Quantum Lyapunov spectrum of the Sachdev-Ye-Kitaev model YITP-YIPQS - - PowerPoint PPT Presentation
Quantum Lyapunov spectrum of the Sachdev-Ye-Kitaev model YITP-YIPQS Workshop, YITP, Kyoto University 27 May 2019 Masaki Tezuka (Kyoto University) Collaborators in this work arXiv:1809.01671 (JHEP04(2019)082) arXiv:1902.11086
YITP-YIPQS Workshop, YITP, Kyoto University 27 May 2019
Hrant Gharibyan arXiv:1809.01671 (JHEP04(2019)082) arXiv:1902.11086 Masanori Hanada Brian Swingle
Հրանտ Ղարիբյան 花田政範
(Stanford University) (University of Southampton) (University of Maryland)
Characterization of many-body quantum chaos The Sachdev-Ye-Kitaev model The quantum Lyapunov spectrum The singular values of two-point correlators Summary
Characterization of many-body quantum chaos The Sachdev-Ye-Kitaev model The quantum Lyapunov spectrum The singular values of two-point correlators Summary
𝜀𝑦 0
𝜀𝑦 𝑢 𝑢
“butterfly effect”
Bounded, nonperiodic dynamics with nonlinearity What happens in quantum mechanics?
by Dan Quinn (on Wikimedia Commons)
Building 24, MIT
𝑗 𝑒 𝑒𝑢 𝜔 = 𝐼 𝜔 𝜔 𝑢 = T exp −𝑗
𝑢
𝐼 𝑢′ 𝑒𝑢 𝜔 𝑢 = 0 = exp −𝑗 𝐼𝑢 𝜔 𝑢 = 0
𝐼 = const.
Linear dynamics Unitary time evolution
𝑦𝑗 𝑢 , 𝑞𝑘 0
PB 2= 𝜖𝑦𝑗 𝑢 𝜖𝑦𝑘 0 2
→ 𝑓2𝜇L𝑢 for large t OTOC: 𝐷𝑈 𝑢 = 𝑋 𝑢 , 𝑊 𝑢 = 0
2
= 𝑋† 𝑢 𝑊† 0 𝑋 𝑢 𝑊 0 + ⋯
Classically, Quantum version:
Correlation between levels, as in random matrices P(s): normalized level separation distribution
Uncorrelated: Poisson (𝑓−𝑡)
limited to very small systems Hard to see exponential time dependence Numerically
correlation
Example: the Sachdev-Ye-Kitaev model
𝐼 = 3! 𝑂3/2
1≤𝑏<𝑐<𝑑<𝑒≤𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒
𝐾𝑏𝑐𝑑𝑒 : Gaussian random ( 𝐾𝑏𝑐𝑑𝑒
2 = 𝐾2 = 1)
𝜓𝑏, 𝜓𝑐 = 𝜀𝑏𝑐
𝜇L =
2𝜌𝑙B𝑈 ℏ
in low 𝑈 limit
(Maldacena-Shenker-Stanford chaos bound)
𝛾, 𝑢 = 𝑎 𝛾, 𝑢 𝑎 𝛾, 𝑢 = 0
2
= 1 𝑎 𝛾, 𝑢 = 0 2
𝑛,𝑜
e−𝛾 𝐹𝑛+𝐹𝑜 ei 𝐹𝑛−𝐹𝑜 𝑢 [Cotler, MT et al., JHEP 1705, 118 (2017)]
𝑋† 𝑢 𝑊† 0 𝑋 𝑢 𝑊 0 ~ 𝐷 + # 𝑓2𝜇L𝑢 Spectral form factor Lyapunov exponent
