Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Dissecting Quantum Many-body Chaos in the Krylov Space (2404.08207v1)

Published 12 Apr 2024 in quant-ph, cond-mat.quant-gas, cond-mat.str-el, and hep-th

Abstract: The growth of simple operators is essential for the emergence of chaotic dynamics and quantum thermalization. Recent studies have proposed different measures, including the out-of-time-order correlator and Krylov complexity. It is established that the out-of-time-order correlator serves as the signature of quantum many-body chaos, while the Krylov complexity provides its upper bound. However, there exist non-chaotic systems in which Krylov complexity grows exponentially, indicating that the Krylov complexity itself is not a witness of many-body chaos. In this letter, we introduce the missing ingredient, named as the Krylov metric $K_{mn}$, which probes the size of the Krylov basis. We propose that the universal criteria for fast scramblers include (i) the exponential growth of Krylov complexity, (ii) the diagonal elements $K_{nn}\sim nh$ with $h\in(0,1]$, and (iii) the negligibility of off-diagonal elements $K_{mn}$ with $m\neq n$. We further show that $h=\varkappa / 2\alpha$ is a ratio between the quantum Lyapunov exponent $\varkappa$ and the Krylov exponent $\alpha$. This proposal is supported by both generic arguments and explicit examples, including solvable SYK models, Luttinger Liquids, and many-body localized systems. Our results provide a refined understanding of how chaotic dynamics emerge from the Krylov space perspective.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (62)
  1. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09, 120, arXiv:0708.4025 [hep-th] .
  2. Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10, 065, arXiv:0808.2096 [hep-th] .
  3. T. Dray and G. ’t Hooft, The gravitational shock wave of a massless particle, Nuclear Physics B 253, 173 (1985).
  4. G. ’t Hooft, The black hole interpretation of string theory, Nuclear Physics B 335, 138 (1990).
  5. Y. Kiem, H. L. Verlinde, and E. P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52, 7053 (1995), arXiv:hep-th/9502074 .
  6. G. ’t Hooft, The Scattering matrix approach for the quantum black hole: An Overview, Int. J. Mod. Phys. A 11, 4623 (1996), arXiv:gr-qc/9607022 .
  7. A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity, Soviet Journal of Experimental and Theoretical Physics 28, 1200 (1969).
  8. S. H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03, 067, arXiv:1306.0622 [hep-th] .
  9. S. H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05, 132, arXiv:1412.6087 [hep-th] .
  10. A. Kitaev, talk given at fundamental physics prize symposium (2014).
  11. D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, JHEP 03, 051, arXiv:1409.8180 [hep-th] .
  12. D. A. Roberts, D. Stanford, and A. Streicher, Operator growth in the SYK model, JHEP 06, 122, arXiv:1802.02633 [hep-th] .
  13. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, JHEP 08, 106, arXiv:1503.01409 [hep-th] .
  14. A. Kitaev, A simple model of quantum holography (part 1), Kavli Institute for Theoretical Physics Program: Entanglement in Strongly-Correlated Quantum Matter (Apr 6 - Jul 2, 2015). (2015).
  15. S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum heisenberg magnet, Phys. Rev. Lett. 70, 3339 (1993).
  16. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016), arXiv:1604.07818 [hep-th] .
  17. A. Kitaev and S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05, 183, arXiv:1711.08467 [hep-th] .
  18. S. Xu and B. Swingle, Scrambling Dynamics and Out-of-Time-Ordered Correlators in Quantum Many-Body Systems, PRX Quantum 5, 010201 (2024), arXiv:2202.07060 [quant-ph] .
  19. X. Chen and T. Zhou, Quantum chaos dynamics in long-range power law interaction systems, Phys. Rev. B 100, 064305 (2019), arXiv:1808.09812 [cond-mat.stat-mech] .
  20. T. Zhou and X. Chen, Operator dynamics in a Brownian quantum circuit, Phys. Rev. E 99, 052212 (2019), arXiv:1805.09307 [cond-mat.str-el] .
  21. J. Maldacena, D. Stanford, and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65, 1700034 (2017), arXiv:1704.05333 [hep-th] .
  22. D. A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04, 121, arXiv:1610.04903 [quant-ph] .
  23. R. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, JHEP 10, 107, arXiv:1707.08570 [hep-th] .
  24. R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97, 066004 (2018), arXiv:1709.00921 [hep-th] .
  25. R. Khan, C. Krishnan, and S. Sharma, Circuit Complexity in Fermionic Field Theory, Phys. Rev. D 98, 126001 (2018), arXiv:1801.07620 [hep-th] .
  26. A. Lucas, Operator size at finite temperature and Planckian bounds on quantum dynamics, Phys. Rev. Lett. 122, 216601 (2019), arXiv:1809.07769 [cond-mat.str-el] .
  27. C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Physical Review Research 5, 033085 (2023), arXiv:2207.13603 [cond-mat.str-el] .
  28. A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102, 085137 (2020), arXiv:1912.12227 [cond-mat.stat-mech] .
  29. J. L. F. Barbón, J. Martín-García, and M. Sasieta, Momentum/Complexity Duality and the Black Hole Interior, JHEP 07, 169, arXiv:1912.05996 [hep-th] .
  30. J. M. Magán and J. Simón, On operator growth and emergent Poincaré symmetries, JHEP 05, 071, arXiv:2002.03865 [hep-th] .
