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One-point functions in integrable coupled minimal models

Pascal Baseilhac

Published Jan 1, 2000• 2 citations
OpenAlex

Abstract

We propose exact vacuum expectation values of local fields for a quantum group restriction of the C (1) 2 affine Toda theory which corresponds to two coupled minimal models. The central charge of the unperturbed models ranges from c = 1 to c = 2, where the perturbed models correspond to two magnetically coupled Ising models and Heisenberg spin ladders, respectively. As an application, in the massive phase we deduce the leading term of the asymptotics of the two-point correlation functions.

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We propose exact vacuum expectation values of local fields for a quantum group restriction of the C (1) 2 affine Toda theory which corresponds to two coupled minimal models. The central charge of the unperturbed models ranges from c = 1 to c = 2, where the perturbed models correspond to two magnetically coupled Ising models and Heisenberg spin ladders, respectively. As an application, in the massive phase we deduce the leading term of the asymptotics of the two-point correlation functions.
Using the exact result for the vacuum expectation values in non-simply laced affine Toda field theory C2(1)C_2^{(1)}C2(1)​, we study short and long distance predictions of the two-point correlation functions in the massive phase of two coupled minimal models. The central charge of the unperturbed models ranges from c=1c=1c=1 to c=2c=2c=2, where the perturbed models correspond to two magnetically coupled Ising models and Heisenberg spin ladders, respectively.
A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed. It is based on the resolution of the quantum inverse scattering problem. The application of this method to theXXZ Heisenberg spin- 12\tfrac{1}{2}21​ in a magnetic field leads to multiple integral representations of its n-point correlation functions. For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra Uq(s^l2){\mathcal{U}_q}(\widehat s{l_2})Uq​(sl2​) together with the corner transfer matrix approach (massive regime). We also discuss further problems, related in particular to long distance asymptotics of these correlation functions.
A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed.It is based on the resolution of the so-called quantum inverse scattering problem, namely, the reconstruction of local operators in terms of the (operator) entries of the quantum monodromy matrix obeying a Yang-Baxter algebra.The application of this method to the XXZ Heisenberg spin-1 2 chain in a magnetic field leads to multiple integral representations of its n-point correlation functions.For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra U q ( ŝl 2 ) together with the corner transfer matrix approach (massive regime).
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.