Spectra and Chaos in the SYK Model

Transcription

1 SYK, Riverhead 2017 p. 1/44 Spectra and Chaos in the SYK Model Jacobus Verbaarschot Riverhead, July 2017

2 SYK, Riverhead 2017 p. 2/44 Acknowledgments Collaborators: Antonio Garcia-Garcia (Jiaotong University)

3 SYK, Riverhead 2017 p. 3/44 References Antonio Garcia-Garcia and J.J.M. Verbaarschot, Spectral and Thermodynamical Properties of the SYK model, Phys. Rev. D94 (2016) [arxiv: ]. Antonio Garcia-Garcia and J.J.M. Verbaarschot, Spectral and Thermodynamical Properties of the SYK model, Phys. Rev. D (accepted for publication) [arxiv: ]. Mario Kieburg, J.J.M. Verbaarschot and Savvas Zafeiropoulos, Dirac Spectra of Two-Dimensional QCD-Like Theories, Phys. Rev. D90 (2014) [arxiv; ]. J.J.M. Verbaarschot and M.R. Zirnbauer, Replica Variables, Loop Expansion and Spectral Rigidity of Random Matrix Ensembles, Ann. Phys. 158, 78 (1984)

4 SYK, Riverhead 2017 p. 4/44 Contents I. Introduction II. The SYK model III. Spectral Density of the SYK model IV. Spectral Correlations V. Conclusions

5 SYK, Riverhead 2017 p. 5/44 Introduction Compound Nucleus Random Matrix Theory Two-Body Random Ensemble

6 SYK, Riverhead 2017 p. 6/44 Quantum States of Black Hole A black hole is a finite system and therefore has discrete quantum states, in fact resonances because they decay. All information that goes into a black hole has been scrambled. Therefore, the information content of these quantum states should be minimized. What is the density of states? What are the correlations of the positions of the resonances? Let us have a look at another physical system with these properties.

7 SYK, Riverhead 2017 p. 7/44 Compound Nucleus n 238 U 239 U 238 U Compound Nucleus n Formation and decay of a compound nuclear are independent. Because the system is chaotic, all information on its formation got lost.

8 SYK, Riverhead 2017 p. 8/44 Quantum Hair of a Compound Nucleus Total cross section versus energy (in ev ). Garg-Rainwater-Petersen-Havens,1964

9 SYK, Riverhead 2017 p. 9/44 Holography in Nuclear Physics Because the compound nucleus is chaotic, the average cross-section, S ab 2 is only depends on the average diagonal S -matrix elements S ab 2 = F universal ( S cc ) with F universal a universal function. JV-Weidenmüller-Zirnbauer-1983, Mello-Pereyra-Seligman-1984 The average diagonal S -matrix is obtained by an energy average S cc (E) = 1 Γ dx π Γ 2 +(E x) 2S cc(x), where the width Γ is much larger than the level spacing. Therefore the average diagonal S matrix is determined by the fast processes, in which a compound nucleus is not formed, i.e. the physics that takes place at the surface of the nucleus.

10 SYK, Riverhead 2017 p. 10/44 Compound Nuclei, Chaos and Black Holes Most likely a compound nucleus saturates the quantum bound on chaos obtained recently by Maldacena, Shenkar and Stanford. Black holes are believe to saturate this bound as well. To some extent, a compound nucleus has no hair, as is the case for a black hole. Bohigas-Giannoni-Schmidt Conjecture: If a system is classically chaotic, its eigenvalues are correlated according to random matrix theory.

11 SYK, Riverhead 2017 p. 11/44 Nuclear Data Ensemble P(S) S Nearest neighbor spacing distribution of an ensemble of different nuclei normalized to the same average level spacing. Bohigas-Haq-Pandey, The eigenvalues of a large class of Random Matrix Ensembles are distributed according to as semi-circle in the limit of very large matrices

12 SYK, Riverhead 2017 p. 12/44 Motivation for the Two-Body Random Ensemble The nuclear level density behaves as e α E. The nuclear interaction is mainly a two-body interaction. Random matrix theory describes the level spacings, but it is and N-body interaction with a semicircular level density. T. von Egidy

