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arXiv:cond-mat/0505367

Finite temperature correlation functions in integrable models: derivation of the semiclassical limit from the formfactor expansion

B. L. Altshuler, Alexei M. Tsvelik

Published May 16, 2005• 1 citation
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Abstract

We propose an approach to the problem of finite temperature dynamical correlation functions in integrable one-dimensional models with a spectral gap. The approach is based on the analysis of the singularities of the operator matrix elements and is not model specific. For the long time, large distance asymptotics of the correlation functions we obtain a formula which, as a particular case, contains the expression for the dynamical susceptibility of Quantum Ising model suggested by Sachdev and Young, Phys. Rev. Lett. {\bf 78}, 2220 (1997).

DOI
10.48550/arxiv.cond-mat/0505367

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We propose an approach to the problem of finite temperature dynamical correlation functions in integrable one-dimensional models with a spectral gap. The approach is based on the analysis of the singularities of the operator matrix elements and is not model specific. For the long time, large distance asymptotics of the correlation functions we obtain a formula which, as a particular case, contains the expression for the dynamical susceptibility of Quantum Ising model suggested by Sachdev and Young, Phys. Rev. Lett. {\bf 78}, 2220 (1997).
In this paper, we demonstrate a recently developed technique which addresses the problem of obtaining nonuniversal prefactors of the correlation functions of one-dimensional (1D) systems at zero temperature. Our approach combines the effective field theory description of generic 1D quantum liquids with the finite-size scaling of form factors (matrix elements) which are obtained using microscopic techniques developed in the context of integrable models. We thus establish exact analytic forms for the prefactors of the long-distance behavior of equal-time correlation functions as well as prefactors of singularities of dynamic response functions. In this paper, our focus is on three specific integrable models: the Calogero-Sutherland, Lieb-Liniger, and XXZXXZXXZ models.
We consider the finite-temperature frequency and momentum dependent two-point functions of local operators in integrable quantum field theories. We focus on the case where the zero temperature correlation function is dominated by a delta-function line arising from the coherent propagation of single particle modes. Our specific examples are the two-point function of spin fields in the disordered phase of the quantum Ising and the O(3) nonlinear sigma models. We employ a Lehmann representation in terms of the known exact zero-temperature form factors to carry out a low-temperature expansion of two-point functions. We present two different but equivalent methods of regularizing the divergences present in the Lehmann expansion: one directly regulates the integral expressions of the squares of matrix elements in the infinite volume whereas the other operates through subtracting divergences in a large, finite volume. Our central results are that the temperature broadening of the line shape exhibits a pronounced asymmetry and a shift of the maximum upwards in energy ("temperature dependent gap"). The field theory results presented here describe the scaling limits of the dynamical structure factor in the quantum Ising and integer spin Heisenberg chains. We discuss the relevance of our results for the analysis of inelastic neutron scattering experiments on gapped spin chain systems such as CsNiCl3 and YBaNiO5.
We consider the finite-temperature frequency and momentum dependent two-point functions of local operators in integrable quantum field theories. We focus on the case where the zero temperature correlation function is dominated by a delta-function line arising from the coherent propagation of single particle modes. Our specific examples are the two-point function of spin fields in the disordered phase of the quantum Ising and the O(3) nonlinear sigma models. We employ a Lehmann representation in terms of the known exact zero-temperature form factors to carry out a low-temperature expansion of two-point functions. We present two different but equivalent methods of regularizing the divergences present in the Lehmann expansion: one directly regulates the integral expressions of the squares of matrix elements in the infinite volume whereas the other operates through subtracting divergences in a large, finite volume. Our central results are that the temperature broadening of the line shape exhibits a pronounced asymmetry and a shift of the maximum upwards in energy ("temperature dependent gap"). The field theory results presented here describe the scaling limits of the dynamical structure factor in the quantum Ising and integer spin Heisenberg chains. We discuss the relevance of our results for the analysis of inelastic neutron scattering experiments on gapped spin chain systems such as CsNiCl3 and YBaNiO5.
We determine the form factor expansion of the one-point functions in integrable quantum field theory at finite temperature and find that it is simpler than previously conjectured. We show that no singularities are left in the final expression provided that the operator is local with respect to the particles and argue that the divergences arising in the non-local case are related to the absence of spontaneous symmetry breaking on the cylinder. As a specific application, we give the first terms of the low-temperature expansion of the one-point functions for the Ising model in a magnetic field.
We present a detailed study of the finite-temperature dynamical properties of the quantum Potts model in one dimension. Quasiparticle excitations in this model have internal quantum numbers, and their scattering matrix deep in the gapped phases is shown to take a simple exchange form in the perturbative regimes. The finite-temperature correlation functions in the quantum critical regime are determined using conformal invariance, while far from the quantum critical point we compute the decay functions analytically within a semiclassical approach of Sachdev and Damle [Phys. Rev. B 57, 8307 (1998)]. As a consequence, decay functions exhibit a diffusive character. We also provide robust arguments that our semiclassical analysis carries over to very low temperatures even in the vicinity of the quantum phase transition. Our results are also relevant for quantum rotor models, antiferromagnetic chains, and some spin ladder systems.
Abstract We propose a method for calculating dynamical correlation functions at finite temperature in integrable lattice models of Yang–Baxter type. The method is based on an expansion of the correlation functions as a series over matrix elements of a time-dependent quantum transfer matrix rather than the Hamiltonian. In the infinite Trotter-number limit the matrix elements become time independent and turn into the thermal form factors studied previously in the context of static correlation functions. We make this explicit with the example of the XXZ model. We show how the form factors can be summed utilizing certain auxiliary functions solving finite sets of nonlinear integral equations. The case of the XX model is worked out in more detail leading to a novel form-factor series representation of the dynamical transverse two-point function.