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Journal of the Physical Society of Japan

Correlation Functions of the Asymmetric XXZ Chain and the Painlevé V Equation

Yasuhiro Fujii, Miki Wadati

Published Oct 1, 1999• 2 citations
Publisher

Abstract

Correlation functions of the asymmetric XXZ chain are studied in the framework of the algebraic Bethe ansatz. In the thermodynamic limit they are expressed by Fredholm's determinants. Based on the τ-function of the Painlevé V equation those asymptotic expressions are explicitly calculated.
DOI
10.1143/jpsj.68.3227

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The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem.Among these solvable models spin 1/2 XXX chain has been investigated for a long time.Even for this model only the nearest neighbor and the second neighbor correlations were known.In 1990's multiple integral formula for the general correlations is derived.But the integration of this formula is also very difficult problem.Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by log 2 and Riemann's zeta functions with odd integer argument ζ(3), ζ(5), ζ(7), . . . .We can calculate density sub-matrix of successive seven sites.Entanglement entropy of seven sites is calculated.These methods can be extended to XXZ chain up to n = 4. Correlation functions are expressed by the generalized zeta functions.Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain.This is quite different with Yang-Yang type TBA equations and contains only one unknown function.This equation is very useful to get the high temperature expansion.In this paper we get the analytic solution of this equation at ∆ = 0.
We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.
Using the algebraic Bethe ansatz, we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin ½ Heisenberg chain, based on the solution of the quantum inverse problem. We construct a representation for the correlation functions on a finite chain for arbitrary spin. Then we show how the string solutions of the Bethe equations can be considered in the framework of this approach in the thermodynamic limit. Finally, a multiple integral representation for the spin 1 zero-temperature correlation functions is obtained in the thermodynamic limit.