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Journal of Physics C Solid State Physics

Phase transitions in two-dimensional antiferromagnetic XY models in external field

Виктор Степанович Доценко, G. Uimin

Published Sep 10, 1985• 11 citations
Publisher

Abstract

A qualitative study has been performed to elucidate the critical properties of an antiferromagnetic planar system in two dimensions. Phase diagrams in the temperature-field plane differ considerably in the cases of square and triangular lattices, resulting in the latter case in coexistence of the Ising-type and vortex unbinding phase transitions.

DOI
10.1088/0022-3719/18/25/019

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A qualitative study has been performed to elucidate the critical properties of an antiferromagnetic planar system in two dimensions. Phase diagrams in the temperature-field plane differ considerably in the cases of square and triangular lattices, resulting in the latter case in coexistence of the Ising-type and vortex unbinding phase transitions.
We consider the zero-temperature properties of four different spin 1/2 models on two-dimensional lattices: the XY ferromagnet, the XY antiferromagnet, the Heisenberg antiferromagnet, and the Dzyaloshinsky–Moriya models. Most of this article is a review of previously published work, but a few previously unpublished results are included. The relation between three of the models on bipartite lattices is described. The properties of the XY ferromagnet in two dimensions, especially those derived from extrapolation of finite lattice results, are reviewed. A numerical factor by which spin-wave and finite-lattice estimates of the long-range order parameter differ is discussed. For frustrated models on the triangular lattice the possibility of a chirality phase transition instead of, or in addition to, a magnetic phase transition is considered.
Abstract We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems ( ). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.
We study the antiferromagnetic {\it XY} model on a triangular lattice by extensive Monte Carlo simulations, focusing on its ordering and critical properties. Our result clearly shows that two separate transitions occur at two distinct temperatures, the one at a higher temperature is associated with a Z2Z_2Z2​-symmetry breaking driven by the chirality, and the one at a lower temperature is associated with the onset of the quasi-long-range order of the {\it XY} spin. We carefully examine the critical properties of each transition to find that the criticality of the chiral transition is consistent with the standard two-dimensional Ising universality class, whereas that of the spin transition might differ from the conventional Kosterlitz-Thouless (KT) one. The observed non-KT nature of the spin criticality is consistent with the most recent simulation result on the fully-frustrated {\it XY} model on a square lattice.
We investigate phase transitions of the XY model on a two-layer square lattice which consists of a Villain plane ( J ) and a ferromagnetic plane ( I ), using Monte Carlo simulations and a histogram method. Depending on the values of interaction parameters ( I , J ), the system presents three phases: namely, a Kosterlitz–Thouless (KT) phase in which the two planes are critical for I predominant over J , a chiral phase in which the two planes have a chiral order for J predominant over I and a new phase in which only the Villain plane has a chiral order and the ferromagnetic plane is paramagnetic with a small value of chirality. We clarify the nature of phase transitions by using a finite size scaling method. We find three different kinds of transitions according to the values of ( I , J ): the KT transition, the Ising transition and an XY -Ising transition with ν = 0.849(3). It turns out that the Ising or XY -Ising transition is associated with the disappearance of the chiral order in the Villain plane.
Antiferromagnetic and asymmetric ppp-state clock models on a square lattice are analyzed with the renormalization group, by studying the effect of ppp-fold symmetry-breaking fields in the planar model. For ppp odd and \ensuremath{\ge}3 (p=3p=3p=3 is the three-state Potts model) these models exhibit two phase transitions with a critical line of continuously varying exponent $\ensuremath{\eta}({\ensuremath{\eta}}_{min}<\ensuremath{\eta}<\frac{1}{4})$ separating them. When the couplings are antiferromagnetic in one direction only, ${\ensuremath{\eta}}_{min}=\frac{2}{{p}^{2}}$, while for the isotropic case ${\ensuremath{\eta}}_{min}=\frac{1}{{p}^{2}}$. In the latter case, the low-temperature phase is characterized by a 2p2p2p-fold degeneracy.