Chaotic-integrable transition
and many-body localization ingeneralized SYK models
17:10 – 17:40, 27 Sep 2018
Anderson Localization and Interactions, MPI-PKS, Dresden

Masaki TEZUKA (Kyoto University)

Contents and collaborators• Chaotic-integrable transition in a generalized SYK modelWith Antonio M. García-García, Bruno Loureiro, and Aurelio Romero-Bermúdez

PRL 120, 241603 (2018) (arXiv:1707.02197)• Many-body localization in a short-range SYK modelWith Antonio M. García-García

arXiv:1801.03204• Quantum Lyapunov spectrum in SYK and XXZ models
With Hrant Gharibyan, Masanori Hanada, and Brian Swingle

arXiv:1809.01671

Part 1: Stability of chaos in the SYK modelAntonio M. García-García, Bruno Loureiro, Aurelio Romero-Bermúdez, and Masaki Tezuka,Phys. Rev. Lett. 120, 241603 (2018) (arXiv:1707.02197)
𝑁

෡SYK 𝑞=4 =𝐻

෍

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑

Various modifications of SYK have been studiede.g.
• Supersymmetric SYK [Fu, Gaiotto, Maldacena, and Sachdev 2016]• Non-random couplings [Witten, 1610.09758]• Higher-dimensional generalizations [Gu, Qi, and Stanford 2017][S. Banerjee and E. Altman 2017][Davison, Fu, Georges, Gu, Jensen, and Sachdev 2017]
Addition of new Fermi species can induce a transition to a Fermi liquid[Banerjee and Altman PRB 95, 134302 (2017)] or MBL transition [S.-K. Jian and H. Yao PRL 119, 206602 (2017)],
additional interaction can induce a metal-insulator transition [C.-M. Jian, Bi, and Xu PRB 96, 115122 (2017)], …

Phys. Rev. Lett. 120, 241603 (2018) (arXiv:1707.02197)

Our motivation and modelQ.: Minimum requirements for chaotic behavior? (→ gravity interpretation?)
Here we study a simple model with analytical + numerical methods

෡=𝐻𝑁

SYK4෍𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑Gaussian random couplings

𝑁SYK2

1≤𝑎<𝑏6𝐽
𝐽𝑎𝑏𝑐𝑑 : average 0, standard deviation 3Τ2𝑁𝐾𝑎𝑏 : average 0, standard deviation𝐾𝑁

SYK4 as unperturbed Hamiltonian,
𝐾 controls the strength of SYK2 (one-body random term, solvable)
Both terms respect charge parity in complex fermion description➔ Full numerical exact diagonalization (ED) of 2N/2-1-dimensional matrix, 𝑁 ≲ 34 possible

𝐽 = 1: unit of energy

In this work:𝑁 ≡ 2 (mod 4)
cf. Nosaka, Rosa, Yoon
Lunkin, Tikhonov, Feigel’man
Yu-Xiang, Ye, Liu

𝑆Τ 𝑘B 𝑁

Entropy per fermion

In SYK4, entropy S per fermion at low T → 0.2324 in large-N limit

SYK4
𝑘B 𝑇Τ𝐽[Cotler, …, and MT, JHEP 1705, 118 (2017)]

Large K (weak SYK4):
S / N almost
independent of N,and vanishing as T → 0𝑁

෡=𝐻෍𝑁𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑𝐽𝑎𝑏𝑐𝑑 : average 0, std. dev.

1≤𝑎<𝑏6𝑁3Τ2𝐾
𝐾𝑎𝑏 : average 0, std. dev.
𝑁Specific heat and large-N analysis
Low T: specific heat C(T) → linear in T

𝑁 = 34, full ED

Slope of C(T)

Large N: replica fields + Hubbard-Stratonovich transf.
[cf. Maldacena and Stanford 2016, Sachdev 2015]← SYK4 limit

𝜋 Τ6
SYK2 limit →Out-of-time order correlator (OTOC)

Chaos bound [Maldacena,Shenker, and Stanford 2016]

SYK4 limit

SYKq + κ SYK2
Large-q limit

𝐾=0non-chaotic𝑒 𝜆L 𝑡1 +𝑡2 Τ2 𝑓 𝑡1 − 𝑡2

𝐾 = 0.2chaotic

0.5
1

Lyapunov exponent is obtained by solving

2𝛽 = 1Τ 𝑘B 𝑇

Deviation from the chaos bound as
SYK2 component is introduced

PRL 120, 241603 (2018) (arXiv:1707.02197)

