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Density propagator for many-body localization: finite size effects, transient subdiffusion, (stretched-)exponentials

Soumya Bera, Giuseppe De Tomasi, Felix Weiner, Ferdinand Evers

Published Oct 10, 2016• 4 citations
OpenAlex

Abstract

We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain ($t{-}V$-model). Our observable is the time-dependent density propagator, $\Pi_{\varepsilon}(x,t)$, calculated in windows of different energy density, $\varepsilon$, of the many-body Hamiltonian and at different disorder strengths, $W$, not exceeding the critical value $W_\text{c}$. The width $\Delta x_\varepsilon(t)$ of $\Pi_\varepsilon(x,t)$ exhibits a behavior $d\ln \Delta x_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)$, where the exponent function $\beta_\varepsilon(t){\lesssim}1/2$ is seen to depend strongly on $L$ at all investigated parameter combinations. (i) We cannot confirm the existence of a region in phase space that exhibits (genuine) subdiffusive dynamics in the sense that $\beta_\varepsilon{<}1/2$ is numerically fixed in the limit of large $L$. Instead, subdiffusion might possibly be transient, only, finally giving way to conventional diffusive behavior with $\beta_\varepsilon{=}1/2$. (ii) Similarly, we cannot confirm the existence of many-body mobility edges deep in the delocalized phase. (iii) (Transient) subdiffusion $0<\beta_\varepsilon(t)\lesssim 1/2$, coexists with an enhanced probability for returning to the origin, $\Pi_\varepsilon(0,t)$, decaying much slower than $1/\Delta x_\varepsilon (t)$. Correspondingly, the spatial decay of $\Pi_\varepsilon(x,t)$ is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.

