Sudden removal of a static force in a disordered system: Induced dynamics, thermalization, and transport

We study the real-time dynamics of local occupation numbers in a one-dimensional model of spinless fermions with a random on-site potential for a certain class of initial states. The latter are thermal (mixed or pure) states of the model in the presence of an additional static force but become nonequilibrium states after a sudden removal of this static force. For this class and high temperatures, we show that the induced dynamics is given by a single correlation function at equilibrium, independent of the initial expectation values being prepared close to equilibrium (by a weak static force) or far away from equilibrium (by a strong static force). Remarkably, this type of universality holds true in both the ergodic phase and the many-body localized regime. Moreover, it does not depend on the specific choice of a unit cell for the local density. We particularly discuss two important consequences. First, the long-time expectation value of the local density is uniquely determined by the fluctuations of its diagonal matrix elements in the energy eigenbasis. Thus, the validity of the eigenstate thermalization hypothesis is not only a sufficient but also a necessary condition for thermalization. Second, the real-time broadening of density profiles is always given by the current autocorrelation function at equilibrium via a generalized Einstein relation. In the context of transport, we discuss the influence of disorder for large particle-particle interactions, where normal diffusion is known to occur in the disorder-free case. Our results suggest that normal diffusion is stable against weak disorder, while they are consistent with anomalous diffusion for stronger disorder below the localization transition. Particularly, for weak disorder, Gaussian density profiles can be observed for single disorder realizations, which we demonstrate for finite lattices up to 31 sites.

Figure 1

Sketch of the setup. (a) At time $t=0$, the disordered system is still in contact with a heat bath at inverse temperature $\beta =1/T$ and a static force leads to an additional potential in the middle of the chain. (b) At times $t>0$, heat bath and static force are both removed and the system evolves unitarily according to the isolated Hamiltonian $\mathcal{H}$. This setup might be seen as a type of quantum quench as well.

Figure 2

Initial expectation value ${\mathcal{N}}_0=\langle n_{L/2}\left(0\right)\rangle$ versus perturbation strength $\varepsilon$ for a high temperature $\beta J=0.01$. Panel (a): Semilogarithmic ($x$ axis) plot. The vertical dashed lines indicate those values of $\varepsilon$ which are used to study dynamics, i.e., $\beta \varepsilon J=0.1,1,3,5$. Panel (b): Same data as in (a) but now in a linear plot. The linear prediction from Eq. (10) is shown. Note that for sufficiently small $\varepsilon$, the response is always small compared to the disorder potential $W$. The other parameters are $L=20$ and $\mathrm{\Delta }=1.5$.

Figure 3

Dynamical expectation value $\langle n_{L/2}\left(t\right)\rangle$ for a high temperature $\beta J=0.01$ and various perturbation strengths $\varepsilon$ up to times $tJ\le 50$. Panels (a) and (b) show data for disorder strengths $W=1$ and $W=4$. Panel (c) shows a collapse of the data according to Eq. (16). The solid line indicates the (normalized) equilibrium correlation function ${\langle n_{L/2}\left(t\right)n_{L/2}\rangle }_{\text{eq}}$. The shaded area indicates the statistical fluctuations $\overline{\langle n_l\left(t\right)\rangle }\pm \mathrm{\Delta }\langle n_l\left(t\right)\rangle$ [cf. Eq. (15)] due to the random potentials. Note that the error of the mean $\mathrm{\Delta }\langle n_l\left(t\right)\rangle /\sqrt{N}$ is significantly smaller. The other parameters are $L=20, \mathrm{\Delta }=1.5$, and $N=300$.

Figure 4

Dynamical expectation value $\langle d_{L/4}\left(t\right)\rangle$ for a high temperature $\beta J=0.01$ and various perturbation strengths $\varepsilon$ up to times $tJ\le 20$. Panel (b) shows a collapse of the data according to Eq. (16) with $d_{\text{eq}}=1$ and $d_{\text{max}}=2$. The other parameters are $L=20,\mathrm{\Delta }=1.5$, and $W=1$.

Figure 5

Comparison between a typical and an untypical state. The other parameters are $L=20,\mathrm{\Delta }=1.5,W=1$, and $\beta J=0.01$.

Figure 6

Panels (a) and (c): Weighted average $\overline{n}\pm \mathrm{\Sigma }$ in the energy range $-3\le E/J\le 3$ for $W=1$ and $W=4$, respectively. Filled symbols are exact-diagonalization results for $L=12$; open symbols are obtained by the typicality-based approach for $L=20$. The microcanonical window around $E/J\approx 0$ corresponds to high temperatures $\beta J\approx 0$. Panels (b) and (d): Distribution of diagonal matrix elements $n_{ii}$ versus eigenenergy $E_i$ in the subspace with $L/4$ fermions ($L=20$). The other parameters are $\delta E/J=0.5$ and $\mathrm{\Delta }=1.5$.

