Abstract
Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs call for calculation of the expectation values of local operators in highly excited energy eigenstates. This can be done efficiently by representing zero modes of these operators in terms of the Virasoro algebra generators. In this paper we present a pedagogical introduction explaining how this calculation can be performed analytically or using computer algebra. We illustrate the computation of zero modes by a number of examples and list explicit expressions for all local operators from the vacuum family with the dimension of less or equal than eight. Finally, we derive an explicit expression for the quantum KdV generator Q7 in terms of the Virasoro algebra generators. The obtained results can be used for quantitative studies of ETH at finite value of central charge.
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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. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854.
E. Ilievski, M. Medenjak and T.z. Prosen, Quasilocal conserved operators in the isotropic Heisenberg spin-1/2 chain, Phys. Rev. Lett. 115 (2015) 120601 [arXiv:1506.05049] [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.
A.C. Cassidy, C.W. Clark and M. Rigol, Generalized thermalization in an integrable lattice system, Phys. Rev. Lett. 106 (2011) 140405.
L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. 2016 (2016) 064007 [arXiv:1604.03990].
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] [INSPIRE].
J. de Boer and D. Engelhardt, Remarks on thermalization in 2D CFT, Phys. Rev. D 94 (2016) 126019 [arXiv:1604.05327] [INSPIRE].
J. Sonner and M. Vielma, Eigenstate thermalization in the Sachdev-Ye-Kitaev model, JHEP 11 (2017) 149 [arXiv:1707.08013] [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].
N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].
N. Lashkari, A. Dymarsky and H. Liu, Universality of quantum information in chaotic CFTs, JHEP 03 (2018) 070 [arXiv:1710.10458] [INSPIRE].
T. Faulkner and H. Wang, Probing beyond ETH at large c, JHEP 06 (2018) 123 [arXiv:1712.03464] [INSPIRE].
F.-L. Lin, H. Wang and J.-J. Zhang, Thermality and excited state Ŕenyi entropy in two-dimensional CFT, JHEP 11 (2016) 116 [arXiv:1610.01362] [INSPIRE].
W.-Z. Guo, F.-L. Lin and J. Zhang, Note on ETH of descendant states in 2D CFT, JHEP 01 (2019) 152 [arXiv:1810.01258] [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].
A. Dymarsky and K. Pavlenko, Generalized Gibbs ensemble of 2d CFTs at large central charge in the thermodynamic limit, JHEP 01 (2019) 098 [arXiv:1810.11025] [INSPIRE].
A. Dymarsky and K. Pavlenko, Exact generalized partition function of 2D CFTs at large central charge, JHEP 19 (2020) 077 [arXiv:1812.05108] [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized eigenstate thermalization hypothesis in 2D conformal field theories, Phys. Rev. Lett. 123 (2019) 111602 [arXiv:1903.03559] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Generalized Gibbs ensemble and the statistics of KdV charges in 2D CFT, JHEP 03 (2019) 075 [arXiv:1810.11054] [INSPIRE].
S. Datta, P. Kraus and B. Michel, Typicality and thermality in 2d CFT, JHEP 07 (2019) 143 [arXiv:1904.00668] [INSPIRE].
M. Be¸sken, S. Datta and P. Kraus, Quantum thermalization and Virasoro symmetry, J. Stat. Mech. 2006 (2020) 063104 [arXiv:1907.06661] [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].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory. II. Q operator and DDV equation, Commun. Math. Phys. 190 (1997) 247 [hep-th/9604044] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory. III. The Yang-Baxter relation, Commun. Math. Phys. 200 (1999) 297 [hep-th/9805008] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Thermal correlation functions of KdV charges in 2D CFT, JHEP 02 (2019) 044 [arXiv:1810.11053] [INSPIRE].
A. Dymarsky, K. Pavlenko and S. Sugishito, Spectrum of the quantum KdV hierarchy from the quasiclassical quantization, in preparation.
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable quantum field theories in finite volume: excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [INSPIRE].
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Dymarsky, A., Pavlenko, K. & Solovyev, D. Zero modes of local operators in 2d CFT on a cylinder. J. High Energ. Phys. 2020, 172 (2020). https://doi.org/10.1007/JHEP07(2020)172
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DOI: https://doi.org/10.1007/JHEP07(2020)172