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Thermal corrections to Rényi entropies for free fermions

  • Christopher P. Herzog
  • Michael Spillane
Open Access
Regular Article - Theoretical Physics

Abstract

We calculate thermal corrections to Rényi entropies for free massless fermions on a sphere. More specifically, we take a free fermion on \( \mathbb{R}\times {\mathbb{S}}^{{{}^d}^{-1}} \) and calculate the leading thermal correction to the Rényi entropies for a cap like region with opening angle 2θ. By expanding the density matrix in a Boltzmann sum, the problem of finding the Rényi entropies can be mapped to the problem of calculating a two point function on an n sheeted cover of the sphere. We follow previous work for conformal field theories to map the problem on the sphere to a conical region in Euclidean space. By using the method of images, we calculate the two point function and recover the Rényi entropies.

Keywords

Conformal and W Symmetry Field Theories in Higher Dimensions Thermal Field Theory 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    T.J. Osborne and M.A. Nielsen, Entanglement in a simple quantum phase transition, Phys. Rev. A 66 (2002) 032110 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [INSPIRE].
  3. [3]
    L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  4. [4]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    C.P. Herzog and M. Spillane, Tracing Through Scalar Entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].ADSGoogle Scholar
  7. [7]
    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].ADSMathSciNetGoogle Scholar
  8. [8]
    T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].MathSciNetGoogle Scholar
  9. [9]
    S. Datta and J.R. David, Rényi entropies of free bosons on the torus and holography, JHEP 04 (2014) 081 [arXiv:1311.1218] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    B. Chen and J.-q. Wu, Single interval Renyi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    C.P. Herzog and T. Nishioka, Entanglement Entropy of a Massive Fermion on a Torus, JHEP 03 (2013) 077 [arXiv:1301.0336] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    J. Cardy and C.P. Herzog, Universal Thermal Corrections to Single Interval Entanglement Entropy for Two Dimensional Conformal Field Theories, Phys. Rev. Lett. 112 (2014) 171603 [arXiv:1403.0578] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    C.P. Herzog, Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres, JHEP 10 (2014) 028 [arXiv:1407.1358] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    C.P. Herzog and J. Nian, Thermal corrections to Rényi entropies for conformal field theories, JHEP 06 (2015) 009 [arXiv:1411.6505] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    P. Candelas and J.S. Dowker, Field Theories On Conformally Related Space-times: Some Global Considerations, Phys. Rev. D 19 (1979) 2902 [INSPIRE].ADSMathSciNetGoogle Scholar
  16. [16]
    M.E. Peskin and D.V. Schroeder, An Introduction to quantum field theory, Addison-Wesley, Reading U.S.A. (1995).Google Scholar
  17. [17]
    F.T.J. Epple, Induced gravity on intersecting branes, JHEP 09 (2004) 021 [hep-th/0408105] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    J. Cardy, Some results on the mutual information of disjoint regions in higher dimensions, J. Phys. A 46 (2013) 285402 [arXiv:1304.7985] [INSPIRE].MathSciNetMATHGoogle Scholar
  19. [19]
    R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].
  20. [20]
    I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. Math. Gen. A 42 (2009) 504003 [arXiv:0906.1663].MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.C.N. Yang Institute for Theoretical Physics, Department of Physics and AstronomyStony Brook UniversityStony BrookU.S.A.

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