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Physical Review Letters

Critical Ising Spin Dynamics on Percolation Clusters

Christopher L. Henley

Published May 6, 1985• 109 citations
Publisher

Abstract

Ferromagnetically interacting Ising spins are placed on a fractal network (such as a critical percolation cluster) with ${T}_{c}=0$, and endowed with a single-spin-flip dynamics. At low temperatures the collective dynamics is determined by thermal activation over energy barriers. The barrier to overturning of the spins in a domain of size $L$ is proportional to $Z\mathrm{ln}L$, where $Z$ is a new geometrical parameter characterizing the fractal. A new "singular" dynamic scaling is found in which the effective dynamic critical exponent diverges at criticality.

DOI
10.1103/physrevlett.54.2030

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Ferromagnetically interacting Ising spins are placed on a fractal network (such as a critical percolation cluster) with Tc=0{T}_{c}=0Tc​=0, and endowed with a single-spin-flip dynamics. At low temperatures the collective dynamics is determined by thermal activation over energy barriers. The barrier to overturning of the spins in a domain of size LLL is proportional to ZlnLZ\mathrm{ln}LZlnL, where ZZZ is a new geometrical parameter characterizing the fractal. A new "singular" dynamic scaling is found in which the effective dynamic critical exponent diverges at criticality.
A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of sss spins is shown to scale as lns\mathrm{ln}slns. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.
In this paper we consider the Ising model (called the dynamical Ising model with dynamical temperature TTT) on the Ising-correlated percolation lattice. To construct this lattice, another Ising model (which is called the quenched Ising model with quench artificial temperature Tq{T}_{q}Tq​) is employed to model the correlations between imperfections. We argue that for each quenched temperature, there is a particular dynamical temperature Tc(Tq){T}_{c}({T}_{q})Tc​(Tq​) for which the dynamical Ising model becomes critical. We present some finite-size arguments (based on the moments of the magnetism and the energy as well as the spanning cluster probability) to extract the critical points and also show that they are compatible with the finite-size scaling of the singular point of the magnetic susceptibility. The model is thoroughly characterized in and out of these points. We find that the critical behaviors of the model change significantly with respect to the regular Ising model as well as the Ising model on the uncorrelated percolation lattice. It is shown that for the critical lattice, ${T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}}\ensuremath{\approx}2.269$, the critical temperature for the dynamical Ising model in the thermodynamic limit is ${T}_{c}({T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}})=1.94\ifmmode\pm\else\textpm\fi{}0.005$, and the fractal dimension of the exterior perimeter of geometrical spin clusters is ${D}_{f}^{T={T}_{c}({T}_{q}=2.269)}=1.408\ifmmode\pm\else\textpm\fi{}0.002$. Many quantities, such as the dynamical critical temperatures, all local and global critical exponents, and the fractal dimension of loops Df{D}_{f}Df​, scale with the quench temperature in a power-law fashion, with some critical exponents that are reported. Significantly we see that ${D}_{f}^{T={T}_{c}({T}_{q})}\ensuremath{-}{D}_{f}^{T={T}_{c}({T}_{q}=2.269)}\ensuremath{\sim}\frac{1}{\sqrt{\ensuremath{\zeta}({T}_{q})}}$ in which $\ensuremath{\zeta}({T}_{q})$ is the correlation length of the quenched Ising model at temperature Tq{T}_{q}Tq​.
An Ising spin system with ferromagnetic nearest-neighbor interactions J is placed on critical percolation clusters on a Cayley tree (i.e., on a Bethe lattice). An energy barrier for a flip between the two ferromagnetically aligned ground states of a cluster of S sites is defined by the process of pushing a domain wall down from the ``head'' of the tree out through its branches. Two independent calculations indicate that the barrier typically scales as 2J(lnS/ln4). This is the same form proposed for finite-dimensional percolation clusters; it leads to ``singular'' dynamic scaling.
The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.
The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability p, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as p increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range p_{c_{1}}<p<p_{c_{2}}, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At p_{c1} and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At p_{c2}, the fractal dimension of clusters and the scaling exponent along p direction are estimated as y_{h2}=1.9580(6) and y_{p2}=0.552(9), of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.
The critical dynamics of Heisenberg spin systems at the percolation threshold is obtained from the transformation of the equations of motion under a length scaling acheived by climination of sites. The method is applied exactly to ddd-dimensional nonrandom fractals representing the infinite-cluster backbone and approximatcly to a truly random twodimensional system to yield dynamic scaling forms for response function and characteristic frequency, and associated static and dynamic critical exponents.