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arXiv:hep-lat/9905026 · Physical Review Letters

Percolation Properties of the 2D Heisenberg Model

B. Allés, Juan J. Alonso, Carlos Criado, Michele Pepe A

Published Nov 1, 1999• 11 citations
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Abstract

We analyze the percolation properties of certain clusters defined on configurations of the two-dimensional Heisenberg model. We find that, given any direction in $\stackrel{\ensuremath{\rightarrow}}{n}$ in $O(3)$ space, spins that are almost perpendicular to $\stackrel{\ensuremath{\rightarrow}}{n}$ form a percolating cluster. This result indicates how the model can avoid a previously conjectured Kosterlitz-Thouless phase transition at a finite temperature $T$.

DOI
10.1103/physrevlett.83.3669

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We analyze the percolation properties of certain clusters defined on configurations of the two-dimensional Heisenberg model. We find that, given any direction in n\ensuremath→\stackrel{\ensuremath{\rightarrow}}{n}n\ensuremath→ in O(3)O(3)O(3) space, spins that are almost perpendicular to n\ensuremath→\stackrel{\ensuremath{\rightarrow}}{n}n\ensuremath→ form a percolating cluster. This result indicates how the model can avoid a previously conjectured Kosterlitz-Thouless phase transition at a finite temperature TTT.
We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K_{perc} from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K_{perc}, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K_{perc}=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K_{BKT}=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.
We define a percolation problem on the basis of spin configurations of the two-dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the existence of percolation transitions when the conducting angle is varied, and we determine the transition point for several values of the XY coupling. It appears that the critical behavior of this percolation model can be well described by the standard percolation theory. The critical exponents of the percolation transitions, as determined by finite-size scaling, agree with the universality class of the two-dimensional percolation model on a uniform substrate. This holds over the whole temperature range, even in the low-temperature phase where the XY substrate is critical in the sense that it displays algebraic decay of correlations.
The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to L=16384L=16384L=16384. Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with LLL. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.
The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability ppp depending on the NN coupling constant JJJ and the temperature TTT, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.
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.
We formulate the Wolff algorithm as a site-bond percolation problem, apply it to the ferromagnetic and antiferromagnetic planar triangular spin models, and study the percolation critical behavior using finite-size scaling. In the former case the Wolff algorithm is successful as an accelerating algorithm, whereas in the latter case it is not. We found the percolation temperatures and the cluster exponents for both models. In the antiferromagnetic model, the percolation temperature is higher than the critical temperature of the spin system. The cluster exponents are found to be the same as the random two-dimensional (2D) percolation. In the ferromagnetic model, the percolation temperature agrees with the critical temperature, and the cluster exponents are different from the random 2D percolation, meaning that they are in different universal classes. For the ferromagnetic model we discuss the mechanism of the cluster growth in the regime of the Kosterlitz-Thouless transition. We also note a relation between the dynamic exponent and the percolation exponents.