Entanglement branes, modular flow, and extended topological quantum field theory. (English) Zbl 1427.81154
Summary: Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the “entanglement brane” we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular ow. As a byproduct we find that the negativity distinguishes between the “log dim R” edge term and the “Shannon” edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.
MSC:
| 81T45 | Topological field theories in quantum mechanics |
| 81P40 | Quantum coherence, entanglement, quantum correlations |
| 83E30 | String and superstring theories in gravitational theory |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 83C80 | Analogues of general relativity in lower dimensions |
Keywords:
1/N expansion; field theories in lower dimensions; topological field theories; two-dimensional gravityReferences:
| [1] | H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci. Kyoto1976 (1976) 809 [INSPIRE]. · Zbl 0326.46031 |
| [2] | H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE]. · Zbl 1152.83361 · doi:10.1088/0264-9381/25/20/205021 |
| [3] | E. Witten, APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory, Rev. Mod. Phys.90 (2018) 045003 [arXiv:1803.04993] [INSPIRE]. · doi:10.1103/RevModPhys.90.045003 |
| [4] | P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and theholographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE]. · doi:10.1016/j.physletb.2008.10.032 |
| [5] | W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE]. |
| [6] | H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE]. |
| [7] | W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav.31 (2014) 214003 [arXiv:1406.7304] [INSPIRE]. · Zbl 1304.81121 · doi:10.1088/0264-9381/31/21/214003 |
| [8] | S. Ghosh, R.M. Soni and S.P. Trivedi, On the entanglement entropy for gauge theories, JHEP09 (2015) 069 [arXiv:1501.02593] [INSPIRE]. · Zbl 1388.81438 · doi:10.1007/JHEP09(2015)069 |
| [9] | L.-Y. Hung and Y. Wan, Revisiting entanglement entropy of lattice gauge theories, JHEP04 (2015) 122 [arXiv:1501.04389] [INSPIRE]. · Zbl 1388.81836 · doi:10.1007/JHEP04(2015)122 |
| [10] | R.M. Soni and S.P. Trivedi, Aspects of entanglement entropy for gauge theories, JHEP01 (2016) 136 [arXiv:1510.07455] [INSPIRE]. · Zbl 1388.81088 · doi:10.1007/JHEP01(2016)136 |
| [11] | J. Lin and D. Radičević, Comments on defining entanglement entropy, arXiv:1808.05939 [INSPIRE]. · Zbl 1473.81045 |
| [12] | A. Blommaert, T.G. Mertens, H. Verschelde and V.I. Zakharov, Edge state quantization: vector fields in rindler, JHEP08 (2018) 196 [arXiv:1801.09910] [INSPIRE]. · Zbl 1396.83012 · doi:10.1007/JHEP08(2018)196 |
| [13] | A. Blommaert, T.G. Mertens and H. Verschelde, Edge dynamics from the path integral: Maxwell and Yang-Mills, JHEP11 (2018) 080 [arXiv:1804.07585] [INSPIRE]. · Zbl 1404.81162 · doi:10.1007/JHEP11(2018)080 |
| [14] | W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP09 (2016) 102 [arXiv:1601.04744] [INSPIRE]. · Zbl 1390.83016 · doi:10.1007/JHEP09(2016)102 |
| [15] | M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math.68 (1989) 175 [INSPIRE]. · Zbl 0692.53053 · doi:10.1007/BF02698547 |
| [16] | W. Donnelly and G. Wong, Entanglement branes in a two-dimensional string theory, JHEP09 (2017) 097 [arXiv:1610.01719] [INSPIRE]. · Zbl 1382.81156 · doi:10.1007/JHEP09(2017)097 |
| [17] | J. Yagi, Branes and integrable lattice models, Mod. Phys. Lett.A 32 (2016) 1730003 [arXiv:1610.05584] [INSPIRE]. · Zbl 1356.81009 |
| [18] | A.D. Lauda and H. Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, math/0510664. · Zbl 1158.57038 |
| [19] | A. Gromov and R.A. Santos, Entanglement entropy in 2D non-abelian pure gauge theory, Phys. Lett. B 737 (2014) 60 [arXiv:1403.5035] [INSPIRE]. · Zbl 1317.81025 · doi:10.1016/j.physletb.2014.08.023 |
| [20] | D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [INSPIRE]. · doi:10.1016/0550-3213(92)90370-Q |
