Abstract
The second law of thermodynamics points to the existence of an ‘arrow of time’, along which entropy only increases. This arises despite the time-reversal symmetry (TRS) of the microscopic laws of nature. Within quantum theory, TRS underpins many interesting phenomena, most notably topological insulators1,2,3,4 and the Haldane phase of quantum magnets5,6. Here, we demonstrate that such TRS-protected effects are fundamentally unstable against coupling to an environment. Irrespective of the microscopic symmetries, interactions between a quantum system and its surroundings facilitate processes that would be forbidden by TRS in an isolated system. This leads not only to entanglement entropy production and the emergence of macroscopic irreversibility7,8,9, but also to the demise of TRS-protected phenomena, including those associated with certain symmetry-protected topological phases. Our results highlight the enigmatic nature of TRS in quantum mechanics and elucidate potential challenges in utilizing topological systems for quantum technologies.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Buy this article
- Purchase on SpringerLink
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
Data sharing is not applicable to this paper as no datasets were generated or analysed during the current study.
References
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008).
Haldane, F. D. M. Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983).
Pollmann, F., Turner, A. M., Berg, E. & Oshikawa, M. Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B 81, 064439 (2010).
von Neumann, J. Proof of the ergodic theorem and the H-theorem in quantum mechanics. Z. Phys. 57, 30–70 (1929).
Goldstein, S., Lebowitz, J. L., Tumulka, R. & Zanghì, N. Long-time behavior of macroscopic quantum systems. Eur. Phys. J. H 35, 173–200 (2010).
Srednicki, M. The approach to thermal equilibrium in quantized chaotic systems. J. Phys. A 32, 1163–1175 (1999).
Chen, X., Gu, Z.-C. & Wen, X.-G. Local unitary transformation, long-range quantum entanglement, wave function renormalization and topological order. Phys. Rev. B 82, 155138 (2010).
Senthil, T. Symmetry-protected topological phases of quantum matter. Annu. Rev. Condens. Matter Phys. 6, 299–324 (2015).
Kitaev, A. Y. Unpaired majorana fermions in quantum wires. Phys. Uspekhi 44, 131–136 (2001).
Alicea, J. New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012).
Das Sarma, S., Freedman, M. & Nayak, C. Majorana zero modes and topological quantum computation. npj Quantum Inf. 1, 15001 (2015).
Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Preprint at https://arxiv.org/pdf/1912.10048.pdf (2019).
Diehl, S., Rico, E., Baranov, M. A. & Zoller, P. Topology by dissipation in atomic quantum wires. Nat. Phys. 7, 971–997 (2011).
Bardyn, C.-E. et al. Topology by dissipation. N. J. Phys. 15, 085001 (2013).
Lieu, S., McGinley, M. & Cooper, N. R. Tenfold way for quadratic Lindbladians. Phys. Rev. Lett. 124, 040401 (2020).
Breuer, H. P. & Petruccione, F. The Theory of Open Quantum Systems (Oxford Univ. Press, 2002).
Buča, B. & Prosen, T. A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains. N. J. Phys. 14, 073007 (2012).
Rainis, D. & Loss, D. Majorana qubit decoherence by quasiparticle poisoning. Phys. Rev. B 85, 174533 (2012).
Yang, J. & Liu, Z.-X. Irreducible projective representations and their physical applications. J. Phys. A 51, 025207 (2017).
Pollmann, F. & Turner, A. M. Detection of symmetry-protected topological phases in one dimension. Phys. Rev. B 86, 125441 (2012).
McGinley, M. & Cooper, N. R. Classification of topological insulators and superconductors out of equilibrium. Phys. Rev. B 99, 075148 (2019).
McGinley, M. & Cooper, N. R. Interacting symmetry-protected topological phases out of equilibrium. Phys. Rev. Res. 1, 033204 (2019).
Goldstein, G. & Chamon, C. Decay rates for topological memories encoded with Majorana fermions. Phys. Rev. B 84, 205109 (2011).
Maciejko, J. et al. Kondo effect in the helical edge liquid of the quantum spin Hall state. Phys. Rev. Lett. 102, 256803 (2009).
Roth, A. et al. Nonlocal transport in the quantum spin Hall state. Science 325, 294–297 (2009).
Schmidt, T. L., Rachel, S., von Oppen, F. & Glazman, L. I. Inelastic electron backscattering in a generic helical edge channel. Phys. Rev. Lett. 108, 156402 (2012).
Budich, J. C., Dolcini, F., Recher, P. & Trauzettel, B. Phonon-induced backscattering in helical edge states. Phys. Rev. Lett. 108, 086602 (2012).
Väyrynen, J. I., Pikulin, D. I. & Alicea, J. Noise-induced backscattering in a quantum spin Hall edge. Phys. Rev. Lett. 121, 106601 (2018).
Van Kampen, N. G. A cumulant expansion for stochastic linear differential equations. I. Physica 74, 215–238 (1974).
Caldeira, A. O. & Leggett, A. J. Influence of dissipation on quantum tunneling in macroscopic systems. Phys. Rev. Lett. 46, 211–214 (1981).
Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Oreg, Y. & von Oppen, F. Majorana zero modes in networks of Cooper-pair boxes: topologically ordered states and topological quantum computation. Annu. Rev. Condens. Matter Phys. 11, 397–420 (2020).
Wong, C. L. M. & Law, K. T. Majorana Kramers doublets in \({d}_{{x}^{2}-{y}^{2}}\)-wave superconductors with Rashba spin–orbit coupling. Phys. Rev. B 86, 184516 (2012).
Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).
De Roeck, W. & Huveneers, F. Stability and instability towards delocalization in many-body localization systems. Phys. Rev. B 95, 155129 (2017).
Acknowledgements
This work was supported by an EPSRC studentship and grants EP/P034616/1 and EP/P009565/1, and by an Investigator Award of the Simons Foundation.
Author information
Authors and Affiliations
Contributions
Both authors contributed to the formulation of the study, interpretation of the results and writing of the manuscript. M.M. developed and performed the calculations.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Physics thanks Zhong Wang and other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
McGinley, M., Cooper, N.R. Fragility of time-reversal symmetry protected topological phases. Nat. Phys. 16, 1181–1183 (2020). https://doi.org/10.1038/s41567-020-0956-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41567-020-0956-z
This article is cited by
-
Entropy Decay for Davies Semigroups of a One Dimensional Quantum Lattice
Communications in Mathematical Physics (2024)
-
Persistence of topological phases in non-Hermitian quantum walks
Scientific Reports (2021)