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The large time asymptotics of the temperature correlation functions of the XXO Heisenberg ferromagnetic: The Reimann-Hilbert approach

Xiaojun Jie

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

The finite temperature correlation functions of the time-dependent local spin operators of the Heisenberg XXO ferromagnet are considered. The Fredholm determinant representation for these correlation functions were obtained in 1992 by Colomo, Izergin, Korepin, and Tognetti. The problem addressed in this dissertation consists of the asymptotic analysis of the XXO correlation functions based on the determinant representations mentioned. The integral kernel involved in the Colomo-Izergin-Korepin-Tognetti determinant representations belongs to the class of “integrable kernels” which usually appear in the theory of exactly solvable statistical mechanics models, and also in the theory of random matrices. This makes it possible to analyze the large time behavior of the correlation functions in the framework of the Riemann-Hilbert approach. In the dissertation, the associated matrix Riemann-Hilbert problem is solved asymptotically via properly modified Deift-Zhou nonlinear steepest descent method. This in turn allows us to determine the structure of the long-time asymptotic series for the correlation functions and to evaluate explicitly the exponential, power-like pre-exponential, and the oscillatory factors of the asymptotics. A recursive procedure for evaluation of the corrections of any order, which is based on the relevant zero-curvature equation, is suggested as well.

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The finite temperature correlation functions of the time-dependent local spin operators of the Heisenberg XXO ferromagnet are considered. The Fredholm determinant representation for these correlation functions were obtained in 1992 by Colomo, Izergin, Korepin, and Tognetti. The problem addressed in this dissertation consists of the asymptotic analysis of the XXO correlation functions based on the determinant representations mentioned. The integral kernel involved in the Colomo-Izergin-Korepin-Tognetti determinant representations belongs to the class of “integrable kernels” which usually appear in the theory of exactly solvable statistical mechanics models, and also in the theory of random matrices. This makes it possible to analyze the large time behavior of the correlation functions in the framework of the Riemann-Hilbert approach. In the dissertation, the associated matrix Riemann-Hilbert problem is solved asymptotically via properly modified Deift-Zhou nonlinear steepest descent method. This in turn allows us to determine the structure of the long-time asymptotic series for the correlation functions and to evaluate explicitly the exponential, power-like pre-exponential, and the oscillatory factors of the asymptotics. A recursive procedure for evaluation of the corrections of any order, which is based on the relevant zero-curvature equation, is suggested as well.
We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance-dependent correlation functions of integrable models described by a six-vertex R-matrix.This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point.Our method builds on a Riemann-Hilbert based analysis.
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.
Using the determinant representation for the field-field correlation functions of impenetrable anyons at finite temperature obtained in a previous paper, we derive a system of nonlinear partial differential equations completely characterizing the correlators. The system is the same as the one for impenetrable bosons but with different initial conditions. The large-distance asymptotic behavior of the correlation functions is obtained from the analysis of the Riemann-Hilbert problem associated with the system of differential equations. We calculate both the exponential and pre-exponential factors in the asymptotics of the field-field correlators. The asymptotics derived in this way agree with those of the free fermions and impenetrable bosons in the appropriate limits, κ→1κ\to 1κ→1 and κ→0κ\to 0κ→0, of the statistics parameter κκκ, and coincide with the predictions of the conformal field theory at low temperatures.
This PhD thesis is devoted to the study of the long-time and large-distance asymptotic behaviour of the dynamical field--field correlation function of the Lieb--Liniger model in the limit of infinite repulsion (the one-dimensional impenetrable Bose gas). Extending previous works we consider a large class of non-thermal equilibrium conditions in addition to the case of thermal equilibrium that had been studied before. Our starting point is an exact representation of the field--field correlation function as the Fredholm determinant of an integrable integral operators. The integrable operator depends parametrically on time t, distance x and on the filling fraction, which is a function that characterizes the thermal or non-thermal equilibrium conditions. Our work is based on the rigorous asymptotic analysis, using Riemann--Hilbert techniques, of a more general integral operator that depends on two more functional parameters. The long-time and large-distance behaviour of the field--field correlation function is derived for two classes of filling fractions. These classes are characterized by the number of poles on the real axis (a generalization of Fermi points) that, together with the unique saddle point, contribute to the asymptotic expansion. For each class, we derive the long-time and large-distance asymptotic behaviour as a series in x^(-1/2) as x and t go to infinity for a fixed ratio x / t. We provide explicit closed-form expressions for the leading and sub-leading terms, logarithmic corrections, and overall constants in terms of special functions and simple integrals. For the impenetrable Bose gas in thermal equilibrium, we verify the derived asymptotic expansions by comparing them with the existing results in the literature and with numerical data. Our analysis is based solely on the analytic properties of the functions parametrizing the kernel of the integrable integral operator. This generalized approach will enable future applications to the long-time and large-distance asymptotic analysis of the Lieb--Liniger model with finite coupling constant.
We study finite temperature dynamical correlation functions of the magnetization operator in the one-dimensional Ising quantum field theory. Our approach is based on a finite temperature form factor series and on a Fredholm determinant representation of the correlators. While for space-like separations the Fredholm determinant can be efficiently evaluated numerically, for the time-like region it has convergence issues inherited from the form factor series. We develop a method to compute the correlation functions at time-like separations based on the analytic continuation of the space-time coordinates to complex values. Using this numerical technique, we explore all space-time and temperature regimes in both the ordered and disordered phases including short, large, and near-light-cone separations at low and high temperatures. We confirm the existing analytic predictions for the asymptotic behavior of the correlations except in the case of space-like correlations in the paramagnetic phase. For this case we derive a new closed form expression for the correlation length that has some unusual properties: it is a non-analytic function of both the space-time direction and the temperature, and its temperature dependence is non-monotonic.