[Kitaev 2015] N mod 8 RMT GOE 2 GUE 4 GSE 6 GUE
Characterization of many-body quantum chaos The Sachdev-Ye-Kitaev model The quantum Lyapunov spectrum The singular values of two-point correlators Summary
𝐼 = 3! 𝑂3/2
1≤𝑏<𝑐<𝑑<𝑒≤𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒
𝐾𝑏𝑐𝑑𝑒 : Gaussian random couplings (𝐾𝑏𝑐𝑑𝑒
2 = 𝐾2 = 1, 𝐾𝑏𝑐𝑑𝑒 = 0)
[A. Kitaev, talks at KITP (2015)]
𝜓𝑏=1,2,…,𝑂: 𝑂 Majorana fermions ( 𝜓𝑏, 𝜓𝑐 = 𝜀𝑏𝑐)
𝐾3567 𝐾1259 𝐾4567 𝐾1348
⋯
O(1) O(N-2)
𝐾𝑗𝑘𝑙𝑚𝐾𝑘𝑙𝑚𝑛 = 𝜀𝑗𝑛
Only “melon-type” diagrams survive 𝐼 = 3! 𝑂3/2
1≤𝑏<𝑐<𝑑<𝑒≤𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒 Analytically solvable in 𝑂 ≫ 1 limit 𝐻 =
𝐻0 𝐻0Σ𝐻0 𝐻0Σ𝐻0Σ𝐻0 𝐻0Σ𝐻0Σ𝐻0Σ𝐻0 + ⋯ = 𝐻0 Σ𝐻0 −1
Figures from [I. Danshita, MT, and M. Hanada: Butsuri 73(8), 569 (2018)]
Σ = 𝐾2𝐻3 𝐻 𝑗𝜕
−1 = 𝑗𝜕 − Σ 𝑗𝜕
𝐼 = 3! 𝑂3/2
1≤𝑏<𝑐<𝑑<𝑒≤𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒 Analytically solvable in 𝑂 ≫ 1 limit
2𝜌𝑙B𝑈 ℏ
in the T0 limit
𝐻 1 − Σ𝐻0 = 𝐻0
Σ = 𝐾2𝐻3
𝐻−1 = 𝐻0
−1 − Σ
𝐻 𝑗𝜕
−1 = 𝑗𝜕 − Σ 𝑗𝜕
Low energy (𝜕, 𝑈 ≪ 𝐾): ignore 𝑗𝜕 and we have
𝑒𝑢𝐻 𝑢1, 𝑢 Σ 𝑢, 𝑢2 = −𝜀 𝑢1, 𝑢2
𝐾2 𝑒𝑢𝐻 𝑢1, 𝑢 𝐻 𝑢, 𝑢2 3 = −𝜀 𝑢1, 𝑢2
𝐻 𝑢 = − 1 4𝜌𝐾2
1/4 sgn 𝑢
𝑢
[S. Sachdev,
041025 (2015)]
Modified SYK model:
Setup:
A double-well
with
6Li
(large recoil energy)
(no degeneracy in the band levels) Sums of two single atom energies
[I. Danshita, M. Hanada, MT: arXiv:1606.02454; PTEP 2017, 083I01 (2017)] (also a proceedings manuscript arXiv:1709.07189)
s: molecular levels See for review including other proposals e.g. using topological insulator, graphene
N Majorana- or Dirac- fermions randomly coupled to each other [Dirac version] [Majorana version]
[A. Kitaev’s talk] [S. Sachdev: PRX 5, 041025 (2015)]
𝐼 = 3! 𝑂3/2
1≤𝑏<𝑐<𝑑<𝑒≤𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒 𝐼 = 1 2𝑂 3/2
𝑗𝑘;𝑙𝑚
𝐾𝑗𝑘;𝑙𝑚 𝑑𝑗
†
𝑑
𝑘 †
𝑑𝑙 𝑑𝑚
[A. Kitaev: talks at KITP
(Feb 12, Apr 7 and May 27, 2015)]
2π𝑙B𝑈 ℏ
Generalizations: q-fermion interactions “SYKq”, supersymmetric SYK, lattice of SYK lands; etc.
