Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 007, 36 pages      arXiv:1010.0858      https://doi.org/10.3842/SIGMA.2011.007
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States

Hermann Boos a, b
a) Fachbereich C – Physik, Bergische Universität Wuppertal, 42097 Wuppertal, Germany
b) Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119991 Moscow, Russia

Received October 07, 2010, in final form January 05, 2011; Published online January 13, 2011

Abstract
We generalize the results of [Comm. Math. Phys. 299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the six-vertex model acting in the so-called Matsubara direction. We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particle-hole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand. As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators. In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987). Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations. On the side of the lattice model we use the excited-state TBA approach. We consider in detail the case of the descendant at level 8.

Key words: integrable models; six vertex model; XXZ spin chain; fermionic basis, thermodynamic Bethe ansatz; excited states; conformal field theory; Virasoro algebra.

pdf (595 kb)   tex (41 kb)

References

  1. Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Hidden Grassmann structure in the XXZ model, Comm. Math. Phys. 272 (2007), 263-281, hep-th/0606280.
  2. Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Hidden Grassmann structure in the XXZ model. II. Creation operators, Comm. Math. Phys. 286 (2009), 875-932, arXiv:0801.1176.
  3. Jimbo M., Miwa T., Smirnov F., Hidden Grassmann structure in the XXZ model. III. Introducing Matsubara direction, J. Phys. A: Math. Theor. 42 (2009), 304018, 31 pages.
  4. Boos H., Jimbo M., Miwa T., Smirnov F., Hidden Grassmann structure in the XXZ model. IV. CFT limit, Comm. Math. Phys. 299 (2010), 825-866, arXiv:0911.3731.
  5. Destri C., de Vega H.J., Unified approach of thermodynamic Bethe ansatz and finite size corrections for lattice models and field theories, Nuclear Phys. B 438 (1995), 413-454, hep-th/9407117.
  6. Bazhanov V., Lukyanov S., Zamolodchikov A., Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Comm. Math. Phys. 177 (1996), 381-398, hep-th/9412229.
  7. Bazhanov V., Lukyanov S., Zamolodchikov A., Integrable structure of conformal field theory II. Q-operator and DDV equation, Comm. Math. Phys. 190 (1997), 247-278, hep-th/9604044.
  8. Bazhanov V., Lukyanov S., Zamolodchikov A., Integrable structure of conformal field theory. III. The Yang-Baxter relation, Comm. Math. Phys. 200 (1999), 297-324, hep-th/9805008.
  9. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  10. Zamolodchikov A.B., Higher order integrals of motion in two-dimensional models of the field theory with a broken conformal symmetry, JETP Lett. 46 (1987), 160-164.
  11. Boos H., Jimbo M., Miwa T., Smirnov F., Completeness of a fermionic basis in the homogeneous XXZ model, J. Math. Phys. 50 (2009), 095206, 10 pages, arXiv:0903.0115.
  12. Suzuki M., Transfer-matrix method and Monte Carlo simulation in quantum spin systems, Phys. Rev. B 31 (1985), 2957-2965.
  13. Boos H., Göhmann F., On the physical part of the factorized correlation functions of the XXZ chain, J. Phys. A: Math. Theor. 42 (2009), 315001, 27 pages, arXiv:0903.5043.
  14. Jimbo M., Miwa T., Smirnov F., On one-point functions of descendants in sine-Gordon model, arXiv:0912.0934.
  15. Jimbo M., Miwa T., Smirnov F., Hidden Grassmann structure in the XXZ model. V. Sine-Gordon model, arXiv:1007.0556.
  16. Dotsenko V.S., Fateev V.A., Conformal algebra and multipoint correlator functions in 2D statistical models, Nuclear Phys. B 240 (1984), 312-348.
  17. Gaudin M., Thermodynamics of the Heisenberg-Ising ring for Δ>~1, Phys. Rev. Lett. 26 (1971), 1301-1304.
    Takahashi M., One-dimensional Heisenberg model at finite temperature, Prog. Theor. Phys. 46 (1971), 401-415.
    Takahashi M., Suzuki M., One-dimensional anisotropic Heisenberg model at finite temperatures, Prog. Theor. Phys. 46 (1972), 2187-2209.
    Woynarovich F., Excitation spectrum of the spin-1/2 Heisenberg chain and conformal invariance, Phys. Rev. Lett. 59 (1987), 259-261.
    de Vega H.J., Woynarovich F., Method for calculating finite size corrections in Bethe ansatz systems: Heisenberg chain and six-vertex model, Nuclear Phys. B 251 (1985), 439-456.
  18. Bazhanov V., Lukyanov S., Zamolodchikov A., Higher-level eigenvalues of Q-operators and Schrödinger equation, Adv. Theor. Math. Phys. 7 (2003), 711-725, hep-th/0307108.
  19. Klümper A., Batchelor M., Pearce P.A., Central charges of the 6- and 19-vertex models with twisted boundary conditions, J. Phys. A: Math. Gen. 24 (1991), 3111-3133.
  20. Klümper A., Free energy and correlation length of quantum chains related to restricted solid-on-solid lattice models, Ann. Physik (8) 1 (1992), 540-553.


Previous article   Next article   Contents of Volume 7 (2011)