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Holographic charged Rényi entropies

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Abstract

We construct a new class of entanglement measures by extending the usual definition of Rényi entropy to include a chemical potential. These charged Rényi entropies measure the degree of entanglement in different charge sectors of the theory and are given by Euclidean path integrals with the insertion of a Wilson line encircling the entangling surface. We compute these entropies for a spherical entangling surface in CFT’s with holographic duals, where they are related to entropies of charged black holes with hyperbolic horizons. We also compute charged Rényi entropies in free field theories.

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References

  1. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].

    Article  ADS  Google Scholar 

  2. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  3. A. Hamma, R. Ionicioiu and P. Zanardi, Ground state entanglement and geometric entropy in the Kitaev model, Phys. Lett. A 337 (2005) 22 [quant-ph/0406202].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 06 (2004) P06002 [hep-th/0405152] [INSPIRE].

    MATH  Google Scholar 

  5. P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory: a non-technical introduction, Int. J. Quant. Inf. 4 (2006) 429 [quant-ph/0505193] [INSPIRE].

    Article  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].

  11. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [arXiv:1005.3035] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. E. Bianchi and R.C. Myers, On the architecture of spacetime geometry, arXiv:1212.5183 [INSPIRE].

  13. R.C. Myers, R. Pourhasan and M. Smolkin, On spacetime entanglement, JHEP 06 (2013) 013 [arXiv:1304.2030] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. V. Balasubramanian, B. Czech, B.D. Chowdhury and J. de Boer, The entropy of a hole in spacetime, JHEP 10 (2013) 220 [arXiv:1305.0856] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. A. Rényi, On measures of information and entropy, in Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, 1, http://digitalassets.lib.berkeley.edu/math/ucb/text/math s4 v1 article-27.pdf, U. of California Press, Berkeley CA U.S.A. (1961), pg. 547.

  16. A. Rényi, On the foundations of information theory, Rev. Int. Stat. Inst. 33 (1965) 1.

    Article  MathSciNet  MATH  Google Scholar 

  17. For example see K. Zyczkowski, Rényi extrapolation of Shannon entropy, Open Syst. Inf. Dyn. 10 (2003) 297 [quant-ph/0305062].

  18. C. Beck and F. Schlögl, Thermodynamics of chaotic systems, Cambridge University Press, Cambridge U.K. (1993).

    Book  MATH  Google Scholar 

  19. H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Rényi entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  21. I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Rényi entropies for free field theories, JHEP 04 (2012) 074 [arXiv:1111.6290] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  22. A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement temperature and entanglement entropy of excited states, arXiv:1305.3291 [INSPIRE].

  24. P. Caputa, G. Mandal and R. Sinha, Dynamical entanglement entropy with angular momentum and U(1) charge, JHEP 11 (2013) 052 [arXiv:1306.4974] [INSPIRE].

    Article  ADS  Google Scholar 

  25. T. Nishioka and I. Yaakov, Supersymmetric Rényi entropy, JHEP 10 (2013) 155 [arXiv:1306.2958] [INSPIRE].

    Article  ADS  Google Scholar 

  26. H. Casini, Entropy inequalities from reflection positivity, J. Stat. Mech. 08 (2010) P08019 [arXiv:1004.4599] [INSPIRE].

    Article  Google Scholar 

  27. B. Swingle, Mutual information and the structure of entanglement in quantum field theory, arXiv:1010.4038 [INSPIRE].

  28. L.Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, in preparation.

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

    ADS  Google Scholar 

  30. A. Roberge and N. Weiss, Gauge theories with imaginary chemical potential and the phases of QCD, Nucl. Phys. B 275 (1986) 734 [INSPIRE].

    Article  ADS  Google Scholar 

  31. M.G. Alford, A. Kapustin and F. Wilczek, Imaginary chemical potential and finite fermion density on the lattice, Phys. Rev. D 59 (1999) 054502 [hep-lat/9807039] [INSPIRE].

    ADS  Google Scholar 

  32. E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  33. R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].

    ADS  Google Scholar 

  34. R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. E. Perlmutter, A universal feature of CFT Rényi entropy, arXiv:1308.1083 [INSPIRE].

  36. H. Osborn and A. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. T. Takayanagi, Entanglement entropies with twisted boundary conditions, unpublished.

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

    MathSciNet  MATH  Google Scholar 

  39. H. Casini, C. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 07 (2005) P07007 [cond-mat/0505563] [INSPIRE].

    Article  Google Scholar 

  40. T. Azeyanagi, T. Nishioka and T. Takayanagi, Near extremal black hole entropy as entanglement entropy via AdS 2 /CF T 1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].

    ADS  Google Scholar 

  41. C.P. Herzog and T. Nishioka, Entanglement entropy of a massive fermion on a torus, JHEP 03 (2013) 077 [arXiv:1301.0336] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. A. Belin, A. Maloney and S. Matsuura, Holographic phases of Rényi entropies, arXiv:1306.2640 [INSPIRE].