SPT phase classification for class BDI: ℤ ℤ8 due to interaction
[L. Fidkowski and A. Kitaev, PRB 2010, PRB 2011]
Classification and random matrix theory
𝐼 = 3! 𝑂3/2
1≤𝑏<𝑐<𝑑<𝑒≤𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒
𝑑
𝑘 =
𝜓2𝑘−1 + i 𝜓2𝑘 2 Introduce 𝑂/2 complex fermions
𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒 respects the complex fermion parity Even ( 𝐼E) and odd ( 𝐼O) sectors: 𝑀 = 2𝑂/2−1 dimensions 𝑌 𝑑
𝑘
𝑌 = 𝜃 𝑑𝑘
†
𝑂 mod 8 2 4 6 𝜃
+1 +1
𝑌2 +1 +1
𝑌 maps 𝐼E to 𝐼E 𝐼O 𝐼E 𝐼O Class AI A+A AII A+A Gaussian ensemble GOE GUE GSE GUE [You, Ludwig, and Xu, PRB 2017] [Fadi Sun and Jinwu Ye, 1905.07694] for SYKq, supersymmetric SYK
Sparse, but energy spectral statistics strongly resemble that of the corresponding (dense) Gaussian ensemble
𝑌 = 𝐿
𝑘=1 𝑂/2
𝑑
𝑘 † +
𝑑
𝑘
𝑂 ≡ 0, 4
(mod 8)
𝑂 ≡ 2, 6 [Cotler, …, MT, JHEP 2017]
𝑏𝑗𝑘 𝑗,𝑘=1
𝐿 𝑏𝑗𝑘 = 𝑏𝑘𝑗
∗
[F. J. Dyson, J. Math. Phys. 3, 1199 (1962)]
Density ∝ 𝑓−𝛾𝐿
4 Tr𝐼2 = exp −
𝛾𝐿 4 𝑗,𝑘 𝐿 𝑏𝑗𝑘 2
Real (β=1): Gaussian Orthogonal Ensemble (GOE) Complex (β=2): G. Unitary E. (GUE) Quaternion (β=4): G. Symplectic E. (GSE)
Gaussian distribution
𝑞 𝑓1, 𝑓2, … , 𝑓𝐿 ∝
1≤𝑗<𝑘≤𝐿
𝑓𝑗 − 𝑓
𝑘 𝛾 𝑗=1 𝐿
𝑓−𝛾𝐿
𝑓𝑗2 4
Joint distribution
Level repulsion
Eigenvalue distribution: semi-circle law
𝑓1 ≤ 𝑓2 ≤ ⋯ ≤ 𝑓𝑀
𝑓𝑘+1−𝑓𝑘 ∆ 𝑓
∆: averaged level separation near 𝑓
𝑘
𝑄 𝑓
GOE/GUE/GSE: 𝑄 𝑡 ∝ 𝑡𝛾 at small 𝑡, has 𝑓−𝑡2 tail Uncorrelated: 𝑄 𝑡 = 𝑓−𝑡 (Poisson distribution)
𝑏𝑗𝑘 𝑗,𝑘=1
𝑀
𝑏𝑗𝑘 = 𝑏𝑘𝑗
∗
Density ∝ 𝑓−𝛾𝐿
4 Tr𝐼2 = exp − 𝛾𝐿
4 𝑗,𝑘 𝐿 𝑏𝑗𝑘 2
