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We study the random transverse field Ising model on a finite Cayley tree.This enables us to probe key questions arising in other important disorderedquantum systems, in particular the Anderson transition and the problem of dirtybosons on the Cayley tree, or the emergence of non-ergodic properties in suchsystems. We numerically investigate this problem building on the cavitymean-field method complemented by state-of-the art finite-size scalinganalysis. Our numerics agree very well with analytical results based on ananalogy with the traveling wave problem of a branching random walk in thepresence of an absorbing wall. Critical properties and finite-size correctionsfor the zero-temperature paramagnetic-ferromagnetic transition are studied bothfor constant and algebraically vanishing boundary conditions. In the latercase, we reveal a regime which is reminiscent of the non-ergodic delocalizedphase observed in other systems, thus shedding some light on critical issues inthe context of disordered quantum systems, such as Anderson transitions, themany-body localization or disordered bosons in infinite dimensions.

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