Quantum percolation and ballistic conductance on a lattice of wires

Motivated by concepts of classical electrical percolation theory, we study the quantum-mechanical electrical conductance of a lattice of wires as a function of the bond-occupation probability p. In the ordered or ballistic case (p=1), we obtain an analytic expression for the energy dispersion relation of the Bloch electrons, which couples all the transverse momenta. We also get closed-form expressions for the conductance gNL of a finite system of transverse dimension N^d-1 and length L (with d=2 or 3). In the limit L{→}{∞}, the conductance is quantized similarly to what is found for the conductance of narrow constrictions. We also obtain a closed-form expression for the conductance of a Bethe lattice of wires and find that it has a band whose width shrinks as the coordination number increases. In the disordered case (p1), we find, in d=3 dimensions, a percolation transition at a quantum-mechanical threshold pq that is energy dependent but is always larger than the classical percolation threshold pc. Near pq (namely, for small values of {}{Δ}{}=={}p-pq{}), the mean quantum-mechanical conductance 〈gL〉 of a cube of length L follows the finite-size-scaling form 〈gL(p)〉{}L^d-2-t/νF({Δ}L^1/ν), where the scaling function F and the critical exponent {ν} are different from their classical analogues.Our numerical estimate of the critical exponents is {ν}=0.75±{}0.1 and t={ν} in accordance with results of nonlinear {σ} models of localization. The distribution of the conductance undergoes a substantial change at threshold. The conductance in the diffusive (metallic) regime in d=3 dimensions follows Ohm's law (it is proportional to L). As p{→}1, the crossover between the metallic and the ballistic regimes is governed by the scaling law 〈gL(p)〉{}L²K(L(1-p)). No percolation transition is found for d=2 but as p{→}1, the crossover between the quasimetallic and the ballistic regimes is governed by a similar scaling law.