 2, let C 2 = {a < b} ⊂ T be a chain with 2 elements and let H be an indecomposable ordered subset of T that properly contains C 2 such that there is no indecomposable ordered subset U of T with C 2 U H. Then H is isomorphic to one of the ordered sets in Figure 2 or H is isomorphic to one of their duals, or (H, a, b) ∈ X .
Proof. Let S = H \ {t ∈ H : a < t < b}. If S is series-decomposable, the statement follows from Lemma 2.2. If S is co-connected, then C 2 is not contained in a nontrivial order-autonomous subset of S (otherwise, H would be decomposable). We conclude that C 2 is contained in an indecomposable subset H ′ of S, and hence of H, that is isomorphic to the index set of the canonical decomposition of S. In particular, this means that H ′ is not a 2chain and, because a is a lower cover of b in S, that a is a lower cover of b in H ′ . Therefore, by Lemma 3.1, C 2 is properly contained in an indecomposable subset of H ′ ⊆ H that is isomorphic to one of the ordered sets in Figure 2 or to one of their duals. The statement now follows from the fact that this subset cannot be properly contained in H.
It can be checked that none of the isomorphism types of the ordered subsets H in Proposition 3.2 can be omitted: One can prove that, for any two chains C = {a < b} and C ′ = {a ′ < b ′ } and for ordered sets H (for C) and H ′ (for C ′ ) as in Proposition 3.2, there is no embedding of H into H ′ that maps a to a ′ and b to b ′ . Such an argument is tedious, but can essentially be done "by inspection." Because of none of the isomorphism types of the ordered subsets H in Proposition 3.2 can be omitted, all bounds in Corollary 3.3 below are sharp. Figure 3.
✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ◗ ◗ ◗ ◗ ◗ ◗ ℓ ℓ a a b b · · · · · · h h d d ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙
The name "indecomposable V-cover" comes from the following facts. Clearly a, b and ℓ form a fence that looks like a "V." Moreover, it is easy to check that every indecomposable V-cover is indeed indecomposable. Finally, we can easily see that, if C is an indecomposable V-cover, then the only indecomposable subset I ⊆ C that properly contains {a, b} is C itself: Because such a subset I must be connected, it must contain ℓ. Now, if any element of F \ {b} were not contained in I, then, with B being the connected component of F ∩ I that contains b, we would have that {a} ∪ B is nontrivially order-autonomous in I, which cannot be. Hence {a, ℓ} ∪ F ⊆ I, and, because {a, ℓ} ∪ F is series-decomposable, we must have d ∈ I and hence C = I. Proposition 4.2 Let T be a finite indecomposable ordered set with |T | > 2 and let A 2 = {a, b} ⊂ T be an antichain with 2 elements. If d(a, b) > 2, then any smallest indecomposable ordered subset I of T that contains A 2 is a fence from a to b with d(a, b) + 1 elements. If d(a, b) = 2, then any smallest indecomposable ordered subset I of T that contains A 2 is either a fence with at least 4 elements, or it is isomorphic or dually isomorphic to an indecomposable V-cover in which a and b are as in Definition 4.1 (see Figure 3) or in which the roles of a and b are interchanged.
Proof. The case d(a, b) > 2 is already discussed at the start of this section. We are left to consider the case d(a, b) = 2.
Let I be a smallest indecomposable ordered subset of T that contains A 2 . If I does not contain a common upper or lower bound of a and b, then an argument similar to the argument at the start of this section shows that I is a fence. (Surprisingly, this case can occur: Consider N = {a < f 2 > f 3 < b} with an additional element ℓ < a, b added.) This leaves the case that I contains a common upper or lower bound of a and b. Without loss of generality, assume that I contains a common lower bound of a and b. (The other case is handled with the dual argument.)
Let L := {x ∈ I : x < a, b} = ∅, let U := {x ∈ I : x > a, b} and let H := {x ∈ I : L < x < U} ⊇ {a, b} be the set of all elements between L and U in I. Let A be the connected component of H that contains a and let B be the connected component of H that contains b. Note that A could be equal to B. If |B| > 1, then, because B cannot be order-autonomous in I, there must be an element in B that has a strict upper or lower bound in I that is not in L ∪ B ∪ U. Similarly, if |A| > 1, there is an element of A that has a strict upper or lower bound in I that is not in L ∪ A ∪ U. Finally, in case |A| = |B| = 1, because {a, b} cannot be order-autonomous in I, a must have a strict upper or lower bound in I that is not in L ∪ A ∪ U, or b must have a strict upper or lower bound in I that is not in L ∪ B ∪ U. Note that all these upper and lower bounds are not in L ∪ H ∪ U. Moreover, note that the existence of a strict upper bound s for an element of A (or B) such that s ∈ L ∪ H ∪ U implies U = ∅, because otherwise s would be in A (or B). Because we can switch the roles of a and b, and because we can work with the dual ordered set if needed, without loss of generality, we can assume that B contains an element that has a strict lower bound in I that is not in L ∪ B ∪ U.
Let s be the shortest distance in B from the element b to an element of B that has a strict lower bound in I that is not in L ∪ B ∪ U. In case a ∈ B, because the roles of a and b can be switched, we can assume that the distance in B from the element a to any element of B that has a strict lower bound in I that is not in L ∪ B ∪ U is at least s. Let F ⊆ B be a fence of length s in B that goes from b to an element h ∈ B that has a lower bound d ∈ I \ (L ∪ B ∪ U). Because a and b do not have common upper or lower bounds in B, F has length s and F is contained in B, we conclude that a is not comparable to any element of F ∪ {d} and that d is not comparable to any element of F \ {h}. Because d ∈ L, no element of L is above d.
Because d < h, we have d < U. Because d < h ∈ B, d ∈ B ⊆ H and B is a connected component of H, the point d cannot be in H. Because d < U, and d ∈ H, the point d cannot be above all elements of L. Therefore, there is an ℓ ∈ L that is incomparable to d. This means that C := {a, b, d, ℓ} ∪ F ⊆ I is an indecomposable V-cover that contains a and b. Because I is smallest possible, we obtain that I = C.
Figure 1 :
1The first four elements (P, a, b) in the class X from Definition 2.1. We have (N, a, b) ∈ X by part 1 of Definition 2.1 and, for H ∈ {X, Y, Z}, (H, a, b) ∈ X is obtained by applying part 2 or 3 of Definition 2.1 to the set to the left of H. The labeling of X is used in the proof of Lemma 2.2.
, but, obviously, runs along similar lines.) Because b is
Figure 2 :
2The forbidden sets in Lemma 3.1, with one possible placement of b and x indicated. The only other possible placement of b and x is obtained by b taking the place of x and vice versa and keeping all other points fixed. The labelings in the sets correspond to cases in the proof of Lemma 3.1.
Corollary 3. 3
3Let S be a finite ordered set and consider the set I of all indecomposable subsets of S, ordered by containment. Let C ⊂ S ba a 2chain and let U be an upper cover of C in I. If C = {a < b} and K is the longest chain from a to b, then |U| ≤ max{2|K|, 9}.Proof. The statement follows from Proposition 3.2.
Figure 3 :
3Visualization of indecomposable V-covers, see Definition 4.1. 4 Upper Covers of 2-Antichains in Sets of Indecomposable Ordered Subsets Definition 4.1 An ordered set C is called an indecomposable V-cover iff C has exactly two minimal elements, ℓ and d, the set ↑ ℓ \ {ℓ} is composed of exactly two connected components, which are a singleton {a} and a fence F from an element b to an element h such that h is maximal in F (with the equality b = h being allowed); and we have ↑ d \ {d} = {h}. The two types of indecomposable V-covers are depicted in
Linear extensions of infinite posets. G Brightwell, Discrete Mathematics. 70G. Brightwell, Linear extensions of infinite posets, Discrete Mathematics 70 (1988), 113-136
Ordered Sets -An Introduction with Connections from Combinatorics to Topology. B Schröder, Birkhäuser VerlagBoston, Basel, Berlinsecond editionB. Schröder, Ordered Sets -An Introduction with Connections from Combinatorics to Topology (second edition), Birkhäuser Verlag, Boston, Basel, Berlin, 2016
Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures. J H Schmerl, W T Trotter, Discrete Mathematics. 113J.H. Schmerl and W.T. Trotter, Critically indecomposable partially or- dered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993), 191-205
Combinatorics and partially ordered sets: dimension theory. W T Trotter, Johns Hopkins University PressBaltimoreW. T. Trotter, Combinatorics and partially ordered sets: dimension theory, Johns Hopkins University Press, Baltimore, 1992
| []
|
[
"γγ decay as a probe of neutrinoless ββ decay nuclear matrix elements",
"γγ decay as a probe of neutrinoless ββ decay nuclear matrix elements"
]
| [
"B Romeo \nDonostia International Physics Center\n20018San SebastiánSpain\n\nLaboratorio Subterráneo de Canfranc\n22880Canfranc-EstaciónSpain\n",
"J Menéndez \nDepartment of Quantum Physics and Astrophysics and Institute of Cosmos Sciences\nUniversity of Barcelona\n08028BarcelonaSpain\n",
"C Peña Garay \nLaboratorio Subterráneo de Canfranc\n22880Canfranc-EstaciónSpain\n\nInstitute for Integrative Systems Biology (I\n\n",
"\nSysBio)\nValenciaSpain\n"
]
| [
"Donostia International Physics Center\n20018San SebastiánSpain",
"Laboratorio Subterráneo de Canfranc\n22880Canfranc-EstaciónSpain",
"Department of Quantum Physics and Astrophysics and Institute of Cosmos Sciences\nUniversity of Barcelona\n08028BarcelonaSpain",
"Laboratorio Subterráneo de Canfranc\n22880Canfranc-EstaciónSpain",
"Institute for Integrative Systems Biology (I\n",
"SysBio)\nValenciaSpain"
]
| []
| We study double gamma (γγ) decay nuclear matrix elements (NMEs) for a wide range of nuclei from titanium to xenon, and explore their relation to neutrinoless double-beta (0νββ) NMEs. To favor the comparison, we focus on doublemagnetic dipole transitions in the final ββ nuclei, in particular the γγ decay of the double isobaric analog of the initial ββ state into the ground state. For the decay with equal-energy photons, our large-scale nuclear shell model results show a good linear correlation between the γγ and 0νββ NMEs. Our analysis reveals that the correlation holds for γγ transitions driven by the spin or orbital angular momentum due to the dominance of zero-coupled nucleon pairs, a feature common to 0νββ decay. Our shell-model findings point out the potential of future γγ decay measurements to constrain 0νββ NMEs, which are key to answer fundamental physics questions based on 0νββ experiments. | 10.1016/j.physletb.2022.136965 | [
"https://arxiv.org/pdf/2102.11101v2.pdf"
]
| 231,986,339 | 2102.11101 | 73d21e3e9f4af970e4fb37f99be28064c9acaae5 |
γγ decay as a probe of neutrinoless ββ decay nuclear matrix elements
B Romeo
Donostia International Physics Center
20018San SebastiánSpain
Laboratorio Subterráneo de Canfranc
22880Canfranc-EstaciónSpain
J Menéndez
Department of Quantum Physics and Astrophysics and Institute of Cosmos Sciences
University of Barcelona
08028BarcelonaSpain
C Peña Garay
Laboratorio Subterráneo de Canfranc
22880Canfranc-EstaciónSpain
Institute for Integrative Systems Biology (I
SysBio)
ValenciaSpain
γγ decay as a probe of neutrinoless ββ decay nuclear matrix elements
We study double gamma (γγ) decay nuclear matrix elements (NMEs) for a wide range of nuclei from titanium to xenon, and explore their relation to neutrinoless double-beta (0νββ) NMEs. To favor the comparison, we focus on doublemagnetic dipole transitions in the final ββ nuclei, in particular the γγ decay of the double isobaric analog of the initial ββ state into the ground state. For the decay with equal-energy photons, our large-scale nuclear shell model results show a good linear correlation between the γγ and 0νββ NMEs. Our analysis reveals that the correlation holds for γγ transitions driven by the spin or orbital angular momentum due to the dominance of zero-coupled nucleon pairs, a feature common to 0νββ decay. Our shell-model findings point out the potential of future γγ decay measurements to constrain 0νββ NMEs, which are key to answer fundamental physics questions based on 0νββ experiments.
