Physical review. B, Condensed matter

Monte Carlo simulation of the planar model using the dual solid-on-solid representation

William J. Shugard, John D. Weeks, George H. Gilmer

Published Jun 1, 1980• 17 citations

Abstract

The results of a Monte Carlo (MC) simulation of the planar ($\mathrm{XY}$) model in the dual "solid-on-solid" representation are reported. Our estimate for the reduced transition temperature ${T}_{c}$ of 0.90 and our finding of a rather sharp peak in the specific heat about 18% above ${T}_{c}$ are in good agreement with recent MC calculations in the spin representation.

DOI: 10.1103/physrevb.21.5309

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Physical review. B, Condensed matter

Monte Carlo simulation of the planar model using the dual solid-on-solid representation

William J. Shugard, John D. Weeks, George H. Gilmer

Published Jun 1, 1980• 17 citations

Publisher

Abstract

The results of a Monte Carlo (MC) simulation of the planar (

\mathrm{XY}

) model in the dual "solid-on-solid" representation are reported. Our estimate for the reduced transition temperature

{T}_{c}

of 0.90 and our finding of a rather sharp peak in the specific heat about 18% above

{T}_{c}

are in good agreement with recent MC calculations in the spin representation.

DOI: 10.1103/physrevb.21.5309

Related papers

Monte Carlo Test of Theories for the Planar Model, the F Model, and Related Systems

William J. Shugard, John D. Weeks, George H. Gilmer

We have performed Monte Carlo simulations for correlation functions in various solid-on-solid models, some of which are equivalent to the $F$ model, the two-dimensional Coulomb gas, and the planar model. Our results for the $F$ model can be quantitatively represented using the theory of Kosterlitz and Thouless. We use this theory to help determine the transition temperatures in other systems.We have performed Monte Carlo simulations for correlation functions in various solid-on-solid models, some of which are equivalent to the $F$ model, the two-dimensional Coulomb gas, and the planar model. Our results for the $F$ model can be quantitatively represented using the theory of Kosterlitz and Thouless. We use this theory to help determine the transition temperatures in other systems.

Physical Review LettersNov 13, 1978112 citations

The specific heat of planar spin models

Martin V. Ferer, A. Hamid Aidinejad, Michael J. Velgakis

In an attempt to understand the discrepancy between renormalization group predictions of a rounded specific heat peak about 38% above the Kosterlitz–Thouless transition temperature Tc and Monte-Carlo results indicating a sharp peak about 7% above Tc, we have undertaken an analysis of our thirteenth-order, high-temperature series for the free energy of the XY model on the triangular lattice. We have performed N fits on our free energy series in order to estimate the dominant singularity. These results are not consistent with either F≂ξ−2 or a transition at temperatures above Tc. Using these estimates of the leading singularity, we have extrapolated partial sums of our specific heat series for temperatures above Tc, thereby locating the specific heat peak.In an attempt to understand the discrepancy between renormalization group predictions of a rounded specific heat peak about 38% above the Kosterlitz–Thouless transition temperature Tc and Monte-Carlo results indicating a sharp peak about 7% above Tc, we have undertaken an analysis of our thirteenth-order, high-temperature series for the free energy of the XY model on the triangular lattice. We have performed N fits on our free energy series in order to estimate the dominant singularity. These results are not consistent with either F≂ξ−2 or a transition at temperatures above Tc. Using these estimates of the leading singularity, we have extrapolated partial sums of our specific heat series for temperatures above Tc, thereby locating the specific heat peak.

Journal of Applied PhysicsMar 15, 19843 citations

Critical behavior of the two-dimensional XY model: A Monte Carlo simulation

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We have performed Monte Carlo (MC) simulations on systems of L\ifmmode\times\else\texttimes\fi{}L classical planar unit spins on square lattices, for L=6, 15, 30, 60, 90, and 200. The interaction between any two given spins S${\ensuremath{\rightarrow}}_{1}$ and S${\ensuremath{\rightarrow}}_{2}$ is given by -JS${\ensuremath{\rightarrow}}_{1}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{2}$ if ${S}_{1}$ and ${S}_{2}$ are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature ( ${T}_{c}$ ). We have found that the diffusion constant for vortex motion is given at ${T}_{c}$ by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically ${10}^{5}$ MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of ${T}_{c}$ =0.89J/ ${k}_{B}$ , the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of ${\ensuremath{\eta}}_{c}$=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.We have performed Monte Carlo (MC) simulations on systems of L\ifmmode\times\else\texttimes\fi{}L classical planar unit spins on square lattices, for L=6, 15, 30, 60, 90, and 200. The interaction between any two given spins S${\ensuremath{\rightarrow}}_{1}$ and S${\ensuremath{\rightarrow}}_{2}$ is given by -JS${\ensuremath{\rightarrow}}_{1}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{2}$ if ${S}_{1}$ and ${S}_{2}$ are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (${T}_{c}$). We have found that the diffusion constant for vortex motion is given at ${T}_{c}$ by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically ${10}^{5}$ MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of ${T}_{c}$=0.89J/${k}_{B}$, the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of ${\ensuremath{\eta}}_{c}$=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.

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Critical behavior of the two-dimensionalXYmodel

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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

Low-temperature expansion of the planar spin model

C Twining

Journal of Physics A Mathematical and GeneralApr 7, 19881 citation

Monte Carlo studies of the quantum XY model in two dimensions

Eugene Loh, Douglas J. Scalapino, Paul Michael Grant

Monte Carlo simulations are performed for the spin-\textonehalf{} $\mathrm{XY}$ model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature ${T}_{c}$ , estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.Monte Carlo simulations are performed for the spin-\textonehalf{} $\mathrm{XY}$ model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature ${T}_{c}$, estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.

Physical review. B, Condensed matterApr 1, 198570 citations

The results of a Monte Carlo (MC) simulation of the planar (

\mathrm{XY}

) model in the dual "solid-on-solid" representation are reported. Our estimate for the reduced transition temperature

{T}_{c}

of 0.90 and our finding of a rather sharp peak in the specific heat about 18% above

{T}_{c}

are in good agreement with recent MC calculations in the spin representation.

We have performed Monte Carlo simulations for correlation functions in various solid-on-solid models, some of which are equivalent to the

F

model, the two-dimensional Coulomb gas, and the planar model. Our results for the

F

model can be quantitatively represented using the theory of Kosterlitz and Thouless. We use this theory to help determine the transition temperatures in other systems.

{S}_{1}

and

{S}_{2}

are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (

{T}_{c}

). We have found that the diffusion constant for vortex motion is given at

{T}_{c}

by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically

{10}^{5}

MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of

{T}_{c}

=0.89J/

{k}_{B}

, the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of ${\ensuremath{\eta}}_{c}$=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.

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.

Monte Carlo simulations are performed for the spin-\textonehalf{}

\mathrm{XY}

model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature

{T}_{c}

, estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.