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 手塚真樹 (Kyoto University)

Collaborators in this work arXiv:1809.01671 (JHEP04(2019)082) arXiv:1902.11086 Hrant Gharibyan 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 Plan of the The quantum Lyapunov spectrum talk The singular values of two-point correlators Summary

Characterization of many-body quantum chaos The Sachdev-Ye-Kitaev model Plan of the The quantum Lyapunov spectrum talk The singular values of two-point correlators Summary

Chaos in deterministic classical dynamics • Sensitivity to initial conditions: exponential growth of initial perturbation 𝜀𝑦 𝑢 𝜀𝑦 0 0 𝑢 “butterfly effect” Bounded, nonperiodic dynamics with nonlinearity What happens in quantum mechanics?

by Dan Quinn (on Wikimedia Commons)

Building 24, MIT

How to characterize quantum chaos? 𝐼 = const. 𝑢 𝑗 𝑒 = = exp −𝑗 𝜔 = 𝐼 𝑢′ 𝑒𝑢 𝐼𝑢 𝜔 𝑢 T exp −𝑗 𝜔 𝑢 = 0 𝜔 𝑢 = 0 𝑒𝑢 𝐼 𝜔 0 Linear dynamics Unitary time evolution • Long time: energy level statistics • Short time: out-of-time correlator Correlation between levels, as in random matrices Classically, 2 2 = 𝜖𝑦 𝑗 𝑢 → 𝑓 2𝜇 L 𝑢 for large t 𝑦 𝑗 𝑢 , 𝑞 𝑘 0 PB 𝜖𝑦 𝑘 0 Quantum version: P ( s ): normalized level 2 𝑋 𝑢 , OTOC: 𝐷 𝑈 𝑢 = 𝑊 𝑢 = 0 separation distribution Uncorrelated: Poisson ( 𝑓 −𝑡 ) 𝑋 † 𝑢 𝑊 † 0 𝑋 𝑢 = 𝑊 0 + ⋯ cf. Bohigas-Giannoni-Schmit conjecture  limited to very small systems  Hard to see exponential time dependence Numerically

Characterization of quantum many-body chaos • Random-matrix like energy level • Exponential Lyapunov growth of out- correlation of-time-order correlators (OTOC) 𝑋 † 𝑢 𝑊 † 0 𝑋 𝑢 ~ 𝐷 + # 𝑓 2𝜇 L 𝑢 𝑊 0 Example: the Sachdev-Ye-Kitaev model 𝜓 𝑏 , 𝜓 𝑐 = 𝜀 𝑏𝑐 3! 𝐼 = 𝐾 𝑏𝑐𝑑𝑒 𝜓 𝑏 𝜓 𝑐 𝜓 𝑑 𝜓 𝑒 𝑂 3/2 𝐾 𝑏𝑐𝑑𝑒 : Gaussian random 2 = 𝐾 2 = 1 ) [Cotler, MT et al., 1≤𝑏<𝑐<𝑑<𝑒≤𝑂 [Kitaev 2015] ( 𝐾 𝑏𝑐𝑑𝑒 JHEP 1705 , 118 (2017)] Spectral form factor Lyapunov exponent 2 N mod 8 RMT 𝑎 𝛾, 𝑢 𝑕 𝛾, 𝑢 = 2𝜌𝑙 B 𝑈 𝑎 𝛾, 𝑢 = 0 0 GOE 𝜇 L = in low 𝑈 limit 1 e −𝛾 𝐹 𝑛 +𝐹 𝑜 e i 𝐹 𝑛 −𝐹 𝑜 𝑢 2 GUE = 𝑎 𝛾, 𝑢 = 0 2 ℏ 𝑛,𝑜 4 GSE (Maldacena-Shenker-Stanford chaos bound) 6 GUE

Characterization of many-body quantum chaos The Sachdev-Ye-Kitaev model Plan of the The quantum Lyapunov spectrum talk The singular values of two-point correlators Summary

The Sachdev-Ye-Kitaev model 3! 𝐼 = 𝐾 𝑏𝑐𝑑𝑒 𝜓 𝑏 𝜓 𝑐 𝜓 𝑑 𝜓 𝑒 𝑂 3/2 cf. Sachdev-Ye model (1993) 1≤𝑏<𝑐<𝑑<𝑒≤𝑂 𝜓 𝑏=1,2,…,𝑂 : 𝑂 Majorana fermions ( 𝜓 𝑏 , 𝜓 𝑐 = 𝜀 𝑏𝑐 ) [A. Kitaev, talks at KITP (2015)] 2 = 𝐾 2 = 1, 𝐾 𝑏𝑐𝑑𝑒 = 0 ) 𝐾 𝑏𝑐𝑑𝑒 : Gaussian random couplings ( 𝐾 𝑏𝑐𝑑𝑒 𝐾 3567 𝐾 1259 𝐾 4567 𝐾 1348 ⋯

