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

SIGMA 3 (2007), 011, 37 pages      hep-th/0611066
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Finite-Temperature Form Factors: a Review

Benjamin Doyon
Rudolf Peierls Centre for Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, U.K.

Received October 09, 2006, in final form December 07, 2006; Published online January 11, 2007

We review the concept of finite-temperature form factor that was introduced recently by the author in the context of the Majorana theory. Finite-temperature form factors can be used to obtain spectral decompositions of finite-temperature correlation functions in a way that mimics the form-factor expansion of the zero temperature case. We develop the concept in the general factorised scattering set-up of integrable quantum field theory, list certain expected properties and present the full construction in the case of the massive Majorana theory, including how it can be applied to the calculation of correlation functions in the quantum Ising model. In particular, we include the ''twisted construction'', which was not developed before and which is essential for the application to the quantum Ising model.

Key words: finite temperature; integrable quantum field theory; form factors; Ising model.

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  1. Kapusta J.I., Finite temperature field theory, Cambridge University Press, Cambridge, 1989.
  2. Doyon B., Finite-temperature form factors in the free Majorana theory, J. Stat. Mech. Theory Exp. (2005), P11006, 45 pages, hep-th/0506105.
  3. Bourbonnais C., Jerome D., The normal phase of quasi-one-dimensional organic superconductors, in Advances in Synthetic Metals, Twenty Years of Progress in Science and Technology, Editors P. Bernier, S. Lefrant and E. Bidan, Elsevier, New York, 1999.
  4. Gruner G., Density waves in solids, Addison-Wesley, Reading (MA), 1994.
  5. Essler F.H.L., Konik R.M., Applications of massive integrable quantum field theories to problems in condensed matter physics, in From Fields to Strings: Circumnavigating Theoretical Physics, Editors M. Shifman, A. Vainshtein and J. Wheater, Ian Kogan Memorial Collection, World Scientific, 2004.
  6. Vergeles S.N., Gryanik V.M., Two-dimensional quantum field theories having exact solutions, Yad. Fiz. 23 (1976), 1324-1334 (in Russian).
  7. Weisz P., Exact quantum sine-Gordon soliton form factors, Phys. Lett. B 67 (1977), 179-182.
  8. Karowski M., Weisz P., Exact form factors in (1+1)-dimensional field theoretic models with soliton behaviour, Nuclear Phys. B 139 (1978), 455-476.
  9. Berg B., Karowski M., Weisz P., Construction of Green's functions from an exact S-matrix, Phys. Rev. D 19 (1979), 2477-2479.
  10. Smirnov F.A., Form factors in completely integrable models of quantum field theory, World Scientific, Singapore, 1992.
  11. Zamolodchikov Al.B., Two-point correlation function in scaling Lee-Yang model, Nuclear Phys. B 348 (1991), 619-641.
  12. Matsubara T.M., A new approach to quantum-statistical mechanics, Progr. Theoret. Phys. 14 (1955), 351-378.
  13. Kubo R., Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12 (1957), 570-586.
  14. Martin C., Schwinger J., Theory of many-particle systems. I, Phys. Rev. 115 (1959), 1342-1373.
  15. Smirnov F.A., Quasi-classical study of form factors in finite volume, hep-th/9802132.
  16. Smirnov F.A., Structure of matrix elements in quantum Toda chain, hep-th/9805011.
  17. van Elburg R.A.J., Schoutens K., Form factors for quasi-particles in c = 1 conformal field theory, J. Phys. A: Math. Gen. 33 (2000), 7987-8012, cond-mat/0007226.
  18. Mussardo G., Riva V., Sotkov G., Finite-volume form factors in semiclassical approximation, Nuclear Phys. B 670 (2003), 464-478, hep-th/0307125.
  19. Bugrij A.I., The correlation function in two dimensional Ising model on the finite size lattice. I, hep-th/0011104.
