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Entanglement entropy for pure gauge theories in 1+1 dimensions using the lattice regularization

Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakechou, Sakyo-ku, Kyoto 606-8502, Japan

Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan

E-mail Address: saoki@yukawa.kyoto-u.ac.jp

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Etsuko Itou

KEK Theory Center, Institute of Particle and Nuclear Physics, High Energy Research Organization (KEK), Ibaraki 305-0801, Japan

E-mail Address: eitou@post.kek.jp

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Keitaro Nagata

KEK Theory Center, Institute of Particle and Nuclear Physics, High Energy Research Organization (KEK), Ibaraki 305-0801, Japan

E-mail Address: knagata@post.kek.jp

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https://doi.org/10.1142/S0217751X1650192XCited by:4 (Source: Crossref)

Abstract

We study the entanglement entropy (EE) for pure gauge theories in 1+1 dimensions with the lattice regularization. Using the definition of the EE for lattice gauge theories proposed in a previous paper,¹ we calculate the EE for arbitrary pure as well as mixed states in terms of eigenstates of the transfer matrix in (1+1)-dimensional lattice gauge theory. We find that the EE of an arbitrary pure state does not depend on the lattice spacing, thus giving the EE in the continuum limit, and show that the EE for an arbitrary pure state is independent of the real (Minkowski) time evolution. We also explicitly demonstrate the dependence of EE on the gauge fixing at the boundaries between two subspaces, which was pointed out for general cases in the paper. In addition, we calculate the EE at zero as well as finite temperature by the replica method, and show that our result in the continuum limit corresponds to the result obtained before in the continuum theory, with a specific value of the counterterm, which is otherwise arbitrary in the continuum calculation. We confirm the gauge dependence of the EE also for the replica method.

Keywords:

