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arXiv:hep-th/9304088 · Physical Review Letters

Critical and topological properties of cluster boundaries in the 3D Ising model

Vladimir S. Dotsenko, Paul Windey, Geoffrey Harris, Enzo Marinari, Emil J. Martinec, Marco Picco

Published Aug 9, 1993• 35 citations
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Abstract

We analyze the ensemble of surfaces surrounding critical clusters at T=${\mathit{T}}_{\mathit{c}}$ in the 3D Ising model. We find that ${\mathit{N}}_{\mathit{g}}$(A), the number of surfaces of genus g and area A, behaves as ${\mathit{A}}^{\mathit{x}(\mathit{g})}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}\mathit{A}}$. We show that \ensuremath{\mu} is constant and x(g) is approximately linear; the sum ${\mathit{tsum}}_{\mathit{g}}$ ${\mathrm{N}}_{\mathit{g}}$(A) scales as a power of A. The cluster volume is proportional to its surface area. We discuss similar reuslts for the ordinary spin clusters of the 3D Ising model and for 3D bond percolation.

DOI
10.1103/physrevlett.71.811

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We analyze the ensemble of surfaces surrounding critical clusters at T=Tc{\mathit{T}}_{\mathit{c}}Tc​ in the 3D Ising model. We find that Ng{\mathit{N}}_{\mathit{g}}Ng​(A), the number of surfaces of genus g and area A, behaves as {\mathit{A}}^{\mathit{x}(\mathit{g})}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}\mathit{A}}. We show that \ensuremath{\mu} is constant and x(g) is approximately linear; the sum tsumg{\mathit{tsum}}_{\mathit{g}}tsumg​ Ng{\mathrm{N}}_{\mathit{g}}Ng​(A) scales as a power of A. The cluster volume is proportional to its surface area. We discuss similar reuslts for the ordinary spin clusters of the 3D Ising model and for 3D bond percolation.
The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the critical behavior of the 3d model. The fractal dimension and the winding angle statistics of the perimeter and external perimeter of the geometric spin clusters at the critical point suggest that, if conformally invariant in the scaling limit, they can be described by the theory of Schramm-Löwner evolution (SLE κ ) with diffusivity of κ=5 and 16/5, respectively, putting them in the same universality class as the interfaces in 2d tricritical Ising model. It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the cross-sections undergo a nontrivial percolation transition, in the same universality class as the ordinary 2d critical percolation.
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.
We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in Z 3. The models are interesting from the statistical mechanical point of view because they have a vanishing bare surface tension. As a result the action depends only on the angles of the discrete surface and not on the area, which is the antithesis of the standard 3d Ising model. The results obtained with the CVM are in good agreement with Monte Carlo simulations for the critical temperatures and the order of the transition as the selfavoidance coupling κ is varied. The value of the magnetization critical exponent β = 0.062 ±0.003, calculated with the cluster variation–Padè approximant method, is also close to the simulation results. 1 The Model In this paper we discuss the use of the cluster variation method (CVM) in mapping out the phase diagram of the gonihedric 3d Ising model. The gonihedric 3d Ising model is a generalization of the usual 3d Ising model where planar Peierls boundaries between + and − spins can be created at zero energy cost. It has been introduced in [1] in relations with string theory. The CVM method in other contexts has shown itself to be an accurate and economical way of describing the phase diagram both for first order and continuous transitions. As both the model and the method may be unfamiliar, we outline both in turn before going on to describe our results. The genesis of the model is in a novel discretized random surface theory, the so-called gonihedric string [1],
In three-dimensional critical percolation we study numerically the number of clusters ${N}_{\ensuremath{\Gamma}}$ which intersect a given subset of bonds $\ensuremath{\Gamma}$. If $\ensuremath{\Gamma}$ represents the interface between a subsystem and the environment, then ${N}_{\ensuremath{\Gamma}}$ is related to the entanglement entropy of the critical diluted quantum Ising model. Due to corners in $\ensuremath{\Gamma}$ there are singular corrections to ${N}_{\ensuremath{\Gamma}}$, which scale as ${b}_{\ensuremath{\Gamma}}ln{L}_{\ensuremath{\Gamma}}$, with ${L}_{\ensuremath{\Gamma}}$ being the linear size of $\ensuremath{\Gamma}$ and the prefactor ${b}_{\ensuremath{\Gamma}}$ is found to be universal. This result indicates that logarithmic finite-size corrections exist in the free energy of three-dimensional critical systems.
Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability ppp between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent yp=0.426(6)y_p = 0.426(6)yp​=0.426(6) and the fractal dimensions of the largest cluster, hull, and shortest path as df=1.8926(20)d_f = 1.8926(20)df​=1.8926(20), dhull=1.663(4)d_{\rm hull} = 1.663(4)dhull​=1.663(4), and dmin=1.080(10)d_{\rm min} = 1.080(10)dmin​=1.080(10), respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.