Characterization of quantum many-body chaoswith quantum Lyapunov exponents
and by two-point correlations:application to a generalized Sachdev-Ye-Kitaev model

APS March Meeting 2019, Boston
6 March 2019

Masaki Tezuka 手塚真樹 (Kyoto University)Collaborators in this work

arXiv:1809.01671
arXiv:1902.11086

Hrant Gharibyan

Masanori Hanada

Brian SwingleՀրանտ Ղարիբյան

花田政範(Stanford University)(University of Southampton)(University of Maryland)Characterization of quantum many-body chaos• Random-matrix like energy levelcorrelation• Exponential Lyapunov growth of outof-time-order correlators (OTOC)෡ † 𝑡 𝑉෠ † 0 𝑊
෡ 𝑡 𝑉෠ 0 ~ 𝐶 + # 𝑒 2𝜆L 𝑡𝑊

Example: the Sachdev-Ye-Kitaev model
෡=𝐻[Cotler, MT et al.,JHEP 1705, 118 (2017)]

3!
𝑁 3/2

෍
1≤𝑎<𝑏<𝑐<𝑑≤𝑁Spectral form factor
2𝑍 𝛽, 𝑡𝑔 𝛽, 𝑡 =𝑍 𝛽, 𝑡 = 01=෍ e−𝛽 𝐸𝑚 +𝐸𝑛 ei 𝐸𝑚 −𝐸𝑛 𝑡2
𝑍 𝛽, 𝑡 = 0𝑚,𝑛𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑N mod 8 RMT0GOE

2GUE4GSE

6GUE𝜒Ƹ 𝑎 , 𝜒Ƹ 𝑏 = 𝛿𝑎𝑏𝐽𝑎𝑏𝑐𝑑 : Gaussian random( 𝐽𝑎𝑏𝑐𝑑 2 = 𝐽2 = 1)

[Kitaev 2015]

Lyapunov exponent
2𝜋𝑘B 𝑇
𝜆L =in low 𝑇 limitℏ(Maldacena-Shenker-Stanford chaos bound)We propose two new characterizations• Quantum Lyapunov spectrum:
Quantum version of finite-time
Lyapunov spectrum• Two-point correlations:෡𝑎𝑏 𝑡 : (anti)commutator of 𝑂෠𝑎 𝑡 and 𝑂෠𝑏 0𝑀
𝑁𝜙
𝐺𝑎𝑏 = 𝜙 𝑂෠𝑎 𝑡 𝑂෠𝑏 0 𝜙 as matrix,log (singular values)

෡𝑗𝑎 𝑡 † 𝑀
෡𝑗𝑏 𝑡𝐿෠ 𝑎𝑏 𝑡 = ෍ 𝑀
𝑗=1

𝜆𝑘 𝑡
𝑠𝑘 𝑡

log 𝑠𝑘 𝑡=for singular values
2𝑡𝑁෠
𝑘=1 of 𝑁 × 𝑁 matrix 𝜙 𝐿𝑎𝑏 𝑡

arXiv:1809.01671

𝜙 .

arXiv:1902.11086

Modified SYK model: Large-N calculation for OTOC
𝑁

෡=𝐻

෍

SYK4𝑁

SYK2𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑Chaos bound [Maldacena,Shenker, and Stanford 2016]

𝐾𝑎𝑏 : standard deviation

𝐾𝑁

1≤𝑎<𝑏

SYK4 limit →SYKq + κ SYK2
Large-q limit(𝐾 = 0)non-chaotic
normalized
Lyapunov exponent

𝐾 = 0.2

chaotic

0.5
1A. M. Garcia-Garcia,
B. Loureiro,
A. Romero-Bermudez,and MT, PRL 120,241603 (2018)

2𝛽 = 1Τ 𝑘B 𝑇 inverse temperature

Deviation from the chaos bound as SYK2 component is introducedGharibyan, Hanada, Swingle, and MT,
submitted (arXiv:1809.01671)

