Abstract
We study the multi-boundary entanglement structure of the states prepared in (1+1) and (2+1) dimensional Chern-Simons theory with finite discrete gauge group G. The states in (1+1)-d are associated with Riemann surfaces of genus g with multiple S1 boundaries and we use replica trick to compute the entanglement entropy for such states. In (2+1)-d, we focus on the states associated with torus link complements which live in the tensor product of Hilbert spaces associated with multiple T2. We present a quantitative analysis of the entanglement structure for both abelian and non-abelian groups. For all the states considered in this work, we find that the entanglement entropy for direct product of groups is the sum of entropy for individual groups, i.e. EE(G1× G2) = EE(G1) + EE(G2). Moreover, the reduced density matrix obtained by tracing out a subset of the total Hilbert space has a positive semidefinite partial transpose on any bi-partition of the remaining Hilbert space.
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References
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] [INSPIRE].
S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids, JHEP05 (2008) 016 [arXiv:0802.3231] [INSPIRE].
G. Salton, B. Swingle and M. Walter, Entanglement from Topology in Chern-Simons Theory, Phys. Rev.D 95 (2017) 105007 [arXiv:1611.01516] [INSPIRE].
V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar, Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants, JHEP04 (2017) 061 [arXiv:1611.05460] [INSPIRE].
S. Dwivedi, V.K. Singh, S. Dhara, P. Ramadevi, Y. Zhou and L.K. Joshi, Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups, JHEP02 (2018) 163 [arXiv:1711.06474] [INSPIRE].
D. Melnikov, A. Mironov, S. Mironov, A. Morozov and A. Morozov, From Topological to Quantum Entanglement, JHEP05 (2019) 116 [arXiv:1809.04574] [INSPIRE].
S. Dwivedi, V.K. Singh, P. Ramadevi, Y. Zhou and S. Dhara, Entanglement on multiple S2boundaries in Chern-Simons theory, JHEP08 (2019) 034 [arXiv:1906.11489] [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys.121 (1989) 351 [INSPIRE].
R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys.129 (1990) 393 [INSPIRE].
D.S. Freed and F. Quinn, Chern-Simons theory with finite gauge group, Commun. Math. Phys.156 (1993) 435 [hep-th/9111004] [INSPIRE].
D.S. Freed, Lectures on Topological Quantum Field Theory, pp. 95–156, Springer Netherlands, Dordrecht (1993) [DOI].
A. Peres, Separability criterion for density matrices, Phys. Rev. Lett.77 (1996) 1413 [quant-ph/9604005] [INSPIRE].
R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, The Operator Algebra of Orbifold Models, Commun. Math. Phys.123 (1989) 485 [INSPIRE].
V. Drinfeld, Quasi-hopf algebras, Leningrad Math. J.1 (1990) 1419.
A. Coste, T. Gannon and P. Ruelle, Finite group modular data, Nucl. Phys.B 581 (2000) 679 [hep-th/0001158] [INSPIRE].
V. Balasubramanian, M. DeCross, J. Fliss, A. Kar, R.G. Leigh and O. Parrikar, Entanglement Entropy and the Colored Jones Polynomial, JHEP05 (2018) 038 [arXiv:1801.01131] [INSPIRE].
S. Stevan, Chern-Simons Invariants of Torus Links, Annales Henri Poincaŕe11 (2010) 1201 [arXiv:1003.2861] [INSPIRE].
A. Brini, B. Eynard and M. Mariño, Torus knots and mirror symmetry, Annales Henri Poincaré13 (2012) 1873 [arXiv:1105.2012] [INSPIRE].
S. Stevan, Knot invariants, Chern-Simons theory and the topological recursion, Ph.D. Thesis, Geneva U. (2014) [INSPIRE].
L.-Y. Hung, Y.-S. Wu and Y. Zhou, Linking Entanglement and Discrete Anomaly, JHEP05 (2018) 008 [arXiv:1801.04538] [INSPIRE].
Y. Zhou, 3d One-form Mixed Anomaly and Entanglement Entropy, JHEP07 (2019) 091 [arXiv:1904.06924] [INSPIRE].
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ArXiv ePrint: 2003.01404
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Dwivedi, S., Addazi, A., Zhou, Y. et al. Multi-boundary entanglement in Chern-Simons theory with finite gauge groups. J. High Energ. Phys. 2020, 158 (2020). https://doi.org/10.1007/JHEP04(2020)158
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DOI: https://doi.org/10.1007/JHEP04(2020)158