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We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain (t−Vt{-}Vt−V-model). Our observable is the time-dependent density propagator, Πε(x,t)\Pi_{\varepsilon}(x,t)Πε​(x,t), calculated in windows of different energy density, ε\varepsilonε, of the many-body Hamiltonian and at different disorder strengths, WWW, not exceeding the critical value WcW_\text{c}Wc​. The width Δxε(t)\Delta x_\varepsilon(t)Δxε​(t) of Πε(x,t)\Pi_\varepsilon(x,t)Πε​(x,t) exhibits a behavior dln⁡Δxε(t)/dln⁡t=βε(t)d\ln \Delta x_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)dlnΔxε​(t)/dlnt=βε​(t), where the exponent function βε(t)≲1/2\beta_\varepsilon(t){\lesssim}1/2βε​(t)≲1/2 is seen to depend strongly on LLL at all investigated parameter combinations. (i) We cannot confirm the existence of a region in phase space that exhibits (genuine) subdiffusive dynamics in the sense that βε<1/2\beta_\varepsilon{<}1/2βε​<1/2 is numerically fixed in the limit of large LLL. Instead, subdiffusion might possibly be transient, only, finally giving way to conventional diffusive behavior with βε=1/2\beta_\varepsilon{=}1/2βε​=1/2. (ii) Similarly, we cannot confirm the existence of many-body mobility edges deep in the delocalized phase. (iii) (Transient) subdiffusion 0<βε(t)≲1/20<\beta_\varepsilon(t)\lesssim 1/20<βε​(t)≲1/2, coexists with an enhanced probability for returning to the origin, Πε(0,t)\Pi_\varepsilon(0,t)Πε​(0,t), decaying much slower than 1/Δxε(t)1/\Delta x_\varepsilon (t)1/Δxε​(t). Correspondingly, the spatial decay of Πε(x,t)\Pi_\varepsilon(x,t)Πε​(x,t) is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.
We investigate charge relaxation in quantum wires of spinless disordered fermions ($t\text{\ensuremath{-}}V$ model). Our observable is the time-dependent density propagator ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$, calculated in windows of different energy density ϵϵϵ of the many-body Hamiltonian and at different disorder strengths WWW, not exceeding the critical value Wc{W}_{c}Wc​. The width ${\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$ of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ exhibits a behavior $d\mathrm{ln}{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)/d\mathrm{ln}t={\ensuremath{\beta}}_{ϵ}(t)$, where the exponent function ${\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ is seen to depend strongly on LLL at all investigated parameter combinations. (i) We confirm the existence of a region in phase space that exhibits subdiffusive dynamics in the sense that ${\ensuremath{\beta}}_{ϵ}(t)<1/2$ in a large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with ${\ensuremath{\beta}}_{ϵ}=1/2$. (ii) We cannot confirm the existence of many-body mobility edges even in regions of the phase diagram that have been reported to be deep in the delocalized phase. (iii) (Transient) subdiffusion $0<{\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ coexists with an enhanced probability for returning to the origin ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(0,t)$, decaying much slower than $1/{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$. Correspondingly, the spatial decay of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ is far from Gaussian, being exponential or even slower. On a phenomenological level, our findings are broadly consistent with the effects of strong disorder and (fractal) Griffiths regions.
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 investigate charge relaxation in disordered and quasiperiodic quantum wires of spinless fermions $(t\ensuremath{-}V$ model) at different inhomogeneity strength WWW in the localized and nearly localized regime. Our observable is the time-dependent density correlation function, $\mathrm{\ensuremath{\Phi}}(x,t)$, at infinite temperature. We find that disordered and quasiperiodic models behave qualitatively similar: Although even at longest observation times the width $\mathrm{\ensuremath{\Delta}}x(t)$ of $\mathrm{\ensuremath{\Phi}}(x,t)$ does not exceed significantly the noninteracting localization length, ${\ensuremath{\xi}}_{0}$, strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. Motivated by our numerical data we discuss a scenario in which the MBL-phase splits into two subphases: in ${\mathrm{MBL}}_{\text{A}}\phantom{\rule{4pt}{0ex}}\mathrm{\ensuremath{\Delta}}x(t)$ diverges slower than any power, while it converges towards a finite value in MBLB{\mathrm{MBL}}_{\text{B}}MBLB​. Within the scenario the transition between MBLA{\mathrm{MBL}}_{\text{A}}MBLA​ and the ergodic phase is characterized by a length scale, $\ensuremath{\xi}$, that exhibits an essential singularity $ln\ensuremath{\xi}\ensuremath{\sim}1/|W\ensuremath{-}{W}_{{\text{c}}_{1}}|$. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.
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.
We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model by \ifmmode \check{Z}\else \v{Z}\fi{}nidari\ifmmode \check{c}\else \v{c}\fi{} et al. [Phys. Rev. Lett. 117, 040601 (2016)]. We also present the exact solution of a semiclassical model of conductors and insulators introduced by Agarwal et al. [Phys. Rev. Lett. 114, 160401 (2015)], which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The critical properties of both models, when passing from diffusion to subdiffusion, are interpreted in terms of ``Griffiths effects.'' We show that the finite-size scaling comes from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. We conjecture that the latter, which grows logarithmically with system size, may potentially be responsible for the fact that heavy-tailed resistance distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.
We study many-body localization for a disordered chain of spin-1/2 fermions. When both down and up components are exposed to the same strong disorder, a power-law growth of the entanglement entropy was observed [P. Prelov\ifmmode \check{s}\else \v{s}\fi{}ek et al., Phys. Rev. B 94, 241104 (2016)] that suggests that many-body localization is not complete; the density (charge) degree of freedom is localized, while the spin degree of freedom is apparently delocalized. We show that this powerlike behavior is only a transient effect and that, for longer times, the growth of the entanglement entropy is logarithmic in time. At the same time, the decay of the spin correlation slows down, showing that the spin transport is strongly reduced at long times. The dynamics of the system is quite similar to the one of many-body localized systems. Because of the limited duration of the numerical experiment, it is not possible to determine whether the system is truly many-body localized or diffusing very slowly at long times.