Journal of Statistical Physics

, Volume 130, Issue 1, pp 129–168 | Cite as

Form Factors of Branch-Point Twist Fields in Quantum Integrable Models and Entanglement Entropy

Article

Abstract

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 nth 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 nth 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.

Keywords

Integrable quantum field theory Entanglement entropy Form factors Twist fields 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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) CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Smirnov, F.: Form factors in completely integrable models of quantum field theory. Adv. Series in Math. Phys., vol. 14. World Scientific, Singapore (1992) MATHGoogle Scholar
  3. 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) Google Scholar
  4. 4.
    Bennet, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046–2052 (1996) CrossRefADSGoogle Scholar
  5. 5.
    Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature 416, 608–610 (2002) CrossRefADSGoogle Scholar
  6. 6.
    Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A 66, 032110 (2002) CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Barnum, H., Knill, E., Ortiz, G., Somma, R., Viola, L.: A subsystem-independent generalisation of entanglement. Phys. Rev. Lett. 92, 107902 (2004) CrossRefADSGoogle Scholar
  8. 8.
    Verstraete, F., Martin-Delgado, M.A., Cirac, J.I.: Diverging entanglement length in gapped quantum spin systems. Phys. Rev. Lett. 92, 087201 (2004) CrossRefADSGoogle Scholar
  9. 9.
    Audenaert, K., Eisert, J., Plenio, M.B., Werner, R.F.: Entanglement properties of the harmonic chain. Phys. Rev. A 66, 042327 (2002) CrossRefADSGoogle Scholar
  10. 10.
    Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003) CrossRefADSGoogle Scholar
  11. 11.
    Latorre, J.I., Rico, E., Vidal, G.: Ground state entanglement in quantum spin chains. Quant. Inf. Comput. 4, 48–92 (2004) MathSciNetGoogle Scholar
  12. 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) CrossRefADSGoogle Scholar
  13. 13.
    Jin, B.-Q., Korepin, V.: Quantum spin chain, Toeplitz determinants and Fisher–Hartwig conjecture. J. Stat. Phys. 116, 79–95 (2004) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lambert, N., Emary, C., Brandes, T.: Entanglement and the phase transition in single-mode superradiance. Phys. Rev. Lett. 92, 073602 (2004) CrossRefADSGoogle Scholar
  15. 15.
    Casini, H., Huerta, M.: A finite entanglement entropy and the c-theorem. Phys. Lett. B 600, 142–150 (2004) CrossRefADSMathSciNetGoogle Scholar
  16. 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) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Weston, R.A.: The entanglement entropy of solvable lattice models. J. Stat. Mech. 0603, L002 (2006) Google Scholar
  18. 18.
    Calabrese, P., Cardy, J.L.: Entanglement entropy and quantum field theory. J. Stat. Mech. 0406, P002 (2004) MathSciNetGoogle Scholar
  19. 19.
    Calabrese, P., Cardy, J.L.: Evolution of entanglement entropy in one-dimensional systems. J. Stat. Mech. 0504, P010 (2005) Google Scholar
  20. 20.
    Holzhey, C., Larsen, F., Wilczek, F.: Geometric and renormalized entropy in conformal field theory. Nucl. Phys. B 424, 443–467 (1994) MATHCrossRefADSMathSciNetGoogle Scholar
  21. 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) Google Scholar
  22. 22.
    Casini, H., Huerta, M.: Entanglement and alpha entropies for a massive scalar field in two dimensions. J. Stat. Mech. 0512, P012 (2005) Google Scholar
  23. 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) CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Cardy, J.L., Mussardo, G.: Form-factors of descendent operators in perturbed conformal field theories. Nucl. Phys. B 340, 387–402 (1990) CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Arafeva, I., Korepin, V.: Scattering in two-dimensional model with Lagrangian L=1/γ(1/2( μ uu)2+m 2(cos u−1)). Pis’ma Zh. Eksp. Teor. Fiz. 20, 680 (1974) ADSGoogle Scholar
  26. 26.
    Vergeles, S., Gryanik, V.: Two-dimensional quantum field theories having exact solutions. Yad. Fiz. 23, 1324–1334 (1976) Google Scholar
  27. 27.
    Schroer, B., Truong, T., Weisz, P.: Towards an explicit construction of the sine-Gordon theory. Phys. Lett. B 63, 422–424 (1976) CrossRefADSGoogle Scholar
  28. 28.
    Arinshtein, A., Fateev, V., Zamolodchikov, A.: Quantum S-matrix of the (1+1)-dimensional Toda chain. Phys. Lett. B 87, 389–392 (1979) CrossRefADSGoogle Scholar
  29. 29.
    Mikhailov, A., Olshanetsky, M., Perelomov, A.: Two-dimensional generalized Toda lattice. Commun. Math. Phys. 79, 473–488 (1981) CrossRefADSMathSciNetGoogle Scholar
  30. 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) CrossRefADSMathSciNetGoogle Scholar
  31. 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) CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Koubek, A., Mussardo, G.: On the operator content of the sinh-Gordon model. Phys. Lett. B 311, 193–201 (1993) CrossRefADSMathSciNetGoogle Scholar
  33. 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) CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Lukyanov, S.L.: Free field representation for massive integrable models. Commun. Math. Phys. 167, 183–226 (1995) MATHCrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Brazhnikov, V., Lukyanov, S.: Angular quantization and form factors in massive integrable models. Nucl. Phys. B 512, 616–636 (1998) CrossRefADSMathSciNetGoogle Scholar
  36. 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) CrossRefADSMathSciNetGoogle Scholar
  37. 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) MATHCrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Peschel, I.: On the entanglement entropy for a XY spin chain. J. Stat. Mech. P12005 (2004) Google Scholar
  39. 39.
    Chung, M.C., Peschel, I.: On density-matrix spectra for two-dimensional quantum systems. Phys. Rev. B 62, 4191–4193 (2000) CrossRefADSGoogle Scholar
  40. 40.
    Peschel, I.: Calculation of reduced density matrices from correlation functions. J. Phys. A 36, L205–L208 (2003) MATHCrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Casini, H., Huerta, M.: Analytic results on the geometric entropy for free fields. arXiv:0707.1300 (2007) Google Scholar
  42. 42.
    Lukyanov, S.L.: Finite temperature expectation values of local fields in the sinh-Gordon model. Nucl. Phys. B 612, 391–412 (2001) MATHCrossRefADSMathSciNetGoogle Scholar
  43. 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) CrossRefADSMathSciNetGoogle Scholar
  44. 44.
    Zamolodchikov, A.: Unpublished Google Scholar
  45. 45.
    Lukyanov, S., Zamolodchikov, A.: Exact expectation values of local fields in quantum sine-Gordon model. Nucl. Phys. B 493, 571–587 (1997) MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • J. L. Cardy
    • 2
    • 3
  • O. A. Castro-Alvaredo
    • 1
  • B. Doyon
    • 3
  1. 1.Centre for Mathematical ScienceCity University LondonLondonUK
  2. 2.All Souls CollegeOxfordUK
  3. 3.Rudolf Peierls Centre for Theoretical PhysicsOxford UniversityOxfordUK

Personalised recommendations