Abstract
We investigate the eigenstate thermalization hypothesis (ETH) of highly excited descendant states in two-dimensional large central charge c conformal field theory. We use operator product expansion of twist operators to calculate the short interval expansions of entanglement entropy and relative entropy for an interval of length ℓ up to order ℓ12. Using these results to ensure ETH of a heavy state when compared with the canonical ensemble state up to various orders of c, we get the constraints on the expectation values of the first few quasiprimary operators in the vacuum conformal family at the corresponding order of c. Similarly, we also obtain the constraints from the expectation values of the first few Korteweg-de Vries charges. We check these constraints for some types of special descendant excited states. Among the descendant states we consider, we find that at most only the leading order ones of the ETH constraints can be satisfied for the descendant states that are slightly excited on top of a heavy primary state. Otherwise, the ETH constraints are violated for the descendant states that are heavily excited on top of a primary state.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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].
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].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [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].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
P. Kraus and A. Maloney, A Cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].
E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].
A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018) 005 [arXiv:1804.08899] [INSPIRE].
Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [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].
B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP 11 (2013) 164 [arXiv:1309.5453] [INSPIRE].
R. Sasaki and I. Yamanaka, Virasoro algebra, vertex operators, quantum sine-Gordon and solvable quantum field theories, in Conformal field theory and solvable lattice models, Elsevier, The Netherlands (1988), pg. 271 [Adv. Stud. Pure Math. 16 (1988) 271].
T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989) 373 [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].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of long-distance AdS physics from the CFT bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro conformal blocks and thermality from classical background fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic entanglement entropy from 2d CFT: heavy states and local quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].
P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum entanglement of localized excited states at finite temperature, JHEP 01 (2015) 102 [arXiv:1410.2287] [INSPIRE].
F.-L. Lin, H. Wang and J.-J. Zhang, Thermality and excited state Rényi entropy in two-dimensional CFT, JHEP 11 (2016) 116 [arXiv:1610.01362] [INSPIRE].
S. He, F.-L. Lin and J.-J. Zhang, Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT, JHEP 08 (2017) 126 [arXiv:1703.08724] [INSPIRE].
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].
S. He, F.-L. Lin and J.-J. Zhang, Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis, JHEP 12 (2017) 073 [arXiv:1708.05090] [INSPIRE].
N. Lashkari, A. Dymarsky and H. Liu, Universality of quantum information in chaotic CFTs, JHEP 03 (2018) 070 [arXiv:1710.10458] [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].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement temperature and entanglement entropy of excited states, JHEP 12 (2013) 020 [arXiv:1305.3291] [INSPIRE].
W.-Z. Guo, F.-L. Lin and J. Zhang, Non-geometric states in a holographic conformal field theory, arXiv:1806.07595 [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, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
X. Dong, The gravity dual of Rényi entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [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].
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
P.H. Ginsparg, Applied conformal field theory, in Les Houches summer school in theoretical physics: fields, strings, critical phenomena, Les Houches, France, 28 June-5 August 1988, pg. 1 [hep-th/9108028] [INSPIRE].
P.D. Francesco, P. Mathieu and D. Sénéchal, Quantum field theory, Springer, New York, U.S.A. (1997) [INSPIRE].
R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory, Lect. Notes Phys. 779 (2009) 1 [INSPIRE].
B. Chen, J. Long and J.-J. Zhang, Holographic Rényi entropy for CFT with W symmetry, JHEP 04 (2014) 041 [arXiv:1312.5510] [INSPIRE].
Z. Li and J.-J. Zhang, On one-loop entanglement entropy of two short intervals from OPE of twist operators, JHEP 05 (2016) 130 [arXiv:1604.02779] [INSPIRE].
B. Chen, J.-B. Wu and J.-J. Zhang, Short interval expansion of Rényi entropy on torus, JHEP 08 (2016) 130 [arXiv:1606.05444] [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].
N. Lashkari, Relative entropies in conformal field theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].
N. Lashkari, Modular Hamiltonian for excited states in conformal field theory, Phys. Rev. Lett. 117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].
B. Chen, Z. Li and J.-J. Zhang, Corrections to holographic entanglement plateau, JHEP 09 (2017) 151 [arXiv:1707.07354] [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].
B. Chen and J.-Q. Wu, Single interval Rényi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].
B. Chen, J.-Q. Wu and Z.-C. Zheng, Holographic Rényi entropy of single interval on torus: with W symmetry, Phys. Rev. D 92 (2015) 066002 [arXiv:1507.00183] [INSPIRE].
G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP 07 (2016) 114 [arXiv:1603.03057] [INSPIRE].
G. Sárosi and T. Ugajin, Relative entropy of excited states in conformal field theories of arbitrary dimensions, JHEP 02 (2017) 060 [arXiv:1611.02959] [INSPIRE].
D. Aharonov, A necessary and sufficient condition for univalence of a meromorphic function, Duke Math. J. 36 (1969) 599.
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: 1810.01258
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. Note on ETH of descendant states in 2D CFT. J. High Energ. Phys. 2019, 152 (2019). https://doi.org/10.1007/JHEP01(2019)152
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2019)152