  31. S.-K. Jian, B. Swingle, and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, JHEP 03, 014, arXiv:2008.12274 [hep-th] .
  32. C.-F. Chen and A. Lucas, Operator Growth Bounds from Graph Theory, Commun. Math. Phys. 385, 1273 (2021), arXiv:1905.03682 [math-ph] .
  33. J. D. Noh, Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field, Phys. Rev. E 104, 034112 (2021), arXiv:2107.08287 [quant-ph] .
  34. P. Caputa and S. Datta, Operator growth in 2d CFT, JHEP 12, 188, [Erratum: JHEP 09, 113 (2022)], arXiv:2110.10519 [hep-th] .
  35. D. Patramanis, Probing the entanglement of operator growth, PTEP 2022, 063A01 (2022), arXiv:2111.03424 [hep-th] .
  36. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022), arXiv:2109.03824 [hep-th] .
  37. A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104, L081702 (2021), arXiv:2104.09514 [hep-th] .
  38. A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2, 043234 (2020), arXiv:1911.09672 [cond-mat.stat-mech] .
  39. C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5, 033085 (2023), arXiv:2207.13603 [cond-mat.str-el] .
  40. B. Bhattacharjee, P. Nandy, and T. Pathak, Operator dynamics in Lindbladian SYK: a Krylov complexity perspective, JHEP 01, 094, arXiv:2311.00753 [quant-ph] .
  41. C. Lv, R. Zhang, and Q. Zhou, Building Krylov complexity from circuit complexity, arXiv e-prints , arXiv:2303.07343 (2023), arXiv:2303.07343 [quant-ph] .
  42. H. Tang, Operator Krylov complexity in random matrix theory, arXiv e-prints , arXiv:2312.17416 (2023), arXiv:2312.17416 [hep-th] .
  43. R. Zhang and H. Zhai, Universal Hypothesis of Autocorrelation Function from Krylov Complexity, arXiv e-prints , arXiv:2305.02356 (2023), arXiv:2305.02356 [cond-mat.stat-mech] .
  44. T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, 2003).
  45. F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, Comptes Rendus Physique 19, 498 (2018), arXiv:1711.03145 [cond-mat.str-el] .
  46. E. Altman and R. Vosk, Universal Dynamics and Renormalization in Many-Body-Localized Systems, Annual Review of Condensed Matter Physics 6, 383 (2015), arXiv:1408.2834 [cond-mat.dis-nn] .
  47. R. Nandkishore and D. A. Huse, Many-Body Localization and Thermalization in Quantum Statistical Mechanics, Annual Review of Condensed Matter Physics 6, 15 (2015), arXiv:1404.0686 [cond-mat.stat-mech] .
  48. M. Serbyn, Z. Papić, and D. A. Abanin, Local conservation laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013).
  49. D. A. Huse, R. Nandkishore, and V. Oganesyan, Phenomenology of fully many-body-localized systems, Phys. Rev. B 90, 174202 (2014).
  50. R. Vosk and E. Altman, Many-body localization in one dimension as a dynamical renormalization group fixed point, Phys. Rev. Lett. 110, 067204 (2013).
  51. Notice that our convention for the Liouvillian superoperator differs from that in [33] by a factor of i𝑖iitalic_i.
  52. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. B 45, 255 (1950).
  53. Z. Liu and P. Zhang, Signature of Scramblon Effective Field Theory in Random Spin Models, Phys. Rev. Lett.  132, 060201 (2024), arXiv:2306.05678 [quant-ph] .
  54. Y. Chen, H. Zhai, and P. Zhang, Tunable Quantum Chaos in the Sachdev-Ye-Kitaev Model Coupled to a Thermal Bath, JHEP 07, 150, arXiv:1705.09818 [hep-th] .
  55. P. Zhang and Z. Yu, Dynamical Transition of Operator Size Growth in Quantum Systems Embedded in an Environment, Phys. Rev. Lett. 130, 250401 (2023).
  56. See supplementary material for: (1). Detailed derivations of the asymptotic behaviors of the Krylov metric; (2). Detailed computations of the operator-size distribution of the Krylov metric.
  57. Y. Huang, Y.-L. Zhang, and X. Chen, Out-of-time-ordered correlators in many-body localized systems, Annalen Phys. 529, 1600318 (2017), arXiv:1608.01091 [cond-mat.dis-nn] .
  58. B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B 95, 060201 (2017), arXiv:1608.03280 [cond-mat.str-el] .
  59. R.-Q. He and Z.-Y. Lu, Characterizing many-body localization by out-of-time-ordered correlation, Phys. Rev. B 95, 054201 (2017), arXiv:1608.03586 [cond-mat.dis-nn] .
  60. Y. Chen, Universal Logarithmic Scrambling in Many Body Localization, arXiv e-prints , arXiv:1608.02765 (2016), arXiv:1608.02765 [cond-mat.dis-nn] .
  61. Y. Gu, A. Kitaev, and P. Zhang, A two-way approach to out-of-time-order correlators, JHEP 03, 133, arXiv:2111.12007 [hep-th] .
  62. P. Zhang and Y. Gu, Operator size distribution in large N quantum mechanics of Majorana fermions, JHEP 10, 018, arXiv:2212.04358 [cond-mat.str-el] .
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets