- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Letter
  • Open Access

Translation symmetry restoration under random unitary dynamics

Katja Klobas1,*, Colin Rylands2, and Bruno Bertini1

  • *Contact author: k.klobas@bham.ac.uk
PDF

Phys. Rev. B 111, L140304 – Published 8 April, 2025

DOI: https://doi.org/10.1103/PhysRevB.111.L140304

Abstract

The finite parts of a large, locally interacting many-body system prepared out of equilibrium eventually equilibrate. Characterizing the underlying mechanisms of this process and its timescales, however, is particularly hard as it requires one to decouple universal features from observable-specific ones. Recently, additional insight was gained by studying how certain symmetries of the dynamics that are broken by the initial state are restored at the level of the reduced state of a given subsystem. This provides a high-level, observable-independent probe. Until now, this idea has been applied to the restoration of internal symmetries, e.g., U(1) symmetries related to charge conservation. Here we show that the same logic can be applied to the restoration of space-time symmetries, and hence can be used to characterize the relaxation of fully generic systems. We illustrate this idea by considering the paradigmatic example of “generic” many-body dynamics, i.e., a local random unitary circuit, where our method leads to exact results. We show that the restoration of translation symmetry in these systems only happens on timescales that are proportional to the subsystem's volume. In fact, for large enough subsystems, the time of symmetry restoration becomes initial-state independent (as long as the latter breaks the symmetry at time zero) and coincides with the thermalization time. For intermediate subsystems, however, one can observe the “quantum Mpemba effect,” where the state of the system restores a symmetry faster if it is initially more asymmetric. We provide an exact characterization of this effect in a nonintegrable system.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (66)