PACS: 03.65.Ud, 11.15.Ha, 11.10.Kk

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References

1. S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba and H. Tasaki, J. High Energy Phys. 1506, 187 (2015), doi:10.1007/JHEP06(2015)187, arXiv:1502.04267 [hep-th]. Web of Science, ADS, Google Scholar
2. P. Calabrese and J. L. Cardy, J. Stat. Mech. 0406, P06002 (2004), doi:10.1088/1742-5468/2004/06/P06002, arXiv:hep-th/0405152. Web of Science, Google Scholar
3. P. Calabrese and J. Cardy, J. Phys. A 42, 504005 (2009), doi:10.1088/1751-8113/42/50/504005, arXiv:0905.4013 [cond-mat.stat-mech]. Google Scholar
4. S. Ryu and T. Takayanagi, J. High Energy Phys. 0608, 045 (2006), arXiv:hep-th/0605073. Web of Science, ADS, Google Scholar
5. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006), arXiv:hep-th/0603001. Web of Science, ADS, Google Scholar
6. V. E. Hubeny, M. Rangamani and T. Takayanagi, J. High Energy Phys. 0707, 062 (2007), arXiv:0705.0016 [hep-th]. Web of Science, ADS, Google Scholar
7. T. Nishioka, S. Ryu and T. Takayanagi, J. Phys. A 42, 504008 (2009). Google Scholar
8. T. Takayanagi, Class. Quantum Grav. 29, 153001 (2012), arXiv:1204.2450 [gr-qc]. Web of Science, ADS, Google Scholar
9. B. Swingle, Phys. Rev. D 86, 065007 (2012), arXiv:0905.1317 [cond-mat.str-el]. Web of Science, ADS, Google Scholar
10. B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:hep-th/1209.3304. Google Scholar
11. M. Van Raamsdonk, Gen. Relativ. Gravit. 42, 2323 (2010) [Int. J. Mod. Phys. D 19, 2429 (2010)], arXiv:1005.3035 [hep-th]. Web of Science, ADS, Google Scholar
12. M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [hep-th]. Google Scholar
13. M. Nozaki, S. Ryu and T. Takayanagi, J. High Energy Phys. 1210, 193 (2012), arXiv:1208.3469 [hep-th]. Web of Science, ADS, Google Scholar
14. A. Mollabashi, M. Nozaki, S. Ryu and T. Takayanagi, J. High Energy Phys. 1403, 098 (2014), arXiv:1311.6095 [hep-th]. Web of Science, ADS, Google Scholar
15. M. Miyaji, S. Ryu, T. Takayanagi and X. Wen, Boundary states as holographic duals of trivial spacetimes, arXiv:hep-th/1412.6226. Google Scholar
16. M. Nozaki, T. Numasawa, A. Prudenziati and T. Takayanagi, Phys. Rev. D 88, 026012 (2013), arXiv:1304.7100 [hep-th]. Web of Science, ADS, Google Scholar
17. J. Bhattacharya and T. Takayanagi, J. High Energy Phys. 1310, 219 (2013), arXiv:1308.3792 [hep-th]. Web of Science, ADS, Google Scholar
18. T. Faulkner, M. Guica, T. Hartman, R. C. Myers and M. Van Raamsdonk, J. High Energy Phys. 1403, 051 (2014), arXiv:1312.7856 [hep-th]. Web of Science, ADS, Google Scholar
19. N. Lashkari, M. B. McDermott and M. Van Raamsdonk, J. High Energy Phys. 1404, 195 (2014), arXiv:1308.3716 [hep-th]. Web of Science, ADS, Google Scholar
20. M. Levin and X. G. Wen, Phys. Rev. Lett. 96, 110405 (2006), arXiv:cond-mat/0510613. Web of Science, ADS, Google Scholar
21. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006), arXiv:hep-th/0510092. Web of Science, ADS, Google Scholar
22. B. Hsu, M. Mulligan, E. Fradkin and E. A. Kim, Phys. Rev. B 79, 115421 (2009), arXiv:0812.0203. Web of Science, ADS, Google Scholar
23. H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008), arXiv:0805.0332 [cond-mat.mes-hall]. Web of Science, ADS, Google Scholar
24. S. T. Flammia, A. Hamma, T. L. Hughes and X.-G. Wen, Phys. Rev. Lett. 103, 261601 (2009), arXiv:0909.3305 [cond-mat.str-el]. Web of Science, ADS, Google Scholar
25. M. B. Hastings, I. Gonzalez, A. B. Kallin and R. G. Melko, Phys. Rev. Lett. 104, 157201 (2010), arXiv:1001.2335 [cond-mat.str-el]. Web of Science, ADS, Google Scholar
26. L. Susskind and J. Uglum, Phys. Rev. D 50, 2700 (1994), arXiv:hep-th/9401070. Web of Science, ADS, Google Scholar
27. D. N. Kabat, Nucl. Phys. B 453, 281 (1995), arXiv:hep-th/9503016. Web of Science, ADS, Google Scholar
28. D. N. Kabat and M. J. Strassler, Phys. Lett. B 329, 46 (1994), arXiv:hep-th/9401125. Web of Science, ADS, Google Scholar
29. N. Shiba, Phys. Rev. D 83, 065002 (2011), arXiv:1011.3760 [hep-th]. Web of Science, ADS, Google Scholar
30. N. Shiba, J. High Energy Phys. 1207, 100 (2012), arXiv:1201.4865 [hep-th]. Web of Science, ADS, Google Scholar
31. T. Nishioka and T. Takayanagi, J. High Energy Phys. 0701, 090 (2007), arXiv:hep-th/0611035. Web of Science, ADS, Google Scholar
32. I. R. Klebanov, D. Kutasov and A. Murugan, Nucl. Phys. B 796, 274 (2008), arXiv:0709.2140 [hep-th]. Web of Science, ADS, Google Scholar
33. A. Lewkowycz, J. High Energy Phys. 1205, 032 (2012), arXiv:1204.0588 [hep-th]. Web of Science, ADS, Google Scholar
34. Y. Nakagawa, A. Nakamura, S. Motoki and V. I. Zakharov, PoS LAT 2009, 188 (2009), arXiv:0911.2596 [hep-lat]. Google Scholar
35. Y. Nakagawa, A. Nakamura, S. Motoki and V. I. Zakharov, PoS LATTICE 2010, 281 (2010), arXiv:1104.1011 [hep-lat]. Google Scholar
36. E. Itou, K. Nagata, Y. Nakagawa, A. Nakamura and V. I. Zakharov, Prog. Theor. Exp. Phys. 2016, 061B01 (2016), doi:10.1093/ptep/ptw050, arXiv:1512.01334 [hep-th]. Google Scholar
37. H. Casini, M. Huerta and J. A. Rosabal, Phys. Rev. D 89, 141 (2008), arXiv:hep-th/1312.1183. Google Scholar
38. D. Radičević, Note on entanglement in Abelian gauge theories, arXiv:hep-th/1404.1391. Google Scholar
39. P. V. Buividovich and M. I. Polikarpov, Phys. Lett. B 670, 141 (2008). Web of Science, ADS, Google Scholar
40. W. Donnelly, Phys. Rev. D 85, 085004 (2012). Web of Science, ADS, Google Scholar
41. W. Donnelly, Class. Quantum Grav. 31 214003 (2014), arXiv:hep-th/1406.7304. ADS, Google Scholar
42. S. Ghosh, R. M. Soni and S. P. Trivedi, J. High Energy Phys. 1509, 069 (2015), doi:10.1007/JHEP09(2015)069, arXiv:1501.02593 [hep-th]. Web of Science, Google Scholar
43. L. Y. Hung and Y. Wan, J. High Energy Phys. 1504, 122 (2015), doi:10.1007/JHEP04(2015)122, arXiv:1501.04389 [hep-th]. Web of Science, ADS, Google Scholar
44. R. M. Soni and S. P. Trivedi, J. High Energy Phys. 1601, 136 (2016), doi:10.1007/JHEP01(2016)136, arXiv:1510.07455 [hep-th]. Web of Science, ADS, Google Scholar
45. J. M. Drouffe and J. B. Zuber, Phys. Rep. 102, 1 (1983), doi:10.1016/0370-1573(83)90034-0. Web of Science, ADS, Google Scholar
46. A. Gromov and R. A. Santos, Phys. Lett. B 737, 60 (2014), doi:10.1016/j.physletb.2014.08.023, arXiv:1403.5035 [hep-th]. Web of Science, ADS, Google Scholar
47. J. Shigemitsu and J. B. Kogut, Nucl. Phys. B 190, 365 (1982). Web of Science, ADS, Google Scholar