Characterization of many-body quantum chaos by quantum Lyapunov spectrum and two-point correlators
Masaki Tezuka (Kyoto University, tezuka@scphys.kyoto-u.ac.jp) in collaboration with Hrant Gharibyan (Stanford), Masanori Hanada (Southampton), and Brian Swingle (Maryland)
Motivation

Frequently used criteria for characterizing quantum chaos:

We try to characterize quantum many-body chaos by• Relation between the two criteria?
• Relation to classical chaos?

• Fine-grained energy level statistics (described by random matrix theory (RMT))• Out-of-time-order correlators (OTOC) (exponential Lyapunov growth)• Generalizing OTOC to define Lyapunov spectrum• Considering simpler quantities more accessible to experimentQuantum Lyapunov spectrum (arXiv:1809.01671)

Singular values of

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 0at finite t: 𝑠𝑘 𝑡

= 𝑒

෡=𝐻𝜆𝑘 𝑡SYK4𝑁

[M. Hanada, H. Shimada, and MT: PRE 97, 022224 (2018)]

෍𝐿 = 𝑥𝑖 𝑡 , 𝑝𝑗 0

OTOC: 𝐶𝑇 𝑡 =PB=2

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 0

1≤𝑎<𝑏<𝑐<𝑑෡ 𝑡 , 𝑉෠ 𝑡 = 0𝑊

2→ 𝑒 2𝜆L𝑡 at large t

෡ † 𝑡 𝑉෠ † 0 𝑊
෡ 𝑡 𝑉෠ 0 + ⋯= 𝑊

SYK2Random hopping term suppresses chaos𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏𝐽𝑎𝑏𝑐𝑑 : average 0, standard deviation

2𝑁1≤𝑎<𝑏6𝐽
𝑁3Τ2

𝐾
𝐾𝑎𝑏 : average 0, standard deviation𝑁𝑧෢෡ = ෍ 𝑆෡𝑖 ∙ 𝑆෢𝐻+
෍ℎ𝑆𝑖+1𝑖 𝑖solvable in large-N limit, strongly coupled at low T
SYK4 limit: maximally chaotic(satisfies Maldacena-Shenker-Stanford chaos bound)𝑖

SYK4 limit →SYKq + κ SYK2Large-q limit

𝑁ℎ𝑖 : uniform in [−𝑊, 𝑊]

𝑖(𝐾 = 0)Standard model of many-body localization (𝑊c ≈ 3.5 )non-chaotic

e.g. [D. J. Luitz, N. Laflorencie, and F. Alet, PRB 91, 081103 (2015)]

𝐾 = 0.2

chaotic+−෢෡𝑎𝑏 𝑡 = 𝑆෢𝑀𝑡,
𝑆𝑎𝑏 00.5
21

inverse temperature 𝛽 = 1Τ 𝑘B 𝑇

෡𝑎𝑏 𝑡 = 𝜒Ƹ 𝑎 𝑡 , 𝜒Ƹ 𝑏 0
𝑀

Quantum Lyapunov spectrum:
𝑁𝐿෠ 𝑎𝑏 𝑡

Comparison with entanglement entropy
For initial state ȁ𝜓 𝑡 = 0 ۧ = ȁ000 … 000ۧ(complex fermion basis)𝑁Τ2

෡𝑎𝑏 𝑡 as (anti)commutator of 𝑂෠𝑎 𝑡 and 𝑂෠𝑏 0Define 𝑀

𝑐𝑗Ƹ =†෡෡= ෍ 𝑀𝑗𝑎 𝑡 𝑀𝑗𝑏 𝑡
𝑗=1

The Lyapunov spectrum is defined as 𝜆𝑘 𝑡log 𝑠𝑘 𝑡
=2𝑡

𝜒2𝑗−1 + i𝜒2𝑗AB.

(Fixed-i) unfolded level separation• Also 𝜆𝑁 and 𝜆1 approach each other → single peak of 𝜆• Chaos bound value expected for strong coupling• Entanglement entropy generation rate ℎKS = sum of positive (all) 𝜆𝑖➔ We conjecture that black holes are not only thefastest scramblers [Sekino and Susskind 2008], but also
the fastest entropy generators.

min 𝑠𝑖+1 −𝑠𝑖 , 𝑠𝑖+2 −𝑠𝑖+1➔ gap distribution 𝑃 𝑠 , adjacent gap ratio 𝑟 =max 𝑠𝑖+1 −𝑠𝑖 , 𝑠𝑖+2 −𝑠𝑖+1

Two-point correlator matrix (arXiv:1902.11086)

𝜆𝑘 = log singular values of

𝑊 = 4: stays close to Poisson

In both models, quantum Lyapunov spectrum distinguishes
chaotic and non-chaotic phases.
Chaotic ➔ Universal random matrix behavior
Non-chaotic ➔ Nearly uncorrelated, exp(-s) level separation

Divide i-th gap 𝑔𝑖 = 𝜆𝑘+1 − 𝜆𝑘 by its sample average Δ𝑖 to obtain unfolded gap 𝑠𝑖

Summary
XXZ spin chain + random field𝜙𝐺𝑎𝑏= 𝜙 𝜒Ƹ 𝑎 𝑡 𝜒Ƹ 𝑏 0 𝜙

𝑊 = 0.5: GUE approachedAveraged gap ratio

Fixed-i unfolding

SYK model

1𝑁2𝜆 𝑡
𝜌A 𝑡 = TrB 𝜌 𝑡 , 𝜌 𝑡 = ȁ𝜓 𝑡 ۧ‫ 𝑡 𝜓ۦ‬ȁ • The difference between 𝜆𝑁 and 𝜆OTOC = 2𝑡 log 𝑁 σ𝑖=1 𝑒 𝑖 ~ 1Τ𝑁

The sum of positive 𝜆𝑖 𝑡 andentanglement entropy increase
with almost the same slope

𝑁
𝑘=1 .

Kolmogorov-Sinai entropy ℎKS= (sum of positive 𝜆 )= entropy production rate

𝑆EE 𝑡 = −Tr𝜌A 𝑡 log 𝜌A 𝑡

2

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

1

almost linear growthof largest exponent

For 𝑁 × 𝑁 matrix 𝜙 𝐿෠ 𝑎𝑏 𝑡 𝜙 , obtain singular values 𝑠𝑘 𝑡𝜙𝐺𝑎𝑏

𝑁[A. M. García-García, B. Loureiro, A. Romero-Bermudez,and MT, PRL 120, 241603 (2018) (arXiv:1707.02197)]

normalized
Lyapunov exponent

Finite-time classical Lyapunov spectrum: obeys RMT statistics for chaos

XXZ spin chain + random field

SYK model=−+𝜙 𝜎ො𝑎 𝑡 𝜎ො𝑏 0𝜙

We have proposed characterization of quantum chaos by• Quantum Lyapunov spectrum➢ from analogy to the classical case
➢ Lyapunov growth for finite time for finite N𝜙𝐺𝑎𝑏• Two-point correlator matrix➢ experimentally more accessible
and have demonstrated their random matrix behavior for
the SYK model and the XXZ spin chain + random field.
𝜙𝑁 mod 8 periodicity understood by symmetry of 𝐺𝑎𝑏(complex symmetric only for 𝑁 ≡ 0 mod 8)

Level statistics of 𝜆𝑘 : GOE after adequate time