Form factors of branch-point twist fields in quantum integrable models and entanglement entropy. (English) Zbl 1134.81043
Summary: In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the “replica trick” whereby the entropy is obtained as the derivative with respect to \(n\) of the trace of the \(n\) th power of the reduced density matrix of the sub-system, evaluated at \(n =1\). The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an \(n\)-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the \(n\) th power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81P15 | Quantum measurement theory, state operations, state preparations |
| 82B10 | Quantum equilibrium statistical mechanics (general) |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
References:
| [1] | Karowski, M., Weisz, P.: Exact S matrices and form-factors in (1+1)-dimensional field theoretic models with soliton behavior. Nucl. Phys. B 139, 455–476 (1978) · doi:10.1016/0550-3213(78)90362-0 |
| [2] | Smirnov, F.: Form factors in completely integrable models of quantum field theory. Adv. Series in Math. Phys., vol. 14. World Scientific, Singapore (1992) · Zbl 0788.46077 |
| [3] | Essler, F.H.L., Konik, R.M.: Applications of massive integrable quantum field theories to problems in condensed matter physics. In: Shifman, M., Vainshtein, A., Wheater, J. (eds.) From Fields to Strings: Circumnavigating Theoretical Physics. World Scientific, Singapore (2004) |
| [4] | Bennet, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046–2052 (1996) · doi:10.1103/PhysRevA.53.2046 |
| [5] | Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature 416, 608–610 (2002) · doi:10.1038/416608a |
| [6] | Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A 66, 032110 (2002) · doi:10.1103/PhysRevA.66.032110 |
| [7] | Barnum, H., Knill, E., Ortiz, G., Somma, R., Viola, L.: A subsystem-independent generalisation of entanglement. Phys. Rev. Lett. 92, 107902 (2004) · Zbl 1171.81315 · doi:10.1103/PhysRevLett.92.107902 |
| [8] | Verstraete, F., Martin-Delgado, M.A., Cirac, J.I.: Diverging entanglement length in gapped quantum spin systems. Phys. Rev. Lett. 92, 087201 (2004) · doi:10.1103/PhysRevLett.92.087201 |
| [9] | Audenaert, K., Eisert, J., Plenio, M.B., Werner, R.F.: Entanglement properties of the harmonic chain. Phys. Rev. A 66, 042327 (2002) · doi:10.1103/PhysRevA.66.042327 |
| [10] | Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003) · doi:10.1103/PhysRevLett.90.227902 |
| [11] | Latorre, J.I., Rico, E., Vidal, G.: Ground state entanglement in quantum spin chains. Quant. Inf. Comput. 4, 48–92 (2004) · Zbl 1175.82017 |
| [12] | Latorre, J.I., Lutken, C.A., Rico, E., Vidal, G.: Fine-grained entanglement loss along renormalisation group flows. Phys. Rev. A 71, 034301 (2005) · doi:10.1103/PhysRevA.71.034301 |
| [13] | Jin, B.-Q., Korepin, V.: Quantum spin chain, Toeplitz determinants and Fisher–Hartwig conjecture. J. Stat. Phys. 116, 79–95 (2004) · Zbl 1142.82314 · doi:10.1023/B:JOSS.0000037230.37166.42 |
| [14] | Lambert, N., Emary, C., Brandes, T.: Entanglement and the phase transition in single-mode superradiance. Phys. Rev. Lett. 92, 073602 (2004) · doi:10.1103/PhysRevLett.92.073602 |
| [15] | Casini, H., Huerta, M.: A finite entanglement entropy and the c-theorem. Phys. Lett. B 600, 142–150 (2004) · Zbl 1247.81021 · doi:10.1016/j.physletb.2004.08.072 |
| [16] | Keating, J.P., Mezzadri, F.: Entanglement in quantum spin chains, symmetry classes of random matrices, and conformal field theory. Phys. Rev. Lett. 94, 050501 (2005) · Zbl 1124.82009 · doi:10.1103/PhysRevLett.94.050501 |
| [17] | Weston, R.A.: The entanglement entropy of solvable lattice models. J. Stat. Mech. 0603, L002 (2006) |
| [18] | Calabrese, P., Cardy, J.L.: Entanglement entropy and quantum field theory. J. Stat. Mech. 0406, P002 (2004) · Zbl 1082.82002 |
| [19] | Calabrese, P., Cardy, J.L.: Evolution of entanglement entropy in one-dimensional systems. J. Stat. Mech. 0504, P010 (2005) |
| [20] | Holzhey, C., Larsen, F., Wilczek, F.: Geometric and renormalized entropy in conformal field theory. Nucl. Phys. B 424, 443–467 (1994) · Zbl 0990.81564 · doi:10.1016/0550-3213(94)90402-2 |
| [21] | Casini, H., Fosco, C.D., Huerta, M.: Entanglement and alpha entropies for a massive Dirac field in two dimensions. J. Stat. Mech. 0507, P007 (2005) |
