Chinese Journal of Natural Medicines

Universal relations in the finite-size correction terms of two-dimensional Ising models.

Y Okabe, N Kawashima

Published Sep 1, 2001• 12 citations

Publisher

Abstract

DOI: 10.1103/physreve.64.035103

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Universal relations in the finite-size correction terms of two-dimensional Ising models

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Critical finite-size-scaling amplitudes of a fully anisotropic three-dimensional Ising model

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Chinese Journal of Natural Medicines

Universal relations in the finite-size correction terms of two-dimensional Ising models.

Y Okabe, N Kawashima

Published Sep 1, 2001• 12 citations

Publisher

Abstract

DOI: 10.1103/physreve.64.035103

Related papers

Universal relations in the finite-size correction terms of two-dimensional Ising models

Yutaka Okabe, Naoki Kawashima

Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.

arXiv:cond-mat/0107514Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 27, 20016 citations

Critical finite-size-scaling amplitudes of a fully anisotropic three-dimensional Ising model

M. A. Yurishchev

A fully anisotropic simple-cubic Ising lattice in the geometry of periodic cylinders n\ifmmode\times\else\texttimes\fi{}n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} is investigated by the transfer-matrix finite-size-scaling method. In addition to the previously obtained critical amplitudes of the inverse correlation lengths and singular part of the free energy [M. A. Yurishchev, Phys. Rev. B 50, 13 533 (1994)], the amplitudes of the usual (``linear'') and nonlinear susceptibilities and the amplitude of the second derivative of the spin-spin inverse correlation length with respect to the external field are calculated. The behavior of critical amplitude combinations (which, in accordance with the Privman-Fisher equations, do not contain in their composition the nonuniversal metric coefficients and geometry prefactor) are studied as a function of the interaction anisotropy parameters. A universality domain for the amplitude ratios is found in the quasi-one-dimensional regime of interactions in the system. In the case of a fully isotropic three-dimensional Ising model for which the high precision values of the critical coupling and critical-point free energy are available, improved estimates are obtained for the following four universal quantities: (1) the amplitude of spin-spin inverse correlation length, (2) the amplitude of singular part of the free energy, (3) the ratio of the amplitude of a second derivative of the spin-spin inverse correlation length with respect to the external field to the usual susceptibility amplitude, and (4) the ratio of the nonlinear susceptibility amplitude to the square of the linear susceptibility amplitude (i.e., for the finite-size counterpart of the four-point renormalized coupling constant).A fully anisotropic simple-cubic Ising lattice in the geometry of periodic cylinders n\ifmmode\times\else\texttimes\fi{}n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} is investigated by the transfer-matrix finite-size-scaling method. In addition to the previously obtained critical amplitudes of the inverse correlation lengths and singular part of the free energy [M. A. Yurishchev, Phys. Rev. B 50, 13 533 (1994)], the amplitudes of the usual (``linear'') and nonlinear susceptibilities and the amplitude of the second derivative of the spin-spin inverse correlation length with respect to the external field are calculated. The behavior of critical amplitude combinations (which, in accordance with the Privman-Fisher equations, do not contain in their composition the nonuniversal metric coefficients and geometry prefactor) are studied as a function of the interaction anisotropy parameters. A universality domain for the amplitude ratios is found in the quasi-one-dimensional regime of interactions in the system. In the case of a fully isotropic three-dimensional Ising model for which the high precision values of the critical coupling and critical-point free energy are available, improved estimates are obtained for the following four universal quantities: (1) the amplitude of spin-spin inverse correlation length, (2) the amplitude of singular part of the free energy, (3) the ratio of the amplitude of a second derivative of the spin-spin inverse correlation length with respect to the external field to the usual susceptibility amplitude, and (4) the ratio of the nonlinear susceptibility amplitude to the square of the linear susceptibility amplitude (i.e., for the finite-size counterpart of the four-point renormalized coupling constant).

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Sladequeiroz Instituto De, Potts Ferromagnets

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Exact Universal Amplitude Ratios for Two-Dimensional Ising Models and a Quantum Spin Chain

N. Sh. Izmailian, Chin‐Kun Hu

Let ${f}_{N}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for ${a}_{k}$ and ${b}_{k}$ in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio ${b}_{k}/{a}_{k}$ is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.Let ${f}_{N}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for ${a}_{k}$ and ${b}_{k}$ in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio ${b}_{k}/{a}_{k}$ is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.

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Jesús Salas

We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a criticalWe compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical

Jan 1, 200120 citations

Let

{f}_{N}

and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for

{a}_{k}

and

{b}_{k}

in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio

{b}_{k}/{a}_{k}

is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.