Abstract
We study the Rényi and entanglement entropies for free 2d CFT’s at finite temperature and finite size, with emphasis on their properties under modular transformations of the torus. We address the issue of summing over fermion spin structures in the replica trick, and show that the relation between entanglement and thermal entropy determines two different ways to perform this sum in the limits of small and large interval. Both answers are modular covariant, rather than invariant. Our results are compared with those for a free boson at unit radius in the two limits and complete agreement is found, supporting the view that entanglement respects Bose-Fermi duality. We extend our computations to multiple free Dirac fermions having correlated spin structures, dual to free bosons on the Spin(2d) weight lattice.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
A. Osterloh, L. Amico, G. Falci and R. Fazio, Scaling of entanglement close to a quantum phase transition, Nature 416 (2002) 608 [quant-ph/0202029].
I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].
A. Riera and J.I. Latorre, Area law and vacuum reordering in harmonic networks, Phys. Rev. A 74 (2006) 052326 [quant-ph/0605112] [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. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
H. Casini, C.D. 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].
J.L. Cardy, O.A. 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].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
M. Headrick, A. Lawrence and M. Roberts, Bose-Fermi duality and entanglement entropies, J. Stat. Mech. 1302 (2013) P02022 [arXiv:1209.2428] [INSPIRE].
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].
C.P. Herzog and T. Nishioka, Entanglement Entropy of a Massive Fermion on a Torus, JHEP 03 (2013) 077 [arXiv:1301.0336] [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, Large Interval Limit of Rényi Entropy At High Temperature, arXiv:1412.0763 [INSPIRE].
B. Chen and J.-q. Wu, Rényi Entropy of Free Compact Boson on Torus, arXiv:1501.0037.
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].
B. Chen and J.-q. Wu, Single interval Renyi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [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].
N. Seiberg and E. Witten, Spin Structures in String Theory, Nucl. Phys. B 276 (1986) 272 [INSPIRE].
J.L. Cardy, Operator content and modular properties of higher dimensional conformal field theories, Nucl. Phys. B 366 (1991) 403 [INSPIRE].
B. Chen and J.-q. Wu, Universal relation between thermal entropy and entanglement entropy in conformal field theories, Phys. Rev. D 91 (2015) 086012 [arXiv:1412.0761] [INSPIRE].
S. Elitzur, E. Gross, E. Rabinovici and N. Seiberg, Aspects of Bosonization in String Theory, Nucl. Phys. B 283 (1987) 413 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer Verlag (1997).
functions.wolfram.com, Elliptic Theta, http://functions.wolfram.com/EllipticFunctions/EllipticTheta1/introductions/JacobiThetas/ 05/ShowAll.html.
V.S. Dotsenko and V.A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691 [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: 1504.01921
On leave from the Tata Institute of Fundamental Research, Mumbai (Sunil Mukhi).
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
Lokhande, S.F., Mukhi, S. Modular invariance and entanglement entropy. J. High Energ. Phys. 2015, 106 (2015). https://doi.org/10.1007/JHEP06(2015)106
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2015)106