Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0201174 · Physical Review Letters

Phase Transition in a Two-Dimensional Heisenberg Model

Henk W. J. Blöte, Wenan Guo, Henk J. Hilhorst

Published Jan 14, 2002• 45 citations
arXivHTMLPDF

Abstract

We investigate the two-dimensional classical Heisenberg model with a nonlinear nearest-neighbor interaction $V(\stackrel{\ensuremath{\rightarrow}}{s},{\stackrel{\ensuremath{\rightarrow}}{s}}^{\ensuremath{'}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2K[(1+\stackrel{\ensuremath{\rightarrow}}{s}\ifmmode \dot{}\else \.{}\fi{}{\stackrel{\ensuremath{\rightarrow}}{s}}^{\ensuremath{'}})/2{]}^{p}$. The analogous nonlinear interaction for the $\mathrm{XY}$ model was introduced by Domany, Schick, and Swendsen, who find that for large $p$ the Kosterlitz-Thouless transition is preempted by a first-order transition. Here we show that, whereas the standard $(p\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)$ Heisenberg model has no phase transition, for large enough $p$ a first-order transition appears. Both phases have only short-range order, but with a correlation length that jumps at the transition.

DOI
10.1103/physrevlett.88.047203

Related papers

Two-dimensional Heisenberg model with nonlinear interactions

Sergio Caracciolo A, Andrea Pelissetto C

We investigate a two-dimensional classical N-vector model with a nonlinear interaction (1+sigma(i) x sigma(j))(p) in the large-N limit. As observed for N=3 by Blöte et al. [Phys. Rev. Lett. 88, 047203 (2002)], we find a first-order transition for p>p(c) and no finite-temperature phase transitions for p<p(c). For p>p(c), both phases have short-range order, the correlation length showing a finite discontinuity at the transition. For p=p(c), there is a peculiar transition, where the spin-spin correlation length is finite while the energy-energy correlation length diverges.

arXiv:cond-mat/0202506•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 24, 2002•17 citations

Existence of Phase Transitions for Anisotropic Heisenberg Models

Jürg Martin Fröhlich, Élliott H. Lieb

The two-dimensional anisotropic, nearest-neighbor Heisenberg model on a square lattice, both quantum and classical, has been shown regorously to have a phase transition in the sense that the spontaneous magnetization is positive at low temperatures. This is so for all anisotropies. An analogous result (staggered polarization) holds for the antiferromagnet in the classical case; in the quantum case it holds if the anisotropy is large enough (depending on the single-site spin).

Physical Review Letters•Feb 21, 1977•45 citations

Novel phase transition in two-dimensional xy-models with long-range interaction

P.G. Maier, Franz Schwabl

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.

Condensed Matter Physics•Jan 1, 2005•4 citations

PHASE TRANSITIONS IN THE TWO-DIMENSIONAL O(3) MODEL

Henk W. J. Blöte, Wenan Guo

We use Monte Carlo, transfer-matrix and finite-size scaling methods to investigate two-dimensional O (n) models with n > 2, in particular the case n = 3 which includes the classical Heisenberg model. Depending on the type of interaction and the lattice structure, two different types of phase transitions are present. One type resembles the hard-hexagon transition and occurs in the loop representation of the honeycomb O (n) model. The other type is a first-order transition which occurs for spin-spin interactions that are strongly nonlinear in the neighbor-spin products. When the nonlinearity is decreased, the first-order line ends in a critical point. The existence of the first-order line is in agreement with mean-field theory as well as with high- and low-temperature approximations.

International Journal of Modern Physics B•Jun 20, 2002•3 citations

Deconfined quantum criticality in the two-dimensional antiferromagnetic Heisenberg model with next-nearest-neighbor Ising exchange

J. Hove, Asle Sudbø

We have considered the $S=1∕2$ antiferromagnetic Heisenberg model in two dimensions, with an additional Ising next-nearest-neighbor interaction. Antiferromagnetic next-nearest-neighbor interactions will lead to frustration, and the system responds with flipping the spins down in the $xy$ plane. For large next-nearest-neighbor coupling the system will order in a striped phase along the $z$ axis, this phase is reached through a first-order transition. We have considered two generalizations of this model, one with random next-nearest-neighbor interactions, and one with an enlarged unit cell, where only half of the atoms have next-nearest-neighbor interactions. In both cases the transition is softened to a second-order transition separating two ordered states. In the latter case we have estimated the quantum critical exponent $\ensuremath{\beta}\ensuremath{\approx}0.25$. These two cases then represent candidate examples of deconfined quantum criticality.

