Abstract
We investigate the Rényi entropy and entanglement entropy of an interval with an arbitrary length in the canonical ensemble, microcanonical ensemble and primary excited states at large energy density in the thermodynamic limit of a two-dimensional large central charge c conformal field theory. As a generalization of the recent work [17], the main purpose of the paper is to see whether one can distinguish these various large energy density states by the Rényi entropies of an interval at different size scales, namely, short, medium and long. Collecting earlier results and performing new calculations in order to compare with and fill gaps in the literature, we give a more complete and detailed analysis of the problem. Especially, we find some corrections to the recent results for the holographic Rényi entropy of a medium size interval, which enlarge the validity region of the results. Based on the Rényi entropies of the three interval scales, we find that Rényi entropy cannot distinguish the canonical and microcanonical ensemble states for a short interval, but can do the job for both medium and long intervals. At the leading order of large c the entanglement entropy cannot distinguish the canonical and microcanonical ensemble states for all interval lengths, but the difference of entanglement entropy for a long interval between the two states would appear with 1/c corrections. We also discuss Rényi entropy and entanglement entropy differences between the thermal states and primary excited state. Overall, our work provide an up-to-date picture of distinguishing different thermal or primary states at various length scales of the subsystem.
Article PDF
Similar content being viewed by others
References
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev.A 43 (1991) 2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev.E 50 (1994) 888.
M. Srednicki, Thermal fluctuations in quantized chaotic systems, J. Phys.A 29 (1996) L75 [chao-dyn/9511001] [INSPIRE].
M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452 (2008) 854 [arXiv:0708.1324].
L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys.65 (2016) 239 [arXiv:1509.06411].
N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech.1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].
A. Dymarsky, N. Lashkari and H. Liu, Subsystem ETH, Phys. Rev.E 97 (2018) 012140 [arXiv:1611.08764] [INSPIRE].
N. Lashkari, A. Dymarsky and H. Liu, Universality of quantum information in chaotic CFTs, JHEP03 (2018) 070 [arXiv:1710.10458] [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys.43 (1975) 199 [Erratum ibid.46 (1976) 206] [INSPIRE].
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev.D 14 (1976) 2460 [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett.B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
T.-C. Lu and T. Grover, Renyi entropy of chaotic eigenstates, Phys. Rev.E 99 (2019) 032111 [arXiv:1709.08784] [INSPIRE].
T. Faulkner and H. Wang, Probing beyond ETH at large c, JHEP06 (2018) 123 [arXiv:1712.03464] [INSPIRE].
W.-Z. Guo, F.-L. Lin and J. Zhang, Distinguishing black hole microstates using Holevo information, Phys. Rev. Lett.121 (2018) 251603 [arXiv:1808.02873] [INSPIRE].
X. Dong, Holographic Rényi entropy at high energy density, Phys. Rev. Lett.122 (2019) 041602 [arXiv:1811.04081] [INSPIRE].
J.R. Garrison and T. Grover, Does a single eigenstate encode the full Hamiltonian?, Phys. Rev.X 8 (2018) 021026 [arXiv:1503.00729] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP11 (2016) 028 [arXiv:1607.07506] [INSPIRE].
X. Dong, The gravity dual of Rényi entropy, Nature Commun.7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
F.-L. Lin, H. Wang and J.-j. Zhang, Thermality and excited state Rényi entropy in two-dimensional CFT, JHEP11 (2016) 116 [arXiv:1610.01362] [INSPIRE].
S. He, F.-L. Lin and J.-j. Zhang, Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT, JHEP08 (2017) 126 [arXiv:1703.08724] [INSPIRE].
S. He, F.-L. Lin and J.-j. Zhang, Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis, JHEP12 (2017) 073 [arXiv:1708.05090] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
B. Chen, Z. Li and J.-j. Zhang, Corrections to holographic entanglement plateau, JHEP09 (2017) 151 [arXiv:1707.07354] [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].
M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev.D 82 (2010) 126010 [arXiv:1006.0047] [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.A. Rajabpour and F. Gliozzi, Entanglement entropy of two disjoint intervals from fusion algebra of twist fields, J. Stat. Mech.1202 (2012) P02016 [arXiv:1112.1225] [INSPIRE].
B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP11 (2013) 164 [arXiv:1309.5453] [INSPIRE].
B. Chen, J.-B. Wu and J.-j. Zhang, Short interval expansion of Rényi entropy on torus, JHEP08 (2016) 130 [arXiv:1606.05444] [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].
B. Chen and J.-q. Wu, Holographic calculation for large interval Rényi entropy at high temperature, Phys. Rev.D 92 (2015) 106001 [arXiv:1506.03206] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
T. Faulkner, The entanglement Rényi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
B. Chen, J. Long and J.-j. Zhang, Holographic Rényi entropy for CFT with W symmetry, JHEP04 (2014) 041 [arXiv:1312.5510] [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].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
R. Sasaki and I. Yamanaka, Virasoro algebra, vertex operators, quantum Sine-Gordon and solvable quantum field theories, Adv. Stud. Pure Math.16 (1988) 271 [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys.177 (1996) 381 [hep-th/9412229] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic entanglement entropy from 2d CFT: heavy states and local quenches, JHEP02 (2015) 171 [arXiv:1410.1392] [INSPIRE].
P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum entanglement of localized excited states at finite temperature, JHEP01 (2015) 102 [arXiv:1410.2287] [INSPIRE].
W.-Z. Guo, F.-L. Lin and J. Zhang, Note on ETH of descendant states in 2D CFT, JHEP01 (2019) 152 [arXiv:1810.01258] [INSPIRE].
G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement temperature and entanglement entropy of excited states, JHEP12 (2013) 020 [arXiv:1305.3291] [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech.1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, 10th anniversary ed., Cambridge University Press, Cambridge, U.K. (2010).
W.-Z. Guo, F.-L. Lin and J. Zhang, Nongeometric states in a holographic conformal field theory, Phys. Rev.D 99 (2019) 106001 [arXiv:1806.07595] [INSPIRE].
M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett.98 (2007) 050405 [cond-mat/0604476].
P. Basu, D. Das, S. Datta and S. Pal, Thermality of eigenstates in conformal field theories, Phys. Rev.E 96 (2017) 022149 [arXiv:1705.03001] [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized Gibbs ensemble of 2d CFTs at large central charge in the thermodynamic limit, JHEP01 (2019) 098 [arXiv:1810.11025] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Thermal correlation functions of KdV charges in 2D CFT, JHEP02 (2019) 044 [arXiv:1810.11053] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Generalized Gibbs ensemble and the statistics of KdV charges in 2D CFT, JHEP03 (2019) 075 [arXiv:1810.11054] [INSPIRE].
A. Dymarsky and K. Pavlenko, Exact generalized partition function of 2D CFTs at large central charge, JHEP05 (2019) 077 [arXiv:1812.05108] [INSPIRE].
E.M. Brehm and D. Das, On KdV characters in large c CFTs, arXiv:1901.10354 [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized eigenstate thermalization in 2d CFTs, arXiv:1903.03559 [INSPIRE].
S. Datta, P. Kraus and B. Michel, Typicality and thermality in 2d CFT, arXiv:1904.00668 [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: 1812.11753
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Guo, Wz., Lin, FL. & Zhang, J. Rényi entropy at large energy density in 2D CFT. J. High Energ. Phys. 2019, 10 (2019). https://doi.org/10.1007/JHEP08(2019)010
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2019)010