Introduction and main result
The observation of the decay of an atomic nucleus emitting only two electrons, neutrinoless double-beta (0νββ) decay, is the process experimentally most feasible to demonstrate that neutrinos are their own antiparticles [1]. Moreover, 0νββ decay is one of the most promising probes of physics beyond the standard model (BSM) of particle physics [2]. For instance, the observation of change in lepton number in 0νββ decay could help to explain the prevalence of matter in the universe [3,4]. Because of this unique potential, a very active program aims to detect 0νββ decay. Currently the most stringent constraints reach half-lives longer than 10 26 years [5][6][7][8][9][10][11][12], and next generation ton-scale experiments are being proposed, among others, for 76 Ge, 100 Mo, 130 Te and 136 Xe nuclei.
Since 0νββ decay changes lepton number-no antineutrinos are emitted to balance the two electrons-its decay rate depends on some unknown BSM parameter(s). In the standard scenario that 0νββ is triggered by the exchange of known neutrinos, this role is played by a combination of absolute neutrino masses and mixing matrix elements, m ββ . The decay rate also depends on a calculable phase-space factor [13,14], and quadratically on the nuclear matrix element (NME) that involves the initial and final nuclear states [15]. Thus, 0νββ NMEs are needed to interpret current experimental half-life limits and to anticipate the reach of future searches. However, typical NME calculations disagree up to a factor 3 [16][17][18][19][20][21][22][23][24][25][26][27][28], about an order of magnitude on the decay rate. Furthermore, first attempts to obtain more controlled 0νββ NMEs using ab initio techniques suggest smaller NME values than most previous studies [29][30][31].
A widely explored approach to reduce the uncertainty in 0νββ analyses is to study related nuclear observables. Nuclear structure [32] or muon capture [33] data are very useful to test nuclear models used to calculate NMEs, but they are not directly related to 0νββ decay. Likewise, ββ decay with neutrino emission shows no apparent correlation with 0νββ, in spite of both being second-order weak processes sharing initial and final states [34]. Useful insights could be gained from nuclear reactions, in the same spirit of the β decay information obtained in charge-exchange experiments [35,36]. Recent efforts include the measurement of nucleon pair transfers [37,38] and double charge-exchange reactions [39]. The good correlation found between 0νββ and double Gamow-Teller transitions [40] could in principle be exploited in double charge-exchange reactions, but the analyses are challenged by tiny cross sections [41,42] and involved reaction mechanisms [43,44].
In this Letter we study the correlation between the NMEs of 0νββ and second-order electromagnetic (EM) transitions emitting two photons (γγ). In fact, the latter were first studied in atoms by Goeppert-Mayer [45,46], and it was the extension to the weak interaction which led her to propose the ββ decay [47]. To ensure that nuclear-structure aspects are as similar as possible in the γγ and ββ sectors, we focus on EM double-magnetic dipole decays-which depend, like the 0νββ operator, on the nuclear spin. In addition, isospin symmetry assures a good correspondence between the γγ and ββ nuclear states if we consider the decay of the double isobaric analogue state (DIAS) of the initial ββ state-an excited state of the final ββ nucleus-into the final ββ state-the ground state (GS) of that nucleus. Our proposal expands the connections between first-order weak and EM transitions involving isobaric analogue states exploited in the past [48][49][50]. Figure 1 summarizes the main result of this work. We find a good linear correlation between γγ and 0νββ NMEs obtained with the nuclear shell model, valid across the nuclear chart. The upper panel presents results for decays in nineteen nuclei comprising titanium, chromium and iron isotopes with nucleon number 46 ≤ A ≤ 60. The lower panel covers twenty five nuclei comprising zinc, germanium, selenium, krypton, tellurium, xenon and barium isotopes with 72 ≤ A ≤ 136. The correlation is independent on the nuclear interaction used. Therefore, our findings call for γγ calculations with other many-body methods to test to what extent the shell-model correlation in Fig. 1 is universal or depends on the theoretical approach. Indeed the 0νββ correlation with double Gamow-Teller matrix elements is common for approaches as different as the nuclear shell model and energy-density functional theory, but it is apparently not fulfilled by the quasiparticle randomphase approximation method.
Second-order EM decays are naturally suppressed with respect to first-order ones. Nevertheless, γγ transitions have been measured between 0 + first-excited states and GSs, where single-γ decay is forbidden [51][52][53], and, recently, among general nuclear states in competition with γ transitions [54,55]. Future DIAS to GS γγ decay measurements, combined with the good linear correlation between NMEs presented in this work, show as a promising tool to give insights on 0νββ NMEs. A linear regression analysis supports this potential. Assuming a ±15% uncertainty in the branching ratio measurement as in Refs. [54,55] would lead to a relatively moderate error around ±(30% − 40%), dominated by the correlation. This would imply a clear improvement over the large spread in current M 0νββ calculations [15] if the same correlation is found to be valid for other many-body methods used to study 0νββ decay.
Electromagnetic DIAS to GS transitions
The γγ decay of a nuclear excited state is an EM process where two photons are emitted simultaneously:
N i (p i ) −→ N f (p f ) + γ λ (k) + γ λ (k ) ,(1)
where N i , N f are the initial and final nuclear states with four-momenta p i and p f , respectively, and photons have four-momenta k, k and helicities λ, λ . The theoretical framework of nuclear two-photon decay is presented in detail in Refs. [52,56,57]. The nonrelativistic interaction Hamiltonian is given bŷ where A µ (x) denotes the EM field,Ĵ µ (x) the nuclear current, andB µν (x, y) is a contact (seagull) operator which represents intermediate nuclear-state excitations not captured by the nuclear model, such as nucleon-antinucleon pairs. Perturbation theory up to second order in the photon field leads to the transition amplitudes
H I = d 4 xĴ µ (x)A µ (x)(2)+ 1 2 d 4 x d 4 yB µν (x, y)A µ (x)A ν (y) , α M γγ (0 DIAS + ⟶0 gs + ) [μ N 2 MeV -1 ] TiM 0νββ (0 gs + ⟶0 gs + ) 72 ≤ A ≤ 136M (1) = δ(k 0 + k 0 + E f − E i ) (3) × n d 3 x d 3 y ε * µλ (k)ε * νλ (k ) e −i(k·x+k ·y) × f |Ĵ µ (x) |n n|Ĵ ν (y) |i E i − k 0 − E n + i + f |Ĵ ν (y) |n n|Ĵ µ (x) |i E i − k 0 − E n + i , M (2) = −(2π)δ(k 0 + k 0 + E f − E i ) (4) × d 3 x d 3 y ε * µλ (k)ε * νλ (k )e −i(k·x+k ·y) f |B µν (x, y) |i ,
where ε µλ (k) is the photon polarization vector. The initial (|i ), intermediate (|n ) and final (|f ) nuclear states have energies E i , E n and E f , respectively. The amplitude M (2) can be neglected for DIAS to GS transitions, in the absence of subleading two-nucleon currents, because it involves a one-nucleon operator in isospin space [52]. It is very useful to perform a multipole decomposition of the γγ amplitude, because nuclear states have good angular momentum. The expansion involves electric (E) and magnetic (M ) multipole operators with angular momentum L, denoted as X. The transition amplitude sums over multipoles, which factorize into a geometrical (phase space) factor and the generalized nuclear polarizability, P J , containing all the information on the nuclear structure and dynamics [52]:
P J (X X; k 0 , k 0 ) = 2π(−1) J f +Ji (2L + 1)(2L + 1) (5) × n,Jn L L J J i J f J n J f || O(X)||J n J n || O(X )||J i E n − E i + k 0 + (−1) Y L L J J i J f J n J f || O(X )||J n J n || O(X)||J i E n − E i + k 0 ,
where the 6j-symbols depend on the total angular momenta of the initial, intermediate, and
final states J i , J n , J f and Y = J − L − L .
The reduced matrix elements of the EM multipole operators involve the photon energy: O(X) ∝ k L 0 , as well as the nucleon radial r, angular spherical harmonics Y L , orbital angular momentum l, and spin s operators.
Double EM and weak decays involve different nuclei:
A Z Y * N → A Z Y N + 2γ vs A Z−2 X N +2 → A Z Y N + 2e − , with N,
Z the neutron and proton number. In order to study the correlation between 0νββ and γγ NMEs, we focus on the γγ decay of the DIAS of the initial ββ state. This is an excited state, with isospin T = T z + 2, of the ββ daughter nucleus with isospin third component T z = (N −Z)/2. The DIAS γγ decay to the GS-the final ββ state-with T = T z , thus connects states with the same isospin structure as ββ decay: an initial state with isospin T i = T f + 2 with a final one with T f = T z . Since isospin symmetry holds very well in nuclei, we expect the nuclear structure aspects of DIAS to GS γγ and 0νββ transitions to be very similar. Altogether, the γγ decay involves the following positive-parity J i = J f = 0 nuclear states:
|0 + i γγ ≡ |0 + i ββ (DIAS) = T − T − K 1/2 |0 + i ββ ,(6)|0 + f γγ ≡ |0 + f ββ ,(7)
with K a normalization constant and T − = A i t − i the nucleus isospin lowering operator, which only changes T z .
Angular momentum and parity conservation impose that transitions between 0 + states just involve the zeromultipole polarizability P 0 , with two EL or M L operators. In the long wave approximation k 0 |x| 1, satisfied when Q = E i − E f ∼ 1 − 10 MeV, dipole (L = 1) decays dominate. Since the nuclear spin is key for 0νββ decay, we focus on double-magnetic dipole (M 1M 1) processes, governed by the operator
M 1 = µ N 3 4π A i=1 g l i l i + g s i s i ,(8)
with µ N the nuclear magneton, and the neutron (n) and proton (p) spin and orbital g-factors: g s n = −3.826, g s p = 5.586, g l n = 0, g l p = 1. For M1M1 transitions Eq. (5) factorizes and the expression can be written in terms of a single NME M γγ (M 1M 1, ∆ε):
P 0 (M 1M 1, k 0 , k 0 ) = 2π 3 √ 3 k 0 k 0 M γγ (M 1M 1, ∆ε) , (9) M γγ (M 1M 1, ∆ε) = n 0 + f ||M 1||1 + n 1 + n ||M 1||0 + i ε n 1 − ∆ε 2 2ε 2 n ,(10)
which depends on the energy difference between the two photons, ∆ε = k 0 −k 0 , and on ε n = E n −(E i +E f )/2. The M 1 operator demands 1 + intermediate states.
