Conductances, conductance fluctuations, and level statistics on the surface of multilayer quantum Hall states

Transport properties on the two-dimensional surface of coupled multilayer heterostructures are studied in integer quantum Hall states. We emphasize the criticality of the surface state and the phase-coherent transport properties in the thermodynamic limit. A stable numerical algorithm for large-scale conductance calculations in the transfer-matrix approach is discussed in detail. It is then applied to a directed network model describing the quantum-mechanical tunneling and impurity scattering of the multilayer edge states. We calculate the two-probe conductance in the direction parallel to the external magnetic field, and its fluctuations and statistical distributions, as functions of the interlayer tunneling strength. Using finite-size scaling, the asymptotic scaling functions of the ensemble-averaged conductance and the conductance fluctuations are calculated for a fixed aspect ratio, and found to be in remarkable agreement with the analytical results obtained using the supersymmetric nonlinear $\sigma$-model approach. The conductance distribution is determined in the quasi-one-dimensional metallic, insulating regime, as well as in the crossover regime where comparisons are made to that at the single-layer quantum Hall transition. We present a detailed study of the level statistics in the eigenvalue spectrum of the transfer matrix. Coexistence of metallic and insulating statistics is observed in the crossover regime, which is attributed to the emergence of a finite-range level repulsion in the crossover regime, separating the metallic (Wigner-surmise) behavior at small level spacings from the insulating (uncorrelated or Poisson) behavior at large level spacings.