| [21] | G. Moore, Lectures on branes, K-theory and RR charges, Clay Mathematical Institute Lectures, U.S.A. (2002). |
| [22] | G.W. Moore and G. Segal, D-branes and k-theory in 2D topological field theory, hep-th/0609042 [INSPIRE]. |
| [23] | J. Baez, This week’s finds in mathematical physics, week 268 (2018). |
| [24] | L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE]. |
| [25] | E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [INSPIRE]. · Zbl 0768.53042 · doi:10.1016/0393-0440(92)90034-X |
| [26] | S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2-D Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. Proc. Suppl.41 (1995) 184 [hep-th/9411210] [INSPIRE]. · Zbl 0991.81585 · doi:10.1016/0920-5632(95)00434-B |
| [27] | I. Runkel and L. Szegedy, Area-dependent quantum field theory with defects, arXiv:1807.08196 [INSPIRE]. · Zbl 1456.81411 |
| [28] | Y. Huang, Computing quantum discord is NP-complete, New J. Phys.16 (2014) 033027 [arXiv:1305.5941]. · Zbl 1451.81107 · doi:10.1088/1367-2630/16/3/033027 |
| [29] | G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE]. |
| [30] | M.B. Plenio, Logarithmic negativity: a full entanglement monotone that is not convex, Phys. Rev. Lett.95 (2005) 090503 [quant-ph/0505071]. |
| [31] | P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum_eld theory, Phys. Rev. Lett.109 (2012) 130502 [arXiv:1206.3092] [INSPIRE]. · doi:10.1103/PhysRevLett.109.130502 |
| [32] | K. Van Acoleyen et al., The entanglement of distillation for gauge theories, Phys. Rev. Lett.117 (2016) 131602 [arXiv:1511.04369] [INSPIRE]. · doi:10.1103/PhysRevLett.117.131602 |
| [33] | D.J. Gross and W. Taylor, Two-dimensional QCD is a string theory, Nucl. Phys.B 400 (1993) 181 [hep-th/9301068] [INSPIRE]. · Zbl 0941.81586 · doi:10.1016/0550-3213(93)90403-C |
| [34] | D.J. Gross and W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys. B 403 (1993) 395 [hep-th/9303046] [INSPIRE]. · Zbl 1030.81518 · doi:10.1016/0550-3213(93)90042-N |
| [35] | J. Polchinski, Combinatorics of boundaries in string theory, Phys. Rev.D 50 (1994) R6041 [hep-th/9407031] [INSPIRE]. |
| [36] | K. Ohmori and Y. Tachikawa, Physics at the entangling surface, J. Stat. Mech.1504 (2015) P04010 [arXiv:1406.4167] [INSPIRE]. · doi:10.1088/1742-5468/2015/04/P04010 |
| [37] | G. Wong, Gluing together modular ows with free fermions, JHEP04 (2019) 045 [arXiv:1805.10651] [INSPIRE]. · Zbl 1415.81039 · doi:10.1007/JHEP04(2019)045 |
| [38] | J. Cano et al., Bulk-edge correspondence in (2 + 1)-dimensional Abelian topological phases, Phys. Rev. B 89 (2014) 115116 [arXiv:1310.5708] [INSPIRE]. · doi:10.1103/PhysRevB.89.115116 |
| [39] | D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys.354 (2017) 865 [arXiv:1607.03901] [INSPIRE]. · Zbl 1377.81040 · doi:10.1007/s00220-017-2904-z |
| [40] | J. Lin, Ryu-Takayanagi area as an entanglement edge term, arXiv:1704.07763 [INSPIRE]. |
| [41] | C. Akers and P. Rath, Holographic Renyi entropy from quantum error correction, JHEP05 (2019) 052 [arXiv:1811.05171] [INSPIRE]. · Zbl 1416.83095 · doi:10.1007/JHEP05(2019)052 |
| [42] | X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, arXiv:1811.05382 [INSPIRE]. · Zbl 1427.83020 |
| [43] | D. Harlow and D. Jafferis, The factorization problem in Jackiw-Teitelboim gravity, arXiv:1804.01081 [INSPIRE]. · Zbl 1435.83127 |
| [44] | A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian theory — A Wilson line perspective, JHEP12 (2018) 022 [arXiv:1806.07765] [INSPIRE]. · Zbl 1405.83040 · doi:10.1007/JHEP12(2018)022 |
| [45] | J. Lin, Entanglement entropy in Jackiw-Teitelboim gravity, arXiv:1807.06575 [INSPIRE]. |
| [46] | A. Blommaert, T.G. Mertens and H. Verschelde, Fine structure of Jackiw-Teitelboim quantum gravity, arXiv:1812.00918 [INSPIRE]. · Zbl 1423.83022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
