- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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

Stochastic strong zero modes and their dynamical manifestations

Katja Klobas1,*, Paul Fendley1,2, and Juan P. Garrahan3,4,2

  • 1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom
  • 2All Souls College, University of Oxford, Oxford, OX1 4AL, United Kingdom
  • 3School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
  • 4Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom

  • *katja.klobas@nottingham.ac.uk
PDF

Phys. Rev. E 107, L042104 – Published 28 April, 2023

DOI: https://doi.org/10.1103/PhysRevE.107.L042104

Abstract

Strong zero modes (SZMs) are conserved operators localized at the edges of certain quantum spin chains, which give rise to long coherence times of edge spins. Here we define and analyze analogous operators in one-dimensional classical stochastic systems. For concreteness, we focus on chains with single occupancy and nearest-neighbor transitions, in particular particle hopping and pair creation and annihilation. For integrable choices of parameters we find the exact form of the SZM operators. Being in general nondiagonal in the classical basis, the dynamical consequences of stochastic SZMs are very different from those of their quantum counterparts. We show that the presence of a stochastic SZM is manifested through a class of exact relations between time-correlation functions, absent in the same system with periodic boundaries.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (36)

  1. J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nat. Phys. 11, 124 (2015).
  2. L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  3. R. Nandkishore and D. A. Huse, Many body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  4. D. A. Abanin and Z. Papić, Recent progress in many-body localization, Ann. Phys. 529, 1700169 (2017).
  5. V. Khemani, R. Moessner, and S. L. Sondhi, A brief history of time crystals, arXiv:1910.10745 (2019)
  6. S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: A review of exact results, Rep. Prog. Phys. 85, 086501 (2022).
  7. A. Y. Kitaev, Unpaired majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001).
  8. P. Fendley, Parafermionic edge zero modes in Zn-invariant spin chains, J. Stat. Mech.: Theory Exp. (2012) P11020.
  9. P. Fendley, Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain, J. Phys. A: Math. Theor. 49, 30LT01 (2016).
  10. J. Alicea and P. Fendley, Topological phases with parafermions: Theory and blueprints, Annu. Rev. Condens. Matter Phys. 7, 119 (2016).
  11. J. Kemp, N. Y. Yao, C. R. Laumann, and P. Fendley, Long coherence times for edge spins, J. Stat. Mech.: Theory Exp. (2017) 063105.
  12. D. V. Else, P. Fendley, J. Kemp, and C. Nayak, Prethermal Strong Zero Modes and Topological Qubits, Phys. Rev. X 7, 041062 (2017).
  13. L. M. Vasiloiu, F. Carollo, and J. P. Garrahan, Enhancing correlation times for edge spins through dissipation, Phys. Rev. B 98, 094308 (2018).
  14. L. M. Vasiloiu, F. Carollo, M. Marcuzzi, and J. P. Garrahan, Strong zero modes in a class of generalized Ising spin ladders with plaquette interactions, Phys. Rev. B 100, 024309 (2019).
  15. L. M. Vasiloiu, A. Tiwari, and J. H. Bardarson, Dephasing-enhanced majorana zero modes in two-dimensional and three-dimensional higher-order topological superconductors, Phys. Rev. B 106, L060307 (2022).
  16. D. J. Yates, F. H. L. Essler, and A. Mitra, Almost strong (0,π) edge modes in clean interacting one-dimensional Floquet systems, Phys. Rev. B 99, 205419 (2019).
  17. D. J. Yates, A. G. Abanov, and A. Mitra, Dynamics of almost strong edge modes in spin chains away from integrability, Phys. Rev. B 102, 195419 (2020).
  18. D. J. Yates, A. G. Abanov, and A. Mitra, Long-lived period-doubled edge modes of interacting and disorder-free Floquet spin chains, Commun. Phys. 5, 43 (2022).
  19. S. Sandow, Partially asymmetric exclusion process with open boundaries, Phys. Rev. E 50, 2660 (1994).
  20. F. H. Essler and V. Rittenberg, Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries, J. Phys. A: Math. Gen. 29, 3375 (1996).
  21. O. Golinelli and K. Mallick, Derivation of a matrix product representation for the asymmetric exclusion process from the algebraic Bethe ansatz, J. Phys. A: Math. Gen. 39, 10647 (2006).
  22. J. de Gier and F. H. L. Essler, Exact spectral gaps of the asymmetric exclusion process with open boundaries, J. Stat. Mech.: Theory Exp. (2006) P12011.
  23. All transitions in the models we consider involve two neighboring sites. This means that in the case of open boudaries, the leftmost set of transitions (hopping/paircondensation/pairannihilation) are between sites 1 and 2, while the rightmost between sites L−1 and L. The local generator (that is, the one for transitions between sites i and i+1) is stochastic in itself. This means that we could even have chosen site dependent rates (see Fig. 1) while keeping the overall generator stochastic. See Eqs. (1) and (2).
  24. R. Stinchcombe, Stochastic non-equilibrium systems, Adv. Phys. 50, 431 (2001).
  25. Note that these two are not necessarily the only stochastic models that exhibit an almost conserved edge mode [11, 12]. However, these are the only two regimes for which the explicit closed-form expression of the edge modes is known, and we rely on that.
  26. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevE.107.L042104 for the additional details on (i) the diagonal form of the free-fermionic generator, (ii) eigenvectors of edge modes, (iii) stationary states, and (iv) dynamical correlations in the stationary state of the interacting generator.
  27. M. D. Grynberg, T. J. Newman, and R. B. Stinchcombe, Exact solutions for stochastic adsorption-desorption models and catalytic surface processes, Phys. Rev. E 50, 957 (1994).
  28. N. Crampe, E. Ragoucy, and M. Vanicat, Integrable approach to simple exclusion processes with boundaries. Review and progress, J. Stat. Mech.: Theory Exp. (2014) P11032.
  29. We remark that Ψ2=1 implies that the spectrum of Ψ consists only of (highly degenerate) eigenvalues 1 and −1.
  30. J. Tailleur, J. Kurchan, and V. Lecomte, Mapping out-of-equilibrium into equilibrium in one-dimensional transport models, J. Phys. A: Math. Theor. 41, 505001 (2008).
  31. F. Spitzer, Interaction of Markov processes, Adv. Math. 5, 246 (1970).
  32. B. Derrida, An exactly soluble non-equilibrium system: The asymmetric simple exclusion process, Phys. Rep. 301, 65 (1998).
  33. R. A. Blythe and M. R. Evans, Nonequilibrium steady states of matrix-product form: A solver's guide, J. Phys. A: Math. Theor. 40, R333 (2007).
  34. M. Grabowski and P. Mathieu, Structure of the conservation laws in quantum integrable spin chains with short range interactions, Ann. Phys. 243, 299 (1995).
  35. L. Faddeev, How algebraic Bethe ansatz works for integrable model, arXiv:hep-th/9605187 (1996).
  36. E. Ilievski, M. Medenjak, T. Prosen, and L. Zadnik, Quasilocal charges in integrable lattice systems, J. Stat. Mech.: Theory Exp. (2016) 064008.

Outline

Information

Phys. Rev. E 107, L042104– Published 28 April, 2023
  • Received 11 July 2022
  • Accepted 6 April 2023
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