arXiv:cond-mat/0605118•Physical Review B•Mar 9, 2007•1 citation

First-order transition in the XY model

H. J. F. Knops

Recent Monte Carlo simulations suggest that changing the shape of the nearest-neighbor $\mathrm{XY}$ interaction might turn the infinite-order Kosterlitz-Thouless transition into a first-order one. In this Communication, an alternative explanation for the Monte Carlo data is proposed which is based on the renormalization pattern that results for this model in a Migdal-Kadanoff approximation.

Physical review. B, Condensed matter•Jul 1, 1984•25 citations
We investigate the two-dimensional classical Heisenberg model with a nonlinear nearest-neighbor interaction V(s\ensuremath→,s\ensuremath→\ensuremath′)=2K[(1+s\ensuremath→\ifmmode˙\else˙\fis\ensuremath→\ensuremath′)/2]pV(\stackrel{\ensuremath{\rightarrow}}{s},{\stackrel{\ensuremath{\rightarrow}}{s}}^{\ensuremath{'}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2K[(1+\stackrel{\ensuremath{\rightarrow}}{s}\ifmmode \dot{}\else \.{}\fi{}{\stackrel{\ensuremath{\rightarrow}}{s}}^{\ensuremath{'}})/2{]}^{p}V(s\ensuremath→,s\ensuremath→\ensuremath′)=2K[(1+s\ensuremath→\ifmmode˙\else˙\fis\ensuremath→\ensuremath′)/2]p. The analogous nonlinear interaction for the XY\mathrm{XY}XY model was introduced by Domany, Schick, and Swendsen, who find that for large ppp the Kosterlitz-Thouless transition is preempted by a first-order transition. Here we show that, whereas the standard (p=1)(p\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)(p=1) Heisenberg model has no phase transition, for large enough ppp a first-order transition appears. Both phases have only short-range order, but with a correlation length that jumps at the transition.
We investigate a two-dimensional classical N-vector model with a nonlinear interaction (1+sigma(i) x sigma(j))(p) in the large-N limit. As observed for N=3 by Blöte et al. [Phys. Rev. Lett. 88, 047203 (2002)], we find a first-order transition for p>p(c) and no finite-temperature phase transitions for p<p(c). For p>p(c), both phases have short-range order, the correlation length showing a finite discontinuity at the transition. For p=p(c), there is a peculiar transition, where the spin-spin correlation length is finite while the energy-energy correlation length diverges.
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 use Monte Carlo, transfer-matrix and finite-size scaling methods to investigate two-dimensional O (n) models with n > 2, in particular the case n = 3 which includes the classical Heisenberg model. Depending on the type of interaction and the lattice structure, two different types of phase transitions are present. One type resembles the hard-hexagon transition and occurs in the loop representation of the honeycomb O (n) model. The other type is a first-order transition which occurs for spin-spin interactions that are strongly nonlinear in the neighbor-spin products. When the nonlinearity is decreased, the first-order line ends in a critical point. The existence of the first-order line is in agreement with mean-field theory as well as with high- and low-temperature approximations.
We have considered the S=1∕2S=1∕2S=1∕2 antiferromagnetic Heisenberg model in two dimensions, with an additional Ising next-nearest-neighbor interaction. Antiferromagnetic next-nearest-neighbor interactions will lead to frustration, and the system responds with flipping the spins down in the xyxyxy plane. For large next-nearest-neighbor coupling the system will order in a striped phase along the zzz axis, this phase is reached through a first-order transition. We have considered two generalizations of this model, one with random next-nearest-neighbor interactions, and one with an enlarged unit cell, where only half of the atoms have next-nearest-neighbor interactions. In both cases the transition is softened to a second-order transition separating two ordered states. In the latter case we have estimated the quantum critical exponent $\ensuremath{\beta}\ensuremath{\approx}0.25$. These two cases then represent candidate examples of deconfined quantum criticality.
Recent Monte Carlo simulations suggest that changing the shape of the nearest-neighbor XY\mathrm{XY}XY interaction might turn the infinite-order Kosterlitz-Thouless transition into a first-order one. In this Communication, an alternative explanation for the Monte Carlo data is proposed which is based on the renormalization pattern that results for this model in a Migdal-Kadanoff approximation.