Abstract
In this paper, we use the replica approach to study the Rényi entropy SL of generic locally excited states in (1+1)D CFTs, which are constructed from the insertion of multiple product of local primary operators on vacuum. Alternatively, one can calculate the Rényi entropy SR corresponding to the same states using Schmidt decomposition and operator product expansion, which reduces the multiple product of local primary operators to linear combination of operators. The equivalence SL = SR translates into an identity in terms of the F symbols and quantum dimensions for rational CFT, and the latter can be proved algebraically. This, along with a series of papers, gives a complete picture of how the quantum information quantities and the intrinsic structure of (1+1)D CFTs are consistently related.
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References
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition, JHEP 09 (2016) 038 [arXiv:1605.08072] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A General Proof of the Quantum Null Energy Condition, arXiv:1706.09432 [INSPIRE].
H. Casini, I.S. Landea and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140 [arXiv:1607.00390] [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].
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].
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].
T. Faulkner and H. Wang, Probing beyond ETH at large c, arXiv:1712.03464 [INSPIRE].
S. He, T. Numasawa, T. Takayanagi and K. Watanabe, Quantum dimension as entanglement entropy in two dimensional conformal field theories, Phys. Rev. D 90 (2014) 041701 [arXiv:1403.0702] [INSPIRE].
W.-Z. Guo and S. He, Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs, JHEP 04 (2015) 099 [arXiv:1501.00757] [INSPIRE].
B. Chen, W.-Z. Guo, S. He and J.-q. Wu, Entanglement Entropy for Descendent Local Operators in 2D CFTs, JHEP 10 (2015) 173 [arXiv:1507.01157] [INSPIRE].
M. Nozaki, T. Numasawa and S. Matsuura, Quantum Entanglement of Fermionic Local Operators, JHEP 02 (2016) 150 [arXiv:1507.04352] [INSPIRE].
M. Nozaki and N. Watamura, Quantum Entanglement of Locally Excited States in Maxwell Theory, JHEP 12 (2016) 069 [arXiv:1606.07076] [INSPIRE].
J.R. David, S. Khetrapal and S.P. Kumar, Universal corrections to entanglement entropy of local quantum quenches, JHEP 08 (2016) 127 [arXiv:1605.05987] [INSPIRE].
T. Zhou, Entanglement Entropy of Local Operators in Quantum Lifshitz Theory, J. Stat. Mech. 1609 (2016) 093106 [arXiv:1607.08631] [INSPIRE].
P. Caputa and M.M. Rams, Quantum dimensions from local operator excitations in the Ising model, J. Phys. A 50 (2017) 055002 [arXiv:1609.02428] [INSPIRE].
P. Caputa and A. Veliz-Osorio, Entanglement constant for conformal families, Phys. Rev. D 92 (2015) 065010 [arXiv:1507.00582] [INSPIRE].
T. Numasawa, Scattering effect on entanglement propagation in RCFTs, JHEP 12 (2016) 061 [arXiv:1610.06181] [INSPIRE].
P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, Evolution of Entanglement Entropy in Orbifold CFTs, J. Phys. A 50 (2017) 244001 [arXiv:1701.03110] [INSPIRE].
A. Jahn and T. Takayanagi, Holographic entanglement entropy of local quenches in AdS 4 /CFT 3 : a finite-element approach, J. Phys. A 51 (2018) 015401 [arXiv:1705.04705] [INSPIRE].
M. Nozaki, T. Numasawa and T. Takayanagi, Holographic Local Quenches and Entanglement Density, JHEP 05 (2013) 080 [arXiv:1302.5703] [INSPIRE].
P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, PTEP 2014 (2014) 093B06 [arXiv:1405.5946] [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].
W.-Z. Guo and F.-L. Lin, Quantum Energy Teleportation in Two-dimensional Conformal Field Theories, arXiv:1708.09544 [INSPIRE].
S. He, Conformal Bootstrap to Rényi Entropy in 2D Liouville and Super-Liouville CFTs, arXiv:1711.00624 [INSPIRE].
Y. Kusuki and T. Takayanagi, Rényi entropy for local quenches in 2D CFT from numerical conformal blocks, JHEP 01 (2018) 115 [arXiv:1711.09913] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].
W.-Z. Guo, Coherent state, local excitation in 2D conformal field theory, arXiv:1510.07142 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333.
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
G.W. Moore and N. Seiberg, Polynomial Equations for Rational Conformal Field Theories, Phys. Lett. B 212 (1988) 451 [INSPIRE].
G.W. Moore and N. Seiberg, Naturality in Conformal Field Theory, Nucl. Phys. B 313 (1989) 16 [INSPIRE].
P. Di Francesco, H. Saleur and J.B. Zuber, Critical Ising Correlation Functions in the Plane and on the Torus, Nucl. Phys. B 290 (1987) 527 [INSPIRE].
P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer, Heidelberg Germany (1998).
V.S. Dotsenko and V.A. Fateev, Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].
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].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators I: partition functions, Nucl. Phys. B 646 (2002) 353 [hep-th/0204148] [INSPIRE].
Z. Wang, CBMS Regional Conference Series in Mathematics. Vol. 112: Topological Quantum Computation, AMS Press, New York U.S.A. (2010).
G. Moore and N. Seiberg, Lectures on RCFT, in Physics, Geometry and Topolog, Springer, Berlin Germany (1990), pg. 263.
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
Y. Gu and X.-L. Qi, Fractional Statistics and the Butterfly Effect, JHEP 08 (2016) 129 [arXiv:1602.06543] [INSPIRE].
S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids, JHEP 05 (2008) 016 [arXiv:0802.3231] [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].
H.-J. Matschull, Black hole creation in (2 + 1)-dimensions, Class. Quant. Grav. 16 (1999) 1069 [gr-qc/9809087] [INSPIRE].
I. Ya. Aref’eva, M.A. Khramtsov and M.D. Tikhanovskaya, Thermalization after holographic bilocal quench, JHEP 09 (2017) 115 [arXiv:1706.07390] [INSPIRE].
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Guo, Wz., He, S. & Luo, ZX. Entanglement entropy in (1+1)D CFTs with multiple local excitations. J. High Energ. Phys. 2018, 154 (2018). https://doi.org/10.1007/JHEP05(2018)154
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DOI: https://doi.org/10.1007/JHEP05(2018)154