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Entanglement entropy of a massive fermion on a torus

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Abstract

The Rényi entropies of a massless Dirac fermion on a circle with chemical potential are calculated analytically at nonzero temperature by using the bosonization method. The bosonization of a massive Dirac fermion to the sine-Gordon model lets us obtain the small mass corrections to the entropies. We numerically compute the Rényi entropies by putting a massive fermion on the lattice and find agreement between the analytic and numerical results. In the presence of a mass gap, we show that corrections to Rényi and entanglement entropies in the limit m gap ≫ T scale as \( {e^{{{{{-{m_{\mathrm{gap}}}}} \left/ {T} \right.}}}} \). We also show that when there is ground state degeneracy in the gapless case, the limits m gap → 0 and T → 0 do not commute.

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References

  1. H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  2. H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].

    ADS  Google Scholar 

  3. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].

  6. S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  7. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    MathSciNet  ADS  MATH  Google Scholar 

  8. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].

    MathSciNet  Google Scholar 

  9. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42 (2009) 504003 [arXiv:0906.1663].

    MathSciNet  Google Scholar 

  10. H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].

    MathSciNet  Google Scholar 

  11. T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].

    MathSciNet  Google Scholar 

  12. T. Takayanagi, Entanglement entropy from a holographic viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. C.P. Herzog and M. Spillane, Tracing through scalar entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].

    ADS  Google Scholar 

  14. H. Casini, C. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 0507 (2005) P07007 [cond-mat/0505563] [INSPIRE].

    Article  Google Scholar 

  15. P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].

    Article  Google Scholar 

  16. V. Korepin, Universality of entropy scaling in one dimensional gapless models, Phys. Rev. Lett. 92 (2004) 96402.

    Article  ADS  Google Scholar 

  17. J. Cardy, O. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. B. Doyon, Bi-partite entanglement entropy in massive two-dimensional quantum field theory, Phys. Rev. Lett. 102 (2009) 031602 [arXiv:0803.1999] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. 26 (2009) 185005 [arXiv:0903.5284] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. T. Azeyanagi, T. Nishioka and T. Takayanagi, Near Extremal Black Hole Entropy as Entanglement Entropy via AdS 2 /CFT 1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  21. N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi surfaces and entanglement entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. S.R. Coleman, The quantum sine-Gordon equation as the massive thirring model, Phys. Rev. D 11 (1975) 2088 [INSPIRE].

    ADS  Google Scholar 

  23. S. Mandelstam, Soliton operators for the quantized sine-Gordon equation, Phys. Rev. D 11 (1975) 3026 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  24. P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer, New York U.S.A. (1997), pg. 890.

    Book  MATH  Google Scholar 

  25. M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].

    ADS  Google Scholar 

  26. W. Fischler, A. Kundu and S. Kundu, Holographic mutual information at finite temperature, arXiv:1212.4764 [INSPIRE].

  27. A.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [INSPIRE].

    ADS  Google Scholar 

  28. S.L. Lukyanov and A.B. Zamolodchikov, Exact expectation values of local fields in quantum sine-Gordon model, Nucl. Phys. B 493 (1997) 571 [hep-th/9611238] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. D. Bernard and A. LeClair, Differential equations for sine-Gordon correlation functions at the free fermion point, Nucl. Phys. B 426 (1994) 534 [Erratum ibid. B 498 (1997) 619-621] [hep-th/9402144] [INSPIRE].

  30. B.-Q. Jin and V.E. Korepin, Quantum Spin Chain, Toeplitz Determinants and the Fisher-Hartwig Conjecture, J. Statist. Phys. 116 (2004) 79 [quant-ph/0304108].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. A.R. Its and V.E. Korepin, The Fisher-Hartwig formula and generalized entropies in XY spin chain, J. Statist. Phys. 137 (2009) 1014 arXiv:0906.4511.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. F. Franchini, A. Its and V. Korepin, Renyi entropy of the XY spin chain, J. Phys. A 41 (2008) 025302 [arXiv:0707.2534] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  33. E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  34. T. Nishioka and T. Takayanagi, AdS bubbles, entropy and closed string tachyons, JHEP 01 (2007) 090 [hep-th/0611035] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. A. Pakman and A. Parnachev, Topological entanglement entropy and holography, JHEP 07 (2008) 097 [arXiv:0805.1891] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794, U.S.A.

    Christopher P. Herzog

  2. Department of Physics, Princeton University, Princeton, NJ, 08544, U.S.A.

    Tatsuma Nishioka

Authors
  1. Christopher P. Herzog
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  2. Tatsuma Nishioka
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Correspondence to Tatsuma Nishioka.

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ArXiv ePrint: 1301.0336

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Herzog, C.P., Nishioka, T. Entanglement entropy of a massive fermion on a torus. J. High Energ. Phys. 2013, 77 (2013). https://doi.org/10.1007/JHEP03(2013)077

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