13 SYK, Riverhead 2017 p. 13/44 Two Body Random Ensemble H = αβγδw αβγδ a αa β a γa δ. French-Wong-1970 Bohigas-Flores-1971 labels of the fermionic creation and annihilation operators run over N single particle states. The Hilbert space is given by all many particle states containing m particles with m = 0,1,,N. The dimension of the Hilbert space is: ( N m ) = 2 N. W αβγδ is Gaussian random. The Hamiltonian is particle number conserving. The matrix elements of the Hamiltonian are strongly correlated. Brody-et-al-1981, Brown-Zelevinsky-Horoi-Frazier-1997, Izrailev-1990,Kota-2001,Benet-Weidenmüller-2002,Zelevinsky-Volya-2004

14 SYK, Riverhead 2017 p. 14/44 First Numerical Results Comparison of the spectral density of the GOE and the two-body random ensemble for the sd-shell. Bohigas-Flores-1971

15 SYK, Riverhead 2017 p. 15/44 The Sachdev-Ye-Kitaev Model The SYK Model Partition Function

16 The Sachdev-Ye-Kitaev (SYK) Model The two-body random ensemble from nuclear physics also has merged into the SYK model, where the fermion creation and annihilation operators are replaced by Majorana operators (in general q of them. For q = 4 the model is Sachdev-Ye-1993,Kitaev-2015 H = α<β<γ<δ W αβγδ χ α χ β χ γ χ δ, q = 4. The fermion operators satisfy the commutation relations {χ α,χ β } = 1 2 δ αβ. The two-body matrix elements are taken to be Gaussian distributed with variance σ 2 = 6 N 3. SYK, Riverhead 2017 p. 16/44

17 SYK, Riverhead 2017 p. 17/44 Representation of Majorana Operators The χ α can be represented as a Clifford algebra. The elements of a Clifford algebra in N dimensions can be constructed recursively out of 2 2 Pauli σ - matrices γ α = σ k1 σ kn. These are 2 N/2 dimensional matrices, which have 2 N/2 nonzero elements. Can be easily constructed and diagonalized numerically up to N = 36.

18 SYK, Riverhead 2017 p. 18/44 Spectrum and Partition Function The partition function of N fermions with Hamiltonian H is given by Z(β) = Tre βh = deρ(e)e βe. The spectral density is thus given by the Laplace transform of the partition function. The partition function can be interpreted as the trace of time evolution operator in imaginary time. Feynman told us how to rewrite the time evolution operator as a path integral, Z(β) = e βf = Tre βh = Dχe β 0 dτ [χ d dτ χ+h(χ) ]. Sachdev-Ye-1992 where the χ are Grassmann valued functions of τ. Kitaev-2015

19 For q = 2 (noninteracting) the zero temperature entropy vanishes, see talk by Subir Sachdev. SYK, Riverhead 2017 p. 19/44 Physical Interpretation The partition function is that of a system of N/2 interacting fermions. The low-temperature expansion is thus given by βf = βe 0 + df dt T d2 F dt 2 = βe 0 +S ct, where E 0 is the ground state energy, S is the entropy and ct the specific heat. E 0, S and c are extensive. The total number of states for N fermions is 2 N/2, so that the noninteracting part of the entropy well away from the ground state is S = N 2 log2.

20 SYK, Riverhead 2017 p. 20/44 Bethe Formula The level density is given by the Laplace transform of the spectral density. ρ(e) = = r+i r i r+i r i dβe βe Z(β) dββ 3/2 e βe e βe 0+S+ c 2β The integral can be done resulting in ρ(e) = sinh( 2cE). This gives the Bethe formula for the nuclear level density. Bethe-1936

21 SYK, Riverhead 2017 p. 21/44 Spectral Density of the SYK Model Large N Limit Leading Corrections Analytical Result for the Spectral Density Bethe Formula

22 SYK, Riverhead 2017 p. 22/44 Spectral Density The spectral density can be obtained from the moments ( ) 2p TrH 2p = Tr W α Γ α with Γ α a product of four gamma matrices. The Gaussian integral is equal to the sum over all pair-wise contractions. When 2p N, the Γ α do not have common gamma matrices and they commute. Since α Γ 2 α = 1 all contractions contribute equally resulting in TrH 2p = (2p 1)!! TrH 2 ) p which gives a Gaussian distribution. Mon-French-1975, Garcia-JV-2016

23 SYK, Riverhead 2017 p. 23/44 Level Density 6 5 N = 34 Gaussian ρ(e) E The center of the spectum is close to Gaussian but the tail deviates strongly. Garcia-JV-2016

