arXiv:cond-mat/9606104 · Nuclear Physics B

Finite temperature correlations in the one-dimensional quantum Ising model

André LeClair, Frédéric Lesage, Subir Sachdev A, Hubert Saleur

Published Dec 1, 1996• 82 citations

Abstract

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.

DOI: 10.1016/s0550-3213(96)00456-7

Related papers

Finite temperature correlations in the one-dimensional quantum Ising model

André LeClair, Frédéric Lesage, Subir Sachdev A +1 more

: 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 [1], 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 spacetime derivatives, of which three are equivalent to the equations obtained earlier in [2],[3]. 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. 04/96 1. Introduction The form-factors approach has been a surprisingly efficient tool for computing correlation functions in 1 + 1 quantum integrable theories at zero temperature (T ) (see eg. [4],[5],[6],[7],[8]). Recall that a quantum integrable theory in 1+1 is conveniently ...

Jan 1, 199660 citations

Dynamical correlation functions in the Ising field theory

István Csépányi, Márton Kormos

We study finite temperature dynamical correlation functions of the magnetization operator in the one-dimensional Ising quantum field theory. Our approach is based on a finite temperature form factor series and on a Fredholm determinant representation of the correlators. While for space-like separations the Fredholm determinant can be efficiently evaluated numerically, for the time-like region it has convergence issues inherited from the form factor series. We develop a method to compute the correlation functions at time-like separations based on the analytic continuation of the space-time coordinates to complex values. Using this numerical technique, we explore all space-time and temperature regimes in both the ordered and disordered phases including short, large, and near-light-cone separations at low and high temperatures. We confirm the existing analytic predictions for the asymptotic behavior of the correlations except in the case of space-like correlations in the paramagnetic phase. For this case we derive a new closed form expression for the correlation length that has some unusual properties: it is a non-analytic function of both the space-time direction and the temperature, and its temperature dependence is non-monotonic.

arXiv:2406.05100SciPost PhysicsDec 10, 20240 citations

Continuum derivation of the Ising model two-point function

David Wilkinson

A new derivation is presented of the exact result of Wu, McCoy, Tracy, and Barouch for the scaling-limit spin-spin correlation function in the two-dimensional Ising model. The starting point is a Fredholm determinant representation which may be obtained either in the Pfaffian formalism, or in the algebraic approach using field theory methods. The connection of this work with the quantum sine-Gordon theory is discussed.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMar 15, 19783 citations

ON ISING CORRELATION FUNCTIONS WITH BOUNDARY MAGNETIC FIELD

Robert Konik, André LeClair, Giuseppe Mussardo

Exact expressions of the boundary state and the form factors of the Ising model are used to derive differential equations for the one-point functions of the energy and magnetization operators of the model in the presence of a boundary magnetic field. We also obtain explicit formulas for the massless limit of the one-point and two-point functions of the energy operator.

arXiv:hep-th/9508099International Journal of Modern Physics AJun 20, 199640 citations

Integral equations and long-time asymptotics for finite-temperature Ising chain correlation functions

Benjamin Doyon, Adam Gamsa

This work concerns the dynamical two-point spin correlation functions of the transverse Ising quantum chain at finite (non-zero) temperature, in the universal region near the quantum critical point. They are correlation functions of twist fields in the massive Majorana fermion quantum field theory. At finite temperature, these are known to satisfy a set of integrable partial differential equations, including the sinh–Gordon equation. We apply the classical inverse scattering method to study them, finding that the 'initial scattering data' corresponding to the correlation functions are simply related to the one-particle finite-temperature form factors calculated recently by one of the persent authors. The set of linear integral equations (Gelfand–Levitan–Marchenko equations) associated with the inverse scattering problem then gives, in principle, the two-point functions at all space and time separations, and all temperatures. From these, we evaluate the long-time asymptotic expansion 'near the light cone', in the region where the difference between the space and time separations is of the order of the correlation length.

arXiv:0711.4619Journal of Statistical Mechanics Theory and ExperimentMar 18, 200829 citations

The Ising model in two dimensions: summary of results

Barry M. McCoy

Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.

Dec 1, 20090 citations