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arXiv:1011.5897 · Advances in Theoretical and Mathematical Physics

Riemann-Hilbert approach to the time-dependent generalized sine kernel

Karol Kajetan Kozlowski

Published Jan 1, 2011• 25 citations
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Abstract

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.

DOI
10.4310/atmp.2011.v15.n6.a3

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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.
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 perform a large-xxx asymptotic analysis of the Fredholm determinant of an integrable integral operator with generalized sine kernel, where xxx controls the strength of the oscillations along the integration contour of the operator and will play the role of the distance variable in applications to the correlation functions of integrable quantum systems. Our generalized sine kernel involves a number of functional parameters that will allow us to adapt it to the analysis of the dynamical correlation functions of the Lieb--Liniger model at finite temperature and for all positive values of the coupling constant. It will also allow us to consider a class of equilibrium correlation functions that are governed by generalized Gibbs ensembles. Our work is based on the analysis of a matrix Riemann--Hilbert problem that is canonically connected with our integrable integral operator.
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.
Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the zero-temperature, time and distance dependent reduced density matrix in the non-linear Schrödinger model. This representation allows one to \textit{read-off} straightforwardly the long-time/large-distance asymptotic behavior of this correlator. This method of analysis reduces the complexity of the computation of the asymptotic behavior of correlation functions in the so-called interacting integrable models, to the one appearing in free fermion equivalent models. We compute explicitly the first few terms appearing in the asymptotic expansion. Part of these terms stems from excitations lying away from the Fermi boundary, and hence go beyond what can be obtained by using the CFT/Luttinger liquid based predictions.
We present a new method allowing us to derive the long-time and large-distance asymptotic behavior of the correlations functions of quantum integrable models from their exact representations. Starting from the form factor expansion of the correlation functions in finite volume, we explain how to reduce the complexity of the computation in the so-called interacting integrable models to the one appearing in free fermion equivalent models. We apply our method to the time-dependent zero-temperature current-current correlation function in the non-linear Schr\"{o}dinger model and compute the first few terms in its asymptotic expansion. Our result goes beyond the conformal field theory based predictions: in the time-dependent case, other types of excitations than the ones on the Fermi surface contribute to the leading orders of the asymptotics.