Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘Monte Carlo studies of the quantumXYmodel in two dimensions’ ’’

Eugene Loh, Douglas J. Scalapino, Paul Michael Grant

Published Apr 1, 1986• 6 citations

Abstract

We agree that our simulations of the two-dimensional quantum XY model differ from those of De Raedt and Lagendijk and argure that our results---a finite specific-heat peak and a conclusively nonzero vortex density at zero temperature---are the correct ones, agreeing with other work on this model.

DOI: 10.1103/physrevb.33.5104

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Comment on ‘‘Monte Carlo studies of the quantumXYmodel in two dimensions’’

Hans A. De Raedt, Ad Lagendijk

We point out that some of the simulation results on the two-dimensional spin-(1/2)XY model of Loh, Scalapino, and Grant are not in agreement with previous work.Received 13 June 1985DOI:https://doi.org/10.1103/PhysRevB.33.5102©1986 American Physical Society

Physical review. B, Condensed matterApr 1, 19863 citations

Reply to ‘‘Comment on ‘Kosterlitz-Thouless transition in the two-dimensional quantumXYmodel’ ’’

Hong Q. Ding, M. S. Makivić

The scaling in the critical region found in the Monte Carlo simulation constitutes clear evidence for the Kosterlitz-Thouless transition. The third dimension in the spin space does not affect universality. Further numerical results at and below the transition provide more evidence. Sznajd's analysis is questioned because his same analysis incorrectly predicts a phase transition in a very similar model, the Heisenberg model.

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Critical behavior of the two-dimensionalXYmodel

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Aji and Goldenfeld Reply:

Vivek Aji, Nigel D. Goldenfeld

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

Peter Olsson

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Monte Carlo studies of the quantum XY model in two dimensions

Eugene Loh, Douglas J. Scalapino, Paul Michael Grant

Physical review. B, Condensed matterApr 1, 198570 citations

Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘Monte Carlo studies of the quantumXYmodel in two dimensions’ ’’

Eugene Loh, Douglas J. Scalapino, Paul Michael Grant

Published Apr 1, 1986• 6 citations

Publisher

Abstract

DOI: 10.1103/physrevb.33.5104

Related papers

Comment on ‘‘Monte Carlo studies of the quantumXYmodel in two dimensions’’

Hans A. De Raedt, Ad Lagendijk

Physical review. B, Condensed matterApr 1, 19863 citations

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Hong Q. Ding, M. S. Makivić

Physical review. B, Condensed matterJan 1, 19928 citations

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

Aji and Goldenfeld Reply:

Vivek Aji, Nigel D. Goldenfeld

This is a response to the comment by J. Lidmar on our work in Phys. Rev. Lett. 87, 197003 (2001), cond-mat/0105622. We show explicitly that, contrary to the statements in the comment, our scaling analysis correctly reproduces the Josephson scaling relation and other known theoretical results, and explains the surprising results of dynamic Monte Carlo simulations in 3D for the vortex loop model. Our explanation, being based on identifying the correct relationship between real and Monte Carlo time, is not relevant for static Monte Carlo simulations, whose results do not need correction.This is a response to the comment by J. Lidmar on our work in Phys. Rev. Lett. 87, 197003 (2001), cond-mat/0105622. We show explicitly that, contrary to the statements in the comment, our scaling analysis correctly reproduces the Josephson scaling relation and other known theoretical results, and explains the surprising results of dynamic Monte Carlo simulations in 3D for the vortex loop model. Our explanation, being based on identifying the correct relationship between real and Monte Carlo time, is not relevant for static Monte Carlo simulations, whose results do not need correction.

arXiv:cond-mat/0208410Physical Review LettersAug 15, 20021 citation

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

Peter Olsson

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.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.

Physical review. B, Condensed matterAug 1, 1995110 citations

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

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.

{\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.

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.