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

Soumya Bera, Giuseppe De Tomasi+2 more|arXiv (Cornell University)|Oct 10, 2016

Quantum many-body systems3 citations

TL;DR

This study investigates charge relaxation in a disordered one-dimensional fermionic Hubbard chain using the time-dependent density propagator $\Pi_\varepsilon(x,t)$ across different energy densities and disorder strengths. It finds no evidence for genuine subdiffusion or mobility edges, but observes transient subdiffusion with $\beta_\varepsilon(t) \lesssim 1/2$ that eventually transitions to diffusion, alongside non-Gaussian, slow-decaying density profiles consistent with strong disorder and Griffiths effects.

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 $W_ ext{c}$. 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.

Motivation & Objective

• To understand the nature of charge transport in the many-body localized phase of disordered fermionic systems.
• To determine whether subdiffusive dynamics persist in the thermodynamic limit or are transient.
• To assess the existence of many-body mobility edges in the delocalized phase.
• To characterize the spatial and temporal decay of the density propagator $\Pi_\varepsilon(x,t)$ under varying disorder and energy density.
• To explore the role of strong disorder and Griffiths regions in shaping relaxation dynamics.

Proposed method

• Numerical computation of the time-dependent density propagator $\Pi_\varepsilon(x,t)$ in energy windows $\varepsilon$ of the many-body Hamiltonian.
• Analysis of the width $\Delta x_\varepsilon(t)$ of $\Pi_\varepsilon(x,t)$ via the exponent function $\beta_\varepsilon(t) = d\ln \Delta x_\varepsilon(t) / d\ln t$.
• Systematic variation of system size $L$, disorder strength $W \leq W_c$, and energy density $\varepsilon$ to probe finite-size effects.
• Comparison of $\Pi_\varepsilon(0,t)$ and $\Delta x_\varepsilon(t)$ to assess return probability and spatial decay behavior.
• Use of phenomenological modeling to interpret results in terms of strong disorder and fractal Griffiths regions.
• Assessment of whether $\beta_\varepsilon(t)$ approaches a fixed value $< 1/2$ in the large-$L$ limit, indicating genuine subdiffusion.

Experimental results

Research questions

• RQ1 Does the system exhibit genuine subdiffusive dynamics with $\beta_\varepsilon < 1/2$ in the thermodynamic limit?
• RQ2 Are there many-body mobility edges present in the delocalized phase of the disordered $t$-$V$ model?
• RQ3 How does the spatial decay of $\Pi_\varepsilon(x,t)$ deviate from Gaussian behavior?
• RQ4 What is the nature of the return probability $\Pi_\varepsilon(0,t)$, and how does it compare to the width $\Delta x_\varepsilon(t)$?
• RQ5 To what extent do strong disorder and Griffiths regions explain the observed transient dynamics?

Key findings

• The exponent $\beta_\varepsilon(t)$ remains dependent on system size $L$ and does not stabilize at a value below $1/2$ in the large-$L$ limit, indicating no evidence for genuine subdiffusion.
• Subdiffusive behavior is likely transient, eventually giving way to diffusive dynamics with $\beta_\varepsilon = 1/2$.
• The return probability $\Pi_\varepsilon(0,t)$ decays slower than $1/\Delta x_\varepsilon(t)$, indicating enhanced localization effects.
• The spatial profile of $\Pi_\varepsilon(x,t)$ decays exponentially or slower, deviating significantly from Gaussian behavior.
• The observed dynamics are broadly consistent with the influence of strong disorder and fractal Griffiths regions.
• No evidence is found for the existence of many-body mobility edges deep within the delocalized phase.