Structure of typical states of a disordered Richardson model and many-body localization

Abstract

We present a thorough numerical study of the Richardson model with quenched disorder (a fully-connected XX-model with longitudinal random fields). We find that for any g > 0 the eigenstates occupy an exponential number of sites on the unperturbed Fock space but that single-spin observables do not thermalize, as tested by a microcanonical version of the Edwards-Anderson order parameter q > 0. We therefore do not observe MBL in this model. We find a relation between the inverse participation ratio, q and the average Hamming distance L between spin configurations covered by a typical eigenstate for which we conjecture a remarkably simple form for the thermodynamic limit L/N = g 2(1+g) . We also studied the random process defined by the spread of a typical eigenstate on configuration space, highlighting several similarities with hopping on percolated hypercube, a process used to mimic the slow relaxation of spin glasses. A nearby non-integrable model is also considered where delocalization is instead observed, although the presence of a phase transition at infinite temperature is questionable.

Many-body localization and quantum ergodicity in disordered long-range Ising models

“…The Hilbert space property D s0 (t) can therefore be obtained from purely local measurements in real space, provided the initial configuration is known. In the context of the Richardson model, a similar relation has been obtained recently, which, however, is restricted to particular initial states and the asymptotic long-time regime 47 . Our Eq.…”

Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems

“…Works [2,3] conjectured that localization in a many-body system survives in the presence of weak interactions. When the strength of the interactions is increased, at some critical value a transition to the delocalized phase -a "many-body localization" transition -takes place, as observed in the numerical simulations [4][5][6][7][8][9][10][11][12][13][14]. …”

Ergodicity breaking and wave-function statistics in disordered interacting systems

“…The composition of this effect over the N spins produces the scaling N γ , γ < 1 of P 2 and a diagonal entropy S < 1. We underline that an analogous mechanism was taking place in the Richardson model [29], where, thanks to integrability, the local conserved charges can be determined analytically. This suggest that the Richardson model can be used as a toy model of the many-body localized phase, albeit due to the fully-connected hopping term, all the considerations about transport are problematic.…”