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Journal of Physics A Mathematical and General

Asymptotic form of the two-point correlation function of theXXZspin chain

Y Fujii and M Wadati

Published Mar 19, 2001• 0 citations
Publisher

Abstract

In equation (3.23), we introduced the function f ( m , α) satisfying the condition f ( m + 1, α) / f ( m , α) = 1 + O( m -1 ). For the calculation of correlation functions, one needs the derivative of this function with respect to α. There is no guarantee that the derivative f '( m , α) = ∂ f ( m , α)/∂α satisfies f '( m + 1, α) / f '( m , α) = 1 + O( m -1 ). Therefore, equation (3.24) is incorrect. This error is crucial because a main result of this paper is incorrect. However, the asymptotic analysis of the Riemann-Hilbert problem is new and correct.

DOI
10.1088/0305-4470/34/12/501

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In equation (3.23), we introduced the function f ( m , α) satisfying the condition f ( m + 1, α) / f ( m , α) = 1 + O( m -1 ). For the calculation of correlation functions, one needs the derivative of this function with respect to α. There is no guarantee that the derivative f '( m , α) = ∂ f ( m , α)/∂α satisfies f '( m + 1, α) / f '( m , α) = 1 + O( m -1 ). Therefore, equation (3.24) is incorrect. This error is crucial because a main result of this paper is incorrect. However, the asymptotic analysis of the Riemann-Hilbert problem is new and correct.
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.