  44. R.-G. Cai and A. Wang, Thermodynamics and stability of hyperbolic charged black holes, Phys. Rev. D 70 (2004) 064013 [hep-th/0406057] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  45. S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. M. Cvetič, S. Nojiri and S.D. Odintsov, Black hole thermodynamics and negative entropy in de Sitter and anti-de Sitter Einstein-Gauss-Bonnet gravity, Nucl. Phys. B 628 (2002) 295 [hep-th/0112045] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. X.–H. Ge, Y. Matsuo, F.-W. Shu, S.-J. Sin and T. Tsukioka, Viscosity bound, causality violation and instability with stringy correction and charge, JHEP 10 (2008) 009 [arXiv:0808.2354] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. D. Anninos and G. Pastras, Thermodynamics of the Maxwell-Gauss-Bonnet anti-de Sitter black hole with higher derivative gauge corrections, JHEP 07 (2009) 030 [arXiv:0807.3478] [INSPIRE].

    Article  ADS  Google Scholar 

  49. A. Belin, L.-Y. Hung, A. Maloney, S. Matsuura, R.C. Myers and T. Sierens, Holographic rotating Rényi entropies, in preparation.

  50. A. Grigor’yan and M. Noguchi, The heat kernel on hyperbolic space, Bull. London Math. Soc. 30 (1998) 643.

    Article  MathSciNet  MATH  Google Scholar 

  51. A. Grigor’yan, Upper bounds on a complete non compact manifold, J. Funct. Anal. 127 (1995) 363.

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Debiard, B. Gaveau and E. Mazet, Théorèmes de comparaison en géométrie riemannienne (in French), Publ. Res. Inst. Math. Sci. Kyoto 12 (1976) 391.

    Article  MathSciNet  MATH  Google Scholar 

  53. R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces, J. Math. Phys. 35 (1994) 4217 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. R. Camporesi, The spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys. 148 (1992) 283 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. A. Lewkowycz, R.C. Myers and M. Smolkin, Observations on entanglement entropy in massive QFT’s, JHEP 04 (2013) 017 [arXiv:1210.6858] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  57. R. Camporesi, The spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys. 148 (1992) 283 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. A.A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Quantum fields and extended objects in space-times with constant curvature spatial section, Phys. Rept. 266 (1996) 1 [hep-th/9505061] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  59. C. Martinez, C. Teitelboim and J. Zanelli, Charged rotating black hole in three space-time dimensions, Phys. Rev. D 61 (2000) 104013 [hep-th/9912259] [INSPIRE].

    ADS  Google Scholar 

  60. I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. L.-Y. Hung and A. Sinha, Holographic quantum liquids in 1 + 1 dimensions, JHEP 01 (2010) 114 [arXiv:0909.3526] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  62. A. Ejaz, H. Gohar, H. Lin, K. Saifullah and S.-T. Yau, Quantum tunneling from three-dimensional black holes, Phys. Lett. B 726 (2013) 827 [arXiv:1306.6380] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. P. Kraus, Lectures on black holes and the AdS 3 /CF T 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].

    ADS  MATH  Google Scholar 

  64. V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  65. R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  66. J.D. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  67. R.C. Myers, Stress tensors and Casimir energies in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 046002 [hep-th/9903203] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].

    Article  ADS  Google Scholar 

  70. D. Chowdhury, S. Raju, S. Sachdev, A. Singh and P. Strack, Multipoint correlators of conformal field theories: implications for quantum critical transport, Phys. Rev. B 87 (2013) 085138 [arXiv:1210.5247] [INSPIRE].

    Article  ADS  Google Scholar 

  71. A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Departments of Physics and Mathematics, McGill University, 3600 rue University, Montréal, Québec, H3A 2T8, Canada

    Alexandre Belin, Alexander Maloney & Shunji Matsuura

  2. Department of Physics, Harvard University, 17 Oxford street, Cambridge, MA, 02138, United States

    Alexandre Belin, Ling-Yan Hung, Alexander Maloney & Shunji Matsuura

  3. Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, N2L 6B9, Canada

    Robert C. Myers & Todd Sierens

  4. Guelph-Waterloo Physics Institute, Department of Physics & Astronomy, University of Waterloo, 200 University Ave. W, Waterloo, Ontario, N2L 3G1, Canada

    Todd Sierens

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  1. Alexandre Belin
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  2. Ling-Yan Hung
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  3. Alexander Maloney
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  4. Shunji Matsuura
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  5. Robert C. Myers
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Correspondence to Shunji Matsuura.

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ArXiv ePrint: 1310.4180

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Belin, A., Hung, LY., Maloney, A. et al. Holographic charged Rényi entropies. J. High Energ. Phys. 2013, 59 (2013). https://doi.org/10.1007/JHEP12(2013)059

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