The SYK model Figures from [I. Danshita, MT, and M. Hanada: Butsuri 73 (8), 569 (2018)] 3! 𝐼 = 𝐾 𝑏𝑐𝑑𝑒 𝜓 𝑏 𝜓 𝑐 𝜓 𝑑 𝜓 𝑒 𝑂 3/2 1≤𝑏<𝑐<𝑑<𝑒≤𝑂 O ( N -2 ) Analytically solvable in 𝑂 ≫ 1 limit O (1) 𝐻 = 𝐻 0 Σ𝐻 0 𝐻 0 𝐾 𝑗𝑘𝑙𝑚 𝐾 𝑘𝑙𝑚𝑛 = 𝜀 𝑗𝑛 Only “melon - type” diagrams survive 𝐻 0 Σ𝐻 0 Σ𝐻 0 −1 = 𝑗𝜕 − Σ 𝑗𝜕 Σ = 𝐾 2 𝐻 3 = 𝐻 0 Σ𝐻 0 −1 𝐻 𝑗𝜕 𝐻 0 Σ𝐻 0 Σ𝐻 0 Σ𝐻 0 + ⋯

The SYK model 3! 𝐼 = 𝐾 𝑏𝑐𝑑𝑒 𝜓 𝑏 𝜓 𝑐 𝜓 𝑑 𝜓 𝑒 𝑂 3/2 1≤𝑏<𝑐<𝑑<𝑒≤𝑂 Analytically solvable in 𝑂 ≫ 1 limit 𝐻 1 − Σ𝐻 0 = 𝐻 0 Low energy ( 𝜕, 𝑈 ≪ 𝐾 ): ignore 𝑗𝜕 and we have −1 − Σ 𝐻 −1 = 𝐻 0 𝑒𝑢𝐻 𝑢 1 , 𝑢 Σ 𝑢, 𝑢 2 = −𝜀 𝑢 1 , 𝑢 2 −1 = 𝑗𝜕 − Σ 𝑗𝜕 Σ = 𝐾 2 𝐻 3 𝐻 𝑗𝜕 𝐾 2 𝑒𝑢𝐻 𝑢 1 , 𝑢 𝐻 𝑢, 𝑢 2 3 = −𝜀 𝑢 1 , 𝑢 2 1/4 sgn 𝑢 1 𝐻 𝑢 = − 4𝜌𝐾 2 𝑢 • Emergent conformal symmetry 2𝜌𝑙 B 𝑈 • Satisfies the “chaos bound” 𝜇 L ≤ in the T  0 limit ℏ

Holographic connection to gravity [S. Sachdev, Phys. Rev. X 5 , 041025 (2015)]

Proposal for experimental realization s : molecular levels [I. Danshita, M. Hanada, MT: arXiv:1606.02454; PTEP 2017 , 083I01 (2017)] (also a proceedings manuscript arXiv:1709.07189) Modified SYK model: Setup: A double-well Sums of two single atom energies See optical lattice (no degeneracy in the band levels) with 6 Li (large recoil energy) for review including other proposals e.g. using topological insulator, graphene

Sachdev-Ye-Kitaev model N Majorana- or Dirac- fermions randomly coupled to each other [Dirac version] [Majorana version] 1 3! † † 𝐼 = 2𝑂 3/2 𝐾 𝑗𝑘;𝑙𝑚 𝑑 𝑗 𝑑 𝑑 𝑙 𝑑 𝑚 𝐼 = 𝐾 𝑏𝑐𝑑𝑒 𝜓 𝑏 𝜓 𝑐 𝜓 𝑑 𝜓 𝑒 𝑘 𝑂 3/2 𝑗𝑘;𝑙𝑚 1≤𝑏<𝑐<𝑑<𝑒≤𝑂 [A. Kitaev: talks at KITP [A. Kitaev’s talk] (Feb 12, Apr 7 and May 27, 2015) ] [S. Sachdev: PRX 5 , 041025 (2015)] • Solvable in the large N limit, Sachdev- Ye “spin liquid” ground state • Nearly conformal symmetric at low temperature (“emergent …”) • Holographically corresponds to a quantum black hole? • Experimental schemes have been proposed • Realizes the Maldacena-Shenker-Stanford chaos bound 𝜇 L = 2π𝑙 B 𝑈 ℏ Generalizations: q - fermion interactions “ SYK q ”, supersymmetric SYK, lattice of SYK lands; etc.

