Abstract
We study the Anderson transition in lattices with the connectivity of a random-regular graph. Our results indicate that fractal dimensions are continuous across the transition, but a discontinuity occurs in their derivatives, implying the non-ergodicity of the metal near the Anderson transition. A critical exponent $\nu = 1.00 \pm0.02$ and critical disorder $W= 18.2\pm 0.1$ are found via a scaling approach. Our data support that the predictions of the relevant Gaussian Ensemble are only recovered at zero disorder.
Highlights
Anderson localization of a single particle is crucial to understand transport in disordered materials [1]
We have shown that fractal dimensions D1, D2, and D∞ are continuous across the Anderson transition but their derivatives are discontinuous
We have not found any evidence of an additional phase transition, one from nonergodic to ergodic metal, so the whole metal is nonergodic for random-regular graphs
Summary
Anderson localization of a single particle is crucial to understand transport in disordered materials [1]. Some groups have observed that metallic wave functions near this transition are not ergodic [13,14,15,16,17,18], while others support their ergodicity [19,20,21,22] and critical level statistics reported near the transition [23] Given this controversy, it is sensible to step down and analyze simpler models that may exhibit similar behavior to that expected in the metallic side of the many-body localization transition. The relatively simple Hamiltonian that may contain a phase of nonergodic metallic states is the one analyzed here, which describes a particle hopping in a random-regular graph. This graph is locally equivalent to a Bethe lattice but has closed loops with lengths that scale with the total number of nodes (see Fig. 1). As the quantity that controls finite-size effects in random graphs
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