- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Rapid Communication
  • Access by Xinjiang University

Braid group and topological phase transitions in nonequilibrium stochastic dynamics

Jie Ren* and N. A. Sinitsyn

  • Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

  • *renjie@lanl.gov
PDF

Phys. Rev. E 87, 050101(R) – Published 10 May, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.050101

Abstract

We show that distinct topological phases of the band structure of a non-Hermitian Hamiltonian can be classified with elements of the braid group. As the proof of principle, we consider the non-Hermitian evolution of the statistics of nonequilibrium stochastic currents. We show that topologically nontrivial phases have detectable properties, including the emergence of decaying oscillations of parity and state probabilities, and discontinuities in the steady state statistics of currents.

View figure in article

Article Text

References (39)

  1. C. Kittel, Introduction to Solid State Physics (Wiley, Singapore, 1996).
  2. D. R. Grempel, R. E. Prange, and S. Fishman, Phys. Rev. A 29, 1639 (1984).
  3. D. Y. H. Ho and J. B. Gong, Phys. Rev. Lett. 109, 010601 (2012).
  4. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, UK, 2011).
  5. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010); X.-L. Qi and S.-C. Zhang, ibid. 83, 1057 (2011).
  6. T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010); N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7, 490 (2011).
  7. L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. Lett. 105, 225901 (2010); Phys. Rev. B 87, 144101 (2013).
  8. M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009); Phys. Rev. B 82, 155418 (2010).
  9. S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Nat. Phys. 7, 971 (2011).
  10. Y. C. Hu and T. L. Hughes, Phys. Rev. B 84, 153101 (2011).
  11. K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Phys. Rev. B 84, 205128 (2011).
  12. P. K. Ghosh, J. Phys.: Condens. Matter 24, 145302 (2011).
  13. C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. Imamoglu, P. Zoller, and S. Diehl, arXiv:1302.5135.
  14. S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, UK, 1995).
  15. Y. Nambu, Rev. Mod. Phys. 81, 1015 (2009).
  16. V. Elgart and A. Kamenev, Phys. Rev. E 70, 041106 (2004).
  17. I. V. Gopich and A. Szabo, J. Chem. Phys. 124, 154712 (2006).
  18. N. A. Sinitsyn and I. Nemenman, Europhys. Lett. 77, 58001 (2007); N. A. Sinitsyn, J. Phys. A: Math. Theor. 42, 193001 (2009).
  19. B. P. English, W. Min, A. M. van Oijen, K. T. Lee, G. Luo, H. Sun, B. J. Cherayil, S. C. Kou, and X. S. Xie, Nat. Chem. Biol. 2, 87 (2006).
  20. S. Pilgram, A. N. Jordan, E. V. Sukhorukov, and M. Büttiker, Phys. Rev. Lett. 90, 206801 (2003).
  21. J. Ren, P. Hänggi, and B. Li, Phys. Rev. Lett. 104, 170601 (2010).
  22. V. Y. Chernyak and N. A. Sinitsyn, J. Stat. Mech. (2010) L07001.
  23. It is worth remarking that out of equilibrium is the necessary condition for the BPT because when detailed balance is respected, the eigenspectra of the non-Hermitian Hamiltonian with counting parameters are always real. To have nontrivial braids, the detailed balance condition at equilibrium must be broken.
  24. H. Qian, Annu. Rev. Phys. Chem. 58, 113 (2007).
  25. R. D. Astumian, Annu. Rev. Biophys. 40, 289 (2011).
  26. Y. Utsumi et al., Phys. Rev. B 81, 125331 (2010).
  27. In fact, if we explore all possible parameters, there are three more phase regions with reversed chirality as well as reversed transport direction compared to the braid phases II, III, and IV, respectively.
  28. S. Nakamura et al. Phys. Rev. Lett. 104, 080602 (2010).
  29. C. Flindt, C. Fricke, F. Hohls, T. Novotný, K. Netočný, T. Brandes, and R. J. Haug, Proc. Natl. Acad. Sci. USA 106, 10116 (2009).
  30. F. Pistolesi, Phys. Rev. B 69, 245409 (2004).
  31. D. B. Gutman, Y. Gefen, and A. D. Mirlin, Phys. Rev. B 81, 085436 (2010).
  32. D. A. Ivanov and A. G. Abanov, Europhys. Lett. 92, 37008 (2010).
  33. A. G. Abanov, D. A. Ivanov, and Y. Qian, J. Phys. A: Math. Theor. 44, 485001 (2011).
  34. C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952); T. D. Lee and C. N. Yang, ibid. 87, 410 (1952); B. H. Zimm, J. Chem. Phys. 19, 1019 (1951).
  35. H. Touchette, Phys. Rep. 478, 1 (2009).
  36. J. P. Garrahan and I. Lesanovsky, Phy. Rev. Lett. 104, 160601 (2010).
  37. H. Ge and H. Qian, J. R. Soc. Interface 8, 107 (2011).
  38. B.-B. Wei and R.-B. Liu, Phys. Rev. Lett. 109, 185701 (2012).
  39. C. Flindt and J. P. Garrahan, Phys. Rev. Lett. 110, 050601 (2013).

Outline

Information

Phys. Rev. E 87, 050101(R)– Published 10 May, 2013
  • Received 18 December 2012
Article part of CHORUS

©2013 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation