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arXiv:2307.04260 · Physical Review Research

Cluster tomography in percolation

Helen S. Ansell, Samuel J. Frank, István A. Kovács

Published Dec 8, 2023• 4 citations
arXivHTMLPDF

Abstract

In cluster tomography, we propose measuring the number of clusters $N$ intersected by a line segment of length $\ensuremath{\ell}$ across a finite sample. As expected, the leading order of $N(\ensuremath{\ell})$ scales as $a\ensuremath{\ell}$, where $a$ depends on microscopic details of the system. However, at criticality, there is often an additional nonlinearity of the form $bln(\ensuremath{\ell})$, originating from the endpoints of the line segment. By performing large scale Monte Carlo simulations of both two- and three-dimensional percolation, we find that $b$ is universal and depends only on the angles encountered at the endpoints of the line segment intersecting the sample. Our findings are further supported by analytic arguments in two dimensions, building on results in conformal field theory. Being broadly applicable, cluster tomography can be an efficient tool for detecting phase transitions and characterizing the corresponding universality class in classical or quantum systems with a relevant cluster structure.

DOI
10.1103/physrevresearch.5.043218

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We study the number of clusters in two-dimensional (2d2d2d) critical percolation, ${N}_{\ensuremath{\Gamma}}$, which intersect a given subset of bonds, $\ensuremath{\Gamma}$. In the simplest case, when $\ensuremath{\Gamma}$ is a simple closed curve, ${N}_{\ensuremath{\Gamma}}$ is related to the entanglement entropy of the critical diluted quantum Ising model, in which $\ensuremath{\Gamma}$ represents the boundary between the subsystem and the environment. Due to corners in $\ensuremath{\Gamma}$ there are universal logarithmic corrections to ${N}_{\ensuremath{\Gamma}}$, which are calculated in the continuum limit through conformal invariance, making use of the Cardy-Peschel formula. The exact formulas are confirmed by large-scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity.
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For the two-dimensional QQQ-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number ${N}_{\ensuremath{\Gamma}}$ of clusters that intersect a given contour $\ensuremath{\Gamma}$. To leading order, ${N}_{\ensuremath{\Gamma}}$ is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of $\ensuremath{\Gamma}$. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition, we find agreement with these predictions from large-scale numerical simulations. For the spin clusters, on the other hand, the cluster numbers are not found to be consistent with the values obtained by analytic continuation, as conventionally assumed.
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.
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Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q → 1 limit of the Q -state Potts model, and analyzing the underlying S Q symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q → 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d = 2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by carrying out extensive Monte Carlo simulations.