Physical review. B, Solid state

Extended universality of the Ising model

Douglas S. Ritchie, Donald D. Betts

Published Apr 1, 1975• 35 citations

Abstract

An extension of the critical-phenomena universality hypothesis assumed previously to be valid for less singular terms is shown to be invalid. A reformulated extension of the universality hypothesis is found to be valid exactly for the spin-1/2 Ising model on the triangular, square, and honeycomb lattices, and, further support is supplied by high- and low-temperature expansions for the susceptibility. The extension fails however for the Ising model on the kagome\r{} lattice and for the spherical model. At present there are insufficient data to test the extended principle for other models or for experimental systems.

DOI: 10.1103/physrevb.11.2559

Related papers

Critical parameters and universal amplitude ratios of two-dimensional spin-SIsing models using high- and low-temperature expansions

P. Butera, M. Comi, Anthony John Guttmann

arXiv:hep-lat/0302010Physical review. B, Condensed matterFeb 5, 200312 citations

Susceptibility amplitudes for the two-dimensional Ising model

Anthony John Guttmann

The recent critical amplitude calculation of Barouch, McCoy, and Wu for the susceptibility of the Ising model on a square lattice is extended to yield amplitudes for the triangular, honeycomb, and kagom\'e lattices. The extension assumes the validity of the generalized law of corresponding states---as first postulated by Betts, Guttmann, and Joyce---for the two-dimensional Ising model.

Physical review. B, Solid stateJun 1, 197414 citations

One—Scale-Factor Universality for Critical Phenomena

Donald D. Betts, Douglas S. Ritchie

Physical Review LettersMar 31, 19753 citations

Updated tests of scaling and universality for spin-spin correlations in the two- and three- dimensional spin- S Ising models using high-temperature expansions

P. Butera, M. Comi

arXiv:hep-lat/0406035Physical Review BMay 19, 20049 citations

The Ising model on the tetrahedron lattice. I. An extended susceptibility series

J Ho-Ting-Hun, J. Oitmaa

Journal of Physics A Mathematical and GeneralDec 1, 197512 citations

Spin- S Ising susceptibility on the triangular lattice

John P. VAN DYKE, William J. Camp

Physical review. B, Solid stateAug 1, 19752 citations

Physical review. B, Solid state

Extended universality of the Ising model

Douglas S. Ritchie, Donald D. Betts

Published Apr 1, 1975• 35 citations

Publisher

Abstract

DOI: 10.1103/physrevb.11.2559

Related papers

Critical parameters and universal amplitude ratios of two-dimensional spin-SIsing models using high- and low-temperature expansions

P. Butera, M. Comi, Anthony John Guttmann

For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann, and Jensen. We have computed various critical parameters and improved the estimates of others. Moreover, the properties of hyperscaling and of universality (spin-S independence) of exponents and of various dimensionless amplitude combinations have been verified accurately. Assuming the validity of the lattice-lattice scaling, from our estimates of critical amplitudes for the square lattice we have also obtained estimates of the corresponding amplitudes for the spin-S Ising model on the triangular, honeycomb, and kagom\'e lattices.For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann, and Jensen. We have computed various critical parameters and improved the estimates of others. Moreover, the properties of hyperscaling and of universality (spin-S independence) of exponents and of various dimensionless amplitude combinations have been verified accurately. Assuming the validity of the lattice-lattice scaling, from our estimates of critical amplitudes for the square lattice we have also obtained estimates of the corresponding amplitudes for the spin-S Ising model on the triangular, honeycomb, and kagom\'e lattices.

arXiv:hep-lat/0302010Physical review. B, Condensed matterFeb 5, 200312 citations

Susceptibility amplitudes for the two-dimensional Ising model

Anthony John Guttmann

Physical review. B, Solid stateJun 1, 197414 citations

One—Scale-Factor Universality for Critical Phenomena

Donald D. Betts, Douglas S. Ritchie

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of ${V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}$ , where ${q}_{X}$ is the coordination number, ${{K}_{C}}^{X}$ is the dimensionless inverse critical temperature, and ${n}_{X}$ , ${g}_{X}$ , and ${l}_{X}$ are the universality scale factors. ${V}_{X}$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{{V}_{X}}{{l}_{X}}$ is an exact invariant.An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of ${V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}$, where ${q}_{X}$ is the coordination number, ${{K}_{C}}^{X}$ is the dimensionless inverse critical temperature, and ${n}_{X}$, ${g}_{X}$, and ${l}_{X}$ are the universality scale factors. ${V}_{X}$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{{V}_{X}}{{l}_{X}}$ is an exact invariant.

Physical Review LettersMar 31, 19753 citations

Updated tests of scaling and universality for spin-spin correlations in the two- and three- dimensional spin- S Ising models using high-temperature expansions

P. Butera, M. Comi

We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin- $S$ Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-$S$ Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.

arXiv:hep-lat/0406035Physical Review BMay 19, 20049 citations

The Ising model on the tetrahedron lattice. I. An extended susceptibility series

J Ho-Ting-Hun, J. Oitmaa

The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.

Journal of Physics A Mathematical and GeneralDec 1, 197512 citations

Spin- S Ising susceptibility on the triangular lattice

John P. VAN DYKE, William J. Camp

The high-temperature susceptibility of the spin- $S$ Ising model on the triangular net is analyzed as a function of $S$ using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.The high-temperature susceptibility of the spin-$S$ Ising model on the triangular net is analyzed as a function of $S$ using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.

Physical review. B, Solid stateAug 1, 19752 citations

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of

{V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}

, where

{q}_{X}

is the coordination number,

{{K}_{C}}^{X}

is the dimensionless inverse critical temperature, and

{n}_{X}

{g}_{X}

, and

{l}_{X}

are the universality scale factors.

{V}_{X}

is an exact invariant for the two-dimensional

S=\frac{1}{2}

Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however,

\frac{{V}_{X}}{{l}_{X}}

is an exact invariant.

We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-

S

Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.

The high-temperature susceptibility of the spin-

S

Ising model on the triangular net is analyzed as a function of

S

using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.