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arXiv:cond-mat/0605040 · Physical Review B

Finite-temperature correlation function for the one-dimensional quantum Ising model: The virial expansion

Sebastián A. Reyes, Alexei M. Tsvelik

Published Jun 22, 2006• 17 citations
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Abstract

We rewrite the exact expression for the finite-temperature two-point correlation function for the magnetization as a partition function of some field theory. This removes singularities and provides a convenient form to develop a virial expansion (expansion in powers of the soliton density).
DOI
10.1103/physrevb.73.220405

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We extend the form-factors approach to the quantum Ising model at finite temperature. The two-point function of the energy is obtained in closed form, while the two-point function of the spin is written as a Fredholm determinant. Using the approach of Korepin et al., we obtain, starting directly from the continuum formulation, a set of six differential equations satisfied by this two-point function. Four of these equations involve only space-time derivatives, of which three are equivalent to the equations obtained earlier. In addition, we obtain two new equations involving a temperature derivative. Some of these results are generalized to the Ising model on the half line with a magnetic field at the origin.
The partition function z, the pair correlation function ρ, and the zero-field susceptibility χ for the one-dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high- and low-temperature expansions are given for z, ρ, and χ. It is shown that the low-temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.
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