多体局域化的密度传播子：有限尺寸效应，瞬时次扩散和指数衰减
Density propagator for many-body localization: finite size effects, transient subdiffusion, and exponential decay
摘要：
研究了无自旋无序费米子量子线（$t{-}V$模型）中的电荷弛豫。我们观察到的是随时间变化的密度传播子，$Π_{\varepsilon}(x,t)$，计算窗口中的不同能量密度，$\varepsilon$，多体哈密顿量和不同的无序强度，$W$，不超过临界值$W_\text{c}$。$Π_\varepsilon(x,t)$的宽度$Δx_\varepsilon(t)$表现出行为$d\ln Δx_\varepsilon(t) / d\ln t {=} β_\varepsilon(t)$，其中指数函数$β_\varepsilon(t){\lesssim}1/2$被认为在所有研究的参数组合强烈依赖于$L$。(i)我们确认存在一个区域在相空间中，表现出亚扩散动力学的意义上，$β_\varepsilon{<}1/2$在大的时间窗口。然而，亚扩散可能只是短暂的，最终让位于$β_{\varepsilon}{=}1/2$的常规扩散行为。(ii)我们不能确认多体迁移率边缘的存在，即使在区域的相图，已被报道为在离域相深。(iii)（瞬时）亚扩散$0<β_\varepsilon(t)\lesssim 1/2$与返回原点的增强概率$Π_\varepsilon(0,t)$共存，$Π_\varepsilon(0,t)$的衰减比$1/Δx_\varepsilon (t)$慢得多。相应地，$Π_\varepsilon(x,t)$的空间衰减与高斯衰减相差甚远，为指数衰减或甚至更慢。在现象学水平上，我们的研究结果与强无序和（分形）格里菲斯地区的影响大致一致。
英文摘要：
We investigate charge relaxation in quantum-wires of spin-less disordered fermions ($t{-}V$-model). Our observable is the time-dependent density propagator, $Π_{\varepsilon}(x,t)$, 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 $Δx_\varepsilon(t)$ of $Π_\varepsilon(x,t)$ exhibits a behavior $d\ln Δx_\varepsilon(t) / d\ln t {=} β_\varepsilon(t)$, where the exponent function $β_\varepsilon(t){\lesssim}1/2$ 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 $β_\varepsilon{<}1/2$ in large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with $β_{\varepsilon}{=}1/2$. (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 $0<β_\varepsilon(t)\lesssim 1/2$, coexists with an enhanced probability for returning to the origin, $Π_\varepsilon(0,t)$, decaying much slower than $1/Δx_\varepsilon (t)$. Correspondingly, the spatial decay of $Π_\varepsilon(x,t)$ 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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