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arXiv:1609.05450 · Physical review. B./Physical review. B

Self-consistent approach to many-body localization and subdiffusion

Peter Prelovšek, J. Herbrych

Published Jul 17, 2017• 40 citations
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Abstract

An analytical theory, based on the perturbative treatment of the disorder and extended into a self-consistent set of equations for the dynamical density correlations, is developed and applied to the prototype one-dimensional model of many-body localization. Results show a qualitative agreement with the numerically obtained dynamical structure factor in the whole range of frequencies and wave vectors, as well as across the transition to nonergodic behavior. The theory reveals the singular nature of the one-dimensional problem, whereby on the ergodic side the dynamics is subdiffusive with dynamical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{\ensuremath{\alpha}}$, i.e., with vanishing dc limit ${\ensuremath{\sigma}}_{0}=0$ and $\ensuremath{\alpha}<1$ varying with disorder, while we get $\ensuremath{\alpha}>1$ in the localized phase.

DOI
10.1103/physrevb.96.035130

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An analytical theory, based on the perturbative treatment of the disorder and extended into a self-consistent set of equations for the dynamical density correlations, is developed and applied to the prototype one-dimensional model of many-body localization. Results show a qualitative agreement with the numerically obtained dynamical structure factor in the whole range of frequencies and wave vectors, as well as across the transition to nonergodic behavior. The theory reveals the singular nature of the one-dimensional problem, whereby on the ergodic side the dynamics is subdiffusive with dynamical conductivity \ensuremathσ(\ensuremathω)\ensuremath∝∣\ensuremathω∣\ensuremathα\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{\ensuremath{\alpha}}\ensuremathσ(\ensuremathω)\ensuremath∝∣\ensuremathω∣\ensuremathα, i.e., with vanishing dc limit \ensuremathσ0=0{\ensuremath{\sigma}}_{0}=0\ensuremathσ0​=0 and \ensuremathα<1\ensuremath{\alpha}<1\ensuremathα<1 varying with disorder, while we get \ensuremathα>1\ensuremath{\alpha}>1\ensuremathα>1 in the localized phase.
An analytical theory, based on the perturbative treatment of the disorder and extended into a self-consistent set of equations for the dynamical density correlations, is developed and applied to the prototype one-dimensional model of many-body localization. Results show a qualitative agreement with numerically obtained dynamical structure factor in the whole range of frequencies and wavevectors, as well as across the transition to the nonergodic behavior. The theory reveals the singular nature of the one-dimensional problem, whereby on the ergodic side the dynamics is subdiffusive with a dynamical exponent varying with the disorder.
The behavior of the dynamical structure factor S(q,ω)S(q,\omega)S(q,ω) is studied numerically in the whole range of wavevectors qqq 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 qqq. An analytical self-consistent approximation based on the perturbation theory is shown to qualitatively account for the nontrivial features of dynamical quantities at all qqq: the emergence of the maximum in dynamical conductivities, nonanalytical low--ω\omegaω 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 many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of random interactions Vij{V}_{ij}Vij​ decaying as power-law ${V}_{ij}/{({r}_{ij})}^{\ensuremath{\alpha}}$ with distance rij{r}_{ij}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{h}_{c}^{r}hcr​, at which ergodic to nonergodic transition takes place, increases with the range of interactions. Time evolution of the density imbalance I(t)I(t)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{h}_{c}^{I}hcI​, above which the system becomes diffusionless (with $\ensuremath{\gamma}\ensuremath{\sim}0$) and transits into the MBL phase, is much larger than hcr{h}_{c}^{r}hcr​. In between hcr{h}_{c}^{r}hcr​ and hcI{h}_{c}^{I}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<hcrh<{h}_{c}^{r}h<hcr​) of the MBL transition, where the eigenstates near the mobility edges are multifractal. For h<h0<hcrh<{h}_{0}<{h}_{c}^{r}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.
Within one-dimensional disordered models of interacting fermions, we perform a numerical study of several dynamical density correlations, which can serve as hallmarks of the transition to the many-body localized state. The results confirm that density-wave correlations exhibit quite an abrupt change with increasing disorder, with a nonvanishing long-time value characteristic for the nonergodic phase. In addition, our results reveal a logarithmic variation of correlations in time in a wide time window, which we can bring in connection with the anomalous behavior of the dynamical conductivity near the transition. Our results support the view that the transition to many-body localization can be characterized by universal dynamical exponents.
We study the real-time dynamics of a two-dimensional Anderson-Hubbard model using nonequilibrium self-consistent perturbation theory within the second-Born approximation. When compared with exact diagonalization performed on small clusters, we demonstrate that for strong disorder this technique approaches the exact result on all available timescales, while for intermediate disorder, in the vicinity of the many-body localization transition, it produces quantitatively accurate results up to nontrivial times. Our method allows for the treatment of system sizes inaccessible by any numerically exact method and for the complete elimination of finite-size effects for the times considered. We show that for a sufficiently strong disorder the system becomes nonergodic, while for intermediate disorder strengths and for all accessible timescales transport in the system is strictly subdiffusive. We argue that these results are incompatible with a simple percolation picture, but are consistent with the heuristic random resistor network model where subdiffusion may be observed for long times until a crossover to diffusion occurs. The prediction of slow finite-time dynamics in a two-dimensional interacting and disordered system can be directly verified in future cold-atoms experiments.
We study the infinite temperature dynamics of a prototypical one-dimensional system expected to exhibit many-body localization. Using numerically exact methods, we establish the dynamical phase diagram of this system based on the statistics of its eigenvalues and its dynamical behavior. We show that the nonergodic phase is reentrant as a function of the interaction strength, illustrating that localization can be reinforced by sufficiently strong interactions even at infinite temperature. Surprisingly, within the accessible time range, the ergodic phase shows subdiffusive behavior, suggesting that the diffusion coefficient vanishes throughout much of the phase diagram in the thermodynamic limit. Our findings strongly suggest that Wigner-Dyson statistics of eigenvalue spacings may appear in a class of ergodic but subdiffusive systems.