Figure 7

Panel (a): Numerical illustration of the validity of Eq. (25). The data points are extracted from the nonequilibrium dynamics $\langle n_{L/2}/\left(t\right)\rangle$ at time $tJ=150$, while the solid lines are calculated according to Eq. (25) with ${\mathrm{\Sigma }}_0=\mathrm{\Sigma }(E/J=0)$. In addition to the cases $W=1$ and $W=4$, we also show data for the case $W=8$ (see Appendix pp2). Panel (b): ${\mathrm{\Sigma }}_0$ versus disorder $W$. For comparison, the lower bound ${\mathrm{\Sigma }}_{\text{min}}=1/\sqrt{4L}$ is indicated, which follows from Eq. (25) and $\langle n_{L/2}(t\to \infty )\rangle \ge ({\mathcal{N}}_0-n_{\text{eq}})/L+n_{\text{eq}}$ in systems of finite size (horizontal line). In all cases, we have $L=20$ and $\mathrm{\Delta }=1.5$.

Figure 8

Panels (a) and (c): Density profile $\langle n_l\left(t\right)\rangle$ for disorder strengths $W=1$ and $W=4$, respectively. The initial state is prepared according to Eq. (4) with $\beta J=0.01$ and $\beta \varepsilon J=3$. Panels (b) and (d): Corresponding width $\sigma \left(t\right)$ for various $\beta \varepsilon$, obtained from Eq. (30). For comparison, we also depict $\sigma \left(t\right)$ from the current autocorrelation function, calculated according to Eq. (31). The dashed vertical line is a guide to the eye. The other parameters are $L=20$ and $\mathrm{\Delta }=1.5$.

Figure 9

Time dependence of the generalized diffusion coefficient $D_q\left(t\right)$ for wave vectors $k=0$ [Eq. (32)] and $k=1,2$ [Eq. (33)], for three disorders $W=0,0.5$, and 1, respectively. For $k=0$, data are always shown for two system sizes: (a) $L=20$ and $L=34$; (b) and (c) $L=20$ and $L=26$. For $k=1,2$, we have $L=20$ in all cases. Other parameters: $\mathrm{\Delta }=1.5$.

Figure 10

Density profile $\langle n_l\left(t\right)\rangle$ for the three disorder strengths (a) $W=0$, (c) $W=0.5$, (e) $W=1$, and a short time $tJ=5$, in a semilogarithmic plot ($y$ axis). For each $W$, a longer time is shown in (b), (d), (f), where boundary effects are still negligibly small. In all (a)–(f), Gaussian fits are indicated for comparison. In (f), an exponential fit to the outer tails is depicted. Other parameters: $\mathrm{\Delta }=1.5$.

Figure 11

Same data as in Fig. 10 but now shown for $W=2$.

Figure 12

Density profile $\langle n_l\left(t\right)\rangle$ for $W=0.5$ and single realizations of the random on-site potential: Two different realizations for (a) $L=30$ and (b) $L=31$ sites. These unaveraged data (symbols) are additionally compared to averaged $N=20$ data (shaded area) from Fig. 10 with $tJ=10$. (c), (d) Corresponding realizations of the random on-site potential. Note that we restrict ourselves to the largest subsector of ($L=30$) or around $(L=31)$ half filling.

Figure 13

Panel (a): Quantity $\gamma \left(t\right)-{\overline{n}}^2$ according to Eqs. (A2) and (A4) for $L=14$ sites, averaged over $N=300$ random realizations of the pure state $|\phi \rangle$. Data are shown for energies $E/J=-3,0,3$. Panel (b): Comparison of the width $\mathrm{\Sigma }\left(E\right)$, as obtained from the exact definition in Eq. (24), with the width $\mathrm{\Sigma }\left(E\right)$, as obtained from Eqs. (A4, A5, A6) with the time interval $[t_{1}J,t_{2}J]=[100,150]$. The other parameters are $\mathrm{\Delta }=1.5$ and $W=0$.

Figure 14

Data collapse $\mathcal{M}\left(\langle n_{L/2}\left(t\right)\rangle \right)$, cf. Eq. (16), for perturbations $\beta \varepsilon J=0.1,1,3,5$ and disorder strengths $W=1,2,4,8$ up to long times $tJ\le 150$. The other parameters are $L=20$ and $\mathrm{\Delta }=1.5$.

Figure 15

Density profile $\langle n_l\left(t\right)\rangle$ for disorder strengths $W=2$ and $W=8$, respectively. The initial state is prepared according to Eq. (4) with the parameters $\beta J=0.01$ and $\varepsilon \beta J=3$. The other parameters are $L=20$ and $\mathrm{\Delta }=1.5$.