- 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
  • Letter
  • Open Access

Integrability is generic in homogeneous U(1)-invariant nearest-neighbor qubit circuits

Marko Žnidarič, Urban Duh, and Lenart Zadnik

PDF

Phys. Rev. B 112, L020302 – Published 7 July, 2025

DOI: https://doi.org/10.1103/tqy8-ynpd

Abstract

Integrability is an exceptional property believed to hold only for systems with fine-tuned parameters. Contrary, we explicitly show that in homogeneous nearest-neighbor qubit circuits with a U(1) symmetry, i.e., circuits that repeatedly apply the same magnetization-conserving two-qubit gate, this is not the case. There, integrability is generic: All such brickwall qubit circuits are integrable, even with a randomly selected gate. We identify two phases with different conservation laws, transport properties, and strong zero edge modes. Experimentally important is the fact that varying any one of the parameters in the generic U(1) gate, one will typically cross the critical manifold that separates the two phases. Finally, we report on an unconventional time-reversal symmetry causing the system with open boundary conditions to be in the orthogonal class, while the one with periodic boundary conditions is in the unitary class.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (64)

  1. J.-S. Caux and J. Mossel, Remarks on the notion of quantum integrability, J. Stat. Mech. (2011) P02023.
  2. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).
  3. P. Calabrese, F. H. L. Essler, and G. Mussardo, Introduction to “Quantum integrability in out of equilibrium systems”, J. Stat. Mech. (2016) 064001, Special Issue on Quantum Integrability in Out of Equilibrium Systems.
  4. P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E. Demler, and W. Ketterle, Spin transport in a tunable Heisenberg model realized with ultracold atoms, Nature (London) 588, 403 (2020).
  5. D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, and J. Zeiher, Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science 376, 716 (2022).
  6. A. Morvan et al., Formation of robust bound states of interacting microwave photons, Nature (London) 612, 240 (2022).
  7. M. K. Joshi, F. Kranzl, A. Schuckert, I. Lovas, C. Maier, R. Blatt, M. Knap, and C. F. Roos, Observing emergent hydrodynamics in a long-range quantum magnet, Science 376, 720 (2022).
  8. C. Destri and H. J. De Vega, Light-cone lattice approach to fermionic theories in 2D: The massive Thirring model, Nucl. Phys. B 290, 363 (1987).
  9. L. D. Faddeev and A. Yu. Volkov, Hirota equation as an example of integrable symplectic map, Lett. Math. Phys. 32, 125 (1994).
  10. V. Gritsev and A. Polkovnikov, Integrable Floquet dynamics, SciPost Phys. 2, 021 (2017).
  11. M. Vanicat, L. Zadnik, and T. Prosen, Integrable Trotterization: Local conservation laws and boundary driving, Phys. Rev. Lett. 121, 030606 (2018).
  12. M. Ljubotina, L. Zadnik, and T. Prosen, Ballistic spin transport in a periodically driven integrable quantum system, Phys. Rev. Lett. 122, 150605 (2019).
  13. P. W. Claeys, J. Herzog-Arbeitman, and A. Lamacraft, Correlations and commuting transfer matrices in integrable unitary circuits, SciPost Phys. 12, 007 (2022).
  14. Y. Miao, V. Gritsev, and D. V. Kurlov, The Floquet Baxterization, SciPost Phys. 16, 078 (2024).
  15. A. Hutsalyuk, Y. Jiang, B. Pozsgay, H. Xu, and Y. Zhang, Exact spin correlators of integrable quantum circuits from algebraic geometry, arXiv:2405.16070.
  16. L. Zadnik, M. Ljubotina, Ž. Krajnik, E. Ilievski, and T. Prosen, Quantum many-body spin ratchets, PRX Quantum 5, 030356 (2024).
  17. S. Gopalakrishnan, Operator growth and eigenstate entanglement in an interacting integrable Floquet system, Phys. Rev. B 98, 060302(R) (2018).
  18. Y. Miao and E. Vernier, Integrable quantum circuits from star-triangle relation, Quantum 7, 1160 (2023).
  19. B. Pozsgay, A Yang–Baxter integrable cellular automaton with a four site update rule, J. Phys. A: Math. Theor. 54, 384001 (2021).
  20. T. Gombor and B. Pozsgay, Integrable spin chains and cellular automata with medium-range interaction, Phys. Rev. E 104, 054123 (2021).
  21. V. K. Singh, A. Sinha, P. Padmanabhan, and V. Korepin, Unitary tetrahedron quantum gates, arXiv:2407.10731.
  22. L. Sá, P. Ribeiro, and T. Prosen, Integrable nonunitary open quantum circuits, Phys. Rev. B 103, 115132 (2021).
  23. C. Paletta and T. Prosen, Integrability of open boundary driven quantum circuits, SciPost Phys. 18, 027 (2025).
  24. P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, UK, 1998).
  25. D. Braun, Dissipative Quantum Chaos and Decoherence (Springer, Berlin, 2001).
  26. K. Sogo, M. Uchinami, Y. Akutsu, and M. Wadati, Classification of exactly solvable two-component models, Prog. Theor. Phys. 68, 508 (1982).
  27. N. Beisert, L. Fievet, M. de Leeuw, and F. Loebbert, Integrable deformations of the XXZ spin chain, J. Stat. Mech. (2013) P09028.
  28. R. S. Vieira, Solving and classifying the solutions of the Yang–Baxter equation through a differential approach. Two-state systems, J. High Energy Phys. 10 (2018) 110.
  29. F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 2010).
  30. Note that the level-spacing distribution (LSD) should not be used as a definition: One can have chaotic LSD in integrable systems [33], or have integrable systems without Poisson statistics [34], or Poisson statistics in chaotic circuits with unresolved space-time symmetries [32].
