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arXiv:cond-mat/9205009

Determinant Representation for time and temperature dependent correlation functions in the isotropic XY model in the transverse magnetic field

Filippo Colomo, Anatoli G. Izergin, V. E. Korepin, V. Tognetti

Published May 16, 1992• 1 citation
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Abstract

Astymptotics of temperature correlations is the most dificult problem of stat. mech. Recently it was solved for the impenetrable Bose Gas. The idea is to represent correlation function as $τ$ function of calssical completely integrable differential equation. Now we start to apply this program to isotropic XY model in the trasnsverse magnetic field. In this paper we are making first step. We representing correlation function as detrminanat of an integral operator.

DOI
10.48550/arxiv.cond-mat/9205009

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Cambridge University Press eBooks•Aug 19, 1993•0 citations
Astymptotics of temperature correlations is the most dificult problem of stat. mech. Recently it was solved for the impenetrable Bose Gas. The idea is to represent correlation function as τττ function of calssical completely integrable differential equation. Now we start to apply this program to isotropic XY model in the trasnsverse magnetic field. In this paper we are making first step. We representing correlation function as detrminanat of an integral operator.
We consider isotropic XY model in the transverse magnetic field on the one dimensional lattice. Another name of the model in Heisenberg XXO model of spin 1/2.We solved long standing problem of evaluation of temperature correlations. We first represent correlation function in the model, by means of completely integrable differential equation. This is famous Ablowitz-Ladik lattice Nonlinear Schrodinger equation.Correlation function is the τττ function of this differential equation. We solved this equation and evaluate asymptotics.
We consider the isotropic XY model in a transverse magnetic field in one dimension. One can alternatively call this model the Heisenberg XXO antiferromagnet. We solve the problem of the evaluation of asymptotics of temperature correlations and explain the physical meaning of our result. To do this we represent the quantum correlation function as a tau function of a completely integrable differential equation. This is the well-known Ablowitz-Ladik lattice nonlinear Schr\"odinger differential equation.
Introduction The natural language for the description of correlation functions in exactly solvable models is the language of differential equations. The determinant representation of correlation functions in the previous chapter can be used to obtain these differential equations. An ordinary differential equation (of the Painlevètype) for the equal-time, zero temperature correlation function of the impenetrable Bose gas was obtained in. In that paper (starting from Lenard's formula for the two-point zero temperature equal-time correlator of impenetrable bosons) it was shown that the correlator is described by an ordinary differential equation (reducible to the Painlevè trascendent), and multipoint zero temperature equal-time correlators were also described as solutions of differential equations. The considerations in that paper were based on the spectral isomonodromic deformation method. Our approach here is different. We begin by considering the logarithm of the determinant of the linear integral operator entering the representation for the correlator as a τ-function for some integrable nonlinear evolutionary system. Then the linear integral equations playing the role of the Gel'fand-Levitan-Marchenko equations for this integrable system are constructed. The differential equations for correlators are derived, in fact, from these integral equations (it is worth mentioning that the idea of obtaining the differential equations from the integral ones can be found in. This approach permits one to include time- and temperature-dependent correlators for impenetrable bosons in the same framework. It is remarkable that space- and time-dependent correlation functions are derived by partial differential equations which almost coincide with the original classical nonlinear Schrödinger equation, the only difference being that complex involution is dropped.
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.
Introduction Impenetrable bosons in one space dimension are a special case (at coupling constant c → ∞) of the one-dimensional Bose gas (NS model) considered in detail in Chapter I. The results obtained there for the NS model can be specialized to the case of impenetrable bosons. It is to be said that all the formulæe are considerably simplified in this case. From the point of view of physics this is due to the fact that the δ-function potentials in the N -particle Hamiltonian (I.1.11) are now infinitely strong and the bosons cannot penetrate one another, so that the wave function should be equal to zero if the bosons' coordinates coincide. In this respect impenetrable bosons are rather similar to free fermions (see Appendix I.I). In this chapter correlation functions of impenetrable bosons are considered and their representations as Fredholm determinants of linear integral operators are given. The (very essential) simplification with respect to the general case is due to the fact that now all the auxiliary quantum fields can be set equal to zero (see IX.6), so that the kernels of these operators do not contain quantum fields. The determinant representation for the time-dependent correlator is easy to obtain in this case. Temperature correlation functions are considered as well as zero temperature ones.