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arXiv:2502.10688 · Physical review. E

Percolation in a three-dimensional nonsymmetric multicolor loop model

Soumya Kanti Ganguly, Sumanta Mukherjee, Chandan Dasgupta

Published Sep 6, 2024• 2 citations
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Abstract

We conduct Monte Carlo simulations to analyze the percolation transition of a nonsymmetric loop model on a regular three-dimensional lattice. We calculate the critical exponents for the percolation transition of this model. The percolation transition occurs at a temperature that is close to, but not exactly, the thermal critical temperature. Our finite-size study on this model yields a correlation length exponent that agrees with that of the three-dimensional XY model with an error margin of 6%.

DOI
10.1103/physreve.110.034117

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We conduct Monte Carlo simulations to analyze the percolation transition of a nonsymmetric loop model on a regular three-dimensional lattice. We calculate the critical exponents for the percolation transition of this model. The percolation transition occurs at a temperature that is close to, but not exactly, the thermal critical temperature. Our finite-size study on this model yields a correlation length exponent that agrees with that of the three-dimensional XY model with an error margin of 6%.
We study order-disorder transitions in three-dimensional multicolored loop models using Monte Carlo simulations. We show that the nature of the transition is intimately related to the nature of the loops. The symmetric loops undergo a first-order phase transition, while the nonsymmetric loops show a second-order transition. The critical exponents for the nonsymmetric loops are calculated. In three dimensions, the regular loop model with no interactions is dual to the XY model. We argue that, due to interactions among the colors, the specific-heat exponent is found to be different from that of the regular loop model. The continuous nature of the transition is altered to a discontinuous one due to the strong intercolor interactions.
After fifty years of lattice gauge theories (LGTs), the nature of the transition between their topological phases (confinement or deconfinement) remains challenging due to the absence of a local order parameter. In this work, we conduct a percolation analysis of Wegner's three-dimensional Z_{2} lattice gauge model using intensive Monte Carlo simulations and finite-size scaling, offering fresh insights into the topological phase transitions of gauge-invariant systems. We demonstrate that, regardless of the connection rules, geometrical loops, constructed by piercing excited plaquettes percolate precisely at the thermal critical point T_{c}, with critical exponents coinciding with those of the loop representation of the dual 3D Ising model. Further, we construct Fortuin-Kasteleyn (FK) clusters in a random-cluster representation, showing that they also percolate at T_{c}, enabling access to all thermal critical exponents. Strikingly, the Binder cumulants of the percolation order parameters for both loops and FK clusters reveal a pseudo-first-order transition. This work sheds new light on the critical behavior of pure LGTs, with potential implications for condensed matter systems and quantum error correction.
The fractal structure of high-temperature graphs of the three-dimensional Ising and XY models is investigated by simulating these graphs directly on a cubic lattice and analyzing them with the help of percolation observables. The Ising graphs are shown to percolate right at the Curie critical point. The diverging length scale relevant to the graphs in the vicinity of the percolation threshold is shown to be provided by the spin correlation length. The fractal dimension of the high-temperature graphs at criticality is estimated to be D=1.7349(65) for the Ising and D=1.7626(66) for the XY models.
Using Monte Carlo methods and finite-size scaling, we investigate surface critical phenomena in the bond-percolation model on the simple-cubic lattice with two open surfaces in one direction. We decompose the whole lattice into percolation clusters and sample the surface and bulk dimensionless ratios Q1 and Qb, defined on the basis of the moments of the cluster-size distribution. These ratios are used to determine critical points. At the bulk percolation threshold pbc, we determine the surface bond-occupation probability at the special transition as p(s)1c = 0.418 17(2), and further obtain the corresponding surface thermal and magnetic exponents as y(s)t1 = 0.5387(2) and y(s)h1 = 1.8014(6), respectively. Next, from the pair correlation function on the surfaces, we determine y(o)h1 = 1.0246(4) and y(e)h1 = 1.25(6) for the ordinary and the extraordinary transition, respectively, of which the former is consistent with the existing result y(o)h1 = 1.024(4). We also numerically derive the line of surface phase transitions occurring at pb < pbc, and determine the pertinent asymptotic values of the universal ratios Q1 and Qb.
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.