Papers
Topics
Authors
Recent
Search
2000 character limit reached

Hilbert Space Diffusion in Systems with Approximate Symmetries

Published 29 May 2024 in cond-mat.stat-mech, cond-mat.str-el, hep-th, and nlin.CD | (2405.19260v4)

Abstract: Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this paper, we describe systematic deviations from RMT behaviour at intermediate time scales in systems with approximate symmetries. At early times, the symmetries allow us to organize the Hilbert space into approximately decoupled sectors, each of which contributes independently to the SFF. At late times, the SFF transitions into the final ramp of the fully mixed chaotic Hamiltonian. For approximate continuous symmetries, the transitional behaviour is governed by a universal process that we call Hilbert space diffusion. The diffusion constant corresponding to this process is related to the relaxation rate of the associated nearly conserved charge. By implementing a chaotic sigma model for Hilbert-space diffusion, we formulate an analytic theory of this process which agrees quantitatively with our numerical results for different examples.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (50)
  1. O. Bohigas, M. J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52 (Jan, 1984) 1–4. https://link.aps.org/doi/10.1103/PhysRevLett.52.1.
  2. Y. Fyodorov, “Random matrix theory,” Scholarpedia 6 no. 3, (2011) 9886.
  3. J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, “Black Holes and Random Matrices,” JHEP 05 (2017) 118, arXiv:1611.04650 [hep-th]. [Erratum: JHEP 09, 002 (2018)].
  4. P. Saad, S. H. Shenker, and D. Stanford, “A semiclassical ramp in SYK and in gravity,” arXiv:1806.06840 [hep-th].
  5. P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th].
  6. A. Altland and J. Sonner, “Late time physics of holographic quantum chaos,” SciPost Phys. 11 (2021) 034, arXiv:2008.02271 [hep-th].
  7. J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A 43 (Feb, 1991) 2046–2049. https://link.aps.org/doi/10.1103/PhysRevA.43.2046.
  8. M. Srednicki, “Chaos and quantum thermalization,” Physical Review E 50 no. 2, (Aug, 1994) 888–901. https://doi.org/10.1103%2Fphysreve.50.888.
  9. L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, “From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Adv. Phys. 65 no. 3, (2016) 239–362, arXiv:1509.06411 [cond-mat.stat-mech].
  10. M. V. Berry, M. Tabor, and J. M. Ziman, “Level clustering in the regular spectrum,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 356 no. 1686, (1977) 375–394. https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1977.0140.
  11. B. Altshuler and B. Shklovskii, “Repulsion of energy levels and conductivity of small metal samples,” Sov. Phys. JETP 64 no. 1, (1986) 127–135. http://jetp.ras.ru/cgi-bin/dn/e_064_01_0127.pdf.
  12. A. V. Andreev and B. L. Altshuler, “Spectral statistics beyond random matrix theory,” Phys. Rev. Lett. 75 (Jul, 1995) 902–905. https://link.aps.org/doi/10.1103/PhysRevLett.75.902.
  13. V. E. Kravtsov and A. D. Mirlin, “Level statistics in a metallic sample: corrections to the wigner-dyson distribution,” arXiv:cond-mat/9408072 [cond-mat].
  14. T. Guhr, A. Muller-Groeling, and H. A. Weidenmuller, “Random matrix theories in quantum physics: Common concepts,” Phys. Rept. 299 (1998) 189–425, arXiv:cond-mat/9707301.
