Physical Review Letters

Critical Dynamics of an Interface in 1+ε Dimensions

Richard Bausch, V. Dohm, H. K. Janssen, Royce K. P. Zia

Published Dec 21, 1981• 134 citations

Abstract

A model describing the relaxation dynamics of an interface of Ising-like systems is introduced. By means of renormalized field theory in $d=1+\ensuremath{\epsilon}$ dimensions the dynamic critical exponent is found as $z=2+\ensuremath{\epsilon}\ensuremath{-}\frac{1}{2}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$. Interpolation with the known result near four dimensions yields good agreement with a high-temperature expansion and with recent real-space and Monte Carlo renormalization-group calculations in two dimensions.

DOI: 10.1103/physrevlett.47.1837

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Non-equilibrium relaxation and interface energy of the Ising model

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Physical Review Letters

Critical Dynamics of an Interface in 1+ε Dimensions

Richard Bausch, V. Dohm, H. K. Janssen, Royce K. P. Zia

Published Dec 21, 1981• 134 citations

Publisher

Abstract

A model describing the relaxation dynamics of an interface of Ising-like systems is introduced. By means of renormalized field theory in

d=1+\ensuremath{\epsilon}

dimensions the dynamic critical exponent is found as

z=2+\ensuremath{\epsilon}\ensuremath{-}\frac{1}{2}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})

. Interpolation with the known result near four dimensions yields good agreement with a high-temperature expansion and with recent real-space and Monte Carlo renormalization-group calculations in two dimensions.

DOI: 10.1103/physrevlett.47.1837

Related papers

Non-equilibrium relaxation and interface energy of the Ising model

Nobuyasu Ito

From the non-equilibrium critical relaxation study of the two-dimensional Ising model, the dynamical critical exponent z is estimated to be 2.165 ± 0.010 for this model. The relaxation in the ordered phase of this model is consistent with exp (−√t/τ) behavior. The interface energy of the three-dimensional Ising model is studied and the critical exponent of the correlation length ν and the critical amplitude of the surface tension σ0 are estimated to be 0.6250 ± 0.0025 and 1.42 ± 0.04, respectively. A dynamic Monte Carlo renormalization group method is successfully applied to the equilibrium properties of the three-dimensional Ising model.From the non-equilibrium critical relaxation study of the two-dimensional Ising model, the dynamical critical exponent z is estimated to be 2.165 ± 0.010 for this model. The relaxation in the ordered phase of this model is consistent with exp (−√t/τ) behavior. The interface energy of the three-dimensional Ising model is studied and the critical exponent of the correlation length ν and the critical amplitude of the surface tension σ0 are estimated to be 0.6250 ± 0.0025 and 1.42 ± 0.04, respectively. A dynamic Monte Carlo renormalization group method is successfully applied to the equilibrium properties of the three-dimensional Ising model.

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Activated dynamic scaling in the random-field Ising model: A nonperturbative functional renormalization group approach

Ivan Balog, Gilles Tarjus

The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions $d$ less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For $d>{d}_{\text{DR}}$ , on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions $d$ less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For $d>{d}_{\text{DR}}$, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.

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Critical dynamics of disordered two-dimensional Ising systems: a Monte Carlo study

Vladimir V. Prudnikov, O. N. Markov

The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.

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Critical Exponents for the Three-Dimensional Ising Model from the Real-Space Renormalization Group in Two Dimensions

Zvi Friedman

The equivalence of the critical behavior of the $d$ -dimensional Ising model at a finite temperature to that of a $d$ - 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.The equivalence of the critical behavior of the $d$-dimensional Ising model at a finite temperature to that of a $d$- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.

Physical Review LettersMay 31, 197651 citations

A model describing the relaxation dynamics of an interface of Ising-like systems is introduced. By means of renormalized field theory in

d=1+\ensuremath{\epsilon}

dimensions the dynamic critical exponent is found as

z=2+\ensuremath{\epsilon}\ensuremath{-}\frac{1}{2}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})

d

less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For

d>{d}_{\text{DR}}

, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.

A novel dynamic Monte Carlo renormalization-group method is devised. Application to the kinetic Ising model in one dimension yields a value of

z

, the dynamic critical exponent, which is in excellent agreement with the exact result. In two dimensions we find $z=2.22\ifmmode\pm\else\textpm\fi{}0.13$ with the most probable value

z=2.17

The equivalence of the critical behavior of the

d

-dimensional Ising model at a finite temperature to that of a

d

- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.