|
SIGMA 3 (2007), 011, 37 pages hep-th/0611066
https://doi.org/10.3842/SIGMA.2007.011
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
Abstract
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.
pdf (469 kb)
ps (311 kb)
tex (42 kb)
References
- Kapusta J.I., Finite temperature field theory,
Cambridge University Press, Cambridge, 1989.
- Doyon B., Finite-temperature form factors in the free Majorana theory,
J. Stat. Mech. Theory Exp. (2005), P11006, 45 pages,
hep-th/0506105.
- 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.
- Gruner G., Density waves in solids, Addison-Wesley, Reading (MA), 1994.
- 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.
- Vergeles S.N., Gryanik V.M.,
Two-dimensional quantum field theories having exact solutions,
Yad. Fiz. 23 (1976), 1324-1334 (in Russian).
- Weisz P., Exact quantum sine-Gordon soliton form factors,
Phys. Lett. B 67 (1977), 179-182.
- Karowski M., Weisz P., Exact form factors in (1+1)-dimensional
field theoretic models with soliton behaviour, Nuclear
Phys. B 139 (1978), 455-476.
- Berg B., Karowski M., Weisz P., Construction of Green's
functions from an exact S-matrix, Phys. Rev. D 19 (1979), 2477-2479.
- Smirnov F.A., Form factors in completely integrable models of
quantum field theory, World Scientific, Singapore, 1992.
- Zamolodchikov Al.B., Two-point correlation function in scaling
Lee-Yang model, Nuclear Phys. B 348 (1991), 619-641.
- Matsubara T.M., A new approach to quantum-statistical mechanics,
Progr. Theoret. Phys. 14 (1955), 351-378.
- 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.
- Martin C., Schwinger J., Theory of many-particle systems. I,
Phys. Rev. 115 (1959), 1342-1373.
- Smirnov F.A., Quasi-classical study of form factors in finite
volume,
hep-th/9802132.
- Smirnov F.A., Structure of matrix elements in quantum Toda chain,
hep-th/9805011.
- 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.
- Mussardo G., Riva V., Sotkov G., Finite-volume form factors in
semiclassical approximation, Nuclear Phys. B 670 (2003),
464-478,
hep-th/0307125.
- Bugrij A.I., The correlation function in two dimensional Ising
model on the finite size lattice. I,
hep-th/0011104.
- Bugrij A.I., Form factor representation of the correlation
function of the two dimensional Ising model on a cylinder,
hep-th/0107117.
- 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.
- Leplae L., Umezawa H., Mancini F., Derivation and application of
the boson method in superconductivity, Phys. Rep. 10 (1974), 151-272.
- Arimitsu T., Umezawa H., Non-equilibrium thermo field dynamics,
Prog. Theoret. Phys. 77 (1987), 32-52.
- Arimitsu T., Umezawa H.,
General structure of non-equilibrium thermo field dynamics, Progr. Theoret. Phys.
77 (1987), 53-67.
- Henning P.A., Thermo field dynamics for quantum fields with
continuous mass spectrum, Phys. Rep. 253 (1995), 235-381,
nucl-th/9311001.
- Amaral R.L.P.G., Belvedere L.V., Two-dimensional thermofield
bosonization,
hep-th/0504012.
- 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.
- 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.
- 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.
- Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse
scattering method and correlation functions, Cambridge
University Press, Cambridge, 1993.
- Doyon B., Gamsa A., Work in progress.
- Balog J., Field theoretical derivation of the TBA integral
equations, Nuclear Phys. B 419 (1994), 480-512.
- Leclair A., Mussardo G., Finite temperature correlation functions
in integrable QFT, Nuclear Phys. B 552 (1999), 624-642,
hep-th/9902075.
- Lukyanov S., Finite-temperature expectation values of local
fields in the sinh-Gordon model, Nuclear Phys. B 612 (2001), 391-412,
hep-th/0005027.
- Essler F., Konik R., Private communication.
- 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.
- 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.
- 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.
- Fonseca P., Zamolodchikov A.B., Ward identities and integrable
differential equations in the Ising field theory,
hep-th/0309228.
- Kadanoff L.P., Ceva H., Determination of an operator algebra for
the two-dimensional Ising model, Phys. Rev. B 3 (1971),
3918-3939.
- 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.
- Doyon B., Lectures on integrable quantum field theory,
http://www-thphys.physics.ox.ac.uk/user/BenjaminDoyon/lectures.pdf.
- van Hove L., Quantum field theory at positive temperature,
Phys. Rep. 137 (1986), 11-20.
- 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.
- McCoy B.M., Wu T.T., The two-dimensional Ising model, Harvard University Press, Cambridge
(MA), 1973.
- Babelon D., Bernard D., From form factors to correlation
functions: the Ising model, Phys. Lett. B 288 (1992),
113-120.
- 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.
- Itzykson C., Drouffe J.-M., Statistical field theory,
Cambridge University Press, Cambridge, 1989.
- Lisovyy O., Tau functions for the Dirac operator on the cylinder,
Comm. Math. Phys. 255 (2005), 61-95,
hep-th/0312277.
|
|