Abstract
We consider entanglement entropy of a cap-like region for a conformal field theory living on a sphere times a circle in d space-time dimensions. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading correction to the entanglement entropy in a low temperature expansion. The correction has a universal form for any conformal field theory that depends only on the size of the mass gap, its degeneracy, and the angular size of the cap. We confirm our result by calculating the entanglement entropy of a conformally coupled scalar numerically. We argue that an apparent discrepancy for the scalar can be explained away through a careful treatment of boundary terms. In an appendix, to confirm the accuracy of the numerics, we study the mutual information of two cap-like regions at zero temperature.
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].
H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].
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].
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].
C.P. Herzog and M. Spillane, Tracing through scalar entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].
J. Cardy and C.P. Herzog, Universal thermal corrections to single interval entanglement entropy for conformal field theories, Phys. Rev. Lett. 112 (2014) 171603 [arXiv:1403.0578] [INSPIRE].
B. Chen and J.-q. Wu, Single interval Renyi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].
I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42 (2009) 4003 [arXiv:0906.1663].
N. Shiba, Entanglement entropy of two spheres, JHEP 07 (2012) 100 [arXiv:1201.4865] [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].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
A.D. Barvinsky and S.N. Solodukhin, Nonminimal coupling, boundary terms and renormalization of the Einstein-Hilbert action and black hole entropy, Nucl. Phys. B 479 (1996) 305 [gr-qc/9512047] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, arXiv:1407.6429 [INSPIRE].
J. Lee, A. Lewkowycz, E. Perlmutter and B.R. Safdi, Renyi entropy, stationarity and entanglement of the conformal scalar, arXiv:1407.7816 [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [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].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
P. Candelas and J.S. Dowker, Field theories on conformally related space-times: some global considerations, Phys. Rev. D 19 (1979) 2902 [INSPIRE].
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: 1407.1358
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. Universal thermal corrections to entanglement entropy for conformal field theories on spheres. J. High Energ. Phys. 2014, 28 (2014). https://doi.org/10.1007/JHEP10(2014)028
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)028