24 SYK, Riverhead 2017 p. 24/44 Moments for LargeN For large N, the moments can be calculated exactly if we ignore correlations between contractions. A product of four Majorana operators satisfies the commutation relations Garcia-Garcia-JV-2016 Γ α Γ β +( 1) p Γ β Γ α = 0, where p is the number of γ-matrices they have in common. α β α β α α β β This results in the suppression factor of intersecting relative to nested contractions ( ) 1 N ( )( ) q N q η N,q = ( 1) p. q p q p p

25 SYK, Riverhead 2017 p. 25/44 Spectral Density at FiniteN If α is the number of intersections, the moments are given by M 2p M p 2 = contractions η α = 1 (1 η) p p k= p ( ) 2p ( 1) k η k(k 1)/2, p+k where the sum has been evaluated by the Riordan-Touchard formula. Erdos-2014, Cotler-et-al-2016, Garcia-Garcia-JV-2017 These are the moments corresponding to the weight function of the Q-Hermite Polynomials. This results in the spectral density ρ QH (E) = c N 1 (E/E0 ) 2 k=1 [1 4 E2 E 2 0 ( 1 2+η k +η k )] with E 2 0 = 4σ2 1 η and σ the variance of the spectral density.

26 SYK, Riverhead 2017 p. 26/44 Accuracy of the Moments The result for the fourth moment is exact. The sixth (15 diagrams) and eigth order moments (105 diagrams) have been calculated exactly. One diagram for the sixth order moment and 20 diagrams for the eighth order moment do not agree with the Q-Hermite result with a difference of only O(1/N 2 ). Example of a sixth order diagram which is evaluated exactly to η 2 α β α γ β γ Sixth order diagram that is not given by the Q-Hermite formula α β γ α β γ

27 SYK, Riverhead 2017 p. 27/44 N Dependence of Sixth and Eight Moments ' " & &'()* +",-./0*- #% #$ #!!!"!!#!!$!! % $ #!!"!!#!!$!!%!! ' & % $!!% &'()* +",-./0*-!! "! #!! #"! $!! $"!!!! "! #!! #"! $!! $"!! $! #!! #" &'()* +",-./0*- '! &!!!"!!#!! #! "!!"!!#!!$!!%!! "! #!! #"! $!! $"!!!! %!!!$!!% $! &'()* +",-./0*-!! "! #!! #"! $!! $"!!

28 SYK, Riverhead 2017 p. 28/44 Comparison with Numerical Results!!!"!"&!"%!"# #"( #"'!"!"#$! &'!!"#$%!!!"!"$ $"( $"' $"&!"!"#$! &'!!"#$% #"& #"% $"% #"#!!"#!#"$ #"# #"$!"#! $"$!!"#!!"$!$"# $"$ $"#!"$!"#! Comparison of the exact spectral density obtained by numerical diagonalization and the Q-Hermite result for the spectral density. Garcia-Garcia-JV-2017

29 Large N Approximation For large N, the Q-Hermite form is up to exponentially small corrections given by ( logη 2q 2 /N ) [ 2arcsin 2 (E/E 0 ) ρ(e) = c N exp logη ]( [ 1 exp 4π logη ( arcsin(e/e 0) π ]) 2 ). For N 1, E not close to ± E 0, this simplifies to Garcia-Garcia-JV-2017 [ 2arcsin 2 ] (E/E 0 ) ρ asym (E) = c N exp. logη Very close to the ground state we can expand arcsin 2 (( E 0 +x)/e 0 ) = π2 4 π 2x/E 0, so that Cotler-et-al-2016, Garcia-JV-2017,Altland-Bagrets-Kamenev-2017 ] ρ(e) = e N 2 log2 N q 2 π 2 4 πn sinh[ 2(1 E/E0 2q 2 ). SYK, Riverhead 2017 p. 29/44

30 SYK, Riverhead 2017 p. 30/44 Comparison to the Bethe Formula #""(%!!" "#'!"!"#$% +""(%!!"!"!"#$%! +,! '()*(! #$%& "#&! %&!!"#$"!!!" *""(%!!" )""(%!!"!!!" "#% (""(%!!" "#$ %!!"#$%!!"#&#!!"#&%!!"#!#!!"#!%!!"#'#!!"#'%! "#"!! "!! Comparison of the exact Q-Hermite result to the Bethe Formula, ρ(e) sinh 2c(E E 0 ). The Bethe formula is valid in the very tail where the density is non-gaussian. Note that the number of states in the tail exp[n/2log2 Nπ 2 /4q 2 ] is exponentially larg but also expontially suppressed with respect to the state in the bulk.