Classification and random matrix theory SPT phase classification for class BDI: 3! ℤ  ℤ 8 due to interaction 𝐼 = 𝐾 𝑏𝑐𝑑𝑒 𝜓 𝑏 𝜓 𝑐 𝜓 𝑑 𝜓 𝑒 𝑂 3/2 [L. Fidkowski and A. Kitaev, PRB 2010, PRB 2011] 1≤𝑏<𝑐<𝑑<𝑒≤𝑂 𝜓 2𝑘−1 + i 𝜓 2𝑘 Introduce 𝑂/2 complex fermions 𝑑 𝑘 = 2 𝜓 𝑏 𝜓 𝑐 𝜓 𝑑 𝜓 𝑒 respects the complex fermion parity 𝐼 O ) sectors: 𝑀 = 2 𝑂/2−1 dimensions 𝑂 ≡ 0, 4 Even ( 𝐼 E ) and odd ( 𝑂/2 (mod 8) † + 𝑌 = 𝐿 𝑑 𝑑 𝑂 mod 8 0 2 4 6 𝑘 𝑘 𝐼 E 𝑘=1 𝜃 -1 +1 +1 -1 † 𝑘 𝑌 𝑑 𝑌 = 𝜃 𝑑 𝑘 𝑌 2 +1 +1 -1 -1 𝐼 E 𝐼 O 𝐼 E 𝐼 O 𝑌 maps 𝐼 E to [You, Ludwig, and Xu, PRB 2017] 𝐼 O Class AI A+A AII A+A 𝑂 ≡ 2, 6 [Fadi Sun and Jinwu Ye, 1905.07694] Gaussian ensemble GOE GUE GSE GUE for SYK q , supersymmetric SYK Sparse, but energy spectral statistics strongly resemble [Cotler, …, MT, JHEP 2017] that of the corresponding (dense) Gaussian ensemble

Gaussian random matrices ∗ 𝑏 𝑗𝑘 = 𝑏 𝑘𝑗 4 Tr𝐼 2 = exp − Density ∝ 𝑓 − 𝛾𝐿 2 𝐿 𝑏 𝑗𝑘 𝛾𝐿 4 𝑗,𝑘 𝐿 [F. J. Dyson, J. Math. Phys. 3 , 1199 (1962)] 𝑏 𝑗𝑘 𝑗,𝑘=1 Real (β=1): Gaussian Orthogonal Ensemble (GOE) Complex (β=2): G. Unitary E. (GUE) Gaussian distribution Quaternion (β=4): G. Symplectic E. (GSE) Eigenvalue distribution: semi-circle law Level repulsion Joint distribution 𝐿 𝛾 𝑓 𝑗2 4 𝑓 −𝛾𝐿 𝑞 𝑓 1 , 𝑓 2 , … , 𝑓 𝐿 ∝ 𝑓 𝑗 − 𝑓 𝑘 𝑄 𝑓 1≤𝑗<𝑘≤𝐿 𝑗=1 𝑓 𝑘+1 −𝑓 𝑘 • 𝑄 𝑡 : Distribution of normalized level separation 𝑡 = ∆ 𝑓 GOE/GUE/GSE: 𝑄 𝑡 ∝ 𝑡 𝛾 at small 𝑡 , has 𝑓 −𝑡 2 tail 𝑓 1 ≤ 𝑓 2 ≤ ⋯ ≤ 𝑓 𝑀 Uncorrelated: 𝑄 𝑡 = 𝑓 −𝑡 (Poisson distribution) ∆ : averaged level separation near 𝑓 𝑘

Gaussian random matrices 4 Tr𝐼 2 = exp − 𝛾𝐿 ∗ Density ∝ 𝑓 − 𝛾𝐿 𝑏 𝑗𝑘 = 𝑏 𝑘𝑗 2 𝐿 𝑏 𝑗𝑘 4 𝑗,𝑘 𝑀 Real (β=1): Gaussian Orthogonal Ensemble (GOE) 𝑏 𝑗𝑘 𝑗,𝑘=1 Complex (β=2): G. Unitary E. (GUE) Quaternion (β=4): G. Symplectic E. (GSE) Gaussian distribution Level repulsion Joint distribution 𝐿 𝛾 𝑓 𝑗2 4 𝑓 −𝛾𝐿 𝑞 𝑓 1 , 𝑓 2 , … , 𝑓 𝐿 ∝ 𝑓 𝑗 − 𝑓 𝑘 1≤𝑗<𝑘≤𝐿 𝑗=1 𝑓 𝑘+1 −𝑓 𝑘 • 𝑄 𝑡 : Distribution of normalized level separation 𝑡 = ∆ 𝑓 GOE/GUE/GSE: 𝑄 𝑡 ∝ 𝑡 𝛾 at small 𝑡 , has 𝑓 −𝑡 2 tail Uncorrelated: 𝑄 𝑡 = 𝑓 −𝑡 (Poisson distribution) • 𝑠 : Average of neighboring gap ratio Uncorrelated GOE GUE GSE 𝑠 = min 𝑓 𝑗+1 − 𝑓 𝑗 , 𝑓 𝑗+2 − 𝑓 𝑗+1 𝑠 2log 2 – 1 = 0.38629… 0.5307(1) 0.5996(1) 0.6744(1) max 𝑓 𝑗+1 − 𝑓 𝑗 , 𝑓 𝑗+2 − 𝑓 𝑗+1 [Y. Y. Atas et al. PRL 2013]  SYK model results: indistinguishable from corresponding Gaussian ensemble

Density of states [Cotler, MT et al., JHEP05(2017)118]