Dynamic Monte Carlo renormalization-group method

Lacasse, Martin D.; Viñals, Jorge; Grant, Martin

doi:10.1103/physrevb.47.5646

Cited by 36 publications

(17 citation statements)

References 44 publications

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“…15 There have been many theoretical and computational studies of critical slowing down for the 2D Ising class magnetic system ͑which belongs to model A in the Hohenberg and Halperin hierarchy of dynamic universality classes 1 ͒. Lacasse et al 16 gave an excellent survey of this work up to 1993, and Nightingale and Blöte 17 listed additional results up to 1996. As simulations and theories became more sophisticated, most current studies placed the value of z between 2.1 and 2.25, with many theoretical results clustered near 2.…”

Section: Introductionmentioning

confidence: 98%

Critical slowing down in the two-dimensional Ising model measured using ferromagnetic ultrathin films

Dunlavy

Venus

2005

Phys. Rev. B

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Power-law scaling of the relaxation time associated with critical slowing down has been experimentally measured in the dynamics of the magnetization of a bilayer of iron grown on top of a W͑110͒ substrate using the complex magnetic ac susceptibility ͑T͒. The observed value of the critical exponent for the slowing down above the Curie transition of this two-dimensional Ising ferromagnetic system is z = 2.09± 0.06 ͑95% confidence͒, in agreement with most contemporary theories and simulations. Further analysis reveals that dynamical effects cause ͑T͒ to deviate from power-law scaling as the temperature is decreased towards T c , whereas the saturation of the correlation length due to finite-size effects ͑on the order of 500 lattice spaces͒ limits the divergence of .

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Section: Introductionmentioning

confidence: 98%

Critical slowing down in the two-dimensional Ising model measured using ferromagnetic ultrathin films

Dunlavy

Venus

2005

Phys. Rev. B

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“…It has been successfully used to tackle problems such as asymptotic scaling behavior in spinodal decomposition [12] and the approach to equilibrium in systems with continuous symmetries, such as XY magnets [13] and liquid crystals [14]. There have also been attempts at using the RG in dynamic Monte Carlo simulations [15,16].…”

Section: Introductionmentioning

confidence: 99%

Renormalization group and perfect operators for stochastic differential equations

Hou

Goldenfeld

McKane³

2001

Phys. Rev. E

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We develop renormalization group (RG) methods for solving partial and stochastic differential equations on coarse meshes. RG transformations are used to calculate the precise effect of small scale dynamics on the dynamics at the mesh size. The fixed point of these transformations yields a perfect operator -an exact representation of physical observables on the mesh scale with minimal lattice artifacts. We apply the formalism to simple nonlinear models of critical dynamics, and show how the method leads to an improvement in the computational performance of Monte Carlo methods.

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“…In particular, the value of the dynamic critical exponent z is still an open question even for the two-dimensional Ising model [3][4][5][6] when dynamics with local flips of spins are considered. The determination of the exponent z for classical models in different lattice dimensions has been done by using several approaches: field-theoretical dynamical renormalization group methods, 2,7 Monte Carlo simulations, [8][9][10] renormalization group methods, [11][12][13][14] damage spreading, 6,15,16 nonequilibrium relaxation, 17,18 and series expansion. 4,19 For the Ising model the various methods obtain in two dimensions 2.10ϽzϽ2.52 and in three dimensions 1.95ϽzϽ2.35.…”

Section: ͓S0163-1829͑97͒00302-0͔mentioning

confidence: 99%

Numerical method to evaluate the dynamical critical exponent

1997

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A finite-size scaling approach is used to show numerically that dynamical scaling occurs for short and long times independently of the initial conditions. Its main idea is to construct particular quantities scaling as L 0 in the thermodynamic limit L→ϱ, L being the linear size of the system. These are the quantities for which the dynamic scaling occurs for short and long times. This approach is applied to obtain the critical dynamical behavior of two-and three-dimensional ferromagnetic Ising models, subjected to Glauber dynamics. ͓S0163-1829͑97͒00302-0͔Although the static critical properties of classical spin systems are well described by renormalization group theories ͑theoretical framework and calculation procedures͒, 1 the dynamic critical properties 2 are not as well understood. In particular, the value of the dynamic critical exponent z is still an open question even for the two-dimensional Ising model 3-6 when dynamics with local flips of spins are considered. The determination of the exponent z for classical models in different lattice dimensions has been done by using several approaches: field-theoretical dynamical renormalization group methods, 2,7 Monte Carlo simulations, 8-10 renormalization group methods, 11-14 damage spreading, 6,15,16 nonequilibrium relaxation, 17,18 and series expansion. 4,19 For the Ising model the various methods obtain in two dimensions 2.10ϽzϽ2.52 and in three dimensions 1.95ϽzϽ2.35. Usually, some of these methods obtain the z exponent from longtime behavior. This limit is hard to be reached because of the critical slowing down that always appears, except for clusters algorithms. 20 Besides, it is also very hard to obtain good statistics in these procedures. Critical slowing down and poor statistics are among the reasons why different calculations produce so many different values for z.Our objective in this work is to present a finite-size dynamical scaling approach which overcomes the usual difficulties pointed out above. The method is then applied to twoand three-dimensional kinetic Ising models with single spin flips. The method can also be applied to systems with more complex ordering behavior subjected to different dynamics.Recently, a method has been proposed 21 to evaluate the z exponent from short-time behavior. It is based on the scaling relation for the dynamics at early times. 22 In this time regime the magnetization initially grows, characterizing a new universal stage of the relaxation of the magnetization, the so called ''critical initial slip.'' However, it turns out that the initial condition ͑zero magnetization and very short correlation length͒ is essential to obtain the dynamical exponent. This happens because the critical initial slip sets right in after a microscopic time scale and eventually crosses over the long-time regime. The characteristic time associated with the critical initial slip is t 0 ϭm 0Ϫz/x , where m 0 is the initial magnetization and x is a new exponent. If m 0 ϭ0, we have that t 0 →ϱ and the early time scaling overlaps with the expected l...

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Dynamic Monte Carlo renormalization-group method

Cited by 36 publications

References 44 publications

Critical slowing down in the two-dimensional Ising model measured using ferromagnetic ultrathin films

Critical slowing down in the two-dimensional Ising model measured using ferromagnetic ultrathin films

Renormalization group and perfect operators for stochastic differential equations

Numerical method to evaluate the dynamical critical exponent

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