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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Some exact results for the two-point function of an integrable quantum field theory

Dennis B. Creamer, H. B. Thacker, David Wilkinson

Published Jun 15, 1981• 53 citations
Publisher

Abstract

The two-point correlation function for the quantum nonlinear Schr\"odinger (one-dimensional $\ensuremath{\delta}$-function gas) model is studied. An infinite-series representation for this function is derived using the quantum inverse-scattering formalism. For the case of zero temperature, the infinitecoupling ($c\ensuremath{\rightarrow}\ensuremath{\infty}$) result of Jimbo, Miwa, M\^ori, and Sato is extended to give an exact expression for the order-$\frac{1}{c}$ correction to the two-point function in terms of a Painlev\'e transcendent of the fifth kind.

DOI
10.1103/physrevd.23.3081

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The two-point correlation function for the quantum nonlinear Schr"odinger (one-dimensional \ensuremathδ\ensuremath{\delta}\ensuremathδ-function gas) model is studied. An infinite-series representation for this function is derived using the quantum inverse-scattering formalism. For the case of zero temperature, the infinitecoupling (c\ensuremath→\ensuremath∞c\ensuremath{\rightarrow}\ensuremath{\infty}c\ensuremath→\ensuremath∞) result of Jimbo, Miwa, M^ori, and Sato is extended to give an exact expression for the order-1c\frac{1}{c}c1​ correction to the two-point function in terms of a Painlev'e transcendent of the fifth kind.
The quantum nonlinear Schrödinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painlève transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrödinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.
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.
Introduction In the previous chapters correlation functions were represented as determinants of a special form of Fredholm integral operator. First, a differential equation was obtained for the time-independent zero temperature field correlation function for the impenetrable Bose gas (it depends only on one variable—distance). By means of an isomonodromic deformation technique A. Jimbo, T. Miwa, Y. Mori and M. Sato wrote down the ordinary differential equation (of Painlevè type). In the previous chapter differential equations for correlation functions in more general situations (with time, temperature and for finite coupling) were constructed. A.R. Its came up with the idea of applying the matrix Riemann problem for the description of these correlation functions (Fredholm determinants). This permits us to write down partial differential equations (completely integrable) for correlation functions and evaluate the asymptotics of correlation functions. Below we follow this line. Integrable nonlinear partial differential equations for correlation functions of the one-dimensional nonrelativistic Bose gas with point-like repulsion between gas particles were given in section XIV. So quantum correlation functions can be expressed in terms of the solutions of classical nonlinear integrable evolution systems. These classical systems have been investigated already by means of the classical inverse scattering method. Our approach allows one, in particular, to solve the particularly interesting problem of calculating the asymptotics of correlators at large time and distance; this is done in the next section.
The authors analyse the nonlinear Schrodinger model (the one-dimensional Bose gas) by means of conformal field theory in conjunction with a newly developed cutoff procedure. This makes it possible to obtain exact expressions for current correlators in the model. A number of new results are presented, including the expression for the two-point current correlator on a finite-size strip and the general n-point current correlator on the plane. All these results have the full time dependence and are presented to order 1/g, where g is the strength of the repulsive delta -function interaction.
A quantum field-theoretical model which describes a spatially nonhomogeneous one-dimensional nonrelativistic repulsive Bose gas in an external harmonic potential is considered. The two-point thermal correlation function of the Bose gas is calculated in the framework of the functional integration approach. The calculations are done in the coordinate representation. The method of successive integration first over the “high-energy” functional variables and then over the “low-energy” ones is used. The effective action functional for the low-energy variables is calculated in one-loop approximation. The functional integral representation for the correlation function is obtained in terms of the low-energy variables and is estimated with the help of stationary phase approximation. The asymptotics of the correlation function is studied in the limit when the temperature tends to zero while the volume occupied by the nonhomogeneous Bose gas increases infinitely. It is demonstrated that the behavior of the thermal correlation function in this limit is power-like and is governed by the critical exponent that depends on the spatial and thermal arguments.