In order to avoid the dependence on the photon energies, which depends on the nucleus, we require that the two photons share the transition energy: k 0 = k 0 = Q/2 and thus ∆ε = 0. In this case we can define the following nuclear matrix element M γγ (M 1M 1) = M γγ (M 1M 1, 0):
M γγ (M 1M 1) = n 0 + f ||M 1||1 + n 1 + n ||M 1||0 + i ε n .(11)
In the following, we calculate the NMEs in Eq. (11). Note that the γγ measurement to constrain these NMEs requires an experimental setup for k 0 k 0 . This is needed because in transitions between the DIAS and the ground state Q and hence ∆ε can easily be of the order of 10 MeV, exceeding the value of ε n .
Nuclear shell model calculations
We calculate M γγ (M 1M 1) for a broad range of 46 ≤ A ≤ 136 nuclei in the framework of the nuclear shell model [58][59][60]. We cover three different configuration spaces spanning the following harmonic oscillator singleparticle orbitals for protons and neutrons: i) 0f 7/2 , 1p 3/2 , 0f 5/2 and 1p 1/2 (pf shell) with the KB3G [61] and GXPF1B [62] effective interactions; ii) 1p 3/2 , 0f 5/2 , 1p 1/2 and 0g 9/2 (pfg space) with the GCN2850 [63], JUN45 [64] and JJ4BB [65] interactions; and iii) 1d 5/2 , 0g 7/2 , 2s 1/2 , 1d 3/2 and 0h 11/2 (sdgh space) with the GCN5082 [63] and QX [66] interactions. All the interactions are isospin symmetric. For our calculations we use the shell model codes ANTOINE [58,67] and NATHAN [58]. The 0νββ NMEs, calculated with the same configuration spaces and nuclear interactions, are taken from Ref. [40].
First, we calculate the final γγ state and the initial ββ one, which we rotate in isospin to obtain its DIAS as in Eq. (6). Next, we build a finite set of intermediate states {1 + n } with the Lanczos strength function method, taking as doorway state the isospin T n = T f + 1 projection of the isovector M 1 operator applied to the final state:
P T =Tz+1 M 1 IV |0 + f .
This guarantees intermediate states with correct angular momentum and isospin.
We evaluate the energy denominator ε n using experimental energies when possible [68,69]. For 48 Ti, E f , E i (DIAS) and also the energy of a T = T z + 1 state (6 + ) are known. We use the latter, together with the calculated energy difference between the 6 + and 1 + states with T = T z + 1, to fix the energy of the intermediate states E n . With this experimental input, M γγ (M 1M 1) only varies the result obtained with calculated energies by 0.2%. Isospin-breaking effects cancel in ε n to a very good approximation [70]. Therefore, in nuclei with unknown energy of the DIAS or T = T z +1 states, we use experimental data on states of the same isospin multiplet in neighboring nuclei: the ββ parent to fix E i , and the ββ intermediate nucleus-when available-for E 1 . Using these experimental energies modifies M γγ (M 1M 1) results by less than 5%.
Results
With these ingredients we evaluate Eq. (11). Figure 2 shows M γγ (M 1M 1) as a function of the excitation energy of the intermediate states, for nuclei covering the three configuration spaces: 48 Ti, 82 Se and 128 Te. The Lanczos strength function gives converged results to ∼ 1% after 50 − 100 iterations. Figure 2 illustrates that, in general, intermediate states up to ∼ 15 MeV can contribute to the double-magnetic dipole NME, and that only a few states dominate each transition. The comparison between weak and EM decays needs to take into account that while 0νββ changes N and Z by two units, they are conserved in γγ decay. This is achieved by comparing isospin-reduced NMEs or, alternatively, by including the ratio of Clebsch-Gordan coefficients dictated by the Wigner-Eckart theorem [71]: Figure 1 shows the good linear correlation between 0νββ NMEs and double-magnetic dipole NMEs obtained with bare spin and orbital g-factors. We observe essentially the same correlation when using effective g-factors that give slightly better agreement with experimental magnetic dipole moments and transitions: g s i (eff) = 0.9g s i , g l p (eff) = g l p +0.1, g l n (eff) = g l n −0.1 in the pf shell [72]; and g s i (eff) = 0.7g s i for pfg nuclei [73]. We have performed a linear regression analysis to the data leading to the correlation in Fig. 1. A fit to the function M 0νββ = a + bM γγ gives best-fit parameters a = 0.872, b = 0.459 for 46 ≤ A ≤ 60 (top panel), and a = 1.29, b = 1.11 for 72 ≤ A ≤ 136 (bottom). The best fit is shown with a solid line, while prediction bands at 90% confidence level (CL) are given in dashed lines. The correlation coefficients for the top and bottom panels are ρ = 0.83 and ρ = 0.84, respectively. The 90% CL bands could be combined with a hypothetical measurement of the γγ M1M1 decay to obtain a M 0νββ NME. A branching ratio measurement with a ±15% uncertainty [54,55] combined with the linear correlation would lead to a relatively moderate error around ±(30% − 40%).
α = 3 2 C T f ,2,T f +2 T f ,2,T f +2 /C T f ,2,T f +2 T f ,0, T f = 1 2 (2 + T f )(3 + 2T f ).
The slope of the linear correlation between γγ and 0νββ NMEs in Fig. 1 only depends mildly on the mass number, being slightly larger in the pf shell than for pfg and sdgh nuclei. This distinct behaviour is due to the energy denominator in M γγ (M 1M 1): when only the numerator in Eq. (10) is considered,M γγ , the same linear correlation is common to all nuclei. Ultimately, this mild dependence on the energy denominator is key for the good correlation between M γγ and M 0νββ .
The small dependence on the energy denominator is illustrated by Fig. 2: the intermediate states that contribute more to M γγ (M 1M 1) lie systematically at lower energies in pf -shell nuclei, compared to A ≥ 72 systems. In fact, the ratio of average energy of the dominant states contributing to M γγ (M 1M 1) in the pf shell over the pfg−sdgh spaces matches very well the ratio of the slopes in the top and bottom panels of Fig. 1. Also, in the bottom panel of Fig. 1 heavier nuclei (Te, Xe, Ba) calculated with the GCN5082 interaction appear in the upper part of the correlation band. This is partly due to a mildly smaller energy denominator and also because of a slightly larger contribution of the orbital angular momentum component of the M 1M 1 operator. We can gain additional insights on the γγ − 0νββ correlation by decomposing the double-magnetic dipole NME into spin, orbital and interference parts. Since the energy denominator plays a relatively minor role, we focus on the changes in the numerator matrix element: M γγ =M γγ ss +M γγ ll +M γγ ls . Figure 3 shows the decomposition for the γγ decay of several nuclei. In some cases like 72 Zn, the spin part dominates. Here, sinceM γγ ss is proportional to the double Gamow-Teller operator, a very good correlation with 0νββ is expected [40]. In contrast, the or-bitalM γγ ll part dominates in 134 Xe or 136 Ba, sdgh nuclei with an l = 5 orbital. Remarkably, these nuclei follow the common trend in Fig. 1, which means that the correlation with 0νββ decay is not limited to operators driven by the nuclear spin. The interferenceM γγ ls is generally smaller, and can be of different sign to the dominant terms. In fact, Fig. 3 also shows that the spin and orbital contributions to γγ decay always have the same sign, preventing a cancellation that would blur the correlation with 0νββ decay. Figure 4 investigates further the relation between spin and orbital γγ contributions, decomposing the NMEs in terms of the two-body angular momenta J of the two nucleons involved in the transition. Analogously to 0νββ NMEs [17,18],M γγ is dominated by the contribution of J = 0 pairs, partially canceled by that of J > 0 ones. This behaviour is common toM γγ ss andM γγ ll , with a more marked cancellation in the spin part, as expected due to the spin-isospin SU(4) symmetry of the isovector spin operator [22,74]. The J = 0 dominance suggests that spin and orbital S = L = 0 pairs are the most relevant in γγ DIAS to GS transitions, implying that s 1 s 2 = (S 2 − 3/2)/2 < 0, and likewise l 1 l 2 < 0. Since the spin and orbital isovector g-factors also share sign, the hierarchy in Fig. 4 explains the absence of cancellations that leads to the γγ correlation with 0νββ decay. The shell-model 0νββ nuclear matrix elements in Fig. 1 have been obtained with axial coupling g A = 1.27. While the nuclear shell model is known to overestimate β [75] and two-neutrino ββ [76][77][78] matrix elements-a feature usually known as "g A quenching"-the need and amount of "quenching" required by shell-model 0νββ NMEs is uncertain. For instance, the larger 0νββ-decay momentum transfer may imply different sensitivity to missing nuclear correlations and two-body currents, the main aspects that cause the deficiencies in β and two-neutrino ββ calculations [15,79]-two-body currents are expected to be less relevant for 0νββ NMEs [80,81]. In contrast, ab initio calculations in light [79,82,83] and middle-mass nuclei [79,84] describe well β-decay matrix elements without additional adjustments. A comparison to the first ab initio 0νββ NMEs [29][30][31] and also to a recently proposed hybrid approach that combines ab initio short-range correlations with the nuclear shell model [85] suggests that 0νββ NMEs in the shell model may be moderately overestimated-by several tens of percent-in a relatively similar way for all ββ emitters. This would imply a similar correlation to the one presented in Fig. 1 but with a and b parameters modified according to the possible overestimation of the shell-model 0νββ NMEs.
Future work includes evaluating two-nucleon current contributions to double-magnetic dipole [86] and 0νββ [79,80] decays, but we do not expect these corrections to alter significantly the NME correlation. In contrast, the recently-proposed leading-order contact contribution could modify sizeably 0νββ NMEs [87][88][89], but note that short-range NMEs can also be correlated to the 0νββ ones in Fig. 1 [90]. In addition, the correlation observed here can be tested with other many-body approaches such as energy-density functional theory [26,27,91], the interacting boson model [23,92], the quasiparticle ramdom-phase approximation (QRPA) [21,93] or ab initio methods [29][30][31]94].
Summary
We have observed a good linear correlation between 0νββ NMEs and γγ ones when the two photons share the energy of the decay. For our shell model calculations, the correlation holds across the nuclear chart, independently on the nuclear interaction used. While the correlation should also be tested with other leading manybody methods used to study 0νββ decay, this suggests a new avenue to reduce 0νββ NME uncertainties if doublemagnetic dipole DIAS to GS γγ transitions can be mea-sured, especially on the most relevant 0νββ nuclei. In fact, first steps in this direction are underway: Valiente-Dobón et al. [95] recently proposed a flagship experiment to determine the conditions of a future program to measure the γγ decay of the 48 Ca DIAS in 48 Ti. Even though these experiments are challenging due to the competition with single-γ, E1E1 γγ and nucleon-emission channels, their potential should not be underestimated. Next generation 0νββ experiments imply a significant investment with the promise to fully cover the inverted neutrino-mass hierarchy region [96], but current NME uncertainties may limit the reach of the proposals under discussion.
Figure 1 :
1Correlation between 0νββ (M 0νββ , from Ref. [40]) and double-magnetic dipole [M γγ (M 1M 1)] NMEs. In the y-axis , α is an isospin factor, see the text. Top panel: 46−58 Ti (blue circles), 50−58 Cr (red diamonds) and 54−60 Fe (black triangles). Bottom panel: 72−76 Zn (light blue circles), 74−80 Ge (violet diamonds), 76−82 Se (green triangles), 82,84 Kr (orange squares), 124−132 Te (violet down triangles), 130−134 Xe (cyan diamonds) and 134,136 Ba (magenta circles). The solid line and dashed band represent the best linear fit and prediction band at 90% confidence level, respectively.
