We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of random interactions
Vij decaying as power-law ${V}_{ij}/{({r}_{ij})}^{\ensuremath{\alpha}}$ with distance
rij. We demonstrate that MBL survives even for $\ensuremath{\alpha}<1$ and is preceded by a broad nonergodic subdiffusive phase. Starting from parameters at which the short-range interacting system shows an infinite temperature MBL phase, turning on random power-law interactions results in many-body mobility edges in the spectrum with a larger fraction of ergodic delocalized states for smaller values of $\ensuremath{\alpha}$. Hence, the critical disorder
hcr, at which ergodic to nonergodic transition takes place, increases with the range of interactions. Time evolution of the density imbalance
I(t), which has power-law decay $I(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ in the intermediate to large time regime, shows that the critical disorder
hcI, above which the system becomes diffusionless (with $\ensuremath{\gamma}\ensuremath{\sim}0$) and transits into the MBL phase, is much larger than
hcr. In between
hcr and
hcI there is a broad nonergodic subdiffusive phase, which is characterized by the Poissonian statistics for the level spacing ratio, multifractal eigenfunctions, and a nonzero dynamical exponent $\ensuremath{\gamma}\ensuremath{\ll}1/2$. The system continues to be subdiffusive even on the ergodic side (
h<hcr) of the MBL transition, where the eigenstates near the mobility edges are multifractal. For
h<h0<hcr, the system is superdiffusive with $\ensuremath{\gamma}>1/2$. The rich phase diagram obtained here is unique to the random nature of long-range interactions. We explain this in terms of the enhanced correlations among local energies of the effective Anderson model induced by random power-law interactions.
The behavior of the dynamical structure factor
S(q,ω) is studied numerically in the whole range of wavevectors
q within the prototype one-dimensional model of many-body localization (MBL). Extracted effective dynamical conductivities and current-relaxation rates confirm strong dependence on disorder but modest variation with
q. An analytical self-consistent approximation based on the perturbation theory is shown to qualitatively account for the nontrivial features of dynamical quantities at all
q: the emergence of the maximum in dynamical conductivities, nonanalytical low--
ω variation in the ergodic phase, and the transition to the nonergodic (MBL) phase. Finite-size scaling also reveals the possibility of the subdiffusive behavior in the ergodic regime.
We study the $t\text{\ensuremath{-}}V$ disordered spinless fermionic chain in the strong-coupling regime, $t/V\ensuremath{\rightarrow}0$. Strong interactions highly hinder the dynamics of the model, fragmenting its Hilbert space into exponentially many blocks in system size. Macroscopically, these blocks can be characterized by the number of new degrees of freedom, which we refer to as movers. We focus on two limiting cases: blocks with only one mover and ones with a finite density of movers. The former many-particle block can be exactly mapped to a single-particle Anderson model with correlated disorder in one dimension. As a result, these eigenstates are always localized for any finite amount of disorder. The blocks with a finite density of movers, on the other side, show a many-body localized (MBL) transition that is tuned by the disorder strength. Moreover, we provide numerical evidence that its ergodic phase is diffusive at weak disorder. Approaching the MBL transition, we observe subdiffusive dynamics at finite timescales and find indications that this might be only a transient behavior before crossing over to diffusion.
We explore thermalization and quantum dynamics in a one-dimensional disordered SU(2)-symmetric Floquet model, where a many-body localized phase is prohibited by the non-Abelian symmetry. Despite the absence of localization, we find an extended nonergodic regime at strong disorder where the system exhibits nonthermal behaviors. In the strong disorder regime, the level spacing statistics exhibit neither a Wigner-Dyson nor a Poisson distribution, and the spectral form factor does not show a linear-in-time growth at early times characteristic of random matrix theory. The average entanglement entropy of the Floquet eigenstates is subthermal, although violating an area-law scaling with system sizes. We further compute the expectation value of local observables and find strong deviations from the eigenstate thermalization hypothesis. The infinite-temperature spin autocorrelation function decays at long times as ${t}^{\ensuremath{-}\ensuremath{\beta}}$ with $\ensuremath{\beta}<0.5$, indicating subdiffusive transport at strong disorders.
We suggest that if a localized phase at nonzero temperature
T>0 exists for strongly disordered and weakly interacting electrons, as recently argued, it will also occur when both disorder and interactions are strong and
T is very high. We show that in this high-
T regime, the localization transition may be studied numerically through exact diagonalization of small systems. We obtain spectra for one-dimensional lattice models of interacting spinless fermions in a random potential. As expected, the spectral statistics of finite-size samples cross over from those of orthogonal random matrices in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. However, these data show deviations from simple one-parameter finite-size scaling: the apparent mobility edge ``drifts'' as the system's size is increased. Based on spectral statistics alone, we have thus been unable to make a strong numerical case for the presence of a many-body localized phase at nonzero
T.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size
L in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.