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The Journal of Chemical Physics

Classical and quantum continuum percolation with hard core interactions

Jeffery G. Saven, James L. Skinner, J. R. Wright

Published May 1, 1991• 16 citations
Publisher

Abstract

We study the classical and quantum percolation of spheres in a three-dimensional continuum. Each sphere has an impenetrable hard core of diameter σ, and two spheres are considered to be directly connected if the distance between their centers is less than d. We calculate the critical percolation density as a function of σ/d. In the classical problem this is the density ρc at which an infinite cluster of connected spheres first forms. In the quantum problem, we study a tight-binding model where the hopping matrix element between two spheres is nonzero only if they are directly connected. In this case the critical density ρq is the density at which the eigenstates of the Hamiltonian first become extended. Our method uses Monte Carlo simulation and finite-size scaling techniques, and for the quantum problem, the concept of quantum connectivity. We find that both ρc and ρq exhibit nonmonotonic behavior as a function of σ/d. We also find that for all values of σ/d, ρq>ρc, although the ratio of the thresholds decreases with increasing σ/d. We argue that a better understanding of this ratio is obtained by considering the average coordination number. We speculate about the nature of both classical and quantum percolation as σ/d approaches 1.

DOI
10.1063/1.460401

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We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.
We introduce the concept of quantum connectivity, which, along with the phenomenological renormalization method, allows us to study the localization of one-electron eigenstates of various disordered tight-binding Hamiltonians. For the site quantum percolation model in three dimensions we find that the quantum percolation threshold is pq{p}_{q}pq​=0.48\ifmmode\pm\else\textpm\fi{}0.01. For the Anderson model we find that the critical disorder is Wc{W}_{c}Wc​/J=15.95\ifmmode\pm\else\textpm\fi{}0.25. For a topologically disordered model with exponential interactions of range a, we find that the dimensionless critical density parameter Rc{\mathit{R}}_{\mathit{c}}Rc​=${\mathrm{\ensuremath{\rho}}}_{\mathit{c}}^{1/3}$a=0.257\ifmmode\pm\else\textpm\fi{}0.014.
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.
The scaling behavior for a binary fragmentation of critical percolation clusters is investigated by a large-cell Monte Carlo real-space renormalization group method in two and three dimensions. We obtain accurate values of critical exponents λ and ϕ describing the scaling of fragmentation rate and the distribution of fragments' masses produced by a binary fragmentation. Our results for λ and ϕ show that the fragmentation rate is proportional to the size of mother cluster, and the scaling relation σ = 1 + λ − ϕ conjectured by Edwards et al. to be valid for all dimensions is satisfied in two and three dimensions, where σ is the crossover exponent of the average cluster number in percolation theory, which excludes the other scaling relations.
Quantum percolation describes the problem of a quantum particle moving through a disordered system. While certain similarities to classical percolation exist, the quantum case has additional complexity due to the possibility of Anderson localisation. Here, we consider a directed discrete-time quantum walk as a model to study quantum percolation of a two-state particle on a two-dimensional lattice. Using numerical analysis we determine the fraction of connected edges required (transition point) in the lattice for the two-state particle to percolate with finite (non-zero) probability for three fundamental lattice geometries, finite square lattice, honeycomb lattice, and nanotube structure and show that it tends towards unity for increasing lattice sizes. To support the numerical results we also use a continuum approximation to analytically derive the expression for the percolation probability for the case of the square lattice and show that it agrees with the numerically obtained results for the discrete case. Beyond the fundamental interest to understand the dynamics of a two-state particle on a lattice (network) with disconnected vertices, our study has the potential to shed light on the transport dynamics in various quantum condensed matter systems and the construction of quantum information processing and communication protocols.
The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random bond dilution is studied using quantum Monte Carlo simulation at the percolation threshold (50% of the bonds removed). Finite-size scaling of the staggered structure factor averaged over the largest connected clusters of sites on L*L lattices shows that long-range order exists within the percolating fractal clusters in the thermodynamic limit. This implies that the order-disorder transition driven by bond-dilution occurs exactly at the percolation threshold and that the exponents are classical. This result should apply also to the site-diluted system.