Small K: RMT-like behavior of energy spectrum
𝑃 𝑠 : level spacing distribution
Ratio of consecutive level spacing 𝐸𝑖+1 − 𝐸𝑖to the local mean level spacing Δ(requires unfolding of the spectrum)

SYK4 limit (small K): RMT(GUE (Gaussian Unitary Ensemble) if 𝑁 ≡ 2 (mod 4))

SYK2 (large K): Poisson (𝑒 −𝑆 )N=30, Central 10 % of eigenvalues
Also see: T. Nosaka, D. Rosa, and J. Yoon, JHEP 1809, 041 (2018) (arXiv:1804.09934) for other symmetry cases
cf. A. V. Lunkin, K. S. Tikhonov, and M. V. Feigel’man, 1806.11211; Y. Yu-Xiang, J. Ye, and W. M. Liu, 1809.07577, …

PRL 120, 241603 (2018) (arXiv:1707.02197)

Small K: RMT-like behavior of energy spectrum𝑟 : average adjacent gap ratio
Average ofmin 𝐸𝑖+1 −𝐸𝑖 ,𝐸𝑖+2 −𝐸𝑖+1
max 𝐸𝑖+1 −𝐸𝑖 ,𝐸𝑖+2 −𝐸𝑖+1

SYK4 (K = 0): RMT(≈ 0.599 for GUE [Y. Y. Atas et al. PRL 2013])

SYK2 (large K): Poisson (2 log 2 − 1 ≈ 0.386)

Central 10 % of eigenvalues

β=0K

Spectral form factor
shows robust ramp reflecting level rigidity

𝑁

Summary of Part 1

෡SYK4 =𝐻෍𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑“Stability of chaos in a generalized Sachdev-Ye-Kitaev model”• Effect of one-body term𝑁

෡=𝐻
෡SYK4 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏𝐻1≤𝑎<𝑏• No longer maximally chaotic• Random-matrix like spectra forweak perturbation• Temperature-dependent transition
to non-chaotic behavior

Phys. Rev. Lett. 120, 241603 (2018)(arXiv:1707.02197)

Interaction with randomness+ locality➔ many-body localization transition?

Other quantities for characterization ofchaotic / non-chaotic phases?

2. Short-range SYK4 + SYK2: many-body localization
Antonio M. García-García and MT, arXiv:1801.03204

For integer 𝐷, 𝑑ሚሚ𝐽𝑖𝑗𝑘𝑙 𝐷 = ቊ𝐽𝑖𝑗𝑘𝑙
0𝑙−𝑖 <𝐷𝑙−𝑖 ≥𝐷

෩𝑖𝑗𝐾෩𝑖𝑗 𝑑 = ൝𝐾0𝑗−𝑖 <𝑑𝑗−𝑖 ≥𝑑෩𝑖𝑗 : Gaussian random𝐽ሚ𝑖𝑗𝑘𝑙 , 𝐾෩ 0<𝐷෩<1 :Also consider fractional 𝐷 = 𝐷 + 𝐷෩ otherwise set to zeroIf 𝑙 − 𝑖 = 𝐷 , use non-zero 𝐽ሚ𝑖𝑗𝑘𝑙 with probability 𝐷,➔ We explore the possibility of a metal-insulator transition as 𝐷 is decreased(following numerical data: for 𝜅 = 1)

Level separation distribution

Gap ratio as a function of D

Crossing at
𝐷 ~ 5.6d = 2: nearest neighbor only

Tail of level separation distribution

Spectral form factor

𝛽 = 10−3

Unfolded spectrum

RMT: 𝑒 −𝑠2

Decay at long distance approaches Poisson
as interaction (SYK4) range D is decreased

t1 ramp disappears as D is decreased
𝑁 = 30Number varianceΣ2 𝐿 : Variance of number of levels within
𝐿 average level separation

Logarithmic increase for quantum chaotic caseLinear with slope 𝜒 < 1 for Anderson localization

𝜒 ~ 0.82 for 𝐷 = 5.6

𝑁 = 30𝜒 ~ 0.27 in 3D cubic lattice,increase and approach 1 for higher dimensions[Zharekeshev and Kramer 1995][Schreiber and Grussbach 1996][García-García and Cuevas 2007]