Figure 2 :
2Contribution (solid lines) and cumulative (dashed lines) values of the M γγ (M 1M 1) NME as a function of the excitation energy of the intermediate states En. The results, for the 0 + DIAS → 0 + gs transition in 48 Ti, 82 Se and 128 Te, are smoothed with a Lorentzian of width 0.1 MeV.
Figure 4 :
4Decomposition of the 136 Ba numerator NMEM γγ , in terms of the two-nucleon angular momenta J : total (T), spinM γγ ss (ss), orbitalM γγ ll (ll) and interferenceM γγ ls (ls) parts.
AcknowledgementWe thank J. J. Valiente-Dobón for illuminating discussions. B. R. warmly acknowledges support from the NEXT Collaboration. This work was supported in part by the "Ramón y Cajal" program with grant RYC-2017-22781, and grants CEX2019-000918-M, PID2020-118758GB-I00 and RTI2018-095979-B-C41 funded by MCIN/AEI/10.13039/501100011033 and, as appropriate, by "ESF Investing in your future".
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| []
|
[
"√ŝ min resurrected",
"√ŝ min resurrected"
]
| [
"Tania Robens \nSchool of Physics and Astronomy\nSUPA\nUniversity of Glasgow\nG12 8QQGlasgowScotland, UK\n\nIKTP, TU\nZellescher Weg 1901069Dresden, DresdenGermany\n"
]
| [
"School of Physics and Astronomy\nSUPA\nUniversity of Glasgow\nG12 8QQGlasgowScotland, UK",
"IKTP, TU\nZellescher Weg 1901069Dresden, DresdenGermany"
]
| []
| We discuss the use of the variable √ŝ min , which has been proposed in order to measure the hard scale of a multi parton final state event using inclusive quantities only, on a SUSY data sample for a 14 TeV LHC. In its original version, where this variable was proposed on calorimeter level, the direct correlation to the hard scattering scale does not survive when effects from soft physics are taken into account. We here show that when using reconstructed objects instead of calorimeter energy and momenta as input, we manage to actually recover this correlation for the parameter point considered here. We furthermore discuss the effect of including W + jets and tt + jets background in our analysis and the use of √ŝ min for the suppression of SM induced background in new physics searches. | 10.1007/jhep02(2012)051 | [
"https://arxiv.org/pdf/1109.1018v2.pdf"
]
| 118,377,028 | 1109.1018 | 5bcd5c21f5ec3c2aea3d4f705b929379b20e5ba9 |
√ŝ min resurrected
1 Mar 2012
Tania Robens
School of Physics and Astronomy
SUPA
University of Glasgow
G12 8QQGlasgowScotland, UK
IKTP, TU
Zellescher Weg 1901069Dresden, DresdenGermany
√ŝ min resurrected
1 Mar 2012
We discuss the use of the variable √ŝ min , which has been proposed in order to measure the hard scale of a multi parton final state event using inclusive quantities only, on a SUSY data sample for a 14 TeV LHC. In its original version, where this variable was proposed on calorimeter level, the direct correlation to the hard scattering scale does not survive when effects from soft physics are taken into account. We here show that when using reconstructed objects instead of calorimeter energy and momenta as input, we manage to actually recover this correlation for the parameter point considered here. We furthermore discuss the effect of including W + jets and tt + jets background in our analysis and the use of √ŝ min for the suppression of SM induced background in new physics searches.
Introduction and motivation
Since the startup and the following successful data taking of the LHC, the LHC experiments have already published a large number of result for exclusion limits for BSM physics, where for many BSM scenarios the actual limits within specific parameter regions have been strongly pushed to higher scales [1]. However, most of these analyses have been performed within specific models, and more generic variables, which provide information about the generic scale of new physics without additional assumptions about the decay topologies or decay chains, have not been fully exploited. Furthermore, many variables which are currently proposed for mass or scale determination for new physics processes only make use of the transverse momentum of the event, thereby neglecting the information which can be obtained by additionally including the longitudinal information 1 . Examples for variables which make use of transverse momentum only are eg M T 2 [4] or M CT [5]. On the other hand, some more traditional variables as eg invarant masses of composite objects as used in edges studies [6,7,8,9] implicitely use all visible information, including the longitudinal momentum of the visible decay products. In [10] a new fully inclusive calorimeter-level variable, √ŝ min , was proposed which promises to give information about the hard scale of the underlying new physics processes without further assumptions or specification of the decay products, and additionally makes use of the longitudinal information of the inclusive event. In that paper, the authors propose a conjecture which relates the peak position of √ŝ min to the actual rise of the hard new physics production cross section. However, this variable was subsequently shown to have a strong dependence on the soft physics in terms of ISR and underlying event. This is mainly caused by the fact that in this case, the energy of the additional soft particles equally enters in the calorimeter-level definition of √ŝ min , which then boosts the variable and its peak position to higher values with respect to the parton-level quantity. In [10], the authors tried to circumvent this problem by introducing a pseudorapidity cut in order to suppress the unwanted effects originating from soft physics from entering √ŝ min . However, the introduction of the cut destroyed the correlation between peak position and hard cross section threshold which holds at parton level; this has explicitely been shown analytically for effects arising from initial state radiation (ISR) [11,12]. In this case, the peak position is basically determined by the value of the pseudorapidity cut. Subsequently, √ŝ min was promoted to √ŝ (reco) min [13], using reconstructed objects at analysis level; for this variable, the correlation between its peak and the threshold of the hard production cross section was recovered, such that a determination of the hard scale of the BSM process was again made possible using experimentally accessible detector level objects.
Apart from providing information about the hard scale of the underlying parton level BSM process, new variables can equally be used as cut parameters for SM background suppression, and several of these variables have already made their way into the current BSM searches at the LHC experiments. It is therefore equally important to determine the use of √ŝ min for SM background suppression. Although this constitutes a slightly weaker use of the variable per se, it is still an important issue to investigate, especially as it has been proposed on a fully inclusive level and can therefore be applied without any further assumptions on the model or the specific decay chains and topology.
In this report, we therefore investigate the properties of √ŝ min at analysis level using reconstructed objects. For this, we use a full sample for the mSugra point SPS1a 2 [15], which contains all strong production as well as all decay chains; for this parameter point, our sample therefore corresponds to the full data set which would be obtained from strongly interacting initial cascade particles in a realization of SPS1a 3 . We include soft physics in terms of initial and final state radiation (ISR/ FSR), as well as a fast detector simulation. We test the correlation between the threshold of the hard process and the peak of √ŝ min on parton level for the inclusive sample as well as exclusive dominant final states. In addition, we show that, for our sample, this relation can be regained on reconstruction level using quite simple analysis object definitions, and that major discrepancies between analysis and parton level quantities can be traced back to uncertainties in the reconstruction of tau jets. We equally comment on the power of the analysis level variable for SM background suppression, and compare to similar variables. All analyses are done for a LHC-like proton proton collider with a center of mass energy of 14 TeV and an integrated luminosity L = 1 fb −1 .
The report is organized as follows: in Section 2, we briefly review the variable definition of √ŝ min and define other kinematic quantities which were used in our study. In Section 3, we describe the data set we use in this study. Section 4 contains the comparison of parton and analysis level √ŝ min , and Section 5 describes the inclusion of SM background and the use of √ŝ min for SM background suppression, as well as a brief comparison with other (transverse) variables. We conclude in Section 6.
Variable definition
In this section, we will briefly review the original variable definition as well as its RECO level version; the interested reader is referred to [10,13] for a more detailed discussion.
In general, √ŝ min is defined on an event by event basis as the minimal value for the partonic center-of-mass energy √ŝ which is in agreement with the events' momentum configuration. It can be derived through a minimization process [10] as
√ŝ min (M inv ) = / E 2 T + M 2 vis + / E 2 T + M 2 inv
(1) 2 We are aware that this parameter point has recently been excluded by ATLAS measurements in the quark/ gluon plus missing transverse energy channel [14]. However, we here want to show that the parton level √ŝ min can actually be recovered with sufficient accuracy from analysis level objects. Spectra which evade current exclusion limits typically exhibit higher initial cascade particle masses, and our arguments are generically not affected by the actual position of the particle production threshold. This is only important in the studies of background suppression presented in Section 5; here, a higher peak value for the BSM induced variable should actually even enhance the SM background suppression which can be obtained using √ŝ min . 3 Gaugino-gaugino initial cascade states, which have not been considered here, would contribute an additional 5% to the total production cross section.
with
M 2 vis = E 2 − − → P 2
being the effective visible mass and the invisible total mass
M inv = invisible m(2)
the sum over all masses of invisible particles. For completeness, we give the minimization procedure leading to Eqn. (1) in Appendix B. Note that M vis and M inv are not defined equivalently, and especially that M inv is not the Lorentz-invariant mass of the total invisible system, but rather the sum over all invisible particles rest masses 4 . In the definition of √ŝ min , M inv is therefore an external input parameter, as it is the only quantity which cannot be measured directly from experiment. Therefore, all results which are derived in the following sections have an implicit dependence on the value of M inv . Throughout our study, we have usually set this to its "true" BSM value
M inv = 2 m χ 0 1 (= 193.4 GeV).
The translation to experimentally accessible quantities is then straightforward and gives
√ŝ min (M inv ) = E 2 − P 2 Z + / E 2 T + M 2 inv .(3)
Note that the use of transverse energy and momentum strongly depends on the definition of the specific quantity; we define
/ − → P T = − − → P T , / E T = | / − → P T |(4)
where E, P are the total energy and four momentum of all visible objects
P µ = vis p µ i , E = P 0 , − → P T = P X P Y(5)
and the z-direction defines the beam-line. In the original proposal, all visible quantities are taken from calorimeters, and soft background is suppressed by a cut in the pseudorapidity η. Subsequently, for the correct value of M inv, true , a conjecture
√ŝ min (M inv, true ) peak ∼ √ s th(6)
is empirically derived, which links the peak position of √ŝ min to the actual threshold √ s th of the hard matrix element process. However, in [11] it was subsequently pointed out that, using different values for the η cut, the peak position for the calorimeter-based variable √ŝ min could actually be arbitrarily shifted around; the same effect has been observed in [16], which applies the original calorimeter-based definition of √ŝ min on our data sample. In answer to this criticism, new reconstruction and subsystem level variables were proposed in [13]. We here use √ŝ min on an inclusive level using reconstructed objects, which basically corresponds to the RECO variable definition given in [13]. In our work, the suppression of effects from the parton shower has been achieved by quite simple object-level definitions given in Table 2. We will show that we obtain the parton-level √ŝ min quite accurately in our sample, and that we indeed observe a similar peak of the RECO-level √ŝ min close to the production threshold for the correct input value of M inv . In addition, we investigate the actual sources of discrepancies between the parton and reconstruction level √ŝ min in more detail. In our work, we equally present the first study of √ŝ min as a variable for SM background suppression in BSM searches.
In this study, we use the term "leptons" for all three SM lepton generations; in cases when we are concerned with tau leptons alone we will mention this explicitly. Equally, the tau jets at analysis/ reconstruction level are defined by the tau jet reconstruction algorithm in Delphes [17] and differ from the parton level tau lepton by the four-momenta of the invisible tau decay products, specifically the associated third generation neutrino. In the following, we use the term "tau" for the parton level and "tau jet" for the reconstruction level quantity, which for an ideal reconstruction of the visible tau decay products four-momenta only differ by the four-momentum of the invisible decay products.