Real (β=1): Gaussian Orthogonal Ensemble (GOE) Complex (β=2): G. Unitary E. (GUE) Quaternion (β=4): G. Symplectic E. (GSE)
Gaussian distribution
𝑞 𝑓1, 𝑓2, … , 𝑓𝐿 ∝
1≤𝑗<𝑘≤𝐿
𝑓𝑗 − 𝑓
𝑘 𝛾 𝑗=1 𝐿
𝑓−𝛾𝐿
𝑓𝑗2 4
Joint distribution
Level repulsion
𝑓𝑘+1−𝑓𝑘 ∆ 𝑓
GOE/GUE/GSE: 𝑄 𝑡 ∝ 𝑡𝛾 at small 𝑡, has 𝑓−𝑡2 tail Uncorrelated: 𝑄 𝑡 = 𝑓−𝑡 (Poisson distribution)
𝑠 = min 𝑓𝑗+1 − 𝑓𝑗 , 𝑓𝑗+2 − 𝑓𝑗+1 max 𝑓𝑗+1 − 𝑓𝑗 , 𝑓𝑗+2 − 𝑓𝑗+1
SYK model results: indistinguishable from corresponding Gaussian ensemble
Uncorrelated GOE GUE GSE 𝑠 2log 2 – 1 = 0.38629… 0.5307(1) 0.5996(1) 0.6744(1) [Y. Y. Atas et al. PRL 2013]
[Cotler, MT et al., JHEP05(2017)118]
Dip-ramp-plateau structure for 𝑂 ≡ 2 mod 8 N = 16, GOE N = 18, GUE N = 20, GSE N = 22, GUE N = 24, GOE N = 14, GUE 𝐻 𝑢 = 𝜓𝑏 𝑢 𝜓𝑏 0
𝛾 =
1 𝑎 𝛾
𝑛,𝑜
e−𝛾𝐹𝑛 𝑛 𝜓𝑏 𝑜
2ei 𝐹𝑛−𝐹𝑜 𝑢
[Cotler, MT et al., JHEP05(2017)118] 𝑂 mod 8 2 4 6 𝑌 maps 𝐼E to 𝐼E 𝐼O 𝐼E 𝐼O even 𝜓 odd finite Gaussian ensemble GOE GUE GSE GUE
𝛾, 𝑢 = 𝑎 𝛾, 𝑢 𝑎 𝛾, 𝑢 = 0
2
= 1 𝑎 𝛾, 𝑢 = 0 2
𝑛,𝑜
e−𝛾 𝐹𝑛+𝐹𝑜 ei 𝐹𝑛−𝐹𝑜 𝑢
𝐻 𝑢 = 𝜓𝑏 𝑢 𝜓𝑏 0
𝛾 =
1 𝑎 𝛾, 𝑢 = 0
𝑛,𝑜
e−𝛾𝐹𝑛 𝑛 𝜓𝑏 𝑜
2ei 𝐹𝑛−𝐹𝑜 𝑢
𝑎 𝛾, 𝑢
𝐾 ~𝑢 −3 2
quickly decays
ramp plateau 𝑎 𝛾, 𝑢 = 𝑎 𝛾 + i𝑢 = Tr e−𝛾
𝐼−i 𝐼𝑢
c 𝛾, 𝑢 = 𝑎 𝛾, 𝑢
2 𝐾 −
𝑎 𝛾, 𝑢
𝐾 2
𝑎 𝛾
𝐾 2
∼ 𝑒𝜇1𝑒𝜇2 𝜀𝜍 𝜇1 𝜀𝜍 𝜇2 𝑓i𝑢 𝜇1−𝜇2
𝑆 𝜇 = 𝜀𝜍 𝜇1 𝜀𝜍 𝜇1 − 𝜇 = − sin2 𝑀𝜇 𝜌𝑀𝜇 2 + 1 𝜌𝑀 𝜀 𝜇
in RMT
𝑢
𝑢 2𝜌𝑀2
2𝑀 𝜌𝑀 −1
dip Exponentially long ramp 𝑢 ~𝑢1 plateau 𝑢 = 𝑂𝐹 𝑎 2𝛾 /𝑎 𝛾 2
GOE 1 GUE 2 GSE 2 GUE 2 GOE 1 GUE 2 GSE 2 GUE 2 GOE 1 GUE 2
𝑂𝐹 ℋSYK =
𝑏,𝑐,𝑑,𝑒 𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒 [Cotler, MT et al., JHEP05(2017)118]; for dip time dependence on N see [Gharibyan-Hanada-Shenker-MT, JHEP07(2018)124].
