Spectral properties of the three-dimensional Anderson model

J. Šuntajs,1 T. Prosen,1 L. Vidmar1,2

1Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia

2Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia

The three-dimensional Anderson model represents a paradigmatic model to understand the Anderson localization transition. Using modern numerical approaches, we investigate the model’s spectral properties, focusing particularly on the quantitative comparison between the level sensitivity statistics and the level statistics. While the former probes the susceptibility of the Hamiltonian eigenlevels towards the introduction of the magnetic flux to the system, the latter deals with the unperturbed energy levels. We define two versions of the dimensionless conductance, the first corresponding to the width of the level curvature distribution relative to the mean level spacing, and the other corresponding to the ratio of Heisenberg and Thouless times obtained from the spectral form factor. We show that both conductances display remarkably similar behaviour around the localization transition, in particular, they predict a nearly identical critical point consistent with other well-established measures of the transition.

Motivated by the pioneering work of Edwards and Thouless [J. Phys. C. 5, 807 (1972)], we then calculate the characteristic energy, defined as the mean change of the eigenlevels upon switching the hopping along one of the lattice edges from periodic to antiperiodic boundary conditions. Making use of the modern hardware to perform a systematic analysis, we obtain results for system sizes much greater than the ones available to the authors of the original study. By accurately pinpointing the location of the critical point, we establish the latter method as a reliable tool for detecting the localization transition in noninteracting systems. In the context of the spectral form factor, we show that at the critical point it enters a broad time-independent regime, in which its value is consistent with the level compressibility obtained from the level variance. Finally, we test the scaling solution of the average level spacing ratio in the crossover regime using the cost function minimization approach introduced recently in [Phys. Rev. B., 102, 064207 (2020)]. The latter approach seeks for the optimal scaling solution in the vicinity of the crossing point, while at the same time allowing for the drift of the crossing point due to finite-size corrections. We find that the extracted transition point and the scalling coefficient accurately agree with those obtained from the literature.

1. J. Šuntajs et. al. Annals of Physics, 168469 (2021)
2. J. Šuntajs et. al. Phys. Rev. B 102, 064207 (2020)
3. J. Edwards, E. Thouless, J. Phys. C. 5, 807 (1972)

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