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Finite temperature correlations in the one-dimensional quantum Ising model

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

Published Jan 1, 1996• 60 citations
OpenAlex

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 [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 ...

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Finite temperature correlations in the one-dimensional quantum Ising model

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

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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.

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Finite-Temperature Form Factors: a Review

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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.

arXiv:hep-th/0611066•Symmetry Integrability and Geometry Methods and Applications•Jan 11, 2007•42 citations
: 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 ...
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.
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.
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.
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.