1. Quantum Lyapunov spectrum

Finite-time classical Lyapunov spectrum: obeys RMT statistics for chaos
Singular values of 𝑀𝑖𝑗 =𝐿 = 𝑥𝑖 𝑡 , 𝑝𝑗 0

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 02
PBOTOC: 𝐶𝑇 𝑡 ==at finite t: 𝑠𝑘 𝑡
𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 02= 𝑒 𝜆𝑘 𝑡→ 𝑒 2𝜆L 𝑡 for large t

෡ 𝑡 , 𝑉෠ 𝑡 = 0𝑊2[Hanada, Shimada, and MT:
PRE 97, 022224 (2018)]

𝛿 𝑥Ԧ 𝑡= 𝑀𝛿 𝑥Ԧ 0𝛿 𝑥Ԧ 00𝑡

෡ † 𝑡 𝑉෠ † 0 𝑊
෡ 𝑡 𝑉෠ 0 + ⋯= 𝑊

෡𝑎𝑏 𝑡 as (anti)commutator of 𝑂෠𝑎 𝑡 and 𝑂෠𝑏 0Quantum Lyapunov spectrum: Define 𝑀𝑁෡𝑗𝑎 𝑡 † 𝑀
෡𝑗𝑏 𝑡
𝐿෠ 𝑎𝑏 𝑡 = ෍ 𝑀
𝑗=1

For 𝑁 × 𝑁 matrix 𝜙 𝐿෠ 𝑎𝑏 𝑡 𝜙 , obtain singular values 𝑠𝑘 𝑡
The Lyapunov spectrum is defined as 𝜆𝑘 𝑡

log 𝑠𝑘 𝑡=2𝑡.

𝑁𝑘=1 .

Quantum Lyapunov spectrum for SYK model + modification𝑁෡=𝐻෍

𝑁𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑1≤𝑎<𝑏𝐽𝑎𝑏𝑐𝑑 : s. d. =𝐾𝑎𝑏 : s. d. =6𝑁3Τ2𝐾𝑁

෡෡• Define 𝐿෠ 𝑎𝑏 𝑡 = σ𝑁
𝑗=1 𝑀𝑗𝑎 𝑡 𝑀𝑗𝑏 𝑡 for time-dependent෡𝑎𝑏 𝑡 = 𝜒Ƹ 𝑎 𝑡 , 𝜒Ƹ 𝑏 0 .
anticommutator 𝑀• Obtain the singular values 𝑎𝑘 𝑡

𝐾
𝑘=1 of• Quantum Lyapunov spectrum: 𝜆𝑘 𝑡

(also dependent on state 𝜙)𝜙 𝐿෠ 𝑎𝑏 𝑡 𝜙log 𝑎𝑘 𝑡
=2𝑡

𝑘=1,2,…,𝐾

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

Spectral statistics of quantum Lyapunov spectrum: SYK

arXiv:1809.01671

Energy eigenstates
N/2 larger exponents

𝐾 = 0.01 (●):

𝐾 = 10 (◆):

Remains GUE for long time

Approaches Poisson

Exponents are nearly constant until
the singular values of 𝜙 𝐿෠ 𝑎𝑏 𝑡 𝜙saturate: Lyapunov growth𝑟 : average ofmin 𝜖𝑖+1 −𝜖𝑖 , 𝜖𝑖+2 −𝜖𝑖+1max 𝜖𝑖+1 −𝜖𝑖 , 𝜖𝑖+2 −𝜖𝑖+1(fixed-i unfolding: unfold each gap
𝜆𝑖+1 − 𝜆𝑖 using its average)

