Toy model for anomalous transport and Griffiths effects near the many-body localization transition

We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the absence of such a coupling the model is localized as expected for the one-dimensional Anderson model, increasing the coupling with the GOE perturbation we find a transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponentials and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the MBL transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.

Figure 1

Pictorial representation of the model (1). Top: The model is made of a 1D chain of $L$ layers coupled along the $x$ direction via the interlayer hopping rates $t$, with random energies ${\epsilon }_i$ (equal on all sites of the layer) extracted from a box distribution of width $W$. Each layer contains $M$ sites. The matrix elements between two sites belonging to a given layer correspond to the adjacency matrix of a random realization of a RRG (different on each layer) times the intralayer hopping rate $\gamma$. One can imagine that each layer of the chain is a pictorial representation of a coarse-grained block of $\ell$ sites of an interacting many-body problem, and that the $M\times M$ RRG matrices are an extreme simplified description of the Hilbert space of the local degrees of freedom on each segment in some specific basis. Bottom: The matrix representation of the Hamiltonian consists of $L$ blocks of $M\times M$ matrices. The diagonal blocks are $(-\gamma$ times) random realizations of the connectivity matrix of a RRG, shifted by random energies ${\epsilon }_i$. Adjacent blocks are connected by blocks of the form $-t\mathcal{I}$.

Figure 2

Sketch of the three kinds of cavity Green's functions defined on the anisotropic RRG.

Figure 3

Pictorial representation of the energy spectrum of the model in the strong disorder limit, $W/t\gg 1$. The DOS is given by a superposition of $L$ semicircles (containing $M$ levels each) centered roughly around the random chemical potentials ${\epsilon }_i$ and of width $4\gamma \sqrt{k}$.

Figure 4

Main panel: Probability that a chain of length $L$ is localized, $P_{\mathrm{loc}}$ (i.e., the probability that the perturbative series in ${\gamma }^2$ converges), as a function of the GOE coupling $\gamma$, multiplied by the system size $L$, for $L=2^{10}$ (circles), $L=2^{12}$ (squares), $L=2^{14}$ (diamonds), $L=2^{16}$ (up triangles), and $L=2^{18}$ (down triangles), and for three values of the disorder strength, $W=0.5$ (green), $W=4$ (blue), and $W=16$ (red). Inset: Nonmonotonic behavior of ${\widehat{\gamma }}_c\left(W\right)$ as a function of $W$. The dashed black curves show the asymptotic behavior ${\widehat{\gamma }}_c\left(W\right)\approx W/(4/\sqrt{k})$ at large disorder and ${\widehat{\gamma }}_c\left(W\right)\approx 1/\left(W^2\sqrt{k}\right)$ at small disorder.

Figure 5

Phase diagram of the model in the plane $\varphi \text{−}{L}^{-1}$. For infinite chains with probability 1 the system is insulating for $\varphi >1$ and conducting for $\varphi <1$. At finite $L$ a broad intermediate region appears where broad insulating and conducting segments coexist. The continuous lines correspond to $P_{\mathrm{loc}}\le \epsilon$ and $P_{\mathrm{loc}}\ge 1-\epsilon$, respectively, with $\epsilon ={10}^{-2},{10}^{-3}$, and ${10}^{-4}$. The light blue squares represent the points where we performed exact diagonalizations (see Sec. 6).

Figure 6

Evolution of the probability distribution of $log\mathrm{Im}\mathcal{G}$ (averaged over many realizations of the disorder) upon increasing the length of the chain for the samples that are in the conducting phase, for $\varphi =1,W=4$, and $\widehat{\gamma }=2$; $L=2^8$ (black), $2^9$ (indigo), $2^{10}$ (violet), $2^{11}$ (blue), $2^{12}$ (magenta), $2^{13}$ (red), and $2^{14}$ (orange).

Figure 7

(Log of the) Imaginary part of ${\mathcal{G}}_i$ in the middle of the system when $\eta$ is set identically to zero on all the layers of the chain except on the first layer, where $\eta =1$, for $\varphi =1,W=4,\widehat{\gamma }=2$, and $L=2^9$. Different colors correspond to different realizations of the disorder. Some of the samples (black, gray, brown, and violet) are in the insulating phase and $log\mathrm{Im}{\mathcal{G}}_i\sim -i/{\xi }_{\mathrm{typ}}$. Conducting samples are characterized instead by regions where $\mathrm{Im}{\mathcal{G}}_i$ decays exponentially separated by few, i.e., $O\left(1\right)$, resonances (“thermal inclusions”) where $\mathrm{Im}{\mathcal{G}}_i$ is of order 1.