31 SYK, Riverhead 2017 p. 31/44 Level Density and Free Energy Z(β) = 2 N/2 [ 2arcsin e βe 2 ] (E/E 0 ) exp. logη For large N this integral can be evaluated by a saddle point approximation. The result is given by with saddle point equation βf = 2 πv πvtan logη 2 (πv)2 2logη βj = πv cos πv 2 and v = (2/π)arcsinĒ/E 0 and J = (E 0 /2)logη. This is exactly the large N large q result obtained by Maldacana and Stanford valid for all β. Maldacena-Stanford-2015.,

32 SYK, Riverhead 2017 p. 32/44 Spectral Correlations of the SYK Model Spectral Correlators Symmetries and Classification

33 SYK, Riverhead 2017 p. 33/44 Upper Bound for Lyapunov Exponent Lyapunov exponent λ (t) (0)e λt Energy-time unertainty relation ( ) t ~ (0) e λ t t E 2 t 1/λ, So we have the bound λ 2πkT E πkt (0) Divergence of trajectories in a stadium at temperature T kt Maldacena-Shenker-Stanford-2015 Of the same type as the η/s bound by Son.

34 SYK, Riverhead 2017 p. 34/44 Spectral Correlations It has been shown hat the SYK model in maximally chaotic, in the sense that the Lypunov exponent saturates the bound of λ L 2πkT. Kitaev-2015, Maldacena-Shenker-Stanford-2016 If this is the case its spectrum should behaves as a quantum chaotic system, i.e. the eigenvalue correlations are given by random matrix theory with the corresponding random matrix ensemble determined by the anti-unitary symmetries.

35 Classification Summary N (C 1 K) 2 (C 2 K) 2 C 1 KC 2 K RMT Matrix Elements iγ 5 GUE Complex Γ 5 GSE Quaternion iγ 5 GUE Complex Γ 5 GOE Real iγ 5 GUE Complex Γ 5 GSE Quaternion Table 1: (Anti-)Unitary symmetries of the SYK Hamiltonian and the corresponding RMT. The symmetries are periodic in N modulo 8 (Bott periodicity). You-Ludwig-Xu-2016, Garcia-Garcia-JV-2016 A similar classification exisits for supersymmetric SYK models. Fu-Gaiott-Maldacena-Sachdev-2017, Li-Liu-Xin-Zhou-2017, Kanazawa-Wettig-2017 SYK, Riverhead 2017 p. 35/44

36 (For long times the sum is dominated by the diagonal terms with k = l) SYK, Riverhead 2017 p. 36/44 Spectral Observables P(S) : the distribution of the spacing of consequetive levels. Σ 2 (L) : the variance of the number eigenvalues in an interval that contains L levels on average. These spectral observables are calculating after mappig the spectrum on one with unit average level density. The mapping function is obtained from the average spectral density. To increase statistics we can perform a spectral average in addition to andensemble average for the calculation of of P(S) and Σ 2 (L). Spectral form factor g(β,t) = 1 Z 2 (β) k,l e (β+it)e k (β it)e l Z(2β) Z 2 (β)

37 SYK, Riverhead 2017 p. 37/44 Spectral Correlations Spectral Density ρ(x) = δ(x E k ). k Two point correlation function ρ 2 (x,y) = δ(x E k )δ(x E l ) kl = δ(x y)ρ(x)+ δ(x E k )δ(x E l ). k l The first term is due to self-correlations. The connected correlator is given by ρ 2c = ρ 2 (x,y) ρ(x)ρ(y).

38 SYK, Riverhead 2017 p. 38/44 Number Variance and Spectral Form Factor Number variance Σ 2 (n) = x0 +n x0 +n dxdyρ 2c (x,y) x 0 x 0 Σ 2 self(n) = n. Unfolded spectral form factor g(β,t) = dxdye (β+it)x (β it)y ρ 2 (x,y). Can be split into a connected part, a disconnected part and a part due to the self correlations. The part due to self correlations is given by g self (t) = dxρ(x)e 2βx = constant.