Summary of Part 2: Short-range SYK4 + SYK2:
many-body localization [1801.03204]
For integer 𝐷, 𝑑
෩ 0<𝐷
෩<1 :Also consider fractional 𝐷 = 𝐷 + 𝐷
෩ , otherwise set to zeroIf 𝑙 − 𝑖 = 𝐷 , use non-zero 𝐽ሚ𝑖𝑗𝑘𝑙 with probability 𝐷𝐽ሚ𝐽ሚ𝑖𝑗𝑘𝑙 𝐷 = ቊ 𝑖𝑗𝑘𝑙
0
෩𝑖𝑗𝐾෩
𝐾𝑖𝑗 𝑑 = ൝0𝑙−𝑖 <𝐷𝑙−𝑖 ≥𝐷
𝑗−𝑖 <𝑑𝑗−𝑖 ≥𝑑Metal → insulator transition as 𝐷 is decreased (𝑑 = 2 : at 𝐷 ~ 5.6 for 𝜅 = 1)
Gap ratio, level separation tail, spectral form factor, number varianceconsistently support a many-body localization

Out-of-Time-Ordered Correlation

3. Quantum Lyapunov spectrum (cf. OTOC)
𝛿𝑥𝑖 𝑡 = 𝑀𝑖𝑗 𝛿𝑥𝑗 0

Canonically conjugate variables 𝑥, 𝑝 at different times

𝜕𝑥 𝑡
2𝑥 𝑡 , 𝑝 0 PB =𝜕𝑥 0

𝑀𝑖𝑗 =2→ 𝑒 2𝜆L𝑡

𝛿𝑥𝑖 𝑡
𝛿𝑥𝑗 0

𝛿𝑥𝑖 𝑡
𝐿=𝛿𝑥𝑗 0at large t

Singular values at finite t: 𝑠𝑘 𝑡
2Hanada, Shimada, and MT: PRE 97, 022224 (2018)
For classical chaos, 𝜆𝑘 𝑡 obeys RMT statistics

෡ 0Corresponding quantity in quantum systems: for bosonic 𝑉, 𝑊 : 𝐶𝑇 𝑡 = − 𝑉෠ 𝑡 , 𝑊
Out-of-time correlator is included

For systems of fermionse.g. Sachdev-Ye-Kitaev (SYK) model?
Anticommutator
෡𝑎𝑏 𝑡 = 𝜒Ƹ 𝑎 𝑡 , 𝜒Ƹ 𝑏 0𝑀𝑁෡𝑗𝑎 𝑡 𝑀෡𝑗𝑏 𝑡
𝐿෠ 𝑎𝑏 𝑡 = ෍ 𝑀
𝑗=1

෡ 0 𝑉෠ 𝑡 𝑊
෡ 0𝑉෠ 𝑡 𝑊
෡=𝐻= 𝑒 𝜆𝑘 𝑡

෍

2etc.

[Norbert Wiener 1938][Larkin & Ovchinnikov 1969]

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁෡𝑎𝑏 𝑡 = 0 = 𝜒Ƹ 𝑎 , 𝜒Ƹ 𝑏 = 𝛿𝑎𝑏𝜒Ƹ 𝑎 : Majorana fermion, 𝑀

Our definition: for state 𝜙 (e.g. eigenstate)

For matrix 𝜙 𝐿෠ 𝑎𝑏 𝑡 𝜙 , obtain singular values 𝑠𝑘 𝑡
and the Lyapunov spectrum is defined as 𝜆𝑘 𝑡𝑁
𝑘=1log 𝑠𝑘 𝑡=.
2𝑡

Other possibilities: see Rozenbaum-Ganeshan-Galitski, 1801.10591; Hallam-Morley-Green: 1806.05204

arXiv:1809.01671 with Hrant Gharibyan, Masanori Hanada, and Brian Swingle

SYK: dependence on the SYK2 coefficient K𝑁

෡=𝐻෍1≤𝑎<𝑏<𝑐<𝑑𝑁𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏
1≤𝑎<𝑏Sample- and state-averaged full Lyapunov
spectrum: time dependence

Almost linear growth oflog(singular values of L)
Close to constant between red lines(20 % and 80 % of the saturated value of 𝜆𝑁 𝑡)

Classical KS entropy vs entanglement entropy production
Initial state: ȁ𝜓 𝑡 = 0 ۧ = ȁ000 … 000ۧ in the complex fermion basis

Coarse-grained entropy= log(# of cells covering the region)~ (sum of positive 𝜆 ) 𝑡
Kolmogorov-Sinai entropy ℎKS= (sum of positive 𝜆 )= entropy production rate𝑒𝜆𝑡

AB𝑁Τ2𝜌A 𝑡 = TrB 𝜌 𝑡 ,𝜌 𝑡 = ȁ𝜓 𝑡 ۧ‫ 𝑡 𝜓ۦ‬ȁ𝑐𝑗Ƹ =𝜒2𝑗−1 + i𝜒2𝑗2𝑆EE 𝑡 = −Tr log 𝜌A 𝑡

Classical KS entropy vs entanglement entropy production
Coarse-grained entropy= log(# of cells covering the region)~ (sum of positive 𝜆 ) 𝑡
Kolmogorov-Sinai entropy ℎKS= (sum of positive 𝜆 )= entropy production rate𝑒𝜆𝑡

Similar time scale for saturation in SYK model; other models?