Figure 8

Top panels: $\langle \mathrm{Im}\mathcal{G}\rangle$ (blue and red) and $e^{\langle log\mathrm{Im}\mathcal{G}\rangle /\alpha L}$ (light blue and orange) vs $\widehat{\gamma }$ for two specific realizations of the disorder (with $\alpha =2.5\times {10}^{-2})$. Bottom panel: $log\langle log\mathrm{Im}\mathcal{G}\rangle$ (blue and red) and $\langle log\mathrm{Im}\mathcal{G}\rangle /\alpha L$ (light blue and orange) vs $log(\widehat{\gamma }-{\widehat{\gamma }}_{\mathrm{loc}})$ for the same samples (with $\alpha \approx {10}^{-2})$. The size of the chain is $L=2^8$ (left panels) and $L=2^{11}$ (right panels) and $W=4,\varphi =1$. The dashed lines correspond to the power-law fits of the critical behavior of Eqs. (12).

Figure 9

Main panel: Average (filled circles) and typical (blue empty squares) correlation functions for $\varphi =1,W=4$, and $\widehat{\gamma }=2$ (a similar behavior is observed for all values of $\widehat{\gamma }$ and $W$ at the critical point). Typical correlations do not depend on the length of the chain, whereas average correlations increase as $L$ is increased $(L=2^8$, maroon; $L=2^9$, violet; $L=2^{10}$, magenta; $L=2^{11}$, red; $L=2^{12}$, orange; $L=2^{13}$, yellow) The dashed lines correspond to fits of the exponential and subexponential decays of $C_{\mathrm{typ}}\left(x\right)$ and $C_{\mathrm{av}}\left(x\right)$, Eqs. (13), with ${\xi }_{\mathrm{typ}}\approx 3.16$. Top inset: Localization length of typical correlation functions, ${\xi }_{\mathrm{typ}}$, as a function of $W$. The green dotted line corresponds to a fit as ${\xi }_{\mathrm{typ}}=A/W^2$, with $A\approx 52.3$. Bottom inset: (Log of the) Fractal exponent $d_{\mathrm{f}}$ extracted from the stretched exponential fits of average correlations as a function of the (log of the) inverse length of the chain. The dotted line corresponds to a power-law fit as $d_{\mathrm{f}}\sim L^{-\kappa }$, with $\kappa \approx 0.02$.

Figure 10

Probability distributions $Q(log|{\mathcal{G}}_{i,i+x}{|}^2)$ at fixed distance $x$ for $\varphi =1,W=4$, and $\widehat{\gamma }=2$. Continuous curves correspond to the PDFs for $x=10\approx 3{\xi }_{\mathrm{typ}}$ varying the system size $(L=2^8$, maroon; $L=2^{10}$, violet; $L=2^{12}$, magenta; $L=2^{14}$, red; $L=2^{16}$, orange). The black dotted line is the power-law fit of the tails as $Q\left(\right|{\mathcal{G}}_{i,i+x}{|}^2)\sim [|{\mathcal{G}}_{i,i+x}{{|}^2]}^{-3/2}$. The dashed curves show the PDFs varying the distance $(x\approx 6.4{\xi }_{\mathrm{typ}}$, brown; $x\approx 1.6{\xi }_{\mathrm{typ}}$, gray) for the largest system size $L=2^{14}$.