correlation
Example: the Sachdev-Ye-Kitaev model
𝐼 = 3! 𝑂3/2
1≤𝑏<𝑐<𝑑<𝑒≤𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒
𝐾𝑏𝑐𝑑𝑒 : Gaussian random ( 𝐾𝑏𝑐𝑑𝑒
2 = 𝐾2 = 1)
𝜓𝑏, 𝜓𝑐 = 𝜀𝑏𝑐
𝜇L =
2𝜌𝑙B𝑈 ℏ
in low 𝑈 limit
(Maldacena-Shenker-Stanford chaos bound)
𝛾, 𝑢 = 𝑎 𝛾, 𝑢 𝑎 𝛾, 𝑢 = 0
2
= 1 𝑎 𝛾, 𝑢 = 0 2
𝑛,𝑜
e−𝛾 𝐹𝑛+𝐹𝑜 ei 𝐹𝑛−𝐹𝑜 𝑢 [Cotler, MT et al., JHEP 1705, 118 (2017)]
𝑋† 𝑢 𝑊† 0 𝑋 𝑢 𝑊 0 ~ 𝐷 + # 𝑓2𝜇L𝑢 Spectral form factor Lyapunov exponent
[Kitaev 2015] N mod 8 RMT GOE 2 GUE 4 GSE 6 GUE
“Randomness and Chaos in Qubit Models” Pak Hang Chris Lau, Chen-Te Ma, Jeff Murugan, and MT, Phys. Lett. B in press (arXiv:1812.04770)
Characterization of many-body quantum chaos The Sachdev-Ye-Kitaev model The quantum Lyapunov spectrum The singular values of two-point correlators Summary
Quantum version of finite-time Lyapunov spectrum
JHEP04(2019)082 (arXiv:1809.01671) arXiv:1902.11086
𝑁𝑏𝑐 𝑢 : (anti)commutator of 𝑃𝑏 𝑢 and 𝑃𝑐 0
𝑀𝑏𝑐 𝑢 =
𝑘=1 𝑂
𝑁
𝑘𝑏 𝑢 †
𝑁
𝑘𝑐 𝑢
𝜇𝑙 𝑢 =
log 𝑡𝑙 𝑢 2𝑢
for singular values 𝑡𝑙 𝑢
𝑙=1 𝑂
𝑀𝑏𝑐 𝑢 𝜚 . 𝐻𝑏𝑐
𝜚 = 𝜚
𝑃𝑏 𝑢 𝑃𝑐 0 𝜚 as matrix, log (singular values)
Modified SYK model: Large-N calculation for OTOC
Deviation from the chaos bound as SYK2 component is introduced
Chaos bound [Maldacena, Shenker, and Stanford 2016] 𝛾 = 1 𝑙B𝑈 inverse temperature
SYK4 limit →
𝐿 = 0.2 0.5 1 2 (𝐿 = 0)
SYKq + κ SYK2 Large-q limit
chaotic non-chaotic
𝐼 =
1≤𝑏<𝑐<𝑑<𝑒 𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒 + 𝑗
1≤𝑏<𝑐 𝑂
𝐿𝑏𝑐 𝜓𝑏 𝜓𝑐
SYK4 SYK2
𝐿𝑏𝑐: standard deviation
𝐿 𝑂
normalized Lyapunov exponent
and MT, PRL 120, 241603 (2018) 𝐿
(2016)] and experimental proposal papers for the SYK model
OTOCs have been intensively studied:
Coarse-grained phase space
𝑢
𝜀𝑦𝑗 𝑢 = 𝑁𝑗𝑘𝜀𝑦𝑘 0
Deviation at t initial infinitesimal deviation
Singular values of 𝑁𝑗𝑘: 𝑏𝑙 𝑢
𝑙=1 𝐿
Time-dependent Lyapunov spectrum
𝜇𝑙 𝑢 = log 𝑏𝑙 𝑢 𝑢
𝑙=1,2,…,𝐿
𝑀 = 𝜀𝑦𝑗 𝑢 𝜀𝑦𝑘 0
2
𝑦 𝑢 , 𝑞 0
PB 2=
𝜖𝑦 𝑢 𝜖𝑦 0
2
→ 𝑓2𝜇L𝑢 𝑁𝑗𝑘 = 𝜀𝑦𝑗 𝑢 𝜀𝑦𝑘 0 = 𝑦 𝑢 , 𝑞 0
PB
𝜀𝑦 0 𝜀𝑦 𝑢 𝑢 𝑓𝜇L𝑢
(Usually 𝑢 → ∞ limit is taken for obtaining 𝜇L) We consider finite 𝑢
[M. Hanada, H. Shimada, and M. Tezuka, PRE 97, 022224 (2018)]
Classical system with 𝐿 degrees of freedom
in several chaotic systems
(without fermions)
Quantum Lyapunov spectrum: Define 𝑁𝑏𝑐 𝑢 as (anti)commutator of 𝑃𝑏 𝑢 and 𝑃𝑐 0
𝑀𝑏𝑐 𝑢 = 𝑁 𝑢 † 𝑁 𝑢
𝑏𝑐 = 𝑘=1 𝑂
𝑁
𝑘𝑏 𝑢 †
𝑁
𝑘𝑐 𝑢
𝑀 = 𝑦𝑗 𝑢 , 𝑞𝑘 0
PB 2= 𝜖𝑦𝑗 𝑢 𝜖𝑦𝑘 0 2
→ 𝑓2𝜇L𝑢 for large t
OTOC: 𝐷𝑈 𝑢 = 𝑋 𝑢 , 𝑊 𝑢 = 0
2 =
𝑋† 𝑢 𝑊† 0 𝑋 𝑢 𝑊 0 + ⋯
For 𝑂 × 𝑂 matrix 𝜚 𝑀𝑏𝑐 𝑢 𝜚 , obtain singular values 𝑡𝑙 𝑢
𝑙=1 𝑂
. The Lyapunov spectrum is defined as 𝜇𝑙 𝑢 =
log 𝑡𝑙 𝑢 2𝑢
.