  20. Bugrij A.I., Form factor representation of the correlation function of the two dimensional Ising model on a cylinder, hep-th/0107117.
  21. Fonseca P., Zamolodchikov A.B., Ising field theory in a magnetic field: analytic properties of the free energy, J. Statist. Phys. 110 (2003), 527-590, hep-th/0112167.
  22. Leplae L., Umezawa H., Mancini F., Derivation and application of the boson method in superconductivity, Phys. Rep. 10 (1974), 151-272.
  23. Arimitsu T., Umezawa H., Non-equilibrium thermo field dynamics, Prog. Theoret. Phys. 77 (1987), 32-52.
  24. Arimitsu T., Umezawa H., General structure of non-equilibrium thermo field dynamics, Progr. Theoret. Phys. 77 (1987), 53-67.
  25. Henning P.A., Thermo field dynamics for quantum fields with continuous mass spectrum, Phys. Rep. 253 (1995), 235-381, nucl-th/9311001.
  26. Amaral R.L.P.G., Belvedere L.V., Two-dimensional thermofield bosonization, hep-th/0504012.
  27. Altshuler B.L., Konik R., Tsvelik A.M., Low temperature correlation functions in integrable models: derivation of the large distance and time asymptotics from the form factor expansion, Nuclear Phys. B 739 (2006), 311-327, cond-mat/0508618.
  28. Sachdev S., The universal, finite temperature, crossover functions of the quantum transition in the Ising chain in a transverse field, Nuclear Phys. B 464 (1996), 576-595, cond-mat/9509147.
  29. Sachdev S., Young A.P., Low temperature relaxational dynamics of the Ising chain in a transverse field, Phys. Rev. Lett. 78 (1997), 2220-2223, cond-mat/9609185.
  30. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge, 1993.
  31. Doyon B., Gamsa A., Work in progress.
  32. Balog J., Field theoretical derivation of the TBA integral equations, Nuclear Phys. B 419 (1994), 480-512.
  33. Leclair A., Mussardo G., Finite temperature correlation functions in integrable QFT, Nuclear Phys. B 552 (1999), 624-642, hep-th/9902075.
  34. Lukyanov S., Finite-temperature expectation values of local fields in the sinh-Gordon model, Nuclear Phys. B 612 (2001), 391-412, hep-th/0005027.
  35. Essler F., Konik R., Private communication.
  36. Wu T.T., McCoy B.M., Tracy C.A., Barouch E., Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13 (1976), 316-374.
  37. Perk J.H.H., Equations of motion for the transverse correlations of the one-dimensional XY-model at finite temperature, Phys. Lett. A 79 (1980), 1-2.
  38. Lisovyy O., Nonlinear differential equations for the correlation functions of the 2D Ising model on the cylinder, Adv. Theor. Math. Phys. 5 (2002), 909-922.
  39. Fonseca P., Zamolodchikov A.B., Ward identities and integrable differential equations in the Ising field theory, hep-th/0309228.
  40. Kadanoff L.P., Ceva H., Determination of an operator algebra for the two-dimensional Ising model, Phys. Rev. B 3 (1971), 3918-3939.
  41. Schroer B., Truong T.T., The order/disorder quantum field operators associated with the two-dimensional Ising model in the continuum limit, Nuclear Phys. B 144 (1978), 80-122.
  42. Doyon B., Lectures on integrable quantum field theory,
  43. van Hove L., Quantum field theory at positive temperature, Phys. Rep. 137 (1986), 11-20.
  44. Doyon B., Two-point functions of scaling fields in the Dirac theory on the Poincaré disk, Nuclear Phys. B 675 (2003), 607-630, hep-th/0304190.
  45. McCoy B.M., Wu T.T., The two-dimensional Ising model, Harvard University Press, Cambridge (MA), 1973.
  46. Babelon D., Bernard D., From form factors to correlation functions: the Ising model, Phys. Lett. B 288 (1992), 113-120.
  47. Leclair A., Lesage F., Sachdev S., Saleur H., Finite temperature correlations in the one-dimensional quantum Ising model, Nuclear Phys. B 482 (1996), 579-612, cond-mat/9606104.
  48. Itzykson C., Drouffe J.-M., Statistical field theory, Cambridge University Press, Cambridge, 1989.
  49. Lisovyy O., Tau functions for the Dirac operator on the cylinder, Comm. Math. Phys. 255 (2005), 61-95, hep-th/0312277.

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