Data sample and event generation
In this report, we have made use of the BSM data samples which have been generated in the course of the 2009 BSM Les Houches mass determination study; first results using these data for studies of various mass determination methods were presented in [16]. We use a SUSY spectrum for the point SPS1a, where the spectrum was generated with the spectrum generator SOFTSUSY [18]. Parton level events have been generated using Madgraph [19,20] for the generation of the heavy initial cascade particles (i.e. the squark-squark, squark-gluino, and gluino-gluino initial states). The heavy pairproduced particles have then been fully decayed according to the respective branching ratios into all possible decay products using Bridge [21] within the Madgraph framework; we therefore consider a complete sample for this parameter point, which contains all possible final states. The SM background has been generated using Alpgen [22]. For parton shower and hadronization, we used Pythia [23], where the parton shower evolution follows the Pythia 6.4 default, ie is Q 2 ordered with additional modifications to guarantee color coherence, as well as matrix element corrections where these are available (Pythia switches are given in Appendix C). The detector simulation has been performed with Delphes [17] in its default mode. For specific input parameters and setups, we refer to the specifications which can be found in the data base for our samples [24]. Data analysis as well has fitting has been done within the ROOT [25,26] framework. All our results have been obtained with a data sample for a center-ofmass energy √ S hadr = 14 TeV and an integrated luminosity L = 1 fb −1 . Table 1 lists the production cross sections for the hard 2 → 2 process; these numbers were obtained using the Madgraph parton level 2 → 2 production cross sections, with the electroweak scale spectrum obtained from SOFTSUSY, convoluted with PDFs to account for the parton to hadron transition for the incoming states. For this study, we X 1 X 2 2 → 2 qq 6.56 qg 19.96 gg 4.53 Table 1: SPS1a production cross sections in pb for p p → X 1 X 2 using Madgraph 2 → 2 parton level production cross sections, convoluted with PDFs, for a hadronic center-of-mass energy of 14 TeV. CTEQ6L1 PDFs [27] were used.
restrict ourselves to the leading order predictions for the hard process in both signal and background simulation 5 .
Parton and analysis level √ŝ min
As already discussed in Section 2, the variable √ŝ min has undergone several developments since its original proposal. Initially defined as a calorimeter-based variable, it was shown to be quite sensitive to effects of soft physics for the respective processes. Especially the original merit of this variable, namely the correlation of the peak position and the threshold of the heavy pair-produced particles at the beginning of the decay chain, is strongly influenced by the soft physics of the event. The original suggestion of the authors was to introduce a cut on the pseudorapidity; however, the authors in [11,12] have shown analytically that the position of the peak position in this case is completely cut-value dependent; similar results have been observed in [16].
In this study, we show that, if √ŝ min is defined at analysis object level rather than on calorimeter level, the parton level variable √ŝ min can be reconstructed quite well using simple object definitions. This recovery of the parton level peak position using a reco-level variable for both inclusive and exclusive final states, however for different parameter points, have equally been presented in [13]. For our sample, we equally observe that the conjectured correlation between the rise of the hard scattering event cross section and the peak position of √ŝ min holds; however, we want to emphasize that this is on the level of a conjecture which has not been systematically studied or proven on an analytic level, although some preliminary studies indicate a kinematic origin which emerges after the convolution with PDFs 6 [28]. Therefore, even on parton level, it is currently unclear whether this conjecture necessarily holds for all BSM parameter points and scenarios. Equally, this conjecture only holds for a correct input value of M inv . 5 A fully differential study including NLO contributions to account for cut effects would require 2 → n event generators for both BSM signal and SM background, which additionally include the matching of parton shower and NLO contribution; although fast progress has been made in this field for SM processes, no fully differential higher order BSM generator is currently publicly available. 6 This result has been obtained with a unit matrix element as well as unit PDFs; in this case, the peak position of √ŝ min arises from the lower PDF integration boundary following from the kinematic lower limit which guarantees that √ s part ≥ √ s threshold . More realistic scenarios with non-uniform PDFs and matrix elements are currently under investigation. object Delphes predefinition additional requirement electron/ position |η| < 2.5 in tracker, p T > 10 GeV isolated muon |η| < 2.4 in tracker, p T > 10 GeV isolated lepton isolation criteria no track with p T > 2 GeV no track with p T > 6 GeV in a cone with dR = 0.5 in a cone with dR = 0.5 around the considered lepton around the considered lepton n leptons --exactly n isolated leptons at detector level taujet p T > 10 GeV -jet p T > 20 GeV p T,jet > 50 GeV, |η| jet < 3 CDF jet cluster algorithm [29], R = 0.7 Missing transverse --E miss T > 100 GeV energy Table 2: Physical object definitions in terms of the single objects pseudorapidity η, absolute value of transverse momentum p T , and (jet) cone radius R for analysis level objects on detector level. We basically adapt the Delphes predefinitions, with slightly more stringent requirements for isolated leptons and jet definitions. We equally set a lower limit E miss T ≥ 100 GeV for events with missing transverse energy.
In the following, we will compare quantities derived on the parton level with the same quantities which have been derived from analysis level objects. For the identification of the former, we consider the hard process, i.e. our data sample after the complete decay to SM particles and the LSP, but before the parton shower, hadronization, and detector simulation. All particles are considered as visible apart from neutrinos and the LSP. The invisible total four-momentum is then the sum of the latter particles' four-vectors P parton invis = ν ′ s, χ 0 1 p i , and the same holds for the missing transverse momentum. At analysis level, we require all physical objects to fulfill the object definition requirements given in Table 2 on detector level; these object definitions basically follow the Delphes predefinitions, where we introduced slightly more stringent requirements for lepton isolation and jet criteria and equally set a lower limit of 100 GeV on the total missing energy 7 . Visible and invisible quantities are then defined according to Eqns. (4) and (5) in Section 2.
We first study the variable √ŝ min for a complete inclusive sample, i.e. we sum over all final states of the hard process. Our main results are shown in Figure 1, where we compare the true √ŝ , parton level √ŝ min , reconstruction level √ŝ min as well as the original calorimeter based variable with and without a cut in pseudo rapidity η, as originally suggested in [10]. We see that the parton level √ŝ min peaks quite close to the actual heavy particle production threshold as suggested in [10]; equally, we observe that the same variable from reconstructed objects again peaks close to the threshold, but is shifted to slightly lower values with respect to the parton level quantity. We will comment on this in more detail below. In contrast, the pure calorimeter based variable exhibits a peak position at quite high values and can therefore not be used for a scale measurement of the new physics process. Restricting the contributions to calorimeter energy deposits with a minimal pseudorapidity improves this behavior and brings the peak closer to lower values; however, this approach suffers from the drawbacks pointed out in [11,12].
For a more accurate determination of the peak position and a viable assessment of the error in its position, we fit the √ŝ min distribution with a Gaussian around its peak, where we use the largest fit region which is still in agreement with
χ 2 /d.o.f ∼ O(1)
. Specifically, we use 600 GeV ≤ √ŝ min ≤ 1400 GeV and 400 GeV ≤ √ŝ min ≤ 1400 GeV to determine the parton level and analysis level peak positions respectively. We then obtain We see that the reconstruction level variable for the overall sample peaks close to the "true" maximum of the parton level variable, the difference being O(100 GeV). In order to pin down the major sources of this shift, we have performed detailed studies for specific final state signatures; we will discuss this in more detail in Section 4.1. To summarise the result of this section, we observe that, in our sample, larger shifts in the peak positions stem from processes with one or more leptons in the final state. One source of this is the imperfect reconstruction of tau jets from parton to analysis level objects. We can test this by taking an "idealistic" approach, where we use the parton level four-vector values for taus in the analysis level objects; this simple "gedankenexperiment" trick, where we assume a perfect reconstruction of tau jets at analysis level, significantly reduces this difference, cf. Figure 2. While such a requirement is in fact not possible in reality, it however shows that our (quite loose) lepton definitions and resulting poor tau reconstruction are a major source of this shift, and more dedicated algorithms might further reduce this discrepancy. Fitting the "new" analysis level distribution within the range 600 GeV ≤ √ŝ min ≤ 1400 GeV, we obtain analysis level √ŝ peak min , τ = τ parton : (1163 ± 4) GeV and we see that the discrepancy with the parton level value of (1152 ± 4) GeV reduces to the permill level, cf. Table 2 are included. True √ŝ (red; solid), parton level √ŝ min (blue; dashed), analysis level √ŝ min (green; dotted), √ s min using calorimeters (pink; dash-dotted) and same with an |η| < 1.4 cut (black; dashdot-dot-dotted). √ S hadr = 14 TeV, L = 1 fb −1 ; corresponds to 31050 events. Shift between parton level and analysis level peak is about 70 GeV. The calorimeter based distribution without a pseudorapidity cut exhibits a peak at much larger √ŝ min values.
error bars 8 . In addition, the object definitions in Table 2 equally allow for an adequate recovery of the parton level distribution shape, and, more specifically, we are able to suppress distribution tails for higher √ŝ min values appearing in the reco-level definition of this variable in [13]. A breakdown in terms of pairs of initially produced particles prior to the cascade decays is given in Table 3.
For illustration purposes and completeness, we also investigate the parton level √ŝ min dependence on the input value for M inv ; similar results have already been presented in [10] and [16] for the analysis level quantity. For R-parity conserving SUSY scenarios, as considered in this study, this corresponds to the guess of the LSP mass, as in this case M inv = 2 m LSP when neutrino masses are neglected. Figure 3 shows the shift of the parton level √ŝ min distribution for different input values M inv . We see that a variation of the M inv mass leads to a shift in the peak of a similar magnitude. Therefore, we again emphasize that the results presented in this study concerning the correlation of the peak of the √ŝ min distribution and the hard scale of the underlying production process have indeed an implicit dependence on the correctness of the guessed input value for M inv , as already discussed in the original proposal of this variable [10], and therefore generically only allow for a measurement of the hard scale as a function of this variable 9 . As before, the parton level distributions and peak positions 8 In this work, we only want to demonstrate that the approximate peak position of the parton level variable can actually be obtained from analysis level objects; for more dedicated analyses, the peak position could also be determined by other means, eg. a fit to a more variable-specific function. are the masses of the heavy initial cascade particles. We see that the peak position from both peak position definitions are close to the actual thresholds; in addition, the effect of imperfect tau reconstruction account for an approximate shift O(100 GeV) for all initial state pairings. Shown are results for M inv = 0 GeV (dark green; long dashed), 200 GeV (black; solid), 400 GeV (pink; dotted), 600 GeV (green; dash-dotted), 800 GeV (dark blue; dash-dot-dot-dotted), 1000 GeV (light blue; short dashed). The corresponding peak positions for increasing M inv input values as given above, using a maximal bin definition, are obtained as √ŝ peak; part min = (1035, 1170, 1305, 1440, 1620, 1845) GeV. As before, the analysis level peak positions using the parton level tau leptons coincide with the parton level peaks (not shown here). could be reproduced using analysis level objects in the idealized version, ie replacing the analysis level tau-jets with parton-level tau leptons in √ŝ ana min .
Signal based searches
In this section, we consider the variable √ŝ min for several exclusive final states. The (parton level) dominant decay modes of our sample are given in Table 4.
