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Physical review. E

Elastic backbone phase transition in the Ising model

M. N. Najafi, Jafar Cheraghalizadeh, Hans Jürgen Herrmann

Published Oct 24, 2019• 5 citations
Publisher

Abstract

The two-dimensional (zero magnetic field) Ising model is known to undergo a second-order paraferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature ${T}_{\text{eb}}<{T}_{c}$ at which the elastic backbone of FK clusters undergoes a second-order phase transition to a dense phase. The corresponding universality class, which is characterized by determining various percolation exponents, is shown to be completely different from directed percolation, which leads us to propose a new anisotropic universality class with $\ensuremath{\beta}=0.54\ifmmode\pm\else\textpm\fi{}0.02$, ${\ensuremath{\nu}}_{||}=1.86\ifmmode\pm\else\textpm\fi{}0.01$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=1.21\ifmmode\pm\else\textpm\fi{}0.04$, and ${d}_{f}=1.53\ifmmode\pm\else\textpm\fi{}0.03$. All tested hyperscaling relations are shown to be valid.

DOI
10.1103/physreve.100.042132

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The two-dimensional (zero magnetic field) Ising model is known to undergo a second-order paraferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature Teb<Tc{T}_{\text{eb}}<{T}_{c}Teb​<Tc​ at which the elastic backbone of FK clusters undergoes a second-order phase transition to a dense phase. The corresponding universality class, which is characterized by determining various percolation exponents, is shown to be completely different from directed percolation, which leads us to propose a new anisotropic universality class with \ensuremathβ=0.54\ifmmode±\else\textpm\fi0.02\ensuremath{\beta}=0.54\ifmmode\pm\else\textpm\fi{}0.02\ensuremathβ=0.54\ifmmode±\else\textpm\fi0.02, \ensuremathν∣∣=1.86\ifmmode±\else\textpm\fi0.01{\ensuremath{\nu}}_{||}=1.86\ifmmode\pm\else\textpm\fi{}0.01\ensuremathν∣∣​=1.86\ifmmode±\else\textpm\fi0.01, \ensuremathν\ensuremath⊥=1.21\ifmmode±\else\textpm\fi0.04{\ensuremath{\nu}}_{\ensuremath{\perp}}=1.21\ifmmode\pm\else\textpm\fi{}0.04\ensuremathν\ensuremath⊥​=1.21\ifmmode±\else\textpm\fi0.04, and df=1.53\ifmmode±\else\textpm\fi0.03{d}_{f}=1.53\ifmmode\pm\else\textpm\fi{}0.03df​=1.53\ifmmode±\else\textpm\fi0.03. All tested hyperscaling relations are shown to be valid.
In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point Tc{T}_{c}Tc​ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability pB<1,{p}_{B}<1,pB​<1, the corresponding site-bond clusters keep on percolating at Tc{T}_{c}Tc​ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability pc{p}_{c}pc​ such that, for any pB> pc,{p}_{B}>~{p}_{c},pB​> pc​, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for pc<pB< 1{p}_{c}<{p}_{B}<~1pc​<pB​< 1 but, for pB=pc,{p}_{B}{=p}_{c},pB​=pc​, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.
The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including SU(N)\mathrm{SU}(N)SU(N) lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.
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 q-state Potts model can be formulated in geometric terms, with Fortuin-Kasteleyn (FK) clusters as fundamental objects. For vanishing external field, the phase transition of the model can be equivalently described as a percolation transition of FK clusters. In this work, we investigate numerically the percolation behaviour along the line of first-order phase transitions of the 3d 3-state Potts model in a non-vanishing external field and find that the percolation strength exhibits a discontinuity along the entire line. The endpoint is also a percolation point for the FK clusters, but the corresponding critical exponents are neither in the Ising nor in the random percolation universality class. (C) 2002 Published by Elsevier Science B.V.
Abstract In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent ν ∼ 1.3 is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent α ∼ − 1.9 . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.