Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1402.6535 · Physical Review B

Corner contribution to percolation cluster numbers in three dimensions

István A. Kovács, Ferenc Iglói

Published May 12, 2014• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.89.174202

Related papers

Corner contribution to percolation cluster numbers

István A. Kovács, Ferenc Iglói, John Cardy

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.

arXiv:1210.4671•Physical Review B•Dec 7, 2012•16 citations

Corner contribution to the entanglement entropy of anO(3) quantum critical point in 2 + 1 dimensions

A B Kallin, E M Stoudenmire, Paul Fendley +2 more

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.

arXiv:1401.3504•Journal of Statistical Mechanics Theory and Experiment•Jun 19, 2014•96 citations

Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2+1 dimensions

Edwin Miles Stoudenmire, Peter Gustainis, Ravi Johal +2 more

In a $D=2+1$ 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 $N$ in the associated $\text{O}(N)$ 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 $\text{O}(2)$ 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.

arXiv:1409.6327•Physical Review B•Dec 2, 2014•62 citations

Universal finite-size scaling functions for percolation on three-dimensional lattices

Chai‐Yu Lin, Chin‐Kun Hu

Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability ${E}_{p},$ the percolation probability $P,$ and the probability ${W}_{n}$ for the appearance of $n$ percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions $L.$ Using the HMCSM, we calculate the existence probability ${E}_{p}$ and the percolation probability $P$ for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of ${E}_{p}$ and $P$ in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that ${E}_{p}$ and $P$ have universal finite-size scaling functions. This implies that the critical ${E}_{p}$ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that ${W}_{n}$ for site and bond percolation on sc lattices have universal finite-size scaling functions.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 1998•50 citations

Finite-size scaling of energylike quantities in percolation

Youjin Deng, Xianqing Yang

We study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds, ${N}_{c}$ and ${N}_{b}$, respectively. These quantities include ${C}_{2c}=(⟨{N}_{c}^{2}⟩\ensuremath{-}{⟨{N}_{c}⟩}^{2})∕{L}^{d}$ and ${C}_{cb}=(⟨{N}_{c}{N}_{b}⟩\ensuremath{-}⟨{N}_{c}⟩⟨{N}_{b}⟩)∕{L}^{d}$, where $L$ is the linear system size and $d$ is the spatial dimensionality. In statistical models with thermal fluctuations, ${C}_{2c}$ and ${C}_{cb}$ are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of ${C}_{2c}$ and ${C}_{cb}$ is described by the thermal critical exponent ${y}_{t}\text{\ensuremath{-}}d$. We also measure quantity ${\ensuremath{\kappa}}_{b}=2⟨{N}_{b}{S}_{2}⟩∕⟨{S}_{2}⟩\ensuremath{-}⟨{N}_{b}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩∕⟨{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩\ensuremath{-}⟨{N}_{b}⟩$ and an analogous quantity ${\ensuremath{\kappa}}_{c}$ for ${N}_{c}$, where ${S}_{2}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}$ are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ diverge as ${L}^{{y}_{t}}$ for $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The analysis of the data of ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ yields ${y}_{t}=1.145(2)$ for the three-dimensional percolation, in good agreement with existing results.

Physical Review E•Jun 13, 2006•4 citations

Corner contribution to cluster numbers in the Potts model

István A. Kovács, Eren Metin Elçi, Martin Weigel +1 more

For the two-dimensional $Q$-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.

arXiv:1311.4186•Physical Review B•Feb 24, 2014•11 citations
In three-dimensional critical percolation we study numerically the number of clusters N\ensuremathΓ{N}_{\ensuremath{\Gamma}}N\ensuremathΓ​ which intersect a given subset of bonds \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ. If \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ represents the interface between a subsystem and the environment, then N\ensuremathΓ{N}_{\ensuremath{\Gamma}}N\ensuremathΓ​ is related to the entanglement entropy of the critical diluted quantum Ising model. Due to corners in \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ there are singular corrections to N\ensuremathΓ{N}_{\ensuremath{\Gamma}}N\ensuremathΓ​, which scale as b\ensuremathΓlnL\ensuremathΓ{b}_{\ensuremath{\Gamma}}ln{L}_{\ensuremath{\Gamma}}b\ensuremathΓ​lnL\ensuremathΓ​, with L\ensuremathΓ{L}_{\ensuremath{\Gamma}}L\ensuremathΓ​ being the linear size of \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ and the prefactor b\ensuremathΓ{b}_{\ensuremath{\Gamma}}b\ensuremathΓ​ is found to be universal. This result indicates that logarithmic finite-size corrections exist in the free energy of three-dimensional critical systems.
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.
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.
Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Ep,{E}_{p},Ep​, the percolation probability P,P,P, and the probability Wn{W}_{n}Wn​ for the appearance of nnn percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions L.L.L. Using the HMCSM, we calculate the existence probability Ep{E}_{p}Ep​ and the percolation probability PPP for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Ep{E}_{p}Ep​ and PPP in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Ep{E}_{p}Ep​ and PPP have universal finite-size scaling functions. This implies that the critical Ep{E}_{p}Ep​ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that Wn{W}_{n}Wn​ for site and bond percolation on sc lattices have universal finite-size scaling functions.
We study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds, Nc{N}_{c}Nc​ and Nb{N}_{b}Nb​, respectively. These quantities include ${C}_{2c}=(⟨{N}_{c}^{2}⟩\ensuremath{-}{⟨{N}_{c}⟩}^{2})∕{L}^{d}$ and ${C}_{cb}=(⟨{N}_{c}{N}_{b}⟩\ensuremath{-}⟨{N}_{c}⟩⟨{N}_{b}⟩)∕{L}^{d}$, where LLL is the linear system size and ddd is the spatial dimensionality. In statistical models with thermal fluctuations, C2c{C}_{2c}C2c​ and Ccb{C}_{cb}Ccb​ are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of C2c{C}_{2c}C2c​ and Ccb{C}_{cb}Ccb​ is described by the thermal critical exponent ${y}_{t}\text{\ensuremath{-}}d$. We also measure quantity ${\ensuremath{\kappa}}_{b}=2⟨{N}_{b}{S}_{2}⟩∕⟨{S}_{2}⟩\ensuremath{-}⟨{N}_{b}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩∕⟨{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩\ensuremath{-}⟨{N}_{b}⟩$ and an analogous quantity ${\ensuremath{\kappa}}_{c}$ for Nc{N}_{c}Nc​, where S2{S}_{2}S2​ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}$ are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ diverge as Lyt{L}^{{y}_{t}}Lyt​ for $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The analysis of the data of ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ yields yt=1.145(2){y}_{t}=1.145(2)yt​=1.145(2) for the three-dimensional percolation, in good agreement with existing results.
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.