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International Journal of Modern Physics B

Renormalization Theory for Ordered Phase of 2D XY Magnet with Dipolar Interaction

А. Кашуба

Published May 30, 1998• 0 citations
Publisher

Abstract

In the ordered phase of 2D XY ferromagnet the dipolar interaction between spins induces a relevant interaction between spin-waves. Although the density of spin-waves is small at low temperature, this long-range interaction correlates two spatially separated spin-waves. Anomalous scaling of spin-wave correlation functions is found in both statics and dynamics cases.

DOI
10.1142/s0217979298000739

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In the ordered phase of 2D XY ferromagnet the dipolar interaction between spins induces a relevant interaction between spin-waves. Although the density of spin-waves is small at low temperature, this long-range interaction correlates two spatially separated spin-waves. Anomalous scaling of spin-wave correlation functions is found in both statics and dynamics cases.
Long-range spin-wave correlation functions in the ordered phase of a 2D XY\mathrm{XY}XY ferromagnet with 2D dipolar forces are governed by the 2D smectic AAA Hamiltonian and possess exact scaling. The latter confirms from a different perspective the exact results obtained recently for the 2D smectic AAA. Exact scaling holds in the ordered phase of the 2D XY\mathrm{XY}XY ferromagnet with 3D dipolar forces, with the dimension of the spatial anisotropy being ${\ensuremath{\Delta}}_{x}=\frac{4}{3}$ and the anomalous dimension of spin being ${\ensuremath{\eta}}_{\ensuremath{\varphi}}=\frac{1}{6}$.
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.
We investigate the low-temperature phase of the two-dimensional dipolar XYXYXY model. Within the framework of the nonlinear $\ensuremath{\sigma}$ model we show that the dipolar XYXYXY 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 XYXYXY model are solely due to spin-wave excitations. In addition to the dXYdXYdXY model we introduce a whole class of long-range XYXYXY 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.
We study the properties of spin systems realized by cold polar molecules interacting via dipole-dipole interactions in two dimensions. Using a spin wave theory, that allows for the full treatment of the characteristic long-distance tail of the dipolar interaction, we find several anomalous features in the ground state correlations and the spin wave excitation spectrum, which are absent in their counterparts with short-range interaction. The most striking consequence is the existence of true long-range order at finite temperature for a two-dimensional phase with a broken U(1) symmetry.
Abstract The crossover between short-range and long-range (LR) universal behaviors remains a central theme in the physics of LR interacting systems. The competition between LR coupling and the Berezinskii–Kosterlitz–Thouless mechanism makes the problem more subtle and less understood in the two-dimensional (2D) XY model, a cornerstone for investigating low-dimensional phenomena and their implications in quantum computation. We study the 2D XY model with algebraically decaying interaction ∼1/r 2+σ . Utilizing an advanced update strategy, we conduct LR Monte Carlo simulations of the model up to a linear size of L = 8192. Our results demonstrate continuous phase transitions into a ferromagnetic phase for σ < 2, which exhibit the simultaneous emergence of a long-ranged order and a power-law decaying correlation function due to the Goldstone mode. Furthermore, we find logarithmic scaling behaviors in the low-temperature phase at σ = 2. The observed scaling behaviors in the low-temperature phase for σ ≤ 2 agree with our theoretical analysis. Our findings request further theoretical understanding and can be of practical application in cutting-edge experiments like Rydberg atom arrays.
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.