  1. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1996), Vol. 2.
  2. A. Altland and B. Simons, Condensed Matter Field Theory, Cambridge Books Online (Cambridge University Press, Cambridge, 2010).
  3. J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs, Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules, Phys. Rev. A 77, 032114 (2008).
  4. G. Gour, I. Marvian, and R. W. Spekkens, Measuring the quality of a quantum reference frame: The relative entropy of frameness, Phys. Rev. A 80, 012307 (2009).
  5. I. Marvian and R. W. Spekkens, Extending Noether's theorem by quantifying the asymmetry of quantum states, Nat. Commun. 5, 3821 (2014).
  6. N. Laflorencie and S. Rachel, Spin-resolved entanglement spectroscopy of critical spin chains and Luttinger liquids, J. Stat. Mech. (2014) P11013.
  7. M. Goldstein and E. Sela, Symmetry-resolved entanglement in many-body systems, Phys. Rev. Lett. 120, 200602 (2018).
  8. J. C. Xavier, F. C. Alcaraz, and G. Sierra, Equipartition of the entanglement entropy, Phys. Rev. B 98, 041106(R) (2018).
  9. R. Bonsignori, P. Ruggiero, and P. Calabrese, Symmetry resolved entanglement in free fermionic systems, J. Phys. A: Math. Theor. 52, 475302 (2019).
  10. J. D. Nardis, B. Doyon, M. Medenjak, and M. Panfil, Correlation functions and transport coefficients in generalized hydrodynamics, J. Stat. Mech. (2022) 014002.
  11. B. Doyon, Hydrodynamic projections and the emergence of linearized Euler equations in one-dimensional isolated systems, Commun. Math. Phys. 391, 293 (2022).
  12. B. Doyon, G. Perfetto, T. Sasamoto, and T. Yoshimura, Ballistic macroscopic fluctuation theory, SciPost Phys. 15, 136 (2023).
  13. S. Gopalakrishnan, A. Morningstar, R. Vasseur, and V. Khemani, Distinct universality classes of diffusive transport from full counting statistics, Phys. Rev. B 109, 024417 (2024).
  14. B. Bertini, P. Calabrese, M. Collura, K. Klobas, and C. Rylands, Nonequilibrium full counting statistics and symmetry-resolved entanglement from space-time duality, Phys. Rev. Lett. 131, 140401 (2023).
  15. S. Murciano, G. D. Giulio, and P. Calabrese, Entanglement and symmetry resolution in two dimensional free quantum field theories, J. High Energy Phys. 08 (2020) 073.
  16. F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nat. Commun. 14, 2036 (2023).
  17. B. Bertini, K. Klobas, M. Collura, P. Calabrese, and C. Rylands, Dynamics of charge fluctuations from asymmetric initial states, Phys. Rev. B 109, 184312 (2024).
  18. V. Khemani, A. Vishwanath, and D. A. Huse, Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws, Phys. Rev. X 8, 031057 (2018).
  19. T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev. X 8, 031058 (2018).
  20. A. J. Friedman, A. Chan, A. De Luca, and J. T. Chalker, Spectral statistics and many-body quantum chaos with conserved charge, Phys. Rev. Lett. 123, 210603 (2019).
  21. U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in U(1) symmetric monitored quantum circuits, Phys. Rev. X 12, 041002 (2022).
  22. F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum circuits, Phys. Rev. Lett. 129, 120604 (2022).
  23. E. B. Mpemba and D. G. Osborne, Cool? Phys. Educ. 4, 172 (1969).
  24. F. Ares, S. Murciano, E. Vernier, and P. Calabrese, Lack of symmetry restoration after a quantum quench: An entanglement asymmetry study, SciPost Phys. 15, 089 (2023).
  25. S. Murciano, F. Ares, I. Klich, and P. Calabrese, Entanglement asymmetry and quantum Mpemba effect in the XY spin chain, J. Stat. Mech.: Theory Expt. (2024) 013103.
  26. B. J. Khor, D. Kürkçüoglu, T. Hobbs, G. Perdue, and I. Klich, Confinement and kink entanglement asymmetry on a quantum Ising chain, Quantum 8, 1462 (2024).
  27. F. Ferro, F. Ares, and P. Calabrese, Non-equilibrium entanglement asymmetry for discrete groups: The example of the XY spin chain, J. Stat. Mech.: Theory Expt. (2024) 023101.
  28. L. Capizzi and M. Mazzoni, Entanglement asymmetry in the ordered phase of many-body systems: The Ising field theory, J. High Energy Phys. 12 (2023) 144.
  29. L. Capizzi and V. Vitale, A universal formula for the entanglement asymmetry of matrix product states, J. Phys. A: Math. Theor. 57, 45LT01 (2024).
  30. C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murciano, and B. Bertini, Microscopic origin of the quantum Mpemba effect in integrable systems, Phys. Rev. Lett. 133, 010401 (2024).
  31. M. Chen and H.-H. Chen, Rényi entanglement asymmetry in (1+1)-dimensional conformal field theories, Phys. Rev. D 109, 065009 (2024).
  32. M. Fossati, F. Ares, J. Dubail, and P. Calabrese, Entanglement asymmetry in CFT and its relation to nontopological defects, J. High Energy Phys. 05 (2024) 059.
  33. F. Caceffo, S. Murciano, and V. Alba, Entangled multiplets, asymmetry, and quantum Mpemba effect in dissipative systems, J. Stat. Mech.: Theory Expt. (2024) 063103.
  34. F. Ares, S. Murciano, L. Piroli, and P. Calabrese, Entanglement asymmetry study of black hole radiation, Phys. Rev. D 110, L061901 (2024).
  35. S. Liu, H.-K. Zhang, S. Yin, and S.-X. Zhang, Symmetry restoration and quantum Mpemba effect in symmetric random circuits, Phys. Rev. Lett. 133, 140405 (2024).
  36. S. Yamashika, F. Ares, and P. Calabrese, Entanglement asymmetry and quantum Mpemba effect in two-dimensional free-fermion systems, Phys. Rev. B 110, 085126 (2024).
  37. F. Ares, V. Vitale, and S. Murciano, The quantum Mpemba effect in free-fermionic mixed states, arXiv:2405.08913.
  38. K. Chalas, F. Ares, C. Rylands, and P. Calabrese, Multiple crossings during dynamical symmetry restoration and implications for the quantum Mpemba effect, J. Stat. Mech.: Theory Expt. (2024) 103101.
  39. X. Turkeshi, P. Calabrese, and A. D. Luca, Quantum Mpemba effect in random circuits, arXiv:2405.14514.
  40. L. K. Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. Kranzl, R. Blatt, P. Zoller, B. Vermersch, P. Calabrese, C. F. Roos, and M. K. Joshi, Observing the quantum Mpemba effect in quantum simulations, Phys. Rev. Lett. 133, 010402 (2024).
  41. M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023).
  42. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  43. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  44. C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018).
  45. T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev. B 99, 174205 (2019).
  46. T. Zhou and A. Nahum, Entanglement membrane in chaotic many-body systems, Phys. Rev. X 10, 031066 (2020).
  47. B. Bertini and L. Piroli, Scrambling in random unitary circuits: Exact results, Phys. Rev. B 102, 064305 (2020).
  48. This follows from our results in Ref. [53].
  49. Symmetries associated to anti-Unitary representations can also be straightforwardly accommodated.
  50. In writing this expression, we assumed that G is a finite group. Our treatment, with obvious modifications, also applies to the case of G being a Lie group.
  51. P. Ruggiero and P. Calabrese, Relative entanglement entropies in 1 + 1-dimensional conformal field theories, J. High Energy Phys. 02 (2017) 039.
  52. Necessarily, we require that L/ν∈N.
  53. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.111.L140304 for details on the solution to the recurrence relation for the quantity Dy, and the discussion of the ν>2 case.
  54. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  55. B. Collins, Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Intl. Math. Res. Notices 2003, 953 (2003).
  56. Note that we chose the first argument equal to −t−1 so that the subsystem A is positioned between sites 1 and 2l; cf. Eq. (16).
  57. The last diagram of (19) is equal to tr[mx+y†mx+ymx†mx], which, in general, differs from γy(x). We consider states for which the two are the same.
  58. A. W. Harrow and R. A. Low, Random quantum circuits are approximate 2-designs, Commun. Math. Phys. 291, 257 (2009).
  59. I. T. Diniz and D. Jonathan, Comment on “Random quantum circuits are approximate 2-designs” by A. W. Harrow and R. A. Low [Commun. Math. Phys. 291, 257 (2009)], Commun. Math. Phys. 304, 281 (2011).
  60. F. G. Brandao and M. Horodecki, Exponential quantum speed-ups are generic, Quantum Inf. Comput. 13, 901 (2013).
  61. M. Žnidarič, Exact convergence times for generation of random bipartite entanglement, Phys. Rev. A 78, 032324 (2008).
  62. W. G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104, 250501 (2010).
  63. F. G. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346, 397 (2016).
  64. F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Efficient quantum pseudorandomness, Phys. Rev. Lett. 116, 170502 (2016).
  65. N. Hunter-Jones, Unitary designs from statistical mechanics in random quantum circuits, arXiv:1905.12053.
  66. J. Haferkamp, Random quantum circuits are approximate unitary t-designs in depth O(nt5+o(1)), Quantum 6, 795 (2022).

Outline

Information

Phys. Rev. B 111, L140304– Published 8 April, 2025
  • Received 2 July 2024
  • Revised 20 January 2025
  • Accepted 5 March 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation