Subdiffusion in a one-dimensional Anderson insulator with random dephasing: Finite-size scaling, Griffiths effects, and possible implications for many-body localization

We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model by Žnidarič et al. [Phys. Rev. Lett. 117, 040601 (2016)]. We also present the exact solution of a semiclassical model of conductors and insulators introduced by Agarwal et al. [Phys. Rev. Lett. 114, 160401 (2015)], which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The critical properties of both models, when passing from diffusion to subdiffusion, are interpreted in terms of “Griffiths effects.” We show that the finite-size scaling comes from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. We conjecture that the latter, which grows logarithmically with system size, may potentially be responsible for the fact that heavy-tailed resistance distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.

Figure 1

A toy model connecting our random dephasing model with the theory of a particle propagating on a random background and in a sequence of “ergodic bubbles” and localized regions (the blue and pink regions, respectively). In the dephasing model, whether a site belongs to an ergodic or localized region is chosen randomly with probability $p$, which defines the distribution of sizes of the ergodic regions.

Figure 2

Scaling properties of the inverse NESS current with system size in the disordered Heisenberg spin chain. Upper panel: $\tilde{R}$ as a function of $L$ for several disorder strengths in the thermalizing phase. The points show the numerical data, and the lines indicate a fit of the form $\tilde{R}=a{L}^{\beta }$ using the three largest system sizes. Lower panel: The (discrete) $\beta$ function of the same data, normalized by its asymptotic value at $L\to \infty$. The solid black line indicates a fit of the form (12) to the $W=0.25$ data. Data reproduced from Ref. [13], for comparison with Fig. 4.

Figure 3

Finite-size effects in the resistance scaling. Upper panel: Median resistance $R_{★}$ as a function of system size $L$ for several disorder strengths $W$ in a system with dephasing on every site ($p=0$). Dotted lines indicate the asymptotic diffusive scaling, and the dashed line shows the exact result in the absence of disorder. Lower panel: Median resistance $R_{★}$ as a function of the rescaled length $L{W}^2$ for several disorder strengths $W$ in a system with no dephasing ($p=1$), showing the scaling of the localization length at weak disorder $\xi \propto W^{-2}$.

Figure 4

The resistance $\beta$ function for various disorder and dephasing values with $p=0$ (discrete derivatives of the data in Fig. 3). We go from $W=4,\gamma =0.2$ which is large disorder and small dephasing, to $W=1,\gamma =0.2$ which is small disorder. Intermediate cases $W=2,\gamma =0.05,0.1$ are shown. Numerical results are fitted with a phenomenological function of the form (14), indicated by the lines of the corresponding color.

Figure 5

Numerical results on the resistance in the random dephasing model with $W=3$. Upper panels: Histograms of the rescaled resistance in the diffusive phase $p=0$ (top left) and $p=0.3$ (top right), and the subdiffusive phase $p=0.7$ (middle left) and $p=0.8$ (middle right). The diffusive rescaling $(R-R_{★})/\sigma$ is compared to a Gaussian distribution (black dotted line) for $p=0$. In the subdiffusive phase the resistance is rescaled as $R/R_{★}$, and the black dotted lines indicate a $R^{-2}$ tail. Lower panel: Scaling of the median resistance $R_{★}$ with system size $L$ for several dephasing fractions $p$. The lines in the corresponding color indicate the numerical fits to the data described in the text. The black dashed line indicates diffusive scaling $R_{★}\propto L$.

Figure 6

Upper panel: Histograms of the rescaled resistance $R/R_{★}$ for $W=3, L=128$ and several dephasing fractions $p$. The black dotted line indicates a heavy $R^{-2}$ tail. Lower panel: Comparison of the resistance scaling exponent $\beta$ (connected filled points) with the predicted value based on the power-law tail of the histogram ${(\nu -1)}^{-1}$ (unconnected unfilled points in the corresponding color). The discrepancy at $p\gtrsim 0.8$ and disorder $W\lesssim 3$ is attributed to the fact that the largest insulating cluster has a size $s_{★}$ which is smaller than the localization length $\xi$; therefore the distribution of resistance does not show the appropriate power-law tail.

Figure 7

Behavior of the resistance $\beta$ function at large $L$, shown for $W=4$ and several values of $p$ near the diffusion-subdiffusion transition. The lines show fits to the data up to linear order in $1/lnL$, with the $y$ intercept equal to the resistance scaling exponent $\beta$ in the thermodynamic limit.

Figure 8

The contours used to evaluate the integral in Eq. (33) for small $\rho$. Poles of the integrand are indicated in red (poles of the $\mathrm{\Gamma }$ function as circles and the other pole as a star), and the integration contour is indicated in purple. Left: The Bromwich contour used in Eq. (33). Right: Deformation of the contour by pushing it to the left, picking up the leading orders in $\rho$ as each pole is enclosed.

Figure 9

Comparison of analytical and numerical results for the semiclassical model. Upper panels: Histograms of the rescaled resistance in the diffusive phase $p=0.1$ (top left), $p=0.3$ (top right), and $p=0.5$ (center left) and the subdiffusive phase $p=0.9$ (center right). The top panels show the rescaling $(R-R_{★})/\sigma$, which is compared to a Gaussian distribution (black dotted line) for $p=0.1$. The existence of a tail in the diffusive phase is clear in the $p=0.3$ histogram. The central panels show the rescaling $R/R_{★}$, with a comparison to the predicted tail for $p=0.5$, and a comparison to the theoretical prediction (9) shown for $p=0.9$ (black dotted lines). Lower panel: Comparison of the numerically determined scaling exponent for the median resistance $\beta$, and also for the interquartile range ${\beta }^{\prime }$, with the analytical prediction ${log}_{1/p}\left(q\right)$. The critical point is $p_c=1/q=2/3$.

Figure 10

The resistance $\beta$ function for the semiclassical model, plotted against $1/lnL$ for a range of $p$ values. The lines indicate fits to linear order in $1/lnL$.

Figure 11

The dynamical phase diagram of the random dephasing model and the semiclassical model, showing a schematic of the finite-size flow of the $\beta$ function. For $p<p_c$ the system is diffusive, with $\beta \to 1$ in the thermodynamic limit, while for $p>p_c$ the system is subdiffusive with $\beta >1$. In the limit $p\to 1$ the system is localized and $\beta \propto L$ diverges in the thermodynamic limit. Above this line, if it existed for this model it would lie the MBL phase, where one has very fast divergence $\beta =\frac{1}{\xi }exp(1/[1/lnL])$ with $\xi$ the many-body localization length.