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

Xiaojun Jie, Purdue University

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.

Degree

Ph.D.

Advisors

Sawyer, Purdue University.

Subject Area

Mathematics

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