| [22] | Casini, H., Huerta, M.: Entanglement and alpha entropies for a massive scalar field in two dimensions. J. Stat. Mech. 0512, P012 (2005) |
| [23] | Yurov, V.P., Zamolodchikov, A.B.: Correlation functions of integrable 2-D models of relativistic field theory. Ising model. Int. J. Mod. Phys. A 6, 3419–3440 (1991) · doi:10.1142/S0217751X91001660 |
| [24] | Cardy, J.L., Mussardo, G.: Form-factors of descendent operators in perturbed conformal field theories. Nucl. Phys. B 340, 387–402 (1990) · doi:10.1016/0550-3213(90)90452-J |
| [25] | Arafeva, I., Korepin, V.: Scattering in two-dimensional model with Lagrangian L=1/{\(\gamma\)}(1/2({\(\mu\)} uu)2+m 2(cosu)). Pis’ma Zh. Eksp. Teor. Fiz. 20, 680 (1974) |
| [26] | Vergeles, S., Gryanik, V.: Two-dimensional quantum field theories having exact solutions. Yad. Fiz. 23, 1324–1334 (1976) |
| [27] | Schroer, B., Truong, T., Weisz, P.: Towards an explicit construction of the sine-Gordon theory. Phys. Lett. B 63, 422–424 (1976) · doi:10.1016/0370-2693(76)90386-5 |
| [28] | Arinshtein, A., Fateev, V., Zamolodchikov, A.: Quantum S-matrix of the (1+1)-dimensional Toda chain. Phys. Lett. B 87, 389–392 (1979) · doi:10.1016/0370-2693(79)90561-6 |
| [29] | Mikhailov, A., Olshanetsky, M., Perelomov, A.: Two-dimensional generalized Toda lattice. Commun. Math. Phys. 79, 473–488 (1981) · Zbl 0491.35061 · doi:10.1007/BF01209308 |
| [30] | Zamolodchikov, A., Zamolodchikov, A.: Factorized S-matrices in two-dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120, 253–291 (1979) · doi:10.1016/0003-4916(79)90391-9 |
| [31] | Fring, A., Mussardo, G., Simonetti, P.: Form-factors for integrable Lagrangian field theories, the sinh-Gordon theory. Nucl. Phys. B 393, 413–441 (1993) · Zbl 1245.81238 · doi:10.1016/0550-3213(93)90252-K |
| [32] | Koubek, A., Mussardo, G.: On the operator content of the sinh-Gordon model. Phys. Lett. B 311, 193–201 (1993) · doi:10.1016/0370-2693(93)90554-U |
| [33] | Delfino, G., Niccoli, G.: The composite operator \(T\bar{T}\) in sinh-Gordon and a series of massive minimal models. J. High Energy Phys. 05, 035 (2006) · doi:10.1088/1126-6708/2006/05/035 |
| [34] | Lukyanov, S.L.: Free field representation for massive integrable models. Commun. Math. Phys. 167, 183–226 (1995) · Zbl 0818.46079 · doi:10.1007/BF02099357 |
| [35] | Brazhnikov, V., Lukyanov, S.: Angular quantization and form factors in massive integrable models. Nucl. Phys. B 512, 616–636 (1998) · Zbl 0947.81031 · doi:10.1016/S0550-3213(97)00713-X |
| [36] | Delfino, G., Simonetti, P., Cardy, J.L.: Asymptotic factorisation of form factors in two-dimensional quantum field theory. Phys. Lett. B 387, 327–333 (1996) · doi:10.1016/0370-2693(96)01035-0 |
| [37] | Rubel, L.A.: Necessary and sufficient conditions for Carlson’s theorem on entire functions. Proc. Natl. Acad. Sci. USA 41(8), 601–603 (1955) · Zbl 0064.31702 · doi:10.1073/pnas.41.8.601 |
| [38] | Peschel, I.: On the entanglement entropy for a XY spin chain. J. Stat. Mech. P12005 (2004) · Zbl 1073.82523 |
| [39] | Chung, M.C., Peschel, I.: On density-matrix spectra for two-dimensional quantum systems. Phys. Rev. B 62, 4191–4193 (2000) · doi:10.1103/PhysRevB.62.4191 |
| [40] | Peschel, I.: Calculation of reduced density matrices from correlation functions. J. Phys. A 36, L205–L208 (2003) · Zbl 1049.82011 · doi:10.1088/0305-4470/36/14/101 |
| [41] | Casini, H., Huerta, M.: Analytic results on the geometric entropy for free fields. arXiv:0707.1300 (2007) · Zbl 1116.81008 |
| [42] | Lukyanov, S.L.: Finite temperature expectation values of local fields in the sinh-Gordon model. Nucl. Phys. B 612, 391–412 (2001) · Zbl 0970.81041 · doi:10.1016/S0550-3213(01)00365-0 |
| [43] | Schroer, B., Truong, T.: The order/disorder quantum field operators associated with the two-dimensional Ising model in the continuum limit. Nucl. Phys. B 144, 80–122 (1978) · doi:10.1016/0550-3213(78)90499-6 |
| [44] | Zamolodchikov, A.: Unpublished |
| [45] | Lukyanov, S., Zamolodchikov, A.: Exact expectation values of local fields in quantum sine-Gordon model. Nucl. Phys. B 493, 571–587 (1997) · Zbl 0909.58064 · doi:10.1016/S0550-3213(97)00123-5 |
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.
