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arXiv:1210.4671 · Physical Review B

Corner contribution to percolation cluster numbers

István A. Kovács, Ferenc Iglói, John Cardy

Published Dec 7, 2012• 16 citations
arXivHTMLPDF

Abstract

We study the number of clusters in two-dimensional ($2d$) 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.

DOI
10.1103/physrevb.86.214203

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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.
Motivated by recent progress on the scaling behavior of entanglement entropy, we study the scaling behavior of the number of clusters crossing the boundary between two subsystems for several classical statistical models in two dimension. This number exhibits a subleading logarithmic dependence of the linear dimension of the boundary when the model is at critical, in analogy to the entanglement entropy of a quantum system. It is shown that the logarithmic scaling of the cluster number originates from the conformal invariance of the critical system. We check this numerically for Ising and Ashkin-Teller models by using Monte Carlo simulations, and show that whether a universal coefficient of the logarithmic term can be observed numerically may strongly depend on the geometry and boundary condictions of the system.
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.
The entanglement entropy for a quantum critical system across a boundary with a corner exhibits a subleading logarithmic scaling term with a scale-invariant coefficient. Using a Numerical Linked Cluster Expansion, we calculate this universal quantity for a square-lattice bilayer Heisenberg model at its quantum critical point. We find, for this 2 + 1 dimensional O (3) universality class, that it is thrice the value calculated previously for the Ising universality class. This relation gives substantial evidence that this coefficient provides a measure of the number of degrees of freedom of the theory, analogous to the central charge in a 1 + 1 dimensional conformal field theory.
In a D=2+1D=2+1D=2+1 quantum critical system, the entanglement entropy across a boundary with a corner contains a subleading logarithmic scaling term with a universal coefficient. It has been conjectured that this coefficient is, to leading order, proportional to the number of field components NNN in the associated O(N)\text{O}(N)O(N) continuum ${\ensuremath{\phi}}^{4}$ field theory. Using density matrix renormalization group calculations combined with the powerful numerical linked cluster expansion technique, we confirm this scenario for the O(2)\text{O}(2)O(2) Wilson-Fisher fixed point in a striking way, through direct calculation at the quantum critical points of two very different microscopic models. The value of this corner coefficient is, to within our numerical precision, twice the coefficient of the Ising fixed point. Our results add to the growing body of evidence that this universal term in the R\'enyi entanglement entropy reflects the number of low-energy degrees of freedom in a system, even for strongly interacting theories.
In cluster tomography, we propose measuring the number of clusters NNN 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 aaa 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 bbb 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.