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 ${\mathit{g}}_{\mathit{N}\mathit{L}}$ of a finite system of transverse dimension ${\mathit{N}}^{\mathit{d}\mathrm{-}1}$ 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 ${\mathit{p}}_{\mathit{q}}$ that is energy dependent but is always larger than the classical percolation threshold ${\mathit{p}}_{\mathit{c}}$. Near ${\mathit{p}}_{\mathit{q}}$ (namely, for small values of ‖Δ‖==‖p-${\mathit{p}}_{\mathit{q}}$‖), the mean quantum-mechanical conductance 〈${\mathit{g}}_{\mathit{L}}$〉 of a cube of length L follows the finite-size-scaling form 〈${\mathit{g}}_{\mathit{L}}$(p)〉≊${\mathit{L}}^{\mathit{d}\mathrm{-}2\mathrm{-}\mathit{t}/\nu }$F(Δ${\mathit{L}}^{1/\nu }$), where the scaling function F and the critical exponent ν are different from their classical analogues.