  31. See Supplemental Material at http://link.aps.org/supplemental/10.1103/tqy8-ynpd for explicit construction of the antiunitary time-reversal symmetry in circuits with open boundary conditions, explicit expressions for the first two local conserved charges, and for additional discussion of the Ruelle-Pollicott spectrum, which includes Ref. [64].
  32. U. Duh and M. Žnidarič, Classification of same-gate quantum circuits and their space-time symmetries with application to the level-spacing distribution, Phys. Rev. Res. 6, 023068 (2024).
  33. L. Benet, F. Leyvraz, and T. H. Seligman, Wigner-Dyson statistics for a class of integrable models, Phys. Rev. E 68, 045201(R) (2003).
  34. M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. R. Soc. Lond. Ser. A 356, 375 (1977).
  35. A. Hurwitz, Ueber die Erzeugung der Invarianten durch Integration, Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl. 1897, 71 (1897).
  36. All matrices are written in a two-qubit basis ordered as {|0102〉,|1102〉,|0112〉,|1112〉}.
  37. 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).
  38. 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).
  39. 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).
  40. Let h(b) span all U(1) invariant Hermitian operators acting on two qubits as parameters b are varied. Assuming it is integrable, we have, e.g., h(b)=∂uŘ(u,vb)|u=0 for some unitary solution Ř(u,v) of Eq. (3), depending on parameters u and v. The challenge now lies in finding the functions τu,v and bu,v, such that Ř(u,v)=exp[iτu,vh(bu,v)], and this matrix spans all MC unitaries when u and v are varied. Put differently, one needs to find an integrable Trotterization [11] of the MC nearest-neighbor Hamiltonian, and then prove that such a Trotterization spans all homogeneous brickwork circuits.
  41. Note that in such Hamiltonians there are only three nontrivial parameters: The staggered field B cancels out due to invariance under one-site translations.
  42. E. Vernier, B. Bertini, G. Giudici, and L. Piroli, Integrable digital quantum simulation: Generalized Gibbs ensembles and Trotter transitions, Phys. Rev. Lett. 130, 260401 (2023).
  43. P. Fendley, Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain, J. Phys. A: Math. Theor. 49, 30LT01 (2016).
  44. E. Vernier, H.-C. Yeh, L. Piroli, and A. Mitra, Strong zero modes in integrable quantum circuits, Phys. Rev. Lett. 133, 050606 (2024).
  45. I. G. Gochev, Spin complexes in a bounded chain, JETP Lett. 26, 127 (1977).
  46. D. Ruelle, Resonances in chaotic dynamical systems, Phys. Rev. Lett. 56, 405 (1986).
  47. M. Pollicott, On the rate of mixing in axiom A flows, Invent. Math. 81, 413 (1985).
  48. T. Prosen, Ruelle resonances in quantum many-body systems, J. Phys. A: Math. Gen. 35, L737 (2002).
  49. M. Žnidarič, Momentum-dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems, Phys. Rev. E 110, 054204 (2024).
  50. T. Mori, Liouvillian-gap analysis of open quantum many-body systems in the weak dissipation limit, Phys. Rev. B 109, 064311 (2024).
  51. J. A. Jacoby, D. A. Huse, and S. Gopalakrishnan, Spectral gaps of local quantum channels in the weak-dissipation limit, Phys. Rev. B 111, 104303 (2025).
  52. C. Zhang, L. Nie, and C. von Keyserlingk, Thermalization rates and quantum Ruelle-Pollicott resonances: insights from operator hydrodynamics, arXiv:2409.17251.
  53. U. Duh and M. Žnidarič, Ruelle-Pollicott resonances of diffusive U(1)-invariant qubit circuits, arXiv:2506.24097.
  54. Y.-L. Zhang and V. Khemani, Asymmetric butterfly velocities in 2-local Hamiltonians, SciPost Phys. 9, 024 (2020).
  55. P. Richelli, K. Schoutens, and A. Zorzato, Brick wall quantum circuits with global fermionic symmetry, SciPost Phys. 17, 087 (2024).
  56. M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56, 889 (1986).
  57. M. Ljubotina, M. Žnidarič, and T. Prosen, Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet, Phys. Rev. Lett. 122, 210602 (2019).
  58. E. Ilievski, J. De Nardis, S. Gopalakrishnan, R. Vasseur, and B. Ware, Superuniversality of superdiffusion, Phys. Rev. X 11, 031023 (2021).
  59. E. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21, 2375 (1988).
  60. N. Keenan, N. F. Robertson, T. Murphy, S. Zhuk, and J. Goold, Evidence of Kardar-Parisi-Zhang scaling on a digital quantum simulator, npj Quantum Inf. 9, 72 (2023).
  61. K. Maruyoshi, T. Okuda, J. W. Pedersen, R. Suzuki, M. Yamazaki, and Y. Yoshida, Conserved charges in the quantum simulation of integrable spin chains, J. Phys. A: Math. Theor. 56, 165301 (2023).
  62. E. Rosenberg et al., Dynamics of magnetization at infinite temperature in a Heisenberg spin chain, Science 384, 48 (2024).
  63. C. Jonay, J. F. Rodriguez-Nieva, and V. Khemani, Slow thermalization and subdiffusion in U(1) conserving Floquet random circuits, Phys. Rev. B 109, 024311 (2024).
  64. J. Bensa and M. Žnidarič, Fastest local entanglement scrambler, multistage thermalization, and a non-Hermitian phantom, Phys. Rev. X 11, 031019 (2021).

Outline

Information

Phys. Rev. B 112, L020302– Published 7 July, 2025
  • Received 14 October 2024
  • Revised 19 June 2025
  • Accepted 23 June 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