Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree

TL;DR

This paper investigates the phase transition and non-ergodic behavior in the random transverse-field Ising model on a Cayley tree, revealing insights relevant to disordered quantum systems like Anderson transitions and many-body localization.

Contribution

It introduces a numerical and analytical study of the model's critical properties, highlighting a non-ergodic delocalized regime similar to phenomena in other disordered quantum systems.

Findings

Agreement between numerics and analytical traveling wave analogy

Identification of a non-ergodic delocalized phase

Finite-size scaling analysis of phase transition

Abstract

We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant and algebraically vanishing boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.