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Physical Review A

Statistical Mechanics of the XY Model. I

Eytan Barouch, Barry M. McCoy, Max Dresden

Published Sep 1, 1970• 556 citations
Publisher

Abstract

The Liouville equation for the $\mathrm{XY}$ model is solved exactly, and the magnetization is computed explicitly. Nonergodic behavior of the magnetization is found for a general class of time-dependent magnetic fields.

DOI
10.1103/physreva.2.1075

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The Liouville equation for the XY\mathrm{XY}XY model is solved exactly, and the magnetization is computed explicitly. Nonergodic behavior of the magnetization is found for a general class of time-dependent magnetic fields.
An exact solution of the Liouville equation for the XY\mathrm{XY}XY model with time-dependent external magnetic fields is obtained. Standard transformations allow a decomposition of the space of states into a direct product of four-dimensional subspaces. The Liouville equation is reduced to a set of coupled differential equations in each subspace. These equations have been explicitly solved for several cases.
The spin-correlation functions of the one-dimensional XYXYXY model are studied in the presence of a constant magnetic field. We find that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.
We present an exact solution of a one-dimensional XY\mathrm{XY}XY model in a random magnetic field in the limit of strong-field pinning. The structure factor exhibits Lorentzian-squared behavior at nonzero temperatures. The scaling behavior of the correlation length as a function of randomness is different from that obtained in the weak-field pinning regime. We also study the dynamical behavior of the system using a simple relaxational model. The magnetization decays anomalously with time as ${t}^{\frac{7}{6}}{e}^{{\ensuremath{-}ct}^{\frac{1}{3}}}$.