  15. A. del Campo, J. Molina-Vilaplana, and J. Sonner, “Scrambling the spectral form factor: unitarity constraints and exact results,” Phys. Rev. D 95 no. 12, (2017) 126008, arXiv:1702.04350 [hep-th].
  16. E. J. Torres-Herrera, A. M. García-García, and L. F. Santos, “Generic dynamical features of quenched interacting quantum systems: Survival probability, density imbalance, and out-of-time-ordered correlator,” Physical Review B 97 no. 6, (Feb., 2018) 5, arXiv:1704.06272 [cond-mat.stat-mech].
  17. P. Kos, M. Ljubotina, and T. Prosen, “Many-body quantum chaos: Analytic connection to random matrix theory,” Phys. Rev. X 8 no. 2, (2018) 021062, arXiv:1712.02665 [nlin.CD].
  18. H. Gharibyan, M. Hanada, S. H. Shenker, and M. Tezuka, “Onset of Random Matrix Behavior in Scrambling Systems,” JHEP 07 (2018) 124, arXiv:1803.08050 [hep-th]. [Erratum: JHEP 02, 197 (2019)].
  19. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, “Quantum chaos challenges many-body localization,” Phys. Rev. E 102 (2020) 062144, arXiv:1905.06345 [cond-mat.str-el].
  20. A. Altland and D. Bagrets, “Quantum ergodicity in the SYK model,” Nucl. Phys. B 930 (2018) 45–68, arXiv:1712.05073 [cond-mat.str-el].
  21. A. Chan, A. De Luca, and J. T. Chalker, “Spectral statistics in spatially extended chaotic quantum many-body systems,” Phys. Rev. Lett. 121 no. 6, (2018) 060601, arXiv:1803.03841 [cond-mat.stat-mech].
  22. A. J. Friedman, A. Chan, A. De Luca, and J. Chalker, “Spectral statistics and many-body quantum chaos with conserved charge,” Physical Review Letters 123 no. 21, (Nov., 2019) 210603. http://dx.doi.org/10.1103/PhysRevLett.123.210603.
  23. A. Altland, D. Bagrets, P. Nayak, J. Sonner, and M. Vielma, “From operator statistics to wormholes,” Physical Review Research 3 no. 3, (Sept., 2021) 033259. http://dx.doi.org/10.1103/PhysRevResearch.3.033259.
  24. M. Winer and B. Swingle, “Hydrodynamic theory of the connected spectral form factor,” Phys. Rev. X 12 no. 2, (2022) 021009, arXiv:2012.01436 [cond-mat.stat-mech].
  25. L. V. Delacrétaz, A. L. Fitzpatrick, E. Katz, and M. T. Walters, “Thermalization and chaos in a 1+1d qft,” Journal of High Energy Physics 2023 no. 2, (Feb., 2023) 45. http://dx.doi.org/10.1007/JHEP02(2023)045.
  26. S. Iida, H. Weidenmüller, and J. Zuk, “Statistical scattering theory, the supersymmetry method and universal conductance fluctuations,” Annals of Physics 200 no. 2, (1990) 219–270. https://www.sciencedirect.com/science/article/pii/000349169090275S.
  27. M. Winer and B. Swingle, “Spontaneous symmetry breaking, spectral statistics, and the ramp,” Phys. Rev. B 105 no. 10, (2022) 104509, arXiv:2106.07674 [cond-mat.stat-mech].
  28. M. Winer, R. Barney, C. L. Baldwin, V. Galitski, and B. Swingle, “Spectral form factor of a quantum spin glass,” JHEP 09 (2022) 032, arXiv:2203.12753 [cond-mat.stat-mech].
  29. M. L. Mehta and A. Pandey, “On some gaussian ensembles of hermitian matrices,” Journal of Physics A: Mathematical and General 16 no. 12, (Aug, 1983) 2655. https://dx.doi.org/10.1088/0305-4470/16/12/014.
  30. A. Pandey and M. L. Mehta, “Gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones,” Communications in Mathematical Physics 87 no. 4, (1983) 449–468. https://doi.org/10.1007/BF01208259.
  31. F. Haake and G. Lenz, “Classical hamiltonian dynamics of rescaled quantum levels,” Europhysics Letters 13 no. 7, (Dec, 1990) 577. https://dx.doi.org/10.1209/0295-5075/13/7/001.