Singular values of 𝑁𝑗𝑘 =
𝜖𝑦𝑗 𝑢 𝜖𝑦𝑘 0
at finite t: 𝑡𝑙 𝑢
= 𝑓𝜇𝑙𝑢 Finite-time classical Lyapunov spectrum: obeys RMT statistics for chaos
[Hanada, Shimada, and MT: PRE 97, 022224 (2018)]
Gharibyan, Hanada, Swingle, and MT, JHEP04(2019)082 (arXiv:1809.01671) 𝜀 𝑦 0 𝜀 𝑦 𝑢 = 𝑁𝜀 𝑦 0 𝑢
𝑀𝑏𝑐 𝑢 = 𝑘=1
𝑂
𝑁
𝑘𝑏 𝑢
𝑁
𝑘𝑐 𝑢 for time-dependent
anticommutator 𝑁𝑏𝑐 𝑢 = 𝜓𝑏 𝑢 , 𝜓𝑐 0 .
𝑙=1 𝐿
𝑀𝑏𝑐 𝑢 𝜚
log 𝑏𝑙 𝑢 2𝑢 𝑙=1,2,…,𝐿
(also dependent on state 𝜚) 𝐼 =
1≤𝑏<𝑐<𝑑<𝑒 𝑂
𝐾𝑏𝑐𝑑𝑒 𝜓𝑏 𝜓𝑐 𝜓𝑑 𝜓𝑒 + 𝑗
1≤𝑏<𝑐 𝑂
𝐿𝑏𝑐 𝜓𝑏 𝜓𝑐
𝐾𝑏𝑐𝑑𝑒: s. d. =
6 𝑂
3 2
𝐿𝑏𝑐: s. d. =
𝐿 𝑂
Other possibilities: see Rozenbaum-Ganeshan-Galitski, 1801.10591; Hallam-Morley-Green: 1806.05204
Sample- and state-averaged
Close to constant between red lines (20 % and 80 % of the saturated value of 𝜇𝑂𝑢)
Gharibyan, Hanada, Swingle, and MT, JHEP04(2019)082 (arXiv:1809.01671)
Almost linear growth of log(singular values of L)
Gharibyan, Hanada, Swingle, and MT, JHEP04(2019)082 (arXiv:1809.01671)
Coarse-grained entropy = log(# of cells covering the region) ~ (sum of positive 𝜇 ) 𝑢 Kolmogorov-Sinai entropy ℎKS = (sum of positive 𝜇 ) = entropy production rate
Coarse-grained entropy = log(# of cells covering the region) ~ (sum of positive 𝜇 ) 𝑢
Kolmogorov-Sinai entropy ℎKS = (sum of positive 𝜇 ) = entropy production rate
A B
Initial state with 𝑇EE = 0: 𝜔 𝑢 = 0 = 000 … 000 in the complex fermion basis 𝑂 2
𝑑
𝑘 = 𝜓2𝑘−1 + i𝜓2𝑘
2
𝑇EE 𝑢 = −Tr 𝜍A 𝑢 log 𝜍A 𝑢
𝜍A 𝑢 = TrB 𝜍 𝑢 , 𝜍 𝑢 = 𝜔 𝑢 𝜔 𝑢 Similar time scales of growth for SYK4 arXiv:1809.01671
𝑓𝜇𝑢
SYK4 limit
1 2𝑢 log 1 𝑂 𝑗=1 𝑂
𝑓2𝜇𝑗𝑢 approach each other; difference decreases as 1 𝑂
all exponent single peak
coupling (low T) limit
~ ℎKS = sum of positive (all) 𝜇𝑗
[conjecture] SYK model: not only the fastest scramblers, but also fastest entropy generators
arXiv:1809.01671
