Spin Correlation Functions of theX−YModel

Abstract

We consider the one-dimensional $X\ensuremath{-}Y$ model of Lieb, Schulz, and Mattis and study the asymptotic behavior of each of the three correlation functions $〈{{\ensuremath{\sigma}}_{0}}^{i}{{\ensuremath{\sigma}}_{N}}^{i}〉={{\ensuremath{\rho}}_{N}}^{i}$, where $i=x$, $y$, or $z$. We study in detail the influence of $X\ensuremath{-}Y$ anisotropy by separately studying the correlation functions in both the isotropic and anisotropic cases at both nonzero and zero temperatures. For nonzero temperature we derive both low- and high-temperature expansions for all three correlations and show that these correlations go to zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The behavior near $T=0$ is studied in the isotropic case by considering the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit with $\mathrm{TN}$ fixed, while in the anisotropic case we must hold ${T}^{2}N$ fixed as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. In this manner we obtain the $T=0$ result that if the interaction is stronger in the $x$ direction, then ${{\ensuremath{\rho}}_{N}}^{x}$ approaches a constant exponentially while ${{\ensuremath{\rho}}_{N}}^{y}$ approaches zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We finally show that in the isotropic case at $T=0$ that ${{\ensuremath{\rho}}_{N}}^{x}={{\ensuremath{\rho}}_{N}}^{y}\ensuremath{\rightarrow}{N}^{\frac{\ensuremath{-}1}{2}}$. In all cases, at least the first two terms of the asymptotic series are explicitly given.