Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

Dynamics of the Ising Model near the Critical Point. I

Masuo Suzuki, Ryogo Kubo

Published Jan 15, 1968• 513 citations
Publisher

Abstract

A simple dynamical model of interacting Ising spins is discussed. Each spin is assumed to flip spontaneously with a transition probability which depends on the temperature and the configuration of surrounding spins, but its functional form is assumed to be the simplest. The frequency-wave number dependent susceptibility χ( q , ω) is given exactly in the one-dimensional case. In two-and three-dimensional cases the model is treated in the molecular field and the generalized approximations.

DOI
10.1143/jpsj.24.51

Related papers

Dynamics of the Ising Model near the Transition Point. II

Ryuzo Abe, Akira Hatano

The Isin spin system is discussed in which the interaction of spins with a heat bath causes the spontaneous flip of spins. A treatment discussed in a previous paper for the square lattice is extended to the honeycomb and triangular lattices. In the case of honeycomb lattice, an additional term to the one studied in the previous paper is calculated explicitly at the transition point and an exact expansion is derived for the dynamic susceptibililty χ(ω) up to the term of the order of 1/ω2. Some discussions are given of the low frequency behavior of χ(ω) on the basis of variational principle.

Progress of Theoretical Physics•Apr 1, 1969•29 citations

Dynamics of the Ising Model near the Transition Point

Ryuzo Abe

The Ising spin system is discussed in which the interaction of spins with heat bath causes the spontaneous flip of spins. It is assumed that the system is in thermal equilibrium up to some instant under a uniform and sufficiently small magnetic field and the magnetic field is switched off at that instant. The initial decay of magnetization in such a situation is studied. In the neighborhood of transition point, it is proved that a relaxation time associated with the initial decay is proportional to the susceptibility, and an exact proportionality constant is calculated for the two-dimensional square lattice with nearest-neighbor interaction.

Progress of Theoretical Physics•Apr 1, 1968•35 citations

Critical Slowing Down in the Kinetic Ising Model

Hideo Yahata, Masuo Suzuki

A simple dynamical model of interacting Ising spins is discussed near the critical point. It is shown that all the moments are positive and never vanish even at the critical point. Lower and upper bounds of the susceptibility are obtained, which yield the non-vanishing of the real part of the susceptibility. This gives a simple proof for the poly-dispersive nature of our system. The method of high temperature expansions for the dynamical susceptibility is presented and is applied to the two-dimensional Ising model up to the ninth order, the results of which show that the critical index of slowing down (the relaxation time τ (1) ∝( T - T c ) - Δ ; Δ =2±0.05) is different from that of the static susceptibility.

Journal of the Physical Society of Japan•Dec 1, 1969•143 citations

Critical behavior of the random-field Ising model

Misha Gofman, Joan Adler, Amnon Aharony A +2 more

We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{〉}}_{\mathit{T}}$〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$${]}_{\mathit{h}}$ and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$${]}_{\mathit{h}}$, where 〈${\mathrm{〉}}_{\mathit{T}}$ indicates a canonical average at temperature T for an arbitrary configuration of random fields and [${]}_{\mathit{h}}$ indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each ${\mathit{h}}_{\mathit{i}}$=\ifmmode\pm\else\textpm\fi{}${\mathit{h}}_{0}$ and a Gaussian distribution in which each ${\mathit{h}}_{\mathit{i}}$ has variance ${\mathit{h}}_{0}^{2}$. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$${\mathit{a}}_{\mathit{n}}$(g,d)(J/T${)}^{\mathit{n}}$, where J is the exchange constant and the coefficients ${\mathit{a}}_{\mathit{n}}$(g,d) are polynomials in g\ensuremath{\equiv}${\mathit{h}}_{0}^{2}$/${\mathit{J}}^{2}$ and in d. We assume that as T approaches its critical value, ${\mathit{T}}_{\mathit{c}}$, one has \ensuremath{\chi}\ensuremath{\sim}(T-${\mathit{T}}_{\mathit{c}}$${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-${\mathit{T}}_{\mathit{c}}$${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(${\mathit{T}}^{2}$)/(${\mathit{gJ}}^{2}$). We find that A approaches a constant value as T\ensuremath{\rightarrow}${\mathit{T}}_{\mathit{c}}$ (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Mar 1, 1996•59 citations

Diffusion Constants near the Critical Point for Time-Dependent Ising Models. I

Kyozi Kawasaki

The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor ${\ensuremath{\chi}}^{\ensuremath{-}1}$, where $\ensuremath{\chi}$ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else \'{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.

