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arXiv:1610.03085 · Physical Review Letters

Density Propagator for Many-Body Localization: Finite-Size Effects, Transient Subdiffusion, and Exponential Decay

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

Published May 8, 2017• 121 citations
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Abstract

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 $ϵ$ of the many-body Hamiltonian and at different disorder strengths $W$, not exceeding the critical value ${W}_{c}$. 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 $L$ 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.

DOI
10.1103/physrevlett.118.196801

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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.
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