Physical Review Letters

Monte Carlo studies of correlation length and hyperscaling in the three-dimensional Ising model

Pik-Yin Lai, K. K. Mon

Published May 29, 1989• 12 citations

Abstract

We report extensive Monte Carlo simulations and finite-size scaling analysis of the correlation length (${\ensuremath{\xi}}_{\mathrm{L}}$) for the spin-spin correlation function and the renormalized coupling (${\mathrm{g}}_{\mathrm{L}}$) of the simple-cubic Ising model with a wide range of lattice sizes (L\ensuremath{\le}96). New methods of analysis which avoid assumptions related to the correlation length were introduced. The exponents \ensuremath{\nu} and 2-\ensuremath{\alpha} were estimated from separate analysis of ${\ensuremath{\xi}}_{\mathrm{L}}$ and ${\mathrm{g}}_{\mathrm{L}}$, respectively. A value of ${\mathrm{\ensuremath{\omega}}}^{\mathrm{*}}$\ensuremath{\approxeq}0.001\ifmmode\pm\else\textpm\fi{}0.07 for the Fisher exponent is obtained. This is consistent with a very small or no violations of hyperscalings.

DOI: 10.1103/physrevlett.62.2608

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Physical Review Letters

Monte Carlo studies of correlation length and hyperscaling in the three-dimensional Ising model

Pik-Yin Lai, K. K. Mon

Published May 29, 1989• 12 citations

Publisher

Abstract

We report extensive Monte Carlo simulations and finite-size scaling analysis of the correlation length (

{\ensuremath{\xi}}_{\mathrm{L}}

) for the spin-spin correlation function and the renormalized coupling (

{\mathrm{g}}_{\mathrm{L}}

) of the simple-cubic Ising model with a wide range of lattice sizes (L\ensuremath{\le}96). New methods of analysis which avoid assumptions related to the correlation length were introduced. The exponents \ensuremath{\nu} and 2-\ensuremath{\alpha} were estimated from separate analysis of

{\ensuremath{\xi}}_{\mathrm{L}}

and

{\mathrm{g}}_{\mathrm{L}}

, respectively. A value of

{\mathrm{\ensuremath{\omega}}}^{\mathrm{*}}

\ensuremath{\approxeq}0.001\ifmmode\pm\else\textpm\fi{}0.07 for the Fisher exponent is obtained. This is consistent with a very small or no violations of hyperscalings.

DOI: 10.1103/physrevlett.62.2608

Related papers

Monte Carlo studies of universal critical amplitudes for the three-dimensional Ising model: Correlation length and renormalized coupling

Pik-Yin Lai, K. K. Mon

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Akai Kurbanovich Murtazaev, I. K. Kamilov, Рамазанов Магомедшейх Курбанович

Physics of the Solid StateJun 1, 200519 citations

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

Alan M. Ferrenberg, Jiahao Xu, David Paul Landau

While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.

arXiv:1806.03558Physical review. EApr 4, 2018191 citations

We report extensive Monte Carlo simulations and finite-size scaling analysis of the correlation length (

{\ensuremath{\xi}}_{\mathrm{L}}

) for the spin-spin correlation function and the renormalized coupling (

{\mathrm{g}}_{\mathrm{L}}

{\ensuremath{\xi}}_{\mathrm{L}}

and

{\mathrm{g}}_{\mathrm{L}}

, respectively. A value of

{\mathrm{\ensuremath{\omega}}}^{\mathrm{*}}

\ensuremath{\approxeq}0.001\ifmmode\pm\else\textpm\fi{}0.07 for the Fisher exponent is obtained. This is consistent with a very small or no violations of hyperscalings.

The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with

{8}^{3}

{16}^{3}

{32}^{3}

, and

{64}^{3}

sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling

{K}_{1}^{c}=0.221654(6)

. After allowing for (i) interpolation to this best estimate for

{K}_{1}^{c}

, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.