Physical Review•May 6, 1966•616 citations

Solution and Critical Behavior of Some "Three-Dimensional" Ising Models with a Four-Spin Interaction

Masuo Suzuki

An exact solution is obtained for some "three-dimensional" Ising models with a four-spin interaction. The spontaneous magnetization is vanishing for all temperatures. The temperature derivative of the susceptibility shows a divergence of the form ${({T}_{c}\ensuremath{-}T)}^{\frac{\ensuremath{-}7}{8}}$ below the critical point. The specific heat shows a logarithmic singularity. Several other related models are introduced with new types of singularities.

Physical Review Letters•Feb 21, 1972•73 citations
A simple dynamical model of interacting Ising spins is discussed. Each spin is assumed to flip spontaneously with a transition probability which depends on the temperature and the configuration of surrounding spins, but its functional form is assumed to be the simplest. The frequency-wave number dependent susceptibility χ( q , ω) is given exactly in the one-dimensional case. In two-and three-dimensional cases the model is treated in the molecular field and the generalized approximations.
The Isin spin system is discussed in which the interaction of spins with a heat bath causes the spontaneous flip of spins. A treatment discussed in a previous paper for the square lattice is extended to the honeycomb and triangular lattices. In the case of honeycomb lattice, an additional term to the one studied in the previous paper is calculated explicitly at the transition point and an exact expansion is derived for the dynamic susceptibililty χ(ω) up to the term of the order of 1/ω2. Some discussions are given of the low frequency behavior of χ(ω) on the basis of variational principle.
The Ising spin system is discussed in which the interaction of spins with heat bath causes the spontaneous flip of spins. It is assumed that the system is in thermal equilibrium up to some instant under a uniform and sufficiently small magnetic field and the magnetic field is switched off at that instant. The initial decay of magnetization in such a situation is studied. In the neighborhood of transition point, it is proved that a relaxation time associated with the initial decay is proportional to the susceptibility, and an exact proportionality constant is calculated for the two-dimensional square lattice with nearest-neighbor interaction.
A simple dynamical model of interacting Ising spins is discussed near the critical point. It is shown that all the moments are positive and never vanish even at the critical point. Lower and upper bounds of the susceptibility are obtained, which yield the non-vanishing of the real part of the susceptibility. This gives a simple proof for the poly-dispersive nature of our system. The method of high temperature expansions for the dynamical susceptibility is presented and is applied to the two-dimensional Ising model up to the ninth order, the results of which show that the critical index of slowing down (the relaxation time τ (1) ∝( T - T c ) - Δ ; Δ =2±0.05) is different from that of the static susceptibility.
We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​]h{]}_{\mathit{h}}]h​ and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​]h{]}_{\mathit{h}}]h​, where 〈〉T{\mathrm{〉}}_{\mathit{T}}〉T​ indicates a canonical average at temperature T for an arbitrary configuration of random fields and []h{]}_{\mathit{h}}]h​ indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each hi{\mathit{h}}_{\mathit{i}}hi​=\ifmmode\pm\else\textpm\fi{}h0{\mathit{h}}_{0}h0​ and a Gaussian distribution in which each hi{\mathit{h}}_{\mathit{i}}hi​ has variance h02{\mathit{h}}_{0}^{2}h02​. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$an{\mathit{a}}_{\mathit{n}}an​(g,d)(J/T)n{)}^{\mathit{n}})n, where J is the exchange constant and the coefficients an{\mathit{a}}_{\mathit{n}}an​(g,d) are polynomials in g\ensuremath{\equiv}h02{\mathit{h}}_{0}^{2}h02​/J2{\mathit{J}}^{2}J2 and in d. We assume that as T approaches its critical value, Tc{\mathit{T}}_{\mathit{c}}Tc​, one has \ensuremath{\chi}\ensuremath{\sim}(T-Tc{\mathit{T}}_{\mathit{c}}Tc​${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-Tc{\mathit{T}}_{\mathit{c}}Tc​${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(T2{\mathit{T}}^{2}T2)/(gJ2{\mathit{gJ}}^{2}gJ2). We find that A approaches a constant value as T\ensuremath{\rightarrow}Tc{\mathit{T}}_{\mathit{c}}Tc​ (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.
The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor ${\ensuremath{\chi}}^{\ensuremath{-}1}$, where $\ensuremath{\chi}$ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else \'{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.