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arXiv:2407.00338 · SciPost Physics

Bethe MMM-layer construction for the percolation problem

Maria Chiara Angelini, Saverio Palazzi, Tommaso Rizzo, Marco Tarzia

Published Jan 23, 2025• 5 citations
arXivHTMLPDF

Abstract

One way to perform field theory computations for the bond percolation problem is through the Kasteleyn and Fortuin mapping to the n+1 n + 1 states Potts model in the limit of n \to 0 n → 0 . In this paper, we show that it is possible to recover the \epsilon ϵ -expansion for critical exponents in finite dimension directly using the M M -layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the M M -layer method to systems where the underlying field theory is unknown or disputed.

DOI
10.21468/scipostphys.18.1.030

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One way to perform field theory computations for the bond percolation problem is through the Kasteleyn and Fortuin mapping to the n+1 n + 1 states Potts model in the limit of n \to 0 n → 0 . In this paper, we show that it is possible to recover the \epsilon ϵ -expansion for critical exponents in finite dimension directly using the M M -layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the M M -layer method to systems where the underlying field theory is unknown or disputed.
This thesis presents and applies the M-layer construction, a recent diagrammatic method for studying the critical behavior of lattice models in finite dimensions. The construction provides a unified tool to compute correlation functions and critical exponents through a perturbative expansion in topological loops, whose leading order corresponds to the behavior on a Bethe lattice. The first application of the method is on well-known problems, such as the Ising model and the percolation problem, where it successfully reproduces standard results, thus validating its efficiency. It is then applied to the Edwards–Anderson spin glass model in a magnetic field at zero temperature, a problem whose finite-dimensional physics remains widely debated. The remarkable new result of the M-layer construction is the identification, for the first time through analytical computations, of a phase transition in a field below the upper critical dimension. The results, obtained through analytical computations and partially checked numerically, show that this approach provides a promising alternative to standard Renormalization Group techniques for investigating phase transitions in disordered and frustrated systems.
The continuous n-state Potts model is reinvestigated. The partition function is cast in a form allowing the limit, n\ensuremath{\rightarrow}1, to be taken prior to perturbation-theoretical expansions. Proceeding in this way, we free the theory of pathologies known to plague other approaches. Several alternatives are presented for a Feynman-diagrammatic expansion of the theory, two of which are identical to those utilized in previous analyses of the mdoel. One of the introduced diagrammatic approaches is used to recover known results of one-loop renormalization calculations. It is also utilized to calculate the partition function of the (1+\ensuremath{\epsilon})-state Potts model, and, hence, the generating function for bond percolation, on a finite lattice. This generating function for cluster statistics exhibits the scaling features proposed by Privman and Fisher for systems of finite size.
The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized" percolation rules correlated to entire blocks of spins. For example these rules allow for percolation on "virtual" bonds between spins not present in the underlying Hamiltonian. As a consequence we can define new random cluster theories each with its own Monte Carlo cluster dynamics that exactly reproduce the Ising model. By tuning parameters on the critical percolation surface, it is demonstrated numerically that cluster algorithms can be formulated for the 2-d and 3-d Ising model that have smaller autocorrelations than the original Swendsen-Wang algorithm.
Abstract In statistical physics, one of the standard methods to study second order phase transitions is the renormalization group that usually leads to an expansion around the corresponding fully connected solution. Unfortunately, often in disordered models, some important finite dimensional second-order phase transitions are qualitatively different or absent in the corresponding fully connected model: in such cases the standard expansion fails. Recently, a new method, the M-layer one, has been introduced that performs an expansion around a different soluble mean field model: the Bethe lattice one. This new method has been already used to compute the upper critical dimension D U of different disordered systems such as the Random Field Ising model or the Spin glass model with field. If then one wants to go beyond and construct an expansion around D U to understand how critical quantities get renormalized, the actual computation of all the numerical factors is needed. This next step has still not been performed, being technically more involved. In this paper we perform this computation for the ferromagnetic Ising model without quenched disorder, in finite dimensions: we show that, at one-loop order inside the M-layer approach, we recover the continuum quartic field theory and we are able to identify the coupling constant g and the other parameters of the theory, as a function of macroscopic and microscopic details of the model such as the lattice spacing, the physical lattice dimension and the temperature. This is a fundamental step that will help in applying in the future the same techniques to more complicated systems, for which the standard field theoretical approach is impracticable.
The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.