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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T.J. Osborne and M.A. Nielsen, Entanglement in a simple quantum phase transition, Phys. Rev. A 66 (2002) 032110 [INSPIRE].
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].
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].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
C.P. Herzog and M. Spillane, Tracing Through Scalar Entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].
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].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
S. Datta and J.R. David, Rényi entropies of free bosons on the torus and holography, JHEP 04 (2014) 081 [arXiv:1311.1218] [INSPIRE].
B. Chen and J.-q. Wu, Single interval Renyi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].
C.P. Herzog and T. Nishioka, Entanglement Entropy of a Massive Fermion on a Torus, JHEP 03 (2013) 077 [arXiv:1301.0336] [INSPIRE].
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].
C.P. Herzog, Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres, JHEP 10 (2014) 028 [arXiv:1407.1358] [INSPIRE].
C.P. Herzog and J. Nian, Thermal corrections to Rényi entropies for conformal field theories, JHEP 06 (2015) 009 [arXiv:1411.6505] [INSPIRE].
P. Candelas and J.S. Dowker, Field Theories On Conformally Related Space-times: Some Global Considerations, Phys. Rev. D 19 (1979) 2902 [INSPIRE].
M.E. Peskin and D.V. Schroeder, An Introduction to quantum field theory, Addison-Wesley, Reading U.S.A. (1995).
F.T.J. Epple, Induced gravity on intersecting branes, JHEP 09 (2004) 021 [hep-th/0408105] [INSPIRE].
J. Cardy, Some results on the mutual information of disjoint regions in higher dimensions, J. Phys. A 46 (2013) 285402 [arXiv:1304.7985] [INSPIRE].
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].
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].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1506.06757
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Herzog, C.P., Spillane, M. Thermal corrections to Rényi entropies for free fermions. J. High Energ. Phys. 2016, 124 (2016). https://doi.org/10.1007/JHEP04(2016)124
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2016)124