Most of the dominant final states can be tracked down to a couple of competing processes, and can be broken down to the following parton-level decay chains •qg, 3 jet channelq
Rg → q RqRqR χ 0 1 → q RqR q R χ 0 1 χ 0 1 (90%)
dependence on an input value for the LSP mass. •qg, 3 jet 2 lepton channel
q Lg → q L τ + τ − χ 0 1 q ′ Rq ′ R χ 0 1 (27%) q Rg → q R χ 0 1 bbτ + τ − χ 0 1 (22%) q Rg → q R χ 0 1 q ′ Lq ′ L τ + τ − χ 0 1 (17%) q Lg → q ′ L τ ν τ χ 0 1 q ′′ Lq ′′′ L τ ν τ χ 0 1 (17%) q Lg → q ′ L τ ν τ χ 0 1 btτ ν τ χ 0 1 (17%)
•qg, 3 jet 1 lepton channel
q Rg → q R χ 0 1 btτ ν τ χ 0 1 (45%, BR ∼ 0.09) q Rg → q R χ 0 1 q Lq ′ L τ ν τ χ 0 1 (30%, BR ∼ 0.06) q Lg → q ′ L τ ν τ χ 0 1 q ′′ Rq ′′ R χ 0 1 (25%, BR ∼ 0.05) •qq, 2 jet 2 lepton channel q Lq ′ L → q ′′ L τ ν τ χ 0 1 q ′′′ L τ ν τ χ 0 1 (36%) q Rq ′ L → q R χ 0 1 q ′ L τ + τ − χ 0 1 (64%)
•qq, 2 jet 1 lepton channel
q Rq ′ L → q R χ 0 1 q ′′ L τ ν τ χ 0 1 (100%)
At the analysis/ reconstruction level, we here require to have a minimal jet multiplicity, which leads to much larger event numbers especially for signatures with a smaller number of leptons. We equally do not apply any dedicated additional channel-based cuts. Figures 4 and 5 show the true √ŝ , parton level √ŝ min , reconstruction level √ŝ min as well as the original calorimeter based variable with and without a cut in the magnitude of the pseudo rapidity |η| < 1.4 for several explicit final states. We observe a similar behavior as in the overall sample, cf. Fig. 1: the parton level variable peaks around the actual production threshold, while there is a shift to lower peak values for the analysis level quantity. In order to understand the origin of this shift, we investigate this for a final state which initially exhibits a large difference between these quantities. We consider the 2 jet 1 tau-lepton channel, where originally the √ŝ min peak positions differ by about 170 GeV. From Eq. (3), we see that the definition of √ŝ min depends on the following independently measured quantities:
E vis , P Z , | − → P T | = P T = E T = / E T .
In order to investigate the origin of the shift between the parton level and analysis level peak positions, we therefore consider each of these variables separately and plot the difference between the respective parton level and analysis level quantity; the results are shown in Figures 6 and 7. We observe that, while the differences between parton and analysis level P Z , P T basically peak around zero, there is an average discrepancy ∼ 50 − 100 GeV between the parton and analysis level total visible energy. However, this discrepancy is accounted for by the fact that when changing from parton to hadron level, we replace the (visible) parton level tau by the (visible) tau-jet and the (invisible) tau-neutrino:
τ part → τ jet + ν τ .
In this transition, we equally shift the four momenta of the tau neutrinos from the visible to the invisible contribution of the definition of √ŝ min (cf. Eq. (3)):
P part vis = ... + p part τ + ..., M part inv ≡ M inv P ana vis = ... + p τ jet + ..., M ana inv = M part inv + m ντ .
We consider the neutrinos to be massless; therefore, we can leave the sum of all invisible particles' masses M inv unchanged. As the original variable definition of √ŝ min and the subsequent correlation in Eqn. (6) only depend on the heavy initial cascade particles, but not on the actual number of visible and invisible decay products, the observed change in the visible energy due to the escaping neutrinos at analysis level should then be compensated by associated changes in P Z , P T on an event by event basis, leading to a similar peak behavior of the √ŝ min distribution at parton and analysis level. Here, in order to assess the overall impact of this shift and a possible poor reconstruction of the tau decay products 10 , we perform a gedankenexperiment and change into a more ideal world where we idealistically reverse the analysis level tau-jet reconstruction and take the parton level tau four-vectors for the analysis level variable. In this case, the shift in the peak position of √ŝ min reduces to roughly 130 GeV. An alternative though less sophisticated way to determine the peak position is to consider the bin which contains a maximal number of entries; using this definition of the distribution peak position, the original shift between parton level and analysis level √ŝ min reduces from 200 GeV to 90 GeV if the parton level tau vectors are used at analysis level. A similar study for the 2 tau lepton 2 jet channel, which originally equally exhibits a quite large shift between the peak positions, shows that the effect of tau misidentification is O(100GeV) for both peak position definitions, reducing to ∼ 100 GeV in both cases when tau misidentification is removed. A similar effect can be observed for other specific final state signatures, cf. Table 5.
Although we still obtain a quite large shift for specific final state signatures, we have seen that, when using parton level taus for the analysis level observable and therefore suppressing possible effects from poor tau reconstruction, the inclusive sample peaks at the same value for both parton and analysis level √ŝ min distributions, cf. Fig. 2. We therefore conclude that, with correctly identified analysis level objects, the peak position of the parton level √ŝ min can indeed be reconstructed from generator level measurements 11 ; however, we want to emphasize that the correlation between the peak position and the actual heavy particle production threshold only exists in the form of a conjecture which lacks a rigorous proof. In case the conjecture proves to hold in all cases, the analysis level √ŝ min variable indeed gives a quite easy grasp on the threshold of the new physics pair-produced particles. Although this analysis was done in a specific scenario, where only certain initial heavy particle spin states are allowed, we saw that our conclusions hold for all possible spin combinations we considered. As our study relies on purely kinematic variables, we are therefore confident that these also hold for other spin combinations both for the heavy initial pair-produced particles as well as the particles in the decay chains, i.e. especially for other (also non-SUSY) BSM scenarios.
Comment on additional soft physics effects
The data set used in this study contains soft physics in the form of initial and final state radiation as described in Section 3, but no simulation of underlying event or pileup. However, the criticism which was expressed by the authors of [11] exactly concerns the dependence of soft physics in terms of ISR, which has been addressed in this work. 11 We want to point out that the reconstruction level objects in Table 2, through their definition by p T and η cuts, still depend on these two parameters; therefore, a recovery of the hard scale from reconstruction level objects will always be obstructed by an implicit dependence on the cut values in the analysis object definitions. However, in contract to the calorimeter-based variable and the cut in pseudorapidity originally proposed in [10], we here use object level definitions which are more optimized to the reconstruct the hard scattering event. We thank B. Webber for bringing this point to our attention.
Additionally, soft physics can enter in the form of minimum bias events, underlying event and pileup. We believe that the cuts in Table 2 are sufficiently hard enough to suppress minimum bias events ( [31], as well as section 6.1 in [32])). Underlying event as well as pileup effects can still distort the overall result for the peak position; however, we believe that these issues should be pursued in an experimental study, in combination with a collaboration internal full detector simulation. We can give a first estimate of the effect of underlying event fake P T contributions by adding ∆P fake T = 10 GeV, which corresponds to a conservative upper limit of the average P T from the underlying event [32,33], in the definition of / E T in Eq. (3); in this case, the best fit value from 400 GeV ≤ √ŝ min ≤ 1400 GeV is again given by In fact, as M inv and / E T appear in the same form in the definition of √ŝ min , the generic effects of additional fake P T s can be estimated from Figure 3. From underlying events, ∆P fake T 100 GeV, and we therefore estimate the uncertainty related to underlying event fake transverse momentum to be generically much smaller than the tau reconstruction effects discussed above. A more realistic investigation of these experimentally dominated effects, which should include a full detector simulation, is beyond the scope of this work 13 .
SM background
In this section, we investigate √ŝ min when SM background is included, as well as its use for the reduction of SM background in new physics searches. As an example, we consider W + jets and tt + jets background. Due to the large cross sections, we applied an additional / P T filter in the generation of the SM data sample, cf. Table 6. The cross sections after these additional cuts are given in Table 7. As before, we investigate the peak position for √ŝ min at parton and analysis level when using the true BSM value M inv = 2 m χ 0 1 . From Fig. 8 we see that in the total number of events after the cuts the peak structure disappears when no further SM cuts are applied. However, assuming an accurate enough (data or Monte Carlo driven) background subtraction, the peak structure is clearly visible again and much larger than the statistical error, cf. Fig. 8. A similar behavior is observed when we vary the input variable M inv : Fig. 9 shows the behavior for M inv = 0 GeV, 1000 GeV respectively after SM background subtraction; we see we obtain a clear BSM signal. We therefore conclude that, at least for the 12 Simulations using more recent tunes for a 14 TeV LHC, as eg the Pythia C4 tune [34], point to additional ∆P fake T ∼ 30 − 40 GeV for similar/ less stringent object definition values [35]; however, following the above considerations the induced error then is certainly ∼ GeV, which again corresponds to a relatively small uncertainty in the determination of √ŝ peak min . 13 In fact the experimental collaborations are already applying algorithms to subtract E T due to underlying event; cf eg [36]. We thank S. Wahrmund for bringing this to our attention.
n leptons | / P T,min | < 2 80 GeV 2 40 GeV > 2 0 GeV Table 6: Additional filters on magnitude of the total missing transverse momenum | / P T | applied for SM background generation, depending on the number of final state leptons n leptons . Leptons are required to obey the cut criterium |p T | > 5 GeV for the magnitude of the transverse momentum and |η| < 3.2 for the magnitude of pseudorapidity. Figure 8: Analysis level √ŝ min after a cut √ŝ min > 700 GeV. Dominant six SM backgrounds after cut (W + 2 j, W + 3 j, W + 4 j, tt, tt + 1 j, tt + 2 j) are included. Left: SM+BSM (green, dotted; 136834 events) and SM only (black, solid; 108017 events). In the sum and without further suppression cuts, the peak structure disappears. Right: Difference between (SM+BSM) and (SM). Assuming the SM background is well-known, the peak structure of the BSM signal is recovered. The difference is much larger than the statistical error. Assuming the SM background is well-known, the peak structure of the BSM signal is recovered. The difference is much larger than the statistical error.
parameter point studied here, with SM background being well-known, √ŝ min can be used as a BSM discovery variable and that for a true input value of M inv , the scale of the new physics can be derived from the peak of both parton level and (properly defined) analysis level quantities.
We furthermore assess the use of √ŝ min as a cut variable for SM background suppression. For this, we investigate the position of the peak for the different SM background channels considered here, where we again use M inv = 2 m χ 0 1 . The respective values are given in Table 8. For a first estimate of these positions, we do not need to perform a more sophisticated fit, and we therefore follow the simplified approach by defining the peak positions according to the bin which has the maximal number of entries. We see that the most dominant SM background channels have distribution peaks around 500 − 700 GeV, while the BSM signals peak at higher values. We therefore apply two different cuts of √ŝ min ≥ 700 GeV and √ŝ min ≥ 800 GeV on all samples; the cross sections after these cuts are summarized in Table 9. We see that, for both cut values, while we only cut out around 10% of the BSM signal, the dominant SM channels are suppressed by factor 3-6. We therefore conclude that √ŝ min can easily be used as a variable for SM background suppression, even for wrong guesses for the total invisible mass M inv . In the previous sections, we discussed how in our sample the peak of the √ŝ min variable is correlated with the real threshold for the hard production cross sections only if the correct value input for M inv is used, cf. Eqn. (6). For the background suppression, however, a correct guess or estimate of this value from other sources is not necessary, and we equally obtain a good SM background suppression with wrong input values for M inv 14 . (7)) and M eff (Eq. (10)) respectively, where M inv = 2 m χ 0 1 . While the maximal suppression factor for the BSM signal is around 1.11, the SM backgrounds are suppressed by factors 2 − 5. Equal results, however, can easily be obtained by a cut on M vis or M eff .
