Document Type : Original Article

Author

Department of Physics, School of Science, Shiraz University, Shiraz, Iran

Abstract

It has been known that by encoding the  Boltzmann weights of a classical Ising model in the amplitudes of the wave function of the ground state of the toric code model, the classical phase transition in Ising model is mapped to a topological phase transition in a perturbed toric code model. Since such topological phase transitions cannot be characterized by any local order parameter, it will be an important challenge to find an order parameter which describes the above topological phase transition. In this paper, using a simple technic based on mapping between classical Ising model and the ground state of the toric code model, we find a non-local order parameter which well reveals the topological nature of the above phase transition. We show that such an order parameter is, in fact, a kind of string order parameter which has been recently introduced for some topological phase transitions.

Keywords

  1. N Goldenfeld, "Lectures on phase transitions and critical phenomena", Perseus Books Publishing, L.L.C., Massachusetts (1992).

  2. L D Landau, Ukr. J. Phys. 11 (1937) 19.

  3. X G Wen, Phys. Rev. B 40  (1989) 7387.

  4. X G.Wen and Q Niu, Phys. Rev. B 41, 13 (1990) 9377.

  5. X G Wen, " Int. J. Mod. Phys. B 5, 1641 (1991).

  6. X. G. Wen, "Topological orders in rigid states, Int. J. Mod. Phys. B 4, 02 (1990) 239.

  7. X G Wen, Adv. Phys. 44  (1995) 405.

  8. N Read and S Sachdev, Phys. Rev. Lett. 66, 13 (1991) 1773.

  9. K Klitzing, G Dorda, and M Pepper, Phys. Rev. Lett. 45, 6 (1980) 494.


10. G Moore and R Nicholas, Nuclear Physics B 360, 2-3 (1991) 362.


11. R Moessner and S L Sondhi, Phys. Rev. Lett. 86 (2001) 1881.


12. J Abouie, M Mogharrar Jahromi, and S Abedinpour, IJPR 17, 5 (2018) 681.


13. H Jiang, Z Wang, and L Balents, Nat. Phys. 8, 12 (2012) 902.


14. M Levin and X G Wen, Phys. Rev. Lett. 96, 11 (2006) 110405


15. A Kitaev and J Preskill, Phys. Rev. Lett. 96, 11 (2006) 110404.


16. S Bravyi, M B Hastings, and S. Michalakis, J. Math. Phys. 51, 9 (2010) 093512.


17. K Duivenvoorden and T Quella. Phys. Rev. B 86, 23 (2012) 235142.


18. R Alicki, M Horodecki, P Horodecki, and R Horodecki, Open Syst. Inf. Dyn. 17  (2008) 1.


19. H Bombin, R W Chhajlany, M Horodecki, and M A Martin-Delgado, New J. Phys. 15, 5  (2013) 055023.


20. J B Brown, D Loss, J K Pachos, C N Self, and J R Wootton, Reviews of Modern Physics 88, 4 (2016) 045005.


21. E Dennis, A Kitaev, A Landahl, and J Preskill, J. Math. Phys. 43, 9 (2002) 4452.


22. H Bombin and M A Martin-Delgado, Phys. Rev. Lett. 97 (2006) 180501.


23. B Yoshida, Ann. Phys. 326, 10 (2011) 2566.


24. H Bombin, New J. Phys. 17, 8 (2015) 083002.


25. S Dusuel, M. Kamfor, R. Orus, K. P. Schmidt, and J.Vidal, Phys. Rev. Lett. 106 (2011) 107203.


26. A Y Kitaev, Ann. Phys. (N.Y.) 2 (2003) 303.


27. S Trebst, P Werner, M Troyer, K Shtengel, and C Nayak, Phys. Rev. Lett. 98 (2007) 070602.


28. J Vidal, S Dusuel, and K P Schmidt, Phys. Rev. B 79 (2009) 033109.


29. J Vidal, R Thomale, K P Schmidt, and S Dusuel, Phys. Rev. B 80 (2009) 081104.


30. I S Tupitsyn, A Kitaev, N V Prokofev, and P C E Stamp, Phys. Rev. B 82 (2010) 085114.


31. F Wu, Y Deng, and N Prokof’ev, Phys. Rev. B 85 (2012) 195104.


32. M H Zarei, Phys. Rev. A 91, 2 (2015) 022319.


33. M H Zarei, Phys. Rev. B 96 (2017) 165146.


34. M H Zarei and A Montakhab, Phys. Rev. A 99 (2019) 052312.


35. C Castelnovo and C Chamon, Phys. Rev. B 77 (2008) 054433.


36. M H Zarei and A Montakhab, arXiv:1907.06216


37. M H Zarei, Phys. Rev. B 100 (2019) 125159.


38. A Hamma, W Zhang, S Haas, D A Lidar, Phys. Rev. B 77, 15 (2008) 155111.


39. S. Gu, Int. J. Mod. Phys. B 24, 23 (2010) 4371.


40. S Yang, S J. Gu, C P Sun, and H Q Lin, Phys. Rev. A 78, 1 (2008) 012304.


41. U Marzolino and P Tomaz, Phys. Rev. B 96, 10 (2017) 104402.


42. A Carollo, S Bernardo, D Valenti, Scienti_c reports 8, 1 (2018) 9852.


43. A Montakhab and A Asadian, Phys. Rev. A 82 (2010) 062313.

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