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Physical Review B

Ferromagnetic ordering in the two-dimensional dipolar XY model

P. G. Maier, Franz Schwabl

Published Oct 29, 2004• 30 citations
Publisher

Abstract

We investigate the low-temperature phase of the two-dimensional dipolar $XY$ model. Within the framework of the nonlinear $\ensuremath{\sigma}$ model we show that the dipolar $XY$ model is renormalizable, despite the anisotropic character of the dipolar interaction. We predict a low-temperature phase with spontaneous magnetization governed by a low-temperature fixed point with infinite dipolar coupling. The critical fixed point shows nonlinear behavior, leading to unusual critical phenomena; e.g., the correlation length and spontaneous magnetization show exponential behavior instead of power laws. The nature and flow diagram of the ferromagnetic transition are strikingly similar to the Kosterlitz-Thouless transition. But while in the Kosterlitz-Thouless transition the exponential behavior is a consequence of the topological excitations, the predicted phenomena in the dipolar $XY$ model are solely due to spin-wave excitations. In addition to the $dXY$ model we introduce a whole class of long-range $XY$ models showing such nonstandard behavior. Parametrizing the divergences in terms of the correlation length we calculate the critical exponents of the transition. These exponents are correct in any loop order. The influence of topological excitations on the ferromagnetic transition is discussed.

DOI
10.1103/physrevb.70.134430

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The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction.We investigate a two-dimensional xy-model extended by the dipolar interaction.Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state.This transition is due to the long-range dipolar interaction.The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling.In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws.The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition.We show that there is a whole class of longrange xy-models leading to such non-standard behavior.Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents.These exponents are correct in any loop order.
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Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with fixed boundary conditions. Using a conformal mapping, it is very easy to deduce the exponent η σ ( T ) of the order parameter correlation function at any temperature in the critical phase of the model. The temperature behaviour of η σ ( T ) is obtained numerically with a good accuracy up to the Kosterlitz-Thouless transition temperature. At very low temperatures, a good agreement is found with Berezinskii's harmonic approximation. Surprisingly, we show some evidence that there are no logarithmic corrections to the behaviour of the order parameter density profile (with symmetry-breaking surface fields) at the Kosterlitz-Thouless transition temperature.
Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional XYXYXY Model with 1/r41/r^41/r4 Interaction
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The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of $\ensuremath{\rho}$, the power of the interaction strength decay. For large enough $\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}$, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\ensuremath{\rho}}_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ${|r|}^{\ensuremath{-}\ensuremath{\rho}}$, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that ${\ensuremath{\rho}}_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for $\ensuremath{\rho}\ensuremath{\ge}4$, and to present a second-order transition for $\ensuremath{\rho}<4$.