Spin Correlation Functions of theX−YModel

We consider the one-dimensional $X-Y$ model of Lieb, Schulz, and Mattis and study the asymptotic behavior of each of the three correlation functions $〈{{\sigma }_0}^i{{\sigma }_N}^i〉={{\rho }_N}^i$, where $i=x$, $y$, or $z$. We study in detail the influence of $X-Y$ 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 $N\to \infty$. The behavior near $T=0\mathrm{}$ is studied in the isotropic case by considering the $N\to \infty$ limit with $\mathrm{TN}$ fixed, while in the anisotropic case we must hold $T^{2}N$ fixed as $N\to \infty$. In this manner we obtain the $T=0\mathrm{}$ result that if the interaction is stronger in the $x$ direction, then ${{\rho }_N}^x$ approaches a constant exponentially while ${{\rho }_N}^y$ approaches zero exponentially as $N\to \infty$. We finally show that in the isotropic case at $T=0\mathrm{}$ that ${{\rho }_N}^x={{\rho }_N}^y\to N^{\frac{-1\mathrm{}}{2}}$. In all cases, at least the first two terms of the asymptotic series are explicitly given.