Figure 11

Top left panel: Return probability as a function of time for $W=16,L=36$, and $M=160$, and for several values of $\gamma$ across the intermediate region (corresponding to the light blue squares in the interval $\varphi \in [0.5,2.2]$ on the phase diagram of Fig. 5). The black dashed curve shows the return probability for the standard 1D Anderson model $\left(\gamma =0\right)$ and the horizontal dotted black line gives the value of the IPR of the unperturbed Anderson localized wave functions. Bottom left panel: Mean-square displacement as a function of $t$ for the same values of $W,L,M$, and $\gamma$. The black dashed curve shows $\langle x^2\left(t\right)\rangle$ for the standard 1D Anderson model $\left(\gamma =0\right)$ and the horizontal dotted black line gives the square of the bare localization length ${\xi }_0$ in 1D. Top right panel: Infinite-temperature optical conductivity $\sigma \left(\omega \right)$ for the same values of $W,L,M$, and $\gamma$. A factor $T$ is implicitly understood. The black dashed curve corresponds to the standard 1D Anderson model $\left(\gamma =0\right)$, for which we recover $\sigma \left(\omega \right)\sim {\omega }^2{log}^2\left(\omega \right)$ (in 1D). Bottom right panel: Numerical values of the exponents $\beta ,{\beta }^{\prime }$, and $\alpha$, obtained by power-law fits of the numerical data, as a function of $\varphi$. The scaling relation $\alpha +2\beta =1$ is roughly satisfied within our numerical accuracy (black diamonds), while we find that ${\beta }^{\prime }\approx 0.6\beta$ [29].

Figure 12

Probability distribution (averaged over many realizations of the disorder) of the log of the imaginary part of ${\mathcal{G}}_{L/2}$ in the middle of the system when $\eta$ is set identically to zero on all the layers of the chain except on the first layer, where $\eta =1$. The results are shown for $\varphi =1,W=4,\widehat{\gamma }=2$, and $L=2^8$ (black circles), $L=2^{10}$ (violet squares), $L=2^{12}$ (magenta diamonds), and $L=2^{14}$ (orange triangles). The dotted curves correspond to $(P_{\mathrm{loc}}$ times) $\tilde{P}(log\mathrm{Im}{\mathcal{G}}_{L/2})$ when the GOE coupling is turned off $(\gamma =0$, i.e., for the standard Anderson tight-binding model in 1D), while the dashed curves show $(1-P_{\mathrm{loc}}$ times) the nonsingular part of the probability distributions $P(log\mathrm{Im}\mathcal{G})$ plotted in Fig. 6 when the imaginary regulator is identically set to $\eta =0^{+}$ on all the sites of the chain.

Figure 13

Top panel: Probability distributions of the (logarithm of the) “susceptibility” $\mathrm{\Sigma }(logS)$ for $\varphi =1,W=8,\widehat{\gamma }=2$, and $L=2^8$ (black), $L=2^9$ (indigo), $L=2^{10}$ (violet), and $L=2^{11}$ (blue). The black dashed line corresponds to the power-law tails $\mathrm{\Sigma }\left(S\right)\sim S^{-2}$. Bottom panel: Probability distributions of the (logarithm of the) “local susceptibility” ${\widehat{\mathrm{\Sigma }}}_W(log{s}^{\left(W\right)})$ for $W=8,\widehat{\gamma }=2$, and $L=2^8$ (black), $L=2^9$ (indigo), $L=2^{10}$ (violet), $L=2^{11}$ (blue), $L=2^{12}$ (magenta), $L=2^{13}$ (red), and $L=2^{14}$ (orange). The same behavior is found for ${\widehat{\mathrm{\Sigma }}}_{\widehat{\gamma }}\left(s^{\left(\widehat{\gamma }\right)}\right)$.

Figure 14

Probability distribution of the local Lyapunov exponents $\lambda$ (averaged over several realizations of the disorder) for $\varphi =1,W=1$, and $\widehat{\gamma }=6$ (green), $W=4$ and $\widehat{\gamma }=2$ (red), and $W=16$ and $\widehat{\gamma }=2.7$ (blue), and for $L=2^{12}$.

Figure 15

Probability distribution of the resistivity $R\left(\rho \right)$ at a fixed sample size in the limit of low frequency $\omega \approx 1/\left(LM\right)$. The data are obtained from exact diagonalizations of finite-size samples with $L=44$ and $M=206$, and several values of the control parameter $\varphi$ across the Griffiths region (light blue squares in the interval $\varphi \in [0.5,2.2]$ on the phase diagram of Fig. 5). As $\varphi$ is increased we observe a crossover between two power-law regimes, $R\left(\rho \right)\sim {\rho }^{-{\tau }^{\prime }}$ at intermediate values of $\rho$, with ${\tau }^{\prime }\approx 1$, followed by $R\left(\rho \right)\sim {\rho }^{-\tau }$ with $\tau \approx 1.5$ at larger values of $\rho$. The first power-law regime extends to larger and larger values of $\rho$ as $\varphi$ is increased, implying that also the average resistivity diverges in the $\gamma \to 0$ limit, recovering the properties of the standard 1D Anderson insulator.