Drude weight fluctuations in many-body localized systems

We numerically investigate the distribution of Drude weights $D$ of many-body states in disordered one-dimensional interacting electron systems across the transition to a many-body localized phase. Drude weights are proportional to the spectral curvatures induced by magnetic fluxes in mesoscopic rings. They offer a method to relate the transition to the many-body localized phase to transport properties. In the delocalized regime, we find that the Drude weight distribution at a fixed disorder configuration agrees well with the random-matrix-theory prediction $P\left(D\right)\propto {({\gamma }^2+D^2)}^{-3/2}$, although the distribution width $\gamma$ strongly fluctuates between disorder realizations. A crossover is observed towards a distribution with different large-$D$ asymptotics deep in the many-body localized phase, which however differs from the commonly expected Cauchy distribution. We show that the average distribution width $〈\gamma 〉$, rescaled by $L\mathrm{\Delta },\mathrm{\Delta }$ being the average level spacing in the middle of the spectrum and $L$ the systems size, is an efficient probe of the many-body localization transition, as it increases (vanishes) exponentially in the delocalized (localized) phase.

Figure 1

Cumulative rescaled Drude weight distribution $F$ for disorder strengths $W$ increasing from 1.5 to 7.5 in steps of 0.5 (left to right data series). Each data set is based on 1000 disorder realizations, each contributing 2554 curvatures from states in the middle of the many-body spectrum. The other system parameters are $U/t=1,L=16$, and $N=8$ particles. For $W\lesssim 2.5$ the system is in the ergodic phase and the distribution is well approximated by the RMT prediction $F_{\mathrm{RMT}}$ (solid thin line) [see Eq. (2)]. Deep in the many-body localized phase ($W/t\gtrsim 5$), the distribution converges towards a different one with longer tails.

Figure 2

Drude weight distribution for $W/t=2$ (left) and $W/t=5.5$ (center), again at $U/t=1$ for the left and the center figures. The main panels are for rescaled distributions of 2574 states in the middle of the many-body spectrum, averaged over 1000 disorder realizations. The insets are for single disorder realizations. The other parameters are chosen as in Fig. 1. For $W/t=2$ both the single-realization and the averaged distributions are in excellent agreement with the RMT prediction. For $W/t=5.5$ a large part of the distribution is well described by a log-normal distribution, whereas the Cauchy distribution of Refs. [36, 37, 53] does not provide a good fit. The right panel (main and inset) shows $F\left(D\right)$ in the absence of interactions, $U=0$. The curvature distribution for a single realization in the inset shows lack of self-averaging.

Figure 3

Main panel: Cumulative distribution functions of the Drudeweight distribution widths $\gamma$ for disorder strength $W/t$ increasing from 1.5 to 8 in steps of 0.5 (right to left data series). We consider ${10}^3$ realizations per disorder value. The solid black lines increase a log-normal fit to the data. Parameters of the simulations are as in Fig. 1. Inset: Average width $〈\gamma 〉$ of the Drude weight distribution (rescaled by $L\mathrm{\Delta },\mathrm{\Delta }$ being the average level spacing in the middle of the spectrum) vs system size $L$ for disorder strength $W/t$ increasing from 1.5 to 5 in steps of 0.5 (top to bottom data series).