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 Nd-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 (p<1), 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 ==p-pq), the mean quantum-mechanical conductance gL of a cube of length L follows the finite-size-scaling form gL(p)Ld-2-t/F(L1/), where the scaling function F and the critical exponent are different from their classical analogues. Our numerical estimate of the critical exponents is =0.75±0.1 and t= in accordance with results of nonlinear 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 Ohms law (it is proportional to L). As p1, 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 p1, the crossover between the quasimetallic and the ballistic regimes is governed by a similar scaling law.