2. Singular value statistics of two-point functions
𝜙𝐺𝑎𝑏 = 𝜙 𝜒Ƹ 𝑎 𝑡 𝜒Ƹ 𝑏 0 𝜙
𝜙𝜆𝑗 = log singular values of 𝐺𝑎𝑏

SYK model
Energy eigenstates

H. Gharibyan, M. Hanada, B. Swingle, and MT, arXiv:1902.11086

2. Singular value statistics of two-point functions
𝜙𝐺𝑎𝑏 = 𝜙 𝜒Ƹ 𝑎 𝑡 𝜒Ƹ 𝑏 0 𝜙

SYK, larger N/2 exponents
𝜙𝜆𝑗 = log singular values of 𝐺𝑎𝑏

GUE at all times

GOE at late timearXiv:1902.11086

min 𝜆𝑖+1 −𝜆𝑖 , 𝜆𝑖+2 −𝜆𝑖+1𝑟 : average of the adjacent gap ratio max
𝜆 −𝜆 , 𝜆 −𝜆𝑖+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]
SYK, larger N/2 exponents
𝜙: energy eigenstates

N mod 8 = 2, 4, 6: GUE
N mod 8 = 0: GOE(the matrix is symmetric)
𝜙𝐺𝑎𝑏 = 𝜙 𝜒Ƹ 𝑎 𝑡 𝜒Ƹ 𝑏 0 𝜙𝜙

𝜆𝑗 = log singular values of 𝐺𝑎𝑏
fixed-i unfolded

arXiv:1902.11086

At late time,Random matrix behavior  chaotic (also for XXZ model + random field)

Summary• Quantum Lyapunov spectrum defined from local operators:
characterizes quantum chaos [1809.01671] ෠𝑁෡• Lyapunov growth• Fastest entropy production in the SYK model?
• Random matrix behavior in chaotic systems

෡𝑗𝑏 𝑡 for𝐿𝑎𝑏 𝑡 = σ𝑗=1 𝑀𝑗𝑎 𝑡 𝑀
෡𝑎𝑏 𝑡 = 𝜒Ƹ 𝑎 𝑡 , 𝜒Ƹ 𝑏 0𝑀
QLS: log(singular values of 𝜙 𝐿෠ 𝑎𝑏 𝑡 𝜙 )/(2t)• Two-point correlation function: singular values exhibit
random matrix behavior in chaotic cases [1902.11086]

𝜙𝐺𝑎𝑏 = 𝜙 𝜒Ƹ 𝑎 𝑡 𝜒Ƹ 𝑏 0 𝜙• Experiments should be possible with phase-sensitive measurements• Both characterizations of chaos demonstrated for the SYK model• Also for XXZ spin chain + random field (see our papers)

Gharibyan, Hanada, Swingle, and MT,submitted (arXiv:1809.01671)

Full Lyapunov spectrum
SYK, N = 16
K = 0.01

K=1Sample- and state-averaged
K = 0.1

K = 10𝜆𝑖 𝑡 : Close to constant between red lines(20 % and 80 % of the saturated value of 𝜆𝑁 𝑡)

arXiv:1809.01671

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

෡SYK4 from an initial state with 𝑆EE = 0:
Time evolution by 𝐻ȁ𝜓 𝑡 = 0 ۧ = ȁ000 … 000ۧ in the complex fermion basis

AB𝑁 Τ2𝑐𝑗Ƹ =2

𝜌A 𝑡 = TrB 𝜌 𝑡 , 𝜌 𝑡 = ȁ𝜓 𝑡 ۧ‫ 𝑡 𝜓ۦ‬ȁ

𝑆EE 𝑡 = −Tr 𝜌A 𝑡 log 𝜌A 𝑡
Similar time scales of growthfor SYK4Kolmogorov-Sinai entropy ℎKS= (sum of positive 𝜆 )= entropy production rate𝜒2𝑗−1 + i𝜒2𝑗

arXiv:1809.01671

Fastest entropy production?
SYK4 limit11• 𝜆𝑁 and 𝜆OTOC = log σ𝑁𝑒 2𝜆𝑖𝑡𝑖=12𝑡𝑁approach each 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➔ [conjecture] SYK model (and black holes): not only thefastest scramblers, but also fastest entropy generators