39 Number Variance Versus Spectral Form Factor Number variance (left) and spectral form factor (right). Σ 2 (L) is calculated starting at the 50th eigenvalue above the ground state.!"#!#!$""!!"!#!$!%!&!!&"! " # $ % &!!"#!$"! +%,,-+$-(! ()*+%,,,-+$-(! $%$&'!!"#!!!"!! #$ %&'%(&)*+%, Garcia-Garcia-JV-arXiv: Cotler-et-al-arXiv: SYK, Riverhead 2017 p. 39/44

40 SYK, Riverhead 2017 p. 40/44 Nearest Neigbor Spacing Distribution 0.8 P(s) N = 32 N = 24 GOE log P(s) P(s) 1 N = 36 N = 34 N = 28 GUE GSE s s s Nearest nieghbor spacing distribution for the bottom (left) and bulk part of the spectrum compared to random matrix theory. Garcia-Garcia-JV-2016, Garcia-Garcia-JV-2017 This is in agreement with results for the distribution of the ratio of consequetive spacings. You-Ludwig-Xu-2016

41 SYK, Riverhead 2017 p. 41/44 Number Variance in the Bulk 1 Σ 2 (L) N = 28 N = 32 N = 34 GUE GSE GOE Σ 2 (L) N = 22 N = 34 GUE L L Garcia-Garcia-JV arxiv: These results have been confirmed by an independent collaboration who calculated the spectral form factor which is the Fourrier transform of the spectral correlator. Cotler-Gur-Ari-Hanada-Polchinski-Saad-Shenker-Streicher-Tezuka-arXiv:

42 SYK, Riverhead 2017 p. 42/44 Tracy-Widom Distribution!!!" "$' "$# "$& "$!! #$%&'!"!"#$!!"!!!!" #! ' & %!! "#$%!!!&*!"!"#$!! '()!! "#$%!!!& "$% $ "$"!! "! #!!! "! #!! #"! $!!! Distribution of the ground state energy compared to the Tracy-Widom distribution of the Gaussian Orhtogonal Ensemble. There is no fitting the parameter of the Tracy-Widom distribution is fixed by equating its expectation value to the numerical one, at the point E = 0, is edge of the spectrum as predicted by the Q-Hermite expresssion. Garcia-Garica-JV-2017

43 SYK, Riverhead 2017 p. 43/44 IV. Conclusions The spectral density of the SYK model is given by a remarkably simple expression that is strongly non-gaussian in the ground state region.

44 SYK, Riverhead 2017 p. 43/44 IV. Conclusions The spectral density of the SYK model is given by a remarkably simple expression that is strongly non-gaussian in the ground state region. This expression reproduces the Bethe formula as well as the free energy obtained from path integral techniques.

45 SYK, Riverhead 2017 p. 43/44 IV. Conclusions The spectral density of the SYK model is given by a remarkably simple expression that is strongly non-gaussian in the ground state region. This expression reproduces the Bethe formula as well as the free energy obtained from path integral techniques. The density of state in the ground state region is exponentially large but expoentially smaller than the density in the bulk.

46 SYK, Riverhead 2017 p. 43/44 IV. Conclusions The spectral density of the SYK model is given by a remarkably simple expression that is strongly non-gaussian in the ground state region. This expression reproduces the Bethe formula as well as the free energy obtained from path integral techniques. The density of state in the ground state region is exponentially large but expoentially smaller than the density in the bulk. It is essential to include subleading corrections to the spectral density.

47 SYK, Riverhead 2017 p. 43/44 IV. Conclusions The spectral density of the SYK model is given by a remarkably simple expression that is strongly non-gaussian in the ground state region. This expression reproduces the Bethe formula as well as the free energy obtained from path integral techniques. The density of state in the ground state region is exponentially large but expoentially smaller than the density in the bulk. It is essential to include subleading corrections to the spectral density. The spectral correlations of the SYK model are given by the corresponding random matrix theory. There are deviations for long range correlations similar to those found in other disordered systems.

48 SYK, Riverhead 2017 p. 44/44 IV. Conclusions Continued The random matrix behavior extends all the way to the ground state.

49 SYK, Riverhead 2017 p. 44/44 IV. Conclusions Continued The random matrix behavior extends all the way to the ground state. These results are consistent with the idea that the SYK model is maximally chaotic.

50 SYK, Riverhead 2017 p. 44/44 IV. Conclusions Continued The random matrix behavior extends all the way to the ground state. These results are consistent with the idea that the SYK model is maximally chaotic. In a sense a black hole is dual to a compound nucleus.