Comparison with other (transverse) variables
The strength of √ŝ min vs other (transverse) variables lies in the conjecture, given by Eqn. (6), about the direct correlation between its peak position (given the correct mass M inv ) and the rise of the parton level production cross section. However, this correlation has so far not risen beyond the status of a conjecture. We therefore briefly discuss two other variables which might serve a similar purpose in background subtraction, namely
M vis = E 2 vis − − → P 2 vis(7)
and
/ E T = | / − → P T |,(8)
where everything is defined at analysis object level. In [10], two more variables, namely E T and H T = E T + / E T , are studied 15 ; however, as we define
/ − → P T = − − → P T(9)
these two variables are only variations of / E T and therefore not discussed here. Figure 10 shows the subtracted (BSM+SM) − (SM) distributions of M vis and / E T respectively; especially the former looks quite promising. Indeed, a cut M inv > 500 GeV reduces the SM background by a factor 6 (W + jets) and 3 (tt+jets), cf. Table 9. We equally compare to the frequently used variable M eff [6,37]
M eff = vis |p T | + / E T ,(10)
which exhibits a similar power for background suppression as M vis , cf. Table 9. We therefore conclude that, for background suppression, both M vis , M eff as well as √ŝ min work in a similar way; the advantage of the latter variable is the (conjectured) M invdependent correlation between the its peak position and the hard (=parton level) process center-of-mass energy, if this can be proven to hold in all cases. Indeed, a similar conjecure of a linear correlation between the SUSY scale and the peak position of M eff [6,37] has recently been shown not to hold in all cases [38]; however, an equivalent systematic study of √ŝ min is still lacking.
Conclusion and outlook
We investigated the variable √ŝ min for a complete BSM sample which includes all strong production as well as all possible decay chains. In our analysis, we include both soft as well as detector effects by including a complete parton shower as well as a generic detector simulation and reconstruction-level objects, which have been defined such that the parton-level variable can be recovered quite accurately. We investigate the variable √ŝ min for a fully inclusive sample which sums over all possible final states of the hard scattering process, as well as for dominant exclusive final states. We see that, on parton level and for a correct input value of M inv , the √ŝ min variable peaks 7)) (left) and / E T (Eq. (8)) (right) distributions for the total inclusive sample. No further cuts were applied. closely to the heavy particle production cross section which is in agreement with the conjecture made by the authors of [10]. In a comparison between parton level and analysis level quantities, we see that in our sample the largest shift between these arises from the transition from tau-leptons which were used for the parton level quantity to the tau-jets and associated invisible neutrinos, which were used at analysis level. In order to asses this effect, we used the true tau-lepton four-vectors in the analysis level quantities. The effect is usually of the order of 100 GeV, and in the totally inclusive sample the shift completely disappears for the idealized case of perfectly reconstructed tau jets. We therefore conclude that for the parameter point considered here, even at analysis level, the parton level √ŝ min peak position can be sufficiently reconstructed 16 . In case the correlation between the threshold of the parton level cross section and the peak of the √ŝ min distribution could be proven rigorously, this would indeed provide a quite elegant and straightforward way to assess the scale of the new physics signal as a function of the total invisible mass of the process. Furthermore, we present the first study which investigates the use of √ŝ min in order to suppress SM background for BSM searches. For this, we considered W + jets as well as tt+jets background. We saw that these backgrounds could be sufficiently reduced by cuts on √ŝ min , leading to suppression factors around 2 − 6, while we retained 90% of the BSM signal. This feature was independent of the input value of the total invisible mass M inv . However, we could achieve similar results by a cut on the total visible mass M vis , which is a simpler variable which additionally does not require the input of M inv . A further comparison with M eff as a cut variable lead to similar results. We therefore conclude that, unless the conjecture about mass particle threshold and peak position of √ŝ min can be rigorously proven, the latter does not exhibit significant advantages over other (transverse) variables. However, if the correlation between the threshold and √ŝ min could be rigorously proven, it would indeed provide a simple and elegant hold on the scale of new physics processes. A further investigation of this relation is in the line of future work.
B Minimization of √ŝ
As stated in Section 2, √ŝ min denotes the minimal center of mass energy √ŝ of an event with a measured visible four-vector P µ vis = (E, − → P T , P Z ) which is still in agreement with total energy momentum conservation as well as onshellness of all outgoing particles. We equally assume the event to be at rest in the transverse plane such that Eqn. (4) holds. The generic expression forŝ in this case is given bŷ
s ( − → P T = − − → / P T ) = E + j E j 2 − P Z + j p jz 2(11)
where the index j goes over the invisible particles in the event with the respective energies E 2 j = m 2 j + p 2 jT + p 2 jz . Here and in the following, we omit the vector notation in p T for simplification, but all transverse quantities should be read as p jT = (p jx , p jy ), / P T = ( / P X , / P Y ), etc.
We use a Lagrange multiplier λ to take the additional constraint for the vector sum of the transverse momenta j p jT = / P T (12) into account. We therefore aim at minimizing
L =ŝ − λ j p jT − / P T ,
ie we try to find the values of the invisible particles' three-momenta − → p i such that
∂L ∂ − → p i = 0.
If we consider the case of n inv invisible particles in the event, we obtain 3 × n inv equations
∂L ∂p iT = 2 E + j E j p iT E i − λ = 0, ∂L ∂p iz = 2 E + j E j p iz E i − 2 P z + j p jz = 0.(13)
Together with the constraint in Eqn. (12), we now have in total 3 n inv + 1 constraints for 3 n inv + 1 unknowns (p ix , p iy , p iz ; λ). From Eqns. (13), we immediately see that
E i E j = p iT p jT = p iz p jz = c ij with c ij = m i m j ≡ const.
Combining this with Eqn. (12), we obtain
m i p iT = j m j / P T ≡ M inv / P T ,(14)
where we defined M inv = j m j to be the sum over the masses of all invisible particles, c.f. Eqn. (2). We therefore have
p iT = / P T M inv m i .
We can now rewrite the second equation in (13) and obtain
E + E i p iT / P T p iz E i − P z + p iz p iT / P T = 0.
Solving this for p iz leads to
p iz = P z m i E 2 − P 2 z 1 + / P 2 T M 2 inv .
We here have reproduced the solutions for p i,z , p i,T given in [10] which minimizeŝ. Inserting these into Eqn. (11) then leads to √ŝ min given by Eqn. (1), which denotes the minimal hard scattering center of mass energy which is allowed by energy momentum conservation for a specific visible total four vector (E, P T , P Z ) obtained from measurement. The only unknown quantity is M inv defined according to Eqn. (2), which has to be treated as an external input parameter for √ŝ min .
We want to comment that the transverse mass variable M T [39,40,41,42] has a functional form similar to √ŝ min as given in Eqn. (1). For a system with visible and invisible total four-vectors P µ vis , P µ inv , this variable is defined as
M 2 T = (E T,vis + E T,inv ) 2 − P T + / P T 2 .
with the transverse energies
E 2 T,vis = M 2 vis + P 2 T , E 2 T,inv = (M ′ inv ) 2 + / P 2 T . Here, M vis , M ′
inv denote the Lorentz-invariant masses of the total visible and invisible system respectively, M 2 vis = P 2 vis , (M ′ inv ) 2 = P 2 inv , which vary on an event by event basis. Assuming Eqn. (4) to hold, it follows that
M T = / E 2 T + M 2 vis + / E 2 T + (M ′ inv ) 2 .
We now see that the functional forms of M T (M ′ inv ) and √ŝ min (M inv ) as given in Eqn. (1) are identical, and differences between the variables only stem from the difference between M ′ inv and M inv . Indeed, for a correct guess of M inv , √ŝ min and M T coincide if for the invisible particles in the event
i = j m i m j = i = j p i · p j .
Componentwise, this equation is only fulfilled if we either have a complete set of massless particles which are all collinear with each other such that cos θ ij = 1 for all (i, j) pairs or for a complete set of massive particles which are all produced at rest. In general, however,
M ′ inv > M (true) inv and therefore M T > √ŝ min M (true) inv
on an event by event basis.
C Pythia 6.4 ISR/ FSR default setup
All switch descriptions here are taken from [23]. We equally refer the reader to section 10 of this reference for a more detailed discussion of the parton shower model and its implementation in Pythia.
MSTP (32) : (D = 8) Q 2 definition in hard scattering for 2 → 2 processes. For resonance production Q 2 is always chosen to beŝ = m 2 R , where m R is the mass of the resonance. The newer options 6-10 are specifically intended for processes with incoming virtual photons. These are ordered from a 'minimal' dependence on the virtualities to a 'maximal' one, based on reasonable kinematics considerations. The old default value MSTP(32) = 2 forms the starting point, with no dependence at all, and the new default is some intermediate choice. Notation is that P 2 1 and P 2 2 are the virtualities of the two incoming particles, p ⊥ the transverse momentum of the scattering process, and m 3 and m 4 the masses of the two outgoing partons. For a direct photon, P 2 is the photon virtuality and x = 1. For a resolved photon, P 2 still refers to the photon, rather than the unknown virtuality of the reacting parton in the photon, and x is the momentum fraction taken by this parton. anomalous photoproduction events, and matching to primordial k ⊥ . = 1 : for anomalous photons, the lower Q 2 cut-off is the larger of PARP(62) 2 and VINT(283) or VINT(284), where the latter is the virtuality scale for the γ → qq vertex on the appropriate side of the event. The VINT values are selected logarithmically even between PARP(15) 2 and the Q 2 scale of the parton distributions of the hard process. = 4 : a stronger damping at large k ⊥ , like dk 2 ⊥ /(k 2 ⊥ + Q 2 /4) 2 with k 0 < k ⊥ < p ⊥min (W 2 ). Apart from this, it works like = 1. = 5 : a k ⊥ generated as in = 4 is added vectorially with a standard Gaussian k ⊥ generated like for VMD states. Ensures that GVMD has typical k ⊥ 's above those of VMD, in spite of the large primordial k ⊥ 's implied by hadronic physics. (Probably attributable to a lack of soft QCD radiation in parton showers.) MSTP(67) : (D = 2) possibility to introduce colour coherence effects in the first branching of the backwards evolution of an initial-state shower in PYSSPA; mainly of relevance for QCD parton-parton scattering processes. = 2 : restrict the polar angle of a branching to be smaller than the scattering angle of the relevant colour flow. Note 1: azimuthal anisotropies have not yet been included. the k 2 ⊥ of the γ → qq vertex. For elastic and diffractive scatterings, m 2 /4 is stored, where m is the mass of the state being diffracted. For clarity, we point out that elastic and diffractive events are characterized by the mass of the diffractive states but without any primordial k ⊥ , while jet production involves a primordial k ⊥ but no mass selection. Both are thus not used at the same time, but for GVMD/anomalous photons, the standard (though approximate) identification k 2 ⊥ = m 2 /4 ensures agreement between the two applications.
