arXiv:cond-mat/0306434

Finite size scaling and equation of state for Ising lattices

J. G. García, Julio A. Gonzalo

Published Jun 17, 2003• 0 citations

Abstract

Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of $\bar h=h/\mid t \mid^{βδ}$ and $\bar τ=t/h^{1/βδ}$. Scaling plots for data on simple cubic Ising lattices are compared with plots of $ML^{βν}$ vs. $\mid t \mid L^{1/ν}$ for increasing $L$ values and with high quality experimental data on $CrBr_{3}$. Finite size effects and the equation of state are discussed.

DOI: 10.48550/arxiv.cond-mat/0306434

Related papers

Equation of State for Ising Lattices

J. García, Manuel I. Marqués, Julio A. Gonzalo

Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of ℏ = h /|t |β δ and = t /h 1/β δ . Scaling plots for data on simple cubic Ising lattices are compared with plots of ML β ν vs. |t |L 1/ν for increasing L values and with high quality experimental data on CrBr 3 . Finite size effects and the equation of state are discussed.

FerroelectricsJun 1, 20050 citations

Finite size scaling study of lattice models in the three-dimensional Ising universality class

Martin Hasenbusch B

arXiv:1004.4486Physical Review BNov 23, 2010333 citations

Numerical computation of finite size scaling functions: An alternative approach to finite size scaling

Jae-Kwon Kim, Adauto J. F. de Souza, David Paul Landau

arXiv:cond-mat/9602060Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsSep 1, 199639 citations

Monte carlo simulations on the simple cubic ising lattice with the dipolar and nearest neighbor interactions

Seizo Watarai, Akira Nakanishi

Monte Carlo simulations are performed on the simple cubic Ising lattice with both the dipolar and the nearest neighbor interactions. Applying the finite size scaling method, cross over effect of critical phenomena, which has been predicted by the renormalization group theory, can be observed from the Monte Carlo data.

FerroelectricsApr 1, 19902 citations

Finite-size behavior of the simple-cubic Ising lattice

David Paul Landau

Physical review. B, Solid stateJul 1, 1976169 citations

Accurate Monte Carlo critical exponents for Ising lattices

J. G. García, Julio A. Gonzalo, Manuel I. Marqués

arXiv:cond-mat/0211270Nov 14, 20020 citations

arXiv:cond-mat/0306434

Finite size scaling and equation of state for Ising lattices

J. G. García, Julio A. Gonzalo

Published Jun 17, 2003• 0 citations

arXiv HTML PDF

Abstract

Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of

\bar h=h/\mid t \mid^{βδ}

and

\bar τ=t/h^{1/βδ}

. Scaling plots for data on simple cubic Ising lattices are compared with plots of

ML^{βν}

vs.

\mid t \mid L^{1/ν}

for increasing

L

values and with high quality experimental data on

CrBr_{3}

. Finite size effects and the equation of state are discussed.

DOI: 10.48550/arxiv.cond-mat/0306434

Related papers

Equation of State for Ising Lattices

J. García, Manuel I. Marqués, Julio A. Gonzalo

FerroelectricsJun 1, 20050 citations

Finite size scaling study of lattice models in the three-dimensional Ising universality class

Martin Hasenbusch B

We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter $D$ on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter $D$ , where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for $D=0.641$ and 0.655 on lattices of a linear size up to $L=360$ we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter $D$ on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter $D$, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for $D=0.641$ and 0.655 on lattices of a linear size up to $L=360$ we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.

arXiv:1004.4486Physical Review BNov 23, 2010333 citations

Numerical computation of finite size scaling functions: An alternative approach to finite size scaling

Jae-Kwon Kim, Adauto J. F. de Souza, David Paul Landau

Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9602060Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsSep 1, 199639 citations

Monte carlo simulations on the simple cubic ising lattice with the dipolar and nearest neighbor interactions

Seizo Watarai, Akira Nakanishi

FerroelectricsApr 1, 19902 citations

Finite-size behavior of the simple-cubic Ising lattice

David Paul Landau

A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large $N$ as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large $N$ as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.

Physical review. B, Solid stateJul 1, 1976169 citations

Accurate Monte Carlo critical exponents for Ising lattices

J. G. García, Julio A. Gonzalo, Manuel I. Marqués

A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.

arXiv:cond-mat/0211270Nov 14, 20020 citations

Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of

\bar h=h/\mid t \mid^{βδ}

and

\bar τ=t/h^{1/βδ}

. Scaling plots for data on simple cubic Ising lattices are compared with plots of

ML^{βν}

vs.

\mid t \mid L^{1/ν}

for increasing

L

values and with high quality experimental data on

CrBr_{3}

. Finite size effects and the equation of state are discussed.

We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter

D

on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter

D

, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for

D=0.641

and 0.655 on lattices of a linear size up to

L=360

we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.

N

as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.