Signatures of a critical point in the many-body localization transition

Abstract

Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered J_1J1-J_2J2 model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.

Non-ergodic extended regime in random matrix ensembles: insights from eigenvalue spectra

“…Here we consider two kinds of power-spectrum functions, both of them have appeared in previous studies. The first one is 30,31,47,48 where Clearly, F(k) measures the averaged Fourier weight of the new eigenvalue spectra r p=3 σ p X (p) mn , which are the spectra after dropping the first two components. And the second one, which bears analytical treatments [49][50][51] , is where The S(k) is the averaged Fourier weight of the new cumulated level spacing {δ n } .…”

Localization challenges quantum chaos in the finite two-dimensional Anderson model

“…In most cases of interest for current studies, the critical parameters of the transition are usually not known in advance. A typical example is the random-field spin-1/2 Heisenberg chain, which has recently experienced a renewed interest to unveil the eventual breakdown of quantum chaos [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60].…”

Modelling sample-to-sample fluctuations of the gap ratio in finite disordered spin chains

“…At small disorder, the quantum many-body system is ergodic; its energy spectrum can be analyzed within the framework of the random matrix theory (RMT), while the eigenstate thermalization hypothesis can describe the relaxation of physical observables of a closed system [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Increasing the disorder strength leads to the ergodicity breakdown that is reflected in the departure from the RMT prediction and can be studied through the behavior of different ergodicity indicators such as the anomalous level statistics and the eigenstate entanglement entropies [17,18], the fidelity susceptibility [22], the anomalous distribution of observable matrix elements [47,48], the opening of the Schmidt gap [49], the gap in the spectrum of the eigenstate one-body density matrix [11], and the correlation-hole time in the survival probability reaching the Heisenberg time t H [50].…”