Subdiffusive Thouless time scaling in the Anderson model on random regular graphs

The scaling of the Thouless time with system size is of fundamental importance to characterize dynamical properties in quantum systems. In this work, we study the scaling of the Thouless time in the Anderson model on random regular graphs with on-site disorder. We determine the Thouless time from two main quantities: the spectral form factor and the power spectrum. Both quantities probe the long-range spectral correlations in the system and allow us to determine the Thouless time as the timescale after which the system is well described by random matrix theory. We find that the scaling of the Thouless time is consistent with the existence of a subdiffusive regime anticipating the localized phase. Furthermore, to reduce finite-size effects, we break energy conservation by introducing a Floquet version of the model and show that it hosts a similar subdiffusive regime.

Figure 1

Sketch of a random regular graph (RRG) with diameter $L=6$ and $N=2^L=64$ nodes and branching number $K=2$, i.e., each node has $K+1=3$ neighbors. The particle can hop along the edges (black lines) of the graph with hopping amplitude $t=1$ and is subject to on-site disorder ${\mu }_i$ (color intensity) on the nodes of the graph (red dots of different intensity).

Figure 2

(a) Spectral form factor and (b) power spectrum of Floquet model Eq. (2) for the system size $L=14$ and disorder strength $W/T=4.0,10.0,16.0$ in units of the Floquet half-period $T$. The Thouless time ${\tau }_{\text{Th}}$ and Thouless momentum $k_{\text{Th}}$ are defined as the point where the spectral form factor or power spectrum fit aligned with the GOE prediction. The black dashed lines are the GOE behavior, $K_{\text{GOE}}\left(\tau \right)\sim \tau$ and $P_{\text{GOE}}\left(k\right)\sim 1/k$. The black solid lines are the Poisson behavior, $K_{\text{Poisson}}\left(\tau \right)\sim O\left(1\right)$ and $P_{\text{Poisson}}\left(k\right)\sim 1/k^2$. The Heisenberg time is the point where the GOE and Poisson values meet at $\tau =1$ and $k=0.5$.

Figure 3

Average consecutive level-spacing ratio $\langle r\rangle$ as a function of disorder $W$ ($W/T$) for different matrix size $N=2^L$. (a) $\langle r\rangle$ of 10% of the Hamiltonian spectrum centered at energy $E=0$ using $2000–5000$ disorder realizations for sizes $L<16$ and $\le 1000$ for $L=17$. The gray line is the critical disorder $W_c=18.17\pm 0.01$ [96]. (b) $\langle r\rangle$ for Floquet model Eq. (2) using $10000$ disorder realizations for $L<13$ and $4000–6000$ for $L=14$. The solid red and blue lines are the GOE and Poisson value, respectively. Error bars are given by $68\%$ bootstrap confidence interval.

Figure 4

Spectral form factor, Eq. (9), for the unfolded Hamiltonian spectrum for different system sizes $L=8$, 10, 12, 14, 16 at (a) $W=4.0$, (b) 8.0, (c) 12.0, (d) 14.0 and for Floquet quasienergies for $L=8$, 10, 12, 14 at (e) $W/T=4.0$, (f) 10.0, (g) 14.0, (h) 16.0. In the static case, only 60% of the spectrum centered at the middle are taken and the disorder average is performed over 2000–5000 disorder realizations The time $\tau \in [{10}^{-4},{10}^1]$ is continuous. In the Floquet case, time $\tau =t/N$ is discrete and given by an integer $t\in [1,n]$ normalized by the matrix size $N$, with $n={10}^6$ for $8<L<14, n={10}^4$ for $L=8$, and $n={10}^7$ for $L=14$.

Figure 5

Power spectrum, Eq. (13), for the unfolded Hamiltonian spectrum for different system sizes $L=8$, 10, 12, 14, 16 at (a) $W=4.0$, (b) 8.0, (c) 12.0, (d) 14.0 and for Floquet quasienergies for $L=8$, 10, 12, 14 at (e) $W/T=4.0$, (f) 10.0, (g) 14.0, (h) 22.0. In the static case, only 60% of the centered at the middle are taken. The “energy momentum” $k$ is normalized by the effective size of the spectrum $\tilde{N}=0.{62}^L$, thus, the arguments of $P_{\text{GOE}}\left(k\right)$ and $P_{\text{Poisson}}\left(k\right)$ are also rescaled by $\tilde{N}$. In the Floquet case, $k$ is rescaled by the matrix size $N=2^L$ such that the Nyquist frequency becomes 0.5.

Figure 6

Thouless time extracted from (a,c) the spectral form factor and (b,d) the power spectrum as the point, where $log(K/K_{\text{GOE}})=0.4$ or $log(P/P_{\text{GOE}})=0.4$ for (a,b) Hamiltonian and (c,d) Floquet cases. Each curve corresponds to the different disorder value shown in the legend. The gray dashed lines show the inverse level spacing (Heisenberg time). The solid gray lines correspond to the diffusive scaling with an arbitrary prefactor. The set of parameters for the computation of the spectral form factor and the power spectrum is the same as the one of Figs. 4 and 5.

Figure 7

Dynamical exponent $\beta$ for the (a) Hamiltonian and (b) Floquet models, extracted by fitting $t_{Th}\sim L^{1/\beta }$ from the spectral form factor (blue) and the power spectrum (red) considering only the five largest system sizes shown in Fig. 6.

Figure 8

Unfolding of Hamiltonian spectra with parameters $L=8$ and $W=6.0$. (a) Average cumulative distribution function (CDF) as a function of eigenvalues $E$ (black) and its fitting using cubic spline (dashed red). Orange points are $E$ values of a single realization, green points are their corresponding unfolded values $\varepsilon /N=\text{CDF}\left(E\right)$. (b) Density of states of the unfolded (folded) eigenvalues $\varepsilon$ ($\tilde{E}$) average over 1000 graph and diagonal disorder realizations. $\tilde{E}=(E-E_{\text{min}})/(E_{\text{max}}-E_{\text{min}})$ is the folded spectrum shifted and rescaled to be between 0 and 1.

Figure 9

Measures to estimate the Thouless time in the Hamiltonian model: $log(F/F_{\text{GOE}})$ with $F$ equal to the (a,c) spectral form factor and (b,d) the power spectrum for the filtering parameter (a,b) $\eta =0.3$ and (c,d) $\eta =0.4$. Disorder is set to $W=6.0$.

Figure 10

Measures to estimate the Thouless time in the Floquet model: $log(F/F_{\text{GOE}})$ with $F$ equal to (a) the spectral form factor and (b) the power spectrum. Disorder is set to $W/T=10.0$.

Figure 11

Thouless time computed from the spectral form factor at periods $T=1.5$, 1.8, 2.2, 2.5 and disorder $W=8$, 12, 16, 20, 24, 28, 32, 36, 40, 44. The procedure for extracting the Thouless time is the same explained in Fig. 6 of the main text.

Figure 12

Dynamical exponent $\beta$ extracted from curves in Fig. 11. The same fitting procedure explained in caption of Fig. 7 of the main text was used.