VDM/ GVDM are acronyms for vector meson dominated/ generalized vector meson dominated events in photo production respectively (cf section 7.7.2 of [23]).
Fig. 2 .Figure 1 :
21The average heavy particle threshold in our sample is true (average) (m 1 + m 2 ) : 1146 GeV which again agrees with the parton level value of √ŝ peak min on permill level within the sqrts(min) Sum ofqq,qg, , andgg initial states, whereqg initial states dominate. All final states which fulfill object definitions from
9 Figure 2 :
92Several other widely used variables, as eg the original definition of M T 2 [4], equally exhibit a sqrts(min) As Fig. 1, where the hard matrix element tau four-vectors were used for the analysis level √ŝ min . With the differences due to tau identification at the analysis level removed, parton and analysis level peak positions agree within error bars. Explicit numbers are given in Section 4.
√ŝ peak;ana,τ = τp min ) in GeV, where the value in the respective first line arises from a Gaussian fit around the peak, while the second corresponds to the more simplified definition of the peak position by maximal number of bin entries. In addition, we give the average threshold value √ s th = (m 1 + m 2 ) for each sample, where m 1,2
Figure 3 :
3Parton level √ŝ part (red; solid), and parton level √ŝ part min dependence on the input value M inv for varying values.
Figure 4 :
4as Fig. 1 L = 1 fb −1 : Left figure for exactly 0 leptons in the final state, 3 hardest jets, 14249 events), and shift between parton level and analysis level √ŝ min is about 100 GeV. Right figure for exactly 2 leptons in the final state, 2 hardest jets (2745 events). Here, the shift between parton level and analysis level √ŝ min is about 200 GeV (reduces to 100 GeV if parton level tau vectors are used) specific final states, specified by the number of visible final state leptons (l), jets (j), and τ -leptons (τ ), corresponding to dominant decay chains in the complete SPS1a sample. Values for the peak position of the analysis level quantity with perfect tau jet reconstruction ( √ŝ peak;ana,τ = τp min ) are also given. For most final states, the effect of the peak shift due to imperfect tau jet reconstruction is O(100 GeV).
Figure 5 :Figure 6 :Figure 7 :
567as Fig. 1 L = 1 fb −1 , exactly 1 tau lepton in the final state, 2 hardest jets (3260 events) Shift between parton level and analysis level peak is 170 GeV(left) and reduces to 130 GeV(right) when parton level tau vectors are used for analysis level objects.Pt(parton)-Pt(ana) Difference between parton level and analysis object level total transverse momentum (left) and tau jet energy (right) for the 1 tau 2 jet channel. While the shift between the two values for the transverse momentum peaks around around zero, the shift between the parton level tau and analysis level tau jet energy is quite large, due to the escaping neutrino in the tau jet reconstruction. The difference in the P Z distribution (not shown here) exhibits a similar behaviour as the P T distribution.E(parton)-E(ana)Difference between parton level and analysis object level total visible energy for the 1 tau 2 jet channel. Left side shows the real difference, while on the right hand side analysis level tau jets were replaced by parton level taus. While we originally observe a large shift between the two values, originating from the escaping tau neutrinos on reconstruction object level and with a peak on the order of O(200 GeV), the distribution of the difference peaks around zero when parton level tau four-vectors are used for the calculation of the analysis level observable.
agrees with the value without the addition of ∆P fake T . Changing the fake additional transverse momentum to ∆P fake T = 100 GeV leads to the result 12 √ŝ peak min (∆P fake T = 100 GeV) : (1093 ± 4) GeV.
n jets W + n jets tt + n
Figure 9 :
9Difference between (BSM+SM) and (SM) for M inv = 0 GeV and a cut √ŝ min > 500 GeV (left; 29815 events) as well as M inv = 1000 GeV and a cut √ŝ min > 1500 GeV (right; 27802 events).
analysis object definitions only; M inv = 2 m χ 0 1 . Last two columns give cross sections σ after √ŝ min cuts respectively. After a minimal analysis level cut on √ŝ min , the W and tt backgrounds are reduced by factors 3 − 6, while we maintain roughly 90 % of the BSM signal.
Figure 10 :
10Difference between total and SM M vis (Eq. (
:
Relevant masses for SPS1a in GeV. u = (u, c), d = (d, s), l = (e, µ).
/2. ensure that the Q 2 scale is always bigger than P 2 . MSTP(62) : (D = 3) level of coherence imposed on the space-like parton-shower evolution. = 3 : Q 2 /p 2 ⊥ values and opening angles of emitted (on-mass-shell or time-like) partons are both strictly ordered, increasing towards the hard interaction. MSTP(63) : (D = 2) structure of associated time-like showers, i.e. showers initiated by emission off the incoming space-like partons in PYSSPA. = 2 : a shower may evolve, with maximum allowed time-like virtuality set by phase space or by PARP(71) times the Q 2 value of the space-like parton created in the same vertex, whichever is the stronger constraint. MSTP(64) : (D = 2) choice of α s and Q 2 scale in space-like parton showers in PYSSPA. = 2 : first-order running α s with argument PARP(64)k 2 ⊥ =PARP(64)(1 − z)Q 2 . MSTP(65) : (D = 1) treatment of soft-gluon emission in space-like parton-shower evolution in PYSSPA. = 1 : soft-gluon emission is resummed and included together with the hard radiation as an effective z shift. MSTP(66) : (D = 5) choice of lower cut-off for initial-state QCD radiation in VMD or
Note 2 :
2for subsequent branchings, MSTP(62) = 3 is used to restrict the (polar) angular range of branchings. MSTP(68) : (D = 3) choice of maximum virtuality scale and matrix-element matching scheme for initial-state radiation. To this end, the basic scattering processes are classified as belonging to one or several of the following categories (hardcoded for each process): = 0 : maximum shower virtuality is the same as the Q 2 choice for the parton distributions, see MSTP(32). (Except that the multiplicative extra factor PARP(34) is absent and instead PARP(67) can be used for this purpose.) No matrix-element correction. = 3 : as = 0, but ME corrections are applied where available. MSTP(69) : (D = 0) possibility to change Q 2 scale for parton distributions from the MSTP(32) choice, especially for e + e − . = 0 : use MSTP(32) scale. MSTP(72) : (D = 1) maximum scale for radiation off FSR dipoles stretched between ISR partons in the new p ⊥ -ordered evolution in PYPTIS. = 1 : the p ⊥max scale of FSR is set as the p ⊥ production scale of the respective radiating parton. Dipoles stretched to remnants do not radiate. The additional switches/ variables appearing above are given by PARP(15) : (D = 0.5 GeV) lower cut-off p 0 used to define minimum transverse momentum in branchings γ → qq in the anomalous event class of γp interactions, i.e. sets the dividing line between the VMD and GVMD event classes. PARP(62) : (D = 1. GeV) effective cut-off Q or k ⊥ value (see MSTP(64)), below which space-like parton showers are not evolved. Primarily intended for QCD showers in incoming hadrons, but also applied to q → qγ branchings. PARP(64) : (D = 1.) in space-like parton-shower evolution the squared transverse momentum evolution scale k 2 ⊥ is multiplied by PARP(64) for use as a scale in α s and parton distributions when MSTP(64) = 2. PARP(67) : (D = 4.) the Q 2 scale of the hard scattering (see MSTP(32)) is multiplied by PARP(67) to define the maximum parton virtuality allowed in Q 2 -ordered space-like showers. This does not apply to s-channel resonances, where the m aximum virtuality is set by m 2 . It does apply to all user-defined processes,however. PARP(71) : (D = 4.) the Q 2 scale of the hard scattering (see MSTP(32)) is multiplied by PARP(71) to define the maximum parton virtuality allowed in time-like showers. This does not apply to s-channel resonances, where the maximum virtuality is set by m 2 . Like for PARP(67) this number is uncertain. VINT(283), VINT(284) : virtuality scale at which a GVMD/anomalous photon on the beam or target side of the event is being resolved. More precisely, it gives
Table 3 :
3Peak positions for separate heavy initial cascade particles for parton level (√ŝ peak;parton
min
) and analysis level (
√ŝ peak;ana
min
) quantities as well as analysis level quantity
for idealized tau jets (
final states, hard matrix element main source N hard N ana0 leptons, 3 jetsqg
4480 14247
2 leptons, 3 jetsqg(97%)
4020 2092
1 lepton, 3 jetsqg(99.99%) 3740 5282
2 leptons, 2 jetsqq
1776 2745
1 lepton, 2 jetsqq
1366 6997
Table 4 :
4Number of events for dominant parton level decay modes, characterized by
specific visible final states, on parton level (N hard ) and at analysis level (N ana ). At
analysis level, the jet number requirement for event selection is changed from an exact
equality to a minimal number of jets. If not stated otherwise, the main source provides
all events with a specific signature on the parton level. Examples for dominant decay
chains leading to the specific parton-level final states are given in Section 4.1.
Table 7 :
7Cross sections in pb for SM background with Alpgen; filters in Tab. 6 were applied in the generation stage.
Table 8 :
8Cross sections σ for SM background processes and
√ŝ
min maximal bin po-
sitions for parton level (
Table 9 :
9last four columns: cuts on M vis (Eq.Total cross sections σ [pb] for BSM as well as W + jets and tt + jets back-
grounds, without and with several cuts on different inclusive quantities. First column:
no cut; second to fifth column: values for
√ŝ
cut = M inv + 500 GeV, with varying M inv
values;
Excellent reviews about different mass determination variables and their use, including advantages and disadvantages, have recently been published in[2,3].
For consistency, we here adopted the notation introduced in [10] for M inv and hope that the potentially misleading nomenclature does not cause confusion in the remainder of our discussion.
These cuts closely follow cuts used in the SUSY analysis studies in[8,30]. Due to the relatively high p T jet cuts, together with a high E miss T cut and lepton isolation criteria, we expect minimum bias events to be sufficiently suppressed ([31], as well as section 6.1 in[32]).
Note that, in case the shift cannot be explained by poor reconstruction of the decay products alone, this equally opens the window to a possible topology-dependence of √ŝ min ; we thank K.Sakurai for pointing this out.
We want to remind the reader that the same value of M inv for the calculation of √ŝ min needs to be used in both SM and BSM samples; the correlation with the threshold however only holds for the sample with the equivalent correct M inv . We thank K. Matchev for reemphasizing this point.
Note that the definition of H T differs in[13], where it basically is set to the variable M eff[6,37].
We point out that, for the scenario considered in this study, even when using analysis tau jets the maximal shift between parton level and analysis level peak positions was ∼ 200 GeV, which effectively leads in an error ∼ 100 GeV in the estimation of the initial heavy particles masses. The magnitude of this effect can of course differ depending on the BSM model and as well as specific model scenario point.
Acknowledgements I want to thank the members of the 2009 Les Houches Mass determination group for providing me with the data as well as parts of the analysis framework used in this study, and specifically J.-R. Lessard and R. Brunelière for answering additional questions. Large parts of this work were inspired by discussions with members of the Cambridge SUSY group, and special thanks goes to B. Allanach, C. Lester, B. Webber, A. Papaefstathiou, and K. Sakurai for many valuable comments. I am equally grateful to K. Matchev for commenting on the manuscript prior to publication, as well as to P. Bechtle, M. Kobel, S. Wahrmund, and especially D. Kar for discussions of underlying event effects at a 14 TeV LHC. This work was supported by the STFC.
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