(fixed-i unfolding: unfold each gap 𝜇𝑗+1 − 𝜇𝑗 using its average) 𝐿 = 10 (◆):
Approaches Poisson
𝐿 = 0.01 (●):
Remains GUE for long time
Exponents are nearly constant until the singular values of 𝜚 𝑀𝑏𝑐 𝑢 𝜚 saturate: Lyapunov growth arXiv:1809.01671
𝑠 : average of
min 𝜗𝑗+1−𝜗𝑗 , 𝜗𝑗+2−𝜗𝑗+1 max 𝜗𝑗+1−𝜗𝑗 , 𝜗𝑗+2−𝜗𝑗+1
Energy eigenstates N/2 larger exponents
𝐼 =
𝑗 𝑂
𝑻𝑗 ∙ 𝑻𝑗+1 +
𝑗 𝑂
ℎ𝑗 𝑇𝑗
𝑨
ℎ𝑗: uniform distribution [−𝑋, 𝑋]
Many-body localization transition at 𝑋 = 𝑋
c ~ 3.6
(though recently disputed; e.g. 𝑋
c ≥ 5 proposed in E. V. H. Doggen et al., [1807.05051]
using large systems with time-dependent variational principle & machine learning) 𝜗c = 0.45 e.g. M. Serbyn, Z. Papic, and D. A. Abanin,
Matrix element of local perturbation Energy separation of neighboring energy eigenstates
𝐼 =
𝑗 𝑂
𝑻𝑗 ∙ 𝑻𝑗+1 +
𝑗 𝑂
ℎ𝑗 𝑇𝑗
𝑨
ℎ𝑗: uniform distribution [−𝑋, 𝑋]
𝑁𝑏𝑐 𝑢 = 𝑇𝑏
+ 𝑢 ,
𝑇𝑐
− 0
W = 0.5 (delocalized) W = 4.0 (Many-body localized)
Quantum Lyapunov spectrum distinguishes chaotic and non-chaotic phases
is not observed, but the statistics approach GUE arXiv:1809.01671
Characterization of many-body quantum chaos The Sachdev-Ye-Kitaev model The quantum Lyapunov spectrum The singular values of two-point correlators Summary
Dip-ramp-plateau structure for 𝑂 ≡ 2 mod 8 N = 16, GOE N = 18, GUE N = 20, GSE N = 22, GUE N = 24, GOE N = 14, GUE 𝐻 𝑢 = 𝜓𝑏 𝑢 𝜓𝑏 0
𝛾 =
1 𝑎 𝛾
𝑛,𝑜
e−𝛾𝐹𝑛 𝑛 𝜓𝑏 𝑜
2ei 𝐹𝑛−𝐹𝑜 𝑢
[Cotler, MT et al., JHEP05(2017)118] 𝑂 mod 8 2 4 6 𝑌 maps 𝐼E to 𝐼E 𝐼O 𝐼E 𝐼O even 𝜓 odd finite Gaussian ensemble GOE GUE GSE GUE
SYK model Energy eigenstates
𝐻𝑏𝑐
𝜚 = 𝜚
𝜓𝑏 𝑢 𝜓𝑐 0 𝜚 for SYK4 + K SYK2
𝜇𝑘 = log singular values of 𝐻𝑏𝑐
𝜚
larger N/2 exponents
𝜇𝑘 = log singular values of 𝐻𝑏𝑐
𝜚
GUE at all times GOE at late time
arXiv:1902.11086
𝐻𝑏𝑐
𝜚 = 𝜚
𝜓𝑏 𝑢 𝜓𝑐 0 𝜚 for SYK4 + K SYK2
𝑠 : average of the adjacent gap ratio
