Physical review. B, Condensed matter

First-order transition in the XY model

H. J. F. Knops

Published Jul 1, 1984• 25 citations

Abstract

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.

DOI: 10.1103/physrevb.30.470

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First-Order Transition in an xy Model with Nearest-Neighbor Interactions

Eytan Domany, Michael Schick C, Robert H. Swendsen

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A generalized XY model with interaction is studied by Monte Carlo renormalization group method on two-dimensional random triangle lattice. For , a line of fixed points has been found. It characterizes that there is a Kosterlitz-Thouless phase transition. For , a first order phase transition has been found. Both of them show the relationship between the nature of phase transition and the class of interactions.

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Physical review. B, Condensed matter

First-order transition in the XY model

H. J. F. Knops

Published Jul 1, 1984• 25 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.30.470

Related papers

First-Order Transition in an xy Model with Nearest-Neighbor Interactions

Eytan Domany, Michael Schick C, Robert H. Swendsen

An $x\ensuremath{-}y$ model with nearest-neighbor interaction $V(\ensuremath{\theta})=2J{1\ensuremath{-}{[{cos}^{2}(\frac{\ensuremath{\theta}}{2})]}^{{p}^{2}}}$ is studied by Monte Carlo simulation. The model has a transition from a paramagnetic to an algebraic (massless) phase. An increase of the parameter $p$ , which changes the shape of the potential, brings about a change in the nature of the transition from continuous to first order.An $x\ensuremath{-}y$ model with nearest-neighbor interaction $V(\ensuremath{\theta})=2J{1\ensuremath{-}{[{cos}^{2}(\frac{\ensuremath{\theta}}{2})]}^{{p}^{2}}}$ is studied by Monte Carlo simulation. The model has a transition from a paramagnetic to an algebraic (massless) phase. An increase of the parameter $p$, which changes the shape of the potential, brings about a change in the nature of the transition from continuous to first order.

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Yukihiro Komura, Yutaka Okabe

We study the two-dimensional generalized XY model that depends on an integer $q$ by the Monte Carlo method. This model was recently proposed by Romano and Zagrebnov. We find a single Kosterlitz-Thouless (KT) transition for all values of $q$ , in contrast with the previous speculation that there may be two transitions, one a regular KT transition and another a first-order transition at a higher temperature. We show the universality of the KT transitions by comparing the universal finite-size scaling behaviors at different values of $q$ without assuming a specific universal form in terms of the KT transition temperature $T_{\rm KT}$ .We study the two-dimensional generalized XY model that depends on an integer $q$ by the Monte Carlo method. This model was recently proposed by Romano and Zagrebnov. We find a single Kosterlitz-Thouless (KT) transition for all values of $q$, in contrast with the previous speculation that there may be two transitions, one a regular KT transition and another a first-order transition at a higher temperature. We show the universality of the KT transitions by comparing the universal finite-size scaling behaviors at different values of $q$ without assuming a specific universal form in terms of the KT transition temperature $T_{\rm KT}$.

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Milan ˇ Zukovič

The effect of inclusion of higher-order interactions in the XY model on critical properties is studied by Monte Carlo simulations. It is found that an increasing number of the higher-order terms in the Hamiltonian modifies the shape of the potential, which beyond a certain value leads to the change of the nature of the transition from continuous to first order. The evidence for the first-order transition is provided in the form of the finite-size scaling and the energy histogram analysis. A rough phase diagram is presented as a function of the number of the higher-order interaction terms.The effect of inclusion of higher-order interactions in the XY model on critical properties is studied by Monte Carlo simulations. It is found that an increasing number of the higher-order terms in the Hamiltonian modifies the shape of the potential, which beyond a certain value leads to the change of the nature of the transition from continuous to first order. The evidence for the first-order transition is provided in the form of the finite-size scaling and the energy histogram analysis. A rough phase diagram is presented as a function of the number of the higher-order interaction terms.

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Critical behavior of the two-dimensionalXYmodel

Rajan Gupta, CLIVE F. BAILLIE

We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat ${\mathit{C}}_{\mathit{v}}$ for the XY model. The simulations are done on ${64}^{2}$ , ${128}^{2}$ , ${256}^{2}$ , and ${512}^{2}$ lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, ${\mathit{T}}_{\mathit{c}}$ =0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat ${\mathit{C}}_{\mathit{v}}$ for the XY model. The simulations are done on ${64}^{2}$, ${128}^{2}$, ${256}^{2}$, and ${512}^{2}$ lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, ${\mathit{T}}_{\mathit{c}}$=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.

Physical review. B, Condensed matterFeb 1, 1992196 citations

Recent Monte Carlo simulations suggest that changing the shape of the nearest-neighbor

\mathrm{XY}

p

, which changes the shape of the potential, brings about a change in the nature of the transition from continuous to first order.

We study the two-dimensional generalized XY model that depends on an integer

q

by the Monte Carlo method. This model was recently proposed by Romano and Zagrebnov. We find a single Kosterlitz-Thouless (KT) transition for all values of

q

, in contrast with the previous speculation that there may be two transitions, one a regular KT transition and another a first-order transition at a higher temperature. We show the universality of the KT transitions by comparing the universal finite-size scaling behaviors at different values of

q

without assuming a specific universal form in terms of the KT transition temperature

T_{\rm KT}

We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat

{\mathit{C}}_{\mathit{v}}

for the XY model. The simulations are done on

{64}^{2}

{128}^{2}

{256}^{2}

, and

{512}^{2}

lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate,

{\mathit{T}}_{\mathit{c}}

=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.