Physical review. B, Condensed matter

Monte Carlo analysis of the two-dimensionalXYmodel. II. Comparison with the Kosterlitz renormalization-group equations

Peter Olsson

Published Aug 1, 1995• 110 citations

Abstract

The temperature scale of relevance for the vortex excitations in the two-dimensional XY model is identified. The size dependence of the helicity modulus at constant vortex temperature is then obtained by Monte Carlo simulations. The Monte Carlo data are then compared with the results for the length-dependent dielectric function from the Kosterlitz renormalization-group equations. The agreement is excellent for temperatures up to and slightly above ${\mathit{T}}_{\mathit{c}}$. The critical temperature is determined to be ${\mathit{T}}_{\mathit{c}}$=0.892 13(10). The temperature dependence of the characteristic length right above ${\mathit{T}}_{\mathit{c}}$ is also obtained and found to be in agreement with recent results from high-temperature series expansions.

DOI: 10.1103/physrevb.52.4526

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Physical review. B, Condensed matter

Monte Carlo analysis of the two-dimensionalXYmodel. II. Comparison with the Kosterlitz renormalization-group equations

Peter Olsson

Published Aug 1, 1995• 110 citations

Publisher

Abstract

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

. The critical temperature is determined to be

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

=0.892 13(10). The temperature dependence of the characteristic length right above

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

is also obtained and found to be in agreement with recent results from high-temperature series expansions.

DOI: 10.1103/physrevb.52.4526

Related papers

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.

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The two-dimensional $S=1/2$ XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to $128 \times 128$ near and below the critical temperature. The critical temperature is estimated as $T_{\rm KT} = 0.3427(2)J$ . The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.The two-dimensional $S=1/2$ XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to $128 \times 128$ near and below the critical temperature. The critical temperature is estimated as $T_{\rm KT} = 0.3427(2)J$. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

arXiv:cond-mat/9803090Journal of the Physical Society of JapanAug 15, 199879 citations

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Physical review. B, Condensed matterNov 1, 1989105 citations

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Peter Olsson

A generalization of periodic boundary conditions to incorporate fluctuations with wavelength larger than the system size is discussed. These boundary conditions are applied to Monte Carlo calculations on the two-dimensional $\mathrm{XY}$ model. The finite-size effect on the vortex-vortex correlation function, and thereby the dielectric function, is shown to be virtually eliminated. These Monte Carlo data are shown to be in excellent agreement with the Kosterlitz renormalization group equations, and the critical temperature is found to be ${T}_{c}=0.8922(2)$ .A generalization of periodic boundary conditions to incorporate fluctuations with wavelength larger than the system size is discussed. These boundary conditions are applied to Monte Carlo calculations on the two-dimensional $\mathrm{XY}$ model. The finite-size effect on the vortex-vortex correlation function, and thereby the dielectric function, is shown to be virtually eliminated. These Monte Carlo data are shown to be in excellent agreement with the Kosterlitz renormalization group equations, and the critical temperature is found to be ${T}_{c}=0.8922(2)$.

Physical Review LettersDec 19, 199449 citations

Logarithmic corrections in the two-dimensional XY model

Wolfhard Janke E

Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square L\ifmmode\times\else\texttimes\fi{}L lattices, the scaling behavior of the susceptibility \ensuremath{\chi} and correlation length \ensuremath{\xi} at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections (ln L${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the finite-size scaling region and (ln \ensuremath{\xi}${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to ${512}^{2}$ we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to \ensuremath{\xi}\ensuremath{\approx}140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square L\ifmmode\times\else\texttimes\fi{}L lattices, the scaling behavior of the susceptibility \ensuremath{\chi} and correlation length \ensuremath{\xi} at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections (ln L${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the finite-size scaling region and (ln \ensuremath{\xi}${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to ${512}^{2}$ we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to \ensuremath{\xi}\ensuremath{\approx}140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.

arXiv:hep-lat/9609045Physical review. B, Condensed matterFeb 1, 199755 citations

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

. The critical temperature is determined to be

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

=0.892 13(10). The temperature dependence of the characteristic length right above

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

is also obtained and found to be in agreement with recent results from high-temperature series expansions.

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.

The two-dimensional

S=1/2

XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to

128 \times 128

near and below the critical temperature. The critical temperature is estimated as

T_{\rm KT} = 0.3427(2)J

. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

\mathrm{XY}

model. The finite-size effect on the vortex-vortex correlation function, and thereby the dielectric function, is shown to be virtually eliminated. These Monte Carlo data are shown to be in excellent agreement with the Kosterlitz renormalization group equations, and the critical temperature is found to be

{T}_{c}=0.8922(2)

{512}^{2}

we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to \ensuremath{\xi}\ensuremath{\approx}140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.