min 𝜇𝑗+1−𝜇𝑗 , 𝜇𝑗+2−𝜇𝑗+1 max 𝜇𝑗+1−𝜇𝑗 , 𝜇𝑗+2−𝜇𝑗+1
Uncorrelated (Poisson): 2 log 2 − 1 ≈ 0.386 Correlated: larger (GOE: 0.5307, GUE: 0.5996 etc. ) [Atas et al., PRL 2013]
At late time,
Random matrix behavior chaotic
N mod 8 = 0: GOE
(the matrix is symmetric)
N mod 8 = 2, 4, 6: GUE
𝐻𝑏𝑐
𝜚 = 𝜚
𝜓𝑏 𝑢 𝜓𝑐 0 𝜚
𝜇𝑘 = log singular values of 𝐻𝑏𝑐
𝜚
fixed-i unfolded
SYK, larger N/2 exponents 𝜚: energy eigenstates arXiv:1902.11086
Random-matrix like for complex fermion number eig igenstates, even for non-chaotic regime
Empty state in complex fermion description: state without long-range entanglement
𝐼 =
𝑗 𝑂
𝑇𝑗 ∙ 𝑇𝑗+1 +
𝑗 𝑂
ℎ𝑗 𝑇𝑗
𝑨 ℎ𝑗 ∈ [−𝑋, 𝑋]
Energy eigenstates (not close to the spectral edges): GOE at short and long times for small W
𝐻𝑏𝑐
𝜚 = 𝜚
𝜏+𝑏 𝑢 𝜏−𝑐 0 𝜚
𝐻𝑏𝑐
𝜚 = 𝜚
𝜏+𝑏 𝑢 𝜏−𝑐 0 𝜚
𝐼 =
𝑗 𝑂
𝑇𝑗 ∙ 𝑇𝑗+1 +
𝑗 𝑂
ℎ𝑗 𝑇𝑗
𝑨 ℎ𝑗 ∈ [−𝑋, 𝑋]
Energy eigenstates GOE at short and long times for small W, close to Poisson at any time for large W
𝐻𝑏𝑐
𝜚 = 𝜚
𝜏+𝑏 𝑢 𝜏−𝑐 0 𝜚
Singular value statistics of two-point correlation function
Model Chaotic (small K / small W) Not chaotic (large K / large W) SYK4 + SYK2 Energy eig. GUE at late time except for 𝑂 ≡ 0 mod 8 : GOE Spin eig. GUE at any time Energy eig. Poisson at any time Spin eig. off from GUE at some time XXZ + random field Energy eig. off from GOE at some time (𝐻𝑏𝑐
𝜚 = 𝜚
𝜏+𝑏 𝑢 𝜏−𝑐 0 𝜚 is symmetric)
Spin eig. converges to GUE (𝐻𝑏𝑐
𝜚 = 𝜚
𝜏+𝑏 𝑢 𝜏−𝑐 0 𝜚 is not symmetric)
Energy eig. close to Poisson Spin eig. approaches Poisson from RMT-like
Pak Hang Chris Lau, Chen-Te Ma, Jeff Murugan, and MT, Phys. Lett. B in press (1812.04770)
characterizes quantum chaos [1809.01671]
random matrix behavior in chaotic cases [1902.11086]
random field
𝑀𝑏𝑐 𝑢 = 𝑘=1
𝑂
𝑁
𝑘𝑏 𝑢
𝑁
𝑘𝑐 𝑢 for
𝑁𝑏𝑐 𝑢 = 𝜓𝑏 𝑢 , 𝜓𝑐 0 QLS: log(singular values of 𝜚 𝑀𝑏𝑐 𝑢 𝜚 )/(2t) 𝐻𝑏𝑐
𝜚 = 𝜚
𝜓𝑏 𝑢 𝜓𝑐 0 𝜚