Conjecture on entropy production
SYK4 limit
11 𝑁• 𝜆𝑁 and 𝜆OTOC = log σ𝑖=1 𝑒 2𝜆𝑖 𝑡2𝑡𝑁approacheach other; difference decreases as 1Τ𝑁• Same for 𝜆𝑁 and 𝜆1 : all exponent → single peak
• All saturate the MSS bound at strong coupling(low T) limit• Growth rate of entanglement entropy ~ ℎKS =sum of positive (all) 𝜆𝑖➔ Black holes: not only the fastest scramblers [Sekino and Susskind 2008],
but also fastest entropy generators

Spectral statistics: SYK

Close to SYK4 :remains GUE even after long time

(fixed-i unfolding:unfold each gap 𝜆𝑖+1 − 𝜆𝑖 by its average)

Strong one-body hopping term:approaches Poisson

Spectral statistics: SYK

min 𝜖𝑖+1 −𝜖𝑖 , 𝜖𝑖+2 −𝜖𝑖+1𝑟 : average of the adjacent gap ratio max
𝜖 −𝜖 , 𝜖 −𝜖𝑖+1𝑖𝑖+2𝑖+1

Uncorrelated (Poisson): 2 log 2 − 1 ≈ 0.386
Correlated: larger (GOE: 0.5307, GUE: 0.5996 etc. ) [Atas et al., PRL 2013]

Spectral statistics: XXZ + random field𝑁𝑁𝑧෢෡ = ෍ 𝑆෡𝑖 ∙ 𝑆෢𝐻+෍ℎ𝑆𝑖+1𝑖 𝑖𝑖ℎ𝑖 : uniform distribution [−𝑊, 𝑊]+−෢෡𝑎𝑏 𝑡 = 𝑆෢𝑀𝑡,𝑆𝑎𝑏 0𝑖

W = 0.5: approaches GUE

W = 4: close to Poisson

Quantum Lyapunov spectrum distinguishes chaotic and non-chaotic phases

New possibility: characterization of chaos bysingular value statistics of two-point functions
SYK, largest 3 exponents
Fixed-i unfolded log of singular values of𝜙
𝐺𝑎𝑏 = 𝜙 𝜓෠𝑎 𝑡 𝜓෠ 𝑏 0 𝜙

Work in progress with Gharibyan, Hanada, Swingle, …

New possibility: characterization of chaos bysingular value statistics of two-point functions
XXZ, all non-trivialexponents
𝜙𝐺𝑎𝑏 = 𝜙 𝜎෢+ 𝑎 𝑡 𝜎ෞ− 𝑏 0 𝜙Preliminary

Work in progressOur other works on localization + interaction
Quasiperiodic site level modulation in 1D U < 0Hubbard model: superfluid-insulator transitionand quench dynamics
MT & A. M. García-García: PRA 82, 043613 (2010)
U = -1Higher effective dimension in 1D
lattice with power-law interaction
A. M. Lobos, MT and A. M. García-García:
PRB 88, 134506 (2013)

1D Hubbard + power-law hopping:stabilization of long-range order
𝑡𝑙𝑚 =𝑡𝑙−𝑚 𝜅Dimensional quench dynamics by tDMRG
MT, A. M. García-García, and M. A. Cazalilla: PRA 90, 053618 (2014)

Remove non-nearest-neighbor hoppings:
quench to 1D from larger effective dimension condensate

PRA 85, 031602R (2012)

Summary of the talk𝑁

෡SYK4 =𝐻෍

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑Part 1: effect of one-body term𝑁෡=𝐻
෡SYK4 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏𝐻
1≤𝑎<𝑏• No longer maximally chaotic• Random-matrix like spectra forweak perturbation• Temperature-dependenttransition to non-chaotic behavior
Phys. Rev. Lett. 120, 241603 (2018)(arXiv:1707.02197)

Part 2: short-range SYK model• Many-body localization
arXiv:1801.03204

Part 3: quantum Lyapunov spectrum• Analogue of time-dependent classicalLyapunov spectrum (Hanada-Shimada-MTPRE 2018)• Defined by out-of-time-ordered correlator• Random-matrix like behavior for SYK4• One-body term destroys RMT behavior
after some time
• Comparison to random-field XXZ model(many-body localization)
arXiv:1809.01671

New possibility: characterization of chaosby two-point functions? (in progress)