  32. G. Lenz and F. Haake, “Transitions between universality classes of random matrices,” Phys. Rev. Lett. 65 (Nov, 1990) 2325–2328. https://link.aps.org/doi/10.1103/PhysRevLett.65.2325.
  33. Springer, 2010. https://doi.org/10.1007/978-3-319-97580-1.
  34. A. Altland, T. Micklitz, and J. T. de Miranda, “Tensor product random matrix theory,” arXiv:2404.10919 [cond-mat.mes-hall].
  35. CRC Press, 2018. https://doi.org/10.1201/9780429493683.
  36. Y. Gu, A. Kitaev, S. Sachdev, and G. Tarnopolsky, “Notes on the complex sachdev-ye-kitaev model,” Journal of High Energy Physics 2020 no. 2, (Feb., 2020) 75. http://dx.doi.org/10.1007/JHEP02(2020)157.
  37. L. Schäfer and F. Wegner, “Disordered system withn orbitals per site: Lagrange formulation, hyperbolic symmetry, and goldstone modes,” Zeitschrift für Physik B Condensed Matter 38 no. 2, (1980) 113–126. https://doi.org/10.1007/BF01598751.
  38. K. Efetov, “Supersymmetry and theory of disordered metals,” Advances in Physics 32 no. 1, (1983) 53–127. https://doi.org/10.1080/00018738300101531.
  39. M. R. Zirnbauer, “Riemannian symmetric superspaces and their origin in random‐matrix theory,” Journal of Mathematical Physics 37 no. 10, (10, 1996) 4986–5018. https://doi.org/10.1063/1.531675.
  40. K. Efetov, Supersymmetry in disorder and chaos. Cambridge University Press, 1995. https://doi.org/10.1017/CBO9780511573057.
  41. J. J. M. Verbaarschot and I. Zahed, “Spectral density of the qcd dirac operator near zero virtuality,” Phys. Rev. Lett. 70 (Jun, 1993) 3852–3855. https://link.aps.org/doi/10.1103/PhysRevLett.70.3852.
  42. J. Verbaarschot, “Spectrum of the qcd dirac operator and chiral random matrix theory,” Phys. Rev. Lett. 72 (Apr, 1994) 2531–2533. https://link.aps.org/doi/10.1103/PhysRevLett.72.2531.
  43. A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Phys. Rev. B 55 (Jan, 1997) 1142–1161. https://link.aps.org/doi/10.1103/PhysRevB.55.1142.
  44. A. Kamenev and M. Mézard, “Wigner-dyson statistics from the replica method,” Journal of Physics A: Mathematical and General 32 no. 24, (Jun, 1999) 4373. https://dx.doi.org/10.1088/0305-4470/32/24/304.
  45. Y. Chen, V. Ivo, and J. Maldacena, “Comments on the double cone wormhole,” arXiv:2310.11617 [hep-th].
  46. A. Belin, J. De Boer, P. Nayak, and J. Sonner, “Charged eigenstate thermalization, Euclidean wormholes and global symmetries in quantum gravity,” SciPost Phys. 12 no. 2, (2022) 059, arXiv:2012.07875 [hep-th].
  47. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91 (May, 2019) 021001. https://link.aps.org/doi/10.1103/RevModPhys.91.021001.
  48. P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, “Many-Body Localization in the Age of Classical Computing,” arXiv:2403.07111 [cond-mat.dis-nn].
  49. C. L. Bertrand and A. M. García-García, “Anomalous thouless energy and critical statistics on the metallic side of the many-body localization transition,” Physical Review B 94 no. 14, (Oct., 2016) 144201, arXiv:1606.08419 [cond-mat.dis-nn]. http://dx.doi.org/10.1103/PhysRevB.94.144201.
  50. A. Bastianello, A. De Luca, and R. Vasseur, “Hydrodynamics of weak integrability breaking,” Journal of Statistical Mechanics: Theory and Experiment 2021 no. 11, (Nov., 2021) 114003, arXiv:2103.11997 [cond-mat.stat-mech]. http://dx.doi.org/10.1088/1742-5468/ac26b2.

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Tweets

Sign up for free to view the 4 tweets with 6 likes about this paper.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up for Free