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Integral equations for the correlation functions of the quantum one-dimensional Bose gas

N. A. Slavnov

Published Jan 1, 1998• 10 citations
OpenAlex

Abstract

The large time and long distance behavior of the temperature correlation functions of the quantum one-dimensional Bose gas is considered. We obtain integral equations, which solutions describe the asymptotics. These equations are closely related to the thermodynamic Bethe Ansatz equations. In the low temperature limit the solutions of these equations are given in terms of observables of the model. 1

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The large time and long distance behavior of the temperature correlation functions of the quantum one-dimensional Bose gas is considered. We obtain integral equations, which solutions describe the asymptotics. These equations are closely related to the thermodynamic Bethe Ansatz equations. In the low temperature limit the solutions of these equations are given in terms of observables of the model. 1
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We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
We investigate the large-distance asymptotic behavior of the static density-density and field-field correlation functions in the one-dimensional Bose gas at finite temperature. The asymptotic expansions of the Bose gas correlators are obtained performing a specific continuum limit in the similar low-temperature expansions of the longitudinal and transversal correlation functions of the XXZXXZXXZ spin chain. In the lattice system the correlation lengths are computed as ratios of the largest and next-largest eigenvalues of the XXZXXZXXZ spin chain quantum transfer matrix. In both cases, lattice and continuum, the correlation lengths are expressed in terms of solutions of Yang-Yang type [C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1151 (1969)] nonlinear integral equations which are easily implementable numerically.
We consider the quantum nonlinear Schrodinger equation in one space and one time dimension. It is equivalent to Bose gas with delta interaction. We are interested in the non-free-fermionic case. We calculate temperature-dependent correlation functions. The determinant representation for correlation functions simplifies in the small mass limit of the Bose particle. In this limit we describe the correlation function in terms of the Maxwell-Bloch differential equation. We solve this equation at large distance and evaluate asymptotics of correlation functions. 1 kojima@kurims.kyoto-u.ac.jp 2 korepin@yukawa.kyoto-u.ac.jp 2 korepin@insti.physics.sunysb.edu ? Research Fellow of the Japan Society for the Promotion of Science. 1 Introduction In this paper we consider correlation functions of exactly solvable models. Our approach is based on the determinant representation of quantum correlation functions [1]. We consider the thermodynamics of Bose gas with delta-interaction at finite temper...
We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin–spin correlation function of the XXZ Heisenberg spin- 1/2 chain